Multiple-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. 

In a multiple dosing regimen involving oral medication, the following equations are useful to find the steady-state parameters ... 

The equations above apply strictly to dmgs administered as a single IV bolus dose, but for drug administered as an infusion or via oral route, or after multiple dosing, the calculation of AUMC must be adjusted to account for drug input [i. e., infusion time (T) or absorption rate constant JCa and extent of bioavailability F], as shown by Straughn [3]. Although, in theory, AUC will not be affected by the route, the AUMC will be overestimated, and this will result in an overestimation of Vss. 

Equation (4) above is the generally applicable expression for a singly labeled diet component when this component also contains a measurable level of isotope 2. For the case where the labeled component has been arranged so that its content of isotope 2 is negligible, as in the case of giving an oral dose of a highly enriched solution, as in the extrinsic tag approach, then Equation (5) is applicable. For situations where more than one component of the diet has been labeled by appropriate use of a selection of different stable isotopes, a more complex set of multiple simultaneous equations can be described (6) for determination of the absorption of zinc from the various labeled diet components. 

Baker (268) compared the responses of volunteers to EA 3580 administered as a dose per man or as a dose per unit of body weight. Four Indicators of effect were used accommodation for near vision, arm-hand steadiness, dynamic flexibility, and manual dexterity The general conclusion was that the use of the dose per unit of body weight may Increase variance, rather than control for extraneous sources of variation, when the purpose of a study is to establish the effects of a substance Itself The use of multiple-regression equations was suggested as an approach to the establishment of definitive Information on effects of chemicals. 

The system of differential equations is integrated using CVODE numerical integration package. CVODE is a solver for stiff and nonstiff ordinary differential equation systems [60]. The fraction of dose absorbed is calculated as the sum of all drug amounts crossing the apical membrane as a function of time, divided by the dose, or by the sum of all doses if multiple dosing is used. 

Any of these compartmental equations can be used to determine serum concentrations after multiple doses. The multiple-dosing factor (1 -e nKr y( i g-Kr where n is the number of doses, K is the appropriate rate constant, and r is the dosage interval, is simply multiplied by each exponential term in the equation, substituting the rate constant of each... 

At steady state, the number of doses becomes large, e approaches zero, and the multiple-dosing factor equals 1/(1 -Therefore, the steady-state versions of the equations are simpler than their multiple-dose counterparts ... 

More recently, Benet has described so-called multiple dosing half-lives, the half-life for a drug that is equivalent to the dosing interval to choose so that plasma concentrations (Equation 17.31) or amounts of drug in the body (Equation 17.32) will show a 50% drop during a dosing interval at a steady state. These parameters are defined in terms of the mean residence time in the central compartment (MRTC)and the mean residence time in the body MRT). 

MRTC in a one-compartment body model is the inverse of the rate constant for elimination. In a multiple-compartment model, where the multiple dosing plasma half-life is useful, MTRC is given by the volume of the central compartment where drug concentrations are measured divided by clearance. MRT in Equation 17.32 is the ratio of AUMC/AUC. 

Equation 7-8 is shown graphically in Figure 7.3a. Concentration in the body compartment rises to some maximum value (which depends on D, k, and ka) and falls with a rate determined by k. Drugs administered in this manner frequently require multiple doses, in an effort to maintain drug concentrations within a therapeutic window for some prolonged time. The outcome of repetitive doses, with each dose assumed to influence the overall concentration independently, is shown in Figure 7.3b. 

Figure 10.2 Pharmacokinetics of intravenous T-20 administration. The graphs show the expected changes in plasma concentration vs time after intravenous administration of a single injection (dashed line) or multiple injections (solid line) of intravenous T-20 (lOOmg/dose). The curves are based on the half-life (1.8 h) and volume of distribution (4.7 L) measured in 17 human volunteers [4] using a one-compartment model (see Equation 7-3). Because of its rapid elimination, multiple doses are needed to maintain the peptide level in the effective range.

The right-hand side of Equation 1.30 also equals the accumulation factor, which is the factor by which accumulation occurs in the body upon multiple dosing, predicting the increase in relative to This factor can also be used to establish a... 

Compartmentel models can be used to simulate or to analyze multiple dose date. When uniform doses are given at uniform dosing intervals, T, the concentration after the fourth IV bolus dose can be calculated using Equation 12.9. 

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. 

At this time, you are urged to consider the similarity between equations for a single intravenous bolus dose (Ch. 3) and multiple doses of intravenous bolus. You may notice an introduction of the term 1-e in the denominator. Otherwise, everything else should appear to be identical. 

Assuming that the fraction of each dose absorbed is constant during a multiple-dosing regimen, the time at which a maximum plasma concentration of dmg at steady state occurs (t max) can be obtained by differentiating the following equation with respect to time and setting the resultant equal to 0. 

The goal is that (Cpj) ji from the loading dose will exactly equal (Cp ) from the multiple-dosing regimen. Hence, equating the two equations ... 

The uses of Equation 4 seems relatively straight-forward for acute single exposures but It also has application to multiple exposures. A simplified application of Equation 4 to multiple doses would predict that the cumulative effect of N multiple doses would be calculated by Equation 9 ... 

Model equations can be augmented with expressions accounting for covariates such as subject age, sex, weight, disease state, therapy history, and lifestyle (smoker or nonsmoker, IV drug user or not, therapy compliance, and others). If sufficient data exist, the parameters of these augmented models (or a distribution of the parameters consistent with the data) may be determined. Multiple simulations for prospective experiments or trials, with different parameter values generated from the distributions, can then be used to predict a range of outcomes and the related likelihood of each outcome. Such dose-exposure, exposure-response, or dose-response models can be classified as steady state, stochastic, of low to moderate complexity, predictive, and quantitative. A case study is described in Section 22.6. 

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