g D from the frequency, g x Gx
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1 These changes wee made /7/2004 to these notes:. Added an eplanato note (in pink! in paentheses on page Coected total deivatives to patial deivatives in Eq. (4 and peceding equation on page 4 and in Eqs. (32 and (33.. Distotion aising om static-ield inhomogeneit o Spin-Echo images.. In-plane eect Assume the usual (spin-wap SE sequence ist with no static-ield inhomogeneit (i.e. D B 0 : ( ( a - i2p ( + gt t p s t A e d d I we add inhomogeneit: ( s t ( a ( p ( a - i2 p + gt + t DB t A e d d ( ( ( a ( p ( a - i2 p + gt + t DB t s t A e d d We note that thee ae two tems in the eponent that involve t. We combine them: - i2 p (( + B gd a( + t t p s t A e d d ( ( a Then we divide and multipl b to pepae o a change o integation vaiable: B( a i2 p D g ö ö - + t + ç ç p t s t A e d d ( ( a The new integation vaiable is ( ( DB a a º + (2 It s meaning is that because o a shit o B( a the should have been pecessing the spins at Ate the change o vaiables we have g D om the equenc g at which a act as i the ae located at a. - i2p ( + gt t p ( ( a( a (3 s t A e d d
2 must eist which means that Eq. (2 poduces a dieent We insist that the unction ( a o eve dieent which in tun means that o ied values o and ( a with. iven that ( inceases a inceases with we know that the deivative > 0 and is theeoe neve equal to eo. This is a easonable assumption as long as BD Since is noneo the deivative going the othe wa eists: ö ç (Note that in addition to the equiement that be noneo we ae using the act that this change o vaiables involves onl one vaiable. I in addition to deining the new vaiable o integation ( we also deine g ( a then the invese elationship will eist i and onl i g 0. Thus the invese would eist o eample i and but not i and. Using ou deinitions o k and k we have - ö - i2p ( g+ k k s( k k A ç a( ( a e d d Taking a Fouie tansom to calculate the value o the image at the position we have whee - F ( { } + i2p ( g+ k k i s k k s k k e d k d k - d ö ö B ç ç d ( ( a a - d ö B ç d ( ( a a - 2
3 ( DB a + d BD + d The thid epession o ( i above tells us that not onl do the spins at a act as i the wee located at d a but also thei image intensit appeas to be divided b d ẋ The eason o the shit om to is as mentioned above the shit in thei equenc caused b. The eason o the change in intensit is moe subtle. Thee ae thee possible situations: D B : Thee is no change in the intensit > : The intensit is deceased. This happens because the signal om the spins is spead out ove a lage potion o the image. < : The intensit is inceased. This happens because the signal om the spins is ocused onto a smalle potion o the image. Discuss Homewok 3: Signiicant igues Conusing and in numbe 2 Simple calculation o numbe 5 (I counted 5 points o o a non-simple calculation Discuss pesentations ive classic papes om which most othe eeences sping: Fowad-evese: Chang and Fitpatick 992 Delaed echo Sekihaa and/o Feig 985 Sumanaweea 993 Jead 995 Show how to ind thei citations on using Web o Knowledge We can ewite the inal equation above o ( i as ollows: 3
