Mathematical Modeling of Physiologically Based Pharmacokinetics: Qualitative Analysis, Numerical Methods and Validation on Experimental Data

The paper builds a system of six nonlinear ordinary differential equations describing the distribution and metabolism of medicinal substances in the mammalian body (PBPK model). A complete qualitative analysis is carried out: the existence and uniqueness of a solution for any non-negative initial conditions are proved, the invariance of the positive orthant is established, and the boundedness of solutions is estimated. A quadratic Lyapunov function was constructed for the zero stationary state, which made it possible to strictly prove the global asymptotic stability and estimate the convergence rate. The Runge–Kutta method of the 4th order with a fixed step is used for numerical integration; the fourth order of convergence is confirmed by the method of solution production. A local sensitivity analysis was performed on six parameters, and the most influential factors were identified. The model was validated on published data for paracetamol in rats: the discrepancy in peak concentration, AUC and half-life does not exceed 6%. The work demonstrates a combination of rigorous mathematical apparatus with practical significance for preclinical research.