### Abstract

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSION
- REFERENCES

Magnetic resonance tagging has proven a valuable tool in the quantification of myocardial deformation. However, time-consuming postprocessing has discouraged the use of this technique in clinical routine. Recently, the harmonic phase (HARP) technique was introduced for automatic calculation of myocardial strain maps from tagged images. In this study, a comparison was made between HARP instantaneous strain maps calculated from single tagged images (SPAMM) and those calculated from subtracted tagged images (CSPAMM). The performance was quantified using simulated images of an incompressible cylinder in the ‘end-systolic’ state with realistic image contrast and noise. The error in the second principal stretch ratio was 0.009 ± 0.032 (mean ± SD) for the SPAMM acquisition, and 0.007 ± 0.016 for CSPAMM at identical contrast-to-noise ratio. Furthermore, differences between the methods were illustrated with in vivo strain maps. Those calculated from CSPAMM images showed fewer artifacts and were less sensitive to the choice of cut-off frequencies in the HARP band-pass filter. A prerequisite for the method to become practical is that the CSPAMM images should be acquired in a single breathhold. Magn Reson Med 46:993–999, 2001. © 2001 Wiley-Liss, Inc.

Since its introduction, MR tissue tagging (1, 2) has been used successfully for the quantification of motion and deformation in both normal and abnormal myocardium. Thus far, however, time-consuming postprocessing has discouraged the use of tagging and strain analysis in clinical routine.

One method that overcomes this limitation is the harmonic phase (HARP) technique (3–8). In the original form, the HARP method uses a narrow band-pass filter to extract the tagging information from the MR image, after which the local frequency of the tag modulation is detected. A key point in the HARP method is the choice of appropriate cut-off frequencies of the band-pass filter. If the filter is too narrow it suppresses the tag modulation in areas with large deformation, while a filter that is too wide introduces artifacts in the strain maps.

In this article, we show that HARP strain maps can be improved by using subtracted tagged images (which we refer to as complementary SPAMM or CSPAMM images [9]). The CSPAMM images are generated by subtraction of a normal SPAMM image and a SPAMM image that has the tagging pattern inverted. As illustrated in Fig. 1, the central part of *k*-space is cancelled and only the tagging signal remains. As a result, the peaks are better separated, giving more freedom to choose the appropriate band-pass filter. The improvements are illustrated with simulated images and with in vivo MR images.

### THEORY

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSION
- REFERENCES

The magnitude of an image *I*_{1}(*x*, *y*) with vertical tag lines, acquired immediately after application of a sinusoidal tag profile (two 45° RF tag pulses, 1-1 SPAMM), is described by multiplication of the nontagged MR image magnitude *I*_{0}(*x*, *y*) and a tag-modulation function *m*_{1}(*x*):

- (1)

For the 1-1 SPAMM image, the modulation directly after application of the tag pulses is given by:

- (2)

where *k*_{1} is the spatial frequency of the (vertical) tag lines and φ_{1} determines the phase at *x* = 0.

Deformation of the tissue will alter the frequency of the tag lines in the heart wall. Ignoring the tag line fading due to *T*_{1} relaxation, we can describe the deformed state by replacing the constant frequency *k*_{1} in Eq. [2] with a frequency that depends on the position (*x*, *y*):

- (3)

A fundamental step in the HARP procedure is the application of a Hilbert transformation for detection of the instantaneous (‘local’) frequency of the tagging signal. The Hilbert transform consists of a Fourier transform, application of a band- or high-pass filter that passes positive frequencies only (*k*_{x} > 0, right half of *k*-space), and an inverse Fourier transform. The result is a complex signal *H*(*x*, *y*) of which the real part equals the original (real) signal without the zero-frequency (offset) term, and of which the imaginary part equals the original signal with a 90° phase shift. In the original HARP procedure (6), the band-pass filter is chosen not only to remove the negative frequencies, but it passes only a narrow frequency range around the tagging frequency.

