Elimination rate first order equations

The rate of elimination is an important characteristic of a drug. Too rapid an elimination necessitates frequent repeated administration of the drug if its concentration is to reach its therapeutic window. Conversely, too slow an elimination could result in the accumulation of the drug in the patient, which might give an increased risk of toxic effects. Most drug eliminations follow first order kinetics (equations (8.1) and (8.2)), no matter how the drug is administered, but there are some notable exceptions, such as ethanol which exhibits zero order kinetics where ... 

For drugs eliminated by first-order kinetics, the relationship between dosing rate and steady-state plasma concentration is given by rearranging Equation 2.3 as follows ... 

Water loss is identified [13] as being diffusion controlled across a dehydration layer of significant thickness that advances into the reactant particles (see Figure 8.1.). The Ca(OH)2 structure is maintained during the greater part of reaction [14]. The rate of this reaction is more deceleratory than the requirements of the contracting volume equation and data satisfactorily fit the first-order equation. The anhydrous residue later recrystallises to CaO but the principal water elimination zone and the phase transformation interface are separated and may advance independently. 

In studying the kinetics of benzene oxidation to maleic anhydride and in order to eliminate diffusion hindrance, Ioffe and Lyubarskii (151) used the flow-circulating method. The rate of this reaction was found to be proportional to benzene concentration to the power of 0.78, and that of high conversion to the power of 0.71. The rate of maleic anhydride oxidation followed a first order equation. Kinetic equations were derived from experimental results ... 

Upon substitution of an appropriate kinetic expression for the rate of generation or consumption of solute within the tissue space, Equation 3-50 can be solved to determine concentration as a function of time and position. Full analytical solutions are generally difficult to obtain, unless both the kinetic expression and the geometry of the system are simple. For example, consider the linear diffusion of solute from an interface where the concentration is maintained constant (as in Figure 3.4d). If the diffusing solute is also eliminated from the tissue, such that the volumetric rate of elimination is first order with a characteristic rate constant k, Equation 3-51 can be reduced to ... 

The chemical meaning of these mathematical equations is that the rate law is first order with respect to the amine base for each reaction (i.e. interconversion of la and Ih and hydrogen bromide elimination). 

The values of the apparent rate constants kj for each temperature and the activation enthalpies calculated using the Eyring equation (ref. 21) are summarized in Table 10. However, these values of activation enthalpies are only approximative ones because of the applied simplification and the great range of experimental errors. Activation entropies were not calculated in the lack of absolute rate constants. Presuming the likely first order with respect to 3-bromoflavanones, as well, approximative activation entropies would be between -24 and -30 e.u. for la -> Ih reaction, between -40 and - 45 e.u. for the Ih la reaction and between -33 and -38 e.u. for the elimination step. These activation parameters are in accordance with the mechanisms proposed above. 

Quantitative estimation of ventilation by indirect methods in mussels requires four assumptions (16) a) reduction of concentration results from uptake, b) constant ventilation (pumping) rate, c) uptake of a constant percentage of concentration (first order process), d) homogeneity of the test solution at all times. Our transport studies have utilized antipy-rine (22, 23) a water soluble, stable chemical of low acute toxicity to mussels. It is readily dissolved in ocean water or Instant Ocean and is neither adsorbed nor volatilized from the 300 ml test system. Mussels pump throughout the 4 hour test period and this action is apparently sufficient to insure homogeneity of the solution. Inspection of early uptake and elimination curves (antipyrine concentration as a function of time) prompted use of Coughlan s equation (16) for water transport. 

In general, taking the ratio of two rate equations eliminates the time variable and gives information on the product distribution. So dividing Eq. 34 by Eq. 32 we obtain the first-order linear differential equation... 

There is a linear relationship between dose and plasma drug levels (i.e., linear or first-order pharmacokinetics) in normal and ultrarapid metabolizers. In these individuals, the earlier equation can be used to predict the daily dose needed to produce a specific plasma drug level once TDM has been done to estimate the patient s elimination rate. In poor metabolizers, TCAs follow nonlinear pharmacokinetics (i.e., disproportionate increases in plasma drug levels with dose increases) because they lack the CYP 2D6 and must use lower affinity enzymes to metabolize these drugs. 

