# Evolution of Brain Tumor and Stability of Geometric Invariants

Authors: K. Tawbe; F. Cotton; L. Vuillon
Journal: International Journal of Telemedicine and Applications
 (2008)

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## Abstract

This paper presents a method to reconstruct and to calculate geometric invariants on brain tumors. The geometric invariants considered in the paper are the volume, the area, the discrete Gauss curvature, and the discrete mean curvature. The volume of a tumor is an important aspect that helps doctors to make a medical diagnosis. And as doctors seek a stable calculation, we propose to prove the stability of some invariants. Finally, we study the evolution of brain tumor as a function of time in two or three years depending on patients with MR images every three or six months.


---

## Body

## 1. Introduction

Cancer is a disease that starts in our cells. Our bodies are made up of millions of cells grouped together to form tissues and organs such as muscles and bones, the lungs, and the liver. Genes inside each cell order it to grow, work, reproduce, and die. Normally, our cells obey these orders, and we remain healthy. But sometimes the instructions get mixed up, causing the cells to form lumps or tumors, or spread through the bloodstream and lymphatic system to other parts of the body. Tumors can be either benign (noncancerous) or malignant
(cancerous). Benign tumor cells stay in one place in the body and are not usually life-threatening. Malignant tumor cells are able to invade nearby tissues and spread to other parts of the body. Cancer cells that spread to other
parts of the body are called metastases.Brain cancer starts in the cells of the brain. The brain is a soft mass of nerves (neurons) and supportive tissue (glial cells), surrounded by membranes (meninges) and protected by the skull. The brain has 3 main areas (see Figure1).Figure 1(i)
The cerebrum is
the largest part of the brain and is made up of the right and left cerebral
hemispheres. It allows us to see, feel, think, speak, and move. The right side
of our brain controls the left side of our body and vice versa.(ii)
The cerebellum
is located in the back of the brain and controls balance and coordination.(iii)
The brain stem
controls our vital bodily functions, like heartbeat, breathing, and reflexes.
It connects the brain to the spinal cord. The skull is
hard and cannot expand, so as a tumor grows the pressure can damage or destroy
delicate brain cells. Brain cancer can involve either the neurons or the glial
cells. Most adult cancers start in the glial cells and are called astrocytomas
or gliomas.Neurological specialists
perform several diagnostic tests for brain tumors. (http://www.medecinenet.com). These
tests include the
following.(i)
Electroencephalogram
(EEG).(ii)
Lumbar
puncture (spinal tap).(iii)
RN
(radionuclide).(iv)
Computerized
axial tomographer (CT or CAT).(v)
MRI.(vi)
PetScan.(vii)
Biopsy. Since the development of tomography,
computers have been used extensively in medical
diagnosis. Unlike classical radiology, tomography requires complex mathematical
calculations in order to obtain a two-dimensional image. Computers are used for
image treatment, visualization, and archiving, and
also for 3D reconstruction. In [1], it is proposed a method to reconstruct a 3D model of
certain organs from a number of 2D cross-sectional images. This method enables
a better understanding of spatial structures, and also open the way to new
applications like radiation therapy planing and surgical planing.In [2], it is proposed a new model to simulate the growth of
glioblastomas multiforma (GBM), the most aggressive
glial tumors. This simulation has a different medical applications, including
an optimized dosimetry in radiotherapy or a better neurosurgical planing in
case of tumor resection.In [3], it is proposed a high-resolution three-dimensional
(3D) connectivity, surface construction, and display algorithms that detect,
extract, and display the surface of a brain from contiguous magnetic resonance
(MR) images. The algorithms identify the external brain surface and create a 3D
image, showing the fissures and surface convolutions of the cerebral
hemispheres, cerebellum, and brain stem. For the purposes of the 3D
reconstruction, it is shown that T1-weighted images
give better contrast between the surface of the brain and the cerebral spinal
fluid than T2-weighted images. 3D reconstruction of MR data provides a
noninvasive procedure for examination of the brain surface and other anatomical
features.In this paper, we reconstruct the tumor from 2D
sections, coming from MRI sequences, for patients having a brain tumor. The
distance between two parallel sections is 5 mm. The tumor has been segmented
semiautomatically by a software provided to us by
(INRIA, Cedex, France). and based on the gradient method. After the
identification of points on the contours, 50 points were spot on each contour.
The reconstruction method is based on these points. Once the  tumor is
reconstructed, we calculate the geometric invariants (volume, area, and
discrete Gauss curvature). The doctors seek a stable calculation that does not
depend on the number of points or their distributions on the contours. This is
why the distribution and the number of points are changed, and the calculation
of geometric invariants is redone to show the stability. The discrete mean
curvature is calculated on the edges of the triangulation surface. However,
this invariant is not  stable according to the distribution and the number of
points on the contours.


