PDEs model oriented to brain tumor therapy Elkhomeini Moulaye Ely - - PowerPoint PPT Presentation

Abstract Description of the Model Introduce the results Proof

University of Sevilla Doc-Course 2010 RESEARCH UNIT 3

PDEs model oriented to brain tumor therapy

Elkhomeini Moulaye Ely With the supervision of : Enrique Fernandez-Cara, Manuel Gonzalez-Burgos, Anna Doubova May 20, 2010

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Abstract :

. In this presentation we study a Parabolic system modeling the brain tumor growth with drug application. We are particularly interested to prove the existence of a solution.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Consideration of the model

Let Ω ⊂ I RN is a connected open, bounded and T > 0. Denote Q = Ω × (0, T) and Σ = ∂Ω × (0, T). We consider the following system, describing the evolution on Ω × (0, T)

• f a brain tumor (glioblastoma):[1] and [2].

         ct − ∇.(D(x)∇c) = f (c) − F(c, β) in Q βt − µ∆β = h(β) − H(c, β) + v1w in Q ∂c ∂n = 0, ∂β ∂n = 0

• nΣ

c(0) = c0, β(0) = β0 inΩ (1)

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Consideration of the model

Ω represents the brain. c(x, t) and beta((x, t) and are respectively the concentrations

• f tumor cells and antibodies (Cytotoxic agents) generated by

the body. D = D(x) is the diffusion coefciente of tumor cells (for simplicity coefciente assumes that the diffusion of antibodies is constant and equal to µ). f and h are functions that determine respectively the rates of reproduction of c and β. The manner in which the interaction of cells and antibodies appears is given by the functions F and H.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Some assumptions

we assume that: D(x) = Dw if x ∈ Ωω Dg if x ∈ Ωg (2)

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Some assumptions

Where 0 < Dw < Dg (Ωω and Ωg are the brain areas

• ccupied respectively by white matter and gray matter).

let Λ : D(Λ) ⊂ L2(Ω) − → L2(Ω) operator diffusion associated, with : D(Λ) =

• w ∈ H1(Ω) : ∇.(D(x)∇w) ∈ L2(Ω)
• ,

Λw = ∇.(D(x)∇w) ∀w ∈ D(Λ) We assume also that the functions f , h, F and H are given by : f (c) = a1c, h(β) = a2β, F(c, β) = b1cβ, H(c, β) = b2cβ (3) where the constants a1, a2, b1 and b2 are positive. in particular, accept that the interaction of antibodies and tumor cells is instantaneous and delay effects scorn, will not take effect).

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Some assumptions

In (1), the initial data must verifiey: c0, β0 ∈ H1(Ω) ∩ L∞(Ω), c0, β0 ≥ 0, c0 ≤ ce (4) Moreover, v = v(x, t) is a control, in practice, We assume that v ∈ L∞(Ω × (0, T)), v ≥ 0 (5) Each function v describes a therapy that is being applied over the time interval (0, T). v is expected to determine the proper increase of antibodies that, in turn, make decreasing through the term −F(c, β), which is in the second member of the first equation of (1).

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Some assumptions

Each function v describes a therapy that is being applied over time interval (0, T) In this model, the therapy is applied only on the points of ω. For any c, we set : ˜ c =    if c > 0 c if c ≤ ce ce if c > ce . We set also: ˜ F(c, β) = b1˜ cβ ˜ H(c, β) = b2˜ cβ

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Introduce the results

For the previous system we have the result below: Theorem We assume that the conditions (2)-(5) are verified. Then (1) has at least one solution (c,β), with : c ∈ L2(0, T; D(Λ)) ∩ L∞(Ω × (0, T)), ct ∈ L2(Ω × (0, T)) β ∈ L2(0, T; H2(Ω)) ∩ L∞(Ω × (0, T)), βt ∈ L2(Ω × (0, T)) (6) In addition, c ≥ 0.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Assumption

We can assume that a1 = 0; Because if it is not the case, we define the following change of variable : u = ce−a1t. The system becames: ut − ∇.(D(x)∇u) = −b1uβ inQ βt − µ∆β = a2β − b2ea1tuβ + v1w inQ (7)

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Method of resolution

Step 1 : We define the solution of the approximate problem Pm. Step 2 : We establish a priori estimates on the solution of Pm. Step 3 : We establish a priori estimates on the time derivative

• f Pm .

