Pharmacokinetic Exercises for Practice

1. Calculate and
Interpret
Pharmacokinetic
Parameter of a
Drug from Given
Data
Dr. Shivankan Kakkar, MD

2. Learning Objectives
After completing this lecture each student should:-
• understand the one compartment model, rapid IV bolus dosing and
linear elimination
• understand the properties of first order kinetics, linear models be
able to define, use, and calculate the parameters:
– Vd (apparent volume of distribution)
– CL (clearance)
– KE (elimination rate constant)
– t1/2 (half-life)
– Cp0 (Plasma drug conc. at zero time after administration)
{as they apply to a one compartment linear model}
• be able to use the integrated equations for a one compartment
linear model to perform various dose and dosing regimen
calculations
Clinical Pharmacokinetics by DrShivMD 2

3. Important Formulas : Arithmetic Vs Graphical
• Vd = Dose / Cp0
• CL = 0.693 * Vd / t1/2
• KE = CL / Vd
• t1/2 = 0.693 / KE
• Cpt= Cp0
* e – KE * t
Common Log Natural Log
Slope of the straight line
of Log C vs t graph is:
Slope = Y2 – Y1 / X2 – X1
or
Slope =
Log Cpt1 – Log Cpt2 / t1 – t2
KE = – Slope * 2.303
Slope of the straight line of
ln C vs t graph is:
Slope = Y2 – Y1 / X2 – X1
or
Slope =
ln Cpt1 – ln Cpt2/ t1 – t2
KE = – Slope
y = mx + b
where
m = slope and
b = y-intercept
Cp0 = Antilogarithm (10^y-
intercept)
y = mx + b
where
m = slope and
b = y-intercept
Cp0= Exponential Function
(e^y-intercept)
Clinical Pharmacokinetics by DrShivMD 3

4. Instructions
Using a scientific calculator for logarithmic and exponential
calculations
Logarithm (log base 10):
To find the logarithm of a
number "N," press the "Log and
N" buttons.
Antilogarithm (10^N):
To find the antilogarithm
(10^N) of a number "N," press
the "Shift" button, then press
the "Log and N" buttons.
Natural Logarithm (ln):
To find the natural logarithm
(ln) of a number "N," press the
"Ln and N" buttons.
Exponential Function (e^N):
To find "e" raised to the power
of "N," press the "Shift" button,
then press the "Ln and N"
buttons.
Clinical Pharmacokinetics by DrShivMD 4

5. Question
• What is the concentration of a drug at
0, 2 and 4 hours after a dose of
500 mg?
• Known pharmacokinetic parameters are
apparent volume of distribution,
Vd is 30 Litre
and the elimination rate constant,
KE is 0.2 hr-1
Problem Set 1
Hint : Use formulas
•Vd = Dose / Cp0
•Cpt= Cp0
* e – KE * t
Clinical Pharmacokinetics by DrShivMD 5

6. Answers
• Cp0 is 16.7 mg/Litre
• Cp2 is 11.2 mg/Litre
• Cp4 is 7.5 mg/Litre
Clinical Pharmacokinetics by DrShivMD 6

7. Question
• If Cp2 at 2 hours is 4.5 mg/Litre and
Cp6 at 6 hours is 3.7 mg/Litre
after a 400 mg IV bolus dose, what are the
values of KE, Cp0, Vd and CL?
Problem Set 2
Hint: Use formulas
• Slope = ln Cpt1 – ln Cpt2/ t1 – t2
KE = – Slope
• Cpt= Cp0
* e – KE * t
• Vd = Dose / Cp0
• KE = CL / Vd
Clinical Pharmacokinetics by DrShivMD 7

8. 1.5
1.3
1.3
1.3
1.4
1.4
1.5
1.5
1.6
0 1 2 3 4 5 6 7
Ln
Conc.
(mg/Litre)
Time (hours)
Ln Conc. Vs Time Plot
Clinical Pharmacokinetics by DrShivMD 8

9. Answers
KE is 0.05 hr-1
Cp0 is 5 mg/Litre
Vd is 80 Litres
CL is 4 Litre/hr
Clinical Pharmacokinetics by DrShivMD 9

10. Data Provided
• Plasma concentration of viomycin after i.v.
bolus administration:
– 10 μg/ml at 2 hours
– 5.5 μg/ml at 4 hours
• Dose administered: 300 mg
Problem Set 3
Clinical Pharmacokinetics by DrShivMD 10

11. Questions
A. Calculate the half-life of the drug (t1/2).
B. Calculate the concentration of drug in plasma
at zero time (Cp0).
C. Calculate the volume of distribution (Vd).
D. Calculate the total systemic clearance (CLT).
Clinical Pharmacokinetics by DrShivMD 11

12. Important Formulas : Arithmetic Vs Graphical
• Vd = Dose / Cp0
• CL = 0.693 * Vd / t1/2
• KE = CL / Vd
• t1/2 = 0.693 / KE
• Cpt= Cp0
* e – KE * t
Common Log Natural Log
Slope of the straight line
of Log C vs t graph is:
Slope = Y2 – Y1 / X2 – X1
or
Slope =
Log Cpt1 – Log Cpt2 / t1 – t2
KE = – Slope * 2.303
Slope of the straight line of
ln C vs t graph is:
Slope = Y2 – Y1 / X2 – X1
or
Slope =
ln Cpt1 – ln Cpt2/ t1 – t2
KE = – Slope
y = mx + b
where
m = slope and
b = y-intercept
Cp0 = Antilogarithm (10^y-
intercept)
y = mx + b
where
m = slope and
b = y-intercept
Cp0= Exponential Function
(e^y-intercept)
Clinical Pharmacokinetics by DrShivMD 12

