### Summary of Pharmacokinetic Calculations

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Harvard-MIT Division of Health Sciences and Technology

HST.151: Principles of Pharmocology

Instructor: Dr. Carl Rosow

1

Summary of Pharmacokinetic Calculations

The following list was extracted from pharmacology lecture notes provided by Dr. Steven

Shafer. It summarizes and embellishes the pharmacokinetic concepts presented:

1. The rate of change (decrease) when drug is injected into a 1 compartment model is

dX

= −kX (first order process)

dt

2. The concentration following that injection is

C(t) = C e −kt

0 where C0 is the initial concentration

3. The half-life, t½ (time required for a 50% decrease), is

0.693

t 1

2

=

k

4. If you know the time required for a 50% decrease, the rate constant, k, is

0.693

k =

t 1

2

5. The definition of concentration is

X

C = , where X is amount and V is volume

V

6. The concentration at time t following a bolus injection will be

X0

−kt where

V

X0

C(t) = is the initial concentration

V

e

7. If ClT is the total clearance (or flow) from a 1 compartment model, the rate at which

drug leaves can be calculated

dX

= C(ClT)

dt

8. Since item 1 and item 7 are the same rate, it follows (after substituting X/V for C)

that 2

ClT

k =

V

Substituting in equation 3, we get this important relationship

0.693(V)

t 1

2

=

ClT

So, as clearance (ClT) increases, k increases, and the half-life decreases. As volume

(V) increases, k decreases, and half-life increases.

9. During an infusion at rate k0, the concentrations are described by the equation

(1 −

−kt C(t) = C e ) where Css is the concentration at steady-state. ss

10. The steady-state concentration can be calculated from infusion rate and clearance

k0

Css =

ClT

11. Half-lives describe the time for a 50% decrease in concentration following a bolus,

and they also describe the time required to reach 50% of the steady-state

concentration during an infusion. Following a bolus, the concentrations will be at

25%, 13%, 6%, and 3% of the initial concentration following 2, 3, 4, and 5 half-lives,

respectively. During a constant-rate infusion, the concentration will reach 75%, 88%,

94%, and 97% of the steady-state concentration in 2, 3, 4, and 5 half-lives,

respectively.

What do you do with this? Well:

1. If you know the amount of drug injected (X0), and the concentration at time 0 (C0),

you can calculate the volume

X0

V =

C0

2. If you know X0, V, and k, then you can calculate the concentration at any given time t

C(t) =

X0

e

−kt

V 3

3. If you know two concentrations,C1 and C2, obtained at times t1 and t2, respectively,

you can calculate k as

ln(C1

) −ln(C2

)

k =

t

2 − t

1

4. If you want to know the clearance (the flow out of the compartment), you can

calculate it as k(V). If k and V are not known, or if there are several values of k

(multicompartment kinetics), you can still calculate

dose

ClT = where AUC is the area under the time vs. concentration curve

AUC

5. If you know the initial target concentration you want to achieve, Ctarget, then you can

calculate Xloading, the intravenous dose required to produce that concentration

Xloading = Ctarget(V)

6. If you want to maintain concentration Ctarget, then you must continuously infuse drug

at the same rate it is leaving. Assuming that you first gave a bolus of Ctarget (V), the

rate at which drug will leave will be Ctarget (ClT). Therefore your maintenance

infusion Xmaintenance will be

X

maintenance = Ctarget(Cl

T) 4

Another exercise from Dr. Shafer:

Dr. Rosow was quite concerned that I wouldn’t explain the basic concepts adequately.

He specifically requested that I make sure that if you are going to give a medication to

“Joe” (must be a friend of his), you can figure out the dose for Joe. I don’t know Joe, but

I do know a few things about a new drug that Carl has started Joe on: cephprololopam,

an antibiotic that has beta blocking and anxiolytic properties:

• The clearance of cephprololopam is 0.2 liters/min

• The volume of distribution of cephprololopam is 20 liters

• The therapeutic concentration is 2 µg/ml.

1. Carl forgot to tell me the half-life of cephprololopam. What is it?

Answer:

⎛ liters ⎞

ClT ⎝

⎜0.2

k = =

min ⎠

⎟

= 0.01min −1

V 20liters

0.693

t 1

2

= = 69min

k

2. What is Joe’s initial dose of cephprololopam?

Answer:

Xloading = Ctarget(V) =

⎛

⎝

⎜2

µ g⎞

20liters = 40mg

ml⎠

⎟

3. How much drug should I give Joe to maintain a cephprololopam concentration of 2

µg/ml?

Answer:

X

maintenance = Ctarget(Cl

T) =

⎝

⎜

⎛

2

µ g ⎞ ⎛0.2liters ⎞

⎠

⎟

= 0.4

mg

ml ⎠

⎟

⎝

⎜

min min

4. I want to put Joe on an oral form of cephprololopam, which he will take every 24 hr.

How much should I give Joe, assuming the the drug is completely absorbed, and I

want Joe’s concentrations, on average, to be at the target?

Answer: Joe will need the same total amount of drug every 24 hr

⎛

0 4 mg

⎠

⎟

⎞

1440min = 576 mg

.

⎝

⎜

min day 5

5. How long will it take Joe to reach steady-state dosing with these repeated oral doses?

Answer: 4-5 half-lives = 276-345 min, i.e., Joe will be at steady state dosing within

the time course of the first dose!