An introduction to the Probabilistic Tractography

Probabilistic fiber tracking is a way of reconstructing white matter connections using diffusion MR imaging. Like deterministic fiber tracking, the probabilistic approach follows the trajectory of a possible pathway step by step starting at a seed, however, unlike deterministic tracking, the tracking direction at each point along the path is chosen at random from a distribution. The distribution at each point is different and depends on the observed diffusion data at that point. The distribution of tracking directions at each point can be represented as a probability mass function (PMF) if the possible tracking directions are restricted to discrete numbers of well distributed points on a sphere.

This example is an extension of the Introduction to Basic Tracking example. We’ll begin by repeating a few steps from that example, loading the data and fitting a Constrained Spherical Deconvolution (CSD) model.

from dipy.core.gradients import gradient_table
from dipy.data import default_sphere, get_fnames
from dipy.direction import peaks_from_model
from dipy.io.gradients import read_bvals_bvecs
from dipy.io.image import load_nifti, load_nifti_data
from dipy.io.stateful_tractogram import Space, StatefulTractogram
from dipy.io.streamline import save_trk
from dipy.reconst.csdeconv import ConstrainedSphericalDeconvModel, auto_response_ssst
from dipy.tracking.stopping_criterion import BinaryStoppingCriterion
from dipy.tracking.streamline import Streamlines
from dipy.tracking.tracker import probabilistic_tracking
from dipy.tracking.utils import seeds_from_mask
from dipy.viz import actor, colormap, has_fury, window

# Enables/disables interactive visualization
interactive = False

hardi_fname, hardi_bval_fname, hardi_bvec_fname = get_fnames(name="stanford_hardi")
label_fname = get_fnames(name="stanford_labels")

data, affine, hardi_img = load_nifti(hardi_fname, return_img=True)
labels = load_nifti_data(label_fname)
bvals, bvecs = read_bvals_bvecs(hardi_bval_fname, hardi_bvec_fname)
gtab = gradient_table(bvals, bvecs=bvecs)

seed_mask = labels == 2
seeds = seeds_from_mask(seed_mask, affine, density=2)

white_matter = (labels == 1) | (labels == 2)
sc = BinaryStoppingCriterion(white_matter)

response, ratio = auto_response_ssst(gtab, data, roi_radii=10, fa_thr=0.7)
csd_model = ConstrainedSphericalDeconvModel(gtab, response, sh_order_max=6)
csd_fit = csd_model.fit(data, mask=white_matter)

The Fiber Orientation Distribution (FOD) of the CSD model estimates the distribution of small fiber bundles within each voxel. We can use this distribution for probabilistic fiber tracking. One way to do this is to represent the FOD using a discrete sphere. This discrete FOD can be used by probabilistic_tracking as a PMF (sf or spherical function) for sampling tracking directions.

fod = csd_fit.odf(default_sphere)
streamline_generator = probabilistic_tracking(
    seeds,
    sc,
    affine,
    sf=fod,
    random_seed=1,
    sphere=default_sphere,
    max_angle=20,
    step_size=0.2,
)

streamlines = Streamlines(streamline_generator)
sft = StatefulTractogram(streamlines, hardi_img, Space.RASMM)
save_trk(sft, "tractogram_probabilistic_sf.trk")

if has_fury:
    scene = window.Scene()
    scene.add(actor.line(streamlines, colors=colormap.line_colors(streamlines)))
    window.record(
        scene=scene, out_path="tractogram_probabilistic_sf.png", size=(800, 800)
    )
    if interactive:
        window.show(scene)

Corpus Callosum using probabilistic tractography from PMF

One disadvantage of using a discrete PMF to represent possible tracking directions is that it tends to take up a lot of memory (RAM). The size of the PMF, the FOD in this case, must be equal to the number of possible tracking directions on the hemisphere, and every voxel has a unique PMF. In this case the data is (81, 106, 76) and small_sphere has 181 directions so the FOD is (81, 106, 76, 181). One way to avoid sampling the PMF and holding it in memory is to use directly from the spherical harmonic (SH) representation of the FOD. By using this approach, we can also use a larger sphere, like default_sphere which has 362 directions on the hemisphere, without having to worry about memory limitations.

streamline_generator = probabilistic_tracking(
    seeds,
    sc,
    affine,
    sh=csd_fit.shm_coeff,
    random_seed=1,
    sphere=default_sphere,
    max_angle=20,
    step_size=0.2,
)

streamlines = Streamlines(streamline_generator)
sft = StatefulTractogram(streamlines, hardi_img, Space.RASMM)
save_trk(sft, "tractogram_probabilistic_sh.trk")

if has_fury:
    scene = window.Scene()
    scene.add(actor.line(streamlines, colors=colormap.line_colors(streamlines)))
    window.record(
        scene=scene, out_path="tractogram_probabilistic_sh.png", size=(800, 800)
    )
    if interactive:
        window.show(scene)

Corpus Callosum using probabilistic tractography from SH

Not all model fits have the shm_coeff attribute because not all models use this basis to represent the data internally. However we can fit the ODF of any model to the spherical harmonic basis using the peaks_from_model function.

peaks = peaks_from_model(
    csd_model,
    data,
    default_sphere,
    0.5,
    25,
    mask=white_matter,
    return_sh=True,
    parallel=True,
    num_processes=1,
)
fod_coeff = peaks.shm_coeff

streamline_generator = probabilistic_tracking(
    seeds,
    sc,
    affine,
    sh=fod_coeff,
    random_seed=1,
    sphere=default_sphere,
    max_angle=20,
    step_size=0.2,
)

streamlines = Streamlines(streamline_generator)
sft = StatefulTractogram(streamlines, hardi_img, Space.RASMM)
save_trk(sft, "tractogram_probabilistic_sh_pfm.trk")

if has_fury:
    scene = window.Scene()
    scene.add(actor.line(streamlines, colors=colormap.line_colors(streamlines)))
    window.record(
        scene=scene, out_path="tractogram_probabilistic_sh_pfm.png", size=(800, 800)
    )
    if interactive:
        window.show(scene)

Corpus Callosum using probabilistic tractography from SH (peaks_from_model)