Gradients and Spheres

This example shows how you can create gradient tables and sphere objects using DIPY.

Usually, as we saw in Getting started with DIPY, you load your b-values and b-vectors from disk and then you can create your own gradient table. But this time let’s say that you are an MR physicist and you want to design a new gradient scheme or you are a scientist who wants to simulate many different gradient schemes.

Now let’s assume that you are interested in creating a multi-shell acquisition with 2-shells, one at b=1000 \(s/mm^2\) and one at b=2500 \(s/mm^2\). For both shells let’s say that we want a specific number of gradients (64) and we want to have the points on the sphere evenly distributed.

This is possible using the disperse_charges which is an implementation of electrostatic repulsion [Jones1999].

import numpy as np
from dipy.core.sphere import disperse_charges, Sphere, HemiSphere

We can first create some random points on a HemiSphere using spherical polar coordinates.

n_pts = 64
theta = np.pi * np.random.rand(n_pts)
phi = 2 * np.pi * np.random.rand(n_pts)
hsph_initial = HemiSphere(theta=theta, phi=phi)

Next, we call disperse_charges which will iteratively move the points so that the electrostatic potential energy is minimized.

hsph_updated, potential = disperse_charges(hsph_initial, 5000)

In hsph_updated we have the updated HemiSphere with the points nicely distributed on the hemisphere. Let’s visualize them.

from dipy.viz import window, actor

# Enables/disables interactive visualization
interactive = False

ren = window.Renderer()
ren.SetBackground(1, 1, 1)

ren.add(actor.point(hsph_initial.vertices, window.colors.red, point_radius=0.05))
ren.add(actor.point(hsph_updated.vertices, window.colors.green, point_radius=0.05))

print('Saving illustration as initial_vs_updated.png')
window.record(ren, out_path='initial_vs_updated.png', size=(300, 300))
if interactive:
    window.show(ren)
../_images/initial_vs_updated.png

Example of electrostatic repulsion of red points which become green points.

We can also create a sphere from the hemisphere and show it in the following way.

sph = Sphere(xyz=np.vstack((hsph_updated.vertices, -hsph_updated.vertices)))

window.rm_all(ren)
ren.add(actor.point(sph.vertices, actor.colors.green, point_radius=0.05))

print('Saving illustration as full_sphere.png')
window.record(ren, out_path='full_sphere.png', size=(300, 300))
if interactive:
    window.show(ren)
../_images/full_sphere.png

Full sphere.

It is time to create the Gradients. For this purpose we will use the function gradient_table and fill it with the hsph_updated vectors that we created above.

from dipy.core.gradients import gradient_table

vertices = hsph_updated.vertices
values = np.ones(vertices.shape[0])

We need two stacks of vertices, one for every shell, and we need two sets of b-values, one at 1000 \(s/mm^2\), and one at 2500 \(s/mm^2\), as we discussed previously.

bvecs = np.vstack((vertices, vertices))
bvals = np.hstack((1000 * values, 2500 * values))

We can also add some b0s. Let’s add one at the beginning and one at the end.

bvecs = np.insert(bvecs, (0, bvecs.shape[0]), np.array([0, 0, 0]), axis=0)
bvals = np.insert(bvals, (0, bvals.shape[0]), 0)

print(bvals)
[    0.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.
  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.
  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.
  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.
  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.
  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.  1000.
  1000.  1000.  1000.  1000.  1000.  2500.  2500.  2500.  2500.  2500.
  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.
  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.
  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.
  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.
  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.
  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.  2500.     0.]
print(bvecs)
[[ 0.          0.          0.        ]
 [-0.80451777 -0.16877559  0.56944355]
 [ 0.32822557 -0.94355999  0.04430036]
 [-0.23584135 -0.96241331  0.13468285]
 [-0.39207424 -0.73505312  0.55314981]
 [-0.32539386 -0.16751384  0.93062235]
 [-0.82043195 -0.39411534  0.41420347]
 [ 0.65741493  0.74947875  0.07802061]
 [ 0.88853765  0.45303621  0.07251925]
 [ 0.39638642 -0.15185138  0.90543855]
                 ...
 [ 0.10175269  0.08197111  0.99142681]
 [ 0.50577702 -0.37862345  0.77513476]
 [ 0.42845026  0.40155296  0.80943535]
 [ 0.26939707  0.81103868  0.51927014]
 [-0.48938584 -0.43780086  0.75420946]
 [ 0.          0.          0.        ]]

Both b-values and b-vectors look correct. Let’s now create the GradientTable.

gtab = gradient_table(bvals, bvecs)

window.rm_all(ren)

We can also visualize the gradients. Let’s color the first shell blue and the second shell cyan.

colors_b1000 = window.colors.blue * np.ones(vertices.shape)
colors_b2500 = window.colors.cyan * np.ones(vertices.shape)
colors = np.vstack((colors_b1000, colors_b2500))
colors = np.insert(colors, (0, colors.shape[0]), np.array([0, 0, 0]), axis=0)
colors = np.ascontiguousarray(colors)

ren.add(actor.point(gtab.gradients, colors, point_radius=100))

print('Saving illustration as gradients.png')
window.record(ren, out_path='gradients.png', size=(300, 300))
if interactive:
    window.show(ren)
../_images/gradients.png

Diffusion gradients.

References

[Jones1999]Jones, DK. et al. Optimal strategies for measuring diffusion in anisotropic systems by magnetic resonance imaging, Magnetic Resonance in Medicine, vol 42, no 3, 515-525, 1999.

Example source code

You can download the full source code of this example. This same script is also included in the dipy source distribution under the doc/examples/ directory.