INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

THE ENDOTHELIAL CELL GLYCOCALYX is an extracellular matrix that is expressed on the luminal surface of the endothelial cells forming the microvessel wall. This matrix is believed to be composed primarily of proteoglycans, glycoproteins, and glycosaminoglycans (14, 18, 19, 21). Because of its distinct location in the transvascular pathway, in conjunction with the intercellular junctions in the cleft between adjacent endothelial cells, the surface glycocalyx is of great importance in determining the microvessel permeability (P) to water and solutes.

Adamson et al. (2) showed that for similar size globular proteins, -lactalbumin (molecular weight = 14,176) and ribonuclease (molecular weight = 13,683), the permeability of the frog mesenteric capillary to positively charged ribonuclease (net charge = +3, including the charge effect from fluorescent probe labeling; P^ribonuclease = 4.3 × 10⁶ cm/s) was twice that of negatively charged -lactalbumin (net charge = 11, including the charge effect from fluorescent probe labeling; P^-lactalbumin = 2.1 × 10⁶ cm/s). Their experiments suggested that the microvessel wall contains negative charges, which enhance the transport of positively charged molecules but retard that of negatively charged molecules. With the use of a Donnan-type model (see below) for electrostatic partitioning, they estimated that the charge density in the microvessel wall was ~11.4 meq/l.

This charge effect of the microvessel wall has also been shown in other experiments. Curry et al. (6) measured P^ribonuclease and P^-lactalbumin in microvessels perfused with orosomucoid in a Ringer-albumin perfusate. They found that P^ribonuclease was six times that of P^-lactalbumin in the presence of orosomucoid. In the presence of orosomucoid, P^-lactalbumin was only about one-half of the value in the absence of orosomucoid. They suggested that these results could be accounted for if orosomucoid increased the net negative charge on microvessel walls in the frog mesentery from 11.2 to 28 meq/l. Huxley and Curry (16) showed that the diffusive solute permeability to -lactalbumin was lower during exposure to plasma [(P^-lactalbumin)^plasma = 1.0 × 10⁶ cm/s] than that during exposure to bovine serum albumin (BSA)-Ringer solution [(P^-lactalbumin)^BSA = 5.0 × 10⁶ cm/s]. Huxley et al. (17) further showed that (P^-lactalbumin)^plasma/(P^-lactalbumin)^BSA = 0.31, whereas there was no change in hydraulic conductivity (L_p). They concluded that the actions of plasma were to confer charge selectivity for anionic solutes and modify the porous pathways of the microvessel wall to a lesser extent. With the use of the same model as in Refs. 2 and 6, they predicted an increase in charge from 11.2 meq/l in the presence of albumin to 34 meq/l in the presence of plasma.

In another line of investigation into the mechanism of decreasing P^-lactalbumin by plasma protein, Adamson and Clough (1) tried to test the hypothesis that plasma protein may modulate surface glycocalyx structural properties. With the use of cationized ferritin staining, they found that the total glycocalyx thickness in the presence of plasma was twice the value of that with BSA-Ringer perfusion. Their interpretation for this was that the increase in the thickness of surface glycocalyx layer is the result of a change in the orientation of surface glycoproteins to which cationized ferritin binds.

Previously, a simple Donnan-type model was proposed to describe the charge effect on microvessel permeability (2, 6, 16, 17). It was based on a Donnan equilibrium distribution of ions, which exists as a result of retention of negative charges on the capillary membrane. It was suggested (8, 9) that the steric and electrostatic exclusions be described in terms of an effective partition coefficient (_eff)

(1)

Here, _steric is the steric partition coefficient describing the size selectivity of the membrane, E is the effective Donnan electrical potential difference across the membrane, Z is the charge on the solute, R is the universal gas constant, F is Faraday's constant, and T is temperature. RT/F is 25.2 mV at 20°C. is the dimensionless electrical potential difference and is equal to EF/RT. With the use of this model, the fixed negative charge in the transport pathway in the frog mesentery (C_m) was estimated to be ~11 meq/l (2, 6, 17). Although this model described the steric and electrostatic partition to a charged solute at the interface between the membrane and the solution, it neglected the thickness of the membrane and thus neglected the steric and electrostatic interactions between the solutes and the membrane components within the membrane.

