We need to look at Neurite Orientation Dispersion and Density Imaging (NODDI) first!
NODDI is a method of quantifying the morphology of neurites (→ Projections of neurons and dendrites, collectively) using branching complexity of the dendritic trees in terms of dendritic density.
Remember that we are looking at dMRI which measures the Displacement Pattern of water molecules undergoing diffusion. NODDI is a Tissue Model which makes use of Orientation Dispersion Index as a Summary Statistic. This helps quantify Angular Variation of Neurite Orientation.
The NODDI tissue model:
So the above diagram has a tiny equation at the three types of microstructural environments. The water diffusion in each of the environments behaves differently and we will look at them in a little more depth below. (BTW: CSF == CerebroSpinal Fluid).
Intra-cellular compartment → space bounded by the membrane of Neurites: Model this as Set of Sticks. The normalized signal for this model is given by:
q → Gradient Direction
b → b-val
f(n)dn → probability of finding sticks along orientation ‘n’
The exponential term → Signal Attenuation due to the intrinsic diffusivity of the sticks
The f(n): the Orientation Distribution function is modeled as a Watson Distribution. This distribution is the simplest distribution that can capture the dispersion in orientations.
M is a confluent hypergeometric function.
𝛍 = mean dispersion
қ = concentration parameter that measures the dispersion about 𝛍
This, as the name suggests takes care of the space around the neurites. The neurites hinder the diffusion, but does not restrict it in any way. Therefore we can model it using a ‘Gaussian Anisotropic Distribution‘.
The signal for this environment of the model looks something like:
Here, D(n) is a cylindrical symmetric tensor with the principal direction of diffusion ‘n’. But, now we need to consider 2 directions of diffusion, perpendicular and parallel diffusivities: d⊥ and d∥.
The parallel diffusivity is the same as the intrinsic
free diffusivity of the intra-cellular compartment; the perpendicular
diffusivity is set with a simple tortuosity model as
d⊥ = d∥ (1 − ν_ic), where ν_ic is the intra-cellular volume fraction.
The CSF compartment models the space occupied by cerebrospinal fluid and is modeled as isotropic Gaussian diffusion with diffusivity d_iso. But in this model we make use of the Orientation Dispersion index instead of қ as follows:
OD = 2/pi * arctan(1/қ)
 Zhang, H., Schneider, T., Wheeler-kingshott, C. A., & Alexander, D. C. (2012). NeuroImage NODDI : Practical in vivo neurite orientation dispersion and density imaging of the human brain. NeuroImage, 61(4), 1000–1016. https://doi.org/10.1016/j.neuroimage.2012.03.072
 Farooq, H., Xu, J., Nam, J. W., Keefe, D. F., Yacoub, E., Georgiou, T., & Lenglet, C. (2016). Microstructure Imaging of Crossing (MIX) White Matter Fibers from diffusion MRI. Scientific Reports, 6(September), 1–9. https://doi.org/10.1038/srep38927
 Ferizi, U., Schneider, T., Panagiotaki, E., Nedjati-Gilani, G., Zhang, H., Wheeler-Kingshott, C. A. M., & Alexander, D. C. (2014). A ranking of diffusion MRI compartment models with in vivo human brain data. Magnetic Resonance in Medicine, 72(6), 1785–1792. https://doi.org/10.1002/mrm.25080