Optimal duration and dosage of BCG intravesical immunotherapy: A free final time optimal control approach

• Published in: Mathematical methods in the applied sciences 2018-03, Vol.41 (5), p.2209-2219
• Main Authors: Alkama, Meryem, Larrache, Abdelilah, Rachik, Mostafa, Elmouki, Ilias
• Format: Article
• Language: English
• Subjects: BCG immunotherapy; Bladder; bladder cancer; Differential equations; Dosage; final gain function; free final time; Immune system; Immunotherapy; Maximum principle; Optimal control; Optimization; Ordinary differential equations; Queuing theory; Time optimal control
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.4745

In this paper, we propose a free final time optimal control approach applied to 4 ordinary differential equations which describe the tumor‐immune interactions after the injection of the bacillus Calmette‐Guérin (BCG) in the bladder of a hypothetical patient. The main goal of this optimal control strategy is to find the optimal dosage amount needed in each instillation of BCG for stimulating the immune‐system and then killing superficial bladder tumors and to determine the optimal duration of treatment, adequate for stopping the intravesical therapy with lesser side‐effects. For this, we introduce into the model of interest, a control function which represents the dose of BCG immunotherapy procedure and we formulate a minimization problem where the final time is considered free (nonfixed). The characterization of the sought optimal control noted u∗ is derived based on Pontryagin's maximum principle, while the formulation of the sought optimal final time noted tfinal∗ is based on formulae of sensitivity which are obtained conditions from the derivative of the objective function with respect to tfinal∗. We investigate the resolution of the free final time optimal control problem in 3 possible cases: (a) when tfinal∗ is quadratic in the final gain function, (b) when the final gain function does not depend on tfinal∗, and (c) when tfinal∗ is linear in the final gain function. Finally, we obtain the sought optimal dose of BCG, and we conclude that in case (a), we obtain an optimal duration which is more beneficial regarding the activation of immune cells while cases (b) and (c) both provide an optimal duration which is more adequate for the minimization of the tumoral population.