Invariants in Experimental Data on Linear Kinetics and the Formulation of Models

Abstract

A new mathematical approach for the analysis of radioactive tracer experiments on compartmentalized systems in steady states is presented. The experimental measurements of the amounts of tracer in one or more compartments are approximated by a sum of exponential functions of time. The coefficients and exponential factors of these functions are shown to represent a set of invariants of the data. These are then related in a concise matrix equation to the compartmental model parameters, which are defined as the transition probabilities per unit time per unit quantity (turn-over rates) of molecules from one compartment (or state) into another. When measurements are incomplete, the freedom in choosing a model is expressed in turns of a minimum number of variables equal to the difference between the number of model parameters and the number of invariants found in the data. It is shown how other information regarding the model parameters or the amounts of material in steady state may be combined with the tracer data to reduce the degrees of freedom of a proposed model. Matrix transformations are worked out to allow the mapping of all mathematically consistent models in the configuration space of the minimum variables. The boundaries in this space corresponding to the limits for physically realizable models are found. Thus, a model is expressed by a set of coordinates and the values of all the parameters are obtained by substituting the values of the coordinates. Investigation of the range of variation of each parameter over the bounded region is also possible.

Thrombolytic and pharmacokinetic properties of a conjugate of recombinant single-chain urokinase-type plasminogen activator with a monoclonal antibody specific for cross-linked fibrin in a baboon venous thrombosis model.

“…The disposition of u-PA-related antigen from plasma was represented by a two-compartment mammillary model composed of one central and one peripheral compartment with elimination occurring from the central compartment. 22 Pharmacokinetic parameters were calculated from these coefficients and exponents with standard formulas derived by Gibaldi and Perrier. 23 The variables A and B were first calculated, assuming steady state at the end of the infusion, using the formulas A=RX0a/kO and B=SX0/kO, where XO is total administered dose, and ko is the rate of infusion.…”

An application of Berman's work on pool-model invariants in analyzing indistinguishable models for whole-body cholesterol metabolism

“…If P can be chosen so that PAP −1 , CP −1 , and Px(0) satisfy the e characteristic equality constraints [Equation (2)] of the second model as well as the inequality constraints that any pool model must satisfy [Equation (1)], the first model is indistinguishable from the second. If the n 2 elements of P cannot be chosen to satisfy the e equations (2) and the general pool-model conditions (1), the first model is distinguishable from the second.…”

“…To illustrate, let the tracer be introduced into pool 1 of a four-pool model, and let x 2 (t) alone be measured (x i (0) is zero except for x 1 . Since , row 4 of P must be .…”

“…Mones Berman's early work on invariants in kinetic data [1], published in 1956, is a major contribution to the theory of linear models of physiological phenomena fitted to experimental data. Berman and Schoenfeld studied unidentifiable models-models that have more parameters than can be uniquely determined from the data-and developed a method for relating the parameters of such a model to those of an identifiable model, through matrix transformations.…”

“…where A ij , C ij , M i , and p i satisfy the following inequality constraints: (1) In matrix notation, where x is a vector with n elements x 1 ,…, x n ; A is an n × n matrix; y is a vector with r elements; C is an r × n matrix; and…”