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Harmonic Phase Magnetic Resonance Imaging

Nael F. Osman and Jerry L. Prince


Tagged cardiac magnetic resonance imaging (MRI) produces images of the heart which can be analyzed to yield detailed maps of motion and mechanical strain, describing regional function of the heart. Harmonic phase (HARP) analysis is a fast and minimally-interactive method for processing tagged MR images to yield a rich collection of functional measures including pathlines, circumferential and radial strains, velocity maps, high-density "synthetic" tags, and principle strains. These measures are useful for visualization and quantifying the heart's performance and are being evaluated for clinical applications in diagnosing and monitoring heart disease. Also, the principles behind HARP have laid the foundation for a variety of new, emerging MR imaging techniques.

Magnetic resonance imaging can be used to create temporary features, called tags, within the body's tissues. These features are not harmful because they are created using the same principles by which images are created in MRI. Several different tag styles are shown in Figure 1. The tags, while visible over their (approximately) one second lifetime, move with the motion of the heart. The pattern of tags can in turn be analyzed to understand the motion of the heart during its contraction and dilation. Usually tags are created at the beginning of a heart beat and data to create images are acquired during systole and part of diastole. Typically, several heart beats are required, over a brief breath-hold, to gather enough data to form a high-resolution MR image sequence of the tag motion. Heart motion is highly repeatable, which makes this method of data acquisition possible.

There have been many methods proposed to analyze the motion of tags within these image sequences. To this date, all methods rely on the localization of "tag lines" and subsequent inferences about what takes place between the tag lines. Because of pixel size, tag size, noise, and tag fading, the research community has had limited success in producing robust, fast, and highly accurate and repeatable algorithms. Important research has been conducted over the last decade or so, typically using painstaking manually-assisted analysis of tagged MR images, pointing to the very high potential for MR tagging to be used clinically for the quantification of regional function and diagnosis and monitoring of heart disease. Unfortunately, slow image acquisition times and slow and tedious processing methods, have prevented migration of this useful research tool into a practical clinical protocol.

Harmonic phase magnetic resonance imaging (HARP-MRI) is a promising new method that overcomes the drawbacks of the other existing methods and makes it possible to use MR tagging possible in clinical applications. HARP-MRI is based on a different perception and understanding of the tagging process. Moreover, HARP-MRI is not limited to an image processing tool, but it paves the way to new imaging techniques based on its principles.

Figure 1: Different tag styles.


The Harmonic Image

Consider the tagged cardiac images shown in Figure 2, showing the same slice at the reference time and later near the end of systole. These images have two-dimensional Fourier transforms that have collections of spectral peaks, as shown in Figure 3 (truncated in the vertical direction). Assuming that the spectral peaks are well separated, it is possible to isolate a single peak in each image and take the inverse Fourier transform of just the spectral contents within this single peak, yielding a harmonic image. The circles shown in Figure 3 represent the pass regions of a bandpass filter used to isolate spectral peaks for HARP analysis. A harmonic image is a complex image, which therefore has real and imaginary images or, equivalently, magnitude and phase images. In HARP analysis, it is necessary to consider both the magnitude and phase of harmonic images.

The Harmonic Magnitude Image

The magnitude of a harmonic image is called a harmonic magnitude image, to distinguish it from the usual magnitude MR image. Figure 4 shows the magnitude images of the harmonic images corresponding to the filtered peaks in Figure 3. These harmonic magnitude images look similar to the tagged images in Figure 2 except that the tag lines are absent, and the images are blurry. The blurriness is due to the filtering process, which reduces the resolution of the harmonic magnitude image. Despite this loss of resolution, harmonic magnitude images can be used for segmentation by thresholding. The segmentation masks shown in Figure 5 roughly separate tissue from non-tissue.

The Harmonic Phase Image

The other component of a harmonic image is called the harmonic phase image, or simply the HARP image. The HARP images corresponding to the harmonic images from Figure 2 are shown in Figure 6. The phase is depicted only on the masks shown in Figure 5 for visualiation purposes. The phase can only be computed between the -180 to 180 degrees, which means that the measured phase is a "wrapped" version of the true harmonic phase. This accounts for the "sawtooth" appearance of the images in Figure 6. In particular, the "crisp" lines that are apparent in these images are due to the changing phase value from 180 to -180 degrees caused by wrapping. Careful observation of these HARP images shows bending of the phase wrapping artifact in the later HARP image. These resemble the tag lines in the original image in Figure 2. It can be shown that the harmonic phase actually depends on the underlying motion as well as the tagging parameters. In fact, the underlying motion is not only represented at the points of phase wrapping, but all other points as well. Mathematically, it can be shown that the phase of the harmonic images is linearly related to the tissue motion in a direction orthogonal to the tag lines.

time = 0
time = 320 ms

Figure 2. Vertical tags at two times.

