Dominant factors that govern pressure natriuresis in diuresis and antidiuresis: a mathematical model

Abstract

We have developed a whole kidney model of the urine concentrating mechanism and renal autoregulation. The model represents the tubuloglomerular feedback (TGF) and myogenic mechanisms, which together affect the resistance of the afferent arteriole and thus glomerular filtration rate. TGF is activated by fluctuations in macula densa [Cl⁻] and the myogefnic mechanism by changes in hydrostatic pressure. The model was used to investigate the relative contributions of medullary blood flow autoregulation and inhibition of transport in the proximal convoluted tubule to pressure natriuresis in both diuresis and antidiuresis. The model predicts that medullary blood flow autoregulation, which only affects the interstitial solute composition in the model, has negligible influence on the rate of NaCl excretion. However, it exerts a significant effect on urine flow, particularly in the antidiuretic kidney. This suggests that interstitial washout has significant implications for the maintenance of hydration status but little direct bearing on salt excretion, and that medullary blood flow may only play a signaling role for stimulating a pressure-natriuresis response. Inhibited reabsorption in the model proximal convoluted tubule is capable of driving pressure natriuresis when the known actions of vasopressin on the collecting duct epithelium are taken into account.

acute elevation of renal perfusion pressure produces large increases in NaCl excretion in volume-expanded animals (1, 16, 45, 63, 84). Antidiuretic animals exhibit a similar response but with a significantly diminished magnitude (33, 34, 82). Natriuresis can also occur without elevations in renal perfusion pressure (9, 10, 25).

A substantial body of evidence suggests that pressure natriuresis is driven, at least in part, by elevated renal interstitial hydrostatic pressure (RIHP) (22, 36, 37). Increased RIHP is linked to elevated medullary blood flow (MBF), which is in turn modulated by vasopressin (23). This may explain why the magnitude of the pressure natriuresis depends on hydration status. However, the mechanisms that cause salt excretion to be sensitive to RIHP and MBF remain insufficiently well characterized.

Renal decapsulation reduces RIHP and significantly diminishes (but does not eliminate) pressure natriuresis (34, 37, 44, 48). Experimental evidence suggests that MBF has a negligible direct effect on salt and water reabsorption in the loop of Henle and the collecting duct (CD) (21, 28, 85). Instead, it has been proposed that elevated MBF initiates signals that mediate pressure-natriuretic responses (21). This signaling cascade includes the inhibition of sodium reabsorption in the proximal convoluted tubule (PCT) (67, 79). This inhibition can be partially countered by infusion of angiotensin II (62, 66, 103). Here we restrict our consideration to the direct effects of elevated MBF on the medullary interstitium (and, therefore, on reabsorption in the loop of Henle, the vasa recta, and the CD) and of inhibited sodium reabsorption in the PCT.

Both elevated MBF and inhibited PCT sodium reabsorption can affect urine flow and NaCl concentration. It remains unclear whether the direct effects of MBF on the interstitium are entirely irrelevant to the pressure natriuresis response and whether the inhibition of NaCl reabsorption in the PCT is sufficient to elevate NaCl excretion to levels observed in volume-expanded animals. To date, there is no experimental technique that allows MBF to be controlled independently of hydration status, meaning that experimental investigations cannot easily address this question. Furthermore, a recent study of a detailed whole kidney model highlighted the importance of the pressure-sensitive inhibition of PCT sodium reabsorption for predicting whole kidney water and NaCl excretion rates but provided only a cursory treatment of the matter (68).

We present a mathematical model of water and NaCl excretion that can independently regulate MBF, glomerular filtration, and proximal tubule NaCl reabsorption. By studying how model excretion rates are affected by MBF and PCT reabsorption, we identify the key factors responsible for driving pressure natriuresis in the model. We show that the model can reproduce acute renal function curves measured in both antidiuretic and volume-expanded rats. We then evaluate whether the differences in model parameters between the antidiuretic and volume-expanded cases offer a plausible explanation for the different magnitudes of pressure natriuresis that are observed in vivo in volume-expanded and volume-depleted animals.

METHODS

The model comprises an existing medullary model of the urine concentrating mechanism that encompasses the tubular and vascular architecture of both the outer and inner medulla (55), a glomerular model that encompasses the afferent and efferent arterioles and the glomerular capillary bed, a model of the myogenic and tubuloglomerular feedback (TGF) responses that regulate afferent blood flow, a PCT segment that exhibits pressure-dependent reabsorption of the filtered load, and a cortical thick ascending limb (cTAL) segment that predicts macula densa delivery, providing the input signal for the TGF response. The model structure is shown in Fig. 1.

Fig. 1.

An overview of the vascular and tubular structure of the model. Only 1 short and 1 long loop of Henle are depicted, whereas the model represents one short loop and as many long loops as there are numerical grid points in the inner medulla (IM). The inner medullary collecting ducts (CDs) undergo successive coalescences. MD, macula densa; TGF, tubuloglomerular feedback; PCT, proximal convoluted tubule; DVR, descending vasa recta; AVR, ascending vasa recta.

Renal blood flow and glomerular filtration.

The glomerular model comprises the afferent and efferent arterioles and a glomerular capillary bed. The net pressure drop across the vasculature is determined by the renal perfusion pressure (P_RPP) and the venous return pressure (P_v = 3.75 mmHg). Afferent arteriolar resistance (R_Aff) is determined by the combined activation of the TGF and myogenic responses, based on the model of Feldberg et al. (30). The activation of these responses varies along the length of the afferent arteriole, where x = 0 is the start of the arteriole and x = 1 is the entry to the glomerulus. The model represents TGF as a sigmoidal response (shown in Fig. 2) whose activation (Ψ_TGF) is determined by the macula densa Cl⁻ concentration (C_Cl, mM) and the TGF-operating point (C_Op, mM):

$${\Psi }_{\text{TGF}}(1)={\Psi }_{\text{MAX}}+\frac{{\Psi }_{\text{MIN}}-{\Psi }_{\text{MAX}}}{1+e^{K_{\text{TGF}}(C_{\text{Cl}}-C_{\text{Op}})}}$$ (1)

This activation is assumed to decrease linearly with increasing distance from the glomerulus (x = 1), disappearing at x = 0.9:

$${\Psi }_{\text{TGF}}(x)=\{\begin{array}{cc}{\Psi }_{\text{TGF}}(1)\cdot (10x-9)& \text{when}\hspace{0.17em}x\ge 0.9\\ \text{0}\hfill & \text{otherwise}\hfill \end{array}$$ (2)

Fig. 2.

TGF response for the superficial and juxtamedullary nephrons.

