Accuracy of Computational Hemodynamics in Complex Arterial Geometries Reconstructed from Magnetic Resonance Imaging

Annals of Biomedical Engineering, Vol. 27, pp. 32 41, 1999 Printed in the USA. All rights reserved. 0090-6964/99/27 1 /32/10/$15.00 Copyright 1999 Biomedical Engineering Society Accuracy of Computational Hemodynamics in Complex Arterial Geometries Reconstructed from Magnetic Resonance Imaging J. A. MOORE, 1 D. A. STEINMAN, 2,3 D. W. HOLDSWORTH, 2,3,4 and C. R. ETHIER 1 1 Department of Mechanical and Industrial Engineering and Institute for Biomedical Engineering, University of Toronto, Toronto, Canada, 2 Imaging Research Laboratories, John P. Robarts Research Institute, London, Ontario, Canada, and Departments of 3 Medical Biophysics and 4 Diagnostic Radiology and Nuclear Medicine, University of Western Ontario, London, Ontario, Canada (Received 4 March 1998; accepted 5 October 1998) Abstract Purpose: Combining computational blood flow modeling with three-dimensional medical imaging provides a new approach for studying links between hemodynamic factors and arterial disease. Although this provides patient-specific hemodynamic information, it is subject to several potential errors. This study quantifies some of these errors and identifies optimal reconstruction methodologies. Methods: A carotid artery bifurcation phantom of known geometry was imaged using a commercial magnetic resonance MR imager. Threedimensional models were reconstructed from the images using several reconstruction techniques, and steady and unsteady blood flow simulations were performed. The carotid bifurcation from a healthy, human volunteer was then imaged in vivo, and geometric models were reconstructed. Results: Reconstructed models of the phantom showed good agreement with the gold standard geometry, with a mean error of approximately 15% between the computed wall shear stress fields. Reconstructed models of the in vivo carotid bifurcation were unacceptably noisy, unless lumenal profile smoothing and approximating surface splines were used. Conclusions: All reconstruction methods gave acceptable results for the phantom model, but in vivo models appear to require smoothing. If proper attention is paid to smoothing and geometric fidelity issues, models reconstructed from MR images appear to be suitable for use in computational studies of in vivo hemodynamics. 1999 Biomedical Engineering Society. S0090-6964 99 01401-0 Keywords Blood flow, Magnetic resonance imaging, Numerical flow modeling, Carotid artery, Three-dimensional, Wall shear stress, Atherosclerosis. INTRODUCTION Address all correspondence to Professor C. Ross Ethier, Department of Mechanical and Industrial Engineering, 5 King s College Road, University of Toronto, Toronto, Ontario M5S 3G8, Canada. Electronic mail: ethier@mie.utoronto.ca 32 Arterial disease, the leading cause of death in western society, is known to be influenced by hemodynamic factors, the most important of which is believed to be wall shear stress. Numerous investigators are attempting to discover the specific link s between hemodynamic features and arterial disease. A powerful way of establishing such links is to combine computational blood flow modeling with three-dimensional 3D medical imaging of arterial geometries in vivo. This virtual vascular laboratory approach 13 has the great advantage of providing patient-specific hemodynamic information in a non invasive manner. There are several methods that can be used to reconstruct morphological structures from medical images. Many of the most popular techniques involve direct surface triangulation e.g., Robb 9. These methods generally produce C 0 polygonal representations of the surface, where the polygon size is determined by the image voxel dimensions. Since the voxel dimensions obtained with magnetic resonance MR can be relatively large, these methods were not found to be suitable for reconstructing small morphological structures such as arteries. Instead, we used a method based on the extraction of arterial cross-sectional profiles, described in detail below. Computational flow modeling in arterial geometries reconstructed in this manner is subject to a number of potential errors. This is particularly true for the computation of wall shear stress, a quantity that is critically dependent on imaged vascular geometry and near-wall flow features. Potential errors can be understood by considering the steps we have used to construct a computational flow model from a MR image of an arterial geometry Fig. 1. After imaging the artery or phantom of interest and obtaining the MR image series steps 1 and 2, the medical images are processed and segmented to produce a series of planar, cross-sectional profiles of the arterial lumen step 3. Step 3 invariably requires user interpretation and intervention, predominantly due to difficulties caused by image resolution, flow artifacts causing nonuniform lumen signal, and partial volume effects. For example, poor image resolution can cause problems determining where, within one pixel, the lumen boundary is located. It can also produce choppy/block-like profiles that do not properly represent the arterial geometry. Image noise can produce artifactual lumenal features, such as pits or bumps. After segmentation, the lumenal profiles are aligned and stacked together step 4, and a