## 2. Detection and Segmentation of Brain Tumors

Detection in MR image with brain tumor is an important image processing technique applied in
radiology for 3D reconstruction. Indeed, contours are rich indexes for any
subsequent interpretation of the image. Contours in image are due to(i)
discontinuities
of the reflectance function,(ii)
discontinuities
of depth (boundaries of the object). The contours
are characterized by the discontinuities of the intensity function. Therefore,
the principle of contours detection is based on the study of the derivatives of
the intensity function in the image. The contours characterize the boundaries
of the objects, and generally they are defined as a transition zones between
two regions of different characteristics presented simultaneously to within a
single digital image.Definition 1.
LetI(x,y)  be the
intensity function of an original image; the gradient of the image is defined
by the vector (1)∇I(x,y)=(∂I(x,y)∂x,∂I(x,y)∂y)t=(I(x+1,y)−I(x,y);I(x,y+1)−I(x,y))t.This gradient is characterized
by a module m and a direction ϕ in the image(2)m=(I(x+1,y)−I(x,y))2+(I(x,y+1)−I(x,y))2,ϕ=arctan(I(x+1,y)−I(x,y)I(x,y+1)−I(x,y)).The direction
of the gradient maximizes the directional
derivative. The basic steps of contours detection are thus to calculate the
derivatives of the intensity function, then to specify the contour
points.The collaboration with Professor François Cotton was a
very important step, especially to locate and to precise the position of the
brain tumor in each 2D sequence for all patients. In our applications, the
boundary of the tumor was detected by François Cotton, and so the final step
before reconstructing the tumor was to segment it by using a software provided
to us by INRIA [4].
The idea of this algorithm is to surround the tumor by a circle, to regard the
minimal and maximal intensities on the boundary of the tumor, and to fix
thresholds for them, then a threshold 				“s” for the gradient is fixed to
determine the tumor's boundary. Once these parameters are fixed, we apply an
ultra classical procedure of minimization to obtain the boundary of the brain
tumor by moving the circle.Once the gradient is evaluated, the points on the
contours which are characterized by local extrema are identified. The idea is
to select the pixels by using the threshold “s” for the norm of the
gradient, that is, all points on contours such that m>s (see Figure 2).Segmentation of the brain tumor and the contour
points. Segmentation validated by François Cotton (professor of
radiology).
(a)(b)(c)(d)(e)(f)(g)(h)


## 3. Surfaces

Asurface is a topological space in which each point has a neighborhood that is
homeomorph to the unit disk {(x,y)∈ℝ2/x2+y2<1} or to the half
unit disk {(x,y)∈ℝ2/x2+y2<1,x≥0}. The boundary of a surface S, denoted by ∂S, is the set of points of this surface that do not have
a neighborhood homeomorph to the unit disk. The interior of S is
complementary to its boundary. A surface is
therefore abstract; the only thing we know is the neighborhoods of each point.


### 3.1. Triangulations

We will now
give some  definitions of discrete surfaces, that is, surfaces defined by a
finite number of points. Indeed, in computer science we work often on such
surfaces, which in generally are approximations of real or ideal surfaces, for example, the rabbit in Figure3—the famous bunny
of stanford—is a mesh surface, created by scanning a real model in a clay
(the scanner detects the geometric position  of some
number of points of the model, and then these points are related three by three
to form triangles, using an adequate
algorithm).(a)Bunny of stanford. (b) Zoom on the ears of
the bunny.
(a)(b)Definition 2.
Lete0,…,en  be a set of  n+1
linearly
independent vectors in the Euclidean space  ℝm,  m≥n≥0. One calls  n-simplex of n
vertices  e0,…,en  the convex hull  σ
of these
points.  n
is the
dimension of  σ.Example 1.
Inℝ3, a  0-simplex is a
vertex, a 1-simplex is an
edge, a   2-simplex is a
triangle, and a  3-simplex is a
tetrahedron (see Figure 4).(a) 0-simplex,
(b) 1-simplex, (c) 2-simplex, and (d) 3-simplex ofℝ3.
(a)(b)(c)(d)Definition 3.
LetS  be a set of
linearly independent vectors and  σ
its convex
hull. Then, the convex hull  τ  of all subset  T  of  S
is a simplex
subset of  σ. One says that  τ  is a face of  σ, and one obtains  τ≤σ.Definition 4.
A simplicial
complex is a finite set of simplicesK={σ0,…,σr}  such that
(i)
ifσi∈K, then all its faces are in K;(ii)
letσi,σj∈K, then σi∩σj=∅ or σi∩σj∈K  (the two
simplices have a common face) (see Figure 5).
The dimension of a simplicial complex is the
dimension of its greatest simplex. We will now establish a link between the topology of a
set of points and combinatorial topology, more precisely between the notions of
topological space and the simplicial complex.Figure 5
Example of
simplicial complex having six vertices, ten edges, and five triangles.Definition 5.
LetK
be a simplicial
complex in  ℝm. The union  |K|  of all the
simplices in  K
with the
topology of the subsets of  ℝm  is called the
polyhedron of  K.Definition 6.
The
triangulation of a topologic spaceX
is a simplicial
complex  K
such that its
polyhedron  |K|
is homeomorph
to  X. If such simplicial complex exists, one said that  X
is
triangulated.


