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and fractional rate constant. In many instances this can only be derived by appropriate pharmacokinetic analysis of a given data set following blood or urine sample collection and appropriate chemical analyses to determine toxicant concentrations in either of these biological matrices. Each of the processes discussed thus far absorption, distribution, and elimination can be described as a rate process. In general, the process is assumed to be rst order in that the rate of transfer at any time is proportional to the amount of drug in the body at that time. Recall that the rate of transport (dC/dt) is proportional to toxicant concentration (C) or stated mathematically: dC = KC, dt where K is the rate constant (fraction per unit time). Many pharmacokinetic analyses of a chemical are based primarily on toxicant concentrations in blood or urine samples. It is often assumed in these analyses that the rate of change of toxicant concentration in blood re ects quantitatively the change in toxicant concentration throughout the body ( rst-order principles). Because of the elimination/clearance process, which also assumed to be a rst-order rate process, the preceding rate equation now needs a negative sign. This is really a decaying process that is observed as a decline of toxicant concentration in blood or urine after intravenous (IV) administration. The IV route is preferred in these initial analyses because there is no absorption phase, but only chemical depletion phase. However, one cannot measure in nitesimal change of C or time, t; therefore there needs to be integration after rearrangement of the equation above: dC dC = kdt becomes = k dt, C C which can be expressed as C = C 0 e kt ,
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where e is the base of the natural logarithm. We can remove e by taking the ln of both sides: ln C t = ln C 0 kt. Note that K is the slope of the straight line for a semilog plot of toxicant concentration versus time (Figure 6.12). In the preceding equation it is the elimination rate constant that is related to the half-life of the toxicant described earlier in this chapter. The derived C 0 can be used to calculate the volume of distribution (Vd ) of the toxicant as follows: Dose Vd = . C0 However, toxicokinetic data for many toxicants do not always provide a straight line when plotted as described above. More complicated equations with more than one exponential term with rate constants may be necessary to mathematically describe the concentration-time pro le. These numerous rate constants are indicative of chemical transport between various compartments in the body and not only to a single central compartment as suggested in the simple equation and semilog plot described in
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ABSORPTION AND DISTRIBUTION OF TOXICANTS
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Cp0 Slope = Kel lnCp K
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Central Kel Time (a) (b)
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Figure 6.12 (a) Semilog plot of plasma concentration (Cp ) versus time. Cp 0 is the intercept on the y-axis, and Kel is the elimination rate constant. (b) Single compartment model with rate constants for absorption, Ka and for elimination, Kel .
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Cp0 A1 = A lnCp
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a = l1 b = l2 Time (a) Central (1) Kel = K10
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K12 K21
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Peripheral (2)
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Figure 6.13 (a) Semilog plot of plasma concentration for (Cp ) versus time representative of a two-compartment model. The curve can be broken down into an or 1 distribution phase and or 2 elimination phase. (b) Two-compartment model with transfer rate constants, K12 and K21 , and elimination rate constant, Kel .
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Figure 6.12. In some instances the data may t to a bi-exponential concentration-time pro le (Figure 6.13). The equation to describe this model is C = Ae t + Be t . In other instances, complex pro les may require a three- or multi-exponential concentration-time pro le (Figure 6.14). The equation to describe the three-pro le case is C = Ae t + Be t + Ce t . In the physiological sense, one can divide the body into compartments that represent discrete parts of the whole-blood, liver, urine, and so on, or use a mathematical model describing the process as a composite that pools together parts of tissues involved in distribution and bioactivation. Usually pharmacokinetic compartments have no anatomical or physiological identity; they represent all locations within the body that have similar characteristics relative to the transport rates of the particular toxicant. Simple rst-order kinetics is usually accepted to describe individual
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