Computational Biology of cardiac arrhythmias: from basic science to application
O.V. Aslanidi, V.N. Biktashev1, M. Chen2, R.H.Clayton3,
A.V. Holden, J.V. Tucker2, H. Zhang4
Computational Biology Laboratory, School of Biomedical University of Leeds, Leeds LS2 9JT
http://www.cbiol.leeds.ac.uk
1
Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL,
2
Department of Computer Science, University of Wales Swansea, Swansea SA2 8PP
3
Department of Computer Science, University of Sheffield, Sheffield, S1 4DP
4
Department of Physics, UMIST, PO Box 88, Manchester M60 1QD
Abstract
Cardiac virtual tissues are biophysically,
histologically
and
anatomically
detailed
computational models that are sufficiently well
validated to be used as a predictive tool, are
currently used in basic research, and are
beginning to be applied to clinical problems.
Virtual cardiac cells and tissues are stiff, high
order ordinary and partial differential equations.
While 1- and 2-D tissues can be run on a single
CPU, 3-D tissues are more suitable for SMP
parallel computation. Human virtual cardiac
tissues have been developed, and clinical
application to individual patients requires faster
patient specific reconstruction from multi-modal
clinical data sets. Implementation of this patient
specific approach will require high bandwidth
access from tertiary clinical centres to teraflop
compute resources. In principal this could be met
by Grid technologies, in practice dedicated HPC
clusters may be required.
Introduction
A virtual tissue is a biophysically, histologically
and anatomically detailed computational model
that is sufficiently well validated to be used as a
predictive tool. Since the essential physiology
and physics of the heart as a pump - CFD,
computational mechanics, excitation nonlinear
wave phenomena in excitable media - are well
understood, cardiac virtual tissues are well
developed. Currently used in basic research,
cardiac virtual tissue engineering is beginning to
be applied to clinical problems, and promises to
produce tools that will allow an order of
magnitude reduction in death rates within a
decade. Virtual cardiac cells and tissues are stiff,
high order ordinary and partial differential
equations. An anisotropic monodomain virtual
cardiac tissue is a computational implementation
of a parabolic reaction-diffusion equation:
Cm
¶V m
= Ñ(DÑVm ) - I ion
¶t
(1)
where Vm is voltage across the cell membrane,
Cm specific membrane capacitance, D a diffusion
tensor and Iion current flow though the cell
membrane per unit area. Combining the
equations for transmembrane ion flows and
intracellular sequestration and binding processes
produces a virtual cell as a nonlinear system of
differential equations. These equations are
typically high order, and stiff (time scales vary
from fractions of a ms to a few s) and may be
numerically solved on a simple single
workstation. While 1- and 2-D tissues can be run
on a single CPU, 3-D excitable tissues in
complicated or moving geometries are more
suitable for SMP parallel computation.
Virtual mammalian cardiac cells and tissues [1]
have been constructed and applied to dissect the
patho-physiology of arrhythmias [2,3], to identify antiarrhythmic pharmacological targets [4],
to design low voltage defibrillation techniques
[5], and to aid the interpretation of signals
recorded during ventricular fibrillation [6]. Their
role as basic research tools is well established.
Some human virtual cardiac tissues have been
developed but their validation is rudimentary.
Clinical application to individual patients - to
facilitate diagnosis, and to predict the effects of
different
pharmacological
and
physical
interventions - requires better validation and
faster algorithmic or atlas-based reconstruction
of cardiac geometry from multi-modal clinical
data sets - electrophysiological mapping,
visualisation (angiography, echo-cardiography,
MRI
and
nuclear
medicine,
and
haemodynamics). This patient specific approach
is being developed within a European multidisciplinary network [7] and will require high
bandwidth access from tertiary clinical centres to
teraflop compute resources.
Figure 1. Construction of virtual mammalian sinoatrial node (A) surface topography, with voltage
isolines showing propagation out from node, and recorded and simulated action potentials (B); molecular
mapping of slice, showing distribution of illustrative proteins in (C) 1-D and (D) 2-D model of
propagation with block zone [9-14].
In principal this could be met by grid
technologies, in practice dedicated HPC clusters
may be required [8].
