Variation of the maximum velocity along the umbilical vein supports the Reynolds pulsometer model.

OBJECTIVE: To challenge, with a modern sonographic approach and a numerical model, the Reynolds's concept which suggests that the vascular structure of the umbilical cord could act as a pulsometer facilitating the venous return to the foetus. METHOD: Forty-five patients between 20 and 28 weeks of gestation were included in the study. The blood maximum velocity in the umbilical vein, measured at both foetal and placental ends, was assessed. Several sonographic parameters of the cord, including the diameter of the umbilical vein at both extremities, cord cross-sectional area and Wharton's jelly section surface were measured. We compare our data with those of a numerical model. RESULTS: A difference in maximum velocity between the two extremities of the umbilical vein (ΔUVVmax) was noted. The maximum velocity was significantly higher at the foetal umbilical end (14.12 +/-3.18 cm/s) than at the placental end (11.93 +/-2.55 cm/s; p < 0.0001). The mean difference is 2.2 +/- 2.3 cm/s. No difference in the umbilical vein diameter was measured at both cord ends (umbilical 4.85 +/-0.9 mm, placental 4.86 +/-0.87 mm, p < 0.0001). There is no significant relationship between ΔUVVmax and the cord cross-sectional area or Wharton's jelly index. CONCLUSION: Modifications of the spatial velocity profile together with the pulsometer model could explain the maximum velocity changes that is measured in the umbilical vein along the cord. This numerical model consolidates the sonographic observations.

Temperature, geometry, and bifurcations in the numerical modeling of the cardiac mechano-electric feedback.

This article characterizes the cardiac autonomous electrical activity induced by the mechanical deformations in the cardiac tissue through the mechano-electric feedback. A simplified and qualitative model is used to describe the system and we also account for temperature effects. The analysis emphasizes a very rich dynamics for the system, with periodic solutions, alternans, chaotic behaviors, etc. The possibility of self-sustained oscillations is analyzed in detail, particularly in terms of the values of important parameters such as the dimension of the system and the importance of the stretch-activated currents. It is also shown that high temperatures notably increase the parameter ranges for which self-sustained oscillations are observed and that several attractors can appear, depending on the location of the initial excitation of the system. Finally, the instability mechanisms by which the periodic solutions are destabilized have been studied by a Floquet analysis, which has revealed period-doubling phenomena and transient intermittencies.

Multiscale model of the human cardiovascular system: Description of heart failure and comparison of contractility indices.

A multiscale model of the cardiovascular system is presented. Hemodynamics is described by a lumped parameter model, while heart contraction is described at the cellular scale. An electrophysiological model and a mechanical model were coupled and adjusted so that the pressure and volume of both ventricles are linked to the force and length of a half-sarcomere. Particular attention was paid to the extreme values of the sarcomere length, which must keep physiological values. This model is able to reproduce healthy behavior, preload variations experiments, and ventricular failure. It also allows to compare the relevance of standard cardiac contractility indices. This study shows that the theoretical gold standard for assessing cardiac contractility, namely the end-systolic elastance, is actually load-dependent and therefore not a reliable index of cardiac contractility.

A comparison between four techniques to measure cardiac output.

Cardiac output is an important variable when monitoring hemodynamic status. In particular, changes in cardiac output represent the goal of several circulatory management therapies. Unfortunately, cardiac output is very difficult to estimate, either in experimental or clinical settings. The goal of this work is to compare four techniques to measure cardiac output: pressure-volume catheter, aortic flow probe, thermodilution, and the PiCCO monitor. These four techniques were simultaneously used during experiments of fluid and endotoxin administration on 7 pigs. Findings show that, first, each individual technique is precise, with a relative coefficient of repeatability lower than 7 %. Second, 1 cardiac output estimate provided by any technique relates poorly to the estimates from the other 3, even if there is only small bias between the techniques. Third, changes in cardiac output detected by one technique are only detected by the others in 62 to 100 % of cases. This study confirms the difficulty of obtaining a reliable clinical cardiac output measurement. Therefore, several measurements using different techniques should be performed, if possible, and all such should be treated with caution.

Importance of metabolism variations in a model of extracorporeal carbon dioxide removal.

