Biot Theory-Based Finite Element Modeling of Continuous Ultrasound Propagation Through Microscale Articular Cartilage

Abstract

Low-intensity ultrasound has shown promise in promoting the healing and regeneration of articular cartilage degraded by osteoarthritis. In this study, a two-dimensional finite element method (FEM) model was developed for solving the Biot theory equations governing the propagation of continuous ultrasound through the cartilage. Specifically, we computed the ultrasound-induced dilatations and displacements in the microscale cartilage that is represented as consisting of four zones, namely, the chondrocyte cell and its nucleus, the pericellular matrix (PCM) that forms a layer around the chondrocyte, and the extracellular matrix (ECM). The chondrocyte–PCM complex, referred to as the chondron, is embedded in the ECM. We model multiple cartilage configurations where in the ECM layer contains chondrons along the depth, as well as laterally. The top surface of the ECM layer is subjected to specified amplitude and frequency of continuous ultrasound. The resulting wave propagation is modeled by numerically solving the two-dimensional Biot equations for seven frequencies in the 0.5 MHz–5 MHz range. It is seen that ultrasound is attenuated in the ECM and the attenuation increases monotonically with frequency. In contrast, manyfold augmentation of the ultrasound amplitude is observed inside the cytoplasm and the nucleus of the chondrocyte. Chondrocytes act as a major sink of ultrasound energy, thereby reducing the depthwise propagation of ultrasound fluctuations. Regions of high dilatations and displacements were found at the ECM–PCM interface, PCM–chondrocyte interface, as well as in the cytoplasm and nucleus of the chondrocyte. We observe that the ultrasound field around a chondron interacts with that around a neighboring chondron located at the same depth in the ECM layer. The qualitative and quantitative insights gained from our study may be relevant to designing ultrasound-based therapies for osteoarthritis.