Diffuse interface models of tumor growth: optimal control and other issues

Transcription

1 Diffuse interface models of tumor growth: optimal control and other issues E. Rocca Università degli Studi di Pavia 4-6 décembre 2017 Laboratoire Jacques-Louis Lions First meeting of the French-German-Italian LIA COPDESC on Applied Analysis joint works with Harald Garcke (Regensburg)-Kei Fong Lam (Hong-Kong) and Sergio Frigeri (Brescia)-Kei Fong Lam (Hong-Kong)-Giulio Schimperna (Pavia) Fondazione Cariplo and Regione Lombardia Grant MEGAsTaR E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

2 Outline 1 Phase field models for tumor growth 2 The optimal control problem 3 First order optimality conditions 4 A multispecies model with velocity 5 Perspectives and Open problems E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

3 Outline 1 Phase field models for tumor growth 2 The optimal control problem 3 First order optimality conditions 4 A multispecies model with velocity 5 Perspectives and Open problems E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

4 Setting Tumors grown in vitro often exhibit layered structures: Figure: Zhang et al. Integr. Biol., 2012, 4, Scale bar 100µm = 0:1mm E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

5 Setting Tumors grown in vitro often exhibit layered structures: Figure: Zhang et al. Integr. Biol., 2012, 4, Scale bar 100µm = 0:1mm A continuum thermodynamically consistent model is introduced with the ansatz: sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species: a diffuse interface separates tumor and healthy cell regions proliferating tumor cells surrounded by (healthy) host cells, and a nutrient (e.g. glucose) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

6 Advantages of diffuse interfaces in tumor growth models Sharp interfaces = narrow transition layers in which tumor and healthy cells are mixed E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

7 Advantages of diffuse interfaces in tumor growth models Sharp interfaces = narrow transition layers in which tumor and healthy cells are mixed The main advantages of the diffuse interface formulation are: E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

8 Advantages of diffuse interfaces in tumor growth models Sharp interfaces = narrow transition layers in which tumor and healthy cells are mixed The main advantages of the diffuse interface formulation are: it eliminates the need to enforce complicated boundary conditions across the tumor/host tissue and other species/species interfaces; it eliminates the need to explicitly track the position of interfaces, as is required in the sharp interface framework; the mathematical description remains valid even when the tumor undergoes toplogical changes (e.g. metastasis) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

9 Optimization over the treatment time: H. Garcke, K.F. Lam, E. Rocca, Applied Mathematics & Optimization, 2017 E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

10 Optimization over the treatment time: H. Garcke, K.F. Lam, E. Rocca, Applied Mathematics & Optimization, 2017 Common treatment for tumors are Chemotheraphy Radiation therapy Surgery E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

11 Optimization over the treatment time: H. Garcke, K.F. Lam, E. Rocca, Applied Mathematics & Optimization, 2017 Common treatment for tumors are Chemotheraphy Radiation therapy Surgery For treatment involving drugs, the patient is given several doses of drugs over a few days, followed by a rest period of 3-4 weeks, and the cycle is repeated. Goal is to shrink the tumor into a more manageable size for which surgery can be applied. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

12 Optimization over the treatment time: H. Garcke, K.F. Lam, E. Rocca, Applied Mathematics & Optimization, 2017 Common treatment for tumors are Chemotheraphy Radiation therapy Surgery For treatment involving drugs, the patient is given several doses of drugs over a few days, followed by a rest period of 3-4 weeks, and the cycle is repeated. Goal is to shrink the tumor into a more manageable size for which surgery can be applied. Unfortunately, cytotoxic drugs also harms the healthy host tissues, and can accumulate in the body. Furthermore, drug clearance may also cause damage to various vital organs (e.g. kidneys and liver). E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

13 Optimization over the treatment time: H. Garcke, K.F. Lam, E. Rocca, Applied Mathematics & Optimization, 2017 Common treatment for tumors are Chemotheraphy Radiation therapy Surgery For treatment involving drugs, the patient is given several doses of drugs over a few days, followed by a rest period of 3-4 weeks, and the cycle is repeated. Goal is to shrink the tumor into a more manageable size for which surgery can be applied. Unfortunately, cytotoxic drugs also harms the healthy host tissues, and can accumulate in the body. Furthermore, drug clearance may also cause damage to various vital organs (e.g. kidneys and liver). Worst case scenario: Cytotoxins may have cancer-causing effects, and tumor cells can mutate to become resistant to the drug. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

14 Optimization over the treatment time: H. Garcke, K.F. Lam, E. Rocca, Applied Mathematics & Optimization, 2017 Common treatment for tumors are Chemotheraphy Radiation therapy Surgery For treatment involving drugs, the patient is given several doses of drugs over a few days, followed by a rest period of 3-4 weeks, and the cycle is repeated. Goal is to shrink the tumor into a more manageable size for which surgery can be applied. Unfortunately, cytotoxic drugs also harms the healthy host tissues, and can accumulate in the body. Furthermore, drug clearance may also cause damage to various vital organs (e.g. kidneys and liver). Worst case scenario: Cytotoxins may have cancer-causing effects, and tumor cells can mutate to become resistant to the drug. Thus, aside from optimising the drug distribution, we should also consider optimising the treatment time. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

15 Cahn Hilliard + nutrient models with source terms The simplest phase field model is a Cahn Hilliard system with source terms for ϕ: the difference in volume fractions (ϕ = 1: tumor phase, ϕ = 1: healthy tissue phase): tϕ = µ + M, µ = Ψ (ϕ) ϕ E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

16 Cahn Hilliard + nutrient models with source terms The simplest phase field model is a Cahn Hilliard system with source terms for ϕ: the difference in volume fractions (ϕ = 1: tumor phase, ϕ = 1: healthy tissue phase): tϕ = µ + M, µ = Ψ (ϕ) ϕ The source term M accounts for biological mechanisms related to proliferation and death. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

17 Cahn Hilliard + nutrient models with source terms The simplest phase field model is a Cahn Hilliard system with source terms for ϕ: the difference in volume fractions (ϕ = 1: tumor phase, ϕ = 1: healthy tissue phase): tϕ = µ + M, µ = Ψ (ϕ) ϕ The source term M accounts for biological mechanisms related to proliferation and death. Introduce a Reaction-diffusion equation for the nutrient proportion σ: tσ = σ S where S models interaction with the tumor cells E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