Experimental data suggest that the relationship between transmural pressure (across the afferent arteriole wall) and the activation of the myogenic response is nearly linear (30). Accordingly, we assume the activation of the myogenic response (Ψ_MYO) is a linear function of the transmural pressure (P):

$${\Psi }_{\text{MYO}}(P)=\{\begin{array}{cc}0\hfill & \text{for}\hspace{0.17em}P<P_{\text{MIN}}\hfill \\ \frac{P-P_{\text{MIN}}}{P_{\text{MAX}}-P_{\text{MIN}}}\hfill & \text{for}\hspace{0.17em}{P}_{\text{MIN}}\le P\le P_{\text{MAX}}\hfill \\ 1\hfill & \text{for}\hspace{0.17em}{P}_{\text{MAX}}<P\hfill \end{array}$$ (3)

The values of P_MIN and P_MAX were derived from experimental measurements (26). The net activation of the smooth muscle is assumed to be the sum of the two responses:

$$\Psi (x)={\Psi }_{\text{TGF}}(x)+{\Psi }_{\text{MYO}}(x)$$ (4)

By treating the wall of the afferent arteriole as rings of smooth muscle and applying Laplace's law (as per Feldberg et al. 30), the transmural pressure P can be written in the form:

$$P={\displaystyle {\int }_{\mathit{1}}^{\eta }\frac{\sigma }{\sqrt{r_p^2-1+z^2}}}dz\hspace{0.17em};\hspace{0.17em}{r}_p=\frac{r_i}{{\rho }_i}\hspace{0.17em}\text{;}\hspace{0.17em}\eta =\frac{{\rho }_o}{{\rho }_i}$$ (5)

where ρ_i and ρ_o are the inner and outer radii of the vessel wall when the transmural pressure P = 0, r_i and r_o are the inner and outer radii of the vessel wall for the transmural pressure P, and σ is the idealized stress of the vascular smooth muscle. This stress comprises passive stress (σ_e) due to the elastic elements of the wall and active stress (σ_a) due to the contraction of the vascular smooth muscle:

$$\sigma ={\sigma }_{\text{e}}+\Psi {\sigma }_{\text{a}}$$ (6)

$${\sigma }_{\text{e}}=C_1(e^{{\alpha }_1\epsilon }-1)+C_2(e^{{\alpha }_2\epsilon }-1)$$ (7)

$${\sigma }_{\text{a}}=\{\begin{array}{cc}{\sigma }_{\text{MAX}}\left(1-\left|\frac{\epsilon -{\epsilon }_{\text{MAX}}}{{\epsilon }_{\text{w}}}\right|\right)\hfill & \text{for}\hspace{0.17em}\left|\epsilon -{\epsilon }_{\text{MAX}}\right|<{\epsilon }_{\text{w}}\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$$ (8)

$$\epsilon =\frac{1}{z}\sqrt{r_p^2-1+z^2-1}$$ (9)

where C₁, C₂, α₁, α₂ are constants, estimated experimentally (20). The common strain ε represents the shortening or lengthening of the elastic and collagen fibers in the vessel wall, and ε_w is the working range of the overlap between the myosin and actin filaments (30).

We model the afferent arteriole as a sequence of n = 100 vascular smooth muscle segments. The resistance of each segment (R_j) was approximated using Poiseuille's law and the inner radius of the segment (r_i):

$$R_j=\frac{k}{r_i^4\cdot n}$$ (10)

where k is a constant that depends on blood viscosity and n is the number of segments. The transmural pressure at the mid-point of each segment was approximated by distributing the pressure drop across the afferent arteriole in proportion to each segment resistance:

$$R_{\text{Aff}}={\displaystyle \sum _{m=1}^n{R}_m}$$ (11)

$$P_0=\frac{2}{3}{P}_{\text{RPP}}$$ (12)

$$P_j=P_{\text{0}}-[P_0-P_{\text{G}}(0)]\frac{0.5R_j+{\displaystyle {\sum }_{m=1}^{j-1}{R}_m}}{R_{\text{Aff}}}$$ (13)

where P₀ and P_G(0) are the hydrostatic pressures at the beginning and end of the arteriole. The initial hydrostatic pressure was defined on the basis that the afferent arteriole is preceded by several levels of renal vasculature. The regulatory range of the model myogenic response is 65 ≤ P₀ ≤ 120 mmHg. By assuming that the renal perfusion pressure (RPP) is reduced by one-third before reaching the afferent arteriole, the myogenic response provides efficient autoregulation of renal blood flow (RBF) for 100 ≤ 180 mmHg. This autoregulatory range is consistent with experimental measurements of glomerular filtration rate (GFR) and RBF. The afferent blood flow is determined by the net afferent resistance and the pressure drop across the arteriole:

$$B_{\text{Aff}}=\frac{P_0-P_{\text{G}}(0)}{R_{\text{Aff}}}$$ (14)

The single-nephron glomerular filtration rate (SNGFR) and efferent blood flow are then calculated by integrating flow and pressure equations along the glomerular capillary bed (0 ≤ x ≤ 1):

$$Q_{\text{Aff}}=B_{\text{Aff}}\times (1-h)$$ (15)

$$B_{\text{G}}(x)=B_{\text{Aff}}-{\displaystyle {\int }_0^x{K}_f[P_{\text{G}}(x)-\mathrm{\Pi }(x)-P_{\text{BS}}]dx}$$ (16)

$$Q_{\text{G}}(x)=Q_{\text{Aff}}-{\displaystyle {\int }_0^{\text{x}}{K}_f[P_{\text{G}}(x)-\mathrm{\Pi }(x)-P_{\text{BS}}]dx}$$ (17)

$$P_{\text{G}}(x)=P_{\text{G}}(0)-{\displaystyle {\int }_0^{\text{x}}{\text{R}}_{\text{G}}{\text{B}}_{\text{G}}(\text{x})\text{dx}}$$ (18)

$$\mathrm{\Pi }(x)=2.1\cdot C_{\mathrm{Pr}}(x)+0.16\cdot C_{\mathrm{Pr}}{(x)}^2+0.009\cdot C_{\mathrm{Pr}}{(x)}^3$$ (19)

$$C_{\mathrm{Pr}}(\text{x})=C_{\mathrm{Pr}}(0)\cdot \frac{Q_{\text{Aff}}}{Q_{\text{G}}(x)}$$ (20)

$$B_{\text{Eff}}=B_{\text{G}}(1)$$ (21)

$$P_{\text{Eff}}=P_{\text{G}}(1)$$ (22)

To solve these equations and determine the renal blood flow and GFR, the inner radius of each afferent arteriole segment and the initial glomerular pressure P_G(0) must be determined. This is achieved with an iterative scheme. Initially, the radius of each segment is set to ρ_i. The resistance of each segment and of the entire arteriole are calculated Eqs. 10 and 11. P_G(0) is then approximated iteratively such that solving Eqs. 14–22 satisfies:

$$\left|P_{\text{G}}(1)-B_{\text{Eff}}{R}_{\text{Eff}}-P_{\text{V}}\right|\le {10}^{-3}$$ (23)

The hydrostatic pressure profile along the afferent arteriole can then be calculated using (13). The myogenic activation Ψ_MYO(x) is then updated and the inner radius of each segment is calculated to satisfy (5). These steps are repeated until the relative change in radius per iteration is <10⁻⁸ for every segment.

Medullary model.

The medullary portion of the model is based on a model that has previously been used to investigate the role that the vascular architecture in the inner medulla (IM) plays in the process of urine concentration (55). The model incorporates recent experimental findings on the relative positions of the tubules and vasa recta, as well as their transport properties (49, 73, 104), and explicitly represents short and long loops of Henle, ascending and descending vasa recta, and a composite CD. Short loops turn at the outer medulla (OM)-IM boundary, while long loops turn at every depth of the IM. Model descending vasa recta (DVR) terminate at every depth of the entire medulla. The model encompasses three solutes: NaCl, urea, and a nonreabsorbable solute that represents all solutes that are neither reabsorbed nor secreted along the nephron tubule and CD. The model predicts fluid flow, solute concentrations, and transepithelial fluxes along each tubule and vessel at every depth of the medulla, and the interstitial solute concentrations for each region. Parameter values for the medullary model can be found in Layton et al. (55).

Cortical tubule segments and delivery to the collecting duct.

The model admits three cortical tubule segments. Reabsorption in the proximal and distal convoluted tubules is controlled by fractional delivery coefficients. The cTAL is represented as an explicit epithelium and is used to determine the macula densa NaCl concentration.

