Synopsis

Using double diffusion encoding it is possible to acquire microstructural and water exchange information. Here, simulations are used to study how restriction effects influence apparent exchange measurements. The simulations indicate that at the chosen experimental parameters the restriction effect can be considerable for large pores and small mixing times, τ_m. In typical exchange rate experiments using clinical MR systems with τ_m > 40 ms, the restriction effect can probably be neglected if pores are small.

PURPOSE

Estimate the effect of restriction on apparent exchange rate measurements.

INTRODUCTION

Double diffusion encoding (DDE) employs two diffusion weightings per acquisition, which can be used to get microstructural information that is not easily available using other non-invasive techniques¹. Recently, DDE has been used to estimate membrane permeability² as well as apparent exchange rate (AXR), both on yeast³ and on humans in vivo^4,5.

In the AXR experiment³, a stimulated echo version of DDE (DDE-STE) is used and the time between the two diffusion weightings, τ_m, is varied (see Fig. 1). The first diffusion weighting acts as a filter, attenuating the signal from fast diffusing spins. The second block is used to calculate a τ_m-dependent apparent diffusion coefficient³, ADC'(τ_m), extracted from S(b,τ_m) = S_f(τ_m) exp{-ADC'(τ_m)b}, where S_f(τ_m) is the signal from an experiment with G⁽²⁾ = 0, b = (ɣδG)² t_d and t_d = Δ – δ/3.

However, previously published results did not consider that an observed τm-dependence of ADC'(τm) can also be due to the diffusion restriction in the sample^1,6.

The typical AXR experiment³ only uses parallel wave-vectors (ψ = 0). Computer simulations⁷ corroborated that at parallel wave-vectors a decrease in diffusion attenuated signal is expected for increasing τ_m due to the restriction effect described by Mitra^1,8.

Using numerical simulation, we studied the restriction effect in terms of ADC'(τ_m) for impermeable spheres to estimate the influence of restriction in AXR experiments.

METHODS

Simulations were performed using MISST v0.93^9-11 with Matlab2015b. In MISST, the diffusion experiment is simulated by successive matrix multiplications.

Parameters have been chosen to be achievable on a clinical MR system (Table 1). Spherical pores without molecular exchange were assumed. Simulated ADC'(τ_m) values were compared with previously published results in yeast³.

RESULTS

Figure 2a shows the signal dependence on ψ for spherical pores of 18 µm diameter. The amplitude of this modulation increases with pore diameter. The modulation amplitude decreases with increasing τ_m and, in this experiment, virtually disappears above τm ≈ 50 ms. The decay of the diffusion signal for ψ = 0 (Fig. 2b) is more noticeable with diameters above 10 µm.

In Fig. 3, the simulated ADC'(τm) of spheres with d ≥ 18 µm exhibits a steep increase with increasing τm and remains constant for τ_m > 50 ms. The ADC'(τm) for spheres with smaller diameters (d ≤ 14 µm) reaches a constant plateau before τ_m = 40 ms.

Impermeable spheres with diameters between 5 and 10 µm have a constant ADC'(τ_m < 10 ms) = 1x10^-4 mm²/s, which corresponds to 18% of ADC'(τ_m = 420 ms) observed in yeast.

DISCUSSION

The MR signal depends on the sphere diameter. The decrease in signal at ψ = 0 when increasing τ_m is more pronounced for spheres with large diameter. Since impermeable spheres are used in this simulation, this decrease in signal is only due to the restriction effect. It leads to an increase in the calculated ADC'(τ_m).

Figure 3 showed that the restriction effect has only very little influence on the increase in ADC'(τ_m) for spheres with diameter ≤ 10 µm. It becomes more significant when increasing the sphere's diameter.

CONCLUSION

AXR experiments require short Δ (in order to avoid exchange during diffusion encoding) and long τ_m. In this situation, the description given by Mitra¹ does not apply.

In experiments on baker's yeast, where cells of diameters between 5 and 10 µm are expected, using experimental parameters comparable to the ones used here, it is safe to say that the effect of restriction in the increase of the ADC'(τ_m) is negligible. Nevertheless, for studying cells with larger diameters (like blood cells), the effects of restriction possibly need to be considered and corrected for. In isotropic tissue, the use of perpendicular wave vectors may be a solution for this issue, since the signal for ψ = π/2 is independent of τ_m (in the absence of exchange). Alternatively, an average of results for ψ = 0 and ψ = π could be used.

Acknowledgements

Patricia Ulloa was supported by the Graduate School for Computing in Medicine and Life Sciences funded by Germany's Excellence Initiative [DFG GSC 235/2-1] and [DFG KO 3389/2-1].

References

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