## Abstract

Vascular gas embolism—bubble entry into the blood circulation - is pervasive in medicine, including over 340,000 cardiac surgery patients in the U.S. annually. The gas–liquid interface interacts directly with constituents in blood, including cells and proteins, and with the endothelial cells lining blood vessels to provoke a variety of undesired biological reactions. Surfactant therapy, a potential preventative approach, is based on fluid dynamics and transport mechanics. Herein we review literature relevant to the understanding the key gas–liquid interface interactions inciting injury at the molecular, organelle, cellular, and tissue levels. These include clot formation, cellular activation, and adhesion events. We review the fluid physics and transport dynamics of surfactant-based interventions to reduce tissue injury from gas embolism. In particular, we focus on experimental research and computational and numerical approaches involving how surface-active chemical-based intervention. This is based on surfactant competition with blood-borne or cell surface-borne macromolecules for surface occupancy of gas–liquid interfaces to alter cellular mechanics, mechanosensing, and signaling coupled to fluid stress exposures occurring in gas embolism. We include a new analytical approach for which an asymptotic solution to the Navier–Stokes equations coupled to the convection-diffusion interaction for a soluble surfactant provides additional insight regarding surfactant transport with a bubble in non-Newtonian fluid.

## Introduction

Anatomically, the vascular space is a closed tubing system filled with blood. Although gases such as oxygen, nitrogen, and carbon dioxide are present, they are dissolved in the plasma or, as also occurs with oxygen, bound to hemoglobin within red blood cells (RBCs). Bulk gas, in the form of gas embolism bubbles, can be introduced into the body both accidentally and deliberately, posing a health threat regardless of the gas source. Gas embolism is pervasive in medicine: it is a fact in all forms of cardiac surgery [1]; it occurs during endoscopy [2], gynecological procedures [3], laparoscopy [4], tissue biopsy [5], neurosurgery [6], central venous line use [7], radiological procedures [8], urological surgery [9], pacemaker placement [10], ophthalmological surgery [11], positive pressure ventilation [12], and with radiology contrast injector equipment use [13]. It occurs in divers [14] and is an occupational hazard for hyperbaric chamber care providers [15].

While very tiny amounts of venous gas embolism rarely have significant physiological consequences, arterial gas bubbles can have dire effects due to their convection into the vasculature of any organ, especially the brain. Cerebral gas embolization is studied most in relation to cardiac surgery [1618]. After coronary bypass surgery, 50% to 70% of patients (>340,000 per annum in the U.S.) have a measurable decrement in brain function. Clinical manifestations of injury are compatible with multifocal cerebrovascular gas embolism [19]. Neurological deficits include stroke, impaired consciousness, seizures, and cognitive impairment limiting function [20]. No treatment strategy has emerged for the large numbers who risk clinically significant gas embolic events. The mortality, morbidity, length of hospitalization, and use of extended care facilities necessitated by adverse cerebral events alone have an enormous human and economic impact [21].

The physiological consequences of gas embolism are the result of biomolecular interactions occurring at the gas–liquid interface. Gas bubbles occlude vessels through binding interactions with endothelial cells which diminish blood flow. Bubbles also interact with proteins present in the blood to initiate blood clotting and inflammatory pathways [22,23]. Minimizing activation of pathophysiological responses evoked by intravascular bubbles is an important strategy to prevent injury from developing. One means of achieving this is by rendering the air-liquid interface biologically inert through rapid adsorption of surfactants that out-compete blood-borne macromolecules (e.g., proteins) for interfacial occupancy. Bubbles in confined flows produce fluid structures that can influence local mixing as well as wall interactions [24]. Surfactants are also well known to alter bubble dynamics including interfacial shape [25]. Surfactant adsorption–desorption characteristics also influence bubble dynamics [26]. In this framework, surfactants can be understood to influence the adverse physiological effects otherwise resulting from vascular gas embolism.

## Gas Embolism, Surfactants, and Blood Clotting

In Ref. [27], Eckmann and Diamond studied the effects of surfactants to reduce activation of blood clotting stimulated by blood exposure to bubbles. They hypothesized that surfactants would preferentially populate gas–liquid interfaces and limit the interfacial area available for occupancy by those moieties stimulating clot formation. They quantified the relative contributions of shear rate, shear duration, and exposure to gas embolism bubbles to the production of thrombin, an end product of activation of blood clotting. They also assessed the extent to which several novel surface-active compounds, added to blood samples before gas embolization, attenuated the clotting response under conditions of imposed shearing. They measured thrombin production in human whole blood samples subjected to venous and arterial levels of shear stress over various durations, with and without the addition of gas embolism bubbles and with and without the addition of a surfactant. A major finding was that the surfactants markedly attenuated thrombin production in the gas-embolized (sparged) samples by ∼30% to 70%, as shown in Fig. 1.

