1Thomas Liu, BE280A, UCSD, Fall 2006Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2006MRI Lecture 4Thomas Liu, BE280A, UCSD, Fall 2006RF ExcitationFrom Levitt, Spin Dynamics, 20012Thomas Liu, BE280A, UCSD, Fall 2006RF ExcitationImage & caption: Nishimura, Fig. 3.2B1 radiofrequency field tuned toLarmor frequency and applied intransverse (xy) plane inducesnutation (at Larmor frequency) ofmagnetization vector as it tipsaway from the z-axis. - lab frame of referenceAt equilibrium, net magnetizaionis parallel to the main magneticfield. How do we tip themagnetization away fromequilibrium?Thomas Liu, BE280A, UCSD, Fall 2006RF Excitationhttp://www.eecs.umich.edu/%7Ednoll/BME516/3Thomas Liu, BE280A, UCSD, Fall 2006Images & caption: Nishimura, Fig. 3.3a) Laboratory frame behavior of M b) Rotating frame behavior of Mhttp://www.eecs.umich.edu/%7Ednoll/BME516/Thomas Liu, BE280A, UCSD, Fall 2006Nishimura 1996! B1(t) = 2B1(t)cos"t( )i= B1(t) cos"t( )i # sin"t( )j( )+ B1(t) cos"t( )i + sin"t( )j( )4Thomas Liu, BE280A, UCSD, Fall 2006! Rotating Frame Bloch EquationdMrotdt= Mrot"#BeffBeff= Brot+$rot#; $rot=00%$& ' ( ( ( ) * + + + Thomas Liu, BE280A, UCSD, Fall 2006! Let Brot= B1(t)i + B0kBeff= Brot+"rot#= B1(t)i + B0$"#% & ' ( ) * kIf "="0=#B0Then Beff= B1(t)i5Thomas Liu, BE280A, UCSD, Fall 2006Nishimura 1996! Flip angle"=#10$%(s)dswhere#1(t) =&B1(t)Thomas Liu, BE280A, UCSD, Fall 2006Nishimura 19966Thomas Liu, BE280A, UCSD, Fall 2006Nishimura 1996Thomas Liu, BE280A, UCSD, Fall 2006Slice Selectionzslicefrect(f/W)W=γGzΔz/(2π)Δzsinc(Wt)7Thomas Liu, BE280A, UCSD, Fall 2006! Let Brot= B1(t)i + B0+"Gzz( )kBeff= Brot+#rot"= B1(t)i + B0+"Gzz $#"% & ' ( ) * kIf #=#0Beff= B1(t)i +"Gzz( )kThomas Liu, BE280A, UCSD, Fall 2006Nishimura 19968Thomas Liu, BE280A, UCSD, Fall 2006Small Tip Angle ApproximationMzM0! "Mxy! For small "Mz= M0cos"# M0Mxy= M0sin"# M0"Thomas Liu, BE280A, UCSD, Fall 2006! Recall that in the rotating frame, flip angle "=#B10$%(s)ds Define &(z) as the Larmor Frequency at each location z referenced to &0The effective field felt in each spin's rotating frame of reference is :B1e(t) = B1(t)exp( j&(z)t)Therefore the flip angle in each spin's frame of reference is"(t,z) =#exp( j&(z)s)B10t%(s)ds With respect to the on - resonance frame of reference, there is alsoa relative phase shift of exp(' j&(z)t), so that "(t,z) =#exp(' j&(z)t) exp( j&(z)s)B10t%(s)ds Applying small angle approximation leads to Mr(t,z) ( jM0"(t,z) = M0#exp(' j&(z)t) exp( j&(z)s)B10t%(s)ds9Thomas Liu, BE280A, UCSD, Fall 2006! Small Tip Angle ApproximationMr(t,z) = jM0exp(" j#(z)t) exp(0t$j#(z)s)#1(s)dsFor symmetric pulse of length %Mr(%,z) = jM0exp(" j#(z)%/2) exp("%/ 2%/ 2$j2&f (z)s)#1(s +%/2)ds = jM0exp(" j#(z)%/2)F#1(t +%/2){ }f =" f (z)="'2&GzzThomas Liu, BE280A, UCSD, Fall 2006! B1(t) = B1rectt "#/2#$ % & ' ( ) Mr#,z( )= jM0exp(" j*(z)#) exp( j*(z)s)0#+*1rects "#/2#$ % & ' ( ) ds= jM0exp(" j*(z)#/2)F1D*1rectt#$ % & ' ( ) $ % & ' ( ) f = ",/ 2-( )Gzz= jM0exp(" j*(z)#/2)*1#sinc f#( )= jM0exp(" j*(z)#/2)*1#sinc,Gz#2-z$ % & ' ( ) Small Tip Angle ExampleNishimura 199610Thomas Liu, BE280A, UCSD, Fall 2006! Mr(3"/2,z) = exp j#(z)"/2( )Mr(",z) = jM0exp j#(z)"/2( )exp $ j#(z)"/2( )F#1(t +"/2){ }f =$%2&Gzz= j M0F#1(t +"/2){ }f =$%2&GzzRefocusingNishimura 1996Thomas Liu, BE280A, UCSD, Fall 2006Slice SelectionGx(t)Gy(t)RFGz(t)Slice select gradientSlice refocusing gradient11Thomas Liu, BE280A, UCSD, Fall 2006Gradient EchoGx(t)Gy(t)RFGz(t)Slice select gradientSlice refocusing gradientADCSpins all inphase at kx=0Thomas Liu, BE280A, UCSD, Fall 2006! B1(t +"/2) = A sinc t /"( )0.5 + 0.46cos2#t"$ % & ' ( ) $ % & ' ( ) = Asinc t /"( )w(t)F(B1(t +"/2)) = A" rect( f") *W ( f )= A" rect+Gzz"2#$ % & ' ( ) *W ,+Gzz2#$ % & ' ( ) Small Tip Angle Example! Width of the rect function is "z =2#$Gz%12Thomas Liu, BE280A, UCSD, Fall 2006Slice Selectionzslicefrect(fτ)Δzsinc(t/τ)! "f =1#=$Gz"z2% Thomas Liu, BE280A, UCSD, Fall 2006Nishimura 199613Thomas Liu, BE280A, UCSD, Fall 2006! Example"z = 5 mm;#= 400 µsec; $=%/2Gz=2%&"z#=1(4257Hz /G)(0.5cm)(400e ' 6)= 1.175 G /cm$(&B1sincs ' T /2#) * + , - . 0T/ds (&B10 area of sinc( )=&B1#B1=$&#=%/22%(4257Hz /G)(400e ' 6)= 0.1468
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