Extracorporeal CO2 Removal device is used in clinics when a patient suffers from a pulmonary insufficiency like Acute Respiratory Distress Syndrome and allows to decarboxylate blood externally. In this work, a model of the respiratory system coupled with such a device is proposed to analyze the decrease of CO2 partial pressure in blood. To validate the model, some parameters are estimated thanks to experimental data. Metabolism is a crucial parameter and we show that its time evolution must be taken into account in order to have correct CO2 partial pressure simulations in arteries and in veins.

Importance of wave-number dependence of Biot numbers in one-sided models of evaporative Marangoni instability: Horizontal layer and spherical droplet.

A one-sided model of the thermal Marangoni instability owing to evaporation into an inert gas is developed. Two configurations are studied in parallel: a horizontal liquid layer and a spherical droplet. With the dynamic gas properties being admittedly negligible, one-sided approaches typically hinge upon quantifying heat and mass transfer through the gas phase by means of transfer coefficients (like in the Newton's cooling law), which in dimensionless terms eventually corresponds to using Biot numbers. Quite a typical arrangement encountered in the literature is a constant Biot number, the same for perturbations of different wavelengths and maybe even the same as for the reference state. In the present work, we underscore the relevance of accounting for its wave-number dependence, which is especially the case in the evaporative context with relatively large values of the resulting effective Biot number. We illustrate the effect in the framework of the Marangoni instability thresholds. As a concrete example, we consider HFE-7100 (a standard refrigerant) for the liquid and air for the inert gas.

A simplified model for mitral valve dynamics.

Located between the left atrium and the left ventricle, the mitral valve controls flow between these two cardiac chambers. Mitral valve dysfunction is a major cause of cardiac dysfunction and its dynamics are little known. A simple non-linear rotational spring model is developed and implemented to capture the dynamics of the mitral valve. A measured pressure difference curve was used as the input into the model, which represents an applied torque to the anatomical valve chords. A range of mechanical model hysteresis states were investigated to find a model that best matches reported animal data of chord movement during a heartbeat. The study is limited by the use of one dataset found in the literature due to the highly invasive nature of getting this data. However, results clearly highlight fundamental physiological issues, such as the damping and chord stiffness changing within one cardiac cycle, that would be directly represented in any mitral valve model and affect behaviour in dysfunction. Very good correlation was achieved between modeled and experimental valve angle with 1-10% absolute error in the best case, indicating good promise for future simulation of cardiac valvular dysfunction, such as mitral regurgitation or stenosis. In particular, the model provides a pathway to capturing these dysfunctions in terms of modeled stiffness or elastance that can be directly related to anatomical, structural defects and dysfunction.

[One-dimensional time-dependent model of the cardiac pacemaker activity induced by the mechanoelectric feedback in a thermo-electro-mechanical background]. / Modèle unidimensionnel instationnaire de l'activité pacemaker cardiaque induite par le feedback mécano-électrique dans un environnement thermo-électromécanique.

AIM OF THE STUDY: In a healthy heart, the mechanoelectric feedback (MEF) process acts as an intrinsic regulatory mechanism of the myocardium which allows the normal cardiac contraction by damping mechanical perturbations in order to generate a new healthy electromechanical situation. However, under certain conditions, the MEF can be a generator of dramatic arrhythmias by inducing local electrical depolarizations as a result of abnormal cardiac tissue deformations, via stretch-activated channels (SACs). Then, these perturbations can propagate in the whole heart and lead to global cardiac dysfunctions. In the present study, we qualitatively investigate the influence of temperature on autonomous electrical activity generated by the MEF. METHOD: We introduce a one-dimensional time-dependent model containing all the key ingredients that allow accounting for the excitation-contraction coupling, the MEF and the thermoelectric coupling. RESULTS: Our simulations show that an autonomous electrical activity can be induced by cardiac deformations, but only inside a certain temperature interval. In addition, in some cases, the autonomous electrical activity takes place in a periodic way like a pacemaker. We also highlight that some properties of action potentials, generated by the mechanoelectric feedback, are significantly influenced by temperature. Moreover, in the situation where a pacemaker activity occurs, we also show that the period is heavily temperature-dependent. CONCLUSIONS: Our qualitative model shows that the temperature is a significant factor with regards to the electromechanical behavior of the heart and more specifically, with regards to the autonomous electrical activity induced by the cardiac tissue deformations.

