A Comprehensive Series for Predicting Bone Dynamics: Forecasting Osseous Tissue Formation Using the Molecular Structure of A

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A Comprehensive Series for Predicting Bone Dynamics:

Forecasting Osseous Tissue Formation

Using the Molecular Structure of a Biomaterial

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree

Doctor of Philosophy in Mechanical Engineering

Mary Elizabeth Kundrat, M.S., B.S.

UNIVERSITY OF DAYTON

Dayton, Ohio

December, 2010

A Comprehensive Series for Predicting Bone Dynamics:

Forecasting Osseous Tissue Formation

Using the Molecular Structure of a Biomaterial

APPROVED BY:

______Khalid Lafdi, PhD Tarun Goswami, D.Sc., PhD Advisory Committee Chairman Committee Member Professor, UDRI Carbon Group Leader Associate Professor Mechanical Engineering Biomedical Engineering University of Dayton Wright State University

______Panagiotis Tsonis, PhD Kevin Hallinan, PhD Committee Member Committee Member Professor, Director of TREND Center Professor and Chairperson Biology Department Mechanical Engineering University of Dayton University of Dayton

______Malcolm W. Daniels, Ph.D. Tony E. Saliba, Ph.D. Associate Dean Dean School of Engineering School of Engineering

ABSTRACT

A COMPREHENSIVE SERIES FOR PREDICTING BONE DYNAMICS:

FORECASTING OSSEOUS TISSUE FORMATION

USING THE MOLECULAR STRUCTURE OF A BIOMATERIAL

Name: Kundrat, Mary Elizabeth University of Dayton

Advisor: Dr. Khalid Lafdi

Tissue engineering, or regenerative medicine, is a novel field that uses various means to develop biological substitutes used to repair or even replace living tissues. Past efforts to restore healing tissues are limited, and their methods lack accuracy. The objective of this research is to simulate and predict behaviors of bone tissue and biomaterials both separately, and collectively, as they restore form and function during tissue regeneration and wound healing processes. Not only will more effective and dependable means of analyzing tissue regeneration be developed, the properties of carbon based biomaterials that prove most advantageous in assisting tissue regeneration will be identified. Through in vitro experiments, primary human osteoblasts and their methods of proliferation were extensively studied. RAMAN spectroscopy, X-RAY diffraction and Atomic Force Microscopy were employed to study the molecular structure of various carbon fibers of interest. Both

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osteoblasts and carbon fibers were then studied collectively to understand how various material properties affected osseous tissue formation potential. An intricate cellular automation based computer program was developed that visually and mathematical predicts osseous tissue formation. A model combining the Logistic and Malthusian Laws was developed to predict both the growth rate and overall cell population of osteoblasts with respect to time. Estimations for cellular parameters such as individual cell volume and mass were constructed and used to calculate tissue density as a function of cell population. Multivariate regression models were formulated to describe cellular behavior in terms of the structural properties of a biomaterial. Additionally, Monte Carlo simulations were performed to provide estimations of developing tissue density with respect to time and material properties. Ultimately a very comprehensive series of theoretical models were successfully developed that can be used separately, or collectively, to provide accurate information pertaining to bone tissue dynamics. Each model’s accuracy, combined with its versatility, provide accurate information pertaining to osseous tissue formation, even when experimental data is unattainable. Through this initiative, the material properties of carbon have proven superior in both structure and performance. Any material’s ability to promote or prevent bone tissue growth can now be promptly examined through the utilization of these mode

DEDICATIONS

For My Parents:

Here we are again…in a place of accomplishment I could have never achieved without your unconditional love. Thank you, and I love you, just don’t seem like enough. God gave me the world when he made you my parents. Thank you for never giving up on me and helping me make my dreams come true. I love you both so very much and am honored to dedicate this work to you.

For My Husband:

You’re my world, my life, my everything. I love you very much and thank you for being there for me with unending faith and love. This accomplishment means so much more with you by my side. I love you and am honored to have you as my husband.

For My Grandparents:

You have always been there for me…very supportive and full of prayer. Your unrelenting faith and love have pulled me through the times when success seemed so far away. I am so blessed to have you in my life. I love you all very much and am honored to devote this research to you.

For Dr. Khalid Lafdi:

You are undoubtedly one of the most brilliant people I have ever met. I feel very honored to have been one of your students and have learned so much from you. Thank you for always believing in me and being there for me.

For My Dissertation Committee:

Your patience and wisdom will always be with me. Thank you for having faith in me and helping me achieve my dream.

For the UDRI Carbon Research Lab:

Thank you for your help and your friendship. I will cherish it always.

For the Professors and Faculty of the University of Dayton:

I am very honored to have worked alongside you and feel so blessed you gave me the opportunity to teach. Thank you for having faith in my ability. I will never forget you or my students.

TABLE OF CONTENTS ABSTRACT ...... iii DEDICATIONS ...... v LIST OF FIGURES ...... xii LIST OF TABLES ...... xvi Chapter 1. Introduction ...... 1 1.1. Research Overview ...... 1 1.2. Problem Introduction and Objectives of Research ...... 2 Chapter 2. Dissertation Statement of Research ...... 4 Chapter 3. Literature Reviews ...... 6 3.1. Bone Tissue Regeneration ...... 6 3.1.1. Cellular Biology ...... 6 3.1.2. Fundamentals of Cellular Dynamics ...... 8 3.1.3. Wound Healing and Regeneration Concepts ...... 11 3.2. Overview of Biomaterials ...... 15 3.2.1. Metals ...... 17 3.2.2. Polymers ...... 21 3.2.3. Biopolymers ...... 24 3.2.4. Animal Inspired Biomaterials ...... 37 3.2.4.1. Self Healing Polymer Composites: Mimicking Nature to Enhance Performance ...... 37 3.2.4.2. Regeneration of ACL Tissue in Large Animal Model ...... 39 3.2.5. Polymers of the Sea ...... 41 3.2.5.1. Material Design Principles of Ancient Fish Armour ...... 42 3.2.5.2. The Transition from Stiff to Compliant Materials in Squid Beaks ... 44 3.2.5.3. Bioinspired Structural Materials ...... 45 3.2.5.4. Mussel Inspired Surface Chemistry for Multifunctional Coatings . 47 3.2.5.5. Stimuli Responsive Polymer Nanocomposites Inspired by the Sea Cucumber Dermis ...... 48 3.2.5.6. Inspirations from Biological Optics for Advanced Photonic Systems ……………………………………………………………………………….50 vii

3.2.5.7. Biomimetics for Next Generation Biomaterials ...... 51 3.2.6. Carbon Based Biomaterials ...... 55 3.3. Techniques of Tissue/Biomaterial Assessment ...... 59 3.3.1. Microscopy Analysis ...... 60 3.3.2. Computed Tomography (CT) Images and X-Rays ...... 61 3.3.3. Magnetic Resonance Imaging (MRI) ...... 61 3.4. Concepts of Cellular Automata (CA) ...... 64 3.4.1. Fundamentals of CA Theory ...... 64 3.4.2. Successful Applications of CA...... 65 3.5. Theory of Mathematical Biology ...... 66 3.5.1. Principle of Population Equation ...... 67 3.5.2. Logistic Law with Growth and Capacity ...... 69 3.5.3. The Malthusian Model ...... 71 3.5.4. Logistic Growth with Carrying Capacity and Harvesting ...... 72 3.5.5. Stochastic Population Theory: The Predator/Prey Model ...... 74 3.5.6. Mathematical Framework for Modeling Tissue Density ...... 75 3.6. Pertinent Statistical Analyses ...... 81 3.6.1. Multivariate Regression ...... 82 3.6.2. The Monte Carlo Method ...... 83 Chapter 4. Drawbacks of Current Tissue/Biomaterial Assessment Methods……...... 86 4.1. Inadequacies of Current Materials and Methods ...... 86 4.1.1. Materials ...... 86 4.1.2. Tissue Analysis Methods ...... 94 4.1.3. Mathematical Methods ...... 95 4.1.4. Experimental Techniques ...... 96 4.2. Benefits of Research Initiative ...... 97 Chapter 5. Research Methods ...... 98 5.1. Theoretical Research Methods ...... 98 5.1.1. Using Cellular Automata to Describe and Predict Cell Growth Rate …………………………………………………………………………………99 5.1.2. Combining Malthusian and Logistic Law Models to Predict Cell Growth ………………………………………………………………………………..101

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5.1.3. Statistical Analyses ...... 102 5.1.3.1. Multivariate Regression to Relate Biomaterial Properties to Cell Growth…...... 102 5.1.3.2. Monte Carlo Simulation to Predict Bone Density with Respect to Significant Biomaterial Properties ...... 104 5.2. Experimental Research Methods ...... 104 5.2.1. Bone: Methods of Growth and Analysis ...... 105 5.2.2. Carbon: Material Properties of Interest ...... 105 5.2.3. Bone-Carbon Systems: Experimentation and Analysis ...... 106 Chapter 6. Dissertation Hypotheses ...... 108 6.1. The Application of Cellular Automation ...... 108 6.2. The Combined Logistic-Malthusian Model and Tissue Density Calculations ...... 108 6.3. Cell Population Predictions using Multivariate Regression ...... 109 6.4. Tissue Density Predictions using Monte Carlo Simulations ...... 110 Chapter 7. Research Techniques ...... 111 7.1. Approaches in Theoretical Research ...... 111 7.1.1. Cellular Automation ...... 111 7.1.2. Mathematical Biology ...... 120 7.1.3. Multivariate Regression ...... 130 7.1.4. Monte Carlo Simulation ...... 131 7.2. Experimental Research Schemes ...... 134 7.2.1. In Vitro Experiments ...... 135 7.2.2. Carbon Material Testing ...... 137 7.2.3. Cell-Carbon System Experiments ...... 141 Chapter 8. Description of Results ...... 144 8.1. Theoretical Research ...... 144 8.1.1. Cellular Automation ...... 144 8.1.2. Mathematical Biology ...... 151 8.1.3. Multivariate Regression ...... 160 8.1.4. Monte Carlo Simulation ...... 163 8.2. Experimental Research ...... 169 8.2.1. In Vitro Experimentation ...... 169

8.2.2. Carbon Material Testing ...... 171 8.2.3. Cell-Carbon System Experimentation ...... 175 8.2.4. Validation of Theoretical Models with Experimental Data ...... 183 Chapter 9. Discussions and Interpretations of Results ...... 187 9.1. Experimental Results ...... 187 9.2. Theoretical Results ...... 203 9.3. Comparisons and Validation ...... 218 Chapter 10. Conclusions and Summary of Achievements ...... 225 10.1. Using Cellular Automation to Predict Bone Cell Growth ...... 225 10.2. The Combined Logistic-Malthusian Model Creates the Most Accurate Means for Assessing Tissue Dynamics ...... 226 10.3. Multivariate Regression Models Supply an Accurate Technique for Predicting Cell Growth Based on the Material Properties of Carbon ...... 227 10.4. Overall Tissue Density Can be Accurately Foreseen by Instilling the Monte Carlo Methods ...... 228 10.5. Collaboration of Theoretical Methods ...... 229 Chapter 11. Comparisons of Objectives with Achievements ...... 231 11.1. Develop More Efficient and Dependable Means for Simulating and Predicting Bone Tissue Regeneration ...... 231 11.2. Determination of Material Properties Most Advantageous to Tissue Growth 233 Chapter 12. Suggestions for Future Research ...... 235 12.1. Predicting Cellular Behavior with Three Dimensional Models ...... 235 12.2. Development of Successful Bioresorbable Implant Materials...... 236 12.3. Porous Implant Materials with the Characteristics of Native Tissues 237 12.4. Understanding How Tissue-Biomaterial Behaviors Change on Differing Scales of Observation ...... 238 REFERENCES ...... 240 Appendix A: Cellular Automation Main Executable Program and Subroutines ...... 246 Appendix B: Cellular Automation Output Export to Excel ...... 322 Appendix C: Mathematical Biology Equations ...... 337 Appendix D: Monte Carlo Formulas: Simulation #1 ...... 344 Appendix E: Monte Carlo Histograms: Simulation #1 ...... 347

Appendix F: Monte Carlo Formulas: Simulation #2 ...... 352 Appendix G: Monte Carlo Histograms: Simulation #2 ...... 354 Appendix H: RAMAN Spectroscopy Data ...... 359 Appendix I: X-RAY Diffraction Calculations ...... 391

LIST OF FIGURES

Figure 3.1.1-1: The Cell Cycle ...... 7 Figure 3.1.3-1: The Phenomenon of Apoptosis ...... 12 Figure 3.1.3-2: Bone Tissue Remodeling Process...... 15 Figure 3.2-1: Sir John Charnley's Hip Replacement ...... 16 Figure 3.2.1-1: Most Established Biomedical Applications of Metals ...... 20 Figure 3.2.1-2: Cross Section of Fixator Screw ...... 21 Figure 3.2.2-1: List of Various Polymers and their Biomedical Applications ...... 23 Figure 3.2.2-2: List of Polymer Based Biomedical Applications ...... 24 Figure 3.2.3-1: Cellulose Nanofibers being Woven and Controlled by Cellulose Creating Bacteria (Virginia Tech Wake Forest University School of Biomedical Engineering)...... 28 Figure 3.2.3-2: Hyaluronic Acid, One of the Main Components in Natural Tissue, Has a Fundamental Role in the Physiologic Processes of Tissue Repair ...... 29 Figure 3.2.3-3: Hyaluronic Acid Working to Repair Cartilage in the Knee (The Pain and Spine Consultants of Pennsylvania) ...... 31 Figure 3.2.3-4: Hybrid Polyester Surgical Mesh and Hydrogel ...... 36 Figure 3.2.4.2-1: Silk Fibroin Based ACL Graft by Serica Technologies ...... 40 Figure 3.2.5.1-1: The Polypterus Senegalus ...... 42 Figure 3.2.5.2-1: The Humboldt Squid Beak is One of Engineering's Most Fascinating Materials because of its Assortment of Mismatched Tissues ... 44 Figure 3.2.5.3-1: Seashells are of Interest because of Their Multi-Layered Structures and the Different Properties of Each ...... 46 Figure 3.2.5.4-1: A Mussel Shell (Nerites, Inc. 2004) ...... 47 Figure 3.2.5.5-1: Skin of the Sea Cucumber can Reversibly Change its Mechanical Properties when Exposed to Some Form of External Stimulus 48 Figure 3.2.5.7-1: Nacreous Shell ...... 52 Figure 3.2.6-1: Dry Carbon Fibers ...... 58 Figure 3.3.2-1: Micro-CT Image of the Proximal Part of a Tibia ...... 61 Figure 3.3.3-1: MRI Equipment and Images ...... 62 Figure 3.4.1-1: Concepts of Cellular Automata ...... 65 Figure 3.6.1-1: Multivariate Linear Regression ...... 83 Figure 4.1.1-1: Micrograph showing pits in the surface of a Co-Cr & UHMWPE Total Knee Replacement that are relatively shallow and measure about 500 μm across. A crack in the Tibial UHMWPE Component is also evident and appears to join with the pits [39]...... 89

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Figure 4.1.1-2: Wear Debris Produced by Artificial Implants (Batchelor, 2004) ...... 90 Figure 4.1.1-3: Example of Wear Debris from an UHMWPE Implant (Batchelor, 2004) ...... 92 Figure 4.1.1-4: Illustration of How Stress Shielding Works (Thelen, Barthelat, & Brinson, 2002) ...... 93 Figure 5.1.1-1: Diagram Depicting the Intricacies of Cellular Dynamics .. 100 Figure 7.2.2-1: A Block Diagram for the Atomic Force Microscope ...... 139 Figure 7.2.3-1: An Inverted Microscope (J. Lawrence Smith)...... 143 Figure 8.1.1-1: Initial Seeding of Osteoblasts on the CA Time-Space Grid...... 145 Figure 8.1.1-2: A Snapshot of the Time-Space CA Grid Just prior to Cells Undergoing Mitosis ...... 146 Figure 8.1.1-3: Between Cycles of Mitosis, the Initial Non-Mineralized Matrix is Identified by the Vibrant Orange Cells within the Grid ...... 147 Figure 8.1.1-4: The CA Grid Just After Another Round of Mitosis Illustrating how the Process of Mitosis Mineralizes the Matrix Components of the Tissue ...... 148 Figure 8.1.1-5: Matrix Continues to be Mineralized as More and More Cells Undergo Mitosis ...... 149 Figure 8.1.1-6: Tissue Continues to Grow. Small Areas of Dead Cellular Debris Can be Seen, Where Osteoclasts have cleaned the Area and Growth Factors have been Developed to Attract New Cells to the Area ...... 150 Figure 8.1.1-7: The Final Snapshot taken of the CA Program Grid. Notice that the Tissue is Now Fully Comprised of Osteocytes, Blood Vessels, Mineralized Matrix and Occasional Voids, which is Very Representative of Actual Healthy Bone Tissue ...... 151 Figure 8.1.2-1: Comparison of the In Vitro Experimental Data and the Logistic-Malthusian Model ...... 153 Figure 8.1.2-2: Comparison of the CA Program and the Logistic-Malthusian Model ...... 153 Figure 8.1.2-3: Comparison between the Cell-Carbon System Experimental Data and the Implementation of the Logistic-Malthusian Model on that Data – For Each Graph: X Axis = Time (hrs) Y-Axis = Cell Population ...... 155 Figure 8.1.2-4: Excerpt of Data from the Tissue Composition Calculations - AS4 Carbon Fiber ...... 158 Figure 8.1.2-5: Tissue Volume with Respect to Time for Each of the Carbon Fiber Bundles. This Volume Calculation Takes into Consideration the Breakdown of Cancellous and Cortical Cells as well as ECM ...... 158 Figure 8.1.2-6: Relative Tissue Density for the Various Carbon Fiber Bundles ...... 159 Figure 8.1.3-1: Regression Analysis for In Vitro Data ...... 160 xiii

Figure 8.1.3-2: Regression Results for the Cell-Carbon System Experimental Data ...... 161 Figure 8.1.3-3: Regression Analysis for Data from CA Program ...... 162 Figure 8.1.4-1: Correlation Matrices for the Various Variables Involved in the MC Simulations ...... 164 Figure 8.1.4-2: The Monte Carlo Setup for 24Hr Time Period. This form of Monte Carlo Simulation was Developed to Understand what the Mean Cell Population would be if the Values of the Various Material Properties Varied ...... 164 Figure 8.1.4-3: Monte Carlo Simulation Cells and Population Mean Results for Each of the Time Frames ...... 165 Figure 8.1.4-4: Histogram for the AS4 8 Hour Sample: The Frequency of Random Numbers Generated During the Monte Carlo Simulation ...... 166 Figure 8.1.4-5: Schematic Illustrating the Second Monte Carlo Setup: All Material Properties were Held Constant and the Results were Based Solely on the Variance Associated with the Regression Model Error ...... 167 Figure 8.1.4-6: Monte Carlo Simulation of the Regression Model Compared to the Actual Experimental Data ...... 168 Figure 8.1.4-7: Histogram for the MC simulations of the AS4-8Hr Regression Model ...... 168 Figure 8.2.1-1: Cell Growth at 8 Hrs into the In Vitro Experiment ...... 170 Figure 8.2.1-2: The Nuclei of Various Osteoblasts after 96 Hrs of Growth . 170 Figure 8.2.1-3: The Cytoplasm of the Nuclei ...... 171 Figure 8.2.2-1: AFM Surface Roughness Analysis of AS4 ...... 172 Figure 8.2.2-2: AFM Surface Roughness Analysis of P25 ...... 172 Figure 8.2.2-3: AFM Surface Roughness Analysis of T650 ...... 173 Figure 8.2.2-4: AFM Surface Roughness Analysis of P120 ...... 173 Figure 8.2.3-1: Cell Population as a Functin of Time on Four Types of Carbon Fibers ...... 176 Figure 8.2.3-2: AS4 Images of Cell Growth for Varying Time Periods ...... 177 Figure 8.2.3-3: P25 Images of Cell Growth at Varying Time Periods ...... 178 Figure 8.2.3-4: T650 Images of Exceptional Cell Growth at Varying Time Periods ...... 179 Figure 8.2.3-5: P120 Images of Exceptional Cell Growth at Varying Time Periods ...... 180 Figure 8.2.3-6: Growth on All Four Carbon Variations at 96 Hours ...... 181 Figure 8.2.3-7: Cells Migrating toward a P25 Carbon Fiber, at 24 Hours Culture ...... 182 Figure 8.2.3-8: Additional Image of Cells Migrating toward a P25 Carbon Fiber for Attachment and Growth. This image was taken at 8 Hours of Culture ...... 183 Figure 8.2.4-1: The Logistic-Malthusian Model being used to Predict the Cell-Carbon System Experimental Data ...... 184 Figure 8.2.4-2: Graphical Summary of All Collected or Calculated Data 185 xiv

Figure 9.1-1: Examples of Cell Size and Shape ...... 188 Figure 9.1-2: Osseous Tissue Formation after 24 Hours of Culture: Extracellular Matrix and Tissue (Left) along with the Nuclei of the Cells that Developed the Tissue (Right) ...... 189 Figure 9.1-3: Comparison of Carbon Fiber Roughnesses and Appearance ...... 192 Figure 9.1-4: A 20μm by 20μm of the AS4 Fiber and 40μm by 40μm Scans of the Other Carbon Fibers ...... 192 Figure 9.2-1: Various Computed Characteristics of Bone Tissue ...... 212 Figure 9.2-2: Monte Carlo and Experimental Data Comparison ...... 215 Figure 9.2-3: Incorporating the MC Method on the Time-Material Regression Model ...... 217 Figure 9.3-1: Comparison of Theoretical Methods for the AS4 Fiber ...... 218 Figure 9.3-2: Comparison of the Theoretical Models for the P25 Fiber ..... 220 Figure 9.3-3: Comparison of Theoretical Models for the T650 Fiber ...... 221 Figure 9.3-4: Comparison of Theoretical Methods for the P120 Fiber ...... 223

LIST OF TABLES

Table 3.2.3-1: Possible Applications of Industrial Polyaspartates ...... 32 Table 7.1.1-1: List of All Possible States for the Cells in the CA Grid ...... 113 Table 7.1.1-2: The Rules of the CA Program that Simulate Bone Tissue Regeneration ...... 113 Table 7.1.1-3: Graphical Representation of Tissue Component Represented in the Cellular Automation Program ...... 114 Table 8.1.2-1: In Vitro Experimental Cell Data ...... 152 Table 8.1.2-2: Cell-Carbon System Experimental Data ...... 154 Table 8.1.2-3: Logistic-Malthusian Model Application to the Cell-Carbon System Data ...... 154 Table 8.1.2-4: Using the GROWTH Function to Predict Cell Populations Beyond the Scope of the Established Experiments: The Orange indicates Projected Values ...... 156 Table 8.1.2-5: Using the GROWTH Function to Predict Cell Populations Beyond the Scope of the Established Logistic-Malthusian Model Approximations: The Orange - Projected Values through a Time of 216 Hours ...... 156 Table 8.1.4-1: Values for Material Roughness and Crystallinity for Each Type of Carbon ...... 163 Table 8.2.1-1: Data Measured from the in vitro Experiments...... 169 Table 8.2.2-1: RAMAN Spectroscopy Measurements of Average Crystal Diameter, La ...... 174 Table 8.2.2-2: Scherrer Equation Calculations for La and Lc ...... 174 Table 8.2.3-1: Cell Populations as a Function of Time on All 4 Carbon Fiber Variations ...... 175 Table 8.2.4-1: Compilation of All Theoretical and Experimental Data Associated with this Research ...... 185 Table 9.1-1: Summary of Surface Roughnesses of Carbon Fibers ...... 190 Table 9.1-2: RAMAN Spectrometer Measurements of Average Crystal Diameter ...... 193 Table 9.1-3: Average Crystal Diameter and Height as Measured Using X- RAY Diffraction ...... 194 Table 9.1-4: Results of Cell-Carbon System In Vitro Experimentation ...... 198 Table 9.2-1: Cell Counts after 96 Hours ...... 206 Table 9.2-2: Results of CA Program after 96 Hours ...... 207 Table 9.2-3: Cell Population Predictions based on the GROWTH Function ...... 209

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Table 9.2-4: Comparison of MC Simulation Results to Experimental Data ...... 216

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Chapter 1. Introduction

1.1. Research Overview

Tissue engineering has evolved into the forefront of the biomedical industry. Its dominance lies in the overall need to develop faster, more efficient ways of maintaining the quality of life to which patients are most accustomed. Since emerging in the late 1980s, tissue engineering has experienced its share of scientific challenges (Lanza, 2007). Among those are the adverse reactions caused by the implementation of biomaterials.

Despite the growing effectiveness of current implants and their materials, the associated successes have proven short term. These materials and their prostheses provide an effective and immediate solution to a patient’s problem, but their outcome is often time-limited. Various studies have shown that these materials tend to degrade, encouraging negative tissue response leading to implant loosening and eventual implant failure

(Batchelor, 2004). Corrosion, wear debris, inflammation, allergic reaction, and numerous other complications have been coupled with the use of these biomaterials (Bonfield, 1996). The foundation for these side effects is based on the fact that the majority of biomaterials being implemented today were developed from engineering practice; they combine the

2 ability to be tolerated by the body, with the mechanical properties sufficient to withstand anticipated physiological stress.

Current research efforts are fixated on developing a second generation of biomaterials; materials that can promote positive tissue response, successfully perform their function, and still maintain their effectiveness in a stressed physiological environment, for the duration of a patient’s life. To help warrant the dependability of more innovative biomaterials, much needs to be learned pertaining to how these materials interact with natural living tissues. Prior to understanding biomaterial-tissue interactions, the tissue itself needs to be fully understood. Many methods have been established to study and analyze native tissues during growth and regeneration processes. The problem however, lies in their ability to provide researchers with complete information regarding tissue behavior, as well as information pertaining to these tissues on a cellular level. This research is intended to eliminate the gaps generated from current methods, to establish a more dependable means of analyzing tissue regeneration processes.

1.2. Problem Introduction and Objectives of Research

The primary objective for all biomedical research is to maintain the proper quality of life for patients by restoring their body’s form and function. The patient’s health and well being should always be

3 maintained as the principal concern. Tissue engineering, also called regenerative medicine, is a novel field that uses various means to develop biological substitutes which are used to repair or even replace living tissues. Ideally, it is important with all research efforts, to focus on improving pre-existing initiatives or develop an original concept through which both patients and scientists will benefit. Because tissue engineering is such a new field, research efforts in this discipline are very novel, and provide many opportunities for improvement and further development.

Past and present efforts to restore form and function to healing tissues have been, for the most part, successful, but leave ample room for improvement. Evidence of this can be seen in the numerous side effects of implemented materials, as well as the short term successes of the methods in which they are utilized. Despite their noteworthy effectiveness, materials used for wound healing require higher biocompatibility, and the methods that incorporate them lack the ability to be successful over a long period of time. Methods used to monitor tissue regeneration processes lack continuity and dependability. It is the main focus of this research to not only develop more effective and dependable means of analyzing tissue regeneration processes, but to also determine which properties of carbon based biomaterials prove the most advantageous when trying to regenerate native bone tissue.

Chapter 2. Dissertation Statement of Research

The objective of this research is to simulate and predict behaviors of bone tissue and biomaterials both separately, and as they work together, to restore form and function during tissue regeneration and wound healing processes. This will yield a better understanding of what is actually taking place when the body’s bones go through the process of repair.

Theoretically based analysis methods are less invasive, and if developed properly, can be just as accurate as anything performed within the body.

These models will prove to be more efficient with respect to both time and cost when compared to patient interactive methods. Additionally, they will have to power to generate just as valuable and precise information without impinging on a patient’s quality of life.

The development of these theoretical models will be based on several established laws and methods in mathematical biology. The use of such methods will provide: (i) means for accurately predicting cell growth rates and proliferation behavior, (ii) ways in which cell population can be predicted based on the material properties of the biomaterials being implemented and (iii) a continuous way in which tissue density at the repair site can be analyzed. Because this research is biomedically

5 based, its focus encompasses various topics in science, engineering and medicine. The comprehension of these, including specifically: Cellular

Dynamics, Bone Tissue Dynamics, Mathematical Biology, Tissue

Regeneration and Wound Healing, and biomaterials research are imperative for this initiative.

Chapter 3. Literature Reviews

3.1. Bone Tissue Regeneration

All natural tissues are composed of cells. Perhaps one of the most dynamic tissues by nature is that of bone. Osseous tissue, or bone tissue, is the major structural and connective tissue of the body. It forms the rigid parts of bones that make up the skeletal system. The tissue itself goes through various calculated events during formation/generation and wound healing/regeneration. In both instances, cells create or recreate bone structure by working in unison, using pre-programmed information and signaling (Lanza, 2007). Full understanding of tissue creation and recreation processes will lead to a more complete perspective of how biomaterials are expected to assist, replace and/or function in intimate contact with living tissue.

3.1.1. Cellular Biology

The cell cycle is a complex series of events that takes place in a cell and leads up to the cells replication. It is the foundation for all forms of tissue formation. The process itself consists of four distinct phases: G1 phase, S phase (synthesis), G2 phase and the M phase (Mitosis)

(Smith & Martin, 1973). The complete cycle cannot advance to the next phase without first properly progressing and completing the previous one

(Elledge, 1996). Particular cells which have temporarily or reversibly stopped dividing are said to have entered the G0 phase, also referred to as a state of quiescence (Hartwell, 2001).

Cells experience the G1 phase first, where they increase in size. The next phase is the S phase, which cells cannot proceed to until the control mechanism of the G1 phase ensures that the cell is prepared for DNA synthesis. During the S phase, DNA replication occurs within the cell. Next the cell goes through the second Gap phase, also called the G2 phase. It is at this point that the cell continues to grow and the G2 phase control mechanism ensures that the cell is prepared for division. The final phase is

Mitosis, or the M phase. It is

here that cell growth stops and

all cellular energy is focused on

the orderly division of the

matured cell into two equal

daughter cells (Hartwell, 2001).

Once cellular division is

complete, each of the

daughter cells begins the

Figure 3.1.1-1: The Cell Cycle various stages of the interphase

(G1, S and G2). Although each of the phases of the cell cycle are not distinguishable morphologically, they do each have a very distinct set of biochemical processes that collectively prepare the cell for division

(Cyclin-Dependent Protein Kinases: Key Regulators of the Eukaryotic Cell

Cycle, 1995). A schematic of the cell cycle is shown in Figure 3.1.1-1.

Cellular division has an assortment of generated byproducts including secreted non-mineralized matrix. The process of cellular division lays the initial matrix groundwork and secretes the proteins necessary for developing the organic and non-organic portions of bone tissue. This explains why its comprehension is essential to the success of any tissue regeneration research.

3.1.2. Fundamentals of Cellular Dynamics

The cell cycle provides the foundation for the formation of osseous tissue (bone tissue). Bones are rigid organs that form part of the endoskeleton of vertebrates. Their main functions are to move, support and protect the various organs of the body, produce red and white blood cells and store minerals (Ham, 1972). Bone tissue is a dense type of connective tissue of various sizes and shapes, with a very complex structure. Their unique design makes them lightweight, yet still hard and strong. Osseous tissue gives bones their rigidity and honeycomb-like structure. There are approximately 270 bones in an infant, and 206 bones

9 in an adult human body (Bramblett, 1988, McDonough, 1934). The human skeleton is almost entirely comprised of bones and it provides the rigid framework of the body, holding it upright, giving it shape, and providing strength. Thus, its reliability is extremely significant, even in times of bone tissue regeneration and wound healing.

Despite popular belief, bone is not a uniformly solid material, but rather a very dynamic tissue involving several crucial elements. It’s this complex structure that makes osseous tissue so difficult to model. The two types of bone tissue are compact (cortical) bone and trabecular

(cancellous) bone. The hard outer layer of bone is composed of compact tissue and accounts for nearly 80% of the total bone mass of an adult skeleton. Its minimal gaps and spaces give bones their smooth and lustery appearance. The interior portion of all bones is composed of trabecular tissue, which is essentially a porous network of rod and plate like cellular elements that make bones lighter and allow room for blood vessels (Boulpaep, 2005).

At the microscopic level, there are several types of cells that constitute bone tissue. Osteoblasts are mononucleate cells responsible for bone formation. There are four main classes of osteoblasts, each associated with a particular cell’s level of maturity: Pre-Osteoblasts, Pre-

Osteoblastic Osteoblasts, Osteoblasts and Osteoblastic Osteocytes. They produce osteiod, which is the organic portion of the matrix of bone tissue,

10 and they are responsible for the mineralization of the osteiod matrix.

Osteoblasts also manufacture hormones, important mineralization enzymes and several additional matrix proteins. They are often referred to as immature bone cells. Bone lining cells are essentially inactive osteoblasts and they cover the entire available bone surface, functioning as a barrier for certain ions. Osteocytes are star-shaped cells that originate from osteoblasts and are the most abundant cell found in compact bone. Osteocytes also have four main classes, again associated with their age and level of maturity: Type II PreOsteocytes,

Type III PreOsteocytes, Young Osteocytes and Old Osteocytes.

Osteocytes are formed when osteoblasts migrate into and become trapped by surrounding bone matrix, which they themselves produce.

Osteocytes are responsible for several processes including reaching out to meet osteoblasts and other osteocytes for the purposes of communication. Some of their more important functions involve the formation of bone, matrix maintenance and calcium homeostasis

(maintaining a constant condition of stability). Studies have also shown that osteocytes are responsible for regulating the bone’s response to stress and mechanical load. Their more common place name is mature bone cells (Boulpaep, 2005). The final major cellular components of osseous tissue are Osteoclasts. These are the cells responsible for bone resorption

(remodeling bone to reduce its volume). Osteoclasts are very large,

11 multinucleated cells located near bone surfaces in what are sometimes called resorption pits. They are equipped with phagocytic like mechanisms which give them the ability to digest dead or dying cellular debris, making room for osteoblasts to migrate to and replenish the area with young bone cells and osteiod (Lanza, 2007, Boulpaep, 2005).

3.1.3. Wound Healing and Regeneration Concepts

Bone tissue regeneration, sometimes referred to as fracture healing, is a proliferating physiological process in which the body’s necessary cellular components facilitate the repair of any form of bone damage or fracture. There are several steps involved in this process, all facilitating the protection and rejuvenation of the areas surrounding the damaged tissue.

The length of the process is dependent upon the extent of the damage or injury. Additionally, the angle of dislocation or fracture can also play a crucial role in healing/regeneration time (Ham, 1972).

Three distinct phases occur during the process of bone healing and regeneration. The reactive phase occurs first. This phase involves the body “reacting” to the injury or problem through a series of protective actions. Initially, blood cells within the tissue adjacent to the injury site make themselves known, while nearby blood vessels work to constrict blood flow and prevent any further bleeding. The blood cells then proceed to form a blood clot. These blood cells, and those in the

12 immediate environment of the clot degenerate and die, and are survived by the surrounding fibroblasts (synthesizing cells). The fibroblasts then form a loose aggregate of cells, which becomes the foundation for the beginnings of preliminary tissue (Ham, 1972, Hunt, 1986).

Next is the reparative phase. This usually occurs days after the fracture or injury, and involves the development of collagen matrix and lamellar bone. The lamellar bone is formed from osteoblasts that have travelled via the vascular channels to the repair site. They penetrate the area and form the precursor to trabecular bone atop mineralized matrix.

Eventually, the precursor is replaced by trabecular bone, restoring the majority of the bone’s original strength (Brighton & Hunt, Early Histologic and Ultrastructural Changes in Microvessels of Periosteal Callus, 1997).

The final phase of regeneration and healing is the remodeling

phase, which is also the

phase in which the

majority of this research

is focused. It is during

this phase that

compact bone is

Figure 3.1.3-1: The Phenomenon of Apoptosis substituted for the previously developed trabecular bone. The trabecular bone is first resorbed by osteoclasts, creating a shallow resorption pit. Osteoblasts

13 then develop and deposit all the necessary components of compact bone including osteiod, mineralized matrix, pertinent enzymes and proteins (Steele & Bramblett, 1988). Depending on the severity of the injury and complexity of the regeneration, the process can take approximately three months, but eventually yields new formed duplicate bone retaining the original bone’s shape and strength (Brighton & Hunt,

Histochemical Localization of Calcium in the Fracture Callus with

Potassium Pyroantimonate: Possible Role of Chondrocyte Mitochondrial

Calcium in Callus Calcification, 1986). The remodeling phase of healing and regeneration is the main focal point of this research initiative, because it also pertains to what happens after apoptosis (programmed cell death) and what attempts to happen after necrosis (traumatic cell death). In the case of apoptosis, the remodeling phase leads to the proper ingestion and/or disposal of cellular debris, without inhibiting the organ’s ability to function. Figure 3.1.3-1 describes the phenomenon of apoptosis.

Necrosis, however, causes a malfunction in the ability of osteoclasts to digest the cellular debris, leading to dead tissue build-up. This phenomenon leaves in its wake very detrimental and sometimes fatal affects. Normal apoptosis confers advantages during an organism’s life cycle. It plays a key role in tissue homeostasis and contributes to the constant replenishment of fresh tissue. Despite its positive attributes,

14 apoptotic processes have been implicated in an extensive variety of diseases, and for this reason, are being researched more and more each day to determine if the underlying cause of the process can be identified and thus, controlled (About Apoptosis, 1999).

Another important aspect of the regeneration process is a phenomenon called Chemotaxis. This phenomenon involves bodily cells directing their movements according to certain chemicals that have been released in their environment. In the instance of tissue remodeling, chemotaxis occurs whenever a cell or aggregation of cells die and their debris is engulfed by osteoclasts. Osteoclasts are formed specifically for the purpose of ingesting dead cellular remains. The growth factors and chemicals released during chemotaxis are the main drivers behind attracting young bone cells (osteoblasts) to the regeneration site in order to repair the area. White Blood Cells play a crucial role in wound healing as well, because together, with proteins that are released during apoptosis and other forms of cell death, they form the osteoclasts that eventually ingest the cellular debris (Lanza, 2007).

A pictorial description of how bone tissue is constantly replenishing itself through cellular dynamics is illustrated below in Figure 3.1.3-2.

Figure 3.1.3-2: Bone Tissue Remodeling Process

The concepts behind the bone remodeling process are very interdependent and quite complex. This healing response is aimed at reconstituting a tissue that closely resembles the original tissue.

Dysfunctions of the different physiological phases of wound healing may be responsible for disorders associated with normal wound healing, and may also be the underlying basis for many fibrotic disorders. This is just one of the many reasons why the complete understanding of cellular biology and dynamics is so crucial when involved with tissue regeneration based research (Ibelgaufts, 2009).

3.2. Overview of Biomaterials

Over the past several decades, many materials have been researched for use in biomedical applications. Only a select few,

16 however, even warrant consideration for prosthetic devices; even fewer being successful in assisting the wound healing process and tissue regeneration.

The beginning of biomaterials closely coincides with the beginnings of surgery. The ancient Egyptians used prostheses to substitute for amputated limbs. It was discovered that back in 1000 BC, a lady was fitted for a wooden prostheses to replace her big toe, which was amputated due to gangrene. Her mummified remains are still currently in existence. During these times, it was also practice to fit prostheses to bodies intended for mummification. During the Veda period of ancient

Indian literature, which dates back to 1500 BC, it is mentioned that artificial legs, eyes, and teeth were used. Once anesthesia was invented,

it became possible to fit implants inside the body. This

allowed metal plates to be attached by screws to the

fractured end of bones, thus repairing complex fractures

(Batchelor, 2004).

Problems of arthritis received attention in the mid

1900s when Philip Wiles, who at the time was working in Figure 3.2-1: Sir John Charnley's Hip Replacement London, developed an articulating metal joint that could be implanted inside the hip. This was followed in 1962 by the design of Sir John Charnley, which combined polymer and metal components.

These hip replacements were designed to closely simulate the original hip

17 joint. They consisted of a separate “cup” and a “ball” joined to a “stem” for securing this part of the prosthesis inside the femur bone (Batchelor,

2004). A picture of Charnley’s hip replacement design is shown in the x- ray in Figure 3.2-1.

Research and technological advancements have led to the increased demand for biomedical implants by patients seeking to maintain a full and vigorous lifestyle for the duration of their existence. The combined demands of patients and government have led to a need for high performance biomedical materials that can last up to thirty years or more, which brings the medical industry to its current biomaterials status

(Biocompatible Polymers, 1993). The succeeding sub-sections of Chapter

3, Section 2 discuss the numerous types of biomaterials and their applications.

3.2.1. Metals

Despite the copious types of metal alloys known to man, only a few merit even preliminary consideration for use as a tissue regeneration or implant material. The body is a relatively corrosive environment which cannot tolerate even minuscule concentrations of most metallic elements. Additionally, most metals cannot perform properly within this environment. Of the remaining possible metals, several such as Tantalum and the noble metals, do not possess suitable mechanical properties for

18 use in orthopaedic applications. This leaves only three major types of metals as candidate materials for tissue repair aides: Titanium Alloys,

Cobalt-Chromium Alloys, and Stainless Steels.

Titanium biomaterials have been one of the most successful families of materials since biomedical device implementation first began.

Attempts to use titanium for general implant fabrication, dates back to the late 1930s. Titanium is the fourth most abundant metallic element in the earth’s crust and it has evolved into a very desirable biomaterial because of the minimal tissue reaction it seems to receive once placed in a physiological environment (Boyan, et al., 1998). Titanium is a light metal with good mechanical and chemical properties, both important features for biomedical implants. The titanium alloy most widely used for manufacturing implants is Ti-6Al-4V. This alloy is composed of Titanium, 5.5

- 6.5 wt.% Aluminum and 3.5 – 4.5 wt.% Vanadium. Another popular titanium alloy is nickel-titanium. This alloy is well known as a shape memory alloy because it possesses the function of returning to its original shape from a plastically deformed shape due to a small temperature change within the material. Nickel-titanium also exhibits superior ductility, fatigue strength biocompatibility and corrosion resistance (Swee, 2004).

Titanium alloys have excellent resistance to stress corrosion cracking and corrosion fatigue in bodily fluids, and have proven to permit some bone growth at bone-implant interfaces (Metals for Implantation, 1999).

Cobalt alloys were originally used for aircraft engines, and were not considered a candidate for biomedical applications until the 1930s. There are basically two types of cobalt chromium alloys. One is the cobalt

CoCrMo alloy, which is usually used to cast a product. The other is the

CoNiCrMo alloy, usually wrought by hot forging. The castable CoCrMo alloy has been used for many decades in dentistry and recently in making artificial joints. The wrought CoNiCrMo alloy is relatively new, and is now being used for making the stems of prostheses for heavily loaded joints such as the knee and the hip. Cobalt-based alloys are highly resistant to corrosion, but as with all metals, galvanic corrosion can occur. They are also excellent in wear and pitting corrosion resistance (Black, 1999).

Within the stainless steel family, austenitic stainless steels, especially types 316 and 316L are the most widely used for biomedical implant fabrication (Lima, Bosch, Lara, Villarreal, & Pina, 2005). They generally contain at least 12% chromium, and because of their low content of impurities as well as their passivated finish, they are somewhat suitable for implantation in the human body. Forged stainless steel has greater yield strength than cast stainless steels, but has lower fatigue strength than other implant alloys. In early hip implants, stainless steel was used because of its good strength, ability to work harden and resistance to pitting corrosion. For economic comparisons, stainless steels are typically

1/10th to 1/15th the price of other typical metallic biomaterials.

Metals are almost exclusively used for load-bearing implants. All three categories of bio-metals (Stainless Steels, Cobalt-Chromium Alloys, and Titanium Alloys) have been used for select components of hip and knee prostheses. Some of these applications are shown in Figure 3.2.1-1.

Figure 3.2.1-1: Most Established Biomedical Applications of Metals

Despite their dependability and popularity, research efforts are always ongoing to improve the biocompatibility of metallic implant materials.

In Figure 3.2.1-1, the elements of the knee replacement include a femoral and tibial component which are composed of Titanium, Stainless

Steel, or a Cobalt-Chromium Alloy, and an articulating component. The articulating component is almost always constructed out of some form of polymer, and takes on the role of cartilage, thus reducing the amount of friction and wear on the artificial joint (Black, 1999). The total hip

21 replacement is somewhat more complex, as it has more components.

The main components include the stem, the socket or cup, and the ball.

The socket and stem need to be made of a material with high strength and mechanical fatigue strength because of their functions. The stem is what is ultimately inserted into the patient’s femur bone and is constructed of one of the three main types of bio-metals. The socket component is also designed out of a metal, with the “ball” or femoral head usually being constructed out of a polymer such as UHMWPE.

In all biomedical applications, the design has a

substantial influence on the selection of the material(s) for

the device, and vice versa (Batchelor, 2004). Another area

in which metals have proven successful for biomedical

Figure 3.2.1-2: applications is dentistry. When implants are required, Cross Section of Fixator Screw relating to the teeth and gums, metals are commonly used to affix or anchor the implant to the surrounding hard or soft tissues.

Fixators, as they are often called, can be items such as pins, needles, blades, discs, or even screws (Batchelor, 2004). An example of a dental implant fixator is shown in Figure 3.2.1-2.

3.2.2. Polymers

Currently, polymers are the most widely used materials in biomedical applications, and have been used in the augmentation and

22 repair of the human body with much success (Black, 1999). Polymers are often found in orthopedic implants working collectively with other materials, usually bio-metals. Polymers and polymer-based materials possess a wide spectrum of properties, which allow them to be used in a diverse range of medical applications. As a scientist or engineer is determining which polymer they feel suitable, their choices are often based upon the body and tissue reactions to the material, the mechanical and thermal properties of the polymer, and whether or not the polymer can be made to fit the application (Black, 1999).

The properties of polymers that make it so compatible with the human body are strongly influenced by material composition, chain structure, and molecular weight. Some of the main polymers used in the biomedical industry today include: Polyethylene (PE), Acrylic, Ultra-High

Molecular Weight Polyethylene (UHMWPE), Polylactic and Glycolic acids,

Polyester, Polyvinyl Chloride, and Silicon Rubber. Each of these will be discussed in upcoming sections, in further detail, with respect to their most common biomedical roles (Dee, 2002).

Polymers are significantly different from other biomaterials in that they have the ability to be deformed to a greater extent before failure.

Most biomaterials do not have the ability to perform in this ductile manner, which is why polymeric biomaterials are so desirable for specific applications. UHMWPE is a particularly interesting material because it is

23 superior to other polymers in the areas of toughness and strength.

UHMWPE is also highly resistant to corrosion at ambient temperatures.

Polymers in general have physical properties most similar to natural tissues, which is why they are so well tolerated by the body. Because the properties of polymers are influenced by their molecular structure, researching and developing these materials seems to show significant promise in developing more successful biomedical materials.

Polymers are often working together with metals to complete a biomedical implant or prosthesis. In the case of a hip replacement, such as the one shown in Figure 3.2.1-1, the femoral head of the implant is almost always formed out of UHMWPE. There are several types of biomedical applications in which polymers play a crucial role. A diagram depicting the different types of polymeric materials, and their prospective applications is shown in Figure 3.2.2-1.

Cardiovascular Implants Orthopaedic Implants Tissue Engineering

Polyethylene UHMWPE Polylactic Acid Polyvinyl Chloride Polymethylmethacrylate Polyglycolic Acid Polyester Polylactide co-glycolide Silicone Rubber Polyethylene Terephthalate PTFE

Figure 3.2.2-1: List of Various Polymers and their Biomedical Applications

Some additional applications that illustrate the wide variety of uses for polymers in the body are listed in Figure 3.2.2-2, along with some illustrations of these examples (Dee, 2002). These successful applications

24 of polymers came from years of research, and will lead to even more successes with polymeric biomaterials.

Figure 3.2.2-2: List of Polymer Based Biomedical Applications

3.2.3. Biopolymers

Biopolymers have become a family of materials in their own right.

More successful integration of biomaterials within the body is dependent upon research that can help identify the differences between naturally and synthetically prepared polymers. In general, the structure and physical properties of polymers remain one of the most active and challenging areas of materials research, particularly when polymers are used within a physiological environment. Ideally, the goal is to learn how

25 to more closely replicate naturally occurring biopolymers in order to create more successful and biocompatible materials and medical devices. A “biopolymer” can be described in one of two ways: a)

Polymers that are produced by biological systems such as microorganisms, plants and animals or b) polymers that are synthesized chemically, but are derived from biological starting materials such as amino acids, sugars, natural fats or oils (Clements, 1993). There are various types of biopolymers that are typically defined on the basis of their chemical structure, which helps identify the functions these polymers serve in living organisms. DNA, proteins, polysaccharides, polyhydroxyalkanoates, polyphenols and polysulfates are all examples of naturally occurring biopolymers (Technology, 1993).

Technological advancements have given scientists more innovative tools to investigate and manipulate an assortment of biological systems.

Genetic engineering is a field that exhibits extraordinary control over various aspects of gene expression (Assessment, 1991). Using this ability to manipulate the genetic blueprint of a natural polymer leads to the development of polymer chains which are virtually uniform in length, composition, stereochemistry and spatial orientation. A good example of this is the protein polymer silk. Silk is produced commercially by silkworms, and can now be made in recombinant microorganisms. The advantages within these methods lie in the possibilities of generating polymers with

26 exceptional structural purity as well as the ability to manipulate the production of biopolymers to create new materials.

For polymers other than proteins, the situation becomes more complicated. Genetic manipulation within these materials permits the control of the production of enzymes that are in turn responsible for the production and polymerization of the building blocks that make up the final polymer. For the majority of polymers, technology such as genetic engineering plays a very preliminary but very significant role that affects the entire production process of the biopolymer. Because of the advances made with such technologies, it is now possible to genetically modify organisms so they produce greater quantities of a particle polymer, as well as modifying organisms to produce novel materials.

Ultimately, it may be possible to construct biological systems for the purpose of producing entirely new classes of polymers (Poirer, 1992).

Biopolymers can be produced through a variety of mechanisms.

Three of the main techniques that produce biopolymers include their derivation from microbial systems, extraction from higher organisms such as plants, or chemical synthesis from basic biological building blocks. In recent years, particular focus has been given to biopolymers that are produced by microbes. This interest stems from the highly publicized disposal problem of the traditional oil-based thermoplastic polymers that are most commonly used (Sikes, 1993). It is in our environment’s best

27 interest to search for a better biodegradable alternative. It is on the microbial level that the tools of genetic engineering can be most readily applied which is why the biopolymers produced by these microorganisms are being actively researched. The three most worked with biopolymers produced by microbial systems are polyesters, proteins and polysaccharides (Technology, 1993).

Within these microbial system polymers are an entire class of natural thermoplastic materials called Polyhydroxyalkanoates (PHAs). These materials are microbial energy reserve materials that accumulate as granules within the cytoplasm of cells. They are genuine polyester thermoplastics with properties similar to oil-based polymers and their mechanical properties can be tailored to resemble anything from elastic rubber to hard crystalline plastic (Doi, 1992). The inherent biocompatibility of PHAs suggests that they have several potential applications in medicine including controlled drug release, surgical sutures, bone plates and wound care. PHAs could also be used as structural materials in personal hygiene products and packaging applications (Seeman, 2003).

Along with PHAs, proteins are a very common variety of microbial system polymers. More than 25 genetically altered proteins have been synthesized in microorganisms and are being transformed into films, gels and fibers. One of the first proteins to be genetically engineered and introduced commercially was ProNectin F™. It was designed to serve as

28 an adhesive coating in cell culture vessels. This polymer has two distinct peptide blocks: one that possesses the strong structural attributes of silk and the other has the cell-binding properties of the human protein fibronectin. ProNectin F™ has exhibited excellent adhesion to plastic surfaces and thus can be used to attach mammalian cells to synthetic substrates (Corp, 1990).

On a larger scale, several natural polymers have been found in plants and other larger organisms. The most common of these is Starch.

Starch is the principal carbohydrate storage product of higher plants.

Starch polymers can be extracted from corn, potatoes, rice, barley, sorghum and wheat. These polymers accumulate in plants as insoluble energy storage granules. Because of its low cost and widespread availability, starch has been incorporated into a variety of products.

Recently it has been

incorporated as a

biodegradable additive in

traditional oil-based

commodity plastics for

packaging and garbage

bags. There are some

Figure 3.2.3-1: Cellulose Nanofibers being Woven and Controlled issues however with its by Cellulose Creating Bacteria (Virginia Tech Wake Forest University School of Biomedical Engineering) ability to degrade once it

29 has been added to synthetic polymers (Technology, 1993).

Plant cellulose is another natural polymer receiving attention.

Cellulose is one of the most abundant constituents of biological matter and is the principal component of plant cell walls. Cotton is made up of approximately 90% cellulose and wood consists of approximately 50% cellulose. Cellulose serves as an important material for many industries including the fabrication of fibers, thickening solutions, gels, detergents, shampoos, toothpaste, skin lotions, paper, ceramics and latex paints.

Figure 3.2.3-1 illustrates one of the many biomedical applications of cellulose working in conjunction with bacteria. A variation of cellulose named hydroxypropylmethylcellulose (HPMC) has shown considerable promise as a useful agent in lowering blood cholesterol levels (High

Molecular Weight Cellulose Derivative Shown to Lower Cholesterol, 1993).

Along with Cellulose, Lignin is a plant derived polymer found in woody and herbaceous plants. Its principle function is to provide

structural support in plant cell walls.

At the present time, most of the

lignin that can be recovered from

plants through a pulping process is

burned as an on-site fuel source

Figure 3.2.3-2: Hyaluronic Acid, One of the Main (Northey, 1992). It is being Components in Natural Tissue, Has a Fundamental Role in the Physiologic Processes of Tissue Repair increasingly used in non-energy

30 applications such as road dust control, binding agents in molding applications and animal feed. The ionic properties of lignin-sulfonates allow them to act as dispersants and therefore be used to prevent mineral buildup in boilers and cooling towers and as thinning agents in oil drilling mud and concrete admixtures (Northey, 1992). Other plant based polymers and polymeric acids with similar origins are being pursued for similar purposes.

