# Continuous and analytical diffusion signal modelling with MAPMRI¶

We show how to model the diffusion signal as a linear combination of continuous functions from the MAPMRI basis [Ozarslan2013]. This continuous representation allows for the computation of many properties of both the signal and diffusion propagator.

We show how to estimate the analytical Orientation Distribution Function (ODF) and a variety of scalar indices. These include rotationally invariant quantities such as the Mean Squared Displacement (MSD), Q-space Inverse Variance (QIV), Return-To-Origin Probability (RTOP) and Non-Gaussianity (NG). Interestingly, the MAP-MRI basis also allows for the computation of directional indices, such as the Return To the Axis Probability (RTAP), the Return To the Plane Probability (RTPP), and the parallel and perpendicular Non-Gaussianity.

The estimation of these properties from noisy DWIs requires that the fitting of the MAPMRI basis is regularized. We will show that this can be done using both constraining the diffusion propagator to positive values [Ozarslan2013] and analytic Laplacian Regularization (MAPL) [Fick2016a].

First import the necessary modules:

```
from dipy.reconst import mapmri
from dipy.viz import window, actor
from dipy.data import fetch_cfin_multib, read_cfin_dwi, get_sphere
from dipy.core.gradients import gradient_table
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable
```

Download and read the data for this tutorial.

MAPMRI requires multi-shell data, to properly fit the radial part of the basis.
to `False`

to only download eddy-current/motion corrected data.
The total size of the downloaded data is 187.66 MBytes, however you only need
to fetch it once.

```
fetch_cfin_multib()
```

`data`

contains the voxel data and `gtab`

contains a `GradientTable`

object (gradient information e.g. b-values). For example, to show the b-values
it is possible to write:

```
print(gtab.bvals)
```

For the values of the q-space indices to make sense it is necessary to
explicitly state the `big_delta`

and `small_delta`

parameters in the
gradient table.

```
img, gtab = read_cfin_dwi()
big_delta = 0.0365 # seconds
small_delta = 0.0157 # seconds
gtab = gradient_table(bvals=gtab.bvals, bvecs=gtab.bvecs,
big_delta=big_delta,
small_delta=small_delta)
data = img.get_data()
data_small = data[40:65, 50:51]
print('data.shape (%d, %d, %d, %d)' % data.shape)
```

The MAPMRI Model can now be instantiated. The `radial_order`

determines the
expansion order of the basis, i.e., how many basis functions are used to
approximate the signal.

First, we must decide to use the anisotropic or isotropic MAPMRI basis. As was
shown in [Fick2016a], the isotropic basis is best used for tractography
purposes, as the anisotropic basis has a bias towards smaller crossing angles
in the ODF. For signal fitting and estimation of scalar quantities the
anisotropic basis is suggested. The choice can be made by setting
`anisotropic_scaling=True`

or `anisotropic_scaling=False`

.

First, we must select the method of regularization and/or constraining the basis fitting.

`laplacian_regularization=True`

makes it use Laplacian regularization [Fick2016a]. This method essentially reduces spurious oscillations in the reconstruction by minimizing the Laplacian of the fitted signal. Several options can be given to select the regularization weight:`regularization_weighting=GCV`

uses generalized cross-validation [Craven1978] to find an optimal weight.`regularization_weighting=0.2`

for example would omit the GCV and just set it to 0.2 (found to be reasonable in HCP data [Fick2016a]).`regularization_weighting=np.array(weights)`

would make the GCV use a custom range to find an optimal weight.

`positivity_constraint=True`

makes it use the positivity constraint on the diffusion propagator [Ozarslan2013]. This method constrains the final solution of the diffusion propagator to be positive at a set of discrete points, since negative values should not exist.- The
`pos_grid`

and`pos_radius`

affect the location and number of constraint points in the diffusion propagator.

- The

Both methods do a good job of producing viable solutions to the signal fitting in practice. The difference is that the Laplacian regularization imposes smoothness over the entire signal, including the extrapolation beyond the measured signal. In practice this results in, but does not guarantee positive solutions of the diffusion propagator. The positivity constraint guarantees a positive solution in a set of discrete points, which in general results in smooth solutions, but does not guarantee it.

A suggested strategy is to use a low Laplacian weight together with the positivity constraint. In this way both desired properties are guaranteed in the final solution.

