Parallel Imaging: The Basics

Transcription

1 Parallel Imagng: The Bascs Peter Kellman Laboratory of Cardac Energetcs, NHLBI, Natonal Insttutes of Health, DHHS, Bethesda, MD, USA Introducton Parallel magng explots the dfference n senstvtes between ndvdual col elements n a receve array to reduce the number of gradent encodng steps requred for magng Parallel magng was orgnally conceved [1-3] as a means of ultrafast magng usng a sngle echo readout, replacng gradent phase encodng entrely wth spatal encodng usng the col senstvtes However, t soon became apparent that there were fundamental as well as practcal lmtatons to the effectve number of encodes that were possble Ths led to more realstc mplementatons [4-6] whch used gradent phase encodng wth a reduced number of steps, referred to as subencodng Ths tutoral wll gve an overvew of the basc parallel MR magng problem formulaton and solutons for both mage and k-space doman mplementatons As shown n the smplfed llustraton of Fgure 1, an array of receve cols wth senstvtes, s(x,y,z) are used to acqure sgnals G from the sample volume wth magnetzaton f(x,y,z) In the early publshed parallel MR jkxx lterature, the sgnals G( kx) = s( x, y) f( x, y) e conssted of a sngle readout (e, frequency encoded but xy, wthout gradent encodng) The encodng (or forward problem) whch was a combnaton of frequency jkxx encodng, e, and spatal col senstvty encodng, s(x,y,z), may be represented n matrx form G= Sfafter measured sgnal dscretzaton The desred magnetzaton mage f(x,y) may vectors, G 1 be reconstructed by solvng the nverse problem, f = S G Unque soluton of the nverse problem requres that the receve number of ndependent equatons (e, unque cols) s equal to the number of unknowns, n ths case number of y- cols values; wth a greater number of equatons (cols) the system of equatons becomes over-determned and has a s (x,y,z) least squares soluton or pseudo-nverse Due to lmtaton f (x,y,z) on the number of cols whch are effectvely ndependent, multple gradent encodng steps are employed n practce sample volume as descrbed n the remander of the tutoral Fgure 1 The chronology of parallel MR magng s summarzed n Fg 2 usng approxmate dates Followng ntal conceptual papers [1-6], practcal mplementatons of parallel MR were presented by Sodckson and Mannng [7] Basc Parallel MR Concept ( ) phase array combnng senstvty encodng & hybrds (sub-encodng) Practcal mplementatons ( ) SMASH: k-space approach SENSE: mage doman Optmzaton (1999-current) auto-calbraton matrx nverse condtonng col optmzaton Extensons (2000-current) generalzed SMASH 2-d SENSE & mult-slce arbtrary k-space Applcatons (2000-current) Fgure 2 Chronology of parallel MR magng and Pruessmann, et al [12] Sodckson, et al [7,8] proposed a method known as SMultaneous Acquston of Spatal Harmoncs (SMASH) whch s a k-space mplementaton of parallel MR magng Pruessmann et al presented a general formulaton and performance analyss of the mage doman senstvty encodng method (SENSE) A number of generalzatons and extensons have been reported The SMASH method has been generalzed to provde talored harmonc fts [9], col-by-col mage reconstructon [10], and a generalzed matrx formulaton [11] SENSE has been appled to arbtrary k-space trajectores [13] and has been extended to 2-d SENSE [14] and mult-slce parallel acqustons [15] General analyss frameworks for comparng mage and k-space doman mplementatons, as well as hybrds, are presented n [16-17] References on array sgnal processng [18-20] formulate the problem as an optmzaton wth nullng constrants

