C W Cryer
P P Lunkenheimer
Mathematical Mo delling of the
Cardiovascular System
Mathematical Mo delli ng
of the
Cardiovascular System
Version
Septemb er
Colin Walker Cryer Paul Peter Lunkenheimer
Abstract
We describ e the problem of mo delling the cardiovascular system and give current refer
ences
AMS Sub ject Classi cation Z C C C
Keywords Mo delling heart ventricular mechanics biomechanics cardiovascular system
mathematics
Institut fur Numerische und instrumentelle Mathematik Westf alische Wilhelms Universit at Einste
instra e Munster e mail cryer math uni muenster de
Exp erimentelle Thorax Herz und Gef a chirurgie Domagkstr Munster
Contents
Prologue
I Biomechanics
The Human Machine
Energetics
Mechanics
II The Heart
Intro duction
The Heart
Visualization
Disease and Pathology
ComparativePhysiology
History of Cardiology
The Cardiac Cycle
Intro duction
The Cardiac Cycle
Ventricular lling
Isovolumetric contraction
Ejection
Isovolumetric Relaxation
Global Motion of the Heart
The Pericardium
Heart Sounds and Murm urs
Concluding Remarks
The Structure of the Myo cardium
Intro duction
Connective Tissue
Previous Mathematical Mo dels
The Pericardium Endo cardium and Epicardium
Exp erimental Observations
The Ventricular Function Curve
In Vivo Stress Measurements
In Vivo Strain Measurements
Coronary Perfusion
Intramyo cardial Pressure
Other Exp erimental Observations
Pericardial E usion
Hyp ervolemia
Oligemia
Arterial Hyp ertension
Aortic Coarctation
Aortic Stenosis
Pulmonary Stenosis
Idoventricular Rhythms
Coronary Occlusion Myo cardial Ischemia
Cardiac Valves
Aortic Regurgitation
Mitral Insu ciency
Mitral Stenosis
III Mathematical Mo delling of the Heart
In tro duction
The Field Equations
The Constitutive Equations
The Goal of Mo delling
Diastolic Expansion
Ventricular Blo o d Pressure
Myo cardial stress
Elastic Recoil
Residual Stress
Ventricular Geometry
Muscle Mechanics
Intro duction
The Phenomological Approach
The Fenn e ect
Feng e ect
Classic Results
Molecular Muscle Mechanics
Ventricular Interaction
PassiveMyo cardium
Exp erimental Metho ds and Results
Constitutive Equations
Strain Energy Functions
Bovendeerd
Smaill and Hunter
Inverse Metho ds for Parameter Estimation
Cardiac Mechanics
Intro duction
The Comp onents of the Heart s Wall
Exp erimental Stress Strain Measurements
ConstitutiveLaws
Hydraulic E ects
Myo cytes
Mo delling the Fibre Architecture of the Myo cardium
Foliations
Top ology
Geo desics
Mo dels with Spherical Geometry
The Law of Laplace
Spherical Mo dels
Analytical Work
Comparison of Mo dels
Mo dels with Cylindrical Geometry
Mo dels with Spheroidal Geometry
Mo dels with Additional Features
The Pericardium
Fibres
Hydrostatic Skeletons and Muscular Hydrostats
Fluid Fibre Mo dels
Systolic Contraction
Fibres
Treatment of the Band Structure
Poro elastic Mo dels
Other Factors
Numerical Mo dels
Numerical Metho ds
Prop erties of Blo o d
Fluid Flow in the heart
Intro duction
Numerical Metho ds
The Immersed Boundary Metho d
Euler Lagrangian Formulation
Vortex and whirl ows
Mo delling the Heart Valves
Intro duction
The Mitral Valve
The Aortic Valve
Architecture of the Aortic Valve Lea ets
Arti cial Valves
Valve Vibration
Mo delling other Ventricular Pro cesses
Electrical and Electrophysiological Activity
Oxygen Di usion
Heat Flow
ComparativePhysiology of the Heart
Intro duction
Sources of Data
Allometric Equations
Di erences Between Mammalian Hearts
Concluding Remarks
Mo delling the Global Motion of the Heart
The Heart as a Mechanical Pump
Cardiac Energetics
Growth
Miscellaneous Topics
Heart Sounds
The Interval Force Relationship
Assistance and Replacement
Anrep E ect
Gregg E ect
IV Mathematical Mo delling of the Circulatory System
Intro duction
The Arterial System
Intro duction
The Pulmonary Circulation
Intro duction
The V enous System
The Micro circulation
V Mathematical Mo delling of the Entire Cardiovascular Sys
tem
Intro duction
Mo dels of the Cardiovascular System
Control
Design Principles of the Cardiovascular System
Optimization of Biological Structures
A The SI System
A Physical Units
A Units
A Useful conversion factors
A Physical Quantities
B Dimensional Analysis
B Dimensional Analysis
C Elementary Concepts from Physical Chemistry
DPhysiology
E Cardiovascular Data
E Man
E Woman
E Dog
F Mathematical Tools
F Di erential geometry
F Geo desics
F Geometric Measure Theory
F Top ology
F Knots
F Partial Di erential Equations
F Singularities
F Chaos
G Mechanics
G Applied Mechanics
G Thermo dynamics
G Contiuum Mechanics
G Elasticity
G Fibre reinforced Materials
G Elastic Solids with Microstructure
H Biomechanics
H General References
H Biomaterials
H Mathematical Biology
H Pattern Formation
H Bio energetics
H Kinesiology
I Computer Graphics
J Numerical Metho ds
J Finite Elements
K Sources
K Sources of Preserved Hearts
K Bo oks Rep orts Theses
K Journals
K So cieties
K Deutsche Literatur
K Deutsch Englisch Verzeichnis
K Diplomthemen
L Worksheets
L Questions
L Availability of Journals etc in Munster
L Literature Searched
L Ph D Theses
L Corresp ondance
M Biomathematics
Prologue
App endix
References
Concepts
List of Figures
The simpli ed circulatory system
The plumbing of the normal heart
The lo cation of the heart
Cross sections of the heart
The cardiac cycle
Simpli ed diagram of isolated dog heart lung preparation of Starling
The ventricular function curve
Hort s Mo del for elastic recoil
Contractile b ehaviour of isolated myo cardial muscle
Stress extension relations for left ventricular midwall sp ecimen during st
th and th cycles of equibiaxial loading Cycle p erio d sec and sp eci
men thickness mm The solid lines indicate the order of loading in the
cross b er direction for clarity these are omitted in the b er direction
The Law of Laplace
Blo o d ow a
Blo o d ow b
Chapter
Prologue
The human heart is a pump which converts metab olic energy to mechanical work and
generates the pressure that drives the blo o d throughout the b o dy s circulation system It
can do so for years or more with a rate varying from ab out litres min in an adult
human male at rest to litres min in Olympic athletes For an organ which is ab out the
size of a st this is a remarkable achievement
In we b egan a longterm pro ject to examine ways of mo delling the heart sp eci cally
its mechanical prop erties This requires several steps
The relevant features of the heart must b e formulated
Quantitative exp erimental data must b e obtained
Appropriate mathematical mo dels must b e constructed
As far as p ossible the mo dels should b e analyzed mathematically
Numerical metho ds must b e used to obtain quantitative results
After comparing theory and exp eriment the mo dels must if necessary b e mo di ed
There is an extensive literature on this sub ject Nevertheless we hop ed to b e able to make
a meaningful contribution by taking advantage of the combined physiological morpho
logical and mathematical knowledge of the authors Particular emphasis was to b e placed
on the bre architecture of the myo cardium and its in uence on ventricular mechanics
Since the mathematician amongst us CWC was a newcomer to the eld it was natural to
b egin by reading the literature and making notes As the breadth of the sub ject b ecame
apparent the notes evolved into the present text whichmay therefore b e particularly
useful to applied mathematicians who wish to contribute to a fascinating area While our
principal in terest lies in the mechanics of the heart this cannot b e considered in isolation
and we include descriptions sometimes extremely brief of the often fascinating problems
connected with other parts of the cardiovascular system
The text is intended for use in a course on the mathematical mo delling of the cardio
vascular system for mathematics students with little knowledge of physiology physics
or chemistry Hence some basic background information with references to additional
sources is given in app endices In the course of acquainting himself with the sub ject
CWC found information such as the addresses of cardiovascular so cieties sources of ma
terials etc to b e very useful and this is also included In this waywehopetohave
provided not just a textb o ok but also a sourceb o ok for anyone entering the eld
The mo delling of the cardiovascular system is the sub ject of several recent b o oks for
engineers and physicists Here we emphasize the mathematical asp ects
By giving sp eci c examples which can then b e sub jected to mathematical and nu
merical analysis
By drawing attention to op en problems
By drawing attention to p otentially useful mathematics
The text di ers from many b o oks on mathematical mo delling in that we assume an
adequate knowledge of partial di erential equations such as could b e provided bya two
semester course on the theory of partial di erential equations and their numerical solution
In the past mathematicians have contributed relatively little to the sub ject and one may
very well ask whether there is anything that they can do which could not b e done b etter
by the physicists biomechanical engineers etc already working in the eld Wewould
like to think so and hop e that this b o ok will help to bring this ab out One source of
encouragement is our exp erience that given the opp ortunity and appropriate help many
mathematics students welcome the opp ortunity to mo del biological pro cesses
We are grateful to Butterworth and Heinemann Ltd for gran ting p ermission to repro duce
several gures and short pieces of text from An Introduction to Cardiovascular Physiology
by J R Levick Lev
The present text would never have b een conceived and written had it not b een for the
enthusiasm of the second author Paul Peter Lunkenheimer who sp entmany hours ex
plaining and demonstrating the workings of the heart to his mathematical colleague
Part I
Biomechanics
Chapter
The Human Machine
In this text we will primarily b e concerned with mo delling one asp ect of the human b o dy
the cardiovascular system and in particular the heart The approach used mathematical
mo delling is in principle straightforward We consider the individual comp onents of the
human b o dy For each comp onent for example the lungs we seek to set up a set of math
ematical equations whichemb o dy the physical pro cesses within the comp onent These
equations which are usually a combination of algebraic ordinary and partial di erential
equations provide a mathematical mo del which can b e used for analysis and prediction
In practice mathematical mo delling is often di cult This is certainly the case in mo d
elling the human b o dy where as one moves from the macroscopic to the microscopic and
then to the molecular level new structures are revealed
This intro ductory part of the text serves two purp oses
To discuss some of the general features of man considered as a machine
Tointro duce the mathematical reader to some basic ideas and to give him or her a
feel for the size of some relevantphysiological quantities
There are some well written non mathematical intro ductions to biomechanics and physi
ology
Life s Devices bySteven Vogel Vog
The Human Machine by Alexander Ale a
There are also several excellent texts on mathematical biology which include sections on
mo delling the human machine
Hopp ensteadt and Peskin Mathematics in Medicine and the Life Sciences HP
Murray Mathematical Biology Mur
Mazumdar An Introduction to Mathematical Physiology and Biology Maz
And now without more ado let us turn to the task in hand
Chapter
Energetics
We b egin by recalling some basic facts ab out the human b o dy considered as a mechanical
device the human machine Itmay b e noted here that there app ears to b e a fundamen
tal di erence b etween mathematical mo dellers and most physiologists A mo deller needs
mo del problems that is problems whcih embody typical features In the presentcontext
the mo deller would liketohave model man or model human machine There is of course no
such thing and physiologists seem to have little need to create it The normal or average
data given b elowhave had to b e extracted by us from several sources
A normal young man weighs W kg Corresp onding to a normal height H
H cm m
His b o dy surface area is ab out m Approximately of the b o dy s weight is made
up of water
A normal young man has an oxygen consumption of
V ml O min
O
breathing on average of ml of air twelve times p er minute
Rasch cite p uses examples from weight lifting to illustrate the p ower and
strength of athletes He also quotes the following maxima for the output of a human
b eing
hp kW kW for a single movement lasting less than second
to hp to kW for exercise lasting b etween and minutes
to hp to kW for steady state work for min to min or more
hp kW for long term work
The b o ok The Fire of Life by Kleib er Kle is a well known intro duction to animal
energetics which contains various interesting details but is also somewhat incoherent
For more details and references see App endix E where the average woman is also describ ed
Chapter
Mechanics
Part II
The Heart
Chapter
Intro duction
The heart is the pump in the cardiovascular system In this chapter the general features of
this system are describ ed In subsequentchapters more detailed information is provided
Wehave drawn heavily on standard texts including Cardiovascular Physiology byJ R
Levick Lev Cardiovascular Physiology by R M Berne and M N Levy BL Cardio
vascular Physiology by Milnor Mil and The Cardiovascular System The Heart edited
by R M Berne Ber
Many exp erimental results are only available for non humans and it is in any case often
useful to compare results for di erent animals When discussing non human physiology
this will b e explicitly stated
The Circulatory System
The mammalian cardiovascular system consists of the heart and blo o d vessels The basic
function of this system is to transp ort blo o d In this pro cess oxygen and other pro ducts
are conveyed to the tissues and waste pro ducts such as carb on dioxide and urea are re
moved
The heart consists of two adjacentintermittantmuscular pumps the rightventricle and
left ventricle Each pump is lled from a reservoir the right atrium or left atrium
resp ectively The designation left or right is from the p ersp ectiv e of the patient the left
ventricle is on the left side of the patient s body Since the heart lies across the chest in a
human the rightventricle and atrium are towards the front anterior while the left ven
tricle and aorta are towards the rear p osterior The rightventricle pumps blo o d through
the lungs to the left atrium the pulmonary circulation while the left ventricle simul
taneously pumps blo o d through the rest of the b o dy and back to the right atrium the
systemic circulation The blo o d is comp elled to follow a circular pathwayby one way
valves lo cated in the heart and veins as was rst established by the London physician
William Harvey and published in Har Har b see Figure
In the pulmonary circulatory system venous blo o d enters the right atrium from
the two ma jor veins the sup erior vena cava and inferior vena cava and the
coronary sinus then ows through a valveinto the rightventricle This ventricle
which is comp osed mainly of cardiac muscle receives the blo o d while it is in a
state of relaxation called diastole Contraction or systole then forces part of the
blo o d out through the pulmonary artery and into the lungs at a low pressure Gases
exchange by di usion in the lung air sacs raising the blo o d oxygen content from
approximately ml l venous blo o d to ml l The oxygenated blo o d returns
through the pulmonary veins to the left atrium and left ventricle
In the systemic circulatory system the left ventricle contracts virtually simultane
ously with the right and ejects approximately the same volume of blo o d but at a
much higher pressure The blo o d ows through the aorta and the branching arterial
system into ne thin walled tub es called capillaries Here the dissolved gases and
nutrients di use b etween the capillary blo o d and the tissue cells The circulation of
the blo o d is completed bythevenous system which conducts blo o d back to the
venae cavae
See App endix C for a discussion of the relevantphysical chemistry The statement that the oxygen
contentofvenous blo o d is ml l means that litre of venous blo o d contains the equivalent of ml
of Oxygen O at STPD Standard Temp erature and Pressure that is C and atmospheric pressure
BRAIN
HEAD and ARMS
S A V O C R LUNGS BRONCHI T A
RIGHT RIGHT LEFT LEFT ATRIUM VENTRICLE ATRIUM VENTRICLE
CORONARY SINUSHEART CORONARY ARTERY A MUSCLE O LIVER R SPLEEN T I GUT A V
C TUBULES GLOMERULUS
TRUNK and LEGS
Figure The simpli ed circulatory system
The ow of blo o d in the cardiovascular system is shown in simpli ed form in Figure
There are four circuits the pulmonary circuit where blo o d ows from the rightventri
cle through the lungs to the left atrium the upp er or sup erior circuit where blo o d o ws
through the aorta to the head and arms returning via the sup erior vena cava SVC the
lower or inferior circuit where blo o d ows through the aorta to the lower b o dy return
ing via the inferior vena cava IVC and the coronary circuit where blo o d o ws through
the coronary artery to the heart muscle returning via the coronary sinus Together the
sup erior inferior and coronary circuits make up the systemic circuits The return of blo o d
from the bronchii is anomalous since it ows into the left atrium Under normal condi
tions the bronchial blo o d ow constitutes at most of the venous return to the heart
In certain pathological states the bronchial circulation may b ecome substantial See Berne
and Levy BL page RIGHT RIGHT VENTRICLE ATRIUM TRISCUPID VALVE PULMONARY VALVE
CORONARY SINUS
HEART LUNGS MUSCLE
CORONARY ARTERY
LEFT LEFT AORTIC VENTRICLE MITRAL ATRIUM
VALVE VALVE
Figure The plumbing of the normal heart
The ow of blo o d through the normal heart can b e represented functionally as in Figure
Eachventricle has an inlet valve the mitral and tricuspid valves resp ectively and an
outlet valve the aortic and pulmonary valves resp ectively These valves op en and close
at appropriate times during the cardiac cycle as describ ed in Section
In Figure the supply of blo o d to the heart muscles through the coronary artery
and coronary sinus is shown separately b ecause this blo o d supply is in uenced by the
pressures in the heart wall and is of imp ortance for the mo delling of the heart
The Heart
In an adult male weighing kg the heart weighs ab out grams and roughly o ccupies
a cub e with sides of length cm The lo cation of the heart is shown in Figure
Figure The lo cation of the heart
The heart lies obliquely across the chest The brotendinous ring black forms the base of
the heart It contains the tricuspid t mitral m aortic and pulmonary valves group ed
in an oblique plane b eneath the sternum The ap ex of the heart is formed by the left
ventricle LV and the anterior surface is formed by the rightventricle RV and right
atrium RA The inferior surface of the heart and the p ericardium not shown rest on the
central tendon of the diaphragm From J R Levick An Introduction to Cardiovascular
Physiology by p ermission
As is shown in Figure a the heart consists of two adjacent pumps the left ventri
cle and rightventricle Blood ows into eachventricle from an antechamb er the left
atrium right atrium through a valve the mitral valve tricuspid valve Blo o d
leaves the left right ventricle through another valve the aortic valve pulmonary
valve The edges of each in ow valve are tethered by strings the chordae tendineae
to twomuscles the papillary muscles which bulge from the ventricle s walls
The wall dividing the left ventricle and atrium from the rightventricle and atrium is
called the septum The four heart valves lie approximately in a plane the intersection
of which with the heart is called the brotendinous ring see Figure
The rightventricle only pumps blo o d to the lungs and back to the left atrium while the
left ventricle must pump blo o d through the whole b o dy As to b e exp ected the outer
wall of the rightventricle is much thinner ab out cm than the outer wall of the left
ventricle ab out cm This is illustrated in Figure b which shows a view of the
lower part of the twoventricles
Visualization
Since the heart is three dimensional it is quite di cult to visualize for a non medical
reader it may help to use a plastic mo del such as can b e obtained from medical b o ok
shops or even b etter the prepared heart of an animal Recently videos showing the
motion of the heart havebecomeavailable
See App endix K for the names and addresses of some suppliers
Figure Cross sections of the heart
a Schematic diagram of an oblique section b Section across the ventricles to illustrate
mo de of emptying c Arrangementofmuscle bres in the ventricle wall RA LA right
and left atrium The op ening just b elow the lab el RA is the coronary sinus RV LV right
and left ventricle T M tricuspid and mitral v alves P papillary muscle with chordae
tendineae A aorta P A PV pulmonary artery and veins SV C IVC sup erior and in
ferior venae cavae From J R Levick An Introduction to Cardiovascular Physiology by
p ermission
Several b o oks provide extensivedrawings and photographs
Anderson and Becker The Heart Structure in Health and Disease AB
M J Davies Colour Atlas of Cardiovascular Pathology Dav
Thomas Geb ert and Hombach Grundlagen der klinischen Medizin Anatomie
Physiologie Pathologie Mikrobiologie Klinik Band Herz und Gefaesse
TGH
Vesalius The Illustrations from the Works of Andreas Vesalius of Brussels
Ves b
For the text of Anderson and Becker an accompanying set of slides is available
Disease and Pathology
An imp ortant justi cation for mo delling the cardiovascular system is that it may suggest
ways to improve the treatment of illness Examples of this approach are given in McDon
ald s Blood Flow in the Arteries pages NO
An immense variety of pathological situations is observed in practice Every part of the
heart can b e a ected either through congenital defects or disease
See
Anderson and Becker The Heart Structure in Health and Disease AB
Acierno The History of Cardiology Aci
Braunwald Heart Disease Bra
Davies Colour A tlas of Cardiovascular Pathology Dav
Julian and Cowan Cardiology JC
Lilly Pathophysiology of Heart Disease Lil
Thomas Geb ert and Hombach Grundlagen der klinischen Medizin Anatomie
Physiologie Pathologie Mikrobiologie Klinik Band Herz und Gefaesse
TGH
White Heart Disease Whi
Pathological situations presentachallenge to mathematical mo delling A mo del should
b e able to correctly repro duce the e ects of any pathology Suchamodelwould b e a
valuable diagnostic to ol No current mo dels have such capabilities We b elieve that this
represents a tremendous opp ortunity
In this connection we observe that the following discuss changes in heart function caused
by disease
Grant Architectonics of the heart Gra
ComparativePhysiology
A great deal can b e learnt ab out the human cardiovascular system by considering the
cardiovascular systems of other animals
Some measurements are more easily made on certain animals
For the development and testing of a go o d theory it is of imp ortance to b e able to
vary parameters e g the size of the heart
The prime purp ose of a cardiovascular system is to transp ort oxygen within the b o dy
Twophysical pro cesses are involved
Di usion which is only e ectiveover short distances b ecause the average time t
required for a particle to di use a distance x is given by
t x D
where D is a constant the di usion co e cient For example for a solution of glucose
in water at D cm s so that the time required to di use cm is
s s h t
that is hours and minutes
Even over short distances nature comp ensates for the slowness of di usion by pro
viding large surfaces
The lungs have a surface of ab out m in a normal adult Guyton
Guy page
The blo o d capillaries o er a surface of m Guyton Guy page
See Cameron Cam and Weib el Wei
Convection which is e ected through the circulatory system and is e ectiveover
long distances
Each animal has its own system of oxygen distribution and nature has provided a mar
vellous range of alternatives some of which are listed b elow
No circulatory system
Op en circulatory systems in which blo o d is pump ed by the heart through a system
of blood vessels but then moves freely b etween the tissues of the b o dy
Closed circulatory systems in which the blo o d is pump ed by the heart through a
system of blo o d vessels which it never leaves
In circulatory systems many p ossible arrangements of the heart s o ccur
One pump e g the frog
One bidirectional pump e g in tunicates see Schmidt Nielsen SN page
Two pumps in one heart mammals and birds
Three separate pumps e g the o ctopus see Steven Vogel Vog page
The great range of cardiovascular mechanisms is clearly reviewed in Chapter of Scheer
Sch while in Chapter of Prosser Pro more details are given including tables of
blood volumes and blo o d pressures in various animals
Other references include
Comparative Animal Physiology by Prosser Pro
Comparative Physiology by Bradley T Scheer Sch
Animal Physiology byKnut Schmidt Nielsen SN
Vital Circuits bySteven Vogel Vog
Biologie in Zahlen by Flindt Fli
History of Cardiology
The history of cardiology is a fascinating topic For the interested reader we can whole
heartedly recommend the translation of Harvey s The Circulation of the Blood Har b
which is b eautifully clear even for to day s reader Foster s Lectures on the History of
Physiology MF is interesting but alas long out of print The recent b o ok The History
by Acierno Aci iswell written and describ es not only historical contri of Cardiology
butions but also quite recentdevelopments Other currently or until recently available
references are
Giovanni Alphonso Borelli On the Movement of Animals Bor
Clendening SourceBook of Medical History Cle
Harris The Heart and Vascular System in Ancient Greece Har
Leib owitz The History of Coronary Heart Disease Lei
Andreas Vesalius Epitome Ves b
Andreas Vesalius The Il lustrations from the Works of Andreas Vesalius of Brussels
Ves a
Willius and KeysClassics of Cardiology previously cal led Cardiac Classics WK
Further references are given by
Acierno Aci
It is of interest to lo ok at old textb o oks such as
Braun Diagnose und Therapie der Herzkrankheiten Bra
Evans Starling s Principles of Human Physiology Eva
Grollman The Cardiac Output of Man in Health and Disease Gro
Tigerstedt Die Physiologie des Kreislaufes Tig
Chapter
The Cardiac Cycle
Intro duction
The basic facts ab out the cardiac cycle are describ ed Some standard texts are
Braunwald Ross jr and Sonnenblick Mechanisms of Contraction of the Normal
and Failing Heart BRS
Baan Arntzenius and Yellin Cardiac Dynamics BAY
Burton Physiology and Biophysics of the Circulation AnIntroductory Text Bur
Berne Handbook of Physiology Section The Cardiovascular System Vol I The
Heart Ber
Frank Zur Dynamik des Herzmuskels Fra
Katz Physiology of the Heart Kat
Noble The Cardiac Cycle Nob
Opie The Heart Physiology and Metabolism Opi
Rushmer Cardiovascular Dynamics Rus This textb o ok for medical students was
a standard reference
ter Keurs and Tyb erg Mechanics of the Circulation tKT
The Cardiac Cycle
The cardiac cycle corresp onding to one full b eat of the human heart lasts ab out sec
onds for a resting p erson The cycle comprises four parts during which the inlet valves
mitral and tricuspid and outlet valves aortic and pulmonary op en and close as appro
priate
The four parts of the cardiac cycle describ ed ab ove are traditionally divided into two
phases systole and diastole During the systolic phase isovolumetric contraction and
ejection the heart is actively pumping blo o d and the heart muscles are tensed During
the diastolic phase isovolumetric relaxation and ventricular lling the heart muscles are
relatively passive However this terminology is somewhat confusing since see e g Mc
Cullo ch and Omens MO p the heart muscle is not fully relaxed even during the
diastolic phase
Typical measurements of pressure and ow on the left side of the human heart adult
resting supine are shown in Figure The following description of events is based on
the b o ok of Levick Lev pages w ehave added some comments in fo otnotes
Figure The cardiac cycle
Diagram of pressure and out ow on the left side of the human heart based on data
from intracardiac catheters and velo city measurements at the aortic ro ot dP dt is
max
the maximum rate of rise of ventricular pressure a measure of myo cardial contractility
The stippled region highlights the pressure gradient which decelerates out ow during the
late ejection phase Note the slight reversal of aortic owataorticvalve closure The
waveforms in the right heart are of similar shap e but the pressures are lower From J R
Levick An Introduction to Cardiovascular Physiology by p ermission
Ventricular lling
Duration s
Inlet valves tricuspid and mitral op en
Outlet valves pulmonary and aortic closed
Initially the atria are in diastole and blo o d ows passively from the great veins through
the op en atrioventricular valves into the ventricles There is an initial phase of rapid ll
ing lasting ab out s which has a curious feature ev en though ventricular volume is
increasing ventricular pressure is falling The reason is that the ventricle wall is recoiling
elastically from the deformation of systole and is in e ect sucking blo o d into the chamb er
As the ventricle reaches its natural volume lling slows down and further lling requires
distension of the ventricle by the pressure of the venous blo o d v entricular pressure now
b egins to rise In the nal third of the lling phase the atria contact and force some addi
tional blo o d into the ventricle PPL fragen In resting sub jects this atrial b o ost is quite
small During exercise however when the time available for passiveventricular lling is
curtailed the atrial b o ost b ecomes imp ortant The volume of blo o d in a ventricle at the
end of the lling phase is called the end diastolic volume or EDV and is typically
around ml in an adult human The corresp onding end diastolic pressure or EDP
is a few mmHg
Isovolumetric contraction
Duration s
Inlet valv es tricuspid and mitral closed
Outlet valves pulmonary and aortic closed
Ventricular systole lasts s and is divided into a brief isovolumetric phase and a longer
ejection phase As so on as ventricular pressure rises fractionally ab ove atrial pressure the
atrioventricular valves are forced shut by the reversed pressure gradient Back ow during
closure is minimal b ecause the cusps are already approximated byvortices b ehind them
in the late lling phase The ventricle is now a closed chamb er and the growing wall
tension causes a steep rise in the pressure of the trapp ed blo o d
The atrioventricular valves are the tricuspid and mitral valves which link the atria and ventricles
Although the ow around the tricuspid and mitral valves has b een extensively studied we are not
aware of a precise justi cation for this assertion Discussed in Section
Ejection
Duration s
Inlet valves tricuspid and mitral closed
Outlet valves pulmonary and aortic op en
When ventricular pressure exceeds arterial pressure the out owvalves are forced op en
and ejection b egins Three quarters of the strokevolume are ejected in the rst half of
the ejection phase s and at rst blo o d is ejected faster than it can escap e out of the
arterial tree As a result much of it has to b e accommo dated by distension of the large
elastic arteries and this drives arterial pressure up to its maximum or systolic level
Vortices b ehind the cusps of the op en aortic valve prevent the cusps from blo cking the
adjacent op enings of the coronary arteries
As systole weakens and the rate of ejection slows down the rate at which blo o d ows