One other natural plant polymer being investigated for various applications is Hyaluronic Acid. This is a very high density polymer, making it very hydrophilic. It is extremely flexible and has a high viscosity, which has gives it its various biomedical applications. It has the ability to detect various diseases such as liver cirrhosis, arthritis and tumors. It has been used as a scaffolding material for ear surgery. During eye surgery, it has been used to protect the corneal tissue and has also been used during retinal reattachment procedures and in glaucoma surgery. Various research efforts have proven its effectiveness stimulating tissue repair for wound healing, as well as its ability to repair flexor tendon lacerations. It is shown in Figure 3.2.3-2 in its more natural state, whereas in Figure 3.2.3-3 it is being shown assisting in the repair of torn meniscus tissue in the human knee joint. Other biomedical applications include anti-adhesion and scar control during various surgical procedures (Technology, 1993).

Figure 3.2.3-3: Hyaluronic Acid Working to Repair Cartilage in the Knee (The Pain and Spine Consultants of Pennsylvania)

Some polymers produced synthetically originate from biological starting materials, and are then chemically polymerized. Even though these polymers aren’t produced by biological systems, they are derived from basic biological building blocks which indicates they yield some of the same properties as microbially or plant-derived biopolymers.

Lactic Acid (lactate) is a very commonly known polymer created using chemical polymerization. Its natural molecule is widely employed in foods as a preservative as well as a flavoring agent. It is also used biomedically in intravenous and dialysis solutions. Lactic acid is the main building block in the polylactide family of polymers, which includes polylactic acids and glycolic acids. Polylactide polymers are the most widely used biodegradable polyester material, making the majority of its uses biomedically related. Some of its biomedical uses include the

32 controlled release of antibiotics and anticancer agents, surgical sutures and experimental orthopedic surgeries to fill bone defects (Langer, 1993).

Polyamino acids are the other main polymers derived from biological starting materials. There are several different types of polyamino acids and therefore several applications for which they are used. Polyamino acid microspheres can be used to encapsulate drugs and agricultural chemicals, ensuring their release in a controlled fashion.

Polyaspartate polymers, derived from aspartic acid are analogues of natural proteins, particularly the rich proteins that can be found in oyster shells. Polymers based on polyamino acids are also known for their corrosion resistance capabilities and biodegradability, leading to a vast number of potential applications, illustrated in Table 3.2.3-1.

Table 3.2.3-1: Possible Applications of Industrial Polyaspartates

Application Function

Water Treatment Antiscaling, anticorrosion, flocculation, Cooling towers, evaporators, desalinators and boilers Dispersants Detergents, paint pigments, drillig mud, portland cement Air Pollution Control Remove sulfur dioxide Ceramics Promotion of crystallization of specific minerals Oil Field Applications Prevent mineralization and corrosion in well holes Fertilizer Preparation Prevent calcification of phosphate slurries Mineral Processing Antiscalants used to keep ores at an optimum size after grinding Textile Industry Addition of crystallization regulators resulting in better fibers Superabsorbants Diapers Dental Treatment Tartar control agents Biomedical Devices Heart valves, prevention of pathological calcification, drug delivery, surface coatin for implants

In recent years, there has been a steady increase in the biopolymer research activities being conducted within governmental agencies. Most of these efforts stem from the need to address the environmental impacts of petroleum-based polymers. With the potential of biopolymer materials

33 being very diverse, several governmental agencies have strengthened their biopolymer research labors.

Within the Department of Defense, the Office of Naval Research has a Biopolymeric Materials Program investigating biologically derived materials for possible use in marine environments. Some of the materials being studied include thermoplastics, coatings, adhesives and elastomers.

The Army Research and Development groups are also actively researching biologically derived materials and the processing mechanisms of biological systems. The ability of biological systems to produce a vast range of materials under mild processing conditions, without creating toxic byproducts, could lead to a new, environmentally sensitive manufacturing method. The army’s three main areas of focus are: advanced biomolecular materials, biodegradable polymers and

“intelligent” materials. Some of their current projects involve: fermentation production and plant sources, purification, chemical modification of natural polymers, blending, processing into brown films, injection molding, biodegradation kinetics and the evaluation of toxicity of materials in marine and soil environments (Technology, 1993).

The National Science Foundation has several programs that either directly or indirectly support biopolymer research. Some of their research focuses on gene transfer, macromolecular structure and function,

34 metabolic pathways and molecular self assembly, all of which have positive spillover effects in biopolymer development.

The Department of Agriculture is supporting a variety of biotechnology programs including gene cloning in microorganisms, nucleic acid hybridization, synthesis of nucleic acids and proteins, and improving the quality of woody plants for the extraction of cellulose (Shih,

2004). One of the biggest efforts in the department of agriculture is researching the use of cornstarch for degradable plastics.

The National Institutes of Health have played a central role in expanding this country’s research in the area of biopolymers. One of its main purposes is expanding knowledge in the fundamental understanding of biological processes at the molecular level. The NIH supports basic biotechnology research in: recombinant DNA techniques, gene mapping and protein engineering. It also supports projects dealing with cellular and molecular biology, biophysics, immunology, virology and pharmacology (Peng Yin, 2008).

The Department of Energy has a large general effort in biotechnology, focusing on projects in structural biology and the human genome. Most of their research is directed toward understanding the structure-function relationships of biological molecules and the synthesis of proteins. A large number of DOE programs are more directly involved with biopolymer research, in particular the National Renewable Energy

Laboratory, who is exploring how cellulosic biomass can be converted to sugars that can then be fermented into ethanol (Technology, 1993).

There are also several areas of research focusing on biopolymers, in which commercial activity is taking place. These are mainly dealing with biodegradable polymers and polymers that can be used for biomedical applications. In the area of biodegradable polymers, starch-based products remain the highly researched. Properties of these materials can be varied depending on the types of starches that are being used and the other biodegradable materials that are being used. Early applications of these polymers include degradable golf tees, loose-fill packaging, compost bags, cutlery, pharmaceutical capsules and agricultural mulch films.

Despite these products constituting a large amount of the biopolymer research, biomedical polymer science research is the most dynamic area of material research. Biologically compatible materials are increasingly being used in a broad range of medical treatments (Zbib,

1993). The three main biomedical market segments are wound management products, polymeric drugs and drug delivery systems, and orthopedic repair devices.

Of those three, the wound management research is the most developed. The wound repair market consists mainly of absorbable sutures, surgical mesh, clips and staples, all of which are currently

dominated by a handful of

companies such as Ethicon,

U.S. Surgical Corporation,

Davis & Geck, Du Pont and

Pfizer. The main polymers

used in these products are

polylactic-polyglycolic acid Figure 3.2.3-4: Hybrid Polyester Surgical Mesh and Hydrogel and related compounds such as polydioxanone. Other biopolymer materials, such as chitin and modified cellulose may also be used as sutures. Figure 3.2.3-4 is a close-up of a hybrid polyester/hydrogel surgical mesh. The principle advantage of utilizing these materials in wound care is that they form natural bonds with surrounding tissue and thereby facilitate the healing process.

In addition to using polyester/hydrogels for surgical purposes, bioadhesives are a new class of materials that could serve as suture enhancements, means for attaching prostheses, and various dental applications. It is a long term goal to use biopolymers to facilitate tissue and organ regeneration and serve as vascular support meshes for blood vessel regeneration (Langer, 1993).

The research focusing on orthopedic repair products faces several technical challenges. In addition to high strength, the biopolymers that may be used for orthopedic applications must have prolonged, well

37 controlled rates of degradation because complete ligament and tendon healing requires up to a 1-2 year time period to completely heal, along with the fact that the degradation end products must be absorbed safely by the body. Even though the end product in most biodegradable orthopedic devices is lactic acid, the FDA is constantly assessing whether these devices cause soft tissue irritation or have toxic side effects. Despite the obstacles this research has already overcome, much is left to be discovered.

3.2.4. Animal Inspired Biomaterials

Mimicking nature to develop more dependable and long lasting biomaterials seems to be the forefront of most biomedical material research. Naturally occurring polymers are providing the framework for which we need to synthetically copy in the most efficient way possible.

Various efforts have focused on learning how mammals and other animals can create such materials, some of which are highlighted in the succeeding sections.

3.2.4.1. Self Healing Polymer Composites: Mimicking Nature to

Enhance Performance

Nature’s ability to heal has inspired new ideas and new mechanisms in the engineering community. From nature’s inspiration,

38 engineers and chemists have proposed different healing concepts that offer the ability to restore the mechanical performance of the material.

Most materials in nature are self-healing composite materials and have offered significant inspiration in the development of self-healing technologies that are currently being developed for fiber reinforced polymeric composites. Most recently, research has attempted to mimic natural healing through the study of mammalian blood clotting and the design of vascular networks found in biological systems.

In most biological organisms, there exists a highly developed multifunctional vascular network that distributes fuel, removes waste and controls internal temperature, effects self-healing and many other biological roles. The reason this network has these capabilities is its fluid distribution and collecting systems have a branching hierarchical network.

This allows for easy access to the fluid for all tissues. The branching design also minimizes the power required to distribute and maintain supporting fluid functions. This system is also reconfigurable in response to various circumstances, through self adjusting vasoconstriction and dilation in mature tissue (Taber, 2001). This phenomenon indicates that the future of the self-healing concept for composite materials relies on the development of a continuous healing network embedded within a composite laminate that delivers healing agents from a reservoir to regions of damage.

Mammalian blood clotting is another phenomenon that has been studied. It evolves around the reactions of a series of active enzymes and their inactive precursors known as clotting factors. Its intrinsic system takes on the form of a cascade or waterfall of reactions and is initiated by damage that breaches the endothelial cells that line the blood vessels.

Recent research has shown that one of the most notable features about this system is that despite how rapidly it takes place, system malfunction is extremely rare. Biomimetic hollow fiber self-healing mimics mammalian self-healing in that a liquid healing agent leaks from a region of mechanical damage. In mammalian blood clotting, rapid response is needed to arrest bleeding, but in biomimetic self-healing, rapid response is needed to restore some degree of structural integrity and prevent crack propagation. A good example of where this technology could be very useful is for an airplane in flight. If at any time after damage might occur, it needs to be able to withstand a maximum limit load while still in flight

(Trask, 2007).

3.2.4.2. Regeneration of ACL Tissue in Large Animal Model

In some cases animal-inspired polymers have lead to the development of synthetic polymers which have been successfully implemented in tissue repair. In this particular instance scientists working for Serica Technologies, Inc., a growth stage medical device company,

40 has received the Cabaud Memorial Award from the American

Orthopedic Society for Sports Medicine because of their pre-clinical research demonstrating the potential of one of their synthetically developed materials to regenerate and help re-grow anterior cruciate ligament (ACL) tissue in the knee in a large animal model.

The study itself provided the first ever

evidence of sustainable ACL tissue

engineering. Current repair options,

which include either an autograft or

allograft have well documented

debilitating side effects. This research

states that by utilizing their silk-based Figure 3.2.4.2-1: Silk Fibroin Based ACL Graft by Serica Technologies biomaterial in a scaffold, it helps encourage the development of functional ACL tissue, thereby avoiding the limitations and lengthy rehabilitation assisted with other existing options. A silk based biomaterial ligament graft is shown in Figure 3.2.4.2-1.

In the goat model that was studied, once the silk scaffold had been implemented, no initial signs of acute inflammation, swelling or scar formation occurred. This research deems it has generated the first successful 12 month ACL regeneration data for a large-animal model with an off-the-shelf product. The natural silk biomaterials associated with the scaffold are designed to help stabilize soft connective tissue structures

41 such as ligaments and tendons, following a surgical repair. The silk itself is from the fiber of the B. mori silkworm. Silk has a proven track record of safety over centuries of human use, and is projected to provide predictable and controlled bioresorption by the body. In pre-clinical studies, 100% of Serica’s silk-based products are shown to be bioresorbed at slower rates than other common structural proteins such as collagen, which facilitates optimal healing. Its products also require no rehydration or advance preparation for surgical implantation (Trask, 2007).

3.2.5. Polymers of the Sea

In addition to mammalian inspirations for developing synthetic polymers, the oceans have been found to be a rich source of proteins, polysaccharides and many other polymeric compounds. Because marine organisms live in a variety of different environments, some of them being extremely harsh, sea animals have developed polymers with a wide range of properties. A good example would be the hard calcium carbonate shell of the abalone, which is held together by glue composed of proteins and sugars. Ocean species in polar climates are able to survive extremely cold temperatures by producing antifreezing proteins.

Other proteins regulate the mineralization processes involved in the creation of shells and crystals, such as the calcite crystals of sea urchin spines. Many other polymers exist within the oceans’ waters, several of

42 which are described in succeeding sections (Technology, 1993). These polymers, along with various others, are opening the doors to natural technologies that can be used to develop more successful biomaterials.

3.2.5.1. Material Design Principles of Ancient Fish Armour

Dermal scales of a fish that first appeared millions of years ago are

showing promise in better

understanding mineralized tissues.

As the fish itself, now called the

“living fossil”, along with its real

name the Polypterus Senegalus, it

evolved over the centuries, and

noted for its scales, or dermal

armor, also evolved in terms of its Figure 3.2.5.1-1: The Polypterus Senegalus multi-layered material structures and overall geometries. This was to help increase its protection against predators as well as increase its ability to perform a number of both mechanical and non-mechanical functions.

Several observations have been made regarding this species’ ability to evolve according to its needs. Several parallels have been made between the evolution of armor in the animal world and the change in design of human designed and engineered body armor. Both designs seem to illustrate a balance between protection and mobility to maximize

43 survivability. The design strategies used by such mineralized biological tissues are of great interest to researchers because of the impact that these strategies could have in various engineering applications. The

Polypterus Senegalus is pictured in Figure 3.2.5.1-1. Thus far, much has been learned about this fish’s dermal armor including: crystal nucleation, growth and morphology, biomacromolecular intercalation and reinforcement, modulation of crystal texture, stabilization of amorphous phases and heterogeneity. The one crucial element that remains uninvestigated is the mechanical properties of the dermal armor plates and scales of this fish. To learn more about these, a sample of these fish, which are still alive today in the fresh water shallows and estuaries of

Africa, was analyzed to determine its armor’s penetration resistance and its elastic and plastic mechanical properties. Instrumented nanoindentation was used to learn more about the four layers that compose the cross-section of one whole scale.

After the various tests were conducted, it was determined that what contributes to the overall penetration resistance of these fish scales is their complex and multiscaled design along with the multiple, distinct reinforcing layers, each of which has its own unique deformation and energy dissipation mechanisms. Each of the reinforcing layers was comprised of different materials, whose sequence was also deemed as important. The sequence of the layers helps to reduce weight while

44 maintaining the required mechanical properties and promotes the most advantageous circumferential cracking mechanism, which helps prevent radial scale cracking. The presence of additional stratified layers that have plywood like structures, serve as a secondary line of defense for the deepest of penetrations. It also prevents catastrophic crack propagation and increases energy dissipation and fracture toughness. Incorporating such multiscale principles into the design of improved engineering biomimetic structural materials has the potential of producing an ultimate protective armor concept for humans (Bruet, 2008).

3.2.5.2. The Transition from Stiff to Compliant Materials in Squid Beaks

The beak of the Humboldt

squid, as shown in Figure 3.2.5.2-

1 represents one of the hardest

and stiffness wholly organic

substances known. The beak

itself is also one of the best Figure 3.2.5.2-1: The Humboldt Squid Beak is One of Engineering's Most Fascinating Materials because of its Assortment of Mismatched Tissues engineering examples of a system with grossly mismatched tissues. Through various experimental techniques including a series of chemical analyses, it was determined that beak stiffness was undoubtedly influenced by both micro-architectural and molecular factors. Stiffness was proved to be closely correlated to

45 the incremental and complementary distributions of two biopolymers: chitin and a family of His- and dopa- containing proteins. Not all of the proteins involved have been fully characterized, but it was definite that the stiffness of the beaks was directly related to the amount of cross- linking and hydration of the proteins.

One of the most significant things about this study is how it illustrates the importance of water in the functional properties of biomolecular materials. It was also proven that the hydrated beak exhibited a large stiffness gradient, spanning almost two orders of magnitude. The benefit of these findings will lead to a knowledgeable process for successfully attaching mechanically mismatched materials in various engineering and biological applications (Miserez, 2008).

3.2.5.3. Bioinspired Structural Materials

The diversity of structural biological materials that exist in nature, even within a single species, as well as the complexity, multifunctionality, and multiscale nature of their structure-property relationships has been studied extensively for many decades. Long term goals of biomedical material engineers involve creating a synthetic material based on the mechanical design principles found in the many biological materials.

If a synthetic material could be developed that took advantage of the mechanical design principles found in nature, that material could

46 significantly transform many fields including materials science, mechanical and civil engineering aeronautics and astronautics (Ortiz,

2008).

Recently, researchers were able to develop a multi-layered alumina platelet-reinforced nanocomposite inspired by the inner nacreous layer of many seashells. The platelets possess very high ultimate tensile strength, with only a plate thickness of approximately 200 nanometers.

Seashells have provided the inspiration for various research initiatives. A seashell’s multilayered structure and prismatic calcite outer layer, along with most seashell’s primary ingredient called nacre, several realizations have been made pertaining to their physical properties.

Figure 3.2.5.3-1: Seashells are of Interest because of Their Multi-Layered Structures and the Different Properties of Each

Recent models and studies of nacre show its complex organic- inorganic structure. Individual platelets within the nacre can each exhibit considerable plasticity before fracture upon penetration (Ortiz, 2008). An assortment of seashells is pictured in Figure 3.2.5.3-1.

It seems as though the theme for understanding the majority of sea related materials is their layered structures, and how each layer provides different properties and orientations to maximize protection and penetration resistance. At the same time, the orientation of these structures optimizes their ability to self heal and provide the most efficient means for survival.

3.2.5.4. Mussel Inspired Surface Chemistry for Multifunctional Coatings

Mussels have the unique ability to adhere to wet surfaces because of the adhesive proteins they secrete. These proteins have inspired researchers to use the same surface chemistry to coat multifunctional polymers in order to increase their adhesive properties. Various polymers

were dip coated in an aqueous solution of

dopamine. The self-polymerization of

dopamine helped form a thin surface-

adherent film on a wide variety of both

organic and inorganic materials including

Figure 3.2.5.4-1: A Mussel Shell (Nerites, noble metals, oxides, polymers, Inc. 2004)

semiconductors and ceramics. The chemical modification of these

materials increased their adhesive properties while at the same time

providing a versatile, secondary platform for surface-mediated reactions

to occur. Figure 3.2.5.4-1 shows a typical mussel. The adhesive properties

of mussels make them a very desirable and realistic coating for many

biomedical implant devices. Despite its novel nature, this two step

method of surface modification is distinctive in its ease of application, use

of simple ingredients and mild reaction conditions. Because of these, this

method, inspired by the adhesive properties of mussels, could be very

successful on many types of materials of many complex shapes, and

have the capacity for multiple end-uses.

3.2.5.5. Stimuli Responsive Polymer Nanocomposites Inspired by the

Sea Cucumber Dermis

Possibly one of the most interesting

animals in the ocean is the Sea Cucumber. Just

like other echinoderms, they have the ability to

rapidly and reversibly alter the stiffness of their

inner dermis. It has been proposed by scientists

that the modulus of this inner dermis is

Figure 3.2.5.5-1: Skin of the Sea controlled by regulating the interactions Cucumber can Reversibly Change its Mechanical Properties when Exposed among collagen fibrils, which help to reinforce to Some Form of External Stimulus

49 a low-modulus matrix. There is a family of polymer nanocomposites that mimic this architecture and display similar mechanic adaptability. Some of these nanocomposites include the family of poly(acrylic acid)-coated carbon Nanotubes (cellulose nanofibers). These fibers are known to exhibit large viscosity changes upon the variation of their pH levels. They also seem to compliment other polymeric systems because of their morphing mechanical behavior. By using additional stimuli, such as heat, even larger mechanical properties changes were witnessed. This simply illustrates that different factors can be manipulated to adjust the material’s mechanical properties as needed. The dermis of the sea cucumber in Figure 3.2.5.5-1 has the ability to change its mechanical properties when exposed to some form of external stimulus.

The architecture and mechanical adaptability of the sea cucumber dermis provided a good model for the creation of similar polymer nanocomposites. The mechanical properties of these mimicking materials can be selectively and reversibly controlled through the formation and decoupling of nano-fibers in response to specific chemical triggers. Additional research is being conducted to determine if such materials have the ability to respond to other triggers such as optical and electrical stimuli (Capadona, 2008).

3.2.5.6. Inspirations from Biological Optics for Advanced Photonic

Systems

There have been various observations made in nature that have allowed humans to develop the necessary technological tools to envision, understand and imitate biology. Biomimetics, in particular, is a field that is highly dependent on the advances made in material science. Recently, creative advances have been made in the field of optical system designs.

Biological optical systems are very unique and normally tailored for the needs of each individual organism. Whether the primary focus is searching for nourishment, evading predators or seeking protection from the elements, the mechanism of sight is perhaps the most valued and most varied aspect of species in the animal kingdom. Tailored for near or far vision, night or day vision, wide or narrow fields; these needs have been the central focus of the relatively new and expanding field of biological optical science. Reconfigurable soft lithography makes the development of three dimensional polymeric optical systems possible, and the configurations are based on the biological designs that are being emulated (Lee, 2005).

Of the many different types of eyes found among the animal kingdom, camera-type eyes and compound eyes are the most common.

Camera-type eyes generally rely on a single lens to focus images onto a retina. The human eye is perhaps the most familiar camera-type optical

51 system. Among the animal kingdom there are many types of animals who have camera type eyes other than humans. The most common are fish, reptiles, birds and amphibians. Despite the differences in each of their structures, they’re all complex and can all be mimicked using soft lithography techniques.

Compound eyes are much different than camera-type optical systems such as the human eye. They are most commonly found in insects, and have proven interesting to scientists because of their complexities and differences to the human eye. Compound eyes can have up to 10 thousand lenslets (or mini-lenses), such as those found in some species of dragon-flies. Within the branch of compound eyes, there are two separate types: superposition and apposition. Each of these types, are also well known to be adapted to the needs of the owner.

Apposition compound eyes have facets that are optically isolated from one another, meaning that this type of eye has the ability to process several images in parallel, because each facet sends signals simultaneously (Lee, 2005).

3.2.5.7. Biomimetics for Next Generation Biomaterials

Many biological tissues and devices boast remarkable engineering properties. Their toughness, strength, and sometimes adhesion properties are just a few of the attractive qualities of these high performance natural

52 materials. Over the course of the last several years, more and more research is being devoted to the study of these materials and how man can accurately duplicate them. In many cases, the design requirements for materials in nature are similar to those in the engineering world, which is why scientists and engineers have begun to mimic nature in its approach to design. The difference however lies in engineers’ approaches to development. As the final product might resemble the natural one, the route taken to manufacture that material is entirely different. This has led to deficiencies in the performance of man-made materials.

Out of all the different biological materials to study, hard biological tissues are the most attractive. These tissues are unique in the way the combine stiffness and strength. Hard tissues serve a variety of functions, including mechanical support, cutting, tearing, crushing and sometimes

even armored protection, like in seashells.

Obviously, the properties and structures

vary greatly depending on the main

purpose of the tissue, which has led to

numerous hard tissue studies. The

combination of the attractive properties,

Figure 3.2.5.7-1: Nacreous Shell all in one material, is what is most interesting and is discussed in succeeding sections.

How nature can combine toughness and stiffness is an intriguing study. The most common route to make materials more still, is by adding minerals, or mineralizing them. There are approximately sixty different types of mineralizing materials, of which calcium carbonate, hydroxyapatite, and silica are the most common. These minerals come in a variety of sizes and shapes, which proves convenient when determining the most appropriate material for each application, or primary function.

Bones, mollusk shells and teeth are of the most stiff of natural materials, and are also on the realm of magnitudes higher in toughness than some of the minerals they contain. Improving a material’s stiffness by orders of magnitude has yet to be done by any man-made based materials

(Barthelat, 2007).

The hardness of natural tissues is also of great interest. The main reason for these tissues having this feature is their hierarchical structure. A good example of this structure is the nacreous shell, shown below in Figure

3.2.5.7-1. This shell is a member of the mollusk family, and the shell is grown to protect the soft body that lies on the inside from any external aggression of predators, rocks or possibly current and wave generated debris. The shells themselves are mostly made up of calcium carbonate and this has made it one of the strongest and toughest natural biological tissues.

Bone is another example of a high-performance biological material.

It combines a soft material (collagen) with a mineral (hydroxyapatite) to achieve the desired stiffness. As with nacre, the structure of bone is organized over several levels of hierarchy. With respect to composition, nanoscopic mineral crystals are embedded into collagen fibrils. The three dimensional arrangement of these fibrils is one of the most important properties of bones, as fibrils are the building block for larger structures such as lamellae and osteons (Barthelat, 2007).

There are several mechanisms at the micro level that help toughen bone, including viscoplastic flow, crack deflection, microcracking and crack bridging. In addition to being a stiff and touch material, bone exhibits remarkable functionality. It continuously regenerates, can adapt to local stress and can also heal itself. Bone itself has a variety of very specialized cells, each that fulfill their own functions; the primary reason for bone being such a remarkable material. The specialization of these cells is highly dependent on what the bone needs, or mechanical stimuli.

If for example a section of bone is experiencing high mechanical strain, the appropriate cell components will come together to adjust the bone stiffness in that area accordingly. This entire process of generation and elimination is continuous, and the average healthy human being has

55 approximately 25% of their entire skeleton regenerated every year

(Barthelat, 2007).

This self healing capability has been recently duplicated in bioinspired, self healing polymers. These polymers contain pockets of a healing agent, along with a catalyst, so if crack propagation occurs, the crack will puncture the healing pockets, thus releasing, mixing and curing the healing agent within the pockets, and eventually gluing together or healing the crack in an epoxy type fashion. While this mechanism is not as sophisticated as bone, it shows promise in the ability to develop self healing materials. These examples of tough and stiff biological materials only touch on a small percentage of the materials that warrant investigation (Barthelat, 2007).

3.2.6. Carbon Based Biomaterials

Carbon based materials have found numerous applications in the biomedical world to date. Carbon, and its many adaptations are attractive as potential biomaterials because they are relatively light weight, but significantly strong, they have high thermal conductivity and they can mimic the dimensions of numerous natural materials in our body

(such as collagen).

Many recent studies have been aimed to address the biocompatibility of carbon (Blazewicz, 2001). Although most carbon

56 based materials have proven successful in working alongside native tissue to assist in wound healing, the inner workings of such materials remain obscure (Price, Ellison, Haberstroh, & Webster, 2004).

Despite the relatively small amount of research aimed at discovering carbon’s molecular structure, several studies have been performed that relate surface roughness to the increase in attachment of osteoblasts (bone cells) (Bacakova, et al., 1996, Boyan, et al., 1998).

Other studies worked to learn how carbon, in particular carbon fibers, responded to the biological environment. It has been shown that the biocompatibility and response of carbon fibers to the physiological environment varies depending upon the temperature at which the fibers were processed. This also has an effect on how well the fibers tend to bond with physiological tissues. Some studies seem to illustrate that there is a relationship between the degree of crystallinity of the carbon fibers, and the amount of integration into the body. Unfortunately, though, this phenomenon of carbon fibers has not been fully addressed to date.

Carbon fiber use as the foundation for scaffolding materials as well as in orthopedic repairs has also been substantially investigated. In some instances, carbon fibers were shown to be an adequate material for total hip replacement as well as internal fracture fixation devices (Edie, 1998).

Carbon fiber reinforced polymer composites have been synthesized that closely match the tensile strength and modulus of bone (Burton, Glasgow,

Lake, Kwag, & Finegan, 2001). This property has the potential to contribute in the production of scaffolds and implants for bone tissue regeneration.

Other forms of carbon that have been customized for biologically based systems include artificial graphite, glass-like carbon, pyrolitic carbon (Pesakova, Smetana, Sochor, Hulejova, & Balik, 2005) carbon fiber reinforced polymers (Brantigan, McAfee, Cunningham, Wang, &

Orbegoso, 1994), and carbon-carbon composites (Balik, Weishauptova,

Glogar, Klucakova, Pesakova, & Adam, 1996). These carbon based composites have the unique ability to be highly biocompatible and at the same time maintain their impressive physical and chemical properties in vivo.

Carbon nanotubes, in particular, are gaining significant attention.

Carbon nanotubes (CNT) are well-ordered, high aspect ratio allotropes of carbon. The two main variants, single-walled carbon nanotubes (SWCNT) and multi-walled carbon nanotubes (MWCNT) both possess a high tensile strength, are ultra-light weight, and have excellent chemical and thermal stability. These materials also possess semi- and metallic-conductive properties. This array of features has led to many proposed applications in the biomedical field, including biosensors, drug and vaccine delivery and the preparation of unique biomaterials such as reinforced and/or conductive polymer nanocomposites (Williams, 2002).

Carbon Fibers have found frequent exploratory uses in biomaterials.

Researchers are looking into their effectiveness as bulk and coated implants, as well as their potential for localized, in vivo drug delivery and various components of biosensors (Webster, 2006). Compared to other metal based implant materials, carbon fibers have shown exceptional promise in promoting osteoblast attachment (Pesakova, Kubies, Hulejova,

& Himmlova, 2007). Not only do cells prefer carbon fibers over these metal based materials, they also tend to proliferate along the carbon, meaning that how and where cells grow could potentially be controlled if a carbon fiber based implant material were employed (Boyan, et al.,

1998, Edie, 1998).

Figure 3.2.6-1: Dry Carbon Fibers

One of the main motivations behind such aggressive research into carbon as a biomaterial is its evidence of promoting higher growth and attachment of various types of cells over other biomaterials including titanium, cobalt chromium and numerous polymers (Pesakova, Smetana,

Sochor, Hulejova, & Balik, 2005). Additionally, because of its abundance not only on earth but within the human body, development of carbon and carbon based biomaterials could prove much more efficient than utilizing other resources that require more expensive mining and fabrication efforts. Dry carbon fibers are highlighted in Figure 3.2.6-1.

3.3. Techniques of Tissue/Biomaterial Assessment

Assessing and analyzing wound healing processes is perhaps the most challenging aspect of tissue regeneration. The ultimate objective is to develop a method in which all inclusive information about a patient’s healing process can be obtained in the most non-invasive manner.

Several methods in which to monitor wound healing have evolved over the course of the past few decades. The most common methods used to investigate the tissue growing process consist of radiological assessments such as various microscopies, micro-CT and X-rays, and Magnetic

Resonance Imaging (MRI) (Lanza, 2007).

3.3.1. Microscopy Analysis

Microscopic methods involve the reflection, diffraction and refraction of electron and/or radiation beams interacting with the subject of study. The subsequent collection of scattered radiation builds up an image of the subject from which further analysis and study is carried out.

Despite its ability to image a variety of objects, this method is limited in the following: (i) the technique itself can only image dark or strongly refracting objects, effectively, (ii) the method of diffraction limits microscopic resolution to approximately 0.2 micrometers, and (iii) out of focus light from points outside the focal plane can reduce the clarity of the image. For the purposes of studying cells and tissue growth, live cells in particular generally lack sufficient contrast to be studied successfully and the internal structures of the cell are colorless and transparent.

Staining methods are useful in increasing the contrast of the structures of microscopic images, but this involves killing and affixing the sample to a medium. Staining a sample can also introduce microscopic artifacts which distort the sample and lead to data collection errors (Smith &

Martin, 1973).

3.3.2. Computed Tomography (CT) Images and X-Rays

More advanced and popular methods of tissue analysis and imaging are micro-CTs (Figure 3.3.2-1) and X-rays (Figure 3.3.2-2). In both in vitro and in vivo applications, these methods exhibit high resolution and contrast of cortical bone. Micro-CTs in particular are excellent in identifying Figure 3.3.2-1: Micro-CT Image of the tissue calcification and bone growth, but Proximal Part of a Tibia unfortunately both micro-CTs and x-rays only seem effective in analyzing cortical bone. The contrast is much less apparent in bone marrow and newly formed bone, making it difficult to study bone growth and regeneration accurately. Micro-CTs and x-rays also demonstrate poor tissue contrast, exhibit poor depth penetration and cause radiation damage to their specimens. These drawbacks have led to the use of

Magnetic Resonance Images (MRIs) as a way of analyzing tissue regeneration (Elledge, 1996).

3.3.3. Magnetic Resonance Imaging (MRI)

The effectiveness of MRIs is due largely in part to the body’s large composition of water molecules, each of which contains two hydrogen protons. When a patient enters the magnetic field of an MRI scanner, the

62 magnetic moments of the protons align with the direction of the

magnetic field. Next a radio frequency

electromagnetic field is turned on, causing

the protons to alter their alignment relative

to the field. When the electromagnetic

field is turned off, the protons return to their

original orientation. These changes in

alignment create a signal which can be

Figure 3.3.3-1: MRI Equipment and Images detected by the MRI scanner. The frequency of the emitted signals depends upon the strength of the magnetic field that they generate. Tumors and diseased and/or abnormal tissues can be easily detected with an MR image because the protons in these tissues return to their original orientations at different rates than normal tissues, thus generating different frequencies (Cyclin-

Dependent Protein Kinases: Key Regulators of the Eukaryotic Cell Cycle,

1995, Steele & Bramblett, 1988).Magnetic resonance imaging has several benefits including its very high resolution, its ability to be applied in vitro and in vivo. It has been used successfully in generating spacial maps of tissue relaxation times, and can help determine the stiffness of developing engineered tissues (Schmiedeler & McDonough, 1934). When contrasting agents are used, MRIs can quantify cell death, assess tissue inflammation and even visualize cell trafficking and gene expression (Boulpaep, 2005).

One of the most beneficial applications of MRIs in tissue engineering is their ability to study the integration of implants with surrounding tissues and check for early signs of implant rejection. MR imaging is also extremely fast (1 image/ms) when capturing data (Ham, 1972). There have been several successful studies utilizing MRIs, in vitro, to measure bone stiffness and strength during regeneration (Brighton & Hunt, Histochemical

Localization of Calcium in the Fracture Callus with Potassium

Pyroantimonate: Possible Role of Chondrocyte Mitochondrial Calcium in

Callus Calcification, 1986).

Despite MR imaging’s many advantages over radiological and other techniques, they still have several drawbacks. MR imaging can only be used for early bone tissue development. Cortical bone is relatively low in hydrogen atoms, making it appear very dark, and is therefore not easily visible in conventional MRIs. The predominant features that are visible in

MR images reside in bone marrow. Because of this, special care must be taken when viewing the interface between bone marrow and surrounding bone tissues; transitions between marrow and cortical bone induce magnetic field artifacts that significantly distort the image

(Brighton & Hunt, 1997). Because transitions between marrow and cortical bone are always, randomly occurring during regeneration, it is difficult to determine how much of the generated MR image is accurate and how much is artifact. Additional drawbacks of MR imaging and the imaging

64 process include its technical difficulty, the fact that it is relatively slow (with respect to gathering sufficient data for study) and it is expensive. The MR imaging techniques that can be used to capture tissue regeneration data are not widely available and require expensive high speed imaging hardware (About Apoptosis, 1999).

3.4. Concepts of Cellular Automata (CA)

While working at the Los Alamos National Laboratory in the 1940s,

Stanisław Ulam, studied the growth of crystals, using a simple lattice network as his model. At the same time, John von Neumann, Ulam's colleague at Los Alamos, was working on the problem of self-replicating systems. Because of the great difficulty and cost associated with Von

Neumann’s research, Ulam suggested that he develop his design around a mathematical abstraction, such as the one Ulam used to study crystal growth. Thus was born the first system of Cellular Automata (Von

Neumann, 2007).

3.4.1. Fundamentals of CA Theory

Cellular Automata is a discrete model or dynamical system. It consists of a grid of cells, each in one of a finite number of states. Time is also discrete, and the state of a cell at time, t, is dependent upon the states of the cells that surround it, or its neighborhood (Wolfram, 2002).

The rules of a CA program are analogous to initial conditions or boundary conditions, in that they determine how each cell changes its state over time. These key principles of CA are illustrated in Figure 3.4.1-1.

Figure 3.4.1-1: Concepts of Cellular Automata

3.4.2. Successful Applications of CA

Since its evolution, cellular automata has been found useful in studying various applications including the behavior of different gases, the crystallization process, forest fire propagation, urban development, erosion, particle aggregation and the Lattice-Boltzmann models for fluid dynamics. In all of these applications, the models developed through CA proved highly successful as long as the rules implemented within the programs accurately depicted the boundary and initial conditions of the

66 problem at hand (Jansenns, 2002). For this particular application of CA, numerous rules were employed to accurately depict each crucial element of the cell cycle, as well as the additional dynamics of the tissue regeneration process.

3.5. Theory of Mathematical Biology

The processes associated with bone growth and wound healing of bone tissue involve a series of sequential, highly dependent cellular events that remain very difficult to understand. It has been established through previous works that cellular activity during tissue regeneration and wound healing is highly regulated by the production and distribution of growth factors and matrix proteins. Mathematical models play a crucial role in providing insights into how growth factors and proteins regulate a tissue’s ability to heal. Additionally, theoretical models, when developed accurately, have the ability to predict bone tissue behavior during and after regeneration, as well as the tissue’s various constituents at any given time. Having access to this information as a researcher, allows for additional parameters of the regeneration process to be accurately studied and modeled including tissue-biomaterial interactions and the mechanical properties of the entire tissue-biomaterial system during wound healing.

Past attempts at studying tissue regeneration have been mainly focused on the study of various cell populations. How do cell populations change with respect to time during a process such as wound healing?

The use of previously established mathematical models for tissue regeneration-based concepts is highlighted in succeeding sections.

3.5.1. Principle of Population Equation

One of the most established math models is the principle population equation. It is a simplified version of what is generally known as the Logistic Equation (discussed in next section) (Baianu, 1987). Its basic form is as follows:

The probability that a birth will occur in the designated time step, ∆t, is:

And the probability that a death will occur in the time step, ∆t, is:

X(t) represents the size of the population at time t, and pj(t)=P(X(t)=j) represents the probability that at time, t, X(t)=j. For this particular equation, it is assumed that only 3 distinct activities could occur in the designated time increment, those being: a) No birth or death occurs, b)

68 a birth occurs and c) a death occurs. Each of these possibilities is represented by the following:

a) No birth or death occurs:

b) Exactly one birth occurs:

c) Exactly one death occurs:

The one particular possibility that is not represented through this model is that both a birth and a death occur within the same time step. To ensure that does not happen, a small enough time step would have to be defined. In most tissue systems, however, it is very possible, especially in a wound healing application, that multiple cells can be born and/or die simultaneously, just in different physical locations of the area in question.

Because this model ignores this possibility, this equation was not researched in depth, however a representation of this equation was developed using the probabilities described above. Combining all the possible probabilities associated with the overall population yields the following:

μ μ

This equation is valid for small time increments only and when simplified becomes:

When simplified still further and combining like terms, and transferring into summations:

The following steps show the implicit solution to the original differential equation:

This equation is what has been used for decades to estimate various types of populations.

3.5.2. Logistic Law with Growth and Capacity

Similar to the last equation, this differential equation models a population density but adds a carrying capacity or limiting factor, which controls the maximum density or population by capping how much it can grow (Chatfield, 1993).

X(t) = Population Density at Time t

K = Environment Carrying Capacity (Or maximum Volume)

r = Reproductive Parameter (i.e. a Growth Rate)

To simplify this equation and make it suitable to solve implicitly, let:

This turns the equation into the following:

Simplifying this equation, it becomes:

with

By using the method of partial fractions, the equation becomes:

By solving the above equation for the coefficients, A = 1 and B = 1. Now the equation can be broken up into the following integrals:

After integration,

Remembering that the initially, y=Yo

After much simplification through several steps of algebra,

After additional simplification, the final version of the equation becomes:

Remembering that:

The following form is that same differential equation but in finite difference form:

Where:

And Xo and to are known

3.5.3. The Malthusian Model

The Malthusian Model is one of the oldest models developed to predict population growth. Despite the model being quite simple, it has provided a solid foundation for the development of several additional means of modeling biological populations. A standard Malthusian model is composed of one variable and one parameter: the variable for the model is the population over time, and the parameter, which can be

72 either constant or changing over time, is the per capita growth rate, denoted by r. The variable, r, is usually defined as:

r = b – d

In this equation, b represents the population birth rate and d represents the population death rate (Freedman, 2005). The main assumption for this model is that additions to the model are made at discrete times, t0, t1, t2,

… With this definition, the Malthusian model can then be written as:

In both of these versions,

3.5.4. Logistic Growth with Carrying Capacity and Harvesting

Despite the simplicity and somewhat effectiveness of the

Malthusian model, it neglects the fact that after long periods of time, populations of biological communities tend to reach a form of “steady state” in which significant changes within the population no longer occur, unless a major environmental change is encountered. For the purposes of studying tissue growth and regeneration, the carrying capacity would simply be the maximum number of cells, or maximum tissue density

73 permitted within the composition of bone tissue. It is important to recognize that in this aptitude, the carrying capacity is described as the population level at which the birth and death rates of the cells match, resulting in a stable population over time. Additionally, the “harvesting” variable comes into play. Harvesting can be perceived as many different things, depending on the type of application and population one is attempting to predict. Harvesting in this purpose refers to taking or killing as a means of controlling a population, or to gather a population when it reaches its maturity. Cells, when reaching maturity, or the end of their life, are recycled, along with their surrounding tissue and remodeled to replenish and rejuvenate the tissue. By expanding the previously mentioned Growth Model with Carrying Capacity, the following equation describes a growth model with carrying capacity and harvesting:

Where:

P(i) = Population at time step, i

r = Growth Rate

K = Carrying Capacity

h = Time increment, with

H = Harvesting Rate (Cell Turnover Rate)

For the purpose of studying biological populations, such as cells, matrix components, growth factors and proteins, H could represent the overall rate that dead cells are recycled and replaced by new cells and new tissue components.

3.5.5. Stochastic Population Theory: The Predator/Prey Model

Also known as the Lotka-Volterra Model, the concept that this model brings is the ability to model how various species (in this case, cellular components) interact (Freedman, 2005). The dynamics of this type of model illustrate the interdependencies between the cells and all of the other physiological substances that compose bone tissue. By developing a model such as this, interactions between different parameters can be monitored and used to develop an even stronger more dependable model that would be primarily based on statistical data.

To start simply, first consider a population consisting of two different components: bone cells and matrix components. If considering standard bone tissue generation, the bone cells will initially grow exponentially, unless held in check by available space, or some other form of regulatory factor. The matrix component population is dependent upon the cell population because every time cellular division occurs, the proteins necessary to mineralize matrix are released. Its population is also affected by trauma, injury and cell turnover because in those entire instances

75 matrix components are broken down and engulfed by osteoclasts, in order to make room for new cells and matrix. The models are as follows:

Assume that: to, Co and Mo are known

Ci = Cell Population during Time Interval i

Mi = Matrix Component Population during Time Interval i

gc = Growth Rate of Cells*

gm = Growth Rate of Matrix Components*

dc = Death Rate of Cells*

dm = Death Rate of Matrix Components*

K = Total Population Carrying Capacity

The * indicates that these variables themselves are dependent upon additional parameters and their prospective functions are yet to be determined. It is known however, that all are dependent upon time and the overall cell and matrix component populations.

3.5.6. Mathematical Framework for Modeling Tissue Density

To understand this model, and its development, one must first understand all of the steps involved in tissue regeneration and wound healing. The three main steps associated with wound healing include:

The Reactive Phase, the Reparative Phase and the Remodeling Phase. In the reactive phase, the tissue reacts to the wound with inflammation, swelling, blood clotting and bone cells dying. During the reparative phase, fibrocartliganous calluses develop over a 3 to 4 week period, allowing capillary flow to areas of clotting blood and phagocytic cells to clean up cellular debris. Succeeding these events, new fibroblasts and osteoblasts migrate to the site and begin the bone reconstruction process. The fibrocartliganous callus is very important because it acts as a splint to the fracture, supporting it until additional bone components can be formed. During the remodeling phase, bony callus will begin to replace the fibrocartliganous callus and become prominent approximately 2 to 3 months after the injury. This is caused by continuous migration and multiplication of osteoblasts and osteocytes. Finally, any excess bony callus is removed and compact bone is laid down in order to reconstruct the bone’s shaft.

Despite this research focusing mainly on the remodeling phase of wound healing, the other two phases lay down the foundation for the entire process and are just as important to the overall tissue regeneration scheme. The foundation for the following model is based on a fracture healing framework previously published that focuses on the activation of chondrocytic and osteoblastic cell differentiation by growth factors. Cell migration, proliferation, differentiation, growth factor production and

77 diffusion and ECM synthesis and degradation are all simulated over time at the site of fracture. For the purpose of this research, this model will have to be modified to include the rate of change of protein production and diffusion over time, as well as the existence of osteoclasts over time, since the model’s dependent variables are relative densities of the tissue components. Expressed in words, the following are the preliminary differential equations (Volokh, 2004):

(1) Mesenchymal Density = Migration + Mitosis – Differentiation

(2) Chondrocyte Density = Mitosis + Differentiation – Endochondral

Replacement

(3) Osteoblast Density = Mitosis + Differentiation – Removal

(4) Connective/Cartilage ECM Density = Synthesis – Degradation

(5) Bone ECM Density = Synthesis – Degradation

(6) Osteogenic Growth Factor Concentration = Diffusion +

Production – Decay

(7) Chondrogenic Growth Factor Concentration = Diffusion +

Production – Decay

The equations for the densities of the different cell types are expressed in terms of variables and shown below (Volokh, 2004):

cm = Mesenchymal Cell Density = cm(x,y,t)

cc = Cartilage Cell Density = cc(x,y,t)

cb = Bone Cell Density = cb(x,y,t)

Am, Ac, Ab = Rates of Proliferation for Each Cell Type

Klm, Klc, Klb = Limiting Cell Densities for Each Cell Type

Bm, Bc, Bb = Constants determined by the Limiting Cell Densitites

F1, F2, F3 = Constants that Relate Cell Differentiation to Growth Factor

Concentration

D, C = Coefficients Related to the Cell Migration Speeds for All Cell Types

The rate at which cells migrate has been proven to be related to the overall extracellular matrix density. The term “Haptotactic” (h) refers to the directional motility or outgrowth of cells, caused by the affects of chemotaxis. The term “Haptokinetic” (k) refers to standard cellular migration without the effect of chemotaxis. The following equations describe the proliferation and migration coefficients (Volokh, 2004):

In the above equations, m represents the matrix density, and:

m = m(x,y,t)

m = mc + mb = mc(x,y,t) + mb(x,y,t)

where mc = Connective Cartilage Matrix Density

and mb = Bone Matrix Density

The chondrocyte and osteoblast parameters are modeled as follows:

The functions relating cell differentiation to the concentrations of existing growth factors are as follows (Volokh, 2004):

In the above equation:

gb = Osteoblast Growth Factors Concentration

gc = Chondrocyte Growth Factors Concentration

Y1, Y2 = Inverse Time Parameter

H1, H2 = Growth Factor Density Production Parameter

The replacement of chondrocytes with osteoblastic and osteocytic tissue is dependent upon the density of connective cartilage present, as well as the concentration of osteoblastic growth factors, yielding a third cell differentiation/growth factor concentration function, F3, defined as

(Volokh, 2004):

Next, the rate of change of the matrix densities, including the bone extracellular matrix and the connective/cartilage, both being a balance between synthesis and degradation, are as follows (Volokh, 2004):

Pbs = Bone Matrix Synthesis Constant

Pcs = Connective/Cartilage Synthesis Constant

Qcd1, Qcd2, Qcd3 = Matrix Degradation Constants

Finally, the only differential equations that still need to be defined are those of the rate of change of the growth factor densities, which are as follows (Volokh, 2004):

Dgc, Dgb = Coefficients of Diffusion

Egc, Egb = Functions Relating Growth Factor Production to Growth Factor

Concentration

dgc, dgb = Constants of Decay

The functions relating the production of growth factors to their prospective concentrations are (Volokh, 2004):

Hgc, Hgb = Growth Factor Production Parameter

This system of equations is obviously very complex and in order to solve, must be non-dimensionalized to reduce the amount of unknowns within the system. Several of the variables included in this model would be very difficult to measure accurately, even within an experimental setting.

Some initial and boundary conditions can be determined through previously performed experiments, however others will have to be given a range of values to try that produce optimum results, and then be narrowed down from there.

3.6. Pertinent Statistical Analyses

Within the realm of this research, it is also important to completely utilize all gathered data. Statistics is the formal science of making effective use of numerical data that relates two groups of information

(Freedman, 2005). This type of science also deals with not only the collection of data but its analysis and interpretation; are there any significant relationships within the collected data? Can those relationships be used productively and accurately for future research efforts?

The two specific statistical methods being utilized for this research involve Multivariate Regression Analysis and the Monte Carlo Simulation

Method.

3.6.1. Multivariate Regression

With respect to statistics, regression analysis involves any techniques meant for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables (Lindley, 1987). Regression analysis helps scientist understand how the typical value of the dependent variable changes when one or more of the independent variables is varied (Cook, 1982). It is also of importance in regression analysis to characterize the variation of the dependent variable with respect to the regression function, which is normally described with a probability distribution (Meade & Islam, 1995).

Several techniques have been developed for carrying out regression analysis. Familiar methods such as linear regression and ordinary least squares regression are parametric in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the known data (Lindley, 1987). Nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite in dimension (Chatfield,

1993).

The performance of a regression analysis method depends on the form of the data-generating process and how it relates to the regression approach being used. Since the true form of the data generating process is not known, regression analysis depends to some extent on making assumptions about the unknown process (Cook, 1982). Classical assumptions associated with regression analysis include: (i) The sample must be representative of the population for the inference prediction, (ii) the error is assumed to be a random variable with a mean of zero conditional on the explanatory variables, (iii) the variables are error free,

(iv) the predictors must be linearly independent and (v) the errors must be uncorrelated. Once a multivariate regression analysis has been performed, the resulting model usually looks like the formula in Figure 3.6.1-

Figure 3.6.1-1: Multivariate Linear Regression

In this model, “y” is the dependent variable, x1 – xn are the independent variables and the unknown parameters are represented by a0 – an.

3.6.2. The Monte Carlo Method

In addition to the regression analysis, another form of statistical science essential to this research initiative is a deterministic means call the

Monte Carlo Method. Monte Carlo methods are a class of computational

84 algorithms that rely on repeated random sampling to compute their results (Metropolis, 1987). Monte Carlo Methods are often used to simulate physical and mathematical systems. Because of their reliance on repeated computation of random or pseudo-random numbers, these methods are most suited for calculation by a computer; they tend to be used when it is unfeasible or impossible to compute an exact result with a deterministic algorithm (Hubbard, 2009).

There is no single Monte Carlo method; instead, the term actually describes a large and widely used class of approaches. The similarities lie in the fact that these approaches usually follow a particular pattern: (i)

Define a domain of possible inputs, (ii) generate inputs randomly from the domain using a certain specified probability function, (iii) perform a deterministic computation using the inputs and (iv) combine the results of the individual computations into the final product (Anderson, 1986).

Monte Carlo simulations are used for a wide variety of applications.

These methods are very important in computational physics and physical chemistry because of their usefulness in modeling phenomena with significant uncertainty in its inputs. It is also useful in these fields because of its ability to study systems with a large number of coupled degrees of freedom (Meade & Islam, 1995). These methods have also been successfully used in business and finance applications as well a telecommunications and games (Berg, 2004). Microsoft Excel’s RiskAMP is

85 a robust tool in which Monte Carlo methods can be implemented within their software. Its strong functionality and efficiency make performing

Monte Carlo Simulations very efficient and productive.

Chapter 4. Drawbacks of Current Tissue/Biomaterial Assessment Methods

4.1. Inadequacies of Current Materials and Methods

Both materials and methods, despite their short term successes, have drawbacks that motivate further research with respect to their applications in tissue engineering. In order to develop more accurate materials, and ways of assessing the wound healing process, these drawbacks and negative aspects of existing tissue analysis methods must first be addressed. These aspects are what supply the need for new and more accurate methods to be produced.

4.1.1. Materials

Although a vast amount of materials have been successfully utilized for tissue regeneration purposes, each still has its negative aspects when implanted within a physiological environment.

With respect to metals, Titanium outperforms any other metal alloy when it comes to biocompatibility and specific strength (strength per density), but has poor shear strength, which makes it less desirable for bone screws, plates and other similar applications. These alloys also

87 possess a very high coefficient of friction. If a piece of bone rubs against a titanium implant, wear particles can form within the bone leading to additional complications of wear debris and implant loosening (Metals for

Implantation, 1999).

For biomedical applications, cobalt alloys are generally cast.

Although cast cobalt chromium alloys excel in wear, pitting, and crevice corrosion resistance, casting them poses some severe problems including coarse grains and grain boundary segregations in the material’s microstructure, making them inferior to non-casting alloys in terms of fatigue strength and fracture toughness. The grain boundary segregations in these materials can eventually lead to aseptic loosening of an implant, and eventual implant failure (Metals for Implantation, 1999).

Although stainless steels are among the most economical solutions for biomedical implant materials, recent studies have shown that the pitting corrosion of stainless steel occurs in saline and chloride environments. Human body fluids, such as tissue fluids, lymph fluids and blood contain approximately 0.9% saline (NaCl) solution, making stainless steel not suitable for permanent implant devices. Stainless steel also has the potential for releasing extremely harmful Ni2+, Cr2+, and Cr6+ ions the body. Thus, stainless steels are now only used in temporary implant devices such as screws, hip nails, and fracture plates (Black, 1999).

Most metallic biomaterials, however, were developed for industrial uses, but are still used for biomedical applications because of their high corrosion resistance. This poses problems when the metals are introduced into a physiological environment because they were not originally intended to perform under the conditions of the human body.

