A numerical investigation of the mechanics of intracranial aneurysms walls Assessing the inﬂuence of tissue hyperelastic laws and heterogeneous properties on the stress and stretch ﬁelds

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A numerical investigation of the mechanics of intracranial aneurysms walls:

Assessing the inﬂuence of tissue hyperelastic laws and heterogeneous properties

on the stress and stretch ﬁelds

I. L. Oliveirad,∗, P. Cardiﬀb, C.E. Baccinc, J.L. Gaschea

aSão Paulo State University (UNESP), School of Engineering, Mechanical Engineering Department

bUniversity College Dublin (UCD), School of Mechanical and Materials Engineering, Dublin, Ireland

cInterventional Neuroradiology/Endovascular Neurosurgery, Beth Israel Deaconess Medical Center, Harvard Medical School,

Boston, MA, US

dSão Paulo State University (UNESP), School of Engineering, Mechanical Engineering Department, Thermal Sciences Building,

Avenida Brasil, 56, Ilha Solteira - SP, Brazil

Abstract

Numerical simulations have been extensively used in the past two decades for the study of intracranial

aneurysms (IAs), a dangerous disease that occurs in the arteries that reach the brain. They may aﬀect up

10 %

of the world’s population, with up to

50 %

mortality rate, in case of rupture. Physically, the blood

ﬂow inside IAs should be modeled as a ﬂuid-solid interaction problem. However, the large majority of those

works have focused on the hemodynamics of the intra-aneurysmal ﬂow, while ignoring the wall tissue’s

mechanical response entirely, through rigid-wall modeling, or using limited modeling assumptions for the

tissue mechanics. One of the explanations is the scarce data on the properties of IAs walls, thus limiting the

use of better modeling options. Unfortunately, this situation is still the case, thus our present study investigates

the eﬀect of diﬀerent modeling approaches to simulate the motion of an IA. We used three hyperelastic laws

— the Yeoh law, the three-parameter Mooney-Rivlin law, and a Fung-like law with a single parameter — and

two diﬀerent ways of modeling the wall thickness and tissue mechanical properties — one assumed that both

were uniform while the other accounted for the heterogeneity of the wall by using a “hemodynamics-driven”

approach in which both thickness and material constants varied spatially with the cardiac-cycle-averaged

hemodynamics. Pulsatile numerical simulations, with patient-speciﬁc vascular geometries harboring IAs,

were carried out using the one-way ﬂuid-solid interaction solution strategy implemented in solids4foam, an

extension of OpenFOAM

, in which the blood ﬂow is solved and applied as the driving force of the wall

motion. We found that diﬀerent wall morphology models yield smaller absolute diﬀerences in the mechanical

response than diﬀerent hyperelastic laws. Furthermore, the stretch levels of IAs walls were more sensitive

to the hyperelastic and material constants than the stress. These ﬁndings could be used to guide modeling

arXiv:2210.05575v1 [physics.flu-dyn] 11 Oct 2022

decisions on IA simulations, since the computational behavior of each law was diﬀerent, for example, with

the Yeoh law yielding the smallest computational time.

Keywords: intracranial aneurysms, hyperelasticity, wall morphology, mechanical response, numerical

simulations

1. Introduction

Intracranial aneurysms (IAs) are pathological dilatations of the human vascular system normally found in

the bifurcations of the cerebral arteries tree. The most common form has a saccular shape, with a prevalence

of up to

90 %

in the brain arteries [

], being a dangerous disease that may aﬀect up to

10 %

of the world’s

population [

] and with up to

50 %

mortality rate, in case of rupture [

]. This pathology has, in the

past three decades, been investigated experimentally [

], which, for example, led to the understanding of

the importance of hemodynamics on its development, but also numerically through Computational Fluid

Dynamics (CFD), which provided detailed information on the hemodynamics [

] — although still being a

debatable topic in the clinical practice [8, 9].

Due to the nature of the pathology, better modeling is continuously sought to allow for more reliable

numerical simulations, for example, through the use of Fluid-Solid Interaction (FSI) modeling [

although numerical techniques to solve FSI problems pose challenging numerical diﬃculties [

Additionally, using an FSI strategy employed for patient-speciﬁc IA geometries requires the modeling of their

wall tissue that also poses diﬃculties hard to overcome, such as the lack of patient-speciﬁc data of the wall

thickness, the constitutive behavior of the tissue and its material properties. This is particularly important

due to the large variability of the disease.

Previous experimental works showed that IAs walls are more likely to have wall thickness and mechanical

and failure properties varying spatially [

]. This local morphology is caused by the natural history of

a particular IA [

]. Meng et al.

