# Analysis of Random Dynamics of Cell Segmented by a Modified Active Contour Method
## 1 Introduction
Diverse segmentation approaches, such as template matching [25; 26], watershed transformations [27; 28], and deformable models [29; 30], have been developed. However, to detect complex cell boundaries, several specific methods such as the level set method [31; 32] and active contour [30] can be employed. Recently, time-lapse living cell data have been analyzed with deformable models considering the evolving contour of a cell from that in the previous frame [31; 33]. The dynamics of cells or biological particles suspended in a solution are usually analyzed as a random process [2; 31; 32]. Based on accurate analysis of particle/cell trajectory, basic random process characteristics can be found using mean square displacement (MSD), which provides information regarding diffusion characteristics. It is well known that normal diffusion shows a linear relationship between MSD and time, and the slope of the linear fit is known as the diffusion coefficient (D). If linearity is disrupted, the process is called abnormal diffusion. When MSD\(\propto\)\(t^{\alpha}\) and \(\alpha>1\), the process is called superdiffusion. When MSD\(\propto\)\(t^{\alpha}\) and \(\alpha>1\), it is called subdiffusion [34]. The velocity autocorrelation function (VACF) can be employed to distinguish other mechanisms for abnormal diffusion [32; 35]. To understand the random dynamic characteristics of particles or cells suspended in a solution, accurate localization of the object in a microscopic image is fundamental and significant. Clinically, it is important to discriminate the cellular type in diagnosis and therapeutics, since the therapeutic strategy and prognosis are strongly dependent on the cellular type [36]. In addition to this, ultimately, we are interested in the application of cellular dynamics to the discrimination of cell condition. For example, metastasis is known to be mediated by the circulating tumor cell (CTC) in the blood of the patient. Thus, it is important to find the specific cell type in the blood sample through simple method such as just observing the cellular dynamics to minimize the cost and time to detect. The method applied in this study could provide the accurate cellular trajectory so that it can be applied to the discrimination of different cellular types through the different cellular random dynamics. Specifically, the image of a cell is different from that of a particle in terms of appearance and intensity profile. Generally, the shape of a cell is more irregular and deformable than those of a particle, and the intensity distribution is noisier. The asymmetric shape and disturbed distribution of intensity might cause significant localization errors, which negatively impact analysis of the random variables estimated based on trajectory. Hence, in this study, we compared three localization methods, namely the radial symmetry algorithm, circular Hough transform method, and modified active contour method with denoising, to determine the most accurate way to localize cellular images. We employed stochastic parameters, such as MSD and VACF, to estimate the accuracy.