Lecture #2 Review of Classical MR

Transcription

1 Lecure #2 Review of Classical MR Topics Nuclear magneic momens Bloch Equaions Imaging Equaion Exensions Handous and Reading assignmens van de Ven: Chapers de Graaf, Chapers 1, 4, 5, 1 (opional). Bloch, Nuclear Inducion, Phys Rev, 7:46-474, 1946 Hisorical Noes Lauerbur, Image Formaion by Induced Local Ineracions: Examples Employing Nuclear Magneic Resonance, Naure 242:19-191, Mansfield and Grannell, NMR Diffracion in solids?, J. Phys. C: Solid Sae Phys., 6:L422-L426,

2 Spin Proons (as well as elecrons and neurons) possess inrinsic angular momenum called spin Spin gives rise o a magneic dipole momen Useful (hough no enirely accurae) o hink of a proon as a spinning or roaing charge generaing a curren, which, in urn, produces a magneic momen. µ 2

3 Nuclear Magneic Momen Consider a poin charge in circular moion: velociy v charge e radius r From EM heory: in he far field a curren loop looks jus like a magneic dipole wih magneic momen µ µ = curren loop area µ = ev 2πr πr2 = e 2m mvr angular momenum L gyromagneic raio γ Thus µ = L 3

4 Gyromagneic Raio γ ofen expressed as = gµ b where µ b = e 2m Planck s consan/2π For proons = gµ n where γ 2π = MHz/T Noe, for elecron spin: µ n = e 2m p γ e γ nuclear magneon = 658 and Bohr magneon (m = elecron mass) and g = spin g facor g g 5.6 Imporan for ESR, NMR conras agens, ec 2 (elecrons) K. Zavoisky 4

5 Nuclear Spin in a Magneic Field In a uniform magneic field, a magneic dipole will experience a orque τ = µ B Example: compass Poenial energy given by: E = µ B = µb cos angle beween µ and B Classically, energy can ake on any value beween ± µb 5

6 Equaion of Moion Newon s Law: d L d = Combining previous equaions: d µ d = µ B 6

7 Physical Picure: Single Spin in a Uniform Magneic Field d µ d = µ B = consan B = B ẑ z θ µ Noe: µ Precession frequency ω γb x y Noe: Some exs use ω = -γb. The Larmor Equaion A compass needle has a magneic momen and sis in he earh s field. Why doesn i precess? Sir Joseph Larmor 7

8 Ne Magneizaion In issue, we are always dealing wih a large number of nuclei. Ne magneizaion:!! M = µ volume Hence, d! M d = γ! M! B Equaion is valid for non-ineracing spins 8

9 Bloch Equaions In order o accoun for iner- and inra-molecular ineracions, we can inroduce exponenial ransverse (T 2 ) and longiudinal (T 1 ) relaxaion ime consans. For! B = B ˆ z d! M d = γ M! B ˆ z M ˆ xx + M y y ˆ ( M z M )ˆ z T 2 T 1 9

10 Le MRI: The Signal Equaion! B = Bˆ z Following RF exciaion (a opic we ll revisi) and using: d! M d = γ M! Bˆ z (ignores relaxaion) each small issue volume looks like a iny oscillaing magneic dipole. x z θ r M y r M xy = M x + im y M xy (x,y,z)e iφ(x,y,z, ) 1

11 MRI: The Signal Equaion Assuming a uniformly sensiive RF coil, he received signal is given by: s() = x Insananeous frequency: ω = dφ/d y z M xy (x, y,z)e iφ(x,y,z, ) dxdydz φ(x, y,z,) = ω(x,y,z, $ )d $ = γ B(x, y,z, #)d # In he presence of linear gradiens: B(x, y,z,) = B + G x ()x + G y ()y + G z ()z P. Lauerbur P. Mansfield s() = e iγb e demodulae a γb x y z M xy (x,y,z)e iγ G x ( $ ( )x +Gy( $ )y +Gz( $ )z)d $ dxdydz 11

12 k-space Comparing... s e () = M xy (x, y,z)e iγ G x ( $ received signal FT of M xy M x y z ( )x +Gy( $ )y +Gz( $ )z )d $ dxdydz (k x,k y,k z ) = M xy (x, y,z)e i2π ( k x x +k y y +k z z ) dxdydz x y z s e () = k-space inerpreaion of MRI. & γ M ( ' 2π G x ( $ )d $, γ 2π G y ( $ )d $, γ 2π ) G z ( $ )d $ + * k x k y k z Gradiens race a rajecory hrough k-space G x G y k z G z k x k y 12

13 Sensiiviy Wha is M? where M = ρ γ 2! 2 B 4kT ρ! = Planck s consan/2π = 1.5 x 1-34 Joule s, k = Bolzmann s consan = 1.38 x 1-23 J/K, T = absolue emperaure. (Laer, we ll do a more complee derivaion) = spins/uni volume (careful wih he erm spin densiy ), Signal (volage induced in coil) signal dm d ωm ρ γ 3! 2 B 2 4kT 13

14 Sensiiviy (con.) Wha abou he noise? MR Receiver Coil L R coil v c 2 C low noise preamp R body v b 2 R coil R body ω 2 ω (due o RF skin deph effecs) (from inducive losses, ignoring dialecric losses) Using Johnson noise specral densiy: v n 2 = 4kTR noise γ 2 B 2 + λ γb 14

15 Sensiiviy (con.) Combining wih he signal equaion: SNR ρ γ 3! 2 B 2 4kT γ 2 B 2 + λ γb For high frequencies where body noise dominaes coil noise (e.g. γb >> 1 Mz) SNR ρ γ 2! 2 B 4kT 15