## 3.1. Triangulations

We will now
give some  definitions of discrete surfaces, that is, surfaces defined by a
finite number of points. Indeed, in computer science we work often on such
surfaces, which in generally are approximations of real or ideal surfaces, for example, the rabbit in Figure3—the famous bunny
of stanford—is a mesh surface, created by scanning a real model in a clay
(the scanner detects the geometric position  of some
number of points of the model, and then these points are related three by three
to form triangles, using an adequate
algorithm).(a)Bunny of stanford. (b) Zoom on the ears of
the bunny.
(a)(b)Definition 2.
Lete0,…,en  be a set of  n+1
linearly
independent vectors in the Euclidean space  ℝm,  m≥n≥0. One calls  n-simplex of n
vertices  e0,…,en  the convex hull  σ
of these
points.  n
is the
dimension of  σ.Example 1.
Inℝ3, a  0-simplex is a
vertex, a 1-simplex is an
edge, a   2-simplex is a
triangle, and a  3-simplex is a
tetrahedron (see Figure 4).(a) 0-simplex,
(b) 1-simplex, (c) 2-simplex, and (d) 3-simplex ofℝ3.
(a)(b)(c)(d)Definition 3.
LetS  be a set of
linearly independent vectors and  σ
its convex
hull. Then, the convex hull  τ  of all subset  T  of  S
is a simplex
subset of  σ. One says that  τ  is a face of  σ, and one obtains  τ≤σ.Definition 4.
A simplicial
complex is a finite set of simplicesK={σ0,…,σr}  such that
(i)
ifσi∈K, then all its faces are in K;(ii)
letσi,σj∈K, then σi∩σj=∅ or σi∩σj∈K  (the two
simplices have a common face) (see Figure 5).
The dimension of a simplicial complex is the
dimension of its greatest simplex. We will now establish a link between the topology of a
set of points and combinatorial topology, more precisely between the notions of
topological space and the simplicial complex.Figure 5
Example of
simplicial complex having six vertices, ten edges, and five triangles.Definition 5.
LetK
be a simplicial
complex in  ℝm. The union  |K|  of all the
simplices in  K
with the
topology of the subsets of  ℝm  is called the
polyhedron of  K.Definition 6.
The
triangulation of a topologic spaceX
is a simplicial
complex  K
such that its
polyhedron  |K|
is homeomorph
to  X. If such simplicial complex exists, one said that  X
is
triangulated.


## 4. Reconstruction Method

The method
presented in this paragraph is a combination of the 2D Delaunay triangulation
of contours and the maximizing volume method.


