Standardized viscosity as a source of error in computational fluid dynamic simulations of cerebral aneurysms

Abstract

Background:

Computational fluid dynamics (CFD) simulations are a powerful tool for studying cerebral aneurysms, capable of evaluating hemodynamics in a way that is infeasible with imaging alone. However, the difficulty of incorporating patient-specific information and inherent obstacles of in vivo validation have limited the clinical usefulness of CFD of cerebral aneurysms. In this work we investigate the effect of using standardized blood viscosity values in CFD simulations of cerebral aneurysms when compared to simulations of the same aneurysms using patient-specific viscosity values derived from hematocrit measurements.

Purpose:

The objective of this work is to determine the level of error, on average, that is, caused by using standardized values of viscosity in CFD simulations of cerebral aneurysms. By quantifying this error, we demonstrate the need for incorporating patient-specific viscosity in future CFD investigations of cerebral aneurysms.

Methods:

CFD simulations of forty-one cerebral aneurysms were conducted using patient-specific boundary conditions. For each aneurysm two simulations were conducted, one utilizing patient-specific blood viscosity derived from hematocrit measurements and another using a standardized value for blood viscosity. Hemodynamic parameters such as wall shear stress (WSS), wall shear stress gradient (WSSG), and the oscillatory shear index (OSI) were calculated for each of the simulations for each aneurysm. Paired t-tests for differences in the time-averaged maps of these hemodynamic parameters between standardized and patient-specific viscosity simulations were conducted for each aneurysm. Bland–Altman analysis was used to examine the cohort-wide changes in the hemodynamic parameters. Subjects were broken into two groups, those with higher than standard viscosity and those with lower than standard viscosity. An unpaired t-test was used to compare the percent change in WSS, WSSG, and OSI between patient-specific and standardized viscosity simulations for the two cohorts.The percent changes in hemodynamic parameters were correlated against the direction and magnitude of percent change in viscosity, aneurysm size, and aneurysm location. For all t-tests, a Bonferroni-corrected significance level of 0.0167 was used.

Results:

63.2%, 41.5%, and 48.7% of aneurysms showed statistically significant differences between patient-specific and standardized viscosity simulations for WSS,WSSG, and OSI respectively.No statistically significant difference was found in the percent changes in WSS, WSSG, and OSI between the group with higher than standard viscosity and those with lower than standard viscosity, indicating an increase in viscosity can cause either an increase or decrease in each of the hemodynamic parameters. On a study-wide level no significant bias was found in either direction for WSS, WSSG, or OSI between the simulation groups due to the bidirectional effect of changing viscosity. No correlation was found between percent change of viscosity and percent change of WSS,WSSG, or OSI, meaning an after-the-fact correction for patient-specific viscosity is not feasible.

Conclusion:

Standardizing viscosity values in CFD of cerebral aneurysms has a large and unpredictable impact on the calculated WSS,WSSG, and OSI when compared to CFD simulations of the same aneurysms using a patient-specific viscosity.We recommend implementing hematocrit-based patient-specific blood viscosity values for all CFD simulations of cerebral aneurysms.

1 |. INTRODUCTION

Evaluating the risk of cerebral aneurysm growth and rupture remains a pressing clinical challenge in both treated and untreated aneurysms. Local hemodynamics and the resulting mechanical forces are implicated in aneurysm pathophysiology, but their small scale and complex nature make them exceedingly difficult to study in vivo.¹ Image-based computational fluid dynamics (CFD) simulations of cerebral aneurysms are used to overcome these limitations,² yielding a long list of valuable insights into the mechanics of cerebral aneurysms, from developing hemodynamic parameters such as oscillatory shear index (OSI) and relative residence time that are predictive of rupture rates to discovering flow phenomena associated with pathological physiology.^3–7 For example, small-scale recirculation zones at the neck of an aneurysm, can drastically increase the risk of rupture, and have been found to correlate with areas of thin lumen walls.⁸ However,CFD has yet to be established as a clinically useful tool, primarily due to a lack of methodological rigor and consistency, and the inherent difficulty of obtaining all of the necessary patient-specific information for accurate simulation.^2,9 Precise and highly resolved patient-specific anatomic geometry^10,11 and boundary conditions^3,12 can improve CFD accuracy, but the effect of standardizing viscosity on the accuracy of CFD of cerebral aneurysms has yet to be satisfactorily established.

