# Digital Eversion of a Hollow Structure: An Application in Virtual Colonography

Authors: Jun Zhao; Liji Cao; Tiange Zhuang; Ge Wang
Journal: International Journal of Biomedical Imaging
 (2008)

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## Abstract

A new methodology is presented for digital eversion of a hollow structure. The digital eversion is advantageous for better visualization of a larger portion of the inner surface with preservation of geometric relationship and without time-consuming navigation. Together with other techniques, digital eversion may help improve screening, diagnosis, surgical planning, and medical education. Two eversion algorithms are proposed and evaluated in numerical simulation to demonstrate the feasibility of the approach.


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## Body

## 1. Introduction

Computed tomography (CT), magnetic
resonance (MR), or other biomedical imaging modalities are frequently used for
inspecting of the inner surfaces of hollow structures such as colon and stomach. For those
purposes, many visualization techniques were developed, some of which have been
adopted in clinical applications [1–7]. However, with traditional visualization techniques only a very
small portion of inner surfaces can be presented in one view, because neither
an operator’s perspective nor a field of view with virtual endoscopy is unlimited. Hence,
the navigation is typically
necessary in a visualization session, which is time-consuming, tedious, and error prone particularly when features of interest are hidden behind folds [8, 9].To overcome the aforementioned difficulties in visualizing anatomical
cavities, Wang et al. proposed a method for unraveling the colon based on the
electrical field model [10–12].  Subsequently, Zhu et al. presented two algorithms for flattening
branched vessels [13]. Other unfolding or flattening methods
were also reported [14–19]. With these unwrapping techniques, the entire inner
surface of a hollow
structure can be mapped into one view. However, such unraveling operations
distort original shapes and relative configurations, compromising the intended
benefits significantly. Fletcher et al. developed a planar virtual pathology (PVP) method, in which a segment
of the colon is cut open to display the luminal surface [20]. The PVP method is
geometrically faithful, keeping the relationship between the colon and neighboring
organs.  Furthermore, this method may display
a larger portion of the luminal surface more efficiently than the conventional virtual endoscopy.  However, the PVP method only handles one opened segment each time, and needs a series of
segments with various cutting planes to examine the whole colon.In this
paper, we present a totally new idea for visualizing the inner surface of a
hollow structure digital eversion meaning to turn inside to outside via
computer [21, 22].  In other words, the
inner surface is converted to the outer surface with the inward normals reserved
outward. The primary advantages of digital eversion over conventional virtual
endoscopy include the direct visualization of a larger portion of the inner
surface and the close correlation to the important features of the global
anatomy, without the need for tedious and time-consuming navigation. Digital
eversion of a hollow structure is complementary or superior to the digital flattening
technique in terms of preserving the involved geometric center, center line,
shape, and spatial relationship. Together with other visualization techniques, digital
eversion may help improve diagnosis, surgical planning, and medical education.In Section2, we present
two algorithms for everting the colon as a major application. One algorithm performs eversion along electrical
force lines formed by simulated charges on the central colon path. The other algorithm
performs eversion along electrical force lines formed by simulated charges on
the reference surface of the colon. In Section 3, we describe experimental results
to show the feasibility of digital eversion of a colon. In Section 4, we discuss
relevant issues and conclude.


## 2. Materials and Methods

After a real
eversion of the colon, the mucosal surface becomes the outer surface while the
serosal surface becomes the inner surface. Similarly, with our digital eversion,
the mucosal surface and serosal surface of the colon are exchanged with respect
to the reference surface that is the average of the mucosal and serosal
surfaces. The reference surface is quite like a curved “mirror” in the eversion
process. Practically, the outer surface of the colon is extremely difficult to
be identified due to the low contrast between the colonic wall and pericolonic tissues.
Clinically, the detection of polyps and masses on the mucosal surface is of primary
interest. Hence, we do not need to be concerned about the serosal surface. Without
a knowledge on the serosal surface, the reference surface can still be generated
by applying 3D mathematical morphology operations to the colon lumen. Specifically,
the colon lumen can be first dilated by a spherical kernel with a relatively
large radius, and then eroded with a small spherical kernel if it is necessary.
The radius of the dilating kernel should be larger than the sizes of folds and
polyps. After these morphologic transforms, the surface wrapping the colon
lumen will be relatively smooth, ready to be used for eversion (see Figure1).Figure 1
A 2D-illustration of the reference surface generating by mathematical morphology.


