the role of acidity in tumours invasion antonio fasano dipartimento di matematica u. dini firenze...
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The role of acidity in tumours invasion
Antonio Fasano
Dipartimento di Matematica U. Dini
Firenze
[email protected] IASI, Roma 11.02.2009
Invasion is not just growth
We will not deal with models describingjust growth
Most recent survey paper:
N. Bellomo, N.K. Li, P.K. Maini, On the foundations of cancer modelling: selected topics, speculations, & perspectives, Math. Mod. Meth. Appl. S. 18, 593-646 (2008)
There are several mechanisms of tumour invasion
intra and extra-vasation metastasis
enzymatic lysis of the ECM + haptotaxis
aggression of the host tissue by increasing acidity
Increase of acidity is originated by the
switch to glycolytic metabolism
Invasive tumours exploit a Darwinianselection mechanism through mutations
The winning phenotypes may exhibit
less adhesion
increased mobility
anaerobic metabolism (favoured by hypoxic conditions)
higher proliferation rate
KREBS cycleMuch more efficient in producing ATPRequires high oxygen consumption
Aerobic metabolism
Glycolytic pathway
Anaerobic metabolism
Anaerobic vs. aerobic metabolism
( 2 ATP)
ATP = adenosine triphosphate. Associated to the “energy level”
acid
The role of ATP production rate in the onset of necrosis in multicellular spheroids has been investigated only very recently
Most recent paper:
A. Bertuzzi, A.F., A. Gandolfi, C. Sinisgalli
Necrotic core in EMT6/Ro tumour spheroids: is it caused by an ATP deficit ? (submitted, 2009)
The level of lactate determines (through acomplex mechanism) the local value of pH :
As early as 1930 it was observed that invasive tumours (may) switch to glycolyticmetabolism (Warburg)
Cells in the glycolitic regime may increase their glucose uptake, thus producing more lactate
lactate − + H +
The prevailing phenotype is acid resistant thanks to compensation mechanisms keeping the internal pH at normal levels
Apoptosis threshold for normal cells: pH=7.1 (Casciari et al., 1992)for tumour cells: pH=6.8 (Dairkee et al., 1995)
And the result is …
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.
invasion front + GAP
(from R.A.Gatenby-E.T.Gawlinski, 1996)
tumour host tissue
H+ ions
The model by Gatenby and Gawlinski consists in a set of equations admitting a solution chacterized by a propagating front with the possible occurrence of a gap
Remark: as it often happens for mathematical models of tumours, it must be taken with some reservation !
Host tissue
tumour
H+ ions
The G.G. model
logistic damage
reduced diffusivity
production decay
Defects: mass conservation? Damage on the tumour? Metabolism? Diffusion as main transport mechanism?…
One space dimension
large
small
s = time
Non-dimensional variables
carrying capacities ions diffusivity
prol. rate host tissue
decay/production
Basic non-dimensional parameters
damage rate
very small
The normalized G.G. model
One space dimension (space coord.x)
Normalized non-dimensional variables: all concentrations vary between 0 and 1
Normalized logistic
growth rate Acidic aggression
production - decay
Normalized diffusivity
Host tissue
tumour
H+ ions
d<<1, b>1
a>0
c>0
Search for a travelling waveSet
with
Z = x t
Solutions of this form, plotted vs. x, are graphs which, as time varies, travel with the speed | | to the right (>0: our case), or to the left (<0)
The system
becomes
)()(
)()(
zutxu
zutxu
x
t
etc.
Conditions at infinity corresponding to invasionNormal cells: max(0,1a) 1
Tumour cells: 1 0
H+ ions : 1 0
For a<1 a fraction of normal cells survives
A. Fasano, M.A. Herrero, M. Rocha Rodrigo:study of all possible travelling waves (2008), To appear on Math. Biosci.
REACTION - DIFFUSION
and travelling waves
M.A. Herrero : Reaction-diffusion systems: a mathematical biology approach.In Cancer modelling and simulation, L. Preziosi .ed , Chapman and Hall ( 2003), 367-420.
The prototype: Fisher, 1930Kolmogorov-Petrovsky-Piskunov, 1937(spread of an advantageous gene)
)1,0( ,0)(" ,0)(
0)0(' ,0)1()0( ,
)()0,( , ),( 2
uuFuF
aFFFCF
xHxuxuFkuu xxt
0 1x
F
0<u<1 for t>0 by the max principle
The solution is asymptotic to a TRAVELLING WAVEwith speed
slope=a
kac 20
1)( ,0)(
),( " '
)1()(
)()(),(
UU
UFkUcU
UaUUF
UctxUtxu
c
Existence requires: kacc 20
0
1
Describes an invasive process
Existence of travelling waves
Fisher’s equation
There are two equilibrium points: U=0, U=1The eigenvalues associated to
0)1( UaUUcUk
linearized near U=0 are the solutions of
02 ack and are both real and positive if kacc 20
so that the equilibrium is unstable
Remark: complex eigenvalues produce oscillations (= sign changes !)
