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Computer Methods and Programs in Biomedicine 14 4 (2017) 135–145
Contents lists available at ScienceDirect
Computer Methods and Programs in Biomedicine
journal homepage: www.elsevier.com/locate/cmpb
A novel method for planning liver resections using deformable Bézier
surfaces and distance maps
Rafael Palomar
a , b , ∗, Faouzi A. Cheikh
a
, Bjørn Edwin
b , e , f
, ˚
Asmund Fretland
b , e , f
,
Azeddine Beghdadi
c
, Ole J. Elle
b , d
a
Department of Computer Science, NTNU, 2815 Gjøvik, Norway
b
The Intervention Centre, Oslo University Hospital, P.O. box 4950 - Nydalen, 0424 Oslo, Norway
c
L2TI, Institut Galilée, Université Paris 13, Avenue J. B. Clément 99, 93430 Villetaneuse, France
d
Department of Informatics, University of Oslo, 0373 Oslo, Norway
e
Department of Hepato-Pancreato-Biliary Surgery, Oslo University Hospital, P.O. box 4950 - Nydalen, 0424 Oslo, Norway
f
Institute of Clinical Medicine, University of Oslo, Norway
a r t i c l e i n f o
Article history:
Received 1 July 2016
Revised 22 February 2017
Accepted 21 March 2017
Keywo rds:
Liver resection
Bézier surfaces
Distance maps
Surgery planning
Visualization
a b s t r a c t
Background and Objective: For more than a decade, computer-assisted surgical systems have been helping
surgeons to plan liver resections. The most widespread strategies to plan liver resections are: drawing
traces in individual 2D slices, and using a 3D deformable plane. In this work, we propose a novel method
which requires low level of user interaction while keeping high flexibility to specify resections. Methods:
Our method is based on the use of Bézier surfaces, which can be deformed using a grid of control points,
and distance maps as a base to compute and visualize resection margins (indicators of safety) in real-
time. Projection of resections in 2D slices, as well as computation of resection volume statistics are also
detailed. Results: The method was evaluated and compared with state-of-the-art methods by a group of
surgeons ( n = 5 , 5–31 years of experience). Our results show that theproposed method presents planning
times as low as state-of-the-art methods (174 s median time) with high reproducibility of results in terms
of resected volume. In addition, our method not only leads to smooth virtual resections easier to perform
surgically compared to other state-of-the-art methods, but also shows superior preservation of resection
margins. Conclusions: Our method provides clinicians with a robust and easy-to-use method for planning
liver resections with high reproducibility, smoothness of resection and preservation of resection margin.
Our results indicate the ability of the method to represent any type of resection and being integrated in
real clinical work-flows.
©2017 The Authors. Published by Elsevier Ireland Ltd.
This is an open access article under the CC BY-NC-ND license.
( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
1. Introduction
Liver cancer is one of the most common causes of cancer death
worldwide and its frequency is increasing in some geographical ar-
eas of historically low incidence rates [1] . Liver resection, which
refers to the surgical removal of a liver tumor, is the only curative
treatment for liver cancer. Planning of liver resections is usually
based on the anatomic division of the liver in segments, as de-
scribed in Couinaud [2] . The Couinaud division, which presents a
wide consensus in the medical community, separates the liver into
8 areas (segments) according to the blood supply, and establishes a
framework for the classification of resections in different types [3] .
∗Corresponding author:
E-mail address: rafael.palomar@ntnu.no (R. Palomar).
Liver cancer is either primary (arising from normal liver tissue)
or secondary (spreading to the liver from cancer located in other
organs). For hepatocellular carcinoma (primary), which accounts
for 70%-80% of the liver cancer cases worldwide [4] , surgical
resection is the treatment of choice and is considered to be po-
tentially curative [5] . Selected patients with metastatic (secondary)
liver tumors—which develop in 50% of the cases of colorectal
cancer—present up to 58% increased 5-year survival rates after
liver resection [6] .
In contemporary liver surgery, pre-operative planning be-
comes increasingly important. New techniques like parenchymal-
sparing [7] can use pre-operative planning to help surgeons
optimizing the resection path, potentially increasing the remnant
liver. Volume expanding techniques (like associating liver partition
and portal split (ALPPS) [8] , and portal vein embolization) can also
http://dx.doi.org/10.1016/j.cmpb.2017.03.019
0169-2607/© 2017 The Authors. Published by Elsevier Ireland Ltd. This is an open access article under the CC BY-NC-ND license.
( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
136 R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145
make use of pre-operative planning to derive the volumetry of
the resection. This can help ensuring that remnant liver is large
enough and with sufficient function to prevent post-operative liver
failure.
For nearly two decades, surgeons and other clinicians have
employed computer-assisted surgical systems to support the
decision-making process for planning and guiding surgical in-
terventions. In the case of liver resections, these systems have
recently been evaluated in the clinical practice and have shown
improvements not only in tumor localization and precision of
surgery planning [9–11] , but also an improved orientation and
confidence of the surgeon during the operation [12] .
