Get: Nuclear (equilibrium) magnetization M 0. (Magnitude dictated by Boltzmann distribution)

Transcription

1 9: Relaaion of nuclear magneiaion. How is he R signal deeced?. Wha is he quanum-mechanical equivalen of he roaing frame? 3. Wha is he roaing frame descripion good for? 4. How can he reurn of he magneiaion o hermodnamic equilibrium described? 5. How is he ime-dependen change of magneiaion described mahemaicall? Afer his course ou. Can describe he principle of R deecion and eciaion. Can eplain how R eciaion is frequenc selecive (resonance) 3. Undersand he principle of relaaion o he equilibrium magneiaion 4. Know wha are he major relaaion imes and how he phenomenologicall affec magneiaion in biological issue, in paricular ha of waer. 5. Can eplain he elemens of he Bloch equaions and FD 6. Undersand he R conras srongl depends on eperimenal parameers Fund Biomag 8 9- Wha do we know abou R so far? Need: Nucleus wih non-ero spin agneic field B Ge: Nuclear (equilibrium) magneiaion (agniude dicaed b Bolmann disribuion) increases wih. Number of spins in voel. agneic field B 3. Gromagneic raio maging H in H O is mos sensiive Thermodnamic equilibrium magneiaion is B d B = does no precess Fund Biomag 8 All his does no generae a measurable signal 9-4

2 9-. How is he R signal deeced? Farada s Law of nducion d da Fund Biomag 8 B B surface R: da=consan db/ B( r, ) da Len s Law induced volage curren magneic field opposes he change in he magneic flu ha produces he curren (Compleel analogous o power generaion!) Bio-avar Law magneic field falls off wih r - agneic flu B B( r ) dl r 4 r 3 agneic field of dipole decreases wih disance : decreases wih disance from magneiaion Roaing frame revisied Equaion of moion for (alwas valid in an reference frame) in presence of B Roaing frame: reference frame roaing abou a frequenc RF Case : non-roaing reference frame ( RF =) Fund Biomag 8 d magneiaion precesses in plane wih frequenc B / Case : roaing frame wih RF = L magneiaion is saionar ( precesses in wih ero frequenc) Equaion of moion is sill valid, i.e. precession frequenc B eff / B eff = B eff Larmor frequenc in he roaing frame: = B eff B o B eff = B o - RF / magneiaion precesses in plane wih frequenc B eff / RF / o L o sin 9-7

3 chrödinger represenaion: d i H ih f H =cons in : e NB. upplemen: Roaing frame Wha are he quanum-mechanical equivalencies? How o deermine < ()> ec? pli H ino ime-invarian and -dependen erms: neracion represenaion (Higher order perurbaion heor) i d e ih / V / Quanum mechanical equivalencies: < >, < >, < > For one spin-/ ( H), i.e. wo energ levels i i For spin: i d H V ( ) H B sin V B cos RF Wha is V () [ RF =B ]? RF V B Quanum mechanical equivalencies: Fund Biomag 8 V ih / / ih e V e B, B,, Wha is he moion of magneiaion when an RF field induces a flip angle? Laboraor frame of reference d B B Roaing frame of reference d B B B Fund Biomag 8 B radiofrequenc field a Larmor frequenc L applied in ransverse () plane for duraion nuaion (a L ) of as i ips awa from he -ais. RF field roaes owards plane Ampliude B deermines how quickl he magneiaion is roaed. cos flip angle = B [rad] sin n R picall B /~.-kh (~ms) 9-9

4 Wha is resonance? Wha range of frequencies can be ecied wih a given RF pulse? A = L - RF (from L ) magneiaion eperiences effecive field srengh B eff B eff B B eff B eff Roaion ais : iled b. on resonance : B >> effecive field B shor RF pulses (<ms) ( B ) ( ) RF field wih ampliude B can ecie a range of frequencies on he order of ±B B B eff agneiaion (ip) / B eff / / off resonance : B << B B eff Quanum mechanical resonance Transiion probabili highes : h = hb / Fund Biomag How is he reurn o equilibrium governed? Relaaion Thermodnamic equilibrium RF pulse(s) B Afer eciaion Relaaion T, T B B Transverse magneiaion: (along and -ais, on resonance) d T d T Eponenial deca of () e T Fund Biomag 8 Equaions formall equivalen o linear aenuaion coefficien (-ra) (same soluion) 9-

