Simultaneous estimation of cortical bone thickness and acoustic wave velocity using a multivariable optimization approach: Bone phantom and in-vitro study

Abstract

The paper presents a method that allows the thickness of a compact bone layer and longitudinal wave velocity in the bone to be determined simultaneously with the use of reflected waves, with particular emphasis on the case of layers when the propagation time through the layer is shorter than the time duration of the interrogating pulse.The proposed method estimates simultaneously the thickness of the cortical bone layer and acoustic wave velocity by fitting the temporal spectrum of the simulated reflected wave to the spectrum of the reflected wave measured experimentally. For the purpose of echo-simulations the model of “soft tissue – compact bone layer – cancellous bone” was developed. Next, the cost function was defined as the least square error between the measured and simulated temporal spectra. Minimization of the cost function allowed us to determine the values of the parameters of the cortical bone layer which best fitted the measurements. To solve the optimization problem a simulated annealing algorithm was used.The method was tested using acoustic data obtained at the frequency of 0.6MHz and 1MHz respectively for a custom designed bone mimicking phantom and a calf femur. For the cortical shell of the calf femur whose thickness varies from 2.1mm to 2.4mm and velocity of 2910m/s, the relative errors of the thickness estimation ranged from 0.4% to 5.5%. The corresponding error of the acoustic wave velocity estimation in the layer was 3.1%. In the case of artificial bone the thickness of the cortical layer was equal to 1.05 and 1.2mm and acoustic wave velocity was 2900m/s. These parameters were determined with the errors ranging from 1.9% to 10.8% and from 3.9% to 4.5% respectively.