One compartment model bolus administration equation Biopharmaceutics and Pharmacokinetics

1. One Compartment Open Model
I.V Bolus Administration
Dr. Prabhjot Singh
(Associate Professor, Pharmaceutics)
CH. DEVILAL COLLEGE OF PHARMACY, HARYANA, INDIA

2. Introduction
• The one-compartment open model offers the simplest way to describe
the process of drug distribution and elimination in the body.
• This model assumes that the drug can enter or leave the body (ie, the
model is "open"), and the body acts like a single, uniform
compartment.
• The simplest route of drug administration from a modeling
perspective is a rapid intravenous injection (IV bolus).
• The simplest kinetic model that describes drug disposition in the body
is to consider that the drug is injected all at once into a box, or
compartment, and that the drug distributes instantaneously and
homogenously throughout the compartment.
• Drug elimination also occurs from the compartment immediately
after injection.

3. Depiction of model
Pharmacokinetic model for a drug administered by rapid intravenous injection.
D B (X) = drug in body;
V D = apparent volume of distribution;
k (KE) = elimination rate constant.
• In the body, when a drug is given in the form of an IV bolus, the entire dose of drug enters the
bloodstream immediately, and the drug absorption process is considered to be instantaneous.
• In most cases, the drug distributes via the circulatory system to potentially all the tissues in
the body.
• Uptake of drugs by various tissue organs will occur at varying rates, depending on the blood
flow to the tissue, the lipophilicity of the drug, the molecular weight of the drug, and the
binding affinity of the drug for the tissue mass.
• Most drugs are eliminated from the body either through the kidney and/or by being
metabolized in the liver.

4. • Because of rapid drug equilibration between the blood and tissue, drug
elimination occurs as if the dose is all dissolved in a tank of uniform fluid (a single
compartment) from which the drug is eliminated.
• The volume in which the drug is distributed is termed the apparent volume of
distribution, VD.
• The apparent volume of distribution assumes that the drug is uniformly
distributed in the body.
• The VD is determined from the preinjected amount of the dose in the syringe and
the plasma drug concentration resulting immediately after the dose is injected.
• The apparent volume of distribution is a parameter of the one-compartment
model and governs the plasma concentration of the drug after a given dose.
• A second pharmacokinetic parameter is the elimination rate constant, k, which
governs the rate at which the drug concentration in the body declines over time.

5. Derivation for one compartment open model
I.V Bolus administration
Rate change of drug in body = Rate of Input - Rate of output
As there is no rate of input involved the equation becomes
Rate change of drug in body= - Rate of output
𝑑𝑥
𝑑𝑡
= −𝐾𝐸 𝑋
𝑑𝑥
𝑋
= −𝐾𝐸 𝑑𝑡
Integrating both sides
𝑑𝑥
𝑋
= −𝐾𝐸 𝑑𝑡

6. Applying Limits LHS = X0----X, RHS = 0-----t
X0
𝑋
𝑑𝑥
𝑋
= −𝐾𝐸
0
𝑡
𝑑𝑡
ln 𝑋 𝑥0
𝑥
= −𝐾𝐸 ln 𝑥 0
𝑡
ln 𝑋 − ln 𝑋0 = −𝐾𝐸 𝑡 − 0
ln 𝑋 − ln 𝑋0 = −𝐾𝐸t
ln 𝑋 = ln 𝑋0 − 𝐾𝐸t
This equation can now be converted into exponential or Common log form

7. Converting to exponential form
𝑋 = 𝑒 𝑙𝑛𝑋0
−𝐾𝐸𝑡
𝑋 = 𝑒𝑙𝑛𝑋0. 𝑒−𝐾𝐸𝑡
𝑋 = 𝑋0 𝑒−𝐾𝐸𝑡
As X = Vd C
Since it is difficult to determine directly the amount of drug in body X advantage
is taken of the fact that a constant relationship exists between drug
concentration in plasma C (easily measurable)
And as Vd is constant equation can be written as
𝐶 = 𝐶0 𝑒−𝐾𝐸𝑡
The above equation shows that disposition of a drug that follows one-
compartment kinetics is monoexponential.

8. Transforming natural log to common log
𝑙𝑜𝑔𝑋 − log 𝑋0 = −
𝐾𝐸𝑡
2.303
𝑙𝑜𝑔𝑋 = log 𝑋0 −
𝐾𝐸𝑡
2.303
As X = Vd C
And as Vd is constant equation can be written as
𝑙𝑜𝑔𝐶 = log 𝐶0 −
𝐾𝐸𝑡
2.303
This is equation of straight line (y = mx + c)

9. Representation of equation on graph

10. Primary Parameters
• KE (Elimination Rate constant) is an additive property of rate constant for
each of these processes and better called as overall elimination rate
constant.
KE = Ke + Km + Kb + ------
• Units are per hour (h-1) or per minute (min-1)
Elimination Half life t ½
It is defined as the time taken for the amount of drug in body as well as
plasma concentration to decline by one half or 50% of its initial value.
Unit is h (hours) or minutes
𝑡
1
2
=
0.693
𝐾𝐸
𝑡
1
2
=
0.693 𝑉𝑑
𝐶𝑙𝑡

11. Volume of distribution
𝑉𝑑 =
𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑑𝑟𝑢𝑔 𝑖𝑛 𝑏𝑜𝑑𝑦 𝑋
𝑃𝑙𝑎𝑠𝑚𝑎 𝑑𝑟𝑢𝑔 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝐶
Clearance and Volume of distribution are two separate independent
pharmacokinetic characteristics of a drug and as they are closely
related with physiologic mechanism in body therefore called as primary
parameters.
Best way to estimate Vd of a drug is to administer it as rapid I.V.
injection.
𝑉𝑑 =
𝑋0
𝐶0