A meta-analysis of cardiac electrophysiology computational models

486 Exp Physiol 94.5 pp 486–495

Experimental Physiology – Research Paper

A meta-analysis of cardiac electrophysiologycomputational models

S. A. Niederer1, M. Fink2, D. Noble2 and N. P. Smith1

1University Computing Laboratory, University of Oxford, Oxford OX1 3QD, UK2Department of Physiology, Anatomy and Genetics, University of Oxford, Oxford OX1 3QX, UK

Computational models of cardiac electrophysiology are exemplar demonstrations of theintegration of multiple data sets into a consistent biophysical framework. These modelsencapsulate physiological understanding to provide quantitative predictions of function. Thecombination or extension of existing models within a common framework allows integrativephenomena in larger systems to be investigated. This methodology is now routinely applied, asdemonstrated by the increasing number of studies which use or extend previously developedmodels. In this study, we present a meta-analysis of this model re-use for two leading modelsof cardiac electrophysiology in the form of parameter inheritance trees, a sensitivity analysisand a comparison of the functional significance of the sodium potassium pump for definingrestitution curves. These results indicate that even though the models aim to represent thesame physiological system, both the sources of parameter values and the function of equivalentcomponents are significantly different.

(Received 13 August 2008; accepted after revision 2 January 2009; first published online 12 January 2009)Corresponding author N. Smith: University Computing Laboratory, University of Oxford, Oxford OX1 3QD, UK.Email: [email protected]

In the postgenomic era, biology is increasinglycharacterized by high-throughput experimental methodsand extensive data sets. Enabled by its availability andmotivated by the requirement to analyse these data,the development of increasingly sophisticated biophysicalmodels has accelerated. This has led to increased re-useof modelling frameworks as subcomponents to coupleacross scales and between scales within larger modellingframeworks. Examples of model re-use include: (1) thecombination of components, such as the inclusion of novelion channels into models of electrophysiology; (2) thecoupling of function, such as the combination of models ofmyocyte electrophysiology and contraction via commoncalcium handing variables; and (3) the embedding ofmodels within larger multiscale frameworks, such as theuse of cellular models to provide inputs for predictingtissue or whole-organ function.

This type of model re-use is a powerful feature of thecomputational modelling approach, in which modelsare parameterized to represent a common system. Inthese cases, model components can be extended orre-used without substantial re-parameterization tostudy an enlarged system of interest. Indeed, thevision of the integration of models at discrete levels

of biological organization across spatial scales tolink physiological mechanisms to function hasbeen organized into global initiatives, such as theInternational Union of Physiological Sciences (IUPS)sponsored Physiome and Virtual Physiological Human(VPH) projects. Products of the ‘Physiome’ effortinclude both databases of computational models(www.cellml.org, www.physiome.org, www.ebi.ac.uk/biomodels/, june.phys.mcw.edu/BioWiki/index.php/Main Page) and underlying experimental measurementsupon which specific model types are based (Popel et al.1998; Goldberg et al. 2004; Ribba et al. 2006).

Within the Physiome Community, the heart is arguablythe most advanced exemplar of a multiscale model of anorgan system. Underpinning this level of sophisticationin many multiscale cardiac models, and the literature ingeneral, is the level of detail incorporated into single-cell models of cardiac electrophysiology. These modelscurrently provide detailed representations of membrane-bound channels and transporter functions, as well asfluxes between the cytosol and intracellular organelles.This level of detail has enabled analysis of the roleeach functional element plays in both health and disease(Terkildsen et al. 2007). Further, they have provided a

DOI: 10.1113/expphysiol.2008.044610 C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

Exp Physiol 94.5 pp 486–495 Computational models of cardiac electrophysiology 487

successful paradigm for integrating individual data setsinto a common framework, such as transmembrane iontransport (Niederer et al. 2008), and for interpreting theensemble behaviour, for example, altered action potentialrecording in perturbed ionic conditions (Shaw & Rudy,1997). Indicative of the success of this class of models isthat component re-use is increasingly being applied acrossboth scales and function (Iyer et al. 2004; ten Tusscher et al.2004; Niederer & Smith, 2007; Ashihara et al. 2008).

However, a difficulty of this type of model re-use is thatthe inheritance of model components, owing to repeatedre-use, now often extends through multiple generationsfor a given class of models. As we demonstrate in thisstudy, this inheritance has paradoxically resulted in anobfuscation of the link between parameter values and theoriginal experimental data sources. We suggest that if thisprocess of re-use proceeds without attempts to link modelparameters to more recent measurements and/or moreappropriate data, the relevance of a model can be eroded.

