Images by Nulear Magneti Resonane.
A Modied Version of the EPI Method
SimoneSouza Ramalho
, Nilson MendesBorges, and Waldemar Wolney Filho
InstitutodeFsia
UniversidadeFederaldeGoias
CaixaPostal131,74001-970, Goi^ania,GO,Brazil
Reeivedon29August,2000
It iswellknownthatdataaquisitiontimesandimage reonstrutionproeduresinonventional
magnetiresonaneimaging(MRI)areusuallylong.Fortimedependentphenomena,suhassample
motion,respiration, bloodowpulsationandothers,fasteraquisiontimesaredesirabletoavoid
ghostingartifats. Overthelast twodeades,agreatnumberoffastandultra-fastMRImethods
have been proposed that greatly redue the aquision times. One of them is the eho-planar
imaging(EPI)methodwhihusesasingleexitationshemetosanentirelythek-spaeinavery
shorttime. Intheinitialsetionsofthisworkwewillsupplythereaderwithabriefdesriptionof
theimagereonstrutionproeduresfortheonventionalMRIsequeneandfourdierentshemes
oftheEPImethod. However,ourmainobjetiveistoproposeamodiedversionofthespin-eho
EPImethod,suggestingasequenewiththeuseofaonstantphaseenodinggradient.
I Introdution
Thepriniplesofthenulearmagnetiresonane
imag-ing (MRI)tehniquewere rstly introdued by
Mans-eld,GrannellandsimultaneouslybyLauterburinthe
yearsof 1973,1974 and1975 [1 4℄
. Sine itsinvention,
theevolutionofthetehniquewasveryfast,havingan
immediate appliation in the medial area as a very
powerful tool for visualization of tissue anatomy and
struture. Alsoithasbeenusedasavaluablediagnosti
method forinvestigatingmalignanttissues,sine
spin-lattie relaxationtime in anerousgrowths is known
tobedierentfrom thatin normaltissue [5℄
. TheMRI
methodhasfurtherfoundappliationsinphysis,
hem-istry, materialsiene,biologyandbiophysis [6℄
.
How-ever, the proess of obtaining onventional magneti
resonane imaging (CMRI) is slow, when ompared
with omputarized tomography (CT), for instane [7℄
.
Thetimes for data aquisition and reonstrution
im-agesare long,typiallyof the order of 10 minutes or
more [8℄
. This happens due to the need of long
wait-ing periods for the repetition of the radio-frequeny
(RF) pulse sequenes, whih are of the order of the
spin-lattie relaxation times T
1
. This eetively
re-dues its appliability in medial linis, where time
dependentphenomena arenormallypresent,asfor
ex-emplein physiologiproessesand involuntarypatient
movement [9℄
. Besidesthedelayintheaquisitiontimes,
in onventional MRI, the physiologiproesses as the
patient's breathing, heartbeats, peristalsis movement,
ardia arrythimia and sanguine ow, an ause
arti-fatsandblurringthatgreatlyaettheimagequality.
Many othertime dependentevents, suh asthe
move-mentofthewallsoftheheartorthedynamisofblood
owanonlybeinvestigatedwiththeuseoftheNMR
tehniqueifthetime ofaquisition data issmall
om-paredtothedurationofsuhevents.
Inthelasttwodeadesagreatnumberofveryfast
pulse sequeneshas beenproposed with the objetive
ofextendingtheappliationsofMRI.Thesesequenes
dier from the onventional ones beausethey redue
thetime ofaquisition tovaluesinferior to100ms,as
the result of the use of a single exitation RF pulse
or sequenes of multiple pulses of low angle to rotate
the magnetization to theplane of the seletedslie [8℄
.
More reently, using trains of low ip angle RF
exi-tation pulses, it beomes possibleto obtain images in
timesoftheorderof10ms(ultra-fastimages) [10℄
. One
of the proesses for obtaining fast images by NMR is
provided by the so-alledEPI (Eho Planar Imaging)
method.
Thismethod andfourofitsvariantswill bebriey
disussedinsetion3. Amodiedversionofthemethod
willbepresentedinsetion4. However,forabetter
un-derstandingofhowthefastandultra-fastimaging
pro-esseswork,rstlywewillpresent,in thenextsetion,
ashortdesriptionoftheproessofdataaquisitionin
theonventional MRImethod.
