PeakLab v1 Documentation Contents AIST Software Home AIST Software Support

Chromatographic Experiments (Tutorial)

For this tutorial, we will fit an extensive chromatographic experiment to study the impact of both concentration and additive level on the fitted peaks. Specifically, we want to see how the system variables of sample concentration and additive level impact the same six peaks in the cation standard we used in the first tutorial.

Importing the Data

Click the **Open** button, the first button in the program's main toolbar. Select the file **CationTutorial2.pfd**
from the program's installed default data directory (\PeakLab\Data).

The sixteen data sets are contained in a single PeakLab PFD data file. This data have already been transformed for a retention or k' capacity factor scale and the baseline subtraction has been done. No IRF has been deconvolved. For this experiment, we will take the additional time to fit the IRF in each of the data sets. We want to know if the sample concentrations and additive levels alter the instrument response function, the IRF.

Visualizing the Effects of Concentration and Additive

Right click the second graph in the grid and choose **Select as Principal Data Set** from the popup
menu. This data set will now have a red background. Click the **Unselect All** button in the dialog.
All of the graphs except the principal data set will now have a gray background. **Left click the the
second graph in each of the three remaining rows** to reselect these three data sets. The 6th, 10th,
and 14th graphs will now have a blue background. Only the principal and selected graphs pass to the visualization
procedure.

Click the **View and Compare Data** button.

For the Format, select **Single Plot**, for the Normalization select **None**, and for the Adjustments,
leave **Force Zmin=0** checked. Using the mouse, box the first set of peaks to zoom in this section
of the data.

As the concentration increases, the first cation peak, Li+, becomes much more strongly fronted. This fronting
or tailing is the chromatographic shape modeled so beautifully by the a_{3} parameter in the HVL,
NLC, GenHVL, and GenNLC models. This a_{3} is a common
chromatographic distortion to all of these models. When a_{3} is negative, a peak is said
to be intrinsically fronted.

**Right click the graph** and select the **Restore Scaling - Undo Zoom** item from the popup menu.
Zoom-in the last set of peaks.

As the concentration increases, the last cation peak, the Mg+ cation in the standard, becomes much more
strongly tailed. When a_{3} is positive, a peak is said to be intrinsically tailed.

Click **OK** to exit the Visualization procedure, and then click the **Select All** button in the
main window. Click the **View and Compare Data** button.

Check **Contour** and **Unit Area**.

At each of the concentrations, the effect of additive level profoundly impacts retention. At a 2x additive level, the Mg+ peak elutes at a retention less than 10. When no additive is used, the retention of this peak occurs at a retention of 26.

Using the mouse, **zoom-in the eight peaks** to the right and then select the **3D Shaded** option.

These are the Ca+ and Mg+ cation peaks for the zero additive case. The concentration will typically influence the peak shape more than any other factor. The additive clearly impacts location. To fully understand the differences, however, we will fit the data to a high level of accuracy.

Fitting the GenNLC<ge> Model

Click **OK** to exit the visualization procedure. Click the **Local Maxima Peaks** button. You will
see the peak placement screen. Ensure the following settings are selected:

Peak Detection

Set Sm
n(1) to 25

Peak Type

Select Chromatography
in the first dropdown

Select GenNLC<ge> as the model in the
second dropdown

Scan

Set the Amp %
threshold to 1.5 %

Leave Use Baseline Segments unchecked

Be sure Use IRF,ZDD is checked

Vary

Leave width a2
and shape a3 checked, all other unchecked

You will note that each of the data sets has an automatic placement of this model with 6 peaks.

Click the **IRF** button. If you made modifications to the IRF
defaults prior to this tutorial for your own fitting, please use the **Save** button to save your
IRF values before resetting the defaults. Click the **Defaults** button. These are based on retention
scale averages of modern IC data for each of the IRFs built into the program. Click **OK**.

We will fit the <ge> IRF, the narrow width component of the IRF a half-Gaussian which may better fit the axial dispersion.

Click the **Peak Fit** button in the lower left of the dialog to open the fit
strategy dialog. Select the **Fit with Reduced Data Prefit, Cycle Peaks, 2 Pass, Lock Shared Parameters
on Pass 1**. Be sure the **Fit using Sequential Constraints** box checked. Click **OK**.

If it is checked, uncheck the **Iteration Update** button to speed up fitting. This stops the graphical
update with each iteration where an improvement in fit occurred.

