Conference PaperPDF Available

A Diffeomorphic Framework for Surrogate-based Motion Estimation in Radiation Therapy: Concept and First Evaluation

Authors:
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: R3Rof 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]×R3;
thus, for a voxel xand 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.
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: ×[0,1] R3param-
eterizing the flow φ: ×[0,1] , a diffeomorphic transformation ϕ: can 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() 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(), 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 φsφt=φs+t= exp ((s+t)v)for scalars s, t, and eventually
exp (v) = exp 1
2Nv2N
.(4)
Thus, under the assumption that exp v(x)/2Nx+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, Ijϕj] + 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] = RP3
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. VjRmwith m= 3n1n2n3and nibe-
ing the image dimension along the i-th image axis; Zjξj). 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(VB0Z) (VB0Z)Ti=VZTZZT1
=ΣVZΣ1
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 v(t)) and the sought motion field is given by ˆu(t) = exp v(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
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 detˆϕ(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).
References
[ACPA06] V. Arsigny, O. Commowick, X. Pennec, and N. Ayache. A Log-Euclidean Framework
for Statistics on Diffeomorphisms. In R. Larsen et al., editors, Medical Image Com-
puting and Computer-Assisted Intervention, MICCAI 2006, volume 4190 of Lecture
Notes in Computer Science, pages 924–931. Springer, 2006.
[Ars06] V. Arsigny. Processing Data in Lie Groups: An Algebraic Approach. Application to
Non-Linear Registration and Diffusion Tensor MRI. Th`
ese de sciences (phd thesis),
´
Ecole polytechnique, November 2006.
[Ash07] J. Ashburner. A fast diffeomorphic image registration algorithm. Neuroimage,
38(1):95–113, Oct 2007.
[BG08] M. N. Bossa and S. Olmos Gasso. A new algorithm for the computation of the group
logarithm of diffeomorphisms. In Workshop Proc. MICCAI 2008: MFCA 2008: Inter-
national Workshop on Mathematical Foundations of Computational Anatomy., 2008.
[BMTY05] M. Faisal Beg, Michael I. Miller, Alain Trouv´
e, and Laurent Younes. Computing
Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms. Inter-
national Journal of Computer Vision, 61(2):139–157, 2005.
[BZO08] M. Bossa, E. Zacur, and S. Olmos. Algorithms for computing the group exponential
of diffeomorphisms: Performance evaluation. In Proc. IEEE Computer Society Conf.
Computer Vision and Pattern Recognition Workshops CVPRW ’08, pages 1–8, 2008.
[DGM98] P. Dupuis, U. Grenander, and M.I. Miller. Variational Problems on Flows of Diffeo-
morphisms for Image Matching. Quarterly of Applied Mathematics, LVI(4):587–600,
Feb 1998.
[EWS+07] J. Ehrhardt, R. Werner, Dennis S., et al. An optical flow based method for improved
reconstruction of 4D CT data sets acquired during free breathing. Medical Physics,
34(2):711–721, Feb 2007.
[EWSRH11] J. Ehrhardt, R. Werner, A. Schmidt-Richberg, and H. Handels. Statistical Modeling of
4D Respiratory Lung Motion Using Diffeomorphic Image Registration. IEEE Trans-
actions on Medical Imaging, 30(2):251–65, Sep 2011.
[Her08] M. Hernandez. Variational techniques with applications to segmentation and reg-
istration of medical images. Phd-thesis, Aragon Institute on Engineering Research,
University of Zaragossa, 2008.
[KLO09] T. Klinder, C. Lorenz, and J. Ostermann. Free-Breathing intra- and intersubject respi-
ratory motion capturing, modeling, and prediction. In J.P.W. Pluim and B.M. Dawant,
editors, SPIE Medical Imaging 2009: Image Processing, volume 7259 of Proc. of
SPIE, pages 72590T–1–11, 2009.
[KMB+06] P. J. Keall, G.S. Mageras, J. M. Balter, et al. The management of respiratory motion in
radiation oncology report of AAPM Task Group 76. Medical Physics, 33:3874–3900,
2006.
[LCC+08] G. Li, D. Citrin, K. Camphausen, et al. Advances in 4D medical imaging and 4D
radiation therapy. Technology in Cancer Research and Treatment, 7(1):67–81, Feb
2008.
[SPH08] C. Schaller, J. Penne, and J. Hornegger. Time-of-flight sensor for respiratory motion
gating. Medical Physics, 35(7):3090–3093, Jul 2008.
