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　　　　　　　(1)=DthDifwmain)ifwmain==DthmaxDith 　　　　　(2)main and Approximately, with the new weights, wiwi,FIF, the doses in the unblocked regions do not subchange and those in the blocked regions reduced by f.(1,)i,FIFwmain, f=,,,f==.d1 and d2 in Figure 4. The side of an MLC leave which had a longer horizontal distance (d1 in Figure 4) than the other side was selected to block the hotspot. The MLC leaves were moved as far as completely covering the hotspot.mainFigure 4　Schematic beam’s eye view of a sub-beam. The gray regions indicate projected hotspotsThe dotted line is the vertical center line of the field of the sub-beam. Rectangles indicate　multileaf collimator leaves, which completely cover the hotspots. d1 (d2) is the horizontal distance from the vertical center line to the outer edges of the hotspot on the left (right) side.The weights of the main beams were reduced to decrease the dose in the hotspot regions as follows:where Dmax is the maximum dose in the original plan, Dth is the dose threshold for the hotspot regions, i is the index for the main beams (i =1, 2), main and wi,FIF are the weights in the original and wiand FIF plans for the main beams, respectively. The weights of the sub-beams (wi,FIF, i =1, 2) were assigned as follows:i,FIFwsubWith the new MLC shapes for the sub-beams and new weights for the main beams and sub-beams, the 3D dose distribution for the FIF plan with the two sub-beams (FIF) was calculated in the TPS and dose indices were evaluated. In this substudy, Dth was fixed at 105% of the prescribed dose at the isocenter.FIF plan in Step 2After Step 1, when the dose coverage of the target is greatly reduced, there is a possibility that the reduction in the dose coverage can be allevi-ated by increasing the number of sub-beams. This is because an increase in the number of sub-beams provides higher degrees of freedom for dose modi-fication. In this study, when the reduction in D95% of the PTV was >1% in Step 1, FIF treatment plan-ning with four sub-beams was performed. The criterion of 1% reduction was arbitrarily adopted as an example. It can be changed to a different criterion easily. In this step, the sub-beams were added sequen-tially. First, two sub-beams were added to the orig-inal plan, as in Step 1. The high-hotspot regions that received a dose greater than the intermediate threshold Dith=(Dmax+Dth)/2 were identified in the dose distribution in the original plan, as shown in Figure 5a. The same procedures as in Step 1 were performed by replacing Dth with Dith. After the procedures, the FIF plan with the two sub-beams was obtained. Second, two sub-beams were newly added. The low-hotspot regions that received a higher dose than Dth were identified in the FIF plan with the two sub-beams. The MLC apertures of the two extra sub-beams were shaped to block the low-hotspot regions in the BEV, as shown in Figure 5b.The weights for the main beams and the first two sub-beams were assigned, as shown in Eqs. (1) and (2), where Dth is replaced by Dith. The weights for the main beams wi,FIF2nd and extra sub-beams wi,FIF2nd in the four sub-beams FIF plan were assigned as follows:where i is the index for the main beams (i =1, 2), and wmax is the maximum dose in the FIF plan with the two sub-beams. The 3D dose distribution was calculated for the FIF plan with four sub-beams (FIF-4SF), and the dose indices were evaluated.ithsubmain379i,FIFwmainmaxi,FIFfw2ndmain(f1i,FIF2ndwmaini,FIF2ndwsub2nd2nd