Analytical Solutions of PBTK Models for Evaluating the Impact of Surface Diffusion Characteristics on the Leaching Profile of Implant Byproducts

Toxicokinetic or pharmacokinetic models, physiologically based or not, offer a unique avenue to understand the transport of toxins or pharmaceuticals in living organisms. The availability of analytical solutions to such models offers the means to engage in a plethora of applications. In the present work, we provide the framework to solve analytically such models using the matrix exponential, and we then apply this method to derive an explicit solution to four-to-five-compartment physiologically based toxicokinetic (PBTK) models considering a single- and an infinite-exponential expression for the amount of mass released from an implantable device. We also offer the conditions that need to be met for analytical solutions to be obtained when the kinetic rates are time-dependent functions. Our analysis compares the computation time between analytical and numerical solutions and characterizes the dependency of the maximum substance mass value and the time it occurs in the various tissue compartments from the material surface diffusion characteristics. Our analytical solutions, which have several advantages over the solutions obtained using numerical solvers, can be incorporated into in silico tools and provide valuable information for human health risk assessment.