18 Cahn Hilliard + nutrient models with source terms The simplest phase field model is a Cahn Hilliard system with source terms for ϕ: the difference in volume fractions (ϕ = 1: tumor phase, ϕ = 1: healthy tissue phase): tϕ = µ + M, µ = Ψ (ϕ) ϕ The source term M accounts for biological mechanisms related to proliferation and death. Introduce a Reaction-diffusion equation for the nutrient proportion σ: tσ = σ S where S models interaction with the tumor cells In [Chen, Wise, Shenoy, Lowengrub (2014)], [Garcke, Lam, Sitka, Styles (2016)]: M = h(ϕ)(pσ A), S = h(ϕ)cσ Here h(s) is an interpolation function such that h( 1) = 0 and h(1) = 1, and h(ϕ)pσ - proliferation of tumor cells proportional to nutrient concentration h(ϕ)a - apoptosis of tumor cells h(ϕ)cσ - consumption of nutrient by the tumor cells A regular double-well potential Ψ, e.g., Ψ(s) = 1/4(1 s 2 ) 2 E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

19 Outline 1 Phase field models for tumor growth 2 The optimal control problem 3 First order optimality conditions 4 A multispecies model with velocity 5 Perspectives and Open problems E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

20 State equations We consider the Cahn Hilliard + nutrient model with linear kinetics and Neumann boundary conditions: tϕ = µ + h(ϕ)(pσ A αu) µ = Ψ (ϕ) ϕ tσ = σ Ch(ϕ)σ E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

21 State equations We consider the Cahn Hilliard + nutrient model with linear kinetics and Neumann boundary conditions: tϕ = µ + h(ϕ)(pσ A αu) µ = Ψ (ϕ) ϕ tσ = σ Ch(ϕ)σ Here h(s) is an interpolation function such that h( 1) = 0 and h(1) = 1, and h(ϕ)pσ - proliferation of tumor cells proportional to nutrient concentration h(ϕ)a - apoptosis of tumor cells h(ϕ)cσ - consumption of nutrient by the tumor cells E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

22 State equations We consider the Cahn Hilliard + nutrient model with linear kinetics and Neumann boundary conditions: tϕ = µ + h(ϕ)(pσ A αu) µ = Ψ (ϕ) ϕ tσ = σ Ch(ϕ)σ Here h(s) is an interpolation function such that h( 1) = 0 and h(1) = 1, and h(ϕ)pσ - proliferation of tumor cells proportional to nutrient concentration h(ϕ)a - apoptosis of tumor cells h(ϕ)cσ - consumption of nutrient by the tumor cells h(ϕ)αu - elimination of tumor cells by cytotoxic drugs at a constant rate α, u acts as a control here. In applications u : [0, T ] [0, 1] is spatially constant, where u = 1 represents full dosage, u = 0 represents no dosage E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

23 Objective functional For positive β T, β u and non-negative β Q, β, β S, we consider τ β Q J(ϕ, u, τ) := 0 2 ϕ ϕ Q 2 β + 2 ϕ(τ) ϕ 2 β τ S β u + (1 + ϕ(τ)) u 2 + β T τ 0 E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

24 Objective functional For positive β T, β u and non-negative β Q, β, β S, we consider τ β Q J(ϕ, u, τ) := 0 2 ϕ ϕ Q 2 β + 2 ϕ(τ) ϕ 2 β τ S β u + (1 + ϕ(τ)) u 2 + β T τ 0 the variable τ denotes the unknown treatment time to be optimised, ϕ Q is a desired evolution of the tumor over the treatment, ϕ is a desired final state of the tumor (stable equilibrium of the system), the term 1+ϕ(τ) 2 measures the size of the tumor at the end of the treatment, the constant β T penalizes long treatment times. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

25 Objective functional For positive β T, β u and non-negative β Q, β, β S, we consider τ β Q J(ϕ, u, τ) := 0 2 ϕ ϕ Q 2 β + 2 ϕ(τ) ϕ 2 β τ S β u + (1 + ϕ(τ)) u 2 + β T τ 0 the variable τ denotes the unknown treatment time to be optimised, ϕ Q is a desired evolution of the tumor over the treatment, ϕ is a desired final state of the tumor (stable equilibrium of the system), the term 1+ϕ(τ) 2 measures the size of the tumor at the end of the treatment, the constant β T penalizes long treatment times. Expectation: An optimal control will be a pair (u, τ ) and we will obtain two optimality conditions. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

26 Regarding the terms appearing in the cost functional J(ϕ, u, τ) := + τ 0 β Q 2 ϕ ϕ Q 2 + β τ S (1 + ϕ(τ)) β 2 ϕ(τ) ϕ 2 β u 2 u 2 + β T τ A large value of ϕ ϕ Q 2 would mean that the patient suffers from the growth of the tumor, and a large value of u 2 would mean that the patient suffers from high toxicity of the drug; E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

27 Regarding the terms appearing in the cost functional J(ϕ, u, τ) := + τ 0 β Q 2 ϕ ϕ Q 2 + β τ S (1 + ϕ(τ)) β 2 ϕ(τ) ϕ 2 β u 2 u 2 + β T τ A large value of ϕ ϕ Q 2 would mean that the patient suffers from the growth of the tumor, and a large value of u 2 would mean that the patient suffers from high toxicity of the drug; The function ϕ can be a stable configuration of the system, so that the tumor does not grow again once the treatment is completed or a configuration which is suitable for surgery; E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

28 Regarding the terms appearing in the cost functional J(ϕ, u, τ) := + τ 0 β Q 2 ϕ ϕ Q 2 + β τ S (1 + ϕ(τ)) β 2 ϕ(τ) ϕ 2 β u 2 u 2 + β T τ A large value of ϕ ϕ Q 2 would mean that the patient suffers from the growth of the tumor, and a large value of u 2 would mean that the patient suffers from high toxicity of the drug; The function ϕ can be a stable configuration of the system, so that the tumor does not grow again once the treatment is completed or a configuration which is suitable for surgery; The variable τ can be regarded as the treatment time of one cycle, i.e., the amount of time the drug is applied to the patient before the period of rest, or the treatment time before surgery; E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

29 Regarding the terms appearing in the cost functional J(ϕ, u, τ) := + τ 0 β Q 2 ϕ ϕ Q 2 + β τ S (1 + ϕ(τ)) β 2 ϕ(τ) ϕ 2 β u 2 u 2 + β T τ A large value of ϕ ϕ Q 2 would mean that the patient suffers from the growth of the tumor, and a large value of u 2 would mean that the patient suffers from high toxicity of the drug; The function ϕ can be a stable configuration of the system, so that the tumor does not grow again once the treatment is completed or a configuration which is suitable for surgery; The variable τ can be regarded as the treatment time of one cycle, i.e., the amount of time the drug is applied to the patient before the period of rest, or the treatment time before surgery; It is possible to replace β T τ by a more general function f (τ) where f : R + R + is continuously differentiable and increasing. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

30 Relaxed objective functional However, we will not study the functional τ β Q J(ϕ, u, τ) := 0 2 ϕ ϕ Q 2 + β τ S + (1 + ϕ(τ)) β 2 ϕ(τ) ϕ 2 β u 2 u 2 + β T τ but a relaxed version - in order to keep a control u just bounded without requiring more regularity E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

31 Relaxed objective functional However, we will not study the functional τ β Q J(ϕ, u, τ) := 0 2 ϕ ϕ Q 2 + β τ S + (1 + ϕ(τ)) β 2 ϕ(τ) ϕ 2 β u 2 u 2 + β T τ but a relaxed version - in order to keep a control u just bounded without requiring more regularity Let r > 0 be fixed and let T (0, ) denote a fixed maximal time in which the patient is allowed to undergo a treatment, we define τ β Q J r (ϕ, u, τ) := 0 2 ϕ ϕ Q τ r + 1 τ β T S (1 + ϕ) + r τ r 2 0 τ r β 2 ϕ ϕ 2 β u 2 u 2 + β T τ E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

32 Relaxed objective functional Let r > 0 be fixed and let T (0, ) denote a maximal time, we define τ β Q J r (ϕ, u, τ) := 0 2 ϕ ϕ Q τ β r τ r 2 ϕ ϕ τ β T S β u (1 + ϕ) + r τ r u 2 + β T τ. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

33 Relaxed objective functional Let r > 0 be fixed and let T (0, ) denote a maximal time, we define τ β Q J r (ϕ, u, τ) := 0 2 ϕ ϕ Q τ β r τ r 2 ϕ ϕ τ β T S β u (1 + ϕ) + r τ r u 2 + β T τ. The optimal control problem is min Jr (ϕ, u, τ) (ϕ,u,τ) subject to τ (0, T ), u U ad = {f L ( (0, T )) : 0 f 1}, and tϕ = µ + h(ϕ)(pσ A αu) in (0, T ) = Q, µ = Ψ (ϕ) ϕ in Q, tσ = σ Ch(ϕ)σ in Q, 0 = nϕ = nσ = nµ on (0, T ), ϕ(0) = ϕ 0, σ(0) = σ 0 in. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

34 Well-posedness of state equations Theorem Let ϕ 0 H 3, σ 0 H 1 with 0 σ 0 1, h C 0,1 (R) L (R) non-negative, and Ψ is a quartic potential, then for every u U ad there exists a unique triplet ϕ L (0, T ; H 2 ) L 2 (0, T ; H 3 ) H 1 (0, T ; L 2 ) C 0 (Q), µ L 2 (0, T ; H 2 ) L (0, T ; L 2 ), σ L (0, T ; H 1 ) L 2 (0, T ; H 2 ) H 1 (0, T ; L 2 ), 0 σ 1 a.e. in Q satisfying the state equations. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

35 Well-posedness of state equations Theorem Let ϕ 0 H 3, σ 0 H 1 with 0 σ 0 1, h C 0,1 (R) L (R) non-negative, and Ψ is a quartic potential, then for every u U ad there exists a unique triplet ϕ L (0, T ; H 2 ) L 2 (0, T ; H 3 ) H 1 (0, T ; L 2 ) C 0 (Q), µ L 2 (0, T ; H 2 ) L (0, T ; L 2 ), σ L (0, T ; H 1 ) L 2 (0, T ; H 2 ) H 1 (0, T ; L 2 ), 0 σ 1 a.e. in Q satisfying the state equations. Key points: Boundedness of σ comes from a weak comparison principle applied to tσ = σ Ch(ϕ)σ and it is an essential ingredient for the existence proof Proof utilises a Schauder fixed point argument E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

36 Existence of a minimiser Using that ϕ L 1 (0, T ; L 1 ), J r is bounded from below: τ β Q J r (ϕ, u, τ) := 0 2 ϕ ϕ Q τ r τ r + 1 τ β T S (1 + ϕ) + r τ r 2 0 β τ S 2r τ r β 2 ϕ ϕ 2 β u 2 u 2 + β T τ ϕ β S 2r ϕ L 1 (0,T ;L 1 ) C. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

37 Existence of a minimiser Using that ϕ L 1 (0, T ; L 1 ), J r is bounded from below: J r (ϕ, u, τ) β τ S ϕ β S 2r 2r ϕ L 1 (0,T ;L 1 ) C. τ r Minimising sequence (u n, τ n) U ad (0, T ), with corresponding state variables (ϕ n, µ n, σ n) such that lim Jr (ϕn, un, τn) = inf J r (φ, w, s). n (φ,w,s) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

38 Existence of a minimiser Using that ϕ L 1 (0, T ; L 1 ), J r is bounded from below: J r (ϕ, u, τ) β τ S ϕ β S 2r 2r ϕ L 1 (0,T ;L 1 ) C. τ r Minimising sequence (u n, τ n) U ad (0, T ), with corresponding state variables (ϕ n, µ n, σ n) such that lim Jr (ϕn, un, τn) = inf Jr (φ, w, s). n (φ,w,s) We extract a convergent subsequence u n u L (Q) and limit functions (ϕ, µ, σ ) satisfying the state equations and ϕ n ϕ in C 0 ([0, T ]; L 2 ) L 2 (Q). Key point: All of the convergence are with respect to the interval [0, T ]. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

39 Existence of a minimiser Using that ϕ L 1 (0, T ; L 1 ), J r is bounded from below: J r (ϕ, u, τ) β τ S ϕ β S 2r 2r ϕ L 1 (0,T ;L 1 ) C. τ r Minimising sequence (u n, τ n) U ad (0, T ), with corresponding state variables (ϕ n, µ n, σ n) such that lim Jr (ϕn, un, τn) = inf J r (φ, w, s). n (φ,w,s) We extract a convergent subsequence u n u L (Q) and limit functions (ϕ, µ, σ ) satisfying the state equations and ϕ n ϕ in C 0 ([0, T ]; L 2 ) L 2 (Q). As {τ n} n N is a bounded sequence, we extract a convergent subsequence τ n τ [0, T ]. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

