quasi-steady-state assumption on a mathematical...

Quasi-Steady-State assumption on a

Mathematical Model of Leukemia

Mikkel Zielinski Ajslev

[email protected] 58224

Hasan M. M. Osman


Stefan Bisgaard


May 29, 2017

i

Abstract

This report deals with mathematical modelling of progression of leukemia, where the

human body is considered a dynamical system of interrelated compartments. The

main goal of the study, is to make a quasi-steady-state approximation of the model by

Andersen et al. (2017a). The purpose of said model, is to describe the dynamics of

leukemia with a system of six coupled differential equations, which we reduce down to

two - that contain information from the full system of six equations. We have however

proposed a slight change to the model by introducing a different density function. We

elaborate on the reason as to why we have introduced this new function, and finally

discuss the consequences of this change. We conclude that the quasi-steady-state

model describes the dynamics of the six dimensional system extremely well.

Resume

Denne raport omhandler matematisk modellering af progression af leukemi, hvor

kroppen er betragtet som et dynamisk system af sammenhængende compartments.

Hoved malet med undersøgelsen er, at udføre en quasi-steady-state approximation

af den foreslaede model af Andersen et al. (2017a). Den førnævnte model beskriver

dynamikken af leukemi ved brug af seks koblede differential ligninger, som vi re-

ducerer til to - som indeholder information fra det fulde system med seks ligninger.

Vi har samtidig foreslaet en lille ændring i modellen, ved at introducere en anden

densitetsfunktion. Vi uddyber ræsonnementet om hvorfor denne nye funktion intro-

duceres, og til sidst diskuteres konsekvenserne af denne ændring. Vi konkluderer at

den quasi-steady-state approximerede model beskriver dynamikken af det fulde, seks

dimensionelle system, rigtigt godt.

Contents

1 Introduction 1

1.1 Research Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theory 5

2.1 Introduction to the Biology . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Mathematical Cancer Model . . . . . . . . . . . . . . . . . . . . 7

2.3 Review of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 System of differential equations . . . . . . . . . . . . . . . . . 12

2.3.2 Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.3 Phase plane analysis . . . . . . . . . . . . . . . . . . . . . . . 13

2.3.4 Null-clines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.5 Quasi Steady State Approximation . . . . . . . . . . . . . . . 16

2.3.6 Model II - in-homogeneous systems . . . . . . . . . . . . . . . 18

3 Introduction and analyses of models 23

3.1 Model III - Including HSC, HMC, LSC and LMC excluding inflamma-

tory feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Model IV - Including Inflammatory response and excluding LSC and

LMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Model V - Including inflammatory response and excluding HSC and

HMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4 Model VI - Quasi Steady State Approximation . . . . . . . . . . . . . 30

ii

CONTENTS iii

4 Determining parameters and initial conditions 33

4.1 Intuitive demands to parameters . . . . . . . . . . . . . . . . . . . . . 34

4.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Parameters estimations based on analyses . . . . . . . . . . . . . . . 37

5 Results 41

5.1 Simulation of model IV . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Simulation of model V . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Simulation of Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4 Simulation of Model VI . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Discussion 51

7 Conclusion 55

Bibliography 57

8 Appendix I

8.1 Elaboration of the inspirational model . . . . . . . . . . . . . . . . . I

8.2 Model I - The full model . . . . . . . . . . . . . . . . . . . . . . . . . III

8.3 Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III

Word List

cytokine any of various proteins, secreted by cells, that carry signals to neighbouring

cells (dictionary.com)

hematopoiesis the formation of blood (dictionary.com).

hematopoiesis the tendency of a system, especially the physiological system of

higher animals, to maintain internal stability, owing to the coordinated response

of its parts to any situation or stimulus that would tend to disturb its normal

condition or function. (dictionary.com)

leukemia any of several cancers of the bone marrow that prevent the normal manu-

facture of red and white blood cells and platelets, resulting in anemia, increased

susceptibility to infection, and impaired blood clotting (dictionary.com)

leukocyte any of various nearly colorless cells of the immune system that circulate

mainly in the blood and lymph and participate in reactions to invading mi-

croorganisms or foreign particles, comprising the B cells, T cells, macrophages,

monocytes, and granulocytes. (dictionary.com)

phagocyte (phagocytic cell) ”Any cell, as a macrophage, that ingests and destroys

foreign particles, bacteria, and cell debris” (dictionary.com)

progenitor cell like a stem cell it has a tendency to differentiate into a specific type

of cell, but is already more specific than a stem cell and is pushed to differentiate

into its ”target” cell (en.wikipedia.org/wiki/)

proliferation the growth or production of cells by multiplication of parts (dictio-

nary.com)

v

vi CONTENTS

stem cell a cell that upon division replaces its own numbers and also gives rise to

cells that differentiate further into one or more specialized types (dictionary.com)

1 Introduction

Mathematics has proved to be an effective tool in understanding the complex mecha-

nisms behind physiological processes. This project investigates a compartment model

of the hematopoietic cells and leukemic mutated cells, in the body - which constitutes

a dynamical system. Thus the model consists of a system of coupled differential

equations. Mathematical models provide efficient tools for examining dynamics of

cancer initiation and progression. Further applications involve helping to propose new

experiments and variations of treatments, since the models grant insight in which

mechanisms that needs to be targeted to gain the most favourable dynamics. An

advantage of this method is that it gives the possibility to predict changes in dynamics

as a result of an intervention, e.g. reducing leukemic cells’ ability to divide. Thus it is

a quantitative method with a more tangible and estimating approach compared to

otherwise qualitative biological evaluations.

This project focuses on analysing a model by Andersen et al. (2017a) that has

successfully implemented an inflammatory response to the already intricate system

of equations. Furthermore the model is simulated such that a representation of the

dynamics in the human body emerges. The exact topic which this report deals with, is

a mathematical model of the growth of cancer caused by myeloproliferative neoplasms

(MPNs). MPN is a group of diseases of the bone marrow that cause excess cell

production (Andersen et al., 2017a). The basic model was introduced by Dingli and

Michor (2006), but we work with an extended model by Andersen et al. (2017a),

which among other things includes an amplification factor describing how one stem

cell divides into multiple mature cells. The model in its entirety can be seen in the

appendix, figure 8.1. However, our mathematical journey begins elsewhere; As we wish

1

2 CHAPTER 1. INTRODUCTION

to account for the basic characteristics of the model, we will build up the full model

from the bottom, gradually adding more mechanisms and making it more advanced.

The last interesting addition to the model is the inflammatory response, which functions

as an up-regulator of stem cell proliferation resulted from cells dying. This complicates

the model with two more differential equations and thus two more dimensions to

account for. A way to come around these implications can be to make sensible

assumptions about the dynamics of the system. We will investigate, whether this can

be applied to the concerned model, by a quasi-steady-state assumption. This process

should reduce the number of differential equations, and thus making an otherwise

complicated system more approachable.

1.1 Research Question

What is the reasoning behind the model in Andersen et al. (2017a). Furthermore will

the dynamics of a quasi-steady-state approximation of the model coincide with the

full model?

1.2 Method

The framework of the report consists of the introductory chapters and sections of the

report: abstract; introduction, chapter 1; research question, section 1.1; and method,

section 1.2. They serve the purpose to widely introduce to our subject; mathematical

modelling of leukemia, and why this subject is interesting. These are followed by

the theoretical and explorational chapter, chapter 2; which serves as an introduction

both to the biological aspect of the report, but also contains a basic review of the

mathematical theory, which is necessary to do an analysis of the dynamics of a system.

Following the theory chapter is the main work of the report, which consists of the

chapters: Introduction and analyses of the models, chapter 3; Determining parameters

and initial conditions, chapter 4; Results, chapter 5. These are separated into three

steps because of the difference in methods used. Chapter 3 contains introductions to

the different models, followed by a pen and paper analysis of each individual model.

1.2. METHOD 3

The main analysis used is the linear stability analysis, but we have also conducted a

quasi steady state approximation. We have included interpretations of the analytical

findings immediately following the respective analyses. We use these analyses and

interpretations in chapter 4 to decide upon which numerical values we will use in

simulations later on. Also included in this chapter is how we find estimates of certain

parameters and initial conditions, which are specific for our full model. Chapter 5,

Results; contains the result of our simulations with Matlab as well as graphs and of

course interpretations of the graphs and dynamics, of the different simulated models.

Lastly we summarize and discuss the findings on our report in chapters 6 Discussion

and 7 Conclusion.

2 Theory

The purpose of this chapter is to briefly explain the backbone of the biological- and

mathematical theory, which is later utilized in the analysis part of chapter 3. We

will introduce to the concepts behind the our full model, model I; which can be seen

in section 8.2 in the appendix, page III. We will do so in multiple steps, gradually

introducing more advanced aspects of the model. Additionally we seek to elucidate the

reasoning behind the choices which were made to abstract the biological knowledge and

concepts into mathematics. Furthermore we will explain the concept of compartment

modelling, and briefly explain the mathematical theory behind our analysis. Lastly

we include an example of an analysis, which is related to our mathematical model.

The result of the analysis will then be interpreted.

2.1 Introduction to the Biology

The goal of the mathematical model we have suggested, inspired by Andersen et al.

(2017a), is to describe the change in the number of cells with respect to time. It includes

four different types of cells: Hematopoietic stem cells (HSC), hematopoietic mature

cells (HMC), MPN-mutated leukemic stem cells (LSC) and leukemic mature cells

(LMC). These four different types of cells are each modelled as a compartment - a box,

as seen on figure 2.1, which is inhabited by only one type of cell. These compartments

are interrelated because the growth of both hematopoietic and leukemic mature cells

are related to the compartments of their respective stem cells. This relation is depicted

by the black arrows, as seen on figure 2.1, between the compartments. These arrows

indicates a flow between the compartments - the positive direction given by the

5

6 CHAPTER 2. THEORY

Figure 2.1: The basic compartments of the model. Flow is represented by arrowsbetween compartments.

arrowhead. Each arrow should also have a rate constant attached, which describes

the rate of flow of that arrow.

The first property of cells we will describe is division. In the model, we work with

either stem cells or mature cells. Stem cells are in the bone marrow, where they renew

themselves. This happens naturally as the organism constantly requires new, healthy

cells. (Reya et al., 2001) Stem cells are the basis of all cells, because they are able

to divide into various functional and specialized cells in the body, including immune

cells. When the stem cell has evolved into a fully functional cell, that cell is called a

mature cell. (Dingli and Michor, 2006) The stem cells have three different ways to

divide (Samuelsen et al., 2016)

• Symmetric self renewal: The stem cell dividing into two daughter stem cells.

These are characterized by the ability that both daughter cells have the same

development potential as their mother cell.

• Asymmetric division: The stem cell divides into a stem cell and a progenitor

cell.

• Symmetric division: The stem cell divides into two progenitor cells.

2.2. THE MATHEMATICAL CANCER MODEL 7

The model does not concern itself with which of the three divisions each individual

cell is undergoing, but fuses them into two different rate constants: the rate constant

of renewal of stem cells, called r, and the rate constant of growth of progenitor cells,

called a. The reason being, that we are dealing with a very large number of cells and

the three variations of division merges into averages, which can be described as rate

constants.

When a stem cell divides to a progenitor cell, it then undergoes a process. It divides

symmetrically (multiple times) eventually becoming mature cells, which do not divide

any further. A first generation progenitor cell divides such that the resulting number of

progenitor cells from a single progenitor cell is 2k, where k is then the average number of

generations. This is based on a assumption than between each generation, the number

of cells is doubled. In the model, the process has been simplified - the intermediate

steps are described as a constant, A, and only the mature cells have been given a

compartment. The positive rate of change, or the inflow, of this compartment is thus

associated to be the amount of first generation progenitor cells created, multiplied with

the constant A - which is called the amplification factor. It describes the intermediate

steps; progression from stem cell to progenitor cell.

Lastly, all the different cells die at some rate. This is described by an outflow arrow

from each of the compartments. The rate constant describing this arrow is d. The

subscript on the constant, refers to which of the four compartments, the rate constant

is referring to.

