Patricia Ulloa^{1}, Vincent Methot^{1}, and Martin A. Koch^{1}

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.

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 techniques^{1}. Recently, DDE has been used to estimate membrane permeability^{2} as well as apparent exchange rate (AXR), both on yeast^{3} and on humans in vivo^{4,5}.

In the AXR experiment^{3}, 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 coefficient^{3}, 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 experiment^{3} only uses parallel wave-vectors (ψ = 0). Computer simulations^{7} 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.

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 yeast^{3}.

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} mm^{2}/s,
which corresponds to 18% of ADC'(τ_{m} = 420 ms) observed in
yeast.

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.

AXR experiments require
short Δ (in order to avoid exchange during diffusion encoding) and
long τ_{m}. In this situation, the description given by
Mitra^{1} 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.

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].

[1] Mitra PP. Multiple wave-vector extensions of the NMR pulsed-field-gradient spin-echo diffusion measurement. Phys. Rev. B 51, 15074–15078 (1995). DOI: 10.1103/PhysRevB.51.15074.

[2] Åslund I, Nowacka A, Nilsson M, Topgaard D. Filter-exchange PGSE NMR determination of cell membrane permeability. J. Magn. Reson. 200, 291–295 (2009). DOI: http://dx.doi.org/10.1016/j.jmr.2009.07.015.

[3] Lasic S, Nilsson M, Lätt J, Ståhlberg F, Topgaard D. Apparent exchange rate mapping with diffusion MRI. Magn. Reson. Med. 66, 356–365 (2011). DOI: 10.1002/mrm.22782.

[4] Nilsson M, Lätt J, van Westen D, Brockstedt S, Lasic S, Ståhlberg F, Topgaard D. Noninvasive mapping of water diffusional exchange in the human brain using filter-exchange imaging. Magn. Reson. Med. 69, 1572–1580 (2013). DOI: 10.1002/mrm.24395.

[5] Lasic S, Oredsson S, Partridge SC, Saal LH, Topgaard D, Nilsson M, Bryskhe K. Apparent exchange rate for breast cancer characterization. NMR Biomed. 29, 631-639 (2016). DOI: 10.1002/nbm.3504.

[6] Shemesh N, Jespersen SN, Alexander DC, Cohen Y, Drobnjak I, Dyrby TB, Finsterbusch J, Koch MA, Kuder T, Laun F, Lawrenz M, Lundell H, Mitra PP, Nilsson M, Örzaslan E, Topgaard D, Westin CF. Conventions and nomenclature for double diffusion encoding NMR and MRI. Magn. Reson. Med., 75, 82–87 (2016). DOI: 10.1002/mrm.25901.

[7] Koch MA, Finsterbusch J. Numerical simulation of double-wave vector experiments investigating diffusion in randomly oriented ellipsoidal pores. Magn. Reson. Med. 62, 247-254 (2009). DOI: 10.1002/mrm.21976.

[8] Özarslan E, Basser PJ. Microscopic anisotropy revealed by NMR double pulsed field gradient experiments with arbitrary timing parameters. J. Chem. Phys. 128, 154511 (2008). DOI: 10.1063/1.2905765.

[9] Drobnjak I, Siow B, Alexader DC. Optimizing gradient waveforms for microstructure sensitivity in diffusion-weighted MR. J. Magn. Reson. 206, 41-51 (2010). DOI: 10.1016/j.jmr.2010.05.017.

[10] Drobnjak I, Zhang H, Hall MG, Alexader DC. The matrix formalism for generalised gradients with time-varying orientation in diffusion NMR. J. Magn. Reson. 210, 151-157 (2011). DOI: 10.1016/j.jmr.2011.02.022.

[11] Ianus A, Siow B, Drobnjak I, Zhang H, Alexander DC. Gaussian phase distribution approximations for oscillating gradient spin echo diffusion MRI. Magn. Reson. 227, 25-34 (2013). DOI: 10.1016/j.jmr.2012.11.021.

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, τ_{m}. (Right)
Signal
decay for the parallel case (ψ
= 0) for different sphere diameters.

Fig 3. ADC'(τ_{m}) dependence for ψ = 0. Black dashed line: minimum τ_{m} 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 δ_{f}). Black stars: experimental results with yeast^{3} (permeable membranes expected). Colored symbols: simulated ADC'(τ_{m}) for impermeable spherical pores, exhibiting a τ_{m} dependence solely based on restriction. Yellow background marks the plot area where simulated ADC'(τ_{m}) 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