A spring-dashpot model of respiratory lung tumour motion: System
development and application to clinical data.
A.E. Cavan1†, P.L. Wilson2, R. I. Berbeco3, J. Meyer1
1
Department of Physics and Astronomy, Private Bag 4800, University of Canterbury,
Christchurch 8140, New Zealand
2
Department of Mathematics and Statistics, Private Bag 4800, University of Canterbury,
Christchurch 8140, New Zealand
3
Department of Radiation Oncology, Brigham and Women's Hospital, Dana-Farber Cancer
Institute and Harvard Medical School, Boston, Massachusetts 02115, USA
†
Corresponding author: Department of Physics & Astronomy, University of Canterbury,
Private Bag 4800, Christchurch 8140, New Zealand, Phone: +64 3 364 2987-7588, Fax: +64
3 364 2469, Email: aec49@student.canterbury.ac.nz.
1
Abstract
The correlation of abdominal motion to lung tumour motion during respiration is modelled by
a one-dimensional spring-dashpot system. Three spring-dashpot models, Maxwell, Voigt and
Standard Linear Solid, are investigated by application to a simulated breathing pattern. The
Voigt model was found to be the most suitable, when applied to clinical data. The Voigt
model achieves an average root mean square error of 0.95 mm between the model and actual
tumour motion in the superior-inferior direction, and has reasonably consistent, patient
specific, parameters over extended treatment periods. Additionally, the model was
determined to be capable of dealing with baseline and frequency drifts, or amplitude
variations of the input data, as well as phase shifts between the input abdominal and output
tumour signals. The Voigt model has potential for clinical application during radiotherapy
treatment of the lung, and steps towards its implementation are described.
Keywords: spring-dashpot model, radiotherapy, lung tumour modelling, respiration
2
Introduction
Lung cancer was the most commonly diagnosed cancer worldwide in 2002, with an incidence
rate of over 1.3 million new cases, or 12 % of the total new cancer diagnoses1. It was also the
most common cause of death from cancer, accounting for 18% of all cancer deaths (1.2
million people), with very low five year survival rates of between 7 and 14%. Radiation
therapy is a common modality for lung cancer treatment, however treatment efficacy is
limited by the motion of the lungs during respiration2, primarily driven by diaphragm motion,
and to a lesser extent by chest motion. The magnitude of the motion depends on patient
breathing and tumour characteristics, and is usually the most significant along the superiorinferior (SI) axis and less important in the anterior-posterior (AP) and lateral directions2. A
study by Onimaru et. al. found SI motion affecting 39.2 % of tumours, with average motion
of 0.50 cm , but up to 3 cm, AP motion affecting 5.4 % of tumours, averaging 0.21 cm and
lateral motion for just 1.8 % of tumours, averaging only 0.12 cm3.
Typically, to ensure sufficient tumour coverage despite target motion the clinical target
volume is enlarged to the internal target volume (ITV)4, but this can result in excessive
irradiation of surrounding healthy tissue, or marginal miss of the tumour5. The high doses
required for tumour control are close to or above the tolerance level for healthy tissues,
resulting in increased side effects, or requiring a reduction in dose to the tumour, which
decreases the tumour control probability (TCP) 6,7. Accounting for the presence of respiratory
motion of lung tumours can improve the therapeutic ratio and thus survival rates8.
Respiratory motion has superficial regularity, but motion tracking is complicated by
variations in baseline, frequency and oscillatory amplitude and shape, which differ
significantly between patients and can alter over the course of treatment6. Methods include
3
abdominal compression to reduce tumour mobility, breath control techniques (active or
passive) and respiratory gating of irradiation to selected breathing phases usually using
external surrogate markers2,6,9-12. All of these techniques have some limitations, such as the
need for active patient cooperation and consistent ability to maintain total lung capacity13, or
a lengthened treatment time for gated therapy due to the required beam-off time.
Modelling of the respiratory-induced lung tumour motion can facilitate dynamic tracking and
compensation for both real-time and gated treatments6,7,12,14. Direct tumour tracking systems
may use portal imaging15 or implanted fiducial markers in the tumour, in conjunction with a
diagnostic x-ray imaging system13, but this continuous imaging can impart a considerable
radiation dose which is not always clinically justifiable16.
