In modeling the dynamics of heart and circulation patient-specifically, a major problem is finding parameters on geometry, myofiber structure, tissue properties and hemodynamics. Measurement of these parameters is often cumbersome, and not realistic in clinical practice.
In heart and blood vessels, geometry and structure is known to adapt to mechanical load, mainly directed to normalize mechanical load in the constituting tissues. Therefore, in stead of performing detailed geometric measurements, we reduce the problem of finding geometric parameters by using adaptation rules, describing geometric or structural responses of tissue to changes in mechanical load.
Measurements of systolic and diastolic arterial blood pressure and echocardiographic measurements provide partial information on global hemodynamic load and size of heart, valves and blood vessels. Adaptation rules for passive elastic tissues of heart and blood vessels are used to determine size of walls and cavities. The active muscular tissue component, like myocytes and smooth muscle cells in blood vessels are 'glued' at the optimal working range within the passive matrix. Thus, adaptation rules for ventricles, atria and blood vessels are assumed to be universal, albeit parameters values are different for the different types of tissues.
Starting from the well-adapted heart and circulation, the model is fine-tuned to additional patients-specific measurements. E.g., myocardial passive stiffness and mitral valve diameter are tuned so that the duration of the early mitral flow peak matches Doppler measurements of mitral flow velocity. As a result, based solely on non-invasive measurements, complete pressure volume curves of ventricles and atria were estimated.
In conclusion, measuring global hemodynamics, adaptation rules are used to determine the most likely cardiovascular geometry and hemodynamic state. A limited amount of additional information is used to fine-tune the simulation patient-specifically. Using non-invasive techniques only, dynamic pressures and flows and wall stresses and strains can be simulated realistically.
Observations on the relationship between cardiac work rate and the levels of energy metabolites ATP, ADP, and phosphocreatine (CrP) have not been satisfactorily explained by theoretical models of cardiac energy metabolism. Specifically, the in vivo stability of ATP, ADP, and CrP levels in response to changes in work and respiratory rate have eluded explanation. Here a previously developed model of mitochondrial oxidative phosphorylation [PLoS Comp. Bio. (2005) 1:e36], which was developed based on data obtained from isolated cardiac mitochondria, is integrated with a spatially distributed model of oxygen transport in the myocardium to analyze data obtained from several laboratories over the past two decades. The model includes the components of the respiratory chain, the F1F0 ATPase, adenine nucleotide translocase, and the mitochondrial phosphate transporter at the mitochondrial level; adenylate kinase, creatine kinase, and ATP consumption in the cytoplasm; and oxygen transport between capillaries, interstitial fluid, and cardiomyocytes. The integrated model is able to reproduce experimental observations on ATP, ADP, CrP, and inorganic phosphate (Pi) levels in canine hearts over a range of workload and during coronary hypoperfusion, and predicts that substrate concentration is a key regulator of the rate of mitochondrial respiration in the heart. Integrated modeling explains metabolite levels observed at low to high work loads and the changes in metabolite levels and tissue oxygenation observed during graded hypoperfusion. These findings suggest that the observed stability of energy metabolites emerges as a property of a properly constructed model of cardiac substrate transport and mitochondrial metabolism. In addition, the validated model provides quantitative predictions of changes in phosphate metabolites during cardiac ischemia.
The Frank-Starling law of the heart describes the interrelationship between end-diastolic volume and cardiac ejection volume, a regulatory system that operates on a beat-to-beat basis. Although the Frank-Starling Law of the Heart constitutes a fundamental property of the heart that has been appreciated for well over a century, the molecular mechanisms that underlie this phenomenon are still incompletely understood. Our research is aimed to enhance our understanding of this important physiological process that controls cardiac performance on a beat-to-beat basis. At the cellular level, sarcomere length dependent myofilament Ca2+ sensitivity underlies this phenomenon (myofilament length dependent activation). How the contractile apparatus transduces the molecular strain information concerning sarcomere length is not known. The overall goal of our research is to elucidate the molecular mechanisms that underlie myofilament length dependent activation. Our work is focused on several molecular aspects of this phenomenon. First, we have found that inter-filament spacing, as measured by small angle x-ray diffraction in a relaxed muscle, does not underlie myofilament length dependent activation. Current experiments are ongoing to test whether such is also the case in activated muscle, that is, whether inter-filament spacing attained in an active muscle underlies myofilament length dependent activation. Second, we have found that troponin, and particularly cardiac troponin-I, plays a pivotal role in myofilament length dependent activation myofilament. In addition, we will present data that suggest that cardiac troponin is sufficient to impart myofilament length dependent activation properties upon striated muscle as demonstrated by experiments in which recombinant troponin subunits are introduced into skinned muscle preparations. We have obtained preliminary data that suggest a pivotal role for the inhibitory domain within cardiac Tn-I to induce myofilament length dependent activation. In addition, cooperative activation mechanisms appear crucial as well. Third, we have obtained preliminary data that suggest that stretch or osmotic compression a muscle causes a radial shift of cross-bridges toward actin, a phenomenon that could underlie myofilament length dependent activation. Small angle x-ray diffraction experiments are used to test this hypothesis, that is, whether radial movement of cross-bridge heads upon sarcomere length stretch is required for myofilament length dependent activation. Finally, detailed stochastic computer modeling is employed to test whether loosely coupled activation schemes can recapitulate the observed phenomenon of myofilament length dependent activation.