A more sophisticated model for steric and diffusion resistance to solute transport in the fiber matrix was proposed by Weinbaum et al. (22). With the use of this theory for the entrance fiber layer, Fu et al. (10-13) developed a three-dimensional model for the interendothelial cleft to describe solute exchange across the microvessel. Whereas this model could successfully explain the size-restricted transport of a solute through the surface glycocalyx and the interendothelial cleft, it did not consider the electrical charge factors of the glycocalyx layer and the solute. Therefore, it can only be applied to describe the transvascular transport of electroneutral molecules. Recently, an electrochemical model was proposed by Stace and Damiano (7, 20) for the transport of charged molecules through the capillary glycocalyx. However, this model did not consider transport through the cleft region.

In the current study, we attempted to develop a two-dimensional model incorporating both size and charge effects so that it will provide, for the first time, a quantitative analysis of various experimental results expected to be associated with negative charges in transvascular pathways. Compared with the model in Fu et al. (13), this model features two new characteristics: 1) the surface glycocalyx contains a negative electric charge, and 2) there is an interface between the surface glycocalyx layer and the cleft entrance (15). This model will help to better understand both the physical and electrochemical mechanisms of selectivity in the endothelial surface glycocalyx layer and therefore provide a new method for controlling transport rates of charged or uncharged molecules in drug delivery.

	MODEL DESCRIPTION

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

Model Geometry

View larger version (26K): [in this window] [in a new window]   Fig. 1.   A: plane view of junction-orifice-fiber entrance layer model of interendothelial cleft. Junction strands with periodic openings lie parallel to the luminal front. Ljun, junction strand thickness; L1 and L3, distances between the junctional strand and luminal and abluminal fronts, respectively; 2D, spacing between adjacent breaks in the junctional strand. At the entrance of cleft on the luminal side, the surface glycocalyx is represented by a periodic square array of cylindrical fibers. a, Radius of these fibers; , gap spacing between fibers; Lf, thickness of the glycocalyx layer. The charge density in the glycocalyx layer is Cm(x). C'i(x) and Ci(x) (x = Lf, 0) are the concentrations of charged ions/molecules from the fiber side and from the lumen/cleft side, respectively, at the interfaces between the fiber layer and the lumen/cleft entrance. E'(x) and E(x) (x = Lf, 0) are the corresponding electrical potentials at the interfaces. B: three-dimensional sketch of single periodic unit of width 2D showing a large orifice of width 2d and height 2B and a narrow slit of height 2bs in the junction strand (revised from Ref. 11).

View larger version (17K): [in this window] [in a new window]   Fig. 2.   Hypothesized charge density profiles Cm(x) in the surface glycocalyx layer. The thickness of the glycocalyx layer under the normal condition is Lf = 100 nm (1, 10-13).

Mathematical Model

Entrance fiber matrix layer. As shown in Fig. 1, the glycocalyx (fiber matrix) layer lies in front of the cleft and covers the entire endothelial surface. The solution in the vessel lumen consists of monovalent cations (C₊) and monovalent anions (C) as well as a small amount of protein (e.g., albumin). The volume flux and flux of solute i are denoted by J_v and J_i, respectively. The concentrations of solute i and the electrical potential within the fiber matrix layer are denoted as C'_i(x) and E'(x). At the interface of the vessel lumen and the fiber layer, C_i(L_f) and E(L_f) represent the concentration and electrical potential at x = L_f from the lumen side and C'_i(L_f) and E'(L_f) from the fiber side. At the interface of the fiber layer and the cleft entrance, C_i(0) and E(0) represent the concentration and electrical potential at x = 0 from the cleft side and C'_i(0) and E'(0) from the fiber side.

(2)

(2a)

7

1

1

6

(3)

(3a)

(3b)

(3c)

(4)

(5)

(6)

Cleft region. Because there is no charge in cleft regions 1 and 3, and, in our case, Pe in the cleft [Pe = K_i,cJ_vL/D_i,c = (K_i,cL_pp/L_jt2B) × L/D_i,c] is in the order of 10² [the retardation coefficient (K_i,c) = 0.99, the diffusion coefficient of the solute in the cleft (D_i,c) = 0.68 × 10⁶ cm²/s, and the cleft depth (L) = 400 nm; Refs. 10, 13, and 22] for the solute of radius 2.01 nm, the governing equation for solute transport in the cleft region can be approximated by a steady two-dimensional diffusion equation averaged over the cleft height (11-13)

(7)

C, where j = 1, 3, is the concentration in regions 1 and 3 of the cleft. Boundary conditions for Eq. 7 are

(7a)

(7b)

(7c)

(7d)

