Calibration using constrained smoothing with applications to mass spectrometry data

Linear regressions are commonly used to calibrate the signal measurements in proteomic analysis by mass spectrometry. However, with or without a monotone (e.g., log) transformation, data from such functional proteomic experiments are not necessarily linear or even monotone functions of protein (or peptide) concentration except over a very restricted range. A computationally efficient spline procedure improves upon linear regression. However, mass spectrometry data are not necessarily homoscedastic; more often the variation of measured concentrations increases disproportionately near the boundaries of the instruments measurement capability (dynamic range), that is, the upper and lower limits of quantitation. These calibration difficulties exist with other applications of mass spectrometry as well as with other broad-scale calibrations. Therefore the method proposed here uses a functional data approach to define the calibration curve and also the limits of quantitation under the two assumptions: (i) that the variance is a bounded, convex function of concentration; and (ii) that the calibration curve itself is monotone at least between the limits of quantitation, but not necessarily outside these limits. Within this paradigm, the limit of detection, where the signal is definitely present but not measurable with any accuracy, is also defined. An iterative approach draws on existing smoothing methods to account simultaneously for both restrictions and is shown to achieve the global optimal convergence rate under weak conditions. This approach can also be implemented when convexity is replaced by other (bounded) restrictions. Examples from Addona et al. (2009, Nature Biotechnology 27, 663-641) both motivate and illustrate the effectiveness of this functional data methodology when compared with the simpler linear regressions and spline techniques.