William F. Stevens

Mathematical simulation of normal nephron function in rat and man

Abstract A mathematical model of the nephron was developed by writing a set of material balance equations for the flow of water, sodium and urea along the length of the nephron. The differential equations derived incorporate the geometrics of an average nephron. The parameters of these equations were calculated using the analytical solutions and the available micro-puncture data. The model yielded piecewise continuous curves for volume flow rate and sodium and urea concentration profiles emphasizing the sequential processing of the filtrate along the consecutive nephron segments. The simulation of normal function in the rat nephron agreed reasonably well with the tubular fluid to plasma ratio data for inulin, urea and sodium along the proximal and distal tubules. Assuming a linear medullary interstitial solute concentration profile, the model generated continuous curves for the luminal concentrations of sodium and urea along the descending and ascending limbs of the loop of Henle. The simulated curves were consistent with available micropuncture data for sodium and suggested a possible mechanism for the handling of urea by the nephron to account for the passive recirculation of this solute and its role in the development of the longitudinal medullary concentration gradient. The simulation of the human nephron, after making certain necessary extrapolations from the rat nephron, gave comparable curves for water, sodium and urea transport.

An Example of the Application of Nonlinear Programming to Chemical-Process Optimization

This paper presents a realistic chemical-process optimization problem in detail, and notes the fact of its equivalence to the general nonlinear programming problem. The eight equality constraints that arise are handled by algebraic substitution, and the resulting four-variable problem is solved by a modified form of the Davidon-Fletcher-Powell algorithm. Details of the method are presented including a FORTRAN computer program, and results of two typical cases are summarized.