The role of acidity in tumours invasion

Antonio Fasano

Dipartimento di Matematica U. Dini

Firenze

fasano@math.unifi.it 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 !