E L S E V I E R Magnetic Resonance Materials in Physics, Biology and Medicine 13 (2002) 158-163

Magnetic Resonance Materials in Physk~, Bioh:,Ry and M ~ i c i ~

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Recent advances in image reconstruction, coil sensitivity calibration, and coil array design for SMASH and generalized

parallel MRI #

Daniel K. Sodickson a,c,,, Charles A. McKenzie b, Michael A. Ohliger c, Ernest N. Yeh ~ Mark D. Price c

Depart, tent of Medichw, C~wa'iovascular Divirion, Beth Israel Deaconess 3,ledical Center and Iqarvard Medical School, One Autumn Street, ,~ifth Floor, Boston, MA 02215, USA

b Department qf Radiology, Beth Israel Deaconess Medical Center and Harvard Medical School, Boston, MA. USA Harvard-MIT DivMon oj" Health Sciences and Technology, Boston, MA, USA

Received 20 August 2001" received in revised form 2 October 2001" accepted 8 October 2001

Abstract

Parallel magnetic resonance imaging (MRI) techniques use spatial information from arrays of radiofrequency (RF) detector coils to accelerate imaging. A number of parallel MRI techniques have been described in recent years, and numerous clinical applications are currently being explored. The advent of practical parallel imaging presents various challenges for image reconstruction and RF system design. Recent advances in tailored SiMultaneous Acquisition of Spatial Harmonics (SMASH) image reconstructions are summarized. These advances enable robust SMASH imaging in arbitrary image planes with a wide range of coil array geometries. A generalized formalism is described which may be used to understand the relations between SMASH and SENSE, to derive typical implementations of each as special cases, and to form. hybrid techniques combining some of the advantages of both. Accurate knowledge of coil sensitivities is crucial for parallel MRI, and errors in calibration represent one of the most common and the most pernicious sources of error in parallel image reconstructions. As one example, motion of the patient and/or the coil array between the sensitivity reference scan and the accelerated acquisition can lead to calibration errors and reconstruction artifacts. Self-calibrating parallel MRI approaches that address this problem by eliminating the need for external sensitivity references are reviewed. The ultimate achievable signal-to-noise ratio (SNR) for parallel MRI studies is closely tied to the geometry and sensitivity patterns of the coil arrays used for spatial encoding. Several parallel imaging array designs that depart from the traditional model of overlapped adjacent loop elements are described, c�9 2002 Elsevier Science B.V. All rights reserved.

Kevwords: Parallel MRI; SMASH; SENSE; RF coil arrays; Rapid imaging

1. Introduction

Parallel magnetic resonance imaging (MRI )use s spa- tial information from an array of radiofrequency (RF) coils to substitute for information normally obtained

* Summary of material presented at the 2001 ISMRM workshop on MRI hardware, Cleveland, OH, USA.

* Corresponding author. Tel.: + 1-617-632-7654; fax: + 1-617-632- 7675.

E-mail ada4"ess: dsodicks@caregroup.harvard.edu (D.K. Sodick- son).

using magnetic field gradients. Acquisition of some port ion of image data in parallel rather than in a traditional sequential order allows acceleration of imag- ing beyond previous physical and physiologic limits associated with gradient switching and RF pulse appli- cation. The first proposals for parallel imaging [1-6] date back to relatively early in the development o f M R multicoil arrays. The modern parallel imaging family includes the SiMultaneous Acquisition of Spatial Har- monics (SMASH) [7] and SENSitivity Encoding (SENSE) [8] techniques, as well as various new ap- proaches currently in development [9-13].

1352-8661/025 - see front matter ~) 2002 Elsevier Science B.V. All rights reserved. PII: S135"~-8661!01)i)0144 -~

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The gains in MR imaging speed and efficiency af- forded by parallel MRI may be used to improve tempo- ral resolution, improve spatial resolution, increase vol- ume coverage per unit time, reduce art ifacts/~om mo- tion or other time-dependent effects, reduce specific absorption rate constraints, improve patient comfort and compliance, etc. In the past few years, a wide range of imaging applications have been explored using SMASH and SENSE. Various methodological and technological challenges will attend the continued de- velopment of high-performance parallel MRI tech- niques for routine clinical use. Several recent developments in SMASH image reconstruction, gener- alized parallel MRI approaches, coil sensitivity calibra- tion, and coil array design are described to follow.

