# A numerical study on the influence of geometry on the rupture risk of
abdominal aortic aneurysms

G R Krishna Chand Avatar<EMAIL_ADDRESS>Department of Aerospace
Engineering, Indian Institute of Science, Bangalore, India  Chinika Dangi
<EMAIL_ADDRESS>Puneet Pushkar<EMAIL_ADDRESS>
###### Abstract

Abdominal aortic aneurysms (AAAs) are local dilatations in the abdominal aorta
occurring due to weakening of arterial wall. The present work investigates the
influence of the ratio of maximum transverse diameter to abdominal height
(DHr) on rupture risk of AAA, using hemodynamics and AAA wall mechanics
simulations. We have considered two idealized AAA geometries, AAA1 of higher
DHr than AAA2. Two constitutive models, namely, Newtonian and Carreau-Yasuda
models have been used for modelling blood as an incompressible fluid.
Additionally, in order to describe the behaviour of AAA wall, three
constitutive models, namely, linear elastic, Saint Venant Kirchhoff elastic
and a phenomenological finite-strain model called Raghavan-Vorp elastic have
been utilised. Numerical simulation of AAA biomechanics have been performed
using solids4Foam, an open source package built on finite volume framework.
Hemodynamic parameteric study reveals that AAA1 has lower time-averaged wall
shear stress (TAWSS) and higher oscillatory shear index (OSI) compared to
AAA2. Thus, AAA1 has increased susceptibility to thrombus deposition and,
therefore, higher risk of AAA rupture. Furthermore, the peak wall stress of
AAA1 is about 8% higher than that of AAA2. It is concluded that higher DHr
leads to greater rupture risk.

##  I Introduction

An arterial aneurysm is a vascular condition characterized by an abnormal
focal dilatation of an artery with respect to its original state Canchi  _et
al._ (2015); Aggarwal  _et al._ (2011); Philip, Patnaik, and Sudhir (2022).
This happens due to localised bulging of blood vessels as a result of the
weakening of the arterial walls Philip, Patnaik, and J. (2020). Aneurysm
shapes could be saccular (asymmetric balloon-like expansions of a portion of
the wall), fusiform (axisymmetric gradual dilatation of the entire
circumference of the artery) or cylindroid Finol, Keyhani, and Amon (2003).
Aortic aneurysms are located either in the thoraric region or in the
infrarenal segment, that is, the abdominal region between the renal arteries
and the iliac bifurcation; the latter is called abdominal aortic aneurysm
(AAA) (Fig. 1) and is nearly 80% of all aortic aneurysms. There are several
risk factors responsible for the genesis, growth and rupture of AAAs, namely,
people aged above 60 years, smoking, males, patients with artherosclerosis,
coronary heart disease and hypertension Ullery, Hallett, and Fleischmann
(2018); Lederle (2000, 1997). AAAs are severe medical conditions often
requiring surgical management. AAAs are often asymptomatic until they suddenly
rupture, which necessitates a measure of rupture risk prediction. Typically,
surgeons tend to use a criterion based on maximum transverse diameter to
decide on the course of treatment. Fusiform AAAs with diameter larger than
5-5.5 cm are recommended for surgical repair while those smaller than 4 cm are
considered to be at low risk of rupture Raut  _et al._ (2013). However, this
criterion has been deemed crude by various researchers Vorp, Raghavan, and
Webster (1998); Finol and Amon (2001); Finol, Keyhani, and Amon (2003); Finol
and Amon (2003); Raut  _et al._ (2013); Boyd  _et al._ (2016) since there have
been instances when aneurysms of diameter smaller than 4 cm ruptured while
those with diameter greater than the critical transverse diameter did not
Venkatasubramaniam  _et al._ (2004); thus necessitating detailed hemodynamics
and wall mechanics studies to identify the parameters directly and indirectly
responsible for the rupture of AAAsMutlu  _et al._ (2023). These parameters
include wall shear stress and its derivatives, and aneurysmal wall stress.

Figure 1: Schematic of normal aorta and abdominal aortic aneurysm. Source:
Canchi  _et al._ (2015)

Bluestein et al. Bluestein  _et al._ (1996) performed computational fluid
dynamics (CFD) simulation of an axisymmetric AAA and deduced that flow
recirculation regions creates suitable conditions encouraging thrombus
formation and the viability of aneurysm rupture. Finol et al. Finol and Amon
(2003) did CFD simulations of patient-specific aneurysm models to quantify
vortex dynamics and flow-induced wall shear stress (WSS) distributions and
found that the vortex dynamics is specific to every individual and is
dependent on the shape and size of the aneurysm. Boyd et al. Boyd  _et al._
(2016) conducted a CFD study on ruptured AAAs and observed that aortic rupture
took place at the sites of low WSS. Philip et al. Philip, Patnaik, and J.
(2020) postulated that high time-averaged wall shear stress (TAWSS) and low
oscillatory shear index (OSI) indicates the initiation of intraluminal
thrombus (ILT) formation. Furthermore, Vorp et al. Vorp, Raghavan, and Webster
(1998) conducted a numerical study on AAAs and found that aneurysm rupture
occurs when the mechanical stress acting on the aneurysm inner wall due to
intraluminal pressure, exceeds the failure strength of the degenerated aortic
tissue. Raghavan et al. Raghavan, Webster, and Vorp (1996) conducted an
experimental study on 78 tissue samples (71 with AAAs), and observed that
yield strength reduced by nearly 50% and ultimate strength reduced by over 50%
for AAAs with respect to the normal aorta. Venkatasubramaniam et al.
Venkatasubramaniam  _et al._ (2004) found that peak wall stress was
significantly higher in the ruptured AAA and also discovered that the area of
peak wall stress correlated with rupture site.

