Schulz
etal. BMCMedicalImaging
2014,
14
:4http://www.biomedcentral.com/14712342/14/4
RESEARCH ARTICLE OpenAccess
Asemiautomatictoolforprostatesegmentationin radiotherapy treatment planning
Jörn Schulz
1*
, Stein Olav Skrøvseth
2
, Veronika Kristine Tømmerås
3
, Kirsten Marienhagen
3
and Fred Godtliebsen
1
Abstract
Background:
Delineation of the target volume is a timeconsuming task in radiotherapy treatment planning, yetessential for a successful treatment of cancers such as prostate cancer. To facilitate the delineation procedure, thepaper proposes an intuitive approach for 3D modeling of the prostate by slicewise best fitting ellipses.
Methods:
The proposed estimate is initialized by the definition of a few control points in a new patient. The methodis not restricted to particular image modalities but assumes a smooth shape with elliptic cross sections of the object.A training data set of 23 patients was used to calculate a prior shape model. The mean shape model was evaluatedbased on the manual contour of 10 test patients. The patient records of training and test data are based on axial T1weighted 3D fastfield echo (FFE) sequences. The manual contours were considered as the reference model.Volume overlap (Vo), accuracy (Ac) (both ratio, range 01, optimal value 1) and Hausdorff distance (HD) (mm, optimalvalue 0) were calculated as evaluation parameters.
Results:
The median and median absolute deviation (MAD) between manual delineation and deformed mean bestfitting ellipses (MBFE) was Vo (0.9
±
0.02), Ac (0.81
±
0.03) and HD (4.05
±
1.3)mm and between manual delineationand best fitting ellipses (BFE) was Vo (0.96
±
0.01), Ac (0.92
±
0.01) and HD (1.6
±
0.27)mm. Additional results show amoderate improvement of the MBFE results after Monte Carlo Markov Chain (MCMC) method.
Conclusions:
The results emphasize the potential of the proposed method of modeling the prostate by best fittingellipses. It shows the robustness and reproducibility of the model. A small sample test on 8 patients suggest possibletime saving using the model.
Keywords:
Delineation, Ellipse model, Empirical Bayes, Prostate, Radiotherapy treatment planning, Statistical shapeanalysis
Background
Prostate cancer is the second most diagnosed canceraccounting for 14 percent of all cancers diagnosed worldwide [1]. It is most common in males over the age of 50,and has the highest incidence rate in the developed countries. Aggressive tumors are usually treated with externradiotherapy or brachytherapy which requires a precisetreatment plan for the target volume. In any type of radiotherapy treatment, radiation of healthy tissue should beminimized while maintaining the desired dose to the target volume. Therefore, a successful treatment of prostate
*Correspondence: jorn.schulz@uit.no1Department of Mathematics and Statistics, University of Tromsø, 9037 Tromsø, NorwayFull list of author information is available at the end of the article
cancer relies on an accurate segmentation of the prostatefrom the surrounding tissue, by imagebased descriptionof the shape and location of the target volume. The volume of interest is characterized by a smooth shape, andfor this reason an algorithmic description of the volume isfeasible.Transrectal ultrasound (TRUS), magnetic resonance(MR) and computed tomography (CT) images are thethree main imaging techniques used in diagnosis, treatment planning and followup examination of prostatecancer. Smith et al. [2] investigated the effects of theseimaging techniques on the properties of the prostate volume. A collection of methods available for prostatesegmentation is reviewed by Ghose et al. [3]. In addition to the methods presented by Ghose et al., alternative
© 2014 Schulz et al.; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the CreativeCommons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, andreproduction in any medium, provided the srcinal work is properly cited.
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etal. BMCMedicalImaging
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approaches are available in the literature, such as themedial or skeleton representation of the prostate [48].The present work proposes a segmentation method whichfalls into the category of deformable meshes in Ghose et al.[3], but refers to the term geometrical parametrization asdescribed in Dryden and Mardia [9]. The main focus of this paper is the development of a statistical shape modelfor the prostate. An overview about this type of models in3D medical image segmentation is presented for exampleby Davies et al. [10] and Heimann and Meinzer [11].
