# Ultra-Weak Variational Method for Ultrasound Modelling

Ultrasound is one of the most widely used medical imaging modalities, but is conventionally used in a qualitative format, giving images related to the scattering of sound waves by dense objects.

Similar to other medical imaging methods such as X-Ray CT, PET and SPECT, and
Electrical or Optical Imaging, quantitative imaging can be developed if a
physical model of wave propogation is employed. In the case of ultrasound
the model is the wave equation, which under Fourier Transform becomes the
Helmholtz equation, a second order elliptic differential equation. Such equations can be solved in several ways such as using Finite Difference Time Domain (FDTD), finite element method (FEM), or boundary element method (BEM). Although these all have their advantages, they suffer from a computational overhead which requires very dense sampling grids to acheive numerical stability.

In this project we will investigate a new method the *Ultra Weak Variational Method* (UWVM) which can be thought of as a combination of BEM and FEM but with a kernel based on plane waves rather than spherical waves which makes their numerical behaviour simpler. The method will be quite easy to implement using the existing UCL TOAST software which allows fast large scale modelling in Matlab. We will test the code using data from the high intenstity focused ultrasound (HIFU) project. If successful the method can also be extended to Photo Acoustic Tomography (PAT). The project will suit someone interested in numerical methods, signal processing and medical imaging.

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### References

Tomi Huttunen, Matti Malinen, Jari P. Kaipio, Phillip Jason White and Kullervo Hynynen, "A full-wave Helmholtz model for continuous-wave ultrasound transmission", *IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control* accepted, 2004.

Tomi Huttunen, Peter Monk, Francis Collino and Jari P. Kaipio, "The ultra-weak variational formulation for elastic wave problems", *SIAM Journal on Scientific Computing* 25(5):1717 - 1742, 2004.

Tomi Huttunen, Peter Monk and Jari P. Kaipio, "Computational aspects of the ultra-weak variational formulation", *Journal of Computational Physics* 182: 27-46 (2002).