# -*- coding: utf-8 -*-
"""
==============================================
Mean signal diffusion kurtosis imaging (MSDKI)
==============================================
Diffusion Kurtosis Imaging (DKI) is one of the conventional ways to estimate
the degree of non-Gaussian diffusion (see :ref:`example_reconst_dki`)
[Jensen2005]_. However, a limitation of DKI is that its measures are highly
sensitive to noise and image artefacts. For instance, due to the low radial
diffusivities, standard kurtosis estimates in regions of well-aligned voxel may
be corrupted by implausible low or even negative values.
A way to overcome this issue is to characterize kurtosis from average signals
across all directions acquired for each data b-value (also known as
powder-averaged signals). Moreover, as previously pointed [NetoHe2015]_,
standard kurtosis measures (e.g. radial, axial and standard mean kurtosis)
do not only depend on microstructural properties but also on mesoscopic
properties such as fiber dispersion or the intersection angle of crossing
fibers. In contrary, the kurtosis from powder-average signals has the advantage
of not depending on the fiber distribution functions [NetoHe2018]_,
[NetoHe2019]_.
In short, in this tutoral we show how to characterize non-Gaussian diffusion
in a more precise way and decoupled from confounding effects of tissue
dispersion and crossing.
In the first part of this example, we illustrate the properties of the measures
obtained from the mean signal diffusion kurtosis imaging (MSDKI)[NetoHe2018]_
using synthetic data. Secondly, the mean signal diffusion kurtosis imaging will
be applied to in-vivo MRI data. Finally, we show how MSDKI provides the same
information than common microstructural models such as the spherical mean
technique [NetoHe2019]_, [Kaden2016b]_.
Let's import all relevant modules:
"""
import numpy as np
import matplotlib.pyplot as plt
# Reconstruction modules
import dipy.reconst.dki as dki
import dipy.reconst.msdki as msdki
# For simulations
from dipy.sims.voxel import multi_tensor
from dipy.core.gradients import gradient_table
from dipy.core.sphere import disperse_charges, HemiSphere
# For in-vivo data
from dipy.data import get_fnames
from dipy.io.gradients import read_bvals_bvecs
from dipy.io.image import load_nifti
from dipy.segment.mask import median_otsu
"""
Testing MSDKI in synthetic data
===============================
We simulate representative diffusion-weighted signals using MultiTensor
simulations (for more information on this type of simulations see
:ref:`example_simulate_multi_tensor`). For this example, simulations are
produced based on the sum of four diffusion tensors to represent the intra-
and extra-cellular spaces of two fiber populations. The parameters of these
tensors are adjusted according to [NetoHe2015]_ (see also
:ref:`example_simulate_dki`).
"""
mevals = np.array([[0.00099, 0, 0],
[0.00226, 0.00087, 0.00087],
[0.00099, 0, 0],
[0.00226, 0.00087, 0.00087]])
"""
For the acquisition parameters of the synthetic data, we use 60 gradient
directions for two non-zero b-values (1000 and 2000 $s/mm^{2}$) and two
zero bvalues (note that, such as the standard DKI, MSDKI requires at least
three different b-values).
"""
# Sample the spherical coordinates of 60 random diffusion-weighted directions.
n_pts = 60
theta = np.pi * np.random.rand(n_pts)
phi = 2 * np.pi * np.random.rand(n_pts)
# Convert direction to cartesian coordinates.
hsph_initial = HemiSphere(theta=theta, phi=phi)
# Evenly distribute the 60 directions
hsph_updated, potential = disperse_charges(hsph_initial, 5000)
directions = hsph_updated.vertices
# Reconstruct acquisition parameters for 2 non-zero=b-values and 2 b0s
bvals = np.hstack((np.zeros(2), 1000 * np.ones(n_pts), 2000 * np.ones(n_pts)))
bvecs = np.vstack((np.zeros((2, 3)), directions, directions))
gtab = gradient_table(bvals, bvecs)
"""
Simulations are looped for different intra- and extra-cellular water
volume fractions and different intersection angles of the two-fiber
populations.
