To validate a method for measuring 3D left ventricular (LV) strain from phase-unwrapped harmonic phase (HARP) images derived from tagged cardiac magnetic resonance imaging (MRI).
To validate a method for measuring 3D left ventricular (LV) strain from phase-unwrapped harmonic phase (HARP) images derived from tagged cardiac magnetic resonance imaging (MRI).
A set of 40 human subjects were imaged with tagged MRI. In each study the HARP phase was computed and unwrapped in each short-axis and long-axis image. Inconsistencies in unwrapped phase were resolved using branch cuts manually placed with a graphical user interface. 3D strain maps were computed for all imaged timeframes in each study. The strain from unwrapped phase (SUP) and displacements were compared to those estimated by a feature-based (FB) technique and a HARP technique.
3D strain was computed in each timeframe through systole and mid-diastole in ≈30 minutes per study. The standard deviation of the difference between strains measured by the FB and the SUP methods was less than 5% of the average of the strains from the two methods. The correlation between peak circumferential strain measured using the SUP and HARP techniques was over 83%.
The SUP technique can reconstruct full 3D strain maps from tagged MR images through the cardiac cycle in a reasonable amount of time and user interaction compared to other 3D analysis methods. J. Magn. Reson. Imaging 2010;31:854–862. ©2010 Wiley-Liss, Inc.
PARAMETERS OF CARDIAC left-ventricular (LV) mechanical function are important for diagnosing and managing patients with heart disease and assessing the efficacy of therapies over time. Imaging methods based on magnetic resonance (MR) tagging (1, 2) and displacement encoding with stimulated-echoes (DENSE) (3, 4) have been used to study the regional functional characteristics of the heart. While recent advances in imaging methods (5–8) and postprocessing techniques (9, 10) have improved the clinical viability of these techniques, the ability to provide dense 3D strain maps and torsion measurements throughout the cardiac cycle in a reasonable amount of time is still an active area of research.
Tagged MRI is an established method for measuring parameters of LV deformation and strain. Tagged MRI spatially modulates the longitudinal magnetization of the underlying tissue before image acquisition. The result is a periodic tag pattern that deforms with the tissue, as shown in Fig. 1a. Several techniques have been developed to measure LV deformation and strain from the deformation of the tag pattern. These techniques include feature-based (FB) methods (11–15), optical-flow and nonrigid registration based methods (16–19), and harmonic phase (HARP)-based methods (7, 10, 20, 21).
In HARP analysis the tag pattern deformation is measured by the local change in phase of the tag pattern. The tag pattern modulates the signal intensity of the image producing a series of peaks in the Fourier domain, as shown in Fig. 1b. One of these peaks is filtered out and inverse Fourier-transformed to produce a HARP phase image, as shown in Fig. 1c,d. The HARP phase at a point in the image is a material property of the underlying tissue and can be tracked through the image sequence or used to compute 2D strain (10, 22) or 3D strain in the middle third of the LV wall (7). The HARP phase, however, is wrapped because it can only be measured modulo 2π. This wrapping can cause tracking problems if a region of the tissue deforms more than one-half tag spacing, ie, a phase shift of more than π between timeframes. Deformations of this magnitude are possible in both healthy and diseased hearts.
In DENSE imaging (4, 23), myocardial tissue displacement is encoded into the phase of the MR image during acquisition. To reconstruct myocardial displacement from the phase data the phase must be unwrapped (3, 23, 24), which involves adding integer multiples of 2π to the wrapped phase so that the unwrapped phase is continuous. Noise and imaging artifacts, however, can cause phase inconsistencies where no integer multiple of 2π can yield a continuous unwrapped phase map. While DENSE phase unwrapping continues to improve (3, 24), it is still possible for phase errors to occur, which can result in errors in deformation and strain estimates.
