Linear extrapolation mode

The linear approach should be used in two distinct circumstances (1) When there are mode-of-action data to indicate that the dose-response curve is expected to have a linear component below the POD. Agents that are generally considered to be linear in this region include agents that are DNA-reactive and have direct mutagenic activity. (2) As a default when the weight of evidence evaluation of all available data is insufficient to establish the mode of action for a tumor site, because linear extrapolation generally is considered to be a health-protective approach. 

To analyze the density-dependent vibrational lifetime data displayed in Fig. 3, it is necessary to separate the contributions to Ti from intramolecular and intermolecular vibrational relaxation. The intermolecular component of the lifetime arises from the influence of the fluctuating forces produced by the solvent on the CO stretching mode. This contribution is density dependent and is determined by the details of the solute-solvent interactions. The intramolecular relaxation is density independent and occurs even at zero density through the interaction of the state initially prepared by the IR excitation pulse and the other internal modes of the molecule. Figure 5 shows the extrapolation of six density-dependent curves (Fig. 3 three solvents, each at two temperatures) to zero density. The spread in the extrapolations comes from making a linear extrapolation using only the lowest density data, which have the largest error bars. From the extrapolations, the zero density lifetime is —1.1 ns. To improve on this value, measurements were made of the vibrational relaxation at zero solvent density. 

The U.S. EPA applies an alternative dose-response evaluation of carcinogens using a low-dose, linear model (EPA 2005). The linear extrapolation is applied under two circumstances (1) when there are data to indicate that the dose-response curve has a linear component below the point of departure or (2) as a default for a tumor site where the mode of action is not established. For a linear extrapolation, a straight line is drawn from the point of departure to the origin. The slope of the line, known as the slope factor, is an upper-bound estimate of risk per increment of dose that can be used to estimate risk probabilities for different exposure levels. The slope factor is equal to 0.01/LEDoi, for example, if the LEDqi is used as the point of departure. The lower hmit on effective doscoi (LEDoi) is the 95% lower confidence hmit of the dose of a chemical needed to produce an adverse effect in 1% of those exposed to the chemical, relahve to control. If, however, there are sufficient data to ascertain that a chemicaTs mode of action supports modeling at low doses, a reference dose or concentrahon may be developed in lieu of a cancer slope factor. 

Generally, mathematical models are used to extrapolate the data on the exposure- or dose-response relationship derived from carcinogenicity bioassays to estimate the risk at concentrations to which the general population is exposed in the absence of more biologically based kinetic or dynamic models. There are numerous uncertainties in such approaches, which often involve linear extrapolation of results over several orders of magnitude, commonly in the absence of relevant data on mode of action for tumor induction or differences in toxico-kinetics and -dynamics between the relevant experimental animal species and humans. 

Figure 5 shows three data points for the linewidths measured for a single mode PhSi-xSe diode laser (indicated by circles) in addition to the linewidth of the laser illustrated in Figure it (indicated by a square). Figure 5 shows the linewidths plotted as a function of the inverse of the single-ended laser power output. A linear extrapolation of the three data points of the single mode laser to zero inverse power indicated negligible linewidth intercept. Hence, the power-independent... 

The values of below the critical temperature, derived from the areas of the spectra, are associated with the local thermal motions of the individual iron atoms relative to their neighbours. The. total mean square displacements of the iron atoms above 7, derived from the areas of the narrow subspectra, were resolved into three components associated with different modes of thermal motions ., s and fc corresponding to local, slow collective and fast collective modes, respectively. This is shown in Figure 6.17 for metmyoglobin and ferritin. The value of ,o,. is obtained by a linear extrapolation from the values of below the critical temperature. The difference between and ioc gives the mean-square collective displacement associated with large-scale motions of parts of the surrounding protein and is resolved into the two parts and fc. The values of correspond to slow... 

Two 0.4-mm-thick free films from the primary and secondary coating materials were run individually in the expansion mode in the TMA (Fig. 4.37). The in situ Tg of both coatings was taken from the intersection of the linear extrapolations of the expansion behavior below and above each glass transition. Note the dashed lines in Fig. 4.37, which help to delineate the beginning... 

Dose-response assessment today is generally performed in two steps (1) assessment of observed data to derive a dose descriptor as a point of departure and (2) extrapolation to lower dose levels for the mmor type under consideration. The extrapolation is based on extension of a biologically based model (see Section 6.2.1) if supported by substantial data. Otherwise, default approaches that are consistent with current understanding of mode of action of the agent can be applied, including approaches that assume linearity or nonlinearity of the dose-response relationship, or both. The default approach is to extend a straight line to the human exposure doses. 

The second step of the dose-response assessment is an extrapolation to lower dose levels, i.e., below the observable range. The purpose of low-dose extrapolation is to provide as much information as possible about risk in the range of doses below the observed data. The most versatile forms of low-dose extrapolation are dose-response models that characterize risk as a probability over a range of environmental exposure levels. Otherwise, default approaches for extrapolation below the observed data range should take into account considerations about the agent s mode of action at each tumor site. Mode-of-action information can suggest the likely shape of the dose-response curve at these lower doses. Both linear and nonlinear approaches are available. 

The calibration mode selected by the laboratory should also be carefully considered, i.e. standard additions, calibration curve and/or use of bracketing standards. All calibration methods suffer from typical sources of error or drawbacks, e.g. for standard additions non-linearity of the calibration curve, extrapolation difficulties, chemical form of calibrant added, etc. for external calibration (calibration curve) changes of the matrix affecting the linearity of the curve for bracketing standards time-consuming procedures for many routine laboratories, etc. (Quevauviller et al., 1996a Quevauviller, 1998b). 

Although, the substituent effect on the cyclopropenium ring must be interpreted in terms of the force constants for a quantitative comparison, the empirical relationship between pKr+ and j>(C+)E is available to estimate the stability of the tri-substituted cyclopropenium ions. If the PKr+ values i>2.i4) are plotted against the frequencies of the E mode, an approximate linear relationship between them is obtained, as shown in Fig. 3. For example, the pKu,+ of trichlorocyclopropenium ion is estimated as —5, and that of triaminocyclopropenium ion ib can be evaluated as 13 by extrapolation. 

A Stark effect for adsorbed sulfate on Pt electrodes has been reported for the 1200 cm symmetric stretching mode of the adsorbed ion [165]. A quadratic dependence of the band center on the applied electric field is observed (Fig. 60). But this field dependence changes with the degree of coverage. The frequency values extrapolated to zero coverage (singleton frequency) present a linear dependence on the applied electric field (Fig. 61). So we conclude that the second-order Stark effect is induced when the ions are close together on the surface. 

The consideration of mode of action in carcinogen risk assessment is becoming standard practice. When data are adequate to demonstrate use of the standard default low dose extrapolation models such as the linearized multistage model is not appropriate, alternate approaches, including threshold approaches are now being used. 

Twenty-one Raman-active modes are expected in A2 a and 48 Raman-active modes in P2 a. The mode at 455 cm is Raman inactive in the high symmetry phase. From the thermal evolution of its scattering intensity (Fig. 7) we find that the low-temperature values extrapolate to zero at 496 K. At higher temperatures, strong scattering intensity persists and decreases linearly with increasing temperature. The extrapolated scattering intensity vanishes at the y- 3 transition point near 825 K. At T < 496 K the quantitative relationship between the intensity and the thermodynamic order parameter Q for the P - a transition is described by... 

Since Eq. (6.152) is quadratic in Rp, the plot of 1/DP vs. Rp is curved, the extent of which varies with the initiator. However, the initial portion of the plot, corresponding to small Rp values, is linear. By extrapolation of this linear part to i p = 0 yields Cm as the intercept. Moreover, the slope of the linear portion is given by 2zkt / k [m] from which the mode of termination or kp/kt may be determined, provided the other is known. For AIBN initiator C is negligibly small as a result Eq. (6.152) is practically linear in Rp even at higher values of Rp. 

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