Center for Computational Medicine, Biotechnology and Bioengineering Center, Department of Physiology, Medical College of Wisconsin, Milwaukee, WI, USA

Abstract

Background

Data on blood flow regulation, renal filtration, and urine output in salt-sensitive Dahl S rats fed on high-salt (hypertensive) and low-salt (prehypertensive) diets and salt-resistant Dahl R rats fed on high-salt diets were analyzed using a mathematical model of renal blood flow regulation, glomerular filtration, and solute transport in a nephron.

Results

The mechanism of pressure-diuresis and pressure-natriuresis that emerges from simulation of the integrated systems is that relatively small increases in glomerular filtration that follow from increases in renal arterial pressure cause relatively large increases in urine and sodium output. Furthermore, analysis reveals the minimal differences between the experimental cases necessary to explain the observed data. It is determined that differences in renal afferent and efferent arterial resistances are able to explain all of the qualitative differences in observed flows, filtration rates, and glomerular pressure as well as the differences in the pressure-natriuresis and pressure-diuresis relationships in the three groups. The model is able to satisfactorily explain data from all three groups without varying parameters associated with glomerular filtration or solute transport in the nephron component of the model.

Conclusions

Thus the differences between the experimental groups are explained solely in terms of difference in blood flow regulation. This finding is consistent with the hypothesis that, if a shift in the pressure-natriuresis relationship is the primary cause of elevated arterial pressure in the Dahl S rat, then alternation in how renal afferent and efferent arterial resistances are regulated represents the primary cause of chronic hypertension in the Dahl S rat.

Background

Animal models of salt- and/or angiotensin II-induced chronic hypertension have revealed shifts in the observed pressure-natriuresis and pressure-diuresis relationships to higher pressures, as well as altered renal blood flow regulation

Guyton and Coleman and coworkers hypothesize that a shift in the pressure-natriuresis relationship to higher pressures is one of the central causal mechanisms of chronic hypertension in salt-sensitive hypertension

Here we analyze data on blood flow regulation, renal filtration, and urine output in SS rats fed on high-salt (hypertensive) and low-salt (prehypertensive) diets and salt-resistant SR rats fed on high-salt diets. We use a simple mathematical model of renal blood flow regulation, glomerular filtration, and solute transport in a nephron to reveal the minimal differences between the three cases necessary to explain the observed data. It is found that the differences in renal blood flow, glomerular filtration, and pressure-diuresis and pressure-natriuresis relationships may be explained based solely on differences in afferent and efferent arteriole regulation in the hypertensive (high-salt) SS compared to the salt-resistant SR and the low-salt SS controls.

Sources of data

Data from the SS and SR rats used for model identification are obtained from Roman

Methods

The mathematical model of renal blood flow, glomerular filtration, and mass transport in nephrons (diagrammed in Figure

A. Diagram of model

**A. Diagram of model. B**. Afferent arterial resistances governed by Equation (8), with parameter values from Table **C**. Efferent arterial conductivities governed by Equations (6), with parameter values from Table

**Dahl-R**

**Dahl-S**

**high-salt**

**low-salt**

**high-salt**

(a) value same as Dahl-R value.

(b) value same as Dahl-S value.

glomerular hydraulic permeability times length

_{
f
}
^{−1}·g^{−1}·mmHg^{−1})

0.0886

a

a

resistance associated with distal tubule

_{
d
} (min·g·mmHg·ml^{−1})

7.4959

a

a

afferent arteriole resistance parameter

_{
a
}
^{0} (ml·min^{−1}·g^{−1}·mmHg^{−1})

37.7673

a

a

minimum afferent arteriole resistance

_{
a
}
^{min} (ml·min^{−1}·g^{−1}·mmHg^{−1})

2.8758

4.88

9.84

afferent autoregulation parameter

_{
i0
} (ml·min^{−1}·g^{−1})

5.5796

4.28

b

afferent autoregulation parameter

_{
i
}

9.5614

a

a

maximum efferent arteriole resistance

_{
e
}
^{max} (ml·min^{−1}·g^{−1}·mmHg^{−1})

7.4185

13.1

9.10

efferent arteriole resistance parameter

^{−1}·g^{−1}·mmHg^{−1})

