Journal of Healthcare Engineering

Volume 2017, Article ID 1848314, 10 pages

https://doi.org/10.1155/2017/1848314

## Quantitative Analysis of Intracellular Motility Based on Optical Flow Model

^{1}College of Electronics and Information Engineering, Hebei University, Baoding 071002, China^{2}School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China

Correspondence should be addressed to Zhiwen Liu; nc.ude.tib@uilwz and Peiguang Wang; nc.ude.ubh@gnawgp

Received 17 February 2017; Accepted 21 May 2017; Published 30 July 2017

Academic Editor: Md. A. R. Ahad

Copyright © 2017 Yali Huang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Analysis of cell mobility is a key issue for abnormality identification and classification in cell biology research. However, since cell deformation induced by various biological processes is random and cell protrusion is irregular, it is difficult to measure cell morphology and motility in microscopic images. To address this dilemma, we propose an improved variation optical flow model for quantitative analysis of intracellular motility, which not only extracts intracellular motion fields effectively but also deals with optical flow computation problem at the border by taking advantages of the formulation based on and norm, respectively. In the energy functional of our proposed optical flow model, the data term is in the form of norm; the smoothness of the data changes with regional features through an adaptive parameter, using norm near the edge of the cell and norm away from the edge. We further extract histograms of oriented optical flow (HOOF) after optical flow field of intracellular motion is computed. Then distances of different HOOFs are calculated as the intracellular motion features to grade the intracellular motion. Experimental results show that the features extracted from HOOFs provide new insights into the relationship between the cell motility and the special pathological conditions.

#### 1. Introduction

Cell morphology and mobility indicate physiological and pathological characters of the organism [1]. It has been demonstrated that quantitative analysis of cell morphology and mobility offers the possibility to improve our understanding of the biological processes at the cellular level [2]. Estimation of live cell mobility for analyzing dynamic properties of biological and pathological phenomena has been extensively used in clinical diagnosis and biological research, including inflammation research, drug test, wound healing, tumor genesis, and immune response [3–7]. Life information under special condition is to be uncovered by quantitative analysis of intracellular motility. However, it is difficult to measure intracellular motility due to irregular complicated cell deformation. Here, we proposed a novel approach for quantitative analysis of intracellular motility based on optical flow model.

Starting from the original work of Horn and Schunck (HS) model as well as Lucas and Kanade (LK) model [8, 9], optical flow method has been widely used in the computer vision applications for estimating the motion of the object, which is also a primary method applied in quantitative motion estimation of biological structures in light microscopy [10–12]. Vig et al. reviewed the main methods for measuring cell-scale flows, including single-particle tracking (SPT), particle image velocimetry (PIV), and optical flow [13]. Moreover, they found that although SPT and PIV techniques have been the principal means for analyzing bioflows in the cellular biophysics, optical flow technique outperforms than the formers for its relatively simple implementation and providing additional biophysical information such as local velocity [13]. Boric et al. applied optical flow method to quantify the movements of populations of cells and detect subtle cell changes in quantitative analysis of cell migration [14]. Guo et al. applied optical flow methods to track red blood cell [15]. Optical flow technique was also used in measurement of blood flow velocity in vivo [16]. All the abovementioned research work mainly regarded the cell as a whole and focus their attention on the movement of the whole cell. Little attention has been to the intracellular motion.

In this paper, we propose an improved optical flow method-based variation model for analyzing intracellular mobility in phase contrast microscopic cell images. The data term in the energy functional of the optical flow model adopts a norm, which is beneficial to extract the smooth velocity field of the intracellular movement. While the smoothness term in the energy functional changes with the regional feature of the image, using norm in the intracellular area and norm nears the edge of the cell according to local features of the image, which is helpful to address the optical flow computation near the edge of the object. Furthermore, histograms of oriented optical flow are used to quantify the intracellular mobility.

The rest of the article is organized as follows. Section 2 reviews related work on optical flow models. Section 3 proposes our scheme: the improved optical flow based on variation model and characterization of intracellular motion based on HOOFs. In Section 4, we present the visualization of intracellular motion based on optical flow and apply the proposed scheme to the synthesized data and the actual data; experimental results are provided. Discussion and conclusions are given in Section 5.

