Computational Chemotaxis Models for Neurodegenerative Disease

The following sections are included:

• Contents
• Preface

This book is predicated on the assumption that a chemotaxis model is a plausible mathematical model for the dynamics of neurodegenerative disease, particularly Alzheimer’s disease (AD). The intent of this book is to provide a generic chemotaxis model for neurodegenerative disease, with emphasis on AD, that can be used to develop new models as experimental observations (data) and understanding through continuing research become available. A central feature of the generic model is an implementation as a series of computer routines written in R² that are explained (documented) in detail…

The model PDEs are first stated in generalized coordinates, which can then be specialized to a particular coordinate system that conforms to the physical system, in the present case, spherical coordinates that conform approximately to a human brain.

In this chapter, we consider the R routines for the numerical method of lines (MOL) solution of eqs. (2.3) to (2.6). Basically, in the MOL, the spatial (boundary value) derivatives in the PDEs are replaced with algebraic approximations, and in the present development, with finite difference (FDs). Then only one independent variable remains, an initial value variable, typically time t, so that a set of ODEs results that approximates the PDEs. This system of ODEs is then integrated numerically by a library initial value ODE integrator.

Since the model MOL/ODEs are central to this approach, we start with the MOL/ODE routine, then consider the main program that calls it.

The model output from the code in Listings 3.1, 3.2 and 3.3 is presented and discussed in this chapter.

A principal advantage of having a computer-based model for a problem system of interest is the opportunity to experiment with the model and observe the resulting solutions. This experimentation can take several forms:

• Variation of the model parameters (usually constants) to observe their effect on the solution. In particular, identifying parameters which are sensitive in the sense that small changes in their values produces significant changes in the solution is important. i.e., the values of those parameters must be established as accurately and reliably as possible.
• Variation of the terms in the model equation. This can include changing the form of the terms, e.g., linear to nonlinear, adding new terms to reflect phenomena that might be considered important in the analysis and deletion of terms which have little or no effect on the solution, or which produce features of the model output that are clearly unrealistic physically.
• Addition or deletion of equations to extend the analysis.
• Formulation of a new model using the methodology implemented in the current model, e.g., the same numerical algorithms and coding (as a prototype)…

We conclude the discussion of chemotaxis model development with some extensions of the preceding models.

The following sections are included:

• Appendix A: Analysis of PDE Chemotaxis Terms
  • Numerical Approximation of PDE Chemotaxis Terms
  • Analysis of the PDE Chemotaxis Terms
• Appendix B: Function dss006

The following section is included:

• Index