Download Creation of a Human Heart Model and its Customisation Using

Creation of a Human Heart Model and its Customisation
using Ultrasound Images
Rolf F. Schulte1,2 , Gregory B. Sands1 , Frank B. Sachse2 , Olaf Dössel2 , Andrew J. Pullan1
1
2
Bioengineering Research Group, The University of Auckland, New Zealand
Institute of Biomedical Engineering, University of Karlsruhe (TH), Germany
Introduction
The inverse problem of electrocardiology might provide
a powerful clinical investigation method for visualising
the electrical activity of the heart. To use this method
one requires accurate models of the human torso and
heart.
The objective of this work was to create an accurate
model of the human ventricles including the valves from
images recorded using Magnetic Resonance Imaging
(MRI). This model is used as a “generic” model, and
is adapted to a given individual with a host mesh fit to
spatially registered Ultrasound (US) images.
Finite Element Method
The cubic Hermite basis function is a high order interpolation function which provides zero and first order
continuity across element boundaries. Accurate models
can often be created with relatively few elements and
nodes. Therefore, it is a powerful surface descriptor.
The scalar value u at a position ξ (0 ≤ ξ ≤ 1) inside
one element is defined by
u(ξ)
=
+
du
dξ
1
du
Ψ02 (ξ) u2 + Ψ12 (ξ)
dξ 2
Ψ01 (ξ) u1 + Ψ11 (ξ)
with the four cubic Hermite basis functions
Ψ01 (ξ) = 1 − 3ξ 2 + 2ξ 3
Ψ02 (ξ) = ξ 2 (3 − 2ξ)
Ψ11 (ξ) = ξ(ξ − 1)2
Ψ12 (ξ) = ξ 2 (ξ − 1)
Two- and three-dimensional elements are created with
a tensor product to form the quadrilateral elements
needed for the representation of surfaces or volumes.
Before any geometry can be fitted, the given data points
are projected onto the surfaces of the mesh using orthogonal projection. Hence pairs of points are created
for the subsequent fitting procedure.
Sobolev smoothing adds a second term to the objective
function on the derivatives of the basis function and
helps to obtain smoother meshes with a more uniform
shape of elements.
Epicardium
Left endocardium
Right endocardium
data
points
2389
933
944
Number of
nodes elements
62
62
62
70
70
70
fixed
nodes
0
14
14
Table 1: Number of data points and various details
about the mesh topology.
Generic Model of Ventricular Surfaces
The first objective was to create a generic model of the
ventricular surfaces based on data classified from the
MR images. The desired application for the solution
of the inverse problem in electro-cardiology with the
Boundary Element Method (BEM) leads to certain restrictions. Only surfaces are used with the boundary
element method, thus a bicubic Hermite interpolation
function is chosen. The model represents the ventricles
and the valves, which are closed off to fulfil the boundary conditions of the problem. Papillary muscles and
trabeculae carneae need to be excluded due to their
unknown electrical properties and uncertain individual
differences.
The images used are taken from the Cardiac MRI
Anatomical Atlas [1]. MR images were obtained from
a healthy 28 year old, and recorded at 5 mm intervals on multiple planes with a T1-weighted dark blood
breath-hold sequence. The images were ECG-gated in
end-diastolic state.
All images are segmented and classified into epicardium, left and right endocardium, and the four
valves, i.e. Aortic, Mitral, Pulmonary, and Tricuspid.
Methods
The three surfaces of the heart are modelled separately
in independent surface meshes: epicardium, left and
right endocardium. Each mesh uses a similar topology
with eight nodal layers in the longitudinal and ten layers in the circumferential direction. The base and apex
are closed with sector elements.
The Initial Mesh is generated from scratch and graphically modified to obtain a favourable arrangement of
nodes and elements. The valves are included into the
left and right endocardial models only. Nodes defining
Iteration
number
1
2
3
Smoothing
∂u
∂ξi
∂2u
∂ξi2
∂2u
∂ξ1 ∂ξ2
0.7
0.5
0.1
0.5
0.1
0.1
1.0
0.2
0.1
RMS error (mm)
epi
lv
rv
3.35
2.94
2.82
2.89
2.74
2.60
3.07
2.67
2.44
Table 2: The root mean square (RMS) errors after each
geometrical fitting and the Sobolev smoothing parameters.
each valve ring are fitted first using a 1D cubic Hermite basis function and are fixed in place during the
subsequent surface fitting procedure.
Figure 1: Host and slave mesh after the gross alignment
(left: RMS error: 7.3 mm) and after the first fit (right:
RMS error: 4.2 mm)
Results
A picture of the surface model is shown in Figure 2
and various facts about the topology and the fitting
are listed in Tables 1 and 2.
Host Mesh Fitting
The main use of the surface model is to obtain individual patient specific models of the human ventricles
for a more accurate solution of the inverse problem in
electro-cardiology.
MRI is the most accurate in-vivo cardiac imaging
method, but it is expensive and the model generation is
time-consuming with plenty of manual interaction required. US is cheaper and more universally available,
but some regions of the heart are barely visible, e.g.
the right ventricular free wall. Thus it is not sufficient
for an accurate model generation. A host mesh fitting
procedure combines the advantages of both investigation methods, as it matches the given US data where
present while still providing shape information in hidden regions.
Methods
All US images are obtained using a standard HP Sonos
5500 US machine with a 4 MHz phased array transducer. The spatial information is obtained with a magnetic receiver and transmitter. Each image is obtained
immediately following the QRS-complex, i.e. at enddiastole. The images are segmented and classified manually into left and right epicardium, left endocardium,
right endocardial septum and free wall, Mitral, Tricuspid and Aortic valves and the left endo- and epicardial
apex.
