Linear algebraic transformations of the bidomain equations: Implications for numerical methods

A mathematical framework is presented for the treatment of the bidomain equations used to model propagation in cardiac tissue. This framework is independent of the model used to represent membrane ionic currents and incorporates boundary conditions and other constraints. By representing the bidomain equations in the operator notation , various algebraic transformations can be expressed as , where P and Q are linear operators. The authors show how previous work fits into this framework and discuss the implications of various transformation for numerical methods of solution. Although such transformations allow many choices of independent variable, these results emphasize the fundamental importance of the transmembrane potential.     

A perturbation solution of the mechanical bidomain model

This research focuses on finding analytical solutions to the mechanical bidomain model for cardiac tissue. In particular, a perturbation expansion is used to analyze the equations, with the perturbation parameter being inversely proportional to the spring constant coupling the intracellular and extracellular spaces. The results indicate that the intracellular and extracellular pressures are not equal and that the two spaces can move relative to each other. This calculation is complicated enough to illustrate the implications of the mechanical bidomain model but is nevertheless simple enough to solve analytically. One application of the calculation is to the mechanical behavior of active cardiac tissue surrounding an ischemic region.     

An approximate solution to the periodic bidomain equations in one dimension

An approximate, computationally tractable solution is proposed for the potentials in the bidomain model with periodic intracellular junctions (the periodic bidomain model). This new approach is based on the one-dimensional rigorous spectral method described previously by Trayanova and Pilkington (IEEE Trans. Biomed. Eng., May 1993). The total solution to the one-dimensional periodic bidomain problem is decomposed in the spectral domain into solutions to (1) the single-fiber classical bidomain problem in which the intracellular conductivity value incorporates the average contribution from cytoplasm and junction and (2) the “junctional” potential problem due to the presence of junctions at discrete locations alone. Solving for the junctional term rigorously requires most of the numerical effort in the solution for the periodic bidomain potentials. Here the junctional potential is found approximately with little numerical effort. A comparison between the rigorous and the approximate solutions serves as a justification for the proposed approximate solution procedure. The procedure outlined in this paper is applicable to higher spatial dimensions where both tissue anisotropy and junctional inhomogeneities play a role in establishing the transmembrane potential distribution.     

A linear system of partial differential equations modeling the resting potential of a heart with regional ischemia

Ischemic ST-segment shift has been modeled using scalar stationary approximations of the bidomain model. Here, we propose an alternative simplification of the bidomain equations: a linear system modeling the resting potential, to be used in determining ischemic TP shift. Results of 2D and 3D simulations show that the linear system model is much more accurate than the scalar model. This improved accuracy is important if the model is to be used for solving the inverse problem of determining the size and location of an ischemic region. Furthermore, the model can provide insight into how the resting potential depends on the variations in the extracellular potassium concentration that characterize ischemic regions.     

The Pinwheel Experiment Revisited

The critical point hypothesis explains the origin of some cardiac arrhythmias, and the bidomain model describes electrical stimulation of the heart. In this paper, the critical point hypothesis is combined with the bidomain model. The result is four new predictions about the pinwheel experiment, a fundamental experiment in cardiac electrophysiology. These are: (1) The duration of the vulnerable period during cathodalS₂stimulation is longer for anS₁wavefront propagating perpendicular to the fibers than for anS₁wavefront propagating parallel to the fibers. (2) For anodalS₂stimulation with theS₁wavefront propagating parallel to the fibers, the vulnerable period splits into two periods with an “invulnerable period” between them. (3) For anodalS₂stimulation with theS₁wavefront propagating perpendicular to the fibers, the vulnerable period consists of only one period. (4) A previously suggested mechanism for the upper limit of vulnerability (S₂is so strong that the entire tissue is depolarized by an amount greater thanS*) is no longer applicable.     

