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2012 | Buch

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Modeling of Physiological Flows

herausgegeben von: Davide Ambrosi, Alfio Quarteroni, Gianluigi Rozza

Verlag: Springer Milan

Buchreihe : MS&A

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book offers a mathematical update of the state of the art of the research in the field of mathematical and numerical models of the circulatory system. It is structured into different chapters, written by outstanding experts in the field. Many fundamental issues are considered, such as: the mathematical representation of vascular geometries extracted from medical images, modelling blood rheology and the complex multilayer structure of the vascular tissue, and its possible pathologies, the mechanical and chemical interaction between blood and vascular walls, and the different scales coupling local and systemic dynamics. All of these topics introduce challenging mathematical and numerical problems, demanding for advanced analysis and efficient simulation techniques, and pay constant attention to applications of relevant clinical interest. This book is addressed to graduate students and researchers in the field of bioengineering, applied mathematics and medicine, wishing to engage themselves in the fascinating task of modeling the cardiovascular system or, more broadly, physiological flows.

Inhaltsverzeichnis

Frontmatter

1. Assumptions in modelling of large artery hemodynamics

Abstract

The last decade has seen tremendous growth in the use of computational methods for simulating large artery hemodynamics. As computational models become more sophisticated and their applications more varied, it is worth (re)considering the simplifying assumptions that are traditionally, and often implicitly, made. This chapter reviews some of the common assumptions about the constitutive properties of the arteries and the blood within, and their potential impact on the computed hemodynamics. It will be seen, for example, that the assumption of rigid walls, while reasonable and expedient, may be questionable for extensive domains and/or heterogeneities in the arterial wall structure and properties, and that this has implications for the way in which prevailing flow conditions are imposed. Simplifying assumptions about the properties of blood are undoubtedly necessary, but the Newtonian/non-Newtonian dichotomy may prove too simplistic, especially as simulations move from laminar flows to unstable and turbulent flows. Rather than dwelling upon the potential limitations arising from these assumptions, this chapter attempts to highlight some of the potentially interesting research opportunities that may arise in investigating and overcoming them.

David A. Steinman

Simplified blood flow model with discontinuous vessel properties: Analysis and exact solutions

Abstract

We formulate a simplified one-dimensional time-dependent non-linear mathematical model for blood flow in vessels with discontinuous material properties. The resulting 3 × 3 hyperbolic system is analysed and the associated Riemann problem is solved exactly, including tube collapse. Our exact solutions constitute useful reference solutions for assessing the performance of numerical methods intended for simulating more general situations. In addition the presented model may be a useful starting point for numerical calculations involving rapid and discontinuous material properties variations.

Eleuterio F. Toro, Annunziato Siviglia

Blood coagulation: A puzzle for biologists, a maze for mathematicians

Abstract

We present a concise summary of mathematical models for the formation and dissolution of blood clots (in other words for the process of hemostasis). For lack of space we restrict our attention to very few models, selected from a very large literature, trying to emphasize the variety of methods and viewpoints. A peculiar aspect concerning hemostasis is the fact that a new interpretation of its extremely complex biological mechanism has been found rather recently, so that most of the mathematical models should be revisited. Also in view of this fact we believed that it was absolutely necessary to write an extensive introduction to the various aspects of hemostasis, including some history, and not disregarding a description of bleeding disorders (another large field of investigation for mathematical modelling), from which much has been learned about the role and importance of each of the numerous elements intervening in hemostasis. We realize that our work is necessarily incomplete. Indeed, our conclusion is that mathematicians are still in front of the huge task of keeping up with the developments of the medical theory and of the therapeutical practice of this multifaceted subject.

Antonio Fasano, Rafael F. Santos, Adélia Sequeira

Numerical simulation of electrocardiograms

Abstract

This chapter presents a concise overview of various mathematical and numerical problems raised by the simulation of electrocardiograms (ECGs). A model for the propagation of the electrical activation in the heart and in the torso is proposed. Some of its mathematical properties are analyzed. This model is not aimed at reproducing the complex phenomena taking place at the microscopic level. It has been devised to produce realistic healthy ECGs, and some pathological ones, with a reasonable level of complexity. Rather, it relies on various assumptions that are carefully discussed through their impact on the ECGs. The coupling between the heart and the torso is a critical numerical issue which is addressed. In particular, efficient coupling strategies based on explicit algorithms are presented and analyzed. The chapter ends with some preliminary results of a reduced order model based on the Proper Orthogonal Decomposition (POD) method.