Blood is a non-Newtonian fluid,¹³ meaning that the viscosity is not constant, but a function of local strain rate. This non-Newtonian behavior is especially difficult to quantify, relying on empirical data or using non-patient-specific assumptions.^14,15 Investigations of CFD models using non-Newtonian viscosity models have been found to overestimate shear thinning effects more than Newtonian models underestimate them.^16,17 At flow rates and velocities relevant to cerebral aneurysms, non-Newtonian effects are small, and a constant viscosity assumption is appropriate.^14,18–20 While extensive work has gone into examining the effect of non-Newtonian viscosity modeling, there has been insufficient investigation into the effect of using a standardized value for the assumed constant viscosity value while also implementing patient-specific boundary conditions.

When treating blood as a Newtonian fluid, the viscosity (μ) can be directly calculated as a function of hematocrit (Hct) using an empirically derived third-order polynomial.²¹ Despite the ready availability of patient-specific values of viscosity, which are easily derived from hematocrit measurements, most studies implementing CFD simulations of cerebral aneurysms have predominantly used a standard, non-patient-specific viscosity value of μ = 0.0035 (Pa · s). We hypothesize that substantial differences in hemodynamic calculations would be observed in CFD simulations of cerebral aneurysms using reference standard constant viscosity compared to patient-specific constant viscosity values derived from hematocrit measurements.

2 |. METHODS

2.1 |. Subject selection

Forty-one patients with unruptured aneurysms who underwent catheter angiography prior to treatment were included in this study, which was approved by the institutional review board. A summary of aneurysm location, maximal size, hematocrit, and calculated viscosity can be found in Table 1.

TABLE 1.

List of aneurysm properties.

Patient no.	Hematocrit	Viscosity (Pa-s)	Size (mm)	Location
1	0.38	0.003098	10.0	Ophthalmic
2	0.45	0.003739	15.0	Cavernous
3	0.36	0.002954	6.6	Paraclinoid
4	0.39	0.003175		Paraclinoid
5	0.37	0.003024	30.0	Cavernous
6	0.38	0.003098	11.5	Pcom
7	0.44	0.003633	11.6	Basilar
8	0.39	0.003175	5.3	Supraclinoid
9	0.37	0.003024	5.3	Paraclinoid
10	0.43	0.003532	3.5	Ophthalmic
11	0.44	0.003633	6.0	Basilar
12	0.39	0.003175	8.0	Ophthalmic
13	0.42	0.003436	6.5	Ophthalmic
14	0.37	0.003024	10.7	Supraclinoid
15	0.42	0.003436	13.0	Supraclinoid
16	0.40	0.003258	5.3	Ophthalmic
17	0.43	0.003532	7.8	Supraclinoid
18	0.41	0.003344	11.8	Ophthalmic
19	0.40	0.003258	8.5	Pcom
20	0.36	0.002954	5.0	Pcom
21	0.36	0.002954	5.7	Paraclinoid
22	0.42	0.003436	5.6	Terminal
23	0.4	0.003258	10	Paraclinoid
24	0.41	0.003344	5.8	Terminal
25	0.41	0.003344	8.0	Ophthalmic
26	0.41	0.003344	8.0	Ophthalmic
27	0.42	0.003436	5.0	Pcom
28	0.32	0.002710	12.0	Supraclinoid
29	0.39	0.003175	4.0	Pcom
30	0.36	0.002954	35.0	Supraclinoid
31	0.41	0.003344	5.0	Cavernous
32	0.41	0.003344	5.0	Supraclinoid
33	0.34	0.002825	5.0	Basilar
34	0.43	0.003532	7.0	Ophthalmic
35	0.37	0.003024	4.8	Ophthalmic
36	0.37	0.003024	5.0	Cavernous
37	0.41	0.003344	7.7	Paraclinoid
38	0.37	0.003024	5.8	Ophthalmic
39	0.35	0.002888	11.0	Pcom
40	0.47	0.003969	10.0	Ophthalmic
41	0.47	0.003969	11.0	Ophthalmic