Indirect tumour tracking systems use external surrogates to obtain tracking signals, such as a
spirometer, strain gauge or abdominal markers, which then require a model to relate the
surrogate and the tumour motions13. This approach requires the relationship between external
surrogate and internal tumour motion to be consistent and well correlated 17, but overcomes
problems such as potential deformation of the relationship of tumour and implanted marker
during radiotherapy14, risky fiducial implantation, extraneous radiation doses or unclearly
defined tumours during simultaneous imaging using x-ray fluoroscopy or other methods.
Most models have been so-called ‘black box’ (no internal behaviour known) or ‘grey box’
(partially known) approaches, which consider the input and output data, but not the physical
relationship. Various approaches include linear correspondence models8,18, composites of
baseline drift, frequency variation, fundamental pattern change and random observation
noise6, characterisation of the motion with a piecewise linear model of defined stages of the
breathing cycle19, least-squares parameter estimation and Systems Identification2 or adaptive
neural networks (can predict motion ahead of time)20. These models generally require a large
4
amount of sample data which encompasses the range of possible relationships between the
input and output states. They tend to deal poorly with irregularities in the breathing pattern,
such as baseline drift or a hysteresis20, a lag or phase shift between internal and external
motions (abdominal motion is most correlated with SI tumour motion)17,21,22 and usually have
a strong dependence on tumour and marker locations, motion dimensions and type of
breathing pattern2,14,18.
A more physical approach models lung motion as a contact problem of elasticity theory by
describing the physiology of breathing using elastic constants to directly model the lung
tissue23. This idea of lungs tissue being modelled elastically was explored by Wilson and
Meyer24 who presented a physical 3D system of springs and dashpots to model the correlation
of an abdominal respiratory signal and the lung tumour motion (rather than directly
modelling the actual lung tissue). Wilson and Meyer showed that it is possible to formally
simplify the three dimensional motions of tumour and abdominal marker to one dimension
each, in the basis of the (usually) small size of the clinically observed motion in the two
minor directions of motion.
This paper presents work done towards improving the Wilson and Meyer model. The
relationship of lung tumour and external abdominal marker movement for the simplified 1D1D system is investigated, considering only the primary dimension of motion for the tumour
(SI axis) and abdominal signal (AP axis).
Material and Methods
Mathematical models
5
The models used are based on the use of springs and dashpots in combination to model
viscoelastic stress and strain.
Three standard spring-dashpot models are applied to the
tumour motion problem25, the Voigt model (consisting of a spring and a dashpot in parallel),
the Maxwell model (a spring and a dashpot arranged in series) and the Standard Linear Solid
model (a spring in parallel with a Maxwell system) as shown in Figure 1. The Maxwell
model is based on that described by Wilson and Meyer24, however they erroneously
envisaged the Maxwell model, whilst basing their system on the Voigt equations. This does
not invalidate their findings.
Figure 2 shows the physical set up of the models. For the mathematical models, the input to
each system, F, is due to the abdominal signal in the y-direction, converted and scaled to the
z-direction. Wilson and Meyer’s approach used a variable length rod to provide a physical
basis for this conversion, however for this study the rod was abandoned in favour of a direct
scaling factor. This reduces the number of parameters from two (position and length) to one,
allowing the model to be more readily optimised, as well as improving the capabilities of the
model by provision of a larger parameter space that is not constrained by the need to remain
physically realisable for the patient’s dimensions, and allowing negative values that can help
to deal with phase shifts in the data. This does however prevent the model from being a
purely physical description of the system, and moves it more firmly into ‘grey box’ territory.
The force from the input abdominal motion causes a displacement, z, and resulting
oscillation of the tumour, attached to the other end of the spring-dashpot system. The springs,
s1/2, provide an oscillation, described in the equations by parameter ω, which was derived
from Hooke’s Law, with spring constant k, and extensions zS1/S2. The dashpot d, provides a
retarding force, parameter λ, derived from damping coefficient, c, and displacement zd. The
6
results are general models, but with parameter sets (ω, λ, and scaling factor) specific to each
patient and tumour site.
