pharmacokinetics: lecture two

Intravenous Bolus

Administration

One compartment Model

Anas Bahnassi PhD RPh

LECTURE’S OBJECTIVES

Upon the completion of this lecture the student should be able to:

• Describe the different pharmacokinetic parameters.

• Determine pharmacokinetic parameters from either plasma or urinary data.

• State the equation for plasma drug concentration as a function of time.

• Calculate the corresponding plasma drug concentration at time t

• Calculate the intravenous bolus dose of a drug that will result in a target (desired) plasma drug concentration at time t.

Anas Bahnassi PhD 2011 2


X

Xu • One-compartment model.

• First-order process. • Passive diffusion.

• No metabolism takes place (elimination is 100% via renal

excretion) • The drug is being monitored in blood

(plasma/serum) and urine.

Assumptions

IV Bolus Equations:


Pharmacokinetic Parameters

• Apparent volume of distribution (Vd).

• Elimination half-life (t1/2).

• Elimination rate constant (K0 or Kel).

• Systemic clearance (Cls).


Apparent volume of distribution

(Vd)

• Concentrations are usually measured not

masses.

• Vd is a proportionality constant whose sole

purpose is to relate the plasma

concentration (Cp) and the mass of drug (X)

in the body at a time.

• It is not a physical volume.


Vd Concept


The concentration of KI is different although the volume of water

in both beakers is the same.

Drug Concentration in Beaker Drug Concentration in Beaker

with charcoal

Dose = 10mg

Cp0 = 20mg/L

Vd= 500mL

Dose = 10mg

Cp0 = 2mg/L

Vd= 5000mL

Calculating Vd


𝑉 =𝑋

𝐶𝑝

Apparent volumes

of distribution are

given in units of

volume (e.g. mL) or

units of volume on a

body weight basis

(Lkg-1 body weight).

Elimination Half life (t1/2)

The time (h, min, day, etc.) at which the mass (or amount) of unchanged drug becomes half (or 50%) of the initial mass of drug.


Semi-logarithmic Paper

Basic Pharmacokinetics: S. Jambhekar , Phillip Breen 2009

Elimination Half life (t1/2)


When Cp = 0.5 (Cp)0

t = t1/2

Elimination Rate Constant (k)


Unit of k in first order process is reciprocal of time (h-1)

𝑘 = 𝑘𝑢 + 𝑘𝑚 Elimination

Rate

Constant

Excretion

Rate

Constant

Metabolism

Rate

Constant

X0=

250mg

M1=

75mg

M2=

50mg

Xu=

125mg

Elimination Rate Constant (k)


𝑘 =0.963

4= 0.173ℎ𝑟−1

%𝒆𝒙𝒄𝒓𝒆𝒕𝒆𝒅 =125

250𝑋100 = 50%

%𝒎𝒆𝒕𝒂𝒃𝒐𝒍𝒊𝒕𝒆𝟏 =75

250𝑋100 = 30%

%𝒎𝒆𝒕𝒂𝒃𝒐𝒍𝒊𝒕𝒆𝟐 =50

250𝑋100 = 20%

𝒌𝒖 = 𝟓𝟎%𝑿𝟎. 𝟏𝟕𝟑 = 𝟎. 𝟎𝟖𝟔𝟔𝒉𝒓−𝟏

𝒌𝒎𝟏 = 𝟑𝟎%𝑿𝟎. 𝟏𝟕𝟑 = 𝟎. 𝟎𝟓𝟏𝒉𝒓−𝟏

𝒌𝒎𝟐 = 𝟐𝟎%𝑿𝟎. 𝟏𝟕𝟑 = 𝟎. 𝟎𝟑𝟒𝟓𝒉𝒓−𝟏

Drawing a best-fit line through the

Data


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X = 61.827e-0.526t

0

5

10

15

20

25

30

35

40

0 1 2 3 4 5 6 7 8 9 10

RL paper


16

From the SL graph

t½= 1.3h

Cp0= 63mg/mL.

𝑉𝑑 =𝑋0

𝐶𝑝0

𝑉𝑑 =600000

63= 9523.8𝑚𝑙

= 9.5238𝑙

𝑲 =𝟎. 𝟗𝟔𝟑

𝟏. 𝟑

Calculating PK Parameters



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Use of Urinary Data

Amount remained

to

be excreted Rate of

Excretion

1. Urine collection is a non-invasive technique. 2. More convenient sample collection 3. Sample size is not restricting. 4. The sampling time reflects cumulative drug concentration

in urine collected over a period of time, rather than a drug concentration at a discrete time.

5. Urinary data allows direct measurement of bioavailability, both absolute and relative, without the need of fitting the data to a mathematical model.


Use of Urinary Data

X

Xu 𝑑𝑥𝑢

𝑑𝑡= 𝑘𝑢𝑋

𝑋𝑢 𝑡 = 𝑋0(1 − 𝑒−𝑘𝑢𝑡)

Cumulative amount In Urine at time (t)

Administered dose of drug

Excretion Rate Constant

𝒊𝒇 𝒕 = ∞

𝑋𝑢 = 𝑋0


Amount Remaining To be excreted

𝑿𝒖 ∞ − 𝑿𝒖 𝒕 = 𝐴𝑚𝑜𝑢𝑛𝑡 𝑅𝑒𝑚𝑎𝑖𝑛𝑖𝑛𝑔 𝑡𝑜 𝑏𝑒 𝑒𝑥𝑐𝑟𝑒𝑡𝑒𝑑 = 𝐴𝑚𝑜𝑢𝑛𝑡 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦

= 𝑿𝒕 Drug Totally Removed Unchanged

Can not calculate Volume of Distribution

Drug Totally Removed Unchanged


Limitations

1. Keep obtaining urine samples until no additional drug practically appears in the urine, (t = 7 t½) 2. Urine samples can not be lost, or urine from any samples used in the determination of Xu (the exact volume of urine at each time interval must be known) 3. A time-consuming method for a drug with a long elimination half life. 4. There is a cumulative build up of error.

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The plot represents the cumulative quantity of the medication from the collected urine

samples vs. the time.

Dose = 80mg


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The plot represents the amount remaining to be excreted of the medication vs. time



𝑘 = 𝑘𝑢

Can not calculate Volume of Distribution


Rate of Excretion

Method

𝒅𝒙𝒖

𝒅𝒕= 𝒌𝒖𝒙

𝒙 = 𝒙𝒐𝒆−𝒌𝒖𝒕

𝒅𝒙𝒖

𝒅𝒕= 𝒌𝒖𝒙𝒐𝒆−𝒌

𝒖𝒕

average rate of excretion

average time between

urine collection


The plot represents average rate of excretion within the time interval between

samples collection vs. average time between urine samples collection

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𝑘 = 𝑘𝑢 =0.693

𝑡½=

0.693

1= 0.693ℎ𝑟−1


Questions:

What is the concentration of drug 0, 2 and 4 hours after a dose of 500 mg. Known pharmacokinetic parameters are apparent volume of distribution, Vd is 30 liter and the elimination rate constant, kel is 0.2 hr-1 ?

From the plot seen calculate all pharmacokinetic parameters that you can calculate

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Anas Bahnassi PhD RPh

Pharmacokinetics

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pharmacokinetics: lecture two

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