Noble2002-Modelling the Heart-Insights Failures and Progress

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Modelling the heart: insights,failures and progressDenis Noble

SummaryMathematical models of the heart have developed overa period of about 40 years. Cell types in all regions ofthe heart have been modelled and they are now beingincorporated into anatomically detailed models of thewhole organ. This combination is leading to the creationof the first virtual organ, which is being used in drugdiscovery and testing, and in simulating the action ofdevices, such as cardiac defibrillators. Simulation is anecessary tool of analysis in attempting to understandbiological complexity. We often learn as much from thefailures as from the successes of mathematical models.It is the iterative interaction between experiment and

simulation that is important. Examples are given wherethis process has been instrumental in some of the majoradvances in the field. BioEssays24:11551163, 2002. 2002 Wiley Periodicals, Inc.

Introduction

Mathematical modelling is widely accepted as an essential

tool of analysis in the physical sciences and engineering, yet

many arestill scepticalaboutits role in biology. Why is this so?

A partial answer would be that many chose to work as biolo-

gistsprecisely because theywere not mathematically inclined.

But I think there is a stronger reason. There is also a wide-

spread belief in the biological sciences that, to be useful,

theory must be correct. Failure is seen as just that, i.e. failure,

rather than as an essential part of a two-way iterative process

between theory and experiment.

In the case of excitable cells, it could also be that the para-

digm model, the Hodgkin Huxley(1) equations for the squid

nerve action potential, was so spectacularly successful that it

created an unrealistic expectation for the rapid application of

the same principles elsewhere. In fact, modelling of the much

more complex cardiac cell has required many years of inter-

action between experiment and theory. In modelling com-

plex biological phenomena, this is precisely what we should

expect(2,3) and it is standard for such interaction to occur over

many years in thephysicalsciencesand engineering.I will first

illustrate this interaction in biological simulation using some

of the cell models that I have been involved in developing.

Since my purpose is didactic, I will be highly selective. A more

complete historical review of cardiac cell models can be found

elsewhere(4) and the volume in which that article appeared is

also a rich sourceof material on modellingthe heart,since that

was its focus.

At each stage, I will highlight the theoretical insights that

came from mathematical modelling as well as the experi-

mental basis. I will then briefly review the use of cell models in

tissue and organ simulations.

Energy conservation during the cardiac cycle:

discovery of various forms of potassium

channels

The first cardiac cell models sought insight into the most

obvious difference between electrical activity in heart and

nerve: the duration of the action potential. A nerve action

potential may last only 1 msec. Its function is to encode infor-

mation as rapidly as possible. In contrast, a human ventricular

action potential may last 400 msec, during which time many

events are triggered that initiate and control mechanical

contraction.

FitzHugh(5)

showed that the HodgkinHuxley model ofthe nerve impulse could generate slow repolarization by

greatly reducing the amplitude and speed of activation of the

delayed potassiumcurrent,IK. These changes notonlyslowed

repolarization; they also created a plateau. This result showed

that there must be some property inherent in the Hodgkin

Huxley formulation of the sodium current that permits a

persistent inward current to occur. The main defect of the

FitzHugh model was that it was a very expensive way of

generating a plateau, with such high ionic conductances that,

during each action potential, the sodium and potassium ionic

gradients would be run down at a rate at least an order of

magnitude too large.

That this was not the case was already evident sinceWeidmanns(6,7) pioneering work showed that the conduc-

tance during the action potential is very low. The experimental

reason for this became clear with the discovery of the inward-

rectifier current, IK1(810) (see Fig. 1, top). The permeability of

this channel falls almost to zero during strong depolarisation.

These experiments were also thefirst to show that there are at

least two K conductances in the heart, IK1 and IK (referred to

BioEssays 24:11551163, 2002 Wiley Periodicals, Inc. BioEssays 24.12 1155

University Laboratory of Physiology, Parks Road, Oxford OX1 3PT,

UK. E-mail: [email protected]

Funding agencies: The British Heart Foundation, The Medical

Research Council, The Wellcome Trust and Physiome Sciences Inc.

