Spatiotemporal Modeling of Stem Cell Differentiation: Partial Differentiation Equation Analysis in R

By William E. Schiesser

Spatiotemporal Modeling of Stem Cell Differentiation: Partial Differentiation Equation Analysis in R covers topics surrounding how stem cells evolve into specialized cells during tissue formation and in diseased tissue regeneration. As the process of stem cell differentiation occurs in space and time, the mathematical modeling of spatiotemporal development is expressed in this book as systems of partial differential equations (PDEs). In addition, the book explores important feature of six PDE model which can represent, for example, the development of tissue in organs. In addition, the book covers the computer-based implementation of example models through routines coded (programmed) in R.

The routines described in the book are available from a download link so that example models can be executed without having to first study numerical methods and computer coding. The routines can then be applied to variations and extensions of the stem differentiation models, such as changes in the PDE parameters (constants) and the form of the model equations.

• 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 for mathematical models and algorithms
• Authored by a leading researcher and educator in PDE models

• Language: English
• Publisher: Academic Press
• Release date: Sep 7, 2021
• ISBN: 9780323914130

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

Spatiotemporal Modeling of Stem Cell Differentiation

Partial Differentiation Equation Analysis in R

William E. Schiesser

Table of Contents

Cover image

Title page

Copyright

Preface

Chapter 1. One PDE stem cell model

1. Introduction

Chapter 2. Implementation of the One PDE model

2. Introduction

Chapter 3. Six PDE model for stem cell differentiation

3. Introduction

Chapter 4. Implementation of the Six PDE model

4. Introduction

Chapter 5. Variations of the six PDE model

5. Introduction

Index

Copyright

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Preface

Stem cells evolve into specialized cells during tissue formation and diseased tissue regeneration. The process of forming specific cells from stem cells is termed differentiation. Since the process of stem cell differentiation occurs in space and time, the mathematical modeling of this spatiotemporal development is expresed in this book as systems of partial differential equations (PDEs).

Stem cell differentiation progresses in the stages of a lineage as explained in [1].

In developing and self-renewing tissues, terminally differentiated (TD) cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit-amplifying, (TA)) cell stages, which ultimately give rise to TD cells.

The mathematical models for stem cell differentiation based on PDEs as presented in this book start with a basic one PDE model for stem cell density as a function of space and time, , and conclude with detailed six PDE models with the following dependent variables:

Biomolecules 1,2,3 regulate the stem cell differentiation. One transit-amplifying cell is considered in the stem cell lineage, but additional transit-amplifying (intermediate) cells can easily be added (through additional PDEs).

An important feature of the six PDE model is the movement of the tissue upper apex boundary as a function of time, t, which can represent, for example, the development of tissue in an organ.

The computer-based implementation of the example models is presented through routines coded (programmed) in R, a quality, open-source scientific computing system that is readily available from the Internet. Formal mathematics is minimized, e.g., no theorems and proofs. Rather, the presentation is through detailed examples that the reader/researcher/analyst can execute on modest computers. The PDE analysis is based on the method of lines (MOL), an established general algorithm for PDEs, implemented with cubic