Cardiovascular disease can alter the mechanical environment of the vascular system, leading to mechano-adaptive growth and remodeling. Predictive models of arterial mechano-adaptation could improve patient treatments and outcomes in cardiovascular disease. Vessel-scale mechano-adaptation includes remodeling of both the cells and extracellular matrix. Here, we aimed to experimentally measure and characterize a phenomenological mechano-adaptation law for vascular smooth muscle cells (VSMCs) within an artery. To do this, we developed a highly controlled and reproducible system for applying a chronic step-change in strain to individual VSMCs with in vivo like architecture and tracked the temporal cellular stress evolution. We found that a simple linear growth law was able to capture the dynamic stress evolution of VSMCs in response to this mechanical perturbation. These results provide an initial framework for development of clinically relevant models of vascular remodeling that include VSMC adaptation.

Introduction

Cardiovascular disease accounts for 17.3 million deaths globally and one of every four deaths in the U.S. each year [1,2]. Regardless of the specific pathology, a common result of cardiovascular disease is a change in the mechanical loads borne by arteries. In this dynamic mechanical environment, arteries must constantly adapt to maintain their functional integrity through mechano-adaptive growth and remodeling. This mechano-adaptation can be functional, as in hypertension, where stress-induced arterial growth and remodeling leads to stiffer and thicker vessels, lowering the wall stress, or dysfunctional, as in aneurysm growth [3,4], vasospasm [5], or cerebral amyloid angiopathy [6], where growth and remodeling can lead to long-term deleterious results. A better understanding of the underlying processes driving mechano-adaptation is important for improved treatment of these diseases.

Theoretical models can be important tools for understanding complex behaviors, like mechano-adaptation, and could be used to develop mathematical model-aided individualized medicine [710]. For example, it has been proposed that biomechanical models could be employed, in conjunction with intravital imaging, to determine the stress in the wall of an aneurysm, and subsequently predict its growth and rupture, and guide treatment [11,12]. But, determining the arterial wall stress is not trivial. Arteries contain residual stresses [13], which evolve spatially and temporally with changes in mechanical load [14], suggesting that the tissue adapts at different rates in different locations in the wall. Growth theory suggests that this is because the artery adapts its own properties to optimize local tissue stress, and that this adaptation can be roughly characterized with a simple phenomenological growth law [1519]. Recently, to develop a more granular understanding of mechano-adaptation, rule-of-mixtures based models have been developed to incorporate constituent-specific phenomena, such as collagen remodeling or cell hypertrophy, on tissue-scale mechanics [2026]. However, the increased power of these models comes with extra burdens, as each constituent's adaptive behavior must be individually characterized. Most mixture models assume that collagen remodeling is the primary mode of artery mechano-adaptation. However, experimental studies of induced hypertension find that early changes in carotid artery mechanics are primarily reflected in their contractile properties, while the extracellular matrix (ECM)-dependent passive properties are relatively unchanged for over a week [27,28]. So, while long-term mechano-adaptation is likely dominated by collagen remodeling, shorter-term adaptation is dependent on VSMC dynamics. To date, VSMC mechano-adaptation has not been directly measured in isolation.

Many cells alter their function in response to changes in their mechanical environment. For example, extracellular material properties influence a range of cellular processes, like stem cell differentiation [29], cell migration rates [30], angiogenesis [31], and tumor metastasis [32]. In VSMCs, migration [30,33,34], contractile function [35], and phenotype expression [36,37] can be influenced by substrate mechanics. Mechanical functions of cells are also influenced by applied loads. Cyclic stretching has been shown to influence cell traction force differentially, depending on substrate mechanics and cell type [3841]. VSMCs produce extracellular matrix [42], reorganize [4345], and alter their phenotype when exposed to cyclic loading [46,47]. High strain rate traumalike loads alter VSMC contractility and phenotype [48]. Several models for cellular responses to altered mechanics have been developed [4953]. However, current cell mechano-adaptation models are not designed to fit within the framework of growth and remodeling theory.

Here, we present an empirically determined model for VSMC mechano-adaptation in response to applied strain. We have developed an experimental system to simultaneously apply chronic alteration of strain to a single micropatterned cell and measure its stress. Using the data from this assay, we fit a simple growth law to characterize the temporal stress evolution of VSMCs. These data provide an initial framework of knowledge that is necessary to develop the next generation of medically relevant mechano-adaptation computational models.

Methods

Construct Fabrication.

Fibronectin (FN) micropatterned polyacrylamide gels were chemically bound to elastomer membranes for controlled cell morphology and strain application using methods adapted from Polio et al., Simmons et al., and Quinlan et al. [5458] (Fig. 1(a)). First, polydimethylsiloxane (PDMS, Sylgard 184) (Dow Corning, Corning, NY) at a 10:1 base:crosslinker ratio was mixed, degassed, and cured on the silinized microfeature wafer, comprised of rectangular islands with an area of 3978 μm2 and an aspect ratio of four (34 μm × 117 μm), to produce the inverted features on PDMS stamps. The micropatterned morphology was determined based on the average area of an unpatterned VSMC [59] and the aspect ratio was chosen to mimic the elongated shape seen in vivo. PDMS stamps were cleaned in 70% ethanol for 30 min and then incubated with 100 μg/ml FN for 1 h. After incubation, excess FN was removed, the stamps were air dried, and placed face-down in conformal contact with clean, plasma-treated, 15 mm glass cover slips for 30 min. During the 30 min transfer of FN from the PDMS stamp to the glass coverslip, a prepolymer solution composed of 10% acrylamide (Sigma-Aldrich, St. Louis, MO), 0.13% bis-acrylamide (Sigma-Aldrich), and 1% 0.2 μm red fluorescent beads (Thermo Fisher Scientific, Waltham, MA) (only added for measured bead displacement experiments) was degassed and clean elastomer membranes (Specialty Manufacturing Inc., Stockwell Elastomerics, Saginaw, MI) secured in custom steel clamps were treated with 10% benzophenone in 35:65 w/w water:acetone for 1 min and rinsed three times with methanol. Note that all the steps involving benzophenone were performed in low light to prevent accidental activation. The benzophenone-treated elastomer membranes were degassed for 30 min and then the chamber was flooded with nitrogen gas. To initiate polymerization, the prepolymer solution, 0.2% tetramethylethylenediamine (TEMED, Sigma-Aldrich), 1M HCl to a pH of 7.0–7.4, 10 μg/ml N-hydroxysuccinimide-acrylic acid ester (Sigma-Aldrich), and finally, 0.05% ammonium persulfate (Sigma-Aldrich) were added to the prepolymer solution. Over 10 μl of activated prepolymer solution was added to the surface of the benzophenone-treated elastomer membrane and the FN stamped glass coverslip was placed face-down on top of the droplet. Then, the polyacrylamide gel was cured under UV light for 30 min (1 in distance from the lamp, Jelight 342 model) and rehydrated in water for 10–15 min allowing for removal of the glass coverslip. The polyacrylamide gels were passivated with 4% bovine serum albumin at 37 °C for 45 min. Prior to any cell seeding, the polyacrylamide gels were incubated in 1×phosphate-buffered saline with 1% penicillin/streptomycin for 48–72 h to remove any residual benzophenone.

Cell Culture.

Human umbilical artery VSMCs were purchased at passage 3 from Lonza (Lot: 7F3794, Walkersville, MD) and cultured at 37 °C in supplemented Medium 199 (Mediatech, Manassas, VA) containing 10% FBS (GIBCO, Grand Island, NY), 50 U/ml penicillin/streptomycin (GIBCO), 1% Nonessential amino acids (GIBCO), 1% HEPES (GIBCO), 3.5 g/L glucose (Sigma-Aldrich), 2 mM L-glutamine (Sigma-Aldrich), and 2 mg/L vitamin B-12 (Sigma-Aldrich). Passages 5–7 were utilized in experiments. To induce an in vivo like contractile phenotype, VSMCs were serum starved for 24 h prior to all the experiments [60].

Cell Structure Quantification.

