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Modeling Renal Hemodynamics E. Bruce Pitman (Buffalo). Harold Layton (Duke) Leon Moore (Stony Brook). The Human Kidneys:. are two bean-shaped organs, one on each side of the backbone represent about 0.5% of the total weight of the body

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Modeling Renal Hemodynamics E. Bruce Pitman (Buffalo)

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Modeling Renal HemodynamicsE. Bruce Pitman (Buffalo)

Harold Layton (Duke)

Leon Moore (Stony Brook)


The Human Kidneys:

  • are two bean-shaped organs, one on each side of the backbone

  • represent about 0.5% of the total weight of the body

  • but receive 20-25% of the total arterial blood pumped by the heart

  • Each contains from one to two million nephrons


In 24 hours the kidneys reclaim:

  • ~1,300 g of NaCl (~97% of Cl)

  • ~400 g NaHCO3 (100%)

  • ~180 g glucose (100%)

  • almost all of the180 liters of water that entered the tubules (excrete ~0.5 l)


Anatomy (approximate)


Water secretion

  • Release of ADH is regulated by osmotic pressure of the blood.

  • Dehydration increases the osmotic pressure of the blood, which turns on the ADH -> aquaporin pathway.

    • The concentration of salts in the urine can be as much as four times that of blood.

  • If the blood should become too dilute, ADH secretion is inhibited

    • A large volume of watery urine is formed, having a salt concentration ~ one-fourth of that of blood


Experimentpressure from a normotensive rat


Experimentpressure spectra from normotensive rats


Anatomy (approximate)


Basics of modeling

In all tubules and interstitium, balance laws for

  • chloride

  • sodium

  • potassium

  • urea

  • water

  • others


Basics of modeling II

Simplifying assumptions

  • infinite interstitial bath

  • infinitely high permeabilities

  • chloride as principal solute driver


Basics of modeling III

  • Macula Densa samples fluid as it passes

  • Feedback relation noted at steady-state

  • We assume the same form in a dynamic model


Basics of modeling IV

  • Single PDE for chloride

  • Empirical velocity relationship: apply steady-state relation to dynamic setting

Flow rate

*

[Cl]


Basics of modeling V


Basics of modeling VI


Model

  • Steady-state solution exists

  • Idea: Linearize about this steady solution

  • Look for exponential solutions


Aside on delay equations


Basic Analysis


Basic Analysis

  • If the real part of λ>0, perturbation grows in time. If Imaginary part of λ≠0, oscillations. [unstable]

  • If the real part of λ<0, perturbation decays in time. [stable]


Bifurcation results


Bifurcation results II


Bifurcation results III


To Be Done

  • Complex perhaps chaotic behavior at high gain

  • Have 2 coupled nephrons. Need full examination of bifurcation

  • Need many coupled nephrons (O(1000))

  • Reduced model


2-nephron model

  • as many as 50% of the nephrons in the late CRA are pairs or triples

  • some evidence of whole organ signal at TGF frequency


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