Introduction

Computational simulations of blood flow in cerebral aneurysm is an active field of research [10, 11, 13, 17, 18, 23, 25, 30–32] motivated by the potential of assessing aneurysmal rupture risk through knowledge of hemodynamic parameters. Of those parameters, wall shear stress (WSS) magnitudes have gained particular interest: a recent study found significant larger wall areas with lower WSS in ruptured versus unruptured aneurysms of the paraclinoid and supraclinoid section of the internal carotid artery [19]. Other studies have also linked low WSS to aneurysm formation or risk of rupture [1, 20, 28]. In particular, the gradient oscillatory index [35] has recently been demonstrated to have a significant correlation with the location of aneurysm formation.

It has been suggested that WSS modulates the endothelial cell phenotype and that a WSS magnitude of less than 1.5 Pa may lead to the degeneration of the arterial wall [27]. While under normal conditions, non-pathological tissue remodeling may occur, the combination of abnormal flow patterns and genetic predisposition could lead to formation and rupture of a cerebral aneurysm [34]. To quantify WSS, computational fluid dynamics (CFD) in combination with geometrical data derived from rotational 3D digital subtraction angiographic (DSA) image data of the aneurysm and the surrounding vasculature are increasingly being used to obtain patient-specific results. The need for patient-specific geometrical data has been demonstrated by studies that reported strong variations in the inflow patterns and distribution of WSS on the aneurysmal wall dependent on the shape of the proximal segment of the parent artery [2, 10, 17, 26, 29, 36].

Besides morphology and geometry of the aneurysm and the parent artery, the volumetric flow waveform is another boundary condition for CFD simulations. Recently, it has been demonstrated that inflow variations into an aneurysm of the anterior communicating artery with two separate inflows may result in WSS changes as high as 43% [24]. The effects of WSS in aneurysm with only one inflow have not yet been systematically assessed. The majority of CFD studies on cerebral aneurysms have relied on averaged or idealized flow waveforms [2, 3, 5–8, 12, 16, 17, 28], and only selected CFD studies so far have used measured flow information [1, 20, 22, 23]. Thus, it appears important to provide an estimate for the uncertainty of WSS made by this simplification.

To explore the importance of patient-specific inflow information in CFD simulations of cerebral aneurysms with single inflow, we quantified differences of the temporal average of the WSS magnitude (<WSS>), its temporal variation (ΔWSS), and the oscillatory shear index (OSI, a measure for the variation of the direction of the WSS vector over time) in CFD simulations on geometrical models of six aneurysms of the paraclinoid and supraclinoid sections of the internal carotid artery (ICA). In contrast to previous studies [24], all aneurysms investigated only had one inflow. Factors such as different strengths between inflow jets or a time difference between inflows (asymmetry in the inflow pattern) will therefore not occur, and potential differences in aneurysm hemodynamics and aneurysm wall shear stress will be due to the shape of the inflow waveform. Two inflow conditions were investigated: (1) idealized volumetric inflow waveform (ID) and (2) patient-specific inflow waveforms (PS) derived from 2D phase contrast magnetic resonance (2D pcMRI) measurements [38].

Materials and methods

Approval of the institutional review board was obtained for this study.

Patient data

Clinical images from diagnostic examinations of six unruptured saccular aneurysms at paraclinoid or supraclinoid sections of the internal carotid artery (ICA) in six patients (mean age 62 years, range 43–76, 1 male, 5 female, range of largest diameter 7.6–37.4 mm) were retrospectively collected for this study. Three of these aneurysms were terminus aneurysms (case 1, case 3, and case 6), and the remaining aneurysms were curvature aneurysms (case 2, case 4, and case 5).

2D phase contrast magnetic resonance imaging

Prior to endovascular treatment, all subjects underwent an MRI exam (1.5 T Sonata, Siemens Medical Solutions, Erlangen). From a 3D surface reconstruction based on time-of-flight localizer images (in-plane resolution 0.43 × 0.43 mm, slice thickness 0.5 mm), a vessel segment proximal to the aneurysm was selected for the flow measurements [39]. Dependent on the length of the cardiac cycle, 12–20 pcMRI images were acquired during the cardiac cycle (in-plane resolution 0.63 × 0.63 mm, slice thickness 5 mm; repetition time ranged from 41 to 82 ms). Volumetric blood flow rates were calculated by multiplying the cross-sectional vessel areas determined in each image with the blood flow velocity. The average volumetric blood flow rate was obtained as average over the cardiac cycle.