Fractional delivery of NaCl (and, therefore, water) from the PCT to the proximal straight tubule (PST) is a sigmoidal function of the renal perfusion pressure, as defined by Eqs. 24 and 25 and shown in Fig. 3. When R_RPP = 100 mmHg the PCT fractional delivery is 1/3.

$$F{D}_{\text{PCT}}=K_1-(K_1-K_0)\cdot \frac{1}{1+e^{K_{\text{PCT}}(P_{\text{RPP}}-K_{\text{Op}})}}$$ (24)

$$F_{\text{PST}}(0)=(Q_{\text{Aff}}-Q_{\text{Eff}})\times F{D}_{\text{PCT}}$$ (25)

Fig. 3.

Fractional delivery in the PCT segment for several values of the slope (K_PCT), shown for the range of perfusion pressures used in the model simulations.

The cortical ascending limbs are represented as explicit epithelia (parameter values are listed in Table 1), and the macula densa NaCl concentration is assumed to be that of the cTAL outflow. The epithelial properties of this segment are identical to those of the outer-medullary ascending limb. The segment is 3-mm long for superficial nephrons and 1-mm long for juxtamedullary nephrons.

Table 1.

cTAL parameters for each nephron

Parameter	Superficial		Juxtamedullary
LcTAL	3 mm		1 mm
Radius		10 μm
Pf		0 cm/s
PNa		1 × 10−5 cm/s
PUr		0.5 × 10−5 cm/s
Km		70 mM
Vmax	12.5 nmol/cm2s		25.9 nmol/cm2s

cTAL, cortical thick ascending limb; see text for additional definitions.

The outer medullary collecting duct (OMCD) delivery is calculated from the outflow of the cortical thick ascending limbs (i.e., the flow at the macula densa) of each nephron (26). The fractional delivery coefficients are listed in Table 2, and the urea concentration is calculated such that the fluid entering the OMCD is isosmotic to blood plasma (307 mosmol/kgH₂O). The delivery to a single OMCD is weighted by the ratio of superficial to juxtamedullary nephrons and scaled to respect the 6.1:1 ratio of nephrons to CDs (51):

$$F_{\text{OMCD}}(0)=6.1\times F_{\text{cTAL}}(L_{\text{cTAL}})\times F{D}_{\text{DCT}}$$ (26)

Table 2.

Hydration-dependent parameter values

Parameter	Antidiuretic	Diuretic
ΨMIN	0.01	0.0433
ΨMAX	0.11	0.0767
COp (superficial)	33 mM	43 mM
COp (juxtamedullary)	42 mM	52 mM
FDDCT(V)	16.5%	36.4%
FDDCT(Na)	11.0%	32.2%
OMCD Pf	463 × 10−5 cm/s	23.15 × 10−5 cm/s
IMCD(x) Pf	750 × 10−4 cm/s	(6 −3.5 · x/L)−1 · 750 × 10−4 cm/s
IMCD PUr	110 × 10−5 cm/s	46 × 10−5 cm/s

OMCD, outer medullary collecting duct; IMCD, inner medullary collecting duct; see text for additional definitions.

The DVR inflow has a reference value of 8 nl/min (as per the original boundary conditions; Ref. 55) and varies as a linear function of renal perfusion pressure:

$$F_{\text{DVR}}(0)=8\hspace{0.17em}+K_{\text{MBF}}(P_{\text{RPP}}-100)$$ (27)

Hydration status and epithelial properties.

Model parameters from Ref. 55 are used to predict water and salt excretion in antidiuresis. To simulate diuresis, we adjusted the epithelial properties of the CD and the fractional reabsorption parameters for the distal convoluted tubule (DCT) to reflect the change in hydration status. The antidiuretic hormone (ADH) concentration is assumed to decrease sufficiently that the CD water permeability and inner medullary collecting duct (IMCD) urea permeability are minimal (see Table 2).

The DCT fractional reabsorption coefficients were adjusted so that the OMCD inflow was increased from 6.5 to 15.0 nl/min (per individual OMCD), assuming the DCT tubular fluid/plasma ratio inulin ratio is ∼9 (2) and that no water reabsorption occurs along the cortical collecting duct. Water permeabilities of the OMCD, initial IMCD, and terminal IMCD were reduced by factors of 20, 6, and 2.5, respectively (89), and urea permeability of the terminal IMCD was reduced from 110 × 10⁻⁵ to 46 × 10⁻⁵ cm/s (89).

The TGF gain was also reduced by a factor of 3 in the diuretic model, and the TGF operating points for the superficial and juxtaglomerular nephrons were increased by 10 mM (Table 2). The other glomerular parameters are not hydration-dependent (see Table 3).

Table 3.

Glomerular parameters

Parameter	Units	Value
ρo	μm	13 ↘ 11.7
ρi	μm	10 ↘ 9
h	—	0.45
Kf	nl·min−1·mmHg−1	1.6*/1.9†
KTGF	mM−1	0.2
PBS	mmHg	15
PV	mmHg	3.75
REff	mmHg·min·nl−1	0.4
RG	mmHg·min·nl−1	0.001
K0	—	3/15
K1	—	7/15
KOp	mmHg	100
PMIN	mmHg	42.3
PMAX	mmHg	86.7
C1	mmHg	20.25
α1	—	2.0
C2	mmHg	1.88
α2	—	12.0
σMAX	mmHg	750
ϵMAX	—	0.25
ϵw	—	0.9
k	mmHg·min·μm	9.406
n	—	100

↘, Decreases linearly. See text for other definitions.

^*Superficial;

^†juxtamedullary.

Regulation of water and salt excretion.

The parameters of interest for exploring how the model regulates water and salt excretion, with regards to renal perfusion pressure, are the slopes of the MBF autoregulatory curve and the pressure-sensitive inhibition of sodium reabsorption in the PCT (hereafter referred to as the “PCT response”):

K_MBF: MBF is well-regulated in antidiuretic rats and poorly regulated in volume expanded rats. ADH presumably plays a significant role since it varies in response to hydration status and has a vasoconstrictive effect on the DVR. If K_MBF = 0, MBF (see Eq. 27) is perfectly autoregulated. As K_MBF increases, MBF autoregulation decreases.

K_PCT: the response is amplified in volume-expanded rats and diminished in antidiuretic rats and is believed to be mediated (at least in part) by the medullary interstitial hydrostatic pressure (which may be sensitive to ADH). Angiotensin II also has an significant effect on PCT reabsorption. The value of K_PCT (see Eq. 24) is the maximal slope of the response.

In particular, the interaction between these two mechanisms and the combined effect they exert on NaCl excretion are the critical determinants of the pressure-natriuresis response of the model.

RESULTS

Adult rats are consistently observed to have GFRs of around 1.0 ml·min⁻¹·g kidney weight⁻¹ (kwt) and whole body GFRs in the range of 2.3–3.0 ml/min (Table 4). Within the autoregulatory range, the model whole body GFR is 2.43 ml/min (assuming that a rat kidney contains 38,000 nephrons; Ref. 99), suggesting the model “single kidney weight” is 1.2 g kwt. To report solute and volume excretion rates in units consistent with experimental studies (μeq·min⁻¹·g kwt⁻¹ and μl·min⁻¹·g kwt⁻¹, respectively), the IMCD volume and solute outflow rates were scaled by 38,000/(1.2·6.1) under the assumptions that a rat kidney contains 38,000 nephrons (99), that the model represents a single kidney of 1.2 g kwt, and that the ratio of nephrons to individual CDs in the model is 6.1:1 (51, 55).

Table 4.

