1. 1 Compartment Model: IV Dosing
Dose = 1000 mg
Time Conc
(hr) (mg/L)
Calculate
Vd, k, half-life,
AUC (0-t & 0 –inf),
and Clearance
Use of Spreadsheets
(Excel®)
to Calculate
Pharmacokinetic
Parameters.
Dose V
-kt
C = e

2. Brief Tutorial on the use of Spreadsheets (Excel®)
A spread sheet is
designed to complete
numerical calculations
which can be
automated
to allow
data analysis,
projections,
reporting and
construction of graphs
and other
visual aid plots.

3. Brief Tutorial on the use of Spreadsheets (Excel®)
Spreadsheets consist
of rows and columns
In Excel the columns are
given alphabetic labels
A, B, C. D …. X, Y, Z, AA, AB…
and the rows are numbered
1, 2, 3 … 6000+
Individual “cells” are
identified by their
row and column location,
e.g. - A1
Cell A1

4. Brief Tutorial on the use of Spreadsheets (Excel®)
Each cell
can contain
a variety of
“types”
of information
Words
Numbers
Dates
Formula
Pictures… etc.
Formula can be created
by the user or the user
may select a formula from
a vast variety of formula
in a pick-list.

5. Brief Tutorial on the use of Spreadsheets (Excel®)
Creating Formula:
Formula can be constructed by
first initiating a formula
with an equal sign ‘=‘
and then using the following operators:
Addition: ‘+’
Subtraction: ‘-’
Division: ‘/ ’
Multiplication: ‘*’
such that (3x4)divided by 2 is would convert to:
=(3*4)/2
Order of operations applies
and so in the previous formula
parenthesis were not required.

6. Brief Tutorial on the use of Spreadsheets (Excel®)
Formula can also
be created
or chosen
from the
pick-list
to complete
any mathematical
calculation:
Financial functions
Statistical functions
Engineering functions
Math/Trig functions
Logical functions
(IF statements)

7. Brief Tutorial on the use of Spreadsheets (Excel®)
Creating Formula:
Column ‘A’ contains
a series of numbers.
In cell A6
we can create a
formula to add all of
the numbers in
column A together.
This can be done
at least two ways:
Brute Force:
= A1+A2+A3+A4+A5
2. Using the Pick-list
of formula:
=SUM(A1:A5)

8. Brief Tutorial on the use of Spreadsheets (Excel®)
Enter the
Concentration-Time
data into Excel®.
Draw
a [ ]-time profile
and
Calculate:
Volume of Distribution
Half-life,
Area Under the Curve
Clearance

9. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
1. Select Chart Wizard

10. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
Select Chart Wizard
Select XY (Scatter)

11. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
Select Chart Wizard
Select XY (Scatter)
you should also
select for no line
although a chart
with data points
connected by a
smooth line
or a straight line
will be acceptable.

12. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
Select Chart Wizard
Select XY (Scatter)
Ensure the Series
is in columns.

13. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
Select Chart Wizard
Select XY (Scatter)
Ensure the Series
is in columns.
4. Select the
data range and
‘click’ the red arrow

14. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
Select Chart Wizard
Select XY (Scatter)
Ensure the Series
is in columns.
4. Select the
data range and
‘click’ the red arrow
5. With the left mouse
button held down
select cells A4 – B10.
Click the red arrow to
return to the wizard

15. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
6. On return to the
Wizard your graph
should look like a
[ ] –time profile.

16. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
6. On return to the
Wizard your graph
should look like a
[ ] –time profile.
7. Enter labels for
the x and y axis.
Other tabs allow further
refinements to the graph.
You could eliminate
the legend.

17. Brief Tutorial on the use of Spreadsheets (Excel®)
Draw
a [ ] - time profile
The final step selects
placement of the
graph in either the
current sheet
beside the data
or as a graph in a
separate sheet.
Select “As Object in” and you will be able to see
the [ ]-Time plot beside the data near where we will add in calculations.

18. Brief Tutorial on the use of Spreadsheets (Excel®)
You have successfully
completed construction
of the
concentration-time
Graph.

19. Brief Tutorial on the use of Spreadsheets (Excel®)
If you place the cursor
on the values of the
y-axis and ‘double click’
you can further modify
the appearance
of your profile.
Under tab “scale”
you may select
‘Logarithmic scale’
and obtain a
semi-log plot
which should show
a linear decline in
the [ ] – Time profile.

20. Brief Tutorial on the use of Spreadsheets (Excel®)
Your sheet should now look similar to this, although relative size may be different.

21. Brief Tutorial on the use of Spreadsheets (Excel®)
Calculate:
Volume of Distribution
Half-life,
Area Under the Curve
Clearance
In the following slides,
you will be given
instructions on how to
complete the procedure
and the ‘answer’
will be shown
on the next slide.