The works of Saroul et al. [12] and Mahdavi et al. [13]
are related to the stacked ellipses parametrization methodused in this paper. Mahdavi et al. [13] proposes a 3Dellipsoid shape of the prostate in warped transrectal 3Dultrasound images based on control points. This methodextends the warping idea proposed in Badiei et al. [14]from 2D to 3D ultrasound images. On the contrary, wefocus on slicewise best fitting ellipses which will introduce more flexibility into the model, e.g., between thepositions and lengths of the first and second axes of theellipses between neighbor slices. The approach of slicewisebestfittingellipseshassimilaritiestoatubularmedialrepresentation [15].Beside the single segmentation of the prostate, severalattempts have been tried out for a joint segmentation of neighbor organ and structure to gain improved segmentation results [1618].To our knowledge, despite the substantial effort in thisarea,nowidelyimplementedalgorithmexists.Inoncology departments this means that the physician has to delineate the prostate slice by slice. This is timeconsumingand inefficient. We propose a less ambitious approachcompared to more sophisticated models, such as skeletal models as discussed above, in that we use a methodthat gives a useful starting point for the physician afterthe definition of few control points. Given the initial estimateofthevolumeofinterest,thephysiciancanadjusttheestimate according to their evaluation of the image ratherthan starting from scratch. By this approach, we obtainthe same accuracy with less effort. The main points in ourapproachareasfollows:First,weacceptthatthealgorithmcannot give a fully precise description of the volume. Ourmain aim is therefore to give a good estimate which canbe used as a starting point for the physician. Second, weuse a simple ellipse model that is easy to interpret andunderstand. Our hypothesis is that a more efficient useof physicians in Radiotherapy Treatment Planning (RTP)of patients with prostate cancer can be obtained by aneasytointerpret semiautomatic tool.Figure 1 shows an example of the initial estimate wetypically obtain for a single image slice. The dashed linein (a) to (e) describes the manual contour while the solidline shows the best fitting ellipse including the two principal axes for the observed data of this slice. Note that thefitted model is very much in agreement with the manualline,indicatingthatthestackedellipsesmodelgivesagooddescription of the object of interest. The solid lines in(f)(j) shows the outcome from our model in this situationtogether with few defined control points.This result shows a typical performance of the method,and that the estimate is close to the best fit we can obtainwith the ellipse model. The full processing demands littlecomputational resources, such that the suggested delineation can be presented immediately. The example is discussed further in the Methods and Results and discussion
section.The rest of the paper is organized as follows. In theMethods section, we introduce the data sources and theproposed stacked ellipses model, and discuss the shapespace and statistics along with constraints and parameters. Results are presented in the Results and discussionsection using a test data set to show the potential of themean shape model, followed by a Conclusions section.Additional file 1 with further detailed discussion is available online.
Methods
Preliminaries
Each prostate must be described by a shape model inorder to calculate statistics, e.g., by stacked ellipses as aparametric shape model. The parameters of a parametric shape model can be estimated from a training set.The training set models also the geometric variability of anatomical structures by a shape probability distribution.The training set contains volume and contour information of segmented prostates from
N
patients. The volumeinformation describes the image modalities (e.g., CT orMR) and the contour information the volume of interestas defined in the following.The volume information of each training set
n
=
1,
...
,
N
is defined by a 3dimensional matrix
V
n
where
V
n
(
i
,
h
)
contains the observed gray level in voxel
(
i
,
h
)
,
i
=
(
i
1
,
i
2
)
∈ {
1,
...
,
I
1
}×{
1,
...
,
I
2
}
are the pixel indicesin a slice, where typically
I
1
=
I
2
, and
h
∈ {
1,
...
,
H
}
isthe number of slices per data set. The number of slices
H
is not necessarily the same for all patients in the trainingdata sets. Therefore, we indicate
H
by
H
n
and in the samemanner
I
1
by
I
n
1
and
I
2
by
I
n
2
, but for simplicity we use
H
,
I
1
and
I
2
if the meaning is clear.In addition to the volume information, each trainingset
n
=
1,
...
,
N
consists of contour information of the prostate, manually drawn by a physician. The contour information can be modeled by a
(
M
×
K
n
)
configuration matrix
X
n
:
=
(
X
n
1
,
...
,
X
nK
n
)
with
X
nk
=
(
x
n
1
k
,
x
n
2
k
,
x
n
3
k
)
T
∈
R
3
,
k
=
1,
...
,
K
n
, where
K
n
definesthe total number of available contour information pointsin a data set and
M
=
3 defines the dimension. Weassume the contour information for an object is defined
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etal. BMCMedicalImaging
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Figure1
SelectedslicesofMRdataset3fromthetestdataset.
(a)(e)
Manual delineation of the prostate (dashed line) and best fitting ellipse(solid line).
