Download Construction of the Heart`s Conduction Tree via Prim`s Algorithm

POSTER 2016, PRAGUE MAY 24
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Construction of the Heart’s Conduction Tree via Prim’s
Algorithm
Leonie KORN
Chair of Medical Information Technology, Helmholtz-Institute for Biomedical Engineering, RWTH Aachen University,
52074 Aachen, Germany
[email protected]
Abstract. The heart’s excitation is a complex process which
is often modelled for several cases of applications. Realtime capability is often postulated but not guaranteed due to
the huge amount of used cells. Therefore, in this paper an
abstract conduction system was developed to reduce complexity. To distinguish between different cardiac regions, a
segmentation of the heart has been performed. The cardiac
conduction system is divided in its regions of tissue in which
the prim’s algorithm has been applied to identify their directed minimal spanning trees for conduction orientation.
As a result an abstract anatomic model which includes the
orientation of the main direction of propagation has been
developed. The model’s potential impact is to reduce complexity for real-time modelling of the heart’s excitation.
Keywords
Heart excitation, Tree, Prim’s algorithm, Modelling
Real-time systems.
prim’s algorithm is given. Section 3 shows the results of
modelling. The paper ends with a discussion.
2. Conduction tree construction
2.1. Modelling the heart’s anatomy
The heart’s anatomy has been modelled using a Computer Tomographical (CT) image series of 76 images with a
resolution of 512x512 (from [2]). A pre-segmentation of this
dataset has been performed by windowing. This was done
according to the Hounsfield scale which describes quantitatively the radiodensity in gray-scale values. It is defined by
the linear attenuation coefficient of water μwater as seen in
Eqn. 1.
μtissue − μwater
(1)
HU = 1000 ∗
μwater
Typical values for different substances can be seen in Tab. 1.
Thus, the heart is separated from surrounding pulmonary
Substance
1. Introduction
The heart’s pumping function is triggered by electric
impulses. These impulses have their origin at the sinoatrial
node and spread over the atrium to the atrioventricular node,
the left and right bundle branches, through the purkinje fibres
into the ventricle’s myocardium [1]. This initiates the essential contraction of the ventricles, so that blood is pumped into
the cardiovascular system. More closely these impulses are
cardiac action potentials, characteristically shaped and generated at every heart cell. The excitation in the heart cells is
directly related to the recorded biosignals in and on the body
volume conductor. The large amount of heart cells result in
a huge complexity regarding biosignal’s modelling but are
significant for the signal’s shapes. That’s why for real-time
applications an abstraction of this complex conduction system is necessary. Therefore, in this paper an abstract model
of the heart’s conduction system has been developed.
This paper is organized as follows. In section 2 the construction of the conduction tree is described. In detail, modelling
of the heart’s anatomy and the description of the modified
Hounsfield sclae [HU]
Air
-1000
-500
Lung
Fat
-100 to -50
0
Water
15
Cerebro-spinal fluid
30
Kidney
Blood
35 to 45
Muscle
10 to 45
Liver
Soft Tissue
Bone
40 to 60
100 to 300
700 to 3000
Tab. 1. Typical values of the Hounsfield scale [4].
tissue, bones and air. For well perfused soft tissue the
Hounsfield scale is not sufficient, since the transitions between various soft tissue is not clear. Therefore, a manual segmentation with ITK-SNAP was done to distinguish
between atrium, ventricles, endocardium and epicardium of
the heart (see Fig. 3). With this tool the Sagittal plane, the
Coronal plane and the Transversal plane are visible anytime.
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L. KORN, CONSTRUCTION OF THE HEART’S CONDUCTION TREE VIA PRIM’S ALGORITHM
CT-Dataset
approx. 40 mio voxel
ITK-SNAP:
Segmentation
Ventricle selection
Surface data
approx. 40.000 vertices
MeshLab:
Smoothing
Decimation
Surface data
approx. 400 vertices
Fig. 1. Steps of the heart segmentation.
z[mm]
Orientation in these three orthogonal planes is simple because of a shared cursor. The manual segmentation is done
due to the atlas of human anatomical cross sections [3]. To
generate a convenient abstraction of the heart’s conduction
system, the surface data from the segmented parts from the
heart was processed with MeshLab. The heart’s surface data
was smoothed with a HC Laplacian Smooth filter and decimated by a Quadric Edge Collapse Decimator. Here, the
heart’s surface consists of triangulations. The separation between endocardium and epicardium as well as the separation
of the left and right ventricle was realised. This is useful for
further processing. Within the segmentation process the CTdataset with approx. 40 million voxels has been reduced to
approx. 400 vertices to create a basis for the heart’s conduction tree (see Fig. 1).
