INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

SEVERAL STUDIES HAVE SHOWN that albumin is present in the medullary interstitium in significant concentrations (11, 12, 15). The mechanisms by which this extravascular pool of albumin is generated and maintained have long remained uncertain. Radiolabeled albumin injected in the medullary interstitium is rapidly cleared (12), but it is unclear how. Lymphatics are absent in the inner medulla and sparse in the outer medulla, and the possibility of clearance of albumin by drainage via prelymphatic channels is not supported by experimental evidence (12).

The most likely mechanism is that of protein clearance by the microcirculation itself. Pallone (14) suggested descending vasa recta (DVR) as the source of interstitial proteins and postulated that accumulation of albumin in the interstitium results from convective transport processes. Because the reflection coefficient of DVR to albumin is higher than that of ascending vasa recta (AVR), it is possible in principle to maintain steady fluxes of albumin from DVR through the interstitium to AVR (12). Wang and Michel (25) recently developed a model of microvascular exchange of fluid, plasma proteins, and small solutes among DVR, AVR, and the medullary interstitial fluid (ISF) to examine this hypothesis. Their results suggest that convection may indeed be the main mechanism by which plasma proteins are transported from DVR to AVR via the interstitium.

Their model, however, does not distinguish between two parallel transport pathways in DVR that differ significantly. The first one consists of aquaporin-1 (AQP1) water channels, which are present in DVR only and are impermeable to all solutes; transcellular volume fluxes across water channels cannot, therefore, carry albumin by solvent drag into the ISF. The second route, the paracellular pathway, appears to have a reflection coefficient to small solutes that is close to zero (16) and to favor water transport from the ISF toward the lumen in most parts of the medulla (4). In those regions where the paracellular flux is directed toward the lumen, the convective transport of albumin from DVR to the interstitium is not possible, even though there is overall volume efflux from DVR. The objective of this work was to reexamine the mechanisms of albumin exchange with a model that accounts for the presence of two separate transport pathways in DVR as well as for concentration polarization.

We first evaluated albumin concentration differences between the bulk interstitium and the interstitial side of vasa recta walls (i.e., immediately adjacent to the capillaries) to calculate accurately the driving forces for transcapillary transport. We then used conservation equations in the interstitium to determine both interstitial protein concentrations and the processes by which proteins are transported across vasa recta (i.e., diffusion and/or convection from AVR to DVR or vice versa). Because the latter mechanisms appear to vary according to the values of vasa recta permeability to proteins and reflection coefficient to small solutes and macromolecules, corresponding parameter sensitivity studies are conducted.

Glossary

Aim	Cross-sectional area of inner medulla
Aint	Cross-sectional area of inner medullary interstitium
AVR	Ascending vasa recta
C, C	Interstitial albumin concentration in bulk and at the vasa recta wall, respectively
C, C, C	Concentration of solute i in plasma, red blood cells, and interstitium, respectively
Chb, C	Molar and molal concentrations of hemoglobin in red blood cells, respectively
D	Vessel diameter
Di	Diffusivity of solute i
DVR	Descending vasa recta
f	Fractional volume of distribution of urea in red blood cells
fp	Fraction of capillary surface occcupied by pores
FVR	Fraction of the inner medullary cross-sectional area occupied by vasa recta
Hi	Hydrodynamic hindrance factor for diffusive transport of solute i
IM	Inner medulla
INT	Interstitium
ISF	Interstitial fluid
Ji	Paracellular molar flux of solute i across capillaries
Jv, J	Volume fluxes across capillaries and red blood cell membranes, respectively
Jvp, Jvt	Paracellular and transcellular volume fluxes across capillaries, respectively
Juc, J	Carrier-mediated transcapillary molar flux of urea, and molar flux of urea across red blood cell membranes
l	Capillary pore length
L	Length of renal medulla
Lim	Length of inner medulla
Lp, Lt	Hydraulic conductivities of paracellular and transcellular pathways, respectively
LR	Hydraulic conductivity of red blood cell membrane
N	Number of vasa recta
Nv	AVR-to-DVR number ratio
OM	Outer medulla
P, PI	Hydraulic pressure in plasma and interstitium, respectively
Pe	Peclet number
Pi	Permeability of capillary wall to solute i
Puc, Pur	Permeability of urea transporter in capillary wall and red blood cell membrane, respectively
QB	Blood flow rate
QP	Plasma flow rate
QR	Red blood cell flow rate
r	Radius
rp	Capillary pore radius
ri	Radius of solute i
RBC	Red blood cell
v	Fluid velocity in interstitium
Xim	Dimensionless inner medullary coordinate, based on length of inner medulla
W	Half-width of slit pore
	Solute distribution coefficient
	Red blood cell-to-vessel surface area ratio
i	Activity coefficient of solute i
i	Oncotic pressure due to solute i
i	Reflection coefficient of the paracellular pathway to solute i
v, Na, u	Generation rate of volume, sodium, and urea, respectively, per unit area of interstitium
a	Albumin
hb	Hemoglobin
Na	Sodium
pr	Plasma protein
ss	Small solute (sodium and urea)
u	Urea