The mathematical descriptions of the models are as follows:
Maxwell Model
𝐹 = 𝐹𝑆1 = 𝐹𝑑
𝑧 = 𝑧𝑆1 + 𝑧𝑑
𝐹𝑆1 = 𝑘1 𝑧𝑆1
dz𝑑
𝐹𝑑 = 𝑐
dt
dz
1 dF𝑆1 𝐹𝑑
1 dF 𝐹
=
+
=
+
dt 𝑘1 dt
𝑐
𝑘1 dt 𝑐
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔:
2
𝑑 𝑧
𝑘
𝑐
𝐹 = 𝑚 2 , 𝜔2 =
𝑎𝑛𝑑 2𝜆 =
𝑚
𝑚
dt
𝑑3𝑧
𝜔2 𝑑 2 𝑧
dz
𝑔𝑖𝑣𝑒𝑠: 3 = −
+ 𝜔2
2
2λ dt
dt
dt
Voigt Model
𝐹 = 𝐹𝑆2 + 𝐹𝑑
𝑧 = 𝑧𝑆2 = 𝑧𝑑
𝐹𝑆2 = 𝑘2 𝑧𝑆2
dz𝑑
𝐹𝑑 = 𝑐
dt
dz
dz
𝐹 = 𝑘2 𝑧𝑆2 + 𝑐 = 𝑘2 𝑧 + 𝑐
dt
dt
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑎𝑠 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙 𝑚𝑜𝑑𝑒𝑙
𝑑2𝑧
dz
𝑔𝑖𝑣𝑒𝑠: 2 = −2𝜆 − 𝜔2 𝑧
dt
dt
Standard Linear Solid (SLS) Model
𝐹 = 𝐹Maxwell + 𝐹𝑆2 = 𝐹𝑆1 + 𝐹𝑆2 = 𝐹𝑑 + 𝐹𝑆2
𝑧 = 𝑧𝑆2 = 𝑧Maxwell = 𝑧𝑆1 + 𝑧𝑑
𝐹𝑆1 = 𝑘1 𝑧𝑆1
dz𝑑
𝐹𝑑 = 𝑐
dt
dz
1 dF𝑆1 𝐹𝑑
1 dF 𝑘2 dz 𝐹 − 𝑘2 𝑧
=
+
=
−
+
dt 𝑘1 dt
𝑐
𝑘1 dt 𝑘1 dt
𝑐
𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑎𝑠 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑀𝑎𝑥𝑤𝑒𝑙𝑙 𝑚𝑜𝑑𝑒𝑙
𝑑 3 𝑧 𝜔12 𝑑 2 𝑧
dz 𝜔12 𝜔22
2
2
𝑔𝑖𝑣𝑒𝑠: 3 =
+
(𝜔
+
𝜔
)
+
𝑧
1
2
2λ dt2
dt
2λ
dt
7
The one dimensional equations of motion for the Voigt and SLS spring-dashpot systems were
reduced to first-order differential equations before being solved computationally. The
Maxwell model required prior substitution of s = dz/dt to be solved, due to the lack of a z
term. The resulting solution was then integrated to solve for z. For all three models the
respective parameters were then optimised for simulated data sets to obtain the final solution.
The results were compared to determine which of the three models was the best. For this
most suitable model, the influence of each parameter was investigated, and then the model
applied to clinical data to evaluate the performance, in particular concerning the impact of
treatment periods of several days and irregularities in the breathing signals.
Clinical data
The clinical data used was as used by Ionascu et.al.17 and consists of 3D internal fiducial
based motion obtained using radiopaque fiducial markers implanted and visualised in realtime using stereoscopic diagnostic x-ray fluoroscopy, collected using a Mitsubishi Real-Time
Radiation Therapy system, at the Nippon Telegraph and Telephone Corporation Hospital,
Sapporo, Japan. This was obtained simultaneously with 1D external abdominal motion,
collected on an independent co-ordinate system using a laser based AZ-733V "RespGate"
made by Anzai Medical, Tokyo, Japan. The data consists of motion results from ten patients
with lung cancer at various sites, for up to ten consecutive treatment fractions and four beam
configurations per fraction(1). The patients are not a random sample of the general lung cancer
population, but were selected on the basis of an estimated internal marker motion of more
than 10 mm peak to peak (SI, lateral or AP direction). The data sets have phase shifts of the
(1)
Patient 5 had eight fractions, then a further four two months later. The analysis was kept consistent.