37.5995

a

a

efferent arteriole resistance parameter

^{−1}·g^{−1}·mmHg^{−2})

0.5269

0.577

0.54

TGF concentration parameter

_{
TGF
} (mM)

25.0

a

a

efferent capillary pressure fitting parameter

_{
e0
} (mmHg)

10.4

a

8.32

efferent capillary pressure fitting parameter

_{
e1
} (mmHg)

15.1

a

a

efferent capillary pressure fitting parameter

_{
e2
} (mmHg)

136.5

a

a

efferent capillary pressure fitting parameter

_{
Pe
} (unitless)

5.93

a

a

sodium permeability of the descending limb

_{
d
} (ml·min^{−1}·g^{−1}·mm^{−1})

7.70 × 10^{−3}

a

a

hydraulic permeability of the descending limb

_{
d
} (ml·min^{−1}·g^{−1}·mmHg^{−1}·mm^{−1})

8.3889 × 10^{−4}

a

a

hydraulic permeability of the collecting duct

_{
c
} (ml·min^{−1}·g^{−1}·mmHg^{−1}·mm^{−1})

1.8777 × 10^{−5}

a

a

maximum sodium reabsorption rate in ascending limb

_{
max
} (ml·mM·min^{−1}·mm^{−1}·g^{−1})

29.172

a

a

apparent Michaelis-Menten constant for sodium reabsorption

_{
m
} (mM)

50.933

a

a

While the treatment of a single nephron as representative of whole kidney is a gross simplification compared to models that capture heterogeneities in loop length and the three-dimensional architecture of the tubules and vasculature

Previous models of renal system that capture overall kidney function, including the pressure-natriuresis and pressure- diuresis phenomena, have been developed

**Governing equations for blood flow and filtration**

Flow and filtration along glomerular capillaries is governed by the conservation equation for flow in a glomerular capillary, _{1}(

where _{
f
} is the hydraulic permeability, Π_{
c
}(_{1} and _{2} are hydrostatic pressures in the capillary and Bowman’s capsule, respectively. This expression assumes that the rate of fluid loss from the capillary is linearly proportional to the pressure difference driving force and that pressure remains effectively constant along the length of the glomerular capillary. Blood enters the capillary at _{
c
}(_{
i
} = 28 mmHg and input flow _{1}(_{
i
}. Assuming a linear relationship between concentration and oncotic pressure, we have

Combining Equations (1) and (2) yields

which is a separable equation that can be solved to yield the following relationship between input flow _{1}(_{
i
} and output flow _{1}(_{
e
}:

where _{1} − _{2}, and _{
i
}/_{
f
} = _{
i
} − _{
e
}.

Blood flow into the glomerulus satisfies the Ohm’s Law relationship

where _{
a
} is the input arterial pressure and _{
a
} is the afferent arterial resistance, which is phenomenologically modeled using the following increasing function of filtration

where _{
a
}
^{
o
}, _{
a
}
^{min},_{
i
}, _{
a0}, and _{
TGF
} are adjustable parameters, and _{
a
}(0) is the sodium concentration in the ascending limb at the location where it feeds into the distal tubule. The factor _{
a
}(0)/(_{
TGF
} + _{
a
}(0)) is employed to account for tubular-glomerular feedback: increasing salt concentration in the distal tubule stimulates vasoconstriction. (The concentration _{
a
}(0) is obtained from the transport component of the model, detailed below.)

Similarly blood flow out of the glomerulus satisfies the Ohm’s Law relationship

where _{
e
} is the efferent capillary pressure and _{
e
} is the efferent arterial resistance, which is phenomenologically modeled using the following decreasing function of _{
1
}, the input pressure into the arteriole.

where _{
e
}
^{max}, _{
1
}. Equations (6) and (8) predict that afferent resistance increases and efferent resistance decreases as renal perfusion pressure is increased, as illustrated in Figure

Roman _{
e
}, efferent capillary pressure. Data on _{
e
} as a function of arterial pressure are used to fit representative function for _{
e
}(_{
a
}):

which invokes four additional adjustable parameters, _{
e0
}, _{
e1
}, _{
e2
}, and _{
pe
}.

