A novel measure of importance of state variables for model reduction: results for the blood coagulation network
Jane Knöchel (1,2), Charlotte Kloft (3), Wilhelm Huisinga (1)
(1) Computational Physiology Group, Institute of Mathematics, Universität Potsdam, Germany (2) PharMetrX Graduate Research Training Program, Freie Universität Berlin and Universität Potsdam, Germany (3) Department of Clinical Pharmacy and Biochemistry, Institute of Pharmacy, Freie Universität Berlin, Germany
Objectives: An increasing understanding of complex processes in biology and pharmacology has led to large-scale mechanistic models. These models, however, are not suitable for the analysis of sparse clinical data due to parameter identifiability issues. A potential solution is to reduce the complexity of the system using model reduction techniques. While many purely computational approaches exist, a quantity that supports the model reduction process by ranking the state variables according to their importance for the systems dynamics is still lacking. The objective was to derive a novel measure of importance for the state variables with a focus on nonlinear dynamical systems.
Methods: By considering the drug or some other stimulus as a model input and the drug effect or some surrogate as the output, we rephrased the problem into a control-theoretical input-output setting. The derivation of the new measure exploited a characterisation of the system based on controllability ('How does the input affect a states?') and observability ('How does a state impact the output?'). The measure is explicitly defined with respect to a reference solution of the system and thereby dependent on the initial state. We used the blood coagulation network model [1] to illustrate our approach.
Results: Based on a generalisation of so-called empirical gramians, we derived a novel index that measures the importance of a state variable for the given input-output relationship as a function of time. A first automated model reduction technique was developed that 'removes' the time-dependence of unimportant state variables by considering them as constant (rather than lumping them). Applied to the prothrombin time (PT) test using the blood coagulation model with tissue factor (TF) concentration as input, we identified different reduced models depending on the TF concentration (high versus low). This confirmed findings from literature that based on the magnitude of the TF concentration different pathways of blood coagulation network become relevant (factor IX and VIII deficiencies cannot be diagnosed with high TF concentrations).
Conclusions: The novel measure of importance is a powerful tool for model reduction of nonlinear models that provides insight into the system dynamics.
References:
[1] Wajima T., Isbister G.K., Duffull S.B. (2009) A Comprehensive Model for the Humoral Coagulation Network in Humans. Journal of Clinical Pharmacology and Therapeutics 86