II MRI Reonstrution in Two
Dimensions
Inathree-dimensionalsample, underequilibrium
on-dition, the magnetizationis alignedwiththediretion
of the stati magnetield H
0
, usually applied along
the z-diretion. If the magnetization is disturbed by
a seletive exitation, thereonstrution of theimage
an be performed in two dimensions, in the plane of
theseletedslie. Theresultingimageisanaverageof
ontributions from all the nulear spinsinside alayer
of thiknessb:The imagereonstrution requires that
thek-spaeissampledandaFouriertransformationis
performed. The sampling requires the appliation of
appropriatedgradients,orientedatwill,intheplaneof
theslie. Suhexibilityisoneofthegreatadvantages
oftheMRImethodovertheomputarizedtomography
byX-ray,forinstane, wheresuh hoieof freedomis
notallowed [11℄
.
Inwhatfollows,wewillturnourattentiontothe
im-agereonstrutionin theslieplane. InCMRImethod,
thesamplingofthek-spaeisperformedusinganite
numberofsamplepoints. Thewaythesepointsare
dis-tributedisknownasraster [9;12℄
. Therastermaybe
de-sribedusingpolarorartersianoordinates. Interms
oftwo-dimensionalFouriertransform,thesimplestway
tounderstandtheproblemistousetheartesianraster,
due toKumar,WeltiandErnest [13;14℄
.
WhentheFID(FreeIndutionDeay)signalis
sam-pled in thepresene ofagradient, thedatapoints are
obtained along a single line in the k-spae. This line
is orientedalong oneof theartesianaxesand the
as-soiated gradient is known as the read gradient. The
x-oordinate isusually asribedto thisdiretion. The
interept of this line along an orthogonal axis anbe
varied by applying another eld gradient orthogonal
to the rst one, by a xed period of time, before the
sampling begins. Sine this gradient imparts a phase
modulation to the signal, it is named phase enoding
gradient. Thediretionasribedtothisgradientisthe
y-oordinate. The MRI tehnique is based upon
en-oding theresonanefrequeny ! of thenulear spins
aording to their spatial oordinates. Considering a
xed lineargradientG, appliedto asample, theloal
resonanefrequenyofthenulearspinsanbedened
as afuntion oftheoordinateras [8;9℄
!(r)=H
0
+G:r; (1)
where is thegyromagnetiratio of theresonant
nu-leus and H
0
is the intensity of the stati magneti
eld. This simple linear relation betweenthe
prees-sion frequeny ! and the nulear spin oordinates, r,
isoffundamental importanefortheunderstanding of
theimagingpriniple. Ifweonsiderthenulearspins
inanelementofvolumed 3
ratpositionr,withaloal
spindensity(r),thenumberosspinsinthiselementof
volume is(r)d 3
r. The NMRsignaloming from this
volumeelementmaybegivenby
dS(G;t)=Ae i(H
0 +G:r)t
e t
T
2
(r)d 3
r; (2)
whereAisaonstantofproporionality. The
exponen-tialterme t
T
2
is theT
2
deay termthat appearsasa
limitingfatorrelatedtothetransverse magnetization
dephasing. This term an be onveniently negleted,
providedthatthedephasingofthetransverse
magneti-zation,duetothespreadintheoresonanefrequeny
G:r, is muh faster than thedeaydue to the
relax-ationtimeT
2 .
When phase-sensitive detetion is used to observe
the nulear signal at the reeiver oil, the
radio-frequeny signal is mixed with a referene osillation
andtheresultisasignalatthedierenefrequeny,a
proess known asheterodyne mixing. Taking the
ref-erenefrequenyequalto !
0
=H
0
,the so-alled
on-resonane ondition, the signal obtained osillates at
frequeny G:r. This will be in the audio-frequeny
range and an be addressed to an analogue-to-digital
onverter (ADC). Relative to the referene frequeny
!
