1Thomas Liu, BE280A, UCSD, Fall 2005Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2005MRI Lecture 3Thomas 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 )xtm(x)s(t)Thomas 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.2Thomas Liu, BE280A, UCSD, Fall 2005Nishimura 1996Thomas Liu, BE280A, UCSD, Fall 2005Spin-WarpGx(t)t1kyGy(t)kxThomas Liu, BE280A, UCSD, Fall 2005Spin-Warp Pulse SequenceGx(t)kykxGy(t)RF3Thomas Liu, BE280A, UCSD, Fall 2005X =*=1/∆kThomas Liu, BE280A, UCSD, Fall 2005Nyquist ConditionsFOVXFOVY1/∆kX1/∆kY1/∆kY> FOVY1/∆kX> FOVXThomas Liu, BE280A, UCSD, Fall 2005Sampling in kykxkyGx(t)Gy(t)RFΔkyτyGyi€ Δky=γ2πGyiτy€ FOVy=1Δky4Thomas Liu, BE280A, UCSD, Fall 2005Sampling in kxxLow passFilterADCxLow passFilterADC€ cosω0t€ sinω0tRF SignalOne I,Q sample every ΔtM= I+jQIQNote: In practice, there are number of ways ofimplementing this processing.Thomas Liu, BE280A, UCSD, Fall 2005Sampling in kxkxky€ Δkx=γ2πGxrΔt€ FOVx=1ΔkxGx(t)t1ADCGxrΔtThomas Liu, BE280A, UCSD, Fall 2005Resolution5Thomas Liu, BE280A, UCSD, Fall 2005Effective Width€ wE=1w(0)w(x) dx−∞∞∫wE€ wE=11sinc(Wkxx)dx−∞∞∫= F sin c(Wkxx)[ ]kx= 0=1WkxrectkxWkx kx= 0=1WkxExample€ 1Wkx€ −1WkxThomas Liu, BE280A, UCSD, Fall 2005Resolution 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= ResolutionThomas Liu, BE280A, UCSD, Fall 2005Resolution€ δx=1Wkx=12kx,max =1γ2πGxrτx€ Wkx€ WkyGx(t)Gxr€ τx€ δy=1Wky=12ky,max =1γ2π2GypτyGy(t)τy€ Gyp6Thomas Liu, BE280A, UCSD, Fall 2005Example€ 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 Teslat1ADCGxrΔtThomas Liu, BE280A, UCSD, Fall 2005Example€ Readout Gradient :δx=1γ2πGxrτxτx=1δxγ2πGxr=10.1cm( )4257 G−1s−1( )0.28675 G/cm( ) = 8.192 ms = NreadΔtwhere Nread=FOVxδx= 256Gx(t)Gxr€ τxThomas Liu, BE280A, UCSD, Fall 2005Example€ 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/cmτyGyi7Thomas Liu, BE280A, UCSD, Fall 2005Example€ 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€ GypThomas Liu, BE280A, UCSD, Fall 2005Sampling€ 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/FOVFOVThomas Liu, BE280A, UCSD, Fall 2005ExampleConsider the k-space trajectory shown below. ADC samples are acquired at the points shownwith € Δt = 10 µsec. The desired FOV (both x and y) is 10 cm and the desired resolution (both xand y) is 2.5 cm. Draw the gradient waveforms required to achieve the k-space trajectory. Labelthe waveform with the gradient amplitudes required to achieve the desired FOV and resolution.Also, make sure to label the time axis correctly.8Thomas Liu, BE280A, UCSD, Fall 2005GE Medical Systems 2003Thomas Liu, BE280A, UCSD, Fall 2005Gibbs Artifact256x256 image 256x128 imageImages from http://www.mritutor.org/mritutor/gibbs.htm*=Thomas Liu, BE280A, UCSD, Fall 2005ApodizationImages 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)9Thomas Liu, BE280A, UCSD, Fall 2005Aliasing and BandwidthxLPFADCADC€ cosω0t€ sinω0tRF SignalIQLPFx*xftxtFOV 2FOV/3Temporal filtering inthe readout directionlimits the readoutFOV. So there shouldnever be aliasing in thereadout direction.Thomas Liu, BE280A, UCSD, Fall 2005Aliasing and BandwidthSlowerFasterxfLowpass filterin the readout direction toprevent aliasing.readoutFOVxB=γGxrFOVxThomas Liu, BE280A, UCSD, Fall 2005GE Medical Systems 200310Thomas Liu, BE280A, UCSD, Fall 2005Slice 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.Thomas Liu, BE280A, UCSD, Fall 2005Slice
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