Linearity and non-linearity question

[bartjoosen]

Hi,

I have a problem with a validation of an HPLC/PAD method.
The method has to be validated for assay and related substances.
But as the response seems to be a 2nd order function of the concentration, I have some problems.

Normally we use a simple linear regression, we prove linearity with a lack of fit test, see if the intercept goes thru 0 and the slope significant different of 0.
For the assay, as we are in the upper region of the 2nd order curve, we can use a first order function, without a lack of fit (as the range is small enough). But the intercept is significant different of 0.
Is it acceptable to use the 1st order function by calibrating on 2 (or 3) reference standards covering the whole range?

For the related substance we can't use a 1st order function, so we have to use a 2nd order function (as we are in the lower, more exponential part of the curve).

Is it acceptable to use for the assay a simple linear regression with an intercept different from 0, but for related substances a 2nd order function?
And if we would validate a 2nd order function, what statistical calculations should we perform? Is a lack of fit test, significance of slope and intercept and R^2 enough?

Thanks for your time,

Bart

PS: below an idea of a 2nd order function:

[AdrianF]

This post has some relevance to your question.
http://www.sepsci.com/chromforum/viewto ... =linearity

It is unacceptable to work in the non linear region as you lose sensitivity and hence accuracy of the assay. The sample needs diluting.
Alternatively when measuring your main peak you could change the wavelength away from the maxima so it comes into a linear region.

[bartjoosen]

The post you mentioned was about HPLC/UV, but we are use pulsed amperometric detection.

And diluting isn't a solution, as the most non-linear part is the region of the related substances, which are measured from just above LOQ (0.1%) to 1.2%.

Thanks anyway


Bart

[JA]

It is unacceptable to work in the non linear region as you lose sensitivity and hence accuracy of the assay. The sample needs diluting.
Alternatively when measuring your main peak you could change the wavelength away from the maxima so it comes into a linear region.

In my experience some of the issues with linear range stem from the necessity to use a sufficiently high concentration (loading) to meet the respective LOQ requirements.

[DR]

^ in these cases, you may well have to run 2 tests - one for assay, and another for RS (where main peak is off the linear portion of curve and therefore not calculated). This assumes that amperometric detection is generally linear.

[AdrianF]

Sorry for barking up the wrong tree! I was not familiar with the abbreviation PAD.

It might be helpful if you gave some data from your experiments.

[bartjoosen]

Some data from experiments:
concentration (mg/100ml) Area
0.0887 88350.00045
0.3549 329411.0028
0.8873 787984.0111
1.4788 1232200.041
2.366 1798722.049
3.5363 2568179.041
4.715 3019678.083
5.89375 3364301.513
7.015 3659163.105
7.891875 3904875.552

Changing concentration isn't possible.
We have different concentrations for assay and related substances.
Lowering de concentration of the assay isn't possible due to accuracy problems.
Lowering the concentration of the RS isn't possible for LOQ reasons.

Bart

[Uwe Neue]

I do not see why one should not be able to use a non-linear curve fit to something that is non-linear. I would use a quadratic or cubic fit forced through 0.

[AdrianF]

I have plotted the data which I assume was the main peak using Excel.
I found it fitted well to a second order polynomial. However if I plotted the first 4 points they fell on a straight line with good correlation.

I don't understand why you can't work in this region of the calibration.

How are you determining your RS? Perhaps you could give more details of your method. It's not clear why you can't measure the RS while overloading the main peak and comparing the RS to the main peak with a 100 fold dilution (1%) of the first injection.

[tom jupille]

First of all, using the same method for assay and related substances is, in my opinion, a bad idea. "One size fits all" usually means "One size fits none"

Using ICH guidelines for validation, linearity for assay only needs to be determined from 80% - 120% of the stated value (I believe it's 70 - 130% for content uniformity). Almost anything is reasonably linear over that kind of restricted range.

The alternative, which would require more arm-twisting, is to simply go log-log. If the response is truly quadratic, you should get a linear plot with a slope of 2. The arm twisting comes in because some reviewer or manager will be shocked at your audacity. You can point out that UV detectors works with log functions in any case (a UV detector measures transmittance; it calculates absorbance)

[bartjoosen]

I have plotted the data which I assume was the main peak using Excel.
I found it fitted well to a second order polynomial. However if I plotted the first 4 points they fell on a straight line with good correlation.