8
internal-external data (one signal lagging the other)
21,22
in the SI direction that are mostly
between 100 and 200 ms.
Modelling
The equations of motion for each model were solved using the MATLAB differential
equation solver ODE45, which is recommended for use for non-stiff problems26. The
resulting model was optimised using the fminsearch function in combination with a cost
function, to find the minimum of the root mean squared error (RMS) (1) between the
measured tumour position data, zTumour, and the model output values, zModel (at each time
instance, n). The algorithm starts with a set of initial parameter estimates (ω, λ and a scaling
factor), and uses a process of unconstrained non-linear optimisation (with a simplex search
method) to find a set of parameters fitting the model to the data. Each optimisation was run
until the cost function changed by less than 0.01 mm over ten iterations.

n
RMS 
i 1
( zTumour  z Model ) 2
n
(1)
Initially for model development and parameter investigation a sine wave (plus noise of 5% of
total amplitude) was used to simulate the abdominal and tumour motion signals. This allowed
faster optimisation and a clear understanding of parameters, studied by setting all three
parameters to one, and then varying each parameter in turn and in combination. The
simulated data was also used to study the ability of the model to handle a phase shift between
the abdominal and tumour signals. Each successive phase shift, between +180° and -180°,
was modelled using initial parameters obtained from the optimisation of the model for the
previous phase shift.
9
The three models were tested by application to the basic and phase shifted sinusoidal data and
compared based on the results of this analysis. The results are presented in the next section.
The model determined to be the most suitable was then applied to a total of 60 clinical data
sets, consisting of all patients (1-10) for beam 1 on every day of their treatment, as well as all
the beam configurations for patients 1-3. Input and output data were corrected to oscillate
independently about their mean positions, removing dependence on the measurement coordinate systems. The actual tumour oscillation data was known, and used for the
optimisation of the system parameters. In the case of data sets that had periods of poor
tracking or missing data points, these points were not considered for the optimisation of the
parameters, but the good tracks were connected with a straight line for use in the modelling.
In particular the consistency of the model over each patient’s treatment period was analysed
by optimising the parameters for the Day 1, Beam 1 data, then applying these (without reoptimisation) to subsequent days, to determine the increase in RMS and investigate how well
the model transferred to previously unseen data. The other focus was on the ability of the
Voigt model to adapt to irregularities in the breathing signal. The model was analysed for ten
data sets, selected because they displayed a range of irregularities, such as missing data
points, baseline drift, amplitude and frequency variation and an inconsistent internal-external
signal relationship2,6,19, to determine the magnitude of change in RMS error due to such
irregularities.
Results and Discussion
Model Selection
10
The Voigt model proved to be the most successful model. The predominant problem with the
Maxwell and SLS models was that they produced outputs phase shifted from the abdominal
motion data by about 30°, regardless of the relative phase between the tumour and the
breathing data. Thus when the phase shift analysis was done for these models it was found
that for a phase shift of 30° (tumour leading abdominal signal) the models provided a good
fit, as the inherent phase shift of the model matched that of the data. For phase shifts of
greater than approximately 40° or less than 20°, when optimised, the model output tended
towards zero at all points. Thus for the clinical data, which had no significant internalexternal phase shift, the solution to the equations was always out of phase with the tumour
motion. Shifting the breathing data by approximately 30° relative to the tumour data, before
the models were solved and optimised, allowed for results of comparable (but slightly higher)
accuracy to the Voigt model, but involved the introduction of additional uncertainty in
determining the appropriate magnitude for the initial phase shift.
Prior to the investigation it was expected that SLS model would be the best, because the
higher number of parameters give it more degrees of freedom to allow a better fit to the
tumour motion, however it is postulated that the spring in series with the dashpot (the
common feature of the two weaker models) may cause a delay in the response of the tumour
to the input force, not present for the system where they are coupled in parallel. There is a
concern that the dependence of the optimisation on the initial estimated parameters (discussed
later) may mean that a suitable solution was not found due to the parameters estimated
(although a range of possible combinations was used to attempt to rule out this problem).
