Coherence gating

Every other page in this section describes one loss mechanism. Coherence gating is different in kind: it is a meta-effect. It adds no new term — it sets when each of the others acts, according to where the magnetization points.

The rule is a property of the coherence state itself: whichever fraction of the magnetization lies transverse accrues the transverse channels (\(T_2\), surface relaxivity, susceptibility, the transverse face of MT); whichever fraction is stored longitudinally accrues only the longitudinal channels (\(T_1\), and their longitudinal siblings). It makes no difference how the magnetization came to be along \(z\) — a deliberate storage pulse or the residue of an imperfect flip. A stimulated echo (PGSTE) is simply the sequence that exploits this on purpose, parking the magnetization along \(z\) for a mixing time \(T_m\) so the transverse channels pause while exchange keeps running — the lever dmipy uses to separate effects a spin echo only sees combined.

Two apparent rates, one gate

Write \(\chi_\perp(t)\in[0,1]\) for the transverse fraction at time \(t\) (the rest, \(1-\chi_\perp\), is stored). In the released idealized-pulse limit (instantaneous, perfect \(90^\circ/180^\circ\) pulses) it is a binary mask: a PGSE spin echo is transverse throughout (\(\chi_\perp\equiv1\)); a PGSTE sets \(\chi_\perp=0\) across \(T_m\) and is transverse only over its two encoding lobes.

Every wall and field mechanism collapses into two apparent rates — one per coherence state. While transverse, the magnetization decays at the apparent transverse rate

\[ \frac{1}{T_2^{\mathrm{app}}} = \underbrace{\frac{1}{T_2}}_{\text{bulk}} + \underbrace{\rho_2\,\frac{S}{V}}_{\text{surface relaxivity}} + \underbrace{R_2'}_{\text{susceptibility}} + \underbrace{k_f}_{\text{MT, transverse}} , \]

while stored, only the apparent longitudinal rate acts:

\[ \frac{1}{T_1^{\mathrm{app}}} = \underbrace{\frac{1}{T_1}}_{\text{bulk}} + \underbrace{\rho_1\,\frac{S}{V}}_{\text{longitudinal surface}} + \underbrace{k_{\mathrm{MT}}^{\parallel}}_{\text{MT, saturation transfer}\;🔬} . \]

The gate is \(\chi_\perp\) selecting between them. As the per-compartment log-weight a walk accumulates,

\[ \log w \;=\; -\!\int_0^{T_E}\!\Big[\;\chi_\perp(t)\,\frac{1}{T_2^{\mathrm{app}}} \;+\;\big(1-\chi_\perp(t)\big)\,\frac{1}{T_1^{\mathrm{app}}}\;\Big]\,dt . \]

The two rates are exact siblings — transverse and longitudinal faces of the same walls — each a bulk term plus one \(S/V\)-weighted (or field) term per mechanism. (🔬 marks the one term not yet in the released public scope — MT's longitudinal saturation transfer \(k_{\mathrm{MT}}^{\parallel}\); the engines carry \(T_2\), \(\rho_2\), \(T_1\), \(\rho_1\), susceptibility's \(R_2'\) and MT's transverse \(k_f\) — see susceptibility and magnetization transfer.)

Diffusion and permeability/exchange are outside the gate. They depend on molecular motion, not on where the magnetization points, so they act in both states: the diffusion grating keeps decaying by displacement through \(T_m\), and walkers keep crossing membranes while stored. That is exactly why storage separates them from the transverse wall sinks.

Magnetization transfer sits on both sides

MT (exchange with a short-\(T_2\) bound pool) has two pathways, and the gate splits them. Its transverse pathway (\(k_f\)) drains the free pool during encoding — the same \(S/V\)-differential \(T_2\) form as surface relaxivity — and is paused by storage. Its longitudinal saturation-transfer pathway (\(k_{\mathrm{MT}}^{\parallel}\)) exchanges \(M_z\) in both states and is not paused. So a stimulated echo removes every transverse wall sink (surface relaxivity and the transverse face of MT) but still pays the longitudinal terms during \(T_m\) — the residual, non-gated confound. The advantage holds to the extent the dominant mechanism is transverse-dephasing rather than a longitudinal population sink.

Why it matters

The headline is separating surface relaxivity from permeability — two rates of the same wall. In a spin echo, surface relaxivity (\(\rho_2\,S/V\) in \(T_2^{\mathrm{app}}\)) erases the wall-adjacent spins that carry the exchange signal, so the two entangle. A stimulated echo pauses the whole \(T_2^{\mathrm{app}}\) bundle over \(T_m\) while exchange accrues, so the exchange time is read far more robustly — the PGSE bias is several times the PGSTE bias at the same relaxivity (worked through in the surface-relaxivity study).

Both sides matter. The transverse rate is what biases \(T_2\)/relaxometry and signal-fraction estimates when ignored — intra- and extra-cellular water carry different \(S/V\), hence different \(T_2^{\mathrm{app}}\), so a \(b{=}0\)-normalized fraction turns TE-dependent. The longitudinal rate opens the \(T_1\)/exchange window. Modeling both compartment-wise keeps recovered microstructure consistent across PGSE, PGSTE and mixed protocols (the inverse side adds the longitudinal interval as the exact sibling of the transverse factor).

Released vs planned

Gating of the released transverse terms (\(T_2\), surface relaxivity, MT's transverse \(k_f\)) with \(T_1\) storage is the newest public capability (idealized-pulse PGSTE), previewed here on the development site. Susceptibility (\(R_2'\)) is now released in dmipy-sim (see its effect page); MT's longitudinal saturation transfer (\(k_{\mathrm{MT}}^{\parallel}\)) is the remaining planned term. The equations above already fix how the gate treats each.