In this Bachelor project we researched the applicability of finite difference methods to solve the multi dimensional Westervelt equation. The Westervelt is an equation describing the propagation of a non-linear wave. The goal of this project was to solve the full Westervelt equation in higher dimensions. We used Julia for our implementations of the finite difference method. The choice of Julia allowed us to easily write high performing code. We were unsuccessful in modelling the full Westervelt equation since we were unable to implement attenuation into our models. We successfully implemented the non-linear wave equation without attenuation for a number of models, including a three dimensional model. The inability to implement attenuation into our models hampered the flexibility of our models. When the non-linearity created a shock front, the top of the wave would
experience numerical errors which quickly made the whole solution unphysical and ultimately unstable. An added implementation of attenuation would dampen the leading edge of the wave preventing it from becoming too steep. Additionally we made two comparisons with the linear wave equation to verify the correctness of our implementations.
We used a uniform grid to discretize the spatial dimensions. After which we used non-stiff FDM solvers with adaptive time stepping algorithms to solve for the remaining time dimension. We analyzed the use of stiff FDM solvers but these proved too computationally expensive to be viable, especially for our 3D model with 64.000 points.

Current research on osteoporosis detection and transcranial imaging suggests that it is necessary to find a method to calculate the speed of sound in curved bones. A popular method to find the speed of sound in bones is by the bidirectional axial transmission technique. Currently, this method assumes that the arrival time of the head wave depends linearly on the source-receiver distance. For curved bones this assumption is not valid. The arrival time of the head wave as a function of the source-receiver distance follows a curved trajectory. In this report, we attempt to find an extension to the current method that also works for curved bones, and evaluate how accurate the results of this extended method are.

This extension is based on the semblance method, that quantifies how well a certain trajectory agrees with the received data. In the first part of our project, theoretical trajectories are calculated for flat and semicircular bones. Furthermore, a numerical method is discussed to find these trajectories for arbitrary geometries. Comparing the trajectories corresponding to different velocities using the semblance methods allows for the determination of the speed of sound.

The accuracy of this extended method is evaluated by computer simulations. In these simulations, a Philips P41 Cardiac Sector Probe consisting of 96 elements was used to perform measurements on three different geometries. These geometries consisted of three different single interfaces between water (simulating soft tissue) and bone. The first interface consists of a straight interface between a 2 mm thick layer of water and a 5 mm thick layer of bone (that continues until the bottom edge of the simulation). The second simulation is the same, but the interface is rotated 2 degrees around its midpoint. In the third and fourth simulation, the interface is a semicircle with a 50 mm and 40 mm radius, respectively. The speed of sound in the bone was taken to be 3 mm/s. The received data was interpolated by a cubic interpolation with a refinement of 4. Both the bidirectional and the extended,
trajectory-based method were applied on the raw and interpolated data.

For the flat interface, the bidirectional method performed slightly better than the trajectory-based method (0.13% and 0.30% error, respectively). The same is true for the tilted interface (0.16% and 0.23% error). Interpolation did not affect the results here. For the 50 mm semicircular interface, the trajectory-based method performed much better than the bidirectional method: it had a 0.10% error compared to a 2% one. This was also true when the interpolation was applied: then the errors were 0.3% and 1.97%. Similar
improvements were seen for the 40 mm radius semicircle. The higher error of the trajectory-based method for the flat and tilted interfaces can be explained by small errors in the additional assumptions that this method makes.

One drawback of the trajectory-based method is that there can be problems interpreting the results. This can be seen in the semblance plots, which plot the semblance coefficient (how well the trajectory fits the data) against the test velocity. For the bidirectional method, these plots clearly have a single biggest peak. For the trajectory-based methods the plot contains multiple large peaks. Furthermore, for the semicircular bone this plot contains a flat peak, which can cause multiple test velocities to have approximately the same semblance. This is a problem, since it means that small changes to the experimental setup can cause big changes in the calculated velocity. Indeed, this can be seen for the trajectory-based method on the 40 mm radius semicircular bone, which has a 1.6% error without interpolation (0.43% with interpolation). Further research could examine solutions to this problem, such as increasing the window size of the semblance method, taking the average velocity over flat peaks, or using an alternative to the semblance method. Finally, we conclude that the trajectory-based method represents an improvement over the bidirectional method when examining curved bones.

This work presents an ASIC designed to be integrated on the tip of an intra-vascular ultrasound catheter, capable of interfacing 64 ultrasonic transducer elements to an imaging system using only one micro-coaxial cable.
A prototype chip has been taped out that can be used to transmit acoustic waves using a small-area transmit switch. The received echo signal is amplified by a phantom powered LNA and read-out at the other end of the micro-coaxial cable by a transimpedance amplifier.