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 encourages the expression of contractile genes and prevents aneurysm development, but does 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.

Appendix

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

$$\begin{aligned}&\sigma _\theta (\lambda _\theta , \lambda _z)=\sum _k{\phi ^{k}(s)\sigma _\theta ^k (\lambda _\theta ^k, \lambda _z^k)},\end{aligned}$$

(6a)

$$\begin{aligned}&\sigma _z (\lambda _\theta , \lambda _z)=\sum _k {\phi ^{k}(s)\sigma _z^k (\lambda _\theta ^k, \lambda _z^k)}, \end{aligned}$$

(6b)

where $\lambda _{\theta }, \lambda _{z}$ are the stretch ratios of the mixture and $\lambda ^{k}_{\theta }, \lambda ^{k}_{z}$ are the stretch ratios of each component. The components have individual homeostatic stretch ratios $(\lambda ^{k}_{\theta h}, \lambda ^{k}_{zh})$ 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 $\lambda _{\theta u}, \lambda _{zu}$. The different stretch ratios are related by

$$\begin{aligned} \lambda _\theta ^k&= \frac{\lambda _\theta \lambda _{\theta h}^k}{\lambda _{\theta u}},\end{aligned}$$

(7a)

$$\begin{aligned} \lambda _z^k&= \frac{\lambda _z\lambda _{zh}^k}{\lambda _{zu}}. \end{aligned}$$

(7b)

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):

Elastin:

$$\begin{aligned} \sigma _\theta ^e (\lambda _\theta ^e, \lambda _z^e)&= 2\lambda _\theta ^e b_1 \left( {1-\frac{1}{\lambda _\theta ^{e^{4}} \lambda _z^{e^{2}} }}\right) ,\end{aligned}$$

(8a)

$$\begin{aligned} \sigma _z^e (\lambda _\theta ^e, \lambda _z^e)&= 2\lambda _z^e b_1\left( {1-\frac{1}{\lambda _\theta ^{e^{2}} \lambda _z^{e^{4}} }}\right) , \end{aligned}$$

(8b)

Collagen:

$$\begin{aligned} \sigma _\theta ^c (\lambda _\theta ^c, \lambda _z^c)&= 2\lambda _\theta ^c b_2 b_3\left( {1-\frac{1}{\lambda _\theta ^{c^{4}} \lambda _z^{c^{2}}}}\right) \exp (Q^{c}(\lambda _\theta ^c ,\lambda _z^c)),\end{aligned}$$

(9)

$$\begin{aligned} \sigma _z^c (\lambda _\theta ^c, \lambda _z^c)&= 2\lambda _\theta ^c b_2 \left[ b_3 \left( {1-\frac{1}{\lambda _\theta ^{c^{2}} \lambda _z^{c^{4}} }}\right) +2b_4 \left( {\lambda _z^{c^{2}} -1}\right) \right] \nonumber \\&\times \exp (Q^{c}(\lambda _\theta ^c, \lambda _z^c)), \end{aligned}$$

(10)

with

$$\begin{aligned} Q^{c}(\lambda _\theta ^c ,\lambda _z^c)&= b_3 \left( {\lambda _\theta ^{c^{2}} +\lambda _z^{c^{2}} +\frac{1}{\lambda _\theta ^{c^{2}} \lambda _z^{c^{2}} }-3}\right) \nonumber \\&+\,\,b_4\left( {\lambda _z^{c^{2}} -1}\right) ^{2}, \end{aligned}$$

(11)

SMCs:

$$\begin{aligned} \sigma _{\theta }^{m}&= \sigma _{\theta ,pas}^{m}+ \sigma _{\theta , act}^{m},\end{aligned}$$

(12a)

$$\begin{aligned} \sigma _{z}^{m}&= \sigma _{z,pas}^{m}, \end{aligned}$$

(12b)

Passive SMCs:

$$\begin{aligned}&\sigma _{\theta ,pas}^m (\lambda _\theta ^m, \lambda _z^m)\nonumber \\&\quad =2\lambda _\theta ^{m^{2}} \left[ {b_5 \left( {1-\frac{1}{\lambda _\theta ^{m^{4}} \lambda _z^{m^{2}}}}\right) +2b_6 b_7 \left( {\lambda _\theta ^{m^{2}}-1}\right) }\right] \nonumber \\&\qquad \times \exp (Q^{m}(\lambda _\theta ^m)), \end{aligned}$$

(13)

with

$$\begin{aligned}&Q^{m}(\lambda _\theta ^m)=b_7 \left( {\lambda _\theta ^{m^{2}}-1}\right) ^{2},\end{aligned}$$

(14)

$$\begin{aligned}&\sigma _{z,pas}^m (\lambda _\theta ^m, \lambda _z^m)=2\lambda _z^{m^{2}} b_5 \left( {1-\frac{1}{\lambda _\theta ^{m^{2}} \lambda _z^{m^{4}}}}\right) , \end{aligned}$$

(15)

Active SMCs:

$$\begin{aligned} \sigma _{\theta , act}^m (\lambda _\theta ^m)=T_{act} \hat{f}(\lambda _\theta ^m), \end{aligned}$$

with

$$\begin{aligned} \hat{{f}}(\lambda _\theta ^m)\!=\!\lambda _\theta ^m \left[ {1-\left( {\frac{\lambda _M -\lambda _\theta ^m }{\lambda _M -\lambda _0}}\right) ^{2}}\right] , T_{act} \!=\! T_{B} \!-\! T_{Q}, \end{aligned}$$

(16)

where $b_{1 - 7}$ are passive material constants that increase 8 % with each developmental time step (Wagenseil 2011). $\lambda _{M}, \lambda _{0}$ 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):

$$\begin{aligned} T_Q&= \frac{1}{\phi ^{m}\hat{f}(\varepsilon _Q^{1/3}\lambda _\theta ^m (0))}(\sigma _{\theta ,pas}^m (s_v )-\sigma _{\theta , \hbox {pas}}^m (0)d)\nonumber \\&\quad +\,\,T_B \left( {1-\frac{\hat{{f}}(\lambda _\theta ^m (0)d)}{\hat{{f}}(\varepsilon _Q^{1/3}\lambda _\theta ^m (0))}}\right) , \end{aligned}$$

(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, $d=\varepsilon _Q^{1/3}h_o /h(s_v)$, 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):

$$\begin{aligned} g^{k}(s)&= 1-\exp [-K_{g}^{k}s/s_{h}],\end{aligned}$$

(18a)

$$\begin{aligned} q^{k}(s)&= \exp [-K_{q}^{k}s/s_{h}], \end{aligned}$$

(18b)

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).