Stochastic Modeling of
Coupled Nephrons
Saziye Bayram*
Bruce E. Pitman**
*SUNY-Buffalo
State College
**SUNY-University at Buffalo
Overview







Anatomy and Physiology of the Kidney
Structural Anatomy and Physiology of Nephrons
Tubuloglomerular Feedback (TGF) Mechanism
Experimental Findings
Earlier Mathematical Models of Nephron’s TGF
Mechanism
Stochastic Models of Nephron’s TGF Mechanism
Goals and Physiological Relevance
KIDNEYS

Filter waste materials out of
the blood & eliminate them as
urine from the body.

Homeostatic (regulates,
balance the state) devices of
the body.

Single human kidney consists
of ~106 nephrons.

Filter 180-200 liters of blood
daily.
Anatomy of Kidney
Cortex and Nephrons
Figure:www.ams.sunysb.edu/.../ SCICOMP/Kidney.index.html
Structural Anatomy of a Nephron
Within 24 hrs, 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
liter only)
Each nephron processes a very small
fraction of the total blood flow to the
kidney, typically 200-300 nl/min for
a rat kidney.
Figure:anatomy.iupui.edu/.../ urinaryf04/urinaryf04.html
TGF System of a Nephron


Regulates tubular fluid flow
of nephron by monitoring
[NaCl] at MD, with a delay.
[NaCl] at MD ↑
Diameter of AA ↓
Blood flow ↓
Pressure in capillaries ↓
Rate of filtration ↓
Transit time ↑
[NaCl] at MD ↓
Figure:ccollege.hccs.cc.tx.us/. ../kidneypict.htm
Macula Densa
EE
AA
Bowman’s
Capsule
Glomerulus
PT
Schematic Diagram of a Nephron
Experimental Findings
(By Just et al., Cupples et al., Leyssac, Holstein-Rathlou et al., and Casellas et al.)

TGF-mediated fluid flow in normotensive rat nephron either
approximates a steady state or exhibits limit cycle oscillations (LCO)
(20-50 mHz).

Irregular and chaotic flow oscillations observed in hypertensive rats.

Evidence of interaction between neighboring nephron: 60-70% of
nephrons occur in pairs and triples.

Sustained oscillations in one nephron can propagate to the coupled
nephron. Resultant oscillations are roughly synchronous.
Interaction between paired nephrons
Types of coupling:
A- Vascular Coupling: Electrotonic in Nature
B- Hemodynamic Coupling: Pressure related
Berne,
R.M., and Levy, M.M. (1996), “Principles of Physiology”, Mosby-Year Book, Inc, MO
The tubular pressure oscillations of a
pair of neighboring nephrons
Single Nephron ODE Model:

 k1  g 3  X 1 , X 2  
X k 
dX 1 1 
k11

  k9  1 3 
  1  k8 1 
dt
k7 
g1  X 2 
k6 
 g 2  X 1 , X 2  

dX 2
 X3
dt
dX 3 g 4  X 1 , X 2   g 7  X 2 , X 6 , Z 

 k19 X 3
dt
k14

dX 4 3  X 1  k3
 
 X 4 
dt
1  k6

dX 5 3
 X 4  X 5 
dt
1
dX 6 3
 X 5  X 6 
dt
1
The first mathematical models of the
TGF system



were deterministic.
were complex but still a simplification of the
real system.
did not capture the irregularities have seen in
the experiments with hypertensive rats.
Deterministic to Stochastic

In reality, there are a variety of factors that
change over time, and can neither be
controlled nor measured but nevertheless
leave their mark on the experimental output.

In this regard, we will model certain
parameters as random processes of some
convenient form (e.g. by adding dynamic
noise).
In our case: Gain, Delay, and Coupling
parameters are of interest because

these are the key parameters in understanding the
stability of the pressure and flow regulation in renal
dynamics

these were the main bifurcation parameters and have
been considered constant in the former models.

computer simulations show that oscillations in the
TGF system occur if the feedback gain is above a
critical value.
Goals and Physiological Relevance

To include noise in models of physiological systems, to provide more realistic
representation of the process under study, and to contribute to a deeper
understanding of the underlying mechanisms.

As a stochastic approach, we will hypothesis that gain, delay and/or coupling
parameters vary randomly with time. (In fact, the gain magnitude is influenced by
a variety of influences, such as arterial blood pressure, which changes over time.)

The constructed SDE will be simulated by the Monte Carlo methods and results
will be compared with the experimental data.

Will estimate one or more of the parameters that determines the dynamics of the
TGF mechanism.

Will be able to estimate the physiological parameters, and perhaps help to identify
the underlying mechanisms of hypertension.