David H. Anderson

Structural properties of compartmental models

Abstract This article treats structural properties of the inverse of a compartmental matrix and how they relate to properties of coefficients of the transfer function of the compartmental system. Newly formulated conditions are presented for certain of these parameters to be zero or positive. Also results are given on the interdependence of transfer function coefficients and how this relates to the identifiability problem. Answers to some questions raised in the recent literature about coefficient dependence are discussed.

Iterative inversion of single exit compartmental matrices

Abstract This article develops an iterative method for solving a system of linear algebraic equations in which the coefficient matrix is a compartmental matrix of the single-exit type. Matrix properties, invertibility, ill-conditioning, algorithm acceleration and error analysis are dicussed.

THE VOLUME OF DISTRIBUTION IN SINGLE-EXIT COMPARTMENTAL SYSTEMS

Publisher Summary This chapter discusses the volume of distribution in single-exit compartmental systems. In recent years, the use of mathematical compartmental models to describe biological phenomena has become more prevalent, particularly with respect to tracer kinetics. In a living organism, the introduced tracer creates an observable transient that will provide information about certain aspects of the system. The chapter focuses on the identification of certain parameters of the system, in particular the volumes of distribution of the tracer, on the basis of these experimental observations. It highlights the class of compartmental systems that is concerned with a problem frequently encountered in physiology—the injection of an isotope by the intravenous route into one compartment at time zero and the sequential sampling of that same compartment to estimate the tracer concentration as a function of time and then utilization of this concentration function to provide information about the characteristics of the system. This chapter describes the development of the mathematical model. It also discusses the identification of the transfer coefficients.

Calculation and utilization of component matrices in linear bioscience models

Component matrices of a given square matrix can be used for a variety of purposes. A new recursive method to find these component matrices is developed. The use of component matrices in linear time-invariant dynamic models of bioscience systems is demonstrated.

Equilibrium points for nonlinear compartmental models

Equilibrium points for nonlinear autonomous compartmental models with constant input are discussed. Upper and lower bounds for the steady states are derived. Theorems guaranteeing existence and uniqueness of equilibrium points for a large collection of system are proved. New information relating to mean residence times is developed. Asymptotic results and a section on stability are included. A recursive process is discussed that generates iterates that converge to steady states for certain types of models. An interesting range of models are included as examples. An attempt is made to provide general qualitative theory for such nonlinear compartmental systems.

Spectral sensitivity in linear biological models

Properties of spectral components of the system matrix of linear time-invariant discrete or continuous models are investigated. It is shown that the entries in these matrices have the interpretation of being the sensitivity of the system matrix eigenvalues with respect to the model parameters. The spectral resolution formula for linear operators is used to get explicit results about component matrices and eigenvalue sensitivity. In biological modeling, particular interest is in the real maximal or minimal roots of the system matrix. Exact formulation of the related spectral components is made in important system matrix cases such as companion, Leslie, ecosystem, compartmental, and stochastic matrices.

Properties of the transfer functions of compartmental models. II

Abstract This paper investigates the nonlinear system of algebraic equations arising from the transfer function of a linear time-invariant compartmental model. Properties of the system and its Jacobian matrix are found. An existence theorem for positive solutions of the system is given. Computation of the model parameters is discussed, based on a combinatorial simplex method and Newtons method. Finally, some sources of nonuniqueness due to symmetry among the equations, and the subsequent effect on the determinant of the system Jacobian, are investigated.

The unilateral circulation compartmental model

Abstract General properties of the linear time-invariant continuous compartmental model in which the system matrix is a unilateral circulation matrix (cycle) are studied. Elements of the inverse matrix can be precisely described and yield improved information on mean residence times. Conditions are found under which eigenvalues of any compartmental matrix are real and distinct. Formulae for spectral sensitivities with respect to flow and excretion rates are presented. Also shown is a simple calculation of the nonzero eigenvalue of a compartmental matrix which has smallest size. Identifiability of the flow and excretion rates for the single-exit, single-input, multiple-output unilateral circulation model is established. Interval identifiability is given for the unidentifiable volumes of the unilateral circulation model representing a series of physical compartments.

Structural Identification of the Model

A central problem in linear compartmental studies is discussed in this section. The question is whether or not the system parameters aij in the model \( {\dot x = Ax(t) + b,\;A \equiv [a_{ij} ]} \) can be uniquely determined from experimental observation or measurement of certain components of the solution vector x. The closely related problem of getting actual numerical estimates of the aijs will be considered later in Section 19.

System Structure and Connectivity

Many properties of our tracer model