One of the more serious conditions involved with lack of biocompatibility is corrosion. A material’s ability to resist various types of corrosion within the confines of the human body is very important aspect of biomaterials (Lima, Bosch, Lara, Villarreal, & Pina, 2005). The cause of corrosion is believed to be a galvanic cell that forms inside the implant alloy. Another potential cause of corrosion is actually the deposit of impurities on the surface of the material during the manufacturing process

(Batchelor, 2004). Additionally, a potential cause of corrosion of biomaterials (once implanted) is the change of the body’s pH level immediately following the surgical procedure. The pH surrounding an implant following the surgery is typically reduced due to the trauma of the surgery. Because of this, infectious microorganisms and crevices can form between components, therefore reducing the oxygen concentration in the area. An example of corrosion pits on an implant is illustrated in Figure

4.1.1-1.

Figure 4.1.1-1: Micrograph showing pits in the surface of a Co-Cr & UHMWPE Total Knee Replacement that are relatively shallow and measure about 500 μm across. A crack in the Tibial UHMWPE Component is also evident and appears to join with the pits [39].

As previously mentioned, human bodily fluids contain approximately 0.9% saline solution, which contributes greatly to the corrosion of material implants. When exposed to this biological environment for an extended period of time, a material’s corrosion, such as that of a cobalt-chrome, leads to harmful metal toxins streaming through the body and possibly causing cancerous tumors (Walker, Blunn,

& Lilley, 2004). Other examples of the harmful effects of corrosion include:

(i) Dermatitis from exposure to nickel, (ii) Chromium induced anemia caused by chromium inhibiting iron from being absorbed into the bloodstream, (iii) Chromium caused ulcers and central nervous system disturbances, and (iv) Epileptic effects and Alzheimer’s disease caused by aluminum present in some implant materials (Lima, Bosch, Lara, Villarreal,

& Pina, 2005). The only apparent solution to the corrosion of biomaterials at this point, lies in choosing better quality materials for these applications.

Along with corrosion, one of the most severe and widely researched side effects of both metallic and polymeric implant materials is their ability to generate wear debris. It remains the primary cause of operational problems with orthopedic prostheses such as hip and knee joints. Wear debris can lead to severe problems involving tissue inflammation, infection, and eventually aseptic loosening and failure of the implant

(Batchelor, 2004). Wear debris in the form of metal or polymer particles can generate severe pathological responses through tissue inflammation.

The size range of the particles is found to strongly influence the intensity of inflammation, with 0.3 – 10 μm generating the strongest response. A picture of wear debris from a titanium implant and a UHMWPE implant are shown in Figure 4.1.1-2.

Figure 4.1.1-2: Wear Debris Produced by Artificial Implants (Batchelor, 2004)

Tissue inflammation involves the secretion of strong oxidants such as hydroxyperoxide and hypochlorite. These secretions are capable of reacting with metal to form dissolved cations (which are simply positively

91 charged ions). Since orthopedic devices are made of materials such as chromium, cobalt, and molybdenum, when these ions are released into the body, problems of toxicity and allergy may arise (McKellop, Clarke,

Markolf, & Amstutz, 1981).

Wear debris from polymers generates an even worse bodily reaction. Wear particles from polymers such as UHMWPE and PTFE create a more prolonged and therefore harmful inflammatory response. Polymer wear particles are less toxic than metal debris; this however, permits large phagocytes to remain active inside the body for a longer period of time.

A phagocyte is a cell that engulfs and digests debris and invading microorganisms (Batchelor, 2004). Active phagocytes then congregate in tissue, causing inflammation, and therefore attracting additional inflammatory cells. Once severe inflammation has commenced, this can lead to joint pain, stiffness, and irreversible tissue damage. Phagocytes also have the ability to attack the bone adjacent to the prosthesis, thus weakening that bone and leading to eventual aseptic loosening of the device. Revision surgery is normally the only option at this stage, but this is a costly and inconvenient operation. An example of polymer wear debris is shown in Figure 4.1.1-3.

Figure 4.1.1-3: Example of Wear Debris from an UHMWPE Implant (Batchelor, 2004)

Less wear debris adds longevity to any implant, thus extending the time before any additional surgery would be needed. Just as in any implant related situation, the ultimate objective for tissue regeneration material selection is to determine which material will optimize the most important mechanical requirements, while at the same time meet the needs and lifestyle of the patient.

Another negative aspect of implementing metals and polymers into biomedical implants is stress or load shielding. In general, bones have a normal, dynamic response to loading. Bone regeneration and repair are also promoted by mechanical loads. A metal, for example, such as

Titanium, is much stiffer than bone. If a solid Titanium implant carried a disproportionate amount of the biological loads, then the surrounding bone is said to be “stress shielded.” This causes the surrounding bone to experience abnormally low levels of stress, which can lead to resorption of the bone and eventual loosening of the implant (Thelen, Barthelat, &

Brinson, 2002). Stress shielding greatly weakens the bone surrounding the

93 implant, making it frail and more susceptible to fracture. An illustration of what stress shielding can do to a patient’s hip is depicted in Figure 4.1.1-4.

Figure 4.1.1-4: Illustration of How Stress Shielding Works (Thelen, Barthelat, & Brinson, 2002)

From the effects of stress shielding, come the demand for biomaterials that have a stiffness closely resembling bone, which would help decrease the amount of load shielding, and also allow the surrounding native bone to regain its natural strength. Even on the small microscopic scale, this remains a very important aspect of biomaterials being used for regeneration and wound healing purposes.

Despite the many success stories and implementations of carbon and carbon based materials, the two most significant drawbacks of carbon are its toxicity during fiber and nanotube fabrication, and the obscurities of its molecular structure. If more could be learned about the inner workings of this material, researchers could be much more confident in its implementation within the human body.

4.1.2. Tissue Analysis Methods

Ideally, the imaging methods and/or theoretical models developed to study bone growth need to provide tissue engineers with complete information on tissue composition, structure and function. Currently, there are no imaging techniques that have the capabilities of accomplishing all three. Probably the most vital weakness of all these methods is that they are all discrete. No matter the resolution, the images resulting from these methods are only a snapshot of the entire regeneration process, containing within them merely a slice of real, continuous objects (Elledge,

1996).

An additional drawback involving image analysis is the detailed and complex image preparation process. This includes the way information is extracted from the images. Despite how the images are acquired, this method involves extracting relevant image information which is a several-step process that includes: sample preparation prior to taking useful images, image pre-processing, segmentation, feature extraction and classification (Lysaght, 2000). Sample preparation is tedious and very detailed. It also involves waiting extended periods of time for implant integration to occur. As with most imaging techniques, some form of sample staining is usually involved in the preparation process, which can lead to unreliable image rendering and unwanted

95 image artifacts. Image pre-processing is the next step and involves image noise reduction and shading to make the image easier to analyze.

Segmentation is the subsequent step and the most difficult because it involves finding the desired objects within the image. After this is the process of feature extraction, which involves extrapolating or measuring feature values and collecting the data. The final step involved in image analysis is classification which involves any statistical analyses and decision making needed to utilize the collected data (Lysaght, 2000).

4.1.3. Mathematical Methods

Not many significant mathematical techniques for analyzing wound healing and regeneration currently exist. Those that do have mostly been simplified to include only the parameters of highest importance: cell population and time. What has been unrealized by most, because of the dynamic nature of tissue regeneration, is that cell population is highly dependent on several environmental parameters including proteins, growth factors, white blood cells and the viability of the tissue. Therefore, in order to accurately predict (not measure) tissue regeneration characteristics, all of these elements must also be included.

4.1.4. Experimental Techniques

With respect to experimental forms of tissue analysis, most are accurate, but still involve a significant amount of risk and uncertainty. First and foremost, standard laboratory experiments are very costly and time consuming. Scientists both want and need dependable means of understanding the growth process in a short and reasonable time frame.

The majority of published experimental data that refers to bone tissue regeneration originates from cadaveric tissues. The dynamics of cadaveric tissues can obviously vary from those of living tissues. To harvest living cells and tissues is also very costly. Additionally, the proper handling of human tissues takes involved training and many forms of medical validation and clearance.

Another drawback of most experimental techniques is their potential for contamination. Most in vitro experiments involve a series of complex steps that help harvest cells, grow tissue, seed cells and/or tissue onto sterilized biomaterials, monitor the growth, etc. Scientists can inadvertently, but very easily, contaminate tissue and experiment samples, leading to useless results and lengthy sterilization processes thereafter. This even furthers the delay of useful data. Some cell samples and tissues will randomly die, which can also ruin an experiment. These uncertainties can cost significant amounts of both time and money. Is it possible to eliminate these drawbacks by developing a theoretical

97 method that is so accurate and dependable, that experiments can be used simply for validation?

4.2. Benefits of Research Initiative

This research effort poses several extraordinary benefits. The theoretical methods developed will have the ability to predict the behavior of bone tissue before, during and after growth and/or regeneration. This can be easily expanded to include other tissues such as muscle, ligaments and eventually intervertebral disc tissue. It is not uncommon to use a prosthetic device to assist in the regeneration of bone tissue; this effort will also have the capability of studying and predicting the interaction characteristics between native bone tissue and biomaterials. In order to develop more effective and efficient biomaterials and devices in the future, this work will also prove useful in that it will help predict the behavior of entire tissue/biomaterial systems, as well as the mechanical properties of various “system” components during the regeneration process.

Chapter 5. Research Methods

Various theoretical and experimental methods will be used to accomplish the previously stated objectives. Upon development of these theoretical models, the experimental approaches will be used to ensure the accuracy of these methods, and validate their efficacy.

5.1. Theoretical Research Methods

There are three main types of theoretical models that will be developed for the prediction of bone tissue during wound healing: (i) A

Cellular Automation Based Model, (ii) Mathematical Biology Models and

(iii) Statistical Based Analyses. Cellular Automation will be used mainly to predict how and where bone cells will grow and proliferate. The methods under Mathematical Biology aim at the mathematical representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques and tools. Mathematical Biology in general has both theoretical and practical applications in biological, biomedical and biotechnology research (Israel, 1988). For example, in cell biology, protein interactions are often represented as "cartoon" models, which, although easy to visualize, do not accurately describe the

99 systems studied. In order to do this, precise mathematical models are required (Baianu, 1987). By describing the systems in a quantitative manner, their behavior can be better simulated, and hence properties can be predicted that might not be evident to the experimenter.

5.1.1. Using Cellular Automata to Describe and Predict Cell Growth Rate

The accuracy of any CA model is based on the rules integrated within the program and how those rules impact the overall state of the model with respect to time. For this research, a computer program incorporating the concepts of cellular automation will be developed. The rules of the CA based program will each pertain to natural recurring phenomena of tissue regeneration and cellular dynamics. All of the highly dependent cellular activities must occur in the proper order for all of the natural phases of tissue regeneration to take place. Phenomenons such as apoptosis, chemotaxis, mitosis and typical cell maturity and death will all be taken into consideration. Figure 5.1.1-1 illustrates the various types of cellular dynamic activities and the order in which they must take place to achieve proper wound healing. In this figure, it can be seen how young bone cells (pre-osteoblasts) mature through several phases in preparation for mitosis (cellular division). Some cells never reach the point of division, but instead go through the programmed cell death phenomena of apoptosis. Additionally, yet a third group of osteoblasts do

100 not undergo mitosis or apoptosis, but instead develop into osteocytes

(mature osteoblasts) that provide the cellular foundation from which the bone’s extracellular matrix will depend. Also within this figure are the complex details of how dead cellular debris is engulfed, which in turn releases growth factors and proteins that are meant to attract young osteoblasts to the appropriate location in order to rejuvenate that particular site with healthy osseous tissue.

Figure 5.1.1-1: Diagram Depicting the Intricacies of Cellular Dynamics

Because it is the intent to validate this computer program using experiments and other, more established theoretical methods, it is crucial that the time space grid constructed for this research is comparable to

101 that of the other experimental and theoretical setups. With this in mind, the time space grid for analysis under cellular automation will be developed to replicate the overall cross sectional area of the average adult’s femur. The femur is the most commonly fractured bone within the human body and was therefore chosen as a vital part of both the theoretical and experimental methods of this research (Gal, Antonia-

Munoz, Muro-Cacho, & Klotch, 2000).

5.1.2. Combining Malthusian and Logistic Law Models to Predict Cell

Growth

Within the domain of mathematical biology, of particular interest was the logistic law with limited growth equation. This framework seemed to provide a somewhat accurate way of predicting cell population with respect to time (Hohler, 2005). The drawback however lies within the ability to develop an accurate intrinsic growth rate, r. Because cell birth and death rates are nonlinear and complex, this variable cannot be easily defined.

It is the intention of this initiative to take a unique approach to predicting tissue growth, by combining the Malthusian and Logistic Growth Models.

This approach is expected to establish a much more accurate intrinsic growth rate value. This can then be used to establish other significant parameters of tissue growth, including accurate cell counts,

102 resulting amounts of extracellular matrix and changes in tissue-biomaterial interactions.

5.1.3. Statistical Analyses

The two forms of statistical analysis being used for this research are

Regression Analysis, specifically multivariate, and the Monte Carlo

Methods. Using these two vastly different approaches will help illustrate the consistency of all the theoretical methods, and their ability to accurately assess cell population and tissue density.

5.1.3.1. Multivariate Regression to Relate Biomaterial Properties to Cell

Growth

Multivariate regression analysis is a very powerful tool that derives relationships between several independent variables and a resulting dependent variable (Cook, 1982). In addition to the many natural materials within the body that impact cell growth and tissue density during regeneration, it is also important to discover how implanted materials and their properties affect the overall integrity of the regenerated tissue. It is essential to learn which properties of the material inhibit and which promote cell growth and proliferation. One of the many objectives of this research is to use multivariate regression to discover how various properties of carbon based materials affect the growth and proliferation

103 capabilities of human osteoblasts when those materials are used in vitro.

Many past publications have illustrated how cell growth is dependent upon surface roughness (Burton, Glasgow, Lake, Kwag, & Finegan, 2001), but it is also important to discover which other material properties prove significant in affecting a tissue’s ability to renew. Additionally, it is crucial to understand how multiple material properties, not just surface roughness, work together to affect (positively or negatively) cell growth. If a statistically significant relationship can developed that relates resulting cell population to a biomaterials mechanical properties, this relationship can be used to predict how a specific material might behave within the body; does it promote or hinder a tissue’s ability to regenerate? Are certain properties more influential than others, maybe accelerating or possible impeding the regeneration process? Changes in surface roughness (Sa), two characteristics of Crystallinity: La – Average Crystal Diameter and Lc

– Average Crystal Height and time will be analyzed to establish how all of these parameters affect bone tissue growth. This method will determine which material properties are not only significant in successful bone tissue regeneration, but yield the ability to predict how long the regeneration process will take when various types of materials are utilized.

104

5.1.3.2. Monte Carlo Simulation to Predict Bone Density with Respect to Significant Biomaterial Properties

Just as important but perhaps more powerful are the Monte Carlo

Methods. These methods use various statistical distributions to generate random numbers from which the behavior of very detailed physical and mathematical and systems can be modeled. Just as multivariate regression can be used to explore how cell population is related to various material properties, the Monte Carlo methods show promise in developing a similar relationship, but allowing for variance in the independent variables. Each of the crucial material properties will be varied within reason. The correlation between these properties with each other and the independent variable (cell population) will be determined based on raw experimental data and the statistical methods used to establish the degree of correlation between two variables. Various simulations will be performed to illustrate how accurately this method coincides with experimental data and to what degree it correlates with multivariate regression techniques.

5.2. Experimental Research Methods

The main focus of this project is to develop accurate, predictive theoretical models to help eliminate the need for expensive and costly experiments. An additional benefit of these models will be their continuity;

105 having no association with the discrete manner of the other methods that are used to predict tissue growth. For validation purposes, a variety of experiments will be conducted during this research. The primary focus is to guarantee that all the characteristics implemented within the mathematical statistical and CA based models are accurate. These experiments have been broken down based on the experiments’ main objectives.

5.2.1. Bone: Methods of Growth and Analysis

Prior to understanding the potential for bone cell growth on synthetic materials, it must first be understood how cells behavior unaccompanied by non-physiological materials. The primary objective when working with human osteoblasts will be to conduct a series of in vitro experiments to simply observe their behavior as they progress through their cell cycle to the point of cellular division. Will the cells grow and divide? Randomly die? Or will they mature into osteocytes? It will be equally important to study any patterns that may present themselves with respect to proliferation, cell morphology and cell size.

5.2.2. Carbon: Material Properties of Interest

As previously mentioned, carbon is a very attractive material candidate for use in the biomedical field. Its ease of manufacture,

106 abundance, and ability to carry such a wide range of material properties indicate its potential for many implant related uses. It has proven its biocompatibility in various forms and shows promise in assisting the tissue regeneration process because of its strength and inertness. Carbon fibers, in particular show promise in tissue regeneration applications because of their established ability to promote cells to grow axially with the fibers

(Czarnecki, 2008).

When working with carbon experimentally, the primary objective is to investigate the differences in material properties of various forms of carbon fibers. Their structure must first be understood before being combined with osteoblasts for bone-carbon system analysis. Surface roughness, fiber orientation, and two measures of crystallinity, crystal diameter and crystal height, will be studies using various analysis techniques. Because most material research on carbon fibers has focused on its surface characteristics, it is a primary objective of this research to understand more about the molecular structure of various carbon fibers and in turn study which of these structural characteristics play a key role in promoting healthy bone tissue regeneration.

5.2.3. Bone-Carbon Systems: Experimentation and Analysis

After studying the viability of osteoblasts under normal in vitro conditions, and learning more about the internal structure of various forms

107 of carbon fibers, the next step will be to determine what types of carbon fibers show the most promise in promoting healthy and efficient bone growth. The objective for studying systems that contain both human osteoblasts and carbon is to seed various types of carbon fibers with human osteoblasts and study them over a period of time to explore their potential in assisting the bone tissue wound healing process.

Each type of carbon will be prepared and sterilized, then seeded with osteoblasts. The osteoblasts will be given ample time to adhere to the carbon and potentially proliferate. Cell population with respect to time will once again be monitored for several hours. The results of the different types of carbon will be analyzed to determine which carbon form exhibits the most promise as a tissue growth material.

The primary goals of these experiments are not only to verify and validate the theoretical models that have been developed, but to also optimize the type of carbon material that enhances tissue growth. This will be accomplished by comparing intrinsic growth rate of cells on carbon materials with the intrinsic growth rate of cells not in the presence of carbon. The differences in growth rates will be evaluated to determine which forms of carbon increase the intrinsic growth rate, r. If cell growth rate can be increased, overall tissue regeneration time will be reduced, making the entire wound healing process more efficient.

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Chapter 6. Dissertation Hypotheses

Ultimately, this research objective has four main purposes.

Numerous topics in science, mathematics and engineering have been studied to construct a set of hypotheses that, through the completion of this research, will be proven and validated.

6.1. The Application of Cellular Automation

Cellular automation has been proven a very powerful tool in a wide assortment of applications. It is proposed that the concepts of cellular automata can be used to successfully predict bone cell population with respect to time, and produce a reasonable approximation of the cell growth rate given a known carrying capacity, or maximum tissue density.

6.2. The Combined Logistic-Malthusian Model and Tissue Density

Calculations

Although the logistic law has been used in the past to predict various forms of populations, its lack of detail pertaining to the growth rate parameter deem it not suitable as a standalone method for predicting bone cell population and growth rate as a function of time. The

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Malthusian model, although somewhat simple can only predict growth rates for small, well described intervals as it stands alone. It is proposed that by combining these two methods, an accurate mathematical model can be developed to predict both the growth rate and overall cell population of osteoblasts with respect to time. Additionally, it is proposed that accurate estimations for various cellular parameters such as individual cell volume and mass can be calculated and used to calculate tissue density as a function of cell population.

6.3. Cell Population Predictions using Multivariate Regression

Multivariate regression is a very robust mathematical method that permits researchers to describe a dependent variable in terms of one or more independent variables with an established amount of accuracy. It is proposed that by studying the material properties of carbon fibers, a regression model can be developed that describes the cellular behavior of human osteoblasts as a function of time and the fibers’ surface roughness and crystallinity. This model will also dictate the overall degree of dependence of cell population on each of these variables and which variables are most significant.

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6.4. Tissue Density Predictions using Monte Carlo Simulations

The Monte Carlo methods are another form of vigorous statistical tools that when used properly can provide essential information pertaining to physically and mathematically complex systems. It is proposed that the

Monte Carlo methods can be used successfully in two ways: (i) The correlation between the material properties of carbon fibers can be used to perform a Monte Carlo simulation and predict cell population and tissue density as a function of time and (ii) the developed multivariate regression model can be used in a Monte Carlo simulation to describe osteoblast behavior with respect to time. This series of simulations will contribute to the overall accuracy of the previously described theoretical methods and provide yet another means of bone tissue assessment through noninvasive and analytically accurate means.

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Chapter 7. Research Techniques

The succeeding sections provide accurate insights into the approaches taken for both the theoretical and experimental techniques employed in this effort.

7.1. Approaches in Theoretical Research

The theoretical methods chosen to complete this research were cellular automation, the Logistic and Malthusian models prevalent in mathematical biology, Multivariate Linear Regression and finally Monte

Carlo methodology.

7.1.1. Cellular Automation

For this particular application of CA, numerous rules were employed to accurately depict each crucial element of the cell cycle, as well as the additional dynamics of the tissue regeneration process.

While compiling all of the necessary parameters that were to be used in the cellular automata program, it was important to establish an appropriate engine for executing and debugging the program. It was determined that MATLAB® would be the most efficient software because

112 of its functionality and ease of use. MATLAB® is a numerical computing environment and programming language maintained by The

MathWorks©, that allows easy matrix manipulation, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs in other languages (The MathWorks, 2009).

After establishing the program methodology and means of execution, considerable research was conducted to outline all the fundamentals involved in the cell cycle and the regeneration of bone tissue. The schematic in Figure 5.1.1-1 helped verify the proper nesting of all MATLAB® functions and sub-functions within the main program.

Each event involved in tissue growth dynamics and the cell cycle was thoroughly researched in order to develop a CA rule pertaining to that particular event. Numerous published sources for each event were referenced to ensure the reliability and truth of the information being implemented within each rule.

Each major event that occurs during tissue growth and regeneration was represented by a unique sub-program or subroutine to ensure each component of the overall process was executed in the proper order. To better understand the clarification of all rules associated with these programs and sub-programs, a complete list of states (the different variables that each individual grid cell can become, depending on the variables in its neighborhood) is shown in Table 7.1.1-1.

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Table 7.1.1-1: List of All Possible States for the Cells in the CA Grid

A table illustrating all of the rules implemented during program execution is shown in Table 7.1.1-2. Additionally, the succeeding paragraphs describe each of the sub-programs, their objectives and their contributions to the overall regeneration process.

Table 7.1.1-2: The Rules of the CA Program that Simulate Bone Tissue Regeneration

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The cellular automation program was developed so that as the program is being executed, each time step, or generation, can be seen within a two-dimensional Matlab© animation of the time-space grid. A color based, graphical representation of each tissue component was designation to make cell state changes easier to distinguish. This representation is illustrated in Table 7.1.1-3.

Table 7.1.1-3: Graphical Representation of Tissue Component Represented in the Cellular Automation Program

In the preceding table, the symbolic and numeric representations are included alongside the graphical representation of variables to illustrate the versatility of the designed program. Not only can the viewer use the animation to witness how the tissue grows with respect to time, but

115 at any given point during execution, both the numeric and symbolic matrices that were designed to represent the time space grid, can be accessed to see what types of tissue components occupy the various cells within the grid.

The unique sub-routines that highlight all of the highly dependent events that occur in tissue regeneration are highlighted in the succeeding paragraphs. For each major event an individual sub-routine was written.

The main program of execution was mainly a placeholder designed to define all of the program’s variables and call all of the sub-routines in the proper order. Each of the sub-routines, along with the main executable program can be viewed in their entirety in Appendix A.

INSERT – Once the original mathematical matrix is developed in

Matlab, this sub-program randomly seeds, or inserts, Pre-Osteoblasts and voids into the matrix. It also places the blood vessels in all necessary locations, being that blood vessels cannot be located more than 0.2mm away from any component within the living tissue (Aarden, 1994, Alberts,

2002).

CELLCYCLE – This sub-program simulates the natural maturation of the young Osteoblasts up to the point in the cell cycle where they are prepared to split, or go through cell division (Hartwell, 2001, Smith &

Martin, 1973). The program was written to illustrate the change in cellular maturity through a change in color.

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OSTEOBLAST – At this point, the cells do one of three things; split into two young osteoblastic daughter cells, transform into an osteocyte because they are surrounded by the matrix components they have secreted, or they die from apoptosis. This sub-program determines the fate of each Osteoblast (Smith & Martin, 1973).

CLEANNEIGHBORHOOD – If an osteoblast or osteocyte happens to die from apoptosis, starvation or because of injury to the area, this sub- program is executed. The first step performed by bone tissue after cells die, is to clear out the surrounding area to prepare for localized tissue regeneration.

PREOSTEOCLAST – This sub-program is performed whenever an

Osteoblast or Osteocyte dies. Its purpose is to search the tissue area for dead cellular debris and secrete the necessary white blood cells and proteins that are needed to produce Osteoclasts (Alberts, 2002).

OSTEOCLASTFINAL – After the necessary osteoclasts components have been secreted in the localized area of dead cells/tissue, the white blood cells and proteins merge to form an osteoclast, whose purpose is to engulf the dead cellular debris (Alberts, 2002).

VOIDCHECK – During cell division, if an osteoblast does not die from apoptosis or injury, it splits into two identical daughter cells; young osteoblasts. This sub-program replaces the mature Osteoblast with a

117 young daughter cell, and searches the original cell’s neighborhood for an available space, or void, in which to place the second daughter cell.

MATRIX – When cellular division does occur, necessary crucial bone matrix components are secreted as a result of the process, which form a significant percentage of bone weight and volume. This program searches the nearby neighborhood for a place in which to secrete this matrix (Ham, 1972).

MINERALIZEDMATRIX – Any proteins secreted during cell division that are not used for other functions are then used to mineralize existing bone matrix. This sub-program searches for newly generated matrix and uses the available proteins to mineralize it (Alberts, 2002).

OSTEOCYTE – If a mature Osteoblast does not go through cellular division, and becomes surrounded by the matrix it secretes, it is transformed into a more mature type of bone cell called an osteocyte.

This sub-program appropriately matures and ages osteocytes. It also keeps track of their age, and when a cell has reached its maximum lifespan (Aarden, 1994).

ENGULF – Once osteoclasts are generated because a cell has died, this sub-program searches the area for dead cellular debris and the osteoclasts engulf that debris. The area previously occupied by an osteoclast becomes a growth factor, because the engulfment processes

118 releases growth factors that attract young osteoblasts to the area and rejuvenate that particular portion of tissue (Smith, 1973, Lanza, 2007).

GROWTHFACTOR – This subprogram searches for available areas within the tissue to release the growth factors once cellular debris has been digested.

CHEMOTAXIS – This sub-program searches the extended neighborhood of growth factors to find the closest young Osteoblast to come help regenerate that tissue area. As soon as an Osteoblast is found, it starts moving toward the growth factor. This phenomenon in which cells orchestrate their movements according to their attraction to certain growth factors within the environment is called chemotaxis. Once the growth factor is reached the Osteoblast sets up camp, and starts maturing at that site (Wilkinson, 1974).

CELLDEATH – If an Osteoblast is older than 100 days it has reached its maximum life span and dies. Osteocytes have the ability to live up to

25 years, but with an average overall bone remodeling rate of 4-10% per year, the life of many osteocytes is much shorter (Frank-Odendaal, 2006).

This program checks the age of all existing cells to determine if they have reached their maximum life span. If this occurs, the series of programs used to create osteoclasts is executed in order to engulf this cellular debris.

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COUNTERCLEAR – In order to accurately keep track of the state of each cell in the grid, and its current value, counters were created that track the age of each matrix cell. This sub-program verifies that only the proper counters are being applied to each matrix cell, and that all other counters are set to zero at that location.

ZCONVERT – This sub-program creates a symbolic matrix of the variables listed previously in order to more easily inspect the grid of tissue after the program has executed.

Because it is intended to determine how bone cells grow and regenerate a cross section similar to that of an average adult femur, the time space grid in the cellular automation program was scaled accordingly. No complete regeneration time was assumed, however a maximum tissue area was assumed; that being the complete cross sectional area of the femur. Once this area was fulfilled on the time- space grid, it was assumed that regeneration was complete. Equilibrium here refers to a section of tissue consisting only of blood vessels, mostly bone matrix, bone lining cells, mature osteocytes and occasional voids which are indicative of the porosity of bone. Because of the phenomenon of apoptosis, singular cells can randomly die at any stage during regeneration or aging. This incident however only affects a very localized area of tissue and is represented accordingly. It can never be guaranteed that once tissue has completely healed, that there won’t be

120 some small incident(s) of apoptosis and cell cleanup occurring, as it is a part bone tissue’s very dynamic nature and need to remodel itself (Franz-

Odendaal, 2006). A specific time for wound healing to be complete cannot be generalized for everyone, since once the cells grow and produce extra-cellular matrix, the finalized section of bone still needs to strengthen and mature.

During program execution, each time step, or generation, represents one standard hour. To illustrate the progression of the program’s animation, hence the progression of projected tissue development, snapshots were taken at varying time steps to illustrate such phenomena as mitosis, apoptosis and chemotaxis. Additional images were captured to show the time-space grid at the completion of the program as well as the initial setup.

7.1.2. Mathematical Biology

The first objective in the realm of mathematical biology was to combine the Malthusian Model with the Logistic Growth Model to determine if this would produce an accurate model for predicting cell growth rate and cell population as a function of time. As previously stated the Malthusian Model consists of the following:

where:

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By rearranging the equation, it can be seen that the intrinsic growth rate is as follows:

The finite difference representation of this equation makes it much easier to implement with experimental data. The logistic growth equation with its own interpretation of growth rate is as follows:

where:

This definition of growth rate is then substituted into the logistic growth model, keeping in mind the limited carrying capacity version of the logistic law. The logistic law then becomes:

This new definition of intrinsic growth, once substituted into the

Logistic Law with Limited Capacity creates the newly formed Logistic-

Malthusian Model.

The Logistic-Malthusian model has vast potential for application in this research effort. After development, it was implemented with the experimental data from not only the in vitro osteoblast experiments, but

122 also with the data gathered from the Cell-Carbon system experiments. Its calculations for cell population were then plotted against the raw data to determine the accuracy of the Logistic-Malthusian model, and determine whether it was a dependable method for the prediction of osteoblast cell populations. Additionally, the results from the Cellular Automation program, the Monte Carlo simulations and the multivariate regression analyses were all compared to the raw data and the results that this method produced to study which overall method seemed to generate the most accurate and dependable results. All results were plotted and analyzed comparatively with each other.

The second objective dealing with mathematical biology is finding a way to relate the experimental data and theoretical calculations of cell population to overall tissue density. Through various published works

(Alberts, 2002, Aarden, 1994, Elledge, 1996, Hartwell, 2001, Ibelgaufts, 2009,

Lanza, 2007), important calculations were performed and crucial parameters were identified to generate realistic boundary and initial conditions for the selected mathematical models. Additionally, these calculations help ensure that all experiments carried out for validation purposes were properly developed and performed. Consistency in all known parameters throughout the mathematical models and their related experiments is crucial to guarantee the results from each are analyzed and evaluated accurately. The calculations were performed in

123 order to determine an estimation of dimensions and other parameters for a section of femur tissue currently undergoing wound healing due to fracture. Once calculations were complete and valuable individual cellular statistics obtained, all calculations were done in comparison to the experimental volume defined in the group of cell-carbon experiments.

The femur bone was chosen because it is the biggest of the long bones, and also the bone most commonly injured. The crucial calculations and parameters that were performed and analyzed consisted of the following:

1) Bone Tissue Component Breakdown

2) Average Dimensions of Sample Femur

3) Cortical and Cancellous Bone Tissue Density

4) Cortical and Cancellous Bone Mass

5) Overall Breakdown of Bone Volume Components

6) Average Cell Populations for a 19mm2 Femur Sample

(Cortical & Cancellous Cell Populations)

7) Average Extra-Cellular Matrix (ECM) Volume per Femur

Sample

Bone Tissue = Cortical Bone Tissue + Cancellous Bone Tissue

Bone Tissue = (Cortical Bone Matrix + Cortical Bone Cells + Cortical Bone

Lacunae) + (Cancellous Bone Matrix + Cancellous Bone Cells +

Cancellous Bone Lacunae)

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Total Typical Sample Volume from a Human Femur

Typical Femur Radius: 2.34 cm = 23.4 mm

Typical Femur Section Thickness: 100 µm = 0.1 mm

After determining the volume of a typical femur sample, it is important to point out that an established density for human bone is:

Because the density is known (Langer, 1993, Lanza, 2007), and the volume of the sample of femur is established, they can be used to calculate the overall bone mass for that section of bone to be:

In normal human bone, cortical bone (compact bone) makes up approximately 80% of overall bone mass, while cancellous bone (spongy bone) makes up approximately 20% of overall bone mass (Ham, 1972).

Using this statistic, it is now possible to calculate the total mass for each type of bone for the sample volume given:

Using these calculations for cortical and cancellous bone mass, the prospective densities of each type of bone can be calculated:

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Cancellous (spongy) bone is approximately only 25% as dense as cortical bone (Ham, 1972), making an equation for cancellous bone to be:

By solving for the cortical density using the previously calculated cortical bone mass, both the cortical and cancellous bone densities can be calculated:

An additional way of calculating ρcancellous is:

They are the major components of bone that contribute to its porosity.

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Within a normal section of femur, cortical lacunae compose approximately 1.5% of the femur’s volume while cancellous lacunae occupy approximately 2.8% (Ham, 1972, Lanza, 2007). Using these percentages, the average lacunae volume for both cortical and cancellous bone within the section of femur volume is as follows:

Since the lacunae are the gaps in which bone cells are housed, only 2/3 of their total volume is actually an open space or void and the remaining

1/3 of their volume is occupied by a bone cell (Langer, 1993, Lanza, 2007).

By multiplying the cortical and cancellous lacunae volume each by (2/3) and subtracting that from the overall femur volume, the net bone volume in the femur can be calculated.

This net bone volume can then be used to calculate net, or actual quantities of bone mass and volume, excluding the volume occupied by porosity.

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This indicates that the difference in mass, (326.839mg – 317.471mg =

9.368mg) is the mass of cellular fluid and other liquid components within bone.

Knowing that the actual cortical bone mass makes up 80% of total bone mass and that cancellous bone mass makes up the other 20%:

These mass calculations can be used along with the density of bone to find the actual net volume of both the cortical and cancellous portions of bone:

Both cortical and cancellous bone can each be broken down even further into two components: cells and matrix. In cortical bone, the extracellular matrix occupies approximately 87.5% of the tissue while cortical bone cells only occupy 12.5% (Ham, 1972). This makes the overall volume of these components of cortical bone to be the following:

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With cancellous bone being more porous, the extracellular matrix only makes up 12.5% of the overall cancellous bone volume and the cells make up approximately 87.5% of the volume.

Knowing that there are approximately 500 cells in every mm3 of cortical bone tissue (Ham, 1972),

Knowing that there are approximately 60 cells in every mm3 of cancellous bone tissue (Ham, 1972),

Since the total number of cells in the tissue sample of the femur are now known, along with the overall volume of both cortical and cancellous bone tissue, the individual volume and radius of both cortical and cancellous cells can also be calculated:

Remembering the definition of density, the mass for each individual cell can also be calculated:

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Knowing that the equation for the volume of a sphere is:

This equation can be manipulated in order to find the average radius of both cortical and cancellous bone cells:

And finally, knowing that individual cells occupy approximately 1/3 of the lacunae in which they are housed, it is possible to determine the approximate volume of the individual lacunae in both cortical and cancellous tissue:

All of these calculations are crucial in developing realistic mathematical models to predict bone tissue behavior during regeneration and wound healing. Additionally, when entire systems of

130 tissue and biomaterials are being analyzed, they will provide a good foundation for comparison purposes to investigate how the presence of biomaterials affects all the different morphological and mechanical parameters of cell, matrix and all other bone tissue components.

All of these calculations as well as the key parameters identified by trusted sources have provided the opportunity to take any value of cell population and convert it to tissue density. Since cells are only a proportion of the overall tissue, the other factors highlighted in this section, that affect the occupied volume and that are included in overall bone density, are included when calculations of tissue density have been made. The preceding calculations can be seen in their entirety in

Appendix C.

7.1.3. Multivariate Regression

Various multivariate regression models were developed to determine if there were any significant relationships between the dependent variable of cell population and any of the independent variables, namely time and the material properties of the carbon fibers.

The initial regression model was simply constructed from the in vitro experimental data to determine any statistical significance that the cell population may have with the time variable.

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The second regression analysis was conducted with the purpose of proving that the data produced by the cellular automation program was accurate and had statistical significance with the experimental data. This result then implies that a mathematical expression, based on both time and the cellular automation program output could be used with confidence to predict cell population.

Lastly, a multivariate regression analysis was performed with respect to the cell-carbon system experimental data and the material properties of the various types of carbons considered for use in bone tissue engineering applications. This analysis was performed to investigate which material properties of carbon fibers prove most significant in affecting overall cell growth. Additionally it was used to develop a model that could predict how well cells would grow and proliferate with respect to time on various types of carbons, each with unique material properties.

This model could also be used to determine the most advantageous combination of material properties that would achieve optimum tissue regeneration. From this prediction, a suitable variation of carbon fiber could then be fabricated.

7.1.4. Monte Carlo Simulation

Similar to the various ways in which regression analysis was utilized, the Monte Carlo methods were also used in a variety of fashions. Initially

132 the Monte Carlo methods were only going to be used to validate the significant relationships between cell population and the various carbon fiber material properties. After developing a regression model that helped define that relationship, it was then determined to utilize these methods to also predict cell population when the regression model parameters had some degree of variation. This was deemed necessary since with any regression based model, there exists a degree of error associated with the model’s parameters. Addressing that error and accounting for its randomness was the purpose for this second model.

Therefore, the first model uses a PERT distribution to define each of the material property variables: surface roughness, crystallinity – crystal height and crystallinity – crystal diameter. A PERT distribution returns and assigns random values to a variable according to a triangular distribution that is described by three bounding points: a minimum value, likely value and a maximum value. In addition to using a PERT distribution to assign values to the material properties, correlation matrices were developed for each of five time periods in which experimental cell data was collected

(8hr, 24hr, 48hr, 72hr and 96hr). A correlation matrix is a statistical technique that can show whether or not, and how strongly, pairs of variables may be related. Correlation values are generated on a scale from 0 to 1, with 0 having no statistical relationship, and 1 being the strongest statistical relationship possible (a variable with itself).

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Once correlation matrices were established to define how strongly specific variables were related, a multivariate normal distribution was generated using these variables and the correlation matrices to predict, just as multivariate regression can, what value will be assigned to cell population; what is the predicted cell population at that particular time.

To gain accuracy and the true effectiveness of such a method, five hundred simulations were performed and the actual cell population values that were recorded were the mean values of these simulations.

These results were then plotted against experimental data to prove accuracy.

The second application of the methods of Monte Carlo involved the regression model that related time, surface roughness and the crystallinity parameters of the carbon fibers to the dependent cell population variable. Once the multivariate regression model had been developed, each variable’s value, along with its prospective standard error were used to develop a normal distribution model. Within this normal distribution model, the actual value of the variable was defined as the mean of the distribution and the standard error was used as the standard deviation. From these normal distribution models, random values were then assigned during the simulations to each of the model parameters, excluding time: surface roughness, crystal height and crystal diameter.

Next, the model derived from multivariate regression was used to

134 calculate the cell population, but used the normal distributions for each variable instead of a singular value. Once again, five hundred simulations were performed to analyze the potential for cell growth (or predicted cell population) (as a function of time) on each of the four carbon fiber materials being studied. The results of these simulations were also plotted not only against experimental data but also against the regression model data to prove the accuracy and dependability of Monte Carlo based methods.

All Monte Carlo simulations were performed using Microsoft Excel’s

RiskAMP® package. This package incorporates a full featured Monte

Carlo Simulation engine with the ability to perform thousands of simulations within minutes (RiskAMP Monte Carlo Add-In, 2005).

7.2. Experimental Research Schemes

To corroborate the diverse analytical methods employed in this effort, a series of experiments have been performed with three separate schemes to monitor and study. These experiments are the foundation of this research in that they illustrate the accuracy of the developed theoretical models and confirm that those models are dependable even without the presence of experimental data. All experiments were performed in a sterile environment to prevent contamination and pure

135 alcohol was used on hands, tools, and all surfaces that were used during the experiments.

7.2.1. In Vitro Experiments

Primary human osteoblasts (ATCC) were maintained in a culture composed of Dulbeccos’s modified Eagle’s medium (Sigma Aldrich, St

Louis MO), also referred to as DMEM. This medium was supplemented with

10% Fetal Bovine Serum, 1% Penn Strep solution (Omega Scientific, Inc.),

0.5mM Sodium Pyruvate (Atlanta Biologicals, Atlanta GA), 2.5mM L- glutamine (Invitrogen, CA) and G418 Sulfate Antibiotic 0.3mg/ml (Fisher

Scientific, NJ). Confluent flasks of osteoblasts were incubated with

Trypsin/HEPES (0.02% Trypsin, 10mM of HEPES in Ca, Mg-free PBS) for 5 minutes, spun down at 1000 RPM for 5 minutes and then re-suspended in fresh culture medium. Osteoblasts were then seeded into five sterile Petri dishes. The cells were incubated in culture medium at 37oC, 5% CO2 until the appropriate time at which samples were removed for staining and analysis. Cells were incubated for up to a period of 96hrs, with one of each of the five samples were removed for staining at 8hrs, 24hrs, 48hrs,

72hrs and 96hrs into the experiment. This staining is what permitted cell growth to be quantified through the microscopic counting of cell populations in each dish, at each time increment.

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Following the incubation period of each sample, the media was removed and the samples were rinsed with PBS to remove any non- adherent cells. Samples were then fixed with 3.7% formaldehyde for 15 minutes and rinsed with PBS. This was followed by a 15-minute permealization step with cold methanol. Samples were then rinsed again in PBS and stained with Hoechst Texas-Red Stain (Invitrogen, CA), 10ug/ml, for 30 min. Finally, the samples were then coated with Prolong Gold

(Invitrogen, CA) and stored in a freezer to prevent any loss of fluorescence.

Once samples were adequately prepared for viewing, structural characterization and analysis of cell populations over the various time periods was carried out using confocal microscopy. A confocal microscope uses point illumination and a pinhole in an optically conjugate plane in front of the detector to eliminate out-of-focus signals.

Since only the light produced by fluorescence very close to the focal plane can be detected, the resolution of the resulting image is much better than that of more common wide-field microscopes (Davidson,

1995-2010). Because these cells were simply monitored for growth in Petri

Dishes, a limited volume was not of consequence here. This experiment was simply derived to determine how cell growth rates perform when not in the presence of a carbon fiber. Standard optical microscopy was also

137 used in between replenishment of media to the samples to ensure the cells remained viable and the samples were not contaminated.

7.2.2. Carbon Material Testing

Past research efforts have proven that various surface characteristics of carbon fibers show promise in promoting cell growth and proliferation. One of the most explored characteristics of carbon fibers are their surface roughness. In addition to studying their surface roughness, four vastly different types of carbon fibers were investigated as potential carriers of human osteoblasts by studying their fiber orientation and their varying degrees of crystallinity. The four types of fibers being investigated are: the AS4, P25, T650 and P120.

The AS4 carbon fiber is a continuous, high strength, high strain PAN based fiber available for manufacture in various bundle sizes. Its relatively ductile nature makes it very applicable in fabric type applications and carbon weaves. The P25 is a continuous pitch based fiber with similar characteristics to the AS4. The fiber however, varies in its crystallinity, in that its ductility is some degree lower than that of the AS4 fiber. The T650 fiber is a high strength, standard modulus carbon fiber that exhibits excellent oxidation resistance in non-physiological environments and performs well in a composite setting. This is a much more crystalline material, which is very evident in its brittle nature. Finally is the P120 fiber.

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This is a continuous filament, ultra-high modulus fiber with high crystallinity and therefore high brittleness. Each of these types of carbon fibers have properties attractive to biomedical applications. The various forms of analysis used to explore these properties determined which materials were more suited for tissue engineering purposes.

All experiments performed on these fibers were executed using samples from the same batch of material to ensure consistency with results. To study the surface roughness of each of these fibers, an Atomic

Force Microscope (AFM) was used to scan various sections of the materials and calculate the overall surface roughness of each section.

AFM is a very high-resolution type of scanning probe microscopy, with demonstrated resolution on the order of fractions of a nanometer and more than 1000 times better than the optical diffraction limit. An actual

Atomic Force Microscope consists of a cantilever with a sharp tip (probe) at its end that is used to scan the specimen surface. The cantilever is typically silicon or silicon nitride with a tip radius of curvature on the order of nanometers. When the tip is brought into proximity of a sample surface, forces between the tip and the sample lead to a deflection of the cantilever according to Hooke's law. A diagram illustrating how an AFM works is shown in Figure 7.2.2-1.

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Figure 7.2.2-1: A Block Diagram for the Atomic Force Microscope

Once the four fiber types were analyzed, the average surface roughness of each piece was recorded and pictures were taken, illustrating their roughness at scales of 40x40μm, 4x4μm and 2x2μm. All images were scanned at a rate of 0.2 Hz and a resolution of 512.

In addition to studying surface roughness, crystallinity was a very important property to investigate because of how greatly each of the materials varied in crystallinity. Additionally, past studies have not been able to truly capture how the molecular structure of carbon fibers influences the ability for cells to grow and proliferate on their surface

(Blazewicz, 2001). Initially a RAMAN spectrometer was used to study the crystallinity of each type of fiber, specifically the fiber’s average crystal diameter, La. RAMAN spectroscopy is a technique used to study vibrational, rotational, and other low-frequency modes in a system. It relies on inelastic scattering, or Raman scattering, of monochromatic light,

140 usually from a laser in the visible, near infrared, or near ultraviolet range.

The laser light interacts with phonons or other excitations in the material, resulting in the energy of the laser photons being shifted up or down. The shift in energy gives information about the phonon modes in the system.

In this particular instance, its purpose is to help characterize the material by finding the crystallographic orientation of the sample. A given solid material has very characteristic phonon modes that can help an experimenter identify it how crystalline the material may be (Gardiner,

1989). Each sample was trimmed to a size appropriate for analyzing within the spectrometer unit and the crystallinity for each was measured utilizing the peak energy shifts within the material, identified by the laser.

To help verify the results gathered from the RAMAN spectrometer, an X-RAY diffractometer was used to calculate the crystallinity of each material. The diffractometer operated at 40V and 30μA, and an x-ray wavelength of =0.154059. Once the material had been scanned, a variation of the Scherrer equation for x-ray diffraction was used to calculate the components of crystallinity within each material. The first variation of this equation was used to calculate the average crystal heights (Lc) of the crystallites on each fiber:

where:

141

The 0.9 factor in the above equation and the 1.84 in the succeeding equation represent the shape factor. A shape factor is used in x-ray diffraction and crystallography to correlate the size of crystallites in a solid to the broadening of a peak in a diffraction pattern (Cullity & Stock, 2001).

For the average diameter of the crystallites of a fiber, La, the following variation of the Scherrer equation was utilized:

where:

Because the Scherrer equation is most accurately applicable to nano-sized particles, it deemed most appropriate for this particular application. From the x-ray scan, all of the necessary parameters were obtained and recorded to calculate both Lc and La for all four types of carbon fibers.

7.2.3. Cell-Carbon System Experiments

The final form of experiments performed for this initiative involved primary human osteoblast systems and each of the individual forms of carbon fibers. Just as with the in vitro experiments described in Section

142

7.2.1, primary human osteoblasts were properly thawed, maintained and cultured for the duration of the experiment. Samples from each batch of the carbon fibers were prepared, measuring 19mm in length and 19mm in width to simulate the cross sectional area of a human femur. These samples were then sterilized using ultraviolet light for a period of 24 hours.

Osteoblasts were then seeded onto each of the samples and incubated in culture medium at 37oC, 5% CO2. The culture medium was replenished every 24 hours.

Five samples for each material were prepared (20 samples total) and seeded with one sample from each type of material being suspended and stained at the following experiment times: 8hrs, 24hrs,

48hrs, 72hrs and 96hrs. When the samples were ready for staining, the media was removed and the samples were rinsed with PBS to remove any non-adherent cells. Samples were then fixed with 3.7% formaldehyde for

15 minutes and rinsed with PBS. This was followed by a 15-minute period of permealization with cold methanol. Samples were then rinsed again in

PBS and stained with Hoechst Texas-Red Stain (Invitrogen, CA), 10ug/ml, for 30 min. Finally, the samples were coated with Prolong Gold

(Invitrogen, CA) and stored in a freezer to prevent any loss of fluorescence.

Each of these samples was viewed for analysis and cell population characterization with an inverted microscope. An inverted microscope is

143 a microscope with its light source and condenser on the top, above the stage pointing down, while the objectives and turret are below the stage pointing up. This type of microscope is very useful for observing living cells or organisms at the bottom of a large container, or tissue culture flask, under more natural conditions than on a glass slide, which is usually the

case with a conventional microscope (Smith J. , 1952).

A picture of an inverted microscope is shown in Figure

7.2.3-1.

With the inverted microscope, each of the samples

was carefully counted and various images obtained

Figure 7.2.3-1: An and properly scaled for reference and additional Inverted Microscope (J. Lawrence Smith) analysis of cell behavior and potential cell proliferation patterns.

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Chapter 8. Description of Results

This chapter details the results of all methods incorporated in this research. Any data, calculations or programs used to achieve these results are shown in their entirety in the appendices.

8.1. Theoretical Research

The results of the theoretical models developed showed excellent promise in their ability to model bone tissue growth and regeneration processes. These results deemed even more impressive when all were compared to each other and the experimental data gathered. Results of the four theoretical methods incorporated will be discussed separately, and in conclusion will be compared with respect to each other for determination of accuracy.

8.1.1. Cellular Automation

The fully functional cellular automata program successfully animated a time space grid with all the components and processes that make up bone tissue. Upon complete execution of the program, the data generated was automatically exported to Microsoft Excel for further

145 analysis and comparison with the other theoretical models. Figure 8.1.1-1 shows the initial setup of the program, where young osteoblasts have been seeded onto the grid, blood cells have been placed no more than

0.2mm apart and the remaining spaces are empty voids.

Figure 8.1.1-1: Initial Seeding of Osteoblasts on the CA Time-Space Grid.

Another snapshot has been taken just prior to this first group of cells undergoing mitosis. As you can see in Figure 8.1.1-2, the cells remain but are now a different color to indicate that they have grown and matured, and are now ready to take on one of three possible fates: mitosis, apoptosis, or becoming an osteocyte. During this portion of the cell cycle, cells can randomly die because of lack of nutrition or

146 overcrowding. This is evident in the minute changes in cell location between Figure 8.1.1-1 and 8.1.1-2.

Figure 8.1.1-2: A Snapshot of the Time-Space CA Grid Just prior to Cells Undergoing Mitosis

During the process of mitosis, important proteins are released that lay down the foundation for the extra-cellular matrix of the tissue; however mitosis will have to occur several times in order for that tissue to be mineralized. Figures 8.1.1-3 and 8.1.1-4 illustrate the non-mineralized matrix produced by this first round of mitosis and the matrix being mineralized as the body of cells experiences mitosis a second time. Cells undergoing

147 mitosis are the key to tissue producing the appropriate amounts of extracellular matrix.

Figure 8.1.1-3: Between Cycles of Mitosis, the Initial Non-Mineralized Matrix is Identified by the Vibrant Orange Cells within the Grid

The striatic patterns of cellular division are interesting in that they indicate that cells prefer to grow in that manner. This is of particular interest since it has been previously established that cells also prefer to grow axially when seeded onto various forms of carbon fibers.

Figure 8.1.1-4 shows the beginning signs of mineralized matrix, indicated by the lighter color of orange. This signifies the presence of

148 necessary proteins and cellular components needed to properly mineralize the extracellular matrix.

Figure 8.1.1-4: The CA Grid Just After Another Round of Mitosis Illustrating how the Process of Mitosis Mineralizes the Matrix Components of the Tissue

As the process progresses, more matrix is develop and mineralized as the cells continue to undergo mitosis. The grid is slowly but surely filled in with cells and matrix, until the only major components present are mature bone cells (osteocytes), blood vessels, mineralized matrix and the occasional void. These voids are important because they represent the naturally occurring porosity of bone. These features are prevalent in

Figures 8.1.1-5 through 8.1.1-7.

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Figure 8.1.1-5: Matrix Continues to be Mineralized as More and More Cells Undergo Mitosis

In Figures 8.1.1-5 and 8.1.1-6, small areas can be seen throughout the already developed tissue, where cells have died, osteoclasts were formed to engulf the dead cellular debris, and growth factors are currently present attempting to attract young osteoblasts to the site for tissue regeneration.

150

Figure 8.1.1-6: Tissue Continues to Grow. Small Areas of Dead Cellular Debris Can be Seen, Where Osteoclasts have cleaned the Area and Growth Factors have been Developed to Attract New Cells to the Area

Figure 8.1.1-7 shows the final grid. Mature osteocytes, extracellular matrix, blood vessels occasion voids now constitute the “cross-section” of tissue.

After achieving this point in the growth/regeneration process the only activities that will continue to occur involve dead cell turnover, in which very small, localized areas of tissue will die and be replenished with new cells. Traumatic injury to the site would be the only other event that could cause mass regeneration, such as what was illustrated by this program, to occur again.

151

Figure 8.1.1-7: The Final Snapshot taken of the CA Program Grid. Notice that the Tissue is Now Fully Comprised of Osteocytes, Blood Vessels, Mineralized Matrix and Occasional Voids, which is Very Representative of Actual Healthy Bone Tissue

Additional information pertaining to the program’s execution, including the computer programming codes is included in Appendix A.

Appendix B contains samples of the program’s output that were automatically exported to Microsoft Excel©.

8.1.2. Mathematical Biology

The Logistic Growth Equation was combined with the Malthusian Model to produce:

152

The in vitro data and cell-carbon system experimental data will be discussed in Sections 8.2.1 and 8.2.3, but the data is included here to illustrate the accuracy of the developed Logistic-Malthusian Model. Table

8.1.2-1 shows the in vitro experimental data collected over a period of 96 hours.

Table 8.1.2-1: In Vitro Experimental Cell Data

Time (Hrs) Cells Rate (Cells/Hr) Cell Popluation (Approximation) 0 10 10 2 35 1.25 35 4 167 1.885714286 166 8 301 0.200598802 295 12 489 0.156146179 473 24 767 0.047375596 728 48 966 0.010810517 922 72 1011 0.001940994 998 96 1205 0.007995384 1148

Using the Logistic-Malthusian Model, this data was approximated.

The Malthusian model was used to calculate the intrinsic growth rate of the cells with respect to time. This value was then plugged into the

Logistic Law to estimate the cell population. This data has been plotted for comparison in Figure 8.1.2-1.

153

Comparison of In Vitro Data and the Logistic- Malthusian Approximation 1400

1200

1000

800

600 Experimental

Cell Population Cell 400 Logistic-Malthusian 200

0 0 20 40 60 80 100 120

Time (hrs)

Figure 8.1.2-1: Comparison of the In Vitro Experimental Data and the Logistic-Malthusian Model

The same type of comparison was made between the results of the CA program and the Logistic-Malthusian model. These are shown in Figure

8.1.2-2.

Time CA Population Growth Rate Approximation (hrs) (Cells) (Cells/Hr) (Cells) 11 5 21 10 0.25 5 32 53 1.070538058 10 Comparison of Cellular Automation and Logistic- 43 62 0.045238095 53 Malthusian Models 53 85 0.088709677 62 64 115 0.09077381 85 4000 75 161 0.100436681 115 3500 85 226 0.10046729 161 96 316 0.098888889 226 3000 107 428 0.088375796 316 117 568 0.082058824 428 2500 128 740 0.0755093 568 139 948 0.070578231 740 2000 Logistic_MalthusianPopulation 150 1173 0.059151194 948 CAPopulation 160 1429 0.054590305 1173 1500 171 1689 0.045510563 1429 182 1956 0.039618707 1689 1000 192 2225 0.034327591 1956 203 2506 0.031539679 2225 500 214 2769 0.02629994 2506 224 3000 0.020799273 2769 0 235 3171 0.01425457 3000 0 50 100 150 200 250 300 246 3300 0.010153895 3171 256 3386 0.006517762 3300 267 3439 0.003937593 3386

Figure 8.1.2-2: Comparison of the CA Program and the Logistic-Malthusian Model

154

Finally, we have the implementation of the Logistic-Malthusian model on the Cell-Carbon System data. As previously notes, this data will be illustrated in its entirety in section 8.2.3, however because it was used in conjunction with the Logistic-Malthusian model, it is also included here.

The cell-carbon system experimental data is shown in Table 8.1.2-2.

Table 8.1.2-2: Cell-Carbon System Experimental Data

Time AS4 P120 T650 P25 (Hrs) Cell Count Cell Count Cell Count Cell Count 8 210 478 258 348 24 121 942 825 257 48 193 1202 907 364 72 271 1476 1242 518 96 414 2190 1847 603

The Logistic-Malthusian model was applied to this data and its results were plotted against the raw data to prove its accuracy. The spreadsheet of calculated data is shown in Table 8.1.2-3. This is followed by Figure 8.1.2-3 which plots the experimental data against the Logistic-

Malthusian data for the four carbon fibers.

Table 8.1.2-3: Logistic-Malthusian Model Application to the Cell-Carbon System Data

155

450 2500 400 350 2000 300 1500 250 AS4 P120 200 AS4_T 1000 P120_T 150

100 500 50 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120

2000 700 1800 600 1600 1400 500 1200 400 1000 T650 P25 300 800 T650_T P25_T 600 200 400 100 200 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120

Figure 8.1.2-3: Comparison between the Cell-Carbon System Experimental Data and the Implementation of the Logistic-Malthusian Model on that Data – For Each Graph: X Axis = Time (hrs) Y-Axis = Cell Population

As an aside, the logistic growth equation was also used to help predict how the cell populations would have continued to grow in the cell-carbon system experiments. To perform these calculations, the

“GROWTH” formula in Microsoft Excel was used. This particular formula calculates predicted exponential growth by using existing data. It returns the y-values for a series of new x-values that have been specified by using existing x-values and y-values. If the cell-carbon system experiments had been carried out for longer than 96 hrs, the growth patterns shown in

Table 8.1.2-4 were predicted to occur based on the laws of exponential growth:

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Table 8.1.2-4: Using the GROWTH Function to Predict Cell Populations Beyond the Scope of the Established Experiments: The Orange indicates Projected Values

Time AS4 P120 T650 P25 (Hrs) Cell Count Cell Count Cell Count Cell Count 8 210 478 258 348 24 121 942 825 257 48 193 1202 907 364 72 271 1476 1242 518 96 414 2190 1847 603 120 457.1451853 3278.034959 3189.783485 718.9738121 144 726.8879507 4158.334158 3914.933802 996.1200412 168 966.2316662 5917.805663 6122.072178 1222.35066 192 1305.9358 8484.612526 9169.335869 1501.382907 216 1756.976844 11615.31076 13332.1912 1934.885112 240 2515.48418 15945.4009 18953.65329 2468.838297 264 3334.030298 22675.99485 28926.31258 3051.938298 288 4592.138173 31408.66568 41866.63598 3878.917006 312 6315.034274 43450.41852 61129.98246 4925.980502 336 8668.288904 60779.52572 90046.35932 6185.192301

This method was also carried out on the theoretical data produced by the

Logistic-Malthusian model through a time of 216 hours. The results are shown in Table 8.1.2-5.

Table 8.1.2-5: Using the GROWTH Function to Predict Cell Populations Beyond the Scope of the Established Logistic-Malthusian Model Approximations: The Orange - Projected Values through a Time of 216 Hours

Theoretical Estimations AS4 P120 T650 P25 Rate Count Rate Count Rate Count Rate Count 2.5 209.4203 5.85 476.6435 3.1 257.2812 4.225 347.0203 -0.02649 126.4174 0.060669 877.7125 0.137355 782.5983 -0.01634 266.1791 0.024793 190.4748 0.0115 1131.009 0.004141 887.3913 0.017348 356.0293 0.016839 266.6365 0.009498 1380.537 0.01539 1153.929 0.017628 501.7519 0.021986 402.7672 0.020156 1884.532 0.020297 1629.2 0.006837 590.2377 0.004342 451.9678 0.020701 2587.369 0.030292 2470.908 0.008014 698.7036 0.024586 691.1455 0.011189 3321.913 0.009472 3244.478 0.016061 938.3633 0.01372 915.8038 0.01763 3797.09 0.023491 3617.492 0.009463 1157.031 0.014649 1210.796 0.018073 4081.753 0.02074 3761.925 0.009511 1402.521 0.014391 1586.243 0.015374 3915.96 0.018917 2268.244 0.012031 1746.232

The very detailed series of equations used to calculate individual cell and tissue characteristics (Section 7.1.2) proved very beneficial in

157 providing a way to relate the cell populations produced by these series of mathematical models, into tissue density. Once a cell population was calculated from either experiment or an analytical method, the relationships highlighted in Section 7.1.2 were used to calculate overall tissue density with respect to time. With respect to the in vitro experimental data, because there was no bounding volume involved with this experiment, the mathematical expressions breaking down the characteristics of both the cells and matrix were not applicable. They are however, applicable with the cell-carbon system data because a volume was defined in which cells could grow and develop tissue. It was advantageous in these experiments to determine from these calculations how long a volume of that size would take to fill with healthy tissue.

The defined volume, or volume in which cells were permitted to grow, was approximately 57mm3. This involved a 19mm x 3mm x 1mm sample of each bundle of carbon fibers. Due to the large size of the data, it is shown in its entirety in Appendix C along with the manual calculations for individualized cell characteristics explained in Section

7.1.2. An excerpt of those calculations and that data is shown here in

Figure 8.1.2-4 for the AS4 Carbon Fiber.

158

Time AS4 Cortical Cells Cancellous Cells Cortical Cell Density Cancellous Cell DensityCortical Cell Volume (Hrs) Cell Count (# cells) (# cells) (cells/mm3) (cells/mm3) (mm3) 8 210 168 42 5.894736842 7 0.336 24 121 96.8 24.2 3.396491228 4.033333333 0.1936 48 193 154.4 38.6 5.41754386 6.433333333 0.3088 72 271 216.8 54.2 7.607017544 9.033333333 0.4336 96 414 331.2 82.8 11.62105263 13.8 0.6624 Cortical Cell Mass Cortical ECM Volume Cancellous Cell Volume Cancellous Cell Mass Cancellous ECM Volume Total Tissue Volume Volume Capactiy Relative Tissue Density (mg) (mm3) (mm3) (mg) (mm3) (mm3) (mm3) (%) 0.6384 2.352 0.7014 1.33266 0.1002 3.4896 57 6.122105263 0.36784 1.3552 0.40414 0.767866 0.057734286 2.010674286 57 3.527498747 0.58672 2.1616 0.64462 1.224778 0.092088571 3.207108571 57 5.626506266 0.82384 3.0352 0.90514 1.719766 0.129305714 4.503245714 57 7.900431078 1.25856 4.6368 1.38276 2.627244 0.197537143 6.879497143 57 12.06929323

Figure 8.1.2-4: Excerpt of Data from the Tissue Composition Calculations - AS4 Carbon Fiber

Shown here in Figure 8.1.2-5 is the tissue volume developed over time for each of the four carbon fiber bundles.

Calculated Tissue Volume with Respect to Time 40

25 ) 3 AS4_TissueVolume 20

(mm P25_TissueVolume 15 T650_TissueVolume P120_TissueVolume 10

0 0 20 40 60 80 100 120

Time (Hours)

Figure 8.1.2-5: Tissue Volume with Respect to Time for Each of the Carbon Fiber Bundles. This Volume Calculation Takes into Consideration the Breakdown of Cancellous and Cortical Cells as well as ECM

To achieve these calculations of tissue volume, the following steps were taken:

(1) Number of Cortical and Cancellous Cells were Calculated from the

Overall Cell Count with Respect to Time

(2) Using the Individual Volume and Mass for a Cortical Cell (0.002 mm3 and

0.0038mg)and a Cancellous Cell, (0.0167 mm3 and 0.03173mg), Overall

159

Cortical and Cancellous Cell Volume and Mass were Calculated with

Respect to Time

(3) Knowing that Cells make up Approximately 12.5% of Cortical Tissue and

87.5% of Cancellous Tissue, these Numbers were used to Calculate the

Produced ECM Volume with Respect to Time.

(4) Knowing Total Cell and Tissue Volume for Both Cancellous and Cortical

Tissue, Total Tissue Volume was Calculated with Respect to Time.

(5) Finally, Knowing the Maximum Carrying Capacity for Each Carbon Fiber

Bundle to be 57mm3, the Relative Tissue Density was Calculated in

Comparison to that Volume to Determine how much Volume still Needed

to Be Filled with Tissue.

Relative Tissue Density 70

40 AS4_RelativeTissueDensity (%) 30 P25_RelativeTissueDensity T650_RelativeTissueDensity 20 P120_RelativeTissueDensity

0 0 20 40 60 80 100 120

Number of Overall Cells

Figure 8.1.2-6: Relative Tissue Density for the Various Carbon Fiber Bundles

Figure 8.1.2-6 illustrates each material’s relative tissue density with respect to time. Additionally, the calculations that identified a cell’s individual volume can also be used to calculate overall tissue volume occupied

160 specifically by cortical and cancellous cells. These values for volume:

Vcort = 0.002 mm3 and Vcanc = 0.0167 mm3 provide a quick means for identifying an approximate tissue volume at any given time.

8.1.3. Multivariate Regression

The first regression analysis performed (Figure 8.1.3-1) was on the in vitro experimental data. The analysis was used to determine a statistical significance between cell population and time existed, which was strongly suspected.

SUMMARY OUTPUT: InVitro Cell Regression Analysis

Regression Statistics Multiple R 0.916975322 R Square 0.840843742 Adjusted R Square 0.801054677 Standard Error 205.5327993 Observations 6

ANOVA df SS MS F Significance F Regression 1 892717.0737 892717.0737 21.13253352 0.010053497 Residual 4 168974.9263 42243.73158 Total 5 1061692

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 246.3052632 131.206382 1.877235386 0.133707772 -117.9820539 610.5925802 -117.9820539 610.5925802 X Variable 1 11.21842105 2.440371547 4.597013544 0.010053497 4.442863417 17.99397869 4.442863417 17.99397869

RESIDUAL OUTPUT PROBABILITY OUTPUT

Observation Predicted Y Residuals Standard Residuals Percentile Y 1 246.3052632 -236.3052632 -1.285426544 8.333333333 10 2 336.0526316 -35.05263158 -0.190675326 25 301 3 515.5473684 251.4526316 1.36782348 41.66666667 767 4 784.7894737 181.2105263 0.985728449 58.33333333 966 5 1054.031579 -43.03157895 -0.234078298 75 1011 6 1323.273684 -118.2736842 -0.643371761 91.66666667 1205

Figure 8.1.3-1: Regression Analysis for In Vitro Data

This analysis yielded the following relationship:

161

The results of the regression indicate a strong significance between cell population and time with an overall model significance of approximately

0.01 and a individual significance of the time variable also of 0.01. The overall accuracy of the model is depicted by the strong R-squared value of 0.84.

The next regression analysis completed involved the data collected from the cell-carbon system experiments. Not only was the cell count data included, but also the experimental measurements of surface roughness and crystallinity, with La =Average Crystal Diameter and Lc =

Average Crystal Height. The results of this analysis are shown in Figure

8.1.3-2.

SUMMARY OUTPUT: Cell-Carbon System Experiments

Regression Statistics Multiple R 0.909950961 R Square 0.828010751 Adjusted R Square 0.782146951 Standard Error 275.0282353 Observations 20

ANOVA df SS MS F Significance F Regression 4 5462362.247 1365590.562 18.05368839 1.33107E-05 Residual 15 1134607.953 75640.53019 Total 19 6596970.2

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -705.5777261 212.0309368 -3.327711215 0.004590485 -1157.510968 -253.6444844 -1157.510968 -253.6444844 Time 9.917036802 1.936396062 5.121388644 0.000125288 5.789706315 14.04436729 5.789706315 14.04436729 La -45.82606223 18.93432345 -2.420264043 0.028665563 -86.18361716 -5.468507304 -86.18361716 -5.468507304 Rough 1.12871642 0.266623129 4.233377738 0.000722546 0.560422675 1.697010165 0.560422675 1.697010165 Lc -8.110939296 1.879739238 -4.314927908 0.00061312 -12.11750862 -4.104369971 -12.11750862 -4.104369971

Figure 8.1.3-2: Regression Results for the Cell-Carbon System Experimental Data RESIDUAL OUTPUT PROBABILITY OUTPUT These results showed just as much promise and dependency, with high Observation Predicted Pop Residuals Standard Residuals Percentile Pop 1 -170.748731 380.748731 1.558089329 2.5 121 overall model2 -12.07614213 significance,133.0761421 high0.544570474 independent variable7.5 statistical193 3 225.9327411 -32.93274112 -0.134766444 12.5 210 4 463.9416244 -192.9416244 -0.78955033 17.5 257 5 701.9505076 -287.9505076 -1.17834303 22.5 258 6 5.451269036 342.548731 1.401768356 27.5 271 7 164.1238579 92.87614213 0.380065156 32.5 348 8 402.1327411 -38.13274112 -0.156045739 37.5 364 9 640.1416244 -122.1416244 -0.499824546 42.5 414 10 878.1505076 -275.1505076 -1.125963227 47.5 478 11 603.251269 -345.251269 -1.412827607 52.5 518 12 761.9238579 63.07614213 0.258118428 57.5 603 13 999.9327411 -92.93274112 -0.380296769 62.5 825 14 1237.941624 4.058375635 0.016607571 67.5 907 15 1475.950508 371.0494924 1.518398376 72.5 942 16 845.051269 -367.051269 -1.502036958 77.5 1202 17 1003.723858 -61.72385787 -0.252584648 82.5 1242 18 1241.732741 -39.73274112 -0.162593214 87.5 1476 19 1479.741624 -3.741624365 -0.015311371 92.5 1847 20 1717.750508 472.2494924 1.932526191 97.5 2190 162 significance and an overall R-squared value over 0.82. This analysis yielded the following relationship:

The final regression model was performed to determine the statistical significance, and overall accuracy of the data extracted from the Cellular Automation Program.

SUMMARY OUTPUT: Cellular Automata Program Data

Regression Statistics Multiple R 0.967368111 R Square 0.935801062 Adjusted R Square 0.933009803 Standard Error 329.0645618 Observations 25

ANOVA df SS MS F Significance F Regression 1 36303270.41 36303270.41 335.2613756 3.28209E-15 Residual 23 2490520.174 108283.4858 Total 24 38793790.59

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -822.088014 135.6767946 -6.059164476 3.52127E-06 -1102.756846 -541.4191819 -1102.756846 -541.4191819 (hrs) 15.64695907 0.85455139 18.31014406 3.28209E-15 13.87918484 17.4147333 13.87918484 17.4147333

Figure 8.1.3-3: Regression Analysis for Data from CA Program

As seen in Figure 8.1.3-3, this analysis proved most accurate thus far, with an extremely high value of significance for both the overall model and the individual independent variable, time. This is reflected in the R- squared value of 0.936. The model developed from this regression analysis is:

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8.1.4. Monte Carlo Simulation

The first Monte Carlo Simulation involved the experimental values of roughness and crystallinity for each of the carbon fibers, as well as the derived correlation matrices showing the strength of relationship between all pairs of variables. Data from the Carbon Material Testing, discussed later in Section 8.2.2, is mentioned here to illustrate the overall effectiveness and means for performing a Monte Carlo simulation. Table

8.1.4-1 shows the data collected for each of the carbon fibers, along with the overall mean and standard deviation of the value of each material property.

Table 8.1.4-1: Values for Material Roughness and Crystallinity for Each Type of Carbon

Roughness La Lc AS4 676.44 6.44 1.59 P25 2083.3806 6.5189 2.046 T650 1418.8407 7.755 175.21 P120 2658.4224 29.974 19.199 Sum 6837.0837 50.6879 198.045 STDev 854.7779452 11.55039042 84.19897943

The correlation matrices, for each of the five times (in hours) that data was collected are shown in Figure 8.1.4-1. Figure 8.1.4-2 illustrates how the

Monte Carlo simulations were prepared. Included in this diagram are the control functions used in Microsoft Excel’s RiskAMP package to randomly generate the various values based on both normal and triangular distribution methods.

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8hrs Population Roughness La Lc 72hrs Population Roughness La Lc Population 1 0.9667 0.87 -0.2903 Population 1 0.6483 0.7319 0.5091 Roughness 0.9667 1 0.7423 -0.1553 Roughness 0.6483 1 0.7423 -0.1553 La 0.87 0.7423 1 -0.1891 La 0.7319 0.7423 1 -0.1891 Lc -0.2903 -0.1553 -0.1891 1 Lc 0.5091 -0.1553 -0.1891 1

24hrs Population Roughness La Lc 96hrs Population Roughness La Lc Population 1 0.5986 0.7014 0.5551 Population 1 0.5897 0.7332 0.5241 Roughness 0.9667 1 0.7423 -0.1553 Roughness 0.5897 1 0.7423 -0.1553 La 0.87 0.7423 1 -0.1891 La 0.7332 0.7423 1 -0.1891 Lc -0.2903 -0.1553 -0.1891 1 Lc 0.524 -0.1553 -0.1891 1

48hrs Population Roughness La Lc Population 1 0.6769 0.7932 0.4308 Roughness 0.6769 1 0.7423 -0.1553 La 0.7932 0.7423 1 -0.1891 Lc -0.2903 -0.1553 -0.1891 1 Figure 8.1.4-1: Correlation Matrices for the Various Variables Involved in the MC Simulations

Simulated Values (500 Random Values Chosen) These values were Based on a Normal Distribution with the Distribution Mean equal to Number of Cells Counted during the Cell-Carbon System Experiments.

24Hr Population Pop Mean Rougness La Lc AS4 515.439471 98.5360433 799.352621 7.0952767 10.85842 P25 641.073381 248.175402 1899.48577 5.939293 23.19322 T650 977.591273 795.329395 1248.26136 5.7509872 201.7046 P120 331.076101 998.32126 3124.55577 32.547383 19.09946

Simulation Means Randomly Generated Values Based on a PERT Distribution, where the Calculated Experimental Value was the Mean Value of the Distribution

MultiNormalValue = (Range of Values, Correlation Matrix, Mean, Standard Deviation) PERTValue = (Minimum, Most Likely, Maximum)

Figure 8.1.4-2: The Monte Carlo Setup for 24Hr Time Period. This form of Monte Carlo Simulation was Developed to Understand what the Mean Cell Population would be if the Values of the Various Material Properties Varied

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All five setups for this first set of Monte Carlo simulations are shown in Figure

8.1.4-3. The four types of carbon are highlighted in yellow, with the material properties of each shown in red. The population is the value for which 500 random variables are generated, and that population’s mean is shown in brown. The mean is the desired parameter because it is the average of the randomly generated values dispensed from via the assigned multi-variable normal distribution. All of the equations utilized during this first set of Monte Carlo Simulations are shown in Appendix D.

8Hr Population Pop Mean Rougness La Lc 72Hr Population Pop Mean Rougness La Lc AS4 399.104235 245.580225 605.66233 6.66899 4.6643 AS4 918.9354135 286.636429 946.2987879 9.073379154 17.12074 P25 526.087176 381.855995 2606.9435 3.456202 12.70088 P25 1869.444093 690.9201474 1574.980052 2.82994516 13.50528 T650 403.88994 293.658439 1245.3465 9.712096 119.5737 T650 2167.503258 1407.867331 1538.985102 16.61885692 90.17457 P120 345.465663 477.040288 1371.1377 33.24773 20.16091 P120 2025.25096 1287.689238 2481.625873 31.706186 2.526842

24Hr Population Pop Mean Rougness La Lc 96Hr Population Pop Mean Rougness La Lc AS4 515.439471 98.5360433 799.35262 7.095277 10.85842 AS4 316.1579086 334.4266448 674.0100302 10.81657716 3.50925 P25 641.073381 248.175402 1899.4858 5.939293 23.19322 P25 50.19509759 462.0992346 2759.809654 10.90204538 2.684369 T650 977.591273 795.329395 1248.2614 5.750987 201.7046 T650 2283.652208 1676.172251 869.9575743 11.93406636 237.4284 P120 331.076101 998.32126 3124.5558 32.54738 19.09946 P120 1538.182087 2172.400147 2593.998537 38.31869902 26.60627

48Hr Population Pop Mean Rougness La Lc AS4 487.423031 242.800175 509.17827 5.775997 21.73402 P25 272.07187 458.488219 2410.1405 8.143145 20.29602 T650 1776.37594 1021.49083 693.29753 4.579035 134.7313 P120 619.205973 1134.69797 2764.0584 37.53073 61.93591

Figure 8.1.4-3: Monte Carlo Simulation Cells and Population Mean Results for Each of the Time Frames

Histograms documenting the frequency of each of the randomly generated numbers within a specific domain are documented fully in

Appendix E, but the histogram for the AS4 Carbon Fiber – 8 hour experiment is shown here in Figure 8.1.4-4.

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Simulation: AS4-8Hr 80

F 70 r 60 e q 50 u 40 e 30 n c 20 y 10

62 22

98 58 18

- - -

102 182 222 302 342 382 422 462 502 582 622 702 142 262 542 662 Randomly Generated Number Values

Figure 8.1.4-4: Histogram for the AS4 8 Hour Sample: The Frequency of Random Numbers Generated During the Monte Carlo Simulation

The regression model developed and based on the material properties of the carbon was used in the second Monte Carlo simulation.

Because each term in the regression model is associated with some error, the error was used as the upper and lower bounds of the randomly generated numbers associated to each material property. Five hundred simulations were performed, where the measured material properties were held constant and the results of the simulation were strictly based on the variance in independent variables associated with the regression model. The setup for this Monte Carlo simulation is shown in Figure 8.1.4-5.

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Measured Data Regression Model Equation Including Experimental Data Error Associated with Each Variable

Pop Time La Rough Lc Regression Model Mean AS4 210 8 6.44 676.44 1.59 -503.1792186 -166.131 AS4 121 24 6.44 676.44 1.59 81.81181728 -45.8388 AS4 193 48 6.44 676.44 1.59 -263.3629318 248.1076 AS4 271 72 6.44 676.44 1.59 531.1168617 464.7932 AS4 414 96 6.44 676.44 1.59 456.7489548 675.6043 P25 348 8 6.5189 2083.381 175.21 -181.339672 -2.42289 P25 257 24 6.5189 2083.381 175.21 -102.7863351 182.4356 P25 364 48 6.5189 2083.381 175.21 216.0964435 381.4873 P25 518 72 6.5189 2083.381 175.21 716.0087509 698.6825 P25 603 96 6.5189 2083.381 175.21 350.6531282 830.6099 T650 258 8 7.755 1418.841 2.046 988.8539524 604.052 T650 825 24 7.755 1418.841 2.046 -33.72603887 769.8844 T650 907 48 7.755 1418.841 2.046 1180.562333 990.89 T650 1242 72 7.755 1418.841 2.046 915.3010549 1227.719 T650 1847 96 7.755 1418.841 2.046 2407.631439 1464.432 P120 478 8 29.974 2658.422 19.199 368.1018897 858.7235 P120 942 24 29.974 2658.422 19.199 1423.730799 1016.888 P120 1202 48 29.974 2658.422 19.199 429.1419118 1200.772 P120 1476 72 29.974 2658.422 19.199 2563.545574 1535.526 P120 2190 96 29.974 2658.422 19.199 1550.416452 1718.868

Simulation Mean

Figure 8.1.4-5: Schematic Illustrating the Second Monte Carlo Setup: All Material Properties were Held Constant and the Results were Based Solely on the Variance Associated with the Regression Model Error

A graph illustrating the consistency between the experimental data and the data from the Monte Carlo simulation of the regression model is shown in Figure 8.1.4-6.

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Monte Carlo Simulation of the Regression Model Compared to the Experimental Data 2500

2000

1500

1000 Experimental RegressionModel

Cell Population Cell 500

0 0 20 40 60 80 100 120 -500 Time (hrs)

Figure 8.1.4-6: Monte Carlo Simulation of the Regression Model Compared to the Actual Experimental Data

As with the previous Monte Carlo simulation, the formulas substantiating this data are shown in their entirety in Appendix F and the histograms documenting the frequency of calculations for this particular model are included in Appendix G. The histogram for the Monte Carlo simulations of the AS4-8hr regression model is shown in Figure 8.1.4-7.

AS4-8Hr 60

Frequency 20

104 192 368 544 632 720 280 456

864 776 688 600 512 424 248 952 336 160

------

1040 - Regression Model Calculation with Variable Variance

Figure 8.1.4-7: Histogram for the MC simulations of the AS4-8Hr Regression Model

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8.2. Experimental Research

Although some of the experimental were already previously mentioned, they are highlighted in their entirety in the succeeding sections.

8.2.1. In Vitro Experimentation

Once the in vitro experiments were complete, characterization was achieved through both optical and confocal microscopy. Since the

Table 8.2.1-1: Data overall objective for these experiments was simply to Measured from the in vitro Experiments. Time (Hrs) Cells observe changes in cell population with respect to 0 10 2 35 time, pictures were only taken to confirm the cell 4 167 8 301 population counts that were measured. The cell 12 489 populations for each time period were measured 24 767 48 966 using confocal microscopy and the results are shown 72 1011 96 1205 in Table 8.2.1-1.

At 8 hours, little if any cellular activity was visible for imaging. This is evident in Figure 8.2.1-1. This, however, changed drastically, as by 96 hours the presence of osseous tissue was definitive, as shown in Figures

8.2.1-2 and 8.2.1-3. These latter two figures are of the same osseous tissue that developed, only the various nuclei are being illustrated in 8.2.1-2 while the corresponding cytoplasms are being highlighted in 8.2.1-3. The

170 two stains were used simply to illustrate that what was being imaged was not staining or microscopy artifact, but in fact actual osseous tissue.

100 μm

Figure 8.2.1-1: Cell Growth at 8 Hrs into the In Vitro Experiment

100 μm

Figure 8.2.1-2: The Nuclei of Various Osteoblasts after 96 Hrs of Growth

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100 μm

Figure 8.2.1-3: The Cytoplasm of the Nuclei

8.2.2. Carbon Material Testing

The first method of carbon material testing involved utilizing the

Atomic Force Microscope (AFM) to measure the surface roughness of each of the carbon fiber bundles. Each material’s calculated surface roughness is shown along with three-dimensional scans of the fiber’s surface. Scans of each fiber bundle were taken at approximately: (i)

40x40μm, (ii) 4x4μm and (iii) 2x2μm to illustrate how the roughness of each material changes on different sized scales. Figures 8.2.2-1 through 8.2.2-4 show the surface roughness results for AS4, P25, T650 and P120, respectively.

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AS4: Average Surface Roughness, Sa = 676 nm

Scan Speed: 0.2 Hz Scan Resolution: 512

Figure 8.2.2-1: AFM Surface Roughness Analysis of AS4

P25: Average Surface Roughness, Sa = 2083 nm

Scan Speed: 0.2 Hz Scan Resolution: 512

Figure 8.2.2-2: AFM Surface Roughness Analysis of P25

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T650: Average Surface Roughness, Sa = 1418.8 nm

Scan Speed: 0.2 Hz Scan Resolution: 512

Figure 8.2.2-3: AFM Surface Roughness Analysis of T650

P120: Average Surface Roughness, Sa = 2658 nm

Scan Speed: 0.2 Hz Scan Resolution: 512

Figure 8.2.2-4: AFM Surface Roughness Analysis of P120

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Upon completion of analysis with the AFM, the next testing method used was RAMAN spectroscopy whose statistical data is shown in Table

8.2.2-1.

Table 8.2.2-1: RAMAN Spectroscopy Measurements of Average Crystal Diameter, La

Average Crystal Diameter

AS4 1.64852 nm P25 3.356818 nm T650 14.74144 nm P120 27.94639 nm

The raw RAMAN data for each of the carbon fiber bundle scans is shown in Appendix H.

RAMAN spectroscopy only yields marginal results with respect to estimating the average crystal diameter of a material. It was therefore deemed necessary to confirm these results with the complimentary method of X-RAY analysis. The X-RAY diffractometer scanned each of the four samples of carbon fibers. This method also permits the calculation of another crystallinity parameter, the average crystallite height, Lc. Utilizing the Scherrer Equation in conjunction with the X-RAY Diffractometer yielded the following results of crystallite size (Lc and La) shown in Table

8.2.2-2.

Table 8.2.2-2: Scherrer Equation Calculations for La and Lc

X-RAY Diffractometer Data La Lc Material (nm) (nm) AS4 6.44 1.59 P25 6.5189 2.046 T650 7.755 175.21 P120 29.974 19.199

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The calculations can be seen in their entirety in Appendix I. The X-

RAY data was used in the development of all theoretical methods.

8.2.3. Cell-Carbon System Experimentation

After studying primary human osteoblasts and carbon fibers separately, the next step was to study how these two substances worked collectively. As previously mentioned, primary human osteoblasts were seeded onto 20 different samples, 5 for each type of carbon. Samples were stained and analyzed at 8hr, 24hrs, 48hrs, 72hrs and 96hrs prospectively. The overall sample size of carbon fiber bundles on which the cells were seeded was approximately 57mm3 in volume. The samples associated with each time stamp were viewed and the cells were counted with the results for each shown in Table 8.2.3-1.

Table 8.2.3-1: Cell Populations as a Function of Time on All 4 Carbon Fiber Variations

Time AS4 P120 T650 P25 (Hrs) Cell Count Cell Count Cell Count Cell Count 8 210 478 258 348 24 121 942 825 257 48 193 1202 907 364 72 271 1476 1242 518 96 414 2190 1847 603

These results are shown graphically in Figure 8.2.3-1.

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Cell Populations on Carbon Fibers Over 96 Hours 2500

2000

1500 AS4 P25 1000

Cell Population Cell T650

500 P120

0 8 24 48 72 96

Time (hrs)

Figure 8.2.3-1: Cell Population as a Functin of Time on Four Types of Carbon Fibers

The AS4 and P25 carbon fibers exhibited the poorest amount of cell growth throughout the duration of the experiment, while the T650 and

P120 carbon fibers demonstrated significantly better growth and proliferation.

Pictures taken with the inverted microscope for AS4 at 8, 24, 48 and

72 hours are shown in Figure 8.2.3-2.

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8 hrs 24 hrs

48 hrs 72 hrs

Figure 8.2.3-2: AS4 Images of Cell Growth for Varying Time Periods

The remaining carbon fibers, P25, T650 and P120 were all characterized in a similar procedure, (Figures 8.2.3-3 through 8.2.3-5 respectively). These micrographs show cell growth at 8, 24, 48 and 72 hours starting with the upper left hand image in the figure and traveling clockwise.

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8 hrs 24 hrs

48 hrs 72 hrs

Figure 8.2.3-3: P25 Images of Cell Growth at Varying Time Periods

The P25 material displays somewhat higher cell growth than the

AS4. This is evident from the data in Table 8.2.3-1 as well as Figure 8.2.3-1.

Additionally, the cells growing on the P25 fibers are more abundant but not as resilient which can be seen by their shrinking appearance and black borders.

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8 hrs 24 hrs

48 hrs 72 hrs

Figure 8.2.3-4: T650 Images of Exceptional Cell Growth at Varying Time Periods

The T650 carbon fibers, as illustrated by these micrographs exhibit excellent cell growth and proliferation compared to AS4 and P25 fibers.

The cells proliferating on the T650 fibers are superior both in size and their resulting ability to secrete and generated extra-cellular matrix. This is evident based on its visibility in the images in Figure 8.2.3-, its non- existence with respect to the cell growth on AS4 and its cloudy uncertain representation on the P25 fiber. The images for P120, shown in Figure

8.2.3-5 were captured for each image. The overall size of the cells growing

180 on the P120 carbon fibers were smaller but the growth was significantly more populous than the other samples.

8 hrs 24 hrs

48 hrs 72 hrs

Figure 8.2.3-5: P120 Images of Exceptional Cell Growth at Varying Time Periods

Despite the higher population of cells on the P120 fiber, the structural integrity of these cells was not as favorable as those proliferating on the T650 fibers. This indicates that an optimum range of material properties exists, that will produce the most superlative environment for bone cells to grow.

181

For comparison purposes, images of cell growth at 96 hours, for all four carbons were placed side by side to visually compare the difference in cell proliferation based on the type of carbon fiber Figure 8.2.3-6. All images within this figure were taken at the same magnification for consistency in assessment.

AS4 P25 T650 P120

Figure 8.2.3-6: Growth on All Four Carbon Variations at 96 Hours

By comparing the cell growth results for all four fibers simultaneously, it is evident that the T650 and P120 carbon fibers contain properties more attractive to osteoblasts and osseous tissue. The key to optimum cell growth would be to utilize these properties in such a way that cell growth will peak, without jeopardizing each cell’s structural stability.

At the beginning stages of growth, the cells seeded on the P25 fibers exhibited interesting behavior, indicating that they preferred to attach to the individual carbon fibers. The beginning stages of this attachment by a small sample of cells is shown in Figure 8.2.3-7.

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Figure 8.2.3-7: Cells Migrating toward a P25 Carbon Fiber, at 24 Hours Culture

The behavior of the cells, as if they have elongated themselves and redirected their movement toward the carbon, indicates their likeness of the carbon environment over the presence of no biomaterials. This phenomenon is also prevalent in Figure 8.2.3-8.

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Figure 8.2.3-8: Additional Image of Cells Migrating toward a P25 Carbon Fiber for Attachment and Growth. This image was taken at 8 Hours of Culture

Despite the small amount of time the cells had been in culture and seeded to their prospective samples, they cells exhibit swift behavior in directing themselves toward a fiber in order to preferentially attach themselves to carbon over attachment to the Petri dish.

8.2.4. Validation of Theoretical Models with Experimental Data

Some comparison of data was illustrated in sections 8.1 while the results for the analysis of the theoretical models were being demonstrated. Additional comparisons of experimental and theoretical

184 data are now shown. Figure 8.2.4-1 compares the accuracy of the

Logistic-Malthusian Model to the experimental data extracted from the cell-carbon system of experiments.

Cell-Carbon System Data Compared to the Logistic- Malthusian Model 2500

2000

AS4_Exp 1500 AS4_LogMal P120_Exp P120_LogMal 1000 Cell Population Cell T650_Exp T650_LogMal

500 P25_Exp P25_LogMal

0 0 20 40 60 80 100 120

Time (hrs)

Figure 8.2.4-1: The Logistic-Malthusian Model being used to Predict the Cell-Carbon System Experimental Data

A generalized comparison of all gathered data, both theoretical and experimental is compiled in the following table and figure. Table

8.2.4-1 consists of all the data accumulated throughout this research initiative. Figure 8.2.4-2 is a graphical representation of how consistent and reliable each of the theoretical methods and data collection methods seems to be with respect to each other.

185

Table 8.2.4-1: Compilation of All Theoretical and Experimental Data Associated with this Research

Time Cell Counts (Hours) Experiment CA AS4 AS4-Math AS4-MC MC-REG P25 P25-Math P25-MC MC-REG T650 T650-Math T650-MC MC-REG P120 P120-Math P120-MC MC-REG 0 10 5 10 10 10 10 8 301 10 210 209 234 2 348 347 368 4 258 257 285 602 478 477 469 894 24 767 115 121 126 256 176 257 266 491 134 825 783 780 758 942 878 1088 1019 48 966 740 193 190 266 215 364 356 280 399 907 887 1207 1009 1202 1131 1478 1256 72 1011 2225 271 266 96 474 518 501 503 646 1242 1153 1308 1210 1476 1381 865 1481 96 1205 3386 414 402 813 704 603 590 877 872 1847 1629 2133 1464 2190 1885 3017 1668

4000 Comparison of All Results

3500

3000

2500

2000

1500

1000

500

0 0 20 40 60 80 100 120 Experiment CA_Scaled AS4_Experiment P25_Experiment T650_Experiment P120_Experiment AS4-Math P25-Math P120-Math AS4_MC P25_MC T650_MC P120_MC T650-Math

Figure 8.2.4-2: Graphical Summary of All Collected or Calculated Data

The visual comparison of all data sets in Figure 8.2.4-2 truly epitomizes the closeness and dependability that the theoretical models have with the collection of experimental data. Despite the drastic differences in the methods of collection, their resemblance in structure indicates the versatility of each of the methods. These results show excellent promise in providing versatility for future research; efforts in which data collection opportunities might be limited. These now

186 established theoretical means provide future studies with a dependable and accurate means of data generation especially in situations where experimental data development is not possible due to lack of resources.

The closeness of these results along with complete analysis on theoretical model development is discussed thoroughly in Chapter 9.

187

Chapter 9. Discussions and Interpretations of Results

Because the theoretical models developed for this research initiative are so highly dependent upon the collected experimental data, discussions pertaining to the experimental results will occur first, followed by the use of the experimental data to further develop the theoretical models. Finally, comparisons will be made to discuss both forms of research and the collaboration of the various theoretical models with the experimental results achieved.

9.1. Experimental Results

The in vitro experiments provided an excellent foundation for understanding standard osseous tissue formation without the presence of a biomaterial. One of the more interesting observations involved the lack of consistent cell size and shape. Some cells appeared simple in shape, either elongated or more spherical such as those in Figure 9.1-1. In these instances, not much can be seem about their internal structure.

188

100 μm 100 μm

Figure 9.1-1: Examples of Cell Size and Shape

The image on the left of Figure 9.1-1 indicates more spherical cells, indicative of cells with poor structural integrity or that have not yet attached to a medium. The image on the right illustrates cells more advanced in their proliferating process. The complexities related to this research involve understanding what differing activities occurred on the molecular level that caused such a noticeable difference in cellular structures. It would be most advantageous to promote cell growth that is consistent and dependable. One of the main focuses of this initiative is to determine if certain molecular properties of biomaterials possess qualities that promote a more consistent cell population and structure. This would provide a means for controlling or at least minimalizing the unpredictability of cell growth during the most vital stages in regeneration.

In addition to cell size and shape, it was difficult to identify any particular patterns of cell growth. It is evident in Figure 9.1-2 that after a period of 24 hours, some osseous tissue formation has occurred. The

189 integrity of that tissue, however, along with the predictability of its structure is completely unknown.

Figure 9.1-2: Osseous Tissue Formation after 24 Hours of Culture: Extracellular Matrix and Tissue (Left) along with the Nuclei of the Cells that Developed the Tissue (Right)

Inconsistency of cell size and shape is also seen in Figure 9.1-2, indicating that not only are different cells at different points in the mitotic cycle, but also that they each have released differing amounts of matrix, proteins and growth factors. The concept of controlling cell morphology is once again presented. If cell size and shape can potentially be controlled by the presence of a biomaterial, perhaps tissue composition and integrity could be more predictable and reliable.

The basis of cell growth and osseous tissue generation exemplifies the need for structure within the process. It is believed that the material properties of carbon could potentially provide that structure since it has already been established that cells prefer to attach to carbon, and do so axially (Czarnecki, 2008). This supplies some structure to where cells will attach and grow. Because carbon has the ability to offer that stability to

190 osseous tissue formation, perhaps the appropriate form of carbon, with optimum material properties could also provide consistency in cell morphology. To better understand this, specific material properties of four different carbons were studied extensively. Surface roughness has already proven to have an effect on cell growth (Bacakova, et al., 1996).

Because the crystallinity and orientation of carbon fibers play such an integral role in determining their mechanical properties, it was decided to focus on these molecular characteristics to determine how they could influence not only the carbon fibers themselves, but also the growth of osteoblasts.

The surface roughness of each carbon fiber was analyzed on three different degrees of magnification. Table 9.1.1 summarizes the surface roughness of each material.

Table 9.1-1: Summary of Surface Roughnesses of Carbon Fibers

Surface Roughness (nm) AS4 676.44 P25 2083.3806 T650 1418.8407 P120 2658.4224

From the images of surface roughness (Figures 8.2.2-1 through 8.2.2-4) it can be seen that the AS4 carbon fibers have the minimum amount of surface roughness. From Figure 8.2.2-1, the 2μm x 2μm cross section appears to be extremely rough with such an abrasive contour, but its overall roughness, measured via average surface area, is much lower

191 than the other specimens. The remaining three specimens all have similar profiles in surface roughness, with much higher average surface areas than the AS4 fiber. The roughnesses on a 2μm x 2μm cross-sectional scale are shown again in Figure 9.1-3. The P25 fiber has a much more grooved, mountainous surface. The T650 fiber incorporates a similar surface appearance, but also maintains a striated pattern of grooves. This actually decreases the fiber’s overall surface area but does provide some organization to the structure. The P120 fiber is unique in that it combines the striatic patterns of the T650 fiber with the mountainous and rough appearance of the P25 fiber. The characteristics of each fiber’s surface roughness appear different on different visual scales. When 40μm x 40μm scans of each fiber were performed, corresponding roughnesses appeared vastly different (Figure 9.1-4.)

192

AS4 P25

T650 P120

Figure 9.1-3: Comparison of Carbon Fiber Roughnesses and Appearance

P25 AS4

T650 P120

Figure 9.1-4: A 20μm by 20μm of the AS4 Fiber and 40μm by 40μm Scans of the Other Carbon Fibers

None of the roughness features associated with any of the fibers is evident on the larger cross sectional scans (40μm2). The smaller cross sectional

193 scans therefore provide much greater insight as to what was actually occurring on each fiber’s surface, and how cellular activity might be affected by this roughness.

In addition to surface roughness, crystallinity also has a significant effect on a material’s strength and performance, meaning it could also impact cellular attachment and growth. Crystallinity has two important parameters: Average crystal height, Lc, and average crystal diameter,

La. RAMAN spectroscopy provides adequate means for measuring a material’s average crystal diameter. Just as with surface roughness, the average crystal diameter was highest in the P120 fiber (Table 9.1-2) and lowest in the AS4 fiber. The substantial difference in crystal diameters is evident in the fiber’s overall structure and appearance. The AS4 fiber bundle is lustery and smooth in appearance while the T650 and P120 fibers appear much more brittle and dull.

Table 9.1-2: RAMAN Spectrometer Measurements of Average Crystal Diameter

Average Crystal Diameter, La

AS4 1.64852 nm P25 3.356818 nm T650 14.74144 nm P120 27.94639 nm

The X-RAY diffractometer was the other more accurate method for obtaining characteristics about each fiber’s crystallinity. In addition to measuring crystal diameter, it also provided enough measured

194 parameters to calculate each material’s average crystal height, Lc.

These values are shown in Table 9.1-3.

Table 9.1-3: Average Crystal Diameter and Height as Measured Using X-RAY Diffraction

X-RAY Diffractometer Data La Lc Material (nm) (nm) AS4 6.44 1.59 P25 6.5189 2.046 T650 7.755 175.21 P120 29.974 19.199

The T650 fiber was actually found to have a significantly higher average crystal height than any of the other carbon fibers, but its diameter was comparable to both the AS4 and P25 fibers. The AS4 and P25 fibers both have short, but fat crystals. The T650 fibers have crystals that are narrow in diameter but extremely tall. This contributes to its overall brittleness.

Finally, the P120 fiber is the only form of carbon that has a good ratio of crystal diameter to crystal height; the crystals are extremely wide in comparison to the others as well as significantly tall.

Because the four types of carbon fibers have vastly different material properties, some interesting comparisons can be made. When comparing the P120 fiber to the P25 fiber, the fiber orientation within the bundles is the same (continuous) but they are vastly different in crystallinity. The P120 fiber is much more crystalline in both crystal height and diameter (Lc = 19.199nm and La = 29.974nm) which is evident in how its behavior differs so greatly from the P25 fiber (Lc = 2.046nm and La =

195

6.52nm). The P25 fiber performs similar to human hair with very low brittleness but a dull appearance. Although the P25 and P120 fiber are exceptionally similar in surface roughness (2083.4nm and 2658.4nm respectively), the drastic difference in crystallinity adds to P120’s shiny appearance and more brittle behavior.

The T650 and AS4 fibers are similar in the parameter of crystallinity, but differ greatly in fiber orientation. The AS4’s continuous fiber orientation and very low crystallinity attributes to its shininess and very high ductility.

This fiber bundle behaves similar to a durable, woven polymer based fiber but with no randomness associated to its orientation. The T650 fiber is coupled with a very random fiber orientation. This is evident in how easy it is to break apart the T650 fiber bundle. Although the AS4 and T650 fibers share very similar crystal heights(LaAS4 = 6.44nm and LaT650 = 7.755nm), they differ greatly in crystal diameter (LcAS4 = 1.59nm and LcT650 = 175.21nm).

They also differ greatly in surface roughness (SaAS4 = 676.44nm and SaT650 =

1418.84nm). This comparison illustrates the overall importance of the average crystal height within a material. This becomes even more evident in how the crystallinity affects the cell growth a proliferation on a biomaterial (which will be discussed in succeeding paragraphs).

The P120 and T650 carbon fibers showed similarities in most crystallinity properties but differed in fiber orientation and surface roughness. The P120 is an all around significantly rough and crystalline

196 material. Its highly continuous fiber orientation, which is much more oriented than the other types of fibers, contributes to its more ductile behavior. The T650 fiber has fairly significant roughness and maintains the extremely high average crystal height, Lc, but with its more random fiber orientation, behaves in a much more brittle manner; its fiber bundles can be disturbed and thus broken much more easily.

The P25 and AS4 fibers, which performed most poorly during the cell-carbon system experiments, share similar fiber orientations and crystallinities (LaP25 = 6.52nm and LaAS4 = 6.44nm, LcP25 = 2.046nm and LcAS4

= 1.59nm), but are immensely different in surface roughness (SaP25 =

2083.4nm versus SaAS4 = 676.44nm). This explains their differing behaviors; both bundles are very ductile in nature, but because the P25 is much more rough than the AS4, its bundles separate very easily, allowing individual fibers to tangle. The AS4 fibers are so smooth and lack that roughness, their bundles are very difficult to spread and break apart.

Perhaps the two most conflicting types of fibers are the P120 and the AS4. The P120, again, has extremely high crystallinity (diameter and height), very high roughness and its fiber orientation is much more organized. Although the AS4 has a continuous fiber orientation, its organization is not nearly as high-quality as the P120. The AS4 is also very smooth, having the lowest surface roughness of all the fibers and very elastic with the lowest crystallinity values of all the fibers. The variation in

197 properties of these materials illustrates the concept that carbon is a very versatile material, whose properties can be tailored to meet the particular application.

Now that the material properties have been discussed, it is important to illustrate how human osteoblasts appeared to behave in the presence of these materials. Because of the vast range of material properties, it is crucial to highlight the differences in cellular behavior, as these differences could correlate to the changes in material roughness, crystallinity and fiber orientation. The results of cell growth on each type of fiber will now be discussed.

The cell-carbon system experiments involved a 96 hour in vitro study in which primary human osteoblasts were seeded onto each type of carbon. With the significant differences in material properties between the carbon fibers, one of the main focuses of this group of experiments was to determine which material, and essentially which “set” of material properties seemed to promote the best cellular response.

The cell growth on each carbon material was monitored at five different time periods: 8 hrs, 24 hrs, 48 hrs, 72 hrs and 96 hrs. The results of these experiments are shown again in Table 9.1-4.

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Table 9.1-4: Results of Cell-Carbon System In Vitro Experimentation

Time AS4 P120 T650 P25 (Hrs) Cell Count Cell Count Cell Count Cell Count 8 210 478 258 348 24 121 942 825 257 48 193 1202 907 364 72 271 1476 1242 518 96 414 2190 1847 603

To be consistent, approximately 10ml of cell-medium mixture was equally divided (0.5ml each) among the twenty total samples (fives samples for each material, one for each time period). Each sample had an initial cell count of approximately 10 cells. After a period of 8 hours, the P120 saw the greatest increase in cell population with 478 cells, down to the AS4 fiber which only had 210 cells. Because those two materials vary greatly in every measured material property, it became evident that cell growth was definitely dependent upon the crystallinity, roughness and fiber orientation of the carbon fibers.

After a period of 24 hours, the AS4 and P25 fibers behaved similarly, in that their cell populations both decreased in size (Table 9.1-4). This decrease could potentially be caused by some degree of toxicity on the surface of the biomaterials. It could also be dependent upon the phenomenon that an abnormal amount of cells, when exposed to these forms of carbon, actually experienced apoptosis. This would minimize the amount of cells still enduring their respective mitotic cycles, and indicate that many fewer cells were indeed going through mitosis. This would result

199 in the cell populations on each of these materials taking a much longer amount of time to actually increase and the resulting tissue would develop at a much slower rate. The T650 and P120 fibers, however, did the opposite with each of their cell populations drastically increasing.

Given that these fibers both have significantly higher crystallinities than the others, it was logical to conclude at this point that crystallinity does indeed play a crucial role in cell growth and proliferation on carbon based materials.

At the 48 hour time frame, all four carbon samples showed an increase in cell population. The AS4 and P25 fibers still seemed to exhibit poor cell growth with increases in cell population of only 72 and 107 cells, respectively. The T650 fiber also exhibited a smaller increase than expected with its population only changing by 82 cells. This small increase could be caused by a similar phenomenon as seen on the AS4 and P25 samples; an unusually large amount of cells went through apoptosis, diminishing the amount that actually experience mitosis and taking away from the tissue’s overall growth rate. The P120 fiber behaved as it did previously, with its cell population increasing in size by approximately 33%.

After 72 hours in culture all four samples exhibited a significant increase in their prospective cell populations with the P120 and T650 fibers showing the most amount of growth (overall populations of 1476 and

1242, respectively). Finally, after 96 hours in culture, there was an obvious

200 difference in the ability of cells to proliferate on the various materials. The

AS4 and P25 fibers had the poorest amount of growth with only 414 and

603 cells, respectively. These two materials are extremely similar in that both have relatively low crystallinity (height and diameter) and their fiber orientations are similar. They differ mainly in their degrees of surface roughness. It seems that since the P25 fiber exhibited more growth, and the two materials are similar in crystallinity, it is important to recognize the role that the surface roughness of the material plays. The only basic difference between P25 and AS4 is that the surface roughness of P25 is exceptionally more. It can therefore be concluded that surface roughness, along with crystallinity play a very important role in cell proliferation and growth.

Both crystallinity and surface roughness contribute to cell growth. If a material has high surface roughness, but low crystallinity, the potential for cell growth is not nearly as promising as that with perhaps moderate values for both properties. A good example of this is the comparison between the P25 and T650 fibers. P25’s surface roughness is extremely high (2083.4nm) but its crystallinity values are low (Lc = 2.046nm and La =

6.52nm). The T650 fiber has a lower surface roughness, 1418.8nm but its crystallinity values are different with Lc = 175.21nm and La = 7.755nm. This example illustrates the importance of a good combination of these two properties to produce the optimum amount of cell growth.

