Methodological considerations on graph theoretical analysis of structural brain networks

Timo Roine^{1}, Ben Jeurissen^{1}, Daniele Perrone^{2}, Jan Aelterman^{2}, Wilfried Philips^{2}, Jan Sijbers^{1}, and Alexander Leemans^{3}

Diffusion-weighted magnetic
resonance imaging (DW-MRI) can be used to noninvasively probe brain
microstructure and connectivity. Recent advances that can deal with complex
fiber configurations^{1-2} have facilitated the investigation of
structural connectivity using fiber tractography methods^{3-6}. However,
the reproducibility of these analyses has not yet been sufficiently studied.
For example, the effect of spherical harmonics (SH) order on the
reproducibility has not been studied earlier, and only limited knowledge exists
about the reproducibility of the structural brain network properties in general^{7-10}.

Here, we studied the reproducibility of
whole-brain structural brain connectivity networks, i.e. connectomes, reconstructed
with constrained spherical deconvolution (CSD)^{1}. Graph theoretical
analysis was used to measure both global and local properties of these complex
networks^{11}. We investigated the reproducibility by calculating
intraclass correlation coefficients (ICC). We selected six network properties:
normalized characteristic path length (nCPL), normalized clustering coefficient
(nCC), normalized global efficiency (nGE), average local efficiency (LE),
betweenness centrality (BC), and small-worldness (SW), and five weights:
binary, number of streamlines, percentage of streamlines, streamline density,
and fractional anisotropy. In addition, the effect of two reconstruction
parameters was studied: spherical harmonics (SH) order was varied from four to ten and
reconstruction density, i.e. the number of streamlines, was varied from 10
million to 100 million. Moreover, correlations between the different network properties
and weights were computed.

**Material and preprocessing**

In addition to T1-weighted
data, we acquired DW-MRI data from 19 healthy subjects in 75 gradient
orientations with b=2800 s/mm^{2} and 2.5 mm isotropic voxel size. Non-DW MRI data were acquired in both forward and reverse
phase-encoding direction. Subject motion, eddy current and echo-planar imaging
induced distortions were corrected using FMRIB Software Library’s (FSL) TOPUP
and EDDY tools^{12-14}. Rigid coregistration was performed to align the
corrected DW data to T1-weighted data.

**Network reconstruction**

Cortical parcellation was performed in
Freesurfer using the Destrieux atlas^{15} and combined with subcortical
structures parcellated with FSL’s FIRST^{16}. Probabilistic streamlines
tractography was performed to reconstruct 10 and 100 million streamlines with CSD
using the iFOD2 algorithm as implemented in MRtrix3^{17-18}. In the tractography, anatomically
constrained tractography was used and streamlines were seeded from the gray
matter-white matter interface^{19}. The number of seed points per voxel
was constant and selected to produce approximately 10 million streamlines per
subject. The maximum spherical harmonics (SH) order was varied from four to
ten. The network reconstruction process is illustrated in Fig. 1.

**Reproducibility analyses**

To investigate reproducibility,
nine additional realizations were generated for each subject using residual
bootstrapping with 8th order SH decomposition^{2}. Then, ICC
was calculated as follows:

$$\frac{{\sigma_\text{inter}}^2}{{\sigma_\text{inter}}^2+{\sigma_\text{intra}}^2}$$

where $$${\sigma_\text{inter}}^2$$$ is the intersubject and $$${\sigma_\text{intra}}^2$$$ the intrasubject variance of the same network
metric.
We calculated the network properties using the
Brain Connectivity Toolbox^{11,20}. Networks were weighted with the
number and percentage of streamlines, streamline density^{21}, and fractional
anisotropy (FA). In addition, binary networks with varying threshold values
were studied.

**Reproducibility analyses**

The results showed that higher SH orders produced more reproducible results than lower SH orders (Fig. 2). In addition, increasing the reconstruction density from 10M to 100M further improved the reproducibility (Fig. 2). Unthresholded binary networks resulted in very irreproducible results for nGE and nCPL. The reproducibility was greatly improved by thresholding the networks with 1000 streamlines (Fig. 3).

**Correlation analyses**

The correlation analyses showed that nCC and SW were highly correlated, as were nGE and LE (Fig. 4). In addition, weighting by the number or percentage of streamlines produced highly correlated results (Fig. 5), although efficiency properties were negatively correlated.

Our
main finding was that SH order plays a significant role in the reproducibility
of structural connectomics. This is most likely caused by the wider peaks of
the fiber orientation distributions when estimated with lower SH orders. Thus,
the variation of the fiber orientations sampled by the tractography algorithm
is larger. We recommend to use at least a SH order eight whenever possible. Further
tests need to be performed to define the most reproducible way to analyze data
acquired with a low b-value or insufficient number of gradient directions. Other
approaches, such as residual bootstrapping based tractography^{2,22},
may result in better reproducibility for the low SH orders. For the binary
networks, thresholding was important especially for nCPL and nGE. A suitable
value, based on Fig. 3, could be 0.01% of the total number of streamlines.

1. Tournier JD, Calamante F, Connelly A. Robust determination of the fibre orientation distribution in diffusion MRI: non-negativity constrained super-resolved spherical deconvolution. NeuroImage. 2007;35(4):1459-1472.

2. Jeurissen B, Leemans A, Tournier JD, et al. Investigating the prevalence of complex fiber configurations in white matter tissue with diffusion magnetic resonance imaging. Hum Brain Mapp. 2013;34(11):2747-66.