Liver resection planning systems are based on the definition of
virtual resections [13] . Virtual resections help clinicians to visualize
the resection (surgical cutting path), affected vessels and resection
margins (safety distance kept between the tumors and the resec-
tion path). In addition, virtual resections allow the computation of
the estimated resected volume.
Simplicity of use and flexibility to specify the virtual resections
are key features of surgery planning systems. Simplicity and
flexibility are often considered as diverging objectives—simple
interactions usually impose constraints on the freedom to describe
virtual resections. The two most common strategies proposed
for specification of virtual resections are: drawing traces in 2D
individual slices [14] ( DS ) and definition of and virtual resections
defined using a deformable cutting plane [15] ( CP ).
1.1. Contribution
In this work, we present a new method for planning liver
resection procedures. The novelty of our method is twofold. On
one hand, our method is based on the use of Bézier surfaces,
which can be deformed in real-time solely by a set of control
points. On the other hand, we propose the use of distance maps to
project the safety margins in real-time onto the resection surface,
thus allowing the user to modify the resection proposal until the
safety requirements are met.
In addition, an implementation of the method based on the
open-source software 3D Slicer [16] is presented. This implementa-
tion includes interaction mechanisms which not only avoid the use
of manual drawing of lines (both in the 3D model as in CP, and
in the 2D slices as in DS), but also presents a flexible yet simple
way to define virtual resections regardless of their type (e.g.,
hemihepatectomy, parenchymal-sparing). Details on visualization
aspects, projection of resection surfaces onto individual 2D slices,
as well as resected volume computation based on our method, are
also detailed.
2. Theoretical background
In this section, we briefly describe the foundations of Bézier
tensor product surfaces and some of their most important proper-
ties. A deeper description can be found in [17–19] . For simplicity
and clarity reasons, in this work we focus on parametric non-
rational Bézier surfaces, however, the methods described in this
work can be easily adapted to other Bézier formulations.
Formally, a parametric non-rational Bézier surface S ∈ R
3 of
degree ( m, n ) can be defined as:
S (u, v ) =
m
i =0
n
j=0
C
i,j
B
i,m
(u ) B
j,n
(v ) , (1)
with u, v ∈ [0, 1]. C
i,j
∈ R
3 are the control points characterizing
the shape of the surface, and the i -th and j -th bases, B
i, m
and
B
j, n
with degrees m and n respectively, are Bernstein polynomials
given by:
B
i,m
(t) =
m
i
(1 −t )
(m −i )
t
i
. (2)
Lemma 1. Let S be a parametric bi-linear Bézier surface of degree ( m,
n ) as described in Eq. (1) . Such surface, has the following properties:
(a) Surface contained in the convex hull CH :
S (u, v ) ∈ CH (C
0 , 0
, . . . , C
m,n
) ∀ (u, v ) . (3)
(b) Affine transformation invariance:
T (S ) =
m
i =0
m
j=0
T (C
i,j
) B
i,p
(u ) B
j,q
(v ) , (4)
where T is an affine transformation (i.e., rotation, reflection,
translation or scaling).
(c) Polyhedral approximation: under triangulation, the net of
control points forms a planar polyhedral approximation of the
surface.
A proof for these properties follows easily from the proof of
the analogous properties for Bézier curves [17] . In the remaining
of the document, we will refer to these properties to justify design
aspects and properties of the proposed method.
3. Materials and methods
3.1. Overview of the method
Regardless of the type of model supporting the definition of vir-
tual resections (i.e., voxel-based or 3D models), models employed
in planning of liver resection ultimately rely on patient-specific
segmented models typically obtained from computed tomography
( CT ).
The approach presented in this work is entirely supported by
patient-specific 3D models. In order to construct these models,
first, a medical image is obtained from CT. Medical images are rep-
resented as scalar fields F : R
3
→ R where the points { p
i
∈ R
3
}
N
i =1
present intensity values F (p
i
) = v . Through segmentation, different
tissues (i.e., vessels, parenchyma
1 and tumors) are separated in
a new scalar field S : R
3
→ { l
1
, . . . , l
k
} with k classes (tissues). Fi-
nally, through isosurface extraction methods, like marching cubes
[20,21] , 3D models of the labeled tissues are obtained.
In computer graphics, surface descriptions such as 3D models
and Bézier surfaces are commonly represented as triangle meshes.
In the remaining of this work, triangular meshes are denoted
as sets M = { V, T } with vertices V = { 0 , 1 , 2 , . . . } and its asso-
ciated positions p
i
∈ R
3
, edges E = { (i, j) | i, j ∈ V } , and triangles
T = { (i, j, k ) | (i, j) , (j, k ) , (k, i ) ∈ E} .
As in CP approaches, the work-flow of our approach ( Fig. 1 ),
consists of two steps: initialization (planar approximation) and
modification of the resection. The differences with other CP
approaches like [15] lie in the underlying representation, and
deformation methods. This not only leads to new properties of the
resection surfaces, but also to different user interaction schemes.
In our method, first, the 3D mesh models M
p (parenchyma) and
M
t (tumor) generated previously, are used to define a planar
contour around the parenchyma. Unlike in CP, user interaction
required to specify the contour is not based on manual drawing,
but on a slicing movable plane ( Section 3.2 ). This contour leads to
the generation of a planar resection surface. In a second step, the
user can deform the planar surface by means of a grid of control
points ( Section 3.3 ).
