São Paulo State University School of Natural Sciences and Engineering Gabriel Bertacco dos Santos Hemodynamic changes in intracranial aneurysms due to stent-induced vascular remodeling Ilha Solteira, Brazil 2018 MECHANICAL ENGINEERING GRADUATE PROGRAM Gabriel Bertacco dos Santos Hemodynamic changes in intracranial aneurysms due to stent-induced vascular remodeling Thesis submitted to the School of Natural Sci- ences and Engineering of the São Paulo State University (UNESP) in partial fulfillment of the requirements for the degree of Master of Sciences in Mechanical Engineering. Field of Thermal Sciences. Prof. Dr. José Luiz Gasche Supervisor Prof. Dr. Julio Militzer Co-supervisor Ilha Solteira, Brazil 2018 dos Santos Hemodynamic changes in intracranial aneurysms due to stent-induced vascular remodelingIlha Solteira2018 82 Sim Dissertação (mestrado)Engenharia MecânicaCiências TérmicasSim . . FICHA CATALOGRÁFICA Desenvolvido pelo Serviço Técnico de Biblioteca e Documentação Santos, Gabriel Bertacco dos. Hemodynamic changes in intracranial aneurysms due to stent-induced vascular remodeling / Gabriel Bertacco dos Santos. -- Ilha Solteira: [s.n.], 2018 82 f. : il. Dissertação (mestrado) - Universidade Estadual Paulista. Faculdade de Engenharia de Ilha Solteira. Área de conhecimento: Ciências Térmicas, 2018 Orientador: José Luiz Gasche Coorientador: Julio Militzer Inclui bibliografa 1. Aneurismas intracranianos. 2. Stents. 3. Remodelagem arterial. 4. Mecânica dos fuidos computacional. 5. Hemodinâmica. S237h Dedicated to my parents: Milton and Solange. ACKNOWLEDGEMENTS I would like to express the sincerest gratitude to my supervisors Dr. José Luiz Gasche—São Paulo State University (UNESP), Ilha Solteira, Brazil—and Dr. Julio Militzer—Dalhousie University, Halifax, Canada—for all their help, support, and guidance during the execution of this research. To Dr. Carlos Eduardo Baccin from the Hospital Israelita Albert Einstein, São Paulo, Brazil, for providing the images of the aneurysms and for all support, advice, and clarifications regarding the medical aspects of this research. This work was conducted with a scholarship funded by CAPES—Brazilian Federal Agency for Support and Evaluation of Graduate Education within the Ministry of Education of Brazil—which is gratefully acknowledged. I would also like to thank the financial support during my exchange period at Dalhousie University, Halifax, Nova Scotia, Canada, provided by the ELAP—Emerging Leaders in the Americas Program—with support from the Government of Canada. This research was partly supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP) and ACENET (Dalhousie University), which are gratefully acknowledged. A very especial gratitude goes out to all friends and colleagues at the Thermal and Fluid Sciences Building at São Paulo State University (UNESP) in Ilha Solteira, Brazil. And also to all friends and colleagues at Dalhousie University in Halifax, Canada for the uncountable laughter and moments of joy during the freezing winter. And finally, my deepest and sincerest gratitude to both my family and my beloved Daniela for providing unconditional support and motivation since day one. ABSTRACT Stents were first designed to act as mechanical barriers, preventing coil herniation into the parent artery. The current generation of self-expanding intracranial stents has recently been shown to change the local vascular geometry, a phenomenon with unclear hemodynamic effects. We carried out numerical simulations to assess the role of stent-induced vascular remodeling in modifying intraaneurysmal hemodynamics. Simulations were performed using the open-source software OpenFOAM. Blood was assumed to behave as an incompressible Newtonian fluid; vessel walls were assumed to be rigid. Wall shear stress, WSS, and oscillatory shear index, OSI, were evaluated to quantify the hemodynamic changes in the aneurysm sac. Four pre- and post-stent patient-specific geometries of intracranial bifurcation aneurysm were used. In one aneurysm at the anterior communicating artery (ACoA) bifurcation, a single stent was deployed, resulting in straightening of the host vessels. After stenting, WSS and OSI increased by approximately 60% and 25%, respectively. In two aneurysms at middle cerebral artery (MCA) bifurcations, two stents in a “Y” configuration were deployed, resulting in straightening of both daughter arteries. The maximum WSS on the aneurysm surface increased by approximately 5% in one case and 22% in the other. In another aneurysm at a bifurcation of the MCA, a single stent was deployed, resulting in straightening of the host vessels. After stenting, WSS and OSI reduced by approximately 13% and 50%, respectively. The results show that hemodynamic changes do occur in intracranial bifurcation aneurysms due to stent-induced vascular remodeling. Contrary to previous suspicions, the effects of stent-induced vascular remodeling in modifying the intraaneurysmal hemodynamics are variable and not always beneficial. In conclusion, stent-assisted treatments of bifurcation aneurysm such as two-step Y-stent coiling may benefit from the novel vascular geometry by placing a single stent in an earlier step. However, a two-step procedure would not be recommended for aneurysms at the ACoA bifurcation, where straightening of the host vessels may cause an adverse hemodynamic environment and increase the risk of rupture. Keywords: Intracranial aneurysms. Stents. Vascular remodeling. Computational fluid dynamics. Hemodynamics. RESUMO Originalmente, stents foram projetados para agir como barreiras mecânicas, impedindo a herniação de coils para a artéria-mãe. Recentemente, estudos mostraram que a atual geração de stents intracranianos auto-expansíveis altera a geometria local das artérias: um fenômeno com efeitos hemodinâmicos em parte incompreendidos. Nós realizamos simulações numéricas para avaliar a influência da remodelagem arterial induzida por stent sobre a hemodinâmica em aneurismas intracranianos. As simulações foram realiza- das utilizando o software open-source OpenFOAM. O sangue foi modelado como fluido Newtoniano incompressível e as paredes arteriais foram consideradas rígidas. Para quan- tificar as alterações hemodinâmicas, avaliamos os parâmetros wall shear stress, WSS, e oscillatory shear index, OSI. Quatro geometrias reais de aneurismas intracranianos em bifurcações foram utilizadas. Em um aneurisma na bifurcação da artéria comunicante anterior (ACoA), um stent foi implantando, levando ao endireitamento das artérias que o receberam. Após o procedimento, os níveis de WSS e OSI aumentaram aproximadamente 60% e 25%, respectivamente. Em dois aneurismas em bifurcações da artéria cerebral média (MCA), dois stents foram implantados em uma configuração em “Y”, resultando em um endireitamento de ambas as artérias-filhas. O WSS máximo na superfície do aneurisma aumentou aproximadamente 5% em um dos casos e 22% no outro. Em outro aneurisma em uma bifurcação da MCA, um stent foi implantado, resultando no endireitamento das artérias que o receberam. Após o procedimento, valores de WSS e OSI sofreram uma redução de aproximadamente 13% e 50%, respectivamente. Os resultados mostram que alterações hemodinâmicas de fato ocorrem em aneurismas intracranianos em bifurcações devido à remodelagem arterial induzida por stents. Em contraste com suspeitas anteri- ores, os efeitos da remodelagem arterial induzida por stents sobre a hemodinâmica de aneurisma apresentam um comportamento variável e nem sempre benéfico. Concluímos, portanto, que a remodelagem arterial induzida por stents pode beneficiar tratamentos de aneurismas intracranianos, como o procedimento em dois tempos de “Y-stent coiling”, através do implante de um único stent em uma etapa prévia. No entanto, procedimentos em dois tempos não seriam recomendado para aneurismas na bifurcação da ACoA, onde o endireitamento das artérias pode causar uma resposta hemodinâmica adversa e aumentar o risco de ruptura do aneurisma. Palavras-chave: Aneurismas intracranianos. Stents. Remodelagem arterial. Mecânica dos fluidos computacional. Hemodinâmica. LIST OF FIGURES Figure 1 – Schematic illustration of (a) bifurcation and (b) sidewall aneurysms. . . 17 Figure 2 – Inferior view of the base of the brain showing an interconnected circula- tory structure composing the circle of Willis—internal carotid arteries (ICA) and their branches: middle cerebral arteries (MCA) and anterior communicating arteries (ACA); the vertebrobasilar system, composed of the vertebral arteries (VA) and the basilar artery (BA); and the anterior communicating artery (ACoA), connecting the left and right branches of the ACA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Figure 3 – Schematic illustration of coiling techniques: (a) stand-alone coil em- bolization and (b) stent-assisted coil embolization. . . . . . . . . . . . . 19 Figure 4 – Schematic illustration of a shape-memory alloy stent subject to a bending angle θ. Fbending is a resulting counteracting force dependent on θ and Fradial is an ever-present outward-directed radial force as a result of the self-expanding design of the stent. . . . . . . . . . . . . . . . . . . . . . 21 Figure 5 – Blood flow rate for the internal carotid artery (ICA) normalized with respect to the temporal mean blood flow rate. . . . . . . . . . . . . . . 24 Figure 6 – Mean blood flow rate through the arteries of the brain as a percentage of the total blood flow rate reaching the brain: (11.95± 2.05) ml/s. . . 25 Figure 7 – Volume rendering representation of a 3DRA image file. Picture taken using the visualization software ParaView. . . . . . . . . . . . . . . . . 26 Figure 8 – Volume rendering representation of a 3DRA image file. Volume of interest (VOI) enclosing the aneurysm and its surrounding vessels. Picture taken using the visualization software ParaView. . . . . . . . . . . . . . . . . 