INTRODUCTION

TOP ABSTRACT INTRODUCTION THE MODEL EQUILIBRIUM CONFIGURATION LINEARIZED SOLUTION NUMERICAL METHODS RESULTS DISCUSSION SUMMARY APPENDIX REFERENCES

The surface glycocalyx on capillary endothelial cells has been the subject of considerable controversy and conjecture in the recent literature on the microcirculation. The focus of much of this attention has been on the mechanical implications of the glycocalyx on microvascular rheology, specifically in terms of its gross effect on capillary tube hematocrit and apparent viscosity (Klitzman and Duling, 1979; Desjardins and Duling, 1990; Vink and Duling, 1996; Damiano et al., 1996; Pries et al., 1997; Damiano, 1998; Secomb et al., 1998). Very little emphasis, however, has been placed on the possible role of the glycocalyx in determining the electrophoretic mobility of charged molecules within capillaries. In light of recent experimental evidence (Vink and Duling, 2000), it appears as if significant electrostatic interactions arise between the glycocalyx and anionic molecular tracers which dramatically influence transport of the tracers. The potential significance of these findings to microvascular permeability and exchange motivates the present analysis of electrochemical molecular transport through the capillary glycocalyx.

Although the composition and structure of the endothelial-cell glycocalyx are not well characterized, insight into its mechanical and electrochemical behavior can be gained from what is known about some of its possible macromolecular constituents. It appears that these constituents include, but are not limited to, heparan sulfate proteoglycan, chondroitin sulfate proteoglycan, and hyaluronic acid (Desjardins and Duling, 1990; Henry and Duling, 1999). In this way, the endothelial-cell glycocalyx is similar to mucopolysaccharide structures arising in other systems (e.g., articular cartilage, tectorial membrane, etc.). This similarity essentially pertains to the fact that these mucopolysaccharide structures are highly hydrated in an electrolytic solution and are rich in proteoglycan, glycoprotein, and glycosaminoglycan (GAG) aggregates, which contain large numbers of solid-bound fixed negative charges. It also appears likely that the molecular composition of the glycocalyx varies across its thickness, from the endothelial-cell surface to its apical region within the capillary lumen. Henry and Duling (1999) found, through enzymatic reduction of the capillary glycocalyx with hyaluronidase, that hyaluronan (and perhaps other constituents that are cleaved by hyaluronidase) may contribute significantly to the apical glycocalyx. This finding was based on the fact the very large dextran molecules, having molecular weights (MW) >580, and red cells remained excluded by the apical glycocalyx; yet smaller dextran molecules, <145 kDa, permeated significantly into the layer after enzymatic reduction with hyaluronidase. They also reported a marked increase in capillary tube hematocrit after hyaluronidase treatment, suggesting that the permeability of the glycocalyx to blood plasma is strongly dependent upon the presence of those constituents that are cleaved by hyaluronidase (Damiano, 1998; Secomb et al., 1998).

Combining intravital brightfield and fluorescence microscopy of the capillary glycocalyx, Duling and co-workers (Vink and Duling, 1996, 2000; Henry and Duling, 1999) have revealed its surprisingly large in vivo dimension, its unexpected permeability properties, and the tenuous nature of its structure. They have also shown that the illumination used to visualize the layer also results in its eradication if epifluorescent exposure is sustained for >3-5 min. Their approach consists essentially of obtaining two imagesone brightfield image of a capillary using transillumination, and one image of fluorescently labeled tracers in the capillary lumen using epifluorescence illumination an instant later. By subtracting the width of the fluorescent tracer column from the anatomical diameter of the capillary imaged under transillumination, one has a measure of either the instantaneous in vivo thickness of glycocalyx, if the tracers are sufficiently large so as to be excluded by the layer, or the extent of diffusion into the layer of tracers small enough to penetrate the glycocalyx pores. Using this technique, Vink and Duling (1996, 2000) concluded that the in vivo thickness of the glycocalyx was ~0.4-0.5 µm. This represents a much more substantial structure than previous estimates derived from electron microscopy studies, which likely underestimate the thickness due to dehydration of the extracellular matrix that inevitably accompanies tissue fixation. Consequently, on the basis of these electron microscopy studies, estimates of the glycocalyx thickness on capillary endothelial cells were on the order of only 50-100 nm. It is for this reason, perhaps, more than any other, that the glycocalyx has been almost entirely overlooked in matters concerning microvascular rheology, permeability, and exchange.

