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Generalized Two-State HVL Models (Experimental)
Extending the Generalized HVL Model Template
We previous described the single density Generalized HVL (Haarhoff VanderLinde) Model Template as follows:
a0 = area of the overall model
a1 = the center value used in the density and cumulatives
a2 = the width value used in the density and cumulatives
a3 = HVL chromatographic distortion parameter (fronted for a3<0, tailed for a3>0)
Density = PDF for the specified ZDD (zero distortion density), area=1
Cumulative = CDF for the specified ZDD, magnitude=1
RevCumulative = CDF complement for the ZDD, magnitude=1
The a3 distortion only appears in the the template and nowhere in the density and cumulative terms.
The above template allows for the PDF, CDF, and CDF complement to be a sum of two distinct components:
Density = frac*Density1 + (1-frac)*Density2
Cumulative = frac*Cumulative1 + (1-frac)*Cumulative2
RevCumulative = frac*RevCumulative1 + (1-frac)*RevCumulative2
In fact, two different densities or ZDDs can be combined in any way you wish, such as convolving the two as opposed to summing them. We limit the number of parameters within a user defined function to 10, so that will represent the limit of the number of densities that enter into the overall Density, Cumulative, and Reverse Cumulative items in the template.
If we deal directly with shapes only, and discount the different parameterization of ZDDs that produce identical shapes, and we look only at the ZDDs used for isocratic analytic peaks, those without gradient compression or overload dilation, and if we restrict models to those where the a1 center is the mean of the deconvolved Gaussian, we have the following ZDDs:
[Y] Generalized Error Model ZDD
[G] Half-Gaussian Modified Gaussian (GMG)
[E] Exponentially-Modified Gaussian (EMG)
[T] Generalized Student's t (Asymmetric)
Further, these ZDDs are not independent. The [Y] is a generalization of both the [Z] and [Q] densities. The [T] is a generalization of both the [Z] and [S] ZDDs. The [V] is a generalization of both the [Z] and [G] ZDDs. The [Y] adds a fourth moment fattening or thinning of tails, the [T] a fourth moment fattening of tails, and the [V] offers two very different third moment skewness adjustments.
What is common to all of these ZDDs is that the a0, a1, and a2 parameters will represent the area, mean, and SD of a pure Gaussian or normal density. In order to meaningfully combine densities, we suspect this is an essential requirement. In other words, the two-component models sum two different ZDDs which are identical Gaussians in their full deconvolutions (identical a1 and a2 parameters).
If a two-component model uses any two of these eight ZDDs, there are n(n-1)/2 or 28 different
permutations. We added six of these as built-in functions to the product:
GenHVL2[Z|E], GenHVL2[Q|E], GenHVL2[Y|E]
GenHVL2[Z|G], GenHVL2[Q|G], GenHVL2[Y|G]
We also built-in the following model:
GenHVL2[Z|GE]
You can view the the GenHVL2[Z|GE] as a three density model, the [Z] ZDD in the first component and an area weighted sum of the [G] and [E] in the second component.
Note the subtle differences in the PeakLab model nomenclature. A Gen2HVL model is twice generalized, often a fourth moment adjustment added to the third moment adjustment in a single ZDD. A GenHVL2 model uses a two-component ZDD where the two different ZDD components are separated by the | symbol, as shown in the models above. A Gen2HVL model is a mainstream chromatographic model where the fourth moment of the peak is addressed. A GenHVL2 two-component model is purely experimental.
The GenHVL2[Z|E] and GenHVL2[Z|G] Models
Our original purpose in creating two-state models was to have a primary ZDD which would model only the core chromatographic peak shape, and an auxiliary ZDD which would model all secondary IRF tailing. If it was in any manner possible, we wanted to find a full closed-form fit which included the IRF.
