Models of Biological Movements: Analysis and Numerical Simulations. Cristiana Di Russo. CNRS, Institut Camille Jordan Université Claude Bernard Lyon 1

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1 Models of Biological Movements: Analysis and Numerical Simulations Cristiana Di Russo CNRS, Institut Camille Jordan Université Claude Bernard Lyon 1 Journées de l'anr MONUMENTALG June 30th, 2011 C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

2 PhD Thesis Analysis & Numerical Approximation: Model of chemotaxis (Cattaneo-Hillen), Model of vasculogenesis (Gamba-Preziosi), [A. Sepe ]. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

3 PhD Thesis Analysis & Numerical Approximation: Model of chemotaxis (Cattaneo-Hillen), Model of vasculogenesis (Gamba-Preziosi), [A. Sepe ]. Modelling & Numerical Approximation: Phototrophic biolms, [F. Clarelli, R. Natalini & M. Ribot]. Inammation during ischemic stroke. [J.B. Lagaert, G. Chapuisat & M.A. Dronne]. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

4 Chemotaxis What is Chemotaxis? a type taxis, a behavioral response by an organism to an external stimulus; the inuence of chemical substances in the environment on the movement of mobile species. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

5 Chemotaxis What is Chemotaxis? a type taxis, a behavioral response by an organism to an external stimulus; the inuence of chemical substances in the environment on the movement of mobile species. Figure: An example of chemotaxis C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

6 Models of Chemotaxis Classical model of Chemotaxis Patlak-Keller-Segel Model (1953/1970) 8 < t u = r (D(u)ru A(u)B()C (r)) + f t = + g (u; ) (1) introduced to describe the aggregation of cellular slide molds like Dictyostelium discoideum, based on a macroscopic approach where the behavior of a population is considered as a whole. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

7 Models of Chemotaxis Classical model of Chemotaxis Patlak-Keller-Segel Model (1953/1970) 8 < t u = r (D(u)ru A(u)B()C (r)) + f t = + g (u; ) (1) introduced to describe the aggregation of cellular slide molds like Dictyostelium discoideum, based on a macroscopic approach where the behavior of a population is considered as a whole. It is a continuous reaction-diusion model: u density of the motile living species, concentration of the chemical species, D(u) motility coecient, A(u)B()C (r) chemotactic sensitivity function. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

8 Models of Chemotaxis Models with nite speed of propagation Experiments show a run and tumble movement of bacteria like E. Coli. Velocity Jump t f + v r x f = T (S; f ) (2) f (t; x; v ) density of cells depending on time t, position x and velocity v. T (S; f ) turning operator. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

9 Models of Chemotaxis Models with nite speed of propagation Experiments show a run and tumble movement of bacteria like E. Coli. Velocity Jump t f + v r x f = T (S; f ) (2) f (t; x; v ) density of cells depending on time t, position x and velocity v. T (S; f ) turning operator. Diusive Scaling t! 2 t, x! x PKS Model. Hydrodynamic Scaling t! t, x! x Hyperbolic Model. [B.Perthame, V. Menez et al., T.Hillen] C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

10 Models of Chemotaxis Hyperbolic Model Cattaneo-Hillen Model 8 >< t u + r v = t v + 2 ru = t = + au b + f (u; ): b(; r)v + h(; r)g (u); (3) u density of the motile species, v ux, concentration of the chemical species. b; h; g have linear growth and f is quadratic. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

11 Models of Chemotaxis One-dimensional case Local and global existence of weak solutions under the assumption of turning rate's boundedness [T.Hillen e A. Stevens]. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

12 Models of Chemotaxis One-dimensional case Local and global existence of weak solutions under the assumption of turning rate's boundedness [T.Hillen e A. Stevens]. Global existence and stability of solution with small initial data for Cauchy problem and perturbation of constant state for Neumann problem [F. Guarguaglini et al.]. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

13 Models of Chemotaxis Multi-dimensional case Global existence and asymptotic behavior of smooth solutions to (3), under the hypothesis of small initial data. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