Because the capillary glycocalyx is at the interface between blood and the luminal endothelial-cell surface, it represents the first barrier to transvascular exchange. It is evident, therefore, that microvascular permeability is dependent upon glycocalyx permeability. To probe this, Vink and Duling (2000) conducted a series of experiments to study glycocalyx permeability within capillaries. They observed that dextran molecules >70 kDa remained excluded from the glycocalyx by virtue of their size for over 3 h, regardless of whether they were labeled with anionic or neutral fluorescent dyes. However, smaller anionic dextrans between 4 and 40 kDa invaded the glycocalyx with size-dependent half-times of between 12 and 60 min, respectively. Even extremely small anionic dyes between 0.4 and 0.6 kDa showed half-times of 11 min. Alternatively, neutral dyes of ~0.4 kDa and neutral dextrans of <40 kDa equilibrated within one capillary transit time. For neutral dextran molecules <40 kDa, the corresponding Fickian diffusion time in plasma over the glycocalyx length scale of ~0.4 µm is <20 ms. Thus, diffusion times for charged molecules could potentially be as much as five orders of magnitude longer than their neutral counterparts. These results suggest an important role for the solid-bound fixed charges of the glycocalyx matrix in capillary permeability.

It is in the midst of this rather unsettled state of affairs that we find ourselves without adequate quantitative explanations for many of these recent experimental findings. In an attempt to close this gap between experimental observation and theoretical understanding, we embark upon an electrochemical analysis of the glycocalyx that is sophisticated enough to address the salient physical phenomena while avoiding contrived specificity. We seek to determine whether a relatively simple electrochemical model of the glycocalyx can account for the disparity in diffusion times between anionic and neutral molecular tracers reported by Vink and Duling (2000). The model assumes that the glycocalyx consists of a multicomponent mixture that includes a fluid constituent (blood plasma), mobile ions (cations and anions), and a solid proteoglycan/glycoprotein/GAG matrix containing fixed negative charges. The negative charges bound to the solid matrix are assumed to have a fixed-charge distribution in the reference configuration given by |z^Fc^F(x, t₀)|, where z^F and c^F are, respectively, the mean valence and concentration distribution associated with the molecular constituents of the glycocalyx. In equilibrium, it is expected that the mobile ions establish a distribution that nearly counterbalances the fixed charges on the solid matrix such that a state of electroneutrality exists throughout the vessel, except for a slight departure localized near the apical glycocalyx, i.e., near the interface between the glycocalyx and vessel lumen. When integrated over the vessel cross section, however, these local charge imbalances should cancel such that global space-charge neutrality exists within the capillary. Therefore, throughout the vessel lumen where there is no glycocalyx, the concentration distributions of mobile anions and cations should be equal. The mobile ions in this region can be thought of as a neutral salt (Lai et al., 1991), which has no net effect on the total charge density within the capillary. However, near the glycocalyx interface, the mobile cation concentration is expected to increase to nearly neutralize the fixed negative charges on the glycocalyx, while the mobile anion concentration should decrease. These concentration gradients in the mobile ion distributions must be supported in equilibrium by the electric field generated by the glycocalyx. As we shall see, this electric field exerts its effect on the diffusing anionic molecular tracers by partially excluding them from the glycocalyx. The degree to which this exclusion occurs is primarily dependent upon the molecular tracer valence and glycocalyx fixed-charge density.

In what follows, a constitutive relationship derived from Fick's and Ohm's laws is proposed for the electrochemical flux of an anionic molecular tracer. Together with the conservation of mass, Gauss's law from electrostatics, and appropriate boundary conditions, a closed model is obtained for electrochemical transport through the capillary glycocalyx. This model is solved numerically for the one-dimensional, axisymmetric, spatiotemporal concentration distributions of the molecular tracer and mobile ions in the blood. Furthermore, a linear analysis is developed which is valid whenever the molecular tracer concentration is small compared with the ion concentration in the blood. From this analysis, a closed-form asymptotic expression is derived for the molecular tracer diffusion time that is valid if either the fixed-charge density is large compared with the ion concentration in the blood or the electronegativity of the anionic molecular tracer is large. Following this is a discussion of analytical results where we consider specific parameter values (e.g., molecular tracer valence, glycocalyx fixed-charge density, and glycocalyx distribution) required to reproduce the recent experimental findings of Vink and Duling (2000). We conclude with a discussion of the model's implications for the system in equilibrium and propose several alternative experimental approaches to finding the glycocalyx fixed-charge density in vivo that are independent of molecular diffusion times or the reaction-diffusion kinetics of the system.

