Notes on MRI, Part III

Transcription

1 BME 483 MRI Noes 3: page 1 Noes on MRI Par III 1D Iaging Frequenc Encoding radiens ac o seup a one-o-one correspondence beween frequenc and spaial posiion. This is nown as frequenc encoding. For eaple: raphicall his is: Low Frequenc B Mag. Field Srengh ω or f 2π High Frequenc Posiion Low Frequenc Objec 1D Iage High Frequenc Posiion As preiousl described o creae an iage we need o ae he 1D forward or inerse FT of he receied signal: Low Frequenc MR Signal ie Fourier Transfor The pulse sequence o produce his igh loo lie his: Objec 1D Iage High Frequenc Posiion RF Daa Acq.

2 BME 483 MRI Noes 3: page 2 Because we hae a consan gradien on while we are acquiring daa he frequenc spaial locaion relaionship holds. The negaie gradien a he beginning is due o he necessi o reerse ie. Preiousl when we defined he relaionship: s F { } M s ep i2πs d s / 2π / 2π We ignored he fac ha we where onl acquiring he FT Ms for posiie ie in 2πs ie begins a he poin of ipping he agneizaion ino he ranserse plane. If we hae a negaie gradien for a period of ie leading o negaie accuulaion of phase hen we hae effeciel urned bac he cloc o a negaie saring ie for phase accuulaion. Alernae Mehods for 1D Localizaion Phase Encoding While he consan gradien is an ecellen ehod for 1D localizaion i is no he onl ehod. I is iporan o noe ha i is he insananeous phase of he agneizaion a each saple ha is iporan and no he insananeous frequenc ha where he agneizaion is poining he cople plane e.g. ro is wha is reall iporan and no how fa i is oing. Recall we are sapling he receied signal: s ro dd Thus we can loo a oher ehods ha will produce he sae disribuion of agneizaion in he roaing frae. Consider he following pulse sequence: RF Daa Acq. In his pulse sequence he sae aoun of phase accuulaion occurs beween saples as in he preious pulse sequence. When he gradien apliude is zero here is zero frequenc offse B ω 0 and hus no phase accuulaion during hese periods here is of course T2

3 BME 483 MRI Noes 3: page 3 deca bu we re ignoring ha for now. If ou looed a a paricular spin i would hae a sopacion ind of appearance: M STOP STOP We could iaging puing a whole bunch of hese fraes ogeher and i igh hae he appearance of a coninuous oeen a a frequenc ω bu in his case i is reall occurring in lile jups. If we ae he FT of hese saple we will sill ge our 1D iage. Consider e an aleraie pulse sequence: RF Daa Acq. Here here is a differing aoun of phase accuulaion afer each RF pulse. In his case i coes fro haing differen alues for he sae aoun of ie bu he ipac is he sae: M STOP STOP Saple 0 Saple 1 Saple 2

4 BME 483 MRI Noes 3: page 4 Again if we ae he FT of hese saple we will sill ge our 1D iage. In boh cases he saples are colleced along a diension ha loos lie ie bu isn quie. Soe in MRI call his diension pseudo-ie. In hese eaples he spaial locaion is no encoded in he frequenc bu raher is encoded ino a specific sequence of phase accuulaion beween saples. This is nown as phase encoding. 2D Iaging One wa o do 2D iaging is o cobine frequenc encoding in one spaial diension e.g. wih phase encoding in anoher spaial diension e.g.. Tae he following pulse sequence: RF Daa Acq. For a gie phase encoding alue if we 1D FT he group of saples we ll ge a 1D iage in. If on he oher hand we ae he n h saple fro each group of saples ha is each of hese differs onl b he phase encoding aoun and ae he 1D FT of his daa we ll ge a 1D iage of he objec in. Thus i we pu he daa in an arra each group along a row and ae he 2D FT of his daa we will ge a 2D iage of he objec! The aboe pulse sequence is nown as he spin-warp pulse sequence is i is he os used pulse sequence iaging ehod in MRI. -Space We will now deelop he general heor for 2D iaging. The receied signal is: s ro dd

