Abstract
We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high‐intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form ppt. However, this should be considered as a low‐order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f(p) pt and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consist of time trace observations of the acoustic pressure at a single point or on a one‐dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of well‐posedness of the resulting pde, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f. Numerical simulations will also be shown to illustrate the efficiency of the methods.
Highlights
The use of ultrasound is well established in the imaging of human tissue and the propagation of high intensity ultrasound is modeled by nonlinear wave equations
Nonlinearity enters the model via the state equation, which is a constitutive relation between acoustic pressure and mass density
A common such model is the Westervelt equation in which the Taylor expansion of this constitutive relation is truncated to its second degree terms and in this case a certain ratio of quantities B/A governs the nonlinearity, cf [7, Chapter 2]
Summary
The use of ultrasound is well established in the imaging of human tissue and the propagation of high intensity ultrasound is modeled by nonlinear wave equations. Nonlinearity enters the model via the state equation, which is a constitutive relation between acoustic pressure and mass density This results in the typical situation of the nonlinear effect appearing as a product of a function of the state variable and its time derivative. In section 2The modelsection. we will give more details on the above models which result from physical laws governing quantities such as the acoustic particle velocity, the acoustic pressure and the mass density Combining these laws while including successive higher order terms, (the Blackstock scheme) one arrives at a succession of more complex partial. We will provide an analysis of the forward problem and of well-definedness of the iteration schemes as well as the results of numerical experiments to demonstrate the effectiveness (and limitations) of these methods for the problem at hand
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