### 4.1. Voronoi Diagram and Delaunay Triangulation

Definition 7.
Given a setS  of  N  sites,  S={pi∈ℝ2/i=(1,…,N)}
such that no
four sites lie on a common circle. One defines  V(i)
as the set of
points closer to site  pi  than to any
other site in  S: (3)V(i)={x∈ℝ2:∀pj∈S,∥x−pi∥≤∥x−pj∥}.V(i) is called the
Voronoi  cell associated to pi. The union over all the V(i) is called the
Voronoi diagram of S (see Figure 6).(a) A Voronoi cell
and (b) the Voronoi diagram of a number of sites.
(a)(b)The boundaries of the  regions are referred to  as
Voronoi edges, the joints of three Voronoi edges are called Voronoi vertices.
Each Voronoi edge is associated with two adjacent Voronoi cells, a Voronoi
vertex is equidistant to three sites (see [5]).If we draw a line segment between each pair of sites
whose Voronoi cells share an edge, we obtain a triangulation of the points inS called the
Delaunay triangulation (see Figure 7).Figure 7
A Voronoi diagram (dashed) and its straight line dual,
the Delaunay triangulation.A Voronoi vertex represents a Delaunay triangle; more
precisely, it is the center of its circumcircle. Each Voronoi edge corresponds
to an edge in the Delaunay triangulation despite the fact that they may not
even intersect. This geometric difference between
Voronoi diagram and Delaunay triangulation becomes important in the
reconstruction issue. Some further properties of the Delaunay triangulation are
the following.(i)
The boundary of
the Delaunay triangulation is the convex hull of its sites.(ii)
The Delaunay
triangulation is unique.(iii)
The number of
triangles in the Delaunay triangulation is at most2N−5, where N is the number
of vertices in the triangulation.(iv)
A Delaunay
triangle does not contain any other site in its circumcircle (empty circle
property).In our
application, the object contours are given as a set of straight line segments,
forming one simple closed polygon. However,  just triangulating the polygons
vertices—or calculating the Voronoi diagram of point sites—may result in a
triangulation where contour segments are not guaranteed to be edges of the
triangulation. Since our goal is to get a 3D polyhedron whose intersection with
the given cross-sections yields the original contours, our triangulation has to
satisfy the following requirement:All contour segments have to appear as Delaunay edges
in the Delaunay triangulation (contour containment condition).To obtain a triangulation that satisfies this
condition, we simply calculate the Delaunay triangulation of the polygons
vertices. Then, we check each contour segment to be contained in the Delaunay
triangulation. Segments that do not appear as Delaunay edges are split into two
parts by adding a new vertex in the middle. We add the new vertex to the
Delaunay triangulation and verify the contour containment condition one more.It can be shown that such a procedure terminates and
yields a Delaunay triangulation that satisfies the containment condition. The
contour shape is not changed, since we add vertices onto contour segments (see
[6]).Once all contour segments  are in the triangulation, we
eliminate the edges which are outside the contour, the vertices added to the
contour, and the edges  related to these vertices without touching the original
segments of the contour (see Figure8).(a)  A contour. It
is evident that trianglee,f,g cannot be a
part of a Delaunay triangulation, since its circumcircle would contain another
site. (b) shows the
Delaunay triangulation of the contour points. (c)  Contour edge e,f is contained
after addition of a vertex to. (d) Elimination of
exterior edges, added vertex, and the edges related to this vertex.
(a)(b)(c)(d)


### 4.2. Triangulation of Two Parallel Sections

Let us consider
two  parallel sections  (contours)C1,C2, with the same number of points on each one. On C1, we have the sequence of points A0,A1,…,Am, and on C2 the points Bm+1,Bm+2,…,Bn. In other words, if m=49, then we have 50 points on C1, as a result n=99, and we have 50 points on C2. Note that the points A0,A1,…,Am, Bm+1,Bm+2,…,Bn on each contour
are distributed in the clockwise direction (see Figure 9).Figure 9
Two parallel contours and 50 points
on each one.(i)
First step. Let A0  be the starting
point in our triangulation, the idea of this method is based on finding the
nearest point to A0 in the adjacent
section. In other words, we calculate the distance(4)d(A0,Bj),j=m+1,…,n.(ii)
Second step. We choose the point Bk,m+1≤k≤n such
that(5)d(A0,Bk)=minj=m+1,…,nd(A0,Bj),where d is the
Euclidean distance between two points.(iii)
Third step. We connect by a segment the point A0 to the nearest
point in the adjacent section C2 which is the point Bk. The first segment of the triangulation will be [A0Bk], then on the section C1 we take the
neighborhood point of A0 in the
clockwise direction, let A1 be this point,
by a segment, we connect A1 to the point Bk in the adjacent
section to form with A0 the first
triangle in the triangulation which has the vertices A0A1Bk (see Figure 10).(iv)
Fourth step.  We take the point A1, and we determine the nearest point to A1 in the adjacent
section without regarding the point Bk. As the sections are parallel, the nearest point will
be Bk+1. By repeating the same procedure in the third step,
we find the second triangle which has the vertices A1A2Bk+1. Automatically, the triangle which has the vertices BkBk+1A1 will form
between the two triangles: A0A1Bk and A1A2Bk+1 (see Figures 11, 12).Figure 10
Starting pointA0, the nearest point in the adjacent contour B50, the nearest neighborhood of A0 in the
clockwise direction A1, and the triangle A0A1B50.Figure 11
TriangleA0A1B50,  triangle A1A2B51, and the triangle B50B51A1 that forms
between them.TriangulationT0 between two
contours (sections).
(a)(b)By repeating these stepsm times, we
obtain the triangulation T0 and the volume V0 delimited
between the two sections C1,C2.


### 4.3. Changing the Starting Point

Let us
initialize the triangulation  and change the starting point, it means instead ofA0 as a starting
point, we take the point A1 and apply steps
1,2,3,4,…  to get the triangulation T1 and the volume V1 between C1,C2.The change of the starting point is donem times to
reconstruct m triangulations
(T0,T1,…,Tm), and m volumes (V0,V1,…,Vm) delimited between C1,C2 (see Figure 13).Figure 13
A0: starting point.


