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To compare generalized autocalibrating partially parallel acquisitions (GRAPPA), modified sensitivity encoding (mSENSE), and SENSE in phase-contrast magnetic resonance imaging (PC-MRI) applications.
Aliasing of the torso can occur in PC-MRI applications. If the data are further undersampled for parallel imaging, SENSE can be problematic in correctly unaliasing signals due to coil sensitivity maps that do not match that of the aliased volume. Here, a method for estimating coil sensitivities in flow applications is described. Normal volunteers (n = 5) were scanned on a 1.5 T MRI scanner and underwent PC-MRI scans using GRAPPA, mSENSE, SENSE, and conventional PC-MRI acquisitions. Peak velocity and flow through the aorta and pulmonary artery were evaluated.
Bland–Altman statistics for flow in the aorta and pulmonary artery acquired with mSENSE and GRAPPA methods (R = 2 and R = 3 cases) have comparable mean differences to flow acquired with conventional PC-MRI. GRAPPA and mSENSE PC-MRI have more robust measurements than SENSE when there is aliasing artifact caused by insufficient coil sensitivity maps. For peak velocity, there are no considerable differences among the mSENSE, GRAPPA, and SENSE reconstructions and are comparable to conventional PC-MRI.
Flow measurements of images reconstructed with autocalibration techniques have comparable agreement with conventional PC-MRI and provide robust measurements in the presence of wraparound.
Quantifying velocity and blood flow is of great importance in the detection of cardiovascular disease. Shunt ratio, which is the ratio of pulmonary artery flow to aortic flow, is an important parameter that assesses the need for closing shunt lesions (1-5). Another parameter is the amount of blood regurgitated into a cardiac chamber divided by the stroke output, or regurgitant fraction (6). Measurements of blood flow in the aorta and pulmonary trunk help characterize these parameters.
Phase-contrast magnetic-resonance imaging (PC-MRI) is a noninvasive technique sensitive to blood flow in the great arteries (7-10). PC-MRI is particularly suited for quantification and has sensitivity that is adjustable to flow velocity. A primary disadvantage of PC-MRI is the lengthy scan time needed to acquire the many flow measurements at adequate temporal and spatial resolution.
Sensitivity encoding (SENSE) (11) is an important parallel imaging technique that significantly reduces acquisition time. Combining SENSE with PC-MRI can either shorten scan time or increase temporal resolution of segmented k-space techniques. Shorter scan time improves patient compliance in breathhold imaging. PC-MRI combined with SENSE has been demonstrated and reported in the literature (12,13).
In PC-MRI the full field of view is not needed for flow quantification, as outer tissue such as the chest wall is irrelevant to the flow measurement. Some spatial aliasing is commonly used in order to maximize the spatial resolution and thus reduce partial volume effects. However, the spatial aliasing may interfere with a SENSE reconstruction. It has been reported that SENSE may be unable to correctly unalias the data due to coil sensitivity maps that do not match that of the aliased tissue in the full field of view image (14,15). The reconstructed images will have artifacts that may compromise the accuracy of flow measurements. One solution to avoid such an artifact is to prescribe a field of view large enough to avoid aliasing. However, this typically requires reduced spatial resolution, causing a loss of accuracy in the flow measurements due to partial volume effects. On the other hand, spatial resolution can be maintained at the cost of a larger imaging matrix, but only at the cost of increased scan time. Also, in certain anatomies, prescribing a slice that avoids aliasing may be difficult.
Autocalibrating sequences like modified SENSE (mSENSE) (16), generalized encoding matrix (GEM) (17), vdSENSE (18), and generalized autocalibrating partially parallel acquisitions (GRAPPA) (19) can avoid the spatial aliasing of a SENSE reconstruction. In image-based schemes, like mSENSE and GEM, the coil sensitivity maps are calculated from lower-resolution images taken from the central k-space profiles. These lower-resolution images possess the same spatial extent, and consequently the same aliasing behavior, of the undersampled image. In k-space schemes like GRAPPA each line of k-space is not affected by the neighboring lines and is independent of the actual field of view.
In this study autocalibrating sequences are applied to PC-MRI for both aortic and pulmonary flow and peak velocity measurements. Different techniques in estimating coil sensitivity maps for PC-MRI application are investigated. The data are reconstructed using the mSENSE and GRAPPA methods and then compared to the results with SENSE. Through Bland–Altman statistical analysis, autocalibrating sequences are shown to have mean differences from conventional PC-MRI that are comparable to SENSE mean differences from conventional PC-MRI in flow and peak velocity measurements. When aliasing is present, autocalibrating sequences are able to remove aliasing and provide accurate measurements.
