1TT Liu BE280A, UCSD Fall 2004Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2004MRI Lecture 3TT Liu BE280A, UCSD Fall 2004Topics• Review signal equation• Sampling requirements• Slice Selection• Gradient Echo and Spin Echo• Image ContrastTT Liu BE280A, UCSD Fall 2004MR 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 )xtm(x)s(t)2TT Liu BE280A, UCSD Fall 2004K-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 2004There is nothing that nuclear spinswill not do for you, as long as youtreat them as human beings. Erwin HahnTT Liu BE280A, UCSD Fall 2004Interpretation∆x 2∆x-∆x-2∆x 0∆B(z)=Gyz€ exp − j2π18Δx x € exp − j2π28Δx x € exp − j2π08Δx x FasterSlower3TT Liu BE280A, UCSD Fall 2004K-space trajectoryGx(t)t€ kx(t) =γ2πGx(τ)dτ0t∫t1t2kxky€ kx(t1)€ kx(t2)TT Liu BE280A, UCSD Fall 2004K-space trajectoryGx(t)tt1t2ky€ kx(t1)€ kx(t2)Gy(t)t3t4kx€ ky(t4)€ ky(t3)TT Liu BE280A, UCSD Fall 2004K-space trajectoryGx(t)tt1t2kyGy(t)kx4TT Liu BE280A, UCSD Fall 2004Spin-WarpGx(t)t1kyGy(t)kxTT Liu BE280A, UCSD Fall 2004Spin-WarpGx(t)t1kyGy(t)kxTT Liu BE280A, UCSD Fall 2004Spin-Warp Pulse SequenceGx(t)kykxGy(t)RF5TT Liu BE280A, UCSD Fall 2004K-space trajectorieskxkykykxEPI SpiralCredit: Larry FrankTT Liu BE280A, UCSD Fall 2004X =*=1/∆kTT Liu BE280A, UCSD Fall 2004Nyquist ConditionsFOVXFOVY1/∆kX1/∆kY1/∆kY> FOVY1/∆kX> FOVX6TT Liu BE280A, UCSD Fall 2004AliasingTT Liu BE280A, UCSD Fall 2004Sampling in kykxkyGx(t)Gy(t)RFΔkyτyGyi€ Δky=γ2πGyiτy€ FOVy=1ΔkyTT Liu BE280A, UCSD Fall 2004Sampling in kxxLow passFilterADCxLow passFilterADC€ cosω0t€ sinω0tRF SignalOne I,Q sample every ΔtM= I+jQIQNote: In practice, there are number of ways ofimplementing this processing.7TT Liu BE280A, UCSD Fall 2004Sampling in kxkxky€ Δkx=γ2πGxrΔt€ FOVx=1ΔkxGx(t)t1ADCGxrΔtTT Liu BE280A, UCSD Fall 2004ResolutionTT Liu BE280A, UCSD Fall 2004Resolution8TT Liu BE280A, UCSD Fall 2004Effective Width€ wE=1w(0)w(x) dx−∞∞∫wE€ wE=11sinc(Wkxx)dx−∞∞∫= F sin c(Wkxx)[ ]kx= 0=1WkxrectkxWkx kx= 0=1WkxExample€ 1Wkx€ −1WkxTT Liu BE280A, UCSD Fall 2004Resolution and spatial frequency€ 2Wkx€ With a window of width Wkx the highest spatial frequency is Wkx/2.This corresponds to a spatial period of 2/Wkx.€ 1Wkx= Effective Width =δx= ResolutionTT Liu BE280A, UCSD Fall 2004Resolution€ δx=1Wkx=12kx,max =1γ2πGxrτx€ Wkx€ WkyGx(t)Gxr€ τx€ δy=1Wky=12ky,max =1γ2π2GypτyGy(t)τy€ Gyp9TT Liu BE280A, UCSD Fall 2004Example€ Goal :FOVx= FOVy= 25.6 cmδx=δy= 0.1 cm€ Readout Gradient :FOVx=1γ2πGxrΔtPick Δt = 32 µsecGxr=1FOVxγ2πΔt=125.6cm( )42.57 ×106T−1s−1( )32 ×10−6s( ) = 2.8675 × 10−5T/cm = .28675 G/cm1 Gauss = 1 ×10−4 Teslat1ADCGxrΔtTT Liu BE280A, UCSD Fall 2004Example€ Readout Gradient :δx=1γ2πGxrτxτx=1δxγ2πGxr=10.1cm( )4257 G−1s−1( )0.28675 G/cm( ) = 8.192 ms = NreadΔtwhere Nread=FOVxδx= 256Gx(t)Gxr€ τxTT Liu BE280A, UCSD Fall 