Step 4 : Take limits as m − → ∞. Step 5 : We Prove that the limit is a solution of the original problem P.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

The approximate problem Pm

We define φj = φ1, φ2, ..., φm, ... a of H1(Ω). Let Sm = [φ1, φ2, ..., φm] be the space spaned by φj, 1 ≤ j ≤ m. The approximate solution is of the form: zm(t) = m

k=1 αk(t)φk

m

k=1 γk(t)φk

• =

cm(t) βm(t)

• =

m

k=1 αk(t)

φk

• + m

k=1 γk(t)

φj

• Elkhomeini MOULAY ELY

PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

The approximate problem Pm

The approximate problem Pm is of the form:    (z

′

m(t), Υ)L2 + a(zm(t), Υ) = (G(zm(t)), Υ)L2 ∀Υ ∈ Sm × Sm

zm(0) = c0m β0m

• (8)

Where : c0m = PSm(c0), β0m = PSm(β0) a(z, Υ) is bilinear form. G(z) = f (z1) + F( ˜ z1, z2) h(z2) + H( ˜ z1, z2)

• Elkhomeini MOULAY ELY

PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

The approximate problem Pm

(8) can be written as follows :                (c

′

m, χ) + (D∇cm, ∇χ) = −b1( ˜

cmβm, χ) ∀χ ∈ Sm (β

′

m, η) + (µβm, ∇η) = a2(βm, η) − b2( ˜

cmβm, η) + (v1w, η) ∀Λ ∈ Sm cm(0) = c0m βm(0) = β0m (9)

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Differential system

(8) an (9) can be writing in the forme: Amλ′(t) + Bmλ(t) = G(λ(t)) λ(0) = λ0 = (λ01, ...., λ0m)T (10) Where Am invertible matrix. By the general theorem of differential equations: Existence of the solution ! ∃Λm, defined for t ∈ [0, tm], where either tm = T or |Λm(t)| − → +∞ as t − → tm, with a priori estmaites we will show that: a)- The solution (cm, βm) are bounded in some spaces. b)- tm = T

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

A priori estimates of (cm, βm)

In the (8) we take χ = cm and η = βm, and we sum the two equations: (c′

m, cm) + (β′ m, βm) +

• Ω D |∇cm|2 + µ
• Ω |∇βm|2 =

−b1

• Ω ˜

cmβmcm + a2

• Ω |βm|2 − b2
• Ω ˜

cm |βm|2 +

• w vβm

⇒ 1

2

d dt [

• Ω |cm|2 +
• Ω |βm|2] +
• Ω(D |∇cm|2 + µ |∇βm|2) ≤

C

• Ω(|cm|2 + |βm|2) − b1
• Ω ˜

cmβmcm +

• w vβm ≤

C

• Ω(|cm|2 + |βm|2) + C
• w |v|2

such that C is a constant that depends on ce. From Gronwall’s lemma, we have that cm, βm remain in a bounded set of L∞(0, T, L2(Ω)) ∩ L2(0, T, H1(Ω)).

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

A priori estimate of the time derivatives

We know that H

′ est dense in L2 which is dense in (H1) ′ the

dual of H1. H1(Ω) ֒ → L2(Ω) ֒ → (H1)

′(Ω)

let ˜ Pm be the projection of (H1)

′(Ω) on Sm.

suppose that the base {φj} is spectral in the sense that ˜ PmL((H1(Ω))′,(H1(Ω))′) ≤ C , where C is a constant that independent of m. This is the case when {φj} is the spectral base assigned to the Laplace.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

A priori estimate of the time derivatives

c

′

m = Pm(∇.(D∇cm) − b1˜

cmβn) c

′

m(H1(Ω))′ ≤ C∇.(D∇cm) − b1˜

cmβm(H1(Ω)′ c

′

m(H1(Ω))′ ≤ C(cm(t)(H1(Ω))+cm(t)(L2(Ω))+βm(t)(L2(Ω)))

c

′

m(H1(Ω))′ ≤ C(cm(H1(Ω)) + βm(L2(Ω)))

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

A priori estimate of the time derivatives

Since cm is bounded in L2(0, T, H1) et L∞(0, T, L2), therefore c

′

m remains in a bounded set of L2(0, T; (H1(Ω))

′).

