Numerical PDE Analysis of Retinal Neovascularization

Mathematical Model Computer Implementation in R

William E. Schiesser

• Publisher: Academic Press
• Year: 2019
• Print ISBN:
  9780128184523
• eBook ISBN:
  9780128184530
• Topic: Biological Sciences
• Subtopic: Biotechnology
• Index: Biological Sciences

Numerical PDE Analysis of Retinal Neovascularization Mathematical Model Computer Implementation in R provides a methodology for the analysis of neovascularization (formation of blood capillaries) in the retina. It describes the starting point—a system of three partial differential equations (PDEs)—that define the evolution of (1) capillary tip density, (2) blood capillary density and (3) concentration of vascular endothelial growth factor (VEGF) in the retina as a function of space (distance along the retina), x, and time, t, the three PDE dependent variables for (1), (2) and (3), and designated as u1(x, t), u2(x, t), u3(x, t), amongst other topics. - Includes PDE routines based on the method of lines (MOL) for computer-based implementation of PDE models - Offers transportable computer source codes for readers in R, with line-by-line code descriptions as it relates to the mathematical model and algorithms - Authored by a leading researcher and educator in PDE models

1

PDE Model Formulation

Abstract

In Chapter 1, a system of three simultaneous, 1D partial differential equations (PDEs) is derived as a model for retinal neovascularization. The dependent variables of the PDE model are (1) capillary tip density, (2) blood capillary density, and (3) vascular endothelial growth factor (VEGF) concentration. The solution of the PDE model gives the dependent variables as a function of distance along the retina and time (the spatiotemporal distribution).

Keywords

Partial differential equations (PDEs); PDE model derivation; Conservation principle; Vascular endothelial growth factor (VEGF); Capillary tip density; Blood capillary density

Acknowledgement

The review and suggestions of Dr. Mark A. Pavilack are gratefully acknowledged.

1.1 Introduction

This book details a mathematical model (system of equations) for the evolution of retinal neovascularization. To explain some of these terms:

• • The retina is the part of the eye that receives light and converts it into an electrical signal, a process termed phototransduction or simply transduction. The electrical signal then travels along the optic nerve to the brain where the sense of vision originates.
• • This remarkable series of events which takes place on a time...

Table of contents

• Cover image
• Title page
• Table of Contents
• Copyright
• Preface
• 1: PDE Model Formulation
• 2: Model Implementation
• 3: Variation of Parameters
• 4: Detailed PDE Analysis
• 5: Oxygen Effect
• 6: Anti-VEGF Drug Therapy
• A1: Functions dss004, dss044
• Index