3.2. Initialization of the resection
The goal of this process is to obtain a first approximation (pla-
nar Bézier) of the resection surface which will be used as a starting
1 In this work, the term parenchyma is used to refer to the part of the liver which
is neither tumor tissue nor blood vessels.
R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145 137
Fig. 1. Flow chart of the proposed method.
point for subsequent modifications. In order to obtain this approx-
imation, the user is first provided with the 3D representation of
the liver and tumor, as illustrated in Fig. 2 a and 2 b. Toge th er with
these anatomic representations, a line connecting the centroid of
the tumor c with an arbitrarily placed 3D end-point p
e (in Fig. 2 a,
b this corresponds to p
end 0
and p
end 1
in different interaction
times) is displayed. This line is associated to an invisible plane P
(in Fig. 2 a this corresponds to P
0
and P
1
at different interaction
times) passing through the middle point of the line connecting c
and a end-point p
e which satisfies the point-normal form:
(p
e
−c )
n
·x −c + p
e
2 = 0 . (5)
The plane P is then used to slice the parenchyma model
M
p
, thus providing a contour representation V
s (ring around the
parenchyma in Fig. 2 b). User interaction takes place by moving the
3D end-point p
e
. The effect of moving this end-point is the mod-
ification of the slicing plane, which effectively creates a contour
(around the parenchyma) moving in real-time. This initialization
process is formally described in Algorithm 1 .
The resulting contour V
s is then used to compute resection
approximation in terms of a planar surface ( Fig. 2 c). Similarly to
[15] , the origin, extent and orientation of this plane is obtained
Algorithm 1 Computation of resection contour.
Precondition: User-defined end point p
e inside parenchyma mesh
M
p
.
1: function Contour ( M
p
, M
t
, p
e )
2: c ← Centroid(M
t
) Centroid of tumor
3:
n ← p
e
−c Normal vector
4: P ← P lane (
p
e
+ c
2
,
n ) Slicing plane P ⊥
n
5: V
s ← Slice (M
p
, P ) Point-based contour
6: return V
s
7: end function
by means of principal component analysis ( PCA ). The orientation
of the initial resection is given by the two eigenvectors
E
1 and
E
2
presenting the larger eigenvalues e
1
and e
2
. These eigenvalues, are
then used to compute the size of the initial resection, in our case:
l
1
= 4
√
e
1
l
2
= 4
√
e
2
. (6)
The election of the lengths l
i
is based on the consideration
of √
e
i
as estimators of the standard deviations of the contour V
s
along the eigenvectors
E
i
. Assuming uniform distribution of the
contour along these eigenvectors, l
i
exceeds the length of the con-
tour, and therefore, the initial plane also exceeds the boundaries
of the parenchyma.
The origin of the plane (center) is computed with respect to
the centroid of the contour V
s
. First the centroid is computed
using all the points that make up the contour:
c =
1
N
N
i =1
p
i , p
i
∈ V
s
. (7)
Then the origin is computed as the translation of the centroid
in the direction ( −
E
1
, −
E
2
) by half the extent of the plane on each
direction, this is:
o = c −l
1
2
E
1
−l
2
2
E
2
. (8)
Once the geometry of the initial resection is computed, we
map a 2D grid of m ×n equally spaced points. This grid of points
will serve as a base to build a deformable Bézier surface—from
Lemma 1.a it follows that, if all control points lie in a plane, the
associated Bézier surface also lies on the same plane. Formally,
this process is described in Algorithm 2 .
3.3. Deformation of Bézier surfaces
Deformation of a Bézier tensor-product surface is performed
through the interactive manipulation of the coordinates of the
control points (distributed in a connected grid). The control points
do not normally lie on the surface (except for the corners, which
always lie in the surface). The fact that the net of control points is
an approximation of the surface (Lemma 1. c ) makes that the defor-
mations of the surface occur in coherence with the manipulation
of the control points.
The number of control points is an important design consid-
eration. On one hand, increasing the number of control points
increases the number of interactions as the user may have to
modify more control points. On the other hand, and as derived
from Eqs. (1) and (2) , the number of control points determines the
degree of the surface, and hence, its representation flexibility. In
this work, surfaces defined by 4 ×4 control points are employed.
The surface resolution has an impact on the performance of
computing Bézier surfaces. For our method, this is a very im-
portant consideration since the computation of Bézier surfaces
138 R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145
Fig. 2. Initialization of the resection: (a) 2D illustration of the initialization process where the initial point p
end
0
(which produces the initial plane P
0
), is moved to p
end
1
,
thus producing the initial plane P
1
; (b) 3D representation initial resection resulting at p
end
1
; (c) Geometry of the initial resection G based on PCA of the contour V
s
.
Algorithm 2 Compute initial resection.
Precondition: Cross-section contour represented as the set of N
3D points V
s
= { p
i
}
N
i =1
. m and n determine the dimensions of
the output control polygon.
