A Comparative Analysis of Physiologically Based Pharmacokinetic Models for Human Immunodeficiency Virus and Tuberculosis Infections.

Physiologically based pharmacokinetic (PBPK) models have gained in popularity in the last decade in both drug development and regulatory science. PBPK models differ from classical pharmacokinetic models in that they include specific compartments for tissues involved in exposure, toxicity, biotransformation, and clearance processes connected by blood flow. This study aimed to address the gaps between the mathematics and pharmacology framework observed in the literature. These gaps included nonconserved systems of equations and compartment concentration that were not biologically relatable to the tissues of interest. The resulting system of nonlinear differential equations is solved numerically with various methods for benchmarking and comparison. Furthermore, a sensitivity analysis of all parameters were conducted to elucidate the critical parameters of the model. The resulting model was fit to clinical data as a performance benchmark. The clinical data captured the second line of antiretroviral treatment, lopinavir and ritonavir. The model and clinical data correlate well for coadministration of lopinavir/ritonavir with rifampin. Drug-drug interaction was captured between lopinavir and rifampin. This article provides conclusions about the suitability of physiologically based pharmacokinetic models for the prediction of drug-drug interaction and antiretroviral and anti-TB pharmacokinetics.

Numerical Convergence Analysis of the Frank-Kamenetskii Equation.

This work investigates the convergence dynamics of a numerical scheme employed for the approximation and solution of the Frank-Kamenetskii partial differential equation. A framework for computing the critical Frank-Kamenetskii parameter to arbitrary accuracy is presented and used in the subsequent numerical simulations. The numerical method employed is a Crank-Nicolson type implicit scheme coupled with a fourth order spatial discretisation as well as a Newton-Raphson update step which allows for the nonlinear source term to be treated implicitly. This numerical implementation allows for the analysis of the convergence of the transient solution toward the steady-state solution. The choice of termination criteria, numerically dictating this convergence, is interrogated and it is found that the traditional choice for termination is insufficient in the case of the Frank-Kamenetskii partial differential equation which exhibits slow transience as the solution approaches the steady-state. Four measures of convergence are proposed, compared and discussed herein.

Nonstandard Finite Difference Method Applied to a Linear Pharmacokinetics Model.

We extend the nonstandard finite difference method of solution to the study of pharmacokinetic-pharmacodynamic models. Pharmacokinetic (PK) models are commonly used to predict drug concentrations that drive controlled intravenous (I.V.) transfers (or infusion and oral transfers) while pharmacokinetic and pharmacodynamic (PD) interaction models are used to provide predictions of drug concentrations affecting the response of these clinical drugs. We structure a nonstandard finite difference (NSFD) scheme for the relevant system of equations which models this pharamcokinetic process. We compare the results obtained to standard methods. The scheme is dynamically consistent and reliable in replicating complex dynamic properties of the relevant continuous models for varying step sizes. This study provides assistance in understanding the long-term behavior of the drug in the system, and validation of the efficiency of the nonstandard finite difference scheme as the method of choice.