Magnetic Resonance Imaging (MRI)

Transcription

1 C. A. Bouman: Digital Image Processing - February 15, 2 1 Magnetic Resonance Imaging (MRI) Can be very high resolution No radiation exposure Very flexible and programable Tends to be expensive, noisy, slow

2 C. A. Bouman: Digital Image Processing - February 15, 2 2 MRI Attributes Based on magnetic resonance effect in atomic species Does not require any ionizing radiation Numerous modalities Conventional anatomical scans Functional MRI (fmri) MRI Tagging Image formation RF excitation of magnetic resonance modes Magnetic field gradients modulate resonance frequency Reconstruction computed with inverse Fourier transform Fully programmable Requires an enormous (and very expensive) superconducting magnet

3 C. A. Bouman: Digital Image Processing - February 15, 2 3 Magnetic Resonance Magnetic Field Procession Atom Atom will precess at the Lamor frequency ω o = LM Quantities of importance M - magnitude of ambient magnetic field ω o - frequency of procession (radians per second) L - Lamor constant. Depends on choice of atom

4 C. A. Bouman: Digital Image Processing - February 15, 2 4 The MRI Magnet Liquid Helium Z axis Megnetic Field X axis Superconducting Magnet Large super-conducting magnet Uniform field within bore Very large static magnetic field of M o

5 C. A. Bouman: Digital Image Processing - February 15, 2 5 Magnetic Field Gradients Magnetic field magnitude at the location (x, y, z) has the form M(x, y) = M o + xg x + yg y + zg z G x, G y, and G z control magnetic field gradients Gradients can be changed with time Gradients are small compared to M o For time varying gradients M(x, y, t) = M o + xg x (t) + yg y (t) + zg z (t)

6 C. A. Bouman: Digital Image Processing - February 15, 2 6 MRI Slice Select Magnetic Field Mo Selected Slice RF Pulse RF Antenna Slope Gz Z RF pulse is emitted at frequency ω o Atoms precess in slice corresponding to M o = ω o /L

7 C. A. Bouman: Digital Image Processing - February 15, 2 7 How Do We Imaging Selected Slice? Y axis Selected Slice RF Antenna X axis Precessing atoms radiate electromagnetic energy at RF frequencies Strategy Vary magnetic gradients along x and y axies Measure received RF signal Reconstruct image from RF measurements

8 C. A. Bouman: Digital Image Processing - February 15, 2 8 Signal from a Single Voxel RF Antenna Voxel of Selected Slice RF signal from a single voxel has the form r(x, y, t) = f(x, y)e jφ(t) f(x, y) voxel dependent weighting Depends on properties of material in voxel Quantity of interest Typically weighted by T1, T2, or T2* φ(t) phase of received signal Can be modulated using G x and G y magnetic field gradients We assume that φ() =

9 C. A. Bouman: Digital Image Processing - February 15, 2 9 Analysis of Phase Frequency = time derivative of phase dφ(t) dt φ(t) = t where we define = L M(x, y, t) ω o = L M o k x (t) = t LG x(τ)dτ k y (t) = t LG y(τ)dτ L M(x, y, τ)dτ = t LM o + xlg x (τ) + ylg y (τ)dτ = ω o t + xk x (t) + yk y (t)

10 C. A. Bouman: Digital Image Processing - February 15, 2 1 Received Signal from Voxel RF Antenna Voxel of Selected Slice RF signal from a single voxel has the form r(t) = f(x, y)e jφ(t) = f(x, y)e j (ω o t+xk x (t)+yk y (t)) = f(x, y)e jωot e j (xk x (t)+yk y (t))

11 C. A. Bouman: Digital Image Processing - February 15, 2 11 Received Signal from Selected Slice Y axis Selected Slice RF Antenna X axis RF signal from the complete slice is given by R(t) = IR IR r(x, y, t)dxdy = IR IR f(x, y)ejω ot e j (xk x (t)+yk y (t)) dxdy = e jω ot IR IR f(x, y)ej (xk x (t)+yk y (t)) dxdy = e jωot F( k x (t), k y (t)) were F(u, v) is the CSFT of f(x, y)

12 C. A. Bouman: Digital Image Processing - February 15, 2 12 K-Space Interpretation of Demodulated Signal RF signal from the complete slice is given by where Strategy F( k x (t), k y (t)) = R(t)e jω ot k x (t) = t LG x(τ)dτ k y (t) = t LG y(τ)dτ Scan spatial frequencies by varying k x (t) and k y (t) Reconstruct image by performing (inverse) CSFT G x (t) and G y (t) control velocity through K-space

13 C. A. Bouman: Digital Image Processing - February 15, 2 13 Controlling K-Space Trajectory Relationship between gradient coil voltage and K-space L x di(t) dt L y di(t) dt using this results in = v x (t) G x (t) = M x i(t) = v y (t) G y (t) = M y i(t) k x (t) = LM x L x k y (t) = LM y L y t t τ v x(s)dsdτ τ v y(s)dsdτ v x (t) and v y (t) are like the accelerator peddles for k x (t) and k y (t)

14 C. A. Bouman: Digital Image Processing - February 15, 2 14 Echo Planer Imaging (EPI) Scan Pattern A commonly used raster scan pattern through K-space Ky Serpintine Scan Kx k x (t) = L t G x(τ)dτ = LM x L x k y (t) = L t G y(τ)dτ = LM y L y t t τ v x(s)dsdτ τ v y(s)dsdτ

15 C. A. Bouman: Digital Image Processing - February 15, 2 15 Gradient Waveforms for EPI Gradient waveforms in x and y look like Gx(t) Gy(t) Voltage waveforms in x and y look like Vx(t) Vy(t)