Notes on MRI, Part III

Transcription

1 oll 6 MRI oe 3: page oe o MRI Par III The 3 rd Deo - Z The 3D gal equao ca be wre a follow: ep w v u w v u M ddd where Muvw he 3D FT of. I he p-warp ehod for D acquo oe le a a e acqured he D Fourer doa or -pace. Th ehod eal eeded o 3D b ug phae ecodg wo deo raher ha ad frequec ecodg he reag deo: Th reul he acquo of a cubc daa e oe le a a e:

2 oll 6 MRI oe 3: page The aplg requree ad paal reoluo requree are he ae a he would be for he D p-warp ehod FOV Δ ; Δ W. If here are ad aple he ad dreco repecvel he he oal e o acqure he 3D volue * *TR. For eaple for 8 ad TR 33 he overall age acquo e 9 raher log! Slce Selecve Ecao The o coo approach for dealg wh he 3 rd deo o ue lce elecve ecao. Th doe b applg a -grade o ha he reoace frequec vare he - dreco ad applg a badpa RF pule o ece ol he hoe p whoe reoa frequec le wh he bad: We wll aga olve he loch equao for h pecfc cae. We wll le be a e-varg agec feld ag a ω. For h aal we ll le he ag frae be a ω frae ω. coω ω j eff ' A -grade appled o he copoe he -dreco : ad he e effecve appled feld :. eff ' eff

3 oll 6 MRI oe 3: page 3 The loch equao for h cae reduce o he followg: dm d M eff M Wha we would le o ow how he ageao M vare a a fuco of poo followg he applcao of he pecfed feld. Th geeral a ver dffcul equao o olve becaue o-lear. Sall Tp Agle Approao Oe parcularl ueful approach o he oluo o he above loch equao o ue he all p agle approao. Here we aue he produce a all e ao agle a d < 3 6 I h cae we ca aue he copoe of he ageao approael equal o durg he RF pule. Eeall we are ag ha: co d d Uder h aupo ad hu. The above loch equao ca d he be rewre a: M d d We ow would le o olve for. We ca he wre a dffereal equao ug for he ravere copoe: d d d d d d

4 oll 6 MRI oe 3: page 4 Oberve ha a coa wh repec o e ad hu we have a fr order dffereal equao wh a drvg fuco. For al codo he oluo o h dffereal equao ca be how be: d e ep ξ ξ ξ Aga we wa he oluo o h dffereal equao a he e of he ed of he RF pule whch we defe a : We ow ae a varable ubuo ξ. We alo aue ha he RF pule ha ercal eve aroud ad ha ero oude of he erval []. The ageao ca ow be decrbed a: { } F e d e d e d e ep ep ep For ercal RF pule he forward ad vere FT are he ae. Thu uder he all p agle approao he lce profle varao of wh relaed o he pecru of he RF pule: { } f F

5 oll 6 MRI oe 3: page 5 ad f he covero bewee pecru ad he locao: We re alo here bu we ll have oe udered phae varao he -dreco ep he ca lead o udered phae deruco whe egraed b he RF col. How h fed? We pl appl a egave for a perod. Th ofe called a lce rephag pule. Th wll reul phae accuulao of: ad hu: 3 ep d Δ ep

6 oll 6 MRI oe 3: page 6 { } 3 F There a -pace pcure o h. For h we aue ha he RF pule occur aaeoul a he ceer of he pule a ad we beg accuulao area - afer ha po. applg a egave grade for he ae durao a he la half of he pule he area cacel ad he -pace locao he dreco reured o he org. oce for a RF pule appled alog he real a he ageao wll ed up alog he agar j a. Alo oberve he flp agle a he ceer of he lce : α d F { } Prevoul we dcued he ha ravere ageao afer a α pule wa α bu for all α α ~ α. So here he ravere ageao alo α. Eaple he c RF pule Coder a RF pule roughl he for: whch ha a pecru: Ac T F f W The ageao wll be: { } ATrec where W T 3 ATrec W α rec Δ

7 oll 6 MRI oe 3: page 7 where he lce hce W Δ ad flp agle α AT. Pug Slce Seleco wh he Sgal Equao Le defe our lce profle fuco: he { } f p F 3 p ow we go bac o he cae where we have a 3D drbuo of ageao b ubug ad pug o he gal equao aga he RF col egrae acro he objec: p ep ddd Here we are perforg D agg whle egrag acro he lce profle. Larger Flp Agle The above aal wa for all flp agle e.g. α < 6 bu ur ou he pule creaed ug he all p agle approao alo perfor well for large flp agle e.g. 9 degree. Here a beer approao for he ravere copoe of he ageao : p 3 alhough due o he o-lear of he loch equao a eac oluo would requre uerc ulao of he RF pule. Mullce Iagg The o coo wa o age 3D volue MRI ue erleaved lce elecve ecao. Here lce eced ad par of he -pace daa are acqured he lce eced ad acqured he lce 3 ad o o. Afer all have bee acqured we coe bac o lce o acqure addoal par of he -pace daa ec. Whe oe lce eced he oher are o perurbed ad hu each lce ha ow T recover e TR. Slce eleco allow he effce ue of loger TR b ulaeoul acqurg a lce:

