Minimum target size for ultrasound power measurements in focussed fields

The minimum target size for power measurements in focussed ultrasonic fields with circular symmetry has been computed; such fields occur in medical diagnostics and in high-intensity therapeutic ultrasound applications.

The time-averaged ultrasonic power emitted by a transducer is one of the relevant key values of medical ultrasonic equipment. The measurement is performed primarily by using a radiation force balance. The ultrasonic beam is directed in water toward a mostly absorbing target, and the force acting on it is measured using an electronic balance and the power is calculated from the force.

An important influencing factor is the cross-sectional size of the target. Due to diffraction at the beam edge, the lateral extent of the field is infinite in theory, even with a transducer of finite size. As there are only finite-sized targets in practice, the question is how large should the target be at least, so that the deviation of the measured radiation force from that experienced by a target of infinite size does not exceed a certain amount. This problem had been solved by the author some time ago for unfocussed fields using theoretical field simulations in line with the 98% criterion that has been established in the literature and in standardization, according to which the target should experience at least 98% of the total radiation force. These calculations have now been extended to focussed fields.

Focussed ultrasonic fields are applied mainly in ultrasonic diagnostics and recently also in the HITU area ("high-intensity therapeutic ultrasound"). HITU is being used for tissue ablation, including the field of cancer therapy. To achieve the required intensities, the transducers in use are very large in comparison with the wavelength.

Field simulation was performed using the Rayleigh integral. It is a double integral intended for the calculation of the field of a planar acoustic source transducer. The focussing is achieved with an additional phase steering, i.e., an appropriate phase distribution of the elementary Huygens wavelets is assumed. The local field quantities describing the radiation force must then be integrated over the target surface; this means that the algorithm was a numerical triple integration. Circular symmetry was assumed.

The necessary target radius b in comparison with the transducer radius a was investigated as a function of the distance z in comparison with the geometric focal length d and depending on two parameters, namely the ka value of the transducer (where k is the circular wavenumber) and the focus (half-)angle ?.

Figure 1 shows an example of the results, here for the parameter values ka = 320 and γ = 36° that are typical of the HITU range. The three solid lines represent the required target size for 99%, 98% and 97% of the radiation force experienced by a target of infinite size. The field constriction in the focal range is obvious. The dashed/dotted line shows the geometric beam edge, i.e., the beam edge under fictitious neglect of diffraction.

Similar curves were obtained with other parameter values, and approximation formulas were derived from them that allow finding the required target radius in a wide range of parameter values. These results will be included in the relevant IEC standards.

Figure 1: Target radius b normalized to the transducer radius a, as a function of the target distance z normalized to the focal length d, calculated for ka = 320, ? = 36°. Ratio of the radiation force acting on the target in question to that on a target of infinite size: Curve 1: 99%, curve 2: 98%, curve 3: 97%. The dashed/dotted line represents the geometric beam edge.

Contact person:

Klaus Beissner, Dept. 1.6, WG 1.62, E-Mail: klaus.beissner@ptb.de