def diffusion_components(dki_params,
sphere='repulsion100',
awf=None,
mask=None):
""" Extracts the restricted and hindered diffusion tensors of well aligned
fibers from diffusion kurtosis imaging parameters [1]_.
Parameters
----------
dki_params : ndarray (x, y, z, 27) or (n, 27)
All parameters estimated from the diffusion kurtosis model.
Parameters are ordered as follows:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the first,
second and third coordinates of the eigenvector
3) Fifteen elements of the kurtosis tensor
sphere : Sphere class instance, optional
The sphere providing sample directions to sample the restricted and
hindered cellular diffusion tensors. For more details see Fieremans
et al., 2011.
awf : ndarray (optional)
Array containing values of the axonal water fraction that has the shape
dki_params.shape[:-1]. If not given this will be automatically computed
using :func:`axonal_water_fraction`" with function's default precision.
mask : ndarray (optional)
A boolean array used to mark the coordinates in the data that should be
analyzed that has the shape dki_params.shape[:-1]
Returns
-------
edt : ndarray (x, y, z, 6) or (n, 6)
Parameters of the hindered diffusion tensor.
idt : ndarray (x, y, z, 6) or (n, 6)
Parameters of the restricted diffusion tensor.
Notes
-----
In the original article of DKI microstructural model [1]_, the hindered and
restricted tensors were definde as the intra-cellular and extra-cellular
diffusion compartments respectively.
References
----------
.. [1] Fieremans E, Jensen JH, Helpern JA, 2011. White matter
characterization with diffusional kurtosis imaging.
Neuroimage 58(1):177-88. doi: 10.1016/j.neuroimage.2011.06.006
"""
shape = dki_params.shape[:-1]
# load gradient directions
if not isinstance(sphere, dps.Sphere):
sphere = get_sphere(sphere)
# select voxels where to apply the single fiber model
if mask is None:
mask = np.ones(shape, dtype='bool')
else:
if mask.shape != shape:
raise ValueError("Mask is not the same shape as dki_params.")
else:
mask = np.array(mask, dtype=bool, copy=False)
# check or compute awf values
if awf is None:
awf = axonal_water_fraction(dki_params, sphere=sphere, mask=mask)
else:
if awf.shape != shape:
raise ValueError("awf array is not the same shape as dki_params.")
# Initialize hindered and restricted diffusion tensors
edt_all = np.zeros(shape + (6, ))
idt_all = np.zeros(shape + (6, ))
# Generate matrix that converts apparent diffusion coefficients to tensors
B = np.zeros((sphere.x.size, 6))
B[:, 0] = sphere.x * sphere.x # Bxx
B[:, 1] = sphere.x * sphere.y * 2. # Bxy
B[:, 2] = sphere.y * sphere.y # Byy
B[:, 3] = sphere.x * sphere.z * 2. # Bxz
B[:, 4] = sphere.y * sphere.z * 2. # Byz
B[:, 5] = sphere.z * sphere.z # Bzz
pinvB = np.linalg.pinv(B)
# Compute hindered and restricted diffusion tensors for all voxels
evals, evecs, kt = split_dki_param(dki_params)
dt = lower_triangular(vec_val_vect(evecs, evals))
md = mean_diffusivity(evals)
index = ndindex(mask.shape)
for idx in index:
if not mask[idx]:
continue
# sample apparent diffusion and kurtosis values
di = directional_diffusion(dt[idx], sphere.vertices)
ki = directional_kurtosis(dt[idx],
md[idx],
kt[idx],
sphere.vertices,
adc=di,
min_kurtosis=0)
edi = di * (1 + np.sqrt(ki * awf[idx] / (3.0 - 3.0 * awf[idx])))
edt = np.dot(pinvB, edi)
edt_all[idx] = edt
# We only move on if there is an axonal water fraction.
# Otherwise, remaining params are already zero, so move on
if awf[idx] == 0:
continue
# Convert apparent diffusion and kurtosis values to apparent diffusion
# values of the hindered and restricted diffusion
idi = di * (1 - np.sqrt(ki * (1.0 - awf[idx]) / (3.0 * awf[idx])))
# generate hindered and restricted diffusion tensors
idt = np.dot(pinvB, idi)
idt_all[idx] = idt
return edt_all, idt_all
def diffusion_components(dki_params, sphere='repulsion100', awf=None,
mask=None):
""" Extracts the restricted and hindered diffusion tensors of well aligned
fibers from diffusion kurtosis imaging parameters [1]_.
Parameters
----------
dki_params : ndarray (x, y, z, 27) or (n, 27)
All parameters estimated from the diffusion kurtosis model.
Parameters are ordered as follows:
1) Three diffusion tensor's eigenvalues
2) Three lines of the eigenvector matrix each containing the first,
second and third coordinates of the eigenvector
3) Fifteen elements of the kurtosis tensor
sphere : Sphere class instance, optional
The sphere providing sample directions to sample the restricted and
hindered cellular diffusion tensors. For more details see Fieremans
et al., 2011.
awf : ndarray (optional)
Array containing values of the axonal water fraction that has the shape
dki_params.shape[:-1]. If not given this will be automatically computed
using :func:`axonal_water_fraction`" with function's default precision.
mask : ndarray (optional)
A boolean array used to mark the coordinates in the data that should be
analyzed that has the shape dki_params.shape[:-1]
Returns
--------
edt : ndarray (x, y, z, 6) or (n, 6)
Parameters of the hindered diffusion tensor.
idt : ndarray (x, y, z, 6) or (n, 6)
Parameters of the restricted diffusion tensor.
Note
----
In the original article of DKI microstructural model [1]_, the hindered and
restricted tensors were definde as the intra-cellular and extra-cellular
diffusion compartments respectively.
References
----------
.. [1] Fieremans E, Jensen JH, Helpern JA, 2011. White matter
characterization with diffusional kurtosis imaging.
Neuroimage 58(1):177-88. doi: 10.1016/j.neuroimage.2011.06.006
"""
shape = dki_params.shape[:-1]
# load gradient directions
if not isinstance(sphere, dps.Sphere):
sphere = get_sphere(sphere)
# select voxels where to apply the single fiber model
if mask is None:
mask = np.ones(shape, dtype='bool')
else:
if mask.shape != shape:
raise ValueError("Mask is not the same shape as dki_params.")
else:
mask = np.array(mask, dtype=bool, copy=False)
# check or compute awf values
if awf is None:
awf = axonal_water_fraction(dki_params, sphere=sphere, mask=mask)
else:
if awf.shape != shape:
raise ValueError("awf array is not the same shape as dki_params.")
# Initialize hindered and restricted diffusion tensors
edt_all = np.zeros(shape + (6,))
idt_all = np.zeros(shape + (6,))
# Generate matrix that converts apparant diffusion coefficients to tensors
B = np.zeros((sphere.x.size, 6))
B[:, 0] = sphere.x * sphere.x # Bxx
B[:, 1] = sphere.x * sphere.y * 2. # Bxy
B[:, 2] = sphere.y * sphere.y # Byy
B[:, 3] = sphere.x * sphere.z * 2. # Bxz
B[:, 4] = sphere.y * sphere.z * 2. # Byz
B[:, 5] = sphere.z * sphere.z # Bzz
pinvB = np.linalg.pinv(B)
# Compute hindered and restricted diffusion tensors for all voxels
evals, evecs, kt = split_dki_param(dki_params)
dt = lower_triangular(vec_val_vect(evecs, evals))
md = mean_diffusivity(evals)
index = ndindex(mask.shape)
for idx in index:
if not mask[idx]:
continue
# sample apparent diffusion and kurtosis values
di = directional_diffusion(dt[idx], sphere.vertices)
ki = directional_kurtosis(dt[idx], md[idx], kt[idx], sphere.vertices,
adc=di, min_kurtosis=0)
edi = di * (1 + np.sqrt(ki * awf[idx] / (3.0 - 3.0 * awf[idx])))
edt = np.dot(pinvB, edi)
edt_all[idx] = edt
# We only move on if there is an axonal water fraction.
# Otherwise, remaining params are already zero, so move on
if awf[idx] == 0:
continue
# Convert apparent diffusion and kurtosis values to apparent diffusion
# values of the hindered and restricted diffusion
idi = di * (1 - np.sqrt(ki * (1.0 - awf[idx]) / (3.0 * awf[idx])))
# generate hindered and restricted diffusion tensors
idt = np.dot(pinvB, idi)
idt_all[idx] = idt
return edt_all, idt_all