gigablochs.bloch.relax

Contents

gigablochs.bloch.relax#

gigablochs.bloch.relax(magnetization, T1, T2, dt, *, M0=(0, 0, 1), extend_shapes=True, match_shapes=True, axis=-1)[source]#

Apply relaxation to the magnetization vector.

Parameters:

  • magnetization (ndarray) – Magnetization vector, can be multidimensional.

  • T1 (ndarray) – Longitudinal relaxation time in seconds.

  • T2 (ndarray) – Transverse relaxation time in seconds.

  • dt (float) – Time step in seconds.

  • M0 (tuple, optional) – Initial magnetization vector. Default is (0, 0, 1), i.e. up along the z-axis.

  • extend_shapes (bool, optional) – If True, the shapes of T1 and T2 are prefixed to the shape of the magnetization array. See match_shapes. Note when False, T1 and T2 should have the same shape as magnetization except along axis, which should be 1. Default is True.

  • match_shapes (bool, optional) – If True, the shapes of T1 and T2 are not prefixed to the shape of the magnetization array when they match the start of its shape. Default is True.

  • axis (int, optional) – Axis along the magnetization array which represents the 2D or 3D spatial magnetization vector. Default is the last axis.

Returns:

updated_magnetization – Updated magnetization vector. The shapes of T1 and T2 are prefixed to the shape of the magnetization array if not already present.

Return type:

ndarray

Raises:

ValueError – If the length of M0 is not equal to the length of the spatial axis: of the magnetization array.

Notes

The relaxation is computed as:

\[\vec{M}_{\text{relaxed}} = \vec{M}_{\text{stressed}} \cdot \left[1 - \frac{\Delta t}{T_2}, 1 - \frac{\Delta t}{T_2}, 1 - \frac{\Delta t}{T_1} \right]^T + \vec{M_0} \left(\frac{\Delta t}{T_1} \right)\]

where \(\vec{M}_{\text{relaxed}}\) is the relaxed magnetization vector, \(\vec{M}_{\text{stressed}}\) is the input magnetization vector, \(\vec{M_0}\) is the initial magnetization vector, \(T_1\) is the longitudinal relaxation time in seconds, \(T_2\) is the transverse relaxation time in seconds, and \(\Delta t\) is the time step in seconds.

Examples

# One line to simulate a relaxation process for 5000 time steps with 3000 parameter combos!
import numpy as np
from gigablochs import bloch

dt = 0.001 # seconds
duration = 5 # seconds
T1 = np.linspace(0.3, 2.3, 20) # seconds
T2 = np.linspace(0.05, 1.1, 30) # seconds
mag = np.random.random((5, 3)) # last axis is the spatial axis
mags = np.array([mag := bloch.relax(mag, T1, T2, dt) for _ in range(round(duration / dt))])
mags.shape # (5000, 20, 30, 5, 3)
Contents