Continuum model for flow diverting stents of intracranial aneurysms

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Thesis

Reference

Continuum model for flow diverting stents of intracranial aneurysms

LI, Sha

Abstract

Aneurysmal subarachnoid hemorrhage remains an important cause of stroke mortality and morbidity. Hemodynamic factors are considered to be a major factor in the progression and rupture of intracranial aneurysms. Flow diverting stent (FDS) is an emerging paradigm for treating traditionally difficult intracranial aneurysms. It is placed in the parent artery to divert the blood flow from the aneurysm sac and promote the progressive thrombosis. In the field of medical FDS, numerical simulation is a tool of high importance for the investigation and development of new flow diverters. It is also used to assist patient-specific decision making, for example during a medical stent placement procedure for the consolidation of an arterial aneurysm. However, fully resolved simulations of the stent are often prohibitively expensive, opening the path for approximate but more efficient simulations framework which model the effect of the FDS through a coarse-grained, macroscopic approach. Porous media model was proposed for this purpose, assuming that FDS can be described as a porous media, obeying Forchheimer's law. However, FDS are not [...]

LI, Sha. Continuum model for flow diverting stents of intracranial aneurysms. Thèse de doctorat : Univ. Genève, 2019, no. Sc. 5307

URN : urn:nbn:ch:unige-1155383

DOI : 10.13097/archive-ouverte/unige:115538

Available at:

http://archive-ouverte.unige.ch/unige:115538

Disclaimer: layout of this document may differ from the published version.

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UNIVERSITE DE GENÈVE Département d’informatique

FACULTE DES SCIENCES Professeur B. Chopard

Continuum Model for Flow Diverting Stents of Intracranial Aneurysms

THÈSE

Présentée à la Faculté des sciences de l’Université de Genève pour obtenir le grade de Docteur ès sciences, mention informatique

Par Sha LI

de

Shijiazhuang (Chine)

Thèse N ^◦5307

GENÈVE

Repro-Mail-Université de Genève 2019

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I would like to dedicate this thesis to my family . . .

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Acknowledgements

Firstly, I would like to express my sincere gratitude to my supervisors, Prof. Bastien Chopard and Dr. Jonas Latt for the continuous support and tremendous help of my Ph.D study and related research, for their patience, motivation and immense knowledge in the last four years.

Their willingness and patience to discuss and analyze the problems greatly helped me all the time. Their tutoring and even accompaniment encouraged me to explore solutions to the difficult problems. Their guidance with expertise and experience always led me to the right direction and saved me from confusion and mistakes. I am deeply grateful to Jonas for all his detailed helps in programming, researching and even writing a research paper in English.

Their supports are far more than I expected, even imagined. Word cannot express my heartfelt appreciation to them.

I would like to thank Dr. Guy Courbebaisse and Prof. Nikolaos Stergiopulos for the positive feedback. Thank them for allowing me to defend my thesis and agreeing to be members of the defence committee.

My sincere thanks also go to all my fellow colleagues of the Scientific and Parallel Computing (SPC) group, Yann, who helped translate the abstract of this thesis into French, Francesco, Christos, Xavier, Orestis, Sebastien, Dimitrios, Aziza, Mohamed, Federico, Jean- Luc, Christophe Charpilloz, Christophe Coreixos, Pierre, Raphael, Jonathan, Anthony, Franck and Remy. Without their precious help, support, comfort and encouragement, it would not be possible to conduct this research. The time we spent together is truly memorable.

I thank all the administrative and IT teams of the CUI (Center Universitaire d’Informatique):

Anne-Isabelle, Daniel, Nicolas and Elie. I am grateful to them for generous and invaluable help in administration and technique equipment.

Last but not the least, I would like to give special thanks to my family: my beloved husband Shuoxing, my parents Shengjuan and Zengchen, and the family of my brother Min for lovingkindness to encourage me to strive towards my goal. I also express my appreciation to my dear friends in Geneva and in all Switzerland; they are my dependence, my family in spirit.

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Abstract

Aneurysmal subarachnoid hemorrhage remains an important cause of stroke mortality and morbidity. Hemodynamic factors are considered to be a major factor in the progression and rupture of intracranial aneurysms. Flow diverting stent (FDS) is an emerging paradigm for treating traditionally difficult intracranial aneurysms. It is placed in the parent artery to divert the blood flow from the aneurysm sac and promote the progressive thrombosis. In the field of medical FDS, numerical simulation is a tool of high importance for the investigation and development of new flow diverters. It is also used to assist patient-specific decision making, for example during a medical stent placement procedure for the consolidation of an arterial aneurysm. However, fully resolved simulations of the stent are often prohibitively expensive, opening the path for approximate but more efficient simulations framework which model the effect of the FDS through a coarse-grained, macroscopic approach. Porous media model was proposed for this purpose, assuming that FDS can be described as a porous media, obeying Forchheimer’s law. However, FDS are not 3D porous media and a solution based on screen model are more adapted. In this thesis, we develop a screen based flow diverter model (SFDM) which circumvents deficiencies of some aspects of the porous media model.

Our framework is developed from the existing screen models which, as opposed to porous media models, are explicitly built to reflect the physics of thin porous structures, like a flow diverter stent. We first review the hydraulic equations of screen in the literatures, then propose a 2D framework which are able to predict the pressure drop and flow deflection across a stent.

The deflection part determined by the drag coefficient, is different from the homogeneous porous media model.

The 2D SFDM is tested on several ideal aneurysms under various conditions, such as different stent placements, varying porosities and pulsatile inlet velocities. The model can effectively reduce the velocity and wall shear stress in the aneurysm sac as a fully resolved stent.

Afterwards, the framework of SFDM is extended to 3D and is applied for actual medical flow diverters in patient-specific aneurysms. The numerical tests show that the 3D SFDM can reproduce the results of direct numerical simulation both qualitatively and quantitatively, and are capable of reducing the simulation time by an order of magnitude or more. Additionally to

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the theoretical framework of the SFDM, this thesis discusses in detail the procedure required to deploy the model for a given stent and artery.

All the simulations above are carried out in the Palabos which is an open source solver based on Lattice Boltzmann method (LBM). The LBM force models and boundary conditions that we utilize for the simulations of stented aneurysms and SFDM are already integrated in the solver.

In summary, we develop, validate and deploy a screen based flow diverter model which can greatly reduce the computational time for the blood flow in stented aneurysm with reasonable accuracy. It opens the possibility to be included in a medical imaging device which is a fast enough simulation mode for helping a clinician to assess the benefit of a given FDS for a specific patient.

