sLecture

↳ Topic 3: Magnetic Resonance Imaging

Excerpt from Prof. Bouman's Lecture

Accompanying Lecture Notes

Magnetic Resonance Imaging (MRI)

Definition

• Can be very high resolution
• No exposure to ionizing radiation
• Very flexible and programmable
• Tends to be expensive, noisy, and slow

Fig 1: The exterior of an MRI scanner

Fig 2: MRI scan of a patient

MRI Attributes

• Based on magnetic resonance effect in atomic species
• Does not requires any ionizing radiation
• Numerous modalitites
  • 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

Magnetic Resonance

Fig 3: Precession of atom in the presence of a magnetic field

• Atom will precess at the Lamor frequency
  

$ \omega_0 = LM \ $
where
$ M $ is the magnitude of the ambient magnetic field
$ \omega_0 $ is the frequency of precession in radians per second
$ L $ is the Lamor constant and its value depends on the choice of atom

The MRI Magnet

Fig 4: The MRI magnet

• Large superconducting magnet
  • Uniform field within bore
  • Very large static magnetic field $ M_0 $

Magnetic Field Gradients

• Magnetic field magnitude at the location $ (x,y,z) $ has the form
  

$ M(x,y,z) = M_0 + 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_0 $

• For the time varying gradients,

$ M(x,y,z,t) = M_0 + xG_x(t) + yG_y(t) + zG_z(t) \ $

MRI Slice Select

Fig 5: MRI slice select

Design RF pulse to excite protons in single slice
- Turn off $ x $ and $ y $ gradients
- Set $ z $ gradient to fix positive value, $ G_z > 0 $
- Use the fact that resonance frequency is given by
$ \omega = L(M_0+zG_z) $

Slice Select Pulse Design

• Design parameters
  • Slice center$ = z_c $
  • Slice thickness $ = \Delta z $

• Slice centered at $ z_c $ ⇒ pulse frequency
  

$ f_c = \frac{LM_0}{2\pi}+\frac{z_cLG_z}{2\pi} = f_0 + \frac{z_cLG_z}{2\pi} $

• Slice thickness $ \Delta z $ ⇒ pulse bandwidth
  

$ \Delta f = \frac{\Delta z LG_z}{2\pi} $

• Using the parameters. the pulse is given by
  

$ s(t) = e^{j2\pi f_ct}sinc(t\Delta f) $

and its CTFT is given by
$ S(f) = rect(\frac{f-f_c}{\Delta f}) $

Imaging the Selected Slice

Fig 6:

• Precessing atoms radiate electromagnetic energy at RF frequencies
• Strategy
  • Vary magnetic gradients along $ x $ and $ y $ axes
  • Measure received RF signal
  • Reconstruct image from RF measurements

Signal from a Single Voxel

Fig 6: Signal from a single voxel

RF signal from a single voxel has the form
$ r(x,y,t) = f(x,y)e^{j\phi(t)} \ $

$ f(x,y) $ voxel dependent weight

• Depends on properties of material in voxel
• Quantity of interest
• Typically "weighted" by T1, T2, or T3*

$ \phi(t) $ phase of received signal

• Can be modulated using $ G_x $ and $ G_y $ magnetic field gradients
• We assume that $ \phi(0) = 0 $

Analysis of Phase

Frequency = time derivative of phase
$ \begin{align} \frac{d\phi(t)}{dt} &= LM(x,y,t) \\ \phi(t) &= \int_0^t LM(x,y,\tau)d\tau \\ &= \int_0^t LM_0 + xLG_x(\tau) + yLG_y(\tau)d\tau \\ &= \omega_0t + xk_x(t) +yky(t) \end{align} $

where we define
$ \omega_0 = LM_0 $
$ k_x(t) = \int_0^t LG_x(\tau)d\tau $
$ k_y(t) = \int_0^t LG_y(\tau)d\tau $

RF signal from a single voxel has the form
$ \begin{align} r(t) &= f(x,y)e^{j\phi(t)} \\ &= f(x,y)e^{j(\omega_0t + xk_x(t) + yk_y(t))} \\ &= f(x,y)e^{j\omega_0t}e^{j(xk_x(t) + yk_y(t))} \end{align} $

Received Signal from Selected Slice

RF signal from the complete slice is given by
$ \begin{align} R(t) &= \int_{\mathbf{R}}\int_{\mathbf{R}}r(x,y,t)dxdy \\ &= \int_{\mathbf{R}}\int_{\mathbf{R}} f(x,y)e^{j\omega_0t}e^{j(xk_x(t) + yk_y(t))}dxdy \\ &= e^{j\omega_0t}\int_{\mathbf{R}}\int_{\mathbf{R}} f(x,y)e^{j(xk_x(t) + yk_y(t))}dxdy \\ &= e^{j\omega_0t}F(-k_x(t),-k_y(t)) \end{align} $

where $ F(u,v) $ is the CSFT of $ f(x,y) $

K-Space Interpretation of Demodulated Signal

RF signal from the complete slice is given by
$ F(-k_x(t),-k_y(t))=R(t)e^{j\omega_0t} $

where
$ k_x(t) = \int_0^t LG_x(\tau)d\tau $
$ k_y(t) = \int_0^t LG_y(\tau)d\tau $

Strategy

• Scan partial frequencies by varying $ k_x(t) $ and $ k_y(t) $
  
• Reconstruct image by performing the (inverse) CSFT
• $ G_x(t) $ and $ G_y(t) $ control velocity through K-space

Controlling K-Space Trajectory

Relationships between gradient coil voltage and K-space
$ \begin{align} L_x\frac{di(t)}{dt} &= v_x(t) \quad G_x(t) = M_xi(t) \\ L_y\frac{di(t)}{dt} &= v_y(t) \quad G_y(t) = M_yi(t) \end{align} $

using this result in
$ \begin{align} k_x(t) &= \frac{LM_x}{L_x}\int_0^t\int_0^{\tau}v_x(s)dsd\tau \\ k_y(t) &= \frac{LM_y}{L_y}\int_0^t\int_0^{\tau}v_y(s)dsd\tau \end{align} $

$ v_x(t) $ and $ v_y(t) $ are like the accelerator peddles for $ k_x(t) $ and $ k_y(t) $.

Echo Planar Imaging (EPI) Scan Pattern

Fig 7: A commonly used raster scan pattern through K-space

$ \begin{align} k_x(t) &= L\int_0^t G_x(\tau)d\tau = \frac{LM_x}{L_x}\int_0^t\int_0^{\tau} v_x(s)dsd\tau \\ k_y(t) &= L\int_0^t G_y(\tau)d\tau = \frac{LM_y}{L_y}\int_0^t\int_0^{\tau} v_y(s)dsd\tau \end{align} $

==Gradient Waveforms for EPI</math>

Fig 8: Gradient waveforms in $ x $ and $ y $

Fig 9: Voltage waveforms in $ x $ and $ y $

References

• C. A. Bouman. ECE 637. Class Lecture. Digital Image Processing I. Faculty of Electrical Engineering, Purdue University. Spring 2013.

• G. Francis. Psy 200. "Introduction to Cognitive Psychology". Class Lecture Notes. Faculty of Psychological Sciences, Purdue University. Spring 2013.

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