Muriel Boulakia, Miguel A. Fernández, Jean-Frédéric Gerbeau, Nejib Zemzemi

Mathematical and numerical methods for reaction-diffusion models in electrocardiology

Abstract

This paper presents a review of current mathematical and numerical models of the bioelectrical activity in the ventricular myocardium, describing cardiac cells excitability and the action-potential propagation in cardiac tissue. The degenerate reaction-diffusion system called the Bidomain model is introduced and interpreted as macroscopic averaging of a cellular model on a periodic assembling of myocytes. The main theoretical results for the cellular and Bidomain models are given. Various approximate models based on some relaxed approaches are also considered, such as Monodomain and eikonal-curvature models. The main numerical methods for the Bidomain and Monodomain models are then reviewed. In particular, we focus on isoparametric finite elements, semi-implicit time discretizations and a parallel iterative solver based on a multilevel Schwarz preconditioned conjugate gradient method. The Bidomain solver is finally applied to the study of the excitation processes generated by virtual electrode response in 3D orthotropic blocks of myocardial tissue.

Piero Colli-Franzone, Luca F. Pavarino, Simone Scacchi

Structurally motivated damage models for arterial walls. Theory and application

Abstract

The mechanical integrity of the arterial wall is vital for the health of the individual. This integrity is in turn dependent on the state of the central load bearing components of the wall: collagen fibres, elastic fibres and smooth muscle. Of these, the elastic fibres, composed largely of the protein elastin, are viewed as responsible for the highly elastic behaviour of the wall at low loads [92]. The collagen fibres are recruited under increasing extension, leading to a highly nonlinear behaviour of the arterial wall [117]. They are responsible for the structural integrity of the wall at elevated physiological loads. Changes in the quantity, distribution, orientation and mechanical properties of these components (the microstructure) are known to occur as part of a healthy response to changing stimuli (e.g. growth and remodelling) as well as during pathological and damage processes in disease and aging. For example, degradation of the elastic fibres is linked to pathological conditions including cerebral aneurysms [12, 15, 20, 65], dissection aneurysms [101], arteriosclerosis [11, 44, 86, 113, 114], and complications from balloon angioplasty [84]. Age related arterial stiffening is attributed to degradation of the elastic fibres, possibly from fatigue failure [11, 30]. The subject of arterial damage is addressed in Sect. 6.4.

Anne M. Robertson, Michael R. Hill, Dalong Li

Arterial growth and remodelling is driven by hemodynamics

Abstract

Experimental observations highlight the importance of altered hemodynamics on arterial function and adaptation [27, 28, 29]. We discuss a class of mechano-biological models for growth and remodelling (G&R) of the arterial wall that describe the intimate interaction between hemodynamics, cell activity, and arterial wall mechanics. For some applications the artery can be described as a thin walled structure: for example, basic adaptations to perturbed pressure and flow, cerebral aneurysms, and vasospasms have been successfully modelled treating the vascular wall as a membrane. A multiple-time scales membrane model is described and illustrative results discussed. Future patient-specific models of large arteries and pathologies as atherosclerosis and abdominal aortic aneurysms require a full 3D model of the interaction between the blood flow and the growing vessel. We discuss the extension of the model to thick walled vessels and some preliminary results.

Luca Cardamone, Jay D. Humphrey

The VPH-Physiome Project: Standards, tools and databases for multi-scale physiological modelling

Abstract

The VPH/Physiome project is developing tools and model databases for computational physiology based on three primary model encoding standards: CellML, SBML and FieldML. For the modelling community these standards are the equivalent of the DICOM standard for the clinical imaging community and it is important that the tools adhere to these standards to ensure that models from different groups can be curated, annotated, reused and combined. This chapter discusses the development and use of the VPH/Physiome standards, tools and databases, and also discusses the minimum information standards and ontology-based metadata standards that are complementary to the markup language standards. Data standards are not as well developed as the model encoding standards (with the DICOM standard for medical image encoding being the outstanding exception) but one new data standard being developed as part of the VPH/Physiome suite is BioSignalML and this is described here also. The PMR2 (Physiome Model Repository 2) database for CellML and FieldML files is also described, together with the Application Programming Interfaces (APIs) that facilitate access to the models from the visualization (cmgui and GIMIAS) or computational (OpenCMISS, OpenCell/OpenCOR and other) software.

Peter Hunter, Chris Bradley, Randall Britten, David Brooks, Luigi Carotenuto, Richard Christie, Alejandro Frangi, Alan Garny, David Ladd, Caton Little, David Nickerson, Poul Nielsen, Andrew Miller, Xavier Planes, Martin Steghoffer, Alistair Young, Tommy Yu

The role of the variational formulation in the dimensionally-heterogeneous modelling of the human cardiovascular system

Abstract

The modelling of the cardiovascular system entails dealing with different phenomena pertaining to different time, constitutive and geometrical scales. Specifically, the problem of integrating various geometrical scales can be understood from a kinematical point of view, which means to integrate models with different kinematics, and in particular different dimensionality. In this context, all the variational machinery can be employed to derive consistent variational formulations according to the underlying kinematical hypotheses that rule over the corresponding models. In this work we discuss the application of variational formulations to model the blood flow in the cardiovascular system making use of heterogeneous representations. Two examples of applications are used to show the capabilities and potentialities of the present approach.