Keywords: intracranial aneurysm; flow diverting stent; CFD; lattice Boltzmann method

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Résumé

Les hémorragies sous-arachnoïdiennes constituent une importante cause de mortalité et de morbidité. On considère que les facteurs hémodynamiques jouent un rôle majeur dans la progression et la rupture des anévrismes intracrâniens. L’endoprothèse (Flow Diverting Stent, FDS) est un paradigme émergeant pour le traitement des anévrismes intracrâniens qui représentent habituellement des difficultés. Un tel dispositif se place dans l’artère parent afin de détourner l’écoulement sanguin du sac anévrismal et de favoriser la thrombose progressive. Dans le domaine des FDS médicaux, les simulations numériques représentent un outil de grande importance pour l’étude et le développement de nouveaux dériveurs d’écoulement. Elles sont également utilisées pour assister, d’une manière spécifique au patient, les prises de décisions, comme par exemple durant la procédure de placement d’un stent médical pour la consolidation d’un anévrisme artériel. Cependant, la simulation d’un stent entièrement résolu s’avère souvent extrêmement dispendieuse en termes de ressources informatiques. Un tel coût ouvre la voie à des modèles plus efficaces utilisant des approximations où l’effet des FDS est modélisé au moyen d’une approche macroscopique. Un modèle de milieu poreux a été proposé dans ce but, en supposant que les FDS peuvent être décrits comme tel, selon la loi de Forchheimer.

Cependant, les FDS ne sont pas des milieux poreux 3D et une description basée sur un modèle d’écran est plus adaptée. Dans cette thèse, nous développons un modèle que nous avons appelé

"screen based flow diverter" (SFDM) et qui pallie à certains défauts du modèle de milieu poreux.

Notre framework est développé à partir de modèles d’écrans existants qui, contrairement aux modèles de milieu poreux, sont explicitement construits pour refléter la physique des structures poreuses fines telles que dans le cas d’un FDS. Une revue de la littérature à propos des équations hydrodynamiques dans le cas des modèles est d’abord effectuée. Dans un second temps, la version 2D du framework est présentée, et nous montrons qu’elle est capable de prévoir correctement le saut de pression et la déviation de l’écoulement dans la région du stent. La partie du modèle qui décrit la déviation du courant, déterminée par le coefficient de frottement, est différente du terme homogène que l’on trouve dans les modèles de milieu poreux classiques.

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Le SFDM 2D est testé sur plusieurs anévrismes idéaux et dans diverses conditions, telles que différents placements de stent, avec des variations de porosité et des vitesses d’entrée pulsatiles. Le modèle peut efficacement réduire la vitesse d’écoulement le taux de cisaillement aux parois au sein du sac anévrismal, en accord avec les résultats des simulations entièrement résolues.

Le framework SFDM est ensuite étendu au cas 3D, puis appliqué à des cas réels (spécifiques aux patients) de stents médicaux dans des anévrismes. Les tests numériques montrent que le modèle 3D peut reproduire les résultats de simulation numérique directe, tant qualitativement que quantitativement. Ainsi, le temps de simulation peut être réduit d’au moins un ordre de grandeur. En plus du framework théorique correspondant au modèle SFDM, cette thèse discute en détail de la procédure nécessaire pour déployer le modèle sur des artères et des stents qui correspondent à des cas réels.

La librairie open source Palabos, dévolue à l’implémentation de la méthode de Boltzmann sur réseaux (LBM), a été utilisée afin de mettre en place les simulations. Les modèles LBM de force et de conditions de bords utilisés pour les simulations de FDS et SFDM sont déjà intégrés à la librairie.

Pour résumer, nous développons, validons et déployons un modèle pour SFDM qui permet de réduire significativement le temps de calcul des écoulements sanguins dans un anévrisme avec stent, tout en conservant une précision raisonnable. Cela ouvre la voie à l’utilisation de cette méthode numérique au sein d’un dispositif d’imagerie médicale qui serait, dès lors, suffisamment rapide pour aider le personnel médical à évaluer les bénéfices d’un FDS donné dans le cas d’un patient spécifique.

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Statement

The present manuscript is a collection of three journal papers

• Li, S., Latt, J., and Chopard, B. (2017). Model for pressure drop and flow deflection in the numerical simulation of stents in aneurysms. International Journal for Numerical Methods in Biomedical Engineering, 34(3):e2949.

• Li, S., Latt, J., and Chopard, B. (2018). The application of the screen-model based approach for stents in cerebral aneurysms.Computers & Fluids, 172:651 – 660.

• Li, S., Chopard, B., and Latt, J. . Continuum model for flow diverting stents in 3D patient- specific simulation of intracranial aneurysms. International Journal for Numerical Methods in Biomedical Engineering(submitted)

completed by an introduction and a conclusion.

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List of figures

1.1 The process of the deployment of balloon [65] . . . 2

1.2 Three types of coiling . . . 3

1.3 Flow diverter stented in aneurysm . . . 4

1.4 Illustration of the streaming process [8] . . . 7

1.5 Illustration of on-grid bounce-back BC . . . 8

2.1 Flow through a screen at an incident angle . . . 14

2.2 Periodic boundary condition . . . 19

2.3 k₀at a porosity of 0.6 . . . 22

2.6 The fitting correlation for the drag coefficientk₀at different porosities. . . 23

2.7 Compare simulatedkⁿwith two models at different incident angles . . . 24

2.8 Two methods for computing the deflection coefficientB . . . 26

2.9 Compare the existing models ofBwith simulation results . . . 27

2.10 The fitting correlation of the deflection coefficient . . . 28

2.11 The velocity reduction coefficient versus Reⁿ_d . . . 29

2.12 The fitting curve of the velocity reduction coefficient . . . 29

2.13 Velocity profile of the straight channel. The unit of the velocity ism/s, while thex−andy−coordinates are measured inm. . . 31

2.14 Representation of the bent artery with aneurysm used in the simulation, parametrized with the radiusR. Units are inmm. . . 32

2.15 Comparison of velocity profiles. The unit of the velocity ism/s, while thex− andy−coordinates are measured inmm. The color palette is defined over a logarithmic scale to emphasize weak flows. . . 33

2.16 Streamlines in aneurysm . . . 33

2.17 Pulsatile inlet flow . . . 35

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3.1 The artery with two outlets. . . 43

3.2 Velocity for the fully resolved simulation at different resolutions (β =0.6). The units for the position and velocity aremmandm/s, respectively. . . 44

3.3 Velocity over the region of the aneurysm for the fully resolved simulation at different resolutions (β =0.6). The units for the position and velocity aremm andm/s, respectively. . . 44

3.4 Velocity obtained from the simulation using our stent model at different resolutions. The units for the position and velocity aremmandm/s, respectively. . 46

3.5 Velocity over the region of the aneurysm of stent 1 (β =0.6). The units for the position and velocity aremmandm/s, respectively. . . 47

3.6 WSS on the wall of aneurysm of stent 1 (β =0.6) . . . 47

3.7 Two outlets: stent 2 . . . 49

3.8 Velocity field with stent 2 (β =0.6). The units for the position and velocity aremmandm/s, respectively. . . 49

3.9 Velocity over the region of the aneurysm with stent 2 (β =0.6). The units for the position and velocity aremmandm/s, respectively . . . 49