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Mathematical Model

Conservation and transport equations in vasa recta plasma. If x is the axial coordinate along the corticomedullary axis, changes in the plasma flow rate (Q^P) in DVR and AVR at steady state are given by the following equation, based on mass conservation

(1)

where J_v and J are the volume fluxes (per unit membrane area) across the capillary wall and the RBC membrane, respectively,  is the cell-to-vessel surface area ratio, N denotes the number of vessels and D their diameter, and + and  apply to AVR and DVR, respectively. J_v is the sum of two contributions, the paracellular (J_vp) and transcellular (J_vt) volume fluxes, which are given by

(2a)

(2b)

where L_p and L_t represent the hydraulic conductivities of the paracellular and transcellular pathways, respectively, P is the transcapillary hydraulic pressure difference, _a and _pr are the transcapillary oncotic pressure differences due to albumin and all plasma proteins, respectively, and _a is the reflection coefficient of the paracellular pathway to albumin. The plasma and interstitial concentrations of solute i are denoted by C and C, respectively; _i is the activity coefficient of i, and _i is the reflection coefficient of the paracellular pathway to i. Note that reflection coefficients are taken to be one for the solute-impermeable transcellular pathway and that J_vt is zero across AVR, where no AQP1 has been found. The oncotic pressures due to albumin and all plasma protein are calculated as, respectively

(3)

where C_a and C_pr are the albumin and total protein concentration, respectively, in grams per deciliter.

(4)

(5)

(6)

(7)

Conservation and transport equations in RBCs. Conservation of mass in RBCs can be expressed as

(8)

where Q^R is the RBC flow rate. If we assume that there is no hydraulic pressure difference across the RBC membrane, J is given by

(9)

where L_R is the RBC membrane hydraulic conductivity, _pr and _hb are the oncotic pressures due to plasma proteins and to hemoglobin in the cells, respectively, and C denotes the RBC concentration of solute i. As described in Edwards and Pallone (5), the oncotic pressure due to hemoglobin in RBCs is calculated as

(10)

where C_hb and C are the molar and molal RBC concentrations of hemoglobin, respectively, and  = 0.75 mg/l is the partial specific volume of hemoglobin.

(11)

(12)

Conservation equations in interstitium. As described in Edwards et al. (4), the deposition of NaCl, urea, and water into the medullary interstitium from the loops of Henle and collecting ducts is simulated with generation rates that undergo spatial variation within the IM interstitium. The interstitial hydraulic pressure (P^I) and small solute concentrations (C and C) are determined by considering that, at any location along the corticomedullary axis, the sum of the fluxes from DVR and AVR, weighted according to their respective surface area, must be equal and opposite to the rate of generation in the interstitium

(13a)

(13b)

(13c)

where A_int is the cross-sectional area of the medullary interstitium (in cm²), and _v, _Na, and _u are the local generation rates of volume, sodium, and urea, respectively, per unit area of interstitium. The latter three terms are taken to be zero in the OM, where in the vascular bundles the exchange of water, sodium, and urea can occur only between vasa recta and interstitium.

(14)

Concentration polarization: annular space model. As water is reabsorbed from the interstitium into AVR, the accumulation of albumin near the AVR wall on the interstitial side, a phenomenon known as concentration polarization, reduces the transcapillary oncotic pressure difference and therefore decreases the driving force for water reabsorption. Conversely, during volume efflux from DVR, the concentration of albumin on the interstitial side of the DVR wall will be lower than that averaged radially over the interstitium; i.e., reverse polarization will occur, leading to a decreased rate of fluid filtration. Polarization (or its reverse) is not expected to be significant within vasa recta, due to the presence of RBCs, which create a circulatory flow that homogenizes plasma concentrations (3).