Because we are fitting 16 data sets each with 6 peaks, we are fitting 96 convolution integrals with a high data sampling density. Even using all of the cores of your processor in a multithreaded fitting, these sixteen fits will require a couple of minutes.

Reviewing the Various Fits

Click **Review Fit**.

The fits range from 4.99 to 20.61 ppm unaccounted variance, superb goodness of fit given that six peaks were fitted rather than one, and the baselines were oscillating and prone to introducing error in the fitting.

Right click any graph and select **Map Identifying Labels to Specific Peaks**.

Use this option to label
peaks in a chromatogram. These have already been added to the data file. The peaks are specified by
sequential order since the x locations vary greatly with additive. Click **OK**.

Right click any graph and select **Map Experimental Process Variables for Fitted Data Sets**.

Use this option to specify
numeric variables associated with an experiment. Here we specify the concentration (ppm) and additive
level (as fraction of recommended amount). These have also been added to the data file. The titles are
shown to help with entry. Click **OK**.

Click **Numeric** if the Numeric Summary is not displayed. In the **Options** menu first choose
**Select Only Fitted Parameters** and then check **Average Multiple Fits**. Scroll to the bottom
of the report.

Average for 16 Fits

Fitted Parameters

r^{2} Coef Det
DF Adj r^{2}
Fit Std Err
F-value
ppm uVar

0.99998974
0.99998972
0.01494899
58,763,437
__10.2583889__

Peak
Type
a_{0}
a_{1}
a_{2}
a_{3}
a_{4}
a_{5}
a_{6}
a_{7}

1
GenNLC<ge>
8.07933699
2.30784582
0.00026474
-0.0099790
__1.11887780__
__0.00737138__
__0.04298459__
__0.63436801__

2 GenNLC<ge> 2.25247000 3.77100711 0.00029000 -0.0016692 1.11887780 0.00737138 0.04298459 0.63436801

3 GenNLC<ge> 2.61798274 4.54185798 0.00027606 -0.0009040 1.11887780 0.00737138 0.04298459 0.63436801

4 GenNLC<ge> 1.18633675 7.21994862 0.00039513 0.00189375 1.11887780 0.00737138 0.04298459 0.63436801

5 GenNLC<ge> 2.52764231 12.3193707 0.00086418 0.01870488 1.11887780 0.00737138 0.04298459 0.63436801

6 GenNLC<ge> 3.58747556 15.2877892 0.00099840 0.02611155 1.11887780 0.00737138 0.04298459 0.63436801

CV Percent for 16 Fits

Fitted Parameters

r^{2} Coef Det
DF Adj r^{2}
Fit Std Err
F-value
ppm uVar

0.0004641% 0.0004651% 90.853670% 62.200140% 45.247606%

Peak
Type
a_{0}
a_{1}
a_{2}
a_{3}
a_{4}
a_{5}
a_{6}
a_{7}

1
GenNLC<ge>
80.621242%
15.246945%
15.044717%
82.950830%
__10.605790%__
__24.030050%__
__3.0855039%__
__1.7568393%__

2 GenNLC<ge> 80.900316% 15.015629% 10.702513% 91.723675% 10.605790% 24.030050% 3.0855039% 1.7568393%

3 GenNLC<ge> 80.982331% 15.015673% 10.937966% 112.55012% 10.605790% 24.030050% 3.0855039% 1.7568393%

4 GenNLC<ge> 80.944661% 15.048280% 11.782047% 84.882322% 10.605790% 24.030050% 3.0855039% 1.7568393%

5 GenNLC<ge> 100.77145% 43.246458% 26.962165% 134.35685% 10.605790% 24.030050% 3.0855039% 1.7568393%

6 GenNLC<ge> 83.137941% 44.156155% 31.510942% 87.437350% 10.605790% 24.030050% 3.0855039% 1.7568393%

The goodness of fit averages 10.2 ppm unaccounted variance and an F-statistic of nearly 59 million. If
we look at the shared parameters held constant across the different peaks, we see the a_{4} ZDD
asymmetry to be somewhat higher than the theoretical value for the NLC (1.11 instead of 0.5), and a CV%
(SD/Mean) of 10.6%. This ZDD non-ideality doesn't vary a huge measure, but enough to give reasonable attention
within the experiment. The <ge> IRF's half-Gaussian component, a_{5}, much narrower in width
(here an SD width), has a 24% CV. This we will give a fair attention in the experimental evaluation. The
a_{6} exponential component of IRF is much wider (this is a time constant), responsible for most
of the secondary tailing in the peaks, and with a 3.09% CV, very close to constant across the sixteen
data sets. We can look for trends, but their impact will be small. The a_{7} parameter which specifies
that 63.4% of the IRF is to consist of the Gaussian has an even narrower 1.76% CV. Again, we can look
for trends, but we would expect little impact upon peak moments and properties.