3.10 WSS on the wall of aneurysm with stent 2 (β =0.6). . . 50

3.11 Velocity over the region of the aneurysm with stent 1 (β =0.8). The units for the position and velocity aremmandm/s, respectively. . . 51

3.12 Velocity over the region of the aneurysm with stent 2 (β =0.8). The units for the position and velocity aremmandm/s, respectively. . . 51

3.13 WSS on the wall of aneurysm (β =0.8) . . . 52

3.14 Velocity over the region of the aneurysm with the heterogeneous stent. The units for the position and velocity aremmandm/s, respectively . . . 53

3.15 Wss on aneurysm wall with an heterogeneous porosity . . . 53

3.16 Measurement points on the aneurysm wall . . . 55

3.17 Velocities at the measurement points . . . 55

3.18 Velocity over the region of the aneurysm of Womersley flow. The units for the position and velocity aremmandm/s, respectively . . . 56

3.19 WSS of the measurement points . . . 57

4.1 Screens studied in 3D . . . 63

4.2 Computational domain for the screen . . . 64

4.3 Drag coefficient for the normal direction . . . 64

4.4 Comparison between the deflection coefficient and the velocity reduction coefficient with the 2D equations . . . 65

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List of figures xix 4.5 The deflection coefficient and the velocity reduction coefficient after fitting

with Equation4.4and4.5 . . . 66 4.6 The vessel and aneurysm of the patients . . . 67 4.7 The flow diverter as deployed in each patient . . . 67 4.8 Illustration of the flow-rate curves for the three patients . . . 68 4.9 Measurement points in the three patient aneurysms: Point A(red), B(blue),

C(green), and D(yellow) . . . 69 4.10 The WSS at different resolutions for Patient 2 . . . 69 4.11 The WSS at different resolutions for Patient 3 . . . 70 4.12 The WSS at different resolutions for Patient 4 . . . 70 4.13 Porosity computation for the Patient 2 stent, and a scale factor of 3 between

the fully resolved and coarse simulations. . . 72 4.14 The illustration for the computation of hydraulic diameter for Raschi’s model 74 4.15 Iso velocity surface with back view (top) and side view (bottom). From left to

right, the results of the fully resolved stent, the SFDM, and the Raschi model are shown. . . 76 4.16 Position of the slices. Right: Slice 1. Left: Slice 2. . . 76 4.17 Velocity norm on two slices inside the aneurysm . . . 77 4.18 WSS in the measurement points. F in the legend means fine resolution, C

coarse resolution . . . 79 4.19 The WSS errors between the SFDM and the fully resolved simulations, at

the measurement points. The four groups 1-4 from left to right correspond to Points A-D . . . 79

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List of tables

2.1 The pressure drops and the average velocities in aneurysm of the three simulation cases . . . 33 3.1 Number of lattice nodes along the diameters of the artery and stent struts at

three different resolutions. . . 44 3.2 RMS and RMSD values of the velocity field for the three resolutions of the

fully resolved stent simulation. . . 45 3.3 RMS of the velocity field from the simulation at the three resolutions using our

stent model. . . 46 3.4 Comparison of the results obtained from the four simulation cases (β =0.6). 48 3.5 Quantitative comparison for stent 2 withβ =0.6 . . . 50 3.6 Results from the simulation using different models for stent 1 withβ =0.8. . 51 3.7 Results from the simulation using different models for stent 2 withβ =0.8. . 53 3.8 RMS velocities in aneurysm with heterogeneous stents . . . 54 3.9 RMS and RMSD of velocities for the Womersley simulation . . . 56 4.1 Characteristic parameters of the three patient arteries simulated in this article 67 4.2 The resolution for all simulations . . . 75 4.3 Characteristic parameter for the patients . . . 81

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Chapter 1 Introduction

1.1 The development of flow diverting stent

Intracranial aneurysms are pathological dilatations of cerebral arteries, leading to an increased risk of rupture with possibly lethal consequences. Before the therapy of neuroradiologic intervention was widely accepted, surgical clipping was the main treatment to prevent the serious complication of rebleeding for the intracranial aneurysms. The endovascular neurosurgery- interventional neuroradiology begins with the first catheterization of intracranial vessels by Luessenhop and Valasquez in 1964 [25,44]. They successfully insert a silastic microcatheter into brain arteries. In the mid-1960s, a new microcatheter called para-operational device(POD) was introduced by Frei, Yodh, Driller, Montgomeryet al[25]. They utilized magnetic field to pull and bend the micromagnet tipped microcatheter, which can inject embolic substances into the arteriovenous malformations to perform endovascular electrothrombosis. However, POD did not become popular due to the later improvement of endovascular microcatheters.

The subsequent development of the treatment of endovascular pathology for cerebral aneurysm can be classified into three stages: balloon era, coil era and flow diverter era.

1.1.1 Balloon era

In 1974, an English article from a Russian neurosurgeon named Fedor Serbinenko appearing inJournal of Neurosurgeryastounded the field of endovascular neurosurgery [61]. He treated more than 300 patients with detachable and non-detachable balloons [25] which were made in a small laboratory [71]. The idea is to fill the balloon with solidifying substance and navigate it into the aneurysm sac. In his 1974 article, Serbinenko suggested the treatment of cerebral aneurysm by employing two balloons, one distal and the other proximal to the orifice of the aneurysm [71]. The process is illustrated in Figure1.1. First, two balloon catheters are delivered

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Fig. 1.1 The process of the deployment of balloon [65]

into the carotid artery(Fig.1.1A). The non-detachable balloon is in front of the detachable one and would guide it into the aneurysm sac. After the detachable balloon is inflated with the silicone polymer, the non-detachable balloon presses it to be positioned in the sac and aids it to be detached from the catheter (Fig.1.1B). Finally, the aneurysm sac is deployed with the detachable balloon (Fig.1.1C).

Over the next 15 years, Serbinenko’s concept was widely applied in neuroendovascular centers [30,18]. With the later development of coils, it was gradually supplanted in the early 1990s [25].

1.1.2 Coil era

In 1988 and 1989, Hilalet alfirst introduced the pushable coil for the endosaccular treatment of aneurysms, but these coils were stiff [25]. The endovascular aneurysm treatment was revolu- tionized by Guglielmiet alwho proposed an electrolytically detachable platinum coils [27,26]

in 1991, as illustrated in Fig.1.2a. The coil which is soft, retrievable and detachable, received approval from US Food and Drug Administration in 1995. In subsequent years, numerous kinds of coils were developed with the improvement of softness, length, shape as well as other aspects.

The standard coil embolization technique is limited by its inability to occlude wide-neck aneurysms [7]. Moret et al proposed a balloon assisted, so called remodeling technique, consisting of a temporary balloon to avoid coil protrusion into the parent artery [50]. The non-detachable inflated balloon is used to block the aneurysm neck during coil placement [74], as shown in Fig.1.2b.