Instantaneous frequency (10) is defined as the spatial derivative of the angle θ(*x*, *y*). Therefore, the instantaneous frequency (vector) can be calculated from the HARP angle ϕ(*x*, *y*) with:

- (5)

where the asterisk denotes removal of the wrapping artifact from the derivative (3). Note that the calculation of the instantaneous frequency is useful only when the zero-frequency (offset) has been removed in the Hilbert transform. Calculation of the instantaneous frequency is identical to techniques used for demodulation of FM signals in communication theory (11). The tag signal intensity (referred to as the harmonic magnitude) is calculated by taking the magnitude of *H*′(*x*, *y*).

The above procedure also holds for the image with horizontal tag lines, from which the instantaneous frequency **k** is calculated using the upper half of *k*-space for the Hilbert transform. Under the condition that the tag frequency of both vertically and horizontally tagged images equals *k*_{1}, the deformation gradient tensor is calculated with:

- (6)

The superscript −*T* stands for inversion of the transposed 2 × 2 matrix built from the two instantaneous frequency vectors. Note that **F** can be related to a material point in the deformed state only, as the displacement between the undeformed and the deformed state is not known (but may be calculated using a tracking algorithm [3]).

Two-dimensional strain parameters are calculated from **F** with the usual mathematics. For example, polar decomposition of **F** separates the rigid body rotation **R** from the stretch **U** (12):

- (7)

The eigenvalues of the stretch tensor **U** are the principal stretch ratios, λ_{1} and λ_{2}, with the convention that λ_{1} ≥ λ_{2}. λ > 1 indicates stretch of the tissue and λ < 1 indicates shortening. The eigenvectors of **U** give the directions of the principal stretches. In this article, we have focused on the eigenvalues of the stretch tensor **U** for evaluation of the HARP method. The eigenvalues are independent of the referenced coordinate system. The functional map of a stretch ratio will be referred to as ‘strain map.’

### DISCUSSION

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSION
- REFERENCES

In this article, we have used an analytical test case to estimate the performance of the HARP method using a high-pass filter to suppress the central peak in *k*-space. Strain maps that were calculated from an in vivo dataset were shown to have similar artifacts as the analytical test case.

The analytical test case has a strong gradient in λ_{1} in the inner part of the cylinder wall, which was systematically underestimated in the strain maps. However, in this respect the cylinder is not likely to be a realistic model of LV heart wall. Approximately the inner 1/3 of the heart wall consists of spongy myocardium, from which blood is ejected into the LV cavity during systole. Therefore the stretch in the subendocardial layers is lower than we modeled with the incompressible cylinder. The systematic underestimation of λ_{1} near the inner part of the cylinder thus represents a ‘worst case’ scenario, and may not be as severe for the in vivo data as the results of our analytical test case suggest.

Comparing the λ_{1} error plot in Fig. 3c (SPAMM) with the plot in Fig. 3g (CSPAMM), it may be observed that the CSPAMM error plot has a cloud which is narrower than that of the SPAMM error plot. This better performance of the CSPAMM data is not represented in the SD of the error in Table 1 because the SD is primarily determined by the systematic underestimation of the steep gradient in the inner layers of the cylinder. To quantify this difference, we also computed the SD of the error using only the pixels with a true λ_{1} < 1.5. The error using SPAMM then equals 0.014 ± 0.054 vs. −0.003 ± 0.039 for the CSPAMM data.

The λ_{1} maps do not give a reliable estimate of the deformation near the edge of the cylinder wall. This is probably caused by the tag lines suddenly being cut off at the edges with the instantaneous frequency vector **k**′ orthogonal to the cylinder wall. Thus, the tag modulation suddenly stops in the ‘direction’ in which the frequency of the tag modulation is detected. Cutting the modulation gives a semirandom phase jump at the edge, which results in a high or low instantaneous frequency at the edge. In our approach, the edge-zone was made as small as possible by using a wide band-pass filter. The median filter we applied to the instantaneous frequency maps eliminated the outliers at the edge. The pixels at the edge are less problematic in the λ_{2} maps, which are mainly determined by tag lines orthogonal to the heart wall.

