Concentration-time data

Titrimetric analysis is a classical method for generating concentration-time data, especially in second-order reactions. We illustrate with data on the acetylation of isopropanol (reactant B) by acetic anhydride (reactant A), catalyzed by A-methyl-imidazole. The kinetics were followed by hydrolyzing 5.0-ml samples at known times and titrating with standard base. A blank is carried out with the reagents but no alcohol. The reaction is... 

Consecutive reactions involving one first-order reaction and one second-order reaction, or two second-order reactions, are very difficult problems. Chien has obtained closed-form integral solutions for many of the possible kinetic schemes, but the results are too complex for straightforward application of the equations. Chien recommends that the kineticist follow the concentration of the initial reactant A, and from this information rate constant k, can be estimated. Then families of curves plotted for the various kinetic schemes, making use of an abscissa scale that is a function of c kit, are compared with concentration-time data for an intermediate or product, seeking a match that will identify the kinetic scheme and possibly lead to additional rate constant estimates. 

Thus, if Ca and Cb can both be measured as functions of time, a plot of v/ca vs. Cb allows the rate constants to be estimated. (If it is known that B is also consumed in the first-order reaction, mass balance allows cb to be easily expressed in terms of Ca-) The rate v(Ca) is the tangent to the curve Ca = f(t) at concentration Ca-This can be determined graphically, analytically, or with computer processing of the concentration-time data. Mata-Perez and Perez-Benito show an example of this treatment for parallel uncatalyzed and autocatalyzed reactions. 

Although a closed-form solution can thus be obtained by this method for any system of first-order equations, the result is often too cumbersome to lead to estimates of the rate constants from concentration-time data. However, the reverse calculation is always possible that is, with numerical values of the rate constants, the concentration—time curve can be calculated. This provides the basis for a curve-... 

Considering the attention that we have given in this chapter to concentrationtime curves of complex reactions, it may seem remarkable that many kinetic studies never generate a comprehensive set of complicated concentration-time data. The reason for this is that complex reactions often can be studied under simplified conditions constituting important special cases for example, whenever feasible one chooses pseudo-first-order conditions, and then one studies the dependence of the pseudo-first-order rate constant on variables other than time. This approach is amplified below. 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

Rather than the use of instantaneous or initial rates, the more usual procedure in chemical kinetics is to measure one or more concentrations over the timed course of the reaction. It is the analysis of the concentrations themselves, and not the rates, that provides the customary treatments. The concentration-time data are fitted to an integrated form of the rate law. These methods are the subjects of Chapters 2, 3, and 4. 

Some reactions follow relatively simple rate expressions. Their time course usually depends on the concentration of only a single species, designated A. We shall learn to recognize these situations and to analyze the resulting concentration-time data. We shall also consider data obtained by instrumental methods and by the method of flooding. 

The method of least squares provides the most powerful and useful procedure for fitting data. Among other applications in kinetics, least squares is used to calculate rate constants from concentration-time data and to calculate other rate constants from the set of -concentration values, such as those depicted in Fig. 2-8. If the function is linear in the parameters, the application is called linear least-squares regression. The more general but more complicated method is nonlinear least-squares regression. These are examples of linear and nonlinear equations ... 

When asked to determine a rate law and rate constant from concentration-time data, we must determine the order of the reaction. The rate law for this reaction has some order x with respect to butadiene (C4 He =A), which we must determine Rate = k [A] We can use the data provided in the problem to test graphically whether the reaction is first or second order in butadiene. If the reaction is... 

Pappas AA, Ackerman BH, Olsen KM, et al. 1991. Isopropanol ingestion A report of 6 episodes with isopropanol and acetone serum concentration time data. Clin Toxicol 21(1) 11-21. 

Assignment of a specific model structure to a real data set should, among other things, take into account the identifiability problem. Simply stated, the identifiability problem occurs when more than one model structure and associated parameter sets are able to describe the actual data, typically drug plasma concentration-time data. In other words, how unique is the parameter set, and what... 

Both the Chen and Gross [48] and the Gallo et al. [49] methods have been applied to eliminating compartments. Both derivation methods are based on the specific mass balance equations for the given model structure. Monte Carlo investigations have demonstrated that both methods provide reasonably accurate and precise estimates of partition coefficients from concentration-time data sets containing error, data one is likely to encounter from in vivo studies. 

A modification of the forcing function approach makes use of linear systems analysis for individual tissue compartments [59], Parametric or nonparamet-ric functions are fitted to observed blood drug concentration-time data and are then combined with tissue drug concentration-time measurements deconvolved... 

A new idea has recently been presented that makes use of Monte Carlo simulations [60,61], By defining a range of parameter values, the parameter space can be examined in a random fashion to obtain the best model and associated parameter set to characterize the experimental data. This method avoids difficulties in achieving convergence through an optimization algorithm, which could be a formidable problem for a complex model. Each set of simulated concentration-time data can be evaluated by a goodness-of-fit criterion to determine the models that predict most accurately. 

Two issues present themselves when the question of PB-PK model validation is raised. The first issue is the accuracy with which the model predicts actual drug concentrations. The actual concentration-time data have most likely been used to estimate certain total parameters. Quantitative assessment, via goodness-of-fit tests, should be done to assess the accuracy of the model predictions. Too often, model acceptance is based on subjective evaluation of graphical comparisons of observed and predicted concentration values. 

From the experimental concentration-time data, determine the reaction rate at various times. 

Differentiation of concentration-time data. Suppose there is only one reactant A, and the rate law is... 

Measurement of Concentration-Time Data and Possible Problems... 

Introduction 257 The Basics of Michaelis-Menten Kinetics 259 Hydrogenation From a Kinetic Viewpoint 263 Measurement of Concentration-Time Data and Possible Problems 263... 

Figure 7 Observed concentration—time data for ISMN from the test extended-release formulations included in the four PK studies. The profile for the reference formulation (a) is represented as an intravenous injection with the same AUC as the reference extended-release formulation (IMDUR) and the literature elimination half-life of 3.77 hr. IVIVC development included the two fast ( ) and one medium (o) batch from Study 194.573 and two slow batches ( ) from Study 372.05/196.638 and external validation included the two medium batches ( ) in Studies 196.581 and 372.02.

In this method, the reaction rate is read directly, say, as the slope of the concentration-time data. Then, the logarithm of the rate is plotted versus the logarithm of the concentration if the data lie along a straight line, the slope is equal to the reaction order. 

Absorption/Distribution - The available plasma concentration time data at steady state in patients showed that anagrelide does not accumulate in plasma after repeated administration. 

Using the original tracer concentration-time data, we find... 

The treatment of the data proceeds as a two step procedure. First, a suitable PK model is fitted to the concentration-time data. Then a PD model is fitted to the data as described by the PK model, simultaneously solving for pharmacodynamic parameters (e.g. max. 50% s) and the effect compartment parameter eO-... 

Given the concentration-time data in Problem 12.8, calculate the average rate of decomposition of cyclopropane during the following time intervals ... 

From the plot of concentration-time data in Figure 12.1, estimate ... 

Consider the following concentration-time data for the decomposition reaction AB — A + B. 

---