Determination of the required accuracy of the beam model parameters
This document describes how the tolerances defined on the dose distribution are translated into tolerances on the GBD beam model parameters. (See section 2.C in the main paper.) We establish the relationship between each beam model parameter x and the relevant property of the dose distribution y. (For example, x = virtual SAD and y = 50% field diameter). The sensitivity of the dose distribution on the beam model parameter is given by the derivative: dy/dx. Using a linear expansion around the nominal x value, the tolerance on the dose-distribution property Δy is translated into a tolerance in the beam model parameter of: Δx = Δy / (dy/dx). For most parameters simple analytical equations can be used. The analytical model that calculates the pristine-peak dose distribution is fairly complex. In this case, we determine the effect of a change in the energy spread empirically: the value of the energy spread is varied in the TPS and the change in calculated dose distribution is determined.
The sensitivity of the dose distribution on the beam model parameters is obviously not constant, but varies over the options and with prescribed clinical parameters within an option (e.g. range, modulation width, field size, and air gap). The tolerances are calculated per option. Within an option, the ‘realistic worst case scenario’ (RWCS) is determined for each beam model parameter. This is the combination of realistic prescribed parameters that results in the largest sensitivity on the specific beam model parameter. For example, the largest field size is chosen to evaluate the required accuracy of the virtual SAD, because the field radius is most sensitive to changes in the virtual SAD at the largest field size. The RWCS is used to translate the tolerance on the dose into a tolerance on the beam model parameter. This means that all (realistic) clinical dose distributions will meet or exceed the required accuracy.
Table 1 shows the tolerance per option for the different GBD parameters. The following sections describe the calculations in more detail.
Table 1 Overview of the tolerances on the various beam model parameters per option of the double-scattering system in the Universal Nozzle.
Option / Min Range / Max Range / Max Diam. / ΔSADvirt / ΔSADeff / Δσsource / ΔΣpristine / Rnozzleg/cm2 / g/cm2 / cm / cm / cm / cm / %
B1 / 4.6 / 5.9 / 24.0 / ±4.8 / ±87 / ±0.32 / ±0.12
B2 / 5.9 / 7.5 / 24.0 / ±4.6 / ±70 / ±0.30 / ±0.12
B3 / 7.5 / 9.6 / 24.0 / ±4.3 / ±55 / ±0.29 / ±0.12
B4 / 9.6 / 11.7 / 24.0 / ±4.0 / ±46 / ±0.27 / ±0.11
B5 / 11.7 / 15.5 / 24.0 / ±3.0 / ±31 / ±0.25 / ±0.12
B6 / 15.5 / 19.8 / 24.0 / ±2.9 / ±23 / ±0.23 / ±0.11
B7 / 19.8 / 23.9 / 24.0 / ±2.6 / ±21 / ±0.21 / ±0.10
B8 / 22.8 / 28.4 / 14.0 / ±4.6 / ±19 / ±0.34 / ±0.11
II.A. Virtual SAD
Using simple geometry and a linear expansion around the nominal virtual SAD, the change in 50%-field diameter (ΔD50%) as function of change in virtual SAD (ΔSADvirt) is found to be
±∆D50%=∓∆SADvirt×Daperture×z-zcollimatorSADvirt-zcollimator2 (II.1)
where Daperture is the physical aperture diameter, zcollimator is the distance from the patient collimator (aperture) to the isocenter plane, and z the distance from the plane in which the field diameter is evaluated to the isocenter plane. The sensitivity is largest for a large distance between the patient collimator and the plane of interest, i.e. for a large air gap and deep plane of interest. A short distance between the collimator plane and source position, i.e. short SSD for fixed air gap, also increases sensitivity. In addition, the sensitivity depends linearly on the aperture diameter. As RWCS we consider for each option the field with the smallest SSD, that is the largest range (Rresidual) and smallest modulation width (M=2.5 g/cm2). The SSD is set such that the middle of the SOBP coincides with the isocenter plane (SSD = SAD – (Rresidual – M/2)). The field diameter is evaluated at the end of the SOBP, at a depth equal to the range (z = -M/2). The air gap is set to 15 cm (zcollimator = SAD – SSD + 15 cm). And the aperture diameter covers the maximum field diameter per option (Daperture = maxD50% x (SADvirt-zcollimator) / SADvirt, with maxD50% = 24.0 cm for options B1- B7; maxD50% = 14.0 cm for option B8).
Table 1 shows the tolerance on the virtual SAD when limiting the error in 50% field diameter (ΔD50%) to ±0.5 mm. The tolerance is most stringent for option B7 (±2.6 cm) because it has the largest field size for the deepest range. A deep range decreases the SSD and therefore the collimator to source distance (for the same air gap). Option B8 is less sensitive because of its smaller field diameter. (Note that for the dashed lines in Figure 1 of the main paper the same calculation is made, with the exception that the range is not equal to the maximum option range, but changes according to the value on the x-axis.)
II.B. Effective SAD
The TPS parameterizes and calculates the pristine-peak depth distributions at ‘infinite SAD’, i.e. as if the beam is not diverging. Next it folds in the effect of the beam divergence by correcting the dose in each point based on its distance from the effective source position. An error in the effective SAD will lead to an error in the shape of the pristine depth dose distributions. If the effective SAD is too short the peak-to-skin ration of the pristine peaks will be too small. Because the TPS optimizes the weights of the SOBP to obtain a flat dose distribution, an error in the effective SSD will only affect the dose distribution in the entrance region, most strongly the dose at skin. (Because the TPS calculates the SOBP weights at nominal SSD (isocenter at a depth equal to range minus half the modulation) an error in effective SAD will also lead to an error in the modeling of the SOBP ‘tilt’ as function of planned SSD. However, this effect is smaller than the effect on the dose in the entrance region. The effective SAD also determines the change in absolute dose as function of SSD shift. Fields that are treated under ‘shifted SSD’, i.e. fields for which the middle of the SOBP does not coincide with isocenter, will have different absolute dose values for the same MU than fields at nominal SSD. Because MU determination is not part of this GBD investigation and assumed to be done outside of the treatment planning algorithm with either measurement or an external model, the effect of changes in effective SAD on the dose-per-MU is not considered here. )
Given that the fluence reduces with inverse of the distance from the effective source squared: Φz=Φ(z=0)SADeffSADeff-z2, the dependence of the error in fluence relative to the fluence at isocenter is given by:
±∆ΦzΦz=0=∓∆SADeff×2zSADeff-z×SADeff
The error is largest for large z, i.e. for points closest to the effective source. As RWCS we consider the error in skin dose for fields with maximum option range (R) and minimal modulation width (pristine peak) because these have the shortest SSD (under nominal setup). The depth of isocenter is taken to be equal to the range (SSD = SAD – R) and the error in fluence (dose) is considered at skin (z = +R). The effective SAD is calculated using the GBD parameters. The maximum allowed error in skin dose is chosen to be ±2%.
The tolerance on the effective SAD varies between 19 cm for option B8 and 87 cm for option B1. The sensitivity of the relative fluence on effective SAD is largest for the highest range, because the difference in distance from the source to the distal peak and from the source to the skin is largest at this range. But even for B8 the effective SAD needs to be known with relatively low accuracy because this difference in distance (~30 cm) is still small compared to the effective SAD (230 cm).
II.C. Effective source size
The lateral penumbra (LP) has two contributions: a geometric contribution that depends on the effective source size and a water-scattering contribution (σH20)
LPz=k×σsource2zcollimator-zSADnom-zcollimator2+σH2o2
Here k is constant translating the sigma of the error function describing the lateral dose profile into the 80%-20% penumbra of the profile (k=1.68), zcollimator is the distance from the patient collimator to the isocenter, SADnom is the nominal position where the source is assumed to be located (SADnom=230 cm). For simplicity the water-scattering contribution is taken to be equal to 2% of the penetration depth[1]: σH2o=0.02×depth=0.02×zcollimator-g-z, where parameter g is the air gap, the distance from the patient collimator to the skin. Derivation of LP gives the error in penumbra as function of error in effective source size as:
±∆LP=±∆σsource×k×σsource×G2σsource2×G2+0.02×zcollimator-g-z2
with: G=zcollimator-zSADnom-zcollimator.
Like in the case of the virtual SAD, the sensitivity of the penumbra on the source size increases with air gap and decreases with distance from the source to the collimator. The worst case scenario is then a large air gap, at a short SSD. Regarding the depth (or distance z) two competing effects occur: with increasing depth the sensitivity of the geometric penumbra on changes in the source size increases, but the water-scattering penumbra will also increase, reducing the sensitivity of the total penumbra on the geometric penumbra. A quick evaluation of the sensitivity as function of depth shows that for the Universal Nozzle, under nominal conditions, the increase in sensitivity of the geometric penumbra dominates over the ‘washing out effect’ of the water scattering. This means the penumbra is most sensitive at the end of the SOBP. As RWCS is then considered: an air gap g of 15 cm, maximum range (R) and minimum modulation (M = 2.5 g/cm2), the SSD such that the middle of the SOBP coincides with the isocenter plan (SSD = SAD – R + M/2), and an evaluation depth at the end of the SOBP (z = SAD - SSD - R = -M/2). The effective source size for given range is determined using the GBD parameters.
When allowing a maximum error in 80%-20% lateral penumbra (ΔLP ) of 0.05 cm the tolerance on the effective source size is found to vary from 0.21 for option B7 to 0.34 cm for option B8. (See Table 1.) B7 has the largest sensitivity because the larger range (depth) makes the drift distance from the collimator to the plane of interest largest, as well as the SSD smallest. Although option B8 has even larger range, its much smaller source size makes the sensitivity small.
II.D. Pristine-peak energy spread
The energy spread of the beam affects the width of the pristine peak. With increasing energy spread the width of the peak increases as well as the dose at skin. Because Eclipse internally optimizes the pristine-peaks weights to obtain a flat SOBP, the shape (energy spread) of the underlying pristine peaks does not affect the dose in the uniform region. The pristine energy spread only (marginally) affects the steepness of the distal fall-off and the shape of the depth dose distribution in the entrance region, most notably the dose at skin. The biggest effect of a change in energy spread is observed at the skin, where the dose increases with increasing energy spread. The magnitude of skin-dose change depends on the number of peaks added to the SOBP, i.e. the modulation width. Considering the change in skin dose as percentage of the skin dose, fields with larger modulation width are less sensitive to changes in energy spread. As the RWCS we investigate the sensitivity of the skin dose for pristine peaks. The sensitivity dDskin/dΣ is determined by varying Σ around its nominal value in the TPS and observing the change in skin dose of the modeled pristine peak. The slope of a linear fit to the change in skin dose as function of the change in energy spread is taken to be the sensitivity. The slope is determined for the lowest, i.e. the most sensitive, range per option. Given the slope for each option and a 2% limit on the error in (skin) dose, the accuracy tolerance on the energy spread is calculated for each option. The tolerance varies from 0.12% for option B1 to 0.10% for option B7. A linear fit to the tolerance as function of range at nozzle entrance is made to create the tolerance limits of Figure 5 in the paper. Because the tolerances are determined for pristine peaks, we expect the dose error to be smaller than the clinical limit for an SOBP. To test this, a B1 SOBP (range of 5.2 g/cm2, modulation of 4/0 g/cm2) was calculated in Eclipse with first the nominal pristine energy spread (1.02%) and then the nominal energy spread plus the calculated tolerance (1.14%). The skin dose was found to change from 80.5% to 80.7% a relative change of +0.3%, well below the 2% limit.
[1] This rough approximation is good enough since we are not interested in the absolute value of the penumbra, but its dependence on the effective source size.