22. Brief Tutorial on the use of Spreadsheets (Excel®)
Calculate:
Volume of Distribution
Half-life,
Area Under the Curve
Clearance
In cell A12 create a label:
‘Volume of Distribution’
and
In cell B12 create
a formula for Volume.
Several possibilities exist to calculate Volume:
Based on the initial concentration of 100 mg/L and a dose of 1000 mg.( V = Dose/ [ ] )
Estimate a back-extrapolated ‘intercept’ using the function ‘intercept’ in the wizard.
This method requires that the concentrations be converted to log values first.

23. Brief Tutorial on the use of Spreadsheets (Excel®)
Calculate:
Volume of Distribution
Half-life,
Area Under the Curve
Clearance
In cell A16 create a label:
‘Half-life’
and
In cell B16 create
a formula for T½.
The best method for calculation of half-life is to place a straight line through the log-linear
portion of the concentration-time profile. This is done using the ‘slope’ function in the
wizard and requires that the concentrations be converted to log values first.
… Visual inspection of the graph & data will also indicate that the half-life is about 6 hrs.

24. Brief Tutorial on the use of Spreadsheets (Excel®)
A Note on Half-life:
What is the correct half –life?
The choice of points to be
included in the terminal
phase is more obvious in a
1C model with IV bolus
administration than
following oral absorption or
any situation where the
first points are uncertain.
Establish the choice of the
number of points in the
terminal phase using linear
regression and the best r-value.
Start with the last two points in the terminal phase. Place a straight line through the points
using linear regression and the ‘slope’ function. Also calculate the correlation coefficient
[‘correl’] using the wizard for this set of points. Repeat this process, each time increasing
the number of points in the terminal phase. Greatest r-value & greatest # point is best.

25. Brief Tutorial on the use of Spreadsheets (Excel®)
A Note on Half-life: What is the correct half –life?
Q1. Please Explain slide 24, what is the correct half-life in the Excel Tutorial slide set?
A. This slide involves some statistics and pharmacokinetics and is included as a more complete evaluation of the kinetics of this data set. Also, it should be pointed out, that the "answer" for the procedure discussed on one slide is shown on the next slide.
Visual inspection of the graph & data from 12 to 18 hours and 18 to 24 hours will indicate that the half-life is about 6 hrs. Equations will also yield a half-life of 6 hours If you use any two concentrations based on times of time zero, 12, 18 or 24 hours.
However, in this data set, and in concentrations drawn from patients, rounding and or inaccuracy in sample timing or sample analysis will result in concentrations which may not all fall exactly on the line. For example, it would appear that I developed this concentration-time profile using the 6 hour half-life (k= hr-1) just calculated.
Using a half-life of 6 hours, concentrations at 1 and 2 hours can be calculated.
C(1) = C(0)x e(-kt)
= 100 e( x 1)
= mg/L
The actual concentration in the table is 89.1 mg/L. This is a minor deviation but serves to show that not all concentrations will lie directly on the line.

26. Brief Tutorial on the use of Spreadsheets (Excel®)
A Note on Half-life: What is the correct half –life?
Similarly for the concentration at 2 hours, a concentration of mg/L is calculated and the excel file lists a concentration of 79.4 mg/L.
Deviations like this and actually much greater than this will occur all the time in pharamcokinetics, but you are usually unable to identify any particular concentration as being in error. Therefore, you must assume that the deviations occur with equal likelihood across all concentrations. As a result, analysis of the half-life is often done using more than just two selected concentrations.
Half-life is usually calculated by finding the "average" or most likely slope running through a number of concentrations. This involves linear regression and the use of the Excel function "SLOPE"
When you are using the SLOPE function you may not realise that you are using linear regression.
As slide 24 points out, start with the last two points in the terminal phase (time 18 and 24 hr). Place a straight line through the points using linear regression and the ‘slope’ function from Excel. Also calculate the correlation coefficient [‘correl’ using Excel] using the wizard for the same set of points. Repeat this process, each time increasing the number of points in the terminal phase. In doing this you must first convert concentrations to their log values. The choice of points to be included in the terminal phase is more obvious in a 1C model with IV bolus administration than following oral absorption or any situation where the first points are uncertain. In this data set and in most 1C model data sets we have started with the last 2 point because these two points are in the terminal phase. While all points are obviously in the terminat phase in this example, following and IV infusion, oral absorption or IV bolus with distribution, the number of points which could potentially be included will vary depending on the observed kinetic results.

