Layton Anita T, Layton Harold E
Department of Mathematics, University of North Carolina, Phillips Hall, Campus Box 3250, Chapel Hill, North Carolina 27599-3250, USA.
Bull Math Biol. 2003 Sep;65(5):859-901. doi: 10.1016/S0092-8240(03)00045-4.
The highly structured organization of tubules and blood vessels in the outer medulla of the mammalian kidney is believed to result in preferential interactions among tubules and vessels; such interactions may promote solute cycling and enhance urine concentrating capability. In this study, we formulate a new model framework for the urine concentrating mechanism in the outer medulla of the rat kidney. The model simulates preferential interactions among tubules and vessels by representing two concentric regions and by specifying the fractions of tubules and vessels assigned to each of the regions. The model equations are based on standard expressions for transmural transport and on solute and water conservation. Model equations, which are derived in dynamic form, are solved to obtain steady-state solutions by means of a stable and efficient numerical method, based on the semi-Lagrangian semi-implicit method and on Newton's method. In this application, the computational cost scales as O(N2), where N is the number of spatial subintervals along the medulla. We present representative solutions and show that the method generates approximations that are second-order accurate in space and that exhibit mass conservation.
哺乳动物肾脏外髓质中高度结构化的肾小管和血管组织被认为会导致肾小管和血管之间的优先相互作用；这种相互作用可能会促进溶质循环并增强尿液浓缩能力。在本研究中，我们为大鼠肾脏外髓质的尿液浓缩机制制定了一个新的模型框架。该模型通过表示两个同心区域并指定分配给每个区域的肾小管和血管的比例来模拟肾小管和血管之间的优先相互作用。模型方程基于跨膜运输的标准表达式以及溶质和水的守恒。以动态形式推导的模型方程通过基于半拉格朗日半隐式方法和牛顿法的稳定且高效的数值方法求解以获得稳态解。在本应用中，计算成本按O(N2) 缩放，其中N是沿髓质的空间子区间数量。我们给出了代表性解，并表明该方法生成的近似值在空间上是二阶精确的并且表现出质量守恒。
