Download New developments in a strongly coupled cardiac electromechanical

Europace (2005) 7, S118eS127
New developments in a strongly coupled cardiac
electromechanical model
David Nickerson, Nicolas Smith, Peter Hunter*
Bioengineering Institute, University of Auckland, Level 6, 70 Symonds Street, Auckland, New Zealand
Submitted 12 January 2005, and accepted after revision 3 May 2005
KEYWORDS
cardiac
electromechanics;
computational model;
finite element
Abstract Aim The aim of this study is to develop a coupled three-dimensional
computational model of cardiac electromechanics to investigate fibre length
transients and the role of electrical heterogeneity in determining left ventricular
function.
Methods A mathematical model of cellular electromechanics was embedded in
a simple geometric model of the cardiac left ventricle. Electrical and mechanical
boundary conditions were applied based on Purkinje fibre activation times and
ventricular volumes through the heart cycle. The mono-domain reaction diffusion
equations and finite deformation elasticity equations were solved simultaneously
through the full pump cycle. Simulations were run to assess the importance of
cellular electrical heterogeneity on myocardial mechanics.
Results Following electrical activation, mechanical contraction moves out
through the wall to the circumferentially oriented mid-wall fibres, producing
a progressively longitudinal and twisting deformation. This is followed by a more
spherical deformation as the inclined epicardial fibres are activated. Mid-way
between base and apex peak tensions and fibre shortening of 40 kPa and 5%,
respectively, are generated at the endocardial surface with values of 18 kPa and
12% at the epicardial surface. Embedding an electrically homogeneous cell model
for the same simulations produced equivalent values of 36.5 kPa, 4% at the
endocardium and 14 kPa, 13.5% at the epicardium.
Conclusion The substantial redistribution of fibre lengths during the early preejection phase of systole may play a significant role in preparing the mid-wall fibres
to contract. The inclusion of transmural heterogeneity of action potential duration
has a marked effect on reducing sarcomere length transmural dispersion during
repolarization.
ª 2005 The European Society of Cardiology. Published by Elsevier Ltd. All rights
reserved.
* Corresponding author. Tel.: C64 9 373 7599; fax: C64 9 367 7157.
E-mail address: [email protected] (P. Hunter).
1099-5129/$30 ª 2005 The European Society of Cardiology. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.eupc.2005.04.009
New developments in coupled cardiac electromechanical model
Introduction
The integration of cardiac electrical and mechanical experimental data and hypothesizes across
multiple spatial scales and functions via mathematical modelling is arguably the most advanced
example of a ‘Physiome’ [1] style framework.
Detailed electrophysiological models of myocyte cellular [2,3] and subcellular processes incorporating genetic mutations [4] are now being
linked to a large body of existing research for
modelling the spread of electrical excitation
throughout cardiac tissue and the whole heart
(for reviews see [5e7]). Numerous methods have
been developed to model the spread of electrical
excitation varying from discrete models in which
individual cells are electrically coupled to their
neighbours via variable resistive elements (gap
junctions) through to continuum based models.
Similarly, detailed finite element continuum models incorporating anatomy, structure and passive
mechanics have been coupled to cellular active
tension (see [5,7] for reviews).
The development of models of these two physical processes has occurred largely independently.
However, coupled models of cardiac electromechanics which trigger mechanical contraction
based on the calculated activation sequence have
recently been developed to investigate the relationship between activation and contraction in the
normal [8,9] and paced heart [10,11]. It is indicative of both the computational load and modelling
complexities that in each of these studies the
activation sequence has been calculated independently and then later used as the excitation wavefront to initiate contraction. Such an approach
negates the possibility of incorporating the effects
of electro-mechanical coupling due to, for example,
length-dependent calcium binding or stretchdependent conductance changes that are characterized in some of the more recently developed
myocyte models [2,12] and have also been shown to
be important in tissue models [13,14].
While we can expect computational performance to continue to improve, in order to incorporate these coupling processes, or indeed new
advances such as signal transduction models [15],
gene regulation models [4] or metabolic models
[16], a method is required which strongly couples
tissue stretch and activation to cellular tension
and membrane potential.
