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KING’S MEDICAL ENGINEERING CENTRE Philip Batchelor’s Work in MR-Reconstruction Tobias Schaeffter Division of Biomedical Engineering and Imaging Sciences King’s College London Wellcome Trust-EPSRC Medical Engineering Centre Reconstruction of KING’S MEDICAL ENGINEERING CENTRE Undersampled Data Magnetic Resonance in Medicine 54:1273–1280 (2005) Matrix Description of General Motion Correction Applied to Multishot Images P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and D. Larkman3 Motion of an object degrades MR images, as the acquisition is rection method could be used to spatially transform the time-dependent, and thus k-space is inconsistently sampled. ghosted image by the transformation corresponding to a This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot, restrictive assumptions on the type of motions, for example, and repeat this operation for all shots (this is a version that it is a translation or rotation, and use special properties of of the method used in (1)). We could then rebuild an k-space for these transformations. Such methods, however, image by inverse Fourier transform. This method is in cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a general incorrect, as shown by the difference between method is presented that can handle general nonrigid motion translations and rotations. Correcting translation repointwise phase changes in k-space. On the models. A general matrix equation gives the corrupted image quires only Magnetic Resonance in Medicine 63:1247–1257 (2010) from the ideal object. Thus, inversion of this system allows us to other hand, correcting rotations requires knowledge of get the ideal image from the corrupted one. This inversion is the data at neighboring k-space positions and these are possible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier the conjugate gradient method. A faster but empirical inversion rotation theorem, we would need to “synchronize” is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions ulated three-dimensional affine data and two-dimensional pul- such as nonrigid deformations cannot have a simple sation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show tion. All examples are where the object moves 2 2 2 2 Freddy Odille,1* multishot Sergio images Uribe, Philip G. Batchelor, Prieto, Tobias Schaeffter, and that itClaudia is possible to correct complicated motions, inbetween shots. The results indicate that it is now possible to 1 David Atkinson correct for nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical many types of patient motion, although computation times re- description of the problems involved; the motion cormain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on This paper describes an acquisition and reconstruction strategy reduce gradient interference (4,6,7), the magnetohydrodythe space of images. Thus, inversion of this matrix Wiley-Liss, Inc. for cardiac cine MRI that does not require the use of electro- namic effect is difficult to model or correct, especiallyisatof correct the motion’s effects. This approach Key words: motion correction; ghosts; multishot; conjugate should cardiogram or breath holding. The method has similarities with high fields, asinterest, it increases the amplitude static Bon theoretical but with its practical value of depends gradient; auto focus 0 self-gated techniques as information about cardiac and respi- field strength. The use of ECG also requires more patient how easily we can find a solution of the linear system. It ratory motion is derived from the imaging sequence itself; here, Motion of an object can degrade MR images and imposes preparation time. Second, holding be a difficult turns out that with somebreath careful linear can algebra, not only by acquiring the center k-space line at the beginning of each constraints on scan parameters that can in turn compro- task are for wemany able to invert or in infants a generalized but alsodue this patients and maysense, be imperfect segment of a balanced steady-state free precession sequence. mise image quality. The cause the degradation is that can be over donetime efficiently in practice. For this we organs drifting (8). Another issue associated However, the reconstruction step isoffundamentally different: a toinversion the acquisition is time-dependent, the Fourier transuse breath the LSQR algorithm, which is a robust implementaholding is that the physiology is modified comgeneralized reconstruction by inversionand of coupled systems is with form of the image seen during acquisition changes due to tion of the conjugate gradient of the normal equation pared to the “normal” free-breathing state. Therefore, it is used instead of conventional gating. By correcting for nonrigid the deformation of themotion, object.generalized This causes inconsistencies (see (4)).relevant to attempt to capture cardiac functional cardiac and respiratory reconstruction by clinically Magnetic Resonance in Medicine 62:1331–1337 (2009) inversion of coupled systems all acquired data, information in k-space and hence ghosts(GRICS) in the uses image. This leaves of finding what motion actually in the freequestion breathing. whereas gatingmotion rejects data acquiredmethods in certainmake motionassumpstates. Standard correction happened. Different costbeen functions have been designed Several strategies have proposed in order to addressto The method relies onofthe processing analysisthat of the tions on the type motions, for and example, it isk-a these quantify how much anreal-time image has been corrupted (1,tech5, 6). issues, including imaging, self-gated space central or linea data: local and information from a 32-channel translation rotation, use formulas on Fourier niques, We explore functions conjunction with and different combinedcost cardiac and in respiratory gating. cardiac coil is in order to automatically extract eigentransforms toused correct the data (1–3). We assume here that Real-time our motion correction. Optimization of suchcardiac cost functions imaging has been shown to allow imagmodes of both cardiac and respiratory motion. In the GRICS these data are acquired in shots. When the data posi- means repeating the matrix inversion iteratively. However, framework, these eigenmodes are used as driving signals of ing during free breathing (9–11) but implies a comprotions at each shot are known, an empirical motion cor- inverting matrices repeatedly may be prohibitive even if it a motion model. The motion model is defined piecewise, so mise between spatiotemporal resolution and signal-tois practicable onItaisone-off basis. We therefore also invesratio (SNR). possible to combine real-time images that each cardiac phase is reconstructed independently. Results noise tigate the useframes, of the using empirical method described above. from six healthy volunteers, with various slice orientations, show different nonrigid image registration in 1 C. Prieto,1 P.G. Batchelor,1 S.from 1 D. Atkinson, 2 H. Eggers, 3 1 Boubertakh, R. Uribe, The matrix equationfor allows us to find when this approxiimproved image quality comparedUniversity to combined and order Medical Physics & Bioengineering, Collegerespiratory London, London, to compensate respiratory motion, and then proUnited Kingdom. 4 M.S. 5 R.S.2010. 1* show three-dimensional random mation is correct. We cardiacSørensen, gating. Magn Reson Hansen, Med 63:1247–1257, © 20101 and T.S. Razavi, T. Schaeffter duce SNR-enhanced cardiac cine images (12). However, 2 DepartmentInc. of Electrical Engineering, Pontifica Universidad Católica de Wiley-Liss, affine and pulsatile nonrigid motion corrections on simu- Model-Based Reconstruction for Cardiac Cine MRI Without ECG or Breath Holding Liver Whole-Heart Imaging Using Undersampled Radial Phase Encoding (RPE) and Iterative Sensitivity Encoding (SENSE) Reconstruction Chile, Chile. this technique still requires the recording of an ECG sig- lated data and an example of nonrigid correction of in vivo Key words: Artifacts; cardiac imaging; gating; motion correc3 Imaging Sciences, Imperial College London, London, United Kingdom. nal during real-time scanning, and its extension to three data (moving tion; navigators; reconstruction Whole-heart isotropic nonangulated cardiac magnetic reso- phology of thelegs). ventricle and great vessels. This removes *Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengi- dimensions remains challenging. Self-gated techniques can nance (CMR) is becoming an important protocol simplifying neering, University College London, Gower Street, LondoninWCIE 6BT, UK, the need for time-consuming slice planning by a skilled remove need for either ECG (13,14) or breath hold E-mail: [email protected] MRI, since it reduces theheart, need referred of cumbersome planning of operator.the The main problem of whole-heart acquisitions is Dynamic imaging of the to as cine imaging, or their combination (16). Self-gating relies on the in part as abstracts at the 2nd International on ParallelTHEORY angulations. However the acquisition of Workshop whole-heart MRI (15), rather long acquisition times, particularly when acquiring isPresented theZürich, reference MRIMIUA, method forSept. thetimes study of cardiac funcMRI, 2004, andto 2004. of information about cardiac and/or respiratory are prohibitive the London, large fields of view (FOVs) and the extraction large imaging volumes at high motion-corrupted image resolutions.MR Recently, tion patientsdue with heart standard technique, We need a method to handle images GrantinSponsor: EPSRC; Grant failure. Numbers:The GR/S30184, AF/001381; Grant motion from the imaging data. This information was shown high spatial resolution required for depicting small structures Sponsor: Chilean FONDECYT; Grant Number: based Cartesian scanning available on clinical scanners, is a 1030570. segmented balanced fast volumetric weretrospective know acquisitions the motion; this is aon necessity to correct for and vessels. To address this problem, we propose a three- towhen allow or prospective synchronization to Received 14 March 2005; revised 8sequence June 2005; acquired accepted 10during June 2005 using parallel imaging techniques (2,3) were introduced steady-state free precession susunknown motions an optimization method. Some dimensional (3D) acquisition scheme that combines Cartesian either the cardiac orwith respiratory cycle. In particular, the DOI 10.1002/mrm.20656 for this purpose (4). However, the radio frequency (RF) coil pended respiration (1). Images from each cardiac phase are MR techniques, navigators, to find the sampling the9readout direction with InterScience an undersampled radial method Published in online September 2005 in Wiley (www.interscience. in Buehrersuch et al.as(16) uses the allow centralus point in the used in thosebut studies only modest accelerareconstructed by retrospective gating, usingundersampling information array motion directly, then allows also require an algorithm to wiley.com). scheme in the phase-encoding plane. Different Magnetic Resonance in Medicine –949 (2007) k-space, which can be thought of as a 57:939 zero-dimensional from electrocardiogram (ECG) (2).with In an some circumtion factors of 2 to 3 and thus the total scan time is still patterns were investigated in combination iterative sen- 1273 © 2005the Wiley-Liss, Inc. navigator signal; the method in Uribe et al. (15) uses the stances however, the method may suffer from several issues sitivity encoding (SENSE) reconstruction and a 32-channel car- rather long (up to 20 min). With the introduction of new center k-space line (CKL), which can be thought of as a associated with the use ofmaps the ECG breath hold. First, MR scanner technology with up to 32 receive channels and diac coil. Noise amplification wereor calculated to compare (1D) coils, navigator signal, producing prothe of distorted the different patterns iterative theperformance ECG signal is during the using MRI scan dueSENSE to the one-dimensional corresponding receive the signal-to-noise ratioa(SNR) the two-dimensional or three-dimensional image reconstruction. The radial effect phase-encoding (RPE) scheme and was jection magnetohydrodynamic (3), to radiofrequency, can be of potentially increased and further acceleration of onto thebecomes frequency-encoding axis.studies With all these implemented a clinical MR switching scanner and(4,5). tested on phantoms to magnetic on field gradient Although sig- content these protocols feasible. Some have althe self-gating canmagnetic be acquired with and volunteers. Thehave proposed method exhibits better nal healthy processing methods been proposed in order to techniques, ready addressed whole-heartsignals coronary resonance 1 2 2 3 1 image quality even for high acceleration factors (up to 12) in minimal distortion of steady-state during a massively balanced Claudia Prieto, Philip G. Batchelor, D.L.G. Hill, Joseph V. Hajnal, Marcelo Guarini, angiography (MRA) in the a single breathhold using comparison to Cartesian acquisitions. Magn Reson Med 62: steady-state free precession sequence. However, combined 1* parallel imaging (5). Nevertheless, these breathholds are and Pablo2009. Irarrazaval 1331–1337, © 2009 Wiley-Liss, Inc. cardiac and respiratory gating has several limitations: (i) Reconstruction of Undersampled Dynamic Images by Modeling the Motion of Object Elements still quite long and the spatial resolution is relatively low. Key words: whole-heart MRI; radial-Cartesian sampling; itera- it is relatively inefficient as data from undesired respiraIn order to image the complete heart with higher spatial tive SENSE reconstruction; parallel 32 channel coil; tory phases are thrown away; (ii) if respiratory motion is Dynamic MRI is restricted due to University the imaging; timeCollege required to obtain 1 missing adata by exploiting the high acquisition spatiotemporal resolution, free-breathing whole-heart using Centre for Medical Image Computing, London, London, the radical phase enough data to encoding reconstruct the image sequence. Several un- correlation United Kingdom. not reproducible from breathing cycle to another, the of dynamic sequences or from prior informaparallel imaging in twoone directions was proposed (6). How2 dersampled reconstruction techniques have been proposed to tion. Division of Imaging Sciences, King’s College London, London, United Kingefficiency decreases further and residual artifacts ever, although a 32-channel coil(iii) array was used, the may optiCardiac resonance (CMR) has become dom. themagnetic reduce acquisition time. In most ofimaging these techniques the a occur as motion within the acceptance window is not corTraditional approaches operate on aindiscrete k-t space mal phase encoding directions result a moderate accelclinically useful tool in noninvasive imaging of cardiovasGrant sponsor: Engineering and Physical Sciences Research Council; Grant nonacquired data are recovered by modeling the temporal inrected, which imposes a tradeoff between quality and either treatof each frame separately or image consider the eration factor 4. number:diseases. UK EP/E001564. cular The widespread of cardiac howformation asEPSRC varying pixel intensitiesuse represented in MRI, time or in and acquisition efficiency (16). Cartesian *Correspondence to: Freddy Odille, CMIC, Malet Place Engineering Build- temporal information as time-varying pixel intensities repAlternatively undersampled acquisitions