Biophysical models of the heart electrical activity

Based on the fundamental knowledge of the space-temporal organization of extracellular electrical fields of the myocardium, a system for 3-D computer modeling of the cardiac electrical activity at different structural levels of the object is being developed at the Institute of Theoretical and Experimental Biophysics. The system is based on the earlier proposed and modified biophysical model of the electrocardiosignal genesis represented by a double electrical layer along the surface of the electrically active myocardium. The system combines the model for activation and repolarization of the heart ventricles; the advanced model for the evaluation of parameters of the cardiac electric field, which makes it possible to derive model electrocardiosignals both in the direct regime of calculation of the potentials and in the regime of calculation of electrocardiosignals from preliminarily determined components of the multipole equivalent electrical heart generator; a database for the model parameters and their combinations in the form of cards of simulated "patients", and a database of modeled electrocardiosignals. In the present paper (first from three within the framework of the problem), simulation methods in electrocardiology are briefly described and a biophysical model of the heart electrical activity is presented, which has made up the basis of the system for computer modeling of forward and inverse problems of the cardiac electric field. The parameters of the model are electrophysiological, anatomical, and biophysical characteristics of the heart.     

Realization of biophysical models of cardiac electrical activity

Considered are the principles of realization of biophysical models of heart ventricle electrical activity in the form of a double electric layer on the surface of the electrically active myocardium (epicardium and endocardium) and the boundary surfaces dividing the model compartments with different electrophysiological characteristics. The model parameters are the electrophysiological and anatomical characteristics of the heart such as the geometry of the ventricles and the specialized His-Purkinje conduction system, the velocity of depolarization spread over myocardium, the ratio of the velocities of excitation transmission through the Myocardium / His / Purkinje elements of the model, the shape of transmembrane action potentials on the boundary surfaces, the orientation of the intrinsic anatomical axes of the heart relative to the initial set of coordinates, and some other biophysical characteristics of the myocardium. This model is the main unit of a computer simulation system, which includes databases of real and simulated electrocardiosignals.     

From Oscillatory Transcranial Current Stimulation to Scalp EEG Changes: A Biophysical and Physiological Modeling Study

Both biophysical and neurophysiological aspects need to be considered to assess the impact of electric fields induced by transcranial current stimulation (tCS) on the cerebral cortex and the subsequent effects occurring on scalp EEG. The objective of this work was to elaborate a global model allowing for the simulation of scalp EEG signals under tCS. In our integrated modeling approach, realistic meshes of the head tissues and of the stimulation electrodes were first built to map the generated electric field distribution on the cortical surface. Secondly, source activities at various cortical macro-regions were generated by means of a computational model of neuronal populations. The model parameters were adjusted so that populations generated an oscillating activity around 10 Hz resembling typical EEG alpha activity. In order to account for tCS effects and following current biophysical models, the calculated component of the electric field normal to the cortex was used to locally influence the activity of neuronal populations. Lastly, EEG under both spontaneous and tACS-stimulated (transcranial sinunoidal tCS from 4 to 16 Hz) brain activity was simulated at the level of scalp electrodes by solving the forward problem in the aforementioned realistic head model. Under the 10 Hz-tACS condition, a significant increase in alpha power occurred in simulated scalp EEG signals as compared to the no-stimulation condition. This increase involved most channels bilaterally, was more pronounced on posterior electrodes and was only significant for tACS frequencies from 8 to 12 Hz. The immediate effects of tACS in the model agreed with the post-tACS results previously reported in real subjects. Moreover, additional information was also brought by the model at other electrode positions or stimulation frequency. This suggests that our modeling approach can be used to compare, interpret and predict changes occurring on EEG with respect to parameters used in specific stimulation configurations.     

Exploring how extracellular electric field modulates neuron activity through dynamical analysis of a two-compartment neuron model

