The use of ultrafast Ultrasound Localization Microscopy (uULM) is a promising technique for obtaining images with a very high resolution. This technique is based on the localization of subwavelength intravascular microbubbles resonating under ultrasound stimulus. Pre-clinical and post-clinical ultrasound applications, such as tumour, microvascular or stroke imaging, are numerous.

This research investigates the theoretical precision limit of uULM in localizing a moving subwavelength scatterer. A 1D transducer array and the scatterer are simulated with the Vantage Research Ultrasound Simulator (Verasonics, Kirkland, WA, USA). A large beam is transmitted through the medium by the transducer array and hits the scatterer. As a sub-wavelength scatterer radiates as an omni-directional pressure field, transducer arrays of finite apertures receive a part of this spherical wave back as parabolas (the radio-frequency data, or RF-data), from which beamformed images (the BF-data) can be reconstructed. The intensity of the RF-data is given in arbitrary units, while the intensity for the BF-data is indicated in decibels. Both the RF-data and BF-data are influenced by different sources of noise, such as jitter and false peaks, which can be modelled with zero-mean white Gaussian noise. In this manuscript, the z-axis is defined to be the direction away from the probe and the x-axis is defined to be colinear to the piezoelectric elements. In Figure 1, an illustration of this process, called Plane Wave Imaging, with the given directions is shown.

The localization precision in the x- and z-directions is computed for different intensities, different signal-to-noise ratios (SNRs), and different depths z for both the RF-data and the BF-data. For the RF-data, there is a clear relation between the depth of the scatterer and the localization precision: if the scatterer is moving further away, the minimum standard deviation decreases due to attenuation in tissues. For example, for a depth of 11 mm and an SNR of 29 dB, one is able to localize the microbubble with a precision of 10 nm in the x-direction and 50 nm in the z-direction based on the RF-data, while this decreases to respectively 1 μm and 0.5 mm for a depth of 21 mm.

The BF-based precision limits are less dependent on depth: for different depths, the limits remain approximately the same, being 0.5 μm in the x-direction and 1 μm in the z-direction for an intensity of 20 dB and an SNR of 29 dB. A remarkable result is that the localization using the radio-frequency data is more precise compared to the beamformed images if the scatterer is close to the transducer array. However, after a certain depth, the BF-based localization surpasses the RF-based one. This difference in precision is due to the beamforming process: to translate radio-frequency data into a readable image, one needs to sum the energy scattered back and select that value as the pixel intensity for the final image. This process, called beamforming, can be done on the fly or in post-process. In this research, it is complex to compare the initial values, being the SNR, the maximum intensity value, and the dependence on depth, between the RF- and BF-data.

The localization precision for the RF-data and for the BF-data reacts similarly to changes in the amount of noise. For a high SNR, the position of the scatterer can be determined more precisely compared to a situation with a low SNR. For example, the RF-data reveals a localization precision of 1 nm in the x-direction for an SNR of 33 dB, while an SNR of 23 dB results in a precision of 10 mm for the same intensity and depth.