away through the arterial system b egins to exceed the ejection rate and pressure b egins
to fall Activeventricular contraction actually ceases ab out two thirds of the way through
the ejection phase but a slow out owcontinues for a while owing to the momentum
of the blo o d As the ventricle b egins to relax ventricular pressure falls b elow arterial
pressure but the outward momentum of the blo o d prevents immediate valve closure Fi
nally a brief back ow comprising less than of strokevolume closes the out owvalve
Valve closure creates a brief pressure rise in the arterial pressure called the dicrotic wave
The ventricle do es not empty completely the a verage ejection fraction in man is
corresp onding to a strokevolume of ml in adults The residual end systolic vol
ume of ab out ml acts as a reserve which can b e utilized to increase strokevolume in
exercise
As of December we do not know of a reference for this statement
Isovolumetric Relaxation
Duration s
Inlet valves tricuspid and mitral closed
Outlet valves pulmonary and aortic closed
With closure of the aortic and pulmonary valves eachventricle once again b ecomes a
closed chamb er Ventricular pressure falls very rapidly owing to the mechanical recoil
of collagen bres within the myo cardium whichwere tensed and deformed by the con
tracting myo cytes When ventricular pressure has fallen just b elow atrial pressure the
atrioventricular valves op en and blo o d o o ds in from the atria whichhave b een re lling
during ventricular systole
Global Motion of the Heart
The heart is not stationary but moves relative to the rest of the b o dy during each cardiac
cycle The rotation of the left ventricular ap ex has b een measured by Buchalter Weiss et
al BWR
The Pericardium
When the chest is op ened the motion of the heart is largely hidden from view b ecause
of the p ericardium a brous uid lled sac in the thoraic cavitywhich encloses and
restrains the heart The p ericardium reduces friction b etween the heart and surrounding
organs and provides a barrier against infection
The p ericardium consists of twolayers a strong outer layer of tough brous tissue the
brous p ericardium and an inner double layer sac the serous p ericardium The
brous p ericardium is rmly attached within the b o dy The serous p ericardium is normally
lled with uid
The in uence of the p ericardium on the motion of the heart has b een discussed by several
authors see McCullo ch and Omens MO p The consensus of opinion seems to b e
that the p ericardium only plays a small role for normal hearts under normal conditions
Indeed humans with no p ericardium as a result of a congenital defect do not usually
showanyunusual symptoms see Bra p However Braunwald Bra p
lists three roles for the p ericardium which deserve consideration by mo dellers
The distribution of hydrostatic forces on the heart
The prevention of acute cardiac dilation
Diastolic coupling of the twoventricles
In the case of illness or injury the p ericardium probably plays an imp ortantroleinre
straining the heart see Braunwald Bra Over a p erio d of time the p ericardium can
adapt to changes in the size of the heart
In health the p ericardium may b e unobtrusive but there are several illnesses asso ciated
with the p ericardium including see Chapter of Braunwald Bra
p ericarditis acute caused by in ammation of the p ericardium
p ericardial e usion caused by an increase in the amount of uid in the p ericardial
space According to Braunwald Bra p the p ericardial space in humans
normally contains b etween and ml of uid If additional uid increases slowly
the p ericardium can stretch and can accommo date up to l of uid without appar
ent discomfort If the uid increases rapidly or the p ericardium is not su ciently
exible then a marked rise of intrap ericardial pressure will o ccur This can lead to
cardiac tamp onade whichischaracterized Braunwald Bra p by
Elevation of intracardiac pressures
Progressive limitation of ventricular diastolic lling
Reduction of strokevolume and cardiac output
p ericarditis constrictive which o ccurs when a brotic thickened and adherent
p ericardium restricts diastolic lling of the heart
Rowell Row p and pages discusses the role of the p ericardium It has b een
demonstrated exp erimentally Rowell Row Figure p that p ericardectomy
the surgical removal of the p ericardium leads to a substantial increase in stroke
volume for exercising dogs Rowell Row p comments on anecdotes that racing dogs
have had the p ericardium cut to improve their p erformance Blomqvist and Salting BS
have suggested that physical conditioning may alter apparentven tricular compliance by
mo difying right and left ventricular p ericardial interactions see Rowell Row p
Tyb erg Misbach Parmley and Glantz TMPG discuss the role of the p ericardium
They suggest see p with con rming exp erimenta data that the p ericardium has a
dominant role in the mechanism of acute shifts in the left ventricular diastolic pressure
volume relationship The authors suggest that the observations can b e explained bya
simple hydraulic mo del As long as the volume of the heart is less than the unstressed
volume of the p ericardium the diastolic pressure in the left ventricle is the same whether
the p ericardium is op en or closed However when the volume of the heart exceeds the
unstressed volume of the p ericardium the ventricular diastolic pressure equals the trans
mural pressure necessary to distend the ventricle plus the p ericardial pressure This implies
that changes in the left ventricular diastolic pressure whenever the unstressed volume of
the p ericardium has b een exceeded The authors p oint out that such shifts in the pressure
volume curve are imp ortant in explaining the resp onse of patients with cardiac failure to
vaso dilators
Heart Sounds and Murmurs
During the cardiac cycle a variety of sounds clicks murmurs etc is pro duced and pro
vides valuable diagnostic information The generation of some of these noises has also
b een analyzed mathematically as will b e discussed later see Peskin Pes p Ped
ley Ped page and page
The terms pericardial space and pericardial sac are used interchangeably
Some common heart sounds are listed b elow For more details and descriptions of other
sounds see Lilly Lil Chapter Luisada Lui a Lui b Luisada and P ortaluppi
LP Guyton Guy Chapter No vey Pencak and Stang NPS
Heart Sounds There are four recognized heart sounds of whichtwo are easily
heard
S The rst heart sound is caused by the closure of the tricuspid and mitral valves
S The second heart sound is caused by the closure of the aortic and pulmonary
valves The mechanism of heart sounds as observed in exp eriments is consid
ered by Luisada and Portaluppi LP p
Heart Murmurs
Heart murmurs are abnormal sounds caused by turbulence which is normally absent
Typically this turbulence is caused bytwotyp es of valve defects see Levick Lev
page
Stenosis isanarrowing of a valve so that a high pressure gradient is needed
to force blo o d through a stenosed valve
Incomp etence is failure of a valve to close tightly thus allowing a regurgi
tation of blood
Korotko Sounds
Korotko sounds are heard when blo o d pressure is measured in the usual way
by placing a cu containing an in atable bag around the upp er arm If the cu
is rst in ated and then the pressure is gradually reduced there comes a p oint
when the blo o d b egins to ow through the brachial artery again at this p oin t the
Korotko sounds can b e heard in a stethoscop e placed against the artery
Concluding Remarks
In this c hapter wehave describ ed the basic observations of the mammalian heart Some
such as the Korotko sounds or the maximum blo o d pressure can b e made by elemen
tary non invasive means and form part of the diagnostic techniqes of every do ctor Oth
ers such as the measurement of the blo o d pressure at the aortic ro ot require the use of
sophisticated invasivetechniques All these observations have b een made and recorded
innumerable times
It is the job of the mo deller to predict and explain these observations To do so requires
much more detailed knowledge In the remainder of this Part we attempt to summa
rize what is known It will b ecome apparent that there are often surprising gaps in this
knowledge
Chapter
The Structure of the Myo cardium
Intro duction
The myo cardium is a complex brous continuum whichcontains not only the muscle
cells or myocytes but also an extensive network of connective tissue We refer to Levick
Lev Chapter for a description of individual myo cytes and their function and to
Lunkenheimer Lunkenheimer and Torrent Guasp LLG Chapter for photographs of
sp ecimens
Connective Tissue
The myo cardium is a complex brous continuum which contains not only the muscle cells
but also an extensivenetwork of connective tissue It is reasonable to supp ose that this
tissue can contribute to the expansion of the ventricles during diastole
This idea has a very long history see Lunkenheimer Lunkenheimer and Torrent Guasp
LLG pages but much more detailed information has now b ecome available
through microscopic analysis
Hunter and Smaill HS review recent ndings and consider the formulation of consti
tutivelaws A Horowitz Y Lanir F C P Yin M Perl I Sheinman and R K Strumpf
HLY consider constitutivelaws for the passivemyo cardium
The individual myocytes have a de nite direction and together form muscle bres which
are clearly visible after preparatory treatment such as b oiling The determination of the
direction of the bres has b een the sub ject of investigation and sp eculation for many
centuries In uential contributions were made byLo wer Pettigrew
MacCullum Mall Flett Fle and Thomas Tho Flett Fle
includes as complete a bibliography a p ossible of the literature as of
The earlier workers recorded their ndings in often very attractivedrawings To our
knowledge the rst detailed quantitative measurements are due to Hort Hor and Hort
Hor Although the individual myo cytes are joined together the determination of the
direction eld of the bres is not easy The literature has therefore tended to split
I On the one hand there are many classical descriptions of the complicated twisted form
taken by the bres As describ ed by Borelli in Bor p
The bres of the heart are not straight and parallel to each other but curved
and coiled They are amazingly intricate Their texture is not similar to that of
a wicker basket as Vesalius b elieved but they are arranged in a more sophis
ticated way Immediately underneath the p ericardium from the basis of the
heart and from the circular tendinous ori ces of the vena cava and pulmonary
auricle and from the origins of the aorta and pulmonary arteries a layer of
eshy bres spreads These bres are parallel to each other and they are orien
tated from the basis straight to the ap ex of the heart where they are diversely
curved and intertwined re ecting towards the cavities of the ventricles This
layer is followed by other layers of bres oblique and descending in spirals
They are more and more inclined and orientated also towards the ap ex of the
heart Before reaching the ap ex they intersect and cover each other and other
ordinary bres Hence they are deviated to the inside and partly re ected in
oblique and transverse spirals like scythes towards the base of the heart Part
of them seem to comp ose internal columns to which the strings of the tricuspid
and mitral valves are attached and others are intertwined transversally and
form the cavity of the rightventricle
Grant Gra brie y and succintly summarizes earlier work on the muscular architecture
of the left v entricle He describ es the work of Lower and Mall and p oints out that Lower
drew attention to the twisting of the bres Grant p compares this with reserva
tions to a twisted rop e Grant also discusses the p ossible role of embryonic development
in the structure of the ventricle see Grant Gra page
Greenbaum Ho Gibson Becker and Anderson GHG brie y summarized previous
work on the bre architecture and rep ort the outcome of a study using dissection and
histology on normal p ostmortem human hearts They conclude p Mo dels based
on uniform myo cardial bre structure cannot explain wall movement in normal sub jects
and are likely to have signi cant limitations if used to investigate left ventricular function
in disease
Lunkenheimer Lunkenheimer and Torrent Guasp LLG and LW have extensive dis
cussions of the bre architecture The latter reference in particular contains a detailed
evaluation of current knowledge
As participants of this conference are well aware Torrent Guasp Gua has extended
this work and p ostulated a mo del based on these observations of twisting bres
II On the other hand many mo dels assume that
The bre directions vary smo othly through the heart wall
The bres run parallel to the inner and outer surfaces
The innermost endo cardial and outermost epicardial bres run longitudinally
from the base to the ap ex the central bres run circumferentially and the other
bres run obliquely
These assumptions are usually justi ed by reference to the pap ers of Streeter and Basset
SB who measured bre orientation in pig heart and Streeter Sp otnitz Patel Ross
and Sonnenblick SSP who measured bre orientation in canine hearts See also
the Ph D thesis of Streeter Str and the article by Streeter Str However this is
an oversimpli cation of the results of Streeter who was well aware of the earlier work
Streeter did measure transmural bre directions but asserted that the asso ciated angle
which he called the imbro cation angle was smaller than ab out
The measurements of Streeter were based on a small numb er of samples at nonrandom
sites The actual situation is much more complex as can b e seen from the pictures of
treated hearts in Lunkenheimer Lunkenheimer and Torrent Guasp LLG pages
Several authors have commented on the shortcomings of the results of Streeter For ex
ample Hunter and Smaill HS page comment
However despite the large numb er of studies there is still a dearth of
quantitative information on ventricular muscle bre orientation No studies
app ear to have measured the muscle bre orientation at more than sites on
a single heart
As rep orted byLunkenheimer p ersonal communication the bres frequently dip into the
surface with an angle of as muchas contradicting assumptions and ab ove
As McLean and Prothero MP b p state
Wehave no a priori reason to assume that the global large scale organization
of the myo b ers in the heart can b e inferred with con dence from a study
restricted to the lo cal organization at one or a few sites in the developing or
adult heart wall
At present several studies are under way which should provide much more detailed infor
mation ab out the bre orientation McLean Ross and Prothero MRP have mapp ed
bre directions in the mouse heart and rep ort that they apparently di er signi cantly
from those in the human heart Nielsen Nie and Nielsen Le Grice Smaill and Hunter
NLSH have made detailed measurements of bre orientation in the canine heart See
also McLean Ross and Prothero MRP
Previous Mathematical Mo dels
Until recently only a few mo dels to ok account of the bre direction A few authers have
considered uid bre mo dels in which the myo cardium is treated as a uid with an
emb edded bre matrix Such mo dels thus ignore the elasticity of the myo cardium but
b ecause of their simplicity can sometimes b e solved analytically See
Ca isch Caf
Fung Fun p
Unfortunately this work has b een discontinued
Skalak Ska
Chadwick und Cha Cha
It seems clear that the results should b e incorp orated into mo dels See
Bovendeerd Huyghe Arts van Camp en and Reneman In uenceofendocardial
epicardial crossover of muscle bers on left ventricular wal l mechanics BHA
Bovendeerd Arts Huyghe van Camp en and Reneman Dendenceoflocal left ven
tricular wal l mechanics on myocardial ber orientation A model study BAH
Yang Tab er and Clark YTC page brie y mention observations on bre orientation
in embryonic chick hearts which they incorp orate into their mo del
Grant Gra brie y and succintly summarizes earlier workonthemuscular architec
ture of the left ventricle He describ es the work of Lower Low and Mall Mal and
p oints out that already Lower had drawn attention to the twisting of the bres Grant
compares this with reservations toatwisted rop e page Grant also discusses the
p ossible role of embryonic development see Grant Gra Finally Grant page also
observes that on its outer surface epicardium the left ventricle has twovortices which
merge b elow the surface The rightven tricle has its own vortex
See also
Pu Puf
Flett Fle
Mall Mal
MacCullum Mac
Thomas Tho
Streeter and Ramon SR
Streeter Daniel Powers Ross and Torrent Guasp SPRG
The Pericardium Endo cardium and Epicardium
For observations on the p ericardium endo cardium and epicardium see
Faller Fal
Chapter
Exp erimental Observations
It is obviously di cult to obtain observations of the heart The layman maywell feel
that direct measurementmust b e nigh to p ossible However the heart is in some ways
remarkably robust as one author CWC rst realized when holding a b eating pig heart
with one hand There is a substantial and rapidly growing b o dy of knowledge which serves
both as a challenge and a guide for mo delling There seems to b e a de nite need to design
exp eriments so as to con rm confound theoretical and numerical predictions
The Ventricular Function Curve
The measurements in Figure refer to a normal working heart However the heart
continues to function outside the b o dy In Otto Frank Fra studied the frog ven
tricle and measured the relation b etween pressure and volume see Knowlton and Starling
KS Markwalder and Starling MS Patterson and Starling PS Patterson Pip er
and Starling PPS as well as Starling s Principles of Human Physiolog Evans Eva
and Levick Lev page This work was extended by Starling and his collab orators
around Figure from Levick p Lev shows the exp erimental setup The
Starling resistor shown in Figure a device to maintain constant arterial pressure is
discussed in detail byFung Fun page
The results of this classical exp eriment are shown in Figure where the strokevolume
is plotted against the lling pressure for an isolated dog heart
A graph whose ordinate is strokevolume or any other measure of contractile energy and
whose abscissa is lling pressure or any other index of resting bre length is called a
ventricular function curve orStarling curve
The upward part of the curve is the basis of the celebrated Frank Starling Law or
Starling s Law of the Heart which asserts that the greater the stretchofthe
ventricle in diastole the greater the strokevolume This law explains a basic control
mechanism of the heart
The frog heart has two atria but only one ventricle
Figure Simpli ed diagram of isolated dog heart lung preparation of Starling
The height of the venous reservoir and the screwclamp regulated central venous pressure
CVP CVP and arterial pressure AP were measured by manometers and AP was held
constantbya variable resistance equivalent to the total p eripheral resistance TPR
Ventricular volume was measured by Henderson s b ell cardiometer an inverted glass
b ell which is attached to the atrioventricular gro oveby a rubb er diaphragm Beat by
b eat volume changes were recorded on a rotating smoked drum Mean cardiac output was
measured directly by collecting blo o d in a measuring cylinder b eyond the resistor After
Knowlton and Starling KS From J R Levick An Introduction to Cardiovascular
Physiology by p ermission
Figure The ventricular function curve
E ect of lling pressure on the strokevolume of an isolated dog heart pumping against a
constant arterial pressure The solid line shows Starling s data for left ventricular stroke
volume LV and left atrial pressure The dashed line shows how the rightventricle has
a similar shap ed curve but a slightly lower lling pressure From J R Levick An Intro
duction to Cardiovascular Physiology by p ermission Lev p
The results of exp eriments such as Starling s are of considerable imp ortance when mo d
elling ventricular mechanics b ecause they provide data ab out the heart as a mechanical
pump when isolated from the atria and the rest of the circulatory system In particular
the e ects due to the elasticity of the aorta are eliminated
Some of Starling s smoked drum recordings are repro duced by Levick Lev p It is
interesting to note that when the lling pressure is changed suddenly the heart adjusts to
the new conditions b oth quickly and smo othly In Figure the strokevolume p eaks at a
pressure of ab out mmHg and then falls This is an abnormal region sometimes called
the descending limb and there may b e exp erimental errors due to leakage through
the atrioventricular valves see Levick Lev p but it would still b e of interest to
repro duce this b ehaviour with a mo del since it is a challenging test for a mo del Katz
Kat p suggests that in some chronically ill patients the heart may op erate up on
this descending limb
For discussions of Starling s Law see
The Physiological Basis of Starling s Law of the Heart by the Ciba Foundation Sta
Sym
Starling s L aw of the Heart Revisited by Keurs and Noble tKN
In Vivo Stress Measurements
In the past various devices have b een implanted in the heart wall and used to measure the
stress but it is di cult to allow for the inevitable e ects of the intrusion Lunkenheimer
Lunkenheimer and Torrent Guasp LLG pages summarize a variety of such
metho ds The stress vector has six comp onents and it is di cult to measure all of these
See also
Huisman Hui
Huisman Sipkema Westerhof and Elzinga HSWE
Huisman Sipkema Westerhof and Elzinga HSWE quoted byHunter and
Smaill HS p
Muss ergaenzt werden PPL fragen
Left ventricular wall motion has b een observed by Greenbaum and Gibson GG
who give further references
Rightventricular wall motion in dogs has b een measured byChuong Sacks Temple
ton Schwiep and Johnson CST
In Vivo Strain Measurements
Avariety of metho ds has also b een used to measure strains in the heart wall Less in
trusion is necessary and recent developments in cardiac imaging promise that accurate
measurements of movements of the surfaces will so on b e available See
Lunkenheimer Lunkenheimer and Torrent Guasp LLG p
Hunter and Smaill HS pages
Waldman Wal
McCullo ch McC
Muss ergaenzt werden PPL fragen
Coronary Perfusion
The suggestion that changes of pressure and ow in the coronary vessels alter the passive
prop erties of the left ventricle has b een studied bymanyworkers Some references are
given by McCullo ch in his Ph D thesis McC p
The term coronary p erfusion refers to the intro duction by normal or arti cial means
of additional blo o d in the coronary arteries This is related to the garden hose e ect
Intramyo cardial Pressure
The extravascular forces in the myo cardium are commonly referred to as the intramy
o cardial pressure
See
Ph D thesis of Rabbany Rab
CWCcheck this de nition
Other Exp erimental Observations
In Chapter of Circulatory Dynamics by Wiggers Wig a varietyof in vivo exp eriments
are describ ed
Pericardial E usion
Wiggers Wig pages discusses the e ect of changes in p ericardial e usion that
is PPL fragen
He quotes several pap ers including
Kuno Kun
Fineb erg Fin
Wiggers Wig
Wiggers Wig Fig p shows graphically the e ect of increasing the p ericardial
pressure the minimum ventricular pressure increases but the maximum ventricular pres
sure decreases For instance if the p ericardial pressure is increased to mmHg the
minimum ventricular pressure rises to mmHg while the maximum ventricular pressure
falls to mmHg PPL fragen
Hyp ervolemia
Oligemia
Arterial Hyp ertension
Aortic Coarctation
Aortic Stenosis
Pulmonary Stenosis
Idoventricular Rhythms
Coronary Occlusion Myo cardial Ischemia
Cardiac Valves
Aortic Regurgitation
Mitral Insu ciency
Mitral Stenosis
Part III
Mathematical Mo delli ng of the
Heart
Chapter
Intro duction
The mo delling of the heart has a long history Already in the seventeenth century Borelli
Bor sketched the arrangementofmuscle bres on the surface of the heart and consid
ered how these could contract
At the end of the nineteenth century Woods Wo o applied the results of Laplace to
relate the pressure in a ventricle to the stress in the heart wall
The Law of Laplace which is derived in section is often quoted in medical texts
One conclusion from this law which nds medical application is that if a diseased heart
increases in size its e ciency drops since R and R increase For a recent discussion
of the Law of Laplace see Moriarty Mor Any realistic mo del must of necessitybe
far more complicated Fortunately mo dern computer developments make it p ossible to
present the results of such mo dels in a readily accessible manner
The longterm goal of mathematical mo delling of the cardiovascular system is to predict
features such as
The blo o d pressure and other characteristics of the blo o d ow
The stresses in and motion of the comp onents of the heart
The general approach is to set up a system of equations which describ es the underlying
physical and biological pro cesses This system of equations will include partial di erential
equations for stresses in the heart wall and blo o d velo cities but may also include ordi
nary di erential equations and algebraic equations This system of equations must then
b e considered b oth mathematically to try to derive general features and numerically to
obtain quantitative results
There are several excellent general references The three volume workofFung Fun
tly app eared as a second edition is a broad Fun Fun one volume of which has recen
and leisurely intro duction to biomechanics The lecture notes of Peskin Pes are often
quoted and emphasize the uid mechanical and mathematical asp ects of mo delling the
heart asp ects which are often somewhat neglected in the literature There has recently
b een a surge of activity in the mo delling of the heart in part owing to the rapid improve
ments in computing facilities The pro ceedings TheoryofHeart edited by Glass Hunter
and McCullo ch GHM is an excellentintro duction to mo dels currently b eing develop ed
and used There is also an extensive review article by Hunter and Smaill HS which
discusses many asp ects of mo delling the heart using continuum mechanics Yin Yin
compares numerical results for di erent geometries and constitutivelaws
Antaki Ant has a lengthyintro ductory section of pages in which mo dels of the
heart are critically discussed and questions such as bre direction p p and p
and compressibility p are addressed with extensive references
Other references include
Part II Ventricular and Myocardial Mechanics inCardiovascular Dynamics and
Models by Brun et al BCL
Cardiovascular Biomechanics by Chandran Cha
Chapter Circulatory System Dynamics by No ordergraaf No o
The Mechanics of the Circulation by Caro Pedley Schroter and Seed CPSS
This is an excellentintro duction to crdiovascular mechanics which tries wherever
p ossible to supply numerical data but also avoids mathematical formulations and
presents mechanics in readily comprehensible terms
The Physics of He art and Circulation by Strackee and Westerhof SW
Engineering Design of the Cardiovascular System of Mammals byDawson Daw
Noble The Cardiac Cycle Nob
Dinnar Cardiovascular Fluid Dynamics Din
Fung A First Course in Continuum Mechanics Fun
Sagawa Maughan Suga and Sunagawa Cardiac Contraction and the Pressure
Volume Relationship SMSS
Sideman and Beyar Analysis and Simulation of the Cardiac System Ischemia SB
Ghista Van Vollenhoven Yang Reul and Bleifeld Cardiovascular Engineering Part
I Model ling GVY
Hopp ensteadt and Peskin Mathematics in Medicine and the Life Sciences HP
The rst step in the mo delling pro cess is to decide whichvariables parameters and pro
cesses signi cantly in uence the problem This decision is partly heuristic and is strongly
in uenced by the need to reduce the complexity to manageable prop ortions In mo delling
the mechanics of the heart twotypical assumptions are
To assume a simpli ed geometry
To assume that heat ow in the heart walls do es not in uence the mechanics of the
heart
Once the signi cantvariables have b een chosen it is necessary to formulate the eld
equations governing these variables There are twotyp es of equations Firstly there are
the equations which represent general physical laws such as conservation of momentum
Secondly there are the constitutivelaws whichembody physical prop erties of the
medium In some cases these can b e simple and uncontroversial as when one can as
sume that heat ow is linearly prop ortional to the temp erature gradient In other cases
the formulation of suitable constitutivelaws is di cult This is the case in ventricular me
chanics where the relationship b etween stress and strain in the heart walls is extremely
complex One frequent cause of di culty is that mo dels nearly always use only one geo
metric scale the constitutiv elaws must re ect pro cesses at smaller scales The dilemma
facing the mo dellist is illustrated bytheLaw of Laplace This law is a useful sim
pli cation but is based on a great numb er of assumptions
The rst mo dels of the heart s wall were develop ed ab out thirtyyears ago and used lin
ear elasticity and or simpli ed geometry While these assumptions were justi ed at the
time the myo cardium is certainly nonlinear and the deformations are so large that the
assumptions of linear elasticity are not tenable With current computing facilities it is
feasible to handle nonlinear material prop erties
The Field Equations
We assume that the myo cardium is incompressible and satis es the nonlinear elasticity
equations
d
div r p for x in
det F
F r
d
Here F denotes the deformation gradient tensor the deviatoric Cauchy stress tensor
and p the pressure These equations are formulated in the deformed b o dy When
solved they must rst b e transformed backonto the undeformed b o dy
The Constitutive Equations
There is a considerable literature on the form of the constitutive equations Following
Bovendeerd Bov we assume the following constitutive equations
S S E S l l t
p a s c
where S represents the passive stress and S represents the active stress created by the
p a
activation of the muscle The terms l and l represent the lengths of the c ontractile ele
c s
ment and sarcomere resp ectively
S is the second Piola Kirchho stress tensor which is related to the Cauchy stress tensor
d
via
d T
S detF F F
It is assumed that the material is hyp erelastic that is that a function W exists the
stored energy function such that
W
S
p
E
where
h i
W E C exp a I a II a E a E E
E
E
and
I E E E
E
II E E E E E E E E E
E
The active stress S is de ned as
a
l T E T l
c a a max s
T T A l B t B t l
max c r s d s s
where the functions T A l B t and B t l are explicitly de ned and l satis es
c r s d s s c
the ordinary di erential equation
dl T T
c a max
v T T
a max
dt T T
a
T T T dl
a max a c
v exp a T T
T a max
dt T T T
a
It is a feature of nonlinear elasticity that standard pure traction traction displacement
and pure displacement problems mayhave more than one solution and this is indeed
desirable see Ciarlet Mathematical Elasticity Vol I Three Dimensional Elasticity Cia
page An example see Gurtin Gur p shows that if an incompressible isotropic
homogeneous cub e is loaded uniformly and the constitutive equation is of the form
T
T I FF
where is a real numb er and is a given constant then there may b e up to di erent
solutions of the tension problem
As shown by Ball see Ciarlet Cia p the displacement traction problem has a
solution if among other conditions the stored energy function W is p olyconvex that
is
W E W x F CofF det F
where W is convex
In the accompanying talk we will try p erhaps provo catively to stimulate discussion on
several questions
Which mo del is appropriate
Whichnumerical metho ds are b est
What should b e the goal of mo delling
The Goal of Mo delling
We conclude with a numb er of questions
