Download Simulations of magnetocardiographic signals using realistic

Adv. Radio Sci., 10, 85–91, 2012
www.adv-radio-sci.net/10/85/2012/
doi:10.5194/ars-10-85-2012
© Author(s) 2012. CC Attribution 3.0 License.
Advances in
Radio Science
Simulations of magnetocardiographic signals using realistic
geometry models of the heart and torso
C. V. Motrescu and L. Klinkenbusch
Institut für Elektrotechnik und Informationstechnik, Christian-Albrechts-Universität zu Kiel, Germany
Correspondence to: L. Klinkenbusch ([email protected])
Abstract. Although the first measurement of the cardiac
magnetic field was reported almost half a century ago magnetocardiography (MCG) is not yet widely used as a clinical
diagnostic technique. With the development of a new generation of magnetoelectric sensors it is believed that MCG will
become widely accepted in the clinical diagnosis. Our goal is
to build a computer-based tool for medical diagnosis and to
use it for the clarification of open electro-physiological questions. Here we present results from modelling of the cardiac
electrical activity and computation of the generated magnetic
field. For the simulations we use MRT-based anatomical
models of the human atria and ventricles where the shape
of the action potential is determined by ionic currents passing through the cardiac cell membranes. The monodomain
reaction-diffusion equation is chosen for the description of
the heart’s electrical activity. This equation is solved for the
transmembrane voltage which is in turn used to calculate current densities at discrete time instants. In subsequent simulations these current densities represent primary sources of
magnetostatic fields arising from a volume conduction problem. In these simulations the heart is placed in a realistic
torso model where the lungs are also considered. Both, the
volume conduction problem as well as the reaction-diffusion
problem are modelled using Finite-Element techniques.
1
the variation in time of the magnetic fields produced by the
heart outside the body and the graphical representation of
the recorded magnetic signals is called magnetocardiogram.
Whereas the ECG is a widely used technique in the clinical
diagnosis, the MCG is not yet. The reasoning for this may lay
in the fact that currently used magnetic sensors (Superconducting QUantum Interference Devices – SQUIDs) require
extremely low temperatures as well as magnetically shielded
environments. Therefore, electrocardiography (ECG) is considerably more attractive for clinicians, though MCG and
ECG supposedly supply disjunctive information and MCG
offers several advantages (e.g. no electrodes needed, magnetic transparency of tissue). Efforts are nowadays directed
to the overcoming of the above mentioned disadvantages of
MCG by using magnetoelectric composite materials (Jahns
et al., 2011) with giant magnetoelectric effect. In the same
time computational tools for modelling the electrical activity of the heart as well as for solving the inverse problem
of magnetocardiography are developed as a framework for
studies on open electrophysiological questions such as the
non-response of some patients to bi-ventricular stimulation.
In this paper we present a modelling of the electrical activity of the heart at both, cellular and organ level and a
method to compute the associated magnetic field produced
by a healthy heart.
Introduction
2
The human body can be seen as a volume conductor where
the electrical activity of the heart gives rise to small electric
currents in the body with associated potential distributions.
The electrocardiography (ECG) records the variation in time
of the electrical potential produced by the heart at the surface
of the body. The graphical representation of the recorded
potentials depicts the well known electrocardiogram. However, the same currents in the body give also rise to weak
magnetic fields. The magnetocardiography (MCG) records
Biological and electro-physiological background
The heart is a hollow muscular organ being structured on four
chambers i.e. two atria and two ventricles. The function of
the heart is to pump blood. This function is achieved by contractions of the muscles composing the atria and ventricles
(working myocardium). The contraction of the cardiac muscles is electrically controlled by the excitation-conduction
system of the heart. This system is made of cells that are
able to fire impulses – a property called autorhythmicity.
Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.