In Fig. 2 the result of the heart’s segmentation, as well as
x[mm]
y[mm]
Fig. 2. Segmented ventricle surface.
the reduced number of vertices on the heart’s surface can be
seen for the left and right ventricle.
2.2. Modified prim’s algorithm
A tree structure was used to model the spatial propagation of the heart’s excitation. Thus, a predefined conduction orientation is desired to reduce complexity, so that realtime capability can be achieved. The conduction system of
the heart includes different tissue areas. In these areas the
shape and the conduction velocity of the cell’s action potentials differ from each other. In Tab. 2 the different conduction velocities of the heart areas are given. These differences
are important since they are directly related to the shape of
Heart area
Sinoatrial node
Myocradium atrium
Atrioventricular node
Conduction velocity in m/s
< 0.01
1.0 - 1.2
0.02 - 0.05
Bundle of His
1.2 - 2.0
Bundle branches
2.0 - 4.0
Purkinje fibres
2.0 - 4.0
Ventricle myocardium
0.3 - 1.0
Tab. 2. Conduction velocities of different heart areas [4].
biosignals measured in the body volume conductor or on the
body volume conductor’s surface. Therefore, the heart is
divided into the following tissue areas: sinoatrial node, left
and right atrium, the atrioventricular node, the bundle of His,
left and right bundle branches and the left and right ventricle’s myocardium. The vertices of the conduction tree were
found manually for the sinoatrial node, left and right atrium,
the atrioventricular node, the bundle of His and the left and
right bundle branches. In contrast the vertices of the left and
right ventricle’s myocardium were found automatically using a modified prim’s algorithm.
The algorithm of prim generates the minimal spanning tree
for a weighted, undirected graph. This minimal spanning
tree includes all vertices, so that the total weight of all edges
is minimized. The graph G = (V, E) contains the amount
of vertices V and the amount of edges E. In E all available
connections between different vertices V are enclosed. Every edge (u, v) ∈ E is dedicated a weight w(u, v). Finding
the minimal spanning tree is an iterative process. Initially
the undirected graph G contains all vertices V . In a first
step, searching for the edge with a minimum weight to the
root r is done and the corresponding vertex is included as
a new vertex into the directed graph A. Within every next
step the lightest edge from a vertex within the vertices from
the directed graph A to one of the vertices from the undirected graph G is determined and added. Only this way it
is ensured that, in the end, graph A involves every vertex.
A pseudo code of the algorithm’s functionality is shown in
Algo. 1 [5]:
POSTER 2016, PRAGUE MAY 24
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Fig. 3. 3D-view of the segmented heart.
G:
Graph
w:
weights
r:
root
V:
Vertices
Q:
Queue of priority
π(u):
predecessor of u
Adj(u):
list of adjacent vertices from u
dist(u):
distance from u to spanning tree
they have the shortest distance to the end points of the right
and left bundle branches. In the next step the modified prim’s
algorithm can be applied, starting at the chosen roots. First
only vertices on the inner surface, the endocardium, are considered for the prim’s algorithm and are included into the
directed graph A. Then the vertices on the epicardium, the
outer surface of the heart, are added to the graph according
to the method of prim’s algorithm. All manual and automatic found minimal spanning trees were merged to get a
predefined orientation of the heart’s conduction tree.
