## 1 Introduction

Double diffusion encoding, also known as double wave vector (DWV) or double-pulsed field-gradient spin-echo (d-PFG), is an extension of conventional diffusion-weighted magnetic resonance imaging (MRI). Here, two pairs of diffusion gradients are applied between excitation and acquisition, separated by a mixing time fi _{m} (see Figure 1). Using this technique, it is possible to acquire micro-structure information [1], such as size and shape of pores, for which no other non-invasive technique exists.

Previous experiments have estimated the cell (or pore) size using the signal difference between antiparallel and parallel diffusion gradient orientation in vivo [2]. However, these results, which assumed cylindrical pore shape, reported pore size estimates well above the size expected in axons in the corticospinal tract (CST). This discrepancy could be explained by a contribution of the extracellular space to the pore size estimate, which can be considerably larger than the intra-axonal compartment.

Measurement of the signal difference between parallel and perpendicular gradient orientations may gather further information about the pore shape [3–5]. In this paper, the DWV’s unique sensitivity to pore shape is used to acquire information on the contribution of the extracellular space to the DWV signal and hence also the pore size results.

## 2 Theory

A perpendicular cross section through densely packed circular cylinders consists of circles, whereas the space between the cylinders resembles an equilateral triangle (Figure 2 A and B). In this particular case we observe a high degree of symmetry of the compartments. Simulations have shown that such an arrangement will not induce a signal difference between parallel and perpendicular gradient orientations in DWV experiments [6]. However, when the cylinders are different in size and not too densely packed, the cross section contains circular pores and the space between the cylinders is irregularly shaped (Figure 2 C). This irregularity can be detected by DWV experiments comparing parallel and perpendicular diffusion gradient directions. Since the space inside the cylinders is regular, the contribution arising from the pores with circular cross section will not exhibit a parallel-perpendicular signal difference in a DWV experiment. Hence, it can be assumed that the parallel-perpendicular signal difference yields information about the shape of the extracellular compartments dominating the DWV pore size estimate in the CST.

The difference between parallel and perpendicular gradient orientations arises from the following reasoning.

In a voxel containing identical eccentric 2 D pores perpendicular to each other, the signal attenuation *E* = *S*/*S*_{0} for parallel and perpendicular gradient orientation is in general not equal

where *D*_{1} and *D*_{2} are the effective diffusion coefficients along the two perpendicular diffusion gradient directions involved and b = *y*^{2}*G*^{2}*δ*^{2}(*Δ* − *δ*/3). Eq. 1 is the basis for eccentricity estimation in DWV experiments.

However, in measurements of the CST, the plane containing the diffusion gradients is not perfectly perpendicular to the fibres. In this case, the cross section of a circular cylinder will be represented as a ellipse (Figure 2 D and E). This eccentric shape also leads to differences between parallel and perpendicular gradient direction in the DWV signal [7]. Nevertheless, the difference arising from this situation can be removed by specific averaging of the signal for different gradient vectors. In a scenario where a voxel contains only a single orientation of eccentric 2 D pores,

can occur. Rotating all gradients by *π*/2 will then yield

Calculating the geometric mean of the signals given in equations (2) and (3), the signal difference between parallel and perpendicular gradient orientations should vanish. This reasoning would apply for both elliptic and tilted circular cylinders. If the difference remains, however, it can be inferred that the signal arises from either eccentric cylinders with more than one orientation or circular cylinders with more than one direction of inclination. Considering the homogeneous structure of the CST, different inclinations in a single voxel are unlikely. It can thus be concluded that the space dominating the signal is irregularly shaped. This would point to the extracellular space as the origin of the DWV signal.

## 3 Methods

Double wave vector experiments were performed on seven healthy volunteers, with no known history of neurological disease. Written informed consent was given prior to the scan. Experiments were performed with a clinical whole body system operating at 3 T magnetic field strength (Ingenia, Philips, Amsterdam), using an 8-channel head coil array (receive only). An in-house implementation of a spin-echo echo planar imaging (EPI) sequence that incorporates two diffusion encoding periods was used (Figure 1). Both refocusing pulses were slice selective and crusher gradients were positioned immediately before and after the refocusing pulses.

Diffusion gradients were applied in the transversal plane (xy) perpendicular to the subjects’ body axes (z) such that the x and y components are equal. Thus, sixteen diffusion gradient directions were used as shown in Figure 3.

