An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

Modelling the Effect of Gap Junctions on Tissue-Level Cardiac Electrophysiology

When modelling tissue-level cardiac electrophysiology, a continuum approximation to the discrete cell-level equations, known as the bidomain equations, is often used to maintain computational tractability. Analysing the derivation of the bidomain equations allows us to investigate how microstructure, in particular gap junctions that electrically connect cells, affect tissue-level conductivity properties. Using a one-dimensional cable model, we derive a modified form of the bidomain equations that take gap junctions into account, and compare results of simulations using both the discrete and continuum models, finding that the underlying conduction velocity of the action potential ceases to match up between models when gap junctions are introduced at physiologically realistic coupling levels. We show that this effect is magnified by: (i) modelling gap junctions with reduced conductivity; (ii) increasing the conductance of the fast sodium channel; and (iii) an increase in myocyte length. From this, we conclude that the conduction velocity arising from the bidomain equations may not be an accurate representation of the underlying discrete system. In particular, the bidomain equations are less likely to be valid when modelling certain diseased states whose symptoms include a reduction in gap junction coupling or an increase in myocyte length.     

The effect of simplifying assumptions in the bidomain model of cardiac tissue: application to ST segment shifts during partial ischaemia

In this study various electrical conductivity approximations used in bidomain models of cardiac tissue are considered. Comparisons are based on epicardial surface potential distributions arising from regions of subendocardial ischaemia situated within the cardiac tissue. Approximations studied are a single conductivity bidomain model, an isotropic bidomain model and equal and reciprocal anisotropy ratios both with and without fibre rotation. It is demonstrated both analytically and numerically that the approximations involving a single conductivity bidomain, an isotropic bidomain or equal anisotropy ratios (ignoring fibre rotation) results in identical epicardial potential distributions for all degrees of subendocardial ischaemia. This result is contrary to experimental observations. It is further shown that by assuming reciprocal anisotropy ratios, epicardial potential distributions vary with the degree of subendocardial ischaemia. However, it is concluded that unequal anisotropy ratios must be used to obtain the true character of experimental observations.     

Space radiation absorbed dose distribution in a human phantom.

The radiation risk to astronauts has always been based on measurements using passive thermoluminescent dosimeters (TLDs). The skin dose is converted to dose equivalent using an average radiation quality factor based on model calculations. The radiological risk estimates, however, are based on organ and tissue doses. This paper describes results from the first space flight (STS-91, 51.65 degrees inclination and approximately 380 km altitude) of a fully instrumented Alderson Rando phantom torso (with head) to relate the skin dose to organ doses. Spatial distributions of absorbed dose in 34 1-inch-thick sections measured using TLDs are described. There is about a 30% change in dose as one moves from the front to the back of the phantom body. Small active dosimeters were developed specifically to provide time-resolved measurements of absorbed dose rates and quality factors at five organ locations (brain, thyroid, heart/lung, stomach and colon) inside the phantom. Using these dosimeters, it was possible to separate the trapped-proton and the galactic cosmic radiation components of the doses. A tissue-equivalent proportional counter (TEPC) and a charged-particle directional spectrometer (CPDS) were flown next to the phantom torso to provide data on the incident internal radiation environment. Accurate models of the shielding distributions at the site of the TEPC, the CPDS and a scalable Computerized Anatomical Male (CAM) model of the phantom torso were developed. These measurements provided a comprehensive data set to map the dose distribution inside a human phantom, and to assess the accuracy and validity of radiation transport models throughout the human body. The results show that for the conditions in the International Space Station (ISS) orbit during periods near the solar minimum, the ratio of the blood-forming organ dose rate to the skin absorbed dose rate is about 80%, and the ratio of the dose equivalents is almost one. The results show that the GCR model dose-rate predictions are 20% lower than the observations. Assuming that the trapped-belt models lead to a correct orbit-averaged energy spectrum, the measurements of dose rates inside the phantom cannot be fully understood. Passive measurements using 6Li- and 7Li-based detectors on the astronauts and inside the brain and thyroid of the phantom show the presence of a significant contribution due to thermal neutrons, an area requiring additional study.     

