Quantification of Vessel Wall Cyclic Strain Using Cine Phase Contrast Magnetic Resonance Imaging

Transcription

1 Annals of Biomedical Engineering, Vol. 30, pp , 2002 Printed in the USA. All rights reserved /2002/30 8 /1033/13/$15.00 Copyright 2002 Biomedical Engineering Society Quantification of Vessel Wall Cyclic Strain Using Cine Phase Contrast Magnetic Resonance Imaging MARY T. DRANEY, 1 ROBERT J. HERFKENS, 2 THOMAS J. R. HUGHES, 1 NORBERT J. PELC, 2 KRISTIN L. WEDDING, 2 CHRISTOPHER K. ZARINS, 3 and CHARLES A. TAYLOR 1,3 1 Department of Mechanical Engineering, Stanford University, Stanford, CA; 2 Department of Radiology, Stanford University, Stanford, CA; and 3 Department of Surgery, Stanford University, Stanford, CA (Received 15 September 2001; accepted 22 July 2002) Abstract In vivo quantification of vessel wall cyclic strain has important applications in physiology and disease research and the design of intravascular devices. We describe a method to calculate vessel wall strain from cine PC-MRI velocity data. Forward backward time integration is used to calculate displacement fields from the velocities, and cyclic Green Lagrange strain is computed in segments defined by the displacements. The method was validated using a combination of in vitro cine PC-MRI and marker tracking studies. Phantom experiments demonstrated that wall displacements and strain could be calculated accurately from PC-MRI velocity data, with a mean displacement difference of mm pixel size 0.39 mm and a mean strain difference of 0.01 strain extent A propagation of error analysis defined the relationship between the standard deviations in displacements and strain based on original segment length and strain magnitude. Based on the measured displacement standard deviation, strain standard deviations were calculated to be validation segment length and typical segment length. To verify the feasibility of using this method in vivo, cyclic strain was calculated in the thoracic aorta of a normal human subject. Results demonstrated nonuniform deformation and circumferential variation in cyclic strain, with a peak average strain of Biomedical Engineering Society. DOI: / Keywords Biomechanics, Wall motion, Aorta, Aneurysm. INTRODUCTION Biomechanical factors have been shown to influence the development and adaptation of the human cardiovascular system. 5,14 Strong correlations between hemodynamic factors and localization of atherosclerotic disease have been demonstrated. 6,26 Wall motion, tensile stress, and strain have also been hypothesized to contribute to the differential localization of occlusive and aneurysmal disease. 1,8,9,15,21 Previous animal experiments have demonstrated that areas of increased wall motion, such as Address correspondence to Charles A. Taylor, PhD, Stanford University Medical Center, Division of Vascular Surgery, 300 Pasteur Drive, Stanford, CA Electronic mail: taylorca@ stanford.edu 1033 those proximal to a coarctation, are associated with increased intimal thickening, 25 while reduction of this wall motion reduces intimal thickening. 22 Better understanding of both normal and abnormal wall motion and strain distributions will contribute to our understanding of disease processes as well as potentially aid in the characterization of vulnerable plaques and aneurysms; for example, changes in wall motion and strain patterns may help predict which aneurysms are more likely to enlarge rapidly and/or rupture. It is not currently feasible to quantify true strain, in vivo, because the unstressed configuration cannot be determined. However, strain over the cardiac cycle relative to a reference configuration, or cyclic strain, can be quantified in vivo. All results and discussion in this article refer to cyclic strain, not true strain. Circumferential cyclic strain has been measured in vivo using invasive techniques such as intravascular ultrasound and sonomicrometer crystals. 7,10,11 Although useful information can be obtained from these methods, strain is computed based purely on diameter changes, and circumferential variations are difficult to measure. Additionally, invasive methods may actually alter motion patterns due to vasospasm or vessel exposure including changes in vessel tethering and external pressure, and are not feasible for studying wall motion and strain in normal human subjects. While some prior investigations used noninvasive techniques to measure cyclic strain in vivo, these methods have significant limitations. Ultrasound has been used 12,13 to quantify strain based on cyclic diameter change, but can only be used for selected subjects and anatomic locations. Time-dependent magnetic resonance MR magnitude images acquired perpendicular to a given vessel can be used to track the luminal boundary of the vessel over time, and the displacement of this boundary can be used to approximate cyclic strain based on the change in radius of the vessel. 18 This method has several shortcomings. First, out-of-plane motions can ap-

2 1034 DRANEY et al. pear as in-plane deformation, and flow effects can artificially change the appearance of the lumen boundary. Both of these factors can produce errors in determining the boundary of the lumen, which in turn lead to errors in computed strain. Second, this method cannot account for nonaxisymmetric deformation or in-plane vessel rotations and is limited to calculations of in-plane strain. An alternate method, MR tagging, 17,27 directly acquires displacement data. However, this method requires the pixel size to be small compared to the organ spatial dimensions, displacements, and heterogeneity of the strain field being studied. Current MR spatial resolution is not adequate to measure displacement and strain accurately in vessel walls using this method. A third MRbased method for quantifying motion and strain utilizes MR phase contrast velocity data. 19,20,24 This method forms the basis of the approach presented to noninvasively measure vessel wall strain in vivo. The use of phase contrast velocity data to quantify vessel strain has several advantages. First, because three components of velocity can be measured, general displacement fields and strain tensors can be calculated. Second, the velocity data are extracted directly from the wall, not the vessel lumen, and thus is less vulnerable to motion-related artifacts. We previously reported a method to quantify cyclic strain from the velocity of the vessel wall acquired using cine PC-MRI data. 2 Although this method yielded promising results with in vitro phantoms, the method was based on the assumption of axisymmetric deformation. We have observed nonaxisymmetric deformations in the human aorta in preliminary investigations, 3,23 indicating that a more general method for noninvasively quantifying vessel strain is needed. We