Simplification of multi-scale systems models for data-driven analyses: what has progressed in these 5 years?
Chihiro Hasegawa (1, 2) and Stephen B. Duffull (1)
(1) Otago Pharmacometrics Group, School of Pharmacy, University of Otago, Dunedin, New Zealand, (2) Clinical Pharmacology, Translational Medicine Center, Ono Pharmaceutical Co., Ltd., Osaka, Japan
Objectives: Bridging multi-scale systems models and pharmacometrics has resulted in models that are highly complex and often not amenable to further exploration via estimation or design. Proper lumping has been used for order reduction of such complicated models. Using the technique, Gulati et al [1] successfully obtained a reduced version of the coagulation network model which describes the time course of fibrinogen recovery after a brown snake bite. However, the process was achieved heuristically since proper lumping cannot be explicitly performed for models described by nonlinear ODEs (common in systems pharmacology models), but the process can be applied directly to linear systems. The aim of this study is to systematically simplify a nonlinear systems model of bone biology [2] and then assess the performance of the simplified model by extrapolating improvement in long-term bone mineral density (BMD) responses from denosumab.
Methods: The methods were performed in two parts: (1) scale reduction of the bone biology model to accommodate the input target or denosumab (a RANKL inhibitor) and (2) the use of this model to analyse BMD data that arose from 1 year treatment with denosumab and then predict BMD of a further 4-year period. Part (1): the original system was first linearised using an inductive approximation in order that proper lumping can be fully applied [3]. Starting with the linearised original system, the best reduced model was searched using proper lumping together with a composite criterion consisting of two opposing indices, i.e. model performance and a penalty for complexity [4]. These were conducted using MATLAB R2015b. An identifiability analysis was conducted on the reduced model to assess parameter estimability. Part (2): the reduced (mechanistic) model and two empirical models were then “trained” (by parameter estimation) to BMD data following administration of denosumab for data over 1 year. Each model was then used to extrapolate the BMD response beyond 1 year. Data were extracted from [5]. Model fitting was conducted using NONMEM 7.3.0.
Results: A linearised version of the original nonlinear bone model was successfully obtained after 20 iterations of the linearisation process. Through proper lumping, the original 28-state original model was reduced to an 8-state model using an automated process based on choice of a weighting criterion value that balances model performance against model complexity. The reduced model described an increase in BMD after denosumab dosing which was indistinguishable from the original nonlinear bone model. The lumping process resulted in some of the states being lumped into either the RANK or RANKL state. Other states, e.g. RANK-RANKL complex and active TGB-beta, remained unlumped as in the original model. Lumping of these states significantly affected performance of the reduced model. Based on the identifiability analysis, 5 parameters were considered estimable. After fitting the reduced model to the BMD data until 1 year, the reduced model was able to be applied to extrapolate long-term BMD responses over 1 year. Both empirical models provided excellent fits to the 1 year BMD data but provided poor predictions when extrapolated beyond 1 year.
Conclusions: A scale of the nonlinear bone biology model was successfully reduced to an 8-state model by inductively linearising the system followed by automatic proper lumping. Importantly both the linearisation and lumping methods are reversible processes so that it is always possible to switch between full and reduced and nonlinear and linear systems. The reduced model described an increase in BMD after denosumab dosing that was equivalent to the full model and was able to accurately predict BMD change when used for extrapolating to long-term responses. The method used in this study is automatic, and can be applied directly to other multi-scale models for developing a mechanism-based structural model for future analyses.
References:
[1] Gulati A, Isbister GK, Duffull SB. Scale reduction of a systems coagulation model with an application to modeling pharmacokinetic-pharmacodynamic data. CPT Pharmacometrics Syst Pharmacol (2014) 3: e90.
[2] Peterson MC, Riggs MM. A physiologically based mathematical model of integrated calcium homeostasis and bone remodeling. Bone (2010) 46: 49-63.
[3] Hasegawa C, Duffull SB. Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations. J Pharmacokinet Pharmacodyn (2017) [Epub ahead of print].
[4] Hasegawa C, Duffull SB. Selection and qualification of simplified QSP models when using model order reduction techniques. AAPS J (2017) 20: 2.
[5] Miller PD, Bolognese MA, Lewiecki EM, McClung MR, Ding B, Austin M et al. Effect of denosumab on bone density and turnover in postmenopausal women with low bone mass after long-term continued, discontinued, and restarting of therapy: a randomized blinded phase 2 clinical trial. Bone (2008) 43: 222-29.