Which if any of the following should b e included in a mo del
Fibre direction
Porous ow
Residual stress
Restoring forces
Arteries veins
Inertia
Epicardium Endo cardium Pericardium
Is it true to say that current mo dels can predict global relationships such as the
pressure volume curve reasonably well
Is the next goal the prediction of lo cal variables particularly in diseased hearts If
so how should one approach this task
Should some standard problems b e develop ed for use in comparing software
Diastolic Expansion
As describ ed in the section the ventricles are closed for a short time ab out s
during which the pressure falls isovolumetric relaxation Then the tricuspid and mitral
valves op en and the ventricles ll with blo o d from the atria ab out s
The mechanisms underlying the motion of the ventricles during diastole have b een the sub
ject of conjecture for manycenturies Lunkenheimer Lunkenheimer and Torrent Guasp
LLG pages summarize the developmentofvarious explanations from Aristotle
to the presentday Gilb ert and Glantz GG give a recent review as do McCullo ch
and Omens MO p Fung Fun p quotes the Chinese b o ok Internal Classic
written ab out BC but the ideas seem very imprecise
wn in Figure In par The variation of blo o d pressure with time during diastole is sho
ticular it can b e seen that initially the blo o d pressure continues to fall From this and
from direct measurements Meesmann Mee and Robinson Factor and Sonnenblick
RFS it is usually concluded that diastolic suction takes place that is that the heart
walls expand and suck blo o d in While this is quite likely it do es not seem to b e essential
one could imagine a system in which the pressure decrease arose b ecause the ventricles
expanded and the pressure dropp ed in accordance with the Law of Laplace see Equation
on page
To explain the observations two basic mechanisms suggest themselves
Ventricular Blo o d Pressure
In Harvey asserted that the blo o d pressure is the sole driving force during diastole
and that the ventricle muscles play no role While this is not completely true the blo o d
pressure is certainly an imp ortant factor Towards the end of the ventricular lling phase
the atria contract thereby forcing more blo o d into the ventricles this is called the atrial
b o ost and is of imp ortance during exercise when the time for passiveventricular lling
is reduced see Levick Lev page
The role of the lling blo o d pressure seems to us to have b een somewhat neglected in
the literature on mo delling So far as weareaware only Peskin and his coworkers have
mo delled the ow of blo o d in the atria and ventricles see PM b These results are
signi cant but they do not so far as we can see provide values for the blo o d pressure
which can b e compared with the observations in Figure
Myo cardial stress
Although the heart is relaxed during diastole the heart walls are not stress free For
example in PPL fragen Dr James Johnson see Philip Phi p found that the
freshly excised heart of a turtle continues to b eat and prop els itself forward when placed
in water where the e ect of lling pressure is absent Three sources of stress have b een
considered
Elastic Recoil
It has b een known but at times forgotten PPL fragen since the work of Vieussenius in
see Vie that the relaxed heart recoils It is p ossible to observe this an empty
ventricle expands after b eing compressed manually see Lunkenheimer Lunkenheimer
and Torrent Guasp LLG p In the past various mechanical mo dels have b een
prop osed to explain the elastic recoil For example in Figure a mechanical mo del for
wn Hort suggests that the intramyo cardial tissue elastic recoil due to Hort Hor sho
between the myo cytes is twisted during systole and then recoils during diastole
Recently scanning electron microscop es have b een used to reveal the detailed structure of
the connective tissue in the myo cardium However this has not yet led to an acceptable
constitutivelaw see Hunter and Smaill HS p
Hort Hor considers the question of reorientation of the bres during systole and di
astole Thus Hort Hor p Aus der Ueb ereinstimmung des Faserverlaufes in
kontrahierenden und dilatierenden Herzen geht hervor dass es in der Herzmuskulatur
eine Mechanismus geb en muss der es den einzelnen Muskelschichten erm oglicht sichver
schiedenen Kammerfuellungen durch Aenderungen der Schichtdicke und Flaechengro esse
anzupassen
Hort Hor p also discusses the idea that energy is stored during systole for release
during diastole and quotes Feneis F enn Benningho Rushmer Chrystal and W agner
See Lunkenheimer There is a mistake here since neither Hort Hor nor Hort Hor has this
gure
Figure Hort s Mo del for elastic recoil
Residual Stress
Fung Fun p vigorously argues the case that at the end of systole the heart has a
residual stress distribution stress in the absence of external forces This idea has b een
develop ed by Omens Ome and McCullo ch and Omens MO p Fung p oints
out that residual stress is well known and used in engineering Both Fung Fun p
and McCullo ch and Omens MO p show examples of living tissue which relaxes
when cut thus proving the existence of residual stress
The di erence b etween residial stress and elastic recoil app ears to b e as follows Resid
ual stress can arise in statically indeterminate elastic b o dies even when the constitutive
equations are linear Elastic recoil presupp oses that while the myo cyte bres are con
tracting during systole other comp onents are b eing compressed in other w ords elastic
recoil presupp oses a nonlinear mechanism in which di erent elastic comp onents act during
systole and diastole CWC check
Residual stress mayplay a role in several ways
If the constitutivelaws are nonlinear then the relationship b etween incremental
stress and incremental strain dep ends up on the total stress and is thus a ected by
any residual stress
If residual stress is included then the total stress distribution is found to b e smaller
as would b e exp ected in an organ which has develop ed b oth through growth and
through evolution The Ph D thesis of Omens see Ome contains a very clear
description of residual stress and its measurement
Residual stress may a ect the coronary blo o d ow and hence in uence the onset of
ischaemia reduced blo o d supply see Ome page
The relationship b etween residual stress and cardiac hyp ertrophy enlargement
of the myo cytes due to an increased workload is investigated by Omens in
Chap er V Ome
Yang Tab er and Clark YTC allow for residual stress in their mo del of the embryonic
chick heart
Ventricular Geometry
As is clear from Figure the heart is a complicated three dimensional ob ject In
Woods Wo o mo delled the left ventricle as a thin walled sphere Later thin walled
ellipsoidal geometry was used by several authors Since the left ventricular wall is of com
parable thickness to the ventricular radius thickwalled mo dels were intro duced For
references see Hunter and Smaill HS p
Only comparatively recently have fully three dimensional measurements b egun to b ecome
available The fully three dimensional ventricular geometry of the dog heart was measured
mechanically by Nielsen Nie and Nielsen Le Grice Smaill and Hunter NLSH The
present authors intend to provide similar information for the human heart in the near
future PPL fragen
So far as we are aware Octob er no workers haveasyet used mo dels with full
three dimensional geometry for the human heart
A large numb er of p ossible factors can b e considered
Geometry
Cylindrical
Spherical
Woods W oo
Ellipsoidal
Wong and Rautaharju WR
Prolate Spheroidal
Mirsky Mir
Shell Theory
Thin
Wo o ds Wo o
Thick
Mirsky Mir
Wong and Rautaharju WR
Constitutive Stress Strain Equations
Linear Elasticity
The symp osium The Physiological Basis of Starling s Law of the Heart Sym page
contains interesting articles and discussions on general topics such as the e ect
of geometry
Chapter
Muscle Mechanics
Intro duction
A muscle is any one of the contractile brous band or bundles that pro duce movement
in animal b o dies A muscle is therefore biological tissue which can b e either relaxed or
active in the latter state force can b e generated F or our purp oses weareinterested in
muscle mechanics the study of the motion of and forces generated bymuscles
The two basic typ es of muscle striated muscle and smo oth muscle di er in their
organization Striated muscle cells are organized very systematically and when viewed
under a microscop e have a de nite banded structure Smo oth muscle cells are organized
much less regularly
Striated muscles o ccur as
skeletal muscles which make up approximately p er cent of the human b o dy
byweight As the name suggests the skeletal muscles are attached to the skeletal
structure of the b o dy and are resp onsible for the movement of arms legs etc
cardiovascular muscles which make up ab out p er cent CWCcheck of the
human b o dy byweight and are the source of the forces in the heart Although similar
in structure to skeletal muscles they also di er in imp ortantways
Approximately p er cent of the human b o dy is smo oth muscle which is found in the
inner organs the eye the blood vessels etc There are two ma jor typ es Guyton Guy
p
Multi unit smo oth muscle which is comp osed of discrete smo oth muscle bres
each of which can contract indep endently of the others
Single unit smo oth muscle which consists of hundreds or millions of muscle bres
that contract together as a single unit This typeofsmoothmuscle is also known
as synctial smo oth muscle or visceral smo oth muscle
The ab ove classi cation of muscles is very broad within eachtyp e of muscle there are
many sp ecial varieties each one adapted to its particular role
For our purp oses we wish to characterize the motion of and the forces generated by mus
cles so that these can b e incorp orated into mathematical mo dels Two basic approaches
can b e followed
The phenomological or engineering approach in which the elastic prop erties
of muscles are measured in the lab oratory and used to formulate constitutivelaws
Under this heading we include the the classical exp eriments on muscles of Blix
Bli Hill Hil
The molecular approach With the development of high p owered electron mi
croscop es and sophisticated chemical metho ds it has b ecome p ossible to elucidate
the structure and mechanisms of individual muscle cells to the extent that one can
for example say that a particular typ e of cell has ab out myosin heads p er m
which form crosslinks with the adjacent actin molecules
Of the two appproaches mentioned ab ove the molecular approach is clearly the more
fundamental However for the forseeable future this approach cannot by itself provide
all the information needed by the mo deller
The molecular approach has concentrated on the contractile units the sarcomeres
But one must also take account of the elastic prop erties of other comp onents of the
muscle
Even when the mechanism of the sarcomere is fully understo o d this is not the end
of the story b ecause the activation is caused by calcium ions whose densitymust
b e mo delled
On the other hand the phenomological approach is also not without di culties
The preparation and testing of sp ecimens is di cult
The stress strain resp onse is highly nonlinear It is not clear how the strain energy
function should b e chosen In at least one case Smaill and Hunter SH p the
b est t material parameters havephysically unacceptable values
The mo deller should therefore combine the information from b oth the phenomological
and the molecular approach The situation is well summarizedby the following quotation
from Smaill and Hunter SH pages
Wehave already stated that any realistic constitutive formulation for pas
sivemyo cardium must incorp orate microstructural information It is worth re
iterating that an appropriate constitutivelaw should b e compact with unique
material parameters Wehave demonstrated that the last of these require
ments cannot easily b e met by phenomenological constitutive formulations
that use highly nonlinear functions However it is unlikely that the rst re
quirement could b e satis ed by a constitutivelaw based completely on cardiac
mircostructure Thus a constitutive formulation that incorp orates the most
advantageous features of b oth phenomenologically and microstructure based
approaches app ears the b est option at present
In the reminder of this chapter we rst summarize these two approaches and then conclude
with some typical mo dels of cardiac muscle
Some general references include
R McNeill Alexander Elastic Mechanisms in Animal Movement Ale
A V Hill First and Last Experiments in Muscle Mechanics Hil
Z He Mathematical models in muscle by periodic stimuli He
Thomas A McMahon Muscles Re exes and Locomotion McM
Rob ert M Miura Lectures on Mathematics in the Life Science Vol Some Math
ematical Questions in Biology Muscle Physiology Miu
ter Keurs and Noble Starling s Law of the Heart Revisited tKN
Hopp ensteadt and Peskin Mathematics in Medicine and the Life Sciences HP
Daniel J Schneck Mechanics of Muscle Sch an engineering approach
Yuan cheng Fung Biomechanics Mechanical Properties of Living Tissues Fun
Chapter Skeletal Muscle Chapter Heart Muscle and Chapter Smo oth
Muscle
The b o ok of McMahon Muscles Re exes and Loc omotion McM is an excellentintro
duction
The Phenomological Approach
Muscle in particular cardiac muscle is a biological material which can b e either relaxed
or passive during diastole or active during systole It is usual to consider these states
separately
Early in this century the term new elastic b o dy was coined to convey the idea that
excitation of muscle instantly changes its elastic state from a compliant level into a sti
level The excited muscle thus b ecomes a new elastic b o dy
Hill and other physiologists likened contracted skeletal muscle to a co cked
spring sp eculating that the amount of energy released by contraction dep ends only on
the preloaded length this w as disproved by the discovery of the Fenn e ect byFenn
Fen see Sagawa Maughan Suga and Sunagawa SMSS p
The Fenn e ect
The Fenn e ect was rst observed byO Fenn in in a skeletal muscle isotonically
contracting at a constant preload the total heat generation plus external work signi cantly
increased and then decreased as the extent of shortening was increased by decreasing the
afterload force from isometric force to nearly zero Sagawa Maughan Suga and Sunagawa
SMSS p See Rall Ral
Feng e ect
Concerning the Feng e ect see
Ciba Foundation The Physiological Basis of Starling s Law of the Heart Sym p
Feng The e ect of length on the resting metabolism of muscle Fen
Classic Results
For the purp oses of biomechanics it is necessary to determine the force generated bya
muscle It turns out that in cardiac muscle these is a complicated relationship b etween
the force develop ed the length of the muscle the velo city of shortening and the time see
e g Brutsaert Bru
To a growing extent this relationship can b e explained at the molecular level as brie y
discussed in Section Here we describ e some classical results
Some brief remarks on early workonmuscle mechanics with esp ecial reference to car
diac muscle are given by Guz Guz and Jewell Jew It was shown bySchwann in
that muscles contract more forcibly when stretched A famous pap er by Blix in
Bli studied the relation b etween length and tension in muscle Early exp erimental
work on the heart is due to Roy Martin Howell and Donaldson
and Roy and Adani The most widely quoted pap ers are those of Otto Frank
Fra on the frog heart and Patterson Pip er and Starling PPS the latter pap er
presenting work p erformed at University College London and the University of Berlin
It is interesting that these work ers b egin their pap ers by quoting work on isolated muscles
and using this as a guideline for their own work on the heart
Atypical exp erimental setup for a single muscle is shown in Fig The muscle is rst
xed at one end and then stretched using a small weight w the pre load Themuscle is
then stimulated electrically As it shortens it picks up a second weight W the afterload
Some basic mo des of testing are
isotonic Here the muscle is allowed to shorten the b earing afterload W The force
is thus known The length of the muscle and sp eed of shortening are measured See
ii and iii in Fig
isometric loading Here the muscle length is kept xed The tension is measured
See i in Fig
As can b e seen from Fig an increase in the initial length of the muscle leads to an
increase of tension velo city and shortening Several comments are due here
Avery plausible explanation of the
In Fig the dep endency on time is not shown
In Fig the length tension curveonlygoesuptothemm ordinate
Mo dels
Wehave not consulted these early pap ers
Figure Contractile b ehaviour of isolated myo cardial muscle
Left Simpli ed diagram of arrangement used to study the contraction of isolated cat
papillary muscle The arrangement shown allows isotonic contraction to o ccur the weight
lab elled afterload b eing picked up as so on as shortening b egins The weight lab elled
preload sets the resting length If the preload is clamp ed in place contraction b ecomes
isometric Right Three fundamental relations i shows isometric contraction at increas
ing lengths ii and iii show isotonic contractions b eginning from two di erent resting
lengths mm and mm Contractile force velo city and shortening are all increased
by stretching the relaxed muscle See Sonnenblick Son From J R Levick An In
troduction to Cardiovascular Physiology by p ermission
Molecular Muscle Mechanics
In Hugh Huxley and J Hanson HH and Huxley and Niedergerke HN inde
p endently prop osed a mo del on the molecular level for
Ventricular Interaction
So far as we are aware February no workers haveasyet mo delled ventricular
interaction The in uence of the rightventricle is discussed by McCullo ch and Omens
MO p and Antaki Ant p who give further references Since the
twoventricles share muscle bres and adjoin one another one must exp ect a signi cant
iteraction see SG and SGL In uence of epicardium etc see Hunter and Smaill
SH p and Fung Fun p and Humphrey and Yin HY
PassiveMyo cardium
Passivemyo cardium is an inert biological material whichcan in principle b e tested in
the lab oratory sothat in principle constitutivelaws can b e exp erimentally determined
There are however many di culties some of which are mentioned b elow
Exp erimental Metho ds and Results
General references on the testing of biological materials will b e found in App endix H
We are not concerned here with hte sp eci c techniques but do wish to draw attention to
some of the di culties see also Smaill and Hunter SH p
Soft biological material can b e damaged on excision It is also necessary to a bu er
solution and to keep the sp ecimen in control the temp erature See also Fung Fun
p
o cardium is three dimensional but so far to our knowledge as of April The my
only one dimensional uni axial and two dimensional bi axial tests have b een car
ried out
Almost all tests have b een carried out on papillary muscle and trab eculae carnae
But the myo cardium is far from b eing homogenous as observed for example by
Feneis Fen and shown in recent electron micrographs see e g Figures A to
D in Smaill and Hunter SH
The elastic prop erties of the passivemyo cardium have b een measured byseveral workers
including Pinto and Fung PF terKeurs Rijnsburger v an Heuningen and Nagelsmit
tKRvHN Yin Strumpf Chew and Zeger YSCZ In Figure w e repro duce the
representative results of Smaill and Hunter SH
Figure illustrates several p oints
The stress strain curves are monotone and convex
We remind the reader that the papillary muscles bulge from the endo cardium and are attached to
the chordae tendineae see Figure Cross sections The trab eculae carnae also bulge from the inner
wall but are not as conspicuous as the papillary muscles
Figure Stress extension relations for left ventricular midwall sp ecimen during st
th and th cycles of equibiaxial loading Cycle p erio d sec and sp ecimen thickness
mm The solid lines indicate the order of loading in the cross b er direction for
clarity these are omitted in the b er direction
There is a p ointbeyond which the material do es not stretch
For the same strains the stress in the bre direction is far higher than that in the
cross bre direction
The crude estimates of Young s mo dulus E are quite low For the bre direction
an extension ratio of corresp onds to a stress of kPa That is
stress kP a
E kP a
strain
For comparison
Material Young s mo dulus E
Steel
stainless steel GPa
Tendon
wet tendon GPa
Thus wet tendon is ab out
times sti er than the passivemyo cardium The great discrepancy b etween tendons
and muscles is simply explained a tendon is a relatively simple material which
must only transmit tension In the context of the heart one sees that relatively thin
chordae tendineae su ce to transmit the forces develop ed by the papillary muscles
Constitutive Equations
Given exp erimental data such as that in Figure one must construct corresp onding
constitutive equations which express the stress as a function of the strain and p ossibly
other prop erties
Strain Energy Functions
Although the stress is known to b e dep endent on time relaxation past history a la
Bowditch and frequency of stimulation see section the ma jorityofworkers have
See Kaye and Laby Physical and Chemical Constants KL and Bennet Ker Dimery and Alexan
der Regional deformation and contractile function in canine right ventricular freewall BKDA Dry
tendon has substantically di erent prop erties see Vincent Structural Biomaterials Vin
We are not aware of measurements of the elastic prop erties of the chordae tendineae and are assuming
that they are mimilar to those of other tendons
chosen to formulate the constitutive equations using a strain energy function That is
W E
S
E
where S is a stress tensor E is a strain tensor and W is a strain energy function
Bovendeerd
Bovendeerd Bov page suggested the following strain energy function
W E
S
E
where S is the second Piola Kirchho stress tensor based on the deviatoric Cauchy stress
tensor and E is the Green Lagrange strain tensor
The myo cardial tissue is assumed to b e transversely isotropic with resp ect to the bre
direction With resp ect to an orthonormal co ordinate system with direction e parallel to
the bre direction and cross bre directions e and e the strain tensor E is written as the
strain matrix E with comp onents E The strain energy function WE for a material that
ij
is transversely isotropic with resp ect to e can b e expressed as a single valued function
of the following set of scalars Malvern Mal Green and Adkins GA page
I E E E
E
II E E E E E E E E E
E
III det E
E
I E
E
II E E
E
ischosen so that W is zero in the unstrained state The functional form of W E
stress is an exp onential function of strain and no stresses are predicted in the unde
formed situation The following strain energy function is consistent with these conditions
h i
W E C exp a I a II a E a E E
E
E
Because of the additional condition of incompressibility of the cardiac tissue the scalar
III is left out of this strain energy function The values of the material parameters are
E
estimated from tting this function to exp erimentally obtained data Firstly according to
the exp erimental ndings of Yin Strumpf Chew and Zeger YSCZ the ratio of second
Piola Kirchho bre to cross bre stress is required to equal under conditions of equal
bre and cross bre stretch This condition is satis ed by setting
a a a
Secondly it is required that the uniaxial stress strain data obtained by ter Keurs Rijns
b erger van Heuningen and Nagelsmit tKRvHN are describ ed reasonably well Thirdly
a is set at zero b ecause of the lack of exp erimental data on the mechanical b ehaviour of
cardiac tissue under conditions of shear stress The resulting parameter values are listed
in table
See App endix G
parameter C a a a a
value
unit kPa
Smaill and Hunter
Smaill and Hunter SH p adapt a suggestion of Fung and use the following strain
energy function for passivemyo cardium
Q Q
W C e C e
where
Q C E C E E
and
Q C E E C E E E
It is hereby assumed that the material is transversely isotropic direction is aligned with
the bre axis The stress is given by
W W
p
ij ij
E E
ij ji
where p is the hydrostatic pressure The b est t parameters for four left ventricular mid
wall sp ecimens are given in Table
C C C C C C
mean
n
Table Best t material parameters for left ventricular midwall sp ecimens
Inverse Metho ds for Parameter Estimation
Given the di culty of direct measurement some authors have constructed mo dels and
then estimated the parameters so as to t the observations
Guccione McCullo ch and Waldman GMW
Sideman
where wehave corrected apparent errors in the mean and standard deviation for C
Chapter
Cardiac Mechanics
Intro duction
The myo cardium is a complex structure containing see Robinson and Kinne RK
Smaill and Hunter HS Horowitz Hor and Lunkenheimer et al LRC b
myo cytes The myo cyte is the heart s muscle cell In shap e it resembles an ellip
soidal cylinder with a diameter of m and length of m The cell is
branched and is attached to adjacent cells in an end to end fashion Eachmyocyte
is connected with several others to form a three dimensional network of cells The
interface b etween adjacent cells is called an intercalated disk
coronary blo o d vessels Within the heart wall is a dense network of coronary
blood vessels One or more capillaries run along the b oundary of eachmyo cyte
parallel to the long axis The coronary arteries and veins act muchlikerivets within
the wall
collagen matrix Individual myo cytes and bundles of myo cytes are connected by
a collagen matrix which consists of a dense network of struts of collagen typ e I
and typ e I I I predominate
connective tissue Additional connective tissue consists of collagen elastin gly
coaminoglycans and glycoproteins
The interstitial space or interstitium between cells is lled by interstitial uid
Since the p ores are so small the uid moves mainly through di usion
The monograph edited by Robinson and Kinne RK Cardiac Myocyte Connective Tis
sue Interactions in Health and Disease Issues Biomed provides a very readable overview
of the structure of the myo cardium Topics considered include
The bio chemistry of the myo cardium
The structure of the connective tissue
The elastic prop erties of myo cytes
In particular Covell Cov discusses the functional dynamics of the myo cardial tissue
Robinson Rob summarizes results obtained using scanning electron microscopy the
light microscopy highvoltage transmission electron microscopy antib o dy lo calization
and in situ hybridization For our purp oses the following observations are of particular
interest
Robinson see page rep orts on signi cant di erences on collagen contentinrats
see Table
Young adults Aged rats
Left ventricle
Rightventricle
Table Collagen content
Robinson Rob p summarizes the o ccurrence of four self similar patterns frac
tal analysis of structures weave branced bre bilt and heliv These patterns o ccur
at several levels
Robinson Rob p sp eculates on the transfer of information b etween myocyte
and broblast
Robinson Rob p raises the question of the relativemechanical strengths of
di erent networks
Robinson Rob p refers to ma jor di erences in cases of stunned myo cardium
or infarct
For mo delling purp oses it is necessary to set up constitutivelaws which mo del the my
o cardium on a larger scale but it is of course imp ortant to take account of the microscopie
structure We therefore brie y describ e the comp onents of the heart s wall in somewhat
greater detail b efore moving on to describ e p ossible constitutivelaws
The Comp onents of the Heart s Wall
Exp erimental Stress Strain Measurements
ConstitutiveLaws
There is an extensive literature on the mechanical prop erties of isolated cardiac muscle
The rst exp eriments were apparently p erformed by Sonnenblick Son who measured
the b ehaviour of isolated myo cardial cat muscle His results are shown in Figure
These results show that stretching the muscle enhances its subsequent contractile energy
in agreement with the Frank Starling Law Levick Lev p discusses the physiologi
cal mechanisms underlying this b ehaviour
This seems very late PPL fragen
Hydraulic E ects
Living tissue is bathed in liquid PPL fragen In addition the heart is criss crossed by
a network of blo o d vessels As a result one may regard the myo cardium as a structure
of bres and blo o d vessels surrounded by uid The pressure in coronary blo o d vessels
is called the p erfusion pressure or coronary p erfusion pressure PPL fragen and
provides a mechanism to explain the expansion of the ventricles There is an analogy with
a hosepip e which straightens as it is lled with water For this reason hydraulic e ects are
sometimes called the garden hose e ect See Lunkenheimer Lunkenheimer and Torrent
Guasp LLG p for further references See McCullo ch and Omens MO pages
and pages and Hunter and Smaill HS pages for an evaluation
of the in uence of coronary p erfusion
McCullo ch McC rep orts on measurement on the e ect of coronary p erfusion on left
ventricle mechanics The pressure in uid outside the myo cardial bres and blo o d vessels
is called the intramyo cardial pressure See Rabbany Rab
Several mo dels take account of the uid in the myo cardium A theory for an elastic struc
ture lled with a uid think of a sp onge was rst formulated by Biot Bio It was
later extended to the case of nite deformations by Biot Bio and was further gen
eralized to a theory of mixtures byBowen Bow Bow These ideas have b een
used CWC check by Bogen see Bog a and Bog b to mo del the swelling of tissue
Huyghe van Camp en Arts and Heethar HvCAH use a p orous medium nite element
mo del to mo del the diastolic left ventricle
Bogen Bog a Bog b considers strain energy descriptions of biological swelling In
the second article multiple uid compartment mo dels are considered together with their
application to cardiac muscle The mo dels are visualized as constisting of networks of
springs or elastic tub es For the case of a single uid with two elastic networks a and b
Bogen Bog b equ suggests the strain energy function
k k k
k k
a a a a
a a
s
s s
k
a
k k k
a a a
k k a
a s
K
a
k k k
k k
b b b
b
b b
s
s s
k
b
k k k k
b b b
k
b
b s
K
b
Bogen draws some general conclusions but do es not give sp eci c applications to the heart
He also considers the puzzling observation byPogatsa Koltai and Grosz PKG that
the ventricular sti ness increase when myo cardial water content is either increased ab ove
or reduced b elow the normal water content PPL fragen
In the rst article Bogen considers various p ossible laws and shows that some are imp os
sible Bog a p
Myo cytes
Concerning myocytes see
Rabbany Funai and No ordergraaf RFN
Chapter
Mo delli ng the Fibre Architecture of
the Myo cardium
In Part I the observations of the brous structure of the myo cardium have b een describ ed
In this chapter we attempt to use geometrical and top ological metho ds to describ e this
structure There seem to b e several p ossibly useful to ols
Foliations
In various works e g Krehl Kre Pettigrew Pet Streeter Str Figure Torrent
Guasp Gua the heart and in particular the left ventricle is depicted as consisting of
nested shells
Mathematically this corresp onds to a two dimensional foliation of the heart or left
ventricle the individual shells b eing called leaves
Top ology
The analysis of three dimensional structures is treated by Winfree Win pages
Chapter and and pages References given include
Bouligand Bou
Gordon and Winfree GW
Thursten and Weeks TW
Delbruck Del
Fuller Ful
Fuller Ful
See App endix Foliations
Winfree Win
Winfree and Strogatz WS
Geo desics
Concerning geo desics see
Thompson On Growth and Form Tho p
Chapter
Mo dels with Spherical Geometry
The Law of Laplace