In contrast to the atrial and ventricular myocytes (of the
working myocardium), the cells belonging to the excitationFig. 1. Schematic representation of a section through the human
heart showing the excitation-conduction system. Picture adapted
conduction
system
of
the
heart
do
not
need
an
external
stimKlinkenbusch: Simulations of Magnetocardiographic Signals
from (Malmivuo and Plonsey , 1995).
ulus for the
depolarization
because
they
are
able
to
depolar86
C. V. Motrescu and L. Klinkenbusch: Simulations of magnetocardiographic signals
ize themselves initiating thus electrical impulses (pacemaker
cells). The transmembrane voltage propagates from a myocyte to the neighboring myocytes in the form of an action
potential. The spread of the action potential is supported by
some rapid intercellular channels called gap junctions which
couple the interiors of adjacent cells. Thus, at the organ level
the excitation propagates in the form of isochronal surfaces
also called wave-fronts. Electrically active cells of the heart
Fig. 2. Stylized form of the normal electrocardiogram.
are located within these wave-fronts and they are usually asFig. 2. Stylized form of the normal electrocardiogram.
sociated to current dipoles. It is these moving current dipoles
with their accompanying conduction currents that give rise to
3 The normal electrocardiogram
the magnetic field of the heart.
gram and the repolarization of the ventricles that determines
120
theThe
T-wave.
excitation-conduction system of the heart (see Fig. 1)
The
atrial
repolarization
place
during the(AV)
QRSconsists
of the
sinoatrial (SA)takes
node, the
atrioventricular
3 The normal electrocardiogram
complex
butcommon
it is masked
much
node, the
bundle by
alsothe
called
the stronger
bundle ofprocess
His, the of
left and right
bundle branches and the Purkinje fibres.
ventricular
depolarization.
The excitation-conduction system of the heart (see Fig. 1)
Under normal conditions, the SA node generates action
consists of the sinoatrial (SA) node, the atrioventricular (AV)
potentials at the rate of about 70 pulses/min (Malmivuo and
4 Plonsey,
Mathematical
modelling
at the cellular
1995) which
spread through
the right level
and left atria
node, the common bundle also called the bundle of His, the
depolarizing
them
and
determining
the
occurrence
of the so
left and right bundle branches and the Purkinje fibres.
125
As
already
mentioned
in
Sect.
2
the
electrical
activity
called
P-wave
in
the
recorded
electrocardiogram
(see
Fig.of
2).the
UnderFig.
normal
conditions,
the SAofnode
generates
1. Schematic
representation
a section
through action
the human
The
only
conduction
path
between
the
atria
and
the
ventricardiac
myocyte
is
mainly
the
result
of
the
ionic
currents
that
potentials
at the
rate representation
of about
70 pulses/min
(Malmivuo
and
heart
showing
the
excitation-conduction
Picture
adapted
Fig.
1. Schematic
of a system.
section
through
the human
cles
is
provided
by
the
AV
node
which
has
a
low
conduction
are crossing the cell membrane through ion channels and
(Malmivuo
and
Plonsey,
1995).the right and left atria
Plonsey
,from
1995)
which
spread
through
heart
showing
the
excitation-conduction
system. Picture adapted
velocity andThese
consequently
introduces
a delay
in the propagatransporters.
currents
determine
the shape
of the acdepolarizing them and determining the occurrence of the so
tion
of
the
excitation
that
corresponds
to
the
PQ-segment
of at
from (Malmivuo and Plonsey , 1995).
tion
potential
of
a
cell.
The
available
cellular
ionic
models
called P-wave in the recorded electrocardiogram (see Fig. 2).
the
electrocardiogram.
Once
the
AV
node
has
been
passed,
Electrical impulses result from ionic activity at the cellular
130
present can be divided into three classes (Clayton and PanThe onlylevel.
conduction
pathcell
between
theofatria
and themuscle
ventri-cells
the excitation propagates through a high conduction-velocity
At rest, the
membrane
the cardiac
filov , 2008):
cles is provided
by
the
AV
node
which
has
a
low
conduction
system (the His-Purkinje system) to the inner sides of the
(myocytes) composing the atria and ventricles, is electrically
velocity polarized
and consequently
introduces
a
delay
in
the
propagawall.
The ventricular
depolarization
in the sense that the inner side of the membrane
simplified
two-variable
models
that reduce determines
the behavior
–ventricular
the
QRS
complex
on
the
electrocardiogram.