### 2.1. Centerline-Based Eversion

This centerline-based
eversion algorithm is suitable for analyzing tubular structures. The center
line of the colon lumen can
be automatically extracted, for example, using the distance transform and
Dijkstra’s shortest path technique [6]. The most straightforward technique for eversion is based on a sequence of planar cross-sections that are orthogonal to
the centerline. However, these cross-sections may intersect, which can cause redundant or
little sampling of the colon surface (Figure 2(a)) [10, 19].Cross-sections
generated for a tubular structure with respect to its centerline. (a) Possible
intersections of planar cross-sections perpendicular to the centerline, which
may lead to redundant or little sampling of the surface, and (b) curved
cross-sections based on an electrical field, in which positive charges are
distributed along the centerline.
(a)(b)Wang et al. used a centerline-based method for unraveling the colon [10]. We adapted their idea for the
digital eversion. Digital eversion of the colon consists of three steps: center line determination,cross-section formation, and colon eversion. Let us assume that positive
charges are distributed along the curvilinear centerline, as shown in Figure 2(b)).
At each point on the centerline, a set of electrical force lines can be found which
are from that point and locally orthogonal to the centerline. The curved cross-sections
based on these electrical force lines cover the space around the inner surface
sufficiently and consistently, even for highly curved colon segments (Figure 2(b)) [10].To reduce
the computational cost, only charges near the current point on the centerline
are used to form the cross-section. We call the corresponding field a partial electrical
field. If only one charge is used, planar cross-sections are generated. The
partial electrical field method is a tradeoff between the performance and the efficiency [10]. An equiangular sampling scheme [10] can be used to form a cross-section. Let Ci and Fij be curved cross-sections and an electrical
force line, where i and j are the indices for the cross-section and the
line, respectively. Denote M, R,
and E as the mucosal surface, the reference surface, and the everted surface, respectively. Let mij stand for intersection of Fij and M, rij stand for intersection of Fij and R,
and eij stand for intersection of Fij and E.
To determine eij,
our criterion is that the length of a curved segment from mij to rij on Fij equals the length of a curved segment from eij to rij on Fij.
The eversion process along one electrical force line is illustrated in Figure 3. After
all eij are found, the everted surface can be formed and rendered.Figure 3
Everting
process with respect to an electrical force line, which is from a centerline
point in a curved cross-section. Along the force line, the length of a curved
segment from point (1) to point (3) equals the length of a curved segment from point (3) to point (2).The everted surfaceE is larger than the mucosal surface M in terms of area, since the distance from E to the centerline is greater than the
counterpart for M.
Thus, polyps would appear larger after eversion than the original. To minimize
the distortion of the polyps, all eij should be scaled down along the electrical
force lines with an appropriate factor, forming a surface denoted as S.
Let sij stand for the intersection of Fij and S.
We can so determine sij that ∑j=1K∑i=1N(dist(sij,mid(sij,mij))+dist(mij,mid(sij,mij))) is minimized, where dist(a,b) is the distance between a and b measured along the electrical force line, mid(c,d) the midpoint between c and d measured along the electrical force line, K and N are the number of the cross sections and the
number of the electrical force lines on a cross section, respectively. This particular correction is to scale down the everted surface E to S so that S and the mucosal surface M are matched as closely as possible. Note that
the scaling of eij to sij is approximately uniform.


