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Adaptive and Robust Radiation Therapy in

the Presence of Drift

Mar PA & Chan TC

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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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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: [email protected]

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.


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,


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.


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.


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.


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


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


(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


(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



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


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


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.


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

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78

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104


Min

imum

tum

or d

ose

(% o

f 72

Gy)



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


Hau

sdor

ff di

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pre

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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


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

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93

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105


Min

imum

tum

or d

ose

(% o

f 72

Gy)



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


Hau

sdor

ff di

stan

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pre

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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


Min

imum

tum

or d

ose

(% o

f 72

Gy)


(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

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14

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18

20


Hau

sdor

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pre

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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.


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.


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.


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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