Introduction

There is an increasing drive to use diffusion spectroscopy to infer the sizes of structures in samples^1-4. Most methods use pulsed gradient spin echo sequences^2,3 which cannot provide short enough diffusion times to probe very small structures. Using oscillating gradient spin echo sequences (OGSE)^5-7 a diffusion spectrum can be obtained which probes the “short-time” or high frequency regime allowing for small structures to be measured. Here we use the apodised cosine OGSE⁷ to infer the surface to volume ratio of a collection of tubes, and thus the tube diameter.

Methods

MRI The sample was imaged using a 7 T Bruker Avance III NMR system with Paravision 5.0 with a BGA6 gradient set with a maximum gradient strength of 1.01 T/m and a 3.5 cm diameter bird cage RF coil (Bruker Biospin). The number of sinusoidal waves in each 40 ms apodised cosine⁶ gradient pulse ranged from n = 1 to 20, in steps of 1. Five gradient strengths were used for each frequency and the gradient pulses were separated by 44.52 ms. The b-values were 0, 60, 120, 180, 240 s/mm². Tubes Capillary tubing (Polymicro Technologies), were used with inner and outer diameters of 5 µm and 151 µm, respectively. Tubing was filled with filtered water, injected using a modified syringe⁸. Tubing was cut into 2 cm pieces (~ 500 in total), packed within a 1.5 mL plastic microcentrifuge tube (Cole Parmer), as shown in Figure 1. The volume of water within the tubes was not large enough to produce a measurable signal. Thus the tubes were soaked in filtered water, before being placed into a bird cage RF coil. Two 750 µm inner diameter tubes filled with filtered water were placed near the microcentrifuge tube and used as a control. A 10 mm thick (2.0 cm)² slice perpendicular to the tubes was chosen. A 64 x 64 matrix was used for 312.5 μm in-plane resolution. Analysis Analysis ROIs were created in the water tubes (ROI #2), within the microcentrifuge tube (ROI #3-6), and in the noise (ROI #1) as shown in Figure 2. The mean $$$\pm$$$ standard deviation of the signal in the ROIs was calculated. The log of the signal versus b-value was fitted to a straight line and the negative of the slope was used as the ADC for each measurement. The mean $$$\pm$$$ standard deviation of the ADC for each frequency was calculated and used in the fit. Monte Carlo Simulation A Monte Carlo computer simulation (N = 57344) was conducted using a hexagonal array of 150 μm diameter cylinders (packing fraction = 0.8) to model the actual tube samples. The diffusion coefficient for water in the interstitial space was 2.0 μm²/ms with no water in the cylinders. All barriers had zero permeability and the time step was 1 μs. Simulations used the same pulse parameters as those in the experiment. 1% Gaussian noise was added to the data. Tube fit The diffusion spectrum D($$$\omega$$$) for the apodised cosine sequence data were fitted to the short-time surface to volume model described in Ref. [9]. The diffusion spectrum at short diffusion times ($$$\Delta$$$ ~ 1/$$$\omega$$$) is $$$D(\omega) = D_0(1 - c(n)c_d(S/V)\sqrt{D_0/\omega})$$$ where D₀ is the free diffusion coefficient, and c(n) and c_d are numerical factors. Both D₀ and the surface to volume ratio were extracted from the fit. We assumed that the tubes were tightly packed in a hexagonal lattice. In a tightly packed hexagonal lattice of cylinders, the surface to volume ratio is related to the cylinder diameter d through the relation $$$d = 4f/[(1-f)(S/V)]$$$, where f is the packing fraction of the tubes. This relation was used to relate the surface to volume ratio to the tube diameter.

Results

ADC vs frequency for both the simulation and a representative ROI (#4 in Figure 1) is plotted in Figure 3. The fits are shown with solid lines. For the simulation, the fitted surface to volume ratio was 0.10 ± 0.02 µm^-1 corresponding to diameters of 160 ± 20 µm. For the real data, the surface to volume ratio was found to be 0.09 ± 0.01 µm^-1. Assuming 80 percent hexagonal packing, this corresponds to tube diameters of 180 ± 30 µm.

Discussion and Conclusion

This work provides experimental evidence for using apodised cosine OGSE to infer the size of small structures. This work lays the foundation for inferring the size of tissue structures, such as axon diameters, in samples using MRI. Future studies will assess optimizing the frequency ranges and gradient strengths for the expected compartment sizes.

Acknowledgements

The authors would like to thank funding from NSERC, CFI and MRIF, and D. Goultz and D. Craig for assistance filling the tubes.