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A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis Sarah Kim

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A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis Sarah Kim Florida State University Libraries 2015 A Mathematical Model of Cerebral Cortical Folding Development Based on a Biomechanical Hypothesis Sarah Kim Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES A MATHEMATICAL MODEL OF CEREBRAL CORTICAL FOLDING DEVELOPMENT BASED ON A BIOMECHANICAL HYPOTHESIS By SARAH KIM A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2015 Copyright c 2015 Sarah Kim. All Rights Reserved. Sarah Kim defended this dissertation on October 29, 2015. The members of the supervisory committee were: Monica K. Hurdal Professor Directing Dissertation Oliver Steinbock University Representative Richard Bertram Committee Member Nick Cogan Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii Soli Deo Gloria iii ACKNOWLEDGMENTS First of all, I would like to express my deepest gratitude to my advisor, Dr. Monica Hurdal, for her excellent guidance, support, and devotion to this research. She always opened the door to listen to my ideas, and meetings with her in and out of her office always inspired me. She helped me navigate the various and unfamiliar territories I experienced when conducting my research, which will be a great asset for my future career. Her letters of recommendation for scholarships always gave me great opportunities. She has been a tremendous mentor and supporter for me. I am so blessed to have her as my advisor at FSU. I would like to thank my committee members: Dr. Richard Bertram, Dr. Nick Cogan, and Dr. Oliver Steinbock. Thank you for your time, brilliant comments and suggestions, and encouraging words throughout my doctoral journey. I would like to thank the FSU Math Department professors, especially, Dr. Xiaoming Wang, Dr. Giray Okten, Dr. Penelope Kirby, and Dr. Steve Bellenot, for providing great lectures, being supportive, and opening new opportunities throughout my graduate career. I would like to thank my FSU Math Department friends who have spent their precious time with me during my academic journey in a strange place, especially, Sevgi, Angela, Daozhi, Mao, Chaoxu, Diego, Justin, Tim, Arij, Qiuping, and Yingyun. Thank you for your friendship, help, and caring. I also would like to thank my seniors, Debbie and Greg, for their work on this topic. Your contributions were a great guide and formed the basis for my study. I would like to thank my family in South Korea, my father, mother, and younger brother, Joseph, for their sincere prayers and encouragement during this journey. Your prayers remind me that he who began a good work in us will carry it on to completion, especially when I am struggling with my weaknesses. Last but not least, I would like to thank my husband, Yoonhwak, and daugther, Claire Taehee, for their love, support, and prayers. Running together with you makes me keep going for this marathon finish line willingly and joyfully. In addition, I was really thankful for researching on the development of the fetal brain with the life inside of me. iv TABLE OF CONTENTS ListofTables.......................................... viii ListofFigures ......................................... ... ix ListofSymbols......................................... xiii List of Abbreviations . xiv Abstract............................................. ... xv 1 Introduction 1 2 Biology Background 4 2.1 Neuroanatomy and Corticogenesis . 4 2.1.1 BrainStructure................................... 4 2.1.2 Early Development and Ventricular System . 5 2.1.3 Neurons and Glial Cells . 7 2.1.4 Anatomical References . 7 2.2 Cortical Folding Development . 12 2.2.1 Magnetic Resonance Imaging of the Brains of Preterm Infants . 14 2.2.2 Summary of Human Fetal Brain Folding Development . 15 2.3 Biological Hypotheses . 21 2.3.1 Biochemical Hypothesis: Intermediate Progenitor Model (IPM) . 21 2.3.2 Biomechanical Hypothesis: Axonal Tension Hypothesis (ATH) . 25 2.3.3 Biomechanical Hypothesis: Differential Growth Hypothesis (DGH) . 25 2.3.4 Debating Hypotheses . 28 2.4 Conclusions........................................ 29 3 The Proposed Mathematical Model of Cerebral Cortical Folding Development 30 3.1 ModelGeometry ..................................... 30 3.2 Directions of the Applied Axonal Tension Forces . 31 3.3 Magnitudes of the Applied Axonal Tension Forces . 33 3.3.1 Turing Reaction-Diffusion System . 35 3.3.2 IPC Self-amplifications to the Axonal Tension Forces . 37 3.4 Displacements..................................... 38 3.4.1 Theory of Elasticity . 39 3.4.2 Finite Element Formulation . 43 3.5 Conclusions........................................ 