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British Journal of Radiology (2005) 78, 316-327
© 2005 British Institute of Radiology
doi: 10.1259/bjr/13015047
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Full Paper

Automatic selection of non-coplanar beam directions for three-dimensional conformal radiotherapy

J Meyer, PhD1,2, S M Hummel, RT(T), CMD1, P S Cho, PhD1, M M Austin-Seymour, MD1 and M H Phillips, PhD1

1 University of Washington Medical Center, Department of Radiation Oncology, Box 356043, Seattle, WA 98195, USA and 2 Universität Würzburg, Klinik für Strahlentherapie, Josef-Schneider-Str. 11, D-97080 Würzburg, Germany


   Abstract
Top
 Abstract
Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
 Results
 Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
 References
 
An algorithm is described, based on ray-tracing and the beam's-eye-view, that exhaustively searches all permitted beam directions. The evaluation of the search is based on a general cost function that can be adapted to the clinical objectives by means of parameters and weighting factors. The approach takes into account the constraints of the linear accelerator by discarding beam directions that are not permitted. A sensitivity analysis was carried out to determine appropriate parameters for different sized organs, and a prostate case was used to benchmark the approach. The algorithm was also applied to two clinical cases (brain and sinus) to test the benefits of the approach compared with manual angle selection. The time to perform a beam direction search was approximately 2 min for the coplanar and 12 min for the non-coplanar beam space. The angles obtained for the prostate case compared well with reports in the literature. For the brain case, the mean dose to the right and left optic nerves was reduced by 12% and 50%, respectively, whilst the target dose uniformity was improved. For the sinus case, the mean doses to the right and left parotid glands were reduced by 54% and 46%, respectively, to the right and left optic nerves by 37% and 62%, respectively, and to the optic chiasm by 39%, whilst the target dose uniformity was also improved. For the clinical cases the plans based on optimized beam directions were simpler and resulted in better sparing of critical structures compared with plans based on manual angle selection. The approach provides a practical alternative to elaborate and time consuming beam angle optimization schemes and is suitable for routine clinical usage.


   Introduction
Top
 Abstract
 Introduction
Methods and materials
 Parameter value selection
 Clinical examples
 Results
 Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
 References
 
The maximum radiation dose that can be deposited in a tumour is limited by the tolerance dose of the surrounding vital organs and critical structures [1]. Inherent objectives for a three-dimensional conformal radiation therapy (3D-CRT) treatment plan are to use beam directions such that the irradiation of the organs at risk (OAR) is minimized so as to reduce normal tissue complications, and to ensure an adequate dose coverage of the planning target volume (PTV).

This work concentrates on conventional 3D-CRT since it still constitutes most clinical treatments despite the increasingly widespread availability of intensity-modulated radiation therapy (IMRT). It was suggested by Webb [2] that only about 30% of clinical problems require the use of IMRT. The impetus of this work is to bridge the gap between 2D treatment planning and computer optimization of IMRT, since the efficacious use of 3D-CRT relies strongly on the judicious selection of non-coplanar beam angles. Rosen estimated in an industrial report in 1999 that only about one-third of facilities performed 3D-CRT regularly [3], but this number has probably increased significantly since then.

The use of non-coplanar 3D-CRT beams, as opposed to coplanar arrangements, has the potential to produce superior plans due to its increased ability to avoid critical structures [4–6]. The beam angle orientation problem has been the subject of a number of studies. The different approaches are based on geometric/volumetric methods [7–13], dose-based objectives [13–16], radiobiological objectives [5, 13] or IMRT specific objectives [17–19].

The geometric and volumetric methods for 3D-CRT are based on the assumption that the avoidance of the OAR is fundamental to the planning process. Further steps, such as optimization of beam weights [20, 21], beam portals [22, 23] or beam intensity modulation [2, 24], can be carried out independently during the next planning stage to achieve target dose homogeneity and to satisfy the dose constraints on the OAR. The work presented in this paper is also based on this principle.

The approaches based on dose and radiobiological objectives as well as IMRT-specific algorithms, are more complex and some of them integrate the beam angle search into the actual plan optimization process [18, 19]. While these approaches have the most potential to find the optimal plan, they are computationally expensive since it is necessary to perform dose calculations during the optimization process. Furthermore, the standard deterministic optimization algorithms are not sufficient to prevent the algorithm becoming trapped in local minima [14, 25–27]. The computational effort required to obtain optimal solutions, e.g. ranging from hours [18] up to days [6], makes some of these approaches prohibitive for routine clinical use, even when sophisticated methods are applied that reduce the computation time by an order of magnitude [28].

For coplanar plans, and only one or two OARs, it is fairly straightforward for an experienced dosimetrist to judge which beam directions are preferable. However, it is not feasible for a human planner to exploit the entire beam space when multiple OARs of different size and radiosensitivity are present. Another difficulty stems from the inability to deliver isocentric beams from any direction owing to limitations relating to the dimensions of the treatment couch, linac design, patient dimensions and the location and size of the tumour. To represent accurately the permitted directions for each individual treatment requires precise modelling of the linac, the couch and the patient. This task is complex since the permitted beam space is a function of the relative location of the tumour [29–31].

Several attempts to automate the process of selecting beam directions have been described in the literature. Chen et al [8] and Myrianthopoulos et al [9] used the beam's-eye-view (BEV) to obtain 2D plots of the percentage volume of OAR irradiated versus the gantry angle as an aid to select beam directions. Rowbottom et al [13] went a step further by combining the BEV volumetric approach with a simple dose model and biological considerations to obtain a single objective score for each beam direction. Pugachev and Xing [17] formulated an IMRT specific score function that was based on the maximum dose deliverable through a portal by means of an intensity modulated beam. The current work is similar to the work of the above authors. The BEV was used as a means to search for beam directions [8, 9], the cost function was introduced to obtain a score for each beam direction [13] and an exhaustive search was conducted to explore the non-coplanar beam space [17]. An exhaustive search circumvents the local minima problem and ensures that the results are reproducible independently of the starting conditions. In addition, the current work takes into account the couch-gantry-patient dimensions, evaluates the optimization by means of logical rules and uses a unique cost function for which optimal parameters were derived.


   Methods and materials
Top
 Abstract
 Introduction
 Methods and materials
Parameter value selection
 Clinical examples
 Results
 Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
 References
 
A beam direction search tool (BDST) was developed and implemented on the University of Washington treatment planning system (TPS) PRISM [32]. This tool evaluates a cost function, Ci for each beam direction, {Phi}i({xi},{psi}), in the domain i=1...t, where t is the total number of beam directions considered, and where each beam direction is defined by the gantry angle, {xi}, and the couch angle, {psi}. The BDST starts with both the gantry angle and couch angle set to zero, and then systematically works through all other permitted directions by increasing first the gantry angle and then the couch angle by multiples of a chosen angular increment, {varpi}. Beam directions that would cause the gantry to collide, either with the couch or with the patient, are excluded from the search. The beam space for the Elekta SL20 series linac (Elekta, Crawley, UK) was modelled for different treatment sites in combination with the Elekta PreciseTM couch and the female RandoTM phantom (The Phantom Laboratory, Salem, NY), by assuming a constant isocentre distance of 100 cm. An example of the beam space for tumours in the head is shown in Figure 1Go, where {Phi}P({xi},{psi}) denotes the permitted directions. Note that PRISM uses different angle conventions to the Elekta accelerator and therefore two axes are plotted in Figure 1Go.



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  		Figure 1. The area that is unshaded denotes the permitted beam space {Phi}P({xi},{psi}). The shaded region represents angles that result in a gantry collision and are, therefore, excluded from the beam search.

 
Each beam portal is shaped to match the projection of the PTV into the BEV and so only the rays represented by the solid circles in Figure 2aGo are considered. For each beam direction the BDST first computes distances along ray paths as illustrated in Figure 2bGo. For ray, j, these distances are the depth from the patient entry surface to the PTV, dj(PTV), the depth from the patient entry surface to each OAR in the beam, dj(OARk), where k is an index for the OAR in question, and the path length through the OAR, lj(OARk). By definition the central ray is the one that passes through the isocentre. The rays are projected onto a Cartesian grid located at a distance of 120 cm from the source (Figure 2Go) such that they are separated by equal increments, s, at the isocentre plane.



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  		Figure 2. Ray-tracing geometry: (a) beam's-eye-view (BEV) illustrating the grid with uniform spacing s and (b) axial slice with planning target volume (PTV) and two organs at risk (OAR).

 
The BDST uses the dj(PTV) values to determine the mean depth to the PTV entry surface for the ith beam direction, di(PTV), and similarly calculates a mean depth to each OARk entry surface, di(OARk). It also estimates the fractional volume of each OARk irradiated, vi(OARk), using the ray path lengths, ray separations s and a pre-calculation of the total OARk volume.

The cost function was designed such that the lower its value the more preferable is the beam direction and it has two components that are added together. The first component, Ci(PTV), is based on the mean depth of the PTV, and the second, Ci(OAR), is a penalty term, which depends on the amount of OAR irradiation, thusGo


{780316E001}

The first component term is computed using the equationGo


{780316E002}

where µ is the linear attenuation coefficient in water for the proposed beam energy. The exponential term, with µ set to a value of 0.0495 cm–1 throughout this paper, corresponds to the attenuation of a 6 MV photon beam in water. Thus, the value of Ci(PTV) approaches zero when the PTV is at shallow depths and approaches unity for very large depths, so favouring those beam directions where the PTV is close to the beam entry surface.

As several different OARs may be included in any particular beam direction, the second component is a weighted average of individual costs computed for each OAR in turn, where different weighting factors, wk, may be assigned to give each different OAR a different relative importance.

Each OAR has both a "depth cost" and a "volume cost". The depth cost for the kth OAR is calculated using the equation:Go


{780316E003}

The same value of µ is used as in Equation 1Go, but, unlike Equation 1Go, this gives higher costs (approaching unity) for the shallower depths where the relative dose to the OAR will be greater. If the OAR is completely outside the beam edge in the BEV the depth cost is set to zero.

The volume cost for each OAR is set equal to the fractional volume irradiated, vi(OARk), which is already conveniently in the range zero to one. In combining the volume and depth costs a factor {lambda} (where 0<{lambda}<1) was introduced so as to control the relative importance of depth and volume, and to ensure that the total cost remains within the range 0 to 1; thus the cost associated with irradiating the kth OAR in the ith beam direction is quantified by:Go


{780316E004}

The angle optimization tool was designed so that the user specified the number of beam directions, nb, required for a plan and the minimum angle separation, {varphi}, between selected angles. To further ensure beam diversity at least one of all nb beams had to be separated by an angle {delta}, with 90°≤{delta}≤120° from the other beams that were selected. An empirical value of {delta}=90° was assumed throughout the work presented in this paper to ensure some dose uniformity in the PTV.

The optimization started by discarding all the beam directions that were not permitted within the specified gantry and couch ranges. The algorithm then iterated through the remaining angle combinations of {xi} and {psi} and created a rank list with the beam directions sorted according to the lowest cost. The direction that topped the list was selected as the first beam. The second direction was selected subject to the criterion that the absolute angle between the first and the second direction was greater or equal to {varphi}. The value of {varphi} depends on the total number of beams and the location of the tumour. However, experience has shown that a minimum separation of 40–45° usually results in satisfactory plans. If the angle was not large enough, the algorithm proceeded to the next position in the rank list until it found an appropriate direction. After the second direction was selected, the algorithm searched for the third direction in the rank list that was separated by a minimum angle {varphi} from any of the previously selected beam directions. The same procedure was carried out for all nb beam directions. For the selection of the last beam direction, the algorithm also examined whether any of the previously selected beam directions and the current candidate direction were separated by at least {delta}, otherwise the angle was discarded and the algorithm proceeded to the next best direction until one meeting the criterion was found.


   Parameter value selection
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
Clinical examples
 Results
 Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
 References
 
Ray-tracing resolution
The trade-off between computation time and a fine spatial resolution was explored by varying the value of s. Since the selection of s depends on the size of the organ, a simple test plan, shown in Figure 3Go, was produced with OAR of different realistic sizes, i.e. Ø(OAR1)=10 mm, Ø(OAR2)=20 mm, Ø(OAR3)=35 mm and Ø(OAR4)=50 mm. The axial length of each OAR was equal to its diameter, as illustrated in Figure 3bGo, and the diameter of the PTV was 25 mm. The diameter of the external contour was 120 mm. For the purpose of this sensitivity analysis, {psi} was set to zero (reducing the search space to coplanar beam arrangements) the gantry range was set to {xi}{0°–360°}, the angle resolution {varpi}=10°, {lambda}=0.2, nb=5, {varphi}=45° and the weighting factors were set to unity. The values of s investigated were 1 mm, 2 mm, 3.3 mm, 5 mm and 10 mm. The variation of the total cost (Equation 1Go) as a function of the ray-trace resolution was analysed. All the computation for this work was carried out on a PC running Linux with a CPU speed of 2.4 GHz.



