Mechanisms driving thoracic aortic aneurysm instability

Abstract

Thoracic aortic aneurysms arise from a combination of biological and mechanical factors. Current clinical guidelines use size and rate of expansion to stratify risk, but such metrics do not necessarily predict if an aneurysm will stabilize, grow, dissect, or rupture. Computational biomechanical models can provide additional insights into mechanisms of aneurysm behavior that would be difficult or impossible to capture in vivo. Here, we use a constrained mixture theory of growth and remodeling to simulate lesion progression while covarying rate-dependent parameters that contribute to the natural history of aneurysm growth. This includes insults to the material properties of the vascular wall and the sensitivity of the vascular cells to their mechanical environment. This framework successfully simulates clinically relevant phenotypes, including cases where lesions with initially similar degrees of dilatation or rates of expansion diverge in behavior later in their progression. By capturing this spectrum of outcomes, our framework lays a foundation for more accurate, patient-specific risk prediction and future integration of machine learning tools to accelerate translation into clinical practice.

1. Introduction

Thoracic aortic aneurysms (TAAs) progressing to dissection and rupture are increasingly responsible for significant morbidity and mortality, with most of these lesions presenting at the aortic root or within the ascending aorta. Fortunately, advances in clinical screening of underlying pathogenetic variants or congenital defects, as well as advances in prosthetic vascular graft technology and surgical technique, have improved treatment outcomes for patients harboring TAAs (Verhagen et al. 2018). Current clinical guidelines recommend prophylactic surgical replacement of the diseased segment of the thoracic aorta when the maximal diameter exceeds a stated threshold value (e.g., 45–55 mm depending on multiple factors including sex and genetics), the rate of enlargement exceeds a stated value (e.g., 3–5 mm/yr), or both (Isselbacher et al. 2022). Nevertheless, many TAAs below recommended values dissect or rupture, while many larger lesions continue to be found incidentally without symptoms or tissue failure. There is, therefore, a need to understand better the natural history of TAAs and which underlying mechanisms determine their clinical destiny.

Computational biomechanical models promise to provide information beyond what is generally available clinically. Such models have advanced to account for many of the complexities of the thoracic aorta, including its complex geometry, nonlinear material properties, and hemodynamic loading. Notwithstanding the dominance of maximum diameter in risk assessment, there is increasingly more attention to rates of increase in lesion size (Oladokun et al. 2016; Henry et al. 2025). Among computational models that can account for rates of change of vascular geometry and material properties, constrained mixture models are particularly applicable. Briefly, the constrained mixture framework specifies three classes of constituent-specific constitutive relations (Humphrey and Rajagopal 2002): one for the rate of mass production, one for the rate of mass removal, and one for the material properties of the existing constituents. The rates of production and removal depend on mechanobiological factors, including the ability of the cells to mechanosense their local environment, deviations between sensed and homeostatic values of mechanical metrics, and mechanosensitivity of the cellular response to these deviations.

The primary purpose of this work is to quantify how rate parameters that represent varying mechanobiological deficiencies dictate whether the growth of a TAA will arrest, continue slowly, or result in a mechanobiological instability indicative of functional or structural failure (Figure 1). Such a quantification provides insight into how vessels with similar levels of initial dilatation and structural damage may lead to divergent outcomes. A secondary objective is to compare the utility of two distinct formulations of constrained mixture models: a fully time-resolved approach versus a mechanobiologically equilibrated (MBE) approach. While both approaches can be coupled to hemodynamic solvers (Latorre and Humphrey 2020a; Schwarz et al. 2023; Pfaller et al. 2024), the MBE approach allows efficient prediction of stable configurations in which an equilibrated state is possible, which provides a benchmark for an idealized, stable outcome. This comparison allows us to identify mechanisms by which stability can be lost and point to critical ratedependent factors that influence the resulting geometry and mechanical vulnerability of the TAA.

Fig. 1. Illustration of possible outcomes of aneurysm progression.

Different insult types contributing to disease progression can result in similar pre-aneurysmal lesions at intermediate times. Despite these initial geometrical similarities, continued progression can result in different outcomes depending on the underlying mechanisms of insult. These outcomes include: (i) stabilization of geometry and wall integrity, (ii) unstable growth with low risk for dissection/rupture, and growth with high risk for dissection/rupture either (iii) below, or (iv) above stated clinical thresholds

2. Methods

2.1. Constrained mixture theory

Constrained mixture theory has been a widely adopted method for simulating aneurysm growth (changes in mass) and remodeling (changes in structure), centered on modeling tissue-level behavior by accounting for the production (deposition of matrix, proliferation of cells) and removal (degradation of matrix, death of cells) of individual vascular constituents. Full details of constrained mixture theory for vascular growth and remodeling (G&R) are described elsewhere (Humphrey 2021). Briefly, this theory posits that structurally significant vascular constituents possess individual natural configurations and exhibit distinct material properties and turnover but are constrained to move with the mixture as a whole. Modeling the aortic wall as a tissue comprised of fibrillar collagens, smooth muscle cells, and elastin, we postulate that the apparent mass density ${\rho }^{\alpha }$ of individual constituents $\alpha$ at current time $s$ can be expressed in terms of constitutive relations for mass production and removal (that is, survival) such that

$${\rho }^{\alpha }\left(s\right)={\rho }^{\alpha }\left(0\right)Q^{\alpha }\left(s\right)+{\int }_0^s{m}^{\alpha }\left(\tau \right)q^{\alpha }\left(s,\tau \right)d\tau ,$$ (1)

where $m^{\alpha }(\tau )$ is the rate of mass density production at time $\tau ,q^{\alpha }(s,\tau )$ is the fraction of constituent $\alpha$ deposited at time $\tau$ that survives until time $s$, and $Q^{\alpha }(s)$ is the fraction of constituent $\alpha$ existing at time 0 that survives until time $s$. Previous studies have demonstrated the utility of stress-mediated relations for mass density production and removal (Humphrey 2021). Traditionally, production $m^{\alpha }(\tau )$ is modeled via a basal value that can be modulated by changes in mixture-level stress, namely

$$m^{\alpha }\left(\tau \right)=m_b^{\alpha }\left(1+K_{\sigma }^{\alpha }\Delta \sigma \left(\tau \right)-K_{{\tau }_w}^{\alpha }\Delta {\tau }_w\left(\tau \right)\right),$$ (2)

where $m_b^{\alpha }$ is a basal production rate and $\Delta \sigma$ and $\Delta {\tau }_w$ represent normalized deviations in intramural and wall shear stress from homeostatic values, respectively, of the form

$$\begin{gathered} \Delta \sigma (\tau )=\sigma (\tau )/{\sigma }_h-1 \\ \Delta {\tau }_w(\tau )={\tau }_w(\tau )/{\tau }_{wh}-1, \end{gathered}$$ (3)

where $\sigma (\tau )$ is a scalar measure of the intramural stress at time $\tau ,{\tau }_w(\tau )$ is the wall shear stress at time $\tau ,{\sigma }_h$ is the homeostatic intramural stress, and ${\tau }_{wh}$ is the homeostatic wall shear stress. $K_{\sigma }^{\alpha }$ and $K_{{\tau }_w}^{\alpha }$ are gain-like parameters that stipulate the level of mechanosensitivity to the particular deviation in stress. Similarly, for mass removal,

$$q^{\alpha }\left(s,\tau \right)=\text{exp}\left(-{\int }_{\tau }^s{k}_b^{\alpha }\left(1+w_{\sigma }^{\alpha }\Delta {\sigma }^2\left(t\right)\right)dt\right),$$ (4)

where $k_b^{\alpha }$ is a basal removal rate and $w_{\sigma }^{\alpha }$ is a gain-like parameter that modulates the removal rate according to the magnitude of the deviation in intramural stress. For simplicity, these gain-like parameters will be assumed to be equal for all the constituents but can be defined on a constituent-specific basis in the context of appropriate data.

