INTRODUCTION

TOP ABSTRACT INTRODUCTION THE MODEL LINEARIZED ANALYSIS NUMERICAL METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

In a recent study, we investigated the influence of the anionic fixed-charge groups bound to the capillary glycocalyx on the electrochemical transport of charged molecules through the glycocalyx (Stace and Damiano, 2001). However, no attempt has been made to analyze the role of these fixed-charge groups on deformations of the glycocalyx matrix such as would arise in the presence of a passing leukocyte through the capillary. In this paper, we build upon the previous electrochemical model and extend the analysis to address transient mechanical deformations of the glycocalyx surface layer. Recent speculation as to the origins of the restoring forces in the capillary glycocalyx has raised attention to a variety of possible sources including elastic-restoring mechanisms, osmotic and oncotic pressures, electrostatic potentials, and fluid dynamical mechanisms (Damiano et al., 1996; Damiano, 1998; Secomb et al., 1998, 2001; Feng and Weinbaum, 2000). New experimental approaches to observing the glycocalyx in vivo have revealed what appears to be the time course of the layer's dynamic response to transient deformations by passing leukocytes (Vink et al., 1999). In the analysis presented here, we develop a mechano-electrochemical model of the glycocalyx, which presumes that, under deformation, electrostatic potentials arising from the fixed charges bound to the solid matrix and concentration gradients arising from a redistribution of the glycocalyx molecules are the predominant mechanisms responsible for the layer's tendency to restore itself to its equilibrium configuration.

The possible relevance of the glycocalyx to microcirculatory function was first considered by Copley and Silberberg (Lahav et al., 1973; Krindel and Silberberg 1979; Copley 1974). Klitzman and Duling (1979) implicated the glycocalyx in accounting for the low capillary tube hematocrits (i.e., the instantaneous volume fraction of red cells resident in the capillary) they observed in capillaries of skeletal muscle. Evidence that the macromolecules of the glycocalyx might interfere with flow in a large plasma layer near the capillary wall was first reported by Desjardins and Duling (1990). After enzyme treatment targeted at cleaving specific proteoglycan molecules within the glycocalyx, they observed a two-fold increase in capillary tube hematocrit. Using a light-dye treatment to remove the glycocalyx, Vink and Duling (1996) observed a similar trend in tube hematocrit and obtained the first estimate of the thickness of the layer in vivo.

Combining network simulations with measurements of blood flow in large-scale microvascular networks, Pries et al. (1994) concluded that the resistance to blood flow in microvessels between 10 and 30 µm in diameter was dramatically higher than in glass tubes of the same diameter. In a more recent study, Pries et al. (1997) found that the resistance to blood flow in microvascular networks decreased markedly after enzyme treatment to remove the glycocalyx. These studies, and those of Duling and coworkers, suggest that the glycocalyx could serve to retard plasma flow near the vessel wall, which, in turn, would result in enhanced resistance to blood flow and lower capillary tube hematocrits in vivo than in smooth glass tubes.

Until recently, all theoretical models of red-cell motion through capillaries in the single-file flow regime completely neglected the endothelial-cell glycocalyx. Most of these models approximated the capillary as a rigid, smooth-walled, uniform circular cylinder. Results of these studies, which were derived from either finite element analyses (Zarda et al., 1977a,b; Özkaya, 1986) or models that invoked the lubrication-theory approximation (Secomb and Gross 1983; Özkaya, 1986; Secomb et al., 1986), were found to be in good agreement with experimental observations of blood flow through narrow glass tubes (Özkaya, 1986; Secomb et al., 1986; Skalak and Özkaya, 1987; Secomb, 1995). None of these models, however, compared favorably with in vivo observations of blood flow in microvascular networks (Pries et al., 1994).