(f)(j)
Manual delineation of the prostate (dashed line), deformed mean shape (solid line) and defined control points in the first, centerand last slice.
in a sequentially sorted number
L
n
of equidistant sliceswhereas each contour slice contains
˜
K
nl
contour points,
l
=
1,
...
,
L
n
. Hence it follows
K
n
=
l
˜
K
nl
and
X
n
=
(
˜
X
n
1
,
...
,
˜
X
nL
n
)
. The image information in slice
l
isdenoted by
S
nl
and
S
n
= {
S
n
1
,
...
,
S
nL
n
} ⊆
V
n
and
˜
X
nl
defines the configuration matrix in slice
S
nl
.In summary, the training population is given by the set
{
V
,
X
}
,withasetofvolumeinformation
V
={
V
1
,
...
,
V
N
}
and configuration matrices
X
={
X
1
,
...
,
X
N
}
. We assume
X
n
defines the configuration matrix for the corresponding data set
V
n
and matches the volume information
V
n
exactly.The contour information is often defined in a Patientbased Coordinate System (PCS) whereas the volumeinformation is defined in an Image based CoordinateSystem (ICS). The ICS can be transformed to PCSby a transformation matrix
DCM
, which transform animage coordinate
p
im
=
(
i
1
,
i
2
,
h
)
T
to patient coordinate
p
p
=
(
x
,
y
,
z
)
T
. The definition of
DCM
and the relationbetween PCS and ICS (see Figure 2) is discussed in detailin the Additional file 1. In addition, we introduce a derotated PCS where volume and contour information arealigned to each other.
Modeling
The prior information inferred from the training set isincorporated into a shape model. We assume a stackedellipse model as a shape prior for the prostate. Specifically,
Figure2
Visualizationofdifferentcoordinatesystemswithexampledata.
PCS:
Patient based coordinate system (manual delineation line).
ICS:
Image coordinate system (volume data).
derotatedPCS:
derotated patient based coordinate system with same scale and srcin as PCS butsame orientation as ICS (derotated best fitting ellipse
dBFE
nl
, derotated control points
dCP
n
).
samplespace:
The transformation matrix
ndCP
mapsthe derotated data
{
dBFE
n
,
dCP
n
}
to
{
BFE
n
,
CP
n
}
in the sample space.
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etal. BMCMedicalImaging
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the prostate outline in slice
S
nl
,
l
=
1,
...
,
L
n
,
n
=
1,
...
,
N
is modeled by a slicewise bestfitting ellipse, as visualized in Figure 3. An ellipse in slice
S
nl
can be uniquely described by
ρ
nl
=
(θ
nl
,
α
nl
,
φ
nl
)
T
∈
R
2
×
R
2
+
×
(
−
π
2
,
π
2
]with
•
position
θ
nl
=
(θ
nl
1
,
θ
nl
2
)
T
∈
R
2
defines the center inslice
S
nl
,
•
length of principle axes
α
nl
=
(α
nl
1
,
α
nl
2
)
T
∈
R
2
+
and
•
rotation angle
φ
nl
∈
(
−
π
2
,
π
2
]
in slice
S
nl
.
The rotation parameter
φ
nl
is defined corresponding tothe ICS with srcin
θ
nl
in slice
S
nl
. The boundary of anellipse
ρ
nl
centered at
θ
nl
∈
R
2
in slice
S
nl
is defined by
C
(ρ
nl
)
=
Rx
+
θ
nl
:
x
21
(α
nl
1
)
2
+
x
22
(α
nl
2
)
2
=
1,
x
∈
R
2
(1)and
R
=
cos
φ
nl
−
sin
φ
nl
sin
φ
nl
cos
φ
nl
is a rotation matrix in
R
2
with rotation angle
φ
nl
and
x
=
(
x
1
,
x
2
)
T
.The shape model described in this section requires thebest fit of an ellipse
C
(ρ
nl
)
to the contour information
˜
X
nl
in each slice, i.e., we model
˜
X
nl
=
C
(ρ
nl
)
+
ǫ
where
ǫ
is anerror with mean zero. The bestfitting ellipses provide uswith a slicebyslice parametrization of the prostate for allslices in each training shape.The problem of fitting an ellipse to geometric featureslike the contour is discussed widely in the literature (e.g.,[19,20]).ThisworkfollowsAhnetal.[19],whoproposeda
leastsquare minimizer for
˜
X
nl
. The nonlinear estimate of parameters
ρ
nl
=
(θ
nl
1
,
θ
nl
2
,
α
nl
1
,
α
nl
2
,
φ
nl
)
T
given
˜
X
nl
mustminimize the error
g
(
ˆ
ρ
nl
)
=
˜
X
nl
− ˜
C
(
ˆ
ρ
nl
)
T
˜
X
nl
− ˜
C
(
ˆ
ρ
nl
)
where
˜
C
(
ˆ
ρ
nl
)
is a set of nearest orthogonal points of
˜
X
nl
to
C
(
ˆ
ρ
nl
)
.