The constituent strain energy can then be written as

$$W^{\alpha }(s)=\frac{{\rho }^{\alpha }(0)}{\rho (s)}{Q}^{\alpha }(s){\hat{W}}^{\alpha }\left({\mathbf{C}}_{n(0)}^{\alpha }(s)\right)+\frac{1}{\rho (s)}{\int }_0^s{m}^{\alpha }\left(\tau \right)q^{\alpha }(s,\tau ){\hat{W}}^{\alpha }({\mathbf{C}}_{n(\tau )}^{\alpha }(s))d\tau ,$$ (5)

where ${\hat{W}}^{\alpha }({\mathbf{C}}_{n(\tau )}^{\alpha }(s))$ is a constituent-specific, mass-averaged strain energy function and ${\mathbf{C}}_{n(\tau )}^{\alpha }(s)={\mathbf{F}}_{n(\tau )}^{\alpha \text{T}}(s){\mathbf{F}}_{n(\tau )}^{\alpha }(s)$ is the constituent-specific right Cauchy–Green tensor that arises from the constituent-specific deformation gradient ${\mathbf{F}}_{n(\tau )}^{\alpha }(s)$, which represents the deformation of a constituent at current time $s$ that was deposited at time $\tau$. This is given as ${\mathbf{F}}_{n(\tau )}^{\alpha }=\mathbf{F}(s){\mathbf{F}}^{-1}(\tau ){\mathbf{G}}^{\alpha }(\tau )$, where $\mathbf{F}(s)$ and $\mathbf{F}(\tau )$ are the deformation gradients of the mixture at G&R times $s$ and $\tau$, respectively, and ${\mathbf{G}}^{\alpha }(\tau )$ is a tensor that accounts for pre-stretch of newly deposited constituents.

We then postulate that the mixture-level strain energy $W(s)$ is given by a simple rule of mixtures such that, conceptually, $W(s)=\sum _{\alpha }{\varphi }^{\alpha }(s){\hat{W}}^{\alpha }(s)$, where ${\varphi }^{\alpha }(s)={\rho }^{\alpha }(s)/\rho (s)$ is the current constituent mass fraction at time $s,{\rho }^{\alpha }(s)$ is the apparent mass density of the constituent, and $\rho (s)$ is the mass density of the mixture as a whole. From this, we define the overall tissue-level Cauchy stress $\mathit{\sigma }$ as

$$\mathit{\sigma }(s)=\frac{2}{\text{det}\mathbf{F}(s)}\mathbf{F}(s)\frac{\partial W(s)}{\partial \mathbf{C}(s)}{\mathbf{F}}^{\text{T}}(s),$$ (6)

which allows efficient computation of the intramural stress in Equation 3, where $\sigma =\frac{1}{3}\text{tr}(\mathit{\sigma })$.

We model the pseudoelastic mechanical behavior of collagen fibers $(\alpha =c)$ and passive smooth muscle cells $(\alpha =m)$ as Fung-type exponentials with preferential fiber directions of the form

$$\begin{gathered} {\hat{W}}^{c,m}\left({\mathbf{C}}_{n(\tau )}^{c,m}(s)\right) \\ =\frac{c_1^{c,m}}{4c_2^{c,m}}\left(\text{exp}\left[c_2^{c,m}{\left({\mathbf{C}}_{n(\tau )}^{c,m}(s):{\mathbf{H}}^{c,m}(\tau )-1\right)}^2\right]-1\right), \end{gathered}$$ (7)

where $c_1^{\alpha }$ and $c_2^{\alpha }$ are material parameters and ${\mathbf{H}}^{c,m}(\tau )={\mathbf{h}}^{c,m}(\tau )\otimes {\mathbf{h}}^{c,m}(\tau ).{\mathbf{h}}^{c,m}(\tau )$ is the orientation of the fiber at time $\tau$, defined by ${\mathbf{h}}^{c,m}(\tau )=\mathbf{F}(\tau ){\mathbf{h}}_0^{c,m}/‖\mathbf{F}(\tau ){\mathbf{h}}_0^{c,m}‖$, where ${\mathbf{h}}_0^{c,m}$ is the fiber direction in the initial reference configuration. This relationship specifies that new fibers are deposited preferentially in the direction of deformation. The deposition tensors are given by

$${\mathbf{G}}^{c,m}(\tau )=\left[\begin{array}{ccc}{G}^{c,m}& 0& 0\\ 0& G^{c,m}& 0\\ 0& 0& G^{c,m}\end{array}\right],$$

where $G^c$ represents the pre-stretch of the collagen fibers at deposition (thought to be accomplished via integrin and actomyosin activity) and $G^m$ is the pre-stretch of the smooth muscle cells (akin to cells spreading to reach a homeostatic mechanical state when placed on a substrate in vitro). In both cases, the deposition tensors act on the oriented fiber vector, ${\mathbf{h}}^{c,m}(\tau )$, resulting in a deposition stretch equal to $G^{c,m}$ in the direction of ${\mathbf{h}}^{c,m}(\tau )$, which is assumed to be within the circumferential-axial plane. Collagen fibers in the aortic wall naturally exhibit a splay of orientations ranging from circumferential to axial. This structural information can be captured by dividing collagen fibers into four subpopulations: circumferential $(\theta )$, axial $(z)$, and symmetric diagonals $(d)$, with associated fractions $\left\{{\beta }^{\theta },{\beta }^z,{\beta }^d\right\}$. The diagonally oriented collagens exhibit an initial fiber angle ${\alpha }_0$ in the reference configuration relative to the axial direction, such that ${\mathbf{h}}_0^{c,d}=\left\{0,\text{sin}{\alpha }_0,\text{cos}{\alpha }_0\right\}$. For computational convenience, we also assume that collagen and smooth muscle cells have a maximum fiber age $A_{\mathit{\text{max}}}$ during their turnover that captures the diminishing structural contribution of fibers and cells after significant removal. Effectively, this means that mass and material behavior from constituents that turn over (e.g., collagen, smooth muscle cells) are considered, and subsequently integrated in Equation 1 and Equation 5, if deposited on the time interval $\left[s-A_{\mathit{\text{max}}},s\right]$. Lastly, we model elastin $(\alpha =e)$ as a neo-Hookean material of the form

$$\hat{W}\left({\mathbf{C}}_{n(\tau )}^e(s)\right)=\frac{c^e}{2}\left({\mathbf{C}}_{n(\tau )}^e(s):\mathbf{I}-3\right),$$ (8)

where $c^e$ is a material parameter and ${\mathbf{G}}^e(\tau )$ is a deposition stretch tensor given by

$${\mathbf{G}}^e\left(\tau \right)=\left[\begin{array}{ccc}\frac{1}{G_{\theta }^e{G}_z^e}& 0& 0\\ 0& G_{\theta }^e& 0\\ 0& 0& G_z^e\end{array}\right].$$

We note, given the normally long half-life of elastic fibers in the aorta (on the order of decades), that elastin is assumed to not undergo natural turnover in the modeling framework. However, as discussed below, its properties can be compromised throughout the progression of TAA.

Linearization of the constrained mixture formulation consistent with these constraints enables efficient implementation within a finite element framework, where simultaneous solution of mechanical and mechanobiological equilibrium (if present) can be ensured across load steps that capture evolving geometries, compositions, and properties of interest for complex boundary value problems (see Schwarz et al. (2023) for details on the full constrained mixture theory within a finite element framework). For this study, this methodology was implemented in the open-source FEBio solver. The source code can be found on GitHub (Schwarz 2025).