describe a new method to track the vessel wall as a function of time using wall velocity data acquired with cine PC-MRI. A forward backward integration method is used to calculate displacement fields from the measured velocities, and a cyclic Green Lagrange finite strain is calculated relative to a reference configuration from the calculated displacements. This method accounts for nonaxisymmetric deformation, rigid body translation and rotation, and circumferential variations in strain. While we assume the vessel behaves as an incompressible membrane undergoing large in-plane deformations, this method is extensible to quantification of strain through the wall thickness and along the vessel axis. This method was validated using in vitro data and was then applied to quantify cyclic strain in the thoracic aorta of a normal human subject. MATERIALS AND METHODS The method for quantifying strain, described herein, has several steps. First, cine PC-MRI is used to acquire velocity data of the vessel wall. A combined forward backward integration method is used to calculate vessel wall displacements and track discrete vessel regions through the cardiac cycle. The time-dependent displacement field is then used to calculate circumferentially varying cyclic Lagrangian strain of the vessel wall over the cardiac cycle, relative to a reference time frame defined as the first time point for any given acquisition. Each of these steps is described in more detail below. Magnetic Resonance Imaging and Measurements Two-dimensional cine PC-MRI was used to acquire in-plane velocity of the wall. Separate velocity encoding values were used for the in-plane directions 3 10 cm/s and the through-plane direction cm/s to improve velocity resolution in-plane and minimize flow artifacts through-plane. Twenty- four time frames through the cardiac cycle were reconstructed. RF spoiling was used to minimize motion artifacts and spatial saturation pulses were used in some acquisitions to minimize flow effects and to enhance visualization of the vessel wall. A 1.5 T GE Signa GE Medical Systems, Milwaukee, WI was used for all scans. A 5 in. receive-only surface coil was used for the in vitro experiments, and a cardiac phased-array coil was used for the in vivo experiments. Specific sequence parameters are presented with the experimental results. Extraction of Velocities and Vessel Regions For each time point, the magnitude and velocity data were read into Matlab The MathWorks, Inc., Natick, MA. The approximate center of the vessel wall was manually defined using a spline curve created with an arbitrary number of user-entered points. Pixels in a userdefined neighborhood of the spline, roughly equivalent to the thickness of the vessel wall, were extracted. Pixels with a magnitude value greater than 40% of the maximum extracted pixel value and with velocity components in the range of one standard deviation of the velocities of adjacent pixels were defined as being in the wall. These wall pixels were then divided into twelve 30 sectors. The centers of each of these sectors with coordinates equal to the mean x and y values of pixels in the sector were used to automatically generate an updated spline curve Fig. 1 a that was smoother and closer to the center of the wall than the original curve. This updated spline curve was found to be relatively insensitive to variations in the initial user-defined spline curve. Pixels in the neighborhood of the updated spline were again extracted and subjected to magnitude and velocity tests. Pixels determined to be in the wall Fig. 1 b subsequent to these tests were stored for each time point along with the corresponding velocities Fig. 1 c. The wall pixels in the reference image, defined as the first temporal frame acquired, were redivided into twelve

3 Quantification of Vessel Wall Strain Using Cine PC-MRI 1035 FIGURE 1. Description of wall segmentation and sector velocities using MR data from an in vitro flow phantom: a updated spline curve approximating center of wall with 12 evenly spaced material points on the reference frame; b centers of wall pixels depicted by dots and updated spline curve; c wall pixel velocity vectors; and d sector boundaries and spatially averaged sector velocities. Note that the radial extent of the sectors exceeds the wall thickness and is large for illustrative purposes only. 30 sectors. The centers of the sectors were defined as the initial material points on the reference image, with velocities equal to the average velocity of the pixels within the sector Fig. 1 d. FIGURE 2. Stages of the forward backward segment tracking algorithm, showing: a the trajectory and position of one sector at the reference and subsequent time frames using forward integration, shown with the wall center splines; b the trajectory and position of the same sector at the reference and previous time frames using backward integration; c the combined trajectory of the sector shown in a and b, and d the combined trajectories of all sectors. The starting points of the trajectories in a c are demarked with a solid black circle. The shade of the sector outline, the center line curve, and the forward and backward trajectory arrows correspond to time; dark gray indicates time frame 1 and lightest gray indicates the last time frame. The size of the trajectory arrowheads relates to the magnitude of displacement within a time interval. Segment Tracking Algorithm Displacement fields, throughout the cardiac cycle, of the material points defined in the reference frame were calculated using a forward backward time integration scheme as described by Pelc et al. 20 This process was implemented to enforce periodicity and to minimize the effects of eddy currents on the tracking process and thus the calculated displacements. The material points in the reference frame were first forward integrated to calculate the displacements of each point and to project each point into the subsequent time frame. Given the velocity, v, and spatial coordinates of a point, f k, at time t k, the position of the point in the subsequent time frame, f k 1, is f k 1 f k v f k,t k t, 1 number of frames, and f 1 are the coordinates of the material points in the reference frame (k 1). At each time point subsequent to the reference frame, a uniformly sized sector (30 ) was defined around each material point, wall pixels within each sector using the previously determined pixels were extracted, and the average sector velocity was used to project the displacement of the point into the next time frame. The forward trajectory, sector locations, and wall center lines are shown for one sector in Fig. 2 a. This integration process was repeated, starting again with the material points in the reference time frame, using velocities to project backwards in time. Given the velocity, v, and spatial coordinates of a point, b k, at time t k, the position of the point in the previous time frame, b k 1,is where k 1,2,...,N 1 is the frame number, t is the duration of the time interval between frames, N is the b k 1 b k v b k,t k t, 2

