Optimizing Beam Selection for Non-Coplanar VMAT Treatment Planning with Simulated Annealing 

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Intensity-modulated radiotherapy (IMRT)

Intensity modulation radiotherapy (IMRT) is a treatment planning technique where beams of non-uniform radiation intensity are used for tumor irradiation in order to achieve a conformal dose distribution. The IMRT technique makes use of radiation beams of modulated intensity as illustrated in figure 1.7. This property differentiates IMRT from 3D-CRT which makes use uniform intensity beams. The modulation of beam intensity is made possible in IMRT by the sub-division of a beam into beamlets. A single beamlet is a subset of a radiation beam created using a grid that subdivides the beam into several smaller units. The smaller the size of the beamlet, the better the amount of shielding to the organs-at-risk during treatment [30] [31]. The increase or decrease of the intensity of individual beamlets is used to achieve modulation. risk avoidance. The different radiation beam profiles are used to target different regions of interest [5].
An improvement in dosimetry has been observed using IMRT compared to 3D-CRT in patient cases where the planning treatment volume is concave and located close to sensitive organs. IMRT enables the delivery of a concave shaped dose distribution around the target tumor cells. In IMRT, the intensity of the rays passing through the sensitive organs are reduced while the intensity of the rays passing through the tumor is increased to achieve modulation. The non-uniform radiation intensities used in IMRT are calculated by an inverse planning process as illustrated in figure 1.8.

Volumetric modulated arc therapy (VMAT)

Volumetric modulated arc therapy (VMAT) is an IMRT delivery technique that involves the continuous rotation of the gantry while the radiation beam is on during patient irradiation [71]. In step-and-shoot IMRT, the gantry is fixed at each orientation where radiation is delivered, leading to an increased usage of monitor units (MU) for treatment. VMAT overcomes this problem by using a continuously rotating gantry during treatment thus delvering lower monitor units on the whole. The lower monitor units recieved by the patient results in a significant improvements in organ-at-risk sparing e.g.a lower dose on the skin.
VMAT allows a variation in different machine parameters such as dose rate, gantry speed, and MLC leaf positions during irradiation. VMAT differs from 3D-CRT in the sense that the treatment beam does not have to conform to a shape of the tumor. The motion of the gantry is approximated as a change in the angular distance X\ between two adjacent radiation beams.
We denote a single radiation beam orientation as a control point. The motion of the MLC leaves between two control points is approximated as a change in the length X3 of the MLC. Restrictions are placed on the motion of the MLC leaf and fluence intensity between two control points so that they are not too dissimilar. The optimization for VMAT is more challenging than other forms of IMRT due to the inclusion of machine parameters in the optimization [72].
VMAT is more time efficient and produces a conformal dose distribution of equal or superior plan quality compared to IMRT. The parameters for the VMAT optimization depends on the target LINAC equipment on which the plan is delivered. Modern Linear accelerators have the ability to vary the dose rate, the gantry speed and the MLC apertures simultaneously thereby lending themselves for use in VMAT delivery. We can distinguish two main methods to performVMAT optimization, aperture-based method and leaf trajectoryVMAT method.

VMAT treatment plan optimization

Given the 3D computed tomography (CT) image of a cancer patient, the entire volume contains a set ofVdiscrete voxels. The incident radiation beam 1 for patient treatment at each orientation can be decomposed into a set of beamlets. The dose received by a voxel 9 2 V is denoted as 39 and the intensity of an incident beamlet 8 2 1 with fluence intensity G8 . B denotes the set of candidate beam angles that are available for the VMAT treatment planning. Such that 1 2 B a set of equally spaced couch-gantry angle pairs that constitute a 4c space. At the initial stage, the infeasible or collision-prone configurations are removed from B. The dose influence matrix, D, expresses the relationship between the dose received by a voxel 39 and a beamlet of unit fluence intensity G8 , such that: Given a dose prescribed by the physician d? to be delivered to the tumor, our objective function minimizes the least-square deviation between the prescribed dose and the actual dose received by the tumor voxels.

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Non-coplanar VMAT planning methodology

Our planning follows the recent state of the art approach [93] with an aim to improve on the beam angle selection. Beam angle selection is performed to obtain optimal beams from the set of all candidate beam angles to serve as control points for the treatment trajectory. A traveling salesman problem is solved to connect all the optimal beam angles obtained from beam selection to form a valid treatment trajectory. Finally a direct leaf trajectory optimization is performed to obtain the final plan using the new trajectory.
In order to improve the beam angle selection step, our approach makes use use of direct leaf trajectory optimization to evaluate the fluence contribution during the beam angle selection instead of fluence map optimization. The direct leaf trajectory optimization is used during the beam angle selection phase to improve comparison between treatment plans and to better account for MLC and machine constraints. Another difference in our approach is that there is no-limitation on the space of the candidate beam angles to be explored during the search for with Simulated Annealing optimal beams. Therefore a very large space of candidate beam orientations is available for selecting the optimal beam orientations.

Beam selection algorithm using simulated annealing

Simulated annealing [123] is a widely used heuristic method that is used to find a global solution to a combinatorial optimization problem. It mimicks the annealing process in metallurgy with the notion of slow cooling interpreted as a slow decrease in the probability of accepting suboptimal solutions as the algorithm proceeds. Simulated annealing has been applied to beam selection for intensity-modulated radiotherapy (IMRT) planning [124] and selecting optimal seed locations during inverse brachytherapy treatment planning [125].

Table of contents :

1 Introduction 
1.1 Cancer
1.2 Radiotherapy
1.3 Types of Radiotherapy
1.4 External Radiotherapy
1.5 Thesis Objective Statement
2 Collision Detection for Non-coplanar VMAT 
2.1 Introduction
2.2 Geometric Setup
2.3 Co-simulation with MATLAB
2.4 Evaluation Study
2.5 Discussion
3 Optimizing Beam Selection for Non-Coplanar VMAT Treatment Planning with Simulated Annealing 
3.1 Introduction
3.2 VMAT treatment plan optimization
3.3 Results
3.4 Discussion
4 A Sampling-based approach for Non-Coplanar VMAT Treatment Planning using RRT 
4.1 Introduction
4.2 Rapidly-exploring random trees (RRT)
4.3 Asymptomatic sub-optimality of the RRT algorithm
4.4 Improved Rapidly-exploring random trees (RRT⇤)
4.5 Motivation for choosing  »)⇤ algorithm
4.6 New  »)⇤ algorithm for Non-coplanar VMAT planning
4.7 Modifications made to  »)⇤ for non-coplanar VMAT planning
4.8 Hardware and software Implementation Details
4.9 Results
4.10 Discussion
Conclusion and Future Work


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