2.2 The Mathematical Cancer Model

The following system of coupled differential equations is a simplified version of the full

system, by Andersen et al. (2017a), seen on figure 8.1 in the appendix; page I. Only

the basic features of the cells; SC self renewal, division into MC’s and death rates have

been included. Furthermore a density function is included on self renewal term of the

two SC compartments, designated φ. All the SCs interact with each other, since they

are limited to the space in the bone marrow, where the cell division is taking place.

Thus the more stem cells the less the value of the function is.

8 CHAPTER 2. THEORY

Figure 2.2: Original by Andersen et al. (2017a), edit by us

The model (see figure 2.2) consists of four compartments of cells denoted x0, x1, y0,

and y1 which are the HSC, HMC, LSC and LMC compartments, respectively. We will,

as Andersen et al. (2017a) do, refer to the hematopoietic cells as x and the leukemic

cells as y. The subscripts 0 and 1 refers to whether the cells are stem cells or mature

cells respectively.

dx0

dt= (rxφx − ax − dx0)x0, x0(0) = x0i (2.1)

dx1

dt= axAxx0 − dx1x1, x1(0) = x1i (2.2)

dy0

dt= (ryφy − ay − dy0)y0, y0(0) = y0i (2.3)

dy1

dt= ayAyy0 − dy1y1, y1(0) = y1i (2.4)

The differential equations describes the dynamics, of the reduced system. Equation 2.1

describes how the change of HSCs depend on the current number of HSCs. Equation

2.2 deals with the part of stem cells that divide into mature cells and these are

designated as x1. Likewise equations 2.3 and 2.4 describes the change in the of LSCs

and LMCs.

The equations each depend on certain parameters that are constant: rx and ry are

the self-renewal rate constants, and the subscripts, x, y, ties it to either the HSC


compartment or the LSC compartment, respectively. ax and ay are rate constants

describing the amount of HSCs and LSCs which divides into HMCs and LMCs

respectively; which is why the term is negative in the differential equations describing

the SC compartments, and the corresponding terms in the equations describing the

MC compartments are positive. d is the final constant, which describes the death of

cells per time.

The density function φ, should constitute the feedback from an increase in the total

amount of SCs, which we model as an effect on the renewal rate constant, of the SCs.

This is because there is a finite amount of room for the SCs to reside in. In literature

this function is a decreasing Hill function (Dingli and Michor, 2006) (Andersen et al.,

2017a), which is a fraction that decreases with increasing interactions. However there

is no data available for this interaction, and we will thus use a mathematically more

suitable expression for φ. In this report, the function is on the following form:

φx = 1− cx(x0 + y0) (2.5)

φy = 1− cy(x0 + y0) (2.6)

The constants cx and cy are dimensionless parameters, simulating the crowding effect

in the bone marrow Dingli and Michor (2006). They differ in value because HSCs

and LSCs are assumed to have interact each other differently. The assumption is that

whereas HSCs only reside in the stroma, the LSCs are more robust and as a result

they occupy both the stroma and the surrounding micro-environment, which leads to

the assumption that cx > cy; i.e. stronger feedback. The φ function used here is a

first order Taylor approximation of the following first order Hill function

1

1 + c(x0 + y0)≈ 1− c(x0 + y0) (2.7)

If c(x0 + y0) = x, and x is a small number, the first order Taylor expansion around 0

becomes

10 CHAPTER 2. THEORY

f(x) =1

1 + xand (2.8)

φ ≈ f(a) + f ′(a)x then (2.9)

f(a) = 1 and (2.10)

f ′(a)x = − x

12= −x thus (2.11)

φ ≈ 1− x (2.12)

Now substituting back, we arrive at the density function which we will work with in

report

φ = 1− c(x0 + y0) (2.13)

Finally we introduce the last two compartments, the dead cells, a, and the cytokines,

s. As a measure of cytokines we use the IL-8 (Interleucim 8), since the concentration

of IL-8 for patients with Polycythemia vera, which is a MPN, has been shown to be

higher than for the average healthy individual Mondet et al. (2015). However, we will

simply refer to the IL-8 as cytokines.

These compartments are designed to model a three-step feedback from the MC

compartments, back to the SC compartments; from the cell compartments to the

dead cell compartment, to the cytokine compartment and finally back to the stem cell

compartments. The equations and dynamics are described by Andersen et al. (2017a).

We will reason about the compartments in an intuitive way.

da

dt= dx0x0 + dx1x1 + dy0y0 + dy1y1 − eaas (2.14)

ds

dt= rsa− ess+ I (2.15)

The rate of change of dead cell compartment, equation 2.14, will be dependent on

the total number of dead cells, which leaves the four cell compartments. Thus the

more cells that dies per time, the more dead cells enters this compartment. The

term which is subtracted is because dead cells are eliminated, and this elimination

process is a consequence of the dead cells being engulfed by phagocytic cells. The

number of phagocytic cells, however, is considered directly proportional to the number


of cytokines, since these are released as a response. Therefore, the elimination of

dead cells is a second order elimination, which is proportional to the amount of dead

cells, a, but also to the amount of cytokines, s, because the more cytokines (and thus

phagocytic cells) and dead cells we have, the more the eliminations occurs. The rate

constant describing this relationship is ea - the dead cell elimination rate, which also

contains information about the cytokine to phagocytic cell ratio. The cytokines in the

body is, in general, connected to the inflammation level. When a human suffers from

tissue damage or infections, cells die more often, and the inflammatory level rises.

Thus the self-renewal of SCs are up-regulated as an immune response. Our model is

however restricted to the leukocytes, and thus the rate of change of the cytokine level

related to these cells, are described by equation 2.15. It is related to the amount of

dead cells in a, multiplied with a ”renewal” rate constant, rs. This is because the

cytokines is a measure of inflammation, and inflammation is a result of dead cells.

Thus more dead cells will result in more inflammation and a higher cytokine level. The

elimination of cytokines happen, when they react with a number of different chemical

compounds. The total amount of different compounds the cytokines can react with,

is very large compared to the amount of cytokines; it is thus assumed that such an

event, which consumes one cytokine and one of compounds, will have a big impact

on the number of cytokines, but a negligible effect on the compounds. This leaves

the number of compounds approximately constant, and it is therefore modelled as a

linear relationship, with the elimination rate, es, which holds information about the

elimination rate constant but also about the likelihood that a cytokine will react with

a compound. Lastly the term I is an added term, giving patient specific information,

about the general change in inflammation level of said patient, potentially from other

inflammatory sources such as smoking. We have however disregarded this term in our

future analyses.

Because the dead cell compartment and the cytokine compartment are connected in

such a way, that both must be used together in the models, we have decided to name

the use of the two compartments for inflammatory feedback.


2.3 Review of Mathematics

This section aims to introduce the reader to basic theory used in our mathematical

modelling, while also introducing to concepts and tool utilized in chapter 3. Included

are stability analysis, application of phase-planes, null-clines and finally the idea behind

the Quasi steady state assumption is explained. These concepts will be introduced for

two-dimensional systems.

2.3.1 System of differential equations

The interactions between two variables for example the evolution of species can be

described by a system of two differential equations.

dX

dt= f(X, Y, δ, µ)

dY

dt= g(X, Y, δ, µ) (2.16)

The functions f and g depend on variables, for example some arbitrary concentrations

(denoted as X and Y) and on some arbitrary parameters (denoted by δ, µ ). This is an

example of a two-dimensional, first order, non-linear system of differential equations

(Gonze and Kaufman, 2015).

The process of analysing the system will in our case be broken down into three main

concepts:

• fixed points

• stability analysis

• null-clines

Finding fixed points is to find the coordinates of the system, where all the differential

equations are equated to zero. The stability aims to study the behaviour of the

system around these fixed point (Gonze and Kaufman, 2015). As a result necessary

2.3. REVIEW OF MATHEMATICS 13

information can be extracted from determining the qualitative behavior. This is done

without having to find the solutions to the system.

2.3.2 Steady State

First and foremost we establish that the system of differential equations is in a steady

state when the system is equal to zero

dX

dt= 0

dY

dt= 0 (2.17)

This shows that the arbitrary compartments (denoted X and Y) will remain constant.

The system then becomes an algebraic expression and the solution can be denoted as

(Xs, Ys) where s stands for steady state

f(Xs, Ys, δ, µ) = 0 (2.18)

g(Xs, Ys, δ, µ) = 0 (2.19)

For general purposes the illustration on figure 2.3.2 shows that the system will change

until it reaches a steady state which satisfies the conditions in equation 2.3.2.

2.3.3 Phase plane analysis

Phase plane analysis is an important technique for studying the behaviour of non-linear

systems, since there is usually no analytical solution to be found for a non-linear

system. The behaviour of the solution of a two dimensional linear system is studied

in the phase plane. Non-linear systems have three qualities that can be examined and

determined (Allen, 2007).

• Non-linear systems will have the same general phase plane behaviour as the

model when linearised around any of its equilibrium points.


Figure 2.3: Variable initially at x0 tending towards the steady state x0s over time(Gonze and Kaufman, 2015)

.

• They often have multiple steady state solutions and the purpose of the phase

plane analysis for a non-linear system is to find the steady state solution in

which a particular set of initial conditions converge to.

• Lastly, the local behaviour when close to any steady state solution can be

understood from a linear phase plane analysis of the particular equilibrium point

dX

dt= a11x+ a12y = f(x, y)

dY

dt= a21x+ a22y = g(x, y) (2.20)

It should be noted that eigenvectors for simple eigenvalues are linearly independent,

this means the solution is linearly independent. The general solution to the system in

this case becomes

x(t) = c1v1eλ1t + c2v2e

λ2t (2.21)

Solutions to a linear system are characterized by the eigenvalues, λ1 and λ2. In the

linear case, the eigenvalues determines the classification of the fixed point of the


system. It is classified as a node, saddle, spiral or centre, depending on the sign of the

eigenvalue, and whether they are a real or complex eigenvalues. Henceforth it can be

determined whether the linear system is then stable or unstable (Allen, 2007).

In case of a non-linear system, the eigenvalues of a fixed point are determined by

linearising around said fixed point. In the case of a two dimensional system - as in

equations 2.3.3, the linearisation is done by evaluating the slope of the tangent plane,

which is essentially determining the Jacobian matrix.

J(x, y) =

[∂f∂x

∂f∂y

∂g∂x

∂g∂y

](2.22)

Thus by evaluating the matrix 2.22 in a fixed point, and solving det((J − λI)) = 0,

yields the eigenvalues. As explained in the following, important knowledge about

stability can be obtained by evaluating the signs of the eigenvalues.

The terms stable or unstable can be defined and distinguished between each other.

For example stable and asymptotically stable. The origin is asymptotically stable if

the eigenvalues of the matrix A are negative or have a negative real part. The origin

is stable if the eigenvalues of the matrix A are non-positive or have a non-positive real

part. The origin is unstable if it turns out that either of the eigenvalues of matrix A

has positive real part. Solutions approach the origin if and when it is asymptotically

stable (Allen, 2007).

1. Node: for the graph to display node qualities, both eigenvalues have the same sign

and may be distinct λ1 ≤ λ2 < 0 or 0 < λ1 ≤ λ2 . Nodes are proper when there

are two distinct eigenvalues, i.e. λ1 6= λ2, as there exists two linearly independent

eigenvectors. The node is called improper if λ1 = λ2.

2. Saddle: eigenvalues λ1 and λ2 have opposite signs, thus λ1 < 0 < λ2 Complex

eigenvalues exist because the system includes factors such as cos(bt) and sin(bt), for

values λ12 = a ± ib, b 6= 0 therefore solutions spiral around the equilibrium. When

a < 0 solutions spiral inward and the spiral outward when, a > 0. Therefore

3. Spiral: eigenvalues have a non-zero real part (a 6= 0)

4. Centre: eigenvalues are purely imaginary a = 0.


Below you will find the illustrations of solutions for an improper, proper node; spiral,

saddle and centre (Allen, 2007).

Figure 2.4: Six cases of trajectory curves around a fixed point (Allen, 2007).

It is so, that non-linear systems will behave as linear systems, close to fixed points of

the system.

2.3.4 Null-clines

When a system contains two variables it’s very useful to resort to the null-clines

representation. The curves are defined by when the system of differential equations

is set to zero. Thus the at the point where two distinct nullclines intersect, i.e. for

instance a y- and x-nullcline, the system has a steady state. In this example, the

different regions are delimited by these null-clines, we can determine the direction of

the changes of the system by studying the sign of dX/dt and of dY/dt. 2.3.4 shows

the different fields in a phase plane and how it moves relative to the equilibrium point.

2.3.5 Quasi Steady State Approximation

Quasi steady state approximation (QSSA) sometimes referred to as pseudo steady

state hypothesis is a tool used to simplify differential equations describing the time


Figure 2.5: Two nullclines intersecting thus creating a fixed point, which is stable inthis case. (Ermentrout, 2002)

dependence of variables. QSSA deals with time dependant differential equations in

which variables are separated into a (fast set) denoted below as vector (y) and a (slow

set) denoted as vector (z) (Pantea, 2014). Below is an example of the general process

of how QSSA works

(fastset) :dy

dt= f(y, z)

(slowset) :dz

dt= g(y, z) (2.23)

The time of interest is between t = 0 and t = tf . Initial values of y and z are known

and set as y(0) and z(0). It is also assumed that the fast set (y) has negligible

rate of change i.e. dy/dt = 0 which means that f(y, z) = 0 is solved to produce an

approximating function.φ0 denotes the approximating variable.

y = φ0(z) (2.24)

equation 2.24 is then substituted back into the slow set equation in 2.3.5 to produce a

differential equation of the form of the QSSA


dz

dt= g(φ0(z), z) (2.25)

The initial value of z is denoted z(0) the original initial value of z and with the

equation below we can obtain the integrated value to be from z which is the QSSA

estimated value of z. the QSSA value for the fast set is then

y = φ0(z) (2.26)

When the QSSA is valid it ignores a short lived period, during which the slow set z is

constant as y changes from y(0) towards a value close to φ0(z(0)). This concludes the

process in which QSSA simplifies models by reducing the dimensions of the system of

equations. Successful application of QSSA requires there to be a separation between

the fast sets and slow sets (Flach and Schnell, 2006).

2.3.6 Model II - in-homogeneous systems

We will now give an example of an analysis, which is related to our further work. The

system which we will analyse is called the simple system. It is a reduced version of

the full system; more specifically it is a system consisting of healthy SCs, and MCs.

Lastly we will add the density function, φ, and once again analyse and interpret the

result, which will be included later in the analysis.

The simple system looks as follows

dx0

dt= (rx − ax − dx0)x0, x0(0) = x0i (2.27)

anddx1

dt= axAxx0 − dx1x1, x1(0) = x1i (2.28)

In this very simple case, the system can actually be completely solved. The equations

are therefore solved for x0(t) and x1(t). The solution for (2.27) can easily be recognized

as the exponential function

x0(t) = Cx0 · ekxt (2.29)


Where the constants has been combined into kx = rx−ax−dx0 and where the constant

Cx0 is given by the initial condition Cx0 = x0i. This equation (2.29) is inserted into

2.28 which yields

dx1

dt= axAxCx0 · ekxt − dx1x1 (2.30)

This equation is linear, which means that it is solvable, and it is inhomogeneous.

Therefore to acquire the full solution, we must find the general solution to the

homogeneous equation and the specific solution to the inhomogeneous equation, and

take the sum of them.

Isolating the function x1 on the left hand side, and the other terms on the right hand

side gives

dx1

dt+ dx1x1 = axAxCx0 · ekxt (2.31)

Putting the right hand side to zero again shows, that the solution to the homogeneous

equation is the exponential

x1,hom = C1,home−dx1t (2.32)

Solving the inhomogeneous equation is also possible, but we will have to guess a

solution. Since we are dealing with the exponential function, a solid first guess would

again be the exponential equation.

x1,inhom = C1,inhomekxt (2.33)

Where C1,inhom is a constant which we can find, but it will be dependent on C1,hom.

Differentiating once and inserting into equation 2.31 allows us to divide out ekxt, and

isolate C1,inhom, which can then be inserted into equation 2.33 which then gives

x1,inhom =axAxCx0

kx + dx1

ekxt (2.34)


Adding the two solutions, equation 2.32 and 2.34 gives the full solution to (2.30),

which is

x1(t) = Chom · e−dx1t +axAxCx0

kx + dx1

ekxt (2.35)

The constant derived from the solution to the homogeneous equation Chom can be

determined with given initial conditions. Evaluating the full solution with respect to

table values given by Andersen et al. (2017a), kx is a very small positive number and

dx1 is a relatively large positive number, which gives the relation dx1 >> kx. As a

result of this it can be concluded that the term Chom · e−dx1t quickly becomes negligible

as time goes and that axAx

k1+dx1≈ axAx

dx1. This further leads to the conclusion that the

value of x1 is related to the value of x0, since it is approximately equal to it with the

factor axAx

dx1multiplied. The fact that the dynamics of x1 rapidly follows that of x0

has important implications for the QSSA as we shall see in section 3.4.

Determining the solutions, equation 2.29 and 2.35, thus allows us to recognize exactly

how the cell divisions depend on time. Another way of analysing the differential

equations is to investigate conditions under which there exist steady state i.e. situations

that fulfill x0(t) = constant and x1(t) = constant. This implies

dx0

dt= 0 ⇔ (rx − ax − dx0)x0 = 0 (2.36)

dx1

dt= 0 ⇔ axAxx0 − dx1x1 = 0 (2.37)

In equation 2.36, disregarding the trivial solution, x0 = 0, we see that rx−ax−dx0 = 0,

which says that the solution does not depend on the initial conditions. If that is the

case, equation 2.36 says that x0 can take any value. Similarly from equation 2.37, x1

can be isolated

x1 =axAxdx1

x0 (2.38)

Again we see that the steady state value of x1 is dependent on x0, which agrees with

the solution. So for a steady state to exist for x1, equation 2.38 implies that x0 must


be in a steady state. As seen in the solution, equation 2.29, for the case of having only

HSCs, the cell division results in exponential growth of cells. This means that there

is no steady state solution for x0, which disagrees with the physiological observation

that a healthy human being is in homeostasis. Therefore we continue the steady state

analysis of the equation describing the rate of change of x0, by accounting for limited

growth, such that the situation includes the feedback mechanism, φx, making the

system non-linear. The reason we do not find the solution to this differential equation

is that it is cumbersome to do, when necessary information can be extracted from the

following steady state analysis. The function φ(x0, y0) is multiplied on rx, since it is

regulating the self-renewal rate depending on the number of existing cells. Thus it is

the following equation:

dx0

dt= (rxφx − ax − dx0)x0 (2.39)

Considering the situation where y0 = 0. Again determining steady state solutions for

x0 involves putting this equation equal to zero:

dx0

dt= 0⇒ (rx(1− cxx0)− ax − dx0)x0 = 0 (2.40)

The trivial solution x0ss = 0 is not relevant, only the remaining is.

rx(1− cxx0ss)− ax − dx0 = 0⇒ x0ss =1

cx

(1− ax + dx0

rx

)(2.41)

This solution gives certain restrictions to the values of the parameters: For rx <

ax+dx0 ⇒ x0ss < 0 and rx = ax+dx0 ⇒ x0ss = 0 are physiologically unrealistic steady

state solutions. Therefore the physiologically interesting case is for rx > ax + dx0.

Given the values of rx, dx0 and ax; the value of cx regulates the steady state value x0ss.

This is sensible, because cx describes how much interactions, internally among the

HSCs, inhibits the overall rate of cell division. The larger cx is, the more influential is

the crowding effect. As a result it also determines what we assume to be the stable

steady state.


The reader should now be familiar with the fundamental mathematical concepts;

namely steady state analysis, stability, nullclines and quasi steady state approximation,

which will allow one to follow the work done in the following chapter 3

3 Introduction and analyses of

models

In this chapter, the different models are introduced and analysed with the mathematical

theory which was explained in chapter 2. The different kinds of analyses which we

put to use are the steady state analysis, stability analysis and lastly a quasi steady

state approximation. We will throughout the chapter comment on the results of the

analyses, and use the results in the discussion chapter 6.

3.1 Model III - Including HSC, HMC, LSC and

LMC excluding inflammatory feedback

In this section we will analyse the model which consists of the differential equations of

the HSC, HMC, LSC and LMC compartments, but without the inflammation feedback

mechanisms from the dead cells and cytokines, and we will do so with a linear stability

analysis. It is the HSC and LSC compartments that is of interest, because there is no

feedback from the MCs, and the steady state of the MCs simply follow their respective

SCs. Therefore model III effectively consists of the following equations

dx0

dt=(rx(1− cx(x0 + y0))− ax − dx0)x0 = f(x0) (3.1)

dy0

dt=(ry(1− cy(x0 + y0))− ay − dy0)y0 = g(y0) (3.2)

The intention is to point out which of the parameters that govern the dynamics of

the SC population, while ensuring that the model gives reasonable results, before it

23

24 CHAPTER 3. INTRODUCTION AND ANALYSES OF MODELS

gets too advanced to deal with analytically. This is done by a steady state analysis.

Solutions to the steady state conditions are

(rx(1− cx(x0 + y0))− ax − dx0)x0 = 0 ⇒

x0,1 = 0

x0,2 =(

1− ax+dx0rx

)1cx− y0

(3.3)

(ry(1− cy(x0 + y0))− ay − dy0)y0 = 0 ⇒

y0,1 = 0

y0,2 =(

1− ay+dy0ry

)1cy− x0

(3.4)

From these solutions the equilibrium points may be deduced as

X1 = (x0,1, y0,1) = [0, 0], (3.5)

X2 = (x0,1, y0,2) =

[0,

(1− ay + dy0

ry

)1

cy

], (3.6)

X3 = (x0,2, y0,1) =

[(1− ax + dx0

rx

)1

cx, 0

](3.7)

X4 = (x0,2, y0,2) =

[(1− ax + dx0

rx

)1

cx− y0,

(1− ay + dy0

ry

)1

cy− x0

](3.8)

However, as we will show, the fixed point X4 and X1, are not biologically relevant

solutions. The trivial solution that is X1 is not a relevant for this model, because we

must have cells. The reasoning behind X4 is a little less obvious, but the symmetry in

the steady state values, x0,2 and y0,2, does give rise to further investigations. If we

suppose that they are sound fixed points, we could insert y0,2 into x0,2

x0 =

(1− ax + dx0

rx

)1

cx−[(

1− ay + dy0

ry

)1

cy− x0

]⇒ (3.9)

x0 =

(1− ax + dx0

rx

)1

cx−[(

1− ay + dy0

ry

)1

cy

]+ x0 (3.10)

Which indicates, since we have x0 on both sides of the equation, that the two parenthesis

must equal to zero

(1− ax + dx0

rx

)1

cx−[(

1− ay + dy0

ry

)1

cy

]= 0 (3.11)

3.1. MODEL III - INCLUDING HSC, HMC, LSC AND LMC EXCLUDINGINFLAMMATORY FEEDBACK 25

And thus they must be equal

(1− ax + dx0

rx

)1

cx=

(1− ay + dy0

ry

)1

cy(3.12)

Which is actually the steady state values of each of the SCs, in the scenario when

the other type is not present. To have such an exact equation sign in a biological

context can be difficult to fulfil. A slight noise in the parameters will offset this point.

Therefore we exclude X4 as a valid fixed point for our model.

In order to account for the stability of the equilibria the Jacobian matrix is evaluated.

The Jacobian matrix is defined as follows:

J(x0, y0) =

[∂f∂x0

∂f∂y0

∂g∂x0

∂g∂y0

]=

[rx(1− cx(2x0 + y0))− ax − dx0 −rxcxx0

−rycyy0 ry(1− cy(x0 + 2y0)− ay − dy0

](3.13)

With respect to the two equilibria, X2 and X3, the matrices become

J(X2) =

[rx

[1− cx

cy

(1− ay+dy0

ry

)]− ax − dx0 0

ay + dy0 − ry ay + dy0 − ry

](3.14)

J(X3) =

[ax + dx0 − rx ax + dx0 − rx

0 ry

[1− cy

cx

(1− ax+dx0

rx

)]− ay − dy0

](3.15)

The Jacobian of both fixed points have a zero in the anti-diagonal. Hence the

eigenvalues are

λX2,1 = rxρ1 − ax − dx0, ρ1 =

[1− cx

cy

(1− ay + dy0

ry

)](3.16)

λX2,2 = ay + dy0 − ry (3.17)

λX3,1 = ax + dx0 − rx (3.18)

λX3,2 = ryρ2 − ay − dy0, ρ2 =

[1− cy

cx

(1− ax + dx0

rx

)](3.19)

In order to determine the stability of each fixed point, we will now explore how the

parameters govern the signs on the eigenvalues. First of all the eigenvalues λX2,2


and λX3,1 (equation 3.17 and 3.18 respectively) indicate stable directions, since both

ry > ay + dy0 and rx > ax + dx0. The latter was deduced from equation 2.41 and there

is reason to believe that it is the case in this context as well. By the assumption that

ry > rx and furthermore ay = ax and dy0 = dx0 the argument is complete. In the

matter of the two other eigenvalues, it is not straight forward recognizing a precise

answer. Both cases has a parameter dependent constant multiplied onto the self

renewal rates, ρ1 and ρ2 respectively, thereby either decreasing or increasing it’s value.

