Mathematical Modelling of Cancer Invasion of Tissue: The Role of the Urokinase Plasminogen Activation System Mark Chaplain and Georgios Lolas Division of Mathematics University 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

The Urokinase Plasminogen Activation System. 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.

The Urokinase Plasminogen Activation System. 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 Urokinase Plasminogen Activation System.

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 high specificity and affinity to uPAR. This binding activates uPA and focuses proteolytic activity to the cell surface where plasminogen is converted to plasmin. Components of the ECM are degraded by plasmin, facilitating cell migration and metastasis. Vitronectin interacts with uPAR leading to the activation of an intracellular signaling cascade. The uPA system.

“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.

ODE Mathematical Model

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

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.

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

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.

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.

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.

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: Linear stability analysis

DISPERSION RELATION Linear stability analysis

Linear stability analysis: Dispersion curve μ=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: