Symbolic PBPK-PDE Modeling Using Open-Source Julia Tools
Authors: Elmokadem Ahmed , Kirouac Daniel , Knab Timothy , Li Yuezhe , Davis Jimena 
Published: February 2025

Objectives: Physiologically based pharmacokinetic (PBPK) models provide a mechanistic characterization of a drug’s distribution in the body. Ordinary differential equations (ODEs) are used for most PBPK models in literature which ignore the spatial distribution of a drug. Spatial distribution, however, can be critical to understand the PK of some drugs like topical preparations, inhaled treatments, and antitumor therapies. As such, investigators have tried to combine PBPK models, represented as ODEs, with partial differential equations (PDEs) to capture the spatial component of drug distribution. PBPK-PDE models are, however, challenging to build. The current work demonstrates a framework to build PBPK-PDE models using the open-source Julia [1] package, ModelingToolkit.jl [2], which can symbolically represent equations and simplify PDE model coding and integration with ODEs. A PBPK-PDE model of naphthalene diffusion from the skin into the body is used as an example of framework. Methods: The PBPK-PDE model used in this work was a simplified version of a previously published naphthalene PBPK-PDE model [3]. The PBPK model described the distribution of topically administered naphthalene from the skin compartment into the circulation and remaining compartments (lung, liver, fat, poorly perfused and richly perfused tissues). The skin compartment was dissected into an outer well where naphthalene was introduced, stratum corneum (SC) and viable epidermis (VE). The diffusion of naphthalene across the one-dimensional space of the SC was represented as a PDE. The Julia open-source package MethodOfLines.jl [4] was used to automatically discretize the PDE problem. Boundary conditions were set to equilibrium conditions between the well and the outermost layer of the SC and between the VE and the innermost layer of the SC.Results: The PBPK-PDE model was able to characterize the diffusion of naphthalene in the different SC layers as well as its penetration into the systemic circulation following dermal administration. The concentration of naphthalene in each of the discretized one-dimensional SC space versus time was demonstrated. Conclusions: A framework using the Julia open-source tool, ModelingToolkit.jl, was developed to build PBPK-PDE models in a simple and intuitive way. A naphthalene PBPK-PDE model was used as a proof-of-concept, while the framework was generally applicable to the variety of pharmacometric models where a spatial component was critical to understanding activity.Citations: 1. Bezanson J, Edelman A, Karpinski S, Shah VB. Julia: A Fresh Approach to Numerical Computing. SIAM Rev. 2017;59: 65–98.2. Ma Y, Gowda S, Anantharaman R, Laughman C, Shah V, Rackauckas C. ModelingToolkit: A Composable Graph Transformation System For Equation-Based Modeling. arXiv [cs.MS]. 2021. Available: http://arxiv.org/abs/2103.052443. Kapraun DF, Schlosser PM, Nylander-French LA, Kim D, Yost EE, Druwe IL. A Physiologically Based Pharmacokinetic Model for Naphthalene With Inhalation and Skin Routes of Exposure. Toxicol Sci. 2020;177: 377–391.4. Creators Jones, Alex W. 1 Hyett, Criston Rackauckas, Chris2 Wellcome Trust Chan Zuckerberg Initiative (United States) Show affiliations 1. SciML 2. MIT. MethodOfLines.jl - Automatic finite difference PDE discretization and solving with Julia SciML. doi:10.5281/zenodo.11186853