Mathematical modeling of human physiopathology is a tremendously ambitious task. It encompasses the modeling of most diverse compartments such as the cardiovascular, respiratory, skeletal and nervous systems, as well as the mechanical and biochemical interaction between blood flow and arterial walls, and electrocardiac processes and electric conduction in biological tissues. Mathematical models can be set up to simulate both vasculogenesis (the aggregation and organization of endothelial cells dispersed in a given environment) and angiogenesis (the formation of new vessels sprouting from an existing vessel) that are relevant to the formation of vascular networks, and in particular to the description of tumor growth.
The integration of models aimed at simulating the cooperation and interrelation of different systems is an even more difficult task. It calls for the setting up of, for instance, interaction models for the integrated cardio-vascular system and the interplay between the central circulation and peripheral compartments, models for the mid-to-long range cardiovascular adjustments to pathological conditions (e.g., to account for surgical interventions, congenital malformations, or tumor growth), models for integration among circulation, tissue perfusion, biochemical and thermal regulation, models for parameter identification and sensitivity analysis to parameter changes or data uncertainty – and many others.
The heart is a complex system in itself, where electrical phenomena are functionally related to wall deformation. In its turn, electrical activity is related to heart physiology. It involves nonlinear reaction-diffusion processes and provides the activation stimulus to heart dynamics and eventually the blood ventricular flow that drives the haemodynamics of the whole circulatory system. In fact, the influence is reciprocal, since the circulatory system in turn affects heart dynamics and may induce an overload depending upon the individual physiopathologies (for instance, the presence of a stenotic artery or a vascular prosthesis).
Virtually all the fields of mathematics have a role to play in this context. Geometry and approximation theory provide the tools for handling clinical data acquired by tomography or magnetic resonance, identifying meaningful geometrical patterns and producing three-dimensional geometric models stemming from the original patient’s data. Mathematical analysis, fluid and solid dynamics, stochastic analysis are used to set up the differential models and predict uncertainty. Numerical analysis and high performance computing are needed to solve the complex differential models numerically. Finally, methods from stochastic and statistical analysis are exploited for the modeling and interpretation of space-time patterns.
Indeed, the complexity of the problems at hand often stimulates the use of innovative mathematical techniques that are able, for instance, to capture accurately those processes that occur at multiple scales in time and space (such as cellular and systemic effects), and that are governed by heterogeneous physical laws.
In this book, the author have collected the contributions of several Italian research groups that are successfully working in this fascinating and challenging field. Each chapter deals with a specific subfield, with the aim of providing an overview of the subject and an account of the most recent research results.
Contents
Chapter 1 Inverse problems in biomedical imaging: modeling and methods of solution
1 Introduction
1.1 X-ray tomography
1.2 Fluorescence microscopy
2 Linearity
3 Ill-posed problems and uncertainty of solution
4 Noise modeling
4.1 Additive Gaussian noise
4.2 Poisson noise
5 The use of prior information
6 Computational issues and reconstruction methods
7 Perspectives
7.1 Electrical Impedence Tomography
7.2 Optical Tomography
7.3 Microwave Tomography
7.4 Magnetoencephalography
Chapter 2 Stochastic geometry and related statistical problems in biomedicine
1 Introduction
2 Elements of stochastic geometry
2.1 Stochastic geometric measures
3 Hazard function
4 Mean densities of stochastic tessellations
5 Interaction with an underlying field
6 Fibre and surface processes
6.1 Planar fibre processes
7 Estimate of the mean density of length of planar fibre processes
7.1 Local mean density of length and the spherical contact distribution function
7.2 Estimate of the local mean density of length
7.3 Numerical results
7.4 Applications
Chapter 3 Mathematical modelling of tumour growth and treatment
1 Introduction
1.1 Why
1.2 How
1.3 What
2 Models including the analysis of stresses
2.1 Applying the mechanics of mixtures to tumour growth: the “two-fluid” approach
2.2 Vascular tumours: models for vascular collapse
3 About tumour morphology and asymptotic behaviour
3.1 Radially symmetric solutions and their stability under radially symmetric perturbations
3.2 Looking for non-radially symmetric stationary solutions
3.3 The general problem of the stability of radially symmetric solutions
3.4 Asymptotic regimes and vascularisation
4 Models with cell age or cell maturity structure for tumour cords
4.1 Tumour cords
4.2 Age and maturity structured models
5 A tumour cord model including interstitial fluid flow
5.1 Cell populations and cord radius
5.2 Extracellular fluid flow and the necrotic region
6 Modelling tumour treatment