Among the earliest analytical attempts to investigate flow through the glycocalyx came with the work of Barry et al. (1991). Their work considered steady and unsteady flow of a Newtonian fluid in a channel lined with a poroelastic wall layer. Wang and Parker (1995) applied mixture theory and two-dimensional lubrication theory to the problem of a sphere falling through a quiescent fluid in a cylindrical tube lined with a deformable porous wall layer. Damiano et al. (1996) provided the first analytical solutions of axisymmetric pressure-driven flow of rigid close-fitting particles in a cylindrical tube lined with a poroelastic wall layer. Damiano (1998) incorporated the axisymmetric model developed by Damiano et al. (1996) into the first realistic model of capillary blood flow that accounts for the effects of the endothelial-cell glycocalyx and the deformation of the red-cell membrane. Secomb et al. (1998, 2001) went further to include the effects of membrane viscosity and membrane bending and shear elasticities in the red cell. Although these models represent a significant improvement over existing theories of capillary rheology, the characterizations of the glycocalyx developed by Damiano (1998) and Secomb et al. (1998, 2001) are rather simplistic and only capture the gross rheological effects of plasma retardation near the capillary wall. In particular, effects of the electrostatic properties of the layer on glycocalyx permeability and deformation have not been addressed in a rigorous analytical model. One of the most compelling reasons to pursue this stems from recent in vivo investigations into the mechanical properties (Vink et al., 1999) and molecular transport characteristics (Henry and Duling, 1999, 2000; Vink and Duling, 2000) of the glycocalyx.

Preliminary experiments of Vink et al. (1999) reveal important information about the mechanical response of the glycocalyx to deformation. Because leukocytes are larger and much stiffer than red blood cells, they occupy more of the capillary lumen. In both in vitro and in vivo studies, extremely thin lubrication layers are observed between the leukocyte membrane and the vessel wall (Needham and Hochmuth, 1990; Vink and Duling, 1996). As a consequence of this, leukocytes travel more slowly through capillaries than do red cells, which likely accounts, in part, for the commonly observed train of red blood cells that often follows a tightly fitting leukocyte (Vink et al., 1999). In capillaries less than 7 µm in diameter, it appears as if the glycocalyx experiences large deformations on the passing of individual leukocytes (Vink et al., 1999). The red cells immediately behind the leukocyte are maximally expanded and fill most of the capillary lumen. As red blood cells from upstream move into the field of view, they become progressively more deformed, presumably as a result of the restoring forces of the glycocalyx as it swells back to its equilibrium configuration. In preliminary studies of Vink et al. (1999), the recovery time of the compressed glycocalyx matrix has been measured in the wake of a passing leukocyte and was reported as being ~1 s. The recovery time was based on the time required for the mean diameter of red cells upstream of the leukocyte to reach a steady-state value.

On the basis of these observations, we seek to develop a mechano-electrochemical model of the glycocalyx that assumes that the layer consists of a multicomponent mixture of an incompressible fluid, an anionic porous deformable matrix, and mobile cations and anions. At time t = t₀, the concentration distribution in the reference configuration is denoted by c^F(X, t₀) = c^F(x, t₀), where the components of x are the spatial coordinates, and the components of X are the reference coordinates of a material point in the field. The fixed-charge density, nc^F(X, t), where n is the mean molecular valence of the glycocalyx, is assumed to be directly proportional to the solid-matrix concentration distribution at any time, t. Although the fixed-charge groups per unit mass of the solid matrix are assumed to be constant, their concentration distribution can change with deformation of the solid matrix. Thus, nc^F depends upon the initial concentration distribution in the reference configuration and the state of deformation of the solid matrix. In equilibrium, the mobile ions establish concentration distributions, c⁺(X, t₀) and c(X, t₀), that nearly counterbalance the fixed charges on the solid matrix. In equilibrium, complete electroneutrality is not achieved everywhere because that would result in large gradients in the ion distributions. Instead, the ion concentrations assume distributions in equilibrium that result in a nonzero electric field and nonzero chemical potential gradients, but which minimize the equilibrium electrochemical potential gradient of the system as a whole. When integrated over the vessel cross section, however, these local charge imbalances cancel such that global space-charge neutrality exists within the capillary. Because of these local charge imbalances and concentration gradients that exist in equilibrium, a state of tension sets up in the matrix that balances the electrochemical forces in the glycocalyx. Thus, the energy required to maintain the static electric field and nonuniform distributions in the ion concentrations is stored as tension in the matrix in equilibrium. Under compression of the layer by a passing leukocyte, an external stress traction is exerted by the cell at the apical end of the glycocalyx and the tension in the matrix is largely relieved. As the layer deforms, an imbalance exists between the externally applied stress traction and the electrochemical forces in the matrix. Upon recovery of the compressed matrix, the electrochemical forces that restore the layer to its equilibrium configuration are resisted by permeation-induced hydraulic drag of the layer as it moves through the plasma.