Accuracy of Hemodynamic Modeling 33 In the second portion of the study, we studied the reconstruction of a carotid artery bifurcation scanned in vivo. The in vivo scan, while not allowing comparison with a known gold-standard geometry, is more representative of the noisy, hard-to-segment images that would be used in a real study. METHODS FIGURE 1. Summary of steps used in this work to construct a finite-element mesh from an arterial geometry. true three-dimensional model of the lumen intimal interface is generated by lofting together the profiles step 5. At this point, errors introduced in step 3 appear as spurious surface features unless suitable profile and surface smoothing techniques have been used. Errors can also arise from other sources: for example, a poor choice of surface splining technique gives twisted surfaces and/or spurious waves in the surface between cross-sectional profiles. 6 After surface creation, the model is discretized to produce a volume-filling computational mesh for flow simulations step 6. In view of the above, we should ask how accurate the overall arterial reconstruction and flow modeling approach is. In a previous publication 7 we showed that surprisingly large errors could occur in this approach, but that such errors can be reduced to an acceptable level if steps are taken to smooth errors in cross-sectional profiles resulting from image noise. However, our previous study used a simple straight-tube model of an artery, and questions therefore remain about the utility of smoothing and image processing techniques for more realistic and complex arterial geometries, such as bifurcations. The objective of this work was therefore to more completely evaluate the accuracy of computational blood flow modeling in complex arterial geometries reconstructed from MR images, and to test several errorreducing smoothing strategies. We placed specific emphasis on errors due to image segmentation and 3D model construction, since they typically dominate the overall error. 7 In the first portion of the study, we quantified errors using a well-characterized, physiologically shaped carotid bifurcation flow phantom incorporating branching, curvature, and cross-sectional area change. 11 In the first portion of the study, a rigid acrylic carotid bifurcation phantom with 8 mm common carotid artery diameter D was used for imaging studies. The geometry of this model is completely specified through a mathematical description, 11 which is used to generate numerically controlled milling instructions for phantom fabrication. A previous study 10 compared the actual asbuilt phantom dimensions with those of the mathematical description, using both vernier caliper measurements at selected cross sections (precision 0.05 mm) and x-ray angiography (precision 0.16 mm). The actual dimensions agreed with the mathematical description to within the precision of the measurement techniques. The phantom was filled with a 40:60 by volume glycerol:water mixture, and MR images were acquired in a General Electric GE Signa 1.5 T scanner using a gradient echo bright blood sequence. Two image series were obtained: a high-resolution series 512 512 image, 16 cm field of view, 32 contiguous 1.5 mm slices, TR 50 ms, TE 8 ms, number of excitations NEX 8, flip angle 30 ], and a low-resolution series same parameters, except 256 256 image, 4 NEX. Slices were oriented normal to the model axis. The highand low-resolution scans gave images with approximately 12.8 and 6.4 pixels across the common carotid radius, respectively. These ratios span the values expected for an in vivo scan. Each image slice in the MR data set was segmented to determine lumenal margin s using grey-scale thresholding as described in Ref. 7. The resulting sets of lumenal profiles were smoothed using a three-phase approach. This three-phase algorithm smoothes lumenal profiles in each imaging plane, profile centroidal locations between imaging planes, and lumenal cross-sectional areas between imaging planes. The smoothing algorithms are described in detail elsewhere. 6,7 To test the image segmentation and model construction steps, five 3D surface models were built using a computer-aided drafting CAD package Geomesh, Fluent Inc., Lebanon, NH, as follows: Model 1 was built from the mathematical description of the carotid phantom. Because of the known fidelity of the physical phantom to the mathematical description, this model was considered a goldstandard description of the phantom.

