Numerical and experimental investigation of impacts of nonlinear scattering encapsulated microbubbles on Nakagami distribution

Purpose The Nakagami statistical model and Nakagami shape parameter m have been widely used in linear tissue characterization and preliminarily characterized the envelope distributions of nonlinear encapsulated microbubbles (EMBs). However, the Nakagami distribution of nonlinear scattering EMBs lacked a systematical investigation. Thus, this study aimed to investigate the Nakagami distribution of EMBs and illustrate the impact of EMBs' nonlinearity on the Nakagami model. Method A group of simulated EMB phantoms and in vitro EMB dilutions with an increasing concentration distribution under various EMB nonlinearities, as regulated by acoustic parameters, were characterized by using the window-modulated compounding Greenwood–Durand estimator. Results Raw envelope histograms of simulated and in vitro EMBs were well matched with the Nakagami distribution with a high correlation coefficient of 0.965 ± 0.021 ( P < 0.005). The mean values and gradients of m parameters of simulated and in vitro EMBs were smaller than those of linear scatterers due to the stronger nonlinearity. These m values exhibited a quasi-linear improvement with the increase in second harmonic nonlinear-to-linear component ratio regulated by pulse lengths and excitation frequencies at low- and high-concentration conditions. Conclusions The Nakagami distribution was suitable for the EMBs characterization but the corresponding m parameter was affected by the EMBs' nonlinearity. These validations provided support and nonlinear impact assessment for the EMBs' characterization using the Nakagami statistical model in the future.