1. Project Macrophage Math Biology Summer School 2008 Jennifer Morrison & Caroline Séguin

2. Introduction To Tumor Growth diffusion of nutrients development of hypoxic zone Avascular development of necrotic core Tumor release of macrophage chemoattractant angiogenesis Vascular Tumor

3. Avascular Tumors Proliferating tumour cells : the crust Hypoxic zone The core

4. 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

5. 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?

6. 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.

7. 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

8. Equations Constant gradient Chemotaxis + diffusion Diffusion Logistic growth

9. Our Model Cellular Automata 2D model which will simulate a cross section of a 3D tumor

10. 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

11. 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

12. 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)

13. CA of Tumour Growth The crust: A(k1,k2)=0 The core: A(k1,k2)=2 Hypoxic zone: A(k1,k2)=1

14. 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) ADD else 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

15. CA Nutrient Concentration Gradient

16. 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)

17. CA Macrophage Chemotaxis

18. 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

19. 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

20. CA Cytotoxic Chemical Release

21. 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.

22. CA: Final Results: Higher Threshold Higher threshold =>Tumour is not destroyed... (hypoxic zone disappears=> no more cytotoxic diffusion!)

23. Results High threshold

24. Comparison : effect of different thresholds

25. 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,...

26. Acknowlegments Special thanks to Gustavo and Andrea Thanks to all other instructors, volunteers and students!