elements of mathematical oncology franco flandoliusers.dma.unipi.it/~flandoli/parts_i_and_ii.pdf10...

Elements of Mathematical Oncology

Franco Flandoli

Contents

Part 1. Stochastic Differential Equations, linear Partial Differential Equations andtheir links 5

Chapter 1. Brownian motion and heat equation 71. Introduction 72. Simulations of Brownian motion 83. About the problem of finding a density from a sample 104. Macroscopic limit of Brownian motions 135. On the weak convergence of measures, in the random case 176. Heat equation as Fokker-Planck and Kolmogorov equation 187. Two-dimensional simulations 19

Chapter 2. SDEs and PDEs 231. Stochastic differential equations 232. Simulation of SDE in dimension one 253. Links between SDEs and linear PDEs 274. Simulations of the macroscopic limit 32

Part 2. Growth and change of species in populations of cells and nonlinear PartialDifferential Equations 37

Chapter 3. Examples of macroscopic systems in Mathematical Oncology 391. An advanced model of invasive tumor with angiogenesis 392. The Fisher-Kolmogorov-Petrovskii-Piskunov model 473. A probabilistic representation 524. Existence, uniqueness, invariant regions 565. The problem of invariant regions 61

Chapter 4. The mathematics of proliferation at the microscopic level 63

Part 3. Interacting systems of cells and nonlinear Partial Differential Equations 65

Bibliography 67

3

Part 1

Stochastic Differential Equations, linear PartialDifferential Equations and their links

CHAPTER 1

Brownian motion and heat equation

1. Introduction

We recall that, given a probability space (Ω,F , P ) and a filtration (Ft)t≥0, namely what we call afiltered probability space (Ω,F ,Ft, P ), a (continuous) Brownian motion is a stochastic process (Bt)t≥0with the following properties:

i) it is a continuous adapted processii) B0 = 0 a.s.iii) for every t ≥ s ≥ 0, the r.v. Bt −Bs is Gaussian N (0, t− s) and it is independent of Fs.If continuity is not prescribed, it can be proved that there is a continuous version. A Brownian

motion in Rd is a stochastic process (Bt)t≥0 with values in Rd, Bt =(B

(1)t , ..., B

(d)t

)such that its

components B(i)t are independent real valued Brownian motions.

We also recall that the heat equation in Rd, with diffusion constant k > 0, is the Partial DifferentialEquation (PDE)

∂ut∂t

= k∆ut, u|t=0 = u0.

Here u = ut = ut (x) denotes the solution, a function u : [0, T ] × Rd → R (we may replace [0, T ] by[0,∞)) and u0 denotes the initial condition, a function u0 : Rd → R.

This section is devoted to the description of a few properties of these objects and their links. Werestrict the attention to k = 1

2 for simplicity of notations and show the modifications in the generalcase at the end of the section.

1.1. Heat kernel. Let pt (x) be the function

pt (x) = (2πt)−d/2 exp(− |x|2 /2t

)defined for t > 0. It is called the heat kernel. We have

∂pt (x)

∂t=

1

2∆pt (x) .

Indeed,

∂pt (x)

∂t= −d/2 (2πt)−d/2−1 2π exp

(− |x|2 /2t

)+ (2πt)−d/2 exp

(− |x|2 /2t

)|x|2 /2t2

= pt (x)

(− d

2t+|x|2

2t2

)7

8 1. BROWNIAN MOTION AND HEAT EQUATION

∂ipt (x) = pt (x) (−xi/t)∂2i pt (x) = pt (x) (−xi/t)2 + pt (x) (−1/t)

= pt (x)

(x2i

t2− 1

t

).

Most of the results of this section depend on this simple computation. However, let us now restartprogressively from Brownian motion and investigate it also at a numerical level.

2. Simulations of Brownian motion

2.1. Simulation of a trajectory of Brownian motion. Let (Bt)t≥0, be a real valued Brownianmotion, on a filtered probability space (Ω,F ,Ft, P ). Let X0 be an F0-measurable random variable.Consider the stochastic process

Xt = X0 +Bt.

It will be the solution of the stochastic differential equation

dXt = dBt, X|t=0 = X0

when this concept will be clarified. The following simple code, written with the free software R,simulates a trajectory, in the spirit of explicit Euler scheme of discretization of stochastic differentialequations (n is the number of time steps; just to introduce one more R command, we use an uniformdistributed initial condition):

n = 10000; dt = 0.01; h = sqrt(dt)X = 1 : nX[1] = runif(1,−1, 1)for (t in 1 : (n− 1)) X[t+ 1] = X[t] + h ∗ rnorm(1)plot(X, type = ”l”, col = 3)lines((1 : n) ∗ 0)

Let us stress the role of the quantity h = sqrt(dt): the random variable corresponding to thesoftware command X[t+ 1]− X[t] is Xt+dt −Xt, which is equal to

Xt+dt −Xt = Bt+dt −Bthence it is, by definition of Brownian motion, a Gaussian N (0, dt). As such, it can be represented inthe form

Bt+dt −Bt =√dtZ

where Z is N (0, 1). This is why, in the code, X[t+ 1]− X[t] is settled equal to h ∗ rnorm(1). Noticefinally that the independence of increments of Brownian motion is reflected, in the code, by the factthat rnorm(1) computes independent values at each iteration of the "for" cycle.

2. SIMULATIONS OF BROWNIAN MOTION 9

2.2. Simulation of several independent Brownian motions. What we are going to do canbe equivalently described as the simulation of several Brownian trajectories, but in the spirit of systemsof many particles we prefer to think as we had several independent Brownian motions and we simulatea single trajectory for each one of them.

Consider a sequence(Bit

)t≥0, i = 1, 2, ... of independent Brownian motions on a filtered probability

space (Ω,F ,Ft, P ).Consider a sequence Xi

0, i = 1, 2, ... of random initial conditions, F0-measurable, independent andidentically distributed with law having density ρ0 (x) with respect to Lebesgue measure on Rd.

Consider the following simple differential equations

dXit = dBi

t, Xi|t=0 = Xi0

having solutionsXit = Xi

0 +Bit.

Let us see a picture with a myriad of colors:N = 500; n = 10000; dt = 0.01; sd0 = 0.5; h = sqrt(dt)X = matrix(nrow = N, ncol = n)X[, 1] = rnorm(N, 0, sd0)for (t in 1 : (n− 1)) X[, t+ 1] = X[, t] + h ∗ rnorm(N)plot(c(0, n), c(−20, 20))for (k in 1 : N) lines(X[k, ], col = k)

First, notice the shape of the envelop, roughly like√t, corresponding to the property V ar

[Bit

]= t.


We may look more closely the distribution of points at some time t. To have a better result, weincrease the number of points and avoid to produce the previous picture which is time consuming:

N=1000; n=10000; dt=0.01; sd0=0.5; h=sqrt(dt)X=matrix(nrow=N,ncol=n)X[,1]=rnorm(N,0,sd0)for (t in 1:(n-1)) X[,t+1]=X[,t]+h*rnorm(N)nn=10hist(X[,nn],30,FALSE)lines(density(X[,nn],bw=sd(X[,nn])/3))Her are two pictures corresponding to the time steps 10 and 1000:

These pictures arise the following question: when the number N of "particles" tends to infinity,does this profile converges to a well defined limit profile? And could we obtain the limit profile by aless time consuming method, for instance by solving a suitable equation?

To get a feeling of the practical problem, take N = 10000. We see that now the software, on anordinary laptop requires a non-negligible amount of time; and the profile is more and more regular. Ina gas or fluid, the number of molecules is of the order of 1020; in a living tissue affected by a cancer,the number of tumor cells may be of the order of 109; both are incredibly larger than N = 10000. Theprofile should be extremely regular, but not affordable by a direct simulation of the particle system.

3. About the problem of finding a density from a sample

In the previous section, given the valuesX1t , ..., X

Nt at some time t, we tried to represent graphically

these points by means of a probability density, which reflects their degree of concentration.

3. ABOUT THE PROBLEM OF FINDING A DENSITY FROM A SAMPLE 11

The histogram is the first easy way: the space is partitioned in cells (here intervals of equal length)and the number of point in each cell is counted. One can plot the histogram giving the number ofpoints per cell or its normalization with area one, to be compared with a probability density.

In order to obtain a more regular probability density function, there are various methods. One ofthem is particularly connected with certain theoretical investigation that we are going to perform inthese lectures: it is the "kernel smoothing" method. It starts from a "kernel", namely a probabilitydensity K (x). Then the kernel is rescaled by the formula (d is the spatial dimension)

Kε (x) := ε−dK(ε−1x

)and, given the points, we have to perform the average

ρN (x) :=1

N

N∑i=1

Kε

(x−Xi

t

).

Notice that ∫Kε (x) dx = 1

hence ∫ρN (x) dx =

1

N

N∑i=1

∫Kε

(x−Xi

t

)dx =

1

N

N∑i=1

∫Kε (x) dx = 1.

In other words, the resulting function ρN (x) is a probability density function.To implement this method one has to choose a kernel K (certain R commands choose a Gaussian

kernel by default), but the result is only mildly affected by this choice. What really makes a differenceis the choice of the rescaling factor ε, also called the "bandwidth". The general idea is that for largeε we get a rather flat and very smooth profile ρN (x); for small ε the profile oscillates. One has tochoose an intermediate value of ε, which is the most diffi cult problem of the implementation of themethod (as it is the choice of the cells in the histogram). Wrong choices really give results that arenot acceptable, too far from reality.

It is quite natural to expect that the value of the bandwidth is related to the standard deviation ofpoints. However, the simple choice ε =sd(data) does not give the best result. Correction by a factorimprove the result, but the choice of the factor is not easy. There are ad hoc rules, quite incredible,like the one at the help page ?bw.nrd or software R. Below we propose to use sd(data)/5, as a trial,but it is not always the best.

Concerning different variants of implementation of the kernel smoothing idea, see the help of R un-der the names ?ksmooth, ?bkde, ?density and for instance the paper at http://vita.had.co.nz/papers/density-estimation.pdf.

The reader is suggested to try the following exercise: a Gaussian sample is generated, the histogramis plotted and, optionally, also the true density that generated the sample is plotted over the histogram.Then, one can over-plot the density given by a method of kernel smoothing, for different values of thebandwidth. Here is an example of code:


Z=rnorm(10000,0,10)hist(Z,50,FALSE)Z0=sort(Z)Y=dnorm(Z0,0,10)lines(Z0,Y)lines(density(Z,bw=sd(Z)/5),col="red")

and here are two examples with wrong choices of the bandwidth (we write only the last line):lines(density(Z,bw=0.1),col="red"), lines(density(Z,bw=10),col="red")

4. MACROSCOPIC LIMIT OF BROWNIAN MOTIONS 13

4. Macroscopic limit of Brownian motions

First of all let us introduce the so called empirical measure

SNt :=1

N

N∑i=1

δXit.

It is a random probability measure on Borel sets, convex combination of random delta Dirac measures.At the position Xi

t of each single particle we put a pointwise mass of size1N . By random probability

measure we may loosely just mean a probability measure depending on ω ∈ Ω; more rigorously wemust explain in which sense we mean the measurability in ω and the simplest way it to say that∫φ (x)SNt (dx) must be measurable, for each φ ∈ Cb

(Rd).

If we imagine thus measure as a bunch of very small point masses, one one side we may get thefeeling that the global mass is more concentrated here than there, but on the other side we do not seeany profile, similar to a Gaussian or others. The only "altitude" we see is 1

N , in a sense.To extract a profile we may mollify the atomic measure SNt by a convolution with a kernel: given

a probability density θ (x), setting

θε (x) := ε−dθ(ε−1x

)we perform the convolution

uN (x) :=(θε ∗ SNt

)(x) :=

∫Rdθε (x− y)SNt (dy) .


The function uN (x) is a regularized version of SNt and it is a probability density (the proof is thesame made above for K). We have

(θε ∗ SNt

)(x) =

1

N

N∑i=1

θε(x−Xi

t

)hence this operation coincides with the kernel smoothing method described above (with K = θ).

After these preliminaries, we ask the following question: does SNt converge, as N →∞, to a limitprobability measure, maybe having a density ρt (x)? Under the previous assumptions, this is true.Dealing with convergence of measures, let us recall that we call weak convergence the property

(4.1) limN→∞

∫Rdφ (x)SNt (dx) =

∫Rdφ (x) ρt (x) dx

when it holds for every continuous bounded function φ.

Theorem 1. Under the assumptions specified at the beginning of Section 2.2,for every test functionφ ∈ Cb

(Rd)we have property (4.1) in the sense of almost sure convergence. Moreover,

ρt (x) =

∫Rdpt (x− y) ρ0 (y) dy

where pt (x) is the density of a Brownian motion in Rd, namely pt (x) = (2πt)−d/2 exp(− |x|2 /2t

).