[18]

, for example, hypothesized two biological pathways, dependent

on diﬀerent local hemodynamic conditions, that would lead to diﬀerent wall phenotypes — the authors

name these two phenotypes as “type-I”, comprising small IAs with thin and translucent walls, and “type-

II”, encompassing large IAs with thick, white or yellow, atherosclerotic walls. Moreover, a spectrum of

∗Corresponding author

Email addresses: iago.oliveira@unesp.br (I. L. Oliveira), philip.cardiff@ucd.ie (P. Cardiﬀ),

cebaccin@gmail.com (C.E. Baccin), jose.gasche@unesp.br (J.L. Gasche)

Preprint submitted to Journal of the Mechanical Behavior of Biomedical Materials October 12, 2022

morphologies would exist between these broad phenotypes, as investigated by Kadasi et al.

[14]

, for example,

who found that 27 % of IAs are type-I, 8 % are type-II, and 65 % contain both patch types.

How a patient-speciﬁc IA grows also inﬂuences its mechanical behavior, classically considered to be

well described by hyperelastic laws [

] — even though particular laws to suitably represent it do not exist

[

]. In the last decade, a few works mechanically characterized samples of IA tissue using uniaxial tests to

failure, obtaining the values of the material constants that appear in hyperelastic laws classically associated

with artery tissue behavior [

]. Typical examples are the Mooney-Rivlin (MR) law [

], the Yeoh law

[

], and an isotropic exponential Fung-like quadratic law [

]. Apart from the mechanical constants, other

properties of IAs tissue have also been reported by Costalat et al.

[22]

, for example, who found that the

tissue of unruptured IAs is stiﬀer than ruptured IAs tissue. Finally, in possession of IA tissue samples, these

works have also measured their average thickness, further conﬁrming that an IA is globally thinner than its

surrounding arteries.

Although it is still a challenge to measure the local morphology, i.e. the local wall thickness and tissue

material properties, for a large number of patient-speciﬁc IAs, some works on the subject exist. Signorelli

et al.

[15]

, for example, used an “indentation device” to measure, in a point-wise manner, the elasticity

modulus of an IA sac sample with a resolution of

1 mm2

. Their ﬁndings suggest that the rupture site is

less stiﬀ, i.e. with a smaller elasticity modulus than the rest of the sac, where they found that stiﬀ regions

were mixed with thinner regions. The technique has the same drawbacks as classical uniaxial tests, though,

because it still requires the aneurysm tissue to be collected, hence in vivo measurements are unfeasible. In

this regard, imaging techniques are thought to be a promising alternative to measuring the local thickness of

a patient-speciﬁc IA sac, as performed by Kleinloog et al.

[25]

through an experimental study in which the

wall thickness of IAs was measured using a 7T Magnetic Resonance Imaging.

Cebral et al.

[26]

used CFD to investigate an IA sac’s local morphological heterogeneity. The authors

subdivided the wall of a sample of IAs into ﬁve regions with speciﬁc phenotypes: atherosclerotic, hyperplastic,

thin, the rupture site, and “normal-appearing” by intraoperative observation and correlated each of them

with local hemodynamics. They found a similar relationship between the local hemodynamics conditions

investigated by Meng et al.

[18]

, so-called “low-ﬂow” and “high-ﬂow” eﬀects, and the wall phenotypes. Their

study is a good example of how numerical simulations could be used to predict the IA sac heterogeneity on a

patient-by-patient basis.

Therefore, accounting for all these modeling requirements makes the modeling of a patient-speciﬁc IA

wall a challenge due to both the scarce experimental data to feed numerical models and the large variability

of the disease, which prevents patient-speciﬁc computations. This is reﬂected in the modeling choices used

by the few numerical works that investigated the mechanical response of IAs. For example, in terms of the

approach to estimating the wall thickness, we found a majority that employed uniform thickness throughout

the IA sac and branches [

], and a small amount that employed a uniform thickness for the

aneurysm sac and a diﬀerent one on the branches [

], or a lumen-diameter thickness [

]. Finally, only a

single work obtained the patient-speciﬁc thickness distribution of the IA sac [

] and compared it with a

whole uniform wall model, nonetheless the authors used micro-computed tomography to scan the aneurysm

sac, a technique that is diﬃcult to apply in a larger cohort of IAs.

Regarding the selection of constitutive law, works that numerically solved the FSI problem with patient-

speciﬁc IA subjects have used several diﬀerent ones. Surprisingly, we found a majority that has chosen the

small-strain Hookean law that, rigorously, should not be used in ﬁnite-deformation motions [

Other works employed the classic neo-Hookean law [

] or more specialized ones, such as exponential laws

[

] and the MR law [

], although in a smaller number, and not always using the material properties of

patient-speciﬁc IA tissue.