[SREWH10] A. Schmidt-Richberg, J. Ehrhardt, R. Werner, and H. Handels. Diffeomorphic Diffu-
sion Registration of Lung CT Images. In B. van Ginneken, K. Murphy, T. Heimann,
V. Pekar, and X. Deng, editors, Workshop Proc. MICCAI 2010: Medical Image Anal-
ysis for the Clinic: A Grand Challenge, pages 55–62, Bejing, China, Sep 2010.
[Tro98] Alain Trouve. Diffeomorphisms Groups and Pattern Matching in Image Analysis.
International Journal of Computer Vision, 28(3):213–221, 1998.
[WESR+10] R. Werner, J. Ehrhardt, A. Schmidt-Richberg, A. Heiss, and H. Handels. Estimation of
motion fields by non-linear registration for local lung motion analysis in 4D CT image
data. International Journal of Computer Assisted Radiology and Surgery, 5(6):595–
605, 2010.
[ZHL+10] Q. Zhang, Y.-C. Hu, F. Liu, K. Goodman, K.E. Rosenzweig, and G.S. Mageras. Cor-
rection of motion artifacts in cone-beam CT using a patient-specific respiratory motion
model. Medical Physics, 37(6):2901–2909, Jun 2010.
Article
Full-text available
Breathing-induced location uncertainties of internal structures are still a relevant issue in the radiation therapy of thoracic and abdominal tumours. Motion compensation approaches like gating or tumour tracking are usually driven by low-dimensional breathing signals, which are acquired in real-time during the treatment. These signals are only surrogates of the internal motion of target structures and organs at risk, and, consequently, appropriate models are needed to establish correspondence between the acquired signals and the sought internal motion patterns. In this work, we present a diffeomorphic framework for correspondence modelling based on the Log-Euclidean framework and multivariate regression. Within the framework, we systematically compare standard and subspace regression approaches (principal component regression, partial least squares, canonical correlation analysis) for different types of common breathing signals (1D: spirometry, abdominal belt, diaphragm tracking; multi-dimensional: skin surface tracking). Experiments are based on 4D CT and 4D MRI data sets and cover intra- and inter-cycle as well as intra- and inter-session motion variations. Only small differences in internal motion estimation accuracy are observed between the 1D surrogates. Increasing the surrogate dimensionality, however, improved the accuracy significantly; this is shown for both 2D signals, which consist of a common 1D signal and its time derivative, and high-dimensional signals containing the motion of many skin surface points. Eventually, comparing the standard and subspace regression variants when applied to the high-dimensional breathing signals, only small differences in terms of motion estimation accuracy are found.
Article
Full-text available
Registration of the lungs in thoracic CT images is required in many fields of application in medical imaging, for example for motion estimation, analysis of pathology progression or the generation of shape atlases. In this paper, we present a robust registration approach that has been optimized for the registration of thoracic CT data. The algorithm con-sists of an initial shape-based adjustment of lung surfaces followed by an intensity-based diffeomorphic image registration. The approach is evaluated based on 20 CT scans provided for the EM-PIRE10 study for pulmonary image registration. A fourth place out of 34 participants suggests a good applicability for the registration of lung CT images.
Article
Full-text available
This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouvé (1995) in which two images I 0, I 1 are given and connected via the diffeomorphic change of coordinates I 0○ϕ−1=I 1 where ϕ=Φ1 is the end point at t= 1 of curve Φ t , t∈[0, 1] satisfying .Φ t =v t (Φ t ), t∈ [0,1] with Φ0=id. The variational problem takes the form \mathop {\arg {\text{m}}in}\limits_{\upsilon :\dot \phi _t = \upsilon _t \left( {\dot \phi } \right)} \left( {\int_0^1 {\left\| {\upsilon _t } \right\|} ^2 {\text{d}}t + \left\| {I_0 \circ \phi _1^{ - 1} - I_1 } \right\|_{L^2 }^2 } \right)
Article
Full-text available
Motivated by radiotherapy of lung cancer non- linear registration is applied to estimate 3D motion fields for local lung motion analysis in thoracic 4D CT images. Reliability of analysis results depends on the registration accuracy. Therefore, our study consists of two parts: optimization and evaluation of a non-linear registration scheme for motion field estimation, followed by a registration-based analysis of lung motion patterns. The study is based on 4D CT data of 17 patients. Different distance measures and force terms for thoracic CT registration are implemented and compared: sum of squared differences versus a force term related to Thirion's demons registration; masked versus unmasked force computation. The most accurate approach is applied to local lung motion analysis. Masked Thirion forces outperform the other force terms. The mean target registration error is 1.3 ± 0.2 mm, which is in the order of voxel size. Based on resulting motion fields and inter-patient normalization of inner lung coordinates and breathing depths a non-linear dependency between inner lung position and corresponding strength of motion is identified. The dependency is observed for all patients without or with only small tumors. Quantitative evaluation of the estimated motion fields indicates high spatial registration accuracy. It allows for reliable registration-based local lung motion analysis. The large amount of information encoded in the motion fields makes it possible to draw detailed conclusions, e.g., to identify the dependency of inner lung localization and motion. Our examinations illustrate the potential of registration-based motion analysis.