Taking V=1U and linearizing near V=0
the equation for the eigenvalues becomes
02 ack
Now the eigenvalues are real but with opposite sign
(U=1 is a saddle point)
A wave has to take from the unstable to the stable equilibrium
In the phase plane p=U’, U
p
U0 1saddleunstable node
connecting heteroclinic
The connecting heteroclinic corresponds to the travelling wave
The solutions of the original p.d.e.’s system with suitable initial data converge to the slowest wave for large time
For their system Gatenby and Gawlinski computed just one wave with speed proportional to d
This is conjectured to describe the large time asymptotic behaviour of the solution of the initial-boundary value problem for the original p.d.e.’s system
The conjecture is based on the similarity to the famous two-population case
The proof is still missing
Good guess !
(due to higher dimensionality)
Two classes of waves: slow waves: = 0d (d<<1): singular perturbation !!!
fast waves: = O(1) as d 0
Technique: matching inner and outer solutions
Take = z/d as a fast variable: look at the front region with a magnifying lens
Slow waves:
Difficult but physical !
A mathematical curiosity ?
u can be found in terms of w
w can be found in terms of v
For all classes of waves
The equation
is of Bernoulli type
The equation
has the integral
Asymptotic convergence rate determined by wave speed
Summary of the results
slow waves: = 0d 0 < ½,
The parameter a decides whether the two cellular species overlap or are separated by a gap
2/1for )1,2/min(
),2/1,0(for 0
0
0
ab
No solutions for >½
similar to Fisher’s case
0 < a 1
1 < a 2overlapping zone
extends to
Thickness of overlapping zone
Normal cells
0
a > 2
gap
Thickness of gap
0
For any a > 0
F solution of the Fisher’s equation
)1,2/min(aD
tumour
H+ ions
limit case
This is the slowest possible wave
Numerical simulations
= ½ , minimal speed
bDd2
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
Survival of host tissue
1 < a 2
Overlapping zone
a > 2
gap
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)
Remarks on the parameters used by G.G.
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
(keeping the same scale)
Fast waves ( = O(1))
No restrictions on > 0
Let
Then the system
has solutions of the form
for a 1
Linear stability of fast waves
Other invasion models are based on a combined mechanism of ECM lysis and haptotaxis
(still based on the analysis of travelling waves)
[ICM Warsaw]
J.Math.Biol., to appear
HSP’s increase cells mobility
Analysis of multicellular spheroids in a host tissue
Acid-mediated invasion
Folkman-Hochberg (1973)
Viable rim
Necrotic core
gap
host tissue
host tissue
Acid is produced in the viable rim and possibly generatesa gap and/or a necrotic core
Evolution described as a quasi-steady process
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.
L. Bianchini, A. Fasano. A model combining acid-mediated tumour invasion and nutrient dynamics, to appear on Nonlinear Analysis: Real World Appl. (2008)
Vascular and avascular case, gap always vascular,no nutrient dynamics (H+ ions produced at constant rateby tumour cells)
Vascularization in the gap affected by acid, acid production controlled by the dynamics of glucose
Many possible cases / Qualitative differences
gap
viable rim
Host tissue sorrounding tumour
Two cases: avascular, vascular
n.c.
Glucose concentration
H+ ions concentration h
The viable rim is divided in a proliferating and in a quiescent region:
>P , <P
For the vascular case we use a factor (h)
reducing the vascular efficiency possibly to zero
]1,0[
P Q
All boundaries are unknown !
gap
r2r1 r3RH
Host tissue (vascular)Necrotic core
rP
gap
quiescent proliferating
The necrotic core, the gap and the quiescent region may or may not exist, depending on the spheroid size r2
The necrotic core may have different origin many different combinations are possible both in the vascular and in the avascular case
spheroid radius
0
r2r1 r3
RH
Host tissue (vascular)Necrotic core
rP
gapquiescent Prol.
= glucose concentration: diffusion-reaction
h = H+ ions concentration: product of metabolism
*
= P
and h flat
Concentrations and fluxes continuous at interfaces
Diffusion of glucose and of H+ ions is quasi-steady
r2r1 r3
RH
Host tissue (vascular)Necrotic core
rP
gapquiescent Prol.
*
= P
and h flat
STRATEGY:
1) suppose r2 is known and compute all other quantities
2) consider some (naive) model for the growth of the spheroid and deduce the value of r2 at equilibrium
By increasing r2 all combinations are encountered
consumption rate
production rate
removal rate by vasculature
The avascular case prol. threshold
death threshold
Non-dimensional variables
r2r1 r3
RH
Host tissue (vascular)Necrotic core
rP
gapquiescent Prol.
1
= P
and h flat
h 0
Conditions are found on r2 for the onset of interfaces
Evolution of the spheroid
A very simplified way of writing volume balance
Volume production in the proliferation rim
Volume removal fromnecrotic core
necrotic gap(same as in Smallbone et al. (2005), but with a degrading gap)
In our case this equation is very complicated.
One of the qualitative results is that the spheroid does not grow to
Vascular case: removal of h, supply of
(h)<1 reduction coefficient
f(r,r2) extra reduction coefficient
We follow the same procedure as for the avascular case …
Of course the problem is much more complicated.
Thank you !