A principal result of the analysis that follows is the reduced momentum equation for the displacement of the glycocalyx solid matrix. It is shown that, under purely radial deformations of the glycocalyx, the kinematically related circumferential and radial principal stretch ratios, and _r = (R)/R, associated with a material point in the glycocalyx solid matrix are governed by

(1)

where the electrochemical potential gradient of the glycocalyx in a monovalent salt solution is given by

(2)

Here, R is the radial coordinate in a Lagrangian cylindrical coordinate frame, where R = R₀ and R = a₀ are, respectively, material points in the reference configuration of the solid matrix at the vessel wall and at the glycocalyx interface with the plasma in the lumen. Through the conservation of mass, the normalized glycocalyx concentration, c^F/c, is kinematically related to the principal stretch ratios (in inverse proportion to the product _r), and by invoking the so-called pseudo-equilibrium approximation, the normalized cation concentration, c⁺/c, is algebraically related to c^F/c. Finally, in Eq. 1, T₀(R) is the spatially varying isotropic tension in the glycocalyx solid matrix in the reference configuration, and k is the local permeability of the glycocalyx to water.

The mechano-electrochemical dynamics described above are embodied in the relatively simple result given by Eq. 1. The restoring force is accounted for by the last term on the right-hand side and is associated with the electrochemical potential gradient of the glycocalyx. In the reference configuration, the term on the left-hand side vanishes and the last term on the right-hand side balances the remaining terms, which are associated with the divergence in the elastic stress tensor of the matrix. During recovery of the matrix after compression, the electrochemical potential gradient dominates over the tension in the matrix and restores the layer to its equilibrium configuration. The term on the left-hand side, associated with permeation-induced drag of the solid matrix through the fluid, resists this recovery and opposes the mechano-electrochemical restoring force. As the layer rehydrates and approaches the reference configuration, the left-hand side tends toward zero until, finally, equilibrium is reestablished when the terms on the right-hand side are once again in balance.

In what follows, the conservation equations for a specialized quaternary mixture will be presented and constitutive relations will be proposed for each of its constituents. In deriving our constitutive equations, we shall assume that the glycocalyx is an extremely diffuse, anionic mucopolysaccharide that is devoid of collagen and has essentially no elastic restoring capability. The only mechanical stress we will assume the matrix itself can support is one of tension. We will therefore assume that the restoring mechanisms of the glycocalyx have their origins in electrochemical rather than elastic forces and that the fixed-charge density and chemical potential of the layer are the fundamental determinants of that restoring force. To address finite deformations of the glycocalyx, the governing equations, together with Gauss's law from electrostatics, will be formulated in Lagrangian variables for the special case of purely radial axisymmetric deformations in the absence of an axial flow through the capillary. We then invoke the assumption that the transport of mobile ions occurs as a pseudo-equilibrium process such that many of the relationships derived by Stace and Damiano (2001) for the case of a static-charge distribution can be adapted to the problem with deformation. The justification for this assumption lies in the realization that the cation concentration in blood is likely to be very large when compared with the glycocalyx fixed-charge density. As such, even large physiologic deformations of the layer result in only small perturbations to the cation concentration field which re-distributes itself quickly relative to the glycocalyx molecules by virtue of the relatively large diffusion coefficient associated with the cations. This approximation significantly simplifies the model, making it much more tractable to numerical and asymptotic analysis. Results of the analysis are presented and discussed in terms of the mechano-electrochemical dynamics of the layer under compression and recovery and the dependence of those dynamics on the parameters in the model. From our results, an estimate of the glycocalyx fixed-charge density is made based on the limited experimental data available. We conclude with a discussion of new experimental approaches inspired by the model that could be taken to obtain more precise bounds on the mechano-electrochemical properties of the structure in vivo.