14 Models of Chemotaxis Multi-dimensional case Global existence and asymptotic behavior of smooth solutions to (3), under the hypothesis of small initial data. Global existence and asymptotic behavior of perturbation of costant state. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

15 Models of Chemotaxis Multi-dimensional case Global existence and asymptotic behavior of smooth solutions to (3), under the hypothesis of small initial data. Global existence and asymptotic behavior of perturbation of costant state. Comparison with the Diusive case. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

16 Models of Chemotaxis Perturbation of Constant States Let us consider the system t ~u + r ~v = 0; t ~v + r~u = ~v + ~ur ~ t ~ = ~ + a~u b ~ ; where (~u; ~v ; ~ ) = (u + u; v ; + ), with (u; 0; ) stationary solution and (u; v ; ) perturbation. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

17 Models of Chemotaxis Perturbation of Constant States Let us consider the system t ~u + r ~v = 0; t ~v + r~u = ~v + ~ur ~ t ~ = ~ + a~u b ~ ; where (~u; ~v ; ~ ) = (u + u; v ; + ), with (u; 0; ) stationary solution and (u; v ; ) perturbation. 8 >< t u + r v = t v + ru = v + (u + t = + au b: (4) C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

18 Models of Chemotaxis Perturbation of Constant States Let us consider the system t ~u + r ~v = 0; t ~v + r~u = ~v + ~ur ~ t ~ = ~ + a~u b ~ ; where (~u; ~v ; ~ ) = (u + u; v ; + ), with (u; 0; ) stationary solution and (u; v ; ) perturbation. 8 >< t u + r v = t v + ru = v + (u + t = + au b: Hyperbolic-parabolic system: the Shizuta-Kawashima coupling condition holds; but dissipativity is not clear, because there is the forcing source term u in the third equation. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24 (4)

19 Models of Chemotaxis Global Existence of Perturbation of Constant States Theorem Let " 0 > 0 such that ku 0 k H s ; ku 0 k L 1 ; kv 0 k H s ; kv 0 k L 1 ; k 0 k H s+1 ; k 0 k L 1 ; u " 0 ; then 9! global solution to the Cauchy problem associated to (4) u 2 C ([0; 1); H s (R n )); v 2 C ([0; 1); H s (R n )); 2 C ([0; 1); H s+1 (R n )); n for s C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

20 Models of Chemotaxis Global Existence of Perturbation of Constant States Theorem Let " 0 > 0 such that ku 0 k H s ; ku 0 k L 1 ; kv 0 k H s ; kv 0 k L 1 ; k 0 k H s+1 ; k 0 k L 1 ; u " 0 ; then 9! global solution to the Cauchy problem associated to (4) u 2 C ([0; 1); H s (R n )); v 2 C ([0; 1); H s (R n )); 2 C ([0; 1); H s+1 (R n )); n for s Moreover ku(t)k L 1 t n 4 ; ku(t)k L 2 t n 4 ; kd k x u(t)k L 2 t k ; kv (t)k L 1 t n 4 ; kv (t)k L 2 t n 4 ; kd k x v (t)k L 2 t k ; k(t)k L 1 t n 4 ; kd 1 x (t)k L1 t n 4 ; k(t)k L 2 t n 4 ; kd k+1 x (t)k L 2 t k ; for k = 0; : : : ; s; where k = min n s 2 ; n 2. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

21 Models of Chemotaxis Sketch of the proof C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

22 Models of Chemotaxis Sketch of the proof Semigroup theory and Fixed point method Local existence. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

23 Models of Chemotaxis Sketch of the proof Semigroup theory and Fixed point method Local existence. Duhamel's Formula & Decay Estimates of Green's functions Boundedness of required norms. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

24 Models of Chemotaxis Sketch of the proof Semigroup theory and Fixed point method Local existence. Duhamel's Formula & Decay Estimates of Green's functions Boundedness of required norms. Continuation principle Global Existence. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