Algorithm 1 Prim’s algorithm(G,w,r)
Q ← V [G]
for all u ∈ Q do
dist[u] ← ∞
π[u] ← 0
dist[r] ← 0
end for
while Q = 0 do
u ← FindMin[Q]
for all v ∈ Adj[u] do
if v ∈ Q and w(u, v) < dist[v] then
π[v] ← u
dist[u] ← w(u, v)
end if
end for
end while
return A
Initializing
3. Results
As a result the abstract anatomy of the heart’s conduction system modelled via the modified prim’s algorithm is
shown in Fig. 4. The different spanning trees of the tissue
areas are represented in various colours. The direction of the
-140
-160
Atrium
Tawara
Ventricle 1, right
Ventricle 2,right
Ventricle 3,right
Ventricle posterior,left
Ventricle anterior,left
-180
In this special case, the prim’s algorithm has been modified for finding the heart’s conduction tree. Therefore, the
distances between all vertices of the heart’s surface data are
determined and saved in an array of weights. A connection
of two vertices is only added as an edge into the array of
edges E, with its determined weight, if the distance stays
under a certain threshold. Only by this it’s guaranteed that
the concave inner surface of the ventricles can be depicted
correctly. The right ventricle has three propagation strands
starting near the apex. The left ventricle has two propagation
strands, one posterior and one anterior passing. Thus, in the
right ventricle there are three independent and in the left ventricle two independent minimal spanning trees implemented.
The roots of those minimal spanning trees are chosen, so that
z [mm]
-200
-220
-240
260
240
40
220
20
200
0
180
-20
-40
160
y [mm]
-60
140
-80
120
x [mm]
-100
Fig. 4. The heart’s conduction tree.
complete conduction tree can be followed from the atrium to
the bundle branches into the left and right myocardium.
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L. KORN, CONSTRUCTION OF THE HEART’S CONDUCTION TREE VIA PRIM’S ALGORITHM
4. Conclusion
The results show an abstraction of the heart’s conduction system in a way that a predefined propagation direction
exists. The characteristic anatomy of the heart is well recognizable. The spread of excitation in the heart has a hierarchical structure. Therefore, choosing a tree structure to model
the spatial expansion of the impulses seems reasonable. The
reason to use a modified prim’s algorithm in this model is
founded by the physiological timing of the impulses’ propagation. First the excitation spreads from the sinoatrial node,
to the atrium into the left and right bundle branches. Initially in the ventricle’s myocardium the excitation starts on
the inner surface of the heart, the endocardium. The main
direction of propagation points from the apex into the direction of the sinoatrial node. Afterwards the excitation transits
to the outer surface, the epicardium, so that the whole ventricle’s myocardium is activated. This characteristic spread
of excitation can be ensured by using the modified prim’s algorithm.
For future work the developed conduction system can be
used to model spatial distribution of action potentials in the
heart. All vertices are representing different heart cells and
the edges can be weighted with area characteristic conduction velocities. With its tree structure the developed model is
well-suited especially for using cellular automata [6]. Further processing of this spatial distribution of action potentials
can result in real-time modelling of biosignals.
Acknowledgements
The research presented in this paper benefited from the
input of Daniel Rüschen, M.Sc. and Univ.-Prof. Dr.-Ing. Dr.
med. Steffen Leonhardt, RWTH Aachen University, who
supervised this work.
References
[1] FOGOROS, R.N. Electrophysiological Testing. John Wiley & Sons,
2012.
[2] OsiriX. DICOM sampla image set MAGIX. http://www.osirixviewer.com/datasets/MAGIX.zip.
[3] MÖLLER, T.B., REIF, E. Taschenatlas der Schnittbildanatomie: Thorax, Herz, Abdomen, Becken. Thieme, 2011.
[4] IAIZZO, P. Handbook of Cardiac Anatomy, Physiology and Devices.
Humana Press, 2009.
[5] CORMEN, T.H., LEISERSON, C.E., RIVEST, R.L., STEIN, C. Minimum Spanning Trees in Introduction To Algorithms. MIT Press, 2001.
[6] YE, P., ENTCHEVA, E., GROSU, R., SMOLKA, S.A. Efficient modeling of excitable cells using hybrid automata. Proc. of CMSB, 2005,
vol. 2, p. 216 - 227.
About Authors. . .
Leonie KORN was born in Aachen, Germany, on May 3,
1990.
She received the M.Sc. degree in Electrical Engineering with
specialisation on Biomedical Engineering from the RWTH
Aachen University, Germany, in October 2015. Currently
she is working towards her doctor degree at the Philips Chair
of Medical Information Technology, RWTH Aachen University. Her research interests include physiological modelling
and biosignal processing with a focus on ventricular assist
devices.