Local Versus Global Mechanical Effects of Intramural Swelling in Carotid Arteries

Short abstract

Glycosaminoglycans (GAGs) are increasingly thought to play important roles in arterial mechanics and mechanobiology. We recently suggested that these highly negatively charged molecules, well known for their important contributions to cartilage mechanics, can pressurize intralamellar units in elastic arteries via a localized swelling process and thereby impact both smooth muscle mechanosensing and structural integrity. In this paper, we report osmotic loading experiments on murine common carotid arteries that revealed different degrees and extents of transmural swelling. Overall geometry changed significantly with exposure to hypo-osmotic solutions, as expected, yet mean pressure-outer diameter behaviors remained largely the same. Histological analyses revealed further that the swelling was not always distributed uniformly despite being confined primarily to the media. This unexpected finding guided a theoretical study of effects of different distributions of swelling on the wall stress. Results suggested that intramural swelling can introduce highly localized changes in the wall mechanics that could induce differential mechanobiological responses across the wall. There is, therefore, a need to focus on local, not global, mechanics when examining issues such as swelling-induced mechanosensing.

Introduction

It has long been known that the passive mechanical behavior of healthy central arteries is dominated by elastin and collagen fibers [1], which in turn has motivated most recent constitutive formulations [2–4]. The related neglect of GAGs in healthy arteries has been justified mainly by their low mass fraction, typically 0.03–0.05. Nevertheless, accumulating experimental and theoretical findings suggest that GAGs play important roles in arterial wall mechanics and mechanobiology in both health and disease. For example, Ref. [5] showed that there is a 33% increase of GAGs in hypertensive when compared to normotensive rats. In Ref. [6], Yang and colleagues showed that an upregulation of TGF-β (transforming growth factor beta) leads to increased lipid-binding proteoglycans, which in turn can accelerate atherosclerosis in diabetic patients. Previous theoretical studies suggested further that increased GAGs can affect the aortic wall in ways ranging from altering residual stresses to increasing the propensity for dissection [7–9]. There is, therefore, a pressing need to investigate further the potential consequences of GAGs on arterial behavior.

Because intramural swelling is the primary mechanical manifestation of accumulated GAGs, we used osmotic loading to explore experimentally the global mechanical and local histological effects of different degrees of swelling of central arteries. An unexpected but interesting finding in mouse common carotid arteries was that swelling occurred primarily in the media, but not necessarily uniformly. Hence, we also employed a novel constrained mixture model to delineate mechanical properties of the medial and adventitial layers and to explore numerically the potential differences in stress distributions arising from two different types of medial swelling. These simulations suggested that while the overall mechanical behavior remains largely unaltered, the local distribution of stresses changes with a nonuniform swelling within the arterial wall.

Experimental Methods

Specimen Preparation.

All animal care and experiments were approved by the Yale University Institutional Animal Care and Use Committee. Following established methods in our laboratory [10], male C57BL/6 mice were euthanized at 10–18 weeks of age via an overdose of Beuthanasia. The right and left common carotid arteries were excised, carefully cleaned of excess perivascular tissue, mounted on custom glass cannula, and secured with 6-O suture. The unloaded length of the cannulated vessels was ∼5 mm.

Osmotic Loading and Mechanical Testing.

The control solution was a Hank's Buffered Salt Solution (GIBCO HBSS2), which maintains the smooth muscle cells (SMCs) viable though in an essentially passive state (cf. Fig. 1 in Ref. [11]). Dilution of the control solution (i.e., 270 mOsm/l) to have 33% or 3.3% of the control sodium chloride (i.e., 90 mOsm/l and 9 mOsm/l, respectively) yielded two hypo-osmotic solutions for swelling the vessels. Pressure–diameter tests were performed in each of these three solutions at room temperature using a modified version of the protocol described by Ref. [10]. Specifically, each cannulated vessel was placed horizontally within a testing chamber (Living Systems Instrumentation, Albans, VT) with one cannula set at a fixed position and the other connected to a custom externally mounted micrometer to control and measure axial lengths. The vessels were then stretched to that length which prevented lateral bending under 140 mmHg of pressure; the associated axial stretch was defined as the ratio of this length to the unloaded length.

Pressure, diameter, and wall thickness were measured online using paired pressure transducers and a video microscope as well as a custom labview program, as described previously [12]. Pressure was controlled manually by quasi-statically raising or lowering a solution-filled syringe connected to the cannula system; values at midvessel were calculated from two pressure transducers connected equidistantly proximal and distal to the cannulae. The unloaded axial length, outer diameter, and wall thickness were first measured in control HBSS (outer diameter was measured online whereas thickness was measured using an interactive edge-detection routine and appropriate backlighting). For the experimental groups, hypo-osmotic testing at 90 and then 9 mOsm/l followed equilibration for 15 min in each new solution. The unloaded geometry was measured again to reveal any overall swelling due to changes in osmolarity. Vessels in all three groups then underwent five preconditioning cycles from 10 to 140 mmHg to obtain repeatable data, and the post-preconditioning unloaded geometry was recorded to delineate possible effects of preconditioning on swelling. Figure 1 (top panel) shows different configurations achieved during testing. Because labview could not track wall thickness automatically, we used interactive software to measure thickness at 20 mmHg increments from 0 to 140 mmHg for two loading and unloading cycles, which were used to obtain pressure–diameter–thickness data for analysis.

Fig. 1.

