Model Predictions of Gas Embolism Growth and Reabsorption during Xenon Anesthesia

The theoretical model we developed to calculate bubble volume over time has been described in detail previously 6,8but is reviewed briefly. The in vivo  bubble shape is modeled as a cylinder of initial length L^owith hemispherical end caps of initial radius R^o. Based on in vivo  findings, 6,7,9we stipulate that as gas moves in or out of the bubble, the cylindrical portion of the embolism elongates or shrinks, respectively, while the hemispherical end cap radius remains fixed. For bubble shrinkage, when the cylindrical portion completely disappears, the remaining hemispherical end caps coalesce to form a sphere. Spherical geometry is maintained for the remainder of bubble absorption. Bubble absorption is solved independently for two separate temporal phases, the cylindrical phase and the spherical phase. In each case, the law of Fick is applied to the specific bubble geometry. Bubble absorption time depends on the initial aspect ratio of the bubble itself, or L^o/R^o. In the case of bubble elongation, the cylindrical portion gains length, and the hemispherical end cap radius remains constant.

During bubble growth or shrinkage, the mathematical expression for the bubble's internal pressure is based on several assumptions. We assume that rapid equilibrium of carbon dioxide and water vapor occurs between the inside of the bubble and the tissue, with maintenance of constant tissue partial pressure of these gases. Capillary perfusion from surrounding vessels is assumed to convect gas to or from tissue near the bubble, with diffusion accounting for gas transport across the bubble interface. The hydrostatic head of blood pressure is ignored because it is assumed to be small compared to the external barometric pressure. Any elastic force exerted by the vessel wall on the bubble is also neglected because changes in vessel diameter would only cause the bubble to lengthen or shorten, without an increase or decrease in internal pressure. Following the law of LaPlace, the pressure inside the bubble is only balanced by a surface tension term proportional to the inverse of the bubble radius and the difference between the external pressure acting on the bubble and the partial pressures of Xe, oxygen, carbon dioxide, and water vapor in the tissue. We consider cases in which complete nitrogen elimination has occurred, representative of a non–air-containing ventilation admixture for cardiac surgery. The internal bubble pressure, therefore, remains fixed during shrinkage or growth of the cylindrical portion because of a constant value of R^o. During absorption of spherical bubbles, the radius decreases. Consequently, the internal pressure increases. The exchangeable gases oxygen and Xe follow their concentration gradients, diffusing into the bubble if the tissue concentration exceeds the bubble concentration, and diffusing out of the bubble if the tissue concentration is lower. Gases moving into and out of the bubble can slow absorption or even promote bubble growth. With this model, we can numerically determine the time-dependent size of an intravascular bubble over a range of geometries for a given initial volume, temperature, and initial gas content in the bubble and the delivered gas admixture. Knowing local tissue characteristics allows further application to specific areas such as the brain.

The governing transport equation for the model is derived from the two initial equations, the diffusion equation, balancing the bubble volume with the flux of gas across the bubble surface, and the law of LaPlace, balancing the pressure difference across the bubble surface with the surface tension and the interfacial curvature (see Branger and Eckmann 6for detail). In this analysis, an accounting for each of the gas species being transported must be made as was done by Branger et al.  8The resultant nonlinear time-dependent governing ordinary differential equation (see Branger and Eckmann 6for explicit derivation) depends on multiple parameters, including the temperature-dependent brain tissue and blood solubility and diffusivity of oxygen and Xe, cerebral blood flow (Q), blood-gas surface tension (ς), atmospheric pressure (P^s), and the partial pressures of carbon dioxide (Ptco²) and water vapor (Pth²o) in the tissues. We used the following values in the computations: ς= 0.0355 mmHg cm, P^s= 760 mmHg, Ptco²= 44 mmHg, Pth²o = 47 mmHg, Q = 0.525 cm³blood · (cm³brain · min)⁻¹at normothermia and mild hyperthermia, and Q = 0.30 cm³blood · (cm³brain · min)⁻¹for hypothermic conditions. 10 

The model requires input parameters of oxygen and Xe solubility and diffusivity in tissue and blood. Many parameter values for biologic gases are available in the literature, but the list is incomplete for purposes of the analysis considered here. Therefore, one major component of this work was the approximation of missing values for diffusion and solubility coefficients of Xe and oxygen. The parameter estimates, along with literature-derived values, were then used for the simulations of embolism bubbles. Estimates of unknown parameter values were incorporated based on published values of other gases similar to Xe 3,11,12and were determined as is described in  appendixes A and B.