	THE MODEL

TOP ABSTRACT INTRODUCTION THE MODEL LINEARIZED ANALYSIS NUMERICAL METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B APPENDIX C APPENDIX D REFERENCES

Using mixture theory, the glycocalyx is modeled as a multicomponent material continuum that consists of four interacting and interpenetrating constituents, which include the interstitial fluid (blood plasma), the electrostatically charged proteoglycan/glycoprotein/glycosaminoglycan solid matrix, and the mobile ions (cations and anions). Mixture theory approximates multicomponent materials as a continuum using a macroscopic field theory description (Truesdell and Toupin, 1960; Bowen, 1976; Lai et al., 1991). Also known as the theory of heterogeneous or superimposed material continua, mixture theory assumes that every spatial point in the field is occupied simultaneously by a material point of all of the constituents in the mixture. In essence, the microscale substructure is smeared or mixed throughout the material. The theory accounts not only for the intrinsic constitutive behavior of each of the components of the mixture, but it also accounts for the transfer of momentum that occurs between constituents. This momentum transfer, characterized by momentum supply forces in the conservation equations, arises from the local interaction between constituents as they move relative to each other. The model proposed here for the glycocalyx is consistent with the so-called "triphasic" theory developed by Lai et al. (1991) in the absence of an elastic-restoring capability by the matrix and in the limit as the solid-volume fraction approaches zero. The validity of invoking a continuum mixture approximation in this context is discussed in Stace and Damiano (2001).

The conservation equations

The glycocalyx will be modeled as an isotropic quaternary mixture of a solid (s), water (w), cationic (+), and anionic () constituent. We denote the velocity vectors of these constituents by v^s, v^w, v⁺, and v, and the partial Cauchy stress tensors by ^s, ^w, ⁺, and . We shall assume that the momentum supply force exerted by one constituent on another is proportional to the relative velocity between the two constituents and that the momentum supply forces acting between two constituents are equal but opposite. The coefficient of proportionality is the frictional drag coefficient between two constituents. For the interaction between the solid and water constituents, the frictional drag coefficient corresponds to the hydraulic resistivity, which is inversely proportional to the matrix permeability to water. The frictional drag coefficient associated with the interaction between the ionic and water constituents is related to the Stokes viscous drag coefficient of an ion in solution. Because we are assuming that the volume fractions of the solid and ionic constituents are negligible, the drag forces associated with ion-ion interactions and ion-matrix interactions will be neglected. All constituents are assumed to be neutrally buoyant, and, therefore, the only body force that arises is due to the electric field, E, induced by the fixed charges bound to the solid matrix. Finally, because the flow through the glycocalyx during physiologic deformations of the layer resides in the extremely low-Reynolds number regime, inertial terms in the momentum equations are negligible compared with surface, body, and momentum supply forces. In light of these assumptions, the momentum conservation equations for the solid, fluid, cationic, and anionic constituents are given respectively by

(3)

(4)

and

(5)

where the flux vectors, J⁺, between the mobile cations and the water constituent, and J, between the mobile anions and the water constituent, are defined such that

(6)

The parameter k in Eqs. 3 and 4 is the permeability of the layer to water, which is inversely proportional to the hydraulic resistivity and may depend locally on the strain field.

