Applied Numerical Mathematics of Structured Multiscale Models of Biological Systems

研究人员提出开发计算方法和软件来解决各种生物系统的多尺度模型中出现的偏微分方程系统。 这些系统的行为不仅取决于时间和空间，还取决于生理特征，如大小或年龄。该方法将时间与生理变量解耦。在这样做时，他们，不像以前的方法，防止经常细分辨率的时间离散膨胀的生理变量中的节点数。 由于近似误差是计算中唯一有意义的误差源，因此这一系列方法仍然是具有最高阶精度的方法。与生物学家的合作是这项研究计划的重要组成部分。该软件将在蒙大拿州立大学的生物膜研究人员和范德比尔特大学的癌症生物学家开发和参数化的模型的背景下开发和使用，期望他们将对理解生物膜生长和肿瘤侵袭的机制产生重大的近期影响。这种合作对于确保方法和软件的相关性和实用性也是至关重要的;在应用真空中开发的方法往往缺乏实用性。研究人员建议开发计算机软件和算法-确定软件如何工作的数学规则-来研究生物膜如何发展以及癌症肿瘤如何在中期和晚期发展，侵入附近组织。 生物膜是生活在自己创造的物质外壳中的细菌。 它们涉及许多最难治疗的感染和工业污染，并用于废水治理。 该软件将处理生物膜或肿瘤内的单个细胞在某些重要方面存在差异的复杂情况。 例如，生物膜中的不同细菌将具有不同的生长和分裂能力，这取决于已经进行的分裂次数。 癌细胞的细胞分裂周期的不同阶段与物理上更大的完整肿瘤有关。 这一点很重要，例如，在研究化疗的总体效果时，使用药物对细胞的影响不同，这取决于它们处于细胞分裂周期的哪一部分。 该提案中的算法背后的数学比以前的算法先进得多，并且仍然是最先进的，从而产生了可以在今天和不久的将来与计算机一起有效使用的软件。 它们也可以被预期用于生物膜和肿瘤以外的问题。 这项研究计划的一个重要部分是研究人员和蒙大拿州博兹曼的蒙大拿州立大学和田纳西州纳什维尔的范德比尔特大学的同事之间的合作。

The investigator proposes to develop computational methods and software to solve systems of partial differential equations that arise in multiscale modelsof various biological systems. The behavior of these systems depends not only on time and space, but also on physiological traits such as size or age. The methods decouple time from the physiological variables. In doing so they, unlike previous methods, prevent the often fine-resolution time discretization from inflating thenumber of nodes in the physiological variables. This family of methods remains the one with the highest-order accuracy due to the fact that approximation error is the only meaningful source of error in the computations. Collaboration with biologists is a critical component of this research plan. The software will be developed and used in the context of models developed and parameterized with biofilm researchers at Montana State University and cancer biologists at Vanderbilt University, with the expectation that they will have a significant and near-term impact on understanding the mechanisms of biofilm growth and tumor invasion. This collaboration is also vital to ensure the relevance and usefulness of the methods and software; methods developed in an application vacuum tend to lack utility.The investigator proposes the development of computer software and algorithms -- the mathematical rules that determine how the software works -- to study howbiofilms develop and how cancer tumors, in middle and later stages of their development, invade nearby tissue. Biofilms are bacteria that live in a material casing of their own creation. They are involved in many of the most difficult to treat infections and in industrial fouling, and are used in wastewater remediation. The software will handle complicated situations where individual cells within the biofilm or tumor differ in some important respect. For example, different bacteria in a biofilm will have different abilities to grow and divide, based on the number of divisions already undertaken. The different stages in the cell-division cycle of cancer cells are linked to the physically larger complete tumor. This is important, for example, in studying the overall effects of chemotherapy when using drugs that affect cells differently depending on what part of the cell-division cycle they are in. The mathematics behind the algorithms in this proposal are significantly more advanced than what came before and remain the state of the art, resulting in software that can be used effectively with the computers of today and the near future. They can also be expected to be useful for problems other than biofilms and tumors. An important part of this research proposal is the collaboration between the investigator and colleagues at Montana State University in Bozeman, MT and Vanderbilt University in Nashville, TN.