### 4.4. Selection of the Best Triangulation

The
triangulation which maximizes the volume of the polyhedronA0,A1,…,Am, Bm+1,Bm+2,…,Bn gives the
optimal approximation of the surface provided by a pair of closed contour
segments (see [7]).Remark 1.
This algorithm is true for convex contours
and concave contours (see Figure14).Figure 14
A24: starting point—selection of the best
triangulation.


## 4.1. Voronoi Diagram and Delaunay Triangulation

Definition 7.
Given a setS  of  N  sites,  S={pi∈ℝ2/i=(1,…,N)}
such that no
four sites lie on a common circle. One defines  V(i)
as the set of
points closer to site  pi  than to any
other site in  S: (3)V(i)={x∈ℝ2:∀pj∈S,∥x−pi∥≤∥x−pj∥}.V(i) is called the
Voronoi  cell associated to pi. The union over all the V(i) is called the
Voronoi diagram of S (see Figure 6).(a) A Voronoi cell
and (b) the Voronoi diagram of a number of sites.
(a)(b)The boundaries of the  regions are referred to  as
Voronoi edges, the joints of three Voronoi edges are called Voronoi vertices.
Each Voronoi edge is associated with two adjacent Voronoi cells, a Voronoi
vertex is equidistant to three sites (see [5]).If we draw a line segment between each pair of sites
whose Voronoi cells share an edge, we obtain a triangulation of the points inS called the
Delaunay triangulation (see Figure 7).Figure 7
A Voronoi diagram (dashed) and its straight line dual,
the Delaunay triangulation.A Voronoi vertex represents a Delaunay triangle; more
precisely, it is the center of its circumcircle. Each Voronoi edge corresponds
to an edge in the Delaunay triangulation despite the fact that they may not
even intersect. This geometric difference between
Voronoi diagram and Delaunay triangulation becomes important in the
reconstruction issue. Some further properties of the Delaunay triangulation are
the following.(i)
The boundary of
the Delaunay triangulation is the convex hull of its sites.(ii)
The Delaunay
triangulation is unique.(iii)
The number of
triangles in the Delaunay triangulation is at most2N−5, where N is the number
of vertices in the triangulation.(iv)
A Delaunay
triangle does not contain any other site in its circumcircle (empty circle
property).In our
application, the object contours are given as a set of straight line segments,
forming one simple closed polygon. However,  just triangulating the polygons
vertices—or calculating the Voronoi diagram of point sites—may result in a
triangulation where contour segments are not guaranteed to be edges of the
triangulation. Since our goal is to get a 3D polyhedron whose intersection with
the given cross-sections yields the original contours, our triangulation has to
satisfy the following requirement:All contour segments have to appear as Delaunay edges
in the Delaunay triangulation (contour containment condition).To obtain a triangulation that satisfies this
condition, we simply calculate the Delaunay triangulation of the polygons
vertices. Then, we check each contour segment to be contained in the Delaunay
triangulation. Segments that do not appear as Delaunay edges are split into two
parts by adding a new vertex in the middle. We add the new vertex to the
Delaunay triangulation and verify the contour containment condition one more.It can be shown that such a procedure terminates and
yields a Delaunay triangulation that satisfies the containment condition. The
contour shape is not changed, since we add vertices onto contour segments (see
[6]).Once all contour segments  are in the triangulation, we
eliminate the edges which are outside the contour, the vertices added to the
contour, and the edges  related to these vertices without touching the original
segments of the contour (see Figure8).(a)  A contour. It
is evident that trianglee,f,g cannot be a
part of a Delaunay triangulation, since its circumcircle would contain another
site. (b) shows the
Delaunay triangulation of the contour points. (c)  Contour edge e,f is contained
after addition of a vertex to. (d) Elimination of
exterior edges, added vertex, and the edges related to this vertex.
(a)(b)(c)(d)