47 4 Numerical Results 48 4.1 Parameters and Gyrification Index . 48 4.1.1 Tissue Elasticity and Geometric Size of the Brain . 49 4.1.2 Direction of the Applied Axonal Tension Force . 49 4.1.3 Strength of the Applied Axonal Tension Force . 50 v 4.1.4 GyrificationIndex(GI) .............................. 51 4.2 DomainSize......................................... 54 4.2.1 Bigger vs. Smaller Brains . 54 4.2.2 Thickervs.ThinnerCortices . 57 4.2.3 Summary ...................................... 57 4.3 Directions of the Applied Axonal Tension Forces . 60 4.3.1 Summary ...................................... 62 4.4 Strengths of the Applied Axonal Tension Forces . 65 4.4.1 Weaker vs. Stronger Pulling Tension Forces . 66 4.4.2 Distance between the Applied Forces . 67 4.4.3 Uneven Magnitudes of the Applied Forces . 70 4.4.4 Summary ...................................... 76 4.5 Conclusions........................................ 77 5 Applications to Human Cerebral Cortical Folding Malformations 78 5.1 Decreased Proliferation . 79 5.1.1 Microcephaly and Microlissencephaly . 79 5.1.2 Modeling Microcephaly and Microlissencephaly . 80 5.2 Increased Proliferation . 81 5.2.1 Hemimegalencephaly . 82 5.2.2 Modeling Hemimegalencephaly . 83 5.3 Undermigration . 87 5.3.1 Classic (Type I) Lissencephaly . 88 5.3.2 Modeling Classic Lissencephaly . 88 5.4 Deranged Organization . 90 5.4.1 Polymicrogyria . 90 5.4.2 Modeling Polymicrogyria . 92 5.5 Conclusions........................................ 94 6 Conclusions 96 6.1 Summary .......................................... 96 6.2 Ongoing and Future Work . 98 6.2.1 Three-dimensionalModel . 98 6.2.2 Dynamic Model . 100 6.2.3 Drug Use Model . 101 6.3 ConcludingRemarks .................................. 101 Appendices A Derivations 103 A.1 MethodsofWeightedResidual . 103 A.2 Linear Quadrilateral Element . 104 A.3 Gauss-Legendre Quadrature Rule . 106 A.4 DerivingTuringConditions . 107 A.4.1 Turing Criterion: Linear Stability in the Absence of Diffusion . 107 A.4.2 Turing Criterion: Diffusion-Driven Instability . 109 vi B Figures 112 B.1 DomainSize........................................ 113 B.1.1 Bigger vs. Smaller Brains . 113 B.1.2 Thickervs.ThinnerCortices . 116 B.2 Directions of the Applied Axonal Tension Forces . 118 B.3 Strengths of the Applied Axonal Tension Forces . 123 B.3.1 Weaker vs. Stronger Pulling Tension Forces . 123 B.3.2 DistancebetweentheAppliedForces . 125 B.4 Preliminary numerical results of a three-dimensional model . 126 References......................................... 128 BiographicalSketch ..................................... 138 vii LIST OF TABLES 4.1 Brain size and the GI when n0=5............................. 59 4.2 Force direction and the GI when n0=3.......................... 61 4.3 Force direction and the GI when n0=5.......................... 62 4.4 Profiles of the applied forces based on the irregular Turing patterns when Cu = 625 . 73 5.1 Causes of cortical folding malformations and simulation parameters . 79 5.2 Profiles of the forces applied in Figures 5.4 and 5.5 . 86 A.1 n-point Gauss-Legendre quadrature weights and points . 107 viii LIST OF FIGURES 2.1 Sixmajorlobes...................................... 5 2.2 Gyriandsulci ...................................... 6 2.3 Serial MR imaging of normal brain growth . 8 2.4 Major gyri and sulci (lateral view) . 9 2.5 Major gyri and sulci (medial view) . 9 2.6 Sixembryoniclayers................................ 10 2.7 Cerebralventricles ................................ 10 2.8 Volumetric changes of the fetal brain, germinal matrix , and lateral ventricles . 11 2.9 Neuron ............................................ 11 2.10 Radial glial cells . 12 2.11 Anatomical references regarding direction . 13 2.12 Sections of the human brain . 13 2.13 MR images of a preterm infant born at the 25th week of GA . 16 2.14 MR images of a preterm infant born at the 26th week of GA . 16 2.15 MR images of a preterm infant born at the 28th week of GA . 17 2.16 MR images of a preterm infant born at the 30th week of GA . 18 2.17 MR images of a preterm infant born at the 32nd week of GA . 19 2.18 MR images of a preterm infant born at the 34th week of GA . 20 2.19 MR images of a preterm infant born at term (approximately 40 weeks of GA) . 20 2.20 Radial unit hypothesis . 22 2.21 Intermediate progenitor hypothesis and model . 23 2.22 Subventricular zone thickness and cortical folding . 24 2.23 Axonal tension hypothesis . 26 2.24 Compactness of cortical wiring and cortical folding . 27 ix 2.25 Camera lucida drawings of brain layers . 27 3.1 Representation of the human brain . 31 3.2 Components of the applied force on the semi-circular model cortex . 32 3.3 The angle φ .........................................
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