Finally, the function ρt (x) is smooth for t > 0 and satisfies the Cauchy problem for the heat equation

∂ρt∂t

=1

2∆ρt, ρ|t=0 = ρ0

(the initial condition is assumed as a limit in L1(Rd)as t→ 0)

Proof. Given t, the r.v. Xit are i.i.d. and thus the same is true for the r.v. φ

(Xit

), with

φ ∈ Cb(Rd); moreover, by the boundedness of φ, the r.v. φ

(Xit

)have finite moments. Therefore, by

the strong Law of Large Numbers,

1

N

N∑i=1

φ(Xit

)→ E

[φ(X1t

)]in the sense of almost sure convergence.

Since Xit = Xi

0 +Bit and since the terms X

i0 and B

it are independent, with densities ρ0 and pt (x)

respectively, then also Xit has density, given by the convolution of ρ0 and pt (x), density we have

denoted above by ρt (x). Moreover

E[φ(X1t

)]=


and thus property (4.1) is fully proved, along with the convolution formula for ρt (x).Finally, it is by the computation of Section 1.1 it easily follows that ρt (x) satisfies the Cauchy

problem for the heat equation.

4. MACROSCOPIC LIMIT OF BROWNIAN MOTIONS 15

At the beginning of this section we have introduced also the regularizations(θε ∗ SNt

)(x). Do they

converge too, to ρt (x)? This question corresponds more closely to the problem stated at the end ofSection 2.2. If we keep ε fixed, the answer is a trivial consequence of the last theorem (under theassumption that θ is also bounded continuous):

limN→∞

(θε ∗ SNt

)(x) = (θε ∗ ρt) (x)

for every x ∈ Rd. We do not get ρt (x), if we keep ε constant.More interesting is to link ε and N , namely examin the limit limN→∞

(θεN ∗ SNt

)(x) for suitable

sequences εN . WhenlimN→∞

(θεN ∗ SNt

)(x) = ρt (x)?

Let us see an example of result. Assume θ ∈ C∞c(Rd), θ ≥ 0,

∫θ (x) dx = 1, to use the most common

results on convergence of mollifiers (we use in particular the fact that θεN ∗f → f uniformly on compactsets if f is continuous), but the next result is true in larger generality using appropriate extensions.

Theorem 2. If

limN→∞

ε−dNN

= 0.

then, for every t > 0 and x ∈ Rd, we have

limN→∞

E[∣∣(θεN ∗ SNt ) (x)− ρt (x)

∣∣2] = 0.

Proof. First, we have (ρt (x) is not the average of θεN(x−Xi

t

))

E[∣∣(θεN ∗ SNt ) (x)− ρt (x)

∣∣2] = E

∣∣∣∣∣ 1

N

N∑i=1

θεN(x−Xi

t

)− ρt (x)

∣∣∣∣∣2

≤ 2E

∣∣∣∣∣ 1

N

N∑i=1

(θεN

(x−Xi

t

)− E

[θεN

(x−Xi

t

)])∣∣∣∣∣2

+ 2

∣∣∣∣∣ 1

N

N∑i=1

E[θεN

(x−Xi

t

)]− ρt (x)

∣∣∣∣∣2

.

About the first term, using the independence between the r.v. θεN(x−Xi

t

)and the property E

[θεN

(x−Xi

t

)− E

[θεN

(x−Xi

t

)]]=

0, we may delete the mixed terms and write

E

∣∣∣∣∣ 1

N

N∑i=1

(θεN

(x−Xi

t

)− E

[θεN

(x−Xi

t

)])∣∣∣∣∣2 =

1

N2

N∑i=1

E[∣∣θεN (x−Xi

t

)− E

[θεN

(x−Xi

t

)]∣∣2]which, being θεN

(x−Xi

t

)equally distributed, is equal to

=1

NE[∣∣θεN (x−X1

t

)− E

[θεN

(x−X1

t

)]∣∣2] .


For the same reason the second term is controlled by

E[∣∣(θεN ∗ SNt ) (x)− ρt (x)

∣∣2]≤ 2

1

NE[∣∣θεN (x−X1

t

)− E

[θεN

(x−X1

t

)]∣∣2]+ 2∣∣E [θεN (x−X1

t

)]− ρt (x)

∣∣2≤ 4

NE[θεN

(x−X1

t

)2]+

4

NE[θεN

(x−X1

t

)]2+ 2

∣∣E [θεN (x−X1t

)]− ρt (x)

∣∣2 .Notice that

E[θεN

(x−X1

t

)]=

∫RdθεN (x− y) ρt (y) dy

(we have proved above that X1t has density ρt). Moreover, we have

limN→∞

∫RdθεN (x− y) ρt (y) dy = ρt (x)

uniformly in x on compact sets, for t > 0. Hence 2∣∣E [θεN (x−X1

t

)]− ρt (x)

∣∣2 converges to zero. Thesequence

E[θεN

(x−X1

t

)]2=

(∫RdθεN (x− y) ρt (y) dy

)2

is, for the same reason, bounded, hence the term 4NE

[θεN

(x−X1

t

)]2 converges to zero. It remainsto deal with the term 4

NE[θεN

(x−X1

t

)2]. We haveE[θεN

(x−X1

t

)2]=

∫Rdθ2εN

(x− y) ρt (y) dy = ε−dN

∫Rdε−dN θ2

(ε−1N (x− y)

)ρt (y) dy

≤ ε−dN ‖ρt‖∞∫Rdε−dN θ2

(ε−1N (x− y)

)dy = ε−dN ‖ρt‖∞

∫Rdθ2 (z) dz

hence4

NE[θεN

(x−X1

t

)2] ≤ ε−dNN

(4 ‖ρt‖∞

∫Rdθ2 (z) dz

).

This term goes to zero if we assume ε−dNN → 0.

Remark 1. The condition limN→∞ε−dNN = 0 has a very simple interpretation. First, the law from

which the particles are extracted is almost compact support (as any probability law). To simplify theargument, assume it has support of linear size 1. If we have N particles, and we thin for simplicitythat they are almost uniformly distributed in the support, the distance between closest neighbors is ofthe order N−1/d. If the bandwidth εN is of this order or smaller, the average performed by the kernelis made only on a finite numebr of particles, sometimes maybe even zero particles, so it fluctuatesrandomly. If, on the contrary, εN is much bigger than N−1/d, then we average over a large number ofparticles and a sort of LLN is active.

5. ON THE WEAK CONVERGENCE OF MEASURES, IN THE RANDOM CASE 17

5. On the weak convergence of measures, in the random case

Above we have proved that, for every φ ∈ Cb(Rd), we have a.s.

limN→∞

∫Rdφ (x)SNt (dx) =

∫Rdφ (x) ρt (x) dx.

The event of zero probability where convergence does not hold may depend on φ. Hence, a priori, wecannot say that, for P -a.e. ω ∈ Ω, the sequence of measures

SNt (ω)

N∈N converges weakly, since

this property precisely means: there exists an event Ω0 ⊂ Ω with P (Ω0) = 1 such that for all ω ∈ Ω0

and all φ ∈ Cb(Rd)we have

limN→∞

∫Rdφ (x)SNt (ω) (dx) =

∫Rdφ (x) ρt (x) dx.

Until now, we have a set Ωφ0 for each φ with a similar property and their intersection is not under

control. The result is true.

Corollary 1. There exists an event Ω0 ⊂ Ω with P (Ω0) = 1 such that, for all ω ∈ Ω0,SNt (ω) (dx) converges weakly to ρt (x) dx.

Proof. The idea is to prove the assertion first for a dense countable set of test functions and thenextend to all test functions by an estimate. However, let us see the details.

Let φαα∈N be a dense sequence in Cc(Rd)(the set of compact support continuous functions),

density measured with respect to the uniform convergence (notice that Cb(Rd), on the contrary, would

not be separable). Since, for each α ∈ N, we have (4.1) with φ = φα, being a countable numebr ofproperties we may say that there exists an event Ω0 ⊂ Ω with P (Ω0) = 1 such that, for all ω ∈ Ω0

and every α ∈ N, we have

(5.1) limN→∞

∫Rdφα (x)SNt (ω) (dx) =

∫Rdφα (x) ρt (x) dx.

For every φ ∈ Cc(Rd)and α ∈ N we have∣∣∣∣∫

Rdφ (x)SNt (ω) (dx)−


∣∣∣∣≤∫Rd|φα (x)− φ (x)|SNt (ω) (dx)

+

∣∣∣∣∫Rdφα (x)SNt (ω) (dx)−

∫Rdφα (x) ρt (x) dx

∣∣∣∣+

∫Rd|φα (x)− φ (x)| ρt (x) dx

≤ ‖φα − φ‖∞(∫

RdSNt (ω) (dx) +

∫Rdρt (x) dx

)+



∣∣∣∣ .


Recall that∫Rd S

Nt (ω) (dx) = 1,

∫Rd ρt (x) dx = 1. Given φ ∈ Cc

(Rd), givern ε > 0, let α ∈ N be such

that ‖φα − φ‖∞ ≤ ε2 . Then∣∣∣∣∫

Rdφ (x)SNt (ω) (dx)−


∣∣∣∣≤ ε+



∣∣∣∣ .Therefore, recalling (5.1), for every ω ∈ Ω0 we have

lim supN→∞

∣∣∣∣∫Rdφ (x)SNt (ω) (dx)−


∣∣∣∣ ≤ ε.Since ε > 0 is arbitrary and the limsup does not depend upon ε, we deduce that the limsup is equalto zero. Finally, recall that, if a sequence of probability measures converges to a probability measures(this is essential, and here it is true) over all test functions of Cc

(Rd), the it converges weakly (namely

over all test functions of Cb(Rd)). The proof is complete.

Remark 2. A similar result is true if we replace a.s. convergence with convergence in probability.

6. Heat equation as Fokker-Planck and Kolmogorov equation

The heat equation is related to Bownian motion in three ways: as macroscopic limit (the theo-rems above), as a Fokker-Planck equation; as a Kolmogorov equation. Let us see here the last twointerpretations.

6.1. Heat equation as Fokker-Planck equation. We shall see below in Section 3.1 the generalmeaning of Fokker-Planck equation. The heat equation is a particular case of it. In Section 3.1 wedeal with measure-valued solutions. Here, due to the simplicity of the particular case, we may dealwith regular solutions of the Fokker-Planck equation.

Consider the equationdXt = dBt, X|t=0 = X0

where X0 has density ρ0 (x). The result is: the density ρt (x) of Xt is solution of the Cauchy problemfor the heat equation

∂ρt∂t

=1

2∆ρt, ρ|t=0 = ρ0.

We have already proved this result in Theorem 1. This section does not present a new result but onlyinsists on the link between SDEs and PDEs which states (under appropriate assumptions) that thelaw of the solution to an SDE satisfies (in a suitable sense) a PDE.

6.2. Probabilistic representation formula. Finally, let us see that, for the heat equation∂ut∂t

=1

2∆ut, u|t=0 = u0

one has the following probabilistic representation formula, in terms of a Brownian motion Bt:

ut (x) = E [u0 (x+Bt)] .

7. TWO-DIMENSIONAL SIMULATIONS 19

Indeed,

E [u0 (x+Bt)] =

∫u0 (x+ y) pt (y) dy

=

∫pt (x− y)u0 (y) dy

and we know that this expression gives a solution of the heat equation. This link is a particular caseof the link between SDEs and Kolmogorov equations, studied below in Section 3.2.

7. Two-dimensional simulations

We complete the first Chapter by a few simulations of Brownian motion in dimension 2, whichwill come back in subsequent chapters.

7.1. Brownian motion in 2D. The next code is a way to simulate a trajectory of a 2D Brownianmotion:

n=10000; dt=0.01; h=sqrt(dt)X=1:n; Y=1:nX[1]=0; Y[1]=0for (t in 1:(n-1)) X[t+1]=X[t]+h*rnorm(1)Y[t+1]=Y[t]+h*rnorm(1)plot(X,Y,type="l", col=3); abline(h=0); abline(0,1000)

The following two variants show a movie, in two different forms:n=10000; dt=0.01; h=sqrt(dt)X=1:n; Y=1:n


X[1]=0; Y[1]=0plot(c(-10,10),c(-10,10))abline(h=0)abline(0,1000)for (t in 1:(n-1)) X[t+1]=X[t]+h*rnorm(1)Y[t+1]=Y[t]+h*rnorm(1)lines(X[1:t],Y[1:t],type="l", col=3)..........................................................n=10000; dt=0.01; h=sqrt(dt)X=1:n; Y=1:nX[1]=0; Y[1]=0for (t in 1:(n-1)) X[t+1]=X[t]+h*rnorm(1)Y[t+1]=Y[t]+h*rnorm(1)plot(c(-10,10),c(-10,10))abline(h=0)abline(0,1000)lines(X[1:t],Y[1:t],type="l", col=3)

7.2. Several Brownian motions in 2D. With several Brownian motions, we cannot plot any-more the full trajectores, the picture would be too full. We may, for instance, plot the final positions:

n=1000; N=1000; dt=0.01; h=sqrt(dt)X=matrix(nrow=N,ncol=n); Y=XX[,1]=rnorm(N,0,1); Y[,1]=rnorm(N,0,1)for (t in 1:(n-1)) X[,t+1]=X[,t]+h*rnorm(N)Y[,t+1]=Y[,t]+h*rnorm(N)plot(c(-10,10),c(-10,10)); lines(X[,n],Y[,n],type="p", col=1)abline(h=0); abline(0,1000)

7. TWO-DIMENSIONAL SIMULATIONS 21

This is a simple example of particle system. With a great degree of abstraction. we could thinkof it as a set of cancer cells, embedded into a tissue or in vitro.