	THE MODEL

TOP ABSTRACT INTRODUCTION THE MODEL EQUILIBRIUM CONFIGURATION LINEARIZED SOLUTION NUMERICAL METHODS RESULTS DISCUSSION SUMMARY APPENDIX REFERENCES

The glycocalyx is modeled here as a continuously distributed anionic matrix made up of proteoglycans, glycoproteins, and GAGs containing fixed-bound negative charges through which a solution of anionic molecular tracers in blood plasma can diffuse. Of fundamental importance to the model is that it account for the presence of ionic salts (Na⁺, Cl, etc.) in the blood. The validity of the continuum approximation used here for the glycocalyx matrix depends not only upon the instantaneous spatial distributions of the matrix and fixed charge groups, but also upon the temporal variations in those distributions arising from Brownian motion of the matrix. In fact, if it were not for these temporal variations, the continuum approximation might not be reasonable. In particular, we assume that in its hydrated state, the glycocalyx is extremely diffuse and resembles other collagen-poor mucopolysaccharide extracellular matrix structures with solid-volume fractions below 1% (Levick, 1987). The instantaneous fixed-charge distribution in such a structure is therefore likely to be quite heterogeneous. Furthermore, the Debye length in normal saline is <0.2 nm, and thus the electric field induced by the fixed charges bound to the glycocalyx is very efficiently shielded by the counter cations in the blood. At such high cation concentrations relative to c^f the strength of the electric field, at a distance of 1 nm or more from one of the fixed-charge groups, would be reduced to <1% of its maximum value if the glycocalyx matrix were a static scaffold with fixed-bound-negative charges. Indeed, if it were not for Brownian motion of the glycocalyx matrix itself, the instantaneous electric field distribution in a system with such a low solid-volume fraction and such a high cation concentration would likely be extremely nonuniform. However, when one accounts for Brownian motion of the proteoglycan/glycoprotein/GAG aggregates at 310 K, the time-averaged spatiotemporal distribution of the electric field would certainly be more uniform than the instantaneous distribution, making the continuum approach more reasonable. Therefore, we assume that variations in the electric field arising from cationic charge shielding and sparsity of the individual fixed charges bound to the glycocalyx matrix are offset, in a time-averaged sense, by Brownian motion of the matrix. We therefore model the glycocalyx as having a continuous concentration distribution and continuous fixed-charge density distribution (with at most a finite number of discontinuities at interfaces) such that spatial variations in the electric field are solely a result of spatial variations in the time-averaged fixed-charge density distribution.

By invoking the continuum approximation, we bring to bear the classical theory of electrochemical ionic transport in solution, which has its origins in the Nernst-Planck equation (Bockris and Reddy, 1970). In the context of this theory, transport is driven principally by chemical gradients and electrostatic forces. In the case of transport of anionic molecular tracers through the glycocalyx, the results of Vink and Duling (2000) suggest a strong dependence on tracer valence; thus, the important contributions to molecular transport that are considered here are derived from chemical and electrostatic potentials.

Conservation of mass

In the absence of chemical reactions, the time rate of change of the concentration, c, of species  is related to its flux, J, relative to a quiescent solvent, by the conservation equation given by

(1)

The various flux contributions mentioned above must be specified by appropriate constitutive relations.

Constitutive flux laws

For any mass transport problem there is a flux, J, associated with the chemical potential of the diffusing species that is proportional to the concentration gradients of the species . Typically, this is modeled using Fick's law of diffusion. To account for the strong charge dependence in the results of Vink and Duling (2000), a flux, J, due to electrostatic interactions of species  with the electric field induced by the glycocalyx is also introduced. The constitutive flux law for electrostatically driven transport is derived from Ohm's law, and is obtained by considering electrostatic and viscous forces that act on a diffusing particle in suspension due to the effect of all charges in the system. Thus, we model the total electrochemical flux, J, of species  as the sum of Fick's and Ohm's laws given by

(2)

(3)

where D is the diffusion coefficient, z is the ionic valence, f is the Stokes drag coefficient, q is the elementary charge, and E is the electric field vector. This expression forms the basis for the model presented here. It should be noted that z represents the effective valence, which provides the correct electrophoretic mobility of the charged molecule in solution. The effect of charge shielding is then accounted for in the valence, which may take on noninteger values.

