An Analysis and Comparison of Convergence and Uniqueness of Time-Independent Bone Adaptation Models

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using ‘close’ and ‘wide’ electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space.

The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required.

The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.     

A solution method for the determination of cardiac potential distributions with an alternating current source

A recently presented solution method for the bidomain model (Johnston et al. 2006), which involves the application of direct current for studying electrical potential in a slab of cardiac tissue, is extended here to allow the use of an applied alternating current. The advantage of using AC current, in a four-electrode method for determining cardiac conductivities, is that instead of using 'close' and 'wide' electrode spacings to make potential measurements, increasing the frequency of the AC current redirects a fraction of the current from the extracellular space into the intracellular space. The model is based on the work of Le Guyader et al. (2001), but is able to include the effects of the fibre rotation between the epicardium and the endocardium on the potentials. Also, rather than using a full numerical technique, the solution method uses Fourier series and a simple one dimensional finite difference scheme, which has the advantage of allowing the potentials to be calculated only at points, such as the measuring electrodes, where they are required. The new alternating current model, which includes intracellular capacitance, is used with a particular four-electrode configuration, to show that the potential measured is affected by changes in fibre rotation. This is significant because it indicates that it is necessary to include fibre rotation in models, which are to be used in conjunction with measuring arrays that are more complex than those involving simply surface probes or a single vertical probe.     

Effect of Twisted Fiber Anisotropy in Cardiac Tissue on Ablation with Pulsed Electric Fields

BackgroundAblation of cardiac tissue with pulsed electric fields is a promising alternative to current thermal ablation methods, and it critically depends on the electric field distribution in the heart.MethodsWe developed a model that incorporates the twisted anisotropy of cardiac tissue and computed the electric field distribution in the tissue. We also performed experiments in rabbit ventricles to validate our model. We find that the model agrees well with the experimentally determined ablation volume if we assume that all tissue that is exposed to a field greater than 3 kV/cm is ablated. In our numerical analysis, we considered how tissue thickness, degree of anisotropy, and electrode configuration affect the geometry of the ablated volume. We considered two electrode configurations: two parallel needles inserted into the myocardium (“penetrating needles” configuration) and one circular electrode each on epi- and endocardium, opposing each other (“epi-endo” configuration).ResultsFor thick tissues (10 mm) and moderate anisotropy ratio (a = 2), we find that the geometry of the ablated volume is almost unaffected by twisted anisotropy, i.e. it is approximately translationally symmetric from epi- to endocardium, for both electrode configurations. Higher anisotropy ratio (a = 10) leads to substantial variation in ablation width across the wall; these variations were more pronounced for the penetrating needle configuration than for the epi-endo configuration.For thinner tissues (4 mm, typical for human atria) and higher anisotropy ratio (a = 10), the epi-endo configuration yielded approximately translationally symmetric ablation volumes, while the penetrating electrodes configuration was much more sensitive to fiber twist.ConclusionsThese results suggest that the epi-endo configuration will be reliable for ablation of atrial fibrillation, independently of fiber orientation, while the penetrating electrode configuration may experience problems when the fiber orientation is not consistent across the atrial wall.     

Experimental study of magnetized plasma rotation in crossed fields

Rotation of magnetized plasma between two coaxial electrodes in crossed electric and magnetic fields was studied experimentally. Three regimes of plasma rotation were observed. In the first regime, the radial electric field is created by a beam?plasma discharge due to the charging of the inner axial electrode by electrons, the outer electrode being grounded. Plasma rotation in this case is accompanied by strong high-frequency current oscillations detected by a Mach probe. When a negative voltage was applied to the coaxial electrodes, the second regime was observed, in which weakly perturbed quasi-stationary plasma rotation occurred at a relatively low radial current. The third regime of plasma rotation was observed upon a spontaneous disruption of the second regime. It is characterized by high currents of ~1 kA, sheared plasma rotation, and excitation of high-frequency perturbations.     