(Top) Schematic representation of the four configurations of interest: an overall unswollen traction-free configuration ${\kappa }_{\mathrm{tf}}$, an osmotically loaded and mechanically preconditioned configuration ${\kappa }_{\mathrm{prec}}^{*}$, an axially stretched and pressurized configuration ${\kappa }_P^{*}$, and an osmotically loaded only (not preconditioned) configuration ${\kappa }_{\mathrm{ol}}^{*}$. (Middle and bottom) Swelling ${\nu }^{*}$measured experimentally between configurations defined in the top panel, evaluated for each solution of interest: control HBSS (i.e., 270 mOsm/l, light gray) and hypo-osmotic HBSS at 33% (i.e., 90 mOsm/l, dark gray) or 3.3% (i.e., 9 mOsm/l, black) content of sodium chloride. In particular, the middle panel shows on the left the swelling due to both the osmotic loading and the mechanical preconditioning, between configurations ${\kappa }_{\mathrm{tf}}$ and ${\kappa }_{\mathrm{prec}}^{*}$, and on the right the swelling associated with axial extension and pressurization of the vessel, between configurations ${\kappa }_{\mathrm{prec}}^{*}$and ${\kappa }_P^{*}$. The bottom panel shows on the left the swelling due to the osmotic loading alone, between configurations ${\kappa }_{\mathrm{tf}}$ and ${\kappa }_{\mathrm{ol}}^{*}$, and on the right the swelling associated with mechanical preconditioning alone, between configurations ${\kappa }_{\mathrm{ol}}^{*}$ and ${\kappa }_{\mathrm{prec}}^{*}$. The asterisks represent statistically significant differences (one way ANOVA, p < 0.05).

For residual stress-related opening angle experiments, a 0.5 mm ring was cut from the central region of each specimen (i.e., where diameter and thickness were measured during testing) following the mechanical testing. The rings were attached using a single suture to a vertical post in control HBSS. If the vessel had been hypo-osmotically loaded during mechanical testing, it was reloaded for 15 min in the appropriate solution prior to introducing a radial cut in the anterior sector of the ring. The near zero-stress state was captured using a video camera after 30 min to determine the opening angle.

Histology.

After mechanical testing and measurement of opening angles, specimens were fixed in the traction-free condition for 2 h in 10% formalin and transferred to 70% ethanol for storage. Subsequently, the arteries were embedded in paraffin, sectioned at 5 μm, and stained with Verhoeff Van Gieson (VVG) to view the elastic fibers and observe changes in intralamellar distances that indicated swelling.

Theoretical Framework

Kinematics.

We employed the framework presented in Ref. [13]. Briefly, we modeled the murine common carotid artery as a bilayered cylindrical structure, the inner layer of which corresponds to the media and the outer layer to the adventitia. The elastic network, SMCs, and most of the amorphous matrix glycosaminoglycans/proteoglycans (GAGs/PGs) reside in the media, which endows the artery with much of its compliance and is thought to carry most of the load within the physiologic range. The adventitial layer consists mainly of a collagenous matrix and is thought to act as a protective sheath against acute pressure overloading [14].

To describe the different mechanical behaviors of the media and adventitia, we employed a constrained mixture theory; namely, each primary constituent of the wall was endowed independently with a constitutive descriptor and allowed to deform separately despite being constrained to move together with all other constituents in the mixture. Let the homeostatic configuration ${\kappa }_{\mathrm{h}}$ (with coordinates $r,\theta ,z$) at mean arterial pressure (MAP ∼93 mmHg) and in vivo axial stretch (${\lambda }_z^{\mathrm{iv}}~1.6$) serve as a computationally and biologically relevant reference configuration [13,14]. Moreover, let each constituent within the wall possess a separate natural, stress-free configuration ${\kappa }_n^{\alpha }$, where $\alpha =e,j,m,\mathit{PG}$ for elastic fibers, j = 1, 2, 3, 4 families of collagen fibers, SMCs, and GAGs/PGs, respectively. Mappings between each natural configuration ${\kappa }_n^{\alpha }$ and the homeostatic configuration ${\kappa }_{\mathrm{h}}$ are defined by a constituent-specific deposition stretch tensor ${\mathbf{G}}^{\alpha }$. This approach allows each constituent to experience a different level of strain in the in vivo homeostatic configuration as well as in any subsequent configuration.

Consequently, from the homeostatic reference configuration one can either apply null boundary conditions, radially and axially, to recover the intact traction-free configuration ${\kappa }_{\mathrm{tf}}$ (with coordinates $\mathrm{\varrho },\vartheta ,\zeta$) or one can numerically apply a radial cut to obtain the nearly stress-free, cut configuration ${\kappa }_{\mathrm{c}}$ (with coordinates $R,\Theta ,Z$). Because the focus of this study was on effects of intramural swelling on in vivo wall mechanics, we also let the intact configuration ${\kappa }_{\mathrm{tf}}$ be mapped into a swollen, intact, traction-free configuration ${\kappa }_{\mathrm{tf}}^{*}$ (with coordinates ${\mathrm{\varrho }}^{*},{\vartheta }^{*},{\zeta }^{*}$). Hence, this intact traction-free swollen configuration could similarly be loaded at homeostatic values of pressure and axial stretch to recover the swollen homeostatic configuration ${\kappa }_{\mathrm{h}}^{*}$ (with coordinates ${\mathrm{\varrho }}^{*},{\vartheta }^{*},{\zeta }^{*}$) or be subjected to a radial cut to recover the nearly stress-free, cut, swollen configuration ${\kappa }_{\mathrm{c}}^{*}$ (with coordinates $R^{*},{\Theta }^{*},Z^{*}$). The deformation gradients $\mathbf{F}$ associated with these mappings are

