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.

Section 1. Compartmental Systems.- 1A. Introduction.- 1B. Preliminary definitions.- 1C. Tracer experiments.- 1D. History of compartmental analysis.- Section 2. Elementary Compartmental Models.- 2A. Drug kinetics.- 2B. Leaky fluid tanks.- 2C. Diffusion.- 2D. Solute mixture.- Section 3. First-Order Chemical Reactions.- Section 4. Environmental Studies.- 4A. Kinetics of lead in the body.- 4B. The Aleut ecosystem.- Section 5. Nonlinear Compartmental Models.- 5A. Continuous flow chemical reactor.- 5B. Reaction order.- 5C. Other nonlinear compartmental models.- Section 6. The General Compartmental Model.- Section 7. Tracer Kinetics in Steady-State Systems.- 7A. The tracer equations.- 7B. Linear compartmental models..- Section 8. Uptake of Potassium by Red Blood Cells.- Section 9. Standard Types of Tracer Experiments.- 9A. Tracer concentration equations.- 9B. Tracer specific activity equations.- Section 10. Analytical Solution of the Tracer Model.- 10A. The general solution of the model.- 10B. Nonnegativity of the solution.- Section 11. System Structure and Connectivity.- 11A. The connectivity diagram.- 11B. Common compartmental systems.- 11C. Strongly connected systems.- Section 12. System Eigenvalues and Stability..- 12A. Nonpositive eigenvalues ..- 12B. The smallest magnitude eigenvalue.- 12C. Symmetrizable compartmental matrices and real eigenvalues.- 12D. Distinct eigenvalues.- 12E. Compartmental model stability.- 12F. Bounds on the extreme eigenvalues.- Section 13. The Inverse of a Compartmental Matrix.- 13A. Invertibility conditions.- 13B. A Neumann series for the inverse matrix.- 13C. Matrix inequalities.- Section 14. Mean Times and the Inverse Matrix.- 14A. Mean residence times.- 14B. The compartmental matrix exponential.- 14C. Further properties of mean residence time.- 14D. System Mean residence time.- Section 15. Solution of the Steady-State Problem for SEC Systems.- 15A. The tracer steady-state problem.- 15B. Ill-conditioned SEC systems.- 15C. An iterative procedure for SEC systems.- 15D. Updating the algorithm.- Section 16. Structural Identification of the Model.- 16A. The system (A, B, C).- 16B. The structural identification problem.- 16C. A simple identification example.- 16D. Realizations of impulse response functions.- 16E. Impulse response function structure.- 16F. Nonlinear identification equations.- 16G. A three compartment model.- 16H. A four compartment model.- Section 17. Necessary and Sufficient Conditions for Identifiability.- 17A. Model identifiability.- 17B. Necessary conditions.- 17C. Sufficient conditions.- Section 18. A Simple Test for Nonidentifiability.- 18A. Counting nonzero transfer function coefficients.- 18B. Coefficient structure.- 18C. Further refinements of formula (18.4).- 18D. The nonidentifiability test.- 18E. Tighter bounds on the number of independent equations.- Section 19. Computation of the Model Parameters.- 19A. Local identifiability.- 19B. Newton’s method and modifications.- 19C. The Kantorovich conditions.- 19D. An example.- Section 20. An Alternative Approach to Identification.- 20A. A new identification method.- 20B. The component matrices of A.- 20C. The identification technique using component matrices.- 20D. Identification of a lipoprotein model.- 20E. The identification technique using modal matrices.- 20F. Identification of a pharmacokinetic system.- 20G. Spectral sensitivity of a linear model.- Section 21. Controllability, Observability, and Parameter Identifiability.- 21A. The control problem.- 21B. Completely controllable systems.- 21C. Completely observable systems.- 21D. Realizations and identifiability.- 21E. A third method of identifiability.- Section 22. Model Identification from the Transfer Function Equations.- 22A. Form of the nonlinear equations.- 22B. Coefficients of the transfer fuction.- 22C. The nonlinear algebraic equations for the identification problem.- 22D. Necessary conditions for positive solutions of the nonlinear system.- 22E. Refined necessary conditions.- 22F. Additional properties of the nonlinear algebraic system.- 22G. An iterative scheme for solving F(?) = 0.- 22H. Triangularization of F(?) = 0.- 22I. Uniqueness of solution F(?) = 0.- Section 23. The Parameter Estimation Problem.- 23A. The basic estimation problem.- 23B. A lipoprotein metabolism model.- 23C. Nonlinear least-squares.- 23D. Initial parameter estimates.- 23E. Method of moments.- 23F. Other methods of parameter estimation.- 23G. Positive amplitudes.- 23H. Curve-fitting sums of exponentials is ill-posed.- 23I. Fitting the differential equation model directly to data.- 23J. Modulating function method.- 23K. An antigen — antibody reaction example.- 23L. Additional literature on fitting of differential equations to data.- Section 24. Numerical Simulation of the Model.- 24A. Compartmental model simulation.- 24B. A three compartment thyroxine model.- 24C. Numerical integration methods and some inadequacies.- 24D. Implicit methods ..- 24E. Determining model stiffness.- Section 25. Identification of Compartment Volumes.- 25A. The basic single exit compartmental model.- 25B. Readily identifiable parameters.- 25C. The catenary single exit system.- 25D. Estimation of compartmental volumes.- 25E. Creatinine clearance model.- 25F. Shock therapy.- 25G. Bounds and approximations on compartmental volumes.- Section 26. A Discrete Time Stochastic Model of a Compartmental System.- 26A. The Markov chain model.- 26B. The liver disease model.- 26C. Simulation of the hepatic system.- 26D. Mathematical analysis of the model.- 26E. Parameter estimation.- 26F. Discussion.- Section 27. Closing Remarks.