The model re-use issue can be further exacerbated inintegrative model frameworks where information is passedbetween dependent model components. Such integrationprovides the ability to account for interactions betweendependent systems, allowing complex physiologicalphenomena to be represented mathematically andanalysed quantitatively. However, the integrative natureof modelling techniques, and indeed the biologicalsystems themselves, means that the introduction ofinappropriate components as a result of componentinheritance inevitably compromises other parts of themodel.

To begin to address the issue of analysing fidelity inwhole-cell models and to provide tangible demonstrationsof the issues associated with model re-use in multiplecases, we have analysed two models (Iyer et al. 2004; tenTusscher et al. 2004). These two models represent leadingcomputational models of the human ventricular myocytein physiological conditions (e.g. body temperature). Thesemodels have been chosen as state-of-the-art Physiomeexemplars whose common system of interest means thatthe formulations and functions can be directly compared.The models are based on extensive experimental dataand achieve the challenging feats of replicating whole-cellintracellular Ca2+ dynamics, action potential restitutioncurves and steady-state ionic concentrations across a rangeof pacing frequencies. The extensive application of thesemodels (Hong et al. 2005; Splawski et al. 2005; Verkerket al. 2005; Keldermann et al. 2008) is testimony to theirversatility and ability to capture a wide range of biologicalfunction.

Using these indicative studies, we analyse the sourceof parameter values and compare model function. Ourgoal is to identify the potential for significant disparityin results between models of the same system, overlapin the use of the scarce physiological human ventricular

myocyte experimental data and deviation of re-usedmodel components from their original characteristics.Furthermore, it is possible to assess the degree to whicha given subcomponent (e.g. channel or exchanger),with different mathematical forms in each model, isresponsible for the same function. The result of thisanalysis provides a measure of functional convergencebetween the subcomponents, which in turn indicates thecurrent potential and future challenges associated withcomponent re-use.

Methods

The origin of experimentally derived parameters andtheir hereditary paths through model componentswere determined through extensive literature searches.Experimental data were classified by the speciesof the protein that was under investigation, thecell type (ventricular myocyte, Purkinji fibre or anexpression system) and temperature. These experimentaldata dependencies were evaluated using ‘Phylogenetic’schematics and data distribution analysis.

Phylogenetic schematics show the inheritance ofinformation from previous modelling and experimentalstudies. Specifically, they show the link between acomponent in a model and the experimental dataused to determine the parameters in that component.These schematics are used to visualize the linksbetween modelling (trapeziums) and experimentalstudies (ellipses). Modelling studies are broken up intocomponents (boxes), where a component is definedas an ionic flux pathway, such as transmembrane ioncarriers/channels or fluxes between subcellular ionicconcentrations, and is linked to its respective model.Connections (arrows) are between components andpublished studies, but when the connection is betweena component and a modelling study the link is implicitlyto the component in the modelling study that shares thename of the original component. The example trees inthe Results do not include the cell-type information, suchthat they do not differentiate between human ventricularmyocyte data and human protein in an expression system.The dependency of the models on different cell types ispresented in the data distribution analysis. Phylogentictrees can be generated that differentiate between cell types,but for the sake of clarity this information is not includedin the example trees in this study.

Additional subtleties in the formulation of theexperimental database include the specific criteria forinclusion in the citation tree. The citation trees showexperimental data or models that were used to determineor define model parameters. Experimental papers usedin the discussion of a modelling study are not includedin the experimental data dependency database. Threefurther types of experimental data dependencies were

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

488 S. A. Niederer and others Exp Physiol 94.5 pp 486–495

not included in the database, as follows: (1) experimentalor modelling studies that motivated model structurebut not the parameter values; (2) referenced studies thatwere not obtainable, specifically symposium/conferenceproceedings published over 20 years ago; and (3) in somecases, experimental data sets are published in multiplestudies. In cases where a component is dependent onmultiple studies, where one study contained only a subsetof data that was published in a second larger study onlythe larger study was included in the experimental datadependency database. In the event of experiments beingperformed at multiple temperatures within one paper andwhen it was not specified which data set was used todetermine model parameters, the higher temperature wastaken. Specific cases where experimental data were notincluded in the experimental data dependency database,as it describes the model structure, are the stochiometryof sodium potassium pump (I NaK) and sodium calciumexchanger (I NaCa). Similarly, although the ten Tuesschercell model uses Hodgkin-and-Huxley-style gating kineticsmodels, this represents the structure of the model and sopapers that motivated these equations are not included inthe experimental data dependency database.

Images of model connectivity are displayed as directed acyclical graphs using the freely available vizgraph package(www.graphviz.org), using the dot language to describethe graphs. Perl scripts were used to probe the citation treedatabase and generate the dot files with nodal attributesconditional on the experimental data classifications.

Figure 1. Phylogenetic schematic for the ten Tusscher et al. (2004; A) and the Iyer et al. (2004; B) cellmodels (B), showing the links between modelling (trapeziums) and experimental studies (ellipses)Modelling studies are broken up into components (boxes), with connections (arrows) between components andpublished studies. ∗ Luo & Rudy (1994) model; and + Jafri et al. (1998) model.