Accurate coil sensitivity maps are essential to maximize signal-to-noise ratio (SNR) and mitigate artifacts. Thunberg et al (13) used a time-resolved gradientecho pulse sequence without flow-encoding to estimate coil sensitivity. However, this requires a separate breathhold scan for every slice of interest. A simple approach to estimating coil sensitivity would be to average low-resolution images across multiple time frames to maximize SNR without cardiac gating. However, coil sensitivity estimations may also be affected by blood flow during the temporal acquisition. In particular, the combination of the different flow-encoding states of PC-MRI affects the reconstruction. In order to understand the effect of flow on coil sensitivity, two pitfalls to consider when combining coil sensitivity maps, flow-encode pairing and averaging over multiple cardiac phases, are presented. The autocalibrated PC-MRI sequence is used in this study.
The autocalibrated PC-MRI sequence is shown in Fig. 1. As in the conventional PC-MRI sequence, the autocalibrated PC-MRI sequence provides an image set temporally resolved across the entire cardiac cycle. For each cardiac phase, two different gradient moments are used to acquire two flow measurements for through-plane flow. In this study the gradient moments were designed as bipolar gradients with positive flow encoding and negative flow encoding states, which are termed flow-up and flow-down. With the autocalibrated PC-MRI sequence there is a low-resolution image collected in every flow state and every cardiac phase. Particularly, for the mSENSE method of reconstruction a combination of the low-resolution images is used to estimate our coil sensitivities. The coil sensitivities also contain velocity information. The coil sensitivities are estimated by dividing the individual surface coil images by the square root of the sum of squares of the individual coil images. Alternatively, coil sensitivities can be estimated from a separate body coil scan.
In flow-encode pairing the choice of combining the flow-encoding states affects the reconstruction accuracy. For purposes of illustration, the mSENSE PC-MRI reconstruction is considered from two ways of estimating the coil sensitivities, 1) singular mapping—the coil sensitivity map is calculated for each flow encode for each cardiac phase, and 2) pairwise mapping—the coil sensitivity map is calculated for each pair of flow encodes for each cardiac phase.
Let the apparent magnetization of a spin density ina flow-up or flow-down acquisition be described as:
where C is the magnitude of the coil sensitivity, ρ is the proton density, ϕflow,up is the flow-up velocity phase, ϕflow,down is the flow-down velocity phase, ϕcoil is the coil sensitivity phase, and ϕother is other phase information like off-resonance phase.
Here a separate coil sensitivity map is calculated for each flow encode for each cardiac phase. (See Appendix for a mathematical derivation of the coil sensitivity maps.) For a conventional SENSE reconstruction without flow, the reconstruction for each aliased set of pixels can be solved in a least squares sense and is represented in matrix notation by the commonly known equation .
The SENSE PC-MRI reconstruction differs from the mSENSE reconstruction by that the coil sensitivity maps are acquired separately from the PC-MRI scan through a calibration scan that does not have flow-encoded phase. Now, comparing to the mSENSE PC-MRI reconstruction, the flow-encoded phase is present in the mSENSE PC-MRI reconstruction:
Similarly, for the flow-down acquisition,
Phase difference processing to obtain the phase contrast images leads to:
As is evident in Eq. , the phase due to flow would be incorrect due to the offset phase term, ϕflow,up − ϕflow,down.
In this situation the coil sensitivity map is calculated for each pair of flow encodes for each cardiac phase. Many methods exist for combining low-resolution images from the flow-up and flow-down encodes to form estimation maps. For convenience, the coil sensitivity is estimated here using the low-resolution images from the flow-down encode only.
The reconstructed signal for the flow-up encode is:
The phase difference is:
As given in Eq. , the phase due to flow is now correct and matches the phase of the conventional SENSE PC-MRI reconstruction. When a common phase offset is introduced into the coil sensitivity maps, which in this case is the flow-down phase, the subtraction operation removes the phase offset and restores the correct velocity.
To verify these results a simulated circular phantom with flow-up phase ranging across 0 to π in the readout direction was created. Then a flow-down phase ranging across 0 to −π was simulated. The phase difference from this simulation, Δflow, results in values from 0 to 2π. The coil sensitivities were created using Biot-Savart properties for a 6-channel circular array. Gaussian noise with variance 5% of the peak signal to the real and imaginary channels was added. Both mapping methods were applied for estimating the coil sensitivity maps and the results were compared to the reconstruction.
