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Adaptive and Robust Radiation Therapy in
the Presence of Drift
Mar PA & Chan TC
Version Post-print/accepted manuscript
Citation (published version)
Mar PA, Chan TC. Adaptive and robust radiation therapy in the presence of drift. Physics in medicine and biology. 2015 Apr 10;60(9):3599.
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The final version of this article is available from Institute of Physics and Engineering in Medicine (IPEM) at https://doi.org/10.1088/0031-9155/60/9/3599.
Copyright/License © 2015 Institute of Physics and Engineering in Medicine
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Adaptive and Robust Radiation Therapy in the
Presence of Drift
Philip Allen Mar and Timothy C.Y. Chan
Department of Mechanical and Industrial Engineering, University of Toronto,
Toronto ON, M5S 3G8, Canada
E-mail: philip.mar@mail.utoronto.ca
Abstract.
Combining adaptive and robust optimization in radiation therapy has the potential
to mitigate the negative effects of both intrafraction and interfraction uncertainty
over a fractionated treatment course. A previously developed adaptive and robust
radiation therapy (ARRT) method for lung cancer was demonstrated to be effective
when the sequence of breathing patterns was well-behaved. In this paper, we examine
the applicability of the ARRT method to less well-behaved breathing patterns. We
develop a novel method to generate sequences of probability mass functions that
represent different types of drift in the underlying breathing pattern. Computational
results derived from applying the ARRT method to these sequences demonstrate that
the ARRT method is effective for a much broader class of breathing patterns than
previously demonstrated.
1. Introduction
Uncertainties in radiation therapy can have a large impact on the quality of the
treatment. Steep dose gradients that are generated by optimization can lead to elevated
healthy tissue dose or underdose in the target if assumptions about the treatment
parameters are violated during treatment delivery. For example, it has been shown
that lung cancer treatments planned with respect to a particular respiratory pattern
can be compromised if a different pattern is exhibited during treatment [Lujan et al.,
2003, Sheng et al., 2006].
Robust optimization is a methodology that can be used to produce treatments that
are desensitized to uncertainties (see Bertsimas et al. [2011] for a general review of the
theory). In intensity-modulated radiation therapy (IMRT), robust optimization has
been applied to problems with organ and patient position uncertainty [Chu et al., 2005,
Olafsson and Wright, 2006], dose matrix calculation uncertainty [Olafsson and Wright,
2006], and organ motion uncertainty [Chan et al., 2006, Bortfeld et al., 2008, Vrancic
et al., 2009, Chan et al., 2014]. In intensity-modulated proton therapy (IMPT), robust
optimization has been used to address both setup and range uncertainty [Unkelbach
et al., 2007, Fredriksson, 2012, Pflugfelder et al., 2008, Chen et al., 2012, Liu et al.,
Adaptive and Robust Radiation Therapy in the Presence of Drift 2
2012b,a, 2013, Cao et al., 2012]. Fredriksson and Bokrantz [2014] compared three
different robust optimization frameworks with varying levels of conservatism. In
addition, the robust methodology used in [Yang et al., 2005] has been applied in [Zhang
et al., 2013] to volumetric modulated arc therapy (VMAT). Stochastic optimization,
which is closely related to robust optimization, has also been used to handle uncertainty
in IMRT [Nohadani et al., 2009] and in IMPT [Unkelbach et al., 2009].
Adaptive radiation therapy (ART) is a paradigm that can be used to address
uncertainty by tapping into the potential of dynamically adjusting or re-optimizing
treatments over a fractionated treatment course [Yan et al., 1997]. These adjustments
are supported by using updated information, often from imaging, in a feedback loop.
Adjustments can take the form of dynamic multi-leaf collimator or couch adjustments
based on updated information about organ position [McMahon et al., 2007, Ruan and
Keall, 2011, Keall et al., 2011, Zhang et al., 2012] or re-optimization with updated
images and related biological information and revised dose limits [Saka et al., 2011, Wu
et al., 2008, Li et al., 2013b,a, Saka et al., 2013, Zhen et al., 2013, Kim et al., 2012].
A recent study developed an integrated framework that combined robust
optimization and adaptive radiation therapy in the context of lung cancer IMRT [Chan
and Misic, 2013, Misic and Chan, 2015], which they referred to as adaptive and robust
radiation therapy (ARRT). This framework was shown to have benefits of both robust
optimization (the ability to mitigate the effects of uncertain intrafraction motion) and
adaptive radiation therapy (the ability to adjust beliefs about the underlying uncertainty
and re-optimize, based on updated motion probability distributions acquired throughout
the treatment). Mathematically, the ARRT approach was proven to be asymptotically
optimal if the sequence of observations of the uncertainty converged. However, it was
also shown that an artificial, pathological sequence of observations could confound the
approach. The question of whether the ARRT method is viable under sequences of
breathing patterns that are neither convergent nor pathological remains open.
Realistic breathing patterns naturally exhibit some amount of variation. Variations
have been observed in the baseline [McNamara et al., 2013, Zhao et al., 2011, Pepin et al.,
2011, Juhler Nøttrup et al., 2007], amplitude [Seppenwoolde et al., 2002, Chan et al.,
2013, Coolens et al., 2008, Mutaf et al., 2011, Juhler Nøttrup et al., 2007] and length of
the breathing period [Coolens et al., 2008].