In this article a new method is presented for measuring 3D deformation and strain from tagged MRI data based on unwrapping HARP images called strain from unwrapped phase (SUP). In contrast to a recently proposed 3D wrapped-phase based method (7), the SUP algorithm can measure strains in the entire LV. The SUP algorithm can also easily measure 3D strains in the entire LV wall over multiple timeframes. The SUP approach is more robust to interframe motion than other techniques (7) that are based on tracking wrapped phase. With unwrapped phase, points can be tracked through an image sequence as long as the average interframe deformation over the entire myocardium is less than one-half tag spacing. Compared to methods based on tracking tag lines (11, 25), displacement measurements from unwrapped HARP phase images are dense, potentially resulting in more accurate estimates of strain.
The SUP algorithm was validated on a cohort of 40 human subjects (10 normal volunteers, 10 patients with myocardial infarction, 10 patients with mitral regurgitation, and 10 patients with hypertension). All human studies were approved by the Institutional Review Boards of both institutions and informed consent was obtained from all participants.
All participants underwent MRI on a 1.5 T MRI scanner (GE, Milwaukee, WI) optimized for cardiac application. Tagged images were acquired in standard views (two and four chamber long axis and short axis) with a fast gradient-echo cine sequence with the following parameters: field of view (FOV) = 300 mm, image matrix = 224 × 256, flip angle = 45, echo time (TE) = 1.82 msec, repetition time (TR) = 5.2 msec, number of cardiac phases = 20, slice thickness = 8 mm. A 2D fast gradient recalled spatial modulation of magnetization (FGR-SPAMM) tagging preparation was done with a tag spacing of 7 pixels. This protocol resulted in two long-axis slices and, on average, 12 short-axis slices per study.
HARP images (10) were obtained by filtering out a single spectral peak in a tagged image and computing the phase in each pixel of the resulting complex-valued image. These phase images are wrapped because phase can only be computed modulo 2π. Phase unwrapping involves adding integer multiples of 2π to each pixel in the wrapped image so that: 1) The unwrapped phase is continuous; 2) If the unwrapped phase is rewrapped, the result is equivalent to the original wrapped image. The relationship between the wrapped phase ψ(x,y) and the unwrapped phase ς(x,y) may be stated as:
where W is the wrapping operator. In this study, phase was unwrapped only inside the LV wall. Pixels outside the myocardium were not processed.
Several phase unwrapping algorithms have been proposed for MRI and other applications (26). In particular, DENSE postprocessing involves unwrapping the DENSE phase maps (3, 4, 27). In this study a path-following method was used that is similar to the one used for DENSE (3, 26), where phase was unwrapped along a path in the image from one pixel to another. If the phase is unwrapped correctly the phase of the destination pixel is independent of the path taken. Two main issues arise with this approach: how to choose the path and how to handle regions of inconsistent phase. The path was chosen by the quality-guided method described below. Phase inconsistencies are called residues and were handled by introducing branch cuts. These methods are described in detail in the following sections.
In path-following methods it is necessary to determine a path that visits each pixel of interest. The phase is unwrapped along this path. Quality-guided algorithms (28) use a quality map, which indicates the pixels' reliability of the wrapped phase map, to guide the integration path. A starting point was arbitrarily chosen and the next pixel visited was the neighbor with the highest quality. This process was repeated until all pixels of interest were processed. Similar to DENSE (3) algorithms, quality in this study was defined as the reciprocal of the phase difference variance (29), which is given by:
where each sum is over a 3 × 3 window, centered at the pixel (m,n). The terms and are the wrapped phase derivatives, and and are the average of the derivatives in the 3 × 3 window.
A fundamental assumption in phase unwrapping algorithms is that the unwrapped phase is smooth and that any large changes in phase between pixels are due to phase wrapping. Noise and artifacts, however, can cause inconsistencies in the phase map where the phase difference between pixels is large and cannot be corrected by adding an integer multiple of 2π. Tag fading, myocardial motion, and the presence of boundaries can also cause phase inconsistencies. Examples of phase inconsistencies are shown in Fig. 2. +45° tag lines disappear before they reach the endocardium at 12 o'clock and 5 o'clock in Fig. 2c. Also, the tag-like phase wrap near the epicardium at 4 o'clock in Fig. 2c is caused by the epicardial boundary. Figure 2d shows a “double wrap” artifact where two tag lines merge (7 o'clock).