The other pitfall involves the combination of multiple images. Two approaches are explained, complex averaging and split averaging. The complex averaging approach simply performs complex averaging over a set of cardiac phases to produce a sensitivity map. Let the estimated coil sensitivity map of the kth coil be the complex addition of a set of coil sensitivity maps from different cardiac phases, , where T is the number of cardiac phases used in the complex averaging. In this example, for simplicity, only the flow-up acquisitions were used. After expanding the coil sensitivity terms and some algebraic operations, the following estimated coil sensitivity map is obtained (see Appendix for a complete mathematical explanation of the coils):
The coil sensitivities will vanish for differences between the cardiac phases of ±π, ±3π, etc., which correspond to multiples of velocity encoding (VENC) values.
To counter this modulation, an alternative method, which we call split averaging, is proposed. An estimate of the magnitude and phase of our coil sensitivity maps is performed separately. The magnitude of the coil sensitivity is estimated by . The phase is estimated as the angle of the complex sum, . With the split averaging scheme the modulation is removed while the images are still averaged to improve SNR in the coil sensitivity estimation.
To verify this, a circular phantom was simulated with two cardiac phases. The first cardiac phase has flow phase ranging across 0 to π in the readout direction. The second cardiac phase has flow phase ranging across 0 to −π. The phase difference from this simulation, Δflow, would result in values from 0 to 2π. The coil sensitivities were created using Biot-Savart properties for a 6-channel circular array. Gaussian noise with variance 5% of the peak signal to the real and imaginary channels was added. Both the complex averaging scheme and the split averaging scheme for coil sensitivity estimation were used and the results were compared.
Simulations were performed with fully sampled flow acquisitions of the aorta. A conventional PC-MRI scan was first acquired. Data from this fully sampled acquisition were then used as the reference. Undersampling was done by removing phase encode profiles from the fully sampled reference. In this manner, comparisons will have no variability from interscan differences. Reduction factors, R, of 2, 3, and 4 were performed. SENSE reconstructions were utilized on the undersampled datasets. In the autocalibrated sequences the undersampling was not performed at the central lines. The reduction factors are specifically referred as the outer reduction factors (ORFs) (20). GRAPPA and mSENSE reconstructions were performed on the undersampled datasets.
The reference scan was acquired using a conventional PC-MRI sequence, namely, a 2D fast GRE segmented k-space cardiac-triggered phase-contrast sequence with retrospective gating. Scanning was performed with a 256 × 160 matrix, flip angle 15°, 10 mm slice, TE ≈ 2 msec, TR ≈ 4 msec, and 62.5 kHz bandwidth. The reference was scanned on a 1.5 T TwinSpeed MRI Scanner (GE Medical Systems, Waukesha, WI) with 150 T/m/sec maximum slew rate and 40 mT/m gradient strength. An 8-channel cardiac phased-array coil (GE Medical Systems) was used.
To calculate the coil sensitivities for mSENSE, the central fully sampled phase encodes of each surface coil data were first apodized with a Hamming window. As a consequence of the two limitations stated previously, pairwise mapping and split averaging are used to compute the coil sensitivities. To maximize SNR, split averaging is performed on all the cardiac phases to estimate the coil sensitivity maps. For ORF = 2, the central 4, 16, and 32 phase encodes were used. For ORF = 3, the central 6, 16, and 32 phase encodes were used. For ORF = 4, the central 8, 16, and 32 phase encodes were used. The coil sensitivities were then calculated by dividing the surface coil images by the square root of the sum-of-squares of the individual coil images. Measurements were taken from the image after removing pixels below a noise threshold. GRAPPA was also applied using the same sampling schemes, where in this case the autocalibration signal, or ACS, lines are the central phase encodes. A block size of 5 was used in the GRAPPA reconstructions. The GRAPPA weights were estimated off the first cardiac phase and flow echo.
For SENSE, a separate fast spin echo (FSE) scan was used to obtain the coil sensitivities. The calibration scan was acquired using a 3D spoiled gradientrecalled echo (SPGR) sequence and was acquired over the full chest volume for all vessels of interest, namely, the aorta and the pulmonary trunk. The sequence was not cardiac gated. A low flip-angle of 5° and a low-resolution matrix of 64 × 64 produced high SNR, low contrast images for the coil sensitivity map. The total calibration scan time was ≈15 seconds.