Figure 3. Central slab of the Fourier transform magnitudes.

Figure 4. Harmonic magnitude images.
Figure 5. Mask derived from the harmonic magnitude images.
Figure 6. Harmonic phase images (masked).
Producing Harmonic Images

The spectrum of a harmonic image is spread over all spatial frequencies; but most of the spectral energy is concentrated around the spectral peaks (Figure 3). The HARP bandpass filter should be properly positioned in order to capture a certain fraction of the spectral energy (for example, 90%). The size and position of this bandpass region will be affected by the underlying motion of the heart as well. For example, a contraction of the heart causes an increase in spectral spreading, a tissue stretching orthogonal to the tag lines will cause a decrease in the magnitude of the central frequency of a spectral peak, and a tissue contraction will cause an increase in the central frequency.

In the heart, the mechanical strain is bound by the local contractility of the myocardial fibers. Thus, the expected size and position of a harmonic peak is limited. After conducting an extensive set of simulations, we found that it is nearly optimal to position the bandpass filter at the spatial tagging frequency itself, and the radius of the bandpass filter should be about X % of the spatial tagging frequency, where X % is the magnitude of the maximum expected strain (typically around 30%).

Other factors affect the computation of harmonic images. For example, the intensity of harmonic images fades in accordance with the tag fading, which reduces the SNR of the harmonic image. Also, interference from other harmonic peaks also produces artifacts. Fortunately, imaging parameters can be found that effectively balance these effects, yielding images that are capable of producing very accurate, reasonably high-resolution regional cardiac function measurements.

Principle of HARP Analysis

Harmonic images are very similar to tagged images, except that they appear to have tags that are sinusoidal patterns, and there are two of them – a cosine pattern and a sine pattern, corresponding to the real and imaginary parts of the harmonic image. When the heart moves, the underlying pattern of each harmonic image changes. For example, compression of the heart muscle causes the crests of the sinusoidal pattern to move closer together while stretching or elongation causes the crests to move farther apart. This means that there is a relationship between the frequency of a harmonic image and the compression and elongation – in other words, the strain – of the heart muscle.

Consider the tag patterns shown in the top row of Figure 7. The initial sinusoidal pattern changes under both fading and deformation so that it has both a lower amplitude and different frequencies. If one looks at the phase of this sinusoid, as shown in the bottom row of Figure 7, it is clear that the phase of the pattern changes as well, but in a more limited way. In fact, the phase of a given point does not change due to motion — the phase is a material property. The slope of the phase change, however, in direct correspondence to the change in frequency of the sinusoid, which in turn reflects the underlying strain.

HARP analysis methods exploit the following two properties: 1) for a given point, harmonic phase is constant with time and 2) the slope of the harmonic phase is linearly related to the underlying mechanical strain.

Figure 7. Principle of HARP.
Motion Tracking Using HARP

Once imposed by the tagging process, the harmonic phase of a point in the tissue is constant throughout a motion or deformation. Motion tracking using HARP exploits this fact by using a phase-based optical flow formulation, which we refer to as CINE-HARP. We use a pair of tagged image sequences, usually with vertical and horizontal tags. These tag orientations provide sufficient information to track points in the plane. Out-of-plane motion is not tracked ordinarily, and therefore the tracking result must be interpreted as an apparent two-dimensional motion. This is consistent with the tag motion that we actually see in two-dimensional images, so HARP tracking can be used to track tag lines, or anatomical features that we see in the images.

Motion tracking uses an iterative algorithm that searches for the point in a second image that has the same two phase values as the point of interest in the first image. This process is repeated throughout the entire tag image sequence, yielding a pathline for the selected point. Points can be tracked both forward and backward in time as well. Figure 8 shows a collection of tracked points overlayed on the product of vertical and horizontal tagged images at end-diastole. The points have been manually selected to correspond to tag intersections, but they could be placed anywhere. Notice that points in the right ventricular myocardium can also be tracked.

Figures 9 and 10 show images of horizontal and vertical tagged images to show like a grid. A collection of points lying on two tag lines have been selected in the first time frame. The points were tracked through an image sequence to the last time frame at endsystole. We observe that, except for the point on the top, the points tracked successfully to lie on the same tag lines. The top point strayed from tagging due to the out-of-plane motion causing some tag lines, respectively the corresponding phase values, to disappear.

