## 2 Materials and Methods
### Design of the Cyclic Stretch System
#### 2.2.2 Developed System  
The system was developed based on the concept as shown in Figure 3. It consists mainly of a laptop PC, a stage driver box, a stage unit, and a microscope. The motorized stages with desired waveforms were driven by the stage controllers according to the digital pulses from PC. The specifications of each system component are listed below:  
* An inverted microscope with 40\(\times\) objective lenses was used for image and video capture (Olympus Inc., Tokyo, Japan). * A laptop PC equipped with the interface card slot with 8 bit Digital I/O, with 1.9 GHz CPU, installed physical memory of 2 GB, and Windows 7 OS as the operation system. * Stage unit: the 5-phase stepping motors with the positioning accuracy of 2 \(\upmu\)m/pulse and drive speed of 15 mm/s (SIGMA KOKI Inc., Tokyo, Japan) are required for both stage 1 and stage 2. A stepper motor with the positioning accuracy of 1 \(\upmu\)m/pulse and drive speed of 2 mm/s is required for stage 3. In this study, we designed the cyclic stretch system based on a commercially available uni-axial stretch chamber (Figure 3c), which can be purchased from STREX Inc. (STREX Inc., Osaka, Japan), and has been used by more than 100 research studies around the world [27; 44; 45; 46]. Because it has been reported that the elastic membrane was uniformly stretched over the whole membrane area and deformation in the direction perpendicular to the stretch axis did not exceed 1% at 20% stretch (4 mm stretch) [4], it is supposed that at most of the points on the stretchable membrane, the displacement in the \(y\)-direction is small enough and can be neglected during the \(x\)-direction extension. In this study,  
Figure 2: Basic concept of the stretching system for real-time compensation. (**a**) Three motorized stages for chamber stretching (stage 1), and compensation in the horizontal (stage 2) and vertical (stage 3) directions, respectively. (**b**) The timing chart for compensation. we only investigated the displacement of targets on the membrane in the horizontal and vertical directions to establish a compensation waveform. Displacement Characteristics in the Horizontal and Vertical Direction for the Design of Compensation Waveforms  
As shown in Figure 2b, for chamber stretching, we used the same method to that of the conventional apparatus. One end of the chamber was fixed, while the other end was stretched by a movable frame driven by stage 1 (Figure 2a) [4]. The displacements of the points on the stretch chamber in the horizontal direction were different from the fixed end to the movable end. In this study, in order to establish a methodology for the sophisticated cyclic stretch system, we used a demonstration system to investigate the displacement characteristics of the chamber exposed to 10% cyclic stretching, the magnitude of which is comparable to that of vascular cells caused by vasodilation [7, 47]. Because processing images during the stretching and relaxation periods (e.g., several seconds) is challenging, it is ideal to control the movement of stage 2 and 3 based on the knowledge about the position shift of the cells on the chamber membrane (feed-forward). Therefore, in order to determine the feasibility of feed-forward control and design the compensation waveforms, we measured the deformation waveforms in the horizontal and vertical directions. As shown in Figure 4a, several markers were drawn on the chamber membrane with an interval of 3 mm for the measurement of displacements. In this study, we focused on and investigated the displacement trajectories of the markers approximately 6 mm from the center of the chamber. The first reason is that the maximum vertical displacement of the chamber fixed at two ends should be located in the center due to the gravity of the culture medium. We want to confirm if the heavy vertical displacement around the center point will cause an unexpected trajectory.