Level Set-Based Integration of Segmentation and Computational Fluid Dynamics for Flow Correction in Phase Contrast Angiography

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1 Level Set-Based Integration of Segmentation and Computational Fluid Dynamics for Flow Correction in Phase Contrast Angiography Masao Watanabe, PhD, Ron Kikinis, MD, Carl-Fredrik Westin, PhD Rationale and Objectives. A novel method to correct flow data from magnetic resonance phase contrast (MR-PC) angiography, based on combining computational fluid dynamics and segmentation in a level set framework, was developed and tested in this study. Materials and Methods. The MR-PC velocity data was used in a partial differential equation-based level set method for vessel segmentation. The results were supplied as the quantitative description of the vessel wall to the flow field solver using computational fluid dynamics, based on the level set method, to obtain a physically meaningful flow. The most significant characteristic of our novel approach is that it requires light computational loads, especially insofar as it avoids generation of complex computational grid system. The integration of segmentation and computational fluid dynamics in a level set framework is shown to be both robust and economic, and yet yields a physically correct velocity field and optimal vessel geometry. Results. The application to the flow field in a straight tube with circular cross section of constant radius demonstrated the validity of out new approach, especially the treatment of the velocity boundary conditions on the solid wall. Simulation of the velocity field in both common carotid artery and bifurcation of basilar and vertebral arteries, based on clinical MR-PC data, provided with smooth and stable results. Conclusion. Applying this procedure to both synthetic and clinical data, significant improvement of the blood velocity field, such as a smooth velocity distribution aligned along the vessels and removal of burst or error vectors, could be observed. This procedure also offers possibilities for improved vessel segmentation. Key Words. Vessel segmentation; computational fluid dynamics; level set method; angiography. AUR, 2003 In 3-dimensional (3D) magnetic resonance phase contrast (MR-PC) angiography sequences, the velocities of blood flow in three orthogonal directions are mapped to phase Acad Radiol 2003; 10: From the Department of Mechanical Engineering Science, Kyushu University, Hakozaki, Higashi-ku, Fukuoka, Japan (M.W.); the Surgical Planning Laboratory (R.K.) and the Laboratory of Mathematics in Imaging (C.F.W.), Brigham and Women s Hospital and Harvard Medical School, Boston, MA. Received May 19, 2003; revision requested August 28; received in revised form September 2, 2003; accepted September 2, Address correspondence to M.W. AUR, 2003 doi: /s (03) differences, which is controlled by a variable known as velocity encoding or venc (1). This sequence results in phase wrapping in areas of flow with greater speed than the venc. As an additional complication, signal quality typically deteriorates because of phase dispersion from turbulence and vortices stemming from pulse or vessel branching. These artifacts impede accurate flow quantification, especially with respect to flow direction. The need to establish vessel diameter is also critical; but because MR-PC sequences produce very weak MR signals in the neighborhood of vessel wall, signal resolution in this region predictably degenerates. 1416

2 Academic Radiology, Vol 10, No 12, December 2003 INTEGRATION OF SEGMENTATION AND CFD Application of the level set method on medical image processing has achieved great success by using the techniques such as active contours, dynamic implicit surfaces (2 4). In MR-PC angiography, vessel geometric structure can be obtained by medical image processing from MR-PC velocity information. Lorigo et al (5) developed level set-based vessel segmentation with a high degree of accuracy. This information can also be supplied to blood flow analysis. Computational approaches based on MR segmentation have previously been applied in arterial biomechanics (6), hemodynamics of carotid artery bifurcations (7), combined computational fluid dynamics (CFD) and MRI studies on the reconstruction of blood flow patterns in a human carotid bifurcation (8). These studies generally make use of complicated, unstructured computational grid system constructed from medical images (9), and use the MRI data only for the segmentation (ie, grid system generation). In general, the available velocity information in MR-PC is in general not fully utilized