We would like to see the motion of the particles. We use a number of tricks:i) we plot only a few times, by means of the commands T=500, if(t%%T==0 ), etc.;ii) we clean the previous points with the command polygon(c(-10,10,10,-10),c(-10,-10,10,10),col="white",

border=NA).Moreover, the method sometimes works poorly, it depends on the values of T compared to the

complexity of the specific code. The example below has been tuned to work.n=100000; N=1000; dt=0.0001; h=sqrt(dt)X=matrix(nrow=N,ncol=n); Y=XX[,1]=rnorm(N,0,1); Y[,1]=rnorm(N,0,1)T=500plot(c(-10,10),c(-10,10),type="n")for (t in 1:(n-1)) X[,t+1]=X[,t]+h*rnorm(N)Y[,t+1]=Y[,t]+h*rnorm(N)if(t%%T==0 )polygon(c(-10,10,10,-10),c(-10,-10,10,10),col="white", border=NA)lines(X[,t+1],Y[,t+1],type="p", col=1)abline(h=0)abline(0,1000)

CHAPTER 2

SDEs and PDEs

1. Stochastic differential equations

1.1. Definitions. We call stochastic differential equation (SDE) an equation of the form

(1.1) dXt = b (t,Xt) dt+ σ (t,Xt) dBt, X|t=0 = X0

where (Bt)t≥0 is a d-dimensional Brownian motion on a filtered probability space (Ω,F ,Ft, P ), X0 isF0-measurable, b : [0, T ] × Rd → Rd and σ : [0, T ] × Rd → Rd×d have some regularity specified caseby case and the solution (Xt)t≥0 is a d-dimensional continuous adapted process. The meaning of theequation is the identity

(1.2) Xt = X0 +

∫ t

0b (s,Xs) ds+

∫ t

0σ (s,Xs) dBs

where we have to assume conditions on b and σ which guarantees that s 7→ b (s,Xs) is integrable,s 7→ σ (s,Xs) is square integrable, with probability one;

∫ t0 σ (s,Xs) dBs is an Itô integral, and more

precisely it is its continuous version in t; and the identity has to hold uniformly in t, with probabilityone. The generalization to different dimensions of B and X is obvious; we take the same dimensionto have less notations.

Even if less would be suffi cient with more arguments, let us assume that b and σ are at leastcontinuous, so that the above mentioned conditions of integrability of s 7→ b (s,Xs) and s 7→ σ (s,Xs)are fulfilled.

In most cases, if X0 = x0 is deterministic, when we prove that a solution exists, we can alsoprove that it is adapted not only to the filtration (Ft) but also to

(FBt), the filtration associated

to the Brownian motion; more precisely, its completion. This is just natural, because the input ofthe equation is only the Brownian motion. However, it is so natural if implicitly we think to have asuitable uniqueness. Otherwise, in principle, it is diffi cult to exclude that one can construct, maybein some artificial ways, a solution which is not B-adapted. Indeed it happens that there are relevantexamples of stochastic equations where solutions exist which are not B-adapted. This is the origin ofthe following definitions.

Definition 1 (strong solutions). We have strong existence for equation (1.1) if, given any filteredprobability space (Ω,F ,Ft, P ) with a Brownian motion (Bt)t≥0, given any deterministic initial condi-tion X0 = x0, there is a continuous Ft-adapted process (Xt)t≥0 satisfying (1.2) (in particular, we maychoose (Ft) =

(FBt)and have a solution adapted to B). A strong solution is a solution adapted to(

FBt).

23

24 2. SDES AND PDES

Definition 2 (weak solutions). Given a deterministic initial condition X0 = x0, a weak solutionis the family composed of a filtered probability space (Ω,F ,Ft, P ) with a Brownian motion (Bt)t≥0 anda continuous Ft-adapted process (Xt)t≥0 satisfying (1.2).

In the definition of weak solution, the filtered probability space and the Brownian motion are notspecified a priori, they are part of the solution; hence we are not allowed to choose (Ft) =

(FBt).

When X0 is random, F0-measurable, the concept of weak solution is formally in trouble becausethe space where X0 has to be defined is not prescribed a priori. The concept of strong solution can beadapted for instance replacing

(FBt)with

(FBt ∨ F0

), or just saying that, if (Xt)t≥0 is a solution on

a prescribed space (Ω,F ,Ft, P ) where X0 and B are defined, then it is a strong solution. If we wantto adapt the definition of weak solution to the case of random initial conditions, we have to prescribeonly the law of X0 and put in the solution the existence of X0 with the given law.

Let us come to uniqueness. Similarly to existence, there are two concepts.

Definition 3 (pathwise uniqueness). We say that pathwise uniqueness holds for equation (1.1)if, given any filtered probability space (Ω,F ,Ft, P ) with a Brownian motion (Bt)t≥0, given any deter-

ministic initial condition X0 = x0, if(X

(1)t

)t≥0

and(X

(2)t

)t≥0

are two continuous Ft-adapted processwhich fulfill (1.2), then they are indistinguishable.

Definition 4 (uniqueness in law). We say that there is uniqueness in law for equation (1.1) if,given two weak solutions on any pair of spaces, their laws coincide.

1.2. Strong solutions. The most classical theorem about strong solutions and pathwise unique-ness holds, as in the deterministic case, under Lipschitz assumptions on the coeffi cients. Assume thereare two constants L and C such that∣∣b (t, x)− b

(t, x′

)∣∣ ≤ L ∣∣x− x′∣∣∣∣σ (t, x)− σ(t, x′

)∣∣ ≤ L ∣∣x− x′∣∣|b (t, x)| ≤ C (1 + |x|)|σ (t, x)| ≤ C (1 + |x|)

for all values of t and x. The second condition on b and σ is written here for sake of generality, but iswe assume, as said above, that b and σ are continuous, it follows from the first condition (the uniformin time Lipschitz property).

Theorem 3. Under the previous assumptions on b and σ, there is strong existence and pathwiseuniqueness for equation (1.1). If, for some p ≥ 2, E [|X0|p] <∞, then

E

[supt∈[0,T ]

|Xt|p]<∞.

Proof. ...

2. SIMULATION OF SDE IN DIMENSION ONE 25

1.3. Weak solutions. Let us see only a particular example of result about weak solutions. As-sume that σ is constant and non-degenerate; for simplicity, assume it equal to the identity, namelyconsider the SDE with additive noise

dXt = b (t,Xt) dt+ dBt.

Moreover, assume b only measurable and bounded (or continuous and bounded if we prefer to maintainthe general assumption of continuity). The key features of these assumptions are: the noise is non-degenerate (hence more restrictive than above for strong solutions) but b is very weak, much weakerthan the usual Lipschitz case. Under such assumption on b, if we do not have the noise dBt in theequation, it is easy to make examples without existence or without uniqueness.

Theorem 4. Under these assumptions, for every x0 ∈ Rd, there exists a weak solution and it isunique in law.

2. Simulation of SDE in dimension one

2.1. Linear example. Consider the equation, with λ > 0,

dXt = −λXtdt+ σdBt, X0 = x.

Its Euler discretization, on intervals of constant amplitude, has the theoretical form

Xtn+1 −Xtn = −λXtndt+ σ√dtBtn+1 −Btn√

dtdt := tn+1 − tn

where the r.v.’s

Zn =Btn+1 −Btn√

dt

are standard Gaussian and independent. The algorithmic form can be the following one; first weconstruct a function, the drift, then we write the main part of the code:

drift=function(x) -xn=10000; dt=0.01; h=sqrt(dt); sig=1X=1:n; X[1]=1for (t in 1:(n-1)) X[t+1]=X[t]+ dt*drift(X[t]) + h*sig*rnorm(1)plot(X,type="l", col=2); abline(h=0); abline(0,1000)

26 2. SDES AND PDES

Exercise 1. Try with other initial conditions and other values of λ and σ.

2.2. Nonlinear example. Consider the equation (called "two-well-potential")

dXt =(Xt −X3

t

)dt+ σdBt, X0 = x.

Its Euler discretization is

Xtn+1 −Xtn =(Xtn −X3

tn

)dt+ σ

√dtZn

dt := tn+1 − tn, Zn =Btn+1 −Btn√

dt.

The code is:drift=function(x) x-x^3n=10000; dt=0.01; h=sqrt(dt); sig=0.5X=1:n; X[1]=1for (t in 1:(n-1)) X[t+1]=X[t]+ dt*drift(X[t]) + h*sig*rnorm(1)plot(X,type="l", col=4); abline(h=0); abline(0,1000)

3. LINKS BETWEEN SDES AND LINEAR PDES 27

Exercise 2. Try with other initial conditions and other values of σ and n.

2.3. Important exercise. In both cases of the two examples above, plot an istogram and afitted (non-parametric) continuous density of the distribution at time t (for instance for t = 1, 10, 50,100).

3. Links between SDEs and linear PDEs

3.1. Fokker-Planck equation. Along with the stochastic equation (1.1) defined by the coeffi -cients b and σ, we consider also the following parabolic PDE on [0, T ]× Rd:

(3.1)∂p

∂t=

1

2

∑∂i∂j (aijp)− div (pb) , p|t=0 = p0

called Fokker-Planck equation. Herea = σσT .

Although in many cases it has regular solutions, in order to minimize the theory it is convenient tointroduce the concept of measure-valued solution µt; moreover we restrict to the case of probabil-ity measures. To give a meaning to certain integrals below, we assume (beside other assumptionsdepending on the result)

b, σ bounded continuous

but it will be clear that more cumbersome results can be done by little additional conceptual effort.We loosely write

(3.2)∂µt∂t

=1

2

d∑i,j=1

∂i∂j (aijµt)− div (µtb) , µ|t=0 = µ0

28 2. SDES AND PDES

but we mean the following concept. By 〈µt, φ〉 we mean∫Rd φ (x)µt (dx). By Cb

([0, T ]× Rd

)we

denote the space of bounded continuous functions ϕ : [0, T ] × Rd → R and by C1,2b

([0, T ]× Rd

)the

space of functions ϕ such that ϕ, ∂φ∂t , ∂iφ, ∂i∂jφ ∈ Cb([0, T ]× Rd

).

Definition 5. A measure-valued solution of the Fokker-Planck equation (3.2) is a family ofBorel probability measures (µt)t∈[0,T ] on Rd such that t 7→ 〈µt, ϕ (t, .)〉 is measurable for all ϕ ∈Cb([0, T ]× Rd

)and

〈µt, φ (t, ·)〉 − 〈µ0, φ (0, ·)〉 =

∫ t

0

⟨µs,

∂φ∂s

+1

2

d∑i,j=1

aij∂i∂jφ+ b · ∇φ

(s, ·)⟩ds

for every φ ∈ C1,2b

([0, T ]× Rd

).

Theorem 5 (existence). The law µt of Xt is a measure-valued solution of the the Fokker-Planckequation (3.2).

Proof. Let φ be of class C1,2b

([0, T ]× Rd

). By Itô formula for φ (t,Xt), we have

dφ (t,Xt) =∂φ

∂t(t,Xt) dt+∇φ (t,Xt) · dXt +

1

2

d∑i,j=1

∂i∂jφ (t,Xt) aij (t,Xt) dt

=∂φ

∂t(t,Xt) dt+∇φ (t,Xt) · b (t,Xt) dt+∇φ (t,Xt) · σ (t,Xt) dBt +

1

2

d∑i,j=1

∂i∂jφ (t,Xt) aij (t,Xt) dt.

We have E∫ T

0 |∇φ (t,Xt) · σ (t,Xt)|2 dt < ∞ (we use here that σ and ∇φ are bounded), henceE∫ t

0 ∇φ (s,Xs) · σ (s,Xs) dWs = 0 and thus (all terms are finite by the boundedness assumptions)

E [φ (t,Xt)]− E [φ (0, X0)] = E

∫ t

0

∂φ

∂s(s,Xs) ds+ E

∫ t

0∇φ (s,Xs) · b (s,Xs) ds

+1

2

d∑i,j=1

E

∫ t

0∂i∂jφ (s,Xs) aij (s,Xs) ds.

Since E [φ (t,Xt)] =∫Rd φ (t, x)µt (dx) (and similarly for the other terms) we get the weak formulation

of equation (3.2). The preliminary property that t 7→ 〈µt, ϕ (t, .)〉 = E [ϕ (t,Xt)] is measurable for allϕ ∈ Cb

([0, T ]× Rd

)is easy.

Remark 3. Under suitable assumptions, like the simple case when aij is the identity matrix, if µ0

has a density p0 then also µt has a density p (t, ·), often with some regularity gained by the parabolicstructures, and thus the Fokker-Planck equation in the differential form (3.1) holds.