Computational fluid dynamic calculations

Diagnostic 3D DSA images (Axiom Artis dBA, Siemens Medical Solutions, Forchheim) were collected retrospectively. A 3D stereolithographic file of the aneurysm and the parent artery was constructed by single-value thresholding after image data were convoluted with a 3D Gaussian kernel to remove image artifacts and noise (Visualization Toolkit, Kitware. Inc.). Guided by mesh-independence analyses obtained with similar aneurysm models, tetrahedral meshes were created (final mesh sizes, case 1, 211,355; case 2, 156,549; case3, 159,419; case 4, 218,810; case 5, 119,359; case 6, 282,536 volume elements). Transient Navier–Stokes equations were solved using a pressure-based, double-precision implicit solver with second-order discretization for pressure and momentum (ANSYS Fluent Inc.) [22]. Blood was modeled as an incompressible fluid with a density of 1,050 kg/m3 and a viscosity of 0.004 kg/ms. Convergence of the simulations was monitored by the continuity residual and the residuals of the velocity components.

Transient simulations were performed for each case with two different waveforms: (1) an idealized waveform (ID) adapted from Ford et al. [9] as an average waveform of 17 young, normal volunteers measured with 2D pcMRI and (2) the measured patient-specific volumetric flow rate waveform from each subject (PS) interpolated by cubic splines (Matlab, Mathworks Inc.). To ensure elimination of initial transients in the CFD simulations, three cardiac cycles were simulated, and results are reported from the third cardiac cycle.

CFD data analysis

From the results of the CFD simulations, the following parameters were calculated for conditions ID and PS:

  1. 1.

    <WSS>: the temporal mean WSS values as the average over the WSS values for one cardiac cycle.

  2. 2.

    ∆WSS: the standard deviation of <WSS> for one cardiac cycle.

  3. 3.

    OSI: the oscillatory shear index is a measure for the variation of the instantaneous WSS vector relative to the direction of the time-averaged WSS vector [14]. The OSI ranges from zero, if the instantaneous WSS vector is collinear with the average WSS vector throughout the cardiac cycle, to 0.5, if the time average of the instantaneous WSS vector is zero (no preferred direction).

Parameters 1–3 were calculated on the surface of the whole computational model. In the comparison analysis, only the aneurysm wall was considered to exclude computational inaccuracies due to inflow or outflow effects at proximal and distal artery segments. For that purpose, the aneurysm was separated from the remaining of the computational mesh by an operator-defined cut plane. At each surface point (node), first parameters 1–3 were calculated for condition ID and for condition PS, and then the difference PS−ID was calculated. Relative differences (RD) (in percent) for each parameter were calculated by subtracting the value obtained with ID from the value obtained with PS and dividing the result by the average value from both simulation and multiplying by 100.

  1. 4.

    Histograms of WSS on the surface of the aneurysm: histograms of WSS have been previously used by Jou et al. to distinguish between stable and an unstable fusiform basilar aneurysms [20]. Histograms were only calculated for the aneurysm wall.

Results

Volumetric inflow waveforms

Magnitude, time of maximum flow, and length of the measured cardiac cycle varied between cases (Table 1). Average over all six cases, the duration of the cardiac cycle, minimum flow, and mean flow were in good agreement (within 1 standard deviation, except for time of maximum flow) with the corresponding values of the idealized volumetric flow waveform (Table 1).

Table 1 Characteristic parameters of the volumetric inflow waveforms proximal to the aneurysms measured with 2D pcMRI and of the idealized waveform
Full size table

Differences in WSS parameters between conditions ID and PS

Absolute mean and maximum values are given in Table 2 for the aneurysm wall. Minimum values were close to zero for all parameters, and cases and are not shown.

Table 2 Mean and maximum values (in parenthesis) of <WSS>, ΔWSS, and OSI on the aneurysm wall obtained from simulations with the patient-specific inflow waveforms (PS) and the idealized waveform (ID)
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RD (Table 3) of mean <WSS> ranged from (−32%) to 11%. RD for mean ∆WSS were even larger ((−100%) to 67%); however, absolute values for this parameter were found to be relatively low compared to <WSS> (Table 2) not exceeding 0.1 Pa. Mean average RD of OSI values ranged from −12% to 40% with absolute mean difference values not exceeding 0.01 (on a scale from 0 to 1). While mean difference values are small, pronounced differences in these parameters occurred spatially localized in regions on the aneurysm dome, (Fig. 1, center row). RD of maximum values for <WSS> averaged over the aneurysm wall ranged from (−35%) to 29% corresponding to several Pascal in absolute units. While RD of maximum values for ∆WSS and OSI were relatively large (Table 3), absolute differences were small (less than 1 Pa for ∆WSS and less than 0.06 Pa for the OSI) (for a graphical presentation, see Fig. 1, center row).

