Noll (2004) MRI Notes 2: page 1. Notes on MRI, Part II

Transcription

1 Noll 4 MRI Nos : pag 1 Nos on MRI Pa II Spaal and poal Vaaons W wll now gnalz ou soluon o h loch quaons o funcons n h objc doan fo apl: z z Plas no h dsncon bwn h subscp whch dnos h dcon of a agnzaon vco and h agun whch dnos h spaal locaon of ha agnzaon vco. W also wll allow h appld agnc fld o b a funcon of boh spac and bu as bfo w wll fs consd h cas wh h appld fld s onl n h z dcon: h loch quaon agan splfs o spaa soluons fo h z and coponns. h quaons s wn h: and 1 d d whch has a soluon: ' ' p d I s also usful o no h spaall vaan vson of h sgnal n a fa oang a : ' ' p d o

2 Noll 4 MRI Nos : pag Kp n nd ha bgns wh ach RF puls h bng agnzaon fo h longudnal as no h ansvs plan wh s obsvabl. L s loo a h apl of a consan lna vaaon n h appld fld nown as a gadn. Spcfcall l h vaaon b h dcon hn h soluon o h loch quaon s: p p wh h spns wll pcss a a fqunc lad o locaon. Fo a gadn n an aba dcon h fqunc-poson laonshp s. and h soluon o h loch quaon s: p p Fo a -vang gadn w hav. and h soluon o h loch quaon s: p ' ' p d wh s a spaall and poall vang phas vaaon. adns adn flds a h pncpl ool fo localzaon n MRI. I s poan o b h gadn flds va along so spaal dcon bu ha fld lns a algnd o h an agnc fld. Fo apl:

3 Noll 4 MRI Nos : pag 3 z h X-adn h Y-adn z

4 Noll 4 MRI Nos : pag 4 h Z-adn z Sgnal Rcpon n MRI h sgnal ha w dc n MRI s a volag nducd n an RF col b changs n agnc flu fo h pcssng agnzaon n h objc. On psson fo h volag nducd n a col s: dφ E d wh Φ s h flu n h col. A coon confguaon s o us h sa RF col o ans 1 flds o h objc and o cv sgnal fo h agnzaon. Assu fo a gvn col confguaon and cun I 1 h RF fld gnad s 1. h pncpl of cpoc h col s cv snsv can b dfnd as C 1 1 I 1.

5 Noll 4 MRI Nos : pag 5 h ncnal volag poducd b agnzaon n an ln d s: C 1 d de h cvd sgnal s hn: V V d de E s C 1 W wll now a a nub of splfng assupons: 1. No ha z vas a a a sla o 1 1 abou 1 Hz bu vas a a a sla o on h od of 1 7 Hz. hus h volag nducd b h - coponns s abou 7 ods of agnud bgg han ha nducd b vaaons n h z coponn. hus w wll consd onl h - coponns of h snsv and h agnzaon.g. n h abov psson.. W wll assu ha col go and placn wll a snsv onl h - coponn of h agnzaon and fuho ha h snsv wll b unfo acoss h volu of ns. hus: 1 C C 1. Obsv ha { } ' ' sn I d Und hs assupons: V d d C s ' ' sn 1

6 Noll 4 MRI Nos : pag 6 wh h dvav of h agnzaon s: ' ' cos 1 ' ' sn d d Fuh splfcaons: 3. h s a h fon of h scond ln of hs psson can b splfd b obsvng ha >> >> and 1 and hus h ladng s can b placd b jus. 4. W wll nglc dca fo now. 5. W wll absob h col snsv no a consan C C 1 and fo convnnc w wll s hs consan o 1: C 1. Now h col snsv s h dvav of h agnzaon s: ' ' cos 1 d C and h cvd sgnal s: V d d s ' ' cos Copl Dodulaon h cvd sgnal s s a al-valud volag. W ansfo hs o a basband sgnal usng a copl dodulao as shown h:

7 Noll 4 MRI Nos : pag 7 W fs loo a h upp and low channls of h copl dodulao fo a sngl coponn of h cvd sgnal a locaon. h upp channl of h dodulao lds: { } [ ] { } cos cos cos cos cos 1 LPF LPF s and h low channl lds: { } [ ] { } sn sn sn sn cos LPF LPF s W can hn consuc h cobnd sgnal s: p 1 s s s