Standard immunofluorescent staining techniques were used to stain single VSMCs for their nucleus and actin cytoskeleton. Briefly, VSMCs seeded onto the micropatterned polyacrylamide-elastomer constructs were fixed in 4% paraformaldehyde, permeabilized in 0.05% Triton, and blocked in 10% bovine serum albumin (Thermo Fisher Scientific). DAPI (Life Technologies, Grand Island, NY) and 488 nm fluorophore-conjugated phalloidin (Thermo Fisher Scientific) were used to stain VSMC nuclei and F-actin, respectively. Z-stack images of F-actin (minimum of eight cells per condition) were taken on an Olympus FluoView FV1000 BX2 upright confocal microscope. Cell thickness and actin alignment were determined from the F-actin z-stacks using custom matlab code as previously published [61,62]. Cell cross-sectional area and cell volume was calculated from the measured cell geometry. Actin alignment was characterized with the orientation order parameter (OOP), such that a value of one represents perfect actin alignment and a value of zero represents a completely unaligned actin structure [62]. Statistical analysis for cross-sectional area, cell volume, and actin alignment was performed using a two-way ANOVA with the Holm–Sidak test for pairwise comparisons. Note that separate cells were used for each time point's structural analysis, and the cells used for structure quantification were not the same cells used for subsequent mechanical experiments.

Chronic Strain Traction Force Microscopy Experiments.

Strain was applied to single VSMCs micropatterned on polyacrylamide-elastomer substrates using a linear motor (Linmot, Elkhorn, WI, PS01-23x80F). The custom steel clamps that secured the polyacrylamide-elastomer substrate along an unidirectional axis were attached to the motor, and the length of the long axis, respective to the micropatterned geometry, was increased at 2% strain per second, until the desired grip strain was achieved (0%, 10%, 20%, 30%) (Fig. 1(b)), as defined by ϵ=(l/L)1, where L is the original length of the membrane and l is the deformed length (Fig. 1(b)). The custom steel clamps were then locked in place at the specified grip strain for the remainder of the experiment. All the reported strains and stretch ratios are grip strains, but measured substrate strain corresponds closely to grip strain (Fig. 1(c)).

At each reported time point after strain application (0, 4, 8, and 24 h), bright field images of the cell of interest and fluorescent images of the bead layer directly below the cell of interest were taken on an Olympus IX81ZDC confocal microscope. An environmental chamber-maintained cell culture conditions throughout imaging and constructs were returned to an incubator between experimental time points. The same single VSMCs were tracked for all the time points. After the final time point, Hoechst dye was used to visualize the cells of interest nuclei to confirm only a single cell was present on each FN island. Finally, the cells were lysed with 100 mM sodium dodecyl sulfate (SDS) and the reference configuration of the fluorescent bead layer was imaged for each cell.

Cell Tractions and Stress Determination.

Pre- and postlysis bead images were compared using a particle image velocimetry to determine the cell-induced substrate deformation [63] (Fig. 1(d)). Young's modulus and Poisson's ratio of the polyacrylamide gel were measured as 13.5 kPa and taken as 0.5, respectively. Traction stress vector fields were determined from bead displacements (Fig. 1(d)) using an unconstrained Fourier transform traction cytometry algorithm [63] with a regularization factor of 1 × 10−9, yielding a grid of n substrate traction stress vectors given by Tn=Txnex+Tyney where ei is the unit vector in the i direction. The total traction force components fx and fy were given as fi=n(TinAn(rin/|rin|)), where i=x,y, An is the area of discrete surface n, and rn=rxnex+ryney is the vector that described the location of surface n with respect to the center of the cell. The Cauchy stresses σi were taken as σi=(fi/2Ai) (Fig. 1(e)). Statistical analysis for total axial and transverse traction force and axial and transverse VSMC Cauchy stress was performed using a two-way ANOVA with the Holm–Sidak test for pairwise comparisons.

Modeling Mechano-Adaptation.

Cells were modeled as pseudo-elastic, incompressible bodies undergoing axisymmetric uniaxial deformation and volumetric growth.

Deformation and Growth.

We assume that the cells undergo mechano-adaptation by volumetric growth as described by Rodriguez et al. [64]. The formulation is based on the concept of an evolving zero-stress configuration, allowing cells to grow and change shape to alter their stress state (Fig. 2).

