Mechanistic parameter estimation

In order to better understand the mechanism of the drug mechanistic models are required. If parameters should be estimated it is often necessary that either different data sets are combined to provide enough information for parameter estimation or a subset of parameters is not estimated and values obtained from other sources (e.g. literature, public databases, former clinical trials) are used for these parameters. In an extreme case all parameters are taken from different sources and it is tested whether the model can describe the data by simulation. If this is not the case one or more hypotheses employed when building the model need to be adjusted. 

The desire to formulate reaction schemes in terms of molecular processes taking place on a catalyst surface must be balanced with the need to express the reaction scheme in terms of kinetic parameters that are accessible to experimental measurement or theoretical prediction. This compromise between mechanistic detail and kinetic parameter estimation plays an important role in the use of reaction kinetics analysis to describe the reaction chemistry for a catalytic process. Here, we discuss four case studies in which different compromises are made to develop an adequate kinetic model that describes the available observations determined experimentally and/or theoretically. For convenience, we selected these examples from our work in this field however, this selection is arbitrary, and many other examples could have been chosen from the literature. 

The process of research in chemical systems is one of developing and testing different models for process behavior. Whether empirical or mechanistic models are involved, the discipline of statistics provides data-based tools for discrimination between competing possible models, parameter estimation, and model verification for use in this enterprise. In the case where empirical models are used, techniques associated with linear regression (linear least squares) are used, whereas in mechanistic modeling contexts nonlinear regression (nonlinear least squares) techniques most often are needed. In either case, the statistical tools are applied most fruitfully in iterative strategies. 

At present these exercises are more interesting than definitive. At the same time, the appearance of the computer has created an opportunity for the application of complex rate expressions in reactor design and in data fitting and parameter estimation. In ways unthinkable before, it has provided us with the means of evaluating the kinetics of complex mechanisms. What computers per se cannot do is provide us with the massive amounts of experimental data required for the fitting of complex mechanistic rate expressions. For that a brand new approach to the measurement of reaction rates is required. 

The kinetics were quantified using >12,000 rate - conversion - temperature (r, X, T) triplets calculated from data obtained using a TS-PFR. Four experiments at various feed compositions were used to improve parameter estimation. After rejecting a number of possible rate expressions and their mechanisms, the results from the TS-PFR were fitted with the remaining two candidate rate equations based on mechanistic considerations ... 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

ModEst (from Model Estimation) is another software (Figure 10.28) that has been designed for parameter estimation of mechanistic mathematical models as well as for experimental design. 

The increase in mechanistic detail of process description is often accompanied by loss of parameter identifiability (Beck, 1999). This makes it extremely important to combine such model extensions with a careful analysis of parameter identifiability. Such an approach can clearly communicate not only the goodness of fit, but also the potential for gaining knowledge about parameter values, the conditionalities of parameters estimated from data, and the remaining degrees of freedom for parameter choices (Brun et al., 2001 Omlin et al., 2001b Reichert and Van-rolleghem, 2001). 

Eor complex reaction and reactor models, the sensitivity analysis and the parameter estimation bv optimization are comppter-time consuming and call for more efficient algorithms and computers. Here clearly, any improvement in the speed of such computations is desirable, and even necessary, for the practical use of fundamental models. The requirements of speedness would be rather increased if a fundamental model, instead of a black box, were used for optimal control purposes. So, we think that supercomputers will be more and more useful for solving the numerical problems involved in the mechanistic noodelling of complex gas phase reactions. 

Ila) appear. On the reverse sweep reduction peak IIIr and, on the second anodic sweep, oxidation peak Ila appear as a reversible couple. Cyclic voltammograms of 20-30 pM 5-HT in 0.01 M HCl or in pH 2.0 phosphate buffer (M = 0.5) at sweep rates up to 5 Vs- (the fastest sweep rate which could be employed where la could be observed above the charging current) did not show a reduction peak coupled to oxidation peak la - This implied that the initial peak la product is unstable on the latter time scale and hence is unavailable for reduction. Sweep rate and concentration studies indicate that the peak current for peak la is strongly controlled by the adsorption of 5-HT at the surface of the PGE. Thus, estimations of voltammetric n-values and other mechanistic parameters from the peak la current and shape were not possible. However, the peak potential (Ep) for peak la is dependent upon pH (Equation 1). Furthermore,... 

Table 3 shows that the small activation enthalpies of the reactions (3) and (4) are clearly affected by the zero point energy corrections. But the relative order of the activation enthalpies remains the same with or without the corrections. The activation entropies have great negative values, which is of mechanistic interest (see part 4.3.1). However, because of their similarity, when comparing the three reactions to one another they have only small importance, e.g. for estimation of copolymerization parameters (see part 4.3.2). 

Activity coefficient models offer an alternative approach to equations of state for the calculation of fugacities in liquid solutions (Prausnitz ct al. 1986 Tas-sios, 1993). These models are also mechanistic and contain adjustable parameters to enhance their correlational ability. The parameters are estimated by matching the thermodynamic model to available equilibrium data. In this chapter, vve consider the estimation of parameters in activity coefficient models for electrolyte and non-electrolyte solutions. 

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