High resolution magnetic images of planar wave fronts reveal bidomain properties of cardiac tissue

We magnetically imaged the magnetic action field and optically imaged the transmembrane potentials generated by planar wavefronts on the surface of the left ventricular wall of Langendorff-perfused isolated rabbit hearts. The magnetic action field images were used to produce a time series of two-dimensional action current maps. Overlaying epifluorescent images allowed us to identify a net current along the wavefront and perpendicular to gradients in the transmembrane potential. This is in contrast to a traditional uniform double-layer model where the net current flows along the gradient in the transmembrane potential. Our findings are supported by numerical simulations that treat cardiac tissue as a bidomain with unequal anisotropies in the intra- and extracellular spaces. Our measurements reveal the anisotropic bidomain nature of cardiac tissue during plane wave propagation. These bidomain effects play an important role in the generation of the whole-heart magnetocardiogram and cannot be ignored.     

Miniature-probe measurements of electric fields and currents induced by a 60-Hz magnetic field in rat and human models

A miniaturized probe was designed and built to provide detailed data on fields induced by a uniform 60-Hz magnetic field in homogeneous models of rat and human. The probe employed three silver wires twisted and potted in an 8-cm hypodermic needle. The exposed tips of the wires formed three sensing electrodes with a centered ground; highly sensitive voltage measurements were enabled by a lock-in amplifier. Tests were conducted in a 1-mT rms field that was uniform within +/- 5%. The models were made by casting 1.5% agar at 1-S/m conductivity into plastic-foam molds. The rat model was scaled 1:1 as an adult (22 cm length; mass about 640 g). The human model was scaled 1:4 as an adult (height = 46.5 cm; mass 1.4 kg). The probe was inserted into each model in several regions, and readings of induced fields were made under different exposure geometries. Maximal strengths of fields induced near the surface of the torso were as high as 120 microV/cm in the laterally exposed rat model. Data extrapolated from the quarter-scale human model revealed that an induced field as high as 700 microV/cm could occur at the torso of a frontally exposed human adult. An overall size-scale factor of about 5 appears to be appropriate for experimental exposures of rats that are intended to simulate currents induced in human beings by magnetic fields. The average strength of electric fields induced in the torso by a 1-mT magnetic field is comparable to that by a vertical electric-field at 60 kV/m and 28 kV/m, respectively, for the rat and human.     

Quantum Attack-Resistent Certificateless Multi-Receiver Signcryption Scheme

The existing certificateless signcryption schemes were designed mainly based on the traditional public key cryptography, in which the security relies on the hard problems, such as factor decomposition and discrete logarithm. However, these problems will be easily solved by the quantum computing. So the existing certificateless signcryption schemes are vulnerable to the quantum attack. Multivariate public key cryptography (MPKC), which can resist the quantum attack, is one of the alternative solutions to guarantee the security of communications in the post-quantum age. Motivated by these concerns, we proposed a new construction of the certificateless multi-receiver signcryption scheme (CLMSC) based on MPKC. The new scheme inherits the security of MPKC, which can withstand the quantum attack. Multivariate quadratic polynomial operations, which have lower computation complexity than bilinear pairing operations, are employed in signcrypting a message for a certain number of receivers in our scheme. Security analysis shows that our scheme is a secure MPKC-based scheme. We proved its security under the hardness of the Multivariate Quadratic (MQ) problem and its unforgeability under the Isomorphism of Polynomials (IP) assumption in the random oracle model. The analysis results show that our scheme also has the security properties of non-repudiation, perfect forward secrecy, perfect backward secrecy and public verifiability. Compared with the existing schemes in terms of computation complexity and ciphertext length, our scheme is more efficient, which makes it suitable for terminals with low computation capacity like smart cards.     

Electric and magnetic fields of the closed depolarization front in the anisotropic myocardium

On the basis of the bidomain model of anisotropic myocardium, mathematical relationships for calculating the characteristics of the extracellular electric and magnetic fields outside a closed or open depolarization front were derived on the assumption that the generator and fields are axially symmetric with respect to the longitudinal direction of the fibres making up the excitable tissue. The spatial configuration of the fields was determined and quantitative estimates were obtained for the electric potential and magnetic induction caused solely by the macroscopic anisotropy of the myocardium tissue as a volume conductor.     

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.     

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.     