40 Existence of minimiser To pass to the limit in: J r (ϕ n, u n, τ n) := + 1 r τn 0 τn τ n r β Q 2 ϕn ϕ Q r β T S (1 + ϕn) τn τ n r β 2 ϕn ϕ 2 β u 2 un 2 + β T τ n, E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

41 Existence of minimiser To pass to the limit in: we make use of J r (ϕ n, u n, τ n) := + 1 r τn 0 τn τ n r β Q 2 ϕn ϕ Q r β T S (1 + ϕn) τn τ n r β 2 ϕn ϕ 2 β u 2 un 2 + β T τ n, χ [0,τn](t) χ [0,τ ](t), ϕ n ϕ Q ϕ ϕ Q strongly in L 2 (Q) to obtain τn lim n 0 τ ϕ n ϕ Q 2 = lim ϕ n ϕ Q 2 χ n [0,τn](t) = ϕ ϕ Q 2. Q 0 E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

42 Existence of minimiser To pass to the limit in: we make use of J r (ϕ n, u n, τ n) := + 1 r τn 0 τn τ n r β Q 2 ϕn ϕ Q r β T S (1 + ϕn) τn τ n r β 2 ϕn ϕ 2 β u 2 un 2 + β T τ n, χ [0,τn](t) χ [0,τ ](t), ϕ n ϕ Q ϕ ϕ Q strongly in L 2 (Q) to obtain τn lim n 0 τ ϕ n ϕ Q 2 = lim ϕ n ϕ Q 2 χ n [0,τn](t) = ϕ ϕ Q 2. Q 0 Weak lower semi-continuity of the L 2 (Q) norm then yields inf (φ,w,s) That is, (u, τ ) is a minimiser. Jr (φ, w, s) lim inf Jr (ϕn, un, τn) Jr (ϕ, u, τ ). n E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

43 Outline 1 Phase field models for tumor growth 2 The optimal control problem 3 First order optimality conditions 4 A multispecies model with velocity 5 Perspectives and Open problems E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

44 Fréchet differentiability with respect to the control We set S(u) = (ϕ, µ, σ) as the solution operator on the interval [0, T ], and introduce the linearized state variables (Φ w, Ξ w, Σ w ) corresponding to w as solutions to tφ = Ξ + h(ϕ)(pσ αw) + h (ϕ)φ(pσ A αu), Ξ = Ψ (ϕ)φ Φ, tσ = Σ C(h(ϕ)Σ + h (ϕ)φσ), with Neumann boundary conditions and zero initial conditions. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

45 Fréchet differentiability with respect to the control We set S(u) = (ϕ, µ, σ) as the solution operator on the interval [0, T ], and introduce the linearized state variables (Φ w, Ξ w, Σ w ) corresponding to w as solutions to tφ = Ξ + h(ϕ)(pσ αw) + h (ϕ)φ(pσ A αu), Ξ = Ψ (ϕ)φ Φ, tσ = Σ C(h(ϕ)Σ + h (ϕ)φσ), with Neumann boundary conditions and zero initial conditions. Theorem For any w L 2 (Q) there exists a unique triplet (Φ, Ξ, Σ) with Φ L (0, T ; H 1 ) L 2 (0, T ; H 3 ) H 1 (0, T ; (H 1 ) ) =: X 1, Ξ L 2 (0, T ; H 1 ) =: X 2, Σ L (0, T ; H 1 ) H 1 (0, T ; L 2 ) L 2 (0, T ; H 2 ) =: X 3, and Φ X1 + Ξ X2 + Σ X3 C w L 2 (Q) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

46 Fréchet differentiability with respect to the control We set S(u) = (ϕ, µ, σ) as the solution operator on the interval [0, T ], and introduce the linearized state variables (Φ w, Ξ w, Σ w ) corresponding to w as solutions to tφ = Ξ + h(ϕ)(pσ αw) + h (ϕ)φ(pσ A αu), Ξ = Ψ (ϕ)φ Φ, tσ = Σ C(h(ϕ)Σ + h (ϕ)φσ), with Neumann boundary conditions and zero initial conditions. Expectation: The Fréchet derivative of S at u U ad in the direction w is D us(u)w = (Φ w, Ξ w, Σ w ). E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

47 Fréchet differentiability with respect to the control We set S(u) = (ϕ, µ, σ) as the solution operator on the interval [0, T ], and introduce the linearized state variables (Φ w, Ξ w, Σ w ) corresponding to w as solutions to tφ = Ξ + h(ϕ)(pσ αw) + h (ϕ)φ(pσ A αu), Ξ = Ψ (ϕ)φ Φ, tσ = Σ C(h(ϕ)Σ + h (ϕ)φσ), with Neumann boundary conditions and zero initial conditions. Expectation: The Fréchet derivative of S at u U ad in the direction w is D us(u)w = (Φ w, Ξ w, Σ w ). Theorem Let U L 2 (Q) be open such that U ad U. Then S : U L 2 (Q) Y is Fréchet differentiable, where [ ] Y = L 2 (0, T ; H 2 ) L (0, T ; L 2 ) H 1 (0, T ; (H 2 ) ) C 0 ([0, T ]; L 2 ) [ ] L 2 (Q) L (0, T ; H 1 ) H 1 (0, T ; L 2 ) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

48 Fréchet differentiability with respect to the control We set S(u) = (ϕ, µ, σ) as the solution operator on the interval [0, T ], and introduce the linearized state variables (Φ w, Ξ w, Σ w ) corresponding to w as solutions to tφ = Ξ + h(ϕ)(pσ αw) + h (ϕ)φ(pσ A αu), Ξ = Ψ (ϕ)φ Φ, tσ = Σ C(h(ϕ)Σ + h (ϕ)φσ), with Neumann boundary conditions and zero initial conditions. Expectation: The Fréchet derivative of S at u U ad in the direction w is D us(u)w = (Φ w, Ξ w, Σ w ). Consequence: For the reduced functional J r (u, τ) := J r (ϕ, u, τ), τ D uj r (u, τ)[w] = β Q (ϕ ϕ Q )Φ w + β uu w + 1 2r 0 τ τ r (β (ϕ ϕ )Φ w + β S Φ w ). Q E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

49 Fréchet differentiability with respect to time For we have J r (ϕ, u, τ) = + 1 r τ 0 τ τ r β Q 2 ϕ ϕ Q r β S (1 + ϕ) + 2 τ τ r T 0 β 2 ϕ ϕ 2 β u 2 u 2 + β T τ, D τ J r (u, τ ) = β T + β Q 2 ϕ(τ ) ϕ Q(τ ) 2 L 2 + β ) ( (ϕ ϕ )(τ ) 2L 2r 2 (ϕ ϕ )(τ r) 2L 2 β S + (ϕ(τ ) ϕ(τ r)). 2r E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

50 Fréchet differentiability with respect to time For we have J r (ϕ, u, τ) = + 1 r τ 0 τ τ r β Q 2 ϕ ϕ Q r β S (1 + ϕ) + 2 τ τ r T 0 β 2 ϕ ϕ 2 β u 2 u 2 + β T τ, D τ J r (u, τ ) = β T + β Q 2 ϕ(τ ) ϕ Q(τ ) 2 L 2 + β ) ( (ϕ ϕ )(τ ) 2L 2r 2 (ϕ ϕ )(τ r) 2L 2 β S + (ϕ(τ ) ϕ(τ r)). 2r Note that the control u does not appear explicitly. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

51 First order optimality conditions Introducing the adjoint system tp = q + Ψ (ϕ )q Ch (ϕ )σ r + h (ϕ )(Pσ A αu )p + β Q (ϕ ϕ Q ) + 1 2r χ (τ r,τ )(t)(2β (ϕ ϕ ) + β S ), q = p, tr = r Ch(ϕ )r + Ph(ϕ )p with Neumann boundary conditions and final time condition r(τ ) = p(τ ) = 0. We have E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

52 First order optimality conditions Introducing the adjoint system tp = q + Ψ (ϕ )q Ch (ϕ )σ r + h (ϕ )(Pσ A αu )p + β Q (ϕ ϕ Q ) + 1 2r χ (τ r,τ )(t)(2β (ϕ ϕ ) + β S ), q = p, tr = r Ch(ϕ )r + Ph(ϕ )p with Neumann boundary conditions and final time condition r(τ ) = p(τ ) = 0. We have Theorem There exists a unique (p, q, r) to the adjoint system such that p L 2 (0, τ ; H 2 ) H 1 (0, τ ; (H 2 ) ) L (0, τ ; L 2 ) C 0 ([0, τ ]; L 2 ), q L 2 (0, τ ; L 2 ), r L 2 (0, τ ; H 2 ) L (0, τ ; H 1 ) H 1 (0, τ ; L 2 ) C 0 ([0, τ ]; L 2 ). E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

53 First order optimality conditions Introducing the adjoint system tp = q + Ψ (ϕ )q Ch (ϕ )σ r + h (ϕ )(Pσ A αu )p + β Q (ϕ ϕ Q ) + 1 2r χ (τ r,τ )(t)(2β (ϕ ϕ ) + β S ), q = p, tr = r Ch(ϕ )r + Ph(ϕ )p with Neumann boundary conditions and final time condition r(τ ) = p(τ ) = 0. We have Theorem The optimal control (u, τ ) satisfy T τ β uu (v u ) h(ϕ )αp(v u ) 0 v U ad, 0 0 and β T + β Q 2 (ϕ ϕ Q)(τ ) 2 L 2 + β S ϕ (τ ) ϕ(τ r) dx 2r + β ) ( (ϕ ϕ )(τ ) 2L 2r 2 (ϕ ϕ )(τ r) 2L 2 = 0. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

54 Issues with the original functional To deal with the original functional: τ β Q J(ϕ, u, τ) = 2 ϕ ϕ Q β τ 2 ϕ(τ) ϕ 2 β u u 2 + β T τ. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

55 Issues with the original functional To deal with the original functional: τ β Q J(ϕ, u, τ) = 2 ϕ ϕ Q β τ 2 ϕ(τ) ϕ 2 β u u 2 + β T τ. Then, the optimality condition for τ is β Q 0 = D τ J (u,τ ) = 2 (ϕ ϕ Q)(τ ) 2 + β 2 (ϕ (τ ) ϕ ) tϕ (τ ) + βu 2 u (τ ) 2 dx + β T. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

56 Issues with the original functional To deal with the original functional: τ β Q J(ϕ, u, τ) = 2 ϕ ϕ Q β τ 2 ϕ(τ) ϕ 2 β u u 2 + β T τ. Then, the optimality condition for τ is β Q 0 = D τ J (u,τ ) = 2 (ϕ ϕ Q)(τ ) 2 + β 2 (ϕ (τ ) ϕ ) tϕ (τ ) + βu 2 u (τ ) 2 dx + β T. Issues: For the above expression to be well-defined, we need ttϕ L 2 (0, T ; L 2 ), u H 1 (0, T ; L 2 ). E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

57 Issues with the original functional To deal with the original functional: τ β Q J(ϕ, u, τ) = 2 ϕ ϕ Q β τ 2 ϕ(τ) ϕ 2 β u u 2 + β T τ. Then, the optimality condition for τ is β Q 0 = D τ J (u,τ ) = 2 (ϕ ϕ Q)(τ ) 2 + β 2 (ϕ (τ ) ϕ ) tϕ (τ ) + βu 2 u (τ ) 2 dx + β T. Issues: For the above expression to be well-defined, we need ttϕ L 2 (0, T ; L 2 ), u H 1 (0, T ; L 2 ). If we define U ad = {u H 1 (0, T ; L 2 ) : 0 u 1, tu L 2 (Q) K} for fixed K > 0, and impose ϕ 0 H 5, σ 0 H 3, then we get ϕ H 2 (0, T ; L 2 ) W 1, (0, T ; H 1 ). However, to require the a-priori boundedness of tu is difficult to verify in applications. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

58 Other control-type results SMC. In [Colli, Gilardi, Marinoschi, E.R., Appl Math Optim, to appear] we introduce a sliding mode control (SMC) law ϱ sign(ϕ ϕ ) in the chemical potential which forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time ϕ ϕ ) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

59 Other control-type results SMC. In [Colli, Gilardi, Marinoschi, E.R., Appl Math Optim, to appear] we introduce a sliding mode control (SMC) law ϱ sign(ϕ ϕ ) in the chemical potential which forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time ϕ ϕ ) Different sources. In the phase field model we introduced tϕ = µ + M, µ = Ψ (ϕ) ϕ tσ = σ S, we can choose different form of M and S: linear phenomenological laws for chemical reactions cf. [Hawkins Daarud, Prudhomme, van der Zee, Oden (2012)], [Frigeri, Grasselli, E.R. (2015)]: M = S = h(ϕ)(σ µ) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

