Modern understanding of hypertonic urine formation is organized around the countercurrent hypothesis. Experimental evidence supports the countercurrent idea, as do solutions of many mathematical models that incorporate a number of simplifying assumptions. Isomorphic models have failed, however, to predict the interstitial concentrations of NaCl and urea that are found in common laboratory animals. A feature common to all simulations solved thus far is that they use only a single independent spatial variable. Anatomical studies suggest a three dimensional ordering of tubules and blood vessels in the renal medulla. The major long term goal of this project is to solve a system of differential equations that describe flows of NaCl, urea, and water in renal medullary structures that are organized in the three dimensional arrangement suggested by anatomists. The hypothesis to be tested is that the three dimensional ordering is essential to achieve experimentally observed hypertonicity. The system of equations is a nonlinear two-point boundary value problem that will be solved with numerical methods that have been adapted to solve this specific problem, and that include quasilinearization, automatic evaluation of partial derivatives, and invariant embedding.