Martufi et al. Martufi  _et al._ (2009) studied 3D axisymmetric AAA and
observed that DHr (ratio of maximum transverse diameter $D_{max}$ to abdominal
height $H$), DDr (ratio of maximum transverse diameter to proximal neck
diameter), Hr (ratio of abdominal height to neck height) and BL (ratio of
bulge height to abdominal height) are important shape parameters in assessing
the aneurysm risk prediction. However, few researchersCappeller  _et al._
(1997); Perez, Rojas-Solorzano, and Finol (2016); Philip, Patnaik, and J.
(2020) have suggested that DHr and DDr could be more accurate parameters and
requires further detailed analysis. Recently, Kumar et al. Kumar, Kumar, and
Rai (2023) investigated the influence of AAA shape and deduced that maximum
dilation diameter and aneurysm neck angle affects local hemodynamic parameters
considerably. Additionally, Faraji et al. Faraji, Sahebi, and SalavatiDezfouli
(2022) studied the role of four different viscosity models on pulsatile blood
flow of AAA and concluded that non-Newtonian model predicted OSI more
accurately. Dalbosco et al. Dalbosco  _et al._ (2023) conducted a multi-scale
computational study on AAA and correlated the growth of AAA with alterations
at microstructural level in the aortic tissue.

To the best of authors’ knowledge, no parametric study has been reported in
literature on AAA biomechanics simulations based on Raghavan-Vorp model. The
present work is a complete study to investigate the influence of DHr on both
hemodynamics and arterial wall mechanics using realistic constitutive models.
We have modeled blood using both Newtonian and Carreau-Yasuda (non-Newtonian)
constitutive relations. The AAA wall has been modeled using both linear
elastic and non-linear elastic (Saint Venant Kirchhoff elastic and Raghavan-
Vorp elastic Raghavan and Vorp (2000)) models. We consider two AAA geometries,
AAA1 and AAA2, whose shape indices are given in table below Philip, Patnaik,
and J. (2020)

| AAA1 | AAA2
---|---|---
DHr | 0.83 | 0.45
DDr | 2 | 2
Hr | 2.4 | 2.4
BL | 0.92 | 0.92

. Hemodynamics and computational solid stress simulations has been performed
using solids4Foam, an OpenFOAM-based open source package for solving continnum
mechanics problems in finite volume framework. Computational models of AAA
lumen geometry were prepared using CAD software Salome. A numerical study
investigating the effect of blood constitutive models, effect of different
shape indices (AAA1 and AAA2), influence of AAA wall contitutive model and DHr
has been presented. The analysis highlights importance of geometry and shape
of AAAs on rupture risk.

##  II Problem formulation

The geometry of the aneurysm problem is denoted by a domain $\Omega$, with
boundary $\Gamma$, which is composed of a solid domain $\Omega^{s}$ and a
fluid domain $\Omega^{f}$, such that $\Omega=\Omega^{s}\cup\Omega^{f}$,
alongwith their corresponding boundaries $\Gamma^{s}$ and $\Gamma^{f}$ as
shown in Fig. 2. The fluid-solid interface is
$\Gamma^{fs}=\Gamma^{f}\cap\Gamma^{s}$ which acts as the wall for fluid
domain).

Figure 2: Fluid domain $\Omega^{f}$ with boundary $\Gamma^{f}$ enclosing it,
solid domain $\Omega^{s}$ with its corresponding boundary $\Gamma^{s}$, and
fluid-solid interface $\Gamma^{fs}$

###  II.1 Fluid dynamical model for Blood

We consider blood to be an incompressible fluid which shows both shear-
thinning and viscoelastic behaviour. Sequeira et al. Sequeira (2018) have
stated that blood flow displays predominantly Newtonian behaviour as it
circulates through large arteries. However, Khanafer et al. Khanafer  _et al._
(2006) have concluded that the blood flow may be affected by non-Newtonian
behaviour of blood in pathological vascular conditions such as AAAs where RBCs
and platelets can aggregate resulting in the change of viscosity. Therefore,
we carry out numerical simulations using both Newtonian and non-Newtonian
Carreau-Yasuda models for the blood with material parameters taken from
Biasetti et al Biasetti, Hussain, and Gasser (2011). The blood density is
$\rho=1050$ kg/m3 and the transport properties for the two models are

  * •

Newtonian model: dynamic viscosity $\mu^{f}=0.0044$ Pa-s

  * •

Carreau-Yasuda model: dynamic viscosity
$\mu^{f}=\mu_{\infty}+(\mu_{0}-\mu_{\infty})[1+(\lambda\dot{\gamma})^{a}]^{(n-1)/a}$
, where $\mu_{0}=0.16$ Pa-s, $\mu_{\infty}=0.0035$ Pa-s, $\lambda=8.2$ s,
$n=0.2128$, $a=0.64$

The governing equations (incompressible Navier-Stokes equations) are derived
from the conservation of mass and balance of linear momentum (neglecting body
forces) and expressed in integral form as

$\begin{split}&\int_{S}\rho^{f}{\bf\it v}^{f}\cdot{\bf\it n}dS=0\\\
&\partialderivative{t}\int_{V}\rho^{f}{\bf\it v}^{f}({\bf\it
x},t)dV+\oint_{S}\rho^{f}{\bf\it v}^{f}{\bf\it v}^{f}\cdot{\bf\it
n}dS=\oint_{S}{\bf\it\sigma}^{f}\cdot{\bf\it n}dS\end{split}$ (1)

The consitutive relation relating the Cauchy stress tensor to the velocity
field for a Newtonian fluid is given as ${\bf\it\sigma}^{f}=-p{\bf\it
I}+\mu^{f}\bigg{(}\gradient{{\bf\it v}^{f}}+(\gradient{{\bf\it
v}^{f}})^{T}\bigg{)}-\frac{2}{3}\mu^{f}(\divergence{{\bf\it v}^{f}}){\bf\it
I}$, where ${\bf\it I}$ is the second order identity tensor, $p$ is the
pressure field.

