Formel für FWHM Berechnung

[Thommy7571]

Zur Analyse eines chromatographischen Peaks wurde von einem Kollegen folgende Formel in Excel verwendet, die ich nicht verstehe und auf der linken Seite des Peaks die Retentionszeit berechnet an der die FWHM gemessen wird..

=(D5-(D6+B7)/2)/(D5-D4)*C4+((D6+B7)/2-D4)/(D5-D4)*C5

Mit folgenden Werten C = Retentionszeiten und D Intensitätswerte des Peaks.

C4: 4.2050 D4: 467441
C5: 4.21 D5: 755786
D6: 1452742
B7 =20000

Dieselbe Formel wird dann noch einmal auf der rechten seite angwandt, um dort die entsprechende Retentionszeit zu berechnen.

Es wird also insgesamt die Breite des Peaks auf halber Höhe berechnet. .
Ich weiß weder, warum die angegebenen Punkte für die Berechnung verwendet wurden noch wie man zu ihnen kommt, noch warum dies mit der angegebenen Formel funktioniert. Der Wert in B7 scheint das Rauschen zu repräsentieren.

Kann mir jemand helfen?
Falls noch Fragen bestehen, stehe ich gerne zur Verfügung.

Danke

Thommy7571

[Hollow]

Hi Thommy7571

as this is an English forum, you should also ask your questions in English.


To be honest, it took me some time and paper, but finally it's just the linear interpolation between two points, in a somewhat "extended" form.

Let's define the measured points as P1 with coordinate (t1/H1) and P2 with (t2/H2) and the point of half-height as P50 with (t50/H50). H50 is calculated as the Apex/2 = H100/2 and lies between P1 and P2.

Assume a linear function (y=ax+b) between P1 and P2, so the slope "a" is (delta H) / (delta t) = (H2-H1)/(t2-t1), but this is the same as (H2-H50)/(t2-t50) and also the same as (H50-H1)/(t50-t1) #take some pen and paper and draw it yourselves

If you then connect the latter two terms and solve for t50 (the time of half-height) you will end up with the formula in your question, with H50 = (Hmax + Noise)/2;

A simpler form would be t50 = t1 + (H50-H1)*(t2-t1)/(H2-H1) #connection of first and second term and solved for t50

[Thommy7571]

Hallo,
Thanks very much for the effort that you made to work on the formula. I understood that there is an interpolation. ChatGPT told me that it is a weighed interpolation. By searching the position of the given points on the peaks I found out that the calculated points are corresponding to the points necessary for the calculation of the FWHM value. However, how are the given points are selected? Why is the result the corresponding value? I hope your findings will help me. I did the same exercise as you and got the result. Thanks very much

[Hollow]

ChatGPT told me that it is a weighed interpolation.

I don't see any weighing. just simple linear algebra.
BTW: I would be careful in using large data models for "learning" unknown things. As the names says, they model things, which does not necessarily mean it's true. To cite George Box (or whoever it said): "Essentially, all models are wrong, but some are useful".
Use them when you can judge and review the output. But that's another topic...

By searching the position of the given points on the peaks I found out that the calculated points are corresponding to the points necessary for the calculation of the FWHM value. However, how are the given points are selected? Why is the result the corresponding value?

FHWM means you want to know the width at half maximum, so you want the difference of two times, the front-t50 and tail-t50. To "model" these times you take the measured data-points just enclosing the H50 height and interpolate at which time the H50 would be reached. It's the simplest way and as close as you can get, without knowing the true curvature like with an analog recorded signal (eg. old stripchart recorders).
And if your sample rate is high enough, the error is probably neglectable for most purposes.

I guess the spreadsheet c4:d5 uses some lookup function to select the correct data-points so that the formula works in this static way.

I hope your findings will help me. I did the same exercise as you and got the result. Thanks very much

Gern geschehen!


For further reading on data-processing, I recommend the page/book of

Tom O'Haver, Professor Emeritus, Department of Chemistry and Biochemistry, University of Maryland:
"A Pragmatic* Introduction to Signal Processing - with applications in scientific measurement:
An illustrated essay with free software and spreadsheet templates to download"

https://terpconnect.umd.edu/~toh/spectrum/TOC.html

[Thommy7571]

Thanks, it is right that in this case I did not understand anything and the answer of ChatGPT was quite confusing. Although I use ChatGPT critically I did not think of the possibility it was wrong.
Many thanks for the hint concerning the signal processing! I hope that will have the time to read the book.
Practically, I understand but MATHEMATICALLY; I wonder realy about the assumption that the .slope is nearly everywhere constant on the rising part of the peak and similarly on the falling part: I wonder if the half height of the peak is identical with the "Scheitelpunkt (where 2nd derivation is zero). There the deriivation is effectively identical coming from both sides.

<<guess the spreadsheet c4:d5 uses some lookup function to select the correct <<data-points so that the formula works in this static way.

I don't know how these values have been determined - there was no formula used... perhaps just manually

<<I hope your findings will help me.
I forgot to delete this phrase Your finding already helped me very much. Without I wouldn't have found the Ansatz!! ChatGPT is generally not good at mathematics.

Thanks very much

[Hollow]

I wonder realy about the assumption that the .slope is nearly everywhere constant on the rising part of the peak and similarly on the falling part

no it's not. one just takes this between two consecutive data points for interpolation. So one has selected P1 (H1) and P2 (H2) so that P50 (H50) is between them, and then one solves for the unknown t50.
It doesn't work with any arbitrary values. That's why I guess the points were pre-selected with a lookup function.

I wonder if the half height of the peak is identical with the "Scheitelpunkt (where 2nd derivation is zero). There the deriivation is effectively identical coming from both sides.

No it's not.
Search for a general book on HPLC theory and Gaussian peak (e.g. Veronika Meyer (in english or in german)), and there will likely be a gaussian peak with the different width at different height drawn in it. maybe even in the help of chromatographic data software.
Or like this article from LC/GC North America, 2017-12: https://www.chromatographyonline.com/vi ... analysis-0

or another articles series from LG/GC 2018
Part IV: https://www.chromatographyonline.com/vi ... t-analysis

[Thommy7571]

Hello,
thanks very much but in general I know the basics.I am afraid that it is a long time ago that I had to handle mathematic problems of this type...
Anyway I would have been astonished. It was just an idea...
This means the points need already to be very close to the desired point.
In fact, I wonder if my practical worker did this by hand for a dozen of peaks, as the calculations of the fwhm value... He did not give a hint and I can't ask him any more...
Thanks very much once more. After this last point everything is clear.

Thanks

Thommy7571