in CFD studies of blood flow. In this article, we propose a novel method for vessel segmentation improvement by integrating CFD and segmentation in the level set framework by using the 3D MR-PC velocity information. The basic idea of our new method is to modify the vessel segmentation with the aid of the information obtained by the comparison between the raw MR-PC velocity field and the CFD results obtained with physical correct model. We have developed a numerical scheme to model blood flow in vessels by solving the incompressible Navier-Stokes equation with vessel geometry segmented by a partial differential equationbased fast local level set method (10). By implementing the level set Ghost Fluid method (11), we have effectively enforced a zero-velocity boundary condition on the vessel wall (the zero level set) without smearing physical properties near the wall. To a great extent, this approach has enabled us not only to reduce the computational loads in generation of the computational grid system a great deal, but also to use a simple structured computational grid with a high degree of accuracy. The improvement in velocity fields is verified for both synthetic and clinical data. MATERIALS AND METHODS Level Set Methods The level set method was originally developed by Osher and Sethian (12) as a simple and versatile method for computing and analyzing the motion of an interface in Figure 1. Schematic diagram of the modification process. two or three dimensions, such as, computing two-phase Navier-Stokes incompressible flows (13). However, the original level set method smears out both the density and the viscosity across the interface to prevent spurious oscillatory solutions at the interface. As explained in (14), the original Ghost Fluid method was developed to solve this problem by populating cells next to the interface with ghost values, and extrapolating values across the interface. On the other hand, the use of partial differential equations in the field of image processing and computer vision, in particular the use of the level set method and dynamic implicit surfaces has increased dramatically. As reviewed previously (15), the field of computer vision was probably the earliest to be influenced significantly by the level set method. The segmented vessel information is supplied to the CFD process on the level set framework, and then the blood flow field in the segmented vessel can be easily calculated on the same points as the MR-PC data acquisition points, which enables us to compare the MR-PC velocity field with those obtained by CFD without numerical manipulations, such as interpolation. Computational fluid dynamics results are sensitive to the vessel structure because the flow path is determined by the boundary structure. Vessel structures, on the other hand, are supplied by the segmentation results, which are based on possibly contaminated MR-PC data. As shown in Figure 1, our concept of improving vessel segmentation lies in implementing feedback from the CFD results. In other words, our approach entails solving the fluid dynamic inverse problem, which presumes the legitimate vessel geometric structure from the incomplete MR-PC velocity field. Vessel Segmentation The vessel segmentation was executed by applying the partial differential equation-based local level set method 1417

3 WATANABE ET AL Academic Radiology, Vol 10, No 12, December 2003 (10) to T1W MR-PC velocity data. In this work, the definition of level set function at calculation point as the distance from the vessel wall was applied. Level set function was negative when the point is inside the vessel, and positive otherwise. To define level set function at any given point in the calculation domain, we calculate the distance function by using the reinitialization technique presented in (13), where the following Hamilton-Jacobi type equation; d S d 0 d 1 0 (1) is integrated in time to steady state, with the initial conditions: d x,0 d 0 x 2.0 l if u M 2.0 l if u M where u M is the T1W MR-PC velocity vector, S is Heaviside function, is pseudo-time, l is the order of x, and is the threshold number. It is sufficient for the level set function (defined as the distance function) to be calculated in the narrow band (10), which significantly reduces the computational loads. It is observed that solving equation 1 with initial conditions 2 provides for thinner vessel geometry, most probably because MR-PC signal tends to be significantly weak in the neighborhood of vessel wall. We then resolve equation 1 with (2) d x,0 d x d 1 x l (3) as the initial conditions instead of equation 2, where d 1 is