Computational Models for the Human Body Special Volume XII.pdf

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General Preface
In the early eighties, when Jacques-Louis Lions and I considered the idea of a
Hand-
book of Numerical Analysis,
we carefully laid out specific objectives, outlined in the
following excerpts from the “General Preface” which has appeared at the beginning of
each of the volumes published so far:
During the past decades, giant needs for ever more sophisticated mathe-
matical models and increasingly complex and extensive computer simula-
tions have arisen. In this fashion, two indissociable activities,
mathematical
modeling
and
computer simulation,
have gained a major status in all aspects
of science, technology and industry.
In order that these two sciences be established on the safest possible
grounds, mathematical rigor is indispensable. For this reason, two compan-
ion sciences,
Numerical Analysis
and
Scientific Software,
have emerged as
essential steps for validating the mathematical models and the computer
simulations that are based on them.
Numerical Analysis
is here understood as the part of
Mathematics
that de-
scribes and analyzes all the numerical schemes that are used on computers;
its objective consists in obtaining a clear, precise, and faithful, representa-
tion of all the “information” contained in a mathematical model; as such, it
is the natural extension of more classical tools, such as analytic solutions,
special transforms, functional analysis, as well as stability and asymptotic
analysis.
The various volumes comprising the
Handbook of Numerical Analysis
will thoroughly cover all the major aspects of Numerical Analysis, by pre-
senting accessible and in-depth surveys, which include the most recent
trends.
More precisely, the Handbook will cover the
basic methods of Numerical
Analysis,
gathered under the following general headings:
–
–
–
–
Solution of Equations in
R
n
,
Finite Difference Methods,
Finite Element Methods,
Techniques of Scientific Computing.
v
vi
General Preface
It will also cover the
numerical solution of actual problems of contempo-
rary interest in Applied Mathematics,
gathered under the following general
headings:
– Numerical Methods for Fluids,
– Numerical Methods for Solids.
In retrospect, it can be safely asserted that Volumes I to IX, which were edited by
both of us, fulfilled most of these objectives, thanks to the eminence of the authors and
the quality of their contributions.
After Jacques-Louis Lions’ tragic loss in 2001, it became clear that Volume IX would
be the last one of the type published so far, i.e., edited by both of us and devoted to some
of the general headings defined above. It was then decided, in consultation with the pub-
lisher, that each future volume will instead be devoted to a single “specific
application”
and called for this reason a “Special
Volume”.
“Specific
applications”
will include Math-
ematical Finance, Meteorology, Celestial Mechanics, Computational Chemistry, Living
Systems, Electromagnetism, Computational Mathematics etc. It is worth noting that the
inclusion of such “specific applications” in the
Handbook of Numerical Analysis
was
part of our initial project.
To ensure the continuity of this enterprise, I will continue to act as Editor of each Spe-
cial Volume, whose conception will be jointly coordinated and supervised by a Guest
Editor.
P.G. C
IARLET
July 2002
Foreword
Computational Models for the Human Body
constitute an emerging and rapidly pro-
gressing area of research whose primary objective is to provide a better understanding
of the physiological and mechanical 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 numerical modeling of physical problems (including the analysis of
existence and uniqueness of solutions for instance), followed by various numerical sim-
ulations.
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 in medicine
will in fact strongly contribute to the development of a more
individualized
and
preven-
tive
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 func-
tional 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), exer-
cising (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, of-
ten acquired from medical imagery. The second level is
physical,
involving mainly the
biomechanical modeling of various tissues, organs, vessels, muscles or bone structures.
The third level is
physiological,
involving a modeling of the functions of the major bio-
logical 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 closely related to each other, and
vii
viii
Foreword
several physiological systems may interact together (e.g., the cardiopulmonary interac-
tion). The choice of the resolution at which each level is described is important, and
may vary from microscopic to macroscopic, ideally through multiscale descriptions.
The first three chapters of this volume study three important physiological models
(vascular, cardiac, and tumoral) from a mathematical and numerical perspective. The
chapter by Alfio Quarteroni and Luca Formaggia addresses the problem of developing
models for the numerical simulation of the human circulatory system, focussing on the
analysis of haemodynamics in arteries. Applications include the prediction (and there-
fore the possible prevention) of stenoses (a local reduction of the lumen of the artery), a
leading cause of cardiovascular accidents. The chapter by Mary Belik, Taras Usyk and
Andrew McCulloch describes computational methods for modeling and simulating the
cardiac electromechanical function. These methods provide tools to predict physiologi-
cal 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 Jesús Ildefonso Díaz and José Ignacio Tello studies the mathe-
matical properties of a simple model of tumor growth. Proofs are given for the existence
and uniqueness of solutions and numerical simulations of the 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,
Stéphane 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 Hervé 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 the
simulation of minimally invasive digestive surgery, including liver surgery.
The last two chapters describe computational models dedicated to image-guided in-
tervention and diagnosis. The chapter by Xenophon Papademetris, Oskar Skrinjar and
James Duncan describes computational models of organs used to predict and track de-
formations of tissues from sparse information acquired through medical imaging. These
models rest on a successful combination of biomechanical modeling with medical im-
age 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 Hervé Delingette, whose
advice was extremely helpful from the very beginning. I wish to thank several col-
leagues for their important help and the many improvements they suggested: Michel
Audette, Chris Berenbruch, Mark Chaplain, Olivier Clatz, Stéphane Lanteri, Denis Lau-
rendeau, Philippe Meseure, Serge Piperno, Jean-Marc Schwartz, Brian Sleeman, Michel
Foreword
ix
Sorine, Matthias Teschner, Marc Thiriet, Marina Vidrascu. I also wish to thank Gilles
Kahn, Scientific Director of INRIA, who has been extremely supportive of this project
originating from our institute.
Finally, 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.
N
ICHOLAS
A
YACHE
Sophia–Antipolis, France
1st November 2003

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