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EMG

EMG Exponentially Modified Gaussian

a_{0} = Area

a_{1} = Center (as mean of Gaussian peak that is convolved by the a_{3} exponential)

a_{2} = Width (SD of the Gaussian peak that is convolved by the a_{3} exponential)

a_{3} = The first order kinetic or exponential decay width convolving the Gaussian (can be negative
to model fronted peaks)

Built in model: EMG

User-defined peaks and view functions: EMG(x,a_{0},a_{1},a_{2},a_{3})

The EMG model was once the most widely used chromatographic model capable of modeling tailing. It was quite extensively studied decades ago. One review paper from that era (M. S. Jeansonne and J. P. Foley, J. Chrom. Sci., 29, p258-266, 1991) documents 127 separate EMG-related references.

The EMG is the mathematical convolution of a Gaussian with a first order decay or exponential response function. There are only two components to this model, a primary Gaussian, and a one-sided exponential response function which convolves or smears the Gaussian as in the profiles above. As this exponential element increases, peaks become more right asymmetric or tailed. Since this model’s components consist of two of the simplest functions in nature, it is not difficult to see why various theoretical underpinnings have been proposed for such.

Insufficiency of EMG Model

Let us look at the GenHVL<ge> or GenNLC<ge> models, the two most useful pre-IRF deconvolution models in the program:

HVL/NLC-Chromatographic-Distortion[ZDD=Gen Normal] Ä IRF(Area Sum of Exponential and Half-Gaussian)

The ZDD is extended with the generalized normal to model the non-idealities in the actual chromatographic
separation. The common
chromatographic distortion, a_{3}, is applied to this ZDD to manage the concentration dependent
fronting and tailing, and this peak is convolved with the <ge> IRF
(it having both a probabilistic and kinetic instrumental distortion). The model is directly fitted as
a convolution integral in the Fourier domain.

From this perspective, the EMG can be seen as a rudimentary specialization of the GenHVL or GenHVL models. We would designate the EMG as a (Gauss Ä exp) or Gauss<e> convolution model. There is no ZDD insofar as there is no intrinsic chromatographic distortion in the model. Further, there is no generalization of the Gaussian to alter the non-Gaussian character of the chromatographic peak. There is the <e> IRF convolution, however, the kinetic decay IRF.

The EMG was attractive since this model was capable of fitting close to to Gaussian peaks with the long tailing once typical of slow detectors. The EMG was simple to compute, the convolution having a closed form solution. The EMG thus makes no accommodation of intrinsic fronting or tailing, nor is the issue of a non-Gaussian shape at infinite dilution anywhere addressed. An external distortion is modeled, but only a first order exponential decay usually attributed to a slow detector, but which would model, generally poorly, every deviation from the Gaussian in the overall peak shape. It was generally useless with the fronting and tailing seen at higher solute concentrations.

In modern chromatography, where the peaks are more sharply defined, the S/N much improved, and slow detectors a thing of the past, the EMG is effectively obsolete. We found it fully so in the course of the research associated with the development of PFChrom v5. Our goal was to produce close to perfect fits with all analytical peaks; the EMG never came close.

The EMG is included in PFChrom for historical reference and for research purposes. For example, it is used as an EMG ZDD in the GenHVL[E] and GenNLC[E] models.

The HVL<irf> and NLC<irf> convolution models are likely to be far more useful than the EMG, and those are inferior to the GenHVL<irf> and GenNLC<irf> models which manage can isolate the intrinsic nonidealities in modern chromatographic data. Given the ease at which these far more effective models can be fitted in PFChrom, and more comprehensive IRFs removed once quantified in a deconvolution step prior to fitting, we cannot see any benefit to a continued use of the EMG for chromatographic modeling.

In the EMG, the intracolumn band broadening processes such as axial diffusion, dispersive effects, mass transfer resistances, and slow kinetics of adsorption of desorption are assumed in the aggregate to distribute or broaden the solute as a Gaussian.