The method we have developed uses cellular
models of electromechanics to drive the dynamic
active material properties of a continuum organ
model while the cellular models themselves get
S119
feedback from both the electrical excitation and
the deforming mechanical model. The mechanical
deformation of the geometric models also feeds into
the excitation model affecting the propagation of
electrical activation and cellular electrophysiology.
In this study we use the developed framework to
examine explicitly the effect of adding transmural
electrical heterogeneity in the embedded cell
models on global myocardial mechanics.
Methods
A computationally efficient cellular model was
developed by coupling the Fenton and Karma [17]
simplified activation model to the cardiac mechanics of Hunter et al. [18] which characterizes the
length and calcium dependence of tension generation. This coupled model is termed below as FKHMT. For the purposes of coupling, a calcium
transient (not explicitly included in the Fenton
and Karma formulation) was calculated from the
slow inward current based on the original intracellular calcium formulation in the Luo and Rudy [19]
model on which the Fenton and Karma is based.
Following the method described by Nickerson et al.
[12], the activation model was then used to provide the calcium transient input required to drive
the active and dynamic development of cellular
tension.
Three different parameter sets were used in this
formulation to characterize the heterogeneous
behaviour of myocardium produced by midmyocardial or M cells, located in the middle of the
ventricular wall [20]. Parameter values of
trZ150 ms for midmyocardial cells, 144 ms for
epicardial and 142 ms for endocardial cells resulted in calculated single cell action potential
durations (APD) of 437 ms, 352 ms and 329 ms,
respectively, matching values found in human
ventricular tissue [21].
This cellular model was then embedded in
a simplified geometric model of the left ventricle
(shown in Fig. 1a) based on a prolate geometry
with 50 mm outer radius (at the base) and 60 mm
longitudinal base apex dimensions represented
using 128 tricubic Hermite finite elements in
rectangular Cartesian coordinates. Wall thickness
was set at 10 mm at the base tapering at the apex.
The simple distribution of the epicardial, midmyocardial, and endocardial cell types prescribed
transmurally in the model is illustrated in Fig. 1b.
The microstructure of the models was defined such
that the fibre angle varied sigmoidally over 120 between the endocardium and the epicardium.
S120
D. Nickerson et al.
Table 1 Tissue conductivities for the mono-domain
activation model based on the values of Hooks et al.
[27]
Figure 1 (a) Simplified geometric model of the left
ventricle and the reference rectangular Cartesian axes.
The red arrows indicate the material fibre axis, the
green arrows are aligned with the sheet axis, and the
blue matches the sheet-normal axis. The mid-septum
region of the left ventricle is indicated by the green
endocardial surface. (b) Distribution of the three cell
types described by the cell model as used in the
heterogeneous left ventricle model. The epicardial cells
are yellow, the midmyocardial cells blue, and the
endocardial cells are drawn in red.
The material properties for the electromechanical stimulations are referred to in this local
embedded microstructure. Recent studies have
shown that conclusive evidence for the use of fully
orthotropic conductivities in tissue level models is
not yet available [22]. Thus, the myocardial model
was treated as transversely isotropic using a ratio
of 10:1 [23], longitudinal:transverse to the fibre
direction; parameters for the mono-domain activation model are shown in Table 1.
The mechanical constitutive properties were
also referenced to the microstructure using the
non-linear pole-zero law [18] with the parameter
values listed in Table 2 taken from Remme et al.
[24]. This parameter set was applied to the regions
of the model shown in Fig. 2 to characterize
increased stiffness at the apex and base due to
higher collagen and valve ring effects, respectively.
Following the work of Tomlinson et al. [25] the
initial activation times at points on the left
Table 2
Material axis
Value (mS mmÿ1)
Fibre conductivity
Sheet conductivity
Normal conductivity
0.263
0.0263
0.0263
ventricular endocardial surface are specified to
simulate Purkinje fibre activation. A stimulus
current is then applied at these points following
the specified activation profile. The applied stimulus distribution is shown in Fig. 3 where the
stimulus times are derived from the initial activation times from the canine ventricular model of
Tomlinson et al. [25]. As shown in Fig. 3, the
earliest activation occurs in the lower mid-septal
wall and the latest activation is toward the base of
the free wall.