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1.1 The development of flow diverting stent 3

(a) Simple coiling (b) Balloon assisted coiling (c) Stent assisted coiling Fig. 1.2 Three types of coiling^a.

ahttp://www.neuroradiologist.com.au/services/intracranial-brain-aneurysm-sydney/

Another similar solution for the treatment of the wide-neck aneurysm is the stent assisted coiling, which became popularized in early 2000s [72]. The first self-expandable stent for intracranial was the Neuroform^TMdevice¹[70]. Stents are first delivered and positioned in the parent artery across the aneurysm neck. Then they act as a scaffold when coils are delivered through a microcatheter navigated into the aneurysm through the stent struts [25]. Fig.1.2c illustrates the supporting the stent assisted coiling .

1.1.3 Flow diverting stent era

Stent was first used as a scaffold to prevent the protrusion of coils into the parent vessel. With the development of technology, intracranial flow diverting stent can be used independently for the endoluminal parent vessel reconstruction, as seen from Fig1.3. It was found that the device deployed in cerebral aneurysm produces hemodynamic and biological effects [70]:

• Flow redirection: the stents could divert flow "away" from the aneurysm "back" into the parent vessel [20].

• Sac thrombosis: clinical observations of broad-based saccular intracranial aneurysms show spontaneous sac thrombosis after stent placement [70,68].

Theoretical in vitro studies suggested that stent porosity is the most important metrics [5], which is defined as thearea percentage of metal over the neck. Low porosity stents lead to a reduction up to 90% of the original flow inside the aneurysm sac [33,43].

1Boston Scientific, Fremont, CA, USA

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(a) Flow diverter (b) Stented aneurysm

Fig. 1.3 Flow diverter stented in aneurysm

Among the multiple choices, research and clinical studies have been focused on two mainly used flow diverting stent (FDS) used in medical center. Pipeline embolization device(PED)²is a self-expanding, flexible cylindrical mesh-like device [53,24]. It is interwoven by 48 struts with a diameter of 25µm. The diameter of the stent is between 2.5 and 5mmand the length is between 10 and 35mm. Silk³is another a self-expanding stent [53,24], which is also braided by 48 struts with a diameter of 35µm, but the material is nitinol. The diameters is 2−5mm and the length is 15−40mm.

1.2 The CFD simulation of stented aneurysm

1.2.1 Research status

Computational fluid dynamics (CFD) provides a great help for the research on the blood flow in aneurysm. It overcomes the limitation that experimental models are largely focused on idealized aneurysm geometries or surgically created aneurysms in animals [10,59,66,67].

CFD models are utilized to study all possible geometries [62] for different purposes. For instance, Lee [38] explored the optimization of the design of stents to determine the most effective arrangement of struts and the gaps between them. The two- and three-dimensional stents are simplified. Anzaiet al.[2,1] used the idealized models in the stent optimization.

Janiga [32] et al carried out automatic CFD-based FDS optimization for patient-specific intracranial aneurysms. They proposed a method which can virtually compress the FDS

2EV3-MTI, Irvine, CA

3Balt, Montmorency, France

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1.2 The CFD simulation of stented aneurysm 5 at different locations automatically. Maet al.[45] developed a virtual flow-diverting stent deployment method using the finite element analysis. They modeled the main steps involved in stent deployment in patient-specific aneurysm to obtain realistic final FD configuration. Kimet al.[33] analysed the effect of stent porosity and strut shape on saccular aneurysm with lattice Boltzmann method. Cebralet al. carried out a series of studies on the CFD simulation of the hemodynamics of cerebral aneurysm [12,14,39,4,11,13]. They proposed an image-based methodology for constructing patient-specific models of the cerebral circulation in [13], studied the hemodynamic characteristics by a methodology of image-based patient-specific analysis in [12], used adaptive embedded unstructured grids [4] and porous media model [55] in for stented aneurysms, and also worked on many other aspects. Most of the CFD simulations related above for the SFD are carried out with the finite element method. This is not an exhaustive review, many works on CFD simulation of stented aneurysm are not covered in this introduction.

1.2.2 The porous media model

Among most of the CFD studies mentioned above, FDS are fully resolved, which results in long computational times due to the large difference in scale between the stent strut on one side, and the blood vessels and the aneurysm on the other side. As a response to this problem, Augsburger [6] proposed an alternative strategy which models a FDS as a statistically isotropic porous media. In the simulation, the porous medium which is assumed to be homogeneous is modeled by the addition of a momentum source term to the standard fluid flow equation.

This model is however relatively simplistic, as it only considers one type of design. In this way, it fails to account for the general important geometrical properties, such as the porosity, the diameter of the stent struts, or to model the local deformations of the stent caused by the deployment procedure. Zhang et al [73] extend the porous media model by adding a centrifugal force term, but they use the same model parameters as [6], and are subject to the same limitations as the latter. Ohtaet al[51] also investigated the tangential drag force of the stent in Augsburger’s way. In their work, the porosity, as the most important factor was not taken into consideration. The aneurysm is also highly idealized in [51] . As a development of the porous-media approach of [6], Raschi [55] proposed and evaluates a method which considers the geometrical parameters of the stent, and test the proposed model with the help of several patient-specific arteries with aneurysms. However, the Raschi model is still statistically isotropic and assumes that the porosity is a uniform, global parameter.

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1.3 Introduction of Lattice Boltzmann method

In the past two decades, the Lattice Boltzmann method(LBM) has gained popularity for the simulation of fluid in both engineering and academic fields. LBM is based on microscopic models and mesoscopic kinetic equations. Besides, it can also be viewed as finite difference method for solving the Boltzmann transport equation [8]. In LBM, the fluid is taken as fictitious particles, which stream along certain directions and collide at lattice sites. The collision and streaming processes are local. Hence, it can be programmed naturally for parallel processing machines [49]. Another advantage of LBM is in dealing with complex boundaries. Through many years of development, LBM has been used in various fields, such as multi-phase flow, chemical reaction, magnetic fluid, biomedical dynamics, porous media as well as other aspects.

The basic concepts and models of LBM will be introduced in the following subsections.

1.3.1 Basic principal and model

In the Boltzmann equation, the fluids can be imagined as consisting a large number of particles.

It describes the time evolution of the particle distribution function as in Eq.1.1[8].

∂f(x,t)

∂t +u·∇f(x,t) =Ω (1.1)

where f(x,t)is the particle distribution function,uis the particle velocity, andΩis the collision operator. Lattice Boltzmann method, inheriting the idea of Lattice Gas Automata, confines the particles on the nodes of lattice and only selected directions of motion are possible.

