Bias in the apparent exchange rate measurements: insight from numerical simulations
Patricia Ulloa1, Vincent Methot1, and Martin A. Koch1

1University of Lübeck, Lübeck, Germany


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, τm. In typical exchange rate experiments using clinical MR systems with τm > 40 ms, the restriction effect can probably be neglected if pores are small.


Estimate the effect of restriction on apparent exchange rate measurements.


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 vivo4,5.

In the AXR experiment3, a stimulated echo version of DDE (DDE-STE) is used and the time between the two diffusion weightings, τm, 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 τm-dependent apparent diffusion coefficient3, ADC'(τm), extracted from S(b,τm) = Sfm) exp{-ADC'(τm)b}, where Sfm) is the signal from an experiment with G(2) = 0, b = (ɣδG)2 td and td = 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 sample1,6.

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 τm due to the restriction effect described by Mitra1,8.

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


Simulations were performed using MISST v0.939-11 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'(τm) values were compared with previously published results in yeast3.


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 τm 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 τm > 50 ms. The ADC'(τm) for spheres with smaller diameters (d ≤ 14 µm) reaches a constant plateau before τm = 40 ms.

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


The MR signal depends on the sphere diameter. The decrease in signal at ψ = 0 when increasing τm 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'(τm).

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


AXR experiments require short Δ (in order to avoid exchange during diffusion encoding) and long τm. 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'(τm) 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 τm (in the absence of exchange). Alternatively, an average of results for ψ = 0 and ψ = π could be used.


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].


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Fig 1. Effective diffusion gradient wave forms. The solid line corresponds to ψ = 0; gradient pulses shown by dashed lines represent ψ = π, where ψ is the angle between diffusion encodings.

Fig 2. (Left) Simulated diffusion-weighted MR signal of water-filled spheres vs. ψ (angle between wave vectors) for different values of mixing time, τm. (Right) Signal decay for the parallel case (ψ = 0) for different sphere diameters.

Fig 3. ADC'(τm) dependence for ψ = 0. Black dashed line: minimum τm achievable in a DDE-STE sequence with parameters in Table 1 (accounting for spoiler gradient pulses before, after and within the longitudinal storage period, and δf). Black stars: experimental results with yeast3 (permeable membranes expected). Colored symbols: simulated ADC'(τm) for impermeable spherical pores, exhibiting a τm dependence solely based on restriction. Yellow background marks the plot area where simulated ADC'(τm) graphs would be expected for spheres between 5 and 10 μm diameter, which is the expected cell diameter in baker's yeast.

Table 1. Parameters used for MISST simulations, as in Lasic et al.3

Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)