MassLynx 4.1Peak Integration and

Quantitation Algorithm Guide

71500122009/Revision A

Copyright © Waters Corporation 2005.All rights reserved.

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Table of Contents

1 Peak Response and Peak Ratio .......................................................... 1-1

Absolute Response ............................................................................................ 1-2 Compound response primary .......................................................................... 1-2 Compound response secondary ....................................................................... 1-2 Compound response both ................................................................................ 1-2

External Response ............................................................................................ 1-2

Internal Response ............................................................................................. 1-3

Primary/Secondary Peak Ratio ..................................................................... 1-3

2 Calibration Curve Calculations .......................................................... 2-1

Weighted Calibration Curves ......................................................................... 2-2

Include Origin .................................................................................................... 2-2

Average RF ......................................................................................................... 2-2

Linear ................................................................................................................... 2-3

Quadratic and Higher Order Curves ........................................................... 2-4

3 Peak Amount Calculations .................................................................. 3-1

User Specified Response Factor .................................................................... 3-2

Average RF Calibration Curve ...................................................................... 3-2

Linear Calibration Curve ................................................................................ 3-2

Quadratic and Higher Order Calibration Curves ..................................... 3-3

User Parameters ................................................................................................ 3-3

4 Calibration Curve Statistics ................................................................ 4-1

Coefficient of Determination ......................................................................... 4-2

Curve Correlation Coefficient ....................................................................... 4-2

Table of Contents iii

RRF mean, SD, and %RSD for Average RF curves .................................... 4-3

5 Totals Compounds ................................................................................. 5-1

Peak Response ................................................................................................... 5-2

External Response ............................................................................................ 5-2

Internal Response ............................................................................................. 5-2

Calibration Curve Calculations ..................................................................... 5-3

Peak Amount Calculations ............................................................................. 5-3

6 Limits of Detection (LOD) and Limits of Quantitation (LOQ) .... 6-1

Chromatogram Noise Calculation ................................................................ 6-2 Manual noise measurement ............................................................................ 6-2 Automatic noise measurement........................................................................ 6-2

Chromatogram Area Noise ............................................................................. 6-3

Response Value for Noise ................................................................................ 6-3

LOD and LOQ Concentrations ....................................................................... 6-4

LOD and LOQ Flags .......................................................................................... 6-4

T-LOD and T-LOQ Concentrations ............................................................... 6-4

7 Peak Statistics ........................................................................................ 7-1

Peak area ............................................................................................................ 7-2

Peak height ......................................................................................................... 7-2

Height / Area ...................................................................................................... 7-2

Baseline width ................................................................................................... 7-3

Width .................................................................................................................... 7-3

Peak sigma (Standard deviation) .................................................................. 7-3

Peak skew (Asymmetry) .................................................................................. 7-4

Peak kurtosis (Peak top flatness) ................................................................. 7-4

iv Table of Contents

Peak quality criteria ........................................................................................ 7-5

Index ..................................................................................................... Index-1

Table of Contents v

vi Table of Contents

1 Peak Response and Peak Ratio

There are two main methods of calculating peak response value: External Standard and Internal Standard. Both are based on the Absolute Response calculated for a peak.Contents:

Topic PageAbsolute Response 1-2External Response 1-2Internal Response 1-3Primary/Secondary Peak Ratio 1-3

1-1

Absolute Response

Absolute Response is based on either peak area or height and the selected combination of compound primary and secondary peaks. The examples below are shown for peak area.

Compound response primary This is the default if no secondary trace is specified.

Abs.Resp. = Primary Area

Compound response secondary Abs.Resp. = Secondary Area

Compound response bothAbs.Resp. = Primary Area + Secondary Area

Where: • Primary Area is the area of a peak calculated by peak detection on the

primary chromatogram trace. • Secondary Area is the area of a peak calculated by peak detection on the

secondary chromatogram trace.

External Response

Peak Response = Abs.Resp Where Abs.Resp is the absolute response of a peak as calculated in Absolute Response on page 1-2.

1-2 Peak Response and Peak Ratio

Internal Response

Where: • Abs.Resp is the absolute response of a peak as calculated in Absolute

Response on page 1-2.• AmountI is the given amount of the Internal Standard in the sample.• Abs.RespI is the area of the internal standard peak as calculated in

Absolute Response on page 1-2.