Table 3 Relative differences (RD) of mean and maximum (in parenthesis) of <WSS>, ΔWSS, and OSI on the surface of the aneurysm obtained from simulations with the patient-specific inflow waveforms (PS) and the idealized waveform (ID)
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Fig. 1
figure 1

Top row: Patient-specific (PS, blue) and idealized (ID, green) volumetric inflow waveforms for cases 1–6 used in the CFD simulations. Blue symbols mark times at which measurements were taken. Time (horizontal axis) is measured in milliseconds, flow rate (vertical axis) in milliliter per minute. For all cases, the measured patient-specific waveform (maximum flow, location of maximum, duration of cardiac cycle) deviated considerably from the idealized waveform. Center row: 3D surface reconstructions of the aneurysms for case 1–6 color coded by the relative differences (relative to the average of the mean values for conditions ID and PS) for <WSS>, ΔWSS, and OSI (patient specific − idealized). Color scale is consistent across cases. For all cases, spatial variations can be appreciated for the results obtained with patient-specific and with idealized flow waveforms. Largest variations are visible for <WSS> and ΔWSS. In particular, for cases 1, 2, 3, and 5, relative differences exceed 10% on large sections of the computational model. Smallest differences can be observed for the OSI; however, sharply localized regions exhibit relative differences well larger than 10% (case 2 and case 4) also at the aneurysm wall. Bottom row: Histograms <WSS>, ΔWSS, and OSI for the aneurysmal wall derived from the simulations using the patient-specific waveform (blue) and the idealized waveform (green). Units of the horizontal axis are Pascal for WSS and ΔWSS, OSI is a unit-less quantity. The units of the vertical axis for all three cases is “counts”. <WSS> showed smallest variations, for cases 1, 2, 4, and 6, histograms appear similar. Largest variations can be appreciated for ΔWSS resulting in entirely different histograms for cases 1 and 6

Full size image

Average RD of mean <WSS> for the group of the terminal aneurysms (case 1, case 3, and case 6) and the group of the curvature aneurysms (case 2, case 4, and case 5) were very similar, −13 ± 21% and −18 ± 15%, respectively, as were the average RD for the mean OSI (9 ± 27% and 6 ± 21%). Larger variation in the results were found between the two groups for the average RD of the mean ΔWSS (3 ± 62% versus −62 ± 36%) and for the maxima RD of each parameter (<WSS> 0.3 ± 32% versus −13 ± 11%, ΔWSS 13 ± 46% versus −65 ± 36%, OSI −6 ± 9% versus 8 ± 9%).

Histograms

The shape of the histograms for WSS, ΔWSS, and the OSI on the aneurysm surface differed in all cases. The presence of large local variations in these parameters resulted in large differences in the histograms (cases 1–6 for ΔWSS and cases 1–4 for OSI). Small differences in the 3D surface plots corresponded to histograms similar in shape (in particular WSS for cases 1–4 and 6, OSI for cases 5 and 6).

Discussion

Differences in the temporal average of the WSS magnitude exceeding 30% and similar differences in its standard deviation and in the OSI were found using patient-specific and idealized flow waveforms of the parent artery in six saccular aneurysms of the ICA. These results emphasize the need for an accurate knowledge of the volumetric inflow waveform for reliable calculations of WSS parameters. While several studies have demonstrated good agreement between in vivo measurements and CFD results [7, 22, 23], the influence of geometrical and physiological boundary conditions on the accuracy of the CFD results remains to be established. Geometrical variations of the proximal segments of the parent artery and of the aneurysm dome have been demonstrated to influence the simulated flow characteristics within the aneurysm [2, 16, 17, 29], and alternations of inflow conditions have shown to modify the results for WSS magnitudes and spatial distributions [3, 21, 37].

Low WSS magnitudes and WSS histograms with predominant low WSS values have been recently correlated with aneurysmal growth [1, 20], and therefore the dependence of these parameters on the CFD boundary conditions are of particular interest. As the majority of CFD studies of cerebral aneurysmal hemodynamics has used non-patient-specific inflow waveforms [2–7, 28, 32], we focused on the differences in average WSS magnitudes, the temporal variations of the WSS, and the histograms of the WSS values on the aneurysm wall in CFD simulations using an idealized inflow waveform and using patient-specific inflow waveforms with otherwise identical boundary conditions and initial parameters. The temporal variation of the WSS magnitude was characterized by the standard deviation ΔWSS computed from all values during the cardiac cycle, and the temporal variation of the direction of the WSS vector was investigated with the OSI, a measure for directional variation [14]. Large relative differences in the temporal average of WSS, corresponding to large absolute differences (several Pascal) and in ∆WSS and OSI as well as in the shape of the histograms for ΔWSS on the aneurysmal wall emphasize the need for accurate inflow boundary conditions to obtain accurate results regarding the temporal variations of the WSS. Simulations with the idealized waveform may result in spatially inaccurate distribution of WSS, which may not be able to reveal the correct location of weak spots on the aneurysmal wall. This may be a potentially serious clinical limitation if CFD simulations should be used in clinical practice for choosing optimal endovascular devices for therapeutic treatment planning (Fig. 1).