In this paper, we test the ARRT approach under sequences of breathing patterns
that exhibit “drift”. In particular, we model the breathing pattern realized by a patient
in any given fraction as a probability mass function (PMF) derived from a variation
of the Lujan model [Lujan et al., 1999]. We then generate a sequence of PMFs by
successively adjusting the parameters in the model and visualize them as a sequence of
points in the probability simplex. The parameters are adjusted in such a way as to model
three types of drift motivated by types of variation observed in the literature: baseline,
amplitude, and breathing phase drift. Finally, we evaluate the dosimetric performance
of the ARRT method on these sequences of PMFs.
Adaptive and Robust Radiation Therapy in the Presence of Drift 3
2. Methods and materials
We begin by briefly reviewing the ARRT framework (Section 2.1) and the Lujan model
(Section 2.2). Then we describe a method to visualize sequences of PMFs (Section 2.3),
our modified version of the Lujan model used to generate PMFs (Section 2.4), and the
setup of our computational experiments (Section 2.5).
2.1. ARRT framework
The ARRT framework [Chan and Misic, 2013] introduces an uncertainty set update
algorithm to the static robust optimization model of Bortfeld et al. [2008]. They optimize
the fluence, w∗k, in each fraction k using the following formulation:
minimize∑v∈N
p′∆vwk
subject to θ ≤ [p′k∆vwk]v∈T ≤ θ, ∀pk ∈ Pk,
wk ≥ 0,
(1)
where θ and θ are upper and lower dose limits on the tumour, p is a nominal PMF
weighting the probability of being in each breathing phase, ∆v is the dose-deposition
matrix for voxel v associated with a phase-beamlet pair, N is the set of healthy lung
voxels and T is the set tumour voxels. The set Pk is a polyhedral uncertainty set, and
is constructed using the uncertainty set from the previous fraction (Pk−1) and the most
recently observed PMF (pk−1) according to:
Pk ← (1− α)Pk−1 + αpk−1. (2)
The parameter α specifies the strength of adaptation – a higher value means that recent
observations receive more weight in the updating process. The method works as follows:
in fraction k−1 we observe a realized PMF pk−1, generate Pk according to (2), solve (1)
to obtain w∗k, and repeat. Assuming m total fractions, the fluence that is delivered in
fraction k is w∗k/m. The end result of this method is a sequence of fluence maps that
is updated in each fraction, in contrast with the static robust method where the same
fluence map w∗1/m is applied in all m fractions. When P1 = {p} is a singleton, the
problem is called the nominal problem and the initial uncertainty set is the nominal
uncertainty set. When P1 is the entire probability simplex (set of all vectors whose
components are nonnegative and add up to 1), the problem is called the margin problem
and the initial uncertainty set is the margin uncertainty set. For all other choices of P1,
we will refer to the problem as a robust problem with a robust uncertainty set.
Although the ARRT method is myopic in that it only considers one fraction at a
time without knowledge of the future, it was proved that if the sequence of PMFs pk
converges to a limiting PMF, then the sequence of optimal solutions w∗k converges to
the optimal solution of (1) with Pk being the singleton limiting PMF [Chan and Misic,
Adaptive and Robust Radiation Therapy in the Presence of Drift 4
2013]. However, it was shown that providing an artificially constructed, pathological
PMF sequence to the ARRT method could result in a sequence of optimal fluences with
poor dosimetric properties, especially for high values of α. A prescient solution was used
as a performance benchmark. The prescient solution arises when Pk is replaced by the
observed pk when solving the robust problem for each fraction k – that is, the fluence
for fraction k is determined with “future” knowledge of how the patient will breathe in
fraction k.
2.2. Generating PMFs
Lujan et al. [1999] models one-dimensional breathing motion in the z-axis using the
following equation in time t:
z(t) = z0 − b cos2s(πt
τ− φ), (3)
where z0 is a vertical translation, b is the amplitude, s is a shape and steepness
parameter, τ is the period of the cycle, and φ is a horizontal translation. PMFs in
n-dimensional space are generated by binning the curve z(t) into n bins that partition
the interval [z0 − b, z0], with each bin corresponding to a phase of the breathing cycle.
We use the binning strategy described in Chan [2007] for the PMF sequences in our
computational experiments below.
2.3. Visualizing PMFs
Next, we describe a method to visualize n-dimensional PMFs using a two-dimensional
regular polygon with n sides (an “n-gon”). Each PMF is represented as a point in the
regular n-gon. Each vertex of the n-gon represents a PMF where all the probability
mass is concentrated at the corresponding phase. Thus, we can represent any PMF as a
convex combination of the vertices of the n-gon. Specifically, if we let {vi}ni=1 ⊂ R2 be
the vertices of the regular n-gon and p be an n-dimensional PMF, then we can represent
p as the point∑n
i=1 pivi ∈ R2 in the n-gon. Figure 1 visualizes three different PMFs in
a regular pentagon.