27 Figure 9 – Volume rendering representation of a 3DRA image; blue surface high- lights the final aneurysm surface reconstructed using VMTK utilities. Picture taken using the visualization software ParaView. . . . . . . . . 28 Figure 10 – Reconstructed surfaces of case 11, case 16, case 17, and case E1. Red lines illustrate where stents were deployed. . . . . . . . . . . . . . . . 29 Figure 11 – Arbitrary polygonal control volume VC with centroid C of a 2D finite volume mesh, and its neighboring cells with centroid Fi. . . . . . . . . 30 Figure 12 – Speedup, s, as a function of the number of processors, N , for the pre-stent geometry of case 11. . . . . . . . . . . . . . . . . . . . . . . . 35 Figure 13 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case 11 at systolic peak; A and B indicate the flow impingement zone at the dome of the aneurysm and a large area of low WSS (WSS < 1.5 Pa), respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surface: 28 Pa and 6.3 versus 29.3 Pa and 7.2, respectively. . 37 Figure 14 – Flow pattern in the (a) pre-stent and (b) post-stent geometries of case 11 at systolic peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Figure 15 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case 11; A and B indicate high OSI values coinciding with the flow impingement zone at the dome of the aneurysm and a large area of low WSS, respectively. Maximum and mean OSI on the pre- and post-stent aneurysm surfaces: 0.46 and 0.042 versus 0.41 and 0.024, respectively. . . . . . . . . . . . . 38 Figure 16 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case 16 at systolic peak; A and B indicate a low WSS region on the pre- and post-stent surfaces, respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surfaces: 47.3 Pa and 15 Pa versus 57.7 Pa and 17.1 Pa, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 17 – Flow pattern in aneurysm case 16 at systolic peak: (a) side view of the pre-stent geometry and (b) frontal view of the post-stent geometry. Points A and B indicate a slow recirculation flow causing a low WSS region on both surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Figure 18 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case 16. Point A indicates elevate OSI levels coinciding with a low WSS region (WSS < 1.5 Pa). Maximum and mean OSI on the pre- and post-stent aneursym surfaces: 0.32 and 0.007 vs 0.40 and 0.008, respectively. . . . 41 Figure 19 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case 17 at systolic peak; A and B indicate areas of low and high WSS on the aneurysm sac, respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surfaces: 49.4 Pa and 11.5 Pa versus 81.2 Pa and 18.5 Pa, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Figure 20 – Flow pattern in the (a) pre-stent and (b) post-stent geometries of case 17 at systolic peak. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Figure 21 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case 17. Maximum and mean OSI on the pre- and post-stent aneursym surfaces: 0.34 and 0.012 versus 0.37 and 0.015, respectively. . . . . . . . . . . . . 42 Figure 22 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case E1 at systolic peak. Point A and B indicate a region of high WSS due to flow impingement against the aneurysm neck and a region of high WSS on the aneurysm dome, respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surfaces: 77.3 Pa and 10.8 Pa versus 47.4 Pa and 9.4 Pa, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Figure 23 – Flow pattern in the (a) pre-stent and (b) post-stent geometries of case E1 at systolic peak. Point A and B indicate a splitting flow region and the flow jet entering the aneurysm; Point C indicates a large recirculation flow zone at the aneurysm fundus. . . . . . . . . . . . . . . . . . . . . . 44 Figure 24 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case E1. Maximum and mean OSI on the pre- and post-stent aneurysm surfaces: 0.43 and 0.026 versus 0.38 and 0.013, respectively. . . . . . . . . . . . . 44 Figure 25 – Mesh and time-step independence analysis for the post-stent geometry of case 11: WSS field on surfaces of meshes with ∼ 400,000, ∼ 750,000, and ∼ 1,500,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Figure 26 – Area-averaged WSS on the post-stent surface of case 11 as a function of time for meshes with ∼ 400,000, ∼ 750,000, and ∼ 1,500,000 control volumes and time-step of 10−5 s. . . . . . . . . . . . . . . . . . . . . . . 58 Figure 27 – Mesh and time-step independence analysis for the pre-stent geometry of case 11: WSS field on surfaces of meshes with ∼ 400,000, ∼ 750,000, and ∼ 1,500,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 28 – Area-averaged WSS on the pre-stent surface of case 11 as a function of time for meshes with ∼ 400,000, ∼ 750,000, and ∼ 1,500,000 control volumes and time-step of 10−5 s. . . . . . . . . . . . . . . . . . . . . . . 59 Figure 29 – Mesh and time-step independence analysis for the post-stent geometry of case 16: WSS field on surfaces of meshes with ∼ 400,000, ∼ 850,000, and ∼ 1,500,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Figure 30 – Area-averaged WSS on the post-stent surface of case 16 as a function of time for meshes with ∼ 400,000, ∼ 850,000, and ∼ 1,500,000 control volumes and time-step of 10−5 s. . . . . . . . . . . . . . . . . . . . . . . 60 Figure 31 – Mesh and time-step independence analysis for the pre-stent geometry of case 16: WSS field on surfaces of meshes with ∼ 400,000, ∼ 850,000, and ∼ 1,500,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 32 – Area-averaged WSS on the pre-stent surface of case 16 as a function of time for meshes with ∼ 400,000, ∼ 850,000, and ∼ 1,500,000 control volumes and time-step of 10−5 s. . . . . . . . . . . . . . . . . . . . . . . 61 Figure 33 – Mesh and time-step independence analysis for the post-stent geometry of case 17: WSS field on surfaces of meshes with ∼ 400,000, ∼ 900,000, and ∼ 1,300,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 34 – Area-averaged WSS on the post-stent surface of case 17 as a function of time for different combinations of mesh and time-step resolution. . . 63 Figure 35 – Mesh and time-step independence analysis for the pre-stent geometry of case 17: WSS field on surfaces of meshes with ∼ 400,000, ∼ 950,000, and ∼ 1,300,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Figure 36 – Area-averaged WSS on the pre-stent surface of case 17 as a function of time for different combinations of mesh and time-step resolution. . . . . 64 Figure 37 – Mesh and time-step independence analysis for the post-stent geometry of case E1: WSS field on surfaces of meshes with ∼ 400,000, ∼ 900,000, and ∼ 1,700,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 38 – Area-averaged WSS on the post-stent surface of case E1 as a function of time for meshes with ∼ 400,000, ∼ 900,000, and ∼ 1,700,000 control volumes and time-steps of 10−5 s. . . . . . . . . . . . . . . . . . . . . . 65 Figure 39 – Mesh and time-step independence analysis for the pre-stent geometry of case E1: WSS field on surfaces of meshes with ∼ 400,000, ∼ 900,000, and ∼ 1,700,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Figure 40 – Area-averaged WSS on the pre-stent surface of case E1 as a function of time for meshes with ∼ 400,000, ∼ 900,000, and ∼ 1,700,000 control volumes and time-steps of 10−5 s. . . . . . . . . . . . . . . . . . . . . . 66 Figure 41 – Arbitrary polygonal element of a finite volume mesh, and its neighboring elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Figure 42 – Over-relaxed decomposition of the normal surface vector sf . Ef lays in the dCF direction and Tf is normal to sf . . . . . . . . . . . . . . . . . 73 Figure 43 – Schematic representation of the assembled linear system of algebraic equations: aφC coefficients are find at the matrix diagonal and aφF coeffi- cients occupy other positions in the line. . . . . . . . . . . . . . . . . . 76 LIST OF TABLES Table 1 – Percentage of ruptured aneurysms based on the aneurysm size and location. 19 Table 2 – Geometric characteristics of the aneurysms in the current study: parent artery and its diameter da, neck diameter dn, dome height hd and diameter dd, aspect ratio Ar, and mean blood flow rate through its parent artery q̄a. 28 Table 3 – Comparison of maximum and mean values of WSS and OSI on the pre- and post-stent aneurysm surfaces of case 11, case 16, case 17, and case E1. 45 Table 4 – Summary of the findings of the current work. WSS and OSI values expressing the percent variation after vascular remodeling. . . . . . . . . 46 LIST OF ABBREVIATIONS 3DRA Three-dimensional rotational angiography ACA Anterior cerebral artery ACoA Anterior communicating artery BA Basilar artery CFD Computational fluid dynamics CV Control volume FIZ Flow impingement zone FVM Finite volume method ICA Internal carotid artery MCA Middle cerebral artery OSI Oscillatory shear index PISO Pressure implicit with splitting operators SACE Stent-assisted coil embolization SAH Subarachnoid hemorrhage SIMPLE Semi-implicit method for pressure linked equations V&V Validation and verification VOI Volume of interest WSS Wall shear stress ETHICS STATEMENT We hereby declare that, to the best of our knowledge, the current study had no influence on any decisions taken by Dr. Carlos Eduardo Baccin regarding the treatment of the aneurysms presented here. CONTENTS 1 INTRODUCTION 17 2 METHODOLOGY 23 2.1 Physical and mathematical modeling . . . . . . . . . . . . . . . . . . . . . 23 2.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.1 Domain discretization . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.2 Discretization of the mathematical model . . . . . . . . . . . . . . . 