Dipartimento di Matematica Corso di Laurea Magistrale in Matematica
Tesi di Laurea
A mathematical model for cancer growth and
an application of Aldous criterion for
a macroscopic limit
Relatore: Laureanda:
Prof. Franco Flandoli Maria Cristina Polito
Controrelatore:
Prof. Maurizio Pratelli
Contents ii
Introduction iii
1 A model for cancer growth 1
1.1 The biological phenomena and some assumptions . . . 1
1.2 The model without treatment and mutation . . . 3
1.2.1 Birth and death: the equation for Nt . . . 4
1.2.2 The equation for the Vascular Endothelial Growth Factor (VEGF) 7 1.2.3 The equation for the angiogenesis . . . 8
1.2.4 Summary of the model without treatment and mutation . . . 9
1.3 Preliminaries related to the treatments . . . 9
1.4 The model under treatment . . . 10
1.4.1 The controlled equations . . . 10
1.4.2 Summary of the model under treatment . . . 12
1.5 The mutation . . . 13
1.5.1 Summary of the complete model . . . 16
1.6 Simulations . . . 18
1.6.1 Model without treatment: parameter calibration and graphical results . . . 18
1.6.2 Model under treatment . . . 19
1.7 Parameter randomization for the Monte Carlo simulation . . . 24
1.8 Results . . . 24
2 The Skorohod space 26 2.1 The space D[0, 1] . . . 26
2.1.1 Definitions and metrics . . . 26
2.1.2 Separability and completeness of (D, d) . . . 31
2.1.3 Compactness . . . 33
2.2 Convergence of probability measures . . . 35
2.2.1 Some recalls . . . 35
2.2.2 Tightness of probability measure on (D[0, 1], D) . . . 36
2.2.3 Stopping times and tightness: Aldous criterion . . . 36
3 A macroscopic limit 41 3.1 Introduction . . . 41
3.2 The model . . . 41
3.3 Useful results . . . 43
3.4 Convergence to the macroscopic limit . . . 47
Introduction
During the past decade, mathematical oncology has received great attention.
By translating biological quantities into mathematical terms, the modeling process de-scribes cancer-related phenomena as a complex set of interactions with the outcome predicted by mathematical analysis. However, there are many difficulties encountered in cancer modeling related to the complexity of the phenomena involved.
In general, tumor models are based on equations that describe, according to the level of detail and sophistication, tumor growth, nutrient evolution, vessel distribution, extra-cellular matrix structure, and so forth. The models require experimental data of various kinds to calibrate the model parameters and to validate the outcome. An increasingly relevant direction in mathematical oncology is the development of a multiscale theory. This is necessary to provide the appropriate framework for predictive mathematical mod-els. The aim of multiscale models is to find links between the different scales at which one can observe the biological phenomenon. There are two main levels we are interested in:
• microscopic, which is the cell level (of the order of a few micron)
• macroscopic, the tissue level (whose scale order is of millimeters or centimeters). On the one hand, this thesis focuses on the development of a model that describes cancer growth at a macroscopic level; on the other, it presents the necessary framework to construct a simple multiscale model.
The first part of this thesis concerns the description of a colorectal cancer growth model before and during a treatment, following specific therapeutic regimens. The gen-eral idea was to prefer simplicity over precision. Nevertheless, the results obtained are reasonable, both from a qualitative and a quantitative point of view. The phenomena involved are modeled by a system of ordinary differential equations (ODEs), which de-scribes the variation of a few number of quantities, summarizing the phenomena. The model is the result of a collaboration with Franco Flandoli, Valeria De Mattei, Marta Leocata, Cristiano Ricci. In particular, we focus on the description of the growth of the primary tumor of metastatic colorectal cancer (mCRC). The choice of this kind of tumor is due to some pattern that it shows during its evolution, as well as to the relative high degree of repetition in the observed clinical results.
The second part of this thesis, instead, is devoted to link a microscopic model to a con-tinuous one. Specifically, the outline is as follows.
In Chapter 1 we present the construction and a detailed description of the a for men-tioned mathematical model. Specifically, we chose the following quantities to describe the tumor growth and related angiogenesis:
Ntnorm : number of normoxic (proliferative) cells, Nthypo : number of hypoxic cells (non proliferative),
Vt : intensity of the Vascular Endothelial Growth Factor (VEGF),
At : level of vascularization due to angiogenesis.
We assumed that tumor grows following a spherical shape. At the beginning, the num-ber of tumor cells grows exponentially; then, the growth slows down in that only the boundary region of the sphere proliferates. The situation is partially restored by the an-giogenesis, which starts when the tumor size is of around 1mm3. Such a model depends on a number of parameters that we calibrated through simulations, in order to keep the biological evidence known in literature. After that, we investigate the effects of the action of the treatment. Finally, the randomization of some parameters carried out through Monte Carlo method, allows us to compute Progression Free Survival (PFS) medians and Kaplain-Meier curves, and to calibrate our model through comparison with clinical data. In Chapter 2 we recall the theoretical framework to study the stochastic processes with paths in D, the set of c`adl`ag functions. In particular, we show the construction and the main results related to the Skorohod space (D[0, 1], D). Finally, we recall an impor-tant criterion, due to Aldous, which produces sufficient conditions for the tightness of a family of probability measures which are the laws of processes whose paths are in D[0, 1]. In Chapter 3 we show a simple connection between microscopic and macroscopic scales, linking a discrete model with a continuous one. Specifically, we show that the differential equation
dNt
dt = λNt,
describing the exponential growth of the tumor in the initial phase, is the macroscopic limit of the model
YtN = 1 N X a∈Λ 1t∈[Ta,N 0 ,T a,N 1 )
as the number of initial cells N goes to infinity. In particular, we show that the family of laws associated to {YtN}N is tight. We observe that the process {YtN}t is a counting process, whose paths are right continuous with left limits. Thus, Aldous’ criterion applied to prove the tightness result.
A model for cancer growth
The aim of this chapter is to present a mathematical model for cancer growth phenomena before and during treatment following specific therapeutic regimens.
To better understand the model construction, first of all we briefly describe the biological problem. Then, we show the construction of the model without treatment, underlining the dependence of some parameters. Some preliminaries about the treat-ments led us to the enrichment of the model with two control functions, representing the drug-effects. The resulting model is not realistic; for this reason we introduce the phenomenon of mutations, which allows us to obtain a model that captures the main features and numerical figures known in the literature, as we show is Simulations section. We end this chapter with the results obtained from the parameter randomization and comparison with clinical data.
We refer to a solid tumor in opposition to the class of liquid ones, as leukemia, which are not treated in this work. In the construction of this model we had in mind the colorectal cancer; nevertheless, the following discussions could be generalized to other kinds of solid cancer for which the spherical approximation of its shape-growth seems reasonable.
1.1
The biological phenomena and some assumptions
In this section we summarize the relevant aspects of a tumor growth. Clearly, it would be a simplified description to underline the main aspects of the modeling.
As everyone knows, cancer is a group of diseases involving abnormal cell growth with the potential to invade or spread to other parts of the body. The growth of a tumor is due to the uncontrolled proliferations of its cells, and it requires a sufficient supply of oxygen. Until the tumor size is less than < 1mm3, the oxygen deriving from pre-existing
vasculature is sufficient to guarantee the proliferation of all cells constituting the tumor mass. In this phase the cancer grows exponentially. Then, the growth slows down, due to the lack of nutrients, oxygen and space. In particular, as the mass grows, some cells do not receive the necessary oxygen, becoming hypoxic in opposition to those one that are sufficiently oxygenated, the normoxic ones. Hypoxic cells release a growth factor, that is a signal protein which stimulates the formation of new blood vessels from the pre-existing ones. This process is called angiogenesis, while the protein’s name is Vascu-lar Endothelial Growth Factor (VEGF). When the angiogenic cascade starts, it slowly recovers a partial form of exponential increase.
An other important aspect is the one related to the mutations. This is a really difficult level to approach, and we are still trying to work on it, but for our discussions we take into consideration only the mutation that converts cancer drug-sensitive cells into cancer drug-resistant cells. As a consequence we have a sub-population of tumor cells that is not affected by the administration of the chemotherapeutic agent.
There are some typical sizes and times for colorectal cancer, which we refer to for the evaluation of the cancer mass growth. In particular, regarding to the size, it is known from literature [31] that
• 1 mm3 is the largest size of an avascular tumor,
• 1 − 2 mm3 is the range cancer size for the angiogenis start,
• 1 cm3 is a threshold size around which cancer is detectable.