(15)

(16)

(17)

(18a)

(18b)

(19)

View larger version (10K): [in this window] [in a new window]   Fig. 1.   Idealized representation of vasa recta and interstitium, used to evaluate the effects of concentration polarization at capillary walls.

(20)

Interstitial space size calculations. Because both the number of AVR and their diameters are greater than those of DVR, the average distance between the bulk of the interstitium and the capillary wall (i.e., r_B  r_A in the model described above) is different for AVR and DVR. At every depth and for each type of vessel, r_B is approximated using the equation

(21)

Because we consider only those vasa recta that are destined for the IM, the number of vasa recta in the OM remains constant. In the IM, the number of DVR and AVR is given by (4)

(22)

where F_VR, the fraction of the inner medullary cross-sectional area occupied by vasa recta, and N_v, the AVR-to-DVR number ratio, are taken to be constant and equal to 0.3 and 2.25, respectively (26). With those assumptions, r_B/r_A varies between 1.3 at the OM-IM junction and 2.4 at the papillary tip for DVR and between 1.1 and 1.5 at the same boundaries for AVR. (Note that, even though r_A is constant and A_int decreases along the corticomedullary axis, the number N of vessels decreases more rapidly, which is why the ratio r_B/r_A increases.) In the OM, we assume that r_B/r_A remains constant and equal to its value at the OM-IM junction. In addition, in those simulations of concentration polarization, the diffusivity of albumin in the interstitium, D_a, is estimated to be ten times smaller than that in water (8).

Albumin interstitial concentration. In our previous approaches (4, 6), the interstitial concentration of albumin was taken to be fixed and constant along the corticomedullary axis. The issue of albumin polarization having been addressed, the transport of albumin may now be modeled more rigorously. An interstitial mass conservation equation can be written to determine the concentration of albumin in the interstitium. If there are no interstitial sources or sinks of albumin (such as transcytosis, proteolysis, or lymphatic drainage), the amount of albumin carried from DVR and AVR should counterbalance

(23)

where J_a is the transcapillary flux of albumin. Implicit in Eq. 23 is the assumption that axial transport in the interstitium is negligible, as discussed earlier. The albumin flux can be written as

(24)

where Pe, the Peclet number, is given by Eq. 6. Note that Eq. 24 includes the interstitial concentration of albumin immediately adjacent to the capillary wall, C, which is related to the bulk interstitial concentration, C, as described in Concentration Polarization. At every depth along the corticomedullary axis, as flow rates and concentrations in plasma are determined, Eq. 23, which relates albumin concentrations in vasa recta to C, is solved to determine interstitial albumin concentrations.

Parameter selection. Parameter values for our model are given in Table 1. The hydraulic pressure P is assumed to remain constant in AVR and IMDVR, with fixed values of 7.8 and 9.2 mmHg, respectively. In OMDVR, P is assumed to decrease linearly from 20 to 9.2 mmHg (5). The fraction of the filtered load recovered by IM vasa recta for water, NaCl, and urea is taken as 1, 1, and 40%, respectively; the filtered load is calculated as described in Edwards et al. (4), based on the values of corticomedullary DVR concentrations and whole kidney glomerular filtration rate (GFR) that are given in Table 1. In the baseline case, the interstitial area-weighted generation rate of water decreases linearly between the OM-IM junction and the papillary tip, whereas those of sodium and urea increase linearly and exponentially, respectively (4).

Permeability of vasa recta to albumin. The permeability of AVR to albumin is <10⁵ cm/s and is therefore too low to be measured by present methods (14). If we assume that the paracellular pathway consists of parallel pores of uniform size, estimates of pore dimensions can be obtained using pore theory and other available measurements, and the permeability of vasa recta to albumin may be calculated using pore theory as well. Calculations are made for both cylindrical and slit pores.

