Importance of incorporating systemic cerebroarterial hemodynamics into computational modeling of blood flow in intracranial aneurysm ＊

Zhi-qiang Zhang, Li-jian Xu, , Rong Liu, Xiao-sheng Liu, Bing-Zhao, Fu-you Liang, ,

1.School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240,China

2.Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai 200240,China

3.Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Hong Kong, China

4.Department of Radiology, Ren Ji Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200127, China

5.Department of Neurosurgery, Ren Ji Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200127, China

6.Institute for Personalized Medicine, Sechenov University, Moscow 119991, Russia

Abstract: The importance of properly treating boundary conditions (BCs) in numerical simulation of hemodynamics in intracranial aneurysm (IA) has been increasingly recognized.In this study, we constructed three types of computational model for each IA to investigate how the outcome of numerical simulation is affected by the treatment of BCs.The first type of model (i.e., Type-A model)was obtained by applying 3-D hemodynamic modeling to the entire cerebral arterial network, with its solution being taken as the reference for evaluating the performance of the other two types of model (i.e., Type-B and Type-C models) in which 3-D modeling was confined to the aneurysm region.In addition, patient-specific 1-D models of the cerebral arterial network were developed to provide hemodynamic information for setting the inflow/outflow BCs of the 3-D models.Numerical tests on three IAs revealed that prescribing the outflow BCs of a localized 3-D aneurysm model based on 1-D model-simulated outflow division (i.e., Type-B model)instead of imposing the free outflow BC on all outlets (i.e., Type-C model) helped to improve the fidelity of the simulation of intra-aneurysmal hemodynamics, but could not guarantee a complete reproduction of the reference solution obtained by the Type-A model.Moreover, it was found that the outcome of hemodynamic simulation was more sensitive to the treatment of BCs when an aneurysm was located at arterial bifurcation rather than sidewall.These findings highlight the importance of taking into account systemic cerebroarterial hemodynamics in computational modeling of hemodynamics in IAs, especially those located at bifurcations.

Key words: Intracranial aneurysm, systemic cerebroarterial hemodynamics, boundary conditions, computational model

Introduction

Intracranial aneurysm (IA) is a common vascular disease that affects 0.5% to 6% of the adult population[1].A major endpoint event of IA is sudden rupture of the pathologically weakened aneurysm wall, which often causes fatal subarachnoid hemorrhage that has been found to result in a mortality rate of 7%-10% and the presence of severe sequelae in 11%-29% of survivors[2].The natural history of an untreated IA is complicated and often exhibits highly patient-specific characteristics, with a population-averaged rupture risk of 1%-2% per year[3].For instance, about 30% of untreated IAs can maintain an unruptured state over decades, while others may suddenly rupture at an unpredictable time[4].Clinically, IAs can be treated by interventional operations (e.g., surgical clipping,endovascular coil embolization) to reduce the risk of rupture[5], however, the balance between clinical benefits and complications associated with invasive interventions remains debatable[6].In this context,identifying aneurysms that have a high risk of rupture and are more likely to benefit from interventional operations has long been a critical issue.Accordingly,a large number of clinical studies have been devoted to exploring the potential value of some specific morphological features (e.g., size ratio, aspect ratio) of aneurysm for predicting the risk of aneurysm rupture[1,7-8].In recent years, computational fluid dynamics (CFD) methods were increasingly being applied to provide additional biomechanical information for assessing the risk of aneurysm rupture,understanding the pathogenesis of aneurysm, or assisting the optimization/design of clinical treatments[9-17].For instance, CFD studies comparing hemodynamics in ruptured vs.unruptured IAs have revealed that high area proportion of low timeaveraged wall shear stress (WSS)[18], high maximum WSS[19]and low pressure loss coefficient[20]associate with increased risk of aneurysm rupture.A major advantage of computational modeling methods is the ability to quantify intra-aneurysmal hemodynamic variables, especially those in the near-wall region or small IAs for which in vivo measurements are either unavailable or subject to unacceptable large errors due to limited spatio-temporal resolutions[21].On the other hand, computational modeling has its own limitations.For instance, most computational studies on IAs confined hemodynamic modeling to local aneurysm regions[9-10], rendering the outcomes of hemodynamic simulations dependent strongly on the treatment of boundary conditions.