2.2. Mechanobiologically equilibrated formulation of constrained mixture theory

Constrained mixture theory for vascular G&R is motivated by the concept of mechanical homeostasis (Humphrey et al. 2015), which aims for balanced constituent turnover to achieve an unchanging mechanical state in which cells deposit new matrix with a preferred pre-stress. We have shown that this constrained mixture framework can be simplified via the assumption of mechanobiological equilibrium at long time scales (Latorre and Humphrey 2018a,b), with homeostatic (steady-state) production balancing removal. Since homeostasis requires that the mixture mass, composition, and properties are unchanging, we observe from Equation 2 that $\left(1+K_{\sigma }^{\alpha }\Delta \sigma (\tau )-K_{{\tau }_w}^{\alpha }\Delta {\tau }_w(\tau )\right)\equiv 1$ in mechanobiologically equilibrated configurations. In other words, $K_{\sigma }^{\alpha }\Delta \sigma (\tau )=K_{{\tau }_w}^{\alpha }\Delta {\tau }_w(\tau )$. Additionally, we define the homeostatic deformation gradient ${\mathbf{F}}_h$ describing the deformation between the initial state and the long-term homeostatic configuration, along with its Jacobian $J_h=\text{det}{\mathbf{F}}_h$. It follows that, for mixture constituents that are assumed to not turn over (e.g., elastin), because their mass remains constant, the homeostatic mass fraction is given by

$${\varphi }_h^e(J_h)=\frac{{\varphi }_o^e}{J_h}.$$ (9)

For constituents that are allowed to turn over (e.g., smooth muscle cells and collagen fibers), their evolved mass fractions can be computed relative to each other using

$$\frac{J_h{\varphi }_h^{{\xi }_i}}{{\varphi }_o^{{\xi }_i}}=\frac{J_h{\varphi }_h^{{\xi }_j}}{{\varphi }_o^{{\xi }_j}},{\xi }_i\ne {\xi }_j.$$ (10)

Since the intramural and wall shear stress contributions to mass production balance each other, one needs only specify the ratio between the gain parameters $\left(K_{{\tau }_w}^{\alpha }/K_{\sigma }^{\alpha }\right)$ as an input, as done previously (Latorre and Humphrey 2020b). This framework has also been implemented in FEBio as a custom material plugin (Latorre and Humphrey 2020a,b), which provides rapid, path-independent estimation of the long-term evolved state of the full constrained mixture model. This serves as a benchmark for a fully stabilized TAA progression, facilitating key comparisons in rate-dependent mechanisms discussed below.

2.3. Mechanobiological insults contributing to aneurysm progression

To model how combined insults to material integrity and mechanobiological sensitivity may potentially impact lesion progression and stability, we model TAAs by prescribing a time-varying mechanobiological insult $\vartheta \in \left[\mathrm{0,1}\right]$ to an initially non-dilated vessel that is pre-loaded to in vivo conditions. The insult profile allows specific control of localized factors that perturb mechanical and mechanobiological equilibrium to initiate aortic dilatation and is defined using a modified expression from Latorre and Humphrey (2020b):

$$\begin{gathered} \vartheta \left(s,z_0,{\theta }_0\right) \\ =\left(1-\text{exp}\left(-k^{\vartheta }s\right)\right)\text{exp}\left(-{\left|\frac{z_0}{z_{\mathit{\text{od}}}}\right|}^5\right)\text{exp}\left(-{\left|\frac{{\theta }_0}{{\theta }_{\mathit{\text{od}}}}\right|}^5\right), \end{gathered}$$ (11)

where $s$ is the current $\text{G}\&\text{R}$ time, $z_0$ and ${\theta }_0$ are axial and circumferential coordinates in the initial configuration, respectively, $k^{\vartheta }$ is a rate parameter representing the rate of insult progression, and $z_{od}$ and ${\theta }_{od}$ represent the spread of the insult in the axial and circumferential directions, respectively, understanding that the maximum insult is centered at $\left\{z_0,{\theta }_0\right\}=\left\{\mathrm{0.0,0.0}\right\}$.

Contributors to TAAs are thought to include disrupted material structure, altered material turnover rates, and compromised sensitivity to changes in the hemodynamic environment. A critical clinical question is whether these factors will lead a particular lesion to stabilize, continue to grow slowly, or progress to dissection or rupture (Figure 1). There is, therefore, a need for a better understanding of the underlying mechanisms. In this study, we focus on aneurysms that are initiated by a structural loss of elastic fiber integrity, whereby the elastin modulus $c^e$ in Equation 8 is reduced by a specified percentage, as done previously to model aneurysm formation (Latorre and Humphrey 2020b; Goswami et al. 2022). We further hypothesize that the natural history of aneurysm progression is modulated by the sensitivity of vascular cells to changes in their local mechanical environment. Compromised integrin and actomyosin activity can lead to aberrant constituent production and removal in response to similar initiating insults, resulting in divergent outcomes. We thus use Equation 11 to parameterize gradual localized insults to elastic fiber integrity $({\vartheta }^e)$ as well as mechanosensitivity to intramural stress (${\vartheta }^p$ for intramural stress-mediated production and ${\vartheta }^r$ for intramural stress-mediated removal) and wall shear stress (${\vartheta }^{{\tau }_w}$ for wall shear stress-mediated production) by using this normalized insult profile to scale multiple contributors simultaneously.

In terms of biological behavior, ${\vartheta }^p$ represents impaired mechanosensitivity related to matrix synthesis. This can occur through disruption of integrin-mediated sensing of the mechanical environment by synthetic cells (e.g., via altered focal adhesion kinase (FAK) and RhoA signaling pathways). It can also occur through disruption of intracellular pathways that further instruct matrix synthesis (e.g., TGF-$\beta$–mediated transcription). ${\vartheta }^r$ represents impaired mechanosensitivity related to matrix degradation. This can reflect disruption in the production or activation of matrix metalloproteinases (e.g., MMP2, MMP9) that are key contributors to matrix degradation in TAAs. In our G&R framework, these biological insults are mathematically modeled as

$${\vartheta }^i(s)={\vartheta }_{max}^i\vartheta \left(s,z_0,{\theta }_0\right)i=e,p,r,{\tau }_w,$$ (12)

where the maximal severity value ${\vartheta }_{\mathit{\text{max}}}^i$ for each contributor is reached as $s\to \infty .k^{\vartheta }$ in Equation 11 can then be chosen so that the insult severity reaches 99% of its maximum value at a desired point within the simulation time-course. Specifically, ${\vartheta }^e$ describes the percent reduction in the elastin modulus from its baseline value, that is, $c^e(\tau )=\left(1-{\vartheta }^e(\tau )\right)c^e(0)$. Compromised mechanosensitivity of constituent production and removal is achieved by modifying the gain-like parameters in Equations 2 and 4 as

$$m^{\alpha }\left(\tau \right)=m_b^{\alpha }\left(1+\left(1-{\vartheta }^p\left(\tau \right)\right)K_{\sigma }^{\alpha }\Delta \sigma \left(\tau \right)-\left(1-{\vartheta }^{{\tau }_w}\left(\tau \right)\right)K_{{\tau }_w}^{\alpha }\Delta {\tau }_w\left(\tau \right)\right)$$ (13)

and

$$q^{\alpha }\left(s,\tau \right)=\text{exp}\left(-{\int }_{\tau }^s{k}_b^{\alpha }\left(1+\left(1-{\vartheta }^r\left(t\right)\right)w_{\sigma }^{\alpha }\Delta {\sigma }^2\left(t\right)\right)dt\right).$$ (14)

For convenience, we assume the combined effects change together both temporally and spatially, governed by $k^{\vartheta }$, such that ${\vartheta }_{\mathit{\text{max}}}^e,{\vartheta }_{\mathit{\text{max}}}^p,{\vartheta }_{\mathit{\text{max}}}^r,{\vartheta }_{\mathit{\text{max}}}^{{\tau }_w}$ are reached at the same simulation time step (Figure 2). The biological and mathematical interpretations of these insults are summarized in Table 1.