2.2 |. Viscosity calculation

Hematocrit values were obtained from each patient’s pre-surgical laboratory analysis on the day of angiography. Patient-specific viscosity was derived from the empirically derived third-order polynomial established in Guyton.²¹

$$\mu =31.694{(\text{Hct})}^3-15.98{(\text{Hct})}^2+5.878(\text{Hct})+1.4175$$ (1)

2.3 |. Verification of the Newtonian assumption

The non-Newtonian behavior of blood is negligible in the range of shear strain rate typical of flow in large cerebral arteries (50 s⁻¹−250 s⁻¹), but on occasion regions of flow through cerebral aneurysms will reach shear rates as low as 10 s⁻¹. Fortunately, recent work has shown that the changes in viscosity between 50 s⁻¹ and 10 s⁻¹ are small compared to the increase in viscosity that occurs below a shear rate of 10 s⁻¹.^20,22,23 To ensure the validity of the Newtonian assumption we calculated the minimum shear rate magnitude for each grid point in each simulation, the average minimum shear rate magnitude across the aneurysm dome, and the percent of the aneurysm dome with a minimum shear rate magnitude below 10 s⁻¹. The average minimum shear rate magnitude across all aneurysms was 87 s⁻¹. No aneurysms had an average minimum shear rate magnitude below 10 s⁻¹. Only 31% (n = 13) of the aneurysms had any location where the shear rate magnitude reached below 10 s⁻¹, and of those aneurysms the average percent volume with sub-10 s⁻¹ was only 11%. Furthermore, the average percentage of the cardiac cycle that a given location with a sub-10 s⁻¹ minimum shear rate magnitude was only 19%; meaning that only ~0.78% of the discrete points in space and time simulated for this work had a shear rate magnitude below 10 s⁻¹. The largest percent volume below 10 s⁻¹ was 23% of the aneurysm dome in patient 30, a massive aneurysm with slow flow, meaning that in the worst-case scenario aneurysm we would expect only 22% of the aneurysm dome to be affected by non-Newtonian behavior for ~19% of the cardiac cycle.

2.4 |. Computational methodology

Our group has extensive experience simulating blood flow through cerebral aneurysms^3,6,12,24 and has previously developed a thorough methodology for conducting CFD simulations of both treated and untreated aneurysms.^6,24–28 Briefly, a commercial finite volume solver (Ansys Fluent) was used to solve the transient 3-D incompressible Navier–Stokes equations.

$$\nabla \cdot \overrightarrow{\mathit{\text{u}}}=0$$ (2)

$$\rho \frac{\partial \overrightarrow{\mathit{\text{u}}}}{\partial t}+\rho (\overrightarrow{\mathit{\text{u}}}\cdot \nabla \overrightarrow{\mathit{\text{u}}})=\mu {\nabla }^2\overrightarrow{\mathit{\text{u}}}-\nabla p+\rho \overrightarrow{\mathit{\text{g}}}$$ (3)

Where $\overrightarrow{\mathit{\text{u}}}$ represents the velocity vector, p represents the pressure field, 𝜌 is the fluid density of blood ($1050\frac{\text{kg}}{{\text{m}}^3}$), μ represents the viscosity and $\overrightarrow{\mathit{\text{g}}}$ represents the acceleration due to gravity. The highly resolved and patient-specific anatomy for each patient was segmented semi-manually from 3D rotational angiography images using the Vascular Modeling Tool Kit (vmtk.org) and cleaned using 3D-slicer. The 3D geometry was discretized using polyhedral elements, with an average element edge size of 0.125 mm, and 6 prism layers at each surface, leading to ~2 million elements per simulation.

Patient-specific inlet boundary conditions are derived from in vivo dual-sensor Doppler guidewires measurements.^3,26 The combination of dynamic and static pressure measurements from the guidewire was used to calculate the maximum velocity waveform for each patient. The velocity waveform was then used to derive a fully resolved Womersley velocity profile at the inlet.^3,12 The outlet boundary conditions were defined as pressure outlets with pressure calculated using a two-element Windkessel model based on vessel diameter and guidewire pressure measurements when multiple outlets are present, and a zero-pressure condition when only one outlet was present, as has been previously established as best practice for CFD of cerebral aneurysms.^3,12 Each simulation was conducted with a time step of 0.001 s and run for a minimum of five cardiac cycles before collecting data to eliminate the effect of initial conditions.

