Numerical methods for three-dimensional models of the urine concentrating mechanism
We recently formulated and solved a successful model of the renal concentrating mechanism, a problem of long-standing interest in mathematical biology [33, 34]. The model is a 39th-order nonlinear two-point boundary-value problem with separated linear boundary conditions. Although quasilinearization and superposition with Gram-Schmidt orthonormalization successfully solved the equations, these methods will probably not be sufficient to solve more detailed models of the urine concentrating mechanism. In this paper we review the current literature on solving two-point boundary-value problems and find a number of promising alternatives and improvements to our current methods. In addition we derive a new numerical method for calculating the parameter sensitivity of the solution to a nonlinear two-point boundary-value problem coupled to a system of algebraic relations. This new method is faster and more accurate than the method previously employed.