Phase shift and parameter influence on the Voigt model.
11
The parameters had mostly predictable effects on the Voigt model of the simulated data
(when their origin in the physical model was considered), as shown in Figure 4, although
there were some exceptions. Predictably, as ω increased in magnitude, the model oscillated
less. Negative ω values were not rejected by the optimisation scheme (and gave results
equivalent to the positive value) because ω is always present as ω2 in the equations, however
this reaffirms that the optimised model is not a completely physical representation of the
system. For consistency, results were constrained to the positive value only. Additionally,
when the absolute value of ω was less than one the model was translated upwards by
approximately the amplitude distance, then decreased over time, indicating the ability to
handle a baseline shift in the data. This shift is postulated to be due to the relative values of
the spring constant and the damping coefficient. ω appeared to have no effect on the
frequency of oscillation of the system, this seemed to be controlled by the λ parameter, with
values close to zero increasing the period and decreasing the regularity. Negative λ caused an
exponential curve with initial oscillations, whilst λ greater than one began to decrease the
amplitude, with limited effects on the frequency. The scale factor controlled the amplitude of
the oscillations.
Phase shifts between -180º and +180º were investigated and it was found that all values of
phase shift investigated allowed a minimum RMS error of less than 1.0 mm, as seen in Figure
5. ω and λ appeared to work in conjunction to correct the shape and position of the
oscillations, and the scale factor opposed any resulting change in amplitude. The
investigation illustrated the need to select suitable initial values for the parameters, as the
points of zero and 180° phase shift (most relevant to the clinical data used) when approached
from two different directions gave considerably different parameter values, although the
RMS value remained between 0.08 and 1.0 mm. The variation between of the parameters is
12
explained by a negative scaling factor adapting for a positive phase shift. This is a limitation
of fminsearch, as the optimisation can become trapped in a local minima, resulting in
different final parameters depending on the initial estimate selection.
Application of the Voigt model to clinical data
Applying the Voigt model to the clinical data to model the lung tumour motion gave
promising results. A representative example of model output is shown in Figure 3. The best
result from the clinical data modelled had an RMS error of 0.38 mm (2 s.f.) (Patient 3, Day 1,
Beam 1), whilst the average RMS error of all 60 data sets modelled was 0.95 mm.
Nearly all of the models showed residuals that were small in amplitude but with an
underlying oscillation which may be due to a failure to fully reach the peaks of the motion.
This may be more significant clinically for those patients whose normal breathing patterns
indicate relatively longer times at the extremes. The residuals in some of the data sets may be
due to an incomplete compensation for any phase shift.
When the Voigt model was applied to clinical data, the magnitude of the parameters
remained reasonably consistent across all 60 data sets, with ω having a mean value of 14.5,
and a standard deviation of 2.9, λ a mean value of 6.3, with a standard deviation of 3.8 and
the scaling factor a mean value of 515, standard deviation 340. Generally, when close to the
optimal result, continuing iterations resulted in very small reductions in λ and/or ω, that were
compensated by increases in the order of the scaling factor, with results of up to 106 attained.
However these drastic increases in the scaling factor were associated with only minimal
improvement in the RMS error (less than 0.01 mm). Whilst the parameters had values
consistently in a specific range, small variations in the values often had a large impact on the
RMS error.
13
Patient specific consistency of parameters
The parameter results obtained from optimising the Day 1, Beam 1 data for each patient were
applied to the data from subsequent beams or days, representative results of this (chosen for
the completeness of the patient’s data sets) are shown in Table 1. The average increase in
RMS error between Beam 1 and subsequent beams (for a given patient and day) was 0.55
mm. For subsequent days, the average increase was 0.54 mm compared to Day 1. When the
results using Day 1, Beam 1 parameters were optimised, there was an average improvement
in the RMS error of 0.43 mm. The maximum increase in the error was seen for Patient 5, Day
6, where the RMS increased by 2.5 mm from the Day 1 error (using Day 1 parameters). This
was not due to gradual erosion of parameter consistency with time, as results on later days
returned to lower RMS values, instead may be due to periods of poor tracking in the data set,
or a fundamentally different breathing pattern on Day 6, perhaps caused by distress.
