Numerical PDE Analysis of Retinal Neovascularization: Mathematical Model Computer Implementation in R

By William E. Schiesser

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

• Language: English
• Publisher: Academic Press
• Release date: Jun 14, 2019
• ISBN: 9780128184530

William E. Schiesser

Dr. William E. Schiesser is Emeritus McCann Professor of Chemical and Biomolecular Engineering, and Professor of Mathematics at Lehigh University. He holds a PhD from Princeton University and a ScD (hon) from the University of Mons, Belgium. His research is directed toward numerical methods and associated software for ordinary, differential-algebraic and partial differential equations (ODE/DAE/PDEs), and the development of mathematical models based on ODE/DAE/PDEs. He is the author or coauthor of more than 16 books, and his ODE/DAE/PDE computer routines have been accessed by some 5,000 colleges and universities, corporations, and government agencies.

Book preview

Numerical PDE Analysis of Retinal Neovascularization

Mathematical Model Computer Implementation in R

First edition

William E. Schiesser

Lehigh University, Bethlehem, PA, United States

Table of Contents

Cover image

Title page

Copyright

Preface

1: PDE Model Formulation

Abstract

Acknowledgement

1.1. Introduction

1.2. Model Specification

References

2: Model Implementation

Abstract

2.1. Introduction

2.2. Method of Lines Routines

2.3. Model Output

2.4. Summary and Conclusions

References

3: Variation of Parameters

Abstract

3.1. Introduction

3.2. MOL Analysis

3.3. Summary and Conclusions

References

4: Detailed PDE Analysis

Abstract

4.1. Introduction

4.2. Analysis of the t Derivatives

4.3. Analysis of PDE RHS Terms

4.4. Summary and Conclusions

5: Oxygen Effect

Abstract

5.1. Introduction

5.2. Four PDE Model

5.3. Summary and Conclusions

References

6: Anti-VEGF Drug Therapy

Abstract

6.1. Introduction

6.2. PDE Model

6.3. Summary and Conclusions

A1: Functions dss004, dss044

A1.1. Function dss004

A1.2. Function dss044

Reference

Index

Copyright

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Preface

W.E. Schiesser     Bethlehem, PA, United States

The intent of this book is to provide a methodology for the analysis of retinal neovascularization. This pathological condition develops as a consequence of inadequate blood flow to the eye so that the oxygen available to the retina is inadequate.

If the retinal blood supply declines, possibly with age or from a visual (ocular) disease, the eye typically responds with the growth of additional capillaries (neovascularization, also termed angiogenesis). The additional blood vessels can develop in the retina and eventually interfere with the phototransduction (conversion of light to electrical signals), leading to impaired vision from, for example, age-related macular degeneration (AMD), and possibly blindness. Insufficient oxygen from adequate blood flow (circulation) is termed hypoxia.

The series of events leading to neovascularization is modeled with a system of partial differential equations (PDEs). Initially, three PDEs are formulated with the dependent variables: capillary tip density, blood capillary density, and vascular endothelial growth factor (VEGF) concentration. The VEGF forms in response to hypoxia, and then leads to capillary tip and blood capillary formation. Later in the book, a fourth PDE is added for oxygen concentration.

reduced to x. The spatial independent variable x is the distance along the retina. The PDEs are dynamic and include variation of the PDE dependent variables with time t.

are approximated by the method of lines (MOL), a procedure for converting PDEs to approximating ordinary differential equations (ODEs). The latter can then be integrated (solved) numerically with a library initial-value ODE integrator.

For each example application, the model PDEs are stated first, including the initial conditions (ICs), boundary conditions (BCs), and model parameters. The coding (programming) of the application is then discussed in terms of documented R routines that are explained in detail, and are available from a download so that the reader/analyst/researcher can use them to confirm the solutions presented in the book, then extend the routines to include additional details and effects that might be of interest. For example, the book concludes with an analysis of anti-VEGF therapy implemented with a term included in the VEGF PDE that decreases the VEGF concentration.

The author would