#### 2. The Variation Model of Optical Flow

Optical flow is defined as the vector field expressing 2-dimensional apparent motion pattern of moving object projected on the screen, and this vector field is also viewed as the velocity field of the moving object, which comprises motion and structure information of the observed object. Optical flow computation is based on the correlation in respect with time and space between the two subsequent images of a video. Although many new concepts have been proposed for dealing with different problems in optical flow models, today’s optical flow is still similar to HS model or LK model [8, 9]. This kind of variation optical flow normally can be expressed as follows: where is the flow vector of a pixel in the 2-dimensional optical flow field; is a data term and is a smooth term, with being a weight between the two terms. Normally, the data terms are composed of some constancy assumptions, such as gray value, gradient constancy assumption, Hessian conservation equations, and Laplacian conservation equations. These constancy assumptions form constraints for the solution of the variation optical flow model. Different constraints of data term have been used in different motion patterns. The smoothness term, which guarantees the existence of a unique of the optical flow model, adopts different smooth strategies in different applications. The optical flow computation based on the variation method is realized by minimizing an energy functional constituted by some data constraints and a smoothness constraint. For example, in the classical HS model, the data term adopts gray value constancy assumption and the smoothness constraint is the square of the magnitude of the gradient of the optical flow velocity. Therefore, the energy functional of HS optical flow model is expressed as follows: where and denote the successive images used to compute optical flow field; denotes the pixel coordinate and is the flow vector of a pixel ( and denote the displacement of a pixel at the horizontal direction and the vertical direction, resp.); indicates the image region. The optical flow field is computed by optimizing the energy functional. That is to say, is obtained by the minimization of .

In the HS optical flow model, both the data term and smooth term are in the form of norm, and the implementation based on norm to the image is equivalent to the isotropic diffusion. This kind of method can avoid piecewise constant in the image but lead to blurry edge and details lost. Thus, Papenberg et al. proposed a novel energy functional, in which the data term and the smooth term were expressed as ( is a small positive number) [17]. To some extent, the minimization of results in approximate implementation based on the norm. Moreover, Pock et al. proposed a total variation energy functional model, named , expressed as follows [18]: where the data term is based on norm and the smooth term adopts total variation (TV). From this energy functional, it is found that the minimization of is equivalent to the optimization based on norm. The formulation based on norm to the image is equivalent to the anisotropic diffusion. This kind of implementation can preserve discontinuities near the edge in the optical flow field but result in piecewise constant in the optical flow field. In order to obtain accurate optical flow field both in the edge and in the intracellular area, we propose a flexible optical flow model functional based on the variation model, and the details are described as follows.

#### 3. Methods

##### 3.1. Adaptive Total Variation Optical Flow Model (Adaptive TV Optical Flow Model)

In order to compute accurately the optical flow field of the single intracellular motion both near the edge and in the intracellular area, we proposed an energy functional of optical flow model as follows:
where is the intensity of the pixel located at in the frame corresponding to time *t*; is a weight between the data term and the smooth term. The vector denotes the vector of a pixel in the 2-dimension optical flow field; and are the horizontal and vertical components of the flow field. It can be concluded that based on the gray value constancy assumptions and Taylor’s formula [8], so we have another expression of (4).
where is an adaptive parameter varying with the features of the image.

Our proposed adaptive TV optical flow model is to minimize the above energy functional in (4-1). That is to say, the solution of optical flow field is obtained by minimizing the energy functional (4-1). Using the calculus of the variations, the Euler-Lagrange equation for this energy functional is obtained as follows [19] (the details of solution are shown in Appendix A): where denote the partial derivatives of image brightness with respect to . We choose according to the local feature of the image, using the following equation: where is the convolution of the image with the Gaussian filter , obtaining smoother image. It is obvious that the value of is high near the edge of the cell. Particularly, if , then . On the contrary, at the intracellular region where far away from the edge, there is , so we obtain . In brief, the adaptive parameter is chosen so that it is smaller near a likely edge and larger away from possible edges, varying with the characteristic of the image. Therefore, the proposed adaptive TV optical flow model is equivalent to the image implementation based on norm near a likely edge when and based on norm far away from possible edges when , which is helpful to deal with the problem of optical flow computation at the border.