DOI 10.1002/bies.10186

Published online in Wiley InterScience (www.interscience.wiley.com).

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as IK2 in early work, but now known to consist of IKr and IKs).

The Noble 1962 model(11,12) (Fig. 1, bottom) was construct-

ed to determine whether this combination of K channels,

togetherwith a Hodgkin Huxleytype(see Box 1 in Marder and

Prinz in this issue of BioEssays) sodiumchannel could explain

all the classical Weidmann experiments on conductance

changes. The model not only succeeded in doing this; it also

demonstrated that an energy-conserving plateau mechanism

was an automatic consequence of the inward-rectifying pro-

perties of IK1. This has featured in all subsequent models,

and it is a very important insight. The main advantage of a low

conductance is minimising energy expenditure.

Unfortunately, however, a low conductance plateau was

achieved at the cost of making the repolarization process

fragile.Pharmaceutical companies today are struggling to deal

with evolutions answer to this problem, which was to entrust

repolarization to the potassium channel iKr. This channel pro-

tein is one of the most promiscuous receptors known: large

ranges of drugs can enter the channel mouth to block it, and

even more interact with the G-protein-coupled receptors thatcontrol it. The consequence can be failed repolarization, and

the triggering of fatal arrhythmias (see http://georgetowncert.

org/qtdrugs_torsades.asp). Computer simulation is now play-

ing a major role in attempting to find a way around this difficult

and seemingly intractable problem.(13)

The main defect of the 1962 model was that it included

only one voltage-gated inward current, INa. There was a good

reason for this. Calcium currents had not then been dis-

covered. There was, nevertheless, a clue in the model that

something important was missing. The only way in which

the model could be made to work was to greatly extend the

voltage range of the sodium window current by reducing

thevoltage dependence of thesodium activation process (seeRef. 12, Figure 15). In effect, the sodium current was made to

serve thefunctionof both thesodium and calcium channels so

far as the plateau is concerned. There was a clear prediction

here: either sodium channels in the heart are quantitatively

different from those in nerve, or other inward current-carrying

channels must exist. Both predictions are correct.

The first successful voltage clamp measurements came in

1964(14) and they rapidly led to the discovery of the cardiac

calcium current.(15) By the end of the 1960s therefore, it was

already clear that the 1962 model needed replacing.

Controversy over the pacemaker

current: the MNT modelIn addition to the discovery of the calcium current, the early

voltage clamp experiments also revealed multiple compo-

nents of IK. Noble & Tsien(16) labelledthem Ix1 and Ix2, but they

are now referred to as IKr and IKs.(17) They also showed that

these slow gated currents in the plateau range of potentials

were quite distinct from those near the resting potential, i.e.

that there were twoseparate voltage rangesin which very slow

Figure 1. Top: Experimental basis of the first classification

of potassium currents. Redrawn from reference (10, Fig. 2B).

The solid line shows the total membrane current recorded in

a Purkinje fibre in a sodium-depleted solution. The inward-

rectifying current was identified as iK1, which is extrapolated

here as nearly zero at positive potentials. The outward-

rectifying current, IK, is now known to be mostly formed by

the component IKr. Thehorizontal arrow indicatesthe trajectory

at the beginning of the action potential, while the vertical

arrow indicates the time-dependent activation of IK which

initiates repolarization. Bottom: Sodium and potassium

conductance changes computed from the 1962 model(12) of

the Purkinje fibre. Two cycles of activity are shown. The con-

ductances are plotted on a logarithmic scale to accommodate

the large changes in sodium conductance. Note the persistentlevel of sodium conductance during the plateau of the action

potential, which is about 2% of the peak conductance. Note

also therapid fall in potassium conductanceat thebeginningof

the action potential. This is attributable to the properties of the

inward rectifier iK1.