A way to evaluate the mass growth is to look at the number of cancer cells assembling the mass. Therefore, we translate volumetric sizes into number of cells. To this end, we idealize the cell with a sphere, assuming that its diameter is 12 µm. This is a reasonable average of realistic values and it allows to have a closely correspondence
1 mm3 ←→ 106cells ,
1 cm3 ←→ 109cells .
Moreover, it is known that the total number of cells in human body is of the order of 1014 cells; therefore, values of the order of 1012, 1013cancer cells are considered incompatible with life.
Translating information deriving from available studies [30, 44] into number of cancer cells, we know that
• 109− 1010cells is a reasonable range values for the number of cancer cells at the
Regarding to the typical time size, one of the quantities we look at for the evaluation of the mass growth is the Doubling Time (DT).
The DT is defined as the amount of time a group of cells takes to double in size. This value varies depending on the specific kind of tumors. Its value is somewhat controver-sial, for various reasons: sometimes it is computed from metastatic lesions, assuming that they grow with a similar rate to the primary tumor (but this is not confirmed); sometimes, it is directly computed from the primary tumor, but using simplified expo-nential rules which do not take into account the different growth regimes.
The colorectal cancer growth is one of the slowest ones. The following information about its growth time are known from literature [6, 15, 36, 42], [4, 38]
• DT ∼ 60 − 180 days,
• 8 years : estimate of the amount of time needed to reach 109cells.
All these information are necessary for the evaluation and the calibration parameters of our model.
1.2
The model construction without treatment and
muta-tion
In this section we present the basic model which approximates the main phenomena described previously. It will be enriched by mutation and controlled by treatment in the following sections. For the discussion that follows we do not take into account these last two phenomena.
In order to highlight some modeling reasons, and to explain the difficulties encoun-tered, we introduce the model in a constructive way. First of all, we set the variables involved in the model and we define our goal. Then, we described separately the equa-tions satisfied by the quantities introduced. The resulting model will depend on a set of parameters.
The aim of our model is to capture the main aspects of cancer mass growth behaviour from a both qualitative and quantitative point of view. The latter is to be understood in terms of orders of magnitude. The general idea of our construction is to prefer simplicity over precision. To this end, we decided to only look at evolution in time, neglecting the spatial structure. Thus, we do not consider the position of cells and the space dependence of variables.
The resulting model is a system of Ordinary Differential Equations (ODEs) which summarizes the variation of a few number of quantities sufficient to describe cancer mass growth and related angiogenesis. Specifically, we set the following variables:
• Nnorm
t as the number of normoxic cells,
• Nthypo as the number of hypoxic cells, • Vt as the intensity of the VEGF,
• At as level of vascularization due to angiogenesis, • Nt= Ntnorm+ N
hypo
t is the total number of cancer cells.
In Table 1.1 we recall the meaning of the quantity introduced. How these quantities are involved in the phenomenon is better explained in the previous section.
The variables just introduced describe the dynamic of our model, as time varies.
Terms Meaning and role
normoxic cell proliferative cell, receiving a sufficient supply of oxygen hypoxic cell non proliferative cell; it is not sufficiently oxygenated
VEGF protein that stimulates the angiogenesis start angiogenesis creation of new vessels from pre-existing ones
Table 1.1: Meaning and role of the model variables.
We assume that the cancer develops in a limited interval of time [0, T ]. Thus, the vari-ables Ntnorm and Nthypo represent real valued functions defined on [0, T ]. The variables Vt, At, instead, are to be understood as a space average at time t; therefore, they are
functions valued in [0, 1].
In the following sections we construct the differential equations satisfied by the quan-tities introduced.
1.2.1 Birth and death: the equation for Nt
A general feature of a living system, as a cancer is, concerns the processes of birth and death: in a cancer there are cells that live and proliferate, and others that for various reasons die (lack of nutrients, apoptosis, mutations, ...). For that reason we start from the Lokta-Volterra model, which well represents this phenomenon [33].
Let us consider
d
dtNt= λN
norm
t − µNt (1.1)
where λ is the proliferation rate, and µ is the death rate.
Remark. Only the normoxic cells can proliferate and contribute to the growth of tumor, while cell lost phenomena affect the whole population. It is important to underline that
proliferating cells die for different reasons than the hypoxic ones. This characteristic should be modeled by different death rates. Nevertheless, this could be included in the λ rate. In fact, if we take
µ1Ntnorm+ µ2Nthypo, then d dtNt= λN norm t − µ1Ntnorm− µ2Nthypo
= λNtnorm− µ1Ntnorm+ µ2Ntnorm− µ2Ntnorm− µ2Nthypo
= eλNtnorm−eµNt
with eλ = λ − µ1+ µ2,µ = µe 2. Hence the form (1.1) is correct.
So, we have to define a suitable expression for Ntnorm and for Nthypo.
Normoxic contribution
We assume that the tumor grows following a spherical shape, and that only the boundary of this sphere can proliferate. When the tumor reaches a certain size, the angiogenic cascade starts and new vessels reach the internal part of the tumor, bringing oxygen. Then, some hypoxic cells receive sufficient oxygen to restart their proliferating activity, becoming normoxic.
This is expressed by the following equation
Ntnorm = Nt 1 − 1 − 1 1 + (Nt)1/3/2η !3 + AtNt 1 − 1 1 + (Nt)1/3/2η !3 . (1.2)
Specifically, the term
Ntnorm,bound:= Nt 1 − 1 − 1 1 + (Nt)1/3/2η !3 (1.3)
represents the contribution of the boundary cells. It results from the following arguments. Let us suppose that the tumor occupies a spherical region and cells have a roughly spherical shape, filling the whole sphere. Let R, r, δ be as follows
R : tumor region radius, r : cell radius,
δ : thickness of the external proliferating layer. The total number of cells in the tumor is
N = 4 3πR 3 4 3πr3 = R r 3 . The volume of the external layer of thickness δ is
4 3πR
3− 4
3π (R − δ)
3
Hence, the number of cells in the layer is Nbound = 4 3πR3− 4 3π (R − δ) 3 4 3πr3 = R r 3 1 − 1 − δ R 3! = N 1 − 1 − δ R 3! . (1.4)
We know that, when a tumor size is very small (< 1mm3), the surrounding oxygen is sufficient for the proliferation of all cells. Therefore, in that situation we would have δ = R. On the contrary, when a tumor is very large, the surrounding oxygen reaches only the boundary cells of the tumor (in absence of angiogenesis). We assume that the thickness of the boundary stabilizes to a certain number η, where
η : number of diameters of cells reached by the surrounding oxygen. Thus, in that case we would have δ = 2ηr.
Therefore, we have that δ R =
(
1 when N is small 2η/N1/3 when N is large
A simple function that catches both these extreme situations is the following δ
R = f (N, η) =
1 1 + N1/3/2η .
Substituting in 1.4, we obtain the expression 1.3.
Remark. When the tumor is large, that is when N1/3/η 1, we have
Nnorm,bound ∼ N 1 − 1 − 2η N1/3 3! ∼ 3N 2η N1/3 = 6ηN 2/3.
For this reason we call this formula the 2/3-formula for external layer of proliferating cells. The 2/3 law often appears in the literature. It is important to underline that this
expression captures the exponential increase, typical of the initial tumor growth; in fact, in that case Nnorm,bound ∼ N . Definitely, this formula allows to take into account the transition between the exponential growth and a slower one, which is due to the lack of nutrients, oxygen and space.
The second term of 1.2
AtNt 1 −
1
1 + (Nt)1/3/2ηsens
!3
(1.5) represents the contribution due to angiogenesis, and its expression is the result of similar arguments we have done for the boundary contribution.
In particular, we consider the number of cells in the internal part of the sphere, the one of radius R − δ. It is 4 3π (R − δ) 3 4 3πrcell3 = R rcell 3 1 − δ R 3 = N 1 − 1 1 + N1/3/η 3 , from which it follows expression 1.5.
Hypoxic contribution
The equation for the number of hypoxic cells can be simply derived from the equation of the total number of cells, once we know the number of normoxic cells. So, we have
Nthypo= Nt− Ntnorm = Nt 1 − 1 − 1 1 + (Nt)1/3/2η !3 + AtNt 1 − 1 1 + (Nt)1/3/2η !3 = (1 − At) Nt 1 − 1 1 + (Nt)1/3/2η !3 (1.6)
1.2.2 The equation for the Vascular Endothelial Growth Factor (VEGF)
The signal protein is produced by the hypoxic cells to stimulate angiogenesis, and it is absorbed by endothelial cells, which constitute the angiogenetic blood vessels. This mechanism is included in the following equation, that we have devised for the variation of VEGF: d dtVt= Chypo→V Nthypo Nt !2/3 − CA←VAt Vt(1 − Vt) (1.7) where
CA←V : parameter for the absorption by angiogenetic endothelial cells.