Electrochemical transport equations

The Stokes drag coefficient, f, is related to the diffusion coefficient according to the Einstein relation, D = k_BT/f, where k_B is Boltzmann's constant and T is the absolute temperature (Reif, 1965). Thus, the flux and species conservation equations become

(4)

(5)

The index  makes explicit the fact that there are different diffusing species in the system. Namely,  may take the values +, , and L, for the mobile cations (Na⁺), the mobile anions (Cl), and anionic molecular tracers, respectively.

It remains now to determine the electric field E due to the presence of charge imbalances in the system. From the outset it should be noted that there will be global charge balance, so that the total charge in the system is zero. There may, however, be local charge imbalances due to gradients in the concentrations of the various species. The electric field due to a system of charges may be determined using Gauss's law from electrostatics. The divergence of the electric field depends on the local charge density, , according to

(6)

where  is the permittivity of the surrounding medium, which is taken to be the same as that of water (1.57 × 10¹¹ F/m).

At this point, the effect of the glycocalyx may be included. The glycocalyx is assumed to be a charged porous matrix of macromolecules, each with valence z^F. The concentration of macromolecules in the glycocalyx is denoted by c^F, which varies over the cross section of the capillary. The glycocalyx fixed-charge density is then denoted by |z^Fc^F|. In this analysis, deformations of the glycocalyx are considered negligible, so the initially specified concentration, c^F, does not vary with time. For convenience, we introduce the quantity  = /q, which represents the local charge imbalance per unit charge and corresponds to the valence-weighted sum of constituent concentrations given by

(7)

where, for convenience, we have introduced the parameters m = z^L and n = z^F. Gauss's law, given in terms of  by  · E = q/, taken together with the three second-order, nonlinear, partial differential equations represented by Eq. 5, provide a system of four scalar equations in the four unknowns, c⁺(x, t), c(x, t), c^L(x, t), and (x, t).

Axisymmetric form of the equations

In all of what follows, axisymmetric conditions will be imposed and axial variations in the field variables will be neglected. With this approximation, all variables depend only on the radial coordinate, r, and time. This simplifies the governing equations substantially since Gauss's law is then integrable. Omitting uniform additive contributions to the electric field in the ê_r, ê, and ê_z directions, Eq. 6 reduces to

(8)

where, by axisymmetry, the electric field has only a radial component such that E = E_r(r t)ê_r. Furthermore, in the axisymmetric case, the electric field must vanish at r = 0. If there is zero net charge in the system, the electric field must also vanish at the system boundary at r = . This global electroneutrality condition requires that

(9)

The conservation equations contained in Eq. 5 may be written in cylindrical coordinates, and upon substitution of Eq. 8 for E_r, they become

(10)

for  = +, , and L. The fourth equation needed for closure is just the definition of  given by Eq. 7. Thus, substitution of Eq. 7 into Eq. 10 provides a coupled system of three scalar, second-order, nonlinear, partial integro-differential equations in the three unknowns, c⁺, c, and c^L.

Boundary conditions

Because the equations represented by Eq. 10 have one time derivative and two space derivatives, an initial condition and two boundary conditions are needed. For a closed system, the flux of any species across the system boundary must vanish. In particular, if the closed boundary of the system is at r = , then the radial component of the flux must satisfy J(, t) = 0. Expressing Eq. 4 for the flux in axisymmetric cylindrical coordinates, this boundary condition takes the form

(11)

Recalling the charge balance requirement of Eq. 9, this boundary condition simplifies to

(12)

The second boundary condition arises from the geometry of the problem. Because the system is presumed to be axisymmetric, odd derivatives of the concentration vanish at r = 0, which provides the second boundary condition given by

(13)

