## 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]

### Significance

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.

### Limitations

1. One-size-fits-all approach ignores the heterogeneity of axonal composition and pathological condition [8].

2. Spurious fibers [8].

3. Inconsistent with histology [9]

## Reference

[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.