Fig. 1
Fig. 1

As a corollary to studying bubbles, surfactants, and thrombin production, Eckmann et al. [28] evaluated surfactant effects on bubble interactions with platelets, which are also important to blood clotting. They used human platelet-rich plasma and analyzed videomicroscopy images of sparged and unsparged specimens drawn through rectangular glass microcapillaries as well as Coulter counter measurements to assess platelet–bubble and platelet–platelet binding. Imaging revealed 60-100 platelets adhered to bubbles in sparged, surfactant-free samples (Fig. 2), but with surfactant added, only a few platelets adhered to bubbles. The numbers of platelet singlets and multiplatelet aggregates not adherent to bubbles were lower in the presence of surfactant compared both with unsparged samples and sparged samples without surfactant. These experiments demonstrated that the introduction of bubbles caused platelets to bind to the bubbles and each other. Surfactants added before sparging attenuate both the platelet–bubble, and platelet–platelet binding.

Fig. 2
Fig. 2

In order to determine the dose dependency of chemically distinct surfactants to attenuate platelet activation and aggregation induced by bubble exposure, Eckmann et al. [29] used two different surfactants at two different concentrations (1%, 10% v/v) in sparged blood. Platelet aggregometry results and flow cytometry measures of expression of CD41, CD62p, and CD63 markers of platelet activation demonstrated dose-dependent and saturable surfactant effects. The system was also addressed computationally using a population-based model for platelet aggregation in the presence of gas bubbles and surfactants. Monte Carlo simulations predicted that the presence of the surfactants inhibited maximal platelet aggregation. The final fractional aggregation was calculated to be decreased by ∼9 to 30%, depending on the surfactant and its concentration (Fig. 3). This agreed favorably with experimental findings.

Fig. 3
Fig. 3

## Gas Embolism, Surfactants, and Endothelial Cells

### Cellular Signaling Responses.

Embolism bubbles transiting the vasculature can enter vessels small enough for the bubble diameter to be larger than the vessel diameter. The bubble deforms into a “sausage” shape (cylindrical section with hemispherical caps) with the air-liquid interface directly contacting surface elements of the endothelial cells lining the vessel lumen. Vascular endothelial cells are a critical structural interface between circulating blood and the adjoining tissues. They play an essential role in mechanosensing within the vasculature and they respond to different levels of hemodynamic shear stress by releasing vasoactive substances to modulate and regulate vascular tone. In an initial study of gas bubble interactions with endothelial cells, Kobayashi et al. [30] contacted individual cells with microbubbles. Cells were loaded with fluorescent dyes indicating calcium- and nitric oxide-signaling and cell viability. The surfactant Pluronic F-127 (PF-127) (Invitrogen, Carlsbad, CA) was added to the culture media in some experiments. Microbubble contact was highly associated with calcium entry into cells and rapid cell death. The presence of the surfactant reduced the lethality of bubble contact by over 75%. This suggested that surface interactions between the bubble interface and plasma- and cell surface-borne macromolecules modulated the mechanism of intracellular calcium trafficking to induce cell death or aberrant cellular function.

In a follow-on study, Sobolewski et al. [31] identified the specific intracellular mechanisms by which bubble contact with the cell surface causes calcium influx into cells. Their experiments showed that calcium influx, which can be lethal to cells, is mediated by a stretch-activated channel. Mechanical forces coupling the gas–liquid interface and the cell membrane produce the response. Additionally, Sobolewski et al. [32] showed that the membrane stretch resulting from bubble/cell mechanical coupling also activates an intracellular signaling pathway that depolarizes mitochondria within cells and causes cellular bioenergetic dysfunction. In an elegant set of molecular studies, Klinger et al. [33] found that the transient intracellular calcium flux was eliminated if either the stretch-activated channel inhibitor, gadolinium, or the transient receptor potential vanilliod family inhibitor, ruthenium red, were present. Also, there was no bubble contact-induced calcium upsurge when cells were exposed to cytochalasin-D, an inhibitor of actin polymerization. These findings reveal the intracellular structures responsible for coupling the extracellular signal of bubble surface/cell membrane to the calcium response described previously. In addition, the biomechanical interactions between the cell and bubble surfaces were also identified through selective digestion of specific macromolecular elements of the endothelial surface layer (ESL). Hyaluran digestion reduced cell-bubble adherence and resulted in more rapid induction of calcium influx after bubble contact. Heparan sulfate depletion significantly decreased the calcium transient amplitude, as did pharmacologically induced sydencan ectodomain shedding. These key molecular constituents involved in the mechanical coupling of bubble interface to endothelial cells from “outside to inside,” which result in mechanotransduction of bubble-cell contact, are depicted in Fig. 4. Finally, Klinger et al. [34] demonstrated that the surfactant perfluorocarbon Oxycyte abolished any bubble-induced calcium transients in a concentration-dependent fashion. Further, no cell death was observed for any concentration of this bioinert surfactant. The surfactant exerted its preferential gas/liquid interface occupancy over that of the mechanosensing ESL components, validating the surfactant mechanism of action to ameliorate cell response to bubble contact.