5 BME 483 MRI Noes 3: page 5 Again please bear in ind ha we are iing coordinae sses here. For eaple in ro he in he arguen refers o phsical locaions in space whereas he in he subscrip refers o a ini-coordinae frae o describe direcion of he agneizaion ecor a each poin in space. Le s consider a spaiall and eporall aring applied agneic fields inroduced b iearing gradien fields: B B 0 The insananeous frequenc a each poin is space is hen: B B 0 which in he roaing frae is: ω and he spaiall arian phase disribuion is: φ d 0 where ie begins wih each RF pulse he bring agneizaion fro he longiudinal ais ino he ranserse plane where i is obserable. The Signal Equaion reisied. For conenience we will le C 1 and we will define 0 0 r. We now ge a reised ersion of he signal equaion: dd d d i dd d i dd i dd s ro ep ep ep φ

6 BME 483 MRI Noes 3: page 6 Finall we define wo quaniies: 2π 2π 0 0 d d And subsiuing ino he aboe signal equaion: s F ep i2π { } M 2D u dd Tha is he signal is equal o he Fourier ransfor of he iniial agneizaion ealuaed a locaions defined b and aboe. Reinder: u i 2π u F2 D{ g } g e dd The signal equaion sas ha saples of he receied signal are equal o saples of he 2D Fourier ransfor of he objec. This ae sense if we hin abou wha eacl he epression for he 2D FT eans he FT a an poin u is he inegral oer he objec odified b a spaiall arian linear roaion in he cople plane. ep-i2π ep-i2π

7 BME 483 MRI Noes 3: page 7 ep-i2π ep-i0 In MRI he inegraion is perfored b he inegraion of olages in he RF coil. The phase ariaion is perfored b he gradiens b shifing he field and hus frequenc in a spaiall linear fashion for a period of ie he agneizaion will roae o a new orienaion in he cople plane. Thus MRI has eacl he sae echaniss as he FT operaion. K-space alwas begins a he origin 00. Wh? Afer he eciaion pulse all spins across he objec are poining in he sae direcion e.g. ep-i0 and he inegral of his is he DC alue of he FT. If we wan o deerine he objec we us full saple is Fourier ransfor. A sequence of saples can be iewed as saples along a pahwa deerined b running inegral under he gradien waefors as defined b and aboe. The final objec can be reconsruced sipl b aing he inerse 2D FT of he sapled Fourier daa -space daa: F { M } 1 2D Does i ae sense ha our saples in ie are acuall saples of spaial frequenc daa? Reeber he 1D case we sapled ie FT ed o ge a specru. Since here was a 1-1 correspondence beween frequenc and spaial posiion he FT of he frequenc daa produces ie-doain daa hus here is 1-1 correspondence beween ie and spaial-frequenc. Coens:

8 BME 483 MRI Noes 3: page 8 - Fourier space is called -space in he MRI lieraure - and conrol he -space rajecories or pahs on which saple locaions fall. - To creae an iage we us saple Mu densel enough o preen aliasing and of a large enough een o hae sufficien spaial resoluion. 1D Iaging We firs eaine he case of a 1D objec rec/w. The receied signal will be he Fourier ransfor Mu W sincwu ealuaed a paricular locaions as dicaed b he inegraion of he gradien waefor. Presened here is pulse sequence for a 1D iaging eperien along wih he -space alues and he receied signal: Recall he in he firs MRI lecure we aled abou aing he FT of he receied signal o ge a 1D iew of he objec. In his case he receied signal is a sinc funcion and hus he 1D FT of he sinc funcion is a rec funcion which is in fac he objec. During he enire ie ha daa is acquired see he Daa Acq. line in he pulse sequence he gradien is consan during his siuaion here is 1-1 correspondence beween frequenc and spaial posiion. This is now as frequenc encoding since spaial locaion is encoded as frequenc.

9 BME 483 MRI Noes 3: page 9 Noice also ha we use a negaie gradien before he posiie gradien. Wihou he negaie gradien we can onl acquire he posiie spaial frequencies or onl ½ of he FT of he objec. This is he difference fro he preious eaple here we oo he FT of s bu assued ha we new s for all ie reall we onl now if for posiie ie. 2D Iaging using Projecions The firs 2D iaging ehod ipleened in MRI b Paul Lauerbur while a SUNY Son Broo used a series of 1D acquisiion wih he gradiens in differen direcions. Please no ha b appling 1D gradiens in and siulaneousl we ge a single 1D gradien a an angle θan -1 /. Thus we can ge 1D iews of he objec or projecions fro an differen angles. We ll discuss in he secion on copued oograph he ehods for reconsrucing iages fro 1D projecions. For now suffice i o sa ha if we acquire enough projecions we can full deerine he underling objec. The pulse sequence used b Lauerbur is gien here:

