Project Macrophage

Math Biology Summer School 2008

Jennifer Morrison & Caroline Séguin

Introduction To Tumor Growth

diffusion of nutrientsdevelopment of hypoxic zone

Avascular development of necrotic coreTumor release of macrophage chemoattractant

angiogenesis

VascularTumor

Avascular Tumors

Proliferating tumour cells : the crust

Hypoxic zone

The core

Macrophages

White blood cell – part of immune system

Main role: phagocytose pathogens

Attracted towards hypoxic region of tumor cells and promote proliferation of cancer cells and angiogenesis

The Plan

• Modify macrophages to release cytotoxic chemical once they get in the hypoxic zone to kill tumour cells

• Goal: new treatment which would target tumour cells

• Study effects of chemical concentration and macrophage concentration on the tumour

• Is this a possible treatment? Under which conditions?

Previous Results

Many models of tumor growth – most using a PDE approach

Owen, M.R. & Sherratt, J.A. 1998. Modelling the macrophage invasion of tumours: Effects on growth and composition. Journal of Mathematics Applied in Medicine & Biology, 15: 165-185.

Normal macrophages both promote proliferation and inhibit early growth of avascular tumors and therefore have no relevant effect on the overall growth.

Developing a new model

Involves chemotaxis and diffusion models

Using: c(r,t)=concentration of nutrients m(r,t)=concentration of macrophages R(t) = Radius of tumor at time t h(r,t)= concentration of chemoattractants x(r,t)=concentration of cytotoxic chemical

Equations

21R

Rdt

dR

2

2

21 r

mD

r

hm

rD

t

mmm

xdmr

xD

t

xx

2

2

Constant gradient

Chemotaxis + diffusion

Diffusion

Logistic growth

22

))((2

),( rtRkkR

cctrc outout

Our Model

Cellular Automata 2D model which will

simulate a cross section of a 3D tumor

Assumptions

Logistic growth of radius and constant proliferating zone

Tumor only contains cancerous cells (no normal cells)

No angiogenesis yet

Macrophages do not proliferate and die when they reach the necrotic core

Toxic chemical concentration is considered to be uniformly released from hypoxic zone

No normal macrophages

Macrophage concentration is uniformly distributed around the tumor

Cytotoxic chemical is diffused: we model it using the solution of the diffusion equation

CA Rules

• Time and space are discrete• We define an (101*101)

matrix , and we assign a value to each element, representing its state.

• Determine the state of a cell at the next time step, from the state of its 8 neighbours and its own.

k1-1, k2+1

Neighbour

k1, k2+1

Neighbour

k1+1,k2+1

Neighbour

k1-1, k2

Neighbour

k1,k2

Cell that we consider

k1+1, k2

Neighbour

k1-1, k2-1

Neighbour

k1, k2-1

Neighbour

k1+1, k2-1

Neighbour

CA of Tumour Growth

k1-1, k2+1

Neighbour

k1, k2+1

Neighbour

k1+1,k2+1

Neighbour

k1-1, k2

Neighbour

k1,k2

Cell that we consider

k1+1, k2

Neighbour

k1-1, k2-1

Neighbour

k1, k2-1

Neighbour

k1+1, k2-1

Neighbour

• Matrix A• A(k1,k2) defines state of

tumour cell at the (k1,k2) element.

• A(k1,k2)=0: proliferating tumour cells

• A(k1,k2)=1: Hypoxic zone• A(k1,k2)=2 :Dead tumour

cells (Necrotic core)• A(k1,k2)=3: No tumour

cells (outside tumour)

CA of Tumour Growth

The crust: A(k1,k2)=0

The core: A(k1,k2)=2

Hypoxic zone:A(k1,k2)=1

CA Nutrient Concentration Gradient

New matrix called “c”

if dist(k1,k2)>R(i+1,2)c(k1,k2)=Cout;elseif dist(k1,k2)<Rn(i+1)

c(k1,k2)=0;elseif

dist(k1,k2)>Rn(i+1)&dist(k1,k2)<(Rn(i+1)+epsilon)

ADDelse if (Rn(i+1)>0) c(k1,k2)=Cout-k_c*(R((i+1),2)-

dist(k1,k2))^2;else c(k1,k2)=0;end

k1-1, k2+1

Neighbour

k1, k2+1

Neighbour

k1+1,k2+1

Neighbour

k1-1, k2

Neighbour

k1,k2

Cell that we consider

k1+1, k2

Neighbour

k1-1, k2-1

Neighbour

k1, k2-1

Neighbour

k1+1, k2-1

Neighbour

CA Nutrient Concentration Gradient

CA Macrophage Chemotaxis

C(k1-1, k2+1)

Neighbour

C(k1, k2+1)

Neighbour

C(k1+1,k2+1)

Neighbour

C(k1-1, k2)

Neighbour

C(k1,k2)

Cell that we consider

C(k1+1, k2)

Neighbour

C(k1-1, k2-1)

Neighbour

C(k1, k2-1)

Neighbour

C(k1+1, k2-1)

Neighbour

New matrix called “macro”: each cell has a value according to the number of macrophages in it

The concentration of chemoattractant is inversely proportional to the concentration of nutrient (because chemoattractant is released in the hypoxic zone)

=> Macrophage moves to neighbouring cell which has the least concentration of nutrient (the highest concentration of chemoattractant)

CA Macrophage Chemotaxis

CA Activated Macrophage

New matrix called “macro_active”

Macrophage is activated (releases the cytotoxic chemical) when it reaches the hypoxic zone

=> in the code, that means when A(k1,k2)=1 and macro(k1,k2)>0

The core: A(k1,k2)=2

The crust: A(k1,k2)=0

Hypoxic zone:A(k1,k2)=1

CA Cytotoxic Chemical Release

• New matrix called “toxic”

• We suppose that the chemical diffuses rapidly compared to our time step: we model it using the solution of the diffusion equation : the Gaussian

CA Cytotoxic Chemical Release

CA: Final Results: Low threshold

Low threshold: necessary concentration of chemical to kill the tumour cell is low: tumour cells are killed easily

=>Tumour is destroyed.

CA: Final Results: Higher Threshold

Higher threshold

=>Tumour is not destroyed...

(hypoxic zone disappears=> no more cytotoxic diffusion!)

Results

High threshold

Comparison : effect of different thresholds

Discussion

Obtained results were not as expected: even if the radius of the tumour reaches steady state, it may not be destroyed... Conditions...

If we suppose that our macrophages can release a sufficient amount of cytotoxic chemical, it would be a possible treatment

Other variables that should be studied : amount of injected macrophages, how many injections, ...

Acknowlegments

Special thanks to Gustavo and Andrea

Thanks to all other instructors, volunteers and students!