34 MOORE et al. TABLE 1. Summary of models built from in vitro and in vivo scans. The long dash symbol means that model building was not applicable for that particular case. See the text for a detailed discussion of how the models were built. Description Model created from in vitro scans Model created from in vivo scans No smoothing A Mathematically specified 1 Augmented sections, approximating splines 2 B Low resolution scan 3 No augmentation, approximating splines 4 Not created No augmentation, interpolating splines 5 C Models 2 5 were built from the MR data sets, using different surface construction approaches Table 1. In models 2 4 we used approximating surface splines to define the three-dimensional lumenal surface. Approximating splines provide a best-fit approximation through cross-sectional contours without being guaranteed to pass through any given contour. They produce a relatively smooth model, i.e., one free of noise-induced surface features, but may compromise overall geometric fidelity, particularly in regions of rapid curvature, expansion, or tapering. In model 5 we used interpolating multiple segment splines to define the lumenal surface. These splines pass through every contour, producing a model that conforms well to the original data set, but that also contains noise-induced surface features. In more detail, models 2 5 were built as follows. Model 2 Augmented : The original highresolution data set was first augmented by the introduction of additional contours. Specifically, three contours, determined by linear interpolation of neighboring smoothed MR contours, were inserted between every pair of MR contours. The surface was then constructed using eighth-order, single segment, approximating surface splines. The optimum order for the approximating surface splines was determined in a separate pilot study. Because the interpolated contours are derived from the original contours, they do not introduce additional information into the data set. However, their presence forces the surface construction approximating splines to closely correspond to the original data, while retaining the desirable smoothing features of such splines. The net result was expected to be a smooth model with good geometric fidelity to the gold standard. Model 3 Low Resolution : This model was constructed from the low-resolution MR data set using techniques identical to that of model 2. Model 4 Smoothed : This model was constructed from the original nonaugmented highresolution data set using eighth-order, single segment, approximating surface splines identical to that used in model 2. This model was expected to be very smooth, but to show poorer geometric fidelity to the original data than model 2. Model 5 Nonsmoothed : This model was constructed from the original nonaugmented highresolution data set using nonsmoothing interpolating, multiple-segment surface splines. This approach was expected to produce a noisy model with good overall geometric fidelity. A cylindrical inlet segment of length 2.5 D was added to the common carotid, while cylindrical outlet segments of length 8 D were added to the ends of the daughter vessels. A volume-filling finite-element mesh consisting of ten-noded tetrahedra was then constructed for each model Geomesh, Fluent, Inc., Lebanon, NH. All meshes had comparable mesh densities, containing approximately 37,000 elements and 60,000 nodes. Both steady and unsteady flow fields and wall shear stress distributions were then computed using a validated finite-element-based solver. 1 This code assumes Newtonian fluid viscosity and rigid vessel walls. For steady flow, the Reynolds number Re D based on common carotid artery diameter and mean velocity, was 250. The no-slip condition was imposed on all walls, a Poiseuille velocity profile was specified at the inlet of the common carotid and at the outlet of the external carotid, and fully developed flow traction-free condition was specified at the internal carotid outlet. For unsteady flow, we used a common carotid flow waveform derived from pulsed Doppler measurements in normal volunteers. 3 This waveform has a mean Re D 275, a peak Re D 1073, and a Womersley parameter of 5.6. Boundary conditions were adjusted to account for unsteady flow, i.e., the common carotid inlet and external carotid outlet velocity profiles were specified as appropriately weighted sums of Womersley harmonics. In all simulations, the flow ratio split was 44:56 external:internal. Due to the large amount of computing time required for unsteady runs, unsteady flow and wall shear stress calculations were only done in models 1 and 2. In a postprocessing step we computed two indices to quantify the variations in wall shear stress: the 3D generalization of the oscillatory shear index OSI 4 and

