Florida State University Libraries 2017 Modeling Cortical Folding Patterns on a Growing Oblate Spheroid Domain Raymond Morie 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 MODELING CORTICAL FOLDING PATTERNS ON A GROWING OBLATE SPHEROID DOMAIN By RAYMOND MORIE A Thesis submitted to the Department of Mathematics in partial fulfillment of the requirements for graduation with Honors in the Major Degree Awarded: Spring 2017 The members of the Defense Committee approve the thesis of Raymond Morie defended on April 26th, 2017. Monica K. Hurdal Thesis Director Eric Klassen Committee Member James Justus Outside Committee Member Richard Bertram Committee Member ii TABLE OF CONTENTS Abstract............................................. ... iv 1 Introduction 1 2 Biology and Models of the Cerebral Cortex 3 2.1 Neuroanatomy and Development . 3 2.1.1 Three- and Five-Vesicle Stages of Development . 5 2.2 Biological Theories of Development . 6 2.2.1 The Radial Unit Hypothesis . 6 2.2.2 Intermediate Progenitor Hypothesis . 7 2.2.3 Intermediate Progenitor Model . 7 2.2.4 Axonal Tension Hypothesis . 8 2.3 Holoprosencephaly and Ventriculomegaly . 9 2.3.1 Characteristics and Development of Ventriculomegaly . 9 2.3.2 Characteristics and Development of Holoprosencephaly . 10 2.4 Mathematical Models of Cortical Folding . 11 2.4.1 StaticModels.................................... 11 2.4.2 Growing Prolate Spheroid . 11 2.5 Conclusions........................................ 12 3 Turing Systems 13 3.1 StaticDomains ....................................... 13 3.2 Kinetics ......................................... 14 3.3 Growing Oblate Spheroid Coordinates . 14 3.4 ExponentialGrowth .................................. 15 3.5 LogisticGrowth...................................... 17 3.6 Conclusions........................................ 18 4 Results 19 4.1 Exponentially Growing Oblate Spheroid Simulations . 20 4.1.1 Comparison to Static and Growing Prolate Spheroid Domain . 20 4.2 Logistically Growing Oblate Spheroid Simulations . 21 5 Conclusions and Future Directions 33 Bibliography .......................................... 35 iii ABSTRACT Many of the mechanisms involved in brain development are not definitively understood. This in- cludes the mechanism by which the folds of the cerebral cortex, called gyri and sulci, are formed. This research adopts the Intermediate Progenitor Model, which emphasizes genetic chemical fac- tor control. This model is based on a hypothesis that activation of radial glial cells produces intermediate progenitor cells that in turn correspond to gyral wall formation. In previous work, Toole and Hurdal developed two biomathematical models that use a Turing reaction-diffusion system on a prolate spheroid domain. These models examined the effect that different domain growth functions had on cortical folding pattern formation. In this research, we modify those models to utilize an oblate spheroid domain. By analyzing the role of various parameters, we seek to use this modified domain to model diseases such as holoprosencephaly and ventriculomegaly, which alter the shape of the lateral ventricle. iv CHAPTER 1 INTRODUCTION Fundamental to the human experience is our ability to discern and model patterns found in our environment. This includes modeling the various phenomena that occur in our body. Fascinated by the way a homogeneous group of cells could differentiate into the various structures of the embryo, Alan Turing developed a system of differential equations capable of producing a variety of patterns formed in nature in his 1952 paper The Chemical Basis of Morphogenesis[19]. In this thesis, we utilize a Turing system to model the folding patterns of the cerebral cortex [17]. Previous work done by Toole and Hurdal created a model of cortical folding on a growing domain. A prolate spheroid, or an ellipsoid created by rotating an ellipse about its major axis, was used as the domain to model the shape the lateral ventricle. The model investigated the role of growth on cortical pattern formation and this was applied to simulate various diseases related to cortical folding. For the purposes of this thesis, we want to further investigate the role of domain shape in cortical pattern formation by using an oblate spheroid, or an ellipse rotating about its minor axis, as our domain. This allows us to model diseases that distort the shape of the lateral ventricles and cerebral cortex. In Chapter 2, we discuss the various structures of neuroanatomy and what is known about the developmental process of the nervous system. We present a number of hypotheses that describe a plausible genetic chemically-driven mechanism for which cortical folds are formed, and a competing hypothesis that emphasizes physical tension. We also present three previous models of cortical folding to illustrate the necessity of this new model. In Chapter 3, Turing systems are discussed in more detail. We consider the effects of domain growth versus the static domain case. Oblate spheroidal coordinates were derived from previous equations utilizing prolate spheroid coordiates. Lastly, two domain growth functions, an exponential function and a logistic function, are presented. The Turing pattern generating capabilities are examined and the connections to biological systems are explained. 