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

∂tn − div(n∇p) = nF(p) n: cell population density, p: pressure, law of state p = P(n), F: proliferation rate,

Mechanical models: examples

     ∂tn − div(n∇p) = nF(p, c) ∂tc − ∆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

     ∂tn1 − div(n1∇p) = n1F1(p, c) + n2G1(p, c) ∂tn2 − div(n2∇p) = n1F2(p, c) + n2G2(p, c) ∂tc − ∆c = −nH(p, c) n1, n2: cell population densities, p: pressure, law of state: p = P(N), with N = n1 + n2, F1, G2: proliferation rates, F2, G1: cross-reaction terms, c: concentration of a generic nutrient (oxygen or glucose) H: consumption rate

Incompressible limit

Purely mechanical model with one species ∂tnγ − div(nγ∇pγ) = nγF(pγ) Law of state p = nγ, with γ ≥ 1

Incompressible limit

Purely mechanical model with one species ∂tnγ − div(nγ∇pγ) = nγF(pγ) Law of state p = nγ, with γ ≥ 1 Equation for the pressure ∂tpγ = γpγ(∆pγ + F(pγ)) + |∇pγ|2

Incompressible limit

Purely mechanical model with one species ∂tnγ − div(nγ∇pγ) = nγF(pγ) Law of state p = nγ, with γ ≥ 1 Equation for the pressure ∂tpγ = γpγ(∆pγ + F(pγ)) + |∇pγ|2 Limit for γ → ∞: pγ, nγ converge strongly to p∞, n∞ that satisfy

Incompressible limit

Purely mechanical model with one species ∂tnγ − div(nγ∇pγ) = nγF(pγ) Law of state p = nγ, with γ ≥ 1 Equation for the pressure ∂tpγ = γpγ(∆pγ + F(pγ)) + |∇pγ|2 Limit for γ → ∞: pγ, nγ converge strongly to p∞, n∞ that satisfy

◮ Limit problem

• ∂tn∞ − div(n∞∇p∞) = n∞F(p∞),

p∞(1 − n∞) = 0.

Incompressible limit

Purely mechanical model with one species ∂tnγ − div(nγ∇pγ) = nγF(pγ) Law of state p = nγ, with γ ≥ 1 Equation for the pressure ∂tpγ = γpγ(∆pγ + F(pγ)) + |∇pγ|2 Limit for γ → ∞: pγ, nγ converge strongly to p∞, n∞ that satisfy

◮ Limit problem

• ∂tn∞ − div(n∞∇p∞) = n∞F(p∞),

p∞(1 − n∞) = 0. ◮ Complementarity relation −|∇p∞|2ζ − 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

• n ∂Ω(t),

V = ∇p∞ · n

• n ∂Ω(t),

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

• n ∂Ω(t),

V = ∇p∞ · n

• n ∂Ω(t),

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

(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

• P. Gwiazda, B. Perthame, A. ´

Swierczewska-Gwiazda: A two-species hyperbolic–parabolic model of tissue growth. Communications in Partial Differential Equations. 44. 1-14. (2019)

(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

• P. Gwiazda, B. Perthame, A. ´

Swierczewska-Gwiazda: A two-species hyperbolic–parabolic model of tissue growth. Communications in Partial Differential Equations. 44. 1-14. (2019)

◮ Two species model: Existence of a weak solution to the model with γ fixed, no incompressible limit

(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

• P. Gwiazda, B. Perthame, A. ´

Swierczewska-Gwiazda: A two-species hyperbolic–parabolic model of tissue growth. Communications in Partial Differential Equations. 44. 1-14. (2019)

◮ Two species model: Existence of a weak solution to the model with γ fixed, no incompressible limit

• F. Bubba, B. Perthame, C. Pouchol, M. Schmidtchen: Hele-Shaw limit for a

system of two reaction- (cross-)diffusion equations for living tissues. Arch. Rational Mech. Anal. (2019)