Comparison of the antidiuretic model to in vivo measurements of whole body GFR in the rat

	Weight, g	GFR, ml·min−1·g kwt−1	Kidneys, g	GFR, ml/min
Model (32 × 103 nephrons) (8)			≈2.05*	2.05
Model (38 × 103 nephrons (99)			≈2.43*	2.43
Sprague-Dawley (male) (81)		1.0	2.54	2.54
Sprague-Dawley (male) (79)	225–300	1.03	2.92	3.01
Dahl (female) (83)	235	1.0	2.30	2.30
Sprague-Dawley (male) (38)	150	1.2
Sprague-Dawley (male) (56)		0.9
Sprague-Dawley (male) (58)		1.0–1.2
Munich-Wistar (60)	200–250	1.2
Charles River (male, hydropenic) (7)	320–350			3.00
Sprague-Dawley (male) (103)	170–230			1.0–1.4

Evaluated within the autoregulatory range of the model, where n is the assumed number of nephrons in a single rat kidney; g kwt, kidney weight in grams. Numbers in parenthesis indicate reference numbers.

^*Calculated by assuming glomerular filtration rate (GFR) ≈ 1.0 ml·min⁻¹·g kwt⁻¹.

Volume and solute transport along portions of individual tubule segments are reported with respect to the medullary depth, where 0 mm refers to the corticomedullary boundary and 7 mm to the papillary tip. Cortical segments are reported with negative depths, from the corticomedullary boundary (0 mm) to macula densa (−1 and −3 mm for the juxtamedullary and superficial cTAL, respectively).

Base case (antidiuresis).

Water and solute flows in the longest loop of Henle, vasa recta, CD, and aggregate superficial/juxtamedullary cTAL segments are shown in Fig. 4. Solute concentrations and osmolalities are shown in Fig. 5. Urine flow and composition are summarized in Table 5. The vast majority of water reabsorption in the descending limbs occurs in the OM. Together the ascending limb segments reabsorb one-half of the NaCl load, the bulk of which is reabsorbed in the OM. Net reabsorption (5–7 mm) and net secretion (2–5 mm) of NaCl both occur along the IM ascending thin limbs.

Fig. 4.

Water and solute flow in the antidiuretic and diuretic models (P_RPP = 100 mmHg). Profiles are shown for the longest loop of Henle and longest vasa recta. Negative depths depict the cortical thick ascending limb (cTAL) segments of the superficial nephron (3 mm in length) and the aggregate juxtamedullary nephron population (1 mm in length). The cTAL profiles are discontinuous at x = 0 since they contain the aggregate flow of all superficial and juxtamedullary nephrons, respectively. ○Flow profiles in ascending limbs of Henle and AVR. *Flow profiles in descending limbs of Henle and DVR. ^aLoop of Henle flow is shown per nephron. ^bVasa recta flow is shown per individual DVR. ^cCD flow is shown per individual CD (6.1 nephrons).

Fig. 5.

Solute concentration and osmolality profiles in the antidiuretic and diuretic models (P_RPP = 100 mmHg). Profiles are shown as per Fig. 4. ○Concentration profiles in ascending limbs of Henle and AVR. *Concentration profiles in descending limbs of Henle and DVR.

Table 5.

Comparison of the model urine in antidiuretic and diuretic states when the perfusion pressure P_RPP = 100 mmHg

Quantity	Antidiuretic	Diuretic
Flow rate, nl·min−1·CD	0.35	8.57
Flow rate, ul·min−1·g kwt−1	1.82	44.49
Osmolality, mosmol/kgH2O	1233	434
[NaCl], mM	241	145
[Urea], mM	582	149
NaCl excretion, ueq·min−1·g kwt−1	0.44	6.45
Urea excretion, ueq·min−1·g kwt−1	1.05	6.62

CD, collecting duct.

Water is rapidly reabsorbed from the CD in the outer stripe. It is then reabsorbed at an approximately linear rate in the inner stripe and upper IM (0.4–4.5 mm). In the deep IM (5–7 mm), CD water reabsorption is enhanced due to the high IMCD urea permeability, and the bulk of the IMCD urea load is reabsorbed into the interstitium. In contrast, there is a small amount of NaCl secretion into the OMCD (where the thick limb active transport of NaCl is highest) and the bulk of the CD NaCl reabsorption occurs in the upper IMCD.

Base case (diuresis).

Key results for a diuretic kidney are shown in Figs. 4 and Fig. 5. Compared with the antidiuretic model, the descending limb of Henle and DVR inflow rates are unchanged and there is a similar degree of water reabsorption in the outer medullary descending limb segments. However, pronounced water secretion into the descending limbs in the IM increases the ascending limb water flow rates. In the IM there is significant NaCl secretion into the ascending limb of Henle, which enhances NaCl delivery to the OM and the cTAL segments. There is also significant water secretion into the DVR in the IM, resulting in papillary DVR flow rates 50% higher than in the antidiuretic model.

Volume delivery to the OMCD in the diuretic model is assumed to be elevated in the diuretic model (6.1 vs. 15.0 nl/min). This results in a much larger load on the urine concentrating mechanism. Combined with the diminished CD water permeability, CD volume reabsorption is greatly reduced. CD fractional water reabsorption decreased from 94 to 43% in the diuretic model. Increased OMCD delivery and diminished CD water reabsorption result in higher NaCl and urea flow rates along the entirety of the CD, although the profiles for both solutes are similar in shape to those observed in the antidiuretic model. Urea reabsorption in the terminal IMCD is diminished due to the lowered IMCD urea permeability, reducing the quantity of urea available for recycling in the IM. CD fractional reabsorption of both NaCl and urea are significantly decreased in the diuretic model, from 84 to 25% (NaCl) and 77 to 13% (urea), even though the absolute water and NaCl reabsorption rates are slightly higher in the diuretic model.

The increased OMCD delivery, decreased urea recycling, and decreased water reabsorption along the medullary CD produce a 25-fold increase in urine flow rate, a 65% decrease in urine osmolality, a 15-fold increase in NaCl excretion, and a 6.3-fold increase in urea excretion, relative to the antidiuretic model (summarized in Table 5).

Medullary blood flow autoregulation in antidiuresis.

Simulations were conducted for several values of K_MBF, 0, 0.005, 0.010, and 0.015 nl·min⁻¹·mmHg⁻¹ to determine the sensitivity of the model excretion rates to MBF. For each simulation, the PCT response was fixed (K_PCT = 0.025) and renal perfusion pressure was incremented from 76 to 194 mmHg. Model SNGFR was well regulated except for P_RPP ≤90 mmHg, where it was significantly diminished. Superficial SNGFR was otherwise maintained between 29 and 33 nl/min and juxtamedullary SNGFR between 35 and 39 nl/min. The urine volume, osmolality and solute concentrations produced by the model for each value of K_MBF are shown in Fig. 6.

Fig. 6.

Effects of different degrees of medullary blood flow (MBF) autoregulation in the antidiuretic model. In response to elevated perfusion pressure: urine flow increases 6- to 12-fold (A); urine osmolality decreases by ∼60% (B); NaCl excretion increases 5- to 6-fold (C); and urea excretion increases 4- to 6-fold (D).

Urine flow rate and solute excretion rates increased in proportion to the slope of the MBF curve (K_MBF), while urine osmolality exhibited an inverse relationship to K_MBF. When renal perfusion pressure was increased from 100 to 200 mmHg, urine flow rate increased by 9–15-fold (depending on K_MBF), urine osmolality decreased by 50–64%, NaCl excretion increased by 620–660%, and urea excretion increased by 400–560%. Urine flow rate, osmolality, and urea excretion rate are all sensitive to K_MBF. In contrast, NaCl excretion exhibits almost complete insensitivity to K_MBF and varied by only 7% at P_RPP = 200 mmHg.

