Fluid flow networks arise in modeling the kidney concentrating mechanism. nc and comprehensive models involve a large n&r of tubes with several solutes in each tube. 'lhe axial and transmural flows are described by nonlinear differential equations. These equations have to be solved numerically. This involves discretization, which leads to several hundred nonlinear algebraic equations. we have developed mathematical theory and algorithms that make use of the physiological connectivity of the tubes in the kidney to solve such systems very efficiently. 'Ihis has lead to considerable savings in ccmputer storage and run times. 'Ihe total system is expressed as a function of a reasonably small subset of variables (called basic, e.g., interstitial and boundary values). A carefully chosen subset of the equations is associated with these basic variables. The rest of the equations -which beccme block bidiagonal -are then solved (in parallel if desired) for assumed values of basic variables. This procedure enables us to express the remaining variables as functions of the variables labeled as basic. The basic equations are solved with quasi-newton type methods. A new hybrid quasi-Newton type method is presented.