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A 1-D model to explore the effects of tissue loading and tissue concentration gradients in the revised Starling principle

¹Departments of Biomedical and Mechanical Engineering, The City College of The City University of New York, New York, New York; and ²Department of Physiology and Membrane Biology, University of California, Davis, California

Submitted 2 November 2005 ; accepted in final form 24 July 2006

	ABSTRACT

TOP ABSTRACT MODEL DESCRIPTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
The recent experiments in Hu et al. (Am J Physiol Heart Circ Physiol 279: H1724–H1736, 2000) and Adamson et al. (J Physiol 557: 889–907, 2004) in frog and rat mesentery microvessels have provided strong evidence supporting the Michel-Weinbaum hypothesis for a revised asymmetric Starling principle in which the Starling force balance is applied locally across the endothelial glycocalyx layer rather than between lumen and tissue. These experiments were interpreted by a three-dimensional (3-D) mathematical model (Hu et al.; Microvasc Res 58: 281–304, 1999) to describe the coupled water and albumin fluxes in the glycocalyx layer, the cleft with its tight junction strand, and the surrounding tissue. This numerical 3-D model converges if the tissue is at uniform concentration or has significant tissue gradients due to tissue loading. However, for most physiological conditions, tissue gradients are two to three orders of magnitude smaller than the albumin gradients in the cleft, and the numerical model does not converge. A simpler multilayer one-dimensional (1-D) analytical model has been developed to describe these conditions. This model is used to extend Michel and Phillips’s original 1-D analysis of the matrix layer (J Physiol 388: 421–435, 1987) to include a cleft with a tight junction strand, to explain the observation of Levick (Exp Physiol 76: 825–857, 1991) that most tissues have an equilibrium tissue concentration that is close to 0.4 lumen concentration, and to explore the role of vesicular transport in achieving this equilibrium. The model predicts the surprising finding that one can have steady-state reabsorption at low pressures, in contrast to the experiments in Michel and Phillips, if a backward-standing gradient is established in the cleft that prevents the concentration from rising behind the glycocalyx.

endothelial glycocalyx; tight junction; capillary permeability; vesicular transport

(1)

where J_V/A is the fluid filtration flux across the capillary wall per unit area, L_p is the hydraulic permeability, _f is the reflection coefficient, and P_L, P_T, _L, and _T are the global values for the hydrostatic and oncotic pressures in the capillary (L) and interstitial (T) compartments, respectively.

Michel (9) and Weinbaum (13) independently proposed that Starling’s hypothesis should be applied locally, just across the thin endothelial glycocalyx layer (EGL), rather than globally, across the entire endothelial layer, between plasma and interstitium, since the EGL is proposed to be the primary molecular sieve for plasma proteins. The primary difficulty in applying the revised Michel-Weinbaum model is that the local Starling forces behind the EGL, P(0) and (0), are spatially varying and unknown because of the large gradients in velocity and protein concentration that are produced by the presence of the tight junction (TJ) strand in the cleft (see Fig. 1A). Typically <10% of the TJ strand is open, with the result that the streamlines and solute flux lines in the cleft depart greatly from one-dimensional (1-D) flow. This creates a highly nonuniform pressure and concentration field throughout the cleft and an important nonlinearity in the resistance to both water and solute, since the presence of the glycocalyx has a substantial effect on the shape of the streamlines and solute flux lines that pass through the discontinuities (orifices) in the TJ strand. The effect on the hydraulic resistance is clearly seen if one were to remove the EGL. The filtration coefficient L_P would rise sharply, not because the integrated average resistance of the glycocalyx is large (typically 10%) but because the EGL diverts the streamlines that would pass directly through the orifice and the resistance of these streamlines is greatly reduced when the EGL is removed. Similarly, there are steep lateral concentration gradients behind the EGL that lead to a significant drop in average concentration across the TJ strand, which depends on the diffusional resistance of the EGL and its thickness. Thus the EGL and the cleft cannot be viewed as two simple linear resistances in series.

View larger version (27K): [in this window] [in a new window]   Fig. 1. Geometric models of the endothelial cleft and endothelial glycocalyx layer (EGL). A: 3-dimensional (3-D) model. Cleft and glycocalyx geometries are based on results summarized in Table 1. PL and L, hydrostatic and oncotic pressures in the vessel lumen; PT and T, hydrostatic and oncotic pressures in the interstitium; LF, thickness of the glycocalyx; L1, depth of the TJ strand from luminal cleft entrance; L2, distance from TJ strand to the abluminal cleft exit; L, mean value of total cleft depth; 2d, mean gap length; 2D, mean gap spacing. B: 1-dimensional (1-D) model. Region F is the EGL of thickness LF. Region A is the cleft region of depth L1 in front of the TJ strand. Region TJ is the tight junction of depth LTJ. Region B is the cleft region of depth L – L1 – LTJ behind the TJ strand. Region T is the tissue space of depth LT and width 2H.