To investigate how extracellular electric field modulates neuron activity, a reduced two-compartment neuron model in the presence of electric field is introduced in this study. Depending on neuronal geometric and internal coupling parameters, the behaviors of the model have been studied extensively. The neuron model can exist in quiescent state or repetitive spiking state in response to electric field stimulus. Negative electric field mainly acts as inhibitory stimulus to the neuron, positive weak electric field could modulate spiking frequency and spike timing when the neuron is already active, and positive electric fields with sufficient intensity could directly trigger neuronal spiking in the absence of other stimulations. By bifurcation analysis, it is observed that there is saddle-node on invariant circle bifurcation, supercritical Hopf bifurcation and subcritical Hopf bifurcation appearing in the obtained two parameter bifurcation diagrams. The bifurcation structures and electric field thresholds for triggering neuron firing are determined by neuronal geometric and coupling parameters. The model predicts that the neurons with a nonsymmetric morphology between soma and dendrite, are more sensitive to electric field stimulus than those with the spherical structure. These findings suggest that neuronal geometric features play a crucial role in electric field effects on the polarization of neuronal compartments. Moreover, by determining the electric field threshold of our biophysical model, we could accurately distinguish between suprathreshold and subthreshold electric fields. Our study highlights the effects of extracellular electric field on neuronal activity from the biophysical modeling point of view. These insights into the dynamical mechanism of electric field may contribute to the investigation and development of electromagnetic therapies, and the model in our study could be further extended to a neuronal network in which the effects of electric fields on network activity may be investigated.     

Computer models of the electrophysiological properties of the heart]

Computer models of the heart can improve the understanding of the electrophysiological processes in healthy and diseased heart. They become more and more important for detailed diagnosis of arrhythmias and for optimization of therapy. Models of myocardium cells known today are described--they are based on the properties of all relevant ion channels in the cell membrane. Then it is demonstrated, how many cells can be joined to form a cell patch and how finally the complete heart can be modelled. A simpler approach is using a so called cellular automation that allows for a significant reduction of calculation time while sacrificing some accordance to reality. Adaptive cellular automations allow for a fast simulation with acceptable accuracy. Using them some results were gained for the simulation of typical arrhythmias, in the field of validation using an animal model and for therapy planning with RF-ablation.     

Mechanisms of electromechanical function disturbances in cardiomyocytes overloaded with calcium. The theoretical study

Arrhythmias and mechanical disturbances are simulated in a mathematical model of cardiomyocyte electromechanical activity. The simulated pattern is similar to that observed for acute heart failure associated with calcium overloading of myocardium cells. Special attention was paid to the calcium overloading resulting from the reduced Na+,K+ pump activity. In the framework of the model, it was shown that mechanical factors could promote arrhythmia initiation when the pump activity reduced. Different approaches to electrical and mechanical function restoration during acute heart failure associated with calcium overloading were suggested and analyzed in the model.     

Electro-Acoustic Behavior of the Mitotic Spindle: A Semi-Classical Coarse-Grained Model

The regulation of chromosome separation during mitosis is not fully understood yet. Microtubules forming mitotic spindles are targets of treatment strategies which are aimed at (i) the triggering of the apoptosis or (ii) the interruption of uncontrolled cell division. Despite these facts, only few physical models relating to the dynamics of mitotic spindles exist up to now. In this paper, we present the first electromechanical model which enables calculation of the electromagnetic field coupled to acoustic vibrations of the mitotic spindle. This electromagnetic field originates from the electrical polarity of microtubules which form the mitotic spindle. The model is based on the approximation of resonantly vibrating microtubules by a network of oscillating electric dipoles. Our computational results predict the existence of a rapidly changing electric field which is generated by either driven or endogenous vibrations of the mitotic spindle. For certain values of parameters, the intensity of the electric field and its gradient reach values which may exert a not-inconsiderable force on chromosomes which are aligned in the spindle midzone. Our model may describe possible mechanisms of the effects of ultra-short electrical and mechanical pulses on dividing cells—a strategy used in novel methods for cancer treatment.     

The direct influence of electromagnetic fields on nerve- and muscle cells of man within the frequency range of 1 Hz to 30 MHz

Summary By using several biophysical approximations and considering man as free space model limiting order-of-magnitude values for external electric and magnetic field strengths which may be hazardous for human beings were calculated. Danger may occur by excitation processes below 30 kHz for field strengths exceeding these limiting values; for frequencies larger than 30 kHz, thermal effects are predominant before excitation occurs. The external electric field strength necessary for causing action potentials in the central nervous system exceeds by far the corona forming level. But excitation is possible by strong alternating magnetic fields.Furthermore, by comparing the electrically and magnetically induced currents with the naturally flowing currents in man caused by the brain's and heart's electrical activity, a lower boundary-line was estimated. Regarding electric or magnetic field strengths undercutting this boundary-line, direct effects on the central nervous system may be excluded. Other mechanisms should be responsible for demonstrated biological effects.     

Myocardial electrodynamics during electrical systole]

A dipole equivalent generator of heart is considered, which represents a circuit containing an active, capacitive and inductive resistance. The model makes it possible to solve a direct diagnostic task for the electric systole. A differential equation for the integral electrical vector of the heart in the dipole approximation was solved. A model vector electrocardiogram and a linear electrocardiogram for the electrical heart parameters are presented.     