Phase inconsistencies can cause any phase unwrapping algorithm to fail. Figure 3a shows an example of a phase image with inconsistencies. Figure 3b shows the unwrapped phase image obtained with the technique described above. The phase inconsistencies result in discontinuities in the unwrapped phase. In this section we present a method for detecting and correcting for phase inconsistencies using residues and branch cuts.
Phase inconsistencies were detected using the principle that the wrapped phase difference around any closed path is equal to zero if the wrapped phase is consistent (26). Consequently, these phase inconsistencies were localized by summing the wrapped phase differences around a 2 × 2 neighborhood of each pixel in the myocardium. It can be shown (26) that this sum is either 0, +2π, or −2π. If the sum is zero, then no phase inconsistency exists. Otherwise, either a positive residue (+2π) or a negative residue (−2π) is present at that pixel. Examples of residues can be seen in Fig. 2e,f.
In this study phase inconsistencies were corrected using the residue compensation method (26), where residues were removed by connecting residues with branch cuts. Branch cuts are lines in the image where an unwrapping path is not allowed to cross. In our implementation, pixels that touch the branch cut line were removed from the myocardium segmentation. Note that the number of pixels removed due to branch cuts is negligible compared to the total number of pixels used in the 3D deformation model fit described below.
Branch cuts either connect a positive to a negative residue or, if only a region of the image is unwrapped, a residue is connected to a point outside the myocardium segmentation. The problem, of course, is which residues to connect. It is common to see positive and negative residues next to each other in the image (see Fig. 2e). These residues are known as dipoles and can be automatically connected by a branch cut.
Connecting nondipole residues (called monopoles), however, is a difficult problem. In cases where there are an equal number of positive and negative residues, automated methods have been proposed that, for example, minimize the average branch cut length (26, 30). Minimizing branch-cut length, however, does not always yield the best unwrapped phase. As a result, the user must place the branch cuts interactively. This was done quickly with a graphical user interface (GUI).
Figure 2g,h shows branch cut placement examples. The phase wraps in the second row of Fig. 2 correspond to the +45 direction tag lines. If the phase inconsistency was caused by a prematurely ending tag line (11 o'clock and 5 o'clock in Fig. 2c) or tag line merging (7 o'clock in Fig. 2d), the branch cuts were placed along the missing or merged tag line. If a phase wrap occurred where there was no tag line (4 o'clock in Fig. 2c) the branch cut was placed to remove the spurious phase wrap from the myocardium segmentation.
Once branch cuts were placed, the phase unwrapping technique described above resulted in a smooth unwrapped phase map as shown in Fig. 3d. Note that after branch cuts are placed the unwrapped phase is independent of the path taken during the unwrapping process.
Instead of unwrapping the HARP phase image, one could unwrap the demodulated HARP phase image. The demodulated HARP phase image is computed by shifting the filtered HARP peak to the center of k-space before computing the inverse Fourier transform. Figure 4a,c shows the HARP phase and demodulated HARP phase images of the tagged image in Fig. 2a. The demodulated HARP phase is similar to the phase obtained with DENSE imaging (31).
At first, it might appear that the demodulated phase is easier to unwrap than the modulated HARP phase because there are fewer phase wraps. Phase wraps, however, are usually not a problem for phase unwrapping algorithms. The challenge in phase unwrapping is how to handle phase inconsistencies.
Since phase inconsistencies are mainly caused by tag lines disappearing, tag lines merging, myocardial boundaries, and other inherent properties of the tissue being imaged, they appear in both the HARP phase and demodulated HARP phase images. For example, Fig. 4b,d shows the residues for both the HARP phase and demodulated HARP phase images. The residues in both images are almost identical.
While either the HARP phase or demodulated HARP phase image could be unwrapped, the HARP phase image was used in this study because the phase wraps correspond to tag lines and provide visual cues that make it easier for the user to place branch cuts, as seen in Fig. 3c.