Following the same way for β: β

′

m

= Pm(−µ∆βm + a2βm − b2 cmβm) = ⇒ β

′

m(H1)′

≤ C − µ∆βm + a2βm − b2 cmβm = ⇒ β

′

m(H1(Ω))′ ≤ C(cm(H1(Ω)) + βm(L2(Ω)))

And as that βm is bounded in L2(0, T, H1) et L∞(0, T, L2), then β

′

m(H1)′ ≤ C such that C is a constant does not

depend time; therefore β

′

m remains in a bounded of

L2(0, T; (H1(Ω))

′). Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

Taking a limits as m − → +∞

Since cm remains in a bounded set of L2(0, T; H1) and c

′

m

remains in a bounded set of L2(0, T, (H1(Ω))

′), then cm

remains in a compact of L2(0, T; L2(Ω)) ∼ = L2(Ω × (0, T)). By extract of a subsequence, and passing to the limit: ∃ subsequence cm, βm such that : cm − → c in L2(0, T; H1(Ω)) weakly. βm − → in L2(0, T; H1(Ω)) weakly. And also: cm − → c in L∞(0, T; L2(Ω)) weakly ∗. βm − → β in L2(0, T; L2(Ω))) weakly ∗. And c

′

m −

→ ct in L2(0, T, (H1(Ω))

′) weakly.

β

′

m −

→ βt in L2(0, T, (H1(Ω))

′) weakly. Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

For the initial conditions

Then : cm − → c in L2(Ω × (0, T)) strongly. βm − → β in L2(Ω × (0, T)) strongly. Since cm remains in a bounded set of L2(0, T, H1(Ω)). and c

′

m

remains in a bounded set of L2(0, T, (H1(Ω))

′), we have that

cm ∈ C 0([0, T]; L2(Ω)) And also since we have that: cm − → c in L2(Ω × (0, T)) βm − → β in L2(Ω × (0, T)) This implies that cm(0) − → c(0) weakly in L2(Ω) c0m − → c0 = ⇒ c(0) = c0.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

The same method for β we get : Since βm remains in a bounded set of L2(0, T, H1(Ω)). and β

′

m

remains in a bounded set of L2(0, T, (H1(Ω))

′), we have that

βm ∈ C 0([0, T]; L2(Ω)) And also since we have that: βm − → β in L2(Ω × (0, T)) βm − → β in L2(Ω × (0, T)) This implies that βm(0) − → β(0) weakly in L2(Ω) β0m − → β0 = ⇒ β(0) = β0.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

To summarize all this, we have: cm − → c in L2(Ω × (0, T)) strongly. βm − → β in L2(Ω × (0, T)) strongly. In other side we have β0m − → β0 c0m − → c0 In other words, when m − → +∞ then (cm, βm) − → (c, β). Therefore, (C, β) are a solution of the problem :          ct − ∇.(D(x)∇c) = −F(˜ c, β) in Q βt − µ∆β = h(β) − H(˜ c, β) + v1w in Q ∂c ∂n = 0, ∂β ∂n

• n Σ

c(0)= c0, β(0) = β0 in Ω (11)

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

To prove that 0 ≤ c ≤ ce

Firstly we prove that c ≤ 0: We set c− = Max(−c, 0) c− = 0 ifc > 0 −c ifc ≤ 0 . We multiply by c− in the frist equation of the system (1) and we integrate on Ω:

• Ω ctc−dt +
• Ω D∇c.∇c−dt = −b1
• Ω ˜

cβc−dt But since

• Ω ˜

cβc−dt = 0, we have that:

1 2

d dt

• Ω |c−|2 +
• Ω(D |∇c−|2 = 0

= ⇒

• Ω |c−|t=0 = 0, from Granwall, we have
• Ω |c−| ≡ 0∀t =

⇒ c− ≡ 0 Therefore c ≥ 0

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

To prove that 0 ≤ c ≤ ce

Secondly we prove that c ≤ ce: We set (c − ce)+ = Max(c − ce, 0) We multiply by (c − ce)+ in the frist equation of the system (1) and we integrate on Ω:

• Ω ct.(c − ce)+dt +
• Ω D∇c.∇(c − ce)+dt = 0

1 2

d dt

• Ω |(c − ce)+|2 +
• Ω(D |∇(c − ce)+|2 = 0

= ⇒

• Ω |(c − ce)+|2 = 0, from Granwall, we have
• Ω |(c − ce)+| ≤ 0∀t =

⇒

• Ω |(c − ce)+|t=0 = 0.

But because we have c0 ≤ ce, than c ≤ ce = ⇒ c− ≡ 0 Therefore c ≥ 0

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof

References

• K. R. Swanson, E. C. Alvord Jr, and J. D. Murray, A

quantitative model for differential motility of gliomas in grey and white matter, Cell Prolif., 33, 317329, 2000.

• K. R. Swanson, E. C. Alvord Jr, and J. D. Murray, Quantifying

Efficacy of Chemotherapy of Brain Tumors (Gliomas) with Homogeneous and Heterogeneous Drug Delivery, Acta Biotheoretica, 50(4): 223-237, 2002.

Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy

Abstract Description of the Model Introduce the results Proof Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy