# Reconstruction of the diffusion signal with the kurtosis tensor model

The diffusion kurtosis model is an expansion of the diffusion tensor model (see Reconstruction of the diffusion signal with the Tensor model). In addition to the diffusion tensor (DT), the diffusion kurtosis model quantifies the degree to which water diffusion in biological tissues is non-Gaussian using the kurtosis tensor (KT) [Jensen2005].

Measurements of non-Gaussian diffusion from the diffusion kurtosis model are of interest because they can be used to charaterize tissue microstructural heterogeneity [Jensen2010]. Moreover, DKI can be used to: 1) derive concrete biophysical parameters, such as the density of axonal fibers and diffusion tortuosity [Fierem2011] (see Reconstruction of the diffusion signal with the WMTI model); and 2) resolve crossing fibers in tractography and to obtain invariant rotational measures not limited to well-aligned fiber populations [NetoHe2015].

The diffusion kurtosis model expresses the diffusion-weighted signal as:

$S(n,b)=S_{0}e^{-bD(n)+\frac{1}{6}b^{2}D(n)^{2}K(n)}$

where $$\mathbf{b}$$ is the applied diffusion weighting (which is dependent on the measurement parameters), $$S_0$$ is the signal in the absence of diffusion gradient sensitization, $$\mathbf{D(n)}$$ is the value of diffusion along direction $$\mathbf{n}$$, and $$\mathbf{K(n)}$$ is the value of kurtosis along direction $$\mathbf{n}$$. The directional diffusion $$\mathbf{D(n)}$$ and kurtosis $$\mathbf{K(n)}$$ can be related to the diffusion tensor (DT) and kurtosis tensor (KT) using the following equations:

$D(n)=\sum_{i=1}^{3}\sum_{j=1}^{3}n_{i}n_{j}D_{ij}$

and

$K(n)=\frac{MD^{2}}{D(n)^{2}}\sum_{i=1}^{3}\sum_{j=1}^{3}\sum_{k=1}^{3} \sum_{l=1}^{3}n_{i}n_{j}n_{k}n_{l}W_{ijkl}$

where $$D_{ij}$$ are the elements of the second-order DT, and $$W_{ijkl}$$ the elements of the fourth-order KT and $$MD$$ is the mean diffusivity. As the DT, KT has antipodal symmetry and thus only 15 Wijkl elemments are needed to fully characterize the KT:

$\begin{split}\begin{matrix} ( & W_{xxxx} & W_{yyyy} & W_{zzzz} & W_{xxxy} & W_{xxxz} & ... \\ & W_{xyyy} & W_{yyyz} & W_{xzzz} & W_{yzzz} & W_{xxyy} & ... \\ & W_{xxzz} & W_{yyzz} & W_{xxyz} & W_{xyyz} & W_{xyzz} & & )\end{matrix}\end{split}$

In the following example we show how to fit the diffusion kurtosis model on diffusion-weighted multi-shell datasets and how to estimate diffusion kurtosis based statistics.

First, we import all relevant modules:

import numpy as np
import matplotlib.pyplot as plt
import dipy.reconst.dki as dki
import dipy.reconst.dti as dti
from dipy.data import get_fnames
from scipy.ndimage.filters import gaussian_filter


DKI requires multi-shell data, i.e. data acquired from more than one non-zero b-value. Here, 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')



Function get_fnames downloads and outputs the paths of the data, load_nifti returns the data as a nibabel Nifti1Image object, and read_bvals_bvecs loads the arrays containing the information about the b-values and b-vectors. These later arrays are converted to the GradientTable object required for Dipy’s data reconstruction.

Before fitting the data, we preform some data pre-processing. We first compute a brain mask to avoid unnecessary calculations on the background of the image.

maskdata, mask = median_otsu(data, vol_idx=[0, 1], median_radius=4, numpass=2,
autocrop=False, dilate=1)


Since the diffusion kurtosis models involves the estimation of a large number of parameters [TaxCMW2015] and since the non-Gaussian components of the diffusion signal are more sensitive to artefacts [NetoHe2012], it might be favorable to suppress the effects of noise and artefacts before diffusion kurtosis fitting. In this example the effects of noise and artefacts are suppress by using 3D Gaussian smoothing (with a Gaussian kernel with fwhm=1.25) as suggested by pioneer DKI studies (e.g. [Jensen2005], [NetoHe2012]). Although here the Gaussian smoothing is used so that results are comparable to these studies, it is important to note that more advanced noise and artifact suppression algorithms are available in DIPY, e.g. the Marcenko-Pastur PCA denoising algorithm (example-denoise-mppca) and the Gibbs artefact suppression algorithm (example-denoise-gibbs).

fwhm = 1.25
gauss_std = fwhm / np.sqrt(8 * np.log(2))  # converting fwhm to Gaussian std
data_smooth = np.zeros(data.shape)
for v in range(data.shape[-1]):
data_smooth[..., v] = gaussian_filter(data[..., v], sigma=gauss_std)


Now that we have loaded and pre-processed the data we can go forward with DKI fitting. For this, the DKI model is first defined for the data’s GradientTable object by instantiating the DiffusionKurtosisModel object in the following way:

dkimodel = dki.DiffusionKurtosisModel(gtab)


To fit the data using the defined model object, we call the fit function of this object:

dkifit = dkimodel.fit(data_smooth, mask=mask)


The fit method creates a DiffusionKurtosisFit object, which contains all the diffusion and kurtosis fitting parameters and other DKI attributes. For instance, since the diffusion kurtosis model estimates the diffusion tensor, all diffusion standard tensor statistics can be computed from the DiffusionKurtosisFit instance. For example, we show below how to extract the fractional anisotropy (FA), the mean diffusivity (MD), the axial diffusivity (AD) and the radial diffusivity (RD) from the DiffusionKurtosisiFit instance.

