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A Diffeomorphic Framework for
Surrogate-based Motion Estimation in Radiation Therapy:
Concept and First Evaluation
Ren´
e Werner, Jan Ehrhardt, Alexander Schmidt-Richberg,
Matthias Wilms, Maximilian Blendowski, Heinz Handels
Institute of Medical Informatics
University of L¨
ubeck
Ratzeburger Allee 160
23538 L¨
ubeck
werner@imi.uni-luebeck.de
Abstract: Respiratory motion is a major obstacle in radiation therapy of thoracic and
abdominal tumors. Techniques to cope with it such as gating and tracking techniques
are based on the use of breathing signals that can be acquired easily and in real-time.
These signals represent only surrogates of the motion of the inner organs and tumors.
Consequently, methods are needed to estimate respiratory motion patterns of the inter-
nal structures based on surrogate measurements.
In this contribution, a diffeomorphic framework based on a multi-linear regression
and the Log-Euclidean framework recently introduced in the context of diffeomorphic
registration is proposed to establish such a correspondence model. The feasibility of
the approach is demonstrated by means of a leave-out evaluation using 4D CT image
sequences of ten lung tumor patients and simulating three different types of breathing
signals: spirometry records, tracking motion of points on the diaphragm, and assessing
the raising/lifting of chest wall points.
1 Introduction
Advances in imaging technologies have opened up new possibilities for diagnosis, treat-
ment planning and image-guided therapy, with radiation therapy (RT) being a typical ex-
ample. Modern image acquisition techniques and resulting images allow the RT-physicists/-
physicians to accurately delineate tumors and organs at risk (OAR), to optimize treatment
plans and dose distributions, to compensate for set-up errors etc.. However, in the thorax
and abdomen respiratory motion still remains a limiting factor. Current 4D(=3D+t) imag-
ing techniques like 4D CT or 4D MRI provide insights into the breathing dynamics of
the individual patient, but are grounded on the application of sophisticated reconstruction
techniques and are consequently not real-time capable as it would be required for image-
based guidance purposes during treatment [LCC+08, KMB+06]. Approaches to cope with
respiratory motion during irradiation – such as gated RT or tumor tracking techniques –
are therefore usually steered by (mainly external) breathing signals acting as surrogates of
Informatik 2012, edited by U. Goltz, et al., Lecture Notes in Informatics, GI,
Bonn, Vol. P-208, 1774-1785, 2012.
Copyright 2012 by Gesellschaft für Informatik
internal motion of tumors and OAR [KMB+06].
While it appears to be natural that a correlation exists between external breathing signals
and internal respiratory motion patterns, the determination of an exact relationship is a
challenging problem, especially when considering factors such as inter-cycle motion vari-
ability or phase shifts between movements of different anatomical structures. Thereby and
further taking into account the complex 3D-nature of internal motion patterns, the reliabil-
ity of simple 1D-surrogates like, e. g., measurements using abdominal belts is considered
to be problematic, and a trend toward the use of more-dimensional surrogates can be ob-
served [SPH08]. To efficiently use them in clinical practice, appropriate correspondence
models between the surrogate signals and internal motion patterns have to be developed
and evaluated.
Placed in that context, in this contribution we present a framework for establishing cor-
respondence between motion patterns of internal structures and surrogate data based on a
multi-variate multi-linear regression (MLR); therein, the internal motion patterns are de-
rived from 4D CT images of lung tumor patients by non-linear registration [WESR+10].
In contrast to existing MLR-based models like in [ZHL+10], we embed the modeling ap-
proach within a diffeomorphic setting exploiting the Log-Euclidean framework proposed
by Arsigny et al. [ACPA06], which in recent years has been proven to be a computation-
ally efficient way for performing statistics on diffeomorphisms [EWSRH11]. Further, we
present a first evaluation of the framework considering three different types of breathing
signals: spirometry records, tracking motion of points on the diaphragm, and imitating a
range imaging device (point/line laser) by evaluating raising/lifting of chest wall points.
