Modelling myocardial ischaemia and reperfusion

Lipid metabolism drives allele-specific early-stage hypertrophic cardiomyopathy

Hypertrophic cardiomyopathy (HCM) results from pathogenic variants in sarcomeric protein genes, that increase myocyte energy demand and lead to cardiac hypertrophy. But it is unknown whether a common metabolic trait underlies the cardiac phenotype at early disease stage. This study characterized two HCM mouse models (R92W-TnT, R403Q-MyHC) that demonstrate differences in mitochondrial function at early disease stage. Using a combination of cardiac phenotyping, transcriptomics, mass spectrometry-based metabolomics and computational modeling, we discovered allele-specific differences in cardiac structure/function and metabolic changes. TnT-mutant hearts had impaired energy substrate metabolism and increased phospholipid remodeling compared to MyHC-mutants. TnT-mutants showed increased incorporation of saturated fatty acid residues into ceramides, cardiolipin, and increased lipid peroxidation, that could underlie allele-specific differences in mitochondrial function and cardiomyopathy.

Characterizing arrhythmia using machine learning analysis of Ca2+ cycling in human cardiomyocytes

Accurate modeling of the heart electrophysiology to predict arrhythmia susceptibility remains a challenge. Current electrophysiological analyses are hypothesis-driven models drawing conclusions from changes in a small subset of electrophysiological parameters because of the difficulty of handling and understanding large datasets. Thus, we develop a framework to train machine learning classifiers to distinguish between healthy and arrhythmic cardiomyocytes using their calcium cycling properties. By training machine learning classifiers on a generated dataset containing a total of 3,003 healthy derived cardiomyocytes and their various arrhythmic states, the multi-class models achieved >90% accuracy in predicting arrhythmia presence and type. We also demonstrate that a binary classifier trained to distinguish cardiotoxic arrhythmia from healthy electrophysiology could determine the key biological changes associated with that specific arrhythmia. Therefore, machine learning algorithms can be used to characterize underlying arrhythmic patterns in samples to improve in vitro preclinical models and complement current in vivo systems.

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Calcium reporter enabled analysis of hPSC-derived CM electrophysiology

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Calcium electrophysiology can be decomposed to train machine learning classifiers

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Multi-class classifier distinguishes different forms of arrhythmias

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Binary-class classifiers identify key electrophysiological changes a priori

A computational model of cardiomyocyte metabolism predicts unique reperfusion protocols capable of reducing cell damage during ischemia/reperfusion

If a coronary blood vessel is occluded and the neighboring cardiomyocytes deprived of oxygen, subsequent reperfusion of the ischemic tissue can lead to oxidative damage due to excessive generation of reactive oxygen species. Cardiomyocytes and their mitochondria are the main energy producers and consumers of the heart, and their metabolic changes during ischemia seem to be a key driver of reperfusion injury. Here, we hypothesized that tracking changes in cardiomyocyte metabolism, such as oxygen and ATP concentrations, would help in identifying points of metabolic failure during ischemia and reperfusion. To track some of these changes continuously from the onset of ischemia through reperfusion, we developed a system of differential equations representing the chemical reactions involved in the production and consumption of 67 molecular species. This model was validated and used to identify conditions present during periods of critical transition in ischemia and reperfusion that could lead to oxidative damage. These simulations identified a range of oxygen concentrations that lead to reverse mitochondrial electron transport at complex I of the respiratory chain and a spike in mitochondrial membrane potential, which are key suspects in the generation of reactive oxygen species at the onset of reperfusion. Our model predicts that a short initial reperfusion treatment with reduced oxygen content (5% of physiological levels) could reduce the cellular damage from both of these mechanisms. This model should serve as an open-source platform to test ideas for treatment of the ischemia reperfusion process by following the temporal evolution of molecular concentrations in the cardiomyocyte.

Intracellular sodium elevation reprograms cardiac metabolism

Intracellular Na elevation in the heart is a hallmark of pathologies where both acute and chronic metabolic remodelling occurs. Here, we assess whether acute (75 μM ouabain 100 nM blebbistatin) or chronic myocardial Na_i load (PLM^3SA mouse) are causally linked to metabolic remodelling and whether the failing heart shares a common Na-mediated metabolic ‘fingerprint’. Control (PLM^WT), transgenic (PLM^3SA), ouabain-treated and hypertrophied Langendorff-perfused mouse hearts are studied by ²³Na, ³¹P, ¹³C NMR followed by ¹H-NMR metabolomic profiling. Elevated Na_i leads to common adaptive metabolic alterations preceding energetic impairment: a switch from fatty acid to carbohydrate metabolism and changes in steady-state metabolite concentrations (glycolytic, anaplerotic, Krebs cycle intermediates). Inhibition of mitochondrial Na/Ca exchanger by CGP37157 ameliorates the metabolic changes. In silico modelling indicates altered metabolic fluxes (Krebs cycle, fatty acid, carbohydrate, amino acid metabolism). Prevention of Na_i overload or inhibition of Na/Ca_mito may be a new approach to ameliorate metabolic dysregulation in heart failure.

The failing heart is characterised by both alterations in mitochondrial metabolism and an elevation of cytosolic sodium. Here, the authors use ²³Na NMR and metabolic profiling to show these are related, and that elevation in intracellular Na reprograms cardiac substrate utilisation via effects on mitochondrial Na/Ca exchange.

Glucose 6-Phosphate Accumulates via Phosphoglucose Isomerase Inhibition in Heart Muscle

RATIONALE

Metabolic and structural remodeling is a hallmark of heart failure. This remodeling involves activation of the mTOR (mammalian target of rapamycin) signaling pathway, but little is known on how intermediary metabolites are integrated as metabolic signals.

OBJECTIVE

We investigated the metabolic control of cardiac glycolysis and explored the potential of glucose 6-phosphate (G6P) to regulate glycolytic flux and mTOR activation.

METHODS AND RESULTS

We developed a kinetic model of cardiomyocyte carbohydrate metabolism, CardioGlyco, to study the metabolic control of myocardial glycolysis and G6P levels. Metabolic control analysis revealed that G6P concentration is dependent on phosphoglucose isomerase (PGI) activity. Next, we integrated ex vivo tracer studies with mathematical simulations to test how changes in glucose supply and glycolytic flux affect mTOR activation. Nutrient deprivation promoted a tight coupling between glucose uptake and oxidation, G6P reduction, and increased protein-protein interaction between hexokinase II and mTOR. We validated the in silico modeling in cultured adult mouse ventricular cardiomyocytes by modulating PGI activity using erythrose 4-phosphate. Inhibition of glycolytic flux at the level of PGI caused G6P accumulation, which correlated with increased mTOR activation. Using click chemistry, we labeled newly synthesized proteins and confirmed that inhibition of PGI increases protein synthesis.

CONCLUSIONS

The reduction of PGI activity directly affects myocyte growth by regulating mTOR activation.

