Exploiting mathematical models to illuminate electrophysiological variability between individuals

Abstract

Across individuals within a population, several levels of variability are observed, from the differential expression of ion channels at the molecular level, to the various action potential morphologies observed at the cellular level, to divergent responses to drugs at the organismal level. However, the limited ability of experiments to probe complex interactions between components has hitherto hindered our understanding of the factors that cause a range of behaviours within a population. Variability is a challenging issue that is encountered in all physiological disciplines, but recent work suggests that novel methods for analysing mathematical models can assist in illuminating its causes. In this review, we discuss mathematical modelling studies in cardiac electrophysiology and neuroscience that have enhanced our understanding of variability in a number of key areas. Specifically, we discuss parameter sensitivity analysis techniques that may be applied to generate quantitative predictions based on considering behaviours within a population of models, thereby providing novel insight into variability. Our discussion focuses on four issues that have benefited from the utilization of these methods: (1) the comparison of different electrophysiological models of cardiac myocytes, (2) the determination of the individual contributions of different molecular changes in complex disease phenotypes, (3) the identification of the factors responsible for the variable response to drugs, and (4) the constraining of free parameters in electrophysiological models of heart cells. Together, the studies that we discuss suggest that rigorous analyses of mathematical models can generate quantitative predictions regarding how molecular-level variations contribute to functional differences between experimental samples. These strategies may be applicable not just in cardiac electrophysiology, but in a wide range of disciplines.

Amrita Sarkar (left) holds a double major in Mathematics and Molecular Biology & Biochemistry from Middlebury College. Her interdisciplinary research interests led her to pursue a PhD program in Computational Biology jointly conducted by the Mount Sinai School of Medicine and New York University. During her Doctorate, she built mathematical models to understand the effects of changes in cardiac ion channel expression on clinically relevant and measurable properties of the heart. Currently, she is a Fellow at the Collège des Ingénieurs in Paris, where she is pursuing a specialized MBA for engineers and scientists while working for a leading French company. David Christini (middle) received a BS degree in electrical engineering from the Pennsylvania State University and MS and PhD degrees in biomedical engineering from Boston University. He is a Professor in the Departments of Medicine and Physiology and Biophysics, Weill Cornell Medical College, New York. He uses computational and experimental methods to study cellular- to organ-level cardiac electrophysiological dynamics, with an emphasis on understanding the mechanisms underlying arrhythmia initiation and in developing new arrhythmia therapies. Eric Sobie (right) is an Associate Professor in the Department of Pharmacology and Systems Therapeutics at Mount Sinai School of Medicine in New York City. He holds a BSE degree from Duke University and a PhD from The Johns Hopkins University, both in biomedical engineering. His research focuses on gaining a greater quantitative understanding of cardiac physiology and pathophysiology by combining experimental studies and simulations performed with mathematical models.

Introduction

Variability between members of the same species runs across different levels of organization. When we consider the electrophysiology of the heart, this variability may manifest itself: (1) at the molecular level as differential expression of membrane ion channels (Gaborit et al. 2007); (2) at the cellular level in the form of variable action potential waveforms (Antzelevitch et al. 1991); (3) at the organ level where there is a range of normal heart rates, ECG metrics and cardiac outputs (Taylor & Lipsitz, 1997); and finally (4) at the organismal level where administration of the same therapy can produce dramatically different outcomes in different individuals (Kannankeril et al. 2010). This variability can be substantial even within a population that would be considered normal or healthy, and it can become significantly more pronounced when a mixed population of healthy and diseased individuals is considered.

Investigators in all fields of physiology, or indeed biological sciences more generally, must confront the fact that different experimental samples will demonstrate variable function. Muscle fibres will have different contraction strengths, neurons of the same type will exhibit different firing patterns, pancreatic β-cells will secrete differing amounts of insulin in response to the same glucose stimulus. The primary message of this review is that newly developed methods for analysing mathematical models can help to understand and predict the mechanisms underlying such functional variability. Since our particular expertise is in cardiac electrophysiology, most of the examples we discuss are from studies in this area. We wish to emphasize, however, that the strategies we advocate for understanding variability are potentially applicable in diverse fields.

In a typical experiment performed by an electrophysiologist in the laboratory, it is quite difficult to treat variability in a rigorous manner. One obvious issue is the small sample sizes that are typical of most studies. However, although investigators often record from numerous preparations and carefully document the response of each sample, that is only the first step in trying to understand the observed variability. Consider, for example, a study that recorded action potential (AP) waveforms from a large number of cells and reported how various measures are distributed across the population. To explain the range of responses one would simultaneously need to know, at a minimum, the differences in expression and function of various ion channels within the same population of cells. Such comprehensive studies are close to impossible with present-day technology.

