2. methods and materials

2. Methods and materials
The treatment plan optimization usually requires the construction of an objective function to establish a mathematical link between the treatment objective and the input beam parameters. This function provides a quantitative measure of the `goodness’ of a plan. The optimization of the function with respect to the beam parameters yields the best possible plans (as judged by the objective function). In this section, we will first give an overview of the optimization process. The dose distribution of a wedged field with an arbitrary wedge angle and orientation is required during the optimization. This problem and related issues will be discussed in Sections 2.2 and 2.3.
2.1. Optimization
Wedge optimization falls into the realm of inverse treatment planning, where the beam parameters are adjusted iteratively under the guidance of an objective function until an optimal solution that minimizes the objective function is obtained [11,14,20]. Given the incident directions and beam energies of J beams, the dose distribution d(n) is a function of the beam weights, wedge angles, and wedge orientations. In this work, a dose-based objective function was chosen and is defined as

Where d0(n) and d(n) are the prescribed and calculated dose distributions, respectively, n is the voxel index, N is the total number of voxels, and rs is the relative importance factor of structure s. The importance factor rs can also be written as a function of n, and its choice is based on clinical considerations. In this paper, a uniform dose of 100% was prescribed to the target and the prescribed dose to all other tissues was set to 0. For the target, different importance factors could be assigned to situations of overdosage and underdosage in order to re¯ect our preference to the two different types of deviations. The objective function in Eq. (1) was designed to achieve a uniform dose to the target while keeping the sum of squares of the

dose to other tissues to a minimum if non-zero importance factors were chosen. Other types of objective functions, such as dose volume histogram-based [19,22] or biological model-based objective functions [1,5], could be implemented similarly.
The objective function in Eq. (1) is a function of beam weights, wedge angles, and wedge orientations of the J beams, and our task is to find the optimal set of the parameters that minimize Fobj. Contrary to previous approaches [11,26], here we optimize the system directly with respect to the variables of beam weights, wedge angles and orientations using the method of simulated annealing [13]. The technique promises to find the global minimum when local minima exist, which is likely the case when the number of beams is large. The algorithm, originally introduced by Metropolis et al. [7], tries to mimic the behavior of a system of interacting particles that are progressively cooled and allowed to maintain thermal equilibrium while reaching the ground state. Our implementation of the simulated annealing algorithm was modified from that of Press et al. [13]. The program starts with a random set of beam parameters (beam weights, wedge angles, and wedge orientations of J beams), which then undergo approximately random changes within the specified ranges to explore different combinations of beam parameters. The starting system `temperature’ was chosen to be larger than the largest value of the objective function from a random set of beam parameters.