Introduction to Magnetic Resonance Imaging

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Larmor frequencies of different nuclei at this field strength:

Nucleus	γ (MHz/T)	Larmor freq at 3.0T
1H	42.576	127.73 MHz
13C	10.708	32.12 MHz
23Na	11.262	33.79 MHz
31P	17.235	51.70 MHz

Transverse magnetization |Mxy| = 1.00 (this is our detectable signal strength) Longitudinal magnetization Mz = 0.00

T₁ relaxation describes the return of longitudinal magnetization to equilibrium. The recovery follows the Bloch equation for ||(M_z||): ||[\frac{dM_z}{dt} = \frac{M_0 - M_z}{T_1}||]with the solution (after a 90° excitation): ||[M_z(t) = M_0 \left(1 - e^{-t/T_1}\right)||]The physical mechanism involves energy exchange between the excited spin system and the surrounding molecular lattice (hence "spin-lattice" relaxation). The efficiency depends on molecular tumbling rates:

• Optimal T₁ relaxation occurs when the molecular tumbling frequency matches the Larmor frequency
• Small, fast-tumbling molecules (like free water) are inefficient → long T₁
• Large, slow-tumbling molecules (like fat) are more efficient → short T₁
• T₁ increases with field strength because the Larmor frequency moves further from typical tumbling frequencies

T₂ = 44 ms* (always shorter than T₂ = 80 ms due to field inhomogeneities)

The Bloch equations describe the time evolution of the magnetization vector ||(\vec{M} = (M_x, M_y, M_z)||) in a magnetic field: ||[\frac{dM_x}{dt} = \gamma (\vec{M} \times \vec{B})_x - \frac{M_x}{T_2}||]||[\frac{dM_y}{dt} = \gamma (\vec{M} \times \vec{B})_y - \frac{M_y}{T_2}||]||[\frac{dM_z}{dt} = \gamma (\vec{M} \times \vec{B})_z - \frac{M_z - M_0}{T_1}||]The cross-product term describes precession, while the decay terms describe relaxation. In matrix form, for a single time step ||(\Delta t||) with only ||(B_0||) along z:

1. Precession: Rotate ||(M||) about z by angle ||(\Delta\phi = \gamma B_0 \Delta t||)
2. T2 decay: Multiply ||(M_x||) and ||(M_y||) by ||(e^{-\Delta t / T_2}||)
3. T1 recovery: Update ||(M_z \rightarrow M_z \cdot e^{-\Delta t / T_1} + M_0 (1 - e^{-\Delta t / T_1})||)

This is exactly how our simulation module implements the Bloch equations -- using rotation matrices for precession and exponential factors for relaxation.

Spin Echo: 90° → wait TE/2 → 180° (refocus) → wait TE/2 → Echo The 180° pulse reverses all static dephasing. The echo signal reflects true T₂ decay only. Used for anatomical imaging (T₁w, T₂w).

Gradient: 0 mT/m → Frequency spread across 30cm FOV: 0.0 kHz (uniform field, no spatial encoding!)

With Gz = 20 mT/m and RF bandwidth = 2000 Hz, the selected slice is 2.3 mm thick. The red segment shows the positions where protons are within the RF bandwidth and will be excited. All other protons are off-resonance and unaffected.

Watch how the spin arrows in each voxel respond to gradients:

• No gradients: all spins precess at the same rate
• Frequency encoding (Gx): columns spin at different rates
• Phase encoding (Gy): rows accumulate different phase offsets
• Both: each voxel gets a unique (frequency, phase) signature

The 2D Fourier transform decodes these signatures back into an image.

The MRI signal at time ||(t||) during readout, for a given phase-encode step with gradient area ||(A_y||), is: ||[S(t) = \int\int \rho(x, y) \cdot e^{-i 2\pi (\gamma G_x x t + \gamma A_y y)} \, dx \, dy||]where ||(\rho(x, y)||) is the spin density (our image). If we define spatial frequencies:

• ||(k_x = \gamma G_x t||) (varies during readout)
• ||(k_y = \gamma A_y||) (set by phase-encode gradient)

then the signal equation becomes: ||[S(k_x, k_y) = \int\int \rho(x, y) \cdot e^{-i 2\pi (k_x x + k_y y)} \, dx \, dy||]This is exactly the 2D Fourier transform of the image! Therefore, the image is simply the inverse Fourier transform of the acquired data: ||[\rho(x, y) = \mathcal{F}^{-1}\{S(k_x, k_y)\}||]The space of ||((k_x, k_y)||) is called k-space.

Progressive fill shows how an MRI scanner acquires k-space line by line, with the image emerging gradually. Try other modes:

• Center only: Blurry but recognizable -- contrast lives in the center
• Periphery only: Only edges visible -- detail lives at the edges
• Undersampled: Aliasing artifacts from skipping lines

The canonical HRF has several notable features:

• Initial dip (small, often not detectable at typical fMRI resolution)
• Peak at ~6s after stimulus
• Post-stimulus undershoot at ~16s
• Returns to baseline after ~25-30s

This sluggish hemodynamic response means fMRI has poor temporal resolution (~seconds) compared to EEG (milliseconds), even though the neural events happen on the millisecond timescale.

Parameter	Value
Spatial Resolution	3.4 mm
Temporal Resolution	1.0 s
Slices per TR	~27
BOLD sensitivity	Good (TE=30ms)
Readout duration	~32 ms