Analysis of Optimal Controls for Biomedical Models of Cancer and HIV

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癌症或艾滋病等具有强烈细胞增殖方面的疾病的化疗中。将继续进行三个研究方向：癌症化疗模型的分析，艾滋病毒感染和艾滋病抗病毒治疗模型的分析，以及结合癌症和艾滋病毒模型的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以前在癌症化疗模型方面的工作经验，并应用基于特征方法和高阶条件的新工具，以实现最优。在癌症化疗的间隔室模型中，通过将细胞周期分成不同的间隔室并使用特定阶段的细胞抑制杀伤剂和阻滞剂来控制细胞周期。其目标是最大限度地增加药物中的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。在本项目中，将构建这些模型的最优控制的综合。此外，还将分析更复杂的数学模型，这些模型将考虑到其他方面，如进化中的耐药性或骨髓破坏。由于艾滋病毒感染的复杂性，艾滋病的数学模型只是最近才开发出来的，而且仍在不断修订和更新，同时考虑到新的医学数据。到目前为止，人们主要关注的是在药物治疗下对病毒和人类免疫系统之间的相互作用进行最佳建模的动力学形式，其中许多模型仍然没有被描述为最优控制问题。研究人员将在最优控制的框架内分析各种这些模型，并比较各种解决方案，以设计最优治疗方案。在该项目中，研究人员将考虑在癌症或艾滋病等具有强烈细胞增殖方面的疾病的化疗中使用数学模型。这些模型将被分析为以药物剂量作为控制参数的最优控制问题，目标是在癌症的情况下最大化药物杀死的癌细胞数量(在艾滋病化疗的情况下分别最大化未感染细胞的数量)，同时最小化化疗的有害副作用和成本。尽管这些问题各自在细节上有很大的不同，但它们也有许多共同之处，可以放在一个包含所有这些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，通过查看这些模型的共同属性，可以获得简化和洞察。这个项目将解决两个方向的问题。对最优控制所满足的条件的生物医学解释将以生物医学人员可理解的方式给出，这些条件将与潜在的生物情况有关。预计这一项目的结果将引起医学界的兴趣，因为它将为分析现有的癌症或艾滋病毒化疗方案提供一些见解，并可能有助于新的改进方案的设计。这些结果对艾滋病毒治疗尤其重要，因为到目前为止，艾滋病毒治疗并不能治愈这种疾病，而只是提供了一个长期的维护计划。

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.