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conductance changes could be observed.(16,18) These experi-

ments formed the basis of the MNT model.(19)

The MNT model reconstructed a much wider range of

experimentalresults, andit didso with great accuracy in some

cases. A good example of this was the reconstruction of the

paradoxical effect of small current pulses on the pacemaker

depolarisation in Purkinje fibres (see Fig. 2)paradoxical

because brief depolarisations slow the process and brief

hyperpolarizations greatly accelerate it. Reconstructing para-

doxical or counter-intuitive results is of coursea major function

of modelling work. This is one of the roles of modelling in

unravelling complexity in biological systems.

But the MNT model also contained the seeds of a spec-

tacular failure. Following the experimental evidence,(18) it

attributed the slow conductance changes near the resting

potential to a slow gated potassium current, IK2. In fact, what

became the pacemaker current, or If, is an inward current

activated by hyperpolarization(20) not an outward current acti-

vated by depolarisation. At thetime, it seemed hard to imagine

a more serious failure than getting both the current directionand the gating by voltage completely wrong. There cannot be

much doubt therefore that this stage in the iterative interaction

between experiment and simulation created a major problem

of credibility. Perhaps cardiac electrophysiology was not really

ready for modelling work to be successful?

This was how the failure was widely perceived. Yet that

perception was a deep misunderstanding of the significance

of what was emerging from this experience. It was no

coincidence that both the current direction and the gating

were wrong, as one follows from the other. And so did much

else in themodelling!Working that outin detailwas theground

on which future progress could be made.

This is thepointat which to emphasise one of theimportant

points about the philosophy of modelling which I highlighted

at the beginning of this article: it is one of the functions of

models to be wrong! (Indeed, a Popperian philosopher of

science would say that one can only ever establish the falsity

of theories.) Of course, there are many ways of being wrong,

and I am not talking here of failing in arbitrary or purely con-

tingent ways, but in ways that advance our understanding by

exploring possible logics of complex systems and determining

which are most accurate. Again, this situation is familiar to

those working in simulation studies in engineering or cosmol-

ogy or in many other physical sciences.And, in fact, thefailure

of the MNT model is one of the most instructive examples

of experiment simulation interaction in physiology, and of

subsequent successful model development. I do not have the

space here to review this issue in all its details. That has

already been done.(21,22) Here I will simply draw the conclu-

sions relevant to modern work.

First, careful analysis of the MNT model revealed that its

pacemaker current mechanism could not be consistent withwhat isknownof theprocessof ionaccumulationanddepletion

in the extracellular spaces between cells. The model itself

was therefore a key tool in understanding the next stage of

development.

Second, a complete and accurate mapping between the

IK2 model and the new If model could be constructed(21)

(see Fig. 2) demonstrating how both models related to the

same experimental results and to each other. Such mapping

between different modelsis rare inbiologicalwork,but itcan be

very instructive.

Third, this spectacular turn-around was the trigger for

the development of models that include changes in ion

Figure 2. Reconstructionof theparadoxical effect ofsmall currentsinjectedduringpacemakeractivity.Left: Computations fromthe MNT

model.(19) Small depolarising and hyperpolarizing currents were applied for 100 msec during the middle of the pacemaker depolarisation.

Hyperpolarizations are followed by an acceleration of the pacemaker depolarisation, whilesubthreshold depolarisations induce a slowing.

Middle: Experimental records from Weidmann (Ref. 6, Fig. 3). Right: Similar computations using the DiFrancesco-Noble (1985)

model.(24) Despite thefundamental differencesbetweenthese twomodels,the featurethat explainsthe paradoxical effects of small current

pulses survives. This kind of detailed comparison was part of the process of mapping the two models onto each other.

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BioEssays 24.12 1157


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concentrations inside and outside the cell and between

intracellular compartments.

Finally, the MNT model was the point of departure for the

ground-breaking work of Beeler and Reuter(23) who developed

the first ventricular cell model. As they wrote of their model:

in a sense, it forms a companion presentation to the recent

publication of McAllister et al. (1975) on a numerical re-

construction of the cardiac Purkinje fibre action potential.

There are sufficiently many and important differences be-

tween these two types of cardiac tissue, both functionally and

experimentally, that a more or less complete picture of mem-

brane ionic currents in the myocardium must include both

simulations. For a recent assessment of this model and the

subsequent Luo-Rudy models see Ref. 4.

