Abstract
Multifocal cerebrovascular gas embolism occurs frequently during cardiopulmonary bypass and is thought to cause postoperative neurological dysfunction in large numbers of patients. We developed a mathematical model to predict the absorption time of intravascular gas embolism, accounting for the bubble geometry observed in vivo. We modeled bubbles as cylinders with hemispherical end caps and solved the resulting governing gas transport equations numerically. We validated the model using data obtained from videomicroscopy measurements of bubbles in the intact cremaster microcirculation of anesthetized male Wistar rats. The theoretical model with the use of in vivo geometry closely predicted actual absorption times for experimental intravascular gas embolisms and was more accurate than a model based on spherical shape. We computed absorption times for cerebrovascular gas embolism assuming a range of bubble geometries, initial volumes, and parameters relevant to brain blood flow. Results of the simulations demonstrated absorption time maxima and minima based on initial geometry, with several configurations taking as much as 50% longer to be absorbed than would a comparable spherical bubble.
 air embolism
 diffusion
 microcirculation
 mathematical model
over 800,000 cardiopulmonary bypass (CPB) surgeries are performed annually in the United States, with ∼6–8% of patients experiencing a serious neurological morbid event, including stroke or death (9, 22). The percentage of patients experiencing lesser postCPB cognitive dysfunction, such as memory loss and attention deficit, has been estimated to be as high as 70% (24). The direct cause of the neurological changes is not clearly defined; however, multifocal cerebrovascular gas embolism is compatible with this type of morbidity and has been frequently implicated (16,20). The number of microemboli (gaseous and particulate) occurring in the cerebral circulation during CPB surgery has been attributed to both the equipment and the surgical maneuvers used during the procedure and is extremely difficult to reduce with present methods (9). The health care costs associated with intravascular gas embolism (IGE) (prolonged hospitalization, increased mortality, and the need for intermediate or longterm care facilities) are, therefore, considerable (22). Developing preventive measures or new treatment methods against the injury suffered as a result of IGE would be extremely beneficial in improving the safety and reducing the cost of patient care.
To develop new methods of prevention or treatment strategies, it is necessary to understand the underlying mechanics behind IGEinduced injury. One explanation for the decrease in neurological function after multiple IGEs is that bubbles lodge in the cerebral blood vessels, blocking blood flow and causing local transient ischemia (16,18). Other studies suggest that the neurological injury is primarily due to secondary thromboinflammatory responses activated by airdamaged endothelium or interactions between blood constituents, such as proteins and platelets, at the gasliquid interface (15, 23). The volume of the embolism decreases as the gas becomes absorbed into the tissue, until the bubble either is completely dissolved or is displaced to a new location downstream. The rate of bubble absorption is, therefore, an important factor in the duration of ischemia and reestablishment of blood flow in the region. Longer bubble absorption times may exacerbate the provocation of thromboinflammatory and endothelial responses.
We have developed a theoretical model specifically to predict the absorption time of in vivo intravascular bubbles. Our model is based on observations of in vivo IGE geometry. We conducted preliminary experiments injecting small volumes of air into the intact rat cremaster circulation. We observed that intravascular bubbles are not spherical when they lodge and occlude blood flow. Rather, the bubbles form elongated “sausagelike” shapes, completely filling the vessel diameter. Similar observations have been reported by other investigators as well (12, 18). One earlier model developed by Hlastala and Van Liew (17) predicts the physiological absorption time of spherical bubbles in tissue. Although that model has been applied to gas bubbles in the circulation, it was created to describe the behavior of extravascular spherical bubbles generated during decompression sickness (11, 17).
Our theoretical model presented was validated by using data from animal experiments conducted with the intact rat cremaster microcirculation. After verification, this model was used to predict the residence time of gas emboli of various volumes and initial geometries.
MATERIALS AND METHODS
Theoretical model.
To derive the governing transport equations for our theoretical model and to solve for the total absorption time of an IGE, we made several initial simplifying assumptions. We assumed rapid equilibrium of the metabolic gases O_{2} and CO_{2} and water vapor between the bubble and the tissue, leaving N_{2}as the principal inert gas for diffusion. This assumption was made on the basis that these metabolic gases constitute a small fraction of the air embolism volume and have a much higher tissue and blood permeability than does the inert gas. The simultaneous exchange of multiple gases in the theoretical model of a spherical bubble has previously been addressed by Burkard and Van Liew (5). The initial rapid efflux of N_{2} into the tissue, at the instant of bubble entrapment, was neglected. In addition, capillary perfusion was assumed to carry away the inert gas in the tissue, preventing buildup and causing a decrease in partial pressure of N_{2} away from the bubble interface.
