Download Modelling of 3D Heart Motion

University of Paderborn
Graduate Seminar: Medical Images
Lecturer: Prof. Dr. Gitta Domik
Modelling of 3D Heart Motion
June 2005
Alexander Stecker
[email protected]
6078721
Michael Karch
[email protected]
6146668
Contents
1 Introduction
1.1 Motivation . . . . . . . . . . .
1.2 Why do we need 3D modelling?
1.3 Dictionary . . . . . . . . . . . .
1.4 Overview . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
2
2
3
3
MRI data
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
3
4
4
6
7
3 Modelling heart motion using Echocardiography
3.1 Shape detection method . . . . . . . . . . . . . . . . . . . . . . .
3.2 Usage in diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . .
8
9
11
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2 Construction of 3D heart motion from tagged
2.1 What is tagged MRI? . . . . . . . . . . . . . .
2.2 3D modelling based on non-rigid registration .
2.3 FEM-model based method using segmentation
2.4 Comparison . . . . . . . . . . . . . . . . . . . .
4 Modelling of a heart phantom with
4.1 NURBS-surface definition . . . . .
4.2 Model generation process . . . . .
4.3 Outlook and Benefit . . . . . . . .
5 Round-up and conclusion
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
NURBS
12
. . . . . . . . . . . . . . . . . 12
. . . . . . . . . . . . . . . . . 14
. . . . . . . . . . . . . . . . . 14
15
1
Introduction
This paper gives an overview of techniques used modelling 3D heart motion.
We will describe two approaches based on tagged Magnetic Resonance Imaging
(MRI) data, an echocardiography based method and the benefit of 3D heart
models based on Non Uniform Rational B-Splines.
1.1
Motivation
Why is it important to do heart related research? The numbers shown in Table
11 will give an answer which is easy to understand. The top three causes of
death in Germany are heart related, what is also a fact for most countries in the
western world. Another interesting aspect concerning non invasive heart analysis techniques is stated by the following quotation: ”A clinically useful method
for the automated 3D analysis of the shape and motion of the whole heart does
not yet exist.” (see [1]). Connecting the benefits of such an automated analysis (as shown in Section 1.2) with upcoming techniques like augmented reality
surgery may help us to reduce the number of deaths caused by cardiovascular
diseases.
cause of death
chronic ischemic heart disease
sudden myocardial infarct
cardiac insufficiency
occurrences
92.673
64.229
59.117
percentage of total deaths
10,9 %
7,5 %
6,9 %
Table 1: Leading cause of death in Germany 2003: cardiovascular diseases
1.2
Why do we need 3D modelling?
As we have shown in Section 1.1 it is important to support the diagnosis of
cardiovascular diseases. According to this, where are the advantages of a 3D
modelling of the heart? You should think about two things improving medical
diagnosis: a better or more exact, in the meaning of correctness, diagnosis and
faster results.
Data used by medical doctors in cardiac diagnosis is often acquired in form
of slices. To give an example MRI produces intersection views of the heart in
different positions. To get an overview you have to take a set of slices in different
positions and at different time steps. This image sets contain the information
you need but they are hard to understand. The slices are independent in time
and space. It’s the work of the operator or the medical doctor to interpret
this slices. At last the quality of the analysis depends on the routine and the
knowledge of the operator. The time needed for the analysis is given by the
amount of data.
In this task-set we would like to apply the (semi) automated 3D modelling
of heart motion. There are some aspects making this technique interesting: the
automated registration of slices reduces the routine you need to understand the
image sets. A perfectly animated 3D view of the heart may allow even untrained
(in the meaning of reading e.g. MRI images) medical doctors to interpret the
1 see:
Statistisches Bundesamt, http://www.destatis.de/basis/d/gesu/gesutab20.php
2
information fast and correct. The amount of time connecting the data will be
reduced by the automation which leads to a faster diagnosis. As a conclusion
you may say: an automated 3D modelling may lead to an easier (probably
better) and faster diagnosis.
