  Axonal damage is the primary pathological correlate of long-term impairment
in multiple sclerosis (MS). Our previous work using our method - diffusion
basis spectrum imaging (DBSI) - demonstrated a strong, quantitative
relationship between axial diffusivity and axonal damage. In the present work,
we develop an extension of DBSI which can be used to quantify the fraction of
diseased and healthy axons in MS. In this method, we model the MRI signal with
the axial diffusion (AD) spectrum for each fiber orientation. We use two
component restricted anisotropic diffusion spectrum (RADS) to model the
anisotropic component of the diffusion-weighted MRI signal. Diffusion
coefficients and signal fractions are computed for the optimal model with the
lowest Bayesian information criterion (BIC) score. This gives us the fractions
of diseased and healthy axons based on the axial diffusivities of the diseased
and healthy axons. We test our method using Monte-Carlo (MC) simulations with
the MC simulation package developed as part of this work. First we test and
validate our MC simulations for the basic RADS model. It accurately recovers
the fiber and cell fractions simulated as well as the simulated diffusivities.
For testing and validating RADS to quantify axonal loss, we simulate different
fractions of diseased and healthy axons. Our method produces highly accurate
quantification of diseased and healthy axons with Pearson's correlation
(predicted vs true proportion) of $ r = 0.99 $ (p-value = 0.001); the one
Sample t-test for proportion error gives the mean error of 2\% (p-value =
0.034). Furthermore, the method finds the axial diffusivities of the diseased
and healthy axons very accurately with mean error of 4\% (p-value = 0.001).
RADS modeling of the diffusion-weighted MRI signal has the potential to be used
for Axonal Health quantification in Multiple Sclerosis.
