Measuring, Reversing, and Modeling the Mechanical Changes Due to the Absence of Fibulin-4 in Mouse Arteries

Abstract

Mice with a smooth muscle cell (SMC) specific deletion of fibulin-4 (SMKO) show decreased expression of SMC contractile genes, decreased circumferential compliance, and develop aneurysms in the ascending aorta. Neonatal administration of drugs that inhibit the angiotensin II pathway encourage expression of contractile genes and prevent aneurysm development, but do not increase compliance in SMKO aorta. We hypothesized that multidimensional mechanical changes in the aorta and/or other elastic arteries may contribute to aneurysm pathophysiology. We found that the SMKO ascending aorta and carotid artery showed mechanical changes in the axial direction. These changes were not reversed by angiotensin II inhibitors, hence reversing the axial changes is not required for aneurysm prevention. Mechanical changes in the circumferential direction were specific to the ascending aorta, therefore mechanical changes in the carotid do not contribute to aortic aneurysm development. We also hypothesized that a published model of postnatal aortic growth and remodeling could be used to investigate mechanisms behind the changes in SMKO aorta and aneurysm development over time. Dimensions and mechanical behavior of adult SMKO aorta were reproduced by the model after modifying the initial component material constants and the aortic dilation with each postnatal time step. The model links biological observations to specific mechanical responses in aneurysm development and treatment.

Introduction

Fibulin-4 (FBLN4) is essential for elastic fiber assembly in the skin, lungs and arteries. FBLN4 mutations in humans are associated with arterial tortuosity and aneurysms (Dasouki et al. 2007). Mice that do not express fibulin-4 (Fbln4−/−) die postnatally from lung and vascular defects (McLaughlin et al. 2006). Mice with a smooth muscle cell (SMC) specific deletion of Fbln4 (SMKO) develop ascending aortic aneurysms. SMKO vascular SMCs exhibit hyperproliferation and loss of a contractile phenotype (Huang et al. 2010). Local activation of angiotensin II (AngII) signaling is a primary cause of SMKO aneurysms. Aneurysms can be prevented with neonatal administration of anti-hypertensive drugs that inhibit angiotensin converting enzyme (ACE), such as captopril (CAP), or that block the angiotensin type I receptor, such as losartan (LOS). Treatment with CAP or LOS encourages expression of SMC contractile genes, and reverses the enlarged diameter, but does not reverse the decreased circumferential compliance in SMKO aorta. Aneurysm prevention is not linked to blood pressure changes alone, because propranolol (PROP), an anti-hypertensive drug that is a non-selective beta-adrenergic receptor blocker, does not prevent aneurysms in SMKO mice (Huang et al. 2013).

The goal of the current study was to further investigate the mechanical behavior of SMKO arteries. We hypothesized that although preventative drug treatment did not reverse the changes in circumferential compliance of the ascending aorta, we may observe alterations in axial mechanical behavior of the ascending aorta and multi-dimensional mechanical behavior of other elastic arteries, such as the carotid artery, that would contribute to aneurysm development in SMKO mice. We also hypothesized that a previously published constrained mixture model of aortic growth and remodeling (Wagenseil 2011) could provide insight into relationships between mechanically-stimulated remodeling and aneurysm development in the growing mouse aorta.

Materials and methods

Mice

129SvEv/C57Bl6 male and female mice with an SMC-specific knockout of the fibulin-4 gene (SMKO) (Huang et al. 2010) and wild-type littermates (CTR) were sacrificed at approximately 6 weeks of age. All protocols were approved by the Institutional Animal Care and Use Committee.

Drug treatment protocols

LOS (0.6 g/L, provided by Merck Inc.), CAP (0.075 g/L, Sigma), and PROP (0.6 g/L, Sigma) were administered to the mice in drinking water ad libitum from age 7 to 43 ± 2 days. Untreated (UNT) groups received plain water. Histology and Western blot data were taken from mice on a different treatment protocol, where the mice were treated from age 7 to 90 (histology) or age 7 to 30 (Western blot) days. Previous results showed no differences between the treatment protocols, as long as LOS was started by 7 days of age (Huang et al. 2013).

Arterial dissection and lengths

Small charcoal particles were placed on the left common carotid artery. The carotid was imaged and the lengths between particles were measured before (l) and after excision (L) to determine the in vivo axial stretch ratio (λ_z^iv = l/L). The ascending aorta was imaged and the in vivo length between the heart base and the innominate artery was measured.

Mechanical testing and unloaded diameters

The left common carotid or the ascending aorta was mounted in a pressure myograph (Danish Myotechnology) for mechanical testing(Le et al. 2011). The artery was immersed in physiologic saline solution at 37° C and stretched to the approximate in vivo length. The axial stretch ratio (λ_z) was calculated by dividing the stretched length by the unstretched length. After preconditioning, the artery was inflated three times from 0–175 mmHg in steps of 25 mmHg with 12 sec between steps while pressure, outer diameter, and axial force were recorded. After testing, the artery was removed, placed in a dish of physiologic saline, cut into 2–3 rings, 0.2–0.3 mm thick, and imaged to measure the unloaded dimensions.

Data analysis

Compliance was calculated as the average change in outer diameter per mmHg for each 25 mmHg pressure step. The loaded inner radius for each artery, r_in, was calculated assuming incompressibility (Faury et al. 1999). The average circumferential stretch ratio, λ_θ, was calculated by:

$${\lambda }_{\theta }=\frac{1}{2}\left(\frac{r_{in}}{R_{in}}+\frac{r_{out}}{R_{out}}\right),$$ (1)

where R_in is the unloaded inner radius and r_out and R_out are the loaded and unloaded outer radii. The average circumferential wall stress, σ_θ, was calculated assuming negligible shear:

$${\sigma }_{\theta }=\frac{p{r}_{in}}{r_{out}-r_{in}},$$ (2)

where p is the measured internal pressure. The incremental elastic modulus in the circumferential direction was calculated as the average change in circumferential stress divided by the average change in circumferential stretch ratio for each 25 mmHg pressure step. The average axial wall stress, σ_z, was calculated for a closed end cylinder:

$${\sigma }_Z=\frac{f+p\pi r_{in}^2}{\pi (r_{out}^2-r_{in}^2)},\text{where}f\text{is the measured axial force}.$$ (3)

Statistics

Data are mean ± SD, unless otherwise noted. ANOVA with a Bonferroni post-hoc test between genotypes for each treatment group was used for all comparisons (SPSS). P < .05 was considered significant.

