Boyan Xu^{1}, Yang Fan^{2}, and Jia-Hong Gao^{1}

The departure from mono-exponential decay of the diffusion-induced signal loss has promoted the research of anomalous diffusion in MRI. It has been found that anomalous diffusion models offer substantial advantages over the conventional method in clinical applications. However, these models require more diffusion weightings for complicated estimation procedure, which prevents its further application. In this study, we demonstrated that machine learning can be applied to accelerate the estimation of anomalous diffusion parameters. Furthermore, feature selection was used to identify the most relevant signals, thus helping to reduce the extensive sets of diffusion weightings.

The fractional motion (FM) model, which is considered to be appropriate
to describe the anomalous diffusion in biological tissues^{10}, was selected for demonstration. The Noah exponent
(*α*) and the Hurst exponent (*H*) characterize the FM model^{11,12}. The FM-based dMRI signal with Stejskal-Tanner
(S-T) pulse gradients can be written as

$$S/S_0=\exp(-\eta \cdot D_{\alpha,H}\cdot\gamma^{\alpha}G^{\alpha}\Delta^{\alpha+\alpha H})$$

where $$$D_{\alpha,H}$$$ is the generalized diffusion coefficient, *γ* is the gyromagnetic radio, *G* is the diffusion gradient amplitude, and *Δ* is the diffusion gradient separation time. *η* is a dimensionless number that can be calculated
with *α*, *H*, *Δ*, and the diffusion
gradient duration (*δ*). The acquisition protocol in previous studies was
evaluated here^{5,6}. Details of the 18
non-zero diffusion weightings can be found in Table 1.

Random
forest (RF) regression was used to learn the mapping between the anomalous
diffusion parameters and the dMRI signals^{13}. The RF regressors were implemented in the
scikit-learn toolkit and each regressor contained 200 trees with maximum depth determined
during training^{14}. To train and validate the regressors, diffusion
signals from 60000 voxels were simulated, with FM-related parameters randomly
selected in the ranges: $$$\alpha\in(1,2]$$$, $$$H\in(0,1)$$$. $$$D_{\alpha,H}$$$ was drawn from a log-norm distribution ($$$\mu=-5.70, \sigma=1.15$$$) to mimic the results in
previous studies^{5,6}. Rician noise was then added and five sets of
signals were generated: noise-free, SNR=50, SNR=40, SNR=30, and SNR=20 (for the
b=0 signals). The RF regressors were trained on 48000 voxels and the remaining 12000
were used for testing. Model fitting was also performed on the test dataset for
comparison.

Figure
1 shows the scatter plots of the Noah exponent *α*
and Hurst exponent *H* against the values calculated by fitting and predicted
by RF. Coefficient
of determination (R^{2}) was used here to measure how well the
estimated values approximate the ground truth. Although there is a one-to-one
correspondence between the fitted and ground truth values when data is
noise-free, The RF outperformed the model fitting method when there is noise.
It should be noted that the fitting method sometimes produced the boundary
values rather than the close values to ground truth, while the RF performed
well in all situations. The computation time is summarized in Table 2. The RF
method completed the estimation much faster than traditional fitting.

Figure 2 illustrates the feature importance in the RF regressors for the 18 diffusion weightings. Only a minority of weightings are decisive. Therefore, irrelevant weightings can be abandoned, which is very helpful to reduce the acquisition time. As Figure 3 indicated, the RF regressors which were constructed on the most important 6 signals performed similarly to those based on all signals.

1. De Santis S, Gabrielli A, Palombo M, et al. Non-Gaussian diffusion imaging: a brief practical review. Magn Reson Imaging 2011;29:1410–6.

2. Kwee TC, Galbán CJ, Tsien C, et al. Intravoxel water diffusion heterogeneity imaging of human high-grade gliomas. NMR Biomed 2010;23:179–87.

3. Sui Y, Xiong Y, Jiang J, et al. Differentiation of low-and high-grade gliomas using high b-value diffusion imaging with a non-Gaussian diffusion model. Am J Neuroradiol 2016;37:1643–1649.

4. Karaman MM, Sui Y, Wang H, et al. Differentiating low- and high-grade pediatric brain tumors using a continuous-time random-walk diffusion model at high b-values. Magn Reson Med 2016;76:1149–57.

5. Xu B, Su L, Wang Z, et al. Anomalous diffusion in cerebral glioma assessed using a fractional motion model. Magn Reson Med 2017;78:1944–9.

6. Xu B, Su L, Wang Z, et al. Anisotropy of anomalous diffusion improves the accuracy of differentiating low- and high-grade cerebral gliomas. Magn Reson Imaging 2018;51:14–9.

7. Golkov V, Dosovitskiy A, Sperl JI, et al. q-Space Deep Learning: Twelve-Fold Shorter and Model-Free Diffusion MRI Scans. IEEE Trans Med Imaging 2016;35:1344–51.

8. Nedjati-Gilani GL, Schneider T, Hall MG, et al. Machine learning based compartment models with permeability for white matter microstructure imaging. NeuroImage 2017;150:119–35.

9. Guyon I, Elisseeff A. An introduction to variable and feature selection. J Mach Learn Res 2003;3:1157–1182.

10. Fan Y, Gao J-H. Fractional motion model for characterization of anomalous diffusion from NMR signals. Phys Rev E 2015;92:012707.

11. Eliazar II, Shlesinger MF. Fractional motions. Phys Rep 2013;527:101–29.

12. Burnecki K, Weron A. Fractional L\’evy stable motion can model subdiffusive dynamics. Phys Rev E 2010;82:021130.

13. Breiman L. Random forests. Mach Learn 2001;45:5–32.14. Pedregosa F, Varoquaux G, Gramfort A, et al. Scikit-learn: Machine Learning in Python. J Mach Learn Res 2011;12:2825–30.

Table 1. Diffusion
weighting parameters for the simulation.

Figure 1. Scatter plots of ground truth versus the fitted and the predicted values. Both model fitting and RF prediction were performed on the test set. Coefficient of determination (R^{2}) was used here to measure how well the estimated values approximate the ground truth.

Table 2. The computation
time on different dataset. Both model fitting and RF prediction were performed
on the test set (n=12000) while RF regressors were trained on the training set
(n=48000).

Figure 2. The feature importance in the RF regressors for
the 18 diffusion weightings. Each diffusion weightings whose importance is above
the average is marked as red.

Figure 3. The performance of
the RF regressors constructed on different subsets of diffusion weightings. According
to the feature importance, diffusion weightings were abandoned one after the other and new
RF regressors were trained and tested based on the remaining dMRI signals. The
performance maintained well even if 12 weightings were reduced.