Time-Concentration Profiles

As it has appeared in recent years that many hmdamental aspects of elementary chemical reactions in solution can be understood on the basis of the dependence of reaction rate coefficients on solvent density [2, 3, 4 and 5], increasing attention is paid to reaction kinetics in the gas-to-liquid transition range and supercritical fluids under varying pressure. In this way, the essential differences between the regime of binary collisions in the low-pressure gas phase and tliat of a dense enviromnent with typical many-body interactions become apparent. An extremely useful approach in this respect is the investigation of rate coefficients, reaction yields and concentration-time profiles of some typical model reactions over as wide a pressure range as possible, which pemiits the continuous and well controlled variation of the physical properties of the solvent. Among these the most important are density, polarity and viscosity in a contimiiim description or collision frequency. 

Kinetic data fitting the rate equation for catalytic reactions that follow the Michaelis-Menten equation, v = k A]/(x + [A]), with[A]0 = 1.00 X 10 J M, k = 1.00 x 10 6 s 1, and k = 2.00 X 10-J molL1. The left panel displays the concentration-time profile on the right is the time lag approach. 

The concentration-time profile for this system was calculated for a particular set of constants k = 1.00X 10 6 s k = 2.00X 10 4 molL 1,and [A]0 = 1.00xl0 3M. The concentration-time profile, obtained by the numerical integration technique explained in Section 5.6, is shown in Fig. 2-11. Consistent with the model, the variation of [A] is nearly linear (i.e., zeroth-order) in the early stages and exponential near the end. 

A new chapter (5) on reaction intermediates develops a number of methods for trapping them and characterizing their reactivity. The use of kinetic probes is also presented. The same chapter presents the Runge-Kutta and Gear methods for simulating concentration-time profiles for complex reaction schemes. Numerical methods now assume greater importance, since useful computer programs are available. The treatment of pH profiles in Chapter 6 is much more detailed. 

Figure 3, Concentration-time profiles for reactants and selected products in the...

Figure 7. Concentration-time profiles during 1-hexyne hydrogenation over 0.4wt.% Pd/ACF. Reaction conditions 0.5kmol/ 1-hexyne in n-heptane (lOOmL) substrate-to-Pd molar ratio of 23,000 303 K pressure of 1.3 MPa 1500 rpm.

Typical concentration-time profiles during the 1-hexyne hydrogenation over 0.4wt.% Pd/ACF catalyst are presented in Figure 7 showing the experimental and simulated curves (Langmuir-Hinshelwood mechanism). Pd/ ACF materials with the same particle size but different Pd loading (0.4, 0.6, 1.2wt.%) show identical initial activity of 0.140 0.004 kmolHj/kgp(j/s. This indicates the absence of diffusion limitations. Selectivity to 1-hexene is 97.1 +0.4% up to 80% conversion, and 95.9 + 0.4% at 90% conversion. 

Figure 5.7. Concentration-time profiles using the parameters as given in the program.

Figure 5.257. Oxygen concentration time profiles at six different positions in the floe.

Colburn, W.A. A time- dependent Volume of distribution term used to describe linear concentration-time profiles. J. Pharmacoki net. Bi opha rm 11(4) 389-400, 1983. 

Traditionally, the ideal extended-release product has been conceived as providing essentially stable blood levels over the whole dosing frequency interval. Thus, unlike the saw-edge blood concentration time profile of a non-controlled-release product that may show rather wild fluctuations between sub- and su-pratherapeutic blood levels, the ideal extended-release product avoids both nontherapeutic blood levels and those likely to have an increased frequency of dose-related side effects. However, in recent years con-trolled-release products that deliberately exploit a pulsatile drug release time profile have also attracted attention. 

Also, if conversion of drug to active metabolite shows significant departure from linear pharmacokinetics, it is possible that small differences in the rate of absorption of the parent drug (even within the 80-125% range for log transformed data) could result in clinically significant differences in the concentration/ time profiles for the active metabolite. When reliable data indicate that this situation may exist, a requirement of quantification of active metabolites in a bioequivalency study would seem to be fully justified. 

To predict oral plasma concentration-time profiles, the rate of drug absorption (Eq. (53)) needs to be related to intravenous kinetics. For example, in the case of the one-compartment model with first-order elimination, the rate of plasma concentration change is estimated as... 

Coupling with its intravenous pharmacokinetic parameters, the extended CAT model was used to predict the observed plasma concentration-time profiles of cefatrizine at doses of 250, 500, and 1000 mg. The human experimental data from Pfeffer et al. [82] were used for comparison. The predicted peak plasma concentrations and peak times were 4.3, 7.9, and 9.3 qg/mL at 1.6, 1.8, and 2.0 hr, in agreement with the experimental mean peak plasma concentrations of... 

To illustrate the subtle differences between the iso-pH and gradient-pH methods, ketoprofen was used in a series of simulation calculations. Figure 3.5a shows the concentration-time profiles for ketoprofen under an iso-pH condition. Consider the case of iso-pH 3, where the molecule is essentially uncharged in solution (pKa... 

Fig. 3.5. Comparison of concentration-time profiles of ketoprofen under (a) iso-pH, and (b) gradient-pH conditions. The sink condition caused by pH gradients lowers the membrane retention from 56% to 9%, and hastens the transport of the weak acid.

Extensions of BCS beyond the oral IR area has also been suggested, for example to apply BCS in the extended-release area. However, this will provide a major challenge since the release from different formulations will interact in different ways with in vitro test conditions and the physiological milieu in the gastrointestinal tract. For example, the plasma concentration-time profile differed for two felodipine ER tablets for which very similar in vitro profiles had been obtained, despite the fact that both tablets were of the hydrophilic matrix type based on cellulose derivates [70], This misleading result in vitro was due to interactions between the gel strength of the matrix and components in the dissolution test medium of no in vivo relevance. The situation for ER formulations would be further complicated by the need to predict potential food effects on the drug release in vivo. 

Fig. 22.3. Mean plasma concentration-time profiles of aciclovir following administration of 1000 mg oral valaciclovir or a 350-mg intravenous infusion of aciclovir over a 1-h period.

Area under the plasma concentration-time profile Chromium-51-labeled ethylenediamine-tetraacetic acid Cytochrome P450, 3A4 isozyme... 

Figure 2 Mean plasma (Cp), CSF (CCSF), and brain (Cb) compound concentration-time profiles (graph) and matrix-specific neuropharmacokinetic parameters (table) of a compound in rats following subcutaneous administration [42]. Abbreviations Cmax, maximal compound concentration Tmax, time of Cmax tV2, compound half-life.

From concentration-time profiles it was seen that some materials did not remove gold from solution regardless of the pH or temperature. On the other hand, some materials removed gold quite effectively from solution. Generally, it was found that a higher temperature caused a faster and more complete gold removal. It was also seen that less in acidic conditions the amount of gold removed from solution increased. 

Ferrous ferric oxide showed a different concentration-time profile with change of pH in comparison to the two other materials - it was linear, while the decline in gold concentration was rather steep for FeOOH and AI2O3 within the first minutes of the experiment. This behavior was observed under all pH conditions. Furthermore, in the case of Fe304 it was seen that an increase in pH resulted in a gradual increase in speed of gold removal from solution, e.g., while it took 90 min to remove all gold present at pH=2 and 3 it took about 60 min at pH=4 and had finished after 5 min at pH=5 and 6, which can also be explained with the mentioned speciation effect in connection with the associated different complex stabilities. 

The toxic effects model uses concentration-time profiles from the respiratory and skin protection models as input to estimate casualty probabilities. Two approaches are available a simple linear dose-effect model as incorporated in RAP and a more elaborate non-linear response model, based on the Toxic Load approach. The latter provides a better description of toxic effects for agents that show significant deviations of simple Haber s law behaviour (i.e. toxic responses only depend on the concentration-time product and not on each quantity separately). 

The E-Z Solve software may also be used to solve Example 12-7 (see file exl2-7.msp). In this case, user-defined functions account for the addition of fiesh glucose, so that a single differential equation may be solved to desenbe the concentration-time profiles over the entire 30-dry period. This example file, with die user-defined functions, may be used as the basis for solution of a problem involving the nonlinear kinetics in equation (A), in place of the linear kinetics in (B) (see problem 12-17). 

The process in Example 12-7 was conducted by initially adding 0.35 mol glucose to the 100-L reactor, and replenishing the reaction with 0.28 mol flesh glucose every 144 h for 30 days. Determine the gluconic acid and glucose concentration-time profiles in the reactor over this period... 

Some aspects of reactor behavior are developed in Chapter 5, particularly concentration-time profiles in a BR in connection with the determination of values of and k2 from experimental data. It is shown (see Figure 5.4) that the concentration of the intermediate, cB, goes through a maximum, whereas cA and cc continuously decrease and increase, respectively. We extend the treatment here to other considerations and other types of ideal reactors. For simplicity, we assume constant density and isothermal operation. The former means that the results for a BR and a PFR are equivalent. For flow reactors, we further assume steady-state operation. 

Fig. 2 Di ssolved oxygen, BOD and nitrate concentration-time profiles (25 °C wastewater).

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