Cells exert forces on the substrate to which they are attached, so that before any strain is applied, they have some basal internal stress. To incorporate this prestress, we first assume that there is a stress-free configuration that the cell could assume while filling the ECM micropattern, which is given by B. To generate stress, the cell could reorganize its cytoskeleton, contract, or change its size or shape. We consider any of these modes of shape change “growth.” To model growth, we first suppose the cell is separated from its substrate to give configuration Bd, which is identical to B but with no external constraints. Growth then occurs in this detached state. Since the cell is unconstrained, it remains stress free. This initial stress-free deformation is given as the growth tensor G(0), indicating the growth prior to time = 0. The grown stress-free configuration is given by Bg(0). The elastic deformation from the grown zero-stress configuration to the true configuration (b(0)) is given by F*(0). Since the cell is constrained to the ECM micropattern, the deformation F(0) between B and b(0) is small and assumed to be F(0)=I. Thus, the elastic deformation at time = 0 is given by F*(0)=G1.

During membrane stretching, external loads deform the cell from b(0) to a new loaded state b(t). This observed deformation is given by F(t). Growth in response to this deformation is treated as stress-free, similar to prestretch growth. The additional growth between time = 0 and time = t is given by G(t), and the total growth is given by G=G(t)G(0). The observed deformation is given by F(t)=F*(t)G, where F*(t) is the elastic deformation at time = t.

We assume that stretched cells undergo shear-free axisymmetric uniaxial deformation, so the deformation can be given by F(t)=diag[λx,λy,λz], where λi is the stretch ratio in the i direction. The x -axis is considered parallel to the long axis of the cell, the y -axis parallel to the short axis of the cell, and the z -axis is normal to the surface of the membrane. The growth tensor is given by G(t)=diag[λgx,λgy,λgz] and the elastic deformation is F*(t)=diag[λx*,λy*,λz*], where λi=λi*λgi. The material is considered incompressible, so λx*λy*λz*=1.

Material Constitutive Laws.

Let λi* be the stretch ratio in the direction i relative to the zero-stress state. For an incompressible, pseudoelastic material, the constitutive equations for the partial Cauchy stresses are given by 
σi=λi*Wλi*p
(1)
where W is the strain-energy density function of the material, and p is a Lagrange multiplier. We assume the cell to be a neo-Hookean solid whose strain energy density function is given by 
W=μ2(I1*3)
(2)

where μ is the shear modulus, taken to be 10 kPa [61], and I1* is the first strain invariant with respect to the zero-stress configuration, given by I1*=λx*2+λy*2+λz*2.

Equilibrium.

We assume axisymmetric uniaxial shear-free stretching with prescribed observed deformations λx and λy. Since the top surface of the cell is free, σz=0, and the in-plane stress is given by 
σi=μ(λi*2λz*2)
(3)

where i=x,y.

Mechano-Adaptation Law.

In tissue-scale models of artery growth and remodeling, it is normally assumed growth acts to maintain the circumferential wall stress (or hoop stress) at some homeostatic preferred stress value [6,25]. VSMCs typically wrap around arteries so that their long axes are circumferentially arranged. Thus, we assume that the maintained stress is σx. There are two basic growth modes by which the cell can modulate its stress. The cell can undergo:

  1. (1)

    axial growth (lengthening), altering the elastic stretch ratio and

  2. (2)

    transverse growth (thickening), altering the cross-sectional area.

We consider both. Let the preferred stress value or “target stress” be given by σo. We let 
λ˙gxλgx=1Tx(σxσo),λ˙gyλgy=λ˙gzλgz=1Tt(σxσo)
(4)

where λ˙gi is the rate of change of the growth stretch ratio λgi, and TxandTt are time constants associated with axial and transverse growth rate, respectively.

Solution Method.

The target stress (σo) was assumed to be the mean control stress for the entire 24 h experimental period. At t=0, we assumed that λgy=λgz=1, and the equilibrium equation (3) was solved to determine the homeostatic value of λgx that gives σx=σo. A step-change in λx (1.1, 1.2, or 1.3) was applied, consistent with the experimental deformation, the mechano-adaptation laws (Eq. (4)) were solved using finite differences, and the cell stress was solved from equilibrium (Eq. (3)). Time constant optimization was performed by comparing predicted stress σx and change in cross-sectional area (λyλz) to those experimentally measured for each of the three applied stretches at the four experimentally measured time points. The parameters that best fit the experimental data, as determined by least squares fitting, was determined to be optimal.