The collision operator was originally a complex matrix. A particularly simple version of collision operator based on Bhatnagar-Gross-Krook(BGK) collision model was independently introduced by several authors [15,34,54]:

f_i(x+c_i∆t,t+∆t)−f_i(x,t) =−1

τ(f_i(x,t)−f_i^eq(x,t)) (1.2) where f_i^eq(x,t)is the equilibrium distribution function.c_iis the particle velocity along the i-th direction andτ is the relaxation time toward local equilibrium. Eq.1.2is updated in the following two steps [52]:

Collision step:

f_i^′(x,t) = f_i(x,t)−1

τ(f_i(x,t)−f_i^eq(x,t)) (1.3) Streaming step:

f_i(x+c_i∆t,t+∆t) = f_i^′(x,t) (1.4)

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1.3 Introduction of Lattice Boltzmann method 7 where f_iand f_i^′denote the pre- and post-collision states of the distribution function. f_i^eqis defined as

f_i^eq(x,t) =ω_iρ

1+c_i·u

c²_s +(c_i·u)² 2c⁴_s − u²

2c²_s

(1.5) whereuis the velocity of the fluid. ω_iis the weight, andc_s=√

RT is related to the speed of sound. The streaming process is illustrated in Fig.1.4.

Fig. 1.4 Illustration of the streaming process [8]

For the two dimensional model, the number of the directions ofc_i can be 5,7 or 9. The most common one is the D2Q9 model, meaning the two-dimensional nine-velocity model. The parameters of D2Q9 are listed as below.

c_i=











(0,0) i=0

(1,0),(0,1),(−1,0),(0,−1) i=1,2,3,4 (1,1),(−1,1),(−1,−1),(1,−1) i=5,6,7,8

(1.6)

ω_i=











4/9 i=0 1/9 i=1,2,3,4 1/36 i=5,6,7,8

(1.7)

c_s= ^√^c

3 and c= ^∆x_∆t is the lattice speed. The relaxation time is related to the kinematic viscosityν by

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ν = 2τ−1 6

(∆x)²

∆t (1.8)

The macroscopic quantities are obtained by the distribution functions and particle velocities as follows:

ρ(x,t) =

8 i=0

∑

f_i(x,t) (1.9)

ρu(x,t) =

8 i=0

∑

f_ic_i (1.10)

p=ρc²_s (1.11)

Through Chapman-Enskog expansion, Eq.1.2can be derived to Navier-Stokes equation.

Three dimensional models (see for instance [48]) is not introduced here for the sake of simplicity.

1.3.2 Boundary conditions

Boundary conditions (BCs) are critical to the stability and the accuracy of any numerical solution. LBM has particular advantage in handling complex geometries. Bounce-back boundary condition is one of the most widely used BCs. It is quite simple and preserves a decent numerical accuracy [8].

Bounce-back model is typically used for the no-slip BCs. The basic idea is that during the streaming step, if f_iencounters a boundary node, it would be reflected and go back to the node where it left. The value of f_iremains same, but its velocity is reversed. The illustration is shown in Fig.1.5.

Collision Streaming

f3

f6

f7

f1 '

f8'

f5'

f1

f8

f5

Wall Wall Wall

Fig. 1.5 Illustration of on-grid bounce-back BC

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1.4 Objectives 9 In this thesis, we use the bounce-back model for the fully resolved stent. Many other boundary conditions are seen in [76,17,28,75].

1.3.3 Palabos

In spite of the particular advantages of LBM, the complicated mathematic models of LBM still cause many difficulty its application. Some CFD-solver based on LBM are developed to simu- late various kinds of problems in engineering and academy. Palabos(http://www.palabos.org) provides a good choice for this purpose. It integrates the most widely used fluid models, boundary conditions, pre- and post-processing as well as other necessary ingredients for the CFD simulation. Programs written with Palabos can be automatically parallelized. Due to the many facilities that Palabos brings, the researchers can concentrate on actual physical problems instead of staying stuck in tedious software development. All the simulations in this thesis are carried out with Palabos.

1.4 Objectives

The overall aim of the present thesis is to develop, explore and exploit a hydrodynamic model which can represent a FDS in aneurysm with fully consideration of its geometrical parameters, like local porosity, strut diameter, deployment effect and so on in a coarse scale. The model needs to be studied and tested from the hydrodynamical prototypes to the ultimate application on patient-specific aneurysms. Specific goals and research questions are:

• Develop a FDS model prototype to replace the fully resolved stent in cerebral aneurysm numerical simulation

From the aspect of the geometrical shape, though porous media has a same porous feature as FDS, it still shows up obvious difference in the thickness. Is there any other porous objects which are more close to the FDS in geometry and whose hydrodynamic characteristics have been well studied?

The expected model should be capable of dealing with various deployed FDSs with specific geometrical parameters.

• Test the FDS model with various conditions

Given that FDS is deployed in various kinds of aneurysms and arteries which may cause the deformation of the FDS, and great different velocity distribution of the blood, the proposed model should be tested and validated with similar environments that FDSs are implanted in.

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• Test the FDS model with specific-patient stented aneurysms

Patient-specific stented aneurysms are always more complicated than the idealized ones due to the geometries of aneurysms of the patient and the actual braided FDS. Therefore, how to implement the model on patient-specific aneurysms? Compared with the fully resolved flow diverters, how does the model performs for the patient-specific aneurysms?

How much computational time can the model save?

To answer the first question, we review the equations for the pressure drop of porous media, including Darcy’s law and Forchheimer’s equation. These equations are then compared with the drag forces of 2D stent-like object obtained by numerical experiments, but the result show that all the porous media equations highly overestimate the pressure drop of the stent. Through many investigations, we find that the hydrodynamic characteristics of the device known as screen is closer to that of the FDS. Therefore, the screen becomes the prototype for the research of the model for FDS in intracranial aneurysms. The result of this part was not included in the incorporated paper, so we can’t show it in the thesis.

Based on this, we develop an effective model which can properly represent, at continuum scale, a stent in an aneurysm. Compared with the porous medium method adopted by previous authors in the literature, we propose the novel idea of applying screen models to the field of stent modeling. Besides, we also take into consideration the deflection effect caused by the stent. This model is explained in detail in Chapter 2.

Then, the performance of the screen based flow diverter model (SFDM) is tested in various conditions such as different stent placements inside an artery, varying homogeneous porosities, and pulsatile inlet velocities in Chapter 3. All the studies above are carried out in two dimensions.

Finally, Chapter 4 extends the framework of screen-based flow diverter model (SFDM) to 3D flows and validates it using actual medical flow diverters in patient specific aneurysms.

The numerical tests show that the 3D SFDM can reproduce the results of direct numerical simulation both qualitatively and quantitatively with high precision, and are capable of reducing the simulation time by an order of magnitude or more.

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Chapter 2 Model for pressure drop and flow

deflection in the numerical simulation of stents in aneurysms

Reprinted from "Sha Li, Bastien Chopard, Jonas Latt, (2017)Int. J. Numer.Meth.Biomed.Engng 34(3):e2949"

Abstract

The numerical simulation of flow diverters like stents contributes to the development and improvement of endovascular stenting procedures, leading ultimately to an improved treatment of intracranial aneurysms. Due to the scale difference between the struts of flow diverters and the full artery, it is common to avoid fully resolved simulations at the level of the stent porosity. Instead, the effect of stents on the flow is represented by a heuristic continuum model.