0
,theamplitudesignalmaybewrittenas
S(t)=A Z Z Z
(r)e i(G:r)t
d 3
r; (3)
wherewenegletedtherelaxationtermduetoT
2 . The
eqn.(3) shows thatthe relationbetween theobserved
signalS(t) and the nulear spindensity funtion (r)
has the form of a Fourier transformation. To make
thingsmorelear,Manseld [4;15;16℄
introduedthe
on-eptofreiproalvetorspae,k,dened by
k= 1
2
Gt: (4)
This equation shows that the k-spae may be
trans-versed by moving either in time or in gradient
mag-nitude. However,the diretion of gradientdetermines
the diretion of the transverse. Using the denition
expressedbyeqn. (3)andtheoneptofFourier
trans-formanditsinverse,wewrite
S(k)=A Z Z Z
(r)e
i(2k:r) t
d 3
r (5)
(r)=A Z Z Z
S(k)e
i(2k:r) t
d 3
k: (6)
TheseequationsshowthatS(k)and(r)aremutually
funtion anbeobtained throughaFouriertransform
oftheNMRsignal,aquiredin thek-spaedomain.
Inpratie,thesamplingofthek-spaetakesplae
whenthesignalissampledatsuessivetimeintervals.
In other words, S(k) is measured in the time domain
whileitsFouriertransformyields(r)inthefrequeny
domain. Inthissense,it isvalidto saythat thereis a
orrespondene between the real spae and frequeny,
andbetweenthereiproalspaeandtime [9℄
. Atually,
eqn. (4)showslearlythatGandtareinvolvedinthe
determinationof thek-spae.
The real samples are three-dimensional objets.
However,only thenulearspins ontainedin one slie
that isseletivelyexited, willontribute totheNMR
signal. The reonstrution of the image is then
per-formed in the two dimensions of the slie plane by
meansofatwo-dimensionalFouriertransform. For
ex-emple,theFIDortheehosignalobtainedforaseleted
slieofthiknessb,utperpendiulartothez-diretion,
atz=z
0
,isgivenby [7;8℄ S(k x ;k y )=A
Z z 0 + b 2 z0 b 2 Z Z
(x;y;z)e
i2[xkx(t)+yky(t)℄
dxdy
dz: (7)
d
In eqn. (7), the outer integral, involving the
z-omponent,justrepresentsanaveragearosstheslie,
and an be onveniently ignored. With this, the
re-onstrution of the averaged planar density funtion
(x;y;z
0
) of the nulear spins into an image is
per-formed by taking the two-dimensional inverse Fourier
transform,eqn. (6),nowwrittenas
(x;y;z
0 )= Z Z S(k x ;k y )e i2[xk x (t)+yk y (t)℄ dk x dk y
=S(t); (8)
d
wherek
x
(t)and k
y
(t)aregivenby:
k x (t)= 1 2 G x t (9) k y (t)= 1 2 G y t: (10)
Sine the magnetization is proportional to the spins
density distributionfuntion, eqn. (8) is also
propor-tional to the distribution of the magnetization in the
seleted slieat thetime of measurement. The
expo-nential term in this equation is periodi, with period
2. This means that the k value desribes a
wave-length=2=k. Thisis thewavelengthofthespatial
harmonisinwhihtherealobjetisdeomposed.The
possiblevaluesofk
x andk
y
are found,respetively,in
theintervals k x;min k x k x;max ; (11) k y;min k y k y;max : (12)
Aording to eqns. (9) and (10), these twoequations
maybewrittenas
( 1 2 )G x T (N r ) 2 k x ( 1 2 )G x T (N r 1) 2 (13) ( 1 )G y T (N p ) k y ( 1 )G y T (N p 1) ; (14) where N r
is the number of points per prole, N
p is
thenumberof phaseenodinggradientsand T is the
sampling-time interval. Thehighestvalueofk
x
deter-mines the smallest value of wavelength and thus, the
resolution. Thelargestwavelengthouringdetermines
whihisalledtheeldofviewFOV,denedas [17℄ FOV= max 2 = 2 k x;min = 2 G x T : (15)
Thesmallestwavelengthintheraster,andsothe
reso-lution,is therefore
min = 4 N r 2 G x T = 4 N r FOV: (16)
Basedontheformalismpresentedabove,Figs1and
2illustrate,repetively,thetimingdiagramandthe
k-spae trajetories for one of the several ways of
pro-duing a two-dimensional magneti resonane image,
the so-alled gradient eho imaging sequene. After
theequilibriummagnetizationofthenulearspins,
on-tainedin agivenslieof thesample, is rotatedto the
xy-plane,withtheuseofaslie-seletiveRFpulse, the
phase enonding and read gradients are applied for a
short period oftime, movingthemagnetization to the
point A in the k-spae. At this moment, the
phase-enondingpulseisswithedoandthereadgradientis
in theread diretion to thepointB. Whilethe signal
is beingsampled, N
r
data points areoleted along a
single line, withonstantk
y
. Thisis named \prole".
