On an adjoint-based numerical approach for time-dependent optimal control problems of biomedical interest

TL;DR

This paper presents a rigorous adjoint-based numerical framework for solving time-dependent optimal control problems governed by PDEs, with applications in biomedical drug delivery systems, demonstrating accuracy, flexibility, and robustness.

Contribution

It introduces a systematic adjoint-based approach for PDE-constrained optimal control problems in biomedical contexts, including verification and real-world applications.

Findings

Accurate and convergent numerical solutions for biomedical control problems.

Flexible handling of complex geometries and heterogeneous parameters.

Robust performance in realistic boundary condition scenarios.

Abstract

This work develops a rigorous numerical framework for solving time-dependent Optimal Control Problems (OCPs) governed by partial differential equations, with a particular focus on biomedical applications. The approach deals with adjoint-based Lagrangian methodology, which enables efficient gradient computation and systematic derivation of optimality conditions for both distributed and concentrated control formulations. The proposed framework is first verified using a time-dependent advection-diffusion problem endowed with a manufactured solution to assess accuracy and convergence properties. Subsequently, two representative applications involving drug delivery are investigated: (i) a light-triggered drug delivery system for targeted cancer therapy and (ii) a catheter-based drug delivery system in a patient-specific coronary artery. Numerical experiments not only demonstrate the accuracy of the approach, but also its flexibility and robustness in handling complex geometries, heterogeneous parameters, and realistic boundary conditions, highlighting its potential for the optimal design and control of complex biomedical systems.