Chromatography Forum

How can we use RRFs to quantitate impurities accurately in whole linear range in Empower?

We used one reference standard and the relative response factors (RRF = slope of reference standard /slope of impurity) of the impurities to quantitate the impurities for drug substances and products. The slopes were got from their calibraton curves. We found some of the calibration curves did not fit linear properly at lower concentration range. The quantitation error is too much. e.g. Deviation is ~40% at 0.05% and ~10% at 0.1% in the calibration range of 0.05%-1% of nominal sample concentration.

The problem is not with Empower, but with the fact that ordinary least squares is really not an appropriate fitting technique for wide-range chromatography data. OLS assumes that the data are "homoscedastic" (the errors are the same across the entire range). Chromatography data tend to be heteroscedastic (the percentage errors are about the same across the entire range, so the absolute errors are larger at the high end). This means that the results of OLS are dominated by errors at the high end and the low end often goes wildly inaccurate as a result.

The remedies are to transform the data into a form that *is* homoscedastic, like log-log and then do OLS on that (which is my preference) or to use weighted least squares (which is what most of the rest of the world does) with weighting factors of 1/x or 1/x^2. There are standard statistical packages that will the do the math; I don't know whether weighted least squares is built in to Empower or not.

The same results were got with repeat test. It's probably not the error problem. It's not convenient to prepare a series of impurities standards for routine test. Non-linear regression can not be used to calculate RRF. Weighted regression can improve one end, but not both ends. Multistep linear regression and multiple RRFs with different boundary conditons for one component may be the solution. Empower has a point to point fit type. All the regression parameters are not editable, except the reference curve and relative factor (RRF).
Thanks.

The same results were got with repeat test. It's probably not the error problem.

I'm not sure how you came to that conclusion.

Weighted regression can improve one end, but not both ends.

I'd argue with that, but I suppose it depends what you mean by "improve". If you know how the error varies with concentration, then weighted least squares will give you the best overall fit. Please note that we're talking about random error here. If you are exceeding the linear range so that you have a systematic error at the high end, then neither OLS nor weighted least squares is appropriate .

Point-to-point fits introduce their own set of errors (for example, you can get large differences in the answer depending on which side of a data point you're on.

It's not convenient to prepare a series of impurities standards for routine test.

Okay, so you're doing "bad science" (taking a short cut) in the interest of convenience. There's no problem with that so long as:
1. you don't try to hide the fact (which you're not), and
2. you're willing to live with the consequences.

As you can tell, I'm not a great fan of point-to-point fits!

Excuse my misleading. I am sure you are right. The practical problem is that the deviation at 0.05% is ~40% in the calibration curve. The recovery at 0.05% is ~100% by comparing the peak area, or 40% less calculated by reference standard curve with RRF.
Thanks.

What is your calibration range?
Make sure you are not doing serial dilutiuons (1% soltn 1:1 to get 0.5%, 0.5% 1:1 to get 0.25%...) but use a single stock solution instead (1ml to 5ml, to 10ml, to 25ml, to 50ml,...)
A 1/x weighting will lead to better fits at the lower end without worsening the upper end to much.
As RRFas are concentration-independend other calibration functions than linear will not work.

Alex