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

Technical Summary: Surface Relaxivity Biases Diffusion and Relaxometry Microstructure Fractions

Problem Statement
Surface relaxivity ($\rho$) and time-dependent diffusion are two observables arising from the same physical process: water molecules colliding with the walls of myelinated axons. While diffusion MRI (dMRI) is often assumed to be "relaxivity-blind" due to $b=0$ normalization, this paper argues that the assumption holds only for a single compartment. In reality, intra-axonal and extra-axonal water reside against different walls with distinct surface-to-volume ratios ($S/V$). Consequently, they experience different transverse relaxation rates ($R_2 = 1/T_2 + \rho S/V$). This differential $T_2$ reweights the signal compartments at the echo time ($TE$) used for normalization, introducing a systematic bias into microstructure estimates. The paper investigates how this unobserved geometric term biases two key metrics: the diffusion intra-axonal signal fraction ($f_{intra}$) and the myelin water fraction (MWF).

Methodology
The study combines analytical derivations with first-principles Monte Carlo (MC) simulations to validate the physics and quantify the bias.

1. Analytical Derivations:

   • Interior Rate: Derived the Brownstein–Tarr surface rate for intra-axonal water over a Gamma-distributed axon caliber population. The rate is the area-weighted average of $\rho(4/d)$.
   • Exterior Rate: Derived the Novikov–Burcaw surface rate for extra-axonal water, which saturates to a plateau determined by the mean exterior surface density ($S_{ext}/V_{ext}$) in the motional-narrowing limit.
   • Closed-Form Ratio: Demonstrated that the ratio of interior to exterior $S/V$ is independent of the caliber distribution and depends solely on the fiber volume fraction ($V_F$) and the $g$-ratio: $(1-V_F)/(g V_F)$.
   • Bias Formulas: Derived closed-form expressions for the bias in $f_{intra}$ (Eq. 9) and the shift in apparent MWF based on the crossing of thin-axon $T_2$ values below the myelin window.

2. Monte Carlo Validation:

   • Developed a wall-counting MC simulator (using a realized-overshoot boundary local-time estimator) to propagate random walkers through explicit packed geometries without analytical relaxivity models.
   • Validated that the MC surface attenuation matches the closed-form Brownstein–Tarr and Novikov–Burcaw rates to within $\sim0.2$ percentage points.
   • Confirmed that thin axons in the MC simulation exhibit $T_2$ values crossing below the 25 ms myelin window, validating the mechanism for MWF bias.

3. Forward Modeling:

   • Diffusion: Simulated multi-shell dMRI signals with surface relaxivity enabled and fitted them using standard spherical-mean techniques (SMT) to quantify the $f_{intra}$ bias.
   • Relaxometry: Simulated multi-echo CPMG decays with a caliber-distributed intra-axonal pool and estimated MWF using standard non-negative least squares (NNLS) with a fixed myelin window.
   • Longitudinal Trajectory: Modeled a within-subject demyelination trajectory (preserving axon lumen, thinning myelin) to test if the bias cancels in longitudinal changes.

Key Results

• Intra-axonal Signal Fraction ($f_{intra}$) Bias:

  • The bias is driven by the differential $S/V$ between the interior and exterior walls.
  • Sign Law: The bias sign is determined by the fiber volume fraction ($V_F$). Below a crossover point $V_F^* = 1/(1+g) \approx 0.59$, intra-axonal water relaxes faster (under-estimation). Above $V_F^*$, extra-axonal water relaxes faster, causing the intra-axonal signal to be over-weighted.
  • Magnitude: In physiologically dense white matter ($V_F \approx 0.65$), the intra-axonal fraction is over-estimated by approximately 12% at a clinical echo time ($TE=80$ ms) using a cited relaxivity value ($\rho = 1.16$ $\mu$m/s). This bias scales linearly with $\rho$ and grows monotonically with $TE$.
  • Nature: This is a first-order, spatially structured systematic error, not zero-mean noise, meaning it does not average out across voxels but shifts regional and cohort means.

• Myelin Water Fraction (MWF) Bias:

  • Mechanism: Surface relaxivity shortens the apparent $T_2$ of intra-axonal water. For the thinnest axons (inner diameter $d < d^* \approx 0.17$ $\mu$m at cited $\rho$), the $T_2$ drops below the standard 25 ms myelin window. This water is misclassified as myelin water.
  • Direction: Fine white matter (high $S/V$) reads as myelin-richer than coarse white matter, even if true myelin volume is identical.
  • Magnitude: At the cited $\rho$, the bias is small (0.33 percentage points in MWF, or ~0.82 pp in inferred myelin content). However, it is super-linear with respect to $\rho$; if $\rho$ is an order of magnitude higher, the bias becomes significant (3.4 pp).
  • Detectability: The bias is buried beneath single-voxel noise but emerges as a systematic offset when averaging over regions of interest (ROIs) with different caliber distributions.

• Longitudinal Demyelination:

  • In a lumen-preserving demyelination trajectory (primary demyelination), the MWF bias is set by the fixed inner axon diameter. Thus, the bias is nearly constant and cancels out when calculating the change ($\Delta$MWF) between time points.
  • Conversely, the $f_{intra}$ bias depends on the difference between interior and exterior $S/V$. As myelin thins, the exterior wall retreats, altering the exterior $S/V$ while the interior remains fixed. Consequently, the $f_{intra}$ bias drifts significantly (e.g., from -5% to -21%) during demyelination, creating a large, TE-dependent artifact in longitudinal diffusion studies.

Significance and Claims
The paper claims to identify a previously unreported source of bias in both diffusion and relaxometry microstructure imaging that stems from the inseparability of bulk $T_2$ and geometric surface relaxivity in multi-echo measurements.

1. Unified Physics: It establishes that surface relaxivity and time-dependent diffusion are dual readouts of the same wall-collision process. The "T2" measured in relaxometry is contaminated by a geometry-dependent rate that diffusion MRI attempts to normalize away but fails to fully eliminate due to compartmental $S/V$ differences.
2. Systematic Errors: The $f_{intra}$ bias is characterized as a first-order, deterministic systematic error that scales with packing density and echo time, potentially confounding cross-site or cross-cohort comparisons if echo times differ.
3. MWF Interpretation: The MWF bias is framed as a structural confound where "myelin water" effectively measures "high-$S/V$ trapped water." This implies that fine white matter regions (e.g., genu of the corpus callosum) may appear to have higher myelin content than coarse regions (e.g., mid-body) simply due to caliber differences, independent of actual myelin volume.
4. Longitudinal Contrast: A key finding is the divergence in how these biases behave longitudinally. While MWF changes in primary demyelination are robust to this bias (as the offset cancels), diffusion-based $f_{intra}$ changes are heavily contaminated by the shifting exterior wall geometry.
5. Limitations: The authors modestly note that the absolute magnitude of the bias depends on the poorly established value of $\rho$ (which may vary by an order of magnitude) and that the idealized geometry (non-touching cylinders) may underestimate effects in realistic, touching fiber packs. They do not propose a new acquisition protocol to resolve this, noting that the degeneracy is structural and requires orthogonal constraints (e.g., high-gradient oscillating gradients) that are currently beyond clinical reach.