Filtrate flow satisfies the Ohm’s Law relationship

where _{
d
} is the distal tubule pressure and _{
d
} is the resistance associated with this pressure drop, assumed constant. In the absence of data on distal tubule hydrostatic pressure, we assume a simple linear proportionality between arterial pressure and _{
d
}:

where _{
Pd
} is set to 0.02, which gives a value of distal tubule pressure of 2.0–3.6 mmHg over a range of renal perfusion pressure of 100 to 180 mmHg.

Equations (4), (5), (7), and (10), invoking 14 adjustable parameters (see Table _{
i
}, _{
e
}, _{
1
}, and _{
2
} to provide model predictions of these flows and pressures, as well as functions of input pressure _{
a
}.

**Governing equations for nephron**

Mass transport in nephrons is represented using a one-dimensionally distributed model accounting for flows and concentrations in a single representative nephron. Thus three-dimensional interactions and the anatomical heterogeneity of loop lengths are not taken into account. Nevertheless, the model is able to effectively match observed pressure-diuresis and pressure-natriuresis relationships. The nephron model, diagramed in Figure _{
d
}, _{
a
}, _{
c
}, and _{
s
}; sodium concentrations are denoted _{
d
}, _{
a
}, _{
c
}, and _{
s
}, where subscripts ‘d’, ‘a’, ‘c’, and ‘s’ indicate descending limb, ascending limb, collecting duct, and interstitial space. After passing through the proximal tubule, filtrate enters the descending limb at spatial position

Fluid transport between the interstitium and the descending limb is assumed to be linearly proportional to the combined mechanical and osmotic pressure driving force, _{
d
} + _{
s
} − _{
s
} + 2_{
d
} − _{
s
}), where _{
d
} is the hydrostatic pressure in the descending limb, _{
s
}is the osmotic pressure in the interstitium, _{
s
} is the hydrostatic pressure in the interstitium, and _{
d
} and _{
s
} are the Na^{+} concentrations in the descending limb and the interstitium. The factor of 2 multiplying the concentration gradient term arises because it is assumed that chloride concentration equals sodium concentration, and sodium plus chloride represent the major contributor to the osmotic gradient. With the hydraulic permeability constant _{
d
} mass conservation yields the equation for _{
d
}, the flow in the descending limb:

where _{
d
} = _{
d
} + _{
s
} − _{
s
}, and interstitial osmotic and hydrostatic pressures are set to _{
s
} = 17 mmHg and _{
s
} = 3 mmHg.

The ascending limb is assumed impermeable to water, and thus flow in the ascending limb, _{
a
}, is constant:

The governing equation for _{
c
}, the flow in the collecting duct is analogous to the equation for _{
d
}.

where _{
c
} = _{
c
} + _{
s
} − _{
s
} and _{
c
} is the hydraulic permeability in the collecting duct. The hydrostatic pressure in the collecting duct is assumed to be 1 mmHg lower than that in the distal tubule: _{
c
} = _{
d
} − 1 mmHg.

Since total volume is conserved

Sodium transport is assumed to be governed by passive permeation in the descending limb and collecting duct and by active transport in the ascending limb. The governing equations for Na^{+} flux in descending limb is given by

where _{
d
} is the descending limb permeability. The transport rate in the ascending limb is given by

where the factor

models a saturable process, with _{
max
} and _{
m
} adjustable parameters. The factor

is applied so that the transport rate goes to zero when concentrations in the nephron exceed an upper limit. Without this factor, concentrations become unbounded when the flow in the collecting duct approaches zero. Physically, this is because the predicted concentration gradient increases as flow through the loop of Henle decreases. Without this factor the solution becomes mathematically unbounded when pressure drops low enough that all of the filtrate is reabsorbed because in this limit _{
a
} and _{
a
}/_{
a
}
_{
a
})/_{
a
} and/or its gradient to become unbounded. Since the concentration gradient drives fluid loss from the descending limb and the collecting duct, increases in the concentration gradient lead to further decreases in flow. The Hill coefficient of 5 in this multiplying factor is also arbitrarily assigned so that transport rapidly approaches zero when _{
a
} exceeds _{
a,max}. The value of the fixed parameter _{
a,max} set to 500 mM, so that the maximal Na^{+} concentration achieved at low flows is approximately 800 mM, associated with an approximately 5-fold magnification of the input concentration of _{
d
}(0) = 150 mM. (See below.) For pressures and flows that result in urine flows that are approximately equal to and greater than the baseline values, _{
a
} remains well below _{
a,max} and the behavior of the model is not sensitive to the values of these fixed parameters.

