Geometric optimization of beam configurations: From 2D to 3D

Eduard Schreibmann, Michael Lahanas and Dimos Baltas

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A main problem in radiotherapy treatment planning is to find an optimal number of beams and their orientation. Commonly a beam configuration is selected based on experience and then the intensity distributions of these beams are optimized.

The aim of beam orientation optimization is to find a configuration of beams such that a desired dose distribution can be achieved. A single beam would deposit a very high dose to the NT. Using more beams it is possible to increase the dose in the tumor, keeping the dose in the surrounding healthy tissue at a sufficient low level, but the treatment complexity increases.

The idea of using geometrical considerations in the cost function was first proposed by Haas et al to obtain an optimum beam configuration. Simplifications such a limitation in 2D, using the most representative 2D computed tomography slice in the plan have been used. The geometric objective functions to be minimized are:

Difference between the area where all M beams overlap and the area of the PTA:


Overlap area between each beam and the j-th OAR:

, (2)

* and are distances shown in Fig. 5 and * a parameter that favors beam entry points further away from OARs.

Overlap from pair wise beam intersections to minimize hot spots:


Figure 1 Definition of geometric parameters used by Haas et al for the solution of the beam orientation problem in radiotherapy. The gantry angle of a field (beam) is shown. The patient body including the normal tissue NT, one organ at risk (OAR) and the planning target area (PTA) which includes the tumor is shown.

A program (ARROW) has been developed that extended the limitations of the 2D approach in 3D, including the possibility of using wedges, shielding etc.


Pruning is important in terms of reducing the size of the problem. The fewer potentially good fields remain the less combinations have to be checked. The optimization of beam directions implies adjustment of the field size for every stationary field, including determination of shielding or beam shaping by a multileaf collimator. There are also beam irradiations positions not feasible in practice due to gantry-table collision or beams passing through regions where we do not have anatomical information from CT.

Beam divergence

By their nature, radiation fields are divergent, which is a property that has to be modeled in the software. One important geometric aspect, we are interested in, is the volume of intersection between the beam cone (given by the source position and the shape of the target volume) and the different organs at risk. For this reason, a simplification by using parallel beam geometry (for computational reasons) is considered inappropriate.

Beam divergence is computed using linear extrusion. Linear extrusion is a modeling algorithm that takes polygonal data as input and generates polygonal data on output. The input dataset is swept according to some extrusion function and creates new polygonal primitives. These primitives form a "skirt" or swept surface. For example, sweeping a line results in a quadrilateral, and sweeping a triangle creates a "wedge".

In Arrow, a collimator is swept along the beam direction, thus producing from a rectangle a pyramid, as in the picture bellow where the beam is viewed from lateral (represented in black). The extrusion is combined with scaling (the scaling factor being proportional with the distance from source) to model beam divergence.

The construction of beam shape follows two possibilities, according to if the user has chosen to use shielding blocks or not.

If we use blocks, the beam shape must conform to PTV.  Beam transversal shape will be equivalent to the shape of the PTV, as seen from the source.

To determine the part of PTV visible from the source, we do a culling of the polygons in PTV, based on their orientation in respect with the source position. The PTV border seen from source is defined by the regions between polygons facing and facing away the source.

Figure 2 Divergence of a beam.

Figure 3 Intersection of beams

Geometrical cost function

For finding the set of beams that would best fit our goals, we define a multi-objective cost function. In terms of geometry, the objectives are formulated as:

* Coverage of the PTV. The geometrical formulation of this objective takes into account the relative volume of the PTV covered by the intersection of the beams which should be maximized. In order to use a minimization algorithm the following objective function is used:


is the intersection volume of the NB beams Bi, i=1,2,…,NB. This function is defined in an analogue way as the Conformal Index COIN introduced by Baltas et al (1998).

* Protection of the NT. This can be achieved by using spatially distributed beams. The objective is to minimize the overlap volume between any pair combination of beams:


Beams having close entrance points will generate a large intersection volume in the overlap region and thus a high value for this objective. As the entrance points are more spaced, the intersection volume between the beams is minimized, ideally tending to the PTV volume.

* Protection of the OARs. The objective is to minimize the exposure of the OARs. This can be formulated by the geometric objective function for the kth OAR which is used to minimize the intersection volume between the beams and the OAR:

, k=1,2,…,NOAR (6)

Bi, i=1,2,…,NB are the NB beams in the plan and NOAR is the number of OARs to be considered. Ideally, there is no intersection between the OAR and any of the beams, and thus the fraction is zero.

The terms are pre-calculated and stored for all possible beams before the optimization. The calculation of these terms during the optimization would considerably increase the optimization time.

The final cost function gTot for the geometrical optimization is a sum of the above objectives:


where , , are the importance factors for the objectives of the NT, PTV and the kth OAR.

For the quality of the dose distribution we use a set of commonly used dosimetric cost function which is the weighted sum of the individual objective functions for the PTV, the NT and the OAR.

The dose based objective functions used are for the PTV the dose variance around the prescription dose for NT the sum of the squared dose values and for each OAR the variance for dose values above a specific critical dose value .




Where , and are the calculated dose values at the jth sampling point for the PTV, the NT and each OAR respectively , and are the corresponding number of sampling points.

The final cost function used by dose optimization algorithms is:


where , dan are the importance factors for the PTV, NT and the kth OAR respectively.

For calculating the dosimetric cost functions, a Clarkson dose computation model was used. The dose value at each sampling point was obtained from the interpolation of tissue phantom ratio (TPR) values and off-center ratio (OCR) values. Such values are presented in tables and are derived from percentage depth dose and beam profile measurements in water phantom.

The graphic library “The Visualization Toolkit” (VTK), a large set of visualization classes as well as associated algorithms, was used to solve the geometrical problems, such as clipping, optimal oriented bounding boxes.


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Update 20-4-2004

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