Accuracy of Hemodynamic Modeling 35 T neg, which is simply the fraction of the cardiac cycle where the local shear, is directed away from the mean direction. 8 At every node on the model surface, we computed the mean-shear direction vector over the cardiac cycle 0,T as n mean 0 T / dt, and then defined the quantity NEG to be 1 if n mean 0, and 0 otherwise. OSI and T neg were then given by the formulas OSI 0 T NEG n mean dt T, T 0 n mean dt neg 0 T NEG dt. T In the second portion of the study, the carotid artery bifurcation of a normal, ostensibly healthy male volunteer was scanned using bilateral 3 in. surface coils in the same 1.5 T GE MR imager used for in vitro phantom imaging. Images were acquired using a cardiac-gated two-dimensional 2D fast spin echo sequence echo train length 4 with saturation bands placed superior and inferior to the volume of slices, to null the signal from flowing blood. Slices were oriented approximately normal to the common carotid axis. To minimize artifacts due to vessel pulsation, cardiac gating was used to ensure that all slices were acquired during diastole. Pulse sequence parameters were: TR 2 R R intervals; TE 16 ms; 16 12 cm field of view; 512 384 acquisition; 313 m in-plane resolution. Eleven contiguous 1.5 mm thick slices were acquired per 4.5 min scan, and three scans were needed to cover the desired 5 cm axial range. Further details regarding in vivo magnetic resonance imaging MRI scanning may be found in Milner et al. 5 The in vivo images were segmented in a manner similar to the in vitro images. Three models were built from these images. Model A was constructed entirely without smoothing, using interpolating surface splines. Model B was constructed using three-phase smoothing, profile augmentation, and approximating splines in precisely the same manner as model 2 above. Model C was constructed using three-phase smoothing, no augmentation, and interpolating surface splines in precisely the same manner as model 5 above. Models A and C were therefore similar, differing only in the presence of three-phase smoothing for model C Table 1. Our rationale for choosing to construct the above three models for the in vivo scans is as follows. Model A is what a naïve user would reconstruct and will demonstrate the need for some sort of smoothing. For models including smoothing, model B represents a model with a fairly large degree of smoothing, while C is a model with minimal smoothing. The augmented data approach used in model B is convenient for controlling surface tangency near seams where different parts of the model are joined in the CAD software. FIGURE 2. Qualitative geometric comparison between models 1 5 in the first portion of the study. Labels: 1 The sharp seam at the apex of the gold-standard geometry is absent in the MR-reconstructed models. 2 The terminus of the carotid sinus is more abrupt in the gold-standard geometry than in the MR-reconstructed models. 3 The tuning fork bend in the external carotid artery is slightly more abrupt in the gold-standard geometry than the MR-reconstructed models. 4 There is a slight pinching in the common carotid artery in the MR-reconstructed models that is absent in the gold-standard geometry. RESULTS Geometric Comparison: Carotid Phantom Qualitative Fig. 2 and quantitative Fig. 3 comparison of the five models indicates very good general agreement between the gold-standard geometry and the MRreconstructed models. However, closer inspection shows several small errors in the MR models. First, in the neighborhood of the apex of the bifurcation, the ideal model showed a seam along the bifurcation, while the MR models lacked this seam and showed a gradual pinching-off at the apex. Second, models 2 5 all tended to smooth out relatively abrupt changes, such as the reduction in cross-sectional area at the carotid bulb terminus and the change in direction of the external carotid

