tumour rd pattern formation

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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_at_maths.dundee.ac.uk http//www.maths.dund
ee.ac.uk/chaplain http//www.simbios.ac.uk
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Talk Overview

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

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The Individual Cancer Cell A Nonlinear Dynamical
System

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The Multicellular Spheroid Avascular Growth

• 10 6 cells
• maximum diameter 2mm
• Necrotic core
• Quiescent region
• Thin proliferating rim

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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 membrane
Cancer cells break through basement membrane
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Turing pre-pattern theory Reaction-diffusion
models
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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)
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Turing pre-pattern theory Reaction-diffusion
models
We consider small perturbations about this steady
state

it can be shown that.
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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
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Dispersion curve
Re ?
k2
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Mode selection dispersion curve
Re ?
k2
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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
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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
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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
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Galerkin Method
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Numerical Quadrature
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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

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Numerical experiments on Schnackenberg system
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Mode selection n2
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Chemical pre-patterns on the sphere
mode n2
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Mode selection n4
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Chemical pre-patterns on the sphere
mode n4
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Mode selection n6
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Chemical pre-patterns on the sphere
mode n6
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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-ß (-ve)
• EGF, TGF-a, bFGF, PDGF, IGF, IL-1a, G-CSF
  (ve)
• TNF-a (/-)

• Experimentally observed interaction (ve, -ve
  feedback) between
• several of the growth factors in many different
  types of cancer

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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

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The Individual Cancer Cell

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Multiple mode selection No isolated mode
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Chemical pre-pattern on sphere no specific
selected mode
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Invasion patterns arising from chemical
pre-pattern
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Growing domain Moving boundary formulation
R(t) 1 at
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Mode selection in a growing domain
t 15
t 9
t 21
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Chemical pre-pattern on a growing sphere
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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
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G. Lolas Spatio-temporal pattern formation and
reaction-diffusion equations. (1999) MSc Thesis,
Department of Mathematics, University of Dundee.
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1D growing domain Boundary growth
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1D growing domain Boundary growth
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Dispersion curve
Re ?
k2
20
90
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Spatial wavenumber spacing
n k2 n(n1) k2 n2 p2 (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
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2D growing domain Boundary growth
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2D growing domain Boundary growth
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2D growing domain Boundary growth
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2D growing domain Boundary growth
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Cell migratory response to soluble molecules
CHEMOTAXIS
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Cell migratory response to local tissue
environment cues HAPTOTAXIS
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The Individual Cancer Cell A Nonlinear Dynamical
System

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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

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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

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Summary