Two benchmark values of K_MBF, 0 and 0.0125, correspond to perfect autoregulation as observed in volume-depleted rats and to weak autoregulation as observed in volume-expanded rats (33, 65). A comparison of flow and solute concentration profiles for these benchmark values (at P_RPP = 140 mmHg) reveals that the increased DVR inflow (from 8 to 12 nl/min) elevated DVR flow. This increased fluid delivery to the IM washed out the IM axial solute gradient. As a result, DVR osmolality was reduced by 150 mosmol/kgH₂O at the papillary tip and urine osmolality fell from 805 to 645 mosmol/kgH₂O.

In the antidiuretic model, the degree of MBF autoregulation exerts significant control over urine flow rate, osmolality, and urea excretion rate, while the NaCl excretion rate is almost completely insensitive to elevated MBF.

Medullary blood flow autoregulation in diuresis.

Similar to the antidiuretic model, increased K_MBF in the diuretic model elevates urine flow rate and urea excretion rate, and decreases urine osmolality (see Fig. 7). The rate of NaCl excretion remained highly insensitive to K_MBF but, in contrast to the antidiuretic model, varied inversely to changes in K_MBF. Comparing the urine at P_RPP = 100 and 200 mmHg shows that the relative increases in excretion rates and the decrease in urine osmolality are significantly smaller in the diuretic model than in the antidiuretic model. For example, NaCl excretion varied by only 3.4% at the maximal perfusion pressure (194 mmHg).

Fig. 7.

Effects of different degrees of MBF autoregulation in the diuretic model. In response to elevated perfusion pressure: urine flow increases 2-fold (A); urine osmolality decreases by around 25% (B); NaCl excretion increases 2-fold (C); and urea excretion increases by 10–70% (D).

The decrease in NaCl excretion as K_MBF increases is due to the large difference in baseline urine flow rate. In the antidiuretic and diuretic models, urine NaCl concentrations are comparable. At P_RPP = 140 mmHg in the antidiuretic model, increased MBF elevates urine flow by 3.9 μl·min⁻¹·g kwt⁻¹, a relative change of 58%. In the diuretic model, the same increase in MBF elevates urine flow by 7.5 μl·min⁻¹·g kwt⁻¹, a relative change of 11%.

In the diuretic model, volume expansion significantly diminishes the sensitivity of the urine flow rate, osmolality, and solute excretion rates to the degree of MBF autoregulation. The additional washout caused by the increased MBF has only a minimal effect on the model urine.

PCT response in antidiuresis.

In these simulations, DVR inflow was fixed at 8 nl/min and the slope of the PCT response (K_PCT) was set to 0.010, 0.025, and 0.040. The values represent diminished, normal, and elevated PCT transport inhibition and affect the PST inflow rates at the corticomedullary boundary. Model urine volume and composition are shown in Fig. 8. Model SNGFR was well regulated and did not differ significantly from the simulations where K_MBF was varied.

Fig. 8.

Effects of different degrees of transport inhibition in the PCT in the antidiuretic model. In response to elevated perfusion pressure: urine flow is significantly regulated by the PCT response (A); urine osmolality is significantly regulated by the PCT response (B); NaCl excretion is significantly regulated by the PCT response (C); and urea excretion is significantly regulated by the PCT response (D).

Increasing K_PCT from 0.010 to 0.025 greatly increased the sensitivity of the urine flow rate, osmolality, and solute excretion rates to the renal perfusion pressure. Increasing K_PCT further to 0.040 had a similar additional effect.

When P_RPP was raised from 100 to 194 mmHg, urine flow rate increased 5- to 10-fold, urine osmolality decreased by 41–52%, NaCl excretion increased 4–7-fold, and urea excretion increased by 290–440%. Comparing the antidiuretic model sensitivity to changes in K_MBF and K_PCT, the most significant difference is that the NaCl excretion rate is much more sensitive to K_PCT than to K_MBF.

The increased PST delivery caused by increases in K_PCT led to elevated CD volume delivery. At P_RPP = 140 mmHg, CD fractional volume reabsorption was diminished from 90% (K_PCT = 0.010) to 84.5% (K_PCT = 0.025) and 80% (K_PCT = 0.040). However, net CD volume reabsorption was increased from 6.52 to 7.11 and 7.47 nl·min⁻¹·CD⁻¹. The difference in urine flow rate for these values of K_PCT was 4.9 μl·min⁻¹·g kwt⁻¹, which was smaller than the difference in OMCD delivery (9.1 μl·min⁻¹·g kwt⁻¹). The increased CD volume delivery also reduced the solute concentrations along the CD and reduced urine osmolality from 986 to 805 and 718 mosmol/kgH₂O. Urine NaCl and urea concentrations varied by 38 and 138 mM as a result of the variations in K_PCT. In contrast, varying K_MBF produced near-identical changes in urine NaCl and urea concentration.

In the antidiuretic model, the slope of the PCT response exerted significant influence on urine flow rate, osmolality, and solute excretion rates. However, elevated MBF produced larger pressure-sensitive variations in urine flow rate, osmolality, and urea excretion rate (compare Fig. 6 to Fig. 8).

PCT response in diuresis.

Variations in the PCT response in the diuretic model are shown in Fig. 9, for the same range of K_PCT values. Urine flow rate and NaCl excretion are significantly elevated (≈10-fold) in the diuretic model, and the urine osmolality is significantly diminished. The most significant difference between the antidiuretic and diuretic model sensitivity to K_PCT is urea excretion, which is highly sensitive to K_PCT in the antidiuretic model and consistently exceeds the NaCl excretion rate. In the diuretic model, the urea excretion rate exhibits almost complete insensitivity to K_PCT and varies by only 4% at P_RPP = 194 mmHg. Urea excretion is also consistently lower than NaCl excretion in the diuretic model.

Fig. 9.

Effects of different degrees of transport inhibition in the PCT in the diuretic model. In response to elevated perfusion pressure: urine flow is significantly regulated by the PCT response (A); urine osmolality is somewhat regulated by the PCT response (B); NaCl excretion is significantly regulated by the PCT response (C); and urea excretion is insensitive to the PCT response (D).

We now compare flow rate and solute concentration profiles for the benchmark values of K_PCT (0.010, 0.025, and 0.040) at P_RPP = 140 mmHg. Increased PCT reabsorption inhibition elevated volume delivery to the descending limbs, which was not negated by an increase in volume reabsorption along the descending limbs. OMCD inflow rate was elevated in proportion to K_PCT, from 17.3 (K_PCT = 0.010) to 19.6 (K_PCT = 0.025) and 21.4 nl·min⁻¹·CD⁻¹ (K_PCT = 0.040). Fractional CD volume reabsorption decreased when K_PCT was increased (from 38 to 33 and 30%) but absolute CD volume reabsorption remained fixed at 6.55 nl·min⁻¹·CD⁻¹. Urine NaCl concentration varied by <1 mM in response to K_PCT, while urine urea concentration decreased by 30 mM. This decrease in urine urea concentration was almost exactly countered by the increase in urine flow rate, resulting in near-constant urea excretion.

In the diuretic model, variations in the slope of the PCT response produced smaller relative changes in urine flow rate, osmolality, and solute excretion rates than those in the antidiuretic model. Urea excretion was near constant, a result of the limited urea recycling in the IM in diuresis. The absolute increases in urine flow rate and NaCl excretion rate, in response to elevated perfusion pressure, were significantly larger in the diuretic model than in the antidiuretic model. The relative increase in these excretion rates were, however, smaller than in the antidiuretic model. This is consistent with the differences between the pressure natriuresis observed in the antidiuretic and the diuretic rat (33, 34, 82).