Reduction to an EMG

By using one of the primary PFChrom models, you are not abandoning the capability of fitting an EMG shape. The following models will reduce to EMG, if an actual EMG shape happens to be present:

HVL<e>, a_{3}=0 (no chromatographic distortion)

HVL<ge>, a_{3}=0, a_{4}=0 and/or a_{6}=0 (no chromatographic
distortion, no probabilistic component to IRF)

GenHVL<e>, a_{3}=0, a_{4}=0 (no chromatographic
distortion, no ZDD asymmetry)

GenHVL<ge>, a_{3}=0, a_{4}=0, a_{5}=0
and/or a_{7}=0 (no chromatographic
distortion, no ZDD asymmetry, no probabilistic component to IRF)

GenNLC<e>, a_{3}=0, a_{4}=0 (no chromatographic
distortion, no ZDD asymmetry)

GenNLC<ge> a_{3}=0, a_{4}=0, a_{5}=0
and/or a_{7}=0 (no chromatographic
distortion, no ZDD asymmetry, no probabilistic component to IRF)

GenHVL[Z]<e>, a_{3}=0, a_{4}=0 (no chromatographic
distortion, no ZDD asymmetry)

GenHVL[Z]<ge>, a_{3}=0, a_{4}=0, a_{5}=0
and/or a_{7}=0 (no chromatographic
distortion, no ZDD asymmetry, no probabilistic component to IRF)

GenNLC[Z]<e>, a_{3}=0, a_{4}=0
(no chromatographic distortion, no ZDD asymmetry)

GenNLC[Z]<ge>, a_{3}=0, a_{4}=0,
a_{5}=0 and/or a_{7}=0
(no chromatographic distortion, no ZDD asymmetry, no probabilistic component
to IRF)

The models that manage the kurtosis of the zero-distortion density, or which furnish a second adjustment of the skew, equally can reduce to the EMG shape.

GenHVL[Y]<e>, a_{3}=0, a_{4}=2, a_{5}=0
(no chromatographic distortion, Gaussian decay, no ZDD asymmetry)

GenHVL[Y]<ge>, a_{3}=0, a_{4}=2, a_{5}=0,
a_{6}=0 and/or a_{8}=0
(no chromatographic distortion, Gaussian decay, no ZDD asymmetry, no probabilistic
component to IRF)

GenNLC[Y]<e>, a_{3}=0, a_{4}=2,
a_{5}=0 (no chromatographic
distortion, Gaussian decay, no ZDD asymmetry)

GenNLC[Y]<ge>, a_{3}=0,
a_{4}=2, a_{5}=0, a_{6}=0
and/or a_{8}=0 (no chromatographic
distortion, Gaussian decay, no ZDD asymmetry, no probabilistic component to IRF)

GenHVL[T]<e>, a_{3}=0, a_{4}=1000000, a_{5}=0
(no chromatographic distortion, infinite Student's t nu, no ZDD asymmetry)

GenHVL[T]<ge>, a_{3}=0, a_{4}=1000000,
a_{5}=0, a_{6}=0
and/or a_{8}=0 (no chromatographic
distortion, infinite Student's t nu, no ZDD asymmetry, no probabilistic component to IRF)

GenNLC[T]<e>, a_{3}=0, a_{4}=1000000,
a_{5}=0 (no chromatographic
distortion, infinite Student's t nu, no ZDD asymmetry)

GenNLC[T]<ge>, a_{3}=0,
a_{4}=1000000, a_{5}=0, a_{6}=0
and/or a_{8}=0 (no chromatographic
distortion, infinite Student's t nu, no ZDD asymmetry, no probabilistic component to IRF)

GenHVL[V]<e>, a_{3}=0, a_{4}=0,
a_{5}=0 (no chromatographic
distortion, zero GMG convolution width, no ZDD asymmetry)

GenHVL[V]<ge>, a_{3}=0, a_{4}=0,
a_{6}=0 and/or a_{8}=0
(no chromatographic distortion, zero GMG convolution width, no ZDD asymmetry,
no probabilistic component to IRF)

GenNLC[V]<e>, a_{3}=0, a_{4}=0,
a_{5}=0 (no chromatographic
distortion, zero GMG convolution width, no ZDD asymmetry)

GenNLC[V]<ge>, a_{3}=0,
a_{4}=0, a_{6}=0 and/or a_{8}=0
(no chromatographic distortion, zero GMG convolution width, no ZDD asymmetry,
no probabilistic component to IRF)

Other models also can reduce to an EMG:

GenHVL[S]<e>, a_{3}=0, a_{4}=1000000
(no chromatographic distortion, infinite Student's t nu)

GenHVL[S]<ge>, a_{3}=0, a_{4}=1000000,
a_{5}=0 and/or a_{7}=0
(no chromatographic distortion, infinite Student's t nu, no probabilistic
component to IRF)

GenHVL[Q]<e>, a_{3}=0, a_{4}=2 (no
chromatographic distortion, Gaussian decay)

GenHVL[Q]<ge>, a_{3}=0, a_{4}=2,
a_{5}=0 and/or a_{7}=0
(no chromatographic distortion, Gaussian decay, no probabilistic component to IRF)

GenHVL[G]<e>, a_{3}=0, a_{4}=0 (no
chromatographic distortion, zero GMG convolution width)

GenHVL[G]<ge>, a_{3}=0, a_{4}=0,
a_{5}=0 and/or a_{7}=0
(no chromatographic distortion, zero GMG convolution width, no probabilistic
component to IRF)

GenHVL[E]<e>, a_{3}=0, a_{4}=0
(no chromatographic distortion, zero EMG convolution width)

GenHVL[E]<ge>, a_{3}=0, a_{4}=0,
a_{5}=0 and/or a_{7}=0
(no chromatographic distortion, zero EMG convolution width, no
probabilistic component to IRF)

EMG Theory

The exponential response function is usually attributed to extracolumn effects. The EMG assumes a simple first order response from a detector and its associated electronics. This response function has a non-zero time constant and is one sided or directionally constrained—a detector does not have an instantaneous response nor can it sense material prior to its arrival at the detector.

Note that no intracolumn effects are modeled by the a_{3} parameter. Indeed, to assume that
any form of intracolumn-originated asymmetry can be independently attributed to something as rudimentary
as a first order convolution decay model seems a vast oversimplification (note the complexity of the kinetic
chromatographic models which directly deal with only a portion of intracolumn dynamics).

When treating the exponential component as a detector response function, only tailed peaks can be
modeled. To enable the EMG to be used as an empirical function for fronted peaks, PFChrom offers the EMG
in a bidirectional form with -a_{3} widths.

In a convolution model, the parameters actually represent a true deconvolution of different physical
system responses. Fitting an analytical form for a convolution product is also the finest form of deconvolution
in that, unlike discrete Fourier procedures, no noise is introduced into the data. Another way to state
the EMG is Gauss(a_{0}, a_{1}, a_{2}) (x) Exp(a_{3}). As such, the fitted
parameters directly produce the deconvolved Gaussian and deconvolved detector response function time constant.
Because convolution of an instrument response function is area invariant, one simply assumes that Gauss(a_{0},
a_{1}, a_{2}) represents the true peak (the peak with a perfect detector and electronics).

The higher moments of the convolved product cannot be assumed to have any significance, but the
moments of the deconvolved Gaussian are as significant as the fit, where a_{1} represents the
centroid or first moment, and a_{2} represents the standard deviation or square root of the second
moment. Assuming the time scale of the data has been pre-transformed with the X=X/t-1 calculation, a_{1}
also represents the true thermodynamic capacity factor k’. As with all models, the degree to which parameters
can be judged significant must be in proportion to the quality of the fit.

The nature of the EMG model, as in all <irf> fits, suggests that a single a_{3} should
be shared across all peaks since it should represent an invariant time constant for the instrument’s detector.
When a_{3} is varied in order to account for peak shape differences across the chromatographic
data, you are in effect fitting an empirical model, and you should no longer assume any significance for
the parameters.