Simulated Pathline Visualization of Computed Periodic Blood Flow Patterns

Transcription

1 Simulated Pathline Visualization of Computed Periodic Blood Flow Patterns David A. Steinman, Ph.D. Imaging Research Labs, John P. Robarts Research Institute, London, Ontario, Canada N6A 5K8 Department of Medical Biophysics, University of Western Ontario, London, Ontario, Canada N6A 5C1 Accepted for Publication in the Journal of Biomechanics Keywords: flow visualization, particle tracking, hemodynamics, computer modeling Address correspondence to: David A. Steinman, Ph.D. Imaging Research Labs John P. Robarts Research Institute 100 Perth Dr., P.O. Box 5015 London, Ontario, Canada, N6A 5K8 Voice: x34113 Fax: Page i

2 Abstract: Improvements in computer hardware and software have made it possible to model pulsatile blood flow in realistic arterial geometries. Such studies produce enormous amounts of velocity data, which are often difficult to interpret and communicate using traditional contour and/or vector field plots. Inspired by in vitro flow visualization techniques such as particle image velocimetry (PIV), we describe a simple and effective method for visualizing periodic three-dimensional velocity data, based on the subdivision and sequential display of computed particle trajectories. Analogous to a PIV experiment, the length and spacing of such simulated particle pathlines are controlled by user-specified shutter-speed and frame rate variables. Strategies for color-coding pathlines to highlight important hemodynamic features such as recirculation zones and branch flow division are presented. 1 Introduction In the last few years, advances in computer hardware and modeling software have made it possible to routinely carry out computational fluid dynamic (CFD) studies pulsatile hemodynamics in realistic, three-dimensional models derived from imaged or cast arteries (Chandran et al, 1996;Milner et al, 1998;Perktold et al, 1998;Moore et al, 1999). Unlike traditional idealized models where contour or vector plots are typically displayed on planes of symmetry realistic models rarely provide such convenient planes for display. The unsteady nature of these flow fields can make it difficult to comprehend the dynamic relationships between primary and secondary flow or to identify the locations of increased blood residence time important factors in the study of hemodynamics and vascular disease using conventional streamline or vector field plots alone. Experimental flow studies based on the introduction of tracer particles into the flow field can provide readily comprehensible visualizations of complex flow patterns. For example, in particle image velocimetry (PIV) studies, reflective particles are seeded into a flow field and illuminated as they are passively-advected through a sheet (or sheets) of laser light. Analog or digital cameras acquire sequential images of the flow field at a specified frame rate and shutter speed. The shutter speed determines the length of individual pathlines, such that long exposures produce longer pathlines. The frame rate controls, for each particle, the spacing between pathlines in successive frames. (This information is then often used to estimate the local velocity field.) Experimental flow visualization studies are, however, difficult to carry out for pulsatile flows in realistic arterial geometries. In this paper, we present a simulated pathline animation technique for visualizing CFDcomputed pulsatile 3-D blood flow patterns. This technique relies on extracting and animating pathlines from computed particle trajectories based on intuitive user-specified frame rate and shutter speed criteria. Using a model of a normal human carotid bifurcation, we demonstrate how color-coding of the pathlines can also be used to enhance important hemodynamic features such as flow recirculation and branch flow division. 2 Methods 2.1 Particle Tracking Given a continuous representation of a time-varying vector velocity field u(x,t), the trajectory of a mass-less, neutrally buoyant and non-diffusing particle can be computed by integrating the time-varying velocity field according to the following equation: Page 1