4 1036 DRANEY et al. where k N 1,N,N 1,...,3 and b N 1 are the coordinates of the material points in the reference frame (k N 1 or k 1) note that b N 1 b 1 f 1 ). The backward trajectory, sector locations, and wall center lines are shown for one sector in Fig. 2 b. The displacements from the forward and backward integrations were combined to produce the timedependent displacement field of the material points, x k : x k w k f k 1 w k b k, 3 where the weighting coefficient, w k, is calculated to minimize the variance in the combined trajectory w k N k 1, 4 N and k 1,2,...,N. The forward, backward, and combined trajectory for one sector is shown in Fig. 2 c, and the combined trajectories for all sectors are shown in Fig. 2 d. Strain Calculation For each time frame, the material points described previously were used as endpoints of adjoining segments around the vessel wall Fig. 3. Cyclic Lagrange strain was then calculated in each segment over time, relative to the segment configuration in time frame 1, from the principal values of the Green Lagrange strain tensor, E, where E 1 2 FT F I. Using the assumptions of incompressibility and a linear variation in displacements around the vessel wall, the principal strain oriented along the axis of the segment, E 1, is piecewise constant and can be calculated in terms of the stretch,, of the membrane element: E , 5 6 FIGURE 3. Linear segments used to calculate cyclic strain, where the segment endpoints are defined by the position on the trajectories at a given time. The solid line is at the reference time frame and the dashed line is at the time of peak expansion. where the stretch, l/(l), is calculated from the original membrane element length the length in time frame 1, L and the final membrane element length, l. It can be shown that in the case of uniform strain, using the definition of a regular polygon, for any length L i.e., number of polygon sides with length L), the calculated Green Lagrange strain is identical to that calculated based on axisymmetric theory and radial displacements. Thus, the number of sectors needed to adequately capture the strain distribution is dependent on the degree of inhomogeneity of the strain field, not on the accuracy of the method. The expected standard deviation of the calculated strain can be analytically derived from the standard deviation of the measured displacements using a propagation of error analysis 16 see the Appendix for details of this derivation, and is a function of the original length, the amount of stretch, and the standard deviation of the displacements: E1 & x L. 7 Validation Two cylindrical phantoms 1 in. inner diameter, were molded out of gadolinium-doped polyvinyl alcohol cryogel 0.5 ml Gd/100 g PVA ; one had a uniform 1 8 in. wall thickness to produce near-uniform expansion and the other had a circumferentially varying wall thickness, ranging from 1 16 to 3 16 in. to produce nonuniform expansion all dimensions are mold dimensions, exact dimensions during the experiment are pressure dependent. Four tantalum wire markers in. diameter were inserted into the wall of each phantom, parallel to the phantom axis Fig. 4. Each phantom was encased in a water-filled box to provide external support and prevent dehydration of the phantom, and was then connected to a computer-controlled pump CardioFlow 1000 MR, Shel-

5 Quantification of Vessel Wall Strain Using Cine PC-MRI 1037 field of the four markers was also used to calculate strain in the segments between the markers; these strain results were compared to corresponding calculations from the velocity-based trajectory to obtain an estimate of the error in strain produced by error in displacement calculations. RESULTS FIGURE 4. Experimental configuration for in vitro validation studies: diagram of PVA phantom with four tantalum wires inserted through the wall parallel to the axis of the tube two of the four markers are depicted ; also shown are the location of the pressure catheter, the imaging plane, and the flow straightener. ley Medical Systems, London, Ontario with rigid tubing. The pump was programmed to produce a physiologicshaped waveform, based on flow in the descending thoracic aorta, with a peak flow of 80 ml/s. The imaging plane was perpendicular to the axis of the phantom and the markers imaging parameters are detailed in the Results section. Pressure waveforms at the outlet of the tube were acquired using a MR-compatible pressure catheter Millar Instruments, Inc., Houston, TX. The location of each marker over time was determined using the center of intensity of a region surrounding the marker Fig. 5, and each marker trajectory was compared to the corresponding trajectory determined by the velocity data, starting at the same location as the marker in the first time frame. The cyclic displacement The comparison of the marker and velocity-based trajectories is shown in Fig. 6 top left for the uniform wall thickness phantom and in Fig. 6 top right for the nonuniform wall thickness phantom. For the uniform wall thickness phantom, the mean distance between marker and velocity-based trajectory points was mm, and the maximum distance between the trajectory points at any given time was 0.86 mm. As a reference, the mean total marker trajectory distance was 3.3 mm. For the nonuniform wall thickness phantom, the mean distance between marker and velocity-based trajectory points was mm, and the maximum distance between the trajectory points at any given time was 0.65 mm. The mean total marker trajectory distance was 2.12 mm. Pixel size for both phantoms was 0.39 mm. Further details of the displacement comparisons, including differences between the x and y components of the marker and velocity-based trajectory points and the distances, are shown in Table 1. Comparisons of strains in segments between adjacent markers using displacement fields from the markers and the velocity-based trajectories are shown below the corresponding displacement comparisons in Fig. 6. The absolute value of the difference between the marker and velocity-based strain calculations average of all four FIGURE 5. Method for determining the marker centers in the validation studies: a magnitude image from PC-MRI scan; b user-defined region surrounding marker; c, top the inverted image of the marker region and the pixels within 80% of the maximum inverted marker intensity; c, bottom the intensity matrix used to calculate the center of intensity; and d the vessel wall with marker pixels and the calculated marker center white.

