Growth Description for Vessel Wall Adaptation: A Thick-Walled Mixture Model of Abdominal Aortic Aneurysm Evolution

Abstract

(1) Background: Vascular tissue seems to adapt towards stable homeostatic mechanical conditions, however, failure of reaching homeostasis may result in pathologies. Current vascular tissue adaptation models use many ad hoc assumptions, the implications of which are far from being fully understood; (2) Methods: The present study investigates the plausibility of different growth kinematics in modeling Abdominal Aortic Aneurysm (AAA) evolution in time. A structurally motivated constitutive description for the vessel wall is coupled to multi-constituent tissue growth descriptions; Constituent deposition preserved either the constituent's density or its volume, and Isotropic Volume Growth (IVG), in-Plane Volume Growth (PVG), in-Thickness Volume Growth (TVG) and No Volume Growth (NVG) describe the kinematics of the growing vessel wall. The sensitivity of key modeling parameters is explored, and predictions are assessed for their plausibility; (3) Results: AAA development based on TVG and NVG kinematics provided not only quantitatively, but also qualitatively different results compared to IVG and PVG kinematics. Specifically, for IVG and PVG kinematics, increasing collagen mass production accelerated AAA expansion which seems counterintuitive. In addition, TVG and NVG kinematics showed less sensitivity to the initial constituent volume fractions, than predictions based on IVG and PVG; (4) Conclusions: The choice of tissue growth kinematics is of crucial importance when modeling AAA growth. Much more interdisciplinary experimental work is required to develop and validate vascular tissue adaption models, before such models can be of any practical use.

Keywords: AAA; G& R; growth; mixture model; soft tissue; vascular adaptation; volume growth.

Figure 1

Illustration of different scenarios for adding mass to individual constituents and a mixture of two constituents. (a) Constituent 1, constant density (CCD): adding the mass $d{m}_1$ of the volume $d{V}_1$ to the initial mass $m_1\left(0\right)$ of the volume $V_1\left(0\right)$ results in increased mass $m_1+d{m}_1$ and volume $V_1+d{V}_1$. However, the density ${\varrho }_1\left(\tau \right)={\varrho }_1\left(0\right)$ remains constant; (b) Constituent 2, constant volume (CCV): adding the mass $d{m}_2$ of the density ${\varrho }_2\left(\tau \right)={\varrho }_2\left(0\right)$ to the initial mass $m_2\left(0\right)$ of density ${\varrho }_2\left(0\right)$ results in increased mass $m_2+d{m}_2$ but unchanged volume $V_2\left(\tau \right)=V_2\left(0\right)$. Hence, the density ${\varrho }_2\left(\tau \right)={\varrho }_2\left(0\right)+\mathrm{d}{\varrho }_2$ increases; (c) Mixture of Constituent 1 (following CCD) and Constituent 2 (following CCV): The tissue of mass m and volume V is composed of Constituent 1 ($m_1,V_1$) and Constituent 2 ($m_2,V_2$). The mass increments $d{m}_1,d{m}_2$ of volumes $d{V}_1,d{V}_2$ are added to each constituent. This results in increased masses of both constituents, increased volume of Constituent 1 and unchanged volume of Constituent 2. Over all the mass $m+dm$ and the volume $V+dV$ of the mixture have been changed.

Figure 2

Kinematics of growth. Deformation gradient $\mathbf{F}(\tau ,t)$ maps reference configuration ${\Omega }_0$ to the current configuration ${\Omega }_{\tau }$. The growth tensor ${\mathbf{F}}^{\mathrm{g}}\left(\tau \right)$ (with $\det {\mathbf{F}}^{\mathrm{g}}=\det \mathbf{F}=\widehat{v}$) connects reference configuration ${\Omega }_0$ to the intermediate stress-free configuration ${\Omega }_{\mathrm{g}}$, which accounts for the volume change at the long time-scale $\tau$. Then, the elastic deformation tensor ${\mathbf{F}}^{\mathrm{e}}\left(t\right)$ connects ${\Omega }_{\mathrm{g}}$ to the current configuration ${\Omega }_{\tau }$. That is, the total deformation gradient $\mathbf{F}\left(\tau \right)$ is split into volumetric growth part ${\mathbf{F}}^{\mathrm{g}}$ and elastic part ${\mathbf{F}}^{\mathrm{e}}$.