3 dx = u( x, t) x( t0 ) = dt x 0 (1) where x 0 defines the initial coordinates of a particle seeded at time t 0. Any number of numerical techniques may used to integrate this equation; for our studies we use a fourth-order Runge-Kutta integration with adaptive time-stepping, and an efficient algorithm for identifying the location of the particle in the finite element mesh (Kunov et al, 1996). 2.2 Pathline Extraction In general, for the i th frame, F i, the pathline of the j th particle represents the portion of the particle trajectory between x j (t i ) and x j (t i + t ss ), where t ss is the shutter-speed and t i is defined as: i 1 t i = (2) f where f is the acquisition frame rate, and we have arbitrarily assigned t=0 as the start time of the first frame (F 1 ). For periodic pulsatile flow, it is sufficient to consider frames for a single cardiac cycle, and repeat them in a loop. However, because particles may reside within the model for more than one cardiac cycle, it is necessary to display multiple pathlines of the same particle in a given frame in order to preserve the appearance of continuity in the animation loop, i.e.: t i 1 i = n + n = 0,1,2, (3) T NF where NF is the integer number of frames per cardiac cycle (i.e. NF = f*t). For convenience, we also define a dimensionless shutter speed, SS, normalized to the period of the cardiac cycle (i.e. SS = t ss /T). These steps are illustrated in Figure Particle Seeding To simulate the uniform concentration of tracer particles in the fluid, it would be necessary to seed particles according to a scheme that accounts for both the spatial and temporal variations in the number of particles entering the model throughout the cardiac cycle. Although there are ways to mimic this computationally (Kunov et al, 1996), for the purposes of flow visualization we have found it sufficient to seed NP particles at NF times equally-spaced throughout the cardiac cycle. The NP particles are distributed randomly across the model inlet, and this random distribution is updated for each seeding time, t i /T, corresponding to the start time of each frame. This results in a total of NP*NF particle trajectories, each of which is computed independently until it has left the model domain (or the desired field-of-view). The coordinates, velocities, and time along the trajectory are written to disk for subsequent extraction of pathlines. 2.4 Animation An individual frame, F i, is constructed by displaying, for each particle, only those entries in the disk file whose times fall between t i /T and t i /T+SS. The pathline coordinates provide the spatial locations for screen display, while the corresponding velocities may be used to color-code the pathlines in a variety of ways (see Results and Discussion). Tecplot software (Amtec Engineering, Inc.; Bellevue WA) is used to display and color-code the pathlines, and each frame is saved as a bitmap image. The resulting series of bitmap images is then converted into a MPEG animation using software supplied with the Indigo/2 workstation (Silicon Graphics Inc.; Mountain View, CA) on which all calculations were carried out. Further details of the animation procedure are omitted, as they are highly specific to the display software and hardware used. We Page 2

4 note, however, that these animation techniques can be implemented with any graphical display software that can plot (and, ideally, color-code) three-dimensional line segments. 3 Results In this section we present frames from visualizations of physiologically pulsatile flow computed in a model of a normal human carotid bifurcation reconstructed from magnetic resonance imaging (Milner et al, 1998). The complete visualizations over the cardiac cycle (which can be found on the Webpage of the Journal of Biomechanics: /inca/publications/store/3/2/1/) were constructed from the time-varying computed velocity field using the steps outlined in the Methods. The frame rate used was NF=60, to adequately resolve the rapid temporal accelerations near peak systole. The number of particles used was NP=40, which was found to provide a sufficient number of pathlines to discern but not obscure flow structures of interest. The images presented in Figures 2-4 represent individual frames from the complete visualizations. Figure 2 illustrates the effect of shutter speed on the display of peak systolic and late diastolic flow patterns. At peak systole, pathlines in the branches largely overlap when using the slower shutter speed, making it difficult to discern the relative magnitudes of the branch velocities (Figure 2a). The faster shutter speed, however, clearly shows the presence of higher velocities in the ECA compared to the ICA (Figure 2b). Conversely, pathlines imaged using SS=1/60 more clearly show the complex recirculating flow patterns proximal to the bifurcation than do those from the SS=1/500 pathlines. This is particularly evident in late diastole, where the secondary flow patterns (i.e. horizontal pathlines) are much more clearly visualized with SS=1/60 (Figure 2c) than with SS=1/500 (Figure 2d). Although information about fluid velocity and direction is contained in the length and orientation of the pathlines, the projected nature of the pathline animations can, as Figure 2 demonstrates, make it difficult to distinguish individual pathlines without the aid of stereoscopic displays. This can be alleviated in part through the use of color-coding to further distinguish velocity magnitudes and identify overlapping pathlines. In Figure 3, for example, pathlines are color-coded according to the axial (Z) velocity component using a pseudo-doppler color map (Vera et al, 1992) to highlight axial velocity magnitude and direction. From this sequence of frames it is much easier to identify the presence of the dynamic recirculation zones at both hips of the bifurcation. The horizontal blue pathlines superimposed on the vertical yellow and red pathlines during flow deceleration indicate that these recirculation zones effectively communicate via the secondary flows around the periphery of the lumen. In the ICA, pathlines at peak systole (Figure 3a) mostly follow the curve of the lumen, indicating uniform flow. As the flow decelerates (Figures 3b-d), however, the progression of helical flow down the ICA branch is clearly evident. A different view of the same flow field is provided in Figure 4, in which a red/blue color-map has been used to distinguish blood exiting the ICA and ECA branches. Inspection of the flow field in the CCA proximal to the bifurcation reveals that blood entering the ECA arises largely from a high-speed core offset from the CCA centerline. In contrast, flow in the ICA branch is fed largely by the slower, secondary flows around the periphery of the CCA. Also evident is the somewhat surprising presence of retrograde flow mixing in the ECA, as evidenced by the presence of ICA flow (i.e. red pathlines) in the ECA downstream of the bifurcation apex during flow deceleration (Figures 4b-d). Page 3