The model incorporating the parameter estimates was solved numerically using the fourth-order Runge-Kutta method in FORTRAN (Lahey Computer Systems, Incline Village, NV) on a Pentium PC. In the case of embolism absorption, time to complete absorption was calculated to within 0.01 min. In the case of bubble growth, doubling time (time required for the bubble to reach twice its initial volume) was calculated to within 0.05 min. All computer simulations were run for bubbles having a 50-nl initial volume entrapped in cerebral vessels. This initial volume is a reasonable size estimate based on in vivo  studies of bubbles in the microcirculation, 13and we have previously simulated bubbles this size. 8Three specific combinations of initial bubble gas composition and delivered gas admixture were simulated: (1) a Xe bubble and oxygen delivery; (2) a Xe bubble and 70% Xe–30% O²delivery; and (3) an oxygen bubble and 70% Xe–30% O²delivery. Conditions of normothermia, cooling, and rewarming during extracorporeal circulation were incorporated by using parameter values (e.g. , solubility, diffusivity) for 37°, 18°, and 39°C, respectively. As an additional test, we simulated a 10-μl oxygen bubble introduced into a large, well-mixed, Xe-saturated water reservoir at constant barometric pressure and 37°C. The bubble was allowed to remain spherical. This is an approximation of the experimental conditions studied by Lockwood. 5 

Estimated parameter values for the diffusivity (D^O2,blood, D^O2,brain, D^Xe,blood, D^Xe,brain) and solubility (L^O2,blood, L^O2,brain, L^Xe,blood, L^Xe,brain) of oxygen and Xe in blood and brain are listed in tables 1 and 2, respectively. The estimated values are close to constants determined experimentally for those cases in which the data exist. For example, our estimates of L^Xe,blood= 0.109 and L^Xe,brain= 0.145 compare closely with the respective published values of 0.115 3and 0.153. 12 

The model predicts rapid efflux of oxygen from the bubble accompanied by rapid initial Xe entry. The bubble radial growth rate decays asymptotically to near zero, yielding a Xe bubble having essentially a fixed volume. The volume reaches 99% of its asymptotic limit of 19.78 μl in 7.4 min. Bubble volume is predicted to increase by 50% in the first 1.4 min.

Figures 1–5illustrate our results of the computational predictions of the gas transport dynamics for bubbles of various initial gas compositions and shapes during delivery of various gas mixtures and at various temperatures. In figures 1–4, we demonstrate our findings for behavior of 50-nl bubbles at 37°C. Figure 1, A  shows results of calculations made for Xe bubbles having initial aspect ratios of 0 (a spherical bubble), 2.6, 10, and 20 (long, slender bubble). Under conditions of oxygen delivery, bubbles are predicted to reabsorb in the range of 15–23 min. The prominent kink present in some curves occurs at the completion of reabsorption of the cylindrical bubble core, with transition to reabsorption of a spherical remnant. Figure 1, B  shows the time course of the individual gas volumes inside the bubble having L^o/R^o= 2.6 shown in figure 1, A . Rapid efflux of Xe is predicted to occur in approximately 11 min. Oxygen influx and subsequent reabsorption extends bubble life to approximately 21 min.

In figure 2, A , we present the predictions of bubble volume over time for Xe bubbles during delivery of a 70% Xe–30% O²mixture. Initial aspect ratios represented are 0, 2.6, 10, and 20. The bubbles seem to shrink initially, but bubble volume then enlarges monotonically. For each of these cases, bubble enlargement occurs by elongation at fixed radius. Figure 2, B  shows the time course of the individual gases volumes inside the bubble having L^o/R^o= 2.6 shown in figure 2, A . The Xe volume initially shrinks, with Xe efflux accompanied by oxygen influx. Over time, both Xe and oxygen enter the bubble to promote continued growth.