Optimal cycling time trial position models: Aerodynamics versus power output and metabolic energy

The aerodynamic drag of a cyclist in time trial (TT) position is strongly influenced by the torso angle. While decreasing the torso angle reduces the drag, it limits the physiological functioning of the cyclist. Therefore the aims of this study were to predict the optimal TT cycling position as function of the cycling speed and to determine at which speed the aerodynamic power losses start to dominate. Two models were developed to determine the optimal torso angle: a ‘Metabolic Energy Model’ and a ‘Power Output Model’. The Metabolic Energy Model minimised the required cycling energy expenditure, while the Power Output Model maximised the cyclists? power output. The input parameters were experimentally collected from 19 TT cyclists at different torso angle positions (0–24°). The results showed that for both models, the optimal torso angle depends strongly on the cycling speed, with decreasing torso angles at increasing speeds. The aerodynamic losses outweigh the power losses at cycling speeds above 46 km/h. However, a fully horizontal torso is not optimal. For speeds below 30 km/h, it is beneficial to ride in a more upright TT position. The two model outputs were not completely similar, due to the different model approaches. The Metabolic Energy Model could be applied for endurance events, while the Power Output Model is more suitable in sprinting or in variable conditions (wind, undulating course, etc.). It is suggested that despite some limitations, the models give valuable information about improving the cycling performance by optimising the TT cycling position.     

An elementary bioelectrical generator of the anisotropic homogeneous myocardium and its extracellular field

On the basis of the bidomain model that takes into account the electric anisotropy of body tissues, analytical relationships were developed for calculating the characteristics of electric and magnetic fields produced by an elementary (dipole) bioelectric generator that arises in the electrogenic excitable tissue of the myocardium. The errors in the identification of intensity and location of the bioelectric generator in the myocardium were estimated from the measurements of its external fields (noninvasive identification of the excited region) using approximate methods based on isotropic models of the physical medium.     

Theory of the anomalous resonant absorption of DNA at microwave frequencies

Aqueous solutions of oligopolymer DNA have been observed by Edwards, Davis, Swicord & Saffer to show structured absorption of microwave energy in the region of several gigahertz characteristic of an ordered series of compressional normal mode vibrations propagating on the polymer chain. Hydrodynamic coupling of such vibrations to the surrounding solvent would preclude the existence of sharp resonances. The inclusion of electromagnetic interactions with surrounding counter ions yields a richer space of possibilities for complex behavior of the combined system. A well defined resonant absorption peak appears when the molecular motion and the nearby solvent motion are even slightly decoupled. The microwave electric fields in the vicinity of the molecule provide a mechanism for such a decoupling not present for the case of electrically neutral solvent.     

Motion detection, noise reduction, texture suppression, and contour enhancement by spatiotemporal Gabor filters with surround inhibition

We study the orientation and speed tuning properties of spatiotemporal three-dimensional (3D) Gabor and motion energy filters as models of time-dependent receptive fields of simple and complex cells in the primary visual cortex (V1). We augment the motion energy operator with surround suppression to model the inhibitory effect of stimuli outside the classical receptive field. We show that spatiotemporal integration and surround suppression lead to substantial noise reduction. We propose an effective and straightforward motion detection computation that uses the population code of a set of motion energy filters tuned to different velocities. We also show that surround inhibition leads to suppression of texture and thus improves the visibility of object contours and facilitates figure/ground segregation and the detection and recognition of objects.     

Interpreting biomagnetic fields of planar wave fronts in cardiac muscle

The recent results of Holzer and co-workers reveal the existence of net currents that flow along the front of a planar wave propagating through cardiac tissue. This is an important contribution toward the better understanding of the physics of biomagnetic fields. However, although the authors claim their results reveal particular bidomain properties, we show in this short letter that the results allow multiple interpretations. For instance, cardiac anisotropy by itself may also explain the existence of a net current along the wave front. Based on our calculations, we suggest additional experiments that would allow distinguishing between these two explanations and thus provide further evidence on the basic physics behind cardiac biomagnetism.     