$$\begin{array}{cc}\multicolumn{1}{c}{}& \begin{array}{cc}\multicolumn{1}{c}{{\kappa }_{\mathrm{h}}\to {\kappa }_{\mathrm{tf}}:}& \hfill \left[{\mathbf{F}}_{\mathrm{tf}}\right]=\mathrm{diag}\left[\begin{array}{ccc}\multicolumn{1}{c}{\frac{\partial \mathrm{\varrho }}{\partial r}}& \hfill \frac{\mathrm{\varrho }}{r}\hfill & \hfill \frac{1}{{\lambda }_z^{\mathrm{iv}}}\hfill \end{array}\right]\hfill \end{array}\hfill \\ \multicolumn{1}{c}{}& \begin{array}{cc}\multicolumn{1}{c}{{\kappa }_{\mathrm{tf}}\to {\kappa }_{\mathrm{c}}:}& \hfill \left[{\mathbf{F}}_{\mathrm{c}}\right]=\mathrm{diag}\left[\begin{array}{ccc}\multicolumn{1}{c}{\frac{\partial R}{\partial \mathrm{\varrho }}}& \hfill \frac{R}{\mathrm{\varrho }}\frac{\pi -{\Phi }_0}{\pi }\hfill & \hfill \frac{1}{{\Lambda }_z}\hfill \end{array}\right]\hfill \end{array}\hfill \\ \multicolumn{1}{c}{}& \begin{array}{cc}\multicolumn{1}{c}{{\kappa }_{\mathrm{tf}}\to {\kappa }_{\mathrm{tf}}^{*}:}& \hfill \left[{\mathbf{F}}^{*}\right]=\mathrm{diag}\left[\begin{array}{ccc}\multicolumn{1}{c}{{\nu }^{*}\left(\mathrm{\varrho }\right)\frac{\partial {\mathrm{\varrho }}^{*}}{\partial \mathrm{\varrho }}}& \hfill \frac{{\mathrm{\varrho }}^{*}}{\mathrm{\varrho }}\hfill & \hfill {\lambda }_{\zeta }^{*}\hfill \end{array}\right]\hfill \end{array}\hfill \\ \multicolumn{1}{c}{}& \begin{array}{cc}\multicolumn{1}{c}{{\kappa }_{\mathrm{tf}}^{*}\to {\kappa }_{\mathrm{h}}^{*}:}& \hfill \left[{\mathbf{F}}_{\mathrm{h}}^{*}\right]=\mathrm{diag}\left[\begin{array}{ccc}\multicolumn{1}{c}{\frac{\partial r^{*}}{\partial {\mathrm{\varrho }}^{*}}}& \hfill \frac{r^{*}}{{\mathrm{\varrho }}^{*}}\hfill & \hfill {\lambda }_z^{\mathrm{iv}*}\hfill \end{array}\right]\hfill \end{array}\hfill \\ \multicolumn{1}{c}{}& \begin{array}{cc}\multicolumn{1}{c}{{\kappa }_{\mathrm{tf}}^{*}\to {\kappa }_{\mathrm{c}}^{*}:}& \hfill \left[{\mathbf{F}}_{\mathrm{c}}^{*}\right]=\mathrm{diag}\left[\begin{array}{ccc}\multicolumn{1}{c}{\frac{\partial R^{*}}{\partial {\mathrm{\varrho }}^{*}}}& \hfill \frac{R^{*}}{{\mathrm{\varrho }}^{*}}\frac{\pi -{\Phi }_0^{*}}{\pi }\hfill & \hfill \frac{1}{{\Lambda }_Z^{*}}\hfill \end{array}\right]\hfill \end{array}\hfill \end{array}$$ (1)

where ${\Phi }_0$ and ${\Phi }_0^{*}$ are opening angles in the cut, nearly stress-free normal and swollen configurations, respectively. Values of the determinant of the deformation gradient represent relative volume changes (equal to 1 or ${\nu }^{*}$, in the case of swelling). Hence, we could evaluate, for each deformation considered, both the value of the stretch in the radial direction and the radial position occupied by each material particle for each circumferential and axial stretch.

Motivated by histological observations to be described in Results, we extended our previous description of the intramural swelling (cf. [13]) to consider two possible transmural swelling distributions, sigmoidal or concentrated peak, each of which localize the swelling within the media. Namely

$${\nu }^{*}={\nu }_{\mathrm{A}}^{*}+\frac{{\alpha }_s({\mathrm{\varrho }}_{\mathrm{MA}}-\mathrm{\varrho })}{\sqrt{1+{\left[\frac{200}{({\nu }_{\mathrm{M}}^{*}+{\nu }_{\mathrm{A}}^{*})}({\mathrm{\varrho }}_{\mathrm{MA}}-\mathrm{\varrho })\right]}^2}}+({\nu }_{\mathrm{peak}}^{*}-{\nu }_{\mathrm{M}}^{*})\exp \left[\frac{-{({\mathrm{\varrho }}_{\mathrm{peak}}-\mathrm{\varrho })}^2}{{\alpha }_p}\right]$$ (2)

