Mathematical Sciences: Parallel Algorithms for Ordinary Differential Equations- Applications in Kidney Modeling

9216614 Tewarson研究者和他的合作者开发了现实和全面的数学模型，以及必要的计算算法，以了解哺乳动物肾脏的集中机制。他们的直接目标是找出肾脏内部（内髓质）是如何浓缩尿液的。研究了变长回路的作用及其并行运算。目前可用的实验数据，以及新数据，当它变得可用时，被用来校准和练习模型。现有的求解串行计算机刚性多点边值微分方程的算法适用于并行超级计算机。针对并行计算问题，提出了基于分裂系统解算器的新算法。哺乳动物将体液的体积和成分维持在非常狭窄的范围内。达到这一目的的基本机制是肾脏——肾脏可以产生比血浆或其他体液浓度更高或更低的尿液。肾脏由大量被称为肾单位的不同功能单位组成，它们并行运作。人体肾脏大约包含一百万个肾单位。肾的纵向横切面显示两个不同的部分：皮质和髓质。鉴于小管的生理特性不同，髓质又分为外髓质和内髓质。数学模型为我们理解尿液浓缩机制提供了许多基本思路。这些模型-使用实验可用的参数-已经确定外髓质使用主动模式（代谢泵）进行浓度，但没有模型能够在内髓质中产生任何显著的浓度。内髓质如何浓缩尿液仍然是肾脏生理学中尚未解决的主要问题之一。不断有人提出假说来回答这个问题。数学模型在检验这种替代假设时是必不可少的。现实和全面的模型涉及成千上万个方程的解。因此，求解代数方程的算法不仅要运行得快，而且要使计算机存储空间最小化。由于肾单位并行运作，能够利用这种生理特性的算法和计算机非常适合肾脏建模。这个跨学科项目的目标是为并行计算机开发有效的方法来了解哺乳动物肾脏的集中机制。这些方法在其他几个重要的生理建模应用中的应用也将被研究。该项目增加了我们对基本和基本生理功能的基本理解，开发了可用于其他生理研究的软件和算法，并可能具有重要的诊断意义。＊＊＊

9216614 Tewarson The investigator and his collaborators develop realistic and comprehensive mathematical models, and the necessary computational algorithms, to understand the concentrating mechanism of the mammalian kidney. Their immediate goal is to find out how the inner part of the kidney (inner medulla) concentrates urine. The role of loops of varying lengths and their parallel operation are studied. Presently available experimental data, as well as new data, as it becomes available, are utilized to calibrate and exercise the models. Current algorithms for solving the stiff multipoint boundary value differential equations for serial computers are adapted for the parallel supercomputers. New algorithms based on split system solvers are developed for parallel computation. Mammals maintain the volume and composition of their body fluids within very narrow limits. A basic mechanism for this purpose is the kidney -- which can produce urine that is either more or less concentrated than plasma or other body fluids. The kidney consists of a large number of distinct functional units called nephrons, operating in parallel. The human kidney contains roughly one million nephrons. A lengthwise cross-section of the kidney reveals two distinct parts: the cortex and the medulla. In view of differing physiological characteristics of the tubules, the medulla is subdivided into the outer and the inner parts. Mathematical models have been responsible for many of the basic ideas leading to our understanding of the urinary concentrating mechanism. These models --- using experimentally available parameters --- have established that the outer medulla uses the active mode (metabolic pumps) for concentration but no model has been able to generate any significant concentration in the inner medulla. How the inner medulla concentrates urine is still one of the major unsolved problems in renal physiology. Hypotheses continue to be proposed to answer this question. Mathematical models are essential in the testing of such alternate hypotheses. Realistic and comprehensive models involve the solution of thousands of equations. Therefore, it is essential that the solution algorithms for the algebraic equations not only run fast and but also minimize computer storage. Since the nephrons operate in parallel, algorithms and computers that are able to utilize this physiological property are eminently suitable for kidney modeling. The goal of this interdisciplinary project is to develop efficient methods for parallel computers to understand the concentrating mechanism of the mammalian kidney. The use of these methods in several other important physiological modeling applications will also be investigated. The project increases our fundamental understanding of basic and essential physiological functions, develops software and algorithms that can be used in other physiological studies, and may have significant diagnostic implications. ***