For treatment modalities in clinical application, optimized leaf ends are in close agreement with actual shapes. It is found that optimal leaf end in the shape of Bézier curve achieves minimal standard deviation, while using B-spline minimum of penumbra mean is obtained. Results show that for circular arc leaf end objective function is convex and convergence to optimal solution is guaranteed using gradient based iterative method. With biobjective function of penumbra mean and variance introduced, genetic algorithm is carried out for approximating the Pareto frontier. Leaf ends represented in parametric forms of circular arc, elliptical arc, Bézier curve, and B-spline are implemented. Compared with Monte Carlo simulation and ray tracing algorithm, our model serves well the purpose of cost-efficient penumbra evaluation. In our study, an analytical method for leaf end induced lateral penumbra modelling is formulated using Tangent Secant Theory. Growing evidence has revealed that, for a single-focused multileaf collimator, lateral penumbra width is leaf position dependent and largely attributed to the leaf end shape. Lateral penumbra of multileaf collimator plays an important role in radiotherapy treatment planning.
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