36 MOORE et al. FIGURE 3. Mean artery radius as a function of position for models 1 5. Mean radii were computed for selected axial slices taken perpendicular to the axis of the common carotid as the square root of cross-sectional area/. It should be noted that due to the very tight construction tolerances for the flow phantom the estimated error in the dimensions of the ideal model are smaller than the plotted symbols. artery. Careful inspection shows that of the MRreconstructed models, model 5 nonsmoothed had the least smoothing at the terminus of the carotid bulb. Third, models 2 5 all showed a slight underestimation of the arterial radii. Over most of the geometry the magnitude of this underestimation was 0.15 mm, although there was slightly more prominent narrowing of the common carotid artery proximal to the bifurcation and of the internal carotid artery just distal to the bifurcation. Note that a geometric error of 0.15 mm is greater than the uncertainty in the gold-standard model, and corresponds to half a pixel in the high-resolution scans. There did not appear to be a significant difference between the models constructed from the low- and high-resolution scans. Despite the differences noted above, the agreement between the MR-reconstructed models and the goldstandard geometry were very good. For example, the maximum error in the shape factor 12 for models 2 5 was approximately 1%, while the maximum local error in the internal external carotid bifurcation angle was approximately 5. 6 We conclude that the MR-scanning/ reconstruction methodology is able to faithfully replicate the gold-standard geometry. Hemodynamic Comparison: Carotid Phantom Velocity profiles for both steady and unsteady flow were qualitatively and quantitatively similar for all models. Very minor differences between models were observed in the shapes and sizes of the steady flow recirculation zones in the internal carotid bulb. For example, the recirculation zone in the gold-standard geometry was slightly longer than in the MR reconstructed models, due to the fact that the gold-standard model s carotid sinus FIGURE 4. Qualitative comparison of wall shear stress distributions for models 1 5 under steady flow conditions at Re D 250. The contoured quantity is wall shear stress magnitude, normalized by the common carotid inlet wall shear stress magnitude in the gold-standard geometry model 1. For a common carotid artery diameter of 8 mm and a blood dynamic viscosity and density of 3.5 cp and 1.05 g/cm 3, this normalizing wall shear stress is 4.2 dyn/cm 2. In vivo, mean shear stresses are higher due to slightly smaller mean common carotid artery diameters and higher Reynolds numbers Ref. 1. was slightly larger and more rounded than the sinuses in models 2 5. However, these differences were not considered significant see the discussion for a precise definition of significant in this context. Comparison of steady flow wall shear stress distributions showed good overall agreement between all models Figs. 4 and 5. Specific features of interest are listed below. In the MR-reconstructed models, the slight narrowing of the common carotid artery approximately three radii upstream of the bifurcation caused a band of elevated shear stress. A sharp wall shear stress peak occurred on the

Accuracy of Hemodynamic Modeling 37 FIGURE 5. Plot of wall shear stresses as a function of axial position along the walls of a cut through the centerplane of models 1 5. The plotted quantity is wall shear stress magnitude, normalized by the common carotid inlet wall shear stress magnitude in the gold-standard geometry model 1. For reference, the error in normalized wall shear stress due to inaccuracies in phantom construction is estimated to lie in the range of 0.038 0.125 normalized shear units. See Fig. 4 caption for discussion of normalizing wall shear stress. external carotid artery wall in the gold-standard geometry point A in Fig. 5. This was due to a slope discontinuity where the external carotid artery joined the common carotid artery. This slope discontinuity was absent in the MR-reconstructed models. All models accurately computed the magnitude of the wall shear stresses in the recirculation region within the carotid sinus. Models 2 5 showed larger wall shear stress within the external carotid artery than was present in the gold-standard geometry, due to the slight underestimation of external carotid artery radius in the MRreconstructed models.

