Multiphysics pharmacokinetic model for targeted nanoparticles

Abstract

Nanoparticles (NP) are being increasingly explored as vehicles for targeted drug delivery because they can overcome free therapeutic limitations by drug encapsulation, thereby increasing solubility and transport across cell membranes. However, a translational gap exists from animal to human studies resulting in only several NP having FDA approval. Because of this, researchers have begun to turn toward physiologically based pharmacokinetic (PBPK) models to guide in vivo NP experimentation. However, typical PBPK models use an empirically derived framework that cannot be universally applied to varying NP constructs and experimental settings. The purpose of this study was to develop a physics-based multiscale PBPK compartmental model for determining continuous NP biodistribution. We successfully developed two versions of a physics-based compartmental model, models A and B, and validated the models with experimental data. The more physiologically relevant model (model B) had an output that more closely resembled experimental data as determined by normalized root mean squared deviation (NRMSD) analysis. A branched model was developed to enable the model to account for varying NP sizes. With the help of the branched model, we were able to show that branching in vasculature causes enhanced uptake of NP in the organ tissue. The models were solved using two of the most popular computational platforms, MATLAB and Julia. Our experimentation with the two suggests the highly optimized ODE solver package DifferentialEquations.jl in Julia outperforms MATLAB when solving a stiff system of ordinary differential equations (ODEs). We experimented with solving our PBPK model with a neural network using Julia's Flux.jl package. We were able to demonstrate that a neural network can learn to solve a system of ODEs when the system can be made non-stiff via quasi-steady-state approximation (QSSA). Our model incorporates modules that account for varying NP surface chemistries, multiscale vascular hydrodynamic effects, and effects of the immune system to create a more comprehensive and modular model for predicting NP biodistribution in a variety of NP constructs.

Para- and Transcellular Transport Kinetics of Nanoparticles across Lymphatic Endothelial Cells

“…Both studies were able to demonstrate that multicompartment models and systems of differential equations were able to recapitulate in vivo data. More so in their studies, they highlighted the predictive power of these models by predicting how nanoparticle formulation parameters could alter transport efficiency. , A key part of these studies is linking theoretical kinetic data with in vivo data. Neubauer et al were able to use MRI as a method to validate three-compartment models in pharmacological studies.…”

Mathematical complexities in radionuclide metabolic modelling: a review of ordinary differential equation kinetics solvers in biokinetic modelling

“…Most of these solvers are connected with external codes, which are compiled either from FORTRAN or C. Following a detailed study of different problem classes with respective ODE solvers, Andersson et al ( 2015 ) proposed to increase the variety of original codes and make them available through the framework provided. Furthermore, a dedicated study on multiphysics pharmacokinetic models demonstrated the need for ODE solvers in compartmental modelling (Glass et al 2022 ). The motivation for this recent study was that physiologically-based pharmacokinetic (PBPK) models use an empirically derived framework that cannot be universally applied to varying nanoparticle constructs and experimental settings.…”

“…According to Glass et al ( 2022 ), two versions of physics-based compartmental models were developed, for which the stiff ODE solving methods used were from MATLAB and Julia (Rackauckas 2017 , Bezanson et al 2017 ) and validated against experimental data. Julia was developed as an alternative to Python and MATLAB.…”

“…Julia was developed as an alternative to Python and MATLAB. For a precise evaluation of the handling of ODE stiffness for both models, Glass et al ( 2022 ) used one stiff MATLAB solver known as Ode15s and five other stiff solvers—such as QNDF, Rodas4, KenCarp4, TRBDF2, and RadauIIA5 from the DifferentialEquations.jl package in Julia. Ode15s from MATLAB was used for solving the system of large and stiff ODEs; however, this resulted in biodistribution solutions for a time interval of 0–1 ms.…”

“…According to Glass et al ( 2022 ), this is due to the nature of the times (small) required for stability in the solver, and thus MATLAB becomes unresponsive if the time steps are increased beyond 1 ns. Moreover, the systems were solved successfully using the stiff packages in Julia for large time points.…”