Multiple tensor models and spherical deconvolution

Multi-tensor model 

Diffusion tensor is known to be limited by its ability to describe restricted diffusion or multiple fiber population. One direct solution is to use the multiple tensor model to model the diffusion signal, allowing for accessing the diffusivity and fiber orientations for multiple fiber populations [1].


1. The introduction of High angular resolution diffusion imaging (HARDI)
2. In the Tuch's paper [1], he showed that the distribution of diffusivity cannot be used as the fiber distribution.


The multi-tensor model has a large number of parameters. The computation often takes a lot of time and prone to overfitting.

Ball-and-stick(s) model

Tim Behrens introduced ball-and-stick model (BSM), a multiple tensor model that handles a partial volume of free water diffusion (ball) and multiple fiber populations (sticks) [2]. The fiber populations are represented by sticks, which has a unique tensor presentation with radial diffusivity removed. The free water diffusion is represented by ball, which is an isotropic tensor.

The BSM can be formulated as follows.

where Di are the "sticks".


The most popular multi-tensor model. BSM substantially reduces the number of parameters and aims for resolving fiber orientation.


BSM assumes that all fiber population shares the same diffusion profile (the sticks), and it cannot provide axial or radial diffusivity information.


BSM is a special case of multi-tensor model. It mainly aims for resolving crossing fibers and estimating their relative volume fractions.

NODDI (neurite orientation dispersion and density imaging) model

Hui Zhang introduced the NODDI model [3] , which can also been viewed as a multiple tensor model, but having more modeling components than restriction than the BSM. NODDI includes three components (1) the intra-cellular components, which can be viewed as a set of sticks, (2) the extra-cellular components, which is specific form of tensor, and (3) the CSF compoment. 

Spherical deconvolution

Tournier et al. [4] proposed spherical deconvolution to calculate the fiber orientation distribution. The method is based on the concept that the distribution of dMRI signals can be represented by fiber orientation distribution convoluted with its signal response. 

The analysis first estimates how the signal distribution of a fiber bundle (termed "response function"). This estimation is often done by (1) calculating the FA map, (2) selecting the regions with the highest FA (usually at the corpus callosum) and (3) estimating their signal distribution in the region. 

The diffusion signals (same b-value) are then parameterized by spherical harmonics (analogy: Fourier series in spherical coordinates) to facilitate deconvolution. The deconvolution then generates a sharpening effect to visualize the axonal directions.

The official tool for spherical deconvolution is MRtrix.

Other variants: constraint spherical deconvolution (ensure non-negativity)[5][6], RL regularized deconvolution (regularized), diffusion decomposition (sparsity-enforced) [7]


Spherical deconvolution introduces the concept of fiber orientation distribution. It is the best approach to achieve high angular resolution, and there is no model fitting. 


1. One-size-fits-all approach ignores the heterogeneity of axonal composition and pathological condition.
2. False fiber problem. Deconvolution can give rise to false fibers that mimics true fibers and thus cannot be effectively removed.
3. The diffusion community in majority believes that higher angular resolution always results in higher accuracy in fiber tracking. A recent open competition result has shown that this may not be true (see the following figure, created from the result at This is due to the fact that higher sensitivity to crossing fibers results in higher sensitivity to noise. The false fibers created by noise can cause substantial error in fiber tracking. 


1. What is the minimum requirement of diffusion images for fitting the multiple-tensor model and BSM?
2. What is the model assumption of the multiple-tensor model, BSM, and spherical deconvolution? 


1. Reconstruct diffusion MRI data using FSL's BSM and MRtrix. The results can be imported in DSI Studio for inspection (see Convert TrackVis, MRtrix, DTI-TK, FSL data to DSI Studio format).


[1] Tuch, David S., et al. "High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity." Magnetic Resonance in Medicine48.4 (2002): 577-582.
[2] Behrens, T. E. J., Woolrich, M. W., Jenkinson, M., Johansen‐Berg, H., Nunes, R. G., Clare, S., ... & Smith, S. M. (2003). Characterization and propagation of uncertainty in diffusion‐weighted MR imaging. Magnetic resonance in medicine,50(5), 1077-1088. [link]
[3] Zhang, Hui, Torben Schneider, Claudia A. Wheeler-Kingshott, and Daniel C. Alexander. "NODDI: practical in vivo neurite orientation dispersion and density imaging of the human brain." Neuroimage 61, no. 4 (2012): 1000-1016.
[4] Tournier, J-Donald, et al. "Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution." NeuroImage 23.3 (2004): 1176-1185.
[5] Tournier, J.D., Calamante, F., Gadian, D.G., Connelly, A., 2004. Direct estimation of the fiber orientation density function from diffusion-weighted MRI data using spherical deconvolution. Neuroimage 23, 1176-1185.
[6] Descoteaux, M., Deriche, R., Knosche, T.R., Anwander, A., 2009. Deterministic and probabilistic tractography based on complex fibre orientation distributions. IEEE Trans Med Imaging 28, 269-286.
[7] Yeh, F.C., Wedeen, V.J., Tseng, W.Y., 2011. Estimation of fiber orientation and spin density distribution by diffusion deconvolution. Neuroimage 55, 1054-1062.