The boundary conditions for simulating the four
phases of the cardiac mechanical cycle were
applied based on the temporal relationship between global contraction and electrical transients
as described by a classical Wigger’s diagram [26].
At all time points throughout the cycle spatially
constant pressures were applied over the entire
endocardial surface. Prior to the initiation of
activation the model was inflated to a cavity of
pressure of 1.0 kPa. The isovolumic phase was
implemented by adding a finite element mesh
which represented the volume of blood in the left
ventricle and was constrained such that these
elements had a constant total volume. This cavity
region was then coupled to the ventricular wall to
act as feedback mechanisms to constrain weakly
the deformation of the ventricular wall. Once
pressure exceeded 10 kPa the ejection phase was
initiated by allowing a central node in the cavity
mesh to displace in the baseeapex axis of the
model, thereby reducing cavity volume. Displacement
Summary of the pole-zero material parameters used in the model [24]
Type
Parameter
Apex
Sub-apex
Normal
Base
Coefficient
k11, k22, k33
k12, k21, k13, k31, k32, k23
a11
a22
a33
a12, a21, a13, a31, a23, a32
b11, b22
b33
b12, b21, b13, b31, b23, b32
2.22 kPa
2.0 kPa
0.136
0.136
0.136
0.3
2.22
2.22
1.5
2.22 kPa
2.0 kPa
0.227
0.227
0.227
0.4
1.67
1.67
1.0
2.22 kPa
1.0 kPa
0.475
0.619
0.943
0.8
1.5
0.442
1.2
2.22 kPa
2.0 kPa
0.423
0.555
0.845
0.4
1.5
0.442
1.0
Pole
Curvature
New developments in coupled cardiac electromechanical model
S121
1. Cellular fields
from activation model
2. Interpolate cellular fields
from influential GBFE nodes
to Gauss points
Figure 2 The apex nodes are shown as red spheres,
the sub-apex as silver spheres, the normal nodes are
gold spheres, and base nodes are green. The material
parameters in Table 3 are specified at these nodes and
linearly interpolated as required at intermediate spatial
locations.
3. Solve cellular model at
Gauss point
5. Solve FE model
of the cavity node was controlled to match the left
ventricular cavity volume transient described by
the Wigger’s diagram, scaled to give a peak ejection fraction of 50%.
The separate solution algorithms of the grid
based finite element method (GBFE) for the monodomain equations (activation) and finite element
method (FEM) of the finite deformations equations
(mechanics), have previously been described elsewhere [27,28]. The solution process for coupling
these two solution techniques together is summarized in Fig. 4. From the activation solutions solved
using a time step of 0.01 ms and an average grid
point spacing 200 mm the cellular fields were
locally interpolated at the lower resolution gauss
Figure 3 Illustration of the Purkinje-like stimulus used
to approximate sinus rhythm in the left ventricle model;
the left panel shows the septal wall and the right is the left
ventricular free wall. The coloured endocardial surface
shows the time of electrical stimulation of the endocardial surface, with red being the earliest activation at time
0 ms and blue the latest site of stimulation at 16 ms (the
contour bands are in 2-ms intervals). The non-coloured
regions of the endocardial surface have no stimulus
current applied. A stimulus current of 150 mA mmÿ3 is
applied to the coloured regions of the endocardial surface
with duration of 1 ms.
Converged?
No
Yes
Figure 4 Schematic diagram of the algorithm developed for the calculation of active tension for a given
Gauss point from the cellular values at the influential
grid points. FE, finite element; GBFE, grid-based finite
element.
points in the mechanics mesh. Using these interpolated values the cell model was solved at
the gauss point to determine regional active
tension, which was then incorporated into the
finite deformation mechanics model. The local
strain was calculated at each activation grid point
from the deformation and used to update the
model cellular lengths. The relatively slower temporal dynamics of myocardial deformation means
that this coupling update process could be implemented at a significantly larger time step, 1 ms,
than the activation solution process.