38 MOORE et al. The peak in wall shear stress at the terminus of the internal carotid sinus in models 2 5 occurred slightly distal to the peak in the gold-standard model point C vs point B in Fig. 5. This difference was due to the slight smoothing at the carotid sinus terminus that occurred in the MR-reconstructed models. Close examination indicates that wall shear stresses in the nonsmoothed and augmented models models 2 and 5 came closest to matching the goldstandard geometry wall shear stress distribution in this region. The peak in wall shear stress on the medial wall of the sinus terminus in the gold-standard geometry point D in Fig. 5 was not well captured in the MR-reconstructed models, again due to smoothing at the carotid sinus terminus. The MR-reconstructed geometries overestimated wall shear stress on the medial wall of the external carotid artery downstream of the tuning fork bend, due to the more abrupt turn in the goldstandard geometry point E in Fig. 5. In order to provide a more quantitative error measurement, the errors in the center plane wall shear stresses Fig. 5 were computed at equally spaced locations, x j, for model i (i 2,...,5) according to the formula model i x j reconstructed, model i x j gold standard x j / max gold standard x j, mean inlet x j. At each location x j, the envelope of maximum pointwise error was then computed as 1 max x j max model i x j ; i 2,...,5, 2 which is clearly an upper bound for the error in any particular model. The value of max (x j ) ranged from 1% to 53%, with a spatially averaged value over the entire model of 15.0% 11% mean standard deviation. In the low-shear regions of the gold-standard geometry where the normalizing factor was mean inlet, the spatially averaged value of max (x j ) was 12.8%. In the high-shear regions where the normalizing factor was gold standard, the spatially averaged value was 15.4%. A comparison of wall shear stress patterns in unsteady flow showed very good agreement between the augmented model model 2 and the gold-standard geometry. Specifically, the distribution of cycle-average wall shear stress, oscillatory shear index, and T neg Ref. 8 showed excellent agreement between the two models. 6 The dynamic nature of the wall shear stress in critical regions such as the carotid sinus also agreed well between the two models. For example, the extent, duration, and intensity of the negative shear stresses caused by flow separation in the carotid sinus was almost identical for models 1 and 2 Fig. 6. The duration and intensity of the systolic high-shear region on the proximal hips of the bifurcation was also well matched Fig. 6. Geometric Comparison: In Vivo Scans Comparison of models A, B, and C reconstructed from the in vivo MR scans showed substantial differences Fig. 7. The presence of imaging noise is clearly distinguishable in model A, which shows highly nonphysiologic distortion and surface irregularities. Model B, constructed with three-phase smoothing, profile augmentation, and approximating splines, shows a much smoother and more physiologic-appearing surface. Model C, constructed with three-phase smoothing and interpolating splines, shows a surface that is intermediate between those of models A and B. Although we do not know the true geometry for this carotid bifurcation, it is clear that model A is unsuitable, since it is highly unlikely that the carotid bifurcation of a normal healthy volunteer would have the geometry shown in Fig. 7 A. Careful examination of model C shows that the surface features are concentrated near the bifurcation, and tend to occur in axial slices normal to the common carotid artery. These slices coincide with the imaging planes, and we therefore infer that the features seen in model C are primarily due to image noise. Based upon these observations, as well as our qualitative expectations about the bifurcation shape in this volunteer, we conclude that model B most reliably represents the in vivo arterial geometry. DISCUSSION AND CONCLUSIONS This study shows that MR imaging, when combined with appropriate smoothing and model construction techniques, can be used to accurately reconstruct complex arterial geometries. It also shows that computed flow fields and wall shear stress distributions in reconstructed models can be accurate. For example, the overall distributions of shear stresses in all four MR-reconstructed models of the carotid bifurcation flow phantom were very similar to that in the gold standard model, with a mean overall wall shear stress error of only 15.0% in the worst reconstructed model. It is important to relate our observed wall shear stress errors to the overall objective of hemodynamic studies, namely, the determination of links between flow features and arterial disease. To do so, we must first define what is meant by a significant difference in wall shear stress. 2 Here, we define a difference to be significant if it would lead to a different conclusion about the cause or location of arterial disease when a reconstructed model is used in

Accuracy of Hemodynamic Modeling 39 FIGURE 6. Variation in normalized wall shear stress magnitude along the lateral centerline of the internal carotid artery sinus under pulsatile flow conditions. Upper panel: gold-standard geometry model 1 ; lower panel: augmented single-segment spline model model 2. The horizontal axis represents distance from the apex, and the vertical axis represents time, normalized by the cardiac period. The shear stress magnitude has been normalized by the temporal mean inlet shear stress for the gold-standard geometry model 1. The inlet flow wave form normalized by the mean inlet flow is shown next to the vertical axis for reference. The heavy black contours outline regions of reversed flow. A reference geometry is shown between the upper and lower panels, with T neg for model 1 shown in contour. Using the assumptions outlined in the caption of Fig. 4, the mean normalizing wall shear stress would be 4.5 dyn/cm 2. a correlation study between hemodynamics and arterial disease. With this definition, the in vitro phantom imaging portion of the study shows that there are no significant differences between the MR-reconstructed models and the gold-standard model, since they all agree in their predictions of the locations and magnitudes of low-shear zones, recirculating areas, zones of high oscillatory shear index, and wall shear stress peaks.

