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CHAPTER OBJECTIVES

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  • Algebraically solve mathematical expressions related to pharmacokinetics.1

  • Express the calculated and theoretical pharmacokinetic values in proper units.

  • Represent pharmacokinetic data graphically using Cartesian coordinates (rectangular coordinate system) and semilogarithmic graphs.

  • Use the least squares method to find the best fit straight line through empirically obtained data.

  • Define various models representing rates and order of reactions and calculate pharmacokinetic parameters (eg, zero- and first-order) from experimental data based on these models.

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1It is not the objective of this chapter to provide a detailed description of mathematical functions, algebra, or statistics. Readers who are interested in learning more about these topics are encouraged to consult textbooks specifically addressing these subjects.

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CALCULUS

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Pharmacokinetic models consider drugs in the body to be in a dynamic state. Calculus is an important mathematic tool for analyzing drug movement quantitatively. Differential equations are used to relate the concentrations of drugs in various body organs over time. Integrated equations are frequently used to model the cumulative therapeutic or toxic responses of drugs in the body.

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Differential Calculus

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Differential calculus is a branch of calculus that involves finding the rate at which a variable quantity is changing. For example, a specific amount of drug X is placed in a beaker of water to dissolve. The rate at which the drug dissolves is determined by the rate of drug diffusing away from the surface of the solid drug and is expressed by the Noyes–Whitney equation:

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where d denotes a very small change; X = drug X; t = time; D = diffusion coefficient; A = effective surface area of drug; l = length of diffusion layer; C1 = surface concentration of drug in the diffusion layer; and C2 = concentration of drug in the bulk solution.

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The derivative dX/dt may be interpreted as a change in X (or a derivative of X) with respect to a change in t.

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In pharmacokinetics, the amount or concentration of drug in the body is a variable quantity (dependent variable), and time is considered to be an independent variable. Thus, we consider the amount or concentration of drug to vary with respect to time.

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EXAMPLE

The concentration C of a drug changes as a function of time t:

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Consider the following data:

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Time (hours) Plasma Concentration of Drug C (μg/mL)
0 12
1 10
2 8
3 6
4 4
5 2

The concentration of drug C in the plasma is declining by 2 μg/mL for each hour of time. The rate of change in the concentration of the drug with respect to time (ie, the derivative of C) may be expressed as

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