Equation 6 is the interface boundary condition, which represents the conservation of mass from the fiber region to the cleft region. Boundary condition Eq. 7a shows the continuity across the junction break, and Eq. 7b indicates the impermeability of the rest part of the junction strand. Equation 7c indicates that concentration is a constant, C_iA, at the tissue side of the cleft. Equation 7d is the symmetric boundary condition. To obtain the solution of Eq. 7 with boundary conditions Eq. 7, a-d, and interface condition Eq. 6, we first found (0) and e^Zi'(x)dx in Eq. 6 by solving modified Nernst-Planck equations for monovalent ion concentrations in the fiber layer. This process is shown in the APPENDIX. A numerical method similar to that in Hu and Weinbaum (15) was applied to solve for C(x, y) in cleft regions 1 and 3. Finally, averaged _i(0) =  C_i(0, y)/2D dy was substituted into Eq. 5 for C_i(0, y) to obtain C(x) in the fiber region. For the case presented in Hu and Weinbaum (15), due to the high filtration pressure (43 cmH₂O), the high plasma oncotic pressure (26 cmH₂O), and larger molecule albumin (radius = 3.55 nm), Pe is highly nonuniform along the y direction behind the surface fiber layer (x = 0). It can be as high as the order of 1 over the break region ( y  < d). Therefore, the contribution from filtration cannot be neglected in their case. However, in our case, the filtration pressure is <10 cmH₂O, the plasma oncotic pressure is ~5 cmH₂O (2, 10, 16, 17), and the largest Pe in the junction break is only ~0.3 for a solute of radius 2.01 nm (11). Neglecting the convection in our case is reasonable. In another work, we will present the convection effect as well as the charge effect when high filtration pressure occurs.

(8)

(9)

Parameter Values

Cleft and fiber layer geometry. Figure 1 shows the three-dimensional model for the interendothelial cleft and the charged surface fiber layer. Model parameters are determined according to experimental data for frog mesenteric capillaries (1, 3), which are the same as those in Fu et al. (13). The total cleft length L = 400 nm. The junction strand is in the middle of the cleft and its thickness, L_jun, which is in the order of 10 nm, can be neglected compared with L. Therefore, L₁ = L₃ = L/2 = 200 nm. The cleft height 2B = 20 nm. The large junction break width 2d = 150 nm, and the average spacing between the adjacent breaks is 2D = 2,640 nm. Because all of the charged solutes in the current study have a diameter of 4.02 nm, which is larger than the width of the small slit 2b_s ~ 2 nm, the small slit is impermeable to these solutes. The total cleft length per unit area (L_jt) = 2,000 cm/cm². In the entrance fiber matrix layer, both periodic and random fiber arrays were examined. We used fiber radius a = 0.6 nm and gap spacing  = 7 nm if periodic fiber arrays exist or volume fraction of fiber matrix S_f = 0.11 if random fiber arrays exist. The values of  or S_f lead to a diffusion coefficient of a solute with radius r_s = 2.01 nm in the fiber matrix of D_i,f = 0.025 × 10⁶ cm²/s (13, 22).

Properties of ions and charged solutes. Two solutions were considered for ion concentration in the vessel lumen. The first was Ringer solution, whose composition was (in mM) 111 NaCl, 2.4 KCl, 1.0 MgSO₄, 1.1 CaCl₂, 0.195 NaHCO₃, 5.5 glucose, and 5.0 HEPES (pH = 7.4). Ringer solution with 10 mg/ml BSA (molecular weight = 69,000) was the perfusate in the experiment (2, 5, 10, 16). At pH 7.4, the molecular charge of the albumin is about 19 (8). The charge density of albumin is 0.144 mM × 19 = 2.7 meq/l, which is negligible compared with the ion charge density (in the order of 100 meq/l). Thus, in Ringer solution, the cation concentration is taken as the same as that of anions: C₊ = C = 118 M.

5

5

6

6

6

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

Electrical Potential Profiles

View larger version (14K): [in this window] [in a new window]   Fig. 3.   Profiles of electrical potential E(x) in the surface glycocalyx layer for the corresponding charge density Cm(x) in Fig. 2. Electrical potential at x = Lf from the lumen side E(Lf) = 0 is the reference point. Case 1 is when Cm(x)/Cm0 = 1, case 2 is when Cm(x)/Cm0 = tanh (1 + x/Lf)/ tanh (1), case 3 is when Cm(x)/Cm0 = tanh (x/Lf)/tanh (1), and case 4 is when Cm(x)/Cm0 = tanh (2 + 2x/Lf)/ tanh (1), Lf < x < Lf/2; Cm(x)/Cm0 = tanh (2x/Lf)/ tanh(1), Lf/2 < x < 0. Cm0 = 25 meq/l for all cases. A: ion concentrations in the lumen C+ = 118 mM and C = 118 mM; B: C+ = 155 mM and C = 138 mM.

