Plateau concentration

Example. A drug has a biologic half-life of 4 hours. Following an IV injection of 100 mg, is found to be 10 pg/mL. Calculate Cmax and Cm n if the 100-mg IV dose is repeated every 6 hour until a plasma concentration plateau is reached. 

Equation (57) can be used to calculate the Cmax and Cmin values on the plasma concentration plateau by substituting values for t that correspond to the peaks and valleys in the Cp versus t curve. Thus, if t = tmax (the time of the peak), Eq. (57) gives Cmax ... 

The speed at which the steady state is reached corresponds to the speed of elimination of the drug. The time needed to reach 90% of the concentration plateau is about 3 times the ti/2 of elimination. 

Phenylbutazone is metabolized by the liver at a rate of about 15-25% per day (B33), but plasma levels do not increase proportionately with increasing doses of the drug. The work of Burns et al. (B33) indicates that above a certain level plasma phenylbutazone concentrations plateau. The concentration at which this occurs varies among individuals and is probably a reflection of the level at which saturation of high-affinity plasma protein binding sites occurs. 

FIGURE 10.18 Illustration of the different types of possible peaks (1) the perturbation peak, (2) the mass peak, and (3) the plateau perturbation peaks, on three concentration plateaus. A single Langmnir model was assumed with a=2.0 and b=0.100. (a) A linear plateau, C=0.05mM. (b) A weakly nonlinear platean, C=0.5mM. (c) A clearly nonlinear plateau, C=5mM. The chromatogram shows the result of an analytical injection of a mixture of labeled and unlabeled molecules on a concentration plateau of unlabeled molecules. The solid line shows the perturbation peak (left scale), the dashed-dotted line shows the plateau perturbation peaks (left scale), and the dotted line shows the mass peak (right scale). Here (mM) is the concentration of unlabeled molecules, Q is the concentration of labeled molecules, and the x axis is time. The mean retention times,, and calculated... 

The method of elution on a plateau was first suggested by Helfferich in Science more than forty years ago [126], In the PP method, the chromatographic column is equilibrated with a constant stream of molecules in the mobile phase and a concentration plateau is established. A perturbation is then accomplished by injecting a sample containing an excess or a deficiency of the molecules as compared to the concentration at the plateau. [118-120, 127], The response at the column outlet will be small peaks, known as perturbation peaks, and their retention times are used to determine the adsorption isotherm parameters. The retention time of the perturbation peak is related to the isotherm through the equation ... 

The purpose of this paper was twofold (i) to investigate how to visualize all four perturbation peaks on a quaternary concentration plateau and to (ii) validate the accuracy of the PP method for determination of isotherm parameters directly from quaternary mixtures of l/l/l/l compositions. For this purpose the technique developed and validated for the binary case in papers IV and V (the Lindholm-technique ) was extended to the multi-component case. The approach is to inject the same excess as deficiency for every second solute. Thus, in a quaternary system the excess of components 1 and 3 is the same as the deficiency of components 2 and 4, or the converse. The concentrations of the components in the sample can be calculated according to Csample i = Chateau i - (-1 ) n, for some number n chosen so that the injected concentrations always are equal to or greater than zero. This technique made all perturbation peaks clearly detectable, although the area were not the same for all of them, see Figure 22. 

Frontal analysis is straightforward when starting from an unloaded column (c1 = 0). A modification to reduce the amount of solute is the stepwise increase of the feed concentration, starting from the unloaded column. This results in successive plateaus. Desorption steps are obtained after the highest concentration plateau has passed through the column, if the concentrations are reduced inversely to the adsorption steps. To consume even less feed mixture, this procedure can be performed in closed-loop or circulation operation (Fig. 6.17). 

The method is advantageously combined with the frontal analysis method, which also requires a concentration plateau and thus shares the disadvantage of high sample consumption if operated in open mode. As indicated in Fig. 6.24, the measurement procedure starts at maximum concentration. This concentration plateau is reduced step-by-step by diluting the solution. To reduce the amount of samples needed for the isotherm determination the experiments can be done in a closed loop arrangement (Fig. 6.17). It is also possible to automate this procedure. 

It is often observed, however, that the actual injection profiles are far from the Dirac model, as illustrated in Figure 2.3b, which compares a rectangular pulse injection of 100 fiL (solid line) and the injection profile recorded with a six-port Valeo valve (Houston, TX) fitted with a 100- L loop [42]. The Dirac injection is an acceptable model only if the width of the experimental injection is small compared to the standard deviation of the band profile under linear conditions. Usually, the experimental injection profile has a sharp front followed by a tailing decay (Figure 2.3b). This profile is also typical of those encoxmtered in preparative chromatography, except that they include a concentration plateau lasting for a certain period of time (see Figure 2.3). 

Contrary to conventional practice, it is not necessary to wait tmtil the steady response of the detector indicates that the eluate concentration is the same as the concentration in the stream of mobile phase pumped into the column to return to a pure mobile phase stream. In fact, a wide rectangular injection is usually sufficient, provided that a concentration plateau is eluted at the column exit. Admittedly, in such an implementation of the method, it becomes difficult to decide whether it is FACP or ECP (see next section) but this is irrelevant. If the width of the rectangular pulse is kept reasonably narrow, the amount of sample needed is comparable to that required in ECP for the determination of the isotherm in the same concentration range. 

