Differential Equation Analysis in Biomedical Science and Engineering ：Ordinary Differential Equation Applications with R

Publication subTitle ：Ordinary Differential Equation Applications with R

Author： William E. Schiesser

Publisher： John Wiley & Sons Inc‎

Publication year： 2014

E-ISBN: 9781118705391

P-ISBN(Paperback): 9781118705483

P-ISBN(Hardback): 9781118705483

Subject： R318 Biomedical

Keyword： Differential Equation Analysis in Biomedical Science and Engineering Ordinary Differential Equation Applications with R William E. Schiesser ODEs/PDEs apoptosis cellular differentiation biomolecular science and engineering ODE analysis and bioreactor dynamics diabetes glucose tolerance test apoptosis programmed cell death dynamic neuron model stem cell differentiation acetylecholine neurocycle tuberculosis with differential infectivity corneal curvature stiff ODE integration

Language： ENG

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Description

Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields

With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-world ODE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the computed numerical solutions and is a valuable resource for dealing with a broad class of linear and nonlinear ordinary differential equations.

The author’s primary focus is on models expressed as systems of ODEs, which generally result by neglecting spatial effects so that the ODE dependent variables are uniform in space. Therefore, time is the independent variable in most applications of ODE systems. As such, the book emphasizes details of the numerical algorithms and how the solutions were computed. Featuring computer-based mathematical models for solving real-world problems in the biological and biomedical sciences and engineering, the book also includes:

R routines to facilitate the immediate use of computation for solving differential equation problems without having to first learn the basic concepts of numerical analysis and programming for ODEs
Models as systems of ODEs with explanations of the associated chemistry, physics, biology, and physiology as well as the algebraic equations used to calculate intermediate variables
Numerical solutions of the presented model equations with a discussion of the important features of the solutions
Aspects of general ODE computation through various biomolecular science and engineering applications

Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R is an excellent reference for researchers, scientists, clinicians, medical researchers, engineers, statisticians, epidemiologists, and pharmacokineticists who are interested in both clinical applications and interpretation of experimental data with mathematical models in order to efficiently solve the associated differential equations. The book is also useful as a textbook for graduate-level courses in mathematics, biomedical science and engineering, biology, biophysics, biochemistry, medicine, and engineering.

Chapter

Chapter 1 Introduction to Ordinary Differential Equation Analysis: Bioreactor Dynamics

1.1 Introduction

1.2 A 7 x 7 ODE System for a Bioreactor

1.3 In-Line ODE Routine

1.4 Numerical and Graphical Outputs

1.5 Separate ODE Routine

1.6 Alternative Forms of ODE Coding

1.7 ODE Integrator Selection

1.8 Euler Method

1.9 Accuracy and Stability Constraints

1.10 Modified Euler Method as a Runge-Kutta Method

1.11 Modified Euler Method as an Embedded Method

1.12 Classic Fourth-Order Runge-Kutta Method as an Embedded Method

1.13 RKF45 Method

References

Chapter 2 Diabetes Glucose Tolerance Test

2.1 Introduction

2.2 Mathematical Model

2.2.1 Glucose Balance

2.2.2 Insulin Balance

2.3 Computer Analysis of the Mathematical Model

2.3.1 ODE Integration by lsoda

2.3.2 ODE Integration by RKF45

2.3.3 ODE Integration with RKF45 in a Separate Routine

2.3.4 h Refinement

2.3.5 p Refinement

2.4 Conclusions

References

Chapter 3 Apoptosis

3.1 Introduction

3.2 Mathematical Model

3.3 Main Program

3.4 ODE Routine

3.5 Base Case Output

3.6 Base Case with Variation in ICs

3.7 Variation in ODEs

3.8 Selection of Units

3.9 Model Solution with RKF45

3.10 Conclusion

Reference

Chapter 4 Dynamic Neuron Model

4.1 Introduction

4.2 The Dynamic Neuron Model

4.3 ODE Numerical Integration

4.3.1 Explicit Euler Integration

4.3.2 Numerical and Graphical Solutions

4.3.3 Evaluation and Plotting of the ODE Derivative Vector

4.3.4 p Refinement

4.4 Conclusions

References

Chapter 5 Stem Cell Differentiation

5.1 Introduction

5.2 Model Equations

5.3 R Routines

5.3.1 Main Program

5.3.2 ODE Routine

5.3.3 Numerical and Graphical Output

5.3.4 Analysis of the Terms in the ODEs

5.3.5 Stable States

5.4 Summary

Reference

Chapter 6 Acetylcholine Neurocycle

6.1 Introduction

6.2 ODE Model

6.3 Numerical Solution of the Model

6.3.1 ODE Routine

6.3.2 Main Program

6.4 Model Output

6.4.1 Equilibrium Solution

6.4.2 Nonequilibrium Solutions

6.4.3 Analysis of the Terms in the ODEs

6.5 ODE/PDE Model

Appendix A1: IC Vector by a Differential Levenberg Marquardt Method

A1.1 ODE Jacobian Matrix

A1.2 Newton's Method

A1.3 Steepest Descent Method

A1.4 The Levenberg Marquardt Method

A1.5 Differential Newton’s Method

A1.6 Differential Steepest Descent Method

A1.7 Differential Levenberg Marquardt Method

A1.8 Solution for the IC Vector of the 8 x 8 ODE System

References

Chapter 7 Tuberculosis with Differential Infectivity

7.1 Introduction

7.2 Mathematical Model

7.3 R Routines for the ODE Model

7.3.1 ODE Routine

7.3.2 Main Program

7.4 Model Output

7.5 Conclusions

References

Chapter 8 Corneal Curvature

8.1 Introduction

8.2 Model Equations

8.3 Method of Lines Solution

8.4 R Routines

8.4.1 Main Program

8.4.2 ODE Routine

8.5 Numerical Solution

8.6 Error Analysis of the Numerical Solution

8.7 Library Routines for Differentiation in Space

8.8 Summary

References

Appendix A1 Stiff ODE Integration

A1.1 Introduction

A1.2 Analytical Solution of Second-Order ODE System

A1.3 Eigenvalue Stability Analysis

A1.4 BDF Methods for Stiff ODEs

A1.5 R Program for First-Order BDF Method

A1.6 Numerical Output from the BDF Integration

A1.7 Alternative Programming of the BDF Integration

A1.8 Second-Order BDF Integration

A1.9 Third-Order BDF Integration

A1.10 Conclusions

References

Index

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