I don't understand why you can't work in this region of the calibration.

How are you determining your RS? Perhaps you could give more details of your method. It's not clear why you can't measure the RS while overloading the main peak and comparing the RS to the main peak with a 100 fold dilution (1%) of the first injection.

from 0 to 1% is a sufficient small range to use a linear fit, but there are impurities allowed up to 6%, and this range 0-6 isn't linear enough.
Although, if you use the first 4 datapoints, and use a linear fit, you get a %Residual of 54% for the first datapoint, which is unacceptable for certain guidelines. If you use a second order polynomal, the maximum %residual is 1.0%.


Maybe I wasn't clear enough, but we are already using 2 different concentrations, one for RS, and one for assay.

Linearity for assay from 80-120% is OK, but the intercept doesn't include 0. Don't know if this is a problem for the guidelines?
And I don't know if it's acceptable to use a linear fit for the assay, and with the same HPLC method a second order polynomal for the RS

[tom jupille]

Linearity for assay from 80-120% is OK, but the intercept doesn't include 0. Don't know if this is a problem for the guidelines?

As I read the ICH, it says nothing about requiring 0 intercept for major component assay. In essence, your standards must bracket the sample concentration. If somebody asks, I would point them at the ICH (or appropriate regulatory guidelines) and ask them to show you where it requires a 0 intercept.

And I don't know if it's acceptable to use a linear fit for the assay, and with the same HPLC method a second order polynomial for the RS

Which is why I would define two separate methods.

Actually, a good case can be made that you should *not* combine high- and low-level calibration data. A standard least squares fit assumes that the absolute errors are constant across the entire range ("homoscedastic"). That's not the case for chromatographic data, where the percentage errors are approximately constant across the whole range ("heteroscedastic"). For the chromatography case, errors at the high end will effectively "swamp" errors at the low and dominate the fit.

This is an issue that comes up all the time in LC-MS/MS of drug metabolites, with lots of discussion about "weighted least squares". It can be sidestepped by either working in a narrow enough range that the data are approximately homoscedastic or by transforming the data set to make it homoscedastic (which is what a log-log fit does).

[Uwe Neue]

Tom,

The log/log fit fixes the problem with the error. It also goes through 0, by definition. However, even on a log-log scale, his data are curved, and it is still a good idea to use a curved fit (second order polynomial) for the log/log data.

To me, this is completely rational, but I am not an ICH expert...

[bartjoosen]

Actually, a good case can be made that you should *not* combine high- and low-level calibration data. A standard least squares fit assumes that the absolute errors are constant across the entire range ("homoscedastic"). That's not the case for chromatographic data, where the percentage errors are approximately constant across the whole range ("heteroscedastic"). For the chromatography case, errors at the high end will effectively "swamp" errors at the low and dominate the fit.

This is an issue that comes up all the time in LC-MS/MS of drug metabolites, with lots of discussion about "weighted least squares". It can be sidestepped by either working in a narrow enough range that the data are approximately homoscedastic or by transforming the data set to make it homoscedastic (which is what a log-log fit does).

Tom,

Does it make sense for the related substances to make 2 linear regression curves: one from 0 to lets say 2%, and a second from 2-6%?
If the impurities are smaller than 2%, we use the first one, if bigger than 2%, we can use the second curve?

This way, the range is small enough to have homoscedastic errors and a linear relationship.

Bart

[tom jupille]

The log/log fit fixes the problem with the error. It also goes through 0, by definition. However, even on a log-log scale, his data are curved, and it is still a good idea to use a curved fit (second order polynomial) for the log/log data.

To me, this is completely rational, but I am not an ICH expert...

Uwe, I'm far from a regulatory expert (I could make a lot more money if I were!) but I agree with you. I have no problem with any reasonable fitting function. If you look over a narrow enough range, everything is approximately linear, and if you look over a wide enough range, nothing is exactly linear. I just "eyeballed" the log/log plot in Excel, and while the quadratic is a better fit, the linear doesn't look too bad.

Does it make sense for the related substances to make 2 linear regression curves: one from 0 to lets say 2%, and a second from 2-6%?
If the impurities are smaller than 2%, we use the first one, if bigger than 2%, we can use the second curve?

This way, the range is small enough to have homoscedastic errors and a linear relationship.

It does make sense, but (to me), not as much as a quadratic fit (which seems to be a more "honest" description of the data), however, I'm not the one you have to convince .