Table Of ContentLectu re Notes in
Biomathematics
Managing Editor: S. Levin
50
David H. Anderson
Compartmental Modeling
and Tracer Kinetics
Springer~Veriag
Berlin Heidelberg New York Tokyo 1983
Editorial Board
W. Bossert H. J. Bremermann J. D. Cowan W. Hirsch S. Karlin
J. B. Keller M. Kimura S. Levin (Managing Editor) R. C. Lewontin R. May
G. F. Oster A. S. Perelson T. Poggio L. A. Segel
Author
Dr. David H. Anderson
Department of Mathematics
Southern Methodist University
Dallas, TX 75275, USA
ISBN 978-3-540-12303-3 ISBN 978-3-642-51861-4 (eBook)
DOI 10.1007/978-3-642-51861-4
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material
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payable to ·Verwertungsgesellschaft Wort", Munich.
© by Springer·Veriag Berlin Heidelberg 1983
2146/3140·543210
PREFACE
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.
Mishe1evich, and A. S. Pere1son. I am also grateful to the Department of Medical
Computer Science of the University of Texas Health Science Center at Dallas, which
provided much of the atmosphere and monetary resources to make this project pos
sible. Special thanks go to my wife, Susan, and to Carolyn Simmons for their
excellent typing of the notes.
David H. Anderson
February 1983
TABLE OF CONTENTS
Section 1. Compartmental Systems.
lAo Introduction.
1
lB. Preliminary definitions 2
lC. Tracer experiments. 3
10. History of compartmental analysis
5
Section 2. Elementary Compartmental Models 7
2A. Drug kinetics 7
2B. Leaky fluid tanks 9
2C. Oiffusi on . 10
20. Solute mixture 11
Section 3. First-Order Chemical Reactions l3
Section 4. Environmental Studies. 15
4A. Kinetics of lead in the body 15
4B. The Aleut ecosystem 18
Secti on 5. Nonl inear Compartmental Model s 22
5A. Continuous flow chemical reactor 22
5B. Reaction order 25
5C. Other nonlinear compartmental models 26
Section 6. The General Compartmental Model 30
Section 7. Tracer Kinetics in Steady-State Systems 33
7A. The tracer equations 33
7B. Linear compartmental models 36
Section 8. Uptake of Potassium by Red Blood Cells 39
Section 9. Standard Types of Tracer Experiments 43
9A. Tracer concentration equations. 43
9B. Tracer specific activity equations 44
Section 10. Analytical Solution of the Tracer Model 45
lOA. The general solution of the model 45
lOB. Nonnegativity of the solution 47
v
Section 11. System Structure and Connectivity 50
11A. The connectivity diagram. 50
11B. Common compartmental systems 51
11C. Strongly connected systems 53
Section 12. System Eigenvalues and Stability 55
12A. Nonpositive eigenvalues .. 55
12B. The smallest magnitude eigenvalue. . . 56
12C. Symmetrizab1e compartmental matrices and real eigenvalues 58
120. Distinct eigenvalues 62
12E. Compartmental model stability 63
12F. Bounds on the extreme eigenvalues. 64
5ection 13. The Inverse of a Compartmental Matrix 69
13A. Invertibi1ity conditions 69
13B. A Neumann series for the inverse matrix 73
13C. r·latrix inequalities 75
Section 14. Mean Times and the Inverse Matrix 79
14P.. t1ean residence times 79
14B. The compartmental matrix exponential 82
14C. Further properties of mean residence time 84
140. System mean residence time 87
Section 15. Solution of the Steady-State Problem for 5EC Systems 90
15A. The tracer steady-state problem 90
15B. Ill-conditioned SEC systems. 90
15C. An iterative procedure for SEC systems 93
150. Updating the algorithm 96
Section 16. Structural Identification of the Model 99
16A. The system (A, B, C) 99
16B. The structural identification problem 101
16C. A simple identification example 103
160. Realizations of impulse response functions 104
16E. Impu1 se response functi on structure 105
16F. Nonlinear identification equations 108
16G. A three compartment model 112
16H. A four compartment model 115
Section 17. Necessary and Sufficient Conditions for Identifiability 120
17A. Model identifiability 120
17B. Necessary conditi ons 120
17C. Sufficient conditions 122
VI
Section 18. A Simple Test for Ilonidentifiability 126
