Ebook: Compartmental Modeling and Tracer Kinetics

This monograph is concerned with mathematical aspects of compartmental an­ alysis. In particular, linear models are closely analyzed since they are fully justifiable as an investigative tool in tracer experiments. The objective of the monograph is to bring the reader up to date on some of the current mathematical prob­ lems of interest in compartmental analysis. This is accomplished by reviewing mathematical developments in the literature, especially over the last 10-15 years, and by presenting some new thoughts and directions for future mathematical research. These notes started as a series of lectures that I gave while visiting with the Division of Applied ~1athematics, Brown University, 1979, and have developed in­ to this collection of articles aimed at the reader with a beginning graduate level background in mathematics. The text can be used as a self-paced reading course. With this in mind, exercises have been appropriately placed throughout the notes. As an aid in reading the material, the e~d of a proof is indicated by ~. Sub­ section titles are utilized to make it easier for the reader to skim over detailed material on a first reading and make the entire manuscript somewhat more accessible, especially to nonmathematicians in the biosciences. The preparation of this monograph has been a long task that would not have been completed without the influence of a number of individuals. I am especially indebted to H. T. Banks, J. W. Drane, J. Eisenfe1d, J. A. Jacquez, D. J.

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Content:
Front Matter....Pages I-VII
Compartmental Systems....Pages 1-6
Elementary Compartmental Models....Pages 7-12
First-order Chemical Reactions....Pages 13-14
Environmental Studies....Pages 15-21
Nonlinear Compartmental Models....Pages 22-29
The General Compartmental Model....Pages 30-32
Tracer Kinetics in Steady State Systems....Pages 33-38
Uptake of Potassium by Red Blood Cells....Pages 39-42
Standard Types of Tracer Experiments....Pages 43-44
Analytical Solution of the Tracer Model....Pages 45-49
System Structure and Connectivity....Pages 50-54
System Eigenvalues and Stability....Pages 55-68
The Inverse of a Compartmental Matrix....Pages 69-78
Mean Times and the Inverse Matrix....Pages 79-89
Solution of the Steady State Problem for SEC Systems....Pages 90-98
Structural Identification of the Model....Pages 99-119
Necessary and Sufficient Conditions for Identifiability....Pages 120-125
A Simple Test for Nonidentifiability....Pages 126-145
Computation of the Model Parameters....Pages 146-156
An Alternative Approach to Identification....Pages 157-170
Controllability, Observability, and Parameter Identifiability....Pages 171-187
Model Identification from the Transfer Function Equations....Pages 188-207
The Parameter Estimation Problem....Pages 208-239
Numerical Simulation of the Model....Pages 240-251
Identification of Compartment Volumes....Pages 252-267
A Discrete Time Stochastic Model of a Compartmental System....Pages 268-285
Closing Remarks....Pages 286-288
Back Matter....Pages 289-304

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Content:
Front Matter....Pages I-VII
Compartmental Systems....Pages 1-6
Elementary Compartmental Models....Pages 7-12
First-order Chemical Reactions....Pages 13-14
Environmental Studies....Pages 15-21
Nonlinear Compartmental Models....Pages 22-29
The General Compartmental Model....Pages 30-32
Tracer Kinetics in Steady State Systems....Pages 33-38
Uptake of Potassium by Red Blood Cells....Pages 39-42
Standard Types of Tracer Experiments....Pages 43-44
Analytical Solution of the Tracer Model....Pages 45-49
System Structure and Connectivity....Pages 50-54
System Eigenvalues and Stability....Pages 55-68
The Inverse of a Compartmental Matrix....Pages 69-78
Mean Times and the Inverse Matrix....Pages 79-89
Solution of the Steady State Problem for SEC Systems....Pages 90-98
Structural Identification of the Model....Pages 99-119
Necessary and Sufficient Conditions for Identifiability....Pages 120-125
A Simple Test for Nonidentifiability....Pages 126-145
Computation of the Model Parameters....Pages 146-156
An Alternative Approach to Identification....Pages 157-170
Controllability, Observability, and Parameter Identifiability....Pages 171-187
Model Identification from the Transfer Function Equations....Pages 188-207
The Parameter Estimation Problem....Pages 208-239
Numerical Simulation of the Model....Pages 240-251
Identification of Compartment Volumes....Pages 252-267
A Discrete Time Stochastic Model of a Compartmental System....Pages 268-285
Closing Remarks....Pages 286-288
Back Matter....Pages 289-304
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