the solution of equation 1 with equation 2. The choice of, typically , will be discussed in a later section. The distance function d obtained by this procedure is used as the level set function determining the flow path geometry for the flow field calculation. Fluid Flow Simulation Blood flow in vessels is considered to be a low-speed flow, which can be assumed to be incompressible. To characterize this flow, we have developed a numerical scheme to model incompressible fluid flow in a tubular flow path, bounded by a rigid solid wall. This approach enables us to define the solid tubular wall boundary as the interface between the incompressible fluid and the rigid solid wall, and to solve a stationary interface problem by applying the proven level set method (16,17). The most significant feature of this numerical method is free from construction of complicated vessel wall fitted unstructured computational grid, as ordinary executed using the finite element method in the field of vessel flow simulation. Instead this method is able to employ level set function as the vessel wall information on the simple structured computational grid. This results in the considerable reduction of computational time. We have four unknowns, ie, pressure and the three components of velocity vector, to be determined at any given calculation point. At each time step, we solve the dimensionless evolution partial differential equations for the velocity and pressure. The most common equation set for the incompressible fluid dynamics is used for the current study, ie, continuity equation 4, which is the mass conservation equation, and Navier-Stokes equation 5, which is the equation of motion (18). u 0 (4) u t u u p 1 Re 2 u (5) where t is time. The dimensionless parameter, Re LU/, used in equation 5 is the Reynolds number, where L and U are the characteristic length and velocity, respectively, and are density and viscosity of blood, respectively. We used the values of kg/m 3 for the liquid density, and kg/ms for the liquid viscosity, respectively. The next step is to specify the initial and boundary conditions. Flow simulation can be carried out by solving governing equations 4 and 5 with appropriate initial and boundary conditions. Because we solve blood flow in vessels, necessary and sufficient boundary conditions are those at inlet and outlet, in other words, at the end of the segmented vessels and on vessel walls. We first consider the inlet and outlet boundary conditions. Calculations are to be performed within a rectangular parallelepiped extracted from the original MR-PC data. At the end of segmented vessel on the surface of the parallelepiped, we need to specify the boundary conditions for both velocity and pressure. The velocity boundary conditions are set equal to the MR-PC velocity data. Pressure boundary conditions for the cross section with maximum inlet velocity are set to zero. For the other inlet and all the outlet 1418

4 Academic Radiology, Vol 10, No 12, December 2003 INTEGRATION OF SEGMENTATION AND CFD cross sections, the pressure gradients normal to the calculated boundary surface are set to vanish. Second, we specify boundary conditions on the vessel wall. To impose the zero velocity boundary condition on the vessel wall, we couple the incompressible Navier- Stokes equation solver with a high degree of accuracy, combining the level set method with a projection method developed by Sussman et al (16), and with the Ghost Fluid method, implemented by using the ghost cells, developed by Fedkiw et al (11). We took full advantage of the level set vessel segmentation, which is described by level set function. The ghost cells are defined in the solidside neighborhood of the fluid-solid interface (ie, vessel wall). We can modify pressure in the ghost cells to satisfy appropriate boundary conditions by using the isobaric fix technique (11), by defining the unit normal vector at every grid point, and by taking spatial derivative of the level set function (15), as N / (6) and then extrapolating values using advection equation. p N p 0 (7) is pseudo-time, same as the one introduced in equation 1. Constant extrapolation of p was obtained by solving equation 7 to steady state. Based on this technique, we have developed the zero-velocity fix on the solid wall, by simple extension of the isobaric fix technique to construct mirror images of velocity u in Ghost Fluid method framework. The concept of mirror image is conventionally used in CFD to impose nonslip boundary condition on solid wall. We consider the new variable v ; v u/ ; First, the constant extrapolation of v is calculated in