5 4 Discussion and Conclusions In the absence of a corresponding experimental flow visualization, it is admittedly difficult to prove the truth of the flow features portrayed in such animations of a complex flow field. However, simulated pathline visualization of pulsatile flow in an idealized stenosed carotid bifurcation model was able to accurately reproduce key flow structures observed in a corresponding DPIV visualization (Steinman et al, 1999). Furthermore, the unanticipated presence of retrograde flow mixing observed in Figure 3 has been hypothesized as a cause of toxic neurological reactions to local anaesthetic agents injected into the external carotid artery branches (Aldrete et al, 1977). We therefore conclude that, within the stated assumptions, our simulated pathline visualization technique provides a faithful representation of the computed flow field. Our experience has been that this dynamic, lagrangian visualization technique has made it easier for us to comprehend computed pulsatile blood flow patterns, but perhaps more importantly to communicate their intricacies to lay audiences and clinical collaborators. Because they represent the history of a given particle, pathlines may also be color-coded according to such variables as time-since-release or accumulated shear exposure, thus allowing the observer to more easily visualize the dynamics of hemolysis or platelet activation. Ultimately, when coupled with parallel computing architectures and high-performance graphic displays, it should be also possible to compute and display pathlines on-the-fly (Kenwright & Lane, 1996), to allow more interactive control over the user-specified variables, model orientation, color-mapping, etc Finally, Lagrangian visualization techniques such as the one described here are not in principle limited to operating on computed velocity data. Medical imaging technologies such as 3-D Doppler ultrasound and magnetic resonance (MR) imaging can also provide measurements of the entire velocity field, albeit typically at lower resolutions than CFD studies. Some attempts have been made to track streamlines (Napel et al, 1992;Buonocore, 1998) and pathlines (Wigstrom et al, 1999) from MR phase contrast velocity measurements, and we are investigating the use of these simulated pathline techniques to animate such non-invasively imaged pulsatile flow fields. Acknowledgments This work was supported by the Medical Research Council of Canada (Group Grant GR ), and the author is supported by a Research Scholarship from the Heart & Stroke Foundation of Canada. The author also thanks Jeffery Kay, Jaques Milner, and Mauro Tambasco for helping to implement and optimize the various steps of the flow visualization process. References Aldrete, J.A., Narang, R., Sada, T., Tan, L.S., and Miller, G.P. (1977) Reverse carotid blood flow a possible explanation for some reactions to local anesthetics. J.Am.Dent.Assoc. 94, Buonocore, M.H. (1998) Visualizing blood flow patterns using streamlines, arrows, and particle paths. Magn.Reson.Med. 40, Chandran, K.B., Vonesh, M.J., Roy, A., Greenfield, S., Kane, B., Greene, R., and McPherson, D.D. (1996) Computation of vascular flow dynamics from intravascular ultrasound images. Med.Eng.Phys. 18, Kenwright, D.N. and Lane, D.A. (1996) Interactive time-dependent particle tracing using tetrahedral decomposition. IEEE Trans.Visualiz.Comput.Graphics 2, Page 4