Results
The membrane potential and myocardial deformation of the finite element mesh are shown in
Fig. 5aeh at specific time points marked on
Fig. 6 through the cardiac cycle to illustrate stages
of activation and contraction.
S122
D. Nickerson et al.
Figure 5 Results from simulations of contraction and ejection in the heterogeneous left ventricular model. The
green lines show the undeformed geometry and the element faces are coloured by the membrane potential spectrum
shown above.
55
20
15
45
10
Cavity pressure
Cavity volume
5
0
0
35
25
100 200 300 400 500 600 700 800 900 1000
Cavity volume (ml)
Cavity pressure (kPa)
At the eight locations in the finite element mesh
shown in Fig. 7a the calculated strain, active tension
and membrane potentials are illustrated through
the cardiac cycle in Fig. 7bed.
Using the Purkinje-like stimulus the smooth
transmural progression of electrical activation is
clear, with full ventricular activation occurring
after 70 ms. The repolarization wave begins at
the apical epicardium approximately at 270 ms and
progresses toward the endocardial surface and
basal plane. The mid-wall at the base of the
ventricle is the last tissue to repolarize approximately at 490 ms. With the simple transmural fibre
variation illustrated in Fig. 1a and the sequence of
contraction solutions given in Fig. 7, we show that
Time (ms)
Figure 6 The imposed cavity volume and calculated
ventricular pressure and through the cardiac cycle (see
text).
the left ventricle initially deforms into a more
spherical shape with just the endocardial region
actively contracting. Then as the mechanical
activation moves out through the wall to the
circumferentially oriented mid-wall fibres, the deformation becomes more longitudinal and the
twisting motion begins. This is followed once more
by a move toward more spherical deformation as
the inclined epicardial fibres are activated and the
twisting increases.
An important feature of this heterogeneous
model is the reversal of the direction in which
the repolarization wave travels through the wall.
During the repolarization phase of the cycle the
model maintains a higher cavity pressure due to
the prolonged action potentials of the midmyocardial cells, giving rise to increased active tension at
the cellular level. The peak pressure achieved by
the model during ejection is consistent with values
reported in similar models [9].
To interpret the sequence of electrical and mechanical events taking place during the pre-ejection
phase of systole, the results shown in Fig. 7 are
reproduced on an expanded 200 ms time scale in
Fig. 8. The electrical activation sequence is endocardium to epicardium with the free wall endocardial
point shown in yellow (B) on the left of the
ventricular model in Fig. 7a being activated 50 ms
New developments in coupled cardiac electromechanical model
S123
1.15
Extension ratio ()
1.1
1.05
1
0.95
0.9
0.85
0
200
400
600
800
1000
Time (ms)
(a)
(b)
20
Membrane potential (mV)
Active tension (kPa)
40
30
20
10
0
0
200
400
600
800
1000
0
-20
-40
-60
-80
0
200
400
600
Time (ms)
Time (ms)
(c)
(d)
800
1000
Figure 7 The temporal variation in cellular signals from locations within the finite element model. The colour of
each trace in (b)e(d) matches the corresponding sphere in (a) which indicates the spatial location of the cell from
which the signal is calculated. Locations in the mid-wall are labelled AeF.
after the septal wall endocardial point shown in grey
(A) on the right. The septal transmural points
(endocardial to epicardial) shown as grey (A), orange
(C) and black (E) are separated by 100 ms intervals
(easiest to see in Fig. 8b) as are the (slightly later)
transmural sequence of free wall points shown as
green (B), light blue (D) and purple (F).