The coil sensitivity maps used for the SENSE reconstruction were computed by first interpolating the 3D calibration scan onto the same prescribed plane of the PC-MRI scan. After interpolation the maps were smoothed by a 3 × 3 kernel. Pixel-wise complex division was performed with a sum-of-squares reconstruction of this map. If the sensitivity maps showed particularly strong noise in the air pixels, a binary mask was used to threshold and to remove these pixels.
All flow measurements were calculated using the CV Flow Analysis software package in an Advantage Workstation (GE Medical Systems). For every cardiac phase the aorta was segmented manually. Flow rate was calculated by summing velocity across the vessel lumen averaged over the cardiac cycle. Postprocessing using linear order correction was performed to eliminate concomitant effects from the Maxwell terms (20).
In addition to the flow measurements, the flow profile of the reference was compared to the flow profile obtained from the different reconstructions. For every point in the cardiac cycle the root-mean-square (RMS) value of the difference between the reference flow measurement and the flow measurement from the other reconstruction was calculated. The difference is expressed as the average flow deviation per cardiac phase from the reference flow measurement. Specifically, for flow F and T cardiac phases we compute our RMS as:
Normal volunteers (n = 5) were scanned on a 1.5 T TwinSpeed MRI Scanner (GE Medical Systems) with 150 T/m/sec maximum slew rate and 40 mT/m gradient strength. Those volunteers underwent PC-MRI scans transverse to the ascending aorta and pulmonary trunk. Institutional Review Board approval was obtained for the study protocol and consent was obtained from all volunteers. Volunteers were asked to breathhold at deep inhalation for both the calibration scan and PC-MRI scans.
For the SENSE reconstruction a calibration scan was acquired first using the same sequence described in the simulations. Afterwards, for each vessel of interest, a conventional PC-MRI was acquired first (256 × 160 matrix, flip angle 15°, 10 mm slice, TE ≈ 2 msec, TR ≈ 4 msec, 62.5 kHz bandwidth, 8-channel cardiac phased-array coil), followed by SENSE PC-MRI with reduction factor of 2 (R = 2) and 3 (R = 3). Finally, the autocalibrated PC-MRI sequence was performed for both ORF = 2 and ORF = 3 with 32 central phase encode lines. Imaging was performed with the slice perpendicular to the flow lumen and flow encoding in the through-plane direction. Ten views per segment were collected. The central phase encodes were grouped together in segments. Scan times were ≈20–30 seconds for the reference set at R = 1, 12–20 seconds for SENSE at R = 2, and 7–10 seconds for SENSE at R = 3. For autocalibrated PC-MRI, it was ≈ 14–22 seconds for ORF = 2 and 9–12 seconds for ORF = 3. Reconstructions for the SENSE, mSENSE, and GRAPPA followed the same procedure as in the simulations.
All flow measurements were calculated using the CV Flow Analysis software package as before. In addition, peak pressure gradient, ΔP, is estimated from the peak velocity, Vp, by the modified Bernoulli equation, (10). Peak velocity measurements were performed using the MatLab software program (MathWorks, Natick, MA). For measurement purposes, the peak velocity is calculated as the average of the top 90% velocities of contiguous pixels within the vessel lumen. This averaging procedure was done to reduce the effect of noise and outliers. The final peak velocity was calculated as the maximum of the averaged peak velocities at systole.
Statistical comparisons were performed for flow and peak velocity measurements between the conventional PC-MRI and the following: SENSE PC-MRI at R = 2, SENSE PC-MRI at R = 3, mSENSE PC-MRI at ORF = 2, mSENSE PC-MRI at ORF = 3, GRAPPA PC-MRI at ORF = 2, and GRAPPA PC-MRI at ORF = 3. For the flow measurements average aortic and pulmonary flow were separately compared for each patient. Peak velocity for the aorta and pulmonary artery were separately compared. Bland–Altman statistics (21) were calculated. These statistics are useful for comparing measurements when the true flow is not known.
In flow-encode pairing the simulations with the coil sensitivity maps verified the need for a common phase. Figure 2 shows the results from the two methods for flow-encoding pairing. Pairwise mapping recovers the phase ramp. There is some noticeable velocity aliasing along the ±π boundary due to the Gaussian noise. On the other hand, singular mapping loses the flow-encoded phase information and zero phase is obtained as expected, leading to inaccurate flow estimates.