Characterization of ischemia-induced loss of epithelial polarity

Summary Total renal ischemia for various time intervals (0–50) min) resulted in the rapid and duration-dependent redistribution of polarized membrane lipids and proteins in renal proximal tubule cells. Following only 15 min of ischemia, apical membrane enrichment of NaK-ATPase, normally a basolateral membrane (BLM) enzyme, had increased (1.6±0.6vs. 2.9±1.2,P<0.01). In vivo histochemical localization of NaK-ATPase showed reaction product throughout the apical microvillar region. PTH-stimulatable adenylate cyclase, another BLM protein, was also found in ischemic but not control apical membrane fractions. One dimensional SDS-PAGE showed four bands, present in control BLM and ischemic apical membranes, which could not be found in control apical membrane fractions. Immunohistochemical localization of leucine aminopeptidase (LAP) showed the enzyme was limited to the apical domain in control cells. Following ischemic injury (50 min), LAP staining could be seen within the cell and along the BLM. Following 24 hr of reperfusion, the BLM distribution of LAP was further enhanced. With cellular recovery from ischemic injury (5 days), LAP was again only visualized in the apical membrane. Duration-dependent alterations in apical and BLM lipids were also observed. Apical sphingomyelin and phosphatidylserine and the cholesterol-tophospholipid ratio decreased rapidly while apical phosphatidylcholine and phosphatidylinositol increased. Taken together, these results indicate renal ischemia causes rapid duration-dependent reversible loss of surface membrane polarity in proximal tubule cells.     

Virtual electrode effects in defibrillation

This modeling study demonstrates that a re-entrant activity in a sheet of myocardium can be extinguished by a defibrillation shock delivered via extracellular point-source electrodes which establish spatially non-uniform applied field. The tissue is represented as a homogeneous bidomain with unequal anisotropy ratios in the cardiac conductivities. Spiral wave re-entry is initiated in the bidomain sheet following an S1-S2 stimulation protocol. The results indicate that the point-source defibrillation shock establishes large-scale changes in transmembrane potential in the tissue (virtual electrodes) that are ‘superimposed’ over regions of various degrees of membrane refractoriness in the myocardium. The close proximity of large-scale shock-induced regions of alternating membrane polarity is central to the ability of the shock to terminate the spiral wave. The new wavefronts generated following anode/cathode break phenomena restrict the spiral wave and render the tissue too refractory to further maintain the re-entry. In contrast, shocks delivered via line electrodes establish, in close proximity to the electrode, changes in transmembrane potential that are of same-sign polarity. These shocks are incapable of terminating the re-entrant activation.     

Alterations in the Production of 14CO2 and [14C]Acetylcholine from [U-14C]Glucose in Brain Subregions Following Transient Forebrain Ischemia in the Rat

Abstract: The production of ¹⁴CO₂ and [¹⁴C]acetylcholine from [U-¹⁴C]glucose was determined in vitro using tissue prisms prepared from the dorsolateral striatum (a region developing extensive neuronal loss following ischemia) and the paramedian neocortex (an ischemia-resistant region) following 30 min of forebrain ischemia and recirculation up to 24 h. Measurements were determined under basal conditions (5 mMK⁺) and following K⁺ depolarization (31 mM K⁺). The production of ¹⁴CO₂ by the dorsolateral striatum was significantly reduced following 30 min of ischemia for measurements in either 5 or 31 mM K⁺ but recovered toward preischemic control values during the first hour of recirculation. Further recirculation resulted in ¹⁴CO₂ production again being reduced relative to control values but with larger differences (20–27% reductions) detectable under depolarized conditions at recirculation times up to 6 h. Samples from the paramedian neocortex showed no significant changes from control values at all time points examined. [¹⁴C]Acetylcholine synthesis, a marker of cholinergic terminals that is sensitive to changes in glucose metabolism in these structures, was again significantly reduced only in the dorsolateral striatum. However, even in this tissue, only small (nonstatistically significant) differences were seen during the first 6 h of recirculation, a finding suggesting that changes in glucose oxidation during this period were not uniform within all tissue components. The results of this study provide evidence that in a region susceptible to ischemic damage there were specific changes during early recirculation in the metabolic response to depolarization. This apparent inability to respond appropriately to an increased need for energy production could contribute to the further deterioration of cell function in vivo and ultimately to the death of some cells.     

