Relaxation. T1 Values. Longitudinal Relaxation. dm z dt. = " M z T 1. (1" e "t /T 1 ) M z. (t) = M 0

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1 Relaxaion Bioengineering 28A Principles of Biomedical Imaging Fall Quarer 21 MRI Lecure 2 An exciaion pulse roaes he magneiaion vecor away from is equilibrium sae (purely longiudinal). The resuling vecor has boh longiudinal M and ranverse M xy componens. Due o hermal ineracions, he magneiaion will reurn o is equilibrium sae wih characerisic ime consans. T 1 spin-laice ime consan, reurn o equilibrium of M T 2 spin-spin ime consan, reurn o equilibrium of M xy Longiudinal Relaxaion T1 Values dm d = " M " M T 1 Gray Maer muscle Whie maer Afer a 9 degree pulse M () = M (1" e " /T 1 ) Due o exchange of energy beween nuclei and he laice (hermal vibraions). Process coninues unil hermal equilibrium as deermined by Bolmann saisics is obained. kidney liver fa The energy ΔE required for ransiions beween down o up spins, increases wih field srengh, so ha T 1 increases wih B. Image, capion: Nishimura, Fig

2 Transverse Relaxaion T2 Relaxaion dm xy d = " M xy T 2 x y x y x y Each spin s local field is affeced by he -componen of he field due o oher spins. Thus, he Larmor frequency of each spin will be slighly differen. This leads o a dephasing of he ransverse magneiaion, which is characeried by an exponenial decay. T 2 is largely independen of field. T 2 is shor for low frequency flucuaions, such as hose associaed wih slowly umbling macromolecules. Afer a 9 degree exciaion M xy () = M e " /T 2 T2 Relaxaion T2 Values x x x x x x x Runners Ne signal x x x x x x x x x x x x x x x x x x x x Tissue T 2 (ms) gray maer 1 whie maer 92 muscle 47 fa 85 kidney 58 liver 43 CSF 4 Table: adaped from Nishimura, Table 4.2 Solids exhibi very shor T 2 relaxaion imes because here are many low frequency ineracions beween he immobile spins. On he oher hand, liquids show relaively long T 2 values, because he spins are highly mobile and ne fields average ou. Credi: Larry Frank 2

3 Example Example T 1 -weighed Densiy-weighed T 2 -weighed Quesions: How can one achieve T2 weighing? Wha are he relaive T2 s of he various issues? Hanson 29 Bloch Equaion Free precession abou saic field dm d = M " #B M xi + M y j ( M M )k T 2 Precession Transverse Relaxaion i, j, k are uni vecors in he x,y, direcions. T 1 Longiudinal Relaxaion dm d = M " #B ˆ i ˆ j ˆ k = # M x M y M B x B y B ( ) i ˆ B M y B y M ( = # ˆ * j ( B M x B x M )* k ˆ & ( B y M x B x M y ) * ) B dμ Μ 3

4 Free precession abou saic field " dm x d " B M y ) B y M dm y d = ( B x M ) B M x # dm d& # B y M x ) B x M y & " B )B y " = ( )B B x # B y )B x & # M x M y M & Precession " dm x d " B " dm y d = ( )B # dm d& # & # Useful o define M " M x + jm y dm d = d d( M x + im y ) = " j#b M Soluion is a ime-varying phasor M x M y M & M x jm y M() = M()e " j#b = M()e " j Quesion: which way does his roae wih ime? Marix Form wih B=B Z-componen soluion " dm x dm y # dm d "(1/T 2 )B " d = ()B 1/T 2 d& # (1/T 1 & # M x M y M " + & # M /T 1 & M () = M + (M () " M )e " /T 1 Sauraion Recovery If M () = hen M () = M (1" e " /T 1 ) Inversion Recovery If M () = "M hen M () = M (1" 2e " /T 1 ) 4

5 M " M x + jm y Transverse Componen Summary 1) Longiudinal componen recovers exponenially. 2) Transverse componen precesses and decays exponenially. ( ) dm d = d d M x + im y = " j (# +1/T 2 )M M() = M()e " j# e " /T 2 Source: hp://mrsrl.sanford.edu/~brian/mri-movies/ Summary 1) Longiudinal componen recovers exponenially. 2) Transverse componen precesses and decays exponenially. Gradiens Spins precess a he Larmor frequency, which is proporional o he local magneic field. In a consan magneic field B =B, all he spins precess a he same frequency (ignoring chemical shif). Gradien coils are used o add a spaial variaion o B such ha B (x,y,) = B +Δ B (x,y,). Thus, spins a differen physical locaions will precess a differen frequencies. Fac: Can show ha T 2 < T 1 in order for M() M Physically, he mechanisms ha give rise o T 1 relaxaion also conribue o ransverse T 2 relaxaion. 5

6 MRI Sysem Imaging: localiing he NMR signal The local precession frequency can be changed in a posiiondependen way by applying linear field gradiens ΔB(x) x Simplified Drawing of Basic Insrumenaion. Body lies on able encompassed by coils for saic field B o, gradien fields (wo of hree shown), and radiofrequency field B 1. Image, capion: copyrigh Nishimura, Fig RF and Gradien Coils Resonan Frequency: ν(x) = γb +γδb(x) Credi: R. Buxon Gradien Fields Inerpreaion B (x, y,) = B + "B "x x + "B "y y + "B " = B + G x x + G y y + G B (x)=g x x x y G = "B " > G y = "B "y > Spins Precess a γb - γg x x (slower) Spins Precess a γb Spins Precess a a γb + γg x x (faser) 6

