New theory for distribution of minimum resolution in multi-component separations with noise/detection limits

Abstract

Equations were proposed recently for computing the distribution of minimum resolution (resolution distribution) of two Gaussian peaks with equal standard deviations, when peak heights in a multi-component separation follow a statistical distribution. The computation depended on the survival function of the peak-height ratio. Previously, an equation was derived for a first-order survival function that excluded peaks with heights less than a noise/detection limit. Here, an equation is derived for a corrected survival function, under the more realistic assumption that two minimally resolved peaks are lost if the height of their shoulder is less than the noise/detection limit. First-order and corrected survival functions and resolution distributions are derived for the exponential and uniform distributions of peak heights, and a corrected survival function and resolution distribution are derived for the log-normal distribution (LND) to complement a previous first-order derivation. Large peak losses (up to 99.3% of the noise/detection limit) are considered to find significant differences between the first-order and corrected resolution distributions. For the LND and exponential distribution, the corrected resolution distribution has slightly greater density in the low-resolution region but otherwise differs little from its first-order counterpart, unless the scale parameter of the LND is small (e.g. 0.75). For the uniform peak-height distribution, the corrected resolution distribution has higher density in the high-resolution region. The first-order and corrected resolution distributions are almost the same as long as the first moment of the first-order resolution distribution is greater than 0.6. The predictions are confirmed by Monte–Carlo simulation.