The initial configuration of the air embolism is modeled as a cylinder with hemispherical end caps, based on in vivo observations from preliminary experiments (Fig. 1,A andB). In vivo experiments also demonstrated that, once the bubble became lodged in the smaller arterioles of the microcirculation, the diameter of the vessels remained essentially constant over the time course of gas absorption. Vasoconstriction occurred immediately on arrival of the bubble in the vessel and persisted for the duration of the embolism resorption. After the initial instant of entrapment, vasoconstriction had no additional effect on lodged bubble geometry. The mathematical equations presented below incorporate these additional in vivo findings by dictating that, as the gas diffuses out of the IGE, the cylindrical portion of the bubble decreases in length, while the radius of the end caps remains fixed (Fig. 1 C). The cylindrical portion eventually disappears, the remaining end caps fuse to form a sphere, and the bubble shrinks as such (Fig.1 D).
At the time of initial bubble entrapment, the bubble dimension in the radial direction is bounded by the vessel wall, whereas the bubble remains free to lengthen, creating an elongated cylinder with hemispherical end caps. The difference between the pressure inside and outside the bubble is balanced by the product of the surface tension and the curvature at the gasliquid interface. This pressure difference (ΔP) is defined by Laplace’s Law as ΔP = 2ς/R, whereR is the bubble radius and ς is the value for surface tension. Because vessel radius showed little or no change during absorption, interfacial shape of the end caps is assumed to remain fixed during collapse of the cylindrical section, and the internal pressure within the bubble also remains constant. Any increase in internal pressure would merely cause the bubble to elongate further. A constant internal pressure creates a uniform driving force for gas diffusion out of the bubble. It is not until the bubble is spherical that the internal pressure changes. As gas diffuses out from the sphere, the bubble radius decreases, causing an increase in bubble internal pressure, an increased driving force for gas diffusion, and, eventually, rapid bubble collapse.
Mathematically, this sequence can be represented in the following manner. At the initial time point when the bubble lodges,t = 0, the length of the cylindrical section of the bubble,l(t), is l(0) =l _{0} (Fig.1 B). Att =T _{1}, the cylindrical portion of the bubble has completely disappeared, i.e.,l(T _{1}) = 0, and the bubble becomes a sphere. During the time period 0 ≤ t ≤T _{1}, the radius of the hemispherical caps,R(t), remains fixed atR(t) = R _{0}. Fort >T _{1},R(t) decreases until time T, when the bubble has completely collapsed,R(T) = 0.
The absorption process can, therefore, be described for two discrete increments in time: a cylindrical phase (0 ≤t ≤T
_{1}) and a spherical phase (T
_{1} ≤t ≤T). Each portion is solved independently. The cylindrical model was solved by using Fick’s Law
Determination of
The driving force for collapse, dP/drat r =R
_{0}, must be solved by using the diffusion equation. To solve for this term inEq. 3
, we begin with
Once the IGE geometry becomes spherical, the transport equations are simpler to solve. The governing equations forT
_{1} ≤t ≤T are, again, Fick’s Law and the diffusion equation, and they yield the time rate of change of the radius, previously defined in Ref. 27 as
Model predictions for cerebrovascular embolism.
We used the computer model and the same physiological parameters listed above to predict the absorption time for cerebrovascular bubbles with identical initial volumes but varying initial geometries by using a range of l
_{0} andR
_{0} to define an aspect ratio, X =l
_{0}/R
_{0}. The total time for bubble absorption,T, isT =T
_{1} +T
_{2}, withT
_{1} andT
_{2} defined as the time taken for the collapse of the cylindrical and spherical portions, respectively. The values forT
_{1} andT
_{2} are calculated from Eqs. 11
and
13
to be
All computer simulations used to obtain absorption times were run under very specific conditions. The embolism was assumed to be entrapped air, at 37°C, in a patient breathing a common postoperative CPB ventilation mixture [inspired O_{2} fraction (
Animal model.
Adult male Wistar rats (n = 5), weighing 200–425 g, were handled according to specifications set forth by the Animal Care and Use Committee and the University of Pennsylvania, in conformance with National Institutes of Health guidelines. Anesthesia was induced by inhalation of halothane in a closed chamber with a flow rate large enough to prevent accumulation of CO_{2}.