In Fig. 3B, the perfusate in the lumen is plasma, in which C₊ = 155 mM and C = 138 mM. These unequal ion concentrations induce a large difference in electrical potential profiles, although the charge density distributions are the same as those shown in Fig. 3A. For case 1 of constant C_m = 25 meq/l, E decreases suddenly from 0 to 0.7 mV at x = L_f, does not change across the entire fiber layer, and suddenly increases from 0.7 to 1.5 mV at x = 0. For case 2, in which C_m(L_f) = 0, there is a sudden increase in E at x = L_f from 0 to 1.5 mV; E then decreases gradually across the fiber layer to 0.7 mV at the fiber layer exit x = 0 and finally jumps to 1.5 mV at the entrance of the cleft x = 0⁺. For case 3, in which C_m(L_f) = C_m0= 25 meq/l, there is a sudden decrease in E at x = L_f from 0 to 0.7 mV; E then gradually increases across the fiber layer to 1.5 mV at x = 0. For case 4, in which C_m(L_f) = 0, there is a sudden increase in E from 0 to 1.5 mV at x = L_f; E then gradually decreases to 0.7 mV at x = L_f/2 and increases to 1.5 mV at x = 0. The difference between the maximum and minimum in E is the same again for all the cases due to the same C_m0. However, the difference is 2.2 mV, when C₊ = 155 mM and C = 138 mM, instead of 2.7 mV, when C₊ = 118 mM and C = 118 mM.

As shown earlier, charges can affect molecular transport in two ways: 1) through the ion migration term in solute flux expression (the second term in Eq. 2), and 2) through electrostatic partitioning of charged solutes at interfaces (Eq. 4). For case 1 of constant C_m, dE'/dx = 0 (or d'/dx = 0), the charge effect on the solute transport comes only from the electrostatic partition at two interfaces of the fiber layer (x = L_f and x = 0). This partition is the same at both interfaces when C₊ = C = 118 mM, whereas the partition is larger at x = 0 when C₊ = 155 mM and C = 138 mM in the lumen. For all cases of C_m = C_m(x), dE'/dx  0 (or d'/dx  0), the charge contributes to the solute migration term in Eq. 2. Under the condition of C₊ = C = 118 mM, there is no partition at both interfaces for case 4, whereas there is a partition at x = L_f for case 3 and a partition for case 2 at x = 0. Under the conditions of C₊ = 155 mM and C = 138 mM, there is a partition at x = L_f for cases 3 and 4 and partitions at two interfaces for case 2.

Concentration Distributions of Solutes

View larger version (30K): [in this window] [in a new window]   Fig. 4.   Dimensionless concentration distributions C(x)/CLumen in the surface glycocalyx layer for a positively charged molecule, ribonuclease (+3), a neutral solute (0), and a negatively charged molecule, -lactalbumin (11) for the four charge density distributions Cm(x) shown in Fig. 2. Cm0 = 25 meq/l for all cases. A: C+ = 118 mM and C = 118 mM; B: C+ = 155 mM and C = 138 mM.

View larger version (18K): [in this window] [in a new window]   Fig. 5.   Ratio of microvessel permeability for charged molecules (ribonuclease and -lactalbumin) to that for a neutral solute with the same size (P/Pneutral) as a function of charge density Cm0 in the fiber matrix layer. Solid lines with circles and dashed lines with triangles are P/Pneutral for constant Cm. Dotted lines with squares and dashed-dotted lines with diamonds are P/Pneutral for varied Cm in the fiber layer, which are the same for all three distributions of Cm(x) shown in Fig. 2. A: C+ = 118 mM and C = 118 mM; B: C+ = 155 mM and C = 138 mM.