If we take a look at ρ1, the innermost bracket; (1− ay+dy0ry

), is a number between 0 and

1. Onto this the fraction cx/cy is multiplied, and from the assertion that cx > cy, the

fraction is above 1. The result of these considerations is very inconclusive, because it

leads to that ρ1 is presumably decreasing rx. It is therefore more useful comparing ρ1

and ρ2. It is noted that it must be the case that ρ2 > ρ1, since it is deduced from the

previous arguments that 1− ay+dy0ry

> 1− ax+dx0rx

and cx/cy > cy/cx. These statements

can thus be summarized to ρ2 > ρ1.

The mathematical statements are equivocal, but they support what we know from

reality. We know that the cancerous state is stable unfortunately, which is supported by

the conclusion found, since ρ1 reduces rx by a larger amount such that rxρ1 < ax +dx0.

From this relation it must be that λX2,1 < 0, making the fixed point stable. With

the healthy state it is the other way around that ρ2 reduces ry by a relatively small

amount, thereby making λX3,2 > 0, since ryρ2 > ay + dy0.

3.2 Model IV - Including Inflammatory response

and excluding LSC and LMC

We wish to investigate the steady states of HSCs and, in the following section 3.3,

steady states of LSC. this is because we assert that there, in model I, exists at least

two steady states; one when no LSC are introduced, which we call model IV; and one

fixed point when there are only cancer cells remaining in the system, model V (see

section 3.3). This is based on the assumption that a healthy person, without cancer,

has a constant number of HCS. We will therefore in this section make a steady state

3.2. MODEL IV - INCLUDING INFLAMMATORY RESPONSE ANDEXCLUDING LSC AND LMC 27

analysis of model IV, which looks as follows

dx0

dt= (rxsφx − ax − dx0)x0 (3.20)

dx1

dt= axAxx0 − dx1x1 (3.21)

da

dt= dx0x0 + dx1x1 − eaas (3.22)

ds

dt= rsa− ess (3.23)

We proceed with a steady state analysis. We quickly see that

dx1

dt= 0 ⇒ (3.24)

axAxx0 − dx1x1 = 0 ⇒ (3.25)

x1 =axAxdx1

x0 (3.26)

is the condition for the MCs to be in steady state. The cytokine and dead cell

compartments are interrelated, and thus takes a few more steps to solve

ds

dt= 0 ⇒ (3.27)

rsa− ess = 0 ⇒ (3.28)

s =rsa

es(3.29)

da

dt= 0 ⇒ (3.30)

dx0x0 + dx1x1 +−eaas = 0 ⇒ (3.31)

a =dx0x0 + dx1x1

eas(3.32)

Which leaves us with two expressions for the unknowns, s and a. This allows us to

insert the solution of s into the solution of a, and vice versa.

a2 = (dx0x0 + dx1x1)esears

(3.33)

a =

√(dx0x0 + dx1x1)

esears

(3.34)


and thus s becomes

s =

√(dx0x0 + dx1x1)

rseaes

(3.35)

Now we only have to find the steady state condition for x0.

dx0

dt= 0 ⇒ (3.36)

(rxs(1− cxx0)− ax − dx0)x0 = 0 (3.37)

The solution x0 = 0 is the trivial solution, and we will continue finding the other

solutions by assuming that x0 6= 0 and then dividing the whole equation with x0.

dx0

dt= 0 ⇒ (3.38)

rxs(1− cxx0)− ax − dx0 = 0 ⇒ (3.39)

1− cxx0 =ax + dx0

srx(3.40)

If we now isolate dx1x1 from equation 3.25 and insert it into equation 3.35 we get the

following

s = βx√x0, βx =

√(dx0 + axAx)

rseaes

(3.41)

Finally we can insert this into equation 3.40 and solve for x0

dx0

dt= 0 = βx

√x0rx(1− cxx0)− ax − dx0 ⇒ (3.42)

ax + dx0 = βx√x0rx (1− cxx0) ⇒ (3.43)

√x0(1− cxx0) = Kx, Kx =

ax + dx0

βxrx(3.44)

By the steady state analysis the system has been reduced to the x0 being a polynomial

containing the steady state values as solutions, excluding the trivial solution. This is

therefore a more complicated description of reality than the stability we saw in section

3.3. MODEL V - INCLUDING INFLAMMATORY RESPONSE ANDEXCLUDING HSC AND HMC 29

3.1. Model II had in this case, for x0, two fixed point: An unstable, which was the

point with zero HSCs; and a stable one, which is the healthy steady state, we assert

that a person should be in. we therefore in model IV, equation 3.44, expect at least

one stable fixed point when x0 6= 0, to continue the assertion of a healthy steady state

healthy person are in.

We will numerically investigate equation 3.44 in the Results chapter, section 5.1; for

fixed point and stability.

3.3 Model V - Including inflammatory response

and excluding HSC and HMC

We find this model interesting as well because we, in the previous section, asserted

that a fixed point exists in the case, when there are no HSC in the model. It is

observable that cancer eventually kills people, which is what indicates this fixed point.

One can consider the two SCs as competing species, and the fittest - the LSC cell, will

eventually outmatch the less resilient HSCs. Because the equations are symmetrical in

model IV and model V, we expect the two situations to look very much alike, which

turns out to be true. We have therefore only included the result of the analysis. Model

V consists of the following equations

dy0

dt= (rysφy − ay − dy0)y0 (3.45)

dy1

dt= ayAyy0 − dy1y1 (3.46)

da

dt= dy0y0 + dy1y1 − eaas (3.47)

ds

dt= rsa− ess (3.48)

As we saw from model IV, the steady state values of all the equations are eventually

determined by x0. Only the result of equation 3.45 is given below in equation 3.49,

because of the symmetry in the equations and in the method, which was used in the

analysis of model IV.


√y0(1− cyy0) = Ky, Ky =

ay + dy0

βyryand βy =

√(dy0 + ayAy)

rseaes

(3.49)

Our assertion says that there should exist at least one stable steady state, excluding

the trivial solution y0 = 0. We will, as for model IV, continue with equation 3.49 and

find the fixed points and stabilities of said points, in section 5.2.

3.4 Model VI - Quasi Steady State

Approximation

In this section we introduce our last model, the Quasi Steady State Approximation

model, model VI. It is a QSSA of model I, because of the hypothesis that the

dynamics of the HSC and LSC are very slow, compared to the dynamics of the other

compartments. The section contains reasoning as to why some of the equations can

be put to zero, as well as a pen and paper analysis, which eventually leads to two

expressions of dx0/dt and dy0/dt. Model I can be seen in the appendix, section 8.2.

First of we put the MC equations to zero. Intuitively, the dynamics of the MCs follow

those of the SCs, and very quickly as well. Thus a change in the SC compartments will

quickly result in a follow-up by the MCs. This intuition is supported by our analysis

of model II, from section 2.3.6.

Putting the differential equations regarding MCs to zero yields x1 and y1.

dx1

dt= axAxx0 − dx1x1 = 0⇒ x1 =

axAxdx1

x0 (3.50)

dy1

dt= ayAyy0 − dy1y1 = 0⇒ y1 =

ayAydy1

y0 (3.51)

We will argue that, from a biological perspective, sickness, inflammation dynamics and

cell deaths are very fast - taking hours to days, compared to cancer cells, which develop

over months and even years. Thus the differential equations regarding cytokines and

dead cells are assumed to be in steady state and are put to zero, which allows us to

3.4. MODEL VI - QUASI STEADY STATE APPROXIMATION 31

solve for rates a and s.

da

dt= dx0x0 + dy0y0 + dx1x1 + dy1y1 − eaas = 0 and (3.52)

ds

dt= rsa− ess = 0 (3.53)

First equation 3.52 is solved for a. The expressions found in equations 3.50 and 3.51

are furthermore inserted in order to get a expressed in terms of x0, y0 and s

da

dt= dx0x0 + dy0y0 + dx1x1 + dy1y1 − eaas = 0⇒ (3.54)

a =1

s

(dx0 + axAx

eax0 +

dy0 + ayAyea

y0

)(3.55)

Solving equation 3.53 for s is easily done

ds

dt= rsa− ess = 0⇒ (3.56)

s =rsesa (3.57)

Equations 3.55 is inserted into equation 3.57 to get s expressed in terms of x0 and y0:

s =rsess

(dx0 + axAx

eax0 +

dy0 + ayAyea

y0

)⇒ (3.58)

s2 =rses

(dx0 + axAx

eax0 +

dy0 + ayAyea

y0

)⇒ (3.59)

s =

√rs(dx0 + axAx)

eseax0 +

rs(dy0 + ayAy)

eseay0 ⇒ (3.60)

s =√αxx0 + αyy0 (3.61)

For simplicity parameter dependent constants has been defined as αx = rs(dx0+axAx)esea

and αy = rs(dy0+ayAy)

esea. This expression for s is inserted into the differential equations

for HSCs and LSCs

dx0

dt= (rx

√αxx0 + αyy0(1− cx(x0 + y0))− ax − dx0)x0 (3.62)


And similarly for the LSCs:

dy0

dt= (ry

√αxx0 + αyy0(1− cy(x0 + y0))− ay − dy0)y0 (3.63)

Now the six differential equations are eventually condensed into two relevant equations.

The equations 3.62 and 3.63 solely describes the dynamics of the competition between

HSCs and LSCs, where the influence from MCs and the inflammation response to

dead cells are implicitly taken into account. These equations still have the steady

state points as described in the previous section.

4 Determining parameters and

initial conditions

In this chapter we will argue for the choice of numerical values, which are used in

the simulation process in chapter 5. Our model is clearly analogous to the model of

Andersen et al. (2017a), which is why we have prioritized their values higher than the

values by Dingli and Michor (2006), although we do consider reasoning from both.

Since we have decided to change the density function, φ, we have decided to estimate

some of the parameters which are more unique fom our model.

Our full model - model I, can be seen in the appendix in section 8.2. It contains a

total of 17 parameters, including cx and cy from the functions φx and φy respectively,

and 6 initial conditions for the 6 compartments. Which of parameters and initial

conditions we have estimated, can clearly be seen on table 4.1.

The method which we apply in this chapter relies mainly on having an analysis, but

is unorthodox in the way which we apply intuitive demands and restrictions to the

analyses. Thereafter we require some numerical values as a starting point, to reduce

the number of unknowns to only the parameter which we wish to determine. This

allows us to find a specific value or at least the orders of magnitude, which fulfils our

demands. Finally the values are ”fine tuned” in a simulation, such that the dynamics

are satisfying - regarding many things such as stability and time scale.

33

34 CHAPTER 4. DETERMINING PARAMETERS AND INITIAL CONDITIONS

4.1 Intuitive demands to parameters

As a consequence of the changes we have made to the original model by Andersen et al.

(2017a), we are not satisfied by the exact same numerical values as has been used by

Andersen et al. (2017b). Therefore we will estimate the parameters which are specific

for our model, before we start simulations, while having certain intuitive demands

for the values. One of our key arguments, which is observable in the real world, is

that untreated cancer eventually kills people. But if there is no cancer, a (grown)

person should have a constant amount of both HSCs and HMCs. This is something

we would demand our model to obey, which eventually restricts our parameters. In

mathematical language: model III should (at least) contain steady states when: there

are no LSC introduced (y0i = 0); but also one when there are no HSC left in the

system (x0final = 0). We have depicted this restriction on the figure 4.1.