25 Models of Chemotaxis Numerical Approximation Finite Dierence Method Waves t u + r v = t v + ru = 0: C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

26 Models of Chemotaxis Numerical Approximation Finite Dierence Method Waves t u + r v = t v + ru = 0: Initial data u 0 (x 1 ; x 2 ) = cos(2x 1 )cos(2x 2 ); v 0 (x 1 ; x 2 ) = 0 C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

27 Models of Chemotaxis Numerical Approximation Finite Dierence Method Waves t u + r v = t v + ru = 0: Initial data u 0 (x 1 ; x 2 ) = cos(2x 1 )cos(2x 2 ); v 0 (x 1 ; x 2 ) = 0 Boundary conditions ru n = 0; v n = 0 Figure: Comparison between the Lax Friedrichs scheme and the Relaxation one [Natalini et al.] C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

] Figure: Chemotaxis of a population of Dictyostelium and numerical solution of the Hillen-Dolak

28 Models of Chemotaxis Numerical Approximation Cattaneo-Hillen Model: Numerical Results I Finite Dierence Method: Relaxation Scheme + AHO [Natalini et al.] Figure: Chemotaxis of a population of Dictyostelium and numerical solution of the Hillen-Dolak model with initial condition u 0(x) 2 [0:5; 0:51] on a square domain [0:20] [0:20]. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

29 Models of Chemotaxis Numerical Approximation Cattaneo-Hillen Model: Numerical Results II Finite Dierence Method: Relaxation Scheme + AHO [Natalini et al.] Figure: Numerical simulations of the population density with dierent initial conditions C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

30 Models of Chemotaxis Numerical Approximation Cattaneo-Hillen Model: Numerical Results II Finite Dierence Method: Relaxation Scheme + AHO [Natalini et al.] Figure: Numerical simulations of the population density with dierent initial conditions To Do Neumann Problem in collaboration with C. Mascia. Proof of blow-up in 2D in collaboration with V. Calvez, M. Ribot. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

31 Model of Vasculogenesis Model of Vasculogenesis [Gamba, Preziosi et al.] 8 >< t + r (u) = t (u) + r (u u) + rp() = u + t = D + a : density of endothelial cells, u velocity of the cells, density of chemoattractant. Figure: Vasculogenesis in vitro and in silico C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

32 Model of Vasculogenesis Model of Vasculogenesis [Gamba, Preziosi et al.] 8 >< t + r (u) = t (u) + r (u u) + rp() = u + t = D + a : density of endothelial cells, u velocity of the cells, density of chemoattractant. D diusion coecient, a rate of release, characteristic degradation time, drift coecient, strength of cell response. Figure: Vasculogenesis in vitro and in silico C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

33 Model of Vasculogenesis Analysis One-dimensional case [D.R.,Sepe] 8 t ~ x (~v ) = t (~v ) x ~v 2 ~ + P(~) = ~@ x ~ ~v ; t ~ = D@xx ~ + a~ ~ ; with P 0 (~) > 0. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

34 Model of Vasculogenesis Analysis One-dimensional case [D.R.,Sepe] 8 t ~ x (~v ) = t (~v ) x ~v 2 ~ + P(~) = ~@ x ~ ~v ; t ~ = D@xx ~ + a~ ~ ; with P 0 (~) > 0. We consider solutions of the form (~; ~v ; ~ ) = ( + ; v ; + ), where (; 0; ) is a constant stationary solution and (; v ; ) a perturbation. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

35 Model of Vasculogenesis Analysis One-dimensional case [D.R.,Sepe] with P 0 (~) > 0. 8 >< t ~ x (~v ) = t (~v ) x ~v 2 ~ + P(~) = ~@ x ~ ~v t ~ = D@xx ~ + a~ ~ ; We consider solutions of the form (~; ~v ; ~ ) = ( + ; v ; + ), where (; 0; ) is a constant stationary solution and (; v ; ) a perturbation. 8 >< t x v = 0; t v 2 x + + P( + ) = ( + )@ x v t = D@ xx + a : (5) C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