1: function InitialResection ( V
s
, m, n )
2: c
c ← Centroid(V
s
) Centroid of contour
3: [ e
1
, e
2
,
E
1
,
E
2
] ← P CA (V
s
)
4: l
1 ← 4
√
e
1
Width of resection plane
5: l
2 ← 4
√
e
2
Height of resection plane
6: o ← c
c
−l
1
2
E
1
−l
2
2
E
2
Plane origin
7: for i ← 1 to m do
8: for j ← 1 to n do
9: C
i,j ← o +
il
1
m
E
1
+
jl
2
n
E
2
10: end for
11: end for
12: C ← { C
i,j
}
m,n
i,j=1
Control polygon
13: return C
14: end function
is followed by other processing stages (
Section 3.4 ) and all the
computations involved must be performed in real-time. In this
work, surfaces of resolution 40 ×40 points are used.
Updating the resection surface when a control point changes its
position requires re-computing the whole extent of the surface—
the reader should notice that this is an inherent property of
the formulation (
Eq. (1 ). Algorithm 3 describes this process for
surfaces of variable number of control points and resolution.
Algorithm 3 Update Bézier Surface.
Precondition: C being the grid of control points of size m ×n , and
r
u
×r
v representing the resolution of the surface.
1: function UpdateBezier ( m, n, r
u
, r
v
, C )
2: for i ← 1 to r
u do
3: u ← i/ (r
u
−1)
4: for j ← 1 to r
v do
5: v ← j/ (r
v
−1)
6: for k ← 1 to m do
7: B
u ←
m
k
u
k
(1 −u )
m −k
8: for l ← 1 to n do
9: B
v ←
n
l
v
l
(1 −v )
n −l
10: S
i , j ← C
i,j
B
u
B
v
11: end for
12: end for
13: end for
14: end for
15: return S
16: end function
R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145 139
Fig. 3. Visualization of the resection path: (a) distance map derived from the tumor model M
t
and the resection surface M
r
. (b) visualization of the resection surface given
by a 3D Bézier surface and thresholding of the distance map using the resection margin; the violation of resection margin is highlighted in yellow (blue contour around); (c)
visualization of the final resection surface where the control points and the resection exceeding
the parenchyma are hidden; (d,e,f) projection of the resection surface into
individual 2D slices with axial, coronal, sagittal orientation respectively.
3.4. 3D Visualization and projection in 2D slices
Togethe r with the visualization of the 3D surface defining the
virtual resection, our approach includes the visualization of the
resection margin—which refers to the safety distance that should
be kept between the resection surface and the tumors. The resec-
tion margin is updated when the resection is modified. In order
to compute the resection margin, we employ the point-to-surface
distance δ:
δ(p ) = min
∀ q
i
∈ V
t
p −q
i
, (9)
where V
t is the set of points of the tumor model M
t and p is a
point belonging to the resection surface model M
r
; ||.|| refers to
the euclidean norm. The point-to-surface distance is computed for
all points of the resection surface which effectively generates a
distance map projected onto the resection surface ( Fig. 3 b). Using
these distance maps, it is possible to determine the validity of the
resection surface in terms of resection margin; for instance, if the
margin set by clinicians is under 10 mm , then the resection would
be valid only if all the points in the surface are further than 10 mm
from the tumor.
For visualization purposes, we avoid the use of a color-map
projected onto the surface. Visually, the color-map contains more
information than clinicians need and all this information can be
distracting. Instead, we threshold the distance map according to
the resection margin. The areas violating the resection margin are
then highlighted in yellow (with blue contour) while the rest of
the surface remains in gray ( Fig. 3 b, 3 c). The part of the Bézier
surface exceeding the liver surface can be hidden as well as the
net of control points ( Fig. 3 c). This facilitates the visualization of
the resection by avoiding occlusions and simplifying the scene.
The projection of the surface onto individual 2D slices
( Fig. 3 d, 3 e, 3 f) is obtained by the intersection of axial, coro-
nal and sagittal planes with the 3D Bézier surface.
140 R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145
3.5. Computation of resected volume
Computation of resection volumetry is a key functionality pro-
vided by existing software solutions for planning liver resections.
Our approach to compute the resected volume consists of three
steps. First, a high-resolution Bézier surface ( r
u
= 300 , r
v
= 300 ) is
generated. Secondly, all the points of this high-resolution surface
are mapped into a segmented image M : R
3
→ { l
b
= 0 , l
p
= 1 , l
t
=
2 , l
r
= 3 } (same dimensions and spacing as the original image
taken from the patient for diagnosis), where the background ( l
b
),
liver parenchyma ( l
p
), target tumors ( l
t
) and resection surface ( l
r
)
are separated by different label values. The mapping of the high-
resolution surface is performed on a basis of a 3 ×3 ×3 voxels
per surface point, which effect is the extrusion (thickening) of the
mapped resection surface. This, together with the high-resolution
construction of the surface, guarantees both continuity of the
mapped surface and a clear boundary between the resected and
the remnant volumes of the liver. Finally, a connected threshold re-
gion growing is applied (low threshold l = l
p and upper threshold
u = l
r
) with a seed point arbitrarily chosen from a target tumor.