The 3-D model in Fig. 1A is far too complicated for most investigators to conveniently use and requires considerable computer time for the numerical solutions to converge. It is also hard to obtain numerical convergence when there are small concentration gradients in the tissue, since there is a 500-fold expansion in the area from the cleft exit to the tissue. To overcome these shortcomings, we have formulated a simpler model in which each of the five regions in Fig. 1B is described by a 1-D convection-diffusion equation that can be solved analytically, subject to simplified interface-matching conditions. Although the velocity and concentration fields are nonlinearly coupled, the unknown constants that appear in the solutions for each region are linearly coupled in all but the Starling equation for the pressure and oncotic force balance across the EGL. This enables one to reduce the entire boundary value problem to a system of algebraic equations for the unknown constants, which is easily solved on the computer. The key simplification is the conversion of the TJ strand with its periodic orifices to a single continuous slit, which provides the same transport area for solutes as the orifice-like breaks and whose resistance to both water and solute is adjusted to provide the same L_P and P_d as in the 3-D model. This simplified model allows one to obtain solutions that closely mimic the full 3-D solutions for J_V/A and the net Starling forces across the EGL.

In the pioneering study by Michel and Phillips (10) on individually perfused frog microvessels, these researchers concluded that, at steady state, no reabsorption is possible at low pressure, and they developed a simplified 1-D model to explain this behavior. This model treats the entire endothelium as a uniform fiber matrix layer without clefts where the tissue is at a uniform concentration (C_i) given by the ratio of the total solute flux J_S/A to water flux J_V/A. The latter condition is approximately satisfied in their experiments because the vessels are occluded, the lumen pressure is nearly uniform, and there are no solute sources in the tissue from vesicular transport or leakage through the mesothelium, as in the case of equilibrating the tissue with superfusate. Levick (6) has shown that, for many tissues, this equilibrium value of C_i in the tissue is 0.4 the lumen concentration (C_L). In this paper, we examine what would happen if the tissue were equilibrated at this concentration. The model predicts the unexpected result that one can obtain steady-state reabsorption at low pressure if the far field is clamped at this average tissue concentration. The model also predicts the lumen pressure where the transition from steady-state filtration to reabsorption occurs and shows that a standing concentration gradient conveying solute toward tissue is established in the cleft and that the latter prevents the rise in concentration behind the EGL, which would cut off reabsorption.

A second major contribution of the present paper is that the new model includes vesicular transport and predicts the rise in the far-field tissue concentration as a function of the ratio of paracellular to transcellular transport. These effects are explored as a function of albumin permeability and the diffusion coefficient of the EGL.

Glossary

A_i

Area of region i in the cleft per unit cleft length; i = F, A, TJ, B, T (nm)

C_i

Albumin concentration in region i; i = F, A, TJ, B, T (mg/ml)

C

Albumin concentration difference across the semipermeable or leaky barrier (mg/ml)

C_i

Average albumin concentration drop across region i; i = A, TJ, B (mg/ml)

D_i

Albumin diffusion coefficient in region i; i = F, A, TJ, B, T (cm²/s)

D_c

Albumin diffusion coefficient in the wide part of the cleft (cm²/s)

D_TJ

Effective albumin diffusion coefficient for the TJ strand in the 1-D model (cm²/s)

G_i1, G_i2

Constants to describe the concentration distribution in region i, i = F, A, TJ, B, T (mg/ml)

J_V/A

Water flux (nm/s)

(J_S)_i

Albumin flux per cleft in region i; i = F, A, TJ, B, T (mg·ml^–1·nm·s^–1)

J_S/A

Albumin flux per unit area of vessel wall (mg·ml^–1·nm·s^–1)

(J_S,D)_i

Albumin diffusive flux per cleft in region i; i = A, TJ, B (mg·ml^–1·nm·s^–1)

K

Coefficient to describe the magnitude of vesicular transport (cm/s)

L

Cleft depth (nm)

L_p

Hydraulic permeability (cm·s^–1·cmH₂O^–1)

P_L

Hydrostatic pressure in the lumen (cmH₂O)

P_T

Hydrostatic pressure in the tissue (cmH₂O)

P_TJ1

Hydrostatic pressure at the TJ entrance (cmH₂O)

P_TJ2

Hydrostatic pressure at the TJ exit (cmH₂O)

P(0)

Average hydrostatic pressure behind the EGL (cmH₂O)

P(0)_c

Corrected hydrostatic pressure at the cleft entrance (cmH₂O)

R

Extra resistance that is required to describe the change in the streamline pattern due to the presence of the EGL (nm·s^–1·cmH₂0)

u_i

Water velocity in region i; i = F, A, TJ, B, T (nm/s)

v(x)

Local velocity in a parabolic profile through a narrow channel (nm/s)

Partition coefficient

_L

Oncotic pressure in the lumen (cmH₂O)

_T

Oncotic pressure in the tissue (cmH₂O)

(0)

Average oncotic pressure behind the EGL (cmH₂O)

_f,i

Albumin reflection coefficient of region i; i = F, A, TJ, B, T

_C

Albumin reflection coefficient in the cleft, same for regions A, TJ and B

µ

Fluid viscosity (Pa·s)

	MODEL DESCRIPTION

TOP ABSTRACT MODEL DESCRIPTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
The 3-D theoretical model for rat mesentery microvessels shown in Fig. 1A is the same as in Ref. 1 except that the cross-bridging proteins in the cleft have been omitted for simplicity. These linker proteins could be cadherin molecules in the adherens junction, which Shulze and Firth (12) observed just distal to the TJ strand. There could also be linker proteins in other parts of the cleft that do not have an ordered quasi-periodic appearance. One expects that neither the periodically spaced cadherins (15 nm spacing) in the adherens junction (12) nor nonordered linker proteins provide significant extra resistance to water or protein transport since they occur in localized regions. Figure 1A is a view along the TJ strand in the centerplane of the cleft.

