A quantitative analysis of cardiac myocyte relaxation: a simulation study

Modelling and measuring electromechanical coupling in the rat heart

Tension-dependent binding of Ca(2+) to troponin C in the cardiac myocyte has been shown to play an important role in the regulation of Ca(2+) and the activation of tension development. The significance of this regulatory mechanism is quantified experimentally by the quantity of Ca(2+) released following a rapid change in the muscle length. Using a computational, coupled, electromechanics cell model, we have confirmed that the tension dependence of Ca(2+) binding to troponin C, rather than cross-bridge kinetics or the rate of Ca(2+) uptake by the sarcoplasmic reticulum, determines the quantity of Ca(2+) released following a length step. This cell model has been successfully applied in a continuum model of the papillary muscle to analyse experimental data, suggesting the tension-dependent binding of Ca(2+) to troponin C as the likely pathway through which the effects of localized impaired tension generation alter the Ca(2+) transient. These experimental results are qualitatively reproduced using a three-dimensional coupled electromechanics model. Furthermore, the model predicts that changes in the Ca(2+) transient in the viable myocardium surrounding the impaired region are amplified in the absence of tension-dependent binding of Ca(2+) to troponin C.

Computational modeling of cardiac disease: potential for personalized medicine

Cardiovascular diseases are leading causes of death, reduce life quality and consume almost half a trillion dollars in healthcare expenses in the USA alone. Cardiac modeling and simulation technologies hold promise as important tools to improve cardiac care and are already in use to elucidate the fundamental mechanisms of cardiac physiology and pathophysiology. However, the emphasis has been on simulating average or exemplar cases. This report describes two classes of cardiac modeling efforts. First, electrophysiological models of channelopathies simulate the altered electrical activity that is thought to promote arrhythmias. Second, electromechanical models attempt to capture both the electrophysiological and mechanical aspects of heart function. One goal of the community is to develop models with sufficient patient customization to assist in personalized treatment planning. Some model aspects can be customized with existing data collection techniques to more closely represent individual patients while other model aspects will likely remain based on generic data. Despite important challenges, cardiac models hold promise to be important enablers of personalized medicine.

Mechanisms of transmurally varying myocyte electromechanics in an integrated computational model

The mechanical properties of myocardium vary across the transmural aspect of the left ventricular wall. Some of these functional heterogeneities may be related to differences in excitation-contraction coupling characteristics that have been observed in cells isolated from the epicardial, mid-myocardial and endocardial regions of the left ventricle of many species, including canine. Integrative models of coupled myocyte electromechanics are reviewed and used here to investigate sources of heterogeneous electromechanical behaviour in these cells. The simulations (i) illustrate a previously unrecognized role of the transient outward potassium current in mechanical function and (ii) suggest that there may also exist additional heterogeneities affecting crossbridge cycling rates in cells from different transmural regions.

Using Physiome standards to couple cellular functions for rat cardiac excitation-contraction

Scientific endeavour is reliant upon the extension and reuse of previous knowledge. The formalization of this process for computational modelling is facilitated by the use of accepted standards with which to describe and simulate models, ensuring consistency between the models and thus reducing the development and propagation of errors. CellML 1.1, an XML-based programming language, has been designed as a modelling standard which, by virtue of its import and grouping functions, facilitates model combination and reuse. Using CellML 1.1, we demonstrate the process of formalized model reuse by combining three separate models of rat cardiomyocyte function (an electrophysiology model, a model of cellular calcium dynamics and a mechanics model) which together make up the Pandit-Hinch-Niederer et al. cell model. Not only is this integrative model of rat electromechanics a useful tool for cardiac modelling but it is also an ideal framework with which to demonstrate both the power of model reuse and the challenges associated with this process. We highlight and classify a number of these issues associated with combining models and provide some suggested solutions.

Effect of bundle branch block on cardiac output: a whole heart simulation study

The heart is an electrically controlled fluid pump which operates by mechanical contraction. Whole heart modelling is a computationally daunting task which must incorporate several subsystems: mechanical, electrical, and fluidic. Numerous feedback mechanisms on many levels, and operating at different scales, exist to finely control behaviour. Understanding these interactions is necessary to understand heart operation, as well as pathologies and therapies. A review of the components in such a model is given. The authors then present a framework for their electro-mechano-fluidic whole heart model based on cable methods. The model incorporates atria and ventricles, and has functioning valves with papillary muscles. The effect of altered propagation due to left and right bundle branch block on cardiac output is examined using the cable-based model. Results are compared to clinically observed phenomena. Good agreement was obtained, but tighter coupling of mechanical and electrical events is needed to fully account for behaviour. Cable-based models offer an alternative to continuum models.

Cardiac-directed parvalbumin transgene expression in mice shows marked heart rate dependence of delayed Ca2+ buffering action

Relaxation abnormalities are prevalent in heart failure and contribute to clinical outcomes. Disruption of Ca2+ homeostasis in heart failure delays relaxation by prolonging the intracellular Ca2+ transient. We sought to speed cardiac relaxation in vivo by cardiac-directed transgene expression of parvalbumin (Parv), a cytosolic Ca2+ buffer normally expressed in fast-twitch skeletal muscle. A key feature of Parv's function resides in its Ca2+/Mg2+ binding affinities that account for delayed Ca2+ buffering in response to the intracellular Ca2+ transient. Cardiac Parv expression decreased sarcoplasmic reticulum Ca2+ content without otherwise altering intracellular Ca2+ homeostasis. At high physiological mouse heart rates in vivo, Parv modestly accelerated relaxation without affecting cardiac morphology or systolic function. Ex vivo pacing of the isolated heart revealed a marked heart rate dependence of Parv's delayed Ca2+ buffering effects on myocardial performance. As the pacing frequency was lowered (7 to 2.5 Hz), the relaxation rates increased in Parv hearts. However, as pacing rates approached the dynamic range in humans, Parv hearts demonstrated decreased contractility, consistent with Parv buffering systolic Ca2+. Mathematical modeling and in vitro studies provide the underlying mechanism responsible for the frequency-dependent fractional Ca2+ buffering action of Parv. Future studies directed toward refining the dose and frequency-response relationships of Parv in the heart or engineering novel Parv-based Ca2+ buffers with modified Mg2+ and Ca2+ affinities to limit systolic Ca2+ buffering may hold promise for the development of new therapies to remediate relaxation abnormalities in heart failure.

Energetic consequences of mechanical loads

In this brief review, we have focussed largely on the well-established, but essentially phenomenological, linear relationship between the energy expenditure of the heart (commonly assessed as the oxygen consumed per beat, oxygen consumption (VO2)) and the pressure-volume-area (PVA, the sum of pressure-volume work and a specified 'potential energy' term). We raise concerns regarding the propriety of ignoring work done during 'passive' ventricular enlargement during diastole as well as the work done against series elasticity during systole. We question the common assumption that the rate of basal metabolism is independent of ventricular volume, given the equally well-established Feng- or stretch-effect. Admittedly, each of these issues is more of conceptual than of quantitative import. We point out that the linearity of the enthalpy-PVA relation is now so well established that observed deviations from linearity are often ignored. Given that a one-dimensional equivalent of the linear VO2-PVA relation exists in papillary muscles, it seems clear that the phenomenon arises at the cellular level, rather than being a property of the intact heart. This leads us to discussion of the classes of crossbridge models that can be applied to the study of cardiac energetics. An admittedly superficial examination of the historical role played by Hooke's Law in theories of muscle contraction foreshadows deeper consideration of the thermodynamic constraints that must, in our opinion, guide the development of any mathematical model. We conclude that a satisfying understanding of the origin of the enthalpy-PVA relation awaits the development of such a model.

Approximate model of cooperative activation and crossbridge cycling in cardiac muscle using ordinary differential equations

We develop a point model of the cardiac myofilament (MF) to simulate a wide variety of experimental muscle characterizations including Force-Ca relations and twitches under isometric, isosarcometric, isotonic, and auxotonic conditions. Complex MF behaviors are difficult to model because spatial interactions cannot be directly implemented as ordinary differential equations. We therefore allow phenomenological approximations with careful consideration to the relationships with the underlying biophysical mechanisms. We describe new formulations that avoid mean-field approximations found in most existing MF models. To increase the scope and applicability of the model, we include length- and temperature-dependent effects that play important roles in MF responses. We have also included a representation of passive restoring forces to simulate isolated cell shortening protocols. Possessing both computational efficiency and the ability to simulate a wide variety of muscle responses, the MF representation is well suited for coupling to existing cardiac cell models of electrophysiology and Ca-handling mechanisms. To illustrate this suitability, the MF model is coupled to the Chicago rabbit cardiomyocyte model. The combined model generates realistic appearing action potentials, intracellular Ca transients, and cell shortening signals. The combined model also demonstrates that the feedback effects of force on Ca binding to troponin can modify the cytosolic Ca transient.