The earliest mechanical response is seen in
Fig. 8a is fibre shortening at A, 15 ms after the
electrical stimulus at A. The mechanical consequence of the active septal endocardial shortening
is stretching of the still passive adjacent mid-wall
fibres at C. The next point to activate is the free
wall endocardial point B which begins contracting
15 ms later and similarly causes substantial lengthening of mid-wall fibres at point D. The third point
to activate is the mid(septal)-wall point C at about
22 ms and 15 ms later its active response d which
follows the ‘pre-stretch’ d can be seen in Fig. 8a
as fibre shortening. The mid-wall fibres at C and D
next begin to shorten (at 35 ms and 45 ms, respectively) and contribute to the very substantial
lengthening of epicardial fibres at E and F which,
following their passive pre-stretch, begin their
contraction about 15 ms after excitation such that
by 60 ms all points are activated. Note, however,
that the fibre shortening at points A and B is
temporarily reversed as the contracting mid-wall
and then epicardial fibres exert their influence.
This substantial redistribution of fibre lengths
during the early pre-ejection phase of systole may
play a significant role in preparing the mid-wall
fibres to contract. By 80 ms ventricular pressure
has risen sufficiently to begin ejection.
To assess the potential influence of cellular
heterogeneity on myocardial deformation, specifically during repolarization, the above simulation
was repeated using a homogeneous distribution of
cellular models (trZ144 ms and APDZ352 ms).
Figs. 9 and 10 contrast the model extension ratios
and active tension transients in the heterogeneous
and homogeneous models and Table 3 provides
data on the transmural APD dispersion in the
septum and free wall of the model. In the heterogeneous model the sequence of repolarization, as
specified by the APD assigned to different regions
through the left ventricular wall, is epicardial first
(points E, F in Fig. 7a), then endocardial (points
S124
D. Nickerson et al.
1.15
1
D
Extension ratio ()
F
Extension ratio ()
C
1.1
1.05
B
1
E
A
0.95
A
0.95
B
C
0.9
D
E
0.9
0.85
F
0
50
100
150
0.85
200
200
300
350
400
450
Time (ms)
(a) Extension ratio of the first 200ms of the cardiac cycle
(a) Calculated extension ratios for the heterogeneous
model during repolarisation
20
1
0
A
Extension ratio ()
Membrane potential (mV)
250
Time (ms)
-20
A B C D E F
-40
-60
0.95
B
C
D
0.9
E
-80
0
50
100
150
200
Time (ms)
(b) Membrane potential for the first 200ms of the cardiac cycle
Figure 8 An enlargement of the results in Fig. 7bed
showing the temporal variation in extension ratio (a) and
membrane potential (b) from the locations labelled AeF
in Fig. 7a for the first 200 ms of the cycle.
A, B), and then mid-wall (points C, D). The initial
epicardial repolarization is reflected in the rapid
early sarcomere length increase from 300 ms on in
the sub-epicardium in Fig. 7b (and in expanded
form in Fig. 9 for both homogeneous and heterogeneous models). The sarcomere lengths at the
mid-wall points C, D begin increasing at about
360 ms for both the homogeneous and heterogeneous models. The major consequence of assuming
homogeneous APDs (Fig. 9b) is a substantially more
dispersed transmural sarcomere length variation.
For example, at 400 ms fibre extension ratio
distribution in the homogeneous model is 0.9 to
0.98, whereas in the heterogeneous model it is
reduced to 0.91 to 0.96. Within the sub-epicardium
and mid-wall region the range is only 0.91 to 0.92.
Discussion
The simulation results presented above demonstrate that the modelling and simulation framework
developed and implemented works well for coupled
electromechanical models using a three-dimensional geometry representative of the cardiac left
0.85
200
250
F
300
350
400
450
Time (ms)
(b) Calculated extension ratios for the homogeneous
model during repolarisation
Figure 9 An enlargement of extension ratios from 200
to 450 ms (during repolarization) in: (a) the heterogeneous model accounting for the transmural variation in
action potential durations; (b) the homogeneous model.