Anatomically, IAs are located in a special arterial territory (i.e., cerebral arterial network) with unique anatomical features, such as the presence of the circle of Willis (CoW) whose anatomical structure differs considerably from one person to another[22].Both experimental and computational studies have demonstrated that anatomical variations of the CoW remarkably alter flow distribution in the cerebral arterial network[23-25].Accordingly, for computational studies aimed to reproduce intra-aneurysmal hemodynamics using models localized to the aneurysm region, the boundary conditions must be prescribed on the basis of a full account of hemodynamic conditions in the entire cerebral arterial network.So far, 1-D modeling, which is computationally inexpensive and permits easy incorporation of many physiological factors (e.g., wall elasticity, tone of distal vessels), has been widely employed to address the large-scale features (e.g., flow distribution, pressure wave propagation) of cerebral hemodynamics[23-26].However, 1-D modeling is limited by the reduced-order representation of vascular and hemodynamic information[27-28],and hence cannot account for the details of blood flow,especially at bifurcations or in arterial segments with pathological changes (e.g., aneurysm, stenosis) where flow patterns are highly complex.The limitation can be compensated by modeling the cerebral arterial network in three dimension, which, however, will incur a high computational cost and therefore has been implemented only in few case studies[29-30].In this context, researchers have turned to exploring new modeling methods capable of integrating the respective advantages of 1-D modeling and 3-D modeling.For instance, Marzo et al.[31]examined the validity of applying the results of 1-D simulations to prescribe the boundary conditions of local 3-D aneurysm models, and Liang et al.[32]developed a multi-scale modeling method to couple a 3-D model of a local arterial segment with a 1-D model of the entire cerebral arterial network.A variety of other methods for coupling a local 3-D model to a reduced-order model of wider vascular region have been reported[33-34], although their applications were not limited to hemodynamic problems in IAs.Despite the useful methodological insights from these studies,it remains unclear what modeling methods can achieve an optimal balance between the reliability of hemodynamic simulation and computational cost.

The purpose of the present study was to investigate to what extent the simulated intraaneurysmal flow patterns are sensitive to modeling strategy.For this purpose, blood flows in IAs present in different regions of the cerebral arterial network were modeled with three methods.The first modeling method applied 3-D hemodynamic modeling to the entire cerebral arterial network, yielding the“reference solution” for verifying the results obtained with the other two (i.e., second and third) methods in which 3-D modeling was confined to the local aneurysm region.The second and third modeling methods differed in the way of setting the outflow boundary conditions.Results obtained with the three modeling methods were compared for each IA in terms of flow waveform, wall shear stress, and flow velocity.Moreover, intercrossing comparisons among the aneurysms were also carried out to investigate whether the location of aneurysm affects the sensitivity of simulated intra-aneurysmal flow patterns to modeling method.

1.Methodology

1.1 Model reconstruction, data extraction and mesh generation

Fig.1 (Color online) Schematic description of image-based model reconstruction, mesh generation and hemodynamic modeling for cerebral arteries and aneurysm in a patient.The radii and lengths of cerebral arteries derived from the medical images are used to construct a patient-specific 1-D hemodynamic model of the cerebral arterial network, with its output being utilized to set the boundary conditions of the 3-D hemodynamic models (which are classified into three types according to the range of 3-D hemodynamic modeling and the ways of setting outflow boundary conditions).The distributions of time-averaged wall shear stress predicted by the three types of 3-D hemodynamic model are illustrated