Fig. 4
Fig. 4

### Surface Adhesion and Bubble Clearance Responses.

One of the hallmarks of vascular gas embolism is that bubbles adhere to endothelial cells as a result of gas–liquid interfacial adsorption of macromolecules tethered to the endothelial cell surface. When a sufficient population of tethering molecules populates the interface, the bubble motion is restrained or even arrested. Suzuki and Eckmann [35] measured the adhesion force per unit surface area of gas–liquid interface using excised microvessels and a perfusion apparatus as shown in Fig. 5. The vessel wall was micropunctured and an embolism bubble was injected. The perfusion apparatus was used to increase the pressure on one side of the bubble until it displaced. Videomicroscopy measurements of bubble dimensions were coupled with the differential pressure across the microbubble recorded at the moment of bubble movement to assess the adhesion force. In a subsequent investigation, Suzuki et al. [35] demonstrated that the presence of a surfactant in the microvessel perfusate before the introduction of the embolism bubble reduced the bubble adhesion force and also preserved endothelial cell structure and vasomotor tone function. Inclusion of the surfactant protected the endothelium from mechanical destruction otherwise caused by large-scale gas perfusion (i.e., a gigantic gas embolism).

Fig. 5
Fig. 5

The fate of intravascular bubbles is for the gas to be reabsorbed over time. This is observable if a bubble's axial motion to transit the vasculature is arrested by adhesion to the vessel wall. Intravascular bubble adhesion and resultant physiological gas resorption have been studied in vivo and modeled. Branger and Eckmann [36] developed a mathematical model of the absorption time of intravascular gas embolism bubbles, accounting for the bubble geometry—a cylinder with hemispherical end caps—observed in vivo. They solved the governing gas transport equations numerically and validated their model using data obtained from videomicroscopy measurements of bubbles in the intact cremaster muscle microcirculation of male rats. The model closely predicted actual absorption times for experimental intravascular gas embolisms, as shown in Fig. 6. Their results demonstrate that embolism bubbles having volumes as small as 10 nanoliters can persist for more than 35 min, a long time to obstruct blood flow and limit oxygen delivery to a region of the brain or other organs.

Fig. 6
Fig. 6

As a translational physiology project, Eckmann and Lomivorotov [37] studied microvascular gas embolization clearance following intravascular administration of a surfactant. With the approach used in Ref. [36], they found that bubbles having smaller volume embolized smaller diameter vessels in the surfactant treated groups. A higher incidence of bubble dislodgement and larger distal intravascular displacement of the bubbles occurred with the surfactant. The time to bubble clearance and restoration of blood flow also decreases. These dynamic events did not occur if surfactant administration occurred after, rather than before, gas embolization, indicating that surfactant transport to the bubble interface was essential for its beneficial value to be realized. A study by Eckmann and Armstead [38] involving cerebrovascular gas embolization amplified the value of having a surfactant circulating in the blood before the gas entry event. Cerebral arterial gas embolism was induced in rats, including groups receiving two different surfactants (PF-127, Oxycyte) intravenously in advance of embolization. Magnetic resonance imaging was performed to determine the size of the resultant brain stroke. Animals underwent cognitive and behavioral testing to assess function following the injury. The gas-embolized rats had significant cognitive and sensorimotor dysfunction that persisted one month after the embolic event. Prophylaxis with either surfactant rendered stroke undetectable by magnetic resonance imaging scanning and also markedly reduced the postembolic deficits in both cognitive and behavioral performance. This study effectively demonstrated on a functional neurological level that the manipulation of the bubble interfacial mechanics translated into protection from gas embolism related brain injury.

## Gas Embolism, Fluid Physics, and Interfacial Transport

### Bubble Motion in a Tube.

The biological studies detailed above provide a rationale and empiricism for utilizing surfactants in a clinical context. However, they do not provide detail regarding the relevant fluid physics and interfacial transport involved in bubble dynamics in flow. These include the coupling of the convection-diffusion interactions occurring with surfactants within the bulk fluid and along with the bubble interface. Much has been done to study the motion of drops and bubbles in the experiment and theory, especially in the case of constant surface tension. Using a numerical approach, Mukundakrishnan et al. [39] evaluated axisymmetric rise and deformation of an initially spherical gas bubble released from rest in a Newtonian liquid-filled, finite circular cylinder as a first approximation of the intravascular bubble problem. Bubble motion and deformation were characterized over a broad range of Morton number, Eötvös number, Reynolds number, Weber number, density ratio, viscosity ratio, and the ratios of the cylinder height and radius to the diameter of the initially spherical bubble. The numerical procedure included a front tracking finite difference method coupled with a level contour reconstruction of the front. This ensured a smooth distribution of the front points and conservation of the bubble volume. Important findings of this work were that bubble shapes at terminal states varied from spherical to intermediate spherical-cap-skirted (see Fig. 7) and that in a thin cylindrical vessel, the motion of the bubble was retarded due to increased total drag. The bubble achieved terminal conditions only a short distance from release. For a fixed volume of the bubble, increasing the cylinder radius resulted in the formation of well-defined rear recirculatory wakes associated with lateral bulging and skirt formation.

