A partial differential equation approach to inhalation PBPK modelling
Elin Boger (1), Oskar Wigström (2)
(1) RIA, IMED Biotech Unit, AstraZeneca R&D Gothenburg, Sweden (2) Automation Research Group, Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden
Inhalation pharmacokinetics (PK) is known to be vastly complex due to the spatial heterogeneity in lung physiology and the range of processes affecting pulmonary drug disposition. Further complexity is added from the difficulty associated with measuring relevant concentrations, i.e. the local concentration(s) driving the pharmacological effect. Hence, it is non-trivial to predict and evaluate how various design decisions will affect the PK, and ultimately the pharmacodynamics (PD) of locally acting inhaled drugs. In the absence of relevant measurements, mechanistic modelling becomes even more critical. However, although some inhalation PBPK models accommodate all main features of inhaled drug disposition, these do not include a high-resolution model of both transport (mucociliary clearance) and shrinkage (dissolution) of polydisperse particles at high resolution. This study therefore aimed to develop the first high resolution model accounting for all main pulmonary drug disposition processes as well as the heterogeneity in physiology and particle size.
The first inhalation PBPK model utilizing partial differential equations (PDEs) for describing dissolving particles of varying size over the depth of the lung was developed. The lung is further described by three states varying over the lung depth: 1) epithelial lining fluid, 2) epithelium, and 3) sub-epithelium. The model mechanistically describes important processes for pulmonary drug disposition, including regional drug deposition, particle dissolution and mucociliary clearance. Furthermore, by reducing the system to a one dimensional PDE and addressing the numerical issues encountered when simulating shrinking particles in other models [1-2], the computational cost is significantly reduced without any loss of accuracy.
In multiple case studies, we demonstrate important features of the model and evaluate how different design decisions affect the target site concentration(s). Furthermore, we also explore how changes in these concentrations are reflected by measurements from observable states. That is, the model can theoretically explore if and how contemporary sampling techniques will reflect the dynamics of the target site concentration(s).
For instance, the particle size distribution (PSD) is highlighted as an important design parameter, both for regional lung-targeting and duration. For poorly soluble compounds, simulations show that larger PSDs provide longer drug coverage at the target as compared to smaller PSDs. However, this comes at the expense of lower drug levels. Smaller PSDs have previously been demonstrated to achieve a higher and earlier plasma peak compared to larger particles , a behavior that was reproduced by our simulations.
In all case studies, a spectrum of free concentrations is predicted to arise along the lung with distinct differences between the epithelial and sub-epithelial layer. Additionally, simulations indicate that the advantage of inhalation can be almost eradicated if the inhaled dose is high enough. Interestingly, this occurs already at dose levels where nonlinearities would be challenging to detect from the measurable plasma concentrations.
The presented model can be used for guiding the design of inhaled molecules and PSDs. Furthermore, it can aid the design and interpretation of preclinical/clinical studies. Its high spatial resolution provides opportunities to explore regional lung-targeting. The importance of highly spatially resolved simulations is further highlighted as predictions indicate that a spectrum of free drug concentrations spans the lung after inhalation. This finding has interesting implications for PK/PD-modelling, as it suggests it is inappropriate to assume a single free concentration driving the pharmacological effect of locally acting inhaled drugs. Equally important, this result emphasizes that it is crucial to identify the pulmonary region(s) relevant for the effect to enable informed design decisions.
Interestingly, the simulations raise concerns about the utility of using plasma PK for evaluating the local effect and therapeutic ratio of inhaled drugs. Thus, prompting the need for utilizing mechanistic modelling and investing in identifying other sampling techniques. Furthermore, the mathematical description is general and may be extended to describe absorption from the gastrointestinal tract, where high level of discretization is still used [1, 4].
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