We use package CVXPY (http://www.cvxpy.org/) to solve convex optimization problems when “positivity_constraint=True”, so we need to first install CVXPY.

For now we will generate the anisotropic models for all combinations.

```
radial_order = 6
map_model_laplacian_aniso = mapmri.MapmriModel(gtab, radial_order=radial_order,
laplacian_regularization=True,
laplacian_weighting=.2)
map_model_positivity_aniso = mapmri.MapmriModel(gtab,
radial_order=radial_order,
laplacian_regularization=False,
positivity_constraint=True)
map_model_both_aniso = mapmri.MapmriModel(gtab, radial_order=radial_order,
laplacian_regularization=True,
laplacian_weighting=.05,
positivity_constraint=True)
```

Note that when we use only Laplacian regularization, the `GCV`

option may
select very low regularization weights in very anisotropic white matter such
as the corpus callosum, resulting in corrupted scalar indices. In this example
we therefore choose a preset value of 0.2, which has shown to be quite robust
and also faster in practice [Fick2016a].

We can then fit the MAPMRI model to the data:

```
mapfit_laplacian_aniso = map_model_laplacian_aniso.fit(data_small)
mapfit_positivity_aniso = map_model_positivity_aniso.fit(data_small)
mapfit_both_aniso = map_model_both_aniso.fit(data_small)
```

From the fitted models we will first illustrate the estimation of q-space indices. For completeness, we will compare the estimation using only Laplacian regularization, positivity constraint or both. We first show the RTOP [Ozarslan2013].

```
# generating RTOP plots
fig = plt.figure(figsize=(10, 5))
ax1 = fig.add_subplot(1, 3, 1, title=r'RTOP - Laplacian')
ax1.set_axis_off()
rtop_laplacian = np.array(mapfit_laplacian_aniso.rtop()[:, 0, :].T,
dtype=float)
ind = ax1.imshow(rtop_laplacian, interpolation='nearest',
origin='lower', cmap=plt.cm.gray)
ax2 = fig.add_subplot(1, 3, 2, title=r'RTOP - Positivity')
ax2.set_axis_off()
rtop_positivity = np.array(mapfit_positivity_aniso.rtop()[:, 0, :].T,
dtype=float)
ind = ax2.imshow(rtop_positivity, interpolation='nearest',
origin='lower', cmap=plt.cm.gray)
ax3 = fig.add_subplot(1, 3, 3, title=r'RTOP - Both')
ax3.set_axis_off()
rtop_both = np.array(mapfit_both_aniso.rtop()[:, 0, :].T, dtype=float)
ind = ax3.imshow(rtop_both, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax3)
cax = divider.append_axes("right", size="5%", pad=0.05)
plt.colorbar(ind, cax=cax)
plt.savefig('MAPMRI_maps_regularization.png')
```

It can be seen that all maps appear quite smooth and similar. Though, it is possible to see some subtle differences near the corpus callosum. The similarity and differences in reconstruction can be further illustrated by visualizing the analytic norm of the Laplacian of the fitted signal.

```
fig = plt.figure(figsize=(10, 5))
ax1 = fig.add_subplot(1, 3, 1, title=r'Laplacian norm - Laplacian')
ax1.set_axis_off()
laplacian_norm_laplacian = np.array(mapfit_laplacian_aniso.norm_of_laplacian_signal()[:, 0, :].T,
dtype=float)
ind = ax1.imshow(laplacian_norm_laplacian, interpolation='nearest',
origin='lower', cmap=plt.cm.gray, vmin=0, vmax=3)
ax2 = fig.add_subplot(1, 3, 2, title=r'Laplacian norm - Positivity')
ax2.set_axis_off()
laplacian_norm_positivity = np.array(mapfit_positivity_aniso.norm_of_laplacian_signal()[:, 0, :].T,
dtype=float)
ind = ax2.imshow(laplacian_norm_positivity, interpolation='nearest',
origin='lower', cmap=plt.cm.gray, vmin=0, vmax=3)
ax3 = fig.add_subplot(1, 3, 3, title=r'Laplacian norm - Both')
ax3.set_axis_off()
laplacian_norm_both = np.array(mapfit_both_aniso.norm_of_laplacian_signal()[:, 0, :].T,
dtype=float)
ind = ax3.imshow(laplacian_norm_both, interpolation='nearest', origin='lower',
cmap=plt.cm.gray, vmin=0, vmax=3)
divider = make_axes_locatable(ax3)
cax = divider.append_axes("right", size="5%", pad=0.05)
plt.colorbar(ind, cax=cax)
plt.savefig('MAPMRI_norm_laplacian.png')
```

