INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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 post-CPB 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 long-term 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 IGE-induced 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 air-damaged endothelium or interactions between blood constituents, such as proteins and platelets, at the gas-liquid 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

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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 O₂ and CO₂ and water vapor between the bubble and the tissue, leaving N₂ 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 N₂ 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 N₂ away from the bubble interface.

View larger version (39K): [in this window] [in a new window]   Fig. 1.   In vivo and model intravascular gas bubble geometries. A: microscopic view of air bubble entrapped in rat cremaster arteriole. Bubble dimensions are length at time 0 (l0) = 368 µm and radius at time 0 (R0) = 33 µm. B: model air embolism geometry at time t = 0. C: model bubble shape for 0 < t < T1, where T1 is time taken for collapse of cylindrical portion, after partial gas absorption. D: model air embolism configuration at t = T1.

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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 l₀ and R₀ to define an aspect ratio, X = l₀/R₀. The total time for bubble absorption, T, is T = T₁ + T₂, with T₁ and T₂ defined as the time taken for the collapse of the cylindrical and spherical portions, respectively. The values for T₁ and T₂ are calculated from Eqs. 11 and 13 to be

(14)

and

(15)

in which  = (2)^1/2. Formally, P_b N2(R^*), the pressure inside the bubble based on the time-averaged mean of the radius of the sphere, R^*, replaces the expression P_b N2[R(t)], which behaves in a nonlinear time-dependent fashion. Initially, with larger values of R, P_b N2[R(t)] is small; however, as the radius becomes very small, P_b N2[R(t)] becomes extremely large. The nonlinear rise of internal pressure does not lend to easy calculation of T₂; therefore, we have made the approximation that R  R^*. We calculated the value of R^* as a fraction, n, of the initial radius: R^* = nR₀. To find the value of n, we used the expression

(16)

Because each of the integrals in Eq. 16 equivalently represents the area under the radius-time curve, we evaluate the latter expression, because we can write T'(R') explicitly as T'(R') = T₂(R₀)  T₂(R'), giving

(17)

Using Eqs. 15 and 17, we obtain the following expression for n

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With this result, we considered a wide range of initial geometries, X, and bubble volumes.

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 CO₂.

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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 of l₀ and R₀. 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 of l₀ and R₀ 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 × 10⁴ cm²/min (2, 29); _ti = 2.103 × 10⁵ ml · cm³ muscle¹ · mmHg¹ (2, 29);  = 0.042 ml blood · cm³ muscle¹ · min¹ (14); and 456 Torr  Pa_N2 500 Torr (11). Those values not listed were assumed to remain the same as those previously defined for the cerebral circulation. From the dimensions of initial bubble geometry, the program calculated the bubble volume and derived a corresponding initial radius for a spherical bubble of the same volume. With this initial radius, the computer simulation then calculated the absorption time for the sphere. For simplicity, bubble dislodgment to a vessel of different diameter was not accounted for in the mathematical model, although this situation occasionally occurred in vivo.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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. 1A.

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 t-test, P < 0.01).

View larger version (13K): [in this window] [in a new window]   Fig. 2.   Measured and predicted absorption times for bubble having initial volume of 2.8 nl. In vivo measurement (large dotted line), prediction using initial in vivo bubble dimensions (dashed line), and prediction assuming spherical bubble of same initial volume (small dotted line) are shown.

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 for X = 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.

View larger version (19K): [in this window] [in a new window]   Fig. 3.   Bubble volume during absorption for different initial aspect ratios (X = l0 /R0). All curves are for an initial volume of 2.5 nl. Aspect ratios shown are spherical bubble X = 0 (solid line), X = 2.5 (long dashed line), X = 10 (short dashed line), and X = 20 (dotted line).

View larger version (18K): [in this window] [in a new window]   Fig. 4.   Predicted absorption time, T, as a function of initial bubble aspect ratio, X. Curves are for fixed initial bubble volume. Dashed line at X = 2.6 defines maximum absorption time, Tmax. Solid lines at X = 13.1 and X = 0.3 are upper and lower bounds, respectively, for absorption times falling within 80% of Tmax.

View larger version (23K): [in this window] [in a new window]   Fig. 5.   Dependence of absorption time on initial surface area, A0(X), for several initial bubble volumes. Spherical bubbles for each volume are designated by solid circles, whereas maximum absorption time is represented by open circles. Lines passing through symbols are X = 0 (dashed line) and X = 2.6 (solid line). See text for explanation of equation.

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	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

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. 1A).

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 2.5-nl bubble has markedly different absorption curves depending on initial geometry. The theoretical model, simulating cerebral IGEs, predicts that there exists a value of X = l₀ /R₀  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.