This document provides guidance on solving a two-compartment pharmacokinetic model problem using plasma concentration data from a patient who received an intravenous injection of meperidine. It explains that a two-compartment model can be described by two first-order processes and outlines the steps to determine the equations for the distribution and elimination phases using the method of residuals. These steps include taking the natural log of concentration values, performing linear regression to calculate rate constants, and subtracting extrapolated from observed concentrations to determine residuals and establish the distribution phase equation.

1. An Nguyen – 9/20/2014
Two-Compartment Model Guide
Two-compartment Model
A patient was given an intravenous injection of 50 mg of meperidine for postoperative pain.
The following plasma concentrations were obtained.
Determine A, alpha, B, beta, Vp, [VD]beta, [VD]SS & CIT.
Time (hr) Concentration (ng/ml)
0.5 420
1.0 290
1.5 220
2.0 180
2.5 150
3.0 125
4.0 96
6.0 60
8.0 38
When approaching a two-
compartment model problem, keep
in mind the equation, what the
graph will look like, and what you
are trying to find.
We are trying find the equation of
a line that describes the
elimination of the drug (red line)
and the distribution of the drug
(green line) so that we can
calculate the plasma concentration
(Cp) of a drug at any time after
administration.

2. An Nguyen – 9/20/2014
Since the drug reaches steady state towards the end of the graph (in the elimination phase) and
shows a linear relationship, we can use the last few points to calculate the equation of the
elimination phase.
Using a TI-83 Calculator:
- First enter the list editor by going to STAT > EDIT…
- In L1, enter the last 3 time entries (4, 6, 8)
- In L3, enter the last 3 concentrations (96, 60, 38)
- Since Two-compartment models are represented by two 1st
order processes, we will need the natural logs of the
concentrations. (Please review 1st
order reactions if this
doesn’t make sense.)
- Highlight L2, the bottom of the calculator should read L2 =
(blank). Enter LN > 2nd
> 3 and hit enter to take the natural
log of the values in L3 (Figure 1)
- Now that you have the Time in L1 and natural log of concentration in L2, go to 2nd
>
MODE to quit the list editor. Then go to STAT > CALC > LinReg (ax +b). This tells the
calculator to figure out the equation of a line using L1 as X-axis and L2 as the Y-axis.
- You should obtain the results in Figure 2. Since this is the
equation for the elimination phase; β = -0.231 and B = e^5.48
= 241 ng/ml (Don’t forget that we entered in the values as
natural log of the concentration, so we have to do the inverse
function, e^, to obtain the actual concentration)
- Putting together what we have so far in the Two-
Compartment model equation:
Cp = Ae-αt
+ 241e-0.231t
Now that we have our equation for the elimination phase, we can use it to find the
equation for the distribution phase. In order to do that, we need to use the Method of
Residuals.
Figure 1: Natural Log of L3
Figure 2: Results of LinReg (ax +b)

3. An Nguyen – 9/20/2014
Since the first few points of the graph do not
express a linear relationship, it is difficult to
calculate a line to describe the distribution phase.
The Method of Residuals attempts to create new
points that better represent a linear relationship by
subtracting observed values (w, x, y, z) from
extrapolated points (x’, x’, y’, and z’).
Using a TI-83 Calculator:
- First, go to the list editor and enter in the first 3 times (0.5, 1, and 1.5) in L1 and
concentrations (420, 290, and 220) in L3.
- To calculate the extrapolated values using the elimination equation, quit the list editor,
enter the elimination equation (241e-0.0.231t
) and substitute t with L1 by doing 2nd
> 1.
- Since we will be using these values later, store these
values in L4 by going to STO >2nd
>4 and hit enter. The
results should look like figure 4.
- To obtain the residual values, enter the list editor,
highlight L5 and enter L3 – L4. This will subtract the
extrapolated values (L4) from the observed values (L3)
and place it in L5. (Figure 5)
- Since we need the natural log of the residuals to calculate
the equation of the line, highlight L2 and enter LN > 2nd
> 5 and hit enter.
- Repeat the same process for calculating the equation for
the elimination phase, go to STAT > CALC > LinReg
(ax+b). From the results you should obtain: α = -1.421, A
= e^6.03 = 415 ng/ml.
Putting it together the equation should be: Cp = 415e-1.421t
+ 241e-0.231t
Figure 3: Method of Residuals – New points are calculated by
subtracting observed points (w, x, y, z) from extrapolated points
x’, x’, y’, and z’.
Figure 4: Calculating extrapolated values
and storing in L4
Figure 5

4. An Nguyen – 9/20/2014
References
1. Applied Biopharmaceutics and Pharmacokinetics: Leon Shargel, S. Wu-Pong and
Andrew Yu. 5th edition, 2005. McGraw-Hill, New York.
2. Concepts in Clinical Pharmacokinetics: J. T. DiPiro. 4th edition, 2005. ASHSP,
Bethesda, MD.