A finite difference model of O2 transport in aortic valve cusps: importance of intrinsic microcirculation

Recent studies have reported the presence of a microcirculation within the tissue of aortic valves. To test the hypothesis that this vascular bed is needed to satisfy the oxygen demands of the cusp tissue, a two-dimensional (2D) finite difference model of oxygen diffusion was developed. The in vivo environment was modeled for vascular and avascular cusps using thickness data from precise radiographic measurements of fresh porcine valves, and O2 diffusivity (DO2) and O2 consumption (VO2) values from experimental data. The location and density of the cusp vasculature were determined by the model to prevent oxygen levels from falling to zero. Validation of the model was performed by simulation of the experimental measurements of cusp DO2 and VO2. For a test cusp with uniform thickness, the model returned simulated DO2 and VO2 measurements within 1.43% and 0.18% difference of the true parameter values, respectively. For native cusps, the simulated DO2 measurements were sensitive to thickness variations (-38 to +21% difference), whereas the VO2 measurements were minimally affected (8% difference). An improved DO2 measurement technique was found to reduce these errors to <5% and is recommended for analysis of experimental data. In the avascular case, the model predicted large regions of hypoxic tissue, whereas in the vascular case, the model predicted vessel locations and densities similar to what was experimentally observed in porcine cusps. Overall, the in vivo model developed in this study confirmed the need for an intrinsic microcirculation in the thicker basal regions of aortic cusps.     

Stochastic model of central synapses: slow diffusion of transmitter interacting with spatially distributed receptors and transporters.

A detailed mathematical analysis of the diffusion process of neurotransmitter inside the synaptic cleft is presented and the spatio-temporal concentration profile is calculated. Using information about the experimentally observed time course of glutamate in the cleft the effective diffusion coefficient Dnet is estimated as Dnet approximately 20-50 nm(2) microseconds(-1), implying a strong reduction compared with free diffusion in aqueous solution. The tortuosity of the cleft and interactions with transporter molecules are assumed to affect the transmitter motion. We estimate the transporter density to be 5170 to 8900 micrometer(-2) in the synaptic cleft and its vicinity, using the experimentally observed time constant of glutamate. Furthermore a theoretical model of synaptic transmission is presented, taking the spatial distribution of post-synaptic (AMPA-) receptors into account. The transmitter diffusion and receptor dynamics are modeled by Monte Carlo simulations preserving the typically observed noisy character of post-synaptic responses. Distributions of amplitudes, rise and decay times are calculated and shown to agree well with experiments. Average open probabilities are computed from a novel kinetic model and are shown to agree with averages over many Monte Carlo runs. Our results suggest that post-synaptic currents are only weakly potentiated by clustering of post-synaptic receptors, but increase linearly with the total number of receptors. Distributions of amplitudes and rise times are used to discriminate between different morphologies, e.g. simple and perforated synapses. A skew in the miniature amplitude distribution can be caused by multiple release of pre-synaptic vesicles at perforated synapses.     

A computational study of the effect of vasomotion on oxygen transport from capillary networks

The objective of this study was to investigate the effect of arteriolar vasomotion on oxygen transport from capillary networks. A computational model was used to calculate blood flow and oxygen transport from a simulated network of striated muscle capillaries. For varying tissue oxygen consumption rates, the importance of the frequency and amplitude of vasomotion-induced blood flow oscillations was studied. The effect of myoglobin on oxygen delivery during vasomotion was also examined. In the absence of myoglobin, it was found that when consumption is high enough to produce regions of hypoxia under steady flow conditions, vasomotion-induced flow oscillations can significantly increase tissue oxygenation and decrease oxygen transport heterogeneity. The largest effect was seen for low-frequency, high-amplitude oscillations (1.5-3 cycles min(-1), 90% of steady-state velocity). By contrast, at physiological tissue myoglobin concentrations, vasomotion did not improve tissue oxygenation. This unexpected finding is due to the buffering effect of myoglobin, suggesting that in highly aerobic muscles short-term storage of oxygen is more important than the possibility of increasing transport through vasomotion.     

Potassium ion accumulation at the external surface of the nodal membrane in frog myelinated fibers.