"""
# Array containing the intra-cellular volume fractions tested
f = np.linspace(20, 80.0, num=7)
# Array containing the intersection angle
ang = np.linspace(0, 90.0, num=91)
# Matrix where synthetic signals will be stored
dwi = np.empty((f.size, ang.size, bvals.size))
for f_i in range(f.size):
# estimating volume fractions for individual tensors
fractions = np.array([100 - f[f_i], f[f_i], 100 - f[f_i], f[f_i]]) * 0.5
for a_i in range(ang.size):
# defining the directions for individual tensors
angles = [(ang[a_i], 0.0), (ang[a_i], 0.0), (0.0, 0.0), (0.0, 0.0)]
# producing signals using Dipy's function multi_tensor
signal, sticks = multi_tensor(gtab, mevals, S0=100, angles=angles,
fractions=fractions, snr=None)
dwi[f_i, a_i, :] = signal
"""
Now that all synthetic signals were produced, we can go forward with
MSDKI fitting. As other Dipy's reconstruction techniques, the MSDKI model has
to be first defined for the specific GradientTable object of the synthetic
data. For MSDKI, this is done by instantiating the MeanDiffusionKurtosisModel
object in the following way:
"""
msdki_model = msdki.MeanDiffusionKurtosisModel(gtab)
"""
MSDKI can then be fitted to the synthetic data by calling the ``fit`` function
of this object:
"""
msdki_fit = msdki_model.fit(dwi)
"""
From the above fit object we can extract the two main parameters of the MSDKI,
i.e.: 1) the mean signal diffusion (MSD); and 2) the mean signal kurtosis (MSK)
"""
MSD = msdki_fit.msd
MSK = msdki_fit.msk
""" For a reference, we also calculate the mean diffusivity (MD) and mean
kurtosis (MK) from the standard DKI.
"""
dki_model = dki.DiffusionKurtosisModel(gtab)
dki_fit = dki_model.fit(dwi)
MD = dki_fit.md
MK = dki_fit.mk(0, 3)
"""
Now we plot the results as a function of the ground truth intersection
angle and for different volume fractions of intra-cellular water.
"""
fig1, axs = plt.subplots(nrows=2, ncols=2, figsize=(10, 10))
for f_i in range(f.size):
axs[0, 0].plot(ang, MSD[f_i], linewidth=1.0,
label='$F: %.2f$' % f[f_i])
axs[0, 1].plot(ang, MSK[f_i], linewidth=1.0,
label='$F: %.2f$' % f[f_i])
axs[1, 0].plot(ang, MD[f_i], linewidth=1.0,
label='$F: %.2f$' % f[f_i])
axs[1, 1].plot(ang, MK[f_i], linewidth=1.0,
label='$F: %.2f$' % f[f_i])
# Adjust properties of the first panel of the figure
axs[0, 0].set_xlabel('Intersection angle')
axs[0, 0].set_ylabel('MSD')
axs[0, 1].set_xlabel('Intersection angle')
axs[0, 1].set_ylabel('MSK')
axs[0, 1].legend(loc='center left', bbox_to_anchor=(1, 0.5))
axs[1, 0].set_xlabel('Intersection angle')
axs[1, 0].set_ylabel('MD')
axs[1, 1].set_xlabel('Intersection angle')
axs[1, 1].set_ylabel('MK')
axs[1, 1].legend(loc='center left', bbox_to_anchor=(1, 0.5))
plt.show()
fig1.savefig('MSDKI_simulations.png')
"""
.. figure:: MSDKI_simulations.png
:align: center
MSDKI and DKI measures for data of two crossing synthetic fibers.
Upper panels show the MSDKI measures: 1) mean signal diffusivity (left
panel); and 2) mean signal kurtosis (right panel).
For reference, lower panels show the measures obtained by standard DKI:
1) mean diffusivity (left panel); and 2) mean kurtosis (right panel).
All estimates are plotted as a function of the intersecting angle of the
two crossing fibers. Different curves correspond to different ground truth
axonal volume fraction of intra-cellular space.
The results of the above figure, demonstrate that both MSD and MSK are
sensitive to axonal volume fraction (i.e. a microstructure property) but are
independent to the intersectiong angle of the two crossing fibers (i.e.
independent to properties regarding fiber orientation). In contrast, DKI
measures seem to be independent to both axonal volume fraction and
intersection angle.