Modeling oxygen requirements in ischemic cardiomyocytes

Heart disease remains the leading cause of death globally. Although reperfusion following myocardial ischemia can prevent death by restoring nutrient flow, ischemia/reperfusion injury can cause significant heart damage. The mechanisms that drive ischemia/reperfusion injury are not well understood; currently, few methods can predict the state of the cardiac muscle cell and its metabolic conditions during ischemia. Here, we explored the energetic sustainability of cardiomyocytes, using a model for cellular metabolism to predict the levels of ATP following hypoxia. We modeled glycolytic metabolism with a system of coupled ordinary differential equations describing the individual metabolic reactions within the cardiomyocyte over time. Reduced oxygen levels and ATP consumption rates were simulated to characterize metabolite responses to ischemia. By tracking biochemical species within the cell, our model enables prediction of the cell's condition up to the moment of reperfusion. The simulations revealed a distinct transition between energetically sustainable and unsustainable ATP concentrations for various energetic demands. Our model illustrates how even low oxygen concentrations allow the cell to perform essential functions. We found that the oxygen level required for a sustainable level of ATP increases roughly linearly with the ATP consumption rate. An extracellular O₂ concentration of ∼0.007 mm could supply basic energy needs in non-beating cardiomyocytes, suggesting that increased collateral circulation may provide an important source of oxygen to sustain the cardiomyocyte during extended ischemia. Our model provides a time-dependent framework for studying various intervention strategies to change the outcome of reperfusion.

The cardiac muscle duplex as a method to study myocardial heterogeneity

This paper reviews the development and application of paired muscle preparations, called duplex, for the investigation of mechanisms and consequences of intra-myocardial electro-mechanical heterogeneity. We illustrate the utility of the underlying combined experimental and computational approach for conceptual development and integration of basic science insight with clinically relevant settings, using previously published and new data. Directions for further study are identified.

Computational model of erratic arrhythmias in a cardiac cell network: the role of gap junctions

Cardiac morbidity and mortality increases with the population age. To investigate the underlying pathological mechanisms, and suggest new ways to reduce clinical risks, computational approaches complementing experimental and clinical investigations are becoming more and more important. Here we explore the possible processes leading to the occasional onset and termination of the (usually) non-fatal arrhythmias widely observed in the heart. Using a computational model of a two-dimensional network of cardiac cells, we tested the hypothesis that an ischemia alters the properties of the gap junctions inside the ischemic area. In particular, in agreement with experimental findings, we assumed that an ischemic episode can alter the gap junctions of the affected cells by reducing their average conductance. We extended these changes to include random fluctuations with time, and modifications in the gap junction rectifying conductive properties of cells along the edges of the ischemic area. The results demonstrate how these alterations can qualitatively give an account of all the main types of non-fatal arrhythmia observed experimentally, and suggest how premature beats can be eliminated in three different ways: a) with a relatively small surgical procedure, b) with a pharmacological reduction of the rectifying conductive properties of the gap-junctions, and c) by pharmacologically decreasing the gap junction conductance. In conclusion, our model strongly supports the hypothesis that non-fatal arrhythmias can develop from post-ischemic alteration of the electrical connectivity in a relatively small area of the cardiac cell network, and suggests experimentally testable predictions on their possible treatments.

Role of Na(+)-H(+) exchange in the modulation of L-type Ca(2+) current during fluid pressure in rat ventricular myocytes

Application of fluid pressure (FP) using pressurized fluid flow suppresses the L-type Ca(2+) current through both enhancement of Ca(2+) release and intracellular acidosis in ventricular myocytes. As FP-induced intracellular acidosis is more severe during the inhibition of Na(+)-H(+) exchange (NHE), we examined the possible role of NHE in the regulation of I(Ca) during FP exposure using HOE642 (cariporide), a specific NHE inhibitor. A flow of pressurized (~16 dyn/cm(2)) fluid was applied onto single rat ventricular myocytes, and the I(Ca) was monitored using a whole-cell patch-clamp under HEPES-buffered conditions. In cells pre-exposed to FP, additional treatment with HOE642 dose-dependently suppressed the I(Ca) (IC(50) = 0.97 ± 0.12 μM) without altering current-voltage relationships and inactivation time constants. In contrast, the I(Ca) in control cells was not altered by HOE642. The HOE642 induced a left shift in the steady-state inactivation curve. The suppressive effect of HOE642 on the I(Ca) under FP was not altered by intracellular high Ca(2+) buffering. Replacement of external Cl(-) with aspartate to inhibit the Cl(-)-dependent acid loader eliminated the inhibitory effect of HOE642 on I(Ca). These results suggest that NHE may attenuate FP-induced I(Ca) suppression by preventing intracellular H(+) accumulation in rat ventricular myocytes and that NHE activity may not be involved in the Ca(2+)-dependent inhibition of the I(Ca) during FP exposure.

Computational approaches to understand cardiac electrophysiology and arrhythmias

Cardiac rhythms arise from electrical activity generated by precisely timed opening and closing of ion channels in individual cardiac myocytes. These impulses spread throughout the cardiac muscle to manifest as electrical waves in the whole heart. Regularity of electrical waves is critically important since they signal the heart muscle to contract, driving the primary function of the heart to act as a pump and deliver blood to the brain and vital organs. When electrical activity goes awry during a cardiac arrhythmia, the pump does not function, the brain does not receive oxygenated blood, and death ensues. For more than 50 years, mathematically based models of cardiac electrical activity have been used to improve understanding of basic mechanisms of normal and abnormal cardiac electrical function. Computer-based modeling approaches to understand cardiac activity are uniquely helpful because they allow for distillation of complex emergent behaviors into the key contributing components underlying them. Here we review the latest advances and novel concepts in the field as they relate to understanding the complex interplay between electrical, mechanical, structural, and genetic mechanisms during arrhythmia development at the level of ion channels, cells, and tissues. We also discuss the latest computational approaches to guiding arrhythmia therapy.

NHE inhibition does not improve Na(+) or Ca(2+) overload during reperfusion: using modeling to illuminate the mechanisms underlying a therapeutic failure

Reperfusion injury results from pathologies of cardiac myocyte physiology that develop when previously ischemic myocardium experiences a restoration of normal perfusion. Events in the development of reperfusion injury begin with the restoration of a proton gradient upon reperfusion, which then allows the sodium-proton exchanger (NHE) to increase flux, removing protons from the intracellular space while importing sodium. The resulting sodium overload drives increased reverse-mode sodium-calcium exchanger (NCX) activity, creating a secondary calcium overload that has pathologic consequences. One of the attempts to reduce reperfusion-related damage, NHE inhibition, has shown little clinical benefit, and only when NHE inhibitors are given prior to reperfusion. In an effort to further understand why NHE inhibitors have been largely unsuccessful, we employed a new mathematical cardiomyocyte model that we developed for the study of ischemia and reperfusion. Using this model, we simulated 20 minutes of ischemia and 10 minutes of reperfusion, while also simulating NHE inhibition by reducing NHE flux in our model by varying amounts and at different time points. In our simulations, when NHE inhibition is applied at the onset of reperfusion, increasing the degree of inhibition increases the peak sodium and calcium concentrations, as well as reducing intracellular pH recovery. When inhibition was instituted at earlier time points, some modest improvements were seen, largely due to reduced sodium concentrations prior to reperfusion. Analysis of all sodium flux pathways suggests that the sodium-potassium pump (NaK) plays the largest role in exacerbated sodium overload during reperfusion, and that reduced NaK flux is largely the result of impaired pH recovery. While NHE inhibition does indeed reduce sodium influx through that exchanger, the resulting prolongation of intracellular acidosis paradoxically increases sodium overload, largely mediated by impaired NaK function.