If the main purpose of a particular study is to demonstrate how one population of cells differs from another (i.e. healthy versus diseased), then considering mean data may be adequate. In the clinical realm, however, treatment decisions are made for one individual at a time, and the pertinent question is therefore not the overall success rate of a given treatment, but whether the treatment is likely to be effective for that particular individual. Variability between individuals must therefore be considered when making clinical decisions, and the entire enterprise of ‘personalized medicine’ is premised on this idea.

Although understanding variability is, for the reasons noted, extremely difficult in experimental and clinical studies, mathematical models can help an investigator to overcome the twin difficulties of limited sample size and unknown differences between individuals. With models it is easy to simulate altered behaviour from one sample to another and, of course, all parameters and conditions are known. However, traditional studies that treat a single model as representative have under-utilised the potential of models to address these questions. More recently, the idea has emerged that considering a variable population of models might be more realistic and informative than examining only a single representative model (Marder & Taylor, 2011), and this review discusses work that we and others have undertaken to consider variability in modelling studies. We discuss how an analysis of mathematical models can establish a framework that allows for the quantitative treatment of variability between individuals. This framework allows one to understand how differences between samples at the molecular level can lead to differences in cellular level and tissue level behaviours, including the response to pharmacological agents. In addition, these considerations lead naturally to new ideas for constraining model parameters based on a systematic comparison of model output to several sets of experimental data. Thus, we argue that recent computational advances are helping investigators in diverse fields to understand and predict the causes of functional variability between experimental samples.

Representations of variability in mathematical models

Figure 1 illustrates how one's thinking must change when variability between subjects is considered. The left panels in this figure represent input or parameter space whereas the right panels represent output or behaviour space. As an illustrative example, we consider a mathematical model of the ventricular myocyte, whereby the location of a cell in the left panel is determined by the values of two ionic conductances, in this case G_Na and G_Ca, while the location in the right panel is given by the action potential upstroke velocity (dV/dt_max) and Ca²⁺ transient amplitude (Δ[Ca²⁺]_i) under a defined set of conditions. Traditionally, since a representative model has defined values of G_Na, G_Ca, dV/dt_max and Δ[Ca²⁺]_i, we think of this model as occupying a single location in both parameter space and output space, as shown in the top panels. Although the set of equations and parameters fully defines the locations in both parameter space and output space, by themselves the equations do not provide insight into how inter-individual variability might affect phenotypic properties. If we wish to consider variability we must represent a model not as a single point but instead as a cloud of points in parameter space, representing many different combinations of G_Na and G_Ca (bottom left). It makes intuitive sense that this collection of different models will also occupy a cloud in output space (bottom right). The problem at this point is that we do not understand the mapping from the parameter space to the output space. Does a point at the top of the cloud in parameter space correspond to a point at the top, at the right or in the centre of the cloud in output space? In general, we do not know. In this particular example, prior knowledge tells us that large G_Na will correspond to fast upstroke velocity and large G_Ca will lead to bigger Ca²⁺ transients, giving us some qualitative idea of the mapping. When dealing with more obscure input parameters or more complicated outputs, however, the correct mapping may be a mystery. Next we describe how parameter sensitivity analysis can be used to make sense of these clouds, thereby allowing a study to consider a population of models. We continue to use examples from cardiac cellular physiology to illustrate concepts that are quite general. For instance, in a model of a biochemical signalling network, the axes in parameter space could be reaction rates and total enzyme concentrations, whereas the outputs could be levels of activated transcription factors.

Figure 1. A comparison of two paradigms in modelling.

A, the traditional approach represents a model as occupying a single point in parameter space (left) and a single point in measurement or output space (right). B, a newer approach towards modelling takes variability into account. Under this paradigm, it is more useful to consider a population of models, each with slightly different parameters. This population occupies regions in both parameter space and output space. Analysis is required, however, to understand how changes in parameters translate to changes in outputs.

To understand how variability in parameter space leads to changes in model outputs, the simplest strategy is to increase or decrease one of the parameters and record the simulation results. When this procedure is performed for several parameters and the outcome is expressed in quantitative terms (i.e. Δoutput/Δparameter), the result is a parameter sensitivity analysis. Traditionally these are performed by altering each parameter individually, and this strategy has provided insight into numerous models of excitable cells (Nygren et al. 1998; Weaver & Wearne, 2008; Romero et al. 2009; Corrias et al. 2011; O’Hara et al. 2011).