8 Noll 4 MRI Nos : pag 8 h Sgnal Equaon. I s lavl as o show ha h abov dodulaon pocss s lna scalng and supposon so n gnal w wll g: s V p d V p ' d' d V o d hus h basband sgnal s can b psnd b h ngal ov h ansvs coponn of h ansvs agnzaon n h oang fa. Ipoan pons! 1. hough copl dodulaon w hav accss o h sgnal n h oang fa wh h fa fqunc s dnd b h local oscllao of h dodulao.. h RF col ngas hs sgnal fo h n objc o fo h pa of h objc o whch h col s snsv. -Spac W fs wll loo a h cas of plana agng ha s w wll l h spaall vaan ansvs agnzaon. h cvd sgnal s hn: s C o dd o o Agan plas ba n nd ha w a ng coodna sss h. Fo apl n o h n h agun fs o phscal locaons n spac whas h n h subscp fs o a n-coodna fa o dscb dcon of h agnzaon vco a ach pon n spac. L s consd a spaall and poall vang appld agnc flds noducd b vang gadn flds:

9 Noll 4 MRI Nos : pag 9 h nsananous fqunc a ach pon s spac s hn: whch n h oang fa s: and h spaall vaan phas dsbuon s: τ τ τ d wh bgns wh ach RF puls h bng agnzaon fo h longudnal as no h ansvs plan wh s obsvabl. h Sgnal Equaon vsd. Fo convnnc w wll l C 1 and w wll dfn. W now g a vsd vson of h sgnal quaon: dd d d dd d dd dd s o p p p τ τ τ τ τ τ τ Fnall w dfn wo quans:

10 Noll 4 MRI Nos : pag 1 τ dτ π τ dτ π And subsung no h abov sgnal quaon: s F p π { } M D u v dd ha s h sgnal s qual o h Fou ansfo of h nal agnzaon valuad a locaons dfnd b and abov. Rnd: u v π u v F D{ g } g dd h sgnal quaon sas ha sapls of h cvd sgnal a qual o sapls of h D Fou ansfo of h objc. hs a sns f w hn abou wha acl h psson fo h D F ans h F a an pon uv s h ngal ov h objc odfd b a spaall vaan lna oaon n h copl plan. p-π p-π

11 Noll 4 MRI Nos : pag 11 p-π p- In MRI h ngaon s pfod b h ngaon of volags n h RF col. h phas vaaon s pfod b h gadns b shfng h fld and hus fqunc n a spaall lna fashon fo a pod of h agnzaon wll oa o a nw onaon n h copl plan. hus MRI has acl h sa chanss as h F opaon. K-spac alwas bgns a h ogn. Wh? Af h caon puls all spns acoss h objc a ponng n h sa dcon.g. p- and h ngal of hs s h DC valu of h F. If w wan o dn h objc w us full sapl s Fou ansfo. A squnc of sapls can b vwd as sapls along a pahwa dnd b unnng ngal und h gadn wavfos as dfnd b and abov. h fnal objc can b consucd spl b ang h nvs D F of h sapld Fou daa -spac daa: F { M } 1 D Dos a sns ha ou sapls n a acuall sapls of spaal fqunc daa? Rb h 1D cas w sapld F d o g a spcu. Snc h was a 1-1 cospondnc bwn fqunc and spaal poson h F of h fqunc daa poducs -doan daa hus h s 1-1 cospondnc bwn and spaal-fqunc. Cons:

12 Noll 4 MRI Nos : pag 1 - Fou spac s calld -spac n h MRI lau - and conol h -spac ajcos o pahs on whch sapl locaons fall. - o ca an ag w us sapl Muv dnsl nough o pvn alasng and of a lag nough n o hav suffcn spaal soluon. 1D Iagng W fs an h cas of a 1D objc cw. h cvd sgnal wll b h Fou ansfo Mu W sncwu valuad a pacula locaons as dcad b h ngaon of h gadn wavfo. Psnd h s puls squnc fo a 1D agng pn along wh h -spac valus and h cvd sgnal: Rcall h n h fs MRI lcu w ald abou ang h F of h cvd sgnal o g a 1D vw of h objc. In hs cas h cvd sgnal s a snc funcon and hus h 1D F of h snc funcon s a c funcon whch s n fac h objc. Dung h n ha daa s acqud s h Daa Acq. ln n h puls squnc h gadn s consan dung hs suaon h s 1-1 cospondnc bwn fqunc and spaal poson. hs s now as fqunc ncodng snc spaal locaon s ncodd as fqunc.