In order to compute volumes using this process, the resection
path must enter and leave the parenchyma completely. This not
only makes sense under the point of view of the application,
but also guarantees a separation between the resected and the
remnant volume.
3.6. User interaction
In order to keep the simplicity of use and flexibility of resection
representation, we introduce two new interaction mechanisms:
global translation of the resection surface and modification of
control points in groups. An example of the possible sequence
of interactions using these mechanisms, and leading to a valid
resection plan is illustrated in Fig. 4 .
Global translation of Bézier surfaces defined by a grid of 4 ×
4 control points requires the modification of all the 16 control
points—which implies a considerable number of user interactions.
To avoid this, we set the control polygon connecting the control
points as an interactive frame that can be moved through drag-
and-drop interactions. Moving the frame produces a translation
transformation on all control points which effectively produces the
translation of the surface (Lemma 1.b).
Resections, regardless of their type, can be defined by a virtual
resection resulting from a resection surface with pseudo-parabolic
shape. For a resection surface defined by a grid of 4 ×4 control
points, this implies the movement of either the 4 inner points of
the grid or the 12 remaining (outer) points. In our implementa-
tion, simple mouse right-click on any of the 4 inner points will
produce translation of all these points together. The same applies
for the 12 outer points. This type of interaction allows the simple
construction of pseudo-parabolic shapes, which can then be refined
by individual modification of the control points. For illustration
purposes we refer to Fig. 2 c, where the 4 inner points are shown
in light-gray, while the 12 outer are shown in black.
4. Evaluation methodology
In order to validate and evaluate the proposed method we
perform a user study which includes a comparison with our own
implementation of CP and DS in 3D Slicer [16] . The implemen-
tation details of CP and DS are described in the following and
summarized in Table 1 .
The implementation of the CP approach is based on [13] and
[15] . This approach uses a surface with a variable mesh resolution
30 ×n square quads, where 30 is the number of quads in the
short axis and n is the number of square quads needed to fill the
extent of the plane ( Eq. (6) ) in the long axis.
Implementation of DS is based on the general principles es-
tablished in [13] combined with design aspects in [14] . In this
implementation, the user can draw and overwrite complete traces
individually over the set of 2D slices. Navigation between traces
was implemented so the user could easily find individual traces
and their corresponding slices. Parametric linear interpolation was
applied to individual traces to obtain a regularly spaced sampled
traces (20 points per trace). The final surface was computed by
means of parametric quadratic interpolation between the traces,
which requires at least 3 traces. Modification of the surface was
allowed on the basis of traces, this is, redrawing of one or more
traces and fast re-computation of the interpolated surface.
Study design and quantitative analysis are performed according
to [22] , which provides a comprehensive guide for the design and
data analysis of experiments similar to the one presented in this
work. In order to compare the different methods we establish
the criteria and their corresponding objective evaluation metrics
described in the following.
Preservation of resection margin. This criteria is concerned with
how accurately the resection margin is preserved. This is measured
by means of the minimum point-to-surface distance between the
tumor and the resection surface derived from Eq. (9) .
Inter-subject reproducibility of results. Surgery planning tools are
essentially geometric modeling methods. This criteria considers
how accurately different users can reach the same resection plan.
In order to measure similarity of resections between users, we
measure the resection volume difference (in %) with respect to the
reference resection volume. Volumetry of resection is computed
using the procedure in Section 3.6 .
Planning time. Integration in the clinical work-flow is of
paramount importance for new computational methods. Therefore
the planning time should improve, or at least be similar, with
respect to state-of-the-art methods.
Smoothness of results. Resection smoothness is a desirable feature.
Smoothness not only helps the interpretation and visualization of
3D models, but also increases the feasibility of performing the
planned resection during surgery (e.g., “curvy” surfaces are more
difficult to perform surgically and sometimes even impossible). As
indicator of surface smoothness we use the mean curvature [23] :
H =
K
1
+ K
2
2
(10)
where K
1 and K
2 are the principal curvatures.
4.1. Study design
Our approach to evaluate and compare the three different plan-
ning methodologies (in the following: Bézier, CP and DS) is the
design of a study where the three planning techniques are used
by the same expert users in different clinical cases. The group of
participants consists of 5 gastro-intestinal surgeons ({5,8,10,11,31}
years of clinical experience).
The evaluation was conducted using a data-set consisting of
5 patient-specific models (obtained from the Oslo-CoMet study
[24] ). This data-set includes CT volumes, segmentation and 3D
modeling of vessels, parenchyma (liver surface) and tumors. From
this data-set, each surgeon generated 8 virtual resections (all
atypical resections). Some of these resections target either single
or multiple tumors. For comparison purposes and in order to
avoid differences in clinical criteria—which could potentially lead
R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145 141
Fig. 4. Instance of liver resection planning using the proposed method including the proposed user interaction techniques. The sequence of interactions (a) to (d) illustrates
the process of obtaining the initial approximation of the resection surface. (e) and (f) show the modification of the group of outer control points. In a similar way, (g) and
(h) show the modification of the inner group of control points. In (i) a rotation of the view is performed. Later, in (j) and (k) the surface is translated (globally). Finally in (l)
the visualization of the final resection is presented. The reader should notice, that
from (d) to (j) the resection presents a yellow (blue contour around) area indicating the
violation of the resection margin (arbitrarily set at 20 mm ). In (k) the resection margin is preserved (no yellow area in the surface) indicating validity of resection in terms
of safety margin.
