Gas bubble dynamics in soft materials.

Epstein and Plesset's seminal work on the rate of gas bubble dissolution and growth in a simple liquid is generalized to render it applicable to a gas bubble embedded in a soft elastic solid. Both the underlying diffusion equation and the expression for the gas bubble pressure were modified to allow for the non-zero shear modulus of the medium. The extension of the diffusion equation results in a trivial shift (by an additive constant) in the value of the diffusion coefficient, and does not change the form of the rate equations. But the use of a generalized Young-Laplace equation for the bubble pressure resulted in significant differences on the dynamics of bubble dissolution and growth, relative to an inviscid liquid medium. Depending on whether the salient parameters (solute concentration, initial bubble radius, surface tension, and shear modulus) lead to bubble growth or dissolution, the effect of allowing for a non-zero shear modulus in the generalized Young-Laplace equation is to speed up the rate of bubble growth, or to reduce the rate of bubble dissolution, respectively. The relation to previous work on visco-elastic materials is discussed, as is the connection of this work to the problem of Decompression Sickness (specifically, "the bends"). Examples of tissues to which our expressions can be applied are provided. Also, a new phenomenon is predicted whereby, for some parameter values, a bubble can be metastable and persist for long times, or it may grow, when embedded in a homogeneous under-saturated soft elastic medium.

Thermodynamic stability in elastic systems: Hard spheres embedded in a finite spherical elastic solid.

We determined the total system elastic Helmholtz free energy, under the constraints of constant temperature and volume, for systems comprised of one or more perfectly bonded hard spherical inclusions (i.e. "hard spheres") embedded in a finite spherical elastic solid. Dirichlet boundary conditions were applied both at the surface(s) of the hard spheres, and at the outer surface of the elastic solid. The boundary conditions at the surface of the spheres were used to describe the rigid displacements of the spheres, relative to their initial location(s) in the unstressed initial state. These displacements, together with the initial positions, provided the final shape of the strained elastic solid. The boundary conditions at the outer surface of the elastic medium were used to ensure constancy of the system volume. We determined the strain and stress tensors numerically, using a method that combines the Neuber-Papkovich spherical harmonic decomposition, the Schwartz alternating method, and Least-squares for determining the spherical harmonic expansion coefficients. The total system elastic Helmholtz free energy was determined by numerically integrating the elastic Helmholtz free energy density over the volume of the elastic solid, either by a quadrature, or a Monte Carlo method, or both. Depending on the initial position of the hard sphere(s) (or equivalently, the shape of the un-deformed stress-free elastic solid), and the displacements, either stationary or non-stationary Helmholtz free energy minima were found. The non-stationary minima, which involved the hard spheres nearly in contact with one another, corresponded to lower Helmholtz free energies, than did the stationary minima, for which the hard spheres were further away from one another.

A fully dynamical theory for the rate of arterial gas embolism growth and dissolution.

An approximation that had been widely used for more than 70 years to estimate the concentration of a dissolved volatile solute at the external surface of a gas bubble was found to be inaccurate for small bubbles. The approximation was to assume that a Henry's law-based partition equilibrium determines the solute's concentration at the bubble's external surface. An alternative model is developed here, wherein solute exchange across the bubble surface, and solute diffusion in the medium next to the bubble, collectively determine the solute's concentration near the bubble's external surface. The previous model was found to predict both gas bubble growth and dissolution to be too rapid, for bubbles with a radius R≤20µ. We found that treating solute exchange across the bubble surface, and solute diffusion in the medium next to the bubble, as a collective dynamical process, was necessary to accurately model the dynamics of an arterial gas embolism (AGE) with a radius R≤20µ. An AGE of this size can block or interfere with blood flow through capillaries and arterioles.

Free energy wells for small gas bubbles in soft deformable materials.