After induction, rats were placed in a supine position on a specialized Plexiglas tray and endotracheally intubated. The animals were ventilated by using a positivepressure pistondriven small animal ventilator with 1.2% halothane in an airO_{2} mixture (0.30 ≤
The cremaster muscle was prepared as previously described (3, 19). Briefly, the cremaster muscle was exposed and separated from the surrounding tissue and organ through a midline incision, first in the scrotum and then the muscle itself. Loose connective tissue was carefully dissected away with forceps. The cremaster was then spread over a transparent pedestal built into the Plexiglas tray. Sutures were attached in five locations to keep the muscle flat on the platform. The cremaster was superfused at 2 l/min with a warmed (34°C), gassed (95% N_{2}5% CO_{2}) Kreb’s buffer containing (in mmol/l) 132 NaCl, 25 NaHCO_{3}, 5 KCl, 1.2 MgCl_{2,} and 2 CaCl_{2}. The cremaster muscle was given 30 min to equilibrate before any experimentation was begun.
The cremaster circulation was visualized at a videomicroscopy workstation consisting of a compound microscope (Leitz Wetzlar), highresolution television camera (Microimage Video CA2063), video image analyzer (Boeckeler VIA150), monitor (Sony PVM1343MD), and videocassette recorder (Panasonic AG1970). The components of this system enable visualization of arterioles 200–10 μm in diameter, determination of vessel dimensions with a video micrometer, observation of erythrocyte motion, and a videotaped recording of the experiment.
An initial record of baseline perfusion was made in the cremaster circulation after the equilibration period to determine control conditions of blood flow. To demonstrate the preservation of vascular responses during the preparation, a 0.75ml bolus of 10^{−4} mol/l ACh, diluted in Kreb’s buffer, was added topically to the muscle, and vasodilation was verified. A discrete series of bubbles, ranging in number from two to eight and in volume from 50 to 500 nl, was created in the femoral artery catheter and injected with a small (∼0.5ml) bolus of 0.9% NaCl, a minimum of 10 min after the application of Ach. The large range of bubble volumes is permissible because individual bubbles break up before reaching the cremaster circulation. After injection, the entrapped bubble was located in the embolized artery and videotaped over time. In the event the entrapped bubble became dislodged, movements into the periphery were continuously tracked and recorded. The bubble was under constant observation until visualization was no longer possible because of either complete absorption or movement to a less favorable viewing area. Occasionally, the bubble failed to enter the cremaster circulation or lodged where visualization was difficult, in which case the procedure of bubble injection was repeated.
Bubble dimensions were measured every 20 s from the videotape by using the video micrometer. The volume of the IGE was calculated assuming a representative geometry (a cylinder with a hemispherical cap on each end) and axisymmetry in the vessel. Only bubbles with volumes between 1.0 and 6.0 nl were examined. Bubbles that lodged where no tissue perfusion from collateral vessels was evident were omitted from the study. Similarly, those bubbles that did not show the cylindrical geometry with hemispherical end caps were also excluded. This included bubbles in direct contact with one another, thereby not exhibiting the hemispherical end caps, or bubbles trapped in vessels with highly nonuniform diameters. Bubbles that continually changed their conformation because of constant dislodgment and reentrapment were also ignored, although occasional movement was permissible.
Validation of theory by experiment.
The absorption times of bubbles from different experiments could not be compared with one another because of the dependence of the absorption time on initial bubble volume and the parameters ofl
_{0} andR
_{0}. No two bubbles in vivo had these same characteristics; therefore, the absorption times for the experimental bubbles were compared with both the theoretical model calculated specifically for the initial geometry in each case and a spherical bubble of the same volume, as shown in Table 1. The theoretical model was used to simulate each in vivo bubble by inserting the measured values ofl
_{0} andR
_{0} and the physiological parameters corresponding to the experimental conditions into the Fortran code. The values for the parameters for this case were estimated to be D = 6.14 × 10^{−4}cm^{2}/min (2, 29); α_{ti} = 2.103 × 10^{−5}ml ⋅ cm^{3}muscle^{−1} ⋅ mmHg^{−1}(2, 29); Q˙ = 0.042 ml blood ⋅ cm^{3}muscle^{−1} ⋅ min^{−1}(14); and 456 Torr ≤
RESULTS
Preliminary observations.