for temporal Here proposenature a new of approach that ever, is frequencies. hampered by thewe complex the multiple ing, University College London, Gower Street, London WC1E 6BT, United In thisinwork, propose an alternative strategy that time we or MR in temporal frequencies. Therefore, contrast-enhanced angiography (7) as well as radial recovers missing dataMR through a motion estimation of need the resented two-dimensional (2D) scanning protocols and the Kingdom.the E-mail: [email protected] is more efficient than combined cardiac-respiratory selfacquisitions for whole-heart MRI were proposed to reduce each pixel is considered in a constant position over time. object elements (“obels,” or pieces of tissue) of the image. This Received 9 June 2009; revised 2 October 2009; accepted 6 November 2009. for highly individualized planning procedures. Wholegatingmethods andtime more generally applicable real-time imagmethod assumes that an obel displacement through the se- These (8,9). In keyhole these acquisitions the readout diinclude (8,9),than reduced encoding DOI 10.1002/mrm.22312 heart isotropic nonangulated CMR is becoming an impor- the scan ing, without requiring the use2D of or either theobtain ECG or breath quence has lower bandwidth than(www.interscience.wiley.com). fluctuations in pixel intensiPublished online in Wiley InterScience rection is changed in either 3D to a (RIGR) limited imaging with generalized-series reconstruction tant protocol in simplifying MRI (1). Subsequent reformat- MR ties caused by the number of projections of the imaging volume. particular © 2010 Wiley-Liss, Inc.motion, and thus it can be modeled with 1247 (10), reduced field of view (rFOV) (11), hybridAtechnique ting of any slice of interest can be obtained from the 3D fewer parameters. Preliminary results show that this technique asset of the imaging radial technique is that the point-spread funcdynamic (12), unaliasing by Fourier-encoding volume for reconstruct qualitative (with assessment ofsquare the complex mor- for can effectively root mean (RMS) errors tionoverlaps (PSF) isusing robustthe with respect dimension to undersampling (10). the temporal (UNFOLD) below 4%) cardiac images and joints with undersampling facAliased signal energy will appear only as slight streaking tors of 8 and 4, respectively. Moreover, in the reconstruction (13), sensitivity encoding incorporating temporal filtering artifact and thus increased pseudonoise, whereas under(TSENSE) (14), k-t broad-use linear acquisition speed-up process an approximation of the motion vectors is obtained for 1King’s College London, British Heart Foundation (BHF) Centre, Division of sampling in a Cartesian acquisition will result in severe Reconstruction of KING’S MEDICAL ENGINEERING CENTRE Compressed Sensing Undersampled Data Magnetic Resonance in Medicine 54:1273–1280 (2005) Matrix Description of General Motion Correction Applied to Multishot Images P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and D. Larkman3 Motion of an object degrades MR images, as the acquisition is rection method could be used to spatially transform the time-dependent, and thus k-space is inconsistently sampled. ghosted image by the transformation corresponding to a This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot, restrictive assumptions on the type of motions, for example, and repeat this operation for all shots (this is a version that it is a translation or rotation, and use special properties of of the method used in (1)). We could then rebuild an k-space for these transformations. Such methods, however, image by inverse Fourier transform. This method is in cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a general incorrect, as shown by the difference between method is presented that can handle general nonrigid motion translations and rotations. Correcting translation repointwise phase changes in k-space. On the models. A general matrix equation gives the corrupted image quires only Magnetic Resonance in Medicine 63:1247–1257 (2010) from the ideal object. Thus, inversion of this system allows us to other hand, correcting rotations requires knowledge of get the ideal image from the corrupted one. This inversion is the data at neighboring k-space positions and these are possible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier the conjugate gradient method. A faster but empirical inversion rotation theorem, we would need to “synchronize” is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions ulated three-dimensional affine data and two-dimensional pul- such as nonrigid deformations cannot have a simple sation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show tion. All examples are where the object moves 2 2 2 2 Freddy Odille,1* multishot Sergio images Uribe, Philip G. Batchelor, Prieto, Tobias Schaeffter, and that itClaudia is possible to correct complicated motions, inbetween shots. The results indicate that it is now possible to 1 David Atkinson correct for nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical many types of patient motion, although computation times re- description of the problems involved; the motion cormain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on This paper describes an acquisition and reconstruction strategy reduce gradient interference (4,6,7), the magnetohydrodythe space of images. Thus, inversion of this matrix Wiley-Liss, Inc. for cardiac cine MRI that does not require the use of electro- namic effect is difficult to model or correct, especiallyisatof correct the motion’s effects. This approach Key words: motion correction; ghosts; multishot; conjugate should cardiogram or breath holding. The method has similarities with high fields, asinterest, it increases the amplitude static Bon theoretical but with its practical value of depends gradient; auto focus 0 self-gated techniques as information about cardiac and respi- field strength. The use of ECG also requires more patient how easily we can find a solution of the linear system. It ratory motion is derived from the imaging sequence itself; here, Motion of an object can degrade MR images and imposes preparation time. Second, holding be a difficult turns out that with somebreath careful linear can algebra, not only by acquiring the center k-space line at the beginning of each constraints on scan parameters that can in turn compro- task are for wemany able to invert or in infants a generalized but alsodue this patients and maysense, be imperfect segment of a balanced steady-state free precession sequence. mise image quality. The cause the degradation is that can be over donetime efficiently in practice. For this we organs drifting (8). Another issue associated However, the reconstruction step isoffundamentally different: a toinversion the acquisition is time-dependent, the Fourier transuse breath the LSQR algorithm, which is a robust implementaholding is that the physiology is modified comgeneralized reconstruction by inversionand of coupled systems is with form of the image seen during acquisition changes due to tion of the conjugate gradient of the normal equation pared to the “normal” free-breathing state. Therefore, it is used instead of conventional gating. By correcting for nonrigid the deformation of themotion, object.generalized This causes inconsistencies (see (4)).relevant to attempt to capture cardiac functional cardiac and respiratory reconstruction by clinically Magnetic Resonance in Medicine 62:1331–1337 (2009) inversion of coupled systems all acquired data, information in k-space and hence ghosts(GRICS) in the uses image. This leaves of finding what motion actually in the freequestion breathing. whereas gatingmotion rejects data acquiredmethods in certainmake motionassumpstates. Standard correction happened. Different costbeen functions have been designed Several strategies have proposed in order to addressto The method relies onofthe processing analysisthat of the tions on the type motions, for and example, it isk-a these quantify how much anreal-time image has been corrupted (1,tech5, 6). issues, including imaging, self-gated space central or linea data: local and information from a 32-channel translation rotation, use formulas on Fourier niques, We explore functions conjunction with and different combinedcost cardiac and in respiratory gating. cardiac coil is in order to automatically extract eigentransforms toused correct the data (1–3). We assume here that Real-time our motion correction. Optimization of suchcardiac cost functions imaging has been shown to allow imagmodes of both cardiac and respiratory motion. In the GRICS these data are acquired in shots. When the data posi- means repeating the matrix inversion iteratively. However, framework, these eigenmodes are used as driving signals of ing during free breathing (9–11) but implies a comprotions at each shot are known, an empirical motion cor- inverting matrices repeatedly may be prohibitive even if it a motion model. The motion model is defined piecewise, so mise between spatiotemporal resolution and signal-tois practicable onItaisone-off basis. We therefore also invesratio (SNR). possible to combine real-time images that each cardiac phase is reconstructed independently. Results noise tigate the useframes, of the using empirical method described above. from six healthy volunteers, with various slice orientations, show different nonrigid image registration in 1 C. Prieto,1 P.G. Batchelor,1 S.from 1 D. Atkinson, 2 H. Eggers, 3 1 Boubertakh, R. Uribe, The matrix equationfor allows us to find when this approxiimproved image quality comparedUniversity to combined and order Medical Physics & Bioengineering, Collegerespiratory London, London, to compensate respiratory motion, and then proUnited Kingdom. 4 M.S. 5 R.S.2010. 1* show three-dimensional random mation is correct. We cardiacSørensen, gating. Magn Reson Hansen, Med 63:1247–1257, © 20101 and T.S. Razavi, T. Schaeffter duce SNR-enhanced cardiac cine images (12). However, 2 DepartmentInc. of Electrical Engineering, Pontifica Universidad Católica de Wiley-Liss, affine and pulsatile nonrigid motion corrections on simu- Model-Based Reconstruction for Cardiac Cine MRI Without ECG or Breath Holding Liver Whole-Heart Imaging Using Undersampled Radial Phase Encoding (RPE) and Iterative Sensitivity Encoding (SENSE) Reconstruction Chile, Chile. this technique still requires the recording of an ECG sig- lated data and an example of nonrigid correction of in vivo Key words: Artifacts; cardiac imaging; gating; motion correc3 Imaging Sciences, Imperial College London, London, United Kingdom. nal during real-time scanning, and its extension to three data (moving tion; navigators; reconstruction Whole-heart isotropic nonangulated cardiac magnetic reso- phology of thelegs). ventricle and great vessels. This removes *Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengi- dimensions remains challenging. Self-gated techniques can nance (CMR) is becoming an important protocol simplifying neering, University College London, Gower Street, LondoninWCIE 6BT, UK, the need for time-consuming slice planning by a skilled remove need for either ECG (13,14) or breath hold E-mail: [email protected] MRI, since it reduces theheart, need referred of cumbersome planning of operator.the The main problem of whole-heart acquisitions is Dynamic imaging of the to as cine imaging, or their combination (16). Self-gating relies on the in part as abstracts at the 2nd International on ParallelTHEORY angulations. However the acquisition of Workshop whole-heart MRI (15), rather long acquisition times, particularly when acquiring isPresented theZürich, reference MRIMIUA, method forSept. thetimes study of cardiac funcMRI, 2004, andto 2004. of information about cardiac and/or respiratory are prohibitive the London, large fields of view (FOVs) and the extraction large imaging volumes at high motion-corrupted image resolutions.MR Recently, tion patientsdue with heart standard technique, We need a method to handle images GrantinSponsor: EPSRC; Grant failure. Numbers:The GR/S30184, AF/001381; Grant motion from the imaging data. This information was shown high spatial resolution required for depicting small structures Sponsor: Chilean FONDECYT; Grant Number: based Cartesian scanning available on clinical scanners, is a 1030570. segmented balanced fast volumetric weretrospective know acquisitions the motion; this is aon necessity to correct for and vessels. To address this problem, we propose a three- towhen allow or prospective synchronization to Received 14 March 2005; revised 8sequence June 2005; acquired accepted 10during June 2005 using parallel imaging techniques (2,3) were introduced steady-state free precession susunknown motions an optimization method. Some dimensional (3D) acquisition scheme that combines Cartesian either the cardiac orwith respiratory cycle. In particular, the DOI 10.1002/mrm.20656 for this purpose (4). However, the radio frequency (RF) coil pended respiration (1). Images from each cardiac phase are MR techniques, navigators, to find the sampling the9readout direction with InterScience an undersampled radial method Published in online September 2005 in Wiley (www.interscience. in Buehrersuch et al.as(16) uses the allow centralus point in the used in thosebut studies only modest accelerareconstructed by retrospective gating, usingundersampling information array motion directly, then allows also require an algorithm to wiley.com). scheme in the phase-encoding plane. Different Magnetic Resonance in Medicine –949 (2007) k-space, which can be thought of as a 57:939 zero-dimensional from electrocardiogram (ECG) (2).with In an some circumtion factors of 2 to 3 and thus the total scan time is still patterns were investigated in combination iterative sen- 1273 © 2005the Wiley-Liss, Inc. navigator signal; the method in Uribe et al. (15) uses the stances however, the method may suffer from several issues sitivity encoding (SENSE) reconstruction and a 32-channel car- rather long (up to 20 min). With the introduction of new center k-space line (CKL), which can be thought of as a associated with the use ofmaps the ECG breath hold. First, MR scanner technology with up to 32 receive channels and diac coil. Noise amplification wereor calculated to compare (1D) coils, navigator signal, producing prothe of distorted the different patterns iterative theperformance ECG signal is during the using MRI scan dueSENSE to the one-dimensional corresponding receive the signal-to-noise ratioa(SNR) the two-dimensional or three-dimensional image reconstruction. The radial effect phase-encoding (RPE) scheme and was jection magnetohydrodynamic (3), to radiofrequency, can be of potentially increased and further acceleration of onto thebecomes frequency-encoding axis.studies With all these implemented a clinical MR switching scanner and(4,5). tested on phantoms to magnetic on field gradient Although sig- content these protocols feasible. Some have althe self-gating canmagnetic be acquired with and volunteers. Thehave proposed method exhibits better nal healthy processing methods been proposed in order to techniques, ready addressed whole-heartsignals coronary resonance 1 2 2 3 1 image quality even for high acceleration factors (up to 12) in minimal distortion of steady-state during a massively balanced Claudia Prieto, Philip G. Batchelor, D.L.G. Hill, Joseph V. Hajnal, Marcelo Guarini, angiography (MRA) in the a single breathhold using comparison