sLecture

↳ Topic 3: Magnetic Resonance Imaging

Excerpt from Prof. Bouman's Lecture

Accompanying Lecture Notes

Magnetic resonance imaging (MRI or MR) is a medical imaging technique that uses the property of nuclear magnetic resonance. We will see later in greater detail that magnetic fields are applied to manipulate the precession of nuclei within the patients body. As they return to their previous states, they emit a signal that can be decoded into a map.

There is, in principal, no upper bound to how high the resolution can be of images produced using MRI. It would of course take a long time to perform such a scan. Generally scans have a spatial resolution in the order of millimeters.

Another advantage of MRI is that it uses no ionizing radiation unlike CT. Additionally, MR is very flexible and programmable. It is basically a software programmable module. MR is however about twice as expensive as CT. One of the reasons it is expensive is that it is slower than CT. The machine itself is also acoustically noisy, which can be uncomfortable for the patient and can interfere with your measurements if for example you are trying to image the part of the brain that is stimulated by sound.

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 require 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

Fig 3: fMRI scan of a patient's brain. It shows the regional cerebral blood volume (rCBV)

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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