Morgan Mercredi^{1}, Sheryl L Herrera^{1}, Richard Buist^{2}, Kant Matsuda^{3}, and Melanie Martin^{1,4}

There is an increasing drive to use diffusion spectroscopy to infer the sizes of structures in samples. We present here the first use of the sine OGSE to infer the effective mean axon diameters in the human corpus callosum and study the effect on accuracy of reducing the number of images used in the inference. Aiming to reduce imaging times, this study examines how the number of frequencies or number of gradients affects accuracy and precision. We found that collecting OGSE data with two gradients gives a difference in results of less than 5% compared to six gradients.

Sample A portion of
normal-appearing corpus callosum from an autopsy human brain was obtained,
which did not demonstrate any pathological changes. The sample was placed in
agarose gel (2% w/v) within a 15 mL
sample tube. MRI Images were acquired with a 2.5 cm diameter RF bird
cage coil (Bruker Biospin), using a 7T Bruker Avance III NMR system (Paravision
5.0), with a BGA6 gradient insert (max strength: 430357 Hz/cm). 20 ms sine
gradient pulses were used ranging from n=1 to 15 sinusodal waves (15 OGSE
frequencies), with six gradient strengths used for each and separated by 24.52
ms. For n=1, g=(0, 22, 32, 39, 45, 50)% of g_{max}.
For n=2-15, g=(0, 44, 61, 76, 88, 99)% of g_{max}. To our knowledge these correspond to
the shortest effective diffusion times (0.5-7.5 ms) used on white matter. The following
imaging parameters were used: NA=2, FOV=2.56 cm^{2}, matrix 64^{2},
TR=1250 ms, TE=50 ms. Analysis The mean ± standard deviation of
the signal in the ROIs (Figure 1) was calculated. The signal was assumed to be
described by a two compartment model of the form,

$$ E(\omega=2\pi n/\sigma,g) = (1-f_{axon})e^{-bD_{h}} + f_{axon}e^{-\beta(D_{i},AxD)}$$

where *f*_{axon}
is the axon packing fraction, *D*_{i}
is the intra-axonal diffusion coefficient, *D*_{h}
is the hindered diffusion coefficient, and *AxD
*is the effective mean axon diameter^{8}. Signals were fitted to the
two compartment model using least squares minimization to extract *AxD*. Higher OGSE frequencies were then
removed and the remaining data was refit to the model to see how fitted
parameters changed to determine if certain images could be excluded from data
collection in clinical settings. Model fitting was also repeated using all
possible combinations of the gradient strengths.

Figure
2 shows the variation in fitted *AxD *as a function of number of frequencies. Between
7 and 15 frequencies, *AxD *are within 1% of each other. The highest and lowest
*AxD *occur when using 2 or 3 frequencies. The smallest fitted *AxD *is 1.9 ±
0.1
um (3 frequencies). The highest fitted *AxD *is 2.6 ± 0.2 µm
(2 frequencies). Compared to *AxD *with 15 frequencies, these are respective
differences of 17% and 3%. Error decreases when more frequencies are used.

Figure
3 shows variation in fitted *AxD *when using only two gradient strengths. With
the exception of the first gradient, fitted *AxD *are within 5% of each other. The
error also increases when smaller gradients are used, with fitted *AxD *values
ranging from 2.40 ± 0.08 µm with the highest gradient strength to 4 ± 6 µm with
the smallest gradient strength.

Figure 4 shows fitted *AxD* for the best combinations with each
number of gradients (these combinations are shown in Table 1). The fitted
values in Figure 4 are all within 3% of each other, while error when using just
two gradients is about 2 times larger than when using all six gradients.

Figure 1: Image of sample showing the
13 regions of interest. Analysis
ROIs were created in the corpus callosum (ROI #1, 3, 5, 7-9), ependymal layer
(ROI #2, 4, 6), cortex (ROI #10-12), and in the agarose (ROI #13).

Figure 2: Fitted *AxD *(µm) (± 95% confidence
bounds) as a function of number of frequencies.The mean and
standard deviation for *AxD *from 7 or more frequencies is 2.35 ± 0.01 µm.

Figure 3: Fitted *AxD *(µm) (± 95% confidence bounds) using
two gradient strengths, g_{0} = 0 and another chosen from between g_{1}
through g_{5}. The error in *AxD* decreases as the non-zero gradient
strength increases.

Figure 4: Best fitted *AxD *(µm) (± 95% confidence bounds)
(as determined from smallest fitted error) when
using 2, 3, 4, 5, or 6 gradient strengths for data fitting. The gradient
subsets are shown in Table 1. The *AxD *values were all consistent with each
other.

Table 1: The gradient combinations for 2, 3, 4, 5,
and 6 gradient strengths that produced the smallest error after fitting. Corresponding
*AxD *shown in Figure 4. Note that all combinations include higher gradient
strengths (g_{5}).