# Analysis of Random Dynamics of Cell Segmented by a Modified Active Contour Method
## 3 Results & Discussion  
### Difference of Trajectory between cPMS Particles and MCF-7 Cells  
Accurate measurement of the reference location of the targeted image is one of the most significant factors when analyzing random dynamics of nano/micro-sized objects. When the object is spherical and non-transparent, there are many traditional methods to find the center or reference point of the object. Unfortunately, our interest is not limited to a spherical hard sphere but is extended to cells or even proteins, which could have diverse irregular shapes and translucent features. Figure 1a,b illustrates representative two-dimensional intensity profiles of carboxylated polystyrene microspheres (cPMS, nominal diameter 16.1 \(\upmu\)m) and the MCF-7 breast cancer cell line, respectively. The diameters of cPMS and MCF-7 are estimated as 34.2 pixels and 32.8 pixels, corresponding to 15.39 \(\upmu\)m and 14.76 \(\upmu\)m, respectively. As shown in Figure 1a, at the center region of the particle, the intensity value reaches 255, which corresponds to the white center region of the particle image in the inset of Figure 1a, and the intensity sharply decreases like a Gaussian point spread function to below 50 as it extends to the edge of the particle. At the edge, a dark circle is formed. At the outside of the edge, a mildly bright circular band is shown, and the background noise around the particle is also observed in Figure 1a. On the other hand, as in Figure 1b, the intensity distribution of MCF-7 is quite different from that of cPMS. An irregular intensity distribution as well as the irregular shape of the cell are observed, even though an edge darker than the background intensity is also shown, like the intensity distribution of the particle edge. However, the intensity value is much higher than those of the cPMS edge, and the uniformity of the intensity along the cellular edge is much more irregular. At the outside of the cellular edge, a bright circular band and the background noise around the particle are also visible. To understand the effect of intensity distribution and the uniformity of image, the comparison of random dynamics between cPMS and MCF-7 is performed. When particles or cells are suspended in a solution, they are continuously contacted by surrounding water molecules due to thermal energy. These interactions result in random motion of particles or cells, as shown in Figure 2a,b, which are the trajectories of cPMS and MCF-7, respectively, as analyzed by the radial symmetry localization method [18]. Both trajectories seem to be random, but there are definite differences between them. The MSD of cPMS in the log-log plot in Figure 2c is linear, while that of MCF-7 in Figure 2d shows  
Figure 1: Intensity profiles of 2-dimensional bright field images of (**a**) carboxylated polystyrene microspheres and (**b**) the MCF-7 breast cancer cell line. nonlinear characteristics over a short time scale (\(\tau\lesssim\)1 s). Comparing the trajectories of cPMS and MCF-7 shows that overall displacement of the particle is much smaller than that of the MCF-7 cell in the experimentally smallest time scale (\(\Delta t\) = 0.04 s), and the movement of MCF-7 appears to change its direction sharply. To characterize the differences in these two random processes, VACF-based analysis is performed, as shown in Figure 2e,f for cPMS and MCF-7, respectively. The VACFs are normalized to a value of \(\tau=0\). A previous report showed that the VACF can provide useful information to distinguish effects on the random process, such as localization errors, confinement, and moderate elasticity. The VACF of the trajectory with localization error has a sharp negative peak over a short time scale, and the peak decays as time \(\delta\) increases. In the case of confined conditions, the VACF decays quickly to zero after it reaches to \(-0.5\) at \(\tau=\delta\)[40]. The VACF of the cPMS is different from that of the MCF-7.