Sodium reabsorption in the collecting duct is not explicitly accounted for in the model and the equation for Na^{+} flux in the collecting duct is

This simplifying assumption is discussed below. Salt transport in the interstitial space combines active transport and passive permeation processes:

Equation (19) assumes that the combined interstitial and vasa recta space gathers the sum of the fluxes from the other structures. Thus, as expressed by Keener and Sneyd,

The boundary conditions for input into the descending limb assume that input concentration is equivalent to plasma sodium concentration of 150 mM and input flow is proportional to _{
f
}, the glomerular filtration rate:

where _{
f
} = _{
i
} − _{
e
} is determined as a function of arterial pressure by the renal blood flow and filtration model component and

The boundary conditions for the ascending limb are obtained from the assumption of continuity of concentration and flow at the turn of the loop of Henle:

Similarly, the ascending limb feeds into the collecting duct

The interstitial flow boundary condition is

Equations (20), (21), (22), and (23) provide eight boundary conditions for the eight first-order differential equations described above. The eighth condition comes from conservation of total fluid flow

Numerical discretization of the nephron model is described in the Appendix.

Since sodium reabsorption occurs in the model only in the ascending limb and the outflow of the ascending limb feeds directly into the collecting duct, the model does not explicitly account for sodium reabsorption in the distal tubule or the collecting duct. Thus all sodium reabsorption processes are represented by the ascending limb sodium transport rate _{
a
}). This simplifying approximation is justified by the fact that during formation of either concentrated or dilute urine, the majority of sodium reabsorption occurs via the ascending limb. Lumping all reabsorption processes into Equation (17) helps keep the model tractable and identifiable. Adding additional processes would add additional uncertainty in parameter values that would not be justified given the available data or yield any additional insight into the operation of the integrated model.

Results

**Model identification**

Predictions of the renal flow and filtration model component are compared in Figure _{
u
} = _{
c
}(_{
c
}(_{
c
}(

Model predictions of renal blood flow and filtration compared to data from Dahl S (SS) rats on high-salt and low-salt diets and Dahl R (SR) rats on high-salt diets

**Model predictions of renal blood flow and filtration compared to data from Dahl S (SS) rats on high-salt and low-salt diets and Dahl R (SR) rats on high-salt diets.** Data on renal flow, filtration rate, glomerular pressure, and pertibular efferent pressure are plotted as circles for SS high-salt fed animals, triangles for SS low-salt fed animals, and filled squares for SR animals. Model predictions based on the parameter values from Table **A** are obtained from Figure **B** from Figure five of Roman **C** and **D** from Figure six of Roman

Predicted pressure-natriuresis and pressure-diuresis relationships

**Predicted pressure-natriuresis and pressure-diuresis relationships.** Urine output (_{u} = _{c}(_{c}(_{c}(

The 19 adjustable parameters invoked in this model are not identifiable for a given experimental group based on the six data sets (renal blood flow, filtration, glomerular pressure, efferent pressure, urine output, and sodium excretion as function of renal perfusion pressure) represented in these figures. However, the combined data set of pressures and flows versus arterial pressure for three experimental cases— prehypertensive (low-salt fed) and hypertensive (high-salt) SS rats and salt-resistant (high-salt) SR rats—provides independent data that can be compared to sixteen model-predicted functions of _{
a
}: _{
i
}, _{
f
}, _{
1
}, _{
e
}, _{
u
}, and _{
u
}· _{
c
}(_{
1
} and _{
e
} as functions of _{
a
} for the SS on low salt are not available; data for the SR were used to parameterize Equation (9) to represent _{
e
}(_{
a
}) for the SS on low salt.) If we assume that most model parameters attain the same values for all three groups, it is possible to determine identifiable parameter sets and to determine the minimal set of differences between the two conditions that is able to explain the observed data.