After the comtion of the
excitation
that
corresponds
to
the
PQ-segment
of
is more electronegative than the outer side of the membrane.
of all the ionic currents to two variables that describe
plete ventricular depolarization, it follows a plateau phase
the electrocardiogram.
the AV
node has been
passed,
This produces theOnce
so called
transmembrane
voltage.
When a
excitation and recovery;
which corresponds to the ST-segment on the electrocardiostrongpropagates
enough stimulus
is applied
to the membrane, certain
the excitation
through
a high conduction-velocity
and the repolarization
of the ventricles
determinesinprotein
channels will
open letting
ionic of
species
biophysically
detailed models
that use that
experimental
–gram
system (the
His-Purkinje
system)
to theparticular
inner sides
the 135to
the
T-wave.
cross
the membrane
along their
electro-chemical
gradients.
formation about the voltage and time dependence of ion
ventricular
wall.
The ventricular
depolarization
determines
The atrial repolarization takes place during the QRSThis
depolarization
of
the
membrane
is
followed
by a rechannel conductances (first generation) as well as inthe QRS complex on the electrocardiogram. After the comcomplex
but it is masked by the much stronger process of
polarization which brings the myocyte in the resting state.
tracellular
ionic concentrations (second generation) in
plete ventricular depolarization, it follows a plateau phase
ventricular
depolarization.
In contrast to the atrial and ventricular myocytes (of the
order
to
reproduce
the action potential;
which
corresponds
to
the
ST-segment
on
the
electrocardioFig. 2. Stylized
of the normal
electrocardiogram.
working form
myocardium),
the cells belonging
to the excitationconduction system of the heart do not need an external stim4 Mathematical modelling at the cellular level
ulus for the depolarization because they are able to depolarize themselves initiating thus electrical impulses (pacemaker
gram andcells).
the The
repolarization
the ventricles
thata mydetermines
As already mentioned in Sect. 2 the electrical activity of
transmembraneof
voltage
propagates from
the cardiac myocyte is mainly the result of the ionic curocyte to the neighboring myocytes in the form of an action
the T-wave.
rents that are crossing the cell membrane through ion chanpotential. The spread of the action potential is supported by
The atrial
repolarization
takes
place
during
the
QRSnels and transporters. These currents determine the shape
some rapid intercellular channels called gap junctions which
of the
complexcouple
but the
it is
masked
by
the
much
stronger
process
of action potential of a cell. The available cellular
interiors of adjacent cells. Thus, at the organ level
ionic
models at present can be divided into three classes
the excitation
propagates in the form of isochronal surfaces
ventricular
depolarization.
(Clayton
and Panfilov, 2008):
also called wave-fronts. Electrically active cells of the heart
are located within these wave-fronts and they are usually as– simplified two-variable models that reduce the behavior
sociated to current dipoles. It is these moving current dipoles
of all the ionic currents to two variables that describe
4 Mathematical
modelling
at thecurrents
cellular
levelrise
with their accompanying
conduction
that gives
excitation and recovery;
to the magnetic field of the heart.
As already mentioned in Sect. 2 the electrical activity of the
– biophysically detailed models that use experimental incardiac myocyte is mainly the result of the ionic currents thatformation about the voltage and time dependence of ion
are crossing the cell membrane through ion channels and
Adv. Radio Sci., 10, 85–91, 2012
www.adv-radio-sci.net/10/85/2012/
transporters.
These currents determine the shape of the action potential of a cell. The available cellular ionic models at
present can be divided into three classes (Clayton and Pan-
phic Signals
220
of 22 years. They are3available in the form of triangulated
surfaces as shown in Fig. 4.
C. V. Motrescu and L. Klinkenbusch: Simulations of magnetocardiographic signals
87
Membrane voltage [mV]
20
0
−20
−40
−60
−80
−100
0
100
200
300
400
500
600
Time [ms]
Fig. 4. MRT-based anatomical mesh models provided with the
freely available anatomical
software code ECGsim
(van Oosterom,
2004). with th
Fig. 4. MRT-based
mesh models
provided
Left: heart and lungs inside the torso. Right: detailed view of the
heart. software code ECGsim (van Oosterom, 2004). Left
Fig. 3. Action potential shape of the Luo-Rudy phase freely
1 cellularavailable
ionic model.
heart and lungs inside the torso. Right: detailed view of the heart.
