Computational Chemotaxis Models For Neurodegenerative Disease

by William E Schiesser

Overview

The mathematical model presented in this book, based on partial differential equations (PDEs) describing attractant-repellent chemotaxis, is offered for a quantitative analysis of neurodegenerative disease (ND), e.g., Alzheimer's disease (AD). The model is a representation of basic phenomena (mechanisms) for diffusive transport and biochemical kinetics that provides the spatiotemporal distribution of components which could explain the evolution of ND, and is offered with the intended purpose of providing a small step toward the understanding, and possible treatment of ND.The format and emphasis of the presentation is based on the following elements:In other words, a methodology for numerical PDE modeling is presented that is flexible, open ended and readily implemented on modest computers. If the reader is interested in an alternate model, it might possibly be implemented by: (1) modifying and/or extending the current model (for example, by adding terms to the PDEs or adding additional PDEs), or (2) using the reported routines as a prototype for the model of interest.These suggestions illustrate an important feature of computer-based modeling, that is, the readily available procedure of numerically experimenting with a model. The current model is offered as only a first step toward the resolution of this urgent medical problem.

Product Details

ISBN-13:	9789813208919
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	04/19/2017
Pages:	172
Product dimensions:	5.90(w) x 9.00(h) x 0.60(d)

Table of Contents

Preface vii

Chapter 1 Introduction 1

Appendix A1.1 Diffusion Models 4

Chapter 2 3-PDE Chemotaxis Model 7

(2.1) 3-PDE Model, Orthogonal Coordinates 7

(2.2) Vector Operators in Spherical Coordinates 8

(2.3) ID, 3-PDE Model, Spherical Coordinates 9

(2.4) Model Parameters 15

Chapter 3 R Routines 17

(3.1) ODE/MOL Routine 17

(3.2) Main Program 23

(3.3) Subordinate Routine 33

Chapter 4 Model Output 35

(4.1) PDE Solution Vector 35

(4.2) PDE Solution Vector with t Derivative Vectors 44

(4.3) PDE Solution Vector with t Derivative Vectors and RHS Terms 56

Chapter 5 Model Variants 85

(5.1) Effect of the Attractor/Repellent 86

(5.2) Alternate Boundary Conditions 104

(5.3) Time-limited Source Function 109

(5.4) Summary and Conclusions 117

Chapter 6 Extensions and Conclusions 119

(6.1) Alternate Production/Depletion Rates 119

(6.2) Nonlinear Diffusion 120

(6.3) Additional Cells and Chemicals 121

(6.4) Conclusions 122

Appendix A Analysis of PDE Chemotaxis Terms 125

(A.1) Numerical Approximation of PDE Chemotaxis Terms 125

(A.2) Analysis of the PDE Chemotaxis Terms 140

Appendix B Function dss006 153

Index 159