In the first-order model, the rate of drug elimination is proportional to the plasma drug concentration such that:

where:

dC = a small change in concentration dt = a small change in time C = concentration

The elimination pathways are not saturated but are gradually recruited as the concentration of drug increases, and vice versa. The rate of change of drug concentration is, therefore, also proportional to the drug concentration. The change in concentration is a lowering of concentration and so the formula has a minus sign. Thus, for first-order kinetics:

This can be calculated from the following derived formula called Key equation 1: where:

kel = elimination rate constant

Figure PK.11 demonstrates this process in which a constant volume is cleared of drug in each time interval resulting in an ever declining rate of fall in drug and drug concentration. Although the drug concentration decreases with time, it approaches but never actually reaches 'zero' drug concentration. The rate of that decrease (the gradient of the slope) also falls with time. This is an exponential decay.

Half Lives and Time Constants (Figure PK.12)

In first-order kinetics, the time taken for the concentration to halve (half life) is a feature of exponential functions of this type. Half lives are hybrid constants that are dependent on primary constants. The time constant, another such feature, is based on the rate of change of concentration (the gradient of the plot). The time constant is the time that it would take for the drug concentration to reach zero if elimination continued at the rate of the chosen starting point. Time constants (t) also apply to exponential functions of the form y = 1 - ex. Figure PK.13 shows the proportion of the initial concentration that exists after a given number of time constants. The initial concentration in this sense may be any point on the plot from which timing is started. As the time constant and half-life are constants for a given exponential function, they must have a constant relationship which is described by the formula:

Dr^g 1000 -nii

Figure PK.10. Effect of zero-order elimination on drug concentration

Figure PK.11. Effect of firstorder elimination on drug concentration constant regardless of the concentration

Half lives

Tims constants

Figure PK.12. Plot of drug concentration against time showing half-lives and time constants (using linear scales)

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