  Axon radius is a potential biomarker for brain diseases and a crucial tissue
microstructure parameter that determines the speed of action potentials.
Diffusion MRI (dMRI) allows non-invasive estimation of axon radius, but
accurately estimating the radius of axons in the human brain is challenging.
Most axons in the brain have a radius below one micrometer, which falls below
the sensitivity limit of dMRI signals even when using the most advanced human
MRI scanners. Therefore, new MRI methods that are sensitive to small axon radii
are needed. In this proof-of-concept investigation, we examine whether a
surface-based axonal relaxation process could mediate a relationship between
intra-axonal T2 and T1 times and inner axon radius, as measured using
postmortem histology. A unique in vivo human diffusion-T1-T2 relaxation dataset
was acquired on a 3T MRI scanner with ultra-strong diffusion gradients, using a
strong diffusion-weighting (i.e., b = 6,000 s/mm$^2$) and multiple inversion
and echo times. A second reduced diffusion-T2 dataset was collected at various
echo times to evaluate the model further. The intra-axonal relaxation times
were estimated by fitting a diffusion-relaxation model to the
orientation-averaged spherical mean signals. Our analysis revealed that the
proposed surface-based relaxation model effectively explains the relationship
between the estimated relaxation times and the histological axon radius
measured in various corpus callosum regions. Using these histological values,
we developed a novel calibration approach to predict axon radius in other areas
of the corpus callosum. Notably, the predicted radii and those determined from
histological measurements were in close agreement.