Fig. 2. Covarying evolution of mechanobiological insult severity functions.

Each insult type ${\vartheta }^i$ is modeled by scaling a common function, $\vartheta \left(s,z_0,{\theta }_0\right)$ (simplified in this figure as $\vartheta (s)$), each with its own maximum severity value ${\vartheta }_{\mathit{\text{max}}}^i$. The rate parameter $k^{\vartheta }$ governs the rate of temporal progression of all insults such that they each reach their maximum severity simultaneously within the simulation time-course. Note that $i=e$ represents insults to the elastic fiber integrity, $i=p$ represents insults to the intramural stress-mediated production mechanosensitivity, $i=r$ represents insults to the intramural stress-mediated removal mechanosensitivity, and $i={\tau }_w$ represents insults to the wall shear stress-mediated production mechanosensitivity

Table 1. Model parameters used to represent biological changes that drive aneurysm formation and progression.

Each parameter ${\vartheta }^i$ represents a distinct biological insult implicated in aneurysm initiation and progression, including degradation of elastic fibers $\left({\vartheta }^e\right)$, impaired matrix production and removal (${\vartheta }^p$ and ${\vartheta }^r$), and altered endothelial shear stress signaling $\left({\vartheta }^{{\tau }_w}\right)$. These insults are implemented mathematically as reductions in material integrity or gains governing stress-mediated constituent mass production and removal. The temporal evolution of each insult is governed by a shared progression rate, $k^{\vartheta }$, which controls the timescale over which the insults approach their maximal severity, ${\vartheta }_{\mathit{\text{max}}}^i$. These parameters enable systematic exploration of how different combinations of insults result in divergent outcomes

Symbol	Biological Motivation	Mathematical Representation
${\vartheta }^e$	Progressive damage or loss of elastic fibers (e.g., inflammation, elastase, or mechanical stress)	Reduction in the material stiffness, $c^e$, of elastin
${\vartheta }^p$	Impaired cell signaling responses that limit new matrix deposition and cell proliferation (e.g., FAK, RhoA, or TGF-β signaling pathways)	Reduction in the sensitivity, $K_{\sigma }^{\alpha }$, of matrix production and cell proliferation to intramural stress
${\vartheta }^r$	Abnormal signaling that disrupts normal matrix and cell turnover (e.g., production or activation of metalloproteinases)	Reduction in sensitivity, $w_{\sigma }^{\alpha }$, of matrix degradation and cell apoptosis to intramural stress
${\vartheta }^{{\tau }_w}$	Impaired endothelial signaling in response to blood flow (e.g., eNOS and NO signaling pathways)	Reduction in the sensitivity, $K_{{\tau }_w}^{\alpha }$, of matrix production and cell proliferation to wall shear stress
$k^{\vartheta }$	Speed of disease progression	Determines how quickly ${\vartheta }^i$ reaches ${\vartheta }_{\mathit{\text{max}}}^i$ over time
${\vartheta }_{\text{max}}^i$	Ultimate extent of structural and biological dysfunction in the vascular lesion	Maximum severity of each insult

We model an illustrative aorta using material parameters (Table 2) based on biaxial biomechanical data of a normal thoracic aorta from a male wild-type mouse (Latorre and Humphrey 2020b). Using the above definitions and parameters as inputs, the finite element solver was used to compute the displacement field for each aortic geometry associated with the mixture-level stress in the context of applied boundary conditions. The aortic domain was modeled as a uniform thick-walled cylinder and meshed using 27-node, fully integrated quadratic hexahedral elements, with 1 × 40 × 160 elements along the radial, circumferential, and axial directions, respectively. Aortas were simulated at a fixed pressure of 105 mmHg, corresponding to an elevated in vivo mean arterial pressure. Where appropriate, we used symmetry conditions to calculate full-domain configurations from domain subsections. We compared outcomes of synthetic TAAs generated using time-resolved and MBE constrained mixture theory, noting that the equilibrated formulation determines the final configuration of the aneurysm directly and therefore needs only to consider the final (maximum) value ${\vartheta }_{\mathit{\text{max}}}^i,i=e,p,r,{\tau }_w$ (Figure 3).

Table 2. Simulation parameters for aortic material and turnover properties.

Initial values are for a normal wild-type male mouse descending thoracic aorta, derived from previous studies (Wilson et al. 2013; Latorre and Humphrey 2018b, 2020b; Goswami et al. 2022).

Material and turnover properties
Initial radius, thickness, length (mm, mm, mm)	$a_0,h_0,l_0$	0.6468, 0.042, 15.0
Initial volume fractions (—,—,—)	${\varphi }_0^e,{\varphi }_0^m,{\varphi }_0^c$	0.34, 0.33,0.33
Directional collagen fractions (—, —, —)	${\beta }^{\theta },{\beta }^z,{\beta }^d$	0.056, 0.067, 0.877
Initial diagonal collagen orientation (°)	${\alpha }_0$	±29.9
Elastin material property (kPa)	$c^e$	89.71
Elastin deposition stretches (—, —, —)	$G_r^e,G_{\theta }^e,G_z^e$	1/($G_{\theta }^e{G}_z^e$), 1.90, 1.62
Collagen material properties (kPa, —)	$c_1^c,c_2^c$	234.9, 4.08
Collagen deposition stretch (—)	$G^c$	1.25
Collagen degradation rate (1/day)	$k_b^c$	0.142
Collagen production gains (—, —)	$K_{\sigma }^c,K_{{\tau }_w}^c$	1.0,0
Collagen degradation gain (—)	$w_{\sigma }^c$	1.0
SMC material properties (kPa, —)	$c_1^m,c_2^m$	261.4, 0.24
SMC deposition stretch (—)	$G^m$	1.20
SMC death rate (1/day)	$k_b^m$	0.142
SMC production gains (—, —)	$k_{\sigma }^m,k_{{\tau }_w}^m$	1.0,0
SMC death gain (—)	$w_{\sigma }^m$	1.0
Maximum fiber age (days)	$A_{max}$	64

SMC = smooth muscle cells

Fig. 3. TAA simulation pipeline and mechanobiological insult profiles.

(a) TAAs initiated by localized mechanobiological insults, defined by two-dimensional insult profiles, can be simulated using time-resolved or mechanobiologically equilibrated (MBE) finite element frameworks, which can be compared. Time-resolved simulations produce a time-course of lesion growth, whereas MBE simulations identify a single stabilized long-term configuration, if such a configuration exists. (b) Illustrative examples of spatially axisymmetric, asymmetric, and random insult profiles. Shown are the prescribed spatial distribution (above, colored by severity) and the corresponding final TAA geometry (below, colored by dilatation)

2.4. Spatial distribution of insult

Using fully time-resolved and MBE constrained mixture formulations, we generate TAAs resulting from combinations of lost elastic fiber integrity and compromised mechanosensitivity for a wide range of insult profile types. To understand relationships among the spatial distribution of insults, their severity, and their rate dependence, we focus here on three classes of insult spatial profiles (Figure 3):

• Axisymmetric profiles vary only axially, assuming full circumferential symmetry by removing dependence on $\theta$, such that Equation 11 simplifies to $\vartheta \left(s,z_0\right)=(1-\left.\text{exp}\left(-k^{\vartheta }s\right)\right)\text{exp}\left(-{\left|\frac{z_0}{z_{od}}\right|}^5\right)$. These are generated with a fixed axial spread $z_{\mathit{\text{od}}}=3.0\text{mm}$ and maximum value (denoted as the insult apex) centered at $z_0=0\text{mm}$.