Pablo J. Blanco, Raúl A. Feijóo

Multiscale modelling of hematologic disorders

Abstract

Parasitic infectious diseases and other hereditary hematologic disorders are often associated with major changes in the shape and viscoelastic properties of red blood cells (RBCs). Such changes can disrupt blood flow and even brain per-fusion, as in the case of cerebral malaria. Modelling of these hematologic disorders requires a seamless multiscale approach, where blood cells and blood flow in the entire arterial tree are represented accurately using physiologically consistent parameters. In this chapter, we present a computational methodology based on dissipative particle dynamics (DPD) which models RBCs as well as whole blood in health and disease. DPD is a Lagrangian method that can be derived from systematic coarse-graining of molecular dynamics but can scale efficiently up to small arteries and can also be used to model RBCs down to spectrin level. To this end, we present two complementary mathematical models for RBCs and describe a systematic procedure on extracting the relevant input parameters from optical tweezers and microfluidic experiments for single RBCs. We then use these validated RBC models to predict the behaviour of whole healthy blood and compare with experimental results. The same procedure is applied to modelling malaria, and results for infected single RBCs and whole blood are presented.

Dmitry Fedosov, Igor Pivkin, Wenxiao Pan, Ming Dao, Bruce Caswell, George E. Karniadakis

Multiscale computational analysis of degradable polymers

Abstract

Degradable materials have found a wide variety of applications in the biomedical field ranging from sutures, pins and screws for orthopedic surgery, local drug delivery, tissue engineering scaffolds, and endovascular stents. Polymer degradation is the irreversible chain scission process that breaks polymer chains down to oligomers and, finally, to monomers. These changes, which take place at the molecular scale, propagate through the space/time scales and not only affect the capacity of the polymer to release drugs, bu also hamper the overall mechanical behaviour of the device, whose spatial scale is denoted as macroscale. A bottom-up multiscale analysis is applied to model the degradation mechanism which takes place in PLA matrices. The macroscale model is based on diffusion-reaction equations for hydrolytic polymer degradation and erosion while the microscale model is based on atomistic simulations to predict the water diffusion as a function of the swelling degree of the PLA matrix. The diffusion coefficients are then passed to the macroscale model. In conclusion, the proposed multiscale analysis is capable to predict the evolution with time of several properties of water/PLA mixtures, according to the change of relevant indicators such as the extent of degradation and erosion of the PLA matrix.

Paolo Zunino, Simone Vesentini, Azzurra Porpora, Joao S. Soares, Alfonso Gautieri, Alberto Redaelli

Applications of variational data assimilation in computational hemodynamics

Abstract

The development of new technologies for acquiring measures and images in order to investigate cardiovascular diseases raises new challenges in scientific computing. These data can be in fact merged with the numerical simulations for improving the accuracy and reliability of the computational tools. Assimilation of measured data and numerical models is well established in meteorology, whilst it is relatively new in computational hemodynamics. Different approaches are possible for the mathematical setting of this problem. Among them, we follow here a variational formulation, based on the minimization of the mismatch between data and numerical results by acting on a suitable set of control variables. Several modeling and methodological problems related to this strategy are open, such as the analysis of the impact of the noise affecting the data, and the design of effective numerical solvers. In this chapter we present three examples where a mathematically sound (variational) assimilation of data can significantly improve the reliability of the numerical models. Accuracy and reliability of computational models are increasingly important features in view of the progressive adoption of numerical tools in the design of new therapies and, more in general, in the decision making process of medical doctors.

Marta D’Elia, Lucia Mirabella, Tiziano Passerini, Mauro Perego, Marina Piccinelli, Christian Vergara, Alessandro Veneziani

Quality open source mesh generation for cardiovascular flow simulations

Abstract

We present efficient algorithms for generating quality tetrahedral meshes for cardiovascular blood flow simulations starting from low quality triangulations obtained from the segmentation of patient specific medical images. The suite of algorithms that are presented in this paper have been implemented in the open-source mesh generator Gmsh [19]. This includes a high quality remeshing algorithm based on a finite element conformal parametrization and a volume meshing algorithm with a boundary layer generation technique. In the result section, we show that the presence of a boundary layer mesh plays an important role to reduce the problem size in cardiovascular flow simulations.

Emilie Marchandise, Paolo Crosetto, Christophe Geuzaine, Jean-François Remacle, Emilie Sauvage

Backmatter

Titel: Modeling of Physiological Flows
herausgegeben von: Davide Ambrosi
Alfio Quarteroni
Gianluigi Rozza
Verlag: Springer Milan
Electronic ISBN: 978-88-470-1935-5
Print ISBN: 978-88-470-1934-8
DOI: https://doi.org/10.1007/978-88-470-1935-5