“Compartment Modelling In Biological Processes:

A Mathematical Perspective”
Submitted in partial fulfilment of the requirements
for the award of the degree of
MASTER OF SCIENCE

IN
MATHEMATICS
By
Sabiya Ali 25999903

Tabinda Majeed 25999910

Arabin Mushtaq 25999911

Ifrah Ashraf 25999914

Under the supervision of

Dr. Nazida Sheikh

JAMMU KASHMIR INSTITUTE OF MATHEMATICAL SCIENCES

Amar Singh College Campus, Gogjibagh SRINAGAR-190008, J&K

1
 CERTIFICATE

This is to certify that the project entitled “Compartment Modelling In

Biological Processes: A Mathematical Perspective” submitted by
“Sabiya Ali(80-IMS-2019), Tabinda Majeed(90-IMS-2019), Arabin
Mushtaq(91-IMS-2019) and Ifrah Ashraf(94-IMS-2019)” in partial
fulfillment of the requirements for the award of Master’s Degree in
Mathematics, of Jammu and Kashmir institute of Mathematical Sciences
during the academic year 2025, is an authenticate work carried out under
my guidance and supervision.

Dr. Nazida Sheikh Prof. T. A. Shikari

(Project Supervisor) (Director JKIMS)

2
 ACKNOWLEGEMENT

The success and final outcome of this project required a lot of guidance and
assistance from many people and we are extremely fortunate to have got this all along
the completion of our project work. Whatever we have done is only due to such
guidance and assistance and we would not forget to thank them. We respect and thank
our project supervisor Dr. Nazida Sheikh for giving us an opportunity to do this
project work and providing us all support and guidance. We are extremely grateful
to her for providing such a nice support and guidance.
Our sincere thanks and appreciation to Prof. Tariq Ahmad Shikari, Director Jammu
and Kashmir Institute of Mathematical Sciences, Srinagar for providing necessary
support. We offer our grateful thanks to our family for always encouraging us in
difficult times. They have always been a source of power and encouragement for us.

Name and Signature of Students

Sabiya Ali (1908)

Tabinda Majeed (1925)

Arabin Mushtaq (1926)

Ifrah Ashraf (1930)

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 CONTENTS
CHAPTER PARTICULARS PAGE
NO.
ABSTRACT 05

01 Introduction to Mathematical Biology 06 - 14

1.1 Mathematical Biology
1.2 Why Mathematical biology
1.3 Key areas in Mathematical Biology
1.4 conclusion

02 Introduction to Mathematical Modelling 15 -23

2.1 Mathematical Modelling
2.2 What is Mathematical Model?
2.3 Types of Mathematical Models
2.4 Formulation of Mathematical Model
2.5 Applications of Mathematical Modelling

03 Compartment Model 24 -30

3.1 Basic Terminology in Compartment Modelling
3.2 Types of Compartment Modelling

04 Three Compartment Model 31- 52

(in biological systems)
4.1 Pharmacokinetics and Drug Distribution
4.2 Applications of Three Compartment Model
(Remifentanil)
4.3 Three Compartment Model for Liver

05 Conclusion 53
Bibliography
 Abstract

Mathematical biology is an interdisciplinary field that applies

mathematical techniques and tools to understand, simulate and
analyse biological systems. From modelling the spread of infectious
disease to understanding neural activity, mathematical biology plays a
crucial role in uncovering patterns and dynamics within living
organisms.
Mathematical modelling, on the other hand, involves constructing and
analysing mathematical representations of real-world systems. In
biology, these models help simplify complex biological processes and
make predictions that can guide experimental and clinical research.
By integrating biology with mathematics, researchers can develop
insights into population dynamics, genetic evolution, epidemiology
and cellular behaviour.
This project explores the foundational concepts of mathematical
biology and various modelling approaches, illustrating how
mathematical methods can be used to solve biological problems and
advance scientific understanding.
Chapter I
1.1 Mathematical Biology:
Mathematical biology is an interdisciplinary field that applies the
principles and techniques of mathematics to understand, analyse, and
model biological systems. It serves as a powerful bridge between
abstract world of mathematical theories and the complex, often
unpredictable, realm of living organisms by formulating biological
problems in the language of mathematics, researchers can gain deeper
insights, predict behaviours, and explore the underlying mechanisms
of life.
As its core, mathematical biology seeks to describe biological
phenomena ranging from the growth of populations and spread of
Infectious Diseases to genetic patterns and neural activity using
mathematical tools such as differential equations, statistics, algebra,
and computational simulations. Unlike purely experimental biology,
which relies heavily on observation and laboratory work,
mathematical biology emphasis modelling and analysis, allowing
scientists to test hypothesis, simulate scenarios, and interpret data
more effectively.
One of the main strengths of mathematical biology is its ability to
simplify and represent complex biological systems through models.
These models can be qualitative or quantitative and may vary in
complexity depending on the level of detail required. Common types
include deterministic models, stochastic models, dynamic systems,
and compartment models. Such models not only help in
understanding the current behaviour of biological systems but also in
making predictions about future states under different conditions.
The field has grown significantly with advancement of computational
technology, enabling the analysis of large data sets and the simulation
of highly intricate biological systems. Today, mathematical biology
plays a vital role in a variety of biological disciplines including

6
epidemiology, ecology, genetics, neuroscience, and cellular biology.
It contributes to solving real world problems like predicting the
spread of diseases, understanding ecosystem dynamics, optimising
drug delivery systems, and even guiding conservation efforts.
In summary, mathematical biology provides a structured and rigorous
framework for exploring life sciences. It empowers scientists and
researchers to go beyond empirical observations and build a
mathematical understanding of life processes, ultimately leading to
more accurate predictions, better decision making, and innovative
solutions to biological challenges.