Action potential shape of the Luo-Rudy phase
1 cellular
[0,1] and
describing the activation, inactivation and recovery
of
the
ion channel and VS being the membrane voltage
channel conductances (first generation) as well as inmodel.
of the species “S” alone. The gating variables are obtained
tracellular ionic concentrations (second generation) in
order to reproduce the action potential;
by solving a system of eight nonlinear ordinary differential
equations (ODEs) of the general form:
7 Method
– reduced cardiac models that simplify the underlying
dxS xS∞ (Vm ) − xS
electrophysiology to a minimum number of currents for
=
(3)
dt
τS (Vm )
The byprocedure
adopted
by us for computing the magneti
which the activation and inactivation is controlled
some empirical variables.
where xby
theheart
steady-state
value of
single gating
variS∞ is
field produced
the
consists
ofa three
subsequent
step
able xS and τS is the time constant for the return of the variwhere
output
one step
is after
useda voltage
as an perturbation.
input for the nex
Choosing the appropriate cellular model for 225
a certain
prob- theable
x to itsof
steady-state
value
lem is a matter of balancing its benefits and limitations. For
voltage
dependencies
of xS∞ and τS are
step. TheThe
steps
are
also differentiated
byexperimentally
the involved tim
this work we have chosen the Luo-Rudy phase 1 model of
determined.
space scales.
mammalian myocytes (Luo and Rudy, 1991). Thisand
cellular
The shape of the action potential as given by the Luo-Rudy
model belongs to the first generation of the biophysically
dephase
1 cellular
and used
in this
work is shownofin electrica
In the first step
wemodel
simulate
the
propagation
tailed models and it is considered that this model offers a
Fig. 3.
excitation
in the isolated heart shown in the right-hand sid
good balance between electrophysiological details and
computational efficiency. Within this model the Hodgkin-Huxley
230
of Fig. 4. As a result we obtain the distribution of the trans
5 Mathematical modelling at the tissue level
type of formalism is used to describe the action potential.
membrane
voltage in the heart at discrete time instants during
The rate of change of the membrane voltage Vm is given
by:
At the tissue level the granular structure may be neglected
a complete
heart beat.
dVm
and the cardiac muscle may be modelled as a functional synCm
= −(Iion + Ist )
(1)
In the second
step the
a volume
problem
cytium in which
membrane conduction
voltage propagates
smoothly. is solved
dt
Under
this
assumption
the
action
potential
can
be
considwhere Cm is the membrane capacitance, Ist is anwhere
applied the driving force is represented by impressed curren
ered to propagate by diffusion in an excitable medium that
stimulus current and Iion is the sum of six ionic
currents:
a
235
densities
in theofheart
areThe
computed
using
consists
one or which
two phases.
bi-domain model
con-the trans
fast sodium current (INa ), a slow inward current (Isi ), a timesiders
the
cardiac
muscle
to
be
a
two-phase
medium
conmembrane
voltage resulted at each time instant considered
dependent potassium current (Ik ), a time-independent
potassisting of intracellular and extracellular spaces separated at
sium current (Ik1 ), a plateau potassium current (Ikp
in) and
thea simulations
the
(first) step.
The
each point in of
space
by previous
the cellular membrane.
Although
thiscomputa
time-independent background current (Ib ). In general, each
is
a
very
detailed
model,
it
is
also
computationally
expentional
current IS carried by an ionic species “S” through
an ion domain consists of the heart and lungs inside the torso
sive. The simpler monodomain model is based on a single
channel is expressed as:
as shown phase
on the
left-hand
ofmuscle
Fig. 4.
output
description
of theside
cardiac
but The
it is still
able of thes
to reproduce
many of the by
experimentally
phenom- and con
240
simulations
is represented
the totalobserved
(impressed
(2)
IS = GS + xs1 xs2 ...xsn (Vm − VS )
ena with a much higher numerical efficiency. The simulation
duction) current
density in the entire torso.
