Alberto De Luca^{1,2}, Filippo Arrigoni^{2}, Alessandra Bertoldo^{1}, and Martijn Froeling^{3}

Pseudo continuous description of the
diffusion MRI (dMRI) signal through multi-compartment deconvolution is a promising technique to disentangle
different water pools in the brain. In this work we verified whether a
deconvolution based approach with L_{2} regularized priors could improve the
repeatability of DTI metrics computed on the brain data of 3 volunteers
acquired twice. Signal fractions of free water and perfusion could reliably be quantified
and removed from the diffusion signal, improving the repeatability of MD
estimation both in gray and white matter.

A dictionary of 300 mono-exponential
Gaussian decaying signals with diffusion coefficients log-spaced in the range
[0,1000]µm^{2}/ms was defined. Membrane restrictions essentially cause
the dMRI signal not to decay at strong diffusion weightings, therefore their
effect can be modeled with the addition of a constant column to the dictionary,
as shown in Figure 1. Deconvolution is inherently noisy, thus we developed a three
stage solution. Firstly, L_{2} regularized Non-Negative Least Squares
(NNLS) were applied to obtain the voxel-wise deconvolution spectra of each
subject. These were averaged and used to define a deconvolution prior(χ_{0}). The second deconvolution step was
performed minimizing voxel-wise the following terms:

$$min||Uχ-S||_2^2+||γχ-χ_0||_2^2$$

$$Uχ≥0$$

where χ is the diffusion spectra, U the deconvolution dictionary, S the dMRI signal and γ a
regularization term. L_{2} regularization may bias the estimation of
the signal amplitudes, thus as last step the diffusion dictionary was voxel-wise
reduced to its minimal components (non-zero diffusion coefficients in the
previous step), then the final NNLS deconvolution was performed. The voxel-wise
diffusion spectra were integrated in specific diffusion ranges to obtain
fractional maps: IVIM for diffusion values in the range [6,500]µm^{2}/ms,
Free Water
(FW) in the range [2.5,6]µm^{2}/ms
and non-Gaussian (NG) in the range [0,0.4] µm^{2}/ms. 3 healthy
controls (HC, 1 male, 27±1 years) were acquired twice at 3T with 7 days
inter-scan. The acquisition protocol included a T_{1W} scan (1mm^{3}
resolution, TE/TR=3.7/8.1ms), a T_{2W} scan (1.5x1.5x1.5mm^{3}
resolution, TE/TR=0.1/4.2s) and a dMRI sequence (2.5x2.5x2.5mm^{3}
resolution, TE/TR=80ms/6.9s). dMRI data was pre-processed with Tortoise[2] using the T_{2W} as reference. Mean
diffusivity was derived from the linear DTI fit of the data at b=5,1000s/mm^{2}
before(MD) and after(MD_{C}) subtraction of the FW and IVIM components
(referred as corrected DTI). T_{1W} data was segmented[3], [4], then boxplots of MD/MD_{C},
NG, FW and IVIM were computed individually within the GM and WM of each
time-point. Repeatability of MD/ MD_{C} was assessed with Intra-Class
Correlation(ICC).

[1] B. Madler, D. R. Hadizadeh, and J. Gieseke, “Assessment of a Continuous Multi-Compartmental Intra-Voxel Incoherent Motion ( IVIM ) Model for the Human Brain,” in International Society for Magnetic Resonance in Medicine, 2013.

[2] C. Pierpaoli, L. Walker, M. O. Irfanoglu, A. S. Barnett, P. J. Basser, L.-C. Chang, C. G. Koay, S. Pajevic, G. K. Rohde, J. Sarlls, and M. T. Wu, “TORTOISE: an integrated software package for processing of diffusion MRI data,” ISMRM 18th Annu. Meet., p. 1597, 2010.

[3] S. M. Smith, “Fast robust automated brain extraction.,” Hum. Brain Mapp., vol. 17, no. 3, pp. 143–55, Nov. 2002.

[4] Y. Zhang, M. Brady, and S. Smith, “Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm.,” IEEE Trans. Med. Imaging, vol. 20, no. 1, pp. 45–57, Jan. 2001.

Figure 1: the
deconvolution dictionary used with NNLS for the dMRI signal decomposition. The
dictionary was built with 300 mono-exponential Gaussian decaying signals with
log-spaced coefficients plus a constant column, shown by the constant yellow
line with value 1.

Figure 2: the
average deconvolution spectra obtained after voxel-wise L_{2}
regularized NNLS of each subject. 8 peaks were observed (one corresponding to
zero diffusion is not shown in the plot), and used to define the population
priors for the subsequent steps.

Figure 3: an
axial slice of the two time-points of each of the three subjects. The first two
rows represent uncorrected (first row) and corrected (second row) MD maps. The
last three rows show example of the fractional maps obtained by integration of
the deconvolution spectra. MD_{C} maps are lower valued and flatter
compared to uncorrected MD. IVIM and FW assumed higher values close to GM and
CSF, while NG was higher in WM.

Figure 4: 25, 50
and 75 percentile of MD/MD_C (first row), NG (middle row),
FW and IVIM (last row) for each time-point. MD was characterized by
higher values in GM than WM. MD_C was still higher in GM than WM, but
the difference was smaller, as was the dispersion around the median value. NG
signal fractions were higher in WM than GM, with median value of 7.6% compared
to 2.3%. FW and IVIM were higher in GM (IVIM up to 5%, FW up to 20%) than WM
(IVIM up to 3%, FW up to 5%).