Afullehosamplingorrespondstoshiftingthe
begin-ningofsamplingtok
x
= 1=4G
x N
r
T;aquiringN
r
read pointsuptok
x
=1=4G
x (N
r
1)T,thesizeof
the prole. After N
p
repetitions of the proess, with
dierent phase enoding gradients, the whole k-spae
is sanned and an array of N
r N
p
data points are
aquired. TE isthetime ofehoformationandTRis
thetimerepetitionofthesequene. InCMRI,thetime
repetitionTR isoftheorderof therelaxationtimeT
1
and atypialarrayis asquarematrix with256256
datapoints.
Figure.1 Gradient-ehoimagingsequene. After the
slie-seletiveexitation90 Æ
pulse,thetransversemagnetization
isphase-enodedwithagradientGyandanehoisformed
in the preseneof a onstant reading gradient Gx. TE is
the timefor ehoformationandTR isthe timerepetition
ofthesequene.
Figure.2 K-spae trajetories of spin-eho. T is the
sam-pling time. G
x and G
y
are gradients responsible for
fre-quenyandphaseenoding,respetively.Therstistermed
readinggradientsinethesignalisaquiredinitspresene.
III Eho Planar Imaging -EPI
sequenes
Theeho-planar imagingtehnique(EPI) and related
variants of the method have the hability of sanning
entirely thespatial frequeny k- spae, using for that
thetransversemagnetization produed byasingle
ex-itation of the nulear spins in the seleted slie [8℄
.
These tehniques are referredto asultra-fast imaging
sequenes. Themainbenetsofthesemethodsinlude
short examination times and thevisualization of
non-yli body motions. The speialized literature
men-tion a variety of sequenes of EPI, among whih we
willdesribefourofthem,aimingabetter
understand-ingof these sequenessineourproposalis to present
a slightly modied version of the method. These
se-quenes are: i) The FLEET (Fast Low-angle
Exi-tation Eho planar Tehnique), that involves a
san-ning through the k -spae, in form of zig-zag [8;18℄
; ii)
The BEST (Blipped Eho planar Single-pulse
Teh-nique),whihisasequenethat sansthek-spaeina
retangular way; iii) The sequene MBEST(Modulus
BEST), whih is amodied version of BEST is more
rapid than FLEET;iv) The last one is the spin-eho
MBEST,whih introduesarefousing180 0
RF pulse
inMBEST,makingitfreefromT
2
,whihisthe
hara-teristirelaxationtimeassoiatedtotheinhomogeneity
ofthestatimagnetieldandotherdisturbanesthat
inreasethetransverserelaxationrate [8;19℄
.
Astheorretrasterinthek-spaeonsistsof
par-allellines,thezig-zagsweeprequiredbyFLEETneeds
appropriate ompensation. This is aomplished by
performingadoublezig-zagsanninginwhihthe
se-ond aquisition is arried out with the raster moving
intotheoppositequadrant,byreversingthesignofthe
startingG
y
gradient. Theseondsanningisperformed
immediatelyaftertherstone,withoutwaitingforthe
restorationofthemagnetizationalongthez-axis,asin
onventional MRI. In order to ensure that the signal
amplitudesarethesameinbothexitations,theangle
ofrotationdue tothe rstRFpulse must be equalto
45 0
x
whiletheseondpulsemustbeequalto90 0
x [18℄
.
