## 4 Discussion  
The aim of the present work was to explore if the fractal analysis could be used to identify benign nodules from malignant ones. The fractal dimensions for the two kinds of thyroid nodules were estimated based on fractal Brownian motion. The calculation results showed that the fractal dimensions of the benign nodules are generally higher than those of malignant ones (Figure 5). A similar phenomenon was observed by [PERSON] et al. [29]. Significant difference was noticed in fractal dimension between B-mode images of the malignant and benign thyroid nodules with an independent nonparametric Mann-Whitney U test (\(p<0.05\)) (Figure 5). Fractal analysis was performed to classifying breast US images and other organs and has shown promising results [27; 28; 29]. Our experimental results proved that the fractal analysis could be used as a useful tool to extract the texture properties of thyroid nodules.  
The AUC value in Figure 6 was 0.792, and corresponding specificity and sensitivity were 86.9% and 64.9%, respectively. Clearly, the single-parameter FD cannot be claimed as a wonderful diagnostic tool to assess thyroid nodules. Therefore, a combination of other features (e.g., texture, morphology, and so on) should be helpful for improving the classification accuracy and also be more adaptable for clinical diagnoses of different types of tissues. For instance, [PERSON] et al. reported that an accuracy higher than 98% was achieved for 20 patients with the combination of texture features with different classifiers [39]. More efforts will be made in the future to explore the combination of FD with other features, and with the help of multi-feature extraction methods.  
In this work, the size of nodules ranged between 4.5 \(\times\) 2 mm\({}^{2}\) to 25 \(\times\) 15 mm\({}^{2}\), and more than 50% of them were small nodules (<10 mm). Compared with the conventional US diagnostic method and elastography [20; 21], the fractal analysis was helpful to determine whether or not a small nodule is suspicious for malignancy. Therefore, fractal dimension might be used to improve the classification performance of existing technology.  
Of course, there are some limitations in the current work. First, since the ROI of thyroid nodule was determined by two senior sonographers, the results might have been affected by their subjective assessment. Thus, in future studies, computer-aided adaptive algorithms are needed to automatically select the ROI with greater objectivity. Second, all the US images studied in this work were obtained from daily diagnosis, without specific conditions set for the patients. The signal-to-noise ratio (SNR) of US images may vary between individual patients. In this case, referring to the methods introduced in a previous work [29], morphology and histogram equalization were applied to achieve noise filtering, shape simplification, and gray level standardization. However, these imaging processing methods were still too general to deal with complicated issues. Therefore, better noise reduction and US image enhancement should be explored in our future work. Meanwhile, since the size of the thyroid nodules varied in a broad range, here we only calculated the FD globally. In our future work, more efforts will be made to explore an appropriate method to estimate D locally. Moreover, there are many other  
Figure 6: ROC analysis was calculated by MATLAB program and AUC was 0.792.  
models that exist to evaluate the FD, such as box counting, fractal interpolation function system, and area measurement [28; 42; 43], while only the fractional Brownian motion model was employed in the present work. In the future, more assessment methods should be tested and compared to find an optimal model and parameters for a typical case, so that better performance of FD analyses can be achieved in classifying benign and malignant lesions in different organs.