Results

Strain-Induced Alteration of VSMC Traction.

To measure VSMC mechano-adaptation in response to an applied strain, we developed a highly controlled and reproducible system for single VSMCs on rectangular islands printed upon a polyacrylamide-elastomer substrate (see Figs. 1(a) and 1(b)). A step-change in strain of 0%, 10%, 20%, or 30% strain was applied and maintained for 24 h (Fig. 3(a)). The cell-induced substrate displacement was measured by tracking flourospheres embedded in the gel. Traction stresses were calculated from the cell-induced substrate displacement and mechanical properties of the gel (Fig. 3(b)). Increased strain resulted in an initial increase in cell-induced axial traction forces, which decreased over the 24 h observed (Fig. 3(c)). The 0 h average axial traction force was 2.07 ± 1.56 μN, 2.17 ± 1.23 μN, 1.59 ± 0.76 μN, and 1.53 ± 0.85 μN for 30%, 20%, 10%, and 0% applied strain, respectively (Fig. 3(c)). Axial forces (fx) (long axis of the cell and direction of stretch) (Fig. 3(c)) were markedly larger than the transverse forces (fy) for all the conditions (Fig. 3(d)). Following this initial increase, axial traction force in the stretched cells temporally approached control values (0% applied strain), as evident by the 24 h mark for all the strain conditions (Fig. 3(c)).

Cellular Microarchitecture in Strained VSMCs.

To elucidate the effects of applied chronic strain on the cellular cytoskeleton, we stained single, micropatterned VSMCs for F-actin at each applied strain and time point (Fig. 4(a)). VSMC thickness was determined from 3D confocal stacks of F-actin (Fig. 4(b)). VSMC cross-sectional area, both axial (Ax) and transverse (Ay), was determined from measured cell width, length, and thickness (Figs. 4(c) and 4(d)). The axial VSMC cross-sectional area (Ax) decreased with increasing substrate strain: 77.8 ± 27.8 μm2, 71.1 ± 15.0 μm2, 70.3 ± 12.2 μm2, and 53.4 ± 10.5 μm2, respectively, for 0%, 10%, 20%, and 30% applied strain immediately following the applied strain (Fig. 4(c)). No significant changes were found for the transverse VSMC cross-sectional area (Ay) (Fig. 4(d)). The VSMC volume percent difference relative to the 0% 0 h volume was calculated to be −13.5%, 2.1%, and −6.4%, respectively, for 10%, 20%, and 30% applied strain immediately following the applied strain (Fig. 4(e)). There was no discernable temporal trend in cell volume following applied strain, suggesting that cell deformation was relatively incompressible (Fig. 4(e)). To determine if cytoskeletal remodeling occurred poststrain, we characterized the F-actin alignment for all the conditions using an OOP [62]. The average alignment increased slightly with increasing stretch (Fig. 4(d)). However, we found no significant temporal changes in the OOP for any applied strain condition, suggesting that cytoskeleton realignment did not occur (Fig. 4(d)).

VSMC Stress Evolution and Mechano-Adaptation Law.

We assumed that mechano-adaptation acts to maintain stress in the cell's long axis, which correlates with the hoop stress in an intact artery. Therefore, we calculated the midplane axial and transverse cell Cauchy stress from the measured axial and transverse cell traction forces and axial and transverse cell cross-sectional areas (Figs. 5(a) and 5(b)). We found that with increasing applied strain, the axial cell Cauchy stress increases, such that at the 0 h time point, the average cell Cauchy stress is 9.8 ± 5.5 kPa, 12.3 ± 5.9 kPa, 18.6 ± 10.5 kPa, and 25.2 ± 18.9 kPa for 0%, 10%, 20%, and 30% applied strain, respectively. Over the measured 24 h, the axial stress in the cells exposed to chronic strain decreased, approaching the axial stress of the unstrained control (Fig. 5(a)). This result demonstrates that the VSMCs' stress temporally re-equilibrates following a step-change in strain. The mean transverse Cauchy stresses were approximately an order of magnitude smaller than the axial Cauchy stresses and there was no significant difference between any of the strains at any one time point (Fig. 5(b)). Individual cell axial Cauchy stress behavior displayed the inherent variability of single cells but was qualitatively consistent (Figs. 5(c)5(f).