to Cartesian acquisitions. Magn Reson Med 62: steady-state free precession sequence. However, combined 1* parallel imaging (5). Nevertheless, these breathholds are and Pablo2009. Irarrazaval 1331–1337, © 2009 Wiley-Liss, Inc. cardiac and respiratory gating has several limitations: (i) Reconstruction of Undersampled Dynamic Images by Modeling the Motion of Object Elements still quite long and the spatial resolution is relatively low. Key words: whole-heart MRI; radial-Cartesian sampling; itera- it is relatively inefficient as data from undesired respiraIn order to image the complete heart with higher spatial tive SENSE reconstruction; parallel 32 channel coil; tory phases are thrown away; (ii) if respiratory motion is Dynamic MRI is restricted due to University the imaging; timeCollege required to obtain 1 missing adata by exploiting the high acquisition spatiotemporal resolution, free-breathing whole-heart using Centre for Medical Image Computing, London, London, the radical phase enough data to encoding reconstruct the image sequence. Several unUnited Kingdom. not reproducible from breathing cycle to another, the correlation of dynamic sequences or from prior informaparallel imaging in twoone directions was proposed (6). How- 2 dersampled reconstruction techniques have been proposed to tion. Division of Imaging Sciences, King’s College London, London, United Kingefficiency decreases further and residual artifacts ever, although a 32-channel coil(iii) array was used, the may optiCardiac resonance (CMR) has become dom. themagnetic reduce acquisition time. In most ofimaging these techniques the a occur as motion within the acceptance window is not corTraditional approaches operate on aindiscrete k-t space mal phase encoding directions result a moderate accelclinically useful tool in imaging cardiovasGrant sponsor: Engineering andnoninvasive Physical Research Council; Grant nonacquired data are recovered by Sciences modeling the of temporal inrected, which imposes a tradeoff between image quality and either treat each frame separately or consider the eration factor of 4. number: UK EPSRC EP/E001564. cular diseases. Thepixel widespread of cardiac howformation as varying intensitiesuse represented in MRI, time or in and acquisition efficiency (16). *Correspondence to: Freddy Odille, CMIC, Malet Place Engineering Build- temporal information as time-varying pixel intensities repAlternatively undersampled Cartesian acquisitions for temporal Here proposenature a new of approach that ever, is frequencies. hampered by thewe complex the multiple ing, University College London, Gower Street, London WC1E 6BT, United In thisinwork, propose an alternative strategy that time we or MR in temporal frequencies. Therefore, contrast-enhanced angiography (7) as well as radial recovers missing dataMR through a motion estimation of need the resented two-dimensional (2D) scanning protocols and the Kingdom.the E-mail: [email protected] is more efficient than combined cardiac-respiratory selfacquisitions for whole-heart MRI were proposed to reduce each pixel is considered in a constant position over time. object elements (“obels,” or pieces of tissue) of the image. This Received 9 June 2009; revised 2 October 2009; accepted 6 November 2009. for highly individualized planning procedures. Wholegatingmethods andtime more generally applicable real-time imagmethod assumes that an obel displacement through the se- These (8,9). In keyhole these acquisitions the readout diinclude (8,9),than reduced encoding DOI 10.1002/mrm.22312 heart isotropic nonangulated CMR is becoming an impor- the scan ing, without requiring the use2D of or either theobtain ECG or breath quence has lower bandwidth than(www.interscience.wiley.com). fluctuations in pixel intensiPublished online in Wiley InterScience rection is changed in either 3D to a (RIGR) limited imaging with generalized-series reconstruction tant protocol in simplifying MRI (1). Subsequent reformat- MR ties caused by the number of projections of the imaging volume. particular © 2010 Wiley-Liss, Inc.motion, and thus it can be modeled with 1247 (10), reduced field of view (rFOV) (11), hybridAtechnique ting of any slice of interest can be obtained from the 3D fewer parameters. Preliminary results show that this technique asset of the imaging radial technique is that the point-spread funcdynamic (12), unaliasing by Fourier-encoding volume for reconstruct qualitative (with assessment ofsquare the complex mor- for can effectively root mean (RMS) errors tionoverlaps (PSF) isusing robustthe with respect dimension to undersampling (10). the temporal (UNFOLD) below 4%) cardiac images and joints with undersampling facAliased signal energy will appear only as slight streaking tors of 8 and 4, respectively. Moreover, in the reconstruction (13), sensitivity encoding incorporating temporal filtering artifact and thus increased pseudonoise, whereas under(TSENSE) (14), k-t broad-use linear acquisition speed-up process an approximation of the motion vectors is obtained for 1King’s College London, British Heart Foundation (BHF) Centre, Division of sampling in a Cartesian acquisition will result in severe IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY Phys. Med. Biol. 56 (2011) N99–N114 doi:10.1088/0031-9155/56/7/N02 NOTE A computationally efficient OMP-based compressed sensing reconstruction for dynamic MRI M Usman1 , C Prieto1 , F Odille2 , D Atkinson2 , T Schaeffter1 and P G Batchelor1 1 King’s College London, Division of Imaging Sciences and Biomedical Engineering, London, UK 2 Centre for Medical Image Computing, University College London, London, UK E-mail: [email protected] Received 12 November 2010, in final form 7 February 2011 Published 2 March 2011 Online at stacks.iop.org/PMB/56/N99 FULL PAPERS Magnetic Resonance in Medicine 000:000–000 (2011) Abstract Compressed sensing (CS) methods in MRI are computationally intensive. Thus, designing novel CS algorithms that can perform faster reconstructions is crucial for everyday applications. We propose a computationally efficient k-t Group Sparse: A Method for Accelerating orthogonal matching pursuit (OMP)-based reconstruction, specifically suited to cardiac MR data. According to the energy distribution of a y–f space Dynamic MRI obtained from a sliding window reconstruction, we label the y–f space as static or dynamic. For staticand y–fP. space images, a computationally efficient masked M. Usman,* C. Prieto, T. Schaeffter, G. Batchelor OMP reconstruction is performed, whereas for dynamic y–f space images, standard OMP reconstruction used.called The‘‘k-t proposed was tested on a Compressed sensing (CS) is a data-reduction technique that is nique sparse’’ method (18) in which the underhas been applied to speed up the numerical acquisition in phantom MRI. How- and sampled data acquisition was doneDepending in the phase encodedynamic two cardiac MR datasets. on the ever, the use of this technique in dynamic MR applications dimension (k-t space) by randomly skipping the field of maximum view composition of thetime imaging data, compared to the standard OMP has been limited in terms of the achievable reducphase encodes for each time frame. The sparsity was tion factor. In general, noise-like artefacts and bad temporal fimethod, reconstruction speedupintroduced factors ranging from 1.5 2.5 areMR achieved. by transforming the to dynamic data using delity are visible in standard CS MRI reconstructions when high reduction factors are used. To increase the maximum a wavelet transform and a Fourier transform along spatial and temporal directions, respectively. Alternatively, achievable reduction factor, additional or prior information can be incorporated in the CS reconstruction. Here, a novel CS Gamper et al. (19) showed that a Fourier transform along FULL PAPER reconstruction method is proposed that exploits the structure the temporal dimension was sufficient to achieve sparMagnetic Resonance in Medicine 000:000–000 (2012) within the sparse representation of a signal by enforcing the sity in x-f space, where x is spatial position and f is temsupport components to be in the form of groups. These groups poral frequency. For low reduction factors (up to 3-fold), act like a constraint in the reconstruction. The information Gamper demonstrated that CS reconstructions exhibit about the support region can be easily obtained from training less error than k-t Broad-use Linear Acquisition Speeddata in dynamic MRI acquisitions. The proposed approach was up Technique (BLAST) reconstructions for highly tested in two-dimensional cardiac cine MRI with both downsampled and undersampled data. Results show that higher dynamic object features. For high reduction factors (>5), high temporal frequency components with low amplitudes acceleration factors (up to 9-fold), with improved spatial and temporal quality, can be obtained with the proposed approach get interspersed with noise in the aliased x-f space and are in comparison to the standard CS reconstructions. Magn not recovered in the CS reconstruction. This leads to noisy C V 2011 Wiley-Liss, Inc. Reson Med 000:000–000, 2011. reconstructions with bad temporal fidelity. Hence, 3 1 * Davidundersampling; Atkinson,2group Muhammad Usman,1sensing; Freddy Odille, Christoph Kolbitsch, improved CS methods for maintaining both reasonable spaKey words: compressed 1 1,4at higher sparsity; l1 minimization;1dynamic MRI tial and temporal1quality of dynamic MR data 1. Introduction In dynamic MRI, the motion of an object is measured by acquiring a series of images at a Motion Corrected Compressed Sensing for high frame rate. However, the time resolution of dynamic MRI is limited by the number of Free-Breathing phase-encoding steps Dynamic that are requiredCardiac for each timeMRI frame. For accelerated acquisition, the data are typically undersampled along the phase encoding and/or time direction, exploiting the correlations in k-space and/or time. Examples include UNFOLD (Madore et al 1999), Keyhole Ghislain Vaillant, Tobias G. Batchelor, andThe Claudia Prieto in these (Vanvaals et al 1993), k–t Schaeffter, BLAST (TsaoPhilip et al 2003), among others. reconstruction reduction factors are very desirable. Several extensions methods a straightforward linear and hence isof very fast. A SENSE few years back, a Dynamic MRI uses applications usually require high formulation spatial and improvements k-t-BLAST/k-t and CS techniand new temporal resolution. speed of MR can Compressed sensing (CS) hasThe beenacquisition demonstrated to‘compressed accelerquessensing’ have been(CS) recently proposed (20–28). be introduced in the MR image reconstruction due data reduction technique called was introduced (Candes 2006, images, however, isbylimited by physical (e.g., gradient ate MRI acquisitions reconstructing sparse images of to One approach improve themovement CS-based MR reconstrucunwanted or to involuntary during acquisiDonoho 2006) and hasphysiological beendata. demonstrated different MR applications (Lustig et al 2007, good qualityand from highly undersampled Motion strength slew rate) and (e.g., during nerve intion. tion can be to exploit cardiac the structure the MR images difin In free-breathing gated of MR acquisitions, MR scans inconsistencies in resultstimulation) constraints (1). thisdata, issue, sev- ferent Jungcanetcause al 2009, JungToetaddress alk-space 2007). According to the CS theory, perfect reconstruction of a sparse representation. example, thea sparse represenk-space profilesFor belonging to specific cardiac ing in reconstruction strong motion artifacts in the reconstructed images. eral techniques have been proposed that phase tation are mayacquired exhibit structure in thebreathing form of the nonzeroor at distinctive positions For CSreconstruct to be usefulMR in these applications, motion correction can images of significant quality from and coefficients occurring inPrinted clusters. dynamic from cardiac ‘‘motion states.’’ The combination of profiles the 0031-9155/11/070099+16$33.00 © 2011 Institute of Physics Engineering in Medicine in theFor UK N99 techniques needacquisitions. to be combined with the undersampled reduced data Following Tsao et al.’s classi- MR data, besides being sparse, the x-f space representareconstruction. Recently, joint motion correction and CS same cardiac phase but different respiratory motion fication (2), these techniques can be classified into those tion tends to be in compact form (29), i.e., the support approaches have been proposed to partially correct for states can result in inconsistencies in k-space, leading to which exploit correlations in the k-space (3–8), in time elements in x-f space intensities aboveInthe noise artifacts in thehaving reconstructed images. addition, effects of motion. However, the main limitation of these motion domain (9–13) or in both k-space and time domains levelunwanted lie together in few Hence, motion can groups. also reduce the incorporating sparsity level approaches is that they can only correct for affine deforma- this (2,14,15). this information could helprepresentation, achieve high reduction factions. In this work, we propose a novel motion corrected CS of MR images in the sparse thus reducing Recently, new data reduction titled torsacceleration in dynamic MRI. framework for afree-breathing dynamic technique cardiac MRI that ‘‘comincor- the factor achievable with CS reconstruction pressed sensing’’ (CS) correction (16,17) has been proposed for CS, the ‘‘block sparse’’ or porates a general motion formulation directly into (7).Recently, Hence, toinbenefit fromconcept the highofacceleration available application in MRI. According to the CScorrect theory,for perfect ‘‘group signals has been introduced in the signal the CS reconstruction. This framework can arbi- from CSsparse’’ methods in these applications, additional flexireconstruction of a signal is possible sampling rates trary affine or nonrigid motion in the CSfrom reconstructed carprocessing literature. This refers to exploiting thewith strucbility is required to combine motion correction the below the Shannon–Nyquist limit,benefiting provided the is ture of sparse signals that have support elements lying in diac images, while simultaneously fromsignal highly CS reconstruction. accelerated acquisition. The transform applicationdomain) of this approach sparse (inMR itself or in some and the groups. Some work has been done on one-dimensional Some approaches to combine CS reconstruction with is measurement demonstrated samples both in simulations andwith in vivo for 2D are obtained an data incoherent signals (such as speech) considering equal or unequal correction techniques have been recently prorespiratory CINE MRI, basis. Forself-gated dynamic free-breathing MRI, Lustig etcardiac al. proposed a CS using tech- motion length groups (30–36). Specifically, these techniques a golden angle radial acquisition. Results show that this posed (7,8). Jung et al. proposed a CS technique ‘‘k-t make a partition of elements within the sparse represenFOCal Underdetermined System Solver (FOCUSS)’’ approach allows for the reconstruction of respiratory motion King’s College London, Division of Imaging Sciences and Biomedical tation into nonoverlapping groups and the group struccorrected CINE Research images Centre with atsimilar to (9–11) that incorporated a motion estimation procedure to Engineering,cardiac NIHR Biomedical Guy’s andquality St Thomas’ ture is enforced in the CS