The Cardiac Biomechanics Laboratory at the San Francisco VA Medical Center (www.cardiacsurgery.ucsf.edu/cbl) uses a combination of mathematical (finite element) modeling and animal experiments to determine the effect of cardiac surgery on left ventricular function and remodeling. The goal of the laboratory is to design optimal operations for heart failure. To date, we have developed finite element models to simulate the mechanical effects of several different novel surgical procedures and devices for the treatment of ischemic heart disease and heart failure, including the Batista procedure or partial left ventriculectomy, the Myocor Myosplint, Surgical Anterior Ventricular Endocardial Restoration or SAVER, and matrix-assisted myocardial stabilization. Our finite element methodology and simulation results will be presented.
Stem cell therapy and tissue engineering will soon offer unprecedented power to remodel or replace infarcted myocardium. However, past interventions in the infarct healing process have often produced unexpected results. In some cases, such as angiotensin converting enzyme (ACE) inhibitors, the benefits were much greater than originally expected; in others, notably the post-infarction administration of steroids, the results were disastrous. In view of these past experiences, we believe it is essential that new interventions be grounded in an understanding of the structural determinants of the mechanical properties of infarcted myocardium at various stages during healing and of the effect of infarct mechanics on global ventricular function.
Over the past 10 years, we have studied infarct mechanics through a combination of in vivo, in vitro, and computational studies. We have shown that healing myocardial scar tissue is highly anisotropic, and that this anisotropy appears to preserve function of the left ventricle. We have formulated structural constitutive models for mature scar tissue composed primarily of collagen. We have used computational models to predict how the presence of an infarct affects endocardial wall motion, an important parameter for clinical diagnosis. However, many challenges remain. The greatest risk of infarct rupture occurs during the first week after infarction, when damaged muscle is undergoing necrosis but little new collagen has been deposited; the determinants of infarct mechanics and rupture during this critical time period are poorly understood. Late in healing, infarct mechanical properties do not correlate with collagen content, suggesting that other factors such as crosslinking must play a role. In order to better understand the structural basis for scar anisotropy, we have developed an in vitro model of myocardial scar tissue. Tissue-engineered analogs will allow us to explore the structural basis for the mechanical properties of healing myocardial infarcts across the full time course of healing and to apply this knowledge to guide the development of new therapeutic and diagnostic techniques.
A change in heart rate is the primary response mechanism to changes in the bodily demand for blood. Cardiac output is the total amount of blood that is pumped by the heart per minute, and is calculated as the product of heart rate and amount of blood ejected per heart beat (stroke volume). Increases in stroke volume are also linked to heart rate; as heart rate goes up in healthy subjects, the force of contraction of the cardiac myocytes increases (force-frequency relationship, or FFR). Not only does force of contraction increase, but the kinetics do as well, resulting in faster contractions, and accelerated relaxation rates at higher heart rates (frequency dependent acceleration of relaxation, or FDAR). This phenomenon of frequency-dependent changes in cardiac function are found in all mammals, including small rodents. The magnitude of the cardiac response to changes in frequency is however heavily dependent on experimental conditions, such as frequency range, species, and temperature. The underlying mechanism of FFR and FDAR is incompletely understood, but changes in the calcium handling of the myocyte have reported to play a major role in FFR and FDAR. With an increase in heart rate, the intracellular calcium transient increases in amplitude, and speeds up. Recent experiments from our lab have now shown that besides significant changes in calcium handling, changes in myofilament responsiveness to the available calcium may be significantly impacted by the stimulation frequency. We were able to assess calibrated calcium transients from isolated rabbit cardiac muscles, and have developed a novel protocol to assess myofilament responsiveness in intact contracting muscle preparations at physiological temperature. Although the molecular mechanism needs to be further unraveled, we describe for the first time the significant impact of myofilament matrix calcium responsiveness on frequency-dependent contractile activation and relaxation.
