## 3 Results
### Simulations of the Response or Air-Filled Ionization Chamber  
The simulations shows that recombination in air ionization chambers with the characteristics of the PTW1500 is not very important for the verification of treatments delivered at doses per pulse of 1 mGy and 2 mGy (typical values employed in FFF radiotherapy). This is illustrated in Figure 4A,B, for a particular position (pixel) in the radiation treatment field. This response is in agreement with the results published in [9] for this detector, which was specifically designed to prevent recombination issues in FFF deliveries. When the maximum dose per pulse is increased to 50 mGy (a value that is not representative of FFF, but closer to those employed in intraoperative radiation therapy (IORT) deliveries), recombination losses become more important, and the Boag formalism shows a clear overestimation of the collection efficiency. The reference dose per pulse for these calculations was 1 mGy, which results in a value \(u_{ref}=0.008\) (collection efficiency 0.996).  
The results for the air ionization chambers with the characteristics of the PTW729, shown in Figure 5, illustrate a greater impact of volume recombination. The reference dose per pulse for these calculations was 1 mGy, which results in a value \(u_{ref}=0.069\) (collection efficiency 0.967). The developed method is able to correct for these effects and reproduce the total integrated charge in the case of 1 mGy and 2 mGy. However, at 50 mGy per pulse, the correction overestimates the real signal as in the case of PTW1500-like air-chamber. The overestimation effect is visibly higher for the case of the PTW1500 detector since the contribution of free electrons is considerably greater (due to lower gap and higher voltage). Therefore, in situations where the collection efficiency departs from the Boag formula, it would be convenient to correct with the model of Equation (3), or a phenomenological characterization of the dependence of the collection efficiency on the dose per pulse, as in Equation (24).  
Figure 3: Simulated readout signal in a pixel of the LIC operating at 1000 V and irradiated by a beam simulating a delivery with a maximum dose per pulse of 2 mGy (roughly corresponding to a delivery at 10 MV-FFF in a TrueBeam linac) and 2.8 ms between pulses (blue line). The corrected signal is also shown (red line), as well as the variation of the dose per pulse during the irradiation. It is observed that the larger the dose per pulse, the more important the difference between raw readout and collected readout (due to increased recombination).  
It should be noted that the integration time in the 1 mGy and 2 mGy cases was set to 200 ms for both chamber types. However, this is not the case for the 50 mGy simulation, where, as the treatment is scaled so that the total dose is maintained, the number of pulses for each configuration is significantly lower. Thus, for this case, the integration time was set to 2.7 ms, a value similar to the distance between pulses, in order to be able to apply the method correctly.  
Figure 4: Effect of recombination and recombination correction in the verification of a dynamic radiotherapy clinical treatment with a PTW1500-like air-chamber. The dynamic dose rate corresponds to that of a lymph node tumor treatment, but scaled up to deliver a maximum dose per pulse of 1 (**A**), 2 (**B**) and 50 mGy (**C**). In the left panels we show for a particular pixel the raw readout current (red lines) and corrected readout current (blue lines) versus time. Notice that the dose per pulse in this pixel does not reach the maximum dose per pulse of the delivery. In the right panels we show a profile of the radiation field (the one shown in Figure 2), including generated charge (solid line), raw collected charge (dashed line), and corrected charge (dash-dot line).