Thomas Jacob Snowden (1) & Piet H. van der Graaf (1)
(1) Certara QSP, Canterbury, CT2 7FG, UK
Introduction:
The increasing use of quantitative systems pharmacology (QSP) and physiologically-based pharmacokinetic (PBPK) modelling in drug development has caused modellers to more commonly confront the issue of model complexity. Due to complexity, such mechanistic models are often difficult to work with , challenging to analyse, and generally not suitable for estimation purposes due to the large number of states and parameters to be handled – even if all unidentifiable parameters were fixed. Methods of model reduction [1], commonly applied in other fields of modelling, can provide a rational framework for addressing issues of complexity and yielding practical, reduced models that retain a high degree of predictive power and a mechanistic basis.
Objectives:
To compare and contrast two recently published methodologies of model reduction via application to an example QSP type model of bone biology. The first such methodology [2] uses inductive linearization and subsequent lumping of the system to obtain a reduced description, the second [3] employs lumping and empirical balanced truncation under the Petrov-Galerkin projection to achieve a reduction. Both methods are compared by being used for the reduction of a QSP type model of bone biology that can describe the effect of denosumab on bone remodelling and osteoclast/osteoblast numbers [4].
Methods:
The bone biology model [4] was implemented and validated in Matlab R2017b as an interacting system of 28 ordinary differential equations (ODEs). Through simulation and analysis, agreement between our implementation and the original publication was attained. Both methods of reduction were then applied to this implementation. The first methodology employs inductive linearization of the original model to produce a time-varying linear system [5], this linear system is then reduced via proper lumping. The second methodology operates entirely on the nonlinear system via the Petrov-Galerkin projection; lumping is employed until the stiffness coefficient is sufficiently reduced to enable the follow-up application of empirical balanced truncation. The resulting models were then compared in terms of accuracy and ease of use with respect to the original system. The primary metric of accuracy was taken to be the maximal relative error between the outputs of the original and reduced models at the various levels of reduction.
Results:
Both methods of model reduction were able to produce significantly simplified systems whilst retaining a high degree of accuracy. A highly accurate reduction from 28 to 8 state-variables was achievable under both methodologies. Notably, however, the lumping and empirical balanced truncation approach was able to yield a 21% lower reduction error at the 7-dimensional reduction, as compared with the linearization and lumping approach. Additionally, whilst the linearization approach is somewhat mathematically simpler and easier to implement, the necessary creation of a time-varying linearization matrix does somewhat obscure the meaning of the reduced model and detract from its mechanistic basis.
Conclusion:
Methods of model reduction can simplify complex systems and enable their more practical application in the context of drug development. When comparing two such methods, we were able to demonstrate that an approach of lumping and empirical balanced truncation was able to produce improved results as over a linearization and lumping methodology when comparing the overall approximation error incurred and the properties of the reduced model.
References:
[1] Snowden TJ et al. Bull Math Biol (2017) 79:1449.
[2] Hasegawa C. Rosa Webinar Series (2018). Retrieved from https://goo.gl/uYKn3e.
[3] Snowden TJ et al. BMC Systems Biology (2017) 11:17
[4] Peterson MC et al. CPT Pharmacometrics Syst Pharmacol. (2012) 1:e14
[5] Hasegawa C & Duffull SB. J Pharmacokinet Pharmacodyn (2017).
Reference: PAGE 27 (2018) Abstr 8647 [www.page-meeting.org/?abstract=8647]
Poster: Methodology - Other topics