These results indicate that it may be feasible to treat patients by collecting internal tumour
motion data and calculating patient specific parameters during the first treatment, then
retaining these for the rest of the treatment period. This approach does allow the possibility of
a large fluctuation in error on a certain day that may not be detected.
Results also appeared to be reasonably patient specific, with the increase in RMS error for the
application of one patient’s parameters to another being up to four times higher than applying
them to the same patient’s results on subsequent days, suggesting parameters that are patient
anatomy and tumour specific rather than dependent on the breathing pattern or some other
factor. However, some combinations of patients did give equivalent accuracy in the results,
this is postulated to be due to those patients having similar tumour sites. The tumour site
14
information for each patient was unfortunately not available, but further work could analyse
whether there is a correlation between tumour location or other patient specific
characteristics, and parameter similarities.
Irregular data sets
The Voigt model was also fitted to ten of the most visibly irregular data sets. The result was
an average RMS error of 1.33 mm, which is reasonable compared to the average value for all
the data sets modelled. An additional consideration is that some irregular data sets were
missing points, which can increase the error by disrupting the ability of the model to fit to the
subsequent point.
Figures 6 and 7 show examples of the Voigt model when applied to data sets with
irregularities. Patient 6, Day 5, Beam 1 presents negative baseline shift of the breathing signal
and some variation in amplitude. Baseline drift is arguably the most important consideration
for modelling lung tumour motion to ensure that errors are not avoidably high. The Voigt
model appears able to deal with baseline shifts of the data with little to no effect on the RMS
errors (compared to data sets with no baseline drift), for this example, the optimised result
had an RMS error of 1.12 mm. Patient 2, Day 1, Beam 1 shows decreasing frequency of
oscillation, with an RMS error for the optimised solution of 1.50 mm.
Further work
In order for clinical implementation to become feasible, several extensions to the current
Voigt modelling process are being implemented. The system needs to be predictive, as there
is a finite time required for the treatment machinery (couch7 or multi-leaf collimator27) to
adjust to the determined change in position. The 1D-1D model presented here needs to be
15
extended to a full three dimensional system which also considers the minor axes of tumour
motion (AP and lateral), for the sake of completeness for the small proportion of patients
where this motion is non-negligible. The optimisation process needs to incorporate a
stochastic search technique such as simulated annealing to remove the dependence of the
result upon the initial parameter estimates, but at the expense of higher computational costs.
Conclusion
The correlation of abdominal motion to lung tumour motion modelled by the 1D springdashpot systems showed that the Voigt model was superior to the Maxwell and Standard
Linear Solid models to model respiration-induced lung tumour motion.
60 clinical data sets were modelled, resulting in an average RMS error between model and
tumour motion (SI direction) of 0.95 mm. Parameters showed reasonable consistency over
treatment periods of up to 12 days, with the use of parameters optimised on the first day
giving increases of 0.55 mm and 0.54 mm to the RMS when applied to subsequent beams or
days respectively. The model also displayed an ability to cope with phase shifts between the
internal and external signals on a set of simulated data.
The Voigt model has shown some promise in overcoming shortcoming of previous models,
including in its ability to deal with baseline and frequency drift, and in its consistency in
modelling the relationship between internal and external signals over time. The model has
potential for clinical application once it has been extended to a three dimensional system,
optimised with a stochastic search technique and has the ability to give predictive output.
16
Acknowledgements
We would like to thank Dr. Seiko Nishioka and Dr. Hiroki Shirato, for the clinical data
provided from the Nippon Telegraph and Telephone Corporation Hospital in Sapporo, Japan.
Thank you to the Canterbury Medical Research Foundation for financial support.
17
Table (from section Results and Discussion: Patient specific consistency of parameters)
Table 1: Voigt model parameters for treatment over several beams (Patient 1) and several
days (Patient 5).
