Risk Anal. 2019 Apr;39(4):940-956. doi: 10.1111/risa.13191. Epub 2018 Sep 25.
The study presents an integrated, rigorous statistical approach to define the likelihood of a threshold and point of departure (POD) based on dose-response data using nested family of bent-hyperbola models. The family includes four models: the full bent-hyperbola model, which allows for transition between two linear regiments with various levels of smoothness; a bent-hyperbola model reduced to a spline model, where the transition is fixed to a knot; a bent-hyperbola model with a restricted negative asymptote slope of zero, named hockey-stick with arc (HS-Arc); and spline model reduced further to a hockey-stick type model (HS), where the first linear segment has a slope of zero. A likelihood-ratio test is used to discriminate between the models and determine if the more flexible versions of the model provide better or significantly better fit than a hockey-stick type model. The full bent-hyperbola model can accommodate both threshold and nonthreshold behavior, can take on concave up and concave down shapes with various levels of curvature, can approximate the biochemically relevant Michaelis-Menten model, and even be reduced to a straight line. Therefore, with the use of this model, the presence or absence of a threshold may even become irrelevant and the best fit of the full bent-hyperbola model be used to characterize the dose-response behavior and risk levels, with no need for mode of action (MOA) information. Point of departure (POD), characterized by exposure level at which some predetermined response is reached, can be defined using the full model or one of the better fitting reduced models.
本研究提出了一种综合、严格的统计方法，通过嵌套的弯曲双曲线模型族，基于剂量反应数据来定义阈值和起点（POD）的可能性。该模型族包括四个模型：完整的弯曲双曲线模型，允许在两个具有不同平滑度水平的线性区域之间进行转换；一个简化为样条模型的弯曲双曲线模型，其中转换固定在一个节点上；一个限制负渐近斜率为零的弯曲双曲线模型，称为带弧的冰球棒（HS-Arc）；以及进一步简化为冰球棒型模型（HS）的样条模型，其中第一段线性斜率为零。使用似然比检验来区分模型，并确定更灵活的模型版本是否提供了更好或显著更好的拟合，而不是冰球棒型模型。完整的弯曲双曲线模型可以同时适应阈值和非阈值行为，可以呈现出各种曲率水平的凸向上和凸向下的形状，可以近似生物化学上相关的米氏门控模型，甚至可以简化为一条直线。因此，使用该模型，阈值的存在与否甚至可能变得无关紧要，而最好的拟合完整的弯曲双曲线模型可以用来描述剂量反应行为和风险水平，而无需进行作用模式（MOA）信息的推断。起点（POD），是指达到某些预定反应的暴露水平，可以使用完整模型或拟合更好的简化模型之一来定义。
