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. Author manuscript; available in PMC: 2022 Apr 5.
Published in final edited form as: Arch Appl Mech. 2021 Jun 3;92(2):447–459. doi: 10.1007/s00419-021-01985-3

A multiphasic model for determination of water and solute transport across the arterial wall: effects of elastic fiber defects

Young Guang 1, Austin J Cocciolone 2, Christie L Crandall 3, Benjamin B Johnston 4, Lori A Setton 5, Jessica E Wagenseil 6
PMCID: PMC8983017  NIHMSID: NIHMS1788626  PMID: 35386426

Abstract

Transport of solute across the arterial wall is a process driven by both convection and diffusion. In disease, the elastic fibers in the arterial wall are disrupted and lead to altered fluid and mass transport kinetics. A computational mixture model was used to numerically match previously published data of fluid and solute permeation experiments in groups of mouse arteries with genetic (knockout of fibulin-5) or chemical (treatment with elastase) disruption of elastic fibers. A biphasic model of fluid permeation indicated the governing property to be the hydraulic permeability, which was estimated to be 1.52×10–9, 1.01×10–8, and 1.07×10–8 mm4/μN.s for control, knockout, and elastase groups, respectively. A multiphasic model incorporating solute transport was used to estimate effective diffusivities that were dependent on molecular weight, consistent with expected transport behaviors in multiphasic biological tissues. The effective diffusivity for the 4 kDA FITC-dextran solute, but not the 70 or 150 kDa FITC-dextran solutes, was dependent on elastic fiber structure, with increasing values from control to knockout to elastase groups, suggesting that elastic fiber disruption affects transport of lower molecular weight solutes. The model used here sets the groundwork for future work investigating transport through the arterial wall.

Keywords: Biphasic, Febio, Permeation, Dextran, Mouse model, Elastin, Elastase, Finite element

1. Introduction

The elastic fiber network is a major extracellular matrix component of the large conduit arteries. Composed of cross-linked elastic fibers made up of elastin and microfibrils, the elastic fiber network is organized in concentric layers called elastic laminae [1]. The internal elastic lamina separates the single layer of endothelial cells at the arterial lumen from the smooth muscle cells in the medial layer. Additional elastic laminae alternate with layers of smooth muscle cells, associated collagen fibers, and interconnecting elastic fibers throughout the medial layer of the arterial wall [2].

Defects in the elastic fiber network are common to many cardiovascular diseases but are a direct factor in the onset, progression, and rupture of aneurysms. Aneurysms are defined as a permanent localized dilation of the vascular diameter at least 50% greater than the normal size and may dissect or rupture leading to severe cardiovascular complications or death [3]. Elastic fiber defects in aneurysmal disease can be caused by genetic mutations that affect elastic fiber assembly [4] or by increased activity of proteases that degrade elastic fibers due to inflammation associated with risk factors such as smoking, alcohol consumption, aging, obesity, and hypertension [5]. Elastic fiber defects manifest as an absence or fragmentation of the elastic laminae, allowing arterial dilation and subsequent wall remodeling.

Pharmacokinetics, or the transport of solutes across the arterial wall associated with progression or treatment of disease, may be altered due to severe tissue remodeling in aneurysms. Previous studies illustrate that both convection and diffusion play a role in governing solute transport across the arterial wall in a manner that is dependent on solute size, porosity, and deformation of the wall, and microstructural components including smooth muscle cells and extracellular matrix components, such as the elastic laminae [6–9]. In work from Tarbell and collaborators, a “mechano-hydraulic model” of the arterial wall as a deformable and porous construct was developed by accounting for the physical presence and organization of smooth muscle cells, elastic fibers, collagen, and proteoglycan constituents [10–14]. This model enabled the prediction that solute, i.e., albumin, transport is dominated by convection rather than diffusion, supporting experimental studies that find the same for medium to high molecular weight solutes [15–17]. This body of work was important for demonstrating that extracellular matrix components can control the mechanics of water and solute transport in arteries, for representing the strain-dependent porosity of arteries, and for incorporating a coupling of fluid to solid phases in these structures. Still, this approach relies upon a priori assumptions of microstructural composition and morphometric information that may be lacking for many arteries, including diseased arteries of interest here.