Tabl e 1
Implementation aspects for DS, CP and Bézier.
Aspect DS CP Bézier
Underlyi ng representation Bi-quadratic polynomial Discrete grid Bi-cubic polynomial
Surface resolutio n 20 ×20 30 ×n 40 ×40
Visualization 2D Slices / 3D Models 3D Models 3D Models
Resection margin visualization ✗ ✗ √
1. Drawing in slices 1. Traces on parenchyma 1. Slicing plane on parenchyma
Interaction (3–5) traces 2.
Local deformation 2. Bézier deformation
to different resection plans for the same tumors—a set of resection
plans was employed as reference. The reference set (median re-
sected volume 208.98 ml ) was generated by the most experienced
surgeon in an earlier pilot study (3 months earlier). All the par-
ticipants were asked to perform the same resection plan as in the
reference. To do this, the participants were allowed to explore ( <
5 minutes) the reference resection plan beforehand. The experts’
comments were recorded after each resection plan (see Section 5 ).
4.2. Procedure
Before starting the experiments (during the same session), the
surgeons were shown the graphical user interface and the process
to obtain resections with the different methods (CP, DS, Bézier).
Surgeons were allowed to use the system to perform a sample
resection as training ( < 1 h ).
The experiment consisted of planning the different cases using
CP, DS and Bézier for all the cases. The cases were ordered for all
the participants, however, the order of the method is a-priori ran-
domized to reduce the impact of confounding factors (i.e., training
or sequence effects). The participants were allowed get help by a
technician on any technical aspect related to the use of the inter-
face whenever needed (due to the short training session). A resec-
tion plan was considered finalized when the participant indicated
(either by obtaining the desired resection or believing the plan
cannot be further improved) and the verification by a technician
142 R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145
Tabl e 2
Descriptive analysis derived from the quantitative evaluation.
Method Time (s) Deviation from Deviation from Median Mean
Margin (mm) Volume (%) Curv. (1/m)
Bézier min 53.00 −3.25 −10.15 0.00
25% 126.50 0.13 −1.8 3 0.01
50% 174. 0 0 0.42 −0.40 0.01
75% 244.00 1.47 1.17 0.02
max 801.00 6.05 3.77 0.02
CP min 42.00 −9.84 −8.97 0.01
25% 129.75 −7.6 5 −4.93 0.03
50% 180 .5 0 −4.75 −2.40 0.03
75% 250.50 −1.0 4 −0.56 0.04
max 748 .0 0 5.46 3.21 0.11
DS min 44.00 −7.9 8 −7. 0 7 0.01
25% 116.75 −3.51 −3.31 0.01
50% 179.00 −1.4 9 −1. 3 9 0.02
75% 345.00 0.12 0.32 0.02
max 757.00 8.53 6.69 0.04
Tabl e 3
Comments from the experts (S1-S5 indicates the participant who provided/expressed the comment).
General comments
[ GC1 ] Undo functionality wou ld be useful (S1, S4).
[ GC2 ] Ability to set transparency of surfaces would be useful (S1, S5)
[ GC3 ] Pre-defined views aligned to surgical way of looking at the liver would be useful (S1).
[ GC4 ] Rotatio n of resection can be useful in some cases, specially in CP and Bézier (S1).
Comments on DS
[ CDS1 ] Poses the steepest learning curve / is the least intuitive method (S1, S2, S3, S5).
[ CDS2 ] Can be difficult to specify resections with high curvature (S1, S5).
[ CDS3 ] Can be adequate
for quasi-planar resections (S1).
[ CDS4 ] Could not reach exactly the desired resection in some cases (S2, S3).
[ CDS5 ] Some resections could be better defined by combination of traces in different views (axial, coronal, sagittal) (S3).
Comments on CP
[ CCP1 ] Resections derived from drawing traces in parenchyma sometimes produce unexpected results in terms of desired curvature (S1, S3).
[ CCP2 ] Modification of resections in CP present more degree of freedom (complexity) than needed. More simplicity would be a benefit (S1,S3).
[ CCP3 ] “Curvy/Bumpy/Wavy” resection plans derived from CP can be difficult to perform surgically (S1, S3, S4, S5).
[ CCP4 ] Local deformations can be useful in particular cases like peripheral metastases (S3 ,S5).
[ CCP5 ] Deformation can be difficult when the initial plane is nearly perpendicular to the screen plane (S4).
Comments on Bézier
[ CB1 ] Visualization of resection margin is an
advantage of this method (S1).
[ CB3 ] In addition to visualization of the margin on the surface, a global warning of resection violation could be useful. Sometimes violation or resection is occluded (S2).
[ CB4 ] Deformation of resection in Bézier does not look obvious (S4).