The investigated aneurysms differed in location. Three (case 1, case 3, and case 6) were terminus aneurysms and three (case 2, case 4, and case 5) were curvature aneurysms. With the small number in each of these groups, an inter-group comparison certainly lacks statistical significance; however, from the negative mean relative deviation of ΔWSS and the negative maximum relative deviations in the <WSS> and ΔWSS, it appears that WSS and its temporal deviation are underestimated in curvature aneurysms when using the idealized waveform. Further studies are warranted to investigate this trend in more detail.

Our models employed for the CFD simulations vary according to the location of the aneurysm and the length and number of the adjacent arteries. In particular, the length of the proximal segment varied (models 1 and 2 do not extend to the cavernous segment, while models 3–6 incorporate variable length of the cavernous segment). The variations are routed in the segmentation routine and the quality of the original 3D DSA image data.

The influence of shape and length of the proximal arterial segment on aneurysm hemodynamics has been investigated previously. A recent study by Sato et al. demonstrated that aneurysms lateral to a curve in the parent artery showed significant higher wall shear stress than aneurysms inside or outside this curve [33]. Another study by this group revealed a strong influence on the steady-state flow in cerebral aneurysms between U-shaped, twisted, or S-shaped arteries [17].

Additional work performed by Castro et al. demonstrated that a truncated parent artery (1 cm upstream from the aneurysm, parent artery replaced by a cylinder) underestimated wall shear stress and resulted in a shift of the impaction zone [2]. The authors of this study conclude that arriving at a theoretic answer to the question of an optimal entry length for complex vascular systems with current knowledge and techniques may be impossible and state that each patient-specific anatomy may have its unique solution.

Previously, we reported about a direct comparison of CFD results and 2D pcMRI flow measurements using intra-aneurysmal flow patterns [22]. Aneurysm models 1 and 2 were used in this comparison. The good agreement between measurement and simulation gives confidence that entry length effects, if at all, play a small role in these particular cases. In particular, the variations of wall shear stress, as reported here, were obtained with identical computational models when using the idealized and the patient-specific waveforms, that is, the length of the parent artery was not changed in between simulations. This fact, as well as the validation of our simulation results with in vivo measurements, provides strong evidence that the reported wall shear stress variations are caused by differences in the inflow waveforms.

Validation and accuracy of CFD simulations remains an important topic before applying results to clinical situations. Approaches include the construction of ex vivo models where pressure and velocity measurements can be performed without the restrictions that exist in vivo. A recent study used particle-imaging velocity in models of ICA and basilar tip aneurysms constructed of clear silicone elastomer [11]. Good agreement in the simulated intra-aneurysmal flow pattern and its complex vortex dynamics with the experiment was found. 2D DSA with high frame rates (up to 15/s) is capable of visualizing intra-arterial and intra-aneurysmal contrast dynamics. Virtual angiograms, based on CFD results that mimick the flow of contrast, have recently been found to be in good agreement with flow pattern observed during intra-arterial contrast injections, in particular in respect to large intra-aneurysmal vortices and outflow effects [7]. These findings could potentially be validated by direct observations or by using intra-operative Doppler probes or novel techniques, such as laser speckle contrast analysis during a surgical intervention [15].

This study is limited by its relatively small sample size for providing representative statistical data (Tables 1, 2, and 3). A larger study including patient-specific waveforms may deviate more from or resemble closer to the here-used idealized waveform. Another potential limitation of obtaining 2D pcMRI measurements exists for ruptured aneurysms, where the need for rapid treatment precludes an additional MRI exam. However, if CFD simulation is aimed for the modeling of hemodynamics in an individual aneurysm, this study clearly demonstrates that not only patient-specific vascular geometry, but also patient-specific inflow boundary conditions, are necessary to obtain near to accurate calculations.

Conclusion

Large relative differences in the temporal variation of the WSS along the wall of saccular cerebral aneurysms can be observed when using idealized waveform and patient-specific waveforms measured with 2D pcMRI. For accurate calculations of WSS and its variations during the cardiac cycle, not only patient-specific aneurysm geometries, but also patient-specific parent artery flow data, is essential for accurate description of aneurysmal hemodynamics calculated with CFD. On the other hand, quantitative results obtained with idealized or standardized waveforms should be interpreted with great caution, in particular, when CFD simulations are used in clinical settings aiming to identify a weak spot in the aneurysmal wall, potentially prone to rupture.