with GS being the maximal conductance of the ion channel,
of a problem with the monodomain model can be by a facxs being gating variables with values varying in the interval
tor of ten
or more
than the field
same problem
simulated
In the third
step
the faster
magnetic
outside
the body pro
ts of one or two phases. The bi-domain model conthe cardiac muscle to be a two-phase medium cong of intracellular and extracellular spaces separated at
point in space by the cellular membrane. Although this
ery detailed model, it is also computationally expenThe simpler monodomain model is based on a single
description of the cardiac muscle but it is still able
roduce many of the experimentally observed phenomith a much higher numerical efficiency. The simulation
roblem with the monodomain model can be by a facten or more faster than the same problem simulated
he bi-domain model (Clayton and Panfilov , 2008). In
ork we consider that the action potential propagates
fusion in the monodomain model of the cardiac tisith local excitation of the membrane voltage. The so
monodomain reaction-diffusion model consists of a
olic partial differential equation (PDE)
to a density in the torso, is computed
duced bycoupled
the total current
www.adv-radio-sci.net/10/85/2012/
Radio Sci., 10, 85–91, 2012
m of ODEs
(Whiteley , 2006):
as a magnetostatic problemAdv.
at each
time instant considered
[
] in the previous (second) step.
∂V
245
C.V. Motrescu and L. Klinkenbusch: Simula
and L. Klinkenbusch: Simulations of Magnetocardiographic Signals
88
C. V. Motrescu and L. Klinkenbusch: Simulations of magnetocardiographic signals
ated
Fig. 5. The spread of electrical excitation in the heart during the
atrial
row), and
ventricular
(bot-the
Fig. 5. depolarization
The spread of(top
electrical
excitation
in depolarization
the heart during
tom row).
the
Left:
rt.
atrial depolarization (top row), and ventricular depolarization (bottom row).
255
etic
eps
next
ime 260
ical
side
ansring 265
ved
rent
ansered 270
utaorso
hese
on275
prouted
ered
with 280
uta-
t
with the bi-domain model (Clayton and Panfilov, 2008). In
this work we consider that the action potential propagates
meshes
by using
the monodomain
”Tetgen” - amodel
freelyofavailable
software
by diffusion
in the
the cardiac
tissuefor
withresearch
local excitation
of the membrane
voltage.(Si,
The2006).
so
code
and non-commercial
purposes
called
reaction-diffusion
model
consists
ofwith
a
The
atriamonodomain
are discretized
with 1.23 millions
mesh
nodes
partial
differential
equation
coupledcontain
to a
an parabolic
average edge
length
of 0.8 mm
and (PDE)
the ventricles
system
of ODEs
2006):
1.43
millions
mesh(Whiteley,
nodes with
an average edge length of 0.95
Fig. 6. Position of the magnetic sensor array (red color) relative to
mm.
∂Vm
∇ · (σ ∇Vm ) = β Cm
(4)
+ I + Ist
The simulations are ∂tbased ion
on the monodomain reaction- the body.
diffusion model−1described by Eq. (4). Each node of the
where σ (Sm ) is the electrical conductivity tensor, Vm
computational domain contains the Luo-Rudy
ionic cellular
(V) is the transmembrane voltage, β (m−1 ) is the surface
model
described
in
Sect.
4.
This
cellular
model
withspecon−2
to volume ratio of the cardiac cells, Cm (Fm ) is the
stant
action
potential
duration
(APD)
is
used
for
both,
atria
−2
cific membrane capacitance, Iion (Am ) is the current flow
andthrough
ventricles.
Theofcardiac
areand
assumed
to−2be) is
homounit area
the cellmuscles
membrane
Ist (Am
an
geneous
isotropic
with the
electricalcurrent
conductivity
appliedand
stimulation
current.
Thescalar
net membrane
Sm−1 . Other parameter values used in the simulaσ = 0.25
∂V
tion
Cmm =m1+µFcm
Imare:
= C
Iion −2 and β = 1400 cm−1 .