The integral equations (Eqn. 1) are discretized using finite volume method.
The resulting discrete system is solved using Pressure Implicit with Splitting
of Operators (PISO) algorithm in a segregated fashion. An unconditionally
stable second-order accurate backward Euler scheme is used to advance the
simulation. The linear systems for velocity components are solved using
preconditioned bi-conjugate gradient (PBiCG) method with tolerance of $1\times
10^{-8}$ while matrix equation for pressure is solved using geometric-
agglomerated algebraic multi-grid (GAMG) solver with a tolerance of $1\times
10^{-6}$.

##### Initial and Boundary conditions

We set uniform values of velocity and pressure field throughout the domain as
initial conditions. The boundary conditions are:

  * •

Inlet: A time-varying paraboloid velocity profile whose cross-sectional
average value is shown in Fig. 3. We use an 8-series Fourier decomposition for
the average velocity. Zero-gradient condition is used for pressure.

  * •

Outlet: We impose a time-varying pressure as shown in Fig. 3 via 8-series
Fourier decomposition for pressure. Zero-gradient condition is used for
velocity.

  * •

Endoluminal surface of AAA wall: No-slip condition is prescribed for velocity
such that ${\bf\it v}={\bf\it 0}$ while zero-gradient is used for the pressure
field.

Figure 3: Waveforms for (a) Average inlet velocity with markers indicating
time instants: initiation of systole A (t = 0.08 s), systolic acceleration B
(t = 0.18 s), peak systole C (t = 0.25 s), systolic deceleration D (t = 0.35
s), beginning of diastole E (t = 0.50 s), late diastole F (t = 0.80 s) (b)
Outlet pressure with marker A (t = 0.32 s) indicating peak pressure. Source:
adapted from Philip, Patnaik, and J. (2020)

###  II.2 Solid dynamical model for the Aneurysm wall

We model the AAA wall as linear elastic Scotti  _et al._ (2005); Philip,
Patnaik, and J. (2020) and Saint Venant Kirchhoff elastic Scotti  _et al._
(2005). We also employ a non-linear isotropic phenomenological constitutive
model for AAAs proposed by Raghavan et al. Raghavan and Vorp (2000); we refer
to this as Raghavan-Vorp elastic model. The mechanical properties of the AAA
wall are assumed to be homogeneous; density $\rho=2000$ kg/m3 and material
constants are

  * •

Linear and Saint Venant Kirchhoff elastic models: $E=2.7\times 10^{6}$ Pa,
$\nu^{s}=0.45$

  * •

Raghavan-Vorp model:, $\alpha=1.74\times 10^{5}$ Pa, $\beta=1.881\times
10^{6}$ Pa, $\nu^{s}=0.45$

The wall thickness is considered to be uniform with a value of $1.2$ mm
Philip, Patnaik, and J. (2020).

We use total Lagrangian formulation Belytschko, Liu, and Moran (2014);
Holzapfel (2000) of the governing equations for solid mechanics, where
deformations and stresses are calculated with respect to the reference or
initial configuration. We represent the deformation by the deformation
gradient tensor ${\bf\it F}$ and the stress by the first Piola-Kirchhoff
stress tensor ${\bf\it P}$. ${\bf\it F}$ is expressed as function of the solid
displacement ${\bf\it u}$ as ${\bf\it F}={\bf\it I}+\gradient_{0}{{\bf\it u}}$
where subscript $0$ indicates differentiation with respect to the intial
configuration. The balance of linear momentum for a deformable solid,
neglecting body force, is

$\displaystyle\rho_{0}^{s}\partialderivative[2]{{\bf\it u}}{t}$
$\displaystyle=\gradient_{0}\cdot{\bf\it P}$ (2)

The first and second Piola-Kirchhoff stress tensors Holzapfel (2000), denoted
as ${\bf\it P}$ and ${\bf\it S}$ respectively, are related to each-other as
${\bf\it P}={\bf\it F}\cdot{\bf\it S}$ and to Cauchy stress tensor
${\bf\it\sigma}$ as

$\displaystyle{\bf\it\sigma}$ $\displaystyle=\frac{1}{J}{\bf\it P}\cdot{\bf\it
F}^{T}=\frac{1}{J}{\bf\it F}\cdot{\bf\it S}\cdot{\bf\it F}^{T}$ (3)

where $J=\det({\bf\it F})$. The Green-Lagrange strain tensor is defined as

$\displaystyle{\bf\it E}$ $\displaystyle=\frac{1}{2}\left({\bf\it
F}^{T}\cdot{\bf\it F}-{\bf\it
I}\right)=\frac{1}{2}\big{(}\underbrace{\gradient_{0}{{\bf\it
u}}+(\gradient_{0}{{\bf\it u}})^{T}}_{\text{linear
part}}+\underbrace{(\gradient_{0}{{\bf\it u}})^{T}\cdot\gradient_{0}{{\bf\it
u}}}_{\text{non-linear part}}\big{)}$ (4)

Now, ${\bf\it E}$ is composed of linear as well as non-linear parts making it
appropriate for measuring large deformations in elastic solids which shows a
non-linear behaviour. A non-linear elastic model used to capture small strains
and finite rotations is the Saint Venant-Kirchhoff model. For an istropic
material, the constitutive relation is given as

$\displaystyle{\bf\it S}$ $\displaystyle=2\mu^{s}{\bf\it
E}+\lambda^{s}\tr({\bf\it E})\mathbf{I}$ (5)

where $\mu^{s}$ (also called shear modulus Chaves (2013)) and $\lambda^{s}$
are Lamé constants which are related to Young’s modulus and Poisson ratio as
Jasak and Weller (2000) $\mu^{s}=\frac{E}{2(1+\nu^{s})}\text{ and
}\lambda^{s}=\frac{\nu^{s}E}{(1+\nu^{s})(1-2\nu^{s})}$

