Model exponential

Model foim. On the basis of the nature of the data, an exponential model was selected initially to represent the trend if = a + he. In this example, the resultant temperature would approach as an asymptotic (a with c negative) the wet-hnlh temperature of the surrounding atmosphere. Unfortunately, this temperature was not reported. 

Rule 2. Eliminate any variable that was not significant in the exponential model. 

Notice that one event has units of per-demand and the others have a per-unit-time dimension. From elementary considerations, the top event can only have dimensions of per-demand (pure probability) or per-unit-time dimensions. Which dimensions they have depends on the application. If the fault tree provides a nodal probability in an event tree, it must have per-demand dimensions, if the fault tree stands alone, to give a system reliability, it must have per-unit-time dimensions. Per-unit-time dimensions can be converted to probability using the exponential model (Section 2.5.2.6). This is done by multiplying the failure rate and the "mission time" to give the argument of the exponential which if small may be... 

At steady-state condition the oxygen concentration profile would be an exponential model ... 

Figure 4.50. Cumulative dissolution results. Two experimental tablet formulations were tested against each other in a dissolution test in which tablets are immersed in a stirred aqueous medium (number of tablets, constructional details and operation of apparatus, and amount of medium are givens). Eighty or more percent of the drug in either formulation is set free within 10 minutes. The slow terminal release displayed by formulation B could point towards an unwanted drug/excipient interaction. The vertical bars indicate ymean - with Sy 3%. A simple linear/exponential model was used to approximate the data for the strength 2 formulation. Strengths I and 3 are not depicted but look very similar.

Figure 3.1 Time course of implanted tumor volume for one experimental subject (Control) and associated fitted model curves (solid line, exponential model dashed line, nonparametric kernel estimate).

This parameter can be obtained by numerical integration, for example using the trapezium rule, between time 0 and the time T when the last plasma sample has been taken. The remaining tail of the curve (between T and infinity) must be estimated from an exponential model of the slowest descending part of the observed plasma curve ((3-phase) as shown in Fig. 39.15. The area under the curve AUC can thus be decomposed into a tmncated and extrapolated part ... 

Steady-state behavior and lifetime dynamics can be expected to be different because molecular rotors normally exhibit multiexponential decay dynamics, and the quantum yield that determines steady-state intensity reflects the average decay. Vogel and Rettig [73] found decay dynamics of triphenylamine molecular rotors that fitted a double-exponential model and explained the two different decay times by contributions from Stokes diffusion and free volume diffusion where the orientational relaxation rate kOI is determined by two Arrhenius-type terms ... 

For statistically stationary isotropic turbulence, the stretched-exponential model has the form... 

Figure 4.11. Reduced chi-square for fitting a single Gaussian distribution function of decays with either a discrete single or double exponential model as a function of dis tribud on width (/f)...

Fig. 5.9 Splitting of the desorption isotherm of dimefuron in the presence of 0.01 M CaCl2 into two other isotherms, corresponding to the two-compartment (linear, exponential) model of desorption isotherms (Barriuso et al. 1992b)...

Among the common semi-variogram models, the exponential model best fits the sample semi-variograms. Therefore, for REF... 

Pharmacokinetics When administered intravenously, ICG rapidly binds to plasma proteins and is exclusively cleared by the liver, and subsequently secreted into the bile [8]. This forms the basis of the use of ICG for monitoring hepatic blood flow and function. Two pharmacokinetics models, a monoexponential decay, which describes the initial rapid clearance of ICG with a half-life of about 3 minutes (Eq. (1)) and a bi-exponential model, which incorporates the secondary phase clearance with a longer half-life (Eq. (2)), describe total clearance of ICG from plasma [ 132]. For real-time measurements by continuous organ function monitoring, the mono-exponential decay is preferred. 

Data from several large collaborative studies were plotted, with results fitting the exponential model (for constant relapse rate) more accurately than a linear model. The relapse rates for these studies were as follows ... 

For line shape calculations, purely exponential dipole models, and exponential models combined with a dispersion term, Dn/R 1, have been used, see Chapter 4 for details. The variation of the line shape with these dipole models has been studied. In all cases, the purely exponential dipole model gives inferior results when compared with the exp-7 models the influence of the dispersion term, while small, is nevertheless significant in line shape analyses moment analyses, in contrast, have reportedly not been able to demonstrate the significance of the dispersion term. It would seem that the accuracy of quantum line shape calculations of the absorption by rare gas pairs has reached the point where further progress must await more accurate experiments at low gas densities and over a wider range of frequencies and temperatures. 

This correlation corresponds to an exponential decay model, k = koe aY. This expression differs from the conventional exponential model often used in continuous-flow systems 22, 23), k = koe at, in that the analog to time in a pulsed reactor is pulse number or its equivalent, cumulative feed introduced. In our case the correlating quantity is cumulative feed converted, Y. If one assumes that deactivation is caused by coke, the amount of which is proportional to hexane actually converted, this... 

Figure 1. Exponential model for deactivation apparent first-order rate constant vs. cumulative hexane converted.

Table IV lists the values of the two parameters, k0 and a, in the exponential decay model for each sample. Too much credence should not be placed in the exact magnitudes of these values since it is known for an exponential model that the covariance of the two parameters is very high (25). It is clear, nevertheless, that the initial activity/ presumably measured by k0, decreases markedly as aluminum is progressively extracted by acid extraction (samples 2 and 3) but increases as sodium is removed by NH4NO3 exchange (samples 4 and 5).

Kinetic data analyzed using a single-phase exponential model. 

Kinetic data analyzed using a two-phase exponential model. The data only show the results of the analysis of the fast component. The slower component was typically two orders of magnitude or more slower than the fast component (cf. Fig. 3). We have attributed the slow component to receptor misfolding (see text for details). Dissociation produced by GTPyS was always faster and to a greater extent than GDP. 

For example, some microcalorimetric data are represented in Figure 3. The exact process that gave rise to these data is not important, but it shall be assumed that they represent the heat output of a partially completed reaction. These data have been fitted to the exponential model shown above to determine the equation parameters that describe them. Once these values have been determined, it is possible to extend the data to such a time as the observed heat flow (power) equals 0 (shown by the dotted line). The area under the dotted line represents the total heat, Q, which would be generated by the reaction if it went to completion. From this information, it is a simple matter to determine the extent of reaction at any time, t. Hence, some useful information can be derived by using nonspecific models. 

Figure 3 The modeling of microcalorimetric data using a nonspecific exponential model and the extrapolation of the data to power = 0.

Pharmacokinetic models. An important advance in risk assessment for hazardous chemicals has been the application of pharmacokinetic models to interpret dose-response data in rodents and humans (EPA, 1996a Leung and Paustenbach, 1995 NAS/NRC, 1989 Ramsey and Andersen, 1984). Pharmacokinetic models can be divided into two categories compartmental or physiological. A compartmental model attempts to fit data on the concentration of a parent chemical or its metabolite in blood over time to a nonlinear exponential model that is a function of the administered dose of the parent. The model can be rationalized to correspond to different compartments within the body (Gibaldi and Perrier, 1982). 

An experimental semivariogram can be modeled by fitting a simple function to the data points. Linear, spherical, or exponential models are often used [KATEMAN, 1987 AKIN and SIEMES.1988]. 

Figure 17.6 A series of transient absorptions recorded at 1661 cm-1 for tRNAphe. Qualitatively, the data at 1620 cm 1 are the same. The initial temperature was held constant at 47 °C and the magnitude of the T-jump was varied up to a maximum of 20 °C. Each transient was fit to a three exponential model from t = 100 ns to the maximum transient absorption (t = 1 ms).

The exponential model fit for continental carbonates, as shown in Figure 10.46, is not good. These rocks dominate the sedimentary carbonate mass older than 100 million years. One can fit the pre-Cretaceous continental carbonate data with a negative exponential function having a decay constant of 0.0025 ma"1, as done by Wilkinson and Walker (1989). However, based on analysis of the mass-age... 

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