Area under the first moment curve AUMC

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). 

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. 

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). ... 

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... 

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... 

AUMC Foo of a CDF Area under the first moment curve... 

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 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. 

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. 

AUMC (area under the [first] moment curve) a synonym for the first statistical moment tf t)dt. It is estimated by the... 

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. 

Finally, the method used to calculate the volume of distribution may be influenced by renal insufficiency. The three most commonly used volume of distribution terms are volume of the central compartment (Ec), volume of the terminal phase (E, E jea). and volume of distribution at steady state (Eis). The central compartment volume is calculated as the intravenous bolus dose divided by the initial plasma concentration. E for many drugs approximates extracellular fluid volume and thus may be increased or decreased by shifts in this physiologic volume. Renal insufficiency, especially oliguric acute renal failure, is often accompanied by fluid overload and a resultant increased Ec due to reduced renal elimination of water and sodium. Uaiea Or E is Calculated as the total body clearance divided by the terminal elimination rate constant (k or /3). This volume term represents the proportionality constant between plasma concentrations in the terminal elimination phase and the amount of drug remaining in the body. E is affected by both distribution characteristics, as well as by the elimination rate constant. The third volume term, the steady-state volume of distribution (Ess), is calculated as (AUMC x dose)/AUC , where AUMC is the area under the first moment of the concentrationtime curve and AUC is the area under the concentration-time curve... 

Vs = dose AUMC/(AUC) where AUMC is area under first moment of plasma concentration-time curve Vs = CVMRT, with MRT the mean residence time Krea = Cl/tcrminal slope... 

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