In recent work, we have extended this idea by varying all parameters at once, as an attempt to capture and understand the intrinsic variability of populations (Sobie, 2009; Sarkar & Sobie, 2010, 2011). This procedure involves creating a randomly generated set of ‘pseudo-data’ in which the important parameters vary according to a pre-defined distribution. For models of heart cell electrophysiology and ion transport, the parameters that are varied usually define the quantities and/or activities of various membrane ion channels, transporters and pumps. With each new set of parameters we: (1) run a series of simulations; (2) record outputs such as the action potential waveforms and Ca²⁺ transients that are specified by each set of parameters; and (3) extract meaningful metrics such as the action potential duration (APD) and the calcium transient amplitude (Δ[Ca²⁺]_i) from the results. Finally, we use multivariable regression to relate the parameters or independent variables to the outputs or dependent variables, thereby defining the mapping between the parameter space and the output space (Sobie, 2009).

The above method is executed as a linear algebraic problem using matrices. Our input matrix, X, consists of the parameters generated by randomly scaling the values in the published model. Each column in X corresponds to a different parameter, and each row constitutes a unique trial, which can be thought of as a distinct cell or individual. The physiologically relevant outputs generated through simulation with each parameter set are collected in the output matrix Y. Again each row is a unique trial, and the columns correspond to common metrics such as APD, resting potential, upstroke velocity, and Δ[Ca²⁺]_i. To render quantities in both matrices unitless, they are converted into z-scores by centreing them on their means and normalising them by their standard deviations. Multivariable linear regression relates the two matrices by calculating a matrix of regression coefficients, which we call B. The regression matrix is calculated in such a way that its product with the input matrix closely approximates the output matrix, i.e. XB = Ŷ≍Y. The elements of B are parameter sensitivities; each indicates how much changing a parameter affects a particular output. In prior work, we have found that linear models relating ionic conductances to physiologically relevant model outputs are generally quite accurate (Sobie, 2009; Sarkar & Sobie, 2010). However, given that ionic concentrations, in particular extracellular potassium, cause non-linear changes in current–voltage relations, these methods may prove less accurate when ionic concentrations vary greatly between individuals.

Even with this caveat, the matrix representation provides a convenient way to think of how changes in ionic conductances influence cellular outputs, i.e. it provides the mapping from the parameter space to the output space, as illustrated in Fig. 2. This figure displays combinations of APD, action potential upstroke velocity (dV/dt_max) and Δ[Ca²⁺]_i generated with a model of the human ventricular action potential (Ten Tusscher et al. 2004). The cell model with published parameters sits at the origin, and each output is represented as a deviation from the control value. To understand the changes in these outputs that will occur in a cell with, in this case, different values of G_Na and G_Ca, we multiply the vector describing the change in inputs and the matrix of parameter sensitivities. The magnitudes of the parameter sensitivities in the regression matrix B indicate how much changing a given parameter affects a particular output, and the signs of these values indicate whether an increase in the parameter causes an increase (positive sign) or decrease (negative sign) in the corresponding output. For example, consider a cell with greater than normal G_Na and less than normal G_Ca. The increased G_Na will shift the cell in the positive direction along the dV/dt_max axis but have virtually no effect in the other directions because the parameter sensitivities corresponding to the effects of G_Na on APD and Δ[Ca²⁺]_i are close to zero. The decreased G_Ca will have essentially no effect on dV/dt_max but will reduce both APD and Δ[Ca²⁺]_i because of these parameter sensitivities. Thus, the difference in location between a control cell and a perturbed cell in output space via any number of intermediate positions can be understood by summing the respective vectors describing the change in position at each step. This general idea can easily be extended to higher dimensions if additional outputs are considered.

Figure 2. Three-dimensional plot showing the transition from control to an altered condition in response to changes in parameter values.

A particular cell, with defined values of 3 physiological outputs, resides in a specific location in output space. The three outputs in this example are action potential duration (APD), Ca²⁺ transient amplitude (Δ[Ca²⁺]_i) and action potential upstroke velocity (dV/dt_max). By convention, the control cell, shown in blue, resides at the origin. When values of two parameters (here G_Na and G_Ca) change, the final location of the cell, shown in red, can be calculated as the sum of two vectors, each of which is the product of a parameter change and the corresponding vector of parameter sensitivities. These are shown to the right of the 3D plot. The first step, due to an increase in G_Na, causes movement only along the upstroke velocity axis because this is the only non-zero parameter sensitivity. The second step, a decrease in G_Ca, leads to a decrease in APD and a decrease in Δ[Ca²⁺]_i, but virtually no change in dV/dt_max.