(5)
∂t
The simulations are performed within ”CHASTE” - the
is givenHeart
by the
cellular
ionic models
such as -the
one
”Cancer,
And
Soft-Tissue
Environment”
a multidescribed
in
Sect.
4.
purpose software library for simulations in the area of biology (Pitt-Francis et al., 2009) provided under the LGPL
license
by a teammodels
from the Computer Science Department
6 Anatomical
of the University of Oxford, the UK. Examples showing the
Anatomical
models
of the heart,
torso used
this
membrane
voltage
distribution
atlungs
threeand
different
timeininstants
work
are
those
provided
with
the
freely
available
software
during the atrial and ventricular depolarization are provided
forDue
simulations
of electrocardiograms
called
“ECGsim”
in code
Fig. 5.
to the constant
action potential
durations
con(van Oosterom, 2004). The models are based on magnetic
sidered in the simulations, the repolarizations follow a simiresonance tomography (MRT) data of a healthy young male
laroftime
course. Rectangular current pulses of 2 ms duration
22 yr. They are available in the form of triangulated surarefaces
usedasfor
the initial
The areas of initial stimshown
in Fig.stimulations.
4.
ulations are taken from the ECGsim (van Oosterom, 2004)
software where depolarization isochrones on the surfaces en7 Method
veloping
the atria and ventricles are obtained from the potentials on the surface of the body by following an inverse
The procedure adopted by us for computing the magnetic
procedure.
The atria
stimulated
inthree
an area
corresponding
field produced
by theare
heart
consists of
subsequent
steps
to where
the position
of
the
SA-node.
In
the
case
the output of one step is used as an input of
forthe
the ventrinext
cles, three areas inside the left ventricle and one area inside
the right ventricle are initially stimulated with some small
Adv. Radio Sci., 10, 85–91, 2012
time delays between them. Animated representations of the
spread of electrical excitation during the depolarization and
step. The steps are also differentiated by the involved time
and space scales.
In the first step we simulate the propagation of electrical
excitation in the isolated heart shown in the right-hand side
of Fig. 4. As a result we obtain the distribution of the transmembrane voltage in the heart at discrete time instants during
a complete heart beat.
In the second step a volume conduction problem is solved
where the driving force is represented by impressed current
densities in the heart which are computed using the transmembrane voltage resulted at each time instant considered
in the simulations of the previous (first) step. The computational domain consists of the heart and lungs inside the
torso as shown on the left-hand side of Fig. 4. The output
of these simulations is represented by the total (impressed
and conduction) current density in the entire torso.
In the third step the magnetic field outside the body produced by the total current density in the torso, is computed
as a magnetostatic problem at each time instant considered
in the previous (second) step.
Thus slowly varying magnetocardiographic signals with time are obtained from a sequence of
stationary computations.
Fig. 6. Position of the magnetic sensor array (red
the body.
repolarization of the atria and ventricles are a
the form of short films.
9 Computation of currents in the body
285
itethe
ghtume 290
The electrical excitation in the heart gives
www.adv-radio-sci.net/10/85/2012/
Magnetocardiographic Signals
5
C. V. Motrescu and L. Klinkenbusch: Simulations of magnetocardiographic signals
89
1
2
3
5
6
7
8
9
10
11
12
13
14
15
16
4
ative to
ded in
so and
Sm−1 ,
8
Examples showing the membrane voltage distribution at
Simulation of the excitation spreading in the heart
hand side of the Fig. 4 are transformed to tetrahedral volume
meshes by using the “Tetgen” – a freely available software
code for research and non-commercial purposes (Si, 2006).
−11
The atria are discretized
x 10with 1.23 millions mesh nodes with
2
an average edge length of 0.8 mm and the ventricles6 contain 1.43 millions mesh nodes with an average edge length
of 0.95 mm.
1
The simulations are based on the monodomain reactiondiffusion model described by Eq. (4). Each node of the
0
computational domain contains the Luo-Rudy ionic cellular
model described in Sect. 4. This cellular model with con−1 duration (APD) is used for both, atria
stant action potential
and ventricles. The cardiac muscles are assumed to be homogeneous and isotropic
with the scalar electrical conduc−2
tivity σ = 0.25 Sm−1 . Other parameter values used in the
simulation are: Cm = 1 µFcm−2 and β = 1400 cm−1 .