2. SMASH with arbitrary image plane orientations and coil array geometries

The SMASH technique [7,14,15] uses linear combina- tions 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 sig- nals in order to generate shifted composite MR signals which can take the place of omitted gradient steps [7,14,15].

When the SMASH technique was originally pre- sented, the requirement to fit accurate spatial harmon- ics led to the concern that the technique would be compatible only with certain image plane orientations and certain coil array configurations. Recent work has shown that such constraints are no longer applicable when appropriately tailored reconstructions are used [15-17]. In place of pure spatial harmonics, tailored harmonics that share some or all of the spatial profile of the coil sensitivities may be used as targets for fitting, resulting in robust reconstructions in oblique or double oblique image planes. The top row of Fig. 1 compares an unaccelerated reference image (Fig. l a )wi th a two- fold accelerated SMASH image (Fig. lb) of the right coronary artery in a double-oblique plane. Both images were obtained using a navigator-gated T2-prepared 3D TFE sequence [18,19], and the SMASH image was reconstructed using the techniques described in [15]. Accurate reconstructions may even be achieved for image planes perpendicular to the coil array, as shown in the axial abdominal images of Fig. l c and d. Robust SMASH imaging may now be performed for coil arrays departing from the traditional linear design used in many previous implementations of SMASH.

3. Beyond SMASH and SENSE: generalized techniques, hybrid techniques, and strategies for high acceleration

Both the constraints and the areas of applicability of modern implementations of SMASH and SENSE are very similar, and in fact the two reconstruction tech- niques may actually be shown to be special cases of a more general approach [13,20,21]. Each acquired MR signal data point may be viewed as a spatial integration of the true image against the joint modulation functions imposed by applied gradients and component coil sensitivities:

S l ( k ) = j~d3.'~p (.'t')Ct(c,:)exp( - i]t " . x ) =_ j~d3.~cp (-~')Bl(tT,:, ,r)

(1) Here l is the array element number, C,, is the sensitiv-

ity of" coil /, p is the distribution of" transverse magne- tization including spin density and relaxation effects, and k is the wavenumber of" a periodic function result- ing from evolution in frequency- and phase-encoding gradients. If the spatial integral in Eq. (1) is approxi- mated with a discrete sum, following one convenient generalized %rmulation [13], we may write the signal S~. as:

Fig. 1. Unaccelerated reference image (a) and 2-fold accelerated SMASH image (b) of a segment of the right coronary artery in a healthy adult volunteer. These slices were taken from double-oblique 3-D volume acquisitions using a navigator-gated T2-prepared 3-D TFE sequence [18,19]. A six-element array was used for imaging. (c. d) Axial T~-weighted abdominal images (unaccelerated reference (c) and two-told accelerated SMASH (d)) demonstrating a renal cyst (arrows) in an adult volunteer, obtained using a four-element coil array. The anterior-posterior phase-encoding direction was perpen- dicular to the principal axis of the array.

160 D.K. Sodickson et al./Magnetic Resonance Materials in Physics, Biology amt Medicine 13 (2002) 158-163

x

or, in matrix form_:

S ~ B . p (3)

B is an encoding matrix, which may be generated from the continuous l:orm in Eq. (1) in various ways. The central step in a parallel image reconstruction now involves inverting the matrix B to recover the magne- tization distribution p.

SMASH may be shown to correspond to inverting one submatrix of a particular representation of the encoding matrix, whereas the image-domain SENSE reconstruction corresponds to inversion of the full ma- trix in another representation [13]. The approximation inherent in SMASH, which entails restricting the com- position of missing/,:-space lines to contributions from nearby lines, can limit overcorrection for noise or error in sensitivity references, can reduce sensitivity calibra- tion requirements and allow faster reconstructions, and can simplify relaxation calculations or other computa- tions that rely on the temporal ordering of k-space data. On the other hand, use of the full encoding matrix information, and provision for complete mixing be- tween measured /,,-space lines, is the most accurate approach when the sensitivities are known with high accuracy and precision. The choice of a particular reconstruction strategy may be dictated by which strat- egy is most conveniently available, and which produces the best images in any given imaging situation.

The sub-matrix structure of the SMASH reconstruc- tion may be extended to form reconstruction ap- proaches which are hybrids between SMASH and SENSE, and may be made more SMASH-like or SENSE-like depending upon particular needs [13]. Nu- merical conditioning may also be added conveniently into generalized reconstructions, in a manner similar to that used recently in SMASH reconstructions[15]. Nu- merical conditioning prevents amplification of noise and errors in regions gor which sensitivity is poorly characterized, such as regions of low spin density on in vivo sensitivity references [13].

