# Compartment ModellingIntravascular 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

• 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

• 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

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

• 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 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

• 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

• 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

• Highly perfused organs – Central compartment

• Poorly perfused organs – peripheral compartment

## 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 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

• 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

• 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

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

R = 2n – 1

R = 2n

where n = number of compartments

## 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

### (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

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

• 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

• 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

Integration of equation 3 yields

In X= In XO – KEt............. (4)

• where XO = Amount of drug at time t = zero i.e the initial amount of drug injected

• Equation 4 in the exponential form as

X = XOe-KEt..............(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 X0 – KEt/ 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 = VdC......................(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 C0 – KEt / 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)

KE = Ke + Km + Kb + KP + ...................(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

Fe = Ke/KE,  ................(10 a)

Fm = Km/KE,  .................(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

t1/2 = 0.693/KE ..................( 11)

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

t1/2 = 0.693 Vd /ClT  ...............(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

Vd = 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

Vd = X0 / C0 = i.v bolus dose / C0.................(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,

Vd = X0 / KEAUC.................(15 a)

• For drugs given by e.v

Vd(area)  = FX0 / KEAUC............(15 b)

Where

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