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Fiber orientation, although important doesn’t seem to be as important as crystallinity and surface roughness. A good example of this is the comparison between T650 and P120. The T650 carbon fibers have a more randomized orientation of fibers, yet cell growth on these fibers was extremely good. The P120 fiber bundle exhibited the best cell growth, but had the most organized continuous fiber orientation of all the fibers. Both materials exhibited higher degrees of crystallinity than the other two, with the T650 having extremely large crystal heights but average crystal diameters and the P120 had a relatively large crystal height (not as large as the T650) as well as crystal diameter. The only other difference was their prospective surface roughnesses. The T650 fiber bundle was not as rough. This seems to be a direct cause of the crystallinity. With such tall and narrow crystals making up the material, their surface should expect to be smoother. With crystal heights more comparable to their diameters, as in the P120 fibers, this produces a rougher, bumpier surface, but with more actual “surface” in which cells can sit and attach. Tall, narrow fibers do not provide as much surface area for attachment, which is why the T650’s surface roughness is somewhat less than the P120 and P25 fibers.

From these observations and experiments, it seems evident that a material’s crystallinity parameters affect its surface roughness. Without some degree of both significant crystal height and diameter, the materials surface will not contain enough roughness on the microscopic scale for

202 which cells can strongly attach and begin to form osseous tissue. It also seems evident that the lower the crystallinity, and hence the lower the surface roughness, the less attractive a fiber becomes to human osteoblasts. The T650 fiber seems to be a rare case in that it has the second lowest surface roughness, but second highest amount of cell growth. This can be explained by the unbalanced crystal height to diameter ratio. The crystals are extremely tall, but very narrow. This results in striations of crystals which develops more of a linear pattern on the fiber’s surface. This is evident in Figure 9.1-3. It is also apparent that the

P12 fiber combines some degree of that striatic pattern with a more balanced crystal height-to-diameter ratio. This results in a striatic but coarse based material surface for cells to attach. The P25 has similar surface roughness to that of the P120 fiber, but lacks the crystallinity necessary to provide some organization to the material’s surface. This seems to be the primary reason for decreased cellular attachment, as it is the only difference between the P25 and P120 fibers.

It can be concluded that the surface roughness and varying parameters of crystallinity within a material play the key roles in determining how well cells seem to attach to that particular material.

These properties however, seem interdependent in that the prospective material needs the proper combination of both for cells to attach and tissue development to occur. This interdependency defines a material’s

203 ability to properly integrate with the developed tissue. Understanding the relationship between crystallinity and surface roughness, and using it in a quantitative sense can help unearth the optimum combination of the two properties that will promote the most successful osseous tissue regeneration.

9.2. Theoretical Results

Several theoretical methods were employed in this research initiative. Of those methods, cellular automation provided a very robust system in which cell growth and proliferation behavior could be forecasted. The key in utilizing this method was to establish accurate and dependable rules from which the program would be based (Table 7.1.1-

2). With these rules being validated by several dependable sources of cellular biology and tissue regeneration, this program offers a very reliable means for assessing not only the quantitative characteristics of osseous tissue formation, but also the visual qualities and patterns that illustrate the means by which tissue is actually formed. The CA program developed, demonstrated a somewhat striatic, diagonal pattern of tissue formation

(Figure 8.1.1-3). The time space grid always seemed to populate itself starting with the upper left corner and finishing with the lower right hand corner. The matrix formation and mineralization occurred horizontally

(Figure 8.1.1-5). This seems realistic being that extra-cellular matrix

204 components are deposited during the process of mitosis and must be deposited in a cell that is not occupied by anything else (blood vessel, osteoblast, osteocyte, protein, etc.) Since the growth patterns of the cells appears diagonal, room lies to the left and right of those cells for matrix deposition. This confirms how dependent matrix formation is on the presence of osteoblasts as well as the fact that the varying types of mature bone cells provide the essential backbone to osseous tissue growth and formation.

The CA program was also devised to allow its generated data to be directly exported to Microsoft Excel for further analysis. One of the key components of this program was to determine its dependability in identifying the amounts of matrix and cells present during and upon the completion of regeneration. Since cells make up approximately 15% of bone tissue’s overall composition (Langer, 1993), it was important to determine whether this program’s rules and execution were accurate.

After a period of 96 hours the experiments conducted without the presence of carbon materials demonstrated a cell population of 1205

(Table 8.2.1-1). To compare the CA program to that data is somewhat unrealistic. Osteoblastic cells normally comprise approximately 15% of bone tissue. This experiment was established with the goal to better understand how and where cells grow; No absolute volume was defined.

This makes it nearly impossible to compare its results to the results from the

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CA program. The actual quantity of cells produced at this time, however, provides a good foundation of what types of cell population numbers to expect. It also allows for an understanding of how fast cells will grow and proliferate without the presence of a biomaterial. The entire purpose of a biomaterial is to assist in the regeneration process. If it can provide structural stability while at the same time increase that growth rate, cells would produce faster, and regenerated would be completed earlier. This in vitro cell population number will be used to compare how much the presence of a biomaterial affects the overall tissue growth rate.

The cell-carbon system experiments, however, had a very well defined volume. Their goal was to not only observe cellular behavior in the presence of these materials, but to analyze how this behavior and the overall tissue growth appears to change when faced with differing material properties. The fixed volume, or maximum carrying capacity for these experiments was 57mm3. Each fiber bundle was affixed to a sterile

Petri Dish and the cells were seeded directly onto the samples. The only cells counted during analysis were those that had attached and grown on the actual carbon fibers.

To be able to compare the defined volume in the experiments to the developed theoretical models accurately, the CA program and other models had to be based on that same volume. Knowing that cancellous cells are approximately 0.0167 mm3, cortical cells are approximately 0.002

206 mm3 and that cells make up approximately 15% of tissue volume, it was calculated that a 57mm3 space, or grid, could be broken up into 23,409 individuals spaces. Any one of these spaces could be occupied by a cell, blood vessel, protein, etc. A square grid of this area results in dimensions of 153 mm x 153 mm. Because of the complexity of the CA program and its resulting execution time, the grid itself was broken up into nine 51mm x

51mm grids. The program was executed using only one of these volumes and its results were multiplied by nine to scale them properly to the appropriate quantities. It takes approximately 8 weeks for the entire foundation of new bone to be formed (Frank-Odendaal, 2006). This indicates that after simply 4 days (96 hours), the regenerated tissue should be approximately 7.1% along in the healing process. Additionally, because cells make up 15% of the tissue composition, there should also be approximately 1672 cells present. The results of the cell-carbon experiments after 96 hours are once again highlighted here in Table 9.2-1 along with the average of the results of the CA program.

Table 9.2-1: Cell Counts after 96 Hours

Time CA Population AS4 P120 T650 P25 (hrs) (Cells) (Cells) (Cells) (Cells) (Cells) 96 1733 414 2190 1847 603

The scaled results of the CA program proved fairly accurate with a cell count average of 1733. Several runs were performed to ensure this accuracy. These results are shown in Table 9.2-2. There should be an

207 estimated 1672 cells present in the grid at 96 hours. It is also important to keep in mind that it has already been proven that the AS4 and P25 fibers hinder cell growth while the T650 and P120 fibers promote it. The standard rule of thumb of tissue taking approximately 8 weeks to heal, and 7.1% of that tissue should be fully developed after 96 hours, is complemented by the results of the CA program. It is also noteworthy to mention that the cell populations the T650 and P120 are more than the estimated 1672 cells, confirming they have indeed promoted cell growth to occur more efficiently. Additionally, the AS4 and P25 cell populations are well below that total, validating that their material properties do indeed hinder cell growth.

Table 9.2-2: Results of CA Program after 96 Hours

Run Time CA Population (#) (hrs) (Cells) 1 96 1539 2 96 2025 3 96 1701 4 96 2673 5 96 729

Despite some variance, the program seems relatively efficient in estimating the number of cells present at any given time. Additional evidence of this is illustrated in Appendix A, where the program was executed for 8 weeks, or approximately 1344 hours to demonstrate the tissue is completely developed. Conclusively, the program takes the collection of highly dependent cellular events that work together to

208 generate bone tissue, and transforms their efforts into understandable and visually explicable data.

The results of the Logistic-Malthusian model were extremely precise.

No matter what the comparison, they proved highly accurate. This model was initially compared to the in vitro experimental data (Figure 8.1.2-1).

The Logistic-Malthusian model was just as accurate when compared to the CA program data and the other experimental data (Figures 8.1.2-2 and 8.1.2-3 respectively). The results for this model are so extraordinarily accurate because of the approach taken with calculating the growth rate. The Malthusian model takes a finite difference approach to calculating growth rate. The only drawback associated with this model is that some data must be present in order to use it. If data were not available, this model could be used in a similar fashion with variables highlighting all of the unknowns in the system. These variables could be given a range of data, and results could be generated based on these ranges.

It has already been established that a relationship does exist between the material properties of a biomaterial and the resulting osteoblast growth on that material. If this relationship can be utilized to determine average cell populations with respect to time based on given material properties, then the Logistic-Malthusian model could be implemented knowing only the surface roughness and crystallinity of a

209 biomaterial. Experimental data would not be necessary to produce accurate approximations of cell population.

It was decided that the GROWTH function be used as an additional tool in order to predict cell growth from existing data. This method is based solely on the logistic law, meaning that very little is understood about the growth rate of the tissue when this method is implemented. The results of this method are shown again in Table 9.2-3.

Table 9.2-3: Cell Population Predictions based on the GROWTH Function

The areas highlighted in yellow help indicate exactly how long it would take the proper amount of cells to form on each particular type of carbon. The target amount of cells is 3450 for a 57mm3 cross sectional area. This is further explained by the following: In the 57mm3 of tissue, 80% of that tissue is cortical bone and the remaining 20% is cancellous. This

210 indicates that there will be 45.6mm3 of cortical bone and 11.4mm3 of cancellous bone at the time of full regeneration. Of the cortical bone volume, 12.5% is occupied by cells while the remaining 87.5% is occupied by matrix. This indicates that cortical cells will eventually occupy 5.7mm3 of space within the tissue. In a similar fashion, with respect to the cancellous bone, approximately 87.5% of its volume is occupied by cancellous cells. This shows that the cancellous cells, upon completion of regeneration will occupy 9.975mm3 of space. Knowing that there are approximately 500 cortical cells/mm3 and 60 cancellous cells/mm3 within the volume of bone, at the end of regeneration, there should be approximately 2850 cortical cells and 600 cancellous cells; 3450 cells total.

By referring to Table 9.2-3, it will take approximately the P120, T650, P25 and AS4 fibers 125 hours, 129 hours, 276 hours and 265 hours to develop that many cells, respectively. This is a range of 5 to 11 days, depending on the material. This time frame is well before the 8 week time period estimated for full regeneration, but this time also does not take into consideration the maturity of these cells, mineralization of the surrounding matrix, and the osteoblast and matrix turnover rate, it which cells wills randomly die and be eventually replaced by healthy cells. The 8 week time period is designated as an average equilibrium period in which the entire system of cells and tissue is, for the most part finished with the regeneration process.

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This tool is simple, but provides a good estimation of predicted cell populations if dependable data is present.

In the preceding paragraphs some of the mathematical biology techniques were implemented to make comparisons between theoretical and actual amounts of tissue present. Because these calculations, and those shown in Section 7.1.2, are based strongly on well known, published data, their utilization in other theoretical techniques such as cellular automata, the logistic law, and the newly developed Logistic-Malthusian model prove very advantageous. These calculations provide an excellent descriptional overview of tissue composition and they can be used to predict several variables as long as the predicted cell population is known. This bridges the gap between understanding just cell population and providing an educated approximation of a tissue’s composition and overall density. The values computed for various tissue components

(shown again here in Figure 9.2-1) are not only accurate according to published standards, but also coincide with proper tissue composition standards (Frank-Odendaal, 2006).

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Tissue Component Characteristics Individual Cell Volume (mm3) Cortical Cell 0.002 Cancellous Cell 0.0167

Individual Cell Mass (mg) Cortical Cell 0.0038 Cancellous Cell 0.03173

Individual Lacunae Volume (mm3) Cortical Lacunae 0.006 Cancellous Lacunae 0.0501

Figure 9.2-1: Various Computed Characteristics of Bone Tissue

These values are very important because they provide a means for understanding how advanced in the regeneration process a tissue might be. Even if experimental cell count data is not available, these computed values can be used in conjunction with one of the other theoretical methods to calculate the approximate amount of both cortical and cancellous cells, the volume of generated matrix, the overall tissue volume and the overall tissue density.

One of the other very dependable theoretical methods instituted in this research was that of regression analysis. This method was used not only on the experimental data (Figure 8.1.3-1), but also on the data generated by the CA program for additional analysis and comparison purposes. The regression model for the in vitro data produced an r- squared value of 0.84 and an overall F-statistic of 0.01. This proves a definite dependency of cell growth on time; this is obviously reasonable and realistic.

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The second regression model (Figure 8.1.3-2) studied the relationship between the surface roughness and crystallinity characteristics of a material and the resulting cell populations as a function of time. Since cell population has already been proved highly dependent on the variable time, it was included in the model to better understand how the growth rate of human osteoblasts tends to change when various types of carbon fibers are introduced to the system. This time-material model produced an r-squared value of 0.828. This is slightly less than the time-only dependent model, but the overall significance is much greater with an F- statistic of 0.0000133. Because the overall significance of the model is much greater and the r-squared value was scarcely affected, it seems advantageous to utilize this model over the time-only dependent model since it takes into account the material properties of the carbon fiber and how those properties affect the overall cell population. An additional benefit of this time-material model is that any material can be tested to determine how its properties would affect potential cell growth if it were implemented as a biomaterial.

The final regression model developed, focused on the data developed by the cellular automation program. As this data proved fairly accurate with respect to experimental data, it was analyzed to establish whether a dependable regression model could be developed and used in the absence of experimental data. This model’s results (Figure 8.1.3-3)

214 produced the highest r-squared value of 0.936 and the overall significance also proved the highest with an F-statistic of 3.3x10-15. With such a reliable relationship between time and resulting cell population, this regression model proves very useful. It can be used to come up with cell populations for situations in which a biomaterial is not present and experimental data cannot be produced. Once the model develops a range of cell populations, the Tissue Component Characteristics (Figure

9.2-1) could then be used to calculate the projected tissue components, all based on a cell population. Utilizing these characteristics on the second regression model would also prove advantageous, as the overall tissue density and breakdown of tissue composition could be predicted simply from knowing what type of biomaterial is being implemented.

The final theoretical method implored in this research was a concept call the Monte Carlo method. As previously mentioned, this method involves a class of computational algorithms that rely on repeated random sampling to compute their results (Metropolis, 1987).

Despite their randomness, these methods have been used with superior confidence to analyze the behavior of many physical and mathematical systems. The correlation matrices employed in this method proved extremely significant as they identified the strength of relation between all of the components of the system in question (time, cell population, surface roughness, La and Lc). Despite the randomness from which the

215 mean cell population values are derived, the results of this method proved highly accurate based on their comparison to the experimental data. Figure 9.2-2 yields a visual representation of the accuracy of this method.

Comparison of Monte Carlo Data to Cell-Carbon Experimental Data 2500

2000

1500

Monte Carlo Data 1000 Cell Population Cell Experimental Data

500

0 0 20 40 60 80 100 120

Time (hrs)

Figure 9.2-2: Monte Carlo and Experimental Data Comparison

A depiction of the data in Figure 9.2-2 is shown in Table 9.2-4. It is evident that the more simulations executed, the higher the accuracy of the method.

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Table 9.2-4: Comparison of MC Simulation Results to Experimental Data

Cell Population 8Hr MC - Mean Experimental AS4 246 210 P25 382 348 T650 294 258 P120 477 478 24Hr AS4 99 121 P25 248 257 T650 795 825 P120 998 942 48Hr AS4 243 193 P25 458 364 T650 1021 907 P120 1135 1202 72Hr AS4 287 271 P25 691 518 T650 1408 1242 P120 1288 1476 96Hr AS4 334 414 P25 462 603 T650 1676 1847 P120 2172 2190

As previously mentioned, these values were derived from the implementation of 500 Monte Carlo simulations. This method proves highly accurate and vastly efficient, with respect to time and compared to some of the supplementary methods employed in this research.

The Monte Carlo methods were also used in conjunction with the time-material based regression model. Because the terms developed from the regression model are associated with some degree of error, it is important to take that deviation of the variables into account when assessing the accuracy of the model. By treating each of the variables as the mean value of a normal distribution of numbers, with a deviation equal to that of the error to which they are associated in the regression

217 model, an accurate Monte Carlo model incorporating the regression model was created. Its overall accuracy can be seen by the comparison of its data to the time-material experimental data in Figure 9.2-3.

Monte Carlo Simulation of the Regression Model Compared to the Experimental Data 3000

2500

2000

1500

1000

500 Experimental

0 MC-Regression Cell Population Cell 0 20 40 60 80 100 120 -500

-1000

-1500

-2000 Time (hrs)

Figure 9.2-3: Incorporating the MC Method on the Time-Material Regression Model

When used unaccompanied, both the Multivariate Regression and the Monte Carlo methods create very accurate depictions of what type of tissue growth to expect when the bone cells are exposed to various materials. When used collectively they provided a range of values that take into account the standard deviation associated with regression.

Monte Carlo simulations, although very probabilistic, provide a very accurate and efficient means for modeling tissue regeneration and growth. One of the superlative qualities associated with this method is experimental data is not needed. As long as a generalized idea exists

218 pertaining to the average values associated with a variable, this method is astonishingly robust. Provided the model’s boundary conditions and numerical distributions are precise and relevant to the complexity of the problem, this method is perhaps the most time efficient means of predicting the behavior of a wound healing process.

9.3. Comparisons and Validation

Several theoretical methods were utilized in this initiative to determine which, if any, would provide accurate means of predicting the regenerative behavior of bone tissue. Figure 8.2.4-2 illustrates just how accurate all of these methods can be. With respect to each individual carbon fiber the accuracy of each theoretical method is illustrated in

Figures 9.3-1 through 9.3-4.

Comparison of Theoretical Methods - AS4 1000

800

600

Experimental Data 400 Logistic-Malthusian MonteCarlo 200

Cell Population Cell Monte Carlo-Regression Regression 0 0 20 40 60 80 100 120

-200

-400 Time (hrs)

Figure 9.3-1: Comparison of Theoretical Methods for the AS4 Fiber

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For the AS4 fiber, the overall cell growth was the worst of the four types of fibers. The experimental growth behavior actually showed a decrease in cell population after 24 hours, just prior to exhibiting a gradual increase throughout the remainder of the experiment. The most accurate theoretical method for this material was the Logistic-Malthusian model because it uses a finite difference approach to determining the cellular growth rate, and directly incorporates the experimental data. The drawback to this method is that experimental data must be available. If not, ranges of data pertaining to each variable in the model could be developed. The regression model, alone yields a very linear approximation of the cell population, which does not seem as accurate considering cell growth is never linear. This is probably caused by the significant error associated with each parameter in the regression model.

The Monte Carlo method, when used independently, does not follow the general flow of the rest of the data. Its predictions are comparable, but the method’s flow needs to replicate the experimental data for situations when no data is known; at least the general behavior of cell growth will be available through the flow of the model, and ranges of data can be used to produce probabilistic data. When it is used in conjunction with the developed regression model, its flow correlates very well with the other data types and presents a realistic representation of cell growth.

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Comparison of Theoretical Methods - P25 1000

900

800

700

600 Experimental 500 Logistic-Malthusian

400 Monte Carlo Cell Population Cell Monte Carlo - Regression 300 Regression 200

100

0 0 20 40 60 80 100 120

Time (hrs)

Figure 9.3-2: Comparison of the Theoretical Models for the P25 Fiber

The P25 fiber demonstrated similar cell growth characteristics as the

AS4, with a decrease in cell population after 24 hours, leading into a gradual population increase throughout the experiment’s duration. The

Logistic-Malthusian model was maintained as the most accurate, but again, experimental data is required to achieve these results. Other than the data at the 8 hour point, the regression model was fairly accurate in depicting the cellular behavior, but still followed a linear pattern, which is not an adequate representation of cell growth. The Monte Carlo method was also somewhat sporadic in nature. For this particular material, the combined Monte Carlo-Regression model proved to linear to accurately represent how well cells tend to proliferate on this material. Other than

221 the Logistic-Malthusian model, the Monte Carlo method provides the most accurate depiction of cell growth. The only conditions of this model that would need to be adjusted are the ranges (means and standard deviations) of the variables (surface roughness and crystallinity) used within the model. Appropriate adjustments to these parameters that reflect the behavior of the experimental data would create a more accurate MC model that could be used to predict P25 fiber cell growth.

Comparison of Theoretical Methods - T650 2500

2000

1500 Experimental Logistic-Malthusian

1000 Monte Carlo Cell Population Cell Monte Carlo - Regression Regression 500

0 0 20 40 60 80 100 120

Time (hrs)

Figure 9.3-3: Comparison of Theoretical Models for the T650 Fiber

The cell proliferation on the T650 fiber was very promising. It was above the expected value of natural cell proliferation (growth without the assistance of a biomaterial) and the cells appeared the healthiest out of all cells imaged in this initiative. The cell bodies were robust and illustrated

222 structural integrity in addition to the amount of cells that grew on this material. The growth on the material did not produce the highest cell population, but the overall integrity of the cells illustrates the strength of the tissue being formed. The Logistic-Malthusian model once again maintained its precision in depicting cell growth, but the other theoretical models showed similar accuracies. The regression model showed minimal error, following the behavior of the cell growth seen by the experimental values. Unlike the AS4 and P25 fibers, utilizing the Monte Carlo method with the regression model did not achieve any better results. This demonstrates the exactness of the developed regression model, without the assist of another method. The Monte Carlo simulation seemed the most radical of the methods on this data in that it followed a cubic polynomial type pattern. Despite this behavior, it seems the most accurate method of prediction other than the Logistic-Malthusian model.

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Comparison of Theoretical Methods - P120 3500

3000

2500

2000 Experimental Logistic-Malthusian

1500 Monte Carlo Cell Population Cell Monte Carlo - Regression 1000 Regression

500

0 0 20 40 60 80 100 120

Time (hrs)

Figure 9.3-4: Comparison of Theoretical Methods for the P120 Fiber

The P120 fiber demonstrated considerable promise in assisting the bone tissue regeneration process. Its experiments yielded the highest cell populations with respect to time and proved that its material properties had the ability to promote cell growth considerably faster than without a biomaterial present. Just as with the other three types of fibers, the

Logistic-Malthusian model proved most accurate. Although fairly accurate, the regression and the Monte Carlo-regression models exhibited very linear behavior, which again, is not indicative of osseous tissue formation. It is important to maintain the curvature of the data, so when experimental data is not an option for comparison, how the growth occurs is still prevalent in the theoretical method chosen for approximation. It is for this reason that the Monte Carlo method seems

224 most accurate aside from the Logistic-Malthusian model. Despite the last two points being somewhat radical from the other data, it maintains the cubic polynomial shape, which is more indicative of cellular growth.

Every theoretical method employed in this research has its benefits as well as its drawbacks. The promise lies in their ability to forecast tissue growth through their individual and sometimes combined predictions of time dependent cell population.

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Chapter 10. Conclusions and Summary of Achievements

Several assumptions pertaining to the formation and development of osseous tissue were explored throughout this proposal of research. Was it possible to simulate and predict behaviors of bone tissue and biomaterials separately, and as they work together, to restore form and function during tissue regeneration and wound healing processes?

10.1. Using Cellular Automation to Predict Bone Cell Growth

Cellular automation is a very robust tool, that when used properly, has the ability to predict the behaviors of many types of physical systems.

This program developed for this research proved successful in providing a means for simulating and predicting the behavior of bone tissue regeneration and wound healing. The program’s versatility permits both visual and mathematical comprehension of bone tissue dynamics in a continuous manner. Additionally, it provides sufficient means for producing accurate cell population data, when experimentation is not an option.

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10.2. The Combined Logistic-Malthusian Model Creates the Most

Accurate Means for Assessing Tissue Dynamics

The logistic law, by itself, has the ability to forecast how a population will behave over a predicted period of time. It lacks the complexity needed to adjust intrinsic growth rate with respect to the nonlinear behavior of osseous tissue formation. The Malthusian model, although simple, carries with it the true definition of intrinsic growth rate; it takes into consideration the dynamics of mitosis (cell birth) as well as cell death. Independently, this model is not adequate in computing cell population with respect to time because no additional variables other than growth rate are considered. When used in conjunction with the logistic law, the combined Logistic-Malthusian model provides a very efficient and effective means for approximating cell population with respect to time. The intermediate step of utilizing the Malthusian model takes into consideration all the interdependent and complex parameters associated with the formation of bone tissue. Its finite difference approach to approximation is the key to its accurate results. The only drawback associated with this method is the need for existing experimental data. Despite this drawback, this method can still be used by applying ranges of values to the involved unknown parameters to produce a cell population model still more accurate than any other discretely based method of analysis.

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10.3. Multivariate Regression Models Supply an Accurate Technique for

Predicting Cell Growth Based on the Material Properties of Carbon

Multivariate regression analysis is an established analytical tool that can be used to relate a dependable variable to several other independent variables of interest. This established means of analysis has proven successful in relating a material’s molecular properties (surface roughness and crystallinity) to the degree of tissue formation on that particular material. Validation of this method indicates that cell population and thus, tissue density can be determined strictly from the properties of the biomaterial of interest. Not only does this deem experiments as primarily an alternative method, but it provides the ability to establish which potential biomaterials would promote the most beneficial cell proliferation. It was proven that independently, both surface roughness and crystallinity were important factors in either promoting or preventing cell growth. It was also proven that adjusting these materials, together, are more influential in the absolute optimization of osseous tissue formation.

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10.4. Overall Tissue Density Can be Accurately Foreseen by Instilling the

Monte Carlo Methods

The Monte Carlo simulation methods are yet another robust means for obtaining important information about various physical and mathematical based systems. Its probabilistic nature permits a degree of variation in the way a system may be analyzed. That degree of variation accounts for the potential errors that most other theoretical models cannot justify. This method was employed successfully in this research initiative and proved most accurate in forecasting the prospected tissue growth on a biomaterial. Its versatility also permitted its use with other established theoretical methods to develop even more accurate depictions of cell population. In comparison to all other experimental and theoretical data, its results were developed more quickly and with less error. The Monte Carlo methods have the ability to incorporate a vast number of variables, determine the strength of dependency between those variables, and use those relationships to generate very reliable data. When used in conjunction with the Tissue Component

Characteristics its projection of tissue density with respect to time was the second most accurate theoretical method, behind the Logistic-

Malthusian model. Its unique advantage over the Logistic-Malthusian model is that experimental data is not needed to generate such dependable and accurate results.

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10.5. Collaboration of Theoretical Methods

Each of the theoretical methods developed for the prediction of bone tissue regeneration and wound healing have their own unique advantages. Perhaps the ultimate benefit they carry is their adaptability to work with each other. If experimental data is known, each of these methods can be used to corroborate this data. If no experimental data is known or given, the cellular automation program or the regression models could generate the data, while the Tissue Component Characteristics could be used to assess the amount of tissue that has developed and the

Monte Carlo methods could assess the generated data with a given range of variation to take into consideration any amount of error associated with the initial generation of the data. If not enough data is known to produce the desired amount of accuracy, data points could be extrapolate and the Logistic-Malthusian model could be used to assess the reliability of that data. Each theoretical method develop here, has the ability to work with every other method in any fashion, to produce the results necessary to meet the needs of the research of interest.

Conclusively, they all provide a more continuous means of assessing bone tissue dynamics even with little or no experimental data present. This collection of models is an excellent foundation from which more detailed

230 assessments and conclusions can be made about the development and repair of bone.

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Chapter 11. Comparisons of Objectives with Achievements

It was the purpose of this research to study the various aspects of biology, mathematics and science in order to develop a better means of understanding bone tissue dynamics. As the research progressed, additional means for analysis became evident and the focus of the research evolved. Not only were methods developed to understand tissue behavior, models were created that can be used to manufacture the optimum biomaterial: a material whose purpose is to promote faster, more structurally sound osseous tissue. Theoretical methods were also created for the purpose of tissue regeneration analysis, even when experimental data is non-existent. This research effort grew into an all- inclusive scheme from which the complete characteristics of a tissue- biomaterial system can be simulated and analyzed.

11.1. Develop More Efficient and Dependable Means for Simulating and

Predicting Bone Tissue Regeneration

Cell growth and bone tissue regeneration concepts were studied in depth and it became evident that current methods of analysis lacked continuity and accuracy. The most up to date ways of assessing the

232 wound healing process all involve discrete forms of analysis; mere snapshots of the whole picture. It was the primary objective of this research to develop better, more dependable and continuous means for assessing bone tissue regeneration and wound healing. Various models were developed and validated through experimentation.

The development of the Cellular Automation program produced a continuous, animated and visually understood means for understanding how complex interdependent cellular components worked together to repair bone.

The Logistic-Malthusian model combines the growth characteristics of the Logistic Law with the complexities of an intrinsic growth rate from the Malthusian model to produce continuous cell population data with no areas for question.

The established relationships of tissue composition were used to compute average values for various cellular components such as individual cell volume and mass. Knowing these parameters adds versatility to the developed theoretical methods by employing a way in which actual tissue volume, density and composition can be calculated, simply from knowing the amount of cells present within the system.

The Monte Carlo simulation methods, although newly discovered, proved to be the most robust of the theoretical models, having the ability to establish dependable cell population data using a probabilistic

233 approach. Their amazing amount of accuracy stems from their ability to account for model variation and error. Besides the Logistic-Malthusian model, which is strongly based on the presence of experimental data, the

Monte Carlo methods, in conjunction with the Tissue Component

Characteristics, provide an excellent means for plotting continuous cell population and tissue density data.

11.2. Determination of Material Properties Most Advantageous to Tissue

Growth

Past methods of analysis did not demonstrate a complete understanding of how a biomaterial’s molecular properties affected a tissue’s ability to grow. This initiative not only focused on gaining a better understanding of the material properties of various carbon fibers, but also on identifying which properties proved most advantageous in assisting in the regeneration of osseous tissue. In vitro experiments were performed to gain initial understanding on cell growth prior to the use of an assisting biomaterial. Additional in vitro experiments in which a system of primary human osteoblasts were permitted to grow on various carbon fibers illustrated carbon’s ability to influence cell growth and proliferation.

RAMAN spectroscopy helped analyze each fiber’s average crystal diameter, while true X-RAY diffraction assisted in the calculation of both integral crystallinity parameters: average crystal diameter and average

234 crystal height. Atomic Force Microscopy identified the roughness of each of the carbon fibers on varying scales of magnification. Combining the efforts of all of these methods paved the way for understanding which material properties proved most significant in promoting tissue growth. In conjunction with regression analysis, a dependable model was developed that has the ability to predict cell population as a function of a material’s surface roughness and crystallinity. Any material’s ability to promote or prevent bone tissue growth can now be promptly examined through the utilization of this model.

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Chapter 12. Suggestions for Future Research

Through the course of this study, several potential avenues of future research were developed. The fields of engineering, tissue regeneration and mathematical biology will only continue to grow through the progression and development of dependable research.

12.1. Predicting Cellular Behavior with Three Dimensional Models

This particular research initiative focused on predicting bone tissue regeneration behavior through the execution of a robust, cellular automation based computer program. Although this program is two- dimensional and is completely reliant on established rules of execution and a time space grid, it maintains its dependability. It would be of particular interest to develop a similar CA based scheme that addresses the patterns associated with cell growth and proliferation, but in a three- dimensional fashion. It is definitely possible to develop a three- dimensional time-space grid and apply the concepts of CA to its execution. The difficulty lies in the multitude of boundary conditions and the magnitude of potential cell growth patterns that would have to be considered.

236

It would also be beneficial to apply the Tissue Component Characteristics developed in this research to a 3D grid. Assign varying volumes and masses to cells depending on how far along they are in the mitotic cycle and be able to successfully represent this in a 3D animation of the program execution. Cell shape would be another avenue that could possibly be represented in a CA based program.

12.2. Development of Successful Bioresorbable Implant Materials

This research effort focused on isolating the specific material properties that have the most impact on bone cell growth. The numerical values for both surface roughness and crystallinity could be optimized and instilled in a bioresorbable implant material that would gradually but safely degrade within the body as it helped native tissue develop. This would ultimately result in the components of the biomaterial being resorbed by the body over time and as the tissue-biomaterial achieves equilibrium, only structurally sound, native tissue would remain. The challenge with this initiative lies in the ability to develop a material that resorbs at the appropriate rate. Since cell growth and regeneration of tissue are nonlinear, a material would have to be developed that could almost sense that behavior and know at different time intervals how to change its resorption rate. As carbon is a structurally inert material, this initiative would have to focus on polymer based materials.

237

12.3. Porous Implant Materials with the Characteristics of Native Tissues

One of the most significant problems with implant materials to date, is their ability to produce the stress shielding effect. As mentioned in

Chapter 3, most implant materials are much stiffer than natural bone.

Bone has a normal dynamic loading response; it depends on various physiological loading to maintain its strength. When used in load-bearing conditions, the implant then takes on a disproportionate amount of the load and thus “shields” the surrounding bone, deeming it weak and more susceptible to fracture. The ability to eliminate this effect through the development of a material more closely resembling bone would be a great achievement in the biomedical field. One way this could be accomplished is through the development of a porous material. A material’s stiffness could be altered simply by adjusting the material’s porosity. Since carbon is already present in the body and can be manufactured with varying degrees of porosity, it would be advantageous to develop a carbon based, porous material, whose stiffness resembled that of natural bone. Bone is very dynamic in nature, therefore the challenge would lie in developing a carbon based material, whose porosity, and thus stiffness could vary within different portions of the material. Because cells have been known to grow axially along various carbon materials, this adjusted porosity would not only maintain that

238 restricted proliferation, but also accommodate the different cell types and tissue components that formulate bone.

12.4. Understanding How Tissue-Biomaterial Behaviors Change on

Differing Scales of Observation

One of the integral reasons why bone is so difficult to understand is because the highly dependent cellular events that orchestrate its development occur on such a small scale. Any biomaterial that comes in contact with bone must also, then, be thoroughly understood on that same scale. If the materials are not analyzed at the appropriate scales, pertinent information pertaining to both tissue and biomaterial behavior could be overlooked. This is evident in the various AFM pictures acquired for this initiative. On a larger scale, or less magnification, some of the materials appeared very smooth in nature. By looking more closely, on a much small, more magnified scale, it was seen that these materials were far from smooth. How does one determine what scale of observation and analysis is necessary to gain dependable data? It would be advantageous to develop a method for analysis in which varying degrees of magnification could be related. Perhaps a mathematical relationship could be developed, for various measurable properties, that highlights how the value of that particular property changes depending on the magnification from which it is being analyzed. This would prove very

239 advantageous in situations, in which the equipment needed to measure a material property on the necessary scale is not attainable.

This research effort focused on developing more accurate theoretical means from which bone tissue regeneration could be analyzed. These proposed initiatives of exploratory research show just as much promise in bringing more knowledge and substance to the world of biomedical engineering.

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Appendix A: Cellular Automation Main Executable Program and Subroutines

Main Program:

% Bone Tissue Development & Regeneration Program % Written By: Mary Kundrat clc clear all close all

n=51; %Grid dimensions syms b1 syms b2 syms b3 syms b4 syms b5 %Osteoblast Variables syms c1 syms c2 syms c3 syms c4 syms c5 %Osteocyte Variables syms M syms MM %Matrix and Mineralized Matrix Variables syms O syms G syms P %Osteoclast, Growth Factor and Protein Variables syms BV syms WBC %Blood Vessel and White Blood Cell Variables syms V syms Z %Void/Dead Cell Variable and Symbolic Matrix nOst=zeros(n,n); nb1=zeros(n,n); nb2=zeros(n,n); nb3=zeros(n,n); nb4=zeros(n,n); nb5=zeros(n,n); %Osteoblast Counters nc1=zeros(n,n); nc2=zeros(n,n); nc3=zeros(n,n); nc4=zeros(n,n); nc5=zeros(n,n); %Osteocyte Counters nM=zeros(n,n); nMM=zeros(n,n); %Matrix and Mineralized Matrix Counters nWBCP=zeros(n,n); nMP=zeros(n,n); %Matrix for White Blood Cells pluz proteins & Matrix + Proteins nG=zeros(n,n); nP=zeros(n,n); %Osteoclast, Growth Factor and Protein Counters nBV=zeros(n,n); nWBC=zeros(n,n); %Blood Vessel & White Blood Cell Counters nV=0; %Void/Dead Cell Counter

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Birth=zeros(n,n); Death=zeros(n,n); Voids=zeros(n,n); Proteins=zeros(n,n); GrowthFactors=zeros(n,n); Matrix=zeros(n,n); Clasts=zeros(n,n); g=0; % The current generation max = 1345; % Maximum number of generations k=0; makemovie=0;

Time=zeros(max-1,1); Birthrate=zeros(max-1,1); Deathrate=zeros(max-1,1); Voidrate=zeros(max-1,1); Proteinrate=zeros(max-1,1); GrowthFactorsRate=zeros(max-1,1); Matrixrate=zeros(max-1,1); Clastsrate=zeros(max-1,1); % %Calls the "Insert" function to set up original matrix % a=1; b=20; X=zeros(n,n); [X,Z, nV, nBV]=insert(X,a, b, n, nV, nBV);

% %Main Loop for animation recording % while(g

[X,Z,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nOst,nMM,nP,n G] = cellcycle(X,Z,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nO st,nMM,nP,nG);

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[X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = NeighborhoodSum(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5); [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = CleanNeighborhood(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5); [X,nWBCP,nb5,nc5,nP] = preosteoclast(X,n,nWBCP,nb5,nc5,nP); [X,nWBCP,nb5,nc5,nP] = OsteoclastFinal(X,n,nWBCP,nb5,nc5,nP); [X,nb5,nc5,nb1] = engulf(X,n,nb5,nc5,nb1); [X,nG,nOst,nb1] = growthfactor2(X,n,nG,nOst,nb1); [X,nG]=chemotaxis(X,n,nG); [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nWBCP,nOst,nP] = osteocyte(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nWBCP,nOst,nP); [X,nb1,nb2,nb3,nb4,nc1,nc2,nc3,nc4,nb5,nc5,nWBCP,nP] = celldeath(X,n,nb1,nb2,nb3,nb4,nc1,nc2,nc3,nc4,nb5,nc5,nWBCP,nP); % [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = CleanNeighborhood(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5); % [X,nWBCP,nb5,nc5,nP] = preosteoclast(X,n,nWBCP,nb5,nc5,nP); % [X,nWBCP,nO,nb5,nc5,nP] = OsteoclastFinal(X,n,nWBCP,nO,nb5,nc5,nP);

[X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nOst,nG,nWBCP,nM,nMM,nMP,nP] =counterclear(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nOst,nG,nWBCP ,nM,nMM,nMP,nP,g); [Z,X,nc1,nc2,nb5,nc5,nMM,nMP] = Zconvert(X,n,nc1,nc2,nb5,nc5,nMM,nMP); %\ for i=1:n for j=1:n if X(i,j)==6 || X(i,j)==7 || X(i,j)==8 || X(i,j)==9 || X(i,j)==11 || X(i,j)==12 || X(i,j)==13 Birth(i,j)=1; elseif X(i,j)==14 Death(i,j)=1; elseif X(i,j)==10 Proteins(i,j)=1; elseif X(i,j)==15 Voids(i,j)=1; elseif X(i,j)==3 GrowthFactors(i,j)=1; elseif X(i,j)==4 || X(i,j)==5 Matrix(i,j)=1; elseif X(i,j)==2 Clasts(i,j)=1; else end end end Time(g+1)=g; Birthrate(g+1)=sum(Birth(:)); Deathrate(g+1)=sum(Death(:)); Voidrate(g+1)=sum(Voids(:)); Proteinrate(g+1)=sum(Proteins(:)); GrowthFactorsRate(g+1)=sum(GrowthFactors(:)); Matrixrate(g+1)=sum(Matrix(:)); Clastsrate(g+1)=sum(Clasts(:));

set(imh,'Cdata',-X)

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Subroutine – “Insert” function [X,Z,nV,nBV,BV] = insert(X,a,b,n,nV,nBV,BV)

% Inserts osteoblasts and voids into original matrix % b1 - Young Osteoblast % V - Void % BV - Blood Vessel % Z - Symbolic Matrix syms b1 syms BV syms V syms Zorig

Xorig=round(a+(b-a)*rand(n)); %Randomnly fill matrix with values from 0-1 for i=1:5:n for j=1:5:n Xorig(i,j)=0; %Inserts blood vessels 0.2mm apart in the matrix X(i,j)=0; Zorig(i,j)=BV; end end

for i=1:n %Loop that inserts osteoblasts and voids into matrix for j=1:n if Xorig(i,j)==0 X(i,j)=0; % Blood Vessels BV = 0 Zorig(i,j)=BV; elseif Xorig(i,j)==1 X(i,j)=6; % Osteoblasts I, b1 = 6 Z(i,j)=b1; else X(i,j)=15; % Voids, V = 15 Z(i,j)=V; nV=nV+1; end end end

% X(5,5)=6; X(5,6)=6; X(5,7)=6; % X(6,5)=6; X(6,7)=6; % X(7,5)=6; X(7,6)=6; X(7,7)=6; X(10,10)=6; X(10,11)=6;

% X(10,12)=6; % X(11,10)=6; X(11,12)=6; % X(12,10)=6; X(12,11)=6; X(12,12)=6; % % X(15,15)=6; X(15,16)=6; X(15,17)=6; % X(16,15)=6; X(16,17)=6; % X(17,15)=6; X(17,16)=6; X(17,17)=6;

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Z=Zorig; imagesc(-X,[-15 0]) %View the matrix colormap(jet); %colorbar; pause;

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Subroutine – “Cell Cycle” function [X,Z,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nOst,nMM,nP,n G] = cellcycle(X,Z,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nO st,nMM,nP,nG) %Performs the maturation of the cells up to cellular division %Cells mature up to the next level every 8 hours, or every 8 generations %and maturation is indicated by the change in color. %b1 = 6 %b2 = 7 %b3 = 8 %b4 = 9 % %c1 = 11 %b5 = 14 %c5 = 14 syms Z syms b1 b2 b3 b4 b5 c1 c2 c3 c4 c5 BV P G WBC V MP M MM

%[X,Z] = BloodVessels(Z,X,n); for i=1:n for j=1:n if X(i,j)==0 %If cell is BV, it stays a BV X(i,j)=0; elseif X(i,j)==6 && nb1(i,j)<=7 %Need to reset the counters or add to the counters here... X(i,j)=6; nb1(i,j)=nb1(i,j)+1; elseif X(i,j)==6 && nb1(i,j)>=8 X(i,j)=7; nb1(i,j)=0; nb2(i,j)=nb2(i,j)+1; elseif X(i,j)==7 && nb2(i,j)<=7 X(i,j)=7; nb2(i,j)=nb2(i,j)+1; elseif X(i,j)==7 && nb2(i,j)>=8 X(i,j)=8; nb2(i,j)=0; nb3(i,j)=nb3(i,j)+1; elseif X(i,j)==8 && nb3(i,j)<=7 X(i,j)=8; nb3(i,j)=nb3(i,j)+1; elseif X(i,j)==8 && nb3(i,j)>=8 X(i,j)=9; nb3(i,j)=0; nb4(i,j)=nb4(i,j)+1; elseif X(i,j)==9 && nb4(i,j)<=7 X(i,j)=9; nb4(i,j)=nb4(i,j)+1; elseif X(i,j)==9 && nb4(i,j)>=8 X(i,j)=9;

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[X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nMM,nP] = osteoblast(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nMM ,nP); elseif X(i,j)==2 && nOst(i,j)<=16 X(i,j)=2; nOst(i,j)=nOst(i,j)+1; elseif X(i,j)==2 && nOst(i,j)> 16 X(i,j)=4; nOst(i,j)=0; nM(i,j)=nM(i,j)+1; elseif X(i,j)==3 && nG(i,j)<=31 X(i,j)=3; nG(i,j)=nG(i,j)+1; elseif X(i,j)==3 && nG(i,j)>=32 X(i,j)=15; nG(i,j)=0; %nb1(i,j)=nb1(i,j)+1; else end end end end

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Subroutine – “Osteoblast” function [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nMM,nP] = osteoblast(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM,nMP,nWBCP,nMM ,nP) % % %Purpose of subfunction is to take mature osteoblasts and give them one of %three fates...(a)It divides into two daugther cells, (b) it becomes an %osteocyte, or (c) it dies. If it dies, proteins are released that are %used, along with White Blood Cells to create Osteoclasts, which engulf %cellular debris. It can only become an osteocyte, if the surrounding area %of matrix has been mineralized, and the cell cannot escape. % % syms b1 b2 b3 b4 b5 syms c1 c2 c3 c4 c5 syms BV WBC P G V syms Z a=1; b=50; XX=round(a+(b-a)*rand(n)); for i=1:n for j=1:n if X(i,j)==0 % If cell is a blood vessel, keep it a blood vessel X(i,j)=0; elseif X(i,j)==9 && XX(i,j)>=17 && XX(i,j)<=18 % 10% of osteoblasts undergo apoptosis X(i,j)=14; %Dead osteoblast = 14 nb4(i,j)=0; nb5(i,j)=nb5(i,j)+1; % [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = CleanNeighborhood(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5); % [X,nWBCP,nb5,nc5,nP] = preosteoclast(X,n,nWBCP,nb5,nc5,nP); % [X,nWBCP,nO,nb5,nc5] = OsteoclastFinal(X,n,nWBCP,nO,nb5,nc5); elseif X(i,j)==9 && XX(i,j)>=1 && XX(i,j)<=5 %If cell is mature osteoblast and XX=2-9 cell splits %80% of mature osteoblasts undergo mitosis nb4(i,j)=0; %Counter is reset X(i,j)=6; nb1(i,j)=0; nb1(i,j)=nb1(i,j)+1; %Cell changes to a young osteoblast %Check neighborhood for open area to put other daughter cell [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM] = VoidCheck(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nM); %Run Matrix program to deposit matrix foundation [X,nM,nb1] = matrix(X,n,nM,nb1);

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%Run Mineralized Matrix program to mineralize existing matrix [X,nM,nMP,nMM,nb1] = mineralizedmatrix2(X,n,nM,nMP,nMM,nb1); %[X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = NeighborhoodSum(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5); else %10% of Cells become osteocytes %[X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = NeighborhoodSum(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5); %If cell is mature osteoblast and XX=5, it becomes an osteocyte, meaning %it is surrounded by matrix %Pre-Osteocyte I = 11 end end end

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Subroutine – “Neighborhood Sum” function [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = NeighborhoodSum(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5)

%This function checks the neighborhood of mature osteoblasts to determine %if they are surrounded by matrix. If they are, they become osteocytes. % % Mature Osteoblast - 9 % Osteocyte - 11 % Matrix - 4 % Mineralized Matrix - 5 % syms b1 b2 b3 b4 b5 c1 c2 c3 c4 c5 BV WBC G P V syms Z for i=1:n for j=1:n if X(i,j)==0 %If cell is a blood vessel, it stays a blood vessel X(i,j)=0; elseif i==1 && j==1 %If cell just split and a neighborhood cell is vacant break elseif i==1 && j>=2 && j<=n-1 sum = X(i,j-1)+X(i+1,j-1)+X(i+1,j)+X(i+1,j+1)+X(i,j+1); if sum >= 15 && sum <=35 && (X(i,j)==9) X(i,j)=11; elseif sum >=36 && sum <=50 && X(i,j)==9 X(i,j)=4; else end elseif i==1 && j==n break elseif i>=2 && i<=n-1 && j==1 sum = X(i-1,j)+X(i-1,j+1)+X(i,j+1)+X(i+1,j+1)+X(i+1,j); if sum >=15 && sum <=35 && (X(i,j)==9) X(i,j)=11; elseif sum >=36 && sum <=50 && X(i,j)==9 X(i,j)=4; else end elseif i==n && j==1 break elseif i==n && j>=2 && j<=n-1 sum = X(i,j-1)+X(i-1,j-1)+X(i-1,j)+X(i-1,j+1)+X(i,j+1); if sum >=15 && sum <=35 && (X(i,j)==9) X(i,j)=11; elseif sum >=36 && sum <=50 && X(i,j)==9 X(i,j)=4; else end elseif i==n && j==n break

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elseif i>=2 && i<=n-1 && j==n sum = X(i-1,j)+X(i-1,j-1)+X(i,j-1)+X(i+1,j-1)+X(i+1,j); if sum >=20 && sum <=45 && (X(i,j)==9) X(i,j)=11; elseif sum>=46 && sum <=55 && X(i,j)==9 X(i,j)=4; else end elseif i>=2 && i<=n-1 && j>=2 && j<=n-1 sum = X(i,j-1)+X(i-1,j-1)+X(i-1,j)+X(i- 1,j+1)+X(i,j+1)+X(i+1,j+1)+X(i+1,j)+X(i+1,j-1); if sum >=41 && sum <=59 && (X(i,j)==9 || X(i,j)==8 || X(i,j)==7) X(i,j)=4; elseif sum >=25 && sum<=40 && (X(i,j)==6) X(i,j)=11; else end end end end

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Subroutine – “Clean Neighborhood” function [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = CleanNeighborhood(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5)

%If a cell dies (osteoblast or osteocyte) its immediate surrounding %neighborhood is eliminated to make room for white blood cells, proteins %and growth factors, etc that help replenish the area by attracting young %osteoblasts to the area. % % Dead Cell - 14 % Void - 15 % White Blood Cell - 1 % Protein - 10 % % Mineralized Matrix - 5 % syms b1 b2 b3 b4 b5 c1 c2 c3 c4 c5 BV WBC G P V syms Z for i=1:n for j=1:n if X(i,j)==0 %If cell is a blood vessel, it stays a blood vessel X(i,j)=0; elseif i==1 && j==1 %If cell just split and a neighborhood cell is vacant break elseif i==1 && j>=2 && j<=n-1 N = [X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && (nb5(i,j)>=2 || nc5(i,j)>=2) N(:,:) = 15; [X] = BloodVessels(X,n); X(i,j)=14; else end elseif i==1 && j==n break elseif i>=2 && i<=n-1 && j==1 N = [X(i-1,j) X(i-1,j+1); X(i,j) X(i,j+1); X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && (nb5(i,j)>=2 || nc5(i,j)>=2) N(:,:) = 15; [X] = BloodVessels(X,n); X(i,j)=14; else end elseif i==n && j==1 break elseif i==n && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1)]; if X(i,j)==14 && (nb5(i,j)>=2 || nc5(i,j)>=2)

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N(:,:) = 15; [X] = BloodVessels(X,n); X(i,j)=14; else end elseif i==n && j==n break elseif i>=2 && i<=n-1 && j==n N = [X(i-1,j-1) X(i-1,j); X(i,j-1) X(i,j); X(i+1,j-1) X(i+1,j)]; if X(i,j)==14 && (nb5(i,j)>=9 || nc5(i,j)>=9) N(:,:) = 15; [X] = BloodVessels(X,n); X(i,j)=14; else end elseif i>=2 && i<=n-1 && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && (nb5(i,j)>=9 || nc5(i,j)>=9) N(:,:) = 15; [X] = BloodVessels(X,n); X(i,j)=14; else end end end end

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Subroutine – “Blood Vessel” function [X] = BloodVessels(X,n)

%Places blood vessels 0.2mm apart in matrix %Need when running the rules, and cells occupied by blood vessels %might get replaced by other variables incidently. syms BV syms Zorig for i=1:5:n for j=1:5:n X(i,j)=0; %Z(i,j)=BV; end end

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Subroutine – “PreOsteoclast” function [X,nWBCP,nb5,nc5,nP] = preosteoclast(X,n,nWBCP,nb5,nc5,nP)

%Once an osteoblast or osteocyte has died, they secrete proteins. These %proteins are used with White Blood Cells to generate osteoclasts that are %then used to engulf the dead cells/cellular debris. % %This program places white blood cells and secretes proteins. % % 0=Blood Vessel 14=Dead Osteoblast 5=Mineralized Matrix % 10=Protein % for i=1:n for j=1:n if X(i,j)==0 %If cell is a blood vessel, it stays a blood vessel X(i,j)=0; elseif i==1 && j==1 %If cell just split and a neighborhood cell is vacant break elseif i==1 && j>=2 && j<=n-1 N = [X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 X(i+1,j)=10; X(i+1,j+1)=1; else end elseif i==1 && j==n break elseif i>=2 && i<=n-1 && j==1 N = [X(i-1,j) X(i-1,j+1); X(i,j) X(i,j+1); X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 X(i,j+1)=10; X(i-1,j+1)=1; else end elseif i==n && j==1 break elseif i==n && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1)]; if X(i,j)==14 X(i-1,j)=10; X(i-1,j+1)=1; else end elseif i==n && j==n break elseif i>=2 && i<=n-1 && j==n N = [X(i-1,j-1) X(i-1,j); X(i,j-1) X(i,j); X(i+1,j-1) X(i+1,j)];

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if X(i,j)==14 X(i,j-1)=10; X(i+1,j-1)=1; else end elseif i>=2 && i<=n-1 && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 X(i-1,j)=10; X(i-1,j+1)=1; else end end end end

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Subroutine – “Osteoclast Final” function [X,nWBCP,nb5,nc5,nP] = OsteoclastFinal(X,n,nWBCP,nb5,nc5,nP)

