Excess risk thresholds in ultrasound safety studies: statistical methods for data on occurrence and size of lesions

Concerns about the safe use of clinical ultrasound (US) at diagnostic pressure levels (below a mechanical index, or MI, = 1.9) have stimulated considerable research in US risk assessment. The objective of the present study was to develop probability-based risk thresholds for US safety studies, to present statistical methods for estimating the thresholds and their standard errors and to compare these methods with the analysis based on a piecewise linear ("hockey stick") model. The excess risk at exposure level x > 0 was defined as the relative increase in the probability of a lesion at that level compared with the background probability of a lesion at exposure x = 0. The risk threshold was then defined as the exposure level at which the excess risk exceeded a specified level (e.g. 5% or 50%). Thus, given pressure-dependent estimates of the excess risk, the thresholds were estimated by solving the risk equation to obtain the pressure at which the target level of excess risk occurs. Threshold estimates of this type have been developed extensively in the literature for incidence (presence or absence) data. Only recently, however, have excess risk threshold estimates been derived for data in which lesion size (depth, surface area) is measured if present and a zero is recorded if the lesion is absent. Tobit regression was used to estimate pressure-dependent percentiles of the size distribution, and the excess risks were estimated from the tobit probability of a positive-valued response. The tobit model provides a well-established approach to modeling data constrained to be nonnegative. Solving the risk equation for the tobit model leads to risk threshold estimates that incorporate the information on size of observed lesions. Results using these probability-based risk estimates were compared with results for a piecewise linear ("hockey stick") model, which has also been used in the US safety literature, although it does not explicitly address the nonnegativity constraint in the sampling model. The comparisons were carried out for data from two previously published studies, from different laboratories, on US-induced lung hemorrhage. The thresholds derived from logistic regression of lesion occurrence and tobit regression of lesion size were quite consistent with each other and within sampling error. The hockey stick thresholds, defined as the exposure level at which the piecewise linear model for the probability of the expected size of a lesion bends upward, corresponded to quite different excess risk values for incidence (lesion occurrence) compared with size (lesion surface area or depth), although these methods have been developed previously for both types of data. The use of probability-based excess risk thresholds is recommended to obtain consistent incidence vs. size thresholds and to ensure that the thresholds are well-defined and interpretable independent of the details of the statistical model.