1.2 Why Mathematical Biology?

Biological systems are often nonlinear, dynamic and involve multiple
interacting components. Traditional descriptive approaches in biology
can fall short when systems become too complex. Mathematics
provides a precise framework to:

• Model population dynamics.

• Understand the spread of diseases.
• Explore the mechanics of neurons.
• Simulate ecosystems.
• Predict evolutionary outcomes.
• Analyses genetic networks.
1.3 Key areas in mathematical biology:
1.Population dynamics:
Mathematical biology plays a critical role in understanding
population dynamics, which is the study of how populations of
living organisms change over time and space. By using
mathematical models, researchers can describe, predict, and
analyse changes in population size, structure and distribution under
various environmental and biological influences.
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 Here are some of the features of population dynamics making use
of mathematics;

1. Modelling population growth: Mathematical biology helps

represent population growth using equations that capture real
world behaviour.
i. Exponential growth:

= RN
where,
N= population size.
R= growth rate.
ii. Logistic growth model:

iii. Predator – prey and competition models:

Mathematical biology describes interactions between species; •
Lotka Volterra model (Predator Prey):

Where, y = predator population parameters

define birth/death and interaction rates.

iv. Competition Models:

8
Two species competing for the same resource can be modelled using
modified logistic equations, showing coexistence or extinction
outcomes.
2. Epidemiology:
Epidemiology is the study of how diseases spread, persist, and can
be controlled in populations. Mathematical biology powerful
toolkit to model, analyse and predict the spread of infectious
diseases. These models help public health authorities design
effective intervention strategies and understand the dynamics of
epidemics.

i. Modelling disease spread:

Mathematical biology uses compartmental models to divide the
population into categories based on disease status. The most
common are;
SIR Model (susceptible infectious recovered);

where,
S = number of susceptible individuals
I = number of infectious individuals
R = number of recovered (immune)
individuals

9
 b = transmission rate

d = recovery rate

ii. Predicts outbreak trends:

This field enables us to forecast how many people might get

infected, when the peak of infection will occur, and how long
an outbreak will last thereby helps in planning hospital
resources and public health responses.
iii. Estimating key epidemiological parameters:
Mathematical biology helps to estimate critical metrics like;
Basic reproduction number(R), which represent average number of
secondary infections caused by one infected individual.
Incubation and infectious periods.
Case fatality rates and herd immunity thresholds.

iv. Designing and evaluating intervention strategies:

Mathematical tools help in testing vaccine efficiency, distribution
strategies, effectiveness of lockdowns, contact tracing, the impact
of partial immunity or waning immunity in the population.
3. Neuroscience:
Mathematical biology significantly enhances neuroscience by
providing tools to model, analyse, and predict the behaviour
of neurons and neural networks. Here are some of the areas in
neuroscience where mathematics plays its role;

i. Modelling neuron activity:

Mathematical models describe how neurons generate and transmit
electrical signals in the form of two models namely “Hodgkin
Huxley model” to describe how action potentials (nerve impulses)

10
 are generated through ion channels and “Fitzhugh Nagumo model”
which is a simplified version used to study excitation and
oscillations in neurons.

ii. Analysing brain rhythms:

The brain shows oscillations like alpha, beta, theta and gamma
waves where mathematics plays its role to analyse EEG data, model
synchronization and desynchronization in neural circuits,
understanding disorders like Parkinson’s disease, where abnormal
oscillations are key symptoms
iii. Imaging and data analysis:
Advanced mathematical techniques are crucial for analysing MRI,
fMRI, PET brain images, extracting meaningful patterns from
large neural datasets (connect omics, brain atlases).
4. Genetics and evolution:
Mathematical biology plays a central role in both genetics and
evolution by providing models and tools to understand how traits
are inherited, how populations change over time, and how genetic
variation evolves.
Here are listed some of the fields of genetics and evolution where
different concepts of mathematics are used;

i. Modelling inheritance patterns:

Maths explains how genes are passed from one generation to the
next. The Mendelian genetics can be expressed using probability
and combinatorics.
Mathematics also helps in predicting outcomes of crosses,
and inherited diseases
ii. Quantitative Genetics:

11
 Mathematical biology links phenotypes (observable traits) to
genotypes (gene combinations) by using linear models and
statistical genetics to estimate heritability, predict trait distribution,
design breeding programs.
5. Molecular and cellular biology:
Mathematical biology plays a vital role in molecular and cellular
biology by helping scientists’ model, quantify and predict the
behaviour of complex biological systems at microscopic scales.
Here’s a breakdown of how math contributes to understanding
molecules and cells;
i. Modelling biochemical reactions:
At the molecular level, math is used to model chemical reactions
inside cells. Its key tools include;
a. Ordinary Differential Equations (ODEs) to model reaction
kinetics (e.g., enzyme activity, gene regulation).
b. Michaelis Menten kinetics for enzyme catalysed reactions.
Mass action kinetics for general chemical reactions.