Ion concentrations, pumps and exchangers:

The DiFrancescoNoble model(24)

The incorporation not only of ion channels (following the

HodgkinHuxley paradigm) but also of ion exchangers, such

as NaK exchange (the sodium pump), NaCa exchange,the SR calcium pump and, more recently, the transporters

involved in controlling cellular pH,(25) was a fundamental

advance since these are essential to the study of some dis-

ease statessuch as congestive heart failure(26) and ischaemic

heart disease.

It was necessary to incorporate the NaK exchange pump

since what made If so closely resemble a potassium channel

in Purkinje fibres was the depletion of K ions in extracel-

lular spaces. This was a key feature enabling the accurate

mapping of the IK2 model (MNT) onto the If model.(21) But, to

incorporate changes in ion concentrations it became neces-

sary to represent the processes by which ion gradients can be

restored and maintained. In a form of modelling avalanche,once changes in one cation concentration gradient (K) had

been introduced, the others (Na and Ca2) had also to be

incorporatedsince thechangesare alllinked viathe NaK and

Na Ca exchange mechanisms. This avalanche of additional

processes was the basis of the DiFrancescoNoble (1985)

Purkinje fibre model.(24)

The greatly increased complexity of the DiFrancesco

Noble model, which for the first time also represented intra-

cellular events by incorporating a model of calcium release

from the sarcoplasmic reticulum, increased both the range of

predictions and the opportunities for failure. Here I will limit

myself to one example of each.

The most influential prediction was that relating to thesodiumcalcium exchanger. In the early 1980s, it was still

widely thought that the original electrically neutral stoichio-

metry (Na:Ca2:1) derived from early flux measurements

was correct. The DiFrancesco-Noble model achieved two

important conclusions. The first was that, with the experimen-

tally known Na gradient, there simply wasnt enough energy

in a neutral exchanger to keep resting intracellular calcium

levels below 1 mM, i.e. at a level low enough to permit relax-

ation to occur. Switching to a stoichiometry of 3:1 readily

allowed resting calcium to be maintained below 100 nM. This

automatically led to the prediction that there must be a current

carried by the NaCa exchanger and that, if this exchanger

was activated by intracellular calcium, it must also be strongly

time-dependent since intracellular calcium varies by an order

of magnitude during each action potential. Even as the model

was being published, experiments demonstrating the current

INaCa were being performed(27) andthe variation of this current

during activity was being revealed either as a late compo-

nent of inward current or as a current tail on repolarization.

This prediction has turned out to have very important con-

sequences for the elucidation of some of the mechanisms of

cardiac arrhythmia in disease states in whichcells accumulate

sodium and calcium, either through loss of energy supply, as

in ischaemia, or as a consequence of reduced activity of the

NaK ATPase (Na pump) as in treatment with cardiac

glycosides. At a critical level of sodiumand calcium accumula-

tion, calcium release occurs spontaneously and becomesrepetitive between sodium concentrations around 13 and

22 mM,(2831) a phenomenon also seen experimentally.

Each calcium release activates inward current carried by

sodium calcium exchange which, if large enough, can trigger

additional (ectopic) action potentials,(32) as shown in Fig. 3.

The main defect of the DiFrancescoNoble model was

that the intracellular calcium transient was far too large, mainly

because the model did not represent the attachment of

calcium to intracellular proteins. This signalled the need to

incorporate intracellular calcium buffering.

Calcium balance: the Hilgemann-Noble model

This deficiency was tackled in the HilgemannNoble(33)

(1987) modelling of the atrial action potential. Although this

was directed towards atrial cells, it also provided a basis for

modelling ventricular cells in species (rat, mouse) with short

ventricular action potentials, and many of its features were

adopted in later ventricular cell models of species with high

plateaus.(3438)

The HilgemannNoble model addressed a number of

important questions concerning calcium balance:

