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Graphing Data on Linear and Semi-Log Graph Paper

A very important part of pharmacokinetic analysis is the ability graph data and interpret the resulting graphs. The graph can provide a useful picture of the data and provide an insight into underlying mathematical model. Graphs should be drawn carefully. Straight lines should be straight --- use a ruler!!

Pharmacokinetic data involves models with rate processes. The most simple of these would be a single first order decline described with a single exponential terms. We may use an equation such as: Cp = 40 * exp(- 0.23*t) to describe drug concentrations after an IV bolus dose. These data are drawn in three graphs on this page. The first figure is a linear or Cartesian) plot of the data versus time. Notice the smooth decline in concentration with time. The second figure is a plot of the natural log (ln) of the concentration values versus time. Now we get a straight line graph. You might try 'rearranging' the equation above to verify that a straight line is to be expected. The third graph is on different paper. This is semi-log graph paper. On this graph paper the scale on the y-axis is proportional to the log of the number not the number itself. Notice that the distance from 1 to 2 is the same as the distance from 2 to 4 or from 4 to 8. Again, we have a straight line but without the need to calculate the natural log of each number.

Figure 2.4.1. Linear Plot of Cp versus Time

Click on the figure to view the Java Applet window

Figure 2.4.2. Linear Plot of ln(Cp) versus Time

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Figure 2.4.3. Semi-log Plot of Cp versus Time

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For many calculations involving rate constants you will be determining the slope of these lines. The second and third graphs (Figure 2.4.2 and 2.4.3) are a lot easier to use in terms of calculating slopes than Figure 2.4.1, since they each represent a straight line. Later we will revisit the equation for this line. Remember semi-log graph paper has a normal x- axis scaling, but the y- axis scaling is proportional to the log of the number not the number itself. It saves you from taking the log of each number before you plot it. Note: Once you take the log of a number you loose the units.

Example use of semi-log graph paper: Plot the data, draw a line"through the data", and calculate the slope of the line.

Table 2.4.1 Example Cp versus Time Data
Time (hr)	1	2	4	8	12
Cp (mg/L)	20	15	6.8	3.2	1.3

Figure 2.4.4. Semi-log Plot of Cp versus Time

A best-fit line is drawn through the data points and extended to the extremes of the graph paper (for better accuracy). Values for Cp¹ and Cp² are read from the y-axis and t¹ and t² values are read from the x-axis. These values can be used to estimate a value for the slope of the line and also the rate constant for the drug elimination. From the Y-axis the first point is 0, 22.8 and from the X-axis the second point is 12.8, 1 (in the format x-value, y-value). Always try to use the extremes of the line for the best accuracy.

From Figure 2.4.4	Value
Cp¹	22.8
Cp²	1.0
t¹	0
t²	12.8

The slope can be calculated using the equation:

Equation 2.4.1 Slope of the line on a semi-log plot

This equation uses logarithms with base 10 for the calculation of slope.

Thus the slope of the line in Figure 2.4.4 is: [log(1.0) - log(22.8)]/(12.8 - 0) = (0.000 - 1.358)/12.8 = - 0.106 hr^-1

In the study of pharmacokinetics first order rate processes and rate constants are common. When the rate constant, k, is calculated from the slope drawn on a semi-log plot, k is found to equal to -slope * 2.303 [Note ln(10) = 2.303]. Thus, the rate constant is calculated as -slope x 2.303. The calculation of k is easier with calculators (etc.) if you use ln (logarithm with base 'e') in the calculation. At the same time you can change the sign by calculating the numerator as ln(Cp¹) - ln(Cp²). Thus k can be calculated using the equation:

Equation 2.4.2 Equation for estimating First Order Rate Constants

Thus k = [ln(22.8) - ln(1.0)]/(12.8 - 0) = (3.127 - 0.000)/12.8 = 0.244 hr^-1

Practice and Assignment Resource

I have developed two exercises to look at drawing straight lines through data on both linear and on semi-log graph paper.

Linear Graph

Semi-log Graph

Try them out if you need practice graphing data and calculating intercept and slope in these graphs.

Drawing a best-fit line through the Data

Drawing a line through the data doesn't mean through just two data points but through all the data points. Be especially careful about picking two adjacent data points. Sometimes the first and last point can work but the last point, the lowest concentration data point will probably be inaccurate. The best approach is to put the line through all the data. There should be points above and below the line.

Calculate kel from the best-fit line

Graph Paper Resources

Graph Paper in Adobe Reader (v5.0) - PDF Format. If you have Adobe Reader installed (or Preview in Mac OS X) clicking on the links below will provide the graph paper indicated.

This file was last modified: Monday 03 Feb 2003 at 09:58 PM