Figure 1B is the simplified 1-D model constructed from Fig. 1A. This view is at right angles to Fig. 1A and shows the cleft in cross-section with its gap height (2h). Note that 2h << 2H, the average spacing between clefts or the average width of the endothelial cell. The model contains five regions. The first region, region F, is an EGL whose thickness is L_F, and the structure is the same as in the 3-D model in Fig. 1A. The EGL covers the entire endothelial surface, including the entrance to the intercellular cleft. The second region, region A, is a cleft entrance region on the lumen side of the TJ strand whose depth is L₁. The third region, region TJ, is a continuous narrow slit of height 2b, which replaces the TJ strand with discontinuities of width 2d and height 2h in the 3-D model; 2b is chosen to be 2h·d/D. This allows the pore area for albumin diffusion to be conserved and the average velocity of the water in the narrow continuous slits to be the same as in the orifice breaks in the 3-D model when L_P is the same in both models. The equating of L_P also accounts for the difference in fluid streamline path lengths in regions A and B of the two models. In contrast to the TJ strand in the 3-D model, which is treated as a zero-thickness layer, the TJ in Fig. 1B has a finite depth, L_TJ, so that it will provide a finite hydraulic resistance in the 1-D model. L_TJ is determined by requiring that the total hydraulic resistance of the cleft be the same as predicted by the 3-D model. The fourth region, region B, is the cleft on the tissue side of the TJ strand whose depth is L – L₁ – L_TJ. The fifth region, region T, is the tissue space of depth L_T and height 2H, the average distance between clefts.

In the 3-D model, the tissue space is divided into two subregions, a near field and a far field. The near field is a region within 5 µm of the cleft exit where the exit jets from the individual junction orifices and adjacent clefts merge with each other and form a uniform flux along the length of the cleft exit. The description of the far field depends on the tissue-loading conditions described at the end of this paragraph. In the 1-D model, we have a continuous narrow slit for the TJ strand, and thus one does not have to deal with the complication introduced by the mixing of discrete exit jets. There is an expansion of the flow from a cleft height 2h to the cleft spacing 2H, but, because L_T >> 2H, this mixing can be neglected. Similarly, the short mixing regions at the interfaces between each cleft region are neglected in the 1-D model. Thus each region can be approximated by a 1-D convection-diffusion equation. At each interface, we assume that the albumin concentration and albumin flux per unit cleft length are continuous. Three different tissue loading conditions are analyzed. The first is a modified Michel and Phillips (10) model in which the tissue concentration is uniform and given by the ratio of the total solute flux to the total water flux. The second is the tissue loading model in Ref. 1 in which the mesothelium is damaged 100 µm from the vessel wall and the tissue concentration is set equal to the superfusate concentration at this location. In both cases, the total water and solute fluxes pass through the paracellular pathway. This loading condition is also an approximation for the far-field concentration that is observed in many tissues, where C_i 0.4 C_L (6). In this case, we require that the tissue concentration approaches this value at 100 µm from the vessel wall. The third loading condition is the model proposed in Refs. 9 and 11 in which transcellular vesicle fluxes of varying strength are added in parallel to the paracellular pathway to see the effect on elevating the tissue concentration and back-diffusion into the cleft.

	METHODS

TOP ABSTRACT MODEL DESCRIPTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
The structural parameters and transport properties for the 1-D and 3-D models for rat mesenteric microvessels are summarized in Tables 1 and 2. All of the other intermediate variables are summarized in the Glossary.

Hydraulic resistance. To convert from a 3-D to a 1-D model, we first have to examine in more detail the effective hydraulic resistance that results from placing a uniform resistance barrier, the EGL, in front of a highly nonuniform cleft with discrete orifice-like breaks in its TJ strand. The key insights can be obtained by analyzing the pressure field for the filtration flow in the 3-D model shown in Fig. 1A. The structural parameters for the cleft are taken from Ref. 1 and are summarized in Table 1. The only unknown in the 3-D model, if the EGL thickness L_F is prescribed, is K_P, which is the Darcy permeability for the EGL. K_P is then determined by satisfying the measured value of L_P =1.3 x 10^–7 cm·s^–1·cmH₂O^–1 for rat mesenteric microvessels (1). Using the structural parameters for the cleft in Table 1 and L_F =150 nm, one finds that K_P = 9.24 nm².

The detailed pressure and velocity vector profiles for pure filtration for the 3-D model are shown in Fig. 2. These profiles scale linearly with P_L. The structural parameters describing the cleft and TJ strand are summarized in Table 1. L_P = 1.3 x 10^–7 cm·s^–1·cmH₂O^–1. The pressure behind the EGL is spatially nonuniform. At the centerline, about one-half of the pressure drop occurs across the EGL and one-half occurs in the cleft, whereas at locations toward the edges, y = ±D, of the periodic unit, the pressure drop across the EGL vanishes and nearly the entire pressure drop occurs across the TJ strand, see Fig. 2A. The pressure drop across the EGL averaged over the entire cleft length is only 10% of the total pressure drop across the endothelial layer. If one naively applies the average pressure behind the EGL as the entrance condition for the cleft, as shown in Fig. 2C, one obtains a much larger water flux than in Fig. 2A (0.034 vs. 0.019 µm/s for P_L = 15 cmH₂O). This difference arises because the fluid streamline patterns in the cleft for the pressure entrance conditions in Fig. 2, A and C, are dramatically different, as observed in the velocity vector profiles at the cleft entrance, x = 0, in Fig. 2, B and D. The 3-D model predicts that the removal of the 150-nm EGL, i.e., using P_L as the entrance condition to the cleft, will lead to a near doubling in L_P. This doubling indicates that approximately one-half of the total pressure drop should occur across the EGL in the region above the orifice. The pressure profiles in Fig. 2A show that this is indeed the case and that the presence of the EGL has doubled the hydraulic resistance on these central streamlines, which provide most of the flow.

View larger version (16K): [in this window] [in a new window]   Fig. 2. 3-D model predictions for pressure in the cleft for pure filtration. The structural parameters describing the cleft and TJ strand are summarized in Table 1. KP = 9.24 nm2. LP = 1.3 x 10–7 cm·s–1·cmH2O–1. LF = 150 nm. Shown are pressure profiles at the cleft entrance (x = 0), in the lumen and tissue sides of the TJ strand (x = L1) when there is an intact 150-nm EGL (A) and when there is no EGL (C). Note that, when the average P(0) from A is applied as the entrance condition for the cleft as in C, the water flux is almost twice that in A (0.019 vs. 0.034 µm/s for PL =15 cmH2O). Velocity vectors at the cleft entrance represent velocity profile at x = 0 when there is an intact 150-nm EGL (B), corresponding to conditions in A and when there is no EGL (D), corresponding to conditions in C. Note that the EGL causes a major shift in the entrance velocity profile, with streamlines being diverted away from the TJ break.

(2)

where u_F is the average velocity in the EGL over the cleft entrance, P(0) is the average pressure behind the EGL, shown in Fig. 2B, and R is the extra resistance that is required to describe the change in the streamline pattern due to the presence of the EGL and µ is the fluid viscosity. The results in Figs. 2 and 3 show that for L_F = 150 nm

(3)

since L_P in Fig. 3 is reduced by a factor of two for this EGL thickness.

View larger version (6K): [in this window] [in a new window]   Fig. 3. 3-D model predictions for LP as a function of LF. Predicted LP are based on the measured structural parameters described in the text and listed in Table 1 for rat mesentery. KP = 9.24 nm2.

Diffusional resistance. Similar to the approach just described for pure filtration, in which we require the hydraulic resistance in the 1-D and 3-D models to be the same, we also require that the two models have the same diffusional resistance in the pure diffusion limit. In this limit, there is no convective flow, and we obtain solutions for the 3-D model that parallel the calculations for the hydraulic permeability, L_P, shown in Fig. 3 for the pure filtration model.

To our knowledge, there are no direct measurements of the diffusion coefficient for albumin in the EGL of rat mesenteric microvessels. Hu and Weinbaum (5) and Hu et al. (4) estimated its value for frog mesenteric microvessels by requiring that their 3-D model predictions provide an optimum fit of the steady-state filtration profile obtained in Michel and Phillips’ experiment (10). They predicted that D_F would need to be 0.001 D_L (P_d 1 x 10^–7 cm/s), if there was to be a sharp bend in the steady-state J_V/A curves in the vicinity of _fL (see Fig. 6). This bend is clearly observed in the experiments in Ref. 10.

View larger version (18K): [in this window] [in a new window]   Fig. 4. 3-D model predictions for pure diffusion. The structural parameters describing the cleft and TJ strand are summarized in Table 1. LP = 1.3 x 10–7 cm·s–1·cmH2O–1. DF = 0.03% DL. Shown are concentration profiles in the lumen, at the cleft entrance (x = 0), in the lumen and tissue sides of the TJ strand (x = L1), and at the cleft exit when there is an intact EGL whose thickness is 30 nm (A), 150 nm (B), and 500 nm (C). D: ratio of the average concentration drop across the TJ, CTJ, to the average concentration drop across the entire cleft, Ccleft, as a function of LF.

View larger version (12K): [in this window] [in a new window]   Fig. 5. Comparison between the predictions of 1-D and 3-D models for steady-state water flux for modified Michel and Phillips model. LP = 1.3 x 10–7 cm·s–1·cmH2O–1. Pd = 0.5 x 10–7 cm/s. The structural parameters in Table 1 are applied. The theoretical predictions from both the 1-D model and the 3-D model correspond closely to the analytic solution in Michel and Phillips (10).

View larger version (15K): [in this window] [in a new window]   Fig. 6. Comparison between the predictions of 1-D and 3-D models for steady-state water flux when tissue is isotonically loaded with superfusate. LP = 1.3 x 10–7 cm·s–1·cmH2O–1. Pd = 0.5 x 10–7 cm/s. The structural parameters in Table 1 are applied. The theoretical predictions from both the 1-D model and the 3-D model correspond more closely to the experimental measurements in Ref. 1 than to the classic Starling relation. The 1-D and 3-D model predictions show good agreement.

