On the KZK-type equation for modulated ultrasound fields

This study introduces a KZK-type model describing the nonlinear wave propagation due to "slowly" modulated ultrasound excitation, where the ratio between the (dominant) modulation frequency and the carrier frequency is o(1). The model is relevant to a variety of diagnostic techniques where the acoustic radiation force, generated by a focused ultrasound beam, is used to palpate tissue locally within the focal region. An indispensable prerequisite for computing the acoustic radiation force, however, is a precise knowledge of the nonlinear wave field generated by the ultrasound transducer at hand. In this class of applications, the MHz carrier signal is commonly modulated with frequencies on the order of 10 ² ÷10 ⁴ Hz which, in the context of the time-domain numerical algorithms, dramatically increases the computational effort necessary to simulate the nonlinear wave field. To alleviate the impediment, this study employs a dual time scale approach towards the development of an intermediate asymptotics solution that fits the gap between the quasi-static approximation and the full solution. Thus obtained modification of the KZK equation, which captures the leading effects of "slow" ultrasound modulation, lends itself to an effective computational treatment that makes use of suitable steady-state solutions. For generality, the asymptotic analysis deploys separate scaling parameters to describe the effects of diffraction and modulation, consequently allowing the model to (i) cover a wide range of transducer geometries, and (ii)be degenerated to the quasi-static approximation as one of its limiting cases. The formulation is accompanied by a set of numerical examples, which illustrate the applicability of the method to both "short" and "long" modulation envelopes while accounting for realistic (frequency-dependent) attenuation in a tissue.