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  		Figure 3. Test plan. (a) Axial slice and (b) lateral view (from left side).

 
Angle resolution
Beam directions were sampled at uniform angle intervals {varpi}, as illustrated in Figure 3Go. For the search to be effective it is essential to keep {varpi} as small as possible. If the resolution is too coarse, potentially good beam directions could be missed. To determine an appropriate value for {varpi}, the beam space for the test plan in Figure 3Go was searched for different values of {varpi}, namely 2°, 5°, 10°, 15° and 30°. The grid increment, s, was set to 3.3 mm and the other parameters remained as described in the previous paragraph.


   Clinical examples
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
Results
 Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
 References
 
The BDST was first applied to a prostate case and the resulting beam arrangement compared with clinically approved beam directions from the literature. The tool was then tested on two clinical cases with tumours in the head.

Prostate comparison
The prostate is well suited for an initial comparison since the anatomy is comparable from patient to patient and coplanar beam arrangements are standard. In our institution, the gantry angles for standard 3D conformal prostate treatments are based on the recommendations by Pickett et al [33]. It is a coplanar 6-field conformal technique with opposing lateral beams and opposed oblique beams angled between 30° and 45° off the lateral beam position. The lateral beams penetrate through the femoral heads and the lateral oblique beams are placed such that they minimize the irradiation of the bladder, the rectum and the femoral heads. The dose to the femoral heads can be controlled by the relative beam weights of the opposed lateral beams.

For the first run, the BDST was applied to an arbitrarily selected prostate case with the parameters for the search set to {xi}{0°–360°}, {psi}{0°}, s=3.3 mm, {varpi}=1°, {varphi}=40°, {lambda}=0.2 and nb=6. Only the rectum and the bladder were considered OAR and the corresponding weighting factors, wk, were set to 10 and 5, respectively.

For the second run nb=5 and {varpi} was increased to 5°. In addition, the femoral heads were considered critical organs. The weighting factors applied were 10 for the rectum, 1 for the bladder and 2 for both femoral heads. These parameters were chosen following the findings by Rowbottom et al [13] who utilized a cost function with volumetric and biological elements to optimize the beam directions for prostate treatment with similar weighting factors.

Clinical case studies
Tumours in the head are especially suited for non-coplanar beam arrangements since numerous critical structures of different importance are to be spared and the permitted beam space is much larger since beams directed through the top of the head are an option. Two cases were studied to test the algorithm. Case I was a patient with a brain tumour and Case II was a patient with a sinus tumour. For each, an experienced senior dosimetrist had produced a 3D-CRT plan, which was used to treat the patient. These will be referred to as the conventional plans. The dosimetrist was confined to non-IMRT beams but could use any means available for 3D-CRT clinical routine planning. This included the use of non-coplanar beams, different energies, optimization of the MLC aperture, customized blocks, electron beams and wedges. These plans were compared, retrospectively, with plans based on beam directions obtained from the beam direction search, and these will be referred to as the optimized plans. The same dosimetrist produced the optimized plans using the same treatment planning tools as previously. Uniform margins of 5 mm for the brain case and 7 mm for the sinus case were applied, to the clinical target volume (CTV) in order to obtain the PTVs and a voxel size of 3.3 mm was used to calculate the final dose distribution. An additional margin of 5 mm was applied to the PTV for the purpose of the beam direction search. This was done to account for the fact that lateral electron transport takes place when the beams are delivered, and without this additional margin, beam directions for which the projection of an OAR is very close to the PTV, but does not overlap with it, might appear to be good. However, when the radiation is delivered the electron transport would result in a high dose to the OAR. The optimized plans were required to use an equal number of beams as the conventional plans. The beam weights and wedge angles were optimized manually for the first case study (brain) and re-optimized with simulated annealing for the second case study (sinus). The plans were compared by means of dose–volume histograms (DVH) and statistical evaluation based on the recommendations for reporting target volume and dose specifications for IMRT plans [34], which is based on ICRU50 [1]. Further evaluation makes use of the Webb and Nahum model [35] for tumour control probability (TCP), the Lyman-Kutcher-Burman model [36, 37] for normal tissue complications probability (NTCP) and the radiation conformity index [38] to the 95% isodose surface (RCI95). The parameters for the NTCP model were based on data published by Emami et al [39].