(25)

(26)

(27)

(28)

(29)

(30)

7

5

8

7

6

6

Numerical Methods

The set of ODEs expressing mass conservation in DVR and AVR is highly coupled. At each point along the corticomedullary axis, evaluating fluxes across DVR requires that values in the interstitium and in AVR be known, and vice versa. However, the ODEs cannot be simply integrated simultaneously along DVR and AVR, because boundary values for flow rates and concentrations in AVR at the papillary tip are not known until differential equations for DVR have been integrated along the entire axis. Hence, we used the following approach.

An initial guess was made for the profiles in AVR of the nine variables along the entire corticomedullary axis. The set of ODEs (Eqs. 1, 4, 5, 8, 11, and 12) was then numerically integrated along DVR; at each step along the corticomedullary axis, algebraic equations were solved to determine the interstitial hydraulic pressure as well as sodium, urea, and albumin interstitial concentrations (Eqs. 13, a-c, and 23). Once papillary tip values were obtained, the same set of differential equations was numerically integrated back up along AVR, and AVR flow rates and concentration values were updated. This process was iterated until the normalized difference between the current and previous estimates of each variable in AVR at any x was <10⁵. Tests demonstrating mass conservation are described in the APPENDIX.

ODEs were integrated along vasa recta by use of Gear's method, which is a self-adaptive, multistep, predictor-corrector method for stiff ODEs. At each value of x, the system of three or four nonlinear algebraic equations (Eqs. 13, a-c, and 23) was solved using a modified Powell hybrid method. This algorithm, which is a variation of Newton's method, uses finite difference approximations to the Jacobian and avoids large step sizes or increasing residuals (13). Simulations were performed on an Alpha PC64 workstation. Convergence was typically achieved in 5 h.

When the effects of concentration polarization are assessed, the incorporation of Eq. 19 into the simulations of medullary microvascular transport is complicated by the fact that v_A, and hence Pe_a and J_a, are themselves functions of C through the interstitial oncotic pressure term in the paracellular and transcellular volume fluxes (Eqs. 2 and 3). At each integration step along DVR and AVR, we first calculated the volume fluxes on the basis of the bulk interstitial concentration of albumin. The albumin interstitial concentration immediately adjacent to the walls was then determined using Eq. 19, and the volume fluxes were calculated anew on the basis of this value. The latter two steps were iterated until convergence was achieved.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

We first examined the extent to which concentration polarization in the medullary interstitium affects flow rates and concentration profiles in vasa recta; for simplicity, the bulk interstitial concentration of albumin was assumed to be constant and known in those calculations. We then eliminated that hypothesis and used instead conservation equations in the interstitium to determine protein interstitial concentrations and the mechanisms by which proteins are exchanged between vasa recta and the interstitium. In the absence of measurements for certain capillary wall permeabilities and reflection coefficients, parameter sensitivity studies were performed in which a range of possible values was explored.

Albumin Concentration Polarization

Results based on the present model of polarization were compared with those obtained in two cases: 1) concentration polarization and its reverse are negligible (the "no-polarization" hypothesis); and 2) the accumulation of albumin on the interstitial side of the AVR wall is so significant that albumin oncotic pressure differences across that barrier vanish entirely (the no-AVR-_a hypothesis). The bulk interstitial concentration of albumin, C, was kept fixed, either at 3.4 g/dl, as measured by Pallone (15) or at 1 g/dl, as reported by MacPhee and Michel (12). To maintain high osmolalities at the papillary tip, we varied only the spatial distributions of the interstitial area-weighted generation rate of urea. As described in the previous section, the set of differential equations (Eqs. 1, 4, 5, 8, 11, and 12) was numerically integrated along vasa recta to obtain flow rates and concentration profiles in plasma; at each step, the algebraic equations (Eq. 13, a-c) had to be solved to yield interstitial values. When polarization is accounted for, the albumin interstitial concentration at the wall was related to that in the bulk through Eq. 19. Results are shown in Table 2.

Reverse polarization at the DVR wall increases the transcapillary albumin oncotic pressure difference, thereby reducing water efflux from DVR; the rise in sodium and urea concentrations along the corticomedullary axis is therefore less accentuated. Polarization at the AVR wall has the same effect: a reduced _a limits water influx into AVR and, hence, efflux from DVR, since the interstitial water balance must be maintained; sodium and urea concentrations thus remain lower. Consequently, as shown in Table 2, the osmolality at the papillary tip is always overestimated when concentration polarization and its reverse are neglected and systematically underpredicted if _a across AVR walls is omitted.

