Correlations of model parameters

Physiological pharmacokinetic models permit assignment of compartments to anatomical regions, and correlation of model parameters to meaningful physiological events. 

The set of coupled second-order partial differential equations (PDEs) was solved with the ACM program package. For these simulation studies correlations of model parameters under steady-state conditions were used (see Table 4.3),... 

Correlation of Model Parameters to Degradation of Starch in the Blends. 

At a later stage, the basic model was extended to comprise several organic substrates. An example of the data fitting is provided by Figure 8.11, which shows a very good description of the data. The parameter estimation statistics (errors of the parameters and correlations of the parameters) were on an acceptable level. The model gave a logical description of aU the experimentally recorded phenomena. 

Ideally, a mathematical model would link yields and/or product properties with process variables in terms of fundamental process phenomena only. All model parameters would be taken from existing theories and there would be no need for adjusting parameters. Such models would be the most powerful at extrapolating results from small scale to a full process scale. The models with which we deal in practice do never reflect all the microscopic details of all phenomena composing the process. Therefore, experimental correlations for model parameters are used and/or parameters are evaluated by fitting the calculated process performance to that observed. 

Correlation of MRI Parameter Changes to the Formation of Vasogenic Edema in Animal Models... 

The correlation of Mossbauer parameters with particle size involves the use of the "shell model that describes the environment of surface nuclei as being of a lower symmetry than those within the particle. The model has been used to rationalize superimposed quadrupole-split spectra in terms of interior and surface iron nuclei. 

It is shown by an example that there are several ways to modify the most commonly used lifetime prediction method, in order to improve its predicting capabilities. These modifications provide an easy and efficient way to perform lifetime predictions. The drawback is, that the modifications have little or no validated physical background. Any correlation between model parameters and macroscopical material properties or damage development characteristics should be established by extensive test programs. This constrains the applicability of these methods to limited situations. 

McGuire and Suffet [728] proposed the calculated net adsorption energy concept which is based on the solubility parameter of the adsorbate. They justified their approach by noting that the interactions involved in the adsorption of nonpolar and polar compounds onto a nonpolar [activated carbon] surface are, for the most part, governed entirely by the dispersion forces. Their results are summarized in Fig. 31. Even on a log-log plot, the r correlation coefficient is only 0.7. The authors cautioned against extrapolating such a correlation to predict the adsorption capacities of other neutral organic compounds. Clearly, incorporation of model parameters that quantify the chemistry of the carbon surface is necessary. 

When mathematical models are used to draw inferences, the values of the model parameters may be a source of uncertainty. As the values of model parameters are not measured by direct observation (they are estimated as part of the model fitting process), the uncertainty of a parameter cannot be characterized by simply recording variability in a series of measurements. However, once the best model criterion has been established, the variability associated with a parameter can be linked to the variability in the data. If standard statistical assumptions are employed, the variability of and correlations among the parameters may be calculated directly. 

Some models invoke multiple surface species involving the specifically adsorbed ion. In these models many combinations of model parameters (stability constants of particular surface species) can fit the experimental data almost equally well. Therefore, the fitted values of stability constants of surface complexes in such models are of limited significance, and so are the correlations involving these stability constants. 

Although it may well be true that the method of least squares is widely misused because of its apparent objectivity and general availability, it was clearly also true that much of the information obtainable from least squares is not used as completely as it could be. The problem of correlation between model parameters illustrates this clearly. High correlation between parameters amounts only to a statement about the data structure as opposed to the data values. The essential issue is the nature of the dependence of the parameter being determined on the data set. If two parameters have similar dependences, then their estimates are going to be correlated. Measuring more data points or a different set of data points would result in a different correlation matrix. The physical limitations of the experimental method, such as the inability to measure spectral characteristics of weak transitions or transitions that fall in inaccessible frequency regions, make it impractical to avoid correlations. 

The initial approach should be to first identify which formulation properties are critical for its in vivo function. Thereafter, all potential physiological factors that may influence this function should be identified and the correlation between the animal and man regarding those factors should be considered. For example, in the case of a pH-dependent enteric-coat formulation, the dissolution of the coating layer will clearly be a critical formulation variable. The pH in the stomach and small intestine, as well as gastric emptying, will all be critical variables. If the correlation of such parameters with man is poor for all available animal models, there is no rationale for performing such studies if the deviations cannot be accounted for when interpreting the results. 

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