Moment curve

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). 

Analysis of most (perhaps 65%) pharmacokinetic data from clinical trials starts and stops with noncompartmental analysis (NCA). NCA usually includes calculating the area under the curve (AUC) of concentration versus time, or under the first-moment curve (AUMC, from a graph of concentration multiplied by time versus time). Calculation of AUC and AUMC facilitates simple calculations for some standard pharmacokinetic parameters and collapses measurements made at several sampling times into a single number representing exposure. The approach makes few assumptions, has few parameters, and allows fairly rigorous statistical description of exposure and how it is affected by dose. An exposure response model may be created. With respect to descriptive dimensions these dose-exposure and exposure-response models... 

Truncated area under the moment curve, by numerical integration AUMCo j. 160.7 mg min L ... 

Extrapolated area under the second moment curve ... 

AUMC = area under the first-moment curve for tissue i AUMCP = area under the first-moment curve for plasma AUCP = area under the plasma concentration-time curve... 

The numerator of Eq. (4) is the integral of the derived function t x f(t), usually called the area under the moment curve (AUMC) (7,8). The denominator is the AUC according to Eq. (2). As visualized by the top plot of Figure 2, the mean as center of gravity represents the time value where the profile (when cut from cardboard) would be in perfect balance. 

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 point we wish to bring out here is that the dipole moment curve around 6-7 bohr is very nearly the value expeeted for one eleetronie charge separated by that distanee. As one moves outward, the avoided erossing region is traversed, the state of the moleeule switches over from Li+ —F to Li-F and the dipole falls rapidly. 

In pharmaceutical research and drug development, noncompartmental analysis is normally the first and standard approach used to analyze pharmacokinetic data. The aim is to characterize the disposition of the drug in each individual, based on available concentration-time data. The assessment of pharmacokinetic parameters relies on a minimum set of assumptions, namely that drug elimination occurs exclusively from the sampling compartment, and that the drug follows linear pharmacokinetics that is, drug disposition is characterized by first-order processes (see Chapter 7). Calculations of pharmacokinetic parameters with this approach are usually based on statistical moments, namely the area under the concentration-time profile (area under the zero moment curve, AUC) and the area under the first moment curve (AUMC), as well as the terminal elimination rate constant (Xz) for extrapolation of AUC and AUMC beyond the measured data. Other pharmacokinetic parameters such as half-life (t1/2), clearance (CL), and volume of distribution (V) can then be derived. 

Traditionally, linear pharmacokinetic analysis has used the n-compartment mammillary model to define drug disposition as a sum of exponentials, with the number of compartments being elucidated by the number of exponential terms. More recently, noncompartmental analysis has eliminated the need for defining the rate constants for these exponential terms (except for the terminal rate constant, Xz, in instances when extrapolation is necessary), allowing the determination of clearance (CL) and volume of distribution at steady-state (Vss) based on geometrically estimated Area Under the Curves (AUCs) and Area Under the Moment Curves (AUMCs). Numerous papers and texts have discussed the values and limitations of each method of analysis, with most concluding the choice of method resides in the richness of the data set. 

Fig. 33. Curve A heat of adsorption of cesium on tungsten, curve B theoretical curve according to Eq. (60) with a constant dipole moment, curve C theoretical curve according to Eq. (60) with a variable dipole moment, but a constant polarizability, a = 10 X 10 curve D is curve C fitted on A at 8 = 0.3.

Singlet-triplet transition moment curves in N2. All b,c,d, results correspond to valence CAS (CAS-1) calculations. 

The Vegard-Kaplan transitions A3E J - A1E+ have been observed both in emission and in absorption [88], and is the most studied T-S system in N2. Shemansky [86] measured the absorption spectrum, identifying seven vibrational bands (6,0)-(12,0) and extracted from the obtained data an absolute transition moment curve in the interval 1.08-1.4 A. The transition moment curve was found to be quite close to linear in the important interval 1.08-1.2 A. The important feature of the curve is that it changes sign at r = 1.173 A, i.e at the vicinity of re 1.1 A. 

In Fig. 7 we recapitulate the spin-averaged Einstein coefficients for the Vegard-Kaplan emission from the lowest vibrational state of the triplet as well as the corresponding values reported by Piper [89]. The relative transition probabilities for different vibronic phosphorescence bands are quite good [26]. The absolute and the relative intensities of the higher vibrations v" are very sensitive to the transition moment curve... 

Spin averaged Einstein coefficients and oscillator strengths for the Ogawa-Tanaka-Wilkinson band B 3E ,Q(v ) <—> X E+(v") for CAS-1 transition moment curve averaged over potentials from Lofthus and Krupenie [88]... 

FIGURE 7.4 Wavepacket absorption spectra of HCl, using (A) a transition dipole moment curve (max absorption at approximately 65300 cm ) and (B) a constant transition dipole moment (max absorption at approximately 63500 cm ). 

The analysis of the dipole moment curves for the motion of the adsorbate perpendicular to the surface provides additional information about the degree of ionicity of a given surface chemical bond. Moreover, the analysis of the dipole moment curves is also related to the interpretation of variations of the surface work function induced by the presence of the adsorbate. However, the response of the surface to the presence of the adsorbate does not permit to extract directly adsorbate charges from the dipole moment curve. A procedure based in the use of frozen densities has been proposed that permits to avoid the effect of the surface polarization on the dipole moment curve. Unfortunately, this method has not been yet extensively used. To close this short discussion about the different procedures commonly used to interpret the chemical bond between chemical species and the surface of a catalyst in terms of net charges we mention the valence bond reading of Hartree-Fock and Configuration Interaction wave functions. This procedure has been used to interpret the electronic correlation effect on the surface chemical bond. ... 

In these equations, the first and second moments, Sq and Sj, are also defined, respectively, as ALfC, area under the curve," and ALJMC, "area under the first moment curve." AUC was introduced in the discussion of bioavailability in Chapter 4, and it and AUMC are the more common expressions in pharmacokinetics and will be used in the following discussions. The second moment, S2, is rarely used and will not be discussed in this chapter. 

Purves RD. Optimum numerical integration methods for estimation of area-under-the-curve and area-under-the-moment-curve. J Pharmacokinet Biopharm 1992 20 211-26. 

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). ... 

AUMC Area under the moment curve IT Median time for death of n% of a... 

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