Calculation of optical signal using three-dimensional bidomain/diffusion model reveals distortion of the transmembrane potential

Optical mapping experiments allow investigators to view the effects of electrical currents on the transmembrane potential, V_m, as a shock is applied to the heart. One important consideration is whether the optical signal accurately represents V_m. We have combined the bidomain equations along with the photon diffusion equation to study the excitation and emission of photons during optical mapping of cardiac tissue. Our results show that this bidomain/diffusion model predicts an optical signal that is much smaller than V_m near a stimulating electrode, a result consistent with experimental observations. Yet, this model, which incorporates the effect of lateral averaging, also reveals an optical signal that overestimates V_m at distances >1 mm away from the electrode. Although V_m falls off with distance r from the electrode as exp(−r/λ)/r, the optical signal decays as a simple exponential, exp(−r/λ). Moreover, regions of hyperpolarization adjacent to a cathode are emphasized in the optical signal compared to the region of depolarization under the cathode. Imaging methods utilizing optical mapping techniques will need to account for these distortions to accurately reconstruct V_m.     

Gene segments encoding membrane domains of the human immunoglobulin gamma 3 and alpha chains

The carboxyterminal region of the heavy chains, according to its hydrophilic or hydrophobic properties, determines whether the immunoglobulin will be secreted or membrane-bound. We have determined the nucleotide sequences of the human IGHG3, IGHA1, and IGHA2 membrane exons isolated from genomic DNA libraries. The IGHG3 M1 and M2 exons are separated by a long intron of 2.1 kilobases (kb) containing an highly repeated motif of 34 base pairs (bp). The IGHA1 and IGHA2 genes, like the mouse Igh-A gene, have a single exon encoding the extracellular, transmembrane, and cytoplasmic regions. For each class of immunoglobulins, the sequences of membrane exons are highly conserved between human and mouse, but no alignment is possible for the flanking regions. In contrast, for a same species, the sequences of the heavy chain membrane exons differ from one class to another. While the hydrophobic profile of the membrane core is well conserved, the cytoplasmic region differs in length and in composition. None of the intracellular domains presents the sequence implied in signal transduction, implying that membrane immunoglobulins need other proteins, which probably interact with the constant or membrane domain, to transmit signals leading to B-cell activation.The nucleotide sequence data reported in this paper have been submitted to the EMBL nucleotide sequence database and have been assigned the accession numbers M35288-91. Address correspondence and offprint requests to: M.-P. Lefranc.     

EFFECTS OF VARIATION IN INTERCELLULAR ELECTRICAL COUPLING ON SYNAPTIC POTENTIALS IN SMOOTH MUSCLE: A COMPUTATIONAL STUDY

We have computationally explored the effect of quantitative variations in the extent of cell-to-cell electrical coupling on the synaptic potentials generated in smooth muscle. Neuronally produced spontaneous excitatory junction potentials (SEJPs) generated in a cubical “bidomain” model of syncytial tissue were simulated computationally. It was found that SEJP properties vary conspicuously as the principal parameter of interest, the cell–to–cell coupling resistance, R_i, is altered. For example, on increasing R_i, SEJP peak amplitudes at node zero (the node of generation) increase dramatically, while amplitudes at nodes 1 and 2 (which are passively depolarized) become progressively lower fractions of the amplitude of the zeroeth-node SEJP. The time to peak of the SEJPs also increases concomitantly when R_i is elevated. These observations indicate the nature of variations in synaptic potentials that would be expected under conditions of altered intercellular electrical coupling in smooth muscle. We discuss their implications in relation to the physiology of syncytial tissue, and in the context of recent experimental observations made in the presence of a putative inhibitor of cell–to–cell electrical coupling, 1-heptanol.     

Reduced expression of NLRP3 and MEFV in human ischemic heart tissue

The innate immune system and, in particular, activation of the multi-protein complex known as the inflammasome complex are involved in ischemic injury in myocardial cells. The nucleotide-binding leucine-rich repeat-containing pyrin receptor 3 (NLRP3) inflammasome has been linked to inflammation and NLRP3 is especially important for increased inflammation in atherosclerosis, which may lead to myocardial infarction. Here we investigated how inflammasome molecules are affected in human ischemic heart tissue. Surprisingly the important member of the inflammasome complex, NLRP3, displayed markedly decreased levels in human ischemic heart tissue compared with non ischemic control heart tissue. However, subsequent gene analysis revealed mutations in NLRP3 in human ischemic heart tissues but not in non-ischemic control tissue. Gene polymorphisms in the NLRP3 inflammasome have been shown to be associated with increased IL-1β and IL-18 production and severe inflammation.The autoinflammatory disorder familial Mediterranean fever (FMF) is associated with decreased expression of the Mediterranean fever gene (MEFV) and increased inflammation. We also observed reduced expression of MEFV in ischemic versus non-ischemic heart tissue. Further analyses showed a mutation in MEFV in human ischemic heart tissue but not in non-ischemic control tissue.Our data show that defects in the inflammasome and associated proteins may be involved in promoting ischemic heart disease.     

