Modeling mechano-driven and immuno-mediated aortic maladaptation in hypertension

Abstract

Uncontrolled hypertension is a primary risk factor for diverse cardiovascular diseases and thus remains responsible for significant morbidity and mortality. Hypertension leads to marked changes in the composition, structure, properties, and function of central arteries; hence, there has long been interest in quantifying the associated wall mechanics. Indeed, over the past 20 years there has been increasing interest in formulating mathematical models of the evolving geometry and biomechanical behavior of central arteries that occur during hypertension. In this paper, we introduce a new mathematical model of growth (changes in mass) and remodeling (changes in microstructure) of the aortic wall for an animal model of induced hypertension that exhibits both mechano-driven and immuno-mediated matrix turnover. In particular, we present a bilayered model of the aortic wall to account for differences in medial versus adventitial growth and remodeling and we include mechanical stress and inflammatory cell density as determinants of matrix turnover. Using this approach, we can capture results from a recent report of adventitial fibrosis that resulted in marked aortic maladaptation in hypertension. We submit that this model can also be used to identify novel hypotheses to guide future experimentation.

Appendix 1: Material model determination

1.1 Progressive nonlinear regression

• 1. Recreate biaxial data ($P-d,f-P$; $f-\lambda ,P-\lambda$) using the given geometry (Table S1) and mean (bulk) mechanical properties (Table S2) from Bersi et al. (2016) for Sham (day 0) and 4wk-Ang II (day 28).

• 2. Use an extended (bilayered, rule-of-mixture-based) elastic arterial model with original (layer-specific) mass fractions $[\phi _{\varGamma o}^{e} ,\phi _{\varGamma o}^{m},\phi _{\varGamma o}^{c}]$ from Bersi et al. (2016) and a calculated medial–adventitial interface radius in the traction-free configuration $r_{MAtf}$ (using relative area fractions) to determine geometry- and mass-related material parameters $\alpha _{0}$, $\beta _{M}^{z}=\beta _{A}^{z}$, and $\beta _{A}^{\theta }$ (with $\beta _{M}^{d}=1-\beta _{M}^{z}$ and $\beta _{A} ^{d}=1-\beta _{A}^{\theta }-\beta _{A}^{z}$) by fitting the biaxial data (at day 0) generated in Step 1, whereupon we obtain (layer- and orientation-specific) original mass fractions

  $$\begin{aligned} \phi _{Mo}&=\left[ \phi _{Mo}^{e},\phi _{Mo}^{m,\theta },\phi _{Mo}^{c,z},\phi _{Mo}^{c,d}\right] \nonumber \\&=\left[ \phi _{Mo}^{e},\phi _{Mo}^{m},\phi _{Mo}^{c}\beta _{M}^{z},\phi _{Mo}^{c} \beta _{M}^{d}\right] \end{aligned}$$

  (21)

  and

  $$\begin{aligned} \phi _{Ao}&=\left[ \phi _{Ao}^{e},\phi _{Ao}^{c,\theta },\phi _{Ao}^{c,z},\phi _{Ao}^{c,d}\right] \nonumber \\&=\left[ \phi _{Ao}^{e},\phi _{Ao}^{c}\beta _{A}^{\theta },\phi _{Ao}^{c}\beta _{A} ^{z},\phi _{Ao}^{c}\beta _{A}^{d}\right] . \end{aligned}$$

  (22)

  Assume a fixed value $\alpha _{0}$ for $s\in [0,28]$ days (Table 1). Because different cohorts of collagen within each layer share the same turnover characteristics (Table 1), $\beta _{M}^{z} =\beta _{A}^{z}$ and $\beta _{A}^{\theta }$ remain constant for $s\in [0,28]$ days as well (cf. Latorre and Humphrey 2018).

• 3. From estimated mass fractions $[\phi _{\varGamma h}^{e},\phi _{\varGamma h} ^{m},\phi _{\varGamma h}^{c}]$ at $s=28$ days from Figure 3b in Bersi et al. (2016), obtain (layer- and orientation-specific) evolved homeostatic mass fractions

  $$\begin{aligned} \phi _{Mh}&=\left[ \phi _{Mh}^{e},\phi _{Mh}^{m,\theta },\phi _{Mh}^{c,z},\phi _{Mh}^{c,d}\right] \nonumber \\&=\left[ \phi _{Mh}^{e},\phi _{Mh}^{m},\phi _{Mh}^{c}\beta _{M}^{z},\phi _{Mh}^{c} \beta _{M}^{d}\right] \end{aligned}$$