Bénard instabilities in a binary-liquid layer evaporating into an inert gas.

A linear stability analysis is performed for a horizontal layer of a binary liquid of which solely the solute evaporates into an inert gas, the latter being assumed to be insoluble in the liquid. In particular, a water-ethanol system in contact with air is considered, with the evaporation of water being neglected (which can be justified for a certain humidity of the air). External constraints on the system are introduced by imposing fixed "ambient" mass fraction and temperature values at a certain effective distance above the free liquid-gas interface. The temperature is the same as at the bottom of the liquid layer, where, besides, a fixed mass fraction of the solute is presumed to be maintained. Proceeding from a (quasi-)stationary reference solution, neutral (monotonic) stability curves are calculated in terms of solutal/thermal Marangoni/Rayleigh numbers as functions of the wavenumber for different values of the ratio of the gas and liquid layer thicknesses. The results are also presented in terms of the critical values of the liquid layer thickness as a function of the thickness of the gas layer. The solutal and thermal Rayleigh and Marangoni effects are compared to one another. For a water-ethanol mixture of 10wt.% ethanol, it appears that the solutal Marangoni effect is by far the most important instability mechanism. Furthermore, its global action can be described within a Pearson-like model, with an appropriately defined Biot number depending on the wavenumber. On the other hand, it is also shown that, if taken into account, water evaporation has only minor quantitative consequences upon the results for this predominant, solutal Marangoni mechanism.

Bifurcation analysis of a periodically forced relaxation oscillator: differential model versus phase-resetting map.

We compare the dynamics of the periodically forced FitzHugh-Nagumo oscillator in its relaxation regime to that of a one-dimensional discrete map of the circle derived from the phase-resetting response of this oscillator (the "phase-resetting map"). The forcing is a periodic train of Gaussian-shaped pulses, with the width of the pulses much shorter than the intrinsic period of the oscillator. Using numerical continuation techniques, we compute bifurcation diagrams for the periodic solutions of the full differential equations, with the stimulation period being the bifurcation parameter. The period-1 solutions, which belong either to isolated loops or to an everywhere-unstable branch in the bifurcation diagram at sufficiently small stimulation amplitudes, merge together to form a single branch at larger stimulation amplitudes. As a consequence of the fast-slow nature of the oscillator, this merging occurs at virtually the same stimulation amplitude for all the period-1 loops. Again using continuation, we show that this stimulation amplitude corresponds, in the circle map, to a change of topological degree from one to zero. We explain the origin of this coincidence, and also discuss the translational symmetry properties of the bifurcation diagram.

Model-based identification and diagnosis of a porcine model of induced endotoxic shock with hemofiltration.

A previously validated cardiovascular system (CVS) model and parameter identification method for cardiac and circulatory disease states are extended and further validated in a porcine model (N=6) of induced endotoxic shock with hemofiltration. Errors for the identified model are within 10% when the model is re-simulated and compared to the clinical data. All identified parameter trends over time in the experiments match clinically expected changes both individually and over the cohort. This work represents a further clinical validation of these model-based cardiovascular diagnosis and therapy guidance methods for use with monitoring endotoxic disease states.

Continuation and bifurcation analysis of a periodically forced excitable system.

The response of an excitable cell to periodic electrical stimulation is modeled using the FitzHugh-Nagumo (FHN) system submitted to a gaussian-shaped pacing, the width of which is small compared with the action potential duration. The influence of the amplitude and the period of the stimulation is studied using numerical continuation and bifurcation techniques (AUTO97 software). Results are discussed in the light of prior experimental and theoretical findings. In particular, agreement with the documented behavior of periodically stimulated cardiac cells and squid axons is discussed. As previously reported, we find many different "M:N" periodic solutions, period-doubling sequences leading to seemingly chaotic regimes, and bistability phenomena. In addition, the use of continuation techniques has allowed us to track unstable solutions of the system and thus to determine how the different stable rhythms are connected with each other in a bifurcation diagram. Depending on the stimulus amplitude, the aspect of the bifurcation diagram with the stimulus period as main varying parameter can vary from very simple to very complex. In its most developed structure, this bifurcation diagram consists of a main "tree" of period-2(P) branches, where the 1:1, 1:0, 2:2, 2:1,... rhythms are located, and of several closed loops made up of period-{N x 2(P)} branches (N>2), isolated from each other and from the main tree. It is mainly on such loops that N:1 rhythms (N>2) on one hand, and N:N-1 or Wenckebach rhythms (N>2) on the other hand, are located. Stable M:N and M:N-1 rhythms (M>or=N) can be found on the same branch of solutions. They are separated by a region of unstable solutions at small stimulus amplitudes, but this region shrinks gradually as the stimulus amplitude is raised, until it finally disappears. We believe that this property is related to the excitability characteristics of the FHN system. It would be interesting to know if it has any correspondence in the behavior of real excitable cells.