A theoretical analysis of the feasibility of a singularity-induced micro-electroporation system

Electroporation, the permeabilization of the cell membrane lipid bilayer due to a pulsed electric field, has important implications in the biotechnology, medicine, and food industries. Traditional macro and micro-electroporation devices have facing electrodes, and require significant potential differences to induce electroporation. The goal of this theoretical study is to investigate the feasibility of singularity-induced micro-electroporation; an electroporation configuration aimed at minimizing the potential differences required to induce electroporation by separating adjacent electrodes with a nanometer-scale insulator. In particular, this study aims to understand the effect of (1) insulator thickness and (2) electrode kinetics on electric field distributions in the singularity-induced micro-electroporation configuration. A non-dimensional primary current distribution model of the micro-electroporation channel shows that while increasing insulator thickness results in smaller electric field magnitudes, electroporation can still be performed with insulators thick enough to be made with microfabrication techniques. Furthermore, a secondary current distribution model of the singularity-induced micro-electroporation configuration with inert platinum electrodes and water electrolyte indicates that electrode kinetics do not inhibit charge transfer to the extent that prohibitively large potential differences are required to perform electroporation. These results indicate that singularity-induced micro-electroporation could be used to develop an electroporation system that consumes minimal power, making it suitable for remote applications such as the sterilization of water and other liquids.     

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.     

A cylindrical model for studying subendocardial ischaemia in the left ventricle

In this paper a mathematical model of a left ventricle with a cylindrical geometry is presented with the aim of gaining a better understanding of the relationship between subendocardial ischaemia and ST depression. The model is formulated as an infinite cylinder and takes into account the full bidomain nature of cardiac tissue, as well as fibre rotation. A detailed solution method (based on Fourier series, Fourier transforms and a one dimensional finite difference scheme) for the governing equations for electric potential in the tissue and the blood is also presented. The model presented is used to study the effect increasing subendocardial ischaemia has on the epicardial potential distribution as well as the effects of changing the bidomain conductivity values. The epicardial potential distributions obtained with this cylindrical geometry are compared with results obtained using a previously published slab model. Results of the simulations presented show that the morphologies of the epicardial potential distributions are similar between the two geometries, with the main difference being that the cylindrical model predicts slightly higher potentials.     

A finite volume method solution for the bidomain equations and their application to modelling cardiac ischaemia

This paper presents an implementation of the finite volume method with the aim of studying subendocardial ischaemia during the ST segment. In this implementation, based on hexahedral finite volumes, each quadrilateral sub-face is split into two triangles to improve the accuracy of the numerical integration in complex geometries and when fibre rotation is included. The numerical method is validated against previously published solutions obtained from slab and cylindrical models of the left ventricle with subendocardial ischaemia and no fibre rotation. Epicardial potential distributions are then obtained for a half-ellipsoid model of the left ventricle. In this case it is shown that for isotropic cardiac tissue the degree of subendocardial ischaemia does not affect the epicardial potential distribution, which is consistent with previous findings from analytical studies in simpler geometries. The paper also considers the behaviour of various preconditioners for solving numerically the resulting system of algebraic equations resulting from the implementation of the finite volume method. It is observed that each geometry considered has its own optimal preconditioner.     

Effects of cavities on EEG dipole localization and their relations with surface electrode positions

Effects of cavities in the human head on EEG dipole localization have been investigated by computer simulation. The human head is represented by a homogeneous spherical conductor including an eccentric spherical cavity which approximates effects of actual cavities inside the head. The homogeneous sphere model is used for assessing the effects caused by neglecting the cavity in the volume conductor model in the inverse dipole fitting procedure. Four electrode configurations have been examined to investigate their relation to the EEG inverse dipole solution. After examination of 2520 dipoles in the brain, the effects of cavities in the human head are found to be negligible when the dipole is located in the cortex or in the subcortex. When the dipole is located in the brain stem, the EEG inverse dipole solution is strongly affected by the cavity and is sensitive to the electrode configuration on the scalp. The EEG inverse dipole solution in the deep brain is sensitive to inhomogeneity in the lower part of the head when a single positive or negative potential pole is observed by the electrodes on the scalp, and at the same time is sensitive to the extent of the scalp covered by the electrodes. In conclusion, the electrodes should cover as much of the upper scalp as possible for deep source localization.     

The effect of boundary conditions on epicardial potential distributions

This study presents a comparison of semi-analytical and numerical solution techniques for solving the passive bidomain equation in simple tissue geometries containing a region of subendocardial ischaemia. When the semi-analytical solution is based on Fourier transforms, recovering the solution from the frequency domain via fast Fourier transforms imposes a periodic boundary condition on the solution of the partial differential equation. On the other hand, the numerical solution uses an insulation boundary condition. When these techniques are applied to calculate the epicardial surface potentials, both yield a three well potential distribution which is identical if fibre rotation within the tissue is ignored. However, when fibre rotation is included, the resulting three-well distribution rotates, but through different angles, depending on the solution method. A quantitative comparison between the semi-analytical and numerical solutiontechniques is presented in terms of the effect fibre rotation has on the rotation of the epicardial potential distribution. It turns out that the Fourier transform approach predicts a larger rotation of the epicardial potential distribution than the numerical solution. The conclusion from this study is that it is not always possible to use analytical or semi-analytical solutions to check the accuracy of numerical solution procedures. For the problem considered here, this checking is only possible when it is assumed that there is no fibre rotation through the tissue.     

