Modelling acid-mediated tumour invasion

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Modelling acid-mediated tumour invasion

• Antonio Fasano
• Dipartimento di Matematica U. Dini, Firenze
• fasano_at_math.unifi.it

Levico, sept. 2008
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K. Smallbone, R.A.Gatenby, R.J.Gilles,
Ph.K.Maini,D.J.Gavaghan. Metabolic changes during
carcinogenesis Potential impact on invasiveness.
J. Theor. Biol, 244 (2007) 703-713.
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General underlying idea
Invasive tumours exploit a Darwinian selection
mechanism through mutations
The prevailing phenotype may be characterized by
a metabolism of glycolytic type resulting in an
increased acidity
Chemical aggression of the host tissue can also
be due to proteases reactions inducing lysis of
ECM
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Anaerobic vs. aerobic metabolism
ATP adenosine triphosphate. Associated to the
energy level
Anaerobic metabolism
Glycolytic pathway
acid
(? 2 ATP)
Aerobic metabolism
KREBS cycle Much more efficient in producing
ATP Requires high oxygen consumption
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The level of lactate determines (through
a complex mechanism) the local value of pH
As early as 1930 it was observed that invasive
tumours switch to glycolytic metabolism (Warburg)
The prevailing phenotype is acid resistant
Apoptosis threshold for normal cells pH7.1
(Casciari et al., 1992) For tumour cells ph6.8
(Dairkee et al., 1995)
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Conclusion
Glycolytic metabolism is very poor from the
energetic point of view, but it provides a
decisive advantage in the invasion process by
raising the acidity of the environment
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Aggressive phenotypes are characterized by low
oxygen consumption, high proliferation rate,
little or no adhesion, high haptotaxis
coefficient
As a result we may have morpholigical
instabilities, i.e. the formation of irregular
structures to which potential invasiveness is
associated
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Hybrid models
A.R.A. Anderson (2005), A hybrid mathematical
model of a solid tumour invasion The importance
of cell adhesion. Math. Med. Biol. 22
163-186. A.R.A. Anderson, A.M. Weaver, P.T.
Cummings, V. Quaranta (2006) , Tumour morphology
and phenotypic evolution driven by selective
pressure from microenvironment. Cell 127,
905-915 P. Gerlee, A.R.A. Anderson (2008) , A
hybrid cellular automaton of clonal evolution in
cancer the emergence of the glycolytic
phenotype, J.Theor.Biol. 250, 705-722
Hybrid means that the model is discrete for the
cells and continuous for other fields. Cells move
on a 2-D lattice according to some unbiased
motility (diffusion) haptotaxis driven by ECM
concentration gradient
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Exploiting inhomogeneities of the ECM
can reproduce irregular shapes of any kind
Anderson et al. 2005
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The role of ATP production in multicellular
spheroids
Venkatasubramanian et al., 2006 Smallbone et
al., 2007
ATP production in multicellular spheroids and
necrosis formation (2008) Bertuzzi-Fasano-Gandolfi
-Sinisgalli
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Acid-mediated invasion
pH lowering in tumours already mentioned by
Fast growing literature, starting from
R. A. Gatenby and E. T. Gawlinski (1996). A
reaction-diffusion model for cancer invasion.
Cancer Res. 56, pp. 57455753.
R. A. Gatenby and E. T. Gawlinski (2003). The
glycolytic phenotype in carcinogenesis and tumour
invasion insights through mathematical
modelling. Cancer Res. 63, pp. 38473854
Tool travelling waves
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G.G. acid-mediated invasion (non-dimensional
variables)
unormal cells conc.
vtumour cells conc.
wexcess H ions conc.
a sensitivity of host tissue to acid
environment b growth rate (with a logistic
term), normalized to the g.r. of normal cells c
H ions production (through lactate) / decay d
tumour cells diffusivity (through gap, i.e. u0)
dltlt1

• Diffusion of v (hindered by u) is the driving
  mechanism
• of invasion
• No diffusion of u (cells simply die)

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• The model has several limitations concerning the
  biological mechanisms involved
• no extracellular fluid
• instantaneous removal of dead cells
• metabolism is ignored

