Area under 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... 

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,... 

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. 

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. 

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... 

The zero moment in the drug plasma concentration-time curve is the total area under the plasma concentration-time curve from f = 0 to f = , (AUC)q". Estimates of the area under this curve are useful in calculating bioavailability as well as drug clearance, which is the ratio of dose over area under the concentration-time curve for an intravenous dose. 

The zeroth moment Mq is simply the area under the curve, represents the normalization constant, and is related to the oscillator strength of the transition. Because e(v) is effectively a probability distribution, it is clear that the first moment for symmetric spectra gives the average frequency... 

This first moment (or, more strictly speaking, according to Yamaoka et al., the unnormalized first moment) is called the AUMC (area under the [first] moment curve). It is estimated by the trapezoidal approximation of the area under the curve having the product of plasma drug concentration multiplied by time on the ordinate and time on the abscissa. AUMC is rarely used per se in pharmacokinetics. However, the ratio of AUMC/AUC is widely used in non-compartmen-tal pharmacokinetic analysis. This ratio, the MRT, is described in considerable detail below. 

The basis for noncompartmental methods for calculation of the parameters of each step of absorption, distribution, and elimination is the theory of statistical moments. The information required is the drug concentration in the central compartment versus time with concentrations taken past the absorptive phase and distributive phase of the curve. The area under the concentration versus time curve (AUC) is the zero moment. The first moment of the AUC is the area under the curve of the product of the concentration times time versus time... 

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

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

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. 

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