Thermodynamic expressions are derived for the system relative Gibbs free energy, and the relative Gibbs free energy per bubble, for all possible equilibrium bubble states that can form in a soft slightly rigid material, initially supersaturated with a dissolved inert gas (N(2)). While the thermodynamic manipulations are exact, the final expressions are approximate, due to an approximation made in deriving the expression for the elastic free energy of a soft material containing more than a single bubble. The expressions predict that provided the shear modulus of the soft material is not negligibly small, free energy wells which stabilize small gas bubbles for finite periods of time exist in such materials. This is consistent with a previous calculation, based solely on the bubble pressure equation, which resulted in the conjecture that bubbles found in soft materials with some rigidity (or shear resistance) are likely to be small. The possible relevance of this to the field of decompression sickness is outlined.

An explicitly multi-component arterial gas embolus dissolves much more slowly than its one-component approximation.

We worked out the growth and dissolution rates of an arterial gas embolism (AGE), to illustrate the evolution over time of its size and composition, and the time required for its total dissolution. We did this for a variety of breathing gases including air, pure oxygen, Nitrox and Heliox (each over a range of oxygen mole fractions), in order to assess how the breathing gas influenced the evolution of the AGE. The calculations were done by numerically integrating the underlying rate equations for explicitly multi-component AGEs, that contained a minimum of three (water, carbon dioxide and oxygen) and a maximum of five components (water, carbon dioxide, oxygen, nitrogen and helium). The rate equations were straight-forward extensions of those for a one-component gas bubble. They were derived by using the Young-Laplace equation and Dalton's law for the pressure in the AGE, the Laplace equation for the dissolved solute concentration gradients in solution, Henry's law for gas solubilities, and Fick's law for diffusion rates across the AGE/arterial blood interface. We found that the 1-component approximation, under which the contents of the AGE are approximated by its dominant component, greatly overestimates the dissolution rate and underestimates the total dissolution time of an AGE. This is because the 1-component approximation manifestly precludes equilibration between the AGE and arterial blood of the inspired volatile solutes (O2, N2, He) in arterial blood. Our calculations uncovered an important practical result, namely that the administration of Heliox, as an adjunct to recompression therapy for treating a suspected N2-rich AGE must be done with care. While Helium is useful for preventing nitrogen narcosis which can arise in aggressive recompression therapy wherein the N2 partial pressure can be quite high (e.g.∼5 atm), it also temporarily expands the AGE, beyond the expansion arising from the use of Oxygen-rich Nitrox. For less aggressive recompression therapy wherein nitrogen narcosis is not a significant concern, Oxygen-rich Nitrox is to be preferred, both because it does not temporarily expand the AGE as much as Heliox, and because it is much cheaper and more conservation-minded.

Generalizations of the Young-Laplace equation for the pressure of a mechanically stable gas bubble in a soft elastic material.

The Young-Laplace equation for the pressure of a mechanically stable gas bubble is generalized to include the effects of both surface tension and elastic forces of its surroundings. The latter are taken to be comprised of a soft isotropic material. Generalizations are derived for conditions of constant external pressure and constant system volume. The derived equations are formally exact for a spherical bubble surrounded by a spherical shell of isotropic material, provided that the bubble is sufficiently large for the surface tension to be treated macroscopically, and that the bubble radius is much larger than the thickness of the bubble/soft material interface. The underlying equations are also used to derive a simple expression for the Gibbs free energy of deformation of an elastic medium that surrounds a gas bubble. The possible relevance of this expression to some recently published ideas on decompression sickness ("the bends") is discussed.

The stability of bubbles formed from supersaturated solutions, and homogeneous nucleation of gas bubbles from solution, both revisited.

The solution of the problem of the relative stability of all possible equilibrium bubble states that can form from a closed, finite, supersaturated gas-liquid solution, maintained at a fixed temperature and a fixed external pressure is given. The supersaturated solution may contain any number of dissolved volatile solutes. The full solution to this problem has remained elusive for decades, because of the complication of pressure inequalities between the bubbles and the constant external (or reservoir) pressure. The method of solution is one that had been used previously to solve the related problem of the stability of a liquid droplet in a supersaturated vapor, where the same complication occurred. The derived equations were found to reduce correctly when simplified; they were consistent with experiment, and the system Gibbs free energy appropriately obeyed the Law of Corresponding States. The expressions were used in the context of transition state theory to provide semiempirical predictions of the rate of homogeneous bubble formation from a supersaturated solution, and the "critical pressure for homogeneous nucleation (P(crit))". The nucleation Gibbs free energy expression derived here had a lower barrier height and resulted in a reduction of P(crit) values, relative to what was obtained from the basis of a pre-existing approximate expression taken from the literature. Applications to chemical engineering and human decompression modeling are briefly described.

