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

9216614特沃森和他的合作者开发了现实而全面的数学模型和必要的计算算法，以了解哺乳动物肾脏的浓缩机制。他们的直接目标是找出肾脏内部(内髓)是如何浓缩尿液的。研究了不同长度回路的作用以及它们的并行运行。目前可用的实验数据，以及新的数据，当它变得可用时，被用来校准和练习模型。现有的求解刚性多点边值问题的串行机算法适用于并行巨型机。提出了基于分裂系统求解器的并行计算新算法。哺乳动物将体液的体积和组成保持在非常有限的范围内。实现这一目的的一个基本机制是肾脏--它可以产生比血浆或其他体液更多或更少浓度的尿液。肾脏由大量不同的称为肾单位的功能单位组成，这些单位是并行运行的。人类肾脏大约含有一百万个肾单位。肾脏的纵向横切面显示出两个截然不同的部分：皮质和髓质。根据肾小管的不同生理特征，延髓分为外、内两个部分。数学模型为我们理解尿液浓缩机制提供了许多基本观点。这些模型-使用实验可用的参数-已经确定外髓使用主动模式(代谢泵)进行集中，但没有一个模型能够在内髓产生任何显著的浓度。内髓如何浓缩尿液仍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。数学模型在检验这些替代假说时是必不可少的。现实而全面的模型涉及数千个方程的解。因此，求解代数方程的算法不仅要运行速度快，而且要尽量减少计算机存储空间。由于肾脏是并行运行的，能够利用这种生理特性的算法和计算机非常适合于肾脏建模。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使用也将被研究。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学研究的软件和算法，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. ***