2.4. Drift
We use a modified version of the Lujan model to generate PMF sequences in this paper:
z(t) = zh + za − b∣∣∣∣cos
(πt
τ− φ)∣∣∣∣2|s| . (4)
We create a sequence of PMFs by iteratively adjusting some of the parameters in
equation (4) and binning the resulting curves. We fix τ = 1 and choose values of
the other parameters zh, za, b, φ and s so that z(t) ∈ [0, 1] and z(t) goes through exactly
one period as t goes from 0 to 1.
Adaptive and Robust Radiation Therapy in the Presence of Drift 5
A
B
C
Figure 1: Visualizing PMFs in an n-gon. Point A is a PMF with equally weighted
phases. Point B is a PMF generated from equation (3) with n = 5, z0 = 1, b = 1, s = 2,
τ = 1, φ = 0. Point C is a PMF with all weight on a single phase.
We consider three different types of drift: baseline, amplitude and breathing phase.
These drifts are controlled by iteratively changing zh, b and s, respectively. The other
parameters not being iteratively changed are set depending on which drift type is being
considered. For each drift type, we produce three different degrees of drift: small,
medium and large. The degree of drift represents the extent of the change in the
breathing pattern between the first and last PMF in the sequence. Finally, we include
a sequence that combines all three individual drift types.
2.4.1. Baseline drift Baseline drift is controlled by iteratively adjusting zh. We fix
za = b = 0.5, φ = 0 and s = 2. Figure 2a shows the initial motion pattern (i.e.,
from the first fraction) corresponding to the small degree of drift and the three final
motion patterns (i.e., from the last fraction) corresponding to the sequences with small,
medium, and large degrees of baseline drift. For each degree of drift, the intermediate
motion patterns are evenly spaced between the initial and final ones, and are not shown.
Figure 2b shows the simplex representation of the PMF sequence with large baseline
drift. The parameters used to generate baseline drift (including the zeroth nominal
PMF) are summarized in Table 1.
Degree zh za b φ s Increment
Small [0, 0.15] 0.5 0.5 0 2 0.005
Medium [0, 0.3] 0.5 0.5 0 2 0.009
Large [0, 0.5] 0.5 0.5 0 2 1/60
Table 1: Baseline drift parameter values.
Adaptive and Robust Radiation Therapy in the Presence of Drift 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Pos
ition
SmallMediumLargeInitialFinal
(a) Degrees of baseline drift (b) Large baseline drift in simplex
Figure 2: Baseline drift. Figure 2a shows the initial breathing pattern used to generate
the PMFs for small baseline drift and the final breathing patterns for small, medium
and large baseline drift. Figure 2b shows the large baseline drift PMF sequence in the
probability simplex.
2.4.2. Amplitude drift Amplitude drift is controlled by iteratively adjusting b. We set
za = b, so za is also iteratively adjusted. We fix zh = 0, φ = 0 and s = 2. Figure 3a shows
the initial motion pattern corresponding to the small degree of drift and the three final
motion patterns corresponding to the sequences with small, medium, and large degrees
of amplitude drift. Figure 3b shows the simplex representation of the PMF sequence
with large amplitude drift. The parameters used to generate amplitude drift (including
the zeroth nominal PMF) are summarized in Table 2.
Degree zh za b φ s Increment
Small 0 b [1, 0.75] 0 2 1/120
Medium 0 b [1, 0.5] 0 2 1/60
Large 0 b [1, 0.25] 0 2 1/40
Table 2: Amplitude drift parameter values.
2.4.3. Breathing phase drift Breathing phase drift is controlled by iteratively adjusting
s. We consider values of s such that |s| ≥ 1 in order to preserve the general shape of
the breathing pattern. If s ≥ 1, then we fix zh = 0, za = b = 1 and φ = 0. If s ≤ −1,
then we set zh = 0, za = 0, b = −1 and φ = π/2. We allow s to be fractional. Thus, we
include absolute values around the cosine in (4) to prevent z(t) from taking on imaginary
values. Figure 4a shows the three initial and final motion patterns corresponding to the
small, medium and large degrees of drift. Figure 4b shows the simplex representation of
Adaptive and Robust Radiation Therapy in the Presence of Drift 7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Pos
ition
SmallMediumLargeInitialFinal
(a) Degrees of amplitude drift (b) Large amplitude drift in simplex
Figure 3: Amplitude drift. Figure 3a shows the initial breathing pattern used to generate
the PMFs for small amplitude drift and the final breathing patterns for small, medium
and large amplitude drift. Figure 3b shows the large amplitude drift PMF sequence in
the probability simplex.
the PMF sequence with large breathing phase drift. The parameters used to generate
phase drift (including the zeroth nominal PMF) are summarized in Table 3.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Pos
ition
SmallMediumLargeInitialFinal
(a) Degrees of breathing phase drift (b) Large breathing phase drift in
simplex
Figure 4: Breathing phase drift. Figure 4a shows the initial and final breathing patterns
used to generate the PMFs for small, medium and large breathing phase drift. Figure 4b
shows the large breathing phase drift PMF sequence in the probability simplex.