Fig. 7
Fig. 7

Given that blood is non-Newtonian, it is also valuable to study other fluid viscosity behaviors. For evaluating bubble motion in non-Newtonian fluid in a tube, Mukundakrishnan et al., have considered a shear-thinning Carreau-Yasuda fluid [40] and a generalized power-law fluid [41]. In each case, conditions relevant to flow in large and small blood vessels were considered, and a nearly occluding bubble was one of the bubble size conditions considered. Results were compared to those for a Newtonian fluid with otherwise matching physical parameters. In the case of the power-law fluid, the results for the Newtonian and the non-Newtonian models agreed well at high shear rates, but there was complete disagreement at lower shear rates. In particular, the bubble residence times at lower shear rates for the non-Newtonian model were predicted to be nearly double those determined in the Newtonian case. This result has significant implications, as detailed above, regarding the fate of endothelial cells. It highlights that to adequately describe the bubble dynamics for all shear rates present within the vasculature, a Newtonian blood viscosity model is inadequate. Utilization of a non-Newtonian model is required. In the case of the Carreau–Yasuda fluid, the drag on the bubble also increased as bubble dimension approached tube occlusion, resulting in retardation of bubble motion. Wall shear stress also significantly increased with increasing occlusion compared to the case of flow without a bubble. These features, which echo those described for the power-law fluid, amplify the need for an appropriate fluid viscosity definition to be incorporated in order to reveal the fluid dynamics associated with gas embolism.

One important fluid physics feature revealed by the analyses in Refs. [40] and [41] and highlighted by Mukundakrishnan et al. [42] was the appearance of time variation of shear stress at a point on the vessel wall due to bubble movement. Figure 8 depicts the velocity vectors and streamlines, plotted for an inertial frame in Figs. 8(a) and 8(b), respectively, for a slightly elongated bubble at a low Reynolds number (=0.2). They are also plotted in the moving frame in Figs. 8(c) and 8(d), respectively. The surface mobility of the bubble produces a Hill's vortex, which largely occupies the volume of the bubble. It is accompanied by two weaker secondary internal vortices, located near the front and rear stagnation points, as shown in Figs. 8(c) and 8(d). Because the sense of rotation of the secondary vortices opposes that of the primary vortex, the corresponding shear stresses have opposite signs. Two distinct bolus formations are evident at either end of the bubble. Two stagnation rings—convergent and divergent—arise on the bubble surface at the points of intersection of the backward flow near the wall and the recirculating boluses, as appear in Fig. 8(d). The recirculating vortex appearing in the streamline plot (Fig. 8(b)) results from fluid entrainment as the bubble moves along the wall. This flow feature exerts important spatial and temporal effects on the distribution of the shear stress and its gradients at the wall, which, in the case of gas embolism, is along the endothelial cell surfaces.

Fig. 8
Fig. 8

The time variation of the shear stress excess, depicted in Fig. 9 at a typical point “A” on the surface of an endothelial cell, behaves like a solitary axial wave that traverses across all points on the vessel lumen as the nearly occluding bubble transits the vasculature. Numbers 1–8 in Fig. 9 (bottom) correspond to spatial locations 1–8 in Fig. 9 (top). These numbers 1–8 in succession describe circumstances at “A” as the bubble approaches, traverses, and passes by this point. While the detail of the fluid physics is described in Ref. [42], these shear stress gradients coupled with sign reversals result in both compression (negative shear) and stretch (increased tension due to positive shear excess) of the cell membrane. These gradients are further amplified as the bubble dimension tends to full cross-sectional occlusion of the vessel, as shown in Ref. [43]. There are multiple potential physiologic implications of this shear stress solitary wave progression across the cell surface, including induction of endothelial cell membrane stretch, activation of mechanotransduction pathways, loss of cell membrane integrity, and even cell membrane stress failure. The biological consequences of these on calcium signaling, mitochondrial bioenergetic failure, and cell death are described above.

Fig. 9
Fig. 9

### Surfactant and Bubble Motion in a Tube: Computational Analysis.

The computational approach required for the analysis performed in Refs. [4043] required a sophisticated computational scheme that accounted for the location of the deformable gas–liquid interface. There was reliance on the front tracking algorithm developed by Zhang et al. [44]. This algorithm was developed to include surfactant transport, which was absent from those initial computational studies previously described. The approach taken relied on the development of a front tracking scheme that accommodated the presence of either a soluble or an insoluble surfactant. There was an emphasis on the dynamic adsorption of the soluble surfactant which nonlinearly alters the surface tension, and this in turn affects the flow and transport in a complex fashion. Moreover, since a gas–liquid interface is being addressed, it was required to accommodate the surfactant concentration jump occurring across the interface in order to properly evaluate both fluid flow and surfactant transport. In the algorithm, the adsorption scheme for the soluble surfactant is meticulously described such that the total mass of the surfactant is well conserved, and the mass flux is then accurately resolved through the implementation of an interface indicator function. Briefly, the numerical procedure consists of the following:

• Given a particular interface position, and bulk and interface concentration distributions, calculate the surface tension forces and distribute them as body forces to the fixed Eulerian grid.