2 A key area of current research has been on auto-calbraton methods for estmatng the n-vvo col senstvtes [21-26] Optmzatons nclude the numercal condtonng or regularzaton of the matrx nverse [27-30], and the desgn of col arrays for parallel MR magng [31-34] Research and development has led to an mproved understandng of performance lmtatons such as tradeoffs between acceleraton, # cols, and SNR degradaton Advances n technology have led to practcal mplementaton and numerous applcatons have been proposed and demonstrated The presentaton wll begn wth the general matrx formulaton of combned senstvty and Fourer (gradent) encodng (the forward problem) and the nverse soluton for arbtrary k-space Ths general formulaton wll then be smplfed to the case of Cartesan k-space acquston Equvalence between mage and k-space doman mplementatons wll be dscussed and the concept of col senstvty spectra [16] wll be presented to motvate how mssng gradent encodes (vews) are replaced by parallel MR The methods of SMASH and harmonc fttng are then presented SNR losses are nherent n nverse problems due to ll-condtonng The varance nflaton factor, or so called SENSE g-factor [12], s explaned The method of 2-d SENSE [14] has been proposed for mproved g- factor for volume magng wth hgh acceleraton factors The 2-d SENSE method s brefly descrbed Thoughout the presentaton, t s assumed that the cols senstvtes, s are known or can be estmated The subject of n vvo estmaton of col senstvtes and auto-calbraton are presented at the concluson 2 Matrx Formulaton The matrx formulaton (Eq [1]) and soluton (followng Pruessmann, et al [12]) ncorporates both Fourer (gradent) and col senstvty encodng for arbtrary k-space acquston In ths formulaton the matrx-vector notaton assumes that 2d varables are ordered n a 1-d vector In other words, the column vector r conssts of n r =N 2 elements over (x,y), and lkewse, k, assumes nk values over (k x,k y ) The column vector G conssts of measured k- space data for nc cols, e, nc nk x1 Eq 1 represents the encodng or forward problem jkr r G( k) = s() r e f() r + n( k) G = Ef + n s (r) complex col senstvtes (B 1 -maps) for col =1,,n c r poston vector r=[r 1,r 2, r nr ] k k-space sample vector k=[k 1,k 2, k nk ] G vector of measured k-space samples (n c n k x 1) E encodng matrx (n c n k x N 2 ) f vector of magnetzaton mage voxels (N 2 x1) nose for col n ˆ H ( ) 1 1 H 1 f = E Ψ E E Ψ G H Ψ= cov( n) = E{ nn } [2] [1] Thoughout the presentaton, t s assumed that the cols senstvtes, s are known or can be estmated wth suffcent accuracy The subject of n vvo estmaton of col senstvtes s very mportant and wll be addressed brefly at the concluson The encodng functons for combned Fourer (gradent) and col senstvty encodng are shown n Fg 3 for llustraton purposes In ths smulated llustraton, there are 4 cols surroundng a cylndrcal phantom and Fourer encodng s shown for (k x,k y )= (0,0), (2,0), (0,2), and (2,2) The real encodng functons (mages) are shown The mage reconstructon or soluton to the nverse problem (Eq [2]) may be estmated by the least squares method provded the total number of encodes nc nk > nr =N 2 Parallel magng may be used to acheve acceleraton by usng fewer gradent encodng steps (nk) and stll mantan the same spatal resoluton The nose covarance (Ψ) weghted least squares soluton optmzes the SNR subject to the constrant of nullng artfacts due to undersamplng The soluton for ths arbtrary k-space formulaton requres an N 2 xn 2 matrx nverson whch s mpractcal to mplement wth drect nverson for typcal mage szes (eg, 256x256) The least square soluton [13] based on the teratve conjugate gradent method may be used to compute solutons convergng wthn reasonable reconstructon tmes An example short axs cardac mage reconstructed usng the teratve conjugate gradent method s shown n Fg 4 for the 1 st 6 teratons In ths example, a varable densty Cartesan k- space acquston was used wth an acquston matrx of 128x60 and 35 phase encodes acqured, every other lne wth 5 added center lnes The acquston used a real-tme true-fisp sequence wthout ECG trggerng, and corresponded to a frame rate of approxmately 15 fps