The air emboli observed in the rat cremaster circulation generally ranged in volume from 0.2 to 6.0 nl. All bubbles lodged having initial radii between 30 and 55 μm and initial lengths from 180 to 550 μm. These dimensions corresponded to an aspect ratio,X, ranging in value from 4.3 to 11.2. An example of a bubble lodged inside a cremaster arteriole is shown in Fig. 1 A.
In no case were the bubbles observed to pass through the capillaries and travel into the venous circulation. If the bubble did travel further downstream after initially lodging in a vessel, bubble movement could usually be described as “stick and slip.” Stick and slip refers to the fact that the bubbles travel at very inconsistent speeds, often becoming reentrapped and dislodging several more times in a vessel of fairly uniform diameter.
Validation of theory by experiment.
To validate our theoretical model, we compared the experimentally measured absorption times with our theoretical predictions using the corresponding physiological values and initial bubble volume for both the in vivo and spherical bubble geometry. The theoretical model calculated longer absorption times for bubbles with the in vivo geometry than for spherical bubbles. The predicted differences in absorption times between the two geometries, in vivo and spherical, however, proved to be quite variable, depending on the initial configuration of the experimental bubble being simulated. In the majority of cases, our model using the in vivo geometry predicted the experimentally measured absorption times more closely than when the model assumed a purely spherical bubble. An example of the various measured and calculated absorption times is presented in Fig.2. Both geometric models tracked the initial minutes (∼4 min) of experimental bubble absorption well, but as the bubble volume became small (<0.5 nl), the in vivo configuration predicted the measured time more closely than did the spherical geometry. The absorption time of the experimental bubble was 22.0 min, and the accompanying theoretical predictions were 21.1 min with the use of the in vivo configuration and 16.4 min assuming a spherical shape. In the five cases examined, the predicted values for absorption times differed overall from the experimental absorption times by 12.1 ± 4.9 (SD)% for the in vivo geometry and by 20.1 ± 12.1% for the spherical configuration. Results from the predictive model were significantly different for the two geometries (Student’s ttest,P < 0.01).
Model predictions for cerebrovascular embolism.
Our theoretical model was used to calculate the time rate of change of bubble volume for a cerebrovascular bubble of a given initial volume under different initial geometric configurations. The bubble volume was constrained at 2.5 nl, and the total absorption time was found forX = 0, 2.5, 10, and 20. The results of the computer simulations are presented in Fig.3. Each bubble geometry gives a unique absorption curve based on initial geometry. The spherical bubble,X = 0, and the highly elongated bubble, X = 20, are predicted to have absorption times of 9.3 and 9.4 min, respectively, whereas a bubble of the same volume with X = 2.5 is predicted to take 13.3 min to be absorbed. The relationship between absorption times and X is, therefore, complex, neither monotonically increasing nor decreasing. The point of discontinuity in the slope in each of the three curves,X = 2.5, 10, and 20, represents the transition between the cylindrical and spherical solutions, which is missing, of course, in the purely spherical case.
Calculations of absorption times were also made at fixed initial bubble volumes (0.5, 2.5, and 10 nl) for varying initial geometries (Fig.4). A maximum absorption time,T _{max}, exists on each curve of initial bubble volume clustered aroundX = 2.6. AtX = 0, the absorption time corresponds to the time required for an initially spherical bubble to be absorbed,T _{sphere}. The predicted T _{sphere}for a 10nl bubble is 22.2 min, whereasT _{max} for the same volume is predicted to be 33.6 min, an increase of 51% fromT _{sphere}.
The time required for a spherical bubble to be absorbed is considered the minimum physiologically realistic absorption time possible. Although shorter absorption times are predicted, they are forX ≫ 20. With this aspect ratio, bubbles would have a diameter smaller than a capillary, a geometry we consider to be physiologically irrelevant. Solid lines atX = 0.3 and 13.1 in Fig. 4 indicate the range of aspect ratios in which bubble absorption time is within 80% of T _{max}. Of note, all of the in vivo experimental bubble configurations fell within this range of aspect ratios.
