Mathematical Modeling of the Chemotherapy, Immunotherapy and Vaccine Therapy of Cancer

研究人员汇集了微分方程和数值、微分几何和最优控制领域的专业知识，目标是开发数学建模和分析工具，应用于创建和测试治疗癌症的化学免疫疗法的新组合。研究人员与他们的学生一起，沿着三条相辅相成的道路开展工作。首先，建立和分析动力系统种群模型，以确定有助于指导联合疗法发展的基本系统行为。其次，微分几何方法允许分析近似球形、柱形、椭圆形和调和球形肿瘤的物理结构的理想化几何形状，这些几何形状可以非常接近活体肿瘤的某些几何形状。第三，最优控制理论被用来揭示新的最优组合疗法的路径，通过实施各种约束来最小化肿瘤负担，同时保持患者的免疫和正常细胞群高于健康阈值。这些研究的结果有可能为改进的针对患者的治疗方案的构建提供指导。对癌症生长和联合治疗策略的数学建模从根本上增加了我们对癌症反应机制的基本理解。研究人员进行的模拟、几何可视化和优化允许运行一系列虚拟实验，这些实验可以快速执行，不会对活人造成风险，但提供的数据可以显著有助于医疗决策。癌症生长和治疗的模型需要多个学科的技能。该项目的一个主要成果是开发了用于模拟和可视化癌症生长和治疗路径的软件，包括化疗和免疫治疗，这可能导致创建新的数学指导的针对患者的联合治疗策略。这项工作是信息技术和生物技术的交汇点，数学是核心。研究人员有能力通过加州医学数学研究小组正在举行的会议，将数学建议的新治疗策略的发现直接传递给合作的临床肿瘤学家。反过来，这个小组中的医生可以通过分享正在进行的临床试验的结果来加强数学模型的开发。调查人员通过研究助学金、课程和独立研究，直接让本科生参与到这一前沿努力的多个方面。这个项目的一个独特方面是三名女调查人员的合作，她们正在吸引更多的女学生进入这一研究领域。作为一个团队，研究人员可以作为抗癌前线的榜样，特别是在影响乳腺癌研究方面处于有利地位，这对人们的健康特别重要。该项目解决数学、计算和生物学难题的成果通过出版物和网上张贴向公众公布。这种合作努力有可能帮助临床医生拯救生命，减少癌症患者的痛苦。

The investigators bring together expertise in the areas ofdifferential equations and numerics, differential geometry, andoptimal control with the goal of developing mathematical modelingand analysis tools applied to the creation and testing of newcombinations of chemo-immunotherapies for treating cancer. Theinvestigators, together with their students, carry out work alongthree complementary paths. First, dynamical systems populationmodels are created and analyzed to determine fundamental systembehavior that helps guide the development of combinationtherapies. Second, differential geometry approaches allow forthe analysis of idealized geometric forms approximating physicalstructures of spherical, cylindrical, elliptic, and harmonicspherical tumors, which can closely approximate certain in vivotumor geometries. Third, optimal control theory is used toreveal paths to new optimal combination therapies withimplementation of a variety of constraints to minimize tumorburden while keeping the patient's immune and normal cellpopulations above a healthy threshold. The results of theseinvestigations have the potential to provide guidance in thestructuring of improved patient-specific treatment protocols. The mathematical modeling of cancer growth and combinationtreatment strategies adds to our basic understanding of cancerresponse mechanisms in fundamental ways. The simulations,geometric visualization, and optimization the investigatorsundertake allow for an array of virtual experiments to be runthat can be performed quickly with no risk to living persons, butthat provide data that can significantly benefit medical decisionmaking. The modeling of cancer growth and treatment requiresskills from multiple disciplines. A major outcome of thisproject is the development of software for simulating andvisualizing cancer growth and treatment pathways, involvingchemotherapy and immunotherapy, which can result in the creationof new mathematically guided patient-specific combinationtreatment strategies. This work lies at the intersection ofinformation technology and biotechnology, with mathematics at thecore. The investigators have the ability to funnel discoveriesfor new treatment strategies suggested by the mathematicsdirectly to collaborating clinical oncologists through theongoing meetings of a California-based Mathematics of MedicineStudy group. In turn, the physicians in this group can enhancethe development of the mathematical models by sharing outcomes ofongoing clinical trials. The investigators directly involveundergraduates in the multiple facets of this cutting-edgeendeavor through research assistantships, courses, andindependent study. A unique aspect of this project is thecollaboration of three women investigators who are attractingmore women students to this area of research. The investigatorsas a team can serve as role models on the forefront of the fightagainst cancer, and in particular are in a good position toaffect breast cancer studies, which are of special importance towomen's health. The results from solving the challengingmathematical, computational and biological problems of thisproject are made publicly accessible through publications andweb-based postings. This collaborative effort has the potentialto help clinicians save lives and reduce the suffering of cancerpatients.