Primary/Secondary Peak Ratio

If primary and secondary traces are specified, the ratio of peaks detected on each trace is calculated as shown below. Ratio is based on either peak area or height: the example below is shown for peak area.

Where: • Primary Peak Area is the area of a peak calculated by peak detection on

the primary chromatogram trace. • Secondary Peak Area is the area of a peak calculated by peak detection

on the secondary chromatogram trace.

Peak Response Abs.Resp AmountI×Abs.RespI

--------------------------------------------------------=

Ratio Primary Peak AreaSecondary Peak Area-----------------------------------------------------------=

Absolute Response 1-3

1-4 Peak Response and Peak Ratio

2 Calibration Curve Calculations

MassLynx can fit several types of calibration curve.Contents:

Topic PageWeighted Calibration Curves 2-2Include Origin 2-2Average RF 2-2Linear 2-3Quadratic and Higher Order Curves 2-4

2-1

Weighted Calibration Curves

Calibration points used when fitting curves can be given a weighted importance: the larger the weighting the more significant a point is treated when the curve is fitted. Weighting (wi) of ith calibration point is calculated using one of the following. All wi are set to 1 for no weighting.

1.

2.

3.

4.Where:

• yi is Y value (response) of ith calibration point. • xi is X value (concentration) of ith calibration point.

Include Origin

If Include Origin is selected as a calibration curve type, an extra point with zero concentration and response is used in the regression. The extra point has the same weighting as the lowest calibration standard unless Force Origin is set, in which case the origin will always be included as a point directly on the calibration curve.

Average RF

The calibration curve formed is linear passing through the origin with a gradient equal to the average response values of the calibration points.

Where:

•

wi yi= 1–

wi yi= 2–

wi xi= 1–

wi xi= 2–

Average RF SwySw

-----------=

Swy yi wi×xi

-----------------∑=

2-2 Calibration Curve Calculations

•

• yi is Y value (response) of ith calibration point.• xi is X value (concentration) of ith calibration point.• wi is weighting of ith calibration point, all set to 1 for no weighting.

Linear

The calibration curve is formed by fitting a line using linear regression to a set of calibration points.

Where:

•

•

•

•

• yi is Y value (response) of ith calibration point • xi is X value (concentration) of ith calibration point • wi is weighting of ith calibration point, all set to 1 for no weighting.

If Force Origin is selected, a line with zero intercept is fitted.

Sw Σwi=

Gradient SwxySwxx--------------=

Intercept yw,mean Gradient xw,mean×–=

yw,meanΣyi wi×Σwi

--------------------=

xw,meanΣxi wi×Σwi

--------------------=

Swxy Σ xi xw,mean–( ) yi yw,mean–( )× wi×=

Swxx Σ xi xw,mean–( )2 wi×=

Gradient Σxi yi× wi×Σxi

2 wi×-------------------------------=

Weighted Calibration Curves 2-3

Quadratic and Higher Order Curves

MassLynx uses a general Least Squares Fit algorithm to regress a polynomial of any order against the calibration points. The method used is outlined below.Polynomial regression can be described as the fitting of m ‘independent’ variables (Xj, j = 0 to m-1) to a single ‘dependent’ variable y. In other words:

y = Xb + eWhere:

• y is the n x 1 vector containing the n y values (yi).• X is the n x m matrix of x values, (xi

j).• b is the m x 1 vector of regression coefficients (bi). • e is the n x 1 vector of residuals from the fit to each yi value.

The familiar least squares solution for the regression coefficients is given by:

Where:• -1 indicates matrix inverse• ' indicates matrix transpose

The above equation can then be solved using Gauss-Jordan elimination. To implement weighted regression X and y are first multiplied by a diagonal n x n matrix P (in other words, X becomes PX and Y becomes PY), before the above equation is solved. Where each element (pij) of P is given by:

for i =j

for i < > j

wi is weighting of ith calibration point, all set to 1 for no weighting.

b X′X( ) 1– X′y=

pij wi1 2/=

pij 0=

2-4 Calibration Curve Calculations

3 Peak Amount Calculations

Contents:

Topic PageUser Specified Response Factor 3-2Average RF Calibration Curve 3-2Linear Calibration Curve 3-2Quadratic and Higher Order Calibration Curves 3-3User Parameters 3-3

3-1

User Specified Response Factor

If a user response factor if selected within the quantitation method, calibration curves are not used. The following calculation is performed to obtain peak amounts:

Where: • Peak Response is the response value calculated for a peak. • Response Factor is the user-entered response factor for that compound.

