A Dynamic Mathematical Model of Single Nephron Glomerular

A Dynamic Mathematical Model of Single Nephron Glomerular Filtration
Rate in Rat Kidneys
Justin Summerville; Ioannis Sgouralis; Anita T. Layton
Department of Mathematics, Duke University
We have developed a dynamic mathematical model of single nephron
glomerular filtration rate (SNGFR) in rat kidneys. The model
implements a previous model by Deen et al. [1] that uses afferent
arteriole pressure and flow rate as inputs. The model consists of 250
identical parallel capillaries discretized into 40 segments. Plasma flow
rate as well as plasma protein concentration are measured along each
segment, and a linear pressure decrease is assumed. Our model
accounts for a compliant Bowman’s space. Our model for Bowman’s
space volume is based on the formula developed by Cortes et al. [2]
that establishes a linear relationship between volume and pressure
inside the Bowman’s space. The model results suggest that a
compliant bowman’s space is the primary mechanism for pressure
damping at high frequency oscillations and not capillary filtration as
previously believed. When choosing the best parameters the amplitude
of high frequency oscillations around 165 mHz are 22% lower than the
amplitude of lower frequency oscillations around 30 mHz. However, this
amount of damping is a result of a large change in bowman’s space
volume. Under these conditions our volume fluctuates nearly 4.0 nL.
This size is not biologically attainable by the glomerulus as the average
size of the bowman’s space has been measured at 1.345 nL. Further
research into bowman space volume and compliance will be required to
formulate a more accurate model. This model can be incorporated into
a larger model of nephron filtration as a replacement for current steady
state models that do not properly damp high frequency oscillations.
Existing mathematical models of glomerular filtration of the kidney,
most notably those by Deen et al. [1], are based on a steady-state
formulation. Experiments in the rat kidney suggest that oscillations in
fluid flow, which are mediated by autoregulatory mechanisms in the
kidney such as the myogenic response and the tubuloglomerular
feedback, are substantially damped during its course through the
proximal tubule and the loop of Henle. Factors that contributed to that
damping include tubular compliance, pressure-dependent tubular
reabsorption, and flow dependent solute transport, etc. Here we
assess the extent to which high and low frequency oscillations are
damped during the glomerular filtration process.
This study presents a dynamic mathematical model of
glomerular filtration in the rat kidney. The model tracks plasma protein
concentration as well as flow rate through the capillaries and, unlike
the steady state models, accounts for a compliant Bowman’s space.
In the model, single nephron glomerular filtration is measured as the
blood plasma flow into the proximal tubule. Initial proximal tubule
pressure is formulated as a function of this flow rate. The model is
used to assess the damping mechanism present in the glomerulus.
Mathematical Model: Single Nephron Glomerular Filtration Rate
The model consists of 250 identical parallel capillaries discretized into 40
sections. At each section flow and protein concentration are measured.
C, Protein Concentration;
Q, Flow;
P, Pressure;
GL, Glomerulus;
AA, Afferent Arteriole;
BS, Bowman’s Space
The flow rate along the capillary is derived by the conservation of
protein mass and is modeled by the following equation
Bowman’s Space Compliance: Alpha Value
Bowman’s Space Volume
Our model allows for a Bowman’s space that can expand and contract
depending on the pressure inside the bowman’s space. When
choosing parameters that best fit experimental data, the model
predicts that the volume expands and contracts by more than 4.0 nL.
Our formula was derived by Cortes et al. [2], however we changed the
alpha and beta values to best match experimental results.
Figure. Volume
fluctuation with best
α = 3.832
s = 0.366
β = -45.155
Capillary Filtration.
In the formulation of Bowman’s space volume we were left with an
unknown parameter and chose it based on what gave us the most
accurate results when compared with experimental values.
α = 7.664
α = 3.832
Bowman’s Space Outflow.
α = 0.5
To model change in concentration over time at a point in the capillary
we used a linear approximation for change in concentration along the
capillary, and used this approximation for dC/dY in the following
Proximal Tubule Pressure Damping
Our Proximal Tubule pressure is formulated as a function of
glomerular filtration
We assume a linear pressure decrease along the capillary and input
pressure is set to be a sinusoidal function. To calculate capillary
filtration we take the difference in flow from the beginning to the end
of the capillary. Our model predicts that this filtration is not the primary
mechanism of damping.
We have developed a dynamic mathematical model of the rat
glomerulus and single nephron glomerular filtration rate
The model predicts significant damping of high frequency
oscillations in afferent arteriole pressure
The primary mechanism of pressure damping is the compliance
of the Bowman’s space .
The model’s final results with the chosen parameters do not truly
represent experimental results as our bowman’s space is
expanding well beyond biological limitations.
Our model predicts that pressure amplitude is damped by 22% from
low pressure oscillations (~30 mHz) to high pressure oscillations
(~165 mHz)
W. M. Deen, C. R. Robertson, and B. M. Brenner. Am J Physiol, 223: 11781183, 1972.
P. Cortes, X. Zhao, B. L. Riser, and R. G. Narins. Am J Physiol Renal
Physiol, 270: F356 – F370, 1996.

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