8 oll 6 MRI oe 3: page 8 For a lce-elecve D p-warp acquo he overall acquo e wll be *TR. For eaple f we are ereed acqurg a T-weghed age wh lce ad a 5 TR ad 8 phae ecodg le -pace he oal acquo e for hee lce *TR ~ ue. Sp Echo Pule Earler we decrbed 8 degree RF pule for purpoe of verg he ageao. Le coder he effec of a 8 degree pule appled o he a he ag frae o a ageao vecor [ ]: For 8 - ad 8 beg he e ju before ad afer he 8 degree pule he ageao wll be:

9 oll 6 MRI oe 3: page 9 ow le loo a a ageao vecor ha lg he ravere plae. Suppoe he vecor wa orgall pooed o he a ad phae φr ha accuulaed due o Δ er. The phae afer he 8 degree pule wll be: φr 8 -φr 8 - or equvalel: 8 8 -* Whe agg he phae er reul fro grade ad ca be wre a: φr. r ad hu: Wha h a ha a 8 degree pule wll ver he locao -pace. Sp-echo 8 degree pule are a e ow a phae reveral or e reveral pule. Sp-echo Sp-warp Pule Sequece Coder h pule equece:

10 oll 6 MRI oe 3: page Here he 8 degree pule ver he poo -pace a how here: Wh do p-echo pule? Magec feld hoogee ca reul ra-voel gal dephag. Coder a agec feld hoogee fuco Δr. The effecve agec feld ag frae wll be: eff. r Δr ad he correpodg phae fuco : φr. r Δωr whe egrag Δωr acro a voel oe gal a be lo. The p-echo pule brg h phae bac ogeher aga. Coder he followg eaple: Igorg he -pace er he phae accuulao a he e of he 8 wll be: ad ju afer he 8 wll be: φr 8 - Δωr 8 φr 8 -Δωr 8

11 oll 6 MRI oe 3: page A h po he phae coue o accuulae ad f we loo a e 8 we wll have a oal phae accuulao of: φr 8 φr 8 ωr 8 -Δωr 8 Δωr 8 Tha all he phae accuulao due o agec feld hoogee cacel o ero a e he e of 8 degree pule or 8. The e of he gal he ravere plae wll loo le h: A how here he gal coe bac ogeher aga a echo a 8. The ore rapd deca of he gal due o T deca plu hoogee effec gve aoher deca er T*. Whe all dephag cacelled b he p-echo however he T deca ll rea. oe MRI Source of oe MRI - Theral oe fro bod heral vbrao of o elecro ec. [Doa ource of oe o MRI e] - Quaao oe he AD devce

12 oll 6 MRI oe 3: page - Preapelecroc oe - Theral oe RF col Soe coe o heral oe: - o relaed o he MR o Pree wh or whou RF rade - Ufor pecral de ear ω whe - Coe fro he whole bod aou of oe deped o he aou of he bod o whch he receve col eve oe Sp-warp agg coder he frequec ecodg grade ad aug ha he feld of vew FOV he he badwdh of he recever wll be e o: W FOV. Δ oe characerc: - oe ero-ea ad addve M - Saple are depede due o he whee of he pecru - aua drbued reul fro large uber of vbrag parcle - -varae - depede oe real agar he chael of he cople deodulaor are orhogoal q

13 oll 6 MRI oe 3: page 3 - oe varace for each aple proporoal o he preaplg badwdh W σ : - A D age recoruced b a po DFT: ep X where X are aple wh depede oe havg varace σ. The oe age pel wll alo be ero-ea addve depede b-varae aua oe bu wh varace σ. The dervao of he varace : [ ] [ ] X E e X X E X E e X e X E E l l l l l l * var σ σ - A D age recoruced b a b vere D DFT. Aga he reula oe he age wll be ero-ea addve depede b-varae aua oe bu wh varace σ. Sgal o oe Rao - The oepel a D age wll be he be: D A T W Δ σ where T AD he oal e he AD aplg.

14 oll 6 MRI oe 3: page 4 - The gal j repree he oal aou of ageao a parcular voel recall ha X j M. Thu he gal proporoal o V where V ΔΔΔ he voel volue ad Δ he lce hce. - The gal o oe rao he: gal SR V T σ proporoal o ρ - he cocerao he ucleu of ere ad. A D Eaple Cae : Suppoe we fd ha we have a age ha oo o o we average ogeher eghborg pel o acheve Δ Δ* all oher deo rea he ae ad Δ ha chaged eher. Sce b averagg age doa we effecvel are dcardg aple -pace T AD T AD ad: Tha we ve proved he SR b qr. T A D SR org SR' ΔΔ Cae : Suppoe we ew advace ha he SR of a age wa oo o ad we copeaed b acqurg a lower reoluo Δ Δ* all oher deo rea he ae bu we ve copeaed o a o preerve he orgal acquo e T AD T AD. Thu: SR' ΔΔ T A SR Fro hee wo eaple we ee ha preferable o acpae he SR ha ecear for a gve age a e he acquo accordgl. We do acheve a good of a SR b oohg he age afer acqured ha f we had acqured a he approprae reoluo orgall. D org Cae 3: Suppoe we average each -pace aple e ad hu creae our age acquo e b a facor of. There he reoluo he ae ad T AD T AD ad: SR' ΔΔ T A SR D org

15 oll 6 MRI oe 3: page 5 Averagg creae he SR b qr where he uber of average. Cae 4: Suppoe we creaed he feld regh b a facor of. The SR' SR org Keepg reoluo coa we ca ue h addoal SR o reduce he uber of average ad hu overall agg e b a facor of 4! Keep d ha edcal agg eoe.