The seond EPI aquisition sheme, the BEST, is
morerapid thanFLEET beausethe double sweep of
k-spaeisavoided. This beomespossiblebyensuring
thattherasterink-spaeisretangularandnotin
zig-zagform. This is ahievedby avoidingthe steady k
x
evolution assoiated with aonstant G
x
gradient. In
this sequene, G
x
onsists of a series of short pulses,
applied soon after eah k
y
blip whih are responsible
for the phase odiation. The total image
aquisi-tiontimeforBEST istypially32 ms,whih isabout
one half the time required for FLEET. However, the
BESTsequenesamplesonlytwoquadrantsofk-spae,
(k
xmin k
x k
xmax
and 0k
y k
ymax
),what
on-fersafatorof1=2penaltyinitsspetrumamplitude [9℄
.
aqui-sitionsheme,knownasMBEST,isshowninFig. 3.In
this sequenethesweepof k-spaeis alsoretangular,
asforBEST,butitsamplesfourquadrantsofk-spae,
asillustrated in Fig. 4. This is ahieved throughthe
appliation of aset ofN
p
short alternate reading
gra-dientpulsesG
x
,appliedsoonaftereahphaseenoder
pulsegradientG
y
. Alsointhissequene,thetransverse
magnetizationisproduedbyasingleslie-seletive
ex-itation90
x
pulse. This magnetizationis usedto san
thek-spaeentirely,aslongasthesanningisfast
om-pared to the relaxationtime T
2
: The transverse
mag-netization,reatedbytheseletivepulse,ispositioned
intheleft-lowerornerofthek-spae,througha
simul-taneous appliation of a phase enonding and a read
gradient for ashort period of time. Thephase
gradi-entis then ut-o while the read gradient is reversed
in order to form the rst eho. One of the aims of
the phase enoder pulses is to perform the hange of
lines when the k-spaeis beingsanned. The N
p
sig-nals of the ehoes train are reated by appliation of
suessivesshortphaseenondingpulses(blips)and
al-ternateread gradients, sothat alllines in k-spae are
aquired within an aquisition window AW. Beause
in this sequene,onseutivelinesin k-spae are
sam-pled in oppositediretions, thealternate data lines in
the array must be reversed prior to Fourier
transfor-mation. Just asthat forBEST, also for MBEST,the
ehoes are formed with appliation of the read
gradi-ents. Like theother twoformersequenes, this oneis
alsosubjetedtoT
2
attenuation. However,withuseof
MBEST, the quality of the image is almost free from
artifats orblurring, anomalies that aet the images
exellene somehow [18℄
. Besides, in this sequene, the
dataaquisitiontimeisinreasedtoabout64ms,sine
nowthefullk-spaeissampled [21℄
.
Figure.3 Eho-Planar Imaging(EPI)Sequene. The
mag-netizationisrotatedtotheslie-planebytheslie-seletive
90 Æ
RFpulseandis realledNptimes by alternatingread
gradientsGx. Thewholegridinthek-spaeissannedwith
asingleexitation. Eahindividual ehoisphaseenoded
byblippingGypriortotheolletiondataofeaheho.
Figure.4 Eho-Planar Imaging (EPI) k-spae raster. The
magnetizationisplaedintheleft-lowerornerofthegrid,
usingthepreparationgradientsGx andGy.
Ingeneralterms,themainproblemsassoiatedwith
EPIand variantmethods, whenompared to the
on-ventional MRI tehnique, are low spatial resolution,
high suseptibility to produe artifats and poor
sig-nal/noise ratio (SNR). These eets are more
aen-tuated in those versions whih use low ip angles to
rotateonlyafrationtheequilibriummagnetizationto
thetransverseplane [11℄
.
Amethodtodereaseoreventoeliminatetheeet
ofT
2
attenuationwasproposedbyRzedzianandPykett
[12;13℄
. Themethodonsistsintheappliationofa180 0
y
RFrefousing pulsein theEPI sequene,applied at a
time after the slie seletivepulse. This pulse
gen-erates an eeitive eho signal at atime 2 after the
appliation of90 0
x
pulse, refousingthespinsthat lost
their phaseohereneinthetransversemagnetization
due to homogeneous and inhomogeneous broadening,
so thatnow,theamplitudeofthespin-ehoesare
sub-jetedonlytoT
2
relaxation.Thissequeneisillustrated
in Fig. 5,whilearepresentationforthek-spaeraster
isshowninFig. 6.