% This program takes the white blood cells and proteins and forms the % osteoclasts. % % 0=Blood Vessel 14=Dead Osteoblast 5=Mineralized Matrix % 10=Protein % for i=1:n for j=1:n if X(i,j)==0 %If cell is a blood vessel, it stays a blood vessel X(i,j)=0; elseif i==1 && j==1 %If cell just split and a neighborhood cell is vacant break elseif i==1 && j>=2 && j<=n-1 N = [X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && X(i+1,j)==10 % && X(i+1,j+1)==1 X(i+1,j)=2; X(i+1,j+1)=15; else end elseif i==1 && j==n break elseif i>=2 && i<=n-1 && j==1 N = [X(i-1,j) X(i-1,j+1); X(i,j) X(i,j+1); X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && X(i,j+1)==10 % && X(i-1,j+1)==1 X(i,j+1)=2; X(i-1,j+1)=15; else end elseif i==n && j==1 break elseif i==n && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1)]; if X(i,j)==14 && X(i-1,j)==10 % && X(i-1,j+1)==1 X(i-1,j)=2; X(i-1,j+1)=15; else end elseif i==n && j==n break elseif i>=2 && i<=n-1 && j==n N = [X(i-1,j-1) X(i-1,j); X(i,j-1) X(i,j); X(i+1,j-1) X(i+1,j)]; if X(i,j)==14 && X(i,j-1)==10 % && X(i+1,j-1)==1 X(i,j-1)=2; X(i+1,j-1)=15; else end

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elseif i>=2 && i<=n-1 && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && X(i-1,j)==10 % && X(i-1,j+1)==1 X(i-1,j)=2; X(i-1,j+1)=15; else end end end end

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Subroutine – “Engulf” function [X,nb5,nc5,nb1] = engulf(X,n,nb5,nc5,nb1) % %After developing the osteoclasts from white blood cells and proteins, this %program searches the area for the dead cellular debris and the osteoclasts %engulf it. % %The area previously occupied by the osteoclasts becomes a growth factor since the engulfing %process releases growth factors to help rejunvenate the site. Once %the debris is eaten, it leaves behind growth factors to attract young %osteoblasts. % % Blood Vessel = 0 % Osteoclast = 2 % Dead Osteoblast = 14 % Dead Osteoclast = 14 % Void = 15 % % for i=1:n for j=1:n if X(i,j)==0 %If blood vessel, it stays blood vessel X(i,j)=0; else if i==1 && j==1 %If cell just split and a neighborhood cell is vacant break elseif i==1 && j>=2 && j<=n-1 N = [X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && X(i+1,j)==2 X(i,j)=2; X(i+1,j)=15; else end elseif i==1 && j==n break elseif i>=2 && i<=n-1 && j==1 N = [X(i-1,j) X(i-1,j+1); X(i,j) X(i,j+1); X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && X(i,j+1)==2 X(i,j)=2; X(i,j+1)=15; else end elseif i==n && j==1 break elseif i==n && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1)]; if X(i,j)==14 && X(i-1,j)==2 X(i,j)=2;

266

X(i-1,j)=15; else end elseif i==n && j==n break elseif i>=2 && i<=n-1 && j==n N = [X(i-1,j-1) X(i-1,j); X(i,j-1) X(i,j); X(i+1,j-1) X(i+1,j)]; if X(i,j)==14 && X(i,j-1)==2 X(i,j)=2; X(i,j-1)=15; else end elseif i>=2 && i<=n-1 && j>=2 && j<=n-1 N = [X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-1) X(i+1,j) X(i+1,j+1)]; if X(i,j)==14 && X(i-1,j)==2 X(i,j)=2; X(i-1,j)=15; else end end end end end

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Subroutine – “Growth Factor” function [X,nG] = growthfactor(X,n,nG) % %After osteoclasts engulf the cellular debris, they release growth factors %that attract new, young cells to the area. % %Blood Vessel = 0 %Void = 15 %Osteoclast = 2 %Growth Factor = 3 % for i=1:n for j=1:n if X(i,j)==0 %If blood vessel, it stays a blood vessel X(i,j)=0; elseif i==1 && j==2 if X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end elseif i==1 && j>=3 && j<=n-2 if X(i,j)==15 && X(i,j-2)==2

268

X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end elseif i==2 && j==n-1 if X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2

269

X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break else end elseif i==1 && j==n if X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break else end elseif i==2 && j==1 if X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2

270

X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end elseif i>=3 && i<=n-2 && j==1 if X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2

271

X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end elseif i==n-1 && j==1 if X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break else end elseif i==n && j==1 if X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2

272

X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break else end elseif i==n && j==2 if X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break else

273

end elseif i==n && j>=3 && j<=n-2 if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break else end elseif i==n && j==n-1 if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2

274

X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break else end elseif i==n && j==n if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break else end elseif i==n-1 && j==n if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2

275

X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break else end elseif i>=3 && i<=n-2 && j==n if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2

276

X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break else end elseif i==2 && j==n if X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break else end elseif i==2 && j==2 if X(i,j)==15 && X(i-1,j-1)==2

277

X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end elseif i==2 && j>=3 && j<=n-2 if X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2

278

X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end elseif i==2 && j==n-1 if X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2

279

X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break else end elseif i>=3 && i<=n-2 && j==2 if X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2

280

X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end elseif i==n-1 && j==2 if X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2

281

X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break else end elseif i==n-1 && j>=3 && j<=n-2 if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2

282

X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break else end elseif i==n-1 && j==n-1 if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2

283

X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break else end elseif i>=3 && i<=n-2 && j==n-1 if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2 X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2

284

X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break else for i=3:n-2 for j=3:n-2 if X(i,j)==15 && X(i-2,j-2)==2 X(i,j)=3; X(i-2,j-2)=round(rand); break elseif X(i,j)==15 && X(i-2,j-1)==2 X(i,j)=3; X(i-2,j-1)=round(rand); break elseif X(i,j)==15 && X(i-2,j)==2 X(i,j)=3; X(i-2,j)=round(rand); break elseif X(i,j)==15 && X(i-2,j+1)==2 X(i,j)=3; X(i-2,j+1)=round(rand); break elseif X(i,j)==15 && X(i-2,j+2)==2 X(i,j)=3; X(i-2,j+2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-2)==2 X(i,j)=3; X(i-1,j-2)=round(rand); break elseif X(i,j)==15 && X(i-1,j-1)==2 X(i,j)=3; X(i-1,j-1)=round(rand); break elseif X(i,j)==15 && X(i-1,j)==2 X(i,j)=3; X(i-1,j)=round(rand); break elseif X(i,j)==15 && X(i-1,j+1)==2

285

X(i,j)=3; X(i-1,j+1)=round(rand); break elseif X(i,j)==15 && X(i-1,j+2)==2 X(i,j)=3; X(i-1,j+2)=round(rand); break elseif X(i,j)==15 && X(i,j-2)==2 X(i,j)=3; X(i,j-2)=round(rand); break elseif X(i,j)==15 && X(i,j-1)==2 X(i,j)=3; X(i,j-1)=round(rand); break elseif X(i,j)==15 && X(i,j+1)==2 X(i,j)=3; X(i,j+1)=round(rand); break elseif X(i,j)==15 && X(i,j+2)==2 X(i,j)=3; X(i,j+2)=round(rand); break; elseif X(i,j)==15 && X(i+1,j-2)==2 X(i,j)=3; X(i+1,j-2)=round(rand); break elseif X(i,j)==15 && X(i+1,j-1)==2 X(i,j)=3; X(i+1,j-1)=round(rand); break elseif X(i,j)==15 && X(i+1,j)==2 X(i,j)=3; X(i+1,j)=round(rand); break elseif X(i,j)==15 && X(i+1,j+1)==2 X(i,j)=3; X(i+1,j+1)=round(rand); break elseif X(i,j)==15 && X(i+1,j+2)==2 X(i,j)=3; X(i+1,j+2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-2)==2 X(i,j)=3; X(i+2,j-2)=round(rand); break elseif X(i,j)==15 && X(i+2,j-1)==2 X(i,j)=3; X(i+2,j-1)=round(rand); break elseif X(i,j)==15 && X(i+2,j)==2 X(i,j)=3; X(i+2,j)=round(rand); break elseif X(i,j)==15 && X(i+2,j+1)==2 X(i,j)=3; X(i+2,j+1)=round(rand); break elseif X(i,j)==15 && X(i+2,j+2)==2 X(i,j)=3; X(i+2,j+2)=round(rand); break else end end end end end end end for i=1:n

286

for j=1:n if X(i,j)==0 X(i,j)=15; elseif X(i,j)==1 X(i,j)=6; else break end end end for i=1:n for j=1:n if X(i,j)==3 nG(i,j)=nG(i,j)+1; end end end for i=1:n for j=1:n if X(i,j)==3 && nG(i,j)>=20 X(i,j)=15; nG(i,j)=0; end end end

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Subroutine – “Chemotaxis” function [X,nG]=chemotaxis(X,n,nG) % %This function looks in the extended neighborhood of growth factors to find %the closest young osteoblast to come and regenerate the area. As soon as %an osteoblast is found the osteoblast starts moving toward the growth %factor, and once the growth factor is reached, the osteoblast sets up came %and matures at that site. % %Blood Vessel = 0 %Pre-Osteoblast I = 6 %Growth Factor = 3 %Void = 15 % for i=1:n for j=1:n if X(i,j)==0 X(i,j)=0; break elseif i==1 && j==2 Neighborhood=[X(i,j-1) X(i,j) X(i,j+1) X(i,j+2); X(i+1,j-1) X(i+1,j) X(i+1,j+1) X(i+1,j+2); X(i+2,j-1) X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==5 || Traveller==7 || Traveller==8) X(i,j)=6; if Traveller==1 %Should nOstt be possible...this should always be a BV X(i,j-1)=6; elseif Traveller==2 X(i+1,j-1)=6; elseif Traveller==5 X(i+1,j)=6; elseif Traveller==7 X(i,j+1)=6; else %Else, Traveller==8 X(i+1,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==3 || Traveller==6 || Traveller==9 || Traveller==10 || Traveller==11 || Traveller==12) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i+1,j-1)~=0 X(i+2,j-1)=6; X(i+1,j-1)=6; elseif Traveller==3 && X(i+1,j-1)==0 X(i+2,j-1)=6; X(i+1,j)=6; %If most convenient space is occupied by BV elseif Traveller==6 && X(i+1,j)~=0 %next closest space is where young cell travels X(i+2,j)=6; X(i+1,j)=6;

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elseif Traveller==6 && X(i+1,j)==0 X(i+2,j)=6; X(i+1,j+1)=6; elseif X(i+1,j+1)~=0 && (Traveller==9 || Traveller==11 || Traveller==12) X(i+1,j+1)=6; if Traveller==9 X(i+2,j+1)=6; elseif Traveller==11 X(i+1,j+2)=6; else %Traveller must be = 12 hence, X(i+2,j+2)~=0 X(i+2,j+2)=6; end elseif X(i+1,j+1)==0 && (Traveller==9 || Traveller==11 || Traveller==12) if Traveller==9 X(i+2,j+1)=6; X(i+1,j)=6; elseif Traveller==11 X(i+1,j+2)=6; X(i-1,j+1)=6; else X(i+2,j+2)=6; X(i+2,j+1)=6; end elseif Traveller==10 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif Traveller==10 && X(i,j+1)==0 X(i,j+2)=6; X(i+1,j+1)=6; end end elseif i==1 && j>=3 && j<=n-2 Neighborhood=[X(i,j-2) X(i,j-1) X(i,j) X(i,j+1) X(i,j+2); X(i+1,j-2) X(i+1,j-1) X(i+1,j) X(i+1,j+1) ... X(i+1,j+2); X(i+2,j-2) X(i+2,j-1) X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==4 || Traveller==5 || Traveller==8 || Traveller==10 || Traveller==11) X(i,j)=6; if Traveller==4 X(i,j-1)=6; elseif Traveller==5 X(i+1,j-1)=6; elseif Traveller==8 X(i+1,j)=6; elseif Traveller==10 X(i,j+1)=6; else X(i+1,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==6 || Traveller==9 || Traveller==12 ... || Traveller==13 || Traveller==14 || Traveller==15) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i,j-1)~=0 X(i,j-2)=15; X(i,j-1)=6;

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elseif Traveller==1 && X(i,j-1)==0 X(i,j-2)=15; X(i+1,j-1)=6; elseif X(i+1,j-1)~=0 && (Traveller==2 || Traveller==3 || Traveller==6) X(i+1,j-1)=6; if Traveller==2 X(i+1,j-2)=6; elseif Traveller==3 X(i+2,j-2)=6; else X(i+2,j-1)=6; end elseif X(i+1,j-1)==0 && (Traveller==2 || Traveller==3 || Traveller==6) if Traveller==2 X(i+1,j-2)=15; X(i,j-1)=6; elseif Traveller==3 X(i+2,j-2)=15; X(i+2,j-1)=6; else X(i+2,j-1)=15; X(i+1,j)=6; end elseif Traveller==9 && X(i+1,j)~=0 X(i+2,j)=15; X(i+1,j)=6; elseif Traveller==9 && X(i+1,j)==0 X(i+2,j)=15; X(i+1,j+1)=6; elseif X(i+1,j+1)~=0 && (Traveller==12 || Traveller==14 || Traveller==15) X(i+1,j+1)=6; if Traveller==12 X(i+2,j+1)=6; elseif Traveller==14 X(i+1,j+2)=6; else X(i+2,j+2)=6; end elseif X(i+1,j+1)==0 && (Traveller==12 || Traveller==14 || Traveller==15) if Traveller==12 X(i+2,j+1)=15; X(i+1,j)=6; elseif Traveller==14 X(i+1,j+2)=15; X(i,j+1)=6; else X(i+2,j+2)=15; X(i+2,j+1)=6; end elseif X(i,j+1)~=0 X(i,j+1)=6; X(i,j+2)=15; elseif X(i,j+1)==0 X(i,j+2)=15; X(i+1,j+1)=6; end else end elseif i==1 && j==n-1 Neighborhood=[X(i,j-2) X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-2) X(i+1,j-1) X(i+1,j) X(i+1,j+1); ... X(i+2,j-2) X(i+2,j-1) X(i+2,j) X(i+2,j+1)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller);

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if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==4 || Traveller==5 || Traveller==8 || Traveller==10|| Traveller==11) X(i,j)=6; if Traveller==4 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==5 && X(i+1,j-1)~=0 X(i+1,j-1)=6; elseif Traveller==8 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==10 && X(i,j+1)~=0 X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==6 || Traveller==9 || Traveller==12) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i,j-2)~=0 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif X(i+1,j-1)~=0 && (Traveller==2 || Traveller==3 || Traveller==6) X(i+1,j-1)=6; if Traveller==2 && X(i+1,j-2) X(i+1,j-2)=6; elseif Traveller==3 && X(i+2,j-2)~=0 X(i+2,j-2)=6; elseif X(i+2,j-1)~=0 X(i+2,j-1)=6; end elseif X(i+1,j)~=0 X(i+1,j)=6; if Traveller==9 && X(i+2,j)~=0 X(i+2,j)=6; elseif X(i+2,j+1)~=0 X(i+2,j+1)=6; end end else end elseif i==2 && j==1 Neighborhood=[X(i-1,j) X(i-1,j+1) X(i-1,j+2); X(i,j) X(i,j+1) X(i,j+2); X(i+1,j) X(i+1,j+1) X(i+1,j+2); X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==3 || Traveller==5 || Traveller==6 || Traveller==7) X(i,j)=6; if Traveller==1 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==2 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==5 && X(i-1,j+1)~=0

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X(i-1,j+1)=6; elseif Traveller==7 && X(i,j+1)~=0 X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==4 || Traveller==8 || Traveller==9 || Traveller==10 || Traveller==11 || Traveller==12) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==4 && X(i+1,j)~=0 && X(i+2,j)~=0 X(i+1,j)=6; X(i+2,j)=6; elseif Traveller==9 && X(i-1,j+1)~=0 && X(i-1,j+2)~=0 X(i-1,j+1)=6; X(i-1,j+2)=6; elseif X(i+1,j+1)~=0 && (Traveller==8 || Traveller==11 || Traveller==12) X(i+1,j+1)=6; if Traveller==8 && X(i+2,j+1)~=0 X(i+2,j+1)=6; elseif Traveller==11 && X(i+1,j+2)~=0 X(i+1,j+2)=6; elseif X(i+2,j+2)~=0 X(i+2,j+2)=6; end elseif X(i,j+1)~=0 && X(i,j+2)~=0 X(i,j+1)=6; X(i,j+2)=6; end else end elseif i>=3 && i<=n-2 && j==1 Neighborhood=[X(i-2,j) X(i-2,j+1) X(i-2,j+2); X(i-1,j) X(i- 1,j+1) X(i-1,j+2); X(i,j) X(i,j+1) X(i,j+2); ... X(i+1,j) X(i+1,j+1) X(i+1,j+2); X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 && X(i,j)~=0 break else if T==0 && X(i,j)==3 && (Traveller==2 || Traveller==4 || Traveller==7 || Traveller==8 || Traveller==9) X(i,j)=6; if Traveller==2 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==4 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==7 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif Traveller==8 && X(i,j+1)~=0 X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; end

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elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==5 || Traveller==6 || Traveller==10 || Traveller==11 || Traveller==12 || Traveller==13 || Traveller==14 || Traveller==15) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-2,j)~=0 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif X(i-1,j+1)~=0 && (Traveller==6 || Traveller==11 || Traveller==12) X(i-1,j+1)=6; if Traveller==6 && X(i-2,j+1)~=0 X(i-2,j+1)=6; elseif Traveller==11 && X(i-2,j+2)~=0; X(i-2,j+2)=6; elseif X(i-1,j+2)~=0 X(i-1,j+2)=6; end elseif Traveller==5 && X(i+2,j)~=0 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif Traveller==13 && X(i,j+2)~=0 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; if Traveller==10 && X(i+2,j+1)~=0 X(i+2,j+1)=6; elseif Traveller==14 && X(i+1,j+2)~=0 X(i+1,j+2)=6; elseif X(i+2,j+2)~=0 X(i+2,j+2)=6; end end end end elseif i==n-1 && j==1 Neighborhood=[X(i-2,j) X(i-2,j+1) X(i-2,j+2); X(i-1,j) X(i- 1,j+1) X(i-1,j+2); X(i,j) X(i,j+1) X(i,j+2); ... X(i+1,j) X(i+1,j+1) X(i+1,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 && X(i,j)~=0 break else if T==0 && X(i,j)==3 && (Traveller==2 || Traveller==4 || Traveller==6 || Traveller==7 || Traveller==8) X(i,j)=6; if Traveller==2 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==4 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==6 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif Traveller==7 && X(i,j+1)~=0 X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; end

293

elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==5 || Traveller==9 || Traveller==10 || Traveller==11 || Traveller==12) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-2,j)~=0 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif X(i-1,j+1)~=0 && (Traveller==5 || Traveller==9 || Traveller==10) X(i-1,j+1)=6; if Traveller==5 && X(i-2,j+1)~=0 X(i-2,j+1)=6; elseif Traveller==9 && X(i-2,j+2)~=0; X(i-2,j+2)=6; elseif X(i-1,j+2)~=0 X(i-1,j+2)=6; end elseif Traveller==11 && X(i,j+1)~=0 && X(i,j+2)~=0 X(i,j+1)=6; X(i,j+2)=6; elseif X(i+1,j+1)~=0 && X(i+1,j+2)~=0 X(i+1,j+1)=6; X(i+1,j+2)=6; end end end elseif i==n && j==2 Neighborhood=[X(i-2,j-1) X(i-2,j) X(i-2,j+1) X(i-2,j+2); X(i-1,j-1) X(i-1,j) X(i-1,j+1) X(i-1,j+2); X(i,j-1) X(i,j) X(i,j+1) X(i,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==2 || Traveller==3 || Traveller==5 || Traveller==8 || Traveller==9) X(i,j)=6; if Traveller==2 && X(i-1,j-1)~=0 X(i-1,j-1)=6; elseif Traveller==3 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==5 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==8 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif X(i,j+1)~=0 X(i,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==4 || Traveller==7 || Traveller==10 || Traveller==11 || Traveller==12) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-1,j-1)~=0 && X(i-2,j-1)~=0 X(i-1,j-1)=6; X(i-2,j-1)=6; elseif Traveller==4 && X(i-1,j)~=0 && X(i-2,j)~=0 X(i-1,j)=6; X(i-2,j)=6; elseif X(i+1,j+1)~=0 && (Traveller==7 || Traveller==10 || Traveller==11)

294

X(i-1,j+1)=6; if Traveller==7 && X(i-2,j+1)~=0 X(i-2,j+1)=6; elseif Traveller==10 && X(i-2,j+2)~=0 X(i-2,j+2)=6; elseif X(i-1,j+2)~=0 X(i-1,j+2)=6; end elseif X(i,j+1)~=0 && X(i,j+2)~=0 X(i,j+1)=6; X(i,j+2)=6; end else end elseif i==n && j>=3 && j<=n-2 Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j) X(i-2,j+1) X(i,j+2); X(i-1,j-2) X(i-1,j-1) X(i-1,j) X(i-1,j+1) ... X(i-1,j+2); X(i,j-2) X(i,j-1) X(i,j) X(i,j+1) X(i,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==5 || Traveller==6 || Traveller==8 || Traveller==11 || Traveller==12) X(i,j)=6; if Traveller==5 && X(i-1,j-1)~=0 X(i-1,j-1)=6; elseif Traveller==6 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==8 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==11 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif X(i,j+1)~=0 X(i,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==7 || Traveller==10 ... || Traveller==13 || Traveller==14 || Traveller==15) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i,j-1)~=0 && X(i,j-2)~=0 X(i,j-1)=6; X(i,j-2)=6; elseif X(i-1,j-1)~=0 && (Traveller==1 || Traveller==2 || Traveller==4) X(i-1,j-1)=6; if Traveller==2 && X(i-1,j-2)~=0 X(i-1,j-2)=6; elseif Traveller==1 && X(i-2,j-2)~=0 X(i-2,j-2)=6; elseif X(i-2,j-1)~=0 X(i-2,j-1)=6; end elseif Traveller==7 && X(i-2,j)~=0 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6;

295

elseif X(i-1,j+1)~=0 && (Traveller==10 || Traveller==13 || Traveller==14) X(i-1,j+1)=6; if Traveller==10 && X(i-2,j+1)~=0 X(i-2,j+1)=6; elseif Traveller==14 && X(i-1,j+2)~=0 X(i-1,j+2)=6; elseif X(i-2,j+2)~=0 X(i-2,j+2)=6; end elseif X(i,j+1)~=0 && X(i,j+2)~=0 X(i,j+1)=6; X(i,j+2)=6; end else end elseif i==n && j==n-1 Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j) X(i-2,j+1); X(i-1,j-2) X(i-1,j-1) X(i-1,j) X(i-1,j+1); ... X(i,j-2) X(i,j-1) X(i,j) X(i,j+1)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==5 || Traveller==6 || Traveller==8 || Traveller==11|| Traveller==12) X(i,j)=6; if Traveller==5 && X(i-1,j-1)~=0 X(i-1,j-1)=6; elseif Traveller==6 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==8 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==11 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif X(i,j+1)~=0 X(i,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==7 || Traveller==10) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i,j-2)~=0 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==7 && X(i-1,j)~=0 && X(i-2,j)~=0 X(i-1,j)=6; X(i-2,j)=6; elseif Traveller==10 && X(i-1,j+1)~=0 && X(i-2,j+1)~=0 X(i-1,j+1)=6; X(i-2,j+1)=6; elseif X(i-1,j-1)~=0 X(i-1,j-1)=6; if Traveller==2 && X(i-1,j-2) X(i-1,j-2)=6; elseif Traveller==1 && X(i-2,j-2)~=0 X(i-2,j-2)=6; elseif X(i-2,j-1)~=0 X(i-2,j-1)=6;

296

end end else end elseif i==n-1 && j==n Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j); X(i-1,j-2) X(i-1,j-1) X(i-1,j); X(i,j-2) X(i,j-1) X(i,j); ... X(i+1,j-2) X(i+1,j-1) X(i+1,j)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break else if T==0 && X(i,j)==3 && (Traveller==6 || Traveller==7 || Traveller==8 || Traveller==710 || Traveller==12) X(i,j)=6; if Traveller==6 && X(i-1,j-1)~=0 X(i-1,j-1)=6; elseif Traveller==7 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==8 && X(i+1,j-1)~=0 X(i+1,j-1)=6; elseif Traveller==10 && X(i-1,j)~=0 X(i-1,j)=6; elseif X(i+1,j)~=0 X(i+1,j)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==5 || Traveller==9) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==9 && X(i-2,j)~=0 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif X(i-1,j-1)~=0 && (Traveller==1 || Traveller==2 || Traveller==5) X(i-1,j-1)=6; if Traveller==1 && X(i-2,j-2)~=0 X(i-2,j-2)=6; elseif Traveller==2 && X(i-1,j-2)~=0; X(i-1,j-2)=6; elseif X(i-2,j-1)~=0 X(i-2,j-1)=6; end elseif Traveller==3 && X(i,j-1)~=0 && X(i,j-2)~=0 X(i,j-1)=6; X(i,j-2)=6; elseif X(i+1,j-1)~=0 && X(i+1,j-2)~=0 X(i+1,j-1)=6; X(i+1,j-2)=6; end end end elseif i>=3 && i<=n-2 && j==n Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j); X(i-1,j-2) X(i-1,j-1) X(i-1,j); X(i,j-2) X(i,j-1) X(i,j); ... X(i+1,j-2) X(i+1,j-1) X(i+1,j); X(i+2,j-2) X(i+2,j-1) X(i+2,j)];

297

Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break else if T==0 && X(i,j)==3 && (Traveller==7 || Traveller==8 || Traveller==9 || Traveller==12 || Traveller==14) X(i,j)=6; if Traveller==7 && X(i-1,j-1)~=0 X(i-1,j-1)=6; elseif Traveller==8 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==9 && X(i+1,j-1)~=0 X(i+1,j-1)=6; elseif Traveller==12 && X(i-1,j)~=0 X(i-1,j)=6; elseif X(i+1,j)~=0 X(i+1,j)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==5 || Traveller==6 || Traveller==10 || Traveller==11 || Traveller==15) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i,j-2)~=0 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==11 && X(i-2,j)~=0 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif Traveller==15 && X(i+2,j)~=0 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif X(i-1,j-1)~=0 && (Traveller==1 || Traveller==2 || Traveller==6) X(i-1,j-1)=6; if Traveller==1 && X(i-2,j-2)~=0 X(i-2,j-2)=6; elseif Traveller==2 && X(i-1,j-2)~=0; X(i-1,j-2)=6; elseif X(i-2,j-1)~=0 X(i-2,j-1)=6; end elseif X(i+1,j-1)~=0 X(i+1,j-1)=6; if Traveller==4 && X(i+1,j-2)~=0 X(i+1,j-2)=6; elseif Traveller==5 && X(i+2,j-2)~=0 X(i+2,j-2)=6; elseif X(i+2,j-1)~=0 X(i+2,j-1)=6; end end end end elseif i==2 && j==n Neighborhood=[X(i-1,j-2) X(i-1,j-1) X(i-1,j); X(i,j-2) X(i,j-1) X(i,j); X(i+1,j-2) X(i+1,j-1) X(i+1,j); X(i+2,j-2) X(i+2,j-1) X(i+2,j)];

298

Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==5 || Traveller==6 || Traveller==7 || Traveller==9 || Traveller==11) X(i,j)=6; if Traveller==5 && X(i-1,j-1)~=0 X(i-1,j-1)=6; elseif Traveller==6 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==7 && X(i+1,j-1)~=0 X(i+1,j-1)=6; elseif Traveller==9 && X(i-1,j)~=0 X(i-1,j)=6; elseif X(i+1,j)~=0 X(i+1,j)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==8 || Traveller==12) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-1,j-2)~=0 && X(i-1,j-1)~=0 X(i-1,j-2)=6; X(i-1,j-1)=6; elseif Traveller==2 && X(i,j-2)~=0 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif X(i+1,j-1)~=0 && (Traveller==3 || Traveller==4 || Traveller==8) X(i+1,j-1)=6; if Traveller==3 && X(i+1,j-2)~=0 X(i+1,j-2)=6; elseif Traveller==4 && X(i+2,j-2)~=0 X(i+2,j-2)=6; elseif X(i+2,j-1)~=0 X(i+2,j-1)=6; end elseif X(i+2,j)~=0 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; end else end elseif i==2 && j==2 Neighborhood=[X(i-1,j-1) X(i-1,j) X(i-1,j+1) X(i-1,j+2); X(i,j-1) X(i,j) X(i,j+1) X(i,j+2); X(i+1,j-1) X(i+1,j) X(i+1,j+1) X(i+1,j+2); X(i+2,j-1) X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==5 || Traveller==7 || Traveller==9 || Traveller==10 || Traveller==11) X(i,j)=6; if Traveller==1 && X(i-1,j-1)~=0 X(i-1,j-1)=6;

299

elseif Traveller==2 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==3 && X(i+1,j-1)~=0 X(i+1,j-1)=6; elseif Traveller==5 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==7 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==9 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif Traveller==10 && X(i,j+1)~=0 X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==4 || Traveller==8 || Traveller==12 || Traveller==13 || Traveller==14 || Traveller==15 || Traveller==16) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==4 && X(i+2,j-1)~=0 && X(i+1,j-1)~=0 X(i+2,j-1)=6; X(i+1,j-1)=6; elseif Traveller==8 && X(i+2,j)~=0 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif Traveller==13 && X(i-1,j+2)~=0 && X(i-1,j+1)~=0 X(i-1,j+2)=6; X(i-1,j+1)=6; elseif Traveller==14 && X(i,j+2)~=0 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; if Traveller==12 && X(i+2,j+1)~=0 X(i+2,j+1)=6; elseif Traveller==15 && X(i+1,j+2)~=0 X(i+1,j+2)=6; elseif X(i+2,j+2)~=0 X(i+2,j+2)=6; end else end end elseif i==2 && j>=3 && j<=n-2 Neighborhood=[X(i-1,j-2) X(i-1,j-1) X(i-1,j) X(i-1,j+1) X(i-1,j+2); X(i,j-2) X(i,j-1) X(i,j) X(i,j+1) ... X(i,j+2); X(i+1,j-2) X(i+1,j-1) X(i+1,j) X(i+1,j+1) X(i+1,j+2); X(i+2,j-2) X(i+2,j-1) X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==5 || Traveller==6 || Traveller==9 || Traveller==11 || Traveller==13 || Traveller==14 || Traveller==15) X(i,j)=6; if Traveller==5 && X(i-1,j-1)~=0

300

X(i-1,j-1)=6; elseif Traveller==6 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==7 && X(i+1,j-1)~=0 X(i+1,j-1)=6; elseif Traveller==9 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==11 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==13 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif Traveller==14 && X(i,j+1)~=0 X(i,j+1)=6; elseif X(i,j+1)~=0 X(i,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==8 || Traveller==12 ... || Traveller==16 || Traveller==17 || Traveller==18 || Traveller==19 || Traveller==20) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-1,j-2)~=0 && X(i-1,j-1)~=0 X(i-1,j-2)=6; X(i-1,j-1)=6; elseif Traveller==2 && X(i,j-2)~=0 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==12 && X(i+2,j)~=0 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif Traveller==17 && X(i-1,j+2)~=0 && X(i-1,j+1)~=0 X(i-1,j+2)=6; X(i-1,j+1)=6; elseif Traveller==18 && X(i,j+2)~=0 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif X(i+1,j-1)~=0 && (Traveller==3 || Traveller==4 || Traveller==8) X(i+1,j-1)=6; if Traveller==3 && X(i+1,j-2)~=0 X(i+1,j-2)=6; elseif Traveller==4 && X(i+2,j-2)~=0 X(i+2,j-2)=6; elseif X(i+2,j-1)~=0 X(i+2,j-1)=6; else end elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; if Traveller==16 && X(i+2,j+1)~=0 X(i+2,j+1)=6; elseif Traveller==19 && X(i+1,j+2)~=0 X(i+1,j+2)=6; elseif X(i+2,j+2)~=0 X(i+2,j+2)=6; else end

301

else end else end elseif i==2 && j==n-1 Neighborhood=[X(i-1,j-2) X(i-1,j-1) X(i-1,j) X(i-1,j+1); X(i,j-2) X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-2) X(i+1,j-1) X(i+1,j) X(i+1,j+1); X(i+2,j-2) X(i+2,j-1) X(i+2,j) X(i+2,j+1)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 && (Traveller==5 || Traveller==6 || Traveller==7 || Traveller==9 || Traveller==11 || Traveller==13 || Traveller==14 || Traveller==15) X(i,j)=6; if Traveller==5 && X(i-1,j-1)~=0 X(i-1,j-1)=6; elseif Traveller==6 && X(i,j-1)~=0 X(i,j-1)=6; elseif Traveller==7 && X(i+1,j-1)~=0 X(i+1,j-1)=6; elseif Traveller==9 && X(i-1,j)~=0 X(i-1,j)=6; elseif Traveller==11 && X(i+1,j)~=0 X(i+1,j)=6; elseif Traveller==13 && X(i-1,j+1)~=0 X(i-1,j+1)=6; elseif Traveller==14 && X(i,j+1)~=0 X(i,j+1)=6; elseif X(i+1,j+1)~=0 X(i+1,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==8 || Traveller==12 || Traveller==16) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-1,j-2)~=0 && X(i-1,j-1)~=0 X(i-1,j-2)=6; X(i-1,j-1)=6; elseif Traveller==2 && X(i,j-2)~=0 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==12 && X(i+2,j)~=0 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif Traveller==16 && X(i+2,j+1)~=0 && X(i+1,j+1)~=0 X(i+2,j+1)=6; X(i+1,j+1)=6; elseif X(i+1,j-1)~=0 X(i+1,j-1)=6; if Traveller==3 && X(i+1,j-2)~=0 X(i+1,j-2)=6; elseif Traveller==4 && X(i+2,j-2)~=0 X(i+2,j-2)=6; elseif X(i+2,j-1)~=0 X(i+2,j-1)=6;

302

end else end end elseif i>=3 && i<=n-2 && j==2 Neighborhood=[X(i-2,j-1) X(i-2,j) X(i-2,j+1) X(i-1,j+2); X(i-1,j-1) X(i-1,j) X(i-1,j+1) X(i-1,j+2); X(i,j-1) ... X(i,j) X(i,j+1) X(i,j+2); X(i+1,j-1) X(i+1,j) X(i+1,j+1) X(i+1,j+2); X(i+2,j-1) X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break else if T==0 && X(i,j)==3 && (Traveller==2 || Traveller==3 || Traveller==4 || Traveller==7 || Traveller==9 ... || Traveller==12 || Traveller==13 || Traveller==14) X(i,j)=6; if Traveller==2 X(i-1,j-1)=6; elseif Traveller==3 X(i,j-1)=6; elseif Traveller==4 X(i+1,j-1)=6; elseif Traveller==7 X(i-1,j)=6; elseif Traveller==9 X(i+1,j)=6; elseif Traveller==12 X(i-1,j+1)=6; elseif Traveller==13 X(i,j+1)=6; else X(i+1,j+1)=6; end elseif T==0 && X(i,j)==3 && (Traveller==1 || Traveller==5 || Traveller==6 || Traveller==10 || ... Traveller==11 || Traveller==15 || (Traveller>=16 && Traveller<=20)) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-1,j-1)~=0 X(i-1,j-1)=6; X(i-2,j-1)=6; elseif Traveller==1 && X(i-1,j-1)==0 X(i-1,j)=6; X(i-2,j-1)=6; elseif Traveller==5 && X(i+1,j-1)~=0 X(i+2,j-1)=6; X(i+1,j-1)=6; elseif Traveller==5 && X(i+1,j-1)==0 X(i+2,j-1)=6; X(i+1,j)=6; elseif Traveller==6 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif Traveller==6 && X(i-1,j)==0 X(i-2,j)=6; X(i-1,j+1)=6; elseif Traveller==10 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif Traveller==10 && X(i+1,j)==0 X(i+2,j)=6; X(i+1,j+1)=6;

303

elseif Traveller==18 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif Traveller==18 && X(i,j+1)==0 X(i,j+2)=6; X(i+1,j+1)=6; elseif X(i-1,j+1)~=0 && (Traveller==11 || Traveller==16 || Traveller==17) X(i-1,j+1)=6; if Traveller==11 X(i-2,j+1)=6; elseif Traveller==16 X(i-2,j+2)=6; else X(i-1,j+2)=6; end elseif X(i-1,j+1)==0 && (Traveller==11 || Traveller==16 || Traveller==17) if Traveller==11 X(i-2,j+1)=6; X(i-1,j)=6; elseif Traveller==16 X(i-2,j+2)=6; X(i,j+2)=6; else X(i-1,j+2)=6; X(i,j+1)=6; end elseif X(i+1,j+1)~=0 && (Traveller==15 || Traveller==19 || Traveller==20) X(i+1,j+1)=6; if Traveller==15 X(i+2,j+1)=6; elseif Traveller==16 X(i+1,j+2)=6; else X(i+2,j+2)=6; end elseif X(i+1,j+1)==0 && (Traveller==15 || Traveller==19 || Traveller==20) if Traveller==15 X(i+2,j+1)=6; X(i+1,j)=6; elseif Traveller==16 X(i+1,j+2)=6; X(i,j+1)=6; else X(i+2,j+2)=6; X(i+1,j+2)=6; end end end end elseif i==n-1 && j==2 Neighborhood=[X(i-2,j-1) X(i-2,j) X(i-2,j+1) X(i-2,j+2); X(i-1,j-1) X(i-1,j) X(i-1,j+1) X(i-1,j+2); ... X(i,j-1) X(i,j) X(i,j+1) X(i,j+2); X(i+1,j-1) X(i+1,j) X(i+1,j+1) X(i+1,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 if (Traveller==2 || Traveller==3 || Traveller==4 || Traveller==6 || ...

304

Traveller==8 || Traveller==10 || Traveller==11 || Traveller==12) X(i,j)=6; if Traveller==2 X(i-1,j-1)=6; elseif Traveller==3 X(i,j-1)=6; elseif Traveller==4 X(i+1,j-1)=6; elseif Traveller==6 X(i-1,j)=6; elseif Traveller==8 X(i+1,j)=6; elseif Traveller==10 X(i-1,j+1)=6; elseif Traveller==11 X(i,j+1)=6; else X(i+1,j+1)=6; end elseif Traveller==1 || Traveller==5 || Traveller==9 || (Traveller>=13 && Traveller<=16) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==1 && X(i-1,j-1)~=0 X(i-2,j-1)=6; X(i-1,j-1)=6; elseif Traveller==1 && X(i-1,j-1)==0 X(i-2,j-1)=6; X(i-1,j)=6; elseif Traveller==5 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif Traveller==5 && X(i-1,j)==0 X(i-2,j)=6; X(i-1,j+1)=6; elseif Traveller==15 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif Traveller==15 && X(i-1,j-1)==0 X(i,j+2)=6; X(i-1,j+1)=6; elseif Traveller==16 && X(i+1,j+1)~=0 X(i+1,j+2)=6; X(i+1,j+1)=6; elseif Traveller==16 && X(i-1,j)==0 X(i+1,j+2)=6; X(i,j+1)=6; elseif X(i-1,j+1)~=0 && (Traveller==9 || Traveller==13 || Traveller==14) X(i-1,j+1)=6; if Traveller==9 X(i-2,j+1)=6; elseif Traveller==13 X(i-2,j+2)=6; else X(i-1,j+2)=6; end elseif X(i-1,j+1)==0 && (Traveller==9 || Traveller==13 || Traveller==14) if Traveller==9 X(i-2,j+1)=6; X(i-1,j)=6; elseif Traveller==13 X(i-2,j+2)=6; X(i-2,j+1)=6; else X(i-1,j+2)=6; X(i,j+1)=6;

305

end end end end elseif i==n-1 && j>=3 && j<=n-2 Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j) X(i-2,j+1) X(i-2,j+2); X(i-1,j-2) X(i-1,j-1) X(i-1,j) X(i-1,j+1) ... X(i-1,j+2); X(i,j-2) X(i,j-1) X(i,j) X(i,j+1) X(i,j+2); X(i+1,j-2) X(i+1,j-1) X(i+1,j) X(i+1,j+1) X(i+1,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 if Traveller==6 || Traveller==7 || Traveller==8 || Traveller==10 || Traveller==12 || ... Traveller==14 || Traveller==15 || Traveller==16 X(i,j)=6; if Traveller==6 X(i-1,j-1)=6; elseif Traveller==7 X(i,j-1)=6; elseif Traveller==8 X(i+1,j-1)=6; elseif Traveller==10 X(i-1,j)=6; elseif Traveller==12 X(i+1,j)=6; elseif Traveller==14 X(i-1,j+1)=6; elseif Traveller==15 X(i,j+1)=6; else X(i+1,j+1)=6; end elseif Traveller==1 || Traveller==2 || Traveller==3 || Traveller==4 || Traveller==5 || ... Traveller==9 || Traveller==13 || (Traveller<=17 && Traveller<=20) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==3 && X(i,j-1)==0 X(i,j-2)=6; X(i-1,j-1)=6; elseif Traveller==4 && X(i+1,j-1)~=0 X(i+1,j-2)=6; X(i+1,j-1)=6; elseif Traveller==4 && X(i+1,j-1)==0 X(i+1,j-2)=6; X(i,j-1)=6; elseif Traveller==9 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif Traveller==9 && X(i-1,j)==0 X(i-2,j)=6; X(i-1,j-1)=6; elseif Traveller==19 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif Traveller==19 && X(i,j+1)==0 X(i,j+2)=6; X(i-1,j+1)=6; elseif Traveller==20 && X(i+1,j+1)~=0

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X(i+1,j+2)=6; X(i+1,j+1)=6; elseif Traveller==20 && X(i+1,j+1)==0 X(i+1,j+2)=6; X(i,j+1)=6; elseif X(i-1,j+1)~=0 && (Traveller==13 || Traveller==17 || Traveller==18) X(i-1,j+1)=6; if Traveller==13 X(i-2,j+1)=6; elseif Traveller==17 X(i-2,j+2)=6; else X(i-1,j+2)=6; end elseif X(i-1,j+1)==0 && (Traveller==13 || Traveller==17 || Traveller==18) if Traveller==13 X(i-2,j+1)=6; X(i-1,j)=6; elseif Traveller==17 X(i-2,j+2)=6; X(i-2,j+1)=6; else X(i-1,j+2)=6; X(i,j+1)=6; end elseif X(i-1,j-1)~=0 && (Traveller==1 || Traveller==2 || Traveller==5) X(i-1,j-1)=6; if Traveller==1 X(i-2,j-2)=6; elseif Traveller==2 X(i-2,j-2)=6; else X(i-2,j-1)=6; end elseif X(i-1,j-1)==0 && (Traveller==1 || Traveller==2 || Traveller==5) if Traveller==1 X(i-2,j-2)=6; X(i-2,j-1)=6; elseif Traveller==2 X(i-1,j-2)=6; X(i,j-1)=6; else X(i-2,j-1)=6; X(i-1,j)=6; end end end end elseif i==n-1 && j==n-1 Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j) X(i-2,j+1); X(i-1,j-2) X(i-1,j-1) X(i-1,j) X(i-1,j+1); ... X(i,j-2) X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-2) X(i+1,j- 1) X(i+1,j) X(i+1,j+1)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 if (Traveller==6 || Traveller==7 || Traveller==8 || Traveller==10 || ...

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Traveller==12 || Traveller==14 || Traveller==15 || Traveller==16) X(i,j)=6; if Traveller==6 X(i-1,j-1)=6; elseif Traveller==7 X(i,j-1)=6; elseif Traveller==8 X(i+1,j-1)=6; elseif Traveller==10 X(i-1,j)=6; elseif Traveller==12 X(i+1,j)=6; elseif Traveller==14 X(i-1,j+1)=6; elseif Traveller==15 X(i,j+1)=6; else X(i+1,j+1)=6; end elseif Traveller==9 || Traveller==13 || (Traveller>=1 && Traveller<=5) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==3 && X(i,j-1)==0 X(i,j-2)=6; X(i+1,j-1)=6; elseif Traveller==4 && X(i+1,j-1)~=0 X(i+1,j-2)=6; X(i+1,j-1)=6; elseif Traveller==4 && X(i+1,j-1)==0 X(i+1,j-2)=6; X(i,j-1)=6; elseif Traveller==9 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif Traveller==9 && X(i-1,j)==0 X(i-2,j)=6; X(i-1,j+1)=6; elseif Traveller==13 && X(i-1,j+1)~=0 X(i-2,j+1)=6; X(i-1,j+1)=6; elseif Traveller==13 && X(i-1,j+1)==0 X(i-2,j+1)=6; X(i-1,j)=6; elseif X(i-1,j-1)~=0 && (Traveller==1 || Traveller==2 || Traveller==5) X(i-1,j-1)=6; if Traveller==1 X(i-2,j-2)=6; elseif Traveller==2 X(i-1,j-2)=6; else X(i-2,j-1)=6; end elseif X(i-1,j-1)==0 && (Traveller==1 || Traveller==2 || Traveller==5) if Traveller==1 X(i-2,j-2)=6; X(i-2,j-1)=6; elseif Traveller==2 X(i-1,j-2)=6; X(i,j-1)=6; else X(i-2,j-1)=6; X(i-1,j)=6;

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end end end end elseif i>=3 && i<=n-2 && j==n-1 Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j) X(i-2,j+1); X(i-1,j-2) X(i-1,j-1) X(i-1,j) X(i-1,j+1); ... X(i,j-2) X(i,j-1) X(i,j) X(i,j+1); X(i+1,j-2) X(i+1,j- 1) X(i+1,j) X(i+1,j+1); X(i+2,j-2)... X(i+2,j-1) X(i+2,j) X(i+2,j+1)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break elseif T==0 && X(i,j)==3 if (Traveller==7 || Traveller==8 || Traveller==9 || Traveller==12 || ... Traveller==14 || Traveller==17 || Traveller==18 || Traveller==19) X(i,j)=6; if Traveller==7 X(i-1,j-1)=6; elseif Traveller==8 X(i,j-1)=6; elseif Traveller==9 X(i+1,j-1)=6; elseif Traveller==12 X(i-1,j)=6; elseif Traveller==14 X(i+1,j)=6; elseif Traveller==17 X(i-1,j+1)=6; elseif Traveller==18 X(i,j+1)=6; else X(i+1,j+1)=6; end elseif Traveller==10 || Traveller==11 || Traveller==15 || ... Traveller==16|| Traveller==20 || (Traveller>=1 && Traveller<=6) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==3 && X(i,j-1)==0 X(i,j-2)=6; X(i+1,j-1)=6; elseif Traveller==11 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif Traveller==11 && X(i-1,j)==0 X(i-2,j)=6; X(i-1,j-1)=6; elseif Traveller==15 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif Traveller==15 && X(i+1,j)==0 X(i+2,j)=6; X(i+1,j+1)=6; elseif Traveller==16 && X(i+1,j+1)~=0 X(i-2,j+1)=6; X(i+1,j+1)=6; elseif Traveller==16 && X(i+1,j+1)==0

309

X(i-2,j+1)=6; X(i-1,j)=6; elseif Traveller==20 && X(i+1,j+1)~=0 X(i+2,j+1)=6; X(i+1,j+1)=6; elseif Traveller==20 && X(i+1,j+1)==0 X(i+2,j+1)=6; X(i+1,j)=6; elseif X(i-1,j-1)~=0 && (Traveller==1 || Traveller==2 || Traveller==6) X(i-1,j-1)=6; if Traveller==1 X(i-2,j-2)=6; elseif Traveller==2 X(i-1,j-2)=6; else X(i-2,j-1)=6; end elseif X(i-1,j-1)==0 && (Traveller==1 || Traveller==2 || Traveller==6) if Traveller==1 X(i-2,j-2)=6; X(i-1,j-2)=6; elseif Traveller==2 X(i,j-1)=6; X(i-1,j-2)=6; else X(i-2,j-1)=6; X(i-1,j)=6; end elseif X(i+1,j-1)~=0 && (Traveller==4 || Traveller==5 || Traveller==10) X(i+1,j-1)=6; if Traveller==4 X(i+1,j-2)=6; elseif Traveller==5 X(i+2,j-2)=6; else X(i+2,j-1)=6; end elseif X(i+1,j-1)==0 && (Traveller==4 || Traveller==5 || Traveller==10) if Traveller==4 X(i+1,j-2)=6; X(i,j-1)=6; elseif Traveller==5 X(i+2,j-2)=6; X(i+2,j-1)=6; else X(i+2,j-1)=6; X(i+1,j)=6; end end end end elseif i>=3 && i<=n-2 && j<=3 && j<=n-2 Neighborhood=[X(i-2,j-2) X(i-2,j-1) X(i-2,j) X(i-2,j+1) X(i-2,j+2); X(i-1,j-2) X(i-1,j-1) X(i-1,j) ... X(i-1,j+1) X(i-1,j+2); X(i,j-2) X(i,j-1) X(i,j) X(i,j+1) X(i,j+2); X(i+1,j-2) X(i+1,j-1) ... X(i+1,j) X(i+1,j+1) X(i+1,j+2); X(i+2,j-2) X(i+2,j-1) X(i+2,j) X(i+2,j+1) X(i+2,j+2)]; Traveller=find(5 < Neighborhood & Neighborhood < 10,1); T=isempty(Traveller); if T==1 break

310

elseif T==0 && X(i,j)==3 if (Traveller==7 || Traveller==8 || Traveller==9 || Traveller==12 || ... Traveller==14 || Traveller==17 || Traveller==18 || Traveller==19) X(i,j)=6; if Traveller==7 X(i-1,j-1)=6; elseif Traveller==8 X(i,j-1)=6; elseif Traveller==9 X(i+1,j-1)=6; elseif Traveller==12 X(i-1,j)=6; elseif Traveller==14 X(i+1,j)=6; elseif Traveller==17 X(i-1,j+1)=6; elseif Traveller==18 X(i,j+1)=6; else X(i+1,j+1)=6; end elseif Traveller==10 || Traveller==11 || Traveller==15 || Traveller==16 ||... (Traveller>=20 && Traveller<=25) || (Traveller>=1 && Traveller<=6) X(i,j)=3; nG(i,j)=nG(i,j)+1; if Traveller==3 && X(i,j-1)~=0 X(i,j-2)=6; X(i,j-1)=6; elseif Traveller==3 && X(i,j-1)==0 X(i,j-2)=6; X(i+1,j-1)=6; elseif Traveller==11 && X(i-1,j)~=0 X(i-2,j)=6; X(i-1,j)=6; elseif Traveller==11 && X(i-1,j)==0 X(i-2,j)=6; X(i-1,j-1)=6; elseif Traveller==15 && X(i+1,j)~=0 X(i+2,j)=6; X(i+1,j)=6; elseif Traveller==15 && X(i+1,j)==0 X(i+2,j)=6; X(i+1,j+1)=6; elseif Traveller==23 && X(i,j+1)~=0 X(i,j+2)=6; X(i,j+1)=6; elseif Traveller==23 && X(i+1,j+1)==0 X(i,j+2)=6; X(i-1,j+1)=6; elseif X(i-1,j-1)~=0 && (Traveller==1 || Traveller==2 || Traveller==6) X(i-1,j-1)=6; if Traveller==1 X(i-2,j-2)=6; elseif Traveller==2 X(i-1,j-2)=6; else X(i-2,j-1)=6; end elseif X(i-1,j-1)==0 && (Traveller==1 || Traveller==2 || Traveller==6) if Traveller==1

311

X(i-2,j-2)=6; X(i-1,j-2)=6; elseif Traveller==2 X(i-1,j-2)=6; X(i,j-1)=6; else X(i-2,j-1)=6; X(i-1,j)=6; end elseif X(i+1,j-1)~=0 && (Traveller==4 || Traveller==5 || Traveller==10) X(i+1,j-1)=6; if Traveller==4 X(i+1,j-2)=6; elseif Traveller==5 X(i+2,j-2)=6; else X(i+2,j-1)=6; end elseif X(i+1,j-1)==0 && (Traveller==4 || Traveller==5 || Traveller==10) if Traveller==4 X(i+1,j-2)=6; X(i,j-1)=6; elseif Traveller==5 X(i+2,j-2)=6; X(i+2,j-1)=6; else X(i+2,j-1)=6; X(i+1,j)=6; end elseif X(i-1,j+1)~=0 && (Traveller==16 || Traveller==21 || Traveller==22) X(i-1,j+1)=6; if Traveller==16 X(i-2,j+1)=6; elseif Traveller==21 X(i-2,j+2)=6; else X(i-1,j+2)=6; end elseif X(i-1,j+1)==0 && (Traveller==16 || Traveller==21 || Traveller==22) if Traveller==16 X(i-2,j+1)=6; X(i-1,j)=6; elseif Traveller==21 X(i-2,j+2)=6; X(i-2,j+1)=6; else X(i-1,j+2)=6; X(i,j+1)=6; end elseif X(i+1,j+1)~=0 && (Traveller==20 || Traveller==24 || Traveller==25) X(i+1,j+1)=6; if Traveller==20 X(i+2,j+1)=6; elseif Traveller==24 X(i+1,j+2)=6; else X(i+2,j+2)=6; end elseif X(i+1,j+1)==0 && (Traveller==20 || Traveller==24 || Traveller==25) if Traveller==20

312

X(i+2,j+1)=6; X(i+1,j)=6; elseif Traveller==24 X(i+1,j+2)=6; X(i,j+1)=6; else X(i+2,j+2)=6; X(i+1,j+2)=6; end end end end end end end for i=1:n for j=1:n if X(i,j)==3 && nG(i,j)>=16 X(i,j)=6; nG(i,j)=0; else end end end