Average RF Calibration Curve

Amounts are calculated using an Average RF calibration as follows:

Where: • Peak Response is the response value calculated for a peak. • Average RF is the average response factor calculated for a set of

calibration points.

Linear Calibration Curve

Amounts are calculated using a linear calibration as follows:

Where: • Peak Response is the response value calculated for a peak.• Intercept is the intercept calculated for the linear calibration. • Gradient is the gradient calculated for the linear calibration.

Amount Peak ResponseResponse Factor---------------------------------------------=

Amount Peak ResponseAverage RF

-----------------------------------------=

Amount Peak Response Intercept–Gradient

-------------------------------------------------------------------------=

3-2 Peak Amount Calculations

Quadratic and Higher Order Calibration Curves

Amounts are calculated by solving the following equation using the Newton-Raphson Method:

Peak Response = P(Amount)Where:

• Peak Response is the response value calculated for a peak.• P( ) is the polynomial function calculated for a set of calibration points.

User Parameters

User parameters can be used to multiply or divide the final quantitation results. These factors are entered per sample in the Sample List. If a factor is not specified, or is zero, it is assumed to be one.

The User Peak Factor is entered per compound in the Quantify Method.

Rules: • User multiplication and divisor factors are not applied to standard

samples.• In MassLynx 3 onwards, Sample List fields can be renamed to user

requirements. When converting from a pre MassLynx V3.0 sample list the following field mappings occur: • 'Initial Amount' becomes 'User Divisor 1' • 'User Factor' becomes 'User Factor 1' • 'Dilution Factor' becomes 'User Factor 2' • 'Extract Volume' becomes 'User Factor 3'

Final Amount Amount User Factor 1 User Factor 2 User Factor 3×××User Divisor 1

-----------------------------------------------------------------------------------------------------------------------------------------------------------=

Final Amount Amount User Peak Factor×=

User Specified Response Factor 3-3

3-4 Peak Amount Calculations

4 Calibration Curve Statistics

Contents:

Topic PageCoefficient of Determination 4-2Curve Correlation Coefficient 4-2RRF mean, SD, and %RSD for Average RF curves 4-3

4-1

Coefficient of Determination

The coefficient of determination is calculated for a regressed calibration curve. In the case of a linear curve it is equivalent to the square of the correlation coefficient and is reported as such. For each data point a value of y (yi,pred) can be predicted from the calibration curve at the position xi. For each data point a residual between the actual and predicted y value can be calculated as (yi - yi,pred), and the weighted residual sum of squares (RSS) can be calculated as:

where wi is the weighting of the ith calibration point.The total variation in the data is reflected in the weighted corrected sum of squares (CSS), calculated as:

where yw,mean is the weighted mean value of y

The model sum of squares (MSS) is the portion of the total variation accounted for by the regression:

MSS = CSS – RSSThe coefficient of determination (R2) is the proportion of the variation accounted for by regression, and is given by the ratio of the model sum of squares to the corrected sum of squares:

R2 = MSS / CSS= (CSS – RSS) / CSS

Curve Correlation Coefficient

In the case of linear curves, the square of the correlation coefficient (r) is equivalent to the coefficient of determination (see Coefficient of Determination on page 4-2) and is reported as such.

RSS Σ yi yi,pred–( )2 wi×=

CSS Σ yi yw,mean–( )2 wi×=

yw,meanΣyi wi×Σwi

--------------------=

4-2 Calibration Curve Statistics

RRF mean, SD, and %RSD for Average RF curves

The calculation of RRF mean, SD, and %RSD are detailed in the following equations:

MeanRRF

RRFiN∑

N---------------------

Abs.ResponseiConci

---------------------------------------N∑

N-----------------------------------------------= =

SDRRF

RRFi MeanRRF–( )2

N∑

N 1–----------------------------------------------------------=

%RSDRRFSDRRF

MeanRRF-----------------------⎝ ⎠⎛ ⎞ 100%×=

Coefficient of Determination 4-3

4-4 Calibration Curve Statistics

5 Totals Compounds

Totals compounds are formed from groups of peaks. These can include Named peaks that have been identified by other method compounds and Unnamed peaks which have not been identified.Compounds are identified as being a part of a Group by a shared group name. This section details calculations performed specifically for Totals compounds.Contents:

Topic PagePeak Response 5-2External Response 5-2Internal Response 5-2Calibration Curve Calculations 5-3Peak Amount Calculations 5-3

5-1

Peak Response

The responses of all peaks that are part of the Totals group are calculated individually.

