mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen...

Mathematical Modelling of Cancer Invasion of Tissue:

The Role of the Urokinase Plasminogen Activation System

Mark Chaplain and Georgios Lolas

Division of MathematicsUniversity of Dundee

SCOTLAND

The Individual Cancer Cell“A Nonlinear Dynamical System”

Multi-Cellular Spheroid

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

Malignant Epithelial Tumour

• Bladder Carcinoma• Typical features :

• Irregular structure• Highly invasive

• Potentially fatal

Metastasis:“A Multistep Process”

The Urokinase Plasminogen Activation System.

• uPA • uPAR • Plasmin• PAI-1• Vitronectin


• uPA released from the cells as a precursor (pro-uPA).

• uPAR is the cell surface receptor of uPA.

• Plasmin is a serine protease that can degrade most ECM proteins.


• PAI-1 is a uPA inhibitor. PAI-1 binds uPA/uPAR complex.

• uPA and PAI-1 are degraded and uPAR is recycled to the cell surface.

• Vitronectin is an ECM protein, involved in the adhesion of cells to the ECM. PAI-1 and uPAR compete for vitronectin binding.

The uPA system.

The uPA receptor (uPAR) is anchored to the surface of a variety of cells including tumor cells.uPA is secreted by normal and tumour cells and binds with highspecificity and affinity to uPAR. This binding activates uPA andfocuses proteolytic activity to the cell surface where plasminogenis converted to plasmin. Components of the ECM are degraded by plasmin, facilitating cellmigration and metastasis.Vitronectin interacts with uPAR leading to the activation of an intracellular signaling cascade.

The uPA system.

The uPA system.

““All models are an approximation, All models are an approximation, and ultimately a falsification, and ultimately a falsification, of reality’’of reality’’

Alan Turing

Mathematical Model at Cell-Receptor Level

• uPA binds to its receptor thus forming a stable complex, namely the uPA/uPAR complex.

• PAI-1 binds with high affinity to uPA.

1/1

/2

1

PAIuPAPAIuPA

uPARuPAuPARuPAk

k

ODE Mathematical Model

productionuPARPAIdecay

productionrecycling

presencePAItoduetiondownregula

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rpa

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)()(

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Steady States

• (i) a steady state where plasminogen activator inhibitor-1 (PAI-1) is in excess over uPA receptor p = 1.12, r = 0.39.

• (ii) a steady state where there is an ‘equality’ of uPAR and PAI-1concentrations: p = 0.62, r =0.72.

• (iii) a steady state where we observe an ‘excess’ of uPAR over PAI-1: r = 4.0, p = 0.1.

Stability of the Steady States

• (i) p = 1.12, r = 0.39, a stable spiral.• (ii) p = 0.62, r =0.72, a saddle point.• (iii) r = 4.0, p = 0.1, a stable node.

Cell Migration in Tissue:Chemotaxis

No ECM

with ECM

ECM + tenascinEC &

Cell migratory response to local tissue environment cues

HAPTOTAXIS

PDE Model: The cancer cells equation

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,

• We assume that they move by linear or nonlinear diffusion (random motility/kinesis).This approach permits us to investigate cell-matrix interactions in isolation.

• We assume that they also move in a haptotactic (VN) and chemotactic (uPA, PAI-1) way. Haptotaxis (chemotaxis) is the directed migratory response of cells to gradients of fixed or bound non diffusible (diffusible) chemicals.

• Proliferation: Logistic growth + cell – matrix signalling.

Vitronectin

• The extracellular matrix is known to contain many macromolecules, including fibronectin, laminin and vitronectin, which can be degraded by the uPA system.

• We assume that the uPA/uPAR complex degrades the extracellular matrix upon contact.

• Proliferation: logistic growth + cell-ECM signalling

• Loss due to PAI-1 binding.

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growth logistic

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The uPA equation.

• Active uPA is produced (or activated) either by the tumour cells or through the cell-matrix interactions.

• The production of active uPA by the tumour cells.• Decay of uPA due to PAI-1 binding.

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t

txuu 313131

2 )(,

• c (x,t) : tumour cell density.• v (x, t) : the extracellular matrix concentration.• u (x, t) : the uPA concentration

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The PAI-1 equation.

• Active PAI-1 is produced (or activated) either by the tumour cells or as a result of uPA/uPAR interaction.

• Decay of PAI-1 due to uPA and VN binding.

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t

txpp 42414141

2 )(,

• c (x,t) : tumour cell density.• v (x, t) : the extracellular matrix concentration.• u (x, t) : the uPA concentration.• p (x, t) : the PAI-1 concentration.

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Turing Type Taxis Instability

• Initially homogeneous steady state evolved into a spatially heterogeneous stable steady state.

• Linearly stable spatially homogeneous steady state at c = 1, v = 0, u = 0.375,

p = 0.8.• The spatially homogeneous steady

state is still linearly stable in Diffusion presence.

Taxis Instability

Since the addition of diffusion did not affect the stability of the aforementioned steady state, our only hope for destabilizing the steady state is the introduction of the chemotaxis term.

Modelling Plasmin Formation.

• c (x,t) : tumour cell density.• v (x, t) : the extracellular matrix concentration.• u (x, t) : the uPA concentration.• p (x, t) : the PAI-1 concentration.• m (x, t): the plasmin concentration.

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txm

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Dynamic Tissue Invasion

Linear stability analysis

Non-trivial steady-state: (c*, v*, u*, p*, m*) (1, 0.07, 0.198, 1.05, 0.29)

linearly stable

Semi-trivial steady state: (0,1,0,0,0)

linearly unstable

We consider small perturbations about the non-trivial steady state:

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where)(),(

form theof

solutions seeking and)...,,(),( denoting

...),,(*,),(*

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ect

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tvvvtuuu

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tk


0)(...)( 2425 khkf

0Re

DISPERSION RELATION


Linear stability analysis: Dispersion curveμ=0.2

Dynamic Tissue Invasion

Linear stability analysis: Dispersion curve: μ=10

Cancer cell density profile: μ=10

Linear stability analysis: Dispersion curve: μ=1

Linear stability analysis: Dispersion curve: μ=0.9

Linear stability analysis: Dispersion curve: μ=0.95

“Stationary” Pattern: μ=0.95

• Relatively simple models generate a wide range of tumour invasion and heterogeneity.

• In line with recent experimental results (Chun, 1997) – plasmin formation results in rich spatio-temporal dynamics and tumour heterogeneity.

• The impact of interactions between tumour cells and the ECM on possible metastasis.

• “taxis”, invasion and signalling are strongly correlated and rely on each other.

• “dynamic” pattern formation through excitation of multiple spatial modes

Conclusions and Future Work:

mathematical modelling of cancer invasion of tissue: the role of the urokinase plasminogen...

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