60 Other control-type results SMC. In [Colli, Gilardi, Marinoschi, E.R., Appl Math Optim, to appear] we introduce a sliding mode control (SMC) law ϱ sign(ϕ ϕ ) in the chemical potential which forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time ϕ ϕ ) Different sources. In the phase field model we introduced tϕ = µ + M, µ = Ψ (ϕ) ϕ tσ = σ S, we can choose different form of M and S: linear phenomenological laws for chemical reactions cf. [Hawkins Daarud, Prudhomme, van der Zee, Oden (2012)], [Frigeri, Grasselli, E.R. (2015)]: M = S = h(ϕ)(σ µ) In [Colli, Gilardi, E.R., Sprekels, Nonlinearity (2017)]: the optimal control with respect to the drug distribution which acts as a control in the nutrient equation E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

61 Outline 1 Phase field models for tumor growth 2 The optimal control problem 3 First order optimality conditions 4 A multispecies model with velocity 5 Perspectives and Open problems E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

62 FLRS: A multispecies model with velocities - with Frigeri, Lam, Schimperna Typical structure of tumors grown in vitro: Figure: Zhang et al. Integr. Biol., 2012, 4, Scale bar 100µm = 0:1mm E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

63 FLRS: A multispecies model with velocities - with Frigeri, Lam, Schimperna Typical structure of tumors grown in vitro: Figure: Zhang et al. Integr. Biol., 2012, 4, Scale bar 100µm = 0:1mm A continuum thermodynamically consistent model is introduced with the ansatz: sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species: a diffuse interface separates tumor and healthy cell regions proliferating and dead tumor cells and healthy cells are present, along with a nutrient (e.g. glucose or oxigene) tumor cells are regarded as inertia-less fluids: include the velocity - satisfying a Darcy type law with Korteveg term E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

64 S. Frigeri, K.-F. Lam, E. R., G. Schimperna, arxiv: (2017) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

65 S. Frigeri, K.-F. Lam, E. R., G. Schimperna, arxiv: (2017) The model is a variant of the one introduced in [Y. Chen, S.M. Wise, V.B. Shenoy and J.S. Lowengrub, Int. J. Numer. Meth. Biomed. Engng. (2014)]: E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

66 S. Frigeri, K.-F. Lam, E. R., G. Schimperna, arxiv: (2017) The model is a variant of the one introduced in [Y. Chen, S.M. Wise, V.B. Shenoy and J.S. Lowengrub, Int. J. Numer. Meth. Biomed. Engng. (2014)]: ϕ p, ϕ d, ϕ h [0, 1]: the volume fractions of the cells: ϕ p: proliferating tumor cell fraction ϕ d : dead tumor cell fraction ϕ h : healthy cell fraction The variables above are naturally constrained by the relation ϕ p + ϕ d + ϕ h = 1 hence it suffices to track the evolution of ϕ p and ϕ d and the vector ϕ := (ϕ p, ϕ d ) lies in the simplex := {y R 2 : 0 y 1, y 2, y 1 + y 2 1} R 2 E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

67 S. Frigeri, K.-F. Lam, E. R., G. Schimperna, arxiv: (2017) The model is a variant of the one introduced in [Y. Chen, S.M. Wise, V.B. Shenoy and J.S. Lowengrub, Int. J. Numer. Meth. Biomed. Engng. (2014)]: ϕ p, ϕ d, ϕ h [0, 1]: the volume fractions of the cells: ϕ p: proliferating tumor cell fraction ϕ d : dead tumor cell fraction ϕ h : healthy cell fraction The variables above are naturally constrained by the relation ϕ p + ϕ d + ϕ h = 1 hence it suffices to track the evolution of ϕ p and ϕ d and the vector ϕ := (ϕ p, ϕ d ) lies in the simplex := {y R 2 : 0 y 1, y 2, y 1 + y 2 1} R 2 n: the nutrient concentration (it was σ before) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

68 S. Frigeri, K.-F. Lam, E. R., G. Schimperna, arxiv: (2017) The model is a variant of the one introduced in [Y. Chen, S.M. Wise, V.B. Shenoy and J.S. Lowengrub, Int. J. Numer. Meth. Biomed. Engng. (2014)]: ϕ p, ϕ d, ϕ h [0, 1]: the volume fractions of the cells: ϕ p: proliferating tumor cell fraction ϕ d : dead tumor cell fraction ϕ h : healthy cell fraction The variables above are naturally constrained by the relation ϕ p + ϕ d + ϕ h = 1 hence it suffices to track the evolution of ϕ p and ϕ d and the vector ϕ := (ϕ p, ϕ d ) lies in the simplex := {y R 2 : 0 y 1, y 2, y 1 + y 2 1} R 2 n: the nutrient concentration (it was σ before) u:=u i, i = 1, 2, 3: the tissue velocity field. We treat the tumor and host cells as inertial-less fluids and assume that the cells are tightly packed and they march together q: the cell-to-cell pressure E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

69 FLRS: the balance law Letting J i, i {p, d, h}, denote the mass fluxes for the cells, then the general balance law for the volume fractions reads as tϕ i + div(ϕ i u) = divj i + S i for i {p, d, h} where we set S h = 0, whereas S p, S d may depend on n, ϕ p and ϕ d E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

70 FLRS: the balance law Letting J i, i {p, d, h}, denote the mass fluxes for the cells, then the general balance law for the volume fractions reads as tϕ i + div(ϕ i u) = divj i + S i for i {p, d, h} where we set S h = 0, whereas S p, S d may depend on n, ϕ p and ϕ d Assume: the tumor growth process tends to evolve towards (local) minima of the free energy functional of Ginzburg Landau type: E(ϕ p, ϕ d ) := F (ϕ p, ϕ d ) ϕp ϕ d 2 dx E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

71 FLRS: the balance law Letting J i, i {p, d, h}, denote the mass fluxes for the cells, then the general balance law for the volume fractions reads as tϕ i + div(ϕ i u) = divj i + S i for i {p, d, h} where we set S h = 0, whereas S p, S d may depend on n, ϕ p and ϕ d Assume: the tumor growth process tends to evolve towards (local) minima of the free energy functional of Ginzburg Landau type: E(ϕ p, ϕ d ) := F (ϕ p, ϕ d ) ϕp ϕ d 2 dx where F = F 0 + F 1 is a multi-well configuration potential, e.g. F 0(ϕ p, ϕ d ):= ϕ p log ϕ p + ϕ d log ϕ d + (1 ϕ p ϕ d ) log(1 ϕ p ϕ d ) F 1(ϕ p, ϕ d ) := χ 2 (ϕ d(1 ϕ d ) + ϕ p(1 ϕ p) + (1 ϕ d ϕ p)(ϕ d + ϕ p)) E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