Upon ethical approval and written patient consent, magnetic resonance angiographic (MRA)data were acquired from three patients diagnosed to have IA.The MRA data of each patient were read into Mimics 16 (Materialise, Belgium) to perform segmentation and construct a 3-D geometrical model of the global cerebral arterial network (see Fig.1).Each 3-D geometrical model was subsequently processed in three ways.Firstly, the geometrical parameters (i.e.,radius and length) of each cerebral artery were extracted using the analysis module of Mimics.Note that arterial radius was extracted discretely along the arterial axis at an interval of 1 mm and that the abrupt changes in lumen area in the vicinity of an aneurysm were not extracted since incorporating such data into a 1-D model will induce numerical instability.Secondly,the aneurysm region was isolated from the global geometrical model to create a local model.Finally, the global and local geometrical models were each imported into ICEM CFD 14.5 (Ansys, Inc., USA) to generate mesh models.Prior to mesh generation, the original geometrical model was modified by adding straight tubes (with their lengths being 10 times of the corresponding lumen diameters) to the proximal and distal ends.Herein, the extension tubes play a role of reducing the influence on the simulation of intraaneurysmal hemodynamics from artifacts introduced by the prescription of boundary conditions (e.g., fixed flow velocity profile that cannot fully represent the 3-D distribution of velocity components in the truncated arterial cross sections).To investigate the hemodynamic effects of varying the length of extension tube, we performed numerical tests on the Type-B model (model types will be described in detail later) of the aneurysm detected in a patient (patient No.3) by generating three geometrical models with different extension tubes (i.e., “model 1”, which has no extension tubes, “model 2” and “model 3”, which have extension tubes with their lengths being ten and twenty times of the corresponding lumen diameters,respectively).Obtained results showed that the value of the time-averaged WSS in the aneurysm simulated with “model 2” differed from that simulated with“model 1” by 9.3%, but was close to that simulated with “model 3” (with their difference being 0.8%),which indicates that setting the lengths of the extension tubes to be ten times of the lumen diameters enables the numerical solution of intra-aneurysmal hemodynamics to be less affected by the artificial prescription of flow velocity distribution at model inlets/outlets.When generating the mesh model, the entire fluid domain was firstly divided by tetrahedral elements with a maximum size of 0.4 mm and a minimum size of 0.2 mm.Subsequently, the mesh model was refined by mapping seven prism layers along the vascular walls to help improve the accuracy of flow computation in the near-wall region[35-36].Numerical tests performed on the model containing an aneurysm at the left internal carotid artery showed that reducing the minimum mesh size from 0.2 mm to 0.1 mm led to less than 2% changes in simulated wall shear stress (averaged spatially over the aneurysm sac).Therefore, the adopted mesh density was considered sufficient to yield numerically acceptable results.The total number of elements contained by each mesh model ranged from 0.7×106to 4.2×106n depending on the size and morphology of the model.

1.2 1-D hemodynamic modeling of the cerebral arterial network

Previous studies have demonstrated that cerebroarterial flow waveforms simulated by a 1-D model of the cerebral arterial network are comparable to those by a 3-D model[37], and suggested that the boundary conditions of a local 3-D aneurysm model, when not acquirable from in vivo measurements, can be predicted by a patient-specific 1-D model of the cerebral arterial network[26,31].Therefore, patient-specific 1-D modeling of the cerebral arterial network was herein carried out to simulate blood flow waveforms in large cerebral arteries that will be used to prescribe the inflow/outflow boundary conditions of the 3-D aneurysm models.

1-D governing equations for blood flow in an artery were obtained by integrating the 3-D Navier-Stokes equations over the artery’s cross section based on a set of assumptions, such as axisymmetric cross section, uniform pressure distribution over the cross section, and cross-sectional velocity profile dominated by the axial components[27,38].

wheretis the time,zthe axial coordinate, andρthe blood density.A,QandPrepresent the cross-sectional area, volume flux and pressure,respectively.αandfare the momentum-flux correction coefficient and the friction force per unit length, and were set respectively to 4/3 and 8πν(νis the kinematic viscosity of blood) by assuming a Poiseuille velocity profile in the cross section of artery.

