# Analysis of Random Dynamics of Cell Segmented by a Modified Active Contour Method
## 3 Results & Discussion
### Analysis of Cellular Random Motion by the Image Analysis Methods  
By comparing the three methods, we concluded that the effect of localization error on particle position should be considered when the image of the cell is analyzed to localize the cellular position. Thus, to compare the effects of localization errors originating from the three image analysis methods, the trajectory of the MCF-7 cells and the MSD and VACF derived from that trajectory are depicted in Figure 4. Figure 4a-c are the trajectories of the MCF-7 cells (N = 36) analyzed by the radial symmetry method, circular Hough transform method, and modified active contour method, respectively. Comparing the three trajectories shows that, even though the trajectory computed by the radial symmetry method (Figure 4a) produces relatively decent trajectory results compared to that derived by the circular Hough transform method (Figure 4b), it still possesses a relatively large amount of localization error in comparison to the modified active contour method presented in Figure 4c.  
To quantify localization errors associated with the three image analysis methods, log-log MSD plots of MCF-7 cells acquired by the radial symmetric method, circular Hough transform method, and modified active contour method are presented in Figure 4d-f, respectively, and the VACFs are shown in Figure 4g-i. Over a short timescale, a nonlinear MSD region is observed, and the time scale of nonlinear MSD is lengthened in the order of the modified active contour method, radial symmetry method, and circular Hough transform method. This is attributed to the characteristics shown in Figure 4a-c such that the trajectory analyzed by the circular Hough transform method is noisier than the other two methods, and the modified active contour method exhibits the least noise. In Figure 4d-f, the fitting  
Figure 4: Random dynamics of MCF-7 cells analyzed by the tracking methods. The (**a–c**) trajectories, (**d–f**) MSDs, and (**g–i**) VACFs \(\delta\) from 0.04 to 2 s calculated by the radial symmetry, circular Hough transform, and modified active contour methods (N = 36). (**j**) The comparison of the representative trajectory achieved from 3 different localization methods.  
results with Equation (3) are expressed along with the experimental results produced by the three methods. From the fitting results, the mean localization error, \(\sigma\), associated with the radial symmetric method, circular Hough transform method, and modified active contour method is estimated as 0.2265, 0.7467, and 0.1153, respectively. As expected from the trajectory results in Figure 4a-c, the circular Hough transform has the largest mean localization error, the modified active contour has the lowest, and the radial symmetric method is in the middle. Figure 4d-f, which present the VACF results acquired by the three methods up to \(\tau\leq 2\) s, show the same characteristics of the methods considered. For the radial symmetric method, the sharp negative peak appears at the smallest \(\delta\) and decreases quickly. In contrast, for the circular Hough transform method, the diminishing rate of the peak value of VACF is much slower than that in the radial symmetric method. The modified active contour shows a very small negative peak relative to the other two methods, which indicates that it is associated with less noise than the others. To compare the trajectory difference according to the localization methods, the representative trajectories are presented in Figure 4j.