4 o whee ( 0 - ö i ç B a - ö i ç i0 ( ( i is the image that would be omed i thee wee no inhomogeneit. (4.2. Though-plane plus in-plane eect. Duing slice-selection spins ae ecited b vaing amounts. The degee o ecitement is measued in tems o the signal-poducing pat o the magnetiation M. Thee is a equenc poile that descibes the sie o the signal poduced b the spins as a unction o the spins equenc o pecession n. The unction is convenientl witten in the om p( n - n whee n is the equenc o the RF pulse. The shape o the unction p( n - n is detemined b the time-dependence o the RF ield B in the otating ame. Detemining p( n - n when given B t when given B t equies that the Bloch equation be solved numeicall. Detemining p n - n equies inveting the Bloch equation. The elationship between these two unctions p n - n is B t which in tun is similal is beond the scope o this couse but o ve small lip angles (< 5 degees appoimatel popotional to a Fouie tansom o popotional to an invese Fouie tansom o is tpicall desied to ecite unioml a ange mean that p( n n a sinc unction s icn( pd tn the shape o p n - n. Theeoe o small angles since it D n gd o equencies which would ect n n B t is made to appoimate - equals the boca unction ( -. Note: The eects that we ae consideing ae pesent egadless o p n - n so at this point we do not need to make an assumption. Below we will assume howeve that both the spin densit and the spatial deivatives o D B ae appoimatel constant ove the noneo ange o p( n - n. I we include slice selection eplicitl in ou epession Eq. ( o the signal om a SE sequence it becomes ( ( n( n - s t v ( ( p - i2 p + gt + t DB t A p e d d d 4
5 whee n ( g D ( n g ( B+ B B 0 whee have named ou selected slice instead o a to o late notational consistenc. Theeoe ( s t ( ö ö ( ö DB ö - i2p + g t + t ç ç DB p A pç g ç + - ( e d d d Now we change integation vaiables o both and : DB ( º + DB ( º + (5 We insist that the unctions ( and ( both eist. This equiement is equivalent to the equiement that the Jacobian o the tansomation om to be noneo. The Jacobian is ö J ç which in this case becomes DON T SHOW EQ. (6. IT IS PART OF THE SOLUTION TO A HOMEWORK PROBLEM! 5
6 + B D B D B D ö J ç 0 0 B D B D B D + (6 Eq. (6 educes to ö BD BD J ç + +. Note that we can view J as being a unction o eithe o since eithe set o vaiables can be detemined om the othe set. Thus ö J ç J J ( ( It is easonable to epect this Jacobian to be noneo as long as BD BD The change o vaiables leads to ( ( g ( ' - ( ( ' s t A p - i2p ( + g t t p - e J d d d ö ( ö ( ' s( k k A( g - p( B D BDd e ç + + whee the aguments o dd ç -i p g k + k 2 D B ae the same as o. Taking a Fouie tansom to calculate the 6
7 value o the image at the position we have - F ( { } + i2p ( g+ k k i s k k s k k e d k d k ( ( ( B ( g p- ( d BD BD + + (7 whee again the aguments D B ae the same as o. At this point we make an appoimation. We assume that both and the deivatives o and ae constant ove the ange o o which p is noneo. Thus ove that ange ( ( ( ( ( ( ( ( ( ( ( ( B D D B B( ( D ( D B ( ( D B With this appoimation ( ( ( ( ( i( B p g- ( d BD BD + + ( ( ( C BD BD + + i0( ( ( BD BD + + (8 whee i 0 is as beoe the image that would be acquied i thee wee no inhomogeneit and 7
8 DB ( º + DB ( º + (9 Since the tansomation in Eqs. (9 is one-to-one we can ewite the last o Eqs. (8 as ( ( i0 ( B( D BD ( i + + (0 whee we ae o notational consistenc also using ( Now we deine a new coodinate sstem otated about the ais elative ou ou cuent sstem - b q º tan(. We designate coodinates that ae eeed to this sstem b putting a : ove them. Thus % c oq+ ss qi n % s- iqn+ c qo s % c o+ q s s qi n % - s iq+ n c qo s whee that s i n 2 2 q + and c o s 2 2 q +. With this tansomation we ind DON T SHOW EQ.(2. IT IS PART OF THE SOLUTION TO A HOMEWORK PROBLEM! 8
