I am trying to perform an impurity linearity on a humanized MAb. The separation is by WCX HPLC. My sample is 92% pure by this method and has six groups of degradant peaks, P1-P6. In my nominal sample P1-P6 range in value from 0.2% to 4.6%. I need to perform a linearity study from 50-150% of the nominal amounts. I collected fractions of each group of peaks from multiple injections and concentrated them. I measured protein concentration by UV and was told to analyze each fraction using the same mass on column by varying injection volumes. Now I have all this data, each fraction has multiple peaks at different % from injections of different volumes. Fractions at different concentrations. I can't figure out how to combine this to get back to my nominal sample with 50-150% of an impurity. I can't figure out how to get to the theoretical percents let alone accuracy. Please help I am so confused!

Dear Anonymous, most critical part is the separtion on the IEX column. Also gradient optimisation is helpful to get best results. As a first step I can forward to you a method for the separation of MABs, so you can compare our method with yours. Just e-mail to me, g.kratz@tosohbiosep.de! Hope this will help you. Regards
Gerhard

1. You have isolated each impurity and have get it concentrated. First you do the analysis of this impurity by UV and fix up the %purity of your impurity. Likewise you need to do for all isolated imputites individualy.

2. Now you may apply the purity factor to the concentration used in linearity, and now you can do all calculations you would like to do.

I think it is more intuitive to work with peak area than percent based on area percent based on different concentrations.

I think the easiest way to solve your equations is to take the peak areas in each fraction and divide by the number of microliters injected. Now you will have area/uL for each peak in each fraction. Now you will need to determine how you need to recombine your fractions to give you the areas you need.

Look at the peak areas for all of the peaks in the nominal sample, this is the 100% level.

When you reconstitute a sample from your fractions they will be linear equations based on fraction volumes. For example let P1 be the total P1 peak area in a reconstituted sample. Let p1f1 be the peak 1 area/uL from fraction 1, and let v1 equal a volume of fraction 1. Now:

P1 = (p1f1)(v1)+(p1f2)(v2)+...+(p1f6)(v6)
P2 = (p2f1)(v1)+(p2f2)(v2)+...+(p6f6)(v6)
.
.
.
P6 = (p6f1)(v1)+(p6f2)(v2)+...+(p6f6)(v6)

You know the peak areas you want P1-P6. You know p1f1 - p6f6 from the analysis of the fractions, many of these will be 0. Now you want to solve for v1 - v6. The set of linear equations above can be rewritten as:

| p1f1 p1f2 p1f3 p1f4 p1f5 p1f6 | (v1) P1
| p2f1 p2f2......................p2f6 | (v2) P2
| p3f1 ...................................| (v3) P3
| p4f1 ...................................| (v4) = P4
| p5f1 ...................................| (v5) P5
| p6f1 .............................p6f6| (v6) P6

This is a matrix times a vector giving you a vector. To solve for your volumes you need to take the inverse of your matrix and multiply by your P1-P6 vector.

v1-v6 = inverse matrix * P1-P6

In Excel enter the matrix in a 6x6 (A1-F6) group of cells. Then select another 6x6 group of cells as the ouput range (A10:F15) and enter the formula =MINVERSE(A1:F6) and press Ctrl+Shift+Enter to tell excel this is an array function. Now enter P1-P6 in H10:H15. Select J10:J15 as the output range and enter the equation = MMULT(A10:F15, H10:H15)and press Ctrl+Shift+Enter. Now you can change the values of P1-P6 to any value you need and the spreadsheet will automatically give you the volumes you need.

Sorry if this is confusing, it would have been faster and clearer if I just posted the spreadsheet. If you can't follow this mess let me know and I will send you the spreadsheet.

To understand what just happened this is covered in an undergrad math text on linear algebra or the Schaum's outline on vector analysis.

As an "end user" of mab, I am really puzzled as to why one needs to know impurity linearity.
We have bought pure antibodies and found impurities even with SEC. Others have uncovered several substances in a pure mab with capillary electrophoresis. Thus, I would be happy if industry would just state: This mab has 92% purity based on ion chromatography and assuming extraeneous peaks to have the same molar absorption coefficient as the alledged mab peak.....
This way we wouldn´t be so apprehensive before infusing them into humans.
Am I missing something important??

My guess is anon works for a pharmaceutical/biotech company and is performing a validation for a stability indicating assay and needs to verify accuracy for different levels of impurities.

Hi Tom,

Thanks for the spreadsheet, it works perfectly. I am too ashamed to admit how many hours(days) I spent trying correction factors and trying to solve the percentage equations by substitution. I tried to stay awake during those math classes :-), honest. Your guess is also right, I was assigned the task of executing someone elses validation protocol. He is in St. Croix and I am here begging for help. Thanks again for the spreadsheet and thanks for the math lesson and the Excel tips - much appreciated.