##### Linear elastic model:

Cauchy infinitesimal strain tensor Fung (1990) defined as
${\bf\it\epsilon}=\frac{1}{2}\big{(}\gradient_{0}{{\bf\it
u}}+(\gradient_{0}{{\bf\it u}})^{T})$ Also, ${\bf\it S}$ reduces to Cauchy
stress tensor ${\bf\it\sigma}$. Thus constitutive relation is the isotropic
Hooke’s law

$\displaystyle{\bf\it\sigma}^{s}$
$\displaystyle=2\mu^{s}{\bf\it\epsilon}+\lambda^{s}\tr({\bf\it\epsilon}){\bf\it
I}$ (6)

##### Raghavan-Vorp elastic model

Raghavan et al. Raghavan and Vorp (2000) obtained the following constitutive
equation relating the Cauchy stress tensor and the elastic deformation

$\displaystyle{\bf\it\sigma}^{s}$ $\displaystyle=-p{\bf\it
I}+2\left[\alpha+2\beta(I_{{\bf\it b}}-3)\right]{\bf\it b}$ (7)

where $p$ is the Lagrange multiplier to enforce incompressibility, $\alpha$
and $\beta$ are material constants obtained after fitting uniaxial loading
experimental data to the proposed model, ${\bf\it b}={\bf\it FF}^{T}$ is left
Cauchy-Green tensor and $I_{{\bf\it b}}$ is its first principal invariant.
Since arterial wall is nearly (but not exactly) incompressible, we developed a
weakly compressible version of Raghavan-Vorp model for which the constitutive
relation is given as

$\displaystyle{\bf\it\sigma}^{s}$
$\displaystyle=\frac{\kappa}{2}\bigg{(}J-\frac{1}{J}\bigg{)}{\bf\it
I}+\text{dev}\bigg{(}\frac{2}{J}\bigg{[}\alpha+2\beta(I_{\overline{{\bf\it
b}}}-3)\bigg{]}\overline{{\bf\it b}}\bigg{)}$ (8)

where $\overline{{\bf\it b}}=J^{-2/3}{\bf\it b}$ and
$\kappa=\frac{2}{3}\mu^{s}+\lambda^{s}$

From Eqn. 3, we have

$\displaystyle{\bf\it\sigma}^{s}$ $\displaystyle=\frac{1}{J}{\bf\it
P}\cdot{\bf\it F}^{T}\implies\qquad{\bf\it P}=J{\bf\it\sigma}^{s}{\bf\it
F}^{-T}$ (9)

Using Eqns. 9 and 8 in Eqn. 2, we obtain the governing equation for Raghavan-
Vorp elastic material which is solved to obtain the solid displacement field
${\bf\it u}$.

The governing equations (Eqn. 2) for solid domain are discretized using finite
volume method. The resultant discretized system of equations for components of
displacement ${\bf\it u}$ are solved using preconditioned conjugate gradient
(PCG) method with tolerance of $1\times 10^{-9}$. Second-order accurate
backward Euler scheme is used for time integration. To enhance convergence,
the displacement equations are under-relaxed.

##### Initial and Boundary conditions (BCs)

The governing equations for solid domain are solved for displacement field. We
set the displacement and velocity fields to be zero throughout the domain as
initial conditions. We impose zero displacement conditions, ${\bf\it
u}={\bf\it 0}$, at the extremities or side faces. The internal (endoluminal)
surface is subjected to Neumann boundary conditions which is the time-varying
pressure loading $p$ corresponding to the cardiac cycle (Fig. 3). The external
surface is traction-free.

###  II.3 Simulation setup

We have the following expression for the idealized fusiform AAA lumen geometry
in terms of the maximum AAA diameter $D_{max}$, proximal neck diameter
$D_{neck2}$ and height of the aneurysm sac $H$ as

$\displaystyle r(z)$
$\displaystyle=\bigg{(}\frac{D_{max}-D_{neck2}}{4}\bigg{)}\bigg{[}1+\sin\bigg(\frac{2\pi
z}{H}-\frac{\pi}{2}\bigg{missing})\bigg{]}+\frac{D_{neck2}}{2}\quad;\quad
0\leq z\leq H$

It is also assumed that the proximal and distal neck diameters are equal
$D_{neck1}=D_{neck2}$. We take $D_{max}=5$ cm.

##### Simulation domain

For proper imposition of inlet and outlet boundary conditions, both inlet and
outlet extensions are taken to be 2$D_{neck2}$ Krishna  _et al._ (2020);
Scotti  _et al._ (2005); Philip, Patnaik, and J. (2020). We used an open-
source computer-aided design (CAD) software Salome Ribes and Caremoli (2007)
for preparing the AAA lumen geometries. The schematic of the AAA geometry is
shown in Fig. 4. We used a fine mesh comprising of prismatic layers to
properly resolve the unsteady boundary layer at the wall for fluid domain and
capture flow dynamics accurately. As suggested in Krishna et al. Krishna  _et
al._ (2020), boundary layer thickness $\delta$ for oscillatory flows is of the
order $\delta\sim\sqrt{\nu^{f}T}$, where $\nu^{f}=\frac{\mu^{f}}{\rho^{f}}$ is
the kinematic viscosity of the blood and $T$ is the time-period of the cardiac
cycle. Using the blood properties, we obtain

$\displaystyle\delta\approx\sqrt{\frac{0.0044}{1050}\cdot 1}=2\times
10^{-3}\text{ m}$

implying that the fluid mesh ought to be refined upto a distance of $2\times
10^{-3}$ m from the wall.