The system of Eqs.(1), (2) was closed by a constitutive equation that describes the pressure-area(P-A) relationship of artery[27-28,38-39]

whereP0is the reference pressure (herein set to 85 mmHg),Eis the Young’s modulus,his the wall thickness,r0is the radius of artery at the reference pressure, andσis the Poisson’s ratio, herein taken to be 0.5 by assuming that the materials of arterial wall are incompressible.

To link hemodynamic variables in adjacent arteries, continuity of mass flux and total pressure was imposed at each arterial junction[27,38].

where the subscript “1” denotes the parent artery, “2”,“3” the corresponding daughter arteries.

It is noted that an aneurysm present in a cerebral arterial segment is usually featured by nonaxisymmetrical lumen shape, sudden changes in lumen area, and complex intra-aneurysmal flow patterns (e.g., vortex and secondary flow) that cannot be fully represented by the 1-D modeling method in which circular lumen shape and fixed hemodynamic distribution in vascular cross section must be assumed.For this reason, all arterial segments with aneurysms have been assumed to have a normal axisymmetrical lumen in 1-D modeling, which amounts to removing the aneurysms from the arterial segments and may lead the simulated results to deviate from the real hemodynamic conditions.Limitations arising from the simplification will be discussed in detail later.

The resulting 1-D model of the cerebral arterial network was subsequently integrated into a global model of the cardiovascular system developed in our previous studies[28,38]by connecting the afferent cerebral arteries to the 1-D model of the systemic arterial tree and the efferent cerebral arteries to lumped-parameter (0-D) models of distal cerebral vasculatures.In this way, the boundary conditions of the 1-D cerebroarterial model could be spontaneously obtained through solving the entire model system.

The radii and lengths of cerebral arteries represented by the 1-D model were assigned patientspecifically based on the geometrical data derived from medical images.With regard to the assignment of parameters used in the global cardiovascular model,since most of them could not be determined in a patient-specific manner based on clinical data available in the present study, they were firstly assigned based on the population-averaged data reported in the literature[23]and subsequently adjusted according to the age of each patient using the method proposed in a previous study[38].Moreover, the vascular resistances distal to the cerebral efferent arteries were tuned with a linear parameter optimization method[24,32]so that the model-simulated total cerebral perfusion and flow division among cerebral efferent arteries both fell in the ranges of the population-averaged data reported in the literature[40-41].

The governing equations (Eqs.(1)-(3)) of the 1-D cerebroarterial model were solved with the two-step Lax-Wendroff method.The equations (Eqs.(4), (5))that govern hemodynamic conditions at arterial junctions were solved by means of a “ghost-point”method implemented in combination with the Newton-Raphson method[42].For more details on numerical methods employed to solve the governing equations of the global cardiovascular model, please refer to a previous study[28].

1.3 3-D hemodynamic modelling

3-D hemodynamic modeling was performed to gain detailed information on blood flow (e.g., wall shear stress, flow velocity) in local arterial segments of interest (e.g., in the vicinity of aneurysm).3-D blood flow was governed by the unsteady Navier-Stokes equations

whereurepresents the flow velocity vector,pis the blood pressure.Herein, blood flow was assumed to be incompressible and Newtonian, and, accordingly,ρa(bǔ)ndνwere fixed at 1 060 kg·m-3and 0.0035 Pa·s, respectively.In addition, all 3-D models were assumed to have rigid walls on which the no-slip flow conditions were imposed.