the direction of the normal to the solid wall in the neighborhood of the wall by solving equation 8. v N v 0 (8) Then the zero velocity fix is completed by setting u v. We follow the discretization methodology and time integration procedure that Sussman et al developed (16). This method implements essentially a non-oscillatory third order method to evaluate the convection term and the fractional time step projection method, while ensuring that the continuity equation 4 is satisfied. These methods guarantee stability in high velocity fields, robustness in complicated geometries, and provide great accuracy without smearing the solution. For detailed explanation, readers are recommended to refer to (16) and references within. In the present study, we neglect the pulsatory flow effects and assume steady flow conditions for the simplicity, and we also employ relatively larger sizes of computational grids, which are based on the MR-PC data acquisition points, as stated above, than those of ordinary blood flow simulation. Therefore, the results show smooth and less vortical flows with medium spatial resolution. We can modify these at the cost of computational efficiency. RESULTS Flow in a Tube: Poiseiulle Flow We first calculated the flow field in a straight tube with circular cross section of constant radius. If pressure gradient along the tube is constant and known, the flow is known as a Poiseiulle flow. We chose this flow to verify the validity of the zero-velocity fix procedure on the solid wall. We compare our results with the following theoretical velocity distribution obtained with the constant pressure gradient dp/dx along tube, as explained in Batchelor (17). u x R2 dp dx 1 r2 U R max 1 r2 (9) R where u x is the velocity component along the tube, r is the distance from the center of tube, R is the tube radius, and U max is the velocity on the center (maximum velocity). The calculation was executed with grid size of 0.5 mm, and dimension of , dp/dx 1,000 Pa/m, R 5 mm. Figure 2 (a) shows the comparison of the velocity distribution between the theoretical and the numerical results. Clearly, both curves agree well, confirming that an enforcement of the zero-velocity condition on the tube wall is a reasonably accurate model. These results strongly suggest that the treatment of both isobar and zero-velocity fixes are valid and effective. Figure 2 (b) shows the pressure distribution along the tube. Our numerical scheme uses the zero gradient condi- 1419

5 WATANABE ET AL Academic Radiology, Vol 10, No 12, December 2003 Figure 3. Numerical simulation with contaminated initial condition for Poiseiulle flow (top) and calculation result (bottom). (a) Initial condition calculated by equation 9 with white Gaussian noise. (b) Calculated result using equations 4 5, after 10 time steps. Figure 2. Comparison between theoretical and numerical results for Poiseiulle flow. Theoretical results are shown as a solid line, calculated results are shown as open circles. (a) Velocity distribution in radial direction. (b) Pressure distribution in axial direction. tion for the outlet pressure; hence the pressure gradient cannot be constant. Discrepancies from theoretical result are therefore inevitable. However, because these discrepancies are sufficiently small, we can assert that our method works well when simulating flow with tubular geometry. We next evaluated our method with both boundary and initial conditions contaminated with noise. Figure 3 (a) shows the synthetic velocity field generated by adding Gaussian white noise to the three components of the velocity field. Figure 3 (b) shows the calculated result after 10 time steps. It can clearly be observed that the velocity field is improved with respect to the flow direction. It should be emphasized that in this CFD scheme, pressure was corrected implicitly, hence the global modification of flow field could be achieved efficiently. As for the inlet and outlet boundaries, no improvement can be attained with the current approach. Furthermore, the vessel boundary has not been altered as it would have resulted in direct smoothing. Magnetic Resonance Phase Contrast Velocity Data In this section, we present results of applying our method to clinical data. The size of the data set is , with a field of view of 240 mm, slice thickness of 1.5 mm, and velocity encoding of 40 cm/s. Figure 4 shows the MIP of this image. In this study, we restricted the application of the method on the partial data set, namely rectangular parallelepiped extracted from the original MR-PC data, because of the computational loads. We show the results obtained by using the data in the vicinity of the common carotid artery, shown as A in Figure 4 (b), and those in the vicinity the bifurcation of basilar and vertebral arteries, shown as B in Figure 4 (b). After the segmentation procedure, described in the previous section, was completed with initial condition of segmentation dependent on î as shown in equation 3, 1420