reconstruction via a ‘‘groupFoundation Trust, London, UnitedMagn Kingdom. breath-held acquisitions. Reson Med 000:000–000, predict different cardiac phases from a fully sampled refsparse/block-sparse’’ formulation. Extensive simulations C 2012 Wiley *Correspondence to:Periodicals, Muhammad Inc. Usman, Ph.D., Division of Imaging V 2012. erence cardiac frame. The knowledge of motion between have demonstrated that group sparse methods have an Sciences, The Rayne Institute, 4th Floor, Lambeth Wing, St Thomas’ Hospital, Key words: compressed sensing; undersampling; motion Reconstruction of KING’S MEDICAL ENGINEERING CENTRE Compressed Sensing Undersampled Data Pharmacokinetic Modeling Magnetic Resonance in Medicine 60:1524 –1530 (2008) Magnetic Resonance in Medicine 54:1273–1280 (2005) Matrix Description of General Motion Correction Applied to Multishot Images P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and D. Larkman3 Motion of an object degrades MR images, as the acquisition is rection method could be used to spatially transform the time-dependent, and thus k-space is inconsistently sampled. ghosted image by the transformation corresponding to a This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot, restrictive assumptions on the type of motions, for example, and repeat this operation for all shots (this is a version that it is a translation or rotation, and use special properties of of the method used in (1)). We could then rebuild an k-space for these transformations. Such methods, however, image by inverse Fourier transform. This method is in cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a general incorrect, as shown by the difference between method is presented that can handle general nonrigid motion translations and rotations. Correcting translation repointwise phase changes in k-space. On the models. A general matrix equation gives the corrupted image quires only Magnetic Resonance in Medicine 63:1247–1257 (2010) from the ideal object. Thus, inversion of this system allows us to other hand, correcting rotations requires knowledge of get the ideal image from the corrupted one. This inversion is the data at neighboring k-space positions and these are possible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier the conjugate gradient method. A faster but empirical inversion rotation theorem, we would need to “synchronize” is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions ulated three-dimensional affine data and two-dimensional pul- such as nonrigid deformations cannot have a simple sation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show tion. All examples are where the object moves 2 2 2 2 Freddy Odille,1* multishot Sergio images Uribe, Philip G. Batchelor, Prieto, Tobias Schaeffter, and that itClaudia is possible to correct complicated motions, inbetween shots. The results indicate that it is now possible to 1 David Atkinson correct for nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical many types of patient motion, although computation times re- description of the problems involved; the motion cormain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on This paper describes an acquisition and reconstruction strategy reduce gradient interference (4,6,7), the magnetohydrodythe space of images. Thus, inversion of this matrix Wiley-Liss, Inc. for cardiac cine MRI that does not require the use of electro- namic effect is difficult to model or correct, especiallyisatof correct the motion’s effects. This approach Key words: motion correction; ghosts; multishot; conjugate should cardiogram or breath holding. The method has similarities with high fields, asinterest, it increases the amplitude static Bon theoretical but with its practical value of depends gradient; auto focus 0 self-gated techniques as information about cardiac and respi- field strength. The use of ECG also requires more patient how easily we can find a solution of the linear system. It ratory motion is derived from the imaging sequence itself; here, Motion of an object can degrade MR images and imposes preparation time. Second, holding be a difficult turns out that with somebreath careful linear can algebra, not only by acquiring the center k-space line at the beginning of each constraints on scan parameters that can in turn compro- task are for wemany able to invert or in infants a generalized but alsodue this patients and maysense, be imperfect segment of a balanced steady-state free precession sequence. mise image quality. The cause the degradation is that can be over donetime efficiently in practice. For this we organs drifting (8). Another issue associated However, the reconstruction step isoffundamentally different: a toinversion the acquisition is time-dependent, the Fourier transuse breath the LSQR algorithm, which is a robust implementaholding is that the physiology is modified comgeneralized reconstruction by inversionand of coupled systems is with form of the image seen during acquisition changes due to tion of the conjugate gradient of the normal equation pared to the “normal” free-breathing state. Therefore, it is used instead of conventional gating. By correcting for nonrigid the deformation of themotion, object.generalized This causes inconsistencies (see (4)).relevant to attempt to capture cardiac functional cardiac and respiratory reconstruction by clinically Magnetic Resonance in Medicine 62:1331–1337 (2009) inversion of coupled systems all acquired data, information in k-space and hence ghosts(GRICS) in the uses image. This leaves of finding what motion actually in the freequestion breathing. whereas gatingmotion rejects data acquiredmethods in certainmake motionassumpstates. Standard correction happened. Different costbeen functions have been designed Several strategies have proposed in order to addressto The method relies onofthe processing analysisthat of the tions on the type motions, for and example, it isk-a these quantify how much anreal-time image has been corrupted (1,tech5, 6). issues, including imaging, self-gated space central or linea data: local and information from a 32-channel translation rotation, use formulas on Fourier niques, We explore functions conjunction with and different combinedcost cardiac and in respiratory gating. cardiac coil is in order to automatically extract eigentransforms toused correct the data (1–3). We assume here that Real-time our motion correction. Optimization of suchcardiac cost functions imaging has been shown to allow imagmodes of both cardiac and respiratory motion. In the GRICS these data are acquired in shots. When the data posi- means repeating the matrix inversion iteratively. However, framework, these eigenmodes are used as driving signals of ing during free breathing (9–11) but implies a comprotions at each shot are known, an empirical motion cor- inverting matrices repeatedly may be prohibitive even if it a motion model. The motion model is defined piecewise, so mise between spatiotemporal resolution and signal-tois practicable onItaisone-off basis. We therefore also invesratio (SNR). possible to combine real-time images that each cardiac phase is reconstructed independently. Results noise tigate the useframes, of the using empirical method described above. from six healthy volunteers, with various slice orientations, show different nonrigid image registration in 1 C. Prieto,1 P.G. Batchelor,1 S.from 1 D. Atkinson, 2 H. Eggers, 3 1 Boubertakh, R. Uribe, The matrix equationfor allows us to find when this approxiimproved image quality comparedUniversity to combined and order Medical Physics & Bioengineering, Collegerespiratory London, London, to compensate respiratory motion, and then proUnited Kingdom. 4 M.S. 5 R.S.2010. 1* show three-dimensional random mation is correct. We cardiacSørensen, gating. Magn Reson Hansen, Med 63:1247–1257, © 20101 and T.S. Razavi, T. Schaeffter duce SNR-enhanced cardiac cine images (12). However, 2 DepartmentInc. of Electrical Engineering, Pontifica Universidad Católica de Wiley-Liss, affine and pulsatile nonrigid motion corrections on simu- Model-Based Reconstruction for Cardiac Cine MRI Without ECG or Breath Holding Liver Whole-Heart Imaging Using Undersampled Radial Phase Encoding (RPE) and Iterative Sensitivity Encoding (SENSE) Reconstruction Chile, Chile. this technique still requires the recording of an ECG sig- lated data and an example of nonrigid correction of in vivo Key words: Artifacts; cardiac imaging; gating; motion correc3 Imaging Sciences, Imperial College London, London, United Kingdom. nal during real-time scanning, and its extension to three data (moving tion; navigators; reconstruction Whole-heart isotropic nonangulated cardiac magnetic reso- phology of thelegs). ventricle and great vessels. This removes *Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengi- dimensions remains challenging. Self-gated techniques can nance (CMR) is becoming an important protocol simplifying neering, University College London, Gower Street, LondoninWCIE 6BT, UK, the need for time-consuming slice planning by a skilled remove need for either ECG (13,14) or breath hold E-mail: [email protected] MRI, since it reduces theheart, need referred of cumbersome planning of operator.the The main problem of whole-heart acquisitions is Dynamic imaging of the to as cine imaging, or their combination (16). Self-gating relies on the in part as abstracts at the 2nd International on ParallelTHEORY angulations. However the acquisition of Workshop whole-heart MRI (15), rather long acquisition times, particularly when acquiring isPresented theZürich, reference MRIMIUA, method forSept. thetimes study of cardiac funcMRI, 2004, andto 2004. of information about cardiac and/or respiratory are prohibitive the London, large fields of view (FOVs) and the extraction large imaging volumes at high motion-corrupted image resolutions.MR Recently, tion patientsdue with heart standard technique, We need a method to handle images GrantinSponsor: EPSRC; Grant failure. Numbers:The GR/S30184, AF/001381; Grant motion from the imaging data. This information was shown high spatial resolution required for depicting small structures Sponsor: Chilean FONDECYT; Grant Number: based Cartesian scanning available on clinical scanners, is a 1030570. segmented balanced fast volumetric weretrospective know acquisitions the motion; this is aon necessity to correct for and vessels. To address this problem, we propose a three- towhen allow or prospective synchronization to Received 14 March 2005; revised 8sequence June 2005; acquired accepted 10during June 2005 using parallel imaging techniques (2,3) were introduced steady-state free precession susunknown motions an optimization method. Some dimensional (3D) acquisition scheme that combines Cartesian either the cardiac orwith respiratory cycle. In particular, the DOI 10.1002/mrm.20656 for this purpose (4). However, the radio frequency (RF) coil pended respiration (1). Images from each cardiac phase are MR techniques, navigators, to find the sampling the9readout direction with InterScience an undersampled radial method Published in online September 2005 in Wiley (www.interscience. in Buehrersuch et al.as(16) uses the allow centralus point in the used in thosebut studies only modest accelerareconstructed by retrospective gating, usingundersampling information array motion directly, then allows also require an algorithm to wiley.com). scheme in the phase-encoding plane. Different Magnetic Resonance in Medicine –949 (2007) k-space, which can be thought of as a 57:939 zero-dimensional from electrocardiogram (ECG) (2).with In an some circumtion factors of 2 to 3 and thus the total scan time is still patterns were investigated in combination iterative sen- 1273 © 2005the Wiley-Liss, Inc. navigator signal; the method in Uribe et al. (15) uses the stances however, the method may suffer from several issues sitivity encoding (SENSE) reconstruction and a 32-channel car- rather long (up to 20 min). With the introduction of new center k-space line (CKL), which can be thought of as a associated with the use ofmaps the ECG breath hold. First, MR scanner technology with up to 32 receive channels and diac coil. Noise amplification wereor calculated to compare (1D) coils, navigator signal, producing prothe of distorted the different patterns iterative theperformance ECG signal is during the using MRI scan dueSENSE to the one-dimensional corresponding receive the signal-to-noise ratioa(SNR) the two-dimensional or three-dimensional image reconstruction. The radial effect phase-encoding (RPE) scheme and was jection magnetohydrodynamic (3), to radiofrequency, can be of potentially increased and further acceleration of onto thebecomes frequency-encoding axis.studies With all these implemented a clinical MR switching scanner and(4,5). tested on phantoms to magnetic on field gradient Although sig- content these protocols feasible. Some have althe self-gating canmagnetic be acquired with and volunteers. Thehave proposed method exhibits better nal healthy processing methods been proposed in order to techniques, ready addressed whole-heartsignals coronary resonance 1 2 2 3 1 image quality even for high acceleration factors (up to 12) in minimal distortion of steady-state during a massively balanced Claudia Prieto, Philip G. Batchelor, D.L.G. Hill, Joseph V. Hajnal, Marcelo Guarini, angiography (MRA) in the a single breathhold using comparison to Cartesian acquisitions. Magn Reson Med 62: steady-state free precession sequence. However, combined 1* parallel imaging (5). Nevertheless, these breathholds are and Pablo2009. Irarrazaval 1331–1337, © 2009 Wiley-Liss, Inc. cardiac and respiratory gating has several limitations: (i) Reconstruction of Undersampled Dynamic Images by Modeling the Motion of Object Elements still quite long and the spatial resolution is relatively low. Key words: whole-heart MRI; radial-Cartesian sampling; itera- it is relatively inefficient as data from undesired respiraIn order to image the complete heart with higher spatial tive SENSE reconstruction; parallel 32 channel coil; tory phases are thrown away; (ii) if respiratory motion is Dynamic MRI is restricted due to University the imaging; timeCollege required to obtain 1 missing adata by exploiting the high acquisition spatiotemporal resolution, free-breathing whole-heart using Centre for Medical Image Computing, London, London, the radical phase enough data to encoding reconstruct the image sequence. Several unUnited Kingdom. not reproducible from breathing cycle to another, the correlation of dynamic sequences or from prior informaparallel imaging in twoone directions was proposed (6). How- 2 dersampled reconstruction techniques have been proposed to tion. Division of Imaging Sciences, King’s College London, London, United Kingefficiency decreases further and residual artifacts ever, although a 32-channel coil(iii) array was used, the may optiCardiac resonance (CMR) has become dom. themagnetic reduce acquisition time. In most ofimaging these techniques the a occur as motion within the acceptance window is not corTraditional approaches operate on aindiscrete k-t space mal phase encoding directions result a moderate accelclinically useful tool in imaging cardiovasGrant sponsor: Engineering andnoninvasive Physical Research Council; Grant nonacquired data are recovered by Sciences modeling the of temporal inrected, which imposes a tradeoff between image quality and either treat each frame separately or consider