External Response

Refer to External Response on page 1-2.

Internal Response

Calculated Internal Response of an unnamed peak is based upon the average response of the Internal Standards of the named compounds in the Group.

Where:

Abs.Resp = Absolute response of Unnamed peak as calculated in Absolute Response on page 1-2.

Ave.Named.RespIS = Average response of group Named peaks Internal Standards.

=

N = Number of found Named compounds in group.n = Named compound index.AbsRespISn = Absolute response of the nth Named Internal

Standard peak as calculated in Absolute Response on page 1-2.

AmountISn = Given amount of the nth Named Internal Stan-dard in the sample.

Peak Response Abs.RespAve.Named.RespIS---------------------------------------------------=

ΣN AbsRespISn AmountISn⁄( )( ) N⁄

5-2 Totals Compounds

Calibration Curve Calculations

The calibration curve for a Totals compound is formed by averaging the calibrations of the Named compounds in the group. The Totals curve is used to calculate the concentration of the Unnamed peaks in the group. Each coefficient of the Totals curve is calculated as follows:

Where:• N is the number of Named compounds in the group. • Ci is the ith coefficient of the Totals curve. • Ci,n is the ith coefficient of the nth Named compound in the group.

Peak Amount Calculations

Totals concentration is the sum of the concentrations of the selected peaks that make up the group. Peaks summed can be specified as only Named peaks, only Unnamed peaks, or both. The concentration of each individual peak in the group is calculated as described in Peak Amount Calculations on page 3-1.

Ci ΣNCi,n( ) N⁄=

Peak Response 5-3

5-4 Totals Compounds

6 Limits of Detection (LOD) and Limits of Quantitation (LOQ)

The calculation of the LOD and LOQ for a compound is dependent upon the noise contained within the chromatogram trace used to quantify that compound.Contents:

Topic PageChromatogram Noise Calculation 6-2Chromatogram Area Noise 6-3Response Value for Noise 6-3LOD and LOQ Concentrations 6-4LOD and LOQ Flags 6-4T-LOD and T-LOQ Concentrations 6-4

6-1

Chromatogram Noise Calculation

Chromatogram noise is calculated as the standard deviation of the noise in the trace (effectively a height), which is multiplied by a user specified factor. Chromatogram noise may be calculated automatically or manually.

Manual noise measurementTo select this method, the user must specify an RT range of the trace which contains noise only. The resulting noise value is the standard deviation of the data in this region, multiplied by the User Factor.Tip: An RT entry of 0.0 may be used to signify either the start or end of the trace if input as the start or end RT respectively. If both the start and end RT are set to 0.0, automatic noise measurement (see page 6-2) will be initiated.

Where:

Automatic noise measurementThis method is initiated by selecting the entire range of the trace for noise measurement (that is, start and end RT equal 0.0). An algorithm is used to determine which regions of the trace consist only of noise, the result being the standard deviation of these regions multiplied by the user specified factor.Rule: The noise detection algorithm relies on a relationship between the median difference between adjacent points and standard deviation which is specific to gaussian deviates. This approximation provides good results when used with raw chromatogram data, but not for smoothed data. As a result, the algorithm measures the noise in the raw trace, to which a scaling factor is then applied to produce the effect of the current smoothing parameters. This smoothing correction is specific to the effect of mean smoothing on gaussian deviates, explained below.

Noise Height SDselected data User Factor×=

SDx

xi Meanx–( )N∑

N 1–-----------------------------------------=

6-2 Limits of Detection (LOD) and Limits of Quantitation (LOQ)

The accuracy of this method is affected: • when Savitzky Golay smoothing is used • to a lesser extent, when the noise regions are not gaussian in nature.

Where:

This equation describes the relationship between the standard deviation of raw and mean smoothed data (for gaussian deviates). Iterations is the number of smooths applied to the raw data and Winsize is the size of the smoothing window. MassLynx requires a half window size to be entered which has the relationship with Winsize described above. A multiplication factor of 3 is commonly used as, for gaussian deviates, this corresponds to a range within which 95% of the data will lie.

Chromatogram Area Noise

The noise value is effectively a measure of noise height: an area value may be calculated that is equivalent to the calculated noise. The Compound’s IS peak is used for this: the ratio of height to area of the corresponding IS peak is calculated, and this is applied to the noise value if the corresponding noise area is required.

Response Value for Noise

The absolute noise values are converted into equivalent response values. If compounds are being quantified using Heights, use the original noise values; if using Areas, use the Area noise values. Calculate an absolute noise response for the compounds based upon the method Secondary Parameters Compound response, Primary, Secondary or Both (in which case sum the noise values from both traces). If a Secondary trace is not specified the Primary is always used. Calculate a noise response value from the absolute noise response by applying the IS response, in the same way as if it were a normal peak.

SDmeansmoothedSDraw data

Winsize0.5 Iterations0.25×---------------------------------------------------------------------=

Winsize 2 WinHalfSize×( ) 1+=

Chromatogram Noise Calculation 6-3

LOD and LOQ Concentrations

The LOD and LOQ concentrations are calculated by applying the compound’s calibration curve to the noise response to obtain a value for the concentration, which is then multiplied by user specified factors for LOD and LOQ respectively. The LOD and LOQ factors are specified as part of the QuanLynx or TargetLynx method and are set for all compounds. The default values are 3 and 8 respectively.Resulting LOD and LOQ concentrations should be stored in Peak Record and be available for display in the Summary. If no value is available output should be blank.

LOD and LOQ Flags

These flags indicate if the calculated concentration of a compound falls outside the LOD or LOQ concentration threshold.If a compound concentration exceeds the threshold the flag is ‘Yes’. If it is less than the threshold the flag is ‘No’.

T-LOD and T-LOQ Concentrations

The Toxic LOD concentration is calculated by multiplying the Toxic Equivalence Factor by the Limit Of Detection. The Toxic LOQ concentration is calculated by multiplying the Toxic Equivalence Factor by the Limit Of Quantification.Resulting T-LOD and T-LOQ concentrations should be stored in the Peak Record and be available for display in the Summary. If no value is available output should be blank.

6-4 Limits of Detection (LOD) and Limits of Quantitation (LOQ)

7 Peak Statistics

Contents:

Topic PagePeak height 7-2Height / Area 7-2Baseline width 7-3Width 7-3Peak sigma (Standard deviation) 7-3Peak skew (Asymmetry) 7-4Peak kurtosis (Peak top flatness) 7-4Peak quality criteria 7-5

7-1

Peak area

The peak area is calculated by summing the areas of each of the segments that make up the peak.

Where:• Xn is the retention time (mins) at which sample n was taken.• Yn is the intensity above the baseline for sample n.

The area of each individual segment is calculated using the trapezium rule.

The advantage of using this method is that the calculated peak area is less dependent on the sampling rate across the peak.

Peak height

The peak top is the retention time with the largest absolute intensity. Peak height is calculated as the distance between this largest intensity and the peak baseline at that retention time.

Height / Area

The peak height divided by the peak area.

Total Area =

X0 X1 X2 X3 X4 XnY1

Y2Y3 Y4

AreaiYi 1+ Yi+( )

2---------------------------- Xi 1+ Xi–( )×=

ΣAreai

n 1–

7-2 Peak Statistics

Baseline width

The difference, in seconds, between the end of the peak baseline and the start of it.

Width

Peak width measured in seconds at 50% of the peak height above the baseline. The value displayed is an approximation calculated as follows.The peak width at the inflection points is actually calculated – this is known as the peak width at 4 (or 4 sigma). The majority of peaks are Gaussian in shape above the inflexion points, since the most variability is at the base of the peak. The peak width at half height is calculated from the equation:

Where:

Peak sigma (Standard deviation)

This is the standard deviation of the data, calculated as the square root of the variance. The weighted variance is calculated as:

Where:• xi is the retention time value for observation i.• wi is the intensity associated with xi.•• is the weighted mean of x whose formula is:

σ

Width at half height 2 2ln(2) σ× 2.35482 σ×==

Width at inflection points 4 σ×=

1d--- wi xi x–( )2∑

d Σwi=x

Σwixi Σwi⁄

Peak area 7-3

The standard deviation is calculated as the square root of the variance:

The variance and standard deviation both require at least two non-missing observations.

Peak skew (Asymmetry)

The peak skewness characterizes the degree of asymmetry of a distribution around its mean (that is, a measurement of the tendency of the deviations to be larger in one direction than in the other).It is non-dimensional. It is a pure number that characterizes only the shape of the distribution, defined as follows:

Where:• is the weighted mean of x.• xi is the retention time value for observation i.• wi is the intensity associated with xi.• is the distribution’s weighted standard deviation defined above.

A positive value of skew signifies a distribution with an asymmetric tail extending out towards more positive x; a negative value signifies a distribution whose tail extends out towards more negative x.Skewness requires at least three non-missing observations.

Peak kurtosis (Peak top flatness)

Peak kurtosis is a non-dimensional quantity. It measures the relative peakedness or flatness of a distribution, relative to a normal distribution. A distribution with positive kurtosis is termed leptokurtic; the outline of a very peaked mountain is an example.

σ x1...xn( ) Var x1...xn( )=

1n--- xi x–( ) σ̂j⁄( )3∑=

1n--- wi

3 2⁄ xi x–( ) σ̂⁄( )3∑=

x

σ

7-4 Peak Statistics

A distribution with negative kurtosis is termed platykurtic; the outline is topped like a plateau. Distributions that are neither leptokurtic nor platykurtic are termed mesokurtic. Weighted kurtosis is computed as follows:

Where:• is • is the weighted mean of x.• xi is the retention time value for observation i.• wi is the intensity associated with xi.• is the distribution’s weighted standard deviation defined above.

The subtraction of 3 in the above expressions normalizes the kurtosis of a Gaussian distribution to zero.Kurtosis requires at least four non-missing observations.

Peak quality criteria

The peak statistics described above are measured against a selection of peak quality criteria (defined in the Method Editor) to generate Pass or Fail status in Peak Quality and a description of any failure in Peak Quality Description.

1n--- xi x–( ) σ̂i⁄( )4 3–∑=

1n--- wi

2 xi x–( ) σ̂⁄( )4 3–∑=

σi2 σ2 wi⁄

x

σ

Peak area 7-5

7-6 Peak Statistics

Index

Symbols%RSD 4-3

AAbsolute Response 1-2asymmetry 7-4Average RF 2-2, 4-3

calibration 3-2

Bbaseline width 7-3

Ccalibration curves 2-1

calculations 5-3Chromatogram noise 6-2coefficient of determination 4-2concentration thresholds 6-4corrected sum of squares 4-2

EExternal Response 1-2, 5-2

FFlags

LOD and LOQ 6-4

GGaussian 7-3, 7-5Gauss-Jordan elimination 2-4Gradient 3-2

HHigher Order curves 2-4, 3-3

IInclude Origin 2-2Intercept 3-2Internal Response 1-3, 5-2

Internal Standards 5-2

Kkurtosis 7-4

LLeast Squares Fit 2-4Limits of Detection 6-1Limits of Quantitation 6-1linear 2-2, 2-3

calibration 3-2LOD 6-1LOQ 6-1

Mmodel sum of squares 4-2

NNamed peaks 5-1Newton-Raphson Method 3-3Noise

measurement 6-2

Ppeak

area 7-2height 7-2kurtosis 7-4named 5-1response 1-1, 3-2, 5-2skewness 7-4top flatness 7-4unnamed 5-1

Peak Amount Calculations 5-3peak quality criteria 7-5Primary Peak Area 1-3

Index-1

QQuadratic curves 2-4, 3-3quantitation

method 3-2results 3-3

Rratio

primary and secondary peaks 1-3regressed calibration curve 4-2residual sum of squares 4-2Response Factor 3-2retention time 7-2RRF mean 4-3

SSavitzky Golay 6-3SD 4-3Secondary Peak Area 1-3standard deviation 7-3sum of squares

corrected 4-2model 4-2residual 4-2

TTotals compounds 5-1

UUnnamed peaks 5-1User parameters 3-3

Vvariance

weighted 7-3

WWeighted Calibration Curves 2-2weighted variance 7-3

Index-2