Notes on MRI, Part II

Transcription

1 BME 483 MRI Noe : page 1 Noe on MRI, Par II Signal Recepion in MRI The ignal ha we deec in MRI i a volage induced in an RF coil by change in agneic flu fro he preceing agneizaion in he objec. One epreion for he volage induced in a coil i: dφ E d where Φ i he flu in he coil. A coon configuraion i o ue he ae RF coil o rani B 1 field o he objec and o receive ignal fro he agneizaion. Aue, for a given coil configuraion and curren I 1, he RF field generaed i B 1. By he principle of reciproci he coil receive eniiviy can be defined a C 1 B 1 /I 1. The increenal volage produced by agneizaion in an eleen dr i: de C 1 r M dr Now uppoe our agneizaion coe fro a preceing pin having agneizaion in he preence of a agneic field of ize B + B i ha a reonan frequency of ω + ω. Tha i: co ω M in ω + ω + ω and uppoe he coil i poiion in he -z plane aking he eniiviy line flu line poin in he y-direcion:

2 BME 483 MRI Noe : page C 1 C The volage induced in he coil, which will becoe our received ignal r, can hen be hown o be: r de ω + ω C co ω + ω Tha i, he volage on he wire will be a coinuoidal variaion a he reonan frequency and wih an apliude proporional o he coil eniivi he reonan frequenc and he ize of he agneizaion. We coonly ake an aupion ha ω i all relaive o ω a good aupion and hu, he ω + ω er in he apliude caling i approiaely a conan and can be aborbed ino C: C co ω + ω If we include T relaaion, he received ignal will look oehing like hi: Thi ignal i known a he Free Inducion Decay or FID free eaning i i no being driven by an RF pule, inducion he acion of a agneic oen preceing around a agneic field wa fir called by Bloch nuclear inducion, and decay eaning T decay. If we ake he Fourier ranfor of hi received ignal we will ge approiaely he following pecru:

3 BME 483 MRI Noe : page 3 where he ize of he pecral peak a +/- ω + ω i proporional o. Now, uppoe we group of pin a differen frequencie and apliude: A: A, ω A B: B, ω B Now he volage induced in he coil will be he u of hee wo group of pin: r de AC co ω + ω A + BC co ω + ω B and he pecru will have peak a frequencie +/-ω + ω Α,Β and apliude proporional o A,B. In general, he volage induced in he coil will be he uaion or inegral over he agneizaion coponen ha coprie he objec we are iaging. Cople Deodulaion The received ignal, r, i a real-valued volage. We ranfor hi o a baeband ignal uing a cople deodulaor, a hown here:

4 BME 483 MRI Noe : page 4 The coponen of hi ye are a local ocillaor ha upplie a coine and a ine wave a frequency ω. The received ignal, r, i uliplied by hee ignal and hen low pa filered LPF in order o produce a wo ignal ha have a reduced bandwidh. Thee ignal are hen apled uing analog o digial A/D converer and hen he apled ignal are cobined on he copuer o creae a cople ignal,. We fir look a he upper and lower channel of he cople deodulaor for a ingle coponen of he received ignal a locaion r. The upper channel of he deodulaor yield: and he lower channel yield: LPF 1 LPF { co ω + ω co ω } { [ co ω + co ω + ω ]} co LPF ω LPF in { co ω + ω in ω } { [ in ω + in ω + ω ]} ω We can hen conruc he cobined ignal, : + i 1 ep ep i ω i ω

5 BME 483 MRI Noe : page 5 Thi i a roaion in he cople plane a frequency ω and he reulan pecru will look like: Uing a iilar arguen, we can deerine he baeband ignal for he cae of wo objec A and B, decribed above a we ve le C 1: A ep i ω + ω A + B ep i ω + ω B Iporan poin! 1. Thee cople ignal are equivalen o he oluion o he Bloch equaion in he roaing frae of reference. Through cople deodulaion, we have acce o he ignal in he roaing frae where he frae frequency i deerined by he local ocillaor of he deodulaor.. Since he ignal ei only on he copuer, i i poible o have a cople ignal. 3. The RF coil u or inegrae hi ignal fro he enire objec or for he par of he objec o which he coil i eniive. Spaial and Teporal Variaion We will now generalize our oluion o he Bloch equaion o funcion in he objec doain, for eaple:, z,, z, i, z, y + y