FA = dkifit.fa
MD = dkifit.md
RD = dkifit.rd


Note that these four standard measures could also be computed from DIPY’s DTI module. Computing these measures from both models should be analogous; however, theoretically, the diffusion statistics from the kurtosis model are expected to have better accuracy, since DKI’s diffusion tensor are decoupled from higher order terms effects [Veraar2011], [NetoHe2012]. For comparison purposes, we calculate below the FA, MD, AD, and RD using DIPY’s TensorModel.

tenmodel = dti.TensorModel(gtab)

dti_FA = tenfit.fa
dti_MD = tenfit.md
dti_RD = tenfit.rd


The DT based measures can be easily visualized using matplotlib. For example, the FA, MD, AD, and RD obtained from the diffusion kurtosis model (upper panels) and the tensor model (lower panels) are plotted for a selected axial slice. DTI’s diffusion estimates present lower values than DKI’s estimates, showing that DTI’s diffusion measurements are underestimated by higher order effects.

axial_slice = 9

fig1, ax = plt.subplots(2, 4, figsize=(12, 6),
subplot_kw={'xticks': [], 'yticks': []})

ax.flat[0].imshow(FA[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=0.7, origin='lower')
ax.flat[0].set_title('FA (DKI)')
ax.flat[1].imshow(MD[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=2.0e-3, origin='lower')
ax.flat[1].set_title('MD (DKI)')
vmin=0, vmax=2.0e-3, origin='lower')
ax.flat[3].imshow(RD[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=2.0e-3, origin='lower')
ax.flat[3].set_title('RD (DKI)')

ax.flat[4].imshow(dti_FA[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=0.7, origin='lower')
ax.flat[4].set_title('FA (DTI)')
ax.flat[5].imshow(dti_MD[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=2.0e-3, origin='lower')
ax.flat[5].set_title('MD (DTI)')
vmin=0, vmax=2.0e-3, origin='lower')
ax.flat[7].imshow(dti_RD[:, :, axial_slice].T, cmap='gray',
vmin=0, vmax=2.0e-3, origin='lower')
ax.flat[7].set_title('RD (DTI)')

plt.show()
fig1.savefig('Diffusion_tensor_measures_from_DTI_and_DKI.png')


Diffusion tensor measures obtained from the diffusion tensor estimated from DKI (upper panels) and DTI (lower panels).

In addition to the standard diffusion statistics, the DiffusionKurtosisFit instance can be used to estimate the non-Gaussian measures of mean kurtosis (MK), the axial kurtosis (AK) and the radial kurtosis (RK).

Kurtosis measures are susceptible to high amplitude outliers. The impact of high amplitude kurtosis outliers can be removed by introducing as an optional input the extremes of the typical values of kurtosis. Here these are assumed to be on the range between 0 and 3):

MK = dkifit.mk(0, 3)
AK = dkifit.ak(0, 3)
RK = dkifit.rk(0, 3)


Now we are ready to plot the kurtosis standard measures using matplotlib:

fig2, ax = plt.subplots(1, 3, figsize=(12, 6),
subplot_kw={'xticks': [], 'yticks': []})

ax.flat[0].imshow(MK[:, :, axial_slice].T, cmap='gray', vmin=0, vmax=1.5,
origin='lower')
ax.flat[0].set_title('MK')
ax.flat[0].annotate('', fontsize=12, xy=(57, 30),
color='red',
xycoords='data', xytext=(30, 0),
textcoords='offset points',
arrowprops=dict(arrowstyle="->",
color='red'))
ax.flat[1].imshow(AK[:, :, axial_slice].T, cmap='gray', vmin=0, vmax=1.5,
origin='lower')
ax.flat[1].set_title('AK')
ax.flat[2].imshow(RK[:, :, axial_slice].T, cmap='gray', vmin=0, vmax=1.5,
origin='lower')
ax.flat[2].set_title('RK')
ax.flat[2].annotate('', fontsize=12, xy=(57, 30),
color='red',
xycoords='data', xytext=(30, 0),
textcoords='offset points',
arrowprops=dict(arrowstyle="->",
color='red'))

plt.show()
fig2.savefig('Kurtosis_tensor_standard_measures.png')


DKI standard kurtosis measures.

The non-Gaussian behaviour of the diffusion signal is expected to be higher when tissue water is confined by multiple compartments. MK is, therefore, higher in white matter since it is highly compartmentalized by myelin sheaths. These water diffusion compartmentalization is expected to be more pronounced prependicularly to white matter fibers and thus the RK map presents higher amplitudes than the AK map.