30 2.3 Hardware support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.4 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 RESULTS 37 3.1 Case 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Case 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3 Case 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Case E1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4 DISCUSSION AND CONCLUSIONS 46 REFERENCES 49 APPENDICES 55 A MESH AND TIME-STEP INDEPENDENCE STUDY 56 A.1 Case 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 A.2 Case 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 A.3 Case 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 A.4 Case E1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 A.5 Final considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 B THE FINITE VOLUME DISCRETIZATION 68 B.1 Spatial discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 B.2 Temporal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 B.3 Pressure-velocity coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 C RESIDUALS IN OPENFOAM 82 17 1 INTRODUCTION Strokes are the main cause of long-term disabilities and the third cause of death in western countries, only behind heart attack and cancer [1, 2]. It consists of the death of brain tissue by poor oxygen supply as a result of two events: brain ischemia and subarachnoid hemorrhage (SAH). The former is caused by insufficient blood flow to the brain [3]; the latter, on the other hand, by bleeding into the subarachnoid space. It has been estimated that 9 out of 100.000 individuals experience SAH each year in western countries [4], 85% of which are caused by the rupture of intracranial aneurysms [5]. Aneurysms are vascular disorders in which an abnormal balloon arises on the arterial wall (Figure 1). These aneurysms can occur at different locations of the human vascular system, more commonly found on the abdominal aorta and on the arteries that reach the brain. Intracranial aneurysms frequently appear on bifurcations of the circle of Willis [1]—an interconnected circulatory structure that supplies blood to the brain (Figure 2). Figure 1 – Schematic illustration of (a) bifurcation and (b) sidewall aneurysms. Parent Artery Daughter Artery Aneurysm Artery Aneurysm (a) (b) Source: prepared by the author. Statistics have shown that 2–5% of the population worldwide carry one or more intracranial aneurysms [7, 8], 2% of which will eventually rupture resulting in subarachnoid hemorrhage [9]. Despite the recent improvements in medicine, subarachnoid hemorrhage remains a serious disease with a 50% death rate [10] and a 12–30% long-term or permanent disabilities rate [11]. Aneurysms are usually asymptomatic, making their discovery merely incidental. Recent improvements and availability of non-invasive neuroradiological imaging techniques Chapter 1. Introduction 18 Figure 2 – Inferior view of the base of the brain showing an interconnected circulatory structure composing the circle of Willis—internal carotid arteries (ICA) and their branches: middle cerebral arteries (MCA) and anterior communicating arteries (ACA); the vertebrobasilar system, composed of the vertebral arteries (VA) and the basilar artery (BA); and the anterior communicating artery (ACoA), connecting the left and right branches of the ACA. Basilar Artery (BA) Vertebral Arteries (VA) Internal Carotid Artery (ICA, right branch) Middle Cerebral Artery (MCA, left branch) Anterior Cerebral Artery (ACA) Anterior Communicating Artery (ACoA) Source: prepared by the author with brain image extracted from KenHub [6]. have made their discovery more frequent. To date, most physicians rely on statistical correlations to determine risk of aneurysm rupture and viability of preventive treatments. According to Wiebers and Unruptured Intracranial Aneurysms Investigators [12], the risk of rupture increases with the aneurysm size (Table 1). Along with aneurysm size, aneurysm aspect ratio—dome height/neck width—has been consistently used to evaluate the risk of rupture. According to Weir et al. [13], the risk of rupture is 20-fold greater for aneurysms with an aspect ratio greater than 3.47 compared to those with an aspect ratio lower than 1.38. However, due to the poor prognosis for subarachnoid hemorrhage, physicians often consider preventive treatments when an Chapter 1. Introduction 19 Table 1 – Percentage of ruptured aneurysms based on the aneurysm size and location. Aneurysms location < 7 mm 7–12 mm 12–24 mm ≥ 25 mm ACA, MCA, and ICA 0% 2.6% 14.5% 40% BA and VA 2.5% 14.5% 18.4% 50% Source: Wiebers and Unruptured Intracranial Aneurysms Investigators [12]. unruptured aneurysm is identified. Currently, the two leading treatment techniques for intracranial aneurysms are surgical intervention—direct clipping, for example—and endovascular treatment—coils placement inside the aneurysm, assisted or not by stents. According to Qureshi et al. [14], post-procedure disabilities are more frequent in patients who undergo surgery compared to those who undergo endovascular treatments (25% versus 8%); also the number of days in intensive care, the length of stay in hospital, and treatment costs are all greater for surgical interventions. Therefore, many physicians have been considering coils, assisted or not by stents (Figure 3), as a primary option to treat unruptured aneurysms. Stent-assisted coil embolization (SACE) of intracranial aneurysms was first de- signed to treat fusiform or wide-neck aneurysms—ratio between aneurysm neck and aneurysm dome height greater than 0.5 or neck size greater than 4 mm—in which other neurovascular treatments were not feasible [15, 16]. However, with gained experience and introduction of specifically-designed self-expanding stents for intracranial use, SACE has been employed to treat a large range of aneurysms [17] in various configurations [18]. Figure 3 – Schematic illustration of coiling techniques: (a) stand-alone coil embolization and (b) stent-assisted coil embolization. (a) Coils (b) Stent Source: prepared by the author. Chapter 1. Introduction 20 Some authors have also proposed using stents as a stand-alone procedure to treat fusiform aneurysms, obtaining gradual aneurysm occlusion [19–22]. The authors speculate that stand-alone stents could create favorable thombogenic conditions by altering intracranial blood flow (hemodynamics), which has been related to aneurysm initiation, development and rupture [23–25]. Out of the various hemodynamic parameters, the wall shear stress (WSS) due to the flow velocity gradient adjacent to the aneurysm wall has been reported to play a major role in the different stages of aneurysm growth [23, 26]. In the human arterial system, high WSS levels can trigger biological mechanisms associated with arterial remodeling, leading to initiation, development, and rupture of aneurysms [24]. On the other hand, it has also been advocated that low WSS have a negative effect on endothelial cells, being an important contributor to aneurysm growth and rupture [23, 27]. Towards an unifying hypothesis, Meng et al. [25] took a step further and proposed that both high and low WSS can drive aneurysm growth and rupture. According to the authors, a combination of low WSS and elevated oscillatory shear index (OSI)—a parameter that measures the changes in the WSS vector direction within a cardiac cycle—triggers a destructive pathway associated with growth and rupture of large aneurysms; whereas a combination of high WSS and positive spatial WSS gradient along the flow triggers a destructive pathway associated with growth of small or secondary bleb aneurysm. In light of the role exerted by hemodynamics in the different stages of aneurysm development, experimental and numerical studies have been conducted to investigate hemodynamic changes induced by stents [4, 9, 18, 28–33]. Cantón et al. [28] measured changes in flow dynamics using digital particle image velocimetry in flexible silicone sidewall aneurysms models after sequential stent placement and concluded that the velocity magnitude of the flow jet entering the aneurysm could be reduced by more than 50%. Tang et al. [29] performed numerical simulations of a bifurcation aneurysm to evaluate the influence of the aneurysm neck size after stenting and concluded that the maximum velocity inside the aneurysm at systolic phase was reduced by 18–24%; and that the volume flow rate entering the aneurysm over the entire cardiac cycle could be reduced by more than 50%. Bernardini et al. [9] performed numerical simulations of a idealized sidewall aneurysm for two stent positions and arrived at similar conclusions to those of Cantón et al. [28] and Tang et al. [29]. To date, we note that most of the studies have been focusing on the hemodynamic changes caused by the presence of the stent itself, i.e., the blockage effect exerted by the struts. The current generation of self-expanding intracranial stents are made of shape- memory nickel-titanium alloy (nitinol), granting flexibility and conformability in tortuous vessels. These stents exert a continuous outward-directed radial force and, in addition, a mild force in resisting bending motion [34] (Figure 4). The former is necessary to maintain Chapter 1. Introduction 21 Figure 4 – Schematic illustration of a shape-memory alloy stent subject to a bending angle θ. Fbending is a resulting counteracting force dependent on θ and Fradial is an ever-present outward-directed radial force as a result of the self-expanding design of the stent. Fradial Fbending(θ) θ Source: prepared by the author. a firm grip on the vessel wall; the latter is a secondary, collateral effect that has been related to immediate and delayed angular remodeling of the local vascular system [18, 35, 36]. Gao et al. [37] evaluated the immediate and delayed stent-induced vascular remod- eling for bifurcation aneurysms at different arteries of the