Regularization of parallel MRI reconstruction using in vivo coil sensitivities

Qi Duan*a, Ricardo Otazoa, Jian Xu b, Daniel K. Sodicksona

a Center for Biomedical Imaging, Department of Radiology, New York University School of Medicine, 660 First Avenue, New York, NY 10016;

b Siemens Medical Solutions USA Inc., 660 First Avenue, New York, NY 10016

ABSTRACT

Parallel MRI can achieve increased spatiotemporal resolution in MRI by simultaneously sampling reduced k-space data with multiple receiver coils. One requirement that different parallel MRI techniques have in common is the need to determine spatial sensitivity information for the coil array. This is often done by smoothing the raw sensitivities obtained from low-resolution calibration images, for example via polynomial fitting. However, this sensitivity post-processing can be both time-consuming and error-prone. Another important factor in Parallel MRI is noise amplification in the reconstruction, which is due to non-unity transformations in the image reconstruction associated with spatially correlated coil sensitivity profiles. Generally, regularization approaches, such as Tikhonov and SVD-based methods, are applied to reduce SNR loss, at the price of introducing residual aliasing. In this work, we present a regularization approach using in vivo coil sensitivities in parallel MRI to overcome these potential errors into the reconstruction. The mathematical background of the proposed method is explained, and the technique is demonstrated with phantom images. The effectiveness of the proposed method is then illustrated clinically in a whole-heart 3D cardiac MR acquisition within a single breath-hold. The proposed method can not only overcome the sensitivity calibration problem, but also suppress a substantial portion of reconstruction-related noise without noticeable introduction of residual aliasing artifacts. Keywords: Parallel MRI, SENSitivity Encoding (SENSE), regularization, in vivo coil sensitivities, denoising, noise reduction, g-factor, artifact correction

1. INTRODUCTION Magnetic Resonance Imaging (MRI) is now a commonly used imaging technique in radiology to visualize anatomical and functional information of the body, given its ability of providing soft-tissue contrast and its non-ionizing feature. In many MRI applications, imaging speed is still of critical importance. For example, imaging cardiovascular system must contend with the challenges of cardiac motion, respiratory motion, and blood flow. Over the decades, many efforts on improving acquisition protocol have been significantly shorten the acquisition time. However, achievable fast acquisition in MRI is still limited by physical and physiological constraints on the switching rate of magnetic field gradient and radiofrequency (RF) pulses.

To circumvent this limitation associated with gradient and RF pulse for fast acquisition, a new class of approach has been developed about ten years ago. This approach, known as “parallel imaging”, utilizes arrays of RF coils as multiple detectors in parallel to multiply imaging speed without increasing gradient switching rate or RF power deposition. For in vivo imaging, SiMultaneous Acquisition of Spatial Harmonics 1, 2, or SMASH, was published in 1997. This SMASH uses linear combinations of component coil signals to emulate directly the effects of encoding gradients, and thereby to reduce the number of time-consuming gradient steps required for image formation. Sinusoidal spatial modulations, or “spatial harmonics,” are formed by linear combination of measured component coil sensitivities, and the same linear combinations are applied to component coil signals in order to generate shifted composite MR signals which can take the place of omitted gradient steps. Subsequent to the introduction of SMASH, the SENSitivity Encoding 3, or SENSE, technique was developed and was also applied successfully to in vivo imaging. In its most common implementations for regularly sampled MR signal data, the SENSE reconstruction operates on component coil images following Fourier transformation, using measured sensitivity calibration information to “unfold” the Nyquist-aliased images which result from omission of gradient steps. In the past few years, a wide range of imaging applications have been explored using

Medical Imaging 2009: Physics of Medical Imaging, edited by Ehsan Samei, Jiang Hsieh,Proc. of SPIE Vol. 7258, 72580Z · © 2009 SPIE

CCC code: 1605-7422/09/$18 · doi: 10.1117/12.812440

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SMASH and SENSE. Meanwhile, various alternative parallel MRI techniques are beginning to be developed. These new techniques in general tend to resemble either the SMASH or the SENSE approach.