2004Example€ Phase - Encode Gradient :FOVy=1γ2πGyiτyPick τy= 4.096 msecGyi=1FOVyγ2πτy=125.6cm( )42.57 ×106T−1s−1( )4.096 ×10−3s( ) = 2.2402 × 10-7 T/cm = .00224 G/cmτyGyi10TT Liu BE280A, UCSD Fall 2004Example€ Phase - Encode Gradient :δy=1γ2π2GypτyGyp=1δy2γ2πτy=10.1cm( )4257 G−1s−1( )4.096 ×10-3s( ) = 0.2868 G/cm =Np2Gyiwhere Np=FOVyδy= 256Gy(t)τy€ GypTT Liu BE280A, UCSD Fall 2004Sampling€ Wkx€ WkyIn practice, an even number(typically power of 2) sample isusually taken in each direction totake advantage of the Fast FourierTransform (FFT) for reconstruction.kyyFOV/41/FOV4/FOVFOVTT Liu BE280A, UCSD Fall 2004Gibbs Artifact256x256 image 256x128 imageImages from http://www.mritutor.org/mritutor/gibbs.htm*=11TT Liu BE280A, UCSD Fall 2004ApodizationImages from http://www.mritutor.org/mritutor/gibbs.htm*=rect(kx)h(kx )=1/2(1+cos(2πkx)Hanning Windowsinc(x)0.5sinc(x)+0.25sinc(x-1)+0.25sinc(x+1)TT Liu BE280A, UCSD Fall 2004Aliasing and BandwidthxLPFADCADC€ cosω0t€ sinω0tRF SignalIQLPFx*xftxtFOV 2FOV/3Temporal filtering inthe readout directionlimits the readoutFOV. So there shouldnever be aliasing in thereadout direction.TT Liu BE280A, UCSD Fall 2004Aliasing and BandwidthSlowerFasterxfLowpass filterin the readout direction toprevent aliasing.readoutFOVxB=γGxrFOVx12TT Liu BE280A, UCSD Fall 2004Slice SelectionRecall, that we can tip spins away from their equilibrium stateby applying a radio-frequency pulse at the Larmor frequency.In the presence of a spatial gradient Gz. spins in an interval -Δz/2 to -Δz/2 have Larmor frequencies ranging fromω0-γGzΔz/2 to ω0 +γGzΔz/2. In order to tip all the spins inthis interval, we can apply an RF pulse with energy that isspaced over this frequency interval.TT Liu BE280A, UCSD Fall 2004Slice Selectionzslicefrect(f/W)W=γGzΔz/(2π)Δzsinc(Wt)TT Liu BE280A, UCSD Fall 2004Slice SelectionGx(t)Gy(t)RFGz(t)Slice select gradientSlice refocusing gradient13TT Liu BE280A, UCSD Fall 2004Gradient EchoGx(t)Gy(t)RFGz(t)Slice select gradientSlice refocusing gradientADCSpins all inphase at kx=0TT Liu BE280A, UCSD Fall 2004Static InhomogeneitiesIn the ideal situation, the static magnetic field is totally uniformand the reconstructed object is determined solely by the appliedgradient fields. In reality, the magnet is not perfect and will notbe totally uniform. Part of this can be addressed by additionalcoils called “shim” coils, and the process of making the fieldmore uniform is called “shimming”. In the old days this wasdone manually, but modern magnets can do this automatically.In addition to magnet imperfections, most biological samplesare inhomogeneous and this will lead to inhomogeneity in thefield. This is because, each tissue has different magneticproperties and will distort the field.TT Liu
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