AltitudeOmics: effect of reduced barometric pressure on detection of intrapulmonary shunt, pulmonary gas exchange efficiency, and total pulmonary resistance.

Blood flow through intrapulmonary arteriovenous anastomoses (QIPAVA) occurs in healthy humans at rest and during exercise when breathing hypoxic gas mixtures at sea level and may be a source of right-to-left shunt. However, at high altitudes, QIPAVA is reduced compared with sea level, as detected using transthoracic saline contrast echocardiography (TTSCE). It remains unknown whether the reduction in QIPAVA (i.e., lower bubble scores) at high altitude is due to a reduction in bubble stability resulting from the lower barometric pressure (PB) or represents an actual reduction in QIPAVA. To this end, QIPAVA, pulmonary artery systolic pressure (PASP), cardiac output (QT), and the alveolar-to-arterial oxygen difference (AaDO2) were assessed at rest and during exercise (70-190 W) in the field (5,260 m) and in the laboratory (1,668 m) during four conditions: normobaric normoxia (NN; [Formula: see text] = 121 mmHg, PB = 625 mmHg; n = 8), normobaric hypoxia (NH; [Formula: see text] = 76 mmHg, PB = 625 mmHg; n = 7), hypobaric normoxia (HN; [Formula: see text] = 121 mmHg, PB = 410 mmHg; n = 8), and hypobaric hypoxia (HH; [Formula: see text] = 75 mmHg, PB = 410 mmHg; n = 7). We hypothesized QIPAVA would be reduced during exercise in isooxic hypobaria compared with normobaria and that the AaDO2 would be reduced in isooxic hypobaria compared with normobaria. Bubble scores were greater in normobaric conditions, but the AaDO2 was similar in both isooxic hypobaria and normobaria. Total pulmonary resistance (PASP/QT) was elevated in HN and HH. Using mathematical modeling, we found no effect of hypobaria on bubble dissolution time within the pulmonary transit times under consideration (<5 s). Consequently, our data suggest an effect of hypobaria alone on pulmonary blood flow. NEW & NOTEWORTHY Blood flow through intrapulmonary arteriovenous anastomoses, detected by transthoracic saline contrast echocardiography, was reduced during exercise in acute hypobaria compared with normobaria, independent of oxygen tension, whereas pulmonary gas exchange efficiency was unaffected. Modeling the effect(s) of reduced air density on contrast bubble lifetime did not result in a significantly reduced contrast stability. Interestingly, total pulmonary resistance was increased by hypobaria, independent of oxygen tension, suggesting that pulmonary blood flow may be changed by hypobaria.

A new class of biophysical models for predicting the probability of decompression sickness in scuba diving.

Interconnected compartmental models have been used for decades in physiology and medicine to account for the observed multi-exponential washout kinetics of a variety of solutes (including inert gases) both from single tissues and from the body as a whole. They are used here as the basis for a new class of biophysical probabilistic decompression models. These models are characterized by a relatively well-perfused, risk-bearing, central compartment and one or two non-risk-bearing, relatively poorly perfused, peripheral compartment(s). The peripheral compartments affect risk indirectly by diffusive exchange of dissolved inert gas with the central compartment. On the basis of the accuracy of their respective predictions beyond the calibration regime, the three-compartment interconnected models were found to be significantly better than the two-compartment interconnected models. The former, on the basis of a number of criteria, was also better than a two-compartment parallel model used for comparative purposes. In these latter comparisons, the models all had the same number of fitted parameters (four), were based on linear kinetics, had the same risk function, and were calibrated against the same dataset. The interconnected models predict that inert gas washout during decompression is relatively fast, initially, but slows rapidly with time compared with the more uniform washout rate predicted by an independent parallel compartment model. If empirically verified, this may have important implications for diving practice.