Applications of sensitivity analysis to understand variability

Parameter sensitivity analysis in particular, and the consideration of variability more broadly, can provide insight into a number of important issues in cardiac physiology. In the remainder of this review we discuss several examples of how such computational techniques have been applied to gain novel understanding. We discuss studies that have addressed four important and largely unresolved questions: (1) Under what conditions do competing models of the same cell type produce divergent predictions? (2) How do changes in multiple ion transport pathways contribute quantitatively to the observed phenotype in disease? (3) How does electrophysiological variability lead to variable responses to potentially pro-arrhythmic drugs? (4) Can understanding variability help to constrain parameters during model development? The insights gained from such studies have important implications not only for building better mathematical models, but possibly also for the diagnosis and treatment of diseases. The focus is on computational studies of cardiac physiology, specifically work performed at the cellular level. The methods discussed can be extended to understand tissue-level and organ-level phenomena, but at present, work at the cellular scale is more advanced. Additionally, we mention several important findings in neuroscience, since neuronal behaviour depends on ligand-gated and voltage-gated ion channels, and many of the same considerations pertain. Moreover, work in neuroscience provides examples of how the exploration of possibilities using mathematical models has led to the formation of new hypotheses, which has then driven experimental work to confirm the model predictions and generate important insight.

Under what conditions do competing models of the same cell type produce divergent predictions?

In the past several years, the number of published mathematical models has grown considerably. Fairly comprehensive lists of published models are provided in recent reviews (Fink et al. 2011; Winslow et al. 2011), and online repositories such as CellML have been developed to annotate and archive these models (e.g. http://models.cellml.org/electrophysiology). As a result, for several cell types two or more competing models from different groups aim to represent the same physiology. This offers opportunities because additional tools are now available, but it also presents challenges, since the strengths and weaknesses of different models are not always apparent.

A few studies have had the explicit goal of comparing models of the same cell type. Cherry and Fenton (2007), for instance, compared two models of the canine ventricular myocyte, those published by Fox et al. (2002) and by Hund & Rudy (2004). This comparison found several important differences between the two models, including the balance of currents responsible for the normal AP, changes in AP morphology with pacing rate, and the stability of re-entrant waves (Cherry & Fenton, 2007). With certain simulation protocols, the differences between the models were so great that the predictions of at least one would have to be considered inconsistent with experimental data. Important differences between competing models have also been observed in comparative studies examining human ventricular myoctes (Ten Tusscher et al. 2006; Bueno-Orovio et al. 2008; Carro et al. 2011), atrial myocytes (Cherry & Evans, 2008; Cherry et al. 2008) and rabbit ventricular myocytes (Romero et al. 2011). Niederer et al. (2009) suggested that such differences may result from the fact that newer models often inherit subcomponents, such as equations describing individual ion channels, from older models. Intracellular Ca²⁺ cycling is an additional factor that can contribute to differences between models. Because Ca²⁺ release in heart cells occurs as the stochastic triggering of thousands of events (Ca²⁺ sparks), most models simplify the representation of this process (Williams et al. 2010; Greenstein & Winslow, 2011), and different simplifications can lead to important differences in model behaviour. Since the implicit goal of all modelling is to generate experimentally correct predictions, the implications of such discrepancies are great, and efforts to develop standards for the publication and validation of models have recently gained momentum (Terkildsen et al. 2008; Cooper et al. 2011).

Many of the aforementioned studies, however, have compared multiple models with respect to a limited number of experimental perturbations. In other words, they could be said to have used a case-by-case approach that is not particularly scalable. Parameter sensitivity analysis, performed using standard methods (Romero et al. 2009) or multivariable regression (Sobie, 2009) offers a means to make model comparison more systematic and complete. This is because each parameter sensitivity is a quantitative and potentially experimentally testable prediction indicating how much altering an ion transport pathway influences a particular cellular behaviour. A side-by-side comparison of two models’ parameter sensitivities therefore enables one to identify fundamental differences in how the models respond to perturbations. For instance, we previously used this approach to examine differences between the Fox–McHarg–Gilmour model (Fox et al. 2002) and the Kurata model of the human AP (Kurata et al. 2005). This analysis demonstrated that inward rectifier current (I_K1) is by far the dominant repolarizing current in the Fox model whereas I_K1, rapid delayed rectifier (I_Kr) and slow delayed rectifier (I_Ks) are essentially equally important in the Kurata model. These insights into differences between models can point to obvious deficiencies that need to be corrected, or they can suggest new experiments that will clarify which model predictions are accurate.

How do changes in multiple ion transport pathways contribute quantitatively to the observed phenotype in disease?

