Monika Twarogowska INRIA Sophia Antipolis - OPALE Project - Team - PowerPoint PPT Presentation

A well-balanced numerical scheme for solution with vacuum to a 1d quasilinear hyperbolic model of chemotaxis Monika Twarogowska INRIA Sophia Antipolis - OPALE Project - Team 14th International Conference on Hyperbolic Problems: Theory, Numerics, Applications Universitá di Padova June 25-29, 2012 Work in collaboration with: Prof. Roberto Natalini (IAC-CNR) Dr. Magali Ribot (Université de Nice-Sophia Antipolis) Monika Twarogowska (INRIA) Padova, 26/06/2012 1 / 22

Outline Chemotaxis: vasculogenesis process 1 2 Analysis of non constant stationary solutions of a quasilinear, hyperbolic model of chemotaxis Numerical approximation 3 4 Numerical tests Monika Twarogowska (INRIA) Padova, 26/06/2012 2 / 22

Outline Chemotaxis: vasculogenesis process 1 2 Analysis of non constant stationary solutions of a quasilinear, hyperbolic model of chemotaxis Numerical approximation 3 4 Numerical tests Monika Twarogowska (INRIA) Padova, 26/06/2012 3 / 22

Biological background Chemotaxis Directed movement of mobile species towards lower/higher concentration of chemical substance present in the surrounding environment Example: Vasculogenesis a process of de novo formation of blood vessels chemotactic factor: VEGF-A released by cells percolative and ”Swiss cheese” transitions depending on the Figure: In vitro experiments of initial mass Vasculogenesis (Serini et. al) Monika Twarogowska (INRIA) Padova, 26/06/2012 3 / 22

Hyperbolic model of vasculogenesis [Gamba A., Preziosi L. et al. (2003)]  ρ t + div ( ρ� u ) = 0  ( ρ� u ) t + div ( ρ� u ⊗ � u ) = χρ ∇ φ − αρ� u − ∇ P ( ρ ) φ t = D ∆ φ + a ρ − b φ  ρ - density of endothelial cells φ - concentration of chemical factor VEGF Monika Twarogowska (INRIA) Padova, 26/06/2012 4 / 22

Hyperbolic model of vasculogenesis [Gamba A., Preziosi L. et al. (2003)]  ρ t + div ( ρ� u ) = 0  ( ρ� u ) t + div ( ρ� u ⊗ � u ) = χρ ∇ φ − αρ� u − ∇ P ( ρ ) φ t = D ∆ φ + a ρ − b φ  ρ - density of endothelial cells φ - concentration of chemical factor VEGF Forces acting on cells: F vol = −∇ P ( ρ ) , where P ( ρ ) = ερ γ , internal force ε > 0 , γ > 1 body force - chemotaxis F chem = χρ ∇ φ, χ > 0 contact force F diss = − αρ u , α > 0 Monika Twarogowska (INRIA) Padova, 26/06/2012 4 / 22

Hyperbolic model of vasculogenesis [Gamba A., Preziosi L. et al. (2003)]  ρ t + div ( ρ� u ) = 0  ( ρ� u ) t + div ( ρ� u ⊗ � u ) = χρ ∇ φ − αρ� u − ∇ P ( ρ ) φ t = D ∆ φ + a ρ − b φ  ρ - density of endothelial cells φ - concentration of chemical factor VEGF Forces acting on cells: F vol = −∇ P ( ρ ) , where P ( ρ ) = ερ γ , internal force ε > 0 , γ > 1 body force - chemotaxis F chem = χρ ∇ φ, χ > 0 contact force F diss = − αρ u , α > 0 ⇒ solutions containing vacuum Di Russo, C. and Sepe, A. - ”Existence and Asymptotic Behavior of Solutions to a Quasilinear Hyperbolic-Parabolic Model of Vasculogenesis” (2011), preprint. Monika Twarogowska (INRIA) Padova, 26/06/2012 4 / 22

Outline Chemotaxis: vasculogenesis process 1 2 Analysis of non constant stationary solutions of a quasilinear, hyperbolic model of chemotaxis Numerical approximation 3 4 Numerical tests Monika Twarogowska (INRIA) Padova, 26/06/2012 5 / 22