However, the commonly used porous media models describe the properties of flow diverters only partially, because they do not explicitly account for the deflection of the flow direction by the stent. We show that this deficiency can be circumvented by adopting the theoretical framework of screen models. The article first reviews existing screen models. It then proposes an explicit formula for the drag and the deflection coefficient, as predicted by each model, for both perpendicular and inclined angles. The results of 2D numerical simulations are used to formulate a generalization of these formulas, to achieve best results in the case of stent modeling. The obtained model is then validated, again through 2D numerical simulation.

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2.1 Introduction

Intracranial aneurysm is a cerebrovascular disorder, consisting of a local dilation of the blood vessel, leading to an increased risk of rupture with possibly lethal consequences [3, 6]. A possible treatment of an unruptured intracranial aneurysm consists of the introduction of a flow diverter, often a stent, into the blood vessel. Stents are flexible, self-expanding porous tubular meshes. In order to improve the design of stents, and in this way improve the success rate of stenting procedures, numerical simulation has proven to be an invaluable tool [3,14]. It allows fundamental investigation of hydrodynamic properties of stents, but also opens the window to patient-specific, personalized simulations. One of the possibilities is to study the alterations to the flow pattern before and after the placement of a stent, and then use this numerical data to determine the chances of success of the intervention [1].

Given the small-scale structure of flow diverters, it is however often prohibitively expensive to fully resolve the flow structure around the coils of a stent [14]. Indeed, the diameters of the struts can differ from the diameter of the artery by more than three orders of magnitude, and it can be very difficult to handle both scales with a unique fluid mesh. This problem is particularly severe for patient-specific simulations, in which the numerical results are expected in a short time frame. As an alternative, numerical simulations are often carried out on a coarse mesh, which cannot capture the coil structure of a stent. Instead, the effect of the stent on the flow is represented through a continuum model.

An approach advocated for example by [6,55] proposes to model the stent like a porous media, using a term similar to Darcy’s law. It is however questionable if the assumptions underlying porous media models remain valid in the case of stents. Such models assume for example that the underlying porous object has a finite thickness, while flow diverters in reality have very thin surfaces. As a consequence, porous media models do not explicitly account for the deflection of the flow through a thin surface.

In the present article, we take a more fundamental standpoint than [6,55] and propose a theoretical framework to formulate, validate, and select continuum stent models. This framework is based on so-called screen models which, as opposed to porous media models, are explicitly built to reflect the physics of thin porous structures. We validate these models by running fully resolved 2D simulations across a stent. We show that while these models provide good results, and in particular are able to predict quite accurately both the pressure drop and the flow deflection across a stent, they also suffer from a major drawback, namely the assumption of a uniform, constant upstream flow in front of the screen. This assumption is not valid for stents deployed in aneurysms, in which case the flow pattern close to the stent surface can be quite complicated.

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2.2 Hydrodynamic models of flow diverters 13 We therefore propose a correction to classical screen models in the form of a velocity reduction coefficient, which relates the presumed asymptotic upstream flow velocity to a local velocity, measured in the vicinity of different points on the stent surface. This model is incorporated into screen models to obtain a globally consistent model for flow diverters.

In a final part, we validate this flow diverter model by comparing fully resolved fluid flow simulations around a flow diverter to coarse-grained simulations using our flow diverter model.

All simulations are carried out in 2D for the sake of simplicity, given that the main purpose of the article is to propose a fundamental framework for a new category of stent models. The authors intend to run 3D simulation and present them in a more application-oriented publication as part of their future research.

2.2 Hydrodynamic models of flow diverters

2.2.1 General principles

Hydrodynamic flow diverter models represent the impact of the flow diverter on the flow pattern in an averaged manner, when the computational mesh is too coarse to fully resolve the structure of the flow diverter. State of the art models represent the impact of a flow diverter in terms of a pressure drop across its surface, and through an additional sink term in the momentum balance equation. Model constants for these two terms are adjusted empirically for specific flow diverters [6,55]. While this approach quite successfully reproduces some aspects of the flow structure, it fails to properly account for the flow deflection through the diverter, which can be particularly important in cases when the flow direction does not align with the normal of the diverter surface.

In this paper, we make the hypothesis that a flow diverter can be approximated as a screen, and we test different screen models for their validity, including their ability to predict flow deflection. The existing literature [37,64] shows that screens are generally used as a kind of fluid distributor to achieve a more uniform velocity field, reduce the intensity of turbulence, or change the flow direction. The relevant hydrodynamic characteristics of screens are the pressure drop and the deviation of the flow direction. The pressure drop caused by the resistance of the screen depends on the properties of both screen and fluid, while the flow deviation depends on the inclination of screen. It is well acknowledged that screens refract the incident flow towards the locally normal direction to the screen surface [47,19].

Considering an incident flow through a screen at an arbitrary angle as shown in Fig.2.1, it is necessary to define three velocities:

• U₁is the velocity of the incident flow far from the screen.

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• U is the velocity at which the flow passes through the screen.

• U₂is the downstream velocity far from the screen.

Here, θ₁,θ, andθ₂ are the angles between the direction of these velocities and the normal direction to the screen. Since the mass flux of fluid through the screen is conserved [21], the normal velocity components are consistent, i.e.u₁=u=u₂. In the tangential direction, the screen refracts the incident flow toward the normal direction of the screen, leading to a corresponding decrease of the tangential velocity components, i.e. (v₁≥v≥v₂).

The resistance generated by the screen gives rise to a pressure difference between the two sides of the screen, and the pressure drop can be considered as a reactive force which drives the fluid through the screen. In hydrodynamics, the pressure drop is expressed by a non-dimensional drag coefficient which is generally defined as:

k_θ₁ = ∆P

1

2ρU₁², (2.1)

where∆Pis the pressure drop andρ the fluid density.

The drag coefficientk_θ₁ is related to the screen’s porosity and geometrical parameters, as well as the Reynolds number

Re= ρU_refL_ref

µ . (2.2)

Here,µ is the dynamic fluid viscosity,U_refis a characteristic velocity, which in most screen models is taken to be the norm of the upstream velocityU₁, andL_refis a characteristic length.

In the models of weaving screens,L_ref is often taken to be equal to the diameters of the screen wires, in which case the corresponding Reynolds number is labeled as Re_d.