A tight coupling between ionic currents, intracellular Ca2+ homeostasis and metabolic fluxes underlies the regulation of cardiac muscle function. Various alterations in the proteins and pathways involved in these coupled processes are now recognized to be primary mechanisms of cardiac dysfunction in a diverse range of common pathologies including cardiac failure during ischemia. As more experimental detail on the biochemistry and biophysics of these complex processes and their interactions accumulates, the intuitive interpretation of the new findings becomes increasingly impractical. The overall goal of our research is to develop biophysically and biochemically detailed computational models that integrate Ca2+ signaling and cell electrophysiology with the main interactions between phosphorylated species (ATP, ADP, AMP, PCr, Cr, Pi) and Lewis cytosolic acids (Na+, K+, Mg2+, H+). Together with experimental measurements, we are using these computational models to test hypotheses regarding excitation-metabolic coupling in ventricular and atrial myocytes under normal and pathological conditions (Michailova and McCulloch, Biophysical Journal 2001, 81: 614-629; Michailova et al., Biophysical Journal 2002, 83: 3134-3151; Michailova et al., Biophysical Journal 2005, 88: 2234-2249; Michailova et al., AJP-Cell Physiology 2006, under revision). In my presentation I will discuss how Ca2+ and Mg2+ buffering and diffusion by ATP and ADP and Mg2+-nucleotides regulate ion channel and pump kinetics and Ca2+ transient during cell excitation, and how the changes in ATP, ADP, AMP, PCr, Cr, Pi and ionic diastolic levels modulate cardiac excitation-metabolic coupling during 20 min of ischemia.
Supported by National Biomedical Computational Resource (NIH Grant P41 RR08605) and Swiss National Science Foundation with grants to A. Michailova and E. Niggli (7BUPJ048575 and 7BUPJ041424).
Cardiac mechanotransduction is the process by which cells of heart covert mechanical signals to chemical signals responsible for cellular function, adaptation and remodeling. When this system is not working properly or cannot meet the demands of increased loading conditions, the cellular response will be abnormal or inadequate, and in many cases the pumping function of the heart will eventually fail. Mechanical signaling and force transmission within and outside the myocyte are important players in the mechanotransduction process, although the cellular components responsible for mechanosensing and the pathways involved in the overall process are not well understood. Mechanical stresses transmitted from the extracellular matrix into the myocytes and from cell to cell are dependent on membrane connections such as integrins and directionality of the external loading patterns. Defects in cytoskeletal components within the myocyte have also been linked to cardiac dilation and failure. The structural and signaling role of these proteins is being investigated, as well as novel loading mechanisms involved with mechanotransduction such a fluid-induced shear stress.
Mitochondrial Ca2+ ([Ca2+]m) plays a key role in the regulation of oxidative phosphorylation and thus contributes to energy supply and demand matching in the heart. Mitochondria take up Ca2+ via the Ca2+-uniporter (MCU), and extrude it through the mitochondrial Na+/Ca2+-exchanger (mNCE). Using a double dye loading method to track mitochondrial and cytosolic Ca2+ ([Ca2+]c) signals in adult cardiac myocytes, we have recently obtained evidence that mitochondria take up Ca2+ rapidly during excitation-contraction coupling but release it slowly, resulting in frequency- and amplitude-dependent modulation of matrix Ca2+. Raising [Na+]i from 5 to 15 mM, which simulates a transition between physiological and pathophysiological [Na+]i, accelerates mitochondrial Ca2+-decay and decreases [Ca2+]m, but enhances cytosolic Ca2+ transients. In response to gradual or abrupt changes of workload, NADH levels are maintained at 5 mM [Na+]i, but at 15 mM, the NADH pool is partially oxidized. The results indicate that mitochondria take up Ca2+ rapidly and contribute to fast buffering during a [Ca2+]c transient and that elevated [Na+]i impairs mitochondrial Ca2+ uptake, with consequent effects on energy supply and demand matching. The results can be simulated using an integrated computational model assuming that mitochondria respond to large Ca2+ spikes in a microdomain near the dyad. The findings are proposed to be relevant to cardiac functional decompensation during heart failure.