where ${\mathrm{\varrho }}_{\mathrm{MA}}$ represents the radial location of the medial–adventitial border in ${\kappa }_{\mathrm{tf}}$, ${\nu }_{\mathrm{M}}^{*}$, and ${\nu }_{\mathrm{A}}^{*}$ represent values of swelling at the inner and outer radius, respectively, and ${\mathrm{\varrho }}_{\mathrm{peak}}$ and ${\nu }_{\mathrm{peak}}^{*}$ represent the radial position in ${\kappa }_{\mathrm{tf}}$ and the value of the maximum swelling for the concentrated distribution. In particular, Eq. (2) can represent a uniform swelling within the media through a sigmoidal function (if ${\nu }_{\mathrm{peak}}^{*}={\nu }_{\mathrm{M}}^{*}\ne {\nu }_{\mathrm{A}}^{*}$, ${\alpha }_s=-100$, and ${\alpha }_p=0$) or a concentrated swelling within a particular intralamellar space within the media (when ${\nu }_{\mathrm{peak}}^{*}\ne {\nu }_{\mathrm{M}}^{*}={\nu }_{\mathrm{A}}^{*}$, ${\alpha }_s=-100$, and ${\alpha }_p=5\times \hspace{1em}{10}^{-5}$, with the maximum located at $\mathrm{\varrho }={\mathrm{\varrho }}_{\mathrm{peak}}$).

Constitutive Behavior.

Within the constrained mixture model, the elastin network was described by a neo-Hookean relation

$$W^e\left({\mathbf{F}}^e\right)=\frac{c^e}{2}(\mathit{tr}{\mathbf{C}}^e-3{\left({\mathrm{\nu }}^{*}\right)}^{1/3})$$ (3)

where $c^e$ is a material parameter having the dimension of stress, ${\mathbf{C}}^e$ is the right Cauchy-Green tensor describing the deformation affecting elastin, namely, ${\mathbf{C}}^e={\left({\mathbf{F}}^e\right)}^T{\mathbf{F}}^e$ with ${\mathbf{F}}^e={\mathbf{FG}}^e$. The passive behavior of the smooth muscle and the behavior collagen fibers were described using Fung-type exponentials

$$\begin{array}{ccc}\multicolumn{1}{c}{W^j\left({\lambda }^j\right)=\frac{c_1^k}{4c_2^k}\{\exp \left[c_2^k{({\left({\lambda }^j\right)}^2-1)}^2\right]-1\},}& \hfill \mathrm{for}\hfill & \hfill {\lambda }^j>1\hfill \end{array}$$ (4)

and

$$\begin{array}{ccc}\multicolumn{1}{c}{W^m({\lambda }^m)=\frac{c_1^m}{4c_2^m}\{\exp [c_2^m{({({\lambda }^m)}^2-1)}^2]-1\},}& \hfill \mathrm{for}\hfill & \hfill {\lambda }^m>1\hfill \end{array}$$ (5)

where $c_1^{\xi }$ and $c_2^{\xi }$ are material parameters, the first with dimension of stress and the second nondimensional, with $\xi =k,m$ for the collagen-dominated and smooth muscle-dominated behaviors, respectively. The Fung-type exponential is a function of stretch in the direction of each fiber family, defined as ${\lambda }^j=\sqrt{{\mathbf{M}}^j·{\mathbf{C}}^j{\mathbf{M}}^j}$ for the jth collagen fiber family (j = 1, 2, 3, 4), where ${\mathbf{C}}^j={\left({\mathbf{F}}^j\right)}^T{\mathbf{F}}^j,{\mathbf{F}}^j={\mathbf{FG}}^j$, and ${\mathbf{G}}^j=G^j{\mathbf{M}}^j\otimes {\mathbf{M}}^j$, with ${\mathbf{M}}^j=(0,\sin {\alpha }_0^j,\cos {\alpha }_0^j)$, where angles ${\alpha }_0^j$ describe the direction of each fiber family with respect to the axial direction (e.g., ${\alpha }_0^1=0(\mathit{deg})$ and ${\alpha }_0^2=90(\mathit{deg})$, for axial and circumferential fibers, respectively, while symmetric diagonal fibers are accounted for by ${\alpha }_0^3=-{\alpha }_0^4={\alpha }_0^k$). Similarly, for the circumferentially oriented smooth muscle ${\lambda }^m=\sqrt{{\mathbf{M}}^m·{\mathbf{C}}^m{\mathbf{M}}^m}$, where ${\mathbf{C}}^m={({\mathbf{F}}^m)}^T{\mathbf{F}}^m$, ${\mathbf{F}}^m={\mathbf{FG}}^m$, and ${\mathbf{G}}^m=G^m{\mathbf{M}}^m\otimes {\mathbf{M}}^m$, with ${\mathbf{M}}^m=(0,1,0)$. Note that both the collagen fibers and the SMCs are endowed with only a tensile stiffness; the mechanical behavior of the amorphous matrix contributes compressive stiffness to the whole mixture.

Finally, the GAGs/PGs were described by a modified Blatz–Ko material as

$$W^{\mathrm{PG}}({\mathbf{F}}^{\mathrm{PG}})=\frac{{\mathrm{c}}^{\mathrm{PG}}}{2}(J_2^{\mathrm{PG}}-3{\left({\mathrm{\nu }}^{*}\right)}^{-2/3})$$ (6)