Combined regulation in antidiuresis.

The sensitivity of the model to simultaneous variations in both MBF and transport inhibition in the PCT was assessed (see Fig. 10). In the antidiuretic rat, MBF is as strongly autoregulated as the cortical blood flow (33, 65). This suggests that the curves for K_MBF = 0 should be representative for the antidiuretic model. In the model, decreased MBF autoregulation elevated urine flow and urea excretion and diminished the urine osmolality, while having little effect on NaCl excretion. On the other hand, the PCT response played a dominant role in driving pressure natriuresis in the model, regardless of MBF autoregulation.

Fig. 10.

Effects of varying both the degree of MBF autoregulation and the degree of transport inhibition in the PCT in the antidiuretic model. In response to elevated perfusion pressure: urine flow is more sensitive to MBF autoregulation than to transport inhibition in the PCT (A); urine osmolality is more sensitive to MBF autoregulation than to transport inhibition in the PCT (B); NaCl excretion is more sensitive to transport inhibition in the PCT than to MBF autoregulation (C); urea excretion is more sensitive to MBF autoregulation than to transport inhibition in the PCT (D).

The increase in K_MBF reduced urea secretion into the long descending limbs by 2–3%, while increasing K_PCT reduced urea secretion by 13–20%. This suggests that MBF autoregulation has a lesser impact than the PCT response on urea recycling in the IM. Net volume reabsorption along the CD was significantly enhanced when MBF was strongly autoregulated, while the PCT response played a greater role in determining the OMCD inflow rate.

Combined regulation in diuresis.

In the diuretic model, PCT reabsorption inhibition and weak MBF autoregulation (as observed in volume-expanded rats) have synergistic effects. Urine flow rate was elevated and urine osmolality was decreased (see Fig. 11). Urea excretion exhibits no sensitivity to K_PCT but was sensitive to MBF. Similar to the antidiuretic model, NaCl excretion was highly sensitive to the PCT response and showed minimal sensitivity to MBF autoregulation. In contrast to the antidiuretic model, decreased MBF autoregulation slightly diminished NaCl excretion (as remarked upon earlier).

Fig. 11.

Effects of varying both the degree of MBF autoregulation and the degree of transport inhibition in the PCT in the diuretic model. In response to elevated perfusion pressure: urine flow is more sensitive to MBF autoregulation than to transport inhibition in the PCT (A); urine osmolality is slightly more sensitive to MBF autoregulation than to transport inhibition in the PCT (B); NaCl excretion is more sensitive to transport inhibition in the PCT than to MBF autoregulation (C); urea excretion is more sensitive to MBF autoregulation than to transport inhibition in the PCT (D).

The DVR flow rate and osmolality, and the NaCl and urea concentrations in the loops of Henle, showed much less sensitivity to both the PCT response and MBF autoregulation than observed in the antidiuretic model. The interstitial washout that accompanies volume expansion limited the effects of these mechanisms. Decreased CD water permeability meant that K_MBF and K_PCT could produce only minimal differences in volume reabsorption along the CD, contrary to the antidiuretic model. Decreased IMCD urea permeability in the diuretic model permitted only minimal urea recycling, regardless of the IMCD urea flow rate. Urea excretion was two to three times higher in the diuretic model than in the antidiuretic model and urine osmolality did not exceed 430 mosmol/kgH₂O when P_RPP ≥ 100 mmHg.

Acute in vivo pressure natriuresis.

The rates of water and NaCl excretion in the antidiuretic and diuretic models were then compared with experimental measurements of acute renal function in the rat (see Fig. 12, A and B). The majority of experimental studies observed strong autoregulation of RBF and GFR over wide ranges of perfusion pressure (from 90 to 200 mmHg) (33, 34, 38, 41, 56, 58, 60, 62, 74, 83), while other studies measured significant increases in RBF and/or GFR (57, 64, 66, 101, 103). There does not appear to be a factor in the chosen cohorts or experimental methods that determines whether RBF and GFR are strongly or weakly regulated. Both RBF and whole kidney GFR were also observed to decrease below control levels at low perfusion pressures (P_RPP ≤90 mmHg) in the antidiuretic kidney.

Fig. 12.

Regulation of water (A) and salt (B) excretion in the antidiuretic model (K_MBF = 0, K_PCT = 0.010, lower curve) and diuretic model (K_MBF = 0.0125, K_PCT = 0.040, upper curve), compared with acute renal function data obtained from studies in the rat (33, 34, 38, 41, 56–58, 60, 62, 64, 66, 74, 83, 101). PCT responses were then fitted (C and D) to a single study of acute pressure-natriuresis in which nitric oxide synthase was inhibited (38).

The rates of water and NaCl excretion in the antidiuretic model compared well to the minimal rates observed across the range of surveyed studies. In contrast, the agreement between the diuretic model and experimental data was less ideal. This is likely attributable to the PCT response and/or the CD epithelial parameters. Thus, in subsequent simulations, we attempted to fit the diuretic model excretion rates to whole kidney excretion data. This was achieved by adjusting the PCT response and by altering the model CD transport parameters.

From the experimental studies shown in Fig. 12, A and B, a single study was chosen and used to fit the model excretion curves. Guarasci and Kline (38) investigated acute pressure natriuresis in the rat following both acute and chronic inhibition of nitric oxide synthase (NOS), which inhibits the regulatory effects of nitric oxide (NO) in the renal medulla. This study was selected on the basis that the animals exhibited strong pressure-natriuresis responses and that the absence of NO from the model formulation matches the experimental conditions. PCT responses were fitted to the two groups of rats: “Control” and “l-NAME.” For each group, model excretion curves were fitted under two assumptions: that there is active NaCl transport along the model CD (as in the antidiuretic model) and that this active transport is inhibited in diuresis (as suggested by some studies; Refs. 50, 52, 70, 94, 96, 98).

In each case, the parameters in Eq. 24 that govern PCT fractional reabsorption (K₀, K₁, and K_Op) were adjusted so that the model NaCl excretion rate resembled the measurements of Guarasci and Kline (38). The resulting PCT response curves are shown in Fig. 13. The agreement between the model urine flow and experimental data was also much improved (compare Fig. 12, A and C).

Fig. 13.

A comparison of the PCT responses fitted to the excretion rates measured by Guarasci and Kline (38), when active transport of NaCl in the model CD is enabled and when it is disabled.

The diuretic model excretion curves provided a good match to the experimental excretion rates when the fitted PCT responses were used. Whether these PCT responses are consistent with experimental observations of reabsorption in the PCT under saline diuresis is another matter. We address this question in the discussion.

DISCUSSION

The model only incorporates a few regulatory mechanisms: the myogenic response of the afferent arteriole, TGF, the direct effects of MBF on medullary solute concentrations, and inhibition of water and NaCl transport in the PCT. The model does not account for the actions of the renin-angiotensin system (RAS), NO, or endothelin, all of which are known to play significant roles in the regulation of renal NaCl excretion. Furthermore, MBF is understood to initiate signaling cascades (via RIHP) that affect sodium reabsorption in the PCT (67). Thus the regulatory mechanisms in the model are both highly simplified and incomplete. This is due, in part, to a shortage of relevant (quantitative) experimental data.

