In response to an infection (e.g., from pathogens such as bacteria and viruses), the immune system can deplete macrophages (specialized white blood cells) and produce cytokines that are pro-inflammatory or anti-inflammatory. This counterproductive autoimmune response is represented mathematically as nonlinear chemotaxis diffusion.
This book is directed to the computer-based modeling of chemotaxis inflammation. The spatiotemporal analysis is
based on a model of three partial differential equations (PDEs).
The three PDE model is coded (programmed) as a set of routines in R, a quality, open-source, scientific programming system. The numerical integration (solution) of the PDEs is by the method of lines (MOL).
The three PDE model can be used for computer-based experimentation, for example, parameter variation and changes in the model equations or alternate models, to enhance a quantitative understanding of a postulated
inflammation.
This experimentation is illustrated by chapters pertaining to: (1) the computation and display of the PDE time derivatives, (2) the RHS terms of the PDEs with emphasis on the chemotaxis terms, (3) parameter variations to demonstrate parameter effects and sensitivities and (4) additonal terms in the PDEs to include PDE coupling and extensions of the basic PDE model.
Author(s): William E. Schiesser
Publisher: CRC Press
Year: 2022
Language: English
Pages: 151
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Contents
Preface
Chapter 1: PDE Chemotaxis Model Formulation
1.1. Introduction
1.2. Three PDE model coordinatefree
1.3. Three PDE model formulation in spherical coordinates
Summary and conclusion
References
Chapter 2: PDE Chemotaxis Model Implementation
2.1. Introduction
2.2. Coding of the chemotaxis model
2.2.1. Main program
2.2.2. ODE/MOL routine
2.2.3. Numerical, graphical output
Summary and conclusion
References
Chapter 3: Analysis of the Chemotaxis Model Time Derivatives
3.1. Introduction
3.2. Analysis of the chemotaxis model time derivatives
3.2.1. Main program
3.2.2. ODE/MOL routine
3.2.3. Numerical, graphical output
Summary and conclusion
References
Chapter 4: Analysis of the Chemotaxis PDE Model Terms
4.1. Introduction
4.2. R routines for PDE RHS, LHS terms
4.2.1. Main program
4.2.2. ODE/MOL routine
4.2.3. Numerical, graphical output
Summary and conclusion
References
Chapter 5: Sensitivity Analysis of the Chemotaxis PDE Model Parameters
5.1. Introduction
5.2. R routines for Case 1
5.3. R routines for Case 2
5.4. R routines for Case 3
Summary and conclusion
References
Chapter 6: Extensions of the Three PDE Chemotaxis Model
6.1. Introduction
6.2. PDE chemotaxis model with anti-inflammatory drug
6.2.1. Main program
6.2.2. MOL/ODE routine
6.2.3. Numerical, graphical output
6.3. Multicomponent extension of PDE chemotaxis model
6.3.1. Main program
6.3.2. ODE/MOL routine
6.3.3. Numerical, graphical output
Summary and conclusion
References
Appendix A1: Functions dss004, dss044
A1.1. dss004 listing
A1.2. dss044 listing
Appendix A2: Accuracy of Numerical PDE Solutions
A2.1. h refinement
A2.2. p refinement
Index