2.4.4. Combined drift The combined drift sequence of PMFs is generated by iteratively
adjusting zh, b and s together. The combined drift sequence combines the large baseline
Adaptive and Robust Radiation Therapy in the Presence of Drift 8
Degree zh za b φ s Increment
Small 0 1 1 0 [1.1, 2.5] 0.1
0 0 -1 π/2 [−2.5,−1] 0.1
Medium 0 1 1 0 [1.5, 8.5] 0.5
0 0 -1 π/2 [−8.5,−1] 0.5
Large 0 1 1 0 [2, 16] 1
0 0 -1 π/2 [−16,−1] 1
Table 3: Breathing phase drift parameter values. There are two rows for each degree of
drift for the two domains of s: s ≥ 1 and s ≤ −1, respectively. The apparent asymmetry
in the s intervals arises from the fact that s = −1 and s = 1 generate the same curve,
so we need to shift the intervals to obtain the sequence.
drift, small amplitude drift, and the medium breathing phase drift.
Figure 5a shows the breathing patterns corresponding to the combined drift.
Figure 5b shows the simplex representation of the PMF sequence. The parameters used
to generate this combined drift (including the zeroth nominal PMF) are summarized in
Table 4.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Pos
ition
InitialFinalPMFs
(a) Degrees of breathing phase drift (b) Combined drift in simplex
Figure 5: Combined drift. Figure 5a shows the breathing patterns used to generate the
PMFs for the combined drift. Figure 5b shows the combined drift PMF sequence in the
probability simplex.
2.5. Experimental setup
For each drift type and degree of drift, we generate a sequence of 31 PMFs – the first one
serves as the nominal PMF and the remaining 30 represent the PMFs that are realized
throughout 30 fractions of treatment. To summarize, there are three types of drift
Adaptive and Robust Radiation Therapy in the Presence of Drift 9
zh zh inc. za b b inc. φ s s inc.
[0, 0.25] 160
b [1/2, 0.375] − 1120
0 [1.1, 2.5] 0.1
[0.25, 0.50] 160
0 −[0.36, 0.25] − 1120
π/2 [−2.5,−1] 0.1
Table 4: Combined drift parameter values. There are two rows for each degree of drift
for the two domains of s: s ≥ 1 and s ≤ −1, respectively. The apparent asymmetry
in the s intervals arises from the fact that s = −1 and s = 1 generate the same curve,
so we need to shift the intervals to obtain the sequence. In the case of b, the values
used decrement from 1/2 to 0.25 by the value 1120
. The case of −[0.36, 0.25] only means
that while we decrement from 0.36 down to 0.25, we take the negative of each value and
input it into equation (4).
(baseline, amplitude and breathing phase); for which there are three different degrees
of drift (small, medium and large); and for which we test based on different strengths
of adaptation (α-value in the algorithm); and for which we have three different initial
uncertainty sets (nominal, robust and margin). The combined drift sequence considers
all three drift types simultaneously.
The type and degree of drift is characteristic of the PMF sequences, whereas the
strength of adaptation and the initial uncertainty set is characteristic of the ARRT
method. For all types of drift, including the combined drift, we test with the strengths
of adaptation α = 0, 0.1, 0.5, 0.9. The case where α = 0 is equivalent to the static, (i.e.,
non-adaptive) robust method of [Bortfeld et al., 2008]. To generate the initial robust
uncertainty set, we use the nominal PMF p and set ui = (1 − pi)β + pi and li = βpiwith β = 0.7. The vectors u and l specify the upper and lower bounds on PMF vectors
that define the robust uncertainty set [Bortfeld et al., 2008].
We evaluate the performance of the ARRT method on the PMF sequences in the
form of curves showing the trade-off between the minimum dose to the tumour and the
mean dose to the left lung. We normalize these values by taking the minimum dose to
the tumour voxels as a percentage of the required 72 Gy and the mean left lung dose
as a percentage of the mean left lung dose in the static (α = 0) case with the margin
uncertainty set. We use Hausdorff distance to calculate the distance between points
on the trade-off curves and the prescient solution. The Hausdorff distance is a metric
between two sets A and B, defined as:
dH(A,B) = max
{supa∈A
infb∈B‖a− b‖2, sup
b∈Binfa∈A‖a− b‖2
}. (5)
3. Results
Figure 6a shows the performance of the ARRT method under large baseline drift.
Observe that curves corresponding to a higher strength of adaptation generally exhibit
higher tumour dose and lower lung dose. Note also that the curves corresponding to
Adaptive and Robust Radiation Therapy in the Presence of Drift 10
α = 0.5 and α = 0.9 exhibit performance similar to the prescient solution. Figures 7a
and 8a illustrate the trade-off curves corresponding to large amplitude drift and large
breathing phase drift, respectively. The performance of the ARRT method on both of
these PMF sequences is similar to that of large baseline drift, in that larger strengths
of adaptation result in similar performance to the prescient solution.
