Journal of Healthcare Engineering

Volume 2018 (2018), Article ID 8504273, 10 pages

https://doi.org/10.1155/2018/8504273

## A Comprehensive Method for Accurate Strain Distribution Measurement of Cell Substrate Subjected to Large Deformation

^{1}College of Materials Science and Engineering, Sichuan University, Chengdu 610065, China^{2}College of Architecture and Environment, Sichuan University, Chengdu 610065, China

Correspondence should be addressed to Yuanwen Zou

Received 26 July 2017; Revised 11 October 2017; Accepted 23 October 2017; Published 8 January 2018

Academic Editor: Wei Yao

Copyright © 2018 Hong He 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

Cell mechanical stretching *in vitro* is a fundamental technique commonly used in cardiovascular mechanobiology research. Accordingly, it is crucial to measure the accurate strain field of cell substrate under different strains. Digital image correlation (DIC) is a widely used measurement technique, which is able to obtain the accurate displacement and strain distribution. However, the traditional DIC algorithm used in digital image correlation engine (DICe) cannot obtain accurate result when utilized in large strain measurement. In this paper, an improved method aiming to acquire accurate strain distribution of substrate in large deformation was proposed, to evaluate the effect and accuracy, based on numerical experiments. The results showed that this method was effective and highly accurate. Then, we carried out uniaxial substrate stretching experiments and applied our method to measure strain distribution of the substrate. The proposed method could obtain accurate strain distribution of substrate film during large stretching, which would allow researchers to adequately describe the response of cells to different strains of substrate.

#### 1. Introduction

Cardiovascular mechanobiology [1–5] is a discipline that focuses on the effects of the mechanical environment on the cardiovascular system and elucidates how mechanical factors produce biological effects that lead to vascular remodeling. Cardiovascular mechanobiology aims to provide biomechanical solutions for the diagnosis, prevention, and rehabilitation of cardiovascular disease. Cell mechanical stretching *in vitro*, vascular stretching *in vitro*, an animal model of stress changes in blood vessel *in vivo*, and numerical simulation method are commonly used in cardiovascular mechanobiology experiments. Cell-substrate stretching technique, extensively used *in vitro* cell mechanical experiments, is an effective method of force transduction [6–8]. Riehl et al. [9] summarized the methods of mechanical stretching in cell-substrate stretching experiments and showed that most of the strain loaded on substrate was large, with the maximum value up to 33%. Besides, strain distribution varies from region to region within a substrate. Thus, it is of great importance to analyze the accurate strain field of substrate under large deformation.

Some commonly used methods for measuring substrate strain include the direct calculation method, the resistance strain gauge measuring method, the phase shift shadow moiré method, and the finite element analysis. In the direct calculation method [10], several marks should be drawn first on the surface of substrate, so that the displacement and strain can be calculated by analyzing the difference of those marks between the images captured before and after deformation. The accuracy of this method is heavily dependent on the distribution of marks. Therefore, the direct calculation method is a rough measurement. The resistance strain gauge measurement method [11] attaches strain sensors to the surface of substrate for strain measurement. However, the use of any contact-type sensor would obstruct the stretching of substrate especially under large deformation and lead to unexpected results. The phase shift shadow moiré method [12] requires complex operation, resulting in large systematic errors, since its accuracy would be influenced by light intensity. The finite element analysis [13, 14] is a computational simulation method, which can investigate many mechanical properties such as displacement and strain by setting material parameters and environmental parameters. However, the finite element analysis is usually based on models under ideal assumptions and cannot replace the actual experiment.

Digital image correlation (DIC) is a contactless full-field displacement and strain measurement technique. Since it was first proposed in the 1980s [15, 16], DIC is now extensively applied in many experimental mechanics researches [17, 18]. Digital image correlation engine (DICe) [19], developed by the Sandia National Laboratory, is an open-source library of DIC. With DICe, DIC would become a convenient and user friendly tool so that users can pay more attention to their researches rather than repeatedly programming the basic and complex procedures of DIC. To the best of our knowledge, to date, DICe has not been used to obtain accurate strain field of substrate in cell-substrate stretching experiments under large strain. The Newton-Raphson (NR) algorithm was used in DICe to calculate the desired deformation. However, in NR algorithm, the calculation result of the previous step is regarded as the start value of iteration for the next step. So the initial guess is of great importance in calculating strain field, especially under large deformation. An unreliable initial guess would seriously impact the results of strain measurement. Pan et al. [20] developed an incremental reliability-guided DIC technique (RG-DIC) for large deformation measurement, in which the recently developed robust RG-DIC technique was combined with an automatic reference image updating scheme, and the reference image for DIC analysis is automatically updated according to the seed point’s zero-mean normalized cross-correlation (ZNCC) coefficient. This method could deal with specimens with irregular geometric shape and/or subjected to discontinuous deformation as well as minimize the accumulated errors in finally estimated displacements. Zhou et al. [21] proposed a fully automated method. In this method, the computer vision technique was used to extract image feature points and to match them between reference and deformed images. The deformation parameters of the seed point are initialized from the affine transform, and then the refined parameters are automatically transferred to adjacent points using a modified quality-guided initial guess propagation scheme. This method can accurately initialize all points of interest for the deformed images even in the presence of large rotation and/or heterogeneous deformation. Zhao et al. [22] proposed the utilization of three well-known population-based intelligent algorithms (PIAs), that is, genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO), and then incorporated standard PIAs with three improving strategies, including Q-stage evolutionary (T), parameter control (C), and space expanding (E) strategies, finally derived of a total of eighteen PIA-based algorithms. They tested these algorithms and found that DE-TCE performed best. Another large deformation measurement scheme, combining improved coarse search method and updating reference image scheme, was proposed by Tang et al. [23]. With this method, not only extremely large deformation can be measured successfully but also the accumulated error introduced by updating reference image could be controlled. Pan et al. [24] proposed an integrated scheme which combined the Fourier–Mellin transform-based cross-correlation (FMT-CC) for seed point initiation with a reliability-guided displacement tracking (RGDT) strategy for the remaining points. This method can provide an accurate initial guess for DIC calculation, even in the presence of large rotations. In order to obtain reliable initial values and then calculate accurate strain field of cell substrate under large strain, a simple method was proposed in this study and was combined with DICe. Compared with methods described above, without the use of feature detection, feature point matching, and other techniques, our method can accurately obtain the initial guess of the NR iteration for large deformation.