Conduction through the inward rectifier potassium channel, Kir2.1, is increased by negatively charged extracellular residues.

Ion channel conductance can be influenced by electrostatic effects originating from fixed "surface" charges that are remote from the selectivity filter. To explore whether surface charges contribute to the conductance properties of Kir2.1 channels, unitary conductance was measured in cell-attached recordings of Chinese hamster ovary (CHO) cells transfected with Kir2.1 channels over a range of K+ activities (4.6-293.5 mM) using single-channel measurements as well as nonstationary fluctuation analysis for low K+ activities. K+ ion concentrations were shown to equilibrate across the cell membrane in our studies using the voltage-sensitive dye DiBAC4(5). The dependence of gamma on the K+ activity (a(K)) was fit well by a modified Langmuir binding isotherm, with a nonzero intercept as a(K) approaches 0 mM, suggesting electrostatic surface charge effects. Following the addition of 100 mM N-methyl-D-glucamine (NMG+), a nonpermeant, nonblocking cation or following pretreatment with 50 mM trimethyloxonium (TMO), a carboxylic acid esterifying agent, the gamma-a(K) relationship did not show nonzero intercepts, suggesting the presence of surface charges formed by glutamate or aspartate residues. Consistent with surface charges in Kir2.1 channels, the rates of current decay induced by Ba2+ block were slowed with the addition of NMG or TMO. Using a molecular model of Kir2.1 channels, three candidate negatively charged residues were identified near the extracellular mouth of the pore and mutated to cysteine (E125C, D152C, and E153C). E153C channels, but not E125C or D152C channels, showed hyperbolic gamma-a(K) relationships going through the origin. Moreover, the addition of MTSES to restore the negative charges in E53C channels reestablished wild-type conductance properties. Our results demonstrate that E153 contributes to the conductance properties of Kir2.1 channels by acting as a surface charge.

Unified modeling of conductance kinetics for low- and high-conductance potassium ion channels.

A kinetics model is proposed for the description of ion conductance of low- and high-conductance potassium ion channels. The model describes ion permeation through the selectivity filter, which is assumed to be the only conductance determining part of the open channel. The filter occupancy can vary from zero to three ions, affecting the ion entry and exit rates. Ion motion between the binding sites inside the filter is assumed fast compared to the latter rates allowing averaging the equilibrium entry and exit rate constants over the possible ion configurations in the filter with a particular occupancy. Averaged rate constants related to a pair of adjacent occupancy states characterize a particular ion permeation mechanism. An expression for the channel conductance as a function of the symmetrical external ion concentration is derived. It comprises a sum of concentration independent conductance amplitudes for different ion permeation mechanisms weighted by the equilibrium filter occupancy probabilities. It is shown that each amplitude (i.e., maximum contribution to the channel conductance from each conductance mechanism) is proportional to an averaged exit rate constant and to quantities characterizing the effect of the applied electric field on the rate constants and the equilibrium ion distribution in the filter. The conductance expression derived provides a good description of the experimentally observed conductance-concentration curves for low-conductance (e.g., Kir2.1) and high-conductance (e.g., KcsA) potassium channels. It enables one to obtain equilibrium ion binding constants at different filter occupancies and to calculate the average number of ions in the selectivity filter for a given external ion concentration. For KcsA this number (2.0 at 200 mM) is in a good agreement with the available experimental value (2.1 at 200 mM). For the high-conductance potassium channels the net negative electrical charge around the selectivity filter increases the ion binding constants, thereby causing the larger occupancy probabilities to occur at smaller external ion concentrations compared to the low-conductance channels. This substantially increases the contributions of the two- and three-ion permeation mechanisms, with the larger conductance amplitudes leading to increased channel conductance compared to the low-conductance channels.

Nonequilibrium molecular dynamics calculation of the conductance of the KcsA potassium ion channel.