7 Roaing Frame of Reference Spins Reference everyhing o he magneic field a isocener. There is nohing ha nuclear spins will no do for you, as long as you rea hem as human beings. Erwin Hahn Phasors G( ) = g(x)exp (" j2# x) dx " " = #2 x Phasor Diagram Imaginary Real " = #2 x =1; x = 2" x = x =1/4 2" x = " /2 x =1/2 2" x = " x = 3/2 2" x = 3" /4 " = " = # /2 " = # " = # /2 " = " = # /2 " = # " = # /2 7

8 Inerpreaion -2 x - x x 2 x ( ( exp " j2# * x* & & 8x ) ) 1 ( ( exp " j2# * x* & & 8x ) ) 2 ( ( exp " j2# * x* & & 8x ) ) Slower B (x)=g x x Faser =; = =; Fig 3.12 from Nishimura Hanson 29 Hanson 29 8

9 Phase wih ime-varying gradien Hanson 29 K-space rajecory Gx() ky Gy() 1 ky ( 4 ) 2 ky ( 3 ) 3 kx 4 kx (1 ) kx ( 2 ) Nishimura

10 G x () K-space rajecory G x () Spin-Warp G y () 1 2 G y () 1 k-space Image space k-space y x Fourier Transform 1

11 k-space RF G x () Spin-Warp Pulse Sequence G y () G y () G x () G y () 1 Spin-Warp Gradien Fields Define G " G xˆ i + G y ˆ j + G ˆ k So ha G x x + G y y + G = G " r r " xˆ i + yˆ j + k ˆ Also, le he gradien fields be a funcion of ime. Then he -direced magneic field a each poin in he volume is given by : B ( r,) = B + G () " r 11

12 Saic Gradien Fields In a uniform magneic field, he ransverse magneiaion is given by: M() = M()e " j# e " /T 2 In he presence of non ime-varying gradiens we have M( r ) = M( r,)e " j#b ( = M( r,)e " j# ( B + G = M( r,)e " j e " j# r ) e " /T 2 ( r ) r ) e " /T 2 ( r ) G r e " /T 2 ( r ) Time-Varying Gradien Fields In he presence of ime-varying gradiens he frequency as a funcion of space and ime is: ( ) = #B ( " r, r,) = #B + # G () r = " + "( r,) Phase Phase = angle of he magneiaion phasor Frequency = rae of change of angle (e.g. radians/sec) Phase = ime inegral of frequency " r, ( ) = "( = ( G " ( "# r, & r,) d = # + " r, ( ) = # ( ( ) Where he incremenal phase due o he gradiens is r,& ) d& r,& ) ) r d& Phase wih consan gradien ( ) = "( 1 "# r, 1 r,&) d& "# r, 2 "# r (, 3 ) = "( 3 ( ) = "( r 2,&) d& = "( r ) 2 if " is non - ime varying. r,&) d& 12

13 Time-Varying Gradien Fields The ransverse magneiaion is hen given by M( r,) = M( r,)e " /T 2 ( r ) e #( r, ) = M( r,)e " /T 2 ( r ) e " j exp " j ( r,)d& = M( r,)e " /T 2 ( r ) e " j exp " j( ( o ) ( G (&) ) r d& o ) Signal from a volume V " " x y Signal Equaion s r () = " M( r,) dv = M(x, y,,)e # /T 2( r ) e # j " exp # j " G (&) r d& dxdyd ( o ) For now, consider signal from a slice along and drop he T 2 erm. Define m(x, y) " M( + / 2 To obain x y r,) d # / 2 s r () = m(x, y)e " j# exp " j ( o ) G () & r d dxdy Signal Equaion Demodulae he signal o obain s() = e j" s r () x y = m(x, y)exp # j y ( G () & r d o ) dxdy ( ) = m(x, y)exp # j o[ G x ()x + G y ()y]d dxdy x x y ( ( )) = m(x, y)exp # j2( ()x + ()y dxdy MR signal is Fourier Transform s() = m(x, y)exp " j2# ()x + ()y dxdy x y ( ) = M (), () [ ] kx ( ), ( ) = F m(x, y) ( ( )) Where () = " 2# () = " 2# G x ()d G y ()d 13

14 Recap Frequency = rae of change of phase. Higher magneic field -> higher Larmor frequency -> phase changes more rapidly wih ime. Wih a consan gradien G x, spins a differen x locaions precess a differen frequencies -> spins a greaer x-values change phase more rapidly. Wih a consan gradien, disribuion of phases across x locaions changes wih ime. (phase modulaion) More rapid change of phase wih x -> higher spaial frequency () = " 2# () = " 2# K-space A each poin in ime, he received signal is he Fourier ransform of he objec s() = M( (), ()) = F[ m(x, y) ] kx ( ), ( ) evaluaed a he spaial frequencies: G x ()d G y ()d Thus, he gradiens conrol our posiion in k-space. The design of an MRI pulse sequence requires us o efficienly cover enough of k-space o form our image. G x () 1 2 K-space rajecory Unis Spaial frequencies (, ) have unis of 1/disance. Mos commonly, 1/cm Gradien srenghs have unis of (magneic field)/ disance. Mos commonly G/cm or mt/m () = " 2# ( 1 ) ( 2 ) G x ()d γ/(2π) has unis of H/G or H/Tesla. () = " 2# G x ( )d = [H /Gauss][Gauss /cm][sec] = [1/cm] 14

15 G x () = 1 Gauss/cm Example 2 =.235ms ( 1 ) ( 2 ) ( 2 ) = " G x ( )d 2# = 4257H /G &1G /cm & (3 s =1 cm (1 1 cm 15