modeling renal hemodynamics e. bruce pitman (buffalo) harold layton (duke) leon moore (stony brook)
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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
• 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
[Cl]
Flow rate
*
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
)exp( )(solution a has )(/)(
0 tututudttdu
2/2/ and )sin( )(solution a has )1(/)(
ttutudttdu
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