Case study I: brain
A 12-year-old paediatric patient with a cavernous meningioma was selected. The PTV was located in-between the temporal lobes and overlapped approximately 25% of the right optic nerve (Figure 4Go). The patient had an acute onset of blindness in his right eye before radiotherapy. The left optic nerve was in close proximity to the PTV. The total dose prescribed to the PTV was 50.4 Gy. Since the tumour was impinging on the optic nerves it was important to reduce the dose to these structures, despite a tolerance dose of approximately 50 Gy, so as to further minimize the impact on the patient's eyesight. This was considered important given the young age of the patient.



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  		Figure 4. Brain case: axial CT slice through a paediatric meningioma.

 
The conventional plan consisted of three wedged beams. Two of the beams were opposed lateral coplanar 6 MV beams ({Phi}1(90°,0°) and {Phi}2(270°,0°)) and the third was a left superior oblique 18 MV beam ({Phi}3(100°,300°)). For the optimized plan the following parameters were applied for the beam direction search: {xi}{0°–360°}, {psi}{0°–100°,260°–360°}, s=0.33, {varpi}=10°, {varphi}=45°, {lambda}=0.2, nb=3 and the weighting factors wk for the OAR were set to 7, 5, 10 and 8, corresponding to the right and left eye, and the right and left optic nerve, respectively. A higher weighting was given to the OAR on the right lateral side compared with the corresponding counterparts on the left because of the overall location of the tumour and the proximity to the right optic nerve. The optic nerves were considered the most important structures, followed by the eyeballs. To allow a fair comparison, both plans were scaled to D95 so that the 95% isodose surface covered the entire PTV.

Case study II: sinus
The second clinical case investigated was a large T4N0M0 sinus adenocarcinoma. A coronal slice through the centre of the tumour is shown in Figure 5Go. The volume of the PTV was approximately 128 cm3 and was close to the brain stem and the parotid glands. Superiorly, the PTV protruded between the eyeballs, which made it difficult to spare the optic nerves and the optic chiasm while at the same time delivering the prescribed dose of 70 Gy to the PTV. It was aimed for tolerance doses of 50 Gy for the optic nerves and the optic chiasm, 52 Gy for the brain stem and 10 Gy to the lens to minimize the risk of cataracts. A secondary priority was to prevent xerostomia by reducing the dose to the parotid glands.



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  		Figure 5. Sinus case: coronal CT slice through tumour.

 
The conventional plan made use of six beams consisting of two anterior oblique 6 MV beams ({Phi}1(335°,0°) and {Phi}2(20°,0°)), a superior oblique 18 MV beam ({Phi}3(90°,310°)), a left lateral 6 MV beam ({Phi}4(90°,0°)), a right lateral 10 MV beam ({Phi}5(270°,0°)) and an anterior 6 MV beam ({Phi}6(0°,0°)). All but the anterior beam made use of a wedge and utilized customized beam blocks.

The following parameters were used for the beam direction search for the optimized plan: {xi}{0°–360°}, {psi}{0°–100°,260°–360°}, s=3.3 mm, {varpi}=10°, {varphi}=45°, nb=6 and the weighting factors wk for the OAR were set to 7, 7, 5, 8, 9, 9, 10, 10, 10, 1 and 1 corresponding to the right and left eye, the spinal cord, the brain stem, the right and left optic nerve, the optic chiasm, the right and left lens and the right and left parotid gland, respectively.


   Results
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
 Results
Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
 References
 
Parameter value selection
The results of the beam direction search for the test plan in Figure 3Go are given as a function of the ray-trace resolution in Table 1Go and as a function of the angular resolution in Table 2Go. With respect to both tables, the time to perform the search is given in column two and the top five angles in the rank list, and the selected angles are given in the third and fourth columns, respectively. Each individual cost term in Equation 1Go was analysed for all of the test cases. Only the total cost is plotted as a function of the ray-trace resolution, s, in Figure 6aGo and as a function of the angle resolution, {varpi}, in Figure 6bGo.


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  		Table 1. The results of the beam direction search as a function of the ray-trace resolution s

 

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  		Table 2. The results of the beam direction search as a function of the angle resolution {varpi}

 


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  		Figure 6. Cost as a function of (a) the ray-tracing resolution, s, and (b) as a function of the angle resolution, {varpi}, for the test case in Figure 3Go.

 

   Prostate comparison
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
 Results
 Prostate comparison
Clinical case studies
 Discussion
 Conclusion
 References
 
The time to perform the first run, for which the femoral heads were not considered OAR, was 112 s. The optimal gantry angles were found to be: 273°, 96°, 56°, 313°, 233° and 136° corresponding to the ranks #1, #3, #149, #158, #163 and #173. The first two angles were virtually opposed lateral fields rotated marginally, mainly to minimize the irradiation of the rectum, but also to reduce irradiation of the bladder. The rest of the beams were angled off the lateral beams by ±40°, which corresponds to the minimum angle separation {varphi}.

The second run lasted 24 s and included the femoral heads as critical structures. The following gantry angles were selected by the BDST: 300°, 65°, 115°, 255° and 340° corresponding to ranks #1, #6, #7, #10 and #45, respectively. Figure 7Go shows a comparison with the cost function vs gantry angle plot obtained by Rowbottom et al [13] (Figure 7aGo) and our results (Figure 7bGo).



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  		Figure 7. Cost function vs gantry-angle plot for prostate: (a) Graph reproduced from Rowbottom et al [13]. (b) Normalized cost corresponding to prostate case. The vertical lines indicate the selected gantry angles. Note that different cost functions were used and therefore the overall magnitude of the graphs is not comparable.

 

   Clinical case studies
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
 Results
 Prostate comparison
 Clinical case studies
Discussion
 Conclusion
 References
 
The times to perform the beam direction searches were 11 min for the brain case and 12 min for the sinus case. The normalized cost distributions over the permitted beam space are depicted in Figure 8aGo for the brain case and in Figure 8bGo for the sinus case. The beam space that was not permitted is depicted with a cost of unity (white). The numbers next to the diamonds denote the overall rankings. For the optimized treatment plan wedges were applied to all three beams for the brain case but no wedges were necessary for the sinus case. The beam energy for all optimized beams was 6 MV.



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  		Figure 8. Normalized cost distribution over the beam space {Phi}P({xi},{psi}). A circle corresponds to the conventional and a diamond to the optimized plan. The grey-scale bar represents the cost values. (a) Brain case and (b) sinus case.

 
DVHs for the brain case are shown in Figure 9Go, and the corresponding statistical evaluation is given in Table 3Go. The graphs corresponding to the critical structures on the left side of the patient are depicted in black and those corresponding to the right side are in grey, and the conventional plan is plotted with dashed lines; whereas the optimized plan is plotted with solid lines. Both plans are within the dose specifications and do not violate the constraints.



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  		Figure 9. Brain case: dose–volume histograms for the conventional plan (dashed) and the optimized plan (solid): (a) planning target volume (PTV) and (b) optic nerves. The vertical lines in (a) denote the ±5% dose range.

 

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  		Table 3. Case I. Brain: statistical evaluation

 
For the sinus case, DVHs are shown in Figure 10Go for the PTV, the brain stem, the parotid glands, the optic nerves and the optic chiasm. The dose to the spinal cord was negligibly low for both the conventional and optimal plans. As the plans could not be normalized to the D95 of 70 Gy, due to the high dose to the optic chiasm for the conventional plan, they were scaled such that the maximum doses in the PTV were comparable. The statistical evaluation, including additional statistical information for the lenses, is given in Table 4Go.



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  		Figure 10. Sinus case: dose–volume histogram for the conventional plan (dashed) and the optimized plan (solid): (a) planning target volume (PTV) and brain stem, (b) parotid glands, (c) optic nerves and (d) optic chiasm. The vertical lines in (a) denote the ±5% dose range.

 

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  		Table 4. Case II. Sinus: statistical evaluation

 

   Discussion
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
 Results
 Prostate comparison
 Clinical case studies
 Discussion
Conclusion
 References
 
Parameter value selection
The weighting factors for this set-up were all set to unity so there was no preference for one OAR over another, and hence the cost for the OAR only reflects the relative penalty for its location and the partial volume irradiated. Owing to the normalization, the cost terms are also independent of the size of the OAR. Assuming that the finest ray-trace resolution and the smallest value of {varpi} result in the most accurate results, the anticipated tendency can be seen. The higher the value of s, the more deviation can be observed. This was most marked for the smallest OAR with a diameter of 10 mm. In Table 1Go it can be seen that the resolution 3.3 mm, 5 mm and 10 mm resulted in the same four beam directions to top the rank list with the first three of the selected beams being the same. However, reducing the resolution to 1 mm and 2.5 mm, the algorithm found beam directions with a lower cost.