"""
"""
Reconstructing diffusion data using MSDKI
=========================================
Now that the properties of MSDKI were illustrated, let's apply MSDKI to in-vivo
diffusion-weighted data. As the example for the standard DKI
(see :ref:`example_reconst_dki`), we use fetch to download a multi-shell
dataset which was kindly provided by Hansen and Jespersen (more details about
the data are provided in their paper [Hansen2016]_). The total size of the
downloaded data is 192 MBytes, however you only need to fetch it once.
"""
fraw, fbval, fbvec, t1_fname = get_fnames('cfin_multib')
data, affine = load_nifti(fraw)
bvals, bvecs = read_bvals_bvecs(fbval, fbvec)
gtab = gradient_table(bvals, bvecs)
"""
Before fitting the data, we preform some data pre-processing. For illustration,
we only mask the data to avoid unnecessary calculations on the background of
the image; however, you could also apply other pre-processing techniques.
For example, some state of the art denoising algorithms are available in DIPY_
(e.g. the non-local means filter :ref:`example-denoise-nlmeans` or the
local pca :ref:`example-denoise-localpca`).
"""
maskdata, mask = median_otsu(data, vol_idx=[0, 1], median_radius=4, numpass=2,
autocrop=False, dilate=1)
"""
Now that we have loaded and pre-processed the data we can go forward
with MSDKI fitting. As for the synthetic data, the MSDKI model has to be first
defined for the data's GradientTable object:
"""
msdki_model = msdki.MeanDiffusionKurtosisModel(gtab)
"""
The data can then be fitted by calling the ``fit`` function of this object:
"""
msdki_fit = msdki_model.fit(data, mask=mask)
"""
Let's then extract the two main MSDKI's parameters: 1) mean signal diffusion
(MSD); and 2) mean signal kurtosis (MSK).
"""
MSD = msdki_fit.msd
MSK = msdki_fit.msk
"""
For comparison, we calculate also the mean diffusivity (MD) and mean kurtosis
(MK) from the standard DKI.
"""
dki_model = dki.DiffusionKurtosisModel(gtab)
dki_fit = dki_model.fit(data, mask=mask)
MD = dki_fit.md
MK = dki_fit.mk(0, 3)
"""
Let's now visualize the data using matplotlib for a selected axial slice.
"""
axial_slice = 9
fig2, ax = plt.subplots(2, 2, figsize=(6, 6),
subplot_kw={'xticks': [], 'yticks': []})
fig2.subplots_adjust(hspace=0.3, wspace=0.05)
im0 = ax.flat[0].imshow(MSD[:, :, axial_slice].T * 1000, cmap='gray',
vmin=0, vmax=2, origin='lower')
ax.flat[0].set_title('MSD (MSDKI)')
im1 = ax.flat[1].imshow(MSK[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=2, origin='lower')
ax.flat[1].set_title('MSK (MSDKI)')
im2 = ax.flat[2].imshow(MD[:, :, axial_slice].T * 1000, cmap='gray',
vmin=0, vmax=2, origin='lower')
ax.flat[2].set_title('MD (DKI)')
im3 = ax.flat[3].imshow(MK[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=2, origin='lower')
ax.flat[3].set_title('MK (DKI)')
fig2.colorbar(im0, ax=ax.flat[0])
fig2.colorbar(im1, ax=ax.flat[1])
fig2.colorbar(im2, ax=ax.flat[2])
fig2.colorbar(im3, ax=ax.flat[3])
plt.show()
fig2.savefig('MSDKI_invivo.png')
"""
.. figure::MSDKI_invivo.png
:align: center
MSDKI measures (upper panels) and DKI standard measures (lower panels).
This figure shows that the contrast of in-vivo MSD and MSK maps (upper panels)
are similar to the contrast of MD and MSK maps (lower panels); however, in the
upper part we insure that direct contributions of fiber dispersion were
removed. The upper panels also reveal that MSDKI measures are let sensitive
to noise artefacts than standard DKI measures (as pointed by [NetoHe2018]_),
particularly one can appriciate that MSK maps always present positive values
in brain white matter regions, while implausible negative kurtosis values are
present in the MK maps in the same regions.