The simulations are performed within “CHASTE” – the
0
200
400
600
800
“Cancer, Heart And Soft-Tissue Time
Environment”
– a multi[ms]
purpose software library for simulations in the area of biology (Pitt-Francis et al., 2009) provided under the LGPL
license by a team from the Computer Science Department of
the University of Oxford, the UK.
B[T]
Fig. 7. Magnetic flux density component
onduring
the chest
threeperpendicular
different time instants
the atrial and ventricuThe spread of excitation in the heart is simulated with Finitelar
depolarization
are
provided
in
Fig.
5. Due to the conrecorded as a function of time at the position of each node of the
Element methods. For this purpose the surface meshes of the
stant action potential durations considered in the simulations,
atria and
ventricles
composing
the heart
shown 6.
on the rightsensor
array
shown
in Fig.
the repolarizations follow a similar time course. Rectangular
B[T]
urrents
heart
7) usscrete
t. The
ned by
cribed
al curhe enng the
analyhysics
Fig. 7. Magnetic flux density component perpendicular on the chest recorded as a function of time at the position of each node of the sensor
array shown in Fig. 6.
current pulses of 2 ms duration are used for the initial stimulations. The areas of initial stimulations are taken from the
ECGsim (van Oosterom, 2004) software where depolariza−12
tionx isochrones
on the surfaces enveloping the atria and ven10
10
tricles are obtained from the potentials
12 on the surface of the
body
by
following
an
inverse
procedure.
The atria are stim8
ulated in an area corresponding to the position of the SAnode.
In the case of the ventricles, three areas inside the left
6
ventricle and one area inside the right ventricle are initially
4
stimulated
with some small time delays between them. Animated representations of the spread of electrical excitation
2
during the depolarization and repolarization of the atria and
ventricles
are also provided in the form of short films.
0
−2
9 Computation of currents in the body
−4
0
200
400
600
800
Time [ms]
The electrical excitation
in the heart gives rise to currents
in the entire body. Impressed current densities in the heart
can be computed as Je = −σ ∇Vm (Aoki et al., 1987) using
the membrane voltage Vm already computed at discrete time
instants during the signals
spread of excitation
in the heart. The asmagnetocardiographic
recorded
sociated conduction currents Jc can then be determined by
Fig. 8. Detailed view of the
at the positions of the sensors denoted by 6 (left) and 12 (right) in
Fig. 7.
www.adv-radio-sci.net/10/85/2012/
Adv. Radio Sci., 10, 85–91, 2012
d
1
,
d
g 330
y
n
L
h
s.
h
or 335
6.
m
re
m- 340
n
−11
2
−12
x 10
10
6
−11
x 10
3
12
375
2
6
4
B[T]
−1
1
2
B[T]
0
0
−2
0
x 10
8
1
B[T]
s
rt
se
e
y
d
rne
ys
Fig. 7. Magnetic flux density component perpendicular on the chest
recorded as a function of time at the position of each node of the
sensor array shown in Fig. 6.
6
C.V. Motrescu
90
C. V. Motrescu and L. Klinkenbusch:
Simulations of magnetocardiographic
signalsand L. Klink
−2
200
400
600
800
−4
0
Time [ms]
200
400
600
800
0
380
−1
Time [ms]
−2
Fig.
8. Detailed
view
of the
magnetocardiographic signals
signals recorded
Fig. 8.
Detailed
view
of the
magnetocardiographic
recorded
at
the
positions
of
the
sensors
denoted
by
6
(left)
and
12
(right)
at the positions of the sensors denoted by 6 (left) and 12 (right)inin
Fig. 7.
Fig. 7.
solving a stationary volume conduction problem described
Fig. 7.
figure∇J
allt the
are
by In
thethis
equation
= 0 magnetocardiographic
where Jt = Je + Jc is thesignals
total current density.
Wesame
compute
the Detailed
total current
density
in the
enrepresented
on the
scale.
views
of the
signals
tire torso
shown
on the6 left-hand
sidegiven
of Fig.
by using
the
recorded
by the
sensors
and 12 are
in4Fig.