Models were described using CellML (Lloyd et al. 2004)and solved within the COR (Garny et al. 2003) modellingenvironment using the CVODE adaptive integrator witha relative tolerance of 10−7, an absolute tolerance of 10−9

and a maximal step size of 1 ms. All simulations weresolved to steady state, achieved after 1000–3000 beats at1 Hz pacing.

Results

Global analysis of cellular models

Figure 1A and B shows phylogenetic schematics for theexperimental data dependencies of the ten Tusscher et al.(2004) and Iyer et al. (2004) cell models, respectively.Figure 2 plots the distribution of data sources containedwithin the two phylogenetic schematics to provide insightinto the distribution of available experimental data for thetwo models. Figure 2A, B and C plots the temperature,species and cell type distribution of the experimentaldata used in each cell model. Both cell models are basedon ∼50% human and ∼25% guinea-pig data, with theremaining ∼25% taken from a wide range of species.Experimental data are commonly acquired at one oftwo temperatures, room temperature (21–25◦C) or bodytemperature (36–37◦C). This is true for both models, withlimited experimental data acquired below 10◦C. The useof ventricle data is the norm in both models, with ∼60%

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

Exp Physiol 94.5 pp 486–495 Computational models of cardiac electrophysiology 489

of all data coming from ventricular cells; this figure risesto ∼80% when experimental data from all heart cells(ventricle, atrial and Purkinje fibre) are considered.

An additional metric of data quality is age.Improvements in recording techniques, cell preparationand capacity of pharmacological blockers have greatlyimproved the accuracy and specificity of availableinformation. Figure 2D shows the distribution of the ageof the model data. The distributions have been separatedinto sources that are unique to a specific model or commonto both. The age of the model data has a bell-shapeddistribution, with peak data used solely for the Iyer and tenTusscher models coming from 1996–2000 and 1991–1995,respectively. Data that were common to both models weremore evenly spread but had an earlier peak, at 1986–1990.

The use of experimental data recorded from the systemand conditions under study (human ventricular myocytedata at approximately body temperature, ≥30◦C) can beexpected to reduce errors introduced into the model thatresult from variability across different temperatures, celltypes and species. Both models rely extensively on humancell data to characterize the model components, as notedabove; however, of the 82 cited experimental sources usedto characterize both these models (Iyer, 48; ten Tusscher,54) only 21% were also consistent in temperature andcell type. The maximal number of these studies eachmodel could contain is therefore 17, whereas each studyonly used 12. This meta-analysis highlights importantissues surrounding the use of experimental data and thedetermination of model parameters.

Figure 2. Plots of the cell type (A),temperature (B), species (C) andpublication date (D) distribution of theexperimental data used in each cellmodelThe filled component of a bar indicates theproportion of the data that is common toboth models.

Sensitivity analysis

Measurment of the change in the action potentialfollowing the perturbation of conductance parametersin cardiac cell models, by magnitudes consistent withexperimental measurements, allows us to test the capacityof a model to simulate the transduction of populationvariation, from the subcellular component scale towhole-cell function. Experimental measurements of theaction potential or component conductance recordgross variation, composed of population variation andexperimental noise. The presence of experimental noisein measurements means that experimentally derivedvariation is an overestimation and can provide only anupper boundary on population variability. Thus, if themodel predicts that changes in a component conductancecause variation in the action potential outside theexperimentally determined upper bounds, this suggeststhat some dampening component is misrepresented orabsent from the model.

This test is demonstrated in the present study bycomparing the variation in transmembrane currentdensities with the variation in the 90% action potentialduration (APD90). The Iyer and ten Tusscher cell models(at 1 Hz pacing) have APD90 values of 314.0 and 268.5 ms,respectively. Human ventricular epicardial myocytes,paced at 1 Hz, have APD90 values of ∼298–351 ms, andvariation of APD90 is ∼5% (Drouin et al. 1995; Li et al.1996, 1998). The current densities of components presentin both models were chosen as the subset of modelparameters to compare and analyse. Variation in the

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

490 S. A. Niederer and others Exp Physiol 94.5 pp 486–495

measurement of current density are commonly reportedto be of the order of 10% across a range of ion channelsand pumps (Stimers et al. 1990; Li et al. 1998, 1999;Wettwer et al. 1998). Thus, our sensitivity analysis is basedon calculating the APD90 for both models at steady statefollowing a ±10% perturbation of the current density.