40 MOORE et al. FIGURE 7. Three reconstructed models of a human carotid artery imaged in vivo. Which reconstruction method was best? For models based on the relatively noise-free in vitro phantom scans, all reconstruction methods models 2 5 gave comparable results with acceptable accuracy. However, for noisier images, we believe that the best reconstruction methodology was that of models 2 and B, in which eighth-order approximating surface splines were combined with augmented linearly interpolated profiles. Our reasoning is summarized below. For both in vitro and in vivo images, the combination of profile augmentation and approximating splines provided a reasonably smooth model, with a surface largely free of features due to imaging noise. In particular, for the in vivo models, model B was the most physiologic in appearance, considering that the scanned subject was a young, healthy individual, believed to be free of significant atherosclerotic lesions. However, it is important to point out that a physiologic appearance is not necessarily a guarantee of fidelity to the true in vivo geometry. The geometric fidelity for in vivo scans remains to be quantified. Additional in vitro phantom studies with a different image set suggested that the interpolating spline method of model 5 could give poor results in the presence of appreciable noise in the MR images. 6 This was confirmed by the results of the in vivo carotid bifurcation reconstructions. In fact, model A also obtained using interpolating surface splines is highly nonphysiologic. This indicates that although interpolating surface splines can give adequate results with high-contrast, low-noise in vitro images, they do not give acceptable results with more realistic images. One point should be made about our quoted wall shear stress errors, since, depending on how it is calculated, wall shear stress error can be a misleading quantity in complex flow fields. One obvious way of calculating error is to use the local wall shear stress in the goldstandard model as a normalizing factor. However, when the wall shear stress in the gold-standard geometry is small, such as in recirculation zones, this gives unrealistically large percentage errors. A second way is to use the mean inlet wall shear stress as the normalizing factor. However, in high-shear stress zones, such as near the bifurcation apex, this gives unrealistically large percentage errors. Equation 1 represents a reasonable compromise between these two approaches, since it uses a wall shear stress normalizing factor appropriate for the specific location of interest. There is another, more subtle, type of error introduced when comparing wall shear stresses between two models with nonidentical geometries. This is the geometry matching problem, i.e., determining where a given location on a reconstructed geometry maps onto the gold-standard model. For example, in this work we compared wall shear stresses between two points at the same axial location, which presumed no error in the axial direction in model reconstruction. Furthermore, we computed the error for the worst reconstructed model. Thus, the quoted overall error of 15.0% overestimates the actual error. Our results give an estimate of how accurately the hemodynamics in a 3D branching arterial geometry can be modeled. It is useful to relate our errors to those expected using in vivo images. In vivo images generally suffer from lower image quality than those obtained in vitro, due to vessel motion artifacts and saturation of slowly moving spins. The first type of artifact blurs wall position, while the latter effect makes it difficult to discriminate between wall tissue and blood near the vessel wall. The in vitro portion of this study was designed to avoid such artifacts by using rigid vessel walls and static fluid, and thus the additional effects of poor in vivo image quality have not been quantified in this work. The difficulties in working with in vivo images are particularly acute when reconstructing diseased arteries, since such vessels typically show very complex, noisy surface features. In such cases, smoothing has a greater potential to remove physiologically relevant and important features. Summarizing all of the above, the mean error quoted above is expected to be a lower bound for the errors expected using in vivo images. Unfortunately, it is very difficult to do a study similar to this one using in vivo arteries, since gold-standard geometries and flow fields are not known in vivo. At present, we are addressing this problem by attempting to quantify the accuracy of arterial reconstruction from in vivo MR scans. 6 In the meantime, our results clearly show the preferred methodology to be used in reconstructing arterial geometries from in vitro and in vivo MR images.