Figure 4: Idealized fusiform AAA geometry for simulation

An open-soure mesh generation tool called cfMesh Juretic and Baburic (2014)
was used to generate fluid mesh for AAA lumen (Fig. 5(a)). On the other hand,
the solid mesh for each AAA geometry was generated by extruding the wall patch
of the fluid mesh along the local wall-normal direction for AAA wall
thickness, and number of layers (Fig. 5(b)).

(a)

(b)

Figure 5: Computational mesh for (a) fluid and (b) solid domains

We use an open-source package called solids4Foam which provides a host of
applications (solvers and utilities) to deal with a wide variety of problems
in fluid dynamics, solid dynamics and fluid-structure interactions.

##  III Hemodynamics simulations of AAA lumen geometries

We perform hemodynamic simulations using Newtonian and Carreau-Yasuda
constitutive models for blood.

For analyzing the blood flow in the AAAs, we consider the variation of wall
shear stress (WSS) and its derived quantities. We also study vortex dynamics
and identify the vortical (coherent) structures using
$\lambda_{2}$-criterionJeong and Hussain (1995); Zhang, Watanabe, and Nagata
(2023), a velocity gradient-based vortex identification method. Wall shear
stress (WSS) vector ${\bf\it\tau}_{w}$ is defined as the tangential component
of the traction vector ${\bf\it t}$ at the wall. Traction vector is obtained
as ${\bf\it t}={\bf\it\sigma}^{f}\cdot{\bf\it n}^{f}$, where
${\bf\it\sigma}^{f}$ is Cauchy stress tensor and ${\bf\it n}^{f}$ is the wall
normal unit vector. WSS vector Iago Lessa de Oliveira (2021) is obtained as
${\bf\it\tau}_{w}={\bf\it t}-({\bf\it t}\cdot{\bf\it n}^{f}){\bf\it n}^{f}$.
We look at the following derived quantities to get a holistic view of the wall
shear stress Browne  _et al._ (2015); Oliveira  _et al._ (2021); Philip,
Patnaik, and J. (2020); Mutlu  _et al._ (2023):

  * •

Time-averaged wall shear stress ($TAWSS$) is calculated as $TAWSS({\bf\it
x})=\frac{1}{T}\int_{0}^{T}\norm{{\bf\it\tau}_{w}({\bf\it x},t)}dt$

  * •

Oscillatory shear index ($OSI$) is a dimensionless metric which characterises
whether the WSS vector is aligned with the $TAWSS$ vector throughout the
cardiac cycle, $OSI({\bf\it
x})=\frac{1}{2}\bigg{(}1-\frac{\norm{\frac{1}{T}\int_{0}^{T}{\bf\it\tau}_{w}({\bf\it
x},t)dt}}{\frac{1}{T}\int_{0}^{T}\norm{{\bf\it\tau}_{w}({\bf\it
x},t)}dt}\bigg{)}$

  * •

Spatial gradient of the mean wall shear stress ($WSSG_{S}$) is calculated as
$WSSG_{S}=\norm{\gradient{(meanWSS)}}$

  * •

Peak systole wall shear stress, $PSWSS({\bf\it
x})=\norm{{\bf\it\tau}_{w}({\bf\it x},t_{ps})}$ where $t_{ps}$ is the time
instant of peak systole

At first, we carry out a comparative study of the effect of constitutive
models on the hemodynamics within AAA1 for wall shear stress (WSS) and other
derived quantities – $PSWSS$, $meanWSS$, $WSSG_{S}$, $TAWSS$, $OSI$. Therefter
we examine the role played by DHr on the flow dynamics within AAA1 and AAA2.

We carry out the simulations for 4 cardiac cycles and perform a mesh
convergence study for AAA1 using three meshes with the number of cells
increasing progressively with a factor of approximately 2 as tabulated in
Table 1. The time-step size is also progressively decreased by a factor of 2
such that a constant maximum Courant-Friedrichs-Levy (CFL) number of 1, that
is CFL $\leq 1$, is maintained throughout every simulation.

| Coarse | Medium | Fine
---|---|---|---
Number of cells (in millions) | 1.82 | 3.66 | 6.8
Number of prismatic layers at the wall | 5 | 9 | 12
Time-step size (in seconds) | $2\times 10^{-4}$ | $1\times 10^{-4}$ | $5\times 10^{-5}$
Table 1: Simulation details

We observe negligible difference in $PSWSS$ amongst the results obtained for
the meshes. Therefore, we choose the medium mesh for further analysis.

###  III.1 Effect of blood constitutive models: Carreau-Yasuda versus
Newtonian

####  III.1.1 Velocity contours

The axial velocity ($v_{z}$) contours plotted at different stations along AAA1
lumen (Fig. 6) for the time instants marked in Fig. 3a suggest that $v_{z}$ at
the central portion of each station is higher for the Newtonian model
throughout the cardiac cycle. This is consistent with the fact that dynamic
viscosity $\mu^{f}$ is higher in the regions facing lower shear rate, which
typically occurs in the core of the flow, and thus decelerating the flow in
those areas.

Figure 6: Velocity contours at different stations: (a) Carreau-Yasuda and (b)
Newtonian models

####  III.1.2 Vortex dynamics

A presentation of vortex dynamics in Fig. 7a displays strong correlation
between the regions of high values of vorticity
${\bf\it\omega}~{}(=\curl{{\bf\it v}}^{f})$ and ring vortices obtained using
$\lambda_{2}$-criterion for a fixed $\lambda_{2}=-75$ s-2 for Carreau-Yasuda
model. As has been discussed in Finol et al. Finol, Keyhani, and Amon (2003);
Finol and Amon (2003), the growth, decay and transport of vortices in the flow
can be divided into the following flow phases alongwith the corresponding time
intervals in the cardiac cycle

  * •

Systolic acceleration ($t=0.08$ s to $t=0.25$ s) involves downstream transport
of the residual vortices, left from the previous cardiac cycle, out of the
aneurysms, resulting in a completely attached flow pattern.