13. Answers
• The half-life of the drug is 2.32 hours.
• The concentration of drug in plasma at zero
time, Cp0 is 18.2 μg/ml.
• The Vd is 16.5 litres.
• The total systemic clearance, CLT is 82.23
ml/min.
Clinical Pharmacokinetics by DrShivMD 13

14. 1
0.74
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Log
Conc.
(mcg/ml)
Time (hours)
Log Conc. Vs Time Plot
Clinical Pharmacokinetics by DrShivMD 14

15. 2.3
1.7
0.0
0.5
1.0
1.5
2.0
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Ln
Conc.
(mcg/ml)
Time (hours)
Ln Conc. Vs Time Plot
Alternatively
Clinical Pharmacokinetics by DrShivMD 15

16. Calculate the half-life of the drug
• Data given: Cpt1 = 10 mcg/ml at t1 = 2 hours
Cpt2 = 5.5 mcg/ml at t2 = 4 hours
• Formula to be used: t1/2 = 0.693/KE
where KE = Elimination rate constant
• So, to calculate t1/2 we have to first determine KE
with the help of given data.
• KE can be determined from the slope of a line
obtained by plotting log conc. on y-axis vs time
on x-axis or by plotting ln conc. on y-axis vs time
on x-axis (first order kinetics)
Clinical Pharmacokinetics by DrShivMD 16

17. 1
0.74
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Log
Conc.
(mcg/ml)
Time (hours)
Log Conc. Vs Time Plot
Clinical Pharmacokinetics by DrShivMD 17

18. Clinical Pharmacokinetics by DrShivMD 18

19. Calculate the half-life of the drug
• KE = – slope * 2.303
Log Cpt1 = Log 10 = 1
Log Cpt2 = Log 5.5 = 0.74
• Slope of the straight line of Log C vs t is:
Slope = Y2 – Y1 / X2 – X1
= 1 – 0.74 / 2 – 4
= – 0.26 / 2 = – 0.13
• KE = – slope * 2.303
= – (– 0.13) * 2.303 = 0.299 hr-1
• t1/2 = 0.693 / KE = 2.32 hours
Common Log- Log Concentration Vs Time Graph
Clinical Pharmacokinetics by DrShivMD 19

20. Calculate the concentration of drug in
plasma at zero time
• Cp0 is the plasma drug concentration at zero time
after administration or initial plasma drug
concentration. This can be obtained from the
graph (Log conc. Vs time) by extrapolating the
straight line and finding the Y-intercept which is
equal to log Cp0
• From the graph, log Cp0 = 1.26 (Also, we can use
the formula: y = mx + b where m = slope and b =
y-intercept and by plugging in the given point
and the slope and we can solve for b.)
• Therefore, Cp0= Anti-log 1.26 = 18.2 mcg/ml
Graphical Method: Common Log- Log Concentration Vs Time Graph
Clinical Pharmacokinetics by DrShivMD 20

21. Calculate the half-life of the drug
• KE = – slope
Ln Cpt1 = Ln 10 = 2.3
Ln Cpt2 = Ln 5.5 = 1.7
• Slope of the straight line of Ln C vs t is:
Slope = Y2 – Y1 / X2 – X1
= 2.3 – 1.7 / 2 – 4
= - 0.6 / 2 = – 0.3
• KE = – slope
= -(- 0.3) = 0.3 hr-1
• t1/2 = 0.693 / KE = 2.31 hours
Alternatively : Natural Log - Ln Concentration Vs Time Graph
Clinical Pharmacokinetics by DrShivMD 21

22. 2.3
1.7
0.0
0.5
1.0
1.5
2.0
2.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Ln
Conc.
(mcg/ml)
Time (hours)
Ln Conc. Vs Time Plot
Alternatively
Clinical Pharmacokinetics by DrShivMD 22

23. Calculate the concentration of drug in
plasma at zero time
• Cp0 is the plasma drug concentration at zero time
after administration or initial plasma drug
concentration. This can be obtained from the
graph (Ln conc. Vs time) by extrapolating the
straight line and finding the Y-intercept which is
equal to ln Cp0
• We can use the formula: y = mx + b where m =
slope and b = y-intercept and by plugging in the
given point and the slope and we can solve for b.)
• Therefore, Cp0 = e^ 2.9 = 18.2 mcg/ml
Graphical Method: Natural Log - Ln Concentration Vs Time Graph
Clinical Pharmacokinetics by DrShivMD 23

24. Calculate the concentration of drug in
plasma at zero time
• Use formula
• Cpt= Cp0
* e – KE * t
10 = Cp0 * e – 0.3 * 2
10 = Cp0 * e – 0.6
10 = Cp0 * 0.54
Cp0 = 10 / 0.54 = 18.5 mcg/ml
Arithmetic Method
Clinical Pharmacokinetics by DrShivMD 24

25. Calculate the volume of distribution
(Vd)
• Vd = Dose / Cp0 where:
Dose = Dose administered intravenously
and
Cp0 = Plasma drug conc. at zero time after
administration
• Therefore,
Vd = 300 * 1000 mcg / 18.2 mcg per ml
= 16,500 ml or 16.5 litres
Clinical Pharmacokinetics by DrShivMD 25

26. Calculate the total systemic clearance
(CLT)
• CLT = KE * Vd
= 0.299 hr-1 * 16.5 litres
= 0.299/60 mins *16.5*1000 ml
= 82.23 ml/min
Clinical Pharmacokinetics by DrShivMD 26