Nonlinear Mechanics of Arterial Growth

TL;DR

A distinctive feature of the model is that the growth variable is not constrained by an explicit upper bound; instead, growth naturally approaches a steady-state value as a consequence of the intrinsic energetic competition.

Abstract

In this paper, we formulate a geometric theory of the mechanics of arterial growth. An artery is modeled as a finite-length thick shell that is made of an incompressible nonlinear anisotropic solid. An initial radially-symmetric distribution of finite radial and circumferential eigenstrains is assumed. Bulk growth is assumed to be isotropic. A novel framework is proposed to describe the time evolution of growth, governed by a competition between the elastic energy and a \emph{growth energy}. The governing equations are derived through a two-potential approach and using the Lagrange-d'Alembert principle. An isotropic dissipation potential is considered, which is assumed to be convex in the rate of growth function. Several numerical examples are presented that demonstrate the effectiveness of the proposed model in predicting the evolution of arterial growth and the intricate interplay among eigenstrains, residual stresses, elastic energy, growth energy, and dissipation potential. A distinctive feature of the model is that the growth variable is not constrained by an explicit upper bound; instead, growth naturally approaches a steady-state value as a consequence of the intrinsic energetic competition.

Figures (7)

• Figure 1: Isotropic growth evolution in a single layer artery. The growth parameter $\mathfrak{g}$ is plotted at the inner ($R = R_i$) and outer ($R = R_o$) radii as a function of time $t$. Results are shown for two values of the growth energy parameter, $k_g = 0.1$ and $0.001$, and for both isotropic and anisotropic elastic responses.
• Figure 2: Evolution of arterial wall thickness, $\Delta r$, under three conditions: normotensive (internal pressure $p_0$), hypertensive (internal pressure $p_0 + \Delta p_0$), and growth under hypertensive loading. Results are shown for isotropic and anisotropic elasticity, considering two values of the growth energy parameter, $k_g$.
• Figure 3: Hoop (circumferential) stress, $\sigma^{\theta\theta}$, radial profile along the wall thickness plotted as a function of the radial coordinate $R$. Results are plotted at three states: normotensive state ($t = 5$), the hypertensive state ($t = 10$), and the post-growth state ($t = 20$). They are shown for isotropic and anisotropic elasticity, considering two values of the growth energy parameter, $k_g$.
• Figure 4: Parametric study assessing the role of fiber orientation angles $(\gamma_1, \gamma_2)$, the material parameter $k_2$, and the pressure increment $\Delta p_0$ on growth parameter $\mathfrak{g}$.
• Figure 5: Growth driven by the evolution in growth energy parameter $k_g$ from 0.1 to 0.001 at $t=20$.

Theorems & Definitions (9)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 3.1
• Remark 3.2
• Remark 3.3