Since 1917, researchers have developed a variety of theoretical models to study gas bubble dynamics in liquids, many of which are summarized in . We structure this section (and ) by starting with the simplest case—that of the free microbubble driven by a low amplitude sound field in an infinite fluid. Clearly, there are many levels of assumptions required to apply this model to the real application—that of encapsulated bubbles traveling within small blood vessels which contain a high concentration of red blood cells and driven by high amplitude sound fields. In order to accurately model the behavior of contrast microbubbles

*in vivo*, a series of increasingly complex models have been developed, still typically including a number of assumptions and simplifications in each case. For example, recent treatments of the behavior of bubbles within a bubble cloud have begun with the unshelled bubble (

Hamilton *et al.*, 2005), while simultaneously others address the behavior of a single micro-bubble non-spherically oscillating within a small vessel (

Qin and Ferrara, 2006).

| **Table 2**Summary of models of gas bubble dynamics |

The fundamental equations of bubble dynamics were developed by Rayleigh and Plesset (

Rayleigh, 1917;

Vokurka, 1985;

Plesset and Prosperetti, 1977;

Plesset, 1949) (), neglecting liquid compressibility effects and assuming that the gas pressure in the bubble is uniform and obeys the polytropic law, which is given by:

where

*R*_{0} is the bubble radius at equilibrium,

and

represent respectively the first- and second-order time derivatives of the bubble radius

*R*,

*p*_{0} is the hydrostatic pressure,

*p*_{i}(

*t*) is the incident ultrasound pressure in the liquid at an infinite distance,

*p*_{g} (

*t*) is the uniform gas pressure within the bubble and

*ρ*,

*σ* and

*η* are the density, surface tension and viscosity of the bulk fluid, respectively. A historical review of the development of this equation was given by Plesset and Prosperetti (

Plesset and Prosperetti, 1977). The gas pressure within the bubble depends on the volumetric change with respect to its equilibrium state and heat diffusion across the bubble wall (

Prosperetti *et al.*, 1988). The polytropic law neglects heat diffusion and relates gas pressure and bubble volume as given by:

where

*κ* is the polytropic exponent,

*R*_{0} is bubble equilibrium radius. If the thermal diffusion length in the gas is greater than the bubble radius, the bubble will behave isothermally (i.e.

*κ* ≈ 1). However, if the thermal diffusion length in the gas is much smaller than the bubble radius and bubble radius is much less than the wavelength of sound in the bubble the bubble will behavior adiabatically (ie.,

*κ* ≈

*γ,* the specific heat ratio of the gas within the bubble) (

Prosperetti, 1982a).

Using a small-amplitude oscillation assumption, the Rayleigh-Plesset equation has been widely applied to study many aspects of bubble dynamics such as bubble natural frequency, acoustic scattering characteristics, thermal damping effects (

Devin, 1959;

Fanelli *et al.*, 1981;

Vokurka, 1985;

Dejong *et al.*, 1994b;

Dejong *et al.*, 1992;

Prosperetti, 1975;

Miller, 1981;

Strasberg, 1956;

Gaunaurd and Uberall, 1978;

Sage *et al.*, 1979;

Allen *et al.*, 2001;

Hu *et al.*, 2004;

Tsamopoulos and Brown, 1983,

1984;

Feng and Leal, 1994) (). Taking into account of the effect of the surface tension, the Minnaert expression for bubble resonance frequency can be given by (

Minnaert, 1933;

Miller, 1981):

The emitted ultrasound pressure at distance *r* from the bubble center is

As verified by numerous experiments, the Rayleigh-Plesset equation works well when the Mach number (the ratio of the velocity of the bubble wall to the sound speed in the liquid,

/

*c*) is small (

Putterman *et al.*, 2001). However, when the incident pressure amplitude increases, the Mach number approaches unity and sound radiation and liquid compressibility become important. For higher pressure amplitudes and larger radial oscillation, extensions of the Rayleigh-Plesset equation have been proposed, including the Keller equation, the Herring equation and the Gilmore equation (