6 Academic Radiology, Vol 10, No 12, December 2003 INTEGRATION OF SEGMENTATION AND CFD Figure 4. Maximum intensity projection (MIP) of a clinical MR-PC data set provided to this work. (a) brain vessel data. (b) Close up in the vicinity of common carotid, basilar and vertebral arteries. Applications of this method to vessels shown as A and B are presented. d in equation 1 was calculated uniquely with selected. As the result, quantitative description of vessel geometric structure was obtained by level set function, and parameter both controlled the vessel wall location and governed the vessel segmentation. Once level set function was determined, the velocity field, u, could then be calculated, with appropriate boundary conditions and MR-PC velocity data u M. It should be emphasized that u depends on, hence, it also depends on parameter. We postulated that the most appropriate, for a given set of MR-PC velocity data, minimizes the discrepancy between the MR-PC data and the calculated results. We used the following expression for this discrepancy, ; Figure 5. (a) Common carotid artery, shown as A in Figure 4, geometric structure segmented by level set method. (b) Computational grid configuration used in level set CFD method. u M u / u M (10) The first data set is a section of the common carotid artery (shown as A in Fig 4 [b]). The flow field in a bending vessel geometry was calculated. This flow was chosen to test both stability and robustness of the method because the geometry was rather easily segmented, and was 3D, which in turn yielded complicated 3D flow fields. The calculation domain was , and the maximum velocity obtained by MR-PC velocity data in this domain was m/s. In Table 1, the effect of Figure 6. Comparison between velocity fields obtained by (a) MR-PC, and (b) level set CFD method, for common carotid artery (a) PCA-MRI and (b) calculated. the level set correction term,, on the velocity calculations is shown. By the optimal evaluation using equation 10, we chose 0.6 because the discrepancy was minimal, and we show the results with this value in the following figures. In Figure 5, segmentation results of the common carotid artery and the computational grid for Table 1 Effect of level set correction term for level set segmentation on discrepancy term defined by equation 10 for common carotid artery

7 WATANABE ET AL Academic Radiology, Vol 10, No 12, December 2003 Table 2 Effect of level set correction term for level set segmentation on discrepancy term defined by equation 10 for bifurcation of basilar and vertebral arteries Figure 7. (a) Bifurcation of basilar and vertebral arteries, shown as B in Figure 4, geometric structure segmented by level set method. (b) Computational grid configuration used in level set CFD method. Figure 8. Comparison between velocity fields obtained by (a) MR-PC, and (b) level set CFD method, for bifurcation of basilar and vertebral arteries. CFD are shown, and CFD results are shown in Figure 6 with 0.6. The location of vessel wall was determined as the zero level set by linear interpolation. Both velocity and pressure were calculated on the computational grid points with the value of level set function of less than triple grid size. The degree of freedom of velocity was 859 in this case. It can be observed that the flow field is significantly improved, especially the direction of velocity vectors which are now naturally aligned along the vessel direction. Notice also that speeds in the original data set, u M s, tend to be greater than those in the calculation, u s, around both elbow and outlet regions. Considering the continuity equation 4, and the velocity distribution around the elbow region shown in Figure 6 (a), it is most possible that the segmentation process provided a thicker vessel diameter around the elbow region. This result also suggests that modification of outlet boundary conditions may provide an improved velocity distribution. The second data set is the bifurcation region of the basilar artery and vertebral arteries (shown as B in Fig 4[b]). These arteries were chosen to test a more complicated flow than the previous one. The calculation domain was , and the maximum velocity obtained by MR-PC velocity data in this domain was m/s. In Table 2, the effect of