Extrapolated concentration

Yoshimura et al. [132] studied the pharmacokinetics of primaquine in calves of 180—300 kg live weight. The drug was injected at 0.29 mg/kg (0.51 mg/kg as primaquine diphosphate) intravenously or subcutaneously and the plasma concentrations of primaquine and its metabolite carboxyprimaquine were determined by high performance liquid chromatography. The extrapolated concentration of primaquine at zero time after the intravenous administration was 0.5 0.48 pg/mL which decreased with an elimination half-life of 0.16 0.07 h. Primaquine was rapidly converted to carboxyprimaquine after either route of administration. The peak concentration of carboxyprimaquine was 0.5 0.08 pg/mL at 1.67 0.15 h after intravenous administration. The corresponding value was 0.47 0.07 pg/mL at 5.05 1.2 h after subcutaneous administration. The elimination half-lives of carboxyprimaquine after intravenous and subcutaneous administration were 15.06 0.99 h and 12.26 3.6 h, respectively. 

Extrapolate the straight line joining the points until it meets the extrapolated concentration axis. The distance between this point and the intersection of the axes represents the concentration of the element (e.gMg, K, Na) being determined in the solution of the substance being examined. 

The method of residuals is used to obtain the individual rate constants (Fig. 5-6). A is determined by extrapolating the terminal slope to the y axis can be obtained by calculating the slope or 0/2 and using the formnlas given for the intravenons bolns case. At each time point in the absorption portion of the curve, the concentration value from the extrapolated line is noted and called the extrapolated concentration. For each point, the actual concentration is subtracted from the extrapolated concentration to compute the residual concentration. When the residual concentrations are plotted on semilogarithmic coordinates, a line with y intercept equal to A and slope equal to -dtJ2.303 is obtained. When these values are calculated, they can be placed into the equation (C = Ae - Ae <, where A = FDk l[V k - k)]) and used to compute the serum concentration at any time after the... 

The residual line is calculated as before using the method of residuals. The terminal line is extrapolated to the y axis, and extrapolated concentrations are determined for each time point. Because actual concentrations are greater in this case, residual concentrations are calculated by subtracting the extrapolated concentrations from the actual concentrations. When plotted on semilogarithmic paper, the residual line has a y intercept equal to A. The slope of the residual line is used to compute a (slope = -a/2.303). With the rate constants (a and /S) and the intercepts (A and B), concentrations can be calculated for any time after the intravenous bolus dose (C = Ae -E or pharmacokinetic constants can be computed Cl = D/[ A/a ) + (S//S)], Vo, = C1//3, Vd.ss = D[(A/ 2) -E (BZ/S )] /[(A/a) -E (B//S)]2. 

High and Danner found that the Koningsveld-Kleintjens expression (Equation 16.39) is superior to the polynomial series in correlating and extrapolating concentration-dependent FH interaction parameters. 

FIGURE 3.6 Compartmental analysis for different terms of volume of distribution. (Adapted from Kwon, Y., Handbook of Essential Pharmacokinetics, Pharmacodynamics and Drug Metabolism for Industrial Scientists, Kluwer Academic/Plenum Publishers, New York, 2001. With permission.) (a) Schematic diagram of two-compartment model for compound disposition. Compound is administrated and eliminated from central compartment (compartment 1) and distributes between central compartment and peripheral compartment (compartment 2). Vj and V2 are the apparent volumes of the central and peripheral compartments, respectively. kI0 is the elimination rate constant, and k12 and k21 are the intercompartmental distribution rate constants, (b) Concentration versus time profiles of plasma (—) and peripheral tissue (—) for two-compartmental disposition after IV bolus injection. C0 is the extrapolated concentration at time zero, used for estimation of V, The time of distributional equilibrium is fss. Ydss is a volume distribution value at fss only. Vj, is the volume of distribution value at and after postdistribution equilibrium, which is influenced by relative rates of distribution and elimination, (c) Time-dependent volume of distribution for the corresponding two-compart-mental disposition. Vt is the starting distribution space and has the smallest value. Volume of distribution increases to Vdss at t,s. Volume of distribution further increases with time to Vp at and after postdistribution equilibrium. Vp is influenced by relative rates of distribution and elimination and is not a pure term for volume of distribution. 

However, before extrapolating the arguments from the gross patterns through the reactor for homogeneous reactions to solid-catalyzed reactions, it must be recognized that in catalytic reactions the fluid in the interior of catalyst pellets may diSer from the main body of fluid. The local inhomogeneities caused by lowered reactant concentration within the catalyst pellets result in a product distribution different from that which would otherwise be observed. 

Using this concept, Burdett developed a method in 1955 to obtain the concentrations in mono-, di- and polynuclear aromatics in gas oils from the absorbances measured at 197, 220 and 260 nm, with the condition that sulfur content be less than 1%. Knowledge of the average molecular weight enables the calculation of weight per cent from mole per cent. As with all methods based on statistical sampling from a population, this method is applicable only in the region used in the study extrapolation is not advised and usually leads to erroneous results. 

