Numerical approximation and analysis of mathematical models arising in cells movement Monika Twarogowska To cite this version: Monika Twarogowska. Numerical approximation and analysis of mathematical models arising in cells movement. Modeling and Simulation. Università degli studi de l’Aquila, 2012. English. tel-00804264 HAL Id: tel-00804264 https://tel.archives-ouvertes.fr/tel-00804264 Submitted on 25 Mar 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITA` DEGLI STUDI DELL’AQUILA Dipartimento di Matematica Pura e Applicata Dottorato in Matematica - XXIV Ciclo Tesi di Dottorato in Matematica Settore Scientifico Disciplinare: MAT/05 Numerical approximation and analysis of mathematical models arising in cells movement Relatori Candidato Prof. Roberto Natalini Monika Twarogowska Dr. Magali Ribot Coordinatore del corso di Dottorato Prof. Anna De Masi ANNO ACCADEMICO 2010/2011 I hereby declare that this Thesis is my own work and effort. Where other sources of information have been used, they have been acknowledged. Acknowledgments Looking back over these three years of the PhD studies I see moments which were full of joy as well as those tough, when everything seemed to be against me and despondency ap- peared. I am very grateful for all of them. They certainly shaped me as a person I am today. One of this joyful moments is now, when I can thank all those without whom this thesis would not be possible. I owe my deepest gratitude to the supervisor of my thesis Roberto Natalini, who took care of my PhD research at a difficult moment time, after the earthquake in L’Aquila. I am thankful for his time, ideas and support that were irreplaceable, for introducing me to the field of numerical analysis and for showing interesting and challenging topics. In addition, I would like to thank for his patience, guidance and dedication in the preparation of this thesis. Through all these years, the joy and enthusiasm he has for Mathematics has been motivational for me. I would also like to express my sincere thanks to Magali Ribot, who became my co-advisor. Her presence and contribution to my research enriched my experience and understanding of numerical methods and implementation of algorithms. Moreover, her openness, kindness and support made the time I spent in France unforgettable. I am heartily thankful to Bruno Rubino for his help, care and encouragement since my first day in L’Aquila. I am really grateful for the support I have received from him during these years and for the ceaseless sensation that in any difficulty I can turn to him. It is so comforting while being far from my family. I wish to thank Federica Di Michele, my room mate for two years and companion at University, for a great friendship and being always available to help, give advice or just listen. I am grateful for her understanding and encouragement, for fruitful discussions, for making time in L’Aquila more interesting, coffees more tasty and running more enjoyable. I devote special thanks to my best friends in Poland. Especially to Joanna Kaminska´ and Weronika Pelc-Garska for irreplaceable encouragement and company in the moments of home- sickness. I would also like to thank Prof. P. Marcati and Marco Di Francesco for introducing me to the field of mathematical modelling and for all the support during my PhD studies. I thank the people in the Laboratoire J. A. Dieudonne´ at Universite´ de Nice-Sophia Antipo- lis for hospitality and for giving me the possibility to broaden my knowledge in the field of numerical analysis. In addition, I would like to thank Christophe Berthon for an invitation to Laboratoire de Mathematiques´ Jean Leray at Universite´ de Nantes and for very interesting and fruitful discussions. I express also my gratitude to Prof. P. Marcati , Prof. L.C. Berselli and Prof. D. Benedetto for accepting to be members of my thesis committee. My time in L’Aquila was made enjoyable in the large part due to many friends that be- come a part of my life. I am grateful for time spent with Federica Di Michele, Bruno Rubino, Donatella Donatelli and Danilo Larivera. It was really a pleasure to meet Vincenzo Caracci- olo, Nicola De Rossi and all those with whom I spent amazing moments while skiing on Campo Imperatore and exploring Abruzzo on bicycle. This thesis would have not been possible unless a constant support of my family, espe- cially my mother and my sister. I would like to thank for their love and encouragement. Despite the distance, their presence in my life has been always significant and moments spent together unforgeable, and for that I am grateful. If not their help, moral support and criticism from time to time arriving to the point where I am now is doubtful. Monika Twarogowska To my Mother and my Sister Contents Introduction I 1 Mathematical modelling of cells movement 1 1.1 Biological background . 