This book presents the state of the art in mathematical research on modelling the mechanics of biological systems – a science at the intersection between biology, mechanics and mathematics known as mechanobiology. The book gathers comprehensive surveys of the most significant areas of mechanobiology: cell motility and locomotion by shape control (Antonio DeSimone); models of cell motion and tissue growth (Benoît Perthame); numerical simulation of cardiac electromechanics (Alfio Quarteroni); and power-stroke-driven muscle contraction (Lev Truskinovsky).
Each section is self-contained in terms of the biomechanical background, and the content is accessible to all readers with a basic understanding of differential equations and numerical analysis. The book disentangles the phenomenological complexity of the biomechanical problems, while at the same time addressing the mathematical complexity with invaluable clarity. The book is intended for a wide audience, in particular graduate students and applied mathematicians interested in entering this fascinating field.
Author(s): Antonio DeSimone, Benoît Perthame, Alfio Quarteroni, Lev Truskinovsky, Davide Ambrosi (editor), Pasquale Ciarletta (editor)
Series: Lecture Notes in Mathematics 2260
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 219
Preface
Abstract
Contents
1 Cell Motility and Locomotion by Shape Control
1.1 Introduction
1.2 Swimming at Low Reynolds Numbers
1.3 Locomotion Principles and Minimal Swimmers
1.3.1 Looping in the Space of Shapes: No Looping? No Party!
1.3.2 Minimal Swimmers With or Without Directional Control
1.3.3 Steering by Modulation of the Actuation Speed
1.3.4 Swimming by Lateral Undulations: Optimality of Traveling Waves of Bending
1.4 Biological Swimmers
1.4.1 Chlamydomonas' Breaststroke
1.4.2 Sperm Cells and Flagellar Beat
1.5 Euglena Gracilis: A Case Study in Biophysics and a Journey from Biology to Technology
1.5.1 Metaboly and Mechanisms for Shape Change, Embodied Intelligence
1.5.2 Flagellar Swimming, Helical Trajectories and a Principle for Self-Assembly
1.6 Shape Control and Gaussian Morphing
1.6.1 Controlling the Shape of Surfaces by Prescribing Their Metric
1.6.2 Axisymmetric Surfaces
1.6.3 Cylinders from Cylinders
1.6.4 Axisymmetric Surfaces with Non-constant Metric
1.6.5 Protruding Necks and Localized Bulges
1.7 Discussion and Outlook
References
2 Models of Cell Motion and Tissue Growth
2.1 Introduction
2.2 Bacterial Movement by Run and Tumble
2.2.1 Modeling Run and Tumble
2.2.2 Existence of Solutions
2.2.3 Derivation of the Patlak/Keller–Segel System
2.2.4 Modulation Along the Path
2.3 Macroscopic Models of Chemotactic Movement
2.3.1 Elementary Properties
2.3.2 Blow-Up in the Keller–Segel System
2.3.3 Keller–Segel System with Prevention of Overcrowding
2.3.4 The Flux Limited Keller–Segel System
2.3.5 Traveling Bands
2.3.6 Instabilities
2.4 Compressible Models of Tissue Growth
2.4.1 A Simple Model with a Single Type of Cells
2.4.1.1 Supersolution with Bounded Support
2.4.1.2 Existence of Solutions and a Priori Bounds
2.4.1.3 A Variant of Aronson-Bénilan Estimate
2.4.2 Single Cell Type Population Model with Nutrient
2.4.3 Models with Two Cell Types
2.4.4 Two Cell Type Model with Different Mobilities
2.4.5 Surface Tension and the Degenerate Cahn–Hilliard Model
2.5 Incompressible Models of Tissue Growth
2.5.1 Single Cell Type Free Boundary Problem
2.5.2 Single Cell Type Model with Nutrient and Free Boundary
2.5.3 Two Cell Types Incompressible Model
2.5.4 Multiphase Models
2.5.4.1 The One Phase Closure
2.5.4.2 The Darcy/Stokes Closure
2.5.4.3 The Single Pressure Closure
2.6 The Incompressible Limit and Stiff Pressure Law
2.6.1 Single Cell Type Model, Incompressible Limit
2.6.1.1 Weak Formulation of the Hele-Shaw Problem
2.6.1.2 The Complementary Relation
2.6.1.3 From the Weak Formulation to the Free Boundary Statement
2.6.2 Single Cell Type with Nutrient, Incompressible Limit
2.6.3 Open Problems
References
3 Segregated Algorithms for the Numerical Simulation of Cardiac Electromechanics in the Left Human Ventricle
3.1 Introduction
3.2 Mathematical Models
3.2.1 Ionic Model and Monodomain Equation
3.2.2 Mechanical Activation
3.2.3 Passive and Active Mechanics
3.2.3.1 Prestress
3.2.4 Cardiac Cycle
3.3 Space and Time Discretizations
3.3.1 Space Discretization
3.3.2 Time Discretization
3.3.2.1 Discretization of the 0D Fluid Model
3.4 Numerical Coupling: Segregated Strategies
3.4.1 Fully Monolithic Strategy (IIEIAIMI)
3.4.2 Partially Segregated Strategy (IIEIAI)–(MI)
3.4.3 Partially Segregated Strategy (ISIESIASI)–(MI)
3.4.4 Fully Segregated Strategy (ISI)–(ESI)–(ASI)–(MI)
3.5 Numerical Results
3.5.1 Preprocessing
3.5.2 Benchmark Problem with Idealized Geometry
3.5.3 Subject-Specific LV: The Full Heartbeat
3.6 Conclusions
References
4 Power-Stroke-Driven Muscle Contraction
4.1 Introduction
4.2 General Ratchet Model
4.3 X-Tilted Ratchet
4.3.1 Typical Cycles
4.3.2 Force-Velocity Relations and Stochastic Energetics
4.4 Y-Tilted Ratchet
4.4.1 Typical Cycles
4.4.2 Force-Velocity Relations and Stochastic Energetics
4.5 XY-Tilted Ratchet
4.5.1 Motor Cycles
4.5.2 Force-Velocity Relations and Stochastic Energetics
4.6 Comparison of the Three Models
4.6.1 Soft Device
4.6.2 Hard Device
4.6.3 Stochastic Energetics
4.7 XY-Tilted Ratchet with a Steric Feedback
4.7.1 The Model
4.7.2 Hysteretic Coupling
4.7.3 Non-potential Models
4.8 Active Rigidity
4.8.1 Macroscopic Problem
4.8.2 Mean Field Model
4.8.3 Non-dimensionalization
4.8.4 Phase Diagrams
4.8.5 Zero Temperature Limit
4.9 Conclusions
References