• Solve the fluid flow equations with the given conditions to obtain the velocity and pressure fields.

• Use the computed velocity field to update the interface and bulk surfactant values.

• Update the position of the interface using the velocity field in a Lagrangian fashion.

• Reconstruct the interface using the level contour reconstruction procedure if the bubble volume changes by more than 0.5% or periodically after every few time steps.

• If the interface is reconstructed, project the surface concentration profile to the new interface in such a way as to conserve the amount of surfactant on the interface.

• Repeat these steps until either steady-state is reached or the bubble reaches the outlet of the computational domain.

Tests of the efficacy of various aspects of the algorithm were conducted and demonstrated to have the flexibility for studying a variety of surfactant adsorption/desorption models and surface tension profiles. These included the Langmuir and the Frumkin models, which have significant practical relevance. The numerical results obtained were qualitatively consistent with results where available. The results presented included an example of Marangoni flow which caused a bubble to propel out of its initial static location due to the development of a surface tension gradient. The method was also used to demonstrate that bubble motion in Poiseuille flow may be significantly retarded due to the presence of a soluble surfactant in the bulk medium. In that case, the Marangoni-induced motion was determined to be in a direction opposite to that driven by the bulk fluid pressure. The application indicated that as the location of the adsorptive interface approaches the tube wall, surfactant depletion in the bulk fluid near the interface may occur. This observation has particular significance in understanding gas embolism and for developing surfactant as a therapeutic.

The algorithm developed in Ref. [44] was implemented by Swaminathan et al. [45] to evaluate the motion of a finite-sized gas bubble in a blood vessel in the presence of a soluble surfactant capable of adsorbing and desorbing from the gas–liquid interface. The bubble, constituting a dispersed phase, was considered to be of sufficient size so as to nearly occlude the vessel. The bulk liquid was modeled as a two-layer fluid made up of a cylindrical core containing a high concentration of RBCs in suspension, and a surrounding annular cell-free layer. The cell-free layer was modeled as a Newtonian fluid having constant viscosity. The core region was modeled as a shear-thinning Casson fluid as expressed by Merrill et al. [46] and Walawender [47] having a viscosity that depended nonlinearly on multiple factors including the asymptotic Newtonian viscosity, the fractional content of the blood that is red blood cells, or the hematocrit, the yield stress, and the local shear rate (see below for further detail). The surface tension of the gas–liquid interface, which separated the continuous and dispersed phases, was spatially and temporally dependent on the surfactant concentration along with the interface following an equation of state for a Langmuir isotherm. Transport of the surfactant included its diffusion and convection in both the bulk fluid and on the interface, with the incorporation of adsorption and desorption from the interface.

The equations of motion and mass conservation for the fluid system coupled to the equations defining the mass flux of surfactant as well as its effects on surface tension were solved computationally as described above. Several cases of vessel size, corresponding to a small artery, a large arteriole, and a small arteriole in normal human anatomy, were considered. A fixed value of 0.45 was used for the hematocrit, and other parameters used for the Casson fluid description of viscosity were selected to correspond to the shear-thinning behavior of human blood. In addition, for arteriolar blood flow, where relevant, the Fahraeus–Lindqvist effect was included. Results demonstrated that, in the absence of any surfactant, the bubble motion caused temporal and spatial gradients of shear stress as previously noted and depicted in Fig. 9. A major finding of this work was that the surfactant altered the local fluid structure. The pressure-driven flow caused the bubble to deform with a bulge at the rear, resulting in a drainage flow that squeezed through the small gap in between the bubble and the vessel wall. This complex flow is a direct determinant of the surface mobility of the bubble and the resultant vortical motions that arise. The strengths of the recirculation and the vertical motions become reduced when the surfactant is present. This has the additional effect of markedly reducing the magnitude of the shear stress gradients imposed on the cell surface in comparison to an equivalent surfactant-free system, as shown in Fig. 10.

Fig. 10
Fig. 10

The general trend of reduction in shear stress illustrated has important implications in the reduction or prevention of endothelial cell injury resulting from nearly occluding embolism bubbles. The phenomenon depicted in Fig. 10 was found to be present for all vessel sizes and was independent of vessel orientation (horizontal or vertical). The magnitude of the reduction in shear stress gradient was dependent on the Reynolds number, with larger effects arising for smaller Reynolds numbers. This indicates that a particular surfactant's effects would be most significant in smaller, rather than in larger, vessels. The effects of surfactants having different transport properties were addressed by Swaminathan et al. [48]. Utilizing the same computational scheme described above, they examined a variety of chemical and physical properties of a model soluble surfactant to determine the significance of specific surfactant properties on the reduction in the level of the shear stress gradients imparted to the cell surface. They found that adsorption, desorption, and maximum possible monolayer interface surfactant concentration were the parameters that most significantly influenced the shear stress levels. The physical properties of bulk and surface diffusivity did not have significant effects. And in a separate study, Mukundakrishnan et al. [49] computationally modeled the effect of blood properties, specifically hematocrit, on the fluid–structure and the shear stress in a surfactant-free system of gas embolism. Interestingly, they found that variations in shear stress, spatial and temporal shear stress gradients, and the size of the gap present bubble and vascular endothelium surfaces were all impacted by hematocrit levels ranging from 0.2 (anemia) to 0.6 (polycythemia). Their results suggest that in the arterial system, the deleterious hydrodynamic effects of gas embolism on endothelial cells were significantly amplified by increasing values of hematocrit. Clinically, this is useful since it directs attention to the potentially deleterious effects of gas embolism in the presence of hemoconcentration while suggesting that mild hemodilution via hydration may be a strategy to minimize arterial gas embolism related endothelial cellular injury with our without a surfactant being used.