ii. Spatial and structural modelling:

Molecular biology is not just about “how much” but also “where”
molecules are. It involves use of Partial Differential Equations to
model how molecules diffuse and react across space.
6. Ecology and ecosystems:
Mathematical biology is essential in ecology and ecosystem
science because it allows researchers to analyse, simulate, and
predict the dynamics of species, populations and environmental
interactions.
Here’s how math contributes in understanding the natural world;

i. Modelling population dynamics:

12
 Math helps to describe how populations of organisms grow, shrink
or stabilizes over time by using some models like Exponential and
Logistic Growth models which describes population growth with
and without limits.

ii. Ecosystem stability and resilience:

Math is used to evaluate ecosystem health, study how systems
respond to disturbances (like fires, deforestation, pollution).

iii. Spatial Ecology:

In this context, Partial Differential Equations (PDEs) and reaction
diffusion models describe spread of species, diseases or pollutants
across landscapes.
iv. Conservation and resource management:
Math helps to estimate extinction risks, optimize harvesting
strategies to ensure sustainability, guide protected area design and
biodiversity conservation.

Thus, after seeing some of the major areas of application of

mathematics with respect to biology, we will sum up the major
mathematical tools and methods that are being employed. These
are illustrated as;
1) Differential equations (dynamics of populations, neurons,
chemicals)
2) Stochastic Processes (Random gene expression,
molecular noise)
3) Statistics and Inference (Data analysis, genomics,
experiments)
4) Dynamical Systems (stability and bifurcations in
biological systems)
5) Linear algebra (networks, age structured models)

13
 6) Graph theory (gene networks, food webs, disease spread)
7) Optimization (drug dosing, ecosystem management)
8) Numerical methods (solving models computationally)
9) Machine learning (Bioinformatics, diagnostics)
10) Probability theory (evolution, epidemiology,
molecular biology)

1.4 Conclusion:
Thus, we can say mathematical biology plays a crucial role in
modern science by providing a quantitative approach in
understanding the complexity of life. As biological data becomes
increasingly abundant and detailed, the integration of mathematical
tools will be ever more essential in uncovering the mechanisms
behind biological processes and solving real world problems in
health, ecology and biotechnology.

14
Chapter II
2.1 Mathematical Modelling:
Mathematical modelling is a powerful tool used to represent real
world systems, phenomena, or problems through mathematical
language and structures. It involves the translation of physical, social,
or biological situations into mathematical form to analyse, interpret,
and make predictions or decisions.

2.2 What is Mathematical Model?

A mathematical model is a representation of a system using
mathematical concepts such as equations, functions, and inequalities.
It can be used to;

• Describe how a system behaves (descriptive modelling)

• Predict future behaviour (predictive modelling) •
Optimize performance (prescriptive modelling)

2.3 Types of Mathematical Models:

Mathematical models can be categorised based on different criteria
such as time, structure, randomness and method of solution below is a
detailed classification of the main types of mathematical models:

1. Deterministic models:
The deterministic models are a type of mathematical model where
the outcome is completely determined by the initial conditions and
the rules of the model - there is no randomness involved. In simple
words, a deterministic model gives the same result every time we
run it with the same starting conditions. It assumes that everything
is predictable and nothing happens by chance.

15
 For example, if we have a model that predicts how a population
grows overtime and we start with 100 people, a growth rate of 5%
and no other changes, then the population size at any future time
will always be the same when we run the model again with those
same values.

2. Stochastic models:
Stochastic models are mathematical models that include
randomness or uncertainty. This means the outcome can be
different each time even if we start with the same conditions. In
simple words, a stochastic model is like a model that includes
“luck” or “chance”. It does not always give the same result,
because it includes random events or unpredictable changes.
For example, imagine a model that predicts how a disease spreads.
Even if we start with same number of sick people, the number of
new infections might be different each time we run the model,
because it depends on chance like who meets whom and whether
they get infected.

3. Static models:
Static models are mathematical models that show a system at a
single point in time. They do not consider changes over time. In
simple words, a static model is like a snapshot - it shows how
things are right now, not how they change in the future or how they
got there.
For example, a static model might show the balance of water in a
tank based on how much is flowing in and out at that moment,
without looking at how it changes overtime.

4. Dynamic models:
Dynamic models are mathematical models that show how a
system change over time. They track how things evolve step by step.

16
In simple words, a dynamic model is like a video - it shows how
things change from one moment to the next, not just a single snapshot.
For example, a dynamic model might show how a population grows
year by year, or how the number of infected people in a disease
outbreak increases and then decreases overtime.

5. Linear models:
Linear models are mathematical models where the relationship
between variables is straight and proportional - when one variable
changes the other changes at a constant rate. In simple words, a
linear model shows a straight-line relationship. If we double one
thing the other also doubles (or halves, depending on the direction)
For example, if we earn Rs 100 for every hour we work, then the
total money we earn increases in a straight line as our hours
increase. That’s a linear model.