One requirement that different parallel imaging techniques have in common is the need to determine spatial sensitivity information for distinct coil array elements. In the SENSE technique, this is usually done by post-processing low-resolution calibration images, e.g. via polynomial fitting 3. This post-processing introduces a possible source of error into the reconstruction. Another important factor in parallel imaging is noise amplification in the reconstruction, which results from poor conditioning of the encoding matrix inversion produced by the spatially correlated coil sensitivity profiles. Generally, regularization approaches, such as Tikhonov 4 and SVD-based methods, are applied to reduce SNR loss, at the price of introducing residual aliasing. In this work, we present a regularization approach using raw images sometimes referred to as “in vivo coil sensitivities”. We demonstrate that this approach can removes noise and preserve image integrity more efficiently that regularization with post-processed sensitivities.

2. METHODOLOGY 2.1 Traditional SENSE acquisition and coil sensitivity estimation

Cartesian SENSitivity Encoding (SENSE) acquisition with regular undersampling 3 can be modeled as a matrix multiplication:

S C X= , (1)

where S is the coil signal vector, representing the Fourier transformation of the acquired signal with undersampled k-

space (i.e. aliased images for each coil), X is the target unfolded image, and C is an encoding matrix composed of complex coil sensitivities at each set of aliased positions in the target image. Here we have used a bra-ket formulation 5 to indicate vector quantities. Fig. 1 illustrates a simulated noise-free SENSE acquisition using a 4-channel array with acceleration factor of 4 using perfect coil sensitivity, with coil signal S (aliased image) for each channel on the left,

corresponding coil sensitivity C for each channel in the middle, and corresponding target unfolded image X on the right.

A common solution to estimation of coil sensitivities 6 has been to subject low-resolution calibration images to various stages of filtering and smoothing, e.g. via local polynomial fitting. This calibration step can introduce a possible source of error into the parallel MRI reconstruction, as it is difficult to ensure that the patient and coil array will be in the same positions during both the calibration scans and the accelerated data acquisitions. Misregistrations or inconsistencies between the calibrated and the “true” sensitivities result in artifacts in the reconstructed images. This is particularly true of real-time imaging, for which motion of the coil array, as well as motion of the anatomy during data acquisition, can render a fixed sensitivity reference inappropriate for the desired reconstruction. The need for calibration scans can be eliminated by auto-calibrating techniques, e.g. AUTO-SMASH 6, by acquiring a small number of extra lines at the center of k-space. Nevertheless, polynomial fitting based approaches for estimating coil sensitivities still may introduce some errors in estimation and may not be appropriate if substantial low-signal anatomical regions (such as lung) exist in the FOV.

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Fig. 1. A simulated noise-free SENSE acquisition using a 4-channel array with acceleration factor of 4 with perfect coil

sensitivities.

2.2 Traditional SENSE reconstruction and regularization

Cartesian SENSE reconstruction with regular undersampling described by equation (1) can be reduced to a straightforward matrix inversion problem, with

invX C S= . (2)

C can be factorized using a singular value decomposition (SVD) as C = USV+, where U and V are unitary matrices and S as a diagonal matrix with singular values. The inverse matrix, or reconstruction matrix, can be written as

11k kinv k k s uC VS U v −− += = Σ , (3)

where kv and ku are columns and rows of the V and U + matrices, respectively, ks is the k-th diagonal element of the diagonal S matrix, and the sum over k runs from 1 to the acceleration factor.

If the matrix C is well conditioned, the inversion can be achieved with minimal amplification of noise. While the encoding matrix can still be inverted even if it is nearly singular, in this ill-conditioned case, small noise perturbations in the measured data can produce large variations in the full-FOV reconstruction. This effect causes noise amplifications in regions of the image where the encoding matrix is ill-conditioned. At high acceleration factors, the encoding matrix C tends to be dominated by several singular values and becomes ill-conditioned, which may result in large noise amplification factor,(or g-factor) for the reconstructed images. Fig. 2 shows a comparison of an unaccelerated reconstruction with SENSE reconstruction with acceleration factor of 4x1 when noise presents. Because of noise amplification (with g-factor map showing on the right panel of Fig. 2), SENSE reconstructed image suffers from noise

