Nonlinear oscillations following the Rayleigh collapse of a gas bubble in a linear viscoelastic (tissue-like) medium

Abstract

In a variety of biomedical engineering applications, cavitation occurs in soft tissue, a viscoelastic medium. The present objective is to understand the basic physics of bubble dynamics in soft tissue. To gain insights into this problem, theoretical and numerical models are developed to study the Rayleigh collapse and subsequent oscillations of a gas bubble in a viscoelastic material. To account for liquid compressibility and thus accurately model large-amplitude oscillations, the Keller-Miksis equation for spherical bubble dynamics is used. The most basic linear viscoelastic model that includes stress relaxation, viscosity, and elasticity (Zener, or standard linear solid) is considered for soft tissue, thereby adding two ordinary differential equations for the stresses. The present study seeks to advance past studies on cavitation in tissue by determining the basic effects of relaxation and elasticity on the bubble dynamics for situations in which compressibility is important. Numerical solutions show a clear dependence of the oscillations on the viscoelastic properties and compressibility. The perturbation analysis (method of multiple scales) accurately predicts the bubble response given the relevant constraints and can thus be used to investigate the underlying physics. A third-order expansion of the radius is necessary to accurately represent the dynamics. Key quantities of interest such as the oscillation frequency and damping, minimum radius, and collapse time can be predicted theoretically. The damping does not always monotonically decrease with decreasing elasticity: there exists a finite non-zero elasticity for which the damping is minimum; this value falls within the range of reported tissue elasticities. Also, the oscillation period generally changes with time over the first few cycles due to the nonlinearity of the system, before reaching an equilibrium value. The analytical expressions for the key bubble dynamics quantities and insights gained from the analysis may prove valuable in the development and optimization of certain biomedical applications.