Mechanical modeling

A published model (Wagenseil 2011) was used to investigate possible mechanisms leading to the remodeling in UNT SMKO and LOS SMKO aorta compared to UNT CTR. These groups were chosen because the adult blood pressures are similar (Huang et al. 2013), which simplifies the model assumptions. The model was previously used to predict changes in dimensions and mechanical behavior during normal postnatal development of the mouse ascending aorta from ages 3 to 30 days in discrete time steps of 4.5 days. The model is based on observations that perturbations in blood pressure, axial forces, or blood flow are counteracted by respective changes in the aortic wall thickness, axial length, or inner radius to maintain homeostatic wall stresses. The stresses include the circumferential and axial stresses (Eqns. 2 and 3), as well as the wall shear stress (τ). For steady, fully-developed flow in a long, straight tube the wall shear stress is:

$$\tau =\frac{4Q\mu }{\pi r_{in}^3},\text{where}Q=\text{blood flow and}\mu =\text{blood viscosity (4 cP)}.$$ (4)

The wall stresses at the current pressure (p_h), deformed length (l_h), flow (Q_h), deformed inner radius (r_in,h) and thickness (h_h) are considered the homeostatic values for each discrete time step. To move to the next developmental time, the pressure, deformed axial length and flow are increased to new values determined from previously published experimental data:

$$p={\epsilon }_p{p}_h,l={\epsilon }_l{l}_h,Q={\epsilon }_Q{Q}_h,\text{where}{\epsilon }_P,{\epsilon }_l\text{and}{\epsilon }_Q\text{are constants for each time step}.$$ (5a,b,c)

When pressure, length and flow are increased, it is assumed that first the aorta instantaneously dilates in an attempt to return to the homeostatic shear stress, then remodeling of the wall procedes in an attempt to restore homeostatic values of circumferential and axial stress. In previous work (Wagenseil 2011), changes in blood flow had to be decoupled from changes in the inner radius to predict postnatal growth of the aorta. A constant 12% increase in inner radius for each time step was assumed, regardless of the change in blood flow, and this assumption is included here. The decoupling of the inner radius from the blood flow implies that the developing mouse aorta does not maintain a homeostatic shear stress and/or that Eqn. 4 is insufficient to describe the shear stresses in the ascending aorta with a complex geometry and pulsatile flows (Van Doormaal et al. 2012). Despite these limitations, the model is able to predict the dimensions and mechanical behavior of a 30 day old mouse aorta after growth and remodeling.

Relevant model equations (Eqns. 6 – 18) are listed in the Appendix. The total stress in the aortic wall is the sum of the component stresses (Eqn. 6). The components include SMCs, elastin and collagen. Each component is produced at a homeostatic stretch ratio (Eqn. 7) and the mechanical behavior is described by a constitutive equation (Eqns. 8 – 17). Starting from the dimensions and mechanical behavior of a 3 day old mouse aorta, stepwise changes in blood flow, pressure and axial length are applied. The arterial wall dilates 12% regardless of the change in blood flow, and then remodeling proceeds as components turnover (Eqn. 18) in an attempt to maintain circumferential and axial stresses near the previous homeostatic values. Remodeling is allowed to reach steady state for each time step and then the final stresses, loaded and unloaded dimensions, stretch ratios, and mass fractions for newly produced components are calculated for use as inputs for the next developmental time step. The solution strategy, model input values, and additional equations are in Wagenseil (2011).

The model is used to qualitatively reproduce the pressure-diameter and circumferential stretch-stress behavior of adult UNT CTR, UNT SMKO and LOS SMKO aortae. The previously published model parameters are used to reproduce data for UNT CTR aorta. Changes to the model parameters, motivated by experimental evidence in Huang et al. (2013), are investigated to determine which changes best reproduce the experimental data for UNT SMKO and LOS SMKO aortae. Starting model inputs are identical for all groups because SMKO aorta is histologically similar to CTR aorta at 1 day old, with slight changes evident by 7 days old. LOS treatment began at 7 days, which is the first time point where alterations to the model parameters have an effect. Adult SMKO aorta shows fragmentation and disorganization of the elastic fibers, with no obvious decrease in the amount of elastic fiber staining, hence we assumed that the elastin contribution to the total wall stress is reduced through a decrease in the SMKO elastin material constant (b₁, Eqn. 8a). LOS treatment partially rescues the elastic fiber fragmentation in SMKO aorta, so LOS SMKO aorta may also have an altered elastin material constant. We reasoned that the collagen contribution to the total wall stress may be increased to compensate for the decreased elastin contribution. In mice lacking fibulin-5, another protein necessary for elastic fiber assembly, the arterial collagen fibers show reduced undulation (Wan et al. 2010). Therefore, we investigated changes to the collagen material parameters (b₂, b₃, b₄, Eqns. 9 – 11) that would increase the collagen stress contribution, as well as change the shape of the stress-stretch curve. Expression of SMC contractile proteins is reduced in SMKO aorta and is rescued by LOS treatment. Altered SMC phenotype may cause changes in the passive SMC material parameters (b₅, b₆, b₇, Eqns. 13 – 15), active SMC constants (λ_M, λ₀, Eqns. 16 – 17), basal SMC tone (T_B, Eqns. 16 – 17), or arterial dilation with each time step. Increased SMC proliferation is also evident in SMKO aorta and is rescued by LOS treatment. This may cause changes in the SMC production constant (K^g_m, Eqn. 18) or total SMC mass fraction (ϕ_m, Eqn. 6). A summary of the altered model parameters is listed in Table 1. For the parameter alterations that best reproduced changes in the dimensions and mechanical behavior of UNT and LOS SMKO aorta compared to UNT CTR, we calculated the physiologic shear stress, circumferential and axial stress, circumferential and axial stretch ratio, the normalized unloaded dimensions and the stress contribution of each wall component.

Table 1.

Model parameters hypothesized to be affected by the loss of SMC-specific Fbln4 and subsequent treatment with LOS. The motivation for each parameter change is based on data from Huang et al. (2013). Relevant model equations and parameter definitions are given in the Appendix, with additional equations and solution strategies in Wagenseil (2011).

