Linear-quadratic model

For radionuclides, NCRP reaffirms use of a best estimate (MLE) of the response probability obtained from a linear or linear-quadratic model as derived from data in humans, principally the Japanese atomic-bomb survivors. This model essentially is linear at the low doses of concern to waste classification. Specifically, for purposes of health protection of the public, NCRP reaffirms use of a probability coefficient for fatal cancers (probability per unit effective dose) of 0.05 Sv 1 (ICRP, 1991 NCRP, 1993a). Although this probability coefficient is less rigorous for intakes of some long-lived radionuclides that are tenaciously retained in the body than for other exposure situations, such as external exposure or intakes of short-lived radionuclides (Eckerman et al., 1999), it is adequate for the purpose of generally classifying waste, especially when the lack of data on cancer risks in humans for most chemicals is considered. 

Figure 4.12 Cell survival fractions SF(D) as a function of absorbed radiation dose D in Gy (top panel). The bottom panel is the so-called reactivity R(D) given by product of the reciprocal dose D-1 and the negative natural logarithm of SF(D), as the ordinate versus D as the abscissa. Experiment (symbols) the mean clonogenic surviving fractions SF(D) (top panel) and R(D) = — (1/D) ln(SF) (bottom panel) for the Chinese hamster cells grown in culture and irradiated by 50 kV X-ray [73]. Theories solid curve - PLQ (Pads Linear Quadratic) model and dotted curve - LQ model (the straight line a + /SD on the bottom panel).

The biodistribution and radiotherapy data, together with required data for Lu-DOTATATE and the tumour size, were incorporated into a response model based on the linear quadratic model. This model provides data on interactions among important radiobiological parameters. 

The formulas derived in the time-independent framework can be easily transferred into the corresponding time-dependent solutions. The formulas in the time-independent linear potential model, for example, provide the formulas in the time-dependent quadratic potential model in which the two time-dependent diabatic quadratic potentials are coupled by a constant diabatic coupling [1, 13, 147]. The classically forbidden transitions in the time-independent framework correspond to the diabatically avoided crossing case in the time-dependent framework. One more thing to note is that the nonadiabatic tunneling (NT) type of transition does not show up and only the LZ type appears in the time-dependent problems, since time is unidirectional. 

Frequently, the relationship between biological activity and log P is curved and shows a maximum [ 18]. In that case, quadratic and non-linear Hansch models have been proposed [19]. The parabolic model is defined as ... 

Draper and Smith [1] discuss the application of DW to the analysis of residuals from a calibration their discussion is based on the fundamental work of Durbin, et al in the references listed at the beginning of this chapter. While we cannot reproduce their entire discussion here, at the heart of it is the fact that there are many kinds of serial correlation, including linear, quadratic and higher order. As Draper and Smith show (on p. 64), the linear correlation between the residuals from the calibration data and the predicted values from that calibration model is zero. Therefore if the sample data is ordered according to the analyte values predicted from the calibration model, a statistically significant value of the Durbin-Watson statistic for the residuals in indicative of high-order serial correlation, that is nonlinearity. 

X-axis. It presents the coefficients of the linear models (straight lines) fitted to the several curves of Figure 67-1, the coefficients of the quadratic model, the sum-of-squares of the differences between the fitted points from the two models, and the ratio of the sum-of-squares of the differences to the sum-of-squares of the X-data itself, which, as we said above, is the measure of nonlinearity. Table 67-1 also shows the value of the correlation coefficient between the linear fit and the quadratic fit to the data, and the square of the correlation coefficient. 

We now develop a mathematical statement for model predictive control with a quadratic objective function for each sampling instant k and linear process model in Equation 16.1 ... 

In some cases when estimates of the pure-error mean square are unavailable owing to lack of replicated data, more approximate methods of testing lack of fit may be used. Here, quadratic terms would be added to the models of Eqs. (32) and (33), the complete model would be fitted to the data, and a residual mean square calculated. Assuming this quadratic model will adequately fit the data (lack of fit unimportant), this quadratic residual mean square may be used in Eq. (68) in place of the pure-error mean square. The lack-of-fit mean square in this equation would be the difference between the linear residual mean square [i.e., using Eqs. (32) and (33)] and the quadratic residual mean square. A model should be rejected only if the ratio is very much greater than the F statistic, however, since these two mean squares are no longer independent. 

The methods used were those of Mitchell ( 1 ), Kurtz, Rosenberger, and Tamayo ( 2 ), and Wegscheider T ) Mitchell accounted for heteroscedastic error variance by using weighted least squares regression. Mitchell fitted a curve either to all or part of the calibration range, using either a linear or a quadratic model. Kurtz, et al., achieved constant variance by a... 

The curves generated here are arbitrary because we just randomly picked the temperature coefficients. To accurately model your resistors, you would need to get a data sheet on the resistors you are using and find out if the temperature dependence is linear, quadratic, or exponential, and also find the correct coefficients. The coefficients used here were just for illustration. 