27. Brief Tutorial on the use of Spreadsheets (Excel®)
A Note on Half-life: What is the correct half –life?
Explanation of correlation
When you have completed this process, inspect your results. Correl will give you a number between 0 and 1. When all points lie on the lie, r = 1, and error = 0:[Error = 1-r^2 (r squared)]. The best choice for the terminal phase is the set of points with the greatest r-value & the greatest # of points. Obviously the using only 2 points gives you an r value of 1 and so balancing greatest r value and greatest number of points may seem subjective. For this data set an r value of is observed for 7 points. On slide 25 this is shaded in. I, therefore, select a K value of , which is equivalent to a half-life of 6.02 hours.
It seems that the very minor rounding procedure that appeared in time points 1 and 2 hours resulted in a deviation of 0.02 hr from the estimated 6 hr half-life (~1 minute!).

28. Brief Tutorial on the use of Spreadsheets (Excel®)
Calculate:
Volume of Distribution
Half-life,
Area Under the Curve
Clearance
In cell A24 create a label:
‘AUC(0-t)’
and
In cell B24 create
a formula for AUC(0-t).
The best method for calculation of AUC involves creating a column (D) where the AUC
from one time point to the next is calculated. This may be done using either raw conc.
values or log concentrations values. In this problem the method using the log of the
concentrations will have less error. The sum of the ‘partial’ areas can be reported in A24.

29. Brief Tutorial on the use of Spreadsheets (Excel®)
Note on Repeating Formula:
In cell D5, the formula for
AUC from time zero to 1 hour
should be placed.
However, this formula is
essentially the same for all
cells in D6 trough D10.
Rather than re-type the formula
you can copy the formula
cells by a copy-paste procedure
or a drag procedure.
Drag procedure. Place the cursor in cell D5. The formula appears in the ‘edit formula’
section (green arrow). In the lower right corner of cell D5 there is a small square. Place
your cursor on this square. Your cursor will turn to a smaller plus (+) sign. Depress the
left mouse button. Drag the square to A10. The formula has been copied.

30. Brief Tutorial on the use of Spreadsheets (Excel®)
Calculate:
Volume of Distribution
Half-life,
Area Under the Curve
Clearance
Calculation of the AUC
from 24 hours to infinity
[AUC ([LP*]  )]
can be calculated by the
pharmacokinetic method.
AUC = [ ]24/k
* Where LP means Last Point
Since the estimate of K or T½ affects this calculation, complete this calculation
for all estimated elimination rate constants in cells F17 F22. Total AUC (0)
can be calculated by adding AUC(0-t) [B24] and the best estimate of AUC (0).

31. Brief Tutorial on the use of Spreadsheets (Excel®)
Note on ‘fixing’ a cell.
When calculating the
AUC ([LP]  )
The concentration of the LP
or the concentration at 24 hours
will be used in 6 calculations.
When drag procedure is used
to copy, the concentration used
in the formula changes
UNLESS the cell is
‘FIXED’ using a ‘$’
The placement of the $ in the formula is important. The formula for AUC (LP)
in call F17 would be: ‘=B10/K17’ Dragging the formula to F18 yield: ‘=B11/K18.
Placing the $ between B and 10 fixes the cell at line 10. [ =B$10/K17]

32. Brief Tutorial on the use of Spreadsheets (Excel®)
Calculate:
Volume of Distribution
Half-life,
Area Under the Curve
Clearance
Clearance can be
calculated two ways
with this data.
1. Clearance = Dose / AUC
2. Clearance = k * Vd
Since we have a variety of estimates for (i) volume (B12 & B14) and we could have
obtained additional estimates using different numbers of points and ; (ii) k; we can
obtain a large number of estimates of clearance. Limit choices to 7 points.

33. Brief Tutorial on the use of Spreadsheets (Excel®)
Additional
Information & Tips
Converting
Raw Concentrations
to log concentration
and back again.
(ii) Using
Natural Logs
(Ln) for kinetic Analysis
(iii) Back Extrapolation

34. Brief Tutorial on the use of Spreadsheets (Excel®)
(i) Converting
Raw Concentrations
to log concentration
and back again.
The log of a
concentration
can be obtained using
the Excel function
‘LOG(##)’.
The value in parenthesis (##)
may be either an actual
number or a cell reference.
Using a Cell Reference allows
the formula to be copied
more easily.

35. Brief Tutorial on the use of Spreadsheets (Excel®)
(i) Converting
Raw Concentrations
to log concentration
and back again.
If you have the log of a number
and wish to convert it
back to the ‘raw’ concentration,
this can be done by computing
the value of 10x
where x is the log value
you wish to convert.
To do this in Excel the format is
10^x
Where ‘^’ is the
Excel operator for power.

36. Brief Tutorial on the use of Spreadsheets (Excel®)
(ii) Using
Natural Logs (Ln)
for kinetic Analysis.
Kinetic data may also be
analysed using
natural logarithms.
The natural log of a
concentration
can be obtained using
the Excel function
‘LN(##)’.

37. Brief Tutorial on the use of Spreadsheets (Excel®)
(ii) Using
Natural Logs (Ln)
for kinetic Analysis
If you have the natural log
of a number and wish to
convert it back to the
‘raw’ concentration,
this can be done by using
the Excel function
‘EXP(##)’.
Where ## is value
to be converted.

38. Brief Tutorial on the use of Spreadsheets (Excel®)
(ii) Using
Natural Logs (Ln)
for kinetic Analysis
If you use Natural logs
all calculation are exactly
the same
(see Worksheet ‘Analysis of IV Bolus using LN)
except that the slope is
the elimination rate constant
(do not multiply by 2.303)
and the
intercept – Initial concentration
must be converted to
concentration
using ‘EXP’ rather than ’10^##
Done properly both
methods yield identical results.

39. Brief Tutorial on the use of Spreadsheets (Excel®)
(iii) Back Extrapolation
In many situations
in pharmacokinetic analysis
you may need to back extrapolate
to determine the concentration
at earlier times.
For example, if a drug
was given by IV bolus
but time zero concentrations
are not available,
back extrapolation is necessary
to determine volume.
There are at least three ways
to do this.
Using the Excel
‘INTERCEPT’ function
(b) Using the Excel ‘SLOPE’ function
(c) Using the formula:
[ ]t2 = [ ]t1 * e(k(t2 – t1))

40. Brief Tutorial on the use of Spreadsheets (Excel®)
(iii) Back Extrapolation
Using the Excel
‘INTERCEPT’ function
When selecting the “appropriate”
number of points in the terminal
phase to determine ‘SLOPE’
you can also determine
the intercept using the
and the same pairs
of conc. & time values.
In the worksheet on the left the
Initial intercept value of 100 was
obtained using the equation in Excel:
=10^INTERCEPT(C$9:C$10,A$9:A$10)
for the last 2 points.

41. Brief Tutorial on the use of Spreadsheets (Excel®)
(iii) Back Extrapolation
(b) Using the Excel ‘SLOPE’ function.
In Excel when the slope is
calculated on log-conc. & time data,
and the line is straight we can
estimate the concentration
anywhere on the line
as it is in the form of y = mx = b.
A concentration at any
time (t1)can be used and the
concentration at another
time (t2) can be determined.
LOG [ ]t2 = LOG [ ]t1 + SLOPE * (t2 – t1)
The log of concentration at t2 (LOG [ ]t2)
can be convert to a raw concentration.

42. Brief Tutorial on the use of Spreadsheets (Excel®)
(iii) Back Extrapolation
(b) Using the Excel ‘SLOPE’ function.
For example, if the concentration
at time zero was to be calculated
from the given data, t2 would = 0.
t1 could be any other given time.
We will use 18 hours.
The concentration at 18 hours
is 12.5 mg/L (as a log:1.097).
LOG [ ]t2= LOG [ ]t1 + SLOPE * (t2 – t1)
= ( * (0-18)
= ( )
= 2.00
and converting to raw concentration
[ ]t2=0 = 10^2.00 =
Deviation of the concentration from the line of best fit
may result in small deviations from the expected value
of 100 if other concentrations and times are used.
This method can be used to calculate a concentration
at any time on the extrapolated line.

43. Brief Tutorial on the use of Spreadsheets (Excel®)
(iii) Back Extrapolation
(c) Using the formula:
[ ]t2 = [ ]t1 * e( - k(t2 – t1))
Similarly, the concentration
at any time on a line taken
from the terminal phase
can be calculated using the
estimated elimination rate constant (k).
A concentration at any
time (t1)can be used and the
concentration at another
time (t2) can be determined.
The difference in time (t2 – t1)
is computed and if t2 occurs
before t1 then difference in time
is negative and this negative sign
cancels the negative sign in the formula e –kt.
If t2 occurs after t1 then the
negative sign remains.

44. Brief Tutorial on the use of Spreadsheets (Excel®)
(iii) Back Extrapolation
(c) Using the formula:
[ ]t2 = [ ]t1 * e( - k(t2 – t1))
For example, if the concentration
at time zero was to be calculated
from the given data, t2 would = 0.
t1 could be any other given time.
We will use 18 hours.
The concentration at 18 hours
is 12.5 mg/L.
[ ]t2 = [ ]t1 * e(k(t2 – t1))
[ ]t2 = 12.5 * EXP( *(0-18)
[ ]t2 = 12.5 * EXP( *(-18)
[ ]t2 = 12.5 * EXP(2.0798)
[ ]t2 = 12.5 * 8.003
[ ]t2 =
Deviation in the concentration from the line of best fit may
result in small deviations from the expected value of100.
This method can be used to calculate a concentration
at any time on the extrapolated line.