The natural frequency of nonlinear oscillation of ultrasound contrast agents in microvessels

Ultrasound contrast agents (UCAs) are under intensive investigation for their applications in physiological and molecular imaging and drug delivery. Prediction of the natural frequency of the oscillation of UCAs in microvessels has drawn increasing attention. To our knowledge, the existing models to predict the natural frequency of oscillation of UCAs in microvessels all apply the linear approximation and treat the blood vessel wall as a rigid boundary. In the potential applications of ultrasound imaging drug and gene delivery, the compliance of small vessels may play an important role in the bubble's oscillation. The goal of this work is to provide a lumped-parameter model to study the natural frequency of nonlinear oscillation of UCAs in microvessels. Three types of the blood vessel conditions have been considered: i.e., rigid vessels, normal compliable vessels and vessels with increasing stiffness that could correspond to tumor vasculature. The corresponding bubble oscillation frequencies in vessels with a radius less than 100 microm are examined in detail. When a bubble with a radius of 4 microm is confined in a compliable vessel (inner radius 5 microm and length 100 microm), the natural frequency of bubble oscillation increases by a factor of 1.7 compared with a bubble in an unbounded field. The natural frequency of oscillation of a bubble in a compliable vessel increases with decreasing vessel size while decreasing with increasing values of vessel rigidity. This model suggests that contrast agent size, blood vessel size distribution and the type of vasculature should comprehensively be considered for choosing the transmitted frequency in ultrasound contrast imaging and drug delivery.