Relationship between MSDKI and SMT2
===================================
As showed in [NetoHe2019]_, MSDKI captures the same information than the
spherical mean technique (SMT) microstructural models [Kaden2016b]_. In this
way, the SMT model parameters can be directly computed from MSDKI.
For instance, the axonal volume fraction (f), the intrisic diffusivity (di),
and the microscopic anisotropy of the SMT 2-compartmental model [NetoHe2019]_
can be extracted using the following lines of code:
"""
F = msdki_fit.smt2f
DI = msdki_fit.smt2di
uFA2 = msdki_fit.smt2uFA
"""
The SMT2 model parameters extracted from MSDKI are displayed bellow: """
fig3, ax = plt.subplots(1, 3, figsize=(9, 2.5),
subplot_kw={'xticks': [], 'yticks': []})
fig3.subplots_adjust(hspace=0.4, wspace=0.1)
im0 = ax.flat[0].imshow(F[:, :, axial_slice].T,
cmap='gray', vmin=0, vmax=1, origin='lower')
ax.flat[0].set_title('SMT2 f (MSDKI)')
im1 = ax.flat[1].imshow(DI[:, :, axial_slice].T * 1000, cmap='gray',
vmin=0, vmax=2, origin='lower')
ax.flat[1].set_title('SMT2 di (MSDKI)')
im2 = ax.flat[2].imshow(uFA2[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=1, origin='lower')
ax.flat[2].set_title('SMT2 uFA (MSDKI)')
fig3.colorbar(im0, ax=ax.flat[0])
fig3.colorbar(im1, ax=ax.flat[1])
fig3.colorbar(im2, ax=ax.flat[2])
plt.show()
fig3.savefig('MSDKI_SMT2_invivo.png')
"""
.. figure::MSDKI_SMT2_invivo.png
:align: center
SMT2 model quantities extracted from MSDKI. From left to right, the figure
shows the axonal volume fraction (f), the intrisic diffusivity (di), and
the microscopic anisotropy of the SMT 2-compartmental model [NetoHe2019]_.
The similar contrast of SMT2 f-parameter maps in comparison to MSK (first panel
of Figure 3 vs second panel of Figure 2) confirms than MSK and F captures the
same tissue information but on different scales (but rescaled to values between
0 and 1). It is important to note that SMT model parameters estimates should
be used with care, because the SMT model was shown to be invalid NetoHe2019]_.
For instance, although SMT2 parameter f and uFA may be a useful normalization
of the degree of non-Gaussian diffusion (note than both metrics have a range
between 0 and 1), these cannot be interpreted as a real biophysical estimates
of axonal water fraction and tissue microscopic anisotropy.
References
----------
.. [Jensen2005] Jensen JH, Helpern JA, Ramani A, Lu H, Kaczynski K (2005).
Diffusional Kurtosis Imaging: The Quantification of
Non_Gaussian Water Diffusion by Means of Magnetic Resonance
Imaging. Magnetic Resonance in Medicine 53: 1432-1440
.. [NetoHe2015] Neto Henriques R, Correia MM, Nunes RG, Ferreira HA (2015).
Exploring the 3D geometry of the diffusion kurtosis tensor -
Impact on the development of robust tractography procedures and
novel biomarkers, NeuroImage 111: 85-99
.. [NetoHe2018] Henriques RN, 2018. Advanced Methods for Diffusion MRI Data
Analysis and their Application to the Healthy Ageing Brain
(Doctoral thesis). Downing College, University of Cambridge.
https://doi.org/10.17863/CAM.29356
.. [NetoHe2019] Neto Henriques R, Jespersen SN, Shemesh N (2019). Microscopic
anisotropy misestimation in spherical‐mean single diffusion
encoding MRI. Magnetic Resonance in Medicine (In press).
doi: 10.1002/mrm.27606
.. [Kaden2016b] Kaden E, Kelm ND, Carson RP, Does MD, Alexander DC (2016)
Multi-compartment microscopic diffusion imaging. NeuroImage
139: 346-359.
.. [Hansen2016] Hansen, B, Jespersen, SN (2016). Data for evaluation of fast
kurtosis strategies, b-value optimization and exploration of
diffusion MRI contrast. Scientific Data 3: 160072
doi:10.1038/sdata.2016.72
.. include:: ../links_names.inc
"""