The selection of the value for the angle resolution {varpi} for the exhaustive search determines how efficiently the beam space is scanned. A value of 10° for {varpi} was acceptable; larger intervals could be used for larger OAR (>20 mm). Based on these results, values of s=3.3 mm and {varpi}=10° were selected for the clinical case studies due to the small size of the optic nerves. A value of {varphi}=45° was chosen for the minimum angle separation for the clinical cases with non-coplanar beam arrangements. A minimum angle separation is required to avoid overlapping beams within the body, and so prevents an unnecessarily high dose to the tissue located between the beam entrance surface and the tumour. In terms of dose homogeneity it is also desirable to have the beams maximally separated in space [11, 40]. The accumulation of all of the beams from one general direction makes it difficult to achieve target dose uniformity and difficult to conform the high dose region to the shape of the tumour.

Prostate comparison
To validate the approach a comparison was drawn with our own prostate technique (first run) and with the extensive work that has been carried out on prostate planning at the Institute of Cancer Research and Royal Marsden Hospital [13] (second run). For the latter, Figure 7Go illustrates that in both cases the gantry angles were selected in the valleys of the graph such that irradiation of the femoral heads was minimized. Based on clinical experience, e.g. symmetry of the beams leading to dose uniformity, Rowbottom et al [13] decided also to select an anterior beam (0°) although the cost was higher than for other candidate gantry angles. The comparison of the cost vs gantry-angle plot revealed that the current methodology can reproduce both a standard 6-field beam arrangement as well as individualized beam angles that correspond well to the findings by other researchers.

Clinical case studies
For each of the clinical test cases, the non-coplanar beam directions were checked by placing the treatment mask on the treatment couch, setting up the isocentre and positioning the couch and the gantry in the acquired positions. None of the beam directions violated any geometrical constraints imposed by the linac.

Case study I: brain
In Figure 8aGo the locations of the angles that were selected manually for the conventional plan are superimposed (circles) on the cost distribution of the beam direction search. The existence of multiple local minima can be observed in the cost distribution, indicated by the low cost regions (dark). The non-coplanar beam with gantry angle 100° and couch angle 300° lies within the low cost area and shows that the dosimetrist was successful in approaching the best non-coplanar beam direction. The dosimetrist might have chosen a slightly different angle for reasons not directly considered in the beam direction search such as dose uniformity in the PTV. The two standard lateral coplanar beams with gantry angles 90° and 270° are not optimal with respect to the cost function used for the beam direction search.

Comparing the DVH for the brain case in Figure 9Go shows that the optimized plan is superior to the conventional plan for both the critical structures and the PTV. The PTV is more uniformly irradiated and the cold volume considerably reduced. This is also reflected in the statistical evaluation in Table 3Go. The NTCP values indicate that both plans do not overdose any critical structures and that the probability of complications is very low. However, any improvements are worthwhile, especially for a paediatric patient. In addition, the improved sparing of OAR without compromising the dose to the PTV is an important foundation for dose escalation, which can only be achieved safely if the OARs are not exposed to toxic dose levels.

Case study II: sinus
The cost distribution in Figure 8bGo shows that the BDST finds two non-coplanar beam directions (ranks #28, #77) of which #77 is located on the edge of the permitted beam space. It is a posterior beam that penetrates through the brain stem. In fact, it is this beam that enables better coverage of the PTV since it complements the other beams. It is also most likely that this beam direction would have not been explored during manual angle selection.

The fact that the best beam with rank #1 is almost identical with one beam selected manually at gantry angle 335° highlights that the dosimetrist again made a good selection. One might wonder why there is no beam positioned in the low cost region on the bottom right and the top left of Figure 8bGo. After the first beam with rank #1 was selected all the beams in close proximity with a low rank were discarded if they were within a certain minimum angle {varphi}. Furthermore, some of the beam angles are redundant. A simple example would be a vertex field that can be delivered using either {Phi}i(90°,270°) or {Phi}i(270°,90°).