Myocardial ischemia, commonly observed when arteries supplying the heart become occluded, results when cardiac tissue receives inadequate blood perfusion. In order to minimize the amount of cardiac damage, ischemic tissue must be reperfused. However, reperfusion can result in deleterious effects that leave the heart muscle sicker than if the ischemia had been allowed to continue. Examples of these reperfusion injuries include lethal arrhythmias and an increased region of cell death. Some of the early events that result in reperfusion injury include changes in pH and an overload of sodium inside the cell. During reperfusion, the sodium-proton exchanger (NHE) removes protons from the cell in an effort to restore normal pH, in turn importing sodium ions. Many strategies have been attempted to prevent reperfusion injury, including inhibition of the NHE, with little clinical effect. Using a mathematical model that we developed to study ischemia and reperfusion in cardiac cells, we found that NHE inhibition produces more severe sodium overload, largely due to adverse consequences of the delayed pH recovery produced by NHE inhibition. These results suggest that NHE inhibition alone may not be a viable strategy, and that therapies which prolong intracellular acidosis may be problematic.

Hydrogen sulfide regulates Na+/H+ exchanger activity via stimulation of phosphoinositide 3-kinase/Akt and protein kinase G pathways

Intracellular pH (pH(i)) is an important endogenous modulator of cardiac function. Inhibition of Na(+)/H(+) exchanger-1 (NHE-1) protects the heart by preventing Ca(2+) overload during ischemia/reperfusion. Hydrogen sulfide (H(2)S) has been reported to produce cardioprotection. The present study was designed to investigate the pH regulatory effect of H(2)S in rat cardiac myocytes and evaluate its contribution to cardioprotection. It was found that sodium hydrosulfide (NaHS), at a concentration range of 10 to 1000 μM, produced sustained decreases in pH(i) in the rat myocytes in a concentration-dependent manner. NaHS also abolished the intracellular alkalinization caused by trans-(±)-3,4-dichloro-N-methyl-N-[2-(1-pyrrolidinyl)-cyclohexyl]benzeneacetamide methane-sulfonate hydrate (U50,488H), which activates NHEs. Moreover, when measured with an NHCl(4) prepulse method, NaHS was found to significantly suppress NHE-1 activity. Both NaHS and cariporide or [5-(2-methyl-5-fluorophenyl)furan-2-ylcarbonyl]guanidine (KR-32568), two NHE inhibitors, protected the myocytes against ischemia/reperfusion injury. However, coadministration of NaHS with KR-32568 did not produce any synergistic effect. Functional study showed that perfusion with NaHS significantly improved postischemic contractile function in isolated rat hearts subjected to ischemia/reperfusion. Blockade of phosphoinositide 3-kinase (PI3K) with 2-(4-morpholinyl)-8-phenyl-4H-1-benzopyran-4-one (LY294002), Akt with Akt VIII, or protein kinase G (PKG) with (9S,10R,12R)-2,3,9,10,11,12-hexahydro-10-methoxy-2,9-dimethyl-1-oxo-9,12-epoxy-1H-diindolo[1,2,3-fg:3',2',1'-kl]pyrrolo[3,4-i][1,6]]enzodiazocine-10-carboxylic acid, methyl ester (KT5823) significantly attenuated NaHS-suppressed NHE-1 activity and/or NaHS-induced cardioprotection. Although KT5823 failed to affect NaHS-induced Akt phosphorylation, Akt inhibitor did attenuate NaHS-stimulated PKG activity. In conclusion, this work demonstrated for the first time that H(2)S produced cardioprotection via the suppression of NHE-1 activity involving a PI3K/Akt/PKG-dependent mechanism.

Mitochondrial energetics, pH regulation, and ion dynamics: a computational-experimental approach

We developed a computational model of mitochondrial energetics that includes Ca(2+), proton, Na(+), and phosphate dynamics. The model accounts for distinct respiratory fluxes from substrates of complex I and complex II, pH effects on equilibrium constants and enzyme kinetics, and the acid-base equilibrium distributions of energy intermediaries. We experimentally determined NADH and ΔΨ(m) in guinea pig mitochondria during transitions from de-energized to energized, or during state 2/4 to state 3 respiration, or into hypoxia and uncoupling, and compared the results with those obtained in model simulations. The model quantitatively reproduces the experimentally observed magnitude of ΔΨ(m), the range of NADH levels, respiratory fluxes, and respiratory control ratio upon transitions elicited by sequential additions of substrate and ADP. Simulation results are also able to mimic the change in ΔΨ(m) upon addition of phosphate to state 4 mitochondria, leading to matrix acidification and ΔΨ(m) polarization. The steady-state behavior of the integrated mitochondrial model qualitatively simulates the dependence of respiration on the proton motive force, and the expected flux-force relationships existing between respiratory and ATP synthesis fluxes versus redox and phosphorylation potentials. This upgraded mitochondrial model provides what we believe are new opportunities for simulating mitochondrial physiological behavior during dysfunctional states involving changes in pH and ion dynamics.

Successes and failures in modeling heart cell electrophysiology

Mathematical models of the electrical activity of the heart using equations for protein ion channels and other transporters began with the Noble 1962 model. These models then developed over a period of about 50 years. Cell types in all regions have been modeled and now are available for download from the CellML website (www.cellml.org). Simulation is a necessary tool of analysis in attempting to understand biological complexity. We often learn as much from the failures as from the successes of mathematical models. It is the iterative interaction between experiment and simulation that is important.

Automaticity in acute ischemia: bifurcation analysis of a human ventricular model

Acute ischemia (restriction in blood supply to part of the heart as a result of myocardial infarction) induces major changes in the electrophysiological properties of the ventricular tissue. Extracellular potassium concentration ([K(o)(+)]) increases in the ischemic zone, leading to an elevation of the resting membrane potential that creates an "injury current" (I(S)) between the infarcted and the healthy zone. In addition, the lack of oxygen impairs the metabolic activity of the myocytes and decreases ATP production, thereby affecting ATP-sensitive potassium channels (I(Katp)). Frequent complications of myocardial infarction are tachycardia, fibrillation, and sudden cardiac death, but the mechanisms underlying their initiation are still debated. One hypothesis is that these arrhythmias may be triggered by abnormal automaticity. We investigated the effect of ischemia on myocyte automaticity by performing a comprehensive bifurcation analysis (fixed points, cycles, and their stability) of a human ventricular myocyte model [K. H. W. J. ten Tusscher and A. V. Panfilov, Am. J. Physiol. Heart Circ. Physiol. 291, H1088 (2006)] as a function of three ischemia-relevant parameters [K(o)(+)], I(S), and I(Katp). In this single-cell model, we found that automatic activity was possible only in the presence of an injury current. Changes in [K(o)(+)] and I(Katp) significantly altered the bifurcation structure of I(S), including the occurrence of early-after depolarization. The results provide a sound basis for studying higher-dimensional tissue structures representing an ischemic heart.

