Exponential parameter estimates

Watts (1994) dealt with the issue of confidence interval estimation when estimating parameters in nonlinear models. He proceeded with the reformulation of Equation 16.19 because the pre-exponential parameter estimates "behaved highly nonlinearly." The rate constants were formulated as follows... 

As introduced in sections 3.1.3 and 4.2.3, the Arrhenius equation is the normal means of representing the effect of T on rate of reaction, through the dependence of the rate constant k on T. This equation contains two parameters, A and EA, which are usually stipulated to be independent of T. Values of A and EA can be established from a minimum of two measurements of A at two temperatures. However, more than two results are required to establish the validity of the equation, and the values of A and EA are then obtained by parameter estimation from several results. The linear form of equation 3.1-7 may be used for this purpose, either graphically or (better) by linear regression. Alternatively, the exponential form of equation 3.1-8 may be used in conjunction with nonlinear regression (Section 3.5). Seme values are given in Table 4.2. 

Noncompartmental analysis is limited in that it is not descriptive or predictive concentrations must be interpolated from data. The appeal of noncompartmental analysis is that the shape of the blood concentration-versus-time curve is not assumed to be represented by an exponential function and, therefore, estimates of metabolic and pharmacokinetic parameters are not biased by this assumption. In order to minimize errors in parameter estimates that are introduced by interpolation, a large number of data points that adequately define the concentration-versus-tie curve are needed. 

The above result is usual in direct adaptive estimation and/or control schemes. The exponential convergence to zero of both the state estimation error and the parameter estimation error is guaranteed only in the presence of the persistency of excitation condition [1,3]. This, in turn, implies that they keep bounded in the presence of bounded uncertainties. However, since persistency of excitation may be difficult to guarantee in practice, a modified parameters update law can be adopted, according to the concept of projection operator [2], In detail, adoption of the following update law instead of (6.28),... 

Third, at the ill-conditioning of numerical problems for parameter estimation with models involving a large number of exponential terms. Wise [299] has developed a class of powers of time models as alternatives to the sums of exponentials models and has validated these alternative models on many sets of experimental data. From an empirical standpoint, Wise [244] reported 1000 or more published time—concentration curves where alternative models fit the data as weU or better than the sums-of-exponentials models. 

The first application of hierarchical SA for parameter estimation included refinement of the pre-exponentials in a surface kinetics mechanism of CO oxidation on Pt (a lattice KMC model with parameters) (Raimondeau et al., 2003). A second example entailed parameter estimation of a dual site 3D lattice KMC model for the benzene/faujasite zeolite system where benzene-benzene interactions, equilibrium constants for adsorption/desorption of benzene on different types of sites, and diffusion parameters of benzene (a total of 15 parameters) were determined (Snyder and Vlachos, 2004). While this approach appears promising, the development of accurate but inexpensive surfaces (reduced models) deserves further attention to fully understand its success and limitation. 

A plot of In Vi as a function of In Xj, while all other X s are held constant at their nominal values in situ, yields a straight line whose slope determines the exponential parameter g,[. Given that there will always be a certain amount of experimental error associated with each assay, it will in general require about 10 data points to obtain a reasonably good estimate of the slope. Thus, we can conclude that 0n assays will be needed to estimate the kinetic parameters of the rate law in the Power-Law Formalism when there are n variables that influence the rate law under consideration. 

Table 1.4 Summary of parameter estimates and goodness of fit statistics after fitting exponential equations to the data in Fig. 1.12.

Numerical identifiability also becomes a problem with a poorly or inadequately designed experiment. For example, a drug may exhibit multi-exponential kinetics but due to analytical assay limitations or a sampling schedule that stops sampling too early, one or more later phases may not be identifiable. Alternatively, if sampling is started too late, a rapid distribution phase may be missed after bolus administration. In these cases, the model is identifiable but the data are such that all the model components cannot be estimated. Attempting to fit the more complex model to data that do not support such a model may result in optimization problems that either do not truly optimize or result in parameter estimates that are unstable and highly variable. 

Whether to model a pharmacodynamic model parameter using an arithmetic or exponential scale is largely up to the analyst. Ideally, theory would help guide the choice, but there are certainly cases when an arithmetic scale is more appropriate than an exponential scale, such as when the baseline pharmacodynamic parameter has no constraint on individual values. However, more often than not the choice is left to the analyst and is somewhat arbitrarily made. In a data rich situation where each subject could be fit individually, one could examine the distribution of the fitted parameter estimates and see whether a histogram of the model parameter follows an approximate normal or log-normal distribution. If the distribution is approximately normal then an arithmetic scale seems more appropriate, whereas if the distribution is approximately log-normal then an exponential scale seems more appropriate. In the sparse data situation, one may fit both an arithmetic and exponential scale model and examine the objective function values. The model with the smallest objective function value is the scale that is used. 

Table 4.4 Parameters used in stretch exponential to estimate the short-time diffusion coefficient...

Additional advantages to resolving AUC and AUMC from curve-fitting the sums of exponentials are the stability of the estimates and that nonlinear regression procedures provide measures of error about the pharmacokinetic parameter estimates. 

Less-experienced researchers may start with trying to estimate the pre-exponentials feo and the activation energies Ea using the standard Arrhenius expression. Our tests showed that three out of five software packages had severe difficulties to obtain a converged solution for the parameter estimation run, which is easily explained by the strong correlation between both parameters. 

The initial estimates for the pre-exponential factors and the activation energies are based on literature [4]. The initial formaldehyde concentrations were given by the experimentalists. For numerical reasons it is better to have the parameters of the same order of magnitude. To obtain this we take the natural logarithm of the pre-exponential factors, and the activation energies are scaled by a factor 1/1000, E = Ei/1000. The scaled initial parameter estimates are listed in the second column of Table 8.2. 

The temperature dependence of A predicted by Eq. (5-11) makes a very weak contribution to the temperature dependence of the rate constant, which is dominated by the exponential term. It is, therefore, not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted dependence of A is observed experimentally. Uncertainties in estimates of A tend to be quite large because this parameter is, in effect, determined by a long extrapolation of the Arrhenius plot to 1/T = 0. 

A more interesting possibility, one that has attracted much attention, is that the activation parameters may be temperature dependent. In Chapter 5 we saw that theoiy predicts that the preexponential factor contains the quantity T", where n = 5 according to collision theory, and n = 1 according to the transition state theory. In view of the uncertainty associated with estimation of the preexponential factor, it is not possible to distinguish between these theories on the basis of the observed temperature dependence, yet we have the possibility of a source of curvature. Nevertheless, the exponential term in the Arrhenius equation dominates the temperature behavior. From Eq. (6-4), we may examine this in terms either of or A//. By analogy with equilibrium thermodynamics, we write... 

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