L. Tugan Muftuler^{1,2}, Daniel V. Olson^{3}, and Volkan Emre Arpinar^{2,4}

Mean Apparent Propagator (MAP) MRI provides a robust analytical framework to estimate the diffusion probability density function (PDF). Several scalar metrics are calculated from the PDF, which might better characterize tissue microstructure compared to conventional diffusion methods. The downside of MAP MRI is the long acquisition times of over an hour. In this study we developed a genetic algorithm (GA) to determine optimal q-space subsampling scheme for MAP MRI that will keep total scan time under 10 minutes, while preserving accuracy. Results show that the metrics derived from the optimized schemes match those from the full set closely.

**Methods:**

A MAP data was acquired from a healthy volunteer (male, 26YO)
and used for optimization. The study was approved by the IRB, and written
informed consents were obtained. The data was acquired with b = {1000, 2000,
3000, 4000, 5000, 6000} s/mm^{2} along 19, 32, 56, 87, 125, and 170
directions, respectively. Single-shot SE-EPI was used to acquire 20 slices with
2.2 x 2.2 x 2mm^{3} voxel resolution (TE=86.8ms, TR=4200ms).

MAP model fitting was performed using DIPy^{3}. Two separate
GA optimizations were performed for return-to-origin probability (RTOP) and
non-Gaussianity (NG). The resulting sampling schemes are denoted GA_{RTOP}
and GA_{NG}. Diffusion signals from 400 randomly selected voxels in the
white matter were used for optimization. 400 voxels were considered to be sufficient for this
purpose^{4}. There were 200 individuals in each population for the GA,
which consisted of 95 b-vectors chosen from the full set. This corresponds to
approximately a 10-minute acquisition. A population size of 200 provided sufficient
diversity to sample solution space^{5}. The GA evaluation function was
the mean squared error (MSE) with respect to the RTOP or NG calculations from
the full set. Selection was fitness-proportionate with Stochastic Uniform
Sampling, as well as sigma scaling to maintain selection pressure. Uniform
crossover was used to increase mixing. Elite survival rate, crossover rate, and
mutation rate were 0.02, 0.8, and 0.01, respectively.

For comparison, the reduced sampling scheme proposed by
Avram et al^{2} was also emulated. The required b-vectors were
subsampled from the full set to include the same number of directions per shell
as proposed.

Initial validations were performed using 1000 voxels in the white matter that were not used in optimization. For each of the reduced sampling schemes, MSE of the MAP metrics with respect to full sampling were computed. A second data set was acquired from a healthy volunteer (female, 26YO) for independent validation and generalizability of the optimal sampling schemes.

Both GA_{RTOP} and GA_{NG} schemes required
more samples from the b = 1000 and 6000 s/mm^{2} shells and fewer
directions on the middle shells as opposed to the monotonically increasing
directions in Avram’s scheme (Fig.1). This finding suggests that preserving
more samples on the innermost and outermost shells are important for accuracy
of MAP metrics.

Initial
validation with the first data showed that both GA schemes reduced the MSE for
all MAP metrics compared to the Avram scheme (Fig.2). GA_{NG} marginally
outperformed GA_{RTOP} in RTOP^{1/3} and had the best
performance for every non-Gaussianity metric.
The results
from the second experiment were similar to those of the first subject, which suggests
generalizability of the optimized schemes (Fig.3). The non-Gaussianity metrics
from the optimized sampling schemes were much closer to those of the full set. Improvements
in RTOP^{1/3}, RTAP^{1/2} and RTPP were more modest.

Qualitatively, each of the reduced sampling schemes
produced sufficiently accurate RTOP^{1/3} estimates (Fig.4). Slight
improvement can be seen in the white matter with the GA sampling schemes. The
GA_{NG} scheme most closely replicated the NG of the full set (Fig.5).
GA_{RTOP} was relatively accurate in the white matter but slightly
overestimated the gray matter NG. Overall, both GA_{RTOP} and GA_{NG}
performed better in highly organized fibers such as the splenium and genu,
producing values much closer to the full set.

1. Özarslan E, Koay CG, Shepherd TM, et al. Mean apparent propagator (MAP) MRI: A novel diffusion imaging method for mapping tissue microstructure. NeuroImage 2013;78:16–32.

2. Avram AV, Sarlls JE, Barnett AS, et al. Clinical feasibility of using mean apparent propagator (MAP) MRI to characterize brain tissue microstructure. NeuroImage 2016;127:422–34.

3. Garyfallidis E, Brett M, Amirbekian B, Rokem A, van der Walt S, Descoteaux M, et al. Dipy, a library for the analysis of diffusion MRI data. Front Neuroinformatics 2014;8:8.

4. Poot DHJ, den Dekker AJ, Achten E, Verhoye M, Sijbers J. Optimal experimental design for diffusion kurtosis imaging. IEEE Trans Med Imaging 2010;29:819–29.

5. Diaz-Gomez PA, Hougen D. Initial population for genetic algorithms: A metric approach. Proc. 2007 Int. Conf. Genet. Evol. Methods, Las Vegas, NV: 2007, p. 43–9.

Fig.1:
Distribution of b-values for three subsampling schemes. Avram scheme (blue) monotonically increases
the number of b-vectors per shell with increasing b-value. The samplings optimized for
RTOP^{1/3} (green) and NG (yellow) acquire more directions on the b=1000 s/mm^{2}
shell, fewer directions in the middle shells, and a high number on the outer
b=6000 s/mm^{2} shell.

Fig.2:
Mean-squared error of MAP metrics in white matter in initial validation.
MSE are relative to metrics calculated from
the full q-space sampling set. Diffusion signals were from 1000 random voxels
in the white matter (FA > 0.3) that were not used in optimization. GA
schemes reduced MSE substantially compared with the Avram scheme. The sampling optimized for
NG provided the best overall results.

Fig.3:
Mean-squared error of MAP metrics in white matter of second subject for
independent validation.
MSE are relative to metrics calculated from
the full q-space sampling set. Diffusion signals were from voxels in the white
matter (FA > 0.3). GA schemes reduced MSE substantially compared to the Avram scheme. The
sampling optimized for NG provided the best overall results.

Fig.4:
RTOP^{1/3} maps from in vivo validation from the second subject.
RTOP^{1/3} metric maps (top) and
absolute difference images (bottom) for each sampling scheme. Reduced sets
performed similarly. The errors were slightly higher with the Avram’s heuristic
subsampling scheme.

Fig.5:
NG maps from in vivo validation from the second subject.
NG maps (top) and difference images (bottom)
for each sampling scheme. GANG most closely matched the full set, especially
in the major white matter tracts. The major errors with Avram’s heuristic subsampling
scheme were in the genu and splenium of corpus callosum.