Lomonosov Moscow State University, Leninskie Gory, MSU, 2nd educational building, Moscow, 119991, Russia; Moscow State University of Railway Engineering, Obraztsova 15, Moscow, 127994, Russia.
Lomonosov Moscow State University, Leninskie Gory, MSU, 2nd educational building, Moscow, 119991, Russia.
Math Biosci. 2017 Dec;294:110-119. doi: 10.1016/j.mbs.2017.10.011. Epub 2017 Oct 23.
In this paper, we study a dynamic optimization problem for a general nonlinear mathematical model for therapy of a lethal form of cancer. The model describes how the populations of cancer and normal cells evolve under the influence of the concentrations of nutrients (oxygen, glucose, etc.) and the applied therapeutic agent (drug). Regulated intensity of the therapy is interpreted as a time-dependent control strategy. The therapy (control) goal is to maximize the viability time, i. e., the duration of staying in a so-called safety region (which specifies safe living conditions of a patient in terms of constraints on the amounts of cancer and normal cells), subject to limited resources of the therapeutic agent. In a specific benchmark case, a novel optimality principle for admissible therapy strategies is established. It states that the optimal trajectories should finally reach a certain corner of the safety region or at least the upper constraint on the quantity of cancer cells. The results of numerical simulations appear to be in good agreement with the proposed principle. Typical qualitative structures of optimal treatment strategies are also obtained. Furthermore, important characteristics of the model are the competition coefficient (describing the negative influence of cancer cells on normal cells), the upper bound in the drug integral constraint, and the ratio between the therapy and damage coefficients (i. e., the ratio between the positive primary and negative side effects of the therapy).
本文研究了一种用于致命性癌症治疗的一般非线性数学模型的动态优化问题。该模型描述了在营养物质（氧气、葡萄糖等）浓度和应用治疗剂（药物）的影响下，癌细胞和正常细胞群体的演化。调节治疗强度被解释为一种时变控制策略。治疗（控制）的目标是使生存时间最大化，即患者在所谓的安全区域（根据癌细胞和正常细胞数量的约束来规定患者的安全生活条件）内停留的持续时间最大化，同时受到治疗剂资源的限制。在一个特定的基准案例中，建立了一种新的可接受治疗策略的最优性原理。它指出，最优轨迹最终应达到安全区域的某个角落，或者至少应达到癌细胞数量的上限约束。数值模拟的结果与提出的原理吻合较好。还获得了最优治疗策略的典型定性结构。此外，模型的重要特征是竞争系数（描述癌细胞对正常细胞的负面影响）、药物积分约束的上限以及治疗和损伤系数之间的比值（即治疗的正初级和负副作用的比值）。