The results of this analysis are summarized inTable 1. The ten Tusscher cell model is insensitive tochanges in current densities, with variations in APD90

being less than those observed experimentally. However,in the case of the Iyer cell model, variation of either theI NaK or the inward rectifier current (I K1) current densitycaused significant increases in the variation of APD90 inexcess of that observed experimentally. In the case ofa 10% decrease in I NaK density, the APD90 fell outsideexperimentally observed values. Assuming that the 5%variation in the APD90 value is due solely to populationvariation, we can calculate an upper boundary for thepopulation variation in the Iyer cell model I NaK and I K1

of 3% and 7%, respectively. Measurements of I NaK densityhave variations of 12–18% (Stimers et al. 1990); althoughthese measurements are not taken from human ventricularcells, a six- to fourfold decrease in variation appearsunlikely given that estimates of I NaK density in the humanventricular myocyte from models varies significantlybetween ∼0.6 and 2.4 μA μF−1 (Courtemanche et al.1998; Iyer et al. 2004). This indicates that the Iyer model ispotentially oversensitive to the I NaK component or systemsthat contain I NaK.

Component analysis

In the large number of other cardiac electrophysiologycell models (including the two under consideration)the ionic flux across the cell membrane is governedby independent ion channels feeding into acompartmentalized intracellular space. Typically, eachion channel is characterized independently from specificexperimental data. Both models under consideration haveapproximately the same combination of ion channels,pumps and exchangers. Hence, it is possible to extendour analysis from focusing on each model independentlyto a comparison of the common model outputs andcomponents that make up each model. Given that bothmodels focus on representing a biophysically basedmodel of the human cardiac myocyte, ideally it shouldbe possible to replace a component of one model witha component from the other. This method provides afunctional analysis of each component’s mathematicaldescription in a comparable system independent of thespecific formulation of parameters and equations. Thisprocess in turn provides the ability to identify cases wherecomponents have equivalent or different characteristicsbut use different sets of equations and/or parameters.

Table 1. Change in APD90 due to a ±10% change in therespective current conductance/maximal flux

Current (±10%) ten Tusscher, �APD90 (ms) Iyer, �APD90 (ms)

INaK +6.2/−7.9 +48.5/−44.9INaCa −2.1/+1.8 −12.1/+11.7IKs −4.5/+4.9 −4.1/+3.9IK1 −2.1/+2.3 −23.0/+27.5ICaL −3.5/+3.2 −12.4/+13.4IKr −3.7/+3.8 −6.1/+5.9INa +0.4/−0.5 +1.7/−1.3

Thenormal values of APD90 for the ten Tusscher et al. (2004) andthe Iyer et al. (2004) models are 268.5 and 314.0 ms, respectively,with a pacing cycle of 1000 ms and a stimulus current of−100 μA μF−1 for 0.5 ms carried by potassium ions.

Since both models were validated against restitutionproperties, this was chosen as the metric for comparison.Equivalent model components were swapped betweenframeworks before the ‘new’models were then solved tosteady state and the restitution curves calculated.

Our working hypothesis was that the output of themodel would not be significantly different for componentsubstitutions, which in turn would validate the philosophyof component re-use. Figure 3 plots the restitution curvefor the Iyer and ten Tusscher cell models with the markedcomponent replaced with the respective component fromthe other model. When the I K1 component of the Iyercell model was replaced with the ten Tusscher variation,the model stopped producing action potentials before itreached steady state and so it is absent from the graph.The apparent model differences in the I K1, the rapidlyactivating and inactivating current (I Kr) and the slowactivating current (I Ks) components in Fig. 3 have beendiscussed previously (Fink et al. 2006, 2008), and thedifferences in I NaCa can be attributed to differences inrepresentations of the dyadic space.

Figure 3 demonstrates the functional differencesbetween components in each model. The effect ofswapping each component is also determined by themodel’s sensitivity to that component (as calculated inTable 1). The Iyer and ten Tusscher models are mostsensitive to the I NaK component, but the effects ofswapping this component are nominal in the ten Tusschermodel and large in the Iyer model, suggesting that theequations and parameters used to describe I NaK in eachmodel are significantly different.

Using I NaK as an example, we analyse the differencesin experimental data dependencies, equations andparameters of this component. Figure 4Aa plots theextracellular Na+ and voltage sensitivity of the Iyer, tenTusscher and Luo & Rudy (1994) I NaK models comparedwith experimental data from Nakao & Gadsby (1989) inthe guinea-pig. The models of the pump are normalizedso that the density of I NaK does not affect these plots. The

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

Exp Physiol 94.5 pp 486–495 Computational models of cardiac electrophysiology 491

original Luo & Rudy model provides a good fit to theexperimental data, whereas the ten Tusscher model showsno sensitivity to extracellular Na+ and the Iyer modelhas significantly different voltage sensitivity. Figure 4Abplots the intracellular Na+ sensitivity which is differentfor both models. Ten Tusscher and Iyer cited the Luo& Rudy I NaK model both directly and indirectly (citingmodels which re-use this model) as the source of their I NaK

model. However, closer inspection of the model equationand parameters reveals that both models used modifiedversions of the Luo & Rudy I NaK model. Iyer et al. alteredthe voltage and [Na+]i dependency of I NaK to reduce Na+

efflux at higher pacing frequencies. The ten Tusscher cellmodel removes the extracellular Na+ function (σ) andalters the [Na+]i and voltage dependency.