6. Nonlinear models:
Non-linear models are mathematical models where the relationship
between variables is not straight or constant - small changes in one
variable can cause big or unpredictable change in another. In
simple words, a non-linear model shows a curved or changing
relationship. Doubling one thing doesn’t always double the other -
the change can be faster, slower, or uneven.
For example, if a disease spreads faster as more people get
infected, the number of infected people might grow slowly at first,
then suddenly very fast. That’s a non-linear model.

7. Continuous models:
Continuous models are mathematical models where changes
happen smoothly and constantly overtime, without certain jumps.

17
In simple words, a continuous model shows how things change
little by little, all the time – like a flowing river, not in steps.
For example, if we model how water fills a tank drop by drop, and
the water level rises smoothly, that’s the continuous model.

8. Discrete models
Discrete models are mathematical models where change happen in
separate steps, not continuously. They deal with values at specific
points in time. In simple words, a discrete model shows changes
that happen one step at a time - like counting numbers or taking
snapshots.
For example, if we count how many people get sick each day
(day1, day2, day3…), that’s a discrete model because it jumps
from one day to the next.

9. Empirical models:
Empirical models are models based on observations and data, not
on theories or known laws. They try to describe what is happening
by looking at real world results. In simple words, an empirical
model is built by using data from experiments or observations - it
shows what usually happens, even if we don’t fully understand
why.
For example, if we measure how a plant grows under different
amounts of sunlight and then create a formula based on those
results, that’s an empirical model.

10. Mechanistic (theoretical) models:

Mechanistic models are the models based on known scientific
principles, laws, or mechanisms. They try to explain why
something happens, not just what happens. In simple words, a
mechanistic model is built using what we already know about how

18
 things work - like physics, chemistry, or biology to explain and
predict behaviour.
For example, if we model how a disease spreads using formulas
that include how people interact, how the virus behaves, and how
the immune system responds, that’s a mechanistic model.

11. Analytical models:

Analytical models are mathematical models that can be solved
exactly using formulas and equations to get a clear answer. In
simple words, an analytical model gives us an exact solution using
math formulas - like solving a puzzle with a clear answer.
For example, if we use a formula to calculate how fast a population
will grow overtime and we get neat answer (like 500 people after
10 years), that’s an analytical model.

12. Numerical models:

Numerical models are models that use step - by - step calculations
or computer simulations to find an approximate solution when an
exact one is too hard or impossible. In simple words, a numerical
model gives an estimated answer by using computers or repeated
calculations - not a perfect formula, but close enough to be useful.
For example, if a disease model is too complex to solve by hand, a
computer can calculate what might happen day by day. That’s
numerical model.

2.4 Steps Involved in the Formulation of Mathematical

Model:
1. Identify the problem: The first step involves to understand and
define the real-world situation or question.

19
2. Make assumptions: This step is concerned in simplifying the
real-world problems by focusing on key variables and
relationships.
3. Formulate the model: In this step we develop mathematical
relationships (equations, inequalities, functions etc) based on
assumptions.
4. Solve the model: In this step, we use mathematical techniques
or methods to analyse or solve the given model.
5. Interpret the results: This step deals with translating the
mathematical results back into the context of the original
problem.
6. Validate the model: In this step, we compute the models’
predictions with real world data and adjust the assumptions or
structure, if needed.
7. Refine and communicate: This is the last step which improves
the model and communicate the findings clearly.
Here is a flowchart depicting the various steps involved in the
formulation of a mathematical model;

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2.5 Applications of Mathematical Modelling:
Mathematical modelling has a wide range of applications across
various fields of science, engineering, social sciences, economics,
and more. Below is a categorised list of key applications of
mathematical modelling:
1. Science and engineering:

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 • Physics: Modelling motion (Newton’s laws),
thermodynamics, quantum mechanics, heat propagation
etc.
• Chemistry: Reaction kinetics, molecular dynamics,
diffusion processes.
• Engineering: Structural analysis, fluid dynamics (CFD),
control systems, signal processing, etc.
• Environmental science: Climate change models,
pollution dispersion, water cycle modelling.
2. Biological sciences and medicine:
• Epidemiology: Spread of diseases for example SIR
models for covid 19, malaria etc.
• Neuroscience: Modelling brain activity neuron dynamics.
• Genetics and evolution. Population genetics, gene
regulation networks.
• Medical imaging: Image reconstruction, tumour growth
modelling.
• Pharmacokinetics: Drug dosage and distribution
modelling in the body.
3. Ecology and ecosystem modelling:
• Predator prey models for example Lotka Volterra
equations.
• Population dynamics.
• Resource management and conservation. • Habitat
fragmentation and biodiversity studies.

4. Economics and finance:

22
 • Macroeconomics: economic growth models inflation
and unemployment analysis.
• Finance: Option pricing (Black- Scholes model), risk
assessment, portfolio optimization.
• Market modelling: Supply demand analysis, game
theory, auction modelling.
5. Social sciences:
• Sociology: Spread of ideas, social behaviour modelling,
opinion dynamics.
• Demography: Population growth and migration
modelling.
• Psychology: Decision making and behaviour models.
6. Computer science and AI:
• Algorithm performance analysis.
• Machine learning and a data fitting model.
• Natural language processing.
• Robotics and path planning.
7. Sports and games:
• Performance prediction.
• Strategy optimization.
• Game theory in competitive settings.
8. Education and learning:
• Curriculum design optimization.
• Learning analytics.
• Knowledge progression models.

