PARAMETER IDENTIFICATION APPLIED TO DETERMINISTIC COMPARTMENTAL BIOLOGICAL SYSTEMS

Abstract

Biological processes are extremly complex, gene rally multivariable and nonlinear with complex patterns of control. This dissertation is a study of compartmen tal models of a large class of biological systems,where the theory of compartmental systems in the area of bio logy and medicine is concerned with the flow of materials among various compartments and environment. In order to gain the maximum possible understanding of the dynamical behaviour of such compartmental models, the system theory concepts are being used. A frequent problem in the compar tmental models of biological systems is to identify the parameters of the models on the basis of experimental observations and to determine the uniqueness of the so lution, as the parameters of the models have a physical significance. Thus, the main aim of this thesis is to study the problem of parameter identification of compart mental model used to study the biological systems. In identification of compartmental models, a phy siologically based structure of the system is chosen a priori by stating explicitly which of the model parame ters are assumed to be different from zero; moreover, the design of the experiment defines the input- output configuration by stating the coefficients fb. .landSc. .1 which are different from zero. As a result, throughout the study, compartmental models associated to a physical system and to an input- output experiment are being considered as fixed structure models. In general, the model parameters can be assumed to be stochastic variables due to a number of factors such as inter-individual variability, drug concentration, mea surement and sampling conditions, environmental effects etc. But the deterministic compartmental models and input output experiments have been and are being widely and successfully used for a quantitative description of several biological systems. Entire study in this thesis is confi ned to deterministic compartmental models only. The basic definitions regarding compartmental models of biological systems such as structural global identifia bility, structural local identifi ability, structural or parameter identification, are first recalled. Structural properties which are strictly related to identifiability i.e. structural controllability, structural observability and input-output connectability of compartmental models are reviewed. To perform an identification problem, three fundamen tal steps are required : - structure determination, para meter identification and model verification. However, before solving the parameter identification problem, one would face the problem of identifiability of parameters. XIV For deterministic fixed structure compartmental models, identifiability criterion - which examines the possibility of evaluating all the unknown para meters through the chosen experiment under the best set of experimental conditions, is developed. This very general identifiability criterion is obtained as an initial-value-problem of a typical formulation of the given model equations. New results are derived for some classes of linear, non linear and time varying compartmental models. For a particular case of linear time- invariant compartmental model a computationally attractive global structural identifiability criterion is extended. Also, a necessary and sufficient condition for structural identifiability is developed using new concept of structural output controllability. A recur sive test for determining structural output controlla bility using only Boolean operations is proposed. Once the model parameters are assumed to be iden tifiable and input - output observations are free of noise an algorithm is developed to determine unknown parameters. It gives a parameter identification scheme from finite input- output sequences generated by a linear discrete time - invariant multi input multi output deterministic compartmental model. Proposed identifica tion algorithm utilizes the properties of generalized Hankel matrices and is computationally simple to apply in practice. Two computer programmes are developed and success fully tested on IBM 1620 and UNIVAC 1100 respectively. First is used to compute Boolean reachability matrix from known interconnection matrix based on a recursive algorithm and Warshall's theorem (for path matrix deter mination). The second programme is used for transforma tion from state-space representation to transfer func tion matrix using Leverrier's algorithm. Both the pro grammes are very useful to test and verify the identi fiability conditions respectively. Lastly, parameter identifiacation techniques for more interesting, inherently stochastic, compartmental models are suggested as future scope of present studies.