Principles of pharmacokinetics

Development Team
Principal Investigator Prof. Farhan J Ahmad
JamiaHamdard, New Delhi
Dr. Vijaya Khader
Former Dean, Acharya N G Ranga Agricultural University
Dr. Javed Ali
Paper Coordinator
JamiaHamdard, New Delhi

Content Writer Dr. Javed Ali

Jamia Hamdard, New Delhi

Dr. Jasjeet Kaur Narang

Content Reviewer Department of Pharmaceutics, Khalsa
College of Pharmacy, Amritsar

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Hello Students Welcome to e PG Pathshala. I am Dr. Javed Ali from Dept.of

Pharmaceutics school of Pharmaceutical Education and Research Jamia

Hamdard. Today we are going to discuss about a module titles as Principles of

Pharmacokinetics under the paper Biopharmaceutics and Pharmacokinetics.

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Contents:
 Introduction
 Rates and order of reaction
 First order, second order, pseudo first order, Half-life for different orders of
reaction
 Pharmacokinetic Models

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Introduction

Pharmacokinetics may be defined as “what the body does to the drug”. Pharmacokinetic deals

with the in vivo fate of drugs which covers important aspects such as absorption, distribution,

and elimination of drugs subsequent to their administration via oral or any other route. In

different ways the pharmacokinetics describe the physicochemical and physiological factors that

influence the absorption of drugs from enteral and parenteral routes of administration, their

distribution within the body, and their routes and mechanisms of elimination. It also explains

how dose, bioavailability, rate of absorption, apparent volume of distribution, total clearance,

and elimination half-life affect the plasma concentrations of a drug after administration of a

single dose. It explains the factors which determine the time-course of systemic accumulation of

a drug administered by infusion or multiple doses. Pharmacokinetic properties of drugs may be

affected by elements such as the site of administration and the dose of administered drug. These

may affect the absorption rate. Pharmacokinetics is often studied in conjunction with

pharmacodynamic, the study of a drug's pharmacological effect on the body.

Rates and order of reaction

The principles of pharmacokinetics are described in terms of mathematical equations which are

used to quantitatively predict the nature of various physiological and pharmacological processes.

The rate of a chemical reaction (process) is the velocity with which it occurs. The rate law is an

expression indicating how the rate depends on the concentrations of the reactants and catalysts.

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The power of the concentration in the rate law expression is called the order with respect to the

reactant or catalyst. For example,

In acidic solutions, hydrogen peroxide and iodide ion react according to the equation:

H2O2 + 2H+ + 3I- = 2 H2O + I3-

In this reaction, the reaction Rate can be expressed as

decreasing Rate of H2O2, - d[H2O2]/dt

decreasing Rate of H+, - d[H+]/dt
decreasing Rate of I-, - d[I-]/dt
increasing Rate of H2O, + d[H2O] /dt
increasing Rate of I3-, d[I3-]/dt

However, from the stoichiometry, we can easily see the following relationship:

d[H2O2] 1 d[H+] 1 d[I-] 1 d[H2O] d[I3-]

- ------- = - - ------ = - - ------- = - ------ = --------
dt 2 dt 3 dt 2 dt dt

In generalize it, let the chemical reaction be represented by,

a A + b B -> c C + d D
then the rate is represented by any one of the following

1 d[A] 1 d[B] 1 d[C] 1 d[D]

rate = - --- ---- = - --- ---- = --- ---- = --- ----
a dt b dt c dt d dt

Usually only the parent (or pharmacologically active) drug is measured experimentally. The

metabolites of the drug or the products of the decomposition of the drug may not be known or

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may be very difficult to quantitate. The rate of the reaction is determined experimentally by

measuring the disappearance of the parent compound at a given time intervals.

The order of a reaction refers to the way in which the concentration of drug or reactants

influences the rate of a chemical reaction or process. Suppose C is the concentration of drug A,

the rate of decrease in C of drug A as it is changed to B can be explained by general expression

as a function of time t.

d[C]
---- = - k [C]n
dt
Where K is the rate constant and n is the order of reaction.

Zero order kinetics

Zero order processes are:

1. Controlled drug delivery systems like osmotic pumps, implants, ocuserts etc.

2. Constant rate intravenous infusion.

3. Protein drug binding/ metabolism/enzyme or carrier mediated transport under saturated

concentration.

If the amount of drug A is decreasing at a constant time interval t, the rate of disappearance of a

drug A is expressed as:

dc/dt= -K

Since n=0 in the equation

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d[C]
---- = - k [C]n (3)
dt
d[C]
---- = - k [C]0
dt
d[C]
---- = - k, where, K=K0
dt

K0= zero order rate constant (mg/min)

dc/dt= - K0

dc= - K0dt

Integration of this equation gives

C-C0= K0t

C=C0- K0t

Where C0 =concentration of drug at time (t)= 0

C= concentration of drug remaining at time t. Figure 1: Graph of zero order kinetics

Based on this expression a graph of C versus t yields a straight line. The y intercept would be
equal to C0 and slope of the line would be equal to K0.