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Chapter III
COMPARTMENT MODEL
A compartment model is a simplified mathematical representation of a
system, often used in fields like pharmacokinetics, epidemiology, and
ecology, where the system is divided into distinct, interconnected
units called “compartments”. These compartments represent groups of
tissues, populations, or environmental zones, and the model uses
equations to describe how materials or individuals move between
these compartments. In other words, we can define a compartment
modeling as a mathematical framework used to describe and analyze
the distribution and movement of substances—such as drugs,
chemicals, or pathogens—within a biological system. It simplifies
complex biological processes into a set of interconnected
compartments, each representing a specific tissue or organ, and
models the transfer of substances between these compartments. Each
compartment represents a distinct state, group, or locations where
substances or entities (like chemicals, populations, or data) are
assumed to be uniformly distributed, like in one compartment model it
assumes that the substance distributes instantaneously and uniformly
throughout the body, in two compartment model it divides the body
into a central compartment (e.g., blood) and a peripheral compartment
(e.g., tissues), allowing for more complex distribution patterns and so
on.

3.1. Basic Terminology in Compartment Modeling

● Compartments: These are simplified representations of
biological tissues or organs. For instance, in pharmacokinetics, a
"central compartment" might represent the bloodstream, while

24
 "peripheral compartments" could represent tissues like muscles or
fat.

● Flow/Transfer Rates: The movement of substances between

compartments is characterized by rate constants (e.g., k₁₂, k₁₀),
which quantify the speed of transfer between compartments.

● Rate constants: Parameters that define the speed of transfer

between compartments.

● Initial conditions: The starting quantities in each compartment.

3.2. Types of Compartment Modeling

i) ONE-COMPARTMENT MODEL:
The one-compartment model is a simplified way to understands how
drugs move and are eliminated in the body. It assumes the body is a
single, well-mixed compartment where the drug rapidly distributes
throughout and is eliminated in a first order process. This model is
useful for drugs that distribute quickly and can help predict drug
behavior and guide dosing. Imagine a drug is injected into the
bloodstream (central compartment) and eliminated over time (kidney
or liver).
Let:

C(t): drug concentration in the compartment at time ‘t’

25
k: elimination rate constant

ODE:

This is a first-order decay equation. The solution is

𝐶(𝑡)= 𝐶0𝑒−𝑘𝑡
Where C0 is the initial concentrations.
A general one-compartment is given as;

ii) TWO-COMPARTMENT MODEL:

26
The two-compartment model divides the body into two interconnected
compartments: a central compartment and a peripheral compartment.
Consider a drug that can distribute between two compartments, the central
compartment represents tissues with rapid drug distribution, like blood and
highly perfused organs, while the peripheral compartment represents tissues
with slower distribution, such as adipose tissue. This model helps understand
how drugs distribute and are eliminated in the body.

Imagine a drug is injected that can be distributed between two

compartments
Let:

C1(t): Drug concentration in the central compartment

C2(t): Drug concentration in the peripheral compartment

k12: Rate constant for transfer from central to peripheral

k21: Rate constant for transfer back from peripheral to

central

ke: Elimination rate from central compartment

ODEs:

- keC1
𝑑𝐶2

27
The first equation describes loss from the central compartment to
peripheral and elimination, plus return flow from the peripheral
compartment.
The second equation captures the exchange between the
compartments.
A general two-compartment model is given as;

28
iii) THREE-COMPARTMENT MODEL:
The three-compartment model is a mathematical framework
commonly used in pharmacokinetics to describe the distribution and
elimination of drugs within the body. It extends the simpler one
compartment and two compartment models by incorporating
additional compartments to more accurately represent the complex
physiological processes involved.

Structure of three-compartment model: In this model, the body

is assumed as consisting of three interconnected compartments:

a) central compartment (C1): Represents the bloodstream and

wellperfumed organs like the heart, liver, and kidneys.

29
b) Peripheral Compartment (C2): This Compartment corresponds to
the tissues with moderate blood flow, such as muscles.

c) Peripheral Compartment (C3): This Compartment accounts for

poorly perfumed tissues, like adipose tissue.

The drug moves between these Compartments through first -order kinetic
processes characterized by rate constants:

● k12, k13: It is the transfer rates from the central to peripheral

compartment.

● k21, k31: It is the transfer rates front peripheral to central

compartments.

● k10: It is the elimination rate from central compartment.

The plasma concentration of the drug over time is typically modeled

using the equation:
C(t) = 𝑨𝒆−𝒂𝒕 + 𝑩𝒆−𝒃𝒕 + 𝑪𝒆−𝒄𝒕
Where a, b, c are the rate constants associated with the drugs
distribution and elimination phases.

30
CHAPTER IV

Three compartment Model In biological systems.

4.1. Pharmacokinetics and Drug distribution
In pharmacokinetics, the three-compartment model is employed to
describe the distribution and elimination of substances Within the body.
The body is divided into three compartments: The central
compartment, and two Peripheral compartments. The drug moves
between these Compartments through first order kinetic Processes.