## 4.2. Triangulation of Two Parallel Sections

Let us consider
two  parallel sections  (contours)C1,C2, with the same number of points on each one. On C1, we have the sequence of points A0,A1,…,Am, and on C2 the points Bm+1,Bm+2,…,Bn. In other words, if m=49, then we have 50 points on C1, as a result n=99, and we have 50 points on C2. Note that the points A0,A1,…,Am, Bm+1,Bm+2,…,Bn on each contour
are distributed in the clockwise direction (see Figure 9).Figure 9
Two parallel contours and 50 points
on each one.(i)
First step. Let A0  be the starting
point in our triangulation, the idea of this method is based on finding the
nearest point to A0 in the adjacent
section. In other words, we calculate the distance(4)d(A0,Bj),j=m+1,…,n.(ii)
Second step. We choose the point Bk,m+1≤k≤n such
that(5)d(A0,Bk)=minj=m+1,…,nd(A0,Bj),where d is the
Euclidean distance between two points.(iii)
Third step. We connect by a segment the point A0 to the nearest
point in the adjacent section C2 which is the point Bk. The first segment of the triangulation will be [A0Bk], then on the section C1 we take the
neighborhood point of A0 in the
clockwise direction, let A1 be this point,
by a segment, we connect A1 to the point Bk in the adjacent
section to form with A0 the first
triangle in the triangulation which has the vertices A0A1Bk (see Figure 10).(iv)
Fourth step.  We take the point A1, and we determine the nearest point to A1 in the adjacent
section without regarding the point Bk. As the sections are parallel, the nearest point will
be Bk+1. By repeating the same procedure in the third step,
we find the second triangle which has the vertices A1A2Bk+1. Automatically, the triangle which has the vertices BkBk+1A1 will form
between the two triangles: A0A1Bk and A1A2Bk+1 (see Figures 11, 12).Figure 10
Starting pointA0, the nearest point in the adjacent contour B50, the nearest neighborhood of A0 in the
clockwise direction A1, and the triangle A0A1B50.Figure 11
TriangleA0A1B50,  triangle A1A2B51, and the triangle B50B51A1 that forms
between them.TriangulationT0 between two
contours (sections).
(a)(b)By repeating these stepsm times, we
obtain the triangulation T0 and the volume V0 delimited
between the two sections C1,C2.


## 4.3. Changing the Starting Point

Let us
initialize the triangulation  and change the starting point, it means instead ofA0 as a starting
point, we take the point A1 and apply steps
1,2,3,4,…  to get the triangulation T1 and the volume V1 between C1,C2.The change of the starting point is donem times to
reconstruct m triangulations
(T0,T1,…,Tm), and m volumes (V0,V1,…,Vm) delimited between C1,C2 (see Figure 13).Figure 13
A0: starting point.


## 4.4. Selection of the Best Triangulation

The
triangulation which maximizes the volume of the polyhedronA0,A1,…,Am, Bm+1,Bm+2,…,Bn gives the
optimal approximation of the surface provided by a pair of closed contour
segments (see [7]).Remark 1.
This algorithm is true for convex contours
and concave contours (see Figure14).Figure 14
A24: starting point—selection of the best
triangulation.


## 5. Surface Area

In our
applications, we calculate an area approximation of a smooth surface by the sum
of triangles area.  To compute
the area of a piecewise linear 2-dimensional space, one divides it in a
partition of triangles, computes the area of each triangle with the familiar
formula—half the product of the basis by the heigh—and then adds all
these areas (see Figure 16). The only point to check is that the result is
independent of the triangulation. In Figure 15, with 50 points on the contours, the area is equal to 149.084, with 100 points, it is
equal to 149.114 and with 200 points, it is
equal to 149.170 (the unit is cm2); see
[8].Reconstruction of brain
tumor: 50 points, 100 points, and 200 points on the contours.
(a)(b)(c)Figure 16
The area ofT is the sum of
the areas t1,t2, and t3.


## 6. Volume of a Domain Limited by a Surface

LetD be a domain in ℝ3 limited by the
surface S.Definition 8.
The volume ofD
is given by (6)vol(D)=∭Ddxdydz=∫−c+cdz∬Dzdxdy,where Dz is the
intersection of D with the plane z=const.Finally, the
Green-Ostrogradsky theorem reduces the calculus of the volume to an integral
surface:(7)vol(D)=∭DdV=13∬∂D(x,y,z)⋅n→dS,where ∂D is the boundary
of D, and n→ is the unit
normal vector at dS oriented to the exterior of D.


## 7. Discrete Gauss Curvature

We will give
the definition of the Gauss curvature at a vertex in a triangulation (see
[9]).LetT be a
triangulation, p is a vertex of T, 𝒯T(p) is the set of
triangles of T that having p as a vertex. We
denote by ST∘ the set of all
interior vertices of T and
by S∂T the set of all
boundary vertices of ∂T. The angle at a vertex p is the real
number defined by(8)αT(p)=∑σ∈𝒯T(p)ασ(p),where ασ(p) is an angle at p of the triangle σ (see Figure 17).Figure 17
The angle atp.The characteristic Euler of a triangulationT is given by(9)𝒳(T)=Ns−Na+Nf,where NS is the number
of vertices of T,  Na is the number
of edges, and Nf is the number
of faces. A simple calculus gives the following formula of Gauss-Bonnet:(10)∑p∈ST∘T(2π−αT(p))+∑p∈S∂T(π−αT(p))=2π𝒳(T). This formula will motivate the following
definitions.(i)
Thediscrete
Gauss curvature at a vertex p∈ST∘ is(11)GT(p)=2π−αT(p).(ii)
Thetotal
Gauss curvature of T is(12)Gint(T)=∑p∈ST∘TGT(p)=∑p∈ST∘T(2π−αT(p)).