As the arrows indicate, if we are living in a healthy state, where cancer does not exist

- on the x0 axis, the fixed point that is x0ss is actually stable. I.e. a person with no

cancer will converge towards a set amount of cells. However if we introduce cancer

as an additional dimension, and pertubate our system just slightly away from the x0

axis - in the y0 direction, the fixed point x0ss is unstable. In this case, the system goes

towards the fixed point y0ss, which must then be stable from all directions.

Another example of an intuitive demand could be, that the death rate constant

regarding mature cells, must be higher than that of stem cells. Stem cells are much

more protected in the bone marrow, while mature cells are functional cells in the

organism and in the blood stream, and thus much more vulnerable; dx1 > dx0. Yet

another example of a demand is, that the self renewal rate constant of LMC is assumed

to be larger than that of HMC; ry > rx. This is not as clear as it may seem, since as

we saw in the previous section the overall stability of the cancerous state also depends

on the relation of for instance cx > cy. However, it is generally recognized that the

statement is correct.

4.2. INITIAL CONDITIONS 35

Figure 4.1: The slope field of model III, with ”fictional” parameter values. The yellowlines are the y0 nullclines, and the red lines are the x = 0 nullclines. The two nullclinesintersect at fixed points, And the possible fixed points are therefore the healthy statex0ss, the cancerous state y0ss and the trivial (0, 0). Directions of the fields are markedwith arrows by the nullclines. Additionally two green solution trajectories has beenplotted

4.2 Initial conditions

Our table values are as mentioned mostly inspired by Andersen et al. (2017a) and

Andersen et al. (2017b). We have decided on leaning on them for the initial conditions

of x0i and x1i.

The initial conditions for the cancer is rather trivial; we have neglected the mutation

term, and therefore our model effectively start at the instant when a HSC has

mutated into 1 LSC. Therefore yi0 = 1 and y1i = 0 are the initial conditions for those

compartments.

We also have to acquire initial conditions for a and s. The only estimation conducted

on the initial conditions is for ai, because the value for si has been chosen based on


Vaidya et al. (2012). It is the average value of IL-8, which was found in a healthy

control group. Therefore we have the value si = 3.2 ≈ 100. Now to estimate ai: From

3.32 we have that

ai =dx0x0 + dx1x1

easi(4.1)

ai can be estimated by inserting the orders of magnitude of the parameters and steady

state values for x0, x1 and si

ai =10−3 · 103 + 102 · 1010

109 · 100≈ 103 (4.2)

4.3. PARAMETERS ESTIMATIONS BASED ON ANALYSES 37

Table 4.1: Parameters and initial conditions.

Initial condition Value Source1 x0i 6, 542 · 103 (Andersen et al., 2017a)*2 x1i 2, 6219 · 1010 (Andersen et al., 2017a)*3 y0i 1 Trivial4 y1i 0 Trivial5 ai 1, 163 · 103 Estimate, see equation 4.2*6 si 2, 9079 (Vaidya et al., 2012)*

Parameters Value Source7 rx 2 · 10−3 Estimated, see equation 2.41*8 ax 1, 1 · 10−5 (Andersen et al., 2017b)9 Ax 4, 7 · 1013 (Andersen et al., 2017b)10 dx0 2 · 10−3 (Andersen et al., 2017b)11 dx1 129* (Andersen et al., 2017b)12 ry 4 · 10−3 Estimated*.13 ay ax (Andersen et al., 2017b), (Dingli and Michor, 2006)14 Ay Ax (Andersen et al., 2017b)15 dy0 dx0 (Andersen et al., 2017b), (Dingli and Michor, 2006)16 dy1 dx1 (Andersen et al., 2017b), (Dingli and Michor, 2006)17 ea 1 · 109 (Andersen et al., 2017b)*18 es 2 (Andersen et al., 2017b)19 rs 5 · 10−3 (Andersen et al., 2017b)*20 cx 10−4 Estimated, see equation 4.421 cy 0, 8cx Estimated

*Values modified by simulations.

4.3 Parameters estimations based on analyses

Of the parameters, most of the values from Andersen et al. (2017b) are inspired by

those of Dingli and Michor (2006), and because of this we find them to be valid

assumptions. Some of their arguments also look alike, for example that dx0 = dy0. We

have decided to imitate the values of Andersen et al. (2017b), as they are adjusted

closer to the model we are working with. The values which we assume to be identical

between the two models - (1) the model by Andersen et al. (2017a) and (2) our model

I, which can be found in the appendix, page I and III respectively; can be seen on


table 4.1. Amongst others, parameters such as d, a and A, are easily borrowed, which

paves the way for us to estimate and modify other, more specific parameters, such

that they fulfil our intuitive and analytic demands.

A reasonable starting point of the estimations is model IV, where we have the following

relation

x1 =axAxdx1

x0 (4.3)

This is because the values of x1 and x0 in a healthy person have been estimated from

experiments to have the orders of magnitude of x1 ≈ 1010 and x0 ≈ 103 (Andersen

et al., 2017a). This implies that the fraction axAx/dx1, must be an order of magnitude

107. We will use values for the parameters ax , dx0 and Ax from Andersen et al.

(2017b), since they give rise to the needed order of both x0 and x1.

From the values dx0 ≈ 10−3, ax ≈ 10−5 and the relation rx > dx0 + ax from equation

2.41, then rx ≈ 10−3 at least.

Finally since HSCs and LSCs are assumed to have similar properties i.e. dy0 = dx0,

ay = ax, then the assumption that ry > rx leads to ry > rx > 10−3. We put ry ≈ 10−3.

Furthermore an approximation for cx can be made from fact that φ > 0, because it

would otherwise result in negative growth. Therefore the following is true for the

healthy steady state

(1− cxx0) > 0⇒ cxx0 < 1⇒ cx <1

x0

(4.4)

x0 healthy steady state is of order of magnitude 103, which is why we consider cx ≈ 10−4

from the statement in equation 4.4. The same can be done for cy, but this requires an

evaluation of the steady state value of y0. Intuitively this y0ss is not expected to be of

a higher order of magnitude than that of x0ss, but just a multiple of for instance 2.

As for cy the same applies, but a simple suggestion could be to assume that cy = 12cx,

which is suggested by (Dingli and Michor, 2006).

Orders of magnitude of every initial value and parameter has been determined and

ready to be used in the simulation process. To finalize the process, we have ”fine

4.3. PARAMETERS ESTIMATIONS BASED ON ANALYSES 39

tuned” the parameters and initial conditions in the simulation. This has been done to

make sure we have the right stability, mainly through parameters; but we also had to

tune the initial conditions so that they started in their respective steady states. This

is because the simulation of model I is supposed to describe the progress of cancer for

a healthy person, who is subject to single mutation of a HSC into a LSC.

5 Results

This chapter contains the results and graphs from the simulation of the models, which

can be seen in the appendix. We will be commenting on our results throughout the

chapter, which will be summarized in the conclusion. Additionally we have included

our thoughts and physiologically interpretations on the different models.

5.1 Simulation of model IV

In this section we will further investigate the system with feedback but no cancer from

section 4.3, where the temporary result can be seen in equation 5.1. We are interested

in the x0 compartment, and the expression we have is analytically difficult to handle.

dx0

dt=√x0(1− cxx0)− ax + dy0

βrx= f(x0), βx =

√dx0 − axAx

esears (5.1)

Therefore we use a matlab script to determine the roots of the polynomial. The script

numerically calculates the values of the polynomial in the interval x0 = [0, 99999].

However the numerical code partitions the interval into a finite number of equidistant

steps, and because of this it is likely, that we will not find the exact values of x0,

where the polynomial is equal to zero. Figure 5.1 shows equation 5.1 plotted as a

function of x0. It is rather difficult to read off the intersections with the x0 axis,

however We can from this plot immediately read off the sign of the eigenvalues. The

sign of the eigenvalue at the first intersection is positive, indicating a unstable fixed

point (specifically a unstable node), while the other eigenvalue is negative (and the

fixed point is a stable node). As we can see from the values of table 5.1, we did

not get an exact value for the steady states. However, we are still satisfied as the

41

42 CHAPTER 5. RESULTS

Figure 5.1: Equation 5.1 plotted as a function of x0.

Table 5.1: Numerically estimated values of x0, the HSC, in search for x0 such thatf(x0) = 0, meaning no change in the HSC compartment.

Name Valuesx0 956 957 6541 6542 6543f(x0) −0, 0050 0, 0066 0, 0068 0, 0009 −0, 0051

value of the f(x0) is extremely close to zero, and does also change sign in this small

interval [−5 · 10−3; 6, 6 · 10−3] and [6, 8 · 10−3;−5, 1 · 10−3] for the first and second

fixed point respectively. The sign change agrees with the stability of the fixed point

as mentioned above; first fixed point being unstable and the second being a stable

fixed point. Combining this with the trivial solution, x0 = 0, we now have the third

and final steady state value of the x0 compartment. Thus in a healthy person or in a

world with no cancer, this model with feedback predicts three different steady state

values of HSC.

This conclusion is further confirmed by running the simulation on the lower and larger

5.2. SIMULATION OF MODEL V 43

side of the point expected to be an unstable fixed point, se the figures 5.2 and 5.3.

Figure 5.2: The simulation is run with x0i = 956, thus being on the lower side of theunstable fixed point and leading to the x0 tending to the zero steady state.

Figure 5.3: The simulation is run with x0i = 958, thus being on the larger side of theunstable fixed point and leading to the x0 tending to the healthy steady state valuex0ss = 6542.

5.2 Simulation of model V

This section contains the results of simulating model V. Unsurprisingly the model

without healthy cells, but with feedback and cancerous cells has the same stability, as


one can see on figure 5.4. The function plotted is equation 5.2.

dy0

dt=√y0(1− cyy0)− ay + dy0

βry= g(y0), βy =

√dy0 − ayAy

esears (5.2)

We have used the same method to numerically calculate these values as for the previous

section. Again combining with the trivial solution y0 = 0 the dynamics of the model

Figure 5.4: Equation 5.2 plotted as a function of y0.

with feedback and LSC but without HSC has been found. Something to notice is that

there is a larger interval in between the two fixed points in this model, compared to

the model with feedback but without cancer.

Table 5.2: Values of y0 for which g(y0), from equation 5.2, is equal to zero.

Name Valuesy0 202 203 10819 10820g(y0) −0, 0012 (0, 0322) 0, 0037 −0, 0040

5.3. SIMULATION OF MODEL I 45

5.3 Simulation of Model I

When beginning the simulation process the parameters are adjusted according to what

is consistent with a healthy patient’s state of hematopoiesis. This involves noting the

value that the cells tend to i.e. the steady states and change the initial conditions to

these. Figure 5.5 shows the steady states with constant values over the time interval.

Figure 5.5: Healthy steady states of x0, x1, a, s. They remain at constant values,shown here over a period of 40 years. There is some insignificant noise from thesimulation for a and s.

Now leukemia is introduced by starting the simulation with y0i = 1. The development

of the LSCs can be seen in figure 5.6.

At time t = 0 one complete mutation has taken place and as a consequence cancer

is initiated and eventually leading to the situation of only LSCs and LMCs being

present after approximately 10 years. When the plateau of the stable y0ss is reached,

the number is somewhat close to double as large as the healthy x0ss. The specific


Figure 5.6: Leukemia starts to develop from one mutated cell, y0i = 1, and it reachesthe steady state of LSC with no remaining HSCs within 10 years. The LSCs arein a better position than HSCs with respect to the overall self renewal and divisionenvironment, which leads to extinction of HSCs.

value of this y0ss is not relevant as long as it fulfils the relation of being larger than

the x0ss. It is more relevant looking at the dynamics and the corresponding time

span. We will continue by looking at the dynamics of all the cells. From figure 5.7 it

is furthermore seen how the number of dead cells and the level of cytokines develop

through the same time interval as in figure 5.6. These clearly increases together with

the increase in LCs. In other words; it seems to be the case that the dynamics of

MCs, a and s quickly follows the dynamics of SCs. This immediately indicate that

the QSSA analysis is valid. Therefore this is further examined by normalising the

dynamics of each compartment, equation 5.3.