In seeking to reproduce the IRF tailing using this second component of the ZDD, we sought to use a Gaussian of identical center and width as the primary component, but which was distorted with either a half-Gaussian right convolution tailing (the closed-form GMG), or a first order exponential convolution tailing (the closed form EMG). It was our hope that, like a true IRF, the tau for the convolution in the closed form EMG, or the SD for the half-Gaussian convolution in the closed form GMG, would be constant for all peaks within an elution, irrespective of location, and would little change with concentration, much as is seen with a true IRF modeled as an external convolution.
Although we did not find either of these models to successfully partition the chromatographic peak shape the and IRF tailing, they are interesting models. We refer you to the two-state model tutorial.
The main difficulty is this partitioning. For strongly tailed peaks with the IRF present, it is almost a tossup whether the GenHVL[Z] or GenHVL[E] best models the peak. Even in the IRF models, the IRF tailing is so smeared into a highly tailed chromatographic shape, the IRF may be hard to fit effectively with a model that is very, very close to the true one. When we use the GenHVL2[Z|E] or GenHVL2[Z|G] models as a closed-form approximation to fitting such strongly tailed shapes with the IRF present, the partitioning becomes difficult. It is an overspecified fitting that occurs. We implemented an area fraction as a base 10 log in these models to account the fact a strongly fronted peak may have a fraction of 1e-9 of the second component, and a strongly tailed peak a .9999 fraction.
The GenHVL[Z|E] experimental model is thus a synched a1 and a2 blend of the [Z] ZDD and [E] ZDD. The GenHVL[Z|G] is a synched a1 and a2 blend of the [Z] ZDD and [G] ZDD. Note that there is a shift in the parameter sequence in these models. The a4 becomes the base 10 fraction of the second component. The other GenHVL primary component parameters then follow. For the [Z|E] and [Z|G] models, the a5 parameter becomes the principal statistical asymmetry in the [Z] component of the density and the a6 parameter will either be the exponential time constant or half-Gaussian SD. An a4 fraction specified as -2 would mean the second component EMG or GMG would be present in the ZDD at a 0.01 fraction of the overall area of the ZDD.
The GenHVL2[Z|GE] Model
The GenHVL2[Z|GE] model offers a second state component that is an area sum of the EMG and GMG. This model was based on the principle that this modification to the ZDD may be able to generate shapes closer to the <ge> IRF which fits the IRF component of chromatographic peaks to near-zero error.
In the GenHVL[Z|GE], the a4 and a5, like the other models, represent the log10 fraction of the second component and the principal statistical asymmetry in the [Z] component of the density. Here the difference rests in the a6- a8 parameters. This is a nine-parameter function, a6 the SD of the Half-Gaussian in the GMG, a7 the exponential time constant in the EMG, and a8 is the area fraction of the GMG in the second component, this as a linear value between 0-1.
This model should be deemed highly experimental.
The Two-State Models in User Functions
This example shows how to create the two-state GenHVL[Z|E] in a user function:
GenHVL[Z|E] parameter count=7
PDF=10^a4*EMG(x,1,a1,a2,a6)+(1-10^a4)*GenNorm(x,1,a1,a2,a5)
CDF=10^a4*EMG_C(x,1,a1,a2,a6)+(1-10^a4)*GenNorm_C(x,1,a1,a2,a5)
CDFc=10^a4*EMG_CR(x,1,a1,a2,a6)+(1-10^a4)*GenNorm_CR(x,1,a1,a2,a5)
Y=MHVL(a0,a1,a2,a3,PDF,CDF,CDFc)
1E-12, PkFnParm(GenHVL,0), 1E+12
1E-12, PkFnParm(GenHVL,1), 1E+12
1E-12, PkFnParm(GenHVL,2), 1E+12
-1, PkFnParm(GenHVL,3), 1
-12, -2, -1E-12
-1, PkFnParm(GenHVL,4), 1
1E-12,.04, 1
You simply add the two components to the PDF, CDF, and CDFc formulas.
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