Modeling of Post-Myocardial Infarction: ODE/PDE Analysis with R

By William E. Schiesser

Modeling of Post-Myocardial Infarction: ODE/PDE Analysis with R presents mathematical models for the dynamics of a post-myocardial (post-MI), aka, a heart attack. The mathematical models discussed consist of six ordinary differential equations (ODEs) with dependent variables Mun; M1; M2; IL10; Ta; IL1. The system variables are explained as follows: dependent variable Mun = cell density of unactivated macrophage; dependent variable M1 = cell density of M1 macrophage; dependent variable M2 = cell density of M2 macrophage; dependent variable IL10 = concentration of IL10, (interleuken-10); dependent variable Ta = concentration of TNF-a (tumor necrosis factor-a); dependent variable IL1 = concentration of IL1 (interleuken-1).

The system of six ODEs does not include a spatial aspect of an MI in the cardiac tissue. Therefore, the ODE model is extended to include a spatial effect by the addition of diffusion terms. The resulting system of six diffusion PDEs, with x (space) and t (time) as independent variables, is integrated (solved) by the numerical method of lines (MOL), a general numerical algorithm for PDEs.

• 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: Aug 23, 2023
• ISBN: 9780443136122

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.

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Preface

Survivors of a myocardial-infaction (MI) are likely to experience a reduced (impaired) cardiac function. This results from a series of post-MI biomolecular reactions that are modeled in this book by systems of ordinary and partial differential equations (ODE/PDEs) [1–3].

Initially, monocytes and myocytes are produced in a post-MI that then react to produce macrophages and cytokines, that may adversely affect the cardiac tissue such as inflammation and reduction (weakening) of the extra cellular matrix (ECM).

The first model developed and analyzed by computer-based numerical methods is a system of six ODEs with time as the independent variable and the following dependent variables:

The system of six ODEs does not include a spatial aspect of a MI in the cardiac tissue. Therefore, the ODE model is extended to include a spatial effect by the addition of diffusion terms. The resulting system of six diffusion PDEs, with x (space) and t (time) as independent variables, is integrated (solved) by the numerical method of lines (MOL), a general numerical algorithm for PDEs.

The infarction of the MI can be spatially variable within the PDE diffusion (cardiac tissue) model. This spatial variation is analyzed through changes of the monocyte and myocyte volumetric generation rates.

The properties of the six PDE model with dependent