6 1038 DRANEY et al. FIGURE 6. Top: trajectory comparison for left-hand side the uniform wall and right-hand side the nonuniform wall phantom. Each marker and velocity-based trajectory is shown actual size on the phantom and enlarged ten-fold inline with the actual trajectory to illustrate differences. The black dotted paths are the marker trajectories and the black vector paths are the velocity-based trajectories. The size of the trajectory arrowheads relates to the magnitude of displacement within a time interval. The black dotted arrows, A D, are the segments used to compare strain calculations using points from the marker and velocity-based trajectories shown below. Bottom left: comparisons of segment principal strains in the uniform wall thickness phantom, as labeled above, calculated from points on the marker and velocity-based trajectories. Of the four segments, strain calculated based on the velocity-based trajectories in segment D had the least difference average 0.006, extent 0.27 when compared to strain calculated based on the marker trajectories comparison, and segment A had the greatest difference average 0.016, extent Bottom right: comparisons of segment principal strains in the nonuniform wall thickness phantom, as labeled above, calculated from points on the marker and velocity-based trajectories. Of the four segments, strain calculated based on the velocity-based trajectories in segment B had the least difference average 0.006, extent 0.14 when compared to strain calculated based on the marker trajectories comparison, and segment A had the greatest difference average 0.017, extent 0.16.

7 Quantification of Vessel Wall Strain Using Cine PC-MRI 1039 TABLE 1. Displacement in the x and y component directions and distance differences between the marker and velocity-based trajectory points. Means for each marker are over time. Uniform wall thickness phantom Nonuniform wall thickness phantom Marker X and Y (mm) Distance (mm) Mean Std Mean Std ALL segments over time for the uniform wall thickness phantom was The maximum instantaneous difference was segment C, t/t 0.38). The average strain extent for the four segments was For the nonuniform wall thickness phantom, the absolute value of the difference between the marker and velocity-based strain calculations average of all four segments over time was The maximum instantaneous difference was segment A, t/t 0.42). The average strain extent for the four segments was The relationship between the standard deviation of measured displacements and the standard deviation of calculated strain, calculated using the propagation of error method, is illustrated in Fig. 7 for different levels of strain related to stretch through Eq. 6. Of note, even for zero strain, any variation in displacement will result in a distribution of calculated strain. The shaded area labeled 4 sectors corresponds to a region of x /L equivalent to that seen in the two phantoms where x 0.18 from Table 1; the standard deviation of all displacement errors in x and y for all markers in both phantoms and L ranged from 16.9 to 22.7 mm, and directly represents the deviation which would be expected in the calculation of strain. The shaded area labeled 12 sectors was calculated in the same way, but using a range of L based on a twelve sector division calculated from the range of four sector lengths using the definition of a regular polygon; L ranged from 6.2 to 8.3 mm. For the four sector strain case, over the range of strain calculated peak-to-peak strain ), the standard deviation in strain as a result of deviation in the displacements was For the twelve sector case, the standard deviation in strain was Strain calculation results from the uniform wall thickness phantoms are shown in Fig. 8 and results from the nonuniform wall thickness are shown in Fig. 9. Figures 8 a and 9 a are the magnitude image from one time frame with sector velocity vectors superimposed, Figs. 8 b and 9 b illustrate the membrane segments at two time points, and Figs. 8 c and 9 c illustrate both the pressure and maximum principal strain circumferential waveforms. Contour plots illustrating the temporal and spatial distribution of principal strain are shown in Fig. 10 a for the uniform wall thickness phantom and Fig. 10 b for the nonuniform wall thickness phantom. These plots illustrate relatively uniform values of strain around the circumference at any given time in the uniform wall thickness phantom, and two spatial regions of increased strain, corresponding to the two thin wall sections, in the nonuniform wall thickness phantom. The descending thoracic aorta and the abdominal aorta of four normal volunteers each were imaged using a phased array cardiac coil. 23 Separate sequences, each 9 min long, were used to quantify both blood flow and wall velocity at the same location. For the wall velocity scan, three components of velocity were acquired and superior inferior spatial saturation was used to minimize flow artifacts and enhance visualization of the vessel wall by darkening the blood signal. For quantifying blood flow, no spatial saturation was used and only the through-plane velocity component was encoded. Specific sequence parameters for both scans and strain calculation results, using the strain method presented herein, for the descending thoracic aorta of one normal male volunteer subject heart rate 58 bpm, age 24, are detailed in Fig. 11. Figure 11 a is the magnitude image from time t/t with the sector velocity vectors superimposed, Fig. 11 b illustrates the membrane segments at two time points, and Fig. 11 c illustrates both the flow and maximum principal strain circumferential waveforms. The FIGURE 7. Relationship between the standard deviation of measured displacements and the standard deviation of calculated strain, calculated using the propagation of error method. The shaded area labeled 4 sectors corresponds to a region of x ÕL equivalent to that seen in the two phantoms; the shaded area labeled 12 sectors was calculated using a range of L equivalent to a twelve sector discretization of the wall.

