Mark Chaplain, The SIMBIOS Centre, Department of Mathematics, University of Dundee, Dundee, DD1 4HN. Mathematical modelling of solid tumour growth: Applications of Turing pre-pattern theory

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

The Individual Cancer Cell “A Nonlinear Dynamical System”

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

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. Basement membraneCancer cells break through basement membrane

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

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 (u 0, v 0 ) i.e. We require this steady state to be (linearly) stable (certain conditions on the Jacobian matrix)

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

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

Dispersion curve Re λ k2k2

Mode selection: dispersion curve Re λ k2k2

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;

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

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 Spherical harmonics: eigenfunctions of Laplace operator on surface of sphere mode 1 patternmode 2 pattern

Galerkin Method

Numerical Quadrature

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, Spectral method of lines, numerical quadrature, FFT reduction from O(N 4 ) to O(N 3 logN) operations

Numerical experiments on Schnackenberg system

Mode selection: n=2

Chemical pre-patterns on the sphere mode n=2

Mode selection: n=4

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

Mode selection: n=6

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

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

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

The Individual Cancer Cell

Multiple mode selection: No isolated mode

Chemical pre-pattern on sphere no specific selected mode

Invasion patterns arising from chemical pre-pattern

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

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

Chemical pre-pattern on a growing sphere

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

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

1D growing domain: Boundary growth

Dispersion curve Re λ k2k2 2090

Spatial wavenumber spacing nk 2 = n(n+1)k 2 = n 2 π 2 (sphere) (1D)

2D growing domain: Boundary growth

Cell migratory response to soluble molecules: CHEMOTAXIS

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

The Individual Cancer Cell “A Nonlinear Dynamical System”

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 Tumour Cell Invasion of Tissue

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 Conclusions

Summary localised avascular solid tumour aggressive invading solid tumour Turing pre-pattern theory