We b egin the Law of Laplace whichwas rst stated byWoods Wo o can b e derived
with the help of Figure see Fung Fun page
Figure The Law of Laplace
Consider a thick walled hemisp erical shell with inner radius r thickness t and outer
radius r t Let the pressure acting on the outside b e p pressure from p ericardium
Consider the equilibrium of forces in the vertical direction One obtains
r t r p r p r t
i
where is the average circumferential wall stress
Hence
p r p r t
i
r r t
When p and t is small we obtain in the limit as t the equation
p r
i
t
for the stress which can also b e written in the form
t
p
i
r
For applications see Woods Wo o Moriartiy Mor and Grossman Gro
The equation is the simplest form of a relation known as the Law of Laplace It
shows that the pressure created by a thin shell is prop ortional to the thickness t prop or
tional to the circumferential stress and inversely prop ortional to the radius r
Laplace s Law is often used to compute the stress in the left ventricular wall in fact the
curvature of the left ventricular is often negative see Greenbaum and Gibson GG
If we wish to obtain expressions for the pressure p the circumferential stress and the
i
thickness t as functions of r it is necessary to derive additional equations
Let the original shell have radius R and thickness T Since the volume of the heart wall
remains nearly constant wehave
R T R r t r
so that
r t r R T R
For small t we obtain
r t R T
Next we need a relationship b etween the wall stretch and the wall stress We use
the relationship prop osed byValanis and Landel VL
C H
where C and H are constants and tting this to the data for rat papillary muscle it is
found Moriarty Mor page that
C
with
The value for C seems to di er greatly from values given byFung Mechanical Properties pages
and
and
C g cm
Pa
Pa
The equations and comprise a system of three equations from which t
and p can b e determined as functions of the radius r or equivalently as functions of the
i
left ventricular volume v r
Spherical Mo dels
We summarize the predictions for the pressure volume P V and pressure stress P S
relations predicted byseveral mathematical mo dels in which the left ventricle LV is
mo delled as a spherically symmetric sphere Wedraw heavily on the article of Moriarty
Mor Chapter of the b o ok of Mazumdar Maz contains a short description of some
mo dels apparently also based on the pap er of Moriarty
Analytical Work
There is clearly relatively little scop e for analytical work Nevertheless suchworkisof
ten valuable since it provides guidelines and viewp oints in a way that numerical results
cannot do
The sort of results which one might hop e to establish analytically would b e of the form
To show that for a certain class of mo dels particular a e g variable the circumfer
ential wall stress increases monotonically from epicardium to endo cardium
To show that for a certain class of mo dels a particular variable e g the wall stress
is maximized for axially symmetric geometry
In greater generality theLaw of Laplace can b e stated as follows
Let P denote the normal force exerted by the heart wall on the blo o d in the ventricle
the Let S b e the tangential stress in the wall w the thickness of the wall and R and R
principal curvatures Then the Law of Laplace states that
P Sw
R R
Woods Woo veri ed this formula exp erimentally Authors using the formula to com
pute the stress in the heart s wall include
Sandler and Do dge SD for patients with heart disease using b oth R and R
Sometimes called the Laplace Young equation since it was discovered indep endently by the Marquis
de Laplace and Thomas Young
Comparison of Mo dels
References in whichvarious mo dels are compared include
Huisman Sipkema Westerhof and Elzinga HSWE
Sandler and Do dge SD use Laplace s Law to estimate wall stress They quote references
to show that the di erence b etween thin walled and thick walled mo dels is of the order
of Sandler and Do dge SD p
Chapter
Mo dels with Cylindrical Geometry
Theo Arts A Mathematical Model of the Dynamics of the Left Ventricle and the
Coronary Circulation Art mo del the left ventricle as a cylinder comp osed of
concentric cylindrical shells
Arts Reneman and Veenstra Amodel of the mechanics of the left ventricle ARV
To ezeren Static analysis of the left ventricle To e
Feit Diastolic pressure volume relations and distribution of pressureand ber ex
tension across the wal l of a model left ventricle Fei
Chadwick The myocardium as a uid ber continuum passive equilibrium con gu
rations Cha and Chadwick Mechanics of the left ventricle Cha
Ca isch Strukturgerechte Analyse der Myokardfaserspannungen Caf
Pelle Ohayon Oddou and Brun Theoretical models in mechanics of the left ventri
cle POOB
Skalak Approximate formulas for myocardial ber stresses Ska
Costa Hunter Wayne Waldman Guccione and McCullo ch A three dimensional
nite element method for large elastic deformations of ventricular myocardium Part
II Prolate spheroidal coordinates CHW
Guccione McCullo ch and Waldman GMW Passive material properties of intact
ventricular myocardium determinedfrom a cylindrical mode construct a cylindrical
mo del which they use to estimate material prop erties
In their second pap er Arts Veenstra and Reneman AVR compare their theoretical
results with exp erimental data for four dogs and obtain go o d agreement CWC check
Chapter
Mo dels with Spheroidal Geometry
Beyar and Sidemean Spatial energy balance within a structural model of the left
ventricle BS
Bovendeerd Huyghe Arts van Camp en and Reneman In uenceofendocardial
epicardial crossover of muscle bers on left ventricular wal l mechanics BHA
also incorp orate transversal bres
Bovendeerd Arts Huyghe van Camp en and Reneman Dependenceoflocal left ven
tricular wal l mechanics on myocardial ber orientation A model study BAH
Beyar and Sideman Acomputer study of the left ventricular performancebased
on the ber structure sarcomere dynamics and transmural electrical propagation
velocity incorporate bres and electrical propagation BS a
Bovendeerd Ph D thesis The Mechanics of the Normal and Ischemic Left Ventricle
During the Cardiac Cycle A Numerical and Experimental Analysis Bov
ter Rogers Guccione Waldman and McCullo ch A three dimensional Costa Hun
nite element method for large elastic deformations of ventricular myocardium
Part I Cylindrical and Spherical Polar Coordinates CHR and Costa Hunter
Wayne Waldman Guccione and McCullo ch A three dimensional nite element
method for large elastic deformations of ventricular myocardium Part II Prolate
Spheroidal Coordinates CHW
Chapter
Mo dels with Additional Features
The Pericardium
Only a few workers have mo delled the p ericardium
Misra and Singh MS incorp orate the p ericardial pressure in their mo del and con
clude MS page that the e ect of the p ericardial pressure which is assumed
to b e one tenth of the left ventricular pressure is quite prominent
Humphrey and Yin HY discuss the e ect of the p ericardium Fung Fun p
also comments on the role of the p ericardium
Fibres
Hydrostatic Skeletons and Muscular Hydrostats
A hydrostatic skeleton see Kier Kie is a uid mechanism that provides a means by
which contractile elements maybeantagonized Hydrostatic skeletons o ccur in a remark
able variety of organisms with examples from not only invertebrates but also vertebrates
Ahydrostatic skeleton is typically considered to include a liquid lled cavity surrounded
byamuscular wall reinforced with connective tissue bres
A muscular hydrostat is a hydrostatic skeleton that consists of a tightly packed three
dimensional arrayofmuscle bres Examples include the tentacles of the o ctopus the
tongues of many animals and the trunk of an elephant
There are obvious similaritiesbetween muscular hydrostats and the brous structure of
the heart wall Thus Lunkenheimer and Whimster LW section use the structure
of the uvular muscle and the tongue to illustrate p ossible mechanisms during diastolic
unfolding
There are some interesting mathematical problems in this area For example Kier Kie
page considers helical muscles wrapp ed around a circular cylinder and claims that
there is a torsional force with no longitudinal force only if the angle of the muscles is
exactly
See also
Johnson and Kier JK
Alexander in Chapter Fibre wound animals of Elastic Mechanisms in Animal
Movement Ale
See also Alexander Ale Clark and Cowey CC and Harris and Crofton HC
For a brief useful accountofvarious biological structures see Chapters and of Life s
Devices byVogel Vog
Fluid Fibre Mo dels
Several authers have considered uid bre mo dels in whichthemyo cardium is treated
as a uid with an emb edded bre matrix Such mo dels thus ignore the elasticity of the
myo cardium but b ecause of their simpicitiy can sometimes b e solved analytically See
Ca isch Caf
Fung Fun p
Skalak Ska
Chadwick und Cha Cha
For example Chadwick Cha considers the bre direction as a given function of x
The Cauchy stress tensor takes the form
p T
ij ij i j
where T is the magnitude of the bre tension The equation of equilibrium is
div
If u x denotes the displacement
x x u x
then since the material is assumed to b e incompressible
div u
Finally the equations are completed by an equation which expresses the relationship
between the bre tension T the time t and the bre deformation
r u
T t E t E t T
where the constants E and E represent the passive and active elasticity mo duli T is a
constant and t isa given function of time re ecting the transition b etween active and
passive states
Systolic Contraction
As describ ed in the section the ventricles are closed for a short time ab out
s during which the pressure rises and the ventricles deform isovolumetric contraction
This is followed by a p erio d lasting ab out s during which the blo o d is ejected
The contraction of the heart muscles is iniated by electrical impulses the propagation of
which is discussed in Section The literature on systolic contraction involving active
muscles seems to b e much smaller than the literature on diastolic expansion involving
passive muscles
Relatively early calculations were p erformed by Ghista and Hamid GH and Chadwick
Cha while Waldman Wal discussed the coupling of the heart to the circulatory
system
Beyar and Sideman BS b consider systolic contraction under the following hyp otheses
The LV left ventricle is a thick walled prolate spheroid with time indep endent
shap e i e constant axis ratio
The LV is incompressible
The bre angle acros the myo cardium varies linearly from to in the un
stressed reference con guration and do es not change during systole
The sarcomere length distribution is uniform at the reference con guration
The electrical activation propagates radially from endo cardium to epicardium
The mechanical activation function is half a sinusoidal wave
The stress strain relationship for the muscle bre is linear
The active stress sarcomere length relationship is linear
The passive stress sarcomere length relationship is exp onential
Trans bre stresses are not included
The afterload aortic pressure is mo delled
Beyar and Sideman BS b givea owchart for their Fortran program which calculated
the various variables through out systole Although their work contains a great many
approximations these are clearly stated The parameters used in the mo del are given and
compared with rep orted values The sensitivity with resp ect to various parameters is also
studied
Fibres
The following include bres in their mo del
Bovendeerd Bov
Yang Tab er and Clark YTC mo delling the chick heart
Beyar and Sideman BS b
Authors using more complex mo dels and allowing for bres parallel to the surface include
Costa Hunter Rogers Guccione Waldman and McCullo ch A three dimensional
nite element method for large elastic deformations of ventricular myocardium Part
I Cylindrical and spherical polar coordinates CHR
Costa Hunter Wayne Waldman Guccione and McCullo ch A three dimensional
nite element method for large elastic deformations of ventricular myocardium Part
II Prolate spheroidal coordinates CHW
Recently in his Ph D thesis Bovendeerd Bov used a mo del which allowed transmural
bres This work has b een extended See
Bovendeerd Huyghe Arts van Camp en and Reneman In uenceofendocardial
epicardial crossover of muscle bers on left ventricular wal l mechanics BHA
Bovendeerd Arts Huyghe van Camp en and Reneman Dependenceoflocal left ven
tricular wal l mechanics on myocar dial ber orientation A model study BAH
Yang Tab er and Clark YTC page brie y mention observations on bre orientation
in embryonic chick hearts which they incorp orate into their mo del as torsion
Treatment of the Band Structure
The incorp oration of a band like structure into a mo del of the heart requires the following
A more detailed description of the orientation of the bres than is currently available
In particular the band like structure implies the existence of a twisted structure
which is more complicated than the usual Streeter structure This in turn will
require the use of much greater numb er of nite elements
The band like structure explicitly involves the rightventricle which should therefore
b e included in the mo del
Poro elastic Mo dels
Several mo dels take account of the uid in the myo cardium and use p oro elastic mo dels
A theory for an elastic structure lled with a uid think of a sp onge was rst formulated
by Biot Bio It was later extended to the case of nite deformations by Biot Bio
and was further generalized to a theory of mixtures byBowen Bow Bow These
ideas have b een used CWC check by Bogen see Bog a and Bog b to mo del
the swelling of tissue Huyghe van Camp en Arts and Heethar HvCAH use a p orous
medium nite element mo del to mo del the diastolic left ventricle Simon and Gaballa
SG a use a p oro elastic nite element mo del to mo del the repsonse of large arteries
See also
Kenyon Ken
For the mathematical theory of p orous ow see Douglas and Hornung DH See also
Sorek and Sideman SS b
Sorek and Sideman SS a
The thesis of Huyghe Huy is extensive The myo cardium is mo delled using a combina
tion of a deformation mo del and a p erfusion mo del CWCmust lo ok further into this
The full cardiac cycle is also mo delled
In his Ph D thesis Huyghe Huy mo dels the myo cardium using a p orous medium nite
deformation theory of blo o d p erfused soft tissue He then derives two sp ecial mo dels
A deformation mo del of the b eating left ventricle
A p erfusion mo del of the left ventricular coronary circulation
Numerical results are given and a complete cycle is simulated
Yang Tab er and Clark YTC mo del the embryonic heart of a chick stage
using a p oro elastic mo del This is a reasonable mo del for the chick heart at this stage of
development
Other Factors
In this pap er we are primarily interested in the role played by the bre direction and have
chosen to neglect a variety of other factors
the p ericardium which encloses the heart and restrains the motion of the heart
particularly during certain abnormal conditions Braunwald Bra comments that
whereas the p ericardial space normally contains at most ml of uid this can
increase to litres in severe cases of p ericardial e usion
So far no mo dels have directly considered the role of the endo cardium and epi
cardium whichmay p ossibly b e signi cant One reference on this is Humphrey
and Yin Biaxial mechanical behaviour of excised epicardium HY
The heart wall contains a large amount of uid in the form of interstitial uid
and blood The ow of this uid has b een mo delled using the equations of Biot by
Huyghe and co authors
The Ph D thesis of Huyghe Huy
Huyghe Arts van Camp en and Reneman Porous medium nite element model
of the beating left ventricle HAvCR
van Camp en Huyghe Bovendeerd and Arts Biomechanics of the heart muscle
vCHBA
However it should b e noted that in this work the e ects of p orosity and elasticity
are uncoupled
In the literature it is customary to neglect the e ects of inertia the pap er of
Moskowitz Mos b eing quoted in justi cation However the pap er of Moskowitz
compares solutions with and without the e ects of inertia in a very simple case In
the opinion of the authors this needs further investigation
Residual stress See e g McCullo ch and Omens MO
As observed by Hort Feneis and other authors the myo cardium is not homogeneous
and there are gaps which can b e interpreted as sliding planes If we accept this
how can it b e mo delled
Chapter
Numerical Mo dels
In preceding chapters wehave considered mo dels classi ed according to their geometry
Here we brie y summarize the developmentofnumerical mo dels See
In his Ph D thesis McCullo ch Deformation and Stress in the Passive Heart McC
develop ed sp ecial nite elements
Yin Yin
Costa Hunter Rogers Guccione Waldman and McCullo ch CHR
Costa Hunter Wayne Waldman Guccione and McCullo ch CHW
Interesting recentwork includes
Authors using more complex mo dels and allowing for bres parallel to the surface include
Costa Hunter Rogers Guccione Waldman and McCullo ch A three dimensional
nite element method for large elastic deformations of ventricular myocardium Part
I Cylindrical and spherical polar coordinates CHR
Costa Hunter Wayne Waldman Guccione and McCullo ch A three dimensional
nite element method for large elastic deformations of ventricular myocardium Part
II Prolate spheroidal coordinates CHW
Recently in his Ph D thesis Bovendeerd Bov used a mo del which allowed transmural
bres This work has b een extended See
Bovendeerd Huyghe Arts van Camp en and Reneman In uenceofendocardial
epicardial crossover of muscle bers on left ventricular wal l mechanics BHA
Bovendeerd Arts Huyghe van Camp en and Reneman Dependenceoflocal left ven
tricular wal l mechanics on myocardial ber orientation A model study BAH
Yang Tab er and Clark YTC page brie y mention observations on bre orientation
in embryonic chick hearts which they incorp orate into their mo del as torsion
Numerical Metho ds
It is prop osed to initially solve the equations using nite elements in particular using the
no de brick element for the displacements and the no de brick element for the pressure
Preliminary calculations were made using the NAG Finite ElementPackage but future
work will probably use Di pack
Chapter
Prop erties of Blood
See
Chapter Biological Fluid Dynamics of An Introduction to Mathematical Physiol
ogy and Biology by Mazumdar Maz
Chapter
Fluid Flow in the heart
Intro duction
See
Sugawara Ka jiya Kitabatake and Matsuo Blood Flow in the Heart and Large Blood
Vessels SKKM
Numerical Metho ds
The Immersed Boundary Metho d
In a series of pap ers Peskin and collab orators used the immersed b oundary metho d
to compute two dimensional ow in the heart Applications included ow patterns around
heart valves Pes Pes blo o d ow in the heart and the design of cardiac valves An
improved version of the metho d which improves the conservation of mass was recently
describ ed byPeskin and Printz PP who give further references Peskin and Printz
PP illustrate the immersed b oundary metho d by considering a two dimensional mo del
of the heart which includes a left atrium a left ventricle and a mitral valve The aortic
valve is mo delled as a sink The entire left heart is mo delled as though it were oating
in a uid The b oundary p oints are mo delled using a mechanical analogue consisting of
elastic plastic and contractile comp onents see McQueen Peskin and Yellin MPY
This mo del is clearly somewhat out of date However it also has interesting features It
is used to mo del the entire cardiac cycle and is run until a p erio dic solution is obtained
There is an interesting discussion of these results and comparisons with exp eriments in
Sugawara Ka jiya Kitabatake and Matsuo SKKM p
Yoganathan Lemmon Kim Walk er Levine and Vesier YLK have used the immersed
b oundary metho d to mo del three dimensional early systolic motion Fluid velo cities in
the out ow tract matched NMR data to within The aortic valvewas not simulated
physiologically The connection of the papillary muscles to the mitral valveby the chordae
tendinea was simulated
See the related work on immersed interface problems of Adams A multigrid algorithm
for immersed interfaceproblems Ada
The authors comment page on de ciencies of the immersed b oundary metho d In
e ect the cardiac tissue is mo delled as an in nitely thin layer page Unfortunately
one of the consequences of a thin walled left ventricle is its tendency to b ecome spherical
at a physiologic left ventricular pressure The other consequence is that a thin walled
structure must pro duce much larger stresses than a thick walled structure to supp ort
the same pressure Toovercome these de ciencies it was artic cially assumed that all
structural p oints representing muscle tissue were tethered to the ventricular centerline
See also the Ph D thesis of Vesier Ves and Vesier and Yoganathan VY
Adams Ada presentsamultigrid algorithm for immersed interface problems See also
LeVeque and Li LL
Euler Lagrangian Formulation
Recently Peskin and McQueen PM b haveintro duced a mixed Eulerian Lagrangian
formulation of the interaction of a viscous incompressible uid with an elastic and p os
sibly active material The elastic part of the system is treated using the Lagrangian
equations while the uid part is treated using the Eulerian equations The use of La
grangian equations for the elastic part is based on ideas of Ebin and Saxton ES ES
Peskin and McQueen PM b use this approach to mo del the three dimensional owof
blo o d in the heart The mo del heart contains the left and rightventricles the left and
right atria the mitral tricuspid aortic and pulmonary v alves and segmen ts of the aorta
main pulmonary artery sup erior and inferior vena cavae and the four pulmonary veins
The authors sp ecify a stress strain law for the m uscle bres and assume that the elastic
energy density dep ends only on the strain in the bre direction To complete the mo del
the paths of the bres are sp eci ed
Several computational results are shown graphically Coloured plates are shown in an
article byPeskin and McQueen PM a Tu and Peskin TP and Mayo and Peskin
MP discuss the stability merits of implicit versus explicit metho ds for computing the
elastic force density from the con guration of the elastic material
The results of Peskin and McQueen PM b are impressive Unfortunately the geometry
is only shown graphically so that the reader cannot repro duce the results
Taylor Okino and Yamaguchi TOY describ e the use of a commercial software package
SCRYU using a nite volume integration of the three dimensional Navier Stokes equa
tions based on the SIMPLE algorithm with a b o dy tted BFC co ordinate system The
time variation of the left ventricle wall was assumed as given
A spherical mo del was used but fully three dimensional computations are planned
That is the material can exert force as for example when muscle bres contract
Vortex and whirl ows
The ow of blood into the left ventricle during diastole is closely connected with the mo
tion of the mitral valve Blo o d entering the ventricle through the mitral valve is directed
towards the ap ex and maybelikened to a jet It is de ected by the ap ex and takes on
the form of a vortex see Fig This vortex like motion plays an imp ortant role in
the closure of the mitral valve see Section In the case of a dilated left ventricle a
whirl ow can b e observed see Figure
Figure Blo o d ow from Strackee and Westerhof The Physics of Heart and Circulation
Figure p SW
Figure Blo o d ow a
Figure Blo o d ow b
See the Bot s Ph D thesis Mathematical Models of Diastolic Blood Flow Patterns in the
Human Left Ventricle Bot and Verburg Bot Strackee and Delemaire VBSD
See also
Bellhouse Bel
Bellhouse and Bellhouse BB
Chapter
Mo delli ng the Heart Valves
Intro duction
The heart has four valves the tricuspid mitral pulmonary and aortic valves The tricus
pid and mitral valves are called atrioventricular valves b ecause they lie b etween the
atria and the ventricles The pulmonary and aortic valves are called arterioventricu
lar valves b ecause they lie b etween the arteries and the ventricles or alternatively the
semilunar valves b ecause their lea ets have the shap e of a half mo on The op eration of
the heart valves has aroused considerable interest since the valves are aps of connective
tissue and are largely or wholly passive
The op eration of the heart valves has nearly always b een considered separately in the
literature The problem was considered by Leonardo da Vinci see Peskin Pes p
Henderson and Johnson HJ probably gave the rst correct explanation Fung Fun
pages describ es the op eration of the di erentvalves
Peskin has made several contributions to the sub ject He rst considered the topic in his
Ph D thesis Pes There is a further discussion in his lecture notes Pes p
Numerical results were later published Pes
For a general discussion of the op eration of the valves see
Section of Fluid Mechanics of Large BloodVessels byPedley Ped
Section of Biodynamics Circulation byFung Fun
Swanson and Clark SC
The Mitral Valve
See
Bellhouse and Bellhouse BB
In German Taschenklapp en i e Pocket valves
Bellhouse Bel
The Ph D thesis of Lee Lee
For measurements see
Sugawara Kagijiya Kitabatake and Matsuo SKKM
The Aortic Valve
The op eration of the mitral valvewas considered by Lee and Talb ot LT Some of their
results are repro duced byFung Fun page See also
Section of Fluid Mechanics of Large BloodVessels byPedley Ped
The Aortic Valve byThubrikar Thu
Section of Fluid Mechanics of Large BloodVessels byPedley Ped
Section of Biodynamics Circulation byFung Fun
The aortic valve consists of three lea ets and three sinuses The lea ets are the mobile
parts of the valve The sinuses are cavities b ehind the lea ets on the aortic downstream
side of the valve Each lea et is p o cket shap ed When the ventricle is lling and the
pressure in the aorta is greater than in the ventricle the p o ckets billow out and together
tricle increases and exceeds that in the aorta close the valve When the pressure in the ven
the lea ets are forced op en
There is a plausible explanation of why three lea ets are used Consider a circular op ening
of radius r closed by n equal lea ets The free edges of the lea ets have the length r
while the length of the circumferential part of the lea et is r n Ifn or then
the free edge is to o short to fold back If n the free edge is excessively long and will
crease If n the free edge can fold back easily without creasing It may b e noted in
this connection that the aortic valve do es not normally op en completely the area when
op en is approximately of the total area
Architecture of the Aortic Valve Lea ets
Peskin and McQueen PM b consider the structure of the aortic valve It is assumed
that the function of the closed valve is to supp ort a uniform pressure load p It is further
assumed that this load is transfered to the aortic wall by a one parameter family of bres
under tension
Let u v b e co ordinates on the surface of the lea et chosen in sucha way that the curves
v constant are the bres with v b eing the free edge of the lea et The co ordinate
u measures arc length along the bres with u along the midline of the lea et The
lea et is given by
x X u v
Let T u v b e the bre tension chosen in sucha waythat T u v dv is the force trans
mitted by the group of bres in the interv al v v dv Let p b e the uniform pressure
applied to the lea et
For the equilibrium of a patch of surface u u v v we obtain
Z Z
u v
X X X
T p dudv
u v
u u u v
It is p ossible to derive certain consequences from equation
The torsion T u v is indep endentofu so that the tension is constantoneach bre
v
X
with equation To see this form the scalar pro duct of
u
X X X X X X T X
T p
u u u u u u v
u
Since
X X
u u
wehave by di erentiation
X X
u
n
Also for general vectors a and b
a a b
We conclude that
T
n
The bres are geo desics on the valve lea et surface
This yields the equation of equilibrium
X X X
T p
u u u v
Equation can b e integrated numerically Because of a singularity at the centre of
the aortic valve the computed bre architecture has a fractal nature
The resulting computed architecture agrees well with the true architecture PPL fragen
Some questions on these results of Peskin and McQueen as of July
To what extent do the assumptions lead to the consequences that
The bre length is constant
The bres are geo desics
Supp ose wehave a cylinder with circular bres supp orting an internal pressure p
Are there transformations di erential geometric which transform this surface into
a lea et preserving the prop erties
What do the numerical results really mean
Do we really just have a problem with inextensible cords
For what classes of surfaces do es a family of geo desics have constant curvature
Arti cial Valves
Concerning ariti cial valves see
Chapter und Chapter in Cardiovascular Biomechanics by Chandran Cha
The Aortic Valve byThubrikar Thu
Blood Flow in Arti cial Organs and Cardiovascular Prostheses by Barb enel Fisher
Gaylor et al BFG
Unsteady ow through prosthetic heart valves Anintegral approach coupled with a
vortex method In Boundary Elements X Vol Heat Transfer Fluid Flow and
Electrical Applications by Cassot Morvan and Tonietto CMT
Valve Vibration
Concerning valve vibration see
Chapter Analysis and Applications of Heart Valve Vibrations of An Introduction
to Mathematical Physiology ad Biology by Mazumdar Maz
Chapter
Mo delli ng other Ventricular
Pro cesses
The mechanical events in the heart are accompanied by electrical chemical and thermal
pro cesses In general it is assumed that the coupling is not very strong so that the
mechanical events can b e treated separately but it is necessary to b e aware of the other
pro cesses It maywell b e that in diseased hearts the e ects of coupling are signi cant
PPL fragen
Electrical and Electrophysiological Activity
The contraction of the heart is activated by electrical impulses These spread rapidly along
the endo cardium inner surface via the Purkinje bre network and more slowly towards
the epicardium outer surface via muscle cell conduction The propagation is complete
after ab out ms Hunter and Smaill HS pages describ e exp erimental re
sults and compare these to numerical mo dels
Sofaraswe are aware Novemb er the electrical activation has not b een mo delled
in conjunction with ventricular mechanics It seems reasonable to assume that the two
asp ects can b e considered separately Itwould however p erhaps b e of interest to deter
mine whether it is of signi cance that most of the endo cardium is activated rst in other
words that a pressure pulse moves outward This might for instance a ect the blo o d
ow in the myo cardium One surprising fact is that a small part of the left ventricular
endo cardium is activated relatively late see Fung Fun p The plot of iso chronal
contours on the epicardium given by Hunter and Smaill HS Figure page also
shows interesting features PPL fragen
The energy for the contraction of heart muscle is provided bychemical reactions Potential
ed to the ow of ionic currents In turn a rise di erences across cell membranes are link
in calcium ion concentration with in a cell activates the contractile proteins in the cell
The electrophysiological activity of the heart is considered in several of the later chapters
in the collection Theory of Heart HSHY to whichwe refer
The underlying mathematical equations are related to the well known Ho dgkin Huxley
equations for conduction and exitation in nerves Much of the emphasis in the mo delling
is on phenomena suchas brillation which are distinct from ventricular mechanics The
equations are also the sub ject of active mathematical research See
Peskin Pes p
Peskin Pes
The intro ductory text of Grinro d Gri
Fife Fif
It has recently b een suggested by Lab and Holden LH that mechano electric feedback
may o ccur
See
Rogers and McCullo ch RM
Frese Fre
In the b o ok When Time Breaks Down The Three Dimensional Dynamics of Electrochem
