Rotational Invariants of the Cumulant Expansion

Table of contents

  1. RICE toolbox (Rotational Invariants of the Cumulant Expansion)
  2. Overview: The cumulant expansion in diffusion MRI
    1. LTE data
    2. Multiple b-tensors
  3. Example use cases

RICE toolbox (Rotational Invariants of the Cumulant Expansion)

Rice is not yet available as part of the TMI toolbox, however its MATLAB implementation is online.

This MATLAB toolbox contains all necessary functions for parameter estimation of the O(b^2) cumulant expansion for arbitrary axially symmetric b-tensors. Check our recent paper for details on this implementation and this book chapter for information on the cumulant expansion in general. Below we provide instructions on how to run the toolbox. See the example_RICE.m script that performs the parameter estimation in an example dataset.

The toolbox also allows the parameter estimation for minimal DKI and minimal RICE protocols proposed in this paper.

Overview: The cumulant expansion in diffusion MRI

Ob2_cumulant_expansion_RICE

LTE data

For conventional dMRI data, linear tensor encoding (LTE), one can represent low-b data with the O(b) cumulant expansion as shown in Eq. (1). This is simply DTI, and it can represent DWIs up to ~b=1200 ms/mm^2. For higher b-values (up to ~b=2500 ms/mm^2), one can represent the DWIs with the O(b^2) cumulant expansion shown in Eq. (2). This is DKI.

Multiple b-tensors

If we consider multiple b-tensor shapes (axially symmetric) as shown in the figure below, where β parametrizes the b-tensor shape. Most common examples are: β=1 for LTE (B has only one nonzero eigenvalue), β=0 for STE (B has 3 equal nonzero eigenvalues), and β=-1/2 for PTE (B has 2 equal nonzero eigenvalues). axSymB We see that for O(b) signals (Eq. (3) ). This representation is still DTI. However, for O(b^2) a new tensor shows up: the diffusion covariance tensor, C, see Eq. (4). C is more general than kurtosis, actually it contains all the information of the kurtosis tensor plus some extra.

Example use cases

The example_RICE.m script shows some examples on how to run the full RICE fitting and also the minimal DKI and minimal RICE ones. We provide example datasets for these, check this link.

Briefly, the basic usage of the code is as follows:

% RICE toolbox parameter estimation example

type = 'fullRICE';  %  Estimate full D and C tensors from LTE + PTE data (WLLS)
CSphase = 1;        % Use Condon-Shortley phase in spherical harmonics definition
nsl_flag = 1;       % Use local nonlinear smoothing for fitting to boost SNR

[b0, tensor_elems, RICE_maps, DIFF_maps] = RICE.fit(DWI, b, dirs, bshape, mask, CSphase, type, nls_flag)

% Compute fiber basis projections (axial and radial diffusivities and kurtosis)
DKI_maps = RICE.get_DKI_fiberBasis_maps_from_4D_DW_tensors(tensor_elems, mask, CSphase);

See the help in RICE.fit and RICE.get_DKI_fiberBasis_maps_from_4D_DW_tensors for more details.

The following options are available in RICE.fit for the input argument ‘type’: (parameter count does not include s0)

  • ‘minimalDTI’: only MD is fit (1 elem, [D00])
  • ‘fullDTI’: full diffusion tensor is fit (6 elem, [D00 D2m])
  • ‘minimalDKI’: full diffusion tensor and MK are fit (7 elem, [D00 D2m S00])
  • ‘minimalDKI_iso’: only MD and MK are fit (2 elem, [D00 S00])
  • ‘fullDKI’: full diffusion and kurtosis tensors are fit (21 elem, [D00 D2m S00 S2m S4m])
  • ‘minimalRICE’: full diffusion tensor + MK + A0 are fit (8 elem, [D00 D2m S00 S2m A00])
  • ‘fullRICE’: full diffusion and covariance tensors are fit (27 elem, [D00 D2m S00 S2m S4m A00 A2m])