Definition 1
(Best fitting ellipse (BFE))
.
A best fitting ellipse for slice S
nl
is defined by the set BFE
nl
:
=
(θ
nl
,
α
nl
,
φ
nl
)
T
∈
R
2
×
R
2
+
×
(
−
π
2
,
π
2
]
, l
=
1,
...
,
L
n
,n
=
1,
...
,
N and minimizes the error function g, i.e., BFE
nl
= ˆ
ρ
nl
with g
(
ˆ
ρ
nl
)
=
min
ρ
nl
∈
R
2
×
R
2
+
×
(
−
π
2
,
π
2
]
g
(ρ
nl
)
. (2)The first and second principal axes must be reorderedafter calculation of
BFE
n
= {
BFE
n
1
,
...
,
BFE
nL
n
}
in orderto establish correspondence between parameters of adjacent slices and across the population. Improved correspondence will support accurate statistics. The basic ideain our reordering procedure is to carry out the reorderingcorresponding to the lowest rotation angle of both principal axes to the first principal axis of the neighbor slicewhere the center slice is chosen as the basis. The rotation between the center slice
M
and an arbitrary slice isconstrained by max
(

φ
i
−
φ
M

)
=
π
,
i
∈ {
1,
...
,
L
}
afterreordering. Therefore, the set
BFE
n
of reordered bestfitting ellipses is an element of
(
R
2
×
R
2
+
×
(
−
π
,
π
]
)
L
n
.A further improvement of correspondence is achievedby the introduction of two additional constraints in theparameter model.First, we relax the rotation parameter
φ
nl
in case of circularity. If both principal axes have the same length, theorientation of an ellipse is undefined. Therefore we penalize
φ
nl
in the case of high circularity by taking
φ
nl
′
fromthe neighboring slices into account. Second, smoothing isperformed between neighboring slices to avoid large forward and backwards rotations between
φ
n
(
l
−
1
)
,
φ
nl
and
φ
n
(
l
+
1
)
. The reordering algorithm and implementation of constraints are described in detail in Additional file 1.
Figure3
Theprostatemodelofstackedslicewisebestfittingellipsesillustratedby
(a)
a3Dviewofthemodeland
(b)
thecorresponding2Dmodeloftheprostatecontourinaslice.
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The current implementation assumes the definition of control points
CP
n
in the training data set
{
V
n
,
X
n
,
BFE
n
}
,where
BFE
n
∈
(
R
2
×
R
2
+
×
(
−
π
,
π
]
)
L
n
is a reordered setof best fitting ellipses,
n
=
1,...,
N
. Furthermore, the control points have to be defined manually by a physicianin a new patient data set. The control points are usedto make the best fitting ellipses
BFE
n
comparable andto transform the parametrized ellipses model to a common position, scale and orientation by a transformationmatrix
ndCP
. The transformation matrix
ndCP
maps thederotated prior data
{
dBFE
n
,
dCP
n
}
to
{
BFE
n
,
CP
n
}
in thesample space, as depicted in Figure 2. In this article, weassume 6 control points in the first, center and last slice atthe boundary of the prostate, i.e.,
CP
n
=
A
n
1
,
...
,
A
n
6
,
P
n
1
,
...
,
P
n
6
,
B
n
1
,
...
,
B
n
6
as visualized in Figure 3. In addition, we have tested alternative control point configurations. They are describedtogether with the construction of
ndCP
in Additionalfile 1.After transformation we have obtained a reordered andcomparable set of best fitting ellipses
BFE
n
=
BFE
n
1
,
...
,
BFE
nL
n
with
BFE
nl
=
(θ
nl
,
α
nl
,
φ
nl
)
T
,
n
=
1,
...
,
N
,
l
=
1,
...
,
L
n
.The statistical analysis of the training data requires anequal number
L
1
=
...