the eration factor of 4. number: UK EPSRC EP/E001564. cular diseases. Thepixel widespread of cardiac howformation as varying intensitiesuse represented in MRI, time or in and acquisition efficiency (16). *Correspondence to: Freddy Odille, CMIC, Malet Place Engineering Build- temporal information as time-varying pixel intensities repAlternatively undersampled Cartesian acquisitions for temporal Here proposenature a new of approach that ever, is frequencies. hampered by thewe complex the multiple ing, University College London, Gower Street, London WC1E 6BT, United In thisinwork, propose an alternative strategy that time we or MR in temporal frequencies. Therefore, contrast-enhanced angiography (7) as well as radial recovers missing dataMR through a motion estimation of need the resented two-dimensional (2D) scanning protocols and the Kingdom.the E-mail: [email protected] is more efficient than combined cardiac-respiratory selfacquisitions for whole-heart MRI were proposed to reduce each pixel is considered in a constant position over time. object elements (“obels,” or pieces of tissue) of the image. This Received 9 June 2009; revised 2 October 2009; accepted 6 November 2009. for highly individualized planning procedures. Wholegatingmethods andtime more generally applicable real-time imagmethod assumes that an obel displacement through the se- These (8,9). In keyhole these acquisitions the readout diinclude (8,9),than reduced encoding DOI 10.1002/mrm.22312 heart isotropic nonangulated CMR is becoming an impor- the scan ing, without requiring the use2D of or either theobtain ECG or breath quence has lower bandwidth than(www.interscience.wiley.com). fluctuations in pixel intensiPublished online in Wiley InterScience rection is changed in either 3D to a (RIGR) limited imaging with generalized-series reconstruction tant protocol in simplifying MRI (1). Subsequent reformat- MR ties caused by the number of projections of the imaging volume. particular © 2010 Wiley-Liss, Inc.motion, and thus it can be modeled with 1247 (10), reduced field of view (rFOV) (11), hybridAtechnique ting of any slice of interest can be obtained from the 3D fewer parameters. Preliminary results show that this technique asset of the imaging radial technique is that the point-spread funcdynamic (12), unaliasing by Fourier-encoding volume for reconstruct qualitative (with assessment ofsquare the complex mor- for can effectively root mean (RMS) errors tionoverlaps (PSF) isusing robustthe with respect dimension to undersampling (10). the temporal (UNFOLD) below 4%) cardiac images and joints with undersampling facAliased signal energy will appear only as slight streaking tors of 8 and 4, respectively. Moreover, in the reconstruction (13), sensitivity encoding incorporating temporal filtering artifact and thus increased pseudonoise, whereas under(TSENSE) (14), k-t broad-use linear acquisition speed-up process an approximation of the motion vectors is obtained for 1King’s College London, British Heart Foundation (BHF) Centre, Division of sampling in a Cartesian acquisition will result in severe IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY Phys. Med. Biol. 56 (2011) N99–N114 doi:10.1088/0031-9155/56/7/N02 NOTE A computationally efficient OMP-based compressed sensing reconstruction for dynamic MRI M Usman1 , C Prieto1 , F Odille2 , D Atkinson2 , T Schaeffter1 and P G Batchelor1 1 King’s College London, Division of Imaging Sciences and Biomedical Engineering, London, UK 2 Centre for Medical Image Computing, University College London, London, UK E-mail: [email protected] Received 12 November 2010, in final form 7 February 2011 Published 2 March 2011 Online at stacks.iop.org/PMB/56/N99 FULL PAPERS Magnetic Resonance in Medicine 000:000–000 (2011) Abstract Compressed sensing (CS) methods in MRI are computationally intensive. Thus, designing novel CS algorithms that can perform faster reconstructions is crucial for everyday applications. We propose a computationally efficient k-t Group Sparse: A Method for Accelerating orthogonal matching pursuit (OMP)-based reconstruction, specifically suited to cardiac MR data. According to the energy distribution of a y–f space Dynamic MRI obtained from a sliding window reconstruction, we label the y–f space as static or dynamic. For staticand y–fP. space images, a computationally efficient masked M. Usman,* C. Prieto, T. Schaeffter, G. Batchelor OMP reconstruction is performed, whereas for dynamic y–f space images, standard OMP reconstruction used.called The‘‘k-t proposed was tested on a Compressed sensing (CS) is a data-reduction technique that is nique sparse’’ method (18) in which the underhas been applied to speed up the numerical acquisition in phantom MRI. How- and sampled data acquisition was doneDepending in the phase encodedynamic two cardiac MR datasets. on the ever, the use of this technique in dynamic MR applications dimension (k-t space) by randomly skipping the field of maximum view composition of thetime imaging data, compared to the standard OMP has been limited in terms of the achievable reducphase encodes for each time frame. The sparsity was tion factor. In general, noise-like artefacts and bad temporal fimethod, reconstruction speedupintroduced factors ranging from 1.5 2.5 areMR achieved. by transforming the to dynamic data using delity are visible in standard CS MRI reconstructions when high reduction factors are used. To increase the maximum a wavelet transform and a Fourier transform along spatial and temporal directions, respectively. Alternatively, achievable reduction factor, additional or prior information can be incorporated in the CS reconstruction. Here, a novel CS Gamper et al. (19) showed that a Fourier transform along FULL PAPER reconstruction method is proposed that exploits the structure the temporal dimension was sufficient to achieve sparMagnetic Resonance in Medicine 000:000–000 (2012) within the sparse representation of a signal by enforcing the sity in x-f space, where x is spatial position and f is temsupport components to be in the form of groups. These groups poral frequency. For low reduction factors (up to 3-fold), act like a constraint in the reconstruction. The information Gamper demonstrated that CS reconstructions exhibit about the support region can be easily obtained from training less error than k-t Broad-use Linear Acquisition Speeddata in dynamic MRI acquisitions. The proposed approach was up Technique (BLAST) reconstructions for highly tested in two-dimensional cardiac cine MRI with both downsampled and undersampled data. Results show that higher dynamic object features. For high reduction factors (>5), high temporal frequency components with low amplitudes acceleration factors (up to 9-fold), with improved spatial and temporal quality, can be obtained with the proposed approach get interspersed with noise in the aliased x-f space and are in comparison to the standard CS reconstructions. Magn not recovered in the CS reconstruction. This leads to noisy C V 2011 Wiley-Liss, Inc. Reson Med 000:000–000, 2011. reconstructions with bad temporal fidelity. Hence, 3 1 * Davidundersampling; Atkinson,2group Muhammad Usman,1sensing; Freddy Odille, Christoph Kolbitsch, improved CS methods for maintaining both reasonable spaKey words: compressed 1 1,4at higher sparsity; l1 minimization;1dynamic MRI tial and temporal1quality of dynamic MR data 1. Introduction In dynamic MRI, the motion of an object is measured by acquiring a series of images at a Motion Corrected Compressed Sensing for high frame rate. However, the time resolution of dynamic MRI is limited by the number of Free-Breathing phase-encoding steps Dynamic that are requiredCardiac for each timeMRI frame. For accelerated acquisition, the data are typically undersampled along the phase encoding and/or time direction, exploiting the correlations in k-space and/or time. Examples include UNFOLD (Madore et al 1999), Keyhole Ghislain Vaillant, Tobias G. Batchelor, andThe Claudia Prieto in these (Vanvaals et al 1993), k–t Schaeffter, BLAST (TsaoPhilip et al 2003), among others. reconstruction reduction factors are very desirable. Several extensions methods a straightforward linear and hence isof very fast. A SENSE few years back, a Dynamic MRI uses applications usually require high formulation spatial and improvements k-t-BLAST/k-t and CS techniand new temporal resolution. speed of MR can Compressed sensing (CS) hasThe beenacquisition demonstrated to‘compressed accelerquessensing’ have been(CS) recently proposed (20–28). be introduced in the MR image reconstruction due data reduction technique called was introduced (Candes 2006, images, however, isbylimited by physical (e.g., gradient ate MRI acquisitions reconstructing sparse images of to One approach improve themovement CS-based MR reconstrucunwanted or to involuntary during acquisiDonoho 2006) and hasphysiological beendata. demonstrated different MR applications (Lustig et al 2007, good qualityand from highly undersampled Motion strength slew rate) and (e.g., during nerve intion. tion can be to exploit cardiac the structure the MR images difin In free-breathing gated of MR acquisitions, MR scans inconsistencies in resultstimulation) constraints (1). thisdata, issue, sev- ferent Jungcanetcause al 2009, JungToetaddress alk-space 2007). According to the CS theory, perfect reconstruction of a sparse representation. example, thea sparse represenk-space profilesFor belonging to specific cardiac ing in reconstruction strong motion artifacts in the reconstructed images. eral techniques have been proposed that phase tation are mayacquired exhibit structure in thebreathing form of the nonzeroor at distinctive positions For CSreconstruct to be usefulMR in these applications, motion correction can images of significant quality from and coefficients occurring inPrinted clusters. dynamic from cardiac ‘‘motion states.’’ The combination of profiles the 0031-9155/11/070099+16$33.00 © 2011 Institute of Physics Engineering in Medicine in theFor UK N99 techniques needacquisitions. to be combined with the undersampled reduced data Following Tsao et al.’s classi- MR data, besides being sparse, the x-f space representareconstruction. Recently, joint motion correction and CS same cardiac phase but different respiratory motion fication (2), these techniques can be classified into those tion tends to be in compact form (29), i.e., the support approaches have been proposed to partially correct for states can result in inconsistencies in k-space, leading to which exploit correlations in the k-space (3–8), in time elements in x-f space intensities aboveInthe noise artifacts in thehaving reconstructed images. addition, effects of motion. However, the main limitation of these motion domain (9–13) or in both k-space and time domains levelunwanted lie together in few Hence, motion can groups. also reduce the incorporating sparsity level approaches is that they can only correct for affine deforma- this (2,14,15). this information could helprepresentation, achieve high reduction factions. In this work, we propose a novel motion corrected CS of MR images in the sparse thus reducing Recently, new data reduction titled torsacceleration in dynamic MRI. framework for afree-breathing dynamic technique cardiac MRI that ‘‘comincor- the factor achievable with CS reconstruction pressed sensing’’ (CS) correction (16,17) has been proposed for CS, the ‘‘block sparse’’ or porates a general motion formulation directly into (7).Recently, Hence, toinbenefit fromconcept the highofacceleration available application in MRI. According to the CScorrect theory,for perfect ‘‘group signals has been introduced in the signal the CS reconstruction. This framework can arbi- from CSsparse’’ methods in these applications, additional flexireconstruction of a signal is possible sampling rates trary affine or nonrigid motion in the CSfrom reconstructed carprocessing literature. This refers to exploiting thewith strucbility is required to combine motion correction the below the Shannon–Nyquist limit,benefiting provided the is ture of sparse signals that have support elements lying in diac images, while simultaneously fromsignal highly CS reconstruction. accelerated acquisition. The transform applicationdomain) of this approach sparse (inMR itself or in some and the groups. Some work has been done on one-dimensional Some approaches to combine CS reconstruction with is measurement demonstrated samples both in simulations andwith in vivo for 2D are obtained an data incoherent signals (such as speech) considering equal or unequal correction techniques have been recently prorespiratory CINE MRI, basis. Forself-gated dynamic free-breathing MRI, Lustig etcardiac al. proposed a CS using tech- motion length groups (30–36). Specifically, these techniques a golden angle radial acquisition. Results show that this posed (7,8). Jung et al. proposed a CS technique ‘‘k-t make a partition of elements within the sparse represenFOCal Underdetermined System Solver (FOCUSS)’’ approach allows for the reconstruction of respiratory motion King’s College London, Division of Imaging Sciences and Biomedical tation into nonoverlapping groups and the group struccorrected CINE Research images Centre with atsimilar to (9–11) that incorporated a motion estimation procedure to Engineering,cardiac NIHR Biomedical Guy’s andquality St Thomas’ ture is enforced in the CS reconstruction via a ‘‘groupFoundation Trust, London, UnitedMagn Kingdom. breath-held acquisitions. Reson Med 000:000–000, predict different cardiac phases from a fully sampled refsparse/block-sparse’’ formulation. Extensive simulations C 2012 Wiley *Correspondence to:Periodicals, Muhammad Inc. Usman, Ph.D., Division of Imaging V 2012. erence cardiac frame. The knowledge of motion between have demonstrated that group sparse methods have an Sciences, The Rayne Institute, 4th Floor, Lambeth Wing, St Thomas’ Hospital, Key words: compressed sensing; undersampling; motion Pharmacokinetic Modeling of Delayed Gadolinium Enhancement in the Myocardium Benjamin R. Knowles,1 Philip G. Batchelor,1 Victoria Parish,1 Matthew Ginks,1 Sven Plein,2 Reza Razavi,1 and Tobias Schaeffter1* Delayed contrast-enhanced magnetic resonance imaging (DCE-MRI) provides prognostic information by delineating regions of myocardial scar. The mechanism of this delayed enhancement in myocardial infarctions (MIs) is hypothesized to result from altered kinetics and changes in the volumes of distribution in the myocardium. Pharmacokinetic models with two and three compartments were fitted to the concentrationtime curves of dynamic contrast-enhanced MRI data obtained from five patients with known MI. Furthermore, the parameter stability was investigated in simulations for the two different models. The transfer constants and volumes of distribution showed a good correlation with imaging findings on early and delayed contrast-enhanced MRI. The two compartment model showed higher parameter stability. The three compartment model allows a more in-depth quantification of myocardial scarring. These models have the potential to improve the diagnosis of myocardial pathologies involving scar, with differing kinetics and volumes of distribution such as infarction or cardiomyopathy. Magn Reson Med 60:1524 –1530, 2008. © 2008 Wiley-Liss, Inc. Key words: pharmacokinetics; delayed contrast enhancement; viability; cardiac MRI; infarction pothesized to result from alterations in wash-in/wash-out kinetics (4) and volume of distribution (5). These parameters are likely to be different in areas with different tissue characteristics. After an acute MI, MRI obtained early after contrast agent administration often lack enhancement at the region of microvascular obstruction (MVO), whereas areas with fibrosis and necrosis are visible as areas of high signal on images obtained at later time points. Such DCE is not specific for MI and can be observed after cardiac interventions and in many other cardiac diseases, e.g., cardiomyopathy and myocarditis, making diagnosis sometimes difficult. Recently, a combination of early and delayed contrast-enhancement MRI has been proposed to differentiate between MI and myocarditis (6), indicating that the pharmacokinetics of the contrast agent retention provides additional information about the underlying pathology. Here we propose the application of pharmacokinetic models for DCE MRI, thus providing a more quantitative approach to the diagnosis of myocardial pathologies. In particular, we have derived compartment models for delayed enhancement that are similar to kinetic models used in positron emission tomography (PET) (7). Myocardial ischemia and myocardial infarction (MI), consequences of coronary artery disease (CAD), are the leading cause of death and highest medical care expense in the THEORY United States and Europe (1). The detection and evaluation of the myocardium damaged during ischemia is of Background and Perfusion Models vital importance for the treatment and prognosis of pa- Pharmacokinetic modeling in MRI is concerned with modFULL PAPER tients with ventricular dysfunction. Other conditions such eling the time course of changes in the concentration of a Magnetic Resonance in Medicine 000:000–000 (2012) as different types of cardiomyopathy also lead to ventric- gadolinium-based contrast agent in a specific tissue of ular failure. The use of magnetic resonance imaging (MRI) interest, (Ct(t)). There are various first-pass perfusion modfor diagnosis of different causes of ventricular failure and els in existence, such as those from Tofts and Kermode (8) treatment monitoring is expanding. In particular, delayed or Larsson et al. (9), and model-independent techniques contrast-enhanced (DCE) cardiac MRI, which was first de- such as in Jerosch-Herold et al. (10). The basis of model scribed more than 10 years ago (2) is becoming the stan- derivation begins by assuming the tissue can be simplified dard for the evaluation of the different patterns of myocar- into compartments, through which the passage of contrast dial scar seen in MI and cardiomyopathies. Localization of agent can be modeled. These models assume two compartscar is performed by the administration of a gadolinium ments, consisting of a blood plasma volume and the extracontrast agent. Retention of contrast agent occurs in areas cellular extravascular space (EES), with respective fracof scarring or fibrosis, and these areas appear as an area of tional volumes of vp and ve, and respective concentrations high signal intensity (3) due Tissue uptake of contrast 1 to the T1 shortening effect1of of contrast agent Cp(t) and Ce(t). * Amedeo Chiribiri, Niloufar L. T. F. Hautvast,2 Masaki Ishida,1 the contrast Zarinabad, agent. The mechanism of DCE in MIs is hy-Gilion agent occurs across a permeable barrier between the blood 1 1 Andreas Schuster,1 Zoran Cvetkovic,3 Philipplasma G. Batchelor, Eike Nagel volume and and the EES. How easily contrast agents can move between compartments is dependent on the 1King’s College London, Division of Imaging Sciences, London, United Kingparameter known as the transfer constant, Ktrans (11). This The purpose of this study is to enable high spatial resolution parameter dom. techniques, including Doppler catheterization andcomcorois dependent on the permeability of the 2voxel-wise analysis myocardial perfusion Academic Unitquantitative of Cardiovascular Medicine,ofUniversity of Leeds, Leeds Gen-in nary sinus thermo areother available for measuring partment barrier in alldilution, conditions than when there is eral Infirmary, Leeds, United Kingdom. dynamic contrast-enhanced cardiovascular MR, in particular by myocardial blood flow in humans. These methvery low blood flow to (MBF) the tissue. Concentration-time *Correspondence to: Tobias Schaeffter, King’s College London, BHF Centre, finding the most favorable quantification algorithm in this context. ods, which variationsinofa indicator dilution methods, curves can bearemeasured tissue from dynamic MR Division of Imaging Science, NHR Biomedical Research Centre at Guy’s and Four deconvolution algorithms—Fermi function modeling, deconSt. Thomas NHS Trust Foundation, London, United Kingdom SE1 7EH. Eare invasive and can only assess average perfusion of volution using B-spline basis, deconvolution using exponential ba- images, and then the model is fitted to these curves. Thus mail: [email protected] whole coronary artery territories. Amongst noninvasive sis, and22 autoregressive moving modeling tested the shape of each curve will be dependent on the transfer Received January 2008; revised 26 average June 2008; accepted —were 7 July 2008. imaging techniques, emission tomography between the bloodpositron volume and the EES (Ktrans), and(PET) the to calculate voxel-wise perfusion estimates. The algorithms were rate DOI 10.1002/mrm.21767 is currently regarded as each a gold standard forMost the quantifivolume size of compartment. imporPublished online Wiley InterScience (www.interscience.wiley.com). developed on insynthetic data and validated against a true gold- respective standard using aInc. hardware perfusion phantom. The accuracy of © 2008 Wiley-Liss, 1524 cation of absolute MBF. However, this technique has several drawbacks including low spatial resolution (makeach method was assessed for different levels of spatial averaging and perfusion rate. Finally, voxel-wise analysis was used to genering it unsuitable for the detection of subtle subendocarate high resolution perfusion maps on real data acquired from five dial perfusion defects), patient radiation exposure, and patients with suspected coronary artery disease and two healthy high cost (2,3). volunteers. On both synthetic and perfusion phantom data, the Compared with PET, dynamic contrast-enhanced carB-spline method had the highest error in estimation of myocardial diovascular magnetic resonance (DCE-CMR) imaging has blood flow. The autoregressive moving average modeling and exseveral potential advantages: superior spatial resolution, ponential methods gave accurate estimates of myocardial blood absence of ionizing radiation, and availability of stable flow. The Fermi model was the most robust method to noise. Both simulations and maps in the patients and hardware phantom and inert contrast agents of low toxicity. Estimation of MBF from DCE-CMR studies has been reported using a showed that voxel-wise quantification of myocardium perfusion is number of different analysis techniques including quanfeasible and can be used to detect abnormal regions. Magn C 2012 Wiley Periodicals, Inc. Reson Med 000:000–000, 2012. V titative and semiquantitative methods (4–14). Although favorable results with semiquantitative techKey words: myocardial perfusion; voxel-wise quantification; accuracy; noise robustness niques such as upslope analysis of the myocardial time– intensity curve have been reported, these methods have shown to underestimate the perfusion parameters INTRODUCTION (15,16). Moreover, semiquantitative analysis relies on a Detection of myocardial ischemia is the key to the diag- ratio which introduce a bias on the data itself and the nosis of coronary artery disease (1). Several invasive relationship between MBF and the semiquantitative methods parameters such as the curve upslope is not as clear-cut as the relationship between MBF and the 1 Division of Imaging Sciences and Biomedical Engineering, King’s College impulse response amplitude which we get from quantitaLondon BHF Centre of Excellence, NIHR Biomedical Research Centre and tive analysis (8,17), whereas using fully quantitative Wellcome Trust and EPSRC Medical Engineering Centre at Guy’s and St. analysis allows the absolute quantification of MBF in Thomas’ NHS Foundation Trust, The Rayne Institute, St. Thomas’ Hospital, London, United Kingdom. units of ml/g/min and may permit more accurate and 2 Philips Healthcare, Imaging Systems–MR, Veenpluis 4-6, The Netherlands. objective assessment of altered myocardial perfusion in 3 DivisionofEngineering,King’sCollegeLondon,Strand,London,UnitedKingdom. patients with heart disease. Grant sponsor: Wellcome Trust and the EPSRC; Grant number: WT Quantitative methods can be further divided into two 088641/Z/09/Z; Grant sponsor: Department of Health via the National Voxel-Wise Quantification of Myocardial Perfusion by Cardiac Magnetic Resonance. Feasibility and Methods Comparison Magnetic Resonance in Medicine 54:1273–1280 (2005) KING’S MEDICAL ENGINEERING CENTRE Matrix Description of General Motion Correction Applied to Multishot Images P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and D. Larkman3 Motion of an object degrades MR images, as the acquisition is time-dependent, and thus k-space is inconsistently sampled. This causes ghosts. Current motion correction methods make restrictive assumptions on the type of motions, for example, that it is a translation or rotation, and use special properties of k-space for these transformations. Such methods, however, cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a method is presented that can handle general nonrigid motion models. A general matrix equation gives the corrupted image from the ideal object. Thus, inversion of this system allows us to get the ideal image from the corrupted one. This inversion is possible by efficient methods mixing Fourier transforms with the conjugate gradient method. A faster but empirical inversion is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves between shots. The results indicate that it is now possible to correct for nonrigid types of motion that are representative of many types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 Wiley-Liss, Inc. Key words: motion correction; ghosts; multishot; conjugate gradient; auto focus Motion of an object can degrade MR images and imposes rection method could be used to spatially transform the ghosted image by the transformation corresponding to a shot, pick the k-space lines corresponding to that shot, and repeat this operation for all shots (this is a version of the method used in (1)). We could then rebuild an image by inverse Fourier transform. This method is in general incorrect, as shown by the difference between translations and rotations. Correcting translation requires only pointwise phase changes in k-space. On the other hand, correcting rotations requires knowledge of the data at neighboring k-space positions and these are acquired at different times. Before applying the Fourier rotation theorem, we would need to “synchronize” neighboring values. Furthermore, complicated motions such as nonrigid deformations cannot have a simple description in Fourier space. Here, however, we show that it is possible to correct complicated motions, including nonrigid motions. We give a full mathematical description of the problems involved; the motion corruption is entirely described by a large matrix acting on the space of images. Thus, inversion of this matrix should correct the motion’s effects. This approach is of theoretical interest, but its practical value depends on how easily we can find a solution of the linear system. It turns out that with some careful linear algebra, not only KING’S MEDICAL ENGINEERING CENTRE Motion during Image Acquisition Lung Liver KING’S MEDICAL ENGINEERING CENTRE Motion during Image Acquisition Lung • Blurring of moving structures Liver KING’S MEDICAL ENGINEERING CENTRE Motion during Image Acquisition Lung • Blurring of moving structures Liver • Image Artifacts KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating 1. Cardiac Motion 2. Breathing Motion KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating ECG KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating ECG Respiration accept reject KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating ECG Respiration accept reject KING’S MEDICAL ENGINEERING CENTRE Cardiac MRI - Motion Gating Efficiency How efficient is Cardiac MRI? 1. Efficient: acquiring during 50% of Time 2. Doing OK: acquiring during 20% of Time 3. Inefficient: acquiring during < 5% of Time KING’S MEDICAL ENGINEERING CENTRE Answer: ECG Respiration accept reject Cardiac Motion: Respiration: 10% 50% Overall: 5% also in other studies that fully overlapping blocks lead to a 958 better motion estimation. The appropriate search range has to be adapted to the largest displacement that depends on the speed of the motion. In the kinetic knee studies the maximal displacement was 23 mm. Using an acquisition time of 230 msec for each subset, the maximal detectable speed was 0.1 m/sec. KING’S MEDICAL ENGINEERING CENTRE Image Reconstruction With Motion Compensation In Figure 2, the motion estimation technique has been applied to the acquired sub-images as described in the Motion Compensated Reconstruction Motion is estimated with respect to the In previous a first study, thesection. feasibility of the motion-compensated FIG. 5. Mean squared difference (MSD), which was used as a quality reconstruction was tested using static MR-images of the measure for the hierarchical motion estimation. a: MSD for various second subset used as a reference state of motion. However, block sizes and different kinetic studies. b: MSD for different knee-joint at four different positions. These static images Magnetic Resonance in Medicine 41:954–963 (1999)the ‘‘gold-standard’’ from which subsets representpixel-offsets between neighboring blocks. An offset of one pixel defined the reference frame can be chosen arbitrarily, and motion ing the various motion states were extracted. In these data, indicates fully overlapping blocks. motion only between the data subsets. The data cantakes beplace estimated with respect to each of the four motion Motion Compensated Projection Reconstruction were subjected to the hierarchical motion compensated states. After determining all displacement fields d!i(x! ), the three resolution levels was applied. The arrows represent reconstruction scheme described above, and the resulting Tobias Schäffter,* Volker Rasche, Ingwer C.for Carlsen theand displacements each block, indicating that the lower high resolution images were compared to the original reconstruction of the high-resolution Motion in Image spaceimage according to limb is swinging to the left. The images demonstrate that images. Figure 6 shows the results of the hierarchical MCthe continuity the motion field is improved if several Over recent years, MRI has shown the capability for real-time of After the motion information has been recorded during the Eq. 1 can simply be modified to compensate for the motion: applications. Although the acquisition timesresolution of fast MRI methreconstruction of one single motion state. The image in levels are used. it is used to reorder all acquired data acquisition, with Motion Compensated MR in Image Space ods have been reduced significantly, patient motion during a respect to the different motion (7). However, Figurethe6a has been reconstructed from the subsets of the The sizes and the overlaps of the blocks of thestates hierarchimagnetic resonance imaging (MRI) examination still causes reconstruction of images at motion is only motion states and shows severe image blur. For different cal algorithm were optimized bydifferent evaluating thestates artifacts in the image. In this paper, the effects of matching motion in MRI R was reconstructed from 256 possible as a post-processing step after sufficient data have comparison, a referenceH image MCIt difference using a radial acquisition scheme are examined. is shown thatimages. In Figure 4 difference images with" " MC been acquired. Finally, corrective reconstruction methods motion can be estimated without the use of additional out (Fig.measure4a) and with motion compensation for different static projections, which is given in Figure 6b. The image i are available, incorporating knowledge about the motion ment, based on the acquired projections only.block A new reconstrucin Figure 6c has been reconstructed using the hierarchical sizes (Fig. 4b,c) are shown for a moving knee. Figure tion technique is introduced that integrates a motion compensa- into the reconstruction process. In spin-warp MR-imaging MC-reconstruction. The artifacts in this image are signifi5 shows theinMSD for different block sizes andhas overlaps. transform correction been introduced (8). tion algorithm into the MR-reconstruction process, resulting a a generalized cantly reduced. However, the overall quality of the MCMSD were normalized to correction the MSD ofwithout motion significant reduction of blurring artifacts in The the reconstructed So far, only the rigid body movement (9) and images. The proposed method is applied tocompensation. different kinds of In linear (Fig. 6c) is worse than that Figure 5a the MSD for different block expansions (10) have been shown to beimage feasible. " of the static image (Fig. motion such as kinetic joint studies. Magn Reson reconstruction methods have also 6b),develdue to imperfections of the motion estimation. sizes Med are41:954– shown. Corrective A minimum in the MSD of 9 pixels is been 963, 1999. ! 