Mathematical Models and Adaptive Algorithms for Tumor Growth

该项目的目标是建立物理上合理的、数学上严格的肿瘤生长扩散界面模型，分析基于这些模型的问题的适定性，设计有效的时间步长方案和有限元离散算法，构建实现这些算法的计算机软件，并开发基于后验误差估计的解决方案验证方法。研究工作的重点将是发展和分析数学模型，这些模型在连续尺度上描述肿瘤的无血管生长，即在没有附近血管的情况下，旨在预测大肿瘤区域的演变，而忽略单个细胞的行为。肿瘤生长的连续介质模型可以通过混合物的连续介质理论从第一性原理导出。混合物理论提供了一个优雅的和一般的框架，用于模拟多组分介质，如活组织，由几种相互作用的成分组成。基于混合物理论的唯象模型的一个显着性质是，当考虑到Helmholtz自由能泛函中的各种组分的浓度梯度时，人们得到了在各种组分之间引入平滑过渡边界的扩散界面模型。所得方程是Cahn-Hilliard型方程组，即非线性时变四阶偏微分方程的复方程组。这类扩散界面肿瘤生长模型是最近才提出的，对这类系统的数学分析和有效离散化的研究还处于起步阶段。本研究的主要目标是解决与发展用于计算机预测肿瘤生长的时空扩散界面相场模型有关的几个重要的开放性问题，包括：（1）发展满足系统物理性质的公式：（2）发展基于梯度流理论的肿瘤生长模型分析的严格数学框架;（3）用半隐式分裂方法发展新的稳定的高阶精度时间推进格式;以及（4）发展有效的面向目标的误差估计算法，用于控制高度非线性时变耦合问题的空间和时间离散化误差，癌症是一种基因组疾病，其特征在于不受控制的细胞生长和侵袭，其每年折磨来自所有年龄类别的数百万美国人。因此，该研究项目的主要动机是关注我们这个时代的重大挑战之一，即了解癌症的机制，以便确定可靠的治疗方法，或者更好的预防措施，以减轻这种疾病对许多人的影响。由于许多原因，这是一项艰难的奋进，但最重要的可能是有一百多种不同类型的癌症，每种类型的原因和影响都发生在广泛的范围内-特定的突变发生在分子尺度上，而肿瘤可能侵入人体的重要部分。认为生物学家或内科医生是癌症研究的主要参与者并不罕见;然而，越来越多的其他学科的科学家，如数学或物理学，正在参与调查肿瘤发展和行为的可能原因。事实上，我们认为数学和计算工具可以提供新的见解，有助于指导生物学家和医学家解决更多的基础研究问题。我们相信计算机模拟代表了进一步发现和获取科学知识的有力手段。这些模拟将有助于在未来探索肿瘤在自然环境中生长的详细机制，而这些机制无法在患者身上直接研究。

The objectives of the proposed project are to develop physically sound and mathematically rig-orous diffuse-interface models for tumor growth, to analyze the well-posedness of problems based onthese models, to design efficient time-stepping schemes and finite-element discretization algorithms,to build the computer software implementing these algorithms, and to develop solution verificationmethods based on a posteriori error estimation. The focus of the research work will be on the devel-opment and analysis of mathematical models that describe at the continuum scale avascular growthof tumors, i.e. in the absence of nearby blood vessels, and aim at predicting the evolution of large tu-morous regions while ignoring the behavior of individual cells. Continuum models of tumor growthcan be derived from first principles through the continuum theory of mixtures. Mixture theory pro-vides an elegant and general framework for modeling multicomponent media such, as living tissue,composed of several species of interacting constituents. A remarkable property of phenomenologicalmodels based on mixture theory is that, when considering the concentration gradients of variousconstituents into the Helmholtz free energy functionals, one obtains diffuse-interface models thatintroduce smooth transitional boundaries between the various constituents. The resulting equa-tions are systems of the Cahn-Hilliard type, namely complex systems of nonlinear time-dependentfourth-order partial-differential equations. Such diffuse-interface tumor-growth models have beenproposed only recently in the literature and the mathematical analysis and development of efficientdiscretizations for such systems are only in the initial stage. The main objectives in this researchproject are thus to address several important open issues related to the development of spatio-temporal diffuse-interface phase field models for computer predictions of tumor growth, including:(1) development of formulations that satisfy thermodynamical properties of the system; (2) devel-opment of a rigorous mathematical framework for the analysis of tumor-growth models based ongradient ow theory; (3) development of new stable and high-order accurate time-stepping schemesby using semi-implicit splitting approaches; and (4) development of efficient goal-oriented errorestimation algorithms for the control of spatial and temporal discretization errors for the highlynonlinear time-dependent coupled problem embodied by the proposed tumor-growth models.Cancer is a disease of the genome, characterized by uncontrolled cellular growth and invasion,that afflicts every year millions of Americans from all age categories. The primary motivationof the research project is thus concerned with one of the grand challenges of our times, thatis, to understand the mechanisms of cancer so that reliable treatments, or better, preventativemeasures, can be determined to relieve the impact this disease has on so many people. It hasturned out to be a difficult endeavor, due to many reasons, but the most important ones couldbe that there are more than one hundred different types of cancer and the causes and effects ofeach type occur on a wide range of scales-specific mutations happen at the molecular scale whiletumors may invade a significant portion of the human body. It is not too uncommon to believethat biologists or medical physicians are the main players in cancer research; however, more andmore scientists from other disciplines, such as mathematics or physics, are getting involved inthe investigation of possible causes of tumor development and behavior. It is in fact our hopethat mathematical and computational tools could provide new insights that could help guide morefundamental research issues to be addressed by biologists and medical physicians. We believe thatcomputer simulations represent powerful means for furthering discovery and acquiring scientificknowledge. These simulations will help in the future explore detailed mechanisms of tumor growthin natural environments that cannot be directly studied in patients.