Integrative modeling of the cardiac ventricular myocyte

Cardiac electrophysiology is a discipline with a rich 50-year history of experimental research coupled with integrative modeling which has enabled us to achieve a quantitative understanding of the relationships between molecular function and the integrated behavior of the cardiac myocyte in health and disease. In this paper, we review the development of integrative computational models of the cardiac myocyte. We begin with a historical overview of key cardiac cell models that helped shape the field. We then narrow our focus to models of the cardiac ventricular myocyte and describe these models in the context of their subcellular functional systems including dynamic models of voltage-gated ion channels, mitochondrial energy production, ATP-dependent and electrogenic membrane transporters, intracellular Ca dynamics, mechanical contraction, and regulatory signal transduction pathways. We describe key advances and limitations of the models as well as point to new directions for future modeling research. WIREs Syst Biol Med 2011 3 392-413 DOI: 10.1002/wsbm.122

A model of Na+/H+ exchanger and its central role in regulation of pH and Na+ in cardiac myocytes

A new kinetic model of the Na(+)/H(+) exchanger (NHE) was developed by fitting a variety of major experimental findings, such as ion-dependencies, forward/reverse mode, and the turnover rate. The role of NHE in ion homeostasis was examined by implementing the NHE model in a minimum cell model including intracellular pH buffer, Na(+)/K(+) pump, background H(+), and Na(+) fluxes. This minimum cell model was validated by reconstructing recovery of pH(i) from acidification, accompanying transient increase in [Na(+)](i) due to NHE activity. Based on this cell model, steady-state relationships among pH(i), [Na(+)](I), and [Ca(2+)](i) were quantitatively determined, and thereby the critical level of acidosis for cell survival was predicted. The acidification reported during partial blockade of the Na(+)/K(+) pump was not attributed to a dissipation of the Na(+) gradient across the membrane, but to an increase in indirect H(+) production. This NHE model, though not adapted to the dimeric behavioral aspects of NHE, can provide a strong clue to quantitative prediction of degree of acidification and accompanying disturbance of ion homeostasis under various pathophysiological conditions.

Modeling cardiac action potential shortening driven by oxidative stress-induced mitochondrial oscillations in guinea pig cardiomyocytes

Ischemia-induced shortening of the cardiac action potential and its heterogeneous recovery upon reperfusion are thought to set the stage for reentrant arrhythmias and sudden cardiac death. We have recently reported that the collapse of mitochondrial membrane potential (DeltaPsi(m)) through a mechanism triggered by reactive oxygen species (ROS), coupled to the opening of sarcolemmal ATP-sensitive potassium (K(ATP)) channels, contributes to electrical dysfunction during ischemia-reperfusion. Here we present a computational model of excitation-contraction coupling linked to mitochondrial bioenergetics that incorporates mitochondrial ROS-induced ROS release with coupling between the mitochondrial energy state and electrical excitability mediated by the sarcolemmal K(ATP) current (I(K,ATP)). Whole-cell model simulations demonstrate that increasing the fraction of oxygen diverted from the respiratory chain to ROS production triggers limit-cycle oscillations of DeltaPsi(m), redox potential, and mitochondrial respiration through the activation of a ROS-sensitive inner membrane anion channel. The periods of transient mitochondrial uncoupling decrease the cytosolic ATP/ADP ratio and activate I(K,ATP), consequently shortening the cellular action potential duration and ultimately suppressing electrical excitability. The model simulates emergent behavior observed in cardiomyocytes subjected to metabolic stress and provides a new tool for examining how alterations in mitochondrial oxidative phosphorylation will impact the electrophysiological, contractile, and Ca(2+) handling properties of the cardiac cell. Moreover, the model is an important step toward building multiscale models that will permit investigation of the role of spatiotemporal heterogeneity of mitochondrial metabolism in the mechanisms of arrhythmogenesis and contractile dysfunction in cardiac muscle.

Modeling the cardiac Na(+)/H (+) exchanger based on major experimental findings

Na(+)-H(+) exchanger (NHE) is the main acid extruder in cardiac myocytes. We review the experimental findings of ion-dependency of NHE activity, and the mathematical modeling developed so far. In spite of extensive investigation, many unsolved questions still remain. We consider that the precise description of NHE activity with mathematical models elucidates the roles of NHE in maintaining ionic homeostasis, especially under pathophysiological conditions.

Modeling transmural heterogeneity of K(ATP) current in rabbit ventricular myocytes

To investigate the mechanisms regulating excitation-metabolic coupling in rabbit epicardial, midmyocardial, and endocardial ventricular myocytes we extended the LabHEART model (Puglisi JL and Bers DM. Am J Physiol Cell Physiol 281: C2049-C2060, 2001). We incorporated equations for Ca(2+) and Mg(2+) buffering by ATP and ADP, equations for nucleotide regulation of ATP-sensitive K(+) channel and L-type Ca(2+) channel, Na(+)-K(+)-ATPase, and sarcolemmal and sarcoplasmic Ca(2+)-ATPases, and equations describing the basic pathways (creatine and adenylate kinase reactions) known to communicate the flux changes generated by intracellular ATPases. Under normal conditions and during 20 min of ischemia, the three regions were characterized by different I(Na), I(to), I(Kr), I(Ks), and I(Kp) channel properties. The results indicate that the ATP-sensitive K(+) channel is activated by the smallest reduction in ATP in epicardial cells and largest in endocardial cells when cytosolic ADP, AMP, PCr, Cr, P(i), total Mg(2+), Na(+), K(+), Ca(2+), and pH diastolic levels are normal. The model predicts that only K(ATP) ionophore (Kir6.2 subunit) and not the regulatory subunit (SUR2A) might differ from endocardium to epicardium. The analysis suggests that during ischemia, the inhomogeneous accumulation of the metabolites in the tissue sublayers may alter in a very irregular manner the K(ATP) channel opening through metabolic interactions with the endogenous PI cascade (PIP(2), PIP) that in turn may cause differential action potential shortening among the ventricular myocyte subtypes. The model predictions are in qualitative agreement with experimental data measured under normal and ischemic conditions in rabbit ventricular myocytes.

Computing work in the ischemic heart

This study analyzes the effect of a poorly perfused (ischemic) region on active tension and strain at a point in a model of the cardiac ventricles. The cellular active mechanics and electrophysiology were calculated using a coupled electromechanical cell model. The deformation of the heart was calculated combining an anisotropic constitutive law with the active tension generated by the cell models. An ischemic cell model was developed by adapting the Noble 98-HMT cell model parameters. The ischemic model exhibits a raised resting potential, shortened action potential duration and reduced amplitude. Ischemic cells were introduced into the heart model in the anterior wall of the left ventricle. The model was solved with and in the absence of the ischemic region. The stress and strains at a point located in the ischemic region were calculated. The ischemic cell model produced negligible amounts of active tension and this reduced the work load of ischemic cells by several orders of magnitude. The ischemic region had a significant effect on pump function in both the left and right ventricles.

Role of Cellular Compartmentation in the Metabolic Response to Stress: Mechanistic Insights from Computational Models

The mechanisms controlling ATP generation in the transition from normal resting conditions to either high work states or ischemia are poorly understood. ATP generation depends upon compartmentation between the mitochondria and cytosol of metabolic pathways and key energy transfer species that cannot be easily assessed experimentally. We developed a multicompartment mathematical model of cardiac metabolism to simulate the metabolic responses to ischemia and increased workload. The model is based on mass balances, transport, and metabolic processes in cardiac tissue, and has three distinct compartments (blood, cytosol, and mitochondria). In addition to distinguishing between cytosol and mitochondria, the model includes a cytosolic subcompartment for glycolytic metabolic channeling. The model simulations predict the rapid activation of glycogenolysis and lactate production at the onset of ischemia, and support the concept of localization of glycolysis to a cytosolic subcompartment. In addition, simulations show that mitochondrial NADH/NAD(+) is primarily determined by oxygen consumption during ischemia, while cytosolic NADH/NAD(+) and lactate production are largely a function of glycolytic flux during the initial phase, and is controlled by mitochondrial NADH/NAD(+) and the malate-aspartate shuttle during the steady state. Finally, the model predicts that metabolic activation with an abrupt increase in workload requires parallel activation of ATP hydrolysis, glycolysis, mitochondrial dehydrogenases, the electron transport chain, and ADP phosphorylation. Taken together, these studies demonstrate the importance of metabolic compartmentation in the regulation of cardiac energetics in response to acute stress, and they highlight the usefulness of computational models in this line of investigation.

Modeling cardiac ischemia

Myocardial ischemia is one of the main causes of sudden cardiac death, with 80% of victims suffering from coronary heart disease. In acute myocardial ischemia, the obstruction of coronary flow leads to the interruption of oxygen flow, glucose, and washout in the affected tissue. Cellular metabolism is impaired and severe electrophysiological changes in ionic currents and concentrations ensue, which favor the development of lethal cardiac arrhythmias such as ventricular fibrillation. Due to the burden imposed by ischemia in our societies, a large body of research has attempted to unravel the mechanisms of initiation, sustenance, and termination of cardiac arrhythmias in acute ischemia, but the rapidity and complexity of ischemia-induced changes as well as the limitations in current experimental techniques have hampered evaluation of ischemia-induced alterations in cardiac electrical activity and understanding of the underlying mechanisms. Over the last decade, computer simulations have demonstrated the ability to provide insight, with high spatiotemporal resolution, into ischemic abnormalities in cardiac electrophysiological behavior from the ionic channel to the whole organ. This article aims to review and summarize the results of these studies and to emphasize the role of computer simulations in improving the understanding of ischemia-related arrhythmias and how to efficiently terminate them.

Late sodium current in the pathophysiology of cardiovascular disease: consequences of sodium-calcium overload

Late sodium current in cardiac cells is very small compared with the fast component, but as it flows throughout the action potential it may make a substantial contribution to sodium loading during each cardiac cycle. Late sodium current may contribute to triggering arrhythmia in two ways: by causing repolarisation failure (early after depolarisations); and by triggering late after depolarisations attributable to calcium oscillations in sodium-calcium overload conditions. Reduction of late sodium current would therefore be expected to have therapeutic benefits, particularly in disease states such as ischaemia in which sodium-calcium overload is a major feature.

Acidosis in models of cardiac ventricular myocytes

The effects of acidosis on cardiac electrophysiology and excitation-contraction coupling have been studied extensively. Acidosis decreases the strength of contraction and leads to altered calcium transients as a net result of complex interactions between protons and a variety of intracellular processes. The relative contributions of each of the changes under acidosis are difficult to establish experimentally, however, and significant uncertainties remain about the key mechanisms of impaired cardiac function. In this paper, we review the experimental findings concerning the effects of acidosis on the action potential and calcium handling in the cardiac ventricular myocyte, and we present a modelling study that establishes the contribution of the different effects to altered Ca2+ transients during acidosis. These interactions are incorporated into a dynamical model of pH regulation in the myocyte to simulate respiratory acidosis in the heart.

Molecular Control of Cardiac Sodium Homeostasis in Health and Disease

INTRODUCTION

Cardiac myocytes utilize three high-capacity Na transport processes whose precise function can determine myocyte fate and the triggering of arrhythmias in pathological settings. We present recent results on the regulation of all three transporters that may be important for an understanding of cardiac function during ischemia/reperfusion episodes.

METHODS AND RESULTS

Refined ion selective electrode (ISE) techniques and giant patch methods were used to analyze the function of cardiac Na/K pumps, Na/Ca exchange (NCX1), and Na/H exchange (NHE1) in excised cardiac patches and intact myocytes. To consider results cohesively, simulations were developed that account for electroneutrality of the cytoplasm, ion homeostasis, water homeostasis (i.e., cell volume), and cytoplasmic pH. The Na/K pump determines the average life-time of Na ions (3-10 minutes) as well as K ions (>30 minutes) in the cytoplasm. The long time course of K homeostasis can determine the time course of myocyte volume changes after ion homeostasis is perturbed. In excised patches, cardiac Na/K pumps turn on slowly (-30 seconds) with millimolar ATP dependence, when activated for the first time. In steady state, however, pumps are fully active with <0.2 mM ATP and are nearly unaffected by high ADP (2 mM) and Pi (10 mM) concentrations as may occur in ischemia. NCX1s appear to operate with slippage that contributes to background Na influx and inward current in heart. Thus, myocyte Na levels may be regulated by the inactivation reactions of the exchanger which are both Na- and proton-dependent. NHE1 also undergo strong Na-dependent inactivation, whereby a brief rise of cytoplasmic Na can cause inactivation that persists for many minutes after cytoplasmic Na is removed. This mechanism is blocked by pertussis toxin, suggesting involvement of a Na-dependent G-protein. Given that maximal NCX1- and NHE1-mediated ion fluxes are much greater than maximal Na/K pump-mediated Na extrusion in myocytes, the Na-dependent inactivation mechanisms of NCX1 and NHE1 may be important determinants of cardiac Na homeostasis.

CONCLUSIONS

Na/K pumps appear to be optimized to continue operation when energy reserves are compromised. Both NCX1 and NHE1 activities are regulated by accumulation of cytoplasmic Na. These principles may importantly control cardiac cytoplasmic Na and promote myocyte survival during ischemia/reperfusion episodes by preventing Ca overload.