An overview of cardiac systems biology

The cardiac system has been a major target for intensive studies in the multi-scale modeling field for many years. Reproduction of the action potential and the ionic currents of single cardiomyocytes, as well as the construction of a whole organ model is well established. Still, there are major hurdles to overcome in creating a realistic and predictive functional cardiac model due to the lack of a profound understanding of the complex molecular interactions and their outcomes controlling both normal and pathological cardiophysiology. The recent advent of systems biology offers the conceptual and practical frameworks to tackle such biological complexities. This review provides an overview of major themes in the developing field of cardiac systems biology, summarizing some of the high-throughput experiments and strategies used to integrate the datasets, and various types of computational approaches used for developing useful quantitative models capable of predicting complex biological behavior.

Computational biology of cardiac myocytes: proposed standards for the physiome

Predicting information about human physiology and pathophysiology from genomic data is a compelling, but unfulfilled goal of post-genomic biology. This is the aim of the so-called Physiome Project and is, undeniably, an ambitious goal. Yet if we can exploit even a small proportion of the rich and varied experimental data currently available, significant insights into clinically important aspects of human physiology will follow. To achieve this requires the integration of data from disparate sources into a common framework. Extrapolation of available data across species, laboratory techniques and conditions requires a quantitative approach. Mathematical models allow us to integrate molecular information into cellular, tissue and organ-level, and ultimately clinically relevant scales. In this paper we argue that biophysically detailed computational modelling provides the essential tool for this process and, furthermore, that an appropriate framework for annotating, databasing and critiquing these models will be essential for the development of integrative computational biology.

Ca(2+) exchange with troponin C and cardiac muscle dynamics

Controversy abounds in the cardiac muscle literature over the rate-limiting steps of cardiac muscle contraction and relaxation. However, the idea of a single biochemical mechanism being the all-inclusive rate-limiting step for cardiac muscle contraction and relaxation may be oversimplified. There is ample evidence that Ca(2+) concentration and dynamics, intrinsic cross-bridge properties, and even troponin C (TnC) Ca(2+) binding and dissociation can all modulate the mechanical events of cardiac muscle contraction and relaxation. However, TnC has generally been thought to play no role in influencing cardiac muscle dynamics due to the idea that Ca(2+) exchange with TnC is very rapid. This definitely is the case for isolated TnC, but not for the more sophisticated biochemical systems of reconstituted thin filaments and myofibrils. This review will discuss the biochemical influences on Ca(2+) exchange with TnC and their physiological implications.

Kinetic mechanism of the Ca2+-dependent switch-on and switch-off of cardiac troponin in myofibrils

The kinetics of Ca²⁺-dependent conformational changes of human cardiac troponin (cTn) were studied on isolated cTn and within the sarcomeric environment of myofibrils. Human cTnC was selectively labeled on cysteine 84 with N-((2-(iodoacetoxy)ethyl)-N-methyl)amino-7-nitrobenz-2-oxa-1,3-diazole and reconstituted with cTnI and cTnT to the cTn complex, which was incorporated into guinea pig cardiac myofibrils. These exchanged myofibrils, or the isolated cTn, were rapidly mixed in a stopped-flow apparatus with different [Ca²⁺] or the Ca²⁺-buffer 1,2-Bis(2-aminophenoxy)ethane-N,N,N′,N′-tetraacetic acid to determine the kinetics of the switch-on or switch-off, respectively, of cTn. Activation of myofibrils with high [Ca²⁺] (pCa 4.6) induced a biphasic fluorescence increase with rate constants of >2000 s⁻¹ and ∼330 s⁻¹, respectively. At low [Ca²⁺] (pCa 6.6), the slower rate was reduced to ∼25 s⁻¹, but was still ∼50-fold higher than the rate constant of Ca²⁺-induced myofibrillar force development measured in a mechanical setup. Decreasing [Ca²⁺] from pCa 5.0–7.9 induced a fluorescence decay with a rate constant of 39 s⁻¹, which was approximately fivefold faster than force relaxation. Modeling the data indicates two sequentially coupled conformational changes of cTnC in myofibrils: 1), rapid Ca²⁺-binding (k_B ≈ 120 μM⁻¹ s⁻¹) and dissociation (k_D ≈ 550 s⁻¹); and 2), slower switch-on (k_on = 390s⁻¹) and switch-off (k_off = 36s⁻¹) kinetics. At high [Ca²⁺], ∼90% of cTnC is switched on. Both switch-on and switch-off kinetics of incorporated cTn were around fourfold faster than those of isolated cTn. In conclusion, the switch kinetics of cTn are sensitively changed by its structural integration in the sarcomere and directly rate-limit neither cardiac myofibrillar contraction nor relaxation.