ventricle. Two specific insights have been provided
by this study. The first is the important role that
electrical heterogeneity may play in reducing the
transmural sarcomere length variation during repolarization. The implications for this effect, including distribution of mechanical work and the
metabolic efficiency of the whole heart, provide
interesting avenues for future investigation via an
electromechanical model. The second insight provided by the model is into the detailed relationship
between activation times, shortening and spatial
location within the myocardium. In particular, the
possibility of endocardial stretch during late isovolumic contraction has not previously been reported. The results in Fig. 8a show a significant
heterogeneity of strain at end isovolumic contraction which is consistent with the model results of
Kerckhoffs et al. [8]. However, this result is inconsistent with experimental data that report
a relatively homogeneous distribution of strain
and work throughout the myocardium [29]. Thus,
possible influences such as geometry, complex fibre
angle variation and a spatial distribution of delay
New developments in coupled cardiac electromechanical model
Active tension (kPa)
40
A
B
30
C
20
E
D
F
10
0
200
250
300
350
400
450
Time (ms)
(a) Calculated active tension in the heterogeneous
model during repolarisation.
Active tension (kPa)
40
B
30
A
C
20
10
E
D
F
0
200
250
300
350
400
450
Time (ms)
(b) Calculated active tension in the homogeneous
model during repolarisation.
Figure 10 An enlargement of calculated active tension from 200 to 450 ms (during repolarization) in: (a)
the heterogeneous model accounting for the transmural
variation in action potential durations; (b) the homogeneous model.
between excitation and activation may ultimately
need to be included in the model. It is also
important to note that the single beat simulated
in this study is effectively modelling a very slow
steady-state heart rate where the cells prior to
stimulation are always at, or close to, the initial
condition set in this model. This may explain the
differences between model and experimental results. However, to characterize more effectively,
these memory effects where ionic concentrations
Table 3 Comparison of the transmural APD dispersion in the homogeneous and heterogeneous models
Spatial
location
Homogeneous
model (ms)
Heterogeneous
model (ms)
A
B
C
D
E
F
362
360
351
350
344
339
437
434
418
417
367
367
The spatial locations of points AeF are illustrated in Fig. 7a.
S125
and gating variables at stimulation are dependent
on heart rate, a more biophysical basis to the cell
model is required.
The most significant drawback in the framework
highlighted by these simulations is the sheer
amount of time taken to obtain these results, the
coupled electromechanics simulations shown in
Figs. 6 and 7 took in the order of three weeks to
achieve 1000 ms of simulation time using an IBM
Regatta P690 high-performance computer implemented in parallel on eight processors. Furthermore, despite the large computational load
imposed by high spatial resolution grids, numerical
artefacts remain. A small error associated with
grid discretization is shown in the brief hyperpolarization prior to upstroke. Also, the sensitivity of
the lengtheactive tension coupling produces rapid
oscillations in calculated active tension during
contraction induced by finite time stepping in the
numerical method. However, on the time scale of
myocardial mechanics these variations have little
effect on ventricular deformation although ultimately physiological based viscous damping will
need to be added completely to eliminate the
oscillations.
It is this computational limitation which currently also precludes the inclusion of a more detailed cellular model. Such a model could
potentially account for mechano-electric feedback
and stretch dependence, which can now be explicitly linked to the tissue simulations via the
strong coupling solution procedure outlined in
Fig. 4. It is this coupling between cellular contraction and activation that Kerckoffs et al. [8]
propose as a possible mechanism to produce a
heterogeneous electromechanical delay that is
required simultaneously to produce both physiological activation and contraction spatio-temporal
sequences.
The framework is sufficiently general, such that
new cell models can be accommodated without
changing the existing software tools. Thus, as
computational resources continue to increase rapidly, we anticipate that in the near future we will
be using this framework with more detailed biophyisically based cell models. This generality has
been implemented using the CellML language
[30,31] to embed the cellular models in the
continuum simulations. A general application programme interface (API) has been developed (freely
available at http://cellml.sourceforge.net) to
provide a layer of abstraction between an application and the implementation of a CellML processing library. Using the API a list of the cellular
equations was generated for the model and a specific maths writer was developed to create a
S126
Fortran subroutine from the cellular equations
which could then be compiled and linked with
our CMISS simulation environment (freely available
for academic use at http://www.cmiss.org). The
CellML model repository (http://www.cellml.org/
examples/repository/index.html) provides an extensive database of cellular models of cardiac
electrophysiology and mechanics. Thus, it is largely
only computational resources that prevent repeating the above study with a detailed biophysically
based cellular model.
