Dynamic properties of cardiovascular systems.

A recently derived mathematical model of an isolated heart is extended here to a closed-loop cardiovascular system. Taking the end-diastolic volume as state variable, the authors show that the closed-loop cardiovascular system can be described by a one-dimensional nonlinear discrete dynamical system that depends on parameters describing the systolic and diastolic properties of the heart, heart rate, total peripheral resistance, and arterial capacitance. Studies of this model show that the system possesses a rich spectrum of dynamical behavior, from stable points through stable cycles to a "chaotic" behavior. It is shown that such an analysis of dynamic behavior yields those domains in the parameter space that correspond to a normal and abnormal beating heart, when the heart ejects time-invariant and time-variant (periodic or aperiodic) stable stroke volumes, respectively. Determination of such domains may lead to better understanding of the specific pathologic mechanism involved in the evolution of an abnormal beating heart.     

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.     

Controllability of non-linear biochemical systems

Mathematical methods of biochemical pathway analysis are rapidly maturing to a point where it is possible to provide objective rationale for the natural design of metabolic systems and where it is becoming feasible to manipulate these systems based on model predictions, for instance, with the goal of optimizing the yield of a desired microbial product. So far, theory-based metabolic optimization techniques have mostly been applied to steady-state conditions or the minimization of transition time, using either linear stoichiometric models or fully kinetic models within biochemical systems theory (BST). This article addresses the related problem of controllability, where the task is to steer a non-linear biochemical system, within a given time period, from an initial state to some target state, which may or may not be a steady state. For this purpose, BST models in S-system form are transformed into affine non-linear control systems, which are subjected to an exact feedback linearization that permits controllability through independent variables. The method is exemplified with a small glycolytic-glycogenolytic pathway that had been analyzed previously by several other authors in different contexts.     

Improved methods for automatic monitoring of contracting heart cells in culture

The present communication describes improved methods for isolating and plating beating heart cells from neonatal rats using collagenase and collagen-coated petri dishes. The amplitude and frequency of contraction are continuously and simultaneously measured under well defined conditions and during prolonged periods of time with a highly sensitive and thermostated instrument. Additions, e.g. drugs and toxic agents, are made through an accessory pump system that involves extensive dilution of the added compound with medium; aliquots of medium can be withdrawn for estimation of metabolites. The system described is reliable and relatively inexpensive and allows a more extensive use of isolated heart cells, especially in studies of heart functions where small changes in amplitude and frequency of beating during prolonged periods of time are important.     

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 discrete model with density dependent fast migration

The aim of this work is to develop an approximate aggregation method for certain non-linear discrete models. Approximate aggregation consists in describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete models with two different time scales, the slow one considered to be linear and the fast one non-linear because of its transition matrix depends on the global variables. In our discrete model the time unit is chosen to be the one associated to the slow dynamics, and then we approximate the effect of fast dynamics by using a sufficiently large power of its corresponding transition matrix. In a previous work the same system is treated in the case of fast dynamics considered to be linear, conservative in the global variables and inducing a stable frequency distribution of the state variables. A similar non-linear model has also been studied which uses as time unit the one associated to the fast dynamics and has the non-linearity in the slow part of the system. In the present work we transform the system to make the global variables explicit, and we justify the quick derivation of the aggregated system. The local asymptotic behaviour of the aggregated system entails that of the general system under certain conditions, for instance, if the aggregated system has a stable hyperbolic fixed point then the general system has one too. The method is applied to aggregate a multiregional Leslie model with density dependent migration rates.     

A generating model of realistic synthetic heart sounds for performance assessment of phonocardiogram processing algorithms

A new model which is capable of generating realistic synthetic phonocardiogram (PCG) signals is introduced based on three coupled ordinary differential equations. The new PCG model takes into account the respiratory frequency, the heart rate variability and the time splitting of first and second heart sounds. This time splitting occurs with each cardiac cycle and varies with inhalation and exhalation. Clinical PCG statistics and the close temporal relationship between events in ECG and PCG are used to deduce values of PCG model parameters.In comparison with published PCG models, the proposed model allows a larger number of known PCG features to be taken into consideration. Moreover it is able to generate both normal and abnormal realistic synthetic heart sounds. Results show that these synthetic PCG signals have the closest features to those of a conventional heart sound in both time and frequency domains. Additionally, a sound quality test carried out by eight cardiologists demonstrates that the proposed model outperforms the existing models.This new PCG model is promising and useful in assessing signal processing techniques which are developed to help clinical diagnosis based on PCG.     

The topology of phase response curves induced by single and paired stimuli in spontaneously oscillating chick heart cell aggregates.

The topological properties of the phase resetting of biological oscillators by an isolated stimulus delivered at various phases of the cycle depend on whether the stimulus is "weak" or "strong." When multiple stimuli are delivered to the oscillator, the response to stimulation also depends on the time between the stimuli, and the rate at which the oscillator returns to an underlying limit cycle attractor. If the time between two consecutive "weak" stimuli is sufficiently short, the effects produced by the pair of stimuli may be characteristic of a single "strong" stimulus. These results are demonstrated in a model experimental system, spontaneously beating aggregates of cells derived from embryonic chick heart, and are illustrated by consideration of a simple theoretical model of nonlinear oscillators, the Poincaré oscillator.     

