Biomedical ultrasound

Personnel: Dr Rosemary Thompson, A/Prof C Macaskill, Dr Barrie Fraser, School of Mathematics & Statistics, University of Sydney, Dr Geoff Aldis, School of Mathematics & Statistics, ADFA, Canberra, Prof Piero Tortoli, Electronic Engineering Department, University of Florence, Italy.

Dr Thompson has been involved in biomedical ultrasound for over 15 years, and has collaborated with clinicians, animal physiologists, and engineers as well as other applied mathematicians, in over 25 publications in the area. Research projects have been supported by the NHMRC, and ARC small grants.

Current mathematical modelling projects

Ultrasound measurements inside thick walled cylindrical vessels

There are many applications of ultrasound where the interior of an approximately cylindrical vessel, or the velocity of a flow within such a vessel, is investigated using an external transducer. When is it reasonable to assume the interior of the vessel is uniformly insonated? The vessel wall is usually acoustically different to its surroundings, more so with in vitro flow phantoms than in vivo. The wall in vivo is however more likely to be irregular, especially in vascular disease where atherosclerotic plaques can develop in the wall. We have modelled the interaction of a simple incident ultrasound beam with a thick cylindrical shell, and now aim to extend the model to include more realistic ultrasound beams, and wall irregularities.

The Doppler spectrum and flow velocity profiles

Range rated pulsed Doppler can be used to make localized velocity measurements within a blood vessel, and a spectral flow profile created by stepping a sufficiently small sample volume across the lumen. Rather than giving "velocity at a point", each Doppler measurement returns a spectrum which is broadened due to a number of different factors. Working out what information about the true velocity can be recovered requires an understanding of the spectrum. Despite the everyday use of ultrasound, no comprehensive mathematical model exists for this very complicated problem, Dr Thompson and Dr Aldis have for a number of years worked on theoretical methods for the calculation of Doppler spectral power density functions. Based on certain simplifying assumptions, the problem can be reduced to the evaluation of sets of beam-intensity weighted triple and quartic integrals. The method can now handle individual flow line spectral spread, together with a very wide range of beam patterns, beam positions and range gating.

These experimental results (figure 1) are from a flow phantom in the laboratory of Professor Tortoli. The flow spectra were obtained from steady flow, where the velocity profile is parabolic. The lack of a simple correspondence between the flow spectra and the velocity profile is because of the spectral broadening associated with small sample volumes, and also because the vessel wall is distorting the intra-luminal intensity. This simulation (figure 2) shows the non uniform distribution over the lumen of a tube, after the beam has passed through a cylindrical interface between two fluids with different acoustic properties.-->