The case of an embolism bubble having an initial aspect ratio of 0, 2.6, 10, or 20 and first composed entirely of oxygen during 70% Xe–30% O²delivery appears in figure 3, A . The graph illustrates that unrelenting bubble growth occurs. The volume increase for each case results in an increase in bubble axial dimension while the radius remains constant. Figure 3, B  shows the time course of the individual gases volumes inside the bubble having L^o/R^o= 2.6 shown in figure 3, A . The rapid ingress of Xe outstrips the initial elimination of oxygen from the bubble. At longer times, entry of both Xe and oxygen into the bubble contributes to its continued growth.

We calculated doubling time, or the time required for a 50-nl bubble to grow to 100 nl. We considered the case in which the bubble is initially comprised of Xe during 70% Xe–30% O²delivery. The data are plotted in figure 4as a function of bubble surface area, rather than aspect ratio, to demonstrate the contribution of surface area to overall gas transport. The single data point represents doubling time for a spherical bubble allowed to grow as a sphere increasing in radius. The curve represents doubling times for bubbles having initial aspect ratios in the range 0 ≥ L^o/R^o≥ 25 but constrained to grow with an elongating cylindrical core and with end caps of fixed dimension. For fixed initial volume, the more slender and elongated bubbles enlarge more rapidly. The growth rate is not linearly proportional to bubble surface area.

The effect of temperature on gas transport and bub-ble size is illustrated in figure 5. For a Xe bubble with L^o/R^o= 10 occurring during oxygen delivery (as in fig. 1, A ), reabsorption is calculated to be only slightly (∼1 min, or < 5%) longer at 39°C than at 18°C, as shown in figure 5, A . Predicted bubble growth of a Xe bubble having L^o/R^o= 10 during 70% Xe–30% O²delivery appears in figure 5, B  (corollary to fig. 2, A ). Bubble volume over time is nearly identical at 37° and 39°C (the two curves overlap) but less at 18°C.

Several unanswered questions pertain to the safety of Xe used as a general anesthetic. One major concern is the potential contribution of Xe to gas bubble enlargement in the case of cerebrovascular gas embolism or bowel obstruction. Gas embolism is well known to occur in the surgical setting of cardiopulmonary bypass. Cardiopulmonary bypass itself may exacerbate the neurologic injury produced by cerebral arterial air embolism. 14Delivery of an anesthetic that further promotes bubble growth could prolong ischemic time, increase the size of the vascular territory affected, and thus amplify the injury.

Gas bubbles have been shown to grow in aqueous solution in the presence of Xe, 5and it is also known that intravascular and extravascular gas bubbles can grow over time in cases of decompression sickness. 8We used a computational scheme based on a previously validated mathematical model 6,7,9to predict the behavior of a 50-nl cerebrovascular gas embolism as a function of specific bubble geometry and initial bubble gas composition, given delivery of a particular gas mixture. We chose initial bubble shapes ranging from spherical to long and slender. The particular value L^o/R^o= 2.6 was selected for study because it was associated with prolonged reabsorption of nitrogen-containing bubbles. 6The model's output does depend on our having used reasonable parameters for the diffusivity and solubility coefficients. Values for coefficients missing from the literature were correlated with and approximated from those constants for gases known and previously published (e.g. , tables 1 and 2). The results indicate bubble shrinkage or growth occurs at rates that depend on the initial conditions of bubble shape, exchangeable gas concentration gradients, and temperature.