Summing Eqs. 3-5, we obtain the conservation of momentum for the mixture as a whole given by

(7)

where

(8)

and

(9)

Electrostatic effects enter as body forces rather than surface tractions as is evident from Eq. 7. In the second of Eq. 9, p accounts for the mechanical and osmotic pressures, and ^E accounts for the tensile stress stored in the solid matrix in equilibrium and during deformation of the glycocalyx. We have neglected the contribution to the total stress tensor, ^T, of viscous stresses associated with ^w. For length scales characteristic of flow within the glycocalyx, previous studies have suggested that viscous drag forces associated with fluid-velocity gradients are likely to be small relative to permeation-induced viscous drag forces associated with fluid motion relative to the solid matrix (Damiano et al., 1996; Damiano, 1998; Secomb et al., 1998; Feng and Weinbaum, 2000). As such, we neglect dissipative losses associated with the deviatoric or viscous stress tensor of the water constituent. This approximation is commonly made in models of articular cartilage based on mixture theory when the prevailing flow direction is normal rather than tangential to the boundaries of the mixture (Mow et al., 1980; Lai and Mow, 1980).

Modeling the water constituent of the glycocalyx as incompressible with a volume fraction near unity, the continuity equation for the water is given by

(10)

The instantaneous system boundaries of the mixture are defined by the deformed configuration of the solid matrix. As such, a flux of water across the system boundaries occurs during compression and recovery of the layer. Thus, we regard the mixture as highly compressible despite the fact that its primary constituent, taken by itself, is incompressible.

For the solid constituent, the mass density, ^s, and molecular concentration, c^F, are governed by the conservation of mass given by

(11)

where t₀ corresponds to a time when the glycocalyx is in its reference configuration and J = det F is the Jacobian of the deformation gradient tensor, F = x/X. In addition to the continuity equations for the solid and fluid constituents, given by Eqs. 10 and 11, we impose the mass conservation equations for the mobile ions, given by

(12)

Using Gauss's law from electrostatics, the electric field is governed by

(13)

where  is given by Eq. 8, the voltage, V, is the scalar electrostatic potential function, q is the elementary charge, and  is the permittivity of water.

Specification of constitutive equations for ^s, ^w, ⁺, , ^E, and k provides a closed system of equations, which includes four scalar continuity equations, Gauss's law, and the definition of  (Eqs. 8, 10, 11, 12, and 13) in the six scalar unknowns, c⁺, c, c^F, p, V, and , and four vector momentum equations (Eqs. 3, 4, and 5), in the four vector unknowns, v^s, v^w, v⁺, and v.

Physicochemical constitutive equations

In addition to the conservation equations for the mixture given above, constitutive relationships are needed to characterize the partial stress tensors of each of the constituents. For ideal solutions, or in the limit of infinite dilution, the van't Hoff equation for the osmotic pressure is analogous to the ideal gas law for the thermodynamic pressure (Katchalsky and Curran, 1965). Because the NaCl concentration in blood plasma is dilute (approximately 0.14 M, which is more than two orders of magnitude lower than the concentration of the solvent), we invoke the van't Hoff equation for the mobile ions and take

(14)

where ^± corresponds to the osmotic pressure of the ions in solution, k_B is Boltzmann's constant, and T is absolute temperature. Under isothermal conditions, then, the ratio of the osmotic pressure to the concentration remains constant. By modeling the cationic and anionic constituents as ideal inviscid fluids, their partial Cauchy stress tensors are given by

(15)

where, for ideal solutions, the chemical potential, µ^±, of the ions is related to the ion concentration according to

(16)

Here, µ, which depends only on temperature, is the chemical potential in the standard state (Katchalsky and Curran, 1965).

