A MATHEMATICAL MODEL OF THE URINE CONCENTRATING MECHANISM (KIDNEY, COUNTER-CURRENT SYSTEM, INTEGRAL EQUATIONS)

After a brief survey of the physiology of the urine concentrating mechanism of the mammalian kidney, a simple model that was developed by Charles Peskin for a single nephron is introduced (F. C. Hoppensteadt and C. S. Peskin, Mathematics in Medicine and the Life Sciences in preparation ). This model is found to have a concentrating limit of a factor of e over blood plasma osmolality, regardless of the length of the nephron or the type of kinetics employed for NaCl reabsorption from the ascending limb. In order to represent the decreasing loop of Henle population as a function of medullary depth, Peskin's model is here extended to a multinephron model, resulting in a system of three highly nonlinear integral equations. The equations exhibit a multiplier effect which greatly enhances concentrating capability. Using the approximate loop distribution of the rat, the model produces a sigmoidal osmolality profile similar to the profiles found in tissue-slice studies of rat kidneys. These model calculations suggest that the decreasing nephron population found in vivo and the corresponding decrease in transverse medullary area as a function of medullary depth are important factors in the concentrating mechanism of the mammalian kidney.;Existence and uniqueness results are obtained for the system of integral equations. Using the Schauder principle it is shown that solutions exist for a general class of NaCl reabsorption kinetics. Uniqueness of solutions is obtained for first-order and Michaelis-Menten kinetics in the case of very low reabsorption rates. Uniqueness of solutions to a prototype problem is established for the case of very high reabsorption rates.