It is important to note that kurtosis estimates might presented negative estimates in deep white matter regions (e.g. red arrow added in the figure above). This negative kurtosis values are artefactual and might be induced by: 1) low radial diffusivities of aligned white matter - since it is very hard to capture non-Gaussian information in radial direction due to it’s low diffusion decays, radial kurtosis estimates (and consequently the mean kurtosis estimates) might have low robustness and tendency to exhibit negative values [NetoHe2012]; 2) Gibbs artefacts - MRI images might be corrupted by signal oscilation artefact between tissue’s edges if an inadequate number of high frequencies of the k-space is sampled. These oscilations might have different signs on images acquired with different diffusion-weighted and inducing negative biases in kurtosis parametric maps [Perron2015], [NetoHe2018].

One can try to suppress this issue by using the more advance noise and artefact suppression algorithms, e.g., as mentioned above, the MP-PCA denoising (example-denoise-mppca) and Gibbs Unringing (example-denoise-gibbs) algorithms. Alternatively, one can overcome this artefact by computing the kurtosis values from powder-averaged diffusion-weighted signals. The details on how to compute the kurtosis from powder-average signals in dipy are described in follow the tutorial (example-reconst-msdki).

## Mean kurtosis tensor and kurtosis fractional anisotropy

As pointed by previous studies [Hansen2013], axial, radial and mean kurtosis depends on the information of both diffusion and kurtosis tensor. DKI measures that only depend on the kurtosis tensor include the mean of the kurtosis tensor [Hansen2013], and the kurtosis fractional anisotropy [GlennR2015]. This measures are computed and illustrated bellow:

MKT = dkifit.mkt(0, 3)
KFA = dkifit.kfa

fig3, ax = plt.subplots(1, 2, figsize=(10, 6),
subplot_kw={'xticks': [], 'yticks': []})

ax.flat[0].imshow(MKT[:, :, axial_slice].T, cmap='gray', vmin=0, vmax=1.5,
origin='lower')
ax.flat[0].set_title('MKT')
ax.flat[1].imshow(KFA[:, :, axial_slice].T, cmap='gray', vmin=0, vmax=1,
origin='lower')
ax.flat[1].set_title('KFA')

plt.show()
fig3.savefig('Measures_from_kurtosis_tensor_only.png')


DKI measures obtained from the kurtosis tensor only.

As reported by [Hansen2013], the mean of the kurtosis tensor (MKT) produces similar maps than the standard mean kurtosis (MK). On the other hand, the kurtosis fractional anisotropy (KFA) maps shows that the kurtosis tensor have different degrees of anisotropy than the FA measures from the diffusion tensor.

## References

Jensen2005(1,2)

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

Jensen2010

Jensen JH, Helpern JA (2010). MRI quantification of non-Gaussian water diffusion by kurtosis analysis. NMR in Biomedicine 23(7): 698-710

Fierem2011

Fieremans E, Jensen JH, Helpern JA (2011). White matter characterization with diffusion kurtosis imaging. NeuroImage 58: 177-188

Veraar2011

Veraart J, Poot DH, Van Hecke W, Blockx I, Van der Linden A, Verhoye M, Sijbers J (2011). More Accurate Estimation of Diffusion Tensor Parameters Using Diffusion Kurtosis Imaging. Magnetic Resonance in Medicine 65(1): 138-145

NetoHe2012(1,2,3,4)

Neto Henriques R, Ferreira H, Correia M, (2012). Diffusion kurtosis imaging of the healthy human brain. Master Dissertation Bachelor and Master Programin Biomedical Engineering and Biophysics, Faculty of Sciences. http://repositorio.ul.pt/bitstream/10451/8511/1/ulfc104137_tm_Rafael_Henriques.pdf

Hansen2013(1,2,3)

Hansen B, Lund TE, Sangill R, and Jespersen SN (2013). Experimentally and computationally393fast method for estimation of a mean kurtosis. Magnetic Resonance in Medicine 69, 1754–1760.394doi:10.1002/mrm.24743

GlennR2015

Glenn GR, Helpern JA, Tabesh A, Jensen JH (2015). Quantitative assessment of diffusional387kurtosis anisotropy. NMR in Biomedicine28, 448–459. doi:10.1002/nbm.3271

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

Perron2015

Perrone D, Aelterman J, Pižurica A, Jeurissen B, Philips W, Leemans A, (2015). The effect of Gibbs ringing artifacts on measures derived from diffusion MRI. Neuroimage 120, 441-455. https://doi.org/10.1016/j.neuroimage.2015.06.068.

TaxCMW2015

Tax CMW, Otte WM, Viergever MA, Dijkhuizen RM, Leemans A (2014). REKINDLE: Robust extraction of kurtosis INDices with linear estimation. Magnetic Resonance in Medicine 73(2): 794-808.

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

NetoHe2018

Neto Henriques R (2018). Advanced Methods for Diffusion MRI Data Analysis and their Application to the Healthy Ageing Brain (Doctoral thesis). https://doi.org/10.17863/CAM.29356

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.