Since, as the next section will outline, no such modelcurrently exists, such provision is the purpose of this paper. Figure 2.Schematics ofsteady-statebubblevolume oscillations vs.applied pressure. The leftcolumn shows the resultfor the inertia-controlledregime, and the rightcolumn corresponds tothe stiffness controlledregime. The four rowscorrespond to conditionswhich are (from topdownwards): linear andlossless; linear and lossy;nonlinear and lossless;nonlinear and lossy.2. Models for propagation through contrast agent populationsIn 1989 Commander and Prosperetti  produced the most widely-used formulation for predictingthe propagation characteristics of an acoustic wave through bubbly liquids. Its applicability forcontrast agents in vivois limited, since the theory assumes linear steady-state bubble pulsations inresponse to a monochromatic driving field. Leighton et al.[5,6] developed a theoretical framework intowhich any single-bubble model could be input, to provide propagation characteristics (e.g.attenuationand sound speed) for a polydisperse bubble cloud (which may be inhomogeneous) incorporatingwhatever 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 mathematicaltools 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 ldPis the change in the pressureapplied to the lthvolume element as a result on an incident ultrasonic field. Dividethe polydispersebubble population into radius bins, such that every individual bubble in the jthbin is replaced byanother bubble which oscillates with radius ( )jRtand volume ( )jVt(about equilibrium values of 0jRand0jV), such that the total number of bubbles Njand total volume of gas ( )jjN Vtin the bin remainunchanged by the replacement. If the bin width increment is sufficiently small, the time history ofevery bubble in that bin should closely resemble 0( )(, )jjVtV Rt(the sensitivity being greatestaround resonance). Hence the total volume of gas in the lthvolume element of bubbly water is0011( ),( ),( )ljljJJgjjcjjjjVtNRt VtVnRt Vt, where 0,ljnRtis the number of bubbles perunit volume of bubbly water within the jthbin. From this scheme Leighton et al.[Error: Referencesource not found] identified a parameter: 2011ljJcwwwjjljccnRdVdP. Crucially thislcprovides a generic framework into which any bubbly dynamics model may be inserted(giving( )( )jldVtdP tappropriate 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 amonodisperse bubble population pulsating in the linear steady-state (Figure 2). If the propagation werelinear and lossless, the graphs of applied pressure (P) against bubble volume (V) would take the form
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