where $c^{\mathrm{PG}}$ is a material parameter having the dimension of stress, $J_2^{\mathrm{PG}}={\mathrm{II}}_{{\mathrm{C}}^{\mathrm{PG}}}/{\mathrm{III}}_{{\mathrm{C}}^{\mathrm{PG}}}$ where $2{\mathrm{II}}_{{\mathrm{C}}^{\mathrm{PG}}}=[{(\mathrm{tr}{\mathbf{C}}^{\mathrm{PG}})}^2-\mathrm{tr}{({\mathbf{C}}^{\mathrm{PG}})}^2]$ and ${\mathrm{III}}_{{\mathrm{C}}^{\mathrm{PG}}}=det{\mathbf{C}}^{\mathrm{PG}}={({\mathrm{\nu }}^{*})}^2$, and ${\mathbf{C}}^{\mathrm{PG}}={({\mathbf{F}}^{\mathrm{PG}})}^T{\mathbf{F}}^{\mathrm{PG}}$ represents the right Cauchy-Green tensor for the GAGs / PGs. ${\mathbf{F}}^{\mathrm{PG}}={\mathbf{FG}}^{\mathrm{PG}}$ describes the deformation gradient experienced by the amorphous matrix and ${\nu }^{*}$ describes the swelling with respect to ${\kappa }_{\mathrm{tf}}$. To describe the whole mixture, we employed a mass averaged strain energy function

$$W={\phi }_{\mathrm{M},\mathrm{A}}^{\mathrm{PG}}{W}^{\mathrm{PG}}({\mathbf{F}}^{\mathrm{PG}})+{\phi }_{\mathrm{M},\mathrm{A}}^e{W}^e({\mathbf{F}}^e)+{\phi }_{\mathrm{M},\mathrm{A}}^m{W}^m({\lambda }^m)+\sum _{j=1}^4{\phi }_{\mathrm{M},\mathrm{A}}^j{W}^j({\lambda }^j)$$ (7)

where superscripts refer to the load bearing constituents (i.e., $\mathit{PG}$ for GAGs/PGs, $e$ for elastin, $m$ for SMCs, and $j$ for each of the four families of collagen fibers), and $W^{\alpha }$ and ${\phi }_{\mathrm{M},\mathrm{A}}^{\alpha }$ are the strain energy density functions and mass fractions of the $\alpha$ constituents that compose the mixture while a subscript $M$ or $A$ refers separately to the medial or adventitial layer, respectively. The material parameters, mass fractions, and deposition stretches were taken from Ref. [13], that is, based on a larger data base for the passive behavior of murine control arteries [10].

Recall that, although the constituents are constrained to deform together within the wall via the deformation gradient $\mathbf{F}$, the constituent-specific natural configurations allow different strains to be experienced by the different constituents, namely ${\mathbf{F}}^{\alpha }={\mathbf{FG}}^{\alpha }$. Since the homeostatic configuration was taken as the reference (i.e., $\mathbf{F}=\mathbf{I}$), the deformation of each constituent in ${\kappa }_{\mathrm{h}}$ equals the constituent specific deposition stretch tensor, ${\mathbf{F}}^{\alpha }={\mathbf{G}}^{\alpha }$, which in turn assures that the stress is homeostatic in the normal in vivo state. In particular, given the assumption of an internally constrained material, where the constraint is $det(\mathbf{F})-J=0$ (where $J$ is equal to 1 or parameterized by ${\mathrm{\nu }}^{*}$), the point-wise Cauchy stress can be written

$$\mathbf{t}=-p\mathbf{I}+{\Sigma }^{\alpha }{\phi }_{\mathrm{M},\mathrm{A}}^{\alpha }{\stackrel{\wedge }{\mathbf{t}}}^{\alpha }$$ (8)

where p is a Lagrange multiplier that enforces the internal constraint and ${\stackrel{\wedge }{\mathbf{t}}}^{\alpha }=(2/J){\mathbf{F}}^{\alpha }(\partial W^{\alpha }/\partial {\mathbf{C}}^{\alpha }){({\mathbf{F}}^{\alpha })}^T$ represents the extra-stress in each load bearing constituent $\alpha$, with $\alpha =\mathit{PG},e,m,j$ for GAGs/PGs, elastin, SMCs, and each one of the four families of collagen fibers, respectively.

Previous experimental results suggest that swelling stiffens arteries in the homeostatic configuration [15]. To understand the effect of swelling on the stiffness in the homeostatic configuration, we used the concept of “small deformations superimposed on large” to find appropriately linearized material properties [4]. Briefly, one first assumes that the deformation is finite from the constituent-specific natural configuration to the homeostatic configuration, and then superimposes, via additional multiplicative deformations, any subsequent deformation throughout the cardiac cycle or during in vitro testing. In particular the Cauchy stress can be written [4]

$$\mathbf{t}={\mathbf{t}}^0+{\mathbf{t}}^{*}$$ (9)

where ${\mathbf{t}}^0=-p^0\mathbf{I}+{\Sigma }^{\alpha }{\phi }_{\mathrm{M},\mathrm{A}}^{\alpha }{\stackrel{\wedge }{\mathbf{t}}}^{\alpha }$ denotes the stress induced by the large deformation and the linearized part is described by (in physical components)

$$t_{\mathit{ij}}^{*}=-p^{*}{\delta }_{\mathit{ij}}+{\mathbb{C}}_{\mathit{ijkl}}{\mathit{\epsilon }}_{\mathit{kl}}$$ (10)

where the linearized stiffness of the whole mixture, ${\mathbb{C}}_{\mathit{ijkl}}={\Sigma }^{\alpha }{\phi }_{\mathrm{M},\mathrm{A}}^{\alpha }{\mathbb{C}}_{\mathit{ijkl}}^{\alpha }$, can be evaluated by

$${\mathbb{C}}_{\mathit{ijkl}}^{\alpha }=2{\delta }_{\mathit{ik}}{F}_{\mathit{lA}}^{\alpha }{F}_{\mathit{jB}}^{\alpha }\frac{\partial W^{\alpha }}{\partial C_{\mathit{AB}}^{\alpha }}+2{\delta }_{\mathit{jk}}{F}_{\mathit{iA}}^{\alpha }{F}_{\mathit{lB}}^{\alpha }\frac{\partial W^{\alpha }}{\partial C_{\mathit{AB}}^{\alpha }}+4F_{\mathit{iA}}^{\alpha }{F}_{\mathit{jB}}^{\alpha }{F}_{\mathit{kP}}^{\alpha }{F}_{\mathit{lQ}}^{\alpha }\frac{{\partial }^2{W}^{\alpha }}{\partial C_{\mathit{AB}}^{\alpha }\partial C_{\mathit{PQ}}^{\alpha }}$$ (11)

with no sum on $\alpha$.

