Single-dose equations

Multiple first-order input and one-compartment disposition (11D1). In a similar manner, MDF can be applied to the I1D1 single-dose equations (Eq. 1.26) to get the multiple-dose equation for Cp (Eq. 1.48) ... 

A multiple dosing equation is derived by inserting the MDF into the single-dose equation as a coefficient to each exponential term in the equation. The MDF can be reduced to a simpler form at steady state or when the amount administered equals the amount being eliminated. When a number of doses are administered, n becomes large enough (i.e., steady state) so that the term,, becomes negligible and Equation 1.29 simplifies to Equation 1.30. 

The Dost ratio permits the determination of the amount and/or the plasma concentration of a drug in the body at any time t (range, t=0 to t — T) following the administration of the nth (i.e. second dose, third dose, fourth dose, etc.) dose by intravascular and/or extravascular routes. In other words, this ratio will transform a single dose equation into a multiple-dosing equation. 

TED50= Tl[2 (IM/H) ln(2 + (Cpeak/CE50) AH) TED90 = 7T/2 (IM/H) ln(10+9 (Cpeak/CE50)A//) For single dose monoexponential kinetics and direct effect conditions, the area under the effect time curve (AUEC) can be derived by integration of the Hill equation. 

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

The most widely used approach to evaluate plasma (total) CL involves IV administration of a single dose of a chug and measuring its plasma concentration at different time points, as shown in Fig. 2.2. In this manner, the calculated clearance will not be confounded by complex absorption and distribution phenomena which commonly occur during oral dosing. Clearance is derived from the equation (Rowland and Tozer, 1995)... 

In this equation, the risk of cancer is assumed to be 100% for an effective equivalent dose of 20 Sv. The equation is based on the probabilities listed in Table 22.10. In order to take into account that a dose dehvered with a relatively low dose rate over a longer period of time has an appreciably smaller effect than a single dose, a dose reduction factor of 2 is recommended for smaller dose rates. However, in the report of the United Nations Scientific Committee on the Effects of Atomic Radiation... 

Fig. 1.11 Semilogarithmic graph showing the decline in plasma drug concentrations (with time) following the intravenous injection of a single dose (10mg/kg). The biexponential equation of the disposition curve is shown (inset). The half-life, U /2 (p), of drug is calculated from the expression U /2 (p) = 0.693/(3, where (S (0.0058 min-1) is the negative logarithm of the slope of the linear terminal portion (elimination phase) of the disposition curve. (Reproduced with permission from Baggot, (1977).)...

Now, given that CL can be defined as given in Equation 17.9 as the relationship between an available single dose and AUC, Equation 17.7c becomes... 

MAT and MDT for a test formulation are determined from drug plasma concentra-tions-time data obtained in a single dose study according to the following equations ... 

Suppose one does a study wherein a single dose of the drug is given both orally and intravenously on two separate occasions and finds that the oral concentrationtime data were best fit using a four-term polyexponential equation, whereas after intravenous administration the concentration-time profile was best fit with a two-term polyexponential equation. In this case there are 27 possible compartmental models to choose from. Whereas most books on pharmacokinetics present the 1- and 2-compartment model, the situation is clearly not that simple (see Wagner s (1993) text for examples). 

The equation for Css ave ends up being the same no matter what type of drug absorption or what type of single-dose model are being used. Finally, consider the... 

Systemic clearance may be determined at steady state by using Equation (1-2). Fora single dose of a drug with complete bioavailability and first-order kinetics of elimination, systemic clearance may be determined from mass balance and the integration of Equation (1-3) over time ... 

The MRT of a drug following administration of a single dose is provided by the following equation ... 

The parameter tp in Equation 1.33 represents the time to peak at steady state, which is slightly less than tp, the time to peak for a single dose. The valne for tp can be determined from Equation 1.33. °... 

It is also worth mentioning here that the pharmacokinetic parameters obtained following the administration of a single dose of a drug, intra-or extravascularly, may prove to be helpful while tackling some equations in multiple-dosing pharmacokinetics. This includes the intercepts of the plasma concentration versus time data, the systemic clearance and the absolute bioavailability of a drug, when applicable. 

Equation 11.11 permits the determination of plasma concentration at any time (t=0 to f=T), following the attainment of steady state. Please note that (Cp)o is the initial plasma concentration (dose/V) that can be obtained following the administration of a single dose. Therefore, if we know the dosing interval and the elimination half life of a drug, we can predict the steady-state plasma concentration at any time t (between 0 and t) (Fig. 11.6). 

Please note, Eqs 11.22 and 11.23 do not require the calculation or the knowledge of the apparent volume of distribution, the elimination rate constant or the dose given every dosing interval. These equations, however, do assume that apparent volume of distribution, the elimination rate constant and the dose are constants over the entire dosing period. In Eq. 11.23, please note that the term (AUC)o is the area under the plasma concentration time curve following the administration of a single dose. Therefore, the dosing... 

Equation 12.3 may be employed to predict the plasma concentration of drug at any time during the dosing interval following the administration of nth dose (second, third, fourth, etc) (Fig. 12.2). However, in order to make such predictions, it is essential to have knowledge of the apparent volume of distribution, the fraction of the administered dose absorbed into the systemic circulation, the apparent first-order rate constant for absorption, the intercept of the plasma concentration versus time profile following the administration of the single dose, and the elimination rate constant. Furthermore, earlier... 

Equation 12.13 permits determination of peak plasma concentration for a drug administered extravascularly provided the intercept value for an identical single dose, peak time, elimination half life and the dosing interval are known. 

It will be very helpful to begin to compare Eq. 11.12 (for an intravenous bolus) and Eq. 12.13 (for extravascularly administered dose) for similarity and differences, if any, and identify the commonality between the two equations. It may he quickly apparent that the information obtained following the administration of a single dose of a drug, either intravenously or... 

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