For diffusion measurements, stimulated-echo acquisition mode (STEAM) has been widely used. To enhance sensitivity to microstructure, previous studies used STEAM to vary the diffusion time by changing the mixing time tM. Here we show that varying tM results in an “apparent” STEAM-measured diffusivity dependence on tM, irrespective of genuine microstructure-specific time dependence. This effect is caused by T1-relaxation and water exchange between myelin water and "free" water (intra- and extra-axonal water). We propose a modified Kärger model considering diffusion+T1-relaxation+exchange, and demonstrate that exchange-induced tM-dependence explains ~20-50% of the total diffusion time dependence, and should be considered while using STEAM.
For human brain WM, we proposed a diffusion+relaxation+exchange model shown in Fig.1a, where Mf and Mm are magnetizations of ”free” and myelin water. It is the K¨arger model [4,5] for a two-component magnetization M∼(MfMm), but with a modified initial condition for the evolution during the tM interval: the observed signal comes only from the “free” pool, i.e. the initial condition ∼(10), and the measurement (at t>tM) projects the evolution back onto the “free” pool, because of short myelin water T2≪TE (Fig.1b) [7]. The result is that the apparent diffusion coefficient (ADC) in the free water acquires tM-dependence even if both compartment diffusivities are constant---in stark contrast to the regular K¨arger model, where exchange does not lead to extra time-dependence [4,5].
In this model (Fig.1a), myelin water has diffusivity Dm, longitudinal relaxation time Tm1, and free water has Df and Tf1, respectively. The exchange rates are rm (from myelin to free) and rf (from free to myelin), conforming to the detailed balance [4,5].
For this modified K¨arger model, the Bloch-Torrey equation is given by [6]∂tM=(D∇2−iγg⋅r−RL−RE)⋅M,(1) where M=(MfMm),D=(Df00Dm),RL=(1/Tf1001/Tm1), and RE=(rf−rm−rfrm).
Solving Eq.(1) with the initial condition ∼(10) based on previous discussion, the non-diffusion weighted signal [6]Sb=0=f+⋅e−λ+⋅tM+f−⋅e−λ−⋅tM,(2) where f++f−=1. Sb=0 is normalized to 1 for tM=0. Fitting parameters (f+,λ+,λ−) are combinations of tissue parameters (Tf1,Tm1,rf,rm), shown in Fig.1c. The myelin water residence time τm=1/rm is roughly estimated by ∼1/√rfrm. Without water exchange (rf=rm=0), Eq.(2) degenerates to a mono-exponential decay S0=e−λ⋅tM.
Solving for the signal at finite q and expanding up to q2, we get our main result: the ADC measured via STEAM DTI is given by Dapp=Df−(Df−Dm)⋅f+⋅1−τtM+(1+τtM)e−2tM/τ1+f+f−e−2tM/τ⋅tMΔ−δ/3,(3) where τ=2/(λ+−λ−), and δ is diffusion gradient pulse width. To stabilize our fitting, we fixed myelin diffusivities: axial D∥m∼0.5μm2/ms and radial D⊥m∼0.2μm2/ms [8]. The case of D∥m=D⊥m=0 is also shown to discuss the influence of Dm on corrections.
We also applied models for time-dependent diffusion [3,9,10] to the measured Dapp and the calculated Df after correction for exchange (Table 1).
Fig.2 shows for each ROI bi-exponential decay of the b=0 signal Sb=0, instead of mono-exponential, with respect to tM (Eq.(2)) with two time scales 1/λ+~65ms and 1/λ−~830ms, and the estimated myelin residence time is τm∼160ms (Table 1a), consistent with previous studies [12], manifesting effects of water exchange and T1-relaxation.
The uncorrected ADC (D∥app,D⊥app) and calculated free water diffusivities (D∥f,D⊥f), based on Eq.(3), with respect to Δ, are both shown in Figs.3 and 4. The fitting parameters of time dependence models are shown in Table 1b-c. In general, ~20-50% of the diffusion time dependence, indicated by strength of restrictions (A∥,A⊥), is contributed to exchange.
Varying tM in STEAM introduces a tM-dependent diffusion in human brain WM, caused by water exchange between myelin water and "free" water with different T1 values. In WM, ~20-50% of the "diffusion time" dependence results from this effect.
Remarkably, STEAM provides an “orthogonal” contrast to monopolar pulsed-gradient spin-echo (PGSE); hence, using both sequences, or providing the correction described above based on the Sb=0 signal, may enable simultaneous mapping of microstructural and relaxational tissue parameters.
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