3 (k x,k y ) (0,0) col 1 col 2 col 3 col 4 teraton 1 teraton 2 teraton 3 (2,0) (0,2) (2,2) real e { } j( kxx+ kyy) { (, ) } j( kxx kyy) real s x y e + Fgure 3 Example encodng functons teraton 4 teraton 5 teraton 6 Fgure 4 Example short axs cardac mages usng varable densty Cartesan k-space acquston, reconstructed usng teratve conjugate gradent SENSE The smplfed matrx formulaton for unform Cartesan k-space acquston s presented next Fgure 5 llustrates the alasng due to unformly undersampled Cartesan acquston for rate R=2 showng mages from a sngle col Gven knowledge of the col senstvtes, the alased mages can be separated by solvng lnear equatons The matrx formulaton and soluton are gven n Equaton 3 S s referred to as the col senstvty matrx, and U s the unmxng matrx Applcaton of the SENSE unmxng matrx may be equvalently descrbed as phased array combnng as llustrated n Fgure 6 Example short axs cardac mages acqured at acceleraton rates 2, 3, and 4 are shown n Fgure 7 before and after SENSE reconstructon These are real-tme free breathng true-fisp wthout ECG trggerng acqured at 15, 225, and 30 fps for rates 2, 3, and 4 respectvely, usng a 128x60 acquston matrx g 1(x,y) s 1(x,y) s 1(x,y FOV/2) n 1(x,y) f(x,y) = + f(x,y FOV/2) g (x,y) s (x,y) s (x,y FOV/2) n (x,y) N N N N g = Sf + n ( ) 1 ˆ H 1 H 1 f = S Ψ S S Ψ g = Ug [3] acqure full k-space data acqure undersampled k-space data FOV s (x,y) f(x,y) + s (x,y) f(x,y) s (x,y-fov/2) f(x,y-fov/2) s (x,y-fov/2) f(x,y-fov/2) Fgure 5 Illustraton of alasng due to undersampled Cartesan k-space acquston

4 reference mages calculate SENSE nverse phased array combnng coeffcents (mag) mult-col k-space data FFT phased array combner (SENSE) Fgure 6 Smplfed dagram of SENSE phased array combnng alased mages R=2 (15 fps) R=3 (225 fps) R=4 (30 fps) Fgure 7 Example short axs cardac SENSE mages SENSE mage (mag) Image doman phased array combnng may equvalently be performed n the k-space doman as shown n Fg 8, where mage doman multplcaton by u (x,y) has been mplemented by k-space doman convoluton wth U (k x,k y ) Ths leads drectly to k-space methods such as SMASH whch fll the mssng k-space usng convoluton wth a truncated kernel composed of a set of 1 or more harmonc fts to the col senstvtes Further nsght s ganed [16] by examnng the spatal spectra of the col senstvtes llustrated n Fg 9 The non-unform col senstvty profle leads to a blurrng n k-space, whch allows parallel magng to solve for the mssng phase encodes u 1 (x,y) mult-col k-space data FFT mult-col mages u N (x,y) mage doman SENSE reconstructon phased array combnng transform equvalents mult-col k-space data * U 1 (k x,k y ) * U N (k x,k y ) FFT k-space doman reconstructon convoluton Fgure 8 Equvalence between mage and k-space doman mplementatons of parallel magng

5 unform senstvty senstvty wth vertcal shadng 1 col senstvty profles no blurrng k-space blurred col senstvty spectra by col senstvty spectra flls n mssng vews (R=4 example) Fgure 9 Col senstvty spectra llustratng k-space vew fllng concept In the formulaton of the k-space parallel magng method for Cartesan k-space acquston, the acqured k- space data may be wrtten as: j( kxx, kyy) G( kx, ky) = s( x, y) e f( x, y) + n( kx, ky ) [4] xy, and the SMASH estmate of the composte k-space data wth mssng vews flled n may be wrtten as: ( ) ˆ(, ) m Gkx ky m ky n G ( kx, ky ) where the coeffcents n = [5] are calculated by a ft to the m-th harmonc jm kyy n s( x, y) e [6] col 1 col 2 col 3 col 4 (1) (1) (1) n 2 n n 2 (1) 1 n 4 composte Fgure 10 SMASH procedure for fllng mssng k-space lnes