Stability analysis of one-dimensional dynamical systems applied to an isolated beating heart

In this paper we propose a new model of an isolated beating heart. The model is described by a one-dimensional non-linear discrete dynamical system which depends on several parameters. Applying stability analysis we investigate the dynamic properties of the non-linear system. We find those domains in the parameter space in which the equilibrium point of the system (the fixed point) and the periodic orbits are attractors and in which they are unstable. These domains correspond to a normal and abnormal beating heart, i.e. when the end diastolic volumes are stable time invariant and time variant, respectively. On transition between these domains there is a bifurcation which gives rise to a pair of attracting points of period 2. This case corresponds to the simplest type of period doubling behavior of an abnormal beating heart, called mechanical alternans. Our results provide qualitative and quantitative predictions which can be used for comprehensive experimental design.     

Dynamic myocardial contractile parameters from left ventricular pressure-volume measurements

A new dynamic model of left ventricular (LV) pressure-volume relationships in beating heart was developed by mathematically linking chamber pressure-volume dynamics with cardiac muscle force-length dynamics. The dynamic LV model accounted for >80% of the measured variation in pressure caused by small-amplitude volume perturbation in an otherwise isovolumically beating, isolated rat heart. The dynamic LV model produced good fits to pressure responses to volume perturbations, but there existed some systematic features in the residual errors of the fits. The issue was whether these residual errors would be damaging to an application where the dynamic LV model was used with LV pressure and volume measurements to estimate myocardial contractile parameters. Good agreement among myocardial parameters responsible for response magnitude was found between those derived by geometric transformations of parameters of the dynamic LV model estimated in beating heart and those found by direct measurement in constantly activated, isolated muscle fibers. Good agreement was also found among myocardial kinetic parameters estimated in each of the two preparations. Thus the small systematic residual errors from fitting the LV model to the dynamic pressure-volume measurements do not interfere with use of the dynamic LV model to estimate contractile parameters of myocardium. Dynamic contractile behavior of cardiac muscle can now be obtained from a beating heart by judicious application of the dynamic LV model to information-rich pressure and volume signals. This provides for the first time a bridge between the dynamics of cardiac muscle function and the dynamics of heart function and allows a beating heart to be used in studies where the relevance of myofilament contractile behavior to cardiovascular system function may be investigated.     

The coronary bed and its role in the cardiovascular system: a review and an introductory single-branch model

To investigate cardiovascular haemodynamics under normal and pathological conditions, a closed-loop model of the cardiovascular system already presented in the literature¹, has been complemented by a model of the coronary bed. Oxygen available to the myocardium is strictly related to the coronary blood flow; we have developed threshold criteria which correlate cardiac output with the coronary flow. The system utilizes control systems related to the cardiac contractility and frequency, and imitates feedback mechanisms peculiar to the heart. The work exemplifies the autoregulation of events that occur when the equilibrium of the system is disturbed. It is suggested that the heart plays an active role in trying to restore the haemodynamic parameters to their physiological values.     

A kinematic approach for efficient and robust simulation of the cardiac beating motion

Computer simulation techniques for cardiac beating motions potentially have many applications and a broad audience. However, most existing methods require enormous computational costs and often show unstable behavior for extreme parameter sets, which interrupts smooth simulation study and make it difficult to apply them to interactive applications. To address this issue, we present an efficient and robust framework for simulating the cardiac beating motion. The global cardiac motion is generated by the accumulation of local myocardial fiber contractions. We compute such local-to-global deformations using a kinematic approach; we divide a heart mesh model into overlapping local regions, contract them independently according to fiber orientation, and compute a global shape that satisfies contracted shapes of all local regions as much as possible. A comparison between our method and a physics-based method showed that our method can generate motion very close to that of a physics-based simulation. Our kinematic method has high controllability; the simulated ventricle-wall-contraction speed can be easily adjusted to that of a real heart by controlling local contraction timing. We demonstrate that our method achieves a highly realistic beating motion of a whole heart in real time on a consumer-level computer. Our method provides an important step to bridge a gap between cardiac simulations and interactive applications.     

A stress-dependent pulsatile cardiovascular model as the basis for regulating artificial hearts

The development and simulation of closed-loop control of blood pumps requires a pulsatile model of the cardiovascular system that takes into account the natural mechanisms of adaptation during exercise. In this article, a model is described which takes account of the baroreceptor reflex, the change in heart rate, contractility of the heart, and the peripheral resistance during exercise, as well as venous tone, and the nonlinearity of vessel compliance. In addition, a model of the artificial heart is presented, in which the nonlinear limitation of the stroke volume is taken into account, and the concept of a pulse frequency modulated (PFM) control system for the circulation with an artificial heart is described.     

A fiber optic sensor system for control of rate-adaptive cardiac pacemakers and implantable defibrillators.

