RICE is a representation-based framework for understanding and estimating diffusion MRI cumulant tensors from arbitrary B-tensor encodings, enabling protocol-independent inference beyond conventional LTE-based DTI and DKI.
This 'MATLAB toolbox' contains all necessary functions for parameter estimation of the O(b^2) cumulant expansion for arbitrary axially symmetric b-tensors1. Check our recent paper for details on this implementation and this book chaper for information on the cumulant expansion in general and its applications to diffusion MRI. Below, we provide instructions on how to run the toolbox. See the example_RICE.m script that performs the parameter estimation in the example datasets.
The toolbox also allows the parameter estimation for iRICE protocols (LTE or LTE+STE) proposed in 1.
First-order expansion, equivalent to diffusion tensor imaging (DTI):
Second-order expansion, equivalent to diffusion kurtosis imaging (DKI) for LTE data or q-space trajectory imaging (QTI) for arbitrary
Conventional dMRI data is acquired with linear tensor encoding (LTE). This means that the diffusion weighting (b) is applied along a given direction (g)
One can represent low-b data with the O(b) cumulant expansion as shown in Eq. (1). This is DTI, and it can represent DWIs up to ~b=1200 ms/mm^2 for the in vivo human brain. 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/QTI depending on the B used.
Different axially symmetric b-tensor shapes are shown in the figure below. Here β parametrizes the tensor shape. Most common examples are: β=1 for LTE (B has only one nonzero eigenvalue), β=0 for spherical tensor encoding (STE) where B has 3 equal nonzero eigenvalues, and β=-1/2 for planar tensor encoding (PTE), where B has 2 equal nonzero eigenvalues.
For O(b) signals, Eq. (1), almost all B probe the entire diffusion tensor; isotropic B (STE) only probe its trace (mean diffusivity).
For O(b^2) things are different: to fully probe the diffusion covariance tensor C, Eq. (2), we need to combine different acquisitions with varying B-shapes. Note that the kurtosis tensor, accessed with LTE, is proportional to the symmetric part of C.
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 = 0; % Use Condon-Shortley phase in spherical harmonics definition - usually for complex basis
ComplexSTF = 0; % Use real-valued spherical harmonics definition
nls_flag = 1; % Use local nonlinear smoothing for fitting to boost SNR
parallel_flag = 1; % Use paralellization
[b0, tensor_elems, RICE_maps, DIFF_maps] = RICEtools.fit(DWI, b, dirs, bshape, mask, CSphase, ComplexSTF, type, nls_flag, parallel_flag);
% Compute fiber basis projections (axial and radial diffusivities and kurtosis)
DKI_maps = RICEtools.get_DKI_fiberBasis_maps_from_4D_DW_tensors(tensor_elems, mask, CSphase, ComplexSTF);
See the help in RICEtools.fit and RICEtools.get_DKI_fiberBasis_maps_from_4D_DW_tensors for more details. In a nutshell, the code uses the SA decomposition for the fitting such that it can handle LTE only data.
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])
If you are interested in the TQ decomposition you can simply convert S,A tensors into T,Q with this linear transformation:
% Computing SA and TQ decompositions
Dlm = tensor_elems(:,:,:,1:6);
Slm = tensor_elems(:,:,:,7:21); % tensor_elems contains Slm and Alm elements of C
Alm = tensor_elems(:,:,:,22:27); % tensor_elems contains Slm and Alm elements of C
% Compute TQ decomposition
Q00 = 5/9 * Slm(:,:,:,1) + 2/9 * Alm(:,:,:,1) ;
T00 = 4/9 * Slm(:,:,:,1) - 2/9 * Alm(:,:,:,1) ;
Q2m = 7/9 * Slm(:,:,:,2:6) - 2/9 * Alm(:,:,:,2:6) ;
T2m = 2/9 * Slm(:,:,:,2:6) + 2/9 * Alm(:,:,:,2:6) ;
T4m = Slm(:,:,:,7:15);
Tlm = cat(4,T00,T2m,T4m);
Qlm = cat(4,Q00,Q2m);
Do not hesitate to reach out to Santiago.Coelho@nyulangone.org (or @santicoelho in Twitter) for feedback, suggestions, questions, or comments1.
A US patent contains some of the related developments.
% Authors: Santiago Coelho (santiago.coelho@nyulangone.org), Els Fieremans, Dmitry Novikov
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% - Coelho, S., Chen J., Szczepankiewicz F., Fieremans, E., Novikov, D.S., Geometry of the cumulant series in diffusion MRI, Nature Communications (2026), https://doi.org/10.1038/s41467-026-70018-w
Footnotes
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Please cite these works if you use the RICE toolbox in your publication:
- Coelho, S., Chen J., Szczepankiewicz F., Fieremans, E., Novikov, D.S., Geometry of the cumulant series in diffusion MRI, Nature Communications (2026), https://doi.org/10.1038/s41467-026-70018-w