In the foreseeable future, tissue and organ level simulations will require parsimonious and computationally efficient representations of the myofilament (MF) system. The most likely implementation will be as sets of ordinary differential equations (ODEs) that compress spatially-dependent interactions via phenomenological or approximate approaches. For example, spatially dependent cooperative interactions and sarcomere-level inhomogeneity can not be directly represented with ODE-based approaches. Here we propose an ODE-based model of the MF system that approximates the behavior of spatially explicit models. Our model differs from others by avoiding a mean-field approximation that produces unphysiological Force-Ca (F-Ca) relations. The mean-field approximation tries to represent the entire ensemble of regulatory proteins by a single number (i.e., the mean activation) that ignores any spatial variability. Briefly, our approach assumes thin filament activation is a steeply nonlinear function of [Ca] to represent phenomenologically the effects of nearest-neighbor interactions along the thin filament. Spatially-explicit modeling of this process supports this assumption. Moreover a novel feature of the model is that Ca binding to troponin is decomposed into "regulatory Ca binding" that activates the thin filament and "apparent Ca binding" that is sensed by the cell. In the real system these binding quantities are equivalent, but here they are separated to avoid the deleterious effects of the mean-field approximation. We have also developed several new approaches to represent the strain-dependent crossbridge state transition rates so that active shortening and lengthening resemble experimental results in force-velocity and work loop protocols. Our MF model can recapitulate experimentally measured F-Ca relations, isometric twitches, work loops, and cell shortening. Additionally the model is coupled to the Chicago rabbit cardiomyocyte model to illustrate the compatibility with existing ODE-based cardiac models and mechanical feedback effects on Ca buffering.
On the most basic functional level, the aortic heart valve is essentially a check-valve that serves to prevent retrograde blood flow from the aorta back into the left ventricle. This seemingly simple function belies the structural complexity, elegant solid-fluid mechanical interaction, and durability necessary for normal aortic valve function. For example, the aortic valve is capable of withstanding 30-40 million cycles per year, resulting in a total of ~3 billion cycles in single lifetime. No valve made from non-living materials has been able to demonstrate comparable functional performance and durability. However, this staggering level of performance can be cut short by aortic valve disease, the most common form being stenosis resulting from calcification. Currently, the treatment of aortic valve disease is usually complete valve replacement. First performed successfully in 1960, surgical replacement of diseased human heart valves by valve prostheses is now commonplace and enhances survival and quality of life for many patients. However, they continue to have significant clinical problems and there is a profound need for new approaches to valve therapies based on sound scientific and engineering principals. The focus of this talk is to present our work on the biomechanical behavior of native and engineered heart valve tissues. In particular, for engineered heart valve tissues many challenges exist to understand the intricate microstructure and the concomitant complexity of mechanical interactions occurring between scaffold, cellular, and extracellular matrix constituents. Mathematical models that simulate the composite mechanical behavior of the scaffold and the developing tissue could potentially facilitate the design of engineered tissues and mechanical conditioning regimens. Such models could thus play a pivotal role in the design and development of an engineered heart valve.
The beta-adrenergic signaling network provides a dominant role in modulation of heart rate and contractility, and also plays a major role in the progression of most cardiac diseases. However, compared with cardiac electrophysiology and mechanics, a quantitative understanding of how molecular components contribute to signaling network function has been lacking. By integrating mechanistic computational models of the beta-adrenergic signaling network with quantitative experiments ranging from live-cell fluorescence to MRI imaging, we have begun to characterize the basis for complex signaling behaviors such as compartmentation, dynamics, and the coordination of the heart's response to neurohumoral stimuli.
Control of cardiac contractile dynamics occurs not only through alterations in cellular Ca-fluxes, but also through alterations in sarcomeric response to Ca2+ (Kobayashi and Solaro, Annu Rev Physiol. 2005; 67:39-67). An important feature of the regulation at the level of the sarcomeres is an influence of the actin-cross-bridge reaction on the state of activation of the thin filament. Sarcomeric response to Ca2+ is also modified by sarcomere length, protein phosphorylation, and alterations in the population of protein isoforms associated with development or linked to familial cardio-myopathies. In the case of protein phosphorylation, there is compelling evidence that cross-bridge kinetics may be modified by post-translational modifications of thin filament proteins. Thus, regulation at the level of the cardiac sarcomeres is prominent both in intrinsic mechanisms by Starling's law and by extrinsic neuro-humoral mechanisms. In spite of the evidence supporting the hypothesis that control of the heart beat can occur via these changes at the levels of the sarcomeres, mathematical models of cellular regulation of cardiac dynamics have generally ignored this mechanism. An understanding of the specific role of the sarcomeres in regulating cardiac dynamics independent of Ca2+ fluxes is important to our understanding of heart disease and the design on new therapeutic approaches (Kass and Solaro, Circulation, 2006; 113:305-315).