Ischemia reperfusion dysfunction changes model-estimated kinetics of myofilament interaction due to inotropic drugs in isolated hearts

BACKGROUND

The phase-space relationship between simultaneously measured myoplasmic [Ca2+] and isovolumetric left ventricular pressure (LVP) in guinea pig intact hearts is altered by ischemic and inotropic interventions. Our objective was to mathematically model this phase-space relationship between [Ca2+] and LVP with a focus on the changes in cross-bridge kinetics and myofilament Ca2+ sensitivity responsible for alterations in Ca2+-contraction coupling due to inotropic drugs in the presence and absence of ischemia reperfusion (IR) injury.

METHODS

We used a four state computational model to predict LVP using experimentally measured, averaged myoplasmic [Ca2+] transients from unpaced, isolated guinea pig hearts as the model input. Values of model parameters were estimated by minimizing the error between experimentally measured LVP and model-predicted LVP.

RESULTS

We found that IR injury resulted in reduced myofilament Ca2+ sensitivity, and decreased cross-bridge association and dissociation rates. Dopamine (8 microM) reduced myofilament Ca2+ sensitivity before, but enhanced it after ischemia while improving cross-bridge kinetics before and after IR injury. Dobutamine (4 microM) reduced myofilament Ca2+ sensitivity while improving cross-bridge kinetics before and after ischemia. Digoxin (1 microM) increased myofilament Ca2+ sensitivity and cross-bridge kinetics after but not before ischemia. Levosimendan (1 microM) enhanced myofilament Ca2+ affinity and cross-bridge kinetics only after ischemia.

CONCLUSION

Estimated model parameters reveal mechanistic changes in Ca2+-contraction coupling due to IR injury, specifically the inefficient utilization of Ca2+ for contractile function with diastolic contracture (increase in resting diastolic LVP). The model parameters also reveal drug-induced improvements in Ca2+-contraction coupling before and after IR injury.

A dynamic model of excitation-contraction coupling during acidosis in cardiac ventricular myocytes

Acidosis in cardiac myocytes is a major factor in the reduced inotropy that occurs in the ischemic heart. During acidosis, diastolic calcium concentration and the amplitude of the calcium transient increase, while the strength of contraction decreases. This has been attributed to the inhibition by protons of calcium uptake and release by the sarcoplasmic reticulum, to a rise of intracellular sodium caused by activation of sodium-hydrogen exchange, decreased calcium binding affinity to Troponin-C, and direct effects on the contractile machinery. The relative contributions and concerted action of these effects are, however, difficult to establish experimentally. We have developed a mathematical model to examine altered calcium-handling mechanisms during acidosis. Each of the alterations was incorporated into a dynamical model of pH regulation and excitation-contraction coupling to predict the time courses of key ionic species during acidosis, in particular intracellular pH, sodium and the calcium transient, and contraction. This modeling study suggests that the most significant effects are elevated sodium, inhibition of sodium-calcium exchange, and the direct interaction of protons with the contractile machinery; and shows how the experimental data on these contributions can be reconciled to understand the overall effects of acidosis in the beating heart.

A role for calcium/calmodulin-dependent protein kinase II in cardiac disease and arrhythmia

More than 20 years have passed since the discovery that a collection of specific calcium/calmodulin-dependent phosphorylation events is the result of a single multifunctional kinase. Since that time, we have learned a great deal about this multifunctional and ubiquitous kinase, known today as calcium/calmodulin-dependent protein kinase II (CaMKII). CaMKII is interesting not only for its widespread distribution and broad specificity but also for its biophysical properties, most notably its activation by the critical second messenger complex calcium/calmodulin and its autophosphorylating capability. A central role for CaMKII has been identified in regulating a diverse array of fundamental cellular activities. Furthermore, altered CaMKII activity profoundly impacts function in the brain and heart. Recent findings that CaMKII expression in the heart changes during hypertrophy, heart failure, myocardial ischemia, and infarction suggest that CaMKII may be a viable therapeutic target for patients suffering from common forms of heart disease.

Ischemia-reperfusion injury changes the dynamics of Ca2+-contraction coupling due to inotropic drugs in isolated hearts

Positive inotropic drugs may attenuate or exacerbate the deleterious effects of ischemia and reperfusion (IR) injury on excitation-contraction coupling in hearts. We 1) quantified the phase-space relationship between simultaneously measured myoplasmic Ca2+ concentration ([Ca2+]) and isovolumetric left ventricular pressure (LVP) using indexes of loop area, orientation, and position; and 2) quantified cooperativity by linearly modeling the phase-space relationship between [Ca2+] and rate of LVP development in intact hearts during administration of positive inotropic drugs before and after global IR injury. Unpaced, isolated guinea pig hearts were perfused at a constant pressure with Krebs-Ringer solution (37 degrees C, 1.25 mM CaCl2). [Ca2+] was measured ratiometrically by indo 1 fluorescence by using a fiber-optic probe placed at the left ventricular free wall. LVP was measured by using a saline-filled latex balloon and transducer. Drugs were infused for 2 min, 30 min before, and for 2 min, 30 min after 30-min global ischemia. IR injury worsened Ca2+-contraction coupling, as seen from decreased orientation and repositioning of the loop rightward and downward and reduced cooperativity of contraction and relaxation with or without drugs. Dobutamine (4 microM) worsened, whereas dopamine (8 microM) improved Ca2+-contraction coupling before and after IR injury. Dobutamine and dopamine improved cooperativity of contraction and relaxation after IR injury, whereas only dopamine increased cooperativity of relaxation before IR injury. Digoxin (1 microM) improved Ca2+-contraction coupling and cooperativity of contraction after but not before ischemia. Levosimendan (1 microM) did not alter Ca2+-contraction coupling or cooperativity, despite producing concomitant increases in contractility, relaxation, and Ca2+ flux before and after ischemia. Dynamic indexes based on LVP-[Ca2+] diagrams (area, shape, position) can be used to identify and measure alterations in Ca2+-contraction coupling during administration of positive inotropic drugs in isolated hearts before and after IR injury.

Regulation of cardiac energetics: role of redox state and cellular compartmentation during ischemia

The heart is capable of altering its metabolic rate during exercise or ischemia. Under most state transitions, the heart maintains the concentration of adenosine triphosphate (ATP) at relatively constant values, in spite of large fluctuations in metabolic rate or in the delivery of fuels and oxygen. However, the mechanisms responsible for the regulation of cardiac energetics under conditions of increased demand or reduced supply are still under debate. To improve quantitative understanding of the regulation of glycolysis and oxidative phosphorylation under physiological and pathological conditions, it is essential to assess the dynamics of cytosolic and mitochondrial nicotinamide adenine dinucleotide (NAD(+)) and its reduced form (NADH) during stress (e.g., ischemia, exercise). However, at present there are no reliable methods to measure the dynamics of redox state in vivo in these subcellular compartments. In the present study, computer simulations with a mathematical model of myocardial energy metabolism are used to investigate the role of cytosolic and mitochondrial redox states in regulating cardiac energetics during reduced myocardial blood flow.