Coil sensitivities Coil signal Unfolded image

Coil 1

Coil 2

Coil 3

Coil 4

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amplification due to the ill-conditioning at the center. All images were acquired on a 1.5-T GE Excite HD-x scanner (GE Healthcare, Waukesha, WI). 2D axial images of the brain were acquired using a head coil array with 8 elements circumferentially distributed. Brain images were acquired with T1-weighted contrast by a spin echo sequence with a single signal average. Acquisition parameters included: 22-cm FOV, 5-mm slice thickness, 256×256 image matrix, ±15.63-kHz acquisition bandwidth, TR/TE for T1-w of 500/20 ms and for T2-w of 2500/85 ms. The coil sensitivity was masked to the region of the brain as suggested by 3.

Fig. 2. An unaccelerated T1-weight brain image (left) and SENSE reconstruction with acceleration factor 4x1 (middle) with

presence of noise. Corresponding g-factor map is shown on the right.

To regularize the noise amplification due to ill-conditioning of the encoding matrix, several regularization approaches have been proposed. The most popular method, Tikhonov regularization 4, provides a framework to stabilize the inversion by solving following energy minimization problem:

( )( )22 2arg min refregX

X X S X XC Aλ= − + − , (4)

with 2λ as the regularization parameter, A as a transform matrix, and Xref as a reference image. It can be shown that with Tikhonov regularization the reconstruction matrix takes the following form if A is identity matrix and a null reference images is chosen, with

1_

12

kinv Tikhonov k k kk

uC VS U v ssλ− +

−

= =⎛ ⎞

+⎜ ⎟⎝ ⎠

Σ . (5)

Large 2λ will improve the noise suppression, but at the same time will introduce reconstruction errors in terms of residual aliasing.

2.3 Regularization using in vivo coil sensitivities

To simultaneously address the coil sensitivity calibration and regularization during reconstruction, we propose using in vivo coil sensitivities 6, 7 combined with an SVD-based regularization approach to achieve reconstructions that are quite robust to noise and residual aliasing in comparison to traditional approaches.

“In vivo coil sensitivities” refers to the direct use of low resolution anatomical images as the coil sensitivities for reconstruction. In this case, the nominal in vivo encoding matrix C is actually a multiplication of the underlying signal and the actual coil sensitivity at each pixel, i.e.:

CC = ρ (6)

g-factor (avg=4.6)

0

2

4

6

8

10

Unaccelerated image SENSE reconstruction with acceleration 4x1

Corresponding g-factor map

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In other words, the true coil sensitivity is modulated by the reference image. In self-calibrating parallel imaging techniques, this low-resolution reference image is obtained without any additional scan from a small portion of the center of k-space sampled during the otherwise accelerated acquisition. Using this new coil sensitivity matrix to perform SENSE reconstruction:

1 1inv invX S S XC C− −= = =ρ ρ (7)

In this case, invC contains information both from the coil sensitivity and from the underlying magnetization density. As a result, after SVD, the signal is more concentrated into a few singular vectors than it is during a standard reconstruction with pure coil sensitivities. Fig. 3 illustrates this point via a mathematical simulation. A SENSE acquisition with acceleration factor of 4 was simulated with a T1 weighted brain image. The sub-image associated with each singular value/vector k in equation (3) is shown. Both a traditional coil-sensitivity based approach and the proposed in vivo sensitivity based approach were tested. The results of Fig. 3 verify the fact that using in vivo sensitivity will result in better separation of signal and noise into different channels of the SVD than a pure coil-sensitivity based approach. This property reduces the residual aliasing artifact when applying regularization.

Fig. 3. Sub-images reconstructed from each singular value/singular vector by using (a) in vivo coil sensitivities and (b) pure

coil sensitivities.

For regularization, we used a shifted-SVD (SSVD) method 8. Shifted-SVD regularization shifts the spectrum of singular values away from zero by adding a small portion of the largest singular value, which significantly reduces the effect of small singular values (noise) in the inverse while maintaining the effect of large singular values (signal). In bra-ket formulation, SSVD can be expressed as:

( )1_

1kinv SSVD k k k uC VS U v s δ− + −

= = +Σ , (8) where δ is a small portion of the largest singular value.

