MRI: Bloch equations simulation

Here are simulations of the Bloch equations. Specifically, we use Matlab ODE45 to solve the differential equations describing the dynamics of the magnetization under different excitation (B₁ field) and spatial encoding (gradient field). Simple relaxation mechanisms are incorporated into the calculation.

1. Exciting the magnetization by tipping it toward the transverse plane

This example calculates the magnetization after transmitting an RF pulse of 0.0587 Gauss for 1 ms. The pulse starts at 1 ms after time 0. Here we calculated the magnetization dynamic between 0 and 10 ms in steps of 0.1 ms. With gyromagnetic ratio of proton (γ =42.58 MHz/T), the theoretical flip angle is 1/4 cycle, i.e., 90 degrees.

This plot shows the waveforms of B₁. Gradient coils were not turned on.

This plot shows the tipping down of the magnetization from m_z to m_y. Note that m_x remained at 0 all the time.

2. Spatially selective excitation: slice selection

This example calculates the magnetization after transmitting a "sinc-like" RF pulse with peak strength at 0.02 Gauss and main lobe width of about 1 ms. The z-gradient with the maximal strength of 0.01 T/m was turned on for 4 ms (from 0.5 ms to 4.5 ms).

This plot shows B₁ and gradient waveforms.

This plot shows the spatial distribution of m_x, m_y, and m_z as a function of the location at the z-axis. Around the center of the z-axis, the magnetization has a big component in m_y with sharp transitions of m_x at both boundaries.

3. Stimulus induced rotatory saturation (SIRS)

This example attempts to replicate Figure 5 in the paper to illustrate how the detectable magnetization varies after locking an oscillatory current source at different frequencies. Specifically, it calculates the detectable (transverse) magnetization after a spin-lock RF pulse. Here we simulated an oscillatory source at 40 Hz with the maximal strength of 50 nT and 1 s duration. The spin-locking field was swept between 1 Hz and 80 Hz in steps of 1 Hz. The spin-locking field was 1-s long.

The simulated pulse sequence is depicted here.

The figure below shows that the maximal attenuated magnetization was observed as the Larmor frequency of the spin-locking field matched the frequency of the oscillatory source.