We have assumed, conventionally, that Vt ∈ (0, 1), and this justifies the multiplicative
terms Vt(1 − Vt). The term Chypo→V Nthypo Nt !2/3
represents the production of VEGF that is proportional to the density of hypoxic cells in the tumor mass. The presence of the exponent 2/3 could be justified by the following arguments.
We think about VEGF as the work needed to move blood vessels from an “infinite” distance to the sphere of hypoxic cells. With this in mind, we can think the internal part of the tumor mass, constituted only by hypoxic cells, as a uniformly “charged” sphere of radius R − δ. It induced the electric field
E(r) = C (R − δ)
3
r2 ,
where C is a suitable constant. The potential on the boundary of the sphere V (R) = C (R − δ)2
is the work done by the electric field to move a point charge from infinity to the boundary of the sphere. Thus, the VEGF would have the form
Vt= C (Rt− δt)2 = eC(Nthypo)2/3.
1.2.3 The equation for the angiogenesis
The generation of new vessels is strictly connected to the quantity of existing vessels itself and the quantity of VEGF that stimulates the phenomena. A simple form that takes into account this is
d
dtAt= CV →A(Vt− Vthrsld) 1Vt>Vthrsld(At+ Apre) (1 − At) , (1.8)
where
CV →A : parameter that modulates the speed of reaction of At to Vt,
Apre = 0.2 : quantity of pre-existing vasculature (not generated from angiogenesis),
Vthrsld = 0.2 : concentration of VEGF necessary to the start of angiogenesis.
Remark. We have rather arbitrarily chosen the value 0.2, anyway the constant CV →A
1.2.4 Summary of the model without treatment and mutation
Summarizing the equations presented in the previous sections, the model for the cancer growth without treatment and mutation assume the following form:
d dtNt= λNt 1 − 1 − 1 1 + (Nt)1/3/2η !3 + λAtNt 1 − 1 1 + (Nt)1/3/2η !3 − µNt Nthypo= (1 − At) Nt 1 − 1 1 + (Nt)1/3/2η !3 d dtVt= Chypo→V Nthypo Nt !2/3 − CA←VAt Vt(1 − Vt) d dtAt= 1Vt>V0.2CV →A(Vt− V0.2) (At+ 0.2) (1 − At) − 2A 10 t 1Vt≤0.2 (1.9) We remark that it depends on the parameters summarized in Table 1.2.
Parameters Meaning
λ growth rate due to cell proliferation µ decay rate due to cell loss
η thickness of proliferating boundary of cells Chypo→V VEGF production rate from hypoxic cells
CA←V absorption rate of VEGF from vasculature
CV →A reaction rate of angiogenesis to VEGF
Table 1.2: Parameters of the model without treatment and mutations.
1.3
Preliminaries related to the treatments
Our goal is to construct a model which reproduces the effects of specific treatments over the cancer mass growth. To this purpose, it would be worth to compare the results with a real and concrete clinical trial.
A relevant study on these subjects was conducted by Kabbinavar, who evaluated the effect of the addition of a monoclonal antibody (the Bevacizumab) to a treatment based on a chemotherapeutic agent (the Fluorouracil) [26].
one population receives a treatment with Fluorouracil alone, the other receives the Flu-orouracil plus the Bevacizumab. The duration of this trial is 96 weeks, divided into 12 cycles, each one of 8 weeks. Patients involved undergo an assessment of tumor status at the beginning of the study (baseline) and at completion of every 8-week cycle (follow-up). Tumor response or progression is determined utilizing the Response Evaluation Criteria in Solid Tumors (RECIST) [13].
One of the outcome measures of this study is Progression Free Survival (PFS): it is defined as the time from random assignment in a clinical trial to disease progression or death from any cause.
At the end of the study, Kaplain Meier curves and PFS medians are calculated for both populations.
In order to test the validity of our model for cancer mass growth comparing with [26], we will enrich the model with two control functions representing the effects of the Fluorouracil and of the Bevacizumab.
For the discussions that follow it is important to know the differences between these two drugs:
• Fluorouracil (5-FU) is a chemotherapeutic agent : it acts on the cells, killing them; • Bevacizumab (BV5) is a monoclonal antibody: it acts on the VEGF, inhibiting its
action in order to stop angiogenesis.
1.4
The model under treatment
The aim of this section is to model the influence of specific treatments over the cancer mass growth. To this purpose, we enrich the model (1.9) with treatment, following specific therapeutic regimens, suggested by [26] and explained in detail into the previous section.
The treatment is modeled by control functions, which act on the specific quantities involved.
1.4.1 The controlled equations
Following the therapeutic regimens described in [26], we suppose that the treatment is based on the administration of a monoclonal antibody, the Bevacizumab (BV5), and a chemotherapeutic agent, the Fluorouracil (5-FU). While the last one acts directly on the cells, killing them, the bevacizumab acts on the VEGF, inhibiting its action in order to stop the stimulation of the angiogenesis.
uF Ut : function that represents the concentration of 5-FU in tissue, uBVt : function that represents the concentration of BV5 in tissue.
It is important to underline that these controls functions express the drug concentration in the tumor tissues and not in plasma.
5-Fluorouracil
Concerning 5-FU, the drug concentration in the tissues decays exponentially in time and is given by the following expression in every week of the drug administration
uF Us = (1 + CF U) exp −log (1 + CF U) NF U s , where s ∈ [0, NF U] and
NF U : constant representing the number of days of action of 5-FU.
We assume that uF Us = 0 in every period the Fluorouracil does not act.
We have in mind that the cell killing by chemotherapy is faster than proliferation. Thus, the controlled equation that we use has the form
d
dtNt= λN
norm
t 1 − uF Ut − µNt, (1.10)
where we notice that at the beginning of every weekly administration, the rate is equal to
λ 1 − uF U0 = λCF U.
Therefore, the constant CF U acts as a multiplier of intensity of the cell killing with
respect to the cell proliferation. A typical value we use is CF U = 20
Remark. At time t = NF U, we have uF Ut = 1, hence the rate at this time is
λ 1 − uF UNF U = 0.
In other words, we prescribe that cell loss occurs until time NF U, in an exponentially
decreasing manner; afterwards, proliferation restarts but not immediately with full rate λ, just with rate λ 1 − uF U
t which is asymptotic to λ for large times. This justifies the
term inside the exponential. Bevacizumab
The Bevacizumab (BV5) is administered once every two weeks in each cycle. Under treatment we assume that
uBVs = exp −log 2 NBV s , (1.11) where s ∈ [0, NBV] and
NBV : half-life of bevacizumab.
We assume that uBV
s = 0 when the drug does not act. The controlled equation for
VEGF assumes the form d dtVt= Chypo→V Nthypo Nt !2/3 − CA←VAt− CBV →V uBVt Vt(1 − Vt) , (1.12) where
CBV →V : parameter for the influence of BV5 on the VEGF.
1.4.2 Summary of the model under treatment
Recalling that Nt= Ntnorm+ N hypo t , Ntnorm = Nt 1 − 1 − 1 1 + (Nt)1/3/2η !3 + AtNt 1 − 1 1 + (Nt)1/3/2η !3 ,
the model of cancer mass growth controlled by treatment administered as in [26], assumes the following expression:
d dtNt= λN norm t 1 − uF Ut − µNt uF Ut = (1 + CF U) exp −log (1 + CF U) NF U t Nthypo= (1 − At) Nt 1 − 1 1 + (Nt)1/3/2η !3 d dtAt= 1Vt>V0.2CV →A(Vt− V0.2) (At+ 0.2) (1 − At) − 2A 10 t 1Vt≤0.2 d dtVt= Chypo→V Nthypo Nt !2/3 − CA←VAt− CBV →V uBVt Vt(1 − Vt) uBVt = exp −log 2 NBV t . (1.13)
In addition to the parameters, which the model without treatment and mutation de-pends on, the model so defined depend on CF U, NF U, NBV CBV →V. The all parameters
Parameters Meaning λ growth rate due to cell proliferation µ decay rate due to cell loss
η thickness of proliferating boundary of cells Chypo→V VEGF production rate from hypoxic cells
CA←V absorption rate of VEGF from vasculature
CV →A reaction rate of angiogenesis to VEGF
CF U intensity of 5FU action
NF U number of days of action of 5-FU (range from 1 to 7)
NBV number of days of action of BV5 (range from 1 to 14)
CBV →V inhibition rate of bevacizumab on VEGF
Table 1.3: Parameters of the model under treatment.