The three different gas concentration gradient cases used were chosen to illustrate various physical extremes, although the clinical conditions and mechanisms for these extremes to occur may only apply under unusual circumstances. In each of these cases, we consider the tissues already to have been denitrogenated by prolonged oxygen or Xe–oxygen delivery before embolization. We have also neglected the gas transport across the bubble interface that occurs between the moment of bubble introduction into the vasculature and its eventual lodging in the microcirculation, assuming this takes only short time relative to bubble lifetime. 7,9 

An oxygen bubble introduced into a Xe-saturated environment is one case of the largest initial Xe and oxygen gradients across the bubble possible. Although this is not physiologically relevant, we simulated this as a test of our model for comparison of results to those reported for experiment. 5Our predicted time of 1.4 min for a 50% volume increase was slightly longer than has been measured (∼45 s), 5but the general features of volume increase are recovered considering the variety of differences between the model and the actual conduct of the experiments.

The other, more germane case of the largest possible initial Xe and oxygen gradients across the bubble is a Xe bubble introduced into an oxygen-saturated environment (fig. 1). In this case, the bubble shrinks for all geometries (fig. 1, A ). As figure 1, B  shows, the transport strongly favors Xe egress from the bubble to surrounding tissue. The initial oxygen flux into the bubble retards bubble reabsorption to a rate below that of Xe transport. After the bubble's Xe supply has been depleted, the remaining oxygen in the bubble reabsorbs.

The phenomena predicted are quite different with Xe (fig. 2) or oxygen (fig. 3) bubbles lodging in the cerebral vasculature during 70% Xe–30% O²delivery. For Xe bubbles, there are initially large Xe and oxygen concentration gradients between the embolism and the cerebral tissue. Although Xe initially passes out of the bubble, oxygen diffuses in, leading to continued dilution of the Xe present in the bubble. Eventually, this reverses the concentration gradient, driving Xe into, rather than out of, the bubble (fig. 2, B ). Concurrent with Xe diffusion into the bubble, the oxygen gradient is preserved, driving oxygen influx. This mass transport mechanism essentially creates a sink that results in continued bubble growth for all initial bubble aspect ratios (fig. 2, A ). Growth occurs by axial elongation of the bubble lodged within the vessel, with the bubble filling a greater length of vessel at fixed diameter. This same phenomenon is observed for oxygen bubbles during 70% Xe–30% O²delivery, although at different growth rates (fig. 3).

One way to assess the relative rates of bubble growth is to calculate the time required for a bubble to double its volume. The time required for a Xe bubble to double in volume during 70% Xe–30% O²delivery is predicted to be greater for bubbles having a more spherical, or slightly elongated aspect ratio (i.e. , slightly bigger than a sphere that fills the vessel, fig. 4). The shape with the minimum surface area (i.e. , a sphere) enlarges faster if the radius is allowed to grow than if it is held fixed and the bubble elongates. The difference between these two growth conditions is that the bubble internal pressure remains constant during cylindrical core growth but falls as the bubble radius increases during spherical growth. A drop in internal pressure increases the gradients driving gas into the bubble and speeds growth. Otherwise, for fixed initial bubble volume, a longer and more slender bubble having a larger surface area doubles its volume in less time than if it were nearly spherical. This same effect can be inferred for oxygen bubbles during 70% Xe–30% O²delivery from figure 3, A . Doubling times are predicted to be in the range of 3–12 min, much faster than the doubling times of comparably shaped Xe bubbles.

The results indicate that both Xe and oxygen bubbles continue to grow during 70% Xe–30% O²delivery, whereas Xe bubbles are completely reabsorbed in the absence of a tissue Xe source. The individual parameters such as gas solubility and diffusivity in tissue and blood used to perform these computations are themselves highly temperature dependent, particularly in the range of temperatures typically used for cardiopulmonary bypass (tables 1 and 2). For the data presented in figure 5, A , we conducted simulations using values of cerebral blood flow, Q, appropriate for the applicable temperature conditions. We did not find data in the literature for Q at 18°C, so we used Q = 0.30 cm³blood · (cm³brain · min)⁻¹, the value determined by Cook et al.  10for 27°C. We also tested model sensitivity using Q = 0.26 cm³blood · (cm³brain · min)⁻¹, half of the value of Q at 37°C. This analysis demonstrated that reducing Q from 0.525 to 0.26 cm³blood · (cm³brain · min)⁻¹while keeping the other parameters constant increased the predicted reabsorption time by 1.9% or less. An explanation of this is relatively straightforward. The parameter Q appears in the governing equations and the method of solution adapted from previous work. 6,8In the model, the value Q^1/2is used in the argument of two highly nonlinear functions. Working with an exchange of two gases, we must calculate values of this argument for both Xe and oxygen. Changing Q from 0.525 to 0.3 reduces the argument for Xe and oxygen by 6.1% and 6.5%, respectively, compared to the values at 37°C. Changing Q from 0.525 to 0.26 reduces the arguments for Xe and oxygen by 12.5% and 13.0%, respectively, from their values at 37°C. The solution is relatively insensitive to changes in Q of this order of magnitude.