The computer model was used to calculateT_{X} , the time required for a bubble of initial volume, V_{0}, to be absorbed as a function of initial surface area,A _{0}(X), over the range 0 ≤ X ≤ 25. The resulting isoinitial volume curves, presented in Fig.5, demonstrate the influence of initial surface area on absorption time. Presenting the data in the surface area domain resulted in similar trends, as were previously noted in Fig. 4. This representation of the data uniquely demonstrates that absorption time is a linear function of initial surface area for a given value of X over a range of initial bubble volumes. Each curve shown has a unique maximum absorption time as well as a point at the far left corresponding toT _{sphere}. The maximum absorption time for each curve occurs atX = 2.6. These maxima form a locus of points linear inA _{0}(X) with a slope ofm _{2.6} = 130.9 mm^{2}/min (R ^{2} = 0.9988). The values ofT _{sphere} on each curve also fall along a straight line having a slope ofm _{0} = 97.5 mm^{2}/min (R ^{2} = 0.9940). The slopes of these two lines were determined by linear regression by using the points illustrated in Fig. 5 and forcing the result through the origin. These two slopes were significantly different (P < 0.001, Student’sttest to evaluate the difference in regressions).
By using the equation included in Fig. 5,T_{X}
=m_{X}A
_{0}(X), the following simple expression can be used to determineT_{X}
based only on initial bubble volume, V_{0}, and the aspect ratio, X
DISCUSSION
Validation of theory by experiment.
IGE is unfortunately a frequent and not completely avoidable complication of CPB surgery that can lead to serious neurological injury or death. To address areas of potential treatments for emboli, one must first understand the behavior of gas bubbles in the vasculature as it relates to the bubble’s in vivo configuration (Fig.1 A).
Air emboli always remained in the arterial circulation until complete absorption, although they would occasionally dislodge and flow downstream. Transcapillary bubble passage has been reported (13, 26); however, we did not observe bubbles lodged in the venous circulation. This is because, at volumes small enough to fit into the capillaries, bubbles collapsed at a rate faster than they could physically move to the venules.
We have only applied our model to arterial air embolism absorption; however, bubbles have also been shown to enter the venous circulation (8), and they have frequently been observed intraoperatively by one of the authors (D. M. Eckmann) in the cephalic vein of surgical patients undergoing placement of a permanent central venous catheter. Those instances have been limited to patients having a distal peripheral iv catheter in the ipsilateral upper extremity, indicating air introduction through the iv catheter.
Our model, however, can easily be adapted to consider venous gas embolism, if desired. In the absence of external or interfacial stresses, surface tension dictates that bubbles would most prefer to assume a spherical shape, to minimize the surface area for a given volume. The increased distensibility of the venous vessels may initially allow bubbles to lodge more closely to their spherical configuration than if entrapped in the arteries. The inability of veins to vasoconstrict to the same degree as arteries makes it less likely that bubble dimensions, and therefore internal pressure, will change after entrapment.
The stickandslip movement observed by bubbles throughout the arterial vasculature has tribological implications of interfacial interactions between various physiological components. A layer of denatured proteins has been seen at the bubbleblood interface in vivo (21, 30). This network of denatured proteins may have some complex adhesive interactions with the glycocalyx or the endothelial cell membrane (10,31). In addition, there may be other types of interactions present at the airbloodvessel interface that affect interfacial mechanics such as contactangle hysteresis with wetting and dewetting phenomena (4,7).
The present theoretical model, taking the specific geometry of the in vivo IGE into account, was validated by using data from our animal experiments. Comparing the absorption times predicted by the theory to the measured absorption times demonstrated that using the in vivo configuration (X ≠ 0) in the model led to predicted times that were significantly different from times calculated assuming a spherical shape (X = 0) and that more accurately matched the experimental data (Table 1).
Despite the improved accuracy, our model still underestimated the time of absorption in a majority of cases, compared with the animal data. This discrepancy may be due to the fact that the bloodborne matter that absorbs to the bubble interface potentially inhibits gas diffusion and prolongs bubble absorption (6, 28) but has not yet been taken into account in our mathematical model.
Model predictions for cerebrovascular embolism.
The initial aspect ratio is an extremely important factor in determining the absorption time of an intravascular bubble, as seen from Eq. 14 . The direct effect of the aspect ratio on the time rate of change of bubble volume is demonstrated in Fig. 3, in which a 2.5nl bubble has markedly different absorption curves depending on initial geometry. The theoretical model, simulating cerebral IGEs, predicts that there exists a value ofX =l _{0} /R _{0}≈ 2.6, which results in a maximum absorption time that is as much as 51% greater than if absorption is calculated for a spherical bubble with the same initial volume. All of the experimental IGEs lodged with initial geometries that were near the peaks of the curves shown in Figs. 4 and 5, suggesting the in vivo bubbles lodge in configurations associated with longer absorption times.