The complete linear models were determined by regression (Table 9). The R2 values for In Y2, T4, In Y5 In Y6 and mainly for Y3 are quite low, indicating that the linear model is inadequate for describing the situation for these variables and that a quadratic model could better fit the data. Nevertheless, Y6 from experiment 12 is a good value, as well as the other responses in this trial and may produce the optimum. 

An even more complex model is possible for this situation. It includes all possible linear, quadratic, and linear-quadratic cross-product terms. 

When the standard curve has been established and the LLOQ and ULOQ validated, the assessment of unknown concentrations by extrapolation is not allowed beyond the validated range. The most accurate and precise estimates of concentration is in the linear portion of the curve even if acceptable quantitative results can be obtained up to the boundary of the curve using a quadratic model. For a linear model, statistic calculations suggest a minimum of six concentrations evenly placed along the entire range assayed in duplicate [5,7,8]. 

The problem with the linear model is that it gives no information about the curvature of fix). This is provided by the more useful local quadratic model... 

To determine the stationary points of the quadratic model we differentiate the model and set the result equal to zero. We obtain a linear set of equations... 

The quadratic model is an improvement on the linear model since it gives information about the curvature of the function and contains a stationary point. However, the model is still unbounded and it is a good approximation to fix) only in some region around xc. The region where we can trust the model to represent fix) adequately is called the trust region. Usually it is impossible to specify this region in detail and for convenience we assume that it has the shape of a hypersphere s <, h where h is the trust... 

To summarize, in the RF approach we make the quadratic model bounded by adding higher-order terms. This introduces n+1 stationary points, which are obtained by diagonalizing the augmented Hessian Eq. (3.22). The figure below shows three RF models with S equal to unity, using the same function and expansion points as for the linear and quadratic models above. Each RF model has one maximum and one minimum in contrast to the SO models that have one stationary point only. The minima lie in the direction of the true minimizer. 

The nominal probability coefficient for radionuclides normally used in radiation protection is derived mainly from maximum likelihood estimates (MLEs) of observed responses in the Japanese atomic-bomb survivors. A linear or linear-quadratic dose-response model, which is linear at low doses, is used universally to extrapolate the observed responses at high doses and dose rates to the low doses of concern in radiation protection. The probability coefficient at low doses also includes a small adjustment that takes into account an assumed decrease in the response per unit dose at low doses and dose rates compared with the observed responses at high doses and dose rates. 

If the time series contains a trend or seasonal fluctuations, these effects may also be smoothed, e.g. the slope of different time ranges is smoothed or the amplitude of periodic fluctuations is smoothed. For model building it is necessary to declare the type of trend (linear, quadratic, or exponential) and the length of the periodicity. 

The non-linear calibration models are also acceptable. Figure 4.6 shows an example of a non-linear calibration approximated with a quadratic fit. Non-linear calibration curves are not acceptable if used to compensate for the detector saturation at a high concentration level. 

All the linear and nonlinear optical properties introduced above are therefore expressed in terms of linear, quadratic and cubic response functions. They can be computed with high efficiency using analytical response theory [9] with a variety of electronic structure models [8],... 

The simplest model that adequately describes the concentration-response relationship should be used, e.g., a linear model is simpler than a quadratic model. At the completion of the validation, evaluation of different regression models must be performed. Justification for using a quadratic regression equation must be documented. 

We have presented various simple scenarios that we are aware of in relevance to the ee amplification of the Soai reaction a quadratic autocatalytic model in a monomer or homodimer system, and a linear autocatalytic model in an antagonistic heterodimer system. All these models can realize ee amplification such that the final value of the ee 0ool depends on, but is larger than, the initial value 0ol> as schematically shown in Fig. 8. The curve in the figure represents 0o - (poo for a given initial ratio qo/c, namely the ratio of the amount qo of the total chiral initiators R and S relative to that of the total reactants c. Amplification is more enhanced if the ratio of the chiral initiator qo/c is smaller. This plot also shows the possibility of increasing the final ee by repeating the reaction. 

The new paradigm of robust optimal control is well on the way to rendering the linear quadratic Gaussian control obsolete (or at least less important) because it can deal explicitly with model error. 

Here, we provide the theoretical basis for incorporating the PE potential in quantum mechanical response theory, including the derivation of the contributions to the linear, quadratic, and cubic response functions. The derivations follow closely the formulation of linear and quadratic response theory within DFT by Salek et al. [17] and cubic response within DFT by Jansik et al. [18] Furthermore, the derived equations show some similarities to other response-based environmental methods, for example, the polarizable continuum model [19, 20] (PCM) or the spherical cavity dielectric... 

Cell surviving fractions after irradiation Pad6-linear quadratic (PLQ) model... 

PADE LINEAR QUADRATIC (PLQ) MODEL for CELL SURVIVAL PROBABILITY... 

An objective judgment of the relative merits of Models 1 and 2 can be made by means of an F test, as discussed in Chapter XXI. This is a statistical test to determine if a decision to include the cT term is justified at the conventional 95 percent confidence level. For the present case, where the degrees of freedom Vj and V2 are 4 and 3 for the linear and quadratic models, respectively, the value is calculated as [see Eq. (XXI-38)]... 

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