Different methods have been proposed to model the J-shaped or U-shaped relationship between a risk factor and mortality so that the optimal risk-factor value (nadir) associated with the lowest mortality can be estimated. The basic model considered is the Cox Proportional Hazards model. Current methods include a quadratic method, a method with transformation, fractional polynomials, a change point method and fixed-knot spline regression. A quadratic method contains both the linear and the quadratic term of the risk factor, it is simple but often it generates unrealistic nadir estimates. The transformation method converts the original risk factor so that after transformation it has a Normal distribution, but this may not work when there is no good transformation to normality. Fractional polynomials are an extended class of regular polynomials that applies negative and fractional powers to the risk factor. Compared with the quadratic method or the transformation method it does not always have a good model interpretation and inferences about it do not incorporate the uncertainty coming from pre-selection of powers and degree. A change point method models the prognostic index using two pieces of upward quadratic functions that meet at their common nadir. This method assumes the knot and the nadir are the same, which is not always true. Fixed-knot spline regression has also been used to model non-linear prognostic indices. But its inference does not account for variation arising from knot selections. Here we consider spline regressions with free knots, a natural generalization of the quadratic, the change point and the fixed-knot spline method. They can be applied to risk factors that do not have a good transformation to normality as well as keep intuitive model interpretations. Asymptotic normality and consistency of the maximum partial likelihood estimators are established under certain condition. When the condition is not satisfied simulations are used to explore asymptotic properties. The new method is motivated by and applied to the nadir estimation in non-monotonic relationships between BMI (body mass index) and all-cause mortality. Its performance is compared with that of existing methods, adopting criteria of nadir estimation ability and goodness of fit.