6 Kunov, M.J., Steinman, D.A., and Ethier, C.R. (1996) Particle volumetric residence time calculations in arterial geometries. J.Biomech.Eng. 118, Milner, J.S., Moore, J.A., Ethier, C.R., Rutt, B.K., and Steinman, D.A. (1998) Computed hemodynamics of normal human carotid artery bifurcations derived from magnetic resonance imaging. J.Vasc.Surg. 28, Moore, J.A., Steinman, D.A., Prakash S., Johnston, K.W., and Ethier, C.R. (1999) A numerical study of blood flow patterns in anatomically realistic and simplified end-to-side anastomoses. J.Biomech.Eng. In Press Napel, S., Lee, D.H., Frayne, R., and Rutt, B.K. (1992) Visualizing three-dimensional flow with simulated streamlines and three-dimensional phase-contrast MR imaging. J.Magn.Reson.Imaging 2, Perktold, K., Hofer, M., Rappitsch, G., Loew, M., Kuban, B.D., and Friedman, M.H. (1998) Validated computation of physiologic flow in a realistic coronary artery branch. J Biomech. 31, Steinman, D.A., Poepping, T.L., Rankin, R.N., and Holdsworth, D.W. (1999) Flow patterns at the stenosed carotid bifurcation: Effect of concentric vs. eccentric stenosis. Submitted to Ann.Biomed.Eng. Vera, N., Steinman, D.A., Ethier, C.R., Johnston, K.W., and Cobbold, R.S. (1992) Visualization of complex flow fields, with application to the interpretation of colour flow Doppler images. Ultrasound Med.Biol. 18, 1-9. Wigström, L., Ebbers, T., Fyrenius, A., Karlsson, M., Engvall, J., Wranne, B., and Bolger, A.F. (1999) Particle trace visualization of intracardiac flow using time-resolved 3D phase contrast MRI. Magn.Reson.Med. 41, Page 5

7 Figure 1: Identification of discrete pathlines from two continuous particle trajectories, using NF=4 and SS=1/4. Changes in grayscale along the trajectories distinguish the pathlines associated with even and odd frames, and the inset flow waveform identifies the respective time intervals corresponding to each frame. Since the particle exiting the ICA branch remains in the model for more than one cardiac cycle, two separate pathlines (corresponding to the pathlines of the first and second cycle) would be displayed in Frame 1. Note that the apparent discontinuity of the ICA pathline in frame 2 is actually a flow-reversal hidden by the adjacent frame 3 pathline. Note also the orientation axis in the top-right corner. (ICA = internal carotid artery; ECA = external carotid artery; CCA = common carotid artery.) Page 6

8 Figure 2: The effect of normalized shutter speed (SS) on the display of pathlines from frames at peak systole and late diastole. (a) Peak systole, SS=1/60. (b) Peak systole, SS=1/500. (c) Late diastole, SS=1/60. (d) Late diastole, SS=1/500. In all cases NF=60, NP=40. Page 7

9 Figure 3: Regions of blood recirculation are highlighted by applying a pseudo-doppler ultrasound color map (Vera et al, 1992) to the axial (Z) velocity component to distinguish forward from reverse flow. Shown are frames 6,8,10, and 12 from an animation with NF=60, SS=1/60, and NP=40. Note the contour legends in each frame, indicating velocities in cm/s. Page 8

10 Figure 4: The complex mixing of flow to the ICA and ECA branches is highlighted by applying a red/blue color map based on branch exit criteria and velocity magnitude. Apart from color map, frames are identical to those in Figure 3. Page 9