Linearisation and automatic scale reduction of bone biology QSP models for data-driven analyses

Background: Bone biology is physiologically complex. Peterson et al [1] developed a QSP model consisting of 28 states that links integrated calcium homeostasis, hormonal control mechanisms and bone remodelling which consists of two continuous processes, (i) bone resorption coordinated by osteoclasts, and (ii) bone formation by osteoblasts. While the model naturally accommodates homeostatic mechanisms it may need to be simplified to apply for estimation or design exploration. Proper lumping has been used for order reduction of complicated linear models. This technique is however not straightforward to apply for nonlinear differential equations that are not uncommon in QSP models which are the basis of the bone biology model.

Aims: To explore (1) a simplification of a nonlinear systems bone biology model by inductively linearising the system followed by automated lumping, and (2) the utility of the reduced model for data-driven analyses.

Methods: An inductive approximation was applied to linearise the original system [2]. Starting with the linearised original system, the best reduced model was searched based on a composite criterion consisting of two opposing indices, i.e. model performance and a penalty for complexity [3]. The reduced model was then applied to extrapolate bone mineral density (BMD) responses beyond 1 year, after being trained on the first 12 months of data [4]. These predictions were compared to those of empirical models developed from data over the same period and extrapolated beyond 1 year.

Results: The original 28-state model was reduced to an 8-state model which described an increase in BMD after denosumab dosing while maintaining physiological meaning. The reduced model was then able to be applied to extrapolate long-term BMD responses over 1 year that were not able to be captured by an empirical approach.

Conclusion: A nonlinear bone biology model was successfully reduced to an 8-state model by inductively linearising the system followed by proper lumping. The reduced model described an increase in BMD after denosumab dosing while maintaining a physiological meaning, and could be used for extrapolating long-term responses. The method used in this study is automatic, and can directly be applied to other multiscale models for developing a mechanism-based structural model for future analyses.


  1. Peterson MC, Riggs MM (2010) Bone 46:49-63
  2. Hasegawa C, Duffull SB (2017) J Pharmacokinet Pharmacodyn [Epub ahead of print]
  3. Hasegawa C, Duffull SB (2017) AAPS J 20:2
  4. Miller PD, Bolognese MA, Lewiecki EM, McClung MR, Ding B, Austin M et al. (2008) Bone 43(2):222-229