the level set correction term,, on the velocity calculations is shown. It should be emphasized that the optimal value of depends on both vessel geometry and the MR-PC signal intensity distribution, which is obvious when compared with the previous results. By the optimal evaluation using equation 10, we chose 0.2 because the discrepancy is minimal. We show the results of this value in the followings. In Figure 7, segmentation results of the bifurcation of basilar and vertebral arteries and the computational grid for CFD are shown, and CFD results are shown in Figure 8 with 0.2. The degree of freedom of velocity was 389 in this case. The surface of segmented vessel shown in Figure 7 was observed less smooth than the one in Figure 5. This is mainly because of the lower spatial resolutions in the case of thinner basilar and vertebral arteries than those in the case of thicker common carotid artery. However, the result is smooth and stable, even with the relatively small number of data points in our example. This result offers yet more evidence of the robustness of this method. Close to the bifurcation, erroneous velocity can be observed, possibly because of phase wrapping (Fig 8 [a]). These errors are successfully suppressed as shown in Figure 8 (b). Considering the strength of velocity from the right vessel at the bifurcation, it is also possible that the vessel diameter is overestimated, and this information can be utilized in the resegmentation of the vessel. 1422

8 Academic Radiology, Vol 10, No 12, December 2003 INTEGRATION OF SEGMENTATION AND CFD CONCLUSION A novel correction procedure of MR-PC velocity data has been developed, by coupling an incompressible Navier-Stokes equation solver with projection level set Ghost Fluid method, to a partial differential equationbased fast local level set vessel segmentation method. Applying this procedure to both synthetic and clinical data, significant improvement of the blood velocity field, such as a smooth velocity distribution aligned along the vessels and removal of burst or error vectors, could be observed. This procedure also offers possibilities for improved vessel segmentation. The authors are aware that additional quantitative validation of this procedure is needed (ie, by using the flow phantom). REFERENCES 1. Leidholdt EM Jr, Bushverg JT, Seibert JA, Boone JM. The essential physics of medical imaging. Baltimore, MD: Williams & Wilkins, Caselles V, Kimmel R, Sapiro G. Geodesic acrive contours. Int J Comput Vision 1997; 22: Kichenassamy S, Kumar A, Olver P, Tannenbaum A, Yezzi A. Conformal curvature flows: from phase transitions to active vision. Arch Rational Mech Anal 1996; 134: Chan T, Vese L. Active contours without edges. IEEE Trans Image Processing 2001; 10: Lorigo LM, Faugeras OD, Grimson WEL, Keriven R, Kikinis R, Nabavi A, Westin CF. CURVES: curve evolution for vessel segmentation. Med Image Analysis 2001; 5: Vorp DA, Steinman DA, Ethier CR. Computational modeling of arterial biomechanics. Comput Sci Eng 2001; 3: Milner JS, Moore JA, Rutt BK, Steinman DA. Hemodynamics of human carotid artery bifurcations: computational studies with models reconstructed from magnetic resonance imaging of normal subjects. J Vasc Surg 1998; 28: Long Q, Xu XY, Ariff B, Thom SA, Hughes AD, Stanton AV. Reconstruction of blood flow patterns in a human carotid bifurcation: a combined CFD and MRI study. J Magn Res Imag 2000; 11: Cebral JR, Lohner R. From medical images to CFD meshes. Proceedings of the 8th International Meshing Roundtable, South Lake Tahoe, CA. October 10 13, Available at: ley/db/conf/imr/imr1999.html Accessed October 21, Peng D, Merriman B, Osher S, Zhao H, Kang M. A PDE-based fast local level set method. J Comput Phys 1999; 155: Fedkiw RP, Aslam T, Merriman B, Osher S. A non-oscillatory eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J Comp Phys 1999; 152: Osher S, Sethian JA. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. J Comput Phys 1988; 79: Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flos. J Comput Phys 1994; 114: Osher S, Fedkiw RP. Level set methods: an overview and some recent results. J Comput Phys 2001; 69: Osher S, Fedkiw RP. Level set methods and dynamic implicit surfaces. New York, NY: Springer-Verlag, Sussman M, Almgren AS, Bell JB, Colella P, Howell LH, Welcom ML. An adaptive level set approach for incompressible two-phase flows. J Comput Phys 1999; 148: Fedkiw RP. Coupling an Eulerian fluid calculation to a lagangian solid calculation with the ghost fluid method. J Comp Phys 2002; 175: Batchelor G.K. An introduction to the fluid dynamics. Cambridge, UK: Cambridge Press,