### Surfactant and Bubble Motion in a Tube: Analytical Approach.

While the work detailed above utilized computational methods, analytical methods can also be used to produce results and derive insights regarding the fluid physics and surfactant transport characteristics relevant to gas embolism. Herein we consider a small spherical gas bubble immersed in a Casson fluid having small yield stress. Both experimental observations and computational investigations have shown that for a bubble with a small volume, surface tension plays a dominant role and the shape of the bubble remains essentially spherical. On this basis, we assume the shape of the bubble to remain spherical in the following analysis. The origin of the spherical coordinate system $(γ,ϑ,φ)$ is setup at the center of the bubble. We also assume that the flow around the bubble is slow, steady, and axisymmetric. The continuous phase, which is an incompressible Casson fluid, contains a soluble surfactant. For this system, the governing equation of motion may be written
$∇p=∇⋅τ¯¯$
(1)
where p is the pressure, and the stress tensor τ for the Casson fluid [46,47] is given by
$τ¯¯=[μ∞12+το12(II2)12]2γ˙¯¯$
(2)
in which the second invariant of the rate of deformation tensor II is defined as
$II=γ˙¯¯:γ˙¯¯$
(3)
In Eq. (3) above, $γ˙$ is the rate of deformation tensor, τ0 is the critical yield stress, μ is the limit viscosity as $γ˙ →∞$. In blood, the quantities μ and τ0 are functions of hematocrit, H, and this relationship is given in Refs. [46] and [47] as
$μ∞12=[μρ(1−H)α]12$
(4)
$TO12=β[(1(1−H))α2−1]$
(5)
in which μp = 1 g/(cm s) is the viscosity for plasma (Newtonian). The parameters α and β for human blood are: α = 2.0 and β = 0.3315 [46]. The normal range of H in healthy adults is 0.36–0.48. Due to the assumed axisymmetry in the problem formulation, the stream function Ψ may be introduced to simplify the equations of motion, and these are expressed by
$vr=−1r2 sin θ ∂Ψ∂θ$
(6)
$vθ=−1r sin θ ∂Ψ∂r$
(7)

in which vr and vθ are the velocity components in the r and θ directions, respectively.

The soluble surfactant molecules can be adsorbed and desorbed at the gas–liquid interface from the bulk phase. We assume that the surfactant is not present in the bubble phase. The presence of the surfactant on the interface induces Marangoni stresses. We now consider cases in which the surface diffusion is negligible compared to surface convection of the adsorbed species, i.e.,
$Pe=URDS →∞$
(8)
in which U is the velocity far from the bubble, Ds is the surface diffusion coefficient for the surfactant, R is the radius of the bubble, and Pe is the mass transport Peclet number. Following Edwards [50], we neglect the interfacial shear viscosity, μs, and the interfacial dilatational viscosity, κs, on the basis that the interface may be regarded as Newtonian. The following boundary conditions apply for the flow field
$vr=0atr=R$
(9)
$τrθ=−1R ∂σ∂θ at r=R$
(10)
$limx→∞ Ψ=−12Ur2 sin2θ far field, r→∞$
(11)

in which σ is surface tension.

The surface tension, σ, is expanded around its equilibrium value
$σ=σeq−EΓeq(Γ−Γeq)+⋯$
(12)
In this relationship, the Gibbs elasticity, E, is defined by
$E=−∂σ∂InΓ$
(13)
and Γ is the surface concentration of the surfactant, which is a slow varying function of θ. To the leading order, Eq. (10) becomes
$τrθ=ERΓeq∂Γ∂θ, at r=R$
(14)
This is related to the desorption of the surfactant through (see Stone and Leal [51])
$k(Γ−Γeq)=1R sin θ∂∂θ(vθΓ sin θ)$
(15)

in which k is the desorption coefficient. Equation (1) must be solved subject to conditions (9), (11), (14), and (15).