The way out of this dilemma is to make measurements at several (nonideal) molarities m and extrapolate the results to a hypothetieal value of at m = 0. In so doing we have extrapolated out the nonideality because at m = 0 all solutions are ideal. Rather than ponder the philosophical meaning of a solution in which the solute is not there, it is better to concentrate on the error due to interionic interactions, which becomes smaller and smaller as the ions become more widely separated. At the extrapolated value of m = 0, ions have been moved to an infinite distance where they cannot interact. 

The nitric acid used in this work contained 10% of water, which introduced a considerable proportion of acetic acid into the medium. Further dilution of the solvent wnth acetic acid up to a concentration of 50 moles % had no effect on the rate, but the addition of yet more acetic acid decreased the rate, and in the absence of acetic anhydride there was no observed reaction. It was supposed from these results that the adventitious acetic acid would have no effect. The rate coefficients of the nitration diminished rapidly with time in one experiment the value of k was reduced by a factor of 2 in i h. Corrected values were obtained by extrapolation to zero time. The author ascribed the decrease to the conversion of acetyl nitrate into tetranitromethane, but this conversion cannot be the explanation because independent studies agree in concluding that it is too slow ( 5.3.1). 

In a standard addition the analyte s concentration is determined by extrapolating the calibration curve to find the x-intercept. In this case the value of X is... 

Several features of equation 6.50 deserve mention. First, as the ionic strength approaches zero, the activity coefficient approaches a value of one. Thus, in a solution where the ionic strength is zero, an ion s activity and concentration are identical. We can take advantage of this fact to determine a reaction s thermodynamic equilibrium constant. The equilibrium constant based on concentrations is measured for several increasingly smaller ionic strengths and the results extrapolated... 

This experiment describes the determination of the stability (cumulative formation) constant for the formation of Pb(OH)3 by measuring the shift in the half-wave potential for the reduction of Pb + as a function of the concentration of OH . The influence of ionic strength is also considered, and results are extrapolated to zero ionic strength to determine the thermodynamic formation constant. 

The response surfaces in Figure 14.2 are plotted for a limited range of factor levels (0 < A < 10, 0 < B < 10), but can be extended toward more positive or more negative values. This is an example of an unconstrained response surface. Most response surfaces of interest to analytical chemists, however, are naturally constrained by the nature of the factors or the response or are constrained by practical limits set by the analyst. The response surface in Figure 14.1, for example, has a natural constraint on its factor since the smallest possible concentration for the analyte is zero. Furthermore, an upper limit exists because it is usually undesirable to extrapolate a calibration curve beyond the highest concentration standard. 

The inherent viscosity (I/C2) In (77/770). A plot of inherent viscosity versus concentration also extrapolates to [77] in the limit of C2 0. That this is the case is readily seen by combining Eq. (9.12) with the definition of the inherent viscosity and then expanding the logarithm ... 

The standard procedure is to measure D at several different initial concentrations, using the procedure just described, and then extrapolating the results to c = 0. We symbolize the resulting limiting value D°. This value can be interpreted in terms of Eq. (9.79), which is derived by assuming 7 -> 1 and therefore requires extreme dilution. It is apparent from Eqs. (9.79) and (9.5) that D° depends on the ratio T/770, as well as on the properties of the solute itself. In order to reduce experimental (subscript ex) values of D° to some standard condition (subscript s), it is conventional to write... 

The concentration dependence of s is eliminated by making measurements at several different concentrations and then extrapolating to zero concentration. The limiting value is given by the symbol s°. This is the sedimentation analog of D°. 

In applying the Debye theory to concentrated solutions, we must extrapolate the results measured at different concentrations to C2 = 0 to eliminate the effects of solute-solute interactions. 

Assuming that concentration effects have been eliminated by extrapolating Kc2/Rg to C2 = 0 (subscript c = 0), we see that Eq. (10.89) is the equation of a straight line if (Kc2/Rg)(,=o plotted against sin (0/2). The characteristic parameters of the line have the following significance ... 

Polymers in Solution. Polyacrylamide is soluble in water at all concentrations, temperatures, and pH values. An extrapolated theta temperature in water is approximately —40° C (17). Insoluble gel fractions are sometimes obtained owing to cross-link formation between chains or to the formation of imide groups along the polymer chains (18). In very dilute solution, polyacrylamide exists as unassociated coils which can have an eUipsoidal or beanlike stmcture (19). Large aggregates of polymer chains have been observed in hydrolyzed polyacrylamides (20) and in copolymers containing a small amount of hydrophobic groups (21). 

This has the advantage that the expressions for the adsotbed-phase concentration ate simple and expHcit, and, as in the Langmuir expression, the effect of competition between sorbates is accounted for. However, the expression does not reduce to Henry s law in the low concentration limit and therefore violates the requirements of thermodynamic consistency. Whereas it may be useful as a basis for the correlation of experimental data, it should be treated with caution and should not be used as a basis for extrapolation beyond the experimental range. 

Halogenated compounds, found in high concentrations in seaweeds consumed in Japan and Hawaii, have been suspected of being carcinogenic, largely based on epidemiological extrapolation (high incidences of hepatic carcinoma). However, direct human causation has not been estabUshed (107). 

---