36 Model of Vasculogenesis Analysis Global Existence of Perturbation of Constant State Theorem [D.R., Sepe] We consider the Cauchy problem associated to (5), with initial data ( 0 ; v 0 ) 2 H 2 (R) and 0 2 H 2 (R). If k( 0 ; v 0 )k H 2 (R), k 0 k H 2 (R) and are suciently small, then 9! global solution (; v ; ) to the system (5) such that: (; v ) 2 C ([0; 1); H 2 (R)); 2 C ([0; 1); H 2 (R)) \ L 2 ([0; 1); H 3 (R)) and, for each T > 0, Z T Z T k(; v )(T )k 2 H 2 + k@ x (; v )( )k 2 H 1 d + kv ( )k 2 H 2 d C k(; v ) 0k 2 H 2 ; 0 0 Z T k(t )k 2 H 2 + k@ x ( )k 2 H 2 d C (k(; v ) 0k 2 H 2 + k 0k 2 H 2 ); where C = C (; k(; v ) 0k H 2 ; k 0k H 2 ) 0 (6) C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

37 Model of Vasculogenesis Analysis Sketch of the proof Kawashima's Theorem Local Existence. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

38 Model of Vasculogenesis Analysis Sketch of the proof Kawashima's Theorem Local Existence. The hyperbolic part is quasilinear to prove the existence no Duhamel formula. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

39 Model of Vasculogenesis Analysis Sketch of the proof Kawashima's Theorem Local Existence. The hyperbolic part is quasilinear to prove the existence no Duhamel formula. Strategy: use the partially dissipative structure of the hyperbolic system, embedding the chemoattract estimates in the Hanouzet-Natalini approach for the hyperbolic systems. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

40 Model of Vasculogenesis Numerical Approximation Gamba Preziosi Model : Numerical results Finite Dierence Method: Relaxation Scheme + AHO [Natalini et al.] Figure: Vasculogenesis in vitro and in silico. Initial datum P (x; 0) = 0: j=1 exp jx xj j C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

Model of Vasculogenesis Numerical Approximation Gamba Preziosi Model : Numerical results Finite Dierence Method: Relaxation Scheme + AHO [Natalini et al.

41 Model of Vasculogenesis Numerical Approximation Gamba Preziosi Model : Numerical results Finite Dierence Method: Relaxation Scheme + AHO [Natalini et al.] Figure: Vasculogenesis in vitro and in silico. Initial datum P (x; 0) = 0: j=1 exp jx xj j To Do Proof of blow-up in 2D in collaboration with V. Calvez, M. Ribot. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

42 Model of Ischemic Stroke Inammation during Ischemic Stroke in collaboration with J.B. Lagaert, G. Chapuisat, M.A. Dronne, T. Dumont, V. Calvez. A stroke is the rapid developing loss of brain function due to disturbance in the blood supply to the brain. This can occours due to ischemia (80%) or due to hemorrhage (20%). Inammatory process is one of the main mechanisms involved in ischemic stroke. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

43 Model of Ischemic Stroke Relations between the dierent cells healthy cells necrosis apoptosis microglia blood C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

44 Model of Ischemic Stroke Relations between the dierent cells healthy cells necrosis apoptosis eliminated microglia activation macrophages blood neutrophils C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

45 Model of Ischemic Stroke Relations between the dierent cells healthy cells necrosis apoptosis eliminated microglia activation macrophages blood neutrophils C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

46 Model of Ischemic Stroke Relations between the dierent cells healthy cells necrosis apoptosis eliminated microglia activation macrophages blood neutrophils C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