Potassium accumulation associated with outward membrane potassium current was investigated experimentally in myelinated fibers and analyzed in terms of two models-three-compartment and diffusion in an unstirred layer. In the myelinated fibers, as in squid giant axons, the three-compartment model satisfactorily describes potassium accumulation. Within this framework the average space thickness, theta, in frog was 5,900 +/- 700 A, while the permeability coefficient of the external barrier, PK, was (1.5 +/- 0.1) X 10(-2) cm/s. The model of ionic diffusion in an unstirred aqueous layer adjacent to the axolemma, as an alternative explanation for ion accumulation, was also consistent with the experimental data, provided that D, the diffusion constant, was (1.8 +/- 0.2) X 10(-6) cm/s and l, the unstirred layer thickness, was 1.4 +/- 0.1 micron, i.e., similar to the depth of the nodal gap. An empirical equation relating the extent of potassium accumulation to the amplitude and duration of depolarization is given.     

Water and deuterium oxide permeability through aquaporin 1: MD predictions and experimental verification

Determining the mechanisms of flux through protein channels requires a combination of structural data, permeability measurement, and molecular dynamics (MD) simulations. To further clarify the mechanism of flux through aquaporin 1 (AQP1), osmotic p(f) (cm(3)/s/pore) and diffusion p(d) (cm(3)/s/pore) permeability coefficients per pore of H(2)O and D(2)O in AQP1 were calculated using MD simulations. We then compared the simulation results with experimental measurements of the osmotic AQP1 permeabilities of H(2)O and D(2)O. In this manner we evaluated the ability of MD simulations to predict actual flux results. For the MD simulations, the force field parameters of the D(2)O model were reparameterized from the TIP3P water model to reproduce the experimentally observed difference in the bulk self diffusion constants of H(2)O vs. D(2)O. Two MD systems (one for each solvent) were constructed, each containing explicit palmitoyl-oleoyl-phosphatidyl-ethanolamine (POPE) phospholipid molecules, solvent, and AQP1. It was found that the calculated value of p(f) for D(2)O is approximately 15% smaller than for H(2)O. Bovine AQP1 was reconstituted into palmitoyl-oleoyl-phosphatidylcholine (POPC) liposomes, and it was found that the measured macroscopic osmotic permeability coefficient P(f) (cm/s) of D(2)O is approximately 21% lower than for H(2)O. The combined computational and experimental results suggest that deuterium oxide permeability through AQP1 is similar to that of water. The slightly lower observed osmotic permeability of D(2)O compared to H(2)O in AQP1 is most likely due to the lower self diffusion constant of D(2)O.     

Mathematical modelling of oxygen transport to tissue

 The equations governing oxygen transport from blood to tissue are presented for a cylindrical tissue compartment, with blood flowing along a co–axial cylindrical capillary inside the tissue. These governing equations take account of: (i) the non–linear reactions between oxygen and haemoglobin in blood and between oxygen and myoglobin in tissue; (ii) diffusion of oxygen in both the axial and radial directions; and (iii) convection of haemoglobin and plasma in the capillary. A non–dimensional analysis is carried out to assess some assumptions made in previous studies. It is predicted that: (i) there is a boundary layer for oxygen partial pressure but not for haemoglobin or myoglobin oxygen saturation close to the inflow boundary in the capillary; (ii) axial diffusion may not be neglected everywhere in the model; (iii) the reaction between oxygen and both haemoglobin and myoglobin may be assumed to be instantaneous in nearly all cases; and (iv) the effect of myoglobin is only significant for tissue with a low oxygen partial pressure. These predictions are validated by solving the full equations numerically and are then interpreted physically. Received: 13 October 2000 / Revised version: 12 June 2001 / Published online: 17 May 2002     

A model of the oscillatory metabolism of activated neutrophils

We present a two-compartment model to explain the oscillatory behavior observed experimentally in activated neutrophils. Our model is based mainly on the peroxidase-oxidase reaction catalyzed by myeloperoxidase with melatonin as a cofactor and NADPH oxidase, a major protein in the phagosome membrane of the leukocyte. The model predicts that after activation of a neutrophil, an increase in the activity of the hexose monophosphate shunt and the delivery of myeloperoxidase into the phagosome results in oscillations in oxygen and NAD(P)H concentration. The period of oscillation changes from >200 s to 10-30 s. The model is consistent with previously reported oscillations in cell metabolism and oxidant production. Key features and predictions of the model were confirmed experimentally. The requirement of the hexose monophosphate pathway for 10 s oscillations was verified using 6-aminonicotinamide and dexamethasone, which are inhibitors of glucose-6-phosphate dehydrogenase. The role of the NADPH oxidase in promoting oscillations was confirmed by dose-response studies of the effect of diphenylene iodonium, an inhibitor of the NADPH oxidase. Moreover, the model predicted an increase in the amplitude of NADPH oscillations in the presence of melatonin, which was confirmed experimentally. Successful computer modeling of complex chemical dynamics within cells and their chemical perturbation will enhance our ability to identify new antiinflammatory compounds.     