72 FLRS: the balance law Letting J i, i {p, d, h}, denote the mass fluxes for the cells, then the general balance law for the volume fractions reads as tϕ i + div(ϕ i u) = divj i + S i for i {p, d, h} where we set S h = 0, whereas S p, S d may depend on n, ϕ p and ϕ d Assume: the tumor growth process tends to evolve towards (local) minima of the free energy functional of Ginzburg Landau type: E(ϕ p, ϕ d ) := F (ϕ p, ϕ d ) ϕp ϕ d 2 dx where F = F 0 + F 1 is a multi-well configuration potential, e.g. F 0(ϕ p, ϕ d ):= ϕ p log ϕ p + ϕ d log ϕ d + (1 ϕ p ϕ d ) log(1 ϕ p ϕ d ) F 1(ϕ p, ϕ d ) := χ 2 (ϕ d(1 ϕ d ) + ϕ p(1 ϕ p) + (1 ϕ d ϕ p)(ϕ d + ϕ p)) The fluxes J i are defined as follows: J i = M i µ i, µ i := δe δϕ i = ϕ i + F,ϕi for i = p, d E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

73 FLRS: the velocity and nutrient evolutions We set J h = J p J d, then upon summing up the three mass balances for i = p, d, h, using the fact that ϕ p + ϕ d + ϕ h = 1 and S h = 0, we deduce the following relation: div u = S p + S d =: S t The velocity field u is assumed to fulfill Darcy s law: u = q ϕ p µ p ϕ d µ d where q denotes the cell-to-cell pressure and the subsequent two terms have the meaning of Korteweg forces E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

74 FLRS: the velocity and nutrient evolutions We set J h = J p J d, then upon summing up the three mass balances for i = p, d, h, using the fact that ϕ p + ϕ d + ϕ h = 1 and S h = 0, we deduce the following relation: div u = S p + S d =: S t The velocity field u is assumed to fulfill Darcy s law: u = q ϕ p µ p ϕ d µ d where q denotes the cell-to-cell pressure and the subsequent two terms have the meaning of Korteweg forces Since the time scale of nutrient diffusion is much faster (minutes) than the rate of cell proliferation (days), the nutrient is assumed to evolve quasi-statically: 0 = n + ϕ pn where ϕ pn models consumption by the proliferating tumor cells E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

75 The goal and the main difficulties of FLRS Goal: to study this multispecies model including different mobilities, singular potential and non-dirichlet b.c.s on the chemical potential. The main problems are: E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

76 The goal and the main difficulties of FLRS Goal: to study this multispecies model including different mobilities, singular potential and non-dirichlet b.c.s on the chemical potential. The main problems are: we have two different Cahn-Hilliard equations with non-zero right hand sides: tϕ i div(m i µ i ϕ i u) = S i and if we do not choose the Dirichlet b.c.s on µ i then we need to estimate the mean values of µ i = ϕ i + F,ϕi containing a multiwell logarithmic type potential F 0 E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

77 The goal and the main difficulties of FLRS Goal: to study this multispecies model including different mobilities, singular potential and non-dirichlet b.c.s on the chemical potential. The main problems are: we have two different Cahn-Hilliard equations with non-zero right hand sides: tϕ i div(m i µ i ϕ i u) = S i and if we do not choose the Dirichlet b.c.s on µ i then we need to estimate the mean values of µ i = ϕ i + F,ϕi containing a multiwell logarithmic type potential F 0 we need the mean values of ϕ i (the proliferating and dead cells phases) to be away from the potential bareers = ad hoc estimate based on ODEs technique E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

78 The goal and the main difficulties of FLRS Goal: to study this multispecies model including different mobilities, singular potential and non-dirichlet b.c.s on the chemical potential. The main problems are: we have two different Cahn-Hilliard equations with non-zero right hand sides: tϕ i div(m i µ i ϕ i u) = S i and if we do not choose the Dirichlet b.c.s on µ i then we need to estimate the mean values of µ i = ϕ i + F,ϕi logarithmic type potential F 0 containing a multiwell we need the mean values of ϕ i (the proliferating and dead cells phases) to be away from the potential bareers = ad hoc estimate based on ODEs technique indeed, integrating the equations for ϕ p and ϕ d we obtain an evolution law for the mean values y i := 1 ϕ i dx for i = p, d E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

79 The goal and the main difficulties of FLRS Goal: to study this multispecies model including different mobilities, singular potential and non-dirichlet b.c.s on the chemical potential. The main problems are: we have two different Cahn-Hilliard equations with non-zero right hand sides: tϕ i div(m i µ i ϕ i u) = S i and if we do not choose the Dirichlet b.c.s on µ i then we need to estimate the mean values of µ i = ϕ i + F,ϕi logarithmic type potential F 0 containing a multiwell we need the mean values of ϕ i (the proliferating and dead cells phases) to be away from the potential bareers = ad hoc estimate based on ODEs technique indeed, integrating the equations for ϕ p and ϕ d we obtain an evolution law for the mean values y i := 1 ϕ i dx for i = p, d such a relation does not involve directly the singular part F 0. Hence, the evolution of y p, y d are not automatically compatible with the physical constraint and this has to be proved by assuming proper conditions on coefficients and making a careful choice of the boundary conditions E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

80 The goal and the main difficulties of FLRS Goal: to study this multispecies model including different mobilities, singular potential and non-dirichlet b.c.s on the chemical potential. The main problems are: we have two different Cahn-Hilliard equations with non-zero right hand sides: tϕ i div(m i µ i ϕ i u) = S i and if we do not choose the Dirichlet b.c.s on µ i then we need to estimate the mean values of µ i = ϕ i + F,ϕi logarithmic type potential F 0 containing a multiwell we need the mean values of ϕ i (the proliferating and dead cells phases) to be away from the potential bareers = ad hoc estimate based on ODEs technique indeed, integrating the equations for ϕ p and ϕ d we obtain an evolution law for the mean values y i := 1 ϕ i dx for i = p, d such a relation does not involve directly the singular part F 0. Hence, the evolution of y p, y d are not automatically compatible with the physical constraint and this has to be proved by assuming proper conditions on coefficients and making a careful choice of the boundary conditions the choice (M i µ i ϕ i u) n = 0 seems essential E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