31
Now, we try to learn some of its applications and explain certain as
follows;

32
4.2. Applications of the Three-Compartment Model

1. Pharmacokinetics of Intravenous Anesthetics (e.g., Propofol)

Helps model rapid onset, redistribution, and slow terminal elimination

phases

2. Complex Drug Disposition

Describes drugs with multiple distribution phases better than simpler

models

3. Drug Dosing Regimens

Assists in designing accurate dosing schedules by accounting for

different tissue kinetics

4. Therapeutic Drug Monitoring

Useful when modeling drugs with a long tail of elimination (e.g.,

aminoglycosides, digoxin)

5. Toxicokinetic

Predicts behavior of toxic substances that may accumulate in different

tissue compartments

6. Drug Development and Clinical Trials

Used in early-phase modeling of pharmacokinetics for new drug

candidates
33
7. Simulation and Modeling Software

Supports tools like NONMEM or MATLAB for

pharmacokinetic/pharmacodynamic (PK/PD) modeling

Remifentanil
: As an intravenous Anesthetic

Remifentanil is a new opioid derivative drug characterized by a fast

onset and by a short time of action, since it is rapidly degraded by
esterases in blood and other tissues. Its pharmacokinetic and
pharmacodynamics properties make remifentanil a very interesting
molecule in the field of 0anesthesia. a complete and versatile
pharmacokinetic description of remifentanil still lacks. In this work a
three-compartmental model has been developed to describe the
pharmacokinetics of remifentanil both in the case in which it is
administered by intravenous constant-rate infusion and by bolus
injection. The model curves have been compared with experimental data
published in scientific papers and the model parameters have been
optimized to describe both ways of administration.
The ad hoc model is adaptable and potentially useful for predictive
purposes.

Keywords: Remifentanil, pharmacokinetics, three-compartmental model

I. OVERVIEW
The topic of the development of new opioid anesthetic agents is mainly
to increase potency, and reduce the cardiovascular toxicity. For this
purpose, recently a new kind of opioid derivative drug, remifentanil, has
34
been synthetized. Remifentanil is an ultrashort acting opioid and it is
subjected to metabolism by esterases in blood and other tissues. In vivo
studies demonstrated an extensive metabolism of this drug by ester
hydrolysis. The primary metabolic pathway experienced by remifentanil
is the formation of a carboxylic acid metabolite (named GI90291)
obtained by desertification. The chemical structures of remifentanil and
its primary metabolite are shown in Fig 1 . It has been demonstrated on
animals that the pharmacodynamics of remifentanil is very similar to the
other opioid drugs, this fact, combined with the reduced effects on the
cardiovascular system makes the use of remifentanil very attractive in
anesthesia. Remifentanil is generally administrated by the intravenous
route. Because of the very short halflife of the drug, usually a bolus
injection is administered to raise the blood concentration immediately,
then a slower intravenous infusion is used to maintain the effective
plasma concentration. Even if it is recommended to infuse remifentanil
only during general anesthesia procedures, the single/repeated bolus
injections could be used in clinical situation in which a brief period of
intense analgesia is required and the set - up of a continuous infusion
pump is difficult (i.e. painful diagnostic and therapeutic procedures
outside the operating theater). For this reason, it is particularly
interesting to model what happen in the plasma concentration of
remifentanil after the bolus injection or continuous infusion
administration. Different kinds of models have been proposed, the
simplest of which is the compartmental one, alternatively, the
physiologically based approach is more complex and potentially more
exhaustive.

Fig 1.
Chemical structures of remifentanil and its metabolite.

The aim of the present work is to develop and validate a new simple
model using the compartmental modeling approach to evaluate the
35
remifentanil plasma concentration in case of bolus injection and
continuous infusion.

II. MODELING
In the three-compartmental modeling, three compartments describe the
fate of a drug once administered: the central compartment, which
represents the plasma; the highly perfused compartment, which
represents the organs and tissues highly perfused by the blood; and the
36
scarcely perfused compartment, which represents the organs and tissues
scarcely perfused by blood. A schematic of the model is shown in Fig 2 .

Fig 2.

A schematic of the three-compartmental model.

To model the pharmacokinetic of remifentanil, the case of intravenous

infusion has been studied. Thus, an amount of drug has been evaluated
as inlet in the central compartment. The processes which cause the
variation of the plasma concentration are: the absorption, the
distribution, and the excretion of the drug. These phenomena have to be
taken into account in the modeling. I(t) is defined as the function which
describes the drug introduction by intravenous infusion (which could be
an infusion at constant rate of administration or a bolus) in the central
compartment. The drug concentration in the compartments could be
evaluated solving the mass balance in the compartments, which could be
written as:

37
a) Central compartment:

3+
−[(𝑘12+𝑘13+𝑘10)⋅𝐶1]⋅𝑉1+𝐼(𝑡)

b) Highly perfused compartment:

c) Scarcely perfused compartment:

In which, C1, C2, and C3 are, respectively, the drug concentrations of the
central, highly perfused, and scarcely perfused compartments. V1, V2,
and V3 are respectively, the volumes of the central, highly perfused, and
scarcely perfused compartments. Cl1, Cl2, and Cl3 are, respectively, the
clearances (rates of drug elimination) of the central, highly perfused, and
scarcely perfused compartments. k12 and k21 are the transport
coefficients between the central and the highly perfused compartments;
k13 and k31 are the transport coefficients between the central and the
scarcely perfused compartments. Finally, k10 is the kinetic constant of
drug elimination from the central compartment. The kinetics of
elimination and transport between the compartments have been

38
considered first order kinetics. These equations have to be solved
coupled with their initial conditions:

𝑡=0, 𝐶1= 0
𝑡=0, 𝐶2=0

𝑡=0, 𝐶3=0

The three equations are inter- dependent, thus, they have to be solved
simultaneously to evaluate the drug concentration in all the
compartments.

Once identified the transport phenomena which take place in the

compartments and defined the differential equations able to solve the
mass balances in the compartments, the value of the parameters has to be
evaluated. To evaluate the parameters value, the model has been used to
fit the experimental data taken in literature, which refer both to
intravenous infusion and bolus.
Defining an error between the experimental data and the model
evaluation:

In which n is the number of experimental data for each experiment, CPi −

CPm (ti, P) is the difference between the experimental plasma
39
concentration and the model prediction at the time ti, P being the vector
of the model parameters. Minimizing this function it is possible to find
the values of the parameters which better approach the experimental
data.
The set of three ODEs constituting the model have been solved by a
code developed using MATLAB, and the same software was used to find
the value of P minimizing the function ɛ (P).

III. RESULTS
The model simulations obtained are shown in Fig 3 and compared with
the experimental data in the case of intravenous constant-rate infusion.
Each curve has been obtained as average values due to the
administration to two subjects. During the 20 minutes infusion, a total of
14 blood samples were taken. After stopping the infusion, 16 blood
samples
were taken, up to 240 min after the stop. Therefore, each history was
described by 30 sample data.

40
41
Fig 3.

42
Comparison between the experimental plasma concentration value and
the model curves in the case of intravenous constant-rate infusion with
an infusion time of 20 minutes. a1) plasma concentration after a dose of
1 μg .kg −1 ·min −1; a2) plasma concentration after a dose of 4 μg .kg −1
·min−1; a3) plasma concentration after a dose of 8 μg ·kg −1 ·min −1.

As could be seen from the graphs, the model reproduces satisfactorily

the experimental data. In particular, the plasma concentration following
the administration of a dose of 1 μg ·kg −1 ·min −1 is well approximated,
however, the concentration of higher doses (4 and 8 μg ·kg −1 ·min −1)
are not well approximated for periods longer than 90 minutes, this is
probably due to the fact that the supposed elimination kinetic could be
still optimized.

Furthermore, the same model has been used to reproduce the

remifentanil plasma concentrations in the case of bolus injection and
compared with the experimental data in Fig 4 . In this case the
administration has been evaluated as a fast infusion (bolus), thus the
plasma concentration immediately rise to a high value. Each curve has
been obtained as the average value over six patients (three men and three
women). Over 360 minutes, 21 blood samples were collected and
assayed for remifentanil. Once again, the model curves satisfactorily
approximate the experimental data.

43
44
45
Fig4

46
 Comparison between the experimental plasma concentration value and the model
curves in the case of fast intravenous infusion (bolus). b 1) plasma concentration
after a dose of 2 μg ·kg −1; b2) plasma concentration after a dose of 5 μg ·kg −1;
Comparison between the experimental plasma concentration value and the model
curves in the case of fast intravenous infusion (bolus). b 3) plasma concentration
after a dose of 15 μg ·kg −1 ; b4) plasma concentration after a dose of 30 μg ·kg −1.
The values of the model parameters obtained after the optimization routine, are
shown in Table 1 . The model developed has been used to evaluate the plasma
concentration both in the case of intravenous constant-rate infusion and
intravenous bolus, which has been reproduced simulating a very fast infusion in
the central compartment. This is a remarkable improvement to the compartmental
modeling: in fact, once the model parameters have been evaluated for a certain
administration, the model is able to predict the drug plasma
concentration varying not only the dose, but also the infusion rate of the
drug.

TABLE I.

Values and dimensions of the three-compartmental model parameters.

paramet Optimize d paramet Optimize d

er value er value

V1 7.88 mL k10 0.172 min-1

47
V2 23.9 mL k12 0.373 min-1

V3 13.8 mL k21 0.103 min-1

Cl1 2.08 mL.min-1 k13

0.0367 min-
1

Cl2 0.828 mL.min-1 k31

0.0124 min-
1

Cl3 0.0784 mL.min-1

IV. CONCLUSIONS

In this work a three-compartmental model has been developed to reproduce the

evolution of remifentanil plasma concentrations after intravenous constant-rate
infusion and intravenous bolus. The main phenomena of absorption,
distribution, and metabolism have been identified and the mass balances for the
three compartments have been written. The model has been then used to
reproduce plasma concentrations taken from literature and the best values of
the model parameters have been found minimizing the error between model
curves and experimental data.
Several studies have been conducted to develop a model which is able to
reproduce the remifentanil pharmacokinetics. The aim of these studies is
48
to compare the measured pharmacokinetic features of remifentanil after
an intravenous infusion to the model prediction. In particular, the
compartmental analysis has been extensively used and compared with
the experimental data, taken after intravenous infusion or bolus. Blood
concentration and time data after a computer-controlled infusion of
remifentanil could be analyzed by nonlinear regression using the
NONMEM program (University of
California) which may produce predicted and individually predicted
values (post hoc Bayesian estimates). The initial two-stage analysis
comparing one-, two-, and three compartmental models found that the
two-compartmental model shows the best fit to the experimental data.
This result was also confirmed by a population analysis. A more
complex analysis of both pharmacokinetics and pharmacodynamics of
remifentanil has been also approached. The
pharmacokinetic/pharmacodynamic relationship has been evaluated
using non-linear regression analysis. The pharmacokinetics have been
described using a one-compartment intravenous infusion model.
Moreover, the pharmacodynamics have been fitted using inhibitory
model. A statistical evaluation of the goodness of the models has been
carried out following the Akaike Information Criterion. According to
this analysis, the decrease of the SSE using a model with a large number
of parameters is useful if and only if it overcomes the increase in the
number of parameters with respect to the use of a model with a limited
number of parameters. In this case, the simple model has shown the best
overall fitting results.

Nevertheless, once the value of the parameters has been evaluated, our
simple model was able to describe the remifentanil concentration on
blood for different ways of administration. This is a remarkable
improvement to the compartmental modelling: in fact once the model
parameters have been evaluated for a certain kind of administration, the
model is able to predict the drug plasma concentration varying not only
49
the dose but also the infusion rate of the drug. This feature makes the
model more versatile than the other available in literature and very
useful for predictive purposes.

4.3. Three compartment model for liver

Liver is a major metabolic organ which performs many essential functions
such as detoxification of the organism, and the synthesis of various
proteins and various other biochemicals necessary for digestion and
growth
The liver plays a central role in the absorption, distribution and
elimination of a drug in the body. A three-compartment model is used
due to its function in drug metabolism and clearance

Example: Hepatic Drug Metabolism – Three-Compartment Model

Compartment Definitions:

i) Central Compartment (Plasma/Blood):

. Site of drug administration and systemic circulation.

. Represents blood and highly perfused tissues.

. Drug concentration is measured here.

ii) Peripheral Compartment (Tissues):

. Represents less perfused tissues where the drug distributes more slowly.
. Acts as a reservoir.

iii) Liver Compartment (Metabolism Site):

. The liver actively metabolizes the drug.

50
 . Blood flows from the central compartment to the liver.
. The liver eliminates the drug through metabolism or biliary excretion.

Diagram

Parameters in the Model:

• k12, k21: Rate constants between central and peripheral compartments.

• k10: Elimination rate from the central compartment (renal clearance or
other).
• Qh: Hepatic blood flow.
• Clh: Hepatic clearance.

Clinical Example: Lidocaine

Lidocaine is an anesthetic drug with high hepatic extraction. It follows a
threecompartment model:

• Central compartment: Rapid distribution to plasma.

• Peripheral compartment: Slower distribution into muscle/fat tissues.
• Liver: Metabolized extensively by cytochrome P450 enzymes
(especially CYP3A4).

51
In this model, the liver's capacity and blood flow greatly influence the drug’s
clearance.

Key points:

• The liver is often modeled as a separate compartment in drugs with

significant hepatic metabolism.
• This allows better prediction of drug concentration-time profiles and
helps in dosage adjustments, especially in liver dysfunction.
• A three-compartment model helps capture both fast and slow distribution
phases and metabolism.

52
CONCLUSION
This project has explored the vital role of mathematical biology with
a particular focus on compartment modelling, in understanding and
analysing complex biological processes. Compartment models
simplify biological systems by dividing them into distinct sections or
‘compartments’ each representing a specific state or group such as
susceptible, infected, and recovered individuals in epidemiological
models.
By using systems of differential equations, compartment models
allow researchers to simulate dynamic chains over time, predict
future behaviour, and assess the impact of interventions. This
approach has proven especially valuable in studying the spread of
infectious diseases, drug kinetics, and ecological interactions.
Through this study it is evident that compartment modelling serves
as a powerful and accessible tool for translating biological
complexity into manageable mathematical frameworks. As
challenges in public health, medicine, and environmental science
continue to evolve, the application of such models will remain
central to data - driven decision making and scientific advancement.
 BIBLIOGRAPHY
1. “MATHEMATICS IN MEDICINE AND THE LIFE
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CHARLES S. PESKIN
2. “MATHEMATICAL MODELS IN BIOLOGY” BY
LEAH EDELSTEIN-KESHET
3. “MATHEMATICAL BIOLOGY” BY J.D. MURRAY 4.
“AN INTRODUCTION TO SYSTEMS BIOLOGY:
DESIGN PRINCIPLES OF BIOLOGICAL CIRCUITS” BY
URI ALON
5. “A BIOLOGIST’S GUIDE TO MATHEMATICAL
MODELING IN ECOLOGY AND EVOLUTION” BY
SARAH P. OTTO AND TROY DAY
6. “INTRODUCTION TO APPLIED MATHEMATICS “BY
GILBERT STRANG
7. “THE NATURE OF MATHEMATICAL MODELING”
BY NEIL GERSHENFELD
8. “MATHEMATICAL MODELS IN POPULATION

54
BIOLOGY AND EPIDEMIOLOGY” BY FRED BRAUER
AND CARLOS
CASTILLO- CHAVEZ

55