Mark Chaplain, The SIMBIOS Centre, Department of Mathematics, University of Dundee,

1. Mathematical modelling of solid tumour growth: Applications of Turing pre-pattern theory Mark Chaplain, The SIMBIOS Centre, Department of Mathematics, University of Dundee, Dundee, DD1 4HN. chaplain@maths.dundee.ac.uk http://www.maths.dundee.ac.uk/~chaplain http://www.simbios.ac.uk

2. Talk Overview • Biological (pathological) background • Avascular tumour growth • Invasive tumour growth • Reaction-diffusion pre-pattern models • Growing domains • Conclusions

3. The Individual Cancer Cell “A Nonlinear Dynamical System”

4. The Multicellular Spheroid: Avascular Growth • ~ 10 6 cells • maximum diameter ~ 2mm • Necrotic core • Quiescent region • Thin proliferating rim

5. Malignant tumours: CANCER Generic name for a malignant epithelial (solid) tumour is a CARCINOMA (Greek: Karkinos, a crab). Irregular, jagged shape often assumed due to local spread of carcinoma. Cancer cells break through basement membrane Basement membrane

6. diffusion reaction ^ n Turing pre-pattern theory: Reaction-diffusion models

7. Turing pre-pattern theory: Reaction-diffusion models Two “morphogens” u,v: Growth promoting factor (activator) Growth inhibiting factor (inhibitor) Consider the spatially homogeneous steady state (u0 , v0 ) i.e. We require this steady state to be (linearly) stable (certain conditions on the Jacobian matrix)

8. spatial eigenfunctions Turing pre-pattern theory: Reaction-diffusion models We consider small perturbations about this steady state: it can be shown that….

9. Turing pre-pattern theory: Reaction-diffusion models ...we can destabilise the system and evolve to a new spatially heterogeneous stable steady state (diffusion-driven instability) provided that: where DISPERSION RELATION

10. Dispersion curve Re λ k2

11. Mode selection: dispersion curve Re λ k2

12. Turing pre-pattern theory…. • robustness of patterns a potential problem (e.g. animal coat marking) • (lack of) identification of morphogens ??? 1) Crampin, Maini et al. - growing domains; Madzvamuse, Sekimura, Maini - butterfly wing patterns; 2) limited number of “morphogens” found; de Kepper et al;

13. Turing pre-pattern theory: RD equations on the surface of a sphere Growth promoting factor (activator) u Growth inhibiting factor (inhibitor) v Produced, react, diffuse on surface of a tumour spheroid

14. Spherical harmonics: eigenfunctions of Laplace operator on surface of sphere mode 1 pattern mode 2 pattern Numerical analysis technique Spectral method of lines: Apply Galerkin method to system of reaction-diffusion equations (PDEs) and then end up with a system of ODEs to solve for (unknown) coefficients

15. Galerkin Method

16. Numerical Quadrature

17. Collaborators • M.A.J. Chaplain, M. Ganesh, I.G. Graham • “Spatio-temporal pattern formation on spherical surfaces: numerical • simulation and application to solid tumour growth.” • J. Math. Biol. (2001) 42, 387- 423. • Spectral method of lines, numerical quadrature, FFT • reduction from O(N 4) to O(N 3 logN) operations

18. Numerical experiments on Schnackenberg system

19. Mode selection: n=2

20. Chemical pre-patterns on the sphere mode n=2

21. Mode selection: n=4

22. mitotic “hot spot” Chemical pre-patterns on the sphere mode n=4

23. Mode selection: n=6

24. mitotic “hot spots” Chemical pre-patterns on the sphere mode n=6

25. Solid Tumours • Avascular solid tumours are small spherical masses of cancer cells • Observed cellular heterogeneity (mitotic activity) on the surface and in • interior (multiple necrotic cores) • Cancer cells secrete both growth inhibitory chemicals and growth • activating chemicals in an autocrine manner:- • TGF-β (-ve) • EGF, TGF-α, bFGF, PDGF, IGF, IL-1α, G-CSF (+ve) • TNF-α (+/-) • Experimentally observed interaction (+ve, -ve feedback) between • several of the growth factors in many different types of cancer

26. Biological model hypotheses • radially symmetric solid tumour, radius r = R • thin layer of live, proliferating cells surrounding a necrotic core • live cells produce and secrete growth factors (inhibitory/activating) • which react and diffuse on surface of solid spherical tumour • growth factors set up a spatially heterogeneous pre-pattern • (chemical diffusion time-scale much faster than tumour growth time scale) • local “hot spots” of growth activating and growth inhibiting chemicals • live cells on tumour surface respond proliferatively (+/–) to distribution of • growth factors

27. The Individual Cancer Cell

29. Multiple mode selection: No isolated mode

30. Chemical pre-pattern on sphere no specific selected mode

31. Invasion patterns arising from chemical pre-pattern

32. r = R(t) radially symmetric growth at boundary spherical solid tumour Growing domain: Moving boundary formulation R(t) = 1 + αt

33. Mode selection in a growing domain t = 9 t = 15 t = 21

34. Chemical pre-pattern on a growing sphere

35. 1D growing domain: Boundary growth Growth occurs at the end or edge or boundary of domain only Growth occurs at all points in domain uniform domain growth

36. G. Lolas Spatio-temporal pattern formation and reaction-diffusion equations. (1999) MSc Thesis, Department of Mathematics, University of Dundee.

37. 1D growing domain: Boundary growth

38. 1D growing domain: Boundary growth

39. Dispersion curve Re λ k2 20 90

40. Spatial wavenumber spacing n k2 = n(n+1) k2 = n2 π2 (sphere) (1D) 2 6 40 3 12 90 4 20 160 5 30 250 6 42 360 7 56 490 8 72 640 9 90 810 10 110 1000

41. 2D growing domain: Boundary growth

42. 2D growing domain: Boundary growth

43. 2D growing domain: Boundary growth

44. 2D growing domain: Boundary growth

46. Cell migratory response to soluble molecules: CHEMOTAXIS

47. No ECM with ECM ECM + tenascinEC & Cell migratory response to local tissue environment cues HAPTOTAXIS

48. The Individual Cancer Cell “A Nonlinear Dynamical System”

49. Tumour Cell Invasion of Tissue • Tumour cells produce and secrete Matrix-Degrading-Enzymes • MDEs degrade the ECM creating gradients in the matrix • Tumour cells migrate via haptotaxis (migration up gradients • of bound - i.e. insoluble - molecules) • Tissue responds by secreting MDE-inhibitors

50. Conclusions • Identification of a number of genuine autocrine growth factors • practical application of Turing pre-pattern theory (50 years on….!) • heterogeneous cell proliferation pattern linked to underlying • growth-factor pre-pattern irregular invasion of tissue • “robustness” is not a problem; each patient has a “different” cancer; • growing domain formulation • clinical implication for regulation of local tissue invasion via • growth-factor concentration level manipulation