To ensure numerical accuracy and eliminate numerical error as a potential source of discrepancy between the standard viscosity and patient-specific viscosity simulation we conducted a grid independence study for patient 19 on both the standardized and patient-specific viscosity simulations.Patient 19 was selected due to the relatively large change (69.5%) in average wall shear stress (WSS) from a relatively small change in viscosity (7.4%). For the mesh independence study the average element edge length was cut in half from the initial simulations, leading to an eightfold increase in element count. There were no statistically significant differences in the time-averaged maps of WSS, WSSG, or OSI as determined by t-tests for the null hypothesis. The refined standardized viscosity simulation led to a change of 0.51% (p = 0.54) 1.04% (p = 0.39), and 0.53% (p = 0.32) in temporally and spatially averaged WSS, WSSG, and OSI respectively, while the refined patient-specific viscosity simulation led to a change of 0.43% (p = 0.62), 0.89% (p = 0.37), and 0.62% (p = 0.46) in WSS, WSSG, and OSI respectively. The mesh independence study assures that the differences in the standardized and patient-specific viscosity are not due to numerical errors.

2.5 |. Data analysis and statistical methods

$$\text{WSS}=\left|\mu \frac{\partial \overrightarrow{\mathit{\text{u}}}}{\partial \overrightarrow{n}}\right|$$ (4)

$$\text{WSSG}=\left|\nabla \cdot \mu \frac{\partial \overrightarrow{\mathit{\text{u}}}}{\partial \overrightarrow{n}}\right|$$ (5)

$$\text{OSI}=\frac{1}{2}\left(1-\frac{\left|{\int }_0^T\mu \frac{\partial \overrightarrow{\mathit{\text{u}}}}{\partial \overrightarrow{n}}\partial t\right|}{{\int }_0^T\left|\mu \frac{\partial \overrightarrow{\mathit{\text{u}}}}{\partial \overrightarrow{n}}\right|\partial t}\right)$$ (6)

For each cerebral aneurysm in each condition (patient-specific versus standard viscosity), we calculated maps of the time average (across the cardiac cycle) of WSS,WSS, gradient (WSSG), and OSI, defined below.

Where $\overrightarrow{\mathit{\text{u}}}$ is the velocity vector, $\overrightarrow{n}$ is the wall’s normal direction, T is the period of a single cardiac cycle, and μ is the viscosity. A paired t-test for the hypothesis that there is a difference in the spatial average of the above values between groups was performed on the time-averaged maps of each of these variables between the standardized and patient-specific simulations of each aneurysm. We use a Bonferroni correction to account for multiple comparisons leading to a significance level of 0.0167.²⁹

Spatial averages across the aneurysm dome were calculated for the time-averaged maps of WSS, WSSG, and OSI. Bland–Altman analysis was performed to evaluate the difference in these spatial averages between the patient-specific and standardized viscosity groups. The percent change in each parameter of interest was calculated between standardized viscosity simulations and patient-specific viscosity simulations, and patients were sorted into two cohorts: those whose patient-specific viscosity was greater than the standard value (the “increased viscosity cohort”), and those whose patient-specific viscosity was lower than the standard value (the “decreased viscosity cohort”). These groups were chosen to test if the directional change of viscosity has an effect on the directional change of the hemodynamic parameters. Unpaired t-tests (as we are now comparing groups of different aneurysms) for the hypothesis that on average the change in hemodynamic parameters in the increased cohort are significantly different than the decreased cohort were performed for each parameter of interest. A linear regression was conducted on the percent change of WSS, WSSG, and OSI between patient-specific and standardized simulations and the percent change in viscosity between the patient-specific and standard values.

