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

Über die Autorin bzw. den Autor

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.

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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. The starting point is 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), designated as u1(x, t), u2(x, t), u3(x, t), respectively, result from the numerical integration (solution) of the three PDEs.

Homogeneous (zero) initial conditions (ICs) and homogeneous Neumann boundary conditions are specified, but these can easily be changed in the R routines that implement the PDEs, ICs and BCs (the programming is in R, a quality, open-source scientific computing system). Initially, the model dependent variables move away from the homogeneous ICs in response to a source term for VEGF in the PDE for u3(x, t). This three PDE model is used to demonstrate the coding and execution of the R routines. The basic numerical algorithm is the method of lines (MOL), a procedure for converting PDEs to systems of approximating ordinary differential equations (ODEs). The latter can then be integrated (solved) numerically with an R library initial-value ODE integrator.

Then the model is extended to four PDEs through the addition of a PDE for oxygen concentration, designated u4(x, t). The model moves away from the homogeneous ICs for u1, u2, u3 in response to a depletion term in the u4(x, t) PDE that represents developing

hypoxia (oxygen deficiency) in the retina. The IC for u4 is a uniform, normal (healthy) level of oxygen and the u4 PDE gives the drop in oxygen concentration from the IC that leads to VEGF production.

Finally, the four PDE model is extended to include the delivery of a therapeutic anti- VEGF drug represented as a source term in the u3(x, t) PDE. The anti-VEGF drug delivery term can be varied as a function of t to represent the scheduled application of the drug therapy, for example, through one injection or repeated injections. The book develops a methodology that can be used to study retinal vascularization. The author intends this to provide a small step toward a better understanding and treatment of conditions such as prenatal neovascularization, diabetic retinopathy and age-related macular degeneration (AMD).