### 2.2. Surface-Based Eversion

The
surface-based eversion is not only suitable for tubular structures but also for
nontubular hollow structures. In this setting, the centerline is no longer
required. Instead, the reference surface plays a more critical role. One simplest
scheme for surface-based eversion is to implement data reflection along each normal
on the reference surface. Calculating normals can be done in the three steps: (1)
extracting triangular elements based on the reference surface; (2) calculating
normals for all the triangular elements; (3) for each point on the reference
surface, estimating the normal from the normals of the neighboring triangular elements.
Note that normals derived from different points on the reference surface may
intersect, as shown in Figure4(a). Distortions may occur if the everting operation
is done along these normals.Normals
and electrical force lines from sampling points on the reference surface. (a)
Normals at different sample points may intersect, which cause distortions in
the eversion, and (b) electrical force lines from sampling points on the reference
surface, which are locally orthogonal to the surface but are never in conflict.
(a)(b)Another surface-based
eversion is based on electric force lines, similar to what has been discussed
in the preceding subsection. This time, positive charges are distributed over the
reference surface. At each sampling point on the surface, an electrical force
line, which is locally orthogonal to the surface, is generated. According to
electrical field theory, these lines never conflict (Figure4(b)).Similar to
the centerline-based eversion, the partial electrical field around each point on
the reference surface can be formed, that
is, charges are only assumed in a prespecified neighborhood of that point to
generate the electrical force line. The next steps have only minor change in
comparison with the centerline-based eversion counterparts.LetFj be an electrical force line, where j is the index. The definitions of M, R, E,
and S remain the same. Let mj stand for the intersection of Fj and M, rj for intersection of Fj and R,
and ej for intersection of Fj and E.
The criterion for determining ej is that the length of a curved segment from mj to rj equals the length of a curved segment from ej to rj on Fj.
The eversion process along one electrical force line is illustrated in Figure 5. Again,
all ej should be scaled down along the electrical
force lines with an appropriate factor to form the surface S.
Let sj stand for intersection of Fj and S.
We can so determine sj that ∑j=1P(dist(sj,mid(sj,mj))+dist(mj,mid(sj,mj))) is minimized, where P is the number of the electrical force lines.Figure 5
Everting
process for a point on the reference surface. Along the curved electrical force
line, the length from (1) to (3) equals the length from (3) to (2).


## 2.1. Centerline-Based Eversion

This centerline-based
eversion algorithm is suitable for analyzing tubular structures. The center
line of the colon lumen can
be automatically extracted, for example, using the distance transform and
Dijkstra’s shortest path technique [6]. The most straightforward technique for eversion is based on a sequence of planar cross-sections that are orthogonal to
the centerline. However, these cross-sections may intersect, which can cause redundant or
little sampling of the colon surface (Figure 2(a)) [10, 19].Cross-sections
generated for a tubular structure with respect to its centerline. (a) Possible
intersections of planar cross-sections perpendicular to the centerline, which
may lead to redundant or little sampling of the surface, and (b) curved
cross-sections based on an electrical field, in which positive charges are
distributed along the centerline.
(a)(b)Wang et al. used a centerline-based method for unraveling the colon [10]. We adapted their idea for the
digital eversion. Digital eversion of the colon consists of three steps: center line determination,cross-section formation, and colon eversion. Let us assume that positive
charges are distributed along the curvilinear centerline, as shown in Figure 2(b)).
At each point on the centerline, a set of electrical force lines can be found which
are from that point and locally orthogonal to the centerline. The curved cross-sections
based on these electrical force lines cover the space around the inner surface
sufficiently and consistently, even for highly curved colon segments (Figure 2(b)) [10].To reduce
the computational cost, only charges near the current point on the centerline
are used to form the cross-section. We call the corresponding field a partial electrical
field. If only one charge is used, planar cross-sections are generated. The
partial electrical field method is a tradeoff between the performance and the efficiency [10]. An equiangular sampling scheme [10] can be used to form a cross-section. Let Ci and Fij be curved cross-sections and an electrical
force line, where i and j are the indices for the cross-section and the
line, respectively. Denote M, R,
and E as the mucosal surface, the reference surface, and the everted surface, respectively. Let mij stand for intersection of Fij and M, rij stand for intersection of Fij and R,
and eij stand for intersection of Fij and E.
To determine eij,
our criterion is that the length of a curved segment from mij to rij on Fij equals the length of a curved segment from eij to rij on Fij.
The eversion process along one electrical force line is illustrated in Figure 3. After
all eij are found, the everted surface can be formed and rendered.Figure 3
Everting
process with respect to an electrical force line, which is from a centerline
point in a curved cross-section. Along the force line, the length of a curved
segment from point (1) to point (3) equals the length of a curved segment from point (3) to point (2).The everted surfaceE is larger than the mucosal surface M in terms of area, since the distance from E to the centerline is greater than the
counterpart for M.
Thus, polyps would appear larger after eversion than the original. To minimize
the distortion of the polyps, all eij should be scaled down along the electrical
force lines with an appropriate factor, forming a surface denoted as S.
Let sij stand for the intersection of Fij and S.
We can so determine sij that ∑j=1K∑i=1N(dist(sij,mid(sij,mij))+dist(mij,mid(sij,mij))) is minimized, where dist(a,b) is the distance between a and b measured along the electrical force line, mid(c,d) the midpoint between c and d measured along the electrical force line, K and N are the number of the cross sections and the
number of the electrical force lines on a cross section, respectively. This particular correction is to scale down the everted surface E to S so that S and the mucosal surface M are matched as closely as possible. Note that
the scaling of eij to sij is approximately uniform.