brain. The authors reported an immediate variation of 20◦ in the bifurcation angle; further variations were observed in up to one year follow-ups with a total angular change of 35◦. According to the authors, the greatest changes were observed within 6 months posttreatment, followed by minor changes until one year follow-up, suggesting an asymptotic behavior. Furthermore, the authors reported the angular remodeling being inversely proportional to the parent artery diameter and proportional to the pretreatment angle; i.e., the smaller the parent vessel or the prestenting vascular angle, the bigger the posttreatment angular modification. More recently, Kono, Shintani, and Terada [38] compared the influence of angular remodeling against the blockage effect of stent struts for sidewall aneurysms. The authors concluded that the stent struts have approximately twice as strong effect on reduction of flow velocity than angular remodeling. Jeong, Han, and Rhee [39] performed a similar study for a group of idealized bifurcation aneurysms, adopting a porous-medium methodology to represent the coils inside the aneurysm sac. According to the authors, angular remodeling may provide an unfavorable hemodynamic environment, which might delay thrombus formation. Despite the valuable contributions of the above-mentioned studies, we still lack information regarding hemodynamic changes caused by stent-induced vascular remodeling in bifurcation aneurysms without the presence of coils. This is of particular interest to stand- alone stent treatment [19–22]; and also to two-step stent-assisted coiling technique [40], in which stenting and coiling are done in separate steps. Chapter 1. Introduction 22 Objectives Using patient-specific models of bifurcation intracranial aneurysms before and after vascular remodeling, we propose to use results of numerical simulations to evaluate the changes in hemodynamic parameters related to aneurysm growth and rupture—WSS and OSI— according to reference values found in the literature and determine whether endovascular treatments could benefit from stent-induced vascular remodeling. Furthermore, we use only open-source tools during all stages of the current work, making our methodology easily available. 23 2 METHODOLOGY Blood flow in aneurysms and arteries may be modeled and solved in different ways. As presented in Chapter 1, we propose to solve this problem numerically, making use of the Finite Volume Method (FVM) [41–43]. Thus, we present throughout this chapter the methodology behind our study: physical modeling, boundary conditions, discretization of the governing equations by the FVM, reconstruction of patient-specific aneurysm geometries from medical images, and data analysis strategy. 2.1 Physical and mathematical modeling Before we present the physical and mathematical modeling, we must define a group of hypothesis. In the literature, hypothesis of rigid vessel walls and blood as an incompressible Newtonian fluid in a isothermal, laminar flow regime are commonly adopted to study blood flow in aneurysms and arteries [2, 7, 33, 44]. Different studies already evaluated the influence of these hypothesis on hemody- namic parameters, such as velocity field and WSS distribution on the arterial wall [26, 45–49]. In general, they have a much lower influence on hemodynamics than changes in the geometry of aneurysms and arteries [45]—the exact subject of this study. Thereby, we adopt the same group of hypothesis in our study, considering a blood density of 1,000 kg/m3 and a kinematic viscosity of 3.3× 10−6 m2/s [50]. In this case, we face a classical problem of fluid dynamics, in which the governing equations are derived from the principle of conservation of mass and the balance of linear momentum, respectively, as follows: ∇ • ( ρv ) = 0 , (2.1) ∂(ρv) ∂t +∇ • ( ρvv ) = −∇p+∇ • τ + fb , (2.2) where v and p are the flow velocity field and the pressure field, respectively; ρ is the blood density; fb represents the body forces field per volume; and τ is the stress tensor, which can be expressed as follows for a Newtonian fluid with constant properties: τ = µ [ ∇v + (∇v)T ] − 2 3µ(∇ • v)I ρ= constant=======⇒ µ= constant τ = µ∇2v , (2.3) where µ is the blood dynamic viscosity and I is the second rank identity tensor. Moreover, it will be useful for us to express the governing equations in their integral form, which is of special interest to the finite volume method. Therefore, Equations 2.1 and 2.2 can be Chapter 2. Methodology 24 written as ∮ S ρv • n dS = 0 (2.4) and ∂ ∂t ∫ V ρv dV + ∮ S ρv(v • n) dS = − ∮ S pn dS + ∮ S µ∇v • n dS , (2.5) where n is the unit normal vector to S pointing outward. In Equations 2.4 and 2.5, we identify the flow velocity and pressure field as the unknowns, both as functions of the spatial coordinates and the time. Therefore, we must define appropriate initial and boundary conditions. At the inlet—the cross section in the aneurysm parent artery—we impose a zero pressure gradient and a time-varying velocity, corresponding to the flow pulse from the beginning of systole until the end of the diastole. The aneurysms in the current study are located in different bifurcations of the cerebral vascular system. Since we do not have patient-specific information on the blood flow rate profile in the aneurysm parent artery, we used available data in the literature to obtain the inlet flow velocity for each case. For that, we first considered a pulsatile normalized blood flow, as shown in Figure 5. Ford et al. [51] measured this profile in the internal carotid artery (ICA) and normalized it with respect to the temporal mean blood flow rate in the same artery. Then, to obtain the dimensional profile we multiplied the normalized profile by the mean blood flow rate in the artery of interest. The mean blood flow rate for different arteries of the cerebral vascular system is provided by Zarrinkoob et al. [52]. The authors measured it as a percentage of Figure 5 – Blood flow rate for the internal carotid artery (ICA) normalized with respect to the temporal mean blood flow rate. 0 0.2 0.4 0.6 0.8 1 0.6 0.8 1 1.2 1.4 1.6 1.8 Time (s) N or m al iz ed flo w ra te Source: prepared by the author with experimental data from Ford et al. [51]. Chapter 2. Methodology 25 the total blood flow rate to the brain in normal subjects, which is (11.95± 2.05) ml/s, as shown in Figure 6. Finally, we computed the inlet flow velocity using the cross sectional area of the inlet boundary. At the outlet—the cross section in the daughter arteries of the vessel bifurcation—we imposed a fixed pressure condition and a zero velocity gradient. Note that for incompressible flows, the pressure field is computed relative to a reference value and commonly set to zero by default. Nevertheless, we adopted a constant pressure equal to the average pressure level in the human body, approximate 100 mmHg or 13.33 Pa. Lastly, at vessel walls we imposed a no-slip condition, assuming a zero relative velocity between flow and vessel walls. Figure 6 – Mean blood flow rate through the arteries of the brain as a percentage of the total blood flow rate reaching the brain: (11.95± 2.05) ml/s. BA: (20 ± 4)% ICA (left and right branches): (36 ± 4)% MCA (left branch): (21 ± 2)% MCA (right branch): (21 ± 3)% ACA (left branch): (12 ± 4)% ACA (right branch): (11 ± 4)% Source: prepared by the author with data from Zarrinkoob et al. [52]. 2.2 Numerical methods To solve the governing equations—Equations 2.1 and 2.2—under the boundary conditions presented in Section 2.1 we chose the Finite Volume Method [41, 43]. We used the software OpenFOAM (Open-source Field Operation and Manipulation) [53, 54], which is the leading free, open-source software for Computational Fluid Dynamics (CFD). In this section we present the domain discretization procedure followed by the discretization procedure of the governing equations. Chapter 2. Methodology 26 2.2.1 Domain discretization The domain discretization was performed splitting the geometry into a finite number of 3D uniform elements, also known as control volumes. To this end, we first needed to obtain the geometry of the aneurysm and surrounding arteries. Geometries of aneurysms were obtained from images of 3D rotational angiography (3DRA). These images were taken using a Philips Allura Xper FD20 and provided by Dr. Carlos Eduardo Baccin (Hospital Israelita Albert Einstein, São Paulo, Brazil). The 3DRA file is a 3D pixelized image that provides a clear visualization of the cerebral vascular system—see Figure 7—and thus used by many interventional neuroradi- ologists. To reconstruct the aneurysm geometry from the 3DRA file, we performed the following steps using the Vascular Modeling Toolkit (VMTK), an open-source collection of libraries and tools for 3D reconstruction, geometric analysis, mesh generation, and surface data analysis for image-based modeling of blood vessels. Figure 7 – Volume rendering representation of a 3DRA image file. Picture taken using the visualization software ParaView. AneurysmICA Source: prepared by the author. First, we selected a volume of interest (VOI), a small portion of the 3DRA image that encloses the aneurysm and its closest vessels, as shown in Figure 8. For that, VMTK provides an utility called vmtkimagevoiselector. This utility extracts the VOI from the 3DRA image and saves it in a separate image file, enabling us to load just the VOI image file during future steps of the reconstruction procedure. Second, to reconstruct the surface of the aneurysm and surrounding vessels inside the VOI we used two utilities: vmtklevelsetsegmentation and vmtkmarchingcubes. The former uses a series of filters and user-defined thresholds to create an initial level set and Chapter 2. Methodology 27 Figure 8 – Volume rendering representation of a 3DRA image file. Volume of interest (VOI) en- closing