1− x0

x0ss

, 1− x1

x1ss

,y0

y0ss

,y1

y1ss

,a− aIass

,s− sIsss

(5.3)

This normalisation ensures the starting values of 0 and end values of 1. Note that

both of the compartments a and s are normalised by forcing them to start with value

5.3. SIMULATION OF MODEL I 47

Figure 5.7: The progress of all compartments within the time period of 30 years.

0, i.e. by taking a− aI and s− sI .These normalised values are plotted together and seen in figure 5.8, and from a visual

evaluation the quasi steady state approximation seems reasonable.

We will continue with simulations of model VI and provide additional validation of

the applicability of the model.


Figure 5.8: The change in number of SCs are rapidly followed by a change in numberof dead cells and the cytokine level. This is illustrated by the lines following the samepattern with respect to the normalized change in dynamics over time.

5.4 Simulation of Model VI

Model VI consists of a system of six differential equations, which constitutes a six-

dimensional space. It can be hard to get an overview of such a system, and because of

our research question, we have conducted a QSSA analysis in section 3.4. The analysis

from said section allows us to draw a phase plane showing the entire slope field, figure

5.9.

dx0

dt= (rx

√αxx0 + αyy0(1− cx(x0 + y0))− ax − dx0)x0 (5.4)

dy0

dt= (ry

√αxx0 + αyy0(1− cy(x0 + y0))− ay − dy0)y0 (5.5)

This phase plane should thus approximately illustrate the dynamics of the of the y0

and x0, with the effects of the other compartments comprised in them. Naturally, the

same fixed points apply to both models. Figure 5.9 shows the solution trajectory to

the initial condition of adding one LSC in a body in homeostasis. This leads to the

stable steady state of having only LSCs. Most of the region in the slope field leads to

5.4. SIMULATION OF MODEL VI 49

Figure 5.9: Slope field of the system 3.62 and 3.63. The small coloured straight linesindicate the direction of the field. The initial conditions are x0i = 6, 542 · 103 andy0i = 1. The solution trajectory goes from x0i to the stable fixed point y0ss.

this point, apart from the fact that there must be a region, within the two unstable

fixed points of x0 and y0, that leads to the stable fixed point of (x0, y0) = (0, 0).

The slope field of model VI thus reveals the same qualitative behaviour of dynamics

as model I. Therefore it is investigated more closely, if the dynamics of these two

models in fact show the same picture. A comparison of the models is provided through

simulation of x0 and y0 over time, i.e. the same plot as made in figure 5.6 with results

of model VI added.

Figure 5.10 illustrates how closely the dynamics of model I and model VI are related.

If it assumed that model I gives a realistic description of the dynamics of cancer grown,

model VI reflects the same realistic dynamics.


Figure 5.10: Comparison of dynamics between simulations of model VI, the dashedline, and model I. In the legend ”*” refers to the results of model VI.

6 Discussion

We try to describe a complicated system of a body, and the studied cells with an

abstracted compartment model. By doing this, we compress a lot of biological

concepts into parameters which characterize the model. Although the model is based

on knowledge of physiological properties, not all the knowledge which we use is exact

verified. For instance, the properties of the HSC: when measuring the number of HSCs

and estimating their various rates, one has to extract a sample from the bone marrow.

This measuring process is complicated and prone to errors, because the extracted cells

are not in their natural equilibrium in the new environment, and secondly, the rate at

which processes outside the body happen might be orders of magnitude off their true

timescale. Thus the properties of the extracted sample no longer reflect the qualities

from when they were part of a body in homeostasis.

If we consider model II - without the density function, which effectively depicts HSCs

dividing inside the bone marrow, it seems reasonable that cells without limitation of

space would grow unhindered. The number of SCs increase exponentially over time,

and the number of MCs follow this growth, as seen by the solutions in equation 2.29

and 2.35.

Adding the limitation of space in the form of the density function φx, the number of

HSCs seek towards a steady state, with the term 1/cx determining the value, equation

2.41. This inverse proportionality between the crowding effect and the steady state

value is characteristic for all the models.

Comparing model III and the analyses of the models IV and V respectively, the

models III and IV have steady states of the healthy steady states on the x0 axis, which

are stable in model IV, but unstable in the y0 direction in model III - which is in

51

52 CHAPTER 6. DISCUSSION

accordance both physiologically and with model V. The steady state of model V has

a value analytically symmetric to that of model IV, but numerically deviant. Apart

from these fixed points which all agrees well with each other, the non-linearity of the

inflammatory feedback gives additional fixed points in the models IV and V. These are

unstable fixed points, whose value lies between the two stable steady states: the ones

mentioned before (physiological homeostasis and death by cancer) and the zero steady

states - who has changed stability. This new unstable fixed point can be interpreted

as the inflammatory feedback adding a minimal value for number of cells. Values of

cells below this value leads to the zero steady state. When considering model IV, this

can be interpreted as either: there must be some minimal number of HSCs in the

system in order to return to homeostasis, and if this boundary value is exceeded, the

system dies out; or that model IV does not depict the physiological reality when the

system is close to or below said boundary value. The interpretation is inconclusive

and the hypotheses hard to prove experimentally, however, it is extremely rare if it is

possible at all, to die from the lack of blood cells - at least if no illness is involved.

Therefore, we suggest the full reality proposed by model IV is incomplete - and by the

argument of symmetry; we argue that the full reality that model V suggests is also

incomplete. However, the inflammatory feedback gives the opportunity to look for

possible bifurcations of the systems, which, physiologically, could prove useful when

treating leukemia. This is something that we did not have the time to do, but could

be exciting to investigate further.

Long term inflammation is considered to be the main facilitator of the development of

leukemia (Andersen et al., 2017a). This is based on the assumption that inflammation

increases the probability of mutations happening when the cells divide. Since we begin

our models, which includes leukemia, with one LSC being present, our system does

not give any information regarding this interaction - LSC increasing the number of

mutations. We only derive information about the competition between leukemic and

hematopoietic cells, where the LSCs has the advantage compared to HSCs, because

of the statements ry > rx and cy < cx. As we saw in chapter 3, when analysing the

models IV and V, the steady states of x0 and y0 are dependent on all parameters,

including those from the inflammatory feedback. Therefore, the contribution from the

53

inflammation feedback to the changes in the dynamics has to be elucidated.

This is done by inspecting model VI:

dx0

dt= (rxs(x0, y0)(1− cx(x0 + y0))− ax − dx0)x0 (6.1)

dy0

dt= (rys(x0, y0)(1− cy(x0 + y0))− ay − dy0)y0 (6.2)

We here refer to s as the inflammation level. s is a function of both x0 and y0, and it

is determined by (equation 3.61)

s =√αxx0 + αyy0 (6.3)

s is a feedback mechanism, where the constants αx and αy express how the number of

each type of cell is going to affect s and thus the rate of change of x0 and y0. These

constants are given by:

αx =rseaes

(dx0 + axAx) , αy =rseaes

(dy0 + ayAy) (6.4)

Here rseaes

is a constant accounting for properties of the immune system, and both

(dx0 + axAx) or (dy0 + ayAy) are determined by the properties of the stem cells and

progenitor cells. Recall that the progenitor cells are the unspecialised intermediate

from stem cells to mature cells. The effect from dx0 and dy0 can be considered

negligible, because they have an order of magnitude of 10−3, while axAx and ayAy are

10−5 · 1013 = 108. Therefore the value of axAx and ayAy have impact on how much the

current number of x0 and y0 respectively, influences the inflammation level, see 6.3.

So far, we have worked with the assumptions that ax = ay and Ax = Ay, thus making

αx = αy. If αx < αy then the number of y0 would result in increasing the s more than

previously asserted. Thus a possible way of slowing the growth of leukemia down is to

inhibit the division of leukemic progenitor cells.

However, the simulations indicate that the interactions between x0 and y0 dominates

the progression. Since the density function φ originate from an approximation, we

will elucidate which problems can occur, when using it instead of a hill function. A

benefit of the decreasing hill function, denoted φhill, is that it can only produce values

between 0 and 1. If we consider the effect it has in the model, then the value of φhill,

54 CHAPTER 6. DISCUSSION

in the ”realistic” limits; 0 < x0 <∞

limx0→ 0

(1

1 + cxx0

)= 1 (6.5)

limx0→∞

(1

1 + cxx0

)= 0 (6.6)

Obviously we cannot obtain a unlimited amount of stem cells, but even if we do allow

it, the value of the decreasing hill function will be 0 ≤ φhill ≤ 1. Our linear response

φ does not allow the model to run to infinity.

limx0→ 0(1− cxx0) = 1 (6.7)

limx0→∞(1− cxx0) = −∞ (6.8)

In section 4.3, equation 4.4, we based our estimate of cx on φx = 1 − cxx0. But

as a consequence of equation6.7, we meet a requirement to restrict our interval of

x0. Because of our φ function, we have to restrict ourselves to the limits where this

function acts as a dampener in the same way as the φhill function. As our function is

linear, we can easily find the transition into negative values

1− cxx0 = 0 ⇒ (6.9)

x0 =1

cx(6.10)

Thus in our no cancer model, we are restricted within the interval 0 ≤ x0 ≤ 1cx

. In the

case of the model I, φx = 1− cx(x0 + y0), the interval is

1− cx(x0 + y0) = 0 ⇒ (6.11)

x0 + y0 =1

cx(6.12)

Since the sum x0 + y0 eventually becomes larger over time, there comes a point, where

the sum is larger than 1/cx thus leading to φx < 0 (same principle applies for φy).

Even though our simulations resulted in satisfactory descriptions of cancer growth, this

limitation of our density function suggests that it is perhaps not a valid approximation

or at least it should give incentive for further investigations of the crowding effect on

stem cells.

7 Conclusion

We have reduced the model by Andersen et al. (2017a) with a quasi-steady-state-

assumption, thereby condensing what previously was a six dimensional system of

differential equations into to a two dimensional system, namely the equations describing

the rates of change of hematopoietic stem cells and leukemic stem cells. By comparing

the dynamics of the model derived from the assumption with the original, it is

ascertained that the overall dynamics of leukemia can be described by the competition

between hematopoietic stem cells and leukemic stem cells.

55

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8 Appendix

8.1 Elaboration of the inspirational model

This model is by Andersen et al. (2017a). It clearly distinguishes how leukemic cancer

would progress when an external factor such as inflammation is introduced to the

equation. The inflammation could come from a variety of sources such as smoking,

chemical fumes etc.

Additionally the rate constant rm describes the possible mutation of HSC into an

Figure 8.1: The model proposed by (Andersen et al., 2017a)

I

II CHAPTER 8. APPENDIX

LSC denoted. This is not to say that it would be just a single mutation but rather a

sequence of mutations that would self-renew, effectively becoming a cancer cell. This

particular mutation was seen as a randomly discrete event, therefore Andersen et al.

(2017a) used the Poisson process where the probability was around 10−7 per year per

cell. An important note would be that not all cells that mutate are cancerous some

just become malignant. We will argue that this mutation rate constant is irrelevant,

because it is very small compared to the other rate constants, and will thus be, to a

good approximation, be zero.

Two equations are introduced: Equation 8.5 takes dead cells, a, into account, it being

a sum of all the death rate terms from the previous equations. The last term eaas is

the result of phagocytic cells consuming the dead cells, since the dead cells have to be

eliminated from the system as well. Thus ea is an elimination rate and s is a measure

of the phagocytic cells and is in fact also a measure of the inflammation level. The

reason for this is that when tissue is damaged, inflammation occurs exerting pressure

on involved cells , which leads to cell death and therefore more phagocytic cells being

produced. These are describes by equation 8.6, where increase in inflammation is

proportional to the number of dead cells, and is eliminated with a rate of es.

dx0

dt= (rxφxs− ax − dx0)x0 − rmsx0, x0(0) = x0i (8.1)

dx1

dt= axAxx0 − dx1x1, x1(0) = x1i (8.2)

dy0

dt= (ryφys− ay − dy0)y0 + rmsx0, y0(0) = y0i (8.3)

dy1

dt= ayAyy0 − dy1y1, y1(0) = y1i (8.4)

da

dt= dx0x0 + dy0y0 + dx1x1 + dy1y1 − eaas, a(0) = ai (8.5)

ds

dt= rsa− ess+ I, s(0) = si (8.6)

Now, going back to when cells die. They go through a series of events where the debris

(dead cells) is engulfed by phagocytic cells all the while anti-inflammatory cytokines

are released. Equation 8.6, which describes the cytokine level, is being up regulated

by the dead cells, which increase the amount of phagocytic cells produced with a rate

8.2. MODEL I - THE FULL MODEL III

constant of rs per dead cell, while they become eliminated with a rate es. This is

based on an assumption that the cytokine to phagocytic cell ratio is proportional.

External factors may add to the inflammatory response represented in figure 8.1 as a

lightning bolt, and as an I in the equation.

A red compartment was created to see the inflammatory level. The amount of

phagocytic is balanced or in equilibrium with the cytokine levels in fixed ratio. This

is because cancer is developed in a time scale of years and inflammatory responses on

time scale of hours or days. Represented as φ(x0, x1) and φ(y0, y1).

8.2 Model I - The full model

This is the full system which we work with

dx0

dt= (rxφxs− ax − dx0)x0, x0(0) = x0i (8.7)

dx1

dt= axAxx0 − dx1x1, x1(0) = x1i (8.8)

dy0

dt= (ryφys− ay − dy0)y0, y0(0) = y0i (8.9)

dy1

dt= ayAyy0 − dy1y1, y1(0) = y1i (8.10)

da

dt= dx0x0 + dy0y0 + dx1x1 + dy1y1 − eaas, a(0) = ai (8.11)

ds

dt= rsa− ess, s(0) = si (8.12)

and these are our density functions, which are allowed, and assumed, to be different

φx = [1− cx(x0 + y0)] (8.13)

φy = [1− cy(x0 + y0)] (8.14)

8.3 Matlab Code

This is the code which we have used to make the graphs and numerical calculations in

the report.

IV CHAPTER 8. APPENDIX

Script to numerically solve model IV and V

1 %% Numerical ly s o l v i n g the func t i on o f the model /w feedback

/w cancer /wo hea l thy c e l l s

2 bb=s q r t ( ( dy0+ay∗Ay) ∗( r s ) /( es ∗ea ) ) ; % Combined exp r e s s i on o f

cons tant s that we need

3 p=ze ro s (100000 ,1) ;

4 r=ze ro s (100000 ,1) ;

5 % Numert ica l ly c a l c u l a t e s the value o f the func t i on ( note x0

i s r ep l aced by

6 % k )

7 f o r k=1:100000

8 j =(k−1) ;

9 q=s q r t ( k )∗(1−cyy∗k )−(ay+dy0 ) /(bb∗ ry ) ;

10 p( k )=q ;

11 r ( k )=j ;

12 end

13 % plo t the f i g u r e

14 f i g u r e (18) ,

15 p lo t ( r , p ) ,

16 g r id on

17 ylim ([−20 4 0 ] ) ;

18 xlim ( [ 0 12000 ] ) ;

19 x l a b e l ( ’ y0 ’ ) ;

20 y l a b e l ( ’ dy0/dt ’ ) ;

21 kk=f i n d ( abs (p)<=0.01) ;% Find the value o f x0 ( k ) where the

func t i on i s very c l o s e to zero

22 %% Numerical ly s o l v i n g the func t i on o f the model /w feedback

/wo cancer

23 bb=s q r t ( ( dx0+ax∗Ax) ∗( r s ) /( es ∗ea ) ) ; % Combined exp r e s s i on o f

cons tant s that we need

24 p=ze ro s (100000 ,1) ;

8.3. MATLAB CODE V

25 r=ze ro s (100000 ,1) ;

26 % Numert ica l ly c a l c u l a t e s the value o f the func t i on ( note x0

i s r ep l aced by

27 % k )

28 f o r k=1:100000

29 j =(k−1) ;

30 q=s q r t ( k )∗(1−cxx∗k )−(ax+dx0 ) /(bb∗ rx ) ;

31 p( k )=q ;

32 r ( k )=j ;

33 end

34 % plo t the f i g u r e

35 f i g u r e (17) ,

36 p lo t ( r , p ) ,

37 g r id on

38 ylim ([−15 1 5 ] ) ;

39 xlim ( [ 0 7600 ] ) ;

40 x l a b e l ( ’ x0 ’ ) ;

41 y l a b e l ( ’ dx0/dt ’ ) ;

42 kk=f i n d ( abs (p)<=0.01) ;% Find the value o f x0 ( k ) where the

func t i on i s very c l o s e to zero

Comparison of the compartments’ dynamics in model I

1 % Argument f o r v a l i d i t y o f qssa − See i f the dynamics o f the

d i f f e r e n t compartments

2 % f o l l o w that o f the x0 and y0

3 c l o s e a l l

4

5 % Load vec to r from each equat ion :

6 % Respec t i v e ly x0 , x1 , y0 , y1 , a , s :

7 h1=cs ( : , 1 ) ;

8 h2=cs ( : , 2 ) ;

9 h3=cs ( : , 3 ) ;

VI CHAPTER 8. APPENDIX

10 h4=cs ( : , 4 ) ;

11 h5=cs ( : , 5 ) ;

12 h6=cs ( : , 6 ) ;

13 % Normal is ing vec to r o f a to be between 0 and 1 , which i s

used in p l o t :

14 h51=(h5−h5 (1 ) ) ;

15 h52=(1/h51 ( end ) )∗h51 ;

16 % Normal is ing vec to r o f s to be between 0 and 1 , which i s

used in p l o t :

17 h61=(h6−h6 (1 ) ) ;

18 h62=(1/h61 ( end ) )∗h61 ;

19

20 % Figures :

21 % x0 , blue ; x1 , cyan :

22 p lo t ( time , (1−h1/ x0I ) , ’−b ’ , time , (1−h2/ x1I ) , ’−−c ’ , ’ LineWidth ’

, 2 ) ;

23 x l a b e l ( ’Time [ Years ] ’ )

24 y l a b e l ( ’ Normalised change ’ )

25 hold on ;

26 % y0 , red ; y1 , ye l low :

27 p lo t ( time , ( h3/h3 ( end ) ) , ’−r ’ , time , ( h4/h4 ( end ) ) , ’−−y ’ , ’

LineWidth ’ , 2 ) ;

28 hold on ;

29 % a , green ; s , magenta :

30 p lo t ( time , h52 , ’−g ’ , time , h62 , ’−−m’ , ’ LineWidth ’ , 2 ) ;

31 l egend ( ’ x0 ’ , ’ x1 ’ , ’ y0 ’ , ’ y1 ’ , ’ a ’ , ’ s ’ ) ;

32

33 a x i s ( [ 0 16 0 1 ] ) ;

Loads the parameters for simulation

1 %% Parameters

2 % This func t i on i n i t i a l i z e s the parameters f o r the model and

8.3. MATLAB CODE VII

3 % s e t s i n i t i a l va lue s f o r the v a r i a b l e s .

4

5 %% Simulat ion time span [ Days ]

6 tspan = [ 0 : 1 : 4 0 ∗ 3 6 5 ] ; %changed from 80 years to 10

7

8 %% ODE parameters

9 % ODE TOL = 1e−6;

10 % DIFF INC = 1e−2;

11

12 %% System parameters [ 1/ year ]

13 rx = 2e−3; %Adjusted from 1e−3

14 ax = 1 .1 e−5;

15 ry = 4e−3; %Adjusted from 1e−3

16 ay = ax ;

17 dx0 = 2e−3;

18 dy0 = dx0 ;

19 dx1 = 129 ;

20 dy1 = dx1 ;

21 c = 1e−4;

22 cxx = c ; %cx

23 cxy = c ;

24 cyx = c ;

25 cyy = 0.8∗ c ; %cy

26 ea = 1e9 ;

27 es = 2 ;

28 r s = 5e−3;

29 rya = 0 ; % Not used

30 ryq = 0 ; % Not used

31 rm = 0 ;

32 Ax = 4.7 e13 ;

33 Ay = 4.7 e13 ;

34 tonse t = 35∗365;

VIII CHAPTER 8. APPENDIX

35 tpause = tonse t +5∗365;

36

37

38 %% I n i t i a l c o n d i t i o n s ( x1I =5∗10ˆ9/kg+2∗10ˆ9/kg=7∗10ˆ9/kg body

weigth , x0I = 8.∗10ˆ7/ kg body weigth )

39 x0I = 6.5421 e3 ; %Adjusted from 1e3 ;

40 x1I = 2.6219 e10 ; %Adjusted from 1e10 ;

41 y0I = 1 ;

42 y1I = 0 ;

43 aI = 1.1631 e3 ; %Adjusted from 1e2 ;

44 s I = 2 . 9 0 7 9 ; %Adjusted from 3 ;

45 yqI = 0 ; % Not used

46 xmI = 0 ; % Not used

Differential equations of model I

1 % This f i l e conta in s the d i f f e r e n t i a l equat ions o f the cancer

model .

2

3 f unc t i on [ c dot ] = Cancer eqs ( t , c , pars )

4

5 % Def ine v a r i a b l e s

6 x0 = c (1) ;

7 x1 = c (2) ;

8 y0 = c (3) ;

9 y1 = c (4) ;

10 a = c (5) ;

11 s = c (6 ) ;

12 yq = c (7) ;

13 xm = c (8) ;

14

15 % Def ine parameters

16 rx = pars ( 1) ;

8.3. MATLAB CODE IX

17 ax = pars ( 2) ;

18 ry = pars ( 3) ;

19 ay = pars ( 4) ;

20 dx0 = pars ( 5) ;

21 dy0 = pars ( 6) ;

22 dx1 = pars ( 7) ;

23 dy1 = pars ( 8) ;

24 cxx = pars ( 9) ;

25 cxy = pars (10) ;

26 cyx = pars (11) ;

27 cyy = pars (12) ;

28 ea = pars (13) ;

29 es = pars (14) ;

30 r s = pars (15) ;

31 rya = pars (16) ;

32 ryq = pars (17) ;

33 rm = pars (18) ;

34 Ax = pars (19) ;

35 Ay = pars (20) ;

36 tonse t = pars (21) ;

37 tpause = pars (22) ;

38 s I = pars (23) ;

39

40 %% Algebra i c r e l a t i o n s

41 phix = 1/(1+( cxx∗x0+cxy∗y0 ) ˆ2) ;

42 phiy = 1/(1+( cyx∗x0+cyy∗y0 ) ˆ2) ;

43 % Other phi func t i on

44 Phix = 1−(cxx∗x0+cxx∗y0 ) ;

45 Phiy = 1−(cyy∗x0+cyy∗y0 ) ;

46

47 % I n f e c t i o n l e v v e l

48 B = 7 ;

X CHAPTER 8. APPENDIX

49 I = B; % Base l ine , no i n f e c t i o n

50 % I = B∗(1+( t>=10∗365) ) ; % I n f e c t i o n

51

52 % Feedback modify s e l f−renewal r a t e s to e f f e c t i v e s e l f−renewal r a t e s

53 rex = rx∗phix∗ s ;

54 rey = ry∗phiy∗ s ;

55

56 % Feedback with other phi func t i on

57 Rex = rx∗ s∗Phix ;

58 Rey = ry∗ s∗Phiy ;

59

60 % E f f e c t i v e mutation ra t e

61 rem = rm∗ s ;

62

63

64 %% D i f f e r e n t i a l equat ions

65

66 % Calcu l a t i on change in c e l l numbers

67 c dot = [ ( Rex−dx0−ax )∗x0 ; . . . % dx0/

dt

68 ax∗Ax∗x0 − dx1∗x1 ; . . .

% dx1/dt

69 (Rey−dy0−ay )∗y0 ; . . . % dy0/dt

70 ay∗Ay∗y0 − dy1∗y1 ; . . .

% dy1/dt

71 dx0∗x0 + dy0∗y0 + dx1∗x1 + dy1∗y1 − ea∗a∗ s ; . . .

% da/dt

72 r s ∗a − es ∗ s ; . . . % ds/dt

73 ryq∗y0 − rya∗yq ; . . .

% dyq/dt

74 rem∗x0 ] ;

8.3. MATLAB CODE XI

% dxm/dt − not used

Figures of model I

1 % Figures ran by Driver Cancer

2

3 %g l o b a l tonse t tpause

4

5 t1 = tonse t /365 ;

6 t2 = tpause /365 ;

7

8 time=t s /365 ; % conver t s time t s in days to time in years

9

10 %f i g u r e (1 )

11 %subplot ( 2 , 1 , 1 )

12 %hold on

13 %plo t ( time , cs ( : , 1 ) , ’−b ’ , time , cs ( : , 3 ) ,’−−r ’ , time , cs ( : , 1 )+cs

( : , 3 ) , ’ : g ’ , ’ LineWidth ’ , 2 )

14 %legend ( ’ x 0 ’ , ’ y 0 ’ , ’ x 0+y 0 ’ , ’ Location ’ , ’ SE ’ ) ;

15 %x l a b e l ( ’ Time [ Years ] ’ )

16 %y l a b e l ( ’ Ce l l s ’ )

17 %hold o f f

18

19 f i g u r e (1 )

20 subplot ( 2 , 1 , 1 )

21 hold on

22 p lo t ( time , cs ( : , 1 ) , ’−b ’ , time , cs ( : , 3 ) , ’−−r ’ , time , cs ( : , 1 )+cs

( : , 3 ) , ’ : g ’ , ’ LineWidth ’ , 2 )

23 l egend ( ’ x 0 ’ , ’ y 0 ’ , ’ x 0+y 0 ’ , ’ Locat ion ’ , ’SE ’ ) ;


25 y l a b e l ( ’ C e l l s ’ )

26 hold o f f

27

XII CHAPTER 8. APPENDIX

28 %For p l o t t i n g only x0 :

29 %f i g u r e (1 )

30 %subplot ( 2 , 1 , 1 )

31 %hold on

32 %plo t ( time , cs ( : , 1 ) , ’−b ’ , ’ LineWidth ’ , 2 )

33 %legend ( ’ x 0 ’ ) ;

34 %x l a b e l ( ’ Time [ Years ] ’ )

35 %y l a b e l ( ’ Ce l l s ’ )

36 %hold o f f

37

38 subplot ( 2 , 1 , 2 )

39 hold on

40 c s s c a l e d=cs ∗2∗max( cs ( : , 2 ) ) /max( cs ( : , 4 ) ) ;

41 p lo t ( time , cs ( : , 2 ) , ’−b ’ , time , c s s c a l e d ( : , 4 ) , ’−−r ’ , time , cs

( : , 2 )+c s s c a l e d ( : , 4 ) , ’ : g ’ , ’ LineWidth ’ , 2 )

42 l egend ( ’ x 1 ’ , ’ y 1 s c a l e d ’ , ’ x 1+y 1 s c a l e d ’ , ’ Locat ion ’ , ’SE ’ ) ;


44 y l a b e l ( ’ C e l l s ’ )

45 hold o f f

46

47 f i g u r e (2 )

48 subplot ( 2 , 2 , 1 )

49 hold on

50 p lo t ( time , cs ( : , 1 ) , ’−b ’ , time , cs ( : , 3 ) , ’−−r ’ , ’ LineWidth ’ , 2 )

51 l egend ( ’ x 0 ’ , ’ y 0 ’ , ’ Locat ion ’ , ’NW’ ) ;


53 y l a b e l ( ’ C e l l s ’ )

54 hold o f f

55 %

56 subplot ( 2 , 2 , 2 )

57 hold on

58 p lo t ( time , cs ( : , 2 ) , ’−b ’ , time , cs ( : , 4 ) , ’−−r ’ , ’ LineWidth ’ , 2 )

8.3. MATLAB CODE XIII

59 l egend ( ’ x 1 ’ , ’ y 1 ’ , ’ Locat ion ’ , ’NW’ ) ;


61 y l a b e l ( ’ C e l l s ’ )

62 hold o f f

63 %

64 subplot ( 2 , 2 , 3 )

65 hold on

66 p lo t ( time , cs ( : , 5 ) , ’−g ’ , ’ LineWidth ’ , 2 )

67 l egend ( ’ a ’ , ’ Locat ion ’ , ’SE ’ ) ;


69 y l a b e l ( ’ C e l l s ’ )

70 hold o f f

71 %

72 subplot ( 2 , 2 , 4 )

73 hold on

74 p lo t ( time , cs ( : , 6 ) , ’−m’ , ’ LineWidth ’ , 2 )

75 hold o f f

76 l egend ( ’ s ’ , ’ Locat ion ’ , ’SE ’ ) ;


78 y l a b e l ( ’ IL−8 conc . [ pg/L ] ’ )

79

80 f i g u r e (3 )

81 subplot ( 2 , 1 , 1 )

82 hold on

83 semi logy ( time , cs ( : , 1 ) , ’−b ’ , time , cs ( : , 3 ) , ’−−r ’ , ’ LineWidth ’

, 2 )

84 l egend ( ’ x 0 ’ , ’ y 0 ’ , ’ Locat ion ’ , ’SW’ ) ;


86 y l a b e l ( ’ l og ( C e l l s ) ’ )

87 hold o f f

88 %

89 subplot ( 2 , 1 , 2 )

XIV CHAPTER 8. APPENDIX

90 hold on

91 semi logy ( time , cs ( : , 2 ) , ’−b ’ , time , cs ( : , 4 ) , ’−−r ’ , ’ LineWidth ’

, 2 )

92 l egend ( ’ x 1 ’ , ’ y 1 ’ , ’ Locat ion ’ , ’SW’ ) ;


94 y l a b e l ( ’ l og ( C e l l s ) ’ )

95 hold o f f

Main script for simulation of model I

1 % −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %

2 % This func t i on Driver Cancer .m s o l v e s the equat ions

s p e c i f i e d

3 % in the f i l e Cancer eqs .m by us ing the bu i l t−in ode−s o l v e r

4 % ode15s or ode45 . I t a l s o saves the s o l u t i o n s and

5 % the time−vec to r in a f i l e c a l l e d r e s u l t s . mat .

6 % −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− %

7

8 c l e a r a l l

9 c l o s e a l l

10

11 %% Parameters

12

13 Parameters ; % loads parameter va lue s from parameters .m

14 pars = [ rx , ax ry ay dx0 dy0 dx1 dy1 cxx cxy cyx cyy ea es r s

rya ryq rm Ax Ay tonse t tpause s I ] ;

15

16

17 %% I n i t i a l va lue s

18

19 I n i t = [ x0I , x1I , y0I , y1I , aI , s I , yqI , xmI ] ; % loads

i n i t i a l va lue s from parameters .m

20

8.3. MATLAB CODE XV

21

22 %% Solve ODE’ s the usua l way

23

24 opt ions = odeset ( ’ RelTol ’ ,1 e−4, ’ AbsTol ’ ,1 e−8) ; %

Tolerances

25 [ ts , c s ] = ode15s ( @Cancer eqs , tspan , In i t , opt ions , pars ) ; %

Solve ODE’ s

26

27

28 %% Figures

29

30 Drive r Cancer F igure s

Euler’s method to solve model VI

1 %Euler s method f o r 1 . order d i f f e r e n t i a l equat ions ;

2

3 c l o s e a l l

4

5 t s t a r t = 0 ; % time , s t a r t

6 t s l u t = 15∗365; % time , end

7 x0 = 6.542 e3 ; % i n i t i a l c ond i t i on f o r x0

8 y0 = 1 ; % i n i t i a l c ond i t i on f o r y0

9 n = 3000 ; % Steps o f i n t e g r a t i o n

10 alphax = ( r s ∗( dx0+ax∗Ax) ) /( ea∗ es ) ; %alpha x constant

11 alphay = ( r s ∗( dy0+ay∗Ay) ) /( ea∗ es ) ; %alpha y constant

12

13 t t a b e l = ze ro s (1 , n+1) ; % I n i t i a l i s i n g vec to r f o r t

14 xtabe l = ze ro s (1 , n+1) ; % x0

15 ytabe l = ze ro s (1 , n+1) ; % y0

16 %ydottabe l=ze ro s (1 , n+1) ;

17 dt = ( t s l u t−t s t a r t ) /n ; %length o f s t ep s

18

XVI CHAPTER 8. APPENDIX

19 t = t s t a r t ;

20 x = x0 ;

21 y = y0 ;

22

23 t t a b e l (1 ) = t ;

24 xtabe l (1 ) = x ;

25 ytabe l (1 ) = y ;

26 %ydottabe l (1 )=t∗y ˆ2 ;

27

28 f o r i = 2 : n+1

29 f = ( rx∗ s q r t ( alphax∗x+alphay∗y )∗(1−cxx ∗(x+y ) )−ax−dx0 )∗x ; %

dx/dt = f

30 g = ( ry∗ s q r t ( alphax∗x+alphay∗y )∗(1−cyy ∗(x+y ) )−ay−dy0 )∗y ;

% dy/dt = g

31 x = x + f ∗dt ;

32 y = y + g∗dt ;

33

34 t=t+dt ;

35

36 t t a b e l ( i )=t ;

37 xtabe l ( i )=x ;

38 ytabe l ( i )=y ;

39 end ;

40

41 t t a b e l i=t t a b e l /365 ;

42

43 %The c a l c u l a t e d va lues are drawn .

44 f i g u r e (1 )

45 p lo t ( xtabel , ytabe l , x tabe l (1 ) , y tabe l (1 ) , ’ o ’ , ’ LineWidth ’ , 2 )

46 %Here x0 and y0 f o r s imu la t i on o f model I and VI are p l o t t e t

r e s p e c t i v e l y .

47 f i g u r e (8 )

8.3. MATLAB CODE XVII

48 p lo t ( time , cs ( : , 1 ) , ’−b ’ , time , cs ( : , 3 ) , ’ r− ’ , ’ LineWidth ’ , 1 . 5 ) %

s imu la t i on p lo t

49 hold on

50 p lo t ( t t a b e l i , xtabe l , ’b−− ’ , t t a b e l i , ytabe l , ’ r−− ’ , ’ LineWidth ’

, 1 . 5 ) %QSSA p lo t

51 l egend ( ’ x 0 ’ , ’ y 0 ’ , ’ x 0∗ ’ , ’ y 0∗ ’ , ’ Locat ion ’ , ’SE ’ ) ;


53 y l a b e l ( ’ C e l l s ’ )

54 a x i s ( [ 2 13 0 12 e3 ] ) ;

55 hold o f f

56

57 %plo t ( time , cs ( : , 1 )−xtabel , ’ b−−’, time , cs ( : , 3 )−ytabel , ’ r−−’, ’

Linewidth ’ , 1 . 5 )

Slope field of model VI

1 %Slope f i e l d o f the f i g u r e from Eulersystem .

2

3 x s t a r t = −10;

4 x s l u t = 0 .8 e4 ;

5 y s t a r t = −10;

6 y s l u t = 1 .2 e4 ;

7

8 anta l = 25 ;

9

10 dx = ( xs lut−x s t a r t ) / anta l ;

11 dy = ( ys lut−y s t a r t ) / anta l ;

12

13 dr = s q r t ( dxˆ2+dy ˆ2) /5 ;

14

15 f i g u r e (1 )

16 hold on

17 a x i s ( [ x s tar t , xs lut , y s ta r t , y s l u t ] ) ;

XVIII CHAPTER 8. APPENDIX

18

19 f o r x = x s t a r t : dx : x s l u t

20 f o r y = y s t a r t : dy : y s l u t

21 f = ( rx∗ s q r t ( alphax∗x+alphay∗y )∗(1−cxx ∗(x+y ) )−ax−dx0 )∗x ; % dx/dt = f

22 g = ( ry∗ s q r t ( alphax∗x+alphay∗y )∗(1−cyy ∗(x+y ) )−ay−dy0 )∗y ; % dy/dt = g

23 ydot = g/ f ; % ydot=(dy/dt ) /( dx/dt ) = g/ f

24 v = atan ( ydot ) ;

25 px = [ x−dr∗ cos ( v ) , x+dr∗ cos ( v ) ] ;

26 py = [ y−dr∗ s i n ( v ) , y+dr∗ s i n ( v ) ] ;

27 p lo t (px , py ) ;

28 x l a b e l ( ’ x0 ’ )

29 y l a b e l ( ’ y0 ’ )

30 end

31 end

quasi-steady-state assumption on a mathematical...

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