Continuum models of fluid–solid interactions have been developed to represent mechanical contributions of deforming extracellular matrices in soft tissues, as well as strain-dependent porosities, hydraulic permeability, and solute diffusivities that may govern the movement of fluid and solute through tissue [18–20]. Here we develop a model of the arterial wall as a continuum mixture of fluid, solid and solute phases to determine intrinsic coefficients of hydraulic permeability and effective diffusivity from experimental data on arterial conductance in denuded (endothelial cell layer removed) mouse arteries [21]. The use of a multiphasic model here enables the study of an aqueous fluid phase as distinct from that of the solute phase, to reveal how fluid movement can impact solute transport in a porous and permeable solid. Using an axisymmetric finite element model of the arterial wall in FEBio [22], we numerically match model predictions for solute flux to measured volume and solute flux for arteries from wild-type control (CTL) and fibulin-5 knockout (KO) mice as a means to determine a role for elastic fiber fragmentation due to genetic mutations on arterial transport. We additionally determine properties of arteries following luminal elastase treatment (ELA) to independently assess a role for elastic fiber disruption due to protease digestion on arterial mechanics [21].

We use our multiphasic mixture model to address the hypothesis that genetic or proteolytic disruption of the elastic fibers increases intrinsic coefficients of hydraulic permeability and effective diffusivity in the arterial wall. These intrinsic coefficients are critical for understanding the transport of solutes that may play a role in the progression or treatment of arterial diseases, such as aneurysms, that are associated with elastic fiber defects.

2. Methods

2.1. Experimental data

Experiments were performed to evaluate perfusion of fluid and solute in mouse carotid arteries as described by Cocciolone and co-workers [21]. In brief, wild-type control (CTL), fibulin-5 knockout (Fbln5−/−) (KO) [22], or elastase-treated (ELA) denuded mouse carotid arteries were mounted on a pressure myograph in phosphate-buffered saline (N = 5–10/group). KO arteries have fragmented elastic fibers due to the loss of fibulin-5, a protein necessary for elastic fiber assembly [23]. ELA arteries have disrupted elastic fibers due to a brief treatment with elastase, a protease that degrades elastic fibers [24]. The carotids were pressurized to 100 mmHg (13.3 kPa), and volumetric flow rate, Q, through the arterial wall was calculated from the steady-state displacement of a bubble in the myograph inlet tubing (Fig. 1). Solute flux of the arterial tissue was determined by adding 2.5 mg/mL of 4, 75, or 150 kDa fluorescein isothiocyanate (FITC)-dextran (Sigma-Aldrich; #46,944, #46,945, and #46,946, respectively) to the arterial lumen (Clum) and measuring the concentration of FITC-dextran in the myograph fluid bath (Cbath) over time (Fig. 1). These data were used by Cocciolone and co-workers to calculate a transmural hydraulic conductance for both fluid and FITC-dextrans [21]; here, they were used to estimate intrinsic hydraulic permeability of the porous and permeable arterial wall, and effective diffusivities governing diffusion of the FITC-dextran solutes, as described below.

Fig. 1.

Fig. 1

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Schematic (a) and photographs (b) of the transport experiments (not to scale). An artery was mounted on a pressure myograph in a bath of phosphate-buffered saline. The myograph inlet was connected to a hydrostatic pressure column (100 mmHg/13.3 kPa) with a reservoir of phosphate-buffered saline that was either solute free or contained 2.5 mg/mL of 4, 70, or 150 kDa FITC-dextran. The direction of flow is from the column toward the arterial lumen and then across the wall of the artery, as the myograph outlet is clamped. Displacement of an air bubble in the inlet tubing was used to determine volumetric fluid flow through the arterial wall and calculate hydraulic conductance. Total solute flux through the arterial wall was measured by recording the starting concentration (Clum) of FITC-dextran within the arterial lumen and the change in FITC-dextran concentration (Cbath) in the external bath over time. Ri and Ro are the inner and outer radii of the artery, respectively, while Rb is the equivalent radius of the 10-mL volume bath. These measurements were used to model the experimental setup in FEBio

2.2. Biphasic computational model prediction of hydraulic flux

A biphasic model of the arterial wall as a porous, fluid-saturated mixture was constructed in FEBio [22] to simulate the flow of fluid under an intraluminal pressure gradient. The arterial wall was modeled as an axisymmetric cylinder with an inner radius Ri = 230μm, outer radius Ro = 269μm, and a length of 5.42 mm, taken from the average dimensions of mouse carotid arteries in the pressurized, axially stretched state [21]. The assumption of axisymmetry with spatial gradients only in the radial direction enabled modeling of a segment of the arterial wall cross section (3° arc) [25] meshed as shown in Fig. 2a using eight-node hexahedral elements refined closer to the inner radius [22]. In preliminary studies, we confirmed that this model of an axisymmetric radial arc gave rise to equivalent model predictions as for the full 360° cross-sectional area, with reduced computational time.