ical Waves and Cardiac Arrhythmias by Winfree Win there are interesting gures and
ideas with reference to cardiac arrhythmias or arrhythmias
See also
Glass and Mackey From Clocks to Chaos The Rhythms of Life GM
Cohen and Hirsch Mathematical developments in Hodgkin Huxley theory and its
approximations CH
Murray Mathematical Biology Mur p
Winfree The Ge ometry of Biological Time Win
Kaplan Ph D Thesis Kap
Winfree Puzzles about excitable media and sudden death Win
Scott Oscil lations waves and chaos in chemical kinetics Sco
Oxygen Di usion
The heart needs a steady supply of oxygen without which misfunction o ccurs the extreme
example b eing a heart attack The demands placed on the coronary circulation system
are high and the uptakeofoxygen from the blo o d in the myo cardium is ab out three times
higher than in the rest of the b o dy Lev Figure and Section The owof
blo o d is in uenced by the pressure in the heart walls During isovolumetric contraction
the coronary blo o d ow ceases and even reverses see Levick Lev p At the time
of writing Novemb er we are not aware of mo delling in this area
Heat Flow
Hunter and Smaill HS pages describ e the mo delling of heat ow in the
heart
According to Ghista Ghi pages the heart pro duces heat at the rate of ab out
kcal day and so pro duces ab out of the b o dy s heat
This gure seems doubtful since a human uses ab out kcal day see Guyton Guy
p Schmidt and Thews ST p and p Schmidt Nielsen SN p CWC
January
One night exp ect the temp erature of the heart to in uence the heart s output since the
following are certainly temp erature dep endent
The blo o d s viscosity and hence the ow of blo o d in the myo cardium
The elasticity parameters of the myo cardium For instance collagen b ecomes pliant
when heated CWC give reference According to Guyton Guy page
Chapter
Comparative Physiology of the
Heart
Intro duction
It is of interest to compare the p erformance of hearts in di erent sp ecies
Out of scienti c curiosity
Since exp eriments on animals are used to obtain information ab out humans
Since a go o d mo del should b e able to predict the p erformance of the hearts of
di erent sp ecies
Nature o ers a great range of circulatory systems and asso ciated pumps Two distinct
groups of animals the mammals and the birds have develop ed closed circulatory systems
with two separate pumps in one unit the heart It is reasonable to assume that this
arrangement has advantages and to try to nd them It is for instance not immediately
obvious whythetwo pumps should b e in the same unit For example Woods Woo
page suggested that it would follow from the Law of Laplace that it would b e
mechanically more e cienttohavetwo separate pumps
Sources of Data
The basis of any comparison of the hearts of di erent sp ecies must b e the systematic
accumulation of data
At the time of writing April weareaware of the following sources of information
Biology Data Book volume III by Altman and Dittmer AD
Comparative Physiology of the Heart by Clark Cla
Merck Veterinary Journal byFraser Fra
The Ecological Implications of Body Size byPeters Pet
Comparative Animal Physiology by Prosser Pro
Animal Physiology byScheer Sch
Scaling Why is Animal Size so Important bySchmidt Nielsen SN
Schmidt Nielsen SN page lists further references
Allometric Equations
Despite the great range of diversity in nature there are also considerable similarities For
example for the mammalian heart the following equation relates the mass of the heart
M in kilograms to the b o dy mass M in kilograms SN page
h b
M M
h
b
Equation is valid over the full range of mammals from shrews with a b o dy weightof
grams to elephants with a b o dy weight of kg Equation states that in general
the mass of the mammalian heart is approximately of the b o dy mass
Equation is an example of an equation of the form
b
y ax
or equivalently
log y loga b log x
with y M and x M
h b
Equations of the form frequently with x equal to the b o dy weight M are called
b
allometric equations in biology In the b o ok The Ecological Implications of Body Size
by R H Peters Pet a large numb er of allometric equations are given
For the study of the cardiovascular system the following allometric equations are of sp e
cial interest
If P denotes the metab olic rate for animals at rest in kcal day then the well known
met
mouse to elephantcurve is expressed by
P M
met
b
This was given by Max Kleib er in but similar equations had long b een known see
Cla Kleib er subsequently suggested that the equation
P M
met
b
would b e equally reasonable
The heartb eat frequency f b eats p er minute is given by K Schmidt Nielsen SN
h
page
f M
h
b
The cardiac output Q ml p er minute is given by Schmidt Nielsen SN page
h
Q M
h
b
Finally the blo o d volume V ml is given by Schmidt Nielsen SN page
b
V M
b
b
so that the blo o d volume accounts for b etween and of the b o dy mass In the b o ok
Engineering Design of the Cardiovascular System of Mammals Dawson Daw system
atically applies scaling laws
The foregoing allometric equations are intrinsically interesting but must b e treated with
some caution Clark Cla quotes many such formulas but often has di erentvalues for
the constants b Clark also gives several graphs which illustrate the not inconsiderable
scatter of the data
Di erences Between Mammalian Hearts
While there are far reaching similarities b etween all mammalian hearts there are also
di erences
Allometric equations represent trends and there are substantial deviations for some
animals For example the masked shrew which has a body weight of grams has
a heart which is times larger than is predicted by equation
There are some qualitative di erences whichmay b e masked by allometric equations
The structure of the myo cardium of the dog heart may b e di erent from that
of man Lunkenheimer p ersonal communication
There may b e qualitative di erences in the bre geometry b etween the mouse
heart and the human heart McLean Rross and Prothero MRP as quoted
byHorowitz Hor page
Pedley Ped page comments on a signi cant di erence b etween the coronary
circulation in the horse and the dog
It has b een suggested see Schmidt Nielsen SN page that the large size of
the shrew heart may arise b ecause there is a physical upp er limit to the frequency
of heart b eats As it is the shrew heart can b eat over times a minute that is
one b eat every milliseconds
Schmidt Nielson SN page comments on a di erence b etween birds and mam
mals For reasons which are not clear some birds have blo o d pressures ab ove
mmHg
Rowell Row p emphasizes the fact that for uprighthumans of the blo o d
volume is b elow the heart level whereas for many animals of the blo o d volume
is at or ab ove heart level
During the Alicante conference di erences b etween human and pig hearts were
rep orted
Concluding Remarks
In summary it seems to one of us CWC at the time of writing April that with
the p ossible exception of very small hearts there are no signi cant qualitative di erences
between mammalian hearts which could b e used to test di erent mathematical mo dels
of the heart
What conclusions can b e drawn which are relevant to the mo delling of the cardiovascular
system
Chapter
Mo delli ng the Global Motion of the
Heart
As mentioned in section the heart as a whole moves quite dramatically during a single
heart b eat This motion has apparently only b een considered by a few authors
Figure b illustrates the motion of the heart walls during systole the ap ex of the
heart moves towards the base the outer walls contract the septum b ecomes more convex
with resp ect to the left ventricle PPL fragen At the same time the ap ex moves forward
and taps against the chest see Figure
Until quite recently itwas not p ossible to accurately measure the motion of the heart
since one had to rely on visual observation in much the same way as William Harvey who
made some of his observations on dying animals whose heartb eat was reduced Harvey
Har chapter
Robinson Factor and Sonnenblick RFS p comment on its implications and refer to
the early technique of ballistic cardiography in which a sub ject lay on a wheeled table
while the motions of the table in resp onse to the motions of the heart were recorded
Related references are Son and YSF The lighthearted b o ok Vital Circuits by
Steven Vogel also contains a brief reference to this sub ject Vog page So far as we
are aware April quantitative estimates of the e ect of the heart s motion on its
p erformance have not b een made
See also
Pedley Ped pages considers momentum
Bo oks on ballistic cardiography see Starr and No ordergraaf No ordergraaf No o
Acierno Aci pages
Rushmer Rus pages
Buchalter Weiss et al BWR measure the rotation of the left ventricular ap ex
using MRI tagging
Beyer and Sideman BS b app endix B consider the e ect of twisting on bre extension
The Heart as a Mechanical Pump
Many authors have considered the heart as a mechanical pump
Burton Bur
Burch Ray and Cronvich BRC
Beyar and Sideman Spatial energy balance within a structural model of the left ven
tricle BS
Peters and Visscher The energy metabolism of the heart failure and the in uenceof
drugs upon it PV
Robinson Factor Sonneblick RFS
Wong Some proposals in cardiac muscle mechanics and energetics Won
Moskowitz Mos estimates inertial e ects
The determinants of myo cardial oxygen consumption M V have b een considered by
O
many authors
Kahler Braunwald Kelminson Kedes Chidsey and Segal E ect of alterations of
coronary blood ow on the oxygen consumption of the nonworking heart KBK
The work W done by the heart is usually estimated as the sum of the mechnical work
external work or volume pressure work and the kinetic energy
W P V mv
where P is the pressure di erence mean ejection pressure minus mena l ling pressure
V is the mass of blo o d ejected and v is the mean ejection velo city of the blo o d see e g
Levick Lev p
For comparison with ordinary mechanical pumps see
Rogers and Mayhew Engineering Thermodynamics Work and Heat Transfer RM
Ryder and Bennett
Cardiac Energetics
Concerning cardiac energetics see
Gibbs Cardiac energetics Gib
Kleib er The Fire of Life Kle
Ryder and Bennett
Chapter
Growth
The human heart likemany other parts of the b o dy grows while maturing but also in
resp onse to changes in loading conditions It is a challenge to determine the mechanisms
which control this growth The study of the embryonic heart is relevanthere growth
and remo delling
Wol s Law see
Wol Das Gesetz der Transformation der Knochen
See
Thompson Tho
Lin and Tab er LT
Ro driguez Hoger and McCullo ch RHM
Yang Tab er and Clark YTC
Currey The Mechanical Adaptation of Bones Cur
Lin and Tab er LT mo del the growth of embryonic chickventricles They use the passive
strain energy function
a
bQ
W e
b
Q I I
where
I
r z
I
r z
are strain invariants intro duced byYang Tab er and Clark YTC
The study of the growth adaptation remo delling and repair of living tissue raises inter
esting questions some of whichhave mathematical counterparts
In the present context one may ask whether knowledge ab out the stress distribution in
an overloaded heart can b e used to explain the developmentofhyp ertrophy see Fung
Fun page See
In Chapter of The Mechanical Adaptations of Bones Currey Cur critically
discusses p ossible mechanisms
The book On Growth and From by D Arcy Thompson Tho is a classic in this
eld
Huxley Hux
Wol
Chapter
Miscellaneous Topics
Heart Sounds
Concerning heart sounds see
The Guide to Heart Sounds Normal and Abnormal NPS
Concerning Korotko sounds see
Pedley Ped pages and
The Interval Force Relationship
Concerning the interval force relationship or force frequency relationship or force
interval relationship or interval strength relationship asso ciated with the name of
Bowditch See
Noble and Seed The Interval ForceRelationship of the Heart Bowditch Revisited
NS b
Wiggers Circulatory Dynamics Wig
Ciba Foundation The Physiological Basis of Starling s Law of the Heart Sym
Bowditch Ueber die Eigenthuemlichkeite n der Reizbarkeit welche die Muskelfasern
des Herzens zeigen Bow and Bowditch On the pecularities of excitability
which the bres of cardiac muscle show Translation of paper of Bow
Assistance and Replacement
See
Chapter Circulatory System Dynamics by No o dergraaf No o
Chapter and Chapter in Cardiovascular Biomechanics by Chandran Cha
Anrep E ect
Concerning the Anrep e ect see
Ciba Foundation The Physiological Basis of Starling s Law of the Heart Sym
Gregg E ect
Concerning the Gregg e ect see
Livingston Halp ering and Yin LHY
Gregg Gre
Part IV
Mathematical Mo delli ng of the
Circulatory System
Chapter
Intro duction
In this Part we attempt to provide an overview of the circulatory system other than the
heart As usual we emphasize the asso ciated mathematical problems
As Lighthill JL p comments
Many of the most imp ortant problems in this eld even if the uid dynamics
in them is considered in isolation are found to raise questions which in the
whole pro ceding history of research on uid dynamics have totally failed to
b e answered or in some cases even to b e asked
Some references include
Circulatory System Dynamics by No ordergraaf No o This is a readible and useful
description of the sub ject with problems and solutions at the end of eachchapter
One drawback is that the mathematics is often only brie y describ ed
Cardiovascular Fluid Dynamics vols edited by Bergel Ber a Ber b
Blood Flow in the Heart and Large BloodVessels by Sugawara Ka jiya Kitabataka
and Matsuo SKKM which describ es many exp erimental techniques and results
Circulation byFolkow and Neil FN
They also discuss the measurement of abnormal ows caused by defects such as
aortic regurgitation in which blo o d reenters the left ventricle from the aorta See
also Braunwald Bra p
shunts involving alternative paths for the circulation of blo o d See Braunwald
Bra p
stenosis the narrowing of a passage in the b o dy e g aortic stenosis See Braun
wald Bra p
Chapter
The Arterial System
The standard reference is McDonald s Blood Flow in Arteries the latest edition of which
is by Nichols and O Rourke NO
Other references include
Part IV Wave Propagation and Arterial Dynamics of Cardiovascular Dynamics
and Models by Brun et al BCL
The Fluid Mechanics of Large BloodVessels byPedley Ped
Chapter of Circulatory System Dynamics by No ordergraaf No o
Arterial System Dynamics by Li Li
Rubinow Rub
Chapter Analysis and Applications of Arterial Flow Dynamics of An Introduction
to Mathematical Physiology and Biology by Mazumdar Maz
Chapter and of Mathematical Bio uiddynamics by Lighthill JL
Chapter of Biodynamics Circulation byFung Fun
Intro duction
The folw of blo o d through the arteries represents an interesting mathematical challenge
Blo o d is a complex non Newtonian uid
The arterial walls are thick elastic containers
The arterial walls are often curved
The arteries branch rep eatedly
The ow is time dep endent
Since the pulsatile ow of the blo o d in the arteries is readily observed it aroused early
interest Early contributors included
Stephen Hales a clergyman who among other contributions rst measured arterial
pressure Hal
Thomas Young physicist and physician who rst calculated the wavevelo city for
thin walled arteries You and You
Poiseuille physician and physicist who studied steady state viscous ow
The brothers W Web er physicist and E H Web er physician WW and
Web
Womersley Wom and Witzig Wit
There are brief historical intro ductions in
McDonald s BloodFlowinArteries Nichols and O Rourke NO pages
Biodynamics Circulation byFung Fun Chapter Fung also includes Table
pages a tabular summary of the contributions of di erent authors
Imp ortant results include the following
The Poiseuille ow of a viscous uid in laminar ow through a circular tub e of radius
a The velo city pro le is
dp
u a r
dx
and the mean velo cityis
a dp
u
m
dx
Turbulent ow in a circular tub e The transition to turbulence is governed by the
Reynolds number
UL
R
where U isatypical velo city L atypical length and the kinematic viscosity
For ow in tub es the ow b ecomes turbulentif R exceeds the critical value
R R
cr it
Surprisingly in the ascending and escending aorta of man and dog Reynolds numb ers
greater than can o ccur Fung Fun p
In Murray Mur used Poiseuille s formula to analyse the optimum design of blo o d
vessel bifurcation Murray found that if a blo o d vessel of radius a splits symmetrically
into twovessels of diameter a making angle with the original blo o d vessel then
a
a
cos
so that a a and These results known as Murray s formula are
roughly in agreement with measurements in the human b o dy
For owinathin walled elastic tub e with Young s mo dulus E of inner radius a and
i
wall thickness h the wave sp eed is found to b e Fung Fun p
s
Eh
c
a
i
This formula was derived byYoung in You but is called the Mo ens Korteweg
formula
More complicated ows were investigated by Witzig Wit and Womersley Wom
The Womersley number
r
D
play an imp ortant role
For numerical computations for ow in curved tub es e g the aorta see Chang and Tar
b ell CT
There are interesting clinical applications of the ab ove results McDonald s Blo o d Flow
in Arteries NO contains numerous examples
The ow of blo o d is in uenced by the sti ening of arteries with age in man and
woman see NO p
It has b een p ostulated that turbulence increases the haemolysis loss of haemoglobin
of red blo o d cells see NO p
Chapter
The Pulmonary Circulation
Intro duction
The pulmonary circulation ist that part of the cardiovascular system in which blood
is pump ed from the rightventricle through the pulmonary artery to the lungs returning
through the pulmonary veins to the left artrium
See
Kowalczyk and Kleib er Model ling and numerical analysis of stresses and strains in
the human lung including tissue gas interaction KK
Berne and Levy BL pages
Pedley The Fluid Mechanics of Large BloodVessels Ped pages and
Nichols and O Rourke in Chapter in McDonald s Blood Flow in Arteries NO
Levick An Introduction to Cardiovascular Physiology Lev
Handbook of Physiology Section Cardiovascular System vol Peripheral Cir
culation
Section Circulatory System Dynamics by No ordergraaf No o
Chapter IV Mathematical Aspects of Heart Physiology byPeskin Pes
The Respiratory Physiology of Animals by Cameron Cam
Chapter Mathematics in Medicine and the Life Sciences by Hopp ensteadt and
Peskin HP
Chapter Respiratory FlowPatterns of Mathematical Bio uiddynamics by Lighthill
JL
Chapter
The Venous System
The venous system has b een studied comparatively little there is no equivalentof Mc
Donalds Blood Flow in the Arteries
References include
Chapter Flow in Col lapsible Tubes ofThe Fluid Mechanics of Large BloodVessels
byPedley Ped
Chapter The Veins ofBiodynamics Circulation byFung Fun
Chapter Circulatory System Dynamics by No ordergraaf No o
Amo ore and Santamore AS
Chapter
The Coronary Circulation
Some references on the coronary circulation are
Part I Myocardial Blood Flow of Cardiovascular Dynamics and Models by Brun et
al BCL
Fluid Mechanics of Large BloodVessels byPedley Ped contains scattered results
An interesting observation is that oscillations have b een observed in the coronary
artery of the horse which are not observed in a dog Ped page
Chapter of McDonald s Blood Flow in Arteries by Nichols and O Rourke NO
Mechanics of the Coronary Circulation by Mates Nerem and Stein MNS
Fulton Ful
Microcirculation of the Heart Theoretical and Clinical Problems by Tillmanns
K ubler and Zeb e TKZ
A series of pap ers on the e ects or coronary owwas
Kahler Braunwald Kelminson Kedes Chidsey and Segal E ect of alterations of
coronary blood ow on the oxygen consumption of the nonworking heart KBK
Ross Klo ck e Kaiser Braunwald E ect of alterations of coronary blood ow on the
oxygen consumption of the working heart RKKB
Sarno Gilmore Skinner Wallace Mitchell Relation between coronary blood ow
and myocardial oxygen consumption SGS
Weisb erg Katz and Boyd In uenceofcoronary ow upon oxygen consumption and
cardiac performance WKB
In his Ph D thesis Huyghe Huy gives a careful discussion of the in uence of myo car
dial mechanics on the coronary blo o d ow pages to and the in uence of the
coronary blo o d owonmyo cardial mechanics pages to
Huyghe Huy brie y mentions the garden hose e ect or erectile prop erties of
the coronary microvasculature and quotes Vogel Apstein Briggs Gaasch and Ahn
VAB
Chapter
The Micro circulatio n
See
Chapter Microcirculation in Biodynamics Circulation byFung Fun
Chapter Mathematical Bio uiddynamics by Lighthill JL
Part V
Mathematical Mo delli ng of the
Entire Cardiovascular System
Chapter
Intro duction
The ultimate goal of mo delling the cardiovascular system is of course to mo del the entire
system as a whole This goal is far from b eing achieved
Some references are
Chapter of McDonald s Blood Flow in the Arteries NO
The conference pro ceedings edited by Yin Yin
The thesis of Waldman Wal
The b o ok of No ordergraaf Circulatory System Dynamics No o esp ecially Chapter
Control and Chapter The Closed Cardiovascular System
No ordergraaf No o quotes work in which mo dels containing hundreds of equations have
b een set up and solved by computer He makes the following comments No o page
The nature of the problems under scrutiny contains a temptation to write
massive systems of equations easily reaching into the hundreds Yet to date
mo dels of the entire circulation have neither answered the questions p osed
ab ove nor many of the other questions whichwere realistically the motivation
for developing the mo dels in the rst place
Chapter
Mo dels of the Cardiovascular
System
A few authors have considered the complete cardiovascular cycle by combining a mo del
of the heart with a simple mo del of the circulatory system load
See
Arts Veenstra and Reneman AVR
Arts Reneman and Veenstra ARV
Huyghe Huy
Chadwick Cha
In two pap ers Perl Horowitz and Sideman PHS and Horowitz Perl Sideman and
Ritman HPSR a complete canine cardiac cycle was mo delled using nite elements
The authors used three dimensional isoparametric elements The left ventricle was
treated as twolayers the ap ex elementshaving no des the reminder having no des
The bres were mo delled as truss elements The mo del allows for large deformations and
nonlinear constitutive equations However the active forces are treated as unknowns and
are chosen during each time step so that the exp erimentally observed ventricular volume
is obtained
An interesting feature of this work is that the authors attempt to assess the imp ortance
of four factors anisotropy geometric nonlinearity material nonlinearity mechanical ac
tivation They do so by dropping each one of these factors in turn and comparing the
results They conclude Horowitz Perl Sideman and Ritman HPSR p that all
factors play a role but the material nonlinearity is most signi cant
Chapter
Control
See
Chapter Circulatory System Dynamics by No ordergraaf No o
Kapp el and Peer KP Based on the four compartment mo del of Gro dins Kapp el
and Peer develop a mo del for the resp onse of the cardiovascular to a short term
submaximal load Basic mechanisms included in the mo del are Starling s lawofthe
heart the Bowditch e ect and autoregulation in the p eripheral regions The mo del
provides a description of data obtained in bicycle ergometer tests
Chapters and Cardiovascular Physiology by Berne and Levy BL
Human Circulation Regulation During Physical Stress byRowell Row
Human Cardiovascular Control byRowell Row Discusses circulatory adjustment
to upright p osture orthostasis and dynamic exercise Detailed but do es not treat
meathematical mo dels
Chapter VI Mathematical Aspects of Heart Physiology byPeskin Pes
Biomechanics and Exercise Physiology by Johnson Joh
Chapter
Design Principle s of the
Cardiovascular System
Almost everyone who studies the cardiovascular system comes to the conclusion that the
system must b e optimal See
Fung
Weib el The Pathway for Oxygen p discusses the question AreAnimals Built
Resonably Wei and intro duces the concept symmorphosis on p and
Weib el Wei Chapter reviews the respiratory system and particular the small
est mammal the Etruscan shrew see p
Talk in Alicante
Optimization of Biological Structures
There are several b o oks in which the optimal design of biological structure is considered
Gordon The New Science of Strong Materials or Why You Don t Fal l Through the
Floor Gor
Vogel Life s Devices The Physical World of Animals and Plants Vog
Wainwright Biggs Currey and Gosline Mechanical Design in Organisms WBCG
Epilogue
In this text wehave attempted to survey the present state of mathematical mo delling of
cardiovascular mechanics
CWC Find a suitable quotation E g Weib el p
A great deal has b een achieved but many questions await de nitive answers At the risk
of arousing criticism and demonstrating ignorance weventure to make some general and
p erhaps provo cative comments
In most numerical work in the literature the results are shown graphically and it is
very di cult for the reader to repro duce the results since neither the data nor the
programs are available A praiseworthy exception is provided byPeter Hunter and
his colleagues NLSH who have made their data available b oth on diskette and
by ftp
During the past few years muchwork has concentrated on mo delling the my
o cardium and the overall view has b een less emphasized Whereas Chadwick Cha
ventured to predict the pressure volume cycle of the left ventricle recentworkers
have taken a muchnarrower view
There is little work whichcombines the computations of the stress in the heart walls
with computations of the blo o d owintheventricles
There seems to b e a need for greater interaction b etween physiological and morpho
logical observation and mathematical mo delling
The cardiovascular system is extremely complex At present most mo delling is
carried out by small groups Ph D students etc Full scale mo delling will require
co ordination of these e ort s
We conclude with an optimistic quotation from Kovacs Kov p
It is extremly likely that in the years to come diagnosis therapy and follow up
of cardiac patients will critically dep end on developments achieved through or
motivated by current mathematical and theoretical metho ds
App endix A
The SI System
A Physical Units
There is a b ewildering range of physical units Here we use with very few exceptions the
SI system
There are a numb er of b o oks which list conversion factors Wehave used Co ok Co o
The following intro duction to units is along the lines of that given bySchmidt Nielsen
SN
See also
Goldman and Bell GB
Mechtly Mec
Symb ols Committee of the Royal So ciety otRS
Physical quantity SI base unit Symbol
length metre m
mass kilogram kg
time second s
thermo dynamic temp erature kelvin K
amount of substance mole mol
electric current amp ere A
luminous intensity candela cd
Table A Base units
This app endix provides names of units and their symb ols a brief list of conversion
factors and a list of some physical quantities and the recommended symb ols for their
designation
A Units
The SI system uses seven base units for seven dimensionally indep endent basic physical
quantities eac h base unit is represented bya symbol Use of these symbols is mandatory
see table A The SI System recognizes several derived units as well as some supple
mentary units Although some of these have sp ecial symb ols they are derived from and
de ned in terms of the base units A few of particular interest in physiology are listed in
Table A The following should b e noted ab out the use of these units and symb ols
Physical quantity SI derived unit Symb ol for unit De nition in
terms of base units
force newton N mkgs
energy work quantity of heat joule J N m m kg s
power watt W J S m kg s
pressure pascal PaN m m kg s
viscosity p oise P Ns m m kg s
electric p otential di erence volt W A V m kg s A
electric resistance ohm V A m kg s A
electric charge coulomb C sA
frequency hertz Hz s
Table A Derived units
The symb ols are printed in roman upright letters The symb ols are not abbreva
tions and are not followed by a p erio d except at the end of a sentence
The plural form of a symb ol is unchanged and the ending s should not b e used
Certain units are derived from prop er names e g Volt the sym b ols for these units
are capital roman upright letters V but the unit names are not capitalized volt
Pre x Symbol Equivalent
tera T
giga G
mega M
kilo k
deci d
centi c
milli m
micro
nano n
pico p
femto f
atto a
Table A Pre xes
There is frequent need for smaller and larger decimal multiples of the units These are
obtained by the use of the pre xes listed in Table A These pre xes may b e attached to
any SI base or derived unit Large multiples of seconds however are rarely expressed this
way The combination of pre x and symb ol is regarded as a single symb ol and comp ound
pre xes are not allowed The decimal sign is usually printed as a p oint in English but
as a comma in French and German In the SI system a p oint or comma should b e used
only for the decimal design To facilitate the reading of long numb ers the digits maybe
divided into groups of three but not separated by commas
Example not
When the decimal sign is placed b efore the rst digit of a numb er a zero should always
b e placed b efore the decimal sign
Example not
Pro ducts or quotients of two or more units are treated according to the rules of math
ematics Note that the use of more than one solidus is ambiguous Thus a sp eci c
rate of oxygen consumption can b e written ml k g s orml kg s or ml k g s or
ml kg s or ml kg s not ml k g s
A Useful conversion factors
Traditional units are commonly used in those elds of physiology that concern energy
e g in the area of metab olic rates and pressure e g in the area of gas exchange
Force
kgf kilogram force N
newton kgf
Energy work force distance
cal J
joule cal
litre O at C mm Hg when used in oxidative metab olism is often equated
with kcal th us liter O kJ
Power work p er unit time
cal s J s W
kcal h W
watt cal s cal min cal h
Pressure force p er unit area
atm standard atmosphere N m kPa
at metric atmosphere kgf Nm kPka mm
Hg at C torr N m Pa
kNm mm Hg atm
Volume
litre is since de ned as exactly equal to dm m th us ml cm
and l mm
Length
inch mm
foot m
Mass
pound avoirdup ois kg
ounce avoirdup ois kg
Temp erature
The temp erature unit of degree Celsius C equals the temp erature unit of
kelvin K
The Celsius temp erature is de ned as the excess of the thermo dynamic temp erature
over K
A Physical Quantities
Physical quantities Symbol Dimensions
Length l L
Height h length
Radius r length
Area A L
Volume V L
Mass m M
Density m V ML
Pressure p ML T
Time t T
Velo city u v LT
Frequency f T
Force F MLT
Work W energy
Energy E W ML T
Power P ML T
Temp erature T
Table A Physical quantities recommended symb ols and dimensions Alternative sym
b ols are separated by commas
The names of some physical quantities and recommended symb ols are given in Table A
The symbolforaphysical quantity should b e a single letter of the Latin or Greek al
phab et and should b e printed in sloping italic typ e When necessary subscripts and
or sup erscripts are used to indicate a sp eci c meaning of a symb ol For example b o dy
temp erature T blood v olume V rate of o xygen consumption V
b bl O
Equations that representphysical or physiological relationships use these and similar
symb ols for physical quantities Symb ols for units m kg s etc should never b e used in
such equations
Also incorrectly known as centigrade
App endix B
Dimensional Analysis
B Dimensional Analysis
When a physical phenomenon is observed exp erimentally and the results are then repre
sented by an equation or law then the measurements must b e made in terms of physical
units such as metres or feet seconds or years but the particular physical units used are
clearly of secondary imp ortance and one may reasonably exp ect to b e able to express the