1 Chapter 4 presents the simulation results of the two models presented in Chapter 3 and discusses the findings with respect to the results of previous models. The parameters utilized and altered are given. Finally, Chapter 5 presents the conclusions to the investigation on domain shape on growing domains. Future directions for models utilizing Turing systems are discussed. 2 CHAPTER 2 BIOLOGY AND MODELS OF THE CEREBRAL CORTEX In order to understand the cortical folding patterns of the cerebral cortex, it is imperative to understand the key structures that are associated with cortical development. In this section, the central nervous system, the ventricular system, the germinal matrix, brain cells and structures and different theories of cortical development will be defined and discussed. In addition, the diseases ventriculomegaly and holoprocencephaly will be considered with respect to their influence on brain size and shape. 2.1 Neuroanatomy and Development The major features of the brain on a macroscopic level are the cerebrum, cerebellum, and brain stem. The cerebrum is separated into two hemispheres (left and right), which are connected by a structure called the corpus callosum [13]. Each hemisphere is made up of four lobes: Frontal, parietal, occipital, and temporal based on their different functions and locations associated to each lobe. The ventricular system is made up of four ventricles. The first two ventricles are called the lateral ventricles, and there is one in each hemisphere. During development the lateral ventricles are shaped approximately like a prolate spheroid, a three-dimensional shape of revolution formed by rotating an ellipse about its major axis. In maturity, these lateral ventricles are c-shaped with posterior horns that extend backwards towards the occipital lobe. These lateral ventricles are attached to a third ventricle by the interventricular formina. The third ventricle is connected to a fourth ventricle which is located in the brainstem by the cerebral aqueduct. The ventricular system produces an iron-rich fluid called cerebral spinal fluid, which brings in nutrients, removes toxins and creates a physical cushion between the brain and the skull. Lining the walls of the lateral ventricles are the ventricular zone and the subventricular zone. These are layers of proliferative cells that are involved in cortical development. Passing from the ventricular 3 Figure 2.1: Major structures of the brain, figure adapted from [20]. zone to the subventricular zone involves continuing radially outward from the inside of one of the lateral ventricles. At about seven weeks gestational age, the subventricular zone produces a structure called the germinal matrix. This term has been used in literature to refer to both the part of the subventricular zone located ventrolateral to the lateral ventricle and extending along its lateral wall [8], and as a synonym for both the ventricular zone and the subventricular zone, referring to the collection of germ cells that are precursors to neurons in the cortex [3]. For the purposes of this thesis, the former definition is used. Through magnetic resonance imaging (MRI) data and computational reconstruction of 3- dimensional images, the growth of the germinal matrix has been examined [8]. The volume of the germinal matrix is found to increase exponentially from 11 − 23 weeks gestational age, and rapidly decrease from 25-28 weeks gestational age. Though there is some discrepancy in the liter- ature on when exactly cortical folding is considered to begin, it is clear that it and other crucial events related to cortical folding development coincide with the period of exponential germinal matrix growth. The two types of cells in the central nervous system are neurons and glial cells, with glial cells accounting for 90% of the total amount. Glial cells support neuron structure and provide insulation 4 which helps messages to be transmitted more efficiently, whereas neurons encode information by electrical impulses called action potentials that are transmitted in between them. The major components of a neuron are the soma (cell body), dendrites,