1.5
The mutation
The model we have introduced until now, produces a non realistic result, in the following sense: fixed all the parameters, the response to the therapy either is good, or is bad from the beginning and health deteriorates, in other words the response is deterministic. This is not realistic because, generally, in a mCRC patient the therapy has a positive effect in the first period, than deteriorates. There could be different reasons for this deterioration, among which:
• side effects,
• problems related to metastasis, • resistance to the therapy.
In order to simplify the model as much as possible, in this work, we investigate only this last one reason. The ability, acquired by some cells, to resist to the drug effects, is the result of genetic modifications. In particular, we can distinguish
i) intrinsic drug resistance, which may appears before the beginning of the therapy, ii) acquired drug resistance, which is the result of an adaptive process: as a con-sequence of the therapy, some cells develop mutation which guarantee them to survive to the therapy.
To take into in account this phenomena, we would introduce a parameter
• p , which represents the cancer cells’ probability to develop a drug-resistant muta-tion during a duplicamuta-tion.
Such parameter summarizes both kinds of drug resistance, and it allows us to introduce mutation in our model.
In this setting, we must to further diversify the cancer cells populations. Therefore, we define the following variables
• Nsens
t : number of cancer cells drug-sensitive,
• Nres
t : number of cancer cell drug- resistant,
with the natural notation for their subpopulations
• Ntsens,norm : the number of cancer cell drug-sensitive and normoxic, • Ntsens,hypo : the number of cancer cell drug-sensitive and hypoxic, • Ntres,norm : the number of cancer cell drug-resistant and normoxic, • Ntres,hypo : the number of cancer cell drug-resistant and hypoxic. Clearly, the total number of cancer cells is
Nt= Ntsens+ Ntres
= Ntsens,norm+ Ntsens,hypo+ Ntres,norm+ Ntres,hypo.
An idea about the distribution of mutated cells
We would like to enrich our model with drug-resistance. The modeling of this phe-nomenon is a quite delicate issue. Specifically, we would determine a growth rule for the subpopulations Ntsens and Ntres. The assumption that the growth follows a spherical shape cannot works simultaneously for both populations, because they are interlaced. From biological considerations, we decide to proceed as follows. Let us distinguish two different situations, before and after treatment.
It seems reasonable to assume that before treatment, it results that Ntsens Ntres.
In that situation we could assume that cancer drug-sensitive cells growth following a spherical shape. Thus, Ntsens would be modeled by the same equation of Nt in (1.9).
Regarding cancer drug-resistant cells, we think they are distributed inside the tumor mass, in a more complex way. We assume that at some time a mutated cell appears, and proliferating (with the same rule of the previous) it generates resistant cells, assembling a kernel of drug-resistant cells. However, this kernel cannot growth indefinitely as the
global tumor mass, because it is partially or completely absorbed into the hypoxic region. Even if a kernel of drug-resistant cells appear in the proliferating region, as the tumor mass growth it would be absorbed into the non oxygenated region.
Definitely, before treatment we think the tumor mass region as a sphere constituting by drug-sensitive cells. The boundary of this region, whose thickness is ηsens, is formed by
proliferative cells. The internal part of this region is formed by non proliferative cells. Into the all spherical region there exist some kernels formed by drug-resistant cells, which have a proliferating boundary whose thickness we call ηres. In that situation we assume
ηres< ηsens.
After treatment, it is more reasonable to approximate the sensitive and resistant popu-lations growth as two separate sphere. The reason is that cell mutated kernels may be left isolated by the loss of surrounding non mutated cells and, to simplify, we assume that drug-resistant mass is dominated by the largest kernel of mutated cells. Thus, the approximation as a single sphere do not deviate too much from reality. So, in that case we take
ηres= ηsens.
The equation of mutated cells
A second peculiarity of mutated cells is that, when the probability of mutation and the number of proliferating cells reach values which trigger the emergence of a first mutated cell, soon others will appear and soon much more (not due to proliferation of existing mutated cells); the number which determines this transition in a small time interval [t, t + ∆t] roughly is
p · λNtsens,norm∆t
because λNtsens,norm∆t is the number of proliferating (mother) cells in [t, t + ∆t], and the average number of them which mutates is the product, p·λNtsens,norm∆t; fluctuations are relatively small, at least for large values of λNtsens,norm∆t. Thus the average number of mutated descendants is
2 · p · λNtsens,norm∆t.
We should not forget that these arguments are true when proliferation freely takes place; under chemotherapy certain proliferating cells are killed. We thus correct the previous formula as
max0, 2 · p · λNtsens,norm 1 − uF Ut ∆t (1.14)
Fitting the idea about cell mutated distribution and the remark just exposed, the variable Ntres satisfies
d dtN res t = max0, 2 · p · λN sens,norm t 1 − uF Ut + λN res,norm t − µNtres
= max0, 2 · p · λNtsens,norm 1 − uF Ut + λNres t 1 − 1 − 1 1 + (Ntres)1/3/2ηres !3 + λAtNtres 1 − 1 1 + (Ntres)1/3/2ηres !3 − µNtres.
The mechanism of mutation is like a change of species, with doubling of the quantity in arrival. As in all change of species equations, we have to delete the same quantity from the delivering population, hence we have to correct the equation for Ntsens as
d dtN sens t = (1 − p) · λN sens,norm t − µNtsens = λ (1 − p) Ntsens 1 − 1 − 1 1 + (Nsens t ) 1/3/2η sens !3 + λ (1 − p) AtNtsens 1 − 1 1 + (Nsens t ) 1/3/2η sens !3 − µNtsens. In addition we have Ntres,hypo= (1 − At) Ntres 1 − 1 1 + (Ntres)1/3/2η res 3 , so we modify the equation for VEGF as
d dtVt= Chypo→V Ntsens,hypo+ Ntres,hypo Nsens t + Ntres !2/3 − CA←VAt− CBV →V uBVt Vt(1 − Vt) .
1.5.1 Summary of the complete model
To conclude this section, we summarize all the discussions we have done, showing the expression of the resulting model.
We recall that Nt= Ntsens,norm+ N sens,hypo t + N res,norm t + N res,hypo t Ntsens,norm= Ntsens 1 − 1 − 1 1 + (Ntsens)1/3/2ηsens !3 + AtNtsens 1 − 1 1 + (Ntsens)1/3/2ηsens !3 Ntres,norm= Ntres 1 − 1 − 1 1 + (Ntres)1/3/2ηres !3 + AtNtres 1 − 1 1 + (Ntres)1/3/2ηres !3 .
Therefore, the model of tumor growth under treatment, which takes into account the mutation phenomenon is represented by the following system:
d dtN sens t = (1 − p) λN sens,norm t 1 − uF Ut − µNtsens d dtN res t = max0 , p λN sens,norm t 1 − uF Ut + λN res,norm t − µNtres Ntsens,hypo = (1 − At) Ntsens 1 − 1 1 + (Nsens t ) 1/3/2η sens !3 Ntres,hypo = (1 − At) Ntres 1 − 1 1 + (Ntres)1/3/2η res 3 d dtAt= 1Vt>V0.2CV →A(Vt− V0.2) (At+ 0.2) (1 − At) − 2A 10 t 1Vt≤0.2 d dtVt= Chypo→V Ntsens,hypo+ Ntres,hypo Nsens t + Ntres !2/3 − CA←VAt− CBV →V uBVt Vt(1 − Vt)
The functions uF Ut and uBVt assume the expressions discussed in Section 1.4.1.
In addition to the parameters summarized in Table 1.3, this model depends on ηsens,
ηres and p.
Parameters Meaning
ηsens thickness of boundary drug-sensitive cells
ηres thickness of boundary drug-resistant kernel
p probability of drug resistant mutation per cell per duplication Table 1.4: Parameters of the model with treatment and mutation.
1.6
Simulations
In this section we show some numerical and graphical results we have obtained imple-menting our model. All the simulations that follow are performed using software Matlab.
1.6.1 Model without treatment: parameter calibration and graphical results
As we discuss in Section 1.2.4, the model without treatments depends on a number of parameters.
Sometimes, the values of the parameters involved in a model could be derived by physi-cal observations, biologiphysi-cal reasons and so on. In our model this was the case for η. In fact, from observation concerning necrotic cells around capillaries, it is possible to guess a value of η of the order of 10.
On the contrary, it is more difficult to have information about the other parameters involved. Therefore, we decided to fix them fitting the number of macroscopic experi-mental evidences mentioned in the first section. In particular,
• roughly 8 years to reach 109 cells,
• angiogenesis starting between 106 and 2 · 106 cells,
• a specific shape of the curve Nt.
