1TT Liu, BE280A, UCSD Fall 2006Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2006MRI Lecture 2TT Liu, BE280A, UCSD Fall 2006GradientsSpins precess at the Larmor frequency, which isproportional to the local magnetic field. In a constantmagnetic field Bz=B0, all the spins precess at the samefrequency (ignoring chemical shift).Gradient coils are used to add a spatial variation to Bzsuch that Bz(x,y,z) = B0+Δ Bz(x,y,z) . Thus, spins atdifferent physical locations will precess at differentfrequencies.2TT Liu, BE280A, UCSD Fall 2006Simplified Drawing of Basic Instrumentation.Body lies on table encompassed bycoils for static field Bo, gradient fields (two of three shown), and radiofrequency field B1.MRI SystemImage, caption: copyright Nishimura, Fig. 3.15TT Liu, BE280A, UCSD Fall 2006Z Gradient CoilB(mT)LCredit: Buxton 20023TT Liu, BE280A, UCSD Fall 2006Gradient Fields! Bz(x, y,z) = B0+"Bz"xx +"Bz"yy +"Bz"zz= B0+ Gxx + Gyy + Gzzz! Gz="Bz"z> 0! Gy="Bz"y> 0yTT Liu, BE280A, UCSD Fall 2006Interpretation∆Bz(x)=GxxSpins Precess atat γB0+ γGxx(faster)Spins Precess at γB0- γGxx(slower)xSpins Precess at γB04TT Liu, BE280A, UCSD Fall 2006Gradient Fields ! Gxx + Gyy + Gzz =r G "r r ! r G " Gxˆ i + Gyˆ j + Gzˆ k ! Bz(r r ,t) = B0+r G (t) "r r Define ! r r " xˆ i + yˆ j + zˆ k So that Also, let the gradient fields be a function of time. Thenthe z-directed magnetic field at each point in thevolume is given by :TT Liu, BE280A, UCSD Fall 2006Static Gradient Fields! M(t) = M(0)e" j#0te"t /T2In a uniform magnetic field, the transverse magnetizationis given by:In the presence of non time-varying gradients we have ! M (r r ) = M (r r ,0)e" j#Bz(r r )te" t /T2(r r )= M (r r ,0)e" j#( B0+r G $r r )te" t /T2(r r )= M (r r ,0)e" j%0te" j#r G $r r te" t /T2(r r )5TT Liu, BE280A, UCSD Fall 2006Time-Varying Gradient FieldsIn the presence of time-varying gradients the frequencyas a function of space and time is: ! "r r ,t( )=#Bz(r r ,t)=#B0+#r G (t) $r r ="0+ %"(r r ,t)TT Liu, BE280A, UCSD Fall 2006PhasePhase = angle of the magnetization phasorFrequency = rate of change of angle (e.g. radians/sec)Phase = time integral of frequency ! "#r r ,t( )= $ "%(r r ,&)0t'd&= $(v G (r r ,&) )r r 0t'd& ! "r r ,t( )= #$(r r ,%)0t&d%= #$0t + '"r r ,t( )Where the incremental phase due to the gradients is6TT Liu, BE280A, UCSD Fall 2006Phase with constant gradient ! "#r r ,t3( )= $ "%(r r ,&)0t3'd& ! "#r r ,t2( )= $ "%(r r ,&)0t2'd&= $"%(r r )t2 if "% is non - time varying. ! "#r r ,t1( )= $ "%(r r ,&)0t1'd&TT Liu, BE280A, UCSD Fall 2006Phase with time-varying gradient7TT Liu, BE280A, UCSD Fall 2006Time-Varying Gradient FieldsThe transverse magnetization is then given by ! M(r r ,t) = M(r r ,0)e"t /T2(r r )e#(r r ,t )= M(r r ,0)e"t /T2(r r )e" j$0texp " j %$r r ,t( )d&ot'( )= M(r r ,0)e"t /T2(r r )e" j$0texp " j(r G (&) )r r d&ot'( )TT Liu, BE280A, UCSD