Quantification of the effects of vasomotion on mass transport to tissue from axisymmetric blood vessels

The process known as vasomotion, rhythmic oscillations in vessel diameter, has been proposed to act as a protective mechanism for tissue under conditions of reduced perfusion, since it is frequently only observed experimentally when perfusion levels are reduced. This could be due to a resultant increase in oxygen transport from the vasculature to the surrounding tissue, either directly or indirectly. It is thus potentially of significant clinical interest as a warning signal for ischemia. However, there has been little analysis performed to quantify the effects of vessel wall movement on time-averaged mass transport. We thus present a detailed analysis of such mass transport for an axisymmetric vessel with a periodically oscillating wall, by solving the non-linear mass transport equation, and quantify the differences between the time-averaged mass transport under conditions of no oscillation (i.e. the steady-state) and varying wall oscillation amplitude. The results show that if the vessel wall alone is oscillated, with an invariant wall concentration, the time-averaged mass transport is reduced relative to the steady-state, but if the vessel wall concentration is also oscillated, then mass transport is increased, although this is generally only true when these oscillate in phase with each other. The influence of Péclet number and the non-dimensional rate of consumption of oxygen in tissue, as well as the amplitude of oscillations, are fully characterised. We conclude by considering the likely implications of these results in the context of oxygen transport to tissue.     

Diffusive coupling and network periodicity: a computational study

Diffusive coupling (nearest-neighbor coupling) is the most common type of coupling present in many systems. Previous experimental and theoretical studies have shown that potassium lateral diffusion coupling (i.e., diffusive coupling) can be responsible for synchronization of neuronal activity. Recent in vivo experiments carried out with anesthetized rat hippocampus suggested that the extracellular potassium could play an important role in the generation of a novel type of epileptiform nonsynaptic activity. Yet, the role of potassium in the generation of seizures remains controversial. We tested the hypothesis that potassium lateral diffusion coupling is responsible for the coupling mechanisms for network periodicity in a nonsynaptic model of epilepsy in vivo using a CA1 pyramidal neuron network model The simulation results show that 1), potassium lateral diffusion coupling is crucial for establishing epileptiform activity similar to that generated experimentally; and 2), there exists a scaling relation between the critical coupling strength and the number of cells in the network. The results not only agree with the theoretical prediction, but strongly suggest that potassium lateral diffusion coupling, a physiological realization of the concept of diffusive coupling, can play an important role in entraining periodicity in a nonsynaptic neural network.     

Computer simulation of damped oscillations during peroxidase-catalyzed oxidation of indole-3-acetic acid

Oscillation patterns in horseradish peroxidase (HRP)-catalyzed oxidation of indole-3-acetic acid (IAA) at neutral pH were studied using computer simulation. Under certain conditions, such as the presence of a reaction promoter and continuous intake of oxygen from the gaseous phase, the simulated system exhibits damped oscillations of the concentrations of oxygen in the aqueous phase, [O(2)](aq), and of all the reaction intermediates. The critical concentration of oxygen in aqueous phase, [O(2)](cr)(aq), was used to describe the nature of the oscillations. The critical concentration is the concentration at which the system abruptly changes its properties. If [O(2)](aq) is higher than [O(2)](cr)(aq) then the reaction develops as an avalanche, otherwise, the reaction stops. The nature of oscillations is accounted for by the interaction of two processes: the consumption/accumulation of oxygen and the accumulation/consumption of reaction intermediates. Oscillations are always damped. Neither HRP or umbelliferone (Umb) deactivation nor IAA consumption can account for the damping. The nature of the damping is determined by the termination reactions of free radical intermediates and ROOH. The three major parameters of oscillations: period of oscillations, initial amplitude of oscillations and the rate of damping were studied as functions of: (i) oxygen concentration in the gaseous phase, (ii) initial oxygen concentration in aqueous phase, (iii) the concentration of IAA and (iv) the initial concentration of HRP.     

Pattern formation in an immobilized bienzyme system. A morphogenetic model.