Parameter Name	Parameter Symbol(s)	Motivation for Change	Original Value(s)	Variation	Results
Elastin material constant	b1	Fragmentation of elastic fibers	35 kPa	x 0.1 – 0.9	Shift press-diam curves left and make stretch-stress more nonlinear
Collagen material constants	b2, b3, b4	Possible compensation by collagen for fragmented elastic fibers	10 kPa, 1.5, .01	x 1.1 – 1.8 all values	Shift press-diam curves left and make stretch-stress more nonlinear
SMC passive material constants	b5, b6, b7	Reduced expression of SMC contractile proteinse	2 kPa, 50, .01	x 1.1 – 1.8 all values	Almost no change in press-diam or stretch-stress relationships
SMC active material constants	ϕM, ϕ0	Reduced expression of SMC contractile proteins	0.6, 2.8	x 0.5 – 2.0 each value independently	Almost no change in press-diam or stretch-stress relationships
SMC basal tone	TB	Reduced expression of SMC contractile proteins	60 kPa	x 0.5 – 2.0	Almost no change in press-diam or stretch-stress relationships
Arterial dilation with each time step	none	Reduced expression of SMC contractile proteins	12%	14 – 20%	Shift press-diam curves up, almost no change in stretch-stress except more nonlinear at highest dilation
SMC production constant	Kgm	Increased SMC proliferation	6.9	x 1.5 – 4.5	Small shift left in press-diam curves and slightly more nonlinear stretch-stress curves
Total SMC mass fraction	ϕ m	Increased SMC proliferation	Decreases with age from 17.3 – 8.4%	+ 1 – 5%, elastin and collagen reduced equally to account for extra SMC fraction	Almost no change in press-diam, small shift to less nonlinear stretch-stress curves

Histology and Western blot analyses

For histology, at least four aortae in each group were harvested, perfusion fixed in 4% paraformaldehyde, and embedded in paraffin. Five-μm sections were stained with Hematoxylin & Eosin (HE) for routine histology, Hart's for visualizing elastic fibers, and Masson-Trichrome for detection of collagen fibers. For Western blot analyses, three aortae in each group were harvested and perivascular adipose tissues were thoroughly removed, then minced in liquid nitrogen by pestle and dissolved into RIPA lysis buffer (Sigma) containing 1% protease inhibitor (Sigma). The lysates were mixed with 3× SDS sample buffer containing 2-mercaptoethanol and boiled at 95°C for 5 min, then subjected to SDS-PAGE. Proteins were transferred to a Western PVDF membrane (Millipore) and immunoblotted with anti-mouse tropoelastin (generous gift from Dr. R. Mecham, 1:1000), anti-collagen I (1:500, Millipore), or anti-GAPDH (1:3000, Cell Signaling) antibodies. Membranes were then incubated with anti-rabbit HRP-conjugated secondary antibody (1:1000, Bio-Rad) and visualized with chemoluminescence kit (Santa Cruz Biotechnology) or SuperSignal West Femto Maximum Sensitivity Substrate (Thermo Scientific).

Results

CAP and LOS reverse increases in SMKO aortic diameter and thickness, but not length

The in vivo length of the ascending aorta in UNT SMKO mice is 93% longer than UNT CTR (p<.001) (Fig. 1A). PROP (p=.015) and CAP (p=.003) SMKO aorta also have increased length, while LOS SMKO shows a trends towards increased length. The unloaded outer diameters of UNT and PROP SMKO aortae are 62 and 84% larger, respectively, than CTR (p<.001) (Fig. 1B). CAP and LOS SMKO aortic diameters are within 19 and 17%, respectively, of CTR, although LOS SMKO diameter is still significantly larger than LOS CTR (p=.029). The unloaded thickness of UNT SMKO aorta is 43% greater than UNT CTR (p=.005) (Fig. 1C). All drug treatments reverse the increased thickness so that SMKO aortic thicknesses are not different from CTR. The in vivo stretch ratio of UNT SMKO left common carotid artery is reduced 18% compared to UNT CTR (p<.001) (Fig. 1D). A decreased stretch ratio may be caused by growth in the axial direction. PROP (p=.049) and CAP (p<.001) SMKO carotid arteries also have decreased stretch ratios, while LOS SMKO carotid shows a trend in the same direction. There are no differences in carotid unloaded dimensions between SMKO and CTR mice for any of the treatment groups (Fig. 1E–F). These results highlight SMKO arterial remodeling in the axial direction that has not previously been reported. These results also confirm that expansion of the SMKO unloaded diameter is restricted to the ascending aorta, with no change in the carotid.

Figure 1.

SMKO mice show artery specific changes in the mean lengths and diameters compared to CTR. The in vivo length of SMKO ascending aorta is longer than CTR and in general this is not reversed by the drug treatments (A). UNT SMKO ascending aorta has a larger unloaded diameter than CTR (B). The large diameter is not normalized by PROP, but is greatly reduced or made statistically insignificant by CAP or LOS. UNT SMKO aorta has an increased thickness compared to CTR and this is reversed with all drug treatments (C). The in vivo stretch ratio of the left common carotid artery is smaller in SMKO mice compared to CTR and in general this is not reversed by the drug treatments (D). There are no differences between SMKO and CTR for the unloaded diameter (E) or thickness (F) of the carotid. N = 6–8 for each group. * = P<.05.

Mechanical behavior is altered in the SMKO ascending aorta, but not in the carotid artery

The outer diameters of UNT and PROP SMKO aortae are 66–151% larger than CTR, depending on the applied pressure (p<.002) (Fig. 2A). CAP and LOS SMKO aortic diameters are not significantly different than CTR, although the shape of the pressure-diameter curve is altered. The axial force in PROP CTR aorta is higher than PROP SMKO aorta (p<.006) (Fig. 2B), but there are no significant differences between genotypes for any other groups. Compliance values for UNT and PROP SMKO aortae are increased 324–577% compared to CTR at 0–25 mmHg (p<.05), while compliance values for CAP and LOS SMKO aortae are comparable to CTR (Fig. 2C). At higher pressures (100–175 mmHg for UNT and PROP aortae; 100–150 mmHg for CAP and LOS aortae), SMKO aortic compliance is reduced 47–79% compared to CTR values (p<.04). While the absolute diameter increases in CAP and LOS SMKO aortae are mostly reversed, the decreased compliance values at physiologic pressures are not. Additionally, at 150–175 mmHg the compliance values of CAP and LOS CTR aortae are decreased 47–64% compared to UNT CTR (p<.01), suggesting that CAP and LOS treatment affect the mechanical behavior of the aortic wall, regardless of Fbln4 deficiency.

Figure 2.