Hyperbolic model of chemotaxis: stationary solutions Problem: We look for non constant stationary solutions of system  ρ t + ( ρ u ) x = 0  ( ρ u ) t + ( ρ u 2 + P ( ρ )) x = − αρ u + χρφ x (1) φ t = D φ xx + a ρ − b φ  defined on a bounded domain Ω = [ 0 , L ] with homogeneous Neumann boundary conditions ρ x | ∂ Ω = 0 , φ x | ∂ Ω = 0 , u | ∂ Ω = 0 � L and the total mass, conserved in time, given by M = 0 ρ ( x , t ) dx . Motivation: description and study of vascular-like networks observed in the in vitro experiments with human, endothelial cells [Serini et.al.] Monika Twarogowska (INRIA) Padova, 26/06/2012 5 / 22

General case: P ( ρ ) = ερ γ , γ > 1 ( ρ u ) x = 0 , ρ u 2 + P ( ρ ) � � = − αρ u + χρφ x , x − D φ xx = a ρ − b φ. Monika Twarogowska (INRIA) Padova, 26/06/2012 6 / 22

General case: P ( ρ ) = ερ γ , γ > 1 ( ρ u ) x = 0 , ρ u 2 + P ( ρ ) � � = − αρ u + χρφ x , x − D φ xx = a ρ − b φ. u | ∂ Ω = 0 ⇒ ρ u = 0 Monika Twarogowska (INRIA) Padova, 26/06/2012 6 / 22

General case: P ( ρ ) = ερ γ , γ > 1 ( ρ u ) x = 0 , ρ u 2 + P ( ρ ) � � = − αρ u + χρφ x , x − D φ xx = a ρ − b φ. u | ∂ Ω = 0 ⇒ ρ u = 0 ⇒ P ( ρ ) x = χρφ x Monika Twarogowska (INRIA) Padova, 26/06/2012 6 / 22

General case: P ( ρ ) = ερ γ , γ > 1 ( ρ u ) x = 0 , ρ u 2 + P ( ρ ) � � = − αρ u + χρφ x , x − D φ xx = a ρ − b φ. u | ∂ Ω = 0 ⇒ ρ u = 0 ⇒ P ( ρ ) x = χρφ x Solutions: 1 . ρ = M φ = aM L , bL , u = 0 Monika Twarogowska (INRIA) Padova, 26/06/2012 6 / 22

General case: P ( ρ ) = ερ γ , γ > 1 ( ρ u ) x = 0 , ρ u 2 + P ( ρ ) � � = − αρ u + χρφ x , x − D φ xx = a ρ − b φ. u | ∂ Ω = 0 ⇒ ρ u = 0 ⇒ P ( ρ ) x = χρφ x Solutions: 1 . ρ = M φ = aM L , bL , u = 0 ρ γ − 1 = χ ( γ − 1 ) 2 . ρ = 0 φ + K or εγ − D φ xx = a ρ − b φ if ρ > 0 φ : = ρ = 0 D φ xx b φ if Monika Twarogowska (INRIA) Padova, 26/06/2012 6 / 22

General case: P ( ρ ) = ερ γ , γ > 1 ( ρ u ) x = 0 , ρ u 2 + P ( ρ ) � � = − αρ u + χρφ x , x − D φ xx = a ρ − b φ. u | ∂ Ω = 0 ⇒ ρ u = 0 ⇒ P ( ρ ) x = χρφ x Solutions: 1 . ρ = M φ = aM L , bL , u = 0 ρ γ − 1 = χ ( γ − 1 ) 2 . ρ = 0 φ + K or εγ − D φ xx = a ρ − b φ if ρ > 0 φ : = ρ = 0 D φ xx b φ if Monika Twarogowska (INRIA) Padova, 26/06/2012 6 / 22

General case: P ( ρ ) = ερ γ , γ > 1 ( ρ u ) x = 0 , ρ u 2 + P ( ρ ) � � = − αρ u + χρφ x , x − D φ xx = a ρ − b φ. u | ∂ Ω = 0 ⇒ ρ u = 0 ⇒ P ( ρ ) x = χρφ x Solutions: 1 . ρ = M φ = aM L , bL , u = 0 ρ γ − 1 = χ ( γ − 1 ) 2 . ρ = 0 φ + K or εγ − D φ xx = a ρ − b φ if ρ > 0 φ : = ρ = 0 D φ xx b φ if Problems in finding an explicit solution I: - number of bumps p ∈ N is not known a priori - for p > 1 : more unknown constants than available equations - for γ > 2 finding φ k is not trivial Monika Twarogowska (INRIA) Padova, 26/06/2012 6 / 22

