## 4 Discussion
In this sense, the inter-BH variations are more similar to interfacial tumor positioning errors than to intrafractional respiratory motion. Our DIBH patterns only included the inter-BH variation in the target position and did not simulate the intra-BH residual motion. Intra-BH motion is a continuous drift of the target position during BH [20], which can be up to 1 cm in real patients [46]. This fact limited our study because intra-BH motion may significantly increase dose distortion due to additional interplay. Inter-BH and intra-BH variations in the target position are very critical for DIBH treatment because they define its stability during beam delivery. In particular, the patient may be considered non-compliant for DIBH treatment if the inter-BH and intra-BH displacements are greater than 8 mm [24] and 5 mm [20], respectively. Individualized CTV-PTV margins [20], single-field optimization [27], and robust optimization [47] are recommended for DIBH treatment. Another problem with DIBH treatment is the increase in irradiation time. In our study, the time for DIBH irradiation was doubled compared to FB irradiation, as was the case for rescanning. Reducing treatment time is a challenge in synchrotron-based facilities due to the limited beam intensity and long energy switching time. The duration of BH is also limited because it may be difficult for the patient to hold their breath for a long time, and the duration of BH is on average 20 s for patients compatible with DIBH treatment [48], as simulated in our study. One of the approaches to reduce the irradiation time in DIBH may be to reduce the number of spots used by [PERSON] et al. [47]. For gated irradiation, the dose distribution was superior and almost indistinguishable from the stationary distribution (Figure 7, blue plot vs. magenta plot) when the film moved regularly. Only for the GPR (3%/2 mm) was a slight decrease of 4% observed compared to the stationary case, but the GPR still met the established threshold. A 10% irregularity in the motion pattern resulted in an 11% decrease in the GPR (3%/2 mm) compared to the stationary case but did not significantly affect other dose metrics. We used the gating window of T40%-T80% and A0%-A20% to achieve the residual motion of less than 2 mm. It may be equivalent to FB irradiation without motion mitigation for the [2 mm-0%] pattern. This was confirmed by the same GPR (3%/2 mm) of 91% for both FB and gated irradiation. Presumably, the irregularities in motion led to an increase in residual motion, resulting in a decrease in the GPR. The irradiation time for gating was the shortest compared to rescanning and DIBH. It was only 25% longer compared to FB irradiation without motion mitigation and did not depend on motion regularity. In clinical practice, they try to use a duty cycle of more than 50% [27], so a further reduction in irradiation time may be possible by widening the gating window. Increasing the window width may result in dose distortion, so the width should be selected based on the residual motion and not vice versa, as the motion curve is patient-specific. In general, gating may be the most preferred active motion mitigation technique for synchrotron-based facilities, because it allows for the synchrotron to operate in its usual pulsed acceleration and beam extraction mode but with an extended cycle equal to the breathing period. In our study, we had to increase an acceleration cycle of the synchrotron from 1.5 s, which is used in normal mode during FB irradiation, to 4.0 s, which was specific to the breathing pattern used. As a result, an average duration of the gate-on/off signal was 1.1 s.  
The present study has several limitations. The water phantom had a simplified geometry without internal heterogeneity compared to the real patient anatomy. Real tumor motion may combine translational and rotational motion and deformation, causing the range effect [8] in addition to the interplay effect, due to which more complex dose distortions may be produced.