Data set
(Patient, Day, Beam)
Parameters
(ω, λ, SF)
RMS error (mm)
(Using Day 1, Beam
1 Parameters)
Best RMS obtained
from further
iterations
(1,1,1)
(1,1,2)
(1,1,3)
(1,1,4)
(5,1,1)
(5,2,1)
(5,3,1)
(5,4,1)
(5,5,1)
(5,6,1)
(5,7,1)
(5,8,1)
(5,9,1)
(5,10,1)
(5,11,1)
(5,12,1)
(13.92, 3.82, 537.40)
(12.37, 9.00, 219.33)
-
1.46
1.96
1.17
1.86
0.64
1.04
0.91
1.02
0.92
3.14
1.42
0.81
1.69
1.04
0.70
0.99
1.56
1.15
1.84
0.85
0.91
0.75
0.69
1.03
0.68
0.76
0.55
0.75
0.48
0.66
18
Figure captions:
Figure 1: (a) Maxwell model (b) Voigt model (c) Standard Linear Solid model. S
denotes a spring and d denotes a dashpot.
Figure 2: The set-up of the spring dashpot system: the relationship of the tumour
motion and the abdominal signal. Note the direction of the axes.
Figure 3: A representative example of the Voigt model results for Patient 6, day 1,
beam 1 and the corresponding residuals. Optimised parameters are ω= 12.5, λ= 1.62
and scaling factor = 340, and RMS error is 0.78 mm.
Figure 4: The effects of the change in parameters on the Voigt model. The parameter
values (ω, λ, scaling factor) are given in the key, and each plot is compared to the set
(1,1,1).
Figure 5: The effects of phase shifted data on the parameters and RMS error, for the
Voigt model.
Figure 6: An example of baseline drift and varying amplitude (Patient 6, Day 5, Beam
1). Parameters are ω=17.5, λ=4.00, scaling factor = 1190, and RMS error is 1.12 mm.
Figure 7: An example of an increasing oscillation period (Patient 2, Day 1, Beam 1).
Parameters are ω=12.7, λ=5.95 and scaling factor = 497, and RMS error is 1.50 mm.
Note the period of poor tracking of abdominal motion, data replaced by a straight line
(at a time of 58 s).
19
Figure 1 (from section Material and Methods: Mathematical Models)
20
Figure 2 (from section Material and Methods: Mathematical Models)
21
Figure 3 (from section Results and Discussion: General Results for Voigt Model)
22
Figure 4 (from section Results and Discussion: Phase shift and parameter influence)
23
Figure 5 (from section Results and Discussion: Phase shift and parameter influence)
24
Figure 6 (from section Results and Discussion: Irregular data sets)
25
Figure 7 (from section Results and Discussion: Irregular data sets)
26
References
1. Bray, F. and Parkin, M. Parkin, World cancer, CancerStats, 2005.
2. Meyer, J., Baier, K., Willbert, J., Guckenberger, M., et al. Three-dimensional spatial
modelling of the correlation between abdominal motion and lung tumour motion with
breathing. Acta Oncol., 45:923-934, 2006.
3. Liu, H., Balter, P., Tutt, T., Choi, B., et al., Assessing respiration-induced tumour motion
and internal target volume using four-dimensional Computed Tomography for radiotherapy
of lung cancer, Int. J. Rad. Onc. Bio. Phys., 68(2):531-540, 2007.
4. Bethesda, M.D., et al. International Commission on Radiation Units and
Measurements:Prescribing, recording, and reporting photon beam therapy, ICRU, 62, 1999.
Supplement to ICRU Report No. 50.
5. Allen, A.M., Siracuse, K.M., Hayman, J.A. and Balter, J.M., Evaluation of the influence of
breathing on the movement and modelling of lung tumours, Int. J. Rad. Onc. Biol. Phys.,
58(4):1251-1257, 2004.
6. Ruan, D., Fessler, J.A., Balter, J.M., and Keall, P.J., Real-time profiling of respiratory
motion: baseline drift, frequency variation and fundamental pattern change, Phys. Med.
Biol., 54:4777-4792, 2009.
7. Wilbert, J., Meyer, J., Baier, K., Guckenberger, M., et al., Tumour tracking and motion
compensation with an adaptive tumour tracking system (ATTS): System description and
prototype testing. Med. Phys., 35(9):3911-3921, 2008.
8. Nioutsikou, E., Seppenwoolde, Y., Richard, J., Heijmen, B., et al., Dosimetric
investigation of lung tumor motion compensation with a robotic respiratory tracking system:
An experimental study, Med. Phys., 35(4):1232-1240, 2008.
9. Rosenzweig, K.E., Hanley, J., Mah, D., Mageras, G., et al., The deep inspiration breathhold technique in the treatment of inoperable non-small-cell lung cancer, Int. J. Rad. Onc.
Biol. Phys., 48(1):81-87, 2000.
10. Partridge, M., Tree, A., Brock, J., McNair, H., et al., Improvement in tumour control
probability with active breathing control and dose escalation: A modelling study, Radiother.
Oncol. 91(3):325-329, 2009.
11. Shimizu, S., Shirato, H., Ogura, S., Kitamura, K., et al., Detection of lung tumour
movement in real-time tumor-tracking radiotherapy, Int. J. Rad. Onc. Biol. Phys., 51(2):304310, 2001.
12. Land, I., Mills, J.A., Young, K., Haas, O., et al., Modelling the effects of lung tumour
motion on planning and dose delivery with gated radiotherapy. Clinical Oncology, 19(3):3650, 2007.
13. Yana, H., Yin, F.F., Zhu, G.P., Ajlouni, M., et al., The correlation evaluation of a tumor
tracking system using multiple external markers, Med. Phys., 33(11):4073-4084, 2006.
14. Onimaru, R., Shirato,H., Fujino, M., Suzuki, K., et al., The effect of tumour location and
respiratory function on tumour movement estimated by real-time tracking radiotherapy
(RTRT) system, Int. J. Rad. Onc. Biol. Phys., 63(1):164-169, 2005.
15. Meyer, J., Richter, A., Baier, K., Wilbert, J., et al., Tracking moving objects with megavoltage portal imaging: A feasibility study, Med. Phys., 33(5):1275-1280, 2006.
16. Ren, Q., Nishioka, S., Shirato, H., and Berbeco, R.I., Adaptive prediction of respiratory
motion for motion compensation radiotherapy, Phys. Med. Biol., 52:6651-6661, 2007.
17. Ionascu, D., Jiang, S.B., Nishioka, S., Shirato, H., et al., Internal-external correlation
investigations of respiratory induced motion of lung tumors, Med. Phys., 34(10):3893- 3903,
2007.
27
18. Kanoulas, E., Aslam, J.A., Sharp, G.C., Berbeco, R.I., et al., Derivation of the tumor
position from external respiratory surrogates with periodical updating of the
internal/external correlation, Phys. Med. Biol., 52(17):5443-5456, 2007.
19. Wu, H., Sandison, G., Zhao, L., Zhao, Q., et al., Correlation between parameters
describing tumour motion and its location in the lungs, Australas. Phys. Eng. Sci. Med.,
30(4):341-344, 2007.
20. Isaksson, M., Jalden, J., and Murphy, M.J., On using an adaptive neural network to
predict lung tumor motion during respiration for radiotherapy applications, Med. Phys.,
32(12):3801-3809, 2005.
21. Hoisak, J.D.P., Sixel, K.E., Tirona, R., Cheung, P., et al., Correlation of lung tumor
motion with external surrogate indicators of respiration, Int. J. Rad. Onc. Biol. Phys.,
60(4):1298-1306, 2004.
22. Chi, P.C., Balter, P., Luo, D., Mohan, R., et al., Relation of external surface to internal
tumor motion studied with cine CT, Med. Phys., 33(9):3116-3123, 2006.
23. Werner, R., Ehrhardt, J., and Handels, H., Modeling of respiratory lung motion as a
contact problem of elasticity theory, In Proceedings of the COMSOL Conference, Hannover,
2008.
24. Wilson, P.L., and Meyer, J., A spring-dashpot system for modelling lung tumour motion
in radiotherapy, Comput. Math. Method. Med., 10(4), 2009.
25. de Haan, Y.M., and Sluimer, G.M., Standard linear solid model for dynamic and time
dependent behaviour of building materials, HERON, 46(1):49-76, 2001.
26. Lagarias, J.C., Reeds, J.A., Wright, M.H., and Wright, P.E., Convergence properties of
the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization,
9(1):112-147, 1998.
27. Sawant, A., Venkat, R., Srivastava, V., Carlson, D., et al., Management of threedimensional intrafraction motion through real-time DMLC tracking, Med. Phys., 35(5):20502061, 2008.
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