Large differences are evident in the mean pressure-diameter and pressure-compliance behavior for SMKO aorta compared to CTR, with no differences for the carotid artery. UNT and PROP SMKO aortae have larger diameters than CTR at all pressures (A). The differences become statistically insignificant after CAP or LOS treatment. The pressure-force behavior is similar for SMKO and CTR aortae in most treatment groups (B). SMKO aortae have reduced compliance at physiologic pressure (100 mmHg) and above, regardless of drug treatment (C). CAP and LOS reduce the compliance of CTR aortae at high pressures. There are no significant differences in the pressure-diameter (D), pressure-force (E), or pressure-compliance (F) behavior of the carotid artery regardless of genotype or drug treatment. Panel A includes a subset of the data from Fig. 4E in Huang et al. (2013). The subset includes all aortae for which unloaded dimensions were also measured. N = 6–8 for each group. * = P<.05 for UNT/PROP SMKO compared to CTR. $ = P<.05 for PROP SMKO compared to CTR. # = P<.05 for all SMKO and CAP/LOS CTR compared to UNT CTR.

In contrast to the large changes in mechanical behavior of the ascending aorta, the left common carotid artery shows no significant differences in the diameter (Fig. 2D), axial force (Fig. 2E), or compliance (Fig. 2F) between genotypes for any group. Although the left common carotid artery is one of the main branches off the ascending aorta, the mechanical behavior is not affected by Fbln4 deficiency. The mechanical behavior is also not affected by the drugs, showing that CAP and LOS specifically target the ascending aorta in both SMKO and CTR mice.

The circumferential stretch between 0–75 mmHg is 10–34% higher in UNT and PROP SMKO aorta compared to CTR (p<.03) (Fig. 3A). The circumferential stress between 25–100 mmHg is 65–205% higher in UNT and PROP SMKO aorta compared to CTR (p<.03) (Fig. 3B). At zero pressure, the axial stress in UNT and PROP SMKO aorta is 75–80% lower than CTR (p<.02) (Fig. 3C). For CAP and LOS SMKO aortae, there are no significant differences from CTR in the circumferential stretch, circumferential stress, or axial stress. There are also no significant differences between SMKO and CTR carotids in any of the groups (Fig. 3D–F).

Figure 3.

Mean stretch ratios and stresses are altered in SMKO aorta, but not in SMKO carotid. The circumferential stretch ratio of SMKO aortae is larger than CTR in the UNT and PROP groups at low pressures (A). This difference becomes statistically insignificant between SMKO and CTR after CAP and LOS treatment. The circumferential stress in SMKO aortae is larger than CTR in the UNT and PROP groups at low pressures (B). This difference becomes statistically insignificant between SMKO and CTR with CAP and LOS treatment. At zero pressures, the axial stress in SMKO aorta is lower than CTR in the UNT and PROP groups (C). There are no significant differences between the circumferential stretch (D), circumferential stress (E), or axial stress (F) for the carotid artery regardless of genotype or drug treatment. Error bars are not shown for clarity. N = 6–8 for each group. * = P<.05 for UNT/PROP SMKO compared to CTR.

Fig. 4A shows the mean circumferential stretch ratio versus stress relationships for the ascending aorta. The curves have a sharp increase in slope, or incremental elastic modulus, at a stretch ratio of 1.4 in SMKO mice compared to a gradual increase in modulus in CTR mice, regardless of treatment group. This is quantified in Fig. 4B and shows that CAP and LOS treatment do not alter the modulus of SMKO aorta, despite preventing aneurysm development. For the carotid artery, the mean circumferential stretch ratio versus stress (Fig. 4C) and incremental modulus (Fig. 4D) are similar between genotypes for all treatment groups. The carotid arteries show a sharp increase in modulus at a stretch ratio of 1.8, which is approximately when the CTR aorta begins to increase in modulus. At a physiologic pressure of 100 mmHg, the ascending aorta is stretched to an average circumferential stretch ratio of 1.5 (Fig. 3A) and the carotid is stretched to an average of 1.7 (Fig. 3D). At this physiologic stretch ratio, all arteries in both genotypes are at a circumferential stress around 150 kPa, which shows the ability of the arteries to maintain a homeostatic stress state despite large changes in the absolute dimensions.

Figure 4.

Mean incremental circumferential modulus is increased in SMKO aorta and is not decreased by drug treatment. Mean circumferential stretch ratio versus stress curves show an increased slope, or incremental modulus, at high stretch in the SMKO aorta compared to CTR for all treatment groups (A). This is quantified by the mean stretch ratio versus incremental modulus curves for the aorta (B). The mean circumferential stretch ratio versus stress (C) and mean incremental modulus (D) are similar for the carotid artery across genotypes and drug treatments. Error bars are not shown for clarity. N = 6–8 for each group.

A constrained mixture model of developing mouse aorta reproduces SMKO aortic mechanics

Table 1 summarizes the results of varying the model parameters assumed to be altered in SMKO aorta. Fig. 5 shows illustrative pressure-diameter and circumferential stretch-stress behavior for remodeled adult aortae with a reduced elastin material constant (b₁) (A, D), increased collagen material constants (b₂, b₃, b₄) (B, E), and increased arterial dilation with each time step (C, F). Results for the other options are not shown because the changes are minor compared to the experimental data. Reductions in elastin or increases in collagen material constants shift the pressure-diameter curves to the left, so that diameter increases occur at lower pressures, as seen in SMKO aorta (Fig. 2A). Alterations in the elastin or collagen material constants reproduce the increased nonlinearity of the circumferential stretch-stress curves for SMKO aorta (Fig. 4B). Increases in the arterial dilation with each time step reproduce the increased diameter for UNT SMKO aorta, with little change in the circumferential stretch-stress behavior. Changing the arterial dilation with each time step is the only parameter variation in Table 1 that results in a large diameter increase.

Figure 5.

Illustrative results for some of the model parameter changes summarized in Table 1. Decreasing the elastin material constant (b₁) or increasing the collagen material constants (b₂, b₃, b₄) shifts the pressure-diameter curves to the left so that diameter changes occur at lower pressures (A, B) and makes the circumferential stretch-stress relationships more nonlinear (D, E), as observed in LOS and UNT SMKO aortae. Increasing the arterial dilation for each developmental time step shifts the pressure-diameter curves upward (C), as observed in UNT SMKO aorta, with little effect on the circumferential stretch-stress relationship (F).

