An SDE Approach to Cancer Therapy Including Stem CellsJulia M. Kroos Münster, August 2014 i Abstract Discrete mathematical models for cancer propagation only describe the behavior

An SDE Approach to Cancer

Therapy Including Stem Cells

Master Thesis

for attainment of the academic degree of

Master of Science

Westfälische Wilhelms-Universität Münster

Department of Mathematics und Informatics

Institute for Applied Mathematics

Supervisors:

Prof. Dr. Martin Burger

Prof. Dr. Christina Surulescu

Dr. Christian Stinner

presented by:

Julia M. Kroos

Münster, August 2014

i

Abstract

Discrete mathematical models for cancer propagation only describe the behavior of

cancer cells to a certain extend. In this thesis we present a stochastic model for can-

cer cells that incorporates intrinsic and extrinsic uncertainties as well as the rather

new concept of cancer stem cells. Here, we derive a model for dierentiated cancer

cells, cancer stem cells and normal tissue cells, prove the existence of solutions and

analyze the behavior of the dierent populations with regards to their persistence

times. Subsequently dierent treatment strategies including chemotherapy and radi-

ation therapy and combinations of these therapies are introduced and incorporated

in the SDE-model. In order to quantify the treatment strategies we introduced two

quality measures: the tumor control probability (TCP) and the normal tissue compli-

cation probability (NTCP). For both measures discrete formulas as well as stochastic

approaches were derived and compared.

ii

Statement of Authorship

I hereby declare that this master thesis has been written only by the undersigned and

without any assistance from third parties.

Furthermore, I conrm that no sources have been used in the preparation of this thesis

other than those indicated in the thesis itself.

Münster, August 2014

Julia Maria Kroos

iii

Contents

1 Introduction 1

2 Biological Introduction 3

2.1 Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 The Origin of Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3 Stem Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4 Cancer Stem Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5 Cancer Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.5.1 Chemotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5.2 Radiation Therapy . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Mathematical Fundamentals 9

3.1 Probability Spaces and Stochastic Processes . . . . . . . . . . . . . . . 9

3.2 The Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Stochastic Dierential Equations . . . . . . . . . . . . . . . . . . . . . 11

3.4 Numerical Solutions for Stochastic Dierential Equations . . . . . . . . 11

3.4.1 The Euler-Maruyama Method . . . . . . . . . . . . . . . . . . . 12

3.4.2 The Milstein Method . . . . . . . . . . . . . . . . . . . . . . . . 12

4 A First Mathematical Model 13

4.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.5 Approximation of the Persistence Time . . . . . . . . . . . . . . . . . . 24

4.5.1 Kolmogorov Backward Equations . . . . . . . . . . . . . . . . . 24

4.5.2 Simulation of the Stochastic Dierential Equations . . . . . . . 27

4.5.3 Comparing the Two Approaches to Compute the Persistence Time 28

Contents iv

5 Cancer Treatment 30

5.1 Continuous Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1.1 Chemotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1.2 Radiation Therapy . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Combining Chemotherapy and Radiation Therapy . . . . . . . . . . . . 40

5.3 Analysis for the Model including Therapies . . . . . . . . . . . . . . . . 41

5.4 Time-discrete Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.5 Persistence Times for the Dierent Therapies . . . . . . . . . . . . . . . 43

5.6 A Short Summary of all Therapies and Parameters . . . . . . . . . . . 44

6 Tumor Control Probability 49

6.1 Statistical Models for TCP . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.2 TCP Models Based on Birth and Death Processes . . . . . . . . . . . . 51

6.3 A TCP Model Based on Stochastic Dierential Equations . . . . . . . . 60


6.4.1 Numerical Simulations for the Discrete Model . . . . . . . . . . 62

6.4.2 Numerical Simulations for the Stochastic Model . . . . . . . . . 66

7 The Model Extended by Normal Cells 69

7.1 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


7.3 Approximation of the Persistence Time . . . . . . . . . . . . . . . . . . 74

7.4 The Eect of Cancer Treatment on Normal Cells . . . . . . . . . . . . 75

8 Normal Tissue Complication Probability 80

8.1 The Lyman Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

8.2 The Critical Volume NTCP Model . . . . . . . . . . . . . . . . . . . . 82

8.3 NTCP Model Based on Birth and Death Processes . . . . . . . . . . . 84

8.3.1 Numerical Solution for the Finite Dimensional System . . . . . 86

8.4 An NTCP Model Based on Stochastic Dierential Equations . . . . . . 87

9 Uncomplicated Tumor Control Probability 89

10 Conclusion and Outlook 91

List of Figures 93

Bibliography 95

1

1 Introduction

Cancer is the disease that represents - after cardiovascular diseases - the second most

important cause of death and morbidity in the European countries [31]. In 2012, 3.45

million new cases of cancer and 1.75 million deaths from cancer were recorded in Europe

[10]. However, the number of adults surviving for at least ve years after diagnosis

varies widely between the European countries depending on the health care system as

well as on the type of cancer occurring [8]. The survival rate for testicular cancer for

example is around 97.3 % whereas the rate of surviving lung cancer amounts to 10.9

% [8]. Consequently increasing the success rate of cancer treatment is an important

research topic.

As the survival of patients depends on several factors, e.g. the stage of cancer when

detected, the patient's constitution or the choice and realization of therapy, it is dicult

to nd the optimal therapy for every patient. The aim is to improve the accuracy

of treatment plans with as high chances of recovery as possible and preferably low

exposure of the patient to radiation or chemical agents.

Till now the choice of therapy and dose of chemicals or radiation depends on empirical

values. Thus mathematical models are a non invasive technique to estimate cancer

growth as well as patient's responses to dierent therapies in order to determine the

optimal treatment schedule.

A common approach are discrete mathematical models that represent the behavior of

cancer cells. They can only be used for a large amount of cells based on the law of mass

action. Nevertheless, for small numbers of cancer cells - which hopefully exist when

the cancer becomes extinct - this law does not account accurately for their behavior.

In this case random movement plays a central role. Other important aspects that are

neglected by the discrete models are inuences of intrinsic and extrinsic uncertainties.

These include for example the movement of organs inside the body or the fact that the

patient does not lie perfectly still during radiation treatment.

Hence, a more accurate and realistic approach are models using stochastic dierential

1 Introduction 2

equations in order to incorporate these random inuences and uncertainties.

A rather new concept for tumor propagation is the inclusion of cancer stem cells. In

contrast to the monoclonal model where the tumor is supposed to be only composed

of dierentiated cancer cells, the concept of cancer stem cells describes a hierarchy.

Here the tumor consists additionally of a few cancer stem cells that have the typical

characteristics of stem cells generating dierentiated cancer cells which enlarge the

tumor by frequent proliferation [28].

In the following we will deduce and analyze a mathematical model combining this rela-

tively new theory of cancer stem cells with the more accurate modeling using stochastic

dierential equations. Our model describes the behavior of cancer stem cells and dif-

ferentiated cancer cells and later also the normal tissue cells after the application of

dierent treatment strategies. These treatment strategies include chemotherapy and

radiation therapy as well as a combination of these two treatments.

For reasons of simplicity we will rst of all start with a simplied model focusing on

cancer stem cells and dierentiated cancer cells excluding the normal tissue cells in

chapter 4. Adding normal tissue cells in chapter 7 makes the model more realistic as

the survival of the surrounding tissue is also an important aspect of cancer treatment

schedules.

In both cases we proceed as follows. Firstly we prove the existence of solutions (in

section 4.2) and analyze the behavior of the populations via the persistence time.

Here we approximate the persistence time with the Kolmogorov backward equation

(in section 4.5.1) and with simulating the numerical solutions of the corresponding

stochastic dierential equation (in section 4.5.2). Secondly we compute the numerical

solutions of our stochastic dierential equations (in section 4.4) and introduce and

compare the dierent therapy strategies to our model in chapter 5.

Following this we introduce dierent quality measures for cancer treatment strategies.

We estimate the probability that the tumor is controlled (TCP) in chapter 6 and the

probability that the healthy tissue is harmed (NTCP) in chapter 8 - both with a discrete

and a stochastic approach. Combining these two values in the uncomplicated tumor

control probability in chapter 9 we can nd the optimal treatment that maximizes the

TCP but minimizes the NTCP at the same time.

3

2 Biological Introduction

2.1 Cancer

The word "'cancer"' derives from the ancient Greek word καρκινoς which was rst

used by Hippocrates to describe tumors. The word tumor in turn originates from the

Latin word for swelling and tumefaction [5]. Cancer has been characterized as a group

of diseases where cells grow uncontrolled and tumors developed invading the tissue of

origin and other organs.

It used to be consensus that once cancer cells have developed this would inevitably lead

to tumor development. Nevertheless latest cancer research showed that early tumor

growth is dependent on intrinsic and external factors such as immune surveillance, the

epithelial-mesenchymal transition or the angiogenetic switch [17].

2.2 The Origin of Cancer

In healthy tissues there is a balance between cell proliferation and programmed cell

death called apoptosis. If this balance is altered for the benet of cell proliferation,

then cancer can develop.

For the proliferation of cells the genetic information, the DNA, has to be replicated and

transmitted completely and without mistakes to the daughter cells. However, DNA can

be corrupted by several factors, by external ones like radiation or chemicals, or internal

ones when a wrong base pair is added by the DNA-polymerase in the replication process

[5]. In general, cells have a repair mechanism that is able to correct small mistakes in

the genetic information. A permanent alteration in the DNA is called mutation and if

a cell displays too many of these mutations, healthy cells activate the apoptosis process

[30]. This programmed cell death is biologically reasonable in order to prevent severe

damages in the DNA. Over time each cell accumulates more and more mutations in

its DNA. If these mutations aect sections of the DNA that regulate growth and cell


division, cancer can develop [30].

Generally it takes more than half a dozen mutations for a cell to become a cancer cell.

These cells reveal special properties that distinguish them from normal cells. They are

Figure 2.1: A lung cancer cell andnormal epithelial lung cells.

able to avoid apoptosis and have an unlimited

replicative potential. Additionally cancer cells can

avoid immune destruction and deregulate their cel-

lular energetics. They are especially characterized

by inducing sustained angiogenesis, tissue invasion

and metastasis, as well as genome instability. Fur-

thermore, cancer cells do not only dier from nor-

mal cells with respect to their properties but also

in their appearance. For example a at, special-

ized epithelial cell in the lung can turn into an

egg-shaped or nearly round lung cancer cell. Malignant cells often possess irregular

structures like nuclei in dierent sizes and shapes [30].

Hence a tumor develops when cells start to multiply autonomously and grow on and

on. These tumors can be benign or malignant. Benign tumors resemble the tissue

they originate from, grow slowly and are restricted to the area where they developed.

Typically, malignant tumors grow initially only inside a certain tissue. Nevertheless,

further mutations can cause the cells to create daughter tumors that migrate to other

organs forming metastases there [19].

2.3 Stem Cells

In general, stem cells are somatic cells that have the ability to dierentiate into various

cell types and tissues. According to the type of stem cells, they have the potential to

turn into any type of cell (embryonic stem cell) or into specied cells (adult stem cell).

The classical characteristics of stem cells are self-renewal and potency. The former

refers to the ability to go through numerous cycles of cell division while maintaining

the undierentiated state, and potency describes the capacity to dierentiate into spe-

cialized cell types. For example, neuronal stem cells generate the main phenotypes of

the nervous system like neurons, astrocytes and oligodendrocytes. Additionally, stem

cells can produce daughter cells that again have the typical properties of a stem cell or

undergo asymmetric cell division resulting in one stem cell and one dierentiated cell.

Whether there is a symmetric or an asymmetric cell division depends on signals from

the biological environment [30].


2.4 Cancer Stem Cells

A relatively new theory for tumor growth is the concept of cancer stem cells rst

proposed by Bonnet and Dick in 1997 [4]. Exploring the origin of leukemia they found

Figure 2.2: A simplied model about the originof the cancer stem cells.

out that cancer stem cells could be

the origin of cancer and possibly

bring about the resistance of malig-

nant tumors and recurrence.

In contrast to the monoclonal model

where the tumor is supposed to be

only composed of dierentiated can-

cer cells, the model involving cancer

stem cells describes a hierarchy. Here

the tumor consists of a few cancer

stem cells that have the typical char-

acteristics of stem cells. They are

able to generate dierentiated cancer

cells that in turn enlarge the tumor by frequent proliferation. These cancer stem cells

could be an explanation for the fact that after therapy the tumor rst disappears and

after some time reappears: as stem cells proliferate less often than dierentiated tumor

cells most therapies do not harm the stem cells as much as they harm the dierentiated

cancer cells.

Nevertheless, the origin of the cancer stem cells is not solved yet. They may develop

when self-renewing normal stem cells acquire mutations and are transformed by alter-

ing only proliferative pathways. It is also possible that the cancer stem cells originate

by multiple mutations in progenitor cells which acquire the capability of self-renewal

[6, 30].

2.5 Cancer Treatment

According to the type, location, size, and grade of a tumor, as well as the patient's

health, certain strategies of treatment are more eective than others. The most com-

mon treatment approaches are surgery, radiation therapy and chemotherapy. Further

treatment strategies are immunotherapy and monoclonal antibody therapy. As the

understanding of the underlying biological processes of cancer has increased, the treat-


ment of cancer has undergone evolutionary changes. Tumor removal surgeries rst

have been documented in ancient Egypt, radiation therapy was developed in 1899 and

chemotherapy and newer targeted therapies are products of the 20th century [26]. As

new information about the biology of cancer emerge, treatments will be developed and

modied to increase eectiveness and precision of therapies as well as survivability and

quality of life for the patients.

Most treatment strategies aim at impairing rapidly dividing cells, however cancer stem

cells are radio-resistant as well as resistant to chemotherapy. The reasons for this re-

sistance are not exactly known yet. One possible reason might be just the fact that

therapies only kill cells with a high proliferation rate and cancer stem cells divide less

often than dierentiated cancer cells. Another possibility might be that cancer stem

cells have genetic mutations that make them resistant to damages from the radio and

chemotherapy. Furthermore, cancer stem cells may be able to repair DNA damage

more rapidly than normal cells [26].

2.5.1 Chemotherapy

Chemotherapy is a treatment for cancer with cytotoxic drugs that mainly aect and

harm cells that have a high proliferation rate. Consequently chemotherapy does not

Figure 2.3: The structure of DNAwith the nucleotides as the small-est units.

only kill cells that divide rapidly, but also harms

cells that divide rapidly under normal circum-

stances, like cells in the bone marrow and hair

follicles [18].

Nevertheless these cytotoxic drugs are not only

poisonous, but they also aect more or less impor-

tant pathways concerning proliferation and growth

of the cells. Some cytotoxic agents are similar to

the natural building blocks of the DNA, the nu-

cleotides, and are inserted instead. Thus they al-

ter the genetic information destroying the DNA or

causing an uneven distribution of the genetic code

to the new daughter cells. If any daughter cells

emerge at all they are not able to proliferate any

more due to the genetic damages. Other cytotoxic

drugs directly interfere in the DNA replication and

the even distribution of the genetic material on the daughter cells.


Some newer anticancer drugs are not indiscriminately cytotoxic, but rather target pro-

teins that are abnormally expressed in cancer cells and that are essential for their

growth. Such treatments are often referred to as targeted therapy. Chemotherapy may

use one drug at a time (single-agent chemotherapy) or several drugs at once (combi-

nation chemotherapy).

All in all modern chemotherapy strategies mainly do not aim at rapidly destroying the

cancer cells but try to prevent further proliferation of the malignant cells and thus to

restrict tumor growth [22].

2.5.2 Radiation Therapy

In radiation therapy ionizing radiation similar to X-rays is used to control or kill ma-

lignant cells.It damages damages the DNA of exposed tissue leading to cellular death

of cancer cells, as - unlikely normal cells - they lack a repair mechanism. In order to

spare normal tissues, shaped radiation beams are sent from several angles to intersect

at the tumor, providing a much larger absorbed dose there than in the surrounding

healthy tissue.

Instead of trying to directly kill the cancer cells modern therapies aim at implicitly

altering important molecules of malignant cells. These changes are supposed to initi-

ate cell death via biological pathways: Radiation leads to the emergence of aggressive

molecules especially in the tumor tissue that is well supplied with blood and oxygen.

Especially oxygen ions attack the genetic information of the cancer cells in the form

of free radicals. Furthermore, these reactive molecules destroy important enzymes

and molecules that play a vital role in the high proliferation rate of the cancer cells.

Consequently, the ability of tumor cells to divide rapidly is sorely aicted and thus

the tumor can no longer expand. Additionally the malignant cells are marked by the

reactive molecules and thus can be detected by the immune system and specically

eliminated.

New treatment strategies fractionate the overall radiation dose so that small dosages of

radiation are applied to the patient, split into many single fractions over a time period

of a few weeks. Thus, the aected healthy tissue gets the opportunity to regenerate and

replace the tumor with scar tissue. The cancer cells on the other hand are devitalized

step-by-step with every new fraction dose of radiation until they nally die as they

lack repair mechanisms. The fractionation makes it possible to apply very high overall

radiation doses to the patient without increasing the risk of long-term consequences.

Common radiation strategies include a daily dose of about two gray from Monday till


Friday for several weeks. Furthermore radiation therapy may also be used as part

of adjuvant therapy preventing tumor recurrence after surgery to remove a primary

malignant tumor [22].

9

3 Mathematical Fundamentals

In this chapter we recall the basic theory of stochastic dierential equations as well as

numerical algorithms to solve these based on [36]. For further details, the reader can

refer to [36] and [27].

3.1 Probability Spaces and Stochastic Processes

The set of possible outcomes in a random experiment is called sample space and is

denoted with Ω. A possible combination of outcomes is called an event and the set of

all events is denoted with A.

Denition 3.1.1 (σ-Algebra). A σ-algebra A is a family of subsets of Ω so that

1. Ω ∈ A,

2. A ∈ A ⇒ Ac ∈ A, with the complement Ac = Ω\A of A in A

3. A1, A2, . . . ∈ A ⇒⋃i≥1Ai ∈ A.

Denition 3.1.2 (Probability measure). Let A be a σ-algebra over Ω. A mapping

P : A → [0, 1] is a probability measure if it satises the following two axioms:

1. P(∅) = 0

2. A1, A2, . . . ∈ A and Ai ∩ Aj = ∅ for i 6= j ⇒ P(⋃

i≥1Ai)

=∑

i≥1 P(Ai).

