Control Theory and Applications Centre, Coventry. University, CV1 5ED, UK. ââ University Hospitals Coventry and Warwickshire NHS. Trust, Coventry CV2 4ED.
MODELLING OF MECHANICAL UNCERTAINTY IN THE GANTRY OF RADIOTHERAPY TREATMENT MACHINE Piotr Skworcow ∗ Olivier C.L. Haas ∗ Keith J. Burnham ∗ John A. Mills ∗∗
∗
Control Theory and Applications Centre, Coventry University, CV1 5ED, UK ∗∗ University Hospitals Coventry and Warwickshire NHS Trust, Coventry CV2 4ED
Abstract: This paper concerns the development of a model to quantify the uncertainty due to limited mechanical accuracy of a radiotherapy treatment machine gantry. A vector-end-effector instead of a point-end-effector approach is proposed to assess deviations of the radiation beam from the ideal nominal values. Results of the measurements performed on 11 clinical machines show advantages of the vector-end-effector approach. A novel concept aiming to achieve a potential reduction in the treatment margins is proposed. Keywords: gantry, isocentre, machine uncertainty, modelling, radiotherapy
1. INTRODUCTION External radiation therapy (RT), employs ionising radiation (X-ray) beams to eradicate cancerous tissues. The goal of radiotherapy treatment is to precisely deliver a lethal dose to tumours while minimizing radiation dose to the surrounding healthy tissues.
work is to reduce the uncertainty due to limited machine accuracy, hence to a potential reduction in the treatment margin. This paper presents the development of a model for gantry deflection of a typical radiotherapy treatment machine.
Every machine exhibits limited accuracy, see e.g. (Mavroidis et al., 1998), and this applies to RT treatment machines. The standard approach to accommodate the uncertainty of radiation beams due to limited accuracy of machines is to add a treatment margin 1 (ICRU, 1993). A larger volume is thus irradiated to ensure adequate dose coverage of the tumour. This causes increased healthy tissue complications and the dose that can be delivered to a tumour is thus limited by the tolerance of the healthy tissue. The aim of this
An analytical model of deflections in robotic manipulators was developed with application to a proton-therapy patient positioning system (Drouet et al., 2002). The purpose of a radiotherapy treatment machine gantry is, however, to facilitate the irradiation of the same point (tumour) from different angles. Hence the endeffector considered here is different to that of a robotic manipulator. There have been a number of experimental methods proposed for the assessment of the gantry accuracy, e.g. (Tsai et al., 1996; Gonzalez et al., 2004; Rosca et al., 2006; Court et al., 2003), nevertheless neither of these researchers explicitly use models of the gantry deflection.
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The remainder of the paper is organised as follows: In Section 2 the problem is formulated. The
Note, that limited machine accuracy is only one of the sources of uncertainties during the RT treatment.
proposed model to quantify uncertainty in the gantry is described in Section 3. Section 4 presents results of simulations. Section 5 contains preliminary analysis of the measurements performed on 11 clinical machines using the proposed methods. Conclusions are given in Section 6.
2. PROBLEM FORMULATION AND EXISTING METHODS A typical gantry-based radiotherapy treatment machine is shown in Figure 1. Ideally, centres of the beams projected from different gantry angles should intersect at a single point, termed the isocentre. In reality, due to finite accuracy of machines, the isocentre is a point which is at the centre of the locus of beam intersections from all gantry angles, henceforth termed isoc ∈ R3 . It is observed that due to the mass of gantry head, the gantry arm tends to ‘bow’ at positions of 0 degree and 180 degree, whilst at angles -90 and 90 degree a ‘twist’ of the gantry arm may occur. This phenomena is generally referred to as gantry deflection.
Fig. 1. Standard radiotherapy treatment machine and coordinate system according to the IEC61217 standard The central axis of the radiation beam is indicated by a light source in the gantry head showing two crossed lines, called the crosswire. The method commonly used in hospitals to assess the mechanical isocentric accuracy utilizes a mechanical pointer mounted onto the accessory tray orthogonal to the head of the gantry, see e.g. (Tsai et al., 1996). The pointer is aligned to the crosswire, hence assumed to indicate the central axis of the radiation beam for all gantry angles. The isocentric accuracy of the gantry is considered at n gantry angles, usually equally distributed, ◦ i.e. every 360 n , typically n is equal to 4 or 8. The ‘size’ of the locus of beam intersection (henceforth termed size of the locus) is defined as: (1) the maximum distance between any two points indicating position of the beam central axis for all considered gantry angles, or
(2) the maximum distance between any point indicating position of the beam central axis and the isoc . Typically, the acceptance criterion for the gantry isocentre is that the size of the locus needs to be smaller than 2mm in radius (IPEM, 1999). 3. MODELLING UNCERTAINTY IN GANTRY 3.1 Point-end-effector and vector-end-effector It is assumed, that due to deflection, at every gantry angle the position and orientation of the radiation head may deviate from theoretical values in all 6 degree-of-freedom (DOF). However, a deviation of position indicated by a physical pointer along the direction of the beam is not considered as an error, since the beam still intersects with the isocentre, see Figure 2. More importantly it should be noted that a single point does not provide information about angle of incidence of the beam. A vector-end-effector is proposed, as a more appropriate approach than the point-end-effector to analyse the mechanical isocentric accuracy of gantry-based cancer treatment machine.
Fig. 2. A beam modelled with point-end-effector (solid) indicates the same error e for both beams a and b, while there is no error using vector-end-effector (dashed) for beam a. Let sri ∈ R3 denote the position of source of radiation for gantry angle i. Let vi = p# i,1 pi,2» : pi,1 , pi,2 ∈ R3 be a vector indicating the beam path for gantry angle i, and V = v1 , . . . , vn , such that: ∀ (i ∈ {1, . . . , n}) vi = const, |p# i,1 sr»i | = const (1) The proposed method to quantify the size of the locus using the vector-end-effector approach, is to choose the isoc such, that the largest of the orthogonal distances between any vi and the isoc is minimal. This is equivalent to the problem of fitting a sphere with minimal radius to V , such that all vectors of V intersect or are tangential to the sphere, and can be expressed as: # » pi,1 isoc × vi (2) r = min 3 max vi isoc ∈R i∈{1,...,n}
where the isoc is the centre of the sphere in question. Note that the radius r, given by (2) is optimal in the sense that it corresponds to the sphere with the min max distance between the isoc and the beam central axes. Due to conditions in (1), it is possible to express the position of a point-end-effector gi ∈ R3 (e.g. the position of a mechanical pointer of length l attached to the gantry head) for gantry angle i, as vi scaled by l: vi gi = pi,1 + · l (3) vi Let G = {g1 , . . . , gn }. With the point-end-effector approach the length of pointer l is chosen such that the size of the locus enclosing all points of G is minimal. This is equivalent to minimisation of the maximum distance between any two points from G, and can be expressed as: # » min max |gi gj | (4) l∈R
3.3 Crosswire projection tracking To obtain the set V , the projection of the crosswire indicating the beam path needs to be measured for n gantry angles. The proposed method to achieve this, is to align markers m1 and m2 to the crosswire, and measure the 3D position of m1 and m2 to obtain pi,1 and pi,2 for different gantry angles. The set V obtained in this way is subject to uncertainties: (i) uncertainty of alignment of m1 and m2 to the crosswire, termed δcm1 and δcm2 , respectively, and (ii) uncertainty of measurement of position of m1 and m2 , termed δmp1 and δmp2 , respectively. Note, that δcm1 and δcm2 are due to ‘thickness’ of markers and of crosswire projection and are assumed to be constant for all gantry angles, while δmp1 and δmp2 are due to finite precision of a measurement device and, in this paper are assumed to be of zero-mean value and normally distributed.
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Once the optimal l has been found, the isoc is: n 1X isoc = gi (5) n i=1 Note, that methods currently adopted typically take only gantry angles 0◦ and 180◦ to find l, whilst all n angles are used to find isoc . 3.2 Compensation vectors The concept proposed here aims to compensate for measurable deviations from the ‘ideal’ isocentre, by calculating the vectors between the isoc and the beam trajectories given by V . The compensation vector vci for gantry angle i is then given by: v 2 u # » u p iso × v 2 i,1 c i u # t pi,1 iso»c − vi # » vci = isoc pi,1 +vi · vi (6) Compensation vectors indicate positional deviations of the radiation beam, from which calculation of angular deviations is straightforward. The positional deviations can potentially be compensated by adjusting the position of the patient according to the compensation vector for given gantry angle. Note, that in a standard machine there are not sufficient degrees of freedom to compensate all possible angular deviations. However, angular deviations can be taken into account at the treatment planning stage to improve the dose prediction.
3.4 Angle-dependent and angle-independent beam deviations Let deviations of a beam vector from the ideal nominal values be split into two components: angle-dependent and angle-independent. Angledependent deviations are caused mostly by elastic deformations in the machine structure (Drouet et al., 2002) of which examples were given in Section 2. Angle-independent deviations are caused mostly by geometric errors (Drouet et al., 2002) and result in a constant offset of beam vector from some ideal nominal value, see Figures 3a, c and e. Note, that without further assumptions, the effect of misalignment of the markers from the crosswire is not distinguishable from angle-independent deviations, c.f. Figures 3a, c, e with b, d, f.
3.5 Correction of lateral markers misalignment / isolation of angle-dependent machine deviations Misalignment of markers or angle-independent beam deviation along the Y-axis (Figures 3 f or e) can change the position of the isoc but does not change the size of the locus, and will be referred to as the Y misalignment/deviation. Lateral misalignment of markers or angle-independent beam deviation (Figures 3 b, d or a, c) can change the size of the locus, but does not change the position of the isoc , and will be referred to as the lateral misalignment/deviation. A method is proposed to remove an offset caused by a lateral misalignment/deviation. This can be considered as: (i) an off-line correction of lateral marker misalignment assuming that there are no
# » pi,1 isoc × vi∗ max min ∗ isoc ∈R3 ,αc ∈R i∈{1,...,n} vi (7) # » where vi∗ = pi,1 rot(pi,2 , vrot , αc ) with rot pi,2 , vrot , αc denoting point pi,2 rotated around vector vrot by angle αc .
4. SIMULATIONS A simulation of gantry rotation was developed to evaluate the impact of measurement uncertainties and efficiency of the developed correction/isolation method. For given input parameters two circular concentric trajectories are generated to define V . The fixed parameters were set at the values corresponding to the actual setup during the measurements, described in Section 5, i.e: n = 24, the radii of trajectories were 234mm and 355mm respectively, the angle between the planes fitted to trajectories was 0.2 degree 2 , which results in the radius of the isocentre locus equal 0.5mm.
Fig. 3. Illustrating the effects of angle-independent beam deviations (sub-plots a, c, e) and marker misalignments (sub-plots b, d, f). Gantry angles are indicated by 0, 90, -90, 180. Circles: m1 – solid, m2 – clear, isocentre locus – striped. Beam path lines: actual – solid, (ideal) nominal – dashed, measured – dotted. Note that on a, c and e real and measured paths are the same, whilst on b, d and f real and (ideal) nominal paths are the same. angle-independent beam deviations, or (ii) isolation of angle-dependent deviations by removing the angle-independent component and simultaneous correction of lateral marker misalignment. An offset is removed by rotation of the whole trajectory of one of the markers (all pi,1 or pi,2 ) by an appropriate correction angle αc around a particular vector vrot . To find αc and vrot to rotate all pi,2 the procedure to follow is: (1) Let P be a plane that fits all pi,2 in the leastsquare-error sense, i.e. minimise: n X
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dist (pi,2 , P )
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where dist (pi,2 , P ) denotes orthogonal distance between point pi,2 and plane P . (2) Vector vrot is normal to the plane P and anchored at the centroid of all pi,2 . (3) Angle αc is found by solving the min-max problem:
4.1 Impact of Y misalignment / deviation on compensation vectors Different misalignment/deviations of one or both markers along the Y axis were simulated to assess their impact on the compensation vectors. It was found, that even for misalignment/deviation of 20mm, which is not likely to occur in reality, the discrepancies in compensation vectors, compared to zero misalignment/deviation were (x,y,z in mm): mean discrepancy (0.0059, 0.0004, 0.0102), maximum discrepancy (0.0083, 0.0006, 0.0167), and therefore are considered to be negligible.
4.2 Lateral misalignment correction / deviation isolation To validate the developed correction/isolation method and assess its impact on results, one marker (δcm1 = 0) or both markers (δcm1 = δcm2 ) were misplaced laterally. The amount of misalignment was between 0 and 5mm with step 0.2mm. The compensation vectors and size of the locus were compared with the reference case (no misalignment/deviation). It was found, that the changes of compensation vectors were smaller than 0.05mm, whilst the changes of size of the locus were smaller than 0.005mm. Note, that the position of isoc did not change. 2
This angle was found to be between 0.18◦ and 0.22◦ for the machines tested
4.3 Impact of measurement noise The repeatability of compensation vectors, the position of the isoc and the size of the locus was assessed for different levels (standard deviations) of measurement noise δmp1 = δmp2 . For each noise level one marker was laterally misaligned by 0 to 1mm with step 0.2mm. For each misalignment step three iterations were performed. The noise was randomly generated independently for each axis of each marker at each iteration, hence each iteration simulates independent measurements on the same machine. RMS of changes in compensation vectors [mm]
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M 2 and M 3) at 4 clinical sites in the UK. Preliminary results of analysis using the model and algorithms described above are presented.
5.1 Methods and materials The gantry tool was designed to be used with a stereoscopic optical tracking system giving the spatial location of infrared markers. The tool was designed such that it can be mounted onto the accessory tray of all the machines assessed. To increase the accuracy of measurement, m1 and m2 were defined using four physical reflective markers, see Figure 6. To measure the 3D position of the markers an NDI Polaris (http://www.ndigital.com/polaris.php) system was used. The accuracy of the Polaris tracking a single physical marker quoted by manufacturer is 0.35mm RMSE. The accuracy of measuring the position of the designed tool was assessed experimentally using industrial milling machine with position sensors with a stated accuracy of 0.005mm. It was found that the accuracy of measuring the position of the designed tool was 0.13mm RMSE.
Fig. 4. Illustrating the changes in compensation vectors due to measurement noise. RMS of changes in isoc and size of the locus [mm]
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Fig. 5. Illustrating the changes in isocentre properties due to measurement noise. For each iteration, the compensation vectors, the position of isoc and the size of the locus were compared with the reference case (no misalignment/deviation and zero noise). Results are presented in Figures 4 and 5. It can be observed, that the resulting uncertainty due to measurement noise and misalignment is relatively small for δmp1 = δmp2 < 0.2mm.
5. EXPERIMENTAL MEASUREMENTS This section describes measurements performed on 11 machines from 3 major manufacturers (M 1,
Fig. 6. Illustrating the design of gantry tool used for measurements. 5.2 Procedure Markers m1 and m2 are aligned against the projected crosswire. After alignment, the position of both markers in respect to the gantry is kept constant to satisfy the requirements given in the Equation (1). The gantry is rotated with stops every 15◦ . The positions of m1 and m2 is logged, pi,1 and pi,2 are obtained only when the gantry is stationary, therefore n = 24. Processing of the data is performed off-line. 5.3 Results In this paper only the angle-dependent deviations were addressed, i.e. any lateral misalign-
ment/deviation was corrected/isolated via the method described in Section 3.5. The size of the locus obtained using vector-endeffector approach, i.e. radius of a sphere fitted to V was between 0.40mm and 0.72mm, with machines from M 1 exhibiting larger size than those from M 2 and M 3. The size of the locus obtained using point-end-effector, i.e. maximum distance between any two points from G was between 0.16mm and 2.56mm, with all machines from M 1 and one 9-year-old machine from M 2 exhibiting significantly larger size. Positions of the isoc obtained with the vector-endeffector and point-end-effector were compared. It was found, that the difference between the positions of both isoc was between 0.16mm and 1.25mm. The largest differences (over 1mm) were exhibited by all machines from M 1 and one 9year-old machine from M 2. The results show, that the isocentre found with methods utilizing a mechanical pointer can be misplaced by over 1mm from the actual point of coincidence of the beams, see Figure 7. Isocentres found with point−end−effector and vector−end−effector −64.5
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Fig. 7. Measurement results: example of the difference between isocentres found with pointend-effector and with vector-end-effector. 6. CONCLUSIONS This paper has presented the development of a model to quantify the uncertainty due to limited mechanical accuracy of a radiotherapy treatment machine gantry. A novel concept of compensation vectors aiming to allow a reduction in treatment margins has been proposed. The influence of measurement uncertainties on the repeatability of the results obtained using the proposed methods has been assessed via simulations. It was found, that for the proposed measurement setup good repeatability of results is achieved, with measurement noise standard deviation smaller than 0.2mm. Experimental measurements revealed, that the mechanical isocentre
found by standard methods utilizing a mechanical pointer can be misplaced by over 1mm from the actual point of coincidence of the beams found with the proposed method.
ACKNOWLEDGMENTS This work is sponsored by the Framework 6 European integrated project Methods and Advanced Equipment for Simulation and Treatment in Radiation Oncology (MAESTRO) CE LSHC CT 2004 503564. The Authors are thankful to Peter Mulholland and to the Physics staff at Addenbrookes, Western General and Newcastle General Hospitals for their help in realising the measurements presented in this paper.
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