For each patient, three types of 3-D model were constructed (see Fig.1).One model accounted for the 3-D flow patterns in the entire cerebral arterial network, herein termed as global model and denoted by “Type-A model”, whereas the other two described blood flow in the vicinity of the aneurysm but with different outflow boundary conditions, herein termed as local models and denoted by “Type-B model” and“Type-C model”, respectively.The inflow conditions of all models were prescribed based on the flow waveforms simulated by the patient-specific 1-D model of the cerebral arterial network.As for the prescription of outflow boundary conditions, two approaches were adopted.The first approach followed from the method proposed in a previous study[43],where the total inflow waveform (obtained by adding up flow waveforms at all inlets) was distributed to the outlets based on the 1-D model-simulated flow division among outflow arteries (except for one outlet to which the free outflow condition (i.e., zero static pressure and zero normal gradient of other flow variables) was imposed to maintain the consistency of numerical simulation).In the second approach, the free outflow condition was imposed on all the outlets,which, in essence, discarded the information of outflow division predicted by the 1-D model of the cerebral arterial network.In this study, the first approach was applied to both Type-A and Type-B models, whereas the second approach was applied solely to the Type-C model.Note that when fixing a flow waveform at the inlet or outlet of a model, the distribution of flow velocities in the normal direction was assumed to obey the Poiseuille law.

The 3-D governing equations of blood flow were numerically solved along with the boundary conditions using a commercial CFD package (ANSYS-CFX 16).Second-order schemes were adopted for both spatial discretization and time integration.The numerical time step was fixed at 0.001 s.At each time step, convergence of numerical solution was judged when the residuals of the mass and momentum conservation equations both went below 10-3.Each set of simulation was continuously run for three cardiac cycles to wash out the effects of initial conditions,with the results obtained in the last cardiac cycle being analyzed and reported.All the numerical simulations were run on a Dell workstation (Precision T5610),with the simulation time varying from 8.5 h to 24.5 h depending on the complexity of each specific model.

1.4 Data analysis

The results from 3-D hemodynamic simulation were analyzed and visualized in terms of wall shear stress, streamline and floe velocity.Moreover, the time-averaged wall shear stress (TAWSS) and oscillatory shear index (OSI) were calculated to evaluate the mean magnitude and oscillation of wall shear stress over a cardiac cycle.

Here WSS represents the wall shear stress vector,Tthe period of a cardiac cycle.

Table 1 Geometrical data of large cerebral arteries extracted from the medical images of three patients.“r ” denotes the radius of a cerebral artery and “L ” the length.Note that r varies along the axis of each artery and therefore is expressed in form of mean±SD

2.Results Results

2.1 Geometrical data of cerebral arteries and 1-D model-simulated flow waveforms

The geometrical data (i.e., radius and length) of large cerebral arteries derived from the medical images of the three patients are summarized in Table 1.Note that the radius of each individual artery is expressed in the form of mean±SD since it varies along the artery axis.It is evident that the geometrical data differ remarkably among the three patients.

Incorporating the geometrical data into the 1-D model of the cerebral arterial network enabled us to patient-specifically simulate blood flows in any cerebral arteries of interest.Figure 2 compares the simulated flow waveforms (herein, flow waveform represents the time-dependent variation of flow rate(Q) during a cardiac cycle) in three representative cerebral arteries (see the lower right panel for their locations) for the three patients.It was observed that the simulated flow waveforms differed remarkably among the patients, indicating that the geometrical parameters of cerebral arteries significantly affect both flow waveforms in individual cerebral arteries and flow distribution in the cerebral arterial network.

2.2 Comparison of simulated hemodynamic variables with three types of 3-D model

The distributions of simulated TAWSS and OSI in the vicinity of aneurysm with the three types of 3-D model (i.e., Type-A model, Type-B model and Type-C model) are illustrated in the upper panels of Figs.3-5 for the three patients, respectively.The Type-B and Type-C models were both observed to yield numerical results deviating considerably from those obtained by the Type-A model.The degree of deviation differed considerably among the three patients.For instance, the smallest deviation was observed for the patient (i.e., patient #2) with a sidewall aneurysm at the left internal carotid artery(see Fig.4), whereas the largest deviation was predicted for the patient (i.e., patient #3) with an aneurysm at the anterior communicating artery (see Fig.5).Moreover, the degrees of differences in simulated TAWSS and OSI by the three types of model were found to vary spatially and exhibit aneurysm-specific characteristics.From the flow streamlines and velocity contour maps (in planes cutting through the aneurysms) visualized at peak inflow velocity (in the lower panels of Figs.3-5),marked differences in simulated flow field by the three types of model appeared in both the aneurysm sac and the adjacent arteries, which may partly account for the inter-model differences in the predictions of TAWSS and OSI.If the results obtained by the Type-B model and Type-C model were compared with respect to the degree of agreement with the ‘reference solution’ (i.e., solution of the Type-A model), the Type-B model exhibited an overall better performance than the Type-C model.