the aneurysm and its surrounding vessels. Picture taken using the visualization software ParaView. AneurysmVOI Source: prepared by the author. evolve it to image gradients; the latter uses these image gradients to generate a triangulated isosurface. Normally the isosurface generated does not have a production-ready surface qual- ity. Therefore, as a third step, we used two utilities to increase the surface quality: vmtksurfaceremeshing and vmtksurfacesmoothing. Remeshing the surface, increases the number of triangulated elements, thereby, improving the efficiency of the smoothing algorithms used by VMTK. Lastly, we used the utility vmtksurfaceclipper to create inlet and outlet bound- aries, clipping the surrounding arteries of the aneurysms, and then yielding the final aneurysm surface. We used this surface to determine the geometric parameters of the aneurysm and its surrounding arteries. Table 2 provides the following parameters of the cases in the current study: aneurysm height hd, neck diameter dn, parent artery name, aneurysm aspect ratio—defined as hd/dn— and mean blood flow rate in the parent artery q̄a—as defined in Section 2.1. We kept the nomenclature of the original 3DRA files as case identifiers. The reconstructed surfaces of each case are presented in Figure 10. Finally, meshes were created from the above-mentioned surface using the utility vmtkmeshgenerator. This utility has numerous options to control mesh quality and size, creating a volume filled with tetrahedral elements in the core and prismatic elements adjacent to the vessel wall. Chapter 2. Methodology 28 Figure 9 – Volume rendering representation of a 3DRA image; blue surface highlights the fi- nal aneurysm surface reconstructed using VMTK utilities. Picture taken using the visualization software ParaView. Source: prepared by the author. Table 2 – Geometric characteristics of the aneurysms in the current study: parent artery and its diameter da, neck diameter dn, dome height hd and diameter dd, aspect ratio Ar, and mean blood flow rate through its parent artery q̄a. Parent artery da (mm) dn (mm) hd (mm) dd (mm) Ar q̄a (ml/s) Case 11 pre-stent MCA 2.7 7.0 8.7 9.6 1.25 2.5 post-stent 2.7 7.4 8.7 9.6 1.17 % variation 0% +5.7% 0% 0% −6.4% Case 16 pre-stent MCA 2.5 6.5 5.4 6.4 1.20 2.5 post-stent 2.9 5.7 5.6 6.0 1.01 % variation +16% −12.3% +3.7% −6.3% −15.8% Case 17 pre-stent ACA 2.3 & 2.1∗ 5.9 5.0 5.1 1.18 1.4 † post-stent (both branches) 1.9 & 1.7∗ 5.6 4.5 4.7 1.24 % variation −17.4% & −19%∗ −5.1% −10% −7.8% +5.1% Case E1 pre-stent MCA 3.3 4.5 5.7 5.9 1.27 2.5 post-stent 3.7 5.1 4.8 5.7 0.94 % variation +12.1% +13.3% −15.8% −3.4% −26% ∗ left & right ACA † per branch Source: prepared by the author. Chapter 2. Methodology 29 Figure 10 – Reconstructed surfaces of case 11, case 16, case 17, and case E1. Red lines illustrate where stents were deployed. Pre-stent Post-stent C as e 11 C as e 16 C as e 17 C as e E1 Parent artery (MCA) Parent artery (MCA) Parent arteries (left and right ACA) Parent artery (MCA) Source: prepared by the author. Chapter 2. Methodology 30 2.2.2 Discretization of the mathematical model Coded in C++, OpenFOAM combines utilities for pre- and post-processing, and solvers for partial differential equations, offering a complete framework for solving problems of Continuum Mechanics. OpenFOAM uses the FVM with cell-centered variable arrangement, i.e., unknown quantities are stored at cell centroids. A typical finite-volume mesh is composed of non-overlapping, arbitrary polygonal elements with flat faces, as shown in Figure 11. Moreover, each face must be shared by only two neighboring cells; exceptions are those composing the boundaries of the domain, which in turn belong to only one cell. All meshes used in this study follow these conditions. Figure 11 – Arbitrary polygonal control volume VC with centroid C of a 2D finite volume mesh, and its neighboring cells with centroid Fi. C Vc F1 F2 F3 F4 F5 F6 f1 f2 f3 f4 f5 f6 dCF Source: prepared by the author. The finite volume discretization procedure is initiated by writing the governing equations as integrals over a control volume VC . Then, Gauss’ divergence theorem is used to transform the volume integrals of the advective and diffusive terms into sums of surface integrals over control volume faces. To evaluate the face integrals, we assume the fields to behave linearly around control volume centroids; thus, using an one-point Gaussian quadrature yields a second-order accurate spatial discretization. The whole procedure is detailed in Appendix B and deeply grounded on the writings of Versteeg and Malalasekera [41], Ferziger and Peric [42], and Moukalled, Mangani, and Darwish [43]. Our main focus here is to apply this procedure to the governing equations: Equations 2.4 and 2.5. The semi-discretized momentum equation for a control volume VC in a finite volume Chapter 2. Methodology 31 mesh is expressed as ρVC 3vn C − 4vn−1 C + vn−2 C 2∆t + ∑ f∼nb(C) ( ṁf vn f ) = ∑ f∼nb(C) [ µ(∇v)nf • sf ] + (∇p)nC VC , (2.6) where f∼nb(C) denotes faces neighboring centroid C — see Figure 11; the subscripts f and C indicate values evaluated at face centroid and control volume centroid, respectively; the superscript n represents the time-step level; sf is a vector with magnitude equal to the face area, normal to face f , and pointing outwards control volume VC ; ṁf = ρ(vf • sf ) is the mass flow rate through face f and thus, must satisfy the discretized principle of conservation of mass: ∑ f∼nb(C) ṁf = ∑ f∼nb(C) ρ(vf • sf ) = 0 . (2.7) Note that because we evaluate the mass flow rate ṁf using known values from the previous iteration, we use no superscript to denote time-step level. To evaluate face values in Equation 2.6, we interpolate values at cell centroids. For the velocity in the advective term, vn f , we adopt a second-order Upwind scheme for unstructured grids, which yields vf = vC + [ 2(∇v)C − (∇v)f ] • dCf , (2.8) where dCf is the vector joining centroid C and face f . Using the least-square method [43] to discretize the velocity in Equation 2.8 yields vf = wvC + (1− w)vF , (2.9) where the subscript F represents a cell centroid, whose face f is shared with element C; and w is a weight factor that depends on the chosen interpolation scheme and accounts for the influence of vC and vF on vf . To interpolate the gradient normal to face f in Equation 2.6, (∇v)f • sf , we used a non-orthogonal corrected scheme as follows: (∇v)f • sf = (∇v)f • Ef︸ ︷︷ ︸ orthogonal contribution + (∇v)f • Tf︸ ︷︷ ︸ non-orthogonal contribution , (2.10) where sf = Ef + Tf , with Ef being in the direction dCF—see Figure 11. Thus, Equa- tion 2.10 can be rewritten as (∇v)f • sf = Ef vF − vC dCF + (∇v)f • Tf , (2.11) Chapter 2. Methodology 32 where the non-orthogonal contribution, (∇v)f • Tf , is evaluated using an over-relaxed approach—see Appendix B, Section B.1 for more details. Combining Equations 2.6, 2.9 and 2.10, we can write an algebraic momentum equation for one control volume in a finite volume mesh as follows: av Cvn C + ∑ F∼NB(C) av Fvn F = bv C − (∇p)CVC , (2.12) where av C , av F , and bv C are functions of the flow velocity. Thus, considering all control volumes in a finite volume mesh, we can assemble a system of algebraic linearized equations, yielding Avvn = bv − VC(∇p)C , (2.13) where A is the matrix of coefficients aC and aF , v is the velocity solution vector, b is the vector of source terms, and ∇p is the vector obtained from the discretization of the pressure gradient. It is important to note that pressure does not appear as a primary variable in either the momentum or continuity equations; furthermore, for incompressible flows, no explicit equation is available to compute the pressure field in the momentum equation, requiring a special treatment for the pressure-velocity coupling. One approach consists of reformulating the governing equations in terms of a momentum and a pressure equation [43], where the pressure equation is a combination of the semi-discretized momentum and continuity equations. To produce the pressure equation, we start considering the discretized continuity equation, Equation 2.7. Then, we evaluate the flow velocity at face centroids from cell centroids values, using the discretized momentum equation, Equation 2.13, which can be rewritten as vC + HC [v] = Bv C −Dv C(∇p)C , (2.14) where HC [v] = ∑ F∼NB(C) av F av C vF , Bv C = bv C av C , and Dv C = VC av C . However, interpolating flow velocities from cell centroids to face centroids using a linear scheme originates the checkboard problem. A solution to this problem was first proposed by Rhie and Chow [55], where a pseudo-momentum equation is constructed at face centroid. For that, the coefficients of the pseudo-momentum equation are linearly Chapter 2. Methodology 33 interpolated from coefficients of momentum equations at cell centroid, yielding vf + Hf [v] = Bv f −Dv f (∇p)f (2.15) or vf = −Hf [v] + Bv f −Dv f (∇p)f . (2.16) Finally, substituting Equation 2.16 into Equation 2.7, we obtain ∑ f∼nb(C) Dv f (∇p)f • sf = ∑ f∼nb(C) { Bv f −Hf [v] } • sf , (2.17) where the gradient (∇p)f • sf is evaluated implicitly using a non-orthogonal-corrected approach—Equation 2.9, more details in Appendix B, Section B.3. After some algebraic manipulations, we are now able to express a linearized algebraic equation for the pressure apCp n C + ∑ F∼NB(C) apFp n F = bpC , (2.18) which represents the discretized pressure equation for one finite control volume VC . Thus, assembling for the whole finite volume mesh, we obtain a linearized system of equations Appn = bp . (2.19) To handle the solution of the governing equations, as well as the pressure-velocity coupling, we adopted the PISO (Pressure-Implicit Split Operators) algorithm [56]. The algorithm first sets the boundary and initial conditions for the velocity and pressure