The Standard Linear Model

We now explore these relationships for the case of the standard linear solid. A unique Poisson s ratio v will be assumed. We have, from (3.1.15), (3.2.12) and (1.6.2P), 

Our task now is to evaluate the summations occurring in (3.11.5) and (3.11.8) for the standard linear model. Consider first the contracting phase. From (2.4.11) and (3.11.9), we have 

The first term in the exponent of Edt) is positive. However, this is always dominated by the second term, so that j.(t) l. Therefore, from (3.11.5) 

Based on these expressions, we write (3.11.4) and (3.11.7) more explicitly. Take t tQ. Then (3.11.7) may be cast in the form 

In this last equation, parameters introduced at the beginning of Sect. 1.6 are used. These must be related to the quantities in (3.11.10) in a manner analogous to (1.6.25, 28-30) see (3.6.15). Relations (1.6.10- 12) or (1.6.2p) are also relevant. Finally, 

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Since the smaller is reciprocal of the variance, the larger is the variance, and vice versa, the consequence is that trials for which the treatment estimate has a smaller variance should be given more weight. Suppose we have a series of perfectly balanced trials, so that for trial i we have rij patients in the experimental and fij in the control group. If we perform a meta-analysis of continuous outcomes using original data and the standard linear model, then we conventionally assume that variances within strata are equal. Where that is so, the variance of the Ith treatment estimate is... 

Consider the following nonlinear models. Linearise them so that linear regression methods can be applied. Explain in which cases the error structure will be that of the standard, linear model (additive error) and where it will not be. 

Problem 7.3.1 Demonstrate the above results explicitly for the standard linear model, discussed in Sect. 1.6. Show in particular that... 

While the standard linear model does not precisely describe creep or stress relaxation behavior because of the assumption of a single relaxation time, the above arguments still ly to actual polymer behavior, where Dc (t) < Dr ) . Thus, for constant load applications, the creep compliance or its inverse, the so-called effective creep modulus should be used, whereas for constant displacement (e.g., a plastic nut and bolt), the relaxation modulus should be used. 

Another model consisting of elements in series and parallel is that attributed to Zener. It is known as the Standard Linear Solid and is illustrated in Fig. 2.41. The governing equation may be derived as follows. 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

Here the term ik is the retardation time. It is given by the product of the compliance of the spring and the viscosity of the dashpot. If we examine this function we see that as t -> 0 the compliance tends to zero and hence the elastic modulus tends to infinity. Whilst it is philosophically possible to simulate a material with an infinite elastic modulus, for most situations it is not a realistic model. We must conclude that we need an additional term in a single Kelvin model to represent a typical material. We can achieve this by connecting an additional spring in series to our model with a compliance Jg. This is known from the polymer literature as the standard linear solid and Jg is the glassy compliance ... 

Recently, Gasteiger et al. [59] reported several models to predict human oral bioavailability using Hou and Wang s data set. A set of ADRIANA.Code and Cerius2 descriptors were calculated, and MLR analysis was performed. The best linear model had r2 of 0.18 and RMSD of 31.15. When a set of subsets was cherry-picked so that each subset had either a common functional group or a similar pharmacological activity, the r2 values were improved and RMSD values dropped. But the performance of those models was still not satisfactory the standard errors were above 20.0 and r2 was lower than 0.6. 

In contrast to noncompartmental analysis, in compartmental analysis a decision on the number of compartments must be made. For mAbs, the standard compartment model is illustrated in Fig. 3.11. It comprises two compartments, the central and peripheral compartment, with volumes VI and V2, respectively. Both compartments exchange antibody molecules with specific first-order rate constants. The input into (if IV infusion) and elimination from the central compartment are zero-order and first-order processes, respectively. Hence, this disposition model characterizes linear pharmacokinetics. For each compartment a differential equation describing the change in antibody amount per time can be established. For... 

The Standard Linear Solid Model combines the Maxwell Model and a like Hook spring in parallel. A viscous material is modeled as a spring and a dashpot in series with each other, both of which other, both of which are in parallel with a lone spring. For this model, the governing constitutive relation is ... 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

The range of application of the integral equation method is not limited to the standard dielectric model. It encompasses the cases of anisotropic dielectrics [8] (liquid crystals), weak ionic solutions [8], metallic surfaces (see ref. [28] and references cited therein),. .. However, it is required that the electrostatic equation outside the cavity is linear, with constant coefficients. For instance, liquid crystals and weak ionic solutions can be modelled by the electrostatic equations... 

The B score (Brideau et al., 2003) is a robust analog of the Z score after median polish it is more resistant to outliers and also more robust to row- and column-position related systematic errors (Table 14.1). The iterative median polish procedure followed by a smoothing algorithm over nearby plates is used to compute estimates for row and column (in addition to plate) effects that are subtracted from the measured value and then divided by the median absolute deviation (MAD) of the corrected measures to robustly standardize for the plate-to-plate variability of random noise. A similar approach uses a robust linear model to obtain robust estimates of row and column effects. After adjustment, the corrected measures are standardized by the scale estimate of the robust linear model fit to generate a Z statistic referred to as the R score (Wu, Liu, and Sui, 2008). In a related approach to detect and eliminate systematic position-dependent errors, the distribution of Z score-normalized data for each well position over a screening run or subset is fitted to a statistical model as a function of the plate the resulting trend is used to correct the data (Makarenkov et al., 2007). 