#### Nondimensionalization.

We scale various quantities as follows:
$r*=rR, vr∗=vrU,vθ∗=vθU,P∗Pμ∞UR, Γ∗ΓΓeq$
(16)
in which the starred variables are dimensionless, and introduce the dimensionless number
$Cn=(TοRUμ∞)12$
(17)
Utilizing this scaling approach, Eq. (1) becomes
$∇∗ρ∗=∇∗⋅ T¯¯∗$
(18)
and the dimensionless stress tensor is given by
$τ*¯¯=[1+Cn(II2∗)12]2γ˙¯¯∗$
(19)
with
$II∗=γ˙¯¯∗:γ˙¯¯∗, γ˙¯¯∗=URγ˙¯¯∗$
(20)
The corresponding boundary conditions become
$vr∗=0 atr∗=1$
(21)
$τrθ∗=−Eσeq∂Γ∗∂θ at r∗=1$
(22)
$limr*→∞ψ*dydx=−12r*2 sin2θ as r*→∞$
(23)
and Eq. (15) may be written as
$Bi(Γ*−1)1 sin θ∂∂θ(vθ*Γ* sin θ)$
(24)

in which Bi is the Biot number defined by Bi = kR/U. In this approach, we are interested in the case where Bi1, which corresponds to a small U. For convenience we omit all the stars in the following development. It must be noted that all the variables, however, are dimensionless.

## Results

### Asymptotic Solutions.

Following standard procedures, we can eliminate the pressure term in Eq. (18) and re-express the problem in terms of a stream function, Ψ. Thus, the equation can be written as
$D4Ψ+∂μ∂r(∂3Ψ∂r3+1r2∂3Ψ∂r∂θ2−cotθr2∂2Ψ∂r∂θ−2r3∂2Ψ∂θ2+2cotθr3Ψθ)+∂μ∂θ(1r2∂3Ψ∂r2∂θ−1r4∂3Ψ∂θ3+cotθr4∂2Ψ∂θ2−1r4 sin2θ∂Ψ∂θ)+∂2μ∂r2F1(Ψ,r,θ)+∂2μ∂θ2F2(Ψ,r,θ)+∂2μ∂r∂θF3(Ψ,r,θ)+∂μ∂rF4(Ψ,r,θ)+∂μ∂θF5(Ψ,r,θ)=0$
(25)
in which
$D2≡∂2∂r2+ sin θr2∂∂θ(1 sin θ∂∂θ)$
(26)
and
$F1(Ψ,r,θ)=∂2Ψ∂r2−2r∂Ψ∂r−1r2∂2Ψ∂θ2+cotθr4∂Ψ∂θ$
(27)
$F2(Ψ,r,θ)=−1r2∂2Ψ∂r2+2r3∂Ψ∂r+1r4∂2Ψ∂θ2−cotθr4∂Ψ∂θ$
(28)
$F3(Ψ,r,θ)=4r2∂2Ψ∂r∂θ−2cotθr∂Ψ∂r−6r3∂Ψ∂θ$
(29)
$F4(Ψ,r,θ)=∂3Ψ∂r3+1r2∂3Ψ∂θ2∂r−2r∂2Ψ∂r2−2r3∂2Ψ∂θ2−cotθr2∂2Ψ∂r∂θ+2r2∂Ψ∂r+2cotθr3∂Ψ∂θ$
(30)
$F5(Ψ,r,θ)=1r4∂3Ψ∂θ3+1r2∂3Ψ∂r2∂θ−cotθr2∂2Ψ∂r2−4r3∂2Ψ∂r∂θ−2cotθr4∂2Ψ∂θ2+2cotθr3∂Ψ∂r+1r4(7+2 cos2θ sin2θ∂Ψ∂θ)$
(31)
The dimensionless viscosity of the Casson fluid is given by
$μ=[1+Cn(II2)12]2$
(32)
In the case of Cn ≪ 1 (small yield stress), we assume that the stream function Ψ takes the form of
$Ψ=Ψ0+CnΨ1+⋯$
(33)
and that, to leading order, Eq. (25) becomes
$D4Ψ0=0$
(34)
with the corresponding boundary conditions
$∂Ψ0∂θ=0 at r=1$
(35)
$limr→∞Ψ0=0 as r=∞$
(36)
The solution to Eq. (34) is
$Ψ0=[−12(r2−r−1)+C(r−r−1)] sin2θ$
(37)
in which C is a constant. The value of C may be determined by applying boundary condition (22), and this yields
$C=12(3+3Ma3+2Ma)$
(38)

in which Ma = E/kRμ is the Marangoni number. To the leading order, the solution coincides with the Newtonian solution.