A high Laplacian norm indicates that the gradient in the three-dimensional signal reconstruction changes a lot - something that may indicate spurious oscillations. In the Laplacian reconstruction (left) we see that there are some isolated voxels that have a higher norm than the rest. In the positivity constraint reconstruction the norm is already smoother. When both methods are used together the overall norm gets smoother still, since both smoothness of the signal and positivity of the propagator are imposed.

From now on we just use the combined approach, show all maps we can generate and explain their significance.

```
fig = plt.figure(figsize=(15, 6))
ax1 = fig.add_subplot(1, 5, 1, title=r'MSD')
ax1.set_axis_off()
msd = np.array(mapfit_both_aniso.msd()[:, 0, :].T, dtype=float)
ind = ax1.imshow(msd, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
ax2 = fig.add_subplot(1, 5, 2, title=r'QIV')
ax2.set_axis_off()
qiv = np.array(mapfit_both_aniso.qiv()[:, 0, :].T, dtype=float)
ind = ax2.imshow(qiv, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
ax3 = fig.add_subplot(1, 5, 3, title=r'RTOP')
ax3.set_axis_off()
rtop = np.array((mapfit_both_aniso.rtop()[:, 0, :]).T, dtype=float)
ind = ax3.imshow(rtop, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
ax4 = fig.add_subplot(1, 5, 4, title=r'RTAP')
ax4.set_axis_off()
rtap = np.array((mapfit_both_aniso.rtap()[:, 0, :]).T, dtype=float)
ind = ax4.imshow(rtap, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
ax5 = fig.add_subplot(1, 5, 5, title=r'RTPP')
ax5.set_axis_off()
rtpp = np.array(mapfit_both_aniso.rtpp()[:, 0, :].T, dtype=float)
ind = ax5.imshow(rtpp, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
plt.savefig('MAPMRI_maps.png')
```

From left to right:

- Mean Squared Displacement (MSD) is a measure for how far protons are able to diffuse. a decrease in MSD indicates protons are hindered/restricted more, as can be seen by the high MSD in the CSF, but low in the white matter.
- Q-space Inverse Variance (QIV) is a measure of variance in the signal, which is said to have higher contrast to white matter than the MSD [Hosseinbor2013]. We also showed that QIV has high sensitivity to tissue composition change in a simulation study [Fick2016b].
- Return to origin probability (RTOP) quantifies the probability that a proton will be at the same position at the first and second diffusion gradient pulse. A higher RTOP indicates that the volume a spin is inside of is smaller, meaning more overall restriction. This is illustrated by the low values in CSF and high values in white matter.
- Return to axis probability (RTAP) is a directional index that quantifies the probability that a proton will be along the axis of the main eigenvector of a diffusion tensor during both diffusion gradient pulses. RTAP has been related to the apparent axon diameter [Ozarslan2013] [Fick2016a] under several strong assumptions on the tissue composition and acquisition protocol.
- Return to plane probability (RTPP) is a directional index that quantifies the probability that a proton will be on the plane perpendicular to the main eigenvector of a diffusion tensor during both gradient pulses. RTPP is related to the length of a pore [Ozarslan2013] but in practice should be similar to that of Gaussian diffusion.