Therefore is goal is simply to show that there is
a mathematcal structure able to reproduce invasion
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Chemical action of the tumour (invasive processes
driven by pH lowering) R.A. Gatenby, E.T.
Gawlinski (1996)
? Travelling wave
gap
Red normal tissue Green tumour Blue H ion
A. Fasano, M.A. Herrero, M. Rocha Rodrigo study
of travelling waves (2008)
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Travelling waves system of o.d.e.s in the
variable z x ? ?t
Conditions at infinity corresponding to invasion
Normal cells max(0,1?a)? 1
Tumour cells 1 ? 0
H ions 1 ? 0
For alt1 a fraction of normal cells survive
G.G. computed just one suitably selected wave. We
want to analyze the whole class of admissible
waves
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• Two classes of waves
• slow waves ? ?0d? (dltlt1)
  singular perturbation
• fast waves ? O(1)

Slow waves
Technique matching inner and outer solutions
Take ? zd? as a fast variable
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For all classes of waves
u can be found in terms of w
w can be found in terms of v
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The equation
is of Bernoulli type
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Summary of the results
slow waves ? ?0d? 0 lt ? ? ½,
No solutions for ?gt½
Related to Fishers equation
The parameter a decides whether the two cellular
species overlap or are separated by a gap
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Normal cells
extends to ??
0 lt a ? 1
1 lt a ? 2
overlapping zone
Thickness of overlapping zone
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a gt 2
gap
Thickness of gap
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For any a gt 0
tumour
?F solution of the Fishers equation
H ions
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Numerical simulations
? ½ , minimal speed
The propagating front of the tumour is very
steep as a consequence of dltlt1
(this is the case treated by G.G.)
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0 lt a ? 1
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1 lt a ? 2
Overlapping zone
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a gt 2
gap
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Remarks on the parameters used by G.G.
Using the data of Gatenby-Gawlinski the resulting
gap is too large
Possible motivation make it visible in the
simulations
Reducing the parameter a from 12.5 (G.G.) to 3
produces the expected value (order of a few cell
diameters)
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a 3
b 10
b 1 (G.G.)
The value of b only affects the shape of the front
b ratio of growth rates, expected to begt1
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Fast waves (? O(1))
No restrictions on ? gt 0
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Linear stability of fast waves
Let
Then the system
has solutions of the form
for a ? 1
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Other invasion models are based on a combined
mechanism of ECM lysis and haptotaxis
(still based on the analysis of travelling waves)
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utumour cells conc.
haptotaxis
proteolysis
cECM conc.
penzyme conc.
Looking for travelling waves
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taking
and eliminating p, the system reduces to
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Travelling waves system
zx-at
The phase plane analysis is not trivial because
of the degeneracy in the first equation
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travelling waves analysis
t.w.
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ICM Warsaw
J.Math.Biol., to appear
to the basic model
tumour cells
diffusion
haptotaxis
ECM
enzyme
diffusion
they add
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the influence of heat shock proteins both on
cells motility and on enzyme activation
I(h)
h
Tumour more aggressive!
TW analysis
h(t) HSP concentration (prescribed)
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Acid is produced in the viable rim and possibly
generate a gap and/or a necrotic core
host tissue
Necrotic core
Viable rim
gap
host tissue
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K. Smallbone, D. J. Gavaghan, R. A. Gatenby, and
P. K. Maini. The role of acidity in solid tumour
growth and invasion. J. Theor. Biol. 235 (2005),
pp. 476484.
Vascular and avascular case, gap always
vascular, no nutrient dynamics (H ions produced
at constant rate by tumour cells)
L. Bianchini, A. Fasano. A model combining
acid-mediated tumour invasion and nutrient
dynamics, to appear on Nonlinear Analysis Real
World Appl. (2008)
Vascularization in the gap affected by acid, acid
production controlled by the dynamics of
glucose Many possible cases (with or without gap,
necrotic core, etc.) Qualitative differences
(e.g. excluding infinitely large
tumours) Theoretical results (existence and
uniqueness)