9 ( ö ( DB D B ö % ç + + c q o+ ç s s q i n % c oqs söqi n B% + ( % D % ç + % ö + B% ( + % % D + ç 2 2 ( ( % B+ % % D % + (2 which becomes % % + B D % % % % (3 whee % is a simple unction o the gadients. DON T SHOW EQS.(4 o (5. THEY ARE PART OF THE SOLUTION TO A HOMEWORK PROBLEM! % + + ( It can easil be shown also that % s iq n c qo s (5 These two igues show cute (but mabe not so useul! geometic elationships among and % : q % q q % Futhemoe 9
10 DON T SHOW EQ.(6. IT IS PART OF THE SOLUTION TO A HOMEWORK PROBLEM! ( ö ( DB D B ö % ç - + s+ q i ç + n c q o s s iqn cö qo s % B+ ( % D% ç ö + % B ( + % % D + ç (6 which becomes % %. Futhemoe DON T SHOW EQ.(7. IT IS PART OF THE SOLUTION TO A HOMEWORK PROBLEM! 0
11 ( ( B D BD + + BD % ö BD % % BD % Bö D ç ç + % % % % B % D % öb D % % ö ç ç + % % BD + ö % ç D + ö + ç B - + % ç B D + ö B+ D % ç % BD + % % (7 Using Eq. (3 in Eq. (7 gives Thus in summa ( ( % + B D + BD % ( % %% DB % % + % % % % % ( ( B D BD % + + % (8 (9 Using Eqs. (9 in Eq. (0 ields - % ö i ç i 0 % ( % %% % % (% (20
12 which looks supisingl like Eq. (4 which ignoed though-plane distotion. Again the intensit can change but this time it is a unction o the deivative in a diection between the and aes..3. Echo-Plana Imaging (EPI We ignoe the eects o slice selection o now and again alte Eq. ( this time to coespond to Spin-Echo EPI. Note that we ae assuming a sequence with onl owad-going eadouts: ( DB( ( g ( s m n A+ p - and the integal ove and can be tansomed as ollows: whee -i2 p D ( mg+ t + tnb D Bp( D a ( + m td n t e d d d -i2 p D ( mg+ t + tnb D Bp( D a ( + m tdn t e d d a ( ö öd ( öd DB a B a t - i2p + m D g + t + t n D ç ç ç p D t DB( ö a D B ö a ( ö e - i2p ç gç + md + t +ç n ( e ( Dt ç ç p e d d e dd D is the blip gadient which is used to povide the phase encoding between ( e eadouts and º Dt td is the eective gadient o the puposes o assessing the p eects o inhomogeneit. It is the gadient that would equied to poduce the same change in phase b being tuned on at a constant ate duing D t as D would b being tuned on o onl onl t p. Since it has so much moe time to act it is much smalle than D. It is impotant to note hee that the time dependence o the blip gadient need not be a squae wave. The impotant thing is the eect caused b the gadient ove the time that it is applied. Its time dependence ma be o an shape without aecting the image so long as the integal has the desied value. Fo this moe geneal case t 0 p D t d b b (2 2
13 p ( e t D 0 b b t t d t D. (This independence o time shape is tue o all phase encoding gadients including those used o the non-epi sequences. As usual we deine spatial equencies o and. Fo EPI the ae as ollows: k m g D t e g D k n t Similal as beoe o spin-wap imaging we can deduce om these epessions that With these deinitions the integal becomes DB a B ö a ö - i2p ç ç + + k +ç ( k ç e ç ( ö D ( ( g X º D t - ( e ( g Y º D t e d. d (22 Now we tansom the integation using - DB ( º + DB ( º + e DB ( º + (23 The Jacobian o this tansomation can be shown to be equal to ö B D B D B D J ç (24 3
14 Using Eqs. (23 and (23 in Eq. (22 and inseting that into the integation that we last saw in Eq. (2 ields ( s k k ö ç ( ö ( ' ( ' - i2p ( g+ k k A ç p( g ( - B d B B e d D D d ç + + D + Taking the invese Fouie tansom gives - F ( { } + i2p ( g+ k i s k k s k k e d k d k ( ( ( ( B ( g -p ( d BD B D B D which is ve simila to Eq. (7. Making the same assumptions that we made ate Eq. (7 but including changes with espect to as well as and ields ( ( ( ( ( ( ( ( ( ( ( ( ( ( i( B p g - ( d BD B D B D C BD B D B D i0 BD B D B D whee i 0 is as beoe the image that would be acquied i thee wee no inhomogeneit and DB( ( º + DB ( º + DB ( º + (27 (25 (26 4