  * •

Systolic deceleration ($t=0.25$ s to $t=0.50$ s) is characterized by flow
separation at the proximal neck of the aneurysm, the growth of a single-vortex
near the midsection which then moves downstream.

  * •

Early diastole ($t=0.50$ s to $t=0.85$ s) involves gradual weakening of the
vortex shed at the end of systole as it translates downstream and the growth
of the translating single-vortex.

  * •

Late diastole ($t=0.85$ s to $t=0.08$ s) is the interval where the most
significant flow disturbance takes place, mainly due to the low-velocity
recirculation region inside the aneurysm towards its distal end. Also, the
downstream vortex translation leads to enclosing of the forward flow stream by
the vortex.

The vortex dynamics for Newtonian case differs slightly from that of the
Carreau-Yasuda in that there is an occurrence of a weak wavy-like ring vortex
of large diameter in all but systolic deceleration phase as seen in Fig. 7a
and 7b.

Figure 7: Vortex dynamics: (a) Vorticity contours and $\lambda_{2}=-75$ s-2
iso-surface for Carreau-Yasuda model, (b) $\lambda_{2}=-75$ s-2 iso-surface
for Newtonian model.

####  III.1.3 WSS and derived quantities

It is clear from Fig. 8(a) that $PSWSS$ values for Newtonian case are higher
than the Carreau-Yasuda case throughout the domain and the difference is
prominently noticeable in the portion towards the distal end of the aneurysm
sac; the maximum $PSWSS$ values reaching 3.19 Pa and 3.01 Pa respectively.
This is so because the dynamic viscosity is lower in the wall region for the
Carreau-Yasuda case which produces lower wall shear stress.

Similar to $PSWSS$, Newtonian model overpredicts $TAWSS$ compared to Carreau-
Yasuda model (see Fig. 8(b)) with higher variations in the distal region of
the aneurysm. While the maximum $TAWSS$ in the case of Carreau-Yasuda model
reaches 0.9 Pa, the corresponding value for Newtonian model is 1.04 Pa; an
increase of $16\%$.

(a)

(b)

Figure 8: (left) $PSWSS$ and (right) $TAWSS$ distributions for (a) Carreau-
Yasuda and (b) Newtonian models for AAA1

We observe from Fig. 9 that blood flow undergoes higher fluctuations in the
aneurysm sac for Carreau-Yasuda case. It is also noticed that while high $OSI$
values are found in the central portion of the aneurysm sac, the high $OSI$
region forming a narrow band shifts proximally in the Newtonian case.

Figure 9: $OSI$ distribution: (a) Carreau-Yasuda and (b) Newtonian models

###  III.2 Effect of DHr: AAA1 versus AAA2

We study how the flow dynamics is influenced by the shape index DHr - AAA1
with a higher DHr and AAA2 with a lower DHr. For both cases, simulations have
been performed using Carreau-Yasuda blood model.

####  III.2.1 Velocity contours

Fig. 10 shows the axial velocity ($v_{z}$) contours plotted at different
stations along AAA2 lumen for various time instants. The velocity at the
maximum bulge location is relatively lower than other stations for all time
instants except t = 0.5 s.

The flow dynamics inside AAA1 forms larger recirculation zones and flow
separation than that of AAA2. The difference in the extent of recirculation
regions are conspicuous at t = 0.08, 0.18, 0.8 s. In such regions, thrombus
finds a fertile environment to settle down and gradually increase in size.
Therefore, there is an increased chance of ILT deposition in AAA1 compared to
AAA2.

Figure 10: Velocity contours at different stations of AAA2 Figure 11: Velocity
vector plot in the longitudinal plane of the aneurysm sac: (a) AAA1 (b) AAA2

####  III.2.2 Pressure contours

We observe from Fig. 12 that the temporal acceleration and deceleration of the
flow has a major influence on the pressure distribution throughout the cardiac
cycle in both AAAs. The flow sees strong favourable pressure gradient in the
initial systolic phase, that is, between $t=0.1$ s to $t=0.25$ s. For the
remaining interval of the cardiac cycle, the flow suffers from adverse
pressure gradient and is strongest around $t=0.35$ s.

The pressure distribution for AAA1 and AAA2 appears to remain fairly similar
except for symmetric pockets of lower pressure found in the aneurysm sac of
AAA1 at $t=0.08$ s, $t=0.5$ s and $t=0.8$ s which are absent in AAA2. These
pockets can be correlated with ring vortices formed in AAA1.

Figure 12: Pressure distribution in the longitudinal plane for (a) AAA1 and
(b) AAA2

####  III.2.3 Vortex dynamics

The vortex dynamics for AAA2 can be categorised as discussed earlier in
Section III.1.2 and display similar behaviour. For $\lambda_{2}=-75$ s-2 iso-
surface shown in Fig. 13, we note the following distinctions pertaining to the
vortex dynamics in the flow phases of AAA1 and AAA2

  * •

Systolic acceleration ($t=0.08$ s to $t=0.25$ s): A weak vortex is observed at
the distal end of AAA2 unlike two vortices found at the mid-way and distal end
of AAA1 sac. In both, the vortices continue to decay till they vanish at peak
systole.

  * •

Systolic deceleration ($t=0.25$ s to $t=0.50$ s): The growth of two vortices
formed at the proximal end and mid-way of AAA1 sac takes place; both
translates downstream. A weak vortex forms in the proximal end of AAA2 sac
which moves downstream too.

  * •

Early diastole ($t=0.50$ s to $t=0.85$ s): While significant flow disturbance
takes place in AAA1 causing growth, decay and translation of multiple
vortices, the situation in AAA2 is relatively quiet.