Fig. 2.

Fig. 2

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Schematic of the model geometry and variables. a The arterial wall is modeled as an axisymmetric wedge of 3° (yellow) with a wall thickness defined by the inner (Ri) and outer (Ro) radii. In addition, the model geometry requires specification of an equivalent radius for the 10-mL fluid bath, Rb. b The table provides the multiphasic model parameters for the arterial wall.

The solid phase for the wall was represented as an isotropic, neo-Hookean solid with uniform modulus (E) and Poisson’s ratio (ν) and with a strain-dependent Holmes–Mow hydraulic permeability k(J) [26] as,

k(J)=k0(J−ϕs1−ϕs)αe12M(J2−1), (1)

where k0 is the reference state hydraulic permeability, J is the Jacobian of the deformation, ϕs is the solid volume fraction, α is the power-law exponent, and M is a coefficient defining the nonlinearity of the strain-dependent effect. In model development, the Poisson’s ratio (ν) was assumed to be 0.4 [27] and the modulus (E), hydraulic permeability in the reference state (k0), and solid volume fraction (ϕs) were varied to test for theoretical solution sensitivity to parameter choice. In the absence of experimental information indicating a nonlinear strain dependence of the hydraulic permeability, the values for α and M were assumed to be unity (Fig. 2b).

The test bath was modeled as a biphasic material and thus was prescribed solid material properties, even though the solid volume fraction (ϕs) is zero to model the fluid phase. As the fluid test bath was fully interfaced with the biphasic model of the arterial wall, parameters for the solid phase of the fluid bath were selected and chosen to represent an incompressible material including a Poisson’s ratio (ν) of 0.49, modulus (E) of 100,000 kPa (much greater than that of the arterial wall), and a constant isotropic hydraulic permeability (ko) of 1×10−9mm4/μN.s (similar to the arterial wall).

Boundary conditions were set to 100 mmHg (13.3 kPa) effective fluid pressure at the inner wall and 0 mmHg on the outer wall, with displacements constrained in the θ- and z-directions. Plots of finite element model (FEM)-predicted steady-state fluid flux were obtained for this range of parameters to reveal a role for E, k0, and ϕs of the arterial wall in contributing to predictions of fluid flux (w) as defined by,

w=−k•(∇p), (2)

where k = k(J)I and ∇p is the gradient of the fluid pressure.

In comparison with experimental data, values for the solid volume fraction (ϕs) and the modulus (E) for the arterial wall were selected from the parametric sensitivity study in model development, and FEM predictions of fluid flux at the inner arterial wall (w) were numerically compared to the experimentally determined values for fluid flux (Jv) [21] as defined by,

|w|=Jv=QAA, (3)

where Q is the volumetric flow rate and AA is the surface area of the inner arterial wall measured in the experimental studies, to determine the intrinsic permeability in the reference state (k0) for each arterial tissue type (CTL, KO, and ELA) (Fig. 2b).

2.3. Multiphasic computational model of solute flux

The transmural solute flux experiment was also modeled in FEBio [22] as described here. Values for the outer arterial radii were obtained from experimentally measured values for each tested artery [21], with an assumed arterial wall thickness of 40 μm to construct a representative axisymmetric mesh of the arterial wall as a porous and permeable, neo-Hookean solid for the FEM. Values for the modulus (E) and solid volume fraction (ϕs) of the arterial wall were chosen from the results of the parametric sensitivity to hydraulic conductance as described in the prior section with values of E = 400 Pa and ϕs = 0.6 for all arteries (Fig. 2b). Poisson’s ratio (ν) was assumed to be 0.4 [27] for all arteries. The hydraulic permeability in the reference state (k0) for each tissue type was estimated from the corresponding data for the hydraulic conductance experiment in Cocciolone et al. [21], as described in the preceding section (Fig. 2b).

