Initial component amounts are calculated in one of three ways: by solving the component's calibration curve (the normal method), by applying a calibration factor (user-supplied or calculated average), or by using the calibration curve of another component (the calibration reference).
Calibration Curves
To solve a calibration curve for amount, the software uses response in the same form as that used to build the curve. That is, if an external standard calibration was
performed, the response used is the peak area or height (whichever has been specified in the method). If an internal standard calibration was performed, the response used is the area ratio or height ratio (component to internal standard).
The initial result of solving a calibration curve is an amount in the same form as that used in building the curve. That is, if an internal standard calibration was performed, the initial amount is an amount ratio. If an alternative amount scaling option was selected, the initial amount is the log of the amount or another variation. The following sections tell how the software solves the various calibration curve types.
Point-to-Point Fit
In a point-to-point fit, each pair of points (calibration levels) is connected by a straight line segment. When the response of the component is examined with respect to the calibration levels, four outcomes are possible:
The response lies below the first calibration level, and the curve has been forced through the origin; or, there is only one calibration level, and the response lies below it. In either case, a straight line is drawn to connect the first point (or the only point) with the origin. The initial amount is calculated by solving the equation of this line.
Quantitation
The response lies between two calibration levels. In this case, the equation of the straight line drawn between these levels is used to calculate the initial amount.
The response lies above the last calibration level. If there is more than one calibration level, a straight line is drawn to connect the last two levels and extended beyond the last level. If there is only one calibration level, a straight line is drawn to connect the origin with the level and is then extended beyond the last level. The initial amount is calculated by solving the equation of this line.
In each case, the software determines the slope and the y intercept of the line and then
solves the following equation to obtain the initial amount Aini:
A ( R ) m ini = − b where R is the response
b is the intercept of the line at the y axis m is the slope of the line
Note that a slope of 0.0 (a horizontal line) prevents an amount from being calculated; thus, two adjacent calibration levels must not have the same response. A slope of infinity (a vertical line) also prevents an amount from being calculated; thus, two adjacent calibration levels must not have the same amount.
First Order Fit
A first order (linear) curve is specified by the equation
y = c + c x0 1
where
c0 is the y intercept c1 is the slope of the line
The software calculates the initial amount by using the same equation as used for point-to-point fits. If the slope of the linear curve is 0.0, the equation cannot be solved.
Second Order Fit
A second order (quadratic) curve is specified by the equation
y c + c x + c x0 1 2 2
=
This equation can be solved to yield two amount (x) values, as follows:
x c c 4c (c y) 2c 1 1 2 0 2 = − + − −
Quantitation
An attempt to solve the equation will have one of four possible results:
• The curve cannot be solved because the square root term is negative. A quadratic curve has a single minimum or maximum. The inability to solve the curve for this reason means that the response (y) is either below the minimum or above the
maximum. The software will produce an error message signaling that the response is outside the domain of the calibration.
• The curve can be solved, and only one of the two versions of the quadratic equation yields a positive result. This is the optimal case. The positive result is taken as the initial amount, and the negative solution is ignored.
• The curve can be solved, and both solutions yield a positive result. The software chooses the amount value closer to the range of the calibration data.
• The curve can be solved, and both solutions yield a negative result. The software always chooses the larger amount value (the lower absolute value). See the discussion on negative amounts on page 18-74.
Third Order Fit
A third order (cubic) equation has the form
y c + c x + c x + c x0 1 2 2 3
3
=
Unlike linear and quadratic equations, cubic equations cannot be conveniently solved directly for x. Instead, TotalChrom uses an iterative process based on Newton's
method of approximating the root of an equation.
The software first computes the solution for a point-to-point fit as a preliminary estimate. Then, by refining that estimate, it trys to converge on the correct solution. This method of solving cubic equations has some limitations. Most notable is the inability to continue converging to a solution if an intermediate estimate lies at a minimum or maximum on the curve. Another problem occurs when successive estimates oscillate around the correct value without converging. In both cases, the software issues a warning message after the analysis report, notifying you of the inability to converge on a solution to the curve.
Negative Amounts
TotalChrom cannot use negative amounts when performing calculations, therefore it interprets negative numbers as zero (0). Occasionally, solving a calibration curve can result in a negative amount value. Although there is certainly no such thing as a negative amount of a component in a sample, negative results are caused by a positive
y intercept value (c0) when the slope of the curve at the intercept is positive. Any
positive response less than the intercept value will correspond with a negative amount.
Physically, a positive intercept can be interpreted to mean that an injection of zero amount of the component causes a positive response. This effect can be caused by instrumental interferences that produce a positive offset (such as stray light in a UV detector). Alternatively, it can arise from calibration data which shows excessive scatter (the positive intercept being merely a consequence of the regression). You can avoid the positive intercept by forcing the curve through the origin. If an amount of zero truly yields a positive response R, then, mathematically, any response less than R
(even if positive) must correspond with a negative amount.
A more probable calibration curve has a negative y intercept. This indicates that at a
point below a certain positive amount value, there is no response from the chromatograph detector. This amount is the detection limit for the equipment.
Quantitation
Constant Calibration Factor
Component amounts are normally determined by solving a calibration curve as described above. However, you have the option to enter into the method a constant calibration factor to be used to calculate a component's amount. If you choose this alternative, the software does not create a calibration curve. Instead, it divides the component's response R by the calibration factor F to obtain an initial amount Aini:
A R F ini =
Reference Calibration
Another alternative to using a component's own calibration curve to determine an unknown amount of it in a sample is to use the curve of another designated component. You choose this alternative by selecting the option Calibrate by Reference when you enter the component information. At that time, you also designate the appropriate reference component. The software calculates a raw amount for the component as though it were the reference component, except that the response of the component and not that of the reference component is used in the calculation.