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

TL;DR

This work develops an adjoint-based Lagrangian framework for time-dependent optimal control problems governed by PDEs, with biomedical applications. It enables efficient gradient computation and derivation of first-order optimality conditions for both distributed and boundary controls, validated on a time-dependent advection-diffusion benchmark and two drug-delivery problems involving light-triggered release and catheter-based delivery in patient-specific geometries. Numerical results show quadratic convergence with mesh refinement, robustness to complex geometries and heterogeneous parameters, and successful steering of drug distributions toward prescribed targets, albeit with notable computational cost. The authors suggest reduced-order modeling as a pathway to dramatic runtime reductions and broader applicability in personalized medicine and hemodynamics.

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.

Figures (13)

• Figure 1: Coronary artery system: sketch of the geometry, including LMCA (Left Main Coronary Artery), LAD (Left Anterior Descending), and LCx (Left Circumflex).
• Figure 2: Academic benchmark: numerical solution $y$ (top row), analytical solution $y_{\text{exact}}$ (second row) and pointwise error $y-y_\text{exact}$ (bottom row) for $\epsilon = 1$ (first column), $\epsilon = 10^{-1}$ (second column), and $\epsilon = 10^{-2}$ (third column) at the final time $T_f=1$. The results were obtained with a mesh of size $h = 1/32$.
• Figure 3: Academic benchmark: numerical solution $\lambda$ (top row), analytical solution $\lambda_{\text{exact}}$ (second row) and pointwise error $\lambda-\lambda_\text{exact}$ (bottom row) for $\epsilon = 1$ (first column), $\epsilon = 10^{-1}$ (second column), and $\epsilon = 10^{-2}$ (third column) at the final time $T_f=1$. The results were obtained with mesh size $h = 1/32$.
• Figure 4: Application 1 - Distributed case: convergence history of the objective function $J$ (top left) and of the contribution $J_{c_{f,T_f}}$ (top right), spatial distribution of the control variable (bottom left) and of the free drug concentration at the final time of the simulation (bottom right) for different values of $\beta_1$ and for $I^0=5$.
• Figure 5: Application 1 - Distributed case: convergence history of the objective function $J$ (top left) and of the contribution $J_{c_{f,T_f}}$ (top right), spatial distribution of the control variable (bottom left) and of the free drug concentration at the final time of the simulation (bottom right) for different values of $\beta_1$ and for $I^0=15$.

Theorems & Definitions (2)

• Remark 2.1
• Remark 2.2