The gradual weakening of the vortex shed at the end of systole as it
translates downstream and the growth of the translating single-vortex.

  * •

Late diastole ($t=0.85$ s to $t=0.08$ s): The downstream vortex translation
leads to enclosing of the forward flow stream by the vortex in AAA1. However,
decay of two vortices at the proximal end of AAA2 takes place accompanied by a
gradual narrowing of the distal vortex.

In a nutshell, AAA1 has richer and more proactive vortex dynamics than AAA2.

Figure 13: Vortex dynamics: $\lambda_{2}=-75$ s-2 iso-surface contours for (a)
AAA1 (b) AAA2.

####  III.2.4 WSS and derived quantities

$PSWSS$ varies from 0.0093 to 3 Pa in the case of AAA1 and 0.33 to 2.3 Pa for
AAA2 (Fig. 14). In both the cases, the maximum $PSWSS$ occurs at the
corresponding distals ends.

It can be observed Fig. 14 that $TAWSS$ varies from 0.097 to 0.9 Pa in AAA1,
AAA2 in the range 0.11 to 0.53 Pa. AAA1 observes lower values of $TAWSS$ than
AAA2 in the region left to its maximum bulge. In either cases, we find that
coexisting high and low $TAWSS$ regions persists towards the proximal end (and
even at the distal end) creating a fertile ground for the thrombus formation
inside the AAAs. This is more apparent in AAA1 case.

(a)

(b)

Figure 14:  $PSWSS$ and $TAWSS$ distributions for (a) AAA1 and (b) AAA2

AAA1 suffers from higher maximum $WSSG_{S}$ (203.1 Pa/m) than AAA2 (85.10
Pa/m); an increase by a factor of 2 than AAA2 (Fig. 15). In either case, the
maximum happens to be located at the proximal and distal ends. The high $OSI$
region in the central portion of AAA1, flanked by lower $OSI$ regions on
either sides (Fig. 15), suggests that the flow velocity undergoes severe
fluctuations for the maximum bulge location. However, the situation is quite
different for AAA2 in which $OSI$ progressively increases from its proximal
end to its distal end.

(a)

(b)

Figure 15:  $WSSG_{S}$ and $OSI$ distributions for (a) AAA1 and (b) AAA2

##  IV Computational solid stress simulations of arterial wall mechanics

To assess stress in the arterial wall, we use von Mises stress and first
principal stress. The state of stress at every material point in the arterial
wall is defined in terms of Cauchy stress tensor ${\bf\it\sigma}^{s}$ which is
symmetric and consists of 6 components, namely, $\sigma_{ij}^{s}$
($i,j=x,y,z$). We quantify the stress state at every point in terms of von
Mises stress and maximum principal stress $\sigma_{1}$. The von Mises stress
$\sigma_{eq}$ is defined as

$\displaystyle\sigma_{eq}=\frac{1}{\sqrt{2}}\sqrt{(\sigma_{xx}-\sigma_{yy})^{2}+(\sigma_{yy}-\sigma_{zz})^{2}+(\sigma_{xx}-\sigma_{zz})^{2}+6(\sigma_{xy}^{2}+\sigma_{yz}^{2}+\sigma_{xz}^{2})}$

while the maximum principal stress $\sigma_{1}$ is the maximum eigenvalue of
${\bf\it\sigma}^{s}$.

We perform computational solid stress (CSST) simulations for both AAA1 and
AAA2 walls. In CSST, a time-varying uniform pressure loading corresponding to
the cardiac cycle (see Fig. 3) is applied on the endoluminal surface of the
aneurysmal walls. We study the influence of solid material models and $DHr$ on
the wall mechanics.

For spatial convergence studies, we consider three different mesh resolutions
for AAA1 wall (see Table 2) using time-step size of $5\times 10^{-3}$ s.

| Number of cells in radial direction | Number of cells
---|---|---
Coarse | 3 | 31,500
Medium | 5 | 73,800
Fine | 8 | 1,63,700
Table 2: Mesh information

It can be inferred from Fig. 16(a) that there is negligible difference between
the maximum von Mises stress corresponding to the medium and fine meshes.

(a)

(b)

Figure 16: AAA1 wall modeled as a linear elastic material: (a) Mesh and (b)
Temporal convergence

For the medium mesh, we consider three different time-step sizes for time
integration, namely, $5\times 10^{-3}$ s, $2.5\times 10^{-3}$ s and $1.0\times
10^{-3}$ s. We infer from Fig. 16(b) that the influence of time-step size on
the solution, here maximum von Mises stress in the wall, is negligible.

On the basis of spatial and temporal convergence studies, we choose the medium
mesh for further simulations on AAA1 wall using a time-step of $2.5\times
10^{-3}$ s. The mesh settings utilised to discretise AAA1 wall is used for
AAA2 wall as well.

Fig. 17(a) shows the total displacement $\norm{{\bf\it u}}$, von Mises stress
$\sigma_{eq}$ and the first principal stress $\sigma_{1}$ for AAA1 wall
modeled as Raghavan-Vorp elastic model at $t=0.32$ s, that is the peak time of
the uniform pressure loading. The displacement is highest ($2.98$ mm) in the
midway of the initiation of curvature and the maximum bulge location. A
similar observation is made regarding $\sigma_{eq}$ and $\sigma_{1}$ whose
maximum values are $3.62\times 10^{5}$ Pa and $3.90\times 10^{5}$ Pa
respectively. The region flanking the central portion of the bulge sees
reduced values for all these variables. The maximum displacement is about 3.37
mm for AAA2 wall (see Fig. 17(b)), higher than the corresponding value for
AAA1 wall. Likewise, the peak values for $\sigma_{eq}$ and $\sigma_{1}$ are
$3.35\times 10^{5}$ Pa and $3.70\times 10^{5}$ Pa respectively, lower than the
corresponding values for AAA1 wall. Since the peak wall stress is higher in
the case of AAA1, this keeps it at a higher risk of rupture than AAA2.