Article
In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and can be used for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. Overview In this article, which is an extended abstract of [1], we focus on the computation of statistics of general diffeomorphisms, i.e. of geometrical deformations (non-linear in general) which are both one-to-one and regular (as well as their inverse). To quantitatively compare non-linear registration algorithms, or in order to constrain them, computing statistics on global deformations would be very useful as was done in [6] with local statistics. The computation of statistics is closely linked to the issue of the parameterization
Thesis
Recently, the need for rigorous frameworks for the processing of non-linear data has grown considerably in medical imaging. In this thesis, we propose several general frameworks to process various types of non-linear data, which all belong to Lie groups. To this end, we rely on the algebraic properties of these spaces. Thus, we propose a general processing framework for symmetric and positive-definite matrices, named Log-Euclidean, very simple to use and which has excellent theoretical properties. It is particularly well-adapted to the processing of diffusion tensor MRI. We also propose several frameworks, called polyaffine, to parameterize locally rigid or affine transformations, in a way that guarantees their invertibility. Their use is illustrated in the case of the locally rigid registration of histological slices and of the locally affine 3D registration of MRIs of the human brain. This led us to propose two general frameworks for computing statistics in finite-dimensional Lie groups: first the Log-Euclidean one, which generalizes our work on tensors, and second a framework based on the novel notion of bi-invariant mean, whose properties generalize to Lie groups those of the arithmetic mean. Finally, we generalize our Log-Euclidean framework to diffeomorphic geometrical transformations, which opens the way to a general and consistent framework for statistics in computational anatomy.
Article
Purpose: Respiratory motion reduction methods to improve cone‐beam CT quality (CBCT) have focused on the thorax, but reduced tissue contrast in abdomen poses additional challenges. We report a method to correct CBCT in abdomen, using a motion model adapted to the patient from a prior respiration‐correlated CT (RCCT) image set. Method and Materials: Model adaptation consists of nonrigid image registration that maps each RCCT image to a reference image in the set, followed by principal component analysis (PCA) to reduce noise in the resultant deformation fields and relate them to diaphragm position and motion (inhalation or exhalation). CBCT projection images are sorted into subsets according to diaphragm position in the images and reconstructed, yielding a set of low‐quality 3‐D images. Model application deforms the CBCT images to a reference CBCT in the set; combining all images yields a high‐quality CBCT image with reduced motion artifacts. We also investigate a simpler correction method, which does not use PCA and correlates motion state with respiration phase. Comparison of contrast‐to‐noise ratios of pixel intensities within kidneys relative to surrounding background tissue provides a quantitative assessment of relative organ visibility. Results: Evaluation of CBCT examples in upper abdomen shows that streaking artifacts and blurring of liver, kidneys, spleen, bowel and implanted fiducial markers are visibly reduced with PCA‐model‐based correction. Phase‐based motion correction without PCA reduces blurring less effectively; in addition, implanted markers appear broken up, indicating inconsistencies in the correction. Model‐based motion correction shows the highest contrast‐to‐noise ratios in the cases examined. Conclusion: Motion correction of CBCT in abdomen is feasible and yields observable improvement. The PCA‐based model is an important component: first, by removing noise; second, by relating deformation to diaphragm position rather than phase, thus accommodating breathing pattern changes between imaging sessions.
Article
Respiration-induced organ motion can limit the accuracy required for many clinical applications working on the thorax or upper abdomen. One approach to reduce the uncertainty of organ location caused by respiration is to use prior knowledge of breathing motion. In this work, we deal with the extraction and modeling of lung motion fields based on free-breathing 4D-CT data sets of 36 patients. Since data was acquired for radiotherapy planning, images of the same patient were available over different weeks of treatment. Motion field extraction is performed using an iterative shape-constrained deformable model approach. From the extracted motion fields, intra-and inter-subject motion models are built and adapted in a leave-one-out test. The created models capture the motion of corresponding landmarks over the breathing cycle. Model adaptation is then performed by examplarily assuming the diaphragm motion to be known. Although, respiratory motion shows a repetitive character, it is known that patients' variability in breathing pattern impedes motion estimation. However, with the created motion models, we obtained a mean error between the phases of maximal distance of 3.4 mm for the intra-patient and 4.2 mm for the inter-patient study when assuming the diaphragm motion to be known.
Article
In a previous paper, it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group. We show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images. Using Lie group ideas, we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations. Then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub-optimal gradient pattern matching. Its links with fluid models are established.
Article
Modeling of respiratory motion has become increasingly important in various applications of medical imaging (e.g., radiation therapy of lung cancer). Current modeling approaches are usually confined to intra-patient registration of 3D image data representing the individual patient's anatomy at different breathing phases. We propose an approach to generate a mean motion model of the lung based on thoracic 4D computed tomography (CT) data of different patients to extend the motion modeling capabilities. Our modeling process consists of three steps: an intra-subject registration to generate subject-specific motion models, the generation of an average shape and intensity atlas of the lung as anatomical reference frame, and the registration of the subject-specific motion models to the atlas in order to build a statistical 4D mean motion model (4D-MMM). Furthermore, we present methods to adapt the 4D mean motion model to a patient-specific lung geometry. In all steps, a symmetric diffeomorphic nonlinear intensity-based registration method was employed. The Log-Euclidean framework was used to compute statistics on the diffeomorphic transformations. The presented methods are then used to build a mean motion model of respiratory lung motion using thoracic 4D CT data sets of 17 patients. We evaluate the model by applying it for estimating respiratory motion of ten lung cancer patients. The prediction is evaluated with respect to landmark and tumor motion, and the quantitative analysis results in a mean target registration error (TRE) of 3.3 ±1.6 mm if lung dynamics are not impaired by large lung tumors or other lung disorders (e.g., emphysema). With regard to lung tumor motion, we show that prediction accuracy is independent of tumor size and tumor motion amplitude in the considered data set. However, tumors adhering to non-lung structures degrade local lung dynamics significantly and the model-based prediction accuracy is lower in these cases. The statistical respir- - atory motion model is capable of providing valuable prior knowledge in many fields of applications. We present two examples of possible applications in radiation therapy and image guided diagnosis.
Article
Respiratory motion adversely affects CBCT image quality and limits its localization accuracy for image-guided radiation treatment. Motion correction methods in CBCT have focused on the thorax because of its higher soft tissue contrast, whereas low-contrast tissue in abdomen remains a challenge. The authors report on a method to correct respiration-induced motion artifacts in 1 min CBCT scans that is applicable in both thorax and abdomen, using a motion model adapted to the patient from a respiration-correlated image set. Model adaptation consists of nonrigid image registration that maps each image to a reference image in the respiration-correlated set, followed by a principal component analysis to reduce errors in the nonrigid registration. The model parametrizes the deformation field in terms of observed surrogate (diaphragm or implanted marker) position and motion (inhalation or exhalation) between the images. In the thorax, the model is obtained from the same CBCT images that are to be motion-corrected, whereas in the abdomen, the model uses respiration-correlated CT (RCCT) images acquired prior to the treatment session. The CBCT acquisition is a single 360 degrees rotation lasting 1 min, while simultaneously recording patient breathing. The approximately 600 projection images are sorted into six (in thorax) or ten (in abdomen) subsets and reconstructed to obtain a set of low-quality respiration-correlated RC-CBCT images. Application of the motion model deforms each of the RC-CBCT images to a chosen reference image in the set; combining all images yields a single high-quality CBCT image with reduced blurring and motion artifacts. Repeated application of the model with different reference images produces a series of motion-corrected CBCT images over the respiration cycle, for determining the motion extent of the tumor and nearby organs at risk. The authors also investigate a simpler correction method, which does not use PCA and correlates motion state with respiration phase, thus assuming repeatable breathing patterns. Comparison of contrast-to-noise ratios of pixel intensities within anatomical structures relative to surrounding background tissue provides a quantitative assessment of relative organ visibility. Evaluation in lung phantom, two patient cases in thorax and two in upper abdomen, shows that blurring and streaking artifacts are visibly reduced with motion correction. The boundaries of tumors in the thorax, liver, and kidneys are sharper and more discernible. Repeat application of the method in one thorax case, with reference images chosen at end expiration and end inspiration, indicates its feasibility for observing tumor motion extent. Phase-based motion correction without PCA reduces blurring less effectively; in addition, implanted markers appear broken up, indicating inconsistencies in the phase-based correction. In structures showing 1 cm or more motion excursion, PCA-based motion correction shows the highest contrast-to-noise ratios in the cases examined. Motion correction of CBCT is feasible and yields observable improvement in the thorax and abdomen. The PCA-based model is an important component: First, by reducing deformation errors caused by the nonrigid registration and second, by relating deformation to surrogate position rather than phase, thus accommodating breathing pattern changes between imaging sessions. The accuracy of the method requires confirmation in further patient studies.