Area under the moment curve

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

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. 

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. 

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. 

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

Generally, comparisons of volume of distribution are made by use of a parameter designated as the volume of distribution at steady state (Vss), which reflects the sum of the volumes of all the pools into which a drug may distribute. Vgg can be calculated from the area under the moment curve (AUMC) and the area under the curve (AUC) as defined by Benet and Galeazzi (11) ... 

The zeroth moment (k = 0) is simply the area under the distribution curve ... 

The first is used for the investigation of EPR signals of VO " -containing bulk phases under reaction conditions (i.e., at elevated temperatures and in the presence of reactants). It is based on calculation of the second and the fourth moment of the EPR absorption signals by using Eq. (3) (24) with n — 2 and n — 4, respectively, where A is the area under the absorption curve, Bj and yj are the resonance field value and amplitude at the /th point of the spectrum, and Bq the resonance field value at the center of the absorption line ... 

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). ... 

The first moment of the plasma concentration-time profile is the total area under the concentrationtime curve resulting from plot of the product of plasma concentration and time (i.e., Cpf) versus time,... 

The above results are plotted in Fig. 15 individually for an applied lateral force and a moment. The trigonometric terms in Eq. 21 suggest that the solution will alternate between tensile and compressive stresses. The exponential decay is so rapid, however, that oscillations beyond the first tension and compression zones are barely evident in graphs of the stresses. For the case of the applied load, the integral of the stresses (over the area) within the adhesive must equal the applied load. For the case of the applied moment, the area under the stress curve must equal zero, and the first moment of the area must equate to the applied couple. The compressive and tensile zones counteract one another so that no net force is present, although they do constitute a couple. Although the areas under these respective portions of the curve are equal, the peak of the region at the end of... 

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... 

The variance of the residence times VRT is derived from the area under the second moment of the plasma concentration curve AUSC ... 

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... 

Experimentally, VDSS is determined by calculating the area under the first moment of the plasma versus time curve (AUMC), which when combined with AUC will yield the mean residence time. 

A better data analysis method, as yet unused, might be to use the area under the step response curve. It can be shown that the moments are related to the step response by... 

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. 

In the framework of the impact approximation of pressure broadening, the shape of an ordinary, allowed line is a Lorentzian. At low gas densities the profile would be sharp. With increasing pressure, the peak decreases linearly with density and the Lorentzian broadens in such a way that the area under the curve remains constant. This is more or less what we see in Fig. 3.36 at low enough density. Above a certain density, the l i(0) line shows an anomalous dispersion shape and finally turns upside down. The asymmetry of the profile increases with increasing density [258, 264, 345]. Besides the Ri(j) lines, we see of course also a purely collision-induced background, which arises from the other induced dipole components which do not interfere with the allowed lines its intensity varies as density squared in the low-density limit. In the Qi(j) lines, the intercollisional dip of absorption is clearly seen at low densities, it may be thought to arise from three-body collisional processes. The spectral moments and the integrated absorption coefficient thus show terms of a linear, quadratic and cubic density dependence,... 

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. 

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. 

First, the area under the curve is determined by the zero moment MOq. In this way the curve can be normalized if necessary. 

Every statistical distribution can be described by its moments. If the distribution is defined by a polynomial expansion, then the coefficients of the polynomial are related to the moments. The peak-like form of the concentration profile suggests that we can define it by its moments. The zeroth moment measures the area under the curve, the first moment gives the mean residence angle of the solute sample and the second moment gives the variance of die peak. The higher moments gives the skewness and flatness. If concentration is denoted by C 9,z) then the moments about the origin of 9 are defined by... 

Stage 1 Pharmacokinetic (PK) data from the healthy subject studies (studies 1 and 2) were analyzed using the statistical moments analysis approach. From the results of the analysis, peak concentration (Cmax) and area under the plasma concentration curve (AUC) were selected for exploring the relationship between exposure and safety data (biomarker elevation). 

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