Input estimation in nonlinear dynamical systems for model – based drug discovery using optimal control techniques

M. Trägårdh, M.J. Chappell, N.D. Evans, M. Jirstrand, J. Gabrielsson, and P. Gennemark. In proceedings of PKUK 2014, November 2014, Bath, UK.


Background: In many pharmacokinetic (PK) applications, it is of interest to determine the input to a dynamical system, based only on sparse and noisy measurements. One typical case is when a drug is administered orally, and where there is a known PK model, but the absorption process is not well understood. When the model of the system is linear and time invariant, input estimation is referred to as deconvolution. Traditional deconvolution methods based on regularised regression can easily be solved in closed form for linear models. However, many PK models are nonlinear, e.g. as a result of saturable elimination. Therefore, being able to handle only the linear case is a severe restriction. Besides, input estimation methods have a much wider applicability than PK, and can be used in any pharmacodynamic (PD) or disease modelling problem where the input to a linear or nonlinear model needs to be determined. As an example, these methods are being considered for use in body weight modelling, estimating the energy intake from body weight measurements.

Aim: To investigate, implement and benchmark techniques for input estimation (deconvolution) for the case when the underlying dynamical system is nonlinear.

Methods: Key techniques from optimal control theory were applied: multiple shooting and collocation in combination with sensitivity analysis and automatic differentiation. The techniques were benchmarked on a previously published dataset measuring the plasma concentration of eflornithine in 26 rats after oral administration. Two choices of regularisation functions were used: Tikhonov (ridge regression) and Maximum Entropy.

Results: The investigated methods worked robustly on the benchmark dataset, even when starting from very poor initial guesses. The multiple shooting methods needed 15 seconds on a standard workstation for a typical dataset, while collocation methods needed about 5 seconds.

Conclusion: Optimal control methods make it possible to use traditional deconvolution methods even when the system is nonlinear.


This work is funded through the Marie Curie FP7 People ITN European Industrial Doctorate (EID) project, IMPACT (Innovative Modelling for Pharmacological Advances through Collaborative Training).

Authors and Affiliations

  • Michael J. Chappell, School of Engineering, University of Warwick
  • Neil D. Evans, University of Warwick
  • Mats Jirstrand, Fraunhofer-Chalmers Centre
  • Johan Gabrielsson, Swedish University of Agricultural Sciences
  • Peter Gennemark, CVMD iMED DMPK AstraZeneca R&D

Photo credits: Nic McPhee