Specifically, if it is assumed that only two afferent arterial flow regulation parameters _{
i
}
^{
o
} and _{
a
}
^{0} and the efferent arterial flow regulation parameters _{
e
}
^{max} are different between the SR and SS (low-salt) cases and that only the parameters _{
a
}
^{0}, _{
e
}
^{max}, _{
e0
} are different between the SS low-salt and high-salt cases, then there are a total of 27 adjustable parameters that can be estimated by matching data to the 16 model-predicted functions in Figures

The predicted trends in afferent conductivity shown in Figure

The acute changes in sodium excretion and urine output in response to changes in renal perfusion pressure plotted in Figure

Predictions of concentration and flow profiles in the nephron, based on the nephron model, are illustrated in Figure

Model-predicted sodium concentration and flow profiles in the nephron model

**Model-predicted sodium concentration and flow profiles in the nephron model.** Simulations are conducted using the parameter values for the Dahl R rat (Table **A**. &**C**. Sodium concentrations as functions of distance along the nephron are plotted for the descending and ascending limb, collecting duct, and interstitium. **B**. &**D**. Flows as functions of distance along the nephron are plotted for the descending and ascending limb and collecting duct. The upper panel (A & B) reports model predictions for the baseline case with _{d}(0) = 0.300 ml·min^{−1}·g^{−1} and _{a} =125 mmHg. The lower panel (**C** &**D**) reports predictions for a lower pressure: _{d}(0) = 0.252 ml·min^{−1}·g^{−1} and _{a} = 95 mmHg. At the lower pressure the concentration gradient steepens and output flow drops to near zero.

To summarize the findings of comparing model predictions to data from Roman

Discussion

**Mechanisms of pressure-natriuresis and pressure-diuresis**

Using a simple mathematical model to simulate blood flow regulation, glomerular filtration, and medullary solute transport in the kidney, we have analyzed data from Dahl S and Dahl R rats to investigate the potential mechanistic underpinnings of renal function observed in these animals. While the pressure-natriuresis and pressure-diuresis relationships illustrated in Figure

Our model simulations, as well as the data analyzed here, are consistent with the pressure-natriuresis and pressure-diuresis phenomena emerging from the mechanical relationships between renal pressure, flow, and filtration. Specifically, in the model increasing arterial pressure causes increased glomerular pressure, which causes increasing filtration rate. For the SS rat data sets, an increase in glomerular pressure of 20–30% over the observed pressure range results in an increase in filtration rate of 30% in the low-salt case and almost 90% in the high-salt case. When pressure increases from 100 to 180 mmHg, filtrate flow increases from 700 to over 900 μl·min^{−1}·g^{−1} while urine output increases from 10 to 60 μl·min^{−1}·g^{−1} in low-salt case. In the high salt case, filtrate flow increases from approximately 480 to 900 μl·min^{−1} while urine output increases from 6 to 68 μl·min^{−1}·g^{−1} over the pressure range of 120 to 200 mmHg. Thus the slope of filtrate flow (_{
f
}) versus arterial pressure can be substantially steeper than the slope of urine output (_{
u
}) versus arterial pressure, even over the pressure range for which blood flow is autoregulated. For these cases the relative change in urine output over the pressure range is much greater than the relative change in filtrate flow because at the lowest pressures nearly all of the filtrate is reabsorbed.

In contrast, the SR data show relatively little increase in filtration over the observed pressure range for the three data points in Figure ^{−1}·g^{−1} is associated with an increase of 65–80 μl·min^{−1}·g^{−1} in urine output. For this case the model is not able to capture the nearly constant _{
f
} as a function of _{
a
} because the glomerular pressure is observed to increase from 44 to 53 mmHg over the same arterial pressure range. Recall that the driving force for filtration is hydrostatic pressure difference minus the oncotic pressure of approximately 28 mmHg. Since the 8 mmHg increase in glomerular pressure over the observed range of arterial pressure represents an approximately 30% increase in driving force for filtration, the model tends to under-represent the slope of _{
1
} versus _{
a
} while over representing the slope of _{
f
} versus _{
a
}. It is unclear how to resolve the substantial differences in driving force for filtration with the apparently constant filtration rate observed in the SR rat. The model predicts that _{
f
} increases roughly 20% over the observed 50 mmHg range of arterial pressure, while measurements in the SS rat and in other rat strains and other species show increases of anywhere from 10% to greater than 20% over the pressure range of autoregulated blood flow