Statistical Analysis

Data are presented as mean ± standard error of the mean and were evaluated using one-way or two-way ANOVA when comparing more than two groups. Specifically, differences between vessels tested in control HBSS solution (n = 6) and in the two hypo-osmotic solutions, namely, 33% (n = 6) or 3.3% (n = 6), were compared using a one-way ANOVA, with significance taken at a level $p<0.05$ using a post-hoc Bonferroni correction.

Results

Figure 1 shows relative changes in volume amongst configurations of interest: values of swelling ${\nu }^{*}$ between configurations ${\kappa }_{\mathrm{tf}}$ and ${\kappa }_{\mathrm{prec}}^{*}$ (middle panel, left), ${\kappa }_{\mathrm{prec}}^{*}$ and ${\kappa }_P^{*}$ (middle panel, right), ${\kappa }_{\mathrm{tf}}$ and ${\kappa }_{\mathrm{ol}}^{*}$ (bottom panel, left), and ${\kappa }_{\mathrm{ol}}^{*}$ and ${\kappa }_{\mathrm{prec}}^{*}$ (bottom panel, right). In particular, the bottom left panel shows that the overall volume increased with a decreased osmolarity of the testing solution, as expected. The middle right panel shows that once preconditioned, the wall volume remained nearly constant, which supports the common assumption of isochoric motions during normal stretching and pressurization. The middle left panel shows that the correlation between osmolarity of the testing solution and swelling was conserved after mechanical preconditioning (cf. bottom left panel), although preconditioning alone caused some swelling (bottom right panel) regardless of osmolarity; this effect was more accentuated, however, if the testing solution was the control (p < 0.05).

Given that we were also interested in quantifying effects of intramural swelling on the distribution of residual stress, it is important to note that prior experimental results showed that the opening angle increases with a decrease in osmolarity in mouse, rat, and pig aortas (e.g., Refs. [7] and [15]). Our preliminary results confirmed these findings for mouse aorta (data not shown), yet this was not the case for mouse carotids. The opening angle decreased when decreasing the osmolarity, even resulting in a negative averaged opening angle for an osmolarity of 90 mOsm/l (i.e., 143% decrease with respect to the control solution).

Figure 2 shows averaged pressure–diameter curves for the control solution (triangle) as well as the 33% (square) and 3.3% (diamond) hypo-osmotic solutions. There was no appreciable difference between the average mechanical behaviors due to osmotic loading, noting that these data were collected incrementally to allow careful measurement of wall thickness at each pressurized state.

Fig. 2.

Averaged experimental data for pressure–diameter responses of mouse common carotid arteries exposed to three different testing solutions: control iso-osmotic solution (i.e., 270 mOsm/l, triangle), and hypo-osmotic solutions with a reduced content of NaCl of 33% (i.e., 90 mOsm/l, square) and 3.3% (i.e., 9 mOsm/l, diamond)

Figure 3 shows VVG-stained cross sections for three representative samples: a vessel in a control solution (top panel) and two vessels fixed in a 3.3% hypo-osmotic solution (middle and bottom panels). Although the same extent of swelling occurred overall in these two cases for the 3.3% solution, note the different transmural distributions of swelling: nearly uniform in the medial layer (middle) or concentrated within an intralamellar space in the outer media (bottom). This histological finding inspired the choice of the distribution function for transmural swelling in Eq. (2), which is illustrated in Fig. 4. For ease of comparison, the overall degrees of swelling (i.e., ${\nu }^{*}=1.09$ for the 33% solution in the top panel and ${\nu }^{*}=1.33$ for the 3.3% solution in the bottom panel) were the same for the sigmoidal (dashed line) and concentrated (solid line) distributions. That is, the integrated extent of each curve corresponded to the overall swelling measured experimentally for that particular testing solution (cf. Fig. 1 from configuration ${\kappa }_{\mathrm{tf}}$ to ${\kappa }_{\mathrm{prec}}^{*}$) normalized by the swelling due to preconditioning in the control solution. This normalization was useful because the reference configuration of the continuum model was the homeostatic configuration recorded experimentally in the control solution, but as implied by Fig. 1 (middle left panel) this configuration was preconditioned. This normalization aimed to standardize the swelling due to preconditioning for all three solutions. Namely, the relative changes in volume were ${\nu }^{*}=1.42/1.30=1.09$ and ${\nu }^{*}=1.77/1.30=1.33$ for an osmolarity of 90 mOsm/l and 9 mOsm/l, respectively.

Fig. 3.

Histological cross sections of osmotically swollen arteries (i.e., Verhoeff-Van Gieson stain). Carotid arteries mechanically tested in (a) iso-osmotic HBSS with no apparent intralamellar swelling, (b) 3.3% hypo-osmotic HBSS displaying a uniform swelling within the medial layer, and (c) 3.3% hypo-osmotic HBSS displaying a swelling concentrated in the outermost intralamellar space.