Experimental and theoretical studies of a reaction-diffusion model of two immobilized enzymes participating in the cellular acid-base metabolism, namely glutaminase and urease, are presented. The system shows an unstable steady state at pH 6.0, where any perturbation will drive the system towards a more alkaline or more acidic pH, owing to the autocatalytic behaviour with respect to pH exhibited by both enzymes. When diffusion is coupled to reaction by means of immobilization, different patterns of the internal pH profile appear across the membrane. If the bienzymic membrane is subjected to a perturbation at its boundaries, of the same amplitude but in opposite directions, the internal pH evolves through an asymmetric pattern to attain a nearly symmetric distribution of pH. The pH value at the final steady state is more acidic or more alkaline than the initial state according to the initial and boundary conditions. The final nearly symmetric state is attained more rapidly when less enzyme is immobilized (1.8 x 10(-4) M.s-1 as against 3.3 x 10(-4) M.s-1 of total enzyme activity in the membrane volume). The experimental results agree rather well qualitatively with numerical predictions of the model equations.     

Mitochondrial coupling in vivo in mouse skeletal muscle

The coupling of mitochondrial ATP synthesis and oxygen consumption (ratio of ATP and oxygen fluxes, P/O) plays a central role in cellular bioenergetics. Reduced P/O values are associated with mitochondrial pathologies that can lead to reduced capacity for ATP synthesis and tissue degeneration. Previous work found a wide range of values for P/O in normal mitochondria. To measure mitochondrial coupling under physiological conditions, we have developed a procedure for determining the P/O of skeletal muscle in vivo. This technique measures ATPase and oxygen consumption rates during ischemia with ³¹P magnetic resonance and optical spectroscopy, respectively. This novel approach allows the independent quantitative measurement of ATPase and oxygen flux rates in intact tissue. The quantitative measurement of oxygen consumption is made possible by our ability to independently measure the saturations of hemoglobin (Hb) and myoglobin (Mb) from optical spectra. Our results indicate that the P/O in skeletal muscle of the mouse hindlimb measured in vivo is 2.16 ± 0.24. The theoretical P/O for resting muscle is 2.33. Systemic treatment with 2,4-dinitrophenol to partially uncouple mitochondria does not affect the ATPase rate in the mouse hindlimb but nearly doubles the rate of oxygen consumption, reducing in vivo P/O to 1.37 ± 0.22. These results indicate that only a small fraction of the oxygen consumption in resting mouse skeletal muscle is nonphosphorylating under physiological conditions, suggesting that mitochondria are more tightly coupled than previously thought. P/O; oxidative phosphorylation; proton leak; optical spectroscopy     

Oscillations in cytosolic free Ca2+, oxygen consumption, and insulin secretion in glucose-stimulated rat pancreatic islets

Insulin secretion in the intact organism, and by the perfused pancreas and groups of isolated perifused islets, is pulsatile. We have proposed a metabolic model of glucose-induced insulin secretion in which oscillations in the ATP/ADP ratio drive alterations in metabolic and electrical events that lead to insulin release. A key prediction of our model is that metabolically driven Ca2+ oscillations will also occur. Using the fluorescent Ca2+ probe, fura 2, digital image analysis, and sensitive O2 electrodes, we investigated cytosolic free Ca2+ responses and O2 consumption in perifused rat islets that had been maintained in culture for 1-4 days. We found that elevated ambient glucose increased the average cytosolic free Ca2+ level, the ATP/ADP ratio, and oxygen consumption, as previously found in freshly isolated islets. Oscillatory patterns were obtained for Ca2+, O2 consumption, and insulin secretion in the presence of 10 and 20 mM glucose. Very low amplitude oscillations in cytosolic free Ca2+ were observed at 3 mM nonstimulatory glucose levels. Evaluation of the Ca2+ responses of a large series of individual islets, monitored by digital image analysis and perifused at both 3 and 10 mM glucose, indicated that the rise in glucose concentration caused more than a doubling of the average cytosolic free Ca2+ value and a 4-fold increase in the amplitude of the oscillations with little change in period. The pattern of Ca2+ change within the islets was consistent with recruitment of responding cells. The coexistence of oscillations with similar periods in insulin secretion, oxygen consumption, and cytosolic free Ca2+ is consistent with the model of metabolically driven pulsatile insulin secretion.     