Specifically, regarding Nt profile, we required that the curve was increasing exponential
at the beginning (up to roughly 105 cells looks reasonable), followed by a period of very slow increase, almost at equilibrium between cell proliferation and loss, then followed by an almost exponential restart due to angiogenesis.
Clearly, there are several sets of parameters reproducing these results. Among these, we chose as standard the values reported in Table 1.5.
A general shape of the cancer mass growth, in terms of number cells Ntand without
therapy, is showed in Figure 1.1. This simulation is produced by the parameters fixed as in Table 1.5.
Parameters Standard fixed value λ 0.05 µ 0.002 ηsens 15 ηres ηsens/1.2 Chypo→V 0.08 CA←V 0.01 CV →A 0.006 p 10−5
Table 1.5: Standard values of parameters in the model without treatment.
3 6 9 I 3 6 9 II 3 6 9 III 3 6 9 IV 3 6 9 V 3 6 9 VI 3 6 9 VII 3 6 9 VIII 3 6 9 IX 3 6 9 X 3 time (months) 1 2 3 4 5 6 7 8 9 10 11 12 13 14
cell number (log10)
1 cm3
angiogenic switch
Figure 1.1: Simulation without therapy produced by parameters fixed as in Table 1.5. Diamonds denote the range106−2 · 106where angiogenesis is expected to start and the first red star is that
initial time. The second red star is when 109cells are reached which should be close to 8 years. In the main picture, blue line denotes total number of cancer cells, green line those having a drug resistant mutation.
1.6.2 Model under treatment
In order to show the drug effects on cancer mass growth following the study conducted by Kabbinavar, we implement the specific therapeutic regimens about 5-FU and 5-FU+BV5
studied in [26]. Moreover, we introduce a specific rule to compute PFS following RE-CIST criteria [13]. Therefore, before showing the results obtained with the model under treatment, we briefly discuss these two topics.
Therapeutic regimens
The regimens we want to reproduce and simulate differ in the addition of the Beva-cizumab. In particular, they are made up of 12 cycles; each cycle corresponds to 8 weeks. Thus, the duration of the study is 96 weeks. Specifically, the regimens are scheduled as follows.
• 5-FU treatment: Fluorouracil is administered weekly for the first 6 weeks of each 8-week cycle;
• 5-FU + BV5 treatment: Fluorouracil is administered as previous, while Beva-cizumab is administered every two weeks for all the duration of the study.
Progression Free Survival and its computation
PFS is the time from assignment in a clinical trial to disease progression or death from any cause. In our simulations we do not take into account death which occurs for different reasons from disease progression (for example for side effects,...). In fact, to be precise we compute the so called Time to Progression (TTP), which is the time from assignment in a trial to disease progression.
Disease Progression (PD) it is defined by the RECIST criteria as
PD = 20% increase in the sum of the longest diameter of target lesions. In order to compute PFS with our model, we proceed as follows.
Let us define the times
• t0 : baseline time (therapy start),
• t0+ 8 k , k = 1, . . . , 12 : follow-up times (assessments of tumor status),
and set
• Nk := Nt0+8 k, k = 1, . . . , 12 as the number of cells at the end of each cycle of therapy. Let us compute Nk0 = min k Nk, and set nk:= Nk− Nk0 Nk0 ∀k > k0.
Finally, we look for the first k1 > k0 such that nk1 > 0.7. Thus, we declare that
progression occurs, setting PFS = k1.
This method is justified by the following arguments: an increasing of the tumor diameter of more than 20% is equivalent to the radius increasing of more than 20%; hence, the volume is more than 1.23 = 1.728 times the previous one. That means Nk & 1.7 · Nk0,
3 6 9 I 3 6 9 II 3 6 9 III 3 6 9 IV 3 6 9 V 3 6 9 VI 3 6 9 VII 3 6 9 VIII3 6 9 IX 3 6 9 X 3 time (months) 1 2 3 4 5 6 7 8 9 10 11 12 13 14
cell number (log10)
1 cm3
angiogenic switch
I II III IV V VI VII VIII IX X 0 2 4 6 8 10 12 14 N
I II III IV V VI VII VIII IX X 0 2 4 6 8 10 12 14 Nhypo
I II III IV V VI VII VIII IX X 0 0.2 0.4 0.6 0.8 1 1.2 A
I II III IV V VI VII VIII IX X 0 0.2 0.4 0.6 0.8 1 V
Figure 1.2: Simulation with therapy, 5-FU plus BV5. Circles denote times of assessment of tumor size. Small subgraphs show the behavior of total number of cells, hypoxic cells, degree of angiogenesis and density of VEGF.
6 9 VIII 3 6 9 IX 3 6 9 time (months) 8 9 10 11 12 13
cell number (log10)
1 cm3
TTP
Figure 1.3: Zoom around 5-FU action. TTP = Time To Progression.
6 9 VIII 3 6 9 IX 3 6 9 time (months) 8 9 10 11 12 13
cell number (log10)
1 cm3
TTP
1.7
Parameter randomization for the Monte Carlo
simula-tion
In order to compare our model results with clinical data presented in [26], we have to represent a population of patients. This result is obtained randomizing some of the model parameters, performing a simple Monte Carlo simulation.
This method relies on repeated random sampling to obtain numerical results. Its essen-tial idea is to use randomness to solve problems that might be deterministic in principle, as our model result every time parameters are fixed.
Therefore, coefficients values are sampled from their range of variation and, then, the deterministic code run with these coefficients.
In this way, we reproduce a sample of patients and we can compute Kaplain-Meier curves and Progression Free Survival medians.
We decide to assign a deterministic value to some of the parameters; this choice is the result of a sensitivity analysis which shows that their variation has no influence on the ouput. In Table 1.6 are summarized all our decisions.
Parameters Value or distribution
λ 0.05
µ 0.002
ηsens 15
ηres ηsens/unif[1, 2]
Chypo→V 0.05 CA←V 0.01 CV →A 0.006 p 10−unif[4,6] Nstart 2 · 10unif[8,11] NF U unif[1, 7] CF U unif[5, 25] NBV unif[1, 14] CBV →V unif[0, 5]
Table 1.6: Parameters of Monte Carlo simulations.
1.8
Results
The Monte Carlo randomization allows us to obtain two sample of patients, one for the trial related to the effect of 5-FU alone, the other for the study on the combined effect
of 5-FU plus BV5 [26].
These two studies are modeled by assuming in the first case that the control function for BV5, uBV, is zero; while in the second, it has the expression (1.11).
Drawing the Kaplain Meier curves, we obtain a good result, comparable with [26]. The differences between the plots can be attributed to the Progression-free survival compu-tation. In fact, PFS is defined as the time from random assignment in a clinical trial to disease progression or death from any cause. In our sample of patients we do not take into account death that may be occurred before observe progression; we only look at progression disease, which is controlled at follow-up time (every 8 weeks).
4 6 8 10 12 14 16 18 20 22 24 time (months) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time to progression FU (Median: 7.5) FU/BV (Median: 9.3)
The Skorohod space
In this chapter we introduce a space that is considered a classic tool in the study of stochastic processes whose paths are c`adl`ag (French “continue droite, limite gauche”), i.e right continuous functions with left limit at each point. This space is obtained consid-ering on the set of these functions a specific topology, which is called Skorohod topology, that is generated by a suitable metric.
This space, called Skorohod space, is the natural one in which we can analyze the Poisson processes and other processes with paths that are discontinuous.
All the results presented in this chapter are taken from [1], [5].
In the first section we define the space with a suitable metric which makes it into a separable, complete space. Then we characterize compact sets, in view of a convergence criterion. In the second section, we look at the Skorohod space, as a space of paths of stochastic processes and we present a really important tightness criterion due to Aldous, which has a great number of applications.
2.1
The space D[0, 1]
After giving the definitions and introducing a suitable metric, we will see that the topol-ogy induced by the latter, the Skorohod topoltopol-ogy, makes this space into a complete separable metric space. Then, it will make sense to wonder about compactness criteria. We will give all the definitions on the interval [0, 1], but the results can be generalized to [0, +∞).
2.1.1 Definitions and metrics
Definition. Denote by
D[0, 1] := {x : [0, 1] −→ R | x is c`adl`ag} 26
the set of real-valued functions on [0, 1] which are right continuous and have left limits: • for 0 ≤ t < 1, x(t+) = lims↓tx(s) exists and x(t+) = x(t);
• for 0 < t ≤ 1, x(t−) = lims↑tx(s) exists.
Sometimes, we will shortly write D instead of D[0, 1], when no confusion can arise. Remark. The space of continuous functions is strictly contained in D[0, 1], i.e.
C[0, 1] ⊂ D[0, 1]. .
