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 R N is a connected open, bounded and T > 0. Let Ω ⊂ I Denote Q = Ω × (0 , T ) and Σ = ∂ Ω × (0 , T ). We consider the following system, describing the evolution on Ω × (0 , T ) of a brain tumor (glioblastoma):[1] and [2]. c t − ∇ . ( D ( x ) ∇ c ) = f ( c ) − F ( c , β ) in Q β t − µ ∆ β = h ( β ) − H ( c , β ) + v 1 w in Q ∂ c (1) ∂β ∂ n = 0 , ∂ n = 0 on Σ c (0) = c 0 , β (0) = β 0 in Ω 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 of 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 w if x ∈ Ω ω D ( x ) = (2) D g if x ∈ Ω g Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy
Abstract Description of the Model Introduce the results Proof Some assumptions Where 0 < D w < D g (Ω ω and Ω g are the brain areas occupied respectively by white matter and gray matter). let Λ : D (Λ) ⊂ L 2 (Ω) �− → L 2 (Ω) operator diffusion associated, with : � � w ∈ H 1 (Ω) : ∇ . ( D ( x ) ∇ w ) ∈ L 2 (Ω) D (Λ) = , Λ w = ∇ . ( D ( x ) ∇ w ) ∀ w ∈ D (Λ) We assume also that the functions f , h , F and H are given by : f ( c ) = a 1 c , h ( β ) = a 2 β, F ( c , β ) = b 1 c β, H ( c , β ) = b 2 c β (3) where the constants a 1 , a 2 , b 1 and b 2 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: c 0 , β 0 ∈ H 1 (Ω) ∩ L ∞ (Ω) , c 0 , β 0 ≥ 0 , c 0 ≤ c e (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 : 0 if c > 0 c = ˜ c if c ≤ c e . c e if c > c e We set also: ˜ F ( c , β ) = b 1 ˜ c β ˜ H ( c , β ) = b 2 ˜ c β 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 ∈ L 2 (0 , T ; D (Λ)) ∩ L ∞ (Ω × (0 , T )) , c t ∈ L 2 (Ω × (0 , T )) β ∈ L 2 (0 , T ; H 2 (Ω)) ∩ L ∞ (Ω × (0 , T )) , β t ∈ L 2 (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 a 1 = 0; Because if it is not the case, we define the following change of variable : u = ce − a 1 t . The system becames: � u t − ∇ . ( D ( x ) ∇ u ) = − b 1 u β inQ (7) β t − µ ∆ β = a 2 β − b 2 e a 1 t u β + v 1 w inQ 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 P m . Step 2 : We establish a priori estimates on the solution of P m . Step 3 : We establish a priori estimates on the time derivative of P m . 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 P m We define φ j = φ 1 , φ 2 , ..., φ m , ... a of H 1 (Ω). Let S m = [ φ 1 , φ 2 , ..., φ m ] be the space spaned by φ j , 1 ≤ j ≤ m . The approximate solution is of the form: � � m � � c m ( t ) � k =1 α k ( t ) φ k � m z m ( t ) = = = k =1 γ k ( t ) φ k β m ( t ) � φ k � � 0 � � m + � m k =1 α k ( t ) k =1 γ k ( t ) 0 φ j Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy
Abstract Description of the Model Introduce the results Proof The approximate problem P m The approximate problem P m is of the form: ′ (z m ( t ) , Υ) L 2 + a ( z m ( t ) , Υ) = ( G ( z m ( t )) , Υ) L 2 ∀ Υ ∈ S m × S m � c 0 m � z m (0) = β 0 m (8) Where : c 0 m = P S m ( c 0 ) , β 0 m = P S m ( β 0 ) a ( z , Υ) is bilinear form. � f ( z 1 ) + F ( ˜ � z 1 , z 2 ) G ( z ) = h ( z 2 ) + H ( ˜ z 1 , z 2 ) Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy
Abstract Description of the Model Introduce the results Proof The approximate problem P m (8) can be written as follows : ′ ( c m , χ ) + ( D ∇ c m , ∇ χ ) = − b 1 ( ˜ c m β m , χ ) ∀ χ ∈ S m ′ ( β m , η ) + ( µβ m , ∇ η ) = a 2 ( β m , η ) − b 2 ( ˜ c m β m , η ) + ( v 1 w , η ) ∀ Λ ∈ S m c m (0) = c 0 m β m (0) = β 0 m (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: � A m λ ′ ( t ) + B m λ ( t ) = G ( λ ( t )) (10) λ (0) = λ 0 = ( λ 0 1 , ...., λ 0 m ) T Where A m invertible matrix. By the general theorem of differential equations: Existence of the solution ! ∃ Λ m , defined for t ∈ [0 , t m ], where either t m = T or | Λ m ( t ) | − → + ∞ as t − → t m , with a priori estmaites we will show that: a)- The solution ( c m , β m ) are bounded in some spaces. b)- t m = T Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy
Abstract Description of the Model Introduce the results Proof A priori estimates of ( c m , β m ) In the (8) we take χ = c m and η = β m , and we sum the two equations: � � Ω D |∇ c m | 2 + µ Ω |∇ β m | 2 = ( c ′ m , c m ) + ( β ′ m , β m ) + � � � � Ω | β m | 2 − b 2 c m | β m | 2 + − b 1 Ω ˜ c m β m c m + a 2 Ω ˜ w v β m � � � d Ω | c m | 2 + Ω ( D |∇ c m | 2 + µ |∇ β m | 2 ) ≤ Ω | β m | 2 ] + ⇒ 1 dt [ 2 � � � Ω ( | c m | 2 + | β m | 2 ) − b 1 C Ω ˜ c m β m c m + w v β m ≤ � � Ω ( | c m | 2 + | β m | 2 ) + C w | v | 2 C such that C is a constant that depends on c e . From Gronwall’s lemma, we have that c m , β m remain in a bounded set of L ∞ (0 , T , L 2 (Ω)) ∩ L 2 (0 , T , H 1 (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 ′ est dense in L 2 which is dense in ( H 1 ) ′ the We know that H dual of H 1 . ′ (Ω) H 1 (Ω) ֒ → L 2 (Ω) ֒ → ( H 1 ) let ˜ P m be the projection of ( H 1 ) ′ (Ω) on S m . suppose that the base { φ j } is spectral in the sense that � ˜ P m � L (( H 1 (Ω)) ′ , ( H 1 (Ω)) ′ ) ≤ C , where C is a constant that independent of m . This is the case when { φ j } is the spectral base assigned to the Laplace. Elkhomeini MOULAY ELY PDEs model oriented to brain tumor therapy
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