Figure 6b shows the Hausdorff distance from each baseline drift trade-off curve
to the prescient solution as a function of the strength of adaptation. The distance
was measured using the data in units of Gy instead of percent. This figure illustrates
that as the strength of adaptation increases, the distance between the trade-off curve
and prescient solution decreases monotonically. Furthermore, the decrease in Hausdorff
distance drops most sharply from the α = 0 to α = 0.1 case. Figures 7b and 8b show
analogous results for amplitude drift and breathing phase drift, respectively.
Figure 9a shows the performance of the ARRT method under the combined drift.
We see that the results are qualitatively very similar to the trade-off curves obtained
for the baseline, amplitude, and breathing phase drifts separately. Similarly, Figure 9b
shows the Hausdorff distance from each combined drift trade-off curve to the prescient
solution as a function of the strength of adaptation. The results are again similar to the
individual drift sequences.
Figure 10 plots the Hausdorff distance as a function of the average separation
between consecutive PMFs in each sequence. The average separation is simply the
average of the Euclidean distance between consecutive PMFs. There are nine columns
of points, with each column corresponding to a combination of drift degree (S, M, L)
and type (Base, Amp, Phase). For the static robust method, the Hausdorff distance
generally increases as the average PMF separation increases. On the other hand, the
ARRT approach with α > 0 maintains its performance regardless of the average PMF
separation, especially for the cases of α = 0.5 and α = 0.9.
Table 5 summarizes all of the computational results for baseline, amplitude, and
breathing phase drifts. The values listed under the “lung” sub-columns are percentages
of the mean left lung dose, relative to the static margin of the respective type and
degree of drift. The values listed under the “tumour” sub-columns are percentages of
the minimum tumour dose, relative to the required dosage of 72 Gy. It can be seen that
the mean lung dose decreases as the strength of adaptation increases, with the minimum
tumour dose staying relatively constant. These are listed under the “Percentages (%)”
column. The corresponding raw data in Gy is presented in the same table under the
“Dose (Gy)” column. This table also presents the tumour dose escalation potential for
each type and degree of drift when the dose is scaled so that the corresponding mean
left lung dose (MLLD) and left lung V20 (LLV20) dose is equal to the static method
with margin uncertainty set. These are listed under the “Scaled minimum tumor dose
(Gy)” column. Finally, Table 6 summarizes the results for combined drift.
Adaptive and Robust Radiation Therapy in the Presence of Drift 11
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102
80
82
84
86
88
90
92
94
96
98
100
102
104
Mean left lung dose (% from static margin)
Min
imum
tum
or d
ose
(% o
f 72
Gy)
StaticPrescientalpha=0.1alpha=0.5alpha=0.9
(a) Large degree of drift trade-off curve
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
18
20
Strength of adaptation
Hau
sdor
ff di
stan
ce to
pre
scie
nt s
olut
ion
Small driftMedium driftLarge drift
(b) Performance versus strength of adaptation
Figure 6: Baseline drift results: trade-off curve and performance versus adaptation
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102
70
72
74
76
78
80
82
84
86
88
90
92
94
96
98
100
102
104
Mean left lung dose (% from static margin)
Min
imum
tum
or d
ose
(% o
f 72
Gy)
StaticPrescientalpha=0.1alpha=0.5alpha=0.9
(a) Large degree of drift trade-off curve
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
18
20
Strength of adaptation
Hau
sdor
ff di
stan
ce to
pre
scie
nt s
olut
ion
Small driftMedium driftLarge drift
(b) Performance versus strength of adaptation
Figure 7: Amplitude drift results: trade-off curve and performance versus adaptation
4. Discussion
Chan and Misic [2013] noted three main insights from their computational results
applying the ARRT method to stable PMF sequences. First, the ARRT method
generally outperformed the static robust method. Second, the ARRT method performed
almost as well as the prescient solution. Third, the ARRT method was fairly insensitive
to the choice of the initial uncertainty set. The results presented in this paper suggest
that these three observations hold even when the ARRT method is applied to a variety
of PMF sequences that exhibit large degrees of drift, and even combinations of different
types of drift. Because the small and medium drift sequences have PMFs that do not
differ as much as in the case of large drift, we expect the performance of ARRT on the
Adaptive and Robust Radiation Therapy in the Presence of Drift 12
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102
75
77
79
81
83
85
87
89
91
93
95
97
99
101
103
105
Mean left lung dose (% from static margin)
Min
imum
tum
or d
ose
(% o
f 72
Gy)
StaticPrescientalpha=0.1alpha=0.5alpha=0.9
(a) Large degree of drift trade-off curve
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
18
20
Strength of adaptation
Hau
sdor
ff di
stan
ce to
pre
scie
nt s
olut
ion
Small driftMedium driftLarge drift
(b) Performance versus strength of adaptation
Figure 8: Breathing Phase drift results: trade-off curve and performance versus
adaptation
84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102
84
86
88
90
92
94
96
98
100
102
Mean left lung dose (% from static margin)
Min
imum
tum
or d
ose
(% o
f 72
Gy)
StaticPrescientalpha=0.1alpha=0.5alpha=0.9
(a) Combined drift trade-off curve
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
2
4
6
8
10
12
14
16
18
20
Strength of adaptation
Hau
sdor
ff di
stan
ce to
pre
scie
nt s
olut
ion
(b) Performance versus strength of adaptation
Figure 9: Combined drift results: trade-off curve and performance versus adaptation
small and medium drift sequences to be at least as good as on the large drift sequences.
