Fitting DTI to a synthetic tensor

Diffusion tensor imaging (DTI) describes the diffusion of water in each voxel with a 3x3 tensor. Its eigenvalues give maps such as fractional anisotropy (FA), which is 0 for diffusion that is the same in every direction and approaches 1 when diffusion follows a single direction, as along a fibre bundle.

This example builds a small DWI image whose signal comes from a known tensor, fits DTI with Larmor, and checks that the fitted FA matches the FA of that tensor.

import numpy as np

from larmor.dwi import DTIModel
from larmor.types import GradientTable, Volume

A tensor that looks like white matter

Fast diffusion along x (1.7e-3 mm^2/s), slow across it (0.3e-3 mm^2/s). Its FA follows from the eigenvalues.

eigenvalues = np.array([1.7e-3, 0.3e-3, 0.3e-3])
tensor = np.diag(eigenvalues)
mean = eigenvalues.mean()
expected_fa = np.sqrt(1.5 * np.sum((eigenvalues - mean) ** 2) / np.sum(eigenvalues**2))

The acquisition: one b=0 image and 30 directions at b=1000 s/mm^2

rng = np.random.default_rng(0)
directions = rng.normal(size=(30, 3))
directions /= np.linalg.norm(directions, axis=1, keepdims=True)
bvals = np.concatenate([[0.0], np.full(30, 1000.0)])
bvecs = np.vstack([[0.0, 0.0, 0.0], directions])

The signal of each measurement is S0 exp(-b g^T D g), the same in every voxel of a small 3x3x3 image.

s0 = 1000.0
signal = s0 * np.exp(-bvals * np.einsum("ni,ij,nj->n", bvecs, tensor, bvecs))
data = np.broadcast_to(signal, (3, 3, 3, len(bvals))).astype(np.float32)

Fitting DTI

The image is a Volume (its data and a voxel-to-world affine, here 1 mm voxels) and the acquisition a GradientTable. Real data comes from files instead: load_volume("dwi.nii.gz") and read_bvals_bvecs("bvals", "bvecs", "dwi.nii.gz").

volume = Volume(data, np.eye(4))
gtab = GradientTable(bvals, bvecs)
fit = DTIModel().fit(volume, gtab, mask=np.ones((3, 3, 3), dtype=bool))

print(f"expected FA: {expected_fa:.4f}")
print(f"fitted FA:   {fit.fa[1, 1, 1]:.4f}")
assert np.allclose(fit.fa, expected_fa, atol=1e-3)

expected FA: 0.7990
fitted FA:   0.7990