Volume 54, Number 6, November-December 2020
|Page(s)||2319 - 2350|
|Published online||11 November 2020|
Mathematical modelling of acoustic radiation force in transient shear wave elastography in the heart
Inria, Université Paris-Saclay, Palaiseau, France
2 LMS, Ecole Polytechnique, CNRS, Université Paris-Saclay, Palaiseau, France
3 Computational Cardiology Laboratory, Institute of Biophysics, Medical University of Graz, Graz, Austria
* Corresponding author: email@example.com
Accepted: 17 March 2020
The aim of this work is to provide a mathematical model and analysis of the excitation and the resulting shear wave propagation in acoustic radiation force (ARF)-based shear wave cardiac elastography. Our approach is based on asymptotic analysis; more precisely, it consists in considering a family of problems, parametrised by a small parameter inversely proportional to the excitation frequency of the probes, the viscosity and the velocity of pressure wave propagation. We derive a simplified model for the expression of the ARF by investigating the limit behaviour of the solution when the small parameter goes to zero. By formal asymptotic analysis – an asymptotic expansion of the solution is used – and energy analysis of the nonlinear elastodynamic problem, we show that the leading-order term of the expansion is solution of the underlying, incompressible, nonlinear cardiac mechanics. Subsequently, two corrector terms are derived. The first is a fast-oscillating pressure wave generated by the probes, solution of a Helmholtz equation at every time. The second corrector term consists in an elastic field with prescribed divergence, having a function of the first corrector as a source term. This field corresponds to the shear acoustic wave induced by the ARF. We also confirm that, in cardiac mechanics, the presence of viscosity in the model is essential to derive an expression of the shear wave propagation from the ARF, and that this phenomenon is related to the nonlinearity of the partial differential equation.
Mathematics Subject Classification: 35B40 / 35Q74 / 74H10 / 74J30
Key words: Shear wave elastography / acoustic radiation force / asymptotic analysis
© EDP Sciences, SMAI 2020
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.