Neurotoxic lipid peroxidation species formed by ischemic stroke increase injury

Stroke is the third leading cause of death in the United States, yet no neuroprotective agents for treatment are clinically available. There is a pressing need to understand the signaling molecules that mediate ischemic cell death and identify novel neuroprotective targets. Cyclopentenone isoprostanes (IsoPs), formed after free radical-mediated peroxidation of arachidonic acid, are used as markers of stress, but their bioactivity is poorly understood. We have recently shown that 15-A_2t-IsoP is a potent neurotoxin in vitro and increases the free radical burden in neurons. In this work, we demonstrate that 15-A_2t-IsoP is abundantly produced in stroke-infarcted human cortical tissue. Using primary neuronal cultures we found that minimally toxic exposure to 15-A_2t-IsoP does not alter ATP content, but in combination with oxygen glucose deprivation resulted in a significant hyperpolarization of the mitochondrial membrane and dramatically increased neuronal cell death. In the presence of Ca²⁺, 15-A_2t-IsoP led to a rapid induction of the permeability transition pore and release of cytochrome c. Taken with our previous work, these data support a model in which ischemia causes generation of reactive oxygen species, calcium influx, lipid peroxidation, and 15-A_2t-IsoP formation. These factors combine to enhance opening of the permeability transition pore leading to cell death subsequent to mitochondrial cytochrome c release. These data are the first documentation of significant 15-A_2t-IsoP formation after acute ischemic stroke and suggest that the addition of 15-A_2t-IsoP to in vitro models of ischemia may help to more fully recapitulate stroke injury.     

Structure and Histochemistry of the Skin of a Torrent Catfish, Liobagrus mediadiposalis

The epidermis of the torrent catfish, Liobagrus mediadiposalis, consists of three layers: the outermost layer, middle layer and stratum germinativum. The epidermis consists of two types of skin glands, small mucus cell and voluminous club cell. The unicellular mucus cell contains acid sulfomucins (some sialomucins) and the club cell, sometimes binucleate, is proteinaceous. Well-developed vascularization is one of the characteristics of epidermis of L. mediadiposalis. Well-developed lymphatic spaces contain lymphocytes in the epidermis. The dermis lacks scales and consists mostly of a thick, dense connective tissue; its superficial region just below the basal membrane is supplied with fine blood capillaries. These histological features of the skin in L. mediadiposalis are consistent with that required for cutaneous respiration.     

A numerically efficient model for simulation of defibrillation in an active bidomain sheet of myocardium

Presented here is an efficient algorithm for solving the bidomain equations describing myocardial tissue with active membrane kinetics. An analysis of the accuracy shows advantages of this numerical technique over other simple and therefore popular approaches. The modular structure of the algorithm provides the critical flexibility needed in simulation studies: fiber orientation and membrane kinetics can be easily modified. The computational tool described here is designed specifically to simulate cardiac defibrillation, i. e., to allow modeling of strong electric shocks applied to the myocardium extracellularly. Accordingly, the algorithm presented also incorporates modifications of the membrane model to handle the high transmembrane voltages created in the immediate vicinity of the defibrillation electrodes.     

