1TT Liu, BE280A, UCSD Fall 2005Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2005MRI Lecture 2TT Liu, BE280A, UCSD Fall 2005GradientsSpins 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 Bz suchthat Bz(x,y,z) = B0+Δ Bz(x,y,z) . Thus, spins at differentphysical locations will precess at different frequencies.TT Liu, BE280A, UCSD Fall 2005Simplified 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. 3.152TT Liu, BE280A, UCSD Fall 2005Z Gradient CoilB(mT)LCredit: Buxton 2002TT Liu, BE280A, UCSD Fall 2005Gradient 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 2005Interpretation∆Bz(x)=GxxSpins Precess atat γB0+ γGxx(faster)Spins Precess at γB0- γGxx(slower)xSpins Precess at γB03TT Liu, BE280A, UCSD Fall 2005Gradient 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 2005Static Gradient Fields€ M(t) = M(0)e− jω0te−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− jγBz(r r )e−t /T2(r r )= M(r r ,0)e− jγ(B0+r G ⋅r r )e−t /T2(r r )= M(r r ,0)e− jω0te− jγr G ⋅r r e−t /T2(r r )TT Liu, BE280A, UCSD Fall 2005Time-Varying Gradient FieldsIn the presence of time-varying gradients the frequencyas a function of space and time is: € ωr r ,t( )=γBz(r r ,t)=γB0+γr G (t)⋅r r =ω0+ Δω(r r ,t)4TT Liu, BE280A, UCSD Fall 2005PhasePhase = angle of the magnetization phasorFrequency = rate of change of angle (e.g. radians/sec)Phase = time integral of frequency € Δϕr r ,t( )= − Δω(r r ,τ)0t∫dτ= −γv G (r r ,τ) ⋅r r 0t∫dτ € ϕr r ,t( )= −ω(r r ,τ)0t∫dτ= −ω0t + Δϕr r ,t( )Where the incremental phase due to the gradients isTT Liu, BE280A, UCSD Fall 2005Phase with constant gradient € Δϕr r ,t3( )= − Δω(r r ,τ)0t3∫dτ € Δϕr r ,t2( )= − Δω(r r ,τ)0t2∫dτ= −Δω(r r )t2 if Δω is non - time varying. € Δϕr r ,t1( )= − Δω(r r ,τ)0t1∫dτTT Liu, BE280A, UCSD Fall 2005Phase with time-varying gradient5TT Liu, BE280A, UCSD Fall 2005Time-Varying Gradient FieldsThe transverse magnetization is then given by € M(r r ,t) = M(r r ,0)e−t /T2(r r )eϕ(r r ,t )= M(r r ,0)e−t /T2(r r )e− jω0texp − j Δωr r ,t( )dτot∫( )= M(r r ,0)e−t /T2(r r )e− jω0texp − jγr G (τ) ⋅r r dτot∫( )TT Liu, BE280A, UCSD Fall 2005Signal EquationSignal from a volume € sr(t) = M(r r , t)V∫dV= M( x, y,z,0)e−t /T2(r r )e− jω0texp − jγr G (τ) ⋅r r dτot∫( )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−Δz / 2z0+Δz /2∫dz € sr(t) = m(x,y)e− jω0texp − jγr G (τ) ⋅r r dτot∫( )y∫x∫dxdyTo obtainTT Liu, BE280A, UCSD Fall 2005Signal EquationDemodulate the signal to obtain € s(t) = ejω0tsr(t)= m(x,y)exp − jγr G (τ) ⋅r r dτot∫( )y∫x∫dxdy= m(x,y)exp − jγGx(τ)x + Gy(τ)y[ ]dτot∫( )y∫x∫dxdy= m(x,y)exp − j2πkx(t) x + ky(t) y( )( )y∫x∫dxdy€ kx(t) =γ2πGx(τ)dτ0t∫ky(t) =γ2πGy(τ)dτ0t∫Where6TT Liu, BE280A, UCSD Fall 2005MR signal is Fourier Transform€ s(t) = m(x, y)exp − j2πkx(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 2005K-space€ s(t) = M kx(t),ky(t)( )= F m ( x, y)[ ]kx(t ),ky(t )€ kx(t) =γ2πGx(τ)dτ0t∫ky(t) =γ2πGy(τ)dτ0t∫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 2005Interpretation∆x 2∆x-∆x-2∆x 0∆Bz(x)=Gxx€ exp − j2π18Δx x € exp − j2π28Δx x € exp − j2π08Δx x FasterSlower7TT Liu, BE280A, UCSD Fall 2005K-space trajectoryGx(t)t€ kx(t) =γ2πGx(τ)dτ0t∫t1t2kxky€ kx(t1)€ kx(t2)TT Liu, BE280A, UCSD Fall 2005K-space trajectoryGx(t)tt1t2ky€ kx(t1)€ kx(t2)Gy(t)t3t4kx€ ky(t4)€ ky(t3)TT Liu, BE280A, UCSD Fall 2005K-space trajectoryGx(t)tt1t2kyGy(t)kx8TT Liu, BE280A, UCSD Fall 2005Spin-WarpGx(t)t1kyGy(t)kxTT Liu, BE280A, UCSD Fall 2005Spin-WarpGx(t)t1kyGy(t)kxTT Liu, BE280A, UCSD Fall 2005Spin-Warp Pulse SequenceGx(t)kykxGy(t)RF9TT Liu, BE280A, UCSD Fall 2005UnitsSpatial 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) =γ2πGx(τ)dτ0t∫= [Hz / Gauss ][Gauss /cm][sec]= [1 / cm]TT Liu, BE280A, UCSD Fall 2005ExampleGx(t) = 1 Gauss/cmt€ kx(t2) =γ2πGx(τ)dτ0t∫= 4257Hz /G ⋅ 1G / cm ⋅0.235 ×10−3s=1 cm−1kxky€ kx(t1)€ kx(t2)t2 = 0.235ms1
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