It is also possible to estimate the amount of non-Gaussian diffusion in every voxel [Ozarslan2013]. This quantity is estimated through the ratio between Gaussian volume (MAPMRI’s first basis function) and the non-Gaussian volume (all other basis functions) of the fitted signal. For this value to be physically meaningful we must use a b-value threshold in the MAPMRI model. This threshold makes the scale estimation in MAPMRI only use samples that realistically describe Gaussian diffusion, i.e., at low b-values.

```
map_model_both_ng = mapmri.MapmriModel(gtab, radial_order=radial_order,
laplacian_regularization=True,
laplacian_weighting=.05,
positivity_constraint=True,
bval_threshold=2000)
mapfit_both_ng = map_model_both_ng.fit(data_small)
fig = plt.figure(figsize=(10, 6))
ax1 = fig.add_subplot(1, 3, 1, title=r'NG')
ax1.set_axis_off()
ng = np.array(mapfit_both_ng.ng()[:, 0, :].T, dtype=float)
ind = ax1.imshow(ng, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax1)
cax = divider.append_axes("right", size="5%", pad=0.05)
plt.colorbar(ind, cax=cax)
ax2 = fig.add_subplot(1, 3, 2, title=r'NG perpendicular')
ax2.set_axis_off()
ng_perpendicular = np.array(mapfit_both_ng.ng_perpendicular()[:, 0, :].T,
dtype=float)
ind = ax2.imshow(ng_perpendicular, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax2)
cax = divider.append_axes("right", size="5%", pad=0.05)
plt.colorbar(ind, cax=cax)
ax3 = fig.add_subplot(1, 3, 3, title=r'NG parallel')
ax3.set_axis_off()
ng_parallel = np.array(mapfit_both_ng.ng_parallel()[:, 0, :].T, dtype=float)
ind = ax3.imshow(ng_parallel, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax3)
cax = divider.append_axes("right", size="5%", pad=0.05)
plt.colorbar(ind, cax=cax)
plt.savefig('MAPMRI_ng.png')
```

On the left we see the overall NG and on the right the directional perpendicular NG and parallel NG. The NG ranges from 1 (completely non-Gaussian) to 0 (completely Gaussian). The overall NG of a voxel is always higher or equal than each of its components. It can be seen that NG has low values in the CSF and higher in the white matter.

Increases or decreases in these values do not point to a specific microstructural change, but can indicate clues as to what is happening, similar to Fractional Anisotropy. An initial simulation study that quantifies the added value of q-space indices over DTI indices is given in [Fick2016b].

The MAPMRI framework also allows for the estimation of Orientation Distribution Functions (ODFs). We recommend to use the isotropic implementation for this purpose, as the anisotropic implementation has a bias towards smaller crossing angles.

For the isotropic basis we recommend to use a `radial_order`

of 8, as the
basis needs more generic and needs more basis functions to approximate the
signal.

```
radial_order = 8
map_model_both_iso = mapmri.MapmriModel(gtab, radial_order=radial_order,
laplacian_regularization=True,
laplacian_weighting=.1,
positivity_constraint=True,
anisotropic_scaling=False)
mapfit_both_iso = map_model_both_iso.fit(data_small)
```

Load an ODF reconstruction sphere

```
sphere = get_sphere('symmetric724')
```

Compute the ODFs.

The radial order `s`

can be increased to sharpen the results, but it might
also make the ODFs noisier. Always check the results visually.

```
odf = mapfit_both_iso.odf(sphere, s=2)
print('odf.shape (%d, %d, %d, %d)' % odf.shape)
```

Display the ODFs.

```
# Enables/disables interactive visualization
interactive = False
r = window.Renderer()
sfu = actor.odf_slicer(odf, sphere=sphere, colormap='plasma', scale=0.5)
sfu.display(y=0)
sfu.RotateX(-90)
r.add(sfu)
window.record(r, out_path='odfs.png', size=(600, 600))
if interactive:
window.show(r)
```

## References¶

[Ozarslan2013] | (1, 2, 3, 4, 5, 6, 7) Ozarslan E. et al., “Mean apparent propagator (MAP) MRI: A
novel diffusion imaging method for mapping tissue microstructure”,
NeuroImage, 2013. |

[Fick2016a] | (1, 2, 3, 4, 5, 6) Fick, Rutger HJ, et al. “MAPL: Tissue microstructure estimation
using Laplacian-regularized MAP-MRI and its application to HCP data.”
NeuroImage (2016). |

[Craven1978] | Craven et al. “Smoothing Noisy Data with Spline Functions.” NUMER MATH 31.4 (1978): 377-403. |

[Hosseinbor2013] | Hosseinbor et al. “Bessel fourier orientation reconstruction (bfor): an analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices. NeuroImage 64, 650-670. |

[Fick2016b] | (1, 2) Fick et al. “A sensitivity analysis of Q-space indices with
respect to changes in axonal diameter, dispersion and tissue composition.
ISBI 2016. |

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.