Fig. 4.

Swelling ${\nu }^{*}$ for two hypo-osmotic solutions considered in this study. Shown are two possible distributions of swelling motivated by the histological images shown in Fig. 3: a sigmoidal distribution concentrating the swelling uniformly within the medial layer (dashed line) and a peak distribution concentrating the swelling mainly within one intralamellar space within the media (solid line). Also noted in each panel is the average normalized change in volume (i.e., $\det ({\mathbf{F}}^{*})\hspace{1em}=\hspace{1em}{\nu }^{*}$, with ${\nu }^{*}\hspace{1em}=\hspace{1em}1.09$ or ${\nu }^{*}\hspace{1em}=\hspace{1em}1.33$), namely, the amount of swelling due to each hypo-osmotic solution and preconditioning (Fig. 1, middle left) normalized by the preconditioning related swelling in control vessels only (Fig. 1, bottom right). The dotted, vertical lines represent the medial–adventitial border in the traction-free intact unloaded configuration.

Using both distributions of swelling from Fig. 4, we simulated the corresponding pressure–diameter behaviors (Fig. 5) and compared experimental results (symbols; present data) and theoretical predictions (lines, with constitutive relations based primarily on data from Ref. [10] augmented with the swelling) for all three extents of swelling. The top panel shows the predicted behavior for the control solution (${\nu }^{*}=1$), the middle panel for the 33% solution (${\nu }^{*}=1.09$), and the bottom panel for the 3.3% solution (${\nu }^{*}=1.33$). The two predictions, one for the sigmoidal swelling (dashed line) and one for the concentrated (solid line) swelling distributions, were essentially superimposed (middle and bottom panels), thus revealing that overall structural behavior is affected more by the total swelling than by the particular distribution through the wall. The gray area in each figure represents the interval of confidence due to uncertainty and specimen-to-specimen variability in the prior experiments [10] that were used for parameter estimation for the baseline model [13]. The figure shows good agreement between the experimental data (with a slight leftward shift due to swelling) and simulations for all the three extents of swelling, particularly in the in vivo range (>80 mmHg). This finding confirms consistency across tests by different investigators within the same lab, while possibly revealing subtle differences between incremental (present) and continuous (prior) pressurization at intermediate levels of pressure.

Fig. 5.

Simulated (solid and dashed lines) and experimental (symbols) pressure–diameter behavior for mouse carotid arteries. The top panel represents the iso-osmotic solution (270 mOsm/l), the middle panel the hypo-osmotic solution at 33% (90 mOsm/l), and the bottom panel the hypo-osmotic solution at 3.3% (9 mOsm/l), respectively. Recall that the baseline model was based on independent biaxial data [10]. The gray area shows the interval of confidence due to the standard error affecting all the geometrical quantities recorded experimentally (e.g., outer diameter and axial stretch in the homeostatic configuration, amount of swelling due to the different osmolarity of the solutions). Goodness of prediction is provided in terms of root mean square error RMSE. Note that the simulations for the sigmoidal (dashed line) and concentrated (solid line) swelling were essentially superimposed; only at the lowest pressure in the bottom panel can they be delineated.

Although we focused primarily on the in vivo configuration, our model allowed us to evaluate the mechanics in every configuration of interest for all cases of swelling considered (e.g., outer diameter and length in the intact, traction-free, swollen configuration; in vivo axial stretch in the homeostatic, swollen configuration; opening angle in the stress-free, cut, swollen configuration). Differences between predicted and measured geometric quantities never exceeded 12%, which was within the experimental error. Interestingly, the predicted values of opening angle showed that uniform swelling within the media increased the opening angle (by 127% and 641% for the 90 and 9 mOsm/l compared to the control solution), while concentrated intralamellar swelling decreased it (by 39% and 98% for the 90 and 9 mOsm/l compared to the control solution). These results suggested that the residual stress distribution is affected not only by the average extent of swelling, but also by its distribution throughout the wall. Because different types of swelling can occur for the same hypo-osmotic solution, one must be careful when averaging measured values.

Finally, Figs. 6 and 7 show, respectively, the calculated distributions of circumferential and axial Cauchy stress at in vivo conditions. This stress analysis was for a MAP of 93 mmHg and in vivo axial stretches ranging from ${\lambda }_z^{\mathrm{iv}*}~1.40$ to $1.60$. The top and bottom panels in each figure show how swelling affected the stress distributions (sigmoidal, dashed line; concentrated, solid line) for the two solutions, 33% and 3.3%, respectively. The control case is shown by a dotted line for comparison. These results suggested that, although the average mechanical behavior may appear unaltered (cf. Fig. 5), different distributions of swelling can result in different in vivo distributions of stresses through the wall consistent with the aforementioned effect on the residual stresses. The homeostatic case (dotted line), represented by the control group, showed a nearly constant stress within each layer (cf. Ref. [14]), with most of the load carried by the medial layer, both circumferentially and axially; such stresses appear favorable mechanobiologically. Increased swelling, even if modest (e.g., ${\nu }^{*}=1.09$ for the 33% solution), increased the stress in the adventitia and amplified the gradient of stress within both layers.

Fig. 6.

Circumferential (hoop) stress distribution for the hypo-osmotic solutions (90 mOsm/l and 9 mOsm/l) for an axial stretch of ${\lambda }_z^{\mathrm{iv}*}~1.60$ and ${\lambda }_z^{\mathrm{iv}*}~1.40$, respectively, and an internal pressure of 93.3 mmHg. The lines correspond to the distributions of swelling, as shown in Fig. 4: a sigmoidal distribution concentrating the swelling within the medial layer (dashed line) and a peak distribution concentrating the swelling in one intralamellar space within the media (solid line). The dotted line represents the circumferential stress for the homeostatic, unswollen, control case.

