1. Project Macrophage: Macrophages on the Move Heather More, Rachel Psutka, Vishaal Rajani

2. Overview Tumour grows quickly Hypoxia Release of chemoattractants Macrophage migration Release of angiogenic factors Blood vessel growth

3. Tumours Abnormal growth of tissue Macrophages migrate via chemotaxis Engineered macrophages activated to release chemicals – kill tumour cells

4. Macrophages in Tumours Movement via chemotaxis – innate behavioural response to the addition of a chemical Tumour cells release chemoattractants under hypoxia

5. Chemotaxis Video Chemotaxis

6. Macrophage Originate from monocytes Phagocytotic, specific and non-specific defense Engulf cellular debris and pathogens Assist in wound healing, migrate to hypoxic regions

7. Macrophage Video Macrophage

8. Blood Vessel Growth Macrophages release angiogenic factors Promotes blood vessel growth

9. Angiogenesis Video Angiogenesis

10. Cells For our 1-dimensional purposes... Necrotic core Hypoxic cells Live cells

12. Building a model for chemotaxis Chemoattractants released by hypoxic cells attract macrophages Taking the diffusion equation of macrophage density at position x and time t; where a is the diffusion coefficient.

13. We add an “attracting” term which describes the attraction of macrophages to positions of highest chemoattractant concentration, becoming: where X denotes the chemotaxis coefficient, c(x,t) denotes the chemoattractant concentration at position x, time t. Diffusion TermAttraction Term

14. Exploring Chemotaxis Hypoxic cells emit chemoattractants What kind of function can we use to model this? length of tumour chemoattractant concentration

15. Exploring Chemotaxis Using a Gaussian profile for the concentration of chemoattractants; we solved the attraction-diffusion equation of macrophages numerically, using MAPLE.

16. t=0

17. What would happen if we introduced bioengineered macrophages that had a killing effect on tumour cells?

18. r a, β β Ф

19. Building the model Chemoattractant concentration spread over space is proportional to hypoxic cells over space... For our purposes.

20. Hypoxic Cells killing effect of macrophages, where β is the proportion of bioengineered macrophages the proportion of living cells that become hypoxic

21. Living Cells killing effect of macrophages, where β is the proportion of bioengineered macrophages living cells that become hypoxic logistic growth of tumour cells What determines r?

22. Becoming Hypoxic When cells are lacking oxygen, they are hypoxic and do not divide. We want a growth rate that applies only to cells that are on the outside layers of the cell. BUT step functions are NOT continuous. Let

23. The Resulting System (Boo Yah!)

24. Initial and Boundary Conditions Initial Conditions Boundary Conditions Of the two possibilities, we used these ones.

25. Observations decreases faster than and/or causes faster decrease of and causes slower decrease of and a large amount of engineered macrophages with enhanced sensitivity to chemoattractant and low diffusion rates are required in order to have the eventual death of all cells

26. Observations β = 0.5 β = 4

27. Observations

28. Assumptions No angiogenesis 1D tumour Killing effect was assumed to be the same for living and hypoxic cells Proliferation of hypoxic cells is negligible

29. Conclusions β has a threshold... a sufficient killing effect is needed to kill the tumour entirely Increasing macrophage sensitivity to chemoattractants would increase the killing effect of bioengineered macrophages Hypoxic cells die faster than living cells with the introduction of bioengineered macrophages

30. Future Projects/Questions Hardcore analysis of β Conservative PDE Solver Angiogenesis Necrotic core...spatial modelling of tumor 2D and 3D models Chemotherapy, Radiotherapy

31. Acknowledgements Byrne, H. The role of mathematics in solid tumour growth. Mathematics Today. April 1999, 48-53. Gerda de Vries, Caroline Bampfylde, Jim Keener, Tomas de-Camino-Beck Colin More (Heather’s brother, for Maple help) All other instructors and volunteers for background knowledge Fellow students for motivation and emotional support

32. The End!