hele shaw limit for a model of tumor growth with nutrients
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Hele-Shaw limit for a model of tumor growth with nutrients Noemi - PowerPoint PPT Presentation

Hele-Shaw limit for a model of tumor growth with nutrients Noemi David (LJLL, Inria) Supervisor: Beno t Perthame (LJLL, Sorbonne Universit e) Co-supervisor: Maria Carla Tesi (Universit` a di Bologna) 28/01/2020 Biological background


  1. Hele-Shaw limit for a model of tumor growth with nutrients Noemi David (LJLL, Inria) Supervisor: Benoˆ ıt Perthame (LJLL, Sorbonne Universit´ e) Co-supervisor: Maria Carla Tesi (Universit` a di Bologna) 28/01/2020

  2. Biological background Cancer is characterized by the unregulated growth and invasion of neoplastic cells

  3. Biological background Cancer is characterized by the unregulated growth and invasion of neoplastic cells Onset : genetic mutations − → uncontrolled division and loss of apoptosis,

  4. Biological background Cancer is characterized by the unregulated growth and invasion of neoplastic cells Onset: genetic mutations − → uncontrolled division and loss of apoptosis, Avascular phase : formation of a quasi-spherical mass with three regions, Figure: Tumor spheroid, from Chaplain and Sherratt, J. Math. Biol. (2001)

  5. Biological background Cancer is characterized by the unregulated growth and invasion of neoplastic cells Onset: genetic mutations − → uncontrolled division and loss of apoptosis, Avascular phase: formation of a quasi-spherical mass with three regions, Angiogenesis : secretion of TAFs − → new blood vessels formation, Figure: Development of tumor angiogenesis, credits: Centre de Recherche des Cordeliers

  6. Biological background Cancer is characterized by the unregulated growth and invasion of neoplastic cells Onset: genetic mutations − → uncontrolled division and loss of apoptosis, Avascular phase: formation of a quasi-spherical mass with three regions, Angiogenesis: secretion of TAFs − → new blood vessels formation, Vascular phase : new blood supply − → fast growth and invasion of the host, Figure: Vascularized tumor, from B. Perthame (2016)

  7. Biological background Cancer is characterized by the unregulated growth and invasion of neoplastic cells Onset: genetic mutations − → uncontrolled division and loss of apoptosis, Avascular phase: formation of a quasi-spherical mass with three regions, Angiogenesis: secretion of TAFs − → new blood vessels formation, Vascular phase: new blood supply − → faster growth and invasion of the host, Metastasis : spread of tumor cells in the vessels − → major clinical problem

  8. Biological background Cancer is characterized by the unregulated growth and invasion of neoplastic cells Onset: genetic mutations − → uncontrolled division and loss of apoptosis, Avascular phase : formation of a quasi-spherical mass with three regions, Angiogenesis: secretion of TAF − → new blood vessels formation, Vascular phase : new blood supply − → fast growth and invasion of the host, Metastasis: spread of tumor cells in the vessels − → major clinical problem

  9. Mechanical tumor growth models Compressible models : systems of PDEs, Incompressible models : free boundary problems

  10. Mechanical tumor growth models Compressible models : systems of PDEs, ◮ Cell proliferation is governed by space availability Incompressible models : free boundary problems

  11. Mechanical tumor growth models Compressible models : systems of PDEs, ◮ Cell proliferation is governed by space availability ◮ Contact inhibition prevents cell division (homeostatic pressure) Incompressible models : free boundary problems

  12. Mechanical tumor growth models Compressible models : systems of PDEs, ◮ Cell proliferation is governed by space availability ◮ Contact inhibition prevents cell division (homeostatic pressure) ◮ Cell motion is driven by the simplified Darcy’s law � v = −∇ p Incompressible models : free boundary problems

  13. Mechanical tumor growth models Compressible models : systems of PDEs, ◮ Cell proliferation is governed by space availability ◮ Contact inhibition prevents cell division (homeostatic pressure) ◮ Cell motion is driven by the simplified Darcy’s law � v = −∇ p Incompressible models : free boundary problems ◮ Describe the geometrical motion of the tumor

  14. Mechanical tumor growth models Compressible models : systems of PDEs, ◮ Cell proliferation is governed by space availability ◮ Contact inhibition prevents cell division (homeostatic pressure) ◮ Cell motion is driven by the simplified Darcy’s law � v = −∇ p Incompressible models : free boundary problems ◮ Describe the geometrical motion of the tumor ◮ The tumor contour is represented as a free boundary

  15. Mechanical tumor growth models Compressible models : systems of PDEs, ◮ Cell proliferation is governed by space availability ◮ Contact inhibition prevents cell division (homeostatic pressure) ◮ Cell motion is driven by the simplified Darcy’s law � v = −∇ p Incompressible models : free boundary problems ◮ Describe the geometrical motion of the tumor ◮ The tumor contour is represented as a free boundary Link : asymptotic analysis, incompressible limit

  16. Mechanical models: examples ∂ t n − div( n ∇ p ) = nF ( p ) n : cell population density, p : pressure, law of state p = P ( n ), F : proliferation rate,

  17. Mechanical models: examples  ∂ t n − div( n ∇ p ) = nF ( p , c )    ∂ t c − ∆ c = − nH ( p , c )  n : cell population density, p : pressure, law of state: p = P ( n ), F : proliferation rate, c : concentration of a generic nutrient (oxygen or glucose), H : consumption rate

  18. Mechanical models: examples  ∂ t n 1 − div( n 1 ∇ p ) = n 1 F 1 ( p , c ) + n 2 G 1 ( p , c )   ∂ t n 2 − div( n 2 ∇ p ) = n 1 F 2 ( p , c ) + n 2 G 2 ( p , c )  ∂ t c − ∆ c = − nH ( p , c )  n 1 , n 2 : cell population densities, p : pressure, law of state: p = P ( N ), with N = n 1 + n 2 , F 1 , G 2 : proliferation rates, F 2 , G 1 : cross-reaction terms, c : concentration of a generic nutrient (oxygen or glucose) H : consumption rate

  19. Incompressible limit Purely mechanical model with one species ∂ t n γ − div( n γ ∇ p γ ) = n γ F ( p γ ) Law of state p = n γ , with γ ≥ 1

  20. Incompressible limit Purely mechanical model with one species ∂ t n γ − div( n γ ∇ p γ ) = n γ F ( p γ ) Law of state p = n γ , with γ ≥ 1 Equation for the pressure ∂ t p γ = γ p γ (∆ p γ + F ( p γ )) + |∇ p γ | 2

  21. Incompressible limit Purely mechanical model with one species ∂ t n γ − div( n γ ∇ p γ ) = n γ F ( p γ ) Law of state p = n γ , with γ ≥ 1 Equation for the pressure ∂ t p γ = γ p γ (∆ p γ + F ( p γ )) + |∇ p γ | 2 Limit for γ → ∞ : p γ , n γ converge strongly to p ∞ , n ∞ that satisfy

  22. Incompressible limit Purely mechanical model with one species ∂ t n γ − div( n γ ∇ p γ ) = n γ F ( p γ ) Law of state p = n γ , with γ ≥ 1 Equation for the pressure ∂ t p γ = γ p γ (∆ p γ + F ( p γ )) + |∇ p γ | 2 Limit for γ → ∞ : p γ , n γ converge strongly to p ∞ , n ∞ that satisfy ◮ Limit problem � ∂ t n ∞ − div( n ∞ ∇ p ∞ ) = n ∞ F ( p ∞ ) , p ∞ (1 − n ∞ ) = 0 .

  23. Incompressible limit Purely mechanical model with one species ∂ t n γ − div( n γ ∇ p γ ) = n γ F ( p γ ) Law of state p = n γ , with γ ≥ 1 Equation for the pressure ∂ t p γ = γ p γ (∆ p γ + F ( p γ )) + |∇ p γ | 2 Limit for γ → ∞ : p γ , n γ converge strongly to p ∞ , n ∞ that satisfy ◮ Limit problem � ∂ t n ∞ − div( n ∞ ∇ p ∞ ) = n ∞ F ( p ∞ ) , p ∞ (1 − n ∞ ) = 0 . ◮ Complementarity relation �� � � −|∇ p ∞ | 2 ζ − p ∞ ∇ p ∞ ∇ ζ + p ∞ F ( p ∞ ) ζ = 0 . (in the sense of distribution) p ∞ (∆ p ∞ + F ( p ∞ )) = 0

  24. Incompressible limit Complementarity relation (in the sense of distribution) p ∞ (∆ p ∞ + F ( p ∞ )) = 0

  25. Incompressible limit Complementarity relation (in the sense of distribution) p ∞ (∆ p ∞ + F ( p ∞ )) = 0 Link between the compressible model and the Hele-Shaw problem: Ω( t ) := { x ; p ∞ ( x , t ) > 0 }  − ∆ p ∞ = F ( p ∞ ) in Ω( t ) ,   p ∞ = 0 on ∂ Ω( t ) ,  V = ∇ p ∞ · � on ∂ Ω( t ) , n 

  26. Incompressible limit Complementarity relation (in the sense of distribution) p ∞ (∆ p ∞ + F ( p ∞ )) = 0 Link between the compressible model and the Hele-Shaw problem: Ω( t ) := { x ; p ∞ ( x , t ) > 0 }  − ∆ p ∞ = F ( p ∞ ) in Ω( t ) ,   p ∞ = 0 on ∂ Ω( t ) ,  V = ∇ p ∞ · � on ∂ Ω( t ) , n  Ω( t ) = { x ; n ∞ ( x , t ) = 1 } is the region occupied by the tumor

  27. (Selected) State of the art B. Perthame, F. Quiros, J.L. Vazquez: The Hele-Shaw Asymptotics for Mechanical Models of Tumor Growth Arch. Rational Mech. Anal. 212 93-127 (2014)

  28. (Selected) State of the art B. Perthame, F. Quiros, J.L. Vazquez: The Hele-Shaw Asymptotics for Mechanical Models of Tumor Growth Arch. Rational Mech. Anal. 212 93-127 (2014) ◮ Mechanical model : Existence and uniqueness of the solution of the limit problem and complementarity relation

  29. (Selected) State of the art B. Perthame, F. Quiros, J.L. Vazquez: The Hele-Shaw Asymptotics for Mechanical Models of Tumor Growth Arch. Rational Mech. Anal. 212 93-127 (2014) ◮ Mechanical model : Existence and uniqueness of the solution of the limit problem and complementarity relation ◮ Model with nutrients : Existence and uniqueness of the limit solution, no complementarity relation

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