In the brain's white matter, the signal travels along axons — long neuronal processes wrapped in an insulating myelin sheath. Water fills the space inside axons and outside, in the narrow gaps between fibers. Two MRI methods — diffusion (which tracks molecular motion) and relaxometry (which measures how quickly the signal decays)—are like two translators listening to the same conversation but hearing different things. Both pick up the impacts of water molecules against the walls, but they interpret them differently.
The authors derived closed-form formulas for these surface effects inside and outside myelinated cylinders. The key parameter is the volume fraction of fibers (VF). When VF exceeds a critical threshold (VF* = 1/(1+g), where g is the ratio of inner to outer surface relaxivity), the intracellular water dominates the signal. For healthy white matter, VF is indeed above this threshold, so the diffusion method at a clinical echo time of 80 ms overestimates the intra-axonal fraction by about 12%. The thinnest axons add to the confusion: their water relaxes so quickly that it falls into the myelin time window, and relaxometry attributes it to the myelin fraction, inflating the estimate by 0.33 percentage points. This error is small but systematic and depends on fiber packing density.
Thus, the same physics of wall collisions distorts two microstructural maps in different ways. For diffusion, it is a first-order error sensitive to packing; for relaxometry, it is a small structural correction that barely changes with primary demyelination and therefore shrinks in longitudinal studies. From now on, these distortions can not only be noticed but also predicted by a simple law.
as written by the authors arXiv:2607.09401
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.
In the white matter of a typical adult human, there are about 150,000 kilometers of axons — enough to wrap around the Earth's equator nearly four times.