First of all, we introduce a metric ρ which makes D into a separable space; unfor-tunately, we will see that this is not enough to make it into a complete metric space. Then, we will modify that metric in such a way that the resulting space is also complete.
Let Λ be the subspace of C[0, 1] defined by:
Λ := {λ : [0, 1] −→ [0, 1] | λ is continuous and strictly increasing, λ(0) = 0, λ(1) = 1} Definition. Let ρ : D[0, 1] × D[0, 1] −→ [0, +∞) defined as
ρ(f, g) = inf{ε > 0 | ∃λ ∈ Λ : sup
t
|λ(t) − t| < ε, sup
t
|f (t) − g(λ(t))| < ε} We remark that the identity function λ(t) = t is in Λ, so ρ is finite.
Proposition 2.1.1. The map ρ : D[0, 1] × D[0, 1] → [0, +∞) is a metric.
Proof. Assume first that ρ(f, g) = 0. Then, there are two possibilities: either f = g, or f (t) = g(t−) for each t. But the functions in D are by definition right continuous with left limits; then, having the same left limits is enough to say that f is actually equal to g.
To prove that ρ is symmetric, let f, g ∈ D. Observe that a function λ in Λ is invertible and its inverse λ−1 is still in Λ: in fact λ−1 is strictly increasing, continuous, and λ−1(0) = 0, λ−1(1) = 1 (because λ is injective).
Setting s := λ−1(t), we have that:
sup t∈[0,1] |λ−1(t) − t| = sup s∈[0,1] |s − λ(s)| and sup t∈[0,1] |f (λ−1(t)) − g(t)| = sup s∈[0,1] |f (s) − g(λ(s))|.
The last step is to prove the triangle inequality. This holds thanks to the inequalities sup t∈[0,1] |λ2◦ λ1(t) − t| ≤ sup t∈[0,1] |λ1(t) − t| + sup s∈[0,1] |λ2(s) − s| and, setting u := λ1(t), sup t∈[0,1] |f (t) − h(λ2◦ λ1(t))| = sup t∈[0,1] |f (t) − g(λ1(t)) + g(λ1(t)) − h(λ2◦ λ1(t))| ≤ sup t∈[0,1] |f (t) − g(λ1(t))| + sup s∈[0,1] |g(u) − h(λ2(u))|.
This concludes that ρ is a metric on the space D[0, 1]. It is important to underline the following two facts: • (D, ρ) is separable.
In fact, we can consider the set of functions x in D for which there exists an integer k such that x restricted to [(i − 1)/k, i/k) is constant and equal to a rational number. The set of this functions is dense in D with respect to the metric ρ, and this conclude.
• (D, ρ) is not complete.
Example. Let {xn} the sequence of functions defined as follows
xn=1[0,1/2n) for every n.
Then, {xn} is a Cauchy sequence (we can easily see that the distance between xn
and xn+1 is bounded by 1/2n). The sequence should therefore converge to 0, but
actually this cannot be true as the distance between xnand 1 is constant and equal
to 1.
We now introduce a slight deformation of the metric ρ, in a way that induces the same notion of convergence, but with the property that the space D, with respect to the new metric, is complete.
The idea is based on the following observation: requiring that λ(t) − t is close to 0 is equivalent to require that the slope λ(t) − λ(s)/(t − s) is close to 1 or, equivalently, that its logarithm has to be close to 0.
With this in mind, we denote by kλk the following number: sup s6=t, s,t∈[0,1] logλ(t) − λ(s) t − s
where λ is a function in Λ, therefore strictly increasing and continuous.
A priori, the quantity defined above on λ could be infinite, but we do restrict our λ to the case in which kλk is bounded.
Definition. Let d : D[0, 1] × D[0, 1] −→ [0, +∞) be defined by d(f, g) := inf{ε > 0 : ∃λ ∈ Λ such that kλk ≤ ε, sup
t∈[0,1]
|f (t) − g(λ(t))| ≤ ε}.
As we did previously for ρ, we can show that d is a metric; in fact, it is true that kλk is preserved by passing to inverses and to compositions:
kλk = kλ−1k
and
kλ1◦ λ2k ≤ kλ1k + kλ2k
Proposition 2.1.2. (D, d) is a metric space.
Proof. The symmetry and the triangle inequality follow from the relations above. Then, we must only prove that
d(f, g) = 0 if and only if f = g. To this end, we firstly compare d to the metric ρ.
Since |u − 1| ≤ e| log u|− 1 for any positive u, we get
sup t∈[0,1] |λt − t| = sup t∈[0,1] t λt − λ0 t − 0 − 1 ≤ ekλk− 1 (2.1)
Now, as v ≤ ev− 1 for each v, we get
ρ(f, g) ≤ ed(f,g)− 1 (2.2)
So, it follows that if d(f, g) = 0, then ρ(f, g) = 0, and this implies that f = g because ρ is a metric. So, d is a metric on the space D[0, 1] too.
We denote with D the topology induced by the metric d, and we refer to it as the Skorohod topology. The space (D, D) is called the Skorohod space.
Our next goal is to prove that the two metrics are equivalent, i.e. they generate the same topology, and so, they have the same convergent sequences.
First of all we notice that, it follows immediately from (2.2) that if d(xn, x) → 0, then
ρ(xn, x) → 0 as well.
The converse’s proof is more complicated and relies on a following lemma.
Before introducing it, we need to define an analogous of the modulus of continuity for c`adl`ag functions.
The modified modulus of continuity
First of all, we recall the definition of modulus of continuity in the space C[0, 1] of con-tinuous functions on the unit interval, endowed with the uniform topology. Then, we show the necessary modifications in the space D[0, 1].
The modulus of continuity of an arbitrary function x ∈ C[0, 1] is defined by the quantity
wx(δ) = sup |s−t|≤δ
|x(s) − x(t)|, 0 < δ ≤ 1 (2.3)
Remark. A necessary and sufficient condition for x to be uniformly continuous is that the limit of the modulus of continuity is zero when δ goes to zero. This is a property of the modulus of continuity that we want to mimic in the Skohorod space.
We assume that our function x is not continuous, but that it belongs to the space D[0, 1], so it has discontinuities of the first kind.
For T ⊆ [0, 1], we can define
wx(T ) = sup s,t∈T
|x(s) − x(t)|. Then, the quantity (2.3), can be written as
wx(δ) = sup 0≤t≤1−δ
wx[t, t + δ]
Definition. A finite sequence of times {ti}, with 0 = t0 < t1 < · · · < tn = 1 is called
δ-sparse if it satisfies the following: sup
1≤i≤n
ti− ti−1> δ
Then, for 0 < δ < 1, we define the modified modulus of continuity of x as w0x(δ) := inf
{ti}
max
1≤i≤nwx[ti−1, ti) (2.4)
where the infimum runs over the δ-sparse sets.
Remark. Also in this case we have that x ∈ D[0, 1] if and only if lim
δ→0w 0
x(δ) = 0.
Remark. One could wonder whether this definition is coherent with the previous one. To see this, we can compare wx0 and wx.
Firstly, we remark that the interval [0, 1) can be split into subintervals [ti−1, ti) such
w0x(δ) ≤ wx(2δ).
Conversely, let j(x) = sup0<t≤1|x(t) − x(t−)| be the maximum jump of x; then, wx(δ) ≤ 2w0x(δ) + j(x).
To see this, let {ti} be a δ-sparse sequence such that wx[ti−1, ti) < wx0(δ) + ε for each
i. If the distance between s and t is less then δ, then they belong to the same interval [ti−1, ti) or to adjacent ones. In the first case, the distance between x(s) and x(t) is at
most w0x(δ) + ε, and in the second case it is at most 2w0x(δ) + ε + j(x). As ε goes to 0, we get wx(δ) ≤ 2w0x(δ) + j(x) and, since j(x) = 0 when x is continuous, we also have
wx(δ) ≤ 2wx0(δ). Then, when x is a continuous function, the two moduli of continuity
are the same. It is also clear that the relation wx(δ) ≤ 2w0x(δ) is false for a general
x ∈ D, therefore the definition extends coherently the previous one.
2.1.2 Separability and completeness of (D, d)
We can now state and prove the lemma that we will use to show the equivalence of the metrics previously defined.
Lemma 2.1.1. If δ ≤ 1/2 and ρ(x, y) ≤ δ2, then d(x, y) ≤ 4δ + w0x(δ), where x ∈ D. Proof. Let us take ε > 0 and a δ-sparse set {ti} such that
wx[ti−1, ti) < w0x(δ) + ε ∀i.
