Interaction of an ultrasound-activated contrast microbubble with a wall at arbitrary separation distances

Both in vitro and in vivo, contrast agent microbubbles move near bounding surfaces, such as the wall of an experimental container or the wall of a blood vessel. This problem inspires interest in theoretical models that predict the effect of a wall on the dynamics of a contrast microbubble. There are models for a bubble at a large distance from a wall and for a bubble adherent to a wall. The aim of the present study is to develop a generalized model that describes the dynamics of a contrast microbubble at arbitrary distances from a wall and thereby make it possible to simulate the acoustic response of the bubble starting from large separation distances up to contact between the bubble and the wall. The wall is assumed to be a plane. Therefore, the developed model applies for in vitro investigations of contrast agents in experimental containers. It can also be used as a first approximation to the case of a contrast microbubble within a large blood vessel. The derivation of the model is based on the multipole expansion of the bubble velocity potential, the image source method, and the Lagrangian formalism. The model consists of two coupled equations, one of which describes the bubble radial oscillation and the second describes the translation of the bubble center. Numerical simulations are performed to determine how the acoustic response of a contrast microbubble depends on the separation distance near walls of different types: rigid, plastic, arterial, etc. The dynamics of the bubble encapsulation is described by the Marmottant shell model. The properties of the plastic wall correspond to OptiCell chambers commonly used in experiments. The results of the simulations show that the bubble resonance frequency near a wall depends on both the separation distance and the wall material properties. In particular, the rigid wall makes the resonance frequency decrease with decreasing separation distance, whereas in the vicinity of the OptiCell wall and the arterial wall, the resonance frequency increases. The theoretical model is validated by comparing with experimental data available in the literature for phospholipid-shelled microbubbles near a compliant agarose gel boundary. The comparison is conducted for two data sets. In the first case, the simulated and measured parameters of the bubble acoustic response (resonance frequency and maximum amplitude of the scattered pressure) are compared as a function of bubble diameter, using the experimental data obtained for bubbles with diameters ranging from 2.3 to 4 μm, insonified at 30 kPa in the frequency range from 4 to 13.5 MHz. In the second case, the simulated and measured values of the maximum amplitude of the scattered pressure are compared as a function of separation distance from the agarose boundary, using the experimental data for a 2.3 μm-diameter bubble insonicated at 69 kPa and 11 MHz. In both cases, the theoretical model correctly predicts the qualitative tendency in the behavior of the measured quantities and their quantitative level.