10 BME 483 MRI Noes 3: page 10 B sweeping hrough angles [02π we acquire he full -space Fourier daa for he objec. There is also a arian on projecion iaging in which he posiie gradien is preceded b a negaie gradien his will allow boh posiie and negaie frequencies o be acquired along a paricular line in -space. One adanage o his approach is ha one onl needs o sweep hrough angles [0π in order o full acquire he -space daa. 2D Spin-Warp Iaging The os coon acquisiion used in MRI oda is nown as he spin-warp acquisiion. The pulse sequence is gien here along wih he corresponding -space rajecor:

11 BME 483 MRI Noes 3: page 11 This is a repeaed pulse sequence wih a differen -gradien alue for each RF eciaion TR ineral. Le s loo a he -gradien as in he case of 1D iaging aboe his gradien encodes he spaial posiion ino frequenc. This is ofen called he freq gradien or in his case is nown as he frequenc direcion. The -gradien is on briefl before each acquisiion bu is no on during daa acquisiion. Thus whaeer encoding perfored b he -gradien is done. In his case he -gradien ses up a spaiall dependen phase disribuion ha reains fied during he frequenc encoding process. In oher words he -gradien encodes spaial posiion ino he phase of he agneizaion direcion of he ecor which is nown as phase encoding. in his case is nown as he phase direcion. Below are depicions of he phase disribuion se up for he and 2 phase encoding seps. These correspond o and 2 ccles of phase across he field of iew respeciel.

12 BME 483 MRI Noes 3: page 12 ep-i-1 T ep-i0 T ep-i1 T ep-i2 T In ers of paraeers described in he aboe pulse sequence we can define seeral paraeers of ineres in he acquired space. The saple spacing and widh of he -space are: read T N W T N W T a π π π π

13 BME 483 MRI Noes 3: page 13 Sapling in -space spaial frequenc doain. Preiousl we discussed sapling of he objec and is effec on he specru. Here we hae he reerse sapling in Fourier doain and is effec on he reconsruced objec. Again we will perfor our sapling b ulipling a funcion ies he 2D cob funcion. Wih saple spacing of and in he and direcions he sapled Fourier daa is: cob ~ n n M n u u u M u M δ The iage space doain equialen is: n n n u n u u u ** cob ** ~ δ Thus sapling in he Fourier doain leads o replicaion in he iage doain. Spacing of he replicaed iage objec is 1/ 1/. The replicaed iages will no oerlap he original iage if he highes spaial posiion in is 2 1 a and he highes spaial posiion in is 2 1 a. If his is no saisfied hen here will be spaial oerlap in he iages or aliasing. The field of iew of an acquisiion is picall defined as one oer he -space saple spacing: FOV 1/ and FOV 1/ and aliasing will no occur if a < ½ FOV and a < ½ FOV.

14 BME 483 MRI Noes 3: page 14 Poin Spread Funcion. Obsere ha he pracical -space is no of infinie een bu raher is liied o W and W. The sapled -space can be wrien as: ~ u u u M u M u rec cob W W which resuls in an iage of he following for: ~ ** W W sinc W sinc W ** cob u The sinc funcions are he poin spread funcion and he cob funcion generaes replicaed ersions of he objec does he aliasing. Obsere ha he sinc funcions hae approiae widhs in and of 1/W and 1/W. This defines in essence he spaial resoluion of an MRI acquisiion in order o ge beer finer spaial resoluion we need o acquire a larger area in -space. Resoluion of he FFT. Mos fors of he FFT wor his wa for an N poin inpu funcion he FFT will produce an N poin oupu. Each oupu poin corresponds o an ineger nuber n of ccles in ep-i2πn across he objec and go fro n [-N/2:N/2-1]. Obsere he DFT of - N/2 and N/2 are he sae and hus his represens he enire unaliased frequenc doain of he objec. Thus F/2 see below ½ of 1/ or F 1/. A siilar arguen can be ade in reerse o ge T 1/f.

15 BME 483 MRI Noes 3: page 15 Resoluion and Objec and Saple Spacing in MRI. For daa acquired on a 2D recilinear grid in -space and reconsruced wih a 2D FFT he spaial resoluion and Field of View relaionships are: FOV 1/ and FOV 1/ 1/W and 1/W