Analysis of Optimal Controls for Biomedical Models of Cancer and HIV

0205093Ledzewicz 在该项目中，将研究最优控制问题，这些问题作为生物医学系统的数学模型出现在具有强烈细胞增殖方面的疾病（例如癌症或艾滋病）的化疗中。将进行三个研究方向：癌症化疗模型分析、艾滋病毒感染和艾滋病抗病毒治疗模型分析以及结合癌症和艾滋病毒模型共同特征的通用数学框架的设计和分析。研究人员将利用他们之前在癌症化疗模型方面的经验，并应用基于特征和高阶条件方法的新工具来实现最优性。在癌症化疗的区室模型中，细胞周期是通过将其阶段聚集到区室中并使用阶段特异性细胞抑制杀伤剂和阻断剂来控制的。目标是最大限度地增加药物中细胞抑制剂杀死的癌细胞数量，同时保持对正常组织的毒性可接受。在该项目中，将为这些模型构建最优控制的综合。此外，还将分析更复杂的数学模型，这些模型考虑了诸如不断变化的耐药性或骨髓破坏等其他方面。由于艾滋病毒感染的复杂性，艾滋病的数学模型直到最近才被开发出来，并且仍在不断地根据新的医学数据进行修订和更新。迄今为止，人们主要关注的是最能模拟药物治疗下病毒与人类免疫系统之间相互作用的动力学形式，但许多模型仍未被表述为最优控制问题。研究人员将在最佳控制的框架内分析各种此类模型，并比较旨在设计最佳治疗方案的解决方案。 在该项目中，研究人员将考虑癌症或艾滋病等具有强烈细胞增殖作用的疾病的化疗中的数学模型。这些模型将被分析为最优控制问题，以药物剂量作为控制参数，目标是在癌症情况下最大化药物杀死的癌细胞数量（在艾滋病化疗情况下分别最大化未感染细胞的数量），同时最大限度地减少化疗的有害副作用和成本。尽管这些问题的具体情况各不相同，但它们也有许多共同点，可以放入一个涵盖所有问题的通用数学模型中。因此，一方面需要单独考虑这些问题，以了解对潜在疾病的影响，另一方面，通过研究这些模型共有的一般属性，可以进行简化并获得见解。该项目将涉及两个方向。最佳控制所满足的条件的生物医学解释将以生物医学人员可以理解的方式给出，并且这些条件将与潜在的生物学情况相关。预计该项目的结果将引起医学界的兴趣，因为它可以对现有癌症或艾滋病毒化疗方案的分析提供一些见解，并可能有助于设计新的改进方案。这些结果对于艾滋病毒治疗尤其重要，迄今为止，艾滋病毒治疗并不能治愈这种疾病，而只能提供长期维持计划。

0205093LedzewiczIn the project, optimal control problems will be investigated which arise as mathematical models for biomedical systems in the chemotherapy of diseases which have a strong cell proliferation aspect such as cancer or AIDS. Three directions of research will be pursued: the analysis of models in cancer chemotherapy, an analysis of models for HIV-infection and anti-viral treatment of AIDS, and the design and analysis of a general mathematical framework which combines common features of both cancer and HIV models. The investigators will use their previous experience working on cancer chemotherapy models and apply new tools based on the method of characteristics and high-order conditions for optimality. In compartmental models for cancer chemotherapy the cell-cycle is controlled by clustering its phases into compartments and using phase specific cytostatic killing and blocking agents. The goal is to maximize the number of cancer cells that the cytostatic agent in the drug kills while keeping the toxicity to the normal tissue acceptable. In this project a synthesis of optimal controls for these models will be constructed. Furthermore, more complex mathematical models that take into account additional aspects such as evolving drug resistance or bone marrow destruction will be analyzed. Due to the complexity of HIV infection, mathematical models for AIDS have only recently been developed and still are constantly being revised and updated taking new medical data into account. With the main attention so far given to the form of the dynamics that best models the interactions between the virus and the human immune system under drug treatment, many of these models still have not been formulated as optimal control problems. The investigators will analyze a variety of these models in the framework of optimal control and compare the solutions aiming at a design of optimal treatment protocols.In the project, the investigators will consider mathematical models in the chemotherapy of diseases that have a strong cell proliferation aspect such a cancer or AIDS. These models will be analyzed as optimal control problems with the drug dosage serving as control parameter with the goal of maximizing the number of cancer cells killed by the drug in the case of cancer (respectively maximizing the number of uninfected cells in the case of chemotherapy for AIDS) while minimizing the harmful side effects and cost of the chemotherapy. Although individually these problems are very different in their specifics, they also have many aspects in common and can be put into one general mathematical model that encompasses them all. Thus, while on one side there is a need to consider these problems separately to understand the implications for the underlying disease, on the other side there are simplifications and insights to be gained by looking at the general properties common to these models. This project will address both directions. A biomedical interpretation of the conditions that optimal controls satisfy will be given in terms understandable by biomedical personnel and the conditions will be related to the underlying biological situation. It is expected that the outcomes of this project will be of interest to the medical community by giving some insights into the analysis of existing chemotherapy protocols for cancer or HIV and possibly aid in the design of new improved protocols. These results are particularly important for HIV treatments which so far do not cure the disease, but only provide a long-term maintenance program.