Concentrating engines and the kidney. I. Central core model of the renal medulla

Mass balance relations, valid for any counterflow system, are derived and applied to a central core model of the renal medulla, in which descending Henle's limbs (DHL), ascending Henle's limbs (AHL), and collecting ducts (CD) exchange with a central vascular core (VC) formed by vasa recta loops, assumed so highly permeable that the core functions as a single tube open at the cortical end, closed at the papillary. Solute supplied to the VC primarily by the water impermeable AHL may either enter the DHL to be recycled or remain in the core to extract water by osmosis from DHL and CD. If concentrations in core and descending flows are nearly equal, then for all degrees of recycling the ratio of entering DHL concentration to loop concentration is given by r = 1/[1 - f(T)(1 - f(U))], where f(T) is the fractional net solute transport out of AHL and f(U) is the ratio of CD flow to the sum of CD and AHL flows. Differential equations for a single solute are derived for core and AHL concentrations. Explicit analytic solutions are given for solute transport out of the AHL governed by Michaelis-Menten kinetics. Finally the energy requirements for concentration are analyzed.