Ths procedure s dagrammed n the llustraton of Fg 10 The harmonc fts for m=0 and m=1 are shown n Fg 11 for smulated complex col profles There s consderable error between the actual and desred harmonc fts The error may be greatly reduced by usng a talored harmonc ft [9] whch fts to a weghted set of harmoncs, as llustrated n Fg 12 whch uses a phased sum of the complex cols as a weghtng functon, as descrbed by Eq 7 jm kyy n s( x, y) s0( x, y) e Nc [7] j s0( x, y) = s( x, y) e φ phased sum = 1 complex col profles m=0 ft m=1 ft ft (real) ft (mag) desred (real) desred (mag) Fgure 11 Harmonc ft to complex col senstvty profles

6 m=0 ft m=1 ft ft (real) ft (mag) desred (real) desred (mag) A further generalzaton of the SMASH method s to use blocks of k-space lnes for calculatng the mssng phase encodes, as shown n Fg 13 Ths mproves the ablty to model col senstvty profles whch have sharper edges, and correspondngly greater spatal bandwdth For example, ths approach s used n the GRAPPA method [23] Fgure 12 Talored harmonc ft to complex col senstvtes profles col 1 col 2 col 3 col 4 n 2 n3 n 1 n4 composte Fgure 13 SMASH procedure usng block of k-space lnes 3 SNR losses Accelerated magng usng the SENSE method ncurs a loss n SNR due to both reduced magng tme and suboptmal col geometry (the so called g-factor) The SNR for accelerated parallel magng usng the SENSE method may be calculated as: SNRSENSE = SNRoptmum g R, [8] where R s the acceleraton factor and SNR optmum s the SNR for B 1 -weghted optmum phased array combnng [18] and the loss n SNR due to varance nflaton, SENSE g-factor [12], s defned as: gk ( x, y) = 1 H -1-1 H -1 Ψ ( k,k) Ψ ( k,k ) ( S S) ( S S), [9] where the subscrpt (k,k) denotes the ndex of the matrx correspondng to the k-th sub-mage The spatally varyng g-factor represents the loss n SNR (nflaton n varance) due to ll-condtonng of the matrx nverse, whch depends on the acceleraton rate, the number of cols, specfc col senstvty profles (szes, shapes, and postons), slce orentaton, and phase encode drecton The g-factor depends strongly on poston (x,y,z) and has several hotspots Fgures 14 and 15 show maps of the g-factor for 4 and 8 cols, respectvely, at varous acceleraton rates B 1 -map magntudes B 1 -map magntudes R=2 R=3 R=4 R=2 R=3 R=4 1<g<12 mean=107 max=12 1<g<3 mean=22 max=31 1<g<5 mean=54 max=86 1<g<12 mean=102 max=105 1<g<3 mean=13 max=17 1<g<5 mean=22 max=31 Fg 14 Example g-factor maps for SENSE wth 4-cols Fg 15 Example g-factor maps for SENSE wth 8-cols

7 The ll-condtonng of the nverse also ncreases the effect of errors n estmates of the complex col senstvtes (B 1 -maps) Errors n the senstvty matrx wll degrade the alas artfact suppresson It s customary to regularze or condton the matrx nverse [27-30] Adaptve regularzaton may be used to spatally vary the degree of condtonng and dramatcally mprove the SENSE reconstructon 4 2-d SENSE In volume magng applcatons usng 2 phase encode dmensons (or spectroscopc magng), t may be preferable to perform accelerated magng n each of the 2 phase encode drectons rather than a hgher rate along a sngle drecton Ths has been referred to as 2-d SENSE [14] The matrx formulaton for the case of R=2 acceleraton along y and z, for an overall R=4 acceleraton, s wrtten as: f(x, y, z) g 1(x, y, z) s 1(x, y, z) s 1(x, y - Dy, z) s 1(x, y, z - D z) s 1(x, y - Dy, z - D z), n 1(x, yz), y f(x, y - D z) = + [10] f(x, y, z - D z) g N(x, y, z) s N(x, y, z) s N(x, y - Dy, z) s N(x, y, z - D z) s N(x, y - Dy, z - D z), f(x, y - Dy, z - D z) n N (x,y z) where D y =FOV y /2 and D z =FOV z /2 Dependng on the specfc col senstvty profles and slce geometry, t may be possble to acheve a g-factor whch s greatly reduced when compared wth R=4 acceleraton along a sngle dmenson, eg, y