8 1040 DRANEY et al. FIGURE 8. Uniform wall thickness phantom results: a sector velocities at time tõtä0.29 the time frame with maximum average velocity ; b vessel segments at time frames of minimum tõtä0.17 and maximum tõtä0.46 average radii; and c mean principal strain and pressure over time. Variation bars on the strain curve represent the standard deviation of the segmental strain around the circumference. Imaging parameters: TRÄ19 ms, TEÄ8.5 ms, Ä20, 2 NEX, 256Ã256 matrix, FOVÄ10 cm 2, slice thicknessä5 mm, inplane v enc Ä3cmÕs, through-plane v enc Ä80 cmõs. FIGURE 9. Nonuniform wall thickness phantom results: a sector velocities at time tõtä0.29 the time frame with maximum average velocity ; b vessel segments at time frames of minimum tõtä0.17 and maximum tõtä0.46 average radii; and c mean principal strain and pressure over time. Variation bars on the strain curve represent the standard deviation of the segmental strain around the circumference. Imaging parameters: TRÄ21 ms, TEÄ13 ms, Ä30, 2 NEX, 256Ã256 matrix, FOVÄ10 cm 2, slice thicknessä5 mm, inplane v enc Ä3cmÕs, through-plane v enc Ä80 cmõs. average flow was 4.52 L/min, and the peak strain, averaged around the circumference, was As in the in vitro examples, error bars on the strain waveform depict the segment-to-segment variation in strain around the circumference at each time frame. The spline curves for each time frame are shown in Fig. 12 a and the trajectories of each of the twelve material points are shown in Fig. 12 b. DISCUSSION Validation studies were conducted to evaluate the accuracy of the displacement fields calculated from the cine PC-MRI velocity data and the resultant error in cyclic strain. Trajectories calculated from the PC-MRI data were compared to displacement trajectories of markers embedded in the phantom walls to evaluate the accuracy of the velocity-derived displacement fields. The shape and extent of all the trajectories in both the uniform and nonuniform wall thickness phantoms were very similar, with a difference in distance between trajectory points of mm for all markers at each time point in the two phantoms. These differences are relative to a pixel size of 0.39 mm. Trajectories with the greatest differences, primarily on the sides of both the phantoms, were in regions containing some visible flow-related image artifacts e.g., the right- and left-hand sides of the phantom shown in Fig. 1 caused by periodic pulsation of the tube. Differences between the trajectories, however, are not entirely due to error in velocity measurement. While significant effort was made to optimize marker imaging, changes in orientation of the markers with respect to the magnetic field can distort the shape of the artifact. This may result in changes, over the cardiac cycle, in the relationship between the center of marker intensity and the actual marker center. Additionally, if the signal dropout from the markers was not surrounded by bright signal from the tube, such as in areas where the artifact was large due to oblique cuts through the marker, segmentation of the marker boundary was challenging. Although the marker validation technique has some limitations, this method still has many advantages. The wire markers produce a signal dropout large enough to image ( 1 pixel) yet themselves are very thin and should minimally affect the expansion of the tube. Although in this experiment the deformation patterns are not being con-

9 Quantification of Vessel Wall Strain Using Cine PC-MRI 1041 FIGURE 10. Principal segment strains plotted as a function of circumference and time inner radius is tõtä0, outer radius is tõtät for a the uniform wall thickness phantom and b the nonuniform wall thickness phantom. These plots illustrate relatively uniform values of strain around the circumference at any given time in the uniform wall thickness phantom, and two spatial regions of increased strain corresponding to the two thin wall sections in the nonuniform wall thickness phantom. The region of largest strain in the nonuniform wall thickness phantom may be an artifact due to marker-induced signal loss in this location. sidered, this will be an important consideration for in vivo animal validation studies where the actual deformation pattern is of interest. Additionally, this method provides a direct comparison to the velocity-based trajectories. Although the displacement fields were very similar, errors in the displacement fields do not directly indicate the magnitude of error in the resulting strain. To evaluate this, as a first approximation, membrane segments were defined between points on the marker and velocity-based trajectories, and the resulting strains of these segments were compared. The absolute value of the difference for all segments at each time point in the two phantoms was , where the mean strain extent of all the segments was The propagation of error analysis illustrates that a finer discretization of the wall results in increased variation in the calculated strain. However, the variation estimated from the propagation or error analysis is still relatively small and demonstrates that PC-MRI velocity can be integrated to produce displacement fields, which can then be used to evaluate circumferential variations in cyclic strain. The uniform wall thickness phantom did not deform uniformly, which resulted in a range of maximum cyclic strain variations on the order of 0.07 around the circumference. Possible sources of this nonuniform strain include effects from clamping the phantoms into the flow chamber, material inhomogeneities, and thickness variations. The nonuniform wall thickness phantom deformed less than the uniform wall thickness phantom but had greater variability in cyclic strain, with a range of 0.1. For both phantoms, the propagation of error results sug- FIGURE 11. In vivo results: descending thoracic aorta of normal male subject. a Sector velocities at time tõt Ä0.125 the time frame with maximum average velocity ; b vessel segments at time frames of minimum tõtä0.0 and maximum tõtä0.25 average radii; and c mean principal strain and pressure over time. Variation bars on the strain curve represent the standard deviation of the segmental strain around the circumference. Imaging parameters: TR Ä29 ms flow TRÄ18 ms, TEÄ8.3 ms flow TEÄ6.1 ms, Ä20, 2 NEX, 256Ã256 matrix, FOVÄ20 cm 2, slice thicknessä5 mm, SÕI saturation strain only, in-plane v enc Ä5 cmõs strain only, through-plane v enc Ä200 cmõs.

