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Towards Mathematical Model of Ultrasound Propagation in Articular Cartilage

Introduction: Quantitative high-frequency ultrasound has been successfully used for the evaluation of articular cartilage e.g. reveal osteoarthritis related changes. During the degenerative process mechano acoustical properties of cartilage change: surface roughness, collagen and proteoglycan contents, and the density are affected. These changes can be detected as changes in variation of ultrasound parameters such as reflection coefficient, speed of sound (SOS) and attenuation. The individual factors affecting to ultrasound parameters are difficult to study experimentally, thus a numerical approach is needed. In the present study a 2D acoustic model, containing chondrocytes, proteoglycans (PG), collagen and water, of articular cartilage was constructed. Pulse echo (PE) and trough transmission (TT) simulations were performed in order to investigate the effects of PG and collagen depletion on acoustical properties of cartilage. Methods: A finite difference time domain model for ultrasound propagation in cartilage was constructed using Wave 3000 Plus software (Cyberlogic Inc., New Your, NY). For simulations a 2D geometry including an osteochondral sample (width=1.0 mm, cartilage thickness=1.5 mm and subchondral thickness=infinite) were created with a custom made software in Matlab 2012a (MathWorks Inc., Natick, MA). The cartilage was divided in 8 depth-wise layers as in a previous study and randomized chondrocyte distribution was implemented [1]. Furthermore, the extracellular cartilage tissue was divided in 50 depth-wise layers and PG, collagen and water distributions (fig. 1) were implemented following the values reported for each component by Maroudas et al. 1980 [2]. For water the elastic parameters were obtained from the material library of Wave 3000 software. The elastic parameters for collagen were calculated by using acoustical parameters reported by Joiner et al. 2001 after PG depletion. Based on Fischer et al. 2004 density of 1350 kg/m3 was used for PG and collagen. Simulations were conducted with intact cartilage and with PG or collagen depleted tissue. The simulation setup is presented in figure 2. A 10 MHz Gaussian-sine input (duration 3 µs) were used and responses were detected with the PE and TT receivers. Finally, SOS in cartilage was determined along with ultrasound reflection from articular surface. Results: In the PE geometry there were over 50% decreases in reflection after PG and collagen depletions. In the TT geometry SOS was decreased by 8.6% after PG depletion and by 16.0% after collagen depletion. Discussion: The present preliminary simulation data indicate decrease in ultrasound reflection at articular surface after collagen depletion. This finding is in harmony with previous experimental …