• Asymmetric profiles exhibit variation in both the axial and circumferential directions as described by Equation 11. The circumferential spread is set to ${\theta }_{\mathit{\text{od}}}=\pi /2$ with the apex at ${\theta }_0=0$, which limits the spread of the insult to only a portion of the full circumference.

• Random insult profiles are generated by superimposing multiple $(N=4)$ asymmetric subprofiles in the aortic domain. We define each subprofile $f_i\left(z_0,{\theta }_0\right)$ as

  $$f_i\left(z_0,{\theta }_0\right)=a_i\text{exp}\left(-{\left|\frac{z_0-b_i}{c_i}\right|}^{d_i}\right)\text{exp}\left(-{\left|\frac{{\theta }_0-e_i}{f_i}\right|}^{g_i}\right),$$ (15)

  where $\left\{a_i,b_i,c_i,d_i,e_i,f_i,g_i\right\}$ are shape parameters independently sampled from a truncated normal distribution $x~𝒩\left({\mu }_x,{\sigma }_x\right)$ subject to $L_x\le x\le U_x$, where $\left\{L_x,U_x\right\}$ denote lower and upper truncation bounds, respectively. Importantly, this method enables repositioning of the maximum insult within the aortic domain ($b_i$ and $e_i$) as well as variation in the spread ($c_i$ and $f_i$) and boundary softness ($d_i$ and $g_i$) of the insult region, all on a per-term basis. To ensure admissible profiles, the parameter bounds are chosen to produce similarly sized insult regions as the axisymmetric and asymmetric cases (i.e., constrained to be positive where appropriate, with axial and circumferential spread distribution means centered on 5 mm and $\pi$, respectively). The final random insult profile ${\vartheta }^{\mathit{\text{rand}}}\left(s,z_0,{\theta }_0\right)$ is computed by combining the terms as

  $${\vartheta }^{\mathit{\text{rand}}}\left(s,z_0,{\theta }_0\right)=\left(1-\text{exp}\left(-k^{\vartheta }s\right)\right)h\sum _{i=1}^N{f}_i\left(z_0,{\theta }_0\right)$$ (16)

  and used in place of Equation 11 in the simulations. $h$ is a final normalizing parameter that imposes the maximum value of ${\vartheta }^{\mathit{\text{rand}}}\left(s,z_0,{\theta }_0\right)$ in the vessel domain to be 1 as $s\to \infty$ (Figure S1). The parameters used to generate the insult profiles are summarized in Table 3.

Table 3. Spatial parameters of insult profiles.

Summary of the shape parameters that define the insult profiles serving as input into the finite element simulations. For randomly generated insult profiles, the means $\left({\mu }_x\right)$, standard deviations $\left({\sigma }_x\right)$, lower bounds $\left(L_x\right)$, and upper bounds $\left(U_x\right)$ of the truncated normal distribution are listed for each parameter

Insult profile		Value
Axisymmetric
Axial spread (mm)	$z_{od}$	5.0
Circ. spread (—)	${\theta }_{od}$	$\infty$
Asymmetric
Axial spread (mm)	$z_{od}$	5.0
Circ. spread (—)	${\theta }_{od}$	$\pi$/2
Random		${\mu }_x,{\sigma }_x,L_x,U_x$
Overall amplitude (—)	$a_i$	1, 0.2, 0, -
Axial position (mm)	$b_i$	0.0, 3.0, -, -
Axial spread (mm)	$c_i$	5.0, 1.0, 0, -
Axial boundary softness (—)	$d_i$	5.0, 1.0, 0, -
Circ. position (—)	$e_i$	0, $\pi$, -, -
Circ. spread (—)	$f_i$	$\pi$, $\pi$/4, 0, -
Circ. boundary softness (—)	$g_i$	5.0, 1.0, 0, -

2.5. Numerical experiments

Because endothelial dysfunction is common in TAAs (Chung et al. 2007), we neglect possible effects of shear stress induced changes in nitric oxide and endothelin-1 which can otherwise play dual roles—affecting vasoactivity, which is not expected to be significant in TAAs, and effects on cell and matrix turnover. We thus focus on exploring the impact of insults to the mechanosensitivity of intramural stress deviations only (that is, $K_{{\tau }_w}^{\alpha }=0$ in Equation 13 and ${\vartheta }^{{\tau }_w}=0$ for the time-resolved constrained mixture model, and $K_{{\tau }_w}^{\alpha }/K_{\sigma }^{\alpha }=0$ for the equilibrated formulation), which allows direct comparison of the time-resolved and MBE results for a given set of input parameters. We performed experiments by varying pairs of insult severity parameters while holding the others constant for each class of insult profile:

• Elastic fiber integrity and production mechanosensitivity: covarying ${\vartheta }_{\mathit{\text{max}}}^e$ and ${\vartheta }_{\mathit{\text{max}}}^p$ while fixing ${\vartheta }_{\mathit{\text{max}}}^r$ and $k^{\vartheta }$.

• Production and removal mechanosensitivity: covarying ${\vartheta }_{\mathit{\text{max}}}^p$ and ${\vartheta }_{\mathit{\text{max}}}^r$ while fixing ${\vartheta }_{\mathit{\text{max}}}^e$ and $k^{\vartheta }$.

• Elastic fiber integrity and insult rate: covarying ${\vartheta }_{\mathit{\text{max}}}^e$ and $k^{\vartheta }$ while fixing ${\vartheta }_{\mathit{\text{max}}}^p$ and ${\vartheta }_{\mathit{\text{max}}}^r$.

For the time-resolved approach, we simulate a 720-day time-course (corresponding to the roughly 2-year lifespan of a mouse) to allow the time-resolved insults to equilibrate, if applicable, for comparison with the MBE framework. Quantities of interest (inner radius, wall thickness, circumferential material stiffness and stress, strain energy, and (scalar) intramural stress, all indications of aortic structure and function) are reported at the location of maximum insult amplitude, corresponding to the location of maximum diameter. Comparisons of progressive luminal expansion and wall thickening are particularly useful given their impact on wall stress as well as potential growth stabilization (Baek et al. 2005). We varied $k^{\vartheta }$ from 0.010 to 0.050 and chose maximum insult values ${\vartheta }_{\mathit{\text{max}}}^i$ separately for each type of insult profile to identify combinations at which TAA stability would be lost and to capture the range of outcomes observed in mice.

3. Results

All insult profiles resulted in fusiform dilatations of the affected region of the model domain. We categorized time-resolved simulation outcomes according to the following four groups motivated by criteria used in clinical decision-making:

• Stabilization: final growth rate at the apex <0.1 mm/year.

• Growing: final growth rate at the apex ≥0.1 mm/year.

• Small-diameter rupture: the simulation exits prior to the full 720-day time-course and the final maximum radius is <1.5 times the original radius.

• Large-diameter rupture: the simulation exits prior to the full 720-day time-course and the final maximum radius is ≥1.5 times the original radius.

Within the time-resolved constrained mixture framework, aneurysm rupture was defined as the loss of the existence of a physically admissible equilibrium solution. Specifically, rupture was identified at the time step for which the constituent-specific removal rate (Equation 4) exceeded the achievable production rate (Equation 2), such that no solution to the governing G&R equations could exist. This condition represents a growing inequality between constituent removal and production, rather than an arbitrary numerical tolerance, and reflects a state in which newly produced material is insufficient to offset the accelerated degradation caused by the applied vascular insults. Under these conditions, the numerical solver necessarily fails to converge because there is no feasible equilibrium configuration. To verify this interpretation, time-resolved production and removal rates were quantified leading up to simulation exit (Figures S2–S4), confirming that numerical non-convergence consistently coincided with an increasing dominance of removal over production. Biologically, this behavior corresponds to catastrophic loss of load-bearing capacity, rapid wall thinning, and escalating stress, consistent with descriptions of aneurysm rupture in mice and humans. Mesh and temporal convergence studies were performed prior to selecting the reported mesh and time step sizes, and the onset of this rupture condition was found to be consistent across reasonable variations in spatial resolution, time increment, and solver tolerances.

