A DYNAMIC NUMERICAL-METHOD FOR MODELS OF THE URINE CONCENTRATING MECHANISM

Dynamic models of the urine concentrating mechanism consist of large s ystems of hyperbolic partial differential equations, with stiff source terms, coupled with fluid conservation relations. Efforts to solve th ese equations numerically with explicit methods have been frustrated b y numerical instability and by long computation times. As a consequenc e, most models have been reformulated as steady-state boundary value p roblems, which have usually been solved by an adaptation of Newton's m ethod. Nonetheless, difficulties arise in finding conditions that lead to stable convergence, especially when the very large membrane permea bilities measured in experiments are used. In this report, an explicit method, previously introduced to solve the model equations of a singl e renal tubule, is extended to solve a large-scale model of the urine concentrating mechanism, This explicit method tracks concentration pro files in the upwind direction and thereby avoids instability arising f rom flow reversal. To attain second-order convergence in space and tim e: the recently developed ENO (essentially non-oscillatory) methodolog y is implemented, The method described here, which has been rendered p ractical for renal models by the emergence of desktop workstations, is adaptable to various medullary geometries and permits the inclusion o f experimentally measured permeabilities. This report describes an imp lementation of the method, makes comparisons with results obtained pre viously by a different method, and presents an example calculation usi ng some recently measured membrane properties.