Dear Asal Asali,

Please find nice discussion about on 'chromforum':

https://www.chromforum.org/viewtopic.php?t=6934

Another discussion @ Bruce Levison should be interesting for you as well.

https://www.researchgate.net/post/Is_it_possible_to_publish_an_LC_MS_MS_method_that_doesnt_give_a_linear_calibration_curve

Additionally I think such paper should answer for your question:

http://www.scielo.org.za/pdf/sajc/v69/27.pdf

Hope it helpful,

Best regards

Tomasz

This goes back to physics! Usually, a UV/PDA/DAD is linear because it obeys Beer's law (A = abc). The injector is linear (you get double the peak area if you double the injection volume). Under these circumstances, the question then is 'How can you not get a linear calibration curve'?

When using a linear calibration, the concentration of sample points outside of the range of the calibration can calculated (within reason). When using a linear regression such as the quadratic you are limited to points within the range of calibration. If your sample concentration falls outside of that range you must either dilute the sample or change the calibration standards.

For most matrices linear calibration methods will work until you approach the saturation limits of the detector. However, matrix interference can make a linear curve difficult to achieve so most regulatory bodies accept a quadratic curve to address this problem.

Dear Asal,

In using least squares method for plotting calibration curve, linearity is assumption.

Given a particular analysis system, from sample handling through delivery of data, there are a variety of processes which can result in the gain and loss of signal through either gain or loss of analyte and interferents or variability in sensitivity which will result in non-ideal sample quantity to signal relationships. In order to reliably determine the quantity of analyte in the original sample, the variability of these behaviors must either be reduced to a level that does not impact the determination of the quantity of analyte, or they must be understood and controlled. For a system which by design has a linear analyte quantity to data relationship, if the signal is non-linear with respect to analyte quantity, then there is a potentially out of control behavior in the system, which will result in an increase in the uncertainty in the data, which may lead to wrong answers.

If the cause of the non-ideal behavior is known, and controlled and the level of control verified to be constant or known in all pertinent cases, then one can specifically address the non-linearity. The same is true for linear response systems that do not go through zero.

Fitting a curve to dataset which should be expected to be linear, may provide a better data fit to the equation, but how do you justify the data? After all, the analysis assumes a known linear concentration behavior, and your better fit of a curve is showing that the system is not performing correctly. The "system" in this case includes the sample handling, sample preparation, sample loading, sample transport, detector operation and performance, as well as the data extraction and delivery.

For me the answer to this question comes down to the minimum number of points required to properly describe the polynomial function represented by the quan cal curve. For a first order linear curve of the form y=mx+b a bare minumum of two points is required for a forced fit and a practical minumum often used in LC and GC/MS is 5 points. For a quadratic curve a forced fit would require three points and you would require more, equally distributed points (perhaps as a many as 8) to ensure accurate desrciption of all possible inflection points in the response curve. Basically that means you'd have to do more injection of different concetration standards in order to ensure accurate quantitation. In most cases you'd prefer to have analytical instrument time spent on analyzing samples rather than standards and so this is one line of reasoning in prefering a first order fit. So in my opinion a second order curve fit is acceptable as long as thoroughly described withe ufficient data points and applied to a system and situation in which non-linearity does not indicate a fault in the system or sample preparation.s

It is also common for non-linearity to indicate overall method issues related to concentration or absolute mass of the an analyte in addition to problems with adsorption, active sites in the chromatographic system and/or ionization issues. These and potentially other related issues would call for remedial action especially for a system which is known to be capable of generating linear response but is not currently doing so.

A final consideration is whether or not non-linear curve fits are allowed within the scope of the method being used especially if for regulatory or forensic reporting purposes.

Hope this helps and complements what the other commenters have already contributed.

Dear Asal Asali,

calibration functions with quadratic fits are allowed according to USP 1736:

"A calibration curve is created by plotting analyte-to-internal standard peak area ratios versus concentration for the reference standards analyzed. For the somewhat narrow calibration range required for Category I and Category II measurements, a simple (nonweighted) linear fit is sufficient to define this response versus concentration function. However, 1/x or 1/x2 weighting also can be used, as well as nonlinear (e.g., quadratic) functions, as supported by appropriate and provided validation data."

https://www.drugfuture.com/Pharmacopoeia/usp38/data/v38332/usp38nf33s2_c1736.html

Due to my experience, most of the people familiar with HPLC-UV sticks to linear calibration functions due to Lambert-Beer law. However, this law does not apply for LC-MS, since LC-MS is not based on (UV)-light absorbtion.

Best regards

Andre

for spectrometer in the liner the Beerُs Law is obeyed

Because is the most simple and precise representation between analytical signal and concentration.

In mass spectrometry analysis the response of the detector is linearly proportional to the number of ions hitting the detector, unless the detector is saturated.

In general a linear fit is preferred over a quadratic fit when possible because there is a greater risk of over fitting your data when using a higher order polynomial; n points can be fit with an n-1 degree polynomial. A quadratic fit will almost always be a better fit than a linear fit, regardless of the accuracy of the calibration.

While there are a number of answers above, that somewhat answer the question, the primary issue is that at every level of measuring a system, the calibration or sample response curve is assuming controlled ideal behavior. In fitting data you are applying a model of the response of the system. If the model that you are applying assumes that the response is linear, and passing through zero, and then if you use a different fit, you are applying a different model, and you need to justify why that model is what you are saying that it is and validate the model. If you have ten points and fit the data to a tenth order polynomial, you can get an exact solution, but what does it mean with respect to your system and your data?

I could give a number of examples where spectrometric data doesn't fit Beer's law. A number of cases where mass spectrometry data is non linear.

When it comes to proving that one's data fits and equation, an old rule of thumb is that one needs 4n*n+1 data points (where n is the number of degrees of freedom) as a minimum to verify that the data fits the equation. You then need to be able to defend what each term in the equation means and that the fitting equation is stable.

Linear calibration curve is easy and it can be obtained with less number of data points. However in the case of saturation which model will fit is not clear. Even saturation may be of different order so it is very difficult to work for a calibration curve which is not following linear relation. This is the reason most of the people like to use linear relation for calibration.