Optimization of drug diffusion in biological tissues using interior point methods

Abstract

Diffusion equations are central tools in modeling transport phenomena in biological systems, including the movement of drugs, nutrients, and signaling molecules within tissues. This paper proposes a novel numerical framework combining primal-dual interior point methods (IPMs) with preconditioned conjugate gradient (PCG) preconditioning for optimizing drug delivery in heterogeneous tumor tissue. It is the first such application achieving O(√M log(1/ε)) complexity independent of PDE conditioning, where κ(AD) = O(1/Δx²). The mathematical model is formulated as a reaction-diffusion PDE-constrained quadratic program and discretized using unconditionally stable implicit finite difference methods. The approach is applied to a pharmacokinetic case study involving the optimization of localized drug release S*(x,y) in a two-dimensional tissue domain containing tumor and healthy tissue interfaces. Numerical results show smooth and stable cytotoxic concentration profiles C*(x,y,t) that effectively target tumor cores (x,y > 0.5) while improving therapeutic efficacy and minimizing side effects (Figures 1–3).