3 |. RESULTS

The p-values obtained from student’s t-test between standardized and patient-specific viscosity simulations for maps of time-averaged WSS, WSSG, and OSI are plotted in Figure 1.Using a Bonferroni correction for multiple comparisons with m = 3 yields a significance level of 0.0167. We found that 63.2%, 41.5%, and 48.7% of aneurysms showed statistically significant differences for WSS, WSSG, and OSI respectively. Figure 1 demonstrates that while there is a general trend of an increase in the significance of differences in simulation results with an increase in difference between standardized and patient-specific viscosity values, the correlation is not significant and even a small change in viscosity (<5%) can lead to a statistically significant change in WSS, WSSG, and OSI.

FIGURE 1.

p-value from t-test on maps of time averaged well shear stress (WSS; above), well shear stress gradient (WSSG; middle), and oscillatory shear index (OSI) (below) between computational fluid dynamics (CFD) simulations of the same aneurysms using patient-specific and standardized values for viscosity plotted against the magnitude of the percent change in viscosity between simulations. The significance level is displayed with dashed red line. Linear regression and confidence interval shown in blue. Each subject is colored by aneurysm size. Shapes represent aneurysm location.

Bland–Altman analysis for the average WSS, WSSG, and OSI is plotted in Figure 2. Each data point is colored by the percent change in viscosity between the two simulations, with “hotter” (red) colors indicating greater positive differences in viscosity, and “cooler” (blue) colors indicating greater negative differences in viscosity. WSS demonstrates unbiased variation between patient-specific and standardized simulations, while the difference in WSSG is biased upward and difference in OSI is biased downward. Due to high variance, there are no statistically significant differences between groups in either direction, as to be expected due to the bidirectional changes in viscosity. While Bland–Altman analysis did not reveal any statistically significant directional differences, there were substantial differences in the WSS, WSSG, and OSI between patient-specific and standardized viscosity simulations. The root mean square (RMS) of percent difference between the patient-specific and standardized viscosity simulations were 33.63%, 42.36%, and 56.54% for WSS,WSSG, and OSI respectively.The RMS of percent difference in viscosity was only 12.26% by comparison, demonstrating that a small change in viscosity can lead to a large change in WSS, WSSG, and OSI.

FIGURE 2.

Bland–Altman plots comparing the differences in wall shear stress (WSS; top), WSS gradient (WSSG; middle), and oscillatory shear index (OSI; bottom) between standardized and patient-specific simulations, with each subject colored by the percent change in viscosity.

Examination of the differences between the increased viscosity cohort and the decreased viscosity cohort demonstrated no trend in directionality for WSS, but an increase in viscosity tended to lead to a decrease in both WSSG and OSI, while a decrease in viscosity tended to increase both WSSG and OSI. However, no significant differences were found between groups for any of the variables (Table 2), indicating that the direction of viscosity change alone is insufficient for predicting the direction of hemodynamic parameter change. To determine the effect of viscosity on the hemodynamic parameters of interest, linear regressions of the percent change in WSS, WSSG, and OSI were plotted against the percent change in viscosity (Figure 3 and Table 3). We found no significant correlation between change in any of the hemodynamic parameters and aneurysm size or aneurysm location. Similarly, there was no correlation between any hemodynamic parameter and the percent change in viscosity.

TABLE 2.

Percent changes in hemodynamic parameters of interest when using patient-specific viscosity, stratified into increased viscosity and decreased viscosity cohorts, and results of an unpaired t-test between the cohorts.

	RMS of % change			Mean of % change (SD)			Unpaired t-test on means
Parameter	All subjects	Increased viscosity	Decreased viscosity	All subjects	Increased viscosity	Decreased viscosity	t-stat	p-Value
Viscosity	12.26	6.61	13.27	−8.09 (11.21)	5.01 (4.61)	−11.27 (7.11)	6.132	3.39E-07
WSS	33.62	13.324	36.90	3.74 (33.82)	4.70 (13.32)	3.50 (37.33)	0.088	0.930
WSSG	42.36	19.683	46.21	1.30 (42.86)	−7.94 (19.26)	3.550 (46.79)	−0.675	0.504
OSI	56.54	16.59	62.49	4.023 (57.10)	−8.84 (15.01)	7.144 (63.04)	−0.706	0.484

Abbreviations: OSI, oscillatory shear index; RMS, root mean square; WSS, wall shear stress; WSSG, WSS gradient.

FIGURE 3.

