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Integrated Equations

The differential equations developed on the previous page provide concise descriptions of the rate of change of drug concentration (dCp/dt) or the elimination rate (dX/dt). However, they can be difficult to use when trying to determine kel or Cl. Measuring the tangent of the Cp versus time plot can not be determined accurately. Later in the Chapter dealing with the Analysis of Urine Data we will describe measuring the rate of excretion directly from the data. However, integrated forms of Equation 5.4.1 and 5.4.2 are generally more useful.

Using Laplace transforms we can start with Equation 5.5.1 (Equation 5.4.1)

Equation 5.5.1 Rate of Change of Concentration versus Concentration

Taking the Laplace of this equation:

Equation 5.5.2 Laplace of Equation 5.5.1

In Equation 5.5.2 L(Cp) and Cp with the bar across the top represent the Laplace of the variable Cp. Rearranging the equation, recognizing that Cp₀ is the initial drug concentration, and solving for the Laplace of Cp gives:

Equation 5.5.3 Laplace of Cp

The back transform can be determine from Laplace tables as described in Chapter 2 or using the finger-print method described in Chapter 7.

Equation 5.5.4 Integrated Equation for Cp versus Time

This equation describes the single exponential decline in drug concentration as a function of time. This fall in plasma concentration is called mono-exponential decay. If we know kel and Cp₀ we could calculate Cp at any time after a single IV bolus dose. However, if still isn't very convenient for estimating a value of kel from concentration versus time data. We could use a non linear regression program such as Boomer however for estimation using graph paper we would prefer a straight line equation. This can be achieved by taking the natural logarithm of both side of Equation 5.5.4

Equation 5.5.5 Ln(Cp) versus Time

This integrated (logarithmic) form of the equation for Cp represents a straight line equation, that is an equation 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 5.5.1 Linear plot of ln(Cp) versus time

NOTICE, there are no UNITS for ln(Cp) in Figure 5.5.1. There units of hour for time (X axis) so slope 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.

Alternately we could use semi-log graph paper. As mentioned earlier the scale on the y-axis are proportional to the log of the number, not the number itself. This plot allows us to calculate the slope and thus kel given Cp versus time data.

Figure 5.5.2. Semi-log plot of Cp versus time

Click on the figure to view the Java Applet window

This file was last modified: Thursday 13 Feb 2003 at 10:30 AM