In this study, the numerical implementation of the proposed adaptive TV optical flow model is obtained based on and , respectively. First, if , we have

Note that is in the denominator, in order to avoid the singularity. It is common to use a slightly perturbed norm to replace , where is a small positive number. In the same way, we used to replace . Then the gradient decent flow of (7) is where ; ; ; ; ; and .

To set up discrete iterative solution, using a finite difference approach on a discrete grid, our iterative solution to (8) is as follows:
where *n* corresponds to discrete time and *γ* denotes the time step for each interaction. Let the indices , , and correspond to , , and . Here, we define some equations as follows:

Substituting (10), (11), and (12) into (9), then we can obtain optical flow by the iteration procedure. Second, if , we have which is the same as the Euler-Lagrange equation in HS model [8]. Similarly, the iterative solution of (13) is as follows:

##### 3.2. Characterization of Intracellular Motion Based on HOOF

Details on the direction and the magnitude of each pixel’s velocity of intracellular motion are expressed in the optical flow field , which is a 2-dimensional vector field. However, the raw optical flow data may be of no use, as it is composed of huge volumes of data. How to obtain quantitative information from the optical flow field is always haunting researchers. A variety of techniques have been devised to address this problem. Chaudhry et al. proposed HOOF for the recognition of human actions [20]. In our study, the HOOF technique is developed to quantify intracellular motion. We perform statistical analysis to the distribution of velocity in the optical flow field. That is to say, the distribution of each pixel’s velocity in the optical flow field is analyzed on the basis of statistics theory.

The feature vector of intracellular motion is extracted as follows. First, our proposed optical flow model is applied to compute optical flow fields from the successive frames of the video. Second, each flow vector in the optical flow field is binned according to its primary angle from the horizontal axis and weighed according to its magnitude. Then, we obtain the histogram of all flow vectors in the optical flow field. The function of the histogram is expressed as follows. where denotes different directions (bins); denotes the number of bins; and is the sum of all velocities of each flow vector in the interval of directions . The direction of each flow vector is computed as , and the velocity of flow vector is obtained by . In our research, in order to extract the quantitative features based on the intracellular motion, the directions of flow vectors are quantified into 16, that is, . So the number of bins in the histogram is 16, and the height of every bin is the sum of all velocities whose angles are in the interval . Lastly, the histogram is normalized. In brief, HOOFs express the features of the distribution of flow vectors in the optical flow field. Furthermore, we compute the distances of the successive HOOFs and then use these series distances as the feature vector of the intracellular mobility.

#### 4. Experiments and Results

Cell microscopic images were acquired through optical phase contrast microscope at a magnification of 16,000x from the peripheral blood samples of clean healthy mice. The animal experiments were conducted by the trained staff in Beijing You’an Hospital, which is affiliated to the Capital Medical University. All the disposals are in accordance with the guideline of animal ethics. Each video lasts for 22–24 seconds, and the frame rate is 25 frames per second (every video includes 550–600 frames). It should be noted that the details of segmentation and tracking of the single cell are presented in our early work [21]. In this article, it is assumed that the cell has been segmented and tracked from microscopic images, so we analyze the intracellular motion directly.

##### 4.1. Visualization of Intracellular Motion Based on Optical Flow

We compute optical flow field from the segmented lymphocyte in video directly, and one frame of optical flow field, which is extracted from two successive images, is shown in Figure 1. The direction of the arrow in the optical flow field denotes the direction of the intracellular motion, and the length of the arrow denotes the magnitude of the intracellular motion. In order to compare the effectiveness of the smooth term with different , we compute three kinds of optical flows according to , , and adaptive , as shown in Figures 1(b), 1(c), and 1(d). The results show that the formulation can preserve discontinuities by applying the robust norm in the smooth term but leads to piecewise constant in the image. On the other side, using norm in the smooth term can extract more details of motion but causes blurry edge. Applying the adaptive changing with the local features of the image can obtain a good balance between preserving discontinuities and optical flow details.