fields. Next, it solves the discretized momentum equation – Equation 2.13 – yielding an intermediate velocity field, which does not necessarily satisfies the continuity. Then, the intermediate velocity field is used to solve the pressure equation – Equation 2.19 – yielding a pressure field that satisfies both the continuity and momentum equations. Lastly, the velocity field is updated using the new pressure field, resulting in a continuity-satisfying velocity field. According to Issa, Gosman, and Watkins [56], two iterations over the pressure equation are able to provide acceptable accuracy. The system of equations presented in Equation 2.13 is solved using a preconditioned bi-conjugated gradient solver for asymmetric matrices, with a diagonal incomplete Cholesky preconditioner for asymmetric matrices and convergence criterion based on a scaled normalized residual tolerance equal to 1× 10−6. For the pressure system, Equation 2.19, we used a geometric agglomerated algebraic multigrid solver, with scaled normalized residual tolerance equal to 1× 10−6. For more details on how OpenFOAM calculates the residuals, see Appendix C. Chapter 2. Methodology 34 2.3 Hardware support In many CFD problems, the solution sequence is very time consuming for a single machine running in serial mode. Considering this, OpenFOAM offers an option to run simulations in parallel on distributed processors. Taking advantage of this feature, all simulations were performed in parallel on the following clusters: • GridUNESP: the computational resource of São Paulo State University (UNESP) runs CentOS 6.9 and has a computational structure of: – 256 nodes and 2,048 CPU cores (Intel Xeon 2.83 GHz); – 4,096 GB of RAM memory (2 GB per core); – InfiniBand interconnection 4X DDR (20 Gbps). • Cedar: integrated into the resources of Compute Canada clusters, it is a heterogeneous cluster suitable for a variety of workloads. The cluster runs CentOS 7.4 and has a computational structure of: – 1,396 nodes and 58,416 CPU cores (Intel Broadwell E5-2650, Intel Broadwell E5-2683, and Intel Skylake Platinum 8160F); – up to 128 GB of RAM memory per node; – Intel OmniPath interconnection (100 Gbps). • Graham: it is also integrated into the resources of Compute Canada clusters. The cluster runs CentOS 7.4 on a computacional structure of: – 1,107 nodes and 35,527 CPU cores (Intel Broadwell E5-2683 and Intel Broadwell E7-4850); – up to 128 GB of RAM memory per node; – Mellanox FDR (56 Gbps) and EDR (100 Gbps) InfiniBand interconnection. Note that all clusters were used separately. OpenFOAM uses a parallel computing method known as domain decomposition, i.e., it breaks the geometry and associated fields into smaller pieces and allocates them to separate processors for solution. For this task, we used a built-in decomposition method known as Scotch. This method attempts to minimize the number of processor boundaries, reducing the amount of data shared between processors. Despite the capability of the clusters, increasing the number of processors does not necessarily mean a reduction in simulation time. Different factors can slow down the process and act as a bottleneck, these include fractions of code that cannot be parallelized and hardware limitations. Therefore, the number of processors were estimated via simulation Chapter 2. Methodology 35 speedup, which measures if a given problem can be solved faster by increasing the number of processors dedicated to that task. The speedup is defined as s = Tserial Tparallel , (2.20) where s is the simulations speedup, Tserial is the time spent to conclude a task running on a single processor and Tparallel is the time spent to conclude the same task running on N processors. Figure 12 shows the speedup measured for the pre-stent geometry of case 11. Note that, despite the fluctuations in the speedup value, it increases rapidily in the beginning until reach a plateau after 100 processors. Figure 12 – Speedup, s, as a function of the number of processors, N , for the pre-stent geometry of case 11. 0 20 40 60 80 100 120 0 10 20 30 40 Number of processors, N Sp ee du p, s Source: prepared by the author. Based on these data, a choice of 104 processors would be reasonable. However, all clusters we used operate on a batch scheduling system with dynamic algorithms to support fair-share access to its resources, i.e., the more processors we ask for, the longer the simulation remains in queue; in the end the simulation may take longer to complete. Therefore, we based our decisions regarding the number of processors on empirical knowledge of the fair-share system. The criterion adopted was 32 cores for meshes smaller than 1,000,000 control volumes and 64 cores for meshes greater than that. In both cases, the average wall clock time spent per simulation was around 18 hours. 2.4 Data analysis Once completed the simulations, we evaluated the following quantities using ParaView, an open-source, multi-platform data analysis and visualization application: Chapter 2. Methodology 36 • Velocity field: to determine flow pattern and impact zone; • WSS on the wall: the WSS is the shear component of the traction due to the stress tensor, τ, of the flow on the wall: WSS = ∥∥∥W SS ∥∥∥ 2 = ∥∥∥ n • τ︸ ︷︷ ︸ total traction − [(n • τ) • n]n︸ ︷︷ ︸ traction normal component ∥∥∥ 2 , (2.21) where n is the unit vector, pointing inwards, normal to the wall surface; • OSI: this index measures changes in the WSS vector direction during the cardiac cycle and it is defined as OSI = 1 2 1− ∥∥∥∥∥ ∫ T 0 (W SS) dt ∥∥∥∥∥∫ T 0 ‖W SS‖2 dt  , (2.22) where T is the cardiac cycle duration. These are the most common hemodynamic parameters related to intracranial aneurysms rupture, as presented in Chapter 1. Besides the above-mentioned parameters we also computed the mean value of both WSS and OSI on the aneurysm surface in order to quantify the hemodynamic changes due to vascular remodeling. These are defined as WSSa = 1 Aa ∥∥∥∥∫ S (W SS) dS ∥∥∥∥ 2 (2.23) and OSIa = 1 Aa ∫ S (OSI) dS , (2.24) where Aa is the aneurysm surface area. 37 3 RESULTS In this chapter, we use the results of the numerical simulations to compare the hemodynam- ics in aneurysms and surrounding arteries for the pre- and post-stent geometries presented in Chapter 2. To ensure numerical accuracy of the simulation data, we performed a mesh and time-step independence study, which is presented in details in Appendix A. Therefore, the following analysis was done using models that were found within an acceptable range of numerical accuracy. 3.1 Case 11 Figures 13 and 14 show the flow pattern and WSS field of the pre- and post-stent geometries of case 11. The lower bound of the WSS scale was set up using a reference value of 1.5 Pa, below which the WSS can degenerate endothelial cells and possibly lead to rupture [23, 25]. Figure 14a shows the flow jet that enters the pre-stent aneurysm sac impinging on the left side of the aneurysm dome, causing a recirculation zone at the opposite side of the aneurysm. The direct impact against the aneurysm wall deflects the flow jet radially, creating a high WSS region—point A in Figure 13a. On the other hand, the low velocity levels of the recirculation flow result in a low WSS area at the right side of the aneurysm sac—point B in Figure 13a. Figure 13 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case 11 at systolic peak; A and B indicate the flow impingement zone at the dome of the aneurysm and a large area of low WSS (WSS < 1.5 Pa), respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surface: 28 Pa and 6.3 versus 29.3 Pa and 7.2, respectively. A AB B (a) (b) Source: prepared by the author. Chapter 3. Results 38 Figure 14 – Flow pattern in the (a) pre-stent and (b) post-stent geometries of case 11 at systolic peak. frontal view frontal view (a) (b) Source: prepared by the author. Figure 15 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case 11; A and B indicate high OSI values coinciding with the flow impingement zone at the dome of the aneurysm and a large area of low WSS, respectively. Maximum and mean OSI on the pre- and post-stent aneurysm surfaces: 0.46 and 0.042 versus 0.41 and 0.024, respectively. A A B B OSImax ∼0.46 OSImax ∼0.41 (a) (b) Source: prepared by the author. For the post-stent geometry (Figure 14b), we note that the direction of the flow jet entering the aneurysm sac has changed, altering the flow pattern inside the aneurysm. Figure 13 points out a displacement of the low and high WSS regions: both following the changes in flow pattern. Despite the rearrangement of both low and high WSS regions, note that the overall WSS distribution remains similar: the pre-stent aneurysm surface presents a maximum WSS of 28 Pa and a mean WSS of 6.3 Pa; whereas the post-stent aneurysm surface presents a maximum WSS of 29.3 Pa and a mean WSS of 7.2 Pa. Chapter 3. Results 39 Figure 15 shows the OSI distribution on the pre- and post-stent aneurysms, another important hemodynamic parameter that has been related to aneurysm rupture, specially when combined with low WSS levels [25, 57]. On the pre-stent aneurysm surface, we identify high levels of OSI matching a region of low WSS—point B in Figures 13a and 15a— these could trigger an inflammatory pathway and lead to rupture of the aneurysm. After stent placement, we readily note that the region of high OSI moved away from its early position—point B in Figure 21b. Furthermore, we note a considerable decrease in the OSI levels on the aneurysm surface. This becomes more evident when comparing the mean OSI on the pre- and post-stent aneurysm surfaces: 0.042 and 0.024, respectively. 