In the model of the solute flux experiment, w is the volumetric flux of solvent relative to the solid and j is the molar flux of solute relative to the solid. In general, w and j are given by [28],

w=−k⋅(∇p+RTdDfree⋅∇c) (4)
j=d⋅(−ϕw∇c+cDfreew) (5)

where k = k(J) I and k(j) is defined in Eq. 1, ∇p is the gradient of the effective pressure (p),

p=p0+RTcΦ (6)

po is the hydrostatic fluid pressure, R is the ideal gas constant and is equal to 8.314 J/mol K, T is the absolute temperature and is equal to 310 K, c is the solute concentration, Φ is the osmotic coefficient and is assumed to be unity, d = Def f I, ∇c is the gradient of the solute concentration, ϕw is the solvent volume fraction, and Dfree is the diffusivity in free solution.

Balance of momentum for the mixture was enforced subject to boundary conditions corresponding to the solute transport experiment. Hydrostatic fluid pressure, po, between the bath and the lumen was 100 mmHg (13.3 kPa). The solute concentration at the inner wall corresponding to the lumen (Clum) was held constant for each solute (4, 70, 150 kDa FITC-dextran). The free diffusivity (Dfree) (Fig. 2b) was estimated for each solute using the approximate Stokes radii for each molecule provided by the manufacturer. FITC-dextran particles were imaged with transmission electron microscopy, as described below, to confirm their approximate size and shape.

The 10-ml test bath used for collection of diffusing solute was modeled as a multiphasic material with solid volume fraction ϕs = 0, representative of a fluid phase at the outer boundary of the arterial wall. The diffusivity for each solute in the bath was modeled with “well-mixed conditions” (i.e., assumed diffusivity 10,000 × higher than free diffusivity) [29, 30]. For these conditions, an effective diffusivity (Deff) for each solute within the arterial wall was obtained by minimizing the difference between the calculated solute flux at the outer wall to that measured experimentally in the test bath with a nonlinear Levenberg–Marquardt optimization algorithm available in FEBio [22]. Following the optimization to determine the effective diffusivity (Deff), the fluid flow (from Eq. 4) and solute flux (Eq. 5) were predicted to estimate the contribution of fluid movement to the total solute flux, as measured.

2.4. Transmission electron microscopy

FITC-dextran particles of each molecular weight were suspended at low density (0.02 mg/ml) in phosphate-buffered saline. Glow-discharged carbon-coated 200 mesh copper grids were placed on a 10 μl drop of the FITC-dextran solution and incubated for 1 min at room temperature. Post-incubation, the grids were washed serially with 5 ddH2O drops and stained with 0.75% uranyl formate for 2 min. Excess uranyl formate was blotted off using filter paper then the grids were air dried. Grids were imaged on a JEOL 1400 transmission electron microscope equipped with an AMT CCD camera operating at 120 kV at 80,000 × nominal magnification resulting in a magnified pixel size of 0.23 nm. Representative images of clearly delineated particles for each molecular weight were captured.

2.5. Statistical analysis

Values for the effective diffusivity (Deff) for each of the three FITC-dextran molecules in the arterial wall were determined by nonlinear optimization to experimental solute flux data for CTL, KO, and ELA carotid arteries on a specimen-specific basis. Tests of normality were performed and differences between tissue types and among solutes were analyzed with a two-way ANOVA followed by Tukey’s post hoc test at a significance level of p < 0.05.

3. Results

Transmission electron microscopy images of the FITC-dextran particles show that they are approximately circular with a radius comparable to the Stokes radii reported by the manufacturer (1.4, 6, and 8.5 nm, respectively for 4, 70, and 150 kDa FITC-dextran) (Fig. 3). Detailed measurements were not taken as size and shape are likely affected by the processing required for imaging. However, the images confirm that Dfree calculations based on a spherical particle with the reported Stokes radii are a reasonable assumption.

Fig. 3.

Fig. 3

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Transmission electron micrographs of individual FITC-dextran particles of 4 (a), 70 (b), and 150 (c) kDa molecular weight. The particles are approximately circular with sizes near the reported Stokes radii from the manufacturer. Scale bar = 10 nm

FEM predictions of fluid flux for a range of parametric values for solid volume fraction, modulus, and hydraulic permeability of the arterial wall are shown in Fig. 4. Fluid flux through the model arterial wall was determined to be insensitive to solid volume fraction (ϕs) for values in the range of 0.6−0.8, contributing less than 4% variation (Fig. 4a) and covering the range of solid volume fractions measured for undeformed and deformed arterial tissue [31]. Thus, we assumed a value of ϕs = 0.6 in further models for estimation of hydraulic permeability. Similarly, the predicted fluid flux was relatively insensitive to values for moduli (E) in the range of 126 kPa–1 MPa, contributing less than 8% variation (Fig. 4b). As the modulus of the mouse arterial wall has been measured to be within this range [32, 33], we chose a value of E = 400 kPa without introducing large variations in predictions of fluid flux, as shown by the parametric study results.

