Predicting dose uncertainty in moving organs at risk
Joep Stroom*, Joos Lebesque*, Marcel van Herk*, and Ben Heijmen
Radiotherapy Division, The Netherlands Cancer Institute - Antoni van Leeuwenhoek Hospital, Amsterdam, The Netherlands.  Department of
Radiation Oncology, Erasmus MC – Daniel den Hoed Cancer Center, Rotterdam, The Netherlands.
*
Abstract
Since the use of safety margins for geometrical uncertainties appears inappropriate for organs at risk (OARs), two methods are
proposed to quickly determine the effect of systematic errors on the DVHs of OARs. Special DVHs are constructed in two different
ways, which indicate the confidence intervals (CIs) on the volumes for each dose bin in the DVHs. The CIs determined by the new
approach were validated with Monte Carlo simulations for the rectum in 40 prostate plans.
Keywords
Organs At Risk (OAR), geometrical uncertainties, confidence interval, systematic errors, dose volume histogram (DVH)
Introduction
The geometrical uncertainties in the position of the clinical
target volume (CTV) during radiotherapy are normally taken
into account by adding a margin to this volume during
planning, hence yielding the planning target volume (PTV) [1].
The margins should be such that the planned dose in the PTV is
representative for the dose in the “moving” CTV. PTV-margin
rules have been developed, which guarantee a low probability
of CTV underdosage [2,3]. For organs at risk (OARs), a similar
approach has been proposed [4,5]. However, since dose
evaluation for OARs depends normally on the percentage of
volume that receives a specific dose, margins appear less
feasible; margin addition normally increases the volume of the
OAR significantly and the dose in the expanded volume hence
cannot be made representative for the moving OAR.
Therefore, other ways of evaluating (during planning) the
effect of geometrical uncertainties on the dose in OARs must
be procured. The most straightforward way is to sample
translations and rotations from their respective error
distributions, moving the OAR accordingly, calculate the dose,
and display the distribution of possible results [6,7]. In Fig. 1 a
volume(%)
100
0
CTV
rectum
dose (%)
100
Figure 1: Example of distribution of possible DVHs due to
systematic errors for a rectum and CTV in a three-field prostate
plan. Thin black lines are original DVHs, grey lines are 100
sampled curves, and black dotted curves the mean. The PTV
margin ensures adequate CTV coverage for alle curves, wheras
the spread in rectum DVHs is obvious.
possible distribution of DVHs is shown as a result of
geometrical errors. However, many samples are required to
obtain a reliable result, which still requires too much time to be
used in practice, especially if this must be incorporated in
inverse planning. Therefore, in this paper two methods to
calculate the spread of possible results due to geometrical
errors will be proposed and compared. At the moment, only
systematic errors are considered, which are of 3 to 4 times
greater importance than the random errors [2,3]. The
systematic errors normally consist of organ motion and set-up
error and are in this abstract represented by a normal
distribution with standard deviation ; non-isotropic error
distributions, deformations, and rotations are ignored in this
abstract.
Material and methods
The coverage probability (CP) method
Previously, a method to quickly evaluate the dose in moving
CTVs was proposed [2]. It appeared that convolution of the
CTV with the distribution of systematic errors and taking the
DVH of this convolved, smeared-out, CTV, yielded the mean
DVH (mDVH, i.e. the mean volume in each dose bin) for the
possible systematic errors (Fig. 2a). The convolved volume
was called the coverage probability (CP) volume because the
voxel values indicate the probability that the particular voxel is
covered by the moving CTV. The mDVH was obtained by
weighing each CTV voxel by its CP-value when filling the
different dose bins. Since the high dose region is normally
planned around the CTV, a CTV motion can only decrease the
CTV dose and the mDVH will give an indication of the
average underdosage.
An OAR is normally not (completely) in the high dose
region. Motion of an OAR can therefore increase as well as
decrease the OAR dose. Consequently, the mDVH can be
almost equal to the original static DVH because individual
DVHs cancel out; no information about the spread of DVHs is
obtained (see Fig. 1). Therefore, the CP-method is extended.
Instead of using the whole error distribution in the convolution,
parts of the distribution in different directions can be used. The
CTV/OAR
-distribution
CP-matrix
(m)DVHs
a)
CTV

OAR
The limited sampling (LS) method
b)
ventral

dorsal
c)

