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4. Integrated equation

We could use the equation

(Equation IV-2 from earlier in this Chapter)

to calculate kel. Plotting - dCp/dt versus Cp would give kel as the slope of the straight line. Howver, in practice, it is difficult to measure dCp/dt (the tangent to the line drawn through the Cp versus time plot) accurately and so this equation is not often used directly (see Figure 7 from earlier in this Chapter).

The problem is the differential term. Mathematically, we get rid of this term by integrating the equation. We can go through this one because it is easy and also to see what is involved in the process, but for other differential equations we'll just use the answer.

Starting with

(Equation IV-2 from earlier)

Rearranging gives

Equation IV-3. Differential Equation Rearranged

Integrating both sides gives

Equation IV-4. Integration of the Rearranged Equation

This gives (by looking the answer in Maths tables) -

Equation IV-5. Rearrange the Integrated Equation

Note that the integral of 1/x is ln(x), when ln represents log to the base e. This leads to the utility of the log to base e which is used extensively in the area of pharmacokinetics.

Expanding Equation IV-5 gives

ln Cpt - ln Cp0 = - kel * t + kel * 0

Equation IV-6. Natural log of Cp versus Time

Or

ln Cpt = ln Cp0 - kel * t

Equation IV-7. Integrated Equation for ln Cp versus Time

Note the use of natural log to base e

Rearranging gives

Equation IV-8. Rearranging the Integrated Equation

Taking the antilog (base e) of both sides gives

Equation IV-9. Another form of the integrated equation

This is a single exponential equation. The fall in plasma concentration is therefore called monoexponential decay

Looking back at the ln (base e) equation; we can convert that into log (base 10), common logs, by dividing by 2.303 giving

Equation IV-10. Log base 10 Cp versus Time

This is another form of the integrated equation

Both of these integrated (logarithmic) forms represent a straight line equation, that is of the form: y = a - m * t with a = intercept and m = slope.

Plotting ln (Cp) versus t should give a straight line with a slope of - kel and an intercept of ln Cp0.

Figure IV-4., Linear plot of Cp versus time, showing intercept and slope. NOTICE, no UNITS for ln (Cp) but units of hour for time (X axis).

Units: Slope has units of time-1 thus kel has units of time-1 e.g. min-1, hr-1.

Now we can measure kel by determining Cp versus time and plotting ln Cp versus time.

On Semi-log graph paper

This plot allows us to calculate kel given Cp at various times.

Figure IV-5. Semi-log plot of Cp versus time, showing slope and intercept


Using a JAVA aware browser you can create your own versions of Figure IV-5.

Plasma Concentration versus Time Plots


Table IV-1. Example Values for Elimination Rate Constant and Half-life for Elimination1

kel, hr-1t1/2, hr
Acetaminophen0.282.5
Diazepam0.02133
Digoxin0.01740
Gentamicin0.352.1
Lidocaine0.431.6
Theophylline0.06311


1 Ritschel, W.A. 1980 Handbook of Basic Pharmacokinetics, 2nd ed., Drug Intelligence Publications, p413-426.

This page was last modified: 12 February 2001

Copyright 2001 David W.A. Bourne


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