The pacemaking system of the heart
Action potentials Vm(t) from different parts of
the heart have different characteristics, in terms
of their maximum rates of rise and fall, duration
and shape, that change as the interval between
action potentials alters. However, they are all
generated by similar processes. Differences in
action potentials from different parts of the heart
result from quantitative differences in the
expression of the different ion transport proteins
[9]. Cardiac cells from different regions have
different densities of different membrane
channels, pumps or exchangers.
In a virtual cardiac tissue the regional changes in
cell properties produced by differential channel
expression can be represented as a gradient in
parameter values for the cell excitation models in
a spatially heterogeneous partial differential
equation model [10]. Fig. 1 illustrates the
construction of cell models for the pacemaker of
the rabbit heart, the sinoatrial node, molecular
mapping that can be used to construct the spatial
variation of parameters in the PDE, and
snapshots of 1- and 2-dimensional solutions. The
detailed cell and membrane experiments that
were necessary for the ab initio construction of
this model are not necessary for the construction
of an analogous model for the human pacemaker:
one can modify the rabbit model, using
molecular mapping to determine the spatial
distribution of membrane channel proteins, and
incorporate any modified channel kinetics that
have been obtained from expressed single
channel studies. The spatial distribution of
expressed proteins provides spatially varying
parameters for the reaction-diffusion equation.
Clinical phenomena - the deterioration of
pacemaking function with age [11], and the
effects of drugs on heart rate [12, 13] can be
explained by such a chimaeric (partly based on
animal, partly based on human) data, and
validated by noninvasive clinical measurements.
magnetic resonance imaging (Fig. 3) coronary
angiography, and radioisotope imaging (Fig. 7).
The 3D+t nature of the heart, different features
of which can be obtained from different imaging
modalities, or computed from virtual cardiac
tissues, requires essentially 3D visualisation
methods that allow combinatorial visual
representations that can extract meaningful
information from different field data sets.
Constructive Volume Geometry provides one
approach, illustrated in Fig. 4, where fibre
bundles have been extracted from the fiber
orientation map, and recombined with whole
ventricle geometry [22, 23].
Figure 2. Virtual tissue engineering of human
atrium: visualisation of human atrium geometry.
Human atrial flutter, fibrillation and
remodelling
Cell models for human tissue are not as well
developed or validated as those for laboratory
animals, but there are two current cell models for
human atrial cell [14, 15]. These are very stiff,
and both, when incorporated into a 2-D excitable
medium show spiral wave breakup [16] due to
excitation dissipation [17, 18]. The twochambered atrium has a complicated geometrythere are junctions with the veins, as well as with
the ventricles, and the walls are thin and
irregular, containing muscle fiber sheets and
bundles. Reconstruction of the moving geometry
from MRI has not been possible, and the
reconstruction from post mortem material on Fig.
2 illustrate the preponderance of surface points.
Numerical solution, by a Cartesian-grid finitedifference approximation with Neumann
conditions on an irregular boundary, combined
with the stiffness of the equations produces
boundary instabilities. As a result, much of the
simulation of atrial electrophysiology uses
tissues with simple geometries - e.g. the 2-D
surface of a coupled spheres with holes. Human
atrial virtual cells and simple tissues can be used
to explore the mechanisms of remodeling – activity induced changes in tissue properties [19].
Ventricular electrophysiology and fibrillation
During ventricular fibrillation (VF), electrical
activation of the ventricles is rapid, self
sustained, and has a complex spatio-temporal
pattern. The rapid ventricular activation during
VF is sustained by re-entry, during which an
excitation wave repeatedly propagates into
recovered tissue, and rotates around a phase
singularity that is a point in 2D and a filament in
3D. There is evidence that VF could be sustained
by either a single re-entrant wave with
fibrillatory conduction [24, 25], or by breakup of
an initial reentrant wave to multiple wavelet reentry. The details of ventricular wall structure are
important, as illustrated in Fig. 5 for re-entrant
propagation, which is stable in a homogenous
ventricular wall, but breaks down in an
anisotropic wall, and breaks down sooner when
there is transmural heterogeneity.
Reconstructing and visualising ventricular
anatomy
Data sets for the canine [20] and rabbit [21]
ventricles are available, from which can be
extracted a Cartesian grid that provides the
ventricular geometry and fiber orientation: this
provides the geometry and diffusion tensor for
ventricular implementation of equation (1).
Human cardiac anatomy can be obtained from
Figure 3. Frames form movies of (a) MRI slice
through ventricles of beating human heart during
diastole (b) Extracted epi- and endocardial
boundaries and (c) surfaces.
Figure 4. Use of constructive volume geometry operations to construct a visualisation of the fibre
bundles within the geometry of the ventricle. The fiber bundles were dissected digitally, by choosing
points at random, and following the same fiber orientation within a tolerance.
The diffusion tensor is computed from the fibre
orientation,
obtained
from
quantitative
histological mapping of fiber angle and sheet
orientation [20]. For the human heart, fiber
Figure 5. Wavefronts and filaments in virtual
canine right ventricular wall during re-entry.
Homogenous (a) isotropic (b) anisotropic and (c)
heterogeneous and anisotropic tissue [28-30].
orientation has been approximated from the NIH
visible female (vhp@nlm.nih.gov), but reliable
public domain detailed digital maps of cardiac
structure are lacking, and fast throughput
methods for mapping post mortem hearts are
needed for anatomical atlas construction.
In principle, anisotropic fiber organization and
orthotropic sheet structure could be obtained by
non-invasive diffusion tensor MRI [26].
Transmural heterogeneity in cell properties has
been obtained from electrophysiological and
molecular mapping techniques for some
mammalian hearts, but quantitative data for the
human ventricle are lacking.
Filaments were detected from the intersection of
Vm = -20 mV and dVm/dt = 0 isosurfaces, and
voxels containing filaments identified. Using this
approach we were able to identify the birth,
death,
bifurcation,
amalgamation
and
continuation of filaments, and we displayed the
dynamics of these interactions as a directed
graph using an approach that has been used to
describe activation on the heart surface [27].
The normal pattern of excitation, following
endocardial activation via Purkinje fibres, is
illustrated in Fig. 6 for the geometry and
anisotropy of [20].
(ms)
1
2
3
4
5
Figure 6. Activation time following endocardial
excitation in virtual ventricle.
Figure 7. Electrophysiological indicators of ischaemia: (a) averaged ECG shows S-T segment depression
(b) ectopics occuring during recovery from exercise but illustrative (c) coronary angiograms and (d)
radiotracer visualisation of myocardial perfusion (stress above rest) apparently normal.
Case study: syndromeX
In syndrome X there are symptoms (angina) and
objective measures (ST depression on exercise,
ectopic ventricular beats) suggesting an
insufficient blood flow to the stressed cardiac
muscle. However, angiography of coronary
arteries, and radiotracer measures of cardiac wall
perfusion show no focal defect: see Fig. 7.
About 10% of patients referred for angiography
after an ECG exercise test have syndrome X. The
clinical data (ECG - multiple time series,
angiography - 2D projections of 3D structures,
and nuclear medicine - 2D sections of a 3D
field), are all obtained as digital data sets and so
could be incorporated into CVG or coupled with
virtual tissues. Fig. 8 is from computational
dissection of the electrophysiology of subendocardial
ischaemia
in
heterogeneous
ventricular wall, showing ST depression and
ectopic activity can be produced by subendocardial ischaemia [31].
using virtual tissues [32]. Such shocks are not
always effective, can damage cardiac tissue, and
be painful. The adoption of implanted intelligent
defibrillators has increased the demands for low
voltage methods of defibrillation. Virtual cardiac
tissues have been used for designing and
exploring two possible low voltage technologies.
The resonant drift method exploits the stability
and symmetries of re-entrant (spiral) waves: a
small perturbation can cause a displacement, or
rotation (phase change) of a spiral, and so
repetitive perturbations can produce a directed
drift if applied at the same phase i.e. at the
1,5
1,0
Low voltage defibrillation
0,5
Re-entrant excitation can break down into
fibrillation that results in haemodynamic
collapse, and VF is quickly lethal unless normal
rhythm can be restored, say by a large amplitude
defibrillating shock. The possible virtual
electrode mechanisms of how such an external
shock defibrillates the heart have been explored
0,0
-0,5
0
200
400
600
800
1000
Figure 8. Space-time plot and electrogram of
transmural propagation in 1-D heterogeneous
virtual ventricular wall. Globally ischaemic,
ectopic initiated in mid M-cell region.
Figure 9. Snapshot after 1.5 s of simulated
electrical fibrillation in geometry of Fig. 6,
showing a voltage isosurface and the associated
filaments.
instantaneous rotation frequency of the spiral.