Figure 8

Figure 9

Figure 10

Measuring Lagrangian Strain

The tracking method can be used to measure Lagrangian strain by observing the change in distance between a pair of tracked points. Because of the special geometry of the heart and its function, it is common to measure the strain in the radial and circumferential directions. For this, we place a collection of points on a circular grid that fits the LV wall. The grid, shown in Figure 11, is three concentric circles representing the endocardium, midwall, and epicardium, Each circle has 16 points uniformly distributed on the circumference. All the points are tracked through the cardiac cycle and by measuring the change in distance between neighboring point on a circle provides a measure of circumferential strain. The distance between points neighboring on the radial direction produce the radial strain.

Figure 11

Figure 12 shows the computed circumferential strain for the different octants of the circles and for each circle. This special experiment represents a mechanically activated canine dog with a pacing lead sewed at location 5 in Figure 11 close to the base. The pacing lead causes early contraction at location 5 accompanied with stretching on the other side (location 1). Soon, the whole LV starts contracting, with less shortening in the early contraction octant 5.

Figure 12. Lagrangian strain in different octants.

Measuring Eulerian Strain

Now we focus on the second principle of HARP: that the slope of the harmonic phase is related to strain. We refer to the measurement of strain made using this technique as Single-shot HARP because it does not require an image sequence. Instead, only one pair of harmonic phase images (vertical and horizontal) are required, both corresponding to a single time in a cardiac cycle. After calculating the slopes of these harmonic phase images at a given point, we can compute a two-dimensional strain tensor at that point. This is repeated over all points (pixels) in an image so that a strain tensor map is produced. Several measurements can be produced from these strain tensors: circumferential strain, radial strain, maximum and minimum strains, and the contraction angle. The contraction angle is the anglebetween the maximum contraction direction and the circumferential direction.

Figure 13 shows the circumferential strain map for a 20 time frames of a mechanically activated canine heart (described above). The darker regions indicates circumferential shortening, while the yellowish color indicates stretching. Starting from the second time frame we can observe a darker spots at the 5 o'clock, the position of the pacing lead. On the other hand, we can observe stretching at the 10 o'clock region.

Figure 13. A sequence of Eulerian strain maps.

In order to quantify the Eulerian strain maps, we used the grid defined in Figure 11, tracked the grid using the previous method, and averaged the Eulerian strain by octant. The resultant plots, shown in Figure 14, are similar to those computed using the Lagrangian tracking method (Figure 12). A minor difference is that at the first time frame in Figure 14 strain can be detected that is not apparent in the first time frame of Figure 12. This is because the Lagrangian frame-of-reference is the first time frame while the Eulerian frame-of-reference is the time in which the tags were place.

Figure 14. Time evolution of Eulerian Strain by Octant.

HARP Analysis Program
We have developed a program using MATLAB that uses HARP ideas to measure myocardial function. The computations are fast and the program requires minimal human intervention. We computed the motion of the mechanically activated using a Pentium II PC 366MHz. Filtering 40 images to produce the harmonic magnitude and HARP images takes 15 seconds. Computing the Eulerian strains takes 3 seconds. Tracking a point takes a fraction of a second, and to compute the Lagrangian strain it takes 10 seconds. The tracking grid requires user intervention to place the circles inside the wall muscle. The total analysis for an image slice can be finished in less than 10 minutes. With MR imaging settings tuned for HARP, the measurements would be available in less than 10 minutes. Development of real-time HARP software written in C/C++ is underway.
HARP is a new method that allows the assessment of cardiac regional function using tagged MRI for clinical applications. So far, HARP has been successful in analyzing motion from conventional tagged MR images. HARP can also be used in conjuntion with special acquisitions for real-time acquisition and processing of myocardial function.

We are grateful to Dr. Elliot R. McVeigh and Tony Z. Faranesh for making these tagged images available for us. This research was supported by the National Institute of Health, National Heart, Lung, and Blood Institute.

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  • S. Sampath, J. A. Derbyshire, N. F. Osman, E. Atalar, and J. L. Prince, "Real-Time Imaging of Cardiac Strain using Ultra-Fast HARP Sequence," Joint Annual Meeting ISMRM-ESMRMB, April 21-27, 2001.
  • D. Kraitchman, S. Sampath, J. A. Derbyshire, A. W. Helman, E. A. Zerhouni, D. A. Bluemke, J. L. Prince, and N. F. Osman, "Detecting the Onset of Ischemia Using Real-time HARP," Joint Annual Meeting ISMRM-ESMRMB, April 21-27, 2001.
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