In the former case, however, the error remains small, <2%, and the lower the C, the smaller the error, because differences between interstitial concentrations in the bulk and near the capillary walls then have less of an effect on oncotic pressure gradients (see Eqs. 2 and 3). If the assumption that _a can be neglected across AVR walls is employed rather than our present approach, the discrepancy can be as high as 15%, suggesting that the no-AVR-_a hypothesis, which we used previously (4), is an overly simplifying assumption.

Given the uncertainty in model geometry and in parameter values such as generation rates and albumin permeability, errors on the order of 2% are not very significant. The annular space model developed here, although based on an idealized representation of the medulla, therefore suggests that the effects of concentration polarization in the renal medulla can be neglected, as they will be in the remainder of this study.

Transport Mechanisms of Plasma Proteins Across Vasa Recta

Paracellular and transcellular volume fluxes. AQP1 water channels in DVR are impermeable to all solutes (18). Because small solutes such as sodium and urea are more concentrated in the medullary interstitium than in DVR, osmotic pressure gradients drive water from DVR toward the interstitium through this transcellular pathway (i.e., J_vt > 0). The reflection coefficient to small solutes of the paracellular pathway (_ss), however, is close or equal to zero (16), so that osmotic pressure gradients have little to no effect on J_vp. Transcapillary protein concentration differences are therefore the dominant driving force across that route, and water moves in the opposite direction through the paracellular pathway, i.e., from the interstitium toward DVR (Eq. 2, a and b).

View larger version (15K): [in this window] [in a new window]   Fig. 2.   Transcapillary volume fluxes across descending (DVR) and ascending vasa recta (AVR) based on the circumference of all vessels (i.e., JvND, as in Eq. 1), as a function of position along the corticomedullary axis, x. L represents the total length of the medulla. The junction between the outer medulla (OM) and the inner medulla (IM) corresponds to x/L = 0.24. The sharp bends at this junction are due to anatomical changes and the sudden reabsorption of water and solutes from the loops of Henle and the collecting duct in the IM. The interstitial concentration of albumin is fixed at 3.4 g/dl. The permeability to albumin of DVR (P) and AVR (P) is taken to be 1 × 107 and 1 × 106 cm/s, respectively, and the reflection coefficient of the paracellular pathways to small solutes (ss) is zero. Because the paracellular flux of volume across DVR is directed mostly toward the capillary lumen, it is unlikely that albumin is carried to the interstitium only by solvent drag from DVR.

Transport of albumin and other plasma proteins. To understand the mechanisms by which albumin appears in the medullary interstitium, albumin concentration in the ISF was then calculated on the basis of interstitial mass conservation (Eq. 23) instead of being specified. The permeability of DVR and AVR to albumin was initially taken as 1 × 10⁷ and 1 × 10⁶ cm/s, respectively; we assumed that _ss = 0, and all other parameters were set to their baseline value (Table 1). The resulting concentration profile is shown in Fig. 3A. Transcapillary fluxes of water and albumin are shown in Fig. 3, B and C, respectively, and Pe values for albumin are given in Fig. 3D. A positive flux of albumin across DVR (or AVR) walls indicates that albumin is carried from DVR (or AVR) into the interstitium, and vice versa. In addition, the greater the absolute value of Pe, the greater the importance of convection relative to diffusion.

View larger version (14K): [in this window] [in a new window]   Fig. 3.   The interstitial concentration of albumin (C) is calculated on the basis of an interstitial balance (Eq. 23), assuming that other plasma protein cannot be exchanged across vasa recta. The permeability to albumin of DVR and AVR is taken to be 1 × 107 and 1 × 106 cm/s, respectively, and ss = 0. Other parameters are those of the baseline case. A: albumin concentration in DVR, AVR, and interstitium, divided by its initial value in DVR at the corticomedullary junction. B: transcapillary volume fluxes across vasa recta, based on the circumference of all vessels. Note that, in the OM vascular bundles, the sum of the fluxes is zero. C: transcapillary albumin fluxes across vasa recta, based on the circumference of all vessels. Because of mass conservation, the fluxes balance each other. D: albumin Peclet number (Pe). After the sign change in the DVR paracellular volume flux near the corticomedullary junction, albumin enters the interstitium by diffusing out of AVR and is then carried by convection into DVR.