# Chapter 4

# One Compartment IV Bolus

return to the Course index

previous | next

## 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 4.5.1 (Eqn 4.4.1) are generally more useful.
Laplace transforms or other methods can be used to integrate Equation 4.5.1.

**Equation 4.5.1 Rate of Change of Concentration ***versus* Concentration

to give

**Equation 4.5.2 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 Cp^{0} we could calculate Cp at any time after a single IV bolus dose. However, it 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 purposes using graph paper we would prefer a straight line equation. A straight line equation can be achieved by taking the natural logarithm of both side of Equation 4.5.2

**Equation 4.5.3 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(Cp^{0}).

**Figure 4.5.1 Linear plot of ln(Cp) ***versus* time

Click on the figure to view the interactive graph

NOTICE, there are no UNITS for ln(Cp) in Figure 4.5.1. There are units of hour for time (X axis) so the 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 4.5.2. Semi-log plot of Cp ***versus* time

Click on the figure to view the interactive graph

return to the Course index

This page was last modified: Wednesday, 29th Nov 2017 at 5:53 pm

Privacy Statement - 25 May 2018

Material on this website should be used for Educational or Self-Study Purposes Only

iBook and pdf versions of this material and other PK material is available

Copyright © 2001-2019 David W. A. Bourne (david@boomer.org)