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

Area of Science:

• Computational mathematics
• Biomedical engineering
• Pharmacokinetics

Background:

• Optimal control problems (OCPs) are crucial for designing efficient biomedical interventions.
• Existing numerical methods may lack the rigor or efficiency for complex drug delivery dynamics.
• Adjoint-based methods offer a systematic way to derive optimality conditions.

Purpose of the Study:

• To develop a robust numerical framework for time-dependent optimal control problems in drug delivery.
• To enable efficient gradient computation and optimality condition derivation.
• To validate the framework's accuracy, flexibility, and robustness for biomedical applications.

Main Methods:

• Adjoint-based Lagrangian methodology for gradient computation.
• Systematic derivation of optimality conditions for distributed and concentrated controls.
• Verification using a time-dependent advection-diffusion equation with a manufactured solution.

Main Results:

• The numerical framework accurately solves time-dependent OCPs.
• The method demonstrates efficiency in gradient computation and optimality condition derivation.
• The approach is validated for drug delivery problems, showing accuracy and robustness.

Conclusions:

• The developed numerical framework is a powerful tool for optimizing drug delivery strategies.
• The adjoint-based approach provides a rigorous and efficient solution for complex biomedical control problems.
• The study confirms the framework's accuracy, flexibility, and robustness for practical applications.