law in a form which is indep endent of the actual physical units used This can b e achieved
byintro ducing dimensionless variables and constants Dimensional analysis provides a
to ol to do this
Dimensional analysis brings additional b ene ts
In simple cases it is p ossible to deduce the general form whichanylawmust take
By ensuring that the dimensionless parameters are not changed one can scale a
problem without changing the form of the solution
As an example with biological applications the nature of the ow of a uid with velo city
u density and dynamic viscosity in a straight tub e of diameter d is determined by
the Reynolds number
ud
Re
For small values of Re the ow is laminar smo oth but if Re the ow can b ecome
turbulent
In animals one would exp ect that laminar ow in blood vessels would b e favoured by
natural selection but in fact the Reynolds numb er is often close to turbulent ow
can o ccur Y C Fung Fun p Fun p K Schmidt Nielsen SN page
Dimensional analysis can b e systematized as follows
Determine all variables and parameters which o ccur in a problem
Determine a complete set of dimensionless pro ducts
Apply Buckingham s Theorem
Theorem An equation is dimensionally homogenous if and only if it can b e ex
pressed in the form F where f g is a complete set
n n
of dimensionless pro ducts
To illustrate the approach consider the pulsating ow of blo o d in an artery As determin
ing variables we take
r radius of artery L
f frequency T
density of blood ML
dynamic viscosity of blood ML T
To determine a complete set of dimensionles pro ducts we consider the pro duct
a b c d
r f
which has the dimensions
a b c c d d d
L T M L M L T
For this pro duct to b e dimensionless wemust have
a c d L
b d T
c d M
from which it follows that
c d
b d
a c d d
Taking for convenience d we see that f g is a base with
s
f
r
is called the Womersley parameter
It is often denoted by
s
f
r
where is the kinematic viscosity
It follows from Buckingham s theorem that any solution must b e of the form
F
In his book Life s Devices Vogel Vog page discusses several examples of scaling
when analysing biological problems
References on dimensional analysis include
Dimensional Analysis byP W Bridgman Bri
Dimensional Analysis in the Biomedical Sciences byB Schepartz Sch a
Chapter in E ects of Scale in Muscles Re exes and Locomotion by Thomas A
McMahon McM
Chapter in AnIntroduction to Mathematical Physiology and Biology by J Mazum
dar Maz
Dimensional Analysis and Theory of Models by Langhaar Lan
Dimensional analysis is a sp ecial case of applications of the mathematical concept of
Lie Group to di erential equations Recent references to this sub ject include P J Olver
Olv and G W Bluman and S Kumei BK
Some applications of dimensional analysis to biological problems are
In his Ph D thesis Peskin Pes pages the ow patterns in the hearts of
di erent mammals are compared
Fung Fun page uses dimensional analysis to consider micro circulation in
the b o dy
Holmes Hol quoted by G W Bluman and S Kumei BK p uses dimen
sional analysis in the mo delling of soft tissue
Pritchard Pri a Pri b has p ointed out that most allometric equations violate
the rules of dimensional analysis He has suggested some new equations which con
form to dimensional analysis These might nd application in the mo delling of the
cardiovascular system
Lee Lee in Chapter discusses applications to hemo dynamics
G unther Gue quoted byPeters Pet
In Chapter of Elastic Mechanisms in Animal Movement by Alexander Ale uses
dimensional analysis to discuss elastic similarity and dynamic similarity
App endix C
Elementary Concepts from Physical
Chemistry
See
Mo ore Physical Chemistry Mo o
Atkins Physical Chemistry Atk
Fenn Engines Energy and Entropy Fen
Atkins Elements of Physical Chemistry Atk
Atkins Physical Chemistry th edition Atk
App endix D
Physiology
See
Guyton Textbook of Medical Physiolog Guy
Glaser Biophysik Gla
Schmidt and Thews Physiologie des Menschen Au age ST
Le Gros The Tissues of the Body WELGC
Duerr and Uehlinger unbek Sau
App endix E
Cardiovascular Data
E Man
We consider a normal young adult man weighing
W kg
The corresp onding normal heigth is
H cm m
If we assume a normal height of H m then according to the formula of DuBois
and DuBois DD he has a b o dy surface area of
S W H m
This formula is quoted by Dinnar Din but do es not app ear in Dub ois and Dub ois
DD See Kleib er Kle p
According to the formula of Carp enter he has a basal metab olic rate BMR of
BMR W H Age kcal h
for a man of age But Dinnar Din quotes kcal h as typical
See
Altman and Dittmer Biology Data Book vol III AD
Altman Dittmer and Greb e Handbook of Circulation ADG
Milnor Cardiovascular Physiology Mil
E Woman
E Dog
See Dinnar Din p
App endix F
Mathematical To ols
See
Mathematical Techniques for Biology and Medicine by Simon Sim
F Di erential geometry
Peskin see e g Peskin Pes has applied di erential geometry in several cases
See
Modern Di erential Geometry of Curves and Surfaces by Gray Gra
Tensor Analysis on Manifolds by Bishop and Goldb erg BG
Di erential Geometry of Curves and Surfaces by do Carmo dC
Riemannian Geometry by do Carmo dC
Di erential Forms with Applications to the Physical Sciences by Flanders Fla
Di erential Geometry by Kreyszig Kre
Di erential Geometry by Stoker Sto
Lectures on Classical Di erential Geometry by Struik Str
An Introduction to Di erential Topology Wil
Schweitzer Hurder dos Santos and Arraut Di erential Top ology Foliations and
Group Actions SHdSA
Eisenhart Introduction to Di erential Geometry with Use of the Tensor Calculus
Eis
Eisenhart Riemannian Geometry Eis
Abraham Marsden and Ratiu Manifolds Tensor Analysis and Applications RAR
Conlon Di erentiable Manifolds A First Course Con
Kosinski Di erential Manifolds Kos
Klingenb erg Riemannian Geometry Kli
F Geo desics
The concept of a geo desic has b een used in several areas
Fibres in the heart Peskin
Chapter Fibre wound animals in Elastic Mechanisms in Animal Movement by
Alexander Ale See also Alexander Ale
Horowitz Perl and Sideman HPS
F Geometric Measure Theory
Concerning geometric measure theory see
Geometric Measure Theory A Beginners Guide by Morgan Mor
F Top ology
The translation of Tamura s b o ok Tam contains a list of recent references on foliations
See
Abraham and Marsden Foundations of Mechanics AM
Armstrong Basic Topology Arm
Arnold Catastrophe Theory Arn
Arnold Singularity Theory Arn
H Blaine Lawson The Quantitative Theory of Foliations Law
Conlon Di er entiable Manifolds A First Course Con
Fuller The writhing number of a space curve Ful
Fuller Decomposition of the linking number of a closedribbon A problem from
molecular biology Ful
Gordon Moriah and Wa jnryb Geometric Topology GMW
Hemp el Manifolds Hem
Lawson The Quantitative Theory of Foliations Law
Nash and Sen Topology and Geometry for Physicists NS a
Schwarz Topology for Physicists Sch
Tamura Topology of Foliations AnIntroduction Tam
Thurston and Weeks The mathematics of three dimensional manifolds TW
Weeks The ShapeofSpace Wee
F Knots
If we regard the bres in the heart wall as twisted then wemaywell ask whether they
form knots See
Adams The Knot Book Ada
Livingston Knot theory Liv
F Partial Di erential Equations
See
Renardy and Rogers RR b
F Singularities
See
Arnold Catastrophe Theory Arn
F Chaos
Some b o oks treating chaos include
Strogatz Singular laments organize chemical waves in three dimensions Wave
Taxonomy WS
App endix G
Mechanics
References on mechanics are
Chetaev Theoretical Mechanics Che
Goldstein Classical Mechanics Gol
Whittaker ATreatise on the Analytical Dynamics of Particles Rigid Bodies Whi
G Applied Mechanics
Under applied mechanics we understand elementary mechanis applied to practical en
gineering problems See
Hillier AppliedMechanics HH
Hibb eler Engineering Mechanics Statics and Dynamics Hib
Meriam and Kraige Engineering Mechanics Dynamics MK
Ryder and Bennet
G Thermo dynamics
Concerning thermo dynamics see
Rogers and Mayhew Engineering Thermodynamics Work and Heat Transfer RM
G Contiuum Mechanics
Standard classical references on continuum mechanics are
Truesdell and Noll The Non Linear Field Theories of Mechanics Handbuch der
Physik TN
Truesdell and Toupin The Classical Field Theories Handbuch der Physik vol III
TT
Truesdell The Elements of Rational Mechanics Tru
Mo delling using continuum mechanics is discussed by
Fung A First Course in Continuum Mechanics Fun
Allen Herrera and Pinder Numerical Modeling in Science and Engineering IHP
Bowen Introduction to Continuum Mechanics for Engineers Bow
Malvern Introduction to the Mechanics of a Continuous Medium Mal
Truesdell A First Course in Rational Continuum Mechanics Tru
Fung Foundations of Solid Mechanics Fun
Smith Constitutive Equations for Anisotropic and Isotropic Materials Smi
G Elasticity
References on elasticity include
Antman Nonlinear Problems of Elasticity Ant
Ciarlet Mathematical Elasticity vol I Cia The standard intro duction for math
ematicians
Ciarletta and Iesan Non classical Elastic Solids CI
Crandall Dahl and Lardner An Introduction to the Mechanics of Solids CDL
Gordon The New Science of Strong Materials or Why You Don t Fal l Through the
Floor Gor
Green and Adkins Large Elastic Deformations and Non Linear Continuum Mechan
ics GA
Green and Zerna Theoretical Elasticity GZ
Gurtin Topics in Finite Elasticity Gur
Love ATreatise on the Mathematical Theory of Elasticity Lov
Malvern Introduction to the Mechanics of a Continuous Medium Mal
Marsden and Hughes Mathematical Foundations of Elasticity MH
Maugin Material in homogeneities in Elasticity Mau
Sp encer Continuum Mechanics Sp e
Stoker Nonlinear Elasticity Sto
Timoshenko and Goodier Theory of Elasticity TG
ory of Anisotropic Elasticity and Applications Wu Ting and Barnett Modern The
WTB
G Fibre reinforced Materials
See
Mallick Fiber reinforced Composites Mal
Derek Hull AnIntroduction to Composite Materials Hul
Green and Adkins chapter VI I Reinforcementby Inextensible Cords Large Elastic
Deformations and Non Linear Continuum Mechanics GA
Christensen Mechanics of Composite Materials Chr
G Elastic Solids with Microstructure
See
Ciarletta and Iesan Non classical Elastic Solids CI
Kunin Elastic Media with MicrostructureI Kun and Kunin Elastic Media with
MicrostructureII Kun
Capriz Continua with Microstructure Cap
Nowacki Theory of Asymmetric Elasticity Now
See also literature on p orous solids Hornung DH
App endix H
Biomechanics
H General References
See
Alexander Animal Mechanics Ale
Alexander Elastic Mechanisms in Animal Movement Ale
Alexander Mechanics of Animal Locomotion Ale a
Biewener Biomechanics Structures and Systems A Practical Approach Bie
Currey The Mechanical Adaptation of Bones Cur
Fung A First Course in Continuum Mechanics Fun
Flindt Biologie in Zahlen Fli
Gans Biomechanics An Approach to Vertebrate Biology Gan
Johnson Biomechanics and Exercise Physiology Joh
McMahon Muscles Re exes and Locomotion McM
Valenta Biomechanics Val
Vincent Structural Biomaterials Vin
Vincent Biomechanics Materials A Practical Approach Vin
Vincent and Currey The Mechanical Properties of Biological Materials Symp Soc
Exp Biol VC
Vogel Steven Life s Devices The Physical World of Animals and Plants Vog
Wainwright Axis and Circumference Wai
Wainwright Biggs Currey and Gosline Mechanical Design in Organisms WBCG
H Biomaterials
See
Brown Structural Materials in Animals Bro
Fung Biomechanics Mechanical Properties of Living Tissues Fun
Silver Biological Materials Structure Mechanical Properties and Modeling of Soft
Tissues Sil
Vincent Structural Biomaterials Vin
Vincent Biomechanics Materials A Practical Approach Vin
Vogel Life s Devices The Physical World of Animals and Plants Vog
Yamada Strength of Biological Materials Yam
H Mathematical Biology
See
Harrison Kinetic Theory of Living Pattern Har a
Levin Lectures on Mathematics in the Life Science Vol Some Mathematical
Questions in Biology Lev
Levin Frontiers in Mathematical Biology Lev
Mathematical Biology Mur Murray
Levin Frontiers in Mathematical Biology Lev
H Pattern Formation
Concerning pattern formation see
Harrison Kinetic Theory of Living Pattern Har a
Murray Mathematical Biology Mur
Concerning phyllotaxis see
Jean Phyl lotaxis A systemic study in plant morphogenesis Jea
Thompson On Growth and Form Tho
The phenomenon of phyllotaxis refers to the arrangement of leaf or oret primordia in
plants according to the Fib onacci sequence f g as seen in more than
of plants
H Bio energetics
See
Kleib er The FireofLife Kle
H Kinesiology
The science of the movement of the b o dy is called kinesiology See
Rasch
App endix I
Computer Graphics
Concerning computer graphics see
Foley van Dam Feiner and Hughes Computer Graphics Principles and Practice
FvDFH
Glassner Graphics GEMS Gla
Holden Reconstructing the heart Hol
Ho ehne D Computergra k in der Medizin Diagnose glasklar Ho e
O Rourke Computational Geometry in C O R
Wickham Jones MathematicaGraphics Techniques and Applications WJ
App endix J
Numerical Metho ds
Concerning numerical metho ds see
Adams MUDPACK Multigrid portable FORTRAN software for the e cient solu
tion of linear el liptic partial di erential equations Ada
Allen Herrera and Pinder Numerical Modeling in Science and Engineering IHP
Bank PLTMG A Software Package for Solving El liptic Partial Di erential Equa
tions Users Guide Ban
Brebbia and Dominguez Boundary Elements An Introductory Course nd edition
BD
Burkett Numerical methods for viscoelasticity Bur
Szab o and Babuska Finite Element Analysis SB
Smith and Cri ths Programming the Finite Element Method SG b
Cris eld Non linear Finite Element Analysis of Solids and Structures Cri b
Zafrany El Techniques of the Boundary Element Method El
Kim and Karrila Microhydrodynamics KK
Hirsch Numerical Computation of Internal and External Flows Vol Computa
tional Methods for Inviscied and Viscous Flows Hir
Glowinski and Le Tallec AugmentedLagrangian and Operator Splitting Methods in
Nonlinear Mechanics GT
Grossmann and Ro os Numerik partiel ler Di erentialgleichungen GR
Langtangen and Nielsen Getting started with Di pack The Di pack Report Series
LN
J Finite Elements
Concerning nite elements see
Dhatt and Touzot The Finite Element Method Displayed DT
MacNeal Finite Elements Their Design and Performance Mac
Reddy AnIntroduction to the Finite Element Method Red
Schwarz Methode der niten Elemente Sch b
Schwarz FORTRAN Programme zur Methode der niten Elemente Sch
Zienkiewicz and Taylor The Finite Element Method Vol Basic Formulation and
Linear Problems ZT a
Zienkiewicz and Taylor The Finite Element Method Vol Solid and Fluid Me
chanics Dynamics and Non Linearity ZT b
App endix K
Sources
K Sources of Preserved Hearts
The following information ab out suppliers of mo dels of hearts is taken from Vogel Vog
page
All of these vendors o er embalmed cow or calf sheep or lamb and pig hearts They
all have horribly exp ensive anatomical mo dels but Fisher and Wards have less costly
ones as well For nonfondlers NASCO has a reasonably priced sheep heart sliced op en
an emb edded in plastic and Wards has freeze dried whole and bisected as opp osed to
dissected sheep hears Addresses
Carolina Biological Supply Co
Burlington NC
NASCO
Janesville Ave Box
Ft Atkinson WI
Fisher Scienti c Company
Educational Materials Division
W LeMoyne Street
Chicago IL
Ward s Natural Science
Establishment
P O Box
Ro chester NY
A go o d source for inexp ensive mo dels but whose catalog maybedangeroustoyour
nancial security is
Edmund Scienti c Co
E Glouchester Pike
Barrington NJ
K Bo oks Rep orts Theses
BMJ Bo okshop P O Box London WC H TE England
Carries sto ck of medical b o oks
National Technical Information Service NTIS
Can provide copies of almost all o cial technical rep orts published in the U S A
University Micro lm UMI
Can provide copies of all Ph D theses listed in the journal
K Journals
The following journals frequently contain information on the cardiovascular system
Cardiovascular Research
K So cieties
Cardiovascular System Dynamics So ciety
The so ciety holds a biannual conference the pro ceedings of which are published
Address
J Yasha Kresh Ph D
Liko Cardiovascular Institute
Hahnemann University Ro om NCB
Broad Vine Streets Mail Stop
Philadelphia PA
Tel
Fax
email kresh cvi hahnemann edu
Europ ean So ciety for Noninvasive Cardiovascular Dynamics
The Europ ean So ciety for Noninvasive Cardiovascular Dynamics ESNICVD was
initiated bythephysicist Prof A C Burger in the Netherland The main goal
tists and professionals whose of the ESNICVD is to unite and to keep together scien
common interest is to promote the knowledge and understanding of cardiovascular
dynamics with noninvasive metho ds in researchaswell as in medical practice The
congresses whichhave b een held all over Europ e for the past decades serve
to keep the scientists from this eld together despite the movementtowards ever
greater sp ecialization
DutchFoundation for Biophysics Verenigung vo or Biofysica This has a subsection
for the Heart and Circulation
Verenigung vo or Biofysika Kring Hart en Circulatie
British Heart Group
K Deutsche Literatur
Bauereisen Physiologie des Kreislaufs Bau
Bowditch Ueber die Eigenthuemlichkeite n der Reizbarkeit welche die Muskelfasern
des Herzens zeigen Bow
Braun Diagnose und Therapie der Herzkrankheiten Bra
Burton Physiologie und Biophysik des Kreislaufs Bur
Ca isch Strukturgerechte Analyse der Myokardfaserspannungen Caf
Culclasure Anatomie und Physiologie des Menschen Lehrprogramme Band
Herz und Kreislaufsystem Cul
do Carmo Di erentialgeometrie von Kurven und Flaechen dC
Dierb erger Finite Elemente Simulat ion der Herzdynamik Das Herzgroessen Schlagvolumen
Konzept Die
Do err et al Pathologische Anatomie des Herzens und seiner Huel len I DSU
Faller Der Faserverlauf im Bindegewebe des Epikards menschlicher Herzen und
funktionel le Deutung Fal
Feneis Das Gefuege des Herzmuskels bei Systole und Diastole Fen
Flindt Biologie in Zahlen Fli
Frese Chaos Theorie und ploetzlicher Herztod Fre
Frank Zur Dynamik des Herzmuskels Fra
Glaser Biophysik Gla
Grossmann and Ro os Numerik partiel ler Di erentialgleichungen GR
Ho ehne D Computergra k in der Medizin Diagnose glasklar Ho e
Hort Untersuchungen ueber die Muskelfaserdehnung und das Gefuege des Myokards
in der rechten Herzkammerwand des Meerschweinchens Hor
Hort Untersuchungen zur funktionel len Morphologie des Bindegewebsgerue stes und
der Blutgefaesse der linken Herzkammerwand Hor
Horvath Physikalische Uebungsaufgaben fuer Biologen und Mediziner Hor
Krehl Beitraege zur Kenntniss der Fuel lung und Entleerung des Herzen Kre
Leonhardt Taschenatlas der Anatomie Band InnereOrgane Au age KLP
Lunkenheimer et al Kardiodynamik Wege zur strukturgerechten Analyse der Myokard
funktion LLG
Mattheck Design in der Natur Mat
Meesmann Nachweis der diastolischen Sogwirkung der Herzkammern und deren
Ein uss auf die intrakardialen Druckablaeufe Mee
Meinzer Raeumliche Bilder des Koerperinneren Mei
Pu Der funktionel le Bau der Herzkammern Puf
Schmidt und Thews Physiologie des Menschen ST
Schwarz Methode der niten Elemente Sch b
Schwarz FORTRAN Programme zur Methode der niten Elemente Sch
Thomas et al Grund lagen der klinischen Medizin Anatomie Physiologie Patholo
gie Mikrobiologie Klinik Band Herz und Gefaesse TGH
Tigerstedt Die Physiologie des Kreislaufes Tig
Torrent Guasp Lunk enheimer Intensivkurs Funktionel le Anatomie des Herzens
Vom bei Dr Torrent Guasp in Spanien Denia Alicante Gespr archsnotizen
GL
Trost Analysis eines raeumlichen Model ls vom Herzmuskel Tro
Wol Das Gesetz der Transformation der Knochen
K Deutsch Englisch Verzeichnis
K Diplomthemen
See
Adams A multigrid algorithm for immersed interfaceproblems Ada
Cameron The Respiratory Physiology of Animals Cam
Costa Hunter Rogers Guccione Waldman and McCullo ch A three dimensional
nite element method for large elastic deformations of ventricular myocardium Part
I Cylindrical and Sherical Polar Coordinate CHR
Costa Hunter Wayne Waldman Guccione and McCullo ch A three dimensional
nite element method for large elastic deformations of ventricular myocardium Part
II Prolate spheroidal coor dinates CHW
Harrison Kinetic Theory of Living Pattern Har a
Kowalczyk and Kleib er Model ling and numerical analysis of stresses and strains in
the human lung including tissue gas interaction KK
LeVeque and Li Immersed interface methods for Stokes ow with elastic boundaries
or surface tension LL
Lin and Tab er AModel for Stress InducedGrowth in the Developing Heart LT
Moskowitz
Rogers and McCullo ch Acol location Galerkin nite element model of cardiac action
potential propagation RM
Yang Tab er and Clark A Nonlinear Poroelastic Model for the Trabecular Embryonic
Heart YTC
App endix L
Worksheets
L Questions
This section contains various questions some of which are p oints to b e discussed with
PPL and some of which are p ossible research topics
Synchronisation of left and rightventricles
It is usually stated that the left and rightventricles work at the same time Is this
true If there is a small time di erence is there an explanation for this
Angular Momentum
The heart swivels during each b eat Is this caused by the angular momentum pro
duced by the out ow of blo o d from the ventricles into the arteries
Global balance
Can one derive information from the global balance of forces See HSHY page
section and Mor
Law of Laplace
The inner wall of the left ventricle the septum is concave see Lev Levick g
c page Can the law of Laplace still apply See Mor
Peskin s Formula
Peskin Comm P Appl Math page compares his theory with results of
Streeter et al Is this correct
Peskin s cross sectional assumption
Peskin Comm P Appl Math page assumes that every bre tub e has
constant cross sectional area Correct
In uence of the p ericardium The p ericardium restrains the heart What is its
role According to Fung Circulation p op ening of the p ericardium has an
imp ortant e ect on the shap e and size of the heart He mentions also p the
di erence b etween surgery carried out with the p ericardium op en and closed
Heart oscillation PPL mentioned the measurement of heart b eats by placing pa
tients on a table
Flowintheventricle How is the ow in the left ventricle in uenced by the papil
lary muscles and by the clearly visible ridges on the inner surface the endo cardium
Stirling s law time dep endence In Stirling s exp eriments the pressure was
changed and the transitionary data were recorded see Lev p The system
is apparently damp ed Can one mo del this
Myo cardial activation In Hunter and Smaill p HS exp erimental and
numerical activation times are compared It would b e interesting to try to analyze
some of the features such as the closed contours in Figure
Heart deformation Are there any results in the literature which mo del the heart
deformation and blo o d pressure as functions of time assuming simple contraction
Systole expansion During diastole the ventrical wall recoils Lev p line
Why
Electrical activity Can the upwards movement of the heart b e explained by the
fact that the muscle is activated at slightly di erent times
Rightventricle and atrium These have not b een considered as extensively even
though they handle the same amountofblood
Biot s equations
Co ordination of veins and right atrium PPL fragen
T ectronic compartments of heart walls PPL fragen
The septum do es not distort but acts as though it were hinged
Wo o ds Role of t
Why
Do surface bres stay on surface
Where do cross bres come from go to
What are the trabi muscles
Are there bre lo ops
Do chordae tendae really do work
What ab out separate hearts
What ab out unrolling
What ab out ap ex
Magnetism
Magnets N S
rotating earth
Can one think up b etter constitutive equations for passivemyocardium What ab out
rational functions
Howdothechorae tendineae transmit the force to the tricuspid and mitral valves
Do the papillary muscles valves distend Are the chordae tendineae optimally
distributed
L Availability of Journals etc in Munster
Amer Heart J
Vol to date MED ZQ
Amer J Physiol
Split up into Series G H etc
MED ZD ie
Institut fr Physiologie
Rob ert Ko ch Str a
Annual Review of Physiology
MED
Anat Rec
MED ZD
Biophysical Journal
Vol to date Zo ologische Institute
Badestr MS
Vol to date UB ZD
Bull math Biology
ZD
Circulation
Vol ZQu
Vol date
Vol date
Vol
Circulation Research
Vol to date Med Klinik Innere Medizin A C
Alb ert Schweizer Str Munster ZC
Vol to date Institute Physiologie I I
Rob ert Ko ch Str Munster
Computer Programs in Biomedicine
Vol Continued as Computer Metho ds and Programs in
Biomedicine NotinM unster
CRC Crit Rev Biomed Eng
Chemical Rubb er Co Not available in Munster
Journal of Biomechanical Engineering
Not available in Munster
Journal of Biomechanics
Vol to date Institut fur
exp Biomechanik
Domagkstr Munster
J Fluid Mechanics
ZD
Journal of Thoracic Surgery
Klinik und Poliklinik fr
Thorax Herz Gefchirurgie
Alb ert Schweizer Str
Literatur Medizin
WG Kardiovaskul ares System in der UB
Math Bioscience
Not in Munster
Progress in Biophysics and molecular biology
Vol to date Zo ologische Institute Badestr Munster
Scand J Thoracic and Cardiov Surg
either ZD all
or b
Scienti c American
ZC
Virchows Archiv
Vol Vol UB ZD
Vol Vol Gerhard Domagk
MED ZD
Institut fur Pathologie
Domagkstr MS
Z fuer Biologie
Vol Vol
Gegenbauers morph J
MED ZD
Journal of Physiology London
MED Z and UB
L Literature Searched
Bull Math Biology
ZD
Cardiovascular Research
Jan June
Circulation Research
Jan June
all
Must copy these pages
p
Dissertation Abstracts DA
No No No No
DA all
Index to Theses UK
Part Part Part
Am J Physiology
Am J Physiology
No Jan
NO Feb
No March
all
L Ph D Theses
The following theses have not b een ordered as yet
Finite analytic numerical solution of laminar and turbulent owinatwo dimensional
arti cial heart mo del
S H Kim Sang Hyun Kim
UniversityofIowa No p B DA
Advisor K B Chandran
The e ects of right heart loading on left ventricular p erformance of in situ ejecting
hearts
E B Lankford
Johns Hopkins p No p B DA
Advisor not given
Hemo dynamics and wall shear rate measurements in the ab dominal aorta of dogs
K C White
Penn State No p B p DA
Advisor John M Tarb ell
Mo dels of cardiac muscle contraction and relaxation
J L Palladino
UniversityofPennsylvania No p B p DA
Advisor Abraham No ordergraaf
Reconstructing uid ow inside the heart from multiple x ray image sequences
N H Cornelius
Carnegie Mellon No p B p DA
Advisor not given
Mathematical mo dels of comp onents of the cardiovascular system Arterial barore
ceptors and heart rate control
T F Itani Northwestern Nov No p B p DA
Advisor not given
Lo ok if theses by following are available
Costa Kevin D
Tab er Larry A
Burkett D A DA No
LeGrice Ian J A Finite Element Model of Myocardial Structur e Implications for
Electrical Activation in the Heart LeG
Rustum Choksi not in DAvol
L Corresp ondance
Prof Dr Charles Samuel Peskin
Professor Dr Charles Samuel Peskin
Courant Institute of Mathematical Sciences
New York University
e mail p eskin cims nyu edu
Prof Dr Peter J Hunter
Prof Dr Peter J Hunter
Department of Engineering Science
Scho ol of Engineering
University of Auckland
New Zealand
Tel
Fax
e mail hunter esvl aukuni ac nz
Prof Dr Karl Heinz H ohne
Prof Dr Karl Heinz H ohne
Universit at Hamburg
Fachb ereich Medizin
Institut fur Mathematik und
Datenverarb eitung in der Medizin
Martinistr
Hamburg
Tel
Prof Dr Udupa
Prof Dr Jayaram K Udupa
UniversityofPennsylvania
Chief Medical Imaging Section and
Asso ciate Professor
Medical Image Pro cessing Group
Department of Radiology
Blo ckley Hall Fourth Flo or
Service Drive
Philadelphia PA
e mail jay jay mipg up enn edu
Prof Dr Poul Michael F nss Nielsen
Biomedical Engineering
Tel
Prof Dr Andrew D McCullo ch
Prof Dr Andrew D McCullo c h
Institute for Biomedical Engineering
Cardiac Mechanics Lab oratory
University of California San Diego
Gilman Drive
La Jolla California
Tel
FAX
e mail mccullo c heart ucsd edu
Prof Dr Ruegsegger
Tel
FAX
Jennifer Rogers
email ja rogers auckland ac nz
Professor William Kier
Department of Biology
The University of North Carolina at Chap el Hall
Coker Hill CB
Chap el Hill NC
Tel
Fax
Professor Abraham No ordergraaf
e mail ano or cardio dyn seas up enn edu
App endix M
Biomathematics
General references on Biomathematics are
Winter On the Stem curve of a tal l palm in a strong wind Win
Biblio graphy
AB Rob ert H Anderson and Anton E Becker The orientation of bres within
the ventricular mass In Churchill Livingstone editor Cardiac anatomy
chapter Gowes Medical Publishing London Quoted by
Greenbaum Ho Gibson Becker and Anderson Get
AB Rob ert H Anderson and Anton E Becker The Heart StructureinHealth
and Disease Gower Medical Publishing London ISBN
Have O K Also have set of slides
Aci Louis J Acierno The History of Cardiology Parthenon Publishers Cast
erton Hall Carnforth Lancs Ordered Blackwells for cwc
Have O K ISBN
AD Philip L Altman and Dorothy S Dittmer editors Biology Data Book
volume I I I Federation of American So cieties for Exp erimental Biol
ogy Bethesda Maryland second edition In Zo ologie Not in BIP
No chmal anschauen
Ada J C Adams MUDPACK Multigrid p ortable FORTRAN software for the
e cient solution of linear elliptic partial di erential equations Appl Math
Comput Get
Ada Loyce Adams Amultigrid algorithm for immersed interface problems
T echnical rep ort Dept of Applied Mathematics UniversityofWashington
Havecwc O K
Ada Colin Conrad Adams The Knot Book W H Freeman New York
ISBN X Havecwc
Ada Adam editor Physikalische Chemie und Biophysik Springer Verlag
From Springer Catalog Lo ok at
ADG Philip L Altman Dorothy S Dittmer and Rudolph M Greb e Handbook
of Circulation W B Saunders Philadelphia In PHYSIOLOGIE I
Not in BIP No chmal anschauen
Ale Rob ert McNeill Alexander Animal Mechanics Blackwell Scienti c Publi
cations Oxford nd edition Quoted in A A Biewener Biomechanics
Structures and Systems A Practical Approach Get In BIP
Not in MED In Whitaker X Ordered Kruep er
Vergri en laut Kruep er vom Vergri en lt Blackwells
Ordered Order UB
Ale Rob ert McNeill Alexander Bending of cylindrical animals with helical
bres in their skin or cuticle Journal of theoretical Biology
Quoted by R McNeill Alexander p Get
Ale Rob ert McNeill Alexander Elastic Mechanisms in Animal Movement
Cambridge University Press Cambridge p Quoted bySchmidt
Nielson Animal Physiology Adaptation and Environment page
Not in MED ISBN Have O K
Ale a Rob ert McNeill Alexander editor Comparative and Environmental Phys
iology Mechanics of Animal Locomotion Springer Verlag ISBN
Have O K
Ale b Rob ert McNeill Alexander The Human Machine Natural History Museum
Publications Have ISBN Havecwc
Ale Rob ert McNeill Alexander Bones The Unity of Form and Function
Macmillan ISBN In BP LC Ordered
Kruep er
AM Ralph Abraham and Jerrold E Marsden Foundations of Mechanics Ben
jamin Cummings nd edition In Math Bibl
AM Roy M Anderson and Rob ert M May Infectious Diseases of Humans
Oxford University Press Have O K
Ant James Francis Antaki An Experimental and Theor etical Model for the Pas
sive Biomechanical Properties of the Intact Heart Volumes I and II PhD
thesis University of Pittsburgh Have UMI No cheinmal
grundlic h lesen Hat mehrere interessante Zitate
Ant Stuart S Antman Nonlinear Problems of Elasticity Springer Verlag
Have ISBN
AR Theo Arts and Rob ert S Reneman The imp ortance of geometry of the
heart to the pump In Henk E D J ter Keurs and Mark I M Noble
editors Starling s Law of the Heart Revisited pages Kluwer
Academic Publishers Have
Arm M A Armstrong Basic Topology Springer Verlag Go o d Intro duc
tion Get In Math Lib
Arm A Armenti Jr editor Physics of Sports AIP Get AMR ISBN
Order for cwc privat Ordered Kruep er
Arn V I Arnold Catastrophe Theory Springer Verlag ISBN
Get
Arn V I Arnold Singularity Theory Cambridge University Press Cambridge
Collection of pap ers including some generating concept of index to
manifolds with b oundary Get
