## 2 Materials and Methods
### Design of the Cyclic Stretch System
#### 2.2.2 Developed System
In addition, researchers tend to focus on the targets (cells) in the center because they generally believe that the deformation of the chamber near the membrane edge may not be uniform. In this study, the points used to investigate the vertical displacements are not always the ones used for horizontal displacement calibration. The reason is that we first considered the symmetry of the chamber design, and selected the representative points at the lower  
Figure 3: Overview of the developed system with the stages. (**a**) System setup. (**b**) Stage unit. (**c**) A stretch chamber with a thin membrane (approximate 200 μm) for on-line cell observation. right corner, and then investigated some points at the upper left corner to randomly increase the testing points. This methodology of compensating for chamber displacements can be used for various studies. Movies of markers were recorded at a frame rate of 30 fps when the chamber was exposed to cyclic stretching. First, the horizontal displacements were investigated. Before stretching, the target was set to the center of the microscope's field of view (FOV). Horizontal movements of the center point (point 3) and the two points on the left (point 2) and right (point 4) sides were recorded by a camera set under the chamber. Displacements were calculated from the captured images with a resolution of approximate 13 \(\upmu\)m/pixel using imagej (NIH, Bethesda, MD, USA), and the interval of image capture for analysis was approximate 0.1 s. From Figure 4b-d, the waveforms were confirmed to be almost linear with high similarity even at different points. In addition, \(\delta\)x (horizontal) was always smaller than the stretch volume. Furthermore, the accurate compensation by stage 2 was not required due to sufficient FOV of the microscope (\(235\times 133\)\(\upmu\)m at 40\(\times\)) compared to the stage positioning accuracy (2 \(\upmu\)m). As a result, the on-line compensation did not require a perfect waveform. In other words, the \(\delta\)x (horizontal) value  
Figure 4: Horizontal displacement trajectories of targets on the chamber membrane. (**a**) The layout of targets on the chamber membrane and the waveform for the control of stage 1/chamber stretching. (**b–d**) Upper: a typical waveform of point \(2,3,\) or \(4\) when the chamber was stretched. Lower: the fit results of the stretching/holding/relaxation phases of the corresponding upper waveform. From (**b**), original data is drawn in light blue. The lines with color in purple, blue, and black are the lines fitting for the lengthening, holding, and relaxation phases, respectively. at the desired measurement point before the experiment can be measured, and compensate with the interpolated waveform in an actual cell experiment. In order to capture the feature of the waveform, a curve fitting was obtained through the Fitting Function Builder of Origin 2017 (OriginLab Corporation, Northampton, MA, USA) with the following C program. The displacement was assumed to be \(\mathrm{x_{d}}\). if(\(\mathrm{x_{d}}<\mathrm{M}\))  
y = A\(\times\)x\({}_{\mathrm{d}}\) + B;  
else if(\(\mathrm{x_{d}}>=\mathrm{M}\) && N \(>=\mathrm{x_{d}}\))  
y = K;  
else  
\[\mathrm{y}=\mathrm{D}\times\mathrm{x_{d}}+\mathrm{E} \tag{1}\]  
M and N are the data number at the end of the stretching process and the starting point of the relaxation phase. A and D are the slope of the lengthening and relaxation phase, respectively, and B and E are the intercept values for each line. K is the value of maximum displacement of a target after the chamber is stretched. Here, we supposed that the stretching and relaxation phases are linear, and further, A = \(-\mathrm{D}\), which simplifies the control of stage 2. That is, if the maximum position shift of a point on the membrane in the horizontal direction is known, the average movement speed of the motor during the stretching process can be calculated. Moreover, the shift can be compensated by the linear movement of the stage. The goodness of the fit is shown in Figure 4b-d (lower figures).