Theorem 6 (uniqueness). Assume that the backward parabolic equation (called backward Kol-mogorov equation)

∂u

∂t+

1

2

d∑i,j=1

aij∂i∂ju+ b · ∇u = 0 on [0, T0]× Rd

u|t=T0 = ψ

has, for every T0 ∈ [0, T ] and ψ ∈ C∞c(Rd), at least one solution u of class C1,2

b

([0, T0]× Rd

). Then

the Fokker-Planck equation (3.2) has at most one measure-valued solution.

Proof. If µt is a measure-valued solution of the Fokker-Planck equation and u is a C1,2b

([0, T0]× Rd

)solution of the Kolmogorov equation, then (from the identity which defines measure-valued solutionsand the identity of Kolmogorov equation)

〈µT0 , ψ〉 − 〈µ0, u (0, ·)〉 .

Then, if µ(i)t , i = 1, 2, are two measure-valued solutions of the Fokker-Planck equation with the same

initial condition µ0, we have ⟨µ

(1)T0, ψ⟩

=⟨µ

(2)T0, ψ⟩.

This identity holds for every ψ ∈ C∞c(Rd), hence µ(1)

T0= µ

(2)T0. The time T0 ∈ [0, T ] is arbitrary, hence

µ(1)· = µ

(2)· .

Obviously the weak aspect of the previous uniqueness result is the assumption, not explicit interms of the coeffi cients. The reason is that there are two main cases when such implicit assumptionis satisfied. One is the case when b and σ are very regular; the other is when σ is non-degenerate. Asan example of the second case, let us mention the fundamental case of heat equation.

Example 1. Consider the case b = 0, σ = Id. The forward Kolmogorov equation

∂v

∂t=

1

2

d∑i,=1

∂2i v on [0, T0]× Rd

v|t=0 = ψ

has the explicit solution

v (t, x) =

∫pt (x− y)ψ (y) dy

which is infinitely differentiable with all bounded derivatives. The function u (t, x) = v (T0 − t, x)is then an explicit and regular solution of the backward Kolmogorov equation above. In this case,therefore, the Fokker-Planck equation has a unique measure-valued solution.

30 2. SDES AND PDES

3.2. Backward Kolmogorov equation. Along with the stochastic equation (1.1) defined bythe coeffi cients b and σ, we consider also the following backward parabolic PDE on [0, T ]×Rd, calledbackward Kolmogorov equation:

∂u

∂t+

1

2

d∑i,j=1

aij∂i∂ju+ b · ∇u = 0, u|t=T = ψ.

To express in full generality the relation with the SDE we have to introduce the SDE on the timeinterval [t0, T ], with any t0 ∈ [0, T ]:

Xt = x+

∫ t

t0

b (s,Xs) ds+

∫ t

t0

σ (s,Xs) dBs, t ∈ [t0, T ] .

Obviously, on [t0, T ], we have the same results as on [0, T ], in particular strong existence and pathwiseuniqueness under Lipschitz assumptions. Assume these conditions and denote the unique solution,defined on some filtered probability space, by Xt0,x

t . The relation with the backward Kolmogorovequation is

u (t, x) = E[ψ(Xt,xT

)].

This relation holds under different assumptions and for solutions u with different degrees of regularity.Let us start with the most elementary result.

Proposition 1. If u is a solution of the backward Kolmogorov equation of class C1,2([0, T ]× Rd

),

with bounded ∇u and σ, then u (t, x) = E[ψ(Xt,xT

)].

Proof. Given t0 ∈ [0, T ], we apply Itô formula to u(t,Xt0,x

t

)on [t0, T ]. The computation is the

same done in the proof of Theorem 5. Since u solves the backward Kolmogorov equation, we get

ψ(T,Xt0,x

T

)= u (t0, x) +

∫ T

t0

∇u(t,Xt0,x

t

)· σ(t,Xt0,x

t

)dBt

where we have used the identity Xt0,xt0

= x. Using the boundedness assumption, ∇u(t,Xt0,x

t

)·

σ(t,Xt0,x

t

)is of class M2, hence

E

[∫ T

t0

∇u(t,Xt0,x

t

)· σ(t,Xt0,x

t

)dBt

]= 0.

The relation u (t0, x) = E[ψ(Xt0,xT

)]follows.

Remark 4. Let us stress the duality of the previous problems. The law µt of Xt acts on testfunctions ψ by

∫ψdµt which is equal to E [ψ (Xt)]; hence there is a form of duality between the law µt

and the expression E [ψ (Xt)]. The PDE satisfied by the law µt and the PDE satisfied by E[ψ(Xt,xT

)]are in duality, as rigorously claimed by Theorem 6. It is a general fact of linear problem that existencefor a problem gives uniqueness for the dual one: think to the fact that full range of a matrix implieskernel zero for the transpose matrix; hence Theorem 6 is natural.


3.3. Macroscopic limit. We may reformulate Theorem 5 as a macroscopic limit of a system ofnon-interacting particles.

Let Bit, i ∈ N, be a sequence of independent Brownian motions in Rd, defined on a filtered

probability space (Ω,F ,Ft, P ). Let Xi0 be independent Rd-r.v.’s, F0-measurable, with the same law

µ0. Let b, σ as in Theorem 5; more precisely, let us assume they are bounded continuous and satisfythe Lipschitz conditions of the classical theorem of existence and uniqueness (one may ask less, sincehere we only need uniqueness in law). Consider the sequence of SDEs in Rd

dXit = b

(t,Xi

t

)dt+ σ

(t,Xi

t

)dBi

t, Xi|t=0 = Xi0.

One can prove that the processes Xit have the same law; denote the marginal at time t of X

it by µt;

we already know that µt is a measure-valued solution of the Fokker-Planck equation.The processes Xi

t are independent, since each Xit is adapted to the corresponding Brownian motion

Bit and initial condition X

i0, which are independent. One can provide rigorous proofs of these facts,

which however are extremely intuitive.Consider, for each N ∈ N, the empirical measure SNt := 1

N

∑Ni=1 δXi

t. Consider also mollifiers

θε (x) = ε−dθ(ε−1x

), with θ ∈ C∞c

(Rd), θ ≥ 0,

∫θ (x) dx = 1.

Theorem 7. For every t ∈ [0, T ], a.s. we have weak convergence of SNt to the solution µt of thePDE (3.2). Moreover, if µt has a density ρt, with ρt ∈ Cb

(Rd)for some t ∈ [0, T ], then for every

x ∈ Rd we havelimN→∞

(θεN ∗ SNt

)(x) = ρt (x)

in mean square (L2 (Ω)-limit), as soon as limN→∞ε−dNN = 0.

Proof. The proof of the first claim is the same done above for the Brownian motions: givenφ ∈ Cb

(Rd), the r.v. φ

(Xit

)are bounded i.i.d., hence by the strong Law of Large Numbers we have,

a.s.,limN→∞

⟨SNt , φ

⟩= E

[φ(X1t

)]= 〈µt, φ〉 .

Then one can find a full probability event Ω0 ⊂ Ω where the same convergence holds for everyφ ∈ Cb

(Rd), by the argument of Corollary 1.

The proof of the second claim is also the same as the one of Theorem 2, where the existence ofthe density ρt is used, along with its continuity and finiteness of ‖ρt‖∞.

For the sequel of these lecture it is very important to extract the following message from thisresult. If a family of cells Xi

t are subject to a drift b(t,Xi

t

)which move them in a certain direction,

at the PDE level the density ρt (x) of those cells will be subject to the transport term

−div (ρb) .

And conversely, if we read such a term in the PDE for a density, the interpretation is a drift whichacts on the particles of that density. The most common case of vector field b is a gradient field of apotential U

b = −∇Uin which case we say that particles move along the gradient of U . The transport term has the formdiv (ρ∇U) that we shall meet several times below in cancer models.

32 2. SDES AND PDES

Similarly, but this is in a sense more obvious, if the cells Xit are subject also to a random motion

described by a Brownian motion dBit, then the density is subject to the diffusion term

12∆ρ. This

relation is generalized to the case of a non homogeneous, non isotropic random motion of the formσ(t,Xi

t

)dBi

t and the corresponding diffusion term∑

ij ∂i∂j (aijρ) in the PDE; however, the homoge-neous isotropic case dBi

t is the rule, in absence of special phenomena.Finally,

4. Simulations of the macroscopic limit

4.1. Simulation of simple PDEs. First of all, let us see a simple code to simulate the heatequation

∂ρ

∂t= k∆ρ on [0, T ]× Rd

ρ|t=0 = ρ0

in dimension d = 1. The method is the finite difference one, with explicit Euler in time. There is afamous and essential rule to know: the time step dt and space step dx must be related by the condition(sometimes called Von Neumann stability condition)

kdt

dx2≤ 1

2.

At the boundary, we have imposed no-flux conditions.

NT= 10000; Nx=500; dx=0.1; K=1; T=100L=dx*Nx; dt=(dx^2)/(4*K)Nx.virt=Nx+1u = matrix(nrow=Nx.virt, ncol=NT)X=seq(0,L,dx)u[,1]=dnorm(X,L/2,5)M=max(dnorm(X,L/2,5))plot(c(0,L),c(-0.01,0.1),type="n")for (t in 1:(NT-1)) for (i in 2:Nx) u[i,t+1] = u[i,t] + K * dt * ((u[i+1,t]-2*u[i,t]+u[i-1,t])) / (dx^2)u[1,t+1]=u[2,t+1]; u[Nx.virt,t+1]=u[Nx,t+1]if(t%%T==0 ) polygon(c(0,L,L,0),c(-0.01,-0.01,0.1,0.1),col="white", border=NA)lines(X,u[,t+1])abline(h=0); abline(h=M)

4. SIMULATIONS OF THE MACROSCOPIC LIMIT 33

Running the simulation for a short while, the effect of the boundary starts to appear and it is notso nice. However, if we set equal to zero at the boundary instead of the no-flux condition, slowly themass disappears, which is even worse in a sense.

We suggest to change dt=(dx^2)/(4*K) into dt=(dx^2)/(2*K) and dt=(dx^2)/(1*K) to see therole of the stability condition.

Now we add a drift to the heat equation, in the Fokker-Plank form

∂ρ

∂t= k∆ρ− div (ρb) on [0, T ]× Rd

ρ|t=0 = ρ0

and we choose a drift b which concentrates mass around a point:

b (x) = −C1 (x− x0) e−C2|x−x0|2

.

NT= 10000; Nx=500; dx=0.1; K=1; T=100L=dx*Nx; dt=(dx^2)/(4*K)Nx.virt=Nx+1u = matrix(nrow=Nx.virt, ncol=NT)X=seq(0,L,dx)b = -(X-L/2-20)*exp(-0.01*abs((X-L/2-20)^2))u[,1]=dnorm(X,L/2,5)M=max(dnorm(X,L/2,5))plot(c(0,L),c(-0.01,0.1),type="n")for (t in 1:(NT-1)) for (i in 2:Nx) u[i,t+1] = u[i,t] + K * dt * ((u[i+1,t]-2*u[i,t]+u[i-1,t])) / (dx^2) - dt*(b[i+1]*u[i+1,t]-b[i]*u[i,t])/dx

34 2. SDES AND PDES

u[1,t+1]=u[2,t+1]; u[Nx.virt,t+1]=u[Nx,t+1]if(t%%T==0 ) polygon(c(0,L,L,0),c(-0.01,-0.01,0.1,0.1),col="white", border=NA)lines(X,u[,t+1])abline(h=0); abline(h=M)

If we want an interpretation, it can be the case of cells which move from their original positionand gather around a blood vessel.

4.2. Particle simulation and comparison with the PDE. We first run the previous simula-tion of the heat equation (black line) together with the simulation of 10000 Brownian particles startingwith the same initial density as the heat equation; we represent the profile of particles by means ofthe "density" R-command.