Many pathological phenotypes result from changes in multiple channels or ion transport pathways. For instance, heart failure is associated with, amongst other changes, downregulation of K⁺ channels (Kaab et al. 1996), decreased pumping of Ca²⁺ into the sarcoplasmic reticulum (SR), leaky SR Ca²⁺ release channels (Marx et al. 2000) and altered triggering of SR Ca²⁺ release (Song et al. 2006). Some changes may exacerbate whereas others may attenuate the observed pathological function. Experimentally determining the quantitative contribution of each change can be difficult when all are occurring simultaneously. To address such questions, mathematical models play an important role, as they easily allow for individual changes to be simulated. Most modelling studies that have explored complicated physiological or pathological phenotypes, however, have either implemented each change one at a time or eliminated each change one at a time to understand the individual effects (Winslow et al. 1999; Saucerman & McCulloch, 2004; Yang et al. 2010). While a useful approach, this may not always allow for a direct quantitative comparison when different pathways are upregulated or downregulated by different amounts.

Parameter sensitivity analysis, and in particular the matrix formulation of our methods (Sobie, 2009; Sarkar & Sobie, 2011), provides a convenient framework for a quantitative understanding of complex phenotypes. If the row vector x represents the extent to which physiological parameters differ from control values in a diseased cell, then ŷ = xB predicts how much functionally important outputs will deviate from the control values. Moreover, the rules of matrix multiplication allow for an easy evaluation of how much each parameter change contributes.

An example is shown in Fig. 3. We simulated heart failure in a mathematical model of the human ventricular myocyte (Grandi et al. 2010) by changing eight parameters (Shannon et al. 2005). After modifying these eight parameters, APD was increased by 35% (Fig. 3A) and Δ[Ca²⁺]_i was reduced by roughly half (Fig. 3B), consistent with experimental observations. To understand the extent to which each parameter change contributes to the overall phenotype, Fig. 3C displays the row vector x (dimensions 1 × 8), the parameter sensitivity matrix B (dimensions 8 × 2), the element-by-element products, and the two outputs (APD and Δ[Ca²⁺]_i) computed from ŷ = xB. This representation demonstrates which parameter changes are most responsible for the observed phenotype. For example, the increase in NCX and the decrease in G_K1 make the most important contributions to AP prolongation whereas the increase in NCX and the increase in SR Ca²⁺ leak (K_leak) are primarily responsible for the decrease in Δ[Ca²⁺]_i.

Figure 3. Multivariable regression applied to predict effects of changes occurring during heart failure.

Eight parameter changes corresponding to heart failure, as previously simulated (Shannon et al. 2005), are implemented in a mathematical model of the human ventricular myocyte (Grandi et al. 2010). A, the heart failure simulation predicted an increase in APD. B, the heart failure simulation predicted a decrease in Δ[Ca²⁺]_i. C, the matrix multiplication approach after performing multivariable regression allows us to correctly predict these changes in APD and Δ[Ca²⁺]_i by summing up the independent contributions of each of the eight parameter changes. This approach also reveals which parameter changes are most responsible for the observed changes in phenotype. For example, NCX and G_K1 both contribute greatly to APD prolongation while NCX and K_leak are predominantly responsible for the decrease in calcium transient amplitude. D, compensatory changes occur to maintain APD at control value when NCX and G_K1 are both reduced to 0.5 and 0.7 times their control values, respectively. For these simulations, the particular parameter changes implemented, represented graphically in C, are as follows: decrease in maximal slow transient outward K⁺ conductance (G_toslow), decrease in maximal fast transient outward K⁺ conductance (G_tofast), decrease in maximal slow delayed rectifier K⁺ current conductance (G_Ks), decrease in maximal inward rectifier K⁺ conductance (G_K1), increase in maximal Na⁺–Ca²⁺ exchange current (NCX), increase in SR Ca²⁺ release (K_s), increase in passive SR Ca²⁺ leak (K_leak), and decrease in maximal rate of SR Ca²⁺ uptake (V_maxSRCaP).

This approach therefore allows for an intuitive dissection of complicated phenotypes in which several electrophysiological properties are altered simultaneously, since it indicates quantitatively how much each alteration contributes to the observed pathological behaviour. Notably, the method can be employed to understand the factors responsible for pathological physiology not only in the cardiovascular system, but also in the case of other systems where the changes between normal and diseased conditions are similarly well characterized.