This intuition is confirmed in Table 5, and in Figures 6b, 7b and 8b.
For a more direct comparison from an iso-lung dose perspective, examine the
MLLD-scaled and LLV20-scaled tumour doses in Table 5. For example, we see that
for large baseline drift with the robust uncertainty set the MLLD-scaled tumour dose‡is 82.76 Gy for α = 0.5 and 79.52 Gy for α = 0. This increase of 3.24 Gy corresponds
to a 4.1% increase in local tumour control, estimated using a relationship between 5-
year local tumour control and tumour dose [Kong et al., 2005, Zhao et al., 2014]. If we
‡ Note that these values are rounded to the nearest hundredths place.
Adaptive and Robust Radiation Therapy in the Presence of Drift 13
0 0.01 0.02 0.03 0.04 0.05 0.060
5
10
15
20
25
Average separation between consecutive PMFs
Hau
sdor
ff di
stan
ce to
pre
scie
nt s
olut
ions
Staticalpha=0.1alpha=0.5alpha=0.9
SBase
MBase
LBase
SAmp
MAmp
LAmp
SPhase
MPhase
LPhase
Figure 10: Performance vs. average of separations
scale the tumour dose using the commonly used lung V20 instead, the increase is 6.74
Gy, which translates to a 8.56% increase in 5-year local control. For large amplitude
drift, the increases in control are 7.24% (MLLD-scaled) and 14.82% (LLV20-scaled). For
large breathing phase drift, the increases in control are 7.06% (MLLD-scaled) and 9.89%
(LLV20-scaled). For the combined drift, the increases in control are 4.10% (MLLD-
scaled) and 8.62% (LLV20-scaled). We note that the gains in local control increase as
α increases, and not necessarily as the degree of drift increases. For example, for small
baseline drift there is an over 11 Gy increase in tumour dose when scaled by LLV20
between α = 0 and α = 0.5 for the robust uncertainty set. Given that the 5-year local
control rates for tumour doses between 74-84 Gy is roughly 35% [Kong et al., 2005], the
gains noted above are non-trivial.
Figures 6b, 7b, and 8b suggest that a little adaptation goes a long way. Additionally,
these figures reinforce the intuition that small degrees of drift can generally be managed
using a smaller value of α, whereas PMF sequences that exhibit larger drift will require a
larger strength of adaptation in order to generate a good solution. Note that the quality
of the ARRT solution for α = 0.5 is quite similar to the case for α = 0.9. Thus most
of the benefit from the ARRT method seems to be derived from the ability to adapt
interfractionally to a moderate extent. A moderate strength of adaptation also reduces
the susceptibility of the method to erratic PMF behaviour [Chan and Misic, 2013].
Furthermore, the performance of the ARRT method on the combined drift sequence is
comparable to its performance on the individual drift sequences, as seen in Figure 9b.
Adaptive and Robust Radiation Therapy in the Presence of Drift 14
We use average PMF separation in Figure 10 as a simple way to characterize how
different consecutive PMFs are in each sequence. We would expect that sequences
with consecutive PMFs that are far apart (large separation) would be more difficult to
manage. Figure 10 illustrates that this is indeed the case for the static robust method,
whose performance degrades as the average separation increases. On the other hand, as
long as there is a moderate amount of adaptation, sequences with large PMF separation
can be managed effectively using ARRT. The distance between PMFs is not a perfect
characterization of what makes certain PMF sequences harder to deal with than others.
For example, we see that large baseline drift has a higher PMF separation than large
phase drift, but the ARRT method performs better on large baseline drift across all
strengths of adaptation. Nevertheless, we see that for moderate to large strengths
of adaptation (α ≥ 0.5), the performance differences are very small across all PMF
sequences, so it may be that only very pathological cases (which are unlikely to be
realized in reality) are the difficult ones to deal with.
The three types of drift considered in this paper are closely related to the
decomposition of real-world breathing patterns into functions of baseline drift, frequency
variation, fundamental pattern change, and additional noise [Ruan et al., 2009] –
amplitude and breathing phase drift can be seen as aspects of the other factors.
If the types of drift considered in this paper are considered “basis functions” that
breathing patterns can be decomposed into, then the ARRT method may be effective
for general breathing patterns that are a combination of different types of drift. This is
demonstrated to some extent with the combined drift results we presented.
There are several other future directions for this research. First, developing an
updating method with a tunable value of α (instead of a fixed one) would provide
even more control to the planner. The strength of adaptation could be increased when
observations indicate the sequence has stabilized, or decreased in erratic parts of the
sequence. Similarly, the uncertainty set can be updated differently. Rather than having
a trade-off between the trailing PMF and the previous uncertainty set, we may be able
to account for distances between consecutive PMFs and even allow the uncertainty set
to grow in a period of instability. Both of these extensions would be enabled by the
measurement of how often the realized PMF lies inside the uncertainty set of a given
fraction.