Michel and Phillips (10) suggested that the EGL is the primary resistance to albumin transport. The 3-D pure diffusion model predicts that, for an EGL whose thickness is 150 nm, 75% of the total diffusional resistance across the entire endothelial layer resides in the EGL if P_d = 0.5 x 10^–7 cm/s, whereas only 56% of the total diffusional resistance resides in the EGL if P_d = 1 x 10^–7 cm/s and D_F =0.1% D_L. For this reason, we have chosen P_d = 0.5 x 10^–7 cm/s for the present calculations, unless otherwise indicated.

Figure 4, A–C, shows the dimensionless albumin concentration profiles in the lumen at the cleft entrance, on the lumen and tissue sides of the TJ strand, and at the cleft exit for L_F = 30, 150, and 500 nm when D_F = 0.03% D_L. One finds that a substantial portion of the concentration drop occurs across the EGL when L_F 150 nm. Less obvious is the relation between the average concentration drop across the TJ and the EGL thickness shown in Fig. 4D, where we have plotted the ratio of the average concentration drop across the TJ, C_TJ, to the average concentration drop across the entire cleft, C_cleft, as a function of the EGL thickness. These results are obtained from our 3-D model for pure diffusion where the local concentration behind the EGL is treated as unknown and determined as part of the coupled boundary value problem for the EGL and cleft. Note that the EGL thickness has a very significant influence on the lateral concentration gradients in the protected region in front of the TJ strand.

1-D Model

Water velocity. In the 1-D model, continuity of water flux at the interface between each region requires that

(4)

(5)

(6)

(7)

where u_i (i = F, A, TJ, B, and T) is the 1-D average velocity. Note that u_F is the velocity in the EGL averaged over the entrance area of the cleft, including the fiber and void phases. Therefore

(8)

Depth of tight junction. The ultrastructural measurements in Ref. 1 show that 9% of the TJ strand is open (2d = 315 nm vs. 2D = 3,590 nm). To satisfy the condition 2b = 2h·d/D, which preserves the TJ transport area, we find that the slit height 2b is 1.58 nm or 9% of the cleft height, 2h = 18 nm.

The average velocity for a parabolic laminar flow in a channel of depth L_TJ and height 2b is

(9)

where P_TJ1 and P_TJ2 are the pressures at the entrance and exit of the TJ slit.

Likewise, in regions A and B

(10)

(11)

where P_T is the pressure in the tissue. We assume that P_T =0.

For pure filtration without albumin in the lumen or in the tissue, one has

(12)

(13)

L_C is the length of cleft per unit area of vessel wall. Therefore, L_C·2h is the area of cleft per unit area of vessel wall. L_C = 1,000 cm of cleft length per cm² vessel wall for rat mesenteric microvessels (1).

Solving Eqs. 3 and 8–13 when L_P = 1.3 x 10^–7 cm·s^–1·cmH₂O^–1, one finds that L_TJ = 1.49 nm. One notices that the depth of region A, L₁, for the 1-D model is the same as in the 3-D model and that because the TJ strand depth, L_TJ, is much smaller than the cleft length, the depth of region B in the 1-D model, L – L₁ – L_TJ, is close to its value in the 3-D model.

One notes that a small channel of height 2b = 1.58 nm and depth L_TJ = 1.49 nm, which satisfies the measured L_P = 1.3 x 10^–7 cm·s^–1·cmH₂O^–1 in rat mesentery, will not allow the passage of albumin (diameter 7 nm). However, in reality, the pore in the TJ strand is 18 nm high and 315 nm wide, and our narrow continuous TJ slit is an artificial construct that preserves the TJ area for albumin diffusion. The full cleft height 2h is used to estimate the reflection coefficient in the TJ strand and cleft regions A and B in the next section.

1-D convection-diffusion. The governing equation for 1-D convection and diffusion in the EGL or any cleft region is given by

(14)

where i = F, A, TJ, or B. Solving Eq. 14 one finds

(15)

where G_i1 and G_i2 are unknown constants that can be determined by applying the boundary and matching conditions in all the regions.

Therefore, the solute flux across any region i per unit cleft length is

(16)

where A_i is the cross-section area of each region and can be replaced by just the height of each region. The reflection coefficient for albumin in the EGL, _f,F, is abbreviated as _f. A typical value is 0.94 for rat mesenteric microvessels (1). One assumes that, in the 1-D model, the reflection coefficients for albumin in the cleft, _f,A = _f,TJ = _f,B, since in a real cleft in rat mesenteric microvessels the height of the TJ is the same as that in the wide part of the cleft. To estimate its value, _C =_f,A = _f,TJ = _f,B, we use the approximate formula for steric exclusion of a solute of radius a in a channel of half height h (3)

(17)

where _C is the retardation coefficient in the cleft and v(x) is the local velocity in a parabolic profile. Equation 17 is the definition of the reflection coefficient for filtration defined in Ref. 3. For circular pores and parallel-walled channels and spherical rigid solutes, it is the same as the reflection coefficient for sieving (2). This definition has recently been shown to be true for a periodic fiber matrix in Ref. 14. The upper limit of integration (h – a) for the solute flux arises from the steric exclusion due to the finite solute size. After evaluating the integrals in Eq. 17, one finds that

(18)

where is the partition coefficient, describing the ratio between the area available to solute and that available to water molecules. Using Eq. 18, one finds that, for albumin, where a = 3.5 nm and h = 9 nm, _C is 0.197.