A further examination of the parameter sensitivities allows for a consideration of compensatory changes. In this example, for instance, the decrease in G_K1 and increase in K_NCX contribute almost equally to the AP prolongation. Hypothetically, however, if G_K1 and K_NCX both decreased, then the effects on APD could cancel each other out due to the opposite signs of their parameter sensitivities. The simulated APs shown in Fig. 3D indicate that if these two changes take place simultaneously, the resulting AP is essentially indistinguishable from the control AP. Evidence for such compensatory changes comes from experiments that have either overexpressed, knocked down or knocked out specific ion channels. Both the neuroscience (MacLean et al. 2003; Swensen & Bean, 2005) and the cardiac (Guo et al. 1999; Zhou et al. 2003) literature show examples in which expression levels of alternative ion transport pathways changed in such a way that electrical activity was similar to that observed without the genetic perturbation. An elegant recent study has combined experiments and modelling to extend this idea. By coupling a biological neuron to a model neuron and systematically varying the synaptic conductance and the model neuron properties, Grashow et al. (2010) demonstrated that the activity of the neuronal circuit could be made identical, even if the biological neurons exhibited highly variable behaviour in isolation.

How does electrophysiological variability lead to variable responses to potentially pro-arrhythmic drugs?

Across a population of individuals, considerable variability may sometimes be seen in physiological metrics such as QT interval (at the cellular level, APD) or contraction strength (at the cellular level, Δ[Ca²⁺]_i). In some circumstances, however, two samples appear essentially identical under baseline conditions, but differences emerge when these samples are subjected to a perturbation. For instance, a collection of neurons from different animals may exhibit similar activity patterns under standard experimental conditions but show variable responses to a neuromodulator (Grashow et al. 2009).

An analogous issue in cardiac biology concerns the so-called drug-induced long QT syndrome (diLQTS). Almost all cases of diLQTS result from agents that block the rapid delayed rectifier current (I_Kr), the product of the gene colloquially known as HERG. Across a population, however, the response to a HERG-blocking drug is generally quite variable (Kannankeril et al. 2010, 2011). Some individuals may experience minimal ECG changes in response to the drug whereas others may experience significant prolongation of the QT interval that can lead to dangerous ventricular tachycardia. Importantly, the response to a HERG-blocking drug appears to be uncorrelated with the individual's baseline ECG characteristics (Kannankeril et al. 2011). It has been hypothesized that patients who react adversely to HERG-blocking agents have reduced ‘repolarization reserve’ (Roden, 1998; Varro & Baczko, 2011), but this term has frequently been used loosely and without rigorous quantification. Several studies have suggested that mathematical modelling may help provide insight into these challenging and unresolved issues (Rodriguez et al. 2010; O’Hara & Rudy, 2011).

At the organismal level, several factors can contribute to drug-induced QT prolongation, including drug pharmacokinetics, cardiac structure and extracellular ionic concentrations. Cellular-level effects are nonetheless extremely important, and we recently simulated a population of cellular models before and after HERG block to obtain new quantitative insight into these variable drug responses (Sarkar & Sobie, 2011). As described above, we randomized parameters and used multivariable regression to relate parameter changes to the model behaviours, but here the critical model output was not APD per se, but the change in APD caused by a HERG-blocking drug (ΔAPD). This allowed us to quantify how each electrophysiological parameter in the model affected the response to the drug. The analysis provided many expected results, but also a few surprises. For instance, in a particular model of the human ventricular myocyte (Ten Tusscher et al. 2004), an increase in L-type Ca²⁺ current caused an increase in baseline APD but a decrease in drug-induced ΔAPD. Because of the differences between models mentioned above, we performed our analysis in five different myocyte models and generated a consensus list of the parameters that were consistently important in determining the response of model cardiac myocytes to HERG block. These parameters represent the factors most likely to determine the susceptibility of individuals to HERG-blocking drugs. Overall this study (Sarkar & Sobie, 2011) represents the most thorough quantification of repolarization reserve performed to date, and it illustrates how variability in a population of models can help to illuminate a clinically relevant issue.

Can understanding variability help in constraining parameters during the development of models?

In cardiac myocyte models, parameters such as maximal conductances are generally determined from voltage clamp experiments that have investigated the individual ionic currents. However, some currents have not been measured in particular cell types, and sometimes results from several studies are inconsistent. Either of these circumstances creates a situation in which a plausible range for a given parameter is established, but the parameter's precise value is unclear. The final stage of model development therefore involves adjusting model parameters so that cellular behaviours such as APs and Ca²⁺ transients more closely match experimental data. After testing several combinations of parameters, a study's authors settle on a single set to represent the baseline mathematical model.