2 Materials and Methods
The study is grounded on 4D CT data sets Ijj∈{1,...,nph},Ij: Ω ⊂R3→Rof 10 lung
tumor patients. Each image sequence consists of 3D CT images of between 10 and 14
breathing phases jand features a spatial resolution of originally approx. 1×1×1.5 mm3;
due to memory and computation time restrictions, the 3D images Ijwere downsampled to
an isotropic resolution of 1.5×1.5×1.5 mm3.
Now, for establishing the sought patient-specific correspondence models, in a first step the
admissible input data of the model / for model generation has to be defined. In our case,
one kind of input data is represented by the surrogate data, assumed to be described by
ξ: [t0, tend]→Rnsur ;
the measurements corresponding to the acquisition times of the CT images Ijare subse-
quently denoted as ξjand the specific types of surrogate signals considered for evaluation
purposes are later detailed in section 2.3. As we finally intend to derive an estimate of mo-
tion patterns of internal structures from the surrogate data, we also have – as a second kind
of input data as well as the output data format of the model – to determine an appropriate
representation of the internal motion. Dealing with respiratory motion and, consequently,
complex deformations, we have chosen to decode internal motion in a general way by
using dense displacement fields. Therefore, we assume w. l. o. g. the 3D CT volume I1
to be the reference representation of the patient’s anatomy being acquired at the phase of
end-inspiration (EI). Then, internal motion is described by fields
u: [t0, tend]×Ω→R3;
thus, for a voxel x∈Ωand the corresponding anatomical point, respectively, the vector
u(t, x) =: ut(x)represents the displacement of the point with regard to its position in I1.
Similar to the surrogate data the displacement fields representing the motion between I1
and the other breathing phases jand 3D images of the 4D image sequence Ijj∈{1,...,nph}
are subsequently denoted as ujj∈{1,...,nph}with u1(x)=0for all x∈Ω.
In the fashion of a multi-linear regression the ujare assumed to be known and serve – to-
gether with the corresponding surrogate measurements ξjj∈{1,...,nph}– as inputs of the
model training phase: They form the basis of the estimation of the relationship between the
surrogate data and internal motion patterns. Aiming at diffeomorphic motion estimation,
which can be considered as a ”natural choice in the study of anatomy as connected sets
remain connected, disjoint sets remain disjoint, smoothness of anatomical features [...] is
preserved, and coordinates are transformed consistently.” [BMTY05], we apply a diffeo-
morphic registration scheme to derive the fields ujfrom the 4D image sequences. The
underlying theory is detailed in section 2.1. The formation of the MLR correspondence
model itself and its application for estimation of internal motion patterns are explained in
section 2.2.
2.1 Estimation of Internal Motion by Diffeomorphic Registration
Diffeomorphic transformations are globally one-to-one and differentiable mappings with
a differentiable inverse [EWSRH11]. They can be modeled as arising from an evolution
equation over unit time t∈[0,1],
∂
∂t φt(x) = v(φt(x), t)with φ0(x) = x. (1)
Thus, for a sufficiently smooth time-dependent velocity field v: Ω ×[0,1] →R3param-
eterizing the flow φ: Ω ×[0,1] →Ω, a diffeomorphic transformation ϕ: Ω →Ωcan be
computed by
ϕ(x) = φ1(x) = φ0(x, 0) + Z1
0
v(φt(x), t)dt (2)
[DGM98, Tro98]. While the time dependence of the velocity field allows for a physically
plausible interpretation, it leads to time and memory consuming algorithms if considered
for image registration purposes [BMTY05, Her08]. Thus, in recent works the restriction
to stationary velocity fields is examined [ACPA06, Ash07, EWSRH11]. In order to define
an efficient algorithm for the time integration in (2), in the case of stationary velocity
fields it can further be exploited that the set of diffeomorphisms Diff(Ω) can be seen as
a differentiable manifold, and (Diff (Ω) ,◦)therefore features, in addition to its general
group structure, a Lie group structure with a Lie algebra g. Now, for a diffeomorphism
parameterized by a stationary velocity field, the velocity field is part of the tangential
space TidDiff (Ω) at the neutral element of Diff(Ω), i. e. the vector space underlying the g
[Ars06]. Since Lie algebra and Lie group are connected by the group exponential map
exp : TidDiff (Ω) →Diff (Ω) ,exp (tv) = φt,
the transformation of (2) can consequently be described by
ϕ(x) = φ1(x) = exp (v(x)) ,(3)
where the velocity field vis called the (group-)logarithm of ϕ,v= log ϕ.