However, these limitations do not necessarily invalidate interpretation of the results presented in this article. In experimental studies of acute pressure natriuresis (such as those shown in Fig. 12), the experimental protocols include measures to keep paracrine levels fixed. This is typically achieved by steady infusion of a “cocktail” of hormones and various inhibitors/antagonists. Short stabilization and collection periods (e.g., 5–20 min) are also used to ensure there is insufficient time for other, slower, regulatory mechanisms to respond.

In short, in the experimental studies of acute pressure natriuresis against which we have compared our model, the very paracrine factors we have been unable to incorporate into our model have been either inhibited or maintained at fixed physiological levels. However, conclusions drawn from the model results may not apply to pressure natriuresis in organisms whose paracrine levels are not tightly controlled. Keeping these limitations in mind, we now compare the model MBF and PCT responses to in vivo data to evaluate the degree to which the model dynamics are consistent with experimental observations.

Medullary blood flow autoregulation.

Medullary blood flow is known to be strongly autoregulated in antidiuretic rats (24, 33, 65) and weakly regulated in diuretic rats (33, 66, 82). The benchmark values for K_MBF used in this study (0 and 0.0125) correspond to the regulatory responses observed in hydropenic and volume-expanded rats (65). Since the experimental techniques used to measure variations in MBF are not capable of measuring absolute flow rates, the only quantitative data against which the model DVR flow rates can be compared are micropuncture measurements at the papillary tip.

Blood flow rate in individual DVR vessels at the papillary tip has been measured in several studies and has consistently been measured at 8–11 nl/min in volume-expanded rats (40, 105, 106). In the model base cases, the papillary DVR plasma flow rate is 5.0 nl/min in the antidiuretic model and 7.3 nl/min in the diuretic model (for an individual DVR vessel). Experimental measurements of papillary DVR hematocrit range from 9 to 21% (40). This would imply blood flow rates of 5.5–6.3 and 8.0–9.2 nl/min in the papillary DVR of the antidiuretic and diuretic models, respectively. The papillary DVR flow rate in the diuretic model is consistent with the experimental observations, while the papillary DVR flow rate in the antidiuretic model (5.5–6.3 nl/min) is lower than that observed by Holliger et al. (40) in young antidiuretic rats (8.83 nl/min).

In the diuretic model, decreased autoregulation of MBF (i.e., interstitial washout) did not stimulate an increase in NaCl excretion. This is consistent with evidence that increased washout is not a dominant factor in pressure natriuresis (21, 28, 85). Likewise, decreased autoregulation of MBF in the antidiuretic model did not increase NaCl excretion, since the increased urine flow rate was countered by an equivalent decrease in urine NaCl concentration (Fig. 6). Thus, while MBF plays a significant role in maintaining the hydration status, the model predictions suggest that its direct effects on the interstitial composition have negligible consequences for pressure natriuresis. It remains unclear whether MBF can trigger phenomena such as reduced PCT reabsorption or if the observed correlation between MBF autoregulation and NaCl excretion is due to a common (unidentified) mechanism, such as elevated RIHP (34, 37, 44, 48). The degree of MBF autoregulation is strongly influenced by circulating levels of vasopressin (23, 33), which indicates that vasopressin might play a role in modulating pressure natriuresis. The model results are consistent with the hydration-dependent magnitude of the pressure-natriuresis response and experimental evidence of the effects of interstitial washout. They cannot provide insight into in vivo signaling and the downstream effects of MBF autoregulation.

NaCl reabsorption in the PCT.

Some studies have suggested that increased NaCl excretion after volume expansion requires alterations in NaCl reabsorption beyond the PCT (50, 52, 70, 94, 96, 98). Sonnenberg (96) observed that “only in the low-salt rats can the natriuresis be explained by inhibition of Na transport in the CD; excretion of NaCl in expanded DOCA rats is uniformly greater than end-distal delivery, indicating net entry of the ion into the terminal nephron segment.”

By contrast, fractional NaCl reabsorption along the CD was consistently 10–20% in the diuretic model. Net NaCl secretion into the CD was not observed. It is therefore important to determine whether the fitted PCT responses (shown in Fig. 13) are consistent with experimental measurements of PCT reabsorption under saline diuresis. Inconsistencies between the model PCT response and experimental data would suggest that some of the diuretic parameter values (Table 2) are inappropriate. Acute volume expansion has been observed to reduce PCT fractional reabsorption from 61 to 42% (perfusion/arterial pressure not reported) (53) and from 62% (P_RPP = 114 mmHg) to 33% (P_RPP = 110 mmHg) (11). Under saline diuresis, increased perfusion pressure has been observed to reduce PCT fractional reabsorption even further, from 65% (P_RPP ≈ 80 mmHg) to as low as 25% (P_RPP ≈ 135 mmHg) (6). An increase of 50% in PCT delivery to the descending limb [as observed by Brenner et al. (11), mentioned above] is consistent with in vivo observations of end proximal tubule fluid velocity and flow rate in response to acute arterial hypertension (105→155 mmHg) in the rat (19). Urine flow rate was also increased ∼2.5-fold in response to the acute hypertension. A smaller degree of hypertension (104→129 mmHg) was observed to increase urine flow by 45% and late proximal flow rate by 30% (18). Clearly, there is significant variation between the baseline reabsorption rates and between the magnitude of the changes in reabsorption in response to volume expansion and acute hypertension.

Guarasci and Kline (38) observed an approximately five- to sixfold increase in urine flow rate and a seven to eightfold increase in NaCl excretion rate in response to an acute perfusion pressure elevation of 50 mmHg (Fig. 4). The PCT responses fitted to the “Control” case, with and without CD NaCl active transport, are consistent the above data. Volume expansion, without elevation in perfusion pressure, was observed in vivo to decrease PCT fractional reabsorption to 30–40%. When combined with an acute elevation of perfusion pressure to 135 mmHg, in vivo PCT fractional reabsorption decreased to 25%. PCT fractional reabsorption for both fits is significantly higher at P_RPP = 100 mmHg than observed in volume-expanded animals. Instead, it is more consistent with levels observed in vivo before volume expansion. The decrease in fractional reabsorption from P_RPP = 125 mmHg to P_RPP = 150 mmHg is larger than observed in vivo, where late proximal flow rate increased by 30% and urine flow rate increased by 45% (18), but the pressure natriuresis observed by Guarasci and Kline (38) was also much larger than that observed by Chou and Marsh (18).

The PCT responses fitted to the “l-NAME” case predicted end-proximal flows and urine flows that were consistent with in vivo measurements in response to an acute elevation from P_RPP = 125 mmHg to P_RPP = 150 mmHg (18). Since PCT fractional reabsorption has been observed as low as 25% at P_RPP = 135 mmHg (6), it seems plausible that it should decrease further when the perfusion pressure is elevated to 175 mmHg. This remains plausible, even when accounting for the shift of the pressure-natriuresis response induced by the administration of nitro-l-arginine methyl ester (l-NAME), as observed by Guarasci and Kline.

Studies have suggested that CD NaCl reabsorption is inhibited under volume expansion (50, 52, 70, 94), and NaCl secretion into the CD has even been observed (96–98). This suggests that the diuretic model is, perhaps, more realistic when active NaCl transport along the CD is inhibited. However, while PCT fractional reabsorption has been measured to be as high as 75% in antidiuretic animals (35), volume expansion is known to significantly reduce PCT reabsorption. Thus PCT fractional reabsorption for the fitted responses is higher than expected at the lowest pressures (P_RPP = 100 mmHg for “Control” and P_RPP = 125 mmHg for “l-NAME”), while being consistent with in vivo data at higher pressures. This suggests that the diuretic model (Table 2) produces a diuresis that is too strong at these low pressures and that PCT reabsorption must be overly elevated to counter this. This could be avoided by scaling the diuresis in proportion to the model perfusion pressure.