Fall 2006Signal EquationSignal from a volume ! sr(t) = M(r r ,t)V"dV= M(x, y,z,0)e#t /T2(r r )e# j$0texp # j%r G (&) 'r r d&ot"( )z"y"x"dxdydzFor now, consider signal from a slice along z and dropthe T2 term. Define ! m(x, y) " M(r r ,t)z0#$z / 2z0+$z / 2%dz ! sr(t) = m(x, y)e" j#0texp " j$r G (%) &r r d%ot'( )y'x'dxdyTo obtain8TT Liu, BE280A, UCSD Fall 2006Signal EquationDemodulate the signal to obtain ! s(t) = ej"0tsr(t)= m(x, y)exp # j$r G (%) &r r d%ot'( )y'x'dxdy= m(x, y)exp # j$Gx(%)x + Gy(%)y[ ]d%ot'( )y'x'dxdy= m(x, y)exp # j2(kx(t)x + ky(t)y( )( )y'x'dxdy! kx(t) ="2#Gx($)d$0t%ky(t) ="2#Gy($)d$0t%WhereTT Liu, BE280A, UCSD Fall 2006MR signal is Fourier Transform! s(t) = m(x, y)exp " j2#kx(t)x + ky(t)y( )( )y$x$dxdy= M kx(t),ky(t)( )= F m(x, y)[ ]kx(t ),ky(t )9TT Liu, BE280A, UCSD Fall 2006Recap• Frequency = rate of change of phase.• Higher magnetic field -> higher Larmor frequency ->phase changes more rapidly with time.• With a constant gradient Gx, spins at different x locationsprecess at different frequencies -> spins at greater x-valueschange phase more rapidly.• With a constant gradient, distribution of phases across xlocations changes with time. (phase modulation)• More rapid change of phase with x -> higher spatialfrequency kxTT Liu, BE280A, UCSD Fall 2006K-space! s(t) = M kx(t),ky(t)( )= F m(x, y)[ ]kx(t ),ky(t )! kx(t) ="2#Gx($)d$0t%ky(t) ="2#Gy($)d$0t%At each point in time, the received signal is the Fouriertransform of the objectevaluated at the spatial frequencies:Thus, the gradients control our position in k-space. Thedesign of an MRI pulse sequence requires us toefficiently cover enough of k-space to form our image.10TT Liu, BE280A, UCSD Fall 2006Interpretation∆x 2∆x-∆x-2∆x 0∆Bz(x)=Gxx! exp " j 2#18$x% & ' ( ) * x% & ' ( ) * ! exp " j2#28$x% & ' ( ) * x% & ' ( ) * ! exp " j2#08$x% & ' ( ) * x% & ' ( ) * FasterSlowerTT Liu, BE280A, UCSD Fall 2006K-space trajectoryGx(t)t! kx(t) ="2#Gx($)d$0t%t1t2kxky! kx(t1)! kx(t2)11TT Liu, BE280A, UCSD Fall 2006K-space trajectoryGx(t)tt1t2ky! kx(t1)! kx(t2)Gy(t)t3t4kx! ky(t4)! ky(t3)TT Liu, BE280A, UCSD Fall 2006Nishimura 199612TT Liu, BE280A, UCSD Fall 2006K-space trajectoryGx(t)tt1t2kyGy(t)kxTT Liu, BE280A, UCSD Fall 2006Spin-WarpGx(t)t1kyGy(t)kx13TT Liu, BE280A, UCSD Fall 2006Spin-WarpGx(t)t1kyGy(t)kxTT Liu, BE280A, UCSD Fall 2006Spin-Warp Pulse SequenceGx(t)kykxGy(t)RF14TT Liu, BE280A, UCSD Fall 2006UnitsSpatial frequencies (kx, ky) have units of 1/distance.Most commonly, 1/cmGradient strengths have units of (magneticfield)/distance. Most commonly G/cm or mT/mγ/(2π) has units of Hz/G or Hz/Tesla.! kx(t) ="2#Gx($)d$0t%= [Hz /Gauss][ Gauss /cm][sec]= [1/cm]TT Liu, BE280A, UCSD Fall 2006ExampleGx(t) = 1 Gauss/cmt! kx(t2) ="2#Gx($)d$0t%= 4257Hz /G &1G /cm &0.235 '10(3s=1 cm(1kxky! kx(t1)! kx(t2)t2 = 0.235ms1
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