Fig. 7.

Axial stress distribution for the hypo-osmotic solutions (90 mOsm/l and 9 mOsm/l) for an axial stretch of ${\lambda }_z^{\mathrm{iv}*}~1.60$ and ${\lambda }_z^{\mathrm{iv}*}~1.40$, respectively, and an internal pressure of 93.3 mmHg. The lines correspond to the distributions of swelling, as shown in Fig. 4: a sigmoidal distribution concentrating the swelling within the medial layer (dashed line) and a peak distribution concentrating the swelling in one intralamellar space within the media (solid line). The dotted line represents the circumferential stress for the homeostatic, unswollen, control case.

Figure 8 compares values of linearized circumferential stiffness in the homeostatic state (luminal pressure equal to 93 mmHg and axial stretch equal to the in vivo axial stretch) for the two hypo-osmotic solutions and the control solution. The top and bottom panels show that swelling increased the linearized stiffness in the adventitia. Moreover, swelling resulted in a global stiffening of the vessel, which can be appreciated by comparing the horizontal arrows in the figure that represent the integral average of stiffness in the wall (i.e., a solid line for an osmolarity of 9 mOsm/l, dashed line for an osmolarity of 90 mOsm/l, and dotted line for an osmolarity of 270 mOsm/l, which represents the control case).

Fig. 8.

Linearized circumferential stiffness for the hypo-osmotic solutions (90 mOsm/l and 9 mOsm/l) for an axial stretch of ${\lambda }_z^{\mathrm{iv}*}~1.60$ and ${\lambda }_z^{\mathrm{iv}*}~1.40$, respectively, and an internal pressure of 93.3 mmHg. The thick lines correspond to the distributions of swelling, as shown in Fig. 4: a sigmoidal distribution concentrating the swelling within the medial layer (dashed line) and a peak distribution concentrating the swelling in one intralamellar space within the media (solid line). The dotted line represents the unswollen, homeostatic (control) case. The arrows on the left represent the integral averages of stiffness within the wall.

Discussion

GAGs are abundant in arteries both in development [16] and disease [8,17] and clearly merit increased attention in these cases. There is increasing recognition, however, that these highly negative charged molecules may yet play important roles in arterial homeostasis as well [7,13]. Because of their high fixed charge density, particularly for versican, which combines with hyaluronan to form large medial aggregates, one of the primary consequences of medial GAGs is intramural swelling.

Previous work on the potential impact of GAG concentration and distribution focused on residual stresses [7,15,18] and showed that GAGs can help homogenize the stress distribution through the wall. These findings are thus important to the understanding of mechanical homeostasis (cf. [13]). In this paper, we showed further that the residual stress distribution is affected both by the overall extent and the distribution of swelling within the wall. In particular, our theoretical results suggest that nonuniform distributions of swelling through the wall can have dramatic effects on the distribution of residual stress, leading to opposite effects on the opening angle (e.g., an increase for sigmoidal swelling but a decrease for concentrated swelling), at least in the mouse common carotid artery. In particular, when GAGs are distributed uniformly within the medial layer, there is a homogenization of the stresses between the two layers and an increased value of the opening angle. When the swelling is distributed nonuniformly in the medial layer, however, it can lead to increased gradients in the stress field distribution within each layer (Figs. 6 and 7), which would be expected to differentially affect mechanobiological responses that could ultimately alter wall structure [8].

Our results confirmed overall acute material stiffening in physiologic ranges of pressure, as reported before by Ref. [15], with both hypo-osmotic solutions resulting in stiffer material behaviors compared with the controls (Fig. 8). Of particular interest, the model predicted that this overall stiffening resulted from a marked stiffening of the adventitial collagen, which would increase its ability to serve as a protective sheath. Indeed, the predicted circumferential stiffness decreased slightly within the media, consistent with the decrease in circumferential stress (Fig. 6), and both effects could be protective to some degree. As shown previously, however, one should also consider carefully the radial deformations resulting from intramural swelling, which could either increase or disable mechanosensing via microfibrillar connections between the elastic laminae and SMCs [13]. Finally, we also note that the HBSS solution maintains the SMCs viable, though in a passive state. Hence, it is unlikely that smooth muscle tone affected the results. The contractile situation in vivo could be different, however, which also merits further attention.

An unexpected finding was the increased swelling due to standard preconditioning (Fig. 1). Although this effect would not be expected to affect standard reports of pressure–diameter behavior (cf. Fig. 2), there is a need to assess whether the in vivo state is truly replicated by standard experimental testing conditions, including the chemical composition of the bathing solutions. At the least, reported values of opening angle would be expected to differ depending on whether ring tests are performed independent of mechanical testing and the associated preconditioning protocols. Whether swollen or not, our results confirmed the utility of the isochoric assumption for computing stress during transient loading.

In summary, we showed for the first time that intramural swelling need not manifest at the macroscale as a dramatic change in pressure–diameter behavior, yet the extent and distribution of medial swelling can have important effects on transmural distributions of residual stress (reflected by different opening angles) as well as in vivo stress and material stiffness. Such differences, in turn, would be expected to affect differentially the mechanobiological responses within the media and adventitia, which could lead to transmural differences in structure and properties. There is, therefore, clear motivation to study further the effects of GAGs in arterial wall mechanics and mechanobiology in morphogenesis, homeostasis, and pathogenesis.