Prosperetti and Lezzi, 1986;

Prosperetti, 1987;

Lezzi and Prosperetti, 1987;

Keller and Kolodner, 1956;

Keller and Miksis, 1980;

Prosperetti *et al.*, 1988;

Trilling, 1952;

Brenner, 1995;

Gilmore, 1952;

Barber *et al.*, 1997;

Lofstedt *et al.*, 1995) (). Prosperetti et al (

Prosperetti *et al.*, 1988;

Prosperetti and Lezzi, 1986) demonstrated that there is a one-parameter family of equations to describe bubble oscillation in unbounded liquids, namely:

where λ is an arbitrary parameter that preserves the first order of accuracy of the equation and

*p*_{B} (

*t*) is the liquid pressure on the external side of the bubble wall, which depends on the internal bubble gas pressure on the wall:

Setting the parameter λ equal to 0 recovers the Keller equation (

Keller and Kolodner, 1956;

Keller and Miksis, 1980), and λ equal to 1 results in the Herring and Trilling equation (

Trilling, 1952). The effects of other factors such as non-uniform pressure within the bubble and heat and gas transfer between the bubble and liquid have also been extensively examined (). The Keller-like equation written in terms of the enthalpy has been verified by extensive experimental results and demonstrated to yield results that are in agreement with full partial differential equation numerical simulations (

Prosperetti and Lezzi, 1986;

Lezzi and Prosperetti, 1987;

Lin *et al.*, 2002).

Ultrasound contrast agents typically have a shell with a thickness from ten to hundred nanometers, motivating studies of the effects of bubble shell viscosity and elasticity (). Roy and his co-workers first treated the bubble shell as a simple viscous liquid and found good agreement between model predictions and

*in vitro* experimental measurements of the cavitation threshold (

Roy *et al.*, 1990). De Jong and his co-workers later (

Dejong *et al.*, 1994a,

b;

Dejong and Hoff, 1993;

Dejong *et al.*, 1992) modeled bubble shells as layers of elastic solids and studied acoustic attenuation and backscatter and nonlinear oscillation, validated by experimental results.

Models that include shell properties using classical mechanical principals have been developed. Church (

Church, 1995) derived a Rayleigh-Plesset-based equation describing the dynamics of encapsulated gas bubbles, assuming that that coating material is a layer of an incompressible, viscous-elastic solid. Hoff et al (

Hoff *et al.*, 2000) then developed a model using viscous and elastic properties of the shell to describe polymeric microbubble behavior. Modified Rayleigh-Plesset equations have been developed for thin and thick viscoelastic-shelled agents, examining shell viscosity and elasticity effects (

Allen *et al.*, 2002;

Morgan *et al.*, 2000). These equations were validated by direct comparison of the predicted bubble radius with optically-measured streak images (

Allen *et al.*, 2002;

Morgan *et al.*, 2000). For a gas bubble encapsulated by a layer of viscous-elastic solid shell or viscous liquid shell, the Rayleigh-Plesset-based equations derived by (

Church, 1995;

Allen *et al.*, 2002) can be written in a simple formula as:

where

*R*_{1} is the inner radius of the agent;

*R*_{2} is its outer radius;

*R*_{10} is its inner radius at equilibrium status;

*σ*_{1} is the surface tension at the inner radius,

*σ*_{2} is the surface tension at the outer radius,

*ρ*_{s} is the shell density.

*G*_{s} equals the Lame constant (the modulus of rigidity) if the agent is a elastic solid shell (

Church, 1995) and

*G*_{s} equals 0 for a viscous liquid shell (

Allen *et al.*, 2002).

