As shown previously (see Reconstruction of the diffusion signal with the Tensor model), the diffusion tensor model is a simple way to characterize diffusion anisotropy. However, in regions near the cerebral ventricle and parenchyma can be underestimated by partial volume effects of the cerebral spinal fluid (CSF). This free water contamination can particularly corrupt Diffusion Tensor Imaging analysis of microstructural changes when different groups of subjects show different brain morphology (e.g. brain ventricle enlargement associated with brain tissue atrophy that occurs in several brain pathologies and ageing).

A way to remove this free water influences is to expand the DTI model to take into account an extra compartment representing the contributions of free water diffusion [Pasternak2009]. The expression of the expanded DTI model is shown below:

\[S(\mathbf{g}, b) = S_0(1-f)e^{-b\mathbf{g}^T \mathbf{D}
\mathbf{g}}+S_0fe^{-b D_{iso}}\]

where \(\mathbf{g}\) and \(b\) are diffusion gradient direction and weighted (more information see Reconstruction of the diffusion signal with the Tensor model), \(S(\mathbf{g}, b)\) is the diffusion-weighted signal measured, \(S_0\) is the signal in a measurement with no diffusion weighting, \(\mathbf{D}\) is the diffusion tensor, \(f\) the volume fraction of the free water component, and \(D_{iso}\) is the isotropic value of the free water diffusion (normally set to \(3.0 \times 10^{-3} mm^{2}s^{-1}\)).

In this example, we show how to process a diffusion weighting dataset using an adapted version of the free water elimination proposed by [Hoy2014].

The full details of Dipy’s free water DTI implementation was published in [Henriques2017]. Please cite this work if you use this algorithm.

Let’s start by importing the relevant modules:

```
import numpy as np
import dipy.reconst.fwdti as fwdti
import dipy.reconst.dti as dti
import matplotlib.pyplot as plt
from dipy.data import fetch_cenir_multib
from dipy.data import read_cenir_multib
from dipy.segment.mask import median_otsu
```

Without spatial constrains the free water elimination model cannot be solved in data acquired from one non-zero b-value [Hoy2014]. Therefore, here we download a dataset that was required from multiple b-values.

```
fetch_cenir_multib(with_raw=False)
```

From the downloaded data, we read only the data acquired with b-values up to 2000 \(s/mm^2\) to decrease the influence of non-Gaussian diffusion effects of the tissue which are not taken into account by the free water elimination model [Hoy2014].

```
bvals = [200, 400, 1000, 2000]
img, gtab = read_cenir_multib(bvals)
data = np.asarray(img.dataobj)
affine = img.affine
```

The free water DTI model can take some minutes to process the full data set. Thus, we remove the background of the image to avoid unnecessary calculations.

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

Moreover, for illustration purposes we process only an axial slice of the data.

```
axial_slice = 40
mask_roi = np.zeros(data.shape[:-1], dtype=bool)
mask_roi[:, :, axial_slice] = mask[:, :, axial_slice]
```

The free water elimination model fit can then be initialized by instantiating a FreeWaterTensorModel class object:

```
fwdtimodel = fwdti.FreeWaterTensorModel(gtab)
```

The data can then be fitted using the `fit`

function of the defined model
object:

```
fwdtifit = fwdtimodel.fit(data, mask=mask_roi)
```

This 2-steps procedure will create a FreeWaterTensorFit object which contains all the diffusion tensor statistics free for free water contaminations. Below we extract the fractional anisotropy (FA) and the mean diffusivity (MD) of the free water diffusion tensor.

```
FA = fwdtifit.fa
MD = fwdtifit.md
```

For comparison we also compute the same standard measures processed by the standard DTI model

```
dtimodel = dti.TensorModel(gtab)
dtifit = dtimodel.fit(data, mask=mask_roi)
dti_FA = dtifit.fa
dti_MD = dtifit.md
```

Below the FA values for both free water elimnantion DTI model and standard DTI model are plotted in panels A and B, while the repective MD values are ploted in panels D and E. For a better visualization of the effect of the free water correction, the differences between these two metrics are shown in panels C and E. In addition to the standard diffusion statistics, the estimated volume fraction of the free water contamination is shown on panel G.

```
fig1, ax = plt.subplots(2, 4, figsize=(12, 6),
subplot_kw={'xticks': [], 'yticks': []})
fig1.subplots_adjust(hspace=0.3, wspace=0.05)
ax.flat[0].imshow(FA[:, :, axial_slice].T, origin='lower',
cmap='gray', vmin=0, vmax=1)
ax.flat[0].set_title('A) fwDTI FA')
ax.flat[1].imshow(dti_FA[:, :, axial_slice].T, origin='lower',
cmap='gray', vmin=0, vmax=1)
ax.flat[1].set_title('B) standard DTI FA')
FAdiff = abs(FA[:, :, axial_slice] - dti_FA[:, :, axial_slice])
ax.flat[2].imshow(FAdiff.T, cmap='gray', origin='lower', vmin=0, vmax=1)
ax.flat[2].set_title('C) FA difference')
ax.flat[3].axis('off')
ax.flat[4].imshow(MD[:, :, axial_slice].T, origin='lower',
cmap='gray', vmin=0, vmax=2.5e-3)
ax.flat[4].set_title('D) fwDTI MD')
ax.flat[5].imshow(dti_MD[:, :, axial_slice].T, origin='lower',
cmap='gray', vmin=0, vmax=2.5e-3)
ax.flat[5].set_title('E) standard DTI MD')
MDdiff = abs(MD[:, :, axial_slice] - dti_MD[:, :, axial_slice])
ax.flat[6].imshow(MDdiff.T, origin='lower', cmap='gray', vmin=0, vmax=2.5e-3)
ax.flat[6].set_title('F) MD difference')
F = fwdtifit.f
ax.flat[7].imshow(F[:, :, axial_slice].T, origin='lower',
cmap='gray', vmin=0, vmax=1)
ax.flat[7].set_title('G) free water volume')
plt.show()
fig1.savefig('In_vivo_free_water_DTI_and_standard_DTI_measures.png')
```

From the figure, one can observe that the free water elimination model produces in general higher values of FA and lower values of MD than the standard DTI model. These differences in FA and MD estimation are expected due to the suppression of the free water isotropic diffusion components. Unexpected high amplitudes of FA are however observed in the periventricular gray matter. This is a known artefact of regions associated to voxels with high water volume fraction (i.e. voxels containing basically CSF). We are able to remove this problematic voxels by excluding all FA values associated with measured volume fractions above a reasonable threshold of 0.7:

```
FA[F > 0.7] = 0
dti_FA[F > 0.7] = 0
```

Above we reproduce the plots of the in vivo FA from the two DTI fits and where we can see that the inflated FA values were practically removed:

```
fig1, ax = plt.subplots(1, 3, figsize=(9, 3),
subplot_kw={'xticks': [], 'yticks': []})
fig1.subplots_adjust(hspace=0.3, wspace=0.05)
ax.flat[0].imshow(FA[:, :, axial_slice].T, origin='lower',
cmap='gray', vmin=0, vmax=1)
ax.flat[0].set_title('A) fwDTI FA')
ax.flat[1].imshow(dti_FA[:, :, axial_slice].T, origin='lower',
cmap='gray', vmin=0, vmax=1)
ax.flat[1].set_title('B) standard DTI FA')
FAdiff = abs(FA[:, :, axial_slice] - dti_FA[:, :, axial_slice])
ax.flat[2].imshow(FAdiff.T, cmap='gray', origin='lower', vmin=0, vmax=1)
ax.flat[2].set_title('C) FA difference')
plt.show()
fig1.savefig('In_vivo_free_water_DTI_and_standard_DTI_corrected.png')
```

- Pasternak2009
Pasternak, O., Sochen, N., Gur, Y., Intrator, N., Assaf, Y., 2009. Free water elimination and mapping from diffusion MRI. Magn. Reson. Med. 62(3): 717-30. doi: 10.1002/mrm.22055.

- Hoy2014(1,2,3)
Hoy, A.R., Koay, C.G., Kecskemeti, S.R., Alexander, A.L., 2014. Optimization of a free water elimination two-compartmental model for diffusion tensor imaging. NeuroImage 103, 323-333. doi: 10.1016/j.neuroimage.2014.09.053

- Henriques2017
Henriques, R.N., Rokem, A., Garyfallidis, E., St-Jean, S., Peterson E.T., Correia, M.M., 2017. [Re] Optimization of a free water elimination two-compartment model for diffusion tensor imaging. ReScience volume 3, issue 1, article number 2

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.