Fig. 4.

Fig. 4

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Parametric study for determination of biphasic model parameters. a Fluid flux plotted against hydraulic permeability for differing values of solid volume fraction and assumed modulus (E = 251.2 kPa). b Fluid flux plotted against hydraulic permeability for differing modulus values and an assumed solid volume fraction (ϕs = 0.6). Hydraulic permeability is the main determinant of fluid flux and fluid flux is largely independent of solid volume fraction and modulus within ranges considered physiologic for the mouse carotid artery

With fixed values for E and ϕs, fluid flux is linearly dependent on hydraulic permeability (Fig. 4). FEM predictions of fluid flux over time were matched to experimental data [21] to obtain values for hydraulic permeability (ko) of 1.52×10−9, 1.01×10−8, and 1.07×10−8 mm4/μN.s for CTL, KO and ELA groups, respectively. These values were then used as model inputs for the solute flux predictions (Fig. 2b).

An effective diffusivity (Deff) for each solute within the arterial wall was obtained by nonlinear optimization of the model predicted solute flux (Eq. 5) to the bath concentration (Cbath) over time measured experimentally for individual artery specimens. Representative FEBio predictions and experimental measurements are shown in Fig. 5. Results of the nonlinear optimization yielded an effective diffusivity (Deff) that showed a dependence on solute molecular weight (Fig. 6a). Deff significantly varied among arterial tissue types for the low molecular weight solute (4 kDa FITC-dextran) (p < 0.05), with the ELA group having the highest Deff. No differences in Deff were observed across tissue types for either the 70 kDa or 150 kDa FITC-dextran; however, there was a trend of increasing Deff with elastic fiber fragmentation for the 70 kDa FITC-dextran that was consistent with that for the 4 kDa FITC-dextran. Further, the highest molecular weight solute (150 kDa FITC-dextran) had the lowest Deff in all tissue groups.

Fig. 5.

Fig. 5

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Representative experimental data (symbols) and model predictions (lines) for the solute transport experiments for individual artery specimens. Model predictions were obtained by nonlinear optimization of the effective diffusivity (Deff) to minimize differences between the experimental and calculated solute flux (Eq. 5). Concentration in the external bath (Cbath) over time was recorded experimentally and predicted computationally for 4 (a), 70 (b), and 150 (c) kDa FITC-dextran after addition to the arterial lumen in wild-type (CTL), Fbln5−/− (KO), and elastase-treated (ELA) mouse carotid arteries

Fig. 6.

Fig. 6

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Effective diffusivity values and simulated fluid flow and solute flux for the multiphasic model. a Effective diffusivity (Deff) values were determined from the model fits to solute flux data on a specimen-specific basis. * = p < 0.05 among all tissue types for 4 kDa FITC-dextran. # = p < 0.05 for 4 kDa compared to 70 and 150 kDa FITC-dextran for KO tissue. & = p < 0.05 for 4 kDa compared to 70 and 150 kDa FITC-dextran for ELA tissue. Significance was determined by two-way ANOVA followed by Tukey’s post hoc test. b Corresponding predictions of relative fluid flow and c solute flux shown on a specimen-specific basis. Fluid flow depends on the imposed pressure gradient, arterial geometry, and hydraulic permeability, and so varies among the CTL, KO and ELA conditions as expected. Solute flux similarly varies among CTL, KO and ELA conditions but is also found to be much higher for the transport of the 4 kDa FITC-dextran due to the increased effective diffusivity values. Significance was not determined for data shown in panels B and C because the values are derived from previously determined values. N = 5–10/group

Simulated volumetric fluid flow through the arterial wall in the solute flux experiment was different among the three groups of arterial tissue types (Fig. 6b). The CTL arteries were predicted to have the lowest fluid flow for all solutes while arteries in the ELA group have the highest fluid flow. The fluid flow behaviors within each arterial tissue type correspond to the hydraulic permeabilities as determined from the hydraulic conductance experiments, vary according to the geometry of each individual arterial specimen, and reflect the presence of a hydraulic pressure gradient across the arterial wall. The simulated solute flux similarly varies across arterial tissue types and according to individual specimen geometry (Fig. 6c) and is much higher for the 4 kDa compared to the 70 and 150 kDa FITC-dextrans, reflecting the differences in effective diffusivities. Statistical analyses were not performed on the simulated fluid flow and solute flux as they are derived values from previously determined parameters.

