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Computational and Mathematical Methods in Medicine
Volume 2016 (2016), Article ID 9515794, 13 pages
Research Article

Lateral Penumbra Modelling Based Leaf End Shape Optimization for Multileaf Collimator in Radiotherapy

Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China

Received 19 February 2015; Accepted 10 May 2015

Academic Editor: Hugo Palmans

Copyright © 2016 Dong Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Lateral penumbra of multileaf collimator plays an important role in radiotherapy treatment planning. 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. In our study, an analytical method for leaf end induced lateral penumbra modelling is formulated using Tangent Secant Theory. Compared with Monte Carlo simulation and ray tracing algorithm, our model serves well the purpose of cost-efficient penumbra evaluation. Leaf ends represented in parametric forms of circular arc, elliptical arc, Bézier curve, and B-spline are implemented. With biobjective function of penumbra mean and variance introduced, genetic algorithm is carried out for approximating the Pareto frontier. Results show that for circular arc leaf end objective function is convex and convergence to optimal solution is guaranteed using gradient based iterative method. 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. For treatment modalities in clinical application, optimized leaf ends are in close agreement with actual shapes. Taken together, the method that we propose can provide insight into leaf end shape design of multileaf collimator.