To assess the computational requirements of
a more detailed cellular model we have run
electromechanics simulations using a 4!4!4 mm
cube of cardiac tissue using a 200 mm grid point
spacing. With a reduced number of grid points it
was possible to compare the computational performance with the simplified model FK-HMT model
used above with a biophysically based coupled
LRd-HMT model [4,18]. Profiling of these simulations showed the additional biophysical detail
produced a threefold increase in memory usage
and a tenfold increase in computational time.
Thus, applying the often quoted Moore’s Law,
a strongly coupled cardiac electromechanics simulation using a detailed cell model will be likely to
be feasible within the next three years. Future
developments within that timeframe also include
solving electromechanical models using more realistic anatomical and microstructural models
[32,33] and coupling three-dimensional ventricular
fluid dynamics solutions to determine the endocardial boundary conditions.
Within the wider scope of the Physiome Project
we aim to enhance further the modelling framework with the use of markup languages such as
CellML (as discussed above) and other ontologies
under development. For example, a markup
language to represent the spatial variation of
fields used in the models (FIELDML: http://www.
physiome.org.nz) is also being developed. In doing
so we aim to improve the accessibility of models
across spatial scales and accelerate the development, validation and application of mathematical
models of the heart and other organs.
Acknowledgements
P.J.H. and D.P.N. acknowledge the support of
the Centre for Molecular Biodiscovery. N.P.S.
acknowledges support from the Marsden Fund
and New Zealand Institute of Mathematics and its
Applications. All authors gratefully acknowledge
contributions by colleagues in the Auckland Bioengineering Institute.
D. Nickerson et al.
References
[1] Hunter PJ, Borg TK. Integration from proteins to organs: the
Physiome Project. Nat Rev Mol Cell Biol 2003;4:237e43.
[2] Noble D, Varghese A, Kohl P, Noble P. Improved guinea-pig
ventricular cell model incorporating a diadic space, IKr and
IKs, and length- and tension-dependent processes. Can J
Cardiol 1998;14(1):123e34.
[3] Hund TJ, Kucera JP, Otani NF, Rudy Y. Ionic charge
conservation and long-term steady state in the Luo-Rudy
dynamic cell model. Biophys J 2001;81(6):3324e31.
[4] Clancy CE, Rudy Y. Na(C) channel mutation that causes
both Brugada and long-QT syndrome phenotypes: a simulation study of mechanism. Circulation 2002;105:1208e13.
[5] Smith NP, Nickerson D, Crampin EJ, Hunter PJ. Computational modelling of the heart. Acta Numer 2004;13:
371e431.
[6] Kleber AG, Rudy Y. Basic mechanisms of cardiac impulse
propagation and associated arrhythmias. Physiol Rev 2004;
84:431e88.
[7] Hunter PJ, Pullan AJ, Smaill BH. Modelling total heart
function. Annu Rev Biomed Eng 2003;5:147e77.
[8] Kerckhoffs RC, Bovendeerd PH, Kotte JC, Prinzen FW,
Smits K, Arts T. Homogeneity of cardiac contraction
despite physiological asynchrony of depolarization: a
model study. Ann Biomed Eng 2003;31(5):536e47.
[9] Usyk TP, Le Grice IJ, McCulloch A. Computational model
of three-dimensional cardiac eletromechanics. Comput
Visual Sci 2002;4:249e57.
[10] Kerckhoffs RC, et al. Timing of depolarization and contraction
in the paced canine left ventricle: model and experiment.
J Cardiovasc Electrophysiol 2003;14:S188e95.
[11] Usyk TP, McCulloch A. Electromechanical model of cardiac
resynchronization in the dilated failing heart with left
bundle branch block. J Electrocardiol 2003;36:57e61.
[12] Nickerson D, Smith NP, Hunter PJ. A model of cardiac
cellular electromechanics. Phil Trans R Soc Lond A 2001;
359:1159e72.
[13] Smith NP, Buist ML, Pullan AJ. Altered T wave dynamics in
a contracting cardiac model. J Cardiovasc Electrophysiol
2003;14:S203e9.