=
L
N
to establish correspondencebetween the parameters of the best fitting ellipses. Therefore, we interpolate the set
BFE
nl
to a common number
L
.When
L
is chosen, interpolation is done by independent cubic interpolation in each dimension, i.e., we findpoints of a onedimensional function that underlies thedata
θ
nl
1
,
θ
nl
2
,
θ
nl
3
,
α
nl
1
,
α
nl
1
and
φ
nl
. The final interpolatedbest fitting ellipses are denoted by
iBFE
n
={
iBFE
n
1
,
...
,
iBFE
nL
}
. (3)These ellipses are used for the statistical analysis andcomputation of a mean shape model. To keep things simple,wedenotesuchareordered,transformedandinterpolated set of bestfitting ellipses by
BFE
nl
=
(θ
nl
,
α
nl
,
φ
nl
)
T
for the number
L
of contour slices with
l
=
1,
...
,
L
and
n
=
1,
...
,
N
. The comparable set of best fitting ellipses
BFE
n
is an element of the shape space
(
R
2
×
R
2
+
×
(
−
π
,
π
]
)
L
.
Statisticalanalyses
After reconstruction of our shape space we estimate theexpectation and variance of the parameters of a meanshape model
µ
BFE
= {
µ
1
BFE
,
...
,
µ
L BFE
}
with
µ
l BFE
=
(µ
l
θ
,
µ
l
α
,
µ
l
φ
)
from the training set
BFE
nl
,
l
=
1,
...
,
L
. Wedenote the mean shape mean best fitting ellipses (MBFE).In addition to the described ellipse parameters we definethe position
θ
nl
=
(θ
nl
1
,
θ
nl
2
,
θ
nl
3
)
T
in terms of a distance vector
η
nl
of
θ
nl
to a center curve defined by the controlpoints. We model
θ
l
=
ξ
l
+
η
l
, where
ξ
l
is analytically defined by
L
intersection points of the curve within eachslice. Thereby, we are describing the mean shape whichis closest to the control points. This approach is reasonable under the assumption that the control points arewell defined. In Additional file 1 we explore various waysof describing the position parameter for different controlpoint methods.The mean curve of the expected location is given by
µ
l
θ
j
=
1
N
N
i
=
1
θ
il j
,
j
∈{
1,2,3
}
, (4)where
µ
l
θ
=
(µ
l
θ
1
,
µ
l
θ
2
,
µ
l
θ
3
)
T
,
l
=
1,
...
,
L
. The varianceand covariance are estimated by
(σ
l
θ
j
)
2
=
1
N
−
1
N
i
=
1
(θ
il j
−
µ
l
θ
j
)
2
,
j
∈{
1,2,3
}
,and (5)
l
θ
=
1
N
−
1
N
i
=
1
(θ
il
−
µ
l
θ
)(θ
il
−
µ
l
θ
)
T
. (6)Thelengthparameterismodeledbyalognormaldistribution because
α
∈
R
2
+
. Thus we estimate the mean and variance of
a
=
log
(α)
∈
R
2
.Theestimationofmeansand variances of the remaining parameters
a
,
φ
,
η
is accordingto (45).
Following Dryden and Mardia [9] we suggest a priordistribution for a new data set as
θ
l
1
∼
N
µ
l
θ
1
,
(σ
l
θ
1
)
2
,
θ
l
2
∼
N
µ
l
θ
2
,
(σ
l
θ
2
)
2
,
a
l
1
∼
N
µ
l a
1
,
(σ
l a
1
)
2
⇐⇒
α
l
1
∼
log
N
µ
l a
1
,
(σ
l a
1
)
2
with
a
l
1
=
log
(α
l
1
)
,
a
l
2
∼
N
µ
l a
2
,
(σ
l a
2
)
2
⇐⇒
α
l
2
∼
log
N
µ
l a
2
,
(σ
l a
2
)
2
with
a
l
2
=
log
(α
l
2
)
,
φ
l
∼
N
µ
l
φ
,
(σ
l
φ
)
2
,
l
=
1,
...
,
L
. If
θ
l
is defined according to the center curvegiven by the control points as described above, we model
η
l i
∼
N
µ
l
η
i
,
(σ
l
η
i
)
2
,
i
=
1,2. Since the rotational parameter is expected to have small variance it is not necessary to apply a circular distribution, and we assume normality.Afterconstructing theshapemodelweestimatethebestfitting ellipse
BFE
l
parametrized by
ρ
l
=
(θ
l
,
α
l
,
φ
l
)
T
,
l
=
l
,
...
,
L
in a new data set given the control points
CP
. This