Direct Evidence for the Involvement of Free Radicals in Ischemic Insult to the Intestine

Ischemia of rat intestine was induced in vivo by occlusion of the superior mesenteric artery (SMA) for 15 min. Sodium salicylate, 100 mg/kg, given IP, 30 min prior to the ischemic event served as a specific trap for hydroxyl radicals and provided direct evidence for the involvement of free radicals during the ischemic insult. Portions of the bowel were sequentially isolated and removed. The hydroxylation products. dihyd-roxybenzoic acid (DHBA) derivatives were isolated, identified and qunatified by HPLC coupled with electrochemical detection (ECD). The level of 2,5-DHBA (Mean ± SE, ng/g tissue) in the preischemic bowel (N = 21) was 241.8 ± 10.0. It rose significantly to 313.3 ± 15.5 in the ischemic specimen (p = 0.0129) and remained unchanged in the reperfusion period (322.8 ± 15.5). The histological examination correlated well with these levels: mild villi damage in the ischemic period with no further damage in the reperfusion period.     

Lipid Peroxide Formation and Membrane Damage in Endotoxin-Poisoned Mice

Lipid peroxide formation and plasma membrane damage in mouse liver following the administration of Salmonella endotoxin were examined. The liver lipoperoxide level was markedly elevated in animals given endotoxin compared with that in the controls, and returned to its normal range after 2 days. On the other hand, superoxide dismutase activity was decreased by 18–48 hr after endotoxin injection, thereafter tending to increase. Glutathione reductase and glutathione peroxidase activities declined in the liver 18 hr after the injection. The endotoxin resulted in much lower lipoperoxide formation in the livers of tolerant mice than in those of the poisoned mice. The lipoperoxide level in endotoxin-poisoned mice after the administration of α-tocopherol was lower than that in the controls, and α-tocopherol administration prevented completely the membrane protein damage that arose from endotoxin challenge. After glutathione administration the membranes of the poisoned mice also returned to almost the normal disk electrophoretic profile. These results suggest that lipid peroxide formation in the liver plasma membrane caused by free radicals might occur in a tissue ischemic state in endotoxicosis.     

Oligomeric Dop1p is Part of the Endosomal Neo1p‐Ysl2p‐Arl1p Membrane Remodeling Complex

Yeast Dop1p is an essential protein that is highly conserved in evolution and whose function is largely unknown. Here, we provide evidence that Dop1p localizes to endosomes and exists in a complex with two other conserved proteins: Neo1p, a P₄‐ATPase and putative flippase, and the scaffolding protein Ysl2p/Mon2p. The latter operates during membrane budding at the tubular endosomal network/trans‐Golgi network (TEN/TGN) in a process that includes clathrin recruitment via adaptor proteins. Consistent with a role for Dop1p during this process, temperature‐sensitive dop1‐3 cells accumulate multivesicular, elongated tubular and ring‐like structures similar to those displayed by neo1 and ysl2 mutants. In further agreement with the concept of Dop1p‐Neo1p‐Ysl2p complex formation and co‐operation, we show that dop1‐3 cells exhibit reduced levels of Neo1p and Ysl2p at steady state. Conversely, mutations or deletions in NEO1 and YSL2 lead to a decrease in Dop1p levels. In addition to binding to Neo1p and Ysl2p, Dop1p can form dimers or multimers. A critical region for dimerization resides in the C‐terminus with leucine zipper‐like domains. Dop1p's membrane association is largely mediated by its internal region, but Ysl2p might not be crucial for membrane recruitment.     

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.     

Sequence–function relationships in MalG, an inner membrane protein from the maltose transport system in Escherichia coli

The maIG gene encodes a hydrophobic cytoplasmic membrane protein which is required for the energy-dependent transport of maltose and maltodextrins in Escherichia coli. The MalG protein, together with MalF and MalK proteins, forms a multimeric complex in the membrane consisting of two MalK subunits for each MalF and MalG subunit. Fifteen mutations have been isolated in malG by random linker insertion mutagenesis. Two regions essential for maltose transport have been identified. In particular, a hydro philic region containing the peptidic motif EAA—G———I-LP, highly conserved among inner membrane proteins from binding protein-dependent transport systems, is essential for maltose transport. The results also show that several regions of MalG are not essential for function. A region (residues 30–50) encompassing the first predicted transmembrane segment and the first periplasmic loop in MalG may be modified extensively with little effect on maltose transport and no effect on the stability and the localization of the protein. A region located at the middle of the protein (residues 153–157) is not essential for the function of the protein. A region, essential for maltodextrin utilization but not for maltose transport, has been identified near the C-terminus of the protein.