Two observables of one wall: how surface relaxivity can bias the diffusion intra-axonal fraction and the myelin water fraction

Abstract

Surface relaxivity and time-dependent diffusion are two readouts of the same wall collisions on one substrate: the transverse rate that microstructure imaging fits is a bulk rate plus a surface rate $\rho\,(S/V)$, and intra- and extra-axonal water carry different $S/V$, so their $T_2$ differ and any compartment estimate that normalises by a TE-weighted $b=0$ is biased. We derive closed forms for the interior (Brownstein-Tarr) and exterior (Novikov-Burcaw) surface rates over myelinated cylinders, validate them with wall-counting Monte Carlo, and quantify the resulting bias on the diffusion intra-axonal signal fraction $f_{\mathrm{intra}}$ and on the myelin water fraction (MWF). The interior/exterior $S/V$ ratio is $(1-\mathrm{VF})/(g\,\mathrm{VF})$, independent of the calibre distribution and crossing unity at fibre volume fraction $\mathrm{VF}^{\ast}=1/(1+g)$. Physiological white matter sits above this crossover, so surface relaxivity over-weights the intra-axonal signal, biasing $f_{\mathrm{intra}}$ by $\approx 12\%$ at clinical PGSE ($\mathrm{TE}=80\,\mathrm{ms}$). That fractions are TE-dependent is known; new here are the surface-relaxivity and packing attribution, a closed-form sign law, and a testable packing-dependent TE drift. The same physics reads through relaxometry as a smaller MWF bias: the thinnest axons' water crosses below the myelin window and is counted as myelin, so fine white matter reads myelin-richer ($\sim 0.33\,\mathrm{pp}$, beneath single-voxel noise but super-linear in the uncertain $\rho$), while in primary lumen-preserving demyelination this bias is nearly constant and cancels in the longitudinal change. One $S/V$ physics thus biases both diffusion and relaxometry microstructure estimates, the $f_{\mathrm{intra}}$ bias being first-order and packing-dependent, the MWF bias small and structural.