NT= 10000; Nx=500; dx=0.1; K=1/2; T=100; N=10000L=dx*Nx; dt=(dx^2)/(4*K); h=sqrt(dt)Nx.virt=Nx+1u = matrix(nrow=Nx.virt, ncol=NT); XMB=matrix(nrow=N,ncol=NT)X=seq(0,L,dx)u[,1]=dnorm(X,L/2,5)XMB[,1]=rnorm(N,L/2,5)M=max(dnorm(X,L/2,5))plot(c(0,L),c(-0.01,0.1),type="n")for (t in 1:(NT-1)) XMB[,t+1]=XMB[,t]+h*rnorm(N)for (i in 2:Nx) u[i,t+1] = u[i,t] + K * dt * ((u[i+1,t]-2*u[i,t]+u[i-1,t])) / (dx^2)

4. SIMULATIONS OF THE MACROSCOPIC LIMIT 35

u[1,t+1]=u[2,t+1]; u[Nx.virt,t+1]=u[Nx,t+1]if(t%%T==0 ) polygon(c(0,L,L,0),c(-0.01,-0.01,0.1,0.1),col="white", border=NA)lines(X,u[,t+1])lines(density(XMB[,t+1],bw=sd(XMB[,t+1])/5),col=2)abline(h=0); abline(h=M)Up to minor differences, the evolution of the profile is the same:

Let us now discuss the second example above. The density is shrinked in a small region, hence thevariations in number of points per space become larger. If one tries with teh same parameters above,the result is modest. The following parameters give on the contrary quite good results:

NT= 1000; Nx=500; dx=0.1; K=1/2; T=100; N=100000L=dx*Nx; dt=(dx^2)/(4*K); h=sqrt(dt)drift=function(x) -(x-L/2-20)*exp(-0.01*abs((x-L/2-20)^2))Nx.virt=Nx+1u = matrix(nrow=Nx.virt, ncol=NT); XMB=matrix(nrow=N,ncol=NT)X=seq(0,L,dx)u[,1]=dnorm(X,L/2,5)XMB[,1]=rnorm(N,L/2,5)M=max(dnorm(X,L/2,5))plot(c(0,L),c(-0.01,0.1),type="n")for (t in 1:(NT-1)) XMB[,t+1]=XMB[,t]+dt*drift(XMB[,t])+h*rnorm(N)for (i in 2:Nx) u[i,t+1] = u[i,t] + K * dt * ((u[i+1,t]-2*u[i,t]+u[i-1,t])) / (dx^2) - dt*(drift(X[i+1])*u[i+1,t]-

drift(X[i])*u[i,t])/dxu[1,t+1]=u[2,t+1]; u[Nx.virt,t+1]=u[Nx,t+1]if(t%%T==0 ) polygon(c(0,L,L,0),c(-0.01,-0.01,0.1,0.1),col="white", border=NA)lines(X,u[,t+1])

36 2. SDES AND PDES

lines(density(XMB[,t+1],bw=sd(XMB[,t+1])/20),col=2)abline(h=0); abline(h=M)

Notice the change in bw=sd(XMB[,t+1])/20 with respect to bw=sd(XMB[,t+1])/5, necessary toappreciate smaller scale structures. This of course causes wider fluctuations in the profile unless thenumber of particles is very high.

Part 2

Growth and change of species in populations ofcells and nonlinear Partial Differential

Equations

CHAPTER 3

Examples of macroscopic systems in Mathematical Oncology

1. An advanced model of invasive tumor with angiogenesis

Let us describe a model introduced by [22]. This model is made of 7 coupled PDE-ODE. Let usimmediately emphasize that this is not "the model" of cancer growth. It is one model, with its owndegree of sophistication, and describing a tumor in a particular phase. It is a model of mechanicaltype, dealing with aspects like random motion, motion along gradients, proliferation, change of type.

1.1. Normoxic, hypoxic and apoptotic cells. Cancer cells are mainly characterized by theirtendency to duplicate. However, when they receive an insuffi cient amount of oxygen, or when they donot have space enough, their proliferation is inhibited.

The model of [22] splits the category of cancer cells in three classes:

(1) normoxic cells: healthy, proliferating tumor cells, with normal oxygen supply(2) hypoxic cells: quiescent tumor cells, with poor oxygen supply(3) apoptotic cells: death or programmed to death tumor cells

The figure below shows schematically the obvious fact that in a three-dimensional tumor mass thehypoxic cells are those inside, with an even smaller core of apoptotic cells.

1.1.1. PDE for normoxic cells. The model prescribes the following PDE for normoxic cells:

39

40 3. EXAMPLES OF MACROSCOPIC SYSTEMS IN MATHEMATICAL ONCOLOGY

∂N∂t

= k1∆N︸︷︷︸background diffusion

(N (t, x) = normoxic cell density)

div (σ (N )∇N )︸︷︷︸crowding-driven diffusion

+ c1N (Vmax − V)︸︷︷︸proliferation

− χ1 div (N∇m)︸︷︷︸transport along ECM gradient

− αN→H1o≤oHN︸︷︷︸normoxic → hypoxic

+ αH→N 1o>oHH︸︷︷︸hypoxic → normoxic

• First, a background diffusion is admitted; and no adhesion constraint is imposed; this is theinvasive phase. The value of the constant k1 is, however, extremely small, the diffusion isextremely slow.• Second, a crowding-driven diffusion is introduced. This is a very interesting term, which willoccupy our effort quite often, but not now, it is too early. Just as a first idea, this termenforces the diffusion when the density of cells is larger.• Third, proliferation. If we neglect space, the time evolution of a complete proliferation isgiven by the differential equation x′ (t) = λx (t), where λ is the proliferation rate. One canconsider a model with time-dependent rate. In the PDE above the time-dependent rate isc1 (Vmax − V): it decreases to zero when the total density of cell (plus ECM)

V = N +H+A+ E +m = total density of cells plus ECM

approaches the threshold Vmax.• Then, transport: normoxic cells, beyond the random motion described by k1∆N , have atendency to move along the gradient of m, the ECM density.• Finally, some normoxic cells become hypoxic (when o≤oH, namely the oxygen is too low) andsome hypoxic cells are restored to the normoxic state (when o>oH).

1.1.2. ODE for hypoxic cells. Hypoxic cells do not move and do not proliferate. Their numberincreases when some normoxic cell deteriorate to the hypoxic state (for o≤oH). And decreases eitherwhen they are restored to the normoxic state (for o>oH) or when they degenerate to apoptotic cells(for o≤oA):

dHdt

= αN→H1o≤oHN︸︷︷︸normoxic → hypoxic

− αH→N 1o≥oHH︸︷︷︸hypoxic → normoxic

− αH→A1o≤oAH︸︷︷︸hypoxic → apoptotic

(H (t, x) = hypoxic cell density)

1.1.3. ODE for apoptotic cells. These cells are programmed to death. It means that they dissolvein a regulated way, not by necrosis and causing infections. The number of apoptotic cells can onlyincrease, due to the hypoxic cells that deteriorate:

1. AN ADVANCED MODEL OF INVASIVE TUMOR WITH ANGIOGENESIS 41

dAdt

= αH→A1o≤oAH︸︷︷︸hypoxic → apoptotic

(A (t, x) = apoptotic cell density)

1.1.4. ODE for ECM. In this model, it is assumed that the Extracellular Matrix can only deteri-orate, due to the invasion of normoxic cells:

dm

dt= − βmN︸︷︷︸

degradation by normoxic cells

(m (t, x) = ExtraCellular Matrix)

1.1.5. Crowding-driven diffusion. In the equation for normoxic cells we have the term

div (σ (N )∇N )

which we have called crowding-driven diffusion. The prescription for σ (N ) in [22] is (up to constants)

σ (N ) = max (N −N0, 0) .

In other words, when the density of normoxic cells passes the threshold N0, an additional diffusionstarts. This term is very intriguing and typical of [22]. It is similar to (up to constants)

σ (N ) = N

which corresponds to

1

2∆N 2

also called porous media diffusion.

1.2. The endothelial cascade. Hypoxic cells need more oxygen to survive. Thus they initiate acascade of cellular interactions. The result is angiogenesis: new vascularization is developed to supplythe tumor (microvessels branching from main vessels in the direction of the tumor).

A messenger from hypoxic cells is sent to endothelial cells: it is called VEGF (Vascular EndothelialGrowth Factor). Here are two pictures taken from the web (see [30] for the second one, which includessome of the complicate molecular details):


See also the movies at the web sites [28], [29].1.2.1. PDEs for the VEGF concentration. VEGF is a density of object at molecular level. These

proteins have always a diffusion (opposite to cells, which diffuse only under special circumstances):

∂g

∂t= k4∆g︸︷︷︸diffusion

(g (t, x) = VEGF concentration)

+ αH→gH︸︷︷︸production by hypoxic cells

− αg→EEg︸︷︷︸uptake by endothelial cells

and the constant k4 is much bigger than k1.


Moreover, the concentration of VEGF is produced by the hypoxic cells, hence increases due to theirpresence, and VEGF is absorbed by endothelial cells (those forming the boundary of blood vessels).

1.2.2. PDEs for endothelial ramification. As the pictures above show, endothelial cells do notdiffuse as isolated individuals but they propagate as microvessels. However, keeping track of theirtopological structure is diffi cult and perhaps not so important (in the opinion of the authors of thismodel), hence the concept of density of endothelial cells, or maybe more precisely density of endothelialramification, is introduced. The PDE is:

∂E∂t

= k2∆E︸︷︷︸diffusion

(E (t, x) = density of endothelial ramification)

− χ2 div (E∇g)︸︷︷︸transport along VEGF gradient

+ c2Eg (Vmax − V)︸︷︷︸proliferation under VEGF presence

where• random motion is considered (however, as for normoxic cells, the value of the constant k1 isextremely small)• endothelial cells move along the VEGF gradient, in order to reach the area occupied byhypoxic cells• in order to build new vessels, they need to proliferate; the rate of proliferation is c2g (Vmax − V),namely it is proportional to VEGF concentration, and is inhibited by the same volume con-straint of normoxic proliferation.

1.2.3. PDEs for oxygen concentration. Oxygen is also molecular-level hence it diffuses, with k3 ismuch bigger than k1 (also bigger than k4)

∂o

∂t= k3∆o︸︷︷︸diffusion

(o (t, x) = oxygen concentration)

+ c3E (omax − o)︸︷︷︸production by endothelial cells

− αo→N ,H,E (N +H+ E) o︸︷︷︸uptake by all living cells

− γo︸︷︷︸oxygen decay

and• oxygen concentration increases proportionally to the density of endothelial ramification, butonly up to omax

• it decreases due to absorption by various cells (not the apoptotic ones)• it decays (slowly).

1.2.4. Summary of variables. It may be useful to summarize the list of variables:N (t, x) = density of normoxic cellsH (t, x) = density of hypoxic cellsA (t, x) = density of apoptotic cellsE (t, x) = density of endothelial cells (or density of vasculature)o (t, x) = oxygen concentrationg (t, x) = angiogenic growth factor (VEGF) concentrationm (t, x) = ECM (ExtraCellular Matrix)


1.2.5. Summary of constants. Diffi culties. The next list of constants is given mostly to emphasizea main diffi culty with this kind of complex models: some of these constants are poorly known andparameter fit is excluded by the impossibility of real experiment.

k1 = background random motility coeffi cient of normoxic cellsk2 = random motility coeffi cient of endothelial cellsk3 = diffusion coeffi cient of oxygenk4 = diffusion coeffi cient of angiogenic factorχ1 = transport coeffi cient of normoxic cells along ECM gradientχ2 = transport coeffi cient of endothelial cells along VEGF gradientVcr = threshold for crowding-driven diffusionVmax = limit to total volume of cells and ECMc1 = proliferation rate of normoxic cellsc2 = proliferation rate of endothelial cellsc3 = production rate of oxygenαN→H = decay rate from normoxic to hypoxic cellsαH→N = restoration rate from hypoxic to normoxic cellsαH→A = decay rate from hypoxic to apoptotic cellsαH→g = production rate of VEGF from hypoxic cellsαo→N ,H,E = uptake rate of oxygen from all living cellsαg→E = uptake rate of VEGF from endothelial cellsomax = maximum oxygen concentrationoH = oxygen threshold for transition normoxic ↔ hypoxicoA = oxygen threshold for transition hypoxic ↔ apoptoticβ = rate of ECM degradationγ = oxygen decay rateTo stress the diffi culties, let us also mention the fact that a tissue is a highly complex environment,

possibly highly heterogeneous. The first picture of Chapter one, like for instance the next one, areexamples of the geometrical complexity of a tissue. This complexity is not considered by the modelabove, at least is this idealized form.


1.2.6. Simulations. In Chapter 3 we shall devote some time to discuss numerical simulation of thefull system and a number of its reductions. However, just to give a first impression, let us see some ofthe pictures reported by the paper [22].

Blue: density of normoxic cells; light blue (green): extracellular matrixDotted black: density of hypoxic cells; red: density of endothelial ramification.


We see a front of normoxic cells propagating in the medium (they degradate ECM), followed by afront of hypoxic cells (it also apparently disappears due to the decay to apoptotic cells, which are notshown). The endothelial ramification is silent for a while, then it is triggered.

1.3. Fragments of microscopic model associated to the PDE system. The full systemof 7 equations above cannot be split into exact sub-systems. But we can consider simplified versionsjust to start understanding the correspondence between Macroscopic and Microscopic. Consider forinstance the closed system between normoxic cell density and ECM, neglecting all other terms andequations:

∂N∂t

= k1∆N︸︷︷︸background diffusion

− χ1 div (N∇m)︸︷︷︸transport along ECM gradient

dm

dt= − βmN︸︷︷︸

degradation by normoxic cells

If m would be a priori given, we could say that N is the weak limit of

SNn (t) :=1

n

n∑i=1

δXNi (t)

where XNi (t), i = 1, 2, ... are the positions of independent normoxic cells subject to the microscopicdynamics

dXNi (t) = χ1∇m(t,XNi (t)

)dt+

√2k1dW

Ni (t) .

But m (t, x) is not given. To simplify, we could decide that the dynamics acts in two steps, likein the numerical methods called slitting up or decomposition: in one step, m is considered as a givenfunction, and the scheme above applies; in the next step, N (t, x) is considered as a given function,and m is computed from equation dm

dt = −βmN ; to be realistic, the two steps should be very short.Otherwise, if we do not want to use a two-step mechanism, we have to couple the two equations.