Systems biology and the heart

Understanding the logic of living systems requires knowledge of the mechanisms involved at the levels at which functionality is expressed. This information does not reside in the genome, nor even in the individual proteins that genes code for. No functionality is expressed at these levels. It emerges as the result of interactions between many proteins relating to each other in multiple cascades and in interaction with the cellular environment. There is, therefore, no alternative to copying nature and computing these interactions to determine the logic of healthy and diseased states. The rapid growth in biological databases, models of cells, tissues and organs and the development of powerful computing hardware and algorithms have made it possible to explore functionality in a quantitative manner all the way from the level of genes to the physiological function of whole organs and regulatory systems. I use models of the heart to demonstrate that we can now go all the way from individual genetic information (on mutations, for example) to exploring the consequences at a whole organ level.

An in silico study of energy metabolism in cardiac excitation-contraction coupling

The heart produces and uses ATP at a high rate. Each step involved in ATP metabolism has been extensively studied. However, functional coupling between ATP production and membrane excitation-contraction coupling, which is the main ATP consumption process, is not yet fully understood because of complicated interactions and the lack of quantitative data obtained in vivo. Computer simulation is a powerful tool for integrating experimental data and for solving their complicated interactions. To investigate the mechanisms underlying cardiac excitation-contraction-energy metabolism coupling, we have developed a computer model of cardiac excitation-contraction coupling (Kyoto model) that includes the major processes of ATP production, such as oxidative phosphorylation that was originally developed for skeletal muscle by Korzeniewski and Zoladz [Biophys Chem 92: 17-34, 2001], creatine kinase, and adenylate kinase. In this review, we briefly summarize cardiac energy metabolism and discuss the regulation of mitochondrial ATP synthesis, using the Kyoto model.

Modeling regulation of cardiac KATP and L-type Ca2+ currents by ATP, ADP, and Mg2+

Changes in cytosolic free Mg(2+) and adenosine nucleotide phosphates affect cardiac excitability and contractility. To investigate how modulation by Mg(2+), ATP, and ADP of K(ATP) and L-type Ca(2+) channels influences excitation-contraction coupling, we incorporated equations for intracellular ATP and MgADP regulation of the K(ATP) current and MgATP regulation of the L-type Ca(2+) current in an ionic-metabolic model of the canine ventricular myocyte. The new model: 1), quantitatively reproduces a dose-response relationship for the effects of changes in ATP on K(ATP) current, 2), simulates effects of ADP in modulating ATP sensitivity of K(ATP) channel, 3), predicts activation of Ca(2+) current during rapid increase in MgATP, and 4), demonstrates that decreased ATP/ADP ratio with normal total Mg(2+) or increased free Mg(2+) with normal ATP and ADP activate K(ATP) current, shorten action potential, and alter ionic currents and intracellular Ca(2+) signals. The model predictions are in agreement with experimental data measured under normal and a variety of pathological conditions.

The heart is already working

Understanding the logic of living systems requires knowledge of the mechanisms involved at the levels at which functionality is expressed. This information resides neither in the genome, nor even in the individual proteins that genes code for. No functionality is expressed at these levels. It emerges as the result of interactions between many proteins relating to each other in multiple cascades and in interaction with the cellular environment. There is therefore no alternative to copying nature and computing these interactions to determine the logic of healthy and diseased states. The rapid growth in biological databases, models of cells, tissues and organs and the development of powerful computing hardware and algorithms have made it possible to explore functionality in a quantitative manner all the way from the level of genes to the physiological function of whole organs and regulatory systems. I use models of the heart to demonstrate that we can now go all the way from individual genetic information (on mutations, for example) to exploring the consequences at a whole-organ level.

Mechanistic model of cardiac energy metabolism predicts localization of glycolysis to cytosolic subdomain during ischemia

A new multidomain mathematical model of cardiac cellular metabolism was developed to simulate metabolic responses to reduced myocardial blood flow. The model is based on mass balances and reaction kinetics that describe transport and metabolic processes of 31 key chemical species in cardiac tissue. The model has three distinct domains (blood, cytosol, and mitochondria) with interdomain transport of chemical species. In addition to distinguishing between cytosol and mitochondria, the model includes a subdomain in the cytosol to account for glycolytic metabolic channeling. Myocardial ischemia was induced by a 60% reduction in coronary blood flow, and model simulations were compared with experimental data from anesthetized pigs. Simulations with a previous model without compartmentation showed a slow activation of glycogen breakdown and delayed lactate production compared with experimental results. The addition of a subdomain for glycolysis resulted in simulations showing faster rates of glycogen breakdown and lactate production that closely matched in vivo experimental data. The dynamics of redox (NADH/NAD+) and phosphorylation (ADP/ATP) states were also simulated. These controllers are coupled to energy transfer reactions and play key regulatory roles in the cytosol and mitochondria. Simulations showed a similar dynamic response of the mitochondrial redox state and the rate of pyruvate oxidation during ischemia. In contrast, the cytosolic redox state displayed a time response similar to that of lactate production. In conclusion, this novel mechanistic model effectively predicted the rapid activation of glycogen breakdown and lactate production at the onset of ischemia and supports the concept of localization of glycolysis to a subdomain of the cytosol.

Integration from proteins to organs: the IUPS Physiome Project

The IUPS Physiome Project is an internationally collaborative open source project intended to provide a public domain framework for computational physiology, including the development of modeling standards, computational tools and web-accessible databases of models of structure and function at all spatial scales and across all organ systems. Here, we illustrate the application of this multi-scale modeling approach to three organ systems: the heart, the lungs and the musculo-skeletal system, and in each case we show how the organ level models incorporate tissue and cell-level physiology. Although the computational physiology framework presented here does not yet incorporate models of ageing processes, the model-based approach is certainly capable of describing ageing and disease-related processes both via parameter changes within the models of normal physiological processes and via models of additional processes added to the framework.

Dimensionality in cardiac modelling

The development of mathematical models of the heart has been an ongoing concern for many decades. The initial focus of this work was on single cell models that incorporate varyingly detailed descriptions of the mechanisms that give rise to experimentally observed action potential shapes. Clinically relevant heart rhythm disturbances, however, are multicellular phenomena, and there have been many initiatives to develop multidimensional representations of cardiac electromechanical activity. Here, we discuss the merits of dimensionality, from 0D single cell models, to 1D cell strands, 2D planes and 3D volumes, for the simulation of normal and disturbed rhythmicity. We specifically look at models of: (i) the origin and spread of cardiac excitation from the sino-atrial node into atrial tissue, and (ii) stretch-activated channel effects on ventricular cell and tissue activity. Simulation of the spread of normal and disturbed cardiac excitation requires multicellular models. 1D architectures suffer from limitations in neighbouring tissue effects on individual cells, but they can (with some modification) be applied to the simulation of normal spread of excitation or, in ring-like structures, re-entry simulation (colliding wave fronts, tachycardia). 2D models overcome many of the limitations imposed by models of lower dimensionality, and can be applied to the study of complex co-existing re-entry patterns or even fibrillation. 3D implementations are closest to reality, as they allow investigation of scroll waves. Our results suggest that 2D models offer a good compromise between computational resources, complexity of electrophysiological models, and applicability to basic research, and that they should be considered as an important stepping-stone towards anatomically detailed simulations. This highlights the need to identify and use the most appropriate model for any given task. The notion of a single and ultimate model is as useful as the idea of a universal mechanical tool for all possible repairs and servicing requirements in daily life. The ideal model will be as simple as possible and as complex as necessary for the particular question raised.

