Leighton and Dumbrell AMUM v 10.doc

Since as the next section will outline no such model

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Since, as the next section will outline, no such model currently exists, such provision is the purpose of this paper. Figure 2. Schematics of steady-state bubble volume oscillations vs. applied pressure. The left column shows the result for the inertia-controlled regime, and the right column corresponds to the stiffness controlled regime. The four rows correspond to conditions which are (from top downwards): linear and lossless; linear and lossy; nonlinear and lossless; nonlinear and lossy. 2. Models for propagation through contrast agent populations In 1989 Commander and Prosperetti [ 4 ] produced the most widely-used formulation for predicting the propagation characteristics of an acoustic wave through bubbly liquids. Its applicability for contrast agents in vivo is limited, since the theory assumes linear steady-state bubble pulsations in response to a monochromatic driving field. Leighton et al. [ 5 , 6 ] developed a theoretical framework into which any single-bubble model could be input, to provide propagation characteristics ( e.g. attenuation and sound speed) for a polydisperse bubble cloud (which may be inhomogeneous) incorporating whatever features ( e.g. bubble-bubble interactions) are included in the bubble dynamics model. Because of the inherent nonlinearity, such a model cannot make use of many familiar mathematical tools of linear acoustics, such as Green’s functions, complex representation of waves, superposition, addition of solutions, Fourier transforms, small-amplitude expansions etc. The crux of this
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model is in the summation of the volume responses of the individual bubbles to the driving pressures. If the contrast agent cloud is divided into volume elements, let l dP is the change in the pressure applied to the l th volume element as a result on an incident ultrasonic field. Divide the polydisperse bubble population into radius bins, such that every individual bubble in the j th bin is replaced by another bubble which oscillates with radius ( ) j R t and volume ( ) j V t (about equilibrium values of 0 j R and 0 j V ), such that the total number of bubbles N j and total volume of gas ( ) j j N V t in the bin remain unchanged by the replacement. If the bin width increment is sufficiently small, the time history of every bubble in that bin should closely resemble 0 ( ) ( , ) j j V t V R t (the sensitivity being greatest around resonance). Hence the total volume of gas in the l th volume element of bubbly water is 0 0 1 1 ( ) , ( ) , ( ) l j l j J J g j j c j j j j V t N R t V t V n R t V t , where 0 , l j n R t is the number of bubbles per unit volume of bubbly water within the j th bin. From this scheme Leighton et al. [Error: Reference source not found] identified a parameter: 2 0 1 1 l j J c w w w j j l j c c n R dV dP . Crucially this l c provides a generic framework into which any bubbly dynamics model may be inserted (giving ( ) ( ) j l dV t dP t appropriate to bubbles in free field or reverberation, in vivo , in structures or sediments, or in clouds of interacting bubbles, etc. as the chosen model dictates). To illustrate this, consider a monodisperse bubble population pulsating in the linear steady-state (Figure 2). If the propagation were linear and lossless, the graphs of applied pressure ( P ) against bubble volume ( V ) would take the form
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