9 - Deterministic Models of Continuous Systems

Abstract

This chapter is related to the process of linking differential equations to the system behavior and data using constant coefficients linear models. Systems are classified by their structure and dimension. By structure, systems are either compartmental or noncompartmental, and the number of system components also determines the dimension of the system. First order multidimensional linear models are represented by differential equations reducible to the form dY/dt = A Y + X, where Y is a set of state variables, A is a matrix of constant coefficients determining the relations among variables, and X is the set of input functions of the system. Compartmental first order linear models are represented by differential equations reducible to the form dY/dt = (A + B) Y + X, where Y is the set of state variables, A is a matrix of constant coefficients determining exchange rates among compartments, B is a matrix defining the system outputs to the outside environment, and X is the set of input functions of the system. The sum of the coefficients of each column of matrix A should always add up to zero. The model represents a closed system if B is a null matrix, otherwise the system is open. Hence, a procedure for fitting linear models to the data of continuous systems is recommended in the chapter. The procedure requires expressing the data as a difference table, using a linear regression procedure to determine the most appropriate model, defining the set of differential equation of the system, determining the state equations, and using a nonlinear curve firing procedure for fine tuning the state equations.