#### 2. Materials and Methods

##### 2.1. DIC Principles

###### 2.1.1. Basic Principles

DIC is an optical-numerical full-field displacement measuring technique with subpixel accuracy. In its basic principle, the measurement is performed by tracking or matching the same points (or pixels) between the two images recorded before and after deformation [17, 25]. The image recorded before and after motion is called reference image and deformed image, respectively. In general, the calculation area, also called the region of interest (ROI), should be specified or defined in the reference image, which is further divided into evenly spaced virtual grids. Moreover, each intersection point of virtual grids is selected as the center of a subset. The subset is the matching unit within the ROI, usually a square of pixels. Then, a correlation criterion should be defined to evaluate the degree of similarity between reference and deformed subsets. The matching procedure is completed through searching the peak position of the distribution of correlation coefficient. Once the correlation coefficient extremum is detected, the position of the deformed subset is determined. The differences in the positions of the reference subset center and the target subset center yield the in-plane displacement vector at the calculate point.

###### 2.1.2. Shape Function

Based on the assumption of deformation continuity of a deformed solid object, a set of neighboring points in a reference subset remains as neighboring points in the target subset. The coordinate of points around the subset center in the reference subset can be mapped to points in the target subset according to the shape function. The first-order shape function that allows translation, rotation, shear, normal strains, and their combinations of the subset is most commonly used. where is the local coordinate of each pixel point in reference subset, is the coordinate in the deformed subset, and are the displacement components, and , , , and are the displacement gradient components.

###### 2.1.3. Correlation Criterion

To evaluate the similarity degree between the reference and deformed subsets, a correlation criterion should be defined in advance before correlation analysis. It is concluded that the zero-normalized cross-correlation (ZNCC) or zero-normalized sum of squared differences (ZNSSD) correlation criterion offers the most robust noise-proof performance and is insensitive to the offset and linear scale in illumination lighting [17].

###### 2.1.4. Subpixel Registration Algorithm

As shown in (1) and (2), the coordinate of points in the deformed subset may locate between pixels. Thus, before evaluating the similarity between reference and deformed subsets using the correlation criterion, the reference image and deformed image must be reconstructed by a certain kind of subpixel registration algorithm to further improve the accuracy of DIC.

###### 2.1.5. DICe

DICe is an open-source library of DIC, which can be used as a module in an external application or as a standalone analysis code. With DICe, DIC would become a more convenient tool and decrease the complexity when users apply it to their researches.

In DICe, the ZNSSD [17] is applied as the correlation criterion for its insensitivity to offset and linear scale in illumination lighting. where and denote the grayscale level of each point in the reference and deformed images, respectively, and and are the mean intensity value of two subsets. is the local coordinate of each pixel point in reference subset, and is the coordinate in the deformed subset.

As a classic algorithm in DIC, the NR algorithm is utilized in DICe as the iterative spatial domain cross-correlation algorithm. In the first-order shape function, the desired mapping parameter vector is , where denotes the displacement components and is the strain of subset. To acquire the desired deformation vector by the NR iteration method, the solution can be written as [17] where is the initial guess of the solution, is the next iterative approximation solution, is the gradients of correlation criteria, and is the second-order derivation of correlation criteria.

##### 2.2. Improved DIC Scheme

As described above, the NR algorithm was used in DICe to calculate the desired deformation vector. However, when it is applied to large strain measurement directly, the iteration divergence would appear, so that the accurate strain cannot be obtained. Figures 1 and 2 show the calculation results of DICe under small and large strain, respectively. Figure 1 is under 0.1% strain and Figure 2 is under 10% strain. As shown in Figure 1, the displacement distribution is continuous and the strain field is uniform. The strain distribution is very close to its setting value 0.1%, suggesting that the DICe calculation is valid and accurate. Nevertheless, as shown in Figure 2, the displacement distribution is not continuous which cannot happen in the actual, indicating that these results are invalid.