5. Conclusion
In this paper, we demonstrated the application of the ARRT method to PMF sequences
that model a variety of drift types in the underlying breathing pattern. Our results
indicate that the ARRT method not only performs well given a well-behaved sequence
of PMFs, but it can also handle breathing patterns that change substantially over a
fractionated treatment course. This suggests that the method is more broadly applicable
than previously demonstrated.
Adaptive and Robust Radiation Therapy in the Presence of Drift 15
Acknowledgements
This research was supported in part by the Natural Sciences and Engineering Research
Council of Canada (NSERC) and the Canadian Institutes of Health Research (CIHR)
through the Collaborative Health Research Projects (CHRP) grant #398106-2011.
Adaptive
andRobu
stRadiation
Therapy
inthe
Presen
ceof
Drift
16Percentage (%) Dose (Gy) Scaled minimum tumor dose (Gy)
Small drift Medium drift Large drift Small drift Medium drift Large drift Small drift Medium drift Large drift
α Lung Tumour Lung Tumour Lung Tumour Lung Tumour Lung Tumour Lung Tumour MLLD LLV20 MLLD LLV20 MLLD LLV20
Baseline
0 Nominal 85.53 98.63 85.58 92.83 85.71 80.95 16.91 71.01 16.96 66.84 17.15 58.28 83.03 94.37 78.10 88.01 68.00 76.06
Robust 89.13 100.00 89.17 99.76 89.27 98.59 17.62 72.00 17.67 71.83 17.87 70.99 80.78 83.34 80.55 82.52 79.52 80.68
Margin 100.00 100.08 100.00 100.13 100.00 100.10 19.77 72.06 19.81 72.10 20.01 72.07 72.06 72.06 72.10 72.09 72.07 72.08
0.1 Nominal 85.65 99.36 86.00 95.75 86.23 91.77 16.93 71.54 17.04 68.94 17.26 66.07 83.52 95.34 80.16 90.08 76.62 83.20
Robust 86.75 99.94 87.06 99.49 87.17 95.98 17.15 71.96 17.25 71.63 17.45 69.10 82.96 92.27 82.27 89.77 79.27 83.36
Margin 89.33 100.07 89.54 100.04 89.48 99.12 17.66 72.05 17.74 72.03 17.91 71.37 80.66 84.61 80.45 83.10 79.75 80.32
0.5 Nominal 85.73 99.82 86.13 98.60 85.40 98.08 16.95 71.87 17.07 70.99 17.09 70.62 83.84 95.54 82.42 91.52 82.69 88.18
Robust 85.94 99.97 86.22 99.28 85.98 98.83 16.99 71.98 17.09 71.48 17.21 71.16 83.76 94.86 82.90 91.62 82.76 87.42
Margin 86.53 99.99 86.86 99.64 86.47 99.34 17.10 71.99 17.21 71.74 17.30 71.53 83.20 93.51 82.59 90.51 82.72 86.70
0.9 Nominal 85.75 99.90 86.11 99.21 85.33 98.95 16.95 71.93 17.06 71.43 17.08 71.24 83.88 95.57 82.95 91.81 83.49 88.68
Robust 85.85 99.98 86.24 99.53 85.40 99.26 16.97 71.99 17.09 71.66 17.09 71.47 83.85 95.12 83.09 91.68 83.69 88.60
Margin 86.27 100.00 86.63 99.65 85.77 99.40 17.05 72.00 17.17 71.75 17.17 71.57 83.46 94.60 82.82 91.35 83.45 88.12
Prescient 85.77 100.00 86.07 100.00 85.24 100.00 16.95 72.00 17.06 72.00 17.06 72.00 83.95 95.60 83.65 92.43 84.47 89.27
Amplitu
de
0 Nominal 84.66 94.05 84.45 78.19 84.37 70.97 17.17 67.72 16.96 56.30 16.90 51.10 79.99 81.43 66.66 68.17 60.56 62.31
Robust 90.09 100.03 89.99 98.50 89.96 97.27 18.27 72.02 18.07 70.92 18.01 70.03 79.94 81.79 78.80 80.69 77.85 80.17
Margin 100.00 100.08 100.00 100.08 100.00 100.11 20.28 72.06 20.08 72.06 20.03 72.08 72.06 72.05 72.06 72.06 72.08 72.08
0.1 Nominal 85.12 98.02 85.55 96.54 85.66 96.29 17.26 70.58 17.18 69.51 17.15 69.33 82.92 85.92 81.24 86.82 80.94 87.96
Robust 87.16 99.89 87.31 99.03 87.27 98.99 17.68 71.92 17.53 71.30 17.48 71.27 82.52 84.64 81.66 86.10 81.67 87.95
Margin 89.92 100.06 89.73 99.86 89.46 99.95 18.24 72.04 18.02 71.90 17.92 71.97 80.12 81.40 80.12 82.76 80.44 84.35