3.2 Case 16 Figures 16 and 17 show the flow pattern and the WSS distribution for the pre- and post-stent geometries of case 16. In Figure 16 we identify a large area of high WSS covering the frontal wall of the parent artery, the aneurysm dome, and daughter arteries. Analyzing Figure 16 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case 16 at systolic peak; A and B indicate a low WSS region on the pre- and post-stent surfaces, respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surfaces: 47.3 Pa and 15 Pa versus 57.7 Pa and 17.1 Pa, respectively. back view back view frontal view frontal view A B (a) (b) Source: prepared by the author. Chapter 3. Results 40 Figure 17 – Flow pattern in aneurysm case 16 at systolic peak: (a) side view of the pre-stent geometry and (b) frontal view of the post-stent geometry. Points A and B indicate a slow recirculation flow causing a low WSS region on both surfaces. A B right side view frontal view (a) (b) Source: prepared by the author. the flow pattern in the pre-stent geometry (Figure 17a) reveals the flow “washing” the frontal wall of the aneurysm, which leads to high velocity gradients adjacent to the wall. Furthermore, we identify a separating flow due to the curved geometry of the parent artery, which causes a recirculation flow zone, resulting in a low WSS area—point A in Figures 16a and 17a. For the post-stent geometry, Figure 16b shows a similar WSS distribution on the aneurysm surface. However, we identify two main differences. First, the low WSS region moved from the back to the left side of the parent artery, indicated by point B in Figure 16b. An analysis of the flow pattern in the post-stent geometry (Figure 17b) shows that the separating flow was dislocated due to straightening of the parent artery. Second, we note a smaller area of high WSS on the aneurysm surface, resulting in a smaller mean WSS on the aneurysm: 17.1 Pa for the pre-stent geometry versus 15 Pa for the post-stent geometry. Despite the reduction in the mean WSS, higher WSS levels are found on the post-stent surface rather than on the pre-stent surface: 57.7 Pa versus 47.3 Pa. Regarding the OSI field, Figure 18 shows the values for both geometries. We easily identify an area of high OSI matching the region of low WSS on both pre- and post-stent geometries, which might increase the risk of rupture [25]. Moreover, in Figure 18 we identify an increase in the maximum OSI values found on the pre- (0.32) versus post-stent (0.40) surfaces. Nonetheless, the mean OSI remains almost unaltered (0.007 versus 0.008). Chapter 3. Results 41 Figure 18 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case 16. Point A indicates elevate OSI levels coinciding with a low WSS region (WSS < 1.5 Pa). Maximum and mean OSI on the pre- and post-stent aneursym surfaces: 0.32 and 0.007 vs 0.40 and 0.008, respectively. A (OSImax ∼0.32) B (OSImax ∼0.40) right side view frontal view (a) (b) Source: prepared by the author. 3.3 Case 17 The WSS field and the flow pattern for the pre- and post-stent geometries of case 17 are presented in Figures 19 and 20, respectively. Comparing the WSS field on the pre- and post-stent surfaces, we note an increase in WSS levels on the aneurysm surface and specially on the surrounding arteries. Such change can be explained analyzing the flow Figure 19 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case 17 at systolic peak; A and B indicate areas of low and high WSS on the aneurysm sac, respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surfaces: 49.4 Pa and 11.5 Pa versus 81.2 Pa and 18.5 Pa, respectively. A B A B (a) (b) Source: prepared by the author. Chapter 3. Results 42 Figure 20 – Flow pattern in the (a) pre-stent and (b) post-stent geometries of case 17 at systolic peak. (a) (b) Source: prepared by the author. Figure 21 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case 17. Maximum and mean OSI on the pre- and post-stent aneursym surfaces: 0.34 and 0.012 versus 0.37 and 0.015, respectively. OSImax ∼0.34 OSImax ∼0.37 (a) (b) Source: prepared by the author. pattern, shown in Figure 20. In the pre-stent geometry, the left parent artery contributes almost exclusively to the blood flow into the aneurysm. The flow “washes” the aneurysm surface and flows into the left daughter artery. This causes a recirculation flow to appear, resulting in a low WSS region. In addition, due to the angular configuration of the arteries, the flow from the right parent artery is smoothly deflected to the right daughter artery. For the post-stent geometry, we note a similar behavior; however, due to the angular remodeling, the new configuration causes the flow from the right parent artery to be deflected in a more abrupt manner, resulting in greater flow activity in the aneurysm and higher levels of WSS—point B in Figure 19a. In sum, the mean WSS increased from 11.5 Pa to 18.5 Pa after stenting. Such increase is not only related to an increase in the area of high WSS but also to higher Chapter 3. Results 43 WSS levels, which increased by approximately 65%: going from a maximum of 49.4 Pa to 81.2 Pa on the high WSS region indicated by point B in Figure 19. Figure 21 shows a comparison of the OSI field on the pre- and post-stent surfaces of case 17. Like the changes in the WSS field, the OSI level increased after stent placement. The maximum OSI on the aneurysm increased after stenting, going from 0.34 to 0.37. The same occurred for the mean OSI, which increase from 0.012 to 0.015. 3.4 Case E1 Figures 22 and 23 show the WSS field and flow pattern for the pre- and post-stent geometries of case E1. Analyzing the WSS distribution on the pre-stent geometry, we note a region of high WSS on the side of the aneurysm indicated by points A and B. From the flow pattern (Figure 23) we identify this region as the FIZ, on which the flow from the Figure 22 – WSS field on the (a) pre-stent and (b) post-stent surfaces of case E1 at systolic peak. Point A and B indicate a region of high WSS due to flow impingement against the aneurysm neck and a region of high WSS on the aneurysm dome, respectively. Maximum and mean WSS on the pre- and post-stent aneurysm surfaces: 77.3 Pa and 10.8 Pa versus 47.4 Pa and 9.4 Pa, respectively. frontal view frontal view back view back view B A B A (a) (b) Source: prepared by the author. Chapter 3. Results 44 Figure 23 – Flow pattern in the (a) pre-stent and (b) post-stent geometries of case E1 at systolic peak. Point A and B indicate a splitting flow region and the flow jet entering the aneurysm; Point C indicates a large recirculation flow zone at the aneurysm fundus. (a) (b) B C A B C A back view back view Source: prepared by the author. Figure 24 – OSI field on the (a) pre-stent and (b) post-stent surfaces of case E1. Maximum and mean OSI on the pre- and post-stent aneurysm surfaces: 0.43 and 0.026 versus 0.38 and 0.013, respectively. frontal view frontal view back view back view OSImax ∼0.43 OSImax ∼0.38 (a) (b) Source: prepared by the author. Chapter 3. Results 45 parent artery impinges and splits into two streams: one flowing into the aneurysm and other into the daughter artery. Furthermore, point C indicates a recirculation flow zone at the aneurysm fundus, resulting in a large region of low WSS on the back of the aneurysm. After stent placement, the vascular remodeling straightened the arteries, increasing the blood flow rate into the daughter artery, therefore, decreasing the blood flow rate into the aneurysm, as seen in Figure 23b. Because of lower flow activity in the aneurysm, we note a reduction in the WSS levels on the aneurysm sac, which becomes more evident when comparing the maximum and mean WSS levels on the pre- and post-stent aneurysm surfaces: 77.3 Pa and 10.8 Pa versus 47.4 Pa and 9.4 Pa, respectively. Regarding the OSI field on the pre- and post-stent geometries (Figure 24), we readily note a reduction in the OSI levels, specially on the aneurysm back. After stenting, the maximum OSI on the aneurysm went from 0.43 to 0.38 and the mean OSI decreased by 50% (0.026 versus 0.013). Note also that while a lower flow activity in the aneurysm sac increased the extent of the low WSS region on the back of the aneurysm, it also decreased the OSI levels on the very same region, which in turn decreases the risk of rupture [25]. Finally, Table 3 shows a summary of maximum and mean values of both WSS and OSI on the pre- and post-stent aneurysm surface of each case presented above. Table 3 – Comparison of maximum and mean values of WSS and OSI on the pre- and post-stent aneurysm surfaces of case 11, case 16, case 17, and case E1. Case WSS (Pa) OSI (-) Maximum Mean Maximum Mean Case 11 pre-stent 28 6.3 0.46 0.042 post-stent 29.3 7.2 0.41 0.024 Case 16 pre-stent 47.3 17.1 0.32 0.007 post-stent 57.7 15.0 0.40 0.008 Case 17 pre-stent 49.4 11.5 0.34 0.012 post-stent 81.2 18.5 0.37 0.015 Case E1 pre-stent 77.3 10.8 0.43 0.026 post-stent 47.4 9.4 0.38 0.013 Source: prepared by the author. 