The experimental data citation tree for I NaK (Fig. 4B)shows that both models are based on the same modelstructure (Luo & Rudy model) and that additional datahave been used to adjust the model each time it was re-used. All additional experimental data that were used tomodify the I NaK model following its creation by Luo &Rudy and prior to its inclusion in either the ten Tusscheror Iyer models have been used to adjust the maximal fluxthrough the pump with the the Na+ or voltage sensitivityremaining unchanged. ten Tusscher and Iyer both refittedthe maximal flux parameter in their respective studiesand so any new information added to the Luo & Rudyformulation by later modelling studies was discarded.The I NaK model is based largely on guinea-pig and sheepexperimental data from two decades ago. The completeabsence of human data provides a strong motivation formaking new recordings of I NaK kinetics in human cardiaccells.

Discussion

In this study, we have investigated the differingexperimental data dependencies, sensitivity to changes inparameters and components, and characteristics of I NaK

in two leading computational models of human cardiacelectrophysiology. This analysis highlights some of thechallenges in model development and in the re-use andadaptation of existing models. We believe these results haveimplications for integrative modelling of physiologicalsystems as well as for specific initiatives, such as thePhysiome Project, which aim to facilitate the constructionof integrated models through a process of model re-usesupported by databases of models and experimental data(Bassingthwaighte, 2000).

The advent of standards to represent mathematicalmodels, in the form of CellML (Lloyd et al. 2004) andSBML (Hucka et al. 2003), and of model repositorieshas significantly improved the efficiency and integrityof cellular and ordinary differential equation-based

biological model development. Additional biophysicalstandards have been proposed to encourage modelsto adhere to conservation of mass, charge and energylaws (Smith et al. 2007). Tools developed in computerscience now allow unit checking of model equations(Cooper & McKeever, 2007) and easy access toleading numerical solvers. This combination of languagestandards and tools for manipulating and runningmodels facilitates model accessibility and use across thebiological science community. However, as demonstratedin this study, there is an important parallel requirementfor improved transparency of mathematical modeldevelopment specifically to define links to experimentaldata, for an effective method to re-use components of

Figure 3. The restitution curves plotted against diastolic interval(DI) for the ten Tusscher et al. (2004; A) and Iyer et al. (2004) cellmodels (B) following the replacement of a component(indicated in the key) from the other modelRestitution curves were calculated using the S1–S2 protocol. Allrestitution curves were calculated using cell models at a 1 Hz pacingfrequency steady state. Action potential duration (APD90) was definedas the time between the maximal upstroke velocity and the time whenthe model was 90% repolarized.

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

492 S. A. Niederer and others Exp Physiol 94.5 pp 486–495

Figure 4. Comparison of sodium potassium pump modelsAa, voltage dependence of Na+−K+ pump models from the Luo & Rudy (1994), ten Tusscher et al. (2004) and Iyeret al. (2004) cell models, with intracellular K+ = 140 mM, extracellular K+ = 5.4 mM, extracellular Na+ = 1.5,50, 100 and 150 mM (increasing from the top curve to the bottom curve) and intracellular Na+ = 50 mM.Ab, intracellular Na+ dependence of Na+−K+ pump model from ten Tusscher and Iyer cell models, withintracellular K+ = 140 mM, extracellular K+ = 5.4 mM, extracellular Na+ = 140 mM and the membrane potentialheld at −40 and −80 mV. B shows a graph of experimental data and model dependency of the ten Tusscher (T04)and Iyer (I4) Na+−K+ pump models. Colours and labels are described in the legend to Fig. 1. ∗ Luo & Rudy (1994)model; and + Nakao & Gadsby (1989) data.

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

Exp Physiol 94.5 pp 486–495 Computational models of cardiac electrophysiology 493

models and for tools/methods for analysing individualmodels and comparing multiple models.

A significant focus of the model development process isthe fitting of model parameters. In papers containing nonew experiments, models are fitted to data extracted fromexisting literature. Analysing the results of this exercisewill become increasingly important as models grow incomplexity. Displaying model data inheritance using adirected acyclical graph shows model re-use, a metric ofage (older data are predominantly towards the bottomof the graph), and can provide insight into what earliermodels a model under investigation is based upon. Theuse of colour to display fields, such as in our example fortemperature (Fig. 1), allows correlations to be observedsimultaneously in the same plot. The ability to view thewhole graph or to view components of the graph using aconsistent format readily allows closer analysis of regionsof interest.

