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The significance of viscous flow properties in the theory of operation of a nephron

Mathematical Biosciences
Publication Date
DOI: 10.1016/0025-5564(69)90007-8
  • Mathematics


Abstract The salient feature of a nephron is its tiny size, 6–10 μ in radius, since the fourth-power law requires a tiny flow rate even from a large pressure gradient. Along closely folded hairpin loops that are present in the nephrons of birds and mammals and are called loops of Henle, a salt concentration gradient forms in the ambient medullary tissue. Urine collects in this tissue in ducts, equilibrates with it osmotically, and produces a final product hypertonic to blood. Other authors explain the mechanism of this loop in terms of a hypothesis of active extrusion of a small amount of sodium from one branch of the loop and operation of a countercurrent multiplication principle. By close attention to realistic physical principles we construct a model that does not use this hypothesis but produces in numerical studies the observed concentration gradient and an amplification of this effect with length. A much weaker assumption than similar ones made in other models, that two salt concentrations take on stationary values, causes a linear initial value problem for a (2×2) first-order ordinary differential equation system to replace a (4×4) first-order partial differential equation system. Basic mechanisms used, such as back diffusion, were of no importance in previous square law models.

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