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 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,
of reality’’
Alan Turing
Mathematical Model at
Cell-Receptor Level
uPA  uPAR
 uPA/ uPAR
k1
uPA  PAI  1 
 uPA/ PAI  1
k2
• 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
a0 p
dr
  k1 (u0  r0  p0  r  p )r  k 1 (r0  r ) 
dt
a1 p  


4
 
1r 


1
recycling
production
downregula tion
due to PAI 1 presence
dp
  k 2 (u0  r0  p0  r  p ) p  k  2 (r0  r ) 
dt
 r p2

decay PAI 1 / uPAR


2
production
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
Cell migratory response to
local tissue environment cues
HAPTOTAXIS
with ECM
ECM + tenascinEC &
PDE Model: The cancer cells equation
cx, t    c   
u
p
v 

D


c


c


 c 
 c
  1 c 1  c   13 c v
c
c
t
x  x  x 
x
x
x 
• 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
vx, t 
   c u v   2 v (1  v)   23 c v   22 p v
   
t
degradation
logistic growth
cell- matrix
signalling
PAI -1/VN
• 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.
u x, t 
2
 Du  u   31c( 31cv )   31up
t
• 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    c   
u
v 
  Dc     c c   c c   1c1  c   13cv
t
x  x  x 
x
x 
vx, t 
 cuv   2 v1  v    23cv
t
u x, t 
 Du  2u   31c  31u
t
• c (x,t) : tumour cell density.
• v (x, t) : the extracellular matrix concentration.
• u (x, t) : the uPA concentration
The PAI-1 equation.
p x, t 
 D p  2 p   41uc ( 41c)   41up   42 vp
t
• 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    c    u
p
v 
  Dc     c c   c c   c   1c1  c   13cv
t
x  x  x 
x
x
x 
vx, t 
 cuv   2v1  v    23cv   22 pv
t
u x, t 
 Du  2u   31c( 31cv)   31up
t
px, t 
 D p  2 p   41uc( 41c)   41up   42 vp
t
•
•
•
•
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    c   
u
p
v 

  cc
  c   1c1  c   13cu
 Dc     c c
t
x  x  x 
x
x
x 
vx, t 
  m v( u m v)   2 v1  v    21up   22 vp
t
u x, t 
 Du  2u   31c   31up   33cu
t
px, t 
 D p  2 p   41m( 41c)   41up   42 vp
t
mx, t 
 Dm 2 m  51 pu   52 pv   53uc
t
•
•
•
•
•
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
Dynamic Tissue Invasion
Dynamic Tissue Invasion
Dynamic Tissue Invasion
Dynamic Tissue Invasion
Dynamic Tissue Invasion
Dynamic Tissue Invasion
Dynamic Tissue Invasion
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
Linear stability analysis
We consider small perturbations about the non-trivial
steady state:
~
~
u  u *   u (x, t ), v  v *   v(x, t ),...
~
~
denoting w (x, t )  ( u , v, ... ) and seeking solutions
of the form
w ( x, t )

t
c
e
 k Wk (x) where
k
2 W  k 2 W  0 , n  W  0
Linear stability analysis
DISPERSION RELATION

5

f (k ) 
2
Re 
4
 ... h(k )  0
2
 0
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
Conclusions and Future Work:
• 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 spatiotemporal 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