8. When
commercially
software
for sensors
Finite-Element
analyplotting
the signalsavailable
recorded
by all the
in a single
figsis
called
COMSOL
Multiphysics
(COMSOL
Multiphysics
ure, the so called butterfly plot is obtained as shown in Fig. 9.
v. 4.2, 2011).
The electrical conductivity values used for the torso and345
the lungs are σtorso = 0.2 Sm−1 and σlung = 0.04 Sm−1 ,
respectively.
11 Conclusions
385
0
200
400
600
800
Time [ms]
Fig.9.9. Butterfly
Butterfly plot
plot of
perFig.
of the
the magnetic
magneticflux
fluxdensity
densitycomponent
component
per- 390
pendicularon
on the
the chest
chest recorded
sensor.
pendicular
recordedatatthe
theposition
positionofofeach
eachpoint
point
sensor.
11 Conclusions
puting
conduction currents in the inhomogeneous and irregularly shaped torso. Future work will focus on the reduction 395
The method used in this work proved to be able to provide
of the number of unknowns in order to reduce the computarealistic values for the computed magnetic fields of the heart.
tion time as well as for more conveniently solving the correWe coupled different state-of-the-art existent software codes
sponding
inverse problem.
and realistic anatomical models in order to compute the magnetic field produced by the heart. We used the magneto-static
approximation to compute slowly varying magnetic field
12
withAcknowledgements
time at successive time instants. The Finite-Element
method (FEM) with tetrahedral elements was used for comWe
areconduction
thankful currents
to all the
persons
who contributed
puting
in the
inhomogeneous
and irreg- to
the
freely
available
software
codes
(ECGsim,
TetGen,
ularly
shaped
torso. Future
work will
focus
on the reduction
CHASTE)
usedofinunknowns
this work.inThis
work
was supported
by the
of the number
order
to reduce
the computation
time
as
well
as
for
more
conveniently
solving
the
correDeutsche Forschungsgemeinschaft (DFG) within the SFB
sponding
inverse problem.
855:
Magnetoelectric
composite material - Future biomag-
Computation
of magnetic
fieldsto
(MCG
signals)
The 10
method
used in this
work proved
be able
to provide
realistic
values for the computed magnetic fields of the heart.
The currents in the torso produce magnetic fields inside and
We coupled
different
existent
software
codes
outside the
body. state-of-the-art
We compute these
magnetic
fields using
and realistic
anatomical
models
in
order
to
compute
the
mag350
Ampere’s law ∇ ×H = Jt at each time instant previously conneticsidered
field produced
by the heart.
Wetotal
used
the magneto-static
in the computation
of the
current
density in the
approximation
to compute
varying
magneticMulfield
torso Jt . These
simulationsslowly
are coupled
in COMSOL
netic interfaces.
v. 4.2,
with those
with tiphysics
time at (COMSOL
successive Multiphysics
time instants.
The2011)
Finite-Element
for
the
solution
of
the
volume
conduction
problems.
All
bimethod (FEM) with tetrahedral elements was used for com- Acknowledgements. We are thankful to all the persons who conological tissues are considered to be non-magnetic with the
relative magnetic permeability µr = 1.
For the results we consider a four by four magnetic sensor
array positioned just in front of the torso as shown in Fig. 6.355
The nodes of the magnetic sensor array are spaced 8.2 cm and
7.2 cm apart from each other on the horizontal and vertical
directions, respectively.
The variation in time of the magnetic field component perpendicular to the chest computed in each node of the mag-360
netic sensor array is shown in Fig. 7. In this figure all the
magnetocardiographic signals are represented on the same
scale. Detailed views of the signals recorded by the sensors 6
and 12 are given in Fig. 8. When plotting the signals recorded
by all the sensors in a single figure, the so called butterfly plot365
is obtained as shown in Fig. 9.
370
Adv. Radio Sci., 10, 85–91, 2012
tributed to the freely available software codes (ECGsim, TetGen, CHASTE) used in this work. This work was supported
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