(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

• P. Gwiazda, B. Perthame, A. ´

Swierczewska-Gwiazda: A two-species hyperbolic–parabolic model of tissue growth. Communications in Partial Differential Equations. 44. 1-14. (2019)

◮ Two species model: Existence of a weak solution to the model with γ fixed, no incompressible limit

• F. Bubba, B. Perthame, C. Pouchol, M. Schmidtchen: Hele-Shaw limit for a

system of two reaction- (cross-)diffusion equations for living tissues. Arch. Rational Mech. Anal. (2019)

◮ Two species model: Incompressible limit and complementarity relation in dimension one

Hele-Shaw limit for a model with nutrients

Model      ∂tn − div(n∇p) = nG(p, c), x ∈ Rd, t ≥ 0 ∂tc − ∆c = −nH(c) + (cB − c)K(p), c(x, t) → cB, for x → ∞ (1)

Hele-Shaw limit for a model with nutrients

Model      ∂tn − div(n∇p) = nG(p, c), x ∈ Rd, t ≥ 0 ∂tc − ∆c = −nH(c) + (cB − c)K(p), c(x, t) → cB, for x → ∞ (1) Difficulties

◮ The function G can change sign: at a fixed pressure, G(p, c) < 0 for c small

Hele-Shaw limit for a model with nutrients

Model      ∂tn − div(n∇p) = nG(p, c), x ∈ Rd, t ≥ 0 ∂tc − ∆c = −nH(c) + (cB − c)K(p), c(x, t) → cB, for x → ∞ (1) Difficulties

◮ The function G can change sign: at a fixed pressure, G(p, c) < 0 for c small ◮ Not possible to recover the standard Aronson-B´ enilan’s estimate (L∞)

Hele-Shaw limit for a model with nutrients

Model      ∂tn − div(n∇p) = nG(p, c), x ∈ Rd, t ≥ 0 ∂tc − ∆c = −nH(c) + (cB − c)K(p), c(x, t) → cB, for x → ∞ (1) Difficulties

◮ The function G can change sign: at a fixed pressure, G(p, c) < 0 for c small ◮ Not possible to recover the standard Aronson-B´ enilan’s estimate (L∞) ◮ Not possible to recover the bound from below for ∂tpγ

Hele-Shaw limit for a model with nutrients

Model      ∂tn − div(n∇p) = nG(p, c), x ∈ Rd, t ≥ 0 ∂tc − ∆c = −nH(c) + (cB − c)K(p), c(x, t) → cB, for x → ∞ (1) Difficulties

◮ The function G can change sign: at a fixed pressure, G(p, c) < 0 for c small ◮ Not possible to recover the standard Aronson-B´ enilan’s estimate (L∞) ◮ Not possible to recover the bound from below for ∂tpγ

Strategy

◮ Find an L3 version of the A.B. estimate for the functional w := ∆pγ + G

Hele-Shaw limit for a model with nutrients

Model      ∂tn − div(n∇p) = nG(p, c), x ∈ Rd, t ≥ 0 ∂tc − ∆c = −nH(c) + (cB − c)K(p), c(x, t) → cB, for x → ∞ (1) Difficulties

◮ The function G can change sign: at a fixed pressure, G(p, c) < 0 for c small ◮ Not possible to recover the standard Aronson-B´ enilan’s estimate (L∞) ◮ Not possible to recover the bound from below for ∂tpγ

Strategy

◮ Find an L3 version of the A.B. estimate for the functional w := ∆pγ + G ◮ Prove an L4 bound for the pressure gradient to gain compactness

Simulation in 1D

Density (black line), pressure (red line) and nutrient concentration (dashed blue line), (made by Xinran Ruan)

Simulation in 2D

Figure: Simulation of the tumor evolution and the development of a necrotic core, from Perthame,

Vauchelet, Tang, Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, (2014)

Preliminary estimates

Proposition (Direct estimates)