or z n ths case A scheme for performng SENSE wth mult-slce parallel acquston s presented n [15] 5 B 1 -Map Estmates and Auto-calbraton Thoughout the presentaton, t s assumed that the cols senstvtes, s are known or can be estmated wth suffcent accuracy The complex col senstvtes, also known as B 1 -maps, depend not only on the col geometry and orentaton, but are stongly nfluenced by the sample volume of nterest whch alters the magnetc feld dstrbuton For ths reason, n-vvo estmaton of col senstvtes s key to achevng accurate estmates In the case of k-space methods such as auto-smash and GRAPPA [21-23], B 1 -maps are not explctly calculated Rather, k- space weghtng coeffcents are based on n vvo k-space reference data drectly, as descrbed brefly at the concluson of ths secton In vvo B 1 -maps may be estmated from separate or nterleaved reference scans, or from the undersampled magng data tself n a varety of schemes referred to as auto-calbratng A basc descrpton follows The complete process may often nvolve several addtonal steps to nclude spatal smoothng and sometmes extrapolaton, the detals of whch are not fully descrbed Frst, the raw senstvtes are estmated from the ndvdual col mages acqured wthout undersamplng artfacts, at ether full or reduced spatal resoluton Normalzaton wth ether a body col mage or root-sum-of-squares (RSS) combned magntude mage s used to remove the mage modulus, as descrbed by Eq [11] or [12], respectvely, where the subscrpt desgnates the col ndex In the case of normalzaton wth a body col mage, both the modulus and phase of the object are removed, assumng that the body col senstvty s unform In the case of normalzaton by RSS magntude mage only the mage modulus s removed leavng the object phase, f/ f Furthermore, the raw senstves wll be weghted by the combned magntude mage leadng to a weghtng on the fnal parallel MR reconstructed mage The object phase may be removed usng an array processng scheme based on the sample correlaton matrx and domnant egenvector [19] The object phase may have rapd spatal varaton, and removal of ths phase enables spatal smoothng of the raw senstvtes Example mages and raw senstvty estmates are shown n Fg 16 for a cardac magng applcaton ˆ s (x,y) f(x,y) s (x,y) ŝ (x,y) = f (x,y) f (x,y) = s (x,y) [11] body col s body col(x,y) f(x,y) s body col(x,y) s (x,y) f(x,y) s (x,y) f(x,y) 2 = ˆ ˆ 2 2 s f(x,y) (x,y) f(x,y) s (x,y) ŝ (x,y) f (x,y) f (x,y) [12] Snce the col senstvtes generally have slower spatal varaton than the object beng maged, several mplementatons acqure the B1-map data at lower spatal resoluton or employ spatal smoothng to reduce nose or fluctuatons If separate unaccelerated scans are used to acqure reference mages, reduced resoluton s often used to acqure the mages n a reasonable tme perod

8 Fgure 16 Indvdual col mages (top row) and raw senstvtes (bottom row) per Eq[12] Auto-calbraton n whch reference data are acqured concdent wth the magng have the beneft of adaptng to changes n poston of col or body that mght otherwse occur between calbraton and magng Several auto-calbratng methods are based on acqurng addtonal k-space center lnes each tme frame [21-24] or by usng non- Cartesan samplng such as radal k-space acquston whch naturally oversamples the center of k-space The central k-space data s then used to reconstruct low resoluton mages for each col whch are used to estmate senstvty maps at each magng tme frame For example, Fg 17 llustrates the lnes of k-space acqured for Cartesan samplng wth varable densty The acquston of center lnes reduces the net acceleraton rate, although they may be used wth a generalzed reconstructon (see Fg 4) to slghtly mprove the mage qualty The lower resoluton of the reference wll lmt the accuracy of the senstvty