10 1042 DRANEY et al. FIGURE 12. Wall motion of the descending thoracic aorta: a The spline curves approximating the wall center at each time point; the shade of the curves corresponds to time dark gray indicates time frame 1 and the lightest gray indicates the last time frame. b Trajectories of the twelve material points in the descending thoracic aorta over time. The image of the vessel is from time tõtä0.42. Both the spline curves and the trajectories illustrate the nonuniform expansion of the vessel. gest that of the strain variation can be attributed to uncertainty in displacement calculations. In the in vivo study, the thoracic aorta in all subjects was observed to expand and contract nonuniformly, with the medial aspect nearest the spine moving very little. The material point trajectories of one subject, shown in Fig. 12 b, demonstrate nonuniform displacements along the circumference of the vessel. Based on these data, it can be inferred that the variation in cyclic strain noted in Fig. 11 c may be due in large part to actual circumferential variations in cyclic strain and not just variations due to experimental error variation in strain could be expected from the propagation of error analysis, assuming the same displacement standard deviation as determined from the phantom studies. The previously published study demonstrated that velocity of the aortic wall can be imaged, and the current analysis indicates that it is possible to use this method in vivo. However, improvements to imaging sequences and coils will likely be needed for accurate and precise quantification of strain around the circumference. Further in vivo investigations are needed to characterize these circumferential variations. Potential sources of error in calculating cyclic Lagrangian strain arise from errors in the calculated displacement fields. In this method, such errors can be introduced during the acquisition of the velocity data, the segmentation of the vessel wall, and the tracking of the material points. Each of these potential sources of error are discussed in more detail below. Errors introduced during the acquisition of the velocity data arise from several sources. First, the precision of phase contrast velocity data is dependent on the signalnoise ratio SNR ; images with high SNR such as the in vitro data will have more reproducible velocities than images with lower SNR such as the in vivo data. Second, the temporal resolution determines whether rapid velocity changes can be captured. Although the apparent temporal resolution for the data presented here is 42 ms (T 1 s, 24 reconstructed time frames, the actual temporal resolution for all of the data presented is 4*TR one reference and three flow encodes ; 80 ms for the in vitro studies and 120 ms for the in vivo study. Any velocity changing more rapidly than the actual temporal resolution, such as during peak systole, will not be accurately measured. This type of error will lead to underestimation of strain during peak systole. 28 Errors introduced during segmentation of the vessel wall are largest in time frames where the boundaries of the vessel wall are blurred. This can be caused by low SNR, blurring from low temporal resolution, and high luminal signal intensity. Luminal intensity can be reduced through the use of spatial saturation pulses, however, these pulses also degrade the temporal resolution. Limited spatial resolution, resulting in pixels which contain a mixture of wall and surrounding tissue or blood partial volume effect, can also cause segmentation errors as well as over- or underestimation of the wall velocity, although blood immediately adjacent to the wall should have the same velocity i.e., no-slip. For the in vitro data, the pixel size was very small compared to the thickness of the wall and thus segmentation errors due to

11 Quantification of Vessel Wall Strain Using Cine PC-MRI 1043 partial volume effects are expected to be minimal. However, at currently achievable spatial resolutions, normal vessel walls are typically only one to two pixels thick, and thus partial volume effects can result in segmentation and velocity measurement errors, and subsequent errors in strain calculations. Although manual identification of the wall was performed in the first segmentation step, as image quality improves including SNR, temporal resolution, and spatial resolution, automatic segmentation may be possible. Assumptions made during velocity integration and material point tracking can also lead to error in the calculated displacement fields. In this implementation, it is assumed that velocities are constant during any given time frame; this may lead to greater error in time frames where the velocity is changing rapidly in comparison to the temporal resolution. It is also assumed that the material points remain in the plane of the image. Although the magnitude data are used to segment the vessel wall and extract velocities, only the velocity data are used in the tracking algorithm. Thus it is possible that the extent of the trajectories are larger or smaller than would be expected from