The mathematical model used to describe our calibration curve was validated by the Pennincky et al. [4] method. At this stage we proved the good fitting properties of the unweighted linear model to our calibration curve. With this treatment we aimed not only at the accuracy but also at the estimation of more realistic sample signal interpolation uncertainties. These uncertainties were obtained by the application of an ISO international standard [5],... 

The quantities p and t are called the model parameters, sometimes referred to as the independent variables. There is one additional parameter in this model which is the standard deviation of the random error 8. It is not explicitly evident from Equation (1) above but is implicit in assumption (a). The model parameters are unknown quantities that must be estimated from the data. The data here are represented by the symbol Y, sometimes referred to as the dependent variable. The relationship between the data and the model parameters is expressed by the linear equation (1), hence the name Linear Model. 

An example of how the model works is shown in Figure 252, in which a typical commercial activation is represented. 273 kg of Cr/silica was charged to an activator. The catalyst was heated to 800 °C at the standard linear ramp rate of 1.4 °C min 1 and an air velocity of 6.4 cm s Then the temperature was held at 800 °C for 12 h. To obtain a predicted conversion, the activation profile (temperature vs. time) was first plotted (Figure 252) and the concentration of water vapor in the gas stream was calculated from the temperatures shown in the plot and a library of laboratory TGA curves that indicate how much water is evolved at each temperature and heat-up rate. The conversion of the chromium to Cr(VI) was calculated at each temperature from the calculated concentration of water vapor by use of the stability curves shown in Figure 251. The Cr(VI) content was found to be high when the temperature reached 500 °C, but it dropped quickly as the temperature was raised, reaching only 0.37% Cr(VI) at 800 °C (Figure 252). 

One of the predictions to emerge from the model is that improved Cr(VI) conversion should result from a "convex bent ramp," usually represented as just a "bent ramp." The object is to accelerate the heat-up process while the temperatures are low, because the Cr(VI) is least sensitive to water vapor in this temperature range. Then, at high temperatures when the sensitivity is greatest, the heat-up rate is slowed so that water is not released as fast. Thus, the catalyst is exposed to a lower water vapor concentration during the most critical period, and the conversion to Cr(VI) is consequently improved. The result is improved conversion to Cr(VI) with no loss in heat-up time, compared with the standard linear thermal ramp. 

Since neither model adequately describes the behavior of real viscoelastic materials, a combination of the classic elements is often made to gain closer representation. The most common configuration is called the standard linear solid4 configuration, and it is illustrated in Figure 6.6. A more accurate representation of actual behavior can be obtained by a composite of multiple elements of the standard linear solid configuration into a multi-element model (Figure 6.7) with an array of coefficients for each element. 

Figure 6.5 Simulated time-effect data where the intercept was normally distributed with a mean of 100 and a standard deviation of 60. The effect was linear over time within an individual with a slope of 1.0. No random error was added to the model—the only predictor in the model is time and it is an exact predictor. The top plot shows the data pooled across 50 subjects. Solid line is the predicted values under the simple linear model pooled across observations. Dashed line is the 95% confidence interval. The coefficient of determination for this model was 0.02. The 95% confidence interval for the slope was —1.0, 3.0 with a point estimate of 1.00. The bottom plot shows how the data in the top plot extended to an individual. The bottom plot shows perfect correspondence between effect and time within an individual The mixed effects coefficient of determination for this data set was 1.0, as it should be. This example was meant to illustrate how the coefficient of determination using the usual linear regression formula is invalid in the mixed effects model case because it fails to account for between-subject variability and use of such a measure results in a significant underestimation of the predictive power of a covariate.

Linear (or first-order) kinetics refers to the situation where the rate of some process is proportional to the amount or concentration of drug raised to the power of one (the first power, hence the name first-order kinetics). This is equivalent to stating that the rate is equal to the amount or concentration of drug multiplied by a constant (a linear function, hence linear kinetics). All the PK models described in this chapter have assumed linear elimination (metabolism and excretion) kinetics. All distribution processes have been taken to follow linear kinetics or to be instantaneous (completed quickly). Absorption processes have been taken to be instantaneous (completed quickly), follow linear first-order kinetics, or follow zero-order kinetics. Thus out of these processes, only zero-order absorption represents a nonlinear process that is not completed in too short of a time period to matter. This lone example of nonlinear kinetics in the standard PK models represents a special case since nonlinear absorption is relatively easy to handle mathematically. Inclusion of any other type of nonlinear kinetic process in a PK model makes it impossible to write the... 

All the standard PK models include a number of inherent assumptions about the ADME processes, including the universal assumption that elimination follows first-order or linear kinetics. 

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