In fact, as shown in figure 5, the nonlinearity of the overall transport phenomena becomes relatively insensitive to any temperature effect. In the case of bubble reabsorption (fig. 5, A ), the bubble residence times are within 5% of each other at 18°, 37°, and 39°C. In the case of Xe bubble growth during 70% Xe-30% O²delivery, growth rates appear superimposed at 37° and 39°C. Slower growth rates are present under hypothermic conditions, with bubbles taking approximately 15–20% longer to reach the same volume at 18°C compared to 37°C (fig. 5, B ).

The mass transport in and out of the bubble cannot be easily inferred by inspection of the complex time-dependent nonlinear governing transport equations. There is a thermal competition between solubility effects (a measure of tissue carrying capacity for the gas) and diffusivity effects (a measure of the displacement tendency of gas molecules). Diffusivity increases with increasing temperature, whereas solubility decreases at warmer temperatures. Overall gas exchange depends greatly on the product of the diffusivity and solubility coefficients at each temperature rather than on either one individually. Calculating these products for oxygen and Xe using the data in tables 1 and 2shows the product for oxygen is 23.6% larger at 18°C than at 37° or 39°C, and the product for Xe is 5.5% smaller at 18°C than at 37° or 39°C. This near constancy of the products, more so for Xe than for oxygen, tracks well with the magnitudes of the predicted behaviors resulting from transport of the two exchangeable gases. In the case of bubble shrinkage, Xe transport dominates the first half of the bubble's existence, and oxygen transport is the dominant mechanism for the second half the bubble's lifetime (fig. 1, B ). In the case of Xe bubble growth during 70% Xe–30% O²delivery, oxygen influx into the bubble (fig. 2, B ) is the driving force for establishing transport gradients throughout the time period of calculated growth. This accounts for the observed differences in figure 5over the range of temperatures explored. In general, predictions for a single temperature (say, normothermia) can be directly applied to more extreme temperature conditions encountered if temperature is relatively static. The results cannot be used to infer elements of bubble growth during periods of rapid temperature change, such as active rewarming, because the model does not accommodate events such as tissues becoming supersaturated with a particular gas.

There have not been any in vivo  studies of gas embolism involving Xe, so it is not clear what the physiologic consequences of bubble growth are after gas embolism lodging during Xe anesthesia. Because bubbles elongate as they grow, the bubble occupies more of the vessel's length over time. This increases the surface area of vascular endothelium in contact with gas, although the cell-damaging effects of this have not been described definitively. Distal bubble elongation should not increase the extent of vascular territory deprived of perfusion because its blood flow will already have been compromised by initial vessel embolization. Bubble growth in the proximal direction could reach to a perfused branch of an arteriolar bifurcation and thus newly deny blood flow into additional tissue. Although Xe has been purported to have neuroprotective qualities, 15,16there are no data to indicate whether neuroprotection conferred by Xe is sufficient to mitigate the local ischemic injury that can be induced by cerebral vascular gas embolization. Based on our findings, this could persist for hours with continued bubble growth. Clearly, bubble shrinkage is desired, because bubble reabsorption is likely to be followed by reperfusion. 7,9 

In conclusion, our results show that Xe can promote bubble growth, which could amplify neurologic injury in cerebral arterial gas embolism. Concentration gradients can be manipulated to favor reabsorption of Xe-containing bubbles. Ultimately, ventilation with oxygen at the conclusion of Xe delivery will lead to total bubble reabsorption as Xe is eliminated from the body. The time required for reabsorption will depend on bubble volume and shape as well as the maintenance of perfusion in tissue surrounding the affected area.