[ CB5 ] Bézier is the most intuitive method (S5).
that the resection was complete (surface exceeds the parenchyma
in all directions). Time to complete the resection plan (excluding
technician assistance in questions related to user interaction and
verification of resection) was recorded, together with the geometry
of the 3D surface models derived from the resection plan.
5. Results
In this section, we present results derived from the use CP, DS
and Bézier by clinicians at Oslo University Hospital, as described
in Section 4 . A descriptive analysis of quantitative results is shown
in Table 2 . Subjective feedback of the participants—which will be
use as a base for discussion in Section 6 —is recorded in Table 3 .
In the same line as [22] , we conduct statistical tests for nor-
mality of data ( Shapiro-Wilk ), difference between methods ( ANOVA,
Friedman ) and pairwise differences between methods ( Wilcoxon,
paired Student’s t-test ) with Bonferroni correction [25] . Due to the
Bonferroni correction, all effects derived from pairwise compar-
isons are reported at a 0.0167 (i.e., one third of the p-value 0.05)
level of significance. Statistical analysis was carried out with the R
statistics software package.
Surgery Planning Time. The surgery planning time was recorded
for every resection performed by the participants ( Fig. 5 a). The
Friedman test reveals no significant difference between methods
in terms of time, with X
2
(2) = 1 . 849 , p = 0 . 39 > 0 . 05 , where the
median completion times were 174 s for Bézier, 179 s for DS and
180 s for CP.
Deviation from Resection Margin. Pairwise comparison between
methods in terms of deviation from resection margin ( Fig. 5 b)
through Wilcoxon signed-rank test yields:
• V = 16 4 , p = 0 . 0 0 06 < 0 . 167 between CP and DS,
• V = 32 , p = 5 . 03 e −09 < 0 . 167 between CP and Bézier,
• V = 75 , p = 8 . 69 e −07 < 0 . 167 between DS and Bézier.
These results show significant differences regarding the devia-
tion of resection margin between methods. The median deviations
from resection margin were 0.42 mm for Bézier, −1 . 49 mm for
CP and −4 . 74 mm for DS. Bézier presents the least deviation
from resection margin. The number of resections violating the
resection margin ( de v < −0 . 01 ) is v
cp
= 31 for CP, v
ds
= 28 for DS
and v
bez
= 1 for Bézier.
Deviation from Reference Volume. Deviation from the reference
volume ( Fig. 5 c was computed as the difference (in %) between
the resected volumes obtained by the participants and their cor-
R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145 143
(a) (b)
(c) (d)
(e)
Fig. 5. Box plots of results derived from the quantitative evaluation: (a) time, (b) deviation from margin which includes the number of resections violating the margin
( de
v < −0 . 01 mm marked in red) and the number of resections preserving the margin ( de v ≥−0 . 01 mm ), (c) deviation from volume and (d) median mean curvature.
responding resected volumes in the reference data-set. Wilcoxon
signed-rank test yields:
• V = 308 , p = 0 . 174 > 0 . 0167 between CP and DS,
• V = 67 , p = 3 . 875 e −07 < 0 . 0167 between CP and Bézier,
• V = 190 , p = 0 . 0025 < 0 . 0167 between DS and Bézier.
Pairwise tests show significant differences between Bézier and
the other two methods. The median volume deviation is −0 . 4% for
Bézier, −1 . 39% for DS and −2 . 40% for CP. Bézier shows the least
deviation with respect to the reference resected volume.
Resection Curvature . Pairwise comparison of resection curvature
between methods ( Fig. 5 d) through Wilcoxon signed-rank test
yields:
• V = 638 , p = 0 . 001 < 0 . 167 between CP and DS,
• V = 654 , p = 0 . 007 < 0 . 167 between CP and Bézier,
• V = 475 , p = 0 . 39 > 0 . 167 between DS and Bézier.
These results show that the difference in curvature for CP is
different from both DS and Bézier. No significant difference was
found between Bézier and DS. Median mean curvature is 0 . 01 m
−1
144 R. Palomar et al. / Computer Methods and Programs in Biomedicine 144 (2017) 135–145
for both Bézier and DS, and 0 . 03 m
−1 for CP. Both Bézier and DS
produce resections with lower curvature than CP.
6. Discussion
Computer-assisted systems for planning and guiding liver
resections have existed for nearly two decades now. Although
some of these systems have made their way into clinical reality,
none of them seem to be established as a gold-standard solution
replacing previous clinical practices. To a great extent, this is
due to the difficulties of generating the patient-specific models
employed by these systems—segmentation, for instance, is still
considered a research problem and a bottleneck for the generation
of patient-specific models. No consensus exists about planning
liver resections—DS and CP approaches currently coexist in the
surgery planning market. New methods for planning liver resec-
tions should, at least, highlight their differences, as well as their
advantages/disadvantages with respect to the existing techniques.
Therefore, in this section, a comparison of our approach with DS
and CP strategies is discussed on the basis of the results presented
in Section 5 .
According to our results, the required time (median) for com-
pletion of a resection plan using the proposed method ( t = 174 s )
is, as low as for the state-of-the-art methods CP ( t = 180 s ) and
DS ( t = 179 s ). This indicates that the adoption of the proposed
method in the clinical routine would not imply any significant
change in the clinical work-flow.