Figure 2: demonstration of the CP-method for the prostate case
from Fig. 1. In a) the volume is convolved with the distribution of
systematic errors yielding mDVHs (dotted) for CTV and rectum.
Original DVHs are solid. In b) only the ventral cone of the
distribution is used yielding a spread in the rectum curves (the
dorsal mDVH curve is also indicated). In c) only the outer 10%
rim of this cone is used, yielding more extreme mDVHs.
DVHs subsequently acquired are the mDVHs for that subgroup
of possible systematic motions (see Fig. 2b,c).
To get an indication of the spread for all dose levels,
subgroup mDVHs in different directions can be combined into
two encompassing DVHs (eDVHs, i.e. the minimum and
maximum volume for each dose bin in the multiple mDVHs).
For instance, the ventral and dorsal mDVHs in Fig. 3 only
reflect the DVH spread for the higher dose levels. Variation at
the lower levels is due to cranio-caudal translations. The
eDVHs in this abstract were constructed from the mDVHs of
the cones in the 6 orthogonal directions and of the 6 cones
rotated 45 degrees around the lateral axis.
The eDVHs can be used to get an indication of a confidence
interval (CI) on volumes for each dose bin in the DVHs.
Crudely, a mDVH representing a subgroup of motions has
about half its curves above and half its curves below the mean.
If the different subgroups used to compose the eDVH each
100
cranial
caudal
volume(%)
represented X % of the total distribution, the CI represented by
the eDVHs should then be about 100-X %. Taking a smaller
part of the cones varies the size of the expected CI (Fig. 2b,c).
Instead of the CP-method, an alternative way to determine
eDVHs that can represent CIs is to sample translations at
specific points in the distribution, move the OAR, and calculate
the DVHs. For instance, a translation T of an OAR equal to 1
implies that about 16% of possible systematic translations are
larger than T (i.e. conform a 1D normal distribution).
Assuming that the dose from a point in the OAR to the high
dose region increases continuously, a larger translation in that
direction will cause a higher dose in the OAR. So, if there is
one dominant direction towards a specific high-dose isosurface D, the volume with at least dose D (V D) corresponding
to T=1 then should have about 16% of possible DVHs above
it. The DVH curve corresponding to T=-1 (i.e. in the opposite
direction) will consequently have about 16% of possible curves
below it.
Now, to cover the 3D translation space effectively,
translations in a limited number of directions will yield
multiple DVHs, which can again be used to construct the
eDVHs. The CI associated with these eDVHs depends on the
size of the selected translations T. For T=1, CI=68%, for
T=2, CI=95%, etc. The eDVHs used in this study were
constructed from 14 isotropically distributed translations; the 6
orthogonals plus the diagonals in all the octants. For example,
the eDVHs corresponding to the 68% CI were constructed
from 14 DVHs with translations ,0,0; 0,,0; 0,0,;
1/3,1/31/3; and all possible combinations with –.
Verification of the methods with Monte Carlo
As a first OAR, the rectum in 3-field prostate plans was
considered for 40 patients. The systematic errors  were
assumed to be 3.5 mm. The various CIs as predicted by the two
methods were compared with Monte Carlo (MC) simulations
of possible translations, each consisting of 200 samples. To
verify the predicted CIs, the number of sampled curves above
and below the eDVHs as determined by the two methods was
counted for dose levels varying from about 30 to 80 Gy (
Dmax). Subsequently, the average (over the dose) difference
between the predicted and simulated CIs was calculated.
ventral
Results
The first results for the LS and CP-method are summarized in
dorsal
0
dose (%)
100
Figure 3: indication that the CP-method using subgroups can
indicate DVH spread due to systematic errors. Original static DVH
in black, 100 MC simulated curves in grey, mDVHs of ventral and
dorsal cones dashed, and mDVHs of cranial and caudal cones
dotted. Each cone represents about 17% of the possible systematic
errors. Hence, the encompassing curves contain about 83% of
curves for each dose level.
T ()
2.5
2
1.5
1
0.5
CI (%)
98.8
95.4
86.6
68.3
38.3
 (%)
-0.3
-0.7
-0.6
-0.3
-0.5
SD (%)
0.8
1.4
2.4
3.5
3.5
Table 1: Summary of the results for rectum motion of 40 prostate
patients with the LS method. T is the translation used to determine
the confidence interval CI,  the mean (of 40 cases) difference
between simulated and expected CI, and SD the standard deviation
of the differences.
F (%)
10
25
50
75
100
 (%)
0.2
0.4
-0.1
0.2
0.4
CI (%)
99.1
97.1
93.0
88.5
82.0
SD (%)
0.7
1.0
1.7
2.3
2.5
Table 2: Summary of the results for rectum motion of 40 prostate
patients with the CP method. F is the (outer) fraction of the cones
that were taken, for the rest see Table 1.
Tables 1 and 2, respectively. Various eDVHs have been
calculated (representing different CIs) by varying the
translation T or outer fraction of the cones F, respectively. For
both methods, the average difference (over 40 cases) between
predicted and simulated CIs are smaller than 1%.
The results for the LS-methods are graphically shown in Fig.
4. The deviations that are seen are partly due to the limited
number of simulations performed and to the low doses used in
the averaging; contrary to the lower doses, the higher doses are
normally located at one spot (around the CTV) so that there is
indeed one dominant direction that determines the spread.
100
be adjusted based on the patient geometry, i.e., the position and
shape of the CTV (or high dose region) with respect to the
OAR. However, initial experience for the spinal cord in lung
and head and neck plans also appears to yield satisfying results
with the current settings.
The eDVHs indicate the CI for each dose bin and can
therefore be used to evaluate the plan. On the other hand,
perhaps not so much the CI for a certain DVH is clinically
important, but the probability that the possible DVHs lay above
a critical DVH point. Only the outer eDVH is required for this.
To use these new methods however, new, probability-based
criteria must be developed.
Conclusion
A method to quickly calculate the effect of systematic errors on
the DVH for a mobile organ at risk has been suggested. The
spread of possible outcomes can be predicted by construction
of special DVHs, which indicate the confidence intervals for
each dose bin in the DVHs. First results for the rectum are
encouraging and other cases will be studied in the future.
T = 2.5
T = 2
References
T = 1.5
[1] International Commission on Radiation Units and
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90
CI (%)
80
70
T = 1
60
50
40
T = 0.5
30
1
4
7
10
13
16
19 22 25
patient nr
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31
34
37
40
Figure 4: the predicted (straight lines) and simulated CIs for
moving rectums  = 3.5 mm) of 40 prostate patients. Different
CIs have been calculated by varying the size of the translations
used to determine the eDVHs.
Discussion
The differences between the new approach and the simulations
will decrease by increasing the number MC samples and the
number of mDVHs to construct the eDVHs. On the other hand,
the optimal directions to predict the spread of the DVH curves
will depend on the dose distribution and on the position and
shape of the OAR. This direction might be automatically
obtained from the treatment geometry, which will reduce the
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For the case studied so far, both methods are able to predict
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distributions (e.g. IMRT) and OARs (e.g. lung). Possibly, the
expected CI associated with values for T and F might have to
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