Mathematical model of oxygen distribution in engineered cardiac tissue with parallel channel array perfused with culture medium containing oxygen carriers

A steady-state model of oxygen distribution in a cardiac tissue construct with a parallel channel array was developed and solved for a set of parameters using the finite element method and commercial software (FEMLAB). The effects of an oxygen carrier [Oxygent; 32% volume perfluorocarbon (PFC) emulsion] were evaluated. The parallel channel array mimics the in vivo capillary tissue bed, and the PFC emulsion has a similar role as the natural oxygen carrier hemoglobin in increasing total oxygen content. The construct was divided into an array of cylindrical domains with a channel in the center and tissue space surrounding the channel. In the channel, the main modes of mass transfer were axial convection and radial diffusion. In the tissue region, mass transfer was by axial and radial diffusion, and the consumption of oxygen was by Michaelis-Menten kinetics. Neumann boundary conditions were imposed at the channel centerline and the half distance between the domains. Supplementation of culture medium by PFC emulsion improved mass transport by increasing convective term and effective diffusivity of culture medium. The model was first implemented for the following set of experimentally obtained parameters: construct thickness of 0.2 cm, channel diameter of 330 mum, channel center-to-center spacingof 700 microm, and average linear velocity per channel of 0.049 cm/s, in conjunction with PFC supplemented and unsupplemented culture medium. Subsequently, the model was used to define favorable scaffold geometry and flow conditions necessary to cultivate cardiac constructs of high cell density (10(8) cells/ml) and clinically relevant thickness (0.5 cm). In future work, the model can be utilized as a tool for optimization of scaffold geometry and flow conditions.     

Diagnostics of plasma oscillations by the field of surface waves guided by a dielectric waveguide

The potentialities of the diagnostic method for determining the plasma parameters by recording the surface waves guided by a dielectric waveguide and scattered by plasma oscillations are discussed. The use of surface (slowed) waves makes it possible to improve both the sensitivity and spatial resolution of measurements. The scattering is the most intense near the waveguide cutoff, at which the dependence of the wave propagation constant on the plasma density is the steepest. It is shown experimentally that the method proposed makes it possible to determine the discharge plasma density and electron energy and to estimate the amplitude of the RF field of the plasma waves forming the discharge and the amplitude of plasma density oscillations in these waves. The data obtained from the measurements of the amplitudes of both high-and low-frequency plasma density oscillations by the proposed method agree satisfactorily with theoretical predictions. The experimental data on the plasma density are confirmed by other diagnostic measurements. The ways of reducing measurement errors are proposed.     

Effect of convection in capillaries on oxygen removal from arterioles in striated muscle

Based on experimental data that show the presence of significant oxygen saturation gradients in precapillary arterioles, it has been suggested that the in vivo permeability to oxygen of resting striated muscle may be significantly higher than the corresponding in vitro value obtained in unperfused tissue samples (Popel et al., 1989b, Adv. expl. Med. Biol. 247, 215). The present study performs two analyses to further compare theoretical predictions with experimental data obtained under control conditions and during hemodilution and hemoconcentration. First, it is shown that, in principle, a capillary-perfused tissue layer with a thickness of a few hundred microns is necessary to convectively carry the experimentally determined amount of oxygen released by precapillary arterioles under control and hemodiluted conditions. This capacity to convect oxygen depends strongly on the resting tissue oxygen tension. Second, a more general version of a previous model (Weerappuli & Popel, 1989, J. Biomech. Eng. 111, 24) is used to examine whether changes made in the model parameters within the physiological range of values can explain the experimentally measured flux. The results show that the theoretical predictions can be made compatible with experimental observations if the in vivo permeability of perfused tissue to oxygen is assumed to be one to two orders of magnitude higher than the in vitro value. Furthermore, the predicted in vivo permeability for perfused tissue surrounding an arteriole varies with the arteriolar luminal oxygen tension and flow. This may be due to simplifying approximations made in the model or possible experimental artifacts. Alternatively, it could also be speculated that this variability indicates the flow dependency of the permeability of perfused tissue to oxygen.     

Recording of spontaneous oscillations in the procerebrum of terrestrial mollusk Helix in free behavior

Procerebrum is the central part of the olfactory system in terrestrial snails. Spontaneous rhythmic oscillations were described in this structure. The role of these oscillations in the mechanisms of odor perception and discrimination is unknown yet. Electrical activity of the Helix procerebrum was recorded in vivo. Changes in spontaneous rhythmic oscillations in response to olfactory stimulation were observed. Within the first 10 s after odor application (cineole) in low concentration, a statistically significant decrease in the frequency and increase in the amplitude of procerebrum oscillations were revealed in freely behaving animals. Timing of those changes corresponded to the time of defensive reaction realization of the tentacle withdrawal. The increase in the amplitude and a tendency to a decrease in the frequency of oscillations in response to odor application in high concentration were observed in time period 11-20 s, which corresponded to an increased duration of tentacle withdrawal. The results suggest an implicit relation of the amplitude and frequency of oscillations in odor perception and discrimination.