In this case we have to solve the system

dm

dt= −βmN

N (t, ·) = limn→∞

SNn (t)

dXNi (t) = χ1∇m(t,XNi (t)

)dt+

√2k1dW

Ni (t) .

Even more realistically, we should not pass to the limit inside the system, but only a posteriori.We mean that we should define some relatively smooth function Nn (t, x) based on SNn (t), like theconvolution with a smooth kernel θn (x) = ε−dn θ

(ε−1n x

):

Nn (t, x) =(θn ∗ SNn (t)

)(x) =

1

n

n∑i=1

θn(x−XNi (t)

)

2. THE FISHER-KOLMOGOROV-PETROVSKII-PISKUNOV MODEL 47

and consider the system

dmn (t, x)

dt= −βmnNn (t, x) = −β

n

n∑i=1

θn(x−XNi (t)

)mn (t, x)

dXNi (t) = χ1∇mn

(t,XNi (t)

)dt+

√2k1dW

Ni (t) .

Now the limit as n → ∞ of the solution of this system has to be investigated ex novo, it is not asimple consequence of the facts seen above on Fokker-Planck equations.

Similar arguments can be applied to the equation for endothelial density,

∂E∂t

= k2∆E︸︷︷︸diffusion

− χ2 div (E∇g)︸︷︷︸transport along VEGF gradient

by introducing the positions of single endothelial cells XEi (t), i = 1, 2, ..., subject to the equations

dXEi (t) = χ2∇g(t,XEi (t)

)dt+

√2k2dW

Ei (t) .

Clearly, to reach a full microscopic description we should be able to describe other diffi cult terms,like nonlinear diffusions

div (σ (N )∇N )︸︷︷︸crowding-driven diffusion

change of typeαH→N 1o≥oHH︸︷︷︸hypoxic → normoxic

proliferationc1N (Vmax − V)︸︷︷︸

proliferation

and so on. The message of this section is that linear diffusion terms and transport ones have a simplemicroscopic counterpart, suggested by the theory of Fokker-Planck equations.

2. The Fisher-Kolmogorov-Petrovskii-Piskunov model

2.1. Introduction. The complexity of the model of Section 1, invasive with angiogenesis, doesnot allow a straightforward mathematical analysis and thus it is convenient to start by understandingan easier model, considered in the earlier literature on oncology and other applications in biology.

It deals with the space-time evolution of a single quantity, the density of tumor cells. It is assumedsubject to diffusion and proliferation:

(2.1)∂u

∂t= D∆u+ ρu (umax − u) , u|t=0 = u0.

The diffusion term is the usual one, based on Fick law, with constant diffusion coeffi cient D. Prolifer-ation happens at each point x. Proliferation rate is ρ (umax − u), ρ a positive constant, hence dampedby the factor (umax − u) which reduces proliferation to zero when u reaches the threshold umax. Wherethe density is higher, proliferation is inhibited.


The dynamical mechanism can be intuitively described as follows: if only the term ρu (umax − u)was present, the density u would increase up to umax at each point x; diffusion redistributes the densityfrom higher to lower zones. Progressively, the density will occupy new and new space (due to diffusion)and at the same time will increase towards umax.

2.2. Scaling transformations. Setting U = u/umax, dividing (2.1) by umax we get

∂U

∂t= D∆U + ρU (1− U)

with ρ = ρumax, hence we may replace umax by 1 with a simple transformation (modifying ρ). SettingV (t, x) = U (λt, µx), with λ, µ > 0, we have

∂V

∂t(t, x) = λ

∂U

∂t(λt, µx) = λD (∆U) (λt, µx) + λρU (λt, µx) (1− U (λt, µx))

= D∆V (t, x) + ρV (t, x) (1− V (t, x))

with D = λDµ2, ρ = λρ. Thus, by a simple transformation, we may arbitrarily change both diffusion

and proliferation constants. For these reasons, it is suffi cient to restrict to the canonical model

(2.2)∂u

∂t=

1

2∆u+ u (1− u) .

Moreover, the function v = 1− u satisfies∂v

∂t= −∂u

∂t= −1

2∆u− u (1− u)

=1

2∆v − uv =

1

2∆v − (1− v) v

hence we may either study (2.2), or∂v

∂t=

1

2∆v + v2 − v.

2.3. Simulations. It is very interesting to see numerically the difference between this model andthe simple heat equation. Let us use the following simple R code (try with C_prol=0 to see thedifference)

NT= 10000; Nx=1000; dx=0.1; K=1; T=20; C_prol=1L=dx*Nx; dt=(dx^2)/(4*K)Nx.virt=Nx+1u = matrix(nrow=Nx.virt, ncol=NT)X=seq(0,L,dx)u[,1]=dnorm(X,L/2,0.5)M=max(dnorm(X,L/2,0.5))plot(c(0,L),c(-0.01,1),type="n")for (t in 1:(NT-1)) for (i in 2:Nx) u[i,t+1] = u[i,t] + K * dt * ((u[i+1,t]-2*u[i,t]+u[i-1,t])) / (dx^2)


u[1,t+1]=u[2,t+1]; u[Nx.virt,t+1]=u[Nx,t+1]for (i in 1:Nx.virt) u[i,t+1] = u[i,t+1] + C_prol * dt * u[i,t+1]*(1-u[i,t+1])if(t%%T==0 ) polygon(c(0,L,L,0),c(-0.01,-0.01,1,1),col="white", border=NA)lines(X,u[,t+1])abline(h=0); abline(h=M, col=3); abline(h=1)

We observe that, even starting from an initial Gaussian profile, the shapes for positive time arenot Gaussian anymore ut more fat; in a sense also more compact support (although this is not truein a strict sense).

2.4. Traveling waves. Simulations with 0 ≤ u ≤ 1 clearly reveal the existence of a front, a wave,which moves. Assume there is a solution of the form

u (t, x) = w (x− ct) .


Substituting into equation (2.2) we get

(2.3) −cw′ = 1

2w′′ + w − w2.

Assume we start with the initial condition u (0, x) = 1x<0 (x). If it happens that it becomes closerand closer to one of these traveling waves (for a suitable value of c), then the function w must fulfill

(2.4) limx→−∞

w (x) = 1, limx→+∞

w (x) = 0.

Let us also ask that w is decreasing.

Theorem 8. If 0 ≤ c <√

2, there are no solutions of (2.3)-(2.4). If c ≥√

2, there exists one andonly one solution, denoted in the sequel by wc (x).

Proof. Let us sketch the proof of [7]. Equation (2.3) is equivalent to the system

w′ = z

z′ = −2cz − 2w + 2w2.

To have a better dynamical intuition, let us write t for x (time) and (x, y) for (w,w′). Hence we studythe system

x′ (t) = y (t)

y′ (t) = −2cy (t)− 2x (t) + 2x (t)2 .

We are looking for solutions (x (t) , y (t)) defined on the whole R, such that

limt→−∞

x (t) = 1, limt→+∞

x (t) = 0.

The fixed points (x, y) of the system satisfy

y = 0

−2cy − 2x+ 2x2 = 0

hence −2x+ 2x2 = 0, x = 0 or x = 1, namely

A = (0, 0) , B = (1, 0) .

Denoting by f (x, y) the vector field on the right of the system, we have

Df (x, y) =

(0 1

−2 + 4x −2c

)hence

Df (A) =

(0 1−2 −2c

), Df (B) =

(0 12 −2c

).

With some computations (here done by Maple) one discovers that Df (A) has eigenvectors

eA1 =

(12

√c2 − 2− 1

2c1

), eA2 =

(−1

2c−12

√c2 − 2

1

)


with eigenvalues

−c−√c2 − 2,

√c2 − 2− c

while Df (B) has eigenvectors

eB1 =

(12c−

12

√c2 + 2

1

), eB2 =

(12c+ 1

2

√c2 + 2

1

)with eigenvalues

−c−√c2 + 2,

√c2 + 2− c.

We always have c ≥ 0, hence both eigenvalues of Df (A) are negative, therefore A is a locally attractivestationary point; while Df (B) has eigenvalues of opposite sign, namely it is an hyperbolic point withunstable manifold tangent to eB2 , which spans a line through the first and third quadrant.

The intuition is the following one: if we have a solution, it arises at time −∞ from B (sincelimt→−∞ x (t) = 1) and converges to A as t→ +∞ (because limt→+∞ x (t) = 0); from B, being forcedto exit along the unstable manifold, and being constrained to have x ∈ [0, 1], it must exit in the thirdquadrant (centered at B), along −eB2 . The next picture shows a numerical simulation of the system,for c =

√2, with an initial condition of the form −εeB2 with very small ε. It goes precisely to A.

The R codes isN=30000; X=1:N; Y=1:N; dt=0.001; c=sqrt(2); eps=0.0001ex=0.5*c+0.5*sqrt(c^2+2); ey=1X[1]=1-eps*ex; Y[1]=-eps*eyfor (i in 1:(N-1)) X[i+1]=X[i]+dt*Y[i]Y[i+1]=Y[i]+dt*(-2*c*Y[i]-2*X[i]+2*X[i]^2)plot(c(0,1),c(-0.6,0),type="n"); lines(X,Y)abline(h=0); abline(0,1000); lines(c(0,1),c(-1/ex,0))A rigorous proof that, for c ≥

√2, a solution exists, requires additional arguments which we omit.

One should for instance consider the triangle shown in the figure, given by the axes and the line


through −eB2 , and prove that the vector field f is directed inside the triangle at every non criticalboundary point.

The non existence for c <√

2 can be understood, on the contrary, from the fact that, for c <√

2,point A attracts spiraling, as shown in the figure (c = 0.8):

which violates the constraint x ∈ [0, 1]. Spiraling can be analytically seen looking for the solutionsof the linearized system x′ = y, y′ = −2cy − 2x, namely x′′ = −2cx′ − 2x, which have the form

C1 exp(−t(c+√c2 − 2

))+ C2 exp

(−t(c−√c2 − 2

)), understanding the exponential also in the

complex case; they are decreasing exponentials for c ≥√

2, but oscillate for c <√

2. Theorem 9. If u (0, x) = 1x<0 (x), the solution converges to the traveling profile with c =

√2.

Preciselylim

t→+∞u (t,mt + x) = w√2 (x)

for every x ∈ R, where mt is the unique point such that u (t,mt) = 1/2.

A probabilistic proof can be found in [10].

Remark 5. One should not think that every initial condition converges to such traveling wave. Onecan prove, for instance, that if u0 ∈ [0, 1] is such that for some b ∈ (0,

√2] the limit limx→+∞ ebx (1− f (x))

exists finite and different from zero, then limt→+∞ u (t, ct+ x) = wc (x), with c = 1/b+ b/2.

3. A probabilistic representation

Consider the equation∂v

∂t=

1

2∆v + λ

(v2 − v

).

McKean proved the following probabilistic representation

u (t, x) = 1− E[Nt∏i=1

u0

(x+Xi

t

)]where the processesXi

t , Nt are defined as follows. First we give an informal definition, then we formalizeit. At time t = 0, from x = 0 a Brownian motion X1

t starts. At the random time T0 ∼ Exp (λ),

3. A PROBABILISTIC REPRESENTATION 53

independent of X1, the process X1 ends its existence and, at position X1T0, two new and independent

Brownian motions start, X1t and X

2t . Each one lives an exponential time Exp (λ) (independent of the

previous objects) then dies and generates two new Brownian motions; and so on. At any time t thereare Nt points alive, that we call Xi

t , i = 1, ..., Nt. They are the random points which appear in theprobabilistic representation formula above.

Let us write a rigorous scheme. On a probability space (Ω,F , P ) assume we have a countable familyof independent exponential times, Exp (λ), indexed by finite sequences of 1,2: Ti1,...,in , ik ∈ 1, 2,k = 1, ..., n, n ∈ N; for n = 0 let us write T0. This family identifies a binary tree with branches ofrandom length, but the tree structure in itself can be maintained behind, hidden, in or description.Set τi1,...,in = T0 + Ti1 + Ti1,i2 + ... + Ti1,...,in , τ0 = T0. Assume moreover that we have a countablefamily of independent Brownian motions (independent among themselves and with respect to therandom times above) Bi1,...,in (t), ik ∈ 1, 2, k = 1, ..., n, n ∈ N; for n = 0 let us write B0 (t). Nowdefine a countable family of processes Xi1,...,in (t), ik ∈ 1, 2, k = 1, ..., n, n ∈ N, taking valuesin R ∪ δ, where δ is an auxiliary point outside R; for n = 0 let us write X0 (t). All processes Xtake the values δ except on a random time interval that we are going to define. For t ∈ [0, τ0) setX0 (t) = B0 (t). For t ∈ [τ0, τi1), i1 = 1, 2, set Xi1 (t) = X0 (τ0) + Bi1 (t− τ0). For t ∈ [τi1 , τi1+i2),i1, i2 ∈ 1, 2, set Xi1+i2 (t) = Xi1 (τi1)+Bi1+i2 (t− τi1). By induction, for t ∈ [τi1,...,in , τi1,...,in,in+1) setXi1,...,in,in+1 (t) = Xi1,...,in (τi1,...,in) + Bi1,...,in,in+1 (t− τi1,...,in). Denote by Λt the set of multi-indices(i1, ..., in) such that Xi1,...,in (t) 6= δ. The precise meaning of the formula above is now

u (t, x) = 1− E[∏a∈Λt

u0 (x+Xa (t))

].