Figure 4, top middle, bottom middle, and bottom, show the cases when C_m = C_m(x), corresponding to those in Fig. 2. In these cases, dE'/dx no longer equals to zero, and there is a contribution from dE'/dx to the transport of the charged solutes within the fiber layer in addition to electrical partitions at the entrance and exit of the fiber layer. In cases 2 and 4, where C_m(L_f) = 0, under condition a, when C₊ = C = 118 mM, there was no electrical partition at x = L_f; the concentration is the same as that in the lumen for all solutes with and without charges. However, under condition b, when C₊ = 155 mM and C = 138 mM, the electrical partition at x = L_f favors the transport of negatively charged -lactalbumin instead of positively charged ribonuclease due to the positive jump in the electrical potential at x = L_f (Fig. 3B). In cases 3 and 4, where C_m(0) = 0, there was no electrical partition at x = 0. Although concentration profiles in the fiber layer are varied for various C_m(x), the concentration at x = 0, the exit of the fiber layer, or the entrance of the cleft, is the same for all cases. Under condition a, the exit concentration for ribonuclease is 0.69; for -lactalbumin, it is 0.49. Under condition b, the exit concentration is 0.57 for ribonuclease but 0.96 for -lactalbumin. The higher the exit concentration, the lower the resistance of the fiber layer to a solute. Because the resistance of the cleft region is the same for ribonuclease, -lactalbumin, and a neutral solute with the same size, the total resistance or permeability of the microvessel wall to a solute is the same for all C_m(x) shown in Fig. 2.

Charge Effect of Surface Glycocalyx Layer on Permeability

View larger version (19K): [in this window] [in a new window]   Fig. 6.   Ratio of permeability of positively charged ribonuclease to that of negatively charged -lactalbumin (Pribonuclease/P-lactalbumin) as a function of charge density Cm0 in the fiber layer. Solid lines with circles and dashed-dotted lines with diamonds are Pribonuclease/ P-lactalbumin for constant Cm. The total charge is the same for these lines, with Cm0 halved and Lf doubled for dashed-dotted lines. Dotted lines with triangles are Pribonuclease/P-lactalbumin for varied Cm, which are the same for all three distributions of Cm(x) shown in Fig. 2. Dashed lines with squares are Pribonuclease/P-lactalbumin from a previous Donnan-type model. A: C+ = 118 mM and C = 118 mM; B: C+ = 155 mM and C = 138 mM.

Comparison of Previous and Current Models

-lactalbumin

Combined Effect of Charge and Thickness of the Glycocalyx Layer

-lactalbumin

View larger version (20K): [in this window] [in a new window]   Fig. 7.   Dimensionless permeability of -lactalbumin (P/PLf = 100 nm, Cm0 = 0) as a function of fiber layer thickness Lf and fiber charge density Cm(x). Cm(x) can be any one of three distributions for varied Cm in Fig. 2. A: C+ = 118 mM and C = 118 mM; B: C+ = 155 mM and C = 138 mM.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL DESCRIPTION RESULTS DISCUSSION APPENDIX REFERENCES

The combined junction-orifice-fiber matrix model in Ref. 13 demonstrated previously that the endothelial surface glycocalyx and the junction strands in the interendothelial cleft provided significant size restriction to the diffusion of intermediate-sized solutes across the walls of continuous microvessels of the frog mesentery. The results of our current model demonstrate that both solute charge and charge carried by the surface glycocalyx also determine the selectivity of the microvessel wall. The model predictions conform to the hypothesis that under previous experimental conditions, the fixed negative charge within transcapillary pathways for intermediate-sized proteins restricts the transport of negatively charged solutes more, relative to positively charged solutes of the same size. However, this effect is counteracted to some extent if the solution on the luminal side contains negatively charged proteins that cannot enter the layer.

From Fig. 6A, to account for the twofold difference in P^ribonuclease and P^-lactalbumin observed in Ref. 2 under Ringer-BSA perfusion, C_m0 would be ~25 meq/l for the case of charge density C_m = constant (case 1 in Fig. 2) and C_m0 would be ~35 meq/l for cases of varied C_m = C_m(x) (cases 2-4 in Fig. 2). These values are significantly larger than 11 meq/l, the prediction from the previous electric partition-only model (2). Under the condition of orosomucoid perfusion, the increased ratio of P^ribonuclease to P^-lactalbumin from two to six and the decrease in P^-lactalbumin to 0.47 of P^-lactalbumin under Ringer-BSA perfusion (6) could be explained if C_m0 increased from ~25 to ~55 meq/l for case 1 and from ~35 to ~75 meq/l for cases 2-4, a roughly twofold increase. The prediction of the previous model was from 11 to 28 meq/l (6). The change in P^-lactalbumin by plasma, to 0.31 of P^-lactalbumin under Ringer-BSA perfusion, observed in Huxley et al. (17) could be accounted for if C_m0 increased from ~25 to ~60 meq/l for case 1 and from ~35 to ~80 meq/l for cases 2-4. The previous model predicted that C_m0 was increased from 11 to 34 meq/l. Because of the thickness of the glycocalyx layer was ignored, the simple Donnan-type model used previously may overestimate the charge effect of the fiber matrix.