1.3
Dictionary
This section will give an short overview of technical terms used in the paper
(especially concerning the heart).
Cardiac simply means heart related.
Long Axis / Short Axis (abbr.: LA / SA) means the view at the heart. The
long axis view is taken from the front or the side of the heart, whereas the
short axis view is taken from the top or the bottom.
Left & Right Ventricle (abbr.: LV, RV) are the technical term in medicine
to name the left and the right chamber of the heart. The left ventricle
(see Figure 3, right part of the image) is nearly an oval, whereas the right
ventricle can be seen as a half-moon shape.
Myocardium is the Latin word for the heart-muscle. The heart muscle denotes
the biggest part of the heart wall.
Diastole names the relaxation of the heart during the cardiac cycle.
Systole is the opposite of Diastole, which means the contraction phase of the
cardiac cycle.
1.4
Overview
The content of this paper is roughly divided into two parts. The first part (see
Chapter 2) done by Alexander Stecker presents two methods dealing with the
modelling of 3D heart motion based on tagged MRI data. Michael Karch describes in the second part (see Chapter 3 and 4) the use of Echocardiography for
modelling 3D heart motion and how to construct a phantom heart model using
NURBS. The two parts are completed by an introduction (see 1, Alexander’s
work) and a round-up (done by Michael).
2
Construction of 3D heart motion from tagged
MRI data
This chapter deals with the construction of 3D heart motion based on data
acquired by tagged MRI. First we will give a short explanation what tagged
MRI is (see 2.1), the following sections describe to different methods to derive a
3D model from this images (see 2.2 and 2.3). At last we will give a conclusion,
comparing the different methods.
3
2.1
What is tagged MRI?
As MRI is described in another paper in this seminar (Tu-Binh Dang, Methods of creating images for medical diagnosis) we only give a short description
of the function of MRI. The second part talks about the extended MRI version
called tagged MRI. Simplified you may say MRI (also called MRT : MR Tomography) uses a magnetic field to excite a resonance response of atomic nuclei in
soft tissue. The rotation of the nuclei can be measured indirect with coils.
The idea behind tagged MRI is based on the fact that a regular perturbation
of the magnetic field will take effect on the resulting images. In Figure 1(a)
you can see a rectangular structure on the myocardium. Due to the distorted
magnetic field the nuclei under the structure wont get stimulated and no rotation
effect can be measured. During the heart cycle the perturbation remains on the
heart and shows its contraction (cf. Figure 1(b) ).
2.2
3D modelling based on non-rigid registration
The description of this method is based upon the paper from Chandrashekara
et al. (cf. [2]). The main idea is to perform an extraction and analysis of
the myocardium an its deformation. The deformation field is constructed using
a non-rigid registration. To achieve a registration a structure called free-form
deformations (FFD) is used. ”A volumetric FFD is defined...by a mesh of nx ×
ny × nz control points Φ.” (cf. [2]). An example of such a mesh can be found
in Figure 2.
To describe the motion of this mesh we would like to have a transformation
T that for a known starting point u at known time t calculates a resulting point
u0 . The formula is shown in 1.
T (u, t) = u0
(1)
Registering the volume image (represented by the FFD) at time t, Vt , to the
segmented volume of the myocardium at time 0, V0 , will result in the deformation field. The registration is done by the optimization of a cost function. This
(a)
(b)
Figure 1: A tagged MRI picture at the end of diastole (a) and systole (b)
4
Figure 2: Example of a Free Form Deformation
cost function basically consist of the sum of the so called ”normalized mutual
information” (NMI, cf. [3]). This NMI can be calculated as shown in Formula
2.
Csimilarity =
H(A) + H(B)
H(A, B)
(2)
H(A) and H(B) represents ”the marginal entropies of the images A and B”
cf. [2]), whereas H(A, B) denotes the joint entropy of both images. Due to fact
that the LA and SA images may consist of a different number of voxels, we need
to do an alignment to fit the weight of the NMI components in the similarity
measure. Such a weight may be calculated as shown in Formula 3 and 4.