## 2.2. Surface-Based Eversion

The
surface-based eversion is not only suitable for tubular structures but also for
nontubular hollow structures. In this setting, the centerline is no longer
required. Instead, the reference surface plays a more critical role. One simplest
scheme for surface-based eversion is to implement data reflection along each normal
on the reference surface. Calculating normals can be done in the three steps: (1)
extracting triangular elements based on the reference surface; (2) calculating
normals for all the triangular elements; (3) for each point on the reference
surface, estimating the normal from the normals of the neighboring triangular elements.
Note that normals derived from different points on the reference surface may
intersect, as shown in Figure4(a). Distortions may occur if the everting operation
is done along these normals.Normals
and electrical force lines from sampling points on the reference surface. (a)
Normals at different sample points may intersect, which cause distortions in
the eversion, and (b) electrical force lines from sampling points on the reference
surface, which are locally orthogonal to the surface but are never in conflict.
(a)(b)Another surface-based
eversion is based on electric force lines, similar to what has been discussed
in the preceding subsection. This time, positive charges are distributed over the
reference surface. At each sampling point on the surface, an electrical force
line, which is locally orthogonal to the surface, is generated. According to
electrical field theory, these lines never conflict (Figure4(b)).Similar to
the centerline-based eversion, the partial electrical field around each point on
the reference surface can be formed, that
is, charges are only assumed in a prespecified neighborhood of that point to
generate the electrical force line. The next steps have only minor change in
comparison with the centerline-based eversion counterparts.LetFj be an electrical force line, where j is the index. The definitions of M, R, E,
and S remain the same. Let mj stand for the intersection of Fj and M, rj for intersection of Fj and R,
and ej for intersection of Fj and E.
The criterion for determining ej is that the length of a curved segment from mj to rj equals the length of a curved segment from ej to rj on Fj.
The eversion process along one electrical force line is illustrated in Figure 5. Again,
all ej should be scaled down along the electrical
force lines with an appropriate factor to form the surface S.
Let sj stand for intersection of Fj and S.
We can so determine sj that ∑j=1P(dist(sj,mid(sj,mj))+dist(mj,mid(sj,mj))) is minimized, where P is the number of the electrical force lines.Figure 5
Everting
process for a point on the reference surface. Along the curved electrical force
line, the length from (1) to (3) equals the length from (3) to (2).