By assumption ρ(x, y) ≤ δ2, so there exists µ ∈ Λ such that sup t |x(t) − y(µ(t))| = sup t x(µ−1(t)) − y(t) < δ2 (2.5) and sup t |µ(t) − t| < δ2.
Our goal is to define λ ∈ Λ such that it is near to µ, but its graph has chords with slopes near to 1.
Let us take λ to agree with µ in ti and to be linear in between. Since the composition
µ−1λ fixes the ti and is increasing, t and µ−1λ(t) always lie in the same subinterval
[ti−1, ti). Then, by 2.5 and the choice of {ti} we have
|x(t) − y(λ(t))| ≤ x(t) − x(µ−1(λ(t))) + x(µ−1(λ(t))) − y(λ(t)) < w0x(δ) + ε + δ2 < 4δ + w0x(δ).
To complete the proof it is enough to prove that kλk≤ 4δ. It results that |λ(ti) − λ(t−1) − (ti− ti−1)| = |µ(ti) − µ(t−1) − (ti− ti−1)|
Since λ is defined by linear interpolation, we have |λ(t) − λ(s) − (t − s)| < 2δ|t − s|, s, t ∈ [0, 1] which leads to λ(t) − λ(s) t − s − 1 ≤ 2δ or, equivalently, log(1 − 2δ) ≤ log λ(t) − λ(s) t − s ≤ log(1 + 2δ) Since | log(1 ± u)| ≤ 2|u| for |u| ≤ 1/2, we can conclude.
Proposition 2.1.3. The metrics ρ and d are equivalent.
Proof. The result follows easily from Lemma 2.1.1 and a previous remark.
In fact we showed that if d(xn, x) → 0 as n → 0, then ρ(xn, x) → 0. The contrary is a
consequence of Lemma 2.1.1.
Remark. Since (D, ρ) is separable and this is a topological property, it follows that (D, d) is a separable space.
We can now prove the following important result. Theorem 2.1.1. The space (D, d) is complete.
Proof. It is sufficient to show that each Cauchy sequence with respect to d contains a subsequence that is d-convergent.
Let (xn) a Cauchy sequence with respect to d, then it contains a subsequence (yk) = (xnk)
such that d(yk, yk+1) < 2−k. This means that for all k there exist λk∈ Λ such that
sup t |yk(t) − yk+1(λk(t))| = sup t yk((λ−1(t))) − yk+1(t) < 2−k kλkk≤ 2−k (2.6)
The problem is to find a function y ∈ D and functions λn ∈ Λ for which kλnk→ 0 and
yn(λ−1(t)) → y(t) uniformly in t.
Since eu− 1 ≤ 2u for 0 ≤ u ≤ 1/2, it follows by 2.1 and 2.6 that for each n
sup t |λn+m+1◦ λm+n◦ · · · ◦ λn(t) − λn+m◦ · · · ◦ λn(t)| = sup t |λn+m+1(s) − s| ≤ 2−(n+m)
Thus, for each n, the sequence (λn+m◦ · · · ◦ λn) (indexed by m) is a sequence of functions
on [0, 1] with respect to the supremum norm on [0, 1]. Therefore, the sequence converges uniformly to a limit νn, which is continuous, non decreasing and such that νn(0) = 0
and νn(1) = 1. If we prove that kνnk is finite, it will follow that νn is strictly increasing
and hence ν ∈ Λ. In fact, it results that
logλn+m◦ · · · ◦ λn(t) − λn+m◦ · · · ◦ λn(s) t − s ≤ kλn+m◦ · · · ◦ λnk ≤ kλn+mk+ · · · + kλnk≤ 2−(n−1). And, as m → ∞, we have logνn(t) − νn(s) t − s ≤ 1 2n−1.
Now, we see that νn= νn+1◦ λ; as a consequence
sup t yk(νk−1(t)) − yk+1(νk+1−1 (t)) = sup t |yk(s) − yk+1(λk(s))| ≤ 2−k.
Therefore, yk◦ νk−1 is a Cauchy sequence on [0, 1] with respect to the supremum norm.
Let y be its limit; since,
sup t yk(νk−1(t)) − y(t) → 0 and kνkk→ 0 as k → ∞, we have that d(yk, y) → 0.
Remark. It is easy to verify that if d(xn, x) → 0 and x ∈ C[0, 1] then the convergence is
uniform, i.e. supt|xn(t) − x| → 0.
So, the topology induced by d on C[0, 1] is the uniform one.
2.1.3 Compactness
We turn now to the problem of characterizing compact sets in D. In view of developing a criterion for relative compactness of probability measures associated with stochastic processes having c`adl`ag paths, we need a characterization of the compact sets in D[0, 1] with the topology generated by the metric d.
Using the definition of the modified modulus of continuity wx0(δ), we can prove an analogue of the Ascoli-Arzel`a theorem.
Theorem 2.1.2. A set A is relatively compact in (D, d) if the following two conditions hold: i) sup x∈A sup t |x(t)| < ∞, ii) lim δ→0x∈Asupw 0 x(δ) = 0.
Remark. The converse implication holds true as well, but we won’t need it.
Proof. Since a complete and totally bounded set in a metric space is compact, it is sufficient to show that A is totally bounded, i.e. for each ε > 0 there exist finitely many balls of radius ε that cover A.
Let η > 0 and k large such that 1/k < η and wx0(1/k) < η for each x ∈ A. Let M = supx∈Asupt|x(t)| and
H = {−M + j/k : j ≤ 2kM }, so that H is a η-net for [−M, M ]. Let
B = {x ∈ D[0, 1] : x constant on each [(i − 1)/k, i/k) and takes values in H}. In particular, x(1) ∈ H. We first prove that B is a 2η-net for A with respect to ρ.
If x ∈ A, there exist t0, . . . , tn such that
t0 = 0, tn= 1, ti− ti−1> 1/k
and
wx[ti−1, ti) < η
for each i. Note that we must have n ≤ k. For each i, choose integers ji such that
ji/k ≤ ti < (ji+ 1)/k.
The ji’s are distinct, since the ti are at least 1/k apart from each other.
Define λ so that
λ(ji/k) = ti
and λ is linear on each interval [ji/k, ji+1/k]. Choose y ∈ B such that
|y(m/k) − x(λ(m/k))|
for each m ≤ k. Observe that each [m/k, (m + 1)/k) lies inside some interval of the form [ji/k, ji+1/k). Since λ is increasing, [λ(m/k), λ((m + 1)/k)) is contained in
[λ(ji/k), λ(ji+1/k)) = [ti, ti+1). The function x does not vary more than η over each
in-terval [ti, ti+1), so x(λ(t)) does not vary more than η over each interval [m/k, (m + 1)/k);
y is constant on each such interval, and hence sup
t
|y(t) − x(λ(t))| < 2η. We have
for each i. By the piecewise linearity of λ, supt|λ(t) − t| < η; thus ρ(x, y) < 2η.
We have proved that, given x ∈ A, there exist y ∈ B such that ρ(x, y) < 2η, or B is a 2η-net for A with respect to ρ.
Now let ε > 0 and choose δ > 0 small so that 4δ + w0x(δ) < ε for each x ∈ A. Set η = δ2/4. Choose B as above to be a 2η-net for A with respect to ρ. By Lemma 2.1.1, if ρ(x, y) < 2η < δ2, then d(x, y) ≤ 4δ + w0
x(δ) < ε. Therefore B is an ε-net for A with
respect to d.
2.2
Convergence of probability measures
We can now introduce some results of weak convergence on the space D[0, 1] with the Skorohod topology D. Since D is separable and complete under the metric d, we can use a characterization of the relative compactness due to the Prokhorov’s theorem, that we recall together with some definitions.
2.2.1 Some recalls
The results and definitions that follows are true for a generic separable, complete metric space.
Let (M, m) be a separable metric space, where M is a set and m a metric. Let M denote the σ-algebra of Borel sets associated with the topology induced on M by m. We recall some important definitions, and the fundamental characterization of relative compactness.
Definitions
• Weakly convergence
A sequence of probability measure (Pn)n∈Non (M, m) converges weakly to a
prob-ability measure P on (M, m) if and only if Z M f dPn−→ Z M f dP as n → ∞ for each bounded continuous function f : M → R.
• Relative compactness
A family of probability measure Π on (M, m) is (weakly) relatively compact if each sequence in Π has a subsequence that converges weakly to a probability measure P on (M, m).
• Tightness
compact set A in M such that
P (A) > 1 − ε for all P ∈ Π .
Theorem 2.2.1 (Prokhorov’s Theorem). A family of probability measures on (M, M) is tight if and only if it is relatively compact.