4. Discussion

Previous investigations have shown the importance of considering the arterial wall as a deformable and porous construct to estimate fluid and solute flux [6–9]. While prior models represented the microstructural elements of smooth muscle cells, elastic fibers, collagen, and proteoglycan constituents as regulators of arterial wall transport [10–14], we adopted a continuum model here that obviates the need to independently estimate microstructural parameters of apparent pore size or biochemical composition. Our multiphasic model is capable of revealing how the aqueous fluid phase can move relative to the solute (also fluid) phase. In this manner, we can understand the contributions of a hydraulic pressure relative to an osmotic pressure in driving transmural solute transport and the impact that solute size, or arterial wall microstructure, might have on those contributions. Our results show that the mobile aqueous phase dominates fluid and solute transport here, particularly for small solutes, as a result of the applied hydraulic pressure gradient that dominates over gradients in chemical potential. A parametric study revealed that the model predictions for fluid flux in this geometry were relatively insensitive to the arterial wall modulus (< 8% variation between E = 125 kPa–1 MPa) and solid volume fraction (< 4% variation between ϕs = 0.6–0.8) within a range of physiologic values, and that the movement of fluid could be governed by a single term corresponding to the undeformed, or reference state hydraulic permeability (k0).

Estimates for k0 given here are the first available for the arterial wall using a biphasic continuum model and show evidence of differences with genetic mutation or proteolytic cleavage of arterial wall constituents, namely fragmentation of elastic laminae as observed in aneurysmal disease. We estimate k0 values on the order of 10−9 mm4/μN.s for control arteries and 10−8 mm4/μN.s for arteries with genetic or proteolytic fragmentation of the elastic laminae (Fig. 2b). Our estimates are of the same order of magnitude as those for bovine cartilage (10−9 mm4/μN.s) [34] and porcine skin (10−8 mm4/μN.s) [35], and lower than those obtained for mouse skin and tendon (10−7 mm4/μN.s) [36]. Additional studies are needed to confirm our values of ko for arterial tissue and investigate nonlinear strain dependence.

An advantage of the continuum mixture approach is the ability to model additional phases representing solvent and solute flux through porous and permeable solid materials. Following the estimation of parameters governing fluid flow in the hydraulic conductance experiment, we incorporated a solvent phase representing FITC-dextran within the fluid to estimate solute flux into the test bath under a concentration gradient. Thus, we were able to estimate an effective diffusivity (Deff) within the arterial wall tissue that showed a dependence upon molecular weight (Fig. 6a), as expected [9, 37]. Our calculated Deff ranges from 14 to 30% (CTL), 10 to 36% (KO), and 41 to 72% (ELA) of the Dfree value for each size FITC-dextran molecule based on a Stokes radius calculation, indicating altered diffusion within the arterial wall tissue and a further dependence on elastic fiber structure. Our calculated Deff for CTL arterial tissue is 5–10 times higher than Deff values for similar sized dextran or albumin molecules within the cell cytoplasm [37, 38], suggesting that the FITC-dextran molecules are moving around (not through) the cells within the arterial wall. Our Deff values are similar to those measured by Hwang and Edelman [9] for transmural transport of albumin across the bovine arterial wall, however their Deff values for planar transport in rectangular tissue samples are 1–2 orders of magnitude higher than albumin diffusion in free solution. We assumed transmural (radial) transport only in our current model. Anisotropy of the multiphasic mechanical behavior should be investigated in future work.

In addition to predicting molecular size-dependent effective diffusivities, the model predicted that Deff was sensitive to elastic fiber structure in the arterial wall for the 4 kDa solute, but not for the 70 or 150 kDa solutes (Fig. 6a). There was a trend toward dependence on elastic fiber structure for the larger molecules in some cases. For example, Deff for low, medium and high molecular weight FITC-dextran was highest in the ELA arteries, as compared to KO and CTL groups. The results indicate that elastic fiber structure affects solute transport in a size-dependent manner. Size-dependent molecular sieving that depends on extracellular matrix content has been well described in cartilage tissue [39]. Studies of solute diffusion under a concentration gradient only, without the imposition of a hydraulic pressure gradient across the wall, would be useful to confirm whether Deff is differentially affected by elastic fiber organization in the arterial wall.