1. When does the calcium that enters during each action

potential return to the extracellular space? Does it do this

during diastole (as most people had presumed) or during

systole itself, i.e. during, not after, the action potential?Hilgemann(39) had done experiments with tetramethylmur-

exide, a calcium indicator restricted to the extracellular

space, showing that the recovery of extracellular calcium

(inintercellular clefts) occurs remarkably quickly (seeFig. 4,

inset). In fact, net calcium efflux is established as soon as

20 msec after thebeginning of theaction potential, which at

that time was considered to be surprisingly soon. Calcium

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activation of efflux via the Na Ca exchanger achieved this

in the model (see Fig. 4compare the computed trace

[Ca]0 with the experimental trace labelled [Ca]0)

2. Where was the current that this would generate and did it

correspond to the quantity of calcium that the exchanger

needed to pump? Mitchell et al(40) had already done

experiments in rat ventricle showing that replacement of

sodium with lithium removes the late plateau. This was the

first experimental evidence that the late plateau in action

potentials with this shape might be maintained by sodium

calcium exchange current. The HilgemannNoble model

showed that this is precisely what one would expect.

3. Could a model of the SR that reproduces at least the

major features of Fabiatos experiments(41,42) showing

calcium-induced calcium release (CICR) be incorporated

into the cell models and integrate in with whatever were

the answers to questions 1 and 2. This was a major

challenge (see reference (33), pp 173174). The model

followed as much of the Fabiato data as possible, but the

conclusions were that the modelling, while broadly con-

sistent with the Fabiato work, could not be based on that

alone. It is an important function of simulation to reveal

when experimental data needs extending.

4. Were the quantities of calcium, free and bound, at each

stage of the cycle consistent with the properties of the

cytosol buffers? The answer here was a very satisfactory

yes. The great majority of the cytosol calcium is bound so

that, althoughmuch morecalciummovement was involved,

the free calcium transients were much smaller, within theexperimental range.

There were however some gross inadequacies in the

calcium dynamics. An additional voltage-dependence of Ca

release was inserted to obtain a fast calcium transient. This

was a compromise that really requires proper modelling of

the subsarcolemmal space where calcium channels and the

Figure 3. Left: Calciumoscillations computedin a model of thePurkinje fibre during therise in intracellular sodium(red) followingblockof

the sodium-potassium ATPase. Each oscillation of intracellular calcium (blue trace below) triggers inward sodium-calcium exchange

current (brown trace top) (from Ref. 25). These computations were done under voltage clamp conditions. Right: Free-running action

potentials in thesame model withintracellularsodiumset at a level corresponding to themiddleof theoscillationperiodin theleft handfigure

(from Ref. 32). For a theoretical analysis of this phenomenon in the DiFrancescoNoble model see Refs. 28,29 and for an analysis of the

more detailed WinslowRiceJafri model(30) see Ref. 31.

Figure 4. The first reconstruction of calcium balance in

cardiac cells. The HilgemannNoble model(33) incor-

porated complete calcium cycling, such that intracellular and

extracellular calcium levels returned to their original state after

each cycleand that the effects of sudden changes in frequency

could be reproduced. Left: Computed action potential (AP),

intracellular calcium transient, contraction (represented by

cross-bridge formation) and extracellular calcium transient.

Inset: Experimental recording of action potential (AP), cell

motion and extracellular calcium transient.

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ryanodine receptors interact, a problem later tackled by Jafri,

Rice & Winslow(43,44) and by Noble et al.(38) Another problem

was how would the conclusions apply to action potentials with

high plateaus. This was tackled both experimentally(45) and

computationally.(35,38) The answer is that the high plateau in

ventricular cells of guinea-pig, dog, human etc greatly delays

the reversal of the sodium calcium exchanger so that net

calcium entry continues for a longer fraction of the action

potential. This property is important in determining the force-

frequency characteristics.

Incorporation of cell models into tissue

and organ models

Although single-cell models of the heart were first constructed

over 40 years ago, it is only relatively recently that it has

become possible to move up to the next levels: tissue and

organ models into which the cell models can be incorporated.