The value of functional component analysis wasdemonstrated in the identification of the Na+−K+ pumpas having a significantly different effect on the restitutioncurve in each model (see Fig. 3). As Fig. 4B shows,both models cite multiple models as the source of theirNa+−K+ pump model, while both in fact use a variant ofa single model developed by Luo & Rudy (1994). The tenTusscher model cites two human cell models as the sourceof the I NaK equations, whereas the original equations werefitted primarily to guinea-pig data, thus demonstratingthe need for improved referencing options for the re-use of single model components. The multiple layers ofcitations, resulting from successive component re-use, thathave ultimately been embedded in both the Iyer and theten Tusscher frameworks, have the potential to obfuscatethe true source of experimental data used to describethe models. The availability of CellML 1.1 (Cuellar et al.2003), which breaks a cell model into a collection ofindividual components, should greatly reduce the chancethat a model component is coded erroneously and/or willhighlight where re-used components have been modifiedand therefore increase the ability of an author to provideappropriate documentation.

Referencing experimental data sources

The two models analysed in this study did not containany new experimental data. Both models achieved thenon-trivial goal of coalescing experimental results, eitherdirectly or indirectly, from previous papers to determinethe model parameters in a consistent framework. Thisis entirely necessary because it is now impossible forone group to perform all of the experiments requiredto characterize a detailed biological model. However, weassert that the experimental data dependencies of a modelrequire clear documentation to underpin confidencein results generated by a model and to facilitate re-

use. Predominantly, the data-parameter dependenciesare described by referencing an earlier modelling orexperimental study. However, in many cases this methodprovides inadequate information to fully characterize therelationship between a model and experimental data.

Specific examples for this study include the intracellularCa2+ dynamics of the ten Tusscher cell model, for whichno references are provided, and the Iyer cell model, whichcites the Jafri et al. (1998) model of Ca2+ dynamicsas the source of its Ca2+ handling equations, but hasparameters inconsistent with this earlier model. Otherexamples include the ambiguity between data sets usedto determine model parameters and those used to validatemodels. A clear delineation is required if validation is tobe considered reliable.

If models are to be re-used with confidence thena user must be easily able to locate the original datasources. This argument leads to the need for modelreferencing standards and tools, such as the experimentaldata visualization tree used in this study, to assist modelusers and experimentalists in identifying where data comefrom. At present, the addition of a column to the table ofparameters, currently used in the majority of modellingpapers, containing the reference to the experimental datapertaining to each model parameter would represent asignificant improvement in the transparency of modeldevelopment.

Inferring parameters from indirect measurements

A strength of computational biology is the capacity toinfer parameters/measurements from indirect measuresbased on a set of assumptions about the system understudy. Typical of the many other electrophysiologicalmodels, both models analysed in this study makeuse of indirect measurements of the action potentialmorphologies, the [Ca2+]i transient and stable limitcycle ionic concentrations to infer unknown modelparameters. Additionally, following the application ofcharge conservation laws, parameters were fitted to achievea desired stable limit cycle at a given pacing frequency.

In our specific example, the I NaK component haddistinctly different characteristics in both models despitebeing based on the same original component. In bothmodels, I NaK was partly fitted to achieve a desirable[Na+]i–pacing frequency relationship at stable limit cycles.This fitting process implicitly assumes that the remainingelements of Na+ regulation are well characterized. Inboth models, Na+ regulation is made up of I NaK,I NaCa, fast inward sodium current (I Na) and backgroundsodium current (I Nab); however, this list excludes theNa+–H+ exchanger, Na+–HCO3

− cotransporter andNa+−K+−2Cl− cotransporter. These additional pathwayshave been shown to make significant contributions to Na+

influx at steady state in both experimental (Despa et al.

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

494 S. A. Niederer and others Exp Physiol 94.5 pp 486–495

2002) and modelling studies (Niederer & Smith, 2007).The absence of these components in the models analysedbrings into question the validity of using steady-state[Na+]i to determine model parameters. The two modelsproduce viable [Na+]i–pacing frequency relationshipsdespite these absent components, which suggests thatsome existing components compensate for the incompleterepresentation of [Na+]i regulation.