Half life of zero order

Half life is defined as the time period required for the concentration of drug to decreased by one
half when T= t/2, C= C0/2 and finally equation becomes

Cо
= Cо − Kо t1/2
2
Cо Cо
t1/2 = = 0.5 (4)
2Kо Kо

Equation (4) shows that t1/2 of a zero order process is not constant but proportional to the zero
order rate constant K0. Since the zero order t1/2 changes with the decline in drug concentration.

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First order kinetics

In equation (3), if n=1 then equation becomes

𝑑[𝐶]
= −𝐾𝐶 (5)
𝑑𝑡

Where K=first order rate constant (time-1).

From equation (5) first order process can be defined as the process whose rate is directly
proportional to the concentration of drug undergoing reaction. Therefore first order process is
said to follow linear kinetics as shown in figure 2.

Figure 2: Linear relationship between rate of reaction and concentration of drug in case of first
order kinetics.

From equation (5) we can get

𝑑[𝐶]
= −𝐾𝑑𝑡 (6)
𝐶

Integration of equation of (6) gives,

ln C= ln Cо − kt (7)

Equation (7) in exponential form gives

𝐶 = Cо e−kt (8)

Since ln =2.303, then

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𝑘𝑡
log C= log C0 - 2.303 (9)

First order Half life

When t= t1/2 and C= C0/2

then equation (9) gives

𝑘t1/2
log C0/2= log C0 - (10)
2.303

on solving it will give equation as

0.693
t1/2 = (11)
𝐾

It is found from equation (11) that for a first order reaction, t1/2 is constant. Irrespective of the

initial amount or concentration of drug, the time required for the total amount of drug to reduce

to one half remain constant. The half life of a first order process is an important pharmacokinetic

parameter. Most pharmacokinetics processes, i.e., absorption, distribution and elimination follow

first order kinetics.

Pseudo first order

A pseudo first order reaction is actually second order but it is assumed to be first order under

special circumstances. For example, a second order reaction of the type

A + B --> C

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It depends on concentrations of both A and B, but monitoring the concentration of two different

reactants at the same time is very difficult. A pseudo first order reaction, chemical reaction

between two reactants, thus a second order reaction, which appears to be first order reaction due

to one of the reactants is in such small quantity that it is not easily noticed.

For a typical second order reaction with rate equation

r = k[A][B] (12)

Suppose the concentration of reactant B is constant then,

r = k[A][B] = k'[A] (13)

Where the pseudo first order rate constant k' = k[B] (14)

The second order rate equation has been reduced to a pseudo first order rate equation, which

makes the treatment to obtain an integrated rate equation much easier.

For example, the hydrolysis of sucrose in acid solution is often cited as a first order reaction with

rate r = k[sucrose].

The actual rate equation is third-order, r = k[sucrose][H+][H2O],

However, concentrations of both the catalyst H+ and the solvent H2O are normally constant, so
that the reaction is pseudo first order.

Half-Life in a Pseudo First order reaction

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Half-life is time required to decrease the concentration of a reactant by one half. Here, [B] will
be the reactant in excess, and its concentration will stay constant. [A]o is the initial concentration
of A; thus half-life concentration of A is 0.5[A]o.

ln(0.5[A]o)−ln[A]o=−kt1/2 (15)

or, ln(0.5[A]o)−ln[A]o=−k′[B]ot1/2

0.5𝐴𝑜
ln = −kt (16)
𝐴0

or, ln (0.5) =−k′[B]ot

or, ln(0.5)=−kt1/2

or, t1/2=(ln0.5)/-k

or, t1/2= (ln0.5)/ -k′[B]o (17)

Pharmacokinetic Models

Pharmacokinetic model is a mathematical modeling technique for predicting the absorption,

distribution, metabolism and excretion (ADME) of synthetic as well as natural substances in

humans and other animal species. It is used in drug development and research, and in health risk

assessment for cosmetics or general chemicals. In biological system drug events often happen

simultaneously. In order to understand a complex biological system, a hypothesis or model is

conceived using mathematical terms, which are concise means of expressing quantitative

relationship. Various mathematical models mimic the rate process of drug absorption,

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distribution and elimination. These mathematical models are useful for estimation of drug

concentration in vivo with respect to time via mathematical equations. A large degree of residual

simplification and empiricism is still present in those models, but they have an extended domain

of applicability compared to that of classical, empirical function based, pharmacokinetic models.

These models may have purely predictive uses, but other uses, such as statistical inference, have

been made possible by the development of Bayesian statistical tools able to deal with complex

models. That is true for both toxicity risk assessment and therapeutic drug development.

Assume a drug which is given by intravenous injection and that rapidly dissolves in the body

fluids. A pharmacokinetic model that would describe this situation would be a tank containing a

volume of fluid which is rapidly equilibrated with the drug. As in the human body, a fraction of

drug would be continually eliminated as a function of time.

The concentration of drug in the tanks after a given dose would be governed by two parameters:

1. The fluid volume of tank and

2. The elimination of drug per unit of time.

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In pharmacokinetics these parameters are assumed to be constants. If a known set of drug

concentrations in the tank, were determined at various time intervals then volume of fluid in the

tank and rate of drug elimination would be established.

Pharmacokinetic models plays significant role in:

1. Predicting drug concentration in various biological fluids (i.e., plasma and urine) and tissue.

2. Calculating the optimum dosage regimen for each patient individually.

3. Correlating drug concentration with pharmacologic or toxicologic activity.

4. Evaluating transformations in the rate or extent of availability between formulations.

5. Determining the alterations in physiology or disease affect the absorption, distribution or

elimination of the drug.

6. Explain drug interactions.

7. Estimate the possible accumulation of drug and/or metabolites.

8. Predict the Multiple dose concentration curves from single dose experiments.

9. Evaluate the risk of toxicity with certain dosage segments.

10. Characterize the behavior of drug in patients.

Since a model is based on a hypothesis and simplifying assumptions which describe biologic

system in mathematical term, a certain degree of caution is needed when relying totally on

pharmacokinetics model to predict drug action.

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There are different approaches to pharmacokinetic analysis of experimental data:

I. Compartment models

a. Mammillary model

b. Caternary model

II. Physiologic model (Flow model)

III. Non-Compartmental analysis.

Compartment models

A Compartment model is often described by decomposition into a number of interacting

subsystems. It should not be assumed as a physical volume. A compartment is not a real

physiologic or anatomic region but is considered as tissue or group of tissues which have similar

blood flow and drug affinity. Within each compartment the drug is considered to be uniformly

distributed. Each compartment may possibly exchange drug with other compartments.

Compartment models are based on linear assumptions using linear differential equations. Rate

constants are used to represent the overall rate processes of drug entry into and exit from the

compartments. The model is an open system since drug can be eliminated from the system. It is

also assumed that the rate of drug movement between the compartments follow first order

kinetics, depending upon whether the compartments are divided into two categories namely;

mammillary and caternary model.

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Mammillary Model

The mammillary model is the most widely applied compartment model in pharmacokinetics. In

this model one or more peripheral compartments connected to a central compartment. Plasma

and highly perfused tissues which rapidly equilibrate with drug represent the central

compartment. The mammillary model may be considered as strongly connected system since one

can estimate the amount of drug in any compartments of the system after drug is introduced into

a given compartment.

The elimination of drug primarily take place from the central compartments because of the

considerable perfusion of organs involved in drug elimination (that is kidney and liver).

Various types of compartment models are depicted in figure 3.

With the help of these models following targets could be achieved:

1. Differentials equations can be framed which would describe drug concentration changes

in each compartment.

2. A visual representation of the rate processes could be done.

3. Calculations of pharmacokinetic constant can be done which are necessary to describe the

process adequately.

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Model-1: One compartment open model- intravenous injection

Model-2: One compartment open model with first order absorption.

Model-3: Two compartment open model-intravenous injection.

Model-4: Two compartment open model with first order absorption.

Model-5: Three compartment open model-intravenous injection.

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Model-6: Three compartment open model-extravascular administration.

Figure 3: Various mammillary compartment models (K a is the first order adsorption rate
constant and Ke is the first order over all elimination rate constant).

Advantages of compartment model

1. It shows how many rate constants are necessary to describe these processes .
2. It gives a visual representation of various rate processes involved in drug disposition.
3. It enables the pharmacokineticist to write differential equation for each of the rate
processes in order to describe drug concentration changes in each compartment.

Disadvantages of compartment model

1. Extensive efforts are required in the development of an exact model that will predict and
describe appropriately the ADME of a certain drug.
2. The model is based on curve fitting of plasma concentration with complex
multiexponential mathematical equations.
3. The compartments and parameters bear no relationship with the physiologic functions or
the anatomic structure of the species. Several assumptions have to be made to facilitate
data interpretation.
4. The model may vary within a study population.
5. The approach can be applied only to a specific drug under study.

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6. The drug behavior within the body may fit different compartmental models depending
upon the route of administration.
7. Difficulties generally arise when using models to interpret the difference between results
from human and animal experiments.

Conclusion

We were able to understand the term Pharmacokinetics

What rate and order of reaction means to us.

We were able to understand First order, second order, pseudo first order in addition Half-life for
different orders of reaction
Pharmacokinetic Models to describe rate processes.

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