A Discrete Isoperimetric Optimal Control Approach for BCG Immunotherapy in Superficial Bladder Cancer: Discussions on Results of Different Optimal Doses

Abstract

Mathematical models devised for the study of the tumor-immune interactions and also, the optimization of clinical agents, are of increasing relevance for applications in cancer research. For our study case, it is important to note that immunotherapies in cancer are often recommended to be followed at discrete-times rather than injecting immunotherapeutic agents continuously. Thus, we are more interested in this present paper to propose a discrete-time mathematical model of superficial bladder cancer (sBCa), with a discrete control incorporated, for seeking the optimal dose of bacillus Calmette–Guerin (BCG) immunotherapy used in the treatment of sBCa. This research article retakes the problem of optimal control studied in Elmouki and Saadi (Int J Dyn Control 1–7, 2014) but in its discrete version, to investigate, the mathematical resolution of such models when an isoperimetric constraint is added, used to fix the total amount of BCG injected in the bladder along the period of BCG intravesical therapy, to an experimental constant dosage amount of BCG, tolerable by patients, based on information from Cheng et al. (ANZ J Surg 74:569–572, 2004). The optimal control characterization is provided based on a discretized maximum principle and the isoperimetric problem is solved numerically using a discrete iterative progressive-regressive scheme combined with the secant method. We include a discussions section in the end of the paper for the analysis of the consequences related to the different administration strategies obtained.