Fig. 2.1 Flow through a screen at an incident angle

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2.2 Hydrodynamic models of flow diverters 15 Concerning the resistance in tangential directions, some authors define an analogous coefficient, named deflection force coefficient [37]:

F_θ₁= τ

1

2ρU₁², (2.3)

whereτ is the tangential stress which causes a change of velocity in the tangential plane of the screen. When the upstream and downstream flows are far away from the screen, Taylor and Bachelor [64] propose that the tangential force could be obtained by measuring the change of direction of the stream passing through the screen, and the deflection force coefficient is expressed as:

F_θ₁ =2 cosθ₁secθ₂sin(θ₁−θ₂). (2.4) In order to express the deflection of velocity in a more direct way, another coefficient named deflection coefficientB[37] is proposed as:

B=1−v₂/v₁. (2.5)

Combining Eqs. (2.4) and (2.5) yields a relationship between these two coefficients:

F_θ₁ =Bsin 2θ1 (2.6)

From Eqs. (2.3) and (2.6), the tangential stress can be expressed directly from the deflection coefficientBand upstream velocities:

τ=Bu₁v₁. (2.7)

2.2.2 The drag coefficient

The computation of the drag coefficient for a screen that is inclined with respect of the incoming flow can be split into two steps. Firstly, the drag coefficient of a screen which is normal to the flow,k₀, withθ₁=0, is computed. In this case, the porosity and Reynolds number are the decisive factors. Secondly, the relationship betweenk₀andk_θ₁, or in other words, the impact of the incident angle on the drag coefficient, is determined.

Kolodziej et al. [35] describe in detail the major theoretical approaches to model the pressure drop across a screen, and they summarize the most important models fork₀.

Wieghardt (as cited in [35]) provides a correlation based on experimental data from the literature:

k₀=61−β

β² (Re_d/β)^−1/3,Re_d=ρU₁d

µ , (2.8)

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whered is the diameter of wires in a weaving screen andβ is the porosity of the screen, defined as:

β =A_open/A_total. (2.9)

Here,A_open is the orthogonally projected open area of the screen andA_total is the total cross- sectional area.

Ehrhardt presents a correlation based on his measurements of oil, water and air flow through individual screens over a range 0.3≤Re_d≤300 (as cited in [35]):

k₀=

0.72+ 49 (Re_d/β)

1−β

β² . (2.10)

Wakeland and Keolian [69] propose their correlation by measuring the drag coefficient of more than 20 kinds of screens with different porosities and wire diameters for both low and intermediate Reynolds numbers.

k₀= 1−β β²

17.0

Re_d +0.55

(2.11) Brundrett [9] proposes a correlation for screens with incident angles:

k_θ₁=cos²θ×1−β² β² ×

σM

σ_KE × 7.125

Re_dcosθ + 0.75

log(Re_dcosθ+1.25)+0.055 log(Re_dcosθ)

(2.12) whereσKE andσM are the correction factors for momentum and kinetic energy. In a uniform flow, the equalityσKE =σM=1.0 holds.

Therefore, according to Brundrett’s model,k₀should take the value k₀=1−β²

β² × σ_M

σ_KE×7.125

Re_d + 0.75

log(Re_d+1.25)+0.055 log(Re_d)

(2.13) From the equations listed above, we can see that the equation for k₀ consists of three elements.The first is proportional to 1/Re_d, and the second is a function of porosity, like (1−β)/β²in Eq.2.11, and the third element is some constants. All the models have a positive correlation with the term of 1/Re_d, while their differences depend on the function of porosity and constant parameters.

Concerning the relationship between k₀ and k_θ₁, Schubauer and Spangenberg [60] plot k_θ₁/cos²θ1against Re_dcosθ1, over a range of angle 0^◦≤θ ≤45^◦. They show that there exists

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2.2 Hydrodynamic models of flow diverters 17 a unique relationship between the two items for all screens and Reynolds numbers. Hence,

k_θ₁

cos²θ₁ = f(Re_dcosθ₁). (2.14)

In the caseθ1=0^◦, we find

k₀= f(Re_d), (2.15)

according to the definition ofk_θ₁ in Eq. 2.1, k_θ₁

cos²θ₁ = ∆P

1

2ρU₁²cos²θ1

= ∆P

1

2ρu². (2.16)

Combining Eq.2.14, Eq.2.15, and Eq.2.16yields k_θ₁

cos²θ₁ =k₀(Re_dcosθ₁). (2.17) To explain more clearly the model of Schubauer and Spangenberg, in which the pressure drop is independent of the incident angleθ₁, we define a new form of the drag coefficient and Reynolds number, which do not explicitly depend onθ₁:

kⁿ= k_θ₁ cos²θ1

= ∆P

1 2ρu², Reⁿ_d=Re_dcosθ₁=ρud

µ ,

(2.18)

wherenstands for the normal direction. Eq.2.17then becomes:

kⁿ=k₀(Reⁿ_d). (2.19)

Eqs2.18and2.19indicate that in case of a deviated flow, the pressure drop through a given screen only depends on the normal component of the velocity. This relation is valid for all screens, independently of the porosity and the incident angle. This equation is widely accepted for the computation of the drag coefficient at an incident angle. Among others, the derivation of Brundrett’s model, Eq.2.13, is based on this relationship.

As an alternative to Schubauer and Spangenberg, Reynolds [56] proposes a theoretical model fork_θ₁ by introducing a local incident angleθ:

kⁿ=k₀(Reⁿ_d)

1+secθ 2

. (2.20)

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In the method of Reynolds, the pressure drop is determined not only by the normal velocity component, but also by the tangential velocity component, which is reflected by the local incident angleθ.

The models cited above can produce significantly different results, depending on the screen type and the range of Reynolds numbers. We therefore executed numerical tests in Section2.4.1 for the various relations, to assess their validity.

2.2.3 The deflection coefficient

Various theoretical and semi-empirical equations are proposed to describe the relationship between deflection coefficient and drag coefficient. In early studies, a relationship betweenk_θ₁ and the deflection force coefficientF_θ₁ was introduced for small incident angles, like in the work of Schubauer and Spangenberg [60]:

B≈F_θ₁

θ₁ = 2k_θ₁

8+k_θ₁, (2.21)

or in the model of Taylor and Batchelor [64]:

B≈F_θ₁

θ₁ =1− 1.1

1+k_θ₁, (2.22)

in whichkⁿis directly related withB. Elder [21] assumes that the flow past a screen is equivalent to a uniform flow past a row of equally spaced vortices of circulation. A correlation is deduced from this assumption:

B=1− 1 p1+√

kⁿ

. (2.23)

Gibblings [23] proposes yet another equation, by assuming that the flow direction takes a value halfway between the upstream and downstream directions:

B=1+kⁿ 4 −

"

kⁿ 4

2

+1

#1/2

. (2.24)

As pointed by Laws and Livesey in their review [37], the evaluation of the deflection coefficient presents considerable difficulties, because the screen is rippled. As a result, no entirely satisfactory form of this relationship has ever been established. The literature on this issue is scarce and has been produced more than thirty years ago. It is therefore necessary to study these models, and individually compare them with experimental data to investigate the relationship between the resistance and the deflection coefficient.

