Chiara Fornari 1, Carmen Pin 1, James W.T. Yates 2, S. Y. Amy Cheung 3, Jerome T. Mettetal 4, Teresa A. Collins 1
1 Safety and ADME Translational Sciences, Drug Safety and Metabolism, IMED Biotech Unit, AstraZeneca, Cambridge, UK; 2 DMPK, Oncology, IMED Biotech Unit, AstraZeneca, Cambridge, UK; 3 Quantitative Clinical Pharmacology, Early Clinical Development, IMED Biotech Unit, Cambridge, UK; 4 Bioscience, Oncology, IMED Biotech Unit, AstraZeneca, Boston, USA
Introduction/ Objectives: Stability analysis, which is often overlooked in the fields of pharmacometrics and quantitative systems pharmacology [1], is an essential tool to explore and understand systems’ behaviour [2]. The semi-mechanistic model developed by Friberg et al. [3] to describe drug-induced haematotoxicity is regularly used to explore the relationship between known physiological behaviors and parameter values, but a formal stability analysis would be essential for the full comprehension of the application of this model to describe haematopoiesis during homeostasis and subjected to perturbations.
To our knowledge, the stability properties of this model [3] have not been assessed before, and here we use stability analysis techniques to gain new insights into the relationship between parameters and system behaviours. Lastly, we discuss these results in the context of non-linear mixed effects modelling, highlighting the consequences in prediction performance, and providing recommendations for future analysis.
Methods: The stability analysis of the semi-mechanistic model developed by Friberg et al. [3] was performed linearizing the system of non-linear differential equations around the steady state of interest (namely the homeostatic values, X*), and then characterizing the long-term behaviour of the linearized system in the neighbourhood of this steady state (X*), [2]. The Routh-Hurwitz criteria [2] were used to investigate the nature of the steady state X*, and Routh-Hurwitz conditions were solved in Mathematica (Wolfram Research, Inc, ver 11.0).
Results: We showed that solutions converging to homeostasis represent drug-induced cytopenia, stable oscillatory solutions may represent cyclic cytopenia [4], or periodic leukemias [5], while unstable solutions do not describe physiological hematopoiesis. In details, we demonstrated that the feedback power parameter (γ) is a critical parameter for this model, which becomes unstable for values of γ greater than the bifurcation threshold (γ*=0.5685…). γ* is called a Hopf bifurcation point [2], and the equilibrium X* loses its stability over time when γ crosses the critical point γ*, exhibiting growing oscillations. This stability condition (γ < γ*) derives from the model structure itself, and it is independent of the other parameter values, such as the size of compartments or the maturation time, and of the type of toxicity or perturbation.
Conclusion: In this analysis, we highlighted the importance of assessing the dynamics of the semi-mechanistic model developed by Friberg et al. [3], and appropriately defining parameter settings when using this model. More precisely, we showed how to properly configure parameter estimation algorithms to guarantee mathematical stability and, hence, avoid a non-physiological behaviour of the system.
Therefore, given the broad usage of this framework in the pharmacometrics and systems pharmacology fields [6], we believe that this analysis could be beneficial for modellers working in this community, and we highly recommend incorporating these results when applying Friberg model [3] to describe drug-induce haematotoxicity data.
References:
[1] Bakshi S et al., Understanding the Behavior of Systems Pharmacology Models Using Mathematical Analysis of Differential Equations: Prolactin Modeling as a Case Study. CPT Pharmacometrics Syst. Pharmacol. 2016
[2] Murray JD. Mathematical Biology?: I . An Introduction , Third Edition. Springer; 2002.
[3] Friberg LE et al., Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. J. Clin. Oncol. 2002
[4] Colijn C, Mackey MC. A mathematical model of hematopoiesis: II. Cyclical neutropenia. J. Theor. Biol. 2005
[5] Colijn C, Mackey MC. A mathematical model of hematopoiesis: I. Periodic chronic myelogenous leukemia. J. Theor. Biol. 2005
[6] Fornari et al., Understanding haematological toxicities with mathematical modelling. Clin. Pharmacol. Ther. 2018
Reference: PAGE 28 (2019) Abstr 8910 [www.page-meeting.org/?abstract=8910]
Poster: Methodology - Model Evaluation