Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany.
Math Biosci. 2011 Jan;229(1):123-34. doi: 10.1016/j.mbs.2010.11.007. Epub 2010 Dec 1.
In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock's direct multiple shooting method. In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.
本文针对化疗药物治疗方案的最优控制问题，研究了文献中的四个不同的数学模型。这些模型基于两组不同的常微分方程，包含化疗、免疫治疗、抗血管生成治疗或它们的组合。基于这些模型提出了最优控制问题的公式，并对其进行了讨论和比较。针对不同的参数集、场景和目标函数，使用 Bock 的直接多重打靶法对最优控制问题进行了数值求解。特别地，我们表明与标准治疗方案以及未经治疗或错误治疗的肿瘤相比，最优控制治疗可以成为肿瘤生长和完全消失的原因。此外，我们还比较了不同的目标函数。最后，我们提出了一个优化驱动的指标，用于评估最优控制的潜在收益。基于该指标，我们表明虽然目前可用的模型由于患者、癌症和治疗特异性因素，还不适合将最优疗法转化为医学实践，但化疗方案仍有很大的优化潜力。