Each image was unwrapped independently and the starting point for phase unwrapping may correspond to a different material point in each image. As a result, the unwrapped phases in two adjacent timeframes may differ by an integer multiple of 2π. This difference was corrected by adding the integer multiple of 2π to all unwrapped phases in the current timeframe. In this study the multiple was chosen to minimize the L1 norm of all displacements between the current and previous timeframes. Note that this correction assumes that the average deformation of the heart between consecutive frames is less than one-half of the tag spacing. Interframe deformation of more than one-half tag spacing in a localized region or regions will not affect the unwrapped phase as long as the average deformation is less than one-half tag spacing.
Once the phases were unwrapped, 1D displacements were measured from each pixel in the myocardium similar to those measured from tag-line tracking data (11). Each 1D displacement measurement is the displacement of a material point back to its reference position in the direction normal to the tag plane. As previously (11, 32, 33), we define the reference time to be the time the tag pattern was applied. Note that this time is several milliseconds before the first image is acquired.
To compute the 1D displacements, an estimate of the reference phase plane must be computed. The reference phase plane was estimated by fitting a plane to the unwrapped phase plane in the first timeframe in a given slice. This computation is analogous to the one performed previously (11, 32, 33) to determine the tag line center and spacing. There can be some motion between the time the tags are applied and the acquisition of the first image, but this is usually small. Since tag lines are parallel when they are applied, the reference phase plane was specified by the 1D linear equation:
where s is a parameter that sweeps out a line perpendicular to the tag line and m and b are parameters of the reference plane fit to the first timeframe unwrapped phase.
To compute the 1D displacement measurements, let sd correspond to a point in a deformed image with phase ϕd(sd). From Eq.  the point in the reference image with the same phase is sr = (ϕd(sd) − b)/m. The displacement measurement is then d = sd − sr. Figure 5 shows an example of displacement estimates measured from a short-axis slice of a normal human volunteer at end-systole. Note that due to through-plane motion a 1D displacement does not establish a one-to-one correspondence between points at the reference and deformed times. Rather, it is a 1D constraint on the 3D displacement of a material point back to its position in the reference time (11, 32, 33).
Segmentation of the myocardium is the first step in most cardiac image analysis procedures (3, 12, 22). Both the phase unwrapping and displacement measurement steps require that the LV wall be segmented in each slice and time of interest. In this algorithm, LV contours were drawn semiautomatically by the user at end-diastole (ED) and end-systole (ES). Nonrigid registration (34) was then used to propagate ED and ES contours to all the timeframes. ED contours are used in almost all tagged MRI analysis algorithms to define a material-point mesh or other set of points of interest. ES contours were included because they improve the ability of the propagated contours to distinguish between papillary muscles and the LV wall.
The 1D displacement measurements and a material-point mesh automatically constructed from the ED contours were used to compute 3D LV deformation and strain in each timeframe. The deformation and strain were reconstructed using the Affine Prolate Spheroidal B-Spline (APSB) method (35), which fits a B-spline deformation model defined in prolate-spheroidal coordinates to the displacement measurements. First, a backward fit was performed in each timeframe that maps each point back to its undeformed position. A series of forward fits were then performed to compute the trajectory of each point in the material-point mesh through the cardiac cycle. 3D strain was computed by spatially differentiating the deformation model.
The SUP technique was validated on the cohort of 40 human subjects described above. In each subject, 3D strain was computed in all imaged timeframes using the SUP technique. 3D strain was computed at ES with the FB technique (35). Both the FB and SUP methods use the APSB deformation model described previously (34). The tags were tracked using the FB technique and then manually corrected by an expert. 2D strain in each timeframe was computed using HARP analysis (10).
To assess the accuracy of 1D displacement measurements obtained from unwrapped phase images, 1D displacement measurements were computed from tag lines tracked in the FB method and compared to 1D measurements computed from the unwrapped phase images at the tracked tag line points.