Initial conditions

We assume a Gaussian radial distribution for the initial concentration, c^L(r, t₀), of molecular tracers. As we will see, the diffusion time is insensitive to the exact form of this initial distribution. However, the initial distributions of the mobile salt ions, Na⁺ and Cl, present a more difficult problem. Because it is assumed that salts in the blood plasma are in equilibrium before molecular tracers are added, the condition that determines the initial c⁺ and c distributions is that the flux vanish identically. Therefore, we must solve Eq. 4 subject to the constraint that J^±  0. The electric field for these equations is given by Eq. 8, but with c^L = 0 in Eq. 7 defining . In cylindrical coordinates, the two zero-flux equations become

(14)

and the expression for  in equilibrium becomes

(15)

As for the unsteady case, the boundary conditions for Eq. 14 are given by Eqs. 12 and 13. From Eqs. 12-15, steady-state solutions can be obtained that represent the equilibrium configurations for c^±(r, t₀) immediately before molecular tracers are added.

Nondimensional form of the equations

So that reasonable order-of-magnitude approximations can be made, we make the relative size of each term in the equations apparent by nondimensionalizing the variables and equations. The parameters that characterize the problem are given in Table 1. Using an asterisk to denote dimensionless variables, the dependent and independent variables are nondimensionalized as follows:

(16)

(17)

Taking D to be constant, and assuming D⁺  D, we have

(18)

(19)

(20)

where we have introduced the following dimensionless groups:

(21)

Henceforth, the nondimensional form of the governing equations will be used, and for convenience, the asterisks will be dropped.

Glycocalyx distribution

The concentrations of proteoglycan, glycoprotein, and GAG macromolecules in the glycocalyx are assumed to increase continuously from zero in the lumen to a nearly constant (but unknown) value near the endothelial-cell wall. Leakage of the tracer molecules from the capillary into the extravascular space might also play an important role in the electrochemical dynamics. The glycocalyx distribution is thus approximated by an expression of the form:

(22)

The dagger indicates that the distribution is symmetrized about r = 0; that is, p(r) = p(r) + p(r), and  is a scaling factor so that max{c^F(r)} = 1. The radii, r_g and r_ec, denote the locations of the lumen-glycocalyx and glycocalyx-endothelial boundaries, respectively, and are known approximately from experimental results of Vink and Duling (1996, 2000). Near r_g, 99% of the rise in c^F occurs over a distance r_g, while 99% of its fall occurs near r_ec over a distance r_ec. Fig. 1 shows a schematic diagram of this distribution. Setting both r_g and r_ec to zero results in a box-shaped distribution, with discontinuities in glycocalyx concentration at r_g and r_ec. In dimensional variables, the maximum concentration of the glycocalyx corresponds to c.

View larger version (12K): [in this window] [in a new window]   FIGURE 1   The assumed form of the cF distribution, as in Eq. 22, showing the parameters rg, rg, rec, and rec. Not to scale.

	EQUILIBRIUM CONFIGURATION

TOP ABSTRACT INTRODUCTION THE MODEL EQUILIBRIUM CONFIGURATION LINEARIZED SOLUTION NUMERICAL METHODS RESULTS DISCUSSION SUMMARY APPENDIX REFERENCES

A few observations can be made about the three equations represented by Eqs. 18 and 19. They are clearly coupled through Eq. 20 for , and are nonlinear. The nonlinearity appears in the flux contribution associated with the electric potential, and is bilinear and quadratic. The integral term complicates the problem considerably because it means that local charge imbalances have global influence. Here we derive some exact and asymptotic relations of the equilibrium solutions, including an asymptotic expression for  as a function of c^F. These results are used in the next section to derive a linearized set of equations.

Product of concentrations at equilibrium

For two different diffusing species at equilibrium, having concentrations c^u and c^v and valences z^u and z^v, the product (c^u)^1/zu(c^v)^1/zv is constant over space. To see this, consider the steady-state version of the flux equations given by

(23)

(24)

Dividing Eq. 23 by z^uc^u and Eq. 24 by z^vc^v and subtracting, we obtain
from which it follows that

(25)

For a salt solution with z^u = +1 and z^v = 1, this result says that the product c⁺c is constant at equilibrium. In dimensional variables, the constant is equal to unity; dimensionally, it is given by c. Furthermore, from Eq. 25 it follows that at any two radial distances, r₁ and r₂,
from which we conclude

(26)

This result becomes useful when solving for the equilibrium distribution of a species because it can be used to uncouple the zero-flux equations.

