Moment theory

It can readily be seen from this example that the contributions of the extrapolated areas to the total areas are relatively more important for the higher order moments. In this example, the contributions are 28, 61 and 72% for AUC, AUMC and AUSC, respectively. Because of this effect, the applicability of the statistical moment theory is somewhat limited by the precision with which plasma concentrations can be observed. The method also requires a careful design of the sampling process, such that both the peak and the downslope of the curve are sufficiently covered. 

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. 

P.R. Mayer and R.K. Brazell, Application of statistical moment theory to pharmacokinetics. J. Clin. Pharmacology, 28 (1988) 481-483. 

Riegelman S, Collier P. The application of statistical moment theory to the evaluation of in vivo dissolution time and absorption time. J Pharmacokinet Biopharm 1980 8 509-534. 

Unfortunately, most of the correlation efforts to date with IR dosage forms have been based on the correlation Level C approach, although there also have been some efforts employing statistical moment theory (Level B). Level A correlation approach is often difficult with IR dosage forms because of the need to sample intensively in the absorptive region of the in vivo study. Thus, Levels B and C are the most practical approaches for IR dosage forms, even though they are not as information-rich and therefore more limited in their application. 

Non-compartmental analysis uses techniques derived from statistical moment theory to... 

To give perspective to /( S ) when S is defined for the continuous p(x) considered before, we mention the following facts, provable by the elementary principles of convexity, calculus of variations, and moment theory. 

Another example from the chemistry of hydrazones (45) represents a complex problem132. The group moment of the heterocycle was taken from theoretical calculations conformation on (0)C—O is evident. One has to determine conformations on N—N and (N)C—O bonds, but the most important problem is the configuration. The final result was a form near to 45 with a torsional angle previous examples, we preferred to express the conformations at this position as an equilibrium of two planar forms or a formally free rotation all these interpretations are indistinguishable within the framework of dipole moment theory. In any case, this example is at the limits of possibilities of this approach. 

Kakutani, T., Yamaoka, K., Hashida, M. and Sezaki, H. (1985) A new method for assessment of drug disposition in muscle application of statistical moment theory to local perfusion systems. J. Pharmacokin. Biopharm., 13, 609-631. 

P.W. Langhoff, Stieltjes-Tchebycheff moment-theory approach to molecular photoionization studies, in T. Rescigno, V. McKoy, B. Schneider (Eds.), Electron-Molecule and Photon-Molecule Collisions, Plenum, New York, 1979 A.U. Hazi, Stieltjes-moment-theory technique for calculating resonance widths, in T. Rescigno, V. McKoy, B. Schneider (Eds.), Electron-Molecule and Photon-Molecule Collisions, Plenum, New York, 1979. 

F. Miiller-Plathe, G.H.F. Diercksen, Molecular photoionization cross sections by moment theory. An introduction, in S. Canuto, J. D Albuquerque e Castro, F.J. Paixao (Eds.), Electronic Structure of Atoms, Molecules and Solids, World Scientific, Singapore, 1990 F. Miiller-Plathe, G.H.F. Diercksen, Perturbative-polarization-propagator study of the photoionization cross section of the water molecule, Phys. Rev. A 40 (1989) 696. 

K. Gokhberg, V. Vysotskiy, L.S. Cederbaum, L. Storchi, F. Tarantelli, V. Averbukh, Molecular photoionization cross sections by Stieltjes-Chebyshev moment theory applied to Lanczos pseudospectra, J. Chem. Phys. 130 (2009) 064104. 

Keywords Ball mill Industrial-scale Moment theory Grinding... 

Intramolecular interactions were introduced for the first time by van der Waals in 1873 he thus attempted to explain the deviation of the real gas from the ideal gas. In order to apply the ideal gas law equation to the behavior of real gases, allowance should be made for the attractive and repulsive forces occurring between molecules. From that time on, the dipole moment theory of Debye (1912) and the dispersion energy or induced dipole theory by London (1930) were the main driving forces of the research about intermolecular interactions. 

The central idea of van Vleck has also been appUed to the field of nuclear quadrupole resonances, still in the solid state. We would also like to mention the well-known work of Gordon,who studied molecular spectra in terms of moments. His investigation provides examples of application of the moment theory in the gas phase. ... 

The Lanczos Method. The use of classical moment theory involves large determinants, thereby implying delicate problems of numerical stability. This instability arises from very severe cancellations in the determinants, which in turn come from the properties of the moments themselves. Whitehead and Watt bypassed this problem by using the Lanczos al-... 

The overall rate constant for the reaction can often be handled (interpolated, extrapolated) over a limited range of temperatures by the empirical methods outlined earlier. However, also as indicated earlier, a great difficulty presented by these reactions is lack of information on the products. Only experimental studies can establish with confidence the product channels and the ways in which branching ratios vary with temperature and pressure, but experiments to quantitatively characterize product channels are usually extremely exacting and, for the moment, theory often offers the only, very limited guidance. For this type of reaction, progress... 

An important limitation of compartment analysis is that it cannot be applied universally to any drug. A simpler approach that is useful in the case of bioequivalency testing is the model independent method. It is based on statistical-moment theory. This approach uses the mean residence time (MRT) as a measure of a statistical half-life of the drug in the body. The MRT can be calculated by dividing the area under the first-moment curve (AUMC) by the area under the plasma curve (AUC). ... 

In Figure 4 our calculated cross sections for the 3cTg level are compared with the results of the Stleltjes moment theory (STHT) approach (25) and of the continuum multiple scattering model (CMSM)... 

Intravenous Drug Disposition. The estimation of primary pharmacokinetic parameters using noncompartmental analysis is based on statistical moment theory [45, 46]. The relationships dehned by this theory are valid under the assumption that the system is linear and time-invariant. For simplicity, we further assume that drug is irreversibly removed only from a single accessible pool (e.g., plasma space). Regardless of the route of administration, the temporal profile of plasma drug concentrations, Cp(t), can represent a statistical distribution curve. As such, the zeroth and first statistical moments (Mo and Mi) are defined as ... 

Statistical moment analysis is a noncompartmental method, based on statistical moment theory, for calculation of the absorption, distribution, and elimination parameters of a drug. This approach to estimating pharmacokinetic parameters has gained considerable attention in recent years. 

The statisticai moment theory offers an attractive alternative for the evaluation and estimation of the absorption data, and even in the absence of intravenous data, this method permits the ranking of several dosage forms, with respect to drug release and absorption, from the available MRT values. 

In recent years, non-compartmental or model-independent approaches to pharmacokinetic data analysis have been increasingly utilized since this approach permits the analysis of data without the use of a specific compartment model. Consequently, sophisticated, and often complex, computational methods are not required. The statistical or non-compartmental concept was first reported by Yamaoka in a general manner and by Cutler with specific application to mean absorption time. Riegelman and Collier reviewed and clarified these concepts and applied statistical moment theory to the evaluation of in vivo absorption time. This concept has many additional significant applications in pharmacokinetic calculations. 

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