In vivo electroporation of skeletal muscle: threshold, efficacy and relation to electric field distribution.

In vivo electroporation is increasingly being used to deliver small molecules as well as DNA to tissues. The aim of this study was to quantitatively investigate in vivo electroporation of skeletal muscle, and to determine the threshold for permeabilization. We designed a quantitative method to study in vivo electroporation, by measuring uptake of (51)Cr-EDTA. As electrode configuration influences electric field (E-field) distribution, we developed a method to calculate this. Electroporation of mouse muscle tissue was investigated using either external plate electrodes or internal needle electrodes placed 4 mm apart, and eight pulses of 99 micros duration at a frequency of 1 Hz. The applied voltage to electrode distance ratio was varied from 0 to 2.0 kV/cm. We found that: (1) the threshold for permeabilization of skeletal muscle tissue using short duration pulses was at an applied voltage to electrode distance ratio of 0.53 kV/cm (+/-0.03 kV/cm), corresponding to an E-field of 0.45 kV/cm; (2) there were two phases in the uptake of (51)Cr-EDTA, the first indicating increasing permeabilization and the second indicating beginning irreversible membrane damage; and (3) the calculated E-field distribution was more homogeneous for plate than for needle electrodes, which was reflected in the experimental results.     

3D Stationary electric current density in a spherical tumor treated with low direct current: an analytical solution

Electrotherapy with direct current delivered through implanted electrodes is used for local control of solid tumors in both preclinical and clinical studies. The aim of this research is to develop a solution method for obtaining a three-dimensional analytical expression for potential and electric current density as functions of direct electric current intensity, differences in conductivities between the tumor and the surrounding healthy tissue, and length, number and polarity of electrodes. The influence of these parameters on electric current density in both media is analyzed. The results show that the electric current density in the tumor is higher than that in the surrounding healthy tissue for any value of these parameters. The conclusion is that the solution method presented in this study is of practical interest because it provides, in a few minutes, a convenient way to visualize in 3D the electric current densities generated by a radial electrode array by means of the adequate selection of direct current intensity, length, number, and polarity of electrodes, and the difference in conductivity between the solid tumor and its surrounding healthy tissue.     

Temperature modulation of electric fields in biological matter

Pulsed electric fields (PEF) have become an important minimally invasive surgical technology for various applications including genetic engineering, electrochemotherapy and tissue ablation. This study explores the hypothesis that temperature dependent electrical parameters of tissue can be used to modulate the outcome of PEF protocols, providing a new means for controlling and optimizing this minimally invasive surgical procedure. This study investigates two different applications of cooling temperatures applied during PEF. The first case utilizes an electrode which simultaneously delivers pulsed electric fields and cooling temperatures. The subsequent results demonstrate that changes in electrical properties due to temperature produced by this configuration can substantially magnify and confine the electric fields in the cooled regions while almost eliminating electric fields in surrounding regions. This method can be used to increase precision in the PEF procedure, and eliminate muscle contractions and damage to adjacent tissues. The second configuration considered introduces a third probe that is not electrically active and only applies cooling boundary conditions. This second study demonstrates that in this probe configuration the temperature induced changes in electrical properties of tissue substantially reduce the electric fields in the cooled regions. This novel treatment can potentially be used to protect sensitive tissues from the effect of the PEF. Perhaps the most important conclusion of this investigation is that temperature is a powerful and accessible mechanism to modulate and control electric fields in biological tissues and can therefore be used to optimize and control PEF treatments.     