Accuracy of Hemodynamic Modeling 41 ACKNOWLEDGMENTS This work has been supported by the Hildegard Doerenkamp Gerhard Zbinden Foundation C.R.E. and the Heart and Stroke Foundation of Ontario, Grant No. NA3197 D.A.S.. REFERENCES 1 Ethier, C. R., D. A. Steinman, and M. Ojha. Comparisons between computational hemodynamics, photochromic dye flow visualization and magnetic resonance velocimetry. In: Hemodynamics of Internal Organs, edited by M. W. Collins and Y. Xu. Ashurst, UK: Computational Mechanics Publications in press. 2 Friedman, M. H., C. B. Bargeron, D. D. Duncan, G. M. Hutchins, and F. F. Mark. Effects of arterial compliance and non-newtonian rheology on correlations between intimal thickness and wall shear. J. Biomech. Eng. 114:317 320, 1992. 3 Holdsworth, D. W., C. J. Norley, R. Frayne, D. Steinman, and B. K. Rutt. Variability in common carotid blood flow waveforms abstract. Radiology 197 P :451, 1995. 4 Ku, D. N., D. P. Giddens, C. K. Zarins, and S. Glagov. Pulsatile flow and atherosclerosis in the human carotid bifurcation: positive correlation between plaque location and low and oscillating shear stress. Arteriosclerosis (Dallas) 5:293 302, 1985. 5 Milner, J., J. A. Moore, B. K. Rutt, and D. A. Steinman. Hemodynamics of human carotid artery bifurcations: computational studies in models reconstructed from magnetic resonance imaging of normal subjects. J. Vasc. Surg. 28:143 156, 1998. 6 Moore, J. A. Computational Blood Flow Modelling In Realistic Arterial Geometries. University of Toronto, Department of Mechanical Engineering: PhD Thesis, 1998. 7 Moore, J. A., D. A. Steinman, and C. R. Ethier. Computational blood flow modelling: Errors associated with reconstructing finite element models from magnetic resonance images. J. Biomech. 31:179 184, 1998. 8 Moore, J. E. J., D. N. Ku, C. K. Zarins, and S. Glagov. Pulsatile flow visualization in the abdominal aorta under differing physiologic conditions: Implications for increased susceptibility to atherosclerosis. J. Biomech. Eng. 114:391 397, 1992. 9 Robb, R. A. Three-dimensional Biomedical Imaging: Principles and Practice. New York: VCH Publishers, 1995. 10 Smith, R. F. Geometric Models of the Stenosed Human Carotid Bifurcation. Dept. Medical Biophysics, U. Western Ontario: MSc Thesis, 1997. 11 Smith, R. F., B. K. Rutt, A. J. Fox, R. N. Rankin, and D. W. Holdsworth. Geometric characterization of stenosed human carotid arteries. Acad. Radiol. 3:898 911, 1996. 12 Sun, H., B. D. Kuban, P. Schmalbrock, and M. H. Friedman. Measurement of the geometric parameters of the aortic bifurcation from magnetic resonance images. Ann. Biomed. Eng. 22:229 239, 1994. 13 Taylor, C. A., T. J. Hughes, and C. K. Zarins. Computational investigations in vascular disease. Comput. Phys. 10:224 232, 1996.