Cardiovascular growth, remodeling, and morphogenesis depend on a dynamic interaction of genetic, molecular, and mechanical factors. Here, we explore the role that mechanics plays in development of the heart and functional adaptation of arteries. We show how linking mathematical modeling to quantitative experimental measurements can provide new insight into these problems. Our models are based on the fundamental principles of continuum mechanics.
The heart is the first functioning organ in the embryo. Cardiac looping is a crucial early morphogenetic process that transforms the initially straight heart tube into a curved tube, creating the basic pattern of the four-chambered heart. Using computational modeling and experiments with chick embryos, we study the initial stages of looping (c-looping), as the heart bends ventrally and twists rightward into a c-shaped tube. A key element of our work is comparing numerical and experimental results for various chemical and mechanical perturbations of looping. The results suggest that bending is driven primarily by actin polymerization within the heart tube, while rotation is driven by external loads exerted by a membrane (splanchnopleure) on the ventral surface of the heart tube and by structures (omphalomesenteric veins) at the caudal end of the tube.
It is well known that arteries grow and remodel in response to changes in blood pressure and flow. In response to acute changes in blood flow rate, many arteries actively contract or dilate to maintain a constant fluid shear stress on the endothelium. Due to a chronic increase in blood pressure, the artery wall thickens, presumably to restore homeostatic wall stress. Long-term changes in arterial structure are achieved primarily through smooth muscle growth and collagen remodeling. We model an artery as a thick-walled tube composed of a mixture of smooth muscle cells, elastin, and collagen, with material properties and spatial distributions of each constituent prescribed according to published data. The analysis includes stress-dependent growth and contractility of the muscle and turnover of collagen fibers. Numerical pressure-radius relations and opening angles (residual stress) show reasonable agreement with published experimental results. In particular, for realistic material and structural properties, the model predicts measured variations in opening angles along the length of the aorta with reasonable accuracy.
Evidence is mounting that Ca2+ leak from the sarcoplasmic reticulum is a major cause of myocardial dysfunction in CHF. The relation between this Ca2+ leak and reduced cardiac force development in Congestive Heart Failure (CHF) is still uncertain. We explore here the subcellular mechanisms leading to decreased force development in trabeculae from Rats with a myocardial infarction. We defined CHF according to clinical and pathological criteria, and compared properties of trabeculae from animals with CHF (cMI) to those of animals with a myocardial scar but without evidence of CHF (uMI), and Sham operated animals.
New findings of this study on properties of cMI trabeculae are that: I) Maximal twitch force following post extra systolic potentiation is unchanged; II) The sensitivity of cMI trabeculae to [Ca2+]o is increased; III) Spontaneous diastolic sarcomere length (SL) fluctuations (SA) are increased in cMI at all levels of SR Ca2+ loading; IV) SA is accompanied by a proportional reduction of Fmax . V) SA may lead to triggered propagated contractions accompanied by arrhythmogenic DADs. The results suggest that the probability of spontaneous diastolic opening of SR Ca2+ channels is increased in CHF. Conclusion: These data provide the basis for a novel mechanism underlying systolic and diastolic dysfunction as well as arrhythmias in hearts in CHF. If SA would be a component of myocardial dysfunction in human CHF, it may profoundly change our thinking about therapy of the patient with CHF.
Work done in collaboration with: A.W. Davidoff, M. Obayashi, B.D.Y. Stuyvers, P.A. BoydenB, Y.M. Zhang, j. Mei, M.L. Zhang, X.C. Wang, and D. Crabbe.
Magnetic Resonance Imaging (MRI) is unique in its ability to provide information on structure, myocardial infarction, and regional tissue deformation. This talk will review methods and application for models derived from MRI data. Firstly, fast and accurate evaluation of mass and volume is enabled by interactive 4D modelling techniques. Analysis of tissue function, strain and strain rate, is can be performed by analysis of displacement encoded images. Mathematical models of regional tissue function and wall motion allow registration between cases and across groups, enabling quantification of multidimensional differences between disease and treatment groups. Combined with late gadolinium enhancement imaging, 3D strain information can be fused with data on 3D infarct geometry. Finally, information on myocardial tissue kinematics can be incorporated into biophysical models of cardiac mechanics, and used to gain an understanding of how physiological tissue parameters such as contractility, ventricular compliance and electrical activation combine to effect whole heart function.