An improved numerical method for strong coupling of excitation and contraction models in the heart

Quantifying the interactions between excitation and contraction is fundamental to furthering our understanding of cardiac physiology. To date simulating these effects in strongly coupled excitation and contraction tissue models has proved computationally challenging. This is in part due to the numerical methods implemented to maintain numerical stability in previous simulations, which produced computationally intensive problems. In this study, we analytically identify and quantify the velocity and length dependent sources of instability in the current established coupling method and propose a new method which addresses these issues. Specifically, we account for the length and velocity dependence of active tension within the finite deformation equations, such that the active tension is updated at each intermediate Newton iteration, within the mechanics solution step. We then demonstrate that the model is stable and converges in a three-dimensional rod under isometric contraction. Subsequently, we show that the coupling method can produce stable solutions in a cube with many of the attributes present in the heart, including asymmetrical activation, an inhomogeneous fibre field and a nonlinear constitutive law. The results show no instabilities and quantify the error introduced by discrete length updates. This validates our proposed coupling framework, demonstrating significant improvement in the stability of excitation and contraction simulations.

Cardiac transgenic and gene transfer strategies converge to support an important role for troponin I in regulating relaxation in cardiac myocytes

Elucidating the relative roles of cardiac troponin I (cTnI) and phospholamban (PLN) in beta-adrenergic-mediated hastening of cardiac relaxation has been challenging and controversial. To test the hypothesis that beta-adrenergic phosphorylation of cTnI has a prominent role in accelerating cardiac myocyte relaxation performance we used transgenic (Tg) mice bearing near complete replacement of native cTnI with a beta-adrenergic phospho-mimetic of cTnI whereby tandem serine codons 23/24 were converted to aspartic acids (cTnI S23/24D). Adult cardiac myocytes were isolated and contractility determined at physiological temperature under unloaded and loaded conditions using micro-carbon fibers. At baseline, cTnI S23/24D myocytes had significantly faster relaxation times relative to controls, and isoproterenol stimulation (Iso) had only a small effect to further speed relaxation in cTnI S23/24D myocytes (delta Iso: 7.2 ms) relative to the maximum Iso effect (31.2 ms) in control. The Ca(2+) transient decay rate was similarly accelerated by Iso in Tg and nontransgenic (Ntg) myocytes. Gene transfer of cTnI S23/24D to myocytes in primary culture showed comparable findings. Gene transfer of cTnI with both serines 23/24 converted to alanines (cTnI S23/24A), or gene transfer of slow skeletal TnI, both of which lack PKA phosphorylation sites, significantly blunted Iso-mediated enhanced relaxation compared with controls. Gene transfer of wild-type cTnI had no effect on relaxation. These findings support a key role of cTnI in myocyte relaxation and highlight a direct contribution of the myofilaments in modulating the dynamics of myocardial performance.

Soft tissue modelling of cardiac fibres for use in coupled mechano-electric simulations

The numerical solution of the coupled system of partial differential and ordinary differential equations that model the whole heart in three dimensions is a considerable computational challenge. As a consequence, it is not computationally practical--either in terms of memory or time--to repeat simulations on a finer computational mesh to ensure that convergence of the solution has been attained. In an attempt to avoid this problem while retaining mathematical rigour, we derive a one dimensional model of a cardiac fibre that takes account of elasticity properties in three structurally defined axes within the myocardial tissue. This model of a cardiac fibre is then coupled with an electrophysiological cell model and a model of cellular electromechanics to allow us to simulate the coupling of the electrical and mechanical activity of the heart. We demonstrate that currently used numerical methods for coupling electrical and mechanical activity do not work in this case, and identify appropriate numerical techniques that may be used when solving the governing equations. This allows us to perform a series of simulations that: (i) investigate the effect of some of the assumptions inherent in other models; and (ii) reproduce qualitatively some experimental observations.

A mathematical model of the slow force response to stretch in rat ventricular myocytes

We developed a model of the rat ventricular myocyte at room temperature to predict the relative effects of different mechanisms on the cause of the slow increase in force in response to a step change in muscle length. We performed simulations in the presence of stretch-dependent increases in flux through the Na(+)-H(+) exchanger (NHE) and Cl(-)-HCO(3)(-) exchanger (AE), stretch-activated channels (SAC), and the stretch-dependent nitric oxide (NO) induced increased open probability of the ryanodine receptors to estimate the capacity of each mechanism to produce the slow force response (SFR). Inclusion of stretch-dependent NHE & AE, SACs, and stretch-dependent NO effects caused an increase in tension following 15 min of stretch of 0.87%, 32%, and 0%, respectively. Comparing [Ca(2+)](i) dynamics before and after stretch in the presence of combinations of the three stretch-dependent elements, which produced significant SFR values (>20%), showed that the inclusion of stretch-dependent NO effects produced [Ca(2+)](i) transients, which were not consistent with experimental results. Further simulations showed that in the presence of SACs and the absence of stretch-dependent NHE & AE inhibition of NHE attenuated the SFR, such that reduced SFR in the presence of NHE blockers does not indicate a stretch dependence of NHE. Rather, a functioning NHE is responsible for a portion of the SFR. Based on our simulations we estimate that in rat cardiac myocytes at room temperature SACs play a significant role in producing the SFR, potentially in the presence of stretch-dependent NHE & AE and that NO effects, if any, must involve more mechanisms than just increasing the open probability of ryanodine receptors.

Mechanism of the Frank–Starling law—A simulation study with a novel cardiac muscle contraction model that includes titin and troponin I

A stretch-induced increase of active tension is one of the most important properties of the heart, known as the Frank-Starling law. Although a variation of myofilament Ca(2+) sensitivity with sarcomere length (SL) change was found to be involved, the underlying molecular mechanisms are not fully clarified. Some recent experimental studies indicate that a reduction of the lattice spacing between thin and thick filaments, through the increase of passive tension caused by the sarcomeric protein titin with an increase in SL within the physiological range, promotes formation of force-generating crossbridges (Xbs). However, the mechanism by which the Xb concentration determines the degree of cooperativity for a given SL has so far evaded experimental elucidation. In this simulation study, a novel, rather simple molecular-based cardiac contraction model, appropriate for integration into a ventricular cell model, was designed, being the first model to introduce experimental data on titin-based radial tension to account for the SL-dependent modulation of the interfilament lattice spacing and to include a conformational change of troponin I (TnI). Simulation results for the isometric twitch contraction time course, the length-tension and the force-[Ca(2+)] relationships are comparable to experimental data. A complete potential Frank-Starling mechanism was analyzed by this simulation study. The SL-dependent modulation of the myosin binding rate through titin's passive tension determines the Xb concentration which then alters the degree of positive cooperativity affecting the rate of the TnI conformation change and causing the Hill coefficient to be SL-dependent.

A computational model of cardiac electromechanics

Driven by the need for a more anatomically realistic ventricular model and with access to greater computational power the Auckland porcine ventricular model has been re-engineered based on the original anatomical recordings. The model retains an accurate representation of the ventricular geometry including the apex and valve rings and also the underlying tissue microstructure. A computational modeling and simulation framework is used to embed biophysically detailed models of cellular electromechanics in the large scale anatomical model. This enables the integrative investigation of feedback mechanisms between the structure and function at the cellular and tissue scales with the macroscopic factors that govern the beating of the heart.