The triplet (Ω,A,P) is called probability space.

Denition 3.1.3 (Stochastic process, trajectory). A stochastic process is a mapping

X : [0, T ]× Ω→ Rn so that


• X(t) = X(t, ·) : Ω→ Rn is a random variable for every t ∈ [0, T ]

• X(·, ω) : [0, T ]→ Rn is called a path, a realization or a trajectory of the stochastic

process for every ω ∈ Ω

In the following we will always simulate several trajectories each representing dierent

patients.

3.2 The Wiener Process

Since the introduction of stochastic analysis by Itô Kiyoshi in the 1940s, the Wiener

process plays a central role in the calculus of time-continuous stochastic processes.

Denition 3.2.1 (Wiener process). A real-valued stochastic process W (·) is called a

Wiener process or a Brownian motion, if:

1. W (0) = 0 almost surely

2. W (t)−W (s) ∼ N (0, t− s) for all 0 ≤ s ≤ t

3. The increments W (t1),W (t2)−W (t1), . . . ,W (tn)−W (tn−1) are independent for

all 0 < t1 < t2 < . . . < tn.

Denition 3.2.2 (History of the Wiener process). The σ-algebra

U(s) := U(W (r) : 0 ≤ r ≤ s)

is called the history of the Wiener process (W (t))t till time s. This implies that U(s)

records all information of our observations of W (r) for all times 0 ≤ r ≤ s.

Denition 3.2.3 (n-dimensional Wiener process). A stochastic process (W(t))t,W : R+ →Rn is an n-dimensional Wiener process (also called n-dimensional Brownian motion),

if it satises:

1. For all i = 1, . . . n the stochastic process (W i(t))t is a one-dimensional Wiener

process with W(t) = (W 1(t), . . . ,W n(t))

2. The σ-algebras W i := B(W i(t) : t ≥ 0) are independent, where B is the σ-algebra

generated by the random variables W i(t), t ≥ 0


3.3 Stochastic Dierential Equations

Let (W(t))t≥0 be an m-dimensional Wiener process and X0 an n-dimensional random

vector that is independent from (W(t))t≥0. Further let

F(t) := U(X0,W(s)), 0 ≤ s ≤ t

be the σ-algebra generated by X0 and the history of the Wiener process till time t.

A stochastic dierential equation on the interval [0, T ] has the form

dX(t) = f(t,X(t))dt+ g(t,X(t))dW(t) (3.1)

for t ∈ [0, T ] and with

X(t) = (X1(t), . . . , Xd(t))t,

W(t) = (W1(t), . . . ,Wm(t))t,

f : [0, T ]× Rd → Rd,

g : [0, T ]× Rd → Rd×m,

where W(t) denotes an n-dimensional Wiener process and X(t) a stochastic process.

The function f is called the drift coecient of the stochastic dierential equation and

g is its diusion coecient.

3.4 Numerical Solutions for Stochastic Dierential

Equations

For constant or linear coecients f , g in equation (3.1) it is possible to compute the

exact solution of this equation. Nevertheless, most realistic models do not have such

simple coecients.

Consequently we will turn to the numerical approximations of the solutions to stochas-

tic dierential equations. First of all we will discuss the Euler-Maruyama method,

which is an extension of the classical Euler method for ordinary dierential equations.

Second we will dene the implicit Milstein method, which is a higher order scheme.


3.4.1 The Euler-Maruyama Method

The Euler-Maruyama method for a stochastic dierential equation (3.1) is dened by:

Xn+1 = Xn + f(tn,Xn)∆t+ g(tn,Xn)∆Wn

with n = 0, . . . , N − 1, Xn ' X(tn), ∆t = TN, ti = i∆t and ∆Wn = W(tn+1)−W(tn) ∼

N (0,∆t). For each component this implies:

Xi,n+1 = Xi,n + fi(tn,Xn)∆t+m∑j=1

gij(tn,Xn)∆Wj,n (3.2)

for i = 1, . . . , d with ∆Wj,n ∼ N (0,∆t) for all j = 1, . . . ,m and n = 0, . . . , N .

3.4.2 The Milstein Method

The implicit Milstein method is dened for each component of Xn by:

Xi,n+1 = Xi,n + fi(tn+1,Xn+1)∆t+m∑j=1

gij(tn,Xn)∆Wj,n +m∑j1=1

m∑j2=1

d∑l=1

glj1∂gij2∂xl

In(j1, j2)

(3.3)

for i = 1, . . . , d and the other parameters dened as in 3.4.1. Additionally, we have

In(j1, j2) =

∫ tn+∆t

tn

∫ s

tn

dWj1(r) dWj2(s).

For j1 = j2 this double Itô-integral can be written as

In(j1, j1) =1

2((∆Wj1,n)2 −∆t).

This double Itô-integral does not have a closed analytical form for j1 6= j2. Nevertheless,

in order to compute it we can approximate this integral by:

In(j1, j1) ' In(j1, j2) =M−1∑j=0

(Wj1(tj,n)−Wj1(t0,n))(Wj2(tj+1,n)−Wj2(tj,n))

with tj,n = tn + j∆tM

for j = 0, . . . ,M .

13

4 A First Mathematical Model

4.1 The Model

Our rst approach is to derive a stochastic model for the dierentiated cancer cells

and the cancer stem cells. Therefore, let c(t) and s(t) be the population sizes of the

dierentiated cancer cells and the cancer stem cells, respectively at time t. We want

to analyze the interactions between these two populations and are going to proceed as

in [36]. We will need a few assumptions:

1. Cancer stem cells are immortal and have an unlimited replicative potential

2. Cancer stem cells are able to divide in various ways:

• into two stem cells (with probability a1),

• into one dierentiated cancer cell and one cancer stem cell (with probability

a2) or

• into two dierentiated cancer cells (with probability a3)

with ai ∈ [0, 1] for i = 1, 2, 3 and∑3

i=1 ai = 1

3. Dierentiated cancer cells are mortal and have a nite potential to divide

4. During proliferation dierentiated cancer cells divide into two cells, each of them

being again a dierentiated cancer cell

These interactions can be described schematically in the following way:


Figure 4.1: The schematic interaction of dierentiated cancer cells c and cancer stemcells s.

The parameters used above have the following meaning:

bc, bs: proliferation rates

ds, dc: death rates

m12, m21: transfer rates between the two populations.

The above assumptions lead to ds = 0, dc > 0 and m12 = 0. For a small interval of

time ∆t we can note the respective transition probabilities for X = (c, s)t:

Changes Probability

∆X(1) = (1, 0)t p1 = bcc∆t+ bsa2s∆t∆X(2) = (−1, 0)t p2 = dcc∆t∆X(3) = (0, 1)t p3 = bsa1s∆t∆X(4) = (0,−1)t p4 = 0 (cancer stem cells are immortal)∆X(5) = (2,−1)t p5 = bsa3s∆t∆X(6) = (−1, 1)t p6 = m12∆t = 0 (cancer cells cannot become cancer stem cells)

∆X(7) = (0, 0)t p7 = 1−∑6

i=1 pi

Now we want to determine the expectation and the covariance matrix for a given time

t:

E(∆X) =7∑i=1

pi∆X(i)

=

(bcc+ bsa2s− dcc+ 2bsa3s

bsa1s− bsa3s

)∆t

E(∆X(∆X)t

)=

7∑i=1

pi∆X(i)(∆X(i)

)t= (bcc+ bsa2s)

(1 0

0 0

)∆t+ dcc

(1 0

0 0

)∆t+ bsa1s

(0 0

0 1

)∆t


+bsa3s

(4 −2

−2 1

)∆t

=

(bcc+ bsa2s+ dcc+ 4bsa3s −2bsa3s

−2bsa3s bsa1s+ bsa3s

)∆t.

We can show that

V =E (∆X(∆X)t)

∆t

is positive-denite. Hence the square root of the matrix V exists which we now dene

as B := V 1/2. Further we dene and compute:

µ :=E(∆X)

∆t=

(bcc+ bsa2s− dcc+ 2bsa3s

bsa1s− bsa3s

),

V :=E (∆X(∆X)t)

∆t=

(bcc+ bsa2s+ dcc+ 4bsa3s −2bsa3s

−2bsa3s bsa1s+ bsa3s

).

Obviously the matrix B is a (2× 2)-matrix, which we can explicitly specify as

B = V 1/2 =

(u v

v w

)1/2

=1

η

(u+ τ v

v w + τ

)

with τ =√uw − v2 and η =

√u+ w + 2τ .

With the previously computed matrix V we obtain the following values for the matrix

B

u = bcc+ bsa2s+ dcc+ 4bsa3s

v = −2bsa3s

w = bsa1s+ bsa3s

τ =√bss(a1(bsa2s+ 4bsa3s+ c(bc + dc)) + a3(bsa2s+ c(bc + dc)))

η = (2√bss(a1(bsa2s+ 4bsa3s+ c(bc + dc)) + a3(bsa2s+ c(bc + dc)))

+bsa1s+ bsa2s+ 5bsa3s+ c(bc + dc))1/2

=√

2τ + bsa1s+ bsa2s+ 5bsa3s+ c(bc + dc).


The stochastic dierential equation for the dynamics of the two cancer cell populations

is then given by

dX = µ(t, c, s)dt+B(t, c, s)dW(t)

with the initial condition X(0) = X0 and W(t) = (W1(t),W2(t))t, where W1(t) and

W2(t) denote independent Wiener processes. For each population we thus obtain

dc(t) = µ1(t, c, s)dt+B11dW1(t) +B12dW2(t)

= (u+ v − 2dcc)dt+1

η(u+ τ)dW1(t) +

v

ηdW2(t)

ds(t) = µ2(t, c, s)dt+B21dW1(t) +B22dW2(t)

= (w + v)dt+v

ηdW1(t) +

1

η(w + τ)dW2(t).

(4.1)

Thus we obtain stochastic processes such that each trajectory describes the tumor

growth dynamics for a specic patient. These stochastic processes are dened on a

random eld (Ω,A,P), where Ω denotes the sample space, i.e. a set of cancer patients

that are considered. Now let p(t, x1, x2) be the probability function at time t for c = x1

and s = x2 which is the discrete counterpart to the density function of continuous

random variables. In this case we can deduce a Fokker-Planck equation for p(t, x1, x2).

With a given expectation vector µ and the square root B of the covariance matrix V

the Fokker-Planck equation is dened by

∂p

∂t(t, x1, x2) = −

2∑i=1

∂

∂xi(µi(t, x1, x2)p(t, x1, x2))

+1

2

2∑i,j=1

∂2

∂xi∂xj

(2∑

k=1

Bik(t, x1, x2)Bjk(t, x1, x2)p(t, x1, x2)

).

Thus the associated Fokker-Planck equation for our system of cancer cell populations

is

∂p

∂t(t, x1, x2) = − ∂

∂x1

(µ1(t, x1, x2)p(t, x1, x2))− ∂

∂x2

(µ2(t, x1, x2)p(t, x1, x2))

+1

2

[∂2

∂x21

((B2

11 +B212

)p(t, x1, x2)

)+

∂2

∂x22

((B2

21 +B222

)p(t, x1, x2)

)]+

∂2

∂x1∂x2

((B11B21 +B12B22) p(t, x1, x2))

= − ∂

∂x1

((u+ v − 2dcx1) p(t, x1, x2))− ∂

∂x2

((w + v)p(t, x1, x2))


+1

2

[∂2

∂x21

(up(t, x1, x2)) +∂2

∂x22

(wp(t, x1, x2))

]+

∂2

∂x1∂x2

(vp(t, x1, x2))

= p(t, x1, x2) (dc − bc + bsa3 − bsa1)

+∂p

∂x1

(t, x1, x2) [bc + dc − 2bsa3 + x1(dc − bc) + x2(bsa2 + 2bsa3)]

+∂p

∂x2

(t, x1, x2) [bsa1 + bsa3 + x2(bsa3 − bsa1)]

+1

2

∂2p

∂x21

(t, x1, x2) [x1(bc + dc) + x2(bsa2 + 4bsa3)]

+1

2

∂2p

∂x22

(t, x1, x2) [x2(bsa1 + bsa3))− ∂2p

∂x1∂x2

(t, x1, x2) (2bsa3x2) .

4.2 Existence of Solutions

Now we want to prove the existence of solutions for our system of stochastic dierential

equations:

dc(t) = µ1(t, c, s)dt+B11(t, c, s)dW1(t) +B12(t, c, s)dW2(t)

ds(t) = µ2(t, c, s)dt+B21(t, c, s)dW1(t) +B22(t, c, s)dW2(t)(4.2)

with

µ1(t, c, s) = u+ v − 2dcc = bs(a2 + 2a3)s+ (bc − dc)c

µ2(t, c, s) = w + v = bs(a1 − a3)s

B11(t, c, s) =1

η(u+ τ)

B12(t, c, s) =v

η= B21(t, c, s)

B22(t, c, s) =1

η(w + τ).

(4.3)

For this purpose we consider the following Theorem (cf. [27] chapt. 5.2 and [32])

Theorem 1. If the coecients of a stochastic dierential equation

dX(t) =µ(t,X(t))dt+ σ(t,X(t))dW (t)

X(0) =X0

(4.4)


with initial conditions X0 are nonanticipative and satisfy the global Lipschitz condition:

|µ(t, x)− µ(t, y)|2 + |σ(t, x)− σ(t, y)|2 ≤ K|x− y|2

for t ∈ [0, T ] and x, y ∈ R with some constant K ≥ 0 and satisfy the linear growth

condition

|µ(t, x)|2 + |σ(t, x)|2 ≤ K(1 + |x|2)

for all t ∈ [0, T ] and x, y ∈ R, then there exists a continuous adapted stochastic process

X which accomplishes the equation (4.4) and is uniformly bounded in the Lebesque

space L2 which implies:

supt∈[0,T ]

E(X2(t)) <∞.

Let Y be another continuous solution of (4.4) that is uniformly bounded in L2(dP ),

then X and Y are indistinguishable.

For our multidimensional problem we use the Euclidean norm instead of the absolute-

value norm.

In the following we want to prove these two conditions for our system of stochastic

dierential equations (4.2) in order to show the existence of solutions. Therefore we

prove that µ and B satisfy the global Lipschitz condition and the linear growth con-

dition respectively. First we will start with the estimations for µ beginning with the

global Lipschitz condition.

For this purpose we have to prove ||µ(t, c1, s1)−µ(t, c2, s2)||22 ≤ K||(c1, s1)t−(c2, s2)t||22:

||µ(t, c1, s1)− µ(t, c2, s2)||22= |µ1(t, c1, s1)− µ1(t, c2, s2)|2 + |µ2(t, c1, s1)− µ2(t, c2, s2)|2

=

b2s(a2 + 2a3)2︸︷︷︸

=:x2

+ b2s(a1 − a3)2︸︷︷︸

=:y2

(s1 − s2)2︸︷︷︸=:s2

+ (bc − dc)2︸︷︷︸=:z2

(c1 − c2)2︸︷︷︸=:c2

+ 2bs(a2 + 2a3)(bc − dc)(s1 − s2)(c1 − c2)

= (x2 + y2)s2 + z2c2 + 2xzsc+ x2s2 − x2s2 + z2c2 − z2c2

= (x2 + y2)s2 + z2c2 + x2s2 + z2c2 − (x2s2 − 2xzsc+ z2c2)︸︷︷︸=(xs−zc)2

=(2x2 + y2

)s2 + 2z2c2 − (xs− zc)2︸︷︷︸

>0


≤ max2x2 + y2, 2z2(c2 + s2

)= max2b2

s(a2 + 2a3)2 + b2s(a1 − a3)2, 2(bc − dc)2︸︷︷︸

=:K

((c1 − c2)2 + (s1 − s2)2

)= K

((c1 − c2)2 + (s1 − s2)2

)= ||(c1, s1)t − (c2, s2)t||22.

The fact that µ fullls the linear growth condition follows analogously:

||µ(t, c, s)||22= |µ1(t, c, s)|2 + |µ2(t, c, s)|2

=

b2s(a2 + 2a3)2︸︷︷︸

=:x2

+ b2s(a1 − a3)2︸︷︷︸

=:y2

s2 + (bc − dc)2︸︷︷︸=:z2

c2 + 2bs(a2 + 2a3)(bc − dc)cs

= (x2 + y2)s2 + z2c2 + 2xzsc+ x2s2 − x2s2 + z2c2 − z2c2

= (x2 + y2)s2 + z2c2 + x2s2 + z2c2 − (x2s2 − 2xzsc+ z2c2)︸︷︷︸=(xs−zc)2

=(2x2 + y2

)s2 + 2z2c2 − (xs− zc)2︸︷︷︸

>0

≤ max2x2 + y2, 2z2︸︷︷︸=:K

(c2 + s2

)︸︷︷︸=||(c,s)t||22

= K||(c, s)t||22≤ K

(1 + ||(c, s)t||22

),

where the last inequality is valid due to the fact that K > 0. Thus µ satises both

conditions from the above theorem 1.

Now we want to prove the two conditions for the matrix B, the square root of the matrix

V . For the linear growth condition we have to prove ||B(t, c, s)||22 ≤ K(1 + ||(c, s)t||22):

||B(t, c, s)||22

=2∑i=1

2∑j=1

|Bij(t, c, s)|2

= B211 +B2

12︸︷︷︸=u

+B221 +B2

22︸︷︷︸=w

= u+ w

= bs(a1 + a2 + 5a3)s+ (bc + dc)c

≤ maxbs(a1 + a2 + 5a3), bc + dc︸︷︷︸=:K

(c+ s)


= K||(c, s)t||1 ≤ Kc2︸︷︷︸=:K

||(c, s)t||2

= K√c2 + s2

≤ K(1 + c2 + s2) = K(1 + ||(c, s)t||22),

where the second last inequality is true because of the equivalence of the p-norms,

that implies c1||x||r ≤ ||x||p ≤ c2||x||r with c1, c2 > 0 and 1 ≤ p ≤ r < ∞. The last

inequality follows with the universal inequality |z| ≤ 1 + z2, that is true for all real

numbers z.