Giving form to the function of the heart: embedding cellular models in an anatomical framework

A computational framework is presented for integrating the electrical, mechanical, and biochemical functions of the heart. The construction of efficient finite element representations of canine and porcine ventricular geometry and microstructure is outlined. Computational techniques are applied to solve large deformation soft tissue mechanics by using orthotropic constitutive laws for myocardial tissue and models of active tension generation embedded at the Gauss points in the finite element mesh. The reaction-diffusion equations governing electrical current flow in the heart are solved on a grid of deforming material points that access systems of ordinary differential equations representing the cellular processes underlying the cardiac action potential. Navier-Stokes equations are solved to predict coronary blood flow in a system of branching blood vessels embedded in the deforming myocardium.

Modeling the Heart

Models of the heart have been developed since 1960, starting with the discovery and modeling of potassium channels. The first models of calcium balance were made in the 1980s and have now reached a high degree of physiological detail. During the 1990s, these cell models were incorporated into anatomically detailed tissue and organ models.

Simulation of ATP metabolism in cardiac excitation-contraction coupling

To obtain insights into the mechanisms underlying the membrane excitation and contraction of cardiac myocytes, we developed a computer model of excitation-contraction coupling (Kyoto model: Jpn. J. Physiol. 53 (2003) 105). This model was further expanded by incorporating pivotal reactions of ATP metabolism; the model of mitochondrial oxidative phosphorylation by Korzeniewski and Zoladz (Biophys. Chem. 92 (2001) 17). The ATP-dependence of contraction, and creatine kinase and adenylate kinase were also incorporated. After minor modifications, the steady-state condition was well established for all the variables, including the membrane potential, contraction, and the ion and metabolite concentrations in sarcoplasmic reticulum, mitochondria and cytoplasm. Concentrations of major metabolites were close to the experimental data. Responses of the new model to anoxia were similar to experimental results of the P-31 NMR study in whole heart. This model serves as a prototype for developing a more comprehensive model of excitation-contraction-metabolism coupling.

Modeling the isolated cardiac myocyte

Computer modeling of cardiac myocytes has flourished in recent years. Models have evolved from mathematical descriptions of ionic channels alone to more sophisticated formulations that include calcium transport mechanisms, ATP production and metabolic pathways. The increased complexity is fueled by the new data available in the field. The continuous production of experimental data has led to the evolution of increasingly refined descriptions of the phenomena by modelers. Integrating the numerous systems involved in cardiac myocyte homeostasis makes the use of computer models necessary due to the unreliability of intuitive approaches. However the complexity of the model should not imply a cumbersome operation of the program. As with any tool, computer models have to be easy to operate or their strength will be diminished and potential users will not benefit fully from them. The contribution of the computer modeler to their respective biological fields will be more successful and enduring if modelers devote sufficient time to implement their equations into a model with user-friendly characteristics.

Parameter estimation in cardiac ionic models

We examine the problem of parameter estimation in mathematical models of excitable cell cardiac electrical activity using the well-known Beeler-Reuter (1977) ionic equations for the ventricular action potential. The estimation problem can be regarded as equivalent to the accurate reconstruction of ionic current kinetics and amplitudes in an excitable cell model, given only action potential experimental data. We show that in the Beeler-Reuter case, all ionic currents may be reasonably reconstructed using an experimental design consisting of action potential recordings perturbed by pseudo-random injection currents. The Beeler-Reuter model was parameterised into 63 parameters completely defining all membrane current amplitudes and kinetics. Total membrane current was fitted to model-generated experimental data using a 'data-clamp' protocol. The experimental data consisted of a default action-potential waveform and an optional series of perturbed waveforms generated by current injections. Local parameter identifiability was ascertained from the reciprocal condition value (1/lambda) of the Hessian at the known solution. When fitting to a single action potential waveform, the model was found to be over-determined, having a 1/lambda value of approximately 3.6e-14. This value improved slightly to approximately 1.4e-10 when an additional 2 perturbed waveforms were included in the fitting process, suggesting that the additional data did not overly improve the identifiability problem. The additional data, however, did allow the accurate reconstruction of all ionic currents. This indicates that by appropriate experimental design, it may be possible to infer the properties of underlying membrane currents from observation of transmembrane potential waveforms perturbed by pseudo-random currents.

Computational studies of the effects of myocardial blood flow reductions on cardiac metabolism

Background

A computational model of myocardial energy metabolism was used to assess the metabolic responses to normal and reduced myocardial blood flow. The goal was to examine to what extent glycolysis and lactate formation are controlled by the supply of glycolytic substrate and/or the cellular redox (NADH/NAD⁺) and phosphorylation (ATP/ADP) states.

Methods

Flow was reduced over a wide range and for a sufficient duration in order to investigate the sequence of events that occur during the transition to a new metabolic steady state.

Results

Simulation results indicated multiple time-dependent controls over both glycolysis and lactate formation.

Conclusions

Changes in phosphorylation state and glucose uptake only significantly affect the initial phase of the glycolytic response to ischemia, while glycogen breakdown exerts control over glycolysis during the entire duration of ischemia. Similarly, changes in the redox state affect the rates of lactate formation and release primarily during the initial transient phase of the response to the reductions in blood flow, while the rate of glycolysis controls the rate of lactate formation throughout the entire period of adaptation.

Model for the onset of fibrillation following coronary artery occlusion

INTRODUCTION

It is the hypothesis of this article that the onset of fibrillation following a coronary artery occlusion is a direct consequence of the spatial inhomogeneity of chemical processes that occur following the occlusion. In particular, the localized increase of extracellular potassium and decrease of ATP availability lead to an increase of resting potential in the affected cells. This difference in potential between affected cells and normal cells drives a current, the "current of injury," which may drive oscillations in the border zone, a "border zone arrhythmia." The border zone arrhythmia may drive a "breakup instability" (related to the action potential duration restitution instability) in the surrounding tissue, leading to self-sustained fibrillation.

METHODS AND RESULTS

In this article, we present a mathematical model demonstrating this transition from normal to fibrillatory dynamics, describing the general conditions under which this transition occurs and showing in a simple ionic model the way in which spatial inhomogeneity alone can initiate self-sustained reentrant activity.

CONCLUSION

Using general arguments and numerical simulations with generic models of excitable media, we have demonstrated that a spatial region with an elevated resting potential surrounded by a spatial region wherein action potentials are foreshortened can drive a breakup instability, leading to the rapid initiation of a fibrillatory state.