Estimates of the diffusion coefficient values not available in the literature were based on known values and their relation to the fractional water content, w , of brain tissue and blood. 11The diffusion coefficient of a particular gas i  in a tissue with fractional water content w , D^i,tissue(w ), is given by 17: in which D^{i, water}is the diffusivity of gas species i  in water and the coefficient v  is derived from experiments. Equation 1holds for 0.65 ≥w ≥ 1. 11Values of v  at 37°C are known for oxygen (4.4), carbon dioxide (3.9), nitrogen (4.3), and hydrogen (4.6). 11A value of v  for xenon must be estimated because a value from experiments has not been published.

First, values for w  in skin, liver, skeletal muscle, brain, lung, spleen, heart, and kidney range from 0.69 to 0.82, 17with w  approximately 0.77 in the adult human brain and w  approximately 0.96 in blood. For rat liver, w  is approximately 0.7. 17Known diffusivities at 37°C include D^{Xe,rat liver}= 0.38 × 10⁻⁵cm²/s, 18D^Xe,water= 1.55 × 10⁻⁵cm²/s, 18and D^O2,water= 3.02 × 10⁻⁵cm²/s. 19From equation 1, a value v = 4.7 is estimated for xenon in tissue. It follows that D^O2,brain= 1.11 × 10⁻⁵cm²/s and D^Xe,brain= 0.54 × 10⁻⁵cm²/s at 37°C.

Lango et al.  12showed an expression that accurately reflects the temperature dependence of diffusion coefficients is: in which T is the temperature (°C), and the constants a  and b  are experimentally determined. Values of a  and b  have been published for many gases, including oxygen (a = 1.40, b = 0.023), but not for xenon. 12Other published values of b  range from 0.019 (nitrous oxide) to 0.027 (argon). Values do not correlate in any discernible fashion with molecular weight, molecular size, or other molecular features. We chose b = 0.023 for xenon because this was the mean of seven published values 12and it falls in the middle of their range. Table 1lists the values of the diffusion coefficients known and estimated using equations 1 and 2and the relevant constants denoted above.

The solubility of oxygen in human plasma, L^O2,plasma, at 37°C is well documented 12and is within 5% of L^O2,blood, the solubility of oxygen in whole blood. We calculated L^O2,blood= L^O2,plasma/0.96 using the published data of oxygen solubility in plasma at the three temperatures of interest. Numerous values of the solubility of various gases in tissues and blood appear in the literature. 20–28The ratio L^i,brain/L^i,bloodfor different gases (denoted by i ) consistently falls in the range of approximately 1.07–1.15. For rabbits, L^O2,brain/L^O2,blood= 1.11. 12,24We calculated L^O2,brain= 1.11 L^O2,blood= 1.156 L^O2,plasmausing known values of L^O2,plasmafor 18°, 37°, and 39°C.

We plotted known Ostwald solubility coefficients for xenon in water, L^Xe,water, against temperature, T, at 0, 20, 30, and 37°C. 3,19We fitted the data to a first-order exponential equation, resulting in a least-squares correlation coefficient r = 0.997. We then calculated values of L^Xe,waterwith equation 3. The same approach used to approximate L^O2,brainand L^O2,bloodwas used to estimate L^Xe,blood. We correlated solubilities of the gas species in different media at the same temperature. 20–28The relation between known values of L^Xe,water, 3,19L^Xe,blood, 3and L^Xe,brain12at 37°C is approximated by: L^Xe,blood= 1.385 L^Xe,water; and L^Xe,brain= 1.33 L^Xe,blood. Estimated values of L^Xe,bloodand L^Xe,brainwere then calculated using values of L^Xe,waterdetermined from equation 3. All the estimated solubility parameter values are listed in table 2.