Raymond Mejia

Spectral localization with optimal pointspread function

Abstract The SLIM experiment ( X. Hu, D. N. Levin, P. C. Lauterbur, and T. Spraggins, spectral localization by imaging, Magn. Reson. Med. 8, 314, 1988 ) promises to be an efficient technique for localized spectroscopy in vivo, allowing simultaneous acquisition of regional spectra from several, arbitrarily shaped compartments. In the present contribution, we evaluate how sensitivity is affected in this experiment, and we identify a source of localization errors stemming from inhomogeneities within the compartments. We present SLOOP (“spectral localization with optimal pointspread function”) as an extension of the SLIM technique. SLOOP optimizes sensitivity and minimizes localization errors by choosing an optimal set of phase-encoding gradients that matches the pointspread function to the shape of the volumes of interest. Experimental results obtained in vitro on a rabbit kidney are shown. SLOOP is also compared to Fourier series localization techniques and to spectroscopic imaging and is presented as a generalization of both of these that samples k space in a nonuniform fashion.

Solution of a Multinephron, Multisolute Model of the Mammalian Kidney by Newton and Continuation Methods

A method of numerically solving the differential equations specifying solute and water flow in a multinephron, multisolute model of the mammalian kidney by a combination of Newton and continuation techniques is described. This method is used to generate a connected component of the steady state solution manifold of the model. A three dimensional section of this manifold is shown to be convoluted, with upper and lower sheets of stable solutions connected by an unstable middle sheet. Two dimensional sections of this surface are followed from a trivial constant profile of concentrations in the nonconcentrating kidney to the profiles of the maximally concentrating kidney. Study of these sections shows that for a given choice of model parameters there may exist no solution, there may be a unique solution, or there may be multiple solutions. A study of the time dependent solutions shows that the dynamic transition from the lower to the upper state and return may be via a hysteresis loop.

Use of sparse matrix techniques in numerical solution of differential equations for renal counterflow systems

Abstract Mammals concentrate urine by intricate counterflow systems in their kidneys. The mathematical models of such systems involve the numerical solution of a system of differential equations. (A preliminary report of this work was given in Notices of the American Mathematical Society21, A498 (1974).) This leads to a system of nonlinear algebraic equations, that are solved by the Newton-Raphson method. The basic step in the Newton-Raphson method is the repeated solution of a system of linear equations with the Jacobian of the nonlinear equations as the coefficient matrix. It is shown that the zero-nonzero structure of the Jacobian can be predicted from the physical connectivity of the model. We exploit this zero-nonzero structure to permute the Jacobian to a bordered block triangular form, which is then used to compute the correction to the solution vector of the nonlinear equations. For this computation a modified form of partial pivoting is used. Results are given that show the high accuracy of the computed solutions, the optimum use of the internal computer storage, and the fast rate of computation.

A test problem for kidney models

There is a need in the literature for standard problems with which investigators may validate the numerical schemes that they apply to solve renal models. We consider the six-tube vasa recta model first described in [I] and used by Farahzad and Tewarson [2] and by Lory in [3] to be a good candidate for inclusion for a number of reasons. First, this model is sufficiently complex to exhibit some of the characteristics of larger models such as discontinuous sources and small flows, while being small enough to be solved on any computer with a FORTRAN compiler. Secondly, it provides a test of an algorithm’s ability to conserve mass and water balance, which is required for accuracy (see [4]). Thirdly, a number of numerical methods have been used to solve it. All of these that express the difference equations in conservative form have obtained essentially the same solution, allowing for variation in the accuracy of each method. (Others have obtained solutions to the difference equations solved, but not to the differential equations.) In [l] a centered-trapezoidal-difference approximation was used for the space derivatives and the resulting nonlinear equations were solved simultaneously using Newton’s method. Farahzad and Tewarson in [2] used the same difference approximations and a sparse-matrix version of Newton’s method for solution of the nonlinear equations. Lory [3] has used a multiple-shooting scheme to solve the problem. We have also solved it using both a partitioning scheme described in [5] for a multinephron model and DDMAD,’ an adaptive finite-difference solver for two-point boundary problems [6,7].

Mathematical modelling of the renal concentrating mechanism

We describe model equations, a method of solution, and show examples of steady-state and transient solutions for the urinary concentrating mechanism. Then we draw some conclusions about the current successes and failures of the models and about future directions in modelling the mammalian concentrating mechanism.