The relationship between sodium excretion and glomerular filtrate rate is further explored in Figure

Relationship between urine output and sodium excretion and glomerular filtration rate

**Relationship between urine output and sodium excretion and glomerular filtration rate. A**. Relative change in urine output _{u} is plotted versus relative change in filtration _{f} over the arterial pressure range studied in Figure **B**. Relative change in urine output _{u} is plotted versus relative change in filtration _{f} . Data from Thompson and Pitts

The mechanistic explanation for the pressure-diuresis and pressure-natriuresis phenomena that emerges is illustrated in Figure _{
a
} = 100 to 160 mmHg. If the slopes are the same over this pressure range, then _{
u
} can be approximated as _{
f
} minus a constant reabsorbed volume. Assuming that sodium concentration remains approximately constant at arterial pressure above the baseline 100 mmHg, the pressure-diuresis relationship of the bottom panel is obtained. While this conceptual model is highly simplified, it does effectively illustrate the basic mechanism that emerges from our mathematical model: since glomerular filtration flow is much larger than urine flow, a relatively small increase in glomerular filtration can cause a relatively large increase in urine output. Thus, this explanation requires that glomerular filtration does increase, albeit slightly, as renal arterial pressure is acutely increased. If, as has been hypothesized, glomerular filtration remains exactly constant as arterial pressure is acutely increased, then this mechanism cannot explain the observed pressure-diuresis and pressure-natriuresis relationships.

Conceptual model for pressure-diuresis and pressure-natriuresis

**Conceptual model for pressure-diuresis and pressure-natriuresis.** Idealized curves are used to illustrate the hypothesized relationships between glomerular filtration flow (_{f}), urine output (_{u}), and sodium excretion following acute changes in renal arterial pressure (_{a}).

The data from Thompson and Pitts, as well as the data of Roman analyzed in Figures _{
f
} versus _{
a
} may be lower in normal animals than that captured by the model and that additional secondary mechanisms may be necessary to satisfactorily explain the pressure-diuresis/natriuresis phenomenon. Clearly, mechanical transduction is not the only mechanism at work in vivo. Yet even without a model simulation, it is apparent from the raw data that a given change in pressure can induce a greater change in glomerular filtration than in urine output over the pressure range for which blood flow is autoregulated. The model reveals the extent to which the relationship between acute changes in arterial pressure and glomerular filtration can explain the observed pressure-natriuresis and pressure-diuresis relationships. These findings highlight the direct effects of pressure on influencing urine production by delivering increased filtrate to the proximal tubule. The ability of the model to match observed relationships between arterial pressure and glomerular pressure, urine flow, and sodium excretion depends on the predicted increase in glomerular filtration with pressure that is not apparent in the SR data set. For the model to capture the phenomenon of constant glomerular filtration over the arterial pressure range of 100 to 150 mmHg observed in the SR group would require the introduction of some (unknown) mechanism that reduces glomerular hydraulic permeability in response to acute increases in pressure. Furthermore, since it does not account for hormonal or nervous factors, or changes in medullary blood flow and transporter activities, the model reveals that these factors are not necessary to explain the acute pressure-natriuresis and pressure-diuresis phenomena, at least in the SS rat on low- and high-salt diets.

**Physiological differences between SS and SR groups**

In addition to revealing insight into how sodium excretion and urine output are influenced by perfusion pressure, model analysis reveals potential mechanistic underpinnings of differences in renal function observed in SR and SS rats when fed on low-salt versus high-salt diets. Our strategy identifies the minimal differences between model parameterizations necessary to explain the data from these groups. Specifically, it was found that differences in five parameters associated with blood flow control (see Table

In developing the model presented here and determining the difference in parameter values necessary to explain the groups, the goal is not to capture all relevant physiological processes impacting renal function and blood pressure regulation in the rat. Indeed several mechanisms important to the renal response to changes in blood pressure, including changes in proximal tubule sodium transport

The different parameterizations used to explain the different experimental groups point to increases in afferent resistance and decreases in sodium transport rate as one moves from lower-pressure to higher-pressure animals. The increase in afferent resistance is able to explain all of the qualitative differences between observed data on renal function—lower flows, filtration rates, and glomerular pressure as well as the shift in the pressure-natriuresis and pressure-diuresis relationships in higher pressure animals. (Since the data analyzed here are obtained from denervated kidneys, the predicted differences in afferent arterial tone cannot be explained based on differences in sympathetic tone, unless chronic differences in sympathetic tone had the effect of chronically altering afferent arterial tone in a way that is reflected in denervated kidneys.)