Hybrid techniques, in combination with numerical conditioning, have been used to achieve accelerations approaching the theoretical maximum factor (equal to the number of coils in the array used for imaging). Fig. 2 shows two examples of double-oblique cardiac images at high accelerations using generalized encoding matrix (GEM) reconstructions.

4. Self-calibrating parallel imaging

The calibration of component coil sensitivities is a critical step in all parallel imaging techniques, since coil sensitivity information is used to distinguish different

Fig. 2. (a) Four-fold accelerated short-axis TrueFISP cardiac image using a hybrid reconstruction with a four-element array. (b) Five-fold accelerated coronary MR image using a six-element array and a generalized reconstruction with numerical conditioning [13].

regions of the reconstructed image. For successful par- allel imaging, the choice of sensitivity calibration strat- egy is in many ways more important than the choice of reconstruction strategy. Volume sensitivity scans may be performed en bloc at the beginning of a study, or targeted in vivo scans may be obtained before or after any particular accelerated scan. The reliance upon cali- bration scans separated in time from the accelerated acquisition results in another potential obstacle for accurate parallel imaging. Artifacts in the reconstructed image may arise not only from motion during a scan, but also from motion between the time of the sensitivity reference and the accelerated scan. This effect has been observed in a clinical study of abdominal imaging using parallel MRI with a flexible coil array [22], and sensitiv- ity misalignment is also a potential limitation for future real-time and interactive applications.

Sensitivity miscalibrations may be avoided through the use of sell-calibrating scans, which contain an inter- nal sensitivity reference. The AUTO-SMASH technique [23] is one example of a self-calibrating approach. In AUTO-SMASH, a small number of additional k-space lines are acquired in addition to the usual undersam- pled acquisition, and these additional lines are used to 'train' a SMASH reconstruction without need of an external reference. The recently described variable-den- sity AUTO-SMASH approach [12] expands and im- proves upon the original AUTO-SMASH concept by acquiring a central block of k-space lines which are used in the reconstructed image itself, as well as in the calibration process.

Another self-calibrating variable-density approach has recently been developed [24], in which an internal sensitivity reference is combined with GEM reconstruc- tions as described above. The fully-sarnpled central k-space lines in a variable-density acquisition are ex- tracted and Fourier-transformed to produce a low-reso- lution image, which may then be used as an in vivo sensitivity reference for most existing parallel image reconstruction techniques, including both SMASH-like

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and SENSE-like approaches. The use of all measured k-space positions in a GEM inversion guarantee s that the central lines will be combined in an optimal fashion. This bootstrapping approach also relegates any residual artifacts, when they occur, to the lower-intensity outer lines of k-space, rather than the dominant central lines. As is the case for variable-density AUTO-SMASH, acquisition of some number of fully-sampled central lines may require sacrificing some of the acceleration which would normally be associated with a regularly undersampled parallel acquisition. This effect may be offset somewhat by the use of a greater degree of undersampling in the outer k-space lines, since in prin- ciple, the total achievable acceleration factor is not affected by the distribution of k-space lines. In practice, to limit noise amplification due to large spacings of outer lines, the outer reduction factor is generally cho- sen to be as small as possible in order to achieve any target total acceleration, while still maintaining on the order of 20 fully-sampled central k-space lines for characterization of coil sensitivities. Of course, the extra time associated with acquisition of a separate sensitivity reference is saved.

The real-time short-axis cardiac images in Fig. 3 illustrate the sensitivity calibration problem and demonstrate a successful example of self-calibration. Fig. 3a shows an unaccelerated image acquired i;or use as an external sensitivity reference. Fig. 3b shows a GEM reconstruction of a 1.5-fold accelerated acquisi- tion, using the external sensitivity reference from Fig. 3a. Since the chest wall and the flexible coil array were moving during the real-time data acquisition, the exter- nal sensitivity reference acquired at one time in the cardiac and respiratory cycles was inappropriate for reconstruction of the accelerated data acquired at an- other time. This miscalibration was the source of arti- facts seen in the reconstructed image. Similar artifacts

were seen in all the frames collected during this experi- ment and reconstructed using the external sensitivity reference. Fig. 3c shows a GEM reconstruction of the same 1.5-fold accelerated variable density acquisition using an extracted internal sensitivity reference. Use of the internal sensitivity reference guaranteed consistency with the accelerated data set, and residual aliasing artifacts were eliminated in all of the reconstructed images.