3. Jeurissen B, Leemans A, Jones DK, et al. Probabilistic fiber tracking using the residual bootstrap with constrained spherical deconvolution. Hum Brain Mapp. 2011;32(3):461-79.

4. Tournier JD, Yeh CH, Calamante F, et al. Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data. NeuroImage. 2008;15;42(2):617-25.

5. Farquharson S, Tournier JD, Calamante F, et al. White matter fiber tractography: why we need to move beyond DTI. J Neurosurg. 2013;118(6):1367-77.

6. Kristo G, Leemans A, Raemaekers M, et al. Reliability of two clinically relevant fiber pathways reconstructed with constrained spherical deconvolution. Magn Reson Med. 2013;70(6):1544-56.

7. Bastiani M, Shah NJ, Goebel R, et al. Human cortical connectome reconstruction from diffusion weighted MRI: the effect of tractography algorithm. NeuroImage. 2012;62(3):1732-1749.

8. Owen JP, Ziv E, Bukshpun P, et al. Test–retest reliability of computational network measurements derived from the structural connectome of the human brain. Brain Connect. 2013;3(2):160-176.

9. Buchanan CR, Pernet CR, Gorgolewski KJ, et al. Test–retest reliability of structural brain networks from diffusion MRI. NeuroImage. 2014;86:231-243.

10. Smith RE, Tournier JD, Calamante F, et al. The effects of SIFT on the reproducibility and biological accuracy of the structural connectome. NeuroImage. 2015;104:253-265.

11. Bullmore E, Sporns O. Complex brain networks: graph theoretical analysis of structural and functional systems. Nat Rev Neurosci. 2009;10(3):186-98.

12. Andersson JL, Skare S, Ashburner J. How to correct susceptibility distortions in spin-echo echo-planar images: application to diffusion tensor imaging. NeuroImage. 2003;20(2):870-88.

13. Smith SM, Jenkinson M, Woolrich MW, et al. Advances in functional and structural MR image analysis and implementation as FSL. Neuroimage. 2004;23 Suppl 1:S208-19.

14. Andersson JL, Sotiropoulos SN. An integrated approach to correction for off-resonance effects and subject movement in diffusion MR imaging. NeuroImage. 2015. doi: 10.1016/j.neuroimage.2015.10.019.

15. Fischl B, van der Kouwe A, Destrieux C, et al. Automatically parcellating the human cerebral cortex. Cereb Cortex. 2004;14(1):11-22.

16. Patenaude B, Smith SM, Kennedy D, et al. A Bayesian Model of Shape and Appearance for Subcortical Brain. NeuroImage. 2011;56(3):907-922.

17. Tournier JD, Calamante F, Connelly A. Improved probabilistic streamlines tractography by 2nd order integration over fibre orientation distributions. In Proc Intl Soc Mag Reson Med. 2009;1670.

18. Tournier JD, Calamante F, Connelly A. MRtrix: diffusion tractography in crossing fiber regions. Int J Imaging Syst Technol. 2012;22(1), 53-66.

19. Smith RE, Tournier JD, Calamante F, et al. Anatomically-constrained tractography: improved diffusion MRI streamlines tractography through effective use of anatomical information. NeuroImage. 2012;62(3):1924-38.

20. Rubinov M Sporns O. Complex network measures of brain connectivity: uses and interpretations. NeuroImage. 2010;52(3):1059-69.

21. Hagmann P, Cammoun L, Gigandet X, et al. Mapping the structural core of human cerebral cortex. PLoS Biol. 2008;6(7):e159.

22. Leemans A, Jeurissen B, Sijbers J, et al. ExploreDTI: a graphical toolbox for processing, analyzing, and visualizing diffusion MR data. In Proc Intl Soc Mag Reson Med. 2009;3536.

Fig. 1. The pipeline for
structural brain network reconstruction. First, whole-brain probabilistic fiber tractography is performed (A), then cortical and subcortical gray matter is
parcellated (B) and then the network (C) is reconstructed by defining the nodes
based on B and the edges based on A.

Fig. 2.
Reproducibility of network properties for varying spherical harmonics (SH)
orders with reconstruction densities of 10 and 100 million streamlines. Networks
were weighted with the number of streamlines. nCC:normalized clustering
coefficient, nCPL:normalized characteristic path length, nGE:normalized
global efficiency, LE:average local efficiency, BC:betweenness centrality,
SW:small-worldness

Fig.
3. Reproducibility of network properties in binary networks reconstructed from
the number of streamlines using varying threshold values. Networks were
reconstructed with spherical harmonics (SH) order 8 and 10 million streamlines. nCC:normalized clustering coefficient, nCPL:normalized characteristic path length, nGE:normalized global efficiency, LE:average local efficiency, BC:betweenness centrality, SW:small-worldness

Fig. 4. Correlation of various
network properties to nCC, nGE and nCPL in networks weighted with the number of
streamlines, and reconstructed with 10 million streamlines using a spherical
harmonics order 8. nCC:normalized clustering coefficient, nCPL:normalized
characteristic path length, nGE:normalized global efficiency, LE:average
local efficiency, BC:betweenness centrality, SW:small-worldness

Fig.
5. Correlation of network properties using various network weights compared to
number of streamlines weighted networks. Networks were reconstructed with 10
million streamlines and a spherical harmonics order 8. nCC:normalized
clustering coefficient, nCPL:normalized characteristic path length, nGE:normalized global efficiency, LE:average local efficiency, BC:betweenness
centrality, SW:small-worldness

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

3437