The fact that changes in ionic currents can partially compensate for each other, however (see Fig. 3), illustrates a potential problem with this approach. Multiple solutions to, for instance, the same AP waveform are often possible. This is illustrated in Fig. 4, which shows results from a classic study in computational neuroscience (Golowasch et al. 2002) and from a more recent study on models of cardiac cells (Sarkar & Sobie, 2010). Two different combinations of electrophysiological parameters can generate voltage traces that look virtually indistinguishable, a phenomenon that has been observed by others (Dokos & Lovell, 2004; Zaniboni et al. 2010). This highlights a problem whereby the model is underconstrained when only a few output characteristics, such as those derived from the AP waveform, are used when selecting model parameters.

Figure 4. Analogous findings in the fields of neuroscience and cardiac electrophysiology.

Virtually indistinguishable electrical activity can result from very different sets of conductances. A, voltage traces from three model neurons look essentially identical. The underlying conductances shown at right, however, are quite different (Golowasch et al. 2002, reprinted with permission) B, similar to the behaviour observed in neuronal models, drastically different conductances in two model variants can produce identical action potentials in a mathematical model of the human ventricular myocyte (Sarkar & Sobie, 2010, reprinted with permission).

Because of this, it can be helpful to consider variability between individuals when constraining model parameters, and several methods exploit variability by constructing a population of candidate models and differentiating between candidates based on how realistically each behaves. Here we discuss three such methods: (1) a database approach pioneered in neuroscience by Marder, Prinz and coworkers (Prinz et al. 2003, 2004; Taylor et al. 2009); (2) genetic algorithms, including a new variant called multiple objective optimization (Druckmann et al. 2007); and (3) a multivariable regression strategy developed in our laboratory (Sarkar & Sobie, 2010).

By independently varying the eight maximal conductances of the lobster stomatogastric neuron, Prinz et al. (2003) developed a database of around 1.7 million models. For each model variant, a number of neuronal properties were calculated including activity type, spike or burst frequency, resting potential, frequency–current relation and phase–response curve. Using this database of models, these authors demonstrated how one could search for a specific combination of a neuron's properties or find a model that exhibited the properties of a specific biological neuron. A conclusion of this work was that very many model variants might be consistent with a single neuronal property, but the set of variants simultaneously consistent with several neuronal properties was much smaller. This general principle was also illustrated in a study that used a database approach to examine a model of Ca²⁺-induced Ca²⁺ release in heart cells (Sobie & Ramay, 2009).

The neuroscience studies also predicted that maximal conductances could vary quite widely between neurons, even if the activity patterns of the neurons were essentially identical. This model prediction was subsequently confirmed using single cell gene expression (Schulz et al. 2006, 2007). These studies obtained the additional interesting finding that expression levels of particular pairs of ion channels could be strongly correlated, thereby providing support for the idea of compensation to maintain roughly constant activity levels. Subsequent modelling and experimental studies have examined potential correlations between ionic conductances in greater detail (Taylor et al. 2009; Tobin et al. 2009; Hudson & Prinz, 2010).

Automated search heuristic approaches, such as genetic algorithms, provide another approach to constrain model parameters based on variability between individuals. Using an iterative optimization protocol that explores many possible solutions simultaneously, these algorithms select the best solutions from each iteration to be passed on to the next, until finding a single solution at the final iteration (Syed et al. 2005; Kherlopian et al. 2011). Using a genetic algorithm to optimize a model of the cerebellar Purkinje cell, Achard & DeSchutter (2006) found that multiple parameter combinations could produce virtually identical activity patterns, similar to the conclusions of Prinz et al. (2003, 2004). Extending this strategy, Druckmann et al. (2007) combined a genetic algorithm with a multiple objective optimization approach that attempted to simultaneously minimize several error functions. By attempting to simultaneously match several features of the spiking response, including spike height and spike rate, the authors were able to uncover tradeoffs between the various objectives.

A third method to constrain free parameters in electrophysiological models of cardiac myocytes was recently developed in our laboratory, as a logical extension of the parameter randomization plus multivariable regression technique described above (Sarkar & Sobie, 2010). Mathematically, this involves swapping the input and output matrices and performing ‘reverse regression’ such that YB_reverse closely approximates the true matrix of parameters X. We demonstrated the feasibility of this approach through simulations with a model (Ten Tusscher et al. 2004) of the human ventricular action potential. We randomly varied the 16 ionic conductances, and with each candidate model we simulated several different experimental conditions and protocols. This allowed us to calculate for each model variant a set of 32 outputs, including not just APD and Δ[Ca²⁺]_i, but also quantities such as the stimulation threshold, the response to changes in pacing rate and the response to changes in potassium concentration. When the output matrix Y contained the complete set of 32 outputs, 12 out of the 16 conductances could be well-constrained through the matrix equation X≍YB_reverse. Interestingly, we found that the same 12 conductances could still be well-constrained using only a subset of 16 outputs, suggesting that the information contained in the remaining 16 outputs was redundant or unnecessary for constraining parameters. Overall, this study demonstrates that considering variability between individuals can assist with constraining parameters during model development.

