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Mathematical Methods to Predict the Dynamic Shape Evolution of Cancer Growth Based on Spatio-Temporal Bayesian and Geometrical Models

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Mathematical Methods to Predict the Dynamic Shape Evolution of Cancer Growth Based on Spatio-Temporal Bayesian and Geometrical Models UNIVERSITY \JAUME I" OF CASTELLON´ Superior School of Technologies and Experimental Sciences DEPARTMENT OF MATHEMATICS MATHEMATICAL METHODS TO PREDICT THE DYNAMIC SHAPE EVOLUTION OF CANCER GROWTH BASED ON SPATIO-TEMPORAL BAYESIAN AND GEOMETRICAL MODELS Author: Advisors: Iulian Teodor Vlad Prof. Dr. Jorge Mateu Prof. Dr. Jos´eJoaquin Gual Arnau Castell´onde la Plana 2015 University \Jaume I" of Castell´on Superior School of Technologies and Experimental Sciences DEPARTMENT OF MATHEMATICS Doctoral Thesis Mathematical Methods to Predict the Dynamic Shape Evolution of Cancer Growth based on Spatio-Temporal Bayesian and Geometrical Models Author: Supervisors: Ing. Iulian Teodor Vlad Prof. Dr. Jorge Mateu Prof. Dr. Jos´eJoaquin Gual Arnau A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in the Computer Mathematics December 2015 Mathematical Methods to Predict the Dynamic Shape Evolution of Cancer Growth based on Spatio-Temporal Bayesian and Geometrical Models Copyright c 2015 - Ing. Iulian Teodor Vlad ALL RIGHT RESERVED Declaration of Authorship I, Ing. Iulian Teodor Vlad, declare that this thesis entitled, 'Mathematical Methods to Predict the Dynamic Shape Evolution of Cancer Growth based on Spatio-Temporal Bayesian and Geometrical Models' and the work presented in it are my own. I confirm that: This work was done wholly or mainly while in candidature for a research degree at this University. Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. Where I have consulted the published work of others, this is always clearly at- tributed. Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. I have acknowledged all main sources of help. Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. DATE: SIGNATURE: ................................................ ................................................. v This dissertation is advanced towards to the University \JAUME I" of Castell´on from Spain, for fulfilment the requirements in order to degree of DOCTOR OF PHILOSOPHY ON MATHEMATICAL COMPUTATION running in the period: oct 2009 - oct 2013, in particular, of fulfill the period of two years (oct 2009 - oct 2011) as well as Formatting Professional Research and another two years (oct 2011 - oct 2013) as Member of the Research Team: Statistical Modeling to Environmental Problems, code 145 coordinated by the Prof. Dr. Jorge Mateu, as requirements of the course 14019 - “Matem´atica Computacional" on the Superior School of Technologies and Experimental Sciences, Department of Mathematics. I certify that I have read this dissertation and that, in my opinion, it is fully ade- quate in scope and quality as a dissertation for the degree of Doctor of Philosophy: Director: Name: Prof. Dr. Jorge Mateu Signature: ........................................ I certify that I have read this dissertation and that, in my opinion, it is fully ade- quate in scope and quality as a dissertation for the degree of Doctor of Philosophy: Co-Director: Name: Prof. Dr. Jos´eJoaquin Gual Arnau Signature: ....................................................... I certify that I have read this dissertation and that, in my opinion, it is fully ade- quate in scope and quality as a dissertation for the degree of Doctor of Philosophy: Committee Member: Name: ........................................ Signature: ........................................ I certify that I have read this dissertation and that, in my opinion, it is fully ade- quate in scope and quality as a dissertation for the degree of Doctor of Philosophy: Committee Member: Name: ........................................ Signature: ........................................ I certify that I have read this dissertation and that, in my opinion, it is fully ade- quate in scope and quality as a dissertation for the degree of Doctor of Philosophy: Committee Member: Name: ........................................ Signature: ........................................ vii Motto: Axiom 5B:\Be good, beyond better, to become the best". I.T. Vlad ix Abstract by Ing. Iulian Teodor Vlad The aim of this research is to observe the dynamics of cancer tumors and to develop and implement new methods and algorithms for prediction of tumor growth. We offer some tools to help physicians for a better understanding and treatment of this disease. Using a prediction method, and comparing with the real evolution a physician can note if the prescribed treatment has the desired effect, and according to this, if necessary, to take the decision of surgical intervention. In this thesis we analyze the spatio-temporal dynamics of shape evolution and we apply these to a particular case of brain tumors. The plan of the thesis is the following. In Chapter 1 we briefly recall some proper- ties and classification of points processes with some examples of spatio-temporal point processes. Chapter 2 presents a short overview of the theory of L´evybases and integration with respect to such basis is given, we recall standard results about spatial Cox processes, and finally we propose different types of growth models and a new algorithm, the Cobweb, which is presented and developed based on the pro- posed methodology. Chapters 3, 4 and 5 are dedicated to present new prediction methods. The implementation in Matlab software comes in Chapter 6. The thesis ends with some conclusion and future research. xi Acknowledgements Special thanks to my advisor Prof. Dr. Jorge Mateu for his collaboration and to University \JAUME I" of Castell´onfor the inestimable help, supporting and invaluable experience offered, an educational institution of study nearby of citi- zens and nature, with a great human and material basis, designed to provide a permanently support to the current development of actual society. Also we want to thank to our colleague Francisco, who offer us the right to use the image acquisitions of the tumor images based on which we have shown the functionality of these algorithms. xiii Contents Title ii Copyright iii Declarationv Approve vii Motto ix Abstract xi Acknowledgements xiii Table of Contents xv List of Figures xix List of Tables xxi Abbreviations xxiii Notations xxv Dedication xxvii 1 Introduction: basics of stochastic processes1 1.1 Temporal processes........................... 12 1.2 Spatial processes............................ 14 1.2.1 Types of spatial processes................... 15 1.3 Point processes............................. 15 1.4 Spatial point processes......................... 27 1.4.1 Regular processes........................ 30 1.4.2 Cluster processes........................ 31 1.4.3 Complete spatial randomness................. 32 1.5 Marked spatial point processes..................... 33 1.6 Poisson processes............................ 35 1.6.1 Homogeneous Poisson process................. 39 xv Contents xvi 1.6.2 Non-homogeneous (inhomogeneous) Poisson process..... 39 1.6.3 Spatial Poisson processes.................... 40 1.7 Cluster processes............................ 41 1.8 Spatio-temporal point processes.................... 46 1.8.1 Earthquake processes...................... 48 1.8.2 Explosion processes....................... 49 1.8.3 Birth-death processes...................... 49 1.8.4 Point patterns sampled in time................ 50 1.8.5 Intensity measures of spatio-temporal point patterns (First- Order Properties)........................ 51 1.8.6 Second-order intensities.................... 54 2 Spatial point processes for tumor growth. Cobweb algorithm1 57 2.1 Spatial Cox point processes...................... 60 2.1.1 L´evy-basedCox processes................... 61 2.1.2 L´evy-basedtumor growth modeling.............. 63 2.2 Modeling tumor growth: a new algorithm............... 69 2.3 Software................................. 76 2.3.1 Input data............................ 76 2.3.2 Procedures that must be fulfilled............... 76 2.4 Real data analysis............................ 78 2.5 Conclusions............................... 81 3 Geometric prediction methods of tumor growth2 83 3.1 Methodology.............................. 86 3.1.1 Shape and growth description................. 86 3.1.2 Normal method......................... 87 3.1.2.1 Curve evolutions................... 93 3.1.3 Radius method......................... 94 3.2 Simulations and application...................... 94 3.2.1 Simulated data: random curves................ 94 3.2.2 Simulated data: parametric curves.............. 97 3.2.3 Real data............................ 98 4 Bayesian prediction of tumor growth3 101 4.1 Data set................................. 103 4.1.1 Image registration........................ 104 4.1.2 Preparing the data....................... 106 4.2 Methodology.............................. 109 4.2.1 Statistical framework...................... 109 4.2.2 Statistical inference....................... 110 1This chapter is based on the published paper: \A geometric approach to cancer growth prediction based on Cox Processes" by Vlad and Mateu (2014)[1] 2This chapter is based on the published paper: \Two handy geometric prediction methods of cancer growth" by Vlad et al. (2015)[2] 3This chapter is based on the published paper: \Bayesian spatio-temporal prediction of cancer dy- namics" by Vlad
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