81 FLRS: The weak notion of solution Definition. (ϕ p, ϕ d, u, q, n) is a weak solution to the problem in (0, T ) if the previous equations hold, for a.e. t (0, T ) and for i = p, d, in the following weak sense: tϕ i, ζ + M i µ i ζ ϕ i u ζ dx = S i ζ dx ζ H 1 (), µ i ζ dx = ϕ i ζ + η i ζ + F 1,ϕi (ϕ p, ϕ d )ζ dx ζ H 1 (), u ξ dx = (S p + S d )ξ dx ξ H0 1 (), u ζ dx = q ζ ϕ p µ p ζ ϕ d µ d ζ dx ζ (L 2 ()) d, 0 = n + ϕ pn a.e. in, η i = F 0,ϕi (ϕ p, ϕ d ) a.e. in, S p = Σ p(n, ϕ p, ϕ d ) + m ppϕ p + m pd ϕ d a.e. in, S d = Σ d (n, ϕ p, ϕ d ) + m dp ϕ p + m dd ϕ d a.e. in. Moreover, there hold the initial conditions ϕ p(x, 0) = ϕ p,0 (x), ϕ d (x, 0) = ϕ d,0 (x) a.e. in, where, denotes the duality pairing between H 1 () and its dual H 1 (). E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

82 FLRS: Assumptions on the mass sources and on the initial data Set Σ(n, ϕ p, ϕ d ) := (Σ p, Σ d ) and M = (m ij ), i, j {p, d}, the matrix of the coefficients of the mass souces in the Cahn-Hilliard equations: (S p, S d ) = Σ + M(ϕ p, ϕ d ) T E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

83 FLRS: Assumptions on the mass sources and on the initial data Set Σ(n, ϕ p, ϕ d ) := (Σ p, Σ d ) and M = (m ij ), i, j {p, d}, the matrix of the coefficients of the mass souces in the Cahn-Hilliard equations: (S p, S d ) = Σ + M(ϕ p, ϕ d ) T Assumption on the mass sources: Σ is globally Lipschitz and E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

84 FLRS: Assumptions on the mass sources and on the initial data Set Σ(n, ϕ p, ϕ d ) := (Σ p, Σ d ) and M = (m ij ), i, j {p, d}, the matrix of the coefficients of the mass souces in the Cahn-Hilliard equations: (S p, S d ) = Σ + M(ϕ p, ϕ d ) T Assumption on the mass sources: Σ is globally Lipschitz and that there exist a closed and sufficiently regular subset 0 contained in the open simplex and constants K p,, K p,+, K d,, K d,+ R, with K p, K p,+ and K d, K d,+, such that Σ(R 3 ) [K p,, K p,+] [K d,, K d,+ ] for any x = (x p, x d ) [K p,, K p,+] [K d,, K d,+ ], there holds (My + x) n < 0 for all y 0, where n denotes the outer unit normal vector to 0 E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

85 FLRS: Assumptions on the mass sources and on the initial data Set Σ(n, ϕ p, ϕ d ) := (Σ p, Σ d ) and M = (m ij ), i, j {p, d}, the matrix of the coefficients of the mass souces in the Cahn-Hilliard equations: (S p, S d ) = Σ + M(ϕ p, ϕ d ) T Assumption on the mass sources: Σ is globally Lipschitz and that there exist a closed and sufficiently regular subset 0 contained in the open simplex and constants K p,, K p,+, K d,, K d,+ R, with K p, K p,+ and K d, K d,+, such that Σ(R 3 ) [K p,, K p,+] [K d,, K d,+ ] for any x = (x p, x d ) [K p,, K p,+] [K d,, K d,+ ], there holds (My + x) n < 0 for all y 0, where n denotes the outer unit normal vector to 0 Assumptions on the initial data : ϕ p,0, ϕ d,0 H 1 () with 0 ϕ p,0, 0 ϕ d,0, ϕ p,0 + ϕ d,0 1 a.e. in, the mean values satisfy ( 1 ϕp,0(x) dx, 1 ϕ d,0(x) dx) int 0 and F 0(ϕ p,0, ϕ d,0 ) L 1 () E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

86 FLRS: Examples of mass sources Examples of mass sources in tϕ i div(m i µ i ϕ i u) = S i for i {p, d} complying with the assumptions in the logarithmic case are: S p = λ M g(n) λ A ϕ p S d = λ A ϕ p λ L ϕ d for positive constants λ M, λ A, λ L (with λ M (λ A + λ L ) < λ A λ L, λ A < 2λ L ) and a bounded positive function g such that 0 < g(s) 1, e.g., g(s) = max(n c, min(s, 1)) for some constant n c (0, 1). E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

87 FLRS: Examples of mass sources Examples of mass sources in tϕ i div(m i µ i ϕ i u) = S i for i {p, d} complying with the assumptions in the logarithmic case are: S p = λ M g(n) λ A ϕ p S d = λ A ϕ p λ L ϕ d for positive constants λ M, λ A, λ L (with λ M (λ A + λ L ) < λ A λ L, λ A < 2λ L ) and a bounded positive function g such that 0 < g(s) 1, e.g., g(s) = max(n c, min(s, 1)) for some constant n c (0, 1). The biological effects we want to model are: the growth of the proliferating tumor cells due to nutrient consumption at a constant rate λ M the death of proliferating tumor cells at a constant rate λ A, which leads to a source term for the necrotic cells the lysing/disintegration of necrotic cells at a constant rate λ L E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42

88 FLRS: Existence of weak solutions The main result of S. Frigeri, K.-F. Lam, E. R., G. Schimperna, arxiv: (2017) Theorem For every T > 0 here exists at least one weak solution (ϕ p, µ p, η p, ϕ d, µ d, η d, u, q, n) to the multi-species tumor model on [0, T ] with the regularity ϕ i H 1 (0, T ; H 1 () ) L (0, T ; H 1 ()) L 2 (0, T ; H 2 ()), with 0 ϕ i 1, ϕ p + ϕ d 1 a.e. in Q, for i = p, d, µ i L 2 (0, T ; H 1 ()), η i L 2 (Q), u L 2 (Q) with div u L 2 (Q), q L 2 (0, T ; H 1 0 ()), n (1 + L 2 (0, T ; H 2 () H 1 0 ())), 0 n 1 a.e. in Q. E. Rocca (Università degli Studi di Pavia) Diffuse interface models of tumor growth 4-6 décembre / 42