Numerical solutions of fractional PDEs for ultrasound waves at viscoelastic fluid-tissue interfaces

State-of-the-art medical ultrasound technology used both in the imaging and incision-free treatment of cancers relies on mathematical models for acoustic wave absorption and dispersion in viscoelastic tissue (e.g. human brain or muscle tissue). The empirically observed power-law like absorption of ultrasound waves in the tissue has to be accurately modelled to allow high precision imaging or targeting and be done efficiently (ideally in real-time).  

In practical terms, this means that one needs to numerically solve fractional differential equations using accurate and efficient (in computational complexity and memory) algorithms on non-homogeneous domains (consider e.g. that the absorption of ultrasound waves will be different in bone, blood, brain tissue as well as the gels used at the body-emitter-sensor interfaces). Fractional differential equations are different from the usual PDEs in that they are typically non-local either in space or time and thus exhibit memory effects which come with both mathematical and computational challenges. 

The aim of this project is to mathematically analyse the current state-of-the-art algorithms as well as work in collaboration with medical engineering collaborators towards developing and implementing efficient and accurate novel approaches, especially for discontinuous fluid-tissue interfaces.