By assuming ideal behavior, Eq. 15 effectively neglects long-range Coulombic interactions between ions, which, nevertheless exist even in dilute solutions (Bockris and Reddy, 1970). To account for this departure from ideal behavior, activity coefficients, ⁺ and (corresponding to the Na⁺ and Cl ions at concentrations equal to that of normal saline), are introduced as multiplicative factors in the argument of the natural logarithm appearing in Eq. 16. These activity coefficients, in general, depend on the local concentration (Robinson and Stokes, 1955). However, by Eq. 16, it is evident that only gradients in the chemical potential appear in the conservation equations governing the mobile ions. For our purposes, then, the departure from ideal behavior is dependent only upon ^± = (^±/c^±)c^±, and is negligible if ^±/c^{± ±}/c^± (Bockris and Reddy, 1970). For normal saline at 310 K, this condition is met if ^±/c^± 5 (Robinson and Stokes, 1955). Recalling our assumption that physiological deformations of the glycocalyx induce only small perturbations in the mobile ion concentrations, it follows that the variation in ^± with respect to c^±, which, in this context, is not likely to exceed 0.01, can safely be neglected. We therefore assert that the approximation implied by Eq. 15 is reasonable despite the long-range Coulombic interactions that undoubtedly arise.

For the solid matrix, we again assume ideal gas behavior and invoke the van't Hoff equation. The chemical potential, µ^s, is thus taken to be

(17)

Introducing an extra stress, denoted by ^E, that accounts for the tension stored in the solid matrix, the partial Cauchy stress tensor, ^s, is given by

(18)

Expressing the divergence of the partial Cauchy stress tensor for the water constituent in terms of the chemical potential, µ^w, we take (Lai et al., 1991)

(19)

Using the constitutive relations given by Eqs. 15, 18, and 19, together with the momentum equation for the mixture given by Eq. 7, provides the Gibbs-Duhem equation for the mixture under isothermal conditions (Tombs and Peacocke, 1974). Upon integration of the Gibbs-Duhem equation, the chemical potential for the water constituent is found to be

(20)

where the concentration of the solvent, c^w, is taken to be constant. Combining Eqs. 19 and 20 provides the constitutive equation for ^w given by

(21)

Reduced form of the equations

Combining the constitutive relationships given by Eqs. 15, 18, and 21 with the conservation equations given by Eqs. 3-5 provides the following reduced form of the momentum equations:

(22)

(23)

and

(24)

where we have introduced the diffusion coefficient, D^±, for the mobile ions by invoking the Stokes-Einstein equation, D^± = k_BT/f. Notice that, when Eqs. 22 and 23 are combined, Eq. 7 is obtained, which corresponds to the conservation of linear momentum for the mixture as a whole.

The equilibrium configuration

The expression for the flux of cations and anions given by Eq. 24 is identical to the electrochemical flux vectors used by Stace and Damiano (2001). In equilibrium, the flux of all constituents vanishes, and thus J⁺ = J = v^s = v^w = 0. In this state, Eqs. 22-24 reduce to

(25)

(26)

and

(27)

Using the definition of  from Eq. 8, we note that Eqs. 26 and 27 combine to give Eq. 25. In the absence of an applied stress traction at the luminal glycocalyx boundary, Eqs. 25 and 27, together with Gauss's law (Eq. 13) and the definition of  (Eq. 8), provide the closed system of equations necessary to determine the equilibrium configuration of the layer corresponding to a specified distribution for c^F(x, t₀). We shall choose this state as the prestressed reference configuration, where ^E(x, t₀) corresponds to the tensile stress distribution within the matrix in equilibrium. Under deformation of the layer, c^F becomes one of the unknown dependent variables.