maps whch may lead to resdual artfacts Reconstructon of lower resoluton reference mages may requre addtonal wndowng to reduce artfacts arsng from Gbb s rngng n the senstvty maps [24] In dynamc magng applcatons, t s possble to aqure a full spatal resoluton reference mage sutable for auto-calbraton by usng a tme nterleaved k-space acquston as llustrated n Fg 18 (rate, R=2 example) Multple undersampled frames may be averaged or low pass temporally fltered to reconstruct a low temporal resoluton reference mage used to calculate col senstvtes Ths method known as TSENSE [25] s dagrammed n Fg 19 Integraton or temporal flterng may be mplemented equvalently n ether k-space or mage domans wth approprate zero-fllng of mssng data central lnes used for low resoluton reference Fg 17 Varable densty Cartesan k-space acquston multple frames ntegrated or temporally fltered for auto-calbraton reference Fg 18 Example of tme nterleaved k-space acquston used for auto-calbratng TSENSE method temporal lowpass flter or ntegrator estmate B 1 -maps compute coeffcents mult-col zero-flled tme nterleaved k-space data FFT array combner (SENSE) Fg 19 Smplfed dagram for auto-calbratng TSENSE method

9 K-space methods such as auto-smash, varable densty auto-smash, and GRAPPA [21-23] are autocalbratng methods whch utlze addtonally acqured central k-space lnes, also known as auto-calbraton sgnal (m) (ACS) lnes, to calculate the coeffcents n used for SMASH reconstructon These coeffcents are calculated drectly from the k-space data wthout explctly calculatng senstvty maps In the orgnal auto-smash formulaton the coeffcents were calculated by least squares ft as: ACS [13] S ( k m k ) = n S ( k ) y y y where S (k y ) s the k-space data for phase encode lne k y and represents the col ndex In varable densty auto- SMASH the ft may be performed for multple pars of ACS lnes, ACS [14] S ( k ( m+ l) k ) = n S ( k l k ) y y, l y y (m) and the multple fts n,l cans be combned wth a weghted average based on the sgnal energy, where l represents the number of avalable fts The GRAPPA method [23] performs parallel MR reconstructons for each col and uses root-sum-of-squares combnng to produce a magntude mage It further generalzes the reconstructon by usng a block (or sldng block) ft for each col j, S ( k ( m+ l) k ) = n S ( k ( br+ l) k ) ACS j y y, jlb,, y y b where R denotes the acceleraton rate, b s the ndex of the undersampled lne wthn each block, and l represents the number of avalable fts GRAPPA, as well as VD auto-smash, use a varable densty samplng to acqure the ACS lnes each tme frame For dynamc magng applcaton, GRAPPA may be mplemented wth a tme nterleaved undersampled acquston and derve the ACS lnes by temporal averagng Ths approach s referred to as TGRAPPA [26] and has the benefts of mproved speed and mage qualty due to ncreased block sze and coeffcent averagng 6 References Early References [1] Carlson JW An algorthm for NMR magng reconstructon based on multple RF recever cols J Magn Reson 1987:74: [2] Hutchnson M, Raff U Fast MRI data acquston usng multple detectors Magn Reson Med 1988:6(1):87-91 [3] Kwat D, Enav S, Navon G A decoupled col detector array for fast mage acquston n magnetc resonance magng Med Phys 1991:18(2): [4] Kelton JR, Magn RL, Wrght SM An algorthm for rapd mage acquston usng multple recever cols Eghth Annual Meetng of the Socety for Magnetc Resonance n Medcne Amsterdam, Netherlands, 1989:1172 [5] Carlson JW, Mnemura T Imagng tme reducton through multple recever col data acquston and mage reconstructon Magn Reson Med 1993:29(5):681-7 [6] Ra JB, Rm CY Fast magng usng subencodng data sets from multple detectors Magn Reson Med 1993:30(1):142-5 SMASH [7] Sodckson DK, Mannng WJ Smultaneous acquston of spatal harmoncs (SMASH): fast magng wth radofrequency col arrays Magn Reson Med 1997:38(4): [8] Sodckson DK, Grswold MA, Jakob PM, Edelman RR, Mannng WJ Sgnal-to-nose rato and sgnal-to-nose effcency n SMASH magng Magn Reson Med 41, (1999) [9] Sodckson DK Talored SMASH mage reconstructons for robust n vvo parallel MR magng Magn Reson Med 44, (2000) [10] McKenze CA, Ohlger MA, Yeh EN, Prce MD, Sodckson DK Col-by-col mage reconstructon wth SMASH Magn Reson Med 2001 Sep;46(3): [11] Bydder M, Larkman DJ, Hajnal JV Generalzed SMASH magng Magn Reson Med 2002 Jan;47(1): SENSE [12] Pruessmann KP, Weger M, Schedegger MB, Boesger P SENSE: Senstvty encodng for fast MRI Magn Reson Med 1999:42(5): [13] Pruessmann KP, Weger M, Bornert P, Boesger P Advances n senstvty encodng wth arbtrary k-space trajectores Magn Reson Med 2001:46(4): [15]

10 2-d SENSE [14] Weger M, Pruessmann KP, Boesger P 2D SENSE for faster 3D MRI MAGMA 2002 Mar;14(1):10-9 [15] Larkman DJ, Hajnal JV, Herlhy AH, Coutts GA, Young IR, Ehnholm G Use of multcol arrays for separaton of sgnal from multple slces smultaneously excted J Magn Reson Imagng 2001 Feb;13(2):313-7 General methods [16] Wang Y Descrpton of Parallel Imagng n MRI usng multple cols Magn Reson Med 2000 Sep;44:495-9 [17] Sodckson DK, McKenze CA A Generalzed Approach to Parallel Magnetc Resonance Imagng Med Phys 2001;28(8): Array Processng [18] Roemer PB, Edelsten WA, Hayes CE, Souza SP, Mueller OM The NMR phased array Magn Reson Med 1990;16: [19] Walsh DO, Gmtro AF, Marcelln MW Adaptve reconstructon of phased array MR magery Magn Reson Med 2000;43: [20] Johnson DH, Dudgeon DE Array sgnal processng: concepts and technques Englewood Clffs, NJ: Prentss- Hall; Auto-calbraton [21] Jakob PM, Grswold MA, Edelman RR, Sodckson DK AUTO-SMASH: a self-calbratng technque for SMASH magng MAGMA 1998;7:42-54 [22] Hedemann RM, Grswold MA, Haase A, Jakob PM VD-AUTO-SMASH magng Magn Reson Med 2001:45(6): [23] Grswold MA, Jakob PM, Hedemann RM, Nttka M, Jellus V, Wang J, Kefer B, Haase A Generalzed autocalbratng partally parallel acqustons (GRAPPA) Magn Reson Med 2002 Jun;47(6): [24] McKenze CA, Yeh EN, Ohlger MA, Prce MD, Sodckson DK Self-calbratng parallel magng wth automatc col senstvty extracton Magn Reson Med 2002 Mar;47(3): [25] Kellman P, Epsten FH, McVegh ER, Adaptve senstvty encodng ncorporatng temporal flterng (TSENSE) Magn Reson Med 2001 May; 45(5): [26] Breuer F, Kellman P, Grswold MA, Jakob PM Dynamc Autocalbrated Parallel Imagng usng TGRAPPA Proc Eleventh Scentfc Meetng Intl Soc Magn Reson Med 2003; 2330 Matrx Condtonng & Regularzaton [27] Kng KF, Angelos L SENSE mage qualty mprovement usng matrx regularzaton Proc Nnth Scentfc Meetng Intl Soc Magn Reson Med 2001;1771 [28] Kellman P, McVegh ER SENSE Coeffcent Calculaton usng Adaptve Regularzaton ISMRM Workshop on Mnmum MR Data Acquston Methods Marco Island, Florda Oct 2001 [29] Lang ZP, Bammer R, J J, Pelc NJ, Glover GH Makng Better SENSE: Wavelet Denosng, Tkhonov Regularzaton, and Total Least Squares Proc Tenth Scentfc Meetng Intl Soc Magn Reson Med 2002; 2388 [30] Tsao J, Pruessmann KP, Boesger P Feedback Regularzaton for SENSE Reconstructon Proc Tenth Scentfc Meetng Intl Soc Magn Reson Med 2002; 739 Cols [31] Grswold MA, Jakob PM, Edelman RR, Sodckson DK A Multcol Array Desgned for Cardac SMASH Imagng MAGMA 2000:10: [32] Bankson JA, Grswold MA, Wrght SM, Sodckson DK SMASH Imagng wth an Eght Element Multplexed RF Col Array MAGMA 2000:10: [33] Weger M, Pruessmann KP, Leussler C, Roschmann P, Boesger P Specfc col desgn for SENSE: a sxelement cardac array Magn Reson Med 2001:45(3): [34] de Zwart JA, Ledden PJ, Kellman P, van Gelderen P, Duyn JH Desgn of a SENSE-Optmzed Hgh- Senstvty MRI Receve Col for Bran Imagng Magn Reson Med 2002 Jun; 47(6):