Commercially available cardiac pacemakers and implantable cardioverters/defibrillators (ICDs) predominantly use an intracardiac-derived electrocardiogram (ECG) for the detection of arrhythmias. To achieve automatic control of the heart frequency in accordance with cardiovascular strain and improved detection of life-threatening arrhythmias, it is desirable to monitor the heart by an input signal correlated with the hemodynamic state. One possible approach to derive such a signal is to measure the inotropy (mechanical contraction strength of the heart muscle). For this purpose, an optoelectronic measurement system has been designed. The fundamental function of the system has been shown in earlier investigations using an isolated beating pig heart. In this paper the design of two algorithms for use in pacemakers and ICDs based on a fiber optic sensor signal is presented.     

Models of Cheyne-Stokes respiration with cardiovascular pathologies

Cheyne-Stokes respiration (CSR) is a periodic breathing pattern, characterized by short intervals of very little or no breathing (apnea), each followed by an interval of very heavy breathing (hyperpnea). This work presents a new compartmental model of the human cardio-respiratory system, simulating the factors that determine the concentrations of carbon dioxide in the compartments of the cardiovascular system and the lungs. The parameter set on which a Hopf bifurcation gives birth to stable CSR oscillations has been determined. The model predicts that the onset of CSR oscillations may result from an increase in any of: ventilation-perfusion ratio, feedback control gain, transport delay, left heart volume, lung congestion, or cardiovascular efficiency. The model is employed to investigate the relationship between CSR and serious cardiovascular pathologies, such as congestive heart failure and encephalitis, as well as the effects of acclimatization to higher altitudes. In all cases, the model is consistent with medical observations.     

Global analysis of dynamical decision-making models through local computation around the hidden saddle

Bistable dynamical switches are frequently encountered in mathematical modeling of biological systems because binary decisions are at the core of many cellular processes. Bistable switches present two stable steady-states, each of them corresponding to a distinct decision. In response to a transient signal, the system can flip back and forth between these two stable steady-states, switching between both decisions. Understanding which parameters and states affect this switch between stable states may shed light on the mechanisms underlying the decision-making process. Yet, answering such a question involves analyzing the global dynamical (i.e., transient) behavior of a nonlinear, possibly high dimensional model. In this paper, we show how a local analysis at a particular equilibrium point of bistable systems is highly relevant to understand the global properties of the switching system. The local analysis is performed at the saddle point, an often disregarded equilibrium point of bistable models but which is shown to be a key ruler of the decision-making process. Results are illustrated on three previously published models of biological switches: two models of apoptosis, the programmed cell death and one model of long-term potentiation, a phenomenon underlying synaptic plasticity.     

A fiberoptic sensor system for cardiac monitoring and electrotherapy.

Commercially available cardiac pacemakers and implantable cardioverters/defibrillators (ICD) predominantly use the intracardiac derived electrocardiogram (ECG) for detection of arrhythmias. To achieve an automatic control of the heart frequency in accordance with cardiovascular strain and an improved detection of life-threatening arrhythmias, it is desirable to monitor the heart by an input signal correlated with the hemodynamic state. One possible approach to derive such a signal, is to measure the inotropy (mechanical contraction strength of the heart muscle). For this purpose an optoelectronic measurement system has been designed. The fundamental function of the system has been shown in earlier investigations using an isolated beating pig heart. In this paper further results showing the correlation of the fiberoptic sensor signal with the left ventricular stroke volume are presented. To make the system useful for implantable devices, further improvements with regard to power consumption and signal quality were achieved.     

Chaotic gene regulatory networks can be robust against mutations and noise

Robustness to mutations and noise has been shown to evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? We provide damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as damage propagation, do not provide much information about robustness to mutations or noise in model gene regulatory networks.     

Shaping the dynamics of a bidirectional neural interface

Progress in decoding neural signals has enabled the development of interfaces that translate cortical brain activities into commands for operating robotic arms and other devices. The electrical stimulation of sensory areas provides a means to create artificial sensory information about the state of a device. Taken together, neural activity recording and microstimulation techniques allow us to embed a portion of the central nervous system within a closed-loop system, whose behavior emerges from the combined dynamical properties of its neural and artificial components. In this study we asked if it is possible to concurrently regulate this bidirectional brain-machine interaction so as to shape a desired dynamical behavior of the combined system. To this end, we followed a well-known biological pathway. In vertebrates, the communications between brain and limb mechanics are mediated by the spinal cord, which combines brain instructions with sensory information and organizes coordinated patterns of muscle forces driving the limbs along dynamically stable trajectories. We report the creation and testing of the first neural interface that emulates this sensory-motor interaction. The interface organizes a bidirectional communication between sensory and motor areas of the brain of anaesthetized rats and an external dynamical object with programmable properties. The system includes (a) a motor interface decoding signals from a motor cortical area, and (b) a sensory interface encoding the state of the external object into electrical stimuli to a somatosensory area. The interactions between brain activities and the state of the external object generate a family of trajectories converging upon a selected equilibrium point from arbitrary starting locations. Thus, the bidirectional interface establishes the possibility to specify not only a particular movement trajectory but an entire family of motions, which includes the prescribed reactions to unexpected perturbations.     

