Concentration-time curves

Delayed action soHd products are designed like conventional dosage forms to release all their dmg contents at one time, but only after a delayed period. Thus, the duration of action and the blood concentration—time curve is like that of a conventional product. However, the onset time is purposely designed to be long. 

The slope of a concentration-time curve to define the rate expression can be determined. However, experimental studies have shown the reaction cannot be described by simple kinetics, but by the relationship ... 

Casado et al. have analyzed the error of estimating the initial rate from a tangent to the concentration-time curve at t = 0 and conclude that the error is unimportant if the extent of reaction is less than 5%. Chandler et al. ° fit the kinetic data to a polynomial in time to obtain initial rate estimates. 

Figure 3-4. Semilogarithmic plot of the concentration-time curves of Fig. 3-2.

In 1950 French " and Wideqvist independently described a data treatment that makes use of the area under the concentration-time curve, and later authors have discussed the method.We introduce the technique by considering the second-order reaction of A and B, for which the differential rate equation is... 

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-... 

Lowry and John studied Scheme XV and discussed the nature of the concentration-time curves, noting that the concentration of B will pass through a maximum if k2 < ki, whereas if 2 > ki, it will not display a maximum. 

One way to examine the validity of the steady-state approximation is to compare concentration—time curves calculated with exact solutions and with steady-state solutions. Figure 3-10 shows such a comparison for Scheme XIV and the parameters, ki = 0.01 s , k i = 1 s , 2 = 2 s . The period during which the concentration of the intermediate builds up from its initial value of zero to the quasi-steady-state when dcfjdt is vei small is called the pre-steady-state or transient stage in Fig. 3-10 this lasts for about 2 s. For the remainder of the reaction (over 500 s) the steady-state and exact solutions are in excellent agreement. Because the concen-... 

The procedure, in analyzing kinetic data by numerical integration, is to postulate a reasonable kinetic scheme, write the differential rate equations, assume estimates for the rate constants, and then to carry out the integration for comparison of the calculated concentration-time curves with the experimental results. The parameters (rate constants) are adjusted to achieve an acceptable fit to the data. Carpen-(ej-48. pp. 76-81 some numerical calculations. Farrow and Edelson and Porter... 

Concentration-time curves. Much of Sections 3.1 and 3.2 was devoted to mathematical techniques for describing or simulating concentration as a function of time. Experimental concentration-time curves for reactants, intermediates, and products can be compared with computed curves for reasonable kinetic schemes. Absolute concentrations are most useful, but even instrument responses (such as absorbances) are very helpful. One hopes to identify characteristic features such as the formation and decay of intermediates, approach to an equilibrium state, induction periods, an autocatalytic growth phase, or simple kinetic behavior of certain phases of the reaction. Recall, for example, that for a series first-order reaction scheme, the loss of the initial reactant is simple first-order. Approximations to simple behavior may suggest justifiable mathematical assumptions that can simplify the quantitative description. 

Area under the Curve (AUC) refers to the area under the curve in a plasma concentration-time curve. It is directly proportional to the amount of drug which has appeared in the blood ( central compartment ), irrespective of the route of administration and the rate at which the drug enters. The bioavailability of an orally administered drug can be determined by comparing the AUCs following oral and intravenous administration. 

AUC is the area under the (diug) concentration time curve. 

Bioavailability is the amount of drug in a formulation that is released and becomes available for absorption or the amount of the drug absorbed after oral administration compared to the amount absorbed after intravenous administration (bioavailability - 100%), judged from areas remaining under plasma drug concentration-time curves. 

As the amount in the body decreases, the concentration decreases by the same law (-dC/d/ = Ke C), i.e., first-order kinetics resulting in an exponential function. The integral is the area under the concentration time curve (ACC). 

To this point we have focused on reactions with rates that depend upon one concentration only. They may or may not be elementary reactions indeed, we have seen reactions that have a simple rate law but a complex mechanism. The form of the rate law, not the complexity of the mechanism, is the key issue for the analysis of the concentration-time curves. We turn now to the consideration of rate laws with additional complications. Most of them describe more complicated reactions and we can anticipate the finding that most real chemical reactions are composites, composed of two or more elementary reactions. Three classifications of composite reactions can be recognized (1) reversible or opposing reactions that attain an equilibrium (2) parallel reactions that produce either the same or different products from one or several reactants and (3) consecutive, multistep processes that involve intermediates. In this chapter we shall consider the first two. Chapter 4 treats the third. 

Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

The progress of precipitation is revealed by the concentration/time curves for zinc and phosphate, since both these species are present initially in solution. There should be maxima for the soluble aluminium, calcium and fluoride which are extracted from the glass, but because of the early onset of precipitation these are not observed. Precipitation is accompanied by an increase in pH when it reaches 1-8, at which juncture 50% of both zinc and phosphate have been precipitated, the cement paste gels (5 minutes after preparation). 

The concentration-time curve can be integrated numerically and yields the so-called area under the curve (AUC) ... 

Compound (1) suffered from an unfavorable pharmacokinetic profile when studied in rats. It is cleared very rapidly from rat plasma (half-life, t 2 — 0.4/z) and is poorly bioavailable F — 2%), as reflected by the low plasma concentration (area under the plasma concentration-time curve, AUCo oo = 0.2pMh) following a single oral dose of 25mg/kg in rats [42]. The main challenge was to further optimize this series to obtain NS3 protease inhibitors with low-nanomolar cell-based potency (EC5q< 10 nM) and with an adequate pharmacokinetic profile for oral absorption. 

The area under the PCP concentration-time curve (AUC) from the time of antibody administration to the last measured concentration (Cn) was determined by the trapezoidal rule. The remaining area from Cn to time infinity was calculated by dividing Cn by the terminal elimination rate constant. By using dose, AUC, and the terminal elimination rate constant, we were able to calculate the terminal elimination half-life, systemic clearance, and the volume of distribution. Renal clearance was determined from the total amount of PCP appearing in the urine, divided by AUC. Unbound clearances were calculated based on unbound concentrations of PCP. The control values are from studies performed in our laboratory on dogs administered similar radioactive doses (i.e., 2.4 to 6.5 pg of PCP) (Woodworth et al., in press). Only one of the dogs (dog C) was used in both studies. 

When we calculated systemic and renal clearance based on the area under the unbound PCP concentration-time curve, we found essentially no change in these parameters compared to the control studies without antibody (Woodworth et al., in press) (figure 6). 

In studies in rats and mink that used more than one dose, the area under the plasma-IMPA concentration time curves indicated that at high doses the principal pathway for the conversion of diisopropyl methylphosphonate to IMPA was saturated (Bucci et al. 1992). In rats, metabolism was saturated at an oral dose of 660 mg/kg, but not at 66 mg/kg in mink, an oral dose of 270 mg/kg caused metabolic saturation which did not occur at 27 mg/kg. 

The significance of P-gp, however, in affecting absorption and bioavailability of P-gp substrate drugs can be seen in studies in knockout mice that do not have intestinal P-gp. The gene responsible for producing that protein has been knocked out of the genetic repertoire. Those animals evidenced a sixfold increase in plasma concentrations (and AUC, area under the plasma concentration-time curve) of the anticancer drug paclitaxel (Taxol) compared to the control animals [54]. Another line of evidence is the recent report... 

Table 2 Peak Plasma Levels and Areas Under Plasma Concentration Time Curves Following Oral and Intravenous Administration to Men...

AUMC = area under the first-moment curve for tissue i AUMCP = area under the first-moment curve for plasma AUCP = area under the plasma concentration-time curve... 

Area under the plasma concentration-time curve... 

Another method of predicting human pharmacokinetics is physiologically based pharmacokinetics (PB-PK). The normal pharmacokinetic approach is to try to fit the plasma concentration-time curve to a mathematical function with one, two or three compartments, which are really mathematical constructs necessary for curve fitting, and do not necessarily have any physiological correlates. In PB-PK, the model consists of a series of compartments that are taken to actually represent different tissues [75-77] (Fig. 6.3). In order to build the model it is necessary to know the size and perfusion rate of each tissue, the partition coefficient of the compound between each tissue and blood, and the rate of clearance of the compound in each tissue. Although different sources of errors in the models have been... 

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