The accuracy of 3D strains at ES was assessed by comparing SUP strains and FB strains with a paired t-test. The accuracy of strains versus time was assessed by comparing peak strains and strain rates computed using SUP and HARP with paired t-tests. In all statistical comparisons, a P-value of 0.05 or less was considered statistically significant.
All studies were processed using the SUP technique described above, which was implemented in MatLab (MathWorks, Natick, MA). Unwrapping all images in a study took less than a minute on a 2.6 GHz Core2 Duo processor with 4 GB of memory. Approximately 30 minutes per study of user interaction were required to resolve residues. Strain reconstruction took ≈15 minutes per study. The total time required to analyze 20 time frames of a typical study was 45 minutes.
Table 1 shows statistics of the differences between 1D displacement measurements computed from unwrapped phase images and tracked tag lines at end-systole. The SUP and tag line methods showed excellent agreement on long-axis images and short-axis images in the proximal and middle thirds of the LV (relative to the base). The mean differences are a few hundredths of a pixel, which means there is no bias toward under- or overestimation of displacement. The difference standard deviation is around one-third of a pixel, which is close to the tag line tracking accuracy (33, 36). In three slices at or next to the apex (out of a total of 429 processed) the maximum difference was large (≈8 pixels) due to partial volume effects that severely diminished the contrast-to-noise ratio (CNR). These slices were removed from the differences in Table 1, but left in for the comparison of strains in subsequent comparisons.
|Slices||Mean difference||Standard deviation||Maximum difference|
|Short-axis – proximal 3rd||−0.0170||0.3572||2.5694|
|Short-axis - middle 3rd||−0.0217||0.3299||2.4348|
|Short-axis – distal 3rd||0.0787||0.5625||2.9898|
Figure 6 shows the maps of 3D circumferential strain (Ecc) in a representative normal human volunteer computed using the SUP method and the FB method. The same material-point mesh and deformation model were used to reconstruct strain in both methods. Note that strain is computed in the entire LV, except the apex, which is typically not included in this type of analysis. The strain maps are similar.
Table 2 shows statistics of the difference between the SUP and FB methods. In all strains in Table 2 the standard deviation of the difference is less than 5% of the average of the two methods. The correlation between the methods is strongest in circumferential strain. Only two long-axis images were acquired, so the correlation in longitudinal strain (Ell) and torsion is lower, but still good. In fact, the SUP method may provide more accurate measurements of Ell, torsion, and other strains than the FB method, but further investigation is needed to support this claim.
|Ecc||0.0009 ± 0.0017||0.9105||0.9613||<0.001||2.037%|
|Ell||−0.0022 ± 0.0019||0.7752||0.9351||<0.001||2.881%|
|Emin||−0.0028 ± 0.0019||0.6717||0.9201||<0.001||1.591%|
|Torsion||−0.3498 ± 0.2251||0.5757||0.8697||<0.001||3.662%|
Table 3 shows statistics of the difference between the 3D strains calculated from SUP and the 2D strains from the HARP method (10). 2D LV torsion using HARP was measured using the described procedure (20). One slice each at the basal and apical levels was used to compute torsion. The basal slice chosen was the slice closest to and below the mitral valve at the ES phase. The apical slice chosen was the slice closest to the apex where the blood pool could be identified. A mesh consisting of three concentric rings and 24 circumferential points was defined in each slice at ED from semiautomatically drawn contours. Each mesh was tracked through all time frames using the improved harmonic phase (HARP) method for motion tracking (21).
