1TT Liu, BE280A, UCSD Fall 2007Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2007MRI Lecture 2TT Liu, BE280A, UCSD Fall 2007GradientsSpins 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.TT Liu, BE280A, UCSD Fall 2007Simplified 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 2007Z Gradient CoilB(mT)LCredit: Buxton 20022TT Liu, BE280A, UCSD Fall 2007Gradient 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 2007Interpretation∆Bz(x)=GxxSpins Precess atat γB0+ γGxx(faster)Spins Precess at γB0- γGxx(slower)xSpins Precess at γB0TT Liu, BE280A, UCSD Fall 2007Rotating Frame of ReferenceReference everything to the magnetic field at isocenter.TT Liu, BE280A, UCSD Fall 2007Interpretation∆x 2∆x-∆x-2∆x 0∆Bz(x)=Gxx! exp " j2#18$x% & ' ( ) * x% & ' ( ) * ! exp " j2#28$x% & ' ( ) * x% & ' ( ) * ! exp " j2#08$x% & ' ( ) * x% & ' ( ) * FasterSlower3TT Liu, BE280A, UCSD Fall 2007Fig 3.12 from Nishimurakx=0; ky=0kx=0; ky≠0TT Liu, BE280A, UCSD Fall 2007Phase with time-varying gradientTT Liu, BE280A, UCSD Fall 2007K-space trajectoryGx(t)tt1t2ky! kx( t1)! kx( t2)Gy(t)t3t4kx! ky( t4)! ky( t3)TT Liu, BE280A, UCSD Fall 2007Nishimura 19964TT Liu, BE280A, UCSD Fall 2007K-space trajectoryGx(t)tt1t2kyGy(t)kxTT Liu, BE280A, UCSD Fall 2007Spin-WarpGx(t)t1kyGy(t)kxTT Liu, BE280A, UCSD Fall 2007Spin-WarpGx(t)t1kyGy(t)kxTT Liu, BE280A, UCSD Fall 2007Spin-Warp Pulse SequenceGx(t)kykxGy(t)RF5TT Liu, BE280A, UCSD Fall 2007Gradient 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 2007Static 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 )TT Liu, BE280A, UCSD Fall 2007Time-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 2007PhasePhase = 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 2007Phase 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 2007Time-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 2007Signal 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 obtainTT Liu, BE280A, UCSD Fall 2007Signal 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$0t%Where7TT Liu, BE280A, UCSD Fall 2007MR signal is Fourier Transform! s(t) = m(x, y)exp " j2#kx(t)x + ky(t)y( )( )y$x$dxdy= M kx(t),ky(t)( )= F m(x, y)[ ]kx(t ),ky(t )TT Liu, BE280A, UCSD Fall 2007Recap• Frequency = rate of change of phase.• Higher magnetic field -> higher Larmor frequency ->phase changes more rapidly with time.• With a constant gradient Gx, spins at different x locationsprecess at different frequencies -> spins at greater x-valueschange phase more rapidly.• With a constant gradient, distribution of phases across xlocations changes with time. (phase modulation)• More rapid change of phase with x -> higher spatialfrequency kxTT Liu, BE280A, UCSD Fall 2007K-space! s(t) = M kx(t),ky(t)( )= F m(x, y)[ ]kx(t ),ky(t )! kx( t) ="2#Gx($) d$0t%ky( t) ="2#Gy($) d$0t%At each point in time, the received signal is the Fouriertransform of the objectevaluated at the spatial frequencies:Thus, the gradients control our position in k-space. Thedesign of an MRI pulse sequence requires us toefficiently cover enough of k-space to form our image.TT Liu, BE280A, UCSD Fall 2007K-space trajectoryGx(t)t! kx( t) ="2#Gx($) d$0t%t1t2kxky! kx( t1)! kx( t2)8TT Liu, BE280A, UCSD Fall 2007UnitsSpatial frequencies (kx, ky) have units of 1/distance.Most commonly, 1/cmGradient strengths have units of (magneticfield)/distance. Most commonly G/cm or mT/mγ/(2π) has units of Hz/G or Hz/Tesla.! kx(t) ="2#Gx($)d$0t%= [Hz /Gauss][Gauss /cm][sec]= [1/cm]TT Liu, BE280A, UCSD Fall 2007ExampleGx(t) = 1 Gauss/cmt! kx(t2) ="2#Gx($)d$0t%= 4257Hz /G &1G /cm & 0.235 '10(3s=1 cm(1kxky! kx( t1)! kx( t2)t2 = 0.235ms1
View Full Document
Unlocking...