1999 Wiley-Liss, Inc. oped i.e. for techniques to projection reconstruction The MC-reconstruction was tested on MR-movies, i.e. found for both cases, a moving related hand and a moving Key words: MRI; motion correction; projection-reconstruction (11–13). To our knowledge, so far no corrective reconstruc- I (x! ) ! ! BP(5p (u! · (x! # d!i(x! )))6i) u ! ! (cos ", sin "). knee. Smaller blocks were found to give poorer results due [4] In this equation, a back-projection is applied to the filtered projection p of the ith subset taking the displacement d!i(x! ) of each pixel with respect to one reference frame into account, i.e. the back-projection is calculated at the position x! # d! (x! ). The motion compensated (MC) highresolution image I (x! ) represents the motion state with respect to one reference frame. As described above, the reference frame can be chosen arbitrarily and for each subset a high-resolution MC-image can be reconstructed using different sets of displacement fields. Thus the MC images show different motion states with high spatial resolution. studies in which motion occurs continuously during the tion technique can correct for complex motion, i.e. motion to a less pronounced maximum in the similarity measure, acquisition process. Although the assumed motion model that cannot be modeled by a rigid body movement or an Patient motion during magnetic resonance imagingfor (MRI) whereas largerexpansion sizes themodel. influence of elastic deforma- is not fully valid, the" MC-reconstruction significantly data acquisition causes artifacts in the reconstructed image reduces tions and rotationsWeofhave the recently blocks shown is unfavorably high. the feasibility of a new correc- image blurring. Figure 7 shows a selection of that obscure anatomical details. The main source of these artifacts is macroscopic motion of organs or extremities, tive reconstruction technique for a projection based reconwhich is either caused by wanted movement as in kinetic struction (14). The method avoids the need for additional studies, or by unwanted physiological motion due to flow, measurements, motion modeling or reordering of data. respiration, peristalsis or cardiac pulsation. The appear- Motion is estimated by means of a block matching algoi ance of these kinds of motion in the final MR-image rithm that can cope with even complex motion. The result HR strongly depends on the trajectory through k-space along of this algorithm is a motion field that represents the MC which the MR data are sampled. In spin-warp imaging, movement of each block by means of a motion vector. motion results in ghost repetitions of the moving structures These estimated motion vectors are used during reconstrucin the phase encoding and blurring in the readout direction tion to either reduce artifacts in real-time MRI or to (1). In spiral and radial acquisition schemes, motion results reconstruct different motion states from one single data set. in blurring of the moving structures with superimposed The purpose of this paper is to demonstrate and discuss radial streaking artifacts or spiral-like artifacts (2,3). Over results obtained from various regions of the human body recent years, MRI has shown the capability for real-time and to outline the advantages and limitations of this applications. Although fast imaging techniques are more approach. immune to most types of macroscopic motion, for some applications like kinetic MRI-studies the acquisition speed of MRI is still too low in relation to motion. METHODS A number of strategies have been developed to degrade This section introduces the new corrective reconstruction. the effects of motion. A first strategy, which requires a high A brief description of the data acquisition and reconstrucdegree of co-operation from the patient, is to suppress the of the knee joint. a: Reconstruction with motion using subsets of four different static positions. b: Reference image without FIG. 6. MR-images tion in projection reconstruction MRI is given. It is shown motion using fixation devices or by means breathcompensated reconstruction. motion. of c: Motion that an interleaved MR-acquisition scheme allows the holding (4). A second strategy is gating, where motion is reconstruction of low-resolution images at different morecorded by navigator echoes (5) and only those data are tion states during the acquisition of one high-resolution accepted that correspond to one single motion state. This image. The influence of motion on the image will be strategy is very effective but has the severe disadvantage of discussed for projection reconstruction MRI. It is shown significantly increased scan time. A third strategy is to that complex motion can be estimated by means of a block reorder the sequence of profiles during acquisition. In matching algorithm. A new reconstruction technique is respiratory ordered phase encoding (ROPE) (6), for exintroduced that integrates estimated motion vectors. Fiample, the knowledge of the respiratory motion is used to nally, an improved version of the motion-compensated apply an acquisition scheme suited to spin-warp imaging. reconstruction is given. Reconstruction With a Hierarchical Motion Compensation Philips Research Laboratories, Division Technical Systems, Hamburg, Ger- According to Eq. 4, an MC-image is reconstructed using displacement fields that are estimated from sub-images, i.e. images reconstructed from one single data subset. The accuracy of the motion estimation is thus limited by the resolution of the sub-images. In the following, a reconstruc- ing the da resolution 3) Step with a h images ha 4) Mot obtained two levels in step 1. 5) The images is resolution the image Due to nique gen can be co ing the ac METHOD All meas Philips G Tesla resp 6000 syst For signa elements w echo sequ The hie ent kinds hierarchic number o each leve the larges the image KING’S MEDICAL ENGINEERING CENTRE Motion Compensated MR in k-Space Magnetic Resonance in Medicine 54:1273–1280 (2005) Matrix Description of General Motion Correction Applied to Multishot Images P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and D. Larkman3 Motion of an object degrades MR images, as the acquisition is time-dependent, and thus k-space is inconsistently sampled. This causes ghosts. Current motion correction methods make restrictive assumptions on the type of motions, for example, that it is a translation or rotation, and use special properties of k-space for these transformations. Such methods, however, cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a method is presented that can handle general nonrigid motion models. A general matrix equation gives the corrupted image from the ideal object. Thus, inversion of this system allows us to get the ideal image from the corrupted one. This inversion is possible by efficient methods mixing Fourier transforms with the conjugate gradient method. A faster but empirical inversion is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves between shots. The results indicate that it is now possible to correct for nonrigid types of motion that are representative of many types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 Wiley-Liss, Inc. Key words: motion correction; ghosts; multishot; conjugate gradient; auto focus Motion of an object can degrade MR images and imposes constraints on scan parameters that can in turn compromise image quality. The cause of the degradation is that the acquisition is time-dependent, and the Fourier transform of the image seen during acquisition changes due to the deformation of the object. This causes inconsistencies in k-space and hence ghosts in the image. Standard motion correction methods make assumptions on the type of motions, for example, that it is a translation or a rotation, and use formulas on Fourier transforms to correct the data (1–3). We assume here that these data are acquired in shots. When the data positions at each shot are known, an empirical motion cor- 1 Medical Physics & Bioengineering, University College London, London, United Kingdom. 2 Department of Electrical Engineering, Pontifica Universidad Católica de Chile, Chile. 3 Imaging Sciences, Imperial College London, London, United Kingdom. *Correspondence to: P. G. Batchelor Room 2.20, Medical Physics & Bioengineering, University College London, Gower Street, London WCIE 6BT, UK, E-mail: [email protected] Presented in part as abstracts at the 2nd International Workshop on ParallelMRI, Zürich, 2004, and MIUA, London, Sept. 2004. Grant Sponsor: EPSRC; Grant Numbers: GR/S30184, AF/001381; Grant rection method could be used to spatially transform the ghosted image by the transformation corresponding to a shot, pick the k-space lines corresponding to that shot, and repeat this operation for all shots (this is a version of the method used in (1)). We could then rebuild an image by inverse Fourier transform. This method is in general incorrect, as shown by the difference between translations and rotations. Correcting translation requires only pointwise phase changes in k-space. On the other hand, correcting rotations requires knowledge of the data at neighboring k-space positions and these are acquired at different times. Before applying the Fourier rotation theorem, we would need to “synchronize” neighboring values. Furthermore, complicated motions such as nonrigid deformations cannot have a simple description in Fourier space. Here, however, we show that it is possible to correct complicated motions, including nonrigid motions. We give a full mathematical description of the problems involved; the motion corruption is entirely described by a large matrix acting on the space of images. Thus, inversion of this matrix should correct the motion’s effects. This approach is of theoretical interest, but its practical value depends on how easily we can find a solution of the linear system. It turns out that with some careful linear algebra, not only are we able to invert in a generalized sense, but also this inversion can be done efficiently in practice. For this we use the LSQR algorithm, which is a robust implementation of the conjugate gradient of the normal equation (see (4)). This leaves the question of finding what motion actually happened. Different cost functions have been designed to quantify how much an image has been corrupted (1, 5, 6). We explore different cost functions in conjunction with our motion correction. Optimization of such cost functions means repeating the matrix inversion iteratively. However, inverting matrices repeatedly may be prohibitive even if it is practicable on a one-off basis. We therefore also investigate the use of the empirical method described above. The matrix equation allows us to find when this approximation is correct. We show three-dimensional random affine and pulsatile nonrigid motion corrections on simulated data and an example of nonrigid correction of in vivo data (moving legs). THEORY We need a method to handle motion-corrupted MR images 78 spatial transformation. This matrix is to be distinguished Motion Ghosting from the inverse matrix of ut, which will not necessarily exist. But, for an image sinduce whose signal in the FOV, Spatial transformations linear remains image transforma! u u s " s. We use lowercase letters for objects to t t the images whose field of view (FOV) related tions on is transimage For space and capital letters for objects to kformed. an image s, the transformed imagerelated is the image space. whose intensities at a pixel are the intensities of s at the KING’S MEDICAL ENGINEERING CENTRE notation). Inversion of this matrix would recover an ob{uwhatever 3: Spatially transform s0. corrupted view, ts0} ject’s image from its motion the 4: Fourier transform. {Futs0} motion and time-sampling pattern in k-space. The matrix 5: Extract the lines corresponding to shot t. {AtFuts0} is large, of size n1n2 % n1n2 for n1%n2 images, but Algo6: Set these lines in S. {S " S # AtFuts0} rithm 1 is an efficient implementation of multiplication by 7: end for of Eq. (1) wouldofbe 8:. A Letk-space s be theversion inverse Fourier transform S. S "{s($ "t A $ttFUHt)S At0Fu"ts0} !S0 where Ut " FutFH is the k-space representation of motion at time t. form. From this algorithm, we obtain a responding matrix Motion Compensated MR in k-Space (inverse) transformed pixels (see (7) 3.3.2). The linear operations on images are defined pixelwise. Aliasing Aliasing is an image space consequence of subsampling Magnetic Resonance in Medicine 54:1273–1280 (2005) Transformation in the Fourier Matrices. domain. As such, it can be represented by Matrix Description of General Correction Applied a matrix. We define a Motion shot to be a subsample of k-space The effect of a spatial transformation on images can be to Multishot andImages make the assumption that all the corresponding represented as a sum of linear image transformations and Fourier components are acquired or in a 1 1 2 1 simultaneously, P. G. Batchelor, Atkinson, P. Irarrazaval, D. L. G. Hill, J. Hajnal,3 and can* D. thus be written as matrix multiplication. In the exD. Larkman3 very short time in comparison to any processes causing treme case, image transformation on a single change toeach the object. In other words operates shots correspond to Motion of an object degrades MR images, as the acquisition is rection method