1. The role of acidity in tumours invasion Antonio Fasano Dipartimento di Matematica U. Dini Firenze fasano@math.unifi.it IASI, Roma 11.02.2009

2. Invasion is not just growth We will not deal with models describing just 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)

3. 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

4. Increase of acidity is originated by the switch to glycolytic metabolism Invasive tumours exploit a Darwinian selection mechanism through mutations The winning phenotypes may exhibit less adhesion increased mobility anaerobic metabolism (favoured by hypoxic conditions) higher proliferation rate

5. 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

6. 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)

7. The level of lactate determines (through a complex mechanism) the local value of pH : As early as 1930 it was observed that invasive tumours (may) switch to glycolytic metabolism (Warburg) Cells in the glycolitic regime may increase their glucose uptake, thus producing more lactate lactate − + H +

8. 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 …

9. 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) tumourhost tissue H + ions

10. 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 !

11. Host tissue tumour H + ions The G.G. model logisticdamage reduced diffusivity production decay Defects: mass conservation? Damage on the tumour? Metabolism? Diffusion as main transport mechanism?… One space dimension large small s = time

12. Non-dimensional variables carrying capacities ions diffusivity prol. rate host tissue decay/production Basic non-dimensional parameters damage rate very small

13. 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 a>0 c>0

14. Search for a travelling wave Set 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)

15. The system becomes etc.

16. 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 survives A. Fasano, M.A. Herrero, M. Rocha Rodrigo: study of all possible travelling waves (2008), To appear on Math. Biosci.

17. 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.

18. The prototype: Fisher, 1930 Kolmogorov-Petrovsky-Piskunov, 1937 (spread of an advantageous gene) 0 1 x F 0 0 by the max principle The solution is asymptotic to a TRAVELLING WAVE with speed slope=a

19. c Existence requires: 0 1 Describes an invasive process Existence of travelling waves Fisher’s equation

20. There are two equilibrium points: U=0, U=1 The eigenvalues associated to linearized near U=0 are the solutions of and are both real and positive if so that the equilibrium is unstable Remark: complex eigenvalues produce oscillations (= sign changes !)

21. Taking V=1  U and linearizing near V=0 the equation for the eigenvalues becomes 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

22. In the phase plane p=U’, U p U0 1 saddleunstable 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

23. For their system Gatenby and Gawlinski computed just one wave with speed proportional to 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)

24. Two classes of waves:  slow waves:  =  0 d  (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 ?

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

26. The equation is of Bernoulli type The equation has the integral Asymptotic convergence rate determined by wave speed

27. 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 similar to Fisher’s case

28. 0 < a  1 1 < a  2 overlapping zone extends to  Thickness of overlapping zone Normal cells 00

29. a > 2 gap Thickness of gap 00

30. For any a > 0  F solution of the Fisher’s equation tumour H + ions limit case This is the slowest possible wave

31. 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.)

32. 0 < a  1 Survival of host tissue

33. 1 < a  2 Overlapping zone

34. a > 2 gap

35. 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.

36. 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)

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

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

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

40. [ICM Warsaw] J.Math.Biol., to appear HSP’s increase cells mobility

41. Analysis of multicellular spheroids in a host tissue Acid-mediated invasion Folkman-Hochberg (1973)

42. Viable rim Necrotic core gap host tissue Acid is produced in the viable rim and possibly generates a gap and/or a necrotic core Evolution described as a quasi-steady process

43. 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 / Qualitative differences

44. 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 PQ All boundaries are unknown ! gap

45. r2r2 r1r1 r3r3 R H  Host tissue (vascular)Necrotic core rPrP gap quiescentproliferating The necrotic core, the gap and the quiescent region may or may not exist, depending on the spheroid size r 2 The necrotic core may have different origin  many different combinations are possible both in the vascular and in the avascular case spheroid radius 0

46. r2r2 r1r1 r3r3 R H  Host tissue (vascular)Necrotic core rPrP gapquiescentProl.  = 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

47. r2r2 r1r1 r3r3 R H  Host tissue (vascular)Necrotic core rPrP gapquiescentProl.   *  *  =  P  and h flat STRATEGY: 1)suppose r 2 is known and compute all other quantities 2)consider some (naive) model for the growth of the spheroid and deduce the value of r 2 at equilibrium By increasing r 2 all combinations are encountered

48. consumption rate production rate removal rate by vasculature The avascular case prol. threshold death threshold

49. Non-dimensional variables

50. r2r2 r1r1 r3r3 R H  Host tissue (vascular)Necrotic core rPrP gapquiescentProl.   1  1  =  P  and h flat h  0 Conditions are found on r 2 for the onset of interfaces

51. Evolution of the spheroid A very simplified way of writing volume balance Volume production in the proliferation rim Volume removal from necrotic 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 

52. Vascular case: removal of h, supply of   (h)<1 reduction coefficient f(r,r 2 ) extra reduction coefficient We follow the same procedure as for the avascular case … Of course the problem is much more complicated.

53. Thank you !