In the context of Lie group theory it can further be shown that for each v∈ TidDiff (Ω)
the corresponding paths φt= exp (tv)are so-called one parameter subgroups of Diff (Ω).
This means especially that φs◦φt=φs+t= exp ((s+t)v)for scalars s, t, and eventually
exp (v) = exp 1
2Nv2N
.(4)
Thus, under the assumption that exp v(x)/2N≈x+v(x)/2Nfor a sufficiently large
N, the time integration of (2) can be substituted/executed by a recursive N-times “squar-
ing” (self-composing) of exp v/2N; this represents the so-called scaling-and-squaring
algorithm [ACPA06, BZO08].
Being interested in the motion fields ujto train the sought MLR model, (3) is now em-
ployed to define a partial differential equation-driven non-linear diffeomorphic framework.
Let therefore I1serve as reference image and the remaining 3D volumes Ijbe the tar-
get images, then for each phase j∈ {2, . . . , nph}we are searching for a transformation
ϕj=id +ujparameterized by a stationary velocity field vjby ϕj= exp (vj)that mini-
mizes an energy functional
J[vj] = D[I1, Ij◦ϕj] + αS[vj].(5)
Instead of defining an explicit image distance measure D, however, we applied active
Thirion forces. Referring to the Euler-Lagrange equations corresponding to (5), these can
be interpreted as a variant of the force term related to the commonly used Sum-of-Squared
Differences (SSD) measure. As the regularization term we chose a diffusion approach,
S[v] = RΩP3
l=1 kvjk2dx; implementation details for the applied registration scheme can
be found in [SREWH10].
2.2 Definition of the Diffeomorphic MLR Correspondence Model
Training phase: Now assuming the motion fields ujj∈{1,...,nph}and surrogate signals
ξjj∈{1,...,nph}corresponding to a 4D CT image sequence Ijj∈{1,...,nph }to be known,
the idea underlying the definition of a diffeomorphic MLR correspondence model is to
work on the Log-Euclidean parametrization of the motion fields, vj= log (id +uj),
instead of the motion fields directly. Appropriate methods to compute the logarithms
and velocity fields, respectively, for diffeomorphic transformations are proposed in, e. g.,
[BG08]; in the current contribution, however, explicit computation of the logarithms is not
necessary because the velocity fields are given as output of the diffeomorphic registration
scheme.
Now, in the context of multivariate statistics, the velocity fields vjand the surrogate sig-
nals ξjare interpreted as random variables Vjand Zj, for which the motion information
is described in a single column vector (i. e. Vj∈Rmwith m= 3n1n2n3and nibe-
ing the image dimension along the i-th image axis; Zj≡ξj). Then, a multi-variate
multi-linear regression can be formulated to estimate the relationship between the ma-
trices V:= Vc
1,...,Vc
nph and Z:= Zc
1,...,Zc
nph , holding the centered variables
Vc
j=Vj−¯
Vwith ¯
V=1
nph Pnph
j=1 Vjand analogously Zc
j, such that
V=BZ (6)
with
B= arg min
B0tr h(V−B0Z) (V−B0Z)Ti=VZTZZT−1
=ΣVZΣ−1
Z.(7)
ΣZis the covariance matrix of the surrogate signal observations Zand ΣVZ denotes the
cross-covariance matrix of Vand Z. Thus, Brepresents an ordinary least squares (OLS)
estimator of the relationship between the surrogate signal observations, interpreted in an
MLR sense as the regressor, and the image-based estimated velocity fields, which are con-
sidered to be the regressand.
Referring to (7), it has to be noted that for more-dimensional breathing signals it is very
likely that the information contributed by the different signal dimensions are highly cor-
related. This poses the problem of multi-collinearities, which in the perfect case leads
to a singular covariance matrix ΣZ. To avoid singularity of the matrix we introduce a
Tikhonov regularization, i. e. we approximate ΣZby ΣZ+γ1with γas a small positive
constant [KLO09].