## 8. Discrete Mean Curvature

LetT be an oriented
triangulation, a is an interior
edge of T, n1→ and n2→ are the unit
oriented normal vectors of the two triangles adjacent to a (see Figure 18).Figure 18
Dihedral angle of the edge.(i)
We calldihedral
angle of T at a the angle(13)β(α)=(n1→,n2→)^∈[0,π[.(ii)
We define theindex
convexity of a the integer i(a)∈{0,1} by (14)−i(a)=0si〈p0p1→,n2→〉≤0,〈p0p2→,n1→〉≤0,−i(a)=1si〈p0p1→,n2→〉≥0,〈p0p2→,n1→〉≥0.(iii)
Themean
curvature of the edgea is(15)HT(a)=12(−1)i(a)l(a)β(a).(iv)
Thetotal
mean curvature of T is(16)H(T)=∑aninterioredgeofTHT(a)=∑anedgeofT12(−1)i(a)l(a)β(a),where l(a) is the length
of a.


## 9. Patients and Images Modality

The following
description precises the different parameters of the images modality, and they
are given by the second author.All patients had partial seizures and a first MRI
compatible with brain tumors. Low-grade gliomas were confirmed with
neurosurgical biopsy, and patients were then followed with serial MRI every 6
months. MR imaging was performed on Philips Intera 1.5 Tesla
MRI system (Philips Medical Systems, Erlangen, The
Netherlands) with a standard head coil. Localizing sagital T1-weighted images
of the brain was obtained initially, followed by
axial FLAIR and T2-weighted images. Thereafter, diffusion- and
perfusion-weighted MR images were obtained (diffusion-weighted image comprised
an echo-planar spin-echo sequence, with the following parameters, TR = 4247,
TE = 95, EPI factor = 77, field of view (FOV) = 230 mm, slice thickness = 5 mm, slice
gap = 1 mm, number of excitations = 1, matrix  =  77× 256, number of
slices = 22, acquisition time = 30 seconds). The images
in diffusion-weighted image (DWI) were acquired for values of b (diffusion
gradient factor) equal to 0 and 1000 s/mm2. An isotropic image was constructed in real time,
pixel by pixel, as an average of the signal intensities of three orthogonal
directions. Apparent diffusion coefficient (ADC) maps were then calculated from
the diffusion-weighted images. Dynamic susceptibility contrast enhanced MRI
(DSC-MRI) using the PRESTO  sequence with the following parameters was employed:
TR = 17, TE = 8, flip angle = 7, EPI factor = 17, FOV) = 220 mm, slice thickness = 3.5 mm,
slice gap = 0 mm, number of excitations = 1, matrix = 64 × 64, number of
slices = 60 series of 22 slices, acquisition time = 77 seconds. The PRESTO
technique is highly sensitive to T2* changes due to its very long TE. Image
acquisition began simultaneously with contrast agent injection. For all
patients, a standard dose (0.2 mL/kg = 0.1 mmol/kg) of gadobenate dimeglumine
(multihance; Bracco Imaging, Milan, Italy) was injected using an automated
power injector, at a flow rate 6 mL/sec followed by a 40 mL saline flush at the
same rate. Contrast agent administrations were in all cases performed using a
20-gauge intravenous catheter. Finally, postcontrast 3D T1-weighted images,
with an isotropic voxel of 0.9 mm, were obtained.


## 10. Results

The
collaboration with professor François Cotton (IRM, Hospices Civils de Lyon,
Lyon, France) consists of the three following steps.(i)
First step. Series of 3 typical biopsy-proved low-grade gliomas followed with serial MRI
every 6 months were first selected for the purpose of the study. FLAIR sequence
was used for the segmentation because it has high sensitivity for the tumor
detection, which typically appears on hypersignal compared to the normal brain
parenchyma. DICOM images including FLAIR sequence were then burned on a CD.(ii)
Second step. The validation of the segmentation was done firstly by comparing the medical
data with the slope of the tumor growth obtained by linear
measurement—secondly according to neuroexperience of the tumor growth. In
medical experience and literature, the mean tumor diameters of
untreated low-grade gliomas (linear measurements)
showed a linear and constant growth before anaplastic transformation with an average
slope of 4 mm per year (2 to 8 mm/year) (see [10–12]).(iii)
Third step. The exchange between mathematicians and specialists in medical imaging was
based on knowledge of MRI sequences, tumor growth, macroscopic and microscopic
structure, microvascularity and angiogenesis, density and contour of brain
tumors, and mathematical model that could be used to analyze these parameters.Our goal is to provide some tools to help the doctor
to make his diagnosis, then to show that some geometric invariants (like the
volume, the area, and the Gauss curvature) are relatively stable by adding
points at the segmentation. Others, like the mean curvature, are sensitive to
the number of used points. Finally, we would like to sensitize the community of
the medical imaging in the importance of calculating geometric invariants in
order to verify that the segmentation and the 3D reconstruction of the tumor
are good from  this point of view.We present the results established on some
patients in Figures19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, and 33.Patient 1. (a) 50 points on the
contours, (b) 100 points on the
contours, (c) 200 points on the
contours, and (d) 400 points on the
contours.
(a)(b)(c)(d)Patient 1. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 1. (a) 50 points on the
contours, (b) 100 points on the
contours, (c) 200 points on the
contours, and (d) 400 points on the
contours.
(a)(b)(c)(d)Patient 2. (a) 50 points on the
contours, (b) 100 points on the
contours, (c) 200 points on the
contours, and (d) 400 points on the
contours.
(a)(b)(c)(d)Patient 2. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 2. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 2. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 2. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 2. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 3. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 3. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 3. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 3. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 3. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)Patient 3. (a) 50 points on the
contours, (b) 100 points on the
contours, and (c) 200 points on the
contours.
(a)(b)(c)