The nonlinearity of absorption times as a function of aspect ratio, seen in Fig. 4, cannot be explained purely as an increase in initial bubble surface area leading to decreased absorption time (Fig. 5). This demonstrates that additional competing mechanisms are at work, such as constant internal bubble pressure. Both the maximum absorption time and the absorption time for a spherical bubble are linear functions of initial surface area over a range of bubble volumes, and the slopes of each of these lines are significantly different from one another. The results of these linear approximations can be used to construct an accurate simplified estimate of absorption time, as shown in Table 2. It is apparent that there is little difference in the bubble absorption times calculated by the full computer model and those times estimated with Eq. 19 . UsingEq. 19 and a value of the aspect ratio at 2.6 or 0, therefore, provides a very simple, accurate means of predicting the maximum and minimum time of bubble absorption, respectively, given the initial volume. This slope equation could be useful in trying to estimate the bounds of residence time of bubbles found entrapped in the cerebral circulation, say by computed tomography or magnetic resonance imaging, and deciding whether or not to invoke a therapy such as hyperbaria.
The theoretical model lends considerable insight into the role of bubble geometry on the residence time of IGEs. In a clinical context, one would desire to minimize this or eliminate gas bubbles from the circulation altogether. Present methods of treatment include hyperbaric therapy and ventilation with gases containing lower concentrations of N_{2}. One avenue that still remains largely unexplored is manipulating bubble geometry, perhaps with the help of surface active agents. Inducing the bubbles to break up (25), or lodge with an elongated shape (X> 15), could potentially minimize the time for bubble absorption and until tissue blood flow is resumed. There are several additional issues associated with bubble geometry that must also be considered when assessing the potential adverse effects of a particular IGE conformation.
Whereas highly elongated (large X) bubbles are predicted to have a shorter residence time in the embolized vessel, a longer initial bubble length increases the contact area between the bubble and the endothelium. Endothelial cells are very susceptible to damage caused by contact with air and have been shown to release from the basement membrane or form gaps after being exposed to bubbles (1, 31). A larger surface area may also potentiate the activation of thromboinflammatory pathways. This induces additional responses, including platelet aggregation and local neutrophil sequestration (1, 21).
Bubbles lodging as spheres may minimize absorption time and endothelial contact area but may have other associated negative effects. Bubbles remaining close to the spherical shape after lodging would have the largest initial radius possible for a given volume. The embolized vessel in this case would be of larger diameter than if the bubble were elongated. Occluding a vessel of larger diameter could block blood flow and O_{2} delivery to a larger vascular territory. The area of tissue ischemia and severity of neurological damage, therefore, would potentially be increased.
Clearly, physics and physiology related to IGEs are extremely complex. The theoretical model presented, based on in vivo bubble geometry, now allows some new issues heretofore ignored to be addressed. We have shown the importance of incorporating bubble geometry when the residence times of intravascular bubbles are predicted. With this theoretical model, we accurately predict the actual bubble absorption times measured in vivo. It is important to stress that these results apply to only one specific set of conditions: cerebral air embolism in a patient with normal blood gases at sea level and 37°C. Situations arise that would change these results and would have to be accounted for in the computer model. These situations include ventilating patients with different gas mixtures such as He and O_{2}, placing patients in a hyperbaric chamber, which increases the external pressure, or cooling or warming patients during CPB. By altering different parameters to simulate clinical situations, use of this model can be extended to predict the effects of various therapeutic maneuvers on their ability to accelerate IGE absorption. Such computer simulations may help form a better understanding of the factors influencing the deposition and absorption of cerebrovascular gas bubbles and lead the way for the development of more effective and innovative strategies in dealing with this important clinical problem.
Acknowledgments
Dr. Alex Loeb and Dareus Conover provided helpful advice.
Footnotes

Address for reprint requests and other correspondence: D. M. Eckmann, Dept. of Anesthesia, Univ. of Pennsylvania, 7Dulles/HUP, 3400 Spruce St., Philadelphia, PA 19104 (Email: deckmann{at}mail.med.upenn.edu).

The Whitaker Foundation and National Heart, Lung, and Blood Institute Grant R01HL60230 were funding sources for this work.

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. §1734 solely to indicate this fact.
 Copyright © 1999 the American Physiological Society