If collagen stress contributions are altered to compensate for the fragmented elastic fibers, changes in the elastin and collagen material constants would likely occur concurrently in SMKO aorta. We examined permutations of the illustrative elastin and collagen parameter variations and compared the resulting pressure-diameter, circumferential stretch-stress behavior and physiologic stress, stretch, and unloaded dimensions. We determined that a 50% reduction of the elastin material constant and 10% increases in the collagen material constants qualitatively reproduce the behavior of LOS SMKO aorta (Fig. 6). Compared to UNT CTR calculated with the original model parameters (Wagenseil 2011), this combination of changes maintains shear stress, decreases circumferential and axial stresses by 30%, decreases circumferential and axial stretch ratios by 4%, and reproduces the trends in the unloaded dimension changes (Table 2). Further decreases in the elastin material constant included in Fig. 5 decrease circumferential and axial stresses at physiologic pressure by 70% and further increases in the collagen parameters increase the circumferential and axial stresses by 65%, which do not match the experimental results.

Figure 6.

Mechanical changes in SMKO aorta can be reproduced by a constrained mixture model of growth and remodeling. Starting from identical inputs for a 3 day old mouse aorta, a previously published model (Wagenseil 2011) was used to reproduce pressure-diameter (A) and circumferential stretch ratio versus stress (B) relationships for a 30-day old mouse aorta after growth and remodeling. Experimental data from Fig. 2A and Fig. 4A are replotted in C and D for comparison to the model results. Original model parameters were used to calculate the relationships for the UNT CTR aorta. For both LOS and UNT SMKO aortae, the elastin material constant was reduced by 50% and the collagen material constants were increased by 10%. Additionally, for UNT SMKO aorta, the arterial dilation for each developmental time step was increased from 12 to 20%. These illustrative changes qualitatively reproduce the behavior of SMKO aorta.

Table 2.

Physiologic values calculated in the model using published parameter values for UNT CTR aorta and parameter variations that reproduce the pressure-diameter and circumferential stretch-stress behavior of LOS and UNT SMKO aortae. SMKO aorta values were obtained by applying a 50% decrease in the elastin material constant and 10% increases in the collagen material constants. For UNT SMKO aorta, the arterial dilation with each developmental time step was increased from 12 to 20% compared to UNT CTR and LOS SMKO. All results start from the same 3 day old mouse aorta and calculate growth and remodeling based on step changes in pressure and axial stretch for developmental time steps of 4.5 days up to a final age of 30 days. Relevant equations are in the Appendix, with additional information in Wagenseil (2011). ND = non-dimensional, final dimensions at 30 days old/starting dimensions at 3 days old.

	Shear stress (Pa)	Circ stress (kPa)	Axial stress (kPa)	Circ stretch ratio	Axial stretch ratio	Unl inner diam (ND)	Unl thickness (ND)	Unl length (ND)
UNT CTR	2.8	188	66	1.93	1.14	2.35	1.92	1.68
SMKO - LOS	2.8	125	46	1.84	1.10	2.45	2.00	2.21
SMKO - UNT	0.8	141	45	1.90	1.07	3.60	2.04	3.06

For UNT SMKO aorta, we used the elastin and collagen material constants determined for LOS SMKO aorta and included a 20% aortic dilation with each time step. The model results for the pressure-diameter and circumferential stretch-stress behavior qualitatively match the experimental data (Fig. 6). Compared to UNT CTR, this combination of parameter changes reduces the shear stress by 71%, reduces the physiologic circumferential and axial stresses by 25 and 32%, respectively, reduces the circumferential and axial stretch by 1 and 6%, respectively, and reproduces the trends in the unloaded dimension changes (Table 2).

Fig. 7 shows the circumferential stress contributions of each component in the model for adult UNT CTR, LOS SMKO, and UNT SMKO aortae. For UNT CTR aorta, elastin contributes primarily at low stretch, collagen contributes at high stretch, and passive and active SMCs contribute very little to the total stress (Fig. 7A). This is consistent with expectations for large elastic arteries (Faury et al. 1999). For LOS SMKO aorta, collagen provides more of the stress contribution, with reduced contributions from elastin (Fig. 7B). This is consistent with our reasoning that fragmented elastic fibers in SMKO aorta have reduced stress contributions, which are partially compensated for by increased collagen stress contributions. For UNT SMKO aorta, collagen and elastin look similar to LOS SMKO aorta, but the passive SMC stress contribution becomes highly nonlinear (Fig. 7C). This behavior does not occur at 18% dilation or lower, which highlights how remodeling over time can compound small changes in the model parameters.

Figure 7.

The constrained mixture model shows variations in the contribution of each wall component in CTR and SMKO aortae. The total circumferential stresses, as well as the contributions of elastin, collagen, passive SMCs, and active SMCs are shown for each group for a 30 day old mouse aorta. The CTR data was calculated using the original model (A). The LOS SMKO data was calculated with a 50% reduction in the elastin material constant and 10% increases in the collagen material constants (B). The UNT SMKO data was calculated with the same material constants as the LOS SMKO aorta, plus an increase in arterial dilation with each developmental time step from 12 to 20% (C). Note the increased collagen contribution in LOS and UNT SMKO aortae and the highly nonlinear contribution of the passive SMCs in the UNT SMKO aorta (arrow). In panels A and B, the passive and active SMC curves overlap and are difficult to distinguish.

Elastin and collagen organization and amounts support the model results

Histological sections show that elastic fibers are fragmented in UNT SMKO and LOS SMKO aorta (Fig. 8A), consistent with a reduction in the elastin contribution to the total wall stress. It also appears that there is a higher amount of and more intense collagen staining in histological sections of UNT SMKO and LOS SMKO aorta, consistent with an increase in the collagen contribution to the total wall stress. Quantification of Western blots for soluble protein amounts shows that tropoelastin is increased 2.5 times in UNT SMKO compared to UNT CTR, but is not increased in LOS SMKO aorta (Fig. 8B). SMCs secrete soluble tropoelastin which is then crosslinked to become insoluble elastin within the elastic fibers. Soluble tropoelastin amounts are difficult to relate to elastin stress contributions. Collagen I, which is soluble under the Western blot protein extraction procedure, is increased 12 times in both UNT SMKO and LOS SMKO compared to UNT CTR, consistent with an increased contribution of collagen to the total wall stress.

Figure 8.

Histological and biochemical data support the model assumptions. Representative histological sections of UNT CTR, UNT SMKO, and LOS SMKO aorta stained with HE (for general tissue organization), Hart's (for elastin) and Masson-Trichrome (for collagen) (A). Elastic fibers are thinner and fragmented in SMKO aorta, even after LOS treatment. Bars are 50 μm. Western blot analyses of the ascending aorta showing tropoelastin (Tropo E), collagen I (Col I), and GAPDH (loading control) (B). Right graphs show quantification. Bars are mean ± SEM. * = P< 0.05. NS; not significant.