Figure 3 shows the results from the pitfall of combining multiple images. Figure 3a shows that complex averaging produces an expected increase in signal for phase differences close to ±π and consequently ±VENC. Using two cardiac phases with flow in the same sign would not lead to vanishing coil sensitivities since VENC is the maximum phase difference. If the cardiac phases had differing signs, like in regurgitant flow or in a flow-up/flow-down combination, then the sensitivity maps may vanish. In the proposed split averaging scheme the modulation of the signal is removed (Fig. 3b).
The images of Fig. 4 illustrate the reference along with the reconstructions of SENSE, mSENSE, and GRAPPA. The SENSE reconstruction shows the aliasing still present at the center of the image due to the lack of coil sensitivity. For this particular subject the aliasing is present at a sizeable portion of the aorta. Qualitatively, both mSENSE and GRAPPA are able to remove this aliasing artifact. Figure 5 shows the flow profile of the different reconstructions of one volunteer. At reduction factors of 2 and 4 the SENSE flow profiles reflect the inaccurate reconstruction. Specifically, for R = 2 there is a 15.83 mL/min RMS difference from the reference flow profile (Table 1) and a difference in 11% in the flow output (Table 2). The flow profiles reconstructed with mSENSE and GRAPPA qualitatively are similar to the reference flow profile.
Figure 6 shows the flow profiles for mSENSE and GRAPPA at ORF = 2 and different number of central phase encodes (4 and 32). GRAPPA was able to fully recover the flow with both number of central phase encodes, while mSENSE was not able to recover accurately with 4 central phase encodes. Table 1 and and22 summarize the RMS and flow measurements for all the simulations. The SENSE reconstruction has a lower RMS value for R = 3 relative to R = 2 due to the strong residual aliasing artifact for the R = 2 case. The GRAPPA reconstructions generally show closer agreement than the mSENSE reconstructions. For each outer reduction factor and the smallest number of central phase encodes the GRAPPA reconstruction produced smaller RMS values than mSENSE, specifically: ORF = 2 and 4 central lines, ORF = 3 and 6 central lines, and ORF = 4 and 8 central lines.
There is generally good correlation between the RMS and flow measurements for the simulations that used 16 and 32 central phase encodes. Because the RMS measure is calculated from the entire cardiac cycle, the measure emphasizes temporal differences. The flow measurements can validly compare the performance among SENSE, mSENSE, and GRAPPA with 16 and 32 central phase encodes.
The Bland–Altman statistics are shown in Fig. 7 and Table 3--6.6. For the aorta the aliasing artifact is observed in 3 of the 5 cases for R = 2 and no cases for R = 3. For the pulmonary artery the aliasing artifact is observed in all 5 cases for R = 2 and 3 of the 5 cases for R = 3. The Bland–Altman statistics for flow in the aorta R = 2 case show that mSENSE and GRAPPA have smaller mean differences than the SENSE reconstruction, which is expected due to the aliasing artifact. The Bland–Altman statistics for flow in the aorta R = 3 case show that mSENSE and GRAPPA have comparable mean differences to the SENSE reconstruction, which has no appreciable aliasing artifact present at the aorta. Pulmonary artery images from one subject are shown in Fig. 8. There is residual wraparound artifact for the SENSE reconstruction. Noise is more noticeably present for the ORF = 3 images of GRAPPA and mSENSE than for the ORF = 2 images. The mSENSE images also show residual artifacts that arise from the sharp cutoff of the coil sensitivity at the edge of the field of view and those that arise at anatomical edges. This is due to the low resolution of autocalibrated coil sensitivities in the mSENSE technique. For the pulmonary artery, statistics show that mSENSE and GRAPPA at ORF = 2 have smaller mean differences than SENSE reconstructions at R = 2 and R = 3. The statistics for the peak velocity, on the other hand, show no considerable differences between the mSENSE, GRAPPA, and SENSE reconstructions and are comparable to the conventional PC-MRI. Here, the mean differences range from −7.3 to −1.5 cm/s for the aorta and −6.9 to −3.7 cm/s for the pulmonary artery for mSENSE, GRAPPA, and SENSE at reduction factors 2 and 3.
With autocalibrated PC-MRI, using pairwise mapping and split averaging, mSENSE and GRAPPA reconstructions were able to perform accurate flow and peak velocity measurements. Regardless of the presence of aliasing artifact, mSENSE and GRAPPA had flow measurements with small mean differences relative to the flow measurements from conventional PC-MRI. When there is no aliasing artifact, then comparable mean differences were obtained between SENSE and autocalibrated PC-MRI. When there is aliasing artifact, measurements from mSENSE and GRAPPA reconstructions had smaller mean differences than SENSE, which indicates there is correlation between the presence of the aliasing artifact and its impact on flow measurement accuracy in SENSE. For the peak velocity the differences between SENSE and autocalibrated PC-MRI are not appreciable. The aliasing may not be as obstructive when dealing with peak velocity measurements because the peak velocity may not lie where the aliasing occurs.