In order to show the global Lipschitz condition for the matrix B, the square root of

the matrix V , we rst of all simplify the particular terms of the matrix and obtain

u = αs+ βc, v = γs, w = δs, τ =√εs2 + ζcs and η =

√θs+ ιc+ 2τ .

Furthermore we need a general theorem to estimate norms.

Theorem 2. Let V ,W be nite-dimensional Banach spaces and let f : G → W with

G ⊂ V be a dierentiable transformation. Let further a, b ∈ G be in a way that the line

segment

ab =: a+ t(b− a)|0 ≤ t ≤ 1

goes into G. Then we obtain

||f(b)− f(a)|| ≤ supx∈ab||Dxf || · ||b− a||,

with the Jacobian matrix Dxf of f .

Now we want to show ||B(t, c1, s1)−B(t, c2, s2)||22 ≤ K||(c1, s1)t− (c2, s2)t||22 and there-

fore we consider the equation:

||B(t, c1, s1)−B(t, c2, s2)||22 =2∑i=1

2∑j=1

|Bij(t, c1, s1)−Bij(t, c2, s2)|2.

Next we apply theorem 2 to the entries of our matrix B meaning f(c, s) := Bij(c, s).

We assume that s and c are bounded, i.e. r ≤ s ≤ R and 0 ≤ c ≤ R for r, R ∈ R+. This

is a sensible assumption because there can only be a limited number of cells present in


a certain area of the human body. For instance for B12(c, s) we obtain:

B12(c, s) =γs√

θs+ ιc+ 2√εs2 + ζcs

and thus

D(c,s)B12(c, s) =

− γs(ζ√εs2+ζcs+ι(cζ+εs))

2(2√εs2+ζcs+ιc+θs)3/2(ζc+εs)

γ((3ζc+2εs)√εs2+ζcs+(2ιc+θs)(ζc+εs))

2(2√εs2+ζcs+ιc+θs)3/2(ζc+εs)

.

As c and s are bounded we get:

|(D(c,s)B12(c, s))1| ≤γ(ζ√ε+ ζ + ιζ + ιε)R2

2(2√ε+ θr)3/2εr

=: c2(γ, ζ, ε, ι, θ, r, R).

Due to the fact that all variables γ, ζ, ε, ι, θ, r and R are positive the constant c2 is

positive and independent of c and s. All other entries of the Jacobian matrix can be

estimated in a similar way because all parameters occurring are positive. Additionally,

the absolute value of every subtraction that occurs can be estimated with the triangle

inequality in a way that all components of the Jacobian matrix are positive. With the

statement of theorem 2 and the strict positivity of s we can deduce the global Lipschitz

condition for the matrix B.

Thus our system of stochastic dierential equations (4.2) fullls all the conditions of

theorem 1 and consequently we obtain the existence of solutions.

4.3 Initial Conditions

In order to solve the stochastic dierential equations (4.1) we need initial conditions

that describe the size of each population at time t = 0. In [15] the choice (c0, s0)t =

(0.3, 0.1)t has been made, but another reasonable assumption is:

y = (y1, y2)t = (c(0), s(0))t = (c0, s0)t (4.5)

= (0.5, 0.4)t.

This implies that we focus on an area that is located closer to the center of the tumor.


4.4 Numerical Simulations

In order to simulate our system of stochastic dierential equations (4.1) we will refer

to Hillen and Bachmann [15] and Powathil [29] as regards to the choice of parameters.

We choose the initial conditions as (c0, s0)t = (0.5, 0.4)t as stated above. Referring

to these papers the mitosis rate of dierentiated cancer cells and cancer stem cells as

bc = bs = ln(2)/3 per day, which implies a cell doubling time of three days. In the

following we will choose a rate that is slightly smaller bc = bs = ln(2)/6. The reason for

this is that high birth and death rates in stochastic dierential equations lead to high

uctuations and thus a lot of trajectories that are zero after a short period of time.

For the dierentiated cancer cells we assume that the death rate equals the mitosis

rate, implying dc = ln(2)/6. The dierent probabilities of the cancer stem cells to

dierentiate are dened as a1 = p > 0.205 for the symmetric division into two cancer

stem cells and a3 = 1 − p for the division into two dierentiated cancer cells. The

asymmetric division of the cancer stem cells can be neglected, which implies a2 = 0.

It is a fact that the model without asymmetric division is equivalent to the complete

model, which was proved by Hillen and Bachmann [15].

Parameter (c0, s0)t bc, bs, dc a1 a2 a3

Value (0.5, 0.4)t ln(2)/6 0.7 0 0.3

Table 4.1: A summary of the model parameters for cancer cells based on [15].

A naive approach would be to perform the simulation with a classical Euler-Maruyama

method as described in section 3.4.1.

This algorithm can lead to negative values for the densities of cancer stem cells and

dierentiated cancer cells. This negativity is due to the fact that the Gaussian incre-

ment is not bounded from below.

To solve this problem we will cut o all trajectories that become negative, setting

them to zero from the rst time when they touch the horizontal axis. Once a tra-

jectory has reached zero it will maintain this status. Some realizations of our system

of stochastic dierential equations and the mean value of 104 trajectories solved with

the Euler-Maruyama method are shown below in gure 4.2. Only in the magnication

we can see the impact of the stochastic part of our dierential equation, the typical

uctuations of stochastic processes.

Further approaches to solve our system numerically are higher order methods like the

implicit Milstein method given in detail in section 3.4.2. The idea is that the supple-


(a) (b)

(c)

Figure 4.2: Dierent realizations of the stochastic dierential equation (4.2) and theirmean value (black line) of 104 trajectories solved with the Euler-Maruyama algorithmwith parameter values like in table 4.1. Figure (c) shows an amplied sector of gure(b).

mentary term in the Milstein method can determine the random term g(tn,Xn)∆Wn.

Trajectories computed with the implicit Milstein method are shown below in gure 4.3.

Comparing theses two solution methods the results are in the same order of magnitude

but the Euler-Maruyama method gives slightly smaller values for the population sizes.

However, one great problem of the Milstein method is the long program execution

time, resulting from the approximations of the derivatives and the double Itô-integral

in equation (3.3).


Figure 4.3: Dierent realizations of the stochastic dierential equation (4.2) and theirmean value (black line) for 104 trajectories solved with the implicit Milstein method.This parameter values are as in table 4.1.

4.5 Approximation of the Persistence Time

An important aspect of two interacting populations is their long-term behavior, es-

pecially the persistence time, i.e. the time it takes for one of the two populations

to become extinct. As we consider a system of stochastic dierential equations it is

sensible to look at the mean time of persistence of our system. Therefore, we can on

the one hand solve the corresponding Kolmogorov backward equation and on the other

hand simulate our stochastic dierential equations numerically.

4.5.1 Kolmogorov Backward Equations

Solving the corresponding Kolmogorov backward equation we can determine the aver-

age persistence time like in [2].

To this aim we assume that each of the two populations has a certain carrying capacity:

K1 for the dierentiated cancer cells and K2 for the cancer stem cells. This means that

there is a maximum population size, that can indenitely be sustained in a certain

habitat [30]. Here we choose K1 = K2 = R.

Now let Z be the random variable for the persistence time. As the latter obviously

depends on the initial conditions y in equation (4.5), we consider Z = Z(y). We will

denote the average persistence time for a population starting in y with T = E(Z(y)).

The function F (t,y) denotes the probability that the persistence time is larger than t


if the initial size of the population was y = (y1, y2)t, meaning

F (t, y1, y2) = P(Z(y) > t). (4.6)

This function satises the Kolmogorov backward equation [36]

∂F

∂t=

2∑k=1

µk(t, y1, y2)∂F

∂yk+

1

2

2∑k,j=1

2∑m=1

Bkm(t,y)Bjm(t,y)∂2F

∂yk∂yj,

were µ and B are as in equation (4.3) and

F (0, y1, y2) = 1, (y1, y2) ∈ (0, K1)× (0, K2),

F (t, 0, y2) = 0, y2 ∈ (0, K2),

F (t, y1, 0) = 0, y1 ∈ (0, K1),

∂F

∂y1

(t,K1, y2) = 0, y2 ∈ (0, K2),

∂F

∂y2

(t, y1, K2) = 0, y1 ∈ (0, K1).

With (4.6) it follows that pZ(y) = −∂F∂t

(y, t), where pZ(y) is the probability density of

Z(y). Consequently, we obtain for the average persistence time

T (y) = E(Z(y))

=

∫ ∞0

tpZ(y)(t) dt

= −∫ ∞

0

t∂F

∂t(y, t) dt

= − tF (y, t)|∞0 +

∫ ∞0

F (y, t) dt

=

∫ ∞0

F (y, t) dt,

so all in all

T (y) =

∫ ∞0

F (y, t) dt.

The populations of cancer cells we discuss here have an initial population size y =

(c0, s0)t and a carrying capacity R like in section 4.2. For our system of stochastic

dierential equations (4.2) we obtain the following Kolmogorov backward equation for


the function F :

∂F

∂t=(u+ v − 2dcc0)

∂F

∂c0

+ (w + v)∂F

∂s0

+1

2

((B2

11 +B212

) ∂2F

∂c20

+(B2

12 +B222

) ∂2F

∂s20

)+ (B11B12 +B12B22)

∂2F

∂c0∂s0

=(u+ v − 2dcc0)∂F

∂c0

+ (w + v)∂F

∂s0

+1

2

(u∂2F

∂c20

+ w∂2F

∂s20

+ 2v∂2F

∂c0∂s0

) (4.7)

with

F (0, c0, s0) = 1, (c0, s0) ∈ (0, R)2,

F (t, 0, s0) = 0, s0 ∈ (0, R),

F (t, c0, 0) = 0, c0 ∈ (0, R),

∂F

∂c0

(t, R, s0) = 0, s0 ∈ (0, R),

∂F

∂s0

(t, c0, R) = 0, c0 ∈ (0, R).

If we now integrate (4.7) with respect to the time t, we obtain:∫ ∞0

∂F

∂t(X(0), t) dt =F (X(0),∞)︸︷︷︸

=0

−F (X(0), 0)︸︷︷︸=1

=(u+ v − 2dcc0)

∫ ∞0

∂F

∂c0

dt︸︷︷︸= ∂T∂c0

+(w + v)∂T

∂s0

+1

2

(u∂2T

∂c20

+ w∂2T

∂s20

+ 2v∂2T

∂c0∂s0

)= ((bc − dc)c0 + bs(a2 + 2a3)s0)

∂T

∂c0

+ (bs(a1 − a3)s0)∂T

∂s0

+1

2

(((bc + dc)c0 + bs(a2 + 4a3)s0)

∂2T

∂c20

+ (bs(a1 + a3)s0)∂2T

∂s20

)− 2bsa3s0

∂2T

∂c0∂s0

(4.8)

with

T (0, s0) = 0, s0 ∈ (0, R),

T (c0, 0) = 0, c0 ∈ (0, R),

∂T

∂c0

(R, s0) = 0, s0 ∈ (0, R),


∂T

∂s0

(c0, R) = 0, c0 ∈ (0, R).

This partial dierential equation with given initial conditions and boundary conditions

can be solved numerically with the nite dierence method. The solution can be

visualized three-dimensionally with the choice of parameters like in table 4.1 and we

Figure 4.4: The solution of the Kolmogorov backward equation (4.8).

can also compute the solution at dierent points, for example:

T ((0.02, 0.02)t) = 0.17889 T ((0.4, 0.5)t) = 2.87678

T ((0.1, 0.1)t) = 0.90253 T ((0.5, 0.5)t) = 3.25905

T ((0.2, 0.2)t) = 1.71561 T ((0.8, 0.2)t) = 3.45338

T ((0.3, 0.1)t) = 1.83700 T ((0.8, 0.8)t) = 3.77572

T ((0.3, 0.2)t) = 2.24212 T ((1, 0.8)t) = 3.88871

T ((0.5, 0.4)t) = 3.26233 T ((1, 1)t) = 3.83908

4.5.2 Simulation of the Stochastic Dierential Equations

In order to compute the average time of persistence numerically, we will simulate the

population behavior via the stochastic dierential equation (4.2) and trace them till

one of the populations becomes extinct. This simulation has to be carried out several

times so that we can average the persistence time.

Concretely this means that we generate for example 1000 trajectories with the Euler-

Maruyama method. These trajectories are stopped whenever we have c ≤ 0 or s ≤ 0

and we note the exact time of this event. Averaging the measured data with the choice

of parameters like in table 4.1 we obtain the following estimations


T ((0.02, 0.02)t) ' 0.18385 T ((0.4, 0.5)t) ' 0.87667

T ((0.1, 0.1)t) ' 0.58479 T ((0.5, 0.5)t) ' 0.9255

T ((0.2, 0.2)t) ' 0.76467 T ((0.8, 0.2)t) ' 0.98805

T ((0.3, 0.1)t) ' 0.86595 T ((0.8, 0.8)t) ' 0.96859

T ((0.3, 0.2)t) ' 0.87392 T ((1, 0.8)t) ' 0.98721

T ((0.5, 0.4)t) ' 0.93701 T ((1, 1)t) ' 0.98359

for an overall time period of [0, 1].

Here it is conspicuous that the persistence times simulated with the Euler-Maruyama

method vary tremendously from the persistence times computed via the Kolmogorov

backward equation. Nevertheless, simulations with the Milstein method result in sim-

ilar values like the ones simulated with the Euler-Maruyama method. The resulting

persistence times of 1000 simulated trajectories computed with the Milstein method

are presented below.

T ((0.02, 0.02)t) ' 0.17828 T ((0.4, 0.5)t) ' 0.88329

T ((0.1, 0.1)t) ' 0.57750 T ((0.5, 0.5)t) ' 0.92460

T ((0.2, 0.2)t) ' 0.76621 T ((0.8, 0.2)t) ' 0.98891

T ((0.3, 0.1)t) ' 0.86007 T ((0.8, 0.8)t) ' 0.96995

T ((0.3, 0.2)t) ' 0.87482 T ((1, 0.8)t) ' 0.98662

T ((0.5, 0.4)t) ' 0.93859 T ((1, 1)t) ' 0.98312

We can see that some of the values for the persistence time computed with the Milstein

method are nearly equal to the values computed with the Euler-Maruyama method,

like for T ((0.8, 0.2)t) and T ((1, 1)t). However, most values slightly deviate from each

other. This could be owing to the fact that we used dierent methods to compute these

values but could also origin from the fact that we only computed 1000 trajectories in

order to determine the mean persistence time. A higher number of trajectories would

have by far exceeded the run-time of the the program computing the values with the

Milstein method.

4.5.3 Comparing the Two Approaches to Compute the

Persistence Time

Till now we have computed the persistence time of the cancer cell populations with the

Kolmogorov backward equation that does not take uncertainties into consideration.


Furthermore we have estimated the persistence time via numerical simulations with

1000 trajectories respectively, that are based on our stochastical model from section

4.1.

We can see that the values for the persistence times with the two dierent approaches

dier. Thus, we can compute the error made by the Kolmogorov backward equation

compared to the approximation with the simulations. In order to determine the relative

and the absolute error we consider

errorrel = ||x− x||

errorabs = ||x−x||||x||

for the exact values x and the approximation x. Using the Frobenius norm and a

discretization of N = 100 we obtain

errorrel = 0.90052

errorabs = 68.3673.

By the examples given above we can see that the Kolmogorov backward equation

basically overestimates the persistence time and the error made is rather big. In the

practical application this implies that the patient is treated longer than necessary.

Generally, the numerical simulations are more accurate and thus oer more realistic

values for the persistence time. However, for very small initial populations sizes the

numerical simulations are less accurate. This is due to the fact that the factor 1η

in equation (4.1) becomes problematic for s and c close to zero and thus an η that

is close to zero. In the Kolmogorov backward equation this problem does not occur

as this approach to compute the persistence time is not discrete and only uses the

deterministic part of equation (4.1) leaving out the stochastic part that includes the

factor 1η.

Consequently, we see that the approach with stochastic dierential equations is more

accurate and can help to spare the patient, as the Kolmogorov backward equation

predicts cancer cells even if there are no more left.

30

5 Cancer Treatment

Cancer stem cells have been identied in many tumors as the driving force behind

cancer growth as well as cancer progression [15]. Consequently eective treatment

schedules and strategies should aect the dierentiated cancer cells as well as the

cancer stem cells.

The eradication of cancer stem cells is rather dicult, as these cells are less sensitive to

radiation or chemical agents in comparison to dierentiated cancer cells [28]. Moreover,

it is dicult to identify these cells in vivo, and they are located everywhere in the tumor

[38].

However, one important consequence of cancer treatment is that it increases the death

rate of cancer stem cells. In the model that we deduced in section 4.1 we assumed

that this death rate ds equals zero. Now we suppose ds 6= 0 and obtain the following

stochastic dierential equations for the dierentiated cancer cells c(t) and the cancer

stem cells s(t) at time t:

dc(t) =(u+ v − 2dcc)dt+1

η(u+ τ)dW1(t) +

v

ηdW2(t)

ds(t) =(w + v − 2dss)dt+v

ηdW1(t) +

1

η(w + τ)dW2(t),

(5.1)

with the initial condition (c(0), s(0))t = X0, independent Wiener processes W1(t) and

W2(t) and


v = −2bsa3s

w = bsa1s+ bsa3s+ dss

τ =√uw − v2

η =√

2τ + bsa1s+ bsa2s+ 5bsa3s+ c(bc + dc) + dss.


5.1 Continuous Therapy

5.1.1 Chemotherapy

In chemotherapy dierent cytotoxic agents aect the death rate of the dierentiated

cancer cells as well as that of cancer stem cells, depending on the treatment dose.

Cancer stem cells are less sensitive to this treatment, resulting in a lower death rate in

comparison to the dierentiated cancer cells [12].