Bézier shows the least deviation from the reference plan in
terms of volume ( −0 . 40% ) compared to DS ( −1 . 39% ) and CP
( −2 . 40% ). In our study, small deviations from resected plans are
expected from all the methods since the median resected volume
for the reference data-sets is relatively small (208.98 ml ); larger
deviations in volume are expected for larger resections (e.g.,
hemihepatectomies).
The comments from the participants show wide consensus on
considering DS the most difficult method to use ( CDS1 ), partic-
ularly for resections exhibiting higher curvature ( CDS2, CDS3 ).
Furthermore, in some cases, DS did not provide satisfactory re-
sults ( CDS4 ); to mitigate these problems, the ability to combine
traces in different views is suggested ( CDS5 ). No consensus was
found on whether CP or Bézier is the most intuitive ( CCP2, CB4,
CB5 ). Considering task completion time as indicator of usability,
and despite the fact that no statistical significance between the
methods was found, the higher variability of DS with respect to
CP and Bézier seems to support that DS is less intuitive than CP
and Bézier, which are comparable in this regard.
As discussed in [15] , continuous visualization of distance from
the resection surface to the tumor is a desirable feature since
it is associated with the preservation of resection margin. Our
results show the visualization technique proposed in Section 3.4 is
an effective mechanism ( CB1 ) to avoid violations of resection
margin ( v
bez
= 1 for Bézier compared to v
cp
= 31 and v
ds
= 28 );
median deviation from resection margin is also lower using Bézier
(0.42 mm ) as compared to using CP ( −4 . 75 mm ) or DS ( −1 . 49 mm ).
Despite the good results in terms of preservation of resection mar-
gin of our proposed method, this was not sufficient to avoid all
the violations of resection margin; occlusions of resection margin
visualization (e.g., by vessels) might lead to unnoticed resection
violations. To avoid this, and in line with the participants’ com-
ments ( CB3 ), an indicator of margin violation external to the
visualization of the surface should be provided (e.g., bi-color state
widget in the GUI or a warning icon).
The shape of the virtual resection is an important aspect since
it relates to the feasibility of performing the resection surgically;
resections presenting wavy resection trajectories might be not
realizable during surgery as they are specified in the virtual plan.
In this sense, resections presenting low curvature are associated
with higher surgical feasibility than resections with high curvature.
According to our results, Bézier and DS provide resections which
are easier to perform surgically (lower curvature) compared to CP.
In this line, and according to the participants’ comments ( CCP3 ),
using CP might lead to resections that are difficult to perform
surgically.
Some of the techniques described in this work can be employed
to improve CP and DS; visualization of resection margin ( CB1 ),
for instance, was already discussed in [15] as a possible improve-
ment. Some other improvements suggested in our experiments
by the experts users, like the possibility of a semi-transparent
visualization of resection surface ( CG2 ) and the possibility to undo
actions ( CG1 ) were also found in [15] and should be considered for
further improvement of all the methods. Rotation of the resection,
particularly for CP and Bézier ( GC4 ), and predefined alignments
of the 3D view to surgical positions (e.g., anterior-posterior axis)
( GC3 ) could be implemented for methods other than Bézier.
Despite that our method showed good performance in terms
of planning time, reproducibility of results, preservation of margin
and curvature, expert users highlight scenarios where the use of
DS and CP could be still advantageous—such as for quasi-planar
resections ( CDS3 ) like hemihepatectomies or small local resections
like peripheral metastases ( CCP4 ). In this regard, and since all
the methods are similar in terms of time, software platforms for
planning liver resections could consider including all the methods
to provide clinicians with greater flexibility to represent resections.
Furthermore, CP and Bézier could even be combined so that local
deformations like in CP are preceded by global deformations like
in Bézier.
7. Conclusion
In this work we propose a novel method for planning liver re-
section procedures. This method is based on the use of deformable
Bézier surfaces for the specification of resection geometry and the
projection of risk areas (representing violations of safety margins)
onto the resection surface through distance maps. Our implemen-
tation of the method includes mechanisms to reduce the number
of interactions making the system easy-to-use by clinicians.
Our experimental results show that the planning time of our
method is as low as state-of-the-art methods, and therefore, can
be integrated in the clinical reality without modifications in the
clinical work-flow. Our method, not only shows superior preserva-
tion of resection margin methods, but also higher reproducibility
of surgery planning results than state-of-the-art . In addition, the
proposed method provides smooth virtual resections presenting
high feasibility to be performed surgically (e.g., absence of sharp
corners and wavy trajectories).
Acknowledgments
This work was supported by the Research Council of Norway
through the Hypercept project (number 221073 ), and The Inter-
vention Centre, Oslo University Hospital (Norway). Authors thank
Leonid Barkhatov, David Aghayan, Sheraz Yaqu b, Mushegh Sa-
hakyan, Kristoffer Lassen and Bård I. Røsok for fruitful discussions
and their contribution to the evaluation of the method proposed
in this work. Authors would like to thank also Xiaoran Lai for
reviewing the data analysis.
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