Moments analysis

One solution to the volume problem was proposed using moment analysis. The steady-state volume of distribution (Vss) can be derived from the area under the curve (AUC) and the area under the first moment curve (AUMC). 

Moment Analysis. The zeroth moment is the molar concentration of polymer and is expressed by Equation 5. The first moment is proportioned to the mass of polymer formed and is related to monomer concentration Equation 3, The second moment WA(t) is expressed by... 

Fig. 39.16. Paradigm for the fitting of sums of exponentials from a compartmental model (c) to observed concentration data (o) as contrasted by the results of statistical moment analysis (s). (After Thom [13].)...

The alternative to compartmental analysis is statistical moment analysis. We have already indicated that the results of this approach strongly depend on the accuracy of the measurement process, especially for the estimation of the higher order moments. In view of the limitations of both methods, compartmental and statistical, it is recommended that both approaches be applied in parallel, whenever possible. Each method may contribute information that is not provided by the other. The result of compartmental analysis may fit closely to the data using a model that is inadequate [12]. Statistical moment theory may provide a model which is closer to reality, although being less accurate. The latter point has been made in paradigmatic form by Thom [13] and is represented in Fig. 39.16. 

Dang, N.D.P., Karrer, D.A. and Dunn, I.J. (1977). Oxygen Transfer Coefficients by Dynamic Model Moment Analysis, Biotechnol. Bioeng. 19, 853. 

S. Yamamura, F. Aida, Y. Momose, and E. Fukuoka, Analysis of mean disintegration time and mean dissolution time by moment analysis microcalorimetric curves, Drug Dev. Ind. Pharm., 26, 1 (2000). 

It has been recently shown [12] that the ELF topological analysis can also be used in the framework of a distributed moments analysis as was done for Atoms in Molecules (AIM) by Popelier and Bader [32, 33], That way, the Mo( 2) monopole term corresponds to the opposite of the population (denoted N) ... 

Moment analysis is one of the simplest types of analysis and is useful for measuring the performance of the chromatography. Moments can be used to measure the same things that are measured in ID chromatographic systems these include the first, second, and third moments, which are more accurate than the related peak maximum, peak width, and peak asymmetry. In 2D, however, these values each have a component in each dimension and this can be easily determined in software-based measurement systems. 

Miyabe, K., Cavazzini, A., Gritti, F., Kele, M., Guiochon, G. (2003). Moment analysis of mass-transfer kinetics in Ci8-silica monolithic columns. Anal. Chem. 75, 6975-6986. 

Esposito, A., Gerritsen, H. C. and Wouters, F. S. (2005). Fluorescence lifetime heterogeneity resolution in the frequency domain by lifetime moments analysis. Biophys. J. 89, 4286-99. 

The most challenging limitation of the REDOR method is that it can only be analytically analyzed, when small spin systems, ideally isolated two-spin-1/2 pairs, are involved. For larger spin systems, the geometry between the spins is important, and, since this is often unknown, accurate distance determinations are precluded for complex materials [176], Bertmer and Eckert have proposed that larger spin systems can still be described, at least semi-quantitatively, by REDOR data in terms of second moment analysis [177],... 

In a typical pulse experiment, a pulse of known size, shape and composition is introduced to a reactor, preferably one with a simple flow pattern, either plug flow or well mixed. The response to the perturbation is then measured behind the reactor. A thermal conductivity detector can be used to compare the shape of the peaks before and after the reactor. This is usually done in the case of non-reacting systems, and moment analysis of the response curve can give information on diffusivities, mass transfer coefficients and adsorption constants. The typical pulse experiment in a reacting system traditionally uses GC analysis by leading the effluent from the reactor directly into a gas chromatographic column. This method yields conversions and selectivities for the total pulse, the time coordinate is lost. 

Tanigawara Y, Yamaoka K, Nakagawa T, Uno T. Moment analysis for the separation of mean in vivo disintegration, dissolution, absorption, and disposition time of ampicillin products. J Pharm Sci 1982 71 1129-1133. 

Level B utilizes the principles of statistical moment analysis. The mean in vitro dissolution time is compared to either the mean residence time or the mean in vivo dissolution time. Like correlation Level A, Level B utilizes all of the in vitro and in vivo data, but unlike Level A it is not a point-to-point correlation because it does not reflect the actual in vivo plasma level curve. It should also be kept in mind that there are a number of different in vivo curves that will produce similar mean residence time values, so a unique correlation is not guaranteed. 

Moment analysis of two station tracer experiments thus allows the Peclet number, uLjD, to be found [63]. For small extents of dispersion, DjuL < approximately 0.01, system RTDs will be essentially symmetrical and of a form given by... 

Silverman, B. D. and Platt, D. E. (1996) Comparative molecular moment analysis (CoMMA) 3D-QSAR without molecular superposition. J. Med. Chem. 39, 2129-2140. 

N=P(Oe) (for a general review see 1969, 26). This implication is in contradiction with indirect conclusions drawn from estimation of Lewis basicity (see Section IV) and from dipole moment analysis/1970,11 A more profound analysis is certainly required in order to elucidate this problem. 

In these equations kei is the elimination rate constant and AUMC is the area under the first moment curve. A treatment of the statistical moment analysis is of course beyond the scope of this chapter and those concepts may not be very intuitive, but AUMC could be thought of, in a simplified way, as a measure of the concentration-time average of the time-concentration profile and AUC as a measure of the concentration average of the profile. Their ratio would yield MRT, a measure of the time average of the profile termed in fact mean residence time. Or, in other words, the time-concentration profile can be considered a statistical distribution curve and the AUC and MRT represent the zero and first moment with the latter being calculated from the ratio of AUMC and AUC. 

The product M(t) W (t) is the integrand in this case and a plot of M(t) W (t) versus t is the moment analysis plot for Mw. Similar plots can be made for the other molecular weight averages. Figure 6 shows moment analysis plots for Mn, Mw and Mz. 

Figure 6 Moment analysis plots for molecular weight averages ...

The precision uncertainty associated with field sampling is generally much larger than that associated with analytical technique, which is roughly 2% to the 67% confidence interval for the two compounds used as conservative and gas tracers. A technique to determine the precision uncertainty associated with field sampling and incorporated into the mean Kl estimate will therefore be propagated with the first-order, second moment analysis (Abernathy et al., 1985) ... 

The final level of complexity is to arrange layers of unidirectional laminae to form the laminate composite, as shown in Figure 5.121. The lower surface of the /th lamina is assigned the coordinates hf relative to the midplane, so the thickness of the /th lamina is /i/ — /i/ 1. The mathematics of stress analysis are an extension of those used in the foregoing description, and they include force and moment analysis on the bending of the laminate. The complete development will not be presented here, but only outlined. The interested reader is referred to two excellent descriptions of the results [9,10]. 

Table 4.2. Empirical induced dipole moments of systems involving molecules m and Is stand for moment analysis and line shape analysis, respectively.

R. W. Hartye, C. G. Gray, J. D. Poll, and M. S. Miller. Moment analysis and quantum effects in collision-induced absorption. Molec. Phys.,... 

Tetrahydro-l,3-oxazines are normally assumed to adopt a chair conformation in which the NH bond has an axial orientation. This view is substantiated by NMR spectroscopy and also by dipole moment analysis <73JCS(P2)325). The ring system is not stable and, for example, when the parent molecule is allowed to stand, it slowly ring opens and then forms a trimer (Scheme 33) (78AF937). In acidic media ring fission is accelerated and the open-chair imines may then hydrolyze. This property has been utilized in the synthesis of aldehydes as previously noted (see Section 2.27.2.2.4(0). 

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