In the following section we will deduce and later on compare two dierent approaches

for chemotherapy models, the so-called dierentiation therapy. Here the aim is to

force the cancer cells to resume the process of maturation. This kind of therapy does

not destroy cancer cells but restrains their growth and allows the application of more

conventional therapies to eradicate the tumor.

Our rst approach to model dierentiation therapy is to increase the sensitivity of

cancer stem cells by adding dierentiation promoting agents, which force the cancer

stem cells to dierentiate and thus become more sensitive to other therapies. Possible

promoters are members of the TGF-β superfamily which are known to aect the char-

acteristics of growing tumors like invasion and immune evasion, and most important,

the increase of stem cell dierentiation [15].

In our model from section 4.1 this would imply to decrease the probability a1 charac-

terizing the dierentiation of a cancer stem cell into two cancer stem cells, and increase

the probability a3 for the dierentiation of a cancer stem cell into two dierentiated

cancer cells.

In the following we will deduce a formula for a1 like in [15] and [38] and obtain accord-

ingly a3 = 1− a1.

The dierentiation therapy can be modeled through a relationship between the prob-

ability for cancer stem cell self-renewal a1 and the average level of the dierentiation

promoter denoted with CF :

a1(t, ψ) = a−1 + (a+1 − a−1 )

(1

1 + ψCF (t)

), (5.2)

where a−1 is the minimum probability of the self-renewal and a+1 is the maximum

probability of self-renewal. If no dierentiation promoter is present the value of a+1 is

attained and a−1 is attained for CF → ∞. In contrast to [38] we will not model the

production of dierentiation promoters by tumor cells. Consequently CF only describes


the level of dierentiation promoter prescribed during the dierentiation therapy. In

order to stress this lack of dierentiation promoters we choose a+1 = 0.505 and a−1 = 0.2

according to [15]. The parameter ψ denotes the sensitivity of the cancer stem cells to

the dierentiation promoter. The dependency of a1(t, ψ) on the sensitivity ψ is shown

in gure 5.1.

Figure 5.1: The rate a1(t, ψ) of the proliferation of cancer stem cells into two cancerstem cells as a function of the sensitivity ψ. With the parameters set as a−1 = 0.2,a+

1 = 0.505 and CF (t) = 1 for all t.

In order to model the level of dierentiation promoter within a spatially homogeneous

tumor as a function of time we assume that the tumor is situated in a spherical region

of tissue and that the dierentiation promoter enters this area through the boundary.

After entering the tumor region from the boundary, the promoter diuses very quickly

and will attain a steady state over this region. To compute the value of CF (t) we

have to solve the problem of diusion over a sphere with radius R and average the

solution over the volume of the sphere. Thus we will use cF (r, t) to describe the radial

symmetric solution of the following boundary value problem:

∂cF∂t

= ω

(∂

∂r

(∂cF∂r

)+

2

r

∂cF∂r

)(5.3)

cF (R, t) = CF0(t), (5.4)

where ω is the eective diusivity of the dierentiation promoter and we will set ω =

10−3cm2/s [29]. Previous to the beginning of the dierentiation therapy CF0(t) is zero

and after the beginning the boundary condition on the sphere is set to CF0(t) = 1 and

the promoter diuses into the sphere. After the dierentiation therapy the boundary


condition is set zero again and the promoter diuses out of the sphere. We set

CF (t) =3

R3

∫ R

0

cF (r, t)r2 dr. (5.5)

Additionally to a change in dierentiation probabilities chemotherapy inuences the

cells's death rate depending on the dose. According to [29] we assume dc = 0.0098 for

the death rate of the dierentiated cancer cells. Due to the fact that cancer stem cells

are less sensitive to this therapy we suppose ds = 0.005.

For numerical simulations we need to solve the partial dierential equation (5.3) with

the corresponding initial conditions (5.13). However, the denition is problematic for

r = 0 as we divide by r.

One way to circumvent this problem is to use the fact that cF is radially symmetric,

implying∂cF∂r

= 0, for r = 0.

Furthermore we use the following second order accurate nite dierence formula using

an explicit scheme and a uniform mesh in space like in [13]:

ui,j+1 − ui,j∆t

=1

r2i∆r

2

(r2i+ 1

2ui+1,j − 2r2

i ui,j + i− 1

2wi−1,j

),

where ri for i = 1, 2, ..., N are the nodal positions along the radial direction and ri± 12

are positions of points located half-way between neighboring nodes. With this we can

numerically compute a solution for cF (see gure 5.2) and thus simulate the above

approach to dierentiation therapy with the Euler-Maruyama method.The results are

shown in gure 5.3. This treatment clearly forces the cancer stem cells to divide into

two dierentiated cancer cells so that the number of dierentiated cancer cells increases

in the beginning. But chemotherapy also raises the death rate of both types of cancer

cells, which leads to an overall decrease in the population size and for the cancer stem

cells even to extinction.

Next we present another approach to model dierentiation therapy. Here we focus on

the fact that the probability a3 characterizing the dierentiation of a cancer stem cell

into two dierentiated cancer cells, depends on the dose of cytotoxic agents. Thus we

get

a3 = a3(δ(t)) =δ(t) + α

1 + δ(t), α > 0, (5.6)

where δ(t) denotes the treatment dose delivered, that depends on the time t and α is

a sensitivity parameter. A common treatment is to deliver 75 mg m−2 per day of a


(a) (b)

(c)

Figure 5.2: The numerical solution of equations (5.3) for cF in gure (a), the behaviorof the average level of dierentiation promotor CF according to equation (5.5) in gure(b) and the rate a1(t, ψ) of the proliferation of cancer stem cells into two cancer stemcells in gure (c) for ψ = 0.5.


(a) (b)

Figure 5.3: Dierent realizations of the stochastic dierential equation (5.1) and themean value of 104 trajectories with the application of chemotherapy solved with theEuler-Maruyama algorithm with parameter values like in table 4.1 and ψ = 0.5.

current cytotoxic agent like temozolomide on 7 days a week [29]. The average total

tumor dimension, e.g. the average diameter of lung cancer is 7.5 cm [11] and thus we

have a treatment dose of δ(t) = 0.05625 mg m−2. For the respective death rates we

choose the values just as in the previous simulations for dierentiation therapy.

Solutions for numerical simulations with the Euler-Maruyama method for dierent

values of the parameter α are shown in gure 5.4. It can be seen that the larger the

value of the parameter α the smaller the population size of the cancer stem cells. As

this chemotherapy mainly aims at the dierentiation rate of the cancer stem cells the

population size of the dierentiated cancer cells is not reduced as much as the number

of cancer stem cells.

Another strategy mostly used in practice is to apply chemotherapy at intervals in turns

with treatment breaks. Here we will apply the last chemotherapy type introduced in

this section for three weeks in turns with a one week treatment break. The outcome

of the simulations is shown in gure 5.5. We can observe that there is a great increase

in the number of cancer stem cells during the treatment break.

5.1.2 Radiation Therapy

In radiation therapy proliferating cells are harmed by altering the genetic material in

these cells. Depending on the cell type and the cell's current position in the cell cycle,

the cell's sensitivity to radiation, the so called radiosensitivity, varies.


(a) (b)

(c) (d)

Figure 5.4: Dierent realizations of the stochastic dierential equation (5.1) and theirmean value for 104 trajectories with the application of chemotherapy solved with theEuler-Maruyama algorithm with parameter values like in tables 5.1 and 4.1. Figure(a) and (b) show the results with α = 0.5 and (c) and (d) the results for α = 0.6.


(a) (b)

(c) (d)

Figure 5.5: Dierent realizations of the stochastic dierential equation (5.1) and theirmean value for 104 simulated trajectories with the application of chemotherapy atintervals solved with the Euler-Maruyama algorithm with parameter values like intables 4.1. Figure (a) and (b) show the results for α = 0.5 and (c) and (d) the resultsfor α = 0.6.


The most common approach for modeling radiation treatment of a tumor is the LQ-

model. This model describes the surviving fraction of cells after radiation with a

specied dose. In this model a cell survives applied doses of radiation if it is able to

act as a progenitor for a signicant line of ospring [7]. The fraction of surviving cells

S(D) after a dose D (in gray) is then given like in [7] as

S(D) = e−(αD+βD2) (5.7)

for parameters α, β that correlate with the cell cycle length. The parameter α can be

interpreted as the lethal damage due to a single track of radiation and the parameter

β can be interpreted as lethal damage due to the misrepair of DNA damage as a result

of two separate tracks of radiation [15]. Tissues with a slow cell cycle correspond

to a small α/β-ratio whereas fast cell cycling tissues that are composed of quickly

proliferating cells correspond to a larger α/β-ratio. In clinical practice the total dose

D is given in n fractions of equal size d [16]. Consequently we obtain new death rates

for equation (5.1):

di(D) =1− S(D(n, d))

=1−[e−(αid+βid

2)]n

=1− e−n(αid+βid2)

=1− e−αiD(

1+ dαi/βi

)for i = c, s

(5.8)

for the cancer stem cells and the dierentiated cancer cells respectively depending

on the single radiation dose d and the number of fractions n. A common radiation

dose is d = 2.53 and 25 days of treatment with weekend breaks leading to an overall

treatment time of 35 days. The other parameter values given in [15] lead to death

rates very close to 1 inducing an instantaneous extinction of the populations. Thus

simulations with these values do not compute a single positive trajectory. Because of

that we adapted the parameter values of βc, βs for a given α = 0.35 (cp. [15]) in a way

that di = 1 − S(D(n, d)) ∈ [0, 1] leads to sensible values. Thus we use the following

parameter values:

Parameter α(in Gy−1) βc(in Gy−2) βs(in Gy−2) d(in Gy) n(in days)

Value 0.35 0.13425 0.13675 2.53 25

Table 5.1: A summary of the model parameters for radiation therapy based on [15].


Unlike in the chemotherapy in the previous section the common treatment with ra-

diation includes weekend breaks. Here again we simulate this therapy strategy with

the Euler-Maruyama method with weekend breaks and a whole week without radiation

after 3 weeks of radiation therapy. Choosing the other parameters like in section 4.4,

table 4.1 we obtain the results that are shown in gure 5.6.

Figure 5.6: Dierent realizations of the stochastic dierential equation (5.1) with theapplication of radiation therapy with the Euler-Maruyama algorithm and their meanvalue for 104 trajectories with parameter values like in tables 5.1 and 4.1.

The outcome of these simulations shows again the increase in the population size during

treatment breaks.

A closely related concept to the LQ-model is the Biological Eective Dose (BED),

which is also called the extrapolated response dose (ERD) [16]. For this quality measure

of radiation treatment we only consider the exponent of the LQ-model obtaining

BED :=− 1

αlog (S(D(n, d)))

=nd

(1 +

d

α/β

)=D

(1 +

d

α/β

)with the dose d per fraction, the number of fractions n and the total dose D[12].

However, this quality measure for radiation therapy does not take into account the

temporal protocol of dose delivery and the cell proliferation [16].


5.2 Combining Chemotherapy and Radiation

Therapy

As most patients rst undergo a surgical procedure for diagnostic and treatment pur-

pose, postoperative treatment strategies like radiation therapy and chemotherapy are

important to reduce further tumor progression. Radiation therapy is known to be an

eective postoperative treatment and chemotherapy is most often used in combination

with radiation therapy. Here chemotherapy can be administered before (neo-adjuvant),

during (concurrent) or after (adjuvant) radiation therapy. For instance recent clinical

trials prove that combining radiation therapy and the cytotoxic agent temozolomide

increase, the survival rate of the patient compared to just the application of radiation

therapy. [29]

For reasons of simplicity we do not incorporate an appropriate form of the LQ model

for radiation therapy into our system of SODEs but simply apply equation (5.7) to

the cancer stem cell and dierentiated cancer cell volume fraction at scheduled times

during simulation. The simulation is stopped at times of radiation treatment, S(D) is

applied to c and s using the corresponding values for α and β, and then the simulation

continues. Coincidentally cytotoxic agents are administered increasing the number of

cancer stem cells dierentiating into two dierentiated cancer cells and thus becoming

more sensitive to radiation.

Simulations of this combined treatment strategy are shown below in gure 5.7. A

(a) (b)

Figure 5.7: Dierent realizations of the stochastic dierential equation (5.1) withthe simultaneous application of radiation therapy and chemotherapy with the Euler-Maruyama algorithm and their mean value for 104 simulated trajectories with param-eter values like in table 5.1 and 4.1 with α = 0.5.


further common way to integrate dierent cancer therapies is to alternately apply

chemotherapy and radiation therapy. For all practical purposes a common procedure

is to apply three to four weeks of chemotherapy followed up by one to two weeks of

radiation therapy. Here we administer cytotoxic drugs to the patient for three weeks

and subsequently treat him with radiation therapy for one week including 2 days break

due to the weekend where we choose all parameters according to the previous sections

5.1.1 and 5.1.2. The results of the corresponding simulations are depicted in gure

5.8. Comparing the two approaches to combine chemotherapy and radiation therapy

(a) (b)

Figure 5.8: Dierent realizations of the stochastic dierential equation (5.1) with the al-ternating application of radiation therapy and chemotherapy with the Euler-Maruyamaalgorithm and their mean value for 104 simulated trajectories with parameter valueslike in table 5.1 and 4.1 with α = 0.5.

we can observe that the simultaneous application leads to smaller population sizes for

both cancer cell types.

5.3 Analysis for the Model including Therapies

Due to the fact that some of our parameters are no longer constant but depend on the

time t we have to take a closer look at the derivation of the Fokker-Plank equations

(cf. 4.1) and the existence of solutions (cf. 4.2). Here we have to check whether

the alterations in our parameters change any important aspects in the proofs and

calculations.

As regards to the biological aspects it makes sense that the proliferation rate bc(t) for

t ∈ [0, T ] is bounded, e.g. bc(t) ∈ [0,Λ] for Λ ∈ R+. In this case we can estimate the


maximum value of bc(t) with Λ and the global Lipschitz condition, as well as the local

growth condition follow in the same way as in 4.2. The same applies to the derivation

of the Fokker-Planck equation in 4.1, where bc is substituted by bc(t).

5.4 Time-discrete Therapy

Up to now we have considered the development of the population sizes of cancer stem

cells and dierentiated cancer cells for continuous therapies. But this does not agree

with reality. Hence we want to introduce a time-discrete therapy, like chemotherapy

or radiation therapy, in a way that it is typically applied to patients. These therapies

are applied to the patient at discrete points in time - for example: once a day except

for the weekends.

As we are considering a time-discrete therapy we select a sequence of points in time

tkk=1,...,N with 0 < t1 < . . . < tN at which the treatment takes place. Further let

ξ1, . . . , ξN be a sequence of i.i.d. random variables that denote the eect of the therapy

on the tumor cells. Thus, the random variable ξk corresponds to the treatment at time

tk. The process Lt describes the eect of the treatment on the dierent cells and can

be given as [35]

Lt =

0 , if 0 ≤ t < t1∑k:tk≤t ξk , if t1 ≤ t.

(5.9)

Next we need stochastic processes Tc(t) and Ts(t) that describe the population sizes of

the dierentiated cancer cells and the cancer stem cells with the application of a ther-

apy. Due to the fact that the cancer stem cells are less sensitive than the dierentiated

cancer cells against the treatment with radiation and most cytotoxic drugs we have:

Tc(t) = c(t)e−0.003Lt (5.10)

Ts(t) = s(t)e−0.002Lt (5.11)

for t ≥ 0. In order to simulate this model we need a few parameters that we choose

similar to section 4.4, table 4.1 and according to [35]. We start the treatment at time

t1 = 1 and continue on every workday. Further we set ξk to be a rescaled non-centered

χ2-distributed random variable

ξk ∼ σ2Dχ

2

(1,αD + βD2

σ2D

)(5.12)


with the parameter values of σD = 0.1, D = 2 Gy, α = 0.145 Gy−1 and β = 0.0353

Gy−2. A simulation of the application of this therapy is depicted in gure 5.9.

(a) (b)

(c)

Figure 5.9: Dierent realizations of (5.1) and their mean value for 104 simulated tra-jectories with a time-discrete therapy solved with the Euler-Maruyama algorithm withparameter values like in table 4.1. Figure (a) and (b) show the results of the simulationfor the dierent populations and gure (c) the process L(t).

Here we can clearly observe a high increase in the number of cancer cells during the

treatment breaks on weekends.

5.5 Persistence Times for the Dierent Therapies

When analyzing the behavior of cancer cell populations with reference to dierent treat-

ment strategies an important aspect is the persistence time. The respective persistence


times can be computed via numerical simulations of the stochastic dierential equa-

tions like in section 4.5.2. Comparing these times can give an insight into the impact

and the eectiveness of the dierent treatment strategies. With the parameters chosen

like in the previous sections of this chapter and a simulation with 104 trajectories we

obtain the following values:

Therapy Persistence Time

(in days)

without any therapy 8.0834

dierentiation therapy (5.1.1, rst approach) 0.19965

dierentiation therapy (5.1.1, second approach, α = 0.5) 5.1056


dierentiation therapy in intervals, α = 0.5 5.1037

dierentiation therapy in intervals, α = 0.6 4.6468

radiation therapy (5.1.2) 1.6754

combining chemotherapy and radiation therapy simultane-

ously (5.2)

1.5821

combining chemotherapy and radiation therapy alternately

(5.2)

4.7456

time-discrete therapy (5.4) 5.6027

Table 5.2: Persistence times of the dierent treatment strategies.

if not stated dierently we used α = 0.5 for these simulations. These persistence times

conrm the simulations of the dierent treatment strategies from the previous sections.

5.6 A Short Summary of all Therapies and

Parameters

In the following we will give a short summary of all therapies and treatment strategies.

This will help the reader to keep track of the eect of the dierent treatment strategies

and the equations and parameters used for the dierent therapies.

As a basis we take the following system of stochastic dierential equations:

dc(t) = (u+ v − 2dcc)dt+ 1η(u+ τ)dW1(t) + v

ηdW2(t)


ds(t) = (w + v − 2dss)dt+ vηdW1(t) + 1

η(w + τ)dW2(t),

with


v = −2bsa3s

w = bsa1s+ bsa3s+ dss

τ =√uw − v2

η =√

2τ + bsa1s+ bsa2s+ 5bsa3s+ c(bc + dc) + dss,

the initial values (c(0), s(0))t = X0 and independent Wiener processesW1(t) andW2(t).