6.1 Spherical tumours
6.2 Tumour cords
6.3 Hyperthermia treatment with geometric model of the patient
7 Conclusions and perspectives
Chapter 4 Modelling the formation of capillaries
1 Vasculogenesis and angiogenesis
2 In vitro vasculogenesis
3 Modelling vasculogenesis
3.1 Diffusion equations for chemical factors
3.2 Persistence equation for the endothelial cells
3.3 Substratum equation
4 In silico vasculogenesis
4.1 Neglecting substratum interactions
4.2 Substratum interactions
4.3 Exogenous control of vascular network formation
5 An angiogenesis model
6 Future perspectives
Chapter 5 Numerical methods for delay models in biomathematics
2 Solving RFDEs by continuous Runge-Kutta methods
2.1 Continuous Runge-Kutta (standard approach) and functional continuous Runge-Kutta methods
3 A threshold model for antibody production: theWaltman model
3.1 The quantitative model
3.2 The integration process
3.3 Tracking the breaking points
3.4 Solving the Runge–Kutta equations
3.5 Local error estimation and stepsize control
3.6 Numerical illustration for theWaltman problem
3.7 Software
4 The functional continuous Runge-Kutta method
4.1 Order conditions
4.2 Explicit methods
4.3 The quadrature problem
Chapter 6 Computational electrocardiology: mathematical and numerical modeling
2 Mathematical models of the bioelectric activity at cellular level
2.1 Ionic current membrane models
2.2 Mathematical models of cardiac cell arrangements
2.3 Formal two-scale homogenization
2.4 Theoretical results for the cellular and averaged models
2.5 Γ -convergence result for the averaged model with FHN dynamics
2.6 Semidiscrete approximation of the bidomain model with FHN dynamics
3 The anisotropic bidomain model
3.1 Boundary integral formulation for ECG simulations
4 Approximate modeling of cardiac bioelectric activity by reduced models
4.1 Linear anisotropic monodomain model
4.2 Eikonal models
4.3 Relaxed nonlinear anisotropic monodomain model
5 Discretization and numerical methods
5.1 Numerical approximation of the Eikonal–Diffusion equation
5.2 Numerical approximations of the monodomain and bidomain models
6 Numerical simulations
7 Conclusions
Chapter 7 The circulatory system: from case studies to mathematical modeling
1 An overview of vascular dynamics and its mathematical features
2 Case studies
2.1 Numerical investigation of arterial pulmonary banding
2.2 Numerical investigation of systemic dynamics
2.3 The design of drug-eluting stents
2.4 Pulmonary and systemic circulation in individuals with congenital heart defects
2.5 Peritoneal dialysis optimization
2.6 Anastomosis shape optimization
3 A wider perspective
Index
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The integration of models aimed at simulating the cooperation and interrelation of different systems is an even more difficult task. It calls for the setting up of, for instance, interaction models for the integrated cardio-vascular system and the interplay between the central circulation and peripheral compartments, models for the mid-to-long range cardiovascular adjustments to pathological conditions (e.g., to account for surgical interventions, congenital malformations, or tumor growth), models for integration among circulation, tissue perfusion, biochemical and thermal regulation, models for parameter identification and sensitivity analysis to parameter changes or data uncertainty – and many others.
The heart is a complex system in itself, where electrical phenomena are functionally related to wall deformation. In its turn, electrical activity is related to heart physiology. It involves nonlinear reaction-diffusion processes and provides the activation stimulus to heart dynamics and eventually the blood ventricular flow that drives the haemodynamics of the whole circulatory system. In fact, the influence is reciprocal, since the circulatory system in turn affects heart dynamics and may induce an overload depending upon the individual physiopathologies (for instance, the presence of a stenotic artery or a vascular prosthesis).
Virtually all the fields of mathematics have a role to play in this context. Geometry and approximation theory provide the tools for handling clinical data acquired by tomography or magnetic resonance, identifying meaningful geometrical patterns and producing three-dimensional geometric models stemming from the original patient’s data. Mathematical analysis, fluid and solid dynamics, stochastic analysis are used to set up the differential models and predict uncertainty. Numerical analysis and high performance computing are needed to solve the complex differential models numerically. Finally, methods from stochastic and statistical analysis are exploited for the modeling and interpretation of space-time patterns.
Indeed, the complexity of the problems at hand often stimulates the use of innovative mathematical techniques that are able, for instance, to capture accurately those processes that occur at multiple scales in time and space (such as cellular and systemic effects), and that are governed by heterogeneous physical laws.
In this book, the author have collected the contributions of several Italian research groups that are successfully working in this fascinating and challenging field. Each chapter deals with a specific subfield, with the aim of providing an overview of the subject and an account of the most recent research results.