To model VSMC mechano-adaptation, we fit our stress data with a set of simple linear growth laws (Eq. (4)) that represent axial elongation (λgx) and cell thickening (λgy,λgz) in the zero-stress configuration (Fig. 6). The model assumes that the cell grows in its stress-free configuration to maintain its stress in the loaded configuration at a specified target stress. We assumed that the stress being maintained was the axial stress (σx) and that the target stress (σo) was the average value of σx in the control cells, 9.7 kPa. The growth law time constants that best fit the data were Tx=1MPah and Tt=10GPah. Consistent with the experimental data, the model exhibited an increase in cell stress immediately after applied strain, followed by a temporal decrease in cell stress approaching the target stress at 24 h (Fig. 6(a)). The model predicted that VSMC growth is primarily axial (Fig. 6(b)), with very little transverse growth (Fig. 6(c)), consistent with the measured cross-sectional areas (see Fig. 4(d)). The axial and transverse deformations, characterized by the observed (λx,λy, or λz) and elastic stretch ratios (λx*,λy*, or λz*), were in agreement with this finding and behaved according to λi=λi*λgi as the axial elastic stretch ratio immediately increased with increasing applied strain and converged with time, while the transverse elastic stretch ratios deviated minimally from the transverse stretch ratio (Figs. 6(d) and 6(e)).

Discussion

A cell's mechanical state is determined by both the extrinsic forces applied to it by its surroundings and its own intrinsic stress generation. Thus, a cell can modify its stress by modulating the load borne by its surrounding extracellular matrix or by altering its intrinsic mechanics. Here, we aimed to directly study intrinsic adaptation by measuring temporal evolution of cellular stress following a mechanical perturbation, with the aim of providing insight necessary to develop better models of arterial growth and remodeling.

Our results suggest that VSMCs have an ideal homeostatic target stress and that the cells act to re-establish this stress following mechanical perturbation. This finding is consistent with long-standing tissue-scale growth theory used to determine the source of residual stress in arteries [15,18,19]. Using a framework originally developed for these tissue models, we simulated a single VSMC as a body undergoing targeted volumetric growth to return the cell to mechanical homeostasis. We assumed that growth could be measured as cell lengthening (axial growth) or thickening (transverse growth). However, we found little change in cell volume over the course of our experiment. This is inconsistent with vessel growth in vivo, where hypertensive vessels show significant cellular hypertrophy [58]. This may simply be because our 24 h assay is too brief to capture slower developing cell hypertrophy, or it may be because our assay does not capture the deformations VSMCs undergo in hypertension. The 0% applied strain cell volume does decrease some at the 24 h time point; however, there is no statistical difference between the 0% 0 h volume and the 0% 24 h volume. We did measure significant mechano-adaptation due to axial growth and found that it can be captured with a simple phenomenological linear growth law. Here, we aggregate all the stress-altering cellular processes, such as cytoskeletal reorganization or changes in cytoskeletal composition into a single growth tensor. Elucidation of the contributions of each of these mechanisms to the mechano-adaptation law requires further study.

Similar studies have found that other cell types and unpatterned VSMCs respond to applied strain much more quickly than the VSMCs we measured here [65,66]. This is unsurprising since VSMCs' primary function is maintaining contractile tone over prolonged periods, which requires slower dynamics. Notably, our cells are maintained in complete serum starvation for 24 h prior to and throughout the experiment, which has been shown to induce an in vivo like contractile phenotype and could also account for a slower temporal adaptation than previously observed [60,66]. In serum, VSMCs switch to a synthetic phenotype, and it is likely that, had the experiments been performed in serum, we would have observed faster adaptation. Mechanical perturbation can also alter VSMC phenotype [4648], suggesting an important role in mechano-adaptation. Modified laws that capture dynamic phenotype switching could provide more accuracy and greater mechanistic insight than the currently described phenomenological laws. Similar studies have also found that under some conditions, the traction forces can overadapt, overshooting their original traction. This response could be modeled using hyper-restoration theory, which is similar to the model used here [6769]. However, we did not observe overshoot in our cells.

