Epidemiologists often build models to analyze the relationship between an exposure and a potential outcome caused by the exposure. In many cases, the exposure may not directly lead to the outcome, but instead, it induces the outcome through a process. Mediation analysis is designed to explain the causal relationship between the exposure and the outcome by examining the intermediate stage, which helps researchers understand the pathway whereby the exposure affects the outcome. The regression-based mediation analysis has been formulated and developed in the last decade, and several papers discussed the situation where the relationship between the mediator and the outcome is curvilinear. In this paper, we develop a method to analytically estimate the direct and indirect effects when we have some prior knowledge on the relationship between the mediator and the outcome (increasing, decreasing, convex or concave) and obtain the asymptotic confidence intervals of those effects via delta method. We illustrate our method using a population-level prenatal screening program data set.