Simultaneous estimation of the nonlinearity parameter and attenuation coefficient with the Gauss-Newton Levenberg-Marquardt algorithm

The nonlinearity parameter (B/A) imaging can complement conventional B-mode imaging for characterization of soft tissue structures. Recently, a method to estimate the B/A in pulse-echo ultrasound using the depletion of the fundamental band and a dual-energy model was presented. However, a requirement of the aforementioned depletion method is to have an estimate of the attenuation coefficient of the assessed medium. Therefore, in the present study, an alternative by simultaneously estimating the B/A and the attenuation coefficient as parameters of a nonlinear model fitting data from the depletion method is developed. The solution of the nonlinear model fitting is conducted using the iterative Gauss-Newton (GN) algorithm. Moreover, to stabilize the GN algorithm, the variant known as Levenberg-Marquardt (LM) with regularization parameter, was used instead. Evaluation of the GN-LM method was performed with data from two numerical samples obtained using the k-Wave toolbox. The simulation settings considered a linear array of 128 elements, 0.3 mm pitch, excited with a pulse with with center frequency of 5 MHz and 22.7% -6 dB bandwidth. The B/A maps obtained using certain ranges of the LM parameter showed to improve the tradeoff of the GN method or an exhaustive search by reducing the standard deviation in about one order of magnitude. Noticeable, the mean and standard deviation of the estimated B/A maps for both phantoms suggests the use of moderate ranges of the LM parameter to prevent convergence towards the initial guess of B/A as well as it highlights the relevance of selecting an initial guess of the attenuation coefficient as accurate as possible to reduce the bias in the reconstructed B/A map. In conclusion, iterative methods do provide a framework to estimate the B/A and the attenuation coefficient simultaneously.