Non-linear PCA: a missing data approach

MOTIVATION: Visualizing and analysing the potential non-linear structure of a dataset is becoming an important task in molecular biology. This is even more challenging when the data have missing values. RESULTS: Here, we propose an inverse model that performs non-linear principal component analysis (NLPCA) from incomplete datasets. Missing values are ignored while optimizing the model, but can be estimated afterwards. Results are shown for both artificial and experimental datasets. In contrast to linear methods, non-linear methods were able to give better missing value estimations for non-linear structured data.Application: We applied this technique to a time course of metabolite data from a cold stress experiment on the model plant Arabidopsis thaliana, and could approximate the mapping function from any time point to the metabolite responses. Thus, the inverse NLPCA provides greatly improved information for better understanding the complex response to cold stress. CONTACT: scholz@mpimp-golm.mpg.de.     

Emergence of Asynchronous Local Clocks in Excitable Media

Excitable media such as the myocardium or the brain consist of arrays of coupled excitable elements, in which the local excitation of a single element can propagate to its neighbors in the form of a non-linear autowave. Since each element has to pass through a refractory period immediately after excitation, the frequency of autowaves is self-limiting. In this work, we consider the case where each element is spontaneously excited at a fixed average rate and thereby initiates a new autowave. Although these spontaneous self-excitation events are modelled as independent Poisson point processes with exponentially distributed waiting times, the travelling autowaves lead collectively to a non-exponential, unimodal waiting time distribution for the individual elements. With increasing system size, a global ‘clock’ period T emerges as the most probable waiting time for each element, which fluctuates around T with an increasingly small but non-zero variance. This apparent synchronization between asynchronous, temporally uncorrelated point processes differs from synchronization effects between perfect oscillators interacting in a phase-aligning manner. Finally, we demonstrate that asynchronous local clocks also emerge in non-homogeneous systems in which the rates of self-excitation are different for all individuals, suggesting that this novel mechanism can occur in a wide range of excitable media.     

Augmentation of the expression of c-myc and c-fos oncogenes as a function of the mechanical activity of the isolated adult rat heart

c-myc and c-fos oncogenes encode nuclear DNA binding proteins, and are involved in both growth regulation and differentiation. Using the molecular hybridization technique and DNA probes complementary to c-myc and c-fos mRNA, we report an increase in c-myc and c-fos expression level in the isolated beating adult rat heart with reference to the arrested isolated heart. This suggests a causal relationship between mechanical activity of the heart and c-myc and c-fos expression. It evidences for the first time a messenger between mechanical factor and adaptational changes in the phenotype which occurs at the beginning of cardiac hypertrophy.     

Towards Active Tracking of Beating Heart Motion in the Presence of Arrhythmia for Robotic Assisted Beating Heart Surgery

In robotic assisted beating heart surgery, the control architecture for heart motion tracking has stringent requirements in terms of bandwidth of the motion that needs to be tracked. In order to achieve sufficient tracking accuracy, feed-forward control algorithms, which rely on estimations of upcoming heart motion, have been proposed in the literature. However, performance of these feed-forward motion control algorithms under heart rhythm variations is an important concern. In their past work, the authors have demonstrated the effectiveness of a receding horizon model predictive control-based algorithm, which used generalized adaptive predictors, under constant and slowly varying heart rate conditions. This paper extends these studies to the case when the heart motion statistics change abruptly and significantly, such as during arrhythmias. A feasibility study is carried out to assess the motion tracking capabilities of the adaptive algorithms in the occurrence of arrhythmia during beating heart surgery. Specifically, the tracking performance of the algorithms is evaluated on prerecorded motion data, which is collected in vivo and includes heart rhythm irregularities. The algorithms are tested using both simulations and bench experiments on a three degree-of-freedom robotic test bed. They are also compared with a position-plus-derivative controller as well as a receding horizon model predictive controller that employs an extended Kalman filter algorithm for predicting future heart motion.     

A note on estimation of dynamics of multiple gene expression based on singular value decomposition

Recently, data on multiple gene expression at sequential time points were analyzed, using singular value decomposition (SVD) as a means to capture dominant trends, called characteristic modes, followed by fitting of a linear discrete-time dynamical system in which the expression values at a given time point are linear combinations of the values at a previous time point. We attempt to address several aspects of the method. To obtain the model we formulate a non-linear optimization problem and present how to solve it numerically using standard MATLAB procedures. We use publicly available data to test the approach. For reader's convenience, we provide a straightforward, ready-to-use, procedure in MATLAB, which employs its standard features to analyze data of this kind. Then, we investigate the sensitivity of the method to missing measurements and its possibilities to reconstruct missing data. Also, we discuss the possible consequences of data regularization, called sometimes 'polishing', on the outcome of analysis, especially when model is to be used for prediction purposes. Summarizing we point out that approximation of multiple gene expression data preceded by SVD provides some insight into the dynamics but may also lead to unexpected difficulties, like overfitting problems.     