Percent change in wall shear stress (WSS), WSS gradient (WSSG), and oscillatory shear index (OSI) plotted against percent change in viscosity. Linear regression and confidence interval shown in blue. Each subject is colored by aneurysm size. Shapes represent aneurysm location.

TABLE 3.

Linear regression table for the correlation between percent change in hemodynamic parameters of interest and percent change in viscosity.

Parameter	Slope	Intercept	r 2	Standard error	p-Value
WSS	0.2914	6.0945	0.0064	0.5792	0.6177
WSSG	0.1408	0.4760	0.0009	0.7513	0.8524
OSI	−0.6298	−1.0763	0.0106	0.9756	0.5224

Abbreviations: OSI, oscillatory shear index; RMS, root mean square; WSS, wall shear stress; WSSG, WSS gradient.

Three representative cases were selected for side-by-side comparisons of WSS in the standardized and patient-specific viscosity simulations (Figure 4). A patient with a small change in viscosity is shown along side a patient with a large increase in viscosity and a patient with a large decrease in viscosity. While patient 22 and patient 41 merely undergo small quantitative changes in aneurysm hemodynamics, the maps of WSS for patient 39 demonstrate the qualitative difference in hemodynamics that can result from using standardized viscosity. As the Navier–Stokes equations are a set of nonlinear partial differential equations they are susceptible to bifurcation points in the set of potential solutions.³⁰ The qualitative changes in the hemodynamics of patient 39 is due to the crossing of a bifurcation point. Due to the complexity of the system, there is no feasible way to determine these bifurcation points prior to simulation, making it nearly impossible to predict the changes in hemodynamics associated with a given change in viscosity.

FIGURE 4.

Contour plots of wall shear stress (WSS) from the standardized and patient-specific viscosity simulations for patients 22, 39, and 41 with change in average WSS and viscosity tabulated above. A small decrease in viscosity leads to a visual indetectable but meaningful reduction in average WSS of 5.43%, while a large increase in viscosity leads to a small but visually detectable reduction in WSS of 4.11% for patient 41. Patient 39 displays a qualitatively different WSS distribution and 125.59% increase in average WSS from a large reduction in viscosity.

4 |. DISCUSSION

While there is an established consensus around the importance of accurately incorporating patient-specific boundary conditions to CFD simulations of cerebral aneurysms,^2,3 there has been little investigation of the error caused by using standardized constant viscosity values. In this work, we have presented an investigation into the potential large and unpredictable inaccuracies in hemodynamic calculations that may result from using standardized viscosity values in CFD simulations of cerebral aneurysms when compared to simulations using patient-specific viscosity values derived from hematocrit measurements.

We compared the differences in time-averaged maps of WSS, WSSG, and OSI between standardized viscosity and patient-specific viscosity simulations via student’s t-tests, revealing that 63.2%,41.5%,and 48.7% of aneurysms showed statistically significant differences for WSS, WSSG, and OSI respectively. The statistical significance of the differences was not highly correlated with the magnitude of the change in viscosity, meaning even a small change in viscosity can lead to a signifi-cant change in hemodynamics (Figure 1).We found that, on average, a percent change in the value of viscosity of just 12.26% led to an average magnitude of percent change of 33.63%, 42.36%, and 56.54% for WSS, WSSG, and OSI respectively.

In addition to finding large differences in aneurysm hemodynamics, we found that these differences are not consistent or predictable in a way that can be corrected post hoc, as demonstrated in Figures 1 and 3. Bland–Altman analysis of WSS, WSSG, and OSI revealed that the effect of using standardized viscosity does not significantly bias the results in a particular direction, but yields unpredictable variation in aneurysm hemodynamics (Figure 2). The variation is less pronounced for WSSG and OSI due to the presence of several large outliers with exceptionally high levels of OSI or WSSG, (Figure 2) but on average the magnitude of the percent change in WSSG and OSI between the standardized and patient-specific viscosity models is larger than that for WSS (Table 2).

