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Table of Contents
Intro; Preface; Contents; 1 Functional Spaces; 1.1 Sobolev Spaces and Embeddings Theorem; 1.2 Poincaré, Wirtinger, and Friedrichs Inequalities; 1.3 Nemytski Operators; 1.4 Subdifferential; Monotonicity of Subdifferentials; Subdifferentials and PDEs; Subdifferential in Hilbert Spaces; References; 2 Biological Background; 2.1 The Mitochondrion; Structure; 2.2 Apoptosis; Pathways; 2.3 Mitochondrial Permeability Transition (MPT); Permeability Transition Pore (PTP); Ca2+ Release; Pharmaceutical Background; Experimental Procedure; References; 3 Model Description; 3.1 Existing Models; Microscale.
MacroscaleFirst Order Kinetics; Several Steps of Calcium Uptake; Second Order Kinetics; 3.2 Spatial Effects; In Vitro Swelling; 3.3 The Mitochondria Model: In Vitro; The Variables; Initial Conditions; Boundary Conditions; Model Function f; Model Function g; Calcium Evolution; References; 4 Mathematical Analysis of Vitro Models; 4.1 Neumann Boundary Conditions; Existence and Uniqueness of Global Solutions; Asymptotic Behavior of Solutions; 4.2 Classification of Partial and Complete Swelling; Convergence Rate; 4.3 Numerical Analysis: In Vitro; The In Vitro Model; Model Parameters.
DiscretizationNumerical Approximation; Initial Values; Simulation; Comparison with Experimental Data; Conclusion; 4.4 Dirichlet Boundary Conditions; Hopf's Maximum Principle; Corresponding Eigenvalue Problem; Wirtinger's Inequality; 4.5 Numerical Simulation; Discussion and Conclusion; References; 5 The Swelling of Mitochondria: In Vivo; 5.1 Increase of Intracellular Ca2+; 5.2 The Cell Membrane; 5.3 Summary; 5.4 Mathematical Analysis of an Vivo Model of Mitochondria Swelling; Well Posedness and Asymptotic Behavior of Solutions; Uniform Convergence of u; Partial Swelling; Complete Swelling.
5.5 Numerical IllustrationsConclusion; References; 6 The Swelling of Mitochondria: Degenerate Diffusion; 6.1 Mathematical Analysis of Mitochondria Swelling in Porous Medium; Finite Propagation Speed; Degeneracy; Well Posedness and Asymptotic Behavior of Solutions; 6.2 Numerical Simulation; References; 7 The Spatial Evolution of Mitochondria: PDE-PDE Systems; 7.1 Well Posedness; 7.2 Asymptotic Behavior of Solutions; 7.3 Numerical Simulations; Conclusion; References.
MacroscaleFirst Order Kinetics; Several Steps of Calcium Uptake; Second Order Kinetics; 3.2 Spatial Effects; In Vitro Swelling; 3.3 The Mitochondria Model: In Vitro; The Variables; Initial Conditions; Boundary Conditions; Model Function f; Model Function g; Calcium Evolution; References; 4 Mathematical Analysis of Vitro Models; 4.1 Neumann Boundary Conditions; Existence and Uniqueness of Global Solutions; Asymptotic Behavior of Solutions; 4.2 Classification of Partial and Complete Swelling; Convergence Rate; 4.3 Numerical Analysis: In Vitro; The In Vitro Model; Model Parameters.
DiscretizationNumerical Approximation; Initial Values; Simulation; Comparison with Experimental Data; Conclusion; 4.4 Dirichlet Boundary Conditions; Hopf's Maximum Principle; Corresponding Eigenvalue Problem; Wirtinger's Inequality; 4.5 Numerical Simulation; Discussion and Conclusion; References; 5 The Swelling of Mitochondria: In Vivo; 5.1 Increase of Intracellular Ca2+; 5.2 The Cell Membrane; 5.3 Summary; 5.4 Mathematical Analysis of an Vivo Model of Mitochondria Swelling; Well Posedness and Asymptotic Behavior of Solutions; Uniform Convergence of u; Partial Swelling; Complete Swelling.
5.5 Numerical IllustrationsConclusion; References; 6 The Swelling of Mitochondria: Degenerate Diffusion; 6.1 Mathematical Analysis of Mitochondria Swelling in Porous Medium; Finite Propagation Speed; Degeneracy; Well Posedness and Asymptotic Behavior of Solutions; 6.2 Numerical Simulation; References; 7 The Spatial Evolution of Mitochondria: PDE-PDE Systems; 7.1 Well Posedness; 7.2 Asymptotic Behavior of Solutions; 7.3 Numerical Simulations; Conclusion; References.