Finally, it is important to emphasize that the twomodels analysed in this study were chosen only asindicative examples largely for reasons of comparisonand because of their ability to represent the state of theart in the field of modelling cardiac electrophysiology.The development of human models, while currentlylimited by the availability of data from non-failing humanmyocytes, is of upmost importance for underpinningfuture clinical application. We expect that the conclusionsof this study would be similar if applied to a largenumber of other models, including our own. Whatthe results of this study do demonstrate are significantdifferences in representations of model componentstogether with a number of unresolved issues surroundingthe description of model construction and the link withexperimental data. Consistent with previous comparativestudies (Cherry & Fenton, 2007; Bueno-Orovio et al.2008), the model comparison shows that models arestill evolving towards a common representation. Thisleads to the specific conclusion that neither one of themodels analysed in this study is superior or, the morelikely scenario, the general conclusion that the re-use ofmodel components is not a currently viable means ofmodel construction. Without new approaches, these areissues which will become more problematic with increasedcomplexity of models and availability of interdependentdata sets. Encouraging authors to publish a citation treeof experimental data and re-used model componentsemployed to create a model using agreed standards withclear descriptions of experimental data dependency wouldprovide a mechanism to improve transparency in modeldevelopment, which in turn could be integrated into anew format for model publication. Standards and markuplanguages to facilitate the linking of model parametersand experimental data are presently being developed buthave yet to be finalized, and the tools necessary to use thisfunctionality are currently limited (Hunter et al. 2008). Viasuch initiatives and further debate and discussion withinthe community, we hope the undoubted potential ofcomputational modelling to represent complex biologicalsystems and provide unique insights will be fully realised.

References

Ashihara T, Constantino J & Trayanova NA (2008). Tunnelpropagation of postshock activations as a hypothesis forfibrillation induction and isoelectric window. Circ Res 102,737–745.

Bassingthwaighte JB (2000). Strategies for the PhysiomeProject. Ann Biomed Eng 28, 1043–1058.

Bueno-Orovio A, Cherry EM & Fenton FH (2008). Minimalmodel for human ventricular action potentials in tissue. JTheor Biol 253, 544–560.

Cherry EM & Fenton FH (2007). A tale of two dogs: analyzingtwo models of canine ventricular electrophysiology. Am JPhysiol Heart Circ Physiol 292, H43–H55.

Cooper J & McKeever S (2007). A model-driven approach toautomatic conversion of physical units. Softw Pract Exp 28,337–359.

Courtemanche M, Ramirez RJ & Nattel S (1998). Ionicmechanisms underlying human atrial action potentialproperties: insights from a mathematical model. Am J PhysiolHeart Circ Physiol 44, H301–H321.

Cuellar AA, Lloyd CM, Nielsen PF, Bullivant DP, Nickerson DP& Hunter PJ (2003). An overview of CellML 1.1, a biologicalmodel description language. Simulation 79,740–747.

Despa S, Islam MA, Pogwizd SM & Bers DM (2002).Intracellular [Na+] and Na+ pump rate in rat and rabbitventricular myocytes. J Physiol 539, 133–143.

Drouin E, Charpentier F, Gauthier C, Laurent K & Lemarec H(1995). Electrophysiologic characteristics of cells spanningthe left-ventricular wall of human heart: evidence forpresence of M-cells. J Am Coll Cardiol 26, 185–192.

Fink M, Giles WR & Noble D (2006). Contributions ofinwardly rectifying K+ currents to repolarization assessedusing mathematical models of human ventricular myocytes.Philos Transact A Math Phys Eng Sci 364,1207–1222.

Fink M, Noble D, Virag L, Varro A & Giles WR (2008).Contributions of HERG K+ current to repolarization of thehuman ventricular action potential. Prog Biophys Mol Biol96, 357–376.

Garny A, Kohl P & Noble D (2003). Cellular OpenResource (COR): a public CellML based environment formodeling biological function. Int J Bifurcat Chaos 12,3579–3590.

Goldberg RN, Tewari YB & Bhat TN (2004). Thermodynamicsof enzyme-catalyzed reactions—a database for quantitativebiochemistry. Bioinformatics 20, 2874–2877.

Hong K, Piper DR, Diaz-Valdecantos A, Brugada J, Oliva A,Burashnikov E, Santos-de-Soto J, Grueso-Montero J,Diaz-Enfante E, Brugada P, Sachse F, Sanguinetti MC &Brugada R (2005). De novo KCNQ1 mutation responsiblefor atrial fibrillation and short QT syndrome in utero.Cardiovasc Res 68, 433–440.

Hucka M, Finney A, Sauro HM, Bolouri H, Doyle JC, Kitano Het al. (2003). The systems biology markup language (SBML):a medium for representation and exchange of biochemicalnetwork models. Bioinformatics 19, 524–531.

Hunter PJ, Crampin EJ & Nielsen PMF (2008). Bioinformatics,multiscale modeling and the IUPS Physiome Project. BriefBioinform 9, 333–343.

Iyer V, Mazhari R & Winslow RL (2004). A computationalmodel of the human left-ventricular epicardial myocyte.Biophys J 87, 1507–1525.

Jafri MS, Rice JJ & Winslow RL (1998). Cardiac Ca2+ dynamics:the roles of ryanodine receptor adaptation and sarcoplasmicreticulum load. Biophys J 74, 1149–1168.