Despite the uncertainty about which law to choose, the assessment of the impact of diﬀerent material

laws on the mechanics of IAs walls has been the subject of even fewer studies. Torii et al.

[28]

performed FSI

simulations for one IA case, by assuming, ﬁrst, the rigid wall assumption — thus “pure” CFD simulations

—, and three elastic laws: the Hookean law (thus, assuming small strains), the St. Venant-Kirchhoﬀ law,

and another hyperelastic law using the exponential strain-energy function proposed by Demiray

[37]

. Their

ﬁndings showed that the displacement proﬁles were qualitatively similar among all the elastic laws, although

the maximum displacement with the exponential hyperelastic law was

36 %

smaller than that for the St.

Venant-Kirchhoﬀ law. Unfortunately, their results on the wall mechanics were limited to the displacement

ﬁeld, no stresses or strain were analyzed because their focus was on the hemodynamics.

An earlier trial to assess diﬀerent constitutive laws in IAs’ mechanical response was conducted by

Ramachandran et al.

[38]

, with patient-speciﬁc IAs geometries. The authors assumed them to be statically

determined, i.e. their mechanical response was independent of the material properties, and investigated

the impact of diﬀerent constitutive laws on the wall stresses and strains by numerically simulating only

the aneurysm sac with Computational Solid Dynamics (CSD) by using a numerical modeling similar to

inﬂation experiments. They used both anisotropic and isotropic versions of Fung-like laws, the Yeoh law

with three parameters, the St. Venant-Kirchhoﬀ law, and Hooke’s law too. Their results suggested that the

aneurysm sac may indeed be statically determined regarding diﬀerent material laws. However, they only

studied the aneurysm sac, i.e. they removed the surrounding arteries portions that may have had an impact on

the aneurysms sac stresses, and their pressure-inﬂation model employed static boundary conditions (BCs),

which limits their conclusions. Indeed, the authors highlighted that these conclusions may not stand when the

full vasculature would be simulated with dynamical BCs that realistically reﬂect the cardiac cycle forces.

In this current scenario, it is clear that it remains broadly unknown what is the average impact of the use

of diﬀerent hyperelastic laws and wall morphology models in the mechanics of IAs, i.e. in the stress and

strain ﬁelds of the IA sac. Therefore, the aim of this work was to assess what could be the impact of choosing

diﬀerent material laws and diﬀerent morphology models to numerically obtain the mechanical response of

IAs. More speciﬁcally, we investigated whether a wall model with uniform thickness and material constants,

for example, would be acceptable to be used, given the dominant heterogeneity existing for this disease. This

is essential in investigations of IA rupture while promising tools that could extract the heterogeneity of the

wall more accurately are not ready to do that for a large cohort of patient-speciﬁc IAs and, also, because

ultimately the rupture event depends on the stress and strain levels on the wall.

2. Numerical Methodology

2.1. Sample Selection and Geometry Preparation

We selected twelve vascular geometries from digital subtraction angiography (DSA) examinations

collected retrospectively. Nine were collected in the Albert Einstein Israelite Hospital, São Paulo, and

approved to be used by the institution’s Research Ethics Committee as also by the Research Ethics Committee

of the Faculty of Medicine of São Paulo State University (UNESP), Campus of Botucatu. The additional

three vascular geometries were obtained from the Aneurisk dataset repository [

], which provides a set of

IAs geometries used during the Aneurisk project and are available under the “CC BY-NC 3.0” license. We

used these additional geometries due to the lack of suﬃcient ruptured cases in the original dataset to build a

representative sample.

The twelve vasculatures harbored thirteen bifurcation IAs, all of them originating from the more common

bifurcation spots of IA occurrences in the brain vessels (the internal carotid artery (ICA) and middle cerebral

artery (MCA)). Seven were unruptured and six ruptured with maximum dome diameter ranging from, roughly,

3 mm

7 mm

, thus categorizing them as small- or medium-sized IAs (mean

standard deviation equals

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Anumericalinvestigationofthemechanicsofintracranialaneurysmswalls:AssessingtheinuenceoftissuehyperelasticlawsandheterogeneouspropertiesonthestressandstretcheldsI.L.Oliveirad,,P.Cardib,C.E.Baccinc,J.L.GascheaaSãoPauloStateUniversity(UNESP),SchoolofEngineering,MechanicalEngineeringDepartmentbUnive...

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A numerical investigation of the mechanics of intracranial aneurysms walls Assessing the inﬂuence of tissue hyperelastic laws and heterogeneous properties on the stress and stretch ﬁelds

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