2.2.2 Tightness of probability measure on (D[0, 1], D)
The Prokhorov’s theorem allows us to focus our attention on the tightness of a family of measures. The characterization of compact sets in D, gives the following result. Theorem 2.2.2. Let {Xn} a sequence of stochastic processes whose paths are in D. Suppose for each ε and η, there exist n0, R and δ such that
i) P[wX0 n(δ) ≥ ε] ≤ η
ii) P[supt∈[0,1]|Xn(t)| ≥ R] ≤ η.
Then the family of probability measures associated to {Xn} is tight with respect to the topology of D[0, 1].
2.2.3 Stopping times and tightness: Aldous criterion
A very useful criterion for tightness is due to Aldous (1978).
It gives a sufficient condition for the tightness of the laws associated with stochastic processes having c`adl`ag paths, in terms of the processes’ behavior after stopping times.
The whole section is dedicated to the proof of the theorem.
Theorem 2.2.3 (Aldous’ Theorem). Let {Xn}n∈Nbe a sequence of stochastic processes with c`adl`ag paths. Then the probability measures induced on D[0, 1] by {Xn}n∈N are tight if the following two conditions hold:
lim
R→∞supn P(|X
n(t)| ≥ R) = 0 (2.7)
and
|Xn(τn+ δn) − Xn(τn)| (2.8)
converges to 0 in probability as n → ∞, where τn are stopping times for Xnwith respect
to the natural σ-fields and take only finitely many values, and δn are reals such that
Proof. We require τn to take values in [0, 1], it is technically convenient to regard each
x ∈ D[0, 1] as extended to [0, 2], by putting x(t) = x(1) for 1 ≤ t ≤ 2: this enables us to write Xn(τn+ δn) instead of Xn(min(1, τn+ δn)).
The proof is split is four steps.
Step 1. We want to see that condition (2.8) implies the following: for a given ε there exist n0, δ such that
P(|Xn(τn+ s) − Xn(τn)| ≥ ε) ≤ ε (2.9)
for each n ≥ n0, s ≤ 2δ, and τn a stopping time for Xn.
To see this, assume we can find an increasing subsequence {nk}, together with
stopping times τnk, and numbers snk ≤ 1/k for which the above relation is not
satisfied. Then, we can set
δnk = snk
Clearly, as (2.9) is not true, |Xn(τn+ δn) − Xn(τn)| cannot converge to 0 in
prob-ability, that is a contradiction.
Step 2. We now want to prove that, for a given ε > 0, and n ≥ n0 fixed, we get:
P(U ≤ T + δ, |Xn(U ) − Xn(T )| ≥ 2ε) ≤ 16ε (2.10) where U and T are two stopping times for Xn.
Let λ be the Lebesgue measure on the real line. Let
AT := {(ω, s) ∈ Ω × [0, 2δ] such that |Xn(T + s) − Xn(T )| ≥ ε}
Observe that, if we replace the stopping times τn by T in the relation 2.9, then,
for each s ≤ 2δ we get
P(ω : (ω, s ∈ AT)) ≤ ε
where we have used (2.9). Let now P×λ be the product measure of the probability P and the Lebesgue measure. Then, P×λ(AT) ≤ 2δε. This relation is of fundamental
importance in the understanding the set AT. In fact, we can use it to integrate on
the product space and getting so the aimed inequality.
Let BT(ω) defined as the slice in AT, that means BT(ω) := {s : (ω, s) ∈ AT}.
Then, by Fubini theorem, we get Z
λ(BT(ω))P(dω) ≤ 2δε
where we have used the result P × λ(AT) ≤ 2δε. Now, if CT is the set {ω :
If we repeat the same argument for the stopping time U , we obtain P(CU) ≤ 8ε,
and therefore P(CT ∪ CU) ≤ 16ε.
We now prove the inequality (2.10). Let ω 6∈ CT∪ CU; then, λ(BT(ω)) ≤ 1/4δ and
λ(BU(ω)) ≤ 1/4δ, by definition of BT and BU. Suppose U ≤ T + δ. Then,
λ{t ∈ [T, T + 2δ] : |Xn(t) − Xn(T )| ≥ ε} ≤ 1/4δ
and
λ{t ∈ [U, U + δ] : |Xn(t) − Xn(U )| ≥ ε} ≤ 1/4δ
Therefore, there exists t ∈ [T, T +2δ]∩[U, U +δ] such that |Xn(t)−Xn(T )| < ε and
|Xn(t) − Xn(U )| < ε, that means |Xn(U ) − Xn(T )| < 2ε. This implies inequality
(2.10).
Step 3 We obtain a bound on wX0 n.
Let Tn,0= 0 and
Tn,i+1= inf{t > Tn,i : |Xn(t) − Xn(Tn,i)| ≥ 2ε} ∧ 2.
Note that we have |Xn(Tn,i+1) − Xn(Tn,i)| ≥ 2ε if Tn,i< 2. We choose n0, δ as in
Step 1. By Step 2 with T = Tn,i and U = Tn,i+1 we have that
P (Tn,i+1− Tn,i< δ , Tn,i< 2) ≤ 16ε. (2.11)
Let K = [2/δ] + 1 and apply 2.9, with ε replaced by ε/K to see that there exist n1≥ n0 and ζ ≤ δ ∧ ε such that if n ≥ n1, s ≤ 2ζ, and τnis a stopping time, then
P (|Xn(τn+ s) − Xn(τn)| > ε/K) ≤ ε/K. (2.12)
By (2.10) with T = Tn,i and U = Tn,i+1 and δ replaced by ζ,
P(Tn,i+1≤ Tn,i+ ζ) ≤ 16ε/K (2.13)
for each i and hence
P(∃ i ≤ K : Tn,i+1≤ Tn,i+ ζ) ≤ 16ε. (2.14)
We have
E[Tn,i− Tn,i−1 , Tn,K < 1] ≥ δP(Tn,i− Tn,i−1≥ δ , Tn,K < 1)
≥ δ[P(Tn,K < 1) − P(Tn,i− Tn,i−1< δ , Tn,K < 1)]
where we used (2.13) in the last inequality. Summing over i from 1 to K, we obtain P (Tn,K < 1) ≥ E [Tn,K , Tn,K < 1] = K X i=1 E [Tn,i− Tn,i−1 , Tn,K < 1] ≥ Kδ[P(Tn,K < 1) − 16ε] ≥ 2[P(Tn,K < 1) − 16ε],
or P(Tn,K < 1) ≤ 32ε. Hence except for an event of probability at most 32ε, we
have wx0n(ζ) ≤ 4ε.
Step 4. In this step we obtain a bound on supt|Xn(t)|. Let ε > 0 and choose δ and n 0 as
in Step 1. Define
DR,n= {(ω, s) ∈ Ω × [0, 1] : |Xn(s)(ω)| > R}
for R > 0. The measurability of DR,nwith respect to the product σ-field F ×B[0, 1],
where B[0, 1] is the Borel σ-field on [0, 1], follows by the fact that Xn is right continuous with left limits. Let
G(R, s) = sup
n P(|X
n(s)| > R).
By (2.7) it follows that G(R, s) → 0 as R → ∞ for each s. Pick R large so that λ(s : G(R, s) > εδ) < εδ. Then Z 1DR,n(ω, s)P(dω) = P(|X n(s)| > R) ≤ ( 1, G(r, s) > εδ εδ, otherwise Integrating over s ∈ [0, 1], we get
P × λ(DR,n) < 2εδ.
If ER,n(ω) = {s : (ω, s) ∈ DR,n and FR,n= {ω : λ(ER,n) > δ/4}, we have
1 4δP(FR,n) = Z FR,n 1 4δP(dω) ≤ Z Z 1 0 1DR,n(ω, s)λ(ds)P(dω) ≤ 2εδ, thus P(FR,n) ≤ 8ε.
Define T = inf{t : |Xn(t)| ≥ R + 2ε} ∧ 2 and define AT, BT and CT as in Step 2.
We have
P(CT ∪ FR,n) ≤ 16ε.
If ω 6∈ CT∪FR,nand T < 2, then λ(ER,n(ω)) ≤ δ/4. Hence, there exists t ∈ [T, T +
2δ] such that |Xn(t)| ≤ R and |Xn(t) − Xn(T )| ≤ ε. Therefore |Xn(T )| ≤ R + ε, which contradicts the definition of T . We conclude that T must equal 2 on the complement of CT ∪ FR,n, or in other words, except for an event of probability at
most 16ε, we have supt|Xn(t)| ≤ R + 2ε, provided, of course, that n ≥ n 0.
This result is useful in a great number of applications; in the following chapter we will show how it is possible to apply this criterion to prove convergence to a macroscopic limit.
