Mechanical Aspects of Tumour and Tissue Growth Luigi Preziosi Dip. Matematica Politecnico di Torino calvino.polito.it/~preziosi calvino.polito.it/~mcrtn
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
Only Tumor Cells in 1D Single population with constant density Greenspan ( 50): ρ = ρ0 = const. X Spherical symmetry Chemical factors and nutrients diffuse 0
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, 305-323 (1997) - H. Byrne & M. Chaplain, Eur. J. Appl. Math. 8, 639-658 (1997) -...
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
Tumours are Multicomponent deformable and degradable ECM extracellular liquid (P. Friedl, K. Wolf) http://jcb.rupress.org/cgi/content/full/jcb.200209006/dc1 host cells and tumour cells
Tumours are Multicomponent Friedl, P. et al. Cancer Res 2008;68:7247-7249 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) www.biochemweb.org/neutrophil.shtml Collective cell motion (P. Friedl, K. Wolf)
ferating proli q ui c escent ne cro ti
ferating proli q ui c escent ne cro ti Modelling Tumour Masses
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, 36-69 (1997) & 15, 1-42 (1998) J. Theor. Med. 1, 171-211 (1999) - C. Breward, H. Byrne & C. Lewis, Eur. J. Appl. Math. 45, 125-152 (2002)
Tumours as Multiphase Systems
Tumours as Multiphase Systems + saturation + diffusion of nutrients & chemical factors D. Ambrosi & L.P., Math. Models Methods Appl. Sci. 12, 737-754 (2002) H. Byrne & L.P., Math. Med. Biol. 20, 341-366 (2004)
Granular Flow in a Porous Medium (Rigid ECM) (P. Friedl) (degenerate parabolic)
Granular Flow in a Porous Medium (Rigid ECM) - E. De Angelis & L. Preziosi, Math. Models Methods Appl. Sci. 10, 379-407 (2000)
Saturated Bi-phasic Models 0 if densities are equal
Satrurated Bi-phasic Models Darcy s law
Bi-phasic Models Tumour as a deformable porous medium - H. Byrne & L. Preziosi, Math. Medicine Biol. 20, 341-366 (2003)
Bi-phasic Models Tumour as a deformable porous medium Poroelastic Model - Breward, Byrne, and Lewis, Bull. Math. Biol. 65: 609-640 (2003). - Araujo and McElwain, SIAM J. Appl. Math. 65: 1261-1284 (2005).
center outside M. Dorie et al, Exp. Cell Res. 141 201-209 (1982)
3 Phases
3 Phases Mechanical effects in: Stress Interaction force Growth
Contact inhibition of growth cadherin switch angiogenic switch
Contact inhibition of growth Epithelial cells growing to confluence Tzukatani et al. (1997)
www.stanford.edu/group/smithlab/smithlab/other_research.html E-cadherin P120 β-catenin α-catenin cytoplasm α-actinin actin Cell membrane www.youtube.com/watch?v=pbuib5jchvo&feature=related
Growth arrest in the G1 phase
Proliferation
X Over - Proliferation proliferation
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
Positive Feedback Loops X Y
Positive Feedback Loops X Z Y +z y y y x,y s u s x x x z
Mutually Inhibited Feedback Loops inactive active x inactive y active
Negative Feedback and Limit Cycles Goodwin (1965) S X YP RP Limit cycle
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
Misperception of stress hyperplasia tumour growth normal cells tumour cells overall volume ratio
Growth to confluence in vitro σ Γ φ Human breast epithelial cells
Generation of normal tissue Stationary values
total volume ratio tumour extracellular matrix normal tissue n a
Tissue Invasion x x tumour cells normal cells
Travelling Wave abnormal normal.1 0 = δ/γ.2 0 = γ δ/.4 0 = γ / δ Ψ
Contact inhibition of growth Monolayer Multilayer Growth to confluence with different clones J. Galle & D. Drasdo
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
Growth of Widr Clones Drasdo & Byrne, JFMHohme, Phys. Biol. 2:133 (2005 See also Drasdo & Hohme, JMB 58: 657 (2009)
Contact inhibition of growth and fibrosis Host Tumour ECM MMP
HyperHypo- } content of ECM i.e. stiffer tissue ECM content in prostate cancer: 7% - 26%
HyperHypo- } content of ECM i.e. stiffer tissue hyper-production of ECM hypo-production of MDEs ECM/cells ~ 0.8 (e.g., fibrosis)
HyperHypo- } content of ECM hypo-production of ECM hyper-production of MDEs ECM/cells ~ 0.125
Osteosarcoma of the Lower Arm Rigid remodelling ECM Cells ECM
Tumor cords
Tumor cords
Tumor cords continuity of velocity stresses nutrient concentration nutrient flux
Tumor cords G. Mersi A. Tosin
Tumor cords
Axial growth of tumor cords Oxygen distribution Volume ratio Oxygen 3 capillaries G. Mersi S. Astanin & A. Tosin, Math. Model. Nat. Phenom., 2: 153-177 (2007)
Glucose metabolism and glycolitic switch S. Astanin & L. P. J. Theor. Biol. 258, 578-590 (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
Glucose metabolism and glycolitic switch S. Astanin Gatenby et al., Brit. J. Cancer 97: 646-653 (2007)