The present study is aimed at comparing the planning and delivery efficiency between three-dimensional conformal radiotherapy (3D-CRT), field-in-field, forward planned, intensity modulated radiotherapy (FIF-FP-IMRT), and inverse planned intensity modulated radiotherapy (IP-IMRT). Treatment plans of 20 patients with left-sided breast cancer, 10 post-mastectomy treated to a prescribed dose of 45 Gy to the chest wall in 20 fractions, and 10 post-breast-conserving surgery to a prescribed dose of 50 Gy to the whole breast in 25 fractions, with 3D-CRT were selected. The FiF-FP-IMRT plans were created by combining two open fields with three to four segments in two tangential beam directions. Eight different beam directions were chosen to create IP-IMRT plans and were inversely optimized. The homogeneity of dose to planning target volume (PTV) and the dose delivered to heart and contralateral breast were compared among the techniques in all the 20 patients. All the three radiotherapy techniques achieved comparable radiation dose delivery to PTV-95% of the prescribed dose covering > 95% of the breast PTV. The mean volume of PTV receiving 105% (V105) of the prescribed dose was 1.7% (range 0-6.8%) for IP-IMRT, 1.9% for FP-IMRT, and 3.7% for 3D-CRT. The homogeneity and conformity indices (HI and CI) were similar for 3D-CRT and FP-IMRT, whereas the IP-IMRT plans had better conformity index at the cost of less homogeneity. The 3D-CRT and FiF-FP-IMRT plans achieved similar sparing of critical organs. The low-dose volumes (V5Gy) in the heart and lungs were larger in IP-IMRT than in the other techniques. The value of the mean dose to the ipsilateral lung was higher for IP-IMRT than the values for with FiF-FP-IMRT and 3D-CRT. In the current study, the relative volume of contralateral breast receiving low doses (0.01, 0.6, 1, and 2Gy) was significantly lower for the FiF-FP-IMRT and 3D-CRT plans than for the IP-IMRT plan. Compared with 3D-CRT and IP-IMRT, FiF-FP-IMRT proved to be a simple and efficient planning technique for breast irradiation. It provided dosimetric advantages, significantly reducing the size of the hot spot and minimally improving the coverage of the target volume. In addition, it was felt that FiF-FP-IMRT required less planning time and easy field placements.

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The existing literature discusses different strategies to solve a scalar ordinary differential equation using Lie point symmetries. We focus on three of these strategies in order to frame methods for finding solutions of non-linear systems of ordinary differential equations. These include Lie's integration theorem, method of successive reduction of order and the method of using the invariants of the admitted symmetry generators. Illustrative examples and those taken from mechanics are presented to highlight the use of these methods.

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