A Mathematical Method to Adjust MLC Leaf End Position for Accurate Dose Calculation in Carbon Ion Beam Radiation Therapy Treatment Planning System

Table 2

The tungsten MLC leaf end offset correction of 400 MeV/n carbon ion beam at different fields.

Field size/2 (cm)	(degree)	(radian)			() (radian)	Projection on the -axis of ()	Projection on the -axis of (light field, cm)	Projection on the -axis of (MLC plan position, mm)	MLC plan position shift to (mm)	Light-radiation agreement (mm)	Penumbra (20%-80%) (mm)
0.5	0.1088	0.0019	0.0019	0.0020	0.0019	0.5050	0.5	0.5081	-0.0301	-0.0504	1.010
1	0.2176	0.0038	0.0039	0.0039	0.0038	1.0110	1	1.0161	-0.0510	-0.11004	1.160
1.5	0.3264	0.0057	0.0058	0.0059	0.0058	1.5165	1.5	1.5241	-0.0763	-0.16515	1.310
2	0.4352	0.0076	0.0077	0.0078	0.0077	2.0197	2	2.0322	-0.1248	-0.19739	1.460
2.5	0.5440	0.0095	0.0096	0.0098	0.0096	2.5230	2.5	2.5403	-0.1729	-0.22978	1.610
3	0.6528	0.0114	0.0116	0.0118	0.0115	3.0285	3	3.0483	-0.1979	-0.28534	1.760
3.5	0.7616	0.0133	0.0135	0.0137	0.0134	3.5322	3.5	3.5564	-0.2421	-0.32171	1.910
4	0.8704	0.0152	0.0154	0.0157	0.0153	4.0368	4	4.0644	-0.2767	-0.36769	2.060
4.5	0.9791	0.0171	0.0174	0.0176	0.0172	4.5414	4.5	4.5725	-0.3112	-0.41367	2.210
5	1.0879	0.0190	0.0193	0.0196	0.0192	5.0460	5	5.0805	-0.3458	-0.45967	2.360
5.5	1.1967	0.0209	0.0212	0.0216	0.0211	5.5525	5.5	5.5886	-0.3606	-0.52544	2.510
6	1.3054	0.0228	0.0232	0.0235	0.0230	6.0598	6	6.0967	-0.3689	-0.59767	2.660
6.5	1.4142	0.0247	0.0251	0.0255	0.0249	6.5618	6.5	6.6047	-0.4288	-0.61831	2.810
7	1.5229	0.0266	0.0270	0.0274	0.0268	7.0690	7	7.1128	-0.4379	-0.68975	2.960
7.5	1.6316	0.0285	0.0289	0.0294	0.0288	7.5736	7.5	7.6208	-0.4724	-0.73581	3.110
8	1.7403	0.0304	0.0309	0.0314	0.0307	8.0782	8	8.1289	-0.5068	-0.78189	3.260
8.5	1.8490	0.0323	0.0328	0.0333	0.0326	8.5828	8.5	8.6369	-0.5413	-0.82798	3.410
9	1.9577	0.0342	0.0347	0.0353	0.0345	9.0874	9	9.1450	-0.5757	-0.87409	3.610