How to mathematically optimize drug regimens using optimal control

Abstract

This article gives an overview of a technique called optimal control, which is used to optimize real-world quantities represented by mathematical models. I include background information about the historical development of the technique and applications in a variety of fields. The main focus here is the application to diseases and therapies, particularly the optimization of combination therapies, and I highlight several such examples. I also describe the basic theory of optimal control, and illustrate each of the steps with an example that optimizes the doses in a combination regimen for leukemia. References are provided for more complex cases. The article is aimed at modelers working in drug development, who have not used optimal control previously. My goal is to make this technique more accessible in the biopharma community.

Keywords: Combination therapy; Constrained optimization; Control theory; Disease modeling; Optimal control.

Fig. 1

Steps in optimization of drug regimens. Evaluation can be performed by running preclinical (animal) or clinical (human) studies and comparing outcomes to the optimal control predictions.

Fig. 2

HIV therapy example: How much better can an optimal control regimen be, in comparison to a standard constant-dose regimen? The solid orange curve represents a protease inhibitor (${\eta }_1$) and the dashed brown curve represents a reverse transcriptase inhibitor. Each has been scaled so that 0 represents no drug administered and 1 represents a level achieving complete efficacy. The solid black curve represents a healthy T cell population (T) and the dashed purple curve represents an infected cell population ($I_{00}$). The dotted green line indicates 200 cells$/\mu L$. Total exposure (area under the curve, AUC) is the same for both regimens, for each drug individually. Both regimens control the infected cell levels, but the optimal regimen gives a better outcome for the patient’s healthy T cell levels. Adapted from [15] (Color figure online)

Fig. 3

Numerical solutions for $u_1$ and $u_2$ for various values of parameter $d_c$. Three different choices for the sensitive parameter $d_c$ give different optimal regimens for the therapies $u_1$ and $u_2$ for a hypothetical patient. From [53]