This was always one of the aims of the single cell work, but

until the development of sufficiently powerful supercomputers

this objective was beyond reach.One of the early examples of massive three-dimensional

ionic tissue models usedintracellular calciumoscillator models

of the kind shown in Fig. 3 to address the question of the

minimal size of an ischaemic focus, simulated by sodium

overload in a defined tissue region, that would be arrhyth-

mogenic.(46) A sodium-overloaded cell group was placed in a

20-cell-diameter spherical volume in the centre of a three-

dimensional network of 12812832 atrial cells using the

EarmNoble single cellmodel(47) (i.e.more than500,000ionic

cell models were used). It was shown that this relatively small

ischaemic focusin real terms, the tissue volume would be

in the order of no more than a cubic millimetreis capable of

triggering a spreading wave of excitation in the whole tissue

model (Fig. 5). This was an important step forward in link-

ing a subcellular mechanism of arrhythmia to multicellular

processes. However, the initiation of an ectopic beat is not,

in itself, sufficient to trigger a life-threatening arrhythmia

(fortunately most ectopic beats are benign!) It still remains to

work out how this mechanism interacts with other features of

ischemic tissue, such as slowed conduction due to potassium

accumulation (4850) and action potential shortening, to trigger

re-entrant arrhythmias that may break down into full-scale

ventricular fibrillation.

On the one hand, this example shows that both qualitative

andquantitativeconclusionsmay be drawnfrom computations

based on multidimensional, uniform grid-structured cardiac

models. On the other hand, the deficiencies in these models

also show that geometrical factors, tissue (in)homogeneity,

cell-to-cell coupling, etc., are critical to an understanding of

detailedphysiology and pathology. Thus, while grid-structured

cardiac tissue models clearly have their place, more sophis-

ticated anatomical models are required to progress be-yond the reproduction of biophysical behaviour and towards

the provision of physiologically and clinically relevant new

insights.

Simulating myocardial excitation in three-dimensional

anatomically detailed models of the heart began with the work

of Panfilov and colleagues(51,52) who mapped finite difference

models of FitzHughNagumo equations to experimentally

determined cardiac geometry and fibre distribution data.(53)

The first solutions of ionic current models used finite

difference techniques and non-deforming anatomical heart

geometry.(54)Later, ionic current models were linked to mate-

rial points of a moving finite element mesh.(55) This allowed

Figure 5. A three-dimensional atrial model with

centrally located small sodium-overloaded ischae-

mic focus. The dimensions of the atrial tissue block

are outlined in orange, while the action potential

wavefront and plateau are colour coded (red and

magenta,respectively).The excitation begins in the

small central region of sodium overload, where a

calcium oscillation generates an action potential by

activating sodium-calcium exchange (see Fig. 3).

If the overload region (top left) is sufficiently large,

the current it generates is capable of exciting the

surrounding normal tissue (top right), so initiating

a propagated wave (bottom left). Reprinted from

Winslow RL, Varghese A, Noble D, Adlakha C,

Hoythya A. Proc Roy Soc B 1990;240:8396, with

permission from The Royal Society.

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simulation of the electromechanical coupling between the

activation process and the mechanically deforming finite

element model in both directions: the solution of the electro-

physiological equations described the calcium current that

initiates contraction, and the shape changes caused by

cardiac contraction influenced electrical propagation.(56)

Extensions of this work have nowbeenmadeto accommodate

more-sophisticated models of cardiac ion channels(3638) and

cellular electromechanical coupling.(57)

At the same time, other important elements of a whole-

heart model are being put in place. Modelling of the coronary

circulation(58) has advanced to the point at which it is possible

to simulate coronary occlusion and so define the areas in the

ventricle that would become ischaemic.(32) Work is also in

progress to incorporate the SA node, atrial and Purkinje

conducting network. A reasonably complete virtual heart is

well on the way to construction. Such a project, with immense

implications for practical use in designing new therapies, (32)

would have been inconceivable until very recently.