313

Subroutine – “Osteocyte” function [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nWBCP,nOst,nP] = osteocyte(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nWBCP,nOst,nP) % % Pre-Osteocyte I = 11, nc1 % Pre-Osteocyte II = 11, nc2 % Old Osteocyte = 13, nc4 % Dead Osteocyte = 14 % Young Osteocyte = 12, nc4 % % syms b1 b2 b3 b4 b5 syms c1 c2 c3 c4 c5 syms BV WBC P V G syms Z a=0; b=1000; YY=round(a+(b-a)*rand(n)); for i=1:n for j=1:n if X(i,j)>=11 && X(i,j)<=13 && YY(i,j)==10 %Random Cell Death X(i,j)=14; nc5(i,j)=nc5(i,j)+1; [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5] = CleanNeighborhood(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5); [X,nWBCP,nb5,nc5,nP] = preosteoclast(X,n,nWBCP,nb5,nc5,nP); [X,nWBCP,nb5,nc5] = OsteoclastFinal(X,n,nWBCP,nb5,nc5); else if X(i,j)==11 && nc1(i,j)< 8760 X(i,j)=11; nc1(i,j)=nc1(i,j)+1; elseif X(i,j)==11 && nc1(i,j)>=8760 X(i,j)=11; nc2(i,j)=nc2(i,j)+1; elseif X(i,j)==11 && nc2(i,j)< 8760 X(i,j)=11; nc2(i,j)=nc2(i,j)+1; elseif X(i,j)==11 && nc2(i,j)>=8760 X(i,j)=12; nc3(i,j)=nc3(i,j)+1; elseif X(i,j)==12 && nc3(i,j)< 8760 X(i,j)=12; nc3(i,j)=nc3(i,j)+1; elseif X(i,j)==12 && nc3(i,j)>=8760 X(i,j)=13; nc4(i,j)=nc4(i,j)+1; elseif X(i,j)==13 && nc4(i,j)< 8760 X(i,j)=13; nc4(i,j)=nc4(i,j)+1; else end end end

314 end Subroutine – “Cell Death” function [X,nb1,nb2,nb3,nb4,nc1,nc2,nc3,nc4,nb5,nc5,nWBCP,nP] = celldeath(X,n,nb1,nb2,nb3,nb4,nc1,nc2,nc3,nc4,nb5,nc5,nWBCP,nP) % %If any osteoblast is older than 300 days (7200 generations), it dies % %Blood Vessel = 0 %b1 = 6 b2 = 7 b3 = 8 b4 = 9 %c1 = 11 c2 = 11 c3 = 12 c4 = 13 %Dead Osteoblast = 14 %Dead Osteocyte = 14 %Void = 15 % for i=1:n for j=1:n if X(i,j)==0 %If blood vessel, it stays blood vessel X(i,j)=0; elseif X(i,j)==6 && nb1(i,j)>=7200 X(i,j)=14; nb1(i,j)=0; nb5(i,j)=nb5(i,j)+1; elseif X(i,j)==7 && nb2(i,j)>=7200 X(i,j)=14; nb2(i,j)=0; nb5(i,j)=nb5(i,j)+1; elseif X(i,j)==8 && nb3(i,j)>=7200 X(i,j)=14; nb3(i,j)=0; nb5(i,j)=nb5(i,j)+1; elseif X(i,j)==9 && nb4(i,j)>=7200 X(i,j)=14; nb4(i,j)=0; nb5(i,j)=nb5(i,j)+1; elseif X(i,j)==11 && nc1(i,j)>=7200 X(i,j)=14; nc1(i,j)=0; nc5(i,j)=nc5(i,j)+1; elseif X(i,j)==11 && nc2(i,j)>=7200 X(i,j)=14; nc2(i,j)=0; nc5(i,j)=nc5(i,j)+1; elseif X(i,j)==12 && nc3(i,j)>=7200 X(i,j)=14; nc3(i,j)=0; nc5(i,j)=nc5(i,j)+1; elseif X(i,j)==13 && nc4(i,j)>=7200 X(i,j)=14; nc4(i,j)=0; nc5(i,j)=nc5(i,j)+1; end end end

315

Subroutine – “Counter Clear” function [X,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nOst,nG,nWBCP,nM,nMM,nMP,nP] =counterclear(X,n,nb1,nb2,nb3,nb4,nb5,nc1,nc2,nc3,nc4,nc5,nOst,nG,nWBCP ,nM,nMM,nMP,nP,g) % %Make sure that only the appropriate counter is counting the value of each %cell % for i=1:n for j=1:n if X(i,j)==6 nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nO(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==7 nb1(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==8 nb2(i,j)=0; nb1(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0;

316

nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==9 nb2(i,j)=0; nb3(i,j)=0; nb1(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==11 && nc1(i,j)>=1 nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nb1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==11 && nc2(i,j)>=1 && nc1(i,j)==0 nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nb1(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==12 nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0;

317

nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nb1(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==13 nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nb1(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==14 && nb5(i,j)>=1 nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb1(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==14 && nc5(i,j)>=1 nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nb1(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0;

318

nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==1 nWBCP(i,j)=nWBCP(i,j)+1; nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nb1(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==2 % nO(i,j)=nO(i,j)+1; nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nb1(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==3 % nG(i,j)=nG(i,j)+1; nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nb1(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==4 nM(i,j)=nM(i,j)+1; nb2(i,j)=0;

319

nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nb1(i,j)=0; nMM(i,j)=0; nMP(i,j)=0; elseif X(i,j)==5 nMM(i,j)=nMM(i,j)+1; nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nb1(i,j)=0; elseif X(i,j)==10 % nP(i,j)=nP(i,j)+1; nb2(i,j)=0; nb3(i,j)=0; nb4(i,j)=0; nb5(i,j)=0; nc1(i,j)=0; nc2(i,j)=0; nc3(i,j)=0; nc4(i,j)=0; nc5(i,j)=0; nOst(i,j)=0; nG(i,j)=0; nWBCP(i,j)=0; nM(i,j)=0; nMM(i,j)=0; nb1(i,j)=0; else end end end

320

Subroutine – “Zconvert” function [Z,X,nc1,nc2,nb5,nc5,nMM,nMP] = Zconvert(X,n,nc1,nc2,nb5,nc5,nMM,nMP) % % Program designed to correctly assign values to the symbolic matrix Z % after all the main program functions have been executed. % syms b1 b2 b3 b4 b5 syms c1 c2 c3 c4 c5 syms M MM MP syms O G P syms BV WBC syms V Z

[X] = BloodVessels(X,n); for i=1:n for j=1:n % Blood Vessel if X(i,j)==0 X(i,j)=0; Z(i,j)=BV; %Void elseif X(i,j)==15 X(i,j)=15; Z(i,j)=V; %Pre-Osteoblast I elseif X(i,j)==6 X(i,j)=6; Z(i,j)=b1; %Pre-Osteoblast II elseif X(i,j)==7 X(i,j)=7; Z(i,j)=b2; %Osteoblast I elseif X(i,j)==8 X(i,j)=8; Z(i,j)=b3; %Osteoblast II elseif X(i,j)==9 X(i,j)=9; Z(i,j)=b4; %Dead Osteoblast elseif X(i,j)==14 && nb5(i,j)>=1 X(i,j)=14; Z(i,j)=b5; %Pre-Osteocyte I elseif X(i,j)==11 && nc1(i,j)>=1 X(i,j)=11; Z(i,j)=c1; %Pre-Osteocyte II elseif X(i,j)==11 && nc2(i,j)>=1 X(i,j)=11; Z(i,j)=c2; %Young Osteocyte

321

elseif X(i,j)==12 X(i,j)=12; Z(i,j)=c3; %Old Osteocyte elseif X(i,j)==13 X(i,j)=13; Z(i,j)=c4; %Dead Osteocyte elseif X(i,j)==14 && nc5(i,j)>=1 X(i,j)=14; Z(i,j)=c5; %Matrix elseif X(i,j)==4 X(i,j)=4; Z(i,j)=M; %Matrix + Protein elseif X(i,j)==5 && nMM(i,j)>=1 X(i,j)=5; Z(i,j)=MM; %Mineralized Matrix elseif X(i,j)==5 && nMP(i,j)>=1 && nMM(i,j)==0 X(i,j)=5; Z(i,j)=MP; %Growth Factor elseif X(i,j)==3 X(i,j)=3; Z(i,j)=G; %Protein elseif X(i,j)==10 X(i,j)=10; Z(i,j)=P; %Osteoclast elseif X(i,j)==2 X(i,j)=2; Z(i,j)=O; elseif X(i,j)==1 X(i,j)=1; Z(i,j)=WBC; end end end

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Appendix B: Cellular Automation Output Export to Excel

Program Results: Run #1

Time Birth Death Void Protein Growth Matrix Clasts 0 8 0 265 0 0 0 0 1 8 0 265 0 0 0 0 2 8 0 265 0 0 0 0 3 8 0 265 0 0 0 0 4 8 0 265 0 0 0 0 5 8 0 265 0 0 0 0 6 8 0 265 0 0 0 0 7 8 0 265 0 0 0 0 8 8 0 265 0 0 0 0 9 8 0 265 0 0 0 0 10 8 0 265 0 0 0 0 11 8 0 265 0 0 0 0 12 8 0 265 0 0 0 0 13 8 0 265 0 0 0 0 14 8 0 265 0 0 0 0 15 8 0 265 0 0 0 0 16 8 0 265 0 0 0 0 17 8 0 265 0 0 0 0 18 8 0 265 0 0 0 0 19 8 0 265 0 0 0 0 20 8 0 265 0 0 0 0 21 8 0 265 0 0 0 0 22 8 0 265 0 0 0 0 23 8 0 265 0 0 0 0 24 8 0 265 0 0 0 0 25 8 0 265 0 0 0 0 26 8 0 265 0 0 0 0 27 8 0 265 0 0 0 0 28 8 0 265 0 0 0 0 29 8 0 265 0 0 0 0 30 8 0 265 0 0 0 0 31 8 0 265 0 0 0 0 32 10 0 259 0 0 3 1

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33 24 0 227 0 0 21 1 34 24 0 227 0 0 21 1 35 23 0 227 0 0 21 2 36 23 0 227 0 0 21 2 37 23 0 227 0 0 21 2 38 23 0 227 0 0 21 2 39 23 0 227 0 0 21 2 40 23 0 228 0 1 20 1 41 23 0 228 0 1 20 1 42 20 0 228 0 1 23 1 43 20 0 228 0 2 23 0 44 20 0 228 0 2 23 0 45 20 0 228 0 2 23 0 46 20 0 228 0 2 23 0 47 20 0 228 0 2 23 0 48 20 0 228 0 2 23 0 49 20 0 228 0 2 23 0 50 20 0 228 0 2 23 0 51 20 0 228 0 2 23 0 52 20 0 228 0 2 23 0 53 20 0 228 0 2 23 0 54 20 0 228 0 2 23 0 55 21 0 228 0 1 23 0 56 21 0 228 0 1 23 0 57 21 0 228 0 1 23 0 58 18 0 228 0 0 26 1 59 18 0 227 0 0 26 2 60 20 0 219 0 0 31 3 61 20 0 219 0 0 31 3 62 20 0 219 0 0 31 3 63 20 0 219 0 0 31 3 64 20 0 219 0 0 31 3 65 20 0 219 0 0 31 3 66 22 0 212 0 1 34 4 67 22 0 213 0 1 34 3 68 22 0 213 0 2 34 2 69 21 0 213 0 2 35 2 70 21 0 213 0 2 35 2 71 21 0 213 0 2 35 2 72 21 0 213 0 2 35 2 73 21 0 213 0 2 35 2 74 21 0 213 0 4 35 0 75 20 0 213 0 4 36 0

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76 18 0 214 0 4 36 1 77 18 0 214 0 4 36 1 78 18 0 214 0 4 36 1 79 18 0 214 0 4 36 1 80 18 0 214 0 4 36 1 81 19 0 214 0 3 36 1 82 19 0 214 0 3 36 1 83 20 0 214 0 2 36 1 84 21 0 214 0 2 36 0 85 19 0 214 0 2 38 0 86 19 0 214 0 2 38 0 87 19 0 214 0 2 38 0 88 19 0 214 0 2 38 0 89 21 0 214 0 0 38 0 90 21 0 214 0 0 38 0 91 21 0 214 0 0 38 0 92 21 0 208 0 0 43 1 93 20 0 208 0 0 43 2 94 20 0 208 0 0 43 2 95 20 0 207 0 0 44 2 96 19 0 207 0 0 44 3 97 19 0 207 0 0 44 3 98 19 0 207 0 0 44 3 99 19 0 207 0 0 44 3

325

Program Results: Run #2

Time Birth Death Void Protein Growth Matrix Clasts 0 7 0 266 0 0 0 0 1 7 0 266 0 0 0 0 2 7 0 266 0 0 0 0 3 7 0 266 0 0 0 0 4 7 0 266 0 0 0 0 5 7 0 266 0 0 0 0 6 7 0 266 0 0 0 0 7 7 0 266 0 0 0 0 8 7 0 266 0 0 0 0 9 7 0 266 0 0 0 0 10 7 0 266 0 0 0 0 11 7 0 266 0 0 0 0 12 7 0 266 0 0 0 0 13 7 0 266 0 0 0 0 14 7 0 266 0 0 0 0 15 7 0 266 0 0 0 0 16 7 0 266 0 0 0 0 17 7 0 266 0 0 0 0 18 7 0 266 0 0 0 0 19 7 0 266 0 0 0 0 20 7 0 266 0 0 0 0 21 7 0 266 0 0 0 0 22 7 0 266 0 0 0 0 23 7 0 266 0 0 0 0 24 7 0 266 0 0 0 0 25 7 0 266 0 0 0 0 26 7 0 266 0 0 0 0 27 7 0 266 0 0 0 0 28 7 0 266 0 0 0 0 29 7 0 266 0 0 0 0 30 7 0 266 0 0 0 0 31 7 0 266 0 0 0 0 32 15 0 250 0 0 8 0 33 19 0 241 0 0 13 0 34 19 0 241 0 0 13 0 35 19 0 241 0 0 13 0 36 19 0 241 0 0 13 0 37 19 0 241 0 0 13 0 38 19 0 241 0 0 13 0 39 19 0 241 0 0 13 0

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40 19 0 241 0 0 13 0 41 20 0 239 0 0 14 0 42 20 0 239 0 0 14 0 43 20 0 239 0 0 14 0 44 20 0 239 0 0 14 0 45 20 0 239 0 0 14 0 46 20 0 239 0 0 14 0 47 20 0 239 0 0 14 0 48 20 0 239 0 0 14 0 49 20 0 239 0 0 14 0 50 20 0 239 0 0 14 0 51 21 0 237 0 0 15 0 52 21 0 237 0 0 15 0 53 21 0 237 0 0 15 0 54 21 0 237 0 0 15 0 55 21 0 237 0 0 15 0 56 21 0 237 0 0 15 0 57 19 0 237 0 0 17 0 58 19 0 237 0 0 17 0 59 19 0 237 0 0 17 0 60 19 0 237 0 0 17 0 61 19 0 237 0 0 17 0 62 19 0 237 0 0 17 0 63 19 0 237 0 0 17 0 64 19 0 237 0 0 17 0 65 26 0 221 0 0 25 1 66 26 0 221 0 0 25 1 67 39 0 178 0 0 54 2 68 39 0 178 0 0 54 2 69 39 0 178 0 0 54 2 70 39 0 178 0 0 54 2 71 39 0 178 0 0 54 2 72 38 0 179 0 0 53 3 73 38 0 180 0 1 52 2 74 36 0 180 0 1 54 2 75 33 0 180 0 2 57 1 76 27 0 180 0 1 64 1 77 26 0 180 0 1 64 2 78 26 0 180 0 1 64 2 79 26 0 180 0 1 64 2 80 26 0 180 0 2 64 1 81 26 0 180 0 2 64 1 82 25 0 180 0 2 65 1

327

83 25 0 180 0 2 65 1 84 25 0 180 0 2 65 1 85 24 0 180 0 3 66 0 86 24 0 180 0 3 66 0 87 24 0 180 0 3 66 0 88 24 0 180 0 3 66 0 89 24 0 180 0 3 66 0 90 25 0 180 0 2 66 0 91 25 0 180 0 2 66 0 92 24 0 180 0 2 67 0 93 24 0 180 0 2 67 0 94 24 0 180 0 2 67 0 95 25 0 180 0 1 67 0 96 25 0 180 0 1 67 0 97 25 0 180 0 1 67 0 98 25 0 180 0 1 67 0 99 25 0 180 0 1 66 1 100 38 1 121 0 1 111 1

328

Program Results: Run #3

Time Birth Death Void Protein Growth Matrix Clasts 0 10 0 263 0 0 0 0 1 10 0 263 0 0 0 0 2 10 0 263 0 0 0 0 3 10 0 263 0 0 0 0 4 10 0 263 0 0 0 0 5 10 0 263 0 0 0 0 6 10 0 263 0 0 0 0 7 10 0 263 0 0 0 0 8 10 0 263 0 0 0 0 9 10 0 263 0 0 0 0 10 10 0 263 0 0 0 0 11 10 0 263 0 0 0 0 12 10 0 263 0 0 0 0 13 10 0 263 0 0 0 0 14 10 0 263 0 0 0 0 15 10 0 263 0 0 0 0 16 10 0 263 0 0 0 0 17 10 0 263 0 0 0 0 18 10 0 263 0 0 0 0 19 10 0 263 0 0 0 0 20 10 0 263 0 0 0 0 21 10 0 263 0 0 0 0 22 10 0 263 0 0 0 0 23 10 0 263 0 0 0 0 24 10 0 263 0 0 0 0 25 10 0 263 0 0 0 0 26 10 0 263 0 0 0 0 27 10 0 263 0 0 0 0 28 10 0 263 0 0 0 0 29 10 0 263 0 0 0 0 30 10 0 263 0 0 0 0 31 10 0 263 0 0 0 0 32 20 0 240 0 0 13 0 33 29 0 221 0 0 23 0 34 29 0 221 0 0 23 0 35 37 0 202 0 0 34 0 36 37 0 202 0 0 34 0 37 37 0 202 0 0 34 0 38 36 0 202 0 0 34 1 39 36 0 202 0 0 34 1

329

40 36 0 202 0 0 34 1 41 35 0 202 0 0 35 1 42 35 0 202 0 0 35 1 43 30 0 202 0 0 40 1 44 26 0 202 0 0 44 1 45 25 0 202 0 0 45 1 46 25 0 202 0 1 45 0 47 25 0 202 0 1 45 0 48 25 0 202 0 1 45 0 49 25 0 202 0 1 45 0 50 22 1 203 0 1 45 1 51 22 0 204 0 1 45 1 52 21 0 204 0 1 46 1 53 21 0 204 0 1 46 1 54 21 0 204 0 1 46 1 55 21 0 204 0 1 46 1 56 21 0 204 0 1 46 1 57 21 0 204 0 1 46 1 58 20 0 204 0 1 47 1 59 20 0 204 0 2 47 0 60 20 0 204 0 2 47 0 61 21 0 204 0 1 47 0 62 21 0 204 0 1 47 0 63 21 0 204 0 1 47 0 64 21 0 204 0 1 47 0 65 16 0 200 0 1 52 4 66 20 0 189 0 1 58 5 67 19 0 189 0 1 58 6 68 17 0 191 0 1 57 7 69 17 0 191 0 1 57 7 70 17 0 191 0 1 57 7 71 17 0 191 0 1 57 7 72 17 0 191 0 1 57 7 73 15 0 193 0 5 57 3 74 13 0 193 0 5 60 2 75 13 0 193 0 6 60 1 76 13 0 193 0 7 60 0 77 14 0 191 0 7 61 0 78 14 0 191 0 7 61 0 79 14 0 191 0 7 61 0 80 14 0 191 0 7 61 0 81 14 0 191 0 7 61 0 82 14 0 191 0 7 61 0

330

83 14 0 191 0 7 61 0 84 14 0 191 0 7 61 0 85 14 0 191 0 7 61 0 86 14 0 191 0 7 61 0 87 14 0 191 0 7 61 0 88 18 0 191 0 3 61 0 89 19 0 191 0 2 61 0 90 20 0 191 0 1 61 0 91 21 0 191 0 0 61 0 92 21 0 191 0 0 61 0 93 21 0 191 0 0 61 0 94 21 0 189 0 0 62 1 95 21 0 189 0 0 62 1 96 21 0 189 0 0 62 1 97 21 0 189 0 0 62 1 98 22 0 185 0 0 64 2 99 23 0 177 0 0 70 3 100 38 1 121 0 1 111 1

331

Program Results: Run #4

Time Birth Death Void Protein Growth Matrix Clasts 0 11 0 262 0 0 0 0 1 11 0 262 0 0 0 0 2 11 0 262 0 0 0 0 3 11 0 262 0 0 0 0 4 11 0 262 0 0 0 0 5 11 0 262 0 0 0 0 6 11 0 262 0 0 0 0 7 11 0 262 0 0 0 0 8 11 0 262 0 0 0 0 9 11 0 262 0 0 0 0 10 11 0 262 0 0 0 0 11 11 0 262 0 0 0 0 12 11 0 262 0 0 0 0 13 11 0 262 0 0 0 0 14 11 0 262 0 0 0 0 15 11 0 262 0 0 0 0 16 11 0 262 0 0 0 0 17 11 0 262 0 0 0 0 18 11 0 262 0 0 0 0 19 11 0 262 0 0 0 0 20 11 0 262 0 0 0 0 21 11 0 262 0 0 0 0 22 11 0 262 0 0 0 0 23 11 0 262 0 0 0 0 24 11 0 262 0 0 0 0 25 11 0 262 0 0 0 0 26 11 0 262 0 0 0 0 27 11 0 262 0 0 0 0 28 11 0 262 0 0 0 0 29 11 0 262 0 0 0 0 30 11 0 262 0 0 0 0 31 11 0 262 0 0 0 0 32 9 0 258 0 0 2 4 33 15 0 243 0 0 10 5 34 15 0 243 0 0 10 5 35 14 0 243 0 0 10 6 36 14 0 243 0 0 10 6 37 14 0 243 0 0 10 6 38 14 0 243 0 0 10 6 39 14 0 243 0 0 10 6

332

40 14 0 243 0 4 10 2 41 14 0 243 0 5 10 1 42 13 0 243 0 5 11 1 43 13 0 243 0 6 11 0 44 13 0 243 0 6 11 0 45 13 0 243 0 6 11 0 46 13 0 243 0 6 11 0 47 13 0 243 0 6 11 0 48 13 0 243 0 6 11 0 49 13 0 243 0 6 11 0 50 13 0 243 0 6 11 0 51 13 0 243 0 6 11 0 52 13 0 243 0 6 11 0 53 13 0 243 0 6 11 0 54 13 0 243 0 6 11 0 55 17 0 243 0 2 11 0 56 18 0 243 0 1 11 0 57 18 0 243 0 1 11 0 58 16 0 243 0 0 14 0 59 16 0 243 0 0 14 0 60 16 0 243 0 0 14 0 61 16 0 243 0 0 14 0 62 16 0 243 0 0 14 0 63 16 0 243 0 0 14 0 64 16 0 243 0 0 14 0 65 17 0 241 0 0 15 0 66 19 0 229 1 0 20 3 67 19 0 229 1 0 20 3 68 19 0 229 1 0 20 3 69 19 0 229 1 0 20 3 70 19 0 229 1 0 20 3 71 19 0 227 1 0 22 3 72 19 0 227 1 0 22 3 73 19 0 227 1 0 22 3 74 20 0 227 1 3 21 0 75 17 0 227 1 3 24 0 76 17 0 227 1 3 24 0 77 17 0 227 1 3 24 0 78 17 0 227 1 3 24 0 79 15 0 227 1 3 26 0 80 15 0 227 1 3 26 0 81 15 0 227 1 3 26 0 82 15 0 227 1 3 26 0

333

83 15 0 227 1 3 26 0 84 16 0 227 1 2 26 0 85 16 0 227 1 2 26 0 86 16 0 227 1 2 26 0 87 16 0 227 1 2 26 0 88 18 0 223 1 2 28 0 89 25 0 213 1 0 33 0 90 33 0 192 1 0 46 0 91 32 0 192 1 0 47 0 92 32 0 192 1 0 47 0 93 32 0 192 1 0 47 0 94 33 0 190 1 0 48 0 95 33 0 190 1 0 48 0 96 33 0 190 1 0 48 0 97 32 0 190 1 0 49 0 98 30 0 190 1 0 51 0

334

Program Results: Run #5

Time Birth Death Void Protein Growth Matrix Clasts 0 7 0 266 0 0 0 0 1 7 0 266 0 0 0 0 2 7 0 266 0 0 0 0 3 7 0 266 0 0 0 0 4 7 0 266 0 0 0 0 5 7 0 266 0 0 0 0 6 7 0 266 0 0 0 0 7 7 0 266 0 0 0 0 8 7 0 266 0 0 0 0 9 7 0 266 0 0 0 0 10 7 0 266 0 0 0 0 11 7 0 266 0 0 0 0 12 7 0 266 0 0 0 0 13 7 0 266 0 0 0 0 14 7 0 266 0 0 0 0 15 7 0 266 0 0 0 0 16 7 0 266 0 0 0 0 17 7 0 266 0 0 0 0 18 7 0 266 0 0 0 0 19 7 0 266 0 0 0 0 20 7 0 266 0 0 0 0 21 7 0 266 0 0 0 0 22 7 0 266 0 0 0 0 23 7 0 266 0 0 0 0 24 7 0 266 0 0 0 0 25 7 0 266 0 0 0 0 26 7 0 266 0 0 0 0 27 7 0 266 0 0 0 0 28 7 0 266 0 0 0 0 29 7 0 266 0 0 0 0 30 7 0 266 0 0 0 0 31 7 0 266 0 0 0 0 32 4 0 264 0 0 1 4 33 4 0 264 0 0 1 4 34 4 0 264 0 0 1 4 35 4 0 264 0 0 1 4 36 4 0 264 0 0 1 4 37 4 0 264 0 0 1 4 38 4 0 264 0 0 1 4 39 5 0 262 0 0 2 4

335

40 6 0 262 0 3 2 0 41 6 0 262 0 3 2 0 42 6 0 262 0 3 2 0 43 6 0 262 0 3 2 0 44 6 0 262 0 3 2 0 45 6 0 262 0 3 2 0 46 6 0 262 0 3 2 0 47 8 0 261 0 2 2 0 48 8 0 261 0 2 2 0 49 8 0 261 0 2 2 0 50 8 0 261 0 2 2 0 51 8 0 261 0 2 2 0 52 8 0 261 0 2 2 0 53 8 0 261 0 2 2 0 54 8 0 261 0 2 2 0 55 10 0 261 0 0 2 0 56 10 0 261 0 0 2 0 57 10 0 261 0 0 2 0 58 10 0 261 0 0 2 0 59 10 0 261 0 0 2 0 60 10 0 261 0 0 2 0 61 10 0 261 0 0 2 0 62 10 0 261 0 0 2 0 63 10 0 261 0 0 2 0 64 11 0 259 0 0 3 0 65 11 0 259 0 0 3 0 66 11 0 259 0 0 3 0 67 11 0 259 0 0 3 0 68 11 0 259 0 0 3 0 69 11 0 259 0 0 3 0 70 11 0 259 0 0 3 0 71 11 0 259 0 0 3 0 72 11 0 259 0 0 3 0 73 12 0 255 0 0 5 1 74 12 0 255 0 0 5 1 75 12 0 255 0 0 5 1 76 12 0 255 0 0 5 1 77 12 0 255 0 0 5 1 78 13 0 254 0 0 5 1 79 13 0 254 0 0 5 1 80 13 0 254 0 0 5 1 81 12 0 254 0 1 5 1 82 11 0 254 0 1 5 2

336

83 11 0 254 0 1 5 2 84 11 0 254 0 1 5 2 85 11 0 254 0 1 5 2 86 11 0 254 0 1 5 2 87 11 0 254 0 1 5 2 88 11 0 254 0 1 5 2 89 10 0 252 0 2 8 1 90 10 0 252 0 3 8 0 91 10 0 252 0 3 8 0 92 10 0 252 0 3 8 0 93 10 0 252 0 3 8 0 94 10 0 252 0 3 8 0 95 9 0 252 0 3 8 1 96 10 0 252 0 2 8 1 97 10 0 252 0 2 8 1 98 9 0 252 0 2 9 1 99 9 0 252 0 2 9 1 100 38 1 121 0 1 111 1

337

Appendix C: Mathematical Biology Equations

Bone Tissue = Cortical Bone Tissue + Cancellous Bone Tissue Bone Tissue = (Cortical Bone Matrix + Cortical Bone Cells + Cortical Bone Lacunae) + (Cancellous Bone Matrix + Cancellous Bone Cells + Cancellous Bone Lacunae)

Total Typical Sample Volume from a Human Femur Typical Femur Radius: 2.34 cm = 23.4 mm Typical Femur Section Thickness: 100 µm = 0.1 mm

After determining the volume of a typical femur sample, it is important to point out that an established density for human bone is:

Because the density is known, and the volume of the sample of femur is established, they can be used to calculate the overall bone mass for that section of bone to be:

In normal human bone, cortical bone (compact bone) makes up approximately 80% of overall bone mass, while cancellous bone (spongy bone) makes up approximately 20% of overall bone mass. Using this statistic, it is now possible to calculate the total mass for each type of bone for the sample volume given:

338

Using these calculations for cortical and cancellous bone mass, the prospective densities of each type of bone can be calculated:

Cancellous (spongy) bone is approximately only 25% as dense as cortical bone, making an equation for cancellous bone to be:

By solving for the cortical density using the previously calculated cortical bone mass, both the cortical and cancellous bone densities can be calculated:

An additional way of calculating ρcancellous is:

Now that overall bone mass, density and volume are known, the two types of bone, cortical and cancellous, can each be broken down into cells, matrix and lacunae, each of which makes its own contribution to bone mass and volume can. In the defined sample of femur bone, there are three major components that make up both the cortical and cancellous portions of the bone: bone cells, extracellular matrix and lacunae. Lacunae are the gaps within bone that house the mature bone cells called osteocytes. They are the major components of bone that contribute to its porosity. Within a normal section of femur, cortical lacunae compose approximately 1.5% of the femur’s volume while cancellous lacunae occupy approximately 2.8%. Using these percentages, the average lacunae volume for both cortical and cancellous bone within the section of femur volume is as follows:

Since the lacunae are the gaps in which bone cells are housed, only 2/3 of their total volume is actually an open space or void and the remaining 1/3 of their volume is occupied by a bone cell. By multiplying the cortical

339 and cancellous lacunae volume each by (2/3) and subtracting that from the overall femur volume, the net bone volume in the femur can be calculated.

This net bone volume can then be used to calculate net, or actual quantities of bone mass and volume, excluding the volume occupied by porosity.

This indicates that the difference in mass, (326.839mg – 317.471mg = 9.368mg) is the mass of cellular fluid and other liquid components within bone.

Knowing that the actual cortical bone mass makes up 80% of total bone mass and that cancellous bone mass makes up the other 20%:

These mass calculations can be used along with the density of bone to find the actual net volume of both the cortical and cancellous portions of bone:

Both cortical and cancellous bone can each be broken down even further into two components: cells and matrix. In cortical bone, the extracellular matrix occupies approximately 87.5% of the tissue while cortical bone cells only occupy 12.5%. This makes the overall volume of these components of cortical bone to be the following:

340

With cancellous bone being more porous, the extracellular matrix only makes up 12.5% of the overall cancellous bone volume and the cells make up approximately 87.5% of the volume.

Knowing that there are approximately 500 cells in every mm3 of cortical bone tissue,

Knowing that there are approximately 60 cells in every mm3 of cancellous bone tissue,

Remembering the definition of density, the mass for each individual cell can also be calculated:

Knowing that the equation for the volume of a sphere is:

This equation can be manipulated in order to find the average radius of both cortical and cancellous bone cells:

341

342

343

Mathematical Calculations in Excel

344

Appendix D: Monte Carlo Formulas: Simulation #1

345

346

347

Appendix E: Monte Carlo Histograms: Simulation #1

8 Hour Histograms:

Simulation: AS4-8Hr Simulation: P25 - 8 Hr 80 80 70 70 60 60 50 50 40 40 30

Frequency 30 Frequency 20 20 10 10 0 0 -118 -46 26 98 170 242 314 386 458 530 602 49 117 185 253 321 389 457 525 593 661 729

Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - 8 Hr Simulation: P120 - 8 Hr 70 80 60 70 50 60 40 50 30 40

Frequency 30 20 Frequency 20 10 10 0 0 -100 -34 32 98 164 230 296 362 428 494 560 145 217 289 361 433 505 577 649 721 793 865 Randomly Generated Number Values Randomly Generated Number Values

348

24 Hour Histograms:

Simulation: AS4 - 24 Hr Simulation: P25 - 24 Hrs 100 120 90 80 100 70 80 60 50 60

40 Frequency Frequency 30 40 20 20 10 0 0

Randomly Generated Number Values Randomly Generated Number Values Simulation: T650 - 24 Hrs Simulation: P120 - 24 Hrs 100 90 120 80 100 70 80 60 50 60

40 Frequency Frequency 40 30 20 20 10 0 0

Randomly Generated Number Values Randomly Generated Number Values

349

48 Hour Histograms:

Simulation: AS4 - 48 Hrs Simulation: P25 - 48 Hrs 70 120

60 100 50 80 40 60

30 Frequency Frequency 40 20 20 10 0

500 100 300 700 900

900 700 100 500 300

28 84

- - - - -

- -

140 196 252 308 364 420 476 532 588 644 700 756

364 308 252 196 140

1100 1300 1900 1500 1700

2100 1500 1900 1700 1300 1100

- - - - -

------Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - 48 Hrs Simulation: P120 - 48 Hrs 90 90 80 80 70 70 60 60 50 50

40 40 Frequency Frequency 30 30 20 20 10 10

0 0

100 300 500 700 900 200 400 600 800

100 800 900 700 500 300 600 400 200

------

1100 1300 1500 1700 2800 1900 2100 2300 2500 2700 1000 1200 1400 1600 1800 2000 2200 2400 2600

1200 1000 1300 1100

- - - - Randomly Generated Number Values Randomly Generated Number Values

350

72 Hour Histograms:

Simulation: AS4 - 72 Hrs Simulation: P25 - 72 Hrs 70 100 90 60 80 50 70 60 40 50

30 40

Frequency Frequency 20 30 20 10 10

0 0

300 900 100 500 700

500 300 900 700 100

- - - - -

103 178 253 328 403 478 553 628 703 778 853 928

422 347 272 197 122

1500 1700 2300 1100 1300 1900 2100

1700 1100 1500 1300

- - - - -

- - - -

1003 1078 Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - 72 Hrs Simulation: P120 - 72 Hrs 80 80 70 70 60 60 50 50

40 40 Frequency Frequency 30 30 20 20 10 10

0 0

500 700 900 100 300 200 400 600 800

800 600 400 700 500 300 100 200

------

1100 1300 1500 1700 1900 2100 2300 3200 2500 2700 2900 3100 3300 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Randomly Generated Number Values Randomly Generated Number Values

351

96 Hour Histograms:

Simulation: AS4 - 96 Hrs Simulation: P25 - 96 Hrs 120 80 70 100 60 80 50

60 40 Frequency Frequency 30 40 20 20 10

0 0

200 400 600 800 200 400 600 800

400 800 800 600 200 600 400 200

------

1000 1600 1800 2800 1200 1400 2000 1000 1200 1400 1600 1800 2000 2200 2400 2600

1800 1600 1000 1200 1000 2000 1400 1200

------Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - 96 Hrs Simulation: P120 - 96 Hrs 80 90 70 80 60 70 60 50 50 40

40 Frequency Frequency 30 30 20 20 10 10

0 0

900 100 400 300 600 700

200 900 600 300

- - - -

1200 1500 1800 2100 2400 2700 3000 3300 3600 5800 3900 4200 4500 4800 5100 1000 1300 1600 1900 2200 2500 2800 3100 3400 3700 4000 4300 4600 4900 5200 5500 Randomly Generated Number Values Randomly Generated Number Values

352

Appendix F: Monte Carlo Formulas: Simulation #2

353

354

Appendix G: Monte Carlo Histograms: Simulation #2

8 Hour Histograms:

Simulation: AS4 - Regression 8 Hr Simulation: P25 - Regression 8 Hr 60 70

50 60 50 40 40 30

30 Frequency Frequency 20 20

10 10

0 0

176 267 358 449 540 631 100 300 500 700 900

643 552 461 370 279 188 916 825 734 900 700 500 300 100

------

2300 1100 1300 1500 1700 1900 2100

1700 1500 1300 1189 1098 1007 1100

------Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - Regression 8 Hrs Simulation: P120 - Regression 8 Hrs 100 70 90 60 80 70 50 60 40 50

40 30

Frequency Frequency 30 20 20 10 10

0 0

400 600 200 800 200 500 800

800 200 600 400 700 400 100

------

1200 1800 2000 3800 1000 1400 1600 2200 1100 1400 1700 2000 2300 2600 2900 3200 3500

1600 1400 2200 1900 1600 1800 1200 1000 1300 1000

------Randomly Generated Number Values Randomly Generated Number Values

355

24 Hour Histograms:

Simulation: AS4 - Regression 24 Hrs Simulation: P25 - Regression 24 Hrs 90 100 80 90 70 80 70 60 60 50 50

40 40

Frequency Frequency 30 30 20 20 10 10 0

600 300 900

300 900 600

- - -

123 218 313 408 503 598 693 788 883 978

352 257 162 922 827 732 637 542 447

1500 1800 2700 1200 2100 2400

3300 2400 1500 1200 3000 2700 2100 1800

------

------Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - Regression 24 Hrs Simulation: P120 - Regression 24 Hrs 90 70

80 60 70 50 60 50 40

40 30 Frequency Frequency 30 20 20 10 10

0 0

100 700 900 600 300 500 300 900

900 100 300 700 500 300 900 600

------

1700 1900 1500 2100 3000 3900 1100 1300 1500 2100 1200 1800 2400 2700 3300 3600

1900 1100 2100 1200 1700 1500 1300 1800 1500

------Randomly Generated Number Values Randomly Generated Number Values

356

48 Hour Histograms:

Simulation: AS4 - Regression 48 Hrs Simulation: P25 - Regression 48 Hrs 60 100 90 50 80 70 40 60 30 50

Frequency Frequency 20 30 20 10 10

0 0

400 100 700

800 500 200

- - -

133 216 299 382 465 548 631 714 797 880 963

531 448 365 282 199 116

1300 2200 2500 3400 1000 1600 1900 2800 3100

2600 1700 2300 2000 1400 1100

------

1046 1129 Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - Regression 48 Hrs Simulation: P120 - Regression 48 Hrs 90 80 80 70 70 60 60 50 50 40

40 Frequency Frequency 30 30 20 20 10 10

0 0

800 100 200 400 600 400 700

800 800 600 400 200 500 200

------

1000 1800 2000 2800 1000 1900 2500 3400 4300 1200 1400 1600 2200 2400 2600 1300 1600 2200 2800 3100 3700 4000

1000 1700 1200 1400 1100

- - - - - Randomly Generated Number Values Randomly Generated Number Values

357

72 Hour Histograms:

Simulation: AS4 - Regression 72 Hrs Simulation: P25 - Regression 72 Hrs 70 100 90 60 80 50 70 60 40 50

30 40

Frequency Frequency 20 30 20 10 10

0 0

400 100 700

800 500 200

- - -

135 230 325 420 515 610 705 800 895 990

435 340 245 150

1300 2200 2500 3400 1000 1600 1900 2800 3100

2600 1700 2300 2000 1400 1100

- - - -

------

1085 1180 1275 1370 1465 Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - Regression 72 Hrs Simulation: P120 - Regression 72 Hrs 90 90 80 80 70 70 60 60 50 50

40 40 Frequency Frequency 30 30 20 20 10 10

0 0

400 800

800 400

- -

600 200 400 800

600 800 400 200

1200 2400 3200 4400 5600 1600 2000 2800 3600 4000 4800 5200

2400 1200 2000 1600

- - - -

1200 1600 2200 2800 1000 1400 1800 2000 2400 2600 3000 3200 Randomly Generated Number Values Randomly Generated Number Values

358

96 Hour Histograms:

Simulation: AS4 - Regression 96 Hrs Simulation: P25 - Regression 96 Hrs 70 70

60 60

50 50

40 40

30 30

Frequency Frequency 20 20

10 10

0 0

300 400 500 600 700 800 900 100 200 200 400 600 800

800 600 400 300 200 100 200

------

1000 1100 1200 3200 1300 1400 1500 1600 1700 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Randomly Generated Number Values Randomly Generated Number Values

Simulation: T650 - Regression 96 Hrs Simulation: P120 - Regression 96 Hrs 90 80 80 70 70 60 60 50 50 40

40 Frequency

Frequency 30 30 20 20 10 10

0 0

100 400 700

500 200 800

- - -

600 800 200 400

600 400 200

1000 1300 2500 2800 4000 4300 1600 1900 2200 3100 3400 3700 4600 4900

1100

- - -

1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 Randomly Generated Number Values Randomly Generated Number Values

359

Appendix H: RAMAN Spectroscopy Data

AS4 – RAMAN Peak Data

0.00029 103410 653.81 1.18681 7 3995.65 Hi.Peak D = 0 0.00133 657.135 5.32832 4 G = 396450 0.00311 660.456 12.4361 2 0.00522 663.774 20.8665 2 Ratio = 2.6084 0.00789 667.089 31.5635 9 La = 1.64852 0.01065 670.401 42.5791 6 0.01400 0.80083 673.71 55.9644 6 1513.05 3199.86 6 0.01700 0.80832 677.015 67.9417 4 1515.69 3229.8 9 0.81459 680.318 82.6307 0.02068 1518.32 3254.85 8 0.02413 0.81871 683.617 96.4328 4 1520.95 3271.31 8 0.82028 686.913 111.677 0.02795 1523.58 3277.56 2 0.03055 0.82148 690.206 122.088 5 1526.2 3282.35 1 0.03327 0.82255 693.496 132.961 6 1528.83 3286.63 2 0.03523 0.82587 696.783 140.788 5 1531.45 3299.9 3 0.03718 0.82861 700.067 148.585 7 1534.07 3310.86 6 0.03877 0.83514 703.348 154.938 7 1536.69 3336.95 6 0.04097 0.84480 706.626 163.731 7 1539.31 3375.53 1 0.04307 0.85468 709.901 172.119 7 1541.93 3415.01 2 0.04613 0.86451 713.173 184.347 7 1544.54 3454.3 5 0.04893 716.441 195.512 1 1547.15 3489.8 0.8734 0.05172 0.88184 719.707 206.685 8 1549.76 3523.54 4 0.05411 0.89238 722.97 216.207 1 1552.37 3565.66 5 726.23 224.969 0.05630 1554.97 3608.39 0.90308

360

3 0.05763 0.91288 729.487 230.303 8 1557.58 3647.56 3 0.05884 0.92244 732.74 235.142 9 1560.18 3685.77 6 0.05989 0.93224 735.991 239.317 4 1562.78 3724.92 4 0.06140 0.93943 739.239 245.35 4 1565.38 3753.66 7 0.94170 742.484 253.925 0.06355 1567.98 3762.74 9 0.06449 0.94232 745.726 257.689 2 1570.57 3765.21 7 0.06565 0.94694 748.965 262.332 4 1573.16 3783.67 7 0.06664 0.95104 752.201 266.276 1 1575.75 3800.05 7 0.06665 0.95343 755.435 266.344 8 1578.34 3809.59 4 0.06615 0.95792 758.665 264.319 2 1580.93 3827.54 7 0.06583 0.96900 761.892 263.063 7 1583.51 3871.79 1 0.06651 0.97872 765.117 265.76 2 1586.1 3910.63 2 0.06837 0.98690 768.338 273.22 9 1588.68 3943.33 6 0.06935 0.99428 771.557 277.127 7 1591.26 3972.82 6 0.07084 774.773 283.075 6 1593.83 3995.65 1 0.07234 0.99890 777.986 289.055 2 1596.41 3991.29 9 0.07292 0.99178 781.196 291.379 4 1598.98 3962.83 6 0.07274 0.98380 784.404 290.668 6 1601.56 3930.93 2 0.07238 0.97888 787.608 289.226 5 1604.13 3911.28 5 0.07277 0.97100 790.81 290.78 4 1606.69 3879.78 1 0.07335 0.96271 794.009 293.112 8 1609.26 3846.69 9 0.07390 0.95233 797.205 295.308 7 1611.82 3805.18 1 0.07486 0.93647 800.398 299.151 9 1614.39 3741.84 8 0.07659 0.91676 803.588 306.029 1 1616.95 3663.06 2 0.89421 806.776 314.097 0.07861 1619.51 3572.96 2 0.08066 0.87219 809.961 322.297 2 1622.06 3485 9 0.08187 0.85140 813.143 327.134 3 1624.62 3401.9 1 816.322 336.104 0.08411 1627.17 3316.08 0.82992

361

7 3 0.08608 0.80998 819.499 343.956 3 1629.72 3236.42 6 0.08915 0.78899 822.672 356.22 2 1632.27 3152.53 1 0.09260 0.76554 825.843 370.026 7 1634.82 3058.86 8 0.09666 0.74191 829.012 386.248 7 1637.37 2964.43 4 0.10017 0.71635 832.177 400.256 3 1639.91 2862.29 2 0.10235 835.34 408.99 9 1642.45 2755.2 0.68955 0.10244 0.66235 838.5 409.35 9 1644.99 2646.53 3 0.63691 841.657 407.278 0.10193 1647.53 2544.88 3 0.09925 0.61441 844.812 396.597 7 1650.07 2454.99 6 0.59415 847.964 388.698 0.09728 1652.6 2374.03 4 0.09620 0.57671 851.113 384.406 6 1655.13 2304.34 2 0.09619 0.55984 854.259 384.353 3 1657.66 2236.96 9 0.09753 0.54188 857.403 389.718 6 1660.19 2165.17 2 0.10047 0.52386 860.544 401.476 8 1662.72 2093.19 7 0.10391 863.683 415.214 7 1665.25 2028.87 0.50777 0.49292 866.819 424.137 0.10615 1667.77 1969.54 1 0.47991 869.952 424.616 0.10627 1670.29 1917.56 2 0.10607 0.46821 873.082 423.841 6 1672.81 1870.83 7 0.10507 0.45793 876.21 419.856 8 1675.33 1829.75 6 0.10442 0.44786 879.335 417.251 6 1677.85 1789.52 7 0.10525 0.43978 882.458 420.558 4 1680.36 1757.24 8 0.10718 0.43477 885.578 428.281 7 1682.87 1737.21 5 0.11018 0.43080 888.695 440.244 1 1685.38 1721.35 6 0.11283 0.42425 891.81 450.833 1 1687.89 1695.19 9 0.11484 0.41474 894.922 458.879 5 1690.4 1657.17 4 0.11644 0.40349 898.031 465.257 1 1692.91 1612.24 9 0.11617 0.39252 901.138 464.212 9 1695.41 1568.39 4 904.243 462.743 0.11581 1697.91 1536.65 0.38458

362

2 1 0.11403 0.38187 907.344 455.64 4 1700.41 1525.84 5 0.11165 910.443 446.131 4 1702.91 1535.65 0.38433 0.11091 0.38660 913.54 443.171 3 1705.41 1544.73 3 0.11037 916.634 441.022 6 1707.9 1544.96 0.38666 0.10978 919.725 438.651 2 1710.39 1534.45 0.38403 0.10914 0.38076 922.814 436.115 7 1712.88 1521.41 7 0.10729 0.37705 925.9 428.714 5 1715.37 1506.58 5 0.10602 0.37349 928.984 423.635 4 1717.86 1492.34 1 0.10295 0.37076 932.065 411.39 9 1720.35 1481.44 3 0.10049 0.36882 935.144 401.551 7 1722.83 1473.69 4 0.09988 0.36502 938.22 399.096 3 1725.31 1458.5 2 0.09890 0.36211 941.294 395.2 8 1727.79 1446.89 6 0.09805 0.36031 944.365 391.797 6 1730.27 1439.71 9 0.09668 0.35961 947.433 386.306 2 1732.75 1436.89 4 0.09482 0.36201 950.499 378.892 6 1735.22 1446.49 6 0.09415 0.36515 953.563 376.21 5 1737.7 1459.04 7 0.09212 0.36927 956.624 368.095 4 1740.17 1475.49 4 0.09070 0.37346 959.682 362.41 1 1742.64 1492.23 4 0.09151 0.37586 962.738 365.651 2 1745.1 1501.81 1 0.09265 0.37616 965.792 370.218 5 1747.57 1503.04 9 0.09456 0.37345 968.843 377.849 5 1750.04 1492.18 1 0.09503 0.36844 971.891 379.727 5 1752.5 1472.17 3 0.09419 0.36598 974.937 376.366 4 1754.96 1462.36 8 0.09257 0.36266 977.981 369.894 4 1757.42 1449.09 7 0.08869 981.022 354.38 1 1759.87 1452.34 0.36348 0.08493 0.36876 984.061 339.362 3 1762.33 1473.47 9 0.08344 0.37567 987.097 333.425 7 1764.78 1501.05 1 990.131 336.238 0.08415 1767.24 1517.64 0.37982

363

1 3 0.08721 0.38249 993.162 348.467 2 1769.69 1528.3 1 0.09022 996.191 360.522 9 1772.14 1534.77 0.38411 0.09353 0.38625 999.218 373.736 6 1774.58 1543.35 8 0.09669 0.38527 1002.24 386.374 9 1777.03 1539.43 6 0.09789 0.38759 1005.26 391.162 7 1779.47 1548.71 9 0.09852 0.39262 1008.28 393.682 8 1781.91 1568.78 2 0.09847 0.39868 1011.3 393.474 6 1784.35 1593.02 9 0.10009 0.40216 1014.31 399.951 7 1786.79 1606.91 5 0.10297 1017.33 411.468 9 1789.23 1626.23 0.407 0.10677 0.41255 1020.34 426.635 5 1791.66 1648.44 9 0.41762 1023.34 448.271 0.11219 1794.09 1668.68 4 0.11775 1026.35 470.495 2 1796.52 1680.41 0.42056 0.12228 0.42231 1029.35 488.602 3 1798.95 1687.44 9 0.12632 0.42361 1032.35 504.735 1 1801.38 1692.61 3 0.12862 0.42417 1035.35 513.947 7 1803.81 1694.84 1 0.13076 0.42388 1038.34 522.485 3 1806.23 1693.7 6 0.13273 0.42568 1041.34 530.365 6 1808.65 1700.9 8 0.13467 0.42898 1044.33 538.106 3 1811.08 1714.08 7 0.43267 1047.31 549.682 0.13757 1813.49 1728.8 1 0.14080 0.43685 1050.3 562.601 3 1815.91 1745.53 8 0.14563 0.43793 1053.28 581.891 1 1818.33 1749.85 9 0.15204 0.43671 1056.26 607.515 4 1820.74 1744.97 7 0.15878 0.43153 1059.24 634.435 1 1823.15 1724.26 4 0.16486 0.42561 1062.22 658.727 1 1825.57 1700.6 3 0.17067 0.42148 1065.19 681.949 3 1827.97 1684.11 6 0.17523 0.42148 1068.17 700.167 2 1830.38 1684.11 6 0.18054 0.42477 1071.14 721.389 4 1832.79 1697.27 9 1074.1 744.047 0.18621 1835.19 1730.15 0.43300

364

4 8 0.19331 0.44018 1077.07 772.422 6 1837.59 1758.82 4 0.20060 0.44481 1080.03 801.548 5 1839.99 1777.32 4 0.20729 0.44459 1082.99 828.293 9 1842.39 1776.46 9 0.21274 0.44218 1085.95 850.044 2 1844.79 1766.8 1 0.21650 0.43974 1088.9 865.076 4 1847.18 1757.08 8 0.21922 0.43832 1091.86 875.931 1 1849.58 1751.38 2 0.22321 0.43683 1094.81 891.879 2 1851.97 1745.45 8 0.43796 1097.76 909.011 0.2275 1854.36 1749.96 6 0.43684 1100.7 930.627 0.23291 1856.75 1745.48 5 0.23749 0.43687 1103.65 948.958 8 1859.14 1745.6 5 0.24233 0.43673 1106.59 968.285 5 1861.52 1745.03 2 0.24638 1109.53 984.476 7 1863.91 1752.97 0.43872 0.24936 0.44256 1112.46 996.387 8 1866.29 1768.32 1 0.25320 0.44565 1115.4 1011.72 5 1868.67 1780.67 2 0.25934 0.44869 1118.33 1036.25 5 1871.05 1792.83 5 0.26614 0.45069 1121.26 1063.42 4 1873.43 1800.82 5 0.27447 0.44761 1124.19 1096.69 1 1875.8 1788.53 9 0.28272 0.44500 1127.12 1129.66 2 1878.17 1778.07 1 0.29102 0.44152 1130.04 1162.85 9 1880.55 1764.18 5 0.29700 0.43814 1132.96 1186.74 8 1882.92 1750.67 4 0.30184 0.43579 1135.88 1206.05 1 1885.29 1741.3 9 0.30562 0.43346 1138.8 1221.17 5 1887.65 1731.98 6 0.30945 0.43385 1141.71 1236.48 7 1890.02 1733.54 7 0.31192 0.43560 1144.62 1246.35 7 1892.38 1740.53 6 0.43658 1147.53 1258.19 0.31489 1894.75 1744.45 7 0.31748 0.43845 1150.44 1268.56 5 1897.11 1751.92 7 0.31963 0.43912 1153.35 1277.16 8 1899.47 1754.59 5 1156.25 1285.82 0.32180 1901.82 1747.93 0.43745