Metallic oxide CdIn2O4 films for the label free electrochemical detection of DNA hybridization

DNA functionalised semiconductor metallic oxide electrodes have been developed for the direct electrochemical detection of DNA hybridization, without labelling or the introduction of a redox couple. Conductive CdIn(2)O(4) thin films with controlled properties were deposited on glass substrates using an aerosol pyrolysis technique. The films exhibit a polycrystalline microstructure with a surface roughness of 1.5 nm (r.m.s.) and an electrical resistivity ranging between 1 and 3 x 10(-3) Omega cm. These electrodes were functionalised using hydroxylation and silanisation steps, to allow the binding of DNA probe sequences (20 bases). The electrical detection of DNA hybridization with complementary sequences has been performed using electrochemical impedance spectrometry (EIS) measuring the variation of impedance before and after hybridization. Two set-ups were used, a standard set-up including three electrodes and a set-up including two symmetrical electrodes. In both configurations, a significant increase of the impedance modulus, more particularly of the real part of the impedance (160-225% according to the electrochemical cell used) has been obtained over a frequency range of 10-10(5)Hz. DNA hybridization has also been systematically confirmed using the fluorescence spectrometry. This study emphasizes the high sensitivity of the CdIn(2)O(4) as a working electrode for the detection of biological events occurring at the electrode surface.     

Effect of Blood Vessel Segmentation on the Outcome of Electroporation-Based Treatments of Liver Tumors

Electroporation-based treatments rely on increasing the permeability of the cell membrane by high voltage electric pulses applied to tissue via electrodes. To ensure that the whole tumor is covered with sufficiently high electric field, accurate numerical models are built based on individual patient anatomy. Extraction of patient''s anatomy through segmentation of medical images inevitably produces some errors. In order to ensure the robustness of treatment planning, it is necessary to evaluate the potential effect of such errors on the electric field distribution. In this work we focus on determining the effect of errors in automatic segmentation of hepatic vessels on the electric field distribution in electroporation-based treatments in the liver. First, a numerical analysis was performed on a simple ''sphere and cylinder'' model for tumors and vessels of different sizes and relative positions. Second, an analysis of two models extracted from medical images of real patients in which we introduced variations of an error of the automatic vessel segmentation method was performed. The results obtained from a simple model indicate that ignoring the vessels when calculating the electric field distribution can cause insufficient coverage of the tumor with electric fields. Results of this study indicate that this effect happens for small (10 mm) and medium-sized (30 mm) tumors, especially in the absence of a central electrode inserted in the tumor. The results obtained from the real-case models also show higher negative impact of automatic vessel segmentation errors on the electric field distribution when the central electrode is absent. However, the average error of the automatic vessel segmentation did not have an impact on the electric field distribution if the central electrode was present. This suggests the algorithm is robust enough to be used in creating a model for treatment parameter optimization, but with a central electrode.     

Design and development of a highly stable hydrogen peroxide biosensor on screen printed carbon electrode based on horseradish peroxidase bound with gold nanoparticles in the matrix of chitosan

The design and development of a screen printed carbon electrode (SPCE) on a polyvinyl chloride substrate as a disposable sensor is described. Six configurations were designed on silk screen frames. The SPCEs were printed with four inks: silver ink as the conducting track, carbon ink as the working and counter electrodes, silver/silver chloride ink as the reference electrode and insulating ink as the insulator layer. Selection of the best configuration was done by comparing slopes from the calibration plots generated by the cyclic voltammograms at 10, 20 and 30 mM K(3)Fe(CN)(6) for each configuration. The electrodes with similar configurations gave similar slopes. The 5th configuration was the best electrode that gave the highest slope. Modifying the best SPCE configuration for use as a biosensor, horseradish peroxidase (HRP) was selected as a biomaterial bound with gold nanoparticles (AuNP) in the matrix of chitosan (HRP/AuNP/CHIT). Biosensors of HRP/SPCE, HRP/CHIT/SPCE and HRP/AuNP/CHIT/SPCE were used in the amperometric detection of H(2)O(2) in a solution of 0.1M citrate buffer, pH 6.5, by applying a potential of -0.4V at the working electrode. All the biosensors showed an immediate response to H(2)O(2). The effect of HRP/AuNP incorporated with CHIT (HRP/AuNP/CHIT/SPCE) yielded the highest performance. The amperometric response of HRP/AuNP/CHIT/SPCE retained over 95% of the initial current of the 1st day up to 30 days of storage at 4 degrees C. The biosensor showed a linear range of 0.01-11.3mM H(2)O(2), with a detection limit of 0.65 microM H(2)O(2) (S/N=3). The low detection limit, long storage life and wide linear range of this biosensor make it advantageous in many applications, including bioreactors and biosensors.     

