caps.diffusion_estimation.py_tensor_estimation.SecondOrderTensorEstimation¶

SecondOrderTensorEstimation¶

class caps.diffusion_estimation.py_tensor_estimation.SecondOrderTensorEstimation(autoexport_nodes_parameters=True, **kwargs)[source]¶

Second Order Tensor Estimation

[+show/hide tensor modelisation]

The diffusion tensor model

A l order tensor models the dependence between diffusion measures and the diffusivity of water molecules [1]. Specifically, the diffusion tensor is the covarience matrix of the displacement of water molecules. Recall that a l order tensor is an l dimensional array which includes the scalar (l=0), the vector (l=1) and the matrices (l=2).

If we denote $\vec{g}=[g_1,g_2,g_3]$ the measure direction, the Stejskal Tanner can be generalized as follows:

$S(\vec{g}) = S_0 exp \left( -b \sum_{i_1=1}^3 \sum_{i_2=1}^3 . . . \sum_{i_l=1}^3 d_{i_1,i_2,...i_l} g_{i_1} g_{i_2} . . . g_{i_l} \right)$

where $d_{i_1,i_2,...i_l}$ is the l order tensor components. The diffusion profile represented by this tensor can be written as follows:

$d(\vec{g}) = \sum_{i_1=1}^3 \sum_{i_2=1}^3 . . . \sum_{i_l=1}^3 d_{i_1,i_2,...i_l} g_{i_1} g_{i_2} . . . g_{i_l}$

Without imposing any restriction on this profile, it is possible to write the following equation:

$d(-\vec{g}) = \left\{ \begin{array}{ll} d(\vec{g}) & \qquad \mathrm{si} \quad l \ pair \\ -d(\vec{g}) & \qquad \mathrm{si} \quad l \ impair \\ \end{array} \right.$

But if we accept the second sub-case (l odd), we allow negative diffusion values which is do not fulfill the undelying physics and do not respect the radial symmetry property of the diffusion process. Thus, only even-order tensor are allowed.

A l order tensor in 3 dimensions is composed of $3^l$ elements. The full symmetry property associated with a diffusion tensor reduces the number of independent components. This property comes from the fact that a diffusion tensor has to link the different components of a vector to the same scalar $d(\vec{g})$ . For instance, when l=2 we have:

$d(\vec{g}) &= \sum_{i_1=1}^3 \sum_{i_2=1}^3 d_{i_1,i_2} g_{i_1} g_{i_2} = \sum_{i_1=1}^3 \sum_{i_2=1}^3 d_{i_1,i_2} g_{i_2} g_{i_1} \nonumber \\ &= \sum_{i_2=1}^3 \sum_{i_1=1}^3 d_{i_2,i_1} g_{i_1} g_{i_2} = \sum_{i_1=1}^3 \sum_{i_2=1}^3 d_{i_2,i_1} g_{i_1} g_{i_2}$

This relation implies $d_{i_1,i_2}=d_{i_2,i_1}$ since it is satisfied for any vector $\vec{g}$ . A similar analysis for a l order tensor gives:

$d_{i_1,i_2,...i_l} = d_{(i_1,i_2,...i_l)}$

where $(i_1,i_2,...i_l)$ represents all the possible permutations of indices. In three dimensions, a symmetric tensor has:

N_l = Gamma_3^l = frac{(l+1)(l+2)}{2}

independent components ( $\Gamma^l_3$ is the number of l-combinations with repetition of 3 elements), where each element $d_k$ ( $k \in [1,\dots, N_l]$ ) is repeated $\mu_k^l$ times, with:

$\mu_k^l = \left( \begin{array}{l} l \\ n_1^l(k) \\ \end{array} \right) \left( \begin{array}{l} l-n_1^l(k) \\ n_2^l(k) \\ \end{array} \right) = \frac{l!}{n_1^l(k)!n_2^l(k)!n_3^l(k)!}$

$n_1^l(k)$, $n_2^l(k)$ and $n_3^l(k)$ are respectivelly the number of index 1, 2 and 3 that defined the kth independant component of the tensor:

$d_k = d_{\underbrace{1,...,1}_{n_1^l(k)},\underbrace{2,...,2}_{n_2^l(k)}, \underbrace{3,...,3}_{n_3^l(k)}}$

with $n_1^l(k)+n_2^l(k)+n_3^l(k)=l$ . Finally, we can rewrite the Steskal-Tanner equation:

$S(\vec{g}) = S_0 exp \left( -b \sum_{k=1}^{N_l} \mu_k^l d_k (g_1)^{n_1^l(k)} (g_2)^{n_2^l(k)} (g_3)^{n_3^l(k)} \right)$

References

[1]	E. Ozarslan et T. Mareci: Generalized diffusion tensor imaging and analytical relationships between diffusion tensor imaging and high angular resolution diffusion imaging. Magnetic Resonance in Medicine, 50:955-965, 2003.

Fit the diffusion tensor model using two strategies:

Ordinary least suquare fit (fast)
Quartic decomposition based fit (return positive semi definite tensors)

Get tensor scalar invariant properties: the fractional anisotropy, the mean diffusivity and the Westion shapes coefficients.

Inputs¶

[Mandatory]

bvals_file: a file name (mandatory)

the the diffusion b-values

bvecs_file: a file name (mandatory)

the the diffusion b-vectors

dwi_file: a file name (mandatory)

an existing diffusion weighted image

mask_file: a file name (mandatory)

a mask image

reference_file: a file name (mandatory)

the referecne b=0 image

[Optional]

Outputs¶

eigen_values_file: a file name

the name of the output eigen values file

eigen_vectors_file: a file name

the name of the output eigen vectors file

fractional_anisotropy_file: a file name

the name of the output fa file

linearity_file: a file name

the name of the fie that contains the linerity coefficients

mean_diffusivity_file: a file name

the name of the output md file

planarity_file: a file name

the name of the fie that contains the planarity coefficients

sphericity_file: a file name

the name of the fie that contains the sphericity coefficients

tensor_file: any value

Pipeline schema¶

../../../_images/caps.diffusion_estimation.py_tensor_estimation.SecondOrderTensorEstimation.png