mark chaplain, the simbios centre, department of mathematics, university of dundee, dundee, dd1 4hn....

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

[email protected]

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


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)


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


...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


mitotic “hot spot”

Mode selection: 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.

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

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

mark chaplain, the simbios centre, department of mathematics, university of dundee, dundee, dd1 4hn....

Documents