Dimensional analysis of the governing equations

Using an asterisk to denote dimensionless variables, the independent and dependent variables are nondimensionalized as follows:

(28)

(29)

where c is the concentration of mobile cations in the blood at the center of the capillary lumen. Henceforth, it will be assumed that c = c. We have taken the capillary radius, R₀, as our characteristic length scale and _c as a characteristic time scale to be assigned later. In addition, the dimensionless permeability, ion fluxes, extra-stress tensor, and osmotic pressure are defined as

(30)

where k₀ is the solid-matrix permeability to water in the reference configuration. Using these definitions, the dimensionless equations governing Gauss's law and the equilibrium configuration are given respectively by

(31)

(32)

(33)

and

(34)

The three dimensionless groups that arise are

(35)

where  denotes the ratio of glycocalyx concentration to cation concentration in blood, and ₀ denotes the ratio of glycocalyx fixed-charge density to the monovalent cationic charge density of blood. In aqueous solutions containing 0.14 M NaCl concentrations, such as normal saline or whole blood, Q  2 × 10⁸ for a characteristic length scale associated with a 5-µm-diameter capillary.

The macromolecular concentrations of mucopolysaccharide structures devoid of collagen are typically low compared with the ion concentration of normal saline. Assuming this to be the case for the glycocalyx, and taking the mean valence, n, of the proteoglycan/glycoprotein/glycosaminoglycan aggregates of the layer to be large compared with unity, we assume that the dimensionless parameters are ordered such that Q¹   ₀ 1.

According to the scaling rules defined above, c*^± ~ O(1), c*^F ~ O(1), and *c*^F ~ O(1). Furthermore, in equilibrium, it can be shown that local departures from electroneutrality are small such that * ~ O(₀/Q) (Stace and Damiano, 2001). We will later show that this assertion is consistent with the analysis that follows. From the dimensionless form of Gauss's law, given by the first of Eq. 31, we must then require that E* = *V* ~ O(₀/Q); and, although z⁺c*⁺ ~ O(1) and zc* ~ O(1), by the definition of * given by the second of Eq. 31, we see that the sum of these two quantities is O(₀). That is,

(36)

Using our result that QE* ~ O(₀) in Eq. 32, we obtain

(37)

If the equation associated with the cations in Eq. 34 is added to the equation associated with the anions, and the result of Eq. 36 is used, we obtain

(38)

Finally, using Eqs. 36 and 37 in Eq. 33, we obtain

(39)

which is consistent with Eq. 38 if the terms O() on either side cancel, which they therefore must. From this we see the self consistency of the ordering we have given.

Axisymmetric equations for purely radial finite deformations of the glycocalyx

In Appendix A, the governing equations are written in the Eulerian cylindrical coordinates (r, , z) and then transformed into the Lagrangian cylindrical coordinates (R, , Z) for the special case of purely radial axisymmetric deformations in the absence of axial flow. There it is shown that, under such conditions, the continuity equation and boundary conditions (see Appendix C for details) for the fluid constituent require that the radial fluid velocity component must also vanish.

In Eulerian variables, the momentum equation for the solid constituent (see Eq. A2) is defined on the unknown domain a(t)  r  R₀, where a(t) is the instantaneous radial position of the glycocalyx interface with the plasma in the lumen, and a_o = a(t₀) is the value of a in the equilibrium configuration. When dealing with large deformations of the glycocalyx, it becomes more convenient to formulate the problem in material coordinates. In Appendix A, we develop the fully nonlinear mechano-electrochemical model of the glycocalyx that applies over the entire range of physiologically relevant radial deformations of the layer. Below, we summarize these one-dimensional equations but leave the details of their development to Appendix A.