Fig.2 (Color online) Comparison of simulated flow waveforms in three representative cerebral arteries (see the lower right panel for the locations of the arteries) by the 1-D cerebroarterial models of three patients (R.ICA1-proximal segment of the right internal carotidartery,L.MCA-left middle cerebral artery, L.ACA1-proximal segment of the left anterior cerebral artery)

To further explore factors underlying the differential outcomes of the three modeling methods,the 3-D model-simulated flow waveforms in the adjacent arteries of the aneurysms were compared (see Fig.6).Overall, the flow waveforms simulated by the local models (i.e., Type-B and Type-C models)differed considerably from those by the global model(i.e., Type-A model).Exceptions were the flow waveforms in some inflow arteries (e.g., R.ICA and L.RCA in patient #1 and patient #2, respectively)where the 1-D model-simulated flow waveforms have been applied directly to set the inflow boundary conditions of all the three types of 3-D model, thereby causing the simulated flow waveforms by the three 3-D models to be always identical.With regard to flow waveforms in outflow arteries, the predictions of the Type-B model were overall closer to those of the Type-A model in comparison with the Type-C model,which may explain why the 3-D hemodynamic quantities predicted by the Type-B model exhibited better agreements with the “reference solution”.

3.Discussion

The significant advancements in medical imaging and computing techniques in the past decades have greatly promoted the application of computational methods to address various hemodynamic problems related to risk assessment or clinical treatment of IAs.In the meantime, a critical issue has emerged as to how and to what extent computational models can reproduce patient-specific in vivo hemodynamics[10,44].So far, while the majority of existing studies focused on blood flows in local aneurysm regions of interest,blood flows in other cardiovascular portions(especially the cerebral circulation), which potentially affect the inflow/outflow conditions of aneurysms via local-global hemodynamic interaction, have been ignored or oversimplified when building computational models.In the present study, we compared the outcomes of three types of computational model in which the inflow/outflow conditions of aneurysm are treated in different degrees of fidelity with respect to the incorporation of systemic hemodynamics.Major findings from the numerical experiments on three patient-specific aneurysms include: (1) in comparison with the imposition of the free outflow condition on all the outlets of a local 3-D aneurysm model (i.e.,Type-C model), prescribing the outflow boundary conditions based on 1-D model-simulated flow division (i.e., Type-B model) helped to improve the prediction of intra-aneurysmal hemodynamics, but was not sufficient to enable the local aneurysm model to completely reproduce the reference results obtained by applying 3-D modeling to the entire cerebral arterial network (i.e., Type-A model), (2) the discrepancies in numerical results obtained by the three types of model not only differed among aneurysms but also varied spatially in the aneurysm region, and (3) the sensitivity of numerical results to modeling method was much higher for the bifurcation aneurysms (in patient #1 and patient #3) than the sidewall one (in patient #2).

Fig.3 (Color online) Comparison of hemodynamic quantities in the vicinity of an aneurysm located at the right ICA-MCA bifurcation (in patient #1) predicted by the three types (i.e., Type-A, Type-B and Type-C) of 3-D hemodynamic model.The upper panels show the distributions of TAWSS (time-averaged wall shear stress) and OSI (oscillatory shear index),and the lower panels show the flow streamlines and velocity contour maps in a plane cutting through the aneurysm sac visualized at peak inflow velocity

