Handbook of Numerical Analysis. Computational Models for the Human Body

Computational Models for the Human Body constitute an emerging and rapidly progressing area of research whose primary objective is to provide a better understanding of the physiological and mechamcal behavior of the human body and to design tools for their realistic numerical simulations. This volume describes concrete examples of such computational models. Although far from being exhaustive, it covers a large range of methods and an illustrative set of applications, and proposes a number of well-defined mathematical and numencal modeling of physical problems (including the analysis of existence and uniqueness of solutions for instance), followed by various numerical simulations.Medical applications are addressed first, because physiological and biomechanical models of the human body already play a prominent role in the prevention, diagnosis and therapy of many diseases. The generalized introduction of such models hi medicine will hi fact strongly contribute to the development of a more individualized and preventive medicine. In effect, through the continuous progress of medical imaging during the past decades, it is currently possible to extract an increasing flow of anatomical or functional information on any individual, with an increasingly accurate resolution in space and time. The overwhelming quantity of available signals and images makes a direct analysis of the data more and more difficult, when not impossible. New computational models are necessary to capture those parameters that are pertinent to analyze the human system under study or to simulate it. There is also a number of important non-medical applications of these computational models which cover numerous human activities, like driving (safer design of vehicles), working (better ergonomy of workplaces), exercising (more efficient training of athletes), entertaining (simulation for movies), etc.There are basically three levels of design for human models. The first level is mainly geometrical and addresses the construction of a digital description of the anatomy, often acquired from medical imagery. The second level is physical, involving mainly the biomechamcal modeling of various tissues, organs, vessels, muscles or bone structures. The third level is physiological, involving a modeling of the functions of the major biological systems (e.g., cardiovascular, respiratory, digestive, hormonal, muscular, central or peripheral nervous system, etc.) or some pathological metabolism (e.g.. evolution of cancerous or inflammatory lesions, formation of vessel stenoses, etc.). A fourth level (not described in this volume) would be cognitive, modeling the higher functions of the human brain. These different levels of modeling are closet)' related to each other, and several physiological systems may interact together {e.g., the cardiopulmonary interaction). The choice of the resolution at which each level is described is important, and may vary from microscopic to macroscopic, ideally dirough multiscale descriptions.The first three chapters of this volume study three important physiological models (vascular, cardiac, and tumoral) from a mathematical and numerical perspecuve. The chapter by Alfio Quarteroni and Luca Formaggia addresses the problem of developing models for the numerical simulation of the human circulator)' system, focussing on die analysis of haemodynamics in arteries. Applications include the prediction (and therefore die possible prevention) of stenoses (a local reduction of ше lumen of die artery), a leading cause of cardiovascular accidents. The chapter by Mazy Belik. Taras Usyk and Andrew McCulloch descnbes computational methods for modeling and simulating die cardiac electromechanical function. These methods provide tools to predict physiological function from quantitative measurements of tissue, cellular or molecular structures. Applications include a better understanding of cardiac pathologies, and a quantitative modeling of their evolution from various sources of measurements, including medical imagery. The chapter by Jesus Ildefonso Diaz and Jose Ignacio Tello studies me madie-matical properties of a simple model of tumor growth. Proofs are given for the existence and uniqueness of solutions and numerical simulations of die model are presented.The next two chapters are dedicated to the simulation of deformations inside the human body in two different contexts. The chapter by Eberhard Haug. Hyung-Yun Choi. Stephane Robin and Muriel Beaugonin describes computational models for crash and impact simulation. It presents the latest generation of virtual human models used to study the consequences of car accidents on organs and important anatomical structures. These models allow the interactive design of safer vehicles with an unrivaled flexibility. The chapter by Herve Delingette and Nicholas Ayache describes computational models of soft tissue useful for surgery simulation. The real-time constraint imposed by the necessary realism of a training system leads to specific models which are applied to die simulation of minimally invasive digestive surgery, including liver surgery.The last two chapters describe computational models dedicated to image-guided intervention and diagnosis. The chapter by Xenophon Papademetris. Oskar Skrinjar and James Duncan describes computational models of organs used to predict and track deformations of tissues from sparse information acquired through medical imaging. These models rest on a successful combination of biomechanical modeling with medical image analysis, with an application to image-guided neurosurgery and an application to the image-based quantitative analysis of cardiac diseases. The chapter by Fred Azar, Dim-itris Metaxas and Mitchell Schnall presents a computational model of the breast used to predict deformations during interventions. The main applications are for image-guided clinical biopsies and for image-guided therapy.Before concluding this introduction, I wish to wholeheartedly thank all the authors for their essential contributions, their patience and confidence during all the genesis process of this book. Special thanks are due to my colleague Herve Delingette. whose advice was extremely helpful from the very beginning. I wish to thank several colleagues for their important help and the many improvements they suggested: Michel Audetie. Chris Berenbmch. Mark Chaplain, Olivier Clatz, Stephane Lanteri. Denis Lau-rendeau. Philippe Meseure, Serge Piperuo, Jean-Marc Schwartz, Bnan Sleeman, Michel Sonne. Matthias Teschner. Marc Thiriet. Manna Vidrascu. I also wish to thank Gilles Kahn. Scientific Director of INRIA. who has been extremely supportive of mis project originating from our institute.Final!)'. I wish to honor the memory of Jacques-Louis Lions, who contacted me for the first time at the end of November 1999 with the proposition to work on this project. The original title changed several times, before finally converging towards its final title after recent discussions with Philippe Ciarlet, to whom will go my final thanks, for his great encouragements and confidence.