Figure 4 shows the dimensionless concentration distribution in the glycocalyx layer for ribonuclease with a net charge +3, -lactalbumin with a net charge 11, and a neutral solute with the same size. When C₊ = C = 118 mM, for cases 1 and 3, there is a charge partition at the entrance of the glycocalyx layer (x = L_f), which favors the passage of positively charged ribonuclease; there is no charge partition for cases 2 and 4. When C₊ = 155 mM and C = 138 mM, the charge partition at the entrance of the glycocalyx layer still favors the transport of ribonuclease in cases 1 and 3; however, in cases 2 and 4, the partition favors the negatively charged -lactalbumin. The reason for this is that the electrical potential E across the interface (x = L_f or x = 0), which determines the charge partition, not only depends on charge density C_m(x) of the glycocalyx layer but also on the concentrations of cations C₊ and anions C in the lumen and cleft (Fig. 3). This dependence of E on C_m(x), as well as C₊ and C, induces the dependence of the solute permeability P on the solute charge and the charge density C_m(x) of the glycocalyx, as well as C₊ and C, which brings us the interesting phenomenon shown in Fig. 5B.

Despite the various concentration distributions (Fig. 4) due to different profiles of charge density C_m(x) within the glycocalyx layer, the concentrations of charged solutes are the same at the entrance of the cleft behind the glycocalyx layer for cases 2-4, provided that these density profiles have the same maximum value C_m0 and the same total charge (Fig. 2). For this sake, the overall solute permeability across the microvessel wall including the surface glycocalyx layer and the cleft region is identical regardless of C_m(x) profiles. If we keep the same maximum value C_m0, the larger the total charge of fibers, the larger the charge effect. In contrast, for the same total charge, the larger the C_m0, the smaller the charge effect. These results are shown in Fig. 6.

Figure 5 shows the ratio of P^ribonuclease to P^{neutral solute} and the ratio of P^-lactalbumin to P^{neutral solute} as a function of C_m0, the maximum value of charge density in the glycocalyx layer. When proteins exist in the plasma, there are unequal concentrations of cations (Na⁺) and anions (Cl) in the lumen, C₊ = 155 mM and C = 138 mM (4). Figure 5B shows that when C_m0 < 35 meq/l for constant C_m, and C_m0 < 55 meq/l for varied C_m(x), the permeability P of negatively charged -lactalbumin is higher than that of positively charged ribonuclease, although the surface glycocalyx carries negative charges. As discussed above, this phenomenon is due to the dependence of P not only on solute charge and charge density of the fiber matrix C_m(x), but also on C₊ and C_. In particular, if C_m0 is small, the negative charge in the layer is not enough to overcome the favored electrical partition to negatively charged -lactalbumin due to the presence of negatively charged proteins in the lumen. This prediction may be used in controlled drug delivery by locally modulating C_m(x), C₊, and C for a certain drug with fixed charge.

In plasma perfusion compared with albumin perfusion, the thickness of the glycocalyx is greatly increased (1). We tested the combined effect of charge and thickness of the glycocalyx on P^-lactalbumin in Fig. 7. When charge density C_m0 is in the range of 25-35 meq/l, as predicted by our model during albumin perfusion, the twofold increase in fiber thickness L_f from 100 nm (albumin perfusion) to 200 nm (plasma perfusion) (1) would only decrease P^-lactalbumin by ~33%, not enough to account for the ~70% decrease observed in Ref. 15. However, if we additionally increase C_m0 by twofold, the combined effect of increasing both charge density C_m0 and the thickness L_f would account for the 70% decrease in P^-lactalbumin.

In summary, we developed a two-dimensional model incorporating the charge effects of the endothelial surface glycocalyx so that it can provide a more detailed quantitative analysis of various experimental results expected to be associated with negative charge in transvascular pathways. This will also help understand the physical mechanisms of glycocalyx selectivity and provide a new method for controlling transport rates of charged solutes in drug delivery. In the current study, we only analyzed the case when the hydraulic conductivity L_p of the microvessel and the pressure difference across the vessel wall p are normal; the convection effect due to enhanced L_p and p will be discussed in our next study.