  (23)

  and

  $$\begin{aligned} \phi _{Ah}&=\left[ \phi _{Ah}^{e},\phi _{Ah}^{c,\theta },\phi _{Ah}^{c,z},\phi _{Ah}^{c,d}\right] \nonumber \\&=\left[ \phi _{Ah}^{e},\phi _{Ah}^{c}\beta _{A}^{\theta },\phi _{Ah}^{c}\beta _{A} ^{z},\phi _{Ah}^{c}\beta _{A}^{d}\right] . \end{aligned}$$

  (24)

• 4. Given deposition stretches for elastin $G_{\theta }^{e}$ and $G_{z}^{e}$, with $G_{r}^{e}=1/(G_{\theta }^{e}G_{z}^{e})$:

  • (a) Use the extended (bilayered, rule-of-mixture) arterial model, now including deposition stretches (cf. Latorre and Humphrey 2018), with original homeostatic mass fractions from Step 2, elastin deformed at the original in vivo state as

    $$\begin{aligned} \mathbf {F}_{\varGamma o}^{e}=\mathbf {G}^{e} \end{aligned}$$

    (25)

    and smooth muscle and collagen with original in vivo equilibrium stresses

    $$\begin{aligned} \hat{\varvec{\sigma }}_{\varGamma o}^{\alpha }=\mathbf {G}_{o}^{\alpha }\hat{\mathbf {S}}_{\varGamma o}^{\alpha }\mathbf {G}_{o}^{\alpha }, \end{aligned}$$

    (26)

    to determine original material parameters (at $s=0$ days) $c_{o}^{e}$, $c_{1o}^{m+}$, $c_{2o}^{m+}$, $c_{1o}^{c+}$, $c_{2o}^{c+}$, $G_{o}^{m}$, and $G_{o}^{c}$, by fitting respective biaxial data generated in Step 1, including only measurements wherein smooth muscle and all collagen fiber families experience tension (Bellini et al. 2014). Fix these values for substep 4b.

  • (b) Determine original material parameters for compressed smooth muscle and collagen fibers/glycosaminoglycans, $c_{1o}^{m-}$, $c_{2o}^{m-}$, $c_{1o}^{c-}$, and $c_{2o}^{c-}$, by fitting all biaxial (original) measurements generated in Step 1 (Bellini et al. 2014; Latorre et al. 2017).

  • (c) Use a G&R-evolved elastic arterial model (cf. Latorre and Humphrey 2018), with evolved homeostatic mass fractions from Step 3, elastin deformed elastically at the new in vivo state as

    $$\begin{aligned} \mathbf {F}_{\varGamma h}^{e}=\mathbf {F}_{\varGamma h}\mathbf {G}^{e} \end{aligned}$$

    (27)

    and smooth muscle and collagen with evolved in vivo equilibrium stresses

    $$\begin{aligned} \hat{\varvec{\sigma }}_{\varGamma h}^{\alpha }=\mathbf {G}_{h}^{\alpha } \hat{\mathbf {S}}_{\varGamma h}^{\alpha }\mathbf {G}_{h}^{\alpha }, \end{aligned}$$

    (28)

    to determine evolved material parameters (at $s=28$ days) $c_{1h}^{m+}$, $c_{2h}^{m+}$, $c_{1h}^{c+}$, $c_{2h}^{c+}$, $G_{h}^{m}$, and $G_{h}^{c}$, with $c_{h}^{e}=c_{o}^{e}\equiv c^{e}$, by fitting respective biaxial data generated in Step 1, including only measurements wherein smooth muscle and all collagen fiber families experience tension. Fix these values for substep 4.d.

  • (d) Determine evolved material parameters for compressed smooth muscle and collagen fibers/glycosaminoglycans, $c_{1h}^{m-}$, $c_{2h}^{m-}$, $c_{1h}^{c-}$, and $c_{2h}^{c-}$, by fitting all biaxial (evolved) measurements generated in Step 1.

  • (e) Compute the associated error between measured and predicted in vivo axial stretches at days 0 and 28.

• 5. Perform an iterative procedure to determine optimal elastin deposition stretches $G_{\theta }^{e}$ and $G_{z}^{e}$ in Step 4 such that a global fitting error in Steps 4b, d, e is minimized according to a predefined objective function. This yields the overall best-fit values in Table 1.