Discussion

Multi-dimensional, artery specific mechanical changes in SMKO mice

We have recently shown that SMKO aorta exhibits upregulation of local AngII signaling and alteration of SMC phenotype that results in the formation of ascending aortic aneurysms (Huang et al. 2013). SMCs are oriented circumferentially in the aortic wall and changes in SMC behavior are expected to predominantly affect the circumferential direction. However, it has been shown that SMCs can generate active axial stress in the mouse aorta when stimulated to contract (Agianniotis et al. 2012) and that changes in matrix protein amounts can lead to remodeling in the axial direction (Jackson et al. 2005; Carta et al. 2009). We show here that SMC-specific Fbln4 deletion leads to changes in the absolute length of the ascending aorta and the axial stretch ratio of the carotid. Our results highlight compensatory responses in the axial direction for SMKO arteries that may affect cardiovascular function. A reduced axial stretch ratio for the carotid artery has also been observed in mouse models of supravalvular aortic stenosis (Wagenseil et al. 2005), muscular dystrophy (Dye et al. 2007), Marfan syndrome (Eberth et al. 2009b), and hypertension (Eberth et al. 2009a).

Loss of Fbln4 would presumably affect elastic fibers in all large arteries, so we measured mechanical properties in the ascending aorta and carotid artery. Although we found a reduced axial stretch ratio as discussed above, we found no changes in the unloaded dimensions or circumferential mechanical behavior for the carotid artery in SMKO mice. Hence, global mechanical changes in the elastic arteries do not contribute to the SMKO aneurysm phenotype. Prompted by this data, we examined histological sections of carotid arteries from adult CTR and SMKO mice (data not shown) and saw no discernible differences in the elastic fiber structure. This is in stark contrast to the disrupted elastic fibers observed in adult SMKO aorta.

The artery-specific changes in SMKO mice are similar to those reported in mouse models of Marfan Syndrome that also develop ascending aortic aneurysms (Bunton et al. 2001; Eberth et al. 2009b). It has been suggested that different embryonic origins of SMCs may account for localized aneurysm development. SMCs in the ascending aorta and the carotid arteries derive from the neural crest (Majesky 2007), so this cannot explain the differences we observed. However, SMCs at the base of the ascending aorta near the aortic valve derive from the secondary heart field and these SMCs may respond differently than neural crest-derived SMCs to genetic, chemical, or mechanical signals and propagate these changes to the rest of the ascending aorta. Alternatively, it has been shown that shear stress, which varies with location in the arterial tree, is an important determinant of arterial function and gene expression (Reneman and Hoeks 2008). Hence, shear stress may contribute to localized aneurysm development.

Artery and direction specific effects of CAP and LOS treatment

Treatment of SMKO mice with CAP or LOS prevents aneurysm development. Although neither of the drugs reverse the decrease in aortic circumferential compliance, decreased compliance may still contribute to aneurysm pathophysiology. For example, in mouse models of Marfan syndrome, combined therapy with LOS and doxycycline is more effective at preventing and treating aneurysms than either drug alone (Yang et al. 2010; Xiong et al. 2012). Doxycycline is a non-specific matrix mettaloproteinase inhibitor. The combination therapy significantly improves elastic fiber organization and decreases the number of elastic fiber breaks, which may increase aortic compliance. In CTR aorta, we found that CAP and LOS treatment actually decrease the aortic compliance compared to UNT CTR at high pressures and the peak compliance shifts to lower pressures. CAP and LOS treatment have no effect on the compliance of the CTR carotid artery. Since CAP and LOS reduce systolic blood pressures in CTR mice (Huang et al. 2013), these drug-treated aortae may have remodeled in response to lower physiologic blood pressure. The peak compliance of the ascending aorta occurs just below the physiologic pressure in developing mice, even if the compliance and physiologic pressure are altered by reduced elastin levels (Le et al. 2011). Aortic remodeling may be an important consideration in the choice of anti-hypertensive treatments, as reduced compliance would not be a desired outcome (Cecelja and Chowienczyk 2009).

CAP and LOS treatment of SMKO arteries does not reverse the mechanical changes in the axial direction. This shows that altering the axial mechanics is not required for aneurysm prevention, but does not exclude relationships between axial mechanics and aneurysm pathophysiology. Doxycycline prevents axial remodeling in arteries with decreased axial stretch (Jackson et al. 2005). Affecting axial mechanics, increasing compliance, and inhibiting Ang II may be mechanisms through which combined doxycycline and LOS therapy improve aneurysm treatment and prevention in Marfan mouse arteries (Xiong et al. 2012; Yang et al. 2010). Persistent increases in axial length and decreases in axial stretch ratio can lead to arterial tortuosity (Jackson et al. 2005). Tortuosity has been linked to flow abnormalities, clot formation, and stroke (Chesnutt and Han 2011) and could develop in individuals where aneurysm formation has been prevented, but arterial axial mechanical behavior has not been addressed.

Models of developmental arterial growth and remodeling

Mechanical models of arterial growth and remodeling have been used to predict general trends in development (Alford et al. 2008; Wagenseil 2011) and in adult arteries with altered pressure or flow (Gleason et al. 2004; Taber and Eggers 1996). To our knowledge, this is the first application of a model to reproduce changes with a specific genetic defect and drug treatment. Our results show that increasing the arterial dilation for each time step, decreasing the elastin material constant, and increasing the collagen material constants qualitatively reproduce the differences in pressure-diameter and circumferential stretch-stress behavior between UNT CTR and UNT SMKO aorta. Behavior of LOS SMKO aorta is reproduced by rescuing the arterial dilation with each time step, but maintaining the changes in elastin and collagen constants. The model provides rationale for how the mechanical properties (determined by the change in elastin and collagen material constants) can be separated from the SMC phenotype and aneurysm development (determined by the altered arterial dilation response) to reproduce the observed differences between UNT SMKO and LOS SMKO aorta.

From the experimental behavior of the composite aortic wall, one may assume that LOS does not change the circumferential stretch-stress behavior of SMKO aorta and consequently, that rescuing the altered mechanical properties of UNT SMKO aorta plays no role in aneurysm prevention. However, the contribution of each wall component in the model shows that the passive SMC stress is different for UNT SMKO compared to LOS SMKO. The distribution of stresses among wall components and increased SMC stress may play a role in aneurysm development. The model results lead to new hypotheses on the interactions between mechanical behavior and aneurysmal disease that cannot be generated from the experimental data alone.