This observed shift (compared to lower pressure controls) of the pressure-natriuresis relationship to higher pressure necessarily occurs in hypertension. This is because net sodium balance, by definition, must occur at a higher pressure in hypertension than in normotension. The view that the chronic pressure-natriuresis relationship (also called the renal function curve) observed in normal animals is effectively infinitely steep implies that the kidney can maintain blood volume and sodium at nearly constant levels in response to small changes in pressure associated with salt-loading and volume expansion

While it is debated whether and/or when renal dysfunction represents the primary cause of chronic hypertension in the SS rat (and other animal models)

**Assumptions and simplifications of the model**

As discussed in the methods section, the developed model for blood flow, glomerular filtration, and sodium transport in the proximal tubule, nephron, and collecting duct developed here is relatively unsophisticated compared to a number of previously developed models of the three-dimensional architecture of the tubules and vasculature

To a certain degree the level of simplification adopted by the model is justified by the nature of the data analyzed here and the specific questions addressed by the model analysis. The appropriate level of complexity represented by a model is the lowest (most simple) that can capture the biophysical processes underlying the phenomena studied. Based on this standard, the present model may be judged as a reasonable, if imperfect, simplification. The most obvious feature that the model does not capture well is the phenomenon of nearly constant filtration rate observed in the SR group. It is not known what anatomical and physiological features not represented in the current model are critical to improve the behavior of the model in comparison to this observation. While the model makes a number of simplifying assumptions, it is not clear that relaxing any one of those simplifying assumptions would explain the apparent disconnect between driving force for filtration and filtration rate in the SR group. What is clear is that the basic model introduced here represents a useful framework for exploring such questions in a systematic matter.

Conclusions

Analysis of data on renal blood flow, filtration, pressure-diuresis and pressure-natriuresis phenomena in Dahl S and Dahl R rats using a simple mathematical model reveals a hypothetical mechanistic explanation for the observed pressure-diuresis and pressure-natriuresis relationships. Idealized curves plotted in Figure _{
f
}), urine output (_{
u
}), and sodium excretion following acute changes in renal arterial pressure (_{
a
}). Increasing pressure is associated with a relatively small increase in glomerular filtration, which increases delivery of filtrate to the nephron, leading to increased urine production. This simplified conceptual model requires that glomerular filtration increases slightly as renal arterial pressure is acutely increased. Furthermore, differences between Dahl salt-sensitive (SS) and salt-resistant (SR) rats in renal filtration and urine production are explained in terms of difference in blood flow regulation.

Appendix: Discretization of nephron equations

Equations (12), (13), and (14) are discretized using finite differences

where _{
d
}
^{0} = _{
d
}(0) is the input flow into the descending limb. The discrete variables _{
d
}
^{<i>}, _{
a
}
^{<i>}, _{
c
}
^{<i>}, and _{
s
}
^{<i>} are numerical approximations for the continuous variables _{
d
}(_{
a
}(_{
c
}(_{
s
}(_{
s
}
^{<i>} is based on mass conservation and the boundary condition _{
s
}
^{<N>} = 0.

The concentrations satisfy numerical approximations of Equations (16) for the descending limb:

Equation (17) for the ascending limb:

Equation (18) or the collecting duct:

and Equation (18) for the interstitium:

The input concentration and flow are obtained from the boundary conditions _{
d
}
^{0} = _{
d
}(0) and _{
d
}
^{0} = _{
d
}(0). Qs^{<N>} and _{
s
}
^{
D
} are the input vasa recta flow and concentration (at _{
s
}
^{<N>} = _{
s
}(_{
s
}
^{<N>} = 0, the value of _{
s
}
^{
D
} is arbitrary.

These equations are solved using an iterative method. Computer codes for the model can be obtained by contacting the author.

Authors’ contributions

DAB and MM developed the model, carried out the simulation studies, and drafted the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by the Virtual Physiological Rat Project funded through NIH grant P50-GM094503.