R.P. Tewarson

A comparison of multinephron and shunt models of the renal concentrating mechanism

Abstract A multinephron model of the inner medullary urinary concentrating mechanism in the mammalian kidney is described. A procedure to represent the distribution of nephrons with a finite number of different nephron types is introduced. The model is compared with a shunt model. These studies show that, for uniform transport permeabilities: (1) 40 properly weighted nephron types are sufficient to represent the distribution of nephrons; (2) the two models give essentially the same results.

An efficient parallel algorithm for solving n-nephron models of the renal inner medulla

A parallel algorithm for solving the multinephron model of the renal inner medulla is developed. The intrinsic nature of this problem supplies sufficient symmetry for a high-level parallelism on distributed-memory parallel machines such as the iPSC/860, Paragon, and CM-5. Parallelization makes it feasible to study interesting models such as the rat kidney with 30,000 nephrons. On a high-end work station, one can study systems with 100 nephrons, while on a 32-node iPSC/860 or Paragon we can handle more than 1000 nephrons.A nearly perfect speedup is achieved by even distribution of load and minimizing the cost of communication.

A quasi-Newton method for solving nonlinear algebraic equations

Abstract An algorithm for solving systems of nonlinear algebraic equations is described. The Jacobian matrix is modified by using a convex combination of Broyden and a weighted update. A q -superlinear convergence theorem and computational evidence exhibiting significant relative efficiency of the proposed method are given.

A quasi-Gauss-Newton method for solving non-linear algebraic equations

Abstract One of the most popular algorithms for solving systems of nonlinear algebraic equations is the sequencing QR factorization implementation of the quasi-Newton method. We propose a significantly better algorithm and give computational results.

Inverse problem for kidney concentrating mechanism

Abstract Mathematical models of the mechanism for making concentrated urine in the mammalian kidney compute the variables V (e.g., volume flows and concentrations) as function of parameters h (e.g., water and solute permeabilites). We consider the inverse problem: given a V , for which h is known to exist, compute h . We give computational evidence that h can be determined well within the round-off error tolerance without any prior information about it.

Renal concentrating mechanism: Central core and vasa recta models☆

Abstract Mathematical models are essential in testing alternative hypotheses to explain the concentrating mechanism of the mammalian kidney. The basis features of a central core and two other vasa recta models, as well as efficient modeling techniques are described. It is shown that, by suitable choice of a few parameters, the two vasa recta models lead to the same osmolality and concentration ratios as the central core model.

Computational techniques for inverse problems in kidney modeling

Abstract In order to understand the concentrating mechanism of the mammalian kidneys, it is necessary to study the relationship between the parameter vector h (permeabilities of water and solutes) and the corresponding vector of concentration profiles V. We consider the inverse problem: determine h from a given V. This problem is ill-posed. Therefore, the regularization methods must be used to circumvent the ill-conditioning. We show how the Levenberg-Tikhonov-Marquard method with the Sobolev norm can be used to handle the inverse problem.

Models of kidney concentrating mechanism: relationship between core concentrations and tube permeabilities

Abstract Using an inner medullary shunt model of the kidney concentrating mechanism, we investigate the relationship between z—salt and urea concentrations in the Central Core, and h—water, salt and urea permeabilities in the Henles loop and the Collecting Duct. Computational results are given comparing (a) the direct problem: given h compute z, to (b) the inverse problem: given z compute h.

Using Quasi-Newton methods for kidney modelling equations

Abstract A method based on a convex combination of the Broyden and another quasi-Newton type update is described. Computational results exhibiting the relative efficiency of the proposed method for kidney models are given.

Preferential interaction and inverse problem algorithms in models of renal concentrating mechanism

Abstract Formulas for transmural transport parameters between two adjacent interacting tubes are derived. Results of computational experiments with a mathematical model of the kidney are given. These results show that only four of optimal parameters, obtained by an inverse algorithm, need to be different from experimental values, to get fairly good concentration profiles. Preferential interaction is shown to result in further improvement to these profiles.