15 Whee these equations ae simila to Eqs. (8 and (9. As beoe we insist that the Jacobian o ou tansomation Eq. (with pimes eplaced b subscipt ones (24 is noneo Since the tansomation in Eqs. (9 is one-to-one we can ewite the last o Eqs. (8 as ( ( i0 ( B( D B D ( BD ( i As beoe we can otate ou coodinate sstem to make things simple. B deining a new coodinate sstem otated so that % % ( ( + ( + ( ( ( + ( (Thee ae an ininite numbe o such coodinate sstems! we can show that (28 ( % %% DB % % + % % % % % ( ( ( B D B D BD % % (29 whee % + + ( With these elationships Eq. (28 becomes ( + ( + ( 5
16 - % ö i ç i 0 % ( % %% % % (% (3 which is eactl the same as Eq. (20!.4. Ignoing small geometic distotions Tpicall in a spin-wap image? and in an echo-plana image ( e? and ( e?. In act in ode to ca out EPI in one shot it is necessa to move ve quickl though k-space paticulal in the diection so is ve lage. As a esult o spin-wap images %» %» - ö» ç DB ( º +» J - ö i» ç i0 ( ( (32 and o echo-plana images 6
17 %» %»» DB + e» J - ö» ç - ö i» ç i0 ( ( (33.5 EPI o back-and-oth eadout EPI ael has owad-going eadouts onl. Tpicall the eadouts go back and oth. The integal in that case has the same om o even values o n but o odd values it looks like this: -i2 p-( md g t + tn + b BDp( D a ( + m t Dn t e d d ( ö ö D ( öd DB a B a t -i2p - m+ g D t + + t n D ç ç ç p D t p e d d e DB( ö a D B ö a ( ö -i2p ç g- ç + m D + t ç + n ( e ( e Dt ç ç DB a ö B a ö -i2p ç ç - + k+ ç + ( k ç e ç ( ö D ( e d. d We alte the signal in this case b evesing the sign o k o odd values o n whee ( e k ng D t. As esult we have ( ( DB a B ö a ö - i2p ç ç ± + k +ç ( k ç e ç ( ö D ( dd s k k e d d whee the ± is a plus o even n and minus o odd n. This change o sign means that ou change o vaiables o no longe woks. The eect on the image is not at all obvious. It causes a ghost image to appea above and below the tue image. The eect is called N-ove-2 ghosting and it is called that because the ghost appeas to be a aint ough acsimile o the unghosted image shited up N/2 piels and anothe vesion shited down N/2 piels. (Fo those who ae inteested in the eason o this eect it is a consequence o the act that even lines ae slightl dieent om odd lines. The 7
18 mathematical desciption o this eect involves multipling in k-space b a unction that changes om one to eo and back as we move upwad (-diection in k-space b one discete unit. Multipling b a modulation unction in k-space is equivalent to convolving with a kenel in image space whee the kenel is the Fouie tansom o the modulation unction. The Fouie tansom o the modulation unction in this space is a kenel that equals eo evewhee ecept at the oigin and at positions that ae above o below the oigin at a distance o ± q N/ whee q is an intege. awa. The value o the kenel at these non-eo positions is o the even lines; o the odd lines it is one o even values o q and - o odd values. Howeve since is ve lage the eect is negligible so we can neglect this tem. As a esult we can use the appoimations given b Eqs. as i the eadout wee alwas owad going. 8
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