Motion estimation: With the computed OLS estimator B, for any measurement ξ(t)≡
Z(t),t∈[t0, tend], a corresponding velocity field ˆv(t)can be derived by
ˆ
V(t) = ¯
V+BZ(t)−¯
Z(8)
and subsequently resorting the entries of ˆ
V(t)appropriately into the field ˆv(t). Again ex-
ploiting (3), the associated diffeomorphic transformation can finally be derived by ˆϕ(t) =
exp (ˆv(t)) and the sought motion field is given by ˆu(t) = exp (ˆv(t)) −id.
2.3 Considered Types of Surrogates
The MLR framework as described above can be applied by using in principle arbitrary
breathing signals. As already mentioned in section 1, in this paper we consider three
different types of surrogates for a first evaluation of the framework:
Spirometry records: The reconstruction process of the 4D CT image sequences con-
sidered was based on spirometry measurements [EWS+07]. These were now used as an
example of an one-dimensional surrogate, ξspiro :t7→ ξspiro (t)∈R.
Tracking motion of points on the diaphragm: For a first demonstration of the potential
of the multi-variate character of the correspondence model, we tracked points xdia ∈Ω
on the diaphragm within the patients’ 4D CT image sequences and interpreted the cor-
responding displacements u1xdia, . . . , unph xdia as regressor measurements. The
motivation of this approach is that, on the one hand, the diaphragm can be considered as
the main motor of breathing motion. On the other hand, the diaphragm is clearly visi-
ble in most medical imaging devices [KLO09] including, e. g., fluoroscopy. Tracking of
diaphragm motion therefore offers the potential to serve as a real-time image-based sur-
rogate. In a first run of the experiments, we identified and used solely the dome of the
left and the right hemi-diaphragm (thus, ξdia 1 :t7→ ξdia 1 (t)∈R2·3). In the second
run, additional 28 points on three concentric circles around the dome are tracked for each
hemi-diaphragm (i. e. ξdia 2 :t7→ ξdia 2 (t)∈R2·29·3).
Simulating range imaging (RI) devices: As a third surrogate type we simulated the use of
a point- and a line-laser for tracking lifting/raising of the chest wall based on the patient’s
4D CT image sequences; point- and line-lasers can be considered as typical examples of
range imaging devices in RT. For a point-laser and each breathing phase j, a ray originat-
ing from a given position above the patient is traced until it intersects with the chest wall
(ray direction: anterior-posterior; air-to-soft tissue threshold: -100 HU; intersection deter-
mined with subvoxel-accuracy); for a line-laser, this procedure is repeated for a series of
points on the line (line orientation: superior-inferior; line points distance = voxel spacing;
scanning range ≈20 cm). Simulating the point laser, in a first run the ray origin is placed
over the sternum (i. e. a standard position for RI-based gating devices; ξRI, sternum :t7→
ξRI, sternum (t)∈R); the corresponding modeling accuracy is compared to an MLR-based
motion detection based on ten laser positions that – considered as individual point lasers
– feature the lowest residual wrt. (7) (ξRI, opt. points :t7→ ξRI, opt. points (t)∈R10 ). Finally,
results are compared to the simulated line laser positioned such that again the residual of
(7) is minimal (ξRI, line :t7→ ξRI, line (t)∈R150 ).
2.4 Experiments
To evaluate the accuracy of the proposed MLR correspondence model and the appropri-
ateness of the OLS estimator to describe the relationship between the surrogate measure-
ments/simulations and the motion fields extracted from the 4D CT image sequences, a
leave-out strategy is applied: Using all breathing phases j∈ {1, . . . , nph }but the phase
of end expiration (EE) and around EE (thus, using in total all but three phases) for train-
ing purposes, capabilities of ”extrapolation” of displacements (i. e. to estimate displace-
ment fields for surrogate measurements not contained in the signal intervals used for the
OLS training) are analyzed by estimating the field uEE between EI and EE based on the
surrogate values at EE. Interpolation capabilities are evaluated similarly by leaving out
mid-respiration phases during training of the OLS estimator and then estimating the cor-
responding fields.
