Analytic prediction of droplet vaporization events to estimate the precision of ultrasound-based proton range verification

Abstract

Background: The safety and efficacy of proton therapy is currently hampered by range uncertainties. The combination of ultrasound imaging with injectable radiation-sensitive superheated nanodroplets was recently proposed for in vivo range verification. The proton range can be estimated from the distribution of nanodroplet vaporization events, which is stochastically related to the stopping distribution of protons, as nanodroplets are vaporized by protons reaching their maximal LET at the end of their range.

Purpose: Here, we aim to estimate the range estimation precision of this technique. As for any stochastic measurement, the precision will increase with the sample size, that is, the number of detected vaporizations. Thus, we first develop and validate a model to predict the number of vaporizations, which is then applied to estimate the range verification precision for a set of conditions (droplet size, droplet concentration, and proton beam parameters).

Methods: Starting from the thermal spike theory, we derived a model that predicts the expected number of droplet vaporizations in an irradiated sample as a function of the droplet size, concentration, and number of protons. The model was validated by irradiating phantoms consisting of size-sorted perfluorobutane droplets dispersed in an aqueous matrix. The number of protons was counted with an ionization chamber, and the droplet vaporizations were recorded and counted individually using high frame rate ultrasound imaging. After validation, the range estimate precision was determined for different conditions using a Monte Carlo algorithm.

Results: A good agreement between theory and experiments was observed for the number of vaporizations, especially for large (5.8 ± 2.2 µm) and medium (3.5 ± 1.1 µm) sized droplets. The number of events was lower than expected in phantoms with small droplets (2.0 ± 0.7 µm), but still within the same order of magnitude. The inter-phantom variability was considerably larger (up to 30x) than predicted by the model. The validated model was then combined with Monte Carlo simulations, which predicted a theoretical range retrieval precision improving with the square-root of the number of vaporizations, and degrading at high beam energies due to range straggling. For single pencil beams with energies between 70 and 240 MeV, a range verification precision below 1% of the range required perfluorocarbon concentrations in the order of 0.3-2.4 µM.

Conclusion: We proposed and experimentally validated a model to provide a quick estimate of the number of vaporizations for a given set of conditions (droplet size, droplet concentration, and proton beam parameters). From this model, promising range verification performances were predicted for realistic perfluorocarbon concentrations. These findings are an incentive to move towards preclinical studies, which are critical to assess the achievable droplet distribution in and around the tumor, and hence the in vivo range verification precision.

Keywords: nanodroplets; proton therapy; range verification; ultrasound.