[14] Li W, Kohl P, Trayanova N. Induction of ventricular
arrhythmias following mechanical impact: a simulation
study in 3D. J Mol Histol 2004;35:679e86.
[15] Saucerman JJ, McCulloch AD. Mechanistic systems models of
cell signaling networks: a case study of myocyte adrenergic
regulation. Prog Biophys Mol Biol 2004;85:261e78.
[16] Smith NP, Crampin EJ. Development of models of active
ion transport for whole-cell modelling: cardiac sodiumpotassium pump as a case study. Prog Biophys Mol Biol
2004;85:387e405.
[17] Fenton F, Karma A. Vortex dynamics in three-dimensional
continuous myocardium with fiber rotation: filament
instability and fibrillation. Chaos 1998;8:20e47.
[18] Hunter PJ, McCulloch AD, ter Keurs HE. Modelling the
mechanical properties of cardiac muscle. Prog Biophys Mol
Biol 1998;69:289e331.
[19] Luo CH, Rudy Y. A model of the ventricular cardiac action
potential. Depolarization, repolarization, and their interaction. Circ Res 1991;68:1501e26.
[20] Antzelevitch C, Sicouri S, Litovsky SH, Lukas A,
Krishnan SC. Heterogeneity within the ventricular wall.
Electrophysiology and pharmacology of epicardial, endocardial, and M cells. Circ Res 1991;69(9):1427e49.
[21] Drouin E, Charpentier F, Gauthier C, Laurent K, Le Marec H.
Electrophysiologic characteristics of cells spanning the left
New developments in coupled cardiac electromechanical model
[22]
[23]
[24]
[25]
[26]
ventricular wall of human heart: evidence for presence of
M cells. J Am Coll Cardiol 1995;26(1):185e92.
Colli-Franzone P, Guerri L, Taccardi B. Modeling ventricular excitation: axial and orthotropic anisotropy effects
on wavefronts and potentials. Math Biosci 2004;188:
191e205.
Roth BJ. Electrical conductivity values used with the
bidomain model of cardiac tissue. IEEE Trans Biomed Eng
2003;44:326e8.
Remme E, Nash MP, Hunter PJ. Distributions of sacromere
stretch, stress and work in a model of the beating ventricles.
In: Kohl P, Franz M, Sachs F, editors. Cardiac mechanoelectric feedback and arrhythmias: from pipette to patient.
Philadelphia: Saunders; 2004. p. 381e91.
Tomlinson KA, Pullan AJ, Hunter PJ. A finite element
method for an eikonal equation model of myocardial
excitation wavefront propagation. SIAM J Appl Math 2002;
63:324e50.
Katz A. Physiology of the heart. 2nd ed. New York: Raven
Press; 1992. p. 361e64.
S127
[27] Hooks DA, et al. Cardiac microstructure: implications for
electrical propagation and defibrillation in the heart. Circ
Res 2002;91:331e8.
[28] Nash MP, Hunter PJ. Computational mechanics of the
heart. J Elasticity 2001;61:113e41.
[29] Prinzen FW, Hunter WC, Wyman BT, McVeigh ER. Mapping
of regional myocardial strain and work during ventricular
pacing: experimental study using magnetic resonance
imaging tagging. J Am Coll Cardiol 1997;33:1735e42.
[30] Lloyd C, Halstead M, Nielsen P. Cell ML: its future,
present and past. Prog Biophys Mol Biol 2004;85:433e50.
[31] Cuellar AA, Lloyd CM, Nielsen PF, Bullivant DP, Nickerson D,
Hunter PJ. An overview of CellML 1.1 a biological model
description language. Simulation 2003;79(12):740e7.
[32] Nielsen PM, Le Grice IJ, Smaill BH, Hunter PJ. Mathematical model of geometry and fibrous structure of the heart.
Am J Physiol 1991;260:H1365e78.
[33] Stevens C, Hunter PJ. Sarcomere length changes in a 3D
mathematical model of the pig ventricles. Prog Biophys
Mol Biol 2003;82:229e41.