## 3. Results

We used the patient dataset acquired at
the Walter Reed Army Medical Center, USA (http://www.gris.uni-tuebingen.de/areas/scivis/volren/datasets/new.html, Study 289 (213), Colon Prone).
It was obtained from an abdominal spiral CT scan with prone orientation after colon
cleansing and insufflating. The image volume was made 512×512×463 voxels of 0.625×0.625×1.0 mm. The
dynamic range was compressed into 256 grey-levels. The mucosal surface of the
colon was segmented via a regional growing algorithm. Two simulated
hemi-ellipsoidal polyps of semi-axis lengths 4, 4, and 6 mm were digitally
implanted.The segmentation and eversion algorithms
were implemented in the Visual Studio platform (Microsoft Corporation) with the
C++ language and ITK library (Insight Segmentation and Registration Toolkit (http://www.itk.org/index.htm)).
The surface rendering was done using the VTK (Visualization Toolkit (http://www.vtk.org)).A small
portion of the colon lumen was chosen for digital eversion (Figure6(a)). It contained
the transverse and descending colon segments and hence had a high curvature
(Figure 6(b)). To present the inner surface of this portion better, a cut plane was
used to “open” the colon, as shown in Figure 6(d). Figure 6(f) shows the reference surface
derived from dilating the lumen with a spherical kernel.Surface rendering of the original colon lumen with two
digitally implanted polyps and its reference surface for digital eversion. (a) An
outside view of the whole colon lumen, in which the portion defined by the
frame is selected for the digital eversion, (b) a magnified view of the
selected colon segment and the simulated polyps (arrows), (c) a magnified view
of (b), (d) a bisected colon to show the inner surface and the simulated polyps
(arrows), (e) a magnified view of (d), and (f) the reference surface for the digital
eversion.
(a)(b)(c)(d)(e)(f)Figure7(a)
shows the digital eversion using the centerline-based partial-electrical-field method
(with 181 neighboring charges). As a result, the mucosal surface of the colon
can be visualized from outside. Figure 7(c) demonstrates the surface-based
eversion using the partial-electrical-field method, in which the sampling
points in a 31×31 neighborhood were assumed on the reference surface. For comparison,
Figure 7(e) is the eversion outcome based on the planar cross-sections locally
orthogonal to the centerline, while Figure 7(f) is the eversion result along the
normals of the reference surface. Like in the case of unfolding the colon with
planar sections [10], both Figures 7(e) and 7(f) exhibit noise and artifacts,
making the polyps difficult to be detected.Digital eversion results using representative
methods.  (a) The centerline-based
eversion using the partial-electrical-field method, (b) a magnified view of
(a), (c) the surface-based eversion using the partial-electrical-field, (d) a
magnified view of (b), (e) the everted mucosal surface based on the planar
cross-sections orthogonal to the centerline, and (f) the mucosal surface everted
along the normals of the reference surface.
(a)(b)(c)(d)(e)(f)


## 4. Discussions and Conclusion

The purpose of this
paper is to present a new idea, digital eversion of a hollow structure, and demonstrate
its feasibility. Indeed, we have demonstrated that the eversion methods seem to
be performing complementary or superior to the existing methods for
visualization of hollow-structures. This approach makes it possible to view the
inner surface of a hollow structure from outside, while keeping relative
positions of key features and the global picture basically unchanged. Although
in this paper we used the colon eversion as an example, the technique can be
applied to any other hollow structures such as gastrointestinal tract,
respiratory tract, urinary tract, blood vessels, spine, and so on, and even
solid organs.The centerline-based
method is more efficient but it can only be applied to tubular structures and
is sensitive to the accuracy of the centerline determination. The surface-based
method can be applied to any hollow structure but it is more time-consuming.
Better eversion results are expected when a hollow structure has a smoother
inner surface such as blood vessels, bladder, uterus, and so on. The eversion
with respect to planar cross-sections or with surface normals is the most
straightforward, and may be applied to a tubular organ with a smooth inner
surface, no folds, and little flexure; such as esophagus, oviduct and urethra.Much work is
needed for refinement of the algorithms and optimization of the parameters. Possibilities
include, but are not limited to, collaboration with radiologists to do extensive
reader studies (using more datasets) with actual polyps, utilization of new
techniques such as conformal mapping, finite element methods and
area-preserving mapping [13–19, 23, 24, 25, 26] to obtain better eversion results, and integration
with other visualization methods; for example, a tortuous tubular structure may
be first straightened completely or to a certain degree, and then be everted.
When certain segments block the
view of other segments, we may remove the front segments to get a good view for
certain areas. The morphology filter with the varied size should be adopted depending
on the size of the haustras and folds and the curvature of the colon.In
conclusion, we have conceptualized a new idea on digital eversion of a hollow
structure. Practical algorithms have been developed according to the electrical
field model, which gives electrical force lines from either a charged centerline
or a charged reference surface. The preliminary experiments have demonstrated that
the eversion of a hollow structure is feasible and promising for inspection of
hollow structures, especially in medical imaging applications.



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*Source: 763028-2008-07-22.xml*