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