1. Chemotherapy: cytotoxic agents aect the death rates of the dierentiated

cancer cells and of the cancer stem cells and increase the dierentiation rate a3

of the cancer stem cells to divide into dierentiated cancer cells.

a) First approach: the relationship between the probability for cancer stem

cell self-renewal a1 and the average level of the dierentiation promoter

(CF (t)) is modeled by

a1(t, ψ) = a−1 + (a+1 − a−1 )

(1

1 + ψCF (t)

).

In order to compute the value of CF (t) we have to solve:

∂cF∂t

= ω

(∂

∂r

(∂cF∂r

)+

2

r

∂cF∂r

)cF (R, t) = CF0(t).

With the radial symmetric solution cF (r, t) we compute the value of CF (t)

with:

CF (t) =3

R3

∫ R

0

cF (r, t)r2 dr.


Parameters Description Value

a−1 minimum probability of the self-renewal 0.2

a+1 maximum probability of self-renewal 0.505

ψ sensitivity of cancer stem cells to the dier-

entiation promoter

0.5

R radius of the tumor (in cm) 1.5

ω eective diusivity of the dierentiation pro-

moter (in cm2/s)

10−3

dc death rate dierentiated cancer cells (in

day−1)

dc + 0.0098

ds death rate cancer stem cells (in day−1) ds + 0.005

Table 5.3: A summary of the model parameters for the rst approach tochemotherapy applied to cancer cells.

b) Second approach: the division of a cancer stem cells into two dierentiated

cancer cells depends on the dose δ(t) of cytotoxic agents

a3 = a3(δ(t)) =δ(t) + α

1 + δ(t), α > 0.


δ(t) dose of cytotoxic agent (in mg m−2) 0.05625

α sensitivity parameter 0.4 - 0.6

Table 5.4: A summary of the model parameters for the second approachto chemotherapy applied to cancer cells.

2. Radiation therapy: application on weekdays and a whole week break after

three weeks of radiation. Proliferating cells are harmed by ionized rays and the

surviving fraction S(D) is given by

S(D) = e−(αD+βD2).

Hence we obtain for the death rates

di(D) = 1− S(D(n, d)) = 1− e−αiD(

1+ dαi/βi

)for i = c, s.



d radiation dose (in Gy) 2.53

n number of treatment fractions 25

D(n, d) total radiation dose n · dαc, αs sensitivity parameters (in Gy−1) 0.35

βc sensitivity parameter for dierentiated cancer cells (in

Gy−2)

0.13425

βs sensitivity parameter for cancer stem cells (in Gy−2) 0.13675

Table 5.5: A summary of the model parameters for radiation therapy appliedto cancer cells.

3. Combining chemotherapy and radiation therapy: combining radiation

therapy and the administration of cytotoxic agents

a) simultaneous application: coincidentally cytotoxic agents are adminis-

tered increasing the number of cancer stem cells dierentiating into two

dierentiated cancer cells and thus becoming more sensitive to radiation

b) alternating application: cytotoxic drugs are administered to the patient

for three weeks and subsequently he is treated with radiation therapy for

one week including a two day break due to the weekend

4. Time-discrete therapy: treatment is applied at discrete points in time - once a

day except for the weekend. The process Lt describes the eect of the treatment

on the dierent cells and is given by:

Lt =

0 , if 0 ≤ t < t1∑k:tk≤t ξk , if t1 ≤ t,

where ξ1, . . . , ξN denotes the treatment eect at the dierent points of time

t1, . . . , tN and the ξk are rescaled non-centered χ2-distributed random variables

dened by

ξk ∼ σ2Dχ

2

(1,αD + βD2

σ2D

).

The stochastic processes Tc(t) and Ts(t) describe the population sizes of the

dierentiated cancer cells and the cancer stem cells with the application of a


therapy:

Tc(t) = c(t)e−0.003Lt

Ts(t) = s(t)e−0.001Lt .


D radiation dose (in Gy) 2

α sensitivity parameter (in Gy−1) 0.145

β sensitivity parameters (in Gy−2) 0.0353

σD 0.1

Table 5.6: A summary of the model parameters for time-discrete therapy ap-plied to cancer cells.

49

6 Tumor Control Probability

The tumor control probability (TCP) is a quality measure for the eectiveness of cancer

treatments. It represents the probability that no malignant cells survive in a certain

region after a treatment.

TCP is used to optimize cancer treatment strategies but only describes the impact on

cancer cells and not on healthy tissue. The aim is to get a TCP that is as close to one

as possible as this implies the extinction of cancer in the respective area.

There are several models to calculate the TCP including statistical models as well as

models that are based on birth-death processes. The statistical models based on cell

survival have on the one hand a high practical relevance due to their simplicity. On

the other hand these models are inaccurate ignoring important cell mechanisms like

cell repair, proliferation and the dierent radiosensitivities of dierent cell types. The

TCP obtained from cell-population models describes the behavior of small numbers of

cells based on stochasticity. Here the change in cell density is prescribed by so called

birth-death processes. As this model is based on individual cell behavior it displays

cell dynamics more accurately. However, these models become more complex and thus

are less used by radiation therapists. However, this deterministic approach is only

adequate for large population sizes. Nevertheless, we would like to analyze the eect

of cancer treatments where the number of cancer cells - hopefully - becomes rather

small. Consequently the use of a deterministic model becomes problematic and thus

including a stochastic component leads to more accurate and realistic values for the

TCP. We will introduce these models based on the works of Dawson and Hillen [7] and

Zaider and Minerbo [39] and a rather new class of models relying on stochastic jump

processes [34] in the following sections.

6.1 Statistical Models for TCP

The most common and simplest model for TCP is a Poisson based model that arises

from the LQ-model dened in section 5.1.2. Considering that, we assume that the initial


cell number n0 is large. Furthermore we postulate that there is a small probability for

cells to survive the treatment. In the following the random variable X denotes the

number of surviving cells. Following the assumptions this random variable is Poisson

distributed. Thus the probability that k cells survive is

P(X = k) =λke−λ

k!.

The observed number of surviving cells, denoted with S(D), is a good estimation for

the survival fraction, hence the expected value of the random variable X is

E(X) = n0S(D) = λ.

Consequently we obtain for the probability that no cancer cells remain:

TCP = P(X = 0) = e−n0S(D) = e−λ.

However, this is only valid if the probability for cell survival is small and the number of

cells surviving the treatment is much less than the initial number of tumor cells which

is mainly the case in radiation therapy.

If we now assume the contrary situation that the number of cells n0 is small at the

beginning of the treatment and the probability for cell survival is high, the random

variableX is binomially distributed to n0 and the survival probability p of an individual

cell. Hence it follows

P(X = k) =

(n0

k

)pk(1− p)n0−k.

Again the observed number of surviving cells is a good estimator for the p, so we obtain

E(X) = n0p = n0S(D).

From this it follows that

TCP = P(X = 0) = (1− S(D))n0 .

These models strongly simplify reality as the TCP does not depend on time, but only

on the dose. As we focus here on a single dose intensity we obtain constant values for

the TCP with the two statistic models. The values of both the Poisson statistics and

the binomial statistics for TCP are listed in table 6.1.


no therapy radiation therapyc s c s

Poisson statistics 0.6424 0.6703 0.7712 0.7333Binomial statistics 0.3399 0 0.6930 0.5503

Table 6.1: TCP values for the dierent cancer cells types computed with the Poissonand the binomial statistics.

6.2 TCP Models Based on Birth and Death

Processes

In the following we will deduce the TCP based on a model involving birth and death

processes with a deterministic approach like in [39]. For this purpose we reduce our

stochastic dierential equation (4.1) in section 4.1 to the ordinary dierential equation:

dc(t)

dt= (bc − dc)c(t) + bsa2s(t) + 2bsa3s(t) (6.1)

ds(t)

dt= bsa1s(t)− bsa3s(t)− dss(t) (6.2)

with the initial conditions (c(0), s(0))t = (c0, s0)t.

First we will deduce the corresponding master equations describing the dynamics of

the transition probabilities. For this purpose we dene Ci(t) as the probability for i

living dierentiated cancer cells at time t and Sj(t) as the probability for j cancer stem

cells at time t with i, j ∈ N. Furthermore we assume Ci(t) = Sj(t) = 0 for all i, j < 0

and with the given initial conditions we get

Cc0(0) = Ss0(0) = 1.

Consequently, the probability that no cancer cells remain after the treatment is TCP(t) =

C0(t) · S0(t). In order to deduce the master equations we will in the following observe

the changes in the two populations in a small interval of time [t, t+ ∆t].

1. Master equation for the dierentiated cancer cells Ci(t)

There are dierent ways to observe i dierentiated cancer cells at time t+ ∆t:

• i+ 1 dierentiated cancer cells are alive at time t and one cell dies due to natural

processes or cancer treatment. Consequently the corresponding probability is

(i+ 1)dcCi+1(t)∆t.

• i − 1 dierentiated cancer cells are alive at time t and either one of these cells


undergoes mitosis with rate bc resulting in two dierentiated cancer cells, or

a cancer stem cell divides into a cancer stem cell and a dierentiated can-

cer cell with rate bsa2. The probability for this event is (i − 1)bcCi−1(t)∆t +

bsa2

∑∞j=0 jSj(t)Ci−1(t)∆t.

• i − 2 dierentiated cancer cells are alive at time t and one cancer stem cell

dierentiates into two dierentiated cancer cells with rate bsa3. Thus we obtain

for the probability

bsa3

∑∞j=0 jSj(t)Ci−2(t)∆t.

• i dierentiated cancer cells are alive at time t and nothing changes. The proba-

bility for this event is (1− (i(dc + bc) + bs(a3 + a2)∑∞

j=0 jSj(t))∆t)Ci(t).

Thus we get the following equation:

Ci(t+ ∆t) = (i+ 1)dcCi+1(t)∆t+ (i− 1)bcCi−1(t)∆t

+bsa2

∞∑j=0

jSj(t)Ci−1(t)∆t+ bsa3

∞∑j=0

jSj(t)Ci−2(t)∆t

+(1− (i(dc + bc) + bs(a3 + a2)∞∑j=0

jSj(t))∆t)Ci(t).

Rearranging this equation we obtain the dierence quotient

Ci(t+ ∆t)− Ci(t)∆t

= (i+ 1)dcCi+1(t) + (i− 1)bcCi−1(t)

+bsa2

∞∑j=0

jSj(t)Ci−1(t) + bsa3

∞∑j=0

jSj(t)Ci−2(t)

−(i(dc + bc) + bs(a3 + a2)∞∑j=0

jSj(t))Ci(t)

and for ∆t→ 0 we obtain the corresponding system of dierential equations

dCi(t)

dt=(i+ 1)dcCi+1(t) + (i− 1)bcCi−1(t)

+ bsa2

∞∑j=0

jSj(t)Ci−1(t) + bsa3

∞∑j=0

jSj(t)Ci−2(t)

− (i(dc + bc) + bs(a3 + a2)∞∑j=0

jSj(t))Ci(t)

(6.3)

for i ∈ N with C−1(t) = 0 and Cc0(0) = 1.


2. Master equation for the cancer stem cells Sj(t)

There are dierent ways to observe j cancer stem cells at time t+ ∆t:

• j+1 cancer stem cells are alive at time t and one cell dies with rate ds due to treat-

ment and natural processes or dierentiates into two dierentiated cancer cells

with a rate bsa3. Thus the probability for this event is (j+1)(ds+bsa3)Sj+1(t)∆t.

• j − 1 cancer cells are alive at time t and one cancer stem cell undergoes mitosis

dierentiating into two cancer stem cells with rate bsa1. The probability for this

event is (j − 1)bsa1Sj−1(t)∆t.

• j cancer stem cells are alive at time t and nothing changes. Thus we obtain the

probability (1− j(bsa1 + ds + bsa3)∆t)Sj(t).

All in all we obtain the following equation for cancer stem cells:

Sj(t+ ∆t) = (j + 1)(ds + bsa3)Sj+1(t)∆t+ (j − 1)bsa1Sj−1(t)∆t

+(1− j(bsa1 + ds + bsa3)∆t)Sj(t).

Rearranging the above equation results in the dierence quotient:

Sj(t+ ∆t)− Sj(t)∆t

= (j+1)(ds+bsa3)Sj+1(t)+(j−1)bsa1Sj−1(t)−j(bsa1+ds+bsa3)Sj(t)

and for ∆t→ 0 we obtain the corresponding system of dierential equations

dSj(t)

dt= (j+ 1)(ds + bsa3)Sj+1(t) + (j− 1)bsa1Sj−1(t)− j(bsa1 + ds + bsa3)Sj(t) (6.4)

for j ∈ N with S−1(t) = 0 and Ss0(0) = 1.

Next we check whether the expected value of c(t) =∑∞

i=0 iCi(t) satises the equation

(6.1). We multiply equation (6.3) by i and sum up the values from zero to innity.

Hence we get

dc(t)

dt=

d

dt

∞∑i=0

iCi(t)

=∞∑i=0

idCi(t)

dt

=∞∑i=0

i(i+ 1)dcCi+1(t) +∞∑i=0

i(i− 1)bcCi−1(t)


+∞∑i=0

ibsa2

∞∑j=0

jSj(t)Ci−1(t) +∞∑i=0

ibsa3

∞∑j=0

jSj(t)Ci−2(t)

−∞∑i=0

i2(dc + bc)Ci(t)−∞∑i=0

ibs(a3 + a2)∞∑j=0

jSj(t)Ci(t)

=∞∑k=1

k(k − 1)dcCk(t) +∞∑

k=−1

k(k + 1)bcCk(t)

+∞∑

k=−1

(k + 1)bsa2

∞∑j=0

jSj(t)Ck(t) +∞∑

k=−2

(k + 2)bsa3

∞∑j=0

jSj(t)Ck(t)

−∞∑k=0

k2(dc + bc)Ck(t)−∞∑k=0

kbs(a3 + a2)∞∑j=0

jSj(t)Ck(t)

= −∞∑k=0

kdcCk(t) +∞∑k=0

kbcCk(t) +∞∑k=0

bsa2

∞∑j=0

jSj(t)Ck(t)

+∞∑k=0

2bsa3

∞∑j=0

jSj(t)Ck(t)

= −dcc(t) + bcc(t) + bsa2s(t) + 2bsa3s(t).

Now we verify that the expected value s(t) =∑∞

j=0 jSj(t) satises equation (6.2) in an

analogous manner. Hence we have

ds(t)

dt=

d

dt

∞∑j=0

jSj(t)

=∞∑j=0

jdSj(t)

dt

=∞∑j=0

j(j + 1)(ds + bsa3)Sj+1(t) +∞∑j=0

j(j − 1)bsa1Sj−1(t)

−∞∑j=0

j2(bsa1 + ds + bsa3)Sj(t)

=∞∑k=1

k(k − 1)(ds + bsa3)Sk(t) +∞∑

k=−1

k(k + 1)bsa1Sk(t)

−∞∑k=0

k2(bsa1 + ds + bsa3)Sk(t)

= −∞∑k=0

k(ds + bsa3)Sk(t) +∞∑k=0

kbsa1Sk(t)

= −dss(t)− bsa3s(t) + bsa1s(t).


In these two calculations we used the fact that the sums vanish for non-positive indices.

In order to calculate the TCP we have to compute C0(t) and S0(t) and thus solve

equation (6.3) and (6.4) for Ci(t) and Sj(t). Nevertheless, the master equations for

Ci(t) and Sj(t) form an innite system of ordinary dierential equations which we

cannot solve directly. In order to compute a solution for these equations we make use

of the moment generating functions dened as:

F (h, t) =∞∑i=0

hiCi(t), G(h, t) =∞∑j=0

hjSj(t).

Like in [7] we assume that these functions and their rst order derivatives exist. First we

solve the system for the dierentiated cancer cells, thus we consider F (h, t). Therefore

we multiply equation (6.3) by hi and sum from zero to innity obtaining:

∞∑i=0

hidCi(t)

dt=

∞∑i=0

hi(i+ 1)dcCi+1(t) +∞∑i=0

hi(i− 1)bcCi−1(t)

+∞∑i=0

hibsa2

∞∑j=0

jSj(t)Ci−1(t) +∞∑i=0

hibsa3

∞∑j=0

jSj(t)Ci−2(t)

−∞∑i=0

hii(dc + bc)Ci(t)−∞∑i=0

hibs(a3 + a2)∞∑j=0

jSj(t)Ci(t)

=∞∑k=1

khk−1dcCk(t) +∞∑

k=−1

khk+1bcCk(t)

+∞∑

k=−1

hk+1bsa2

∞∑j=0

jSj(t)Ck(t) +∞∑

k=−2

hk+2bsa3

∞∑j=0

jSj(t)Ck(t)

−∞∑k=0

khk(dc + bc)Ck(t)−∞∑k=0

hkbs(a3 + a2)∞∑j=0

jSj(t)Ck(t)

=∞∑k=0

khk−1dcCk(t) +∞∑k=0

khk+1bcCk(t)

+∞∑k=0

hk+1bsa2

∞∑j=0

jSj(t)Ck(t) +∞∑k=0

hk+2bsa3

∞∑j=0

jSj(t)Ck(t)

−∞∑k=0

khk(dc + bc)Ck(t)−∞∑k=0

hkbs(a3 + a2)∞∑j=0

jSj(t)Ck(t)

= dc∂F (h, t)

∂h+ h2bc

∂F (h, t)

∂h+ hbsa2

∞∑j=0

jSj(t)F (h, t)


+h2bsa3

∞∑j=0

jSj(t)F (h, t)− h(dc + bc)∂F (h, t)

∂h

−bs(a3 + a2)∞∑j=0

jSjF (h, t)

=(dc + h2bc − hbc − hdc

) ∂F (h, t)

∂h+(hbsa2 + h2bsa3 − bs(a3 + a2)

)s(t)F (h, t)

= (h− 1)(hbc − dc)∂F (h, t)

∂h+ (h− 1) (bsa2 + (h+ 1)bsa3) s(t)F (h, t).