The host mesh uses a tricubic Hermite basis function,
as the shape of the elements can be retained with the
Sobolev smoothing. The topology is a simple cubic
shape with equidistant refinement to give between 1
and 4 elements per side.
The first step is to embed the slave mesh into the host
mesh. This attaches the nodal coordinates and derivatives of the slave (ventricular) mesh to the ξ-positions
of the host mesh. Next, a gross alignment of both the
host and slave meshes is performed which incorporates
a translation and rotation using several identifiable control points: the epicardial, left and right endocardial
centroids, the left endo- and epicardial apices and the
centres of the Aortic and the Mitral valves.
Following this, the US data are projected orthogonally
onto their corresponding surfaces of the ventricular
mesh, which also defines their ξ-location in the host
mesh. The host mesh is fitted by minimising the errors
between the data and projection points. Finally, the
slave mesh is updated, which means calculating new
nodal positions and derivatives of slave mesh from the
deformed host mesh. The data point projections onto
the surfaces can then be performed again, and the fit
repeated to gain greater accuracy.
Results
The host mesh fitting procedure yields fairly low RMS
errors and preserves an anatomically realistic shape in
the presence of sparse US data, with few distortions as
visible in Figure 1. The adaption to US data of the same
person leads to the smaller RMS error of 3.5 mm in
contrast to 4.9 mm for Patient 1 or 4.2 mm for Patient
2 (single fitting step with a host mesh refinement of
two).
Extensive comparisons were made for obtaining proper
parameters for the host mesh fitting procedure.
Volumetric Model of Ventricular Walls
An additional objective of this work was the creation
of a volumetric model of human ventricles for electrical
and mechanical simulations, which require consistency
of the ξ-directions to be maintained. This leads to a
complicated topology especially in the area of the valve
plane. Furthermore it is necessary to close off these
valves.
The data points are the same as used for the creation
of the surface model.
Methods
A tricubic Hermite basis function is chosen together
with a three-dimensional rectangular Cartesian coordi-
data
points
4096
nodes
296
Number of
elements
volume
surface
214
262
fixed
nodes
50
Table 3: Number of data points and various details
about the mesh topology.
Iteration
number
1
2
Figure 2: Surface model Figure 3:
Volumetric
from anterior view.
model from anterior view.
Smoothing
∂u
∂ξi
∂2u
∂ξi2
∂2u
∂ξ1 ∂ξ2
1.4
0.5
1.0
0.1
2.0
0.2
RMS error
(mm)
4.00
3.83
Table 4: The root mean square errors after each geometrical fitting and the Sobolev smoothing parameters.
Discussion and Outlook
nate system. The model consists of eight nodal layers (plus the apex) in the longitudinal direction, eleven
layers circumferentially and three layers transmurally.
The mid-myocardial layer bifurcates to model the right
endocardium.
All valves are included into the model by a separate
fit beforehand. During the main fitting procedure the
nodal positions and the ξ1 -derivatives are fixed. An
offset in the x-direction yields a thickness of 3 mm for
the valve plane.
The Mitral valve lies in the imaginary centre of the
model and is conveniently closed off with sector elements. The three other valves are closed off with four
new elements, which are connected to new nodes at
their centroids. One side is collapsed manually to make
use of different versions for the nodal derivatives.
A proper initial mesh is obtained in several steps. The
first mesh is generated from the surface model and uses
a similar topology to the canine heart model [2, 3].
Three more layers are added to model the valve plane.
The whole mesh is then refined twice in the longitudinal and once in the circumferential direction. Manual graphical modification leads to a favourable initial
mesh.
The first step for the geometric fitting is the orthogonal projection of data points onto the corresponding
surfaces. The main fit minimises the errors of epicardial, right and left endocardial surfaces only. This is
achieved in one step by connecting up the nodes into
two-dimensional elements with a bicubic Hermite interpolation function.
Finally, this surface mesh is transformed into a truly
volumetric mesh by linearly interpolating the nodes inside the left ventricular mid-myocardium and updating
the transmural ξ3 derivative.
The developed surface model provides a suitable model
for the desired application in the solution of the inverse
problem in electro-cardiology using BEM.
The host mesh fitting yields reasonable results, with individual customisation not distorting the heart model
unrealistically. However, maybe the most important
verification for the model is the comparison to MR images of another person. First, the generic heart needs
to be adapted to the US data of that person. MR images of this person need to be segmented and classified
and compared to the host mesh fitted generic model.
The volumetric model is designed to be used for electrical and mechanical simulations. The fibre and sheet
information still needs to be included either by rules or
from measurements on a human heart.
References
[1] A Young and B Cowan,
http://www.scmr.org/.
“Cardiac atlas,”
[2] PMF Nielsen, IJ LeGrice, PJ Hunter, and
BH Smaill, “Mathematical model of geometry and
fibrous structure of the heart,” Am J Physiol, vol.
260, pp. H1365–H1378, 1991, Heart Circ Physiol.
29.
[3] PJ Hunter, BH Smaill, PMF Nielsen, and
IJ LeGrice, Computational Biology of the Heart,
chapter 6: A Mathematical Model of Cardiac
Anatomy, pp. 171–215, John Wiley & Sons Ltd,
1997, ISBN 0-471-96020-9.
Acknowledgements
I would like to thank Alistair Young and Brett Cowan for
kindly providing the Cardiac Atlas and Steve Thrupp for
segmenting and classifying these MR images.
Results
A picture of the volumetric model is shown in Figure 3
and various facts about the topology and the fitting are
listed in Tables 3 and 4.