For a first-order reaction, in either reactant, the combination of these equations and the elimination of the surface and liquid-phase concentrations lead to the formulation of an overall rate, expressed as a function of the bulk gas-phase concentration. This procedure is essentially the same as the one presented analytically in Section 3.1.2 for the derivation of an overall rate in three-phase systems. 

The rates of movement of foreign compound into and out of the central compartment are characterized by rate constants kab and kei (Fig. 3.23). When a compound is administered intravenously, the absorption is effectively instantaneous and is not a factor. The situation after a single, intravenous dose, with distribution into one compartment, is the most simple to analyze kinetically, as only distribution and elimination are involved. With a rapidly distributed compound then, this may be simplified further to a consideration of just elimination. When the plasma (blood) concentration is plotted against time, the profile normally encountered is an exponential decline (Fig. 3.24). This is because the rate of removal is proportional to the concentration remaining it is a first-order process, and so a constant fraction of the compound is excreted at any given time. When the plasma concentration is plotted on a logio scale, the profile will be a straight line for this simple, one compartment model (Fig. 3.25). The equation for this line is... 

The opposite of an addition to a double bond is a 1,2-elimination reaction. In solution, where the reaction is promoted by solvent or by base, the most common eliminations (and those to which we shall limit our discussion) are those that involve loss of HX, although loss of X2 from 1,2-dihalides and similar reactions are also well known. The mechanisms of eliminations of HX are of three main types (1) The Ex (elimination, first-order), shown in Equation 7.22, which is the reverse of the AdE2 reaction and which has the same first, and rate-determining,... 

The rates of a large number of eliminations are (1) second-order, first-order each in base and in substrate (2) decreased if /3-deuterium is substituted for /3-hydrogen and (3) strongly dependent on the nature of the leaving group. The mechanism of these reactions (shown-in Equation 7.24), in which G—H and C—X bond breaking are concerted, is E2. 

Then the differences in rate caused by the electronic effect of the substituent are correlated by the Hammett equation log(kz/kH) = poz, where kz is the rate constant obtained for a compound with a particular meta or para substituent, ku is the rate constant for the unsubstituted phenyl group, and crz is the substituent constant for each substituent used. The proportionality constant p relates the substituent constant (electron donating or wididrawing) and the substituent s effect on rate. It gives information about the type and extent of charge development in the activated complex. It is determined by plotting log(kz/kQ) versus ov for a series of substituents. The slope of the linear plot is p and is termed the reaction constant. For example, the reaction shown above is an elimination reaction in which a proton and the nosy late group are eliminated and a C-N n bond is formed in their place. The reaction is second order overall, first order in substrate, and first order in base. The rate constants were measured for several substituted compounds ... 

After the absence of film diffusion effects has been verified and if the reaction order n is known, the expression for the rate equation r = r) kcat[E][S]buik/KM (first-order reaction assumed) can be inserted into the definition for 7j and the unknown rate constant k can be eliminated (Weisz, 1954) [Eq. (5.66)]. 

The relative rates for the p-fluoro substituent deviate even more seriously. At first glance, the discrepancies appear far greater than could be tolerated in a simple first-order treatment (Fig. 45). Considerable scatter from a line defined by the average value of greater than for m-t-butyl (Fig. 42). Moreover, the absence of a relationship between log pf and the reaction constants is also certain. More careful inspection indicates that the data define a rather narrow band of reactivity. The behavior of this group is more clearly portrayed by another test equation which eliminates hydrogen as the abscissa (33). 

If we let K = (D Sa Pc/d), then, since A is present in the equation, n must equal 1, so we have a first-order rate process. Fick s law of diffusion, which is important for quantitating rates of absorption, distribution, and elimination, is thus the basis for using first-order kinetics in most pharmacokinetic models. 

Landrum et al. (1992) developed a kinetic bioaccumulation model for PAHs in the amphipod Diporeia, employing first-order kinetic rate constants for uptake of dissolved chemical from the overlying water, uptake by ingestion of sediment, and elimination of chemical via the gills and feces. In this model, diet is restricted to sediment, and chemical metabolism is considered negligable. The model and its parameters, as Table 9.3 summarizes, treat steady-state and time-variable conditions. Empirically derived regression equations (Landrum and Poore, 1988 and Landrum, 1989) are used to estimate the uptake and elimination rate constants. A field study in Lake Michigan revealed substantial differences between predicted and observed concentrations of PAHs in the amphipod Diporeia. Until more robust kinetic rate constant data are available for a variety of benthic invertebrates and chemicals, this model is unlikely to provide accurate estimates of chemical concentrations in benthic invertebrates under field conditions. 