### 10.1. Evolution Curves

We present the
evolution curves of the brain tumor as a function of time in Figures34, 35, and 36.Figure 34
Patient 1. Evolution of the tumor with respect to time.Figure 35
Patient 2. Evolution of the tumor with respect to time.Figure 36
Patient 3. Evolution of the tumor with respect to time.


### 10.2. Interpretation and Conclusion

In this section, we give an interpretation of the
results previously shown. Patients 1 and 3 have curves tumor growth which agree
with the expectations of doctors and the standard models of growth tumor. The
curve of the second patient is characterized by the volume evolution of the
tumor after a surgical operation. After a phase of growth, the two last dates
show that the tumor has been partially removed and that a remission phase
appears.Results are consistent with the clinical history and
natural course of low-grade gliomas. 3D segmentation is more sensitive and
consistent than linear measurement. A fully automatic method of volume and
contour segmentation will be of great interest for the clinician and patient
management. It is now recognizing that all low-grade gliomas (except type 1
such as pilocytic astrocytoma) will present an anaplastic transformation which
highly correlates with tumor growth.After this study, we remark that some geometric
invariants are stable if we increase the number of points, or if we change the
distribution of points. The Gauss curvature is always equal to4π, which is normal, because the triangulated surface is
a deformable sphere without holes. However, the mean curvature depends heavily
on the number of points. Once we increase the number of points, the surface
becomes increasingly disturbed.Our study helps to precise the importance of the
geometric invariants during the reconstruction of brain tumors in medical
imaging. It should be noted that another algorithm was tested to reconstruct
the tumor but topological problems were found, especially for the Gauss
curvature for some 3D objects. It seems essential to verify the stability and
the values of geometric invariants in order to give valid estimations to the
doctors.


## 10.1. Evolution Curves

We present the
evolution curves of the brain tumor as a function of time in Figures34, 35, and 36.Figure 34
Patient 1. Evolution of the tumor with respect to time.Figure 35
Patient 2. Evolution of the tumor with respect to time.Figure 36
Patient 3. Evolution of the tumor with respect to time.


## 10.2. Interpretation and Conclusion

In this section, we give an interpretation of the
results previously shown. Patients 1 and 3 have curves tumor growth which agree
with the expectations of doctors and the standard models of growth tumor. The
curve of the second patient is characterized by the volume evolution of the
tumor after a surgical operation. After a phase of growth, the two last dates
show that the tumor has been partially removed and that a remission phase
appears.Results are consistent with the clinical history and
natural course of low-grade gliomas. 3D segmentation is more sensitive and
consistent than linear measurement. A fully automatic method of volume and
contour segmentation will be of great interest for the clinician and patient
management. It is now recognizing that all low-grade gliomas (except type 1
such as pilocytic astrocytoma) will present an anaplastic transformation which
highly correlates with tumor growth.After this study, we remark that some geometric
invariants are stable if we increase the number of points, or if we change the
distribution of points. The Gauss curvature is always equal to4π, which is normal, because the triangulated surface is
a deformable sphere without holes. However, the mean curvature depends heavily
on the number of points. Once we increase the number of points, the surface
becomes increasingly disturbed.Our study helps to precise the importance of the
geometric invariants during the reconstruction of brain tumors in medical
imaging. It should be noted that another algorithm was tested to reconstruct
the tumor but topological problems were found, especially for the Gauss
curvature for some 3D objects. It seems essential to verify the stability and
the values of geometric invariants in order to give valid estimations to the
doctors.



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*Source: 210471-2009-03-23.xml*