In summary, in the EGL

(19)

(20)

in region A

(21)

(22)

in region TJ

(23)

(24)

and in region B

(25)

(26)

The diffusion coefficients for regions A and B in the 1-D model are the same as those in the wide part of the cleft in the 3-D model. These coefficients are described by the restricted diffusion of a spherical molecule in a parallel-walled channel. The effective diffusion coefficient for the artificial TJ construct in the 1-D model will be described later using the 3-D solutions shown in Fig. 4.

The governing equation in the tissue is

(27)

Solving Eq. 27 one finds

(28)

where G_T1 and G_T2 are unknown constants that can be determined by the boundary and matching conditions. The solute flux leaving the cleft is

(29)

Boundary and matching conditions.
MODIFIED MICHEL AND PHILLIPS MODEL At the entrance of the EGL, the albumin concentration is the same as the lumen concentration. Therefore,

(30)

and from Eqs. 8 and 19

(31a)

At the interfaces between each region, the solute flux per unit cleft length is continuous. Therefore,

(31b)

(31c)

(31d)

At the interfaces between each region, the concentration is the same. Therefore,

(31e)

(31f)

(31g)

At the cleft exit, the concentration is the ratio of the albumin to water flux. Therefore,

(32)

Using Eqs. 22 and 25, one obtains

and from Eqs. 31c and 31d

(33)

There are eight unknown constants, G_i1 and G_i2 (i = F, A, TJ, and B), and four unknown water velocities, u_i (i = F, A, TJ, and B). The four water velocities can be related to each other by Eq. 8; therefore, they can be expressed in terms of only one unknown velocity, which we choose to be u_A. Thus there are nine unknowns. However, there are only eight equations, Eqs. 31a–31g and 33. For the ninth equation, we apply the Starling principle across the EGL

(34)

The oncotic force behind the glycocalyx, (0) (cmH₂O), is related to the albumin concentration, C_A(0) (mg/ml), by the nonlinear empirical relation in Eq. 4 of Ref. 8. One notes that Eq. 4 of Ref. 8 is for 35°C (rabbit knee experiments). This can be corrected to 37°C for our rat experiments by the Van’t Hoff equation. C_A(0), in turn, can be expressed as a function of G_A1, G_A2, and u_A.

The water flow in regions A, TJ, and B can be approximated by Poiseuille channel flow, whose velocities are described by Eqs. 10, 9, and 11, respectively. Solving Eqs. 8-11, one obtains

(35)

One can relate P(0) and P(0)_C with the use of Eq. 2. Therefore, there are two new unknowns, P(0) and P(0)_C, but three more equations, Eqs. 2, 34, and 35. In total, there are 11 unknowns: G_i1 and G_i2 (i = F, A, TJ, and B), one water velocity, u_A, and two pressures, P(0) and P(0)_C; there are 11 equations (Eqs. 2, 31a–31g, and 33-35). Solving all of these 11 equations simultaneously, one obtains all the unknowns.

ADAMSON ET AL. (1): TISSUE EQUILIBRATING WITH SUPERFUSATE. In our model for tissue equilibrating with superfusate described in Ref. 1, the boundary and matching conditions in the lumen and in the cleft described by Eqs. 31a–31g still apply. However, the tissue is equilibrated with superfusate 100 µm from the vessel wall, where the mesothelium is damaged, and we assume the tissue concentration is the same as in the superfusate. At the cleft exit, the albumin flux and concentration are continuous. Thus

(36a)

(36b)

At the tissue loading site, the albumin concentration is prescribed and equal to that in the superfusate C_i,

(37)

and from Eqs. 8 and 28

(36c)

The matching conditions for the water flux at the cleft entrance and in the cleft (Eqs. 2, 34, and 35) still apply. Therefore, there are 13 unknowns: G_i1, G_i2, (i = F, A, TJ, B, and T), u_A, P(0) and P(0)_C; there are 13 equations: Eqs. 2, 31a–31g, 34, 35, and 36a–36c. Solving these 13 equations simultaneously, one determines all of the unknowns.

COMBINED TRANSENDOTHELIAL-PARACELLULAR TRANSPORT. In the case of combined transendothelial and paracellular transport, the boundary and matching conditions in the lumen, at the cleft entrance, at the TJ entrance and exit (described by Eqs. 2, 31a–31g, 34, 35, and 36b) are still valid. At the cleft exit, the concentration is continuous, but the flux is not, since there is a source for albumin due to the transcellular pathway. Therefore, at the cleft exit

(38a)

Using Eqs. 8, 26, and 29, one finds that

(39a)

where the vesicular flux is described by the second term in Eq. 38a. In this term, K is a coefficient, which depends on the vesicular volume that passes from the plasma to the interstitium per unit time and the partition of solute molecules in the vesicular plasma. Note that, in Eq. 38a, the transcellular flux can cross all along the basal membrane of the endothelial cell whose width is 2H, the average separation of the clefts. Because 2H >> 2h, the height of the cleft is neglected in estimating the basal dimension of the transendothelial pathway.