An ephaptic transmission model of CA3 pyramidal cells: an investigation into electric field effects

Extracellular electric fields existing throughout the living brain affect the neural coding and information processing via ephaptic transmission, independent of synapses. A two-compartment whole field effect model (WFEM) of pyramidal neurons embedded within a resistive array which simulates the extracellular medium i.e. ephapse is developed to study the effects of electric field on neuronal behaviors. We derive the two linearized filed effect models (LFEM-1 and LFEM-2) from WFEM at the stable resting state. Through matching these simplified models to the subthreshold membrane response in experiments of the resting pyramidal cells exposed to applied electric fields, we not only verify our proposed model’s validity but also found the key parameters which dominate subthreshold frequency response characteristic. Moreover, we find and give its underlying biophysical mechanism that the unsymmetrical properties of active ion channels results in the very different low-frequency response of somatic and dendritic compartments. Following, WFEM is used to investigate both direct-current (DC) and alternating-current field effect on the neural firing patterns by bifurcation analyses. We present that DC electric field could modulate neuronal excitability, with the positive field improving the excitability, the modest negative field suppressing the excitability, but interestingly, the larger negative field re-exciting the neuron back into spiking behavior. The neuron exposed to the sinusoidal electric field exhibits abundant firing patterns sensitive to the input frequency and intensity. In addition, the electrical properties of ephapse can modulate the efficacy of field effect. Our simulated results are qualitatively in line with the relevant experimental results and can explain some experimental phenomena. Furthermore, they are helpful to provide the predictions which can be tested in future experiments.     

Extension of the neuronal membrane model to account for suppression of the action potential by a constant magnetic field

A biophysical explanation of the reduced excitability in neurons exposed to a constant magnetic field is based on an extended neuronal membrane model. In the presence of a constant magnetic field, reduced excitability is manifested as an increase in the excitation threshold and a decrease in the frequency of action potentials. The proposed explanation for the reduced excitability rests on the well-known Hall effect. The separation of charges resulting from the Lorentz force exerted on moving intracellular ions leads to the formation of a Hall electric field in a direction perpendicular to that of action-potential transmission. Consequently, the ion current for discharging the membrane capacitance is reduced in the presence of a magnetic field, thereby limiting initiation of the action potential. The validity of the proposed biophysical explanation is justified analytically and verified by simulations based on the Hodgkin and Huxley model for the electrical excitability of a neuron. Based on derivation of the current segregation ratio α characterizing the reduction in the stimulating current from first principles, the equivalent circuit model of the neuronal membrane is extended to account for the reduced excitability of neurons exposed to a constant magnetic field.     

Relationship of the maximum QRS vector with the left ventricular myocardial mass during the development of hypertrophy: Computer simulation and empirical data

A model of electrical activity of the heart has been used to demonstrate that, all other conditions remaining the same, the spatial vector of the heart changes, as a first approximation, proportionally to the increase in the surface area of the heart ventricles (rather than the myocardial mass) during proportional homothetic changes in the sizes of the heart and His-Purkinje system. In contrast, the electrocardiosignal (ECS) amplitude is determined, at any given moment, by the area of depolarized regions of the epicardium and endocardium, which agrees with the model of a double electrical layer on the surface of an electrically active myocardium.     

Physically corrected orthogonal electrocardiographic lead systems in macaques and baboons

The adequacy of various physically corrected electrocardiographic lead systems for lower primates was compared with the aid of physical models of the cardiac electrical field. Electrolytic tanks fashioned from plaster casts of the thorax of young adult male and female macaques and baboons were used. A dipole source situated at different points in the heart region simulated the electrical activity of the heart. Statistical evaluation and reciprocal comparison of the resultant parameters showed that the proposed modification of the McFee-Parungao lead system for macaques and baboons was most satisfactory, followed by the human variant of the same lead system and then by the variant for dogs. The greatest variability was displayed by parameters determined from measurements made with Frank's lead system, which was ranked last in the evaluation.     

基于能量融合积分模型的心脏生理物理活动研究

为了更深入地了解目前靠生理实验及临床手段无法洞察的心脏三维空间的电生理运行机制,分析和表现心脏复杂的电生理活动,从而揭示心脏的生理物理特性,本研究通过人类心肌细胞的动作电位传导数学模型,结合基于心脏解剖数据所建立的真实心脏组织结构的三维空间模型,构建出精细的心脏生物物理融合模型,并将心脏在三维空间中的生物物理活动表现出来.实验结果表明,基于心脏动作电位传导的融合模型,不同时刻的动作电位传导在非匀质性组织内的三维空间中的传播位置、空间关系以及生物物理过程被清晰地显示出来,心脏研究人员从而能够以视觉感知的方式认识和深入理解人类心脏电生物物理系统的功能机制,并有助于进一步推测心脏的生理和病理反应.     