18A. Counting nonzero transfer function coefficients 126
18B. Coefficient structure. . . . . . 130
18C. Further refinements of formula (18.4). 133
18D. The nonidentifiabi1ity test.. 138
18E. Tighter bounds on the number of independent equations 142
Section 19. Computation of the Model Parameters 146
19A. Local identifiability. 146
19B. Newton's method and modifications 147
19C. The Kantorovi ch conditi ons 152
19D. An example 154
Section 20. An Alternative Approach to Identification 157
20A. A new identification method. 157
20B. The component matrices of A . 157
20C. The identification technique using component matrices 161
20D. Identification of a lipoprotein model 163
20E. The identification technique using modal matrices 165
20F. Identification of a pharmacokinetic system 166
20G. Spectral sensitivity of a linear model 168
Section 21. Controllability, Observabi1ity, and Parameter Identifiability. 171
21A. The control problem 171
21B. Completely controllable systems 171
21C. Completely observable systems . 174
21D. Realizations and identifiability 175
21E. A third method of identifiabil ity 182
Section 22. r1ode1 Identification fro!ll the Transfer Function Equations 188
22A. Form of the nonlinear equations 188
22B. Coefficients of the transfer fuction . 188
22C. The nonlinear algebraic equations for the
identification problem 192
22D. Necessary conditions for positive solutions of the
nonlinear system 193
22E. Refi ned necessary conditi ons 196
22F. Additional properties of the nonlinear algebraic system 198
22G. An iterative scheme for solving F(a) 0 200
22H. Triangu1arization of F(a) = 0 . 201
221. Uniqueness of solution F(a) = 0 204
Section 23. The Parameter Estimation Problem 208
23A. The basic estimation problem 208
23B. Ali poprotei n metabo 1i sm model 210
23C. Nonlinear 1east-squ~res . 215
23D. Initial parameter estimates 217
23E. Method of moments . 219
VII
23F. Other methods of parameter estimation 224
23G. Positive amplitudes . . . . . . . . 225
23H. Curve-fitting sums of exponentials is ill-posed . . . 226
231. Fitting the differential equation model directly to data 226
23J. Modulating function method . . . 231
23K. An antigen - antibody reaction example . . 235
23L. Additional literature on fitting of differential equations
to data 237
Section 24. Numerical Simulation of the Model 240
24A. Compartmental model simulation . 240
24B. A three compartment thyroxine model . . . . . 241
24C. Numerical inteoration methods and some inadequacies 242
240. Implicit methods . 245
24E. Determining model stiffness . 250
Section 25. Identification of Compartment Volumes 252
25A. The basic single exit compartmental model 252
25B. Readily identifiable parameters 253
25C. The catenary single exit system . . 254
250. Estimation of compartmental volumes . 257
25E. Creatinine clearance model 262
25F. Shock therapy. ..... 263
25G. Bounds and approximations on compartmental volumes 264
Section 26. A Discrete Time Stochastic ~odel of a Compartmental System 268
26A. The Markov chain model 268
26B. The liver disease model . . . 269
26C. Simulation of the hepatic system 274
260. Mathematical analysis of the model 275
26E. Parameter estimation 279
26F. Discussion 283
Section 27. Closing Remarks 286
Bibliography . 289
SECTION 1. COMPARTMENTAL SYSTE~S
lAo Introduction
In this monograph we shall discuss the analysis of biological systems by means
of compartmental modeling and tracer studies. This type of modeling has application
in a variety of areas such as drug kinetics in pharmacology, studies of metabolic
systems, analysis of ecosystems, and chemical reaction kinetics. The fundamental
approach in this style of modeling is to analyze a system by separating it into a
finite number of component parts, called compartments or states, which interact
through the exchange of material. The mathematical theory for the behavior of these
systems of compartments is called compartmental analysis, and it has been found to
be quite useful in the analysis of experiments in many branches of the biosciences.
