Niklas Hartung 1, Wilhelm Huisinga 1
1 University of Potsdam (Potsdam, Germany)
OBJECTIVES
Physiologically-based pharmacokinetic (PBPK) models predict blood and tissue concentrations through a complex physiological parametrization and compartmentalization. Proper lumping allows to simplify such models, while retaining interpretability of lumped compartments [1]. Employing a lumping matrix and a pseudoinverse of the lumping matrix, a reduced system can be obtained for general linear systems [2]. While any pseudoinverse yields the same reduced system, the choice of pseudoinverse does matter when extracting the original states from lumped states (unlumping). Different means of addressing this non-uniqueness have been proposed in the literature. First, the lumped model performance can be evaluated in comparison to lumped data. While this approach does not require any unlumping, it drastically limits applicability since it requires tissue data to be available for all compartments that are lumped together [1,3]. Another approach consists of not lumping observable states, which also avoids unlumping during model evaluation, but again strongly limits the possibilities for lumping [4]. Finally, to directly address rather than circumvent the unlumping problem, lumping conditions have been formulated [5,6]. This latter method, which we call mechanistic lumping, corresponds to a particular choice of pseudoinverse of the lumping matrix, generally different from the Moore-Penrose pseudoinverse suggested in some articles [4,7]. Since this method derives an algebraic structure in the model (the lumping conditions), it is applicable across substances. However, to date the applicability of mechanistic lumping is limited to a few models (e.g., well-stirred model for small-molecule drugs or an extravasation-based model for monoclonal antibodies) since the lumped models must be derived manually for each set of equations. A generalization to a larger class of models (such as stable linear systems) is still lacking. In this work, we aim to extend mechanistic lumping to general stable linear systems.
METHODS
We propose to generalize the formulation of lumping conditions via a global quasi-steady state assumption of model states with respect to a reference state. Importantly, dynamic differences can still be kept for lumped models – the approach only informs the unlumping step. To efficiently operationalize this method for large PBPK models, we propose to exploit the sparse connectivity of a PBPK model topology by decomposing its linear compartmental system into a graph encoding model topology and an algebraic representation of local changes (mass transfer rates and elimination) and exploiting efficient graph-theoretical algorithms [8]. More precisely, lumping conditions are derived via the following scheme: (1) selection of a reference state inducing a pruned graph without incoming connections to the selected state; (2) construction of the corresponding condensation graph; (3) topological traversal and symbolic determination of local quasi-steady states, including incoming edges to the reference state and elimination.
RESULTS
The proposed algorithm was implemented in the compphysiol R package [9]. When applied to the previously studied well-stirred and extravasation-limited models, it yielded the same lumping conditions as derived in the literature, but in a fully automated manner. Furthermore, lumping conditions for a permeation-based model with extracellular and cellular organ subcompartments were derived. The choice of venous blood as a reference state (rather than arterial blood) was made explicit in this construction, as it has not been mentioned previously. Importantly, specific parameters such as the hepatic extraction ratio are automatically obtained with this method, without any need for manual derivation. Furthermore, the formalization of our approach sheds insights into what makes mechanistic lumping structurally complex. Whereas convective processes, e.g., blood supply of the organs, simply propagate local quasi-steady state conditions through the graph, diffusive exchange produces clusters in the condensation graph necessitating symbolic solutions of multidimensional systems, resulting in complex expressions in lumping conditions.
CONCLUSIONS
Many variations of PBPK models have been used in the literature, in particular mechanistic tissue submodels involving different tissue layers. By formalizing and extending mechanistic lumping and implementing an efficient algorithm to calculate lumping conditions, the range of applicability of this method has been extended significantly, lowering the barrier to use this methodology, potentially also in commercial and open-source PBPK software tools.
References:
[1] Nestorov et al., J. Pharmacokinet. Biopharm. 26(1):21-46, 1998
[2] Li and Rabitz, Chem. Eng. Sci. 44(6):1413-30, 1989
[3] Dokoumetzidis and Aarons, J. Pharmacokinet. Pharmacodyn. 36(6):613-28, 2009
[4] Pan and Duffull, J. Pharmacokinet. Pharmacodyn. 46(6):361-70, 2019
[5] Pilari and Huisinga, J. Pharmacokinet. Pharmacodyn. 37(4):365-405, 2010
[6] Fronton et al., Pharmacokinet. Pharmacodyn. 41(2):87-107, 2014
[7] Li and Rabitz, Chem. Eng. Sci. 45(4):977-1002, 1990
[8] Tarjan, SIAM J. Comput. 1(2), 1972
[9] Hartung N. (2026). compphysiol: Computational Physiology Toolbox. [online] https://github.com/niklhart/compphysiol-R
Reference: PAGE 34 (2026) Abstr 12298 [www.page-meeting.org/?abstract=12298]
Poster: Methodology - New Modelling Approaches