Models beyond tensor

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 (bedpostx in FSL), 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 a 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 ignores disease conditions.


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

What is the minimum requirement of diffusion images for fitting the multiple-tensor model and BSM?

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 [8][10].
2. Spurious fibers [8].
3. Inconsistent with histology [9]


What is the minimum requirement of diffusion images for spherical deconvolution? 

NODDI (neurite orientation dispersion and density imaging) model


Hui Zhang introduced the NODDI [3] to model three components (1) the intracellular components, which can be viewed as a set of sticks, (2) the extracellular components, which is a specific form of the tensor, and (3) the CSF component. 

What is the minimum requirement of diffusion images for fitting NODDI?


Reconstruct diffusion MRI data using FSL's BSM, NODDI, 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.
[8] Parker, G. D., Marshall, D., Rosin, P. L., Drage, N., Richmond, S., & Jones, D. K. (2013). A pitfall in the reconstruction of fibre ODFs using spherical deconvolution of diffusion MRI data. Neuroimage, 65, 433-448.
[9] Schilling, K., Janve, V., Gao, Y., Stepniewska, I., Landman, B. A., & Anderson, A. W. (2016). Comparison of 3D orientation distribution functions measured with confocal microscopy and diffusion MRI. Neuroimage, 129, 185-197.
[10] Schilling, K. G., Gao, Y., Stepniewska, I., Janve, V., Landman, B. A., & Anderson, A. W. (2019). Histologically derived fiber response functions for diffusion MRI vary across white matter fibers—An ex vivo validation study in the squirrel monkey brain. NMR in Biomedicine, e4090.