Mathematical Modelling Cancer Cell Invasion of Tissue:The Roles of Cell Adhesion

Vivi Andasari and Mark A.J. Chaplain

Division of Mathematics, Mathematical Biology Research Group

Introduction

Several key steps in metastatic cascade:I loss of cell-cell adhesion that causes

disaggregation of tumour cells from primarytumour mass,

I tissue degradation, by over-expression ofproteolytic enzymes, i.e., urokinase-typeplasminogen activator (uPA),

I active cell migration through theextracellular matrix (ECM) by enhancementcell-matrix adhesion.

We consider mathematical models in anattempt to understand better the roles ofcell adhesion involved in cancer invasion.

Continuous Modelling

We use Partial Differential Equations formodelling in cell density level. The focus ison the roles of urokinase-typeplasminogen activator (uPA) for ECMdegradation and cell adhesion for cellmigration through the ECM.

Multi-scale Modelling

Here we model cancer cell detachment bymodelling intracellular interactions mainlybetween E-Cadherin and β-Catenin insideeach of individual cell. The interactionmodel is adopted from an article byRamis-Conde et al., 2008.

[Ec] GGGGAcontact

[Em]

[Em] + [β]ν

GGGGGA[E/β]

[E/β] GGGGGGGGAdetachment

[Ec] + [β]

[β] + [P]k+

GGGGGGA

k−[β/P]

k2GGGGGGA [P] + [ω]

References

N.J. Armstrong, K.J. Painter, J.A. Sherratt.A Continuum Approach to Modelling Cell-cell Adhesion.Journal of Theoretical Biology, 243:98–113, 2006.

M.A.J. Chaplain, G. Lolas.Mathematical Modelling of Cancer Cell Invasion of Tissue: The Role of TheUrokinase Plasminogen Activation System.Mathematical Models and Methods in Applied Sciences, 15:1685–1734,2005.

I. Ramis-Conde, D. Drasdo, A.R.A. Anderson, M.A.J. Chaplain.Modeling the Influence of the E-Cadherin-β-Catenin Pathway in Cancer CellInvasion: A Multiscale Approach.Biophysical Journal, 95:155–165, 2008.

Acknowledgment

I The Northern Research PartnershipI Dr Alf Gerisch, Technische Universitat DarmstadtI Dr Maciej Swat, Biocomplexity Institute, Indiana University, BloomingtonI Ryan Roper, University of Washington, Seattle

Continuous Modelling: Partial Differential Equations

cells 1 :∂c1

∂t= Dc1∇2c1︸ ︷︷ ︸

diffusion

−∇ ·[c1

1R

∫ R

−Rf (Scc1, Scv1, c1, c2, v) Ω(r ) dr

]︸ ︷︷ ︸

cell-cell & cell-matrix adhesion

−∇ · (χuc1∇u)︸ ︷︷ ︸uPA chemotaxis

−∇ · (χpc1∇p)︸ ︷︷ ︸PAI-1 chemotaxis

+µ1c1(1− c1 − c2)︸ ︷︷ ︸proliferation

−λc1F (t)︸ ︷︷ ︸conversion

cells 2 :∂c1

∂t= Dc2∇2c2︸ ︷︷ ︸

diffusion

−∇ ·[c2

1R

∫ R

−Rf (Scc2, Scv2, c1, c2, v) Ω(r ) dr

]︸ ︷︷ ︸

cell-cell & cell-matrix adhesion

−∇ · (χuc2∇u)︸ ︷︷ ︸uPA chemotaxis

−∇ · (χpc2∇p)︸ ︷︷ ︸PAI-1 chemotaxis

+µ2c2(1− c1 − c2)︸ ︷︷ ︸proliferation

+λc1F (t)︸ ︷︷ ︸conversion

ECM :∂v∂t

= −δvm︸ ︷︷ ︸degradation

+ φ21pu︸ ︷︷ ︸growth due uPA/PAI-1

− φ22pv︸ ︷︷ ︸neutralization by PAI-1

+µ2v(1− v)︸ ︷︷ ︸remodelling

uPA :∂u∂t

= Du∇2u︸ ︷︷ ︸diffusion

− φ31pu︸ ︷︷ ︸removal by PAI-1

−φ33(c1 + c2)u︸ ︷︷ ︸removal by cells

+α31(c1 + c2)︸ ︷︷ ︸production

PAI-1 :∂p∂t

= Dp∇2p︸ ︷︷ ︸diffusion

− φ41pu︸ ︷︷ ︸loss due uPA

− φ42pv︸ ︷︷ ︸loss due VN

+α41m︸ ︷︷ ︸production

plasmin :∂m∂t

= Dm∇2m︸ ︷︷ ︸diffusion

+ φ52pv︸ ︷︷ ︸production by PAI-1/VN

+φ53(c1 + c2)u︸ ︷︷ ︸production by cells

−φ54m︸ ︷︷ ︸decay

Continuous Modelling: Simulation Results

Sequence of profiles showing the growth and invasion of multiple sub-populations of cancer cells of extracellular matrix (bottom

plot) in 1 dimension. The less aggressive cancer cells (top plot) have stronger cell-cell adhesion and weaker cell-matrix adhesion

than the more aggressive cancer cells (middle plot), which have stronger cell-matrix adhesion and weaker cell-cell adhesion.

Same as the above, these figures show 2 dimensional numerical simulations. All were performed using Matlab R©.

Multi-scale Modelling: Ordinary Differential Equations

Cell attachment:d [E/β]

dt= ν[β][Em]− di(t)[E/β]− α[E/β]

d [β]

dt= −ν[β][Em] + di(t)[E/β]− k+[β][P] + k−[β/P] + km

Cell detachment:d [E/β]

dt= −(α + di(t))[E/β]

d [β]

dt= (α + di(t))[E/β]− k+[β][P] + k−[β/P] + km

Multi-scale Modelling: Simulation Results

This simulations show a sheet of cells that produces a wave of down-regulated E-Cadherin-β-Catenin complex concentration, after

which the cells detach from the main tumour mass. The down-regulation starts from the left bottom cell and propagates radially.

Detached cells are marked with black. The simulations were performed using CompuCell3D and SOSlib.

http://www.maths.dundee.ac.uk/ vivi@maths.dundee.ac.uk