A mathematical model of human heart including the effects of heart contractility varying with heart rate changes

Incorporating the intrinsic variability of heart contractility varying with heart rate into the mathematical model of human heart would be useful for addressing the dynamical behaviors of human cardiovascular system, but models with such features were rarely reported. This study focused on the development and evaluation of a mathematical model of the whole heart, including the effects of heart contractility varying with heart rate changes. This model was developed based on a paradigm and model presented by Ottesen and Densielsen, which was used to model ventricular contraction. A piece-wise function together with expressions for time-related parameters were constructed for modeling atrial contraction. Atrial and ventricular parts of the whole heart model were evaluated by comparing with models from literature, and then the whole heart model were assessed through coupling with a simple model of the systemic circulation system and the pulmonary circulation system. The results indicated that both atrial and ventricular parts of the whole heart model could reasonably reflect their contractility varying with heart rate changes, and the whole heart model could exhibit major features of human heart. Results of the parameters variation studies revealed the correlations between the parameters in the whole heart model and performances (including the maximum pressure and the stroke volume) of every chamber. These results would be useful for helping users to adjust parameters in special applications.     

A second-generation computational modeling of cardiac electrophysiology: response of action potential to ionic concentration changes and metabolic inhibition

Background

Cardiac arrhythmias are becoming one of the major health care problem in the world, causing numerous serious disease conditions including stroke and sudden cardiac death. Furthermore, cardiac arrhythmias are intimately related to the signaling ability of cardiac cells, and are caused by signaling defects. Consequently, modeling the electrical activity of the heart, and the complex signaling models that subtend dangerous arrhythmias such as tachycardia and fibrillation, necessitates a quantitative model of action potential (AP) propagation. Yet, many electrophysiological models, which accurately reproduce dynamical characteristic of the action potential in cells, have been introduced. However, these models are very complex and are very time consuming computationally. Consequently, a large amount of research is consecrated to design models with less computational complexity.

Results

This paper is presenting a new model for analyzing the propagation of ionic concentrations and electrical potential in space and time. In this model, the transport of ions is governed by Nernst-Planck flux equation (NP), and the electrical interaction of the species is described by a new cable equation. These set of equations form a system of coupled partial nonlinear differential equations that is solved numerically. In the first we describe the mathematical model. To realize the numerical simulation of our model, we proceed by a finite element discretization and then we choose an appropriate resolution algorithm.

Conclusions

We give numerical simulations obtained for different input scenarios in the case of suicide substrate reaction which were compared to those obtained in literature. These input scenarios have been chosen so as to provide an intuitive understanding of dynamics of the model. By accessing time and space domains, it is shown that interpreting the electrical potential of cell membrane at steady state is incorrect. This model is general and applies to ions of any charge in space and time domains. The results obtained show a complete agreement with literature findings and also with the physical interpretation of the phenomenon. Furthermore, various numerical experiments are presented to confirm the accuracy, efficiency and stability of the proposed method. In particular, we show that the scheme is second-order accurate in space.

     

Rigidity matching between cells and the extracellular matrix leads to the stabilization of cardiac conduction

Biomechanical dynamic interactions between cells and the extracellular environment dynamically regulate physiological tissue behavior in living organisms, such as that seen in tissue maintenance and remodeling. In this study, the substrate-induced modulation of synchronized beating in cultured cardiomyocyte tissue was systematically characterized on elasticity-tunable substrates to elucidate the effect of biomechanical coupling. We found that myocardial conduction is significantly promoted when the rigidity of the cell culture environment matches that of the cardiac cells (4 kiloPascals). The stability of spontaneous target wave activity and calcium transient alternans in high frequency-paced tissue were both enhanced when the cell substrate and cell tissue showed the same rigidity. By adapting a simple theoretical model, we reproduced the experimental trend on the rigidity matching for the synchronized excitation. We conclude that rigidity matching in cell-to-substrate interactions critically improves cardiomyocyte-tissue synchronization, suggesting that mechanical coupling plays an essential role in the dynamic activity of the beating heart.     