The DVH in Figure 10Go shows that dose uniformity in the PTV was improved in the optimized plan. More importantly, the cold spot was greatly reduced which is reflected in the increase in the TCP value from 73% to 94%. The mean dose to the brainstem is higher for the optimized plan, which is due to the posterior beam. However, the maximum dose is lower, which is reflected in the similar NTCP values for both plans. All the other statistical values in Table 4Go are clearly in favour of the optimized plan with greatly reduced mean and maximum doses for the parotid glands, the optic nerves and the optic chiasm. The most important difference between the two plans is the much improved sparing of both parotid glands. It has been reported that a mean parotid dose of more than 26 Gy slows salivary flow [41] and doses over 40 Gy may result in total dysfunction of salivary production [42]. The mean dose for the right parotid gland was reduced from 42 Gy to 20 Gy (improvement of 54%) for the optimized plan, which resulted in a reduction of the NTCP from 40% to <0.1%. For the left parotid gland, the improvement in the NTCP was approximately 46%. This indicates that both parotid glands would be virtually complication free. The sixth beam selected (rank #77) went straight through the brain stem but avoided all the other critical structures. This selection was due to the design of the cost function and the choice of the weighting factors, which all had a similar magnitude. The weighting factors were based on common sense, but were chosen somewhat arbitrarily insofar as the magnitudes, as well as the relative difference could have been selected differently. Selecting weighting factors that are similar in magnitude can be considered as a soft constraint, and this is necessary if multiple objectives, in this case multiple OAR, are present. If one weighting factor is much higher than all the others, then this could result in a solution where only the one particular OAR would be spared. It should also be remembered that the individual cost terms are normalized to the volume of the respective organ, requiring the entire organ volume to be contoured. The beam weight for the beam that went straight through the brain stem was set relatively low by the beam weight optimization procedure. This is an interesting observation as it paves the way for the application of the BDST for IMRT planning [43]. If this particular beam were replaced by an IMRT beam it would have probably received a much higher overall weighting since IMRT has the ability to spare critical structures by modulating the intensity across the field.

The main advantage of the BDST is its ability to search the permitted beam space within a time frame that is acceptable for clinical practice and to select beam directions in a rule-based manner. The approach is inferior to beam angle optimization techniques that integrate dose calculations, optimize wedge angles, etc. Nevertheless the results of this work have shown that the BDST is able to suggest beam directions that result in plans with greatly improved sparing of critical structures and is, therefore, a good compromise to more elaborate techniques. In this work, all the suggestions in terms of selected beam directions by the BDST were accepted. In clinical routine, however, the dosimetrist might only select some of the beam directions suggested by the BDST and combine them with standard beam directions or beam directions that have a high cost but when the portal is split into two sub-fields (e.g. one on the left and one on the right side of the spinal cord), or if the intensity is modulated, result in preferable beam directions.

The BDST has shown that small changes in the beam directions (see clinical case I) can already result in considerable improvements in the reduction of dose to the normal tissues. It is noteworthy that none of the beams for the clinical case II made use of wedges or different energies, which highlights that the conventional plan was more complex and simpler plans might be possible.

It would be of further interest to evaluate the trade-offs between the speed and the quality of a plan based on the current approach and more sophisticated techniques that include dose calculations during the optimization procedure. The cost maps obtained could be used to predetermine and separate the "good" beam directions from the "bad" ones. This could then be used as an input for more complex stochastic optimization procedures similar to the approach described by Pugachev et al [28] who used prior knowledge to reduce the optimization time.


   Conclusion
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
 Results
 Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
References
 
A general algorithm has been presented that explores the entire permitted beam space to determine beam directions that minimize the irradiation of healthy tissues and critical structures. The algorithm can be adjusted to different clinical goals and is a valuable tool for 3D-CRT treatment planning. The beam directions obtained for the prostate case compared well with standard beam directions and other work described in the literature. For the brain and the sinus cases the optimized plans were simpler and better than conventional plans. It is anticipated that the plans could be further improved if the intensity of the conformal beams could be modulated. Indeed, the fact that one beam penetrated through the brain stem in order to spare the other OAR makes it an interesting starting point for further research on the use of the BDST for IMRT, especially with regard to forward planned IMRT [44].


   Acknowledgments
 
The author would also like to thank Dr Homayon Parsai for his assistance with LaTeX during the preparation of this manuscript.


   Footnotes
 
This work was partially supported by a grant from Elekta Oncology Systems. Back

Received for publication April 16, 2004. Revision received September 9, 2004. Accepted for publication November 29, 2004.


   References
Top
 Abstract
 Introduction
 Methods and materials
 Parameter value selection
 Clinical examples
 Results
 Prostate comparison
 Clinical case studies
 Discussion
 Conclusion
 References
 

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