DMRI Reconstruction Models, Scalar Maps, and Spatial Normalization

For each subject, dMRI microstructural measures were computed from four different reconstruction models: DTI, TDF, NODDI and FW. Five measures were extracted from these models: FA and mean diffusivity (MD) from DTI, fractional anisotropy from TDF (FA-TDF), the orientation dispersion index (OD) from NODDI and the free water index (FW). We will not describe the well known DTI based FA and MD here, but will briefly describe the other three models:

The Tensor Distribution Function (TDF) represents the diffusion profile as a probabilistic mixture of tensors [2] allowing the reconstruction of multiple underlying fibers per voxel, together with a distribution of weights. We compute the voxel-wise TDF as the probability distribution function P(D) defined on all feasible 3D Gaussian diffusion processes in tensor space D:

where S is the measured intensity signal, q = rSG, where r, l, and G are the gyromagnetic ratio, the duration of the diffusion sensitization, and the applied magnetic gradient vector, respectively. The number of detected peaks is estimated by examining the local maxima of the tensor orientation distribution (TOD), defined in the unit sphere along directions в:

where X are the eigenvalues. The TDF-averaged eigenvalues of each fiber were calculated by computing the expected values along the principal direction of the fiber. From these eigenvalues a scalar TDF anisotropy (FA-TDF) is calculated as an extension of the standard FA formula:

The Neurite Orientation Dispersion and Density Imaging (NODDI) is a

composite model that takes into account three compartments that affect water diffusion in the brain: the intracellular compartment, the extracellular compartment, and the cerebrospinal fluid (CSF) [5]. The intracellular compartment is modeled as cylinders with a radius of zero that represent the axons and dendrites of the brain tissue, which are jointly called neurites. The ODF of the intracellular compartment is modeled as a Watson distribution that can capture the dispersion orientation of coherent central white matter bundles as well as the incoherent neurites of the grey matter. The normalized intracellular compartment A_{ic} is modeled as:

Here, q represents the gradient directions, b the b-value of the diffusion weighting, n are the orientations of the cylinders with parallel diffusivity d along which the signal is attenuated and f (n) is the Watson distribution, which has two parameters (д, K) and is defined as:

Here, the distribution tends to be symmetric around the mean orientation д, and M is Kummer’s confluent hypergeometric function. K is called the concentration parameter. For K > 0, as K increases the density along д tends to concentrate.

Once K is estimated the orientation dispersion index (OD) is calculated as:

OD goes from 0 to 1, the higher the value the more dispersed the neurites in a particular voxel. In our analyses below we used only the OD maps. The intracellular and extracellular volume fractions as well as the isotropic CSF volume fraction are not taken into account in our analyses. Zhang et al. demonstrated that the latter measures require more than one shell in order to be reliable, whereas the OD can be computed reliably with single shell data even with standard clinical acquisition b-values of b = 1000 s/mm^{2} [5]. OD may be more informative than DTI, in areas with less organized patterns such as areas of multiple fiber crossings as well as towards the gray/white matter boundaries.

Free-Water Imaging (FW) estimates the contribution of freely diffusing water molecules to the diffusion signal with a bi-tensor model [13]. The first component of the model is the so-called tissue compartment that represents either grey matter or a bundle of the white matter. The second component reflects the free-water compartment, which is said to be proportional to the amount of CSF contamination, especially in areas of the white matter that are close to the ventricles. The free-water component is also expected to increase with neuroinflammation due to edema. The full model is defined as:

where S is the attenuated signal, q are the applied diffusion gradient directions, b is the b-value of the diffusion weighting, D is the diffusion tensor and f is the fractional volume of the tissue compartment (0 1). The second term is a fully isotropic tensor, where d_{w} is the bulk diffusivity of water, which is constant at body temperature (3 x 10“^{3} mm^{2}/s).

Voxel-wise maps of all five measures—FA, MD, FA-TDF, OD, and FW—were created for all 102 subjects; all subjects’ maps were spatially normalized to a custom ADNI- derived minimal deformation template (MDT). Template creation and spatial normalization was performed according to previously published voxelwise ADNI- DTI analyses [7].