5 Wha are he mechanisms of relaaion? Tumbling of olecule (Brownian moion) Creaes local oscillaing/flucuaing magneic field Flucuaing magneic field depends on orienaion of he whole molecule & correlaion ime c (=ime for molecule o roae rad) ources of flucuaing magneic field: Dipolar coupling beween nuclei and solven ineracion beween nuclear magneic dipoles Fund Biomag 8 Correlaion funcion G Describes degree of correlaion of moion sec apar Correlaion ime c : : viscosi k: Bolmann consan r: sie of molecule G( ) e 4r c 3kT 3 / c 9- Wha is he cause of loss of ransverse agneiaion? flucuaing microscopic magneic fields B T : phenomenological ime consan Range μs (bone) several s (waer) ransverse relaaion, T relaaion Cause: olecular dnamics and spin-spin ineracions Hisoricall : spin-spin relaaion loss of signal in plane emor relaaion Rule of humb for issue waer: The less issue (bone, solues, proeins, membranes) is in conac wih bulk waer, he longer bulk waer T Afer eciaion () e T random precession of nuclei dephasing of spins wih ime consan T Phase accrued over c : =B ib( r, ) c c ( r ) e dv voel c large (immobile spins): large phase differences shor T Fund Biomag 8 bone, membranes, proeins are R- invisible 9-3

6 Fund Biomag 8 How does reurn o equilibrium? Longiudinal relaaion T Afer deca of b T : Afer T relaaion Longiudinal Relaaion (along -ais) d T / T / T ( e ) () e Afer 9 eciaion: = echanisms: ncoheren molecular flucuaions on populaion disribuion corresponds o T=: he order of he Larmor frequenc L m=-/ possibili of energ ransfer maching frequenc h E E B Hisoricall : spin-laice relaaion N kt e (hea los o he surroundings) N m=/ T ~.5-5s (waer) Rule of humb for waer: The less issue is in conac wih bulk waer (bone, solues, proeins, membranes), he longer bulk waer T Bolmann disribuion re-esablished b energ (phoon) ransfer from spins o ssem (laice). os efficien when energ levels of ssem and nuclear spins mach, i.e. c ~ T minimal (bone: c >> T ~ s o min) Wha equaions describe he change in magneiaion? Bloch Equaions add relaaion erms (T, T ) o he fundamenal Eq of moion of magneiaion: d [ B B ] d [ B B ] d [ B B ] Fund Biomag 8 - B ubsiuing =-B + RF (B =B is no ime-dependen) ields: T () T T () Roaing reference frame B B ( )] [ B B along along along B Feli Bloch Phsics 95 B : RF field in roaing frame 9-5

7 Wha characeries he basic R signal? Free inducion deca: Precession and relaaion (afer RF pulse) Transverse magneiaion () e i e / T T T Longiudinal magneiaion (afer 9 RF eciaion) / T ( e ) T Fund Biomag NB. can never eceed T T 9-6 How can T changes be measured? repeiive pulsing ulipulse eperimen wih RF pulses applied ever TR seconds TR =9 (signal) TR / T ( TR) ( e ) The effec of T (and T ) on he signal depends on how i is measured Convers o TR (ms) Fund Biomag 8 ( ) cos / T ( e ) () / T ( e ) e / T Opimal TR o deec changes in T? Use noise error propagaion calculaion (Lesson ) F=ma: T e / df / T / T F e e 3 T T T T / T e TR op = T =TR op : T T 9-7

8 ummar agneic resonance so far agneic field B Equilibrium magneiaion ncreases wih. number of spins in voel. aic magneic field B 3. gromagneic raio RF field B (applied on-resonance i.e. L ) ils magneiaion ino ransverse plane Precession of is deeced T and T relaaion eponenial deca of ep. reurn of o reflec molecular environmen source of conras. Onl mobile spins (e.g. waer) are deeced. reflecs amoun of nuclei and hus waer conen [Waer conen varies 7-ml/g in bod (poor conras)] 3. Effec of T and T changes on image conras depend srongl on eperimenal parameers (RF pulse iming and flip angle) Fund Biomag 8 9-9