The effective transverse relaxation rate (R 2 *) is sensitive to the microstructure of the human brain, e.g. the g-ratio characterising the relative myelination of axons. However, R 2 * depends on the orientation of the fibres relative to the main magnetic field degrading its reproducibility and that of any microstructural derivative measure. To decipher its orientation-independent part (R 2,iso *), a second-order polynomial in time (M2) can be applied to single multi-echo gradient-recalled-echo (meGRE) measurements at arbitrary orientation. The linear-time dependent parameter, β 1 , of M2 can be biophysically related to R 2,iso * when neglecting the signal from the myelin water (MW) in the hollow cylinder fibre model (HCFM). Here, we examined the effectiveness of M2 using experimental and simulated data with variable g-ratio and fibre dispersion. We showed that the fitted β 1 effectively estimates R 2,iso *when using meGRE with long maximum echo time (TE max ≈ 54 ms) but its microscopic dependence on the g-ratio was not accurately captured. This error was reduced to less than 12% when accounting for the MW contribution in a newly introduced biophysical expression for β 1 . We further used this new expression to estimate the MW fraction (0.14) and g-ratio (0.79) in a human optic chiasm. However, the proposed method failed to estimate R 2,iso * for a typical in-vivo meGRE protocol (TE max ≈ 18 ms). At this TE max and around the magic angle, the HCFM-based simulations failed to explain the R 2 *-orientation-dependence. In conclusion, estimation of R 2,iso * with M2 in vivo requires meGRE protocols with very long TE max ≈ 54 ms.