Art Theo Arts A Mathematical Model of the Dynamics of the Left Ventricle and
the Coronary Circulation PhD thesis University of Limburg Maastricht
The Netherlands Ordered UB Wrote Jan to Theo Arts
NL Ordered UB No ch nichts erhalten UB
ARV Theo Arts Rob ert S Reneman and Pieter C Veenstra A mo del of the
mechanics of the left ventricle Annals of Biomedical Engineering
Have
AS John N Amo ore and William P Santamore Venous collapse and the res
piratory variability in systemic venous return Cardiovascular Research
Have O K
Ask Michael J Askew Advances in Bioengineering American So ciety of Me
chanical Engineers New York Have ISBN
Ata Fatihcan M Atay A di erential delay equation arising in the model ling of
the pupil light re ex PhD thesis Brown University Rho de Island
Get
Atk Peter William Atkins Elements of Physical Chemistry Oxford University
Press Oxford ISBN X Have
Atk Peter William Atkins Physical Chemistry th edition Oxford University
Press Oxford Have ISBN
Atl SatyaN Atluri Alternate stress and conjugate strain measures and mixed
variational formulations involving rigid rotations for computational anal
yses of nitely deformed solids with applications to plates and shells I
Theory Computers and Structures Quoted byYang
Tab er and Clark Have O K
AVR Theo Arts Pieter C Veenstra and Rob ert S Reneman Epicardial defor
mation and left ventricular wall mechanics during ejection in the dog Amer
ican Journal on Physiology Heart and Circulatory Physiology H
H Have O K
AY Loyce Adams and Z Yang editors Acomparison of techniques for solving
il l conditionedproblems arising from the immersedboundary method Pro
ceedings of Symp osia in Applied Mathematics Vancouver CA Aug
Get Quoted byLoyce Adams
Baa Jan Baan editor Ventricular Function and PressureVolume Analysis
the Role of Conductance in Car diology Pro ceedings Conference in Nuern
b erg August Europ ean Heart J vol Suppl E Academic
Press London Not in BIP In MED Lesesaal MED ZD No chmal
anschauen
BAH Peter Hermanus Marie Bovendeerd Theo Arts Jacques Maria R J
Huyghe DickH van Camp en and Rob ert S Reneman Dep endence of lo
cal left ventricular wall mechanics on myo cardial b er orientation A mo del
study Journ Biomech Have
Ban Randolph E Bank PLTMG A Software Package for Solving El liptic Par
tial Di erential Equations Users Guide So ciety for Industrial and
Applied Mathematics Philadelphia Have ISBN
Bau E Bauereisen Physiologie des Kreislaufs Band und Springer Verlag
Berlin In PHYSIOLOGIE I Not in print
BAY Jan Baan Alexander C Arntzenius and Edward L Yellin editors Car
diac Dynamics Proc rd International Conference Cardiovascular System
Dynamic Society Kluwer Academic Press Dordrecht p
ISBN Have O K
BB B J Bellhouse and F H Bellhouse Fluid mechanics of a mo del mitral
valve and left ventricle Circulation Research In MED
ZC Checkvol year
BB R D Bauer and R Busse The Arterial System Dynamics Control Theory
and Regulation Springer Verlag New York Get Not in BIP Ordered
UB Zurueck Vorh in MS UB G o der G
BCL Philipp e Brun Richard S Chadwick and Bernard I Levy editors Cardio
vascular Dynamics and Models Proceedings of NIH INSERM workshops
Bethesda September Paris volume of Col loques INSERM
Les Editions INSERM Paris Have ISBN
BD C A Brebbia and J Dominguez Boundary Elements An Introductory
Course nd edition Computational Mechanics Publications and McGraw
Hill Bo ok Company Have
BE D Bluestein and S Einav Transition to turbulence in pulsatile ow
through heart valves a mo di ed stability approach Transactions of the
ASME Journal of Biomechanical Engineering Have
Bel B J Bellhouse The uid mechanics of heart valves In D H Bergel
editor Cardiovascular Fluid Dynamics volume pages Academic
Publishers New York Have
Ben Jan E W Beneken Some computer mo dels in cardiovascular research
In D H Bergel editor Cardiovascular Fluid Dynamics volume pages
Academic Publishers New York Have
Ber a D H Bergel editor Cardiovascular Fluid Dynamics volume Aca
demic Press New York ISBN Ordered Kruep er
Rep orted out of Print Dillons Not available Have Xerox
copy
Ber b D H Bergel editor Cardiovascular Fluid Dynamics volume Academic
Press New York Have Xerox copy ISBN
Ber Rob ert M Berne editor Handbook of Physiology Section The Car
diovascular System Vol I The Heart American Physiological So ciety
Bethesda distributed by Oxford University Press Oxford Have
O K ISBN Not in MED In PHYSIOLOGIE I I Weitere
Baende Galloway
BFG J C Barb enel A C Fisher J D S Gaylor W J Angerson and C D
Sheldon editors Blood Flow in Arti cial Organs and Cardiovascular Pros
theses volume of OxfordMedical Engineering Series Clarendon Press
Oxford p ISBN Have Xeroxcopy O K
BG Richard L Bishop and Samuel I Goldb erg Tensor Analysis on Manifolds
Dover Have O K
BHA Peter Hermanus Marie Bovendeerd Jacques Maria R J Huyghe Theo
Arts DickH van Camp en and Rob ert S Reneman In uence of
endo cardial epicardial crossover of muscle b ers on left ventricular wall
mechanics Journ Biomech Have
Bie A A Biewener editor Biomechanics Structures and Systems A Practical
Approach Oxford University Press Have ISBN
Bio M A Biot unb ek J Anat Physiol Get Medizin ZD
MED
Bio M A Biot Theory of nite deformations of p orous solids Indiana Univ
Math J Have O K
Bio M A Biot Ho dograph metho d of nonlinear stress analysis of thick walled
cylinders and spheres including p orous materials Int J Solids Structures
Quoted byAntman p Get Order UB
BK Paul J Blatz and William L Ko Application of nite elasticity to the
deformation of rubb ery materials Transactions of the Society of Rheology
Quoted byYang Tab er and Clark Have O K
BK George W Bluman and Sukeyuki Kumei Symmetries and Di erential
Equations Springer Verlag New York Have
BKDA M B Bennet R F Ker N J Dimery and Rob ert McNeill Alexander
Mechanical prop erties of various mammalian tendons J of Zoology A
Get UB ZD
BL Rob ert M Berne and Matthew Levy Cardiovascular Physiology Mosby
Year Bo ok St Louis sixth edition Have O K
Bli Blix Get Volume I I I p Volume IV p
Bli Blix Get Volume V pp and Volume VI p
BN L F Bishop and J Neilson Jr editors History of Cardiology Medical
Life Press New York Ordered UB Get Not in BIP Not in MED
Not available in Germany
BNR Jan Baan A No ordergraaf and J Raines editors Cardiovascular System
usetts Institute of Technology Cambridge Dynamics MIT Press Massach
Have O K
Bog a Daniel K Bogen Strain energy descriptions of biological swelling I Single
uid compartment mo dels Transactions of the ASME Journal of Biome
chanical Engineering Have UB O K
Bog b Daniel K Bogen Strain energy descriptions of biological swelling I I Mul
tiple uid compartment mo dels Transactions of the ASME Journal of
Biomechanical Engineering Have UB O K
Bol L Bolis editor Comparative physiology Locomotion respiration transport
and blood Proc Intern Congress Aquasparta Elsevier Out
of Print Kruep er Get ISBN Not in MED Not in
UB Dillons Not available Order UB
Bor Giovanni Alfonso Borelli On the Movement of Animals Translated
by P Maquet Springer Verlag New York Quoted byFung A rst
course in continuum mechanics pages and ISBN
Have O K
Bos Amitabha K Bose Existence and stability of travel ling waves for couples
nerve axon equations PhD thesis Brown University Rho de Island
Get
Bot Hans Bot Mathematical Mo dels of Diastolic Blood Flow Patterns in the
Human Left Ventricle PhD thesis University of Amsterdam Have
current address Have
Bou Y Bouligand Twisted brous arrangements in biological materials and
cholesteric mesophases Tissue and Cel l Quoted by Win
free Get ZOOLOG
Bov Peter Hermanus Marie Bovendeerd The Mechanics of the Normal and
Ischemic Left Ventricle During the Cardiac Cycle A Numerical and Ex
perimental Analysis PhD thesis Rijksuniversiteit Limburg te Maastricht
Have
Bow H P Bowditch Ueb er die Eigenthuemlichkeiten der Reizbarkeit welche
die Muskelfasern des Herzens zeigen Abhand lungen der mathematisch
physischen Classe der Koeniglich Saechsischen Gesel lschaft der Wis
senschaften Leipzig Have
Bow RayM Bowen Theory of mixtures In Ahmed Cemal Eringen editor
Continuum Physics volume pages Academic Press Get
UB G Lesesaal
Bow RayM Bow en Incompressible p orous media mo dels by use of the theory
of mixtures Int J Enging Sci Have UB O K
Bow RayM Bowen Introduction to Continuum Mechanics for Engineers
Plenum Press Have O K ISBN
Bow H P Bowditch On the p ecularities of excitability which the bres of
cardiac muscle show translation of pap er of In Mark I M No
ble and W A Seed editors The Interval ForceRelationship of the Heart
Bowditch Revisited pages Cambridge University Press Cambridge
Have
Bra Ludwig Braun Diagnose und Therapie der Herzkrankheiten Urban und
Schwarzenb erg Berlin nd edition Havecwc
Bra A J Brady Passive elastic prop erties of cardiac myo cytes relative to in
tact cardiac tissues In T F Robinson and R K H Kinne editors Car
diac Myocyte Connective Tissue Interactions in Health and Disease Issues
Biomed volume Karger Basel Have O K
Bra Eugene Braunwald editor Heart Disease W B Saunders Philadelphia
th edition Available as or vols ISBN A
standard reference for clinicians Excellentvalue for money Have O K
BRC G E Burch C T Ray and J A Cronvich Certain mechanical p eculiari
ties of the human cardiac pump in normal and diseased states Circulation
Hav e O K
Bre C A Brebbia editor Boundary Elements X Vol Heat transfer uid
ow and electrical applications Computational Mechanics Publications and
Springer Verlag Have
Bri Percy Williams Bridgman Dimensional Analysis Yale Press New Haven
revised edition Have Xerox copy
Bro C H Brown Structural Materials in Animals Pitman Publishing
Havecwc O K ISBN
BRS Eugene Braunwald John Ross Jr and Edmund H Sonnenblick Mecha
nisms of Contraction of the Normal and Failing Heart Little Brown and
Company Boston second edition Quoted by Braunwald in Hand
book of physiology The heart page Have Xeroxcopy Excellent and
detailed O K
Bru Brutsaert See Sagawa b o ok for other pap ers Get
BS J R Bo oker and J C Small An investigation of the stabilityofnumeri
cal solution of Biot s equations of consolidation Int J Solids Structures
Quoted by Huyghe Order UB Get
BS C Gunnar Blomqvist and Bengt Saltin Cardiovascular adaptations to
physical training Annual R ev Physiology Quoted by
Rowell p Have
BS a Rafael Beyar and Samuel Sideman A computer study of the left ventricular
p erformance based on the b er structure sarcomere dynamics and trans
mural electrical propagation velo city Circulation Research
Have O K
BS b Rafael Beyar and Samuel Sideman A mo del for left ventricular contraction
combining the force length velo city relationship with the time varying elas
tance theory Biophys Journ Get Quoted by BS a ZOOLOG
UB ZD
BS Rafael Beyar and Samuel Sideman Spatial energy balance within a struc
tural mo del of the left ventricle Annals of Biomedical Engineering
Have O K
BS O O Blumenfeld and S Seifter Bio chemistry of connective tissue with sp e
cial emphasis on the heart In T F Robinson and R K H Kinne editors
Cardiac Myocyte Connective Tissue Interactions in Health and Disease Is
sues Biomed volume Karger Basel Have O K
BS Susanne C Brenner and L Ridgway Scott The Mathematical Theory of
Finite Element Methods Springer Verlag Have ISBN
BT T K Borg and L Terracio Interaction of the extracellular matrix with
cardia myo cytes during development and disease In T F Robinson and
R K H Kinne editors Cardiac Myocyte Connective Tissue Interactions
in Health and Disease Issues Biomed volume Karger Basel
Have O K
Bur Alan C Burton The imp ortance of the shap e and size of the heart Amer
ican Heart Journal Have O K
Bur Alan C Burton Physiologie und Biophysik des Kreislaufs Schattauer
Verlag Stuttgart In PHYSIOLOGIE
Bur Alan C Burton Physiology and Biophysics of the Circulation An Intro
ductory Text Bo oks on Demand nd edition Year Bo ok Medical
Publishers Chicago Available from LC ISBN
pap er Have O K
Bur Daniel Andrew Burkett Numerical methods for viscoelasticity PhD thesis
Carnegie Mellon University Pennsylvania Get DA B Order
UMI
BWR Maurice B Buchalter James L Weiss Walter J Rogers Elias A Zer
houni Myron L Weisfeld Rafael Beyar and Edward P Shapiro Non
invasivequanti cation of left ventricular rotational deformation in normal
humans using magnetic resonance imaging myo cardial tagging Circulation
Have O K
Caf Amadeo Ca isch Strukturgerechte Analyse der Myokardfaserspannungen
Have Seite fehlt O K
Cam James N Cameron The Respiratory Physiology of Animals Oxford Uni
versity Press Have ISBN O K
Cap G Capriz Continua with microstructure In C Truesdell editor Springer
Tracts in Natural Philosophy Springer Verlag New York Get
Quoted by Ciarletta and Iesan Non classical Elastic Solids Pitman Re
search Notes in Mathematics UB b estellen
Car Gail A Carp enter editor Lectures on Mathematics in the Life Science Vol
Some Mathematical Questions in Biology Circadian Rhythms Ameri
can Mathematical So ciety Providence Have
CB James B Caul eld and T K Borg The collagen network of the heart
Laboratory Investigation Have O K
CC R B Clark and J B Cowey Factors controlling the change of shap e of
certain nemertean and turb ellarian worms Journal of experimental Biology
Quoted by R McNeill Alexander p Get ZQu
UB XEROX
CC Gill Jeong Cheon and K B Chandran Transientbehavior analysis of a
mechanical monolea et heart valve prosthesis in the closing phase Trans
actions of the ASME Journal of Biomechanical Engineering
Have
CCMH H J Chiel P Crago J M Mansour and K Hathi Biomechanismsofa
muscular hydrostat a mo del of lapping by a reptilian tongue Biological
Cybernetics Quoted by Johnson and Keir ZC
ZS ALL Get UB XEROX
CCPC P S Chen Y M Cha B B Peters and L S Chen E ects of my
o cardial b er orientation on the electrical induction of v entricular brilla
tion American Journal on Physiology Heart and Circulatory Physiology
H H Get Have Medline abstract
CDL Stephen H Crandall Norman C Dahl and Thomas J Lardner editors
An Introduction to the Mechanics of Solids McGraw Hill Have O
K ISBN
CF C J Chuong and Yuan cheng Fung Three dimensional stress distribution
in arteries Transactions of the ASME Journal of Biomechanical Engi
neering Quoted by J H Omens p Have
O K
CF C J Chung and Yuan cheng Fung Residual stress in arteries In G W
Schmidt Scho enb ein S L Wo o and B W Zweifach editors Frontiers in
Biomechanics Springer Verlag New York Have Check
CH Cohen and Hirsch Mathematical developments in Ho dgkin Huxley theory
and its approximations In Simon A Levin editor Some Mathematical
Questions in Biology volume pages American Mathematical So
ciety Providence Have
Cha Richard S Chadwick The myo cardium as a uid b er continuum passive
equilibrium con gurations Advances in Bioengineering pages
Have O K
Cha Richard S Chadwick Mechanics of the left ventricle Journ Biomech
Have O K
Cha Eric L Charnov Life history invariants Get ISBN X
Order Kruep er
Cha K B Chandran Cardiovascular Biomechanics NewYork University Press
New York Ordered He ers ISBN Have
O K Text for use in course Emphasizes blo o d ow but also discusses imag
ing myo cardial mechanics and assist devices Tends to summarize results
and refer to literature Has some exercises no solutions
Che Nikolai G Chetaev Theoretical Mechanics Springer Verlag Have
O K
Chh R P Chhabra Bubbles Drops and Particles in Non Newtonian Flu
ids CRC Press Get ISBN Ordered Kruep er
Cho Rustum Choksi Two topics on the equations of solid mechanics The in
compressible limit in nonlinear elasticity and conservation laws in bre
reinforced materials PhD thesis Brown University Rho de Island
Get Not in DAVol Order UMI
Chr Richard M Christensen Mechanics of Composite Materials Krieger Mal
abar Have O K ISBN
CHR Kevin D Costa Peter J Hunter Jack M Rogers Julius M Guccione Jr
Lewis K Waldman and Andrew M McCullo ch A three dimensional nite
element metho d for large elastic deformations of v entricular myo cardium
Part I Cylindrical and spherical p olar co ordinates Transactions of the
ASME Journal of Biomechanical Engineering Have
CHW Kevin D Costa Peter J Hunter Jennifer S Wayne Lewis K Waldman
Julius M Guccione Jr and Andrew M McCullo ch A three dimensional
nite element metho d for large elastic deformations of ventricular my
o cardium Part I I Prolate spheroidal co ordinates Transactions of the
ASME Journal of Biomechanical Engineering Have preprint
CI M Ciarletta and D Iesan Non classical Elastic Solids volume of
Pitman Research Notes in Mathematics Longman Scienti c Technical
Have ISBN X
Cia Philipp e G Ciarlet Mathematical Elasticity Vol I Three Dimensional
Elasticity North Holland Amsterdam Havecwc O K ISBN
X
Cia Philipp e G Ciarlet Mathematical Elasticity Vol II Lower Dimensional
Theories of Plates and Rods North Holland Amsterdam Ordered
Cia Philipp e G Ciarlet Mathematical Shel l Theory Wiley Get O K
ISBN Ordered Kruep er Publication Dec
CKK John A Callahan Thomas E Keys and JackD Key editors Classics
of Cardiology former title Cardiac Classics volume Krieger
Have ISBN
CKT Kenneth B Campb ell Rob ert D Kirkpatrick Anthony H Tobias
Hesameddin Taheri and Sanjeev G Shro Series coupled non contractile
elements are functionally unimp ortant in the isolated heart Cardiovascular
Research Have O K
Cla A J Clark Comparative Physiology of the Heart Cambridge University
Press Comment A clear summary of the state of knowledge in
Includes references to earlier work Has numerous tables and graphs Two
chapters are of particular relevances Chapter The work of the Heart
and Chapter The In uence of Chemical Environment Not in ZOOL
Have Xeroxcopy O K
Cle Logan Clendening SourceBook of Medical History Dover New York
Quoted by Acierno page re Vesalius ISBN Have
cwc O K
CMK John A Callahan Dwight C McGo on and JackD Key editors Classics
of Cardiology former title Cardiac Classics volume parts Krieger
Have ISBN
CMMR Jos e E Castillo Greg McDermott Marc McEachern and John Richardson
A comparative analysis of numerical techniques applied to a mo del of the
otolith membrane Compters Math Applic Have O K
CMT F Cassot D Morvan and G Tonietto Unsteady ow through pros
thetic heart valves Anintegral approach coupled with a vortex metho d In
Boundary Elements X Vol Heat Transfer Fluid Flow and Electrical
Applications pages Computational Mechanics Publications and
Springer Verlag Have O K
Con Lawrence Conlon Di erentiable Manifolds A First Course Birkhaeuser
Boston ISBN Havecwc O K
Co o James L Co ok Conversion Factors Oxford University Press Have
cwc
Cou Coussy Mechanics of Porous Continua Wiley Get ISBN
Order Kruep er
Cov James W Covell Functional dynamics of the myo cyte connective tissue
unit In T F Robinson and R K H Kinne editors Cardiac Myocyte
Connective Tissue Interactions in He alth and Disease Issues Biomed vol
ume Karger Basel Have O K
CPSS Colin Gerald Caro T J Pedley R C Schroter and W A Seed The
Mechanics of the Circulation Oxford University Press Have Xerox
copy ISBN O K
Cri a Cris eld Non Linear Finite Element Analysis Wiley Get ISBN
Ordered Kruep er
Cri b M A Cris eld Non linear Finite Element Analysis of Solids and Struc
tures Vol Essentials Wiley ISBN Get
Cry Colin Walker Cryer A comparison of the three dimensional consolidation
theories of Biot and Terzaghi Quarterly Journal of Mechanics and Applied
Mathematics Part Have
CST C J Chuong M S Sacks G Templeton F Schwiep and R L John
son Regional deformation and contractile function in canine rightventric
ular free wall Am J Physiol Heart and Circulatory Physiology
H Have Medline abstract
CT L J Chang and John M Tarb ell Numerical simulation of fully develop ed
sinusoidal and pulsatile physiological ow in curved tub es J Fluid Mech
Have O K
Cul David F Culclasure Anatomie und Physiologie des Menschen Lehrpro
gramme Band Herz und Kreislaufsystem VCH Verlagsgesellschaft
Weinheim ISBN Have O K
Cur John D Currey The Me chanical Adaptation of Bones Princeton University
Press Princeton Quoted by Biewener Biomechanics Structures and
Systems A Practical Approach Have pap er copy Not in BIP
O K
Cus J H Cushman editor Dynamics of Fluids in Hierarchial Porous Media
Academic Press San Diego Get AMR ISBN
CvD Angela Y Cheer and Cornelis P van Dam editors Fluid Dynamics in Biol
ogy Proceedings of an AMS IMS SIAM Joint Summer Research Conference
held July with support from the National ScienceFoundation and
NASA Headquarters volume of Contemporary Mathematics American
Mathematical So ciety Providence Have ISBN
CW Carleton B Chapman and Eugene Wasserman On the dynamics of cardiac
muscle American Heart Journal Translation Otto Frank Zur
Dynamik des Herzmuskels
CW James W Covell and Lewis K Waldman Ventricular wall motion In Henk
E D J ter Keurs and J V Tyb erg editors Mechanics of the Circulation
chapter pages Martinus Nijho Publishers Have
Dav M J Davies Colour Atlas of Cardiovascular Pathology Oxford University
Press Have O K ISBN
Dav Paul Davis The art of the heart valve SIAM News Have
O K
Daw Thomas H Dawson Engineering Design of the Cardiovascular System of
Mammals Prentice Hall Ordered He ers p
ISBN Have
dC Manfredo P do Carmo Di erentialgeometrie von Kurven und Flaechen
Vieweg Braunschweig corrected edition Abbreviated translation
ISBN X Havecwc O K
dC Manfredo Perdig ao do Carmo Riemannian Geometry Birkhaeuser
ISBN Have O K
dC Manfredo Perdig ao do Carmo Di erential Geometry of Curves and Sur
faces Prentice Hall Englewo o d Cli s Get Order Kruep er for
cwc
DD Dela eld DuBois and Eugene F DuBois Clinical calorimetry V The mea
surement of the surface area of man Archivs of internal Medicine
Quoted by Dinnar in b o ok edited by Ghista as source of formula
of DuBois and Dub ois Have O K
DD Dela eld DuBois and Eugene F DuBois Clinical calorimetry a formula
to estimate the approximate surface area if height and weightbeknown
Archivs of internal Medicine Get
Del M Delbruck Knotting problems in biology mathematical problems in the
biological sciences Proc Symp Appl Math Quoted by
Winfree Get Math
DF John Kenneth Dickinson and P A Forsyth Preconditioned conjugate gra
dient metho ds for three dimensional linear elasticity Tec hnical rep ort
Department of Computer Science UniversityofWaterlo o Obtained
by ftp from Waterlo o Rep ort CS Have
DH Jim Douglas Jr and Ulrich Hornung editors Flow in Porous Media
Proceedings of the Oberwolfach Conference June Birkhaeuser
Verlag Basel Have O K ISBN
DHN Angela J Drake Holland and Mark I M Noble editors Cardiac
Metabolism Wiley Interscience LC pap ISBN
Bks Demand Not in MED Ordered Kruep er
Nachfragen b ei Kruep er
DHW A S Douglas W C Hunter and M D Wiseman Inhomogeneous defor
mation as a source of error in strain measurements derived from implanted
markers in the canine left ventricle Journ Biomech
Get O K XEROX Biomech
Dic John Kenneth Dickinson Preconditioned conjugate gradient metho ds for
three dimensional linear elasticity Technical rep ort Department of Com
puter Science UniversityofWaterlo o Obtained by ftp from Water
lo o Rep ort CS Have
Die Bernhard Dierb erger Finite Elemente Simulation der Herzdynamik
Das Herzgroessen Schlagvolumen Konze pt PhD thesis Eb erhard Karls
Universitaet Tuebingen Have
Din Uri Dinnar Cardiovascular Fluid Dynamics CRC Press p
ISBN Not in MED Have pap er copy
DMT Sid DeutschandEvangelia Micheli Tzanakou Neuroelectric Systems New
York University Biomedical Engineering Series New York University Press
From K B Chandran Cardiovascular Biomechanics Ordered
Kruep er
DROH A S Douglas Edward K Ro driguez W O Dell and W C Hunter Unique
strain history during ejection in canine left ventricle American Journal on
Physiology Heart and Circulatory Physiology H H
Have O K
DS Paul Doucet and Peter B Slo ep Mathematical Modeling in the Life Sci
ences Ellis Horwo o d in Mathematics and its Applications Ellis Horwood
Have ISBN X
DSLT Jean G Dumesnil Rashad M Shoucri Jean Louis Laurenceau and Jules
Turcot A mathematical mo del of the dynamic geometry of the intact left
ventricle und its application to clinical data Circulation
Have O K
DSU Do err Seifert and E Uehlinger Pathologische Anatomie des Herzens und
seiner Huellen I Get UB
DT Gouri Dhatt and Gilb ert Touzot The Finite Element Method Displayed
Wiley Chichester Have Xeroxcopy ISBN LC
In BP Kruep er Not available
dTtK Pieter P deTomb e and Henk E D J ter Keurs Force and velo cityof
sarcomere shortening in trab eculae from rat heart e ects of temp erature
Circulation Research Have
EH C Lovatt Evans and H Hartridge editors Starling s Principles of Human
Physiology Lea and Febiger Philadelphia Havecwc
Eis Luther Pfahler Eisenhart Introduction to Di erential Geometry with Use
of the Tensor Calculus Princeton University Press Princeton Get
ISBN Bo oks on Demand Ordered Kruep er
Vergri en lt Kruep er
Eis Luther Pfahler Eisenhart Riemannian Geometry Princeton University
Press revised edition ISBN Have O K
El Zafrany El Techniques of the Boundary Element Method Ellis Horwood
Get in BIP Ordered Kruep er
ELW J Eisenfeld D S Levine and M Witten editors Biomedical Modeling
and Simulation North Holland Amsterdam Have
ES David G Ebin and Ralph A Saxton The initial value problem for elas
to dynamics of incompressible b o dies Arch Rat Me ch Anal
Have O K
ES David G Ebin and Ralph A Saxton The equations of incompressible elas
ticity In Barbara Lee Key tz and Herb ert C Kranzer editors Contempo
rary Mathematics Nonstrictly Hyperbolic Conservation Laws volume
pages American Mathematical So ciety Providence Have
O K
ETvdBW Gijs Elzinga G P To orop G C van den Bos and Nicolaas Westerhof
Ventricular pump function and arterial resistance In Henk E D J ter
Keurs and J V Tyb erg editors Mechanics of the circulation chapter
pages Martinus Nijho Publishers Have
Eul L Euler Principia pro motu sanguins p er arterias determinadeo Opera
posthuma mathematica et physica EdideruntP H
Fuss et N Fuss Petrop oli Apud Eggers et so cios Get Quoted byFung in
Bio dynamics p
Eva C Lovatt Evans Starling s Principles of Human Physiology Lea and
Febiger Havecwc
EvdHW Gijs Elzinga G J van der Horn and Nicolaas Westerhof Cardiac pump
function and ventricular dimensions In Henk E D J ter Keurs and Mark
I M Noble editors Starling s Law of the Heart Revisited pages
Kluwer Academic Publishers Have
EW Eghbali Webb editor Mole cular Biology of Col lagen Matrix in the Heart
Springer Verlag From Springer Catalog Lo ok at
Fac Stephen M Factor Pathological alterations of myo cyte connective tissue
interactions in cardiovascular disease In T F Robinson and R K H
Kinne editors Cardiac Myocyte Connective Tissue Interactions in Health
and Disease Issues Biomed volume Karger Basel Have O K
Fal Adolf Faller Der Faserverlauf im Bindegeweb e des Epikards menschlicher
Herzen und funktionelle Deutung Gegenbauers morphologisches Jahrbuch
Heft Have O K
Fei Theo dore S Feit Diastolic pressure volume relations and distribution of
pressure and b er extension across the wall of a mo del left ventricle Journ
Biophys Have O K
Fen Wallace O Fenn A quantitative comparison b etween the energy lib erated
and the work p erformed by the isolated sartorius muscle of the frog J
Physiol London Get UB Z
Fen T P Feng The e ect of length on the resting metab olism of muscle J
Physiol London Quoted by Ciba Foundation p
Get UB Z Ordered MED
Fen H Feneis Das Gefuege des Herzm uskels b ei Systole und Diastole Gegen
baurs morph Jb Quoted byPaul Peter Lunkenheimer
A Lunkenheimer and Francisco Torrent Guasp Kardio dynamik Wege zur
strukturgerechten Analyse der Myokardfunktion Get ZD Not
in MED Ordered Interlib Heft nichtinMSvorhanden Order UB
Fen J B Fenn Engines Energy and Entropy W H Freeman New York
Get Not in UB Institut fr Mineralogie Quoted byAtkins p
Mineralogie
FG Piero Colli Franzone and Luciano Guerri Mo dels of the spreading of ex
citation in myo cadial tissue In Theo C Pilkington Bruce Loftis Jo e F
Thompson Savio L Y Wo o Thomas C Palmer and Thomas F Budinger
editors High Performance Computing in Biomedical Research CRC Press
Inc Have
Fif P C Fife Mathematical Aspects of Reacting and Di using Systems
Springer Verlag Berlin Get ISBN pap er NU
MER
Fin M H Fineb erg Functional capacity of the normal p ericardium American
Heart Journal Have O K
Fla Harley Flanders Di erential Forms with Applications to the Physical Sci
ences Dover Have O K
Fle R L Flett The musculature of the heart with its application to physiology
and a note on heart rupture J Anat Has extensive
bibliography Quoted by Greenbaum Ho Gibson Becker and Anderson
Have O K
Fli Rainer Flindt Biologie in Zahlen Gustav Fischer Verlag th edition
Have O K
FN Bj rn Folkow and Eric Neil Circulation Oxford University Press
New York In PHYSIOL I I C Not in BIP No chmal an
schauen
Fog Aaron L Fogelson A mathematical mo del and numerical metho d for study
ing platelet adhesion and aggregation during blo o d clotting Journal of
Computational Physics Have
Fra Otto Frank Zur Dynamik des Herzmuskels Z Biol
A translation was published by Carleton B Chapman and Eugene Wasser
man Am Heart J and Have O K HaveTrans
lations
Fra Otto Frank Die Grundform des arteriellen Pulses Z Biol
Get
Fra Clarence M Fraser editor The Merck Veterinary Manual Rahway N J
th edition Have O K ISBN Merck
Fre Walter Frese Chaos Theorie und plo etzlicher Herzto d Naturwis
senschaftliche Rundschau Hav e
Ful F Bro ckFuller The writhing numb er of a space curve Proc Natl Acad
Sci U S A Quoted by Winfree Have O K
Ful F Bro ckFuller Decomp osition of the linking numb er of a closed ribb on A
problem from molecular biology Proc Natl Acad Sci U S A
Quoted by Winfree Have O K
Ful W F M Fulton Morphology of the myo cardial micro circulation In H Till
manns W Kuebler and H Zeb e editors Microcirculation of the Heart
pages Springer Verlag Have O K
Fun Yuan cheng Fung Foundations of Solid Mechanics Prentice Hall Engel
wo o d Cli s Get ISBN Ordered Kruep er
Fun Yuan cheng Fung Biodynamics Circulation Springer Verlag New York
Have O K
Fun Yuan cheng Fung Biomechanics Motion Flow Stress and Growth
Springer Verlag New York Have O K
Fun Yuan cheng Fung Biomechanics Mechanical Properties of Living Tissues
Springer Verlag second edition Have O K
Fun Yuan cheng Fung A First Course in Continuum Mechanics Prentice Hall
Englewo o d Cli s N J rd edition ISBN HaveO K
FvDFH James D Foley Andries van Dam Steven K Feiner and John F Hughes
Computer Graphics Principles and Practice Addison Wesley Publishing
Company second edition Have
FY J S FrankandR Yung The myo cyte connective tissue interface In T F
Robinson and R K H Kinne editors Cardiac Myocyte Connective Tissue
Interactions in Health and Disease Issues Biomed volume Karger
Basel Have O K
GA Alb ert E Green and J E Adkins Large Elastic Deformations and Non
Linear Continuum Mechanics Clarendon Press Oxford second edition
Not in BIP not in Whitaker Have Xerox copy
Gal Galileo Galilei Dialogues Concerning Two New Sciences Translatedby
Henry Crew and Alfonso de Salvio Translation of original italian edition
of New York Dover Have ISBN
Gan Carl Gans Biomechanics An Approach to Vertebrate Biology University
of Michigan Press Quoted byVogel in Lifes Devices Have O K
Gar Rob ert H Gardner editor Some Mathematical Questions in Biology Pre
dicting Spatial E ects in ecological systems American Mathematical So ci
ety Providence Have O K ISBN
GB David T Goldman and R J Bell editors The International System of
Units SI NBS Sp ecial Publ Washington D C Quoted by
Schmidt Nielson Animal Physiology Adaptation and Environment
Available from National Technical Information Service Spring eld VA
Have O K
GB William Grossman and Donald S Bain editors Cardiac Catheterization
Angiography and Intervention Lea and Febiger Philadelphia th edition
Get Quoted byFung page and Parmley and Talb ot page
as a source ISBN X Have O K
GBF William H Gaasch Oscar H L Bing Alvin Franklin David Rho des
Sheilah A Bernard and Ronald M Weintraub The in uence of acute alter
ations in coronary blo o d ow on left ventricular diastolic compliance and
wall thickness European Journal of Cardiology Suppl
