Nonlinear scaling relationships

In addition to nonlinear lipophilicity relationships for the transport and distribution of drugs, nonlinear relationships on molar refractivity are frequently observed in QSAR studies of enzyme inhibition data (provided that MR values are scaled by a factor of 0.1, as usual) [60,63,64,66-68]. Two such examples are given in Eq. (63) (Escherichia coli DHFR) and Eq. (64) (Lactobacillus casei DHFR) [101]. The differences between both models could be explained after the 3D structure of the enzyme became known. Whereas all substituents of a benzyl ring contribute to biological activities in E. coli DHFR, only the 3- and 4-substituents show up in the QSAR model for L. casei DHFR but not the 5-substituents. This results from a narrower binding pocket in L. casei DHFR a (3-branched leucine hinders the accommodation of 5-substituents, whereas a more flexible methionine in the same position of E. coli DHFR opens a wider binding pocket [101] ... 

Most of the envisioned practical applications for nonlinear optical materials would require solid materials. Unfortunately, only gas-phase calculations have been developed to a reliable level. Most often, the relationship between gas-phase and condensed-phase behavior for a particular class of compounds is determined experimentally. Theoretical calculations for the gas phase are then scaled accordingly. 

Nonlinear pharmacokinetics. Nonlinear pharmacokinetics simply means that the relationship between dose and Cp is not directly proportional for all doses. In nonlinear pharmacokinetics, drug concentration does not scale in direct proportion to dose (also known as dose-dependent kinetics). One classic drug example of nonlinear pharmacokinetics is the anticonvulsant drug phenytoin.38 Clinicians have learned to dose pheny-toin carefully in amounts greater than 300 mg/day above this point, most individuals will have dramatically increased phenytoin plasma levels in response to small changes in the input dose. 

A second, even more speculative point is that the mathematical framework of nonlinear dynamics may provide a basis to begin to bridge the gap between local microstructural features of a fluid flow or transport system and its overall meso- or macroscale behavior. On the one hand, a major failure of researchers and educators alike has been the inability to translate increasingly sophisticated fundamental studies to the larger-scale transport systems of traditional interest to chemical engineers. On the other hand, a basic result from theoretical studies of nonlinear dynamical systems is that there is often an intimate relationship between local solution structure and global behavior. Unfortunately, I am presently unable to improve upon the necessarily vague notion of a connection between these two apparently disparate statements. 

Figure 2 Hypothetical E—i polarization resistance data for hypothetical corroding interfaces with Rp = 100, 1000, and 10,000 ohms (assumed 1 cm2) and [T, [ic = 60 mV/ decade. The three cases produce corrosion current densities of 130.4, 13.0, and 1.3 XA/ cm2, respectively. Plots (a) and (b) of the same data provide different current scales to indicate the nonlinearity in each case. Plot (c) shows the linear relationship between log///,) and log(/corr). (From Ref. 8.)...

Because pressure transducers from different manufacturers can vary significantly, it is important to understand their performances such as accuracy. An ideal device would have a direct linear relationship between pressure and output voltage. In reality, there will always be some deviations this is referred to as nonlinearly. The best straight line is fitted to the nonlinear curve. The deviation is quoted in their specifications and expressed as a percent of full scale. The nonlinear calibration curve is determined in ascending direction from zero to full rating. This pressure will be slightly different from the pressure measured in descending mode. This difference is termed hysteresis it can be reduced via electrical circuits. 

This result implies that the energy equipartition relationship of Eq. (2.S) applies as well as the general definitions of Chapter I. Note that for Af m the variable turns out to be coupled weakly to the thermal bath. This condition generates that time-scale separation which is indispensable for recovering an exponential time decay. To recover the standard Brownian motion we have therefore to assiune that the Brownian particle be given a macroscopic size. In the linear case, when M = w we have no chance of recovering the properties of the standard Brownian motion. In the next two sections we shall show that microscopic nonlinearity, on the contrary, may allow that the Markov characters of the standard Brownian motion be recovered with increasing temperature. 

Regression analysis is considered an appropriate approach to evaluating the stability data for a quantitative attribute and establishing a shelf life. The nature of the relationship between an attribute and time will determine whether data should be transformed for linear regression analysis. The relationship can be represented by a linear or nonlinear function on an arithmetic or logarithmic scale. In some cases, a nonlinear regression can better reflect the true relationship. An appropriate approach to shelf life estimation is to analyze a quantitative attribute (e.g., assay, degradation products) by... 

There have been attempts to deal with the issue of nonlinearity in data sets. Detrended principal components (DPC) use a polynomial expression to remove the nonlinear relationships from the PCA axes. DPC are useful for data sets of moderate nonlinearity. Detrended correspondence analysis uses a more complex algorithm to eliminate the nonlinearity but requires a more complex computation. Nonmetric multidimensional scaling (NMDS) is a robust method that deals with nonlinearities by using ranks. 

We present a pediatric population PK (PPK) model development example to illustrate the impact that the model development approach to scaling parameters by size can have on pediatric PPK analyses a typical pediatric study is included. It is intuitive that patient size will affect PK parameters such as clearance, apparent volume, and intercompartmental clearance and that the range of patient size in most pediatric PPK data sets is large. Thus, it is expected that in most pediatric PPK studies subject size will affect multiple PK parameters. However, because there are complex interactions between covariates and parameters in pediatric populations, there are also intrinsic pitfalls of stepwise forward covariate inclusion. Selection of significant covariates via backward elimination has appeal in nonlinear model building however, it requires knowledge of the relationship between the covariate and model parameters (linear vs. nonlinear impact) and can encounter numerical difficulties with complex models and limited volume of data often available from pediatric studies. Thus, there is a need for PK analysis of pediatric data to treat size as a special covariate. Specifically, it is important to incorporate it into the model, in a mechanistically appropriate manner, prior to evaluations of other covariates. 

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