Similarly, we found no discernable trend or significant correlation between change in hemodynamic parameters of interest and change in viscosity,aneurysm size,or aneurysm location, further demonstrating that the effect of using patient-specific viscosity is unpredictable and cannot be accounted for after the fact (Figure 3 and Table 3). This trend holds even when accounting for the directionality of viscosity change, as we found no statistically significant differences between the increased viscosity and decreased viscosity cohorts in the percent change of WSS, WSSG, or OSI between the standardized and patient-specific viscosity simulations; this demonstrates that predicting even the direction of the effect on aneurysm hemodynamics of a patient-specific viscosity, that is, higher or lower than the standard value is not feasible on a patient-by-patient basis. (Table 2)

The fact that up to 63% of CFD simulations of cerebral aneurysms may be significantly impacted by using standardized viscosity, small changes in viscosity can lead to 33.63%, 42.36%, and 56.54% changes in WSS, WSSG, and OSI respectively, and that it cannot be easily accounted for after the fact, calls into question the predictive power of simulations using standardized viscosity. While viscosity may not be as critical as patient-specific boundary conditions and anatomy in calculating accurate hemodynamics,^2,3,9,31,32 this work suggests that patient-specific viscosity must be included in the effort to make CFD of cerebral aneurysms as accurate as possible.

There are several limitations to this work. The first is the inherent uncertainty in CFD of cerebral aneurysms. While CFD is currently the only way to quantitatively investigate aneurysm hemodynamics, the lack of available in vivo hemodynamic data means we cannot directly validate CFD. Additionally, there are assumptions built into the CFD that may affect the accuracy. In this work, we assume blood is a Newtonian fluid, and while this assumption is appropriate in the vast majority of the domain^19,20,23 there are regions of aneurysm domes with very low shear rates that are likely affected by the non-Newtonian properties of blood. We also assume rigid walls for the simulation, while recent work has shown that aneurysm walls undergo deformation across the cardiac cycle.³³ While it is not currently feasible to accurately incorporate these phenomena, for the most accurate CFD simulations future research should strive to include wall deformation and non-Newtonian effects.

The complex nature of aneurysm hemodynamics and obstacles to in vivo validation are significant challenges for the clinical applicability of CFD of cerebral aneurysms,⁹ so it is important to include as much patient-specific detail into the simulations as possible to ensure accuracy. The growing consensus over the need to standardize computational methodology, eliminate human error from anatomic geometry segmentation, and implement patient-specific boundary conditions has helped make significant strides on the course to clinical applicability of CFD of cerebral aneurysms,^1,8,34 but a definitive relationship between hemodynamics and risk of aneurysm growth and rupture has yet to be developed.

The results of this work suggest that using standardized viscosity may have played a meaningful role in the inability of past studies to develop predictive metrics. With limited cohort sizes and the potential that complex and nonlinear interactions contribute to aneurysm growth and rupture, even a small reduction in accuracy could eliminate any predictive power of a given dataset.In this work we have demonstrated the ability for small changes in viscosity to lead to large differences in WSS,WSSG,and OSI.Fortunately, adjusting for patient-specific viscosity is not difficult, as most empirical work has found that while viscosity of blood is dependent on a variety of parameters including fibrinogen and temperature, it can be estimated relatively accurately using hematocrit alone.^{13,16,20,35,36} As hematocrit measurements are clinically available and typically attained for most aneurysm patients,such measurements should be easy to incorporate into both retro- and prospective CFD studies of aneurysm hemodynamics.

5 |. CONCLUSIONS

By conducting CFD simulations of forty-one cerebral aneurysms using both patient-specific viscosity and a standardized viscosity value we found that using standardized values of viscosity leads to substantial errors in hemodynamic parameters, with 63.2%, 41.5%, and 48.7% of aneurysms demonstrating statistically significant differences for WSS, WSSG, and OSI respectively. On a cohort wide basis an average change in viscosity of only 12.26% led to an average change of 33.63%, 42.36%, and 56.4% for WSS, WSSG, and OSI respectively. Furthermore, we found there is no significant correlation between the percent change in viscosity and the percent change in WSS, WSSG, or OSI, indicating that the effect of implementing patient-specific viscosity on aneurysm hemodynamics is not predictable and must be accounted for beforehand. These findings demonstrate that standardizing viscosity leads to avoidable errors in capturing aneurysm hemodynamics via CFD.