All in all we obtain

∂F (h, t)

∂t= (h− 1)(hbc − dc)

∂F (h, t)

∂h+ (h− 1) (bsa2 + (h+ 1)bsa3) s(t)F (h, t) (6.5)

with the initial conditions F (h, 0) = hc0 as we presume to have c0 dierentiated cancer

cells at the beginning of the treatment implying Cc0(0) = 1 and Ci(0) = 0 for all i 6= c0.

Analogously for the cancer stem cells we multiply equation (6.4) by hj and sum from

zero to innity. Thus we obtain

∞∑j=0

hjdSj(t)

dt=

∞∑j=0

hj(j + 1)(ds + bsa3)Sj+1(t) +∞∑j=0

hj(j − 1)bsa1Sj−1(t)

−∞∑j=0

hjj(bsa1 + ds + bsa3)Sj(t)

=∞∑k=1

khk−1(ds + bsa3)Sk(t) +∞∑

k=−1

khk+1bsa1Sk(t)

−∞∑k=0

khk(bsa1 + ds + bsa3)Sk(t)

=∞∑k=0

khk−1(ds + bsa3)Sk(t) +∞∑k=0

khk+1bsa1Sk(t)

−∞∑k=0

khk(bsa1 + ds + bsa3)Sk(t)

= (ds + bsa3)∂G(h, t)

∂h+ h2bsa1

∂G(h, t)

∂h− h(bsa1 + ds + bsa3)

∂G(h, t)

∂h

= (1− h)(ds + bsa3 − hbsa1)∂G(h, t)

∂h.


All in all we obtain

∂G(h, t)

∂t= (1− h)(ds + bsa3 − hbsa1)

∂G(h, t)

∂h(6.6)

with the initial condition G(h, 0) = hs0 because of Ss0(0) = 1 and Sj(0) = 0 for all

j 6= s0.

Next we want to solve equation (6.5) with the corresponding initial conditions via the

method of characteristics. The equations of the characteristic curve are given by:

− dh

(h− 1)(hbc − dc)=dt

1=

dF

F (h− 1) (bsa2 + (h+ 1)bsa3) s.

Hence we have dFdt

= F (h− 1) (bsa2 + (h+ 1)bsa3) s and

dh(t)

dt= (1− h(t))(h(t)bc − dc) = (1− h(t))(bc − dc)− bc(1− h(t))2 (6.7)

with h(0) = hc0 . We choose y(t) = 11−h(t)

in order to transform equation (6.7) into a

linear equation of y(t):

dy(t)

dt=

1

(1− h(t))2

dh

dt= (bc − dc)y(t)− bc, y(0) =

1

1− h0

.

The solution of this ordinary dierential equation can be computed with the help of

the method of variation of parameters and is:

y(t) =

(y(0)− bc

∫ t

0

e−∫ τ0 (bc−dc)dξdτ

)e∫ t0 (bc−dc)dξ

=

(y(0)− bc

∫ t

0

Γc(τ)dτ

)Γc(−t)

with Γc(t) := e−∫ t0 (bc−dc)dξ = e−t(bc−dc). Substituting the denition of y(t) and the

corresponding initial conditions into the previous equation we obtain

1

1− h(t)=

(1

1− h0

− bc∫ t

0

Γc(τ)dτ

)Γc(−t).

If we solve the equation for h0 we get:

h0 = 1− 1Γc(t)

1−h(t)+ bc

∫ t0

Γc(τ)dτ. (6.8)


Now we consider the ordinary dierential equation: dFdt

= F (h−1) (bsa2 + (h+ 1)bsa3) s.

This homogenous linear ODE can again be solved with the help of the method of vari-

ation of parameters resulting in:

F (h(t), t) =F (h(0), 0) exp

(∫ t

0

(h(τ)− 1)(bsa2 + (h(τ) + 1)bsa3)s(τ)dτ

)=hc00 exp

(∫ t

0

(h(τ)− 1)(bsa2 + (h(τ) + 1)bsa3)s(τ)dτ

).

(6.9)

Next we want to nd a new expression for h(τ) − 1 and as the right hand side of

equation (6.8) depends on the full characteristic path, we can replace h(τ) by h(t),

obtaining:

h0 = 1− 1Γc(t)

1−h(t)+ bc

∫ t0

Γc(τ)dτ= 1− 1

Γc(τ)1−h(τ)

+ bc∫ τ

0Γc(ξ)dξ

and thus

h(τ)− 1 = − Γc(τ)Γc(t)

1−h(t)+ bc

∫ tτ

Γc(ξ)dξ

after a few transformations. Inserting this in equation (6.9) this results in an explicit

formula for F (h, t):

F (h, t) =

[1− 1

Γc(t)1−h(t)

+ bc∫ t

0Γc(τ)dτ

]c0·

exp

[∫ t

0

(− Γc(τ)

Γc(t)1−h(t)

+ bc∫ tτ

Γc(ξ)dξ

)·(

bsa2 +

(− Γc(τ)

Γc(t)1−h(t)

+ bc∫ tτ

Γc(ξ)dξ+ 2

)bsa3

)s(τ)dτ

].

(6.10)

Now we want to solve equation (6.6) with the corresponding initial conditions forG(h, t)

with the help of the methods of characteristics. Similar to computing the solution for

F (h, t) the characteristic equations of our partial dierential equation are given by:

dh(t)

dt= (1− h(t))(h(t)bsa1− ds− bsa3) = (1− h(t))(bsa1− ds− bsa3)− bsa1(1− h(t))2

(6.11)

with h(0) = h0 anddG(h(t), t)

dt= 0 (6.12)

with initial conditions G(h0, 0) = hs00 . Thus G(h, t) is a constant function. If we now


choose again y(t) = 11−h(t)

to transform equation (6.11) into an equation for y(t), we

get

dy(t)

dt=

1

(1− h(t))2

dh(t)

dt= (bsa1 − ds − bsa3)y(t)− bsa1, y(0) =

1

1− h0

.

The solution of this ordinary dierential equation is given by

y(t) =

(y(0)− bsa1

∫ t

0

Γs(τ)dτ

)Γs(−t)

with Γs(t) = e−∫ t0 (bsa1−ds−bsa3)dξ = e−t(bsa1−ds−bsa3). Substituting the denition of y(t)

and corresponding initial conditions into the previous equation we obtain

1

1− h(t)=

(1

1− h0

− bsa1

∫ t

0

Γs(τ)dτ

)Γs(−t).

If we solve this for h0 we get

h0 = 1− 1Γs(t)

1−h(t)+ bsa1

∫ t0

Γs(τ)dτ.

All in all if we substitute this in equation (6.12) we obtain the solution for G(h, t):

G(h, t) = hs00 =

[1− 1

Γs(t)1−h(t)

+ bsa1

∫ t0

Γs(τ)dτ

]s0. (6.13)


Combining equations (6.10) and (6.13) we nally obtain the TCP formula:

TCP(t) =C0(t)S0(t) = F (0, t)G(0, t)

=

[1− 1

Γc(t) + bc∫ t

0Γc(τ)dτ

]c0 [1− 1

Γs(t) + bsa1

∫ t0

Γs(τ)dτ

]s0·

exp

[∫ t

0

(− Γc(τ)

Γc(t) + bc∫ tτ

Γc(ξ)dξ

)·(

bsa2 +

(− Γc(τ)

Γc(t) + bc∫ tτ

Γc(ξ)dξ+ 2

)bsa3

)s(τ)dτ

]

=

[1− 1

Γc(t) + bc∫ t

0Γc(τ)dτ

]c0 [1− 1

Γs(t) + bsa1

∫ t0

Γs(τ)dτ

]s0·

exp

[bs

(a3

∫ t

0

s(τ)

(Γc(τ)

Γc(t) + bc∫ tτ

Γc(ξ)dξ

)2

dτ

+ (a2 − 2a3)

∫ t

0

s(τ)

(Γc(τ)

Γc(t) + bc∫ tτ

Γc(ξ)dξ

)dτ

)]

(6.14)

with Γc(t) := e−∫ t0 (bc−dc)dξ = e−t(bc−dc) and Γs(t) := e−

∫ t0 (bsa1−ds−bsa3)dξ = e−t(bsa1−ds−bsa3).

6.3 A TCP Model Based on Stochastic Dierential

Equations

Till now we have derived a TCP formula based on the deterministic model but in the

following we will deduce a TCP formula including stochastic components.

We will denote by τc, τs and τc+s the random times indicating the moment where c(t),

s(t) and (c+ s)(t) rst become zero, respectively. Then the probability that the cancer

becomes extinct is given by

TCP(t) = P(τc+s ≤ t) , t ≥ 0. (6.15)

This implies that the TCP is the cumulative distribution function of the random vari-

able τc+s. As our system of dierential equations is quite complicated we cannot

explicitly compute a formula for the TCP but we can approximate it numerically with

the help of simulations.

Based on (6.15) we can compute the TCP upon simulating a larger number Q of sim-


ulations for the process (c + s)(t) denoting the total number of cancer cells. Thus we

obtain

TCP(t) =number of simulations with τc+s ≤ t

Q, (6.16)

which is a probability that converges to TCP(t) for Q→∞.


In the previous sections we derived two dierent formulas to measure the probability of

tumor control. Now we want to simulate a general TCP-curve for the dierent thera-

pies and treatment strategies and the dierent approaches and analyze their impact on

tumor death. First we will use the TCP formula derived in 6.2 based on the determin-

istic model and simulate the discretization and later on we will compute simulations

of the model including stochastic components.

For reasons of clarity and comprehensibility we shortly review the meaning and values

of the dierent parameters used in the TCP formula. For our standard model without

any therapy they are given in the following table

Parameter Description Value

bc birth rate dierentiated cancer cells (in day−1) ln(2)/6

bs birth rate cancer stem cells (in day−1) ln(2)/6

dc death rate dierentiated cancer cells (in day−1) ln(2)/6

ds death rate cancer stem cells (in day−1) 0

a1 probability for cancer stem cells to dierentiate into two cancer

stem cells

0.7

a2 probability for cancer stem cells to dierentiate into one cancer

stem cell and one dierentiated cancer cell

0

a3 probability for cancer stem cells to dierentiate into two dieren-

tiated cancer cells

0.3

c0 initial fraction of dierentiated cancer cells 0.5

s0 initial fraction of cancer stem cells 0.4

Including therapies in our model alters a few parameters e.g. the death rates become

time dependent and increase due to radiation therapy and the dierentiation rates

of the cancer stem cells are modied by chemotherapy. A summary of all treatment

strategies and parameters can be found in 5.6.


6.4.1 Numerical Simulations for the Discrete Model

In order to simulate a general TCP-curve we need to discretize and thus evaluate the

TCP function (6.14) at sampling times ti where we choose ∆t = ti − ti−1 for all i.

The implementation of the discrete TCP formula turned out to be rather complex so

it needs some preparation to understand the dierent elements of the code. First of all

we will assume that the parameters do not depend on the time variable, which is true

for the second approach for chemotherapy and the system without any therapy. Later

on we will treat the other cases separately.

We will rst of all give the discretization of the functions c(t), s(t), Γc(t) and Γs(t) and

later on derive a discrete formula for the TCP with that. As equations (6.1) and (6.2)

form a system of homogenous ordinary dierential equations we can compute:

c(t) =

(c0 + s0

bs(a2 + 2a3)(e(bs(a1−a3)−ds−bc+dc)t − 1)

bs(a1 − a3)− ds − bc + dc

)e(bc−dc)t

s(t) = s0ebs(a1−a3)t−dst

and like in the previous section we have

Γc(t) = e−∫ t0 bc−dcdξ = e(dc−bc)t

Γs(t) = e−∫ t0 bs(a1−a3)−dsdξ = e(bs(a3−a1)+ds)t.

With this we can now derive the discretization for the TCP formula (6.14):

TCP(ti) =

[1− 1

Γc(ti) + bc∫ ti

0Γc(τ)dτ

]c0 [1− 1

Γs(ti) + bsa1

∫ ti0

Γs(τ)dτ

]s0·

exp

[bs

(a3

∫ ti

0

s(τ)

(Γc(τ)

Γc(ti) + bc∫ tiτ

Γc(ξ)dξ

)2

dτ

+ (a2 − 2a3)

∫ ti

0

s(τ)

(Γc(τ)


Γc(ξ)dξ

)dτ

)]=: exp (log V (ti) + logW (ti)) .

With this denition the problem can be splitted into two subfunctions V (ti) andW (ti).

We begin with an examination of the rst one:

log V (ti) = c0 log

(1− 1

Γc(ti) + bc∫ ti

0Γc(τ)dτ

)


+s0 log

(1− 1

Γs(ti) + bsa1

∫ ti0

Γs(τ)dτ

)=: c0 log (1− χ1(ti)) + s0 log (1− χ2(ti))

χ1(ti) =1

Γc(ti) + bc∫ ti

0Γc(τ)dτ

=:1

Γc(ti) + bczc(ti)

zc(ti) =

∫ ti

0

Γc(τ)dτ =

∫ ti

0

e(dc−bc)τdτ

=e(dc−bc)ti − 1

dc − bcχ2(ti) =

1

Γs(ti) + bsa1

∫ ti0

Γs(τ)dτ

=:1

Γs(ti) + bsa1zs(ti)

zs(ti) =

∫ ti

0

Γs(τ)dτ

=e(bs(a3−a1)+ds)ti − 1

bs(a3 − a1) + ds.

Now we continue with the second exponent in the TCP formula

logW (ti) = bs

(a3

∫ ti

0

s(τ)

(Γc(τ)


Γc(ξ)dξ

)2

dτ

+ (a2 − 2a3)

∫ ti

0

s(τ)

(Γc(τ)


Γc(ξ)dξ

)dτ

)=: bs(a3A(ti) + (a2 − 2a3)B(ti))

A(ti) =

∫ ti

0

s(τ)

(Γc(τ)


Γc(ξ)dξ

)2

dτ

= s0(dc − bc)2

∫ ti

0

e(bs(a1−a3)−ds)τ(

e(dc−bc)τ

dce(dc−bc)ti − bce(dc−bc)τ

)2

dτ

= s0(dc − bc)2

i∑j=1

∫ tj

tj−1


e(dc−bc)τ


)2

dτ

=: s0(dc − bc)2

i∑j=1

Ci(tj)

Ci(tj) =

∫ tj

tj−1


e(dc−bc)τ


)2

dτ


B(ti) =

∫ ti

0

s(τ)

(Γc(τ)


Γc(ξ)dξ

)dτ

= s0(dc − bc)∫ ti

0

e(bs(a1−a3)−ds)τ e(dc−bc)τ

dce(dc−bc)ti − bce(dc−bc)τdτ

= s0(dc − bc)i∑

j=1

∫ tj

tj−1


dce(dc−bc)ti − bce(dc−bc)τdτ

=: s0(dc − bc)i∑

j=1

Di(tj)

Di(tj) =

∫ tj

tj−1


dce(dc−bc)ti − bce(dc−bc)τdτ.

Focusing on the application of cancer therapies some parameters are no longer constants

but depend on the time. Consequently we cannot solve the dierential equations and

the integrals as easily as before.

For the solution of the ordinary dierential equations for our cancer cell populations

c(t) and s(t) we use the implicit Euler method to compute the solution numerically

and the integrals that are too complicated to be solved analytically are approximated

with ∫ b

a

f(t)dt ≈ b− an

n∑i=0

f(ti)

for a = t0 < t1 < . . . < tn = b and ti+1 − ti = 1/n for all i = 0, 1, . . . , n.

For the therapies at intervals we dene the corresponding parameters via indicator

functions. The results are shown in gure 6.1 and 6.2.


(a) (b)

(c) (d)

Figure 6.1: The TCP for the dierent therapy strategies computed with the discretiza-tion like demonstrated above and with the parameters from table 4.1. Figures (a)-(c)show the application of chemotherapy from section 5.1.1, where (a) displays the rstapproach with eq. (5.2) - (5.5), (b) the second one with eq. (5.6) and (c) the secondapproach applied at intervalls. Figure (d) shows the application of radiation therapyfrom section 5.1.2 with eq. (5.7) and (5.8).


(a) (b)

(c)

Figure 6.2: The TCP for the dierent therapy strategies computed via the discretizationlike demonstrated above and with the parameters from table 4.1. Figure (a) and(b) display a combination of chemotherapy and radiation therapy with simultaneousand alternating application, respectively, like in section 5.2 and gure (c) shows theapplication of the time-discrete therapy in section 5.4 with eqs. (5.9) - (5.12).

The TCP is computed at certain times and with the corresponding parameter values.

This leads to the dierent TCP values at intervals for the therapies that are applied

at intervals.

6.4.2 Numerical Simulations for the Stochastic Model

If we now compute simulations for the TCP model based on stochastic dierential

equations like in the previous section 6.3 and with the parameters given in table 4.1

we obtain the results presented in gure 6.3.


(a)

(b)

Figure 6.3: The TCP for the dierent therapy strategies based on stochastic dierentialequations with 104 simulations.

With the discrete formula the TCP converges to values between 0.9 and 1 within about

20-60 days, depending on the type of treatment. The TCP is computed at certain times

and with the corresponding parameter values but does not take the parameter changes

during the therapy into consideration. This leads to the dierent TCP values at in-

tervals for the therapies that are applied at intervals. In the stochastic approach the

TCP continuously converges to 1 in a very short period of time but takes the changes

of parameter values into account.

Comparing the two approaches we can see that both describe the tendencies of the


TCP properly and as it was expected from the simulations of the dierent treatment

strategies in chapter 5. Nevertheless, the discrete approach does not take the behavior

of the dierentiated cancer cells c into consideration due to the set up of the formula.

Thus the stochastic approach is more accurate, as it comprises the change in param-

eters, the behavior of the dierentiated cancer cells and it is based on the solution of

the stochastic dierential equations instead of the ordinary dierential equations.