In general, the tissue concentration in the steady state is given by

(38b)

and from Eqs. 8, 28 and 39a

(39b)

There are 13 unknowns: G_i1, G_i2 (i = F, A, TJ, B, and T), u_A, P(0), and P(0)_C; there are 13 equations: Eqs. 2, 31a–31g, 34, 35, 36b, and 39a–39b to determine all of the unknowns.

Effective diffusion coefficient, D_TJ, for the TJ strand. The 1-D model is unable to describe the lateral concentration gradients in front of and behind the TJ strand (see Fig. 4, A–C). Therefore, an effective diffusion coefficient, D_TJ, needs to be chosen for the 1-D model that can correct for the added diffusional resistance of the TJ strand. D_TJ is determined by requiring that the diffusional resistance of the cleft in the 1-D model be the same as that in the 3-D model in the pure diffusion limit. Therefore, we require in the 1-D model that

(40)

(41)

and

(42)

where J_(S,D)TJ, J_(S,D)A, and J_(S,D)B are the albumin diffusive fluxes across the TJ strand and across regions A and B and C_TJ, C_A, and C_B are the corresponding average concentration drops across the TJ strand and regions A and B, respectively. Because J_(S,D)A = J_(S,D)TJ = J_(S,D)B,

(43)

Therefore,

(44)

and

(45)

In Eq. 45, C_cleft is the concentration drop across the entire cleft and C_TJ/C_cleft is obtained from Fig. 4D for each EGL thickness.

	RESULTS

TOP ABSTRACT MODEL DESCRIPTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
1-D Model Prediction for Water Flux

Figure 5 compares the 1-D and 3-D predictions for the modified Michel and Phillips model with the analytical solution in Ref. 10 (cleft neglected) for steady-state filtration in which the tissue concentration is set by the condition that C_T = J_S/J_V and P_d = 0.5 x 10^–7 cm/s. In this model, where all water and solutes pass through a paracellular pathway, the J_V/A curve bends sharply near _fL, and there is no steady-state reabsorption even at very low lumen pressure. The L_P used in Fig. 5, 1.3 x 10^–7 cm·s^–1·cmH₂O^–1 (1), is satisfied by both the 1-D and 3-D models. Figure 5 shows that the J_V/A curve for the present 1-D model corresponds closely with the Michel and Phillips model in Ref. 10, where the cleft is neglected, at all pressures. The J_V/A curve for the 3-D model is in close agreement with both 1-D models at pressures below 40 cmH₂O but diverges slightly at higher pressures. However, the maximum differences are <10%. This result shows that the resistance to water and solute fluxes under the EGL can be accounted for using additional hydraulic and diffusional resistances and that the methods to average the oncotic and hydrostatic pressure distributions in the 3-D model are well described by the 1-D model.

A second comparison of the 3-D and 1-D model predictions for J_V/A is shown in Fig. 6 for the case where the albumin concentration in the superfusate is set equal to the albumin concentration of individually perfused rat mesenteric microvessels. The mesothelium was damaged at a distance of 100 µm on both sides of the perfused microvessels to allow the albumin concentrations in the interstitial space of the mesentery to rise to the superfusate concentration of 50 mg/ml at this site. A steady-state filtration is then set up at a series of capillary pressures. Here, the predicted water flux is determined not only by the resistance of the EGL/TJ break size and distribution but by the interaction between albumin concentration gradients in the tissue and those in the interendothelial cleft under the EGL. The results in Fig. 6 also show that the 1-D model closely approaches the 3-D model at low pressures and has a maximum deviation at all pressures of <10%. The excellent agreement at low pressures is due to our requirement that both models have the same limiting behavior in the case of pure diffusion.

Figure 7 shows the most important new result predicted by the 1-D model. The striking result is for the case where the albumin interstitial concentration is clamped at 20 mg/ml instead of 50 mg/ml at a distance of 100 µm from the vessel wall. As already noted, this albumin interstitial concentration is broadly representative of interstitial fluid protein concentrations found in many tissues in Ref. 6. There is steady-state reabsorption when P_L is 17 cmH₂O or lower. Our detailed concentration profiles presented in the next section show that this behavior will occur if diffusional gradients of albumin in the cleft behind the EGL, which are accompanied by very small concentration gradients in the interstitial space, overwhelm the tendency of albumin to accumulate behind the EGL as it is carried up to the base of the EGL by convective flows during reabsorption. It is important to stress that the shallow gradients responsible for the maintenance of the albumin flux away from the base of the EGL and in the tissue are best described by the analytic solution for diffusion in the 1-D model. It is difficult and inefficient to describe these gradients by numerical solutions of the 3-D model because the length scales for diffusion in the cleft (tens of nm) are so much smaller than the length scales for diffusion in the tissue (tens of µm). The dashed curve in Fig. 7 shows the range over which the 3-D numerical solutions converge. One notes that the deviation between the 1-D and 3-D models is <10% in this range.

View larger version (11K): [in this window] [in a new window]   Fig. 7. Comparison between the predictions of 1-D and 3-D models for steady-state water flux when tissue is loaded with superfusate at 20 mg/ml 100 µm from the vessel wall. Pd = 1.3 x 10–7 cm·s–1·cmH2O–1. Pd = 0.5 x 10–7 cm/s. The structural parameters in Table 1 are applied. The theoretical predictions for steady-state water flux of 3-D and 1-D models are in close agreement.