365

5 8 0.32564 0.43700 1159.15 1301.16 4 1904.18 1746.13 8 0.43569 1162.05 1315.09 0.32913 1906.53 1740.88 4 0.33333 0.43396 1164.95 1331.89 5 1908.89 1733.99 9 0.33737 0.43395 1167.84 1348.02 2 1911.24 1733.94 7 0.34228 0.43368 1170.73 1367.66 7 1913.59 1732.84 2 0.34633 0.43255 1173.62 1383.84 7 1915.94 1728.35 8 0.35134 0.43257 1176.51 1403.85 5 1918.28 1728.41 3 0.35721 0.43391 1179.4 1427.31 6 1920.63 1733.79 9 0.36529 0.43812 1182.28 1459.6 7 1922.97 1750.6 6 0.37218 0.44350 1185.16 1487.13 7 1925.31 1772.09 5 0.37743 0.44729 1188.04 1508.1 5 1927.65 1787.22 1 0.38049 0.44907 1190.92 1520.34 9 1929.99 1794.34 3 0.38298 0.44670 1193.79 1530.29 9 1932.33 1784.86 1 0.44236 1196.66 1534.69 0.38409 1934.66 1767.52 1 0.38602 0.43994 1199.53 1542.42 5 1937 1757.88 8 0.43886 1202.4 1552.95 0.38866 1939.33 1753.55 5 0.39388 0.44066 1205.27 1573.81 1 1941.66 1760.73 2 0.39925 0.44505 1208.13 1595.3 9 1943.99 1778.3 9 0.40399 0.44713 1210.99 1614.24 9 1946.32 1786.61 9 0.40847 0.44639 1213.85 1632.11 2 1948.64 1783.65 8 0.41505 0.44314 1216.71 1658.43 9 1950.97 1770.65 4 0.42169 1219.57 1684.94 4 1953.29 1755.65 0.43939 0.42908 1222.42 1714.47 4 1955.61 1745.3 0.4368 0.43729 0.43670 1225.27 1747.28 6 1957.93 1744.91 2 0.44504 0.43941 1228.12 1778.23 1 1960.25 1755.74 3 0.45289 1230.96 1809.61 5 1962.56 1764.36 0.44157 0.45998 0.44211 1233.81 1837.93 3 1964.88 1766.54 6 1236.65 1863.03 0.46626 1967.19 1763.87 0.44144

366

5 8 0.47396 0.44098 1239.49 1893.79 3 1969.5 1762.01 2 0.48131 0.43953 1242.33 1923.15 1 1971.81 1756.24 8 0.48965 0.43829 1245.16 1956.49 5 1974.12 1751.28 7 0.49816 0.43967 1248 1990.5 7 1976.43 1756.8 8 0.50736 1250.83 2027.27 9 1978.74 1762.84 0.44119 0.51836 0.44108 1253.66 2071.19 1 1981.04 1762.42 5 0.52902 0.44086 1256.49 2113.8 5 1983.34 1761.53 2 0.53817 0.44041 1259.31 2150.37 8 1985.64 1759.74 4 0.54748 0.44421 1262.13 2187.55 3 1987.94 1774.91 1 0.55580 0.44704 1264.95 2220.81 7 1990.24 1786.25 9 0.56366 0.44841 1267.77 2252.22 8 1992.54 1791.72 8 0.57099 1270.59 2281.51 8 1994.83 1798.64 0.45015 0.58073 0.45031 1273.4 2320.43 9 1997.13 1799.29 2 0.59348 0.44896 1276.22 2371.37 8 1999.42 1793.89 1 0.60551 0.44547 1279.03 2419.43 6 2001.71 1779.95 2 0.61643 0.44188 1281.84 2463.05 3 2004 1765.62 6 0.62740 0.44272 1284.64 2506.89 5 2006.29 1768.99 9 0.63675 0.44104 1287.45 2544.26 7 2008.57 1762.27 7 0.64483 0.43636 1290.25 2576.54 6 2010.86 1743.55 2 0.65309 0.43334 1293.05 2609.53 3 2013.14 1731.49 4 0.66316 1295.84 2649.77 4 2015.42 1717.77 0.42991 0.67648 0.42846 1298.64 2703 6 2017.7 1712 6 0.68846 0.42690 1301.43 2750.87 6 2019.98 1705.76 4 0.42689 1304.22 2800.11 0.70079 2022.26 1705.72 4 0.71255 0.42968 1307.01 2847.12 5 2024.54 1716.87 5 0.72364 0.43133 1309.8 2891.42 2 2026.81 1723.47 7 0.73481 0.43153 1312.59 2936.08 9 2029.08 1724.26 4 1315.37 2985.55 0.7472 2031.35 1731.9 0.43344

367

6 0.43291 1318.15 3040.85 0.76104 2033.63 1729.77 3 0.77493 0.43393 1320.93 3096.37 5 2035.89 1733.85 4 0.78602 0.43297 1323.71 3140.67 2 2038.16 1730.01 3 0.43057 1326.48 3178.14 0.7954 2040.43 1720.41 1 0.80109 0.42784 1329.25 3200.91 9 2042.69 1709.53 8 0.80445 0.42698 1332.02 3214.33 7 2044.95 1706.1 9 0.80706 0.42670 1334.79 3224.76 8 2047.22 1704.98 9 0.42707 1337.56 3243.27 0.8117 2049.48 1706.46 9 0.81931 0.42680 1340.32 3273.7 6 2051.73 1705.37 7 0.82693 0.42745 1343.09 3304.15 7 2053.99 1707.95 2 0.83560 0.42498 1345.85 3338.77 1 2056.25 1698.08 2 0.84617 0.41992 1348.6 3381.02 5 2058.5 1677.87 4 0.85437 1351.36 3413.77 2 2060.75 1651.8 0.4134 0.85870 0.40892 1354.11 3431.08 4 2063.01 1633.92 5 0.86253 1356.87 3446.4 8 2065.26 1615.36 0.40428 0.86647 0.39935 1359.62 3462.14 7 2067.51 1595.68 4 0.86997 0.39673 1362.36 3476.1 1 2069.75 1585.22 6 0.39292 1365.11 3482.33 0.87153 2072 1569.99 5 0.87457 0.38585 1367.85 3494.49 4 2074.24 1541.75 7 0.87955 0.37886 1370.6 3514.38 2 2076.49 1513.8 2 0.88208 0.37156 1373.34 3524.51 7 2078.73 1484.64 4 0.88071 0.36781 1376.07 3519.02 3 2080.97 1469.67 8 0.87994 0.36426 1378.81 3515.97 9 2083.21 1455.48 6 0.87671 0.36176 1381.54 3503.03 1 2085.45 1445.49 6 0.87305 0.36222 1384.27 3488.42 4 2087.68 1447.34 9 0.86877 0.35970 1387 3471.32 5 2089.92 1437.25 4 0.86575 1389.73 3459.26 7 2092.15 1415.02 0.35414 1392.46 3456.01 0.86494 2094.38 1393.92 0.34885

368

3 9 0.86454 0.34185 1395.18 3454.43 8 2096.61 1365.92 2 0.86255 0.33722 1397.9 3446.48 8 2098.84 1347.45 9 0.85975 0.32880 1400.62 3435.27 2 2101.07 1313.8 8 0.85243 0.32018 1403.34 3406.02 2 2103.3 1279.34 3 1406.06 3371.65 0.84383 2105.52 1250.36 0.31293 0.83275 0.30572 1408.77 3327.38 1 2107.75 1221.58 7 0.82054 0.29961 1411.48 3278.6 2 2109.97 1197.15 3 0.81200 0.29498 1414.19 3244.48 3 2112.19 1178.66 6 0.80850 0.29133 1416.9 3230.51 7 2114.41 1164.07 4 0.80807 1419.6 3228.77 1 2116.63 1158.82 0.29002 0.80878 0.28313 1422.31 3231.63 7 2118.85 1131.3 3 0.80838 0.27404 1425.01 3230.03 7 2121.07 1095 8 0.80604 0.26390 1427.71 3220.67 4 2123.28 1054.47 4 0.80232 0.25398 1430.41 3205.8 3 2125.49 1014.85 9 0.79787 0.24523 1433.1 3188.03 5 2127.71 979.86 2 0.79500 0.23589 1435.79 3176.57 7 2129.92 942.563 7 0.79454 0.22925 1438.49 3174.72 4 2132.13 916.015 3 0.79395 0.22364 1441.18 3172.37 6 2134.34 893.622 9 0.79227 0.21532 1443.86 3165.66 7 2136.54 860.353 2 0.78799 0.20744 1446.55 3148.56 7 2138.75 828.877 5 0.78194 0.19832 1449.23 3124.37 3 2140.95 792.423 1 0.77836 0.18933 1451.91 3110.09 9 2143.16 756.532 9 0.77717 0.18078 1454.59 3105.33 8 2145.36 722.371 9 0.77703 1457.27 3104.76 5 2147.56 681.459 0.17055 0.77720 0.16201 1459.95 3105.43 3 2149.76 647.372 9 0.77662 0.15254 1462.62 3103.13 7 2151.96 609.524 7 0.77521 0.14382 1465.29 3097.5 8 2154.16 574.663 2 0.77096 0.13622 1467.96 3080.52 8 2156.35 544.318 8

369

0.76831 0.12654 1470.63 3069.93 8 2158.55 505.624 4 0.76829 0.11780 1473.3 3069.84 6 2160.74 470.7 3 0.10894 1475.96 3076.25 0.7699 2162.93 435.312 6 0.77190 0.09840 1478.62 3084.28 9 2165.12 393.189 4 0.77288 0.08834 1481.28 3088.18 6 2167.31 352.987 3 0.77546 1483.94 3098.5 8 2169.5 305.068 0.07635 0.77751 0.06563 1486.6 3106.66 1 2171.69 262.239 1 0.77913 1489.25 3113.16 7 2173.87 219.88 0.05503 0.78137 0.04477 1491.9 3122.11 7 2176.06 178.908 6 0.78320 1494.55 3129.4 2 2178.24 146.64 0.0367 0.78325 0.02927 1497.2 3129.6 2 2180.43 116.962 2 0.78248 0.02129 1499.85 3126.55 8 2182.61 85.0842 4 0.78134 1502.49 3121.99 7 2184.79 57.4569 0.01438 0.78386 0.00753 1505.13 3132.04 2 2186.97 30.1157 7 0.78740 0.00295 1507.78 3146.18 1 2189.14 11.8045 4 0.79475 1510.41 3175.58 9

P25 – RAMAN Peak Data

0.05755 653.81 125.634 1 2182.99 Hi.Peak D = 228280 0.05734 657.135 125.192 9 G = 178208 0.05670 660.456 123.794 8 0.05695 1.28097 663.774 124.326 2 Ratio = 5 3.35681 667.089 127.313 0.05832 La = 8 0.06046 670.401 131.988 2 0.06128 0.34240 673.71 133.788 7 1513.05 747.459 1 677.015 134.63 0.06167 1515.69 752.223 0.34458

370

2 4 0.34558 680.318 135.498 0.06207 1518.32 754.414 7 0.06401 0.34783 683.617 139.736 1 1520.95 759.329 9 0.06486 0.35557 686.913 141.596 3 1523.58 776.215 4 0.06563 0.36163 690.206 143.279 4 1526.2 789.441 3 0.36666 693.496 148.115 0.06785 1528.83 800.429 6 0.06940 0.37095 696.783 151.512 6 1531.45 809.786 3 0.06912 0.38015 700.067 150.907 9 1534.07 829.877 6 0.06879 0.38600 703.348 150.185 8 1536.69 842.653 9 0.06848 0.39083 706.626 149.496 2 1539.31 853.197 9 0.07030 0.39372 709.901 153.474 4 1541.93 859.495 4 0.07059 0.40755 713.173 154.105 4 1544.54 889.694 8 0.07133 0.42282 716.441 155.726 6 1547.15 923.026 6 0.07244 0.43799 719.707 158.144 4 1549.76 956.147 9 0.07271 0.46061 722.97 158.729 2 1552.37 1005.51 1 0.07216 0.48429 726.23 157.528 2 1554.97 1057.22 9 0.07153 0.50825 729.487 156.158 4 1557.58 1109.51 2 0.07230 0.53184 732.74 157.833 1 1560.18 1161.02 9 0.07365 0.55065 735.991 160.787 4 1562.78 1202.08 8 0.07453 0.58439 739.239 162.712 6 1565.38 1275.73 6 0.07614 0.62789 742.484 166.228 7 1567.98 1370.68 1 0.07703 745.726 168.166 5 1570.57 1486.9 0.68113 0.07696 0.74518 748.965 168.006 1 1573.16 1626.73 4 0.07704 0.82739 752.201 168.191 6 1575.75 1806.2 7 0.07663 0.90618 755.435 167.29 3 1578.34 1978.2 8 0.07798 758.665 170.239 4 1580.93 2112.02 0.96749 0.08019 0.99537 761.892 175.062 4 1583.51 2172.9 8 0.08025 765.117 175.205 9 1586.1 2182.99 1 768.338 179.103 0.08204 1588.68 2133.21 0.97719

371

5 6 0.08339 0.93772 771.557 182.059 9 1591.26 2047.04 3 0.08376 0.88744 774.773 182.85 1 1593.83 1937.28 3 0.08492 0.84845 777.986 185.389 4 1596.41 1852.16 1 0.08544 0.80967 781.196 186.533 8 1598.98 1767.51 4 0.08721 0.77519 784.404 190.381 1 1601.56 1692.24 4 0.08648 0.74267 787.608 188.799 6 1604.13 1621.25 4 0.08384 0.71508 790.81 183.041 9 1606.69 1561.03 8 0.08459 0.69180 794.009 184.673 6 1609.26 1510.2 3 0.08356 0.67369 797.205 182.419 4 1611.82 1470.66 1 800.398 186.035 0.08522 1614.39 1427.02 0.6537 0.08903 0.63707 803.588 194.365 6 1616.95 1390.73 6 0.09303 0.62525 806.776 203.101 8 1619.51 1364.93 7 0.09509 0.61190 809.961 207.583 1 1622.06 1335.78 4 0.09429 0.59483 813.143 205.837 1 1624.62 1298.51 1 0.09418 0.57010 816.322 205.608 6 1627.17 1244.53 3 0.09542 0.54279 819.499 208.305 2 1629.72 1184.92 7 0.51265 822.672 203.499 0.09322 1632.27 1119.13 9 0.09548 0.47889 825.843 208.451 9 1634.82 1045.43 8 0.09700 0.44901 829.012 211.755 2 1637.37 980.195 5 0.09828 0.42051 832.177 214.554 4 1639.91 917.983 6 0.09703 0.39863 835.34 211.835 9 1642.45 870.209 2 0.09656 0.37582 838.5 210.796 3 1644.99 820.418 3 0.09595 0.35250 841.657 209.471 6 1647.53 769.513 4 0.09671 0.33259 844.812 211.133 7 1650.07 726.044 2 0.09552 0.31570 847.964 208.524 2 1652.6 689.189 9 0.09622 0.29987 851.113 210.049 1 1655.13 654.618 2 0.09507 0.28517 854.259 207.548 5 1657.66 622.528 2 0.09550 0.26755 857.403 208.49 7 1660.19 584.062 1

372

0.09368 0.25379 860.544 204.505 1 1662.72 554.025 2 0.09547 0.23908 863.683 208.415 2 1665.25 521.918 4 0.09423 866.819 205.712 4 1667.77 496.063 0.22724 0.09492 0.21740 869.952 207.229 9 1670.29 474.584 1 0.09596 0.20917 873.082 209.484 2 1672.81 456.619 1 0.09478 0.20476 876.21 206.913 4 1675.33 446.999 5 0.09288 0.19797 879.335 202.761 2 1677.85 432.17 2 0.09321 0.18849 882.458 203.487 5 1680.36 411.488 7 0.09275 0.18363 885.578 202.493 9 1682.87 400.882 9 0.09456 0.18000 888.695 206.442 8 1685.38 392.946 4 0.09402 0.17644 891.81 205.254 4 1687.89 385.168 1 894.922 206.576 0.09463 1690.4 375.605 0.17206 0.09512 0.16821 898.031 207.658 5 1692.91 367.21 4 0.09504 901.138 207.483 5 1695.41 357.814 0.16391 0.15779 904.243 207.799 0.09519 1697.91 344.46 3 0.09628 0.15322 907.344 210.193 7 1700.41 334.494 7 0.09705 0.15157 910.443 211.876 8 1702.91 330.882 3 0.10038 0.14953 913.54 219.138 4 1705.41 326.443 9 0.10121 0.14808 916.634 220.95 4 1707.9 323.263 3 0.10157 0.14824 919.725 221.734 4 1710.39 323.622 7 0.10192 0.14662 922.814 222.505 7 1712.88 320.079 4 0.10236 0.14284 925.9 223.458 3 1715.37 311.82 1 0.09981 0.14007 928.984 217.886 1 1717.86 305.785 6 0.09780 0.13855 932.065 213.498 1 1720.35 302.466 6 0.09582 0.13776 935.144 209.178 2 1722.83 300.732 2 0.09759 0.13593 938.22 213.051 6 1725.31 296.749 7 0.09544 0.13442 941.294 208.359 7 1727.79 293.452 7 0.09674 0.13535 944.365 211.198 7 1730.27 295.473 2 947.433 216.91 0.09936 1732.75 296.228 0.13569

373

4 8 0.10167 0.13461 950.499 221.95 2 1735.22 293.87 8 0.10293 0.13258 953.563 224.701 3 1737.7 289.428 3 0.10329 0.13124 956.624 225.491 5 1740.17 286.506 5 0.10274 0.12826 959.682 224.297 8 1742.64 280.008 8 0.10175 962.738 222.127 4 1745.1 270.036 0.1237 0.09994 0.11996 965.792 218.178 5 1747.57 261.88 4 0.09953 0.11900 968.843 217.286 6 1750.04 259.794 8 0.12241 971.891 218.365 0.10003 1752.5 267.222 1 0.09903 0.12378 974.937 216.2 8 1754.96 270.216 3 0.10080 0.12474 977.981 220.05 2 1757.42 272.324 8 0.09953 0.12751 981.022 217.282 4 1759.87 278.366 6 0.09975 0.12835 984.061 217.759 3 1762.33 280.2 6 0.09768 0.12732 987.097 213.24 3 1764.78 277.953 7 0.09699 0.12678 990.131 211.743 7 1767.24 276.776 8 0.09681 0.12433 993.162 211.342 3 1769.69 271.427 7 0.09739 0.12428 996.191 212.614 6 1772.14 271.322 9 0.09538 0.12106 999.218 208.226 6 1774.58 264.277 2 0.09860 0.12076 1002.24 215.244 1 1777.03 263.621 1 0.09966 0.12143 1005.26 217.568 5 1779.47 265.089 4 0.10311 0.12224 1008.28 225.097 4 1781.91 266.863 7 0.10525 0.12250 1011.3 229.766 3 1784.35 267.437 9 0.10714 0.12247 1014.31 233.892 3 1786.79 267.352 1 0.10885 0.12084 1017.33 237.622 2 1789.23 263.81 8 0.10980 0.12195 1020.34 239.7 4 1791.66 266.222 3 0.11020 0.12084 1023.34 240.575 4 1794.09 263.801 4 1026.35 241.898 0.11081 1796.52 267.547 0.12256 0.10993 0.12236 1029.35 239.979 1 1798.95 267.13 9 0.11104 0.12341 1032.35 242.41 5 1801.38 269.415 6

374

1035.35 245.434 0.11243 1803.81 268.18 0.12285 0.11281 0.11866 1038.34 246.272 4 1806.23 259.035 1 0.11261 0.11682 1041.34 245.835 4 1808.65 255.03 6 0.11412 0.11828 1044.33 249.137 6 1811.08 258.221 8 0.11575 0.11837 1047.31 252.698 8 1813.49 258.421 9 0.11550 0.11918 1050.3 252.14 2 1815.91 260.173 2 0.11423 0.12065 1053.28 249.377 6 1818.33 263.379 1 0.11355 0.12145 1056.26 247.889 5 1820.74 265.132 4 0.11490 0.11972 1059.24 250.83 2 1823.15 261.352 2 0.11772 1062.22 251.481 0.1152 1825.57 256.995 6 0.12008 1065.19 257.003 0.11773 1827.97 262.145 5 0.12121 0.12276 1068.17 264.61 4 1830.38 268.004 9 0.12328 0.12478 1071.14 269.129 5 1832.79 272.398 2 0.12575 0.12338 1074.1 274.521 5 1835.19 269.351 6 0.12658 1077.07 276.329 3 1837.59 271.848 0.12453 0.12676 0.12338 1080.03 276.718 1 1839.99 269.344 3 0.12685 0.12275 1082.99 276.92 4 1842.39 267.966 2 0.12760 0.12276 1085.95 278.568 8 1844.79 267.985 1 0.12957 0.12472 1088.9 282.87 9 1847.18 272.279 8 0.12925 0.12707 1091.86 282.168 8 1849.58 277.41 8 0.13192 0.12895 1094.81 287.997 8 1851.97 281.517 9 0.13685 0.12603 1097.76 298.752 4 1854.36 275.128 3 0.13790 0.12377 1100.7 301.054 9 1856.75 270.2 5 0.13963 0.12331 1103.65 304.815 2 1859.14 269.195 5 0.14036 0.12273 1106.59 306.415 5 1861.52 267.921 1 0.14269 0.12115 1109.53 311.495 2 1863.91 264.479 4 0.14262 1112.46 311.351 6 1866.29 265.473 0.12161 0.14202 0.12335 1115.4 310.042 6 1868.67 269.289 8 1118.33 311.129 0.14252 1871.05 271.947 0.12457

375

4 5 0.14740 0.12453 1121.26 321.775 1 1873.43 271.855 3 0.12579 1124.19 318.084 0.14571 1875.8 274.616 8 0.14862 0.12868 1127.12 324.454 8 1878.17 280.927 9 0.14795 1130.04 322.975 1 1880.55 280.012 0.12827 0.14785 0.12721 1132.96 322.771 7 1882.92 277.704 3 0.14782 0.12703 1135.88 322.709 9 1885.29 277.32 7 0.14878 0.12495 1138.8 324.804 9 1887.65 272.784 9 0.15116 0.12450 1141.71 330 9 1890.02 271.788 3 0.15788 0.12472 1144.62 344.659 4 1892.38 272.272 4 0.15841 0.12674 1147.53 345.819 5 1894.75 276.686 6 0.16173 0.12583 1150.44 353.075 9 1897.11 274.696 5 0.16181 0.12589 1153.35 353.242 6 1899.47 274.826 4 0.16345 0.12517 1156.25 356.824 7 1901.82 273.248 1 0.16301 0.12773 1159.15 355.863 6 1904.18 278.849 7 0.16480 1162.05 359.777 9 1906.53 278.964 0.12779 0.16686 0.12785 1164.95 364.256 1 1908.89 279.097 1 0.16792 0.12308 1167.84 366.57 1 1911.24 268.695 6 0.16639 0.12207 1170.73 363.238 5 1913.59 266.494 8 0.16385 0.11721 1173.62 357.692 4 1915.94 255.877 4 0.16339 0.11915 1176.51 356.691 6 1918.28 260.117 6 0.12000 1179.4 364.123 0.1668 1920.63 261.96 1 0.16804 0.12484 1182.28 366.841 5 1922.97 272.526 1 0.17377 0.12695 1185.16 379.35 5 1925.31 277.143 6 0.17849 0.12807 1188.04 389.645 1 1927.65 279.579 2 0.18410 0.12399 1190.92 401.908 9 1929.99 270.673 2 0.18732 0.12227 1193.79 408.932 7 1932.33 266.929 7 0.18283 0.11871 1196.66 399.128 5 1934.66 259.159 7 1199.53 398.286 0.18245 1937 262.817 0.12039

376

3 0.18383 1202.4 401.303 2 1939.33 263.859 0.12087 0.18455 0.12053 1205.27 402.89 9 1941.66 263.124 4 0.18915 1208.13 412.919 3 1943.99 266.783 0.12221 0.19334 0.12405 1210.99 422.065 3 1946.32 270.804 2 0.19718 0.12514 1213.85 430.448 3 1948.64 273.183 2 0.20092 1216.71 438.618 5 1950.97 274.423 0.12571 0.19968 0.12540 1219.57 435.905 3 1953.29 273.764 8 0.20073 0.12658 1222.42 438.203 5 1955.61 276.336 6 0.20280 0.12518 1225.27 442.721 5 1957.93 273.268 1 0.20437 0.12223 1228.12 446.145 3 1960.25 266.837 5 0.20884 0.12410 1230.96 455.899 2 1962.56 270.922 6 0.21300 0.12483 1233.81 464.984 3 1964.88 272.512 4 0.21543 0.12596 1236.65 470.293 5 1967.19 274.976 3 0.21916 0.12676 1239.49 478.437 6 1969.5 276.734 8 0.22249 0.12503 1242.33 485.704 5 1971.81 272.954 7 0.22368 0.12705 1245.16 488.31 9 1974.12 277.365 7 0.22700 0.12709 1248 495.558 9 1976.43 277.445 4 0.23026 0.12867 1250.83 502.67 7 1978.74 280.896 5 0.23463 0.12979 1253.66 512.211 7 1981.04 283.333 1 0.23896 0.13126 1256.49 521.665 8 1983.34 286.544 2 0.24621 0.13351 1259.31 537.493 9 1985.64 291.465 6 0.25153 0.13305 1262.13 549.095 3 1987.94 290.467 9 0.13225 1264.95 561.204 0.25708 1990.24 288.714 6 0.26362 0.13260 1267.77 575.495 7 1992.54 289.466 1 0.27034 0.13374 1270.59 590.17 9 1994.83 291.958 2 0.27987 0.13375 1273.4 610.965 5 1997.13 291.98 2 0.28631 0.13124 1276.22 625.022 5 1999.42 286.512 8 1279.03 639.451 0.29292 2001.71 286.718 0.13134

377

4 2 0.30359 0.13330 1281.84 662.747 6 2004 290.995 1 0.31067 0.13181 1284.64 678.195 3 2006.29 287.743 1 0.31676 0.13413 1287.45 691.486 1 2008.57 292.808 2 0.32684 0.13181 1290.25 713.498 4 2010.86 287.758 8 0.33749 0.13134 1293.05 736.748 5 2013.14 286.719 2 0.35185 0.12686 1295.84 768.088 1 2015.42 276.949 7 0.36311 0.12639 1298.64 792.673 3 2017.7 275.921 6 0.12544 1301.43 810.632 0.37134 2019.98 273.846 5 0.38750 1304.22 845.919 5 2022.26 274.074 0.12555 0.40446 1307.01 882.94 4 2024.54 272.196 0.12469 0.42097 0.12605 1309.8 918.986 6 2026.81 275.182 7 0.43616 0.12461 1312.59 952.146 6 2029.08 272.041 9 0.45712 0.12489 1315.37 997.901 6 2031.35 272.636 1 0.47929 0.12251 1318.15 1046.29 2 2033.63 267.453 7 0.50143 0.12677 1320.93 1094.63 6 2035.89 276.745 3 0.52252 0.12674 1323.71 1140.66 2 2038.16 276.675 1 0.54777 0.12949 1326.48 1195.78 2 2040.43 282.691 7 0.57214 0.12839 1329.25 1248.99 6 2042.69 280.276 1 0.59569 0.12759 1332.02 1300.39 2 2044.95 278.537 4 0.61832 0.12689 1334.79 1349.8 6 2047.22 277.02 9 0.64275 0.12784 1337.56 1403.12 1 2049.48 279.075 1 0.66657 0.12838 1340.32 1455.12 2 2051.73 280.259 3 0.69087 0.13003 1343.09 1508.18 8 2053.99 283.862 4 0.71833 0.12754 1345.85 1568.11 1 2056.25 278.427 4 0.73640 0.12814 1348.6 1607.56 3 2058.5 279.747 9 0.74823 0.12511 1351.36 1633.38 1 2060.75 273.126 6 0.75527 0.12341 1354.11 1648.75 1 2063.01 269.408 2 1356.87 1653.18 0.75730 2065.26 263.657 0.12077

378

1 8 0.74743 0.12179 1359.62 1631.64 4 2067.51 265.887 9 0.73923 0.12344 1362.36 1613.75 8 2069.75 269.488 9 0.72892 0.12455 1365.11 1591.23 2 2072 271.898 3 0.71723 0.12394 1367.85 1565.71 2 2074.24 270.575 7 0.12384 1370.6 1520.89 0.6967 2076.49 270.356 7 0.66860 0.12350 1373.34 1459.55 1 2078.73 269.603 2 0.64220 0.12306 1376.07 1401.92 2 2080.97 268.644 2 0.61533 1378.81 1343.27 5 2083.21 268.552 0.12302 0.12268 1381.54 1278.36 0.5856 2085.45 267.829 9 0.56593 0.12398 1384.27 1235.43 5 2087.68 270.662 7 0.54780 0.12323 1387 1195.86 8 2089.92 269.013 1 0.53043 0.12362 1389.73 1157.93 3 2092.15 269.864 1 0.51922 0.12268 1392.46 1133.46 4 2094.38 267.814 2 0.50282 0.12104 1395.18 1097.66 4 2096.61 264.245 7 0.48970 0.11974 1397.9 1069.03 9 2098.84 261.394 1 0.47909 0.11890 1400.62 1045.85 1 2101.07 259.578 9 0.46256 0.11419 1403.34 1009.78 7 2103.3 249.281 2 0.44636 1406.06 974.403 2 2105.52 249.407 0.11425 0.43064 1408.77 940.091 4 2107.75 245.258 0.11235 0.11416 1411.48 910.787 0.41722 2109.97 249.215 2 0.41063 0.11460 1414.19 896.417 7 2112.19 250.181 5 0.39869 0.11718 1416.9 870.35 6 2114.41 255.818 7 0.39433 0.11883 1419.6 860.835 8 2116.63 259.42 7 0.38734 0.11838 1422.31 845.579 9 2118.85 258.424 1 0.11582 1425.01 822.332 0.3767 2121.07 252.84 3 0.11406 1427.71 807.531 0.36992 2123.28 249.003 5 0.35855 0.11201 1430.41 782.716 2 2125.49 244.537 9 1433.1 772.036 0.35366 2127.71 239.951 0.10991

379

9 0.34888 0.11029 1435.79 761.622 9 2129.92 240.782 9 0.34411 0.11084 1438.49 751.203 7 2132.13 241.967 2 0.34481 0.11018 1441.18 752.728 5 2134.34 240.527 2 0.33823 0.11109 1443.86 738.356 2 2136.54 242.522 6 0.33597 0.11096 1446.55 733.437 8 2138.75 242.239 7 0.33673 0.10863 1449.23 735.091 6 2140.95 237.152 6 0.32990 0.10569 1451.91 720.189 9 2143.16 230.725 2 0.32892 0.10457 1454.59 718.044 7 2145.36 228.284 4 0.32662 0.10582 1457.27 713.017 4 2147.56 231.019 7 0.32328 0.10645 1459.95 705.721 2 2149.76 232.395 7 0.31832 0.10643 1462.62 694.901 5 2151.96 232.339 2 0.31351 0.10773 1465.29 684.4 5 2154.16 235.182 4 0.31502 0.10658 1467.96 687.706 9 2156.35 232.666 1 0.10392 1470.63 690.437 0.31628 2158.55 226.871 7 0.31934 1473.3 697.125 4 2160.74 216.203 0.09904 0.32246 0.09636 1475.96 703.933 3 2162.93 210.355 1 0.32679 0.09466 1478.62 713.391 5 2165.12 206.644 1 0.32549 0.09624 1481.28 710.559 8 2167.31 210.107 7 0.32105 0.09698 1483.94 700.851 1 2169.5 211.723 8 0.32119 0.10033 1486.6 701.158 2 2171.69 219.029 4 0.32234 0.10132 1489.25 703.668 1 2173.87 221.19 4 0.32457 0.10087 1491.9 708.541 4 2176.06 220.202 2 0.32988 0.09936 1494.55 720.14 7 2178.24 216.911 4 0.33190 1497.2 724.55 7 2180.43 210.179 0.09628 0.33251 0.09295 1499.85 725.878 5 2182.61 202.913 2 0.33342 0.09065 1502.49 727.857 2 2184.79 197.9 5 0.33447 0.08888 1505.13 730.155 5 2186.97 194.027 1 1507.78 738.456 0.33827 2189.14 193.857 0.08880

380

7 3 0.33862 1510.41 739.209 2

T650 - RAMAN Peak Data

0.00059 653.81 1.70228 8 2847.54 Hi.Peak D = 24654.8 0.00180 657.135 5.129 1 G = 84522.6 0.00349 660.456 9.95807 7 0.00557 0.29169 663.774 15.877 6 Ratio = 5 0.00786 14.7414 667.089 22.3837 1 La = 4 670.401 28.9888 0.01018 0.01258 0.22277 673.71 35.8449 8 1510.41 634.349 1 0.01516 0.22221 677.015 43.1926 8 1513.05 632.767 5 0.01782 680.318 50.7645 7 1515.69 636.939 0.22368 0.02047 0.22692 683.617 58.2966 3 1518.32 646.19 9 0.02300 0.23130 686.913 65.4979 2 1520.95 658.642 2 0.23585 690.206 72.3265 0.0254 1523.58 671.614 8 0.02765 0.23907 693.496 78.7427 3 1526.2 680.765 1 0.02993 0.24108 696.783 85.2379 4 1528.83 686.505 7 0.24208 700.067 91.6344 0.03218 1531.45 689.344 4 0.03438 0.24310 703.348 97.9202 8 1534.07 692.243 2 0.03641 0.24524 706.626 103.699 7 1536.69 698.333 1 0.03811 0.24875 709.901 108.541 7 1539.31 708.339 5 0.03963 0.25376 713.173 112.858 4 1541.93 722.608 6 0.04109 0.26030 716.441 117.02 5 1544.54 741.222 3 0.04267 0.26806 719.707 121.512 3 1547.15 763.332 7 722.97 126.833 0.04454 1549.76 789.144 0.27713

381

1 2 0.04648 0.28709 726.23 132.364 4 1552.37 817.506 2 0.04823 0.29845 729.487 137.35 5 1554.97 849.875 9 0.04967 0.31248 732.74 141.46 8 1557.58 889.812 4 0.05065 0.33068 735.991 144.252 8 1560.18 941.631 2 0.05143 0.35672 739.239 146.472 8 1562.78 1015.79 5 0.05232 0.39626 742.484 148.992 3 1565.38 1128.38 5 0.45498 745.726 152.372 0.05351 1567.98 1295.6 9 0.05501 0.53566 748.965 156.646 1 1570.57 1525.34 9 0.05646 0.63626 752.201 160.78 3 1573.16 1811.8 8 0.05757 0.74883 755.435 163.936 1 1575.75 2132.33 2 0.05832 0.85880 758.665 166.094 9 1578.34 2445.48 4 0.05879 0.94724 761.892 167.412 2 1580.93 2697.33 9 0.05916 0.99757 765.117 168.484 8 1583.51 2840.64 7 0.05957 768.338 169.639 4 1586.1 2847.54 1 0.05994 0.95430 771.557 170.686 2 1588.68 2717.43 8 0.06032 0.86913 774.773 171.771 3 1591.26 2474.9 6 0.06059 0.76027 777.986 172.554 8 1593.83 2164.92 7 0.06084 0.64577 781.196 173.269 9 1596.41 1838.86 1 0.06120 0.54049 784.404 174.276 2 1598.98 1539.07 1 0.06197 0.45295 787.608 176.483 7 1601.56 1289.81 6 0.06318 0.38642 790.81 179.927 7 1604.13 1100.35 1 0.06474 0.33965 794.009 184.354 1 1606.69 967.184 6 0.06654 0.30945 797.205 189.499 8 1609.26 881.189 6 0.06853 0.29142 800.398 195.154 4 1611.82 829.845 5 0.07039 803.588 200.447 3 1614.39 803.092 0.28203 0.07195 806.776 204.885 2 1616.95 793.011 0.27849 0.27824 809.961 207.585 0.0729 1619.51 792.307 3 813.143 209.106 0.07343 1622.06 792.065 0.27815

382

4 8 0.27600 816.322 209.693 0.07364 1624.62 785.927 2 0.07392 0.27079 819.499 210.516 9 1627.17 771.096 4 0.07460 0.26253 822.672 212.438 4 1629.72 747.589 9 0.07591 0.25208 825.843 216.182 9 1632.27 717.831 8 0.07757 0.24113 829.012 220.904 7 1634.82 686.652 9 0.07936 0.23158 832.177 225.999 6 1637.37 659.455 8 0.08085 0.22472 835.34 230.237 5 1639.91 639.919 7 0.08195 0.22049 838.5 233.373 6 1642.45 627.865 4 0.08275 0.21830 841.657 235.643 3 1644.99 621.629 4 0.08356 0.21799 844.812 237.956 5 1647.53 620.737 1 0.08461 0.21824 847.964 240.932 1 1650.07 621.464 6 0.08631 0.21835 851.113 245.793 8 1652.6 621.765 2 0.21752 854.259 251.608 0.08836 1655.13 619.418 7 0.09064 0.21566 857.403 258.126 9 1657.66 614.125 9 0.09281 0.21295 860.544 264.297 6 1660.19 606.408 9 0.09449 0.21002 863.683 269.073 3 1662.72 598.066 9 0.09548 0.20755 866.819 271.894 4 1665.25 591.029 8 0.09557 0.20691 869.952 272.156 6 1667.77 589.194 3 0.09481 0.20777 873.082 269.981 2 1670.29 591.65 6 0.09357 0.20990 876.21 266.462 6 1672.81 597.714 5 0.09188 0.21213 879.335 261.656 8 1675.33 604.073 9 0.09012 0.21373 882.458 256.64 7 1677.85 608.622 6 0.08870 0.21410 885.578 252.584 3 1680.36 609.676 6 0.08794 0.21369 888.695 250.415 1 1682.87 608.515 8 0.21301 891.81 250.242 0.08788 1685.38 606.578 8 0.08834 0.21314 894.922 251.565 5 1687.89 606.943 6 898.031 253.717 0.0891 1690.4 610.057 0.21424 0.08988 0.21644 901.138 255.953 6 1692.91 616.325 1

383

0.09021 0.21924 904.243 256.897 7 1695.41 624.309 5 0.08989 0.22207 907.344 255.972 2 1697.91 632.355 1 0.22423 910.443 254.029 0.08921 1700.41 638.521 6 0.08858 0.22549 913.54 252.244 3 1702.91 642.117 9 0.08811 0.22558 916.634 250.9 1 1705.41 642.352 1 0.08823 0.22468 919.725 251.25 4 1707.9 639.802 6 922.814 253.545 0.08904 1710.39 634.916 0.22297 0.09028 0.22062 925.9 257.08 1 1712.88 628.245 7 0.09135 0.21809 928.984 260.143 7 1715.37 621.022 1 0.09175 0.21595 932.065 261.283 7 1717.86 614.934 3 0.09146 0.21515 935.144 260.461 9 1720.35 612.656 3 0.09049 0.21628 938.22 257.68 2 1722.83 615.882 6 0.08882 0.21940 941.294 252.928 3 1725.31 624.753 1 0.08733 0.22424 944.365 248.68 2 1727.79 638.543 4 0.08618 0.23012 947.433 245.411 4 1730.27 655.295 7 0.08511 0.23554 950.499 242.372 6 1732.75 670.725 5 0.08382 0.23969 953.563 238.683 1 1735.22 682.55 8 0.08203 0.24251 956.624 233.601 6 1737.7 690.565 3 0.08001 0.24463 959.682 227.842 4 1740.17 696.619 9 0.07777 0.24623 962.738 221.474 7 1742.64 701.172 8 0.07554 0.24750 965.792 215.129 9 1745.1 704.78 5 0.07413 0.24891 968.843 211.106 6 1747.57 708.788 2 0.07345 971.891 209.176 8 1750.04 713.309 0.2505 0.07319 0.25141 974.937 208.426 5 1752.5 715.913 5 0.25186 977.981 208.668 0.07328 1754.96 717.205 8 0.07358 0.25268 981.022 209.546 8 1757.42 719.534 6 0.07427 0.25408 984.061 211.513 9 1759.87 723.516 5 0.07489 0.25558 987.097 213.274 8 1762.33 727.789 5 990.131 214.131 0.07519 1764.78 733.097 0.25744

384

9 9 0.07523 0.25981 993.162 214.245 9 1767.24 739.827 3 0.07465 0.26203 996.191 212.587 6 1769.69 746.158 6 0.07327 0.26356 999.218 208.655 6 1772.14 750.52 8 0.07146 0.26495 1002.24 203.502 6 1774.58 754.458 1 0.06983 0.26711 1005.26 198.851 3 1777.03 760.61 1 0.06914 0.26953 1008.28 196.902 8 1779.47 767.519 8 0.06956 0.27187 1011.3 198.095 7 1781.91 774.165 2 0.07130 0.27462 1014.31 203.05 7 1784.35 782.003 4 0.07420 0.27725 1017.33 211.306 7 1786.79 789.491 4 0.07729 0.27898 1020.34 220.113 9 1789.23 794.43 8 0.07948 0.27991 1023.34 226.331 3 1791.66 797.077 8 0.08007 0.28049 1026.35 228.005 1 1794.09 798.719 4 0.07879 0.28133 1029.35 224.361 1 1796.52 801.116 6 0.07632 0.28206 1032.35 217.33 2 1798.95 803.183 2 0.07349 1035.35 209.287 7 1801.38 805.483 0.28287 0.07172 0.28435 1038.34 204.238 4 1803.81 809.722 8 0.28597 1041.34 204.738 0.0719 1806.23 814.327 6 0.07410 0.28778 1044.33 211.029 9 1808.65 819.472 2 0.07763 0.29012 1047.31 221.061 2 1811.08 826.139 4 0.08142 0.29305 1050.3 231.861 5 1813.49 834.492 7 0.08413 0.29647 1053.28 239.58 6 1815.91 844.216 2 0.08548 0.29953 1056.26 243.416 3 1818.33 852.945 7 0.08547 0.30143 1059.24 243.382 1 1820.74 858.338 1 0.08502 0.30188 1062.22 242.122 8 1823.15 859.634 7 0.08492 0.30020 1065.19 241.821 3 1825.57 854.839 3 0.08594 0.29744 1068.17 244.721 1 1827.97 846.981 3 0.08813 0.29458 1071.14 250.961 3 1830.38 838.831 1 1074.1 259.522 0.09113 1832.79 834.241 0.29296

385

9 9 0.09405 0.29336 1077.07 267.813 1 1835.19 835.378 8 0.09651 1080.03 274.825 3 1837.59 841.162 0.2954 0.09834 0.29812 1082.99 280.039 4 1839.99 848.918 3 0.09975 0.30086 1085.95 284.046 1 1842.39 856.718 2 0.30256 1088.9 287.459 0.10095 1844.79 861.572 7 0.10264 0.30354 1091.86 292.293 8 1847.18 864.367 9 0.10519 0.30324 1094.81 299.551 6 1849.58 863.507 7 0.10860 0.30230 1097.76 309.265 8 1851.97 860.821 3 0.11212 0.30112 1100.7 319.289 8 1854.36 857.474 8 0.11507 0.30002 1103.65 327.678 4 1856.75 854.334 5 0.11707 1106.59 333.365 1 1859.14 853.065 0.29958 0.11782 0.30024 1109.53 335.506 3 1861.52 854.958 4 0.11769 0.30242 1112.46 335.13 1 1863.91 861.178 9 0.11752 1115.4 334.66 6 1866.29 871.376 0.30601 0.11818 1118.33 336.539 6 1868.67 880.887 0.30935 0.12000 0.31182 1121.26 341.711 2 1871.05 887.924 1 0.31306 1124.19 347.828 0.12215 1873.43 891.456 2 0.12408 0.31298 1127.12 353.329 2 1875.8 891.237 5 0.31198 1130.04 356.712 0.12527 1878.17 888.394 6 0.31033 1132.96 356.711 0.12527 1880.55 883.698 7 0.12463 0.30897 1135.88 354.912 8 1882.92 879.809 2 0.12402 0.30787 1138.8 353.16 3 1885.29 876.688 6 0.12414 1141.71 353.509 5 1887.65 872.77 0.3065 0.12511 0.30487 1144.62 356.267 4 1890.02 868.132 1 0.12645 1147.53 360.088 6 1892.38 863.971 0.30341 1150.44 365.254 0.12827 1894.75 860.212 0.30209 0.12999 0.30097 1153.35 370.174 8 1897.11 857.05 9 0.13134 0.30024 1156.25 374.017 7 1899.47 854.953 3

386

0.30031 1159.15 378.067 0.13277 1901.82 855.168 8 0.13435 0.30135 1162.05 382.593 9 1904.18 858.122 6 0.13590 0.30313 1164.95 387.002 7 1906.53 863.195 7 0.13716 0.30505 1167.84 390.582 5 1908.89 868.659 6 0.13810 0.30694 1170.73 393.25 2 1911.24 874.042 6 0.13891 0.30815 1173.62 395.577 9 1913.59 877.471 1 0.13931 0.30803 1176.51 396.696 2 1915.94 877.13 1 0.13947 1179.4 397.158 4 1918.28 873.453 0.30674 0.13978 1182.28 398.044 5 1920.63 867.645 0.3047 0.14014 0.30269 1185.16 399.064 3 1922.97 861.929 2 0.14004 0.30088 1188.04 398.778 3 1925.31 856.788 7 0.13951 0.29872 1190.92 397.274 5 1927.65 850.625 3 0.13880 0.29675 1193.79 395.256 6 1929.99 845.016 3 0.13818 0.29488 1196.66 393.487 5 1932.33 839.699 6 0.29297 1199.53 392.619 0.13788 1934.66 834.262 6 0.13831 0.29165 1202.4 393.849 2 1937 830.501 6 0.13948 0.29138 1205.27 397.187 4 1939.33 829.734 6 0.14111 1208.13 401.837 7 1941.66 832.25 0.29227 0.14242 1210.99 405.569 8 1943.99 836.352 0.29371 0.14280 0.29448 1213.85 406.645 6 1946.32 838.565 8 0.14227 0.29499 1216.71 405.143 8 1948.64 840.006 4 0.29502 1219.57 401.703 0.14107 1950.97 840.105 8 0.13979 0.29482 1222.42 398.07 4 1953.29 839.538 9 0.13919 0.29490 1225.27 396.375 9 1955.61 839.763 8 0.13949 0.29551 1228.12 397.214 4 1957.93 841.493 6 0.14077 0.29634 1230.96 400.857 3 1960.25 843.843 1 0.14250 0.29719 1233.81 405.788 5 1962.56 846.285 9 0.14396 0.29745 1236.65 409.941 3 1964.88 847.011 4

387

0.14522 0.29767 1239.49 413.535 5 1967.19 847.652 9 0.14605 0.29823 1242.33 415.888 2 1969.5 849.237 5 0.14658 0.29947 1245.16 417.41 6 1971.81 852.778 9 0.14688 0.30130 1248 418.257 4 1974.12 857.975 4 0.30336 1250.83 418.047 0.14681 1976.43 863.847 6 0.14649 0.30480 1253.66 417.152 6 1978.74 867.949 7 0.14638 0.30588 1256.49 416.842 7 1981.04 871.019 5 0.14695 0.30612 1259.31 418.47 8 1983.34 871.714 9 0.14866 1262.13 423.341 9 1985.64 871.091 0.30591 0.15109 0.30607 1264.95 430.249 5 1987.94 871.573 9 0.15373 0.30684 1267.77 437.756 1 1990.24 873.765 9 0.15549 1270.59 442.773 3 1992.54 876.986 0.30798 0.15583 0.30880 1273.4 443.749 6 1994.83 879.341 7 0.15464 1276.22 440.35 2 1997.13 877.753 0.30825 0.15297 0.30681 1279.03 435.602 5 1999.42 873.661 3 0.15195 0.30437 1281.84 432.709 9 2001.71 866.715 3 0.30140 1284.64 433.681 0.1523 2004 858.253 2 0.15384 0.29919 1287.45 438.091 9 2006.29 851.976 7 0.15616 1290.25 444.683 4 2008.57 849.508 0.29833 0.15822 0.29917 1293.05 450.557 7 2010.86 851.915 6 0.15989 0.30116 1295.84 455.313 7 2013.14 857.572 2 0.30315 1298.64 458.255 0.16093 2015.42 863.252 7 0.30499 1301.43 461.13 0.16194 2017.7 868.48 3 0.16335 0.30600 1304.22 465.155 3 2019.98 871.353 2 0.16539 0.30529 1307.01 470.978 8 2022.26 869.343 6 0.16757 0.30328 1309.8 477.172 3 2024.54 863.623 7 0.16944 0.30061 1312.59 482.51 8 2026.81 856.002 1 0.29793 1315.37 486.501 0.17085 2029.08 848.379 4

388

0.17246 1318.15 491.096 3 2031.35 841.021 0.29535 0.29287 1320.93 497.352 0.17466 2033.63 833.984 9 0.17823 0.29059 1323.71 507.521 1 2035.89 827.488 7 0.18337 0.28817 1326.48 522.179 9 2038.16 820.596 7 0.19006 0.28503 1329.25 541.212 3 2040.43 811.66 9 0.19732 0.28160 1332.02 561.883 2 2042.69 801.882 5 0.20442 0.27853 1334.79 582.12 9 2044.95 793.132 2 0.21105 0.27590 1337.56 600.987 5 2047.22 785.647 4 0.21721 0.27361 1340.32 618.539 9 2049.48 779.125 3 0.22287 0.27177 1343.09 634.644 4 2051.73 773.887 4 0.22824 0.27022 1345.85 649.947 9 2053.99 769.474 4 0.23308 0.26886 1348.6 663.722 6 2056.25 765.614 9 0.23733 0.26795 1351.36 675.819 4 2058.5 763 1 0.24039 0.26750 1354.11 684.542 8 2060.75 761.721 1 0.24188 0.26780 1356.87 688.788 9 2063.01 762.584 4 0.24157 0.26830 1359.62 687.9 7 2065.26 764.004 3 0.23923 0.26804 1362.36 681.242 9 2067.51 763.262 3 0.23506 0.26648 1365.11 669.347 1 2069.75 758.826 5 0.22928 0.26321 1367.85 652.897 5 2072 749.506 2 0.22235 0.25844 1370.6 633.161 4 2074.24 735.934 6 0.21496 0.25272 1373.34 612.117 3 2076.49 719.657 9 0.20770 0.24629 1376.07 591.45 6 2078.73 701.338 6 0.20113 0.24057 1378.81 572.742 6 2080.97 685.04 3 0.19566 0.23573 1381.54 557.163 5 2083.21 671.277 9 0.19114 0.23168 1384.27 544.302 8 2085.45 659.736 6 0.18780 0.22812 1387 534.77 1 2087.68 649.585 1 0.18553 0.22443 1389.73 528.331 9 2089.92 639.088 5 0.18426 0.22086 1392.46 524.712 9 2092.15 628.928 7

389

0.18355 0.21679 1395.18 522.671 2 2094.38 617.34 8 0.18336 0.21203 1397.9 522.138 5 2096.61 603.772 3 0.18395 0.20760 1400.62 523.816 4 2098.84 591.166 6 0.18511 0.20350 1403.34 527.112 1 2101.07 579.494 7 0.18629 0.20027 1406.06 530.473 2 2103.3 570.28 1 0.18736 0.19709 1408.77 533.536 7 2105.52 561.234 4 0.18839 0.19345 1411.48 536.464 6 2107.75 550.86 1 0.18909 0.19009 1414.19 538.446 2 2109.97 541.314 9 0.18911 0.18638 1416.9 538.516 6 2112.19 530.738 5 0.18251 1419.6 537.131 0.18863 2114.41 519.708 1 0.18829 0.17877 1422.31 536.188 9 2116.63 509.069 5 0.18794 0.17515 1425.01 535.172 2 2118.85 498.751 2 0.18747 0.17152 1427.71 533.837 3 2121.07 488.434 8 0.18689 0.16642 1430.41 532.202 9 2123.28 473.89 1 0.18684 0.15996 1433.1 532.042 3 2125.49 455.514 8 0.18765 1435.79 534.364 8 2127.71 436.585 0.15332 0.18935 0.14668 1438.49 539.198 6 2129.92 417.69 5 0.19166 1441.18 545.779 7 2132.13 401.617 0.14104 0.19473 0.13648 1443.86 554.527 9 2134.34 388.643 4 0.19794 0.13290 1446.55 563.646 1 2136.54 378.448 3 0.20073 0.12904 1449.23 571.609 8 2138.75 367.457 4 0.20265 0.12356 1451.91 577.066 4 2140.95 351.851 3 0.20383 0.11644 1454.59 580.441 9 2143.16 331.59 8 0.20462 0.10804 1457.27 582.676 4 2145.36 307.662 5 0.09852 1459.95 582.636 0.20461 2147.56 280.555 5 0.20377 0.08930 1462.62 580.265 8 2149.76 254.303 6 0.20311 0.08106 1465.29 578.385 7 2151.96 230.826 2 0.20308 0.07462 1467.96 578.284 2 2154.16 212.507 8

390

0.20418 0.06989 1470.63 581.422 4 2156.35 199.023 3 0.20648 0.06697 1473.3 587.976 6 2158.55 190.723 8 0.20935 0.06565 1475.96 596.138 2 2160.74 186.96 7 0.21214 0.06497 1478.62 604.083 2 2162.93 185.025 7 0.21362 0.06362 1481.28 608.311 7 2165.12 181.163 1 0.21354 0.06109 1483.94 608.079 5 2167.31 173.968 4 0.21286 0.05659 1486.6 606.149 8 2169.5 161.163 7 0.21235 0.05027 1489.25 604.697 8 2171.69 143.155 3 0.21329 1491.9 607.377 9 2173.87 120.85 0.04244 0.21600 0.03396 1494.55 615.095 9 2176.06 96.7083 2 0.21958 0.02570 1497.2 625.275 4 2178.24 73.1993 6 0.22306 0.01811 1499.85 635.18 3 2180.43 51.5811 4 0.22537 0.01145 1502.49 641.771 7 2182.61 32.6313 9 0.22561 0.00633 1505.13 642.45 6 2184.79 18.036 4 0.22444 0.00274 1507.78 639.123 7 2186.97 7.81407 4 0.00065 2189.14 1.86721 6

391

Appendix I: X-RAY Diffraction Calculations

AS4 Calculations

Measured:

Given:

Calculations:

392

P25 Calculations

Measured:

Given:

Calculations:

393

T650 Calculations

Measured:

Given:

Calculations:

394

P120 Calculations

Measured:

Given:

Calculations: