The Power model

Dose proportionality was assessed using the power model. AUC0 24 and Cmax did not change over day of administration, but did increase with increasing dose (p <0.0001 see Fig. 13.3). For AUC0 24, the 90% Cl for the slope related to dose was 0.96, 1.18 with a point estimate of 1.07. The 90% Cl for the slope related to Cmax was 1-02, with a 90% Cl of 0.92, 1.13. Hence, the 90% Cl for both AUC0 24 and Cmax contained the value 1.0 and both parameters were dose-proportional. The between-subject variability for AUC0 24 and Cmax was 32% and 40%, respectively. 

If the scale-up factor didn t have to be that small and models were not so expensive to build, experiments could be run with models of various sizes in water at the same value of Fr and then the results could than be extrapolated to Nex at Re-p In view of the powerful model reduction and the resulting extreme differences in the Reynolds number... 

The simplest model to describe shear thinning is the power model ... 

The viscosity and the shear dependency of the viscosity both increase with growing effective volume fraction of particles. For very low shear rates there is usually a low shear plateau jo and for high shear rates a high shear plateau joo exists. These features are not described by the power model, but a more elaborate model such as the cross model is needed ... 

A new mathematical model was developed to predict TPA behaviors of hydrocarbons in an adsorber system of honeycomb shape. It was incorporated with additional adsorption model of extended Langmuir-Freundlich equation (ELF). LDFA approximation and external mass transfer coefficient proposed by Ullah, et. al. were used. In addition, rate expression of power law model was employed. The parameters used in the power model were obtained directly from the conversion data of hydrocarbons in adsorber systems. To get numerical solutions for the proposed model, orthogonal collocation method and DVODE package were employed. 

The introduction of Valence Bond theory has motivated the search for structural regularities that can be interpreted by models of local electronic features, such as the powerful model of Valence Shell Electron Pair Repulsion [93,94] theory. Alternative approaches, based on Molecular Orbital theory, have led to the discovery of important rules, such as the Woodward-Hoffmann orbital symmetry rules [95] and the frontier orbital approach of Fukui [96,97], As a result of these advances and the spectacular successes of ab initio computations on molecular... 

With the assumed population PK parameters, the PK model, and the exposure-MAE relationship, a clinical trial simulation was performed in S-Plus (Insightful Corporation, Seattle, WA). Two thousand four hundred profiles were simulated for each design and analyzed in S-Plus. Dose proportionality was estimated using the power model (26) and mixed effects modeling in S-Plus. Population PK parameter estimates were obtained using nonlinear mixed effects modeling in S-Plus. 

The similar efficiency of the designs in estimating dose proportionality was not unexpected given the fact that it was an implicit assumption in the simulation study. The power model implicitly assumes that there is a linear relationship between dose and exposure (e.g., AUC). 

Variations in DOC quality (DOC - specific attenuation and absorption) warrant further discussion. These variations can be divided into two related categories changes in specific attenuation correlated with DOC concentrations, and changes in specific attenuation correlated with lake and watershed characteristics. The immediate basis for the power relationships between K io [DOC] (Table 3) comes directly from the power relationship between ad32o and [DOC] (Table 2) for the data of Morris et al. [60]. But why should DOC quality change together with DOC concentration over the scale of lakes in this study What determines the exponent in the power model relating specific absorption and DOC concentration ... 

Lastly, another model referred to as the power model of dose proportionality is to expand Eq. (5.7) to... 

For the power model, the predicted geometric mean at the high dose is given by e9oh01 while for the geometric mean for the low dose is given by e9ol01. Dose proportionality then implies that... 

Power models are linear on a Ln-Ln scale. After Ln-transformation the power model becomes... 

Representative line plots for the power model and after Ln-Ln transformation are shown in Fig. 7.3. Power models are often used because of the allometric relationship between many physiological parameters and body weight. Adolph (1949) proposed that anatomic and physiologic variables were related to body weight by power functions. For example, brain weight is related to total body weight by the model... 

An example of the above consideration is the equivalence of the power model proposed by Colombo et al. (1994) and Clausen s model (1993) for emission controlled by internal diffusion in the source. The assumptions made in the latter - more physically based - model lead to a final description of the emission rate at the surface of the source which is equivalent to the description of the former - purely empirical - model. The equivalence of the models is valid when the parameter C of the empirical model takes the value 1. This can readily be seen if we compare the mathematical equations of the two models ... 

The hypothesis of Bmnauer [63] is based on the Powers model with the assumption that the C-S-H gel consists of the small 10 nm particles, separated with the water layers about 0.5 nm thick. The C-S-H particles are composed of the two or three layers the thickness of one layer is about 1 nm. There are two types of pores in the gel stracture the micropores of the size 0.3-0.4 nm and the laiger pores, not exceeding 2 nm [63]. 

Elvik, R., P. Christensen and A. Amundsen (2004). Speed and road accidents An evaluation of the Power Model. TOI report 740/2004. Institute of Transport Economics, Norway. 

It is well known that speed is a crucial road safety factor. Many implemented safety measures aim to induce road users to reduce their speed and comply with speed limits. With respect to the change in the mean speed, the impacts on road safety in terms of number of accidents and the number of injured and killed people are well known. For example, this relationship is described by the Power model [ELV 04, NIL 04], which is often used to estimate the traffic safety effects of speed changes. However, it is not certain that only the mean speed is affected by particular traffic safety measures measures such as the 85th percentile, standard deviation of speed and shape of the speed distribution can also be affected. 

The changes in accident risks are calculated using the Power model [ELV 04, NIL 04], according to which the relative speed changes affect the number of... 

Table 11.5 shows the estimated change in the nnmber of fatalities, severe and minor injuries. The changes in risk are calcnlated nsing the Power model [ELV 04], and the powers nsed are d= 4.5 for the nnmber of fatal, d = 3.0 for the number of severe and r/ = 1.5 for the number of minor injuries. 

Table 11.5. Change in risk of injured of differing severities with three traffic safety measures speed limit change from 110 to 100 km/h on roads without speed cameras, speed limit change from 90 to 80 km/h on roads with speed cameras and introduction of new speed cameras on roads with a speed limit of 90 km/h. Risk changes calculated using the Power model...

ELY 13] Elvik R., A re-parameterisation of the Power Model of the relationship between the speed of traffic and the number of accidents and accident victims . Accident Analysis and Prevention, vol. 50, pp. 854-860,2013. 

NIL 04] Nilsson G., Traffic safely dimensions and the power model to describe the effect of speed on safety. Bulletin 221, Lund, Sweden Lund Institute of Technology, Department of Technology and Society, Traffic Engineering, 2004. 

Let nb acc(v) be the injury accident count at speed v, and let Vb be the before speed the Power model gives ... 

Exponent a and (5% confidence interval) of the power model Lane close to the hard shoulder Middle lane Median Lane (fast lane)... 

Table 12.1. Calibration of the exponents of the Power model -Marius network, exduding rainy periods and accidents...

Because of the non-linearity of the power model, the decrease in type 2 accidents is not additive , i.e. equal to the sum of decreases obtained when the Power model is applied independently on the two parts (urban and interatban) of the network. A better apphcation would have been to apply the Power model by speed class. 

Elvik, R. The power model of the relationship between speed and road safety Update and new analyses. 2009, Institute of Transport Economics, Norwegian Centre for Transport Research, Oslo, Norway. 

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