Thus can be achieved by feedback, and such
resonant drift under feedback control using
appropriately timed small amplitude shocks can
produce drift velocity of cm/s in virtual cardiac
tissue, driving the core of the spiral to the
boundary and extinguishing re-entry in a few
seconds [33, 34].
Experiments using isolated perfused hearts have
show that periodic (5-20 Hz), low amplitude
sinusoidal forcing can establish standing waves
that eliminate fibrillation: computations show
that the mechanism for these standing waves
depends on extracardiac, as well as intracardiac
extra- and intracellular current pathways [35].
Conclusions
Virtual cardiac tissues were originally
constructed as a research project, now developed
and validated, they are being applied as routine
laboratory tools. Such applications produces a
quantitative shift, from customised to mass
throughput, that is being accelerated by the
introduction of high throughput, quantitative
techniques into biomedical sciences. This
increases demand for HPC resources.
A simple but approximate unit for quantifying
the computational load is the Euler heart beat;
the number of floating point operations that
would be necessary to compute one heartbeat.
The resting human heart rate is about 70/min., so
a heartbeat lasts about 1s. Computing 1 s of
electrical activity, with a time step of 0.01 ms
and a space step of 0.1 mm would require some
1014 floating point operations using fixed time
and space steps. As a rough illustration, an
arrhythmia may take a few such Euler heart
beats; systematic investigation of the effects of
one drug on such an example may take 103 Euler
heart beats. There is an ever increasing demand
for compute performance, as virtual tissues are
applied to pharmacological prescreening and
defibrillation methods.
The incorporation of further mechanisms into
cardiac virtual tissues produce a continuing
inflation in the computational exchange rate for
an Euler heart beat. Although the computational
electromechanics and fluid dynamics (coronary
perfusion and blood ejection) is a grand
challenge projects, more demanding and useful
challenges emerge as virtual tissues are applied
to real clinical problems, leading to intermittant
needs for real time solutions.
Multiple runs of 2- and 3-dimensional slab
computations, as illustrated in Fig. 5, are run on
our grid, while whole ventricle computations, as
illustrated in Fig 9, are run on our SMP machine.
Cardiac virtual tissues are not suitable for
parallelization over a large number of thin nodes,
but many problems may be run in batch mode
over an assembly of separate processors. This
provides a massive increase in throughput, as it
is production line rather than developmental
scientific computation, and has produced a
visulisation bottleneck. This is being solved by
grid-enablement of visualisation tools, to allow
computational guidance of multiple runs and
steering of whole ventricle computations.
However, all grid based approaches assume high
bandwidth data exchange between clinical and
HPC facilities: these are currently prevented by
cultural rather than technical firewalls, but there
are local approaches towards solving these
problems, e.g. Leeds Interagency for Sharing
Information protocol between health and social
care agencies.
Even for small slabs of virtual tissue,
computational guidance requires extracting of
aspects of the solution from the full 3dimensions+time computational output, that can
be fully explored in virtual reality through a vrml
browser. This is illustrated in Fig. 5, that
Figure 10. 2-D ventricular virtual tissue: reentrant spiral, with tip and meandering tip
trajectory in 8 cm square medium. Tip
trajectories under resonant drift under feedback
control, applied at four different delays/phases:
in all cases the spiral is driven to the boundary
and extinguished.
Figure 11. Frames from movie showing response of re-entry to periodic low-amplitude sinusoidal
forcing, resulting in elimination of re-entry via a standing wave.
displays surface views and filaments from frames
from a movie of activity in a slab. For the canine
ventricular geometry illustrated in Fig. 9, 1 s of
activity could be simulated in about 3.5 hours
with OpenMP parallel computation on 8 750
MHz Sun Ultrasparc III processors. For
currently funded research projects in our
laboratory - virtual prescreening, mapping the
pacemaker of the heart and on the mechanisms of
ventricular fibrillation - a sustained 24/7 compute
demand of 1-2 teraflop is anticipated within 3
years. In the UK alone there are several other
centres with similar research requirements.
Acknowledgements. Research in the CBL is
supported by project and programme grants from
the BHF, EPSRC and MRC, on equipment
provided by HEFC/JREI and Sun Microsystems
AEGs.
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