69

7 The Model Extended by Normal

Cells

Up to now we have only considered a model for cancer cells without incorporating

healthy tissue cells. Nevertheless, these cells from the surrounding non cancerous tissue

are very important for a realistic model. On the one hand, these cells interact with the

cancer cells and are being killed by the malignant cells and on the other hand the eect

of treatment strategies on normal cells is important because the aim is to preserve as

many normal cells as possible by reducing the side eects for the patient.

Thus we will in the following deduce a stochastic model for cancer cells interacting

with normal tissue cells, by extending the model from section 4.1.

Therefore let n(t) be the number of normal cells at time t and c(t) and s(t) the number

of dierentiated cancer cells and cancer stem cells at time t, respectively. Furthermore,

we need a few assumptions for the behavior of normal cells:

1. There are no mutations, which implies that no normal cells turn into cancer stem

cells or dierentiated cancer cells.

2. The birth rate of normal tissue cells is limited by their carrying capacity.

3. The death rate of normal cells depends on the interactions with the cancer cells.

4. The number of normal tissue cells is bounded from below.

Thus, the behavior of the normal cells does not depend on the cancer cells except for

the death rate that in turn depends on (c + s)(t). Nevertheless, the equation for n(t)

is decoupled from the other equations for c(t) and s(t), so that we can deduce the one

for n(t) separately.

For the birth rate bn and the death rate dn of n(t) and a small interval of time ∆t we

can note the respective transition probabilities for n(t):


Changes Probability

∆n(1) = 1 p1 = bnn∆t

∆n(2) = −1 p2 = dnn∆t

∆n(3) = 0 p3 = 1− p1 − p2

With this we can derive analogously to section 4.1 the stochastic dierential equation

for the normal cells:

dn(t) = (bn − dn)n(t)dt+√

(bn + dn)n(t)dW (t) (7.1)

with initial conditions n(0) = n0 and the Wiener process W (t). The death rate dn is

dened as

dn(t) = h(t) + δ(t)(c(t) + s(t)) (7.2)

consisting of the death rate due to radiation and natural death and the interaction with

the interaction factor δ(t) between the normal cells and the tumor cells. The birth rate

depends on an organ specic carrying capacity M [12]:

bn(i) =

µ(1− i

M

), if i = 1, . . . ,M

0, otherwise, (7.3)

where bn(i) denotes the birth rate if i normal cells are alive. This implies that the

normal cell growth is limited by growth factors like space and nutrient supply. Due

to this denition an increasing population size leads to a decreasing birth rate. With

this choice of bn(i) the number of normal cells would always stay below the carrying

capacity M if we would only examine the deterministic model. However, including the

stochastic component with the Wiener process can also lead to a lager population size.

As we have used volume fractions for the cancer populations c(t) and s(t) we also have

to consider volume fractions for the normal cells n(t). Therefore we nondimensionalize

equation (7.1) and the corresponding parameters. With n(t) = n(t)N

for a reference size

N we obtain

dn(t) = (bn − dn)n(t)dt+

√(bn + dn)n(t)dW (t)

with bn = µ

(1− n(t)

M

), dn(t) = h(t) + δ(t)(c(t) + s(t))

and µ = Nµ, M =M

N, h = Mh and δ = Mδ.

(7.4)


These equations become particularly simple for N = M . With this choice we obtain

the following nondimensionalized parameters

µ = Mµ, M = 1, h = Mh, δ = Mδ. (7.5)

For reasons of simplicity we will drop the tilde in the notations of equations (7.4) and

(7.5) in the following.

Now let p(t, n) be the probability function at time t for the normal tissue cells n(t)

which is the discrete counterpart of the density function of continuous random vari-

ables. Analogously to section 4.1 we can deduce a Fokker-Planck equation for p(t, n),

obtaining

∂p

∂t(t, n) = − ∂

∂n((bn − dn)n(t)p(t, n)) +

1

2

(∂2

∂n2((bn + dn)n(t)p(t, n))

).

7.1 Existence of Solutions

Like in chapter 4 where we deduced the model for cancer stem cells and dierentiated

cancer cells we also have to prove the existence of a solution for our newly derived

stochastic dierential equation (7.1) for normal tissue cells.

Therefore we refer to theorem 1 in section 4.2 and will prove the two conditions for

equation (7.1). We will start with the global Lipschitz condition.

First we want to prove ||(bn− dn)n1− (bn− dn)n2||2 ≤ K||n1− n2||2 for a constant K:

||(bn − dn)n1 − (bn − dn)n2||2 = (bn − dn)2n21 + (bn − dn)2n2

2 − 2(bn − dn)2n1n2

= (bn − dn)2(n21 + n2

2 − 2n1n2)

= (bn − dn)2︸︷︷︸=:K

(n1 − n2)2

= K||n1 − n2||2

where K is bounded because bn and dn are bounded and c(t), s(t) ≤ R, which we also

assumed in section 4.2. Next we will prove the global Lipschitz condition for the second

part of our stochastic dierential equation with the help of theorem 2 in section 4.2.

The theorem states that

||f(b)− f(a)|| ≤ supx∈ab||Dxf || · ||b− a||,


with the Jacobian matrix Dxf of f . Here we have f(x) :=√

(bn + dn)x and we want

to prove ||√

(bn + dn)n1 −√

(bn + dn)n2||2 ≤ K||n1 − n2||2. If we now have a closer

look at the Jacobian matrix, we obtain

Dxf =1

2

√bn + dnx

.

As the variables bn, dn are positive and the number of normal cells is bounded from

below, implying n ≥ ε for ε > 0, Dxf is positive and independent from t. All in all we

obtain

||√

(bn + dn)n1 −√

(bn + dn)n2||2 ≤bn + dn

4ε||n1 − n2||2.

For the linear growth condition we have to show that ||(bn−dn)n||2+||√

(bn + dn)n||2 ≤K(1 + ||n||2) for a constant K:

||(bn − dn)n||2 + ||√

(bn + dn)n||2 = (bn − dn)2︸︷︷︸≥0

n2 + (bn + dn)n

≤ (bn − dn)2n2 + (bn − dn)2 + (bn + dn)n

= (bn − dn)2(1 + n2) + (bn + dn)n

≤ (bn − dn)2(1 + n2) + (bn + dn)(1 + n2)

≤ max(bn − dn)2, bn + dn︸︷︷︸=:K

(1 + n2)

= K(1 + ||n||2).

where we used |z| ≤ 1 + z2. Thus our stochastic dierential equation (7.1) fullls all

the conditions of theorem 1 and consequently we obtain the existence of solutions.


In order to simulate our stochastic dierential equation (7.1) we will refer to [33] for the

choice of parameters. However, the parameters listed below vary from those in [33] due

to the nondimensionalization in equations (7.4) and (7.5) and as we slightly changed

the birth-death rate ratio because the given values led to a instantaneous extinction of

the population even without the impact of cancer cells.



µ constant maximum birth rate (in day−1) 0.05

h natural death rate (in day−1) 0.003

δ interaction factor 0.01

n0 initial number of cells 1

M carrying capacity 500

M nondimensionalized carrying capacity 1

Table 7.1: A summary of the model parameters for normal tissue cells based on [33].

To compute numerical solutions we apply again the Euler-Maruyama method as de-

scribed in section 3.4.1 and the results are shown in gure 7.1. These simulations show

Figure 7.1: Dierent realizations of the stochastic dierential equation (7.1) and theirmean value (black line) of 104 trajectories solved with the Euler-Maruyama algorithmwith parameter values like in table 7.1. Figure (a) shows the behavior of normal tissuecells without the impact of cancer cells and (b) includes the cancer cells model fromchapter 4. The dotted grey line describes the threshold population.

that without any therapy the presence of cancerous cells increases the death rate of

the healthy cells.

Even though cancer treatment also reduces the number of normal tissue cells, most

organs are still able to work properly with only a fraction of the initial number of cells.

As we choose parameter values for head and neck cancer in the previous sections these

organs function properly with 60-70 % (we choose 65%) of healthy tissue cells.

Thus the organ infested with cancer cannot work properly any more because the num-

ber of tissue cells drops below the threshold after about 45 days. This leads to a

function loss and severe problems for the patient.


7.3 Approximation of the Persistence Time

An important aspect of the behavior of populations is their long-term performance, in

particular their persistence time. Like in section 4.5 for the system of cancer cells we

are going to have a closer look at the mean time of persistence of the normal tissue

cells.

Analogously to that we can approximate the persistence time by solving the correspond-

ing Kolmogorov backward equation or simulate numerical solutions of our stochastic

dierential equations. As we have already noted in subsection 4.5.3, the discrete ap-

proach with the corresponding Kolmogorov backward equation is not as exact as the

numerical approach and the error made is rather large. Consequently, we are only

focusing here on the more exact simulations of numerical solutions to our stochastic

dierential equations.

In order to compute the average persistence time numerically, we will simulate the

population behavior via the stochastic dierential equation (7.1) and trace it till the

density falls below the threshold where the organ cannot function properly any more.

For head and neck cancer this threshold is 65% of the maximal cell number for normal

tissue cells. Generating 104 trajectories with the Euler-Maruyama method and aver-

aging the measured persistence times with the choice of parameters like in table 7.1,

we obtain the following estimations depending on the initial population size:

T (0.65) = 4.6762 T (0.9) = 25.6802

T (0.7) = 8.6839 T (1) = 33.0047 .

T (0.8) = 17.8969

Comparing the persistence time T (1) = 33.0047 with the simulation in gure 7.1 we

can observe a discrepancy between the persistence times. In the simulation the number

of normal cells drops below the threshold γ = 0.65 after approximately 45 days. This

variation is caused by the fact that the computation of the persistence time is stopped

as soon as a trajectory rst reaches the threshold. In the simulations we suppose that

an organ can recover, which means that trajectories that have once dropped below

the threshold γ, but remained positive, have the chance to reach values above the

threshold again. This raises the mean value and explains the higher persistence time

in the simulations.


7.4 The Eect of Cancer Treatment on Normal Cells

Cancer treatment does not only aect cancer cells but can also harm normal tissue cells,

depending on the treatment strategy. Chemotherapy that aims at the destruction of

fast-growing cells has also a negative impact on normal cells whereas chemotherapy that

only inuences the dierentiation should not harm the normal tissue cells. Radiation

therapy aects all types of cells.

Without any treatment the number of tissue cells will fall below the threshold γ = 0.65

after a short period of time, which can be observed in gure 7.1. The averaged time

where the organs with an initial population size of 1 cannot function properly any more

is 32.7937 days and it was computed like the persistence time in the previous section.

The application of chemotherapy does not explicitly alter our model for normal cells

because we only focus on dierentiation therapy for caner stem cells here. But the

dierentiation indirectly aects the normal cells, as this therapy reduces the number

of cancer cells and thus decreases the death rate of the normal cells. Simulations of

the behavior of normal cells with the applications of chemotherapy and the dierent

approaches from section 5.1.1 are shown in gures 7.2 and 7.3.

Figure 7.2: Dierent realizations of the stochastic dierential equation (7.1) and theirmean value (black line) for 104 trajectories with the application of the rst approach tochemotherapy solved with the Euler-Maruyama algorithm with parameter values likein tables 7.1.


(a) (b)

(c) (d)

Figure 7.3: Dierent realizations of the stochastic dierential equation (7.1) and theirmean value (black line) for 104 trajectories with the application of the second approachto chemotherapy solved with the Euler-Maruyama algorithm with parameter valueslike in tables 7.1. Figure (a) shows simulations for the therapy from section 5.1.1 forα = 0.5, (b) for α = 0.6 and (c) and (d) the chemotherapy applied at intervals againfor α = 0.5 and α = 0.6, respectively.

For all kinds of chemotherapy the patient survives and the organ maintains its function.

Depending on the type and the intensity of the treatment the number of surviving

normal cells varies after 100 days of therapy. As chemotherapy at intervals leads to a

higher number of cancer cells after 100 days, this treatment strategy causes a reduction

of the normal cells when compared to the continuous application of chemotherapy.

Radiation therapy increases the death rate of the normal tissue cells, adding a third

component to (7.2) [7]. Hence if we add the death rate due to radiation r(t) (also


known as hazard rate) we obtain

dn(t) = h(t) + δ(t)(c(t) + s(t)) + r(t)

r(t) = (α + 2βdt)d(7.6)

for the death rate of the normal cells with the sensitivity parameters α and β and the

radiation dose D.

For the simulations we use the parameter values from table 7.1 and the following values

which are based on [12]:


α sensitivity parameter (in Gy−1) 0.003

β sensitivity parameter (in Gy−2) 0.001

d treatment dose (in Gy) 2.5

γ number of cells the organ needs to work properly 0.65 ·M

Table 7.2: The model parameters for the radiation of normal tissue cells based on [12].

Figure 7.4 shows a simulation for the behavior of the normal cells with radiation therapy

applied as well as the dierent combinations of chemotherapy and radiation therapy

and the time-discrete therapy. The decrease of normal tissue cells due to radiation

can be clearly observed in these simulations. The loss of normal cells depends on

the treatment strategy. Here radiation therapy and the simultaneous application of

chemotherapy and radiation therapy lead to critical values.

In order to compare the impact and the eectiveness of the dierent treatment strate-

gies we have a closer look at the respective persistence times, in this case the time when

the number of normal cells comes below the threshold γ. These times can be computed

via numerical simulations of the stochastic dierential equations like in section 4.5.2.

The persistence times we computed with the parameters chosen like in the previous

sections of this chapter and a simulation with 104 trajectories are displayed below in

table 7.3.


(a) (b)

(c) (d)

Figure 7.4: Dierent realizations of the stochastic dierential equation (7.1) and theirmean value (black line) of 104 trajectories solved with the Euler-Maruyama algorithmwith parameter values like in tables 7.1 and 7.2. Figure (a) shows the impact ofradiation therapy, (b) the simultaneous application of chemotherapy and radiationtherapy, (c) the alternating application of those therapies and (d) the application ofthe time-discrete therapy.


Therapy Persistence Time

without any therapy 32.7937

dierentiation therapy (5.1.1, rst approach) 43.8361



dierentiation therapy in intervals 36.6619

radiation therapy (5.1.2) 31.1387

combining chemotherapy and radiation therapy simul-

taneously (5.2)

29.0166

combining chemotherapy and radiation therapy alter-

nately (5.2)

34.1356

time-discrete therapy (5.4) 35.0925

Table 7.3: Persistence times of the normal tissue cells for the dierent treatment strate-gies.

If not stated dierently we used α = 0.5 for these simulations. These persistence times

conrm the simulations of the dierent treatment strategies from the previous pages.

80

8 Normal Tissue Complication

Probability

The aim of an eective cancer treatment is not only to eradicate the cancer cells but

also to preserve the normal, healthy tissue cells.

In the previous chapter we have discussed models for the tumor control probability but

now we will focus on models that describe the damaging impact of treatment strategies

on the surrounding healthy tissue.

A quality measure for this damaging eect is the Normal Tissue Complication Prob-

ability (short NTCP) which describes the probability that severe complications occur

in the surrounding normal tissue [21]. Thus the overall goal of an optimal treatment is

to maximize the TCP and simultaneously minimize the NTCP. These conicting aims

can be illustrated by dose-response curves for the cancerous and the healthy tissue

respectively which is visualized in gure 8.1 [21]. Here an increasing radiation dose

leads to a rise in the TCP and with an even higher dose to an increase in the NTCP.

Both values converge to one for a rising dose but the TCP value always remains above

the NTCP value due to the higher radiosensitivity and a lack of repair mechanisms

of the cancer cells. Combining these two values by multiplying TCP and 1-NTCP we

obtain an expression that describes the probability for tumor control and no normal

tissue complication. The maximum of this expression is the optimal dose to maximize

the TCP and simultaneously minimize the NTCP. In the following we will review com-

mon NTCP models like the Lyman model and the critical volume NTCP model. Later

on we will derive a formula for the NTCP via the birth and death processes and one

formula based on a model involving stochastic dierential equations.

8.1 The Lyman Model

The model proposed by Lyman in 1985 is one of the simplest models for NTCP. It is

a method to estimate the complication probability of normal tissue is developed from


Figure 8.1: The dose-response curve for TCP and NTCP and the probability for tumorcontrol without normal tissue complications (dotted line).

dose-volume histograms and data from cancer treatment patients [24].

Therefore organ specic tolerance doses TDi are introduced which describe the dose

that would result in i % complication probabilities after 5 years. These doses are

functions depending on the percentage volume of the organ irradiated and the absorbed

dose received by this volume [24]. For a given organ the tolerance dose of a fractional

part V ∈ [0, 1] is given by

TDi(V ) = TDi(1)V −n for i ∈ [0, 100]

where TDi(V ) is the tolerance dose for a given fractional volume V , TDi(1) is the

tolerance dose for the full volume and n ∈ [0, 1] is a tted parameter [24]. In the

Lyman model the complication probability for a uniform irradiation of a normal tissue

volume V is a dose dependent integral of a normal distribution:

NTCPLyman(D) =1

σ√

2π

∫ D

−∞e−

(t−µ)2

2σ2 dt (8.1)

with the mean value µ = TD50 and the standard deviation approximated by σ =

mTD50, where m is a parameter that governs the slope of the function. Rescaling by

z = D−TD50

σ, equation (8.1) is reduced to a standard normal distribution:

NTCPLyman(D) =1√2π

∫ D−TD50σ

−∞e−

z2

2 dz. (8.2)

It is obvious that this formula for NTCP is completely dened by the parameters

TD50(1), n and m. A simulation for the irradiation of the heart where we set TD50 =


41.8, n = 0.5 and m = 0.1 like in [24] is shown in gure 8.2. As the volume V is varying

we obtain a three-dimensional surface.

Figure 8.2: The probability of complication for the heart depending on the dose andthe partial volume that is uniformly irradiated.

However the model parameters are obtained from literature search and clinical experi-

ence [9], where only a few tolerance doses were used to t the model parameters. Thus,

the derived model has to be treated with caution. Furthermore, the normal tissue will

not be uniformly irradiated in clinical practice, due to the emergence of new tech-

nologies like three-dimensional CT scanned images enabling computerized treatment

planning.