Twenty transversal slices of 3 mm thickness were acquired using 3 x 3 mm^{2} nominal in-plane resolution, fat suppression using Spectral Presaturation with Inversion Recovery (SPIR), repetition time TR = 4.4 s for the first volunteer and TR = 6.5 s for the other six, echo time TE = 180 ms for the first six volunteers and TE = 200 ms on volunteer number 7 to allow variation of mixing time. Gradient duration = 10 ms, gradient amplitudes **G** = 44 mT m^{−1}, diffusion time *Δ* = 62 ms, gradient rise time *t _{r}* = 900

*µ*s, mixing time

*τ*=

_{m}*δ*+

*t*for all seven volunteers, corresponding to a total diffusion weighting of

_{r}*b*= 2 x 812 s mm

^{−2}and 15 repetitions. An extra experiment was performed on volunteer number 7 using

*τ*=

_{m}*δ*+15 ms, so that a decrease of the modulation amplitude is expected.

_{r}In addition, T1-weighted and diffusion tensor images (b = 800 s mm^{−2} and 32 gradient directions) were acquired for radiology control and fibre direction estimation, respectively. The acquisition time for the DWV experiment measurement was approximately 30 min, total measurement was not longer than 2 hours.

The pore sizes were calculated using the mean squared radius of gyration 〈*R*^{2}〉

which allows comparison of in vivo results with [2]. For a region-of-interest (ROI) analysis both left and right corticospinal tracts (CST) were delineated manually from a thresholded diffusion-weighted image.

The images were realigned to the first non-diffusion weighted image (b = 0 s mm^{−2}) to correct for subject motion, using the Diffusion toolbox extension (http://sourceforge.net/projects/spmtools) for SPM8 in Matlab R2012b (The MathWorks, Inc., Natick, Massachusetts), and then averaged over all repetitions and directions independently. First, the geometric mean is calculated over equivalent gradient orientations (*θ* = 0, *π*/2, *π* and 3*π*/2) separately. Then, the difference between parallel and perpendicular gradient orientation is given by

## 4 Results

After calculating the geometric mean over the four different orientations of the gradient vectors in the laboratory system (i.e. over a column in Figure 3), signal intensities from parallel orientations are significantly larger than those from perpendicular ones (*p* < 0.05, Wilcoxon signed-rank test). The differences between parallel and perpendicular diffusion gradient orientations yielded positive results ((*S*(*q*, 0) − *S*(*q*, *π*/2)) > 0) in the ROI covering both CSTs. Figure 4 shows a coloured map of this difference in volunteer no. 2. Table 1 shows the mean difference between parallel and perpendicular gradient orientations for all seven volunteers.

Parallel-perpendicular signal attenuation difference for all volunteers. Mean over the ROIs covering both CSTs ± standard deviation in the same region. Experiment using TE = 200 ms for volunteer 7.

Subject | Parallel - perpendicular |
---|---|

1 | 0.0403 ± 0.013 |

2 | 0.0469 ± 0.020 |

3 | 0.0371 ± 0.013 |

4 | 0.0474 ±0.017 |

5 | 0.0466 ±0.016 |

6 | 0.0403 ± 0.013 |

(7) | 0.0427 ±0.021 |

mean | 0.0430 ±0.016 |

Experiment on volunteer no. 7, where a longer *τ _{m}* = 25.9 ms was used did not a show significant difference with the acquisition at short

*τ*= 10.9 ms (

_{m}*p*< 0.05, Wilcoxon rank-sum test)

The mean 〈*R*^{2}〉_{est} derived from the antiparallel-parallel signal difference in vivo was (4.11 ± 14.5 %)*µ*m^{2}, which corresponds to an average pore diameter of 2r_{av} ≈ 5.23 *µ*m for ellipsoidal pores. These results are in the same order of magnitude as in previous published results [2].

## 5 Discussion and conclusion

Here, we proposed the geometric mean between four different gradient vectors in the laboratory coordinate system as a method to obtain the signal difference between parallel and perpendicular gradient orientations in a DVW experiment. We obtained consistent results in seven volunteers showing that this signal difference bears information on the shape of the extracellular compartments dominating the DWV pore size estimate in the CST.

In one volunteer, two experiments using different mixing times have been performed. The difference between parallal and antiparallel orientations should vanish for long *τ _{m}* if it is due to restriction effects. Here, no signifi-cant difference was found between short

*τ*and long

_{m}*τ*acquisition. However, this may be explained by the chosen mixing time, that might be not long enough to detect a difference between experiments.

_{m}In conclusion, it has been shown that the difference between parallel and perpendicular DWV diffusion-weighted signals yields positive results in vivo in the CST. This can be attributed to an irregularly shaped compartment contributing considerably to the DWV diffusion-weighted signal.

This new technique may become very helpful in the field of neuroscience. Investigating subtle changes in the pore size and shape in white matter, the characterization of brain development and disease would be improved.

P.U. was supported by the Graduate School for Computing in Medicine and Life Sciences funded by Germany’s Excellence Initiative [DFG GSC 235/1].

## Funding

German Research Foundation, grant KO3389/2-1.

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Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.

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