Area under the first 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). 

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 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. 

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). ... 

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... 

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... 

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... 

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 ... 

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 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 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... 

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 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... 

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... 

Stiffness (1) The load per unit area required to elongate the film 1% from the first point in the stress-strain curve where the slope becomes constant. (2) A term relating to the ability of a material to resist bending while under stress. Resistance to the bending is called flexural stiffness, and may be defined as the product of the modulus of elasticity and the moment of inertia of the section. Compare Rigid Plastic. 

Two-dimensional phase diagrams, i.e. r A) curves at various temperatures, are one of the basic means for studying the rich behaviour, whether the layers are gas-like, llquid-llke, solid-like, or mixed, including the structural transitions that monolayers can exhibit. Let us for the moment assume that such ti[A] isotherms have been properly measured, that hysteresis is under control, etc. By way of introduction to the types of phases, emd their succession that may be observed, consider first the schematic picture of fig. 3.6. This way of plotting is phenomenological A is the total area. However, in describing specific examples, it is more useful to use the average area per molecule (oj, in nm ), obtained as A/N°. Here, N must be inferred from the amount of surfactant spread. If so desired, the surface concentration, = n /A (in mole m" ) can also be reported. When there is only one component, the subscript i may be dropped. 

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