In this sequene, the timing is adjusted in suh a
waythattheenteroftheehotrainTE(whihisalso
the enter of k-spae and the aquisition window) is
positionedattime2 afterthe90 0
x
RFpulse. As
previ-ouslydesribed,thetotaltimeofsanningthek-spae
in the EPIsequenes, depends on thetime for the
ef-fetiveehoformationTE andalsoonthetimelength
oftheaquisition windowAW. Completeimageswith
anarraywith128128datapointshavebeenaquired
intimesthatlastbetween25and100ms [12℄
. Thisvery
short imaging time makes funtional studies possible
and of ourse almost eliminates the eets of
impre-ditable patientmovementsonspatialresolution [22℄
Figure.5Eho-PlanarImaging(EPI)sequene. An180 Æ
RF
pulse is appliedat time after the 90 Æ
RF exitation pulse
torefoustheinhomogeneousbroadeningofthe
magnetiza-tion.
Figure.6 Eho-PlanarImaging(EPI)raster. The
magneti-zationisplaeintheleft-lowerornerofthegrid,usingthe
preparation gradientsG
x , G
y
and therefousing 180 Æ
RF
pulse.
In omparison with the onventional MRI
teh-niques, the EPI method, unfortunately introdues at
least two main tehnial drawbaks [7℄
. One is that
the alternating gradientapplied as series of
retangu-lar pulses are often diÆult to be implemented when
therequiredgradientpowerandfrequenyarehigh [14℄
.
Seondly,theimplementationofthistehniquein
stan-dard ommerialequipments,requesttheuseof
gradi-entoilsapabletogenerateeldgradientsoftheorder
of 1G=m; with slew rates faster than 200 s=G=m;
and a fast A/D onverter apable of handling
band-widthsinexessof300kHZ [8℄
. Besides,thefast
swith-ing of the gradients G
x and G
y
, using high urrents,
indues eddyurrentsinthemetalli strutures ofthe
equipmentwhih,byitsturn,generatesartifats,
harm-ing the qualityof the image. However, improvements
ingradientdrivingiruitryandtheuseoflarger
mag-netields,haveresultedinaninreaseinimagearray
size from 3232; in the infany of the tehnique, to
size of 128128, at present days, produing
aept-ableimagequality [8℄
. Inpartiular,theeetausedin
theimage quality due to theappearane of eddy
ur-rentsanbesubstantiallyreduedwith improvements
ingradientoils design [18℄
.
IV Modied version of EPI
se-quene
As it was observed in the previous setion, the spin
ehoEPIsequeneismarkedbytheappliationofquite
shortpulsesG
y
forphaseodiation. Anotherfeature
ofBESTandMBESTsequenesistherapidgradients
swithing. Theraisingandfallingtimesofthese
gradi-entsareof theorderof100s;whihonferto thema
prolesimilartotrapezesandnotaperfetsquareform
astheyareshowninFigs. 3and5. Thiseet anbe
ompensated for by using non linear signal sampling
alongthek
y axis
[9℄
:
As mentionedbefore, one ofthe main inovenients
in theimplementationof the EPImethod in
ommer-ialsystemsis the demand for produing gradients of
veryhighmagnetields(1to40G=m)andveryshort
swithingtimes(210 3
to510 4
s). Besides,there
isalso theproblem of theeddy urrentsin the
metal-listrutureofthespetrometer. MihaelStehlingand
o-workers [23℄
, show that when very fast raising and
fallingtimesarerequested,gradientoilsoflow
indu-tane (' 100H) must be used. This demands alot
fromthesystemofthegradientiruits,whihhaveto
supplyurrentsupto1000Aat320V [23℄
.