(a)

(b)

Figure 17: Total displacement (top), von Mises stress (middle), first
principal stress (bottom) for (a) AAA1 and (b) AAA2 walls modeled as Raghavan-
Vorp elastic material at $t=0.32$ s under peak uniform pressure loading

Since the trend of variation of $\norm{{\bf\it u}}$, $\sigma_{eq}$ and
$\sigma_{1}$ differ only quantity-wise for each material model but not
quality-wise, we shall plot the peak values of these quantities over a cardiac
cycle for both AAA1 and AAA2 cases.

####  IV.0.1 Influence of constitutive models for solid

The maximum displacement of the AAA1 wall in the Raghavan-Vorp elastic model
($\approx 3$ mm) is about 1.4 times that of linear elastic ($\approx 2.1$ mm)
and approximately 1.8 times that of Saint Venant Kirchhoff elastic models
(Fig. 18). Maximum von Mises stress in AAA1 wall for Raghavan-Vorp elastic
model ($\approx 3.60\times 10^{5}$ Pa) is 12.4 % higher than the corresponding
value of Saint Venant Kirchhoff and 28 % higher than the linear elastic models
(Fig. 19(a)). Similarly, maximum first principal stress for Raghavan-Vorp
model ($\approx 3.90\times 10^{5}$ Pa) is 11.4 % and 18.2 % higher than the
Saint Venant Kirchhoff and linear elastic models (Fig. 19(b)). It is important
to note that while maximum $\sigma_{eq}$ and $\sigma_{1}$ is higher for Saint
Venant Kirchhoff model as compared to linear elastic case, it is the otherwise
for $\norm{{\bf\it u}}_{max}$.

Figure 18: Variation of maximum total displacement in AAA1 wall over a cardiac
cycle for linear, Saint Venant Kirchhoff and Raghavan-Vorp elastic models

(a)

(b)

Figure 19: Variation of (a) maximum principal stress and (b) maximum von Mises
stress in AAA1 wall over a cardiac cycle for linear, Saint Venant Kirchhoff
and Raghavan-Vorp elastic models

####  IV.0.2 Influence of DHr

Since we have already observed how material model influences the wall
mechanics, we shall now consider the results when the AAA1 and AAA2 walls are
modeled using the more realistic Raghavan-Vorp elastic model. $\norm{{\bf\it
u}}_{max}$ of the AAA2 wall (value is $\approx 3.4$ mm) is about 13.3 % higher
than that of AAA1 wall (value is $\approx 3.0$ mm) (see Fig. 20). Maximum
$\sigma_{eq}$ in AAA1 wall ($\approx 3.60\times 10^{5}$ Pa) is 7.5 % higher
than the corresponding value of AAA2 wall (Fig. 21(a)). Likewise, maximum
$\sigma_{1}$ of AAA1 case ($\approx 3.90\times 10^{5}$ Pa) is 5.4 % higher
than that of AAA2 (Fig. 21(b)).

Figure 20: Variation of maximum total displacement in AAA2 wall over a cardiac
cycle for linear, Saint Venant Kirchhoff and Raghavan-Vorp elastic models

(a)

(b)

Figure 21: Variation of (a) maximum principal stress and (b) maximum von Mises
stress in AAA2 wall over a cardiac cycle for linear, Saint Venant Kirchhoff
and Raghavan-Vorp elastic models

##  V Conclusion

We have investigated the role of hemodynamic parameters ($TAWSS,WSSG_{S},OSI$)
and wall stress ($\sigma_{eq}$) in assessing rupture risk of AAAs with respect
to the geometric shape index DHr. We draw the following conclusions:

  * •

Blood, modeled as a Newtonian fluid, overpredicts WSS and time-averaged WSS
($TAWSS$) throughout the aneurysm sac, primarily in the distal end, and has
lower WSS fluctuations leading to lower oscillatory shear index ($OSI$) as
compared to non-Newtonian Carreau-Yasuda fluid.

  * •

For two AAA geometries, AAA1 and AAA2 of different DHr (AAA1 with higher DHr),
the flow inside AAA1 forms larger recirculation regions and has richer vortex
dynamics than AAA2. On comparing AAA1 and AAA2, we found that AAA1 has lower
$TAWSS$ and higher $OSI$ in its proximal end alongwith higher WSS spatial
gradient $WSSG_{S}$ than AAA2.

  * •

From AAA wall mechanics simulations, we observe that for a particular AAA,
linear elastic and Saint Venant Kirchhoff elastic models underpredict peak
wall stress and also the maximum wall displacement as compared to Raghavan-
Vorp elastic model. Saint Venant Kirchhoff elastic model gives higher peak
wall stress but a lower maximum wall displacement as compared to the linear
elastic model. In every case, the peak wall stress is in the midway of the
ends of the aneurysm sac and the central bulge location. The peak wall stress
of AAA1 is about 8 % higher than that of AAA2. suggesting that AAA of higher
DHr is at a greater risk of rupture.

  * •

The analysis show that hemodynamics simulations point to an increased
susceptibility to thrombus deposition in the case of AAA1 which is a major
predisposing factor for AAA rupture. Overall, DHr is an important determinant
for rupture risk prediction.

###### Acknowledgements.

The authors are thankful to Indian Institute Science, Bangalore for
fellowship.

## Author Declarations

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

G R Krishna Chand Avatar: Conceptualization; Data curation; Formal analysis;
Methodology; Software; Validation; Visualization; Writing – original draft,
review & editing. Chinika Dangi: Data curation; Software; Validation; Writing
– review & editing. Puneet Pushkar: Conceptualization; Formal analysis;
Methodology; Software; Writing – review & editing.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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