THE FORMULATION AND TESTING OF MODELS

The term “model” is used in both a physical and mathematical sense to represent any set of equations or functions that describe the response of a system to a stimulus. In a mathematical model, functions or differential equations are employed without regard to any mechanistic aspects of the system. On the other hand a physical model implies certain mechanisms or entities that have physiological, biochemical, or physical significance. The parameters of a model are the arbitrary constants of the functions or equations and apply to both physical and mathematical models. Procedures for the formulation of models to interpret kinetic data involve a number of quantitative and qualitative phases. It is the purpose of this paper to discuss from a single point of view some of these phases and the way they may fit into a formalism for the formulation and testing of models. It is first necessary to choose the type or class of models applicable. Frequently a choice can be based on theoretical considerations of the system studied. For example, linear compartmental analysis is often justifiable for tracer kinetics. Other types of models may be chosen for their conceptual simplicity, for the relative ease with which they can be treated mathematically, or for their simplicity in predicting the experimental data. To choose a model from a given type or class of models it is necessary to consider its complexity. Two aspects contribute to this: (1) the “order” of the wodel, namely, the number of independent functions or equations necessary to describe its response, and (2) the number of parameters for a given order. In the case of linear compartmental models, the order is equivalent to the number of compartments (or the number of exponential components), and the number of parameters is equivalent to the number of interconnections or transition paths between compartments. It is not possible to derive the order of a model on the basis of the data alone because any model that is compatible with the data can always be interpreted as a degenerate case for one of higher order. One can, however, derive a minimal order below which, based on some criterion, the data cannot be predicted satisfactorily, and this is usually the order chosen for a model. For compartmental models, this means that a model with the smallest number of compartments compatible with the data is usually chosen, unless other criteria are introduced. Given the number of compartments, it is necessary next to determine