Implication for tracer exclusion in equilibrium

The results of the previous section have important implications for the exclusion of molecular tracers from regions where the glycocalyx fixed-charge density is large. To illustrate this, we initially restrict consideration to z⁺ = 1 and z = 1. Subsequently, we will give an argument to generalize the results to arbitrary values of z⁺ and z.

Using Eq. 26, and letting r₁ = 0, r₂ = r, u = +, and v = L (so that z⁺ = 1 and z^L = m), we obtain

(27)

Similarly,

(28)

Recalling the definition of  from Eq. 20, we note that B  4000 in blood plasma, and also that nF is presumed to be large compared with m, which is <~5 in the experiments of Vink and Duling (2000). Because all concentrations are nondimensional, they are of order unity so mc^L is negligible compared with the other terms. Also, because modest charge imbalances result in large forces,  is generally assumed to be very small. Therefore,

(29)

In the lumen, c^F is also negligible because the glycocalyx is assumed not to extend across the entire vessel (Vink and Duling, 1996, 2000). Since c^F(0) = 0, then at the center of the lumen z⁺c⁺(0)   zc(0). Using this in Eq. 28, and recalling the assumptions that z⁺ = +1 and z = 1, Eq. 29 becomes

(30)

from which it follows that the nonnegative value of c⁺(r) is given by

(31)

Now, for convenience we define

(32)

and substitute Eq. 31 into Eq. 27 to obtain

(33)

for z⁺ = z = 1. Because the concentrations are nondimensional, c⁺(0) = 1, and if r = r₀ is chosen where the glycocalyx is most concentrated, then c^F(r₀) = 1. Thus, ₀ = nF/(z⁺B) measures the ratio of the glycocalyx fixed-charge density to the luminal concentration of free salts, and

(34)

This quantity will be referred to as the exclusion factor because it gives the factor by which anionic molecular tracers are suppressed within the glycocalyx compared with the lumen. It is plotted against ₀ for several different values of m in Fig. 2. If ₀ and m are small, then c^L(r₀)/c^L(0) approaches unity. If ₀ and m are large, then c^L(r₀)/c^L(0) approaches zero, implying that the molecular tracers are excluded from the glycocalyx. Thus, if the fixed-charge density of the glycocalyx is large compared with the concentration of free salts in the blood, or if m is large, then anionic tracers are excluded from the glycocalyx. This plays a very important role in suppressing the flux of tracers through the glycocalyx, and thereby lengthening the diffusion time.

View larger version (20K): [in this window] [in a new window]   FIGURE 2   The dependence of the exclusion factor, Eq. 34, on 0 for several different values of m.

Equilibrium distribution for

The equilibrium configuration for  can be determined from the zero-flux equation for c⁺ corresponding to the dimensionless form of Eq. 14 given by

(35)

We divide the zero-flux equation by c⁺, which vanishes nowhere, and differentiate with respect to r to find

(36)

This immediately gives  in terms of the distribution of c⁺ such that

(37)

Using Eq. 31, and recalling that c⁺(0) = 1, we substitute for c⁺ in Eq. 37 to obtain

(38)

(39)

where

(40)

Equation 38 shows that  is related to the second derivative of the glycocalyx distribution, c^F. Since Q = 72300 for a typical capillary,  is a small quantity, as assumed in the derivation of Eq. 29, making the assumption self-consistent. Based on dimensional considerations discussed earlier, this estimate for  should be accurate to within one part in z⁺c_blood/(mc) = B/m  1000 even during the diffusion process. Next it will be shown that f is closely related to the voltage field induced in the neighborhood of the glycocalyx.

Induced voltage

Although electroneutrality has been imposed globally, concentration of the glycocalyx on the endothelial-cell wall gives rise to small departures from electroneutrality near the interface between the glycocalyx and the plasma in the lumen. The resulting electric field that arises from these charge imbalances can be calculated directly from  using Eq. 8. Substituting the approximate analytic expression for , given by Eq. 39, into Eq. 8, and redimensionalizing, gives the electric field explicitly:

(41)

The voltage is then given by:

(42)

where V₀ is a constant of integration. Substituting for f in this expression from Eq. 40 gives

(43)

The reference voltage has been chosen to make the voltage zero within the lumen. This expression agrees with the interface potential quoted by Masaki et al. (2000).

	LINEARIZED SOLUTION

TOP ABSTRACT INTRODUCTION THE MODEL EQUILIBRIUM CONFIGURATION LINEARIZED SOLUTION NUMERICAL METHODS RESULTS DISCUSSION SUMMARY APPENDIX REFERENCES

In this section we invoke the approximation that B/m 1 and use the results of the previous section to decouple and linearize the governing equations. This permits an eigenfunction solution to the problem. The value of the first nonzero eigenvalue, ₁, is shown to have important implications for the diffusion time, and an asymptotic expression for ₁ is found for a box-shaped distribution of the glycocalyx.

Quasi-static approximation for

The key observation is that although mc^L might significantly exceed  throughout the system,  varies only slightly as c^L changes with time. The physical justification for this is somewhat subtle. Before the addition of anionic molecular tracers, c⁺ and c have been in the system sufficiently long to attain equilibrium with the glycocalyx molecules. This means that the electric field set up by the fixed charges bound to the glycocalyx is already supporting concentration gradients in c^± near the glycocalyx. When tracers are added, they also diffuse through the glycocalyx; however, because c c_blood the presence of tracers introduces only a small perturbation to the free salt ion concentration. Thus, c^± does not change significantly, and neither does the electric field, which continues to support concentration gradients in c^±. Therefore,  is perturbed only slightly around its equilibrium value and remains nearly unchanged throughout the diffusion process. Essentially, the presence of anionic molecular tracers may only perturb  by around one part in B/m  1000.

This result can be derived formally by considering Eq. 37 in conjunction with the requirement of global charge balance. Because the additional cations must globally balance the charge on the anionic molecular tracers, we have

(44)

where c⁺(r, t₀) is the equilibrium distribution of cations before the tracers are added. Thus, the temporal variation of c⁺ must be of order m/B, so we write c⁺(r, t_f) = c⁺(r, t₀) + (m/B)c⁺(r), where c⁺(r, t_f) is the cation equilibrium distribution after the tracers have equilibrated, and c⁺(r) is of order unity (in the sense that its volumetric integral on 0  r  1 is unity). Substituting this expression into Eq. 37 gives

(45)

(46)

(47)

Thus, consistent with the qualitative arguments made previously, the fractional variation in  is of order m/B  1/1000, and is therefore negligible. Thus,  can be considered quasi-static, and is well-approximated by the steady-state result given by Eq. 39.

Eigenvalue problem for static

Using the quasi-static approximation for  in Eq. 39, the governing equations can be linearized and decoupled. Recalling Eq. 19, the conservation of mass for c^L is given by

(48)

Substituting Eq. 39 for  and integrating provide

(49)

This equation is linear in c^L and homogeneous, and is uncoupled from the conservation equations governing the other species. Separation of variables provides a series solution in terms of orthogonal functions. Thus, we seek multiplicatively separable solutions of the form R(r)T(t) that satisfy Eq. 49 and the boundary conditions. Using standard methods (Boyce and DiPrima, 1992), a linear superposition of such solutions will provide an eigenfunction expansion that will be made to satisfy the initial condition at t = t₀. Substituting into Eq. 49 and separating r and t dependence we find

(50)

where  is a real constant. In this form it is evident that R(r) depends only on  and f(r), which in turn depend only on m, ₀, and c^F(r). Because Eq. 49 is homogeneous, we conclude from the second of Eq. 50 that T(t) = e^2t/D.

Because the flux must vanish at r = 0 and r = 1, according to Eqs. 12 and 13, the boundary conditions on R(r) are simply R'(0) = R'(1) = 0. Imposing these boundary conditions provides a denumerable infinite set of eigenvalues, {_n}, corresponding to the set of eigenfunctions, {R_n(r)}, where each R_n(r) must satisfy the ordinary differential equation (ODE) given by the first of Eq. 50. For ₀ = 0, an analytic expression for R₀(r) is found to within a multiplicative constant to be

(51)

This corresponds to the equilibrium distribution for c^L after transients have decayed. To within a multiplicative constant factor, it is the same as Eq. 33. For the remaining eigenfunctions, R_n(r), however, there is no analytic expression for general f(r).