A mathematical model of the ductal carcinoma by reected Brownian motion

A mathematical model of the ductal carcinoma by reected Brownian motion

Chiara Carmignani

Abstract

The goal of this work is to develop a model to describe the evolution of the ductal carcinoma. The ductal carcinoma is a cancer that originates in the female breast duct. We can imagine the duct as cylinder, with walls made of epithelial cells and its interior empty, in normal conditions. We call the walls of the duct basement membrane. If, during the normal proliferation of epithelial cells, an abnormal cell is generated, it originates a ductal carcinoma and is called cancer cell. The cancer cell can proliferate and generate other cancer cells, free to move. Initially the cells are forced to stay inside the duct and this stage is called carcinoma in situ. In the next phase, called invasive cancer, the cancer cells get out of the duct and give rise to metastases. To describe a model of this cancer, we study the reected Brownian motion. We assume, in fact, that the movement of cancer cells is modeled by the Brownian motion. When the cells meet the basament membrane, the Brownian motion is rejected, i.e. reected.

In the rst chapter we dene the reected Brownian motion. We analyze deeply the theory of the Local Time and Itô formula in dimension 1 and then we generalize to the case of arbitrary dimension d. The reected Brownian Motion ˜ B is dened in a limited domain D ⊂ R ^d with smooth boundary ∂D and we can write

B ˜ t = ˜ B 0 + W t + L t ,

|L| _t = Z t

0

1 { ˜ B

∈∂D} d|L| _s , L _t = Z t

0

n( ˜ B _s )d|L| _s ,

where W is a Brownian motion in R ^d , n(x) is the inward normal unit vector on the boundary ∂D in x ∈ ∂D and |L| t è the total variation of L up to time t. We dene the reected Brownian motion with potential. For this scope, we consider a function f n := exp((nδ(x)) ⁻¹ ) , in which δ(x) indicates the distance between x ∈ D and ∂D, and a squence of stochastic processes 

X ⁽ⁿ⁾ 

n∈N such that dX _t ⁽ⁿ⁾ = − 1

2 ∇f n

 X _t ⁽ⁿ⁾ 

dt + dW t ,

with W Brownian Motion in R ^d . This sequence of stochastic processes is dened in D and his weak limit is the reected Brownian motion ˜ B . We want to simulate the reected Brownian motion and we can assume that, for n large enough, the behavior of the process X ⁽ⁿ⁾ is very similar to the reected Brownian motion ˜ B . Thanks to this analogy between the reected Brownian motion and the sequence X ⁽ⁿ⁾ we have

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a stochastic process dened in D, which simulation is easier.

In the second chapter we study the link that exists between the reected Brownian motion and the heat equation with Neumann boundary conditions. We see that the reected Brownian motion has density and that its density is a solution of the heat equation with Neumann boundary condition. In addition we study the macroscopic limit to understand the behavior of a very large set of cancer cells and the link between the motion of these cells and the heat equation.

In the third chapter we give a short biological introduction of the ductal carcinoma and we dene its model. In addition we do some simulations of the model with Matlab. Initially we consider the Brownian motion with potential and we dene a function to regulate cell interactions, the repulsive cell interaction that prevents the overlap of cells and the attractive cell interaction. Then we substitute the potential function with a function that regulates the repulsive force of the the epithelial cells on the tumoral cells to mantein tumoral cells inside the duct. We dene a complete model of the evolution of the carcinoma, in which the tumoral cells can break the basement membrane and leave the duct. The model is the following

dX _t ⁱ = 1 N

N

X

j=1

K X _t ⁱ − X _t ^j
dt + 1 Z

Z

X

j=1

H X _t ⁱ − E _t ^j
dt + σdB _t ⁱ ,

dE _t ^j = 1 Z

Z

X

k=1

Q E _t ^j − E _t ^k
dt + 1 N

N

X

k=1

P E _t ^j − X _t ^k
dt,

where X t ⁱ indicates the position of the i-th tumoral cell at the time t and E t ^j indicates the position of the j-th epithelial cell at the time t. We simulate the model with Matlab and we generate the congurations of the cells of the carcinoma at dierent times. Finally, we show that the congurations generated by the model are matching the biological images of the ductal carcinoma.

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