Jane Knöchel (1,2,4), 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 (4) Current address: AstraZeneca R&D, Mölndal, Sweden
Objectives:
A growing understanding of complex biological processes has led to large-scale mechanistic, nonlinear models of pharmacologically relevant processes. These models are increasingly used to study the response of the system to a given input or stimulus, e.g., after drug administration. Understanding the input-response relationship, however, is often a challenging task due to the complexity of the interactions between its constituents (nonlinearity) and the size of the models. An approach that quantifies the importance of the different state variables for a given input-response relationship and that allows to reduce the dynamics to its essential features (allowing for parameter estimation in the statistical analysis of clinical data) is therefore highly desirable. The objective was first to derive a novel measure of importance for state variables with a focus on typically nonlinear systems pharmacology models and second to reduce the systems pharmacology models based on the introduced measure of importance to its essential functionalities for the given input-response relationship.
Methods:
We rephrased the input-response characterisation into a control-theoretical input-output setting, by considering the drug or some other entity as a model input and the drug effect or some surrogate as the output. We used a characterisation of each state variable of the system in terms of controllability (’How does the input affect a state variable?’) and observability (’How does a state variable impact the output?’).
We have chosen the blood coagulation network model [1,2] to illustrate our approach. The blood coagulation model describes the coordinate activation of different proteins (so-called coagulation factors) upon stimulation that eventually result in the activation of fibrinogen. The detailed mathematical model [1,2] was previously used to predict the effect of the Australian elapid venoms on the blood coagulation. The model consists of 62 state variables and 178 parameters.
Results:
We derived a novel state and time-dependent quantity of the system called the input-response (ir) index that quantifies the importance of state variables for a given input-response relationship at a particular time. We linked the ir indices to the product of two sensitivity coefficients. The first coefficient quantifies the impact of the input on a given state variable (“controllability”) and the second coefficient provides a measure of how a state variable impacts the output (“observability”). The measure is explicitly defined with respect to a reference solution of the system and thereby dependent on the initial state (this is an important feature of the measure). In application to systems pharmacology models, the ir indices give insight into the coordinated action of specific constituents and about those constituents that contribute only little to the response. We subsequently used the ir indices to reduce these large-scale models in a two-step automated procedure: (i) elimination of states whose dynamics have only negligible impact on the input-response relationship and (ii) proper lumping of the remaining (lower order) model.
Applying this procedure to the brown snake venom-fibrinogen system resulted in a reduction from 62 to 8 state variables in the first step, and finally a reduction to 5 state variables [3]. We further illustrate that the sequence, in which a recursive algorithm eliminates and/or lumps state variables, has an impact on the final reduced model. The ir indices are particularly suited to determine an informed sequence, since they are based on the dynamics of the original system (as opposed to the dynamics of some already partially reduced system in the context of iterative greedy algorithms).
Conclusions:
The novel input-response index as a measure of importance of state variables provides a powerful tool for understanding the complex dynamics of large-scale systems in the context of a specific drug-response relationship. Furthermore, the indices provide a means for a very efficient model order reduction and thus an important step towards translating insight of the biological process incorporated in detailed systems pharmacology models to the population analysis of clinical data.
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
[2] Gulati A., Isbister G.K., Duffull S.B. (2014) Scale reduction of a systems coagulation model with an application to modeling pharmacokinetic-pharmacodynamic data. CPT: Pharmacometrics & Syst. Pharmacol. 3(1)
[3] Knöchel J., Kloft C., Huisinga W. (2017) Understanding and reducing complex systems pharmacology models based on a novel input-response index. Journal of Pharmacokinetics and Pharmacodynamics; https://doi.org/10.1007/s10928-017-9561-x
Reference: PAGE 27 (2018) Abstr 8798 [www.page-meeting.org/?abstract=8798]
Poster: Methodology - New Modelling Approaches