Given that the slope of the whole kidney pressure natriuresis response is commonly observed to be around four to six times steeper in diuretic animals than in antidiuretic animals (34, 60, 64, 74, 78, 80, 82, 103), it is reasonable to suggest that the model PCT response in the diuretic model should also be four to six times steeper than in the antidiuretic model. The slope of the fitted PCT responses in the diuretic model varied from 0.09 to 0.10, which is four times larger than the slope considered reasonable for the antidiuretic model (0.025).

Plasma ADH is known to increase the water permeability of the late distal tubule and CD (12, 27, 31, 42), to increase the urea permeability of the IMCD (42, 88, 90), and to stimulate active NaCl transport along the CD (39, 42, 61, 77). The epithelial changes introduced in the diuretic model are consistent with the known effects of the decrease in circulating ADH that is caused by volume expansion. Therefore, the model results suggest that the known actions of decreased ADH levels and increased inhibition of reabsorption in the PCT are sufficient to drive pressure natriuresis to levels consistent with those observed in experimental studies. The mechanisms by which PCT reabsorption is inhibited under volume expansion remain insufficiently quantified and cannot be investigated with this model.

Regulation of urea excretion.

In all mammals investigated, urea excretion has been observed to vary with the protein content of the diet (91), which is beyond the scope of this model. Urea excretion is also regulated by ADH (86, 87), which stimulates urea reabsorption in the terminal IMCD, and so we should expect the model urea excretion rate to be elevated in the diuretic kidney. Model excretion rates were 1–6 μeq·min⁻¹·g kwt⁻¹ in the antidiuretic model and 7–11 μeq·min⁻¹·g kwt⁻¹ in the diuretic model (compare Figs. 10 and 11). Model urine urea concentrations were 220–580 mM in the antidiuretic kidney and 70–150 mM in the diuretic kidney. In both cases, the concentration was maximal at low perfusion pressures and decreased in response to elevations in perfusion pressure.

Urea excretion has been measured in various studies in the rat. In male Sprague- Dawley rats an acute elevation of perfusion pressure by 37 mmHg doubled the urea excretion rate from 3.7 to 7.5 μeq·min⁻¹·g kwt⁻¹ (81). In response to acute elevations of around 60 mmHg in male Wistar-Kyoto rats, urea excretion increased from 2.7 to 5.5 μeq·min⁻¹·g kwt⁻¹ (80). In male spontaneously hypertensive rats, the same elevation increased urea excretion from 4.1 to 5.0 μeq·min⁻¹·g kwt⁻¹ (80). Water diuresis in male Wistar rats produced a rapid increase in urea excretion that returned to nondiuretic levels after 6–8 h (5). Osmotic diuresis also produced a rapid increase in urea excretion before decreasing to levels around 50% higher than in the nondiuretic rat (5). Saline diuresis in male Wistar rats elevated the urea excretion rate from 2.7 to 7.5 μeq·min⁻¹·g kwt⁻¹ in dehydrated rats, from 4.0 to 6.7 μeq·min⁻¹·g kwt⁻¹ in “normal” rats, and from 2.7 to 5.4 in hydrated rats (4). In response to ADH infusion in male Wistar rats, it was observed that “plasma urea concentrations were lowest in the diuretic groups, where the high urine flows were associated with the highest urinary urea outputs and urea clearances” (3).

In the antidiuretic model, urea excretion rate was increased from 1.1 to 3.1 μeq·min⁻¹·g kwt⁻¹ in response to acute elevations of 40 mmHg (K_MBF = 0, K_PCT = 0.040) and 60 mmHg (K_MBF = 0, K_PCT = 0.025), similar to the observations in Sprague-Dawley and Wistar-Kyoto rats. Acute elevations in the diuretic model produced smaller fractional increases in urea excretion rate, from 6.6 to 8.4 μeq·min⁻¹·g kwt⁻¹ (40 mmHg) and 9.2 μeq·min⁻¹·g kwt⁻¹ (60 mmHg), comparable to the observations in spontaneously hypertensive rats. At P_RPP = 100 mmHg the urea excretion rates in the antidiuretic and diuretic models were 1.1 and 6.6 μeq·min⁻¹·g kwt⁻¹, respectively, which is comparable to the effects of saline diuresis in dehydrated male Wistar rats (from 2.7 to 7.5 μeq·min⁻¹·g kwt⁻¹). In all model simulations it was observed that increased urine flow was accompanied by decreased urine urea concentration and increased urea excretion, consistent with the behavior observed in Wistar rats undergoing ADH infusion. This suggests that the model urine concentrating mechanism degrades in a realistic manner.

Anatomical considerations.

Juxtamedullary afferent arterioles exhibit efficient blood flow autoregulation in the rat (14, 100), even in response to volume expansion (29, 43, 79, 95). If the DVR inflow rate is determined solely by the efferent flow rate of the juxtamedullary nephron population, this implies that MBF must be well-regulated in diuresis, which appears to contradict experimental measurements of MBF in diuretic rats (32, 33, 65). Roman et al. (82) suggested that the vascular resistance of the DVR might decrease when RPP is elevated. In the model simulations presented here, we have ignored this issue and varied the DVR inflow independently of the juxtamedullary SNGFR, but we can speculate as to how these two quantities may vary independently of each other in vivo.

Preglomerular shunts have been observed in the rat kidney (15), although the role of these vessels remains uncertain (72). If we assume that such shunts exist in a small fraction of the juxtamedullary nephron population and that their endothelium is similar to that of the initial DVR (46), then it seems plausible that such vessels could be closed and admit no flow of plasma or renal blood cells when the circulating levels of ADH are relatively high and that such vessels could be open and admit the flow of plasma and renal blood cells when the circulating levels of ADH are relatively low (69, 71). This would explain how juxtamedullary SNGFR and DVR can be regulated independently and why the rate of MBF is dependent upon the hydration status of the animal.

Conclusions.

The model has been used to predict pressure-sensitive changes in NaCl excretion in response to variations in MBF autoregulation and inhibition of PCT reabsorption. Changes in epithelial permeability, representing physiological effects of volume expansion, were shown to increase model NaCl and water excretion rates, but these increases are insensitive to the perfusion pressure.

The results also suggest that the direct effects of MBF on the interstitial composition cannot stimulate pressure natriuresis. In both the antidiuretic and diuretic models, MBF autoregulation exerted a stronger effect on urine osmolality and flow rate than did the PCT response. This suggests that efficient MBF autoregulation, as observed in the antidiuretic rat, is important for maintaining the whole body hydration status.

Model PCT responses were fitted to produce water and solute excretion rates that matched experimental measurements of pressure natriuresis in the diuretic rat. These responses were consistent with experimental observations of PCT reabsorption inhibition, suggesting that inhibition of PCT reabsorption is sufficient to drive pressure natriuresis to levels observed in vivo. The signaling mechanisms that mediate this inhibition are presumably key factors responsible for stimulating pressure natriuresis.

GRANTS

This research was supported by National Institute of Diabetes and Digestive and Kidney Diseases Grant DK-89066 (to A. T. Layton).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

AUTHOR CONTRIBUTIONS

Author contributions: R.M. and A.T.L. conception and design of research; R.M. performed experiments; R.M. and A.T.L. analyzed data; R.M. and A.T.L. interpreted results of experiments; R.M. prepared figures; R.M. drafted manuscript; R.M. and A.T.L. edited and revised manuscript; R.M. and A.T.L. approved final version of manuscript.