Figure 8 shows the theoretical predictions for the concentration profiles for our 1-D tissue model in which the interstitial albumin concentration is clamped at 50 mg/ml (Fig. 8A) and at 20 mg/ml (Fig. 8B) when P_d = 0.5 x 10^–7 cm/s. The convective and diffusive fluxes at several key interfaces in the cleft are summarized in Tables 3 and 4. As observed in the experiments of Adamson et al. (1), where C_L = 50 mg/ml and the superfusate is maintained at the same concentration, our 1-D model predicts that the albumin concentration 5 µm from the vessel wall is relatively insensitive to the filtration rate (or the lumen pressure). Our model predicts that, at P_L = 60 cmH₂O, the albumin concentration at x = 5 µm is 0.8 C_i. However, the convective flux of solute through the TJ slit exceeds the backward diffusive flux, with the result that the concentration behind the EGL is only 0.1 C_i (5 mg/ml). When the superfusate concentration at the tissue-loading site is reduced to 20 mg/ml, one observes in Table 4 a crossover in behavior at P_L of 17 cmH₂O. For P_L > 17 cmH₂O, there is filtration across the EGL, whereas for P_L < 17 cmH₂O, there is reabsorption. At low lumen pressures, the concentration at the back of the EGL does not rise high enough to arrest the reabsorption (see Fig. 8B). One notices that the concentration gradient in the tissue is very small (Fig. 8B, inset) and that the relative magnitudes of the diffusive and convective fluxes of albumin in the tissue are comparable at both low and high lumen pressures in Table 4. From the modeling point of view, the transition from a net diffusion gradient into the cleft region in Fig. 8B when the pressure is above 19 cmH₂O and P_d = 0.5 x 10^–7 cm/s to a net diffusion gradient out of the cleft at lower pressures is poorly described by numerical methods in the 3-D model. The analytic solutions shown in Fig. 8B provide a more accurate description of these gradients. Additional calculations have been performed for P_d = 1 x 10^–7 cm/s (results not shown). There is a small shift in the value of P_L for reabsorption to occur from 17 to 15 cmH₂O.

View larger version (23K): [in this window] [in a new window]   Fig. 8. Concentration profiles predicted by tissue loading model. LP = 1.3 x 10–7 cm·s–1·cmH2O–1. Pd = 0.5 x 10–7 cm/s. The entire solute flux passes through the paracellular pathway. The tissue is equilibrated with superfusate at 50 mg/ml (A) and at 20 mg/ml (B). A: albumin concentration at the cleft exit is high and relatively insensitive to the filtration rate (or the lumen pressure). B: superfusate concentration is 0.4 CL (6). When the lumen pressure is low, the concentration at the back of the EGL does not rise high enough to arrest the reabsorption.

View larger version (13K): [in this window] [in a new window]   Fig. 9. 1-D model prediction for the convective and diffusive solute fluxes at the cleft exit as a function of lumen pressure when there is a tissue sink Ci = 20 mg/ml at x = 100 µm.

Figure 10 shows the theoretical predictions for the concentration profiles for the modified Michel and Phillips model when P_d =0.5 x 10^–7 cm/s. For all the lumen pressure conditions, there is always a very small concentration gradient in the cleft, 2% of the concentration gradient across the EGL or less. At high lumen pressures, P_L = 40 or 60 cmH₂O, the concentrations at the back of the EGL (the cleft entrance) and in the cleft approach the convection limit, (1 – _f)C_L and are insensitive to the lumen pressure. This result simply reflects the main assumption in this model that the determinants of the concentration distal to the EGL are only the paracellular fluxes of water and solutes and that tissue concentration gradients are negligible. The flat profiles in the cleft in Fig. 10 reflect the fact that there are no diffusion gradients back into the cleft when there is no exchange of water and solutes across the mesothelium or into any sink in the tissue such as a lymphatic. At P_L+ = 25 cmH₂O, a typically average capillary pressure, the albumin concentration behind the EGL is 7 mg/ml. The nonlinear effects of convection are dominant for this low value of P_d. This convective effect substantially reduces the albumin concentration behind the EGL at the higher filtration pressures.

View larger version (14K): [in this window] [in a new window]   Fig. 10. Concentration profiles predicted by modified Michel and Phillips model. LP = 1.3 x 10–7 cm·s–1·cmH2O–1. Pd = 0.5 x 10–7 cm/s. The entire solute flux passes through the paracellular pathway. There is a very small concentration gradient in the cleft when C = 0.197.

With the use of our 1-D model for the combined transendothelial-paracellular transport model, the magnitude of the vesicular flux, K, is determined by requiring that at P_L = 25 cmH₂O, a typical average capillary pressure, the tissue concentration, C_T, is 20 mg/ml, or 40% of the lumen concentration as estimated in Ref. 6 for a number of tissues. One finds that K = 3.43 x 10^–7 cm/s when P_d = 0.5 x 10^–7 cm/s. Equation 38b requires that the solute in the tissue is well mixed so that there is no diffusive flux in the tissue. For this value of C_T and P_d = 0.5 x 10^–7 cm/s, the vesicular albumin flux is 80% of the total flux through the combined transendothelial-paracellular pathway.

	DISCUSSION

TOP ABSTRACT MODEL DESCRIPTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
There is a growing body of evidence in support of the Michel-Weinbaum model for a re