## 2 Materials and Methods
### Motion Patterns  
The 4D-WPh was positioned relative to the nozzle so that the motion of the holder with the inserted film was perpendicular to the direction of the beam delivery and there was no variation in the water equivalent thickness (WET) (Figures 1a and 3b). The time-correlated position of the holder was determined by various motion patterns, including the FB and BH patterns. The triangular FB patterns (Figure 4a,b) provided idealized motion similar to respiration-induced motion but described by linear functions. They consisted of an inhalation phase and an exhalation phase. The exhalation phase contained a dynamic state when the film was moving and a pause state when the film was stationary. The exhalation phase occupied approximately 60% of the motion cycle, and the pause state was approximately 20% of the exhalation phase. The triangular FB patterns were asymmetric with respect to the mean line. The asymmetric pattern is more similar to real breathing patterns than the most commonly used sinusoidal pattern and may have a dosimetric effect [39]. The regular and irregular FB patterns were introduced in this study. For the regular FB patterns (Figure 4a), the film moved with a constant period of 4 s and a constant pk-pk amplitude of 2 mm, 6 mm, 12 mm, and 20 mm, resulting in 4 patterns of [2 mm-0%], [6 mm-0%], [12 mm-0%], and [20 mm-0%] (pk-pk amplitude-variation). These patterns were used for FB irradiation without any motion mitigation techniques. For the irregular FB patterns (Figure 4b), the same values were used for a period and a pk-pk amplitude, but they varied randomly by 10% (SD) from cycle to cycle around fixed means, resulting in 4 patterns of [2 mm-10%], [6 mm-10%], [12 mm-10%], and [20 mm-10%] (pk-pk amplitude-variation). These patterns were simulated for FB irradiation with rescanning to minimize the interference effect between the film motion and the scan pattern observed with overly regular motion [12; 13; 14]. Two FB patterns, including the regular [12 mm-0%] (Figure 4a) and irregular [12 mm-10%] (Figure 4b) patterns, were applied for gated irradiation to determine the stability of the gating device for irregular motion. The random variations had a Gaussian distribution for all the patterns. The pk-pk residual motion was less than 2 mm for the gating window of T40%-T80% and A0%A20%. Figure 3: (**a**) The dose distribution in the axial view calculated by the TPS, (**b**) the geometry of the 4D-WPh in the axial view for the film-based measurement where the holder with the film was moved relative to the stationary PTV perpendicular to the beam delivery angle, and (**c**) the normalized dose distribution, presented as a color map in the beam’s eye view, obtained from the stationary film. ROI1 and ROI2 were used for the quantitative dose analysis, and a magenta horizontal line defined a lateral profile of the dose distribution. Abbreviations: 4D-WPh = 4D water phantom; PTV = planning target volume; TPS = treatment planning system; and ROI = region of interest. The triangular BH patterns (Figure 4c) provided idealized motion similar to respiration-induced motion with DIBH guidance but described by linear functions. They consisted of alternating FB and BH phases, each lasting 20 s. During the FB phase, the film moved under the triangular regular FB pattern of [12 mm-0%], and during the BH phase, the film was stationary. The transition from the FB phase to the BH that occurred through a deep breath had a pk-pk amplitude of 20 mm. The random variations of \(\pm\)5% (SD), \(\pm\)10%, and \(\pm\)15%  
Figure 4: An example of the normalized irradiation diagrams for gating with (**a**) the regular and (**b**) irregular FB patterns and (**c**) for DIBH. The diagrams include the motion signal (blue), the gate-on/off signal (green), and the beam-on/off signal (magenta). The motion and gate-on/off signals were generated by the ORGS, and the beam-on/off signal was received from a dose monitoring unit integrated into a nozzle.