6 BME 483 MRI Noe : page 6 Pleae noe he diincion beween he ubcrip, which denoe he direcion of a agneizaion vecor, and he arguen, which denoe he paial locaion of ha agneizaion vecor. We alo will allow he applied agneic field o be a funcion of boh pace and ie, bu a before, we will fir conider he cae where he applied field i only in he z direcion: B, z, B + B, z, k In iaging, we are ypically dealing wih ju wo of hee paial dienion. I can be any of hee wo, bu by convenion we will ue and y. Thu, we will ypically ju ue: B, B + B, k If B i conan, he oluion o he Bloch equaion will hen be: y,, yep i ω + ω, y where ω, y B, y. In he roaing frae, he oluion i:,, yep i ω, y y, ro In ro,, he,y in he arguen refer o phyical,y locaion in pace, wherea he y in he ubcrip refer o a ini-coordinae frae o decribe direcion of he agneizaion vecor a each poin in pace. For a ie-varying B field, he oluion will ake on a for iilar o wha we have een before in fir e of noe on NMR: y,, yep iω ep i B, ' d' and again, in he roaing frae, he oluion i:, ro,, yep i B, ' d' y

7 BME 483 MRI Noe : page 7 The Signal Equaion. Above, we decribed he volage induced in a coil and furher conruced a baeband ignal ha gave a repreenaion of he ignal in roaing frae. A dicued, he ignal will be he u or inegral of all he pin he coprie he objec. For a ulidienional objec, he ignal equaion i he inegral over he agneizaion in he roaing frae again, we will le he coil eniivi C 1: ro, ddy, yep i ω, y dd or for a ie - varying field : y, ep i B y d,, ' ' ddy Fro here onward, we will oly ju conider he agneizaion in he roaing frae. Magneic Field Non-Uniforiy There any hing ha can affec he agneic field. Thee include ain Magneic field inhoogeneiy hi reflec our inabiliy o ake he field perfecly hoogeneou. Mo agne are hied o abou.5 par per illion over he ize of a huan head. Magneic ucepibiliy hi i he agneizaion of iue ielf. Differen iue, bone and he urrounding air all have agneic ucepibiliy difference of everal par per illion. The ne field i given a B B 1+χ, where χ i he agneic ucepibiliy χ air i nearly, χ waer i abou 91-6 or 9 pp. Cheical hif Differen cheical pecie have differing hielding of he nucleu fro he urrounding elecron cloud. Here he ne field i B B 1-σ, where σ i he cheical hif a poiive cheical hif iplie hielding of he nucleu or a downward hif in he field. A coon cheical hif he hif beween waer proon bonded o O and fa proon bonded o C: σ wf i abou 3.35 pp. A 1.5 T, hi reul in a hif of he reonan frequency of abou 15 Hz. Below i a proon pecru in he huan head wih cheical hif along he -ai wih he bigge 3 peak being N-aceeyl aparae NAA, Creaine, and Choline waer ha been uppreed and here i no fa in

8 BME 483 MRI Noe : page 8 he iddle of he head: radien Thee are inenional linear variaion in he agneic field. radien radien field are he principle ool for localizaion in MRI. I i iporan o reeber he gradien field vary along oe paial direcion, bu ha field line are aligned o he ain agneic field. For eaple: z The -radien y

9 BME 483 MRI Noe : page 9 The Y-radien z y The Z-radien z y 1D Localizaion Le look a he eaple of a conan, linear variaion in he applied field known a a gradien. Specificall le he variaion be he direcion, oluion o he Bloch equaion i: B, y,, hen he

10 BME 483 MRI Noe : page 1 ro,, yep i, yep i ω where he pin will prece a a frequency relaed o locaion, ω or f Iporan! Noice in he preceding epreion ha frequency ha a one-o-one correpondence o paial locaion in. The ignal equaion for hi eaple i: Now, le define, yep i ω, yep i ddy, y dy a funcion ha repreen he inegral over y of all agneizaion a each locaion. Here he ignal i: ep i d ddy Now, if we ubiue, we can ee ha i really ju he 1D FT of : ep i d F { } M / / Now, if we wan o deerine recall, our goal in MRI i o ake iage of he agneizaion, hen i ee logical o ake he invere FT of he received ignal. Rewriing he above relaionhip we ge: M /

11 BME 483 MRI Noe : page 11 and now: { } F M F 1 1 Recalling, everal of our FT relaionhip, we can alo how ha: { } / f F Thi i he ae relaionhip beween frequency and paial poiion decribed before. The negaive ign coe fro he fac he pin prece in he negaive direcion e.g. he negaive ign in ep-iω. Le look a an eaple wih rec/. The received ignal will now be: { } F inc inc / /

12 BME 483 MRI Noe : page 1 The 1D FT of will be: { } f F F f f f rec rec rec inc / / / ae a he original objec.