N (VSA,0 )
N (VSA,0 ) + N (VLA,0 )
N (VLA,0 )
=
N (VSA,0 ) + N (VLA,0 )
wSA =
(3)
wLA
(4)
N (x) gives the number of voxels of an image. Applying this formulas to the
similarity measure in a combination of LA and SA images will result in Formula
5.
C(Φ)
= Csimilarity (VSA,0 , T (VSA,t ), VLA,0 , T (VLA,t ))
(5)
H(VLA,0 ) + H(T (VLA,t ))
H(VSA,0 ) + H(T (VSA,t ))
+ wLA
= wSA
H(VSA,0 , T (VSA,t )
H(VLA,0 , T (VLA,t )
The values in Φ describes the parameters of the transformation T .
5
To obtain a continuing result, we have to describe the transformation over
the complete sequence of images. With one transformation called Tlocal for one
time step, you can describe the complete transformation as follows.
T (u, t) =
t
X
h
Tlocal
(u)
(6)
h=1
The resulting transformation may be visualised as a plot of arrows, pointing
in the movement direction of each voxel. Comparing the results of this automated registration and motion detection with a manually analysed sequence, we
get a root mean square error (rms error) below 3mm. In most cases the error
was smaller than a voxel, which means that a better result depends on better
input data.
2.3
FEM-model based method using segmentation
The techniques presented in the second part are published in the papers of
Metaxas et al. (see [1] and [4]). This second approach differs from the first
in several ways. Instead of 7 images per time step as used in Section 2.2 we
only need 3 slices, 2 SA images and 1 LA image. This lack of information is
compensated by a priori knowledge, added to the data set in form of a Finite
Element Methode model. This FEM-model describes a healthy heart, as is can
be seen in Figure 3. Another difference becomes obvious while looking at the
picture. The FEM-model based method is able to model the motion of both, left
and right ventricle. In addition you may even derive the behaviour of the heart
wall (containing the myocardium). Further reading can be found in chapter 4
of this paper (modelling of a 3D heart phantom).
Figure 3: FEM model of the heart
The following task is to match the phantom with the taken images. To
obtain the integrity of the phantom it is not direct manipulated but deformed
by the adjust of certain parameters. This set of parameters is connected to
the phantom and allows the algorithm to adjust the structure in meanings of
6
length, weight, thickness, etc. As a result you will always get a correct model
of a heart, manipulated by parameters derived from the MRI data.
(a)
(b)
Figure 4: Image of a heart with segmentation of tag-lines, LV, RV and myocardium
The data is analysed by an automated segmentation algorithm, supporting
the work of the anatomist. A visualised result can be found in Figure 4. Image
4(a) shows the heart at end-diastole. The tag-lines, denoted by small circles,
are almost parallel (due to the perturbation of the magnetic field at the time of
end-diastole). The left and right ventricle and the myocardium are surrounded
by thin lines showing the border of each part. Image 4(b) shows the same heart
at end-systole. You can see the change of tag-lines in some areas an the smaller
circles around LV and RV due to the contraction of the heart. In a next step the
algorithm gathers information described by the segmentation result, calculates
parameters and applies them to the FEM-model of the heart. As done for every
time step you get a discrete version of the heart motion. Interpolation between
the different FEM-models will result in a fluent motion, applicable for easy
diagnosis.