I will finish this article with a spectacular example of thiskind of use of the virtual heart. Figure 6 (top) shows the

anatomical model of the canine ventricles(59) witha 1 mmthick

slice in which the fibre orientations are shown. The lower part

of the figure shows three frames from a simulation of arrhy-

thmia (called a tachycardia) in which a rotating spiral wave

(at the back), triggered by the collision of two stimulated

waves, is generating successive waves that propagate

through the structure of the slice. This simulation uses a

modified version of the BeelerReuter ventricular cell equa-

tions.(60) It is part of a major study not only of the conditions for

initiating arrhythmia but also of the mechanisms by which

electrical defibrillation may work. The model is the first to use

bi-domain simulation (in which the electrical field outsidethe cells is reconstructed as well as the potentials across

the cell membranes), withdetailedfibre geometry and detailed

cellular electrophysiology.(61,62)

Such simulations require enormous computing resources,

currently available only on large supercomputers (which is

the main reason why the simulation is restricted to a slice of

the ventriclecomputations of the whole canine ventricle

would, at present, be too demanding).

Modularity in biological systems

A recurring theme in these developments is what I have

called a modelling avalanche in which incorporation of one

new element has required incorporation of a set of relatedprocesses. Biological modelling often exhibits this form of

modularity, making it necessary to incorporate a group of

protein components together. It will be one of the major chal-

lenges of mathematical biology to use simulation work to

unravel the modularity of nature. Groups of proteins co-

operating to generate a function and therefore being selected

together in the evolutionary process will be revealed by this

approach. This piecemeal approach to reconstructing the

logic of life, which is the strict meaning of the word phy-

siology,(63) could also be theroutethroughwhich a systematic

theoretical biology could eventually emerge.(64,65)

Figure 6. Top: Diagram of the canine ventricular model(59)

showing the fibre structure of the slice in which the computa-

tions shown below were carried out. The slice is represented

as a full bi-domain model, i.e. the electric current flow in

the extracellular space is also computed. This enables such

models to be used to investigate the effects of massive electric

shocks of the kind used in defibrillation. Bottom: Rotating

spiral wave generating successive propagated waves com-

putedwithin theslice modelusing a modification ofthe Beeler

Reuter ventricular cell ionic current equations (from Refs.

61,62).

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Future challenges and the nature

of biological simulation

Progress in this field was slow and spasmodic in the period

between 1960 and 1990. This reflected the lack of experi-

mental data (a problem that was overcome at the cellular level

with the introduction of the patch clamp technique and its

application to isolated cells), andthe lack of computing power.

But since 1990 there has been an extraordinary explosion of

modelling work on the heart.(66) There are multiple models of

all the cell types, and I confidently predict that there will be

many more to come. Why do we have so many? Couldnt

we simply standardise the field and choose the best? To

some extent, that is happening. None of the historical models

described in this article is now used much in its original form.

One of the major reasons for the multiplicity of models is

that there will always be a compromise between complexity

and computability. A good example here is the modelling of

calcium dynamics. As we understand these dynamics in ever

greater detail,(31) models become more accurate and they

encompass more biological data, but they also becomecomputationally demanding. This was the motivation behind

the simplified dyadic space model of Noble et al. (38) which

achieves many of the required features of the initiation of

calcium signalling with only a modest (10%) increase in

computation time, an important consideration when import-

ing such models into models of the whole heart. But no-one

would use that model to study the fine properties of calcium

dynamics at the subcellular level. That was not its purpose.

There will probably therefore be no unique model that does

everything at all levels. Any of the boxes at one level could be

deepened in complexity at a lower level, or fused with other

processes at a higher level. In any case, all models are only

partial representations of reality. One of the first questions toaskof a modeltherefore is what questionsdoes it answerbest.

It is through the iterative interaction between experiment and

simulation that we will gain that understanding.

It is however already clear that incorporation of cell models

into tissue and organ models is capable of spectacular

insights. The incorporation of cell models into anatomically

detailed heart models, as shown in Fig. 6, hasbeen an exciting

development. The goal of creating an organ model capable of

spanning the whole spectrum of levels from genes(6771) to

the electrocardiogram(13,32) is within sight, and is one of the

challenges of the immediate future. The potential of such

simulations for teaching, drug discovery, device development

and, of course, for pure physiological insight is only beginningto be appreciated.

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