Table of Contents
- Utilize nuclear magnetic resonance
1. Components
- Magnets
- Create a homogeneous magnetic field \(\vec{B}_0\) as strong as 0.5 T.
- Gradient Coils
- Create gradient in the magnetic field, in the \(\vec{B}_0\) direction with a desired amount.
- RF Coils
- Magnetic field of the RF signal is called \(B_1\).
- Send and receive RF signals in the traverse direction.
- Not spacially selective.
2. Procedure
- Utilize the Hydrogen nuclei.
- Excite the nuclei with a radio wave of Larmor frequency
- Measure the signal after echo time (TE) when the difference between tissues is at its maximum.
- Re-exite the nuclei after repetition time (TR).
\[ S(\mathrm{TR}, \mathrm{TE})= (1-e^{-\mathrm{TR}/T_1})M_0e^{-\mathrm{TE}/T_2} \]
3. T₂ Weighted Image
\(\mathrm{TR}\gg T_1\) and \(\mathrm{TE} \sim T_2\)
4. T₁ Weighted Image
\(\mathrm{TR}\sim T_1\) and \(\mathrm{TE} \ll T_2\)
5. Spin Density Image
- \(\mathrm{TR}\gg T_1\) and \(\mathrm{TE} \ll T_2\)
- MRI measured with the echo time of zero which only reflects the proton density.
6. Hyperpolarization
- Happen with ^{3}He, ^{13}C, ^{129}Ze, …
- It can be used to increase the signal strength significantly.
7. Gradient Recalled Echo
- How MRI Works - Part 4 - The Gradient Recalled Echo (GRE) - YouTube
- GRE. Sometimes, just Gradient Echo.
7.1. Magnetic Field Gradient
\[ \frac{\partial B_z}{\partial x_i} = G_{x_i} \] Now \(\omega(\mathbf{x}) = \gamma (B_0+\mathbf{x}\cdot \nabla B_z)\)
Spins whose Larmor frequency matches the RF signals will experience excitation.
7.2. Slice Selection
Send a \(\rm sinc\) pulse that has transmit bandwidth \(\Delta \omega\), then the slice thickness is \[ \Delta z = \frac{\Delta \omega}{\gamma G_z} \]
A pulse looks like \[ B_1(t)=e^{i\omega_0 t}\mathrm{sinc}(\Delta \omega t) \] where \(\omega_0\) is the Larmor frequency at the center of the slice.
This is the ((6539064f-e2f8-45fb-b576-4a1a894f9a87)).
- The signal from the 2D slice \(R\) is \[ S(t)=\iint_R M(x,y)e^{i\phi(x,y,t)}\,dxdy \] where \(M(x,y)\) is the magnetization at TE, and \[ \phi(x,y,t)=-\gamma\left[\int_0^t G_x(u)x\,du + \int_0^{t} G_y(u) y\, du\right] \]
7.3. Phase Encoding
Impose gradient \(G_y\) for time \(\tau\), and have \[ \phi(x,y,t)=-\gamma\left[\int_0^t G_x(u)x\,du + \tau G_y y\right] \]
7.4. Frequency Encoding
Impose gradient \(G_x\) while doing the read out, and get \[ S(t)=\iint_R M(x,y)e^{-i\gamma (tG_x x+\tau G_yy)}\, dxdy \] where \(t\) starts at the start of \(G_x\).
This is a line of known phasor in \(k\)-space.