Mathematical Simulation of the Rabbit Heterogeneous Intact Sinoatrial Node Using Similitude Theory

Biophysics - Abstract—In this work, the electrical activity of the heterogeneous intact sinoatrial node at interaction with the atrial myocardium is simulated based on a detailed approach....     

Effect of Direct-Current Electric Field on Enzymatic Activity and the Concentration of Laccase

This work investigates the effect of direct-current electric field on the extracellular enzymatic activity, concentration and other experimental parameters of laccase from Trametes versicolor. The results showed that laccase could significantly contribute to the change of pH at the end of graphite electrode. In addition, it increased the electrical conductivity of the water. In the experiment, the optimum pH and catalytic pH range for laccase activity were 3.0 and pH 2.5–4.0. The application of 6 V direct current showed significant effects on the laccase enzyme activity. The activity of laccase was enhanced in the anodic region, but at the same time was strongly inhibited at the cathode. The electric charge characteristics of laccase were changed when exposed to electric field, and some laccases molecules moved to the anode, which produced a slight migration phenomenon. This study is the basis of combination of laccase and electrical technology, at the same time, providing a new direction of enhancing laccase activity. Compared to immobilization, using electric field is simple, no chemical additives, and great potential.     

Establishment of a Novel In Vitro Test Setup for Electric and Magnetic Stimulation of Human Osteoblasts

When large defects occur, bone regeneration can be supported by bone grafting and biophysical stimuli like electric and magnetic stimulation (EMS). Clinically established EMS modes are external coils and surgical implants like an electroinductive screw system, which combines a magnetic and electric field, e.g., for the treatment of avascular bone necrosis or pseudarthrosis. For optimization of this implant system, an in vitro test setup was designed to investigate effects of EMS on human osteoblasts on different 3D scaffolds (based on calcium phosphate and collagen). Prior to the cell experiments, numerical simulations of the setup, as well as experimental validation, via measurements of the electric parameters induced by EMS were conducted. Human osteoblasts (3 × 10⁵ cells) were seeded onto the scaffolds and cultivated. After 24 h, screw implants (Stryker ASNIS III s-series) were centered in the scaffolds, and EMS was applied (3 × 45 min per day at 20 Hz) for 3 days. Cell viability and collagen type 1 (Col1) synthesis were determined subsequently. Numerical simulation and validation showed an adequate distribution of the electric field within the scaffolds. Experimental measurements of the electric potential revealed only minimal deviation from the simulation. Cell response to stimulation varied with scaffold material and mode of stimulation. EMS-stimulated cells exhibited a significant decrease of metabolic activity in particular on collagen scaffolds. In contrast, the Col1/metabolic activity ratio was significantly increased on collagen and non-sintered calcium phosphate scaffolds after 3 days. Exclusive magnetic stimulation showed similar but nonsignificant tendencies in metabolic activity and Col1 synthesis. The cell tests demonstrate that the new test setup is a valuable tool for in vitro testing and parameter optimization of the clinically used electroinductive screw system. It combines magnetic and electric stimulation, allowing in vitro investigations of its influence on human osteoblasts.     

Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart

The electrical activity of the heart may be modeled with a system of partial differential equations (PDEs) known as the bidomain model. Computer simulations based on these equations may become a helpful tool to understand the relationship between changes in the electrical field and various heart diseases. Because of the rapid variations in the electrical field, sufficiently accurate simulations require a fine-scale discretization of the equations. For realistic geometries this leads to a large number of grid points and consequently large linear systems to be solved for each time step. In this paper, we present a fully coupled discretization of the bidomain model, leading to a block structured linear system. We take advantage of the block structure to construct an efficient preconditioner for the linear system, by combining multigrid with an operator splitting technique.     

An electromechanical model of cardiac tissue: constitutive issues and electrophysiological effects

We present an electromechanical model of myocardium tissue coupling a modified FitzHugh-Nagumo type system, describing the electrical activity of the excitable media, with finite elasticity, endowed with the capability of describing muscle contractions. The high degree of deformability of the medium makes it mandatory to set the diffusion process in a moving domain, thereby producing a direct influence of the deformation on the electrical activity. Various mechano-electric effects concerning the propagation of cylindrical waves, the rotating spiral waves, and the spiral breakups are discussed.