46 4 DISCUSSION AND CONCLUSIONS Throughout the last decade, stent-assisted coil embolization of intracranial aneurysms have achieved widespread acceptance, with different variants being constantly proposed. In the two-step Y-stent coiling of bifurcation aneurysms, stenting and coiling are done in separate steps; in between these steps, stent-induced vascular remodeling has been speculated to play an important role, altering the hemodynamics in the aneurysm sac. However, it is still unknown whether this two-step approach could benefit from the vascular remodeling caused by the placement of stents. The current work sought to elucidate the role of stent-induced vascular remodeling in modifying the hemodynamics of bifurcation aneurysms. Table 4 shows a summary of our findings. Our results support previous suspicions reported in the literature, showing that in fact stent-induced vascular remodeling alters the hemodynamics in the aneurysm sac; moreover, vascular remodeling showed a variable effect in modifying the hemody- namics. The number of cases presented in the current study is not sufficient to draw definitive conclusions regarding the effects of stent-induced vascular remodeling on the hemodynamics of bifurcation aneurysms; therefore, interpretation of these findings requires caution. Nonetheless, an analysis of these cases may provide a better understanding of the consequences of hemodynamic changes in the aneurysm sac. Table 4 – Summary of the findings of the current work. WSS and OSI values expressing the percent variation after vascular remodeling. Aneurysm location Parent artery Stent configuration Maximum Mean Maximum Mean WSS (%) WSS (%) OSI (%) OSI (%) ACoA bifurcation left and right ACA single stent 80.8 60.2 8.8 25 MCA bifurcation MCA Y-stent 4.6 14.2 -10.9 -42.9 MCA bifurcation MCA Y-stent 21.9 -12.3 25 14.3 MCA bifurcation MCA single stent -38.7 -12.9 -11.6 -50 Source: prepared by the author. Since the first clinical reports on stent-induced vascular remodeling, authors have speculated that vessel straightening could promote a favorable hemodynamic environment in the aneurysm sac, contributing to the aneurysm healing process. However, in the present study one of the cases—at the ACoA bifurcation—showed an adverse hemodynamic response to the novel vascular geometry, contradicting those suspicions. ACoA bifurcation aneurysms are under the influence of two parent arteries: left and right branches of the ACA, both arriving at the ACoA bifurcation parallel to each other. By straightening the vessels, the novel vascular geometry may cause the parent arteries to lay in a perpendicular- like configuration. In our study, this was observed to induce a greater flow instability in Chapter 4. Discussion and conclusions 47 the aneurysm sac, increasing hemodynamic parameters associated with aneurysm rupture. Nonetheless, further investigation is necessary to determine whether this is a common behavior for aneurysms at the ACoA bifurcation. Recent studies have shown that a hands-up vascular configuration in bifurcation aneurysms—i.e., a Y-like vascular geometry—leads to rupture more often than other vascular geometries. In our study, two aneurysms at bifurcations of the MCA were treated with Y-stent technique, resulting in a Y-like vascular geometry. A Y-like geometry aligns the parent artery and aneurysm ostium; thus, producing a straight flow jet that impinges directly on the aneurysm dome. In both cases, this was observed to increase the maximum WSS levels at the flow impingement zone. An increase in WSS levels may accelerate its destructive effect on the aneurysm surface, increasing the risk of rupture. Both aneurysms treated with Y-stenting showed an adverse hemodynamic response to the novel Y-like vascular geometry. However, our results do not consider the presence of coils. In the literature, clinical reports have shown that Y-stent coiling is effective in healing the aneurysm, reducing rates of recanalization and recurrence. Moreover, numerical studies have shown that the effectiveness of Y-stent coiling is associated with a combined effect of vascular remodeling and the presence of a coil mass inside the aneurysm sac: the Y-like vascular geometry redirects the flow jet towards the aneurysm ostium, where the coil mass is located—an inert, nonbiologically active surface that is not affected by the destructive effect of high WSS levels. Therefore, considering these scenarios, it would not be recommended to perform Y-stenting and coiling in separate steps. One of the aneurysms at the MCA bifurcation was initially treated with a single stent. In bifurcation aneurysms, placement of single stent leads to straightening of the parent-daughter artery segment hosting the stent, which approximates the novel vascular geometry to the one of a sidewall aneurysm. In sidewall aneurysms, stent-induced vascular remodeling has been shown to reduce the blood flow rate into the aneurysm, which reduces flow activity in the aneurysm sac. In this case, we observed a similar effect. Straightening of the daughter artery led to a reduction in the blood flow rate into the aneurysm, which reduced the flow activity in the aneurysm sac. Moreover, due to lower flow activity in the aneurysm sac, WSS and OSI levels decreased by up to 50%, which may delay the destructive effect of the WSS and OSI on the aneurysm surface. One of the limitation of the present study is related to the absence of stent struts covering the aneurysm ostium. Stent struts act as momentum sinks, reducing flow velocity inside the aneurysm sac; therefore, modifying the intraaneurysmal hemodynamics. Another limitation of the present study is related to the assumption of rigid vessel walls, which have been proven to influence the intraaneurysmal hemodynamics. However, due to the presence of a continuous outward-directed radial force, stents may increase stress in the vessel wall and interfere in wall functions—specially wall movement. Therefore, further investigation is necessary to elucidate the influence of stents on vessel wall movement. Chapter 4. Discussion and conclusions 48 Future works should explore the presence of stent struts covering the aneurysm ostium in order to achieve a better understanding of the combined influence of stent-induced vascular remodeling and stent struts on intraaneurysmal hemodynamics. In addition, the assumptions of fixed outlet pressure and blood as Newtonian fluid should be re-evaluated. Different assumptions—such as non-Newtonian fluid models and time-varying pressure conditions—have been proposed in the literature and should be explored. Despite the limitations of the present study, we were able to elucidate aspects related to stent-induced vascular remodeling that were previously unknown. In conclusion, hemodynamics change in intracranial aneurysms due to stent-induced vascular remodeling. Therefore, stent-assisted treatments may benefit—or not—from the novel vascular remod- eling. In two-step Y-stent coiling, placement of a single stent in an earlier step reduces hemodynamic parameters related to aneurysm rupture during the “healing-in” process of the stent. Another important conclusion of the current work regards the variable effect of stent-induced vascular remodeling in modifying the hemodynamics. 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Patankar. “Numerical heat transfer and fluid flow”, CRC press, 1980. https://doi.org/10.3174/ajnr.A2461 https://doi.org/10.3174/ajnr.A2461 https://doi.org/10.3174/ajnr.A2461 https://doi.org/10.1016/S0376-0421(02)00005-2 https://doi.org/10.1016/S0376-0421(02)00005-2 https://doi.org/10.1016/j.jbiomech.2012.07.030 https://doi.org/10.1016/j.jbiomech.2012.07.030 https://doi.org/10.3174/ajnr.A3894 https://doi.org/10.3174/ajnr.A3793 https://doi.org/10.3174/ajnr.A3793 https://doi.org/10.3174/ajnr.A3793 https://doi.org/10.3174/ajnr.A4263 https://doi.org/10.3174/ajnr.A4263 https://doi.org/10.3174/ajnr.A4263 https://isbndb.com/search/books/9780891165224 Appendices 56 A – MESH AND TIME-STEP INDEPENDENCE STUDY In a numerical simulation, we must assess the accuracy and reliability of the simulation results. To this end, validation and verification (V&V) present a number of recommenda- tions and guidelines [58]. In V&V, validation refers to physical test of the mathematical model adopted to describe the problem. Unfortunately, we were unable to validate the results due to lack of experimental data on aneurysms; we leave the task of setting up an physical test for blood flow in aneurysms as a suggestion for future works. On the other hand, verification aims to assess the accuracy and correctness of the mathematical model implementation, i.e., temporal and spatial resolution, discretization method of the governing equations, and code implementation. Regarding code imple- mentation, OpenFOAM uses the FVM, a well established and consolidated method for discretizing partial differential equations in the form of algebraic equations. Furthermore, OpenFOAM has an active community of developers and is widely used in both academic and industrial research, being constantly audited, tested, and benchmarked. For these reasons, we consider the code verified. To assess numerical accuracy of temporal and spatial resolution, we carried out a mesh and time-step independence study considering two parameters: the spatial distribution of WSS on the surface at systolic peak, which represents the instant of maximum blood flow rate in the parent artery; and the area-averaged WSS over the wall surface, defined as WSSS = 1 AS ∥∥∥∥∥ ∫ S W SS dS ∥∥∥∥∥ 2 , (A.1) where S is the wall surface and AS its area. Further, because of the unique flow pattern due to geometric features of the aneurysm [45], we share the idea that each patient-specific geometry requires its own mesh and time-step independence study [59]. Therefore, we performed a separate analysis for each one of the cases in our research, as presented in the following sections. A.1 Case 11 Post-stent Figure 25 shows the WSS field on the post-stent surface of case 11 for meshes with ∼ 400,000, ∼ 750,000, and ∼ 1,500,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. Observing the time-steps 10−4 s and 10−5 s, we note qualitative differences in the WSS field; on the other hand, we see the same WSS field when comparing time-steps A. Mesh and time-step independence study 57 Figure 25 – Mesh and time-step independence analysis for the post-stent geometry of case 11: WSS field on surfaces of meshes with ∼ 400,000, ∼ 750,000, and ∼ 1,500,000 control volumes and time-steps of 10−4 s, 10−5 s, and 5× 10−6 s. 10 − 4 s 10 − 5 s 5 × 10 − 6 s ∼400,000 CVs ∼750,000 CVs ∼1,500,000 CVs Source: prepared by the author. 10−5 s and 5× 10−6 s, suggesting that a time-step of 10−5 is adequate to represent the blood flow in this geometry. Regarding spatial resolution, Figure 25 shows a similar WSS field when comparing the meshes with ∼ 400,000, ∼ 750,000, and ∼ 1,500,000 for a tim