Modelling acid-mediated tumour invasion

1. Modelling acid-mediated tumour invasion Antonio Fasano Dipartimento di Matematica U. Dini, Firenze fasano@math.unifi.it Levico, sept. 2008

2. 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.

3. 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

4. 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

5. 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)

6. 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

7. Aggressive phenotypesare 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

8. 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

9. Exploiting inhomogeneities of the ECM can reproduce irregular shapes of any kind Anderson et al. 2005

10. 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

11. 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

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

13. 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

14. 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)

15. 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

16. 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

17. For all classes of waves u can be found in terms of w w can be found in terms of v

18. The equation is of Bernoulli type

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

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

21. a > 2 gap Thickness of gap

22. For any a > 0 tumour Fsolution of the Fisher’s equation H+ ions

23. 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.)

24. 0 < a  1

25. 1 < a  2 Overlapping zone

26. a > 2 gap

27. 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)

28. 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 be>1

29. Fast waves ( = O(1)) No restrictions on  > 0

30. Linear stability of fast waves Let Then the system has solutions of the form for a  1

31. Other invasion models are based on a combined mechanism of ECM lysis and haptotaxis (still based on the analysis of travelling waves)

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

33. taking and eliminating p, the system reduces to

34. Travelling waves system z=x−at The phase plane analysis is not trivial because of the degeneracy in the first equation

35. travelling waves analysis t.w.

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

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

38. 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

39. 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)