Visualizing outcomes of the covarying parametric simulations in a classification map allows identification of multiple stability regimes within the parameter spaces (see below). This provides a framework for relating theoretical mechanobiological parameters to clinical quantities of interest and patient outcomes. In addition, the time-resolved approach provides critical knowledge on intermediate states of unstable TAAs. These are discussed below, focusing first on axisymmetric insult profiles.

3.1. Elastic fiber insult increases low-risk TAA size, while production mechanosensitivity insult increases rupture risk

Dilatation increased with higher ${\vartheta }_{\mathit{\text{max}}}^e$ and ${\vartheta }_{\mathit{\text{max}}}^p$ (Figure 4), with ${\vartheta }_{\mathit{\text{max}}}^r=0$ and $k^{\vartheta }=0.010$ held constant across the axisymmetric simulations. For cases where ${\vartheta }_{\mathit{\text{max}}}^p=0$, the time-resolved framework converged asymptotically toward the MBE predicted values for all metrics, supporting the validity of using the MBE approach for TAA evolutions with non-impaired mechanosensitivity (Figure 4a–c, top left). For moderate elastic fiber insult $\left({\vartheta }_{\mathit{\text{max}}}^e=0.36\right)$, both formulations resulted in vessel dilatation stabilizing at the threshold for aneurysmal classification (>1.5 times the original radius) with modest thickening (Figure 4a), as well as increased circumferential stiffness (Figure 4b), decreased strain energy, and well-regulated intramural stress (Figure 4c)—all commonly observed in mouse models of stable TAAs (Humphrey and Tellides 2019). Once the elastic fiber integrity insult surpassed 70%, stability of the TAA was lost in the time-resolved model, giving rise to unbounded growth, for which no corresponding solution existed in the MBE framework (Figure 4a–c, bottom left).

Fig. 4.

Effects of covarying elastic fiber insult (${\vartheta }_{\mathit{\text{max}}}^e$) and production mechanosensitivity insult (${\vartheta }_{\mathit{\text{max}}}^p$). (a) Dilatation and thickness, (b) circumferential stiffness and stress, and (c) strain energy and intramural stress are shown over time for the fully time-resolved (solid lines) and MBE (dashed lines) models, evaluated at the location of maximum insult severity. Removal mechanosensitivity insult is fixed at ${\vartheta }_{\mathit{\text{max}}}^r=0$, and insult rate is fixed at $k^{\vartheta }=0.010$. Increasing ${\vartheta }_{\mathit{\text{max}}}^e$ with low ${\vartheta }_{\mathit{\text{max}}}^p$ eventually results in an unstable growing aneurysm in which stiffness and stress no longer approach steady-state values but rather increase indefinitely. Increased ${\vartheta }_{\mathit{\text{max}}}^p$ can destabilize a moderate ${\vartheta }_{\mathit{\text{max}}}^e$ that is otherwise stable when ${\vartheta }_{\mathit{\text{max}}}^p=0$. This results in increasing stiffness and stress over time that triggers the rupture exit threshold (red crosses). Both moderate and severe ${\vartheta }_{\mathit{\text{max}}}^e$ reach this rupture exit threshold, though they demonstrate strikingly different trends in strain energy at the exit point. Results are shown for an axisymmetric insult distribution. See Supplemental Figures S5–S8 for similar time-course results for different cases

For cases where production mechanosensitivity was severely compromised, the time-resolved model exited before halfway through the time-course, even at moderate elastic fiber insult values (Figure 4a–c, top right). This effect was exacerbated with increasing values of ${\vartheta }_{\mathit{\text{max}}}^e$, with the simulations reaching the rupture exit threshold within a quarter of the full time-course (Figure 4a–c, bottom right). Additionally, the computed metrics no longer approached those predicted by the MBE framework, frequently overestimating the stable value. Aneurysms with compromised production mechanosensitivity $({\vartheta }_{\mathit{\text{max}}}^p=0.82)$ exhibited uncontrolled increase in circumferential stiffness and stress as compared to their counterparts with non-impaired mechanosensitivity. Additionally, in the case of moderate elastic fiber insult $\left({\vartheta }_{\mathit{\text{max}}}^e=0.36\right)$, there was an increase in strain energy prior to exiting that differed strikingly from the expected MBE behavior (Figure 4c), though in the case with a severe elastic fiber insult, strain energy decreased before exiting.

3.2. Production and removal mechanosensitivity insults modulate the rate of unstable growth

To examine the relationship between production and removal mechanosensitivity in axisymmetric lesions, we fixed ${\vartheta }_{\mathit{\text{max}}}^e=0.36$ and $k^{\vartheta }=0.050$ while covarying ${\vartheta }_{\mathit{\text{max}}}^p$ and ${\vartheta }_{\mathit{\text{max}}}^r$ between 0 (non-impaired mechanosensitivity) and 0.82 (severe mechanosensitivity loss). For non-impaired mechanosensitivity, we obtained the expected convergence of the time-resolved solution to the MBE prediction. As long as production mechanosensitivity remained intact, the two frameworks continued to produce consistent predictions, even with severely compromised removal mechanosensitivity (Figure S5a–c, top rows). The time-resolved simulations showed high sensitivity to insults in production mechanosensitivity, such that any simulation with a sufficiently high value of ${\vartheta }_{\text{max}}^p$ rapidly lost convergence.

Prescribing high insults to both production and removal allowed the simulation to progress further in time by slowing disease progression and delaying rupture. This resulted in overall larger aneurysms that demonstrated higher stresses and stiffnesses before reaching the rupture exit threshold (Figure S5a–c, bottom right). All cases of severe production mechanosensitivity insults resulted in the rupture exit threshold being reached despite identical magnitudes of elastic fiber insults resulting in small, stable TAAs in the non-impaired mechanosensitivity and MBE simulations (Figure S5a–c, bottom rows). As before, we also observed unbounded increases in stiffness and stress exceeding the MBE predictions (Figure S5b, bottom right), as well as increases in strain energy along with poorly regulated intramural stress (Figure S5c, bottom right).

3.3. Reduced insult rate can recover TAA stability for otherwise unstable insult conditions

Finally, we investigated the relationship between the severity of elastic fiber insult and the insult rate in axisymmetric lesions by covarying ${\vartheta }_{\mathit{\text{max}}}^e$ and $k^{\vartheta }$ while fixing the production and removal mechanosensitivity insults at ${\vartheta }_{\mathit{\text{max}}}^p=0.82$ and ${\vartheta }_{\mathit{\text{max}}}^r=0.41$, which are expected to result in unstable outcomes (Figure 5). Decreasing the rate at which elastic fiber and mechanosensitivity insults occurred $(k^{\vartheta }=0.010)$ stabilized TAA progression compared to simulations with identical elastic fiber and mechanosensitivity insults but a more rapid rate of insult accumulation (Figure S5). Indeed, the moderately compromised evolution with a slow insult rate was shown to gradually converge to the MBE prediction (Figure S6a–c, top left). However, the time-resolved model continued to show that elastic fiber insult values beyond 70% could not reach a stable configuration even with a significantly lower insult rate (Figure S6, bottom rows), for which no MBE equivalent existed.

Fig. 5.

Covarying biomechanical insults produce varying regimes of stability, growth, and rupture. Mechanobiological insult parameters ${\vartheta }_{\mathit{\text{max}}}^e,{\vartheta }_{\mathit{\text{max}}}^p,{\vartheta }_{\mathit{\text{max}}}^r$, and $k^{\vartheta }$ are varied in pairs for axisymmetric insults, and simulation outcomes are categorized according to the final lesion size and growth rate for converged simulations, or by whether the simulation reached the rupture exit threshold. Green regions represent stabilized vascular outcomes whose configurations were simulated equally well using the MBE framework. (a) ${\vartheta }_{\mathit{\text{max}}}^e\in \left[\mathrm{0.12,0.72}\right]$ versus ${\vartheta }_{\mathit{\text{max}}}^p\in \left[\mathrm{0.00,0.82}\right]$ with ${\vartheta }_{\mathit{\text{max}}}^r=0$ and $k^{\vartheta }=0.025$. Increased elastic fiber insult leads to unbounded growth, while increased production mechanosensitivity insult leads to rupture. (b) ${\vartheta }_{\mathit{\text{max}}}^p\in \left[\mathrm{0.00,0.82}\right]$ versus ${\vartheta }_{\mathit{\text{max}}}^r\in \left[\mathrm{0.00,0.82}\right]$ with ${\vartheta }_{\mathit{\text{max}}}^e=0.36$ and $k^{\vartheta }=0.05$. Rupture potential is primarily driven by production rather than removal mechanosensitivity insult. (c) ${\vartheta }_{\mathit{\text{max}}}^e\in \left[\mathrm{0.12,0.72}\right]$ versus $k^{\vartheta }\in \left[\mathrm{0.010,0.050}\right]$ with ${\vartheta }_{\mathit{\text{max}}}^p=0.6$ and ${\vartheta }_{\mathit{\text{max}}}^r=0$. Higher insult rate increases rupture potential

3.4. Effects of insult profile asymmetry

Variations in the spatial distribution of the insult profile exhibited the same qualitative trends, with differences emerging in the insult magnitudes at which stability would be lost.

3.4.1. Asymmetric Insults

Asymmetric insult profiles are preferentially applied to one side of the aortic domain, as represented by Equation 11. This resulted in non-circumferentially uniform dilatation that is representative of in vivo TAAs which often display circumferential asymmetry. It has previously been shown that limiting the circumferential spread of insults enables higher maximum elastic fiber insults to be prescribed before stability is lost (Latorre and Humphrey 2020b). Within the present framework, this allows an insult severity of ${\vartheta }_{\mathit{\text{max}}}^e=0.72$ before the MBE formulation is unable to provide a stabilized solution. This higher threshold for instability was reflected in the outcomes of the time-resolved simulations. Unlike the symmetric insult profiles, a stable configuration was reached at high production mechanosensitivity insults with moderate elastic fiber insults in the asymmetric insult profiles (Figure S7a–c, top right). However, high elastic fiber insults displayed similar trends to those in symmetric insult profiles, which in the absence of a mechanosensitivity insult, resulted in an unstable growing aneurysm that did not reach the rupture threshold criteria during the simulation. At high production mechanosensitivity insult, high elastic fiber insult reached the rupture threshold criteria approximately halfway through the simulation. Higher removal insults and lower insult rates prolonged the time-course and final magnitude of unstable aneurysm growth prior to reaching the rupture threshold criteria (Figure S7). Like the outcomes of symmetrical aneurysm insults, stable outcomes of asymmetric insults agreed with MBE predictions.

3.4.2. Random Insults

Random insult profiles resulted in irregular, more natural TAA geometries. Aneurysms initiated with these insults tended to lose stability over a similar time-course as the idealized asymmetric cases. This similarity demonstrates that a lack of circumferential symmetry may result in similar stability behavior, even in complex lesions (Figure S8). Random insult profiles with high elastin insult resulted in unstable growth. At high production mechanosensitivity insult, high elastic fiber insult resulted in the rupture threshold criteria being reached. High removal mechanosensitivity insult and low insult rate both resulted in a prolonged time-course and magnitude of aneurysm growth, resulting in overall greater dilatation prior to rupture. Stable outcomes in vessels with random insult profiles agreed with MBE predictions.

3.5. Summary

Numerical results from Figure 4 and Supplemental Figures S5–S8 show effects of rate-dependent insults superimposed on underlying insults to elastic fiber integrity for many cases, including axisymmetric, asymmetric, and random insults. Together they reveal how different combinations can lead to growth arrest (stability), continuing growth, or growth leading to structural failure. These and similar results are summarized for the axisymmetric case in Figure 5, which reveals in general that increasing elastic fiber insults are expected to lead to unbounded growth whereas rupture potential is driven mainly by compromised mechanosensitivity to stimuli that otherwise drive production, noting that increased removal relative to production can lead to structural failure. As expected, increases in the insult rates tend to increase disease progression and potential for rupture. Figure 5 provides a visual summary of these and other insult-dependent changes.

4. Discussion

Not all TAAs are the same. About 20% arise from predisposing pathogenetic variants (e.g., FBN1 variants that give rise to Marfan syndrome, MFS) or congenital defects (primarily bicuspid aortic valve, BAV), with the remaining ~80% arising sporadically, often in older individuals. A 2016 meta-analysis of growth rates of TAAs and associated risk factors assessed data from 11 publications (1,383 patients total) and found overall mean rates of lesion enlargement ranging from 0.2 to 4.2 mm/yr, with accelerated growth noted in larger lesions as well as in MFS and BAV patients (Oladokun et al. 2016). The authors concluded, nonetheless, that there was “a shortfall in the understanding of TAA expansion rates.” A more recent meta-analysis based on 55 publications further showed the importance of evaluating growth in different classes of lesions, with growth rates of 0.81 mm/yr in Loeys-Dietz syndrome, 0.45 mm/yr in MFS, 0.37 mm/yr in BAV patients, 0.33 mm/yr in sporadic TAAs, and 0.25 mm/yr in Turner syndrome (Henry et al. 2025). Clearly, there is a need to understand better the underlying mechanisms that drive such different growth rates in different patient populations. It is suggested herein that computational models can contribute in this regard.

Most prior computational studies of TAAs have focused on quantifying hemodynamics (Caballero and Laín 2013; Febina et al. 2018), calculating aneurysm wall stresses using finite element analyses (Martin et al. 2015; Callaghan et al. 2015; Wang et al. 2021), or modeling the coupled fluid–solid interactions of the lesion hemodynamics and vascular wall (Pasta et al. 2013; Campobasso et al. 2018; Mendez et al. 2018; Vignali et al. 2021; Mousavi et al. 2021). These studies, and many others like them, have provided significant insights into the biomechanical complexities of TAAs but have not provided detailed information on mechanisms that drive progressive lesion enlargement versus arrest. One counterexample, however, is a finite element study wherein biaxial wall stresses were computed in TAAs for patients based on serial medical imaging over 1–6 year periods (Zamirpour et al. 2024). This study suggested that evolving values of wall stress are better predictors of all-cause mortality than evolving values of aortic diameter, but there were few details provided on the assumed material properties and their likely evolution. A more natural way to study progressive lesion enlargement is to use computational G&R models. Such data-informed models have been proposed for TAAs both for patients (Mousavi et al. 2019; Ghavamian et al. 2020) and for mice (Li et al. 2023; Bazzi et al. 2025), though without detailed parameter sensitivity studies that could be useful in understanding potential effects of underlying contributors to disease (Baek et al. 2005; Valentin et al. 2011; Brandstaeter et al. 2021).