A mathematical view on the decoupled sites representation

The decoupled sites representation (DSR) is a theoretical instrument which allows to regard complex pH titration curves of biomolecules with several interacting proton binding sites as composition of isolated, non-interacting sites, each with a standard Henderson–Hasselbalch titration curve. In this work, we present the mathematical framework in which the DSR is embedded and give mathematical proofs for several statements in the periphery of the DSR. These proofs also identify exceptions. To apply the DSR to any molecule, it is necessary to extend the set of binding energies from ${\mathbb{R}}$ to a stripe within ${\mathbb{C}}$ . An important observation in this context is that even positive interaction energies (repulsion) between the binding sites will not guarantee real binding energies in the decoupled system, at least if the molecule has more than four proton binding sites. Moreover, we show that for a given overall titration curve it is not only possible to find a corresponding system with an interaction energy of zero but with any arbitrary fix interaction energy. This result also effects practical work as it shows that for any given titration curve, there is an infinite number of corresponding hypothetical molecules. Furthermore, this implies that—using a common definition of cooperative binding on the level of interaction energies—a meaningful measure of cooperativity between the binding sites cannot be defined solely on the basis of the overall titration. Consequently, all measures of cooperativity based on the overall binding curve do not measure the type of cooperativity commonly defined on the basis of interaction energies. Understanding the DSR mathematically provides the basis of transferring the DSR to biomolecules with different types of interacting ligands, such as protons and electrons, which play an important role within electron transport chains like in photosynthesis.     

Thermal sensitivity mapping - warmth and cold detection thresholds of the human torso

Skin as the largest organ of the human body accomplishes many important functions, including thermoregulation. In this context, investigating cold (CDT) and warmth detection thresholds (WDT) constitutes an important research branch, and investigating thermal thresholds has a significant impact on the clothing and fabric textile industry. In this regard, not only the extremities, but also torso regions are of high relevance. However, only few examinations have conducted detailed mapping studies of the human torso. Additionally, some of these studies show certain methodological limitations. Furthermore, the issue of whether cutaneous thermal sensitivity is gender-dependent is still controversial. Therefore, the present study investigated the cutaneous thermal sensitivity (CDT, WDT) of 42 male and female young and healthy subjects. Measurements were taken at 11 anatomical regions. We found that gender plays an important role when investigating thermal thresholds: Females tended to be more sensitive than males. We also found considerable differences between the tested regions, even within the anterior torso, for example. We identified locations which were constantly sensitive (lower back), while others were consistently insensitive (e.g. scapula). We also detected greater data variability for males compared to females, and for WDT compared to CDT. Furthermore, mainly for WDT, we found a proximal-to-distal increase of thermal torso and upper arm sensitivity. In line with previous investigations, our subjects were more sensitive to cold than to warmth. The findings of this study have important implications. First, our data may complement basic research, e.g. in terms of reference data of body regional maps. Second, our data provides important insights that could be leveraged in the textile industry, and also used to optimize current broadly applicable test methods and tools, like thermal manikins and thermophysiological models.     

Torso receptor activity is regulated by a diffusible ligand produced at the extracellular terminal regions of the Drosophila egg.

torso encodes a receptor tyrosine kinase (torso) required for anterior and posterior terminal development of the Drosophila embryo. Injecting eggs with in vitro synthesized torso mRNAs revealed that torso activation is governed by an extracellular molecule, presumably the torso ligand, produced at terminal regions of the egg during early embryogenesis. In the absence of torso, the ligand shows no apparent localization, indicating that it is diffusible and normally bound by an excess of torso receptor at the egg poles. Mutant ligand-binding torso proteins can suppress telson formation in a dominant negative manner, suggesting that the ligand is limited in amount. Analysis of torso mutations indicates that torso functions as a tyrosine kinase and that gain-of-function mutations causing ligand-independent activation are located in the extracellular domain.     

Coupled and uncoupled Markov systems: a possible way to distinguish between them

The conditional distributions of openings and closings are computed for Markov schemes with two open and two closed states and with different pathways connecting the open and closed aggregates. The computation is performed for uncoupled schemes by directly applying the probability laws and by using a convolution algorithm for coupled schemes. The results show that, for coupled schemes, conditional distributions can be nonmonotonic functions of the dwell time duration. Simulations, illustrating how the difference between coupled and uncoupled models can be detected, are also reported.     

Electric and magnetic fields from two-dimensional anisotropic bisyncytia.

Cardiac tissue can be considered macroscopically as a bidomain, anisotropic conductor in which simple depolarization wavefronts produce complex current distributions. Since such distributions may be difficult to measure using electrical techniques, we have developed a mathematical model to determine the feasibility of magnetic localization of these currents. By applying the finite element method to an idealized two-dimensional bisyncytium with anisotropic conductivities, we have calculated the intracellular and extracellular potentials, the current distributions, and the magnetic fields for a circular depolarization wavefront. The calculated magnetic field 1 mm from the tissue is well within the sensitivity of a SQUID magnetometer. Our results show that complex bisyncytial current patterns can be studied magnetically, and these studies should provide valuable insight regarding the electrical anisotropy of cardiac tissue.