16 Bloch equaion d! M d Summary = γ M! B ˆ z M ˆ xx + M y y ˆ ( M M z )ˆ z T 2 T 1 Signal equaion s e () = x y z M xy (x, y,z)e iγ! ( G r! )d % dxdydz Wha s missing? 16

17 T 1 and T 2 Can be included as k-space weighings: bu... j r s e () = M xy (x, y,z,t 1 j,t 2 j )e /T 2 j e iγ G rd % Wha are he underlying mechanisms? Why do differen issues have differen T 1 s and T 2 s? How do conras agens work? dr 17

18 Chemical Shif Ineracion beween elecron cloud and B Looks like a new k-space axis ω = γb eff = γb (1 σ) shielding consan Depends on: elecron densiy molecular geomery, ec s e () = ω r s e () = M xy (x, y,z,ω)e & γ M ( ' 2π G x ( $ )d $, γ 2π iγ G rd & G y ( $ )d $, e iω drdω γ 2π G z ( $ )d $, ) + 2π * k x k y k z k ω Noe, same equaion also holds for B inhomogeneiy (due o magneic suscepibiliy, ec) 18

19 Bu wha abou Coupling beween spins Chemical exchange Nuclei wih spin 1/2 ec. 19

20 Nex Lecure: Inroducion o Quanum Mechanics 2

21 Biography: Sir Joseph Larmor Joseph Larmor ( ) was educaed a he Royal Belfas Academical Insiuion and he Queens College Belfas. He hen ook anoher degree a S. Johns College Cambridge, as was common for promising young sudens from provincial universiies. He won op prize a he final mahemaical examinaion in Cambridge. This was he second year in a row ha a suden from Belfas had been crowned "senior wrangler". Larmor hen reurned o Ireland as Professor of Naural Philosophy a Queens College Galway. He held his posiion for five years bu hen reurned o Cambridge o ake up a new Mahemaics posiion and he was laer appoined o he presigious Lucasian Chair of Mahemaics. Larmor is well known for his conribuions o he heory of elecromagneism, in paricular he elecron heory of maer. Larmor published his colleced papers on elecromagneism in 19 in a famous book eniled "Aeher and Maer". Larmor's work, hough rooed in he classical physics in which he had been rained, evenually led o he breakdown of classical physics and he rise of relaiviy heory and quanum mechanics. He was described as 'one who rekindled he dying embers of he old physics o prepare he adven of he new'. Larmor saw himself as par of an Irish scienific radiion and was involved in ediing he colleced works of a number of Irish scieniss. Larmor spen mos of his career in Grea Briain bu reurned o Ireland mos summers and moved back permanenly afer his reiremen from he Lucasian chair. He was commied o he Union of Ireland wih Grea Briain and his led him o serve in Parliamen as a member for Cambridge Universiy from 1911 o

22 Biography: Sir Peer Mansfield (born Ocober 9, 1933, London, England) English physicis who, wih American chemis Paul Lauerbur, won he 23 Nobel Prize for Physiology or Medicine for he developmen of magneic resonance imaging (MRI), a compuerized scanning echnology ha produces images of inernal body srucures, especially hose comprising sof issues. Mansfield received a Ph.D. in physics from he Universiy of London in Following wo years as a research associae in he Unied Saes, he joined he faculy of he Universiy of Noingham, where he became professor in Mansfield was knighed in Mansfield's prize-winning work expanded upon nuclear magneic resonance (NMR), which is he selecive absorpion of very high-frequency radio waves by cerain aomic nuclei subjeced o a srong saionary magneic field. A key ool in chemical analysis, i uses he absorpion measuremens o provide informaion abou he molecular srucure of various solids and liquids. In he early 197s Lauerbur laid he foundaions for MRI afer realizing ha if he magneic field was deliberaely made nonuniform, informaion conained in he signal disorions could be used o creae wo-dimensional images of a sample's inernal srucure. Mansfield ransformed Lauerbur's discoveries ino a pracical echnology in medicine by developing a way of using he nonuniformiies, or gradiens, inroduced in he magneic field o idenify differences in he resonance signals more precisely. He also creaed new mahemaical mehods for quickly analyzing informaion in he signal and showed how o aain exremely rapid imaging. Because MRI does no have he harmful side effecs of X-ray or compued omography (CT) examinaions and is noninvasive, he echnology proved an invaluable ool in medicine. 22

23 Biography: Paul Lauerbur American chemis (born May 6, 1929, Sidney, Ohio died March 27, 27, Urbana, Ill.) won he Nobel Prize for Physiology or Medicine in 23, ogeher wih Briish physicis Sir Peer Mansfield, for he developmen of magneic resonance imaging (MRI), a compuerized scanning echnology ha produces images of inernal body srucures, especially hose comprising sof issues. Lauerbur received a Ph.D. (1962) in chemisry from he Universiy of Pisburgh. He served as a professor a he Sae Universiy of New York a Sony Brook from 1969 o 1985, when he acceped he posiion of professor a he Universiy of Illinois a Urbana-Champaign and direcor of is Biomedical Magneic Resonance Laboraory. In he early 197s Lauerbur began using nuclear magneic resonance (NMR), which is he selecive absorpion of very-high-frequency radio waves by cerain aomic nuclei subjeced o a srong saionary magneic field. NMR is a key ool in chemical analysis, using he absorpion measuremens o provide informaion abou he molecular srucure of various solids and liquids. Lauerbur realized ha if he magneic field was deliberaely made nonuniform, informaion conained in he signal disorions could be used o creae wo-dimensional images of a sample's inernal srucure. This discovery laid he groundwork for he developmen of MRI as Mansfield ransformed Lauerbur's work ino a pracical medical ool. Noninvasive and lacking he harmful side effecs of X-ray and compued omography (CT) examinaions, MRI became widely used in medicine. 23