Relating model results to biological signaling

PROP, an anti-hypertensive medication that does not act through the AngII pathway, does not prevent aneurysms in SMKO mice (Huang et al. 2013). ACE, which converts AngI to AngII, is upregulated in SMKO ascending aorta and the ACE-inhibitor CAP prevents SMKO aneurysms. Treatment with the AngII receptor type I blocker LOS increases expression of SMC contractile genes and prevents aneurysms in SMKO mice. Our model results suggest that increased local AngII signaling and the resultant decrease in SMC contractile genes are directly related to arterial dilation with each developmental time step. Increased arterial dilation may be caused by a dysregulated response to flow changes in SMKO aorta. In vitro, endothelial cells and SMCs respond to flow by upregulating ACE (Gosgnach et al. 2000), which combined with the already increased ACE expression in SMKO aorta may exaggerate SMKO arterial dilation. AngII is a potent vasoconstrictor, so it is counterintuitive that increases in AngII would increase arterial dilation. However, the vascular response to AngII depends on the secondary pathway. Specifically, hydrogen peroxide (H₂O₂) is induced by AngII and can exert either a contraction or dilation response depending on the vascular bed, contractile state, and cellular source (Nguyen Dinh Cat and Touyz 2011).

Remodeling of LOS SMKO aorta can be reproduced by rescuing the arterial dilation with no changes in elastin and collagen material constants compared to UNT SMKO, implying that LOS blocks local AngII signaling, but cannot repair the elastic fibers. Because elastic fibers are assembled during late embryonic and early postnatal development, it is reasonable that postnatal LOS treatment cannot prevent SMKO elastic fiber fragmentation. The elastic fiber fragmentation and subsequently reduced elastin material constant for UNT and LOS SMKO aorta are supported by histological evidence. Although our model shows that the elastin material constants can be the same, it is likely that there are some differences between LOS and UNT SMKO aorta. Western blot analyses show that there are differences between soluble tropoelastin amounts in UNT SMKO and LOS SMKO aorta. H₂O₂ solubilizes elastin (Umeda et al. 2001), which may increase soluble tropoelastin in UNT SMKO aorta. This may be prevented in LOS SMKO aorta by blocking AngII-induced H₂O₂ generation. Quantification of insoluble elastin amounts, elastin crosslinking and fiber fragmentation would add support to the model assumption of a reduced elastin stress contribution in SMKO aorta.

It was assumed that the collagen material constants would increase to compensate for the reduced elastin stress contribution in SMKO aorta. This would increase the collagen stress contribution, as well as the nonlinearity of the stretch-stress relationship. Although difficult to quantify, histological evidence supports an increased amount of collagen in UNT and LOS SMKO aorta. Western blot analyses show a dramatic increase in collagen I amounts in UNT and LOS SMKO compared to UNT CTR. Because of the nonlinear behavior of collagen in the model, stress contribution cannot be directly related to collagen amounts. Physical characteristics of the collagen fibers such as size, orientation, undulation, and crosslink density will influence the mechanical behavior. Changes in any of these factors in SMKO aorta could increase the nonlinearity of the collagen stretch-stress relationship. The current model results provide new avenues for further research on the mechanical and biological factors in SMKO aneurysm development.

Limitations

We use two models in the current work: a mouse model of human disease and a mechanical model of growth and remodeling in the aortic wall. Mice offer a convenient and well-controlled model where genetic and phenotypic changes can be linked. However, they often do not recapitulate many factors of the related human diseases and may not respond to drug treatments in the same manner as humans. Mechanical models offer a convenient and well-controlled method for investigating factors, such as the contribution of different wall components, that cannot easily be separated in the experimental data. One limitation is the number of assumptions and parameters that must be included in the model equations. In this work, we did not try to fit model parameters to the experimental data or make changes to the previous model assumptions (Wagenseil 2011). The previous model has several limitations, including the separation of the radial growth from the blood flow and step changes in pressure and length, that must be addressed in future work. Our goal in this work was to use the previous model for UNT CTR aorta and to investigate experimentally-motivated changes in the model parameters to provide illustrative examples that qualitatively reproduce experimental data for UNT and LOS SMKO aorta. Lastly, we provide histological images and protein quantification as supporting evidence for the model assumptions, but more work is needed to relate the changes in stress contributions to physical changes in elastin and collagen amount and organization.

Conclusions

SMKO mice show axial remodeling in the ascending aorta and carotid artery, which is not reversed by AngII inhibitors that prevent aneurysms. However, the axial remodeling may be linked to additional cardiovascular complications, such as tortuosity. SMKO mice show circumferential changes in the pressure-diameter behavior and stretch-stress behavior that are specific to the ascending aorta. Hence, circumferential mechanical changes in the carotid artery do not contribute to aortic aneurysm development. A previously published constrained mixture model for developing mouse aorta qualitatively reproduces the pressure-diameter and circumferential stretch-stress relationships after growth and remodeling for the adult CTR aorta. Increasing the arterial dilation with each developmental time step from 12 to 20%, along with a 50% decrease in the elastin material constant and a 10% increase in the collagen material constants in the model reproduce the behavior of UNT SMKO aorta. The behavior of LOS SMKO aorta can be reproduced by rescuing the arterial dilation with each time step, while maintaining the altered elastin and collagen material constants. Histological and biochemical data support the model assumptions. The model results link biological observations to mechanical responses and suggest new hypotheses on the relationships between mechanical behavior and aneurysm development.

Appendix

The aorta is considered a constrained mixture of wall components (k) where the total mean Cauchy stress (σ) in the circumferential (θ) and axial (z) direction is the sum of the stresses in each component (${\sigma }_{\theta }^k$, ${\sigma }_z^k$) multiplied by the mass fraction (ϕ^k) of each component at time, s:

$${\sigma }_{\theta }({\lambda }_{\theta },{\lambda }_z)=\sum _k{\varphi }^k\left(s\right){\sigma }_{\theta }^k({\lambda }_{\theta }^k,{\lambda }_z^k),{\sigma }_z({\lambda }_{\theta },{\lambda }_z)=\sum _k{\varphi }^k\left(s\right){\sigma }_z^k({\lambda }_{\theta }^k,{\lambda }_z^k),$$ (6a,b)

where λ_θ, λ_z are the stretch ratios of the mixture and λ^k_θ, λ^k_z are the stretch ratios of each component. The components have individual homeostatic stretch ratios (λ^k_θh, λ^k_zh) at which they are produced and these values increase 3% in the circumferential direction and decrease 3% in the axial direction with each developmental time step (Wagenseil 2011). The unloaded stretch ratios of the mixture when the components are produced are λ_θu, λ_zu. The different stretch ratios are related by:

$${\lambda }_{\theta }^k=\frac{{\lambda }_{\theta }{\lambda }_{\theta h}^k}{{\lambda }_{\theta u}},{\lambda }_z^k=\frac{{\lambda }_z{\lambda }_{zh}^k}{{\lambda }_{zu}}.$$ (7a,b)

The component stresses are defined by constitutive equations for elastin (e), collagen (c) and SMCs (m). SMCs have both passive (pas) and active (act) stress contributions (Gleason and Humphrey 2004; Gleason et al. 2004):

$$\text{Elastin}:{\sigma }_{\theta }^e({\lambda }_{\theta }^e,{\lambda }_z^e)=2{\lambda }_{\theta }^e{b}_1\left(1-\frac{1}{{\lambda }_{\theta }^{e^4}{\lambda }_z^{e^2}}\right),{\sigma }_z^e({\lambda }_{\theta }^e,{\lambda }_z^e)=2{\lambda }_z^e{b}_1\left(1-\frac{1}{{\lambda }_{\theta }^{e^2}{\lambda }_z^{e^4}}\right),$$ (8a,b)

$$\text{Collagen}:{\sigma }_{\theta }^c({\lambda }_{\theta }^c,{\lambda }_z^c)=2{\lambda }_{\theta }^c{b}_2{b}_3\left(1-\frac{1}{{\lambda }_{\theta }^{c^4}{\lambda }_z^{c^2}}\right)\text{exp}\left(Q^c\right({\lambda }_{\theta }^c,{\lambda }_z^c\left)\right),$$ (9)

$${\sigma }_z^c({\lambda }_{\theta }^c,{\lambda }_z^c)=2{\lambda }_{\theta }^c{b}_2\left[b_3\left(1-\frac{1}{{\lambda }_{\theta }^{c^2}{\lambda }_z^{c^4}}\right)+2b_4({\lambda }_z^{c^2}-1)\right]\text{exp}\left(Q^c\right({\lambda }_{\theta }^c,{\lambda }_z^c\left)\right),$$ (10)

$$\text{with}{Q}^c({\lambda }_{\theta }^c,{\lambda }_z^c)=b_3\left({\lambda }_{\theta }^{c^2}+{\lambda }_z^{c^2}+\frac{1}{{\lambda }_{\theta }^{c^2}{\lambda }_z^{c^2}}-3\right)+b_4{({\lambda }_z^{c^2}-1)}^2,$$ (11)

$$\text{SMCs}:{\sigma }_{\theta }^m={\sigma }_{\theta ,pas}^m+{\sigma }_{\theta ,act}^m,{\sigma }_z^m={\sigma }_{z,pas}^m,$$ (12a,b)

$$\text{Passive SMCs}:{\sigma }_{\theta ,pas}^m({\lambda }_{\theta }^m,{\lambda }_z^m)=2{\lambda }_{\theta }^{m^2}\left[b_5\left(1-\frac{1}{{\lambda }_{\theta }^{m^4}{\lambda }_z^{m^2}}\right)+2b_6{b}_7({\lambda }_{\theta }^{m^2}-1)\right]\text{exp}\left(Q^m\right({\lambda }_{\theta }^m\left)\right),$$ (13)

$$\text{with}{Q}^m\left({\lambda }_{\theta }^m\right)=b_7{\left({\lambda }_{\theta }^{m^2}-1\right)}^2,$$ (14)

$${\sigma }_{z,\mathit{pas}}^m({\lambda }_{\theta }^m,{\lambda }_z^m)=2{\lambda }_z^{m^2}{b}_5\left(1-\frac{1}{{\lambda }_{\theta }^{m^2}{\lambda }_z^{m^4}}\right),$$ (15)

$$\text{Active SMCs}:{\sigma }_{\theta ,\mathit{act}}^m\left({\lambda }_{\theta }^m\right)=T_{\mathit{act}}\widehat{f}\left({\lambda }_{\theta }^m\right),\text{with}\widehat{f}\left({\lambda }_{\theta }^m\right)={\lambda }_{\theta }^m\left[1-{\left(\frac{{\lambda }_M-{\lambda }_{\theta }^m}{{\lambda }_M-{\lambda }_0}\right)}^2\right],T_{\mathit{act}}=T_B-T_Q,$$ (16)

where b_{1 – 7} are passive material constants that increase 8% with each developmental time step (Wagenseil 2011). λ_M, λ₀ are active SMC material constants, T_B = basal SMC tone constant and T_Q = SMC activation caused by changes in flow. T_Q can be calculated according to (Gleason et al. 2004):

$$T_Q=\frac{1}{{\varphi }^m\widehat{f}\left({{\epsilon }_Q}^{1∕3}{\lambda }_{\theta }^m\right(0\left)\right)}\left({\sigma }_{\theta ,\mathit{pas}}^m\right(S_v)-{\sigma }_{\theta ,\mathit{pas}}^m(0\left)d\right)+T_B\left(1-\frac{\widehat{f}\left({\lambda }_{\theta }^m\right(0\left)d\right)}{\widehat{f}\left({{\epsilon }_Q}^{1∕3}{\lambda }_{\theta }^m\right(0\left)\right)}\right),$$ (17)

where the SMC stretch ratios and stresses are functions of the time elapsed since each step change in pressure, length and flow. At time = 0, the aorta is at its homeostatic state before the step change occurs and at time = s_v, the instantaneous dilation response occurs. Additionally, $d={{\epsilon }_Q}^{1∕3}{h}_o∕h\left(s_v\right)$, where h_o = initial wall thickness at time = 0.

The components are continually produced with each developmental time step. SMCs and collagen are also continually degraded, but elastin is not because of its long half-life. Kinetic functions for the production (g) and the degradation (q) of each component are (Gleason and Humphrey 2004; Gleason et al. 2004):

$$g^k\left(S\right)=1-\text{exp}\left[-K_g^{k}S∕S_h\right],q^k\left(S\right)=\text{exp}\left[-K_q^{k}S∕S_h\right],$$ (18a,b)

where K^k_g and K^k_q are the associated rate constants for each component and s_h is the homeostatic time at which remodeling is complete. A rate constant of 6.9 allows almost complete turnover with about 0.1% of the original component remaining. Total mass fractions (original + new components) at each time step are determined from previously published experimental data (Wagenseil 2011).