From this formula one can immediately see a few facts, like the property u (t, x) ∈ [0, 1] whenu0 ∈ [0, 1], or the fact that the function x 7→ u (t, x) is decreasing if u0 is decreasing. In [10] it is usedto prove several facts stated above on traveling waves.

Let us finally see the proof of the formula. We disintegrate the expected value with respect to T0:

v (t, x) :=

E

[∏a∈Λt

u0 (x+Xa (t))

]= P (T0 > t)E

[∏a∈Λt

u0 (x+Xa (t)) |T0 > t

]

+

∫ t

0λe−λsE

[∏a∈Λt

u0 (x+Xa (t)) |T0 = s

]ds

= e−λtE [u0 (x+B0 (t))]

+

∫ t

0λe−λsE

∏a∈Λ1t

u0 (x+Xa (t))∏a∈Λ2t

u0 (x+Xa (t)) |T0 = s

ds


where Λkt is the set of elements (i1, ..., in) ∈ Λt such that i1 = k, k = 1, 2

= e−λt(e12

∆tu0

)(x) +

∫ t

0λe−λsE

E ∏a∈Λt−s

u0

(x′ +Xa (t− s)

)2

x′=x+B0(s)

ds=(e(

12

∆−λ)tu0

)(x) +

∫ t

0

(e(

12

∆−λ)sλv2 (t− s, ·))

(x) ds

and this is the PDE in the mild sense.

3.1. Is FKPP the macroscopic equation of the tree structure? The previous probabilisticformula is based on a particle system which seems a reasonable microscopic model of cells: eachcell moves like a Brownian motion and at a random time duplicates. However, FKPP is not themacroscopic limit of this particle system. To see this, let us discover the true macroscopic limit of thisparticle system.

In Chapter 1 we have shown (in greater generality) that given a family of independent Brownianmotions Bi

t, w.r.t. a filtration Ft, and i.i.d. F0-adapted initial conditions Xi0, the empirical measure

SNt := 1N

∑Ni=1 δXi

tof the particle system

Xit = Xi

0 +Bit

weakly converges to a measure-valued solution of the heat equation∂u

∂t=

1

2∆u

with initial condition given by the law of X10 . In this particular case we also have the representation

formulau (t, x) = E

[u0

(X1t

)]but this coincidence should not be translated to more non-linear mechanisms.

Consider now the process defined in the previous section and let us investigate its macroscopiclimit. What do we send to infinity? As in the case just recalled of independent Brownian motions Bi

t,let us consider an family of N independent processes of that form, with i.i.d. initial conditions Xi

0,independent of the Brownian motions and exponential times of the process. Consider the empiricalmeasure

SNt :=1

N

N∑i=1

∑a∈Λit

δXia(t).

Theorem 10. SNt weakly converges to a measure-valued solution of the equation

(3.1)∂u

∂t=

1

2∆u+ λu.

Proof. Let us apply also here the LLN:⟨SNt , φ

⟩=

1

N

N∑i=1

∑a∈Λit

φ(Xia (t)

)→ E

[∑a∈Λt

φ (Xa (t))

]=: 〈ut, φ〉 .

3. A PROBABILISTIC REPRESENTATION 55

Now we have to find the equation satisfied by the time-dependent probability measure ut. Introduceρφ (t, x):

ρφ (t, x) := E

[∑a∈Λt

φ (x+Xa (t))

]

= P (T0 > t)E

[∑a∈Λt

φ (x+Xa (t)) |T0 > t

]+

∫ t

0λe−λsE

[∑a∈Λt

φ (x+Xa (t)) |T0 = s

]ds

= e−λtE [φ (x+B0 (t))] +

∫ t

0λe−λsE

∑a∈Λ1t

φ (x+Xa (t)) +∑a∈Λ2t

φ (x+Xa (t)) |T0 = s

ds= e−λt

(e12

∆tφ)

(x) +

∫ t

0λe−λsE

2E

∑a∈Λt−s

φ(x′ +Xa (t− s)

)x′=x+B0(s)

ds=(e(

12

∆−λ)tφ)

(x) + 2

∫ t

0

(e(

12

∆−λ)sλρφ (t− s, ·))

(x) ds.

This implies

∂

∂tρφ (t, x) =

1

2∆ρφ (t, x)− λρφ (t, x) + 2λρφ (t, x)

=1

2∆ρφ (t, x) + λρφ (t, x)

ρφ (0, x) = φ (x) .

We aim to prove that ut satisfies∂ut∂t

=1

2∆ut + λut

knowing that ρφ (t, 0) = 〈ut, φ〉 satisfies the equation written above. We have ρφ (t, x) = 〈ut, φ (x+ ·)〉hence

∂

∂t〈ut, φ (x+ ·)〉 =

1

2∆ 〈ut, φ (x+ ·)〉+ λ 〈ut, φ (x+ ·)〉

but ∆ 〈ut, φ (x+ ·)〉 = 〈ut, (∆φ) (x+ ·)〉 hence∂

∂t〈ut, φ (x+ ·)〉 =

1

2〈ut, (∆φ) (x+ ·)〉+ λ 〈ut, φ (x+ ·)〉

hence (x = 0)∂

∂t〈ut, φ〉 =

1

2〈ut,∆φ〉+ λ 〈ut, φ〉

that is the PDE we wanted, in weak form. Remark 6. Equation (3.1) is not FKPP.

Remark 7. The solutions equation (3.1) increase exponentially. The solutions of FKPP cannotovercome a threshold.


Remark 8. In order to find a microscopic model of FKPP we have to modulate the birth rateby the factor 1 − u (t, x). But u (t, x) is not known at the microscopic level. One could then use1−

(θεN ∗ SNt

)(x), where θεN is a mollifier which goes to the delta Dirac at zero. It allows one to talk

of "SNt at point x". We shall deal with this problem in Chapter 4.

4. Existence, uniqueness, invariant regions

Consider the equation (we drop 1/2 since we do not use probabilistic arguments in this section)

(4.1)∂u

∂t= ∆u+ u (1− u)

for t ∈ [0, T ] and x ∈ Rd. We are interested in solutions which are bounded, precisely u ∈ [0, 1], andof class C1,2 (u ∈ C1,2 if the partial derivatives ∂u

∂t , ∂xiu, ∂xj∂xiu exist and are continuous); we shallcall them classical solutions.

About the regularity of the initial condition u0, maybe we ask a little bit too much. Let UCb(Rd)

be the space of uniformly continuous and bounded functions f : Rd → R. We assume that u0 ∈ C2(Rd)

with u and its first and second derivatives in UCb(Rd); we write u0 ∈ UC2

b

(Rd). We address [19],

Chapter 14, Section A for a discussion of the advantages of UCb(Rd)in this framework.

When we write expressions like u0 ∈ [0, 1], u ∈ [0, 1], we mean that the property holds for allvalues of the independent variables x, (t, x) etc.

Theorem 11. Equation (4.1) with UC2b

(Rd)-initial condition u0 ∈ [0, 1], has one and only one

classical solution u ∈ [0, 1] (uniqueness holds in the larger class of mild solutions, see the proof; thesolution is also UCb

(Rd)at every time, plus other regularity properties).

Proof. Let us only sketch the main steps, some of which will be expanded in Section ??.Step 1. We prove the result for the auxiliary equation

(4.2)∂u

∂t= ∆u+ u (1− u) + h (u)

where h (·) is a smooth function, strictly negative for u = 1, strictly positive for u = 0. This step isdivided in three sub-steps.

Notice that the introduction of the auxiliary term h (u) is not always needed. It depends on theproof we make of the invariance of the region [0, 1] (Step 1c). We present two proofs; the first onedoes not need h (u), the second (deeper) one requires h (u).

Step 1a. For a general equation of the form

(4.3)∂u

∂t= ∆u+ f (u)

with smooth f , one can prove local existence and uniqueness of mild solutions (bounded continuousfunctions satisfying the mild form of the equation). If we then prove an a priori bound on solutionsin uniform norm, this will lead to global existence (of mild solutions).

Step 1b. One can prove that mild solutions are also classical ones, when u0 is regular.Step 1c. For the particular equation (4.2) an a priori bound in uniform norm is proved (this is

the main conceptual step). Precisely, we prove that, if u is a classical solution with u0 (x) ∈ [0, 1],then u (t, x) ∈ [0, 1]. Putting together the three sub-steps, one gets the theorem for equation (4.2).

4. EXISTENCE, UNIQUENESS, INVARIANT REGIONS 57

Step 2. (This step is needed only if we had to introduce h (u) above.) Consider now the problem

∂uδ

∂t= ∆uδ + b (t, x) · ∇uδ + uδ

(1− uδ

)− δh

(uδ)

where h (u) is like above. We have uδ ∈ [0, 1] by Step 1. One can prove that uδ converges uniformlyon compact sets (first locally in time, then globally), as δ → 0. Then one can prove that the limit uis a solution of the original equation. Therefore there exists a solution u ∈ [0, 1].

Step 3. Uniqueness holds locally by Step 1a. This completes the proof.See Section ?? for additional details. In the sequel we give some elements of the proof of this theorem.

4.1. Homogeneous and non-homogeneous heat equation. Consider equation

(4.4)∂u

∂t= ∆u+ g, u|t=0 = u0

where u0 ∈ UC2b

(Rd), g : [0, T ]×Rd → R is of class C

([0, T ] ;UC2

b

(Rd)). We say that u : [0, T ]×Rd →

R is a classical solution if it is of class C1,2 and satisfies the equation pointwise, on [0, T ] × Rd. Onecan check that, for g = 0 and u0 ∈ UC2

b

(Rd), there exists one and only one classical solution, given

by

u (t, x) =

∫RdG (t, x− y)u0 (y) dy

G (t, x) = (4πkt)−d/2 exp

(−|x|

2

4kt

).

That this expression defines a function of class C1,2 and that this function satisfies the equation can bechecked by several but elementary computations, that we do not repeat. Conversely, if u is a classicalsolution, we multiply the equation by G (t, x′ − x), integrate on Rd, integrate by parts and get theformula.

Introduce the operator Pt : Cb(Rd)→ Cb

(Rd)given by

(Ptv) (x) =

∫RdG (t, x− y) v (y) dy, v ∈ Cb

(Rd).

One can check that, for g ∈ C([0, T ] ;UC2

b

(Rd)), there exists one and only one classical solution of

equation (4.4), given by

u (t, x) = (Ptu0) (x) +

∫ t

0(Pt−sg (s)) (x) ds.

4.2. Mild and regular solutions (inside Step 1b).

Definition 6. We say that u : [0, T ]×Rd → R is a mild solution of equation (4.3) if it is of classCb(Rd)(bounded continuous) and satisfies pointwise the identity

u (t, x) = (Ptu0) (x) +

∫ t

0(Pt−sf (u (s))) (x) ds.


Proposition 2. If u is a classical solution, then it is a mild solution. If u is a mild solution andu0 ∈ UC2

b

(Rd), then it is a classical solution.

4.3. Local existence and uniqueness. Assume that f , in equation (4.3), is a locally Lipschitzcontinuous function.

Theorem 12. Given u0 ∈ UCb(Rd), there exists one and only one mild solution of equation (4.3)

on some interval [0, τ ]. The size τ depends only on the uniform norm of u0.

Proof. Consider the Banach space C([0, τ ] ;UCb

(Rd))and let Xτ,θ ⊂ C

([0, τ ] ;UCb

(Rd))be

the set of all functions v such that

‖v‖0 := sup(t,x)∈[0,τ ]×Rd

|v (t, x)| ≤ θ.

Given v ∈ Xτ,θ, set

(Γv) (t, x) = (Ptu0) (x) +

∫ t

0(Pt−sf (v (s))) (x) ds

for t ∈ [0, τ ]. The function Γv is well defined and of class C([0, τ ] ;UCb

(Rd)). We have

‖Γv‖0 ≤ sup(t,x)∈[0,τ ]×Rd

∣∣∣∣∫ t

0(Pt−sf (v (s))) (x) ds

∣∣∣∣ ≤ Cf,θτwhere

Cf,θ = max|u|≤θ

|f (u)| .

Moreover we have

‖Γv − Γw‖0 ≤ sup(t,x)∈[0,τ ]×Rd

∣∣∣∣∫ t

0(Pt−s (f (v (s))− f (w (s)))) (x) ds

∣∣∣∣≤∫ τ

0Lf,θ ‖v − w‖0 ds = τLf,θ ‖v − w‖0

whereLf,θ = max

|u|≤θ|∇f (u)| .

It is now easy to verify that, for τ small enough, the set Xτ,θ is invariant under Γ and on such set Γis a contraction. This gives us local existence and uniqueness, by the contraction principle.

4.4. Invariant regions (Step 1c and completion of Step 1).4.4.1. Approach by linear equations. Let u be a local classical solution. Just for notational sim-

plicity, we assume it is define on [0, T ]. Setting λ (t, x) = 1− u (t, x), we have

∂u

∂t= ∆u+ λu.

For this kind of equations it is well known that:

Lemma 1. If u is a classical solution on [0, T ] with u0 ≥ 0, then u ≥ 0 on [0, T ].

4. EXISTENCE, UNIQUENESS, INVARIANT REGIONS 59

Proof. Among different proof let us choose a probabilistic one: by the well-known Feynman-Kacformula

u (t, x) = E[e∫ t0 λ(t−s,x+

√2Bs)dsu0

(x+√

2Bt

)]we immediately deduce the result. Here Bt is a Brownian motion in Rd. Let us recall the proof ofFeynman-Kac formula in this simple case. Given T0 > 0, by Itô formula for t ∈ [0, T0] we have

du(T0 − t, x+

√2Bt

)= −∂u

∂t

(T0 − t, x+

√2Bt

)dt+∇u

(T0 − t, x+

√2Bt

)·√

2dBt

+ ∆u(T0 − t, x+

√2Bt

)dt

= −λu(T0 − t, x+

√2Bt

)dt+∇u

(T0 − t, x+

√2Bt

)·√

2dBt

hence the process ξt := u(T0 − t, x+

√2Bt

)satisfies dξt = −λtξtdt+dMt, where λt = λ

(T0 − t, x+

√2Bt

),

Mt =∫ t

0 ∇u(T0 − s, x+

√2Bs

)·√

2dBs, whence

d(e∫ t0 λsdsξt

)= λt

(e∫ t0 λsdsξt

)dt+ e

∫ t0 λsds (−λtξtdt+ dMt) = e

∫ t0 λsdsdMt.

This implies

E[e∫ T00 λsdsu0

(x+√

2BT0

)]= u (T0, x)

which is the desired formula.

Now let us set, as we have done above, v = 1− u. We have∂v

∂t= ∆v − v (1− v)

hence∂v

∂t= ∆v + λv

where λ = v − 1 = −u. By the lemma we deduce

Corollary 2. If u is a classical solution on [0, T ] with u0 ≤ 1, then u ≤ 1 on [0, T ].

Proof. The lemma applied to the equation for v gives us v ≥ 0 on [0, T ]. Hence u ≤ 1 on[0, T ].

4.4.2. Approach by the theory of invariant regions. The previous approach is very fast but it hasthe disadvantage to be applicable only when the algebraic structure of the terms is particularly simple.In order to deal with the full system described in Chapter 1, we think it is convenient to know thefollowing somewhat deeper approach. We address to [19], Chapter 14 for more details on this approach.

Our aim is to prove that the interval [0, 1] is an invariant region for equation (4.2) (it will be alsofor equation (4.1)).

Let u be a classical solution of∂u

∂t= ∆u+ u (1− u) + h (u)


on some interval [0, τ ]. We want to prove that, if u0 ∈ [0, 1], then u ∈ [0, 1] on [0, τ ]. For simplicity ofnotation we take τ = T .

Let us argue by contradiction; hence, assume that u0 ∈ [0, 1] but [0, 1] is not invariant (there issome (t′, x′) with u (t′, x′) /∈ [0, 1]).

Assume that, due to the lack of invariance, we can prove that there is t0 ∈ [0, T ] with the followingproperties:

i) t0 > 0ii) u (t, x) ∈ [0, 1] for every x ∈ Rd and every t ∈ [0, t0]iii) u (t0, x0) ∈ 0, 1 for some x0 ∈ Rd.Let us analyze the case u (t0, x0) = 1. We deduce:a) ∇u (t0, x0) = 0 and ∆u (t0, x0) ≤ 0 (because u (t0, x) ≤ 1 for all x and u (t0, x0) = 1, and u (t0, ·)

is C2)b) ∂u

∂t (t0, x0) < 0 (from the equation, because ∆u (t0, x0) ≤ 0, u (t0, x0) (1− u (t0, x0)) = 0,h (u (t0, x0)) < 0)

c) ∂u∂t (t0, x0) ≥ 0 (because u (t, x0) ≤ 1 for t ∈ [0, t0] and u (t0, x0) = 1).

Thus there is a contradiction. In the case u (t0, x0) = 0 we havea’) ∇u (t0, x0) = 0 and ∆u (t0, x0) ≥ 0 (because u (t0, x) ≥ 0 for all x and u (t0, x0) = 0, and

u (t0, ·) is C2)b’) ∂u

∂t (t0, x0) > 0 (from the equation, because ∆u (t0, x0) ≥ 0, u (t0, x0) (1− u (t0, x0)) = 0,h (u (t0, x0)) > 0)

c’) ∂u∂t (t0, x0) ≤ 0 (because u (t, x0) ≥ 0 for t ∈ [0, t0] and u (t0, x0) = 0)

hence again a contradiction.Thus it remains to prove that t0 ∈ [0, T ] exists with properties (i)-(iii). Intuitively this is reason-

able, so we may split the proof in two parts: the main conceptual part is the one just given; a secondpart deals with the proof of existence of such t0. This second part is complicated by two potentialdiffi culties.

Let us start with the definition of t0:

t0 = inft′ ∈ [0, T ] : ∃x′ such that u

(t′, x′

)/∈ [0, 1]

or equivalently

(4.5) t0 = sup t ∈ [0, T ] : ∀x we have u (t, x) ∈ [0, 1] .The first diffi culty comes from the fact that t0 may be equal to zero, violating property (i). To

avoid this, one could argue as follows: we first assume u0 ∈ [ε, 1− ε] for some ε > 0. In this case, sincewe know from the existence theorem that t 7→ u (t) is continuous in the space UCb

(Rd), there is some

small time interval [0, δ] such that that u (t, x) ∈ [ε/2, 1− ε/2] for t ∈ [0, δ]; therefore t0 defined aboveby (4.5) is strictly positive. Later, when the proof of invariance is done for initial conditions satisfyingu0 ∈ [ε, 1− ε], given a general u0 ∈ [0, 1] it is suffi cient to take un0 =

(1− 1

n

)u0 + 1

2n and prove thatthe solutions un with initial conditions un0 converge uniformly on compact sets to u. The continuousdependence on initial conditions can be established similarly to the proof of existence by contractionprinciple (or the proof of continuous dependence on δ in step 2 above).

The second diffi culty is related to lack of compactness. Given the definition above of t0, withproperties (i)-(ii), can we say that (iii) holds? We know that for every ε > 0 there is t′ε ∈ (t0, t0 + ε)

5. THE PROBLEM OF INVARIANT REGIONS 61

and x′ε ∈ Rd such that u (t′ε, x′ε) /∈ [0, 1], so by continuity there is t′′ε ∈ [t0, t0 + ε) and x′′ε ∈ Rd such

that u (t′′ε , x′′ε ) ∈ 0, 1. Obviously limε→0 t

′′ε = t0, but what about the family x′′ε ?

One way to solve this problem is to prove that there is a compacts set K ⊂ Rd such that u (t, x) ∈(0, 1) for all t ∈ [0, T ] and x /∈ K. When this is proved, we know that x′′ε ⊂ K, hence we may extracta sequence which converges to a point x0, so that u (t0, x0) ∈ 0, 1.

The existence ofK requires some technical work but, conceptually, it is based on simple ingredients:the property of decay to zero at infinity is preserved by the heat semigroup, the image of the heatsemigroup is made of functions which are strictly positive everywhere (infinite speed of diffusion).Notice that this way of solving the problem restricts the set of initial conditions: they have to decayto zero at infinity. Using other arguments one can treat more general u0 ∈ [0, 1].

4.5. Continuous dependence on h (Step 2). This part can be done in two ways. One is byestimating the distance between u and uδ, by triangle inequality and suitable estimates. The otherby proving that Ascoli-Arzelà theorem can be applied, and then passing to the limit.

5. The problem of invariant regions

CHAPTER 4

The mathematics of proliferation at the microscopic level

63

Part 3

Interacting systems of cells and nonlinearPartial Differential Equations

Bibliography

[1] A. R. A. Anderson, A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion,Mathematical Medicine and Biology (2005) 22, 163—186.

[2] M. Bodnar, J. J. L. Velazquez, Derivation of macroscopic equations for individual cell-based models: a formalapproach, Math. Methods Appl. Sci. 28 (2005), no. 15, 1757—1779.

[3] H. M. Byrne, Mathematical Biomedicine and modeling avascular tumor growth, OCCAM Preprint Number 12/96.[4] H. Byrne, D. Drasdo, Individual-based and continuum models of growing cell populations: a comparison, J. Math.

Biol. (2009) 58:657—687.[5] V. Capasso, D. Morale, Stochastic modelling of tumour-induced angiogenesis, J. Math. Biol. 58 (2009), no. 1-2,

219—233.[6] D. Drasdo, S. Hoehme, M. Block, On the role of Physics in the growth and pattern formation of multi-cellular

systems: what can we learn from individual-cell based models? J. Stat. Physics 128 (2007), n.1/2, 287-345.[7] Jean-Pierre Françoise, Oscillations en biologie: Analyse qualitative et modèles, Springer 2006.[8] I.I. Gikhman, A.V. Skorohod, The Theory of Stochastic Processes I, Springer, Berlin 1974.[9] A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame, Effects of space sctructure and combination

therapies on phenotypic heterogeneity and drug resistance in solid tumors, arxhiv 2013.[10] H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure

Appl. Math. XXVIII (1975), 323-331[11] D. Morale, V. Capasso, Vincenzo, K. Oelschläger, An interacting particle system modelling aggregation behavior:

from individuals to populations, J. Math. Biol. 50 (2005), no. 1, 49—66.[12] K. Oelschläger, A law of large numbers for moderately interacting diffusion processes, Z. Wahrsch. Verw. Gebiete

69 (1985), no. 2, 279—322.[13] K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately inter-

acting stochastic processes, Probab. Theory Related Fields 82 (1989), no. 4, 565—586.[14] K. Oelschläger, Large systems of interacting particles and the porous medium equation, J. Differential Equations

88 (1990), no. 2, 294—346.[15] S. Olla, S. R. S.Varadhan, H.-T. Yau, Hydrodynamical limit for a Hamiltonian system with weak noise, Comm.

Math. Phys. 155 (1993), no. 3, 523—560.[16] R. Philipowski, Interacting diffusions approximating the porous medium equation and propagation of chaos, Sto-

chastic Process. Appl. 117 (2007), no. 4, 526—538.[17] R. Philipowski, Stochastic interacting particle systems and nonlinear partial differential equations from fluid me-

chanics, RIMS Kôkyûroku Bessatsu B6 (2008), 201—211.[18] M. Scianna, L. Preziosi, A Hybrid Model Describing Different Morphologies of Tumor Invasion Fronts, Math. Model.

Nat. Phenom. Vol. 7 (2012), No. 1, pp. 78-104.[19] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Second Ed., Springer, New-York, 1994.[20] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, Berlin 1991.[21] A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-

particle systems, SIAM J. Appl. Math. 61 (2000), no. 1, 183—212.[22] P. Hinow, P. Gerlee, L. J. McCawley, V. Quaranta, M. Ciobanu, S. Wang, J. M. Graham, B. P. Ayati, J. Claridge, K.

R. Swanson, M. Loveless, A. R. A. Anderson, A spatial model of tumor-host interaction: application of chemoterapy,Math. Biosc. Engin. 2009.

67

68 BIBLIOGRAPHY

[23] A.-S. Sznitman, Topics in propagation of chaos, Ecole d’Eté de Probabilités de Saint-Flour XIX – 1989, LectureNotes in Mathematics Volume 1464, 1991, pp 165-251.

[24] Youshan Tao, Jianjun Paul Tian, Mathematical Oncology, Springer, Berlin 2010.[25] C. Tremoulet, Hydrodynamic limit for interacting Ornstein-Uhlenbeck particles, Stochastic Process. Appl. 102

(2002), no. 1, 139—158.[26] S.E. Wang, P. Hinow, N. Bryce, A. M. Weaver, L. Estrada, C. L. Arteaga, G. F. Webb, A mathematical model

quantifies proliferation and motility effects of TGF-β on cancer cells, Comput. Math. Methods in Medicine, 2013.[27] S. R. S. Varadhan, Scaling limits for interacting diffusions, Comm. Math. Phys. 135 (1991), no. 2, 313—353.[28] http://www.cancerresearchuk.org/cancer-info/cancerandresearch/all-about-cancer/what-is-cancer/grow-and-

spread/growing-a-blood-supply/growing-a-blood-supply[29] http://www.cancerresearchuk.org/about-cancer/cancers-in-general/what-is-cancer/grow/how-a-cancer-gets-its-

blood-supply[30] http://galleryhip.com/tumor-angiogenesis.html

elements of mathematical oncology franco flandoliusers.dma.unipi.it/~flandoli/parts_i_and_ii.pdf10...

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