Stahl67# bmp

THE BOLE OF MODELS IN THEORETICAL BIOLOGY 187

their simulation on a Computer is provided by Berman et dl. (1962a,b). Berman describes a Computer program which will fit a wide variety of compartment models in automatic or semiautomatic fashion, on the basis of statistical best-fit methods. In earlier work, Berman and Schoen-feld (1956) discussed the naturę of uwariants arising in linear kinetics models. These are typically the coefficients of exponential terms in the solution of the goveming system of differential eąuations and usually represent ratios of rates, concentrations, or time constants. Morę com-plex invariants relate to the properties of the coefficient matrices for these systems and are of a relational or topological naturę, rathei than being simple numerical uwariants. For example, an invariant topological property is implied by all models having precisely the form that there is input only into one compartment, which then interchanges materials reversibly with only two others, and all excretion occurs from one of the latter.

Berman (1963a,b) considers linear or nonlinear kinetics simulations based on differential eąuations to be a very broad and important class of biological models and has obtained many useful results with them. They apply to such diverse phenomena as movement of metabolites, drugs, and radioactive tracers; enzyme systems; celi proliferation kinetics; ecological and genetic models; and certain psychological phenomena. It is elear from Table I however, that kinetic models represent only one class of all known biological models and should not ordinarily be con-sidered as physical analogs governed by dimensionless numbers.

B. Metabolic and Compartment Models

Hazelrig et al. (1963, 1964) describe Computer techniąues for fitting kinetic models pertaining to body metabolism, such as distribution of thyroid hormones. It is found that direct fitting of experimental data to preselected eąuations is sometimes impossible and iterative approximate curve-fitting techniąues must be used. In other reports Ackerman apphes the same methods for analysis of metabolic control mechanisms. An extensive numerical Computer model for distribution of ions and oxygen in the blood is reported by DeHaven and DeLand (1963), who have used thermodynamic principles to simulate establishment of a complex simultaneous eąuilibrium involving dozens of substances in the body fluids. The similarity criteria for metabolic compartment models of this type would include ratios of rates, time constants, thermodynamic parameters or concentrations, and also certain relational criteria.

Many other reports dealing with compartment models are available.