mark chaplain, the simbios centre, department of mathematics, university of dundee,
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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
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