Figure 3

Effect of Isotropic Volume Growth (IVG), in-Plane Volume Growth (PVG), and in-Thickness Volume Growth (TVG) kinematics on the stress-stretch properties of the neo-Hookean material under equi-biaxial extension. First Piola-Kirchoff Stress (P) versus stretch ($\lambda$) is depicted for constant $\widehat{v}=1.0$ (solid curves), decreasing $\widehat{v}=0.5$ (dashed curves), and increasing $\widehat{v}=1.5$ (dash-dotted curves) tissue volumes.

Figure 3

Effect of Isotropic Volume Growth (IVG), in-Plane Volume Growth (PVG), and in-Thickness Volume Growth (TVG) kinematics on the stress-stretch properties of the neo-Hookean material under equi-biaxial extension. First Piola-Kirchoff Stress (P) versus stretch ($\lambda$) is depicted for constant $\widehat{v}=1.0$ (solid curves), decreasing $\widehat{v}=0.5$ (dashed curves), and increasing $\widehat{v}=1.5$ (dash-dotted curves) tissue volumes.

Figure 4

Predicted shapes during Abdominal Aortic Aneurysm (AAA) expansion. (a) Normal aorta at homeostasis with inner diameter $d_0=23.0$ (mm); Configurations (b–d) show AAAs at maximum inner diameters of $2d_0$, $3d_0$, and $4d_0$, respectively.

Figure 5

Effect of collagen net growth (parameter $\beta$) on the predicted Abdominal Aortic Aneurysm (AAA) expansion over time $\tau$. Isotropic Volume Growth (IVG), in-Plane Volume Growth (PVG), in-Thickness Volume Growth (TVG) and No Volume Growth (NVG) describe growth kinematics. (a) AAA expansion ${\widehat{d}}_{\mathrm{i}}=d_{\mathrm{i}}/d_0$; (b) AAA growth per year; (c) collagen stretch ${\lambda }_{\mathrm{c}}$; (d) collagen mass change ${\widehat{m}}_{\mathrm{c}}$; and (e) local tissue volume change $\widehat{v}$ at the site of the maximum diameter $d_{\mathrm{i}}$; (f) Total tissue volume change over time. Clinically used AAA repair indication of $55.0$ (mm) and $10.0$ (mm/year) is shown by the dotted line in (a) and (b), respectively.

Figure 6

Predicted Abdominal Aortic Aneurysm (AAA) properties at twofold expansion, $d_i/d_0=2$. Collagen growth is specified by $\beta =1.25$ (year⁻¹), and Isotropic Volume Growth (IVG), in-Plane Volume Growth (PVG), in-Thickness Volume Growth (TVG) and No Volume Growth (NVG) describe growth kinematics. Transmural plots are shown for (a) collagen stretch ${\lambda }_{\mathrm{c}}$; (b) local tissue volume change $\widehat{v}$; (c) Cauchy hoop stress ${\sigma }_{\phi \phi }$; (d) Vessel wall thickness, normalized to the thickness $h_0$ of aorta at homeostasis, along AAA length.

Figure 7

Effect of the initial volume fraction of elastin (a) and collagen (b) on the evolution of Abdominal Aortic Aneurysm (AAA) diameter ${\widehat{d}}_{\mathrm{i}}=\left(d_{\mathrm{i}}\right)/d_0$ over time $\tau$. Isotropic Volume Growth (IVG), in-Plane Volume Growth (IVP), in-Thickness Volume Growth (TVG) and No Volume Growth (NVG) describe growth kinematics. (a) AAA expansion with initial elastin volume fractions of ${\varphi }_{\mathrm{e}}=0.12$ (solid curves) and ${\varphi }_{\mathrm{e}}=0.18$ (solid curves with circles); (b) AAA expansion with initial collagen volume fractions of ${\varphi }_{\mathrm{c}}^i=0.75$ (solid curves) and ${\varphi }_{\mathrm{c}}^i=0.15$ (solid curves with circles) prescribed in the media.