Characterizations of the mathematical models of Ultrasound Contrast Agents: a review

Abstract

The typical contrast agents of ultrasound imaging composed of microbubbles
of smaller than 7 μm in diameter this microbubbles consist of gas coated with
a protein, lipid or polymer layer it is acting as very powerful scatterers.
There are several models describe the dynamic behavior of these
microbubbles under ultrasound filed, the Lord Rayleigh’s model is the oldest
one and the basis for all other models, this model was modified by plesset by
added the driving sound field, resulting in an equation called Rayleigh-Plesset
equation which describes bubble oscillating due to the driving sound field in
an inviscid and incompressible fluid of constant density, RP model is a second
order nonlinear ODE, The RP equation may be modified by Noltingk,
Neppiras and Poritsky Whose added the effects of the surrounding field ,
this led to an equation called the RPNNP equation which is considered the the
first step to construct a shelled bubble model. Later researchs Take into
account the liquid compressibility effect on the bubble dynamics, the major
developments occurred in this area by Keller and Miksis. The microbubbles
which used in such contrast agents are normally stabilized by a thin
shell ,the stiffness and viscosity of this thin shell add a another foctors to the
acoustical behavior of the bubble, several models exist for the encapsulating
shell, the common models of encapsulated bubble are Hoff model and
Marmottant Model, these models are an extension of the RP equation.
Understanding the behavior of the microbubbles under ultrasound filed gives
us a good tool to predict it’s dynamic motion, which help in designing a new
and good contrast agent. In this article we review the linear and non-linear
behavior of the microbubbles and its mathematical models