Intercellular power transmission along trichomes of cyanobacteria

An attempt at demonstrating lateral power transmission over millimeter distances along a coupling membrane has been undertaken. Trichomes of the multicellular filamentous cyanobacteria Phormidium uncinatum were illuminated with a very narrow light beam forming a light spot that covered only 4–5% of a 1–2 mm long cyanobacterial trichome. Such illumination was found to support motility (gliding along agar surface) of the trichome under conditions when the light was the only energy source. It was also shown that illumination with the light spot caused rotation of rings of slime (accompanying the operation of the ‘motors’ responsible for the motility of cyanobacteria) not only in the illuminated, but also in the distal, nonilluminated part of the trichome. Electric potential transmission along trichomes was revealed by means of the extracellular electrode technique. The light spot was found to induce generation of an electric potential difference between two electrodes in the dark region of the trichomes, which were placed at different distances from the illuminated end. Cutting the trichomes between the light spot and the closest ‘dark’ electrode abolished this effect. Valinomycin + K⁺ and carbonyl cyanide p-trifluoromethoxyphenylhydrazone affected the potential difference formation between two ‘dark’ electrodes much stronger than that between a light and a dark electrode. All the light spot-induced effects develop in the seconds time scale. Both the amplitudes and the kinetics of the potential difference measured with four electrodes placed along the trichome prove to be in good agreement with the theoretical curves computed on the basis of the electric cable equation. It is concluded that transcellular power transmission in the form of Δψ takes place along trichomes of cyanobacteria. This confirms the hypothesis about the biological function of Δψ as a transportable form of energy.     

Correlated Motion and Oscillation of Neighboring Cells in Vitro

It has long been realized that fibroblastic and epithelial cells establish recognizable patterns in tissue culture. This behavior implies that neighboring cells interact with one another to produce organized populations. Interaction between cells that are separated by many intervening cells is also possible and is demonstrated here using a special configuration of a biosensor referred to as electric cell-substrate impedance sensing (ECIS). Normally the electrical impedance of a single electrode covered with a confluent cell layer is measured, and the morphological changes of the cells are reflected in the impedance. In this case the cells are cultured on two closely spaced electrodes whose impedances are measured independently as a function of time, and communication between the cell populations is revealed as a correlation between these two time series. We also report for the first time another striking manifestation of dynamic cell interaction, where confluent layers of Madin-Darby canine kidney cells (MDCK) on a single electrode are observed to oscillate in synchrony with a period of approximately 2.5 h.     

Correlated motion and oscillation of neighboring cells in vitro

It has long been realized that fibroblastic and epithelial cells establish recognizable patterns in tissue culture. This behavior implies that neighboring cells interact with one another to produce organized populations. Interaction between cells that are separated by many intervening cells is also possible and is demonstrated here using a special configuration of a biosensor referred to as electric cell-substrate impedance sensing (ECIS). Normally the electrical impedance of a single electrode covered with a confluent cell layer is measured, and the morphological changes of the cells are reflected in the impedance. In this case the cells are cultured on two closely spaced electrodes whose impedances are measured independently as a function of time, and communication between the cell populations is revealed as a correlation between these two time series. We also report for the first time another striking manifestation of dynamic cell interaction, where confluent layers of Madin-Darby canine kidney cells (MDCK) on a single electrode are observed to oscillate in synchrony with a period of approximately 2.5 h.     

Electrical impedance spectroscopy measurements using a four-electrode configuration improve on-line monitoring of cell concentration in adherent animal cell cultures

This paper describes the improvement in the use of electrical impedance spectroscopy (EIS) for animal cell concentration monitoring of adherent cultures by using a four-electrode configuration instead of the commonly used two-electrode configuration. This four-electrode configuration prevents cell concentration measurements from external masking effects such as the electrode covering ratio, the degree of cellular adherence to the electrodes and the impedance of the measuring electrodes. Cell concentration was monitored using both four-electrode and two-electrode configurations in vero cell and human mesenchymal stem cell cultures in order to analyze the attained improvement in two cell lines with opposite growth characteristics. The experiments performed with vero cell cultures evidenced that the four-electrode configuration enables cell concentration measurements along all culture phases, even once the culture reached cell confluence (over 2×10(5) cells/cm(2)), confirming that this configuration is less effected by all the external influences. The experiments performed with human mesenchymal stem cells demonstrated good sensitivity of the measurement at very low cell concentrations, as well as a very good robustness all over the 12-days experiment. Finally, off-line cell measurements during cell cultures proved good accuracy of impedance measurements carried out with a four-electrode configuration along all cell growth phases, enabling determination of relevant cell growth parameters.