An important limitation of the experimental study, as well as the FEM prediction, is the absence of the intact endothelial cell layer that is a key regulator of fluid and solute transport in healthy arterial walls [40]. However, local loss of or increased permeability of the endothelial cell layer is a hallmark of many cardiovascular diseases, including aneurysms. An advantage of the approach used here is the ability to incorporate layers corresponding to the endothelial cells and concentric layers of smooth muscle cells and elastic laminae that could be used to model different disease processes in future work. As data required to independently determine the full set of physical properties were not available even for the simple denuded homogeneous arterial wall used here, we chose to focus on assessing the hydraulic permeability and effective diffusivity as a first step toward this goal. The continuum model allows separation of diffusivity and fluid flow, which is important for understanding different contributions of diffusive and advective transport. Our modeling results suggest that convection plays a dominant role in this experimental configuration (Figs. 6b and c), consistent with prior studies in large arteries with and without an intact endothelial cell layer [15–17].

Our continuum model begins with a volume-averaged approach that assumes a homogeneous and continuous fluid phase throughout the arterial wall that is not impacted by local variations in wall architecture at the nanometer level. These features could be incorporated with representations of solid phase microstructure in future work with evidence that fluid–solid interactions at the nanometer-level impact fluid transport beyond the porosity and permeability. Many different features contribute to the microstructure of the solid phase (e.g., elastic fibers and collagen) and also the composition of the aqueous fluid phase (i.e., charged macromolecules). Additional features of solute or matrix charge that could affect binding of the molecules during transport, or cellular uptake, could also be included in future models. There are numerous examples of elastic laminae binding with macromolecules in the arterial wall [41–43]. Hydrophobic molecules, such as paclitaxel and sirolimus, routinely used to target the arterial wall for therapeutic purposes [44] can locally partition resulting in wall concentrations higher than applied levels [45]. Degradation of elastic laminae with aneurysmal disease may affect molecular binding or partitioning within the arterial wall.

Despite the benefits of the continuum model approach, the predictions rely upon assumed parameters that have not been independently measured for mouse arteries, and that would need to be experimentally determined for the CTL, KO and ELA groups. However, the parameters (namely the solid volume fraction and Young’s modulus of the arterial wall) used in the current study are within physiologic ranges and our sensitivity study showed that variations over a broader range had negligible effects on fluid flux. Additional studies could be undertaken to determine nonlinear strain-dependent material properties, solid volume fraction, and their changes with experimental treatment. Further studies with more complex, structurally based constitutive models may reveal additional insight into how fluid–solid interactions impact transport across the arterial wall. The modeling developed here lays out an approach that can be used to simulate transport under different boundary conditions or microstructure to reveal how genetic mutations in or proteolytic degradation of elastic fibers affect specific features of arterial mechanics relevant for progression or treatment of human disease.

Funding

This work was supported by grants from the American Heart Association 19TPA-34910047 (JEW) and the National Institute of Arthritis and Musculoskeletal and Skin Diseases of the NIH under award number R01AR070975 (LAS). The FEBio software suite has been developed with partial support from the NIH. Transmission electron microscopy was performed by the Washington University Center for Cellular Imaging supported by Washington University School of Medicine, the Children’s Discovery Institute of Washington University, St. Louis Children’s Hospital (CDI-CORE-2015-505 and CDI-CORE-2019-813), and the Foundation for Barnes-Jewish Hospital (3770 and 4642).

Footnotes

Code availability The model was developed in FEBio, an open-source software. Code for the model is available on request from the corresponding author.

Conflict of interest None.

Contributor Information

Young Guang, Department of Biomedical Engineering, Washington University, St. Louis, MO, USA.

Austin J. Cocciolone, Department of Biomedical Engineering, Washington University, St. Louis, MO, USA

Christie L. Crandall, Department of Mechanical Engineering and Materials Science, Washington University, St. Louis, MO, USA

Benjamin B. Johnston, Department of Biomedical Engineering, Washington University, St. Louis, MO, USA

Lori A. Setton, Department of Biomedical Engineering, Washington University, St. Louis, MO, USA

Jessica E. Wagenseil, Department of Mechanical Engineering and Materials Science, Washington University, St. Louis, MO, USA.

Availability of data

All data are included in the published manuscript or references.

References

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