2.4
Comparison
We have shown the function of two different approaches trying to model the
motion of the heart in 3D. As based on the same kind of input data it would be
of interest to make a comparison. We will use FFD and FEM to label the two
different techniques. Starting with the input data you may denote that FFD
needs up to 7 slices of a time step where FEM is able to produce results with
even 3 slices. The kind of tagging also differs in both methods. FFD is free in
the choose of a pattern whereas the FEM algorithm is applied for one special
pattern (it’s possible to change this implementation, but there exists no generic
approach). With the a priori knowledge of the FEM approach it is possible
to model the behaviour of the whole heart compared to FFD allowing only to
model the left ventricle. The RMS error for FFD is below 3mm, FEM is not
mentioned. According to the faster analysis (in comparison to a ”hand made”
segmentation) and the unknown error of FEM it would be applicable in a rapid
diagnosis or as a intuitive representation of MRI data for non MRI experts.
FFD achieves an error rate most times smaller than a voxel size which means
the results may be useful for diagnosis and therapy.
7
3
Modelling heart motion using Echocardiography
The modelling method introduced in [5] uses echocardiographic images to construct a three dimensional model of the left ventricle of the human heart. The
model is represented by a surface mesh. Echocardiography is usable to scan
non-rigid objects, like the heart. However we do not model the whole heart,
see Figure 5, but it’s left ventricle. Medical diagnosis focuses on left ventricle
to detect coronary artery diseases or Cardiomyopathy. Cardiomyopathy is a
serious disease in which the heart muscle becomes inflamed and does not work
as well as it should. There may be multiple causes including viral infections.
Which benefits can be achieved using a three-dimensional view for diagnosis is
described in Chapter 3.2.
The method described uses two-dimensional ultra sonic images, taken from
different angles and in different axis, to gather the data necessary to form a
three-dimensional model. Those images were used in diagnosis back in 2002,
the year the paper [5] was published. The images were taken from a position
above the chest, camera facing the patient. The therapist then had to interpret
a time series of those images to create a diagnosis. The problem for him was
that image quality is varying with the patients. This can be compensated with
a priori knowledge about the heart’s shape and movement during Diastole and
Systole.
Figure 5: Sketch of the human heart
There are two ways of how to acquire two-dimensional image slices angled
around the heart to use them for this method. One is to move the scanner head
subsequently a few degrees to the side after each slice was taken. The downside
is that this method takes roughly a few minutes to cover a scan angle of 180
degrees. The heart and the patient itself are moving during this period of time.
8
To solve this unsatisfactory approach parallel scanners were made available.
With them all slices can be recorded at one point of time. Such scanners are a
technological challenge and were not widely spread in 2002. The trend to use
such parallel scanners was clear back then and it was assumed they would be
ubiquitous in the near future.
To model the left ventricle we use a so called Two Simplex Mesh. It combines
the advantage of using relatively few vertices with an accurate reproduction of
shape. Next section we give us the idea how the LV’s shape is detected and
transformed into the model. The scans of nine men and three women build the
data source for this paper. All of them were in good health and aged 25 to 35.
3.1
Shape detection method
As already mentioned, a priori knowledge about the shape of the heart and it’s
movement during the cardiac cycle is essential to gain a high level of robustness
and speed up the whole procedure. The normal shape of the left ventricle allows
us to roughly position the model there where we suspect it and thus keeps the
shape modification step short.
Figure 6: Initial position of the mesh (a sphere), before edge detection
The workflow of the shape detection method is the following: At first the
therapist or operator manually marks four landmarks on the images from the
first time frame. The shape detection and refitting of the initial mesh begins.
In all following time frames the shape detection does not start with the already
defined initial mesh or with another landmark setup, but with the previous
mesh.
In Figure 6 you see the an initial mesh positioned by the four landmarks of
which three are located on the mitral valve and one on the apex. In this figure
however a sphere mesh, instead of a predefined LV mesh, was used. Using a LV
mesh would increase the quality of initialisation.
9
The mesh used to model this deformable closed-surface model is a Two
Simplex Mesh. Each vertex is connected to exactly three neighbours. A vertex
is defined by three barycentric coordinates and a simplex angle defining the
elevation of it’s place. The surface normal is also the search line which you can
see in Figure 7. The search for the subsequent position of the according vertex
happens along this search line. The resolution of the mesh, meaning the count
of vertices used, is dynamic. It can be modified manually or automatically by
the implementation of this method. Usually 500 vertices are sufficient to give a
good description of the myocardium.