In the method of elution by characteristic points [134,135] the isotherm is derived from the rear part of the overloaded elution profile obtained upon injection of a large sample (Figme 3.37a [140]). The isotherm is calculated using Eq. 3.93. The method of ECP is very sirmlar to FACP. In ECP, the isotherm is determined from the rear part of an overloaded elution profile, and in FACP it is determined from the rear of a wide rectangular band or from the profile obtained when washing a concentration plateau off the column (these are equivalent). Both methods are thus based on the demonstration that a velocity can be associated with a concentration and that this velocity depends only on the concentration, through the isotherm (see Chapter 7, Section 7.2.1). These methods are based on the determination of Uz and the integration of Eq. 7.4 (Chapter 7). 

The method of elution on a plateau was first suggested by Reilley et al. [147], A steady stream of a solution of the studied component in the mobile phase is pumped through the column until equilibrium is reached, Le., until the breakthrough of the constant concentration plateau has been reached. Then a small pulse of the component is injected. The velocity and the retention time of that pulse are related to the isotherm through the equations... 

Figure 3.38 Typical chromatograms obtained in the determination of equilibrium isotherms by pulse chromatographic methods, (a) Injection on a concentration plateau, and response of a selective detector for a tracer, (b) Response of a nonselective detector for the tracer injection made in (a), (c) Individual profiles of the labeled tracer (gray line) and the unlabeled component (black line).

The method of elution of an isotopic pulse on a plateau was developed by Helfferich and Peterson [136] (see Figure 3.38). If labeled and unlabeled molecules are injected simultaneously on a concentration plateau, the unlabeled molecules travel at the velocity associated with the plateau concentration (Eq. 3.94), while the labeled molecules travel at the velocity associated with the concentration shock ... 

It can be shown that if a wide rectangular pulse of a binary mixture is injected and the pulse width is sufficient for a concentration plateau at the feed composition to appear in the elution profile of the mixed band, a plot of the local concentration of one component versus that of the other one is made of two lines which intersect at the point of the graph having for coordinates the composition of the feed (and that of the intermediate plateau) [106]. 

Note that the velocity of a shock, given by Eq. 7.7, is also the velocity of the molecules in a volume of mobile phase at concentration C. Figure 7.3b illustrates how the injection of a plug of labeled molecules on a concentration plateau permits the determination of both concentration and shock velocities in the same experiment, provided selective detectors are available for the labeled and unlabeled molecules (see Chapter 3, Section 3.5.4). The concentration perturbation moves at the velocity associated with the concentration, Uz (positive peak on the gray Une), while the labeled molecules (solid line, lower trace) move at the shock velocity. Us- The retention time of the tracer peak is therefore ... 

The time to,s is the relaxation time of the shock formation [23]. It is proportional to f , hence inversely proportional to the slope of the concentration gradient. It is also inversely proportional to the product bCp that characterizes the degree of nonlinear behavior of the isotherm at the end of the experiment, when the concentration plateau becomes equal to Cq. 

In most cases, chromatography is performed with a simple initial condition, C(f = 0,z) = q t = 0,z) = 0. TTie column is empty of solute and the stationary and mobile phases are under equilibrium. There are some cases, however, in which pulses of solute are injected on top of a concentration plateau (see Chapter 3, Section 3.5.4). The behavior of positive concentration pulses injected xmder such conditions is similar to that of the same pulses injected in a column empty of solute and they exhibit similar profiles. Even imder nonlinear conditions (high plateau concentration), a pulse that is sufficiently small can exhibit a quasi-linear behavior and give a Gaussian elution profile. Its retention time is linearly related to the slope of the isotherm at the plateau concentration. Measuring this slope is the purpose of the pulse method of measurement of isotherm data. Large pulses may also be injected and they will give overloaded elution profiles similar to those obtained with a column empty of solute. 

On a concentration plateau, however, another kind of perturbations can be made. Negative concentration pulses or vacancies can be injected. Because the velocity associated with a concentration varies in opposite directions, depending on whether the concentration increases or decreases, the shape of a vacancy profile will be the converse of that of a positive concentration pulse. If the isotherm... 

The theory of simple waves applies to large-volume injections, i.e., to the profiles obtained upon injection of rectangular profiles which are so wide that the injection plateau has not been entirely eroded when the band elutes. Then, simplifications of the solution occur because there is a constant state, the concentration plateau. This solution is not valid in overloaded elution chromatography when the injection volume is sufficiently small that the injection plateau has eroded and disappeared by the time the band elutes from the column. It is important to discuss this solution, however, because it takes a finite time for the profile of even a narrow rectangular injection to decay, and the band profile during that period is given by the simple wave solution. Also, this solution is the basis for a method of determination of competitive equilibrium isotherms (Chapter 4, Section 4.2.4). 

The last point on the concentration plateau for the first component is also the point at concentration Cj on a continuous profile, where it moves at the velocity 1, as given by Eq. 8.6a. Using the competitive Langmuir isotherm equations, the directional derivatives (Eq. 8.7a) become... 

Equations 8.25 and 8.26a give the rear continuous profiles of the two components behind the feed concentration plateau in the mixed zone. They are valid as long as concentration Q is different from 0, which eventually happens at time t. This time is given by writing that Q is zero in Eq. 8.26a [14] ... 

This is also the time when the elution of the concentration plateau, C, ends. 

In summary, the wide rectangular profile is characterized by two concentration shocks, at times and fR,2/ for the first and the second component, respectively, by a residual of the injection plateau, and by a concentration plateau at C having a length At2- These characteristics define the three zones of the elution chromatogram of a binary mixture (Figure 8.1) the pure first component zone, the mixed zone, and the pure second component zone. Analytical solutions are provided to calculate the individual band profiles for a binary mixture. Table 8.1. We now study the elution profile of a narrow injection pulse, when Eq. 8.36 is no longer verified. 

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