2 1.2 Macroscopic modelling: mixture theory . 4 1.3 Parabolic models of chemotaxis . 6 1.4 Hyperbolic models of chemotaxis . 8 2 Asymptotic stability of constant stationary states for a 2 × 2 reaction-diffusion system arising in cancer modelling 11 2.1 Introduction . 11 2.2 Preliminaries and results . 15 2.2.1 Constant stationary states . 15 2.2.2 Results . 17 2.3 Stability of a linearised model . 18 2.4 Proof of the main theorem . 22 2.5 Conclusion . 26 3 Background on numerical methods 27 3.1 Introduction . 27 3.2 Finite differences methods . 29 3.2.1 Conservative schemes . 30 3.2.2 High resolution schemes: the flux-limiter approach . 32 3.2.3 θ-method . 34 3.2.4 Advanced explicit schemes based of the relaxation technique . 34 3.3 Systems of conservation laws . 36 3.3.1 Extension of scalar 3-point schemes to systems . 38 3.3.2 Isentropic gas dynamics . 40 3.3.3 Godunov type schemes: approximate Riemann solvers . 43 3.4 Approximation of the source term . 53 3.4.1 Upwinding the sources . 53 3.4.2 Well-balancing . 55 I 3.5 Boundary conditions . 62 4 Numerical approximation: parabolic models with nonlinear diffusion and chemo- taxis 65 4.1 Introduction . 65 4.2 Numerical schemes . 66 4.2.1 Splitting schemes . 66 4.2.2 Explicit scheme . 70 4.3 Simulations . 73 4.3.1 Porous medium equation . 73 4.3.2 Angiogenesis model . 75 4.3.3 Keller-Segel type model for chemotaxis with nonlinear diffusion of the porous medium type . 77 5 Numerical study of 1D Euler equations for isentropic gas dynamics with damping 86 5.1 Introduction . 86 5.2 Existing mathematical theory. 87 5.2.1 Waiting time phenomenon . 88 5.2.2 Physical boundary condition . 91 5.3 Numerical schemes . 95 5.3.1 Detection of the waiting time . 96 5.4 Numerical simulations . 100 6 Numerical approximation: some hyperbolic models of chemotaxis 116 6.1 Introduction . 116 6.2 Failure of the standard, explicit scheme . 118 6.2.1 Semilinear model . 118 6.2.2 Quasilinear model . 119 6.3 Well-balanced scheme . 125 7 Numerical study of some hyperbolic models of chemotaxis in 1D 135 7.1 Introduction . 135 7.2 Stationary solutions . 137 7.2.1 Analytic results . 138 7.2.2 Simulations . 150 7.3 Linear vs. Nonlinear pressure . 165 7.4 Parabolic vs. Hyperbolic model . 170 7.4.1 Logistic Sensitivity function and linear pressure γ = 1 . 171 7.4.2 Constant sensitivity function and quadratic pressure γ = 2 . 173 7.5 Discussion . 174 Bibliography 178 Introduction The aim of this thesis is to investigate some mathematical models arising in cells move- ment, from both analytical and numerical point of view. Migration of cells in the fibrous environment is an essential feature of normal and patho- logical, biological phenomena such as embryonic morphogenesis, wound healing, angiogenesis or tumour invasion. In the simplest case we can consider a tissue as a tridimensional structure consisting of fibres (Extracellular matrix ECM) and cells attached to them. Many experimental studies (see for instance [42]) give a good insight into how the cells move. It is known that they interact with the tissue matrix as well as with other cells using different biological and physical mechanisms. The forces driving the cells motion are generated by controlled remodelling of the actin network within the cell as a response to mechanical and chemical signals. The movement itself is strongly influenced by the spatial and temporal configuration of the fibres. Moreover, it depends on the contact-guidance phenomenon and attached-detached processes. Also their behaviour can be altered by the presence of some stimulus giving rise to the so called taxis movements. Basically it means to change the direction, in which the cell moves, for example, according to the gradient of some quantities such as adhesion forces (haptotaxis), electric field (galvanotaxis), chemical substances (chemotaxis), oxygen concentration (aerotaxis). In this thesis we focus our attention to the chemotaxis that is a directed movement of mobile species towards the lower/higher concentration of a chemical substance in the surrounding environ- ment. Since the first description of chemotaxis in bacteria by T.W.Engelmann (1881) and W.F.Pfeffer (1884), our knowledge about this phenomenon has been continuously expanding.