Area under curve measurements

In the total plasma response approach, the bioavailability of a compound is determined by measuring its plasma concentration at different times (up to weeks) after single or long-term ingestion of the compound from supplements or food sources. Generally, a plasma concentration-versus-time plot is generated, from which is determined the area-under-curve (AUC) value used as an indicator of the absorption of the componnd. Here, the term relative bioavailability is more appropriate since AUC valnes of two or more treatments are usually compared. This is in contrast to absolnte bioavailability for which the AUC value of the orally administered componnd is compared to that obtained with intravenous administration taken as a reference (100% absorption). 

Figure 7. Comparison of (a, solid) electrochemical and (b, dashed) UHV measurements of the H, coverage/potentiaI differential versus potential on Pt(lll).1.) cathodic sweep (25 mV/s) voltammogram in 0.3 M HF from Ref. 20, constant double layer capacity subtracted, b.) dB/d(A ) versus A plot derived from A versus B plot of Ref. 26. Potential scales aligned at zero coverage. Areas under curves correspond to a.) 0.67 and b.) 0.73 M per surface Pt atom.

From day 6 onwards, the slope of the curve corresponds to the effective half-life of 131I on herbage, namely 5 d. Assuming that this continues indefinitely, the area under curve A in Fig. 3.6 is 1.4 m2 d l-1. This is equivalent to the transfer factor km, defined by equation (2.12). Values of Fm for 131I and 137Cs are about the same, but the radioactive decay of 131I reduces km compared with that for137Cs (Table 2.19). Also shown in Fig. 3.6 are values of C/ as deduced from measurements near... 

Fig. 4. Various ways of quantitation of TAC in inhibition assays measurement of induction time, absorbance (fluorescence) after fixed time, and area the kinetic curve of time course of changes in absorbance or fluorescence. Dashed line, reference solid line, sample measured. Differences between the areas under curves for sample and reference (protection area) indicated only.

Another group of methods relies on the destruction of an indicator by an oxidant, leading to a decrease of its characteristic absorbance or fluorescence. Usually this decrease is not linear with time, so the method of choice for the analysis of data is the either the lag time measurement or the area-under-curve approach, that is, comparing the areas under curves of absorbance or fluorescence versus time for a reference sample and a sample studied. Antioxidants protecting the oxidizable substrate increase the area under the kinetic curve (Fig. 4). An absorptiometric method used the bleaching of the carotenoid crocin induced by ABAP (L23, T7). However, crocin is rather expensive and the recommended method of its preparation (by extraction of saffron) may lead to extracts of different composition and properties varying from batch to batch. 

Area under Curve—It is obvious that the integral of either Eq (23-7) or (23-8) gives the area under the curve. Since all eventualities are included within the abscissa and the curve, the area must be equal to unity. Similarly the integral to any value of the abscissa yields the frequency of particles up to that diameter (if we remain concerned with particle measurement). 

The concept of curvature was developed by Isaac Newton in the middle of the 17th century, as a natural extension to his work on the calculus. At that time, the determination of the perimeter of planar curves and the area under curves were major problems. In particular, Newton s new analytical tools allowed him to determine the "quadrature" (area) of a circle. It occurred to Newton that the radius of the circle of best fit to an arbitrary planar curve at all points on the curve was a useful measure, for which he coined the term "crookedness"[2]. This is curvature (Fig. 1.1). 

Ae with degradation is a useful measure of the accumulation of polar groups in the polymer. However, there appears to be considerable overlapping of multiple peaks, and in the early stages of the reaction, Ac, which Is determined from areas under curves in the e" vs 1/T plots is not readily accessible. The relaxation peaks are fairly S3immetrlcal and e" values at the frequency of maximum loss have been used as alternative data to follow reactions. 

The calibration factor can therefore be calculated from the quotient of the exchanged heat and the area under the measured curve. The reliability of a calorimeter depends essentially on the repeatability of this factor in case of a variation of other experimental parameters such as type and amount of sample and the nature and pressure of the gas used. 

From Table 2.26b the area under the normal curve from — 1.5cr to -I- 1.5cr is 0.866, meaning that 86.6% of the measurements will fall within the range 30.00 0.45 and 13.4% will lie outside this range. Half of these measurements, 6.7%, will be less than 29.55 and a similar percentage will exceed 30.45. In actuality the uncertainty in z is about 1 in 15 therefore, the value of z could lie between 1.4 and 1.6 the corresponding areas under the curve could lie between 84% and 89%. 

Note that /4 = 0 when capillary condensation is complete.) Integration by measurement of the area under the curve of ln(p°/p) against n between the stated limits therefore gives the value of A, which is the area of the walls of the cores, not of the pores (cf. Fig. 3.28). 

For any one ion type (e.g., Cs ), measurement of its abundance in a sample requires the sample to be evaporated over a period of time. The total yield of ions is obtained by integrating the area under the ion-yield curve (Figure 7.8c). 

The abihty of a fiber to absorb energy during straining is measured by the area under the stress—strain curve. Within the proportional limit, ie, the linear region, this property is defined as toughness or work of mpture. For acetate and triacetate the work of mpture is essentially the same at 0.022 N/tex (0.25 gf/den). This is higher than for cotton (0.010 N/tex = 0.113 gf/den), similar to rayon and wool, but less than for nylon (0.076 N/tex = 0.86 gf/den) and silk (0.072 N/tex = 0.81 gf/den) (3). 

A fiber that is strained and allowed to recover releases a portion of the work absorbed during straining. The ratio of the work recovered to the total work absorbed, measured by the respective areas under the stress—strain and stress—recovery curves, is designated as resiUence. 

The median particle diameter is the diameter which divides half of the measured quantity (mass, surface area, number), or divides the area under a frequency curve ia half The median for any distribution takes a different value depending on the measured quantity. The median, a useful measure of central tendency, can be easily estimated, especially when the data are presented ia cumulative form. In this case the median is the diameter corresponding to the fiftieth percentile of the distribution. 

Normalization is a preprocessing method often appHed to spectral data. It makes the lengths of all of the data vectors the same. Thus the sum of the squares of the elements of the data vectors is constant for all samples in the set. If is this sum for the unnormalized sample /, then to normalize the data vectors to the constant m, each element of the data vector would be multiphed by vnj.yj. A common example of this method is normalizing the area under a set of curves to unit area. AppHcation of this method effectively removes the variance in a data set because of arbitrary differences in magnitudes of a set of measurements when such variation is not meaningful and would obscure the significant variance. 

For measurement data, probability is defined by the area under the curve between specified limits. A density function always must have a total area of 1. 

Figure 9.3. Stress-strain curves for (a) rigid amorphous plastics material showing brittle fracture and (b) rubbery polymer. The area under the curve gives a measure of the energy required to break the...

Total area under tracer concentration (or a quantity proportional to it) curve versus time as measured at the outlet... 

Due to its nature, random error cannot be eliminated by calibration. Hence, the only way to deal with it is to assess its probable value and present this measurement inaccuracy with the measurement result. This requires a basic statistical manipulation of the normal distribution, as the random error is normally close to the normal distribution. Figure 12.10 shows a frequency histogram of a repeated measurement and the normal distribution f(x) based on the sample mean and variance. The total area under the curve represents the probability of all possible measured results and thus has the value of unity. 

As illustrated in Figure 44.42, a resonance peak represents a large amount of energy. This energy is the result of both the amplitude of the peak and the broad area under the peak. This combination of high peak amplitude and broad-based energy content is typical of most resonance problems. The damping system associated with a resonance frequency is indicated by the sharpness or width of the response curve, ci) , when measured at the half-power point. i MAX is the maximum resonance and Rmax/V is the half-power point for a typical resonance-response curve. 

The tensile strengths are about 55 MN/m, the elongations at break usually less than 10% and the modulus of elasticity about 2-7 GN/m Since the area under the curve provides a measure of the energy required to break the bonds, and since this area is small such polymers will have a low impact strength (which is closely related to energy to break) and will break with a brittle fracture. 

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