examination of the magnitude data alone, such as by looking at the wall center spline curves shown in Fig. 12 a. The precision of the material point tracking will suffer as the number of pixels in each sector decreases, i.e., as the number of sectors around the circumference increases. The degree to which the precision is affected will depend heavily on the SNR and the quality of the velocity data. Future experiments will examine this question. In the case of uniform expansion, it can be shown using the definition of a regular polygon that any segment length will exactly represent the circumferential strain. However, in the nonuniform expansion case, accuracy will improve as the sector size decreases. Determination of the maximum acceptable sector size to capture strain nonuniformities will depend on the vessel being analyzed, and will require the equivalent of a mesh refinement study. The method presented to calculate cyclic Lagrangian strain was implemented assuming a linear variation in displacements around the circumference. This results in piecewise constant strain around the circumference of the vessel, clearly seen in the spatial/temporal strain plots in Fig. 10. Although this may be adequate for describing circumferential variations in strain, a higher order approximation of vessel displacements may be necessary if smoother or more detailed strain fields are required. In this case, the simplified principal strain calculation given in Eq. 6 will not hold. The method of calculating strain from velocity-derived displacements can be extended to quantify two- and three-dimensional strain fields i.e., through the wall thickness and along the vessel axis using two or more planes of data. As vessel geometry becomes tortuous, the strain components involving the longitudinal direction will be increasingly important. Alternatively, 4D MR techniques time-resolved volume data may be used in the future to provide these data. Although the method presented used velocity data, cyclic Lagrangian strain is ultimately calculated from displacements. Therefore, if tagging methods gain resolution sufficient to capture displacements of vessel walls, this method is readily extensible. MR sequence optimization is currently in progress to increase spatial resolution, temporal resolution, and SNR in the in vivo scans. Further improvements in SNR may be achieved through the use of specialized MR coils, improved gradients, and higher field strength magnets. In vivo validation of the method described in this article is in progress. 4 CONCLUSIONS We have described a noninvasive, in vivo method to quantify circumferentially varying vessel wall cyclic strain. Phantom experiments demonstrated that wall displacements and cyclic strain could be calculated accurately from PC-MRI velocity data, with a mean displacement difference of 0.20 mm relative to a pixel size of 0.39 mm and a mean strain difference 0.01 relative to a strain extent of Quantification of cyclic strain in the thoracic aorta of a normal human subject demonstrated nonuniform circumferential expansion and cyclic strain and verified the feasibility of using this method in vivo. This method could enable the characterization of wall motion and strain fields in both normal and diseased subjects and may provide data to test hypothesis on biomechanical factors in aneurysmal and atherosclerotic disease processes. The magnitude and variation of wall motion and strain may ultimately provide clinical information to enable the prediction of aneurysm growth and rupture vulnerability. ACKNOWLEDGMENT This research was supported in part by the Lucas Foundation, NIH P41RR09784, and NIH R01HL APPENDIX: PROPAGATION OF ERROR The standard deviation of a calculated parameter can be calculated from the standard deviation of measured variables through a propagation of error analysis. The general expression for calculating propagation of error is 2 z z 2 x 1 2 x1 z 2 x 2 2 x2 z 2 x 3 2 x3...,

12 1044 DRANEY et al. where z(x n ) is the calculated parameter and x n are the measured variables used in the parameter calculation. Error in these measured variables is assumed to be independent. This formula can be applied to calculate the standard deviation of the principal values of strain ( E ) given the measured standard deviations in the position of segment endpoints ( xij ). Only the positions in each of the current time configurations are included since, by definition, error is zero in the reference time configuration: 2 Ei E i 2 x11 E i 2 x12 E i 2 x21 2 x 11 E 2 i x 22 2 x22. 