An alternative to counter the aliasing would be to prescribe the field of view to avoid this wraparound effect. In the in vivo examples of the aorta this was certainly possible for some of the cases where the slice can be shifted to avoid wraparound. An autocalibrated approach would, however, reduce the chance of operator error in prescribing these slices. Also, most of the cases for the pulmonary artery were prescribed in a fashion such that aliasing could not be avoided. The obliquity of the slice may prevent anatomy, such as parts of the torso, to be entirely excluded from extending greater than the field of view in the phase encoding direction and thus causing wraparound. In addition, another solution would be to prescribe a larger field of view. The image matrix can be preserved although at the cost of decreased spatial resolution. Additionally, the image matrix can be increased at the cost of increased scan time.
The effect of the aliasing artifact may cause underestimation of velocity. The effect is more pronounced in the systolic phases of the heart, as seen in Fig. 4. If one were to treat the aliased tissue as if it were static, then the effect would be analogous to the effect resulting from partial volume. Assuming that the magnitudes for the two flow echoes are approximately equal and that bipolar gradients are used for the two flow-encoding steps, the static tissue will cause the phase difference to be smaller than expected and the velocity will be underestimated.
The RMS values with GRAPPA are smaller than the values with mSENSE, although the flow measurements are comparable. The simulations indicate that GRAPPA is more robust when using four central lines for ORF = 2 and 8 central lines for ORF = 4. In the in vivo experiments the mean differences are comparable in mSENSE and GRAPPA for both flow and peak velocity. Despite some residual aliasing artifact from inaccuracies in the mSENSE coil sensitivity maps, there is still good agreement between mSENSE and the conventional PC-MRI images. In addition, integrating the central phase-encodes into a reconstruction such as vdSENSE or GEM may be problematic with PC-MRI. These techniques are similar to SENSE and require coil sensitivity maps to reconstruct images. They would also have to address the issue raised in flow-encode pairing.
An advantage of autocalibrated PC-MRI is that patient motion is minimized between the calibration scans and data acquisition scans. The patient may shift positions between the calibration scan and SENSE PC-MRI scan. The position of the chest wall may slightly differ with different breathholds, possibly leading to inaccurate coil sensitivity maps and inaccurate measurements. When this type of artifact becomes noticeable the calibration scan must be repeated. Autocalibrated PC-MRI would be insensitive to these types of movement.
In conclusion, a practical autocalibrated PC-MRI sequence has been implemented and tested in normal volunteers. Flow velocity data derived from autocalibrated PC-MRI reconstructed with mSENSE and GRAPPA are found to be comparable with conventional PC-MRI in the imaging of the great arteries. GRAPPA and mSENSE PC-MRI are shown to have more robust measurements than SENSE when there is aliasing artifact caused by insufficient coil sensitivity maps. The peak velocity measurements show no significant differences. Thus, autocalibrated PC-MRI may be a valuable and powerful tool for enhancing the performance of PC-MRI.
Contract grant sponsor: National Institute of Health (NIH); Contract grant numbers: P41RR09784, R01EB002711; Contract grant sponsor: GE Medical Systems; Contract grant sponsor: Center of Advanced MR Technology at Stanford; Contract grant sponsor: Whitaker Foundation; Contract grant sponsor: Lucas Foundation.
For the flow-up encode, the coil sensitivity for the kth coil of an N-channel array is estimated as:
The subscript indicating low-resolution data has been dropped for convenience.
With this method of estimation the characteristic multiple-coil weighting in conventional phased-array sum-of-squares reconstruction is present. A low spatial resolution acquisition of the flow encoding phase is also present.
For the flow-down encode the coil sensitivity is similarly expressed:
The phase inhomogeneity and sum-of-squares normalization is dropped with the understanding that the mSENSE reconstruction will reflect this weighting.
Let the estimated coil sensitivity map of the kth coil be the complex addition of a set of coil sensitivity maps from different cardiac phases, , where T is the number of cardiac phases used in the complex averaging.
For flow-up acquisitions, after expanding coil sensitivity maps:
The common phase can be factored out of the summation.