Mathematical model of heart rate variability

The study presents a mathematical model of non-linear dynamics of the heart rate variability (HRV). The model is based on quantitative characteristics of pulse conduction in the heart conducting system: the delays of sinoatrial (SA) and atrioventricular (AV) pulse conduction and refractors periods of the SA and AV nodes. The model predicts heart rate disturbances in fast electric activity of the atria, increase in the delay of the AV conduction, the critical value of atrial period where transition to non-linear dynamics of the heart rate variability starts. The correlation between indexes of HRV and period of stimulation of atria for 1-contour cardiac control model has been demonstrated.     

Heart and scaphognathite beat behaviour in laboratory-held Crangon crangon (L.)

Heart and scaphognathite beating activities of Crangon crangon (L.) have been monitored for several days under a fixed photoperiod regime. Freshly-captured animals, in particular, spend much of the light period buried in the substratum, and these periods are characterized by low heart rates and high scaphognathite beating rates. During the dark period, animals emerge from the sand. Swimming and walking excursions are most common during the first hours of the dark period, and such times are characterized by high heart rates and higher scaphognathite rates (both compared with daytime, buried rates). Periods of low activity, with the animals resting on the surface of the sand, extend over much of the remainder of the dark period, and at these times the animals had high heart rates but scaphognathite rates lower than those of buried animals. The increase in scaphognathite rates associated with the buried condition may be shown to be due to the gill ventilation system adopted by buried animals.     

Model of the sino-atrial and atrio-ventricular nodes of the conduction system of the human heart.

The coupled sino-atrial and atrio-ventricular nodes of the heart are discussed using a dedicated non-linear oscillator model. Several modes by which the oscillations cease in the system are obtained (asystole models). The oscillations of the model are compared with heart rate variability in heart block patients.     

Cell culture of bivalves: tool for the study of the effects of environmental stressors.

Spontaneous beating cells can be isolated from the heart of the oyster (Crassostrea gigas) and cultured for more than two months. They form adherent contractile networks in culture conditions. They show muscarinic and beta-adrenergic reactivity thus showing that they are functional cardio-myocytes: Acetylcholine induced a dose dependent decrease in spontaneous beating rate via an increase in potassium conductance, this effect being blocked by atropine. Epinephrine induced a dramatic increase in calcium conductance which was blocked by high concentrations of propranolol but not by sotalol and reversed by verapamil. Tributyltin and cadmium induced a dose and time dependent decrease mainly in inward ionic conductances, leading to a decrease or even a total suppression of the beating rate. Present study indicates that this model could be used as a sensitive test to study the effects of some marine pollutants at the cellular level in molluscs.     

Segmentation of heart sounds based on dynamic clustering

The heart sound signal is first separated into cycles, where the cycle detection is based on an instantaneous cycle frequency. The heart sound data of one cardiac cycle can be decomposed into a number of atoms characterized by timing delay, frequency, amplitude, time width and phase. To segment heart sounds, we made a hypothesis that the atoms of a heart sound congregate as a cluster in time–frequency domains. We propose an atom density function to indicate clusters. To suppress clusters of murmurs and noise, weighted density function by atom energy is further proposed to improve the segmentation of heart sounds. Therefore, heart sounds are indicated by the hybrid analysis of clustering and medical knowledge. The segmentation scheme is automatic and no reference signal is needed. Twenty-six subjects, including 3 normal and 23 abnormal subjects, were tested for heart sound signals in various clinical cases. Our statistics show that the segmentation was successful for signals collected from normal subjects and patients with moderate murmurs.     

Specific impairment of cardiogenesis in mouse ES cells containing a human chromosome 21

Down syndrome (DS) leads to cardiac defects which are common and significant in babies with DS. We recently generated chimeric mice carrying a human chromosome (hChr) 21. The contribution ratio of embryonic stem (ES) cells containing a hChr 21 was specifically low in the heart, compared to other organs, and cardiovascular malformations were observed, suggesting that an additional copy of hChr 21 also disrupts the normal development of heart in mice. Here we describe that the presence of hChr 21 in ES cells delays the appearance of beating cardiomyocyte during differentiation, whereas differentiation into other cell types is not disrupted. Furthermore, the defect in cardiogenesis was restored following the deletion of a specific region of hChr 21. Therefore, we conclude that the imbalance of specific gene(s) on hChr 21 may lead to the disturbance of cardiogenesis and that this may be a useful system to model and investigate the cardiac defects of human DS.     

Photoelectric monitoring of single beating heart cells in culture

This paper characterizes a photoelectric recording system which monitors individual beating heart cells in vitro. A culture procedure which supports continued beating for up to three months without confluent overgrowth or aggregation is described.