The assay presented here aims to recreate the changes in VSMC stress in a mechanically perturbed state like hypertension. Our simplified mechanical perturbation is a first step at characterizing the complex in vivo VSMC mechano-adaptation. However, it does not precisely recreate the mechanical environment of a VSMC in a hypertensive vessel. For example, a step-change in pressure in the artery may not result in a simple step-change in VSMC strain and vascular pressure changes are more dynamic than the monotonic strain applied here. Previous studies have found that cell spread area, traction force, and orientation are altered by cyclic stretching [40,41]. In addition, altering cyclic strain frequency has been found to alter adenosine triphosphate (ATP) production in VSMCs [45]. Additionally, the change in endothelial shear stress caused by a step-change in vessel strain would initiate endothelial cell signaling aimed at returning the shear to homeostasis, in part, by endothelin-1-mediated VSMC contraction [70]. Thus, our assay underestimates the complexity that exists in vivo. We micropatterned the VSMCs with an aspect ratio of four to mimic the elongated and highly organized structure of VSMCs in vivo. However, VSMCs do not have uniform architecture. Notably, in cerebral small artery bifurcations, where aneurysms often form, VSMCs have a range of architectures [71]. Previous studies have shown that VSMCs' architecture can affect their functional contractility [62,72,73]. It is possible that architecture similarly affects mechano-adaptation rates in VSMCs, but we did not test for that here.

The model captures the experimental data well; however, there are several assumptions made in the model that could affect its accuracy. First, we assume a neo-Hookean strain energy density for a VSMC, which, given the highly anisotropic organization of the actin cytoskeleton, is unlikely to accurately represent a micropatterned VSMC. A neo-Hookean strain energy density was selected over a strain-stiffening strain energy density function, as our data suggest a linear relationship between stress and strain at the initial time point (see Fig. 5(a)). An anisotropic and nonlinear constitutive description would likely change the parameters of the mechano-adaptation law, but the qualitative results would likely be consistent. For the purposes of this model, we chose to represent a single VSMC as an axisymmetric, pseudo-elastic, incompressible body undergoing uniaxial deformation and volumetric growth. While we recognize that there are spatial differences in stress generation within a single cell that could be elucidated with finite-element modeling, we present this model as a simple description of the holistic VSMC mechano-adaptation. Currently, it is unclear whether force, stress, or strain is the mechanotransducer. Here, we used stress to be consistent with previous targeted growth models [15,18,26,74]. While the experimental transverse stresses are not zero, they are small relative to the axial stresses, so the model assumes that they are zero. For the purposes of our initial mechano-adaptation law, we wanted to model the cell as simply as possible but recognize this could affect our model parameters. Additionally, the model provides insight into the time-response and main direction of adaptation, axial (see Fig. 6(c)), but the parameters Tx and Tt, do not aid in interpreting the biological or physical mechanisms behind the observed temporal stress evolution, as they are aggregated parameters of all growth and remodeling mechanisms. However, future studies could tease apart these mechanisms and their effect on Tx and Tt by singling out and silencing cellular contractile mechanisms or including the known behavior of VSMC phenotype switching.

A primary goal of computational modeling of vascular mechano-adaptation is to develop patient-specific models to aide physicians making treatment decisions. Though current models are improving [7577], the VSMCs' temporal dynamics are not yet well enough understood to capture their contribution to maladaptive growth and remodeling in disease. The methods and results presented here represent a first step toward development of more in-depth theory for use in models of artery mechanics.

Acknowledgment

We gratefully acknowledge the imaging equipment at the University of Minnesota University Imaging Center and the fabrication resources at the University of Minnesota Nano Center. We gratefully acknowledge our funding from the National Science Foundation CMMI-1553255 (PWA, JLH), National Institute of Health 1R03EB016969 (PWA), and American Heart Association AHA15PRE21790000 (KES) and AHA16PRE27770112 (ZW).

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