Exploring the Parametric Relationships

Click **Explore**. Select **3D Shaded**. Enter **Conc** for x in the first dropdown, **Additive**
for the y in the second dropdown, and **Parameters** for the z in the third dropdown. Change the Peaks
to view from Peak **1** to Peak **1- we** will start with just the Lithium peak, the one most strongly
fronted.

This is the visualization of the experiment. The concentration of the solutes in the standard is on the
X-axis, and the additive level is on the Y-axis. The eight plots are for the GenNLC<ge> parameters
a_{0}-a_{7} for the first cation peak (Lithium). We will use this visualization to understand
as many of the relationships of chromatographic modeling as we can.

a_{0} Area

**Double click the first graph**. This will produce a plot of just the Area. Select **3D Gradient**.

If you have a wheel-mouse, spin the wheel forward and backward to rotate the surface by the angle in the XY plane. To change the Z angle, press the tilt wheel left or right. Click the wheel to restore the XY and Z angle settings before the changes were made.

The **Modify Graph View** button in the graph's toolbar can also be used to adjust the different 3D
settings.

At the higher concentration, there is clear diminishing of retained solute with increasing additive. The colors in the gradient correspond with the Z area scaling. If you look closely at the colors, you will see some indication that there is also an effect of diminished peak area even at the lower concentrations.

Click **2D Contour**. You are now looking straight down at the surface from above.

The impact of additive on the area of the peaks at the four different concentrations is now sharply visualized. The effect at the 25 and 50 ppm solute concentrations is clear. The effect at the two lower concentrations is less evident.

Using the mouse, zoom-in the contour by boxing just the lower concentration points.

Here we are looking at a zone of the 3D interpolated surface formed from 16 points. One must be careful in reading too much into areas where no data is present. In this case, it is clear that even at low concentrations, the additive reduces the amount of the Lithium cation retained.

Right click the graph and select **Restore Scaling - Undo Zoom or Custom Scaling** from the popup menu.
Double click the graph to restore all of the graphs.

a_{1} Center

In the **GenNLC** model, a_{1} is the deconvolved mean of the zero distortion density, in this
case a generalized normal.

**Double click the second graph**.

The contour illustrates the massive dependency of location upon the additive level. The greater the additive, the earlier the elution. The concentration also has a small impact. Higher concentrations elute slightly earlier.

**Double click the graph** to restore all plots.

a_{2} Width

For the GenNLC
models, a_{2} is a first order time constant, the inverse of the rate in the Giddings density.

Select **3D Gradient**. **Double click the third graph**.

If you have a wheel mouse, adjust the XY angle by spinning the wheel up and down to where the angle covers the shallow depression in the interpolated surface.

The angles chosen (XY=85, Z=5) show only modest increases with additive until the 50 ppm concentration (likely having a measure of overload) occurs. One would expect to see a consistent increase in width with concentration, not the well suggested by the interpolation.

Select **2D Contour**.

Here we see the three 10 ppm concentrations with additive comprising this valley or minimum in the response
surface. It is likely real; the fitted a_{2} widths are actually smaller at 10 ppm on the additive
samples. Without additive, the peaks elute later and have higher widths.

**Double click the graph** to restore all plots. Select **3D Gradient**.

a_{3} Chromatographic Distortion

**Double click the fourth graph**.

Click the **Start/Stop 3D Animation** button in the graph's toolbar to begin a 3D animation of the
surface.

If you have too fast or too slow an animation, you can adjust the angular increments and program in the Modify 3D Animation Settings option.

Note that overall relationship of the a_{3} intrinsic chromatographic distortion appears linear
at each of the additive levels and the difference is mainly in the slope.

Click the **Start/Stop 3D Animation** button in the graph's toolbar to stop the animation. Click **2D
Contour**.

This is the parameter which addresses the unique skewness or third moment behavior which determines the
principal fronting or tailing in a chromatographic shape. As you animated the 3D surface, you were animating
the default 3D rendering, which is an interpolant. Any deviation will be seen in a lack of local planar
shape in the rendered surface. The science behind the Haarhoff-VanderLinde and Wade-Thomas common chromatographic
distortion is every bit as good as this a_{3} surface suggests.

We call the a_{3} parameter the "intrinsic" chromatographic distortion because it is
the true shape, the true fronting or tailing, absent the IRF and absent the non-idealities in the underlying
density. When you fit a GenNLC or GenHVL model, you deconvolve a true chromatographic shape, the one that
would ideally exist for the given concentration of the solutes. Although the a_{3} distortion
in the NLC and HVL models have very different theoretical derivations (one associated with the rate constants
of adsorption and desorption, the other with the adsorption isotherm), the a_{3} distortion operator
for both models produces identical chromatographic shapes for a given zero-distortion (infinite dilution)
density. In fact, for the GenHVL and GenNLC, these were adjusted to be identical in value.

a_{4} ZDD Zero-Distortion Density Asymmetry

The parameters from a_{4}-a_{7}, the a_{4} ZDD asymmetry, and the a_{5}-a_{7}
IRF, are shared across all peaks. The next graphs will be identical irrespective of which peak is being
inspected. Although these four parameters were shared, and increased the overall count of fitted parameters
from 24 to 28, the difference in fit is immense. We will look closely at the ZDD and IRF parameters for
any trends.

**Double click the graph** to restore all plots. Select **3D Gradient**.

At this point note that the a_{4} and a_{5} plots show approximately the same overall
shape of surface. In statistical terms, this means the parameters will be correlated and more difficult
to fit with accuracy. Indeed, these are the two parameters with the least significance (the lowest absolute
t-value in the Parameter Statistics in the Numeric
Summary). We will keep this correlation in mind as we survey the relationships.

**Double click the fifth graph**.

Click the **Start/Stop 3D Animation** button in the graph's toolbar to begin a 3D animation of the
surface. **Click the left mouse button on the graph** when the there is increasing concentration to
the right. This pauses the animation at the present position.

The fitted a_{4} ZDD asymmetry is far more sensitive to the concentration, increasing from 5 to
10 ppm, and then decreasing at 25 and further decreasing at 50 ppm. The obvious inference is that ZDD
asymmetry is close to the theoretical (though still higher) at the lowest concentration, that such then
increases, and that the subsequent decrease is perhaps associated with a small measure of overload. Since
the additive only marginally impacts a_{4}, we have close to four replicates at each concentration.
This does suggest the asymmetry does have a concentration dependency. In all 16 fits, we are well above
the 0.5 a_{4} theoretical asymmetry of the pure NLC and its Giddings zero-distortion density.

**Click the left mouse button on the graph** to restart the animation.

Click the **Start/Stop 3D Animation** button to stop the animation. **Double click the graph** to
restore all plots.

a_{5} Fast IRF Component - Half Gaussian SD

**Double click the sixth graph**.

If you have a wheel-mouse, **spin the mouse wheel** to rotate the surface by the angle in the XY plane
until you see increasing concentration, similar to the a_{4} plot.

Unlike the prior plot, the a_{5} fast half-Gaussian component of the IRF, increases somewhat with
increasing additive level. The values are narrow, the trend reasonably confirmed. This fast component
of the IRF little impacts tailing, and is subtle, more difficult to fit. This was the parameter determined
with least accuracy in the fitting, though still strongly significant with respect to statistical significance
tests.

**Double click the graph** to restore all plots.

a_{6} Slow IRF Component - Exponential Time Constant

**Double click the seventh graph**.

Amazingly, the exponential plots with a strong response surface despite only small differences in the
a_{6} value across the sixteen different fits. The exponential width is impacted by concentration,
decreasing with increasing concentration. The impact of additive, in general, is to increase the a_{6}
tau.

**Double click the graph** to restore all plots.

a_{7} Half-Gaussian IRF Component Area Fraction

**Double click the seventh graph**.

If you have a wheel-mouse, **spin the mouse wheel** to rotate the surface by the angle in the XY plane
until you see a similar surface to the one below:

The half-Gaussian component of the IRF is important, fitting to better than half the area in the IRF.
It is much narrower in width, however, and thus far less visible with respect to introduced tailing. The
a_{7} half-Gaussian fraction decreases with additive, especially at lower concentrations. All
of values are very close to the 5/8 half-Gaussian, 3/8 exponential assumption which was made in the first
tutorial. At higher concentrations, the impact of additive is less pronounced. The relationship with concentration
is less determinate.

Exploring the Deconvolved Moments

Moments as Fitted

**Double click the graph** to restore all plots. Change the z variable to **Moments. **If you have
a wheel-mouse, **spin the mouse wheel** to rotate the surface by the angle in the XY plane until you
see a respectable view of all five surfaces.

If you do not have a wheel-mouse, click the **Modify Graph View** button in the graph's toolbar and
change the XY angle to **220** degrees, the Z angle to **20** degrees.

Unlike the parameter relationships, there is a consistent and clear defined surface for all of the moments. The third and fourth moments track one another since the predominate influence upon the thickness of the tails is the skew arising from the IRF. In the above surfaces, we are looking at the moments of the peaks as registered by the instrument. We will now take advantage of the deconvolution within the fitting to see the deconvolved moments. We will look at two levels of deconvolution, the peaks with the IRF removed, and the peaks with the IRF and ZDD removed. It is the latter that we deem most important. We continue to look at just the first peak of the six in the cation standard, the strongly fronted Lithium peak.

Moments with IRF Removed

Change the z variable to **Moments DC1**. This plots the moments for the first level of deconvolution,
the GenNLC peaks
with the IRF removed. We still see the intrinsic non-ideality within the chromatography, but we remove
the instrumental/system distortion. To do this, we use the first five GenNLC<ge> fitted parameters
in the GenNLC, the model with no IRF.

With the IRF removed, the third moment, the skewness, and the fourth moment, the kurtosis, look very different. For one, the skewness values are now all negative, indicative of the intrinsic fronting (left skew) for this first peak. The kurtosis now shows the tailing of the GenNLC thinning with concentration, so much so the kurtosis is well below the 3.0 of a Gaussian. On strongly fronted chromatographic peaks, the kurtosis can drop to such low values.

Moments with IRF and ZDD Removed - the Pure Chromatographic Peak

Change the z variable to **Moments DC2**. This plots the moments for the second level of deconvolution,
the NLC
peaks with both the IRF and the ZDD non-ideality removed:

Here we plot the pure NLC, the theoretical peaks with only the chromatographic a_{3} parameter
impacting the higher moments. The IRF and ZDD deconvolution are implicit in the GenNLC<ge> fit.
We only need to use the principal a_{0}-a_{3} parameters to extract the pure NLC chromatographic
peaks. Here we see the same <3 kurtosis as the concentration increases.

We will not do this as part of the tutorial, but if we were to select **IRF Deconv.** in the first
dropdown of the main Review window, and **Partial Deonconv.** in the second, and zoom in the first
peak in the first of the 50 ppm fits, we would see the GenNLC<ge> peak as fitted in the white curve,
the GenNLC in
the red, and the NLC
in the green. Absent an IRF, a strongly fronted GenHVL, GenNLC, HVL, or NLC, can have an exceptionally
sharp decay with a kurtosis well below 3.0. In general, however, the kurtosis of the deconvolved peaks
will be very close to 3.0.

Smoothing Instead of Interpolating 3D Surfaces

**Double click the Skewness plot**, the fourth graph, in the Explore option.

Click the **Specify Surface Modeling** button in the graph's toolbar. Select **Fast Smooth** and
**C1 Smooth Surface**. Set the Model Order to **2**, and the Neighbor Count to **13**. This produces
a locally smoothed quadratic surface instead of an interpolated one. For difficult relationships of parameters,
moments, and properties, you may need this local smoothing to visualize the trends.

When the points do not lie on the surface, you can click on the Modify Point Format in the graph toolbar
and select the **Visible Only** option. This will hide the points which would not be visible, those
hidden by the surface. This is particularly useful for animating smoothed surfaces.

Click **Cancel**. We will leave the surface as interpolant. Click **OK** in the the **Explore**
Window to restore all graphs and click **OK** again to close the Explore option.

Click **OK** to close the Review window. Check **Save updated information to the current data file
when adding fits** and click **OK**, accepting the default name for the fit. Click **OK** to confirm
and click **OK** one last time to leave the placement screen and return to main screen.

In the third tutorial, we will subtract the average IRF from these fits from each of the data sets, and then fit a closed form model. We will see how much accuracy is lost when removing a system average IRF prior to fitting.