Generalisation of continuous time random walk to anomalous diffusion MRI models with an age-related evaluation of human white matter

Diffusion MRI measures of the human brain provide key insight into microstructural variations across individuals and into the impact of central nervous system diseases and disorders. One approach to extract information from diffusion signals has been to use biologically relevant analytical models to link millimetre scale diffusion MRI measures with microscale influences. The other approach has been to represent diffusion as an anomalous process and infer information from the different anomalous diffusion equation parameters. How parameters of various anomalous diffusion models vary with age in the human brain white matter has not been investigated. We perform the tests within a general continuous time random walk framework, in which temporal jump lengths and structural barriers to diffusion are considered. We outline the formulations for the super-diffusion (i.e., stretched exponential), sub-diffusion, quasi-diffusion and fractional Bloch-Torrey models with respect to the continuous time random walk expression. We also provide a direct link between diffusion kurtosis and the sub-diffusion model and demonstrate mathematical equivalence between parameters. 7T diffusion weighted MRI data in the age range 19-67 was used in this study, particularly focusing on the corpus callosum. Anomalous diffusion model parameters were found to show trends with aging, and interestingly, our investigations led to a new robust technique for generating maps of diffusion kurtosis.