Dynamic numerical method for models of the urine concentrating mechanism

Dynamic models of the urine concentrating mechanism consist of large systems of hyperbolic partial differential equations, with stiff source terms, coupled with fluid conservation relations. Efforts to solve these equations numerically with explicit methods have been frustrated by numerical instability and by long computation times. As a consequence, most models have been reformulated as steady-state boundary value problems, which have usually been solved by an adaptation of Newton's method. Nonetheless, difficulties arise in finding conditions that lead to stable convergence, especially when the very large membrane permeabilities measured in experiments are used. In this report, an explicit method, previously introduced to solve the model equations of a single renal tubule, is extended to solve a large-scale model of the urine concentrating mechanism. This explicit method tracks concentration profiles in the upwind direction and thereby avoids instability arising from flow reversal. To attain second-order convergence in space and time, the recently developed ENO (essentially non-oscillatory) methodology is implemented. The method described here, which has been rendered practical for renal models by the emergence of desktop workstations, is adaptable to various medullary geometries and permits the inclusion of experimentally measured permeabilities. This report describes an implementation of the method, makes comparisons with results obtained previously by a different method, and presents an example calculation using some recently measured membrane properties.