Figure 7: Searchline used for possible wall detection
The implementation searches for LV walls by scanning the intensity of the
given image. A large change in intensity from black to white is the decision
factor for wall detection. This process is described as the 3D active objectbased segmentation (3DAO).
As mentioned before we do not want to start with the initial mesh for each
time frame. We use the mesh generated in the previous time frame and use
information about the cardiac cycle. The knowledge about the cardiac cycle
lets us determinate contraction and rotation of the heart in the next time frame
and was collected from a mean heart. There are four deformation parameters
known for the whole heart cycle: deformation in all three axis and the apical
rotation angle. They are represented as indexed time curves, Figure 8, starting
end-Systole where the index is set to 1. To be able to know at which point of
the cardiac cycle the next time frame is location we need to work with tagged
data. Gerard et al. used MRI tagged data obtained by CSPAMM2 .
Figure 8: Diagram on the left shows the apical rotation, while diagram on the
right side shows deformation factor in all three spatial directions
2 CSPAMM:
complimentary spatial modulation of magnetization
10
The given parameters let us estimate the change between two time frames
which can be quite significant. We apply the deformation and rotation to our
model generated in the previous time frame and use the result to initiate the
subsequent time frame. The result is a deformable surface model as shown in
Figure 9. The model can be shaded to use it more intuitively in diagnosis.
Figure 9: Resulting 3D surface model on top of respective echocardiographic
heart scan
3.2
Usage in diagnosis
This model can be used for wall motion analysis. Hence it represents the LV’s
walls it is intuitively reasonable to use it for this kind of analysis. If we know
the position of the walls, we are able to derive from this knowledge the volume enclosed by them. This enables us to study the LV volume during the
cardiac cycle. Another directly connected method in diagnosis is to detect the
local extent of contraction and relaxation as well as the largest motion between
them. There exist automated tools implemented for all the described usage in
diagnosis.
The popular bull’s eyes view among therapists can also be generated from
this model. The 16 regions of the polar map are located by the landmarks we
selected in the very first step of this procedure. Their colour is determined by
the weight of the faces of the mesh in their region. The weight of a face is equal
to it’s surface area. Finally the total weight is matched to a colour scale which
can be manually adapted.
The main application is the 3D view of the beating heart, actually it’s left
ventricle, over the whole cardiac cycle. Figure 10 shows the difference between
the traditional and the 3D method. You can see the visual improvement of the
3D representation which is located on top of the image slices used to generate
it.
11
Figure 10: 3D model compared to 2D image slices
4
Modelling of a heart phantom with NURBS
In this chapter we will study a method, introduced in [6], to generate a computer
model of a heart phantom. A phantom is an artificial organ used for research
purposes. We saw the use of phantoms in other talks during this seminar. They
were used to compare actual gathered patient data to a healthy patient data or
as a source of knowledge about shape and movement of an organ. This was the
case in previous Chapter 3.
The goal of Segars et al. was not to generate a 3D model only, but to extend
it into the fourth dimension. The fourth dimension is the time component which
allows us to see the heart model in different states of the cardiac cycle. Also this
model is more flexible then other e.g. pixel-based or solid models. To begin we
will review the NURBS definition, then see how the model itself was created.
At the end of this chapter we will discuss the advantages of the method used.
4.1
NURBS-surface definition
First we should have a look at the definition of non-uniform rational B-Splines,
called NURBS. It is a mathematical representation of a curve or, in this case, of
a surface. The surface spreads in u and v dimension, the elevation is calculated
by the given formula:
12
n P
m
P
S(u, v)
=
Ni,p (u)Nj,q (v)wi,j Pi,j
i=0 j=0
n P
m
P
(7)
Ni,p (u)Nj,q (v)wi,j
i=0 j=0
with 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1
It uses the basic functions N , given by recursive definition (8) and (9). The
degree in u direction is given by p, respectively by q in v direction. Controlpoints are given by Pi,j as illustrated in Figure 11. Each control-point has a
weight wi,j assigned.