We have developed an efficient method for the atomistic determination of the conductance of a biological ion channel model by applying an external field to the conducting ions only. The underlying theory is discussed and demonstrated in a simple test system consisting of two ions in a box of water. Finally, the theory is applied to the experimentally determined structure of the KcsA potassium channel from which a conductance in reasonable agreement with the experimental result is predicted.

Non-Michaelis-Menten kinetics model for conductance of low-conductance potassium ion channels.

A reduced kinetics model is proposed for ion permeation in low-conductance potassium ion channels with zero net electrical charge in the selectivity filter region. The selectivity filter is assumed to be the only conductance-determining part of the channel. Ion entry and exit rate constants depend on the occupancy of the filter due to ion-ion interactions. The corresponding rates are assumed slow relative to the rates of ion motion between binding sites inside the filter, allowing a reduction of the kinetics model of the filter by averaging the entry and exit rate constants over the states with a particular occupancy number. The reduced kinetics model for low-conductance channels is described by only three states and two sets of effective rate constants characterizing transitions between these states. An explicit expression for the channel conductance as a function of symmetrical external ion concentration is derived under the assumption that the average electrical mobility of ions in the selectivity filter region in a limited range of ion concentrations does not depend on these concentrations. The simplified conductance model is shown to provide a good description of the experimentally observed conductance-concentration curve for the low-conductance potassium channel Kir2.1, and also predicts the mean occupancy of the selectivity filter of this channel. We find that at physiological external ion concentrations this occupancy is much lower than the value of two ions observed for one of the high-conductance potassium channels, KcsA.

Decompression sickness in breath-hold diving, and its probable connection to the growth and dissolution of small arterial gas emboli.

We solved the Laplace equation for the radius of an arterial gas embolism (AGE), during and after breath-hold diving. We used a simple three-region diffusion model for the AGE, and applied our results to two types of breath-hold dives: single, very deep competitive-level dives and repetitive shallower breath-hold dives similar to those carried out by indigenous commercial pearl divers in the South Pacific. Because of the effect of surface tension, AGEs tend to dissolve in arterial blood when arteries remote from supersaturated tissue. However if, before fully dissolving, they reach the capillary beds that perfuse the brain and the inner ear, they may become inflated with inert gas that is transferred into them from these contiguous temporarily supersaturated tissues. By using simple kinetic models of cerebral and inner ear tissue, the nitrogen tissue partial pressures during and after the dive(s) were determined. These were used to theoretically calculate AGE growth and dissolution curves for AGEs lodged in capillaries of the brain and inner ear. From these curves it was found that both cerebral and inner ear decompression sickness are expected to occur occasionally in single competitive-level dives. It was also determined from these curves that for the commercial repetitive dives considered, the duration of the surface interval (the time interval separating individual repetitive dives from one another) was a key determinant, as to whether inner ear and/or cerebral decompression sickness arose. Our predictions both for single competitive-level and repetitive commercial breath-hold diving were consistent with what is known about the incidence of cerebral and inner ear decompression sickness in these forms of diving.

The lifetimes of small arterial gas emboli, and their possible connection to Inner Ear Decompression Sickness.

We solved both the Diffusion and Laplace equations which predicted very similar results for the problem of a dissolving small gas bubble suspended in a liquid medium. These bubbles dissolved both because of surface tension and solute concentration effects. We focused on predicting bubble lifetimes ("td"), and dissolution dynamics - radius vs time (R vs t) for these contracting bubbles. We also presented a direct comparison of the predicted results, obtained by applying either Dirichlet or Neumann boundary conditions, to the bubble/medium interface. To the best of our knowledge, this is the first direct comparison that has ever been published on the application of these different boundary conditions to a moving gas/liquid boundary. We found that the results obtained by applying either Dirichlet or Neumann boundary conditions were very similar for small, short-lived bubbles (R0<25 µ,td<40s), but diverged considerably for larger, longer-lived bubbles. We applied our expressions to the timely problem of Inner Ear Decompression Sickness, where we found that our predictions were consistent with much of what is known about this condition.