The application of compartmental analysis is like that of general mathematical
modeling. One goal is to develop a plausible mathematical representation of a par
ticular biological phenomenon. This model will be a simplification of that phenom
enon, but it should be an adequate representation for the purpose in mind. It
requires considerable depth of knowledge in the field from which the problem is
taken to formulate a meaningful and useful mathematical model. This is usually best
accomplished by finding someone knowledgeable in the area to provide the appropriate
background and justification for the use of a compartmental model. Another goal is
to develop and expand the theory of compartmental systems. In this book a consider
able amount of space is devoted to structural properties of compartmental systems,
especially in the linear case. A third goal in compartmental analysis is to deal
effectively with the inverse problem, which consists of the three phases: model
specification, structural identifiability, and parameter estimation. Here model
specification refers to the determination of the number of compartments and the
interconnections between compartments which allow the exchange of material. In the
structural identifiability aspect of this problem, we inquire as to whether or not
the system parameters are uniquely defined under error free observations, given that
the model is completely specified including the compartments receiving inputs and
those which are observed. We will observe that the experimental information found
2
and its relation to the model parameters are given by a set of nonlinear algebraic
equations, and as a consequence, the structural identification problem reduces to
the question of whether or not a nonlinear algebraic system of equations possesses a
unique solution. Provided the structural identification of the system obtains, the
practical problem of computing the "best" values for these model parameters-
parameter estimation--still remains. We will discuss analytical and numerical
methods of calculating these best values.
lB. Preliminary Definitions
Kinetics is that branch of dynamics that pertains to the turnover of specific
particles in a biological system. A compartment is an amount of material which acts
kinetically in a homogeneous distinct way. The compartment to which a particle
belongs characterizes both its physical-chemical properties and its environment.
A compartment may not be an actual physical volume; however, the amount of some
material in a physiological space is often treated as a compartment in certain
clinical studies. The particles of each compartment are influenced by forces which
cause the particles to transfer from one compartment to another. All particles in
a particular compartment have the same probability of transition since within the
compartment all particles are well-mixed and considered indistinguishable by the
system. The transition from one compartment to another occurs by passing through
some physical barrier or by undergoing some physical or chemical transformation.
Associated with each compartment is its size. Sometimes the terms volume and size
are used interchangeably.
A compartmental system consists of two or more compartments, interconnected in
the sense that among certain compartments there is exchange of material. The com-
partmental system will be primarily modeled in a continuous deterministic manner by
a collection of ordinary differential equations, each equation describing the time
rate of change of amount of material in a particular compartment. These rates of
change are dictated by the physical-chemical laws that govern the material exchange
between interacting compartments, e.g., diffusion, temperature, chemical reactions,
etc. It will be assumed that the set of equations model average properties of the
3
system for very large numbers of particles. From the idealized physical model it is
hypothesized as to the exact nature of the interconnections that occur between com
partments. In the usual representation for a compartmental system, a box denotes a
compartment, and an arrow indicates the transfer of material into or out of a com
partment (see Figure 1.1). There also may be inputs from the outside environment
into one or more compartments (vertical arrows pointing into the tops of boxes), and
there can be excretion of material from some of the compartments to the outside
environment (vertical arrows pointing out of the bottoms of boxes). Should there be
no exchange of material to the outside environment, the compartmental system is
referred to as closed; otherwise it is said to be an open system. Realistically,
many compartmental systems are open, for some material is lost by excretion,
metabolism, etc.
2
Figure 1.1
le. Tracer Experiments
Higher-level organisms tend to maintain characteristics such as temperature
within fairly uniform ranges, even with slight disturbances that occasionally occur
