Introduction

The permeability of cell membranes is an important biological parameter, which might be changed in pathologies. Recently, apparent exchange rate (AXR) mapping has been introduced, which might allow gaining access to this quantity non-invasively^1,2. However, AXR is not only related to membrane permeability but also to the packing density of cells so that the contribution of these parameters might be unclear. In this study, we investigated the influence of the packing density and the permeability on the AXR using theoretical analysis, simulations and measurements with yeast cells.

Methods

Experiments were carried out on a BRUKER 600 MHz spectrometer. A schematic representation of the used AXR sequence can be found in Figure 1. Yeast suspension phantoms were produced using fresh baker’s yeast which was dissolved in tap water in 3/1, 2/1, 1/1 and 1/2 water/yeast weight ratios. The measurements were performed at 280 K in order to prevent the yeast from budding and keep the ratio stable. To evaluate the AXR for each sample, apparent diffusion coefficient (ADC) values for different mixing times t_m (Fig. 1) ranging from 28 ms to 491 ms were determined. For each mixing time, five b-values ranging from 37 s/mm² to 930 s/mm² were acquired. Gradient duration for the filter block was set to δ_F = 7.3 ms with an echo time of T_F = 21.1 ms while the diffusion weighting was set to δ_D = 7.8 ms with T_D = 22.4 ms. The time between the onsets of the filter and diffusion gradients was Δ_F = 9.7 ms and Δ_D = 10.2 ms, respectively. The b-value of the filter was set to 1989 s/mm². A spoiler gradient with G_S = 0.07 T/m and a duration of 5.3 ms was used. The gradients to choose the right echo path were run with G_C = 0.05 T/m for a duration of 1.3 ms. The used fit function yielding the AXR values is $$ADC = ADC_{\mathrm{eq}} (1-\sigma \exp(-AXR\,t_m))\quad(1)$$ where ADC_eq is the equilibrium ADC observable in the limit of very long t_m and σ the filter efficiency.
Simulations were carried out using a GPU accelerated Monte Carlo simulation approach³. Yeast cells were represented as ellipsoids with the semi-axes a = 2.725 µm and b = c = 1.875 µm, which was roughly estimated from microscopic images of the used yeast suspensions. The ellipsoids were randomly distributed within a volume of (100 µm)³. Diffusion coefficients were set to D_i = 0.625 µm²/ms for intracellular space and D_e = 1.25 µm²/ms for the extracellular fraction. The step size of the simulation was set to dt = 1 µs, the number of simulated particles was set to 75000 to roughly match the SNR of the measurements. For each packing density, 4 averages with 8 mixing times were simulated with the membrane permeabilities ρ = 0.007 µm/ms, ρ = 0.005 µm/ms and ρ = 0.003 µm/ms. Packing densities characterized by the extracellular volume fraction f_e ranged from f_e = 0.93 to f_e = 0.71, which roughly matches the yeast samples, assuming an amount of 6.2*10⁹ cells per gram baker’s yeast. The theoretical reference AXR values were calculated using $$AXR_{\mathrm{reference}}=\rho\frac{S_{\mathrm{cell}}}{V_{\mathrm{cell}}}\frac{1}{f_{e}} \quad (2) $$ where S_cell is the surface and V_cell the volume of the cell and $$$f_{e}=(V_{\mathrm{total}}-V_{\mathrm{intracellular}})/V_{\mathrm{total}}$$$.

Results

The obtained experimental ADC curves for each considered weight ratio are shown in Figure 2, the obtained AXR values in Table 1. As expected, the ADC value is decreasing with increasing amount of yeast which is corresponding to a smaller extracellular fraction f_e. For low yeast concentrations, only very small ADC changes occur making AXR determination unreliable for the lowest yeast concentration (3/1 in Fig. 2). The AXR values for the remaining three ratios do not exhibit significant differences within the 2 sigma confidence interval. Simulated and experimental AXR values with the corresponding theoretical values obtained from Eq. (2) are shown in Figure 3. The measured AXR values match the order of magnitude obtained from theory and simulations with very rough assumptions on packaging density and permeability. For low concentration, i.e. high f_e, the dependence of AXR on f_e is rather weak while the slope of the curve is considerably higher for small f_e. In accordance with the experiment, the simulated values for the lower yeast concentrations also depict large errors and discrepancies from the theoretical values.

Discussion and Conclusion

It could be shown that the AXR is mainly influenced by permeability changes for low packaging density of cells, while changes in the cell concentration lead to moderate variation, which was also observed for the cell experiments. In this experimental range, it could be possible to relate AXR changes mainly to permeability changes, while, for low f_e, a high sensitivity to volume fraction changes is to be expected which might be useful to detect e.g. tumors implying changes in f_e. Limitations include the very rough estimation of f_e in the yeast samples with basic assumptions, which might be resolved by flow cytometry. Further, especially for small t_m, undesired coherence pathways may disturb the signal. The very small ADC changes for low concentrations may lead to unreliable AXR measurements requiring very high SNR.

Acknowledgements

Financial support by the DFG (Grant No. KU 3362/1-1) is gratefully acknowledged.