In this setion, in an attempt of minimizing these
problems,weareproposingaslightlymodiedversion
for the spin-eho MBEST sequene. As illustratedin
Fig.7,inthisnewversion,thegradientofthephase
en-odingpulse G
y
issmallinmagnitudeandmaintained
onstant during the whole readout time. A onstant
timedelayisplaedbetweentwoonseutive
alternat-ing read gradients, in order to perform the hange of
linesinthek-spaearray. Thesamplingofthek-spae
isshowninFig. 8. Sinenowthephasegradientpulse
isonstant,thek-spaegridisproduedbyslightly
in-linedlines,informofzig-zag. Inallotheraspetsthe
version is similar to the spin-eho EPI sequene. In
whatfollows,theimpliationoftheonstantphase
en-odinggradientontheimagereonstrutionproedure
Figure.7ModiedversionoftheEho-PlanarImaging(EPI)
sequene. The phase enoding is performed using a
on-stantgradientGy ofsmallmagnitude.An180 Æ
RFpulseis
appliedattimeaftertheslie-seletive90 Æ
RFpulseto
re-foustheinhomageneousbroadeningofthemagnetization.
Asitisshowningures7and8,theappliationofa
onstant phase-enonding gradient G
y
simultaneously
with the read gradient G
x
leads to a non-orthogonal
sampling in k-spae and therefore to modiations in
the reonstrution of the image by two-dimensional
Fouriertransform. Due to this kindof sheareet on
thegrid,thetrajetoriesin k-spaewillform ansmall
angle=artan(G
y =G
x
)withrespettox axis.
A-ording to eqn.(8), (x;y) represents the planar spin
density of the objet and S(k
x; k
y
); the value of the
NMR signal that would bemeasured at a given
posi-tion in k-spae. For a non-retangular grid, like the
oneshowningure8,theeqn.(8)maybemodiedand
written as
Figure.8ModiedversionoftheEho-PlanarImaging(EPI)
k-spaeraster. Theinlinedlinesisduetotheuseofa
on-stantphaseenodinggradientGy.
0
(x;y;z
0 ) =
Z Z
S(k 0
x (t);k
0
y (t))e
i2[xk 0
x (t)+yk
0
y (t)℄
dk 0
x dk
0
y
= Z Z
S[k
x (t)(1+
G 2
y
G 2
x )
1
2
;k
y (t)℄
e
2i[xkx(t)(1+ G
2
y
G 2
x )
1
2
+yky(t)℄
dk 0
x dk
0
y
; (17)
whihorrespondstothespindensitydistortedbytheshearinthereadoutdiretion. AstheNMRsignalissampled
at anite samplingrate, sothat N
x
pointsis obtainedperprole,weneedtohangeeqn. (17)toadigitalform.
Inordertoahievethis,letusdenetheplanar-retangularspindensity(x;y)by [24℄
l;m
=(lx;my); (18)
wherel;marepixelindiesandx;yarepixelsdimensions. Foradatasetmeasuredonaretangulargridspaed
byk
x =1=N
x
xand k
y =1=N
y
y,thevalueoftheNMRsignalthat wouldbemeasuredatagivenposition
inthek-spaeis
S
l;m =S(
1
2 G
x lT;
1
2 G
y
mT) (19)
andforfullk-spaesampling,adisreteFouriertransformationmaybewrittenas
l;m =
1
2 N 1
X
l= 1
N 1
2 N 1
X
m= 1
N S
l;m e
2i( lnx
N
x +
mn
y
N
y )
; n
x ;n
y =
1
2 N;:::;
1
2
However, in ourmodiedEPI version,the data set is sampled on a non-retangulargrid, based ona vetork
r ;
havingomponents
k 0
x (t)=
1
2 G
x lT(1+
G 2
y
G 2
x )
1
2
; k 0
y (t)=
1
2 G
y
mT (21)
TheFouriertransformofthisdatasetgivestheeetiveimage
0
l;m =
1
2 N 1
X
l= 1
2 N
1
2 N 1
X
m= 1
2 N
S[k 0
x (l;T);k
0
y
(m;T)℄e [
lnx
N
x (1+
G 2
y
G 2
x )
1
2
+ mn
y
N
y ℄
n
x ;n
y =
1
2 N;:::;
1
2
N 1 (22)
d
whih orresponds to the spin density distorted by a
shear in the readoutdiretion. Forsingle shot
experi-ments,onsideringG
y
jG
x
jandwithonstant
mag-nitude, wean make theapproximationk
x 0
=k
x (1+
G 2
y
G 2
x )
1
2
'k
x ;k
0
y =k
y
. Withthis,aordingtoeqns.(20)
and(22),wewillhave 0
(x;y;z
0
)'(x;y;z
0
);resulting
in a shear eet negligibly small in the image
reon-strution proedure.