13 Noll 4 MRI Nos : pag 13 Noc also ha w us a ngav gadn bfo h posv gadn. Whou h ngav gadn w can onl acqu h posv spaal fquncs o onl ½ of h F of h objc. D Iagng usng Pojcons h fs D agng hod plnd n MRI b Paul Laubu whl a SUNY Son oo usd a ss of 1D acquson wh h gadns n dffn dcons. Plas no ha b applng 1D gadns n and sulanousl w g a sngl 1D gadn a an angl θan -1. hus w can g 1D vws of h objc o pojcons fo an dffn angls. W ll dscuss n h scon on copud oogaph h hods fo consucng ags fo 1D pojcons. Fo now suffc o sa ha f w acqu nough pojcons w can full dn h undlng objc. h puls squnc usd b Laubu s gvn h:

14 Noll 4 MRI Nos : pag 14 swpng hough angls [π w acqu h full -spac Fou daa fo h objc. h s also a vaan on pojcon agng n whch h posv gadn s pcdd b a ngav gadn hs wll allow boh posv and ngav fquncs o b acqud along a pacula ln n -spac. On advanag o hs appoach s ha on onl nds o swp hough angls [π n od o full acqu h -spac daa. D Spn-Wap Iagng h os coon acquson usd n MRI oda s nown as h spn-wap acquson. h puls squnc s gvn h along wh h cospondng -spac ajco:

15 Noll 4 MRI Nos : pag 15 hs s a pad puls squnc wh a dffn -gadn valu fo ach RF caon R nval. L s loo a h -gadn as n h cas of 1D agng abov hs gadn ncods h spaal poson no fqunc. hs s ofn calld h fqunc ncodng gadn o n hs cas s nown as h fqunc dcon. h -gadn s on bfl bfo ach acquson bu s no on dung daa acquson. hus whav ncodng pfod b h -gadn s don. In hs cas h -gadn ss up a spaall dpndn phas dsbuon ha ans fd dung h fqunc ncodng pocss. In oh wods h -gadn ncods spaal poson no h phas of h agnzaon dcon of h vco whch s nown as phas ncodng. n hs cas s nown as h phas dcon. low a dpcons of h phas dsbuon s up fo h 1 1 and phas ncodng sps. hs cospond o 1 1 and ccls of phas acoss h fld of vw spcvl.

16 Noll 4 MRI Nos : pag 16 p--1 p- p-1 p- In s of paas dscbd n h abov puls squnc w can dfn sval paas of ns n h acqud spac. h sapl spacng and wdh of h -spac a: ad N W N W a π π π π

17 Noll 4 MRI Nos : pag 17 Saplng n -spac spaal fqunc doan. Pvousl w dscussd saplng of h objc and s ffc on h spcu. H w hav h vs saplng n Fou doan and s ffc on h consucd objc. Agan w wll pfo ou saplng b ulplng a funcon s h D cob funcon. Wh sapl spacng of and n h and dcons h sapld Fou daa s: cob ~ n n M v n u v u v u M v u M δ h ag spac doan quvaln s: n n v n u v n u v u v u ** cob ** ~ δ hus saplng n h Fou doan lads o plcaon n h ag doan. Spacng of h plcad ag objc s 1 1. h plcad ags wll no ovlap h ognal ag f h hghs spaal poson n s 1 a and h hghs spaal poson n s 1 a. If hs s no sasfd hn h wll b spaal ovlap n h ags o alasng. h fld of vw of an acquson s pcall dfnd as on ov h -spac sapl spacng: FOV 1 and FOV 1 and alasng wll no occu f a < ½ FOV and a < ½ FOV.

18 Noll 4 MRI Nos : pag 18 Pon Spad Funcon. Obsv ha h paccal -spac s no of nfn n bu ah s ld o W and W. h sapld -spac can b wn as: ~ u v u u M u v M u vc cob W W whch suls n an ag of h followng fo: ~ ** W W snc W snc W ** cob u v h snc funcons a h pon spad funcon and h cob funcon gnas plcad vsons of h objc dos h alasng. Obsv ha h snc funcons hav appoa wdhs n and of 1W and 1W. hs dfns n ssnc h spaal soluon of an MRI acquson n od o g b fn spaal soluon w nd o acqu a lag aa n -spac. Rsoluon of h FF. Mos fos of h FF wo hs wa fo an N pon npu funcon h FF wll poduc an N pon oupu. Each oupu pon cosponds o an ng nub n of ccls n p-πn acoss h objc and go fo n [-N:N-1]. Obsv h DF of - N and N a h sa and hus hs psns h n unalasd fqunc doan of h objc. hus F s blow ½ of 1 o F 1. A sla agun can b ad n vs o g 1f.

19 Noll 4 MRI Nos : pag 19 Rsoluon and Objc and Sapl Spacng n MRI. Fo daa acqud on a D clna gd n -spac and consucd wh a D FF h spaal soluon and Fld of Vw laonshps a: FOV 1 and FOV 1 1W and 1W