|Peak strain||−0.0223 ± 0.0005||0.004||0.8361||<0.001||3.909%|
|Systolic rate||−0.1004 ± 0.0024||0.011||0.8387||<0.001||3.494%|
|Early-diastolic rate||−0.0494 ± 0.0068||0.502||0.6615||<0.001||8.100%|
|Peak strain||−0.0217 ± 0.0005||<0.001||0.7023||<0.001||4.926%|
|Systolic rate||0.0657 ± 0.0042||0.132||0.6240||<0.001||6.407%|
|Early-diastolic rate||−0.2091 ± 0.0111||0.052||0.5739||<0.001||12.771%|
|Peak||−0.0036 ± 0.1092||0.998||0.6567||<0.001||6.999%|
|Systolic rate||−9.4941 ± 0.5079||0.069||0.649||<0.001||7.053|
|Early-diastolic rate||3.9996 ± 0.8812||0.533||0.2578||0.108||10.408%|
The peak strain and systolic strain rate of the measured circumferential strain show a high correlation. Correlation in longitudinal strain is lower because the SUP method uses 3D deformation model fitting to compute longitudinal strain from the motion of all tag points, whereas HARP computes strain from two long-axis images. Also, the tag line CNR decreases through the cycle due to T1 decay of the tag pattern, so early diastolic correlations are lower than correlations for peak strain and systolic strain rate. It was seen that HARP in general underestimated the strains as compared to the SUP method. This could be attributed to the improved accuracy of tracking using the SUP method as a result of removal of inconsistencies in the HARP image using branch cuts.
A few degrees of difference was seen in comparison of peak torsion. The 3D method corrects for the through-plane motion, but the 2D method tracks points in a given slice through time. In the 2D method the tracked points in the basal slice start out near mid-ventricle and become more basal through systole. Since the base typically rotates more than the mid-ventricle, though-plane motion can distort the 2D rotation and torsion curves.
Figure 7 shows plots of torsion versus time averaged over 10 normal human volunteers, 10 patients with myocardial infarction, 10 patients with hypertension, and 10 patients with mitral regurgitation. The hypertensive patients showed an increase in torsion, which has been reported in the clinical literature (37). The untwisting in early-diastole was slower for patients with mitral regurgitation as compared to normal human volunteers, which is consistent with previously reported results (38).
The SUP method was presented for measuring 3D strain from tagged MRI. Strains computed using SUP demonstrate excellent agreement with a previously published FB method (35). Some user interaction is also required to resolve phase-unwrapping residues (especially later in the cycle), but the SUP method can compute strains in the entire LV instead of just the middle third.
The time required for the SUP measured 3D strains in the entire LV over 20 timeframes is ≈45 minutes per study. The same analysis using the FB method would require ≈3 hours.
While the tagged imaging protocol used in this study is commercially available and is widely used clinically, advanced tagged imaging techniques such as complementary SPAMM can yield a higher tag CNR throughout the cardiac cycle. Higher CNR images will result in fewer phase inconsistencies and less user interaction will be required to correct them.
The use of residues to automatically detect phase inconsistencies in wrapped phase images and the use of branch cuts to correct them can also be used in unwrapping the phase in DENSE imaging.
Spottiswoode et al (3) used a spatiotemporal method of phase unwrapping to maintain interframe phase consistency in DENSE images where a group of pixels are selected in the first timeframe and these pixels are sequentially unwrapped in each successive timeframe. This technique works well in DENSE images because there are only a few phase wraps across the myocardium. The SUP method, however, is based on HARP images, which make it easier to place branch cuts. HARP images have 6–10 phase wraps across the myocardium. Temporal phase unwrapping in HARP images is sensitive to large (greater than one-half tag spacing) interframe displacements. Therefore, we used a method based on the phase of the entire myocardium over time to maintain phase consistency. This allows for local tissue to have displacements greater than one-half tag-spacing between timeframes as long as the average deformation of tissue between timeframes is less than one-half tag-spacing.
The SUP technique reconstructs full 3D strain maps from tagged MR images through the cardiac cycle and takes into account the through-plane motion of the heart. Slice-following DENSE (23), zHARP (5), and HARP-SENC (6) can account for through-plane motion, but these techniques require two breath-holds per slice. In contrast, tagged images can be acquired with multiple slices per breath-hold, which allows the entire LV to be imaged in significantly less time.
In conclusion, the SUP method can compute 3D strains in the LV through the cardiac cycle in a reasonable amount of time and user interaction compared to other 3D analysis methods.