Mathematical models of acid

Modelling acid-mediated
tumour invasion
Antonio Fasano
Dipartimento di Matematica U. Dini, Firenze
[email protected]
Levico, sept. 2008
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.
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
Anaerobic vs. aerobic metabolism
ATP = adenosine triphosphate. Associated to the “energy level”
Anaerobic metabolism
( 2 ATP)
Aerobic metabolism
KREBS cycle
Much more efficient
in producing ATP
Requires high oxygen
consumption
Glycolytic pathway
acid
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: pH=7.1 (Casciari et al., 1992)
For tumour cells: ph=6.8 (Dairkee et al., 1995)
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
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
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
Exploiting inhomogeneities of the ECM can
reproduce irregular shapes of any kind
Anderson et al. 2005
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
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. 5745–5753.
R. A. Gatenby and E. T. Gawlinski (2003). The glycolytic
phenotype in carcinogenesis and tumour invasion: insights
through mathematical modelling. Cancer Res. 63, pp. 3847–
3854
Tool: travelling waves
G.G. acid-mediated invasion (non-dimensional variables)
u=normal cells conc.
v=tumour cells conc.
w=excess 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. u=0)
d<<1
Diffusion of v (hindered by u) is the driving mechanism
of invasion
No diffusion of u (cells simply die)
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
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)
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 a<1 a fraction of normal cells survive
G.G. computed just one suitably
selected wave.
We want to analyze the whole class of
admissible waves
Two classes of waves:
 slow waves:  = 0d
(d<<1): singular perturbation
 fast waves:  = O(1)
Slow waves
Technique: matching inner and outer solutions
Take  = zd as a fast variable
For all classes of waves
u can be found in terms of w
w can be found in terms of v
The equation
is of Bernoulli type
Summary of the results
slow waves:  = 0d
0 <   ½,
No solutions for >½
0  0 for   (0,1 / 2),
0  b min( a / 2,1) for   1 / 2
Related to Fisher’s equation
The parameter a decides whether the two cellular
species overlap or are separated by a gap
Normal cells
0<a1
extends to 
1<a2
overlapping zone
Thickness of overlapping zone
a>2
gap
Thickness of gap
For any a > 0
tumour
F solution of the Fisher’s equation
D  b min( a / 2,1)
H+ ions
Numerical simulations
 = ½ , minimal speed
  2 bDd
The propagating front of the tumour is very steep
as a consequence of d<<1
(this is the case treated by G.G.)
0<a1
1<a2
Overlapping zone
a>2
gap
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)
a=3
b = 1 (G.G.)
b = 10
The value of b only affects the shape of the front
b = ratio of growth rates, expected to be>1
Fast waves ( = O(1))
No restrictions on  > 0
Linear stability of fast waves
Let
Then the system
has solutions of the form
for a  1
Other invasion models are based on a combined
mechanism of ECM lysis and haptotaxis
(still based on the analysis of travelling waves)
u=tumour cells conc.
haptotaxis
proteolysis
c=ECM conc.
p=enzyme conc.
Looking for travelling waves …
taking
and eliminating p, the system reduces to
Travelling waves system
z=x−at
The phase plane analysis is not trivial
because of the degeneracy in the first equation
travelling waves analysis
t.w.
[ICM Warsaw]
J.Math.Biol., to appear
to the basic model
tumour cells
diffusion
haptotaxis
ECM
enzyme
diffusion
they add …
the influence of heat shock proteins both on
cells motility and on enzyme activation
  ( I ( h )u )
I(h)
h
Tumour more aggressive!
TW analysis
h(t) = HSP concentration (prescribed)
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
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. 476–484.
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)