Cross-Contributions Between Analyte and Internal Standard - a Need for Nonlinear Regression

2c Cross-Contributions Between Analyte and Internal Standard - a Need for Nonlinear Regression 

It is important to note that all of the calibration and measurement procedures involving a VIS and SIS discussed thus far have assumed that there is no cross-talk between the mass spectrometric signals for analyte and 

Most elements on planet Earth exist naturally as several isotopes (see Table 1.4) as a more-or-less homogeneous mixture. Therefore when a molecule is assembled from its constituent atoms, the likelihood that one isotope (e.g., C) will be incorporated rather than another ( C) is a matter of random probabihty controlled by the relative abundances of the two isotopes. (The situation is exactly analogous to a typical problem of elementary statistics in which a very large stockpile containing different percentages of black and white balls is sampled to give several collections each containing n balls.) 

Consider a molecule that contains n atoms of an element with two isotopes with percentage abundances a and X (where a + x= 100). For example, if the element is carbon, a = 98.93 for C and x=1.07 for C. Then the relative probabilities that molecules will contain atoms plus n — kf C atoms are given by the terms in the Binomial Expansion of (a + x) = 2[n /k (n — k) ]a where the summation is over = 0 to n so that there are (m + 1) terms. This is particularly useful in cases where x a (as is often the case for relative abundances of isotopes of an element, the main exceptions being chlorine and bromine see Table 1.4), since in such cases the first few terms provide a good approximation to the complete expression. 

The coefficients [n /k (n-k) ] can be calculated directly, or given by the famous Pascal Triangle (named after the 17th century French mathematician Blaise Pascal). 

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