Next, the first-order equation becomes
$4D4Ψ1=1II0∂II0∂r(∂3Ψ0∂r3+1r2∂3Ψ0∂r∂∂θ2−cotθr2∂2Ψ0∂r∂θ−2r3∂2Ψ0∂θ2+2cotθr3Ψ0θ)+1II0∂II0∂r(1r2∂3Ψ0∂r2∂θ−1r4∂3Ψ0∂θ3+cotθr4∂2Ψ0∂θ2−1r4 sin2θ∂Ψ0∂θ)+1II0[∂2II0∂r2−54II0(∂II04II0)2]F1(Ψ0,r,θ)+1II0[∂2II0∂θ2−54II0(∂II0∂θ)2]F2(Ψ0,,r,θ)+1II0[∂2II0∂r∂θ−54II0∂II0∂r∂II0∂θ]F3(Ψ0,r,θ)+1II0∂II0∂rF4(Ψ0,r,θ)+1II0∂II0∂θF5(Ψ0,r,θ)$
(39)
We can obtain the first-order correction Ψ1 to satisfy all the proper boundary conditions. Here, we assume that Ma is small, so that to the leading order C ∼ 1/2. Then the right-hand side of the inhomogeneous term becomes separable, and the first-order correction is given by
$Ψ1=−14[16(r−1−r)+r In(r)]$
(40)

Figure 11 shows the streamlines of the first order solution for a Casson fluid with Ma =0.04. Streamlines II0 are also demonstrated for Newtonian and power-law fluids [52]. It is easily recognized that the fluid motion represented by the streamlines for the Casson fluid, which is highly representative of blood, can hardly be appreciated to deviate from the streamlines for the Newtonian fluid. The streamlines for the power-law fluid depart a little more modestly from those of the other two fluids, but maintain the basic appearance and form, indicating the presence of a similar underlying flow structure being present in that case. This example gives the impression that any of these viscosity representations describe vascular gas embolism accurately at this small value of Ma.

Fig. 11
Fig. 11

The results presented in Fig. 12 show the streamlines of the first order for Newtonian, Casson, and power-law fluids with Ma=0.4. Again it is obvious that the fluid motion represented by the streamlines for the Casson fluid deviates only slightly toward the midline from the streamlines for the Newtonian fluid. The streamlines for the power-law fluid, in this case, take a completely different form, indicating the presence of a very different underlying flow structure being present in that case. This example demonstrates that the fluid mechanics is quite sensitive to the viscosity representations, which must capture the behavior of blood properly for the fluid dynamics behaviors resulting from vascular gas embolism to be captured accurately at this larger value of Ma.

Fig. 12
Fig. 12
We compute the drag force from
$FD=−πR2∫0π(τγγ+p)|γ=R sin(2θ)dθ+2πR2∫0π(τγθ)|γ=R sin2θdθ$
(41)
using a similar asymptotic approach for FD
$FD FD0=+CnFD1+⋯$
(42)
in which
$FD0=−2πRμ∞U(1+3+3Ma3+2Ma)$
(43)
$FD1=−πRμ∞U(1112+34Ma3+2Ma)$
(44)
In addition, it is convenient to compute the drag coefficient, CD, through the expression
$CD=(2F/A)/(ρU2)$
(45)
which yields
$CD∼1Re[1+3+3Ma3+2Ma+Cn(1124+17Ma3+2Ma)].$
(46)

Here the Reynolds number, Re, is defined as Re = ρUR/μ. A contour plot of the drag coefficient, CD, is shown in Fig. 13 as a function of dimensionless parameters Ma and Cn. In Fig. 13, Re=0.01, and CD increases due to the increasing presence of the surfactant, i.e., as Ma increases. The shear thinning effect is evident by observing the variation of the drag coefficient versus Cn for a fixed Ma. For a fixed Ma, CD increases as Cn increases. These results as well as those presented in Figs. 11 and 12 demonstrate the value of the analytical approach to provide some insights regarding the fluid dynamics and surfactant transport interactions in vascular gas embolism.

Fig. 13
Fig. 13

## Concluding Remarks

In summary, the biological response to vascular gas embolism involves events occurring at multiple length scales. These include biomacromolecular events incurred at the endothelial cell surface which are coupled to transmembrane signaling mechanisms, and which result in intracellular responses at the atomic level (e.g., calcium molecule release) and at the cell organelle level (e.g., loss of mitochondrial potential). These events can culminate in the loss of endothelial cell regulatory inputs into modulation of vascular tone or they can outright trigger cell death. Additionally, vascular gas embolism bubbles interact directly with elements of the blood clotting system. Experiments and numerical simulations have demonstrated how the presence of a gas–liquid interface in the blood can provoke platelet activation and thrombin formation. Moreover, the convection of embolism bubbles within the bloodstream can be arrested, resulting in vessel occlusion and obstruction of blood flow caused by adhesion interactions between the endothelial surface layer and the gas–liquid interface.

One approach to ameliorate all of these adverse responses to gas embolism is to render the gas–liquid interface effectively biologically inert via surface occupancy with a surfactant. The experiments, computational approaches, and analytical evaluations described above represent a compendium of studies directed at demonstrating the utility of surfactants to reduce or prevent the undesired biological responses to gas embolism-related events. In particular, the fluid mechanics analyses reveal the underlying mechanistic details of how surfactant transport to, and on, the bubble interface alters the local fluid–structure and modifies the local forces to which the endothelial cell surface is subjected. The reduction in shear stress gradients that is facilitated by surfactant transport has direct implications for altering cell membrane stretch-related injury. The relevant surfactant transport mechanics also provide insights regarding the therapeutic or prophylactic effects of intravascular surfactant for the treatment or prevention of gas-bubble related biological damage.

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