Fig. 4 shows an example of T1-weighted brain image reconstructed with 8-ch encircling head array with acceleration factor 4x1. Direct SENSE reconstruction, SENSE reconstruction with in vivo SSVD, and SENSE reconstruction with SSVD are shown, with unaccelerated image as the reference. Both regularization methods with SSVD used δ=5%. In vivo SSVD method exhibited good noise reduction and much less residue artifact compared with SSVD using coil sensitivity.

(b)(a)

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Fig. 4. Comparison of SENSE, SENSE with in vivo SSVD, SENSE with SSVD, and unaccelerated images.

3. RESULTS AND DISCUSSIONS 3.1 Clinical application: single breath-hold whole heart 3D acquisition

The feasibility of our proposed method was demonstrated in a whole-heart 3D acquisition within single breathhold with a 4x2 two-dimensional acceleration using a 32-element coil array. A separate fully-sampled low resolution scan was performed for generation of in vivo coil sensitivities.

Fig. 5 shows one slice of the whole-heart 3D reconstruction using SENSE with in vivo coil sensitivities. The shifted-SVD regularization method substantially reduced the noise from the non-regularized reconstruction without visible residual aliasing artifacts.

Fig. 5. Representative axial slice from a SENSE-reconstructed 3D whole-heart acquisition with 4x2 acceleration using in-vivo coil sensitivities. (a) no regularization and (b) shifted-SVD regularization with a shift of 10% of the maximum singular value.

3.2 Sensitivity to SSVD parameter setting

Similar to other regularization methods, in vivo SSVD also has a trade-off between noise reduction and residue aliasing. Fig. 6 illustrates the variation of SSVD and in vivo SSVD reconstruction respective to changes in δ. The artifact levels for both methods are increases with increase in δ. However, in comparison with SSVD, in vivo SSVD is much less sensitive to the parameter selection. Even at high level of threshold, e.g. 10%, the artifact level is still acceptable, whereas SSVD starts to degrade at δ =2%.

Unaccelerated image SENSE (4x1) SENSE + in vivo SSVD SENSE + SSVD

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This insensitivity to shift threshold for in vivo SSVD is expected since it has better separation of signal and noise components by utilizing in vivo sensitivities. This feature is also favorable in clinical application when an absolute “optimal” parameter setting is hard to define.

Fig. 6. SSVD and in vivo SSVD reconstruction with different parameter settings.

3.3 Sensitivity to resolution of reference images

The ability of in vivo SSVD in separating signal and noise components depends on the amount of signal information contained in the in vivo sensitivity. Thus the resolution of the reference images in terms of number of lines acquired for the center of k-space will affect the performance of in vivo SSVD. Fig. 7 illustrates the variation in in vivo SSVD performance with changes in number of reference lines in the center of k-space for a 128x128 image and δ=5%. As expected, the residue artifact reduces with increase in number of reference lines.

Fig. 7. in vivo SSVD reconstruction with various reference lines for a 128x128 image and δ=5%.

4. CONCLUSION This paper presents preliminary results and demonstrates the technical feasibility of regularization of Parallel MRI reconstruction using in vivo coil sensitivities. This new approach not only combines the coil sensitivity calibration and regularization of the inverse problem, but also provides better performance than a traditional two-step scenario in terms of signal-to-noise ratio without adding residual aliasing artifacts. Demonstrated in simulation and clinical application, the proposed in vivo SSVD has much better performance than SSVD without in vivo coil sensitivity. Although the residue artifact would increase with SSVD parameter, the proposed method is much less sensitive to the parameter selection to regular SSVD, which makes it much more robust in clinical applications when optimal parameter settings are hard to define. The performance of in vivo SSVD also depends on the resolution of reference images. Ultimately, this

δ=1%

in vivo SSVD

SSVD

δ=3% δ=5% δ=10% δ=50%

32 lines 48 lines 64 lines 96 lines 128 lines

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approach could be applied to any SENSE-based parallel MRI reconstruction, especially at high acceleration factor and when coil sensitivity profiles are difficult to estimate.

REFERENCES

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