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society

Exp Physiol 94.5 pp 486–495 Computational models of cardiac electrophysiology 495

Keldermann RH, ten Tusscher KHWJ, Nash MP, Hren R,Taggart P & Panfilov AV (2008). Effect of heterogeneousAPD restitution on VF organization in a model of the humanventricles. Am J Physiol Heart Circ Physiol 294, H764–H774.

Li GR, Feng JL, Yue LX & Carrier M (1998). Transmuralheterogeneity of action potentials and I-to1 in myocytesisolated from the human right ventricle. Am J Physiol HeartCirc Physiol 275, H369–H377.

Li GR, Feng JL, Yue LX, Carrier M & Nattel S (1996). Evidencefor two components of delayed rectifier K+ current inhuman ventricular myocytes. Circ Res 78, 689–696.

Li GR, Yang BF, Feng JL, Bosch RF, Carrier M & Nattel S(1999). Transmembrane I-Ca contributes to rate-dependentchanges of action potentials in human ventricular myocytes.Am J Physiol Heart Circ Physiol 276, H98–H106.

Lloyd CM, Halstead MDB & Nielsen PF (2004). CellML: itsfuture, present and past. Prog Biophys Mol Biol 85, 433–450.

Luo CH & Rudy Y (1994). A dynamic model of the cardiacventricular action potential. I. Simulations of ionic currentsand concentration changes. Circ Res 74, 1071–1096.

Nakao M & Gadsby DC (1989). [Na] and [K] dependence ofthe Na/K pump current-voltage relationship in guinea-pigventricular myocytes. J Gen Physiol 94, 539–565.

Niederer SA & Smith NP (2007). A mathematical model of theslow force response to stretch in rat ventricular myocytes.Biophys J 92, 4030–4044.

Niederer SA, Swietach P, Wilson DA, Smith NP &Vaughan-Jones RD (2008). Measuring and modelingchloride-hydroxyl exchange in the guinea-pig ventricularmyocyte. Biophys J 94, 2385–2403.

Popel AS, Greene AS, Ellis CG, Ley KF, Skalak TC & TonellatoPJ (1998). The Microcirculation Physiome Project. AnnBiomed Eng 26, 911–913.

Ribba B, Tracqui P, Boix J-L, Boissel J-P & Thomas SR (2006).QxDB: a generic database to support mathematicalmodelling in biology. Philos Transact A Math Phys Eng Sci364, 1517–1532.

Shaw RM & Rudy Y (1997). Electrophysiologic effects of acutemyocardial ischemia: a theoretical study of altered cellexcitability and action potential duration. Cardiovasc Res 35,256–272.

Smith NP, Crampin EJ, Niederer SA, Bassingthwaighte JB &Beard DA (2007). Computational biology of cardiacmyocytes: proposed standards for the physiome. J Exp Biol210, 1576–1583.

Splawski I, Timothy KW, Decher N, Kumar P, Sachse FB, BeggsAH, Sanguinetti MC & Keating MT (2005). Severearrhythmia disorder caused by cardiac L-type calciumchannel mutations. Proc Natl Acad Sci USA 102,8089–8096.

Stimers JR, Shigeto N & Lieberman M (1990). Na/K pumpcurrent in aggregates of cultured chick cardiac myocytes. JGen Physiol 95, 61–76.

ten Tusscher KHWJ, Noble D, Noble PJ & Panfilov AV (2004).A model for human ventricular tissue. Am J Physiol HeartCirc Physiol 286, H1573–H1589.

Terkildsen JR, Crampin EJ & Smith NP (2007). The balancebetween inactivation and activation of the Na+-K+ pumpunderlies the triphasic accumulation of extracellular K+during myocardial ischemia. Am J Physiol Heart Circ Physiol293, H3036–H3045.

Verkerk AO, Wilders R, Schulze-Bahr E, Beekman L, BhuiyanZA, Bertrand J, Eckardt L, Lin D, Borggrefe M, Breithardt G,Mannens MMAM, Tan HL, Wilde AAM & Bezzina CR(2005). Role of sequence variations in the humanether-a-go-go-related gene (HERG, KCNH2) in the Brugadasyndrome. Cardiovasc Res 68, 441–453.

Wettwer E, Himmel HM, Amos GJ, Li Q, Metzger F & RavensU (1998). Mechanism of block by tedisamil of transientoutward current in human ventricular subepicardialmyocytes. Br J Pharmacol 125, 659–666.

Acknowledgements

This work has been supported by the United KingdomEngineering and Physical Sciences Research Council GrantEP/F059361/1 and the the European Community’s SeventhFramework Programme (FP7/2007-2013) under grantagreement no. FP7-2007-ICT-224495 (euHeart project).

C© 2009 The Authors. Journal compilation C© 2009 The Physiological Society