Quoted by J H Omens p Have O K
Geo P L George editor Automatic Mesh Generation Wiley Have
GG R A Greenbaum and D G Gibson Regional non uniformity of left ven
tricular wall movement in man Br Heart Journal Quoted
by Greenbaum Ho Gibson Becker and Anderson Have O K
GG John C Gilb ert and Stanton A Glantz Determinants of left ventricular
lling and of the diastolic pressure volume relation Circulation Research
Hav e O K
GGBM Phillip Gould Dhanjo o N Ghista Lawrence Bromb olich and Israel Mirsky
In vivo stresses in the human left ventricular wall analysis accounting for
the irregular dimensional geometry and comparison with idealised geom
etry analyses Journ Biomech Have Quoted by J H
Omens p O K
GH Dhanjo o N Ghista and M Sahul Hamid Finite element stress analysis
of the human left ventricle whose irregular shap e is develop ed from single
plane cineangio cardiogram Computer Programs in Biomedicine
Have p o or copy Last page missing According to UB In Inst
f Med Informatik und Biomathematik Domagkstr
GH Leon Glass and Peter J Hunter There is a theory of heart PhysicaD
Have O K
GHG R A Greenbaum Siew Yen Ho D G Gibson Anton E Becker and
Rob ert H Anderson Left ventricular bre architecture in man British
Heart J Have O K
Ghi Dhanjo o N Ghista Theoretical Foundations of Cardiovascular Processes
Advances in cardiovascular physics vol Karger Basel UB
Freihand G Have Xeroxcopy ISBN Hav e
O K
GHM Leon Glass Peter J Hunter and Andrew D McCullo ch editors Theory of
Heart Springer Verlag New York Have O K ISBN
Gib C L Gibbs Cardiac energetics In Henk E D J ter Keurs and J V
Tyb erg editors Mechanics of the Circulation chapter pages Mar
tinus Nijho Publishers Have
GJ Arthur C Guyton and E C Jones editors Cardiovascular Physiology
Butterworth and Co Ltd London Get UB G Not in BIP
GKJ Gauer Kramer and Jung Physiologie des Menschen Herz und Kreis
lauf Urban Schwarzenb erg Muenchen Get Ordered Kruep er
Vergri en laut Kruep er
GL Francisco Torrent Guasp and Paul Peter Lunkenheimer Intensivkurs
Funktionelle Anatomie des Herzens Vom b ei Dr Torrent
Guasp in Spanien Denia Alicante Gespr archsnotizen Have
GL J Guckenheimer and I S Lab ouriau Bifurcation of the Ho dgkin and
Huxley equations A new twist Bul l Math Biology
Have O K
Gla Roland Glaser Biophysik Fischer Stuttgart rd edition Get Phys
E In ZOOL DM Ordered Kruep er
Gla Andrew S Glassner Graphics GEMS Academic Press Cambridge MA
ISBN Have
GM Leon Glass and Michael C Mackey From Clocks to Chaos The Rhythms of
Life Princeton University Press Princeton Have O K pap erback
Bestellen fur cwc ISBN
GMW Julius M Guccione Jr Andrew D McCullo ch and Lewis K Waldman
Passive material prop erties of intact ventricular myo cardium determined
from a cylindrical mo del Transactions of the ASME Journal of Biome
chanical Engineering Have O K
GMW Cameron Gordon Yoav Moriah and BronislawWa jnryb editors Geomet
ric Topology volume of Contemporary Mathematics American Math
So c Providence Rho de Island ISBN Have O K
Gol Herb ert Goldstein Classical Mechanics Addison Wesley nd edition
ISBN Have O K
Gol B Goldstein editor Cel l Biology American Mathematical So ciety Provi
dence Have O K ISBN
Gor J E Gordon The New Science of Strong Materials or Why You Don t Fal l
Through the Floor Harmondsworth UK Penguin Bo oks second edition
Reprinted Penguin Bo oks ISBN Quoted by
Vogel in Life s Devices Have O K
Gor J E Gordon editor Structures or Why Things Don t Fal l Down Penguin
Bo oks Harmondsworth Middlessex Have O K ISBN
Quoted byVogel in Life s Devices
GR Christian Grossmann and Hans Go erg Ro os Numerik partiel ler Di eren
tialgleichunge n B G Teubner Stuttgart Have ISBN
Gra Rob ert P Grant Architectonics of the heart American Heart Journal
Quoted by Greenbaum Ho Gibson Becker and Anderson
Have O K
Gra Rob ert P Grant Notes on the muscular architecture of the left ventricle
Circulation Quoted by Greenbaum Ho Gibson Becker
and Anderson Have O K
Gra Alfred Gray Modern Di erential Geometry of Curves and Surfaces CRC
Press Bo ca Raton Have ISBN
Gre Donald E Gregg E ect of coronary p erfusion pressure or coronary owon
oxygen usage of the myo cardium Circulation Research
Have O K
Gri Peter Grindro d Patterns and Waves The Theory and Applications of
Reaction Di usion Equations Clarendon Press Oxford Have In
NUMERIK Bib O K
Gro Arthur Grollman The Cardiac Output of Man in Health and Disease
Charles C Thomas Spring eld IL Havecwc
Gro W Grossman Cardiac Hyp ertrophy Useful Adaptation or Pathologic Pro
cess American Journal of Medicine Get Quoted by
Lin and Taylor MED
Gro Louis J Gross editor Some Mathematical Questions in Biology Plant
Biology American Mathematical So ciety Providence Have O K
ISBN
GT Roland Glowinski and PatrickLeTallec AugmentedLagrangian and
Operator Splitting Methods in Nonlinear Mechanics So ciety for Industrial
and Applied Mathematics Philadelphia Have ISBN
Gua Francisco Torrent Guasp El Musculo Cardiaco Publicaciones De La Fun
dacion Juan March Guadarrama Havecwc O K ISBN
Gua Francisco Torrent Guasp The Cardiac Muscle Juan MarchFoundation
ISBN Havecwc
Gua Francisco Torrent Guasp La estructuraci on macrosc opica del mio cardio
entricular Revista Espa nola de Cardiologia Have O K v
Gua Francisco Torrent Guasp Anatomia Funcional del Corazon GRASS
Get p ISBN pta Ordered Kruep er
Gua Francisco Torrent Guasp Macroscopical structure of the heart Have
O K
Gue B Guenther Dimensional analysis and theory of biological similarity Phys
iological Reviews Get UB ZD
Gur Morton E Gurtin Topics in Finite Elasticity So ciety for Industrial and
Applied Mathematics Philadelphia Have ISBN
Guy Arthur C Guyton Textbook of Medical Physiology W B Saunders
Philadelphia th edition Have O K ISBN
Guz A Guz Chairman s intro duction In Ciba Foundation Sta editor The
Physiological Basis of Starling s Law of the Heart pages Elsevier
Amsterdam Have
GVY Dhanjo o N Ghista Eric Van Vollenhoven Wen Jei Yang Helmut Reul and
Walter Bleifeld editors Cardiovascular Engineering Part I Model ling S
Karger Advances in cardiovascular physics vol p
ISBN Have O K
GW H Gordon and Arthur T Winfree A single spiral artifact in arthrop o d
cuticle Tissue and Cel l Quoted by Winfree Get
MED
GWM a Julius M Guccione Jr Lewis K Waldman and Andrew D McCullo ch
Mechanics of active contraction in cardiac muscle Part I Constitutive
relations for b er stress that describ e deactivation T ransactions of the
ASME Journal of Biomechanical Engineering Have
GWM b Julius M Guccione Jr Lewis K Waldman and Andrew D McCullo ch
Mechanics of active contraction in cardiac muscle Part I I Cylindrical
mo dels of the systolic left ventricle Transactions of the ASME Journal of
Biomechanical Engineering Have
GZ Alb ert E Green and Wolfgang Zerna Theoretical Elasticity Dover New
York Havecwc ISBN O K
GZL Francisco Torrent Guasp Pedro Zarco Paul Peter Lunkenheimer
A Lunkenheimer P Aragoncillo Ballestros J Fari na Gonz alez Daniel D
Streeter Jr and Ceon Ram on Estructuraymec anicadelcoraz on GRASS
Ediciones Barcelona Have O K
Hal Stephen Hales Statical essays Containing haemastatics History of
Medicine Series Library of New York Academy of Medicine No
Hafner Publishing New York Get Not in BIP Not in UB
Har William Harvey Exercitatio Anatomica de Moto Cordis et Sanguinis in
Animalibus Frankfurt Get
Har C R Harris The Heart and the Vascular System in Ancient Greek Medicine
from Alcmaeon to Galen Oxford University Press Get Ordered
He ers ISBN In BIP Out of Print
He ers In MED WG Dillons Not available
Har a Lionel G Harrison Kinetic Theory of Living Pattern Cambridge Univer
sity Press Have O K ISBN
Har b William Harvey The Circulation of the Blood And Other Writings Trans
lated from the Latin by Kenneth J Franklin J M Dent London
ISBN Have O K
HAvCR Jacques Maria R J Huyghe Theo Arts DickH van Camp en and
Rob ert S Reneman Porous medium nite element mo del of the b eat
ing left ventricle American Journal on Physiology Heart and Circulatory
Physiology pages H H Have O K
HB G M Hutschins and B H Bulkley Catenoid shap e of the interventricular
septum p ossible cause of idiopathic hyp ertrophic subaortic stenosis Cir
culation Quoted by Greenbaum Ho Gibson Becker and
Anderson Get
HBH J I E Ho man R W Baer F L Hanley L M Messina and M T
Grattan Regulation of transmural myo cardial blo o d ow In R E Mates
R M Nerem and P D Stein editors Mechanics of the Coronary Circu
lation pages American So ciety of Mechanical Engineers New York
Quoted by Huyghe Havecwc
HC J E Harris and H D Crofton Structure and function in nemato des
internal pressure and cuticular structure in ascaris Journal of experimental
Biolo gy Quoted by R McNeill Alexander p
ZQu Get
HCL Peter J Hanley Patricia J Co op er and Denis S Loiselle E ect of hyp er
osmotic p erfusion on rate of oxygen consumption of isolated guinea pig and
rat hearts during cardioplegia Cardiovascular Research
Have O K
He Zhixiong He Mathematical models in muscle by periodic stimuli PhD
thesis Marquette University Wisconsin Get DA Diss
Abst No B Order UMI
Hem Hemp el Manifolds Princeton University Press Princeton The
bible on manifolds
HEWS Ro elof Martinus Huisman Gijs Elzinga Nicolaas Westerhof and Pieter
Sipkema Measurement of left ventricular wall stress Cardiovascular Re
search Have
HH Hugh E Huxley and J Hanson Changes in the cross striations of muscle
during contraction and stretch and their structural interpretation Nature
Get
HH J Hannah and M J Hillier AppliedMechanics Longman Have O
K
HHOT A K Head S D Howison J R Ockendon and S P Tighe An equilibrium
e O theory of dislo cation continua Siam Review Hav
K
Hib R C Hibb eler Engineering Mechanics Statics and Dynamics Prentice
Hall Have O K
Hij Henk J A M Hijmans Mathematical morphology A mo dern approach
in image pro cessing based on algebra and geometry Siam Review
Have O K
Hil Archibald Vivian Hill editor The heat of shortening and the dynamic con
stants of muscle volume B Pro c Royal So c London Get
Hil Archibald Vivian Hill First and Last Experiments in Muscle Mechanics
Cambridge University Press Cambridge Have Xerox copy
Hir C Hirsch Numerical Computation of Internal and External Flows Vol
Computational Methods for Inviscied and Viscous Flows Wiley Get
ISBN Ordered Kruep er
HJ Yandell Henderson and F Elmer Johnson Two mo des of closure of the
heart valves Heart Have O K
HJvC Jacques Maria R J Huyghe J D Janssen and DickH van Camp en ed
itors Poroviscoelasticity of passive myocardial tissue Biomechanics
Symposium volume ASME Have
HKT Hideyuki Honda Yoshiro Koiwa KazuhikoTakagi Jun Ichi Kikuchi
Nobuo Hoshi Tamotsu Takishima and James P Butler Noninvasivemea
surement of left ventricular myo cardial elasticity American Journal on
Physiology Heart and Circulatory Physiology H H
Have O K
HLM S M Horner Max J Lab C F Murphy D J Dick B Zhou and F G
Harrison Mechanically induced changes in action p otential duration and
left ventricular segment length in acute regional ischaemia in the in situ
p orcine heart Cardiovascular Research Have O K
HLY Arie Horowitz Y Lanir Frank C P Yin M Perl I Sheinman and
Rob ert K Strumpf Structural three dimensional constitutivelaw for the
passivemyo cardium Transactions of the ASME Journal of Biomechanical
Engineering Have
HMNS Peter J Hunter Andrew D McCullo ch Poul Michael F nss Nielsen and
Bruce H Smaill A nite element mo del of passiveventricular mechanics
In Rob ert L Spilker and Bruce R Simon editors Computational Methods
in Bioengineering BED volume pages American So cietyof
Mechanical Engineers New York Have
HN Hugh E Huxley and R Niedergerke Structural changes in muscle during
contraction Nature Get UB
HNS Peter J Hunter Poul Michael F nss Nielsen Bruce H Smaill Ian J
LeGrice and Ian W Hunter An anatomical heart mo del with applications
to myo cardial activation and ventricular mechanics In Theo C Pilkington
Bruce Loftis Jo e F Thompson Savio L Y Wo o Thomas C Palmer and
Thomas F Budinger editors High Performance Computing in Biomedical
Research pages CRC Press Inc Have
Ho e Karl Heinz Ho ehne D Computergra k in der Medizin Diagnose glasklar
Bild der Wissenschaft Have Sonderdruck
Hof J I E Ho man The e ect of intramyo cardial forces on the distribution of
intramyo cardial blo o d ow Transactions of the ASME Journal of Biome
chanical Engineering Quoted by Huyghe Get Order
UB
Hol M H Holmes Comparison theorems and similarity solution approximations
for a nonlinear di usion equation arising in the study of soft tissue SIAM
Journal on Applied Mathematics Get Artikel kopieren
Hol M H Holmes Finite deformation of soft tissue Analysis of a mixture
mo del in uniaxial compression Transactions of the ASME Journal of
Biomechanical Engineering Quoted byFung p
Get Order UB
Hol Arun V Holden Reconstructing the heart Physics World Feb
Have
Hor Waldemar Hort Untersuchungen ueb er die Muskelfaserdehnung und
das Gefuege des Myokards in der rechten Herzkammerwand des Meer
schw einchens Virchows Arch path Anat Quoted by
Paul Peter Lunkenheimer A Lunkenheimer and Francisco Torrent Guasp
Kardio dynamik Wege zur strukturgerechten Analyse der Myokardfunktion
XEROX MED ZD Have O K
Hor Waldemar Hort Untersuchungen zur funktionellen Morphologie des
Bindegewebsgeruestes und der Blutgefaesse der linken Herzkammerwand
Virchows Arch path Anat Physiol Quoted byPaul
Peter Lunkenheimer A Lunkenheimer and Francisco Torrent Guasp Kar
dio dynamik Wege zur strukturgerechten Analyse der Myokardfunktion
ZD Have O K
Hor Arie Horowitz Structural considerations in formulating material laws for
the myo cardium In Leon Glass Peter J Hunter and Andrew D Mc
Cullo ch editors Theory of Heart chapter pages Springer Verlag
Have O K
Hor Helmuth Horvath Physikalische Uebungsaufgaben fuer Biologen und Medi
haftsverlag Have ISBN ziner B I Wissensc
HOvC Jacques Maria R J Huyghe Gees W Omens and DickH van Camp en
Low Reynolds numb er steady state ow through a branching network of
rigid vessels I I A nite element mixture mo del Biorheology
Have
HOvCH Jacques Maria R J Huyghe Gees W Omens DickH van Camp en and
Rob ert M Heethaar Low Reynolds numb er steady state ow through a
branching network of rigid vessels I A mixture theory Biorheology
Have
HP Frank C Hopp ensteadt and Charles Samuel Peskin Mathematics in
Medicine and the Life Sciences Springer Verlag New York Have
O K
HPR R M Heethaar Y C Pao and E L Ritman Computer asp ects of three
dimensional nite element analysis of stresses and strains in the intact
heart Comput Biomed Res Quoted byPerl Horowitz
and Sideman Get
HPS Arie Horowitz M Perl and Samuel Sideman Geo desics as a mechanically
optimal b er geometry for the left vetnricle Basic Res Cardiol
Have Medline abstract ZD
HPSR Arie Horowitz Mordechai Perl Samuel Sideman and Eric L Ritman Com
prehensive mo del for the simulation of left ventricle mechanics Part Im
plementation and results analysis Medical and Biological Engineering and
Computing Have O K Has biographies of authors For
Part see Perl Horowitz and Sideman
HS Peter J Hunter and Bruce H Smaill The analysis of cardiac function a
contin uum approach Prog Biophys mol Biol Have
O K
HSHY Jay D Humphrey Rob ert K Strumpf Henry R Halp erin and Frank
C P Yin Toward a stress analysis in the heart In Leon Glass Peter J
Hunter and Andrew D McCullo ch editors Theory of Heart pages
Springer Verlag Have O K
HSL a Arie Horowitz I Scheinman Y Lanir M Perl and Samuel Sideman Non
linear incompressible nite element for simulating loading of cardiac tissue
Part I Two dimensional formulation for thin myo cardial strips Trans
actions of the ASME Journal of Biomechanical Engineering
Have O K
HSL b Arie Horowitz I Sheinman and Y Lanir Nonlinear incompressible nite
elementforsimulating loading of cardiac tissue Part I I Three dimensional
formulation for thickventricular wall segments Transactions of the ASME
Journal of Biomechanical Engineering Have O K
HSWE Ro elof Martinus Huisman Pieter Sipkema Nicolaas Westerhof and Gijs
Elzinga Wall stress measurements in non b eating canine hearts appli
cation of a metho d tested in rubb er mo dels Cardiovascular Research
Have O K
HSWE Ro elof Martin us Huisman Pieter Sipkema Nicolaas Westerhof and Gijs
Elzinga Comparison of mo dels used to calculate left ventricular wall force
Medical Biolo gical Engineering Computing Have
O K
HTRR William P Ho o d Jr Walter J Thomson Charles E Rackley and Ellis L
Rolett Comparison of calculations of left ventricular wall stress in man
from thin walled and thick walled ellipsoidal mo dels Circulation Research
Have O K ZC
Hui Ro elof Martinus Huisman Forces in the Wal l of the Left Ventricle
PhD thesis Vrije Universiteit Amsterdam The Netherlands Have
O K
Hul Derek Hull An Introduction to Composite Materials Cambridge University
Press ISBN Havecwc O K
Hun Peter Hunter Finite element analysis of cardiac muscle mechanics PhD
thesis Oxford University Get Quoted byPedley in Fluid Mechanics
of Large Blo o d Vessels p
Hux Julian S Huxley Problems of Relative Growth Johns Hopkins Univer
sity Press Havecwc ISBN Ordered Kruep er
Huy Jaques Maria R J Huyghe Non Linear Finite Element Models of the
Beating Left Ventricle and the Intramyo cardial Coronary Circulation PhD
thesis Eindhoven UniversityofTechnology Eindhoven The Netherlands
Have O K
HvCAH Jacques Maria J R Huyghe DickH van Camp en Theo Arts and
Rob ert M Heethaar The constitutive b ehaviour of passive heart muscle
tissue A quasi linear visco elastic formulation Journ Biomech
Have Quoted by Huyghe et al A two phase nite element
mo del of the diastolic left ventricle
HvCW R Huiskes DickH van Camp en and J R De Wijn Biomechanics Prin
ciples and Applications volume Martinus Nijho Publ Get
HW Gab or T Herman and Dallas Webster editors Surfaces of organs in dis
crete three dimensional space volume of Lect Notes Med Inf Ob erwol
fach Pro c Have Summary
HW Gab or T Herman and Dallas Webster A top ological pro of of a surface
tracking algorithm Compt Vision Graphics Image Process
Have Summary
HY Jay D Humphrey and F rank C P Yin Biaxial mechanical b ehaviour of
excised epicardium Transactions of the ASME Journal of Biomechani
cal Engineering Have O K Discusses e ect of p eri
cardium
HY Jay D Humphrey and Frank C P Yin Constitutive relations and nite
deformations of passive cardiac tissue I I Stress analysis in the left ventricle
Circulation Research Have O K
HYMW Anna R Hashima Alistair A Young Andrew D McCullo ch and Lewis K
Waldman Nonhomogeneous analysis of epicardial strain distributions dur
ing acute myo cardial ischemia in the dog Journ Biomech
Have
IB David Ingram and Ralph Blo ch editors Mathematical Methods in
Medicine Part I Statistical and Analytical Techniques Wiley Have
O K ISBN
IB David Ingram and Ralph Blo ch editors Mathematical Methods in
Medicine Part II Applications in Clinical Specialities Wiley Have
O K ISBN X
Ib e A S Ib erall Growth form and function on mammals Ann N Y Acad
Sci Get Quoted by Li page MED
IHP Myron B Allen I I I Ismael Herrera and George F Pinder Numerical
Modeling in Science and Engineering John Wiley Sons New York
Have O K ISBN
Ile A Ileb ekk Mechanics of the interventricular septum In Henk E D J ter
Keurs and J V Tyb erg editors Mechanics of the circulation chapter
pages Martinus Nijho Publishers Hav e
Ing Ingels Circulation Get Quoted in talk in Alicante
Jac William Jaco Lectures on three manifold top ology American Mathemat
ical So ciety Providence MATH
Jal Jose Jalife Mathematical Approaches to Cardiac Arrhythmias Annals of
the New York Academy of Sciences Have O K Only the table
of contents
JC Desmond G Julian and J Campb ell Cowan Cardiology Bailliere Tindall
London th edition Have O K ISBN
Jea Roger V Jean Phyl lotaxis A systemic study in plant morphogenesis Cam
bridge University Press Have O K
Jew B R Jewell Intro duction The changing face of the length tension relation
In Ciba Foundation Sta editor The Physiological Basis of Starling s Law
of the Heart pages Elsevier Amsterdam Have
JJ Go jmir Ju zni c and Susara Ju zni c editors Noninvasive cardiovascular func
tion Lectures and communications Workshop on Cardiovascular Dynam
ics Tainach Tinje October Europ ean So ciety for Noninva
ve sive Cardiovascular Dynamics Ljubljana Medicinski Razgledi Ha
YU ISSN
JK S Johnson and William M Kier Intramuscular crossed connective tissue
bres Skeletal supp ort in the lateral ns of squid and cuttle sh Mollusca
Cephalop o da J Zool London Have
JL Sir James Lighthill Mathematical Bio uiddynamics Society for Industrial
and Applied Mathematics Philadelphia Havecwc O K ISBN
JM Christopher R Johnson and Rob ert S MacLeo d Nonuniform spatial mesh
adaptation using a p osteriori error estimates applications to forward and
inverse problems Applied Numerical Mathematics Have
JMM Christopher R Johnson Rob ert S MacLeo d and Michael A Metheson
Computational medicine Bio electric eld problems IEEE October pages
Have
Joh Arthur T Johnson Biomechanics and Exercise Physiology John Wiley
ISBN Have O K
Jon Kerry N Jones Geometric structures on branched covers over universal
links Contemporary Mathematics In Math Bibl Have
Kap Daniel Theo dore Kaplan The Dynamics of Cardiac Electrical Instability
PhD thesis MIT Have DA O K
Kat Arnold M Katz The descending limb of the Starling curve and the failing
heart Circulation Have O K
Kat Arnold M Katz Physiology of the Heart Raven Press New York second
edition ISBN Have O K
Kau Louis H Kau man Knots and Applic ations World Scienti c Have
O K
KBK Richard L Kahler Eugene Braunwald Leslie L Kelminson Laurence
Kedes Charles A Chidsey and Stanton Segal E ect of alterations of
coronary blo o d owontheoxygen consumption of the nonworking heart
Circulation Research Have O K
KBP Sandor J Kovacs Jr Benico Barzilai and Julio E Perez Evaluation
of diastolic function with Doppler echo cardiography The PDF formalism
Am J Physiol Heart and Circulatory Physiology Vol H
H Have O K
Kei A Keith The functional anatomy of the heart Br Med J i
Quoted by Greenbaum Ho Gibson Becker and Anderson Get UB
ZC
Ken D E Kenyon The theory for an incompressible solid uid mixture Archive
for Rational Mechanics and Analysis Quoted byYang
Tab er and Clark Get
Kie William M Kier The Functional Morphology of the Musculatureofthe
Arms and Tentacles of Cephalopods PhD thesis Duke University
Get Order No DEU Diss Abstr p Order UMI
Kie William M Kier The musculature of squid arms and tentacles Ultrastruc
tural evidence for functional di erences Journal of Morphology
Have
Kie William M Kier The functional morphology of the tentacle musculature of
nautilus p ompilius In W Bruce Saunders and Neil H Landman editors
Nautilus Plenum Publishing Corp oration Have
Kie William M Kier The n musculature of cuttle sh and squid mollusca
cephalop o da morphology and mechanics J Zool Lond
Have
Kie William M Kier Squid cross striated muscle The evolution of a sp ecialized
muscle b er typ e Bul letin of Marine Science Have
Kie William M Kier Hydrostatic skeletons and muscular hydrostats In An
drew A Biewener editor Biomechanics Structures and Systems A Prac
tical Approach Oxford University Press Oxford Have
KK Barbara Lee Key tz and Herb ert C Kranzer editors Contemporary math
ematics Nonstrictly hyperbolic conservation laws Proceedings of an AMS
special session held January volume Providence Rho de Island
American Mathematical So ciety Providence Have
KK Sangtae Kim and Sapp o J Karrila Microhydrodynamics Butterworth
Heinemann ISBN Have O K
KK P Kowalczyk and M Kleib er Mo delling and numerical analysis of stresses
and strains in the human lung including tissue gas interaction Eur J
Mech A Have O K
KL G W C Kaye and T H Laby editors Tables of Physical and Chemical
Constants and Some Mathematical Functions Wiley London th edition
Get ISBN Order Kruep er
Kle Max Kleib er The Fire of Life Krieger revised edition Der
Energiehaushalt von Mensch und Haustier German translation in UB
G LC p ISBN Quoted bySchmidt
Nielson Have O K
Kli Wilhelm P A Klingenb erg Riemannian Geometry Walter de Gruyter
second revised edition Have O K
KLM M K Kwan W M Lai and V C Mow A nite deformation theory for
cartilage and other soft hydrated connective tissues I Equilibrium results
Journ Biomech Get Quoted by Huyghe et al A two
phase nite element mo del of the diastolic left ventricle and quoted
byFung p
KLP Werner Kahle Helmut Leonhardt and Werner Platzer Taschenatlas der
Anatomie Band InnereOrgane Au age Thieme Stuttgart
Have ISBN
KMM William M Kier John B Messenger and Jaleel A Miyan Short commu
nication Mechanoreceptors in the ns of the cuttle sh sepia o cinalis J
exp Biol Have
Kor D J Korteweg Ueb er die fortp anzungsgeschwindigkeit des schalles in
elastischen rohren Ann der Physik u Chemie Get
Quoted byFung in Bio dynamics p
Kos Anton A Kosinski Di erential Manifolds Academic Press New York
ISBN Go o d Bulletin Review O K Have O K
Kov Sandor J Kovacs A clinical p ersp ective on theory of heart In Leon
Glass Peter J Hunter and Andrew D McCullo ch editors Theory of Heart
chapter pages Springer Verlag Have O K
KP F Kapp el and R O Peer A mathematical mo del for fundamental regu
lation pro cesses in the cardiovascular system J Math Biol
Have
Kre L Krehl Beitraege zur Kenntniss der Fuellung und Entleerung des Herzens
Abhand lungen der mathematisch physischen Classe der Koeniglich Saech
sischen Gesel lschaft der Wissenschaften Leipzig Mit VI I Tafeln
Quoted by MacCallum Have Micro che O K
Kre Erwin Kreyszig Di erential Geometry Dover Have O K
KRK C Cetin Kiris Stuart Rogers Do chan Kwak and I Dee Chang Computation
of incompressible viscous ows through arti cial heart devices with moving
b oundaries In Angela Y Cheer and Cornelis P van Dam editors Fluid
Dynamics in Biology volume of Contemporary Mathematics pages
Have O K
Kro A Krogh The Comparative Physiology of Respiratory Mechanisms Uni
versity of Philadelphia Press Philadelphia Reprinted byDover Pub
lications New York Quoted bySchmidt Nielsen in Animal Physiology
Chapter Not in BIP Not in UB Get
KS F P Knowlton and Ernest H Starling The in uence of variations in tem
p erature and blo o d pressure on the p erformance of the isolated mammalian
heart J Physiol London Have O K
KS William M Kier and Kathleen K Smith Tongues tentacles and trunks
The biomechanics of movementinmuscular hydrostats Zoological J of the
Linnean Soc Have
KS William M Kier and Andrew M Smith The morphology and mechanics
of o ctopus suckers Biol Bul l Have
KS William M Kier and FrederickH Schachat Bio chemical comparison of
fast und slo wcontracting squid muscle J exp Biol
Have
KSM William M Kier Kathleen K Smith and Jaleel A Miyan Electromyogra
phy of the n musculature of the cuttle sh sepia o cinalis J exp Biol
Have
Kun Yas Kuno The mechanical e ect of uid in the p ericardium on the function
of the heart J Physiol London Have Z
Kun I A Kunin Elastic Media with MicrostructureI volume of Springer
Series in Solid State Sciences Springer Verlag Quoted by Ciarletta
and Iesan Non classical Elastic Solids Pitman Research Notes in Mathe
matics Lo ok at
Kun I A Kunin Elastic Media with MicrostructureII volume of Springer
Series in Solid State Sciences Springer Verlag Quoted by Ciarletta
and Iesan Non classical Elastic Solids Pitman Research Notes in Mathe
matics Lo ok at
KW Thomas E Keys and Fredrick A Willius editors Classics of Cardiology
formerly titled Cardiac Classics Vol and Dover Publications
Have O K
Lam J W Lamb ert On the nonlineartities of uid ow in nonrigid tub es
Lam H Lamb On the velo city of sound in a tub e as a ected by the elasticity
of the walls In Phil Soc of Manchester Memoires and Proc volume
pages Get Quoted byFung in Bio dynamics p
Lan Henry L Langhaar Dimensional Analysis and The ory of Models Rob ert
E Krieger Publishing Company Malabar Florida Have
Law H Blaine Lawson Jr The Quantitative Theory of Foliations American
Mathematical So ciety Providence ISBN In Math
Bibl Quoted by Abraham and Marsden Ordered AMS July
Lax Peter D Lax editor Hyperbolic Systems of Conservation Laws and
the Mathematical Theory of Shock Waves Regional Conference Series in
Applied Mathematics So ciety for Industrial and Applied Mathematics
Philadelphia Have ISBN
LCRR Paul Peter Lunkenheimer Colin Walker Cryer K Redman and K D
Richter The watch spring like in spiral structure of the left ventricular
myo cardium Manuscript Have
Lee Chee Shuen Francis Lee Fluid Mechanical Studies of Mitral Valve Motion
PhD thesis University of California Berkeley Quoted byParmley
and Talb ot p in page reference is made to a system
of o des DDJ Ordered UMI Have
Lee Euiwo o Lee Uniqueness results for bursting solutions in a model of excitable
membranes PhD thesis Ohio State University Get Order UMI
LeG Ian J LeGrice A Finite Element Model of Myocardial Structure Impli
cations for Electrical Activation in the Heart PhD thesis Universityof
Auckland Get Ordered UB Quoted in article by LeGrice et al
Amer J of Physiol
Lei J O Leib owitz The History of Coronary Heart Disease volume XVI I I
Publications of the Wellcome Institute of the History of Medicine
Have O K
Lev Simon A Levin editor Lectures on Mathematics in the Life Science vol
Some Mathematical Questions in Biology American Mathematical So ciety
Providence Have O K ISBN
Lev Simon A Levin editor Lectures on Mathematics in the Life Science Vol
Some Mathematical Questions in Biology American Mathematical So ciety
Providence Have O K ISBN
Lev J Ro dney Levick An Introduction to Cardiovascular Physiology
Butterworth Heinemann Oxford Have O K
Lev Simon A Levin Mathematics and Biology The Interface Technical rep ort
Lawrence Berkeley Lab oratory Get
Lev Simon A Levin Frontiers in Mathematical Biology Springer Verlag
ISBN Have O K
LG M Lorange and R M Gulranjani A computer heart mo del incorp orating
anisotropic propagation I Mo del construction and simulation of normal
activation J Electrocariol Get Have Medline abstract
LH Max J Lab and Arun V Holden Mechanically induced changes in elec
trophysiology In Leon Glass Peter J Hunter and Andrew D McCullo ch
editors Theory of Heart pages Springer Verlag Have
O K
LHS Ian J LeGrice Peter J Hunter and Bruce H Smaill Laminar structure of
the heart I I Mathematical mo del American Journal on Physiology Heart
and Circulatory Physiology Have
LHY Julie Z Livingston Henry R Halp erin and Frank C P Yin Accounting for
the Gregg e ect in tetanised coronary arterial pressure ow relationships
Cardiovascular Research Have
Li John K J Li Arterial System Dynamics NewYork University Press New
York p ISBN X Have O K
Li Z Li The immersed interface method a numerical approach for partial
di er ential equations with interfaces PhD thesis UniversityofWashington
Department of Applied Mathematics Seattle WA Get Quoted
byLoyce Adams
Lig M James Lighthill Comments on Dr Prange s pap er In T J Pedley
editor Scale E ects in Animal Locomotion volume lo c cit Get
Lil Leonard S Lilly editor Pathophysiology of Heart Disease Lea Febinger
Pennsylvania Have O K ISBN X
Liv Charles Livingston Knot theory The Mathematical Asso ciation of Amer
ica Get
LL Randall J Leveque and Zhilin Li Immersed interface metho ds for stokes
ow with elastic b oundaries or surface tension Available from
ftp amath washington edu pub rjl pap ers rjl li stokes Have O K
LLG Paul Peter Lunkenheimer A Lunkenheimer and Francisco Torrent Guasp
Kardiodynamik Wege zur strukturgerechten Analyse der Myokardfunktion
p erimed Fachbuch Verlagsgesellschaft mbH Erlangen HaveO K
ISBN
LMK Paul Peter Lunkenheimer R P Mueller Chr Konermann A Lunken
heimer and F Ko ehler Architecture of the myo cardium in computed
tomography Investigative R adiology No Have
LN Hans Petter Langtangen and Bjrn Fredrik Nielsen Getting started with
Di pack The Di pack Rep ort Series Technical rep ort University of Oslo
Department of Mathematics Have O K
Lov A E H Love ATreatise on the Mathematical Theory of Elasticity Dover
New York fourth edition Have O K
Low Richard Lower Tractatus de Corde item de Motu et Colore Sanguinis et
Chyli in Transitