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). 

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 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... 

The first moment and synonyms is the location at which the curve, if cut out, would balance on a knife-edge. 

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. 

Both Ksec <1 pore size distribution have been measured experimentcilly for hard-sphere column packing materials (9), but for soft gel packing materials there does not seem to be ciny reliable information presumably because the accepted method of pore structure characterisation in porous materials, mercury porosime-try, cannot be used. However, Ep Ccin be measured for gels without great difficulty from the column calibration curve (as is mcinife-st from Equation 12) provided the calibration is made on the basis of the peak mean position, i.e. the first moment of the peak... 

Test for the RTD curve. Proper RTD curves must satisfy the material balance checks (calculated zero and first moments should agree with measured values)... 

Figure 3.42 shows the measurements (dots, etc.), at various temperatures of the spectral moment yo of H2-H2 pairs for the fundamental band, v = 0 —> v = 1, at woi = 4161.1 cm-1. Also shown are computations of that quantity from first principles (curve) [281]. The agreement is well within the experimental uncertainties. Figures 3.43 and 3.44 show similarly the spectral moments of H2-He pairs. Again, measurement and... 

The second method is an indirect method, based on the liquid s average residence time evaluated with the tracer injection technique. From the first moment of the RTD curve the total external liquid hold-up can be calculated. 

As a concrete example, we now consider phase separation from parent distributions of the form p o) oc exp(ycr) (for — 1 < a < 1, otherwise zero). The shape parameter y is then a fixed function of the parental first moment density p, = J do op(° o). Figure 9 shows the exact coexistence curve for pf = 0.2, along with the predictions from our moment free energy with n moment densities (pf — J do oip(o), i = 1... n) retained. Comparable results are found for other p, . Even for the minimal set of moment densities (n = 1)... 

Most studies have assumed equation (3) to apply, so that equation (1) takes the form of Fick s law, with the composite (effective) diffusion taking account of both bulk and Knudsen diffusion. For the stealy state operation of the Wicke-Kallenbach cell, this can often be a reasonable assumption. Smith et al (18) have also used this description of the transport processes to analyze the situation when a pulse of the trace component is applied at z=0 and the concentration is monitored at z=L. For sufficiently high flow rates of the carrier gas, the first moment of the response curve to a pulse input is ... 

In another version of a dynamic experiment proposed first by Gibilaro et al. [21] an impulse of the tracer is introduced into the carrier gas flowing through the first chamber, and the response CA(L,t) in the second chamber is recorded. A suitable analysis is based on the normalized first moment of the response curve, which is defmed as... 

Models with varying degrees of complexity have been employed to analyze the experimental results by a variety of techniques. The most comprehensive models include terms to account for axial dispersion in the packed bed, external mass transfer, intraparticle diffusion in both macropore and micropore regions of the pellet and a finite rate of adsorption. Of the several methods of analysis, the most popular ones are based on the moments of the response curve. The first moment of the chromatogram is defined by Equation 5.25 in which the concentration now is taken at the outlet of the column. The second central moment is calculated from equation... 

For a short pulse input of a nonadsorbable tracer and a catalyst characterized by a unidisperse pore structure, the following relationships are obtained for the first moment and second central moment of the effluent curve of the bed [24] ... 

Here, C (t) is the concentration of the tracer at time t on the RTD curve at position i, U is the real mean axial velocity of the phase being considered, L is the distance between two measuring points, fit is the first moment and zrfjs the second moment of the RTD curve. Since the first moment of the response curve is essentially the mean of that curve, the average residence time of the tracer can be calculated by taking the difference of the first moment of the response curves... 

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