Modelling acid-mediated tumour invasion Antonio Fasano Dipartimento di Matematica U. Dini, Firenze 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)

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

KREBS cycle Much more efficient in producing ATP Requires 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 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 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, 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, 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

Venkatasubramanian et al., 2006 Smallbone et al., 2007 ATP production in multicellular spheroids and necrosis formation (2008) Bertuzzi-Fasano-Gandolfi-Sinisgalli The role of ATP production in multicellular spheroids

Acid-mediated invasion 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 pH lowering in tumours already mentioned by

G.G. acid-mediated invasion (non-dimensional variables) 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 w=excess H+ ions conc. v=tumour cells conc. u=normal cells conc.  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) Red: normal tissue Green: tumour Blue: H+ ion A. Fasano, M.A. Herrero, M. Rocha Rodrigo: study of travelling waves (2008)  Travelling wave gap

Travelling waves system of o.d.e.’s in the variable z = x   t Normal cells: max(0,1  a)  1 Tumour cells: 1  0 H + ions : 1  0 Conditions at infinity corresponding to invasion 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:  =  0 d  (d<<1): singular perturbation  fast waves:  = O(1) Technique: matching inner and outer solutions Take  = zd  as a fast variable Slow waves

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

Summary of the results slow waves :  =  0 d  0 <   ½, The parameter a decides whether the two cellular species overlap or are separated by a gap No solutions for  >½ Related to Fisher’s equation

0 < a  1 1 < a  2 overlapping zone extends to  Thickness of overlapping zone Normal cells

a > 2 gap Thickness of gap

For any a > 0  F solution of the Fisher’s equation tumour H + ions

Numerical simulations  = ½, minimal speed 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

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

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)

haptotaxis proteolysis Looking for travelling waves … u=tumour cells conc. c=ECM conc. p=enzyme conc.

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.

tumour cells ECM enzyme diffusionhaptotaxis to the basic model J.Math.Biol., to appear they add … diffusion [ICM Warsaw]

h I(h) the influence of heat shock proteins both on cells motility and on enzyme activation h(t) = HSP concentration (prescribed) Tumour more aggressive! TW analysis

Viable rim Necrotic core gap host tissue Acid is produced in the viable rim and possibly generate a gap and/or a necrotic core

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 rate by tumour cells) 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)