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

chaplain@maths.dundee.ac.uk

http://www.maths.dundee.ac.uk/~chaplain

http://www.simbios.ac.uk

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) tumouris a CARCINOMA (Greek: Karkinos, a crab). Irregular, jagged shape often assumed due to localspread of carcinoma.

Basement membraneCancer cells break through basement membrane

Turing pre-pattern theory:Reaction-diffusion models

vdvugv

uvufu

t

t

2

2

),(

),(

reaction diffusion

n̂

given)0,(,)0,(

on 0..

,

xx

nn

vu

vu

vu

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.

0),(),( 0000 vugvuf

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:

0Wn0WW

xWxw

xw

xx

,

where)(),(

form theof solutions seeking and),(),( denoting

),(,),(

22

~~

~

0

~

0

k

ect

vut

tvvvtuuu

kk

tk

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:

k

kt

k ect )(),( xWxw

where0)()( 222 khkf

0Re

DISPERSION RELATION

Dispersion curve

Re λ

k2

21k 2

mk

Mode selection: dispersion curve

Re λ

k2

21k 2

mk

2ik

2jk

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

),(

),(

*

*

vugvdv

vufuu

t

t

Growth promoting factor (activator) uGrowth inhibiting factor (inhibitor) vProduced, react, diffuse on surface of a tumour spheroid

Numerical analysis technique

Spectral method of lines:

22 ,,,,,,,,,,,

,,,,,,,,,,,

)()(),(),(

)()(),(),(

1987654321

11

21

22

12

02

12

22

11

01

11

00

1

0

1

0

NN

NN

NN

N

n

n

nm

mn

mnN

N

n

n

nm

mn

mnN

UUUUUUUUUUU

YYYYYYYYYYY

YtVtvtv

YtUtutu

xxx

xxx

Apply Galerkin method to system of reaction-diffusion equations (PDEs) andthen end up with a system of ODEs to solve for (unknown) coefficients

)(and)( tVtU mn

mn

Spherical harmonics:eigenfunctions of Laplace operator onsurface of sphere

mode 1 pattern mode 2 pattern

Galerkin Method

)),((2

),),(()()1()(

}1,...0,||,)(exp)(cos)({

),),((),(),)((

2/

11

||

*

qp

S

M

pp

M

q

mnNN

mn

mn

mn

mn

mnN

NNNNNNtN

wM

YvufUnnU

NnnmimPcYG

GvufGuGu

x

Numerical Quadrature

)),(())),,((),),(((2

)),,((

),),(()()1()(

),),((),(),)((

)),(()),((2

),(

2/

11

*

_2/

11

qpm

nqpNqpN

M

pp

M

qM

mnNN

Mm

nNNmn

mn

MNNNNNNtN

qpqp

M

pp

M

qM

YtvufwM

Yvuf

YvufUnnU

GvufGuGu

vuwM

vu

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

Numerical experiments on Schnackenberg system

modeselect chosen to,18,0056.17

,69.0,2.1,1,2.0

sin

1sin

sin

1

)(

)(

00

2

2

*

2*

2*

dd

vuba

uuu

vubvdv

vuuauu

c

t

t

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

100,25

,69.0,2.1,1,2.0 00

d

vuba

Chemical pre-pattern on sphere no specific selected mode

Invasion patterns arising from chemical pre-pattern

Growing domain: Moving boundary formulation

),()]([

1

),()]([

1

*2

*2

vugvdtR

v

vufutR

u

t

t

spherical solid tumour

r = R(t)

radially symmetricgrowth 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. LolasSpatio-temporal pattern formation and reaction-diffusion equations. (1999) MSc Thesis, Department of Mathematics, University of Dundee.

1D growing domain: Boundary growth

1D growing domain: Boundary growth

Dispersion curve

Re λ

k2

21k 2

mk

20 90

Spatial wavenumber spacing

n k2 = n(n+1) k2 = n2 π2

(sphere) (1D)

2 6 403 12 904 20 1605 30 2506 42 3607 56 4908 72 6409 90 81010 110 1000

2D growing domain: Boundary growth

2D growing domain: Boundary growth

2D growing domain: Boundary growth

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