g D from the frequency, g x Gx

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Rosamund Davidson
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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

In many engineering and other applications, the. variable) will often depend on several other quantities (independent variables).

In many engineering and other applications, the. variable) will often depend on several other quantities (independent variables). II PARTIAL DIFFERENTIATION FUNCTIONS OF SEVERAL VARIABLES In man engineeing and othe applications, the behaviou o a cetain quantit dependent vaiable will oten depend on seveal othe quantities independent