THEORETICAL ANALYSIS OF SOLUTE & WATER TRANSPORT IN MAMMALIAN KIDNEYS

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The overall goal of this work is to synthesize experimental data at the membrane and molelcular level into predictive mathematical models of the mammalian kidney that are useful in understanding both its normal and diseased function. The mammalian kidney consists of a large number of units, the nephrons, operating in parallel. Each nephron is a tube approximately 1 cm long and 10 to the -3 power cm in diameter. The closed end is wrapped around a specialized knot of capillaries to form the glomerulus; the open ends merge to empty into the ureter and thence the bladder. The first step in urine formation is the expression of a protein and cell-free filtrate of blood. As this glomerular filtrate flows down the nephron it is modified by the selective reabsorption of most of the solutes and water and the selective secretion of other solutes to form the final urine. By varying the composition of the final urine, the composition of the interstitial fluid bathing the cells of the body is maintained within the narrow limits compatible with life. From experiments on isolated perfused tubules, a great deal is known about the transport properties of the individual renal tubules, and from whole animal experiments a great deal is known about overall function of the kidney. Lacking is a coherent theory that links microscopic function with overall function. The general method we have devised for modeling this intricate system is to solve the differential equations describing volume and solute flow in the individual tubules against assumed values of solute concentrations and hydrostatic pressure in the vascular interstitial space. Transmural solute and water fluxes computed for the tubes are then substituted into the differential equations for the space. If these are satisfied to some specified tolerance, we have a solution. If not, a correction to the assumed concentrations is computed by a Newton type method. The scheme is iterated until a satisfactory solution is obtained. Our current research is evolving toward more realistic models along two parallel paths: l) Incorporation of more details of the renal anatomy. Here we are moving from models in which the interstitial space is assumed to be radially well mixed toward models in which both the radial and axial distribution of structures are considered. 2) Incorporation of more details of transtubular transport. Our current kidney models describe fluxes of solute and water from tubular lumen to surrounding interstitial space by treating the tubular wall as a single membrane and describing fluxes through it by the phenomenology of irreversible thermodynamics. In models of individual segments this description has been replaced by one that considers the details of transport through cells and intracellular space. These detailed tubular models are being introduced into the whole kidney models.