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2.3 Numerical approach 19

2.3 Numerical approach

2.3.1 Direct numerical simulation

There exist various types of screens, such as perforated plates, arrays of cylinders and wire- gauzes. In the present work, a stent is represented by an array of 2D cylinders (the stent struts), and formally modelled as a screen. The drag coefficient and the deflection coefficient of the stent are measured through numerical simulation and compared with the predictions of the screen models in Section2.2. Direct numerical simulation in 2D is launched with the help of the open-source fluid solver Palabos (http://www.palabos.org) based on lattice Boltzmann method (see for instance Chen and Doolen [16]).

Fig. 2.2 Periodic boundary condition

As shown in Fig.2.1, the fluid is set up to flow through the stent, respectively screen, at an incident angle. It is assumed that the screen and flow field extend infinitely in the direction parallel to the stent, an effect which is achieved through periodic boundary condition on the top and bottom boundaries, as shown in Fig.2.2. The fluid enters the numerical domain from the left at a uniform velocity and with an approaching angle which varies from 0^◦to 85^◦. A pressure boundary condition is imposed at the outlet, on the right. A no-slip boundary condition is applied on the surface of the struts. Both the inlet and outlet boundaries are far away from the stent, to guarantee that the upstream and downstream velocities have constant values at a sufficient distance from the stent. The density and viscosity of the fluid are similar to the ones of human blood:ρ =1000kg/m³andµ =3.3×10⁻³Pa·s. The diametersd of the cylinders are taken to be 20µm,30µm, and 40µm, corresponding respectively to a porosity of 0.8, 0.7, and 0.6. The distancelbetween is constant. The Reynolds number Re_d varies from 0.1 to 40,

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and is adjusted by changing the flow velocity at the inlet. The fluid mesh is uniform, and is resolved by 1000×200 nodes, with a discrete space stepδxof 0.75µm.

The program measures directly both the normal and tangential components of the drag force on the cylinders, and the pressure drop∆Pand tangential component stressτ are derived from them according to the following relationship:

∆P= F_n S , τ= F_t

S,

(2.25)

whereF_nandF_tare the normal and tangential components of the total drag force exerted on the struts, andSrepresents the cross-sectional area of the stent. Since the upstream flow is uniform and the top and bottom boundaries are periodic, the value of∆Pandτ is the same for all the struts.

2.3.2 Coarse-grained force model

Force model

In [6,55], a coarse-grained model for the stents is proposed by means of a corrective source term to the momentum balance equations. In the present work, we follow the convention of screen models and use a body force term, which is correlated in a more sophisticated way to the local flow quantities, to reproduce both the expected pressure drop and velocity deviation.

In fully resolved simulations, the body force is applied to all mesh nodes that are located inside stent struts. It is directly related to the upstream velocity, and to the parameterskⁿandB predicted by the validated screen models, as follows:

f_n= 1

∆L 1 2kⁿρu²₁, f_t = 1

∆L 1

2F_θ₁ρU₁²= 1

∆LBu₁v₁,

(2.26)

where f_nand f_t are the normal and tangential components of the body force, and∆Lis the thickness of the volume over which the body force is applied. In the coarse grained case, we apply the force on one layer of nodes.

Improved force model

Screen models, as they are presented in the literature, are only of limited use for our purpose, because they assume that the upstream velocity is known and is uniforma, as shown in Fig.2.1.

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2.4 Results 21 For a stent placed in an artery, these assumptions are likely to be too restrictive. Since both the blood vessel and the stent are most often bent and follow a complex shape, the upstream velocity does not converge to a well-defined, asymptotic value. Furthermore, in an artery, just like in any other interior flow, the velocity exhibits a non-constant profile, varying from a zero velocity close to the walls to maximal velocity near the center.

It is therefore more adequate to relate the body force to a local value of the velocityU(x,y), which is measured in the vicinity of the stent and can have different values along the surface of the stent, instead of a constant valueU₁. We therefore propose a novel phenomenological law in Section2.4.3which relatesU(x,y)toU₁according to our numerical observations.

2.4 Results

In this section, we investigate the drag coefficientsk₀andkⁿ, for angles in the range 0^◦≤θ₁<

90^◦, as well as the deflection coefficientB. The boundary conditions are periodic, to mimic a large system. The coefficientsu₁and v₁, as defined in Section2.3.1, are first measured in a fully resolved simulation of a stent with 2D cylindrical struts, and then compared to the various screen models.

2.4.1 Drag coefficient

Drag coefficient at normal incidencek₀

The simulated drag coefficient at normal incidencek₀ is compared with the following four empirical correlations listed in Section 2.2.2: the model by Wieghardt (Eq. 2.8), as cited in [35], the model by Ehrhardt (Eq.2.10), as cited in [35], the model by Wakeland and Keolian (Eq.2.11) proposed in [69], and the model by Brundrett (Eq.2.13) proposed in [9].

Fig.2.3to2.5compare the numerical data with model predictions at porosities of 0.6,0.7 and 0.8. Fig. 2.3 shows that the equation of Ehrhardt provides a best match for the drag coefficient at a screen porosity of 0.6, but Fig.2.4and Fig.2.5show that it does not work well at porosites of 0.7 and 0.8. For porosities of 0.7 and 0.8, it can be seen that the model of Wakeland et al and the model of Brundrett are more consistent with the simulation results. However, while these two models provide good preditions fork₀at low Reynolds number (Re_d<5), they fail at higher Reynolds numbers (5<Re_d<40). In the latter regime, Wieghardt’s model agrees much better with the numerical data. All in all, none of the models, taken by itself, provides satisfying results for the full range of Reynolds numbers and porosities of relevance to stents in aneurysms. But we notice that while the curve of Wieghardt’s model does not exhibit the expected slope, the slope of the three other model curves is in accordance with the simulation

Continuum model for flow diverting stents of intracranial aneurysms

Thesis

Reference

Continuum model for flow diverting stents of intracranial aneurysms

Continuum Model for Flow Diverting Stents of Intracranial Aneurysms

THÈSE

Acknowledgements

Abstract

Résumé

Statement

Table of contents

List of figures

List of tables

Chapter 1 Introduction

1.1 The development of flow diverting stent

1.1.1 Balloon era

1.1.2 Coil era

1.1.3 Flow diverting stent era

1.2 The CFD simulation of stented aneurysm

1.2.1 Research status

1.2.2 The porous media model

1.3 Introduction of Lattice Boltzmann method

1.3.1 Basic principal and model

∑

∑

1.3.2 Boundary conditions

1.3.3 Palabos

1.4 Objectives

Chapter 2

Model for pressure drop and flow

deflection in the numerical simulation of stents in aneurysms

Abstract

2.1 Introduction

2.2 Hydrodynamic models of flow diverters

2.2.1 General principles

2.2.2 The drag coefficient

2.2.3 The deflection coefficient

2.3 Numerical approach

2.3.1 Direct numerical simulation

2.3.2 Coarse-grained force model

2.4 Results

2.4.1 Drag coefficient