**Compartment Modelling**

**Intravascular and
Extravascular**

__Contents__

• Introduction

• Compartment modelling

• One compartment open model

• One compartment open model for IV bolus

__Session
objectives__

By the end of this lecture, students will be able to:

• Explain the concept of modeling in bio-pharmaceutics and
pharmacokinetics

• Discuss the advantages and disadvantages of different
compartment modeling

• Estimate pharmacokinetic parameters for intravascular and
extravascular administration from single and multi-compartment models

• Describe the differential equations for a simple
pharmacokinetic model for open IV bolus administration

__Mathematical
model__

__Mathematical model__

• A model is a mathematical description of biologic system
and used to express quantitative relationships.

• Mathematical models are a collection of mathematical
quantities, operations and relations together with their definitions and they
must be realistic and practical.

• A model is a hypothesis that employs mathematical terms to
concisely describe quantitative relationships.

**Why model the data?**

There are three main reasons due to which the data is
subjected to modelling:

**1. Descriptive:**
to describe the drug kinetics in a simple way

**2. Predictive:** to
predict the time course of the drug after multiple dosing based on single dose
data, to predict the absorption profile of the drug from the iv data.

**3. Explanatory:**
to explain unclear observations.

__Qualities
of a mathematical model__

__Qualities of a mathematical model__

**Validity:** It
should have practical applicability and should be valuable in describing events
chosen accurately with high precision

**Prediction ability:**
These models predict the qualitative and quantitative changes in these
parameters that are rate constants and half-lives of drugs

**Computability**

**Consistency of
results:** Reproducibility is an important quality of a mathematical model

__Applications
of pharmacokinetic model__

__Applications of pharmacokinetic model__

• Characterizing the behavior of drugs in patients

• Calculating the optimum dosage regimens for individual
patients

• Evaluating the bioequivalence between different
formulations of same drug

• Determining the influence of altered physiology or disease
state on drug ADME

• Explaining the drug interactions

__Compartment
models__

__Compartment models__

• Compartment Models also called as Empirical models

• Compartmental analysis is the traditional and most
commonly used approach to pharmacokinetic characterization of a drug

• These models simply interpolate the experimental data and
allow an empirical formula to estimate the drug concentration with time

• A compartment is not real but an imaginary &
hypothetical one

• Body is composed of several compartments that are
connected reversibly with each other

• The kinetics of drugs can be explained by modeling

• The rate of drug movement b/w compartment follows first
order kinetics

**Advantages**

• Visual representation of various rate process involved in
drug disposition

• Possible to derive equations describing drug concentration
changes in each compartment

• Possibility of estimating amount of drug in any
compartment of the system after the drug is introduced into a given compartment

**Disadvantages**

• Concentration of some drugs cannot be predicted well
though blood levels can be predicted easily

• Compartment behavior may change with route of
administration

__Types of
compartments__

__Types of compartments__

• Highly perfused organs – Central compartment

• Poorly perfused organs – peripheral compartment

__Assumptions
of compartmental models__

__Assumptions of compartmental models__

• The body is represented as a series of compartment
arranged in series or parallel to each other

• The rate of drug movement between compartments is
described by first order kinetics

• Rate constants are used to represent rate of entry into
and exit from compartment

• A statistical analysis of plasma concentration time data
is another method used to find out no.of compartments

• Within each compartment, the drug is considered to be
rapidly and uniformly distributed i.e. the compartment is well-stirred

__Pharmacokinetic
Modelling__

__Pharmacokinetic Modelling__

__Pharmacokinetic
models__

__Pharmacokinetic models__

• Means of expressing mathematically or quantitatively, time
course of drug throughout the body and compute meaningful pharmacokinetic
parameters.

**Useful in:**

• Characterize the behavior of drug in patient.

• Predicting conc. of drug in various body fluids with
dosage regimen.

• Calculating optimum dosage regimen for individual patient.

• Evaluating bioequivalence between different formulations.

• Explaining drug interaction.

• Pharmacokinetic models are hypothetical structures that
are used to describe the fate of a drug in a biological system following its
administration.

**Model**

• Mathematical representation of the data.

• It is just hypothetical

• Depending upon whether the compartments are arranged parallel
or in a series, compartment models are divided into two categories —

Mammillary model

Catenary model

__Mammillary model__

__Mammillary model__

• Most commonly used compartment model

• Consists of 1 or more peripheral compartments connected to
the central compartment in a manner similar to connection of satellites to a
planet (parallel to the central compartment)

• The central compartment comprises of plasma and highly per
fused tissues (lungs, liver, kidneys) which rapidly equilibrate with the drug

• The drug is directly absorbed into this compartment (i.e.
blood). Elimination too occurs from this compartment (liver and kidneys)

• The peripheral compartments (denoted by numbers 2,3..) are
those with low vascularity and poor perfusion

• Distribution of drugs to these compartments is through
blood. Movement of drug between compartments is defined by characteristic
first-order rate constants denoted by letter K

__Caternary
model__

__Caternary model__

• In this model, the compartments are joined to one another
in a series like compartments of a train.

• This is however not observable
physiologically/anatomically as the various organs are directly linked to the
blood compartment.

• Hence this model is rarely used

__Catenary
model__

__Catenary model__

• The number of rate constants which will appear in a
particular compartment model is given by R.

• For intravenous administration

**R = 2n – 1**

For extra vascular administration,

**R = 2n**

where n = number of compartments

__Applications
of Pharmacokinetic Models__

__Applications of Pharmacokinetic Models__

• Characterizing the behaviour of drugs in patients

• Predicting the multiple-dose concentration curves from
single dose experiments

• Calculating the optimum dosage regimen for individual patients

• Evaluating the risk of toxicity with certain dosage
regimens. Correlating plasma drug concentration with pharmacological response

• Estimating the possibility of drug and/or metabolite(s)
accumulation in the body

• Explaining drug interactions

Assumptions

__One
Compartment Open Model__

__One Compartment Open Model__

**(Instantaneous
Distribution Model) **

• The body is considered as a single, kinetically homogenous
unit that has no barriers to the movement of drug

• Simplest model

• Final distribution equilibrium is attained between the
drug in plasma & other body fluids is attained instantaneously &
maintained at all times.

• Model applies only to those drugs which are rapidly
distributed throughout the body

• Drugs move dynamically in (absorption) out (elimination)

• Elimination is a first order process with first order rate
constant

• Rate of absorption (input) > Rate of elimination
(output)

• Reference compartment is plasma i.e., drug concentration
in plasma is respective to drug concentration in body fluids & tissues

• Change in plasma concentration is proportional to the
change in drug concentration in tissues throughout the body

• The term open indicates that input (availability) & output
(elimination) are unidirectional

• If the drug is not removed from the body then model refers
as closed model

• Generally used to describe plasma levels of a single dose
of a drug

__Different
forms of one compartment__

__Different forms of one compartment__

**Open models based on
the rate of input**

• One compartment open model IV bolus administration

• One compartment open model for continuous IV infusion

• Extravascular administration Zero order absorption

• Extravascular administration first order absorption

__One
Compartment Open Model – Intravenous (IV) Bolus Administration__

__One Compartment Open Model – Intravenous (IV) Bolus Administration__

• Drug administered by i.v route rapidly distributes throughout
the body.

• It takes 1 to 3 min for complete circulation

• The rate of absorption is neglected, only distribution is taken
into consideration

• Rate of drug presentation in the body

**dx / dt = rate in (availability) – rate out (elimination) ...(1)**

• Since rate in or absorption is absent in the equation (1),
it becomes

**dx / dt = - rate out ...........(2)**

• If the rate order follows first order kinetics, then the
above equation becomes

**dx / dt = - KEX ............(3)**

KE= first order elimination rate constant

X = amount of drug in the body at any time t remaining to be
eliminated

__Estimation
of Pharmacokinetic Parameters – IV Administration__

__Estimation of Pharmacokinetic Parameters – IV Administration__

• The drug that follows one compartment kinetics administered
as rapid i.v injection, the decline in plasma drugs level is only due to elimination
of drug from the body (not due to distribution), the phase is called
elimination phase

• Elimination phase is characterised by 3 parameters

- Elimination rate constant

- Elimination half life

- Clearance

__Elimination
Rate Constant__

__Elimination Rate Constant__

Integration of equation 3 yields

**In X= In X _{O }– K_{E}t............. (4)**

• where X_{O} = Amount of drug at time t = zero i.e
the initial amount of drug injected

• Equation 4 in the exponential form as

**X = X _{O}e-^{K}E^{t}..............(5)**

• The above equation shows that disposition of a drug that
follows one compartment kinetics is monoexponential

• Transforming equation 4 into common logarithms:

**log X = log X _{0} – K_{E}t/ 2.303 ...............(6 )**

• Since it is difficult to determine directly the amount of
drug in the body X, direct relationship exists b/w plasma C and X, thus

**X = V _{d}C......................(7 )**

• Where Vd = proportionality constant or apparent volume of
distribution

• If the PK parameter allows the use of plasma drug conc. in
place of amount of drug in the body, the equation becomes

**log C = log C _{0} – K_{E}t / 2.303 ................(8)**

• C0 = plasma drug concentration immediately after i.v
injection

**The elimination of
the drug from the body is the sum of:**

• Urinary excretion

• Metabolism

• Biliary excretion

• Pulmonary excretion

• Other mechanisms involved.

**Overall elimination
rate constant (KE)**

**K _{E} = K_{e} + K_{m} + K_{b} + K_{P}
+ ...................(9)**

• Drug eliminated by particular route can be evaluated if
the number of rate constants are involved & this values are known.

**Ex:** If drug is
eliminated by urinary excretion & metabolism only, then fraction of drug excreted
& fraction metabolised Fe, Fm can be given as

**F _{e} = K_{e}/K_{E}, ................(10 a)**

**F _{m }= K_{m}/K_{E}, .................(10 b)**

**Elimination Half Life**

• Also called as biological half life

• It is one of the oldest, best known, important PK
parameter of the drug

• It is defined as the amount of drug in the body as well as
plasma concentration to decline by one half or 50 % of its initial value

• Half-life is related to elimination rate constant by the
following equation

**t _{1/2 }= 0.693/K_{E} ..................( 11)**

• But half-life is a secondary parameter depends upon
primary parameters like clearance & apparent volume of distribution

**t _{1/2 }= 0.693 V_{d} /Cl_{T} ...............(12)**

**Apparent Volume of
Distribution**

• Apparent volume of distribution & Clearance are
related with the physiological mechanisms in the body hence they are called as
primary parameters

**V _{d} = X / C ..............(13)**

• Vd is the measure of the extent of distribution of drug
expressed in litres

• Simplest way to determine Vd by rapid i.v injection is
determining the plasma conc. immediately

**V _{d }= X_{0} / C_{0} = i.v bolus dose / C_{0}.................(14)**

• It can be used for drugs that obey one compartment
kinetics. But non ompartmental model can be applied to many compartmental model
for estimating Vd

• For drugs given as i.v bolus,

**V _{d} = X_{0} / K_{E}AUC.................(15 a)**

• For drugs given by e.v

**V _{d(area) }= FX_{0}
/ K_{E}AUC............(15 b)**

**Where **

X0 = dose administered

F = fraction of drug absorbed into the systemic circulation

F = 1, when drug is administered by i.v route

__Summary__

• Body is composed of several compartments that are
connected reversibly with each other

• The kinetics of drugs can be explained by modeling

• Types of compartments are Central compartment and
peripheral compartment

• One compartment applies only to those drugs which are
rapidly distributed throughout the body

• The term open indicates that input (availability) &
output (elimination) are unidirectional

• If the drug is not removed from the body then model refers
as closed model

• Different forms of one compartment open models based on
the rate of input are one compartment open model IV bolus administration, one
compartment open model for continuous IV infusion, extravascular administration
Zero order absorption and extravascular administration first order absorption

## 0 Comments