Standing waves in the FitzHugh-Nagumo model of cardiac electrical activity.

When excitable media are submitted to appropriate time dependent boundary conditions, a standing wavelike pattern can be observed in the system, as shown in recent experiments. In the present analysis, the physical mechanism explaining the occurrence of such space-time patterns is shown to be a competition between Ohmic diffusion and an action potential propagation across the system, coupled with the existence of refractory states for excitable media.

Experiments on the role of gas height in the Rayleigh-Marangoni instability problem.

Experimental evidence is provided to show the effect of gas phase dynamics on the onset of thermal convection and on the accompanying patterns in a silicone oil-air convecting bilayer. Very good agreement with three-dimensional calculations for linearized stability is obtained mostly for small and large gas heights. Reasons for this agreement as well as the results at intermediate gas heights are qualitatively explained from the perspective of well-established nonlinear analysis.

Improved 1.5-sided model for the weakly nonlinear study of Bénard-Marangoni instabilities in an evaporating liquid layer.

Bénard-Marangoni instability, with coupled gravity and surface tension effects, in an evaporating liquid layer surmounted by its vapor and an inert gas is investigated theoretically. We show that this system can be described by a model that consists in the liquid layer equations plus the diffusion equation for the vapor in the gas (the so-called 1.5-sided model) and that this model is equivalent to a one-sided model when the vapor mass fraction field can be considered as quasi-stationary, provided that the equivalent Biot number is a nonlocal function of the interface temperature. A comparison of weakly nonlinear results for the 1.5-sided model with a previous one-sided model [M. Dondlinger, J. Margerit, P.C. Dauby, J. Colloid Interface Sci. 283 (2) (2005) 522-532] that considered a Biot number depending on the wavenumber evaluated at the threshold is performed. Very good agreement is found between both models. For this reason, the present analysis can also be considered as a detailed theoretical justification for the use of a one-sided model in the study of evaporative thermoconvection.

Weakly nonlinear study of Marangoni instabilities in an evaporating liquid layer.

We propose a theoretical study of Marangoni driven convection in an evaporating liquid layer surmounted by an inert gas-vapor mixture. After reduction of the full two-layer problem to a one-sided model we use a Galerkin-Eckhaus method leading to a finite set of amplitude equations for the weakly nonlinear analysis of the problem. We analyze the stability of the roll, square, and hexagonal patterns emerging above the linear stability threshold for a water-air and for an ethanol-air system.

Influence of a nonlinear reference temperature profile on oscillatory Bénard-Marangoni convection.

We analyze oscillatory instabilities in a fluid layer of infinite horizontal extent, heated from above or cooled from below, taking into account the nonlinearity of the reference temperature profile during the transient state of heat conduction. The linear stability analysis shows that a nonlinear reference temperature profile can have a strong effect on the system, either stabilizing or destabilizing, depending on the relative importance of buoyancy and surface tension forces. For the nonlinear analysis we use a Galerkin-Eckhaus method leading to a finite set of amplitude equations. In the two-dimensional (2D) case, we show the solution of these amplitude equations are standing waves.

Amplitude equations for Rayleigh-Bénard convective rolls far from threshold.

An extension of the amplitude method is proposed. An iterative algorithm is developed to build an amplitude equation model that is shown to provide precise quantitative results even far from the linear instability threshold. The method is applied to the study of stationary Rayleigh-Bénard thermoconvective rolls in the nonlinear regime. In particular, the generation of second and third spatial harmonics is analyzed. Comparison with experimental results and direct numerical calculations is also made and a very good agreement is found.