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 doublescattering system in the Universal Nozzle.
Option
B1
B2
B3
B4
B5
B6
B7
B8
Min
Range
Max
Range
Max
Diam.
ΔSADvirt
ΔSADeff
Δσsource
ΔΣpristine /
Rnozzle
g/cm2
g/cm2
cm
cm
cm
cm
%
4.6
5.9
7.5
9.6
11.7
15.5
19.8
22.8
5.9
7.5
9.6
11.7
15.5
19.8
23.9
28.4
24.0
24.0
24.0
24.0
24.0
24.0
24.0
14.0
±4.8
±4.6
±4.3
±4.0
±3.0
±2.9
±2.6
±4.6
±87
±70
±55
±46
±31
±23
±21
±19
±0.32
±0.30
±0.29
±0.27
±0.25
±0.23
±0.21
±0.34
±0.12
±0.12
±0.12
±0.11
±0.12
±0.11
±0.10
±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
𝑧−𝑧𝑐𝑜𝑙𝑙𝑖𝑚𝑎𝑡𝑜𝑟
±∆𝐷50% = ∓∆𝑆𝐴𝐷𝑣𝑖𝑟𝑡 × 𝐷𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒 × (𝑆𝐴𝐷
2
𝑣𝑖𝑟𝑡 −𝑧𝑐𝑜𝑙𝑙𝑖𝑚𝑎𝑡𝑜𝑟 )
(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 =
max
D50% x (SADvirt-zcollimator) / SADvirt, with
max
D50% = 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-perMU is not considered here. )
Given that the fluence reduces with inverse of the distance from the effective source squared:
2
Φ(𝑧) = Φ(𝑧 = 0)(𝑆𝐴𝐷𝑒𝑓𝑓 ⁄(𝑆𝐴𝐷𝑒𝑓𝑓 − 𝑧)) , the dependence of the error in fluence relative to
the fluence at isocenter is given by:
Φ(𝑧)
2𝑧
±∆ (
) = ∓∆𝑆𝐴𝐷𝑒𝑓𝑓 ×
Φ(𝑧 = 0)
(𝑆𝐴𝐷𝑒𝑓𝑓 − 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)
2
𝑧𝑐𝑜𝑙𝑙𝑖𝑚𝑎𝑡𝑜𝑟 − 𝑧
2
𝐿𝑃(𝑧) = 𝑘 × √𝜎𝑠𝑜𝑢𝑟𝑐𝑒
(
) + 𝜎𝐻22 𝑜
𝑆𝐴𝐷𝑛𝑜𝑚 − 𝑧𝑐𝑜𝑙𝑙𝑖𝑚𝑎𝑡𝑜𝑟
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 depth1: 𝜎𝐻2 𝑜 = 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:
𝐺2
±∆𝐿𝑃 = ±∆𝜎𝑠𝑜𝑢𝑟𝑐𝑒 × 𝑘 × 𝜎𝑠𝑜𝑢𝑟𝑐𝑒 ×
2
√𝜎𝑠𝑜𝑢𝑟𝑐𝑒
× 𝐺 2 + (0.02 × (𝑧𝑐𝑜𝑙𝑙𝑖𝑚𝑎𝑡𝑜𝑟 − 𝑔 − 𝑧))
2
with: 𝐺 = (𝑧𝑐𝑜𝑙𝑙𝑖𝑚𝑎𝑡𝑜𝑟 − 𝑧)⁄(𝑆𝐴𝐷𝑛𝑜𝑚 − 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
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.
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/cm 2,
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.