47 Model of Ischemic Stroke A model of inammation during ischemic stroke (II) in collaboration with G.Chapuisat, M.A. Dronne, V. Calvez, T. tn = k cv cv + k nv tv = k cv cv k nv nv ; k mn(m i + m a + )n; m tm i = k nmn(m i + ) 1 M tm a = D ma m a r ma m tc = D c c + k mcf ((m i + m a)n) k mi m i ; m a M macro )rc k cc; k ma m a; n; v density of dead and live cells, m i ; m a density of activated microglia and macrophages, c concentration of the cytokines. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

48 Model of Ischemic Stroke A model of inammation during ischemic stroke (II) in collaboration with G.Chapuisat, M.A. Dronne, V. Calvez, T. tn = k cv cv + k nv tv = k cv cv k nv nv ; k mn(m i + m a + )n; m tm i = k nmn(m i + ) 1 M tm a = D ma m a r ma m tc = D c c + k mcf ((m i + m a)n) k mi m i ; m a M macro )rc k cc; k ma m a; n; v density of dead and live cells, m i ; m a density of activated microglia and macrophages, c concentration of the cytokines. To do Fit parameters with experimental data. Study the phenomenon under dierent conditions and dierent geometries. 3D simulation. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

49 References Di Russo, C. Analysis and Numerical Approximation of Hydrodynamical Models of Biological Movements PhD Thesis, Mar Di Russo, C., Natalini, R. and Ribot, M. Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis. Communications in Applied and Industrial Mathematics, Vol 1, (2010). Di Russo, C. Global existence of smooth solutions and time asymptotic behavior for the Cattaneo-Hillen model of chemotaxis. In preparation. Di Russo, C. and Sepe, A. Global existence and time asymptotic behavior for some quasilinear hyperbolic models of chemotaxis.in preparation. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

50 References Di Russo, C. Analysis and Numerical Approximation of Hydrodynamical Models of Biological Movements PhD Thesis, Mar Di Russo, C., Natalini, R. and Ribot, M. Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis. Communications in Applied and Industrial Mathematics, Vol 1, (2010). Di Russo, C. Global existence of smooth solutions and time asymptotic behavior for the Cattaneo-Hillen model of chemotaxis. In preparation. Di Russo, C. and Sepe, A. Global existence and time asymptotic behavior for some quasilinear hyperbolic models of chemotaxis.in preparation. Di Russo, C., Lagaert, J-B., Chapuisat, G. and Dronne, M-A. A mathematical model of inammation during ischemic stroke. ESAIM: Proceedings, CEMRACS 2009: Mathematical Modelling in Medicine, Vol. 30 (2010). C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

51 References Di Russo, C. Analysis and Numerical Approximation of Hydrodynamical Models of Biological Movements PhD Thesis, Mar Di Russo, C., Natalini, R. and Ribot, M. Global existence of smooth solutions to a two-dimensional hyperbolic model of chemotaxis. Communications in Applied and Industrial Mathematics, Vol 1, (2010). Di Russo, C. Global existence of smooth solutions and time asymptotic behavior for the Cattaneo-Hillen model of chemotaxis. In preparation. Di Russo, C. and Sepe, A. Global existence and time asymptotic behavior for some quasilinear hyperbolic models of chemotaxis.in preparation. Di Russo, C., Lagaert, J-B., Chapuisat, G. and Dronne, M-A. A mathematical model of inammation during ischemic stroke. ESAIM: Proceedings, CEMRACS 2009: Mathematical Modelling in Medicine, Vol. 30 (2010). Clarelli, F., Di Russo, C., Natalini, R. and Ribot, M. Mathematical models for biolms on the surface of monuments. APPLIED AND INDUSTRIAL MATHEMATICS IN ITALY III, proceedings of SIMAI Conference Clarelli, F., Di Russo, C., Natalini, R. and Ribot, M. A uid-dynamic model for the growth of phototrophic biolms. Submitted. Clarelli, F., Di Russo, C., Natalini, R. and Ribot, M. Multidimensional biolm growth: inuence of light and temperature. In preparation. C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

52 Thank you for your attention! :) C.Di Russo (Univ. Lyon 1) Biological Models Jun 30, / 24

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