Although applying the free outflow boundary condition to all the outlets of a local aneurysm model can largely simplify the modeling work and has been widely adopted in previous studies, it essentially discards the hemodynamic interaction between the aneurysm and distal vasculatures and hence may cause the simulated hemodynamic quantities to deviate significantly from their in vivo counterparts,especially for bifurcation aneurysms where the intraaneurysmal blood flow mingles with blood flows in multiple outflow arteries.The results presented in Fig.6 clearly show that the Type-C model-predicted flow waveforms in the outflow arteries of the aneurysms differ remarkably from those predicted by the Type-A model.Some researchers have recognized the limitations of the free outflow boundary condition and in turn proposed the use of the resistance outflow boundary condition derived from the Murray’s law[45].The resistance boundary condition, though expected to more reasonably incorporate the influence of distal vasculatures on outflow division compared with the free outflow boundary condition, cannot account for blood flow distribution in the cerebral arterial network[43], and therefore may not be well suited to aneurysm models whose outflow arteries terminate in the circle of Willis.In this regard, setting the outflow division of a local aneurysm model based on the flow distribution predicted by a patient-specific 1-D model of the entire cerebral arterial network may offer a more reasonable approach, although the obtained results still deviate from the “reference solution” due to the inherent limitations of 1-D modeling.

Fig.4 (Color online) Comparison of hemodynamic quantities in the vicinity of a sidewall aneurysm located at the left ICA (in patient #2) predicted by the three types (i.e., Type-A, Type-B and Type-C) of 3-D hemodynamic model.The upper panels show the distributions of TAWSS and OSI, and the lower panels show the flow streamlines and velocity contour maps in a plane cutting through the aneurysm sac visualized at peak inflow velocity

Fig.5 (Color online) Comparison of hemodynamic quantities in the vicinity of an aneurysm located at the ACoA (in patient #3)predicted by the three types (i.e., Type-A, Type-B and Type-C) of 3-D hemodynamic model.The upper panels show the distributions of TAWSS and OSI, and the lower panels show the flow streamlines and velocity contour maps in a plane cutting through the aneurysm sac visualized at peak inflow velocity

The discrepancies in simulated intra-aneurysmal flow patterns by the Type-A model and Type-B model are attributable primarily to the differential performances of the 1-D model and 3-D model (i.e., Type-A model) of the cerebral arterial network in simulating flow waveforms in the adjacent arteries of aneurysm(see Fig.6).1-D modeling differs significantly from 3-D modeling in the level of detail in representing arterial geometry and hemodynamic variables.For instance, when building a 1-D model, one often needs to introduce the simplification of arterial geometry(e.g., circular shape of arterial lumen, ignorance of vascular tortuosity) and assumptions on spatial hemodynamic distribution (e.g., uniform pressure distribution and fixed profile of axial velocity in the cross section, absence of flow velocity components in the radial and circumferential directions)[23,27].Moreover,the anatomically existing aneurysm has been artificially removed when constructing a 1-D model for the cerebral arterial network in the present study because the complex flow patterns (e.g., vortex,secondary flow) and abrupt lumen-area changes in the aneurysm region are beyond the capability of 1-D modeling.These factors will together compromise the fidelity of the 1-D model in simulating flow waveforms in cerebral arteries, especially those adjacent to the aneurysm.Therefore, applying the results of 1-D simulation to set the boundary conditions of a local 3-D aneurysm model has limitations in essence.Such limitations can be overcome by modeling the aneurysm along with the entire cerebral arterial network in a fully 3-D manner(i.e., the modeling strategy employed by the Type-A model), although the timing-consuming model construction/meshing works and high computational cost associated with 3-D modeling might be an issue of concern, especially for studies involving a large number of patients.On the other hand, the marked discrepancies in simulated flow waveforms in the adjacent arteries of some aneurysms between the aneurysm-removed 1-D model and the intact 3-D model imply that the formation of cerebral aneurysm may alter not only local flow patterns but also flow distribution among cerebral arteries and even the state of distal resistance vasculature.In this sense, it would be interesting to investigate the reciprocal influence between aneurysm growth and local/global hemodynamic alterations in follow-up studies.