In this paper, we extended prior G&R simulations that focused on potential mechanobiological drivers of TAA (Latorre and Humphrey 2020b) through inclusion of rate-based parameters within constrained mixture models of vascular G&R. We focused on the striking influence of mechanosensitivity factors on TAA phenotype, stability, and outcome. Our idealized axisymmetric and asymmetric insult profiles illustrated general trends in how structural (${\vartheta }^e$), mechanobiological (${\vartheta }^p$ and ${\vartheta }^r$), and rate-based ($k^{\vartheta }$) parameters impacted aneurysm growth and stability. Not surprisingly, greater elastic fiber insult resulted in larger aneurysms. At very high insult values, there was no achievable stable configuration. Although this was similarly predicted by the MBE framework, the time-resolved framework showed the temporal progression of this instability where the aneurysm continued to grow past the simulation endpoint. Despite the unbounded growth due to large elastic fiber insults, an additional insult to production mechanosensitivity was still required to reach a rupture threshold within our simulation time-course. At large production mechanosensitivity insults, larger dilatations, stresses, and stiffnesses were produced for the same magnitudes of elastic fiber insult compared to their counterparts with non-impaired mechanosensitivity. Even moderate elastic fiber insults that would otherwise result in stable, non-aneurysmal outcomes could be destabilized by a severe production mechanosensitivity insult to achieve rupture exit threshold outcomes. By simultaneously varying the removal mechanosensitivity insult, this behavior could be tuned to achieve rupture at either a small diameter or a large diameter, demonstrating how variations in underlying cellular processes can result in different rupture conditions. This observation held in the asymmetric and random insult patterns, although the reduced circumferential spread of the lesion tended to allow more severe insults and longer time-courses to be simulated before reaching the rupture exit threshold. Given that the trends from the computationally less expensive axisymmetric insult profile held in the more complex lesion geometries, the insight gained from axisymmetric profiles could offer a path forward for more computationally tractable parametric studies of aneurysm G&R without as great a need for high-fidelity geometric representations.

These simulations also allow quantification of the biomechanical behavior of aneurysms that may not be feasible to measure in vivo and may identify potential targets for further experimental investigation. For example, experimental studies of aneurysms in $Fbn{1}^{C1041G/+}$ and $Fbn{1}^{mgR/mgR}$ mice showed higher circumferential stiffness and lower strain energy in TAAs (Bellini et al. 2016; Cavinato et al. 2021). In our computational study, stable and growing dilatations that did not rupture agreed with this observation. However, our simulations also explored a class of lesions with moderate elastic fiber insult and high production mechanosensitivity insult that exhibited increased stiffness, stress, and strain energy before eventual rupture. Although ruptured aortas cannot be biomechanically evaluated experimentally, this modeling framework offers some predictions of the prerupture biomechanical phenotype, though this would be difficult to validate. Nevertheless, this may suggest that certain trends in decreasing strain energy could be indicative of underlying mechanisms of aneurysm progression and risk of rupture.

Establishing a three-dimensional framework capable of generating time-resolved simulations of (i) stable aortic dilatation, (ii) progressive dilatations, (iii) small-diameter rupture, and (iv) large-diameter rupture (Figure 1) is of significant clinical value. These categories parallel clinical decision-making, where distinguishing between stable and unstable cases—even among lesions with similar intermediate geometries—is central to risk assessment and treatment planning. Demonstrating that these outcomes can be reliably produced in a time-resolved framework for both idealized and complex insult profiles establishes a robust foundation for a variety of in silico studies. Looking forward, whereas regression models may prove increasingly useful in predicting which individuals are at risk of developing TAA (Mori et al. 2021), or perhaps even which patients are susceptible to unstable lesion growth, the ability to generate large amounts of physiologically-relevant synthetic data is particularly relevant to large data models in the clinical setting. Notably, machine learning models often rely on large highquality datasets on which to train. Such clinical datasets are rarely available, but a robust simulation platform can help overcome this limitation by generating synthetic datasets that mimic in vivo behavior. This approach has been used in transfer learning, where simulated data augment limited clinical data to substantially improve neural operator performance (Goswami et al. 2022; Li et al. 2025).

We also showed that the more computationally efficient MBE formulation indeed offers sound predictions of time-resolved outcomes when used in cases where a mild insult severity and gradual rate, coupled with robust mechanosensitivity, may eventually stabilize the growth. This is the first time such a comparison has been made across three-dimensional finite element models; it demonstrates that the MBE framework identifies unique stable configurations, as long as such states exist. This comparison is relevant to future computational studies of aneurysm G&R as the MBE framework is significantly less computationally expensive (~0.1 CPU hours per simulation) than the time-resolved framework (~1000 CPU hours per simulation). The distinction between these costs demonstrates the utility of knowing the boundaries of the equilibrated versus time-resolved approaches and choosing the appropriate framework, supporting the use of the MBE framework when stable configurations under robust mechanosensing and mechanoregulation are expected, while using the time-resolved framework when rate-dependent parameters that the MBE framework is not capable of capturing are expected to contribute to aneurysm outcomes.

Notwithstanding new insight gleaned from the simulations reported herein, many additional considerations remain. We evaluated time-course changes in geometric and mechanical metrics of interest in the region of maximum insult to evaluate the possible stabilization of lesion growth, but we did not perform a formal mechanobiological stability analysis (Cyron and Humphrey 2014; Latorre and Humphrey 2019); such analysis of TAA growth could provide additional insight (cf. Cyron et al. (2014)). We also did not consider initial regional variations in material properties, though such differences exist in both humans (Davis et al. 2016) and mice (Bersi et al. 2019) and methods are available for regional assessments (e.g., Seshaiyer and Humphrey (2003)). There is a pressing need for information on regional properties and how they may evolve with lesion growth (cf. Cavinato et al. (2021)), and the present three-dimensional finite element platform is well suited for such investigations. Although the present study employs an idealized initial cylindrical geometry, the underlying framework is not restricted to this configuration. The governing G&R equations are fully three-dimensional and formulated independently of vessel geometry, enabling direct application to curved and patient-specific domains. Indeed, the fluid-solid-growth methodology on which this work builds on has previously been used in fully threedimensional geometries, including patient-specific aortic models (Schwarz et al. 2023). Despite this, extending the present aneurysm framework to more complex domains introduces several practical challenges. Most notably, constrained mixture theory accounts for multiple constituent families, each with potentially distinct stress-free configurations, which are difficult to infer in complex geometries. Several studies have already sought to address these challenges in other G&R formulations, with varying approaches to these problems. Future work will build on these approaches to first extend this framework to curved geometries and then apply it to patient-specific aortic models.

Additionally, we focused solely on phenomenological constrained mixture models of vascular G&R. Such models have proven effective in describing and predicting tissue-level changes in diverse cases of vascular disease (Humphrey 2021); nevertheless, there are considerable advantages to coupling tissue-level G&R models and cellular signaling models (Irons et al. 2021) to consider both fundamental mechanisms and potential therapeutic targets. Such coupling becomes particularly attractive given our finding that evolving changes in the sensitivity of the cells to perceived changes in stress from homeostatic values likely reflects changes in receptor-ligand interactions, including downregulation of receptors.

5. Conclusion

Computational simulations have the potential to aid clinical decision-making for TAAs by revealing how diverse biomechanical factors influence growth with or without stabilization and rupture. By incorporating additional rate-based parameter insults into the constrained mixture theory of G&R, we developed a computational framework capable of simulating clinically relevant outcomes of TAAs. This approach enables identification of biomechanical trends that precede each outcome type and examination how combinations of insult types produce distinct phenotypes. Furthermore, we directly compared outcomes from timeresolved constrained mixture simulations to MBE simulations, demonstrating that the MBE framework robustly predicts stabilized configurations when such outcomes exist. These findings motivate use of the MBE approach—which is orders of magnitude more computationally efficient—for identifying and simulating stable outcomes, while reserving the time-resolved framework for cases when the evolution or instability of vessels must be captured. Altogether, this work establishes a foundation for clinically relevant in silico studies with translational potential.

Supplementary Material

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s10237-026-02054-w.

Availability of data and material

Data used in this study are included in the manuscript body.

Code Availability

Code is available in a public repository as cited in the paper Schwarz (2025).