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: Benoˆ ıt Perthame (LJLL, Sorbonne Universit´ e) Co-supervisor: Maria Carla Tesi (Universit` a di Bologna) 28/01/2020

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

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

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)

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

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)

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

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

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

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

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

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

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

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

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

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,

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

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

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

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

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

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 .

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

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

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 

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

(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)

(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

(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