Thus,with theelimination ofthesuessionof the
N
p
fastpulsesG
y
(blips); inthe onventionalEPI, we
aim toeliminate,at leastpartially,theproblem of
ir-ulantingurrentsinthemetalli stutureofthe
mag-netandonsequently,reduingtheartifatsoriginated
from thefastswithing ofthegradientoils relatedto
the phase enoding pulses. Beause the reading
gra-dients G
x
, in this new version, has the sameform as
in spin-ehoEPIsequene,theproblemofirulanting
urrents is minimized but not ompletely eliminated.
Extra improvementsouldbeahievedwiththe
devel-opmentofsreenedgradientoils [25;26℄
.
Besidestheadvantagesalreadymentioned,obtained
with thepartial elimination ofthe inonvenienes
re-ated by the irulating urrents, we an still mention
others that are related to the safety of the resonane
exams in medial linis. As already mentioned, the
tehnique of EPI diers of onventional MRI by the
fatthatitrequestsgradientswithhighmagnetields
and very fast swihting times. Aording to Sethling
et al [23℄
,these ause highvariationtaxesin the
gradi-entoils,50to100T/s, partiularlywhenimages with
betterresolutiondegreearerequested. Assoiatedwith
thesefators,inthesamereferene,therearereportsof
reentinvestigationsofsensationsinpatients,astikles
andevenpain,whensubmittedtolinialexams. This
ouldbealledollateraleetsinexamswithmagneti
resonane in humans. Using the proedure proposed
here, these eets should be diminished, beause the
gradient phase enoder magnitude is small and
main-tainedonstantduringthewholesamplingtime.
How-ever,onlyanexperimentalimplementationofthisnew
version, will garantee us wether the method is viable
and ifthe several improvements pointed outould be
onrmedin pratie.
V Conlusions
Trying to ahievea better understanding of themain
harateristisof the eho-planarimaging(EPI)
teh-nique, in therstsetion ofthis work,wepresenteda
briefdisussionoftheonventionalMRIproeduresfor
imagereonstrution. Theseondsetion is dediated
to the presentation of the main aspets of four
vari-antsof the EPImethod: namely, the FLEET,BEST,
MBEST and another one that ould be alled
Spin-Eho MBEST, trying to show their advantages and
disadvantages, asompared to the onventional MRI,
intheproessoffastimagereonstrutionproedures.
Finally, in the last setion, weintrodued a modied
version of the Spin-Eho EPI sequene. In this new
version, the fast swithing phase enoding gradients
blips were eliminated. This modiation seems
per-tinentandeasiertobeimplementedthanotherexistint
EPImethods.. Othertwobenetsthat weare
expet-ingtoahievewiththeuseofaonstantphaseenoding
gradient,are: i)A possibleimprovementin the image
quality, due to the redution of ghosting artifats
re-latedtotheirulatingurrentsinthemetalistruture
ofthespetrometer. ii)Theredutionoftheunpleasant
sensationsinpatientsinmediallinis,assoiatedwith
the use fast swithing time phase enoding gradients.
Asdisussedin setion4,apossiblepratiallimitation
of this modied version is theintrodution of a shear
eetinthek-spaegrid,whihmaybeonsidered
om-pletelynegligible, provided that themagnitude of the
gradientG
y
issmall omparedwiththereading
gradi-ent G
x
: Anothersoure of ghostingartifats, inherent
to EPI methods, is the use of two opposite polarities
readout gradients for signal aquision. The modied
proedureproposedheredoesnotoveromethissortof
diultyand proposalsfor additionalimprovementsof
Aknowlegments
The authors thank Dr. Alberto Tannus of the
In-stituto de Fsia de S~ao Carlos - Universidade de S~ao
Paulo for helpful disussions and assistane and Dr.
Orlando Afonso Valle do Amaral of the Instituto de
FsiadaUniversidadeFederaldeGoiasforreadingthe
manusript. One of us (S.S.R) thanks the support of
the Brazilian Ageny CAPES and The Universidade
Federal deGoias.
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