1.2 Additional estimations

• 6. Observing that evolved variables at 2 and 4 weeks after Ang II infusion (Figure 2 in Bersi et al. 2016) are almost the same, we can assume that the G&R response is, in practice, immunomechanobiologically adapted at 2 weeks. Knowing that adaptations following increases in pressure are “forgiving” (when compared to adaptations following increases in flow rate or axial stretch, see Latorre and Humphrey 2018), we can estimate a characteristic time for collagen and smooth muscle turnover of the order $s_{\mathrm{G} \& \mathrm{R}}$ $\lesssim 2$ weeks. Thus, because of the lack of additional experimental data (i.e., measured between $s=0$ and $s=14$ days, when the G&R evolution takes place), we estimate $k_{o}^{m}=k_{o}^{c}=7$ day$^{-1}$. Recall from Latorre and Humphrey (2018) that the gain parameters $K_{\varGamma \sigma }^{\alpha }$ and $K_{\varGamma \tau }^{\alpha }$ also affect the adaptation process. Again, due to the lack of experimental data describing the evolution process, we estimate values $K_{M\sigma }^{c}=2$ and $K_{M\tau }^{c}=2.5$ for medial collagen. Comparing geometries and mass fractions in Steps 2 and 3, while considering model-consistent relations for the evolution of smooth muscle and collagen mass densities (Latorre and Humphrey 2018), we can estimate $\eta _{\varUpsilon }^{m}=0.8$ and $\eta _{\varUpsilon }^{c}=1.667$, hence we obtain $K_{M\sigma }^{m}=\eta _{\varUpsilon }^{m}K_{M\sigma }^{c}=1.6$ and $K_{M\tau } ^{m}=\eta _{\varUpsilon }^{m}K_{M\tau }^{c}=2$, as well as $K_{A\sigma }^{c} =\eta _{\varUpsilon }^{c}K_{M\sigma }^{c}=3.33$ and $K_{A\tau }^{c}=\eta _{\varUpsilon }^{c}K_{M\tau }^{c}=4.17$. Finally, from the immunomechanobiological equilibrium condition $\varUpsilon _{Mh}^{c}=1$ (cf. Eq. (20a)), we obtain

  $$\begin{aligned} K_{M\sigma }^{c}\varDelta \sigma _{h}-K_{M\tau }^{c}\varDelta \tau _{wh}+K_{M\varphi } ^{c}\varDelta \varrho _{\varphi h}=0 \end{aligned}$$

  (29)

  whereupon, with $\varDelta \varrho _{\varphi h}=1$, $\varDelta \sigma _{h}<0$, and $\varDelta \tau _{wh}>0$ also known (cf. Eq. (20b)),

  $$\begin{aligned} K_{M\varphi }^{c}=K_{M\tau }^{c}\varDelta \tau _{wh}-K_{M\sigma }^{c}\varDelta \sigma _{h}=1.74>0 \end{aligned}$$

  (30)

  so $K_{M\varphi }^{m}=\eta _{\varUpsilon }^{m}K_{M\varphi }^{c}=1.39$ and $K_{A\varphi }^{c}=\eta _{\varUpsilon }^{c}K_{M\varphi }^{c}=2.90$.

• 7. Due to the lack of additional experimental data during the actual G&R evolution, we assume an evolution of material parameters $c_{1}^{m}$, $c_{2}^{m}$, $c_{1}^{c}$, $c_{2}^{c}$, $G^{m}$, and $G^{c}$ in terms of inflammatory cell level, say $\varsigma ^{i}\left( \varDelta \varrho _{\varphi }\right)$, common for all parameters [recall, e.g., Eq. (19)]

  $$\begin{aligned} \varsigma \left( \varDelta \varrho _{\varphi }\right) =\varsigma _{o}+f(\varDelta \varrho _{\varphi })\left( \varsigma _{h}-\varsigma _{o}\right) \end{aligned}$$

  (31)

  where $f(\varDelta \varrho _{\varphi })=(\varDelta \varrho _{\varphi })^{1/3}$ proved useful to illustrate some qualitative results including inflammatory effects in the evolution (Figs. 2, 3, 4, 5). This nonlinear evolution, common for all evolving parameters, is a hypothesis that should be tested against (or determined from) additional experimental data obtained during actual G&R evolutions. Indeed, note that each parameter $\varsigma ^{i}$ could evolve between its corresponding values $\varsigma _{o}^{i}$ and $\varsigma _{h}^{i}$ independent of other parameters.