Fig.6 (Color online) Flow waveforms in the adjacent arteries of three aneurysms (in three patients respectively) predicted by the three types (i.e., Type-A, Type-B and Type-C) of 3-D model.In the cases of Type-B model, the 3-D model-predicted flow waveforms were identical to the flow waveforms prescribed at the model inlets/outlets based on the 1-D model simulations.In patient #1 and patient #2, the three types of 3-D model predict the same flow waveforms in the ICAs because these waveforms have been prescribed as the inflow conditions of the models based directly on the 1-D model simulations.In patient #3, however, the inflow waveforms (in the L.ACA1 and R.ACA1) predicted by the Type-A model differ significantly from those by the Type-B and Type-C models because the flow waveforms in the L.ACA1 and R.ACA1 are obtained by solving 3-D blood flows in the entire cerebral arterial network in the case of Type-A model rather than by artificial prescription based on the 1-D model simulations in the cases of Type-B and Type-C models

Our study is subject to certain limitations.Firstly,the models were not fully validated using patientspecific in vivo data (such as flow rates and/or flow waveforms in large cerebral arteries measured with quantitative flow MRI[46]), which may have considerably compromised the fidelity of patient-specific hemodynamic simulation.Secondly, the elastic deformation of vascular/aneurysmal wall has been ignored in 3-D modeling by assuming all the walls to be rigid.Major reasons for introducing the assumption are twofold: (1) solving the fluid-structure interaction problem is extremely expensive and often incurs difficulties in numerical convergence when the computational domain covers the entire cerebral arterial network, (2) in vivo data necessary for patientspecifically assigning the parameters of vascular/aneurysmal walls are not available.The assumption may cause the numerical results to further deviate from in vivo conditions, thereby compromising the ability of the Type-A model to provide an accurate‘reference solution’ for evaluating the performance of the Type-B and Type-C models.Nevertheless, it has been demonstrated that the magnitude of the pulsatile motion of cerebral aneurysm wall is small (estimated to be ＜15% of the aneurysm height)[47], and that ignoring wall motion in hemodynamic simulation does not significantly alter the simulated characteristics of flow patterns in cerebral aneurysms although it can cause 8%-50% overestimates of the maximum wall shear stress in some local regions[48-49].In addition, the limitations associated with the rigid-wall assumption are expected to exert comparable influence on the outcomes of the three types of 3-D model, and hence would not substantially alter the major findings of the present study regarding the sensitivity of the simulation of intra-aneurysmal hemodynamics to the treatment of inflow/outflow boundary conditions.Finally, it is worthy of note that due to the small number of aneurysms involved, it remains unclear whether blood flows in aneurysms at the same site have the same degree of sensitivity to modeling method as reported in the present study.In particular, given that intra-aneurysmal flow patterns are sensitive to both flow distribution within the cerebral arterial network and the geometry of aneurysm that usually differ significantly among patients, larger-scale studies would be needed to further address the issue.

4.Conclusion

Three modeling methods have been compared with respect to their influence on simulated hemodynamic quantities in three IAs.It was found that the “reference solution” obtained by a fully 3-D model (i.e., global model) of the entire cerebral arterial network could not be completely reproduced by a local aneurysm model.Despite the inherent limitations of local aneurysm model, prescribing the outflow boundary conditions based on the outflow division predicted by a 1-D model of the cerebral arterial network was demonstrated to yield results closer to the “reference solution” than applying the traditional free outflow boundary condition.Moreover,the discrepancies between the outcomes of local and global models were found to be larger for the bifurcation aneurysms than the sidewall one.These findings suggest that systemic hemodynamic conditions in the cerebral arterial network have considerable influence on the local flow patterns in an IA, and such influence should be carefully addressed when building computational models to simulate blood flows in IAs, especially those located at bifurcations of cerebral arteries.

Acknowledgements

This work was supported by the Clinical Research Plan of SHDC (Grant Nos.16CR3031A,16CR2045B), the SJTU Medical-Engineering Crosscutting Research Foundation (Grant Nos.YG2015MS53, YG2017MS45).