A broader conclusion of all of the above studies is that model parameters may be poorly constrained when the model is required to match only a few experimental results, but parameters are constrained more robustly when additional experimental information is considered. In cardiac myocyte models in particular, a careful consideration of changes in membrane input resistance during the action potential can help greatly in constraining model parameters (Zaniboni, 2011), but unfortunately only a limited number of recently developed models (Corrias et al. 2011) consider this physiologically relevant variable.

Conclusions and future directions

The studies discussed above lead to two important general conclusions. The first is that considering a population of models with different characteristics may be more informative than examining just a single, representative model (Marder & Taylor, 2011; Davies et al. 2011). The second is that parameter sensitivity analysis is a powerful method for understanding the behaviour of the population quantitatively. In addition, these results highlight several unresolved questions that can be addressed through a combination of computational and experimental approaches.

Since the work described above was performed at the cellular level, an obvious next step is to use such methods to examine variability at the tissue and organ levels. Here the parameters considered would include variables such as gap junctional conductance, fibre structure and heterogeneity of cell types through the myocardium. A few studies have looked at the relative effects of cell-to-cell coupling versus specific ionic currents in simplified systems (Viswanathan & Rudy, 2000; Spitzer et al. 2006); strategies such as those discussed here can help make the comparisons more thorough and systematic. We should also note that intracellular signalling models deal with related issues regarding the relative importance of biochemical versus geometric parameters (Neves & Iyengar, 2009), so similar techniques should also be of use this area.

Although variability in ionic conductances has been measured in several experimental studies in neuroscience (Schulz et al. 2006, 2007; Tobin et al. 2009), such issues have not been addressed to the same extent in cardiac electrophysiology, and many basic questions remain unresolved. In a population of healthy individuals, for instance, how much variability in maximal ionic conductances is typical? Is the degree of variability observed in inbred species such as laboratory rodents less than the amount observed in outbred species such as humans and dogs? Do all ion transport pathways (channels, pumps and exchangers) show similar variability (in relative terms), or are some experimentally measurable values more tightly constrained than others? The answer to this last question may have important implications for our understanding of regulatory mechanisms. If a particular parameter is tightly constrained across a population of individuals, this may imply that cells actively attempt to maintain its value within a particular range.

Just as we presently have limited information about how electrophysiological parameters vary from cell to cell in healthy myocytes, we know even less about whether particular ionic conductances are correlated. Correlations in ion channel mRNA levels have been observed in neurons (Schulz et al. 2006, 2007; Tobin et al. 2009), but analogous studies examining ionic conductances in cardiac myocytes are just beginning to be performed (Banyasz et al. 2011, 2012). If such correlations are found to be robust, then this will lead to the following question. Do correlations between conductances provide insight into the mechanisms responsible for maintaining these correlations? One possibility is that cells attempt to maintain a particular AP or Ca²⁺ transient morphology, and increases in one ionic conductance lead to changes in others that act to keep cellular behaviour relatively constant (Rosati & McKinnon, 2004). The results from several knockout or knockdown experiments seem consistent with this idea (Guo et al. 1999; MacLean et al. 2003; Zhou et al. 2003; Swensen & Bean, 2005), since reductions in one channel altered expression of other channels to partially offset the initial perturbation. Mechanistically, however, an alternative possibility is that correlations simply result from channels being regulated by similar sets of transcription factors, and do not explicitly relate to the regulation of physiological outputs (Marder, 2011). To differentiate between these hypotheses, modelling studies that consider variability can be helpful by determining what combinations are feasible for observing particular experimental behaviours.

A final important issue concerns the idea of individual-specific predictions. As the examples presented have illustrated, variability implies that different individuals may exhibit both different baseline physiology and divergent responses to a perturbation such as a drug. Indeed, the concept of repolarization reserve (Roden, 1998) was put forward to help explain the clinical experience with potentially pro-arrhythmic drugs, and recent work has shown how considering a population of models can help to illuminate such behaviour (Sarkar & Sobie, 2011). An obvious goal, then, is to be able to predict, rather than simply understand, such divergent responses. If innovative computational approaches such as those described here are combined with experiments, then we may soon reach a day when an investigator can examine cells from two healthy animals, and test quantitative physiological predictions specific to each animal. The hope is that if these strategies are successful, then similar approaches can be applied to guide rational, mechanism-based, patient-specific clinical decisions.