## 2 Materials and Methods
### Design of the Cyclic Stretch System
#### 2.2.2 Developed System
Here, when the R-squared value is larger than 0.95, the fitting is regarded as an appropriate one. From the results, it was confirmed that the movement of cells on the membrane can be compensated by the linear movement of stage 2 in the horizontal direction. The method to measure the maximum position shift will be demonstrated later. Second, the design of the vertical compensation waveforms of stage 3 was conducted based on the analysis of the vertical shift of these markers on the chamber membrane. By using a laser displacement sensor (KEYENCE Inc., Osaka, Japan) fixed on a robust aluminum structure under the chamber, the vertical displacement of the chamber at the points can be measured during the stretching and relaxation processes. Further, the position under each marker was painted with a silver permanent marker for the reflection of the laser beam. Firstly, we measured the shift of the membrane without adding culture medium into the chamber. Here, the sampling interval was set to be 3 ms. As shown in Figure 5b-e, when the chamber was exposed to stretching of a frequency of approximate 0.17 Hz, the vertical displacement waveforms at points 3, 4, 5, and 6 showed a repeatability feature similar to the stretching waveform of stage 1, although there may be slight non-linearity when compared with the displacement waveforms in the horizontal direction. Analysis of the displacement waveforms in the vertical direction was conducted using Origin 2017 (OriginLab). In order to cut off the noise, a fast Fourier transformation (FFT) filter was used to obtain data smoothing with points of the window at 10. Next, as mentioned above, because the displacement waveforms possess a repeatability feature, a stretching-holding-relaxation cycle was extracted and used for the design of compensation waveforms. We expected that a linear waveform can also be used to compensate for the most part of the vertical shift, although there may be some instantaneous position aberrations during a cycle when the compensation waveform is linear. Therefore, similar to the curve fitting process in the horizontal direction, Formula (1) was used to perform the curve fitting. Figure 5b-e (lower) shows the waveforms at position 3, 4, 5, and 6 before and after smoothing using FFT filters. Curve fitting results of a stretching-holding-relaxation cycle are shown under the corresponding displacement waveforms. Here, a high R-square of above 0.95 was obtained for each displacement waveform, which reveals that the trendline was reliable and could be used as the rough trajectory of the movement of stage 3. Figure 5: Vertical 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–e**) Upper: a typical waveform of points 3–6 when the chamber was stretched. Lower: the fit results of the stretching/holding/relaxation phases of the corresponding upper waveform. From (**b**) to (**e**), original data is in light blue and filtered data in orange. The lines with color in purple, blue, and black are the lines fitting for the lengthening, holding, and relaxation phases, respectively. To further confirm that the vertical shift waveform during the stretching and relaxation phase can be fitted using a linear approximation method, the displacement waveform in the vertical direction at a different frequency (0.25 Hz) was investigated. As shown in Figure 6b,c, a linear function fit well to the set of data of a stretching or relaxation phase of the vertical shift at point 1 or 2. Further, in order to mimic realistic conditions close to an actual cell experiment with 1-mL culture medium, 1 mL of water was added into the stretch chamber before measuring the displacement of targets. The vertical displacement waveform can be also fitted with linear waveforms at the stretching or relaxation phase (Figure 7), which demonstrates practical application in cell monitoring when exposed to cyclic stretching.