As quantitative measures, the accuracy of the tumor mass center motion as estimated by
the MLR approach is considered (only motion estimation between EI and EE; ground
truth: manual tumor segmentations within the CT images at EE and EI). Furthermore, a
target registration error (TRE) is computed based on inner lung landmark correspondences
determined manually within the CT data at the different breathing phases (70 landmarks
per patient and breathing phase).
Results are computed for all surrogate types and runs described above and the proposed
diffeomorphic MLR framework as well as for a ”standard” MLR with the regression di-
rectly performed on the fields ujj∈{1,...,nph}and the surrogate signals ξjj∈{1,...,nph }.
3 Results
The results of the leave-out tests are summarized in Tables 1 and 2. Referring to the accu-
racy of the different types of surrogates, no significant differences can be observed between
the use of the spirometry records and tracking of diaphragm points for the extrapolation
scenarios (p>0.05 for both tumor motion estimation and the landmark-based TRE values;
paired t-test); only in the case of interpolation purposes, the more-dimensional diaphragm
motion offers a slightly, but significantly higher accuracy (p<0.01). In comparison, track-
ing only a single point of the chest wall leads to a significantly decreased accuracy of the
MLR models (p<0.01). However, using more points and eventually simulating line track-
ing improves the accuracy with the resulting measures being slightly lower, but in a similar
order than in the case of using spirometry measurements or tracking the diaphragm (dif-
ferences still significant, p<0.01). This demonstrates the potential of information fusion
and the use of multi-variate methods for the given field of application.
The results of Tables 1 and 2 additionally show that almost no differences can be ob-
served between the diffeomorphic MLR framework and the application of the standard
MLR-based motion estimation when referring to the accuracy measures considered. The
potential of the diffeomorphic framework – i. e. avoiding singularities in the estimated
motion fields – becomes obvious especially in the case of extrapolation. A clinically mo-
tivated example is described in figure 1; the figure also demonstrates the importance of
the choice of the number of self-compositions performed in the context of the transition
between the velocity fields and the transformations and motion fields, respectively (i. e.
the choice of Nin the scaling-and-squaring algorithm, cf. (4)).
4 Discussion
Current techniques to cope with respiratory motion in radiation therapy of thoracic and
abdominal tumors like gating or tumor tracking techniques are usually grounded on the
use of breathing signals (internal motion surrogates) that can be acquired easily and fast
during treatment. Taking into account a trend toward the use of more-dimensional signals
Table 1: Target registration error values computed for the diffeomorphic MLR-based estimation of
inner lung motion as part of the leave-out tests (EE = end expiration, EI = end inspiration, MI = mid
inspiration, ME = mid expiration), listed for the different surrogate types and contrasted to values
obtained by a standard (= non-diffeomorphic) MLR-based motion estimation. Given are the mean
values obtained for the ten patients considered and the corresponding standard deviations.
Target-Registration-Error [mm]
Approach used for Motion Estimation EI →EE EI →MI EI →ME
No motion estimation 6.8±1.8 4.9±1.2 2.5±0.6
Intra-patient registration 1.6±0.2 1.6±0.1 1.5±0.2
Diffeomorphic framework:
MLR: surrogate = spirometry, ξspiro 2.0±0.3 2.0±0.3 1.8±0.3
MLR: surrogate = diaphragm motion, ξdia 1 2.1±0.4 1.8±0.2 1.7±0.3
MLR: surrogate = diaphragm motion, ξdia 2 2.0±0.3 1.8±0.2 1.6±0.2
MLR: surrogate = chest wall motion, ξRI, sternum 4.7±1.4 2.6±0.9 2.4±0.7
MLR: surrogate = chest wall motion, ξRI, opt. points 2.7±0.7 2.0±0.2 2.0±0.4
MLR: surrogate = chest wall motion, ξRI, line 2.1±0.4 1.9±0.2 1.8±0.2
Standard framework:
MLR: surrogate = spirometry, ξspiro 2.0±0.3 2.0±0.3 1.8±0.3
MLR: surrogate = diaphragm motion, ξdia 1 2.1±0.5 1.8±0.2 1.7±0.3
MLR: surrogate = diaphragm motion, ξdia 2 2.0±0.3 1.8±0.2 1.7±0.2
MLR: surrogate = chest wall motion, ξRI, sternum 4.7±1.4 2.6±0.9 2.4±0.7
MLR: surrogate = chest wall motion, ξRI, opt. points 2.8±0.7 2.0±0.2 1.9±0.4
MLR: surrogate = chest wall motion, ξRI, line 2.0±0.4 1.9±0.2 1.7±0.2
Table 2: Accuracy of the MLR-based estimation of tumor motion between end inspiration (EI) and
end expiration (EE) as obtained during the leave-out tests and based on manual tumor segmentations
in the EI and EE CT data.