couldone be usedlocation to spatially transform the pixel, moving the pixel value from to another. time-dependent, and time thus k-space is inconsistently sampled. ghosted steps. A partition of image k-space in ns shots leads by the transformation corresponding to a to n s This causes ghosts. Current motion correction methods make shot, pick the k-space lines corresponding to that shot, We write uAt for the matrix acting on image space that restrictive assumptions on the type of motions, for example, and repeat matrices to the corresponding this operation for all shots (this is shot a versionposit that project that it is a translation or rotation, and use special properties of of deformation the method used in (1)). We could then rebuild an corresponds to a spatial dependent on time t. k-space for these transformations. Such methods, however, tions and sum to theimage identity matrix when k-space is by inverse Fourier transform. This method is in cannot be generalized easily to nonrigid types of motions, and ! general incorrect, as shown by the difference between If the spatial transformation can be inverted, we write u even rotations in multiple shotscovered. can be a problem. Here, aaliasing matrices a are defined to be t fully The t translation remethod is presented that can handle general nonrigid motion translations and rotations. Correcting Hequation quires only pointwise phase changes inby k-space. Oninverse the for the matrix of the transformation induced the models. A general matrix gives the corrupted image F A F, where Fourier transformation is represented by t of this system allows us to other hand, correcting rotations requires knowledge of from the ideal object. Thus, inversion data atmatrix neighboring k-space positions and these are spatial transformation. This is to(hermitian be distinguished get the ideal image from corrupted inversion is the F the and FHone.isThisthe conjugate transpose conjupossible by efficient methods mixing Fourier transforms with acquired at different times. Before applying the Fourier the conjugate gradient method. A faster but empirical inversion from the inverse matrix of u , which will not necessarily rotation theorem, we would need to “synchronize” gate) of F. t is discussed as well as methods to determine the motion. Sim- neighboring values. Furthermore, complicated motions ulated three-dimensional affine data and two-dimensional pul- such exist. But, for an image s whose signal remains ina the as nonrigid deformations cannot have simple For rectangular FOVs, and subsampling along oneFOV, spasation data and in vivo nonrigid data are used for demonstradescription in Fourier space. Here, however, we show tion. All examplesu are ! multishot images where the object moves u s " s. We use lowercase letters for objects related to tial dimension, the computation factors along dimensions, that it is possible to correct complicated motions, int t between shots. The results indicate that it is now possible to correct for nonrigid types of motion that are representative of cluding nonrigid motions. We give a full mathematical image space and capital letters for objects related to kand the matrices a have a sparse block structure corret description of the problems involved; the motion cormany types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 ruption is entirely described by a large matrix acting on sponding to the standard ghosts. the space of images. Thus, inversion of this matrix Wiley-Liss, Inc. space. Key words: motion correction; ghosts; multishot; conjugate gradient; auto focus Motion of an object can degrade MR images and imposes constraints on scan parameters that can in turn compromise image quality. The cause of the degradation is that the acquisition is time-dependent, and the Fourier transform of the image seen during acquisition changes due to the deformation of the object. This causes inconsistencies in k-space and hence ghosts in the image. Standard motion correction methods make assumptions on the type of motions, for example, that it is a translation or a rotation, and use formulas on Fourier transforms to correct the data (1–3). We assume here that these data are acquired in shots. When the data positions at each shot are known, an empirical motion cor- should correct the motion’s effects. This approach is of theoretical interest, but its practical value depends on how easily we can find a solution of the linear system. It turns out that with some careful linear algebra, not only are we able to invert in a generalized sense, but also this inversion can be done efficiently in practice. For this we use the LSQR algorithm, which is a robust implementation of the conjugate gradient of the normal equation (see (4)). This leaves the question of finding what motion actually t happened. Different cost functions have been designed to quantify how much an image has been corrupted (1, 5,t6). We explore different cost functions in conjunction with our motion correction. Optimization of such cost functions means repeating the matrix inversion iteratively. However, 0 may be prohibitive even if it inverting matrices repeatedly is practicable on a one-off basis. We therefore also investigate the use of the empirical method described above. The matrix equation allows us to find when this approximation is correct. We show three-dimensional random affine and pulsatile nonrigid motion corrections on simulated data and an example of nonrigid correction of in vivo s data (moving legs). Motion during acquisition Aliasing We are now to space give the exact formulaoffor the effect of Aliasing is angoing image consequence subsampling acquisition Suppose u deinany the motion Fourierduring domain. As such,init k-space. can be represented by scribes the spatial transformation at time t, and A is the a matrix. We define a shot to be a subsample of k-space sampling at this time. observed image s is and make of thek-space assumption that The all the corresponding related to the ideal object image s by the sequence Fourier components are acquired simultaneously, or inin a Algorithm 1. The expressions in curly braces are the corvery short time in comparison to any processes causing Medical Physics & Bioengineering, University College London, London, change to the object. In other words shots correspond to United Kingdom. Department of Electrical Engineering, Pontifica Universidad Católica de Chile, Chile. time steps. A partition of k-space in n shots leads to ns Imaging Sciences, Imperial College London, London, United Kingdom. *Correspondence to: matrices P. G. Batchelor Room 2.20, Physicsproject & BioengiAMedical that to the corresponding shot posit London neering, University College London, Gower Street, WCIE 6BT, UK, E-mail: [email protected] tions sum to the THEORY identity matrix when k-space is Presented in part as abstracts at the and 2nd International Workshop on ParallelMRI, Zürich, 2004, and MIUA, London, Sept. 2004. We need a matrices method to handle motion-corrupted MR imagesto be Grant Sponsor: EPSRC; Grant Numbers: GR/S30184, AF/001381; fully covered. The Grant aliasing a are defined 1 2 3 simple matrix description of the motion corruption: the Empirical Inversesimage is a superposition of aliased views motion corrupted of the object in different positions, namely, The empirical motion correction method mentioned in the Motion inamounts K-space Introduction to Algorithm 1 with the motions ut ! ! n s !u1t . We write replaced by their inverses for the corresponding matrix, swhich weat u call the empirical inverse.[1] " s " s . t 0 0 This algorithm is incorrect in the sense that in general t"0 ! s " ! s0 & s0. The mathematical reason is that a sum of not the inverse of the sum. physical Theinverses formulais here is the key insight as it The introduces the n s !1 reason is that the shot lines corresponding to t were indeed matrix g ! $t"0 atut that entirely describes the effect of transformed a known spatial but only motions on by signals acquired intransformation, the Fourier domain. We these lines, and this algorithm applies the transformation call this matrix the ghosting matrix (hence the choice of to the whole image. It is easy to see that if the aliasing notation). Inversion of this matrix would recover an obmatrices at and the spatial transformations matrices ut, ject’s image from its motion corrupted view, whatever the commute for all t,t', then ! inverts (assuming ut!uts0 " motion and time-sampling pattern in k-space. The matrix Inverse sEmpirical ). is0large, of size n1n2 % n1n2 for n1%n2 images, but Algorithm 1 is an efficient implementation of multiplication by ns ! 1 ns ! 1 ns ! 1 . A k-space version of Eq. (1) would be S " ($t AtUt)S0 " ! ! ! H s " a ,u a u s " a u a ut s0 t t t t 0 t' t' !S0 where Ut " FutF is the k-space trepresentation of t' " 0 t"0 t,t' " 0 motion at time t. !" # !" # " " n s !1 at' at ut' !ut s0 ! Empirical"Inverses t,t' " 0 " " ns ! 1 Batchelor ! et al. t"0 at ut ut s0 " " ns ! 1 at s0 " s0. t"0 The empirical motion correction method mentioned in the FIG. 2. Nonrigid example (simulation), Introduction amounts to whose Algorithm 1 with the motions Spatial transformations transformation matrices ut which simulates a pulsation!in 16 shots. (a)! replaced by their inverses u . We write for the commute with aliasing matrices are then special in correthat t (b) the correcThe motion-corrupted image; sponding matrix, which we inverse. they allow for a fast exact reconstruction. This means that $ call the empirical tion by empirical inverse ; (c) the correcaliasing first and transforming the Thistion algorithm isthen incorrect in (d) thethemust sense that in to general by the LSQR algorithm; gold-amount ! ! s s" & s . The mathematical reason is that a sum standard0image. 0 (e) The deformation at each of inverses is not the inverse of the sum. The physical of the 16 shots on a square checkerboard reason is that the shot lines corresponding to t were indeed image. transformed by a known spatial transformation, but only these lines, and this algorithm applies the transformation KING’S MEDICAL ENGINEERING CENTRE Motion Compensated Cardiac MRI Magnetic Resonance in Medicine 54:1273–1280 (2005) Knowledge [v.5.6] Matrix Description of General Motion Correction Applied http://cm.webofknowledge.com/viewCitationPrint.do to Multishot Images P. G. Batchelor,1* D. Atkinson,1 P. Irarrazaval,2 D. L. G. Hill,1 J. Hajnal,3 and D. Larkman3 Print | Close Motion of an object degrades MR images, as the acquisition is time-dependent, and thus k-space is inconsistently sampled. This causes ghosts. Current motion correction methods make restrictive assumptions on the type of motions, for example, that it is a translation or rotation, and use special properties of k-space for these transformations. Such methods, however, cannot be generalized easily to nonrigid types of motions, and even rotations in multiple shots can be a problem. Here, a method is presented that can handle general nonrigid motion models. A general matrix equation gives the corrupted image from the ideal object. Thus, inversion of this system allows us to get the ideal image from the corrupted one. This inversion is possible by efficient methods mixing Fourier transforms with the conjugate gradient method. A faster but empirical inversion is discussed as well as methods to determine the motion. Simulated three-dimensional affine data and two-dimensional pulsation data and in vivo nonrigid data are used for demonstration. All examples are multishot images where the object moves between shots. The results indicate that it is now possible to correct for nonrigid types of motion that are representative of many types of patient motion, although computation times remain an issue. Magn Reson Med 54:1273–1280, 2005. © 2005 Wiley-Liss, Inc. Matrix description of general motion correction applied to multishot images ©2012 rection method could be used to spatially transform the Tuesday, September 18 2012 ghosted image by the transformation corresponding to a shot, pick the k-space lines corresponding to that shot, and repeat this operation for all shots (this is a version of the method used in (1)). We could then rebuild an image by inverse Fourier transform. This method is in general incorrect, as shown by the difference between translations and rotations. Correcting translation requires only pointwise phase changes in k-space. On the other hand, correcting rotations requires knowledge of the data at neighboring k-space positions and these are acquired at different times. Before applying the Fourier rotation theorem, we would need to “synchronize” neighboring values. Furthermore, complicated motions such as nonrigid deformations cannot have a simple description in Fourier space. Here, however, we show that it is possible to correct complicated motions, including nonrigid motions. We give a full mathematical description of the problems involved; the motion corruption is entirely described by a large matrix acting on the space of images. Thus, inversion of this matrix should correct the motion’s effects. This approach is of theoretical interest, but its practical value depends on how easily we can find a solution of the linear system. It turns out that with some careful linear algebra, not only are we able to invert in a generalized sense, but also this inversion can be done efficiently in practice. For this we use the LSQR algorithm, which is a robust implementation of the conjugate gradient of the normal equation (see (4)). This leaves the question of finding what motion actually happened. Different cost functions have been designed to quantify how much an image has been corrupted (1, 5, 6). We explore different cost functions in conjunction with our motion correction. Optimization of such cost functions means repeating the matrix inversion iteratively. However, inverting matrices repeatedly may be prohibitive even if it is practicable on a one-off basis. We therefore also investigate the use of the empirical method described above. The matrix equation allows us to find when this approximation is correct. We show three-dimensional random affine and pulsatile nonrigid motion corrections on simulated data and an example of nonrigid correction of in vivo Motion Compensated Cardiac MRI: Motion of an object can degrade MR images and imposes Michael Schacht Hansen (NIH) constraints on scan parameters that can in turn compromise image quality. The cause of the degradation is that Johannes Sebastian Kozerke (ETH Zurich) the acquisitionSchmidt is time-dependent,… and the Fourier transform of the image seen during acquisition changes due to the deformation of Usman the object. This causes Muhammad …inconsistencies Claudia Prieto (KCL) in k-space and hence ghosts in the image. Standard motion correction methods make assumpFreddy Odille,Maria Filipovic, … Jacques Felblinger (Nancy) tions on the type of motions, for example, that it is a Key words: motion correction; ghosts; multishot; conjugate gradient; auto focus translation or a rotation, and use formulas on Fourier transforms to correct the data (1–3). We assume here that these data are acquired in shots. When the data positions at each shot are known, an empirical motion cor- 1 Medical Physics & Bioengineering, University College London, London, United Kingdom. 2 Department of Electrical Engineering, Pontifica Universidad Católica de Chile, Chile. 3 Imaging Sciences, Imperial College London, London, United Kingdom.