0.5 Nominal 85.40 99.51 85.52 99.23 85.34 99.16 17.32 71.65 17.18 71.44 17.09 71.40 83.90 87.71 83.54 90.19 83.67 91.81
Robust 85.99 99.92 86.05 99.74 85.83 99.60 17.44 71.94 17.28 71.81 17.19 71.71 83.66 87.22 83.45 90.19 83.55 91.84
Margin 86.58 99.97 86.67 99.86 86.50 99.80 17.56 71.98 17.41 71.90 17.32 71.86 83.13 86.56 82.96 89.22 83.08 90.89
0.9 Nominal 85.47 99.72 85.53 99.57 85.30 99.52 17.34 71.80 17.18 71.69 17.08 71.66 84.00 87.98 83.82 90.70 84.00 92.16
Robust 85.76 99.90 85.86 99.78 85.63 99.72 17.39 71.93 17.24 71.84 17.15 71.80 83.87 87.68 83.67 90.35 83.85 91.80
Margin 86.14 99.90 86.25 99.81 85.94 99.75 17.47 71.93 17.32 71.87 17.21 71.82 83.50 87.06 83.32 89.79 83.57 91.38
Prescient 85.54 100.00 85.56 100.00 85.33 100.00 17.35 72.00 17.18 72.00 17.09 72.00 84.17 88.44 84.15 91.30 84.38 92.66
Breath
ingPhase
0 Nominal 86.72 88.89 86.34 79.66 86.52 76.60 17.54 64.00 17.52 57.35 17.58 55.15 73.80 82.38 66.43 74.20 63.74 70.76
Robust 89.89 99.88 89.43 97.75 89.95 96.48 18.18 71.91 18.14 70.38 18.28 69.47 80.00 80.93 78.70 81.22 77.23 80.55
Margin 100.00 100.09 100.00 100.08 100.00 100.07 20.22 72.07 20.29 72.06 20.32 72.05 72.07 72.07 72.06 72.06 72.05 72.05
0.1 Nominal 86.81 94.79 86.37 90.89 86.26 89.18 17.56 68.25 17.52 65.44 17.53 64.21 78.61 85.81 75.77 82.70 74.44 80.96
Robust 87.29 98.41 87.28 94.78 87.29 93.77 17.65 70.85 17.71 68.24 17.74 67.51 81.18 86.38 78.18 83.61 77.34 82.95
Margin 89.91 99.93 89.80 98.99 89.70 98.23 18.18 71.95 18.22 71.27 18.23 70.73 80.02 80.93 79.37 80.12 78.85 79.83
0.5 Nominal 86.38 98.68 85.44 97.93 85.32 97.70 17.47 71.05 17.33 70.51 17.34 70.34 82.26 87.91 82.52 88.60 82.45 88.62
Robust 86.66 99.33 85.71 98.71 85.69 98.53 17.53 71.52 17.39 71.07 17.41 70.94 82.53 87.53 82.92 88.57 82.79 88.34
Margin 87.20 99.68 86.38 99.15 86.28 98.97 17.63 71.77 17.52 71.39 17.53 71.26 82.31 86.54 82.64 87.41 82.59 87.30
0.9 Nominal 86.31 99.26 85.33 98.85 85.22 98.72 17.46 71.47 17.31 71.17 17.32 71.08 82.80 88.15 83.41 89.39 83.40 89.49
Robust 86.46 99.56 85.42 99.18 85.34 99.08 17.48 71.68 17.33 71.41 17.34 71.34 82.91 88.01 83.59 89.43 83.60 89.54
Margin 86.81 99.64 85.87 99.26 85.80 99.15 17.56 71.74 17.42 71.47 17.43 71.39 82.64 87.58 83.23 88.68 83.20 88.72
Prescient 86.22 100.00 85.22 100.00 85.10 100.00 17.44 72.00 17.29 72.00 17.29 72.00 83.51 88.59 84.49 90.28 84.60 90.36
Table 5: Baseline, amplitude, and breathing phase drift: minimum tumor dose versus left lung dose in various metrics, all values
rounded to hundredths place.
REFERENCES 17
Percentage (%) Dose (Gy) Scaled min tumor dose (Gy)
α Lung Tumour Lung Tumour MLLD LLV20
Combined
α = 0 Nominal 86.25 85.45 17.15 61.53 71.33 80.23
Robust 89.73 98.48 17.84 70.90 79.02 82.42
Margin 100.00 100.16 19.88 72.12 72.12 72.12
α = 0.1 Nominal 86.09 92.11 17.12 66.32 77.04 86.25
Robust 87.06 96.71 17.31 69.63 79.98 87.96
Margin 89.71 99.63 17.84 71.74 79.97 82.43
α = 0.5 Nominal 85.97 97.78 17.09 70.40 81.89 89.63
Robust 86.14 98.41 17.13 70.85 82.25 89.22
Margin 86.82 98.83 17.26 71.16 81.97 88.34
α = 0.9 Nominal 85.89 98.71 17.08 71.07 82.75 90.20
Robust 85.99 99.05 17.10 71.32 82.93 90.24
Margin 86.36 99.12 17.17 71.37 82.64 89.44
Prescient 85.77 100.00 17.05 72.00 83.94 91.18
Table 6: Combined drift: minimum tumor dose versus left lung dose in various metrics,
all values rounded to hundredths place.
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