Eduard Schreibmann, Michael Lahanas and Dimos Baltas

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:

(1)

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:

_{
}(3)

**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.

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:

_{
}(4)

_{
}is the intersection volume
of the *N _{B}* beams B

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:

_{
}(5)

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 k^{th} OAR which is used to minimize the
intersection volume between the beams and the OAR:

_{
}, k=1,2,…,*N _{OAR}*
(6)

B_{i},
i=1,2,…,*N _{B}* are the

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 *g _{Tot}* for the geometrical
optimization is a sum of the above objectives:

_{}
(7)

where
_{
},
_{
,
}
are the importance factors for the objectives of the NT, PTV and the
k^{th} 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
.

(8)

(9)

(10)

Where
_{
,
}and _{
}are the calculated dose values at the j^{th} 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:

(11)

where
_{
}, _{
}dan _{
}are the importance factors for the PTV, NT and the k^{th}
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.

**References**

E.
Schreibmann, R. Uricchio, M. Lahanas, K. Theodorou, C. Kappas, N.
Zamboglou and D. Baltas, A geometry based optimisation algorithm for
conformal external beam orientation. Phys. Med.
Biol. **48** 1825-1841, 2003 Abstract
**Preprint**

Cho
B J, Roa W H, Robinson D, Murry B. The development of target-eye-view
maps for selection of coplanar or noncoplanar beams in conformal
radiotherapy treatment planning, *Med.* *Phys. 26*
2367–2372

O.
C. L. Haas, K. J. Burnham and J. A. Mills Optimization of beam
orientation in radiotherapy using planar geometry *Phys. Med. Biol.*
**43** 2179—2193 (1998)

E.
Schreibmann, M. Lahanas, L. Xing and D. Baltas, Multiobjective
evolutionary optimization of number of beams, their orientations and
weights for IMRT, submitted for publication to *Phys. Med. Biol*.
2003

W.
J. Schroeder. K. Martin, W. E. Lorensen. *The Visualization
Toolkit: An Object-Oriented Approach to 3D *

*Graphics.
*Prentice Hall, Upper Saddle River, NJ, 1996

W. J. Schroeder, W. E. Lorensen, G. D. Montanaro and B. Yamrom, A Run-Time Interpreted Environment for Rapid Application Development.” GE Corporate R&D Report No. 91CRD266. January, 1991

W.
J. Schroeder, K. M. Martin, and W. E. Lorensen, The design and
implementation of an object-oriented toolkit for 3d graphics and
visualization, in R. Yagel and G. M. Nielson, editors, *IEEE
Visualization ’9*6, pages 93–100. IEEE CS Press, 1996

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