Comparing the in vivo strain maps with earlier reports, the results appear credible. For example, the transmural gradient in λ_{2} is in agreement with the results of Waldman et al. (16) and Moore et al. (17). Also, the largest stretch may be expected to occur in the lateral free wall, as confirmed by the λ_{1} map. Furthermore, the stretch ratio in the papillary muscle near the lateral wall is almost 1 (no deformation) in both maps, which agrees with this muscle contracting mainly in the longitudinal direction.

A general drawback of the instantaneous HARP method is that the rigid body displacement cannot be calculated from the instantaneous frequency maps. Thus, the strain can be mapped only onto the deformed heart, while it cannot be mapped onto the undeformed heart. This drawback is overcome by the application of HARP-tracking (3). Provided that the tracking algorithm performs correctly, the errors for the tracking method are expected to be equal to the errors in the instantaneous strain maps.

#### Filter Cut-off Frequency

A key point in the HARP method is the choice of an appropriate filter to isolate the tagging peak in *k*-space from the central peak. In this article, we have given examples of strain maps produced with a high-pass filter to suppress the central peak. The cut-off frequency of ν = 20 was established by evaluation of strain maps calculated from simulated and in vivo CSPAMM images, using a range of cut-off frequencies between ν = 0 and ν = 40. For the simulated images, all strain maps with cut-off frequency ν < 25 were equal. Choosing a low cut-off frequency of ν = 10 resulted in light artifacts in the in vivo end-systolic strain maps. These artifacts are due to imperfect cancellation of the central peak in the CSPAMM image, occurring if the SPAMM image is not exactly reproduced with inverted tag lines in the second SPAMM image.

The artificial stripes in the strain maps in Fig. 4a,b arise from overlap in the peaks in *k*-space in the SPAMM images. A low filter cut-off frequency passes the tail of the central peak, causing the striping artifact. In the analytical test case, the striping artifact is less visible because the central peak is less spread out in *k*-space compared to the in vivo data. Increasing the cut-off frequency reduces the striping artifact, but also filters out the tag signal in areas with large stretch, as it rejects the lowest tag frequencies (see Fig. 2f and 4f). Using CSPAMM images provides a solution to this problem: the resulting strain maps are less sensitive to small alterations in the filter cut-off frequency.

#### SPAMM vs. CSPAMM Acquisition

The strain maps calculated from the SPAMM acquisition (requiring half the acquisition time of the CSPAMM acquisition) display either artificial lines (cut-off at ν = 20) or underestimate the first principal strain (cut-off at ν = 30). HARP does allow correct estimation of the second principal strain from SPAMM images. Note that we simulated the SPAMM images with the CNR of a double acquisition (requiring the same acquisition time as the CSPAMM acquisition). In practice, however, one would use just a single acquisition, which would increase the noise in the strain map compared to our analytical test case.

The CSPAMM method requires two SPAMM acquisitions. Subtraction of the SPAMM images is sensible only if the two single breathhold acquisitions are identical (except for the sign in the tag modulation). The drawback is that there can be substantial variation in position between breathholds. The consequence is that images in two different breathholds often do not match, resulting in large gradients in image intensity at the edges of the heart wall. These gradients interfere with the tag signal and affect the estimated strain near the edges of the heart wall.

The CSPAMM-HARP method therefore seems practical only if both acquisitions can be measured in one single breathhold (19).

### CONCLUSION

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSION
- REFERENCES

An analytical test case was used to quantify the error in the HARP instantaneous myocardial strain maps from SPAMM and CSPAMM images with sinusoidal modulation using a wide band-pass filter in the Hilbert transform. Best results were obtained with the CSPAMM images. A major advantage of the CSPAMM acquisition is that the central peak in *k*-space is effectively suppressed, resulting in more accurate strain maps that are less sensitive to the choice of the HARP band-pass filter.

The drawback of CSPAMM acquisition is the double acquisition time compared to the SPAMM acquisition. A prerequisite for the method to become practical is that the CSPAMM images should be acquired in one single breathhold. Fast imaging techniques, such as segmented echo planar imaging, should bring this within reach (19).