While the above discussion demonstrates that substantial effort has been applied to the development of models for shelled microbubbles, accurately representing the shell properties before and after insonation continues to be a challenging problem. Given the small thickness of the shell (typically nanometers) and small diameter of the bubble (typically microns), evaluating the mechanical properties of the shell material

*in situ* is challenging. Moveover, the molecular scale properties of lipid membranes cannot be simply incorporated within models that describe only the radial component of oscillation. Lipid molecules self-assemble at the gas-liquid interface. When the lipid membrane is compressed (as might be expected during the compressional half-cycle), the monolayer can reversibly buckle or shed lipid into solution (

Ridsdale *et al.*, 2001). During rarefaction, the lipid membrane expands, potentially leaving patches of uncoated gas-liquid interface. Application of the classical mechanic theory for these scenarios is very limited. Therefore specialized approaches for each shell material are required. Further, in order to translate these models to a blood and tissue environment, investigations of the effect of tissue elasticity and non-uniform blood viscosity on oscillations of ultrasound contrast agents have been undertaken and efforts have also been made to develop theoretical models for radial oscillations of gas bubbles in non-Newtonian liquids ().

After intravenous bolus injection, low amplitude ultrasound pulses can be administered to deflect the drug-coated bubbles toward the blood vessel wall and facilitate imaging or drug delivery. A model was developed to predict the radial oscillation and simultaneous translational displacement of an encapsulated gas bubble after insonation by clinically-applicable megaHertz ultrasound pulses (

Dayton *et al.*, 2002;

Zhao *et al.*, 2004) (). As with the free bubble, the translational displacement of the shelled bubble was shown to be determined by the combination of the driving force, the translational added mass, the oscillatory added mass and the quasistatic drag force.

Oscillations of ultrasound contrast agents within the blood pool are constrained by blood vessels, and this effect is greatest within small vessels. The boundary can be simplified and image theory or boundary integral methods then applied in order to calculate bubble oscillations (). This simplification works well when the ratio of bubble diameter to the tube diameter is small. Alternatively, a bubble oscillating within a small tube has been described by analytical approximations (

Hu *et al.*, 2005;

Yuan *et al.*, 1999) or by numerical computations (

Ory *et al.*, 2000;

Qin and Ferrara, 2006;

Qin *et al.*, 2006;

Ye and Bull, 2006;

Qin and Ferrara, 2007), (). Our investigation has demonstrated that oscillations of a bubble within a small vessel depend on the vessel and bubble diameter and mechanical properties of vessel and connecting tissues (

Qin and Ferrara, 2006,

2007). The natural frequency of the bubble within a small rigid vessel is substantially decreased with decreasing vessel size. However, for the same sized vessels, the natural frequency of a bubble increases with decreasing vessel rigidity (

Qin and Ferrara, 2007). For a bubble constrained in a small vessel, the non-radial component of oscillation is significant (

Hu *et al.*, 2005;

Qin and Ferrara, 2006) () and should be included as the effect of the oscillation on the vessel wall is evaluated. With a transmitted PRP of 0.5 MPa and a center frequency of 1 MHz, the expansion of a microbubble (with an initial diameter of 3 μm) within a compliant vessel (of diameter 8 μm) can dilate the vessel wall by a distance as large as a few microns (

Qin and Ferrara, 2006) ().

In drug delivery, the ultrasound pressure (PRP > 0.5 MPa) can be substantially larger than the pressure used in traditional imaging (PRP ~50 kPa). When insonified by a high ultrasound pressure, bubbles become unstable, rapidly collapse and fragment; thus, models for stability and mode analysis are summarized in . Further, given that drug delivery studies often use high microbubble concentrations, models of bubble-bubble interaction, bubble-bubble-vessel interaction and bubble cloud/density effects have been developed (), often requiring substantial computational resources to solve partial differential equations. In limited cases, solutions have been obtained assuming a uniform formulation or using image theory.

In summary, understanding and accurately predicting microbubble oscillation

*in vivo* is a challenging problem. While a great deal of progress has been made in accurately representing the radial oscillation of the shelled microbubble in an infinite fluid, still refinements and improvements of such models continue, e.g. (

Doinikov and Dayton, 2006). Moreover, modeling of nonlinear and aspherical oscillation, modeling of bubble clouds and modeling of bubble interaction with neighboring cells and vessel walls require additional development. Therefore, gaining a perspective on the hierarchy of approaches to this problem (as in ) can be of value.