Machine learning can boost the acquisition and estimation of anomalous diffusion model
Boyan Xu1, Yang Fan2, and Jia-Hong Gao1

1Center for MRI Research, Peking University, Beijing, China, 2MR Research China, GE Healthcare, Beijing, China


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.


In recent years, anomalous diffusion models have been proposed to address the phenomenon that diffusion-induced MRI signals deviate from the mono-exponential form1. Clinical feasibility studies have been conducted and those anomalous diffusion models have been found to outperform the conventional diffusion MRI (dMRI) method2–6. However, numerous diffusion weightings and complicated estimation procedures are required to obtain those models. It leads to long acquisition and estimation time. Recently, machine learning techniques have been applied in dMRI researches7,8. Inspired by these studies, here we propose a machine learning based framework to efficiently estimate anomalous diffusion parameters. Moreover, the most important diffusion weightings was identified with the feature selection technique9, which is useful to prune the extensive sets of diffusion weightings.


The fractional motion (FM) model, which is considered to be appropriate to describe the anomalous diffusion in biological tissues10, was selected for demonstration. The Noah exponent (α) and the Hurst exponent (H) characterize the FM model11,12. 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 here5,6. 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 signals13. The RF regressors were implemented in the scikit-learn toolkit and each regressor contained 200 trees with maximum depth determined during training14. 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 studies5,6. 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 (R2) 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.

Discussion and Conclusion

In this study, we applied random forest regression to accelerate the estimation of anomalous diffusion parameters. Compared with conventional model fitting method, the RF regressors can estimate the anomalous diffusion parameters faster and more accurately. Although, there are other machine learning methods (e.g. deep learning) can be applied to this issue, the random forest is proven to identify the most relevant signals, which is useful to reduce the extensive sets of diffusion weightings.


No acknowledgement found.


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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 (R2) 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.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)