PDE Modeling of Tissue Engineering and Regenerative Medicine: Computer Analysis in R

By William E. Schiesser

PDE Modeling of Tissue Engineering and Regenerative Medicine: Computer Analysis in R presents the formulation and computer implementation of mathematical models for the forefront research areas of tissue engineering and regenerative medicine. The mathematical model discussed in this book consists of a system of eight partial differential equations (PDEs) with dependent variables. The computer-based example models are presented through routines coded 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. Includes detailed examples that the reader can execute on modest computers.

• 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 20, 2022
• ISBN: 9780443187414

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

Preface

This book is directed to the computer-based modeling of tissue engineering and regenerative medicine (TERM). As explained in [2],

The goal of tissue engineering is to assemble functional constructs that restore, maintain, or improve damaged tissues or whole organs.

Regenerative medicine is a broad field that includes tissue engineering but also incorporates research on self-healing – where the body uses its own systems, sometimes with help foreign biological material to recreate cells and rebuild tissues and organs. The terms tissue engineering and regenerative medicine have become largely interchangeable, as the field hopes to focus on cures instead of treatments for complex, often chronic, diseases.

The TERM mathematical model discussed in this book consists of a system of eight partial differential equations (PDEs) with dependent variables , , , , , , , [1]. These variables are explained in the following table:

The eight PDE model is coded (programmed) as a set of routines in R, a quality, open-source, scientific programming system [3]. r, z, t are the PDE independent variables. r, z are two spatial variables in cylindrical coordinates reflecting the geometry of the scaffold and seeded stem cells of the tissue construct (TC), and t is time. These independent variables therefore define the spatiotemporal basis of the PDE model. is of particular interest since it expresses the level of the terminally differentiated (TD) cell density, the final goal/product of tissue engineering.

The TERM 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 tissue engineering and regenerative medicine procedure.

The author would welcome reader impressions of the computer-based TC models discussed in this book. Comments can be directed to [e-mail].

References

[1] C-S. Chou, et al., Spatial dynamics of multistage cell lineages in tissue stratification, Biophysical Journal November 2010;90:3145–3154.

[2] https://www.nibib.nih.gov/science-education/science-topics/tissue-engineering-and-regenerative-medicine.

[3] K. Soetaert, J. Cash, F. Mazzia, Solving Differential Equations in R. Heidelberg, Germany: Springer-Verlag; 2012.

1: One PDE model development

Abstract

This book is directed to the computer-based modeling of tissue engineering and regenerative medicine (TERM). In Chapter 1, a one PDE model defining the O2 level in stem cells seeded on a scaffold is derived in detail, including (1) the diffusion PDE, second order in space and first order in time, (2) the initial condition (IC) and (3) the boundary conditions (BCs).

Keywords

Tissue engineering; regenerative medicine; seeded stem cells; tissue construct scaffold; mathematical model; partial differential equation (PDE); initial condition (IC); boundary condition (BC)

Introduction

This book is directed to the computer-based modeling of tissue engineering and regenerative medicine. These areas offer the possibility of implantable, synthetic tissues and organs. If this research and development (R&D) works out (succeeds), the replacement of damaged and diseased tissues and organs will be feasible (and dependency on donor organs will