2 x 12 2 x 21 The four displacement errors the error in each direction for each endpoint of a given segment at the current time: x11, x12, x21, and x22 ) are equal to the displacement differences between the markers and the velocity-based displacements, xij x. Although the displacement errors are set to be equal, it is assumed that they are independent, i.e., the displacement error in one direction does not predispose a change in error in the other direction. The standard deviation of the principal strain aligned along the segment axis, E1, is related to the standard deviation of the displacements, x, the original length, L and the amount of stretch, : E1 & xl L 2 & x L, thus, the principal strain can be expressed as where and l/l. E & x L, l x 21 x 11 2 x 22 x /2, L X 21 X 11 2 X 22 X /2, REFERENCES 1 Bayliss, W. M. On the local reactions of the arterial wall to changes of internal pressure. Proc. Physiol. Soc. 26: , Draney, M. T., K. L. Wedding, C. A. Taylor, and N. J. Pelc. An in vivo method for measuring vessel wall motion and cyclic strain using magnetic resonance imaging. Proceedings of the 1st Joint BMES/EMBS Conference, Atlanta, GA, Draney, M. T. Biomechanics of the human aorta and aortic aneurysms. Proceedings of the Frontiers in Vascular Disease Conference, Pebble Beach, CA, Draney, M. T., F. R. Arko, M. T. Alley, M. Markl, R. J. Herfkens, N. J. Pelc, and C. K. Zarins. In vivo quantification of porcine aortic wall motion using cine PC-MRI. Proceedings of the 10th Annual International Society for Magnetic Resonance in Medicine Conference, HI, Dzau, V. J., and G. H. Gibbons. Vascular remodeling: Mechanisms and implications. J. Cardiovasc. Pharmacol. 21 Suppl. 1, S1 S5, Friedman, M. H., G. M. Hutchins, and C. B. Bargeron. Correlation between intimal thickness and fluid shear in human arteries. Atherosclerosis 39:425, Fronek, K., G. Schmid-Schoenbein, and Y. C. Fung. A noncontact method for three-dimensional analysis of vascular elasticity in vivo and in vitro. J. Appl. Physiol. 40: , Fung, Y. C., and S. Q. Liu. Change of residual strains in arteries due to hypertrophy caused by aortic constriction. Circ. Res. 65: , Glagov, S., C. K. Zarins, D. P. Giddens, and D. N. Ku. Hemodynamics and atherosclerosis. Insights and perspectives gained from studies of human arteries. Arch. Pathol. Lab. Med. 112: , Hansen, B., A. H. Menkis, and I. Vesely. Longitudinal and radial distensibility of the porcine aortic root see comments. Ann. Thorac. Surg. 60:S384 S390, Hardt, S. E., A. Just, R. Bekeredjian, W. Kubler, H. R. Kirchheim, and H. F. Kuecherer. Aortic pressure-diameter relationship assessed by intravascular ultrasound: Experimental validation in dogs. Am. J. Physiol. 276:H1078 H1085, Hokanson, D. E., D. J. Mozersky, D. S. Summer, and D. E. Strandness, Jr. A phase-locked echo tracking system for recording arterial diameter changes in vivo. J. Appl. Physiol. 32: , Imura, T., K. Yamamoto, K. Kanamori, T. Mikami, and H. Yasuda. Noninvasive ultrasonic measurement of the elastic properties of the human abdominal aorta. Cardiovasc. Res. 20: , Kamiya, A., and T. Togawa. Adaptive regulation of wall shear stress to flow change in the canine carotid artery. Am. J. Physiol. 239:H14 H21, Lyon, R. T., A. Runyon-Hass, H. R. Davis, S. Glagov, and C. K. Zarins. Protection from atherosclerotic lesion formation by reduction of artery wall motion. J. Vasc. Surg. 5:59 67, Meyer, S. L. Propagation of error and least squares. In: Data Analysis for Scientists and Engineers. New York: Wiley, 1975 pp Moore, C. C., E. R. McVeigh, and E. A. Zerhouni. Quantitative tagged magnetic resonance imaging of the normal human left ventricle. Top. Magn. Reson. Imaging. 11: , Moreno, M. R., J. E. Moore, Jr., and R. Meuli. Crosssectional deformation of the aorta as measured with magnetic resonance imaging. J. Biomech. Eng. 120:18 21, Pelc, N. J., R. J. Herfkens, A. Shimakawa, and D. R. Enzmann. Phase contrast cine magnetic resonance imaging. Magn. Reson. Q. 7: , Pelc, N. J., M. Drangova, L. R. Pelc, Y. Zhu, D. C. Noll, B.

13 Quantification of Vessel Wall Strain Using Cine PC-MRI 1045 S. Bowman, and R. J. Herfkens. Tracking of cyclical motion using phase contrast cine MRI velocity data. J. Magn. Reson. Imaging. 5: , Thubrikar, M. J., and F. Robicsek. Pressure-induced arterial wall stress and atherosclerosis. Ann. Thorac. Surg. 59: , Tropea, B. I., S. P. Schwarzacher, A. Chang, C. Asvar, P. Huie, R. K. Sibley, and C. K. Zarins. Reduction of aortic wall motion inhibits hypertension-mediated experimental atherosclerosis. Arterioscler. Thromb. Vasc. Biol. 20: , Wedding, K. L., M. T. Draney, R. J. Herfkens, C. K. Zarins, C. A. Taylor, and N. J. Pelc. Measurement of vessel wall strain using cine phase contrast MRI. J. Magn. Reson. Imaging. 15: , Wedeen, V. J. Magnetic resonance imaging of myocardial kinematics. Technique to detect, localize, and quantify the strain rates of the active human myocardium. Magn. Reson. Med. 27:52 67, Xu, C., S. Glagov, M. A. Zatina, and C. K. Zarins. Hypertension sustains plaque progression despite reduction of hypercholesterolemia. Hypertension 18: , Zarins, C. K., D. P. Giddens, B. K. Bharadvaj, V. S. Sottiurai, R. F. Mabon, and S. Glagov. Carotid bifurcation atherosclerosis: Quantitative correlation of plaque localization with flow velocity profiles and wall shear stress. Circ. Res. 53: , Zerhouni, E. A., D. M. Parish, W. J. Rogers, A. Yang, and E. P. Shapiro. Human heart: Tagging with MR imaging: A method for noninvasive assessment of myocardial motion. Radiology 169:59 63, Zhu, Y., M. Drangova, and N. J. Pelc. Fourier tracking of myocardial motion using cine-pc data. Magn. Reson. Med. 35: , 1996.