Contents Chapter 1 Inverse problems in biomedical imaging: modeling and methods of solution
1 Introduction
1.1 X-ray tomography
1.2 Fluorescence microscopy
2 Linearity
3 Ill-posed problems and uncertainty of solution
4 Noise modeling
4.1 Additive Gaussian noise
4.2 Poisson noise
5 The use of prior information
6 Computational issues and reconstruction methods
7 Perspectives
7.1 Electrical Impedence Tomography
7.2 Optical Tomography
7.3 Microwave Tomography
7.4 Magnetoencephalography
Chapter 2 Stochastic geometry and related statistical problems in biomedicine
1 Introduction
2 Elements of stochastic geometry
2.1 Stochastic geometric measures
3 Hazard function
4 Mean densities of stochastic tessellations
5 Interaction with an underlying field
6 Fibre and surface processes
6.1 Planar fibre processes
7 Estimate of the mean density of length of planar fibre processes
7.1 Local mean density of length and the spherical contact distribution function
7.2 Estimate of the local mean density of length
7.3 Numerical results
7.4 Applications
Chapter 3 Mathematical modelling of tumour growth and treatment
1 Introduction
1.1 Why
1.2 How
1.3 What
2 Models including the analysis of stresses
2.1 Applying the mechanics of mixtures to tumour growth: the “two-fluid” approach
2.2 Vascular tumours: models for vascular collapse
3 About tumour morphology and asymptotic behaviour
3.1 Radially symmetric solutions and their stability under radially symmetric perturbations
3.2 Looking for non-radially symmetric stationary solutions
3.3 The general problem of the stability of radially symmetric solutions
3.4 Asymptotic regimes and vascularisation
4 Models with cell age or cell maturity structure for tumour cords
4.1 Tumour cords
4.2 Age and maturity structured models
5 A tumour cord model including interstitial fluid flow
5.1 Cell populations and cord radius
5.2 Extracellular fluid flow and the necrotic region
6 Modelling tumour treatment
6.1 Spherical tumours
6.2 Tumour cords
6.3 Hyperthermia treatment with geometric model of the patient
7 Conclusions and perspectives
Chapter 4 Modelling the formation of capillaries
1 Vasculogenesis and angiogenesis
2 In vitro vasculogenesis
3 Modelling vasculogenesis
3.1 Diffusion equations for chemical factors
3.2 Persistence equation for the endothelial cells
3.3 Substratum equation
4 In silico vasculogenesis
4.1 Neglecting substratum interactions
4.2 Substratum interactions
4.3 Exogenous control of vascular network formation
5 An angiogenesis model
6 Future perspectives
Chapter 5 Numerical methods for delay models in biomathematics
2 Solving RFDEs by continuous Runge-Kutta methods
2.1 Continuous Runge-Kutta (standard approach) and functional continuous Runge-Kutta methods
3 A threshold model for antibody production: theWaltman model
3.1 The quantitative model
3.2 The integration process
3.3 Tracking the breaking points
3.4 Solving the Runge–Kutta equations
3.5 Local error estimation and stepsize control
3.6 Numerical illustration for theWaltman problem
3.7 Software
4 The functional continuous Runge-Kutta method
4.1 Order conditions
4.2 Explicit methods
4.3 The quadrature problem
Chapter 6 Computational electrocardiology: mathematical and numerical modeling
2 Mathematical models of the bioelectric activity at cellular level
2.1 Ionic current membrane models
2.2 Mathematical models of cardiac cell arrangements
2.3 Formal two-scale homogenization
2.4 Theoretical results for the cellular and averaged models
2.5 Γ -convergence result for the averaged model with FHN dynamics
2.6 Semidiscrete approximation of the bidomain model with FHN dynamics
3 The anisotropic bidomain model
3.1 Boundary integral formulation for ECG simulations
4 Approximate modeling of cardiac bioelectric activity by reduced models
4.1 Linear anisotropic monodomain model
4.2 Eikonal models
4.3 Relaxed nonlinear anisotropic monodomain model
5 Discretization and numerical methods
5.1 Numerical approximation of the Eikonal–Diffusion equation
5.2 Numerical approximations of the monodomain and bidomain models
6 Numerical simulations
7 Conclusions
Chapter 7 The circulatory system: from case studies to mathematical modeling
1 An overview of vascular dynamics and its mathematical features
2 Case studies
2.1 Numerical investigation of arterial pulmonary banding
2.2 Numerical investigation of systemic dynamics
2.3 The design of drug-eluting stents
2.4 Pulmonary and systemic circulation in individuals with congenital heart defects
2.5 Peritoneal dialysis optimization
2.6 Anastomosis shape optimization
3 A wider perspective
Index
Book Details
- Paperback: 306 pages
- Publisher: Springer; Softcover reprint of hardcover 1st edition (December 21, 2010)
- Language: English
- ISBN-10: 8847015553
- ISBN-13: 978-8847015555
- Product Dimensions: 9.2 x 6.1 x 0.7 inches