Distance of Jaccard index
tumor mass for tumor
Approach used for Motion Estimation centers [mm] propagation
No motion estimation 6.9±6.1 0.50 ±0.26
Intra-patient registration 0.9±0.5 0.78 ±0.06
Diffeomorphic framework:
MLR: surrogate = spirometry, ξspiro 1.5±0.9 0.74 ±0.10
MLR: surrogate = diaphragm motion, ξdia 1 1.7±0.9 0.71 ±0.11
MLR: surrogate = diaphragm motion, ξdia 2 1.6±1.3 0.72 ±0.14
MLR: surrogate = chest wall motion, ξRI, sternum 4.7±4.3 0.52 ±0.25
MLR: surrogate = chest wall motion, ξRI, opt. points 1.7±1.3 0.70 ±0.13
MLR: surrogate = chest wall motion, ξRI, line 1.4±1.0 0.74 ±0.11
Standard framework:
MLR: surrogate = spirometry, ξspiro 1.5±1.0 0.73 ±0.11
MLR: surrogate = diaphragm motion, ξdia 1 1.6±0.8 0.71 ±0.10
MLR: surrogate = diaphragm motion, ξdia 2 1.5±1.2 0.71 ±0.12
MLR: surrogate = chest wall motion, ξRI, sternum 4.7±4.3 0.52 ±0.25
MLR: surrogate = chest wall motion, ξRI, opt. points 1.8±1.3 0.70 ±0.13
MLR: surrogate = chest wall motion, ξRI, line 1.4±0.9 0.74 ±0.11
Figure 1: In the left figure, a period of the spirometry record of a lung tumor patient is shown.
The zero volume indicates the measured volume for a mid-inspiration CT of the patient’s 4D CT
image sequence, which in the current case was chosen for treatment planning. The dashed lines
represent the volumes for the 3D CT data at end inspiration (EI) and end expiration (EE). The MLR
framework was applied to estimate internal motion patterns with the focus on spirometry volumes
and movements not being represented as part of the 4D CT, i. e. for extrapolation purposes. In the
right figure, the number of singularities in the estimated fields (voxels xwith det∇ˆϕ(x)<0) is
visualized as a function of spirometry volume and the inter-/extrapolation factor; this demonstrates,
on the one hand, the potential and advantage of the diffeomorphic framework proposed (please note
that a factor of 1.0 means a maximum motion of ≈15 mm, while for the extreme factor of 4.0
– usually, factors of >2are hardly observed – maximum voxel displacements of >50 mm are
computed). On the other hand, the importance of the choice of the number Nof self-compositions
during the scaling-and-squaring algorithm becomes also obvious.
and exploiting the Log-Euclidean framework for computing statistics on diffeomorphisms,
in the current contribution we proposed a diffeomorphic MLR framework for establish-
ing correspondence between the surrogate measurements and motion patterns of internal
anatomical and pathological structures. The results demonstrate feasibility and potential
of the proposed approach. In the next steps, however, thorough evaluations based on ad-
ditional data sets are required; especially the use of follow-up 4D image sequences would
be of interest in order to evaluate the capabilities of the OLS estimator in the presence of
intra- and inter-session breathing variations.
Acknowledgments:
This study is funded in part by the German Research Foundation DFG (HA 2355/9-2).
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