Mechanical Aspects of Tumour and Tissue Growth

Transcription

1 Mechanical Aspects of Tumour and Tissue Growth Luigi Preziosi Dip. Matematica Politecnico di Torino calvino.polito.it/~preziosi calvino.polito.it/~mcrtn

2 Plan of the Lectures 1st generation models: Mass balance models with suitable closures (Few words) 2nd generation models: Multiphase models with fluid-like constitutive equations Including mechanics in the growth term, 3rd generation models (?): Cell-ECM adhesion Including solid-like properties

3 Only Tumor Cells in 1D Single population with constant density Greenspan ( 50): ρ = ρ0 = const. X Spherical symmetry Chemical factors and nutrients diffuse 0

4 Only Tumor Cells in 3D 1. Constant density X 0 2. Potential flow n ΦT = volume ratio - H. Byrne, IMA J. Math. Appl. Med. Biol., 14, (1997) - H. Byrne & M. Chaplain, Eur. J. Appl. Math. 8, (1997) -...

5 Only Tumor Cells in 3D Macklin & Lowengrub JTB (2008) Compact Cell motility Fingering Fragmenting Radiotherapy Nutrient diffusion Original movies at biomathematics.shis.uth.tmc.edu/multimedia.php

6 Tumours are Multicomponent deformable and degradable ECM extracellular liquid (P. Friedl, K. Wolf) host cells and tumour cells

7 Tumours are Multicomponent Friedl, P. et al. Cancer Res 2008;68: Growing ensemble of glioma cells (T. Demuth, M. Berens) jcs.biologists.org/cgi/content/abstract/116/21/4409 Cell-cell interaction (neutrophil chasing a bacterium) (D. Rogers) Collective cell motion (P. Friedl, K. Wolf)

8 ferating proli q ui c escent ne cro ti

9 ferating proli q ui c escent ne cro ti Modelling Tumour Masses

10 More Cell Populations in 1D 1. Saturation 2. Radial Symmetry 3. Proportionality vj = αj v with given αj e.g., αj=0 means fixed some αj=1 means moving together all αj=1 means constrained mixtures - J. Ward & J. King, IMA J. Math. Appl. Med. Biol. 14, (1997) & 15, 1-42 (1998) J. Theor. Med. 1, (1999) - C. Breward, H. Byrne & C. Lewis, Eur. J. Appl. Math. 45, (2002)

11 Tumours as Multiphase Systems

12 Tumours as Multiphase Systems + saturation + diffusion of nutrients & chemical factors D. Ambrosi & L.P., Math. Models Methods Appl. Sci. 12, (2002) H. Byrne & L.P., Math. Med. Biol. 20, (2004)

13 Granular Flow in a Porous Medium (Rigid ECM) (P. Friedl) (degenerate parabolic)

14 Granular Flow in a Porous Medium (Rigid ECM) - E. De Angelis & L. Preziosi, Math. Models Methods Appl. Sci. 10, (2000)

15 Saturated Bi-phasic Models 0 if densities are equal

16 Satrurated Bi-phasic Models Darcy s law

17 Bi-phasic Models Tumour as a deformable porous medium - H. Byrne & L. Preziosi, Math. Medicine Biol. 20, (2003)

18 Bi-phasic Models Tumour as a deformable porous medium Poroelastic Model - Breward, Byrne, and Lewis, Bull. Math. Biol. 65: (2003). - Araujo and McElwain, SIAM J. Appl. Math. 65: (2005).

19 center outside M. Dorie et al, Exp. Cell Res (1982)

20 3 Phases

21 3 Phases Mechanical effects in: Stress Interaction force Growth

22 Contact inhibition of growth cadherin switch angiogenic switch

23 Contact inhibition of growth Epithelial cells growing to confluence Tzukatani et al. (1997)

24 E-cadherin P120 β-catenin α-catenin cytoplasm α-actinin actin Cell membrane

25 Growth arrest in the G1 phase

26 Proliferation

27 X Over - Proliferation proliferation

28 Growth Term Feedback loops in protein cascades Bi-stability All-or-none response activity restriction point threshold stimulus + Stochastic effects Growth term mollifier of step function

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30 Positive Feedback Loops X Y

31 Positive Feedback Loops X Z Y +z y y y x,y s u s x x x z

32 Mutually Inhibited Feedback Loops inactive active x inactive y active

33 Negative Feedback and Limit Cycles Goodwin (1965) S X YP RP Limit cycle

34 Hypothesis Cells replicate if they sense there is sufficient space If not, they enter a quiescent state ready to re-activate if, f.i., a neighboring cell dies Cells move preferentially toward regions with lower stress Cells constantly produce ECM and MDE

35 Misperception of stress hyperplasia tumour growth normal cells tumour cells overall volume ratio

36 Growth to confluence in vitro σ Γ φ Human breast epithelial cells

37 Generation of normal tissue Stationary values

38 total volume ratio tumour extracellular matrix normal tissue n a

39 Tissue Invasion x x tumour cells normal cells

40 Travelling Wave abnormal normal.1 0 = δ/γ.2 0 = γ δ/.4 0 = γ / δ Ψ

41 Contact inhibition of growth Monolayer Multilayer Growth to confluence with different clones J. Galle & D. Drasdo

42 Growth of Widr Clones Volume ratio Growth rates J. Galle, & L. P. Appl. Math. Letters Clones on the left are more motile è Faster stress relaxation

43 Growth of Widr Clones Drasdo & Byrne, JFMHohme, Phys. Biol. 2:133 (2005 See also Drasdo & Hohme, JMB 58: 657 (2009)

44 Contact inhibition of growth and fibrosis Host Tumour ECM MMP

45 HyperHypo- } content of ECM i.e. stiffer tissue ECM content in prostate cancer: 7% - 26%

46 HyperHypo- } content of ECM i.e. stiffer tissue hyper-production of ECM hypo-production of MDEs ECM/cells ~ 0.8 (e.g., fibrosis)

47 HyperHypo- } content of ECM hypo-production of ECM hyper-production of MDEs ECM/cells ~ 0.125

48 Osteosarcoma of the Lower Arm Rigid remodelling ECM Cells ECM

49 Tumor cords

50 Tumor cords

51 Tumor cords continuity of velocity stresses nutrient concentration nutrient flux

52 Tumor cords G. Mersi A. Tosin

53 Tumor cords

54 Axial growth of tumor cords Oxygen distribution Volume ratio Oxygen 3 capillaries G. Mersi S. Astanin & A. Tosin, Math. Model. Nat. Phenom., 2: (2007)

55 Glucose metabolism and glycolitic switch S. Astanin & L. P. J. Theor. Biol. 258, (2009). glucose Glycolysis 2 pyruvate + 2 ATP 2 lactic acid 2 Acetyl-CoA + 2 CO2 anaerobic condition aerobic condition citric acid cycle 4 CO2 + 4 H2O + 34 ATP

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60 Glucose metabolism and glycolitic switch S. Astanin Gatenby et al., Brit. J. Cancer 97: (2007)