Equation 7.2 highlights the first-order nature of Figure 7.1. The rate of elimination is directly proportional to Cp. If a patient has twice as much drug in her bloodstream, then that patient s rate of elimination will be twice as high. Because drug elimination so frequently behaves in a first-order fashion, drugs tend to display a regular half-life (ti/2). A half-life is the time interval required for Cp to decrease by a factor of 0.5. The relationship between r i/2 and el is shown in Equation 7.3. 

In contrast to noncompartmental analysis, in compartmental analysis a decision on the number of compartments must be made. For mAbs, the standard compartment model is illustrated in Fig. 3.11. It comprises two compartments, the central and peripheral compartment, with volumes VI and V2, respectively. Both compartments exchange antibody molecules with specific first-order rate constants. The input into (if IV infusion) and elimination from the central compartment are zero-order and first-order processes, respectively. Hence, this disposition model characterizes linear pharmacokinetics. For each compartment a differential equation describing the change in antibody amount per time can be established. For... 

In other words, for elimination processes that follow first order kinetics tm is a constant, characteristic of the drug and the biological system. Consequently, for drugs that exhibit first order elimination kinetics both and ke and t]/2 are used as indicators of the rate of elimination of a drug from the system. Half-lives are normally quoted in the literature and kel values are calculated as required using equation (8.5). 

Since the rate of infusion is normally maintained at a constant value, infusion will usually follow zero order kinetics (Equation (8.4)). Therefore, assuming that elimination processes exhibit first order kinetics, it follows that ... 

The changes in a drug s plasma concentration with time may be calculated for a specific pharmacokinetic model by substituting of the appropriate rate expressions into Equation (8.29). For example, for a one compartment model in which the drug exhibits first order absorption and elimination (Figure 8.10), it is possible to show that ... 

The classic extrusion model gives insight into the screw extrusion mechanism and first-order estimates. For more accurate design equations, it is necessary to eliminate a long series of simplifying assumptions. These, in the order of significance are (a) the shear rate-dependent non-Newtonian viscosity (b) nonisothermal effects from both conduction and viscous dissipation and (c) geometrical factors such as curvature effects. Each of these... 

In the reactive case, r is not equal to zero. Then, Eq. (3) represents a nonhmoge-neous system of first-order quasilinear partial differential equations and the theory is becoming more involved. However, the chemical reactions are often rather fast, so that chemical equilibrium in addition to phase equilibrium can be assumed. The chemical equilibrium conditions represent Nr algebraic constraints which reduce the dynamic degrees of freedom of the system in Eq. (3) to N - Nr. In the limit of reaction equilibrium the kinetic rate expressions for the reaction rates become indeterminate and must be eliminated from the balance equations (Eq. (3)). Since the model Eqs. (3) are linear in the reaction rates, this is always possible. Following the ideas in Ref. [41], this is achieved by choosing the first Nr equations of Eq. (3) as reference. The reference equations are solved for the unknown reaction rates and afterwards substituted into the remaining N - Nr equations. 

Reactions were studied under the pseudo first-order condition of [substrate] much greater than [initial dihydroflavin]. Under these conditions, the reactions are characterized by a burst in the production of Flox followed by a much slower rate of Flox formation until completion of reaction. The initial burst is provided by the competition between parallel pseudo first-order Reactions a and b of Scheme 3. These convert dihydroflavin and carbonyl compound to an equilibrium mixture of carbinolamine and imine (Reaction a), and to Flox and alcohol (Reaction b), respectively. The slower production of Flox, following the initial burst, occurs by the conversion of carbinolamine back to reduced flavin and substrate and, more importantly, by the disproportionation of product Flox with carbinolamine (Reaction c followed by d). Reactions c and d constitute an autocatalysis by oxidized flavin of the conversion of carbinolamine back to starting dihydroflavin and substrate. In the course of these studies, the contribution of acid-base catalysis to the reactions of Scheme 3 were determined. The significant feature to be pointed out here is that carbinolamine does not undergo an elimination reaction to yield Flox and lactic acid (Equation 25). The carbinolamine (N(5)-covalent adduct) is formed in a... 

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