8.2 The Critical Volume NTCP Model

A further deterministic NTCP model that is closely related to the Lyman model in

the previous section is the critical volume NTCP model rst introduced in 1992 by

Niemierko et al. [25] based on the work of Withers et al. [37].

Here it is assumed that tissue can be described as a composure of functional subunits

(FSUs) that are dened either structurally or functionally. An example for structural

FSUs is the kidney that is composed of a large number of nephrons that are identical

in their setup and function independently [37]. FSUs are the maximum volume or area

that can be repopulated by only one clonogenic cell [37].

In the following we assume that the organ of interest consists of N FSUs which are


identical, independent from each other and uniformly distributed through the organ.

Furthermore each FSU consists of N0 cells. With PFSU(D) we denote the probability

that one FSU is killed after the application of cancer treatment of dose D. To calculate

this probability we use the complement of the LQ-model from section 5.1.2 to describe

the cell death within one FSU:

PFSU(D) =(

1− e−(αD+βD2))N0

.

If we now assume fractionated radiation with x fractions of dose d we obtain

PFSU(D) =(

1− e−αD(1+ dα/β

))N0

with∑d = D. As a single clonogenetic cell is able to regenerate the FSU of its origin,

all N0 cells have to be destroyed in order to eradicate one FSU.

The random variable describing the probability that i FSUs are killed after applying

a dose D is binomial distributed to the number of FSUs N in the organ and the

probability PFSU(D) leading to

Pbin(i) =

(N

i

)PFSU(D)i(1− PFSU(D))N−i. (8.3)

Next we assume that the organ loses its normal functionality if more than M −1 FSUs

are killed. For example the kidney the normal renal function can be maintained with

30 % - 50 % of healthy nephrons [25].

The probability that at least M of the N FSUs are destroyed is given by:

NTCPcv(D) =N∑i=M

Pbin(i) (8.4)

=N∑i=M

(N

i

)PFSU(D)i(1− PFSU(D))N−i. (8.5)

The above expression is the cumulative binomial probability. For a large number of

FSUs the distribution approaches the normal distribution according to the central limit

theorem [3]. Thus we obtain that (8.3) converges to

Pnorm(i) =1

σ√

2πe−

12( i−µσ )

2

(8.6)


with µ = EX = NPFSU(D) and σ = VX = NPFSU(D)(1 − PFSU(D)). With this we

can approximate the NTCP formula (8.4) as follows

NTCPcv(D) =N∑i=M

Pbin(i) ≈∫ ∞M

Pnorm(i)di

=1

σ√

2π

∫ ∞M

e−12( i−µσ )

2

di.

Rescaling with t = i−µσ

we get

NTCPcv(D) =1√2π

∫ M−µσ

−∞e−

t2

2 dt (8.7)

with the symmetry of the normal distribution. This is the same formula we had in the

Lyman model, compare equation (8.2).

If organs have a serial architecture which implies that the organ can only survive when

all FSUs survive, we have the special case of M = 1. Equation (8.7) then reduces to

the critical element NTCP

NTCPce(D) =N∑i=1

Pbin(i) = 1− Pbin(0) = 1− (1− PFSU(D))N .

Another special case is when the organ survives if at least one FSU survives, implying

M = N . In this case equation (8.7) reduces to

NTCPcv(D) = PFSU(D)N .

8.3 NTCP Model Based on Birth and Death

Processes

Analogously to the TCP model in section 6.2 we now want to derive an NTCP formula

where the probability for cell survival is based on birth and death processes. The fol-

lowing derivation is the analogue to the TCP by Zaider and Minerbo [39] and is based

on Gong [12].

Therefore we assume that all healthy tissue cells are identical and independent through-

out the organ and that an organ works properly if it consists of more than L intact

cells. By Pi(t) we denote the probability that i normal cells are alive at time t. The


NTCP, the probability that an organ cannot function properly anymore is thus given

by

NTCPZM(t) =L∑i=0

Pi(t) for i, t ≥ 0.

Like in section 6.2 we can derive the master equation for Pi(t) and obtain

dP0(t)

dt=dn(t)P1(t)

dPi(t)

dt=bn(i− 1)(i− 1)Pi−1(t) + dn(t)(i+ 1)Pi+1(t)− (bn(i) + dn(t))iPi(t)

(8.8)

with the birth rate bn(i) for i ≥ 1, a time dependent death rate dn(t) and the initial

values Pn0(0) = 1 and Pi(0) = 0 for i 6= n0 where n0 denotes the initial population size

of the normal tissue cells at time t = 0.

In the following we will use the denition (7.3) for the birth rate and (7.2) for the death

rate.

For the TCP in the previous chapter 6.2 we were only interested in P0(t) which we

computed by solving the system of ordinary dierential equations analytically. Now

for the NTCP we are interested in the solutions for Pi(t) for i = 1, . . . ,M . Previously

to solving this system analytically or numerically, we have to make sure that the system

is nite and thus solvable. Like in [12] we rst prove that the system is nite by proving

the following lemma.

Lemma 8.3.1. Assume bn(i) is given by (7.3) and provided the series∑∞

i=0 iPi(t)

converges for all t ≥ 0. If Pi(0) = 0 for i ≥ M + 1, then Pi(t) = 0 for i ≥ M + 1 for

all t > 0 which implies that the system is nite.

Proof. We dene Rj(t) :=∑∞

i=j Pi(t) for j ≥M+1. Because of Pi(0) = 0 for i ≥M+1

we also get Rj(0) = 0 for j ≥M + 1. Since bn(M) = 0 we obtain

dRM+1(t)

dt=

∞∑i=M+1

dPi(t)

dt= bn(M)MPM(t)− dn(t)(M + 1)PM+1(t) ≤ 0.

From this and the fact that Rj(0) = 0 it follows that RM+1(t) ≤ 0. But RM+1(t) is

also a sum of probabilities and thus RM+1(t) ≥ 0. All in all we have RM+1(t) = 0.

Similarly, we can prove that Rj(t) = 0 for j > M + 1. Putting this together we obtain

Pj(t) = Rj(t)−Rj+1(t) = 0 for j ≥M + 1.


We can now prove that bn(i) will make the solution of the mean eld equation obey

logistic growth under certain assumptions.

Theorem 3. Assume bn(i) is dened as in (7.3). Provided the series

N(t) =∞∑i=0

iPi(t)

converges, then N(t) is the mean eld function of the system (8.8) and satises the

dierential equation

dN(t)

dt= µN(t)

(1− N(t)

M

)− dn(t)N(t)− µ

MV(X) (8.9)

where V(X) is the variance of the normal tissue cells dened by V(X) = E(X−N(t))2

and N(t) = E(X).

Proof. The proof is analogous to the one for theorem 6.3.2 in [12].

8.3.1 Numerical Solution for the Finite Dimensional System

Now that we have proven that the system of dierential equations (8.8) is nite, we

can compute the results numerically. Like in [12] we dene P (t) = (P0(t), . . . , PM(t))t

so that the system (8.8) can be written as

dP (t)

dt= AP (t) (8.10)

with the matrix A given as

A =

0 dn 0 . . . . . . 0

0 −(bn(1) + dn) 2dn 0...

0 bn(1) −2(bn(2) + dn) 3dn. . .

......

. . . . . . . . . 0...

. . . . . . Mdn

0 · · · · · · 0 (M − 1)bn(M − 1) −M(bn(M) + dn)

.

The solution of the ODE-system can be computed with parameter values chosen ac-

cording to tables 7.1 and 7.2. We used the matlab-solver ode45 in order to solve this

ODE-system and the solution ist shown in gure 8.3.


Figure 8.3: NTCP-curve for the numerical solution of the nite dimensional system(8.10).

8.4 An NTCP Model Based on Stochastic

Dierential Equations

Up to now we have only derived an NTCP formula based on the deterministic model

but in the following we will deduce, like in section 6.3 for TCP, an NTCP formula

including stochastic components.

We will denote by τn the random times indicating the moment when n(t) rst drops

below a specic threshold value γ. This threshold γ describes the minimum number of

normal cells that are necessary for a specic organ to function properly. If the number

of normal cells drops below γ the organ loses its function. Then the probability that

severe complications occur in the healthy tissue is given by:

NTCP(t) = P(τn ≤ t) t ≥ 0. (8.11)

This implies that the NTCP is the cumulative distribution function of the random

variable τn. As our dierential equations are quite complicated we cannot explicitly

compute a formula for the NTCP but we can approximate it numerically.

Based on (8.11) we can compute the NTCP upon simulating a larger number Q of

trajectories for the process n(t) denoting for the number of healthy tissue cells. Thus

we obtain

NTCP(t) =number of simulations with τn ≤ t

Q, (8.12)


which converges to NTCP(t) for Q → ∞. Simulating the NTCP via the model based

on stochastic dierential equations with the parameters given in tables 7.1 and 7.2 we

obtain the results presented in gure 8.4. These simulations correspond to the numer-

Figure 8.4: The NTCP for the dierent therapy strategies based on stochastic dier-ential equations with 104 simulations.

ical simulations of the treatment eect on normal cells in chapter 7.4. Chemotherapy

reduces the complications of normal tissue because this therapy only reduces the num-

ber of cancer cells but does not harm the normal cells. Radiation therapy and any

combinations of it also aect normal cells and reduce their number.

89

9 Uncomplicated Tumor Control

Probability

In chapter 6 and 8 we have computed and estimated values for the TCP and NTCP with

dierent models. For an eective treatment we do not only have to increase the TCP

but also keep the NTCP at a minimum. Therefore we introduce the uncomplicated

tumor control probability (UTCP) which is a general expression for the probability of

achieving complication-free tumor control [20].

In general, the UTCP can be described as the probability for tumor control (denoted

with P (C)) minus the probability that the patient suers severe injuries (denotes with

P (I)) but is controlled:

UTCP = P (C)− P (C ∩ I) (9.1)

where P (C ∩ I) denotes the probability that the patient is both controlled and suers

severe injuries [1]. With the multiplication law of statistics we can rewrite (9.1) as

UTCP = P (C)− P (C)P (I|C) = P (C)(1− P (I|C))

where P (I|C) is the conditional probability for injury provided the tumor has been

controlled. When the event of an injury I and the event of tumor control C are

statistically independent we obtain P (I|C) = P (I). Thus we obtain

UTCP = P (C)(1− P (I))

which is

UTCP = TCP (1− NTCP) (9.2)

with our notations from the previous chapters. UTCP just summarizes the results form

the simulations in chapter 6 for TCP and chapter 8 for NTCP. Obviously radiation

therapy and the combination with chemotherapy result in the highest UTCP of about

0.68 but only for an application period of approximately 20 days. Longer treatment

9 Uncomplicated Tumor Control Probability 90

Figure 9.1: The UTCP for the dierent treatment strategies based on the simulationsof the stochastic dierential equation with 104 simulations.

times cause a great damage by killing normal cells. For chemotherapy the UTCP

varies between 0.35 and 0.42 at the peak which is reached after 12 days. For a longer

treatment the UTCP value drops but remains above 0.15.

As we can observe in gure 9.1 the radiation therapy and the simultaneous combination

with chemotherapy result in the highest UTCP. However, the treatment time is a very

sensible issue for these strategies. So a good choice of treatment would be to apply

radiation therapy for a period of 10-18 days and then switch to another therapy, like the

alternating application of chemotherapy and radiation therapy. With this treatment

schedule we would obtain the optimal treatment eect with the maximal UTCP for

the dierent possibilities we considered here.

91

10 Conclusion and Outlook

In this thesis we derived and analyzed a stochastic model for cancer progression includ-

ing the time-dependent behavior of cancer stem cells, dierentiated cancer cells and

normal tissue cells.

In the rst part (chapter 4) we derived and analyzed the model only for the two types

of cancer cells. Here we proved the existence of a solution of the SDE-system and

performed numerical simulations with the Euler-Maruyama and the Milstein (a higher

order) method. These simulations turned out to be slightly problematic because not all

computed trajectories were positive. But this problem was solved by setting all trajec-

tories, that become negative, equal to zero. Once a trajectory has reached zero it will

maintain this status. Subsequently we analyzed the SDE-model with respect to the per-

sistence time. Therefore we compared the computations of the deterministic approach

with the Kolmogorov backward equation with the results of the numerical approxima-

tion of the persistence time. The error made by the Kolmogorov backward equation is

rather big and the persistence time is basically overestimated in this approach. Thus

the stochastic approach is more accurate and can help to prevent unnecessarily long

cancer treatment of the patient.

Next we introduced various therapies and treatment strategies including dierent ap-

proaches to chemotherapy, radiation therapy, two common ways of combining these

two and a radiation therapy that is applied at discrete points in time. In chapter 5 we

discussed these treatment strategies in detail and studied their impact on cancer stem

cells and dierentiated cancer cells with the help of simulations and persistence times.

In chapter 6 we gave an overview of existing TCP models. We derived and solved

dierent discrete models of increasing complexity resulting in a two compartment model

that was based on birth and death processes. This model was then compared to

the stochastic approach. We also simulated TCP-curves for the dierent treatment

strategies and again noticed that the stochastic approach to TCP is more accurate and

also takes the change of parameters during the treatment into consideration.

10 Conclusion and Outlook 92

After focusing on the behavior of cancer cells and the impact of therapy on them we

also derived a stochastic model for normal tissue cells. This model was studied in

chapter 7 analogously to the SDE-system of the cancer cells in chapter 4 with respect

to the existence of a solution, numerical simulations, persistence time and the impact

of the dierent treatment strategies introduced in chapter 5 on the normal cells. In the

subsequent chapter we transferred all TCP cases to healthy tissue to obtain the NTCP.

A detailed overview about the existing deterministic models was given, including the

NTCP based on Lyman and the NTCP critical volume model. We also reviewed how

Gong adapted the TCP model from Zaider and Minerbo to normal tissue cells. For all

these approaches to NTCP we computed numerical simulations and compared them to

the stochastic approach to NTCP.

Finally we combined the TCP and the NTCP in the UTCP in order to nd the optimal

treatment time achieving complication-free tumor control in chapter 9.

What is left to do is to account for spacial heterogeneity in the tumor cell migration.

Other aspects we did not take into consideration were the presence of non cancerous

stem cells that contribute to the recovery of healthy tissues and interactions with the

immune system.

93

List of Figures

2.1 Lung cancer cell and normal epithelial lung cells, source: http://www.

sciencephoto.com/media/254134/enlarge . . . . . . . . . . . . . . . 4

2.2 A simplied model about the origin of the cancer stem cells, source:

http://www.cancerci.com/content/7/1/9/figure/F2?highres=y . . 5

2.3 The structure of DNA with the nucleotides as the smallest units, source:

http://pubs.niaaa.nih.gov/publications/aa86/aa86.htm . . . . . 6

4.1 The schematic interaction of dierentiated cancer cells and cancer stem

cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Dierent realizations of the SDE with the Euler-Maruyama algorithm,

computed with Matlab 30.06.2014 . . . . . . . . . . . . . . . . . . . . . 23

4.3 Dierent realizations of the SDE with the implicit Milstein method,


4.4 The solution of the Kolmogorov backward equation (4.8), computed with

Matlab 29.07.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.1 The rate a1(t, ψ) as a function of the sensitivity ψ, computed with Mat-

lab 10.04.2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Numerical solutions of cF and CF , computed with Matlab 30.07.2014 . 34

5.3 A simulation with the application of chemotherapy (dierentiation ther-

apy), computed with Matlab 31.07.2014 . . . . . . . . . . . . . . . . . . 35

5.4 A simulations with the application of chemotherapy, computed with

Matlab 31.07.2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.5 A simulation with the application of chemotherapy including therapy

breaks, computed with Matlab 31.07.2014 . . . . . . . . . . . . . . . . 37

5.6 A simulation with the application of radiation therapy, computed with

Matlab 31.07.2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.7 A simulation with the simultaneous application of radiation therapy and

chemotherapy, computed with Matlab 31.07.2014 . . . . . . . . . . . . 40

http://www.sciencephoto.com/media/254134/enlarge

http://www.sciencephoto.com/media/254134/enlarge

http://www.cancerci.com/content/7/1/9/figure/F2?highres=y

http://pubs.niaaa.nih.gov/publications/aa86/aa86.htm

List of Figures 94

5.8 A simulation with the alternating application of radiation therapy and

chemotherapy, computed with Matlab 31.07.2014 . . . . . . . . . . . . 41

5.9 Simulation with a time-discrete therapy, computed with Matlab 31.07.14 43

6.1 The TCP for chemotherapy and radiation therapy computed via the

discretization, computed with Matlab 01.08.2014 . . . . . . . . . . . . . 65

6.2 The TCP for the dierent therapy strategies computed via the discretiza-

tion, computed with Matlab 01.08.2014 . . . . . . . . . . . . . . . . . . 66

6.3 Simulations for the TCP model based on SODEs, computed with Matlab

24.08.2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.1 Dierent realizations of the SDE for normal cells with the Euler-Maruyama

algorithm, computed with Matlab 02.08.2014 . . . . . . . . . . . . . . . 73

7.2 Dierent realizations of the SDE for normal cells with the application of

the rst approach to chemotherapy computed with the Euler-Maruyama

algorithm, computed with Matlab 02.08.2014 . . . . . . . . . . . . . . . 75

7.3 Dierent realizations of the SDE for normal cells with the application

of chemotherapy computed with the Euler-Maruyama algorithm, com-

puted with Matlab 02.08.2014 . . . . . . . . . . . . . . . . . . . . . . . 76

7.4 Dierent realizations of the SDE for normal cells with the application of

radiation therapy computed with the Euler-Maruyama algorithm, com-

puted with Matlab 02.08.2014 . . . . . . . . . . . . . . . . . . . . . . . 78

8.1 The dose-response curve for TCP and NTCP, computed with Matlab

14.05.2014 based on [21] . . . . . . . . . . . . . . . . . . . . . . . . . . 81

8.2 The complication probability for the heart based on [24], computed with

Matlab 14.05.2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3 NTCP-curve for the numerical solution of the nite dimensional system,


8.4 The NTCP for the dierent treatment strategies, computed with Matlab

24.08.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

9.1 The UTCP for the dierent treatment strategies, computed with Matlab

24.08.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

95

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