Self-calibrating parallel imaging approaches may be of particular use for future real-time and/or interactive applications, in which the position and orientation of ira, age planes may vary rapidly, and for which it may be inconvenient or inaccurate to register images with a stored reference set.

5. Coil array design for parallel imaging

The ultimate achievable image quality and signal-to- noise ratio (SNR) for parallel MR images is closely tied to the geometry and sensitivity patterns of the coil arrays used for spatial encoding. However, coil arrays optimized for SNR in traditional sequential imaging do not necessarily afford optimal performance for spatial encoding. In particular, effective parallel MRI requires that the sensitivity patterns of different array elements be sufficiently distinct. One consequence of insufficient distinctness is that, due to the poor conditioning of the encoding matrix when sensitivities from different array elements are similar, noise may multiplied out of pro- portion to signal in parallel MRI reconstructions. The pattern of noise amplification, which in general varies spatially across any target image plane, may be calcu- lated for a given set of coil sensitivities.

Coil arrays designed with parallel imaging in mind musL therefore, take into account the potential SNR

Fig. 3. Real-time True-FISP short-axis cardiac images demonstrating the sensitivity calibration problem, and a successful example of sell-calibration. (a) Position of the heart and chest wall at the time of in vivo sensitivity reference acquisition. (b) GEM reconstruction using the external sensitivity reference from (a) for a 1.5-fold accelerated variable density acquisition. Since the chest wall and coil array were moving during the real-time data acquisition, the external sensitivity reference was inappropriate for reconstruction of the accelerated data. resulting in miscalibration artii:acts. (c) GEM reconstruction of the same 1.5-/-.)ld accelerated variable density acquisition using an extracted internal sensitivity reference. Since the internal sensitivity reference is consistent with the accelerated data set. the residual aliasing artifacts seen in (b) have been eliminated.

162 D.K. Sodickson et al . / Magnetic Resommce Materials in Physics, Biology and Medicine 13 (2002) 158-163

Fig. 4. Conductor arrangement l\)r a prototype concentric array designed for parallel MRI [30]. (a-d) Individual array elements. (e) Schematic depiction o1" concentric arrangement of elements. (t3 Photograph of prototype array etched onto PVC substrate, showing elements (a) and (c) on the topmost substrate and elements (b) and (d) beneath.

cost of parallel image reconstruction. Simulations were used to predict the encoding capabilities of initial linear arrays designed explicitly for parallel imaging [25,26]. Optimization of array element positioning using calcu- lated noise amplifications for target image planes has also been described [27].

Several candidate parallel imaging array designs that depart from the traditional model of adjacent loop elements are currently being explored. The planar strip array design [28] uses appropriately terminated con- ducting strips as its basic elements. A two-element quadrature array taking advantage of the relative phases of component coil sensitivities has also been proposed [29]. Fig. 4 illustrates a concentric array de- sign for parallel MRI [30] which is a generalization of the quadrature concept. Concentric arrays like the one shown in the figure use current paths with varying numbers of crosses to ensure that coil sensitivities are substantially independent for a wide range of shared fields of view. Suitable geometrical arrangements also provide automatic inductive decoupling between ele- ments [30]. Simulations show favorable performance of concentric arrays for parallel MRI. In vivo investiga- tions are underway.

The use of tailored array element shapes for im- proved spatial encoding is in some ways reminiscent of gradient coil design, and indeed one might envision a

role for target field approaches in the design of coil arrays for parallel MRI. For example, a birdcage de- sign with sinusoidal sensitivities has been proposed [31], which would render parallel image reconstruction triv- ial, would simplify or eliminate sensitivity calibration, and would result in minimal noise amplification for selected fields of view.

6. Conclusions

Recent advances in image reconstruction algorithms, coil sensitivity calibration procedures, and coil array designs have improved the practical robustness of par- allel MRI techniques. While fundamental questions about the maximum achievable acceleration factor and the 'optimal' RF system design remain unanswered, the use of tailored reconstructions and dedicated coil arrays promises to continue to improve the practical capabili- ties of parallel MRI approaches.

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