Assumption: p = 1 , P ( ρ ) = ερ 2 Lateral bump a χ 2 ε D − b π If α = D > 0 and L > √ α then there exists a unique, positive solution of the form cos ( √ α x ) φ ( x ) = 2 εβ K ρ ( x ) = χ x ) − aK [ 0 , ¯ x ] , cos ( √ α ¯ 2 εφ ( x ) + K on α D , αχ β L ) sinh ( √ β ( x − L )) φ ( x ) = 2 ε bK � [¯ x , L ] , tan ( cosh ( √ β (¯ ρ ( x ) = 0 on x − L )) , χ √ α ] π 1 given by the smallest ¯ x ∈ 2 , π [ satisfying � α tan ( √ α ¯ β � x ) = tanh ( β (¯ x − L )) (2) α M and K equal to K = x . √ α tan ( √ α ¯ β x ) − β ¯ Monika Twarogowska (INRIA) Padova, 26/06/2012 7 / 22

Assumption: p = 1 , P ( ρ ) = ερ 2 Lateral bump χ � 2 ε φ ( x ) + K x ∈ [ 0 , ¯ x ] ρ ( x ) = 0 x ∈ (¯ x , L ] Monika Twarogowska (INRIA) Padova, 26/06/2012 8 / 22

Assumption: p = 1 , P ( ρ ) = ερ 2 Lateral bump χ � 2 ε φ ( x ) + K x ∈ [ 0 , ¯ x ] ρ ( x ) = 0 x ∈ (¯ x , L ] Centered bump  x ∈ [ 0 , ¯ x ) 0  χ ρ ( x ) = 2 ε φ ( x ) + K x ∈ [¯ x , L − ¯ x ] x ∈ ( L − ¯ x , L ] 0  Solution is SYMMETRIC Monika Twarogowska (INRIA) Padova, 26/06/2012 8 / 22

Assumption: p = 1 , P ( ρ ) = ερ 2 Lateral bump χ � 2 ε φ ( x ) + K x ∈ [ 0 , ¯ x ] ρ ( x ) = 0 x ∈ (¯ x , L ] Centered bump  x ∈ [ 0 , ¯ x ) 0  χ ρ ( x ) = 2 ε φ ( x ) + K x ∈ [¯ x , L − ¯ x ] x ∈ ( L − ¯ x , L ] 0  Solution is SYMMETRIC Problems in finding an explicit solution II: existence of interface points ¯ x k in the case p > 1 is an open problem Monika Twarogowska (INRIA) Padova, 26/06/2012 8 / 22

Outline Chemotaxis: vasculogenesis process 1 2 Analysis of non constant stationary solutions of a quasilinear, hyperbolic model of chemotaxis Numerical approximation 3 4 Numerical tests Monika Twarogowska (INRIA) Padova, 26/06/2012 9 / 22

Numerical scheme for a 1d quasilinear model of vasculogenesis  ρ t + ( ρ u ) x = 0  ( ρ u ) t + ( ρ u 2 + P ( ρ )) x = − αρ u + χρφ x φ t = D φ xx + a ρ − b φ  Monika Twarogowska (INRIA) Padova, 26/06/2012 9 / 22

Numerical scheme for a 1d quasilinear model of vasculogenesis  ρ t + ( ρ u ) x = 0  ( ρ u ) t + ( ρ u 2 + P ( ρ )) x = − αρ u + χρφ x φ t = D φ xx + a ρ − b φ ⇒ Standard FDM  Monika Twarogowska (INRIA) Padova, 26/06/2012 9 / 22

Numerical scheme for a 1d quasilinear model of vasculogenesis  ρ t + ( ρ u ) x = 0  ( ρ u ) t + ( ρ u 2 + P ( ρ )) x = − αρ u + χρφ x  Requirements for a numerical scheme: consistency with the original system preservation of the non negativity of densities and concentrations preservation of the total mass treatment of vacuum states good approximation of non constant steady states low numerical viscosity Monika Twarogowska (INRIA) Padova, 26/06/2012 9 / 22