Purely radial, axisymmetric, finite deformations of the layer, in which material points are displaced only along radial coordinate lines, constitute a principal state of strain. Under this loading configuration, the Lagrangian and Eulerian coordinates are related according to r*(R*, t*) = R* + U^*_R(R*, t*),  = , and z* = Z* where U^*_R is the dimensionless radial displacement of the solid matrix and is the only nonvanishing component of the solid matrix displacement field. The principal stretch ratios are then given by _r = r*/R*,  = r*/R*, and _z = 1. In dimensionless axisymmetric Lagrangian variables, for purely radial deformations, the dimensionless nonlinear diffusion equation and the ion mass conservation and flux equations are given, respectively, by

(40)

and

where

(43)

(44)

(45)

₀ = a₀/R₀, and D = k₀k_BTc. For the deformations considered here, the continuity equation for the solid matrix given by Eq. 11 reduces to Eq. 45 where f(R*) corresponds to the normalized shape of the glycocalyx concentration distribution in the reference configuration and J = det F = _r. In general, f(R*) = 0 for R* < ₀, and increases sharply to a maximum of 1 for R* > ₀. For computational purposes, it will be taken to be a unit-step function, f(R*) = H(R*  ₀), but this is not assumed in the derivation of the equations that follow. The parameter D represents the effective Fickian diffusion coefficient, in the reference configuration, if the glycocalyx matrix were hydrated in a nonelectrolytic solution.

The pseudo-equilibrium approximation for monovalent ionic transport

Although physiologic deformations of the glycocalyx can typically be large, if   ₀ 1, these deformations induce only small perturbations in the ion concentrations away from their initial equilibrium distributions at t₀. However, we shall not assume that the ion distributions, c*⁺(R*, t*) and c*(R*, t*), remain equal to their initial equilibrium distributions, but rather that c*⁺(R*, t*) and c*(R*, t*) change in such a way as to continuously correspond to the equilibrium distributions associated with each instantaneous glycocalyx distribution, c*^F(R*, t*). This is equivalent to assuming that the transport of monovalent mobile ions occurs as a pseudo-equilibrium process. Significant simplifications in the analysis can be made by invoking this pseudo-equilibrium approximation because many of the relationships derived by Stace and Damiano (2001) for the case of static * can be adapted to the problem with deformation. Physically, the motivation for this assumption lies in the realization that free unbalanced electric charges generate immense electrostatic forces that drive the system back to local charge neutrality very rapidly. Even small charge imbalances give rise to large electromotive forces. Furthermore, because the characteristic Fickian diffusion time, R/D⁺, associated with the mobile ions is many orders of magnitude smaller than the characteristic Fickian diffusion time, R/D, associated with the glycocalyx molecules, the cations experience very little resistance to their motion through the solvent. Thus, strong electromotive forces combined with low ionic drag result in a rapid and continuous redistribution of ions into their instantaneous and nearly electroneutral equilibrium configurations.

We now establish this formally. Suppose we choose _c such that _t*(R*) on the left-hand side of Eq. 40 is O(1). On the basis of Eq. 36, we make the assertion, which we will later show to be self-consistent, that the total variation in c*^± is O(₀). Because the variation in R* is O(1), the quantity _t*c*^±/_t*(R*) is O(₀), and, therefore, if we divide the left-hand side of Eq. 41 by the left-hand side of Eq. 40, we see that this ratio is O(₀D^F/D^±), which is a very small quantity because D^F D^±. We also note that _c cancels out of this ratio, so the choice of _c made above is not restrictive. However, the terms on the right-hand side of Eq. 40 are O(1 + /), so the left-hand side must be O(1 + /) or smaller. Therefore, if / is O(1) or smaller, the left-hand side of Eq. 41 is O(₀D^F/D^±) or smaller.

In contrast, the right-hand side of Eq. 41 is O(₀), and we have just established that the left-hand side is O(₀D^F/D^±) or smaller, thus making it O(D^F/D^±) smaller than the right-hand side. (More precisely, the arguments made here lead to the same approximation even if / 1, as long as (D^F/D^±) (/) 1.) We therefore neglect the left-hand side in comparison to the right-hand side and set the latter to zero, from which we conclude that R*J^*_r^±(R*, t*) is spatially uniform, and, further, that the ionic flux, J^*_r^±, must vanish identically to match the zero-flux boundary condition at R* = 1 (see Appendix C).