Given (nγ, pγ, cγ) a weak solution of the system (1) for γ > 1, and T > 0, there exists a constant C(T), independent of γ, such that for all 0 ≤ t ≤ T 0 ≤ nγ ≤ nH, 0 ≤ pγ ≤ pH, 0 ≤ cγ ≤ cB nγ(t)L1(Rd) ≤ C(T), pγ(t)L1(Rd) ≤ C(T), cγ(t) − cBL1(Rd) ≤ C(T), ∇cγ(t)L2(Rd) ≤ C(T), ∆cγL2(QT ) ≤ C(T),

• ∂cγ

∂t

• L2(QT )

≤ C(T)

• ∂nγ

∂t

• L1(QT )

≤ C(T)

• ∂pγ

∂t

• L1(QT )

≤ C(T), ∇cγL4(QT ) ≤ C(T), ∇pγ(t)L2(Rd) ≤ C(T).

Results

We define w := ∆p + G(p, c)

Theorem (Aronson-B´ enilan estimate in L3)

Given T > 0, let ΩT be a compact domain in Rd, independent of γ, such that supp(pγ(t)) ⊂ ΩT for all 0 ≤ t ≤ T. Then, we have T

• ΩT

|w|3

− ≤ C(T) and

• ΩT

|∆pγ(t)| ≤ C(T). where C depends on T and previous bounds and is independent of γ.

Results

Theorem (L4 estimate on the pressure gradient)

Given T > 0, then T

• Ω

pγ|∆pγ + G|2 + T

• Ω

pγ

• i,j

(∂2

i,jpγ)2 ≤ C(T),

and T

• Ω

|∇pγ|4 ≤ C(T), where C depends on T and previous bounds and is independent of γ.

Results

Theorem (Complementarity relation)

We set Q := Rd × (0, ∞). For all test functions ζ ∈ D(Q), the limit pressure p∞ satisfies

• Q
• −|∇p∞|2ζ − p∞∇p∞∇ζ + p∞G(p∞, c∞)ζ
• = 0.

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p),

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p), where

◮ ΦP and ΦN are the densities of proliferating and necrotic cells

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p), where

◮ ΦP and ΦN are the densities of proliferating and necrotic cells ◮ p = (ΦP + ΦN)γ is the pressure

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p), where

◮ ΦP and ΦN are the densities of proliferating and necrotic cells ◮ p = (ΦP + ΦN)γ is the pressure

Difficulties:

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p), where

◮ ΦP and ΦN are the densities of proliferating and necrotic cells ◮ p = (ΦP + ΦN)γ is the pressure

Difficulties: For the existence of a weak solution for the two species model a technical assumption was needed in the paper by Perthame, Gwiazda and Swierczewska-Gwiazda (2019): |G(0, c)|+ = 0 for all c

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p), where

◮ ΦP and ΦN are the densities of proliferating and necrotic cells ◮ p = (ΦP + ΦN)γ is the pressure

Difficulties: For the existence of a weak solution for the two species model a technical assumption was needed in the paper by Perthame, Gwiazda and Swierczewska-Gwiazda (2019): |G(0, c)|+ = 0 for all c Impossible!

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p), where

◮ ΦP and ΦN are the densities of proliferating and necrotic cells ◮ p = (ΦP + ΦN)γ is the pressure

Difficulties: For the existence of a weak solution for the two species model a technical assumption was needed in the paper by Perthame, Gwiazda and Swierczewska-Gwiazda (2019): |G(0, c)|+ = 0 for all c Impossible! Strategy:

Model with necrotic cells

Model      ∂tΦP − div(ΦP∇p) = ΦPG(p, c), ∂tΦN − div(ΦN∇p) = ΦP|G(p, c)|−, ∂tc − ∆c = −ΦPH(c) + (cB − c)K(p), where

◮ ΦP and ΦN are the densities of proliferating and necrotic cells ◮ p = (ΦP + ΦN)γ is the pressure

Difficulties: For the existence of a weak solution for the two species model a technical assumption was needed in the paper by Perthame, Gwiazda and Swierczewska-Gwiazda (2019): |G(0, c)|+ = 0 for all c Impossible! Strategy: Adapting the L4 method used in the case with nutrients

Thank you!