Modeling epithelial cell homeostasis: Assessing recovery and control mechanisms

Critical to epithelial cell viability is prompt and direct recovery, following a perturbation of cellular conditions. Although a number of transporters are known to be activated by changes in cell volume, cell pH, or cell membrane potential, their importance to cellular homeostasis has been difficult to establish. Moreover, the coordination among such regulated transporters to enhance recovery has received no attention in mathematical models of cellular function. In this paper, a previously developed model of proximal tubule (Weinstein, 1992, Am. J. Physiol. 263, F784–F798), has been approximated by its linearization about a reference condition. This yields a system of differential equations and auxiliary linear equations, which estimate cell volume and composition and transcellular fluxes in response to changes in bath conditions or membrane transport coefficients. Using the singular value decomposition, this system is reduced to a linear dynamical system, which is stable and reproduces the full model behavior in a useful neighborhood of the reference. Cost functions on trajectories formulated in the model variables (e.g., time for cell volume recovery) are translated into cost functions for the dynamical system. When the model is extended by the inclusion of linear dependence of membrane transport coefficients on model variables, the impact of each such controller on the recovery cost can be estimated with the solution of a Lyapunov matrix equation. Alternatively, solution of an algebraic Riccati equation provides the ensemble of controllers that constitute optimal state feedback for the dynamical system. When translated back into the physiological variables, the optimal controller contains some expected components, as well as unanticipated controllers of uncertain significance. This approach provides a means of relating cellular homeostasis to optimization of a dynamical system.     

Alterations in cellular calcium handling as a result of systemic calcium deficiency in the developing chick embryo: II. Ventricular myocytes.

We have previously shown that cardiovascular anomalies, such as hypertension and tachycardia, develop in Ca(2+)-deficient, shell-less (SL) chick embryos cultured ex ovo, accompanied by elevated circulating catecholamines and higher alpha-adrenergic sensitivity of cardiovascular functions. Results described in the preceding work, using erythrocytes as an experimental system, show that cellular Ca2+ handling properties are also altered as a result of long-term calcium deficiency. To examine the relevance of these findings to cells of the cardiovasculature, we have analyzed and compared the Ca2+ handling characteristics of the heart cells of SL and normal (NL) embryos. For this study, isolated and cultured ventricular myocytes of SL and NL embryos were loaded with Fura-2 via transient membrane damage with glass beads. Compared to Fura-2/AM, bead loading yielded similar values and kinetic profiles of [Ca2+]i-dependent differential fluorescence and, in addition, did not affect cell viability and beating activity. The Fura-2 loaded ventricular myocytes were washed in Ca(2+)-free buffer and then analyzed by ratiometric fluorescence (350 nm/380 nm) microscopy for kinetic changes in [Ca2+]i (R350/380 values) as a function of [Ca2+]o and adrenergic modifiers. At 0.5 and 1.0 mM [Ca2+]o, SL cells showed significantly higher [Ca2+]i, higher beating rates, and faster rate of increase in [Ca2+]i compared to NL cells. At higher [Ca2+]o (3.5 mM), there was no significant difference in [Ca2+]i and beating rate between NL and SL cells. Treatment with norepinephrine (NE; 0.01-1 microM) at 1 mM [Ca2+]o substantially increased [Ca2+]i in both NL and SL cells. In the former, the NE effect was completely inhibited by beta-blockade (1 microM propranolol). In contrast, in SL cells, NE remained effective after beta-blockade, and combined alpha-blockade (1 microM prazosin) and beta-blockade was needed to inhibit completely the NE effect. In both NL and SL cells, treatment with NE substantially increased beating rates in a similar manner. Taken together, these findings suggest that Ca2+ handling and adrenergic regulation of the heart cells are significantly altered in the SL embryos, and that these alterations may be related to the development of impaired cardiovascular functions resulting from systemic Ca2+ deficiency.     

Compact energy metabolism model: Brain controlled energy supply

The regulation of the energy metabolism is crucial to ensure the functionality of the entire organism. Deregulations may lead to severe pathologies such as obesity and type 2 diabetes mellitus. The decisive role of the brain as the active controller and heavy consumer in the complex whole body energy metabolism is the matter of recent research. Latest studies suggest that the brain's energy supply has the highest priority while all organs in the organism compete for the available energy resources. In our novel mathematical model, we address these new findings. We integrate energy fluxes and their control signals such as glucose fluxes, insulin signals as well as the ingestion momentum in our new dynamical system. As a novel characteristic, the hormone insulin is regarded as central feedback signal of the brain. Hereby, our model particularly contains the competition for energy between brain and body periphery. The analytical investigation of the presented dynamical system shows a stable long-term behavior of the entire energy metabolism while short time observations demonstrate the typical oscillating blood glucose variations as a consequence of food intake. Our simulation results demonstrate a realistic behavior even in situations like exercise or exhaustion, and key elements like the brain's preeminence are reflected. The presented dynamical system is a step towards a systemic understanding of the human energy metabolism and thus may shed light to defects causing diseases based on deregulations in the energy metabolism.