Figure 11: Modeling a NURBS surface with control-points in 3D
Ni,0 (u)
=
Ni,p (u)
=
1 if ui ≤ u ≤ ui+1
0 otherwise
u − ui
ui+p+1 − u
Ni,p−1 (u) +
Ni+1,p−1 (u)
ui+u − ui
ui+p+1 − ui+1
(8)
(9)
The definition of N (v) is almost alike. Simply substitute u to v and p to q.
The both essential knot-vectors U and V , Equation 10 and 11, must contain a
sequence of monotone increasing elements.




U = 0, . . . , 0, up+1 , . . . , ur−p−1 , 1, . . . , 1
| {z }
| {z }
p+1
(10)
p+1




V = 0, . . . , 0, vq+1 , . . . , vs−q−1 , 1, . . . , 1
| {z }
| {z }
q+1
q+1
with r = n + p + 1 and s = m + q + 1
13
(11)
4.2
Model generation process
The basic approach to create a NURBS surface is to define a set of controlpoints. The control-points were manually picked from MRI3 tagged data. The
MRI data was acquired by scanning a normal patient. The heart cycle was
divided into 15 time frames. Each frame made up of multiple 1cm thick MRI
images with a pixel size of 1.56mm by 1.56mm.
About 200 control-points were selected for every relevant part of the heart:
Atria, Ventricles, Inner-walls and Outer-walls. This selection process was accomplished with a software called SURFdriver, which was used to display the
images and to choose the control-points. This software was also used in the
following step to reconstruct a surface from the defined control-points.
By applying the same procedure to every single time frame we return 15
models of a phantom. The final step is to enter the fourth dimension. This is
described as the skinning procedure. In this procedure data-points are spread
over the NURBS surface. The result is a set of 4D data-points. Between those
a surface is interpolated for each point of time.
The resulting model proved to be valid in the evaluation which was undertaken. Heart volume and muscle mass were tested. The heart volume was
similar to an already adopted phantom model. The fluctuation in volume are a
result of the single patient source for the NURBS model compared to multiple
patient source for the compared model. The test on muscle mass returned a 2%
deviation, which is acceptable.
4.3
Outlook and Benefit
Why is this model a step into the right direction? Usual pixel-based models are
fixed to the anatomy of the patient who was the data source for that specific
model. Also pixel-based models are bound to the resolution they were created
with. However they are more realistic because they match exactly with the
source anatomy. Geometry-based models lack this realism but are more flexible.
Our NURBS model incorporates both realism and flexibility. This advantage
is visible by comparing the NURBS phantom to the geometry-based MCAT
phantom as shown in Figure 12. The MCAT phantom was developed to meet
the same demands but lacks realism.
Figure 12: Comparing geometry-based phantom on the left to NURBS phantom
on the right
Another advantage of the NURBS model is the richness of details we are
3 MRI
: Magnetic Resonance Imagining
14
able to achieve. The developers however want to introduce more details into
the model by a more granular modelling, what can be achieved by a better
MRI scan resolution (The MRI slices taken as source data for this model were
1cm thick), but also adding more details, e.g. more valves, to the heart. It is
intended to use NURBS to create realistic models of other organs or even of the
whole human body.
5
Round-up and conclusion
Finally we want to summarize the four methods introduced in this paper.
Both FFD and FEM based methods, see Chapter 2, enable the therapist to
run an automated analysis of the heart. An automated analysis allows faster
results. Based on the exact data of a MRI scanner in combination with the
tagging of images it is possible to do a high level registration of the images.
The obtained data can be used for a fast diagnosis and may allow even non
MRI experts to interpret the informations. In a conclusion you may say that
3D modelling of heart motion increases the benefit of MRI data.
The modelling of left ventricle using Echocardiography, compare Chapter 3,
takes only four minutes on a standard PC with the method described in Chapter
3. Those four minutes already include the manual landmark detection, needed
to start the computation procedure. This is an enormous speed-up compared
to other methods which take several hours. Also this result helps to create a
more reliable and robust diagnosis in a shorter time.
The downside is the need of a priori knowledge of a mean heart and it’s
movement during time. Because the mesh from a previous time frame is needed
to create the mesh for the current frame it is not possible to parallelise this
method by assigning each time frame data to one computing node.
In Chapter 4 we saw a method to generate a phantom using NURBS. Modelling with NURBS is a well studied topic in computer graphics. NURBS were
first used in the 1980s. They are very flexible to define almost all possible kind
of shapes and can be calculated fast and accurate by numerically stable algorithms. One further advantage is their small need for storage space compared
to e.g. voxel-based models, since they store their control-points only. The most
significant point mentioned in [6] was the fact that NURBS models are not limited by resolution. Control-points can be relocated by affine transformation and
the model can be exported to a pixel-based model at any resolution.
Because the control-points selection method requires a lot of user interaction
this method cannot be used in diagnosis for every patient efficiently. It seems
this set was very time consuming and is therefore only applicable to generate a
phantom, what is done only once. This phantom can be used in medical studies
or for testing and training. Since it is a computerized model it is more flexible
then a solid phantom.
15
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
A tagged MRI picture at the end of diastole (a) and systole (b) .
Example of a Free Form Deformation . . . . . . . . . . . . . . . .
FEM model of the heart . . . . . . . . . . . . . . . . . . . . . . .
Image of a heart with segmentation of tag-lines, LV, RV and
myocardium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sketch of the human heart . . . . . . . . . . . . . . . . . . . . . .
Initial position of the mesh (a sphere), before edge detection . . .
Searchline used for possible wall detection . . . . . . . . . . . . .
Diagram on the left shows the apical rotation, while diagram on
the right side shows deformation factor in all three spatial directions
Resulting 3D surface model on top of respective echocardiographic
heart scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3D model compared to 2D image slices . . . . . . . . . . . . . . .
Modeling a NURBS surface with control-points in 3D . . . . . .
Comparing geometry-based phantom on the left to NURBS phantom on the right . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5
6
7
8
9
10
10
11
12
13
14
References
[1] Kyoungju Park, Albert Montillo, Dimitris Metaxas, and Leon Axel. Volumetric heart modeling and analysis. COMMUNICATIONS OF THE ACM,
48(2):43–48, February 2005.
[2] Raghavendra Chandrashekara, Raad H. Mohiaddin, and Daniel Rueckert.
Analysis of 3-d myocardial motion in tagged mr images using nonrigid image registration. IEEE Transactions on Medical Imaging, 23(10):1245–1250,
October 2004.
[3] C. Studholme, D. L. G. Hill, and D. J. Hawkes. An overlap invariant entropy
measure of 3d medical image alignment. Pattern Recognition, 32(1):71–86,
1998.
[4] Idith Haber, Dimitris Metaxas, and Leon Axel. Using tagged mri to reconstruct a 3d heartbeat. IEEE Computational Science and Engineering,
2(5):18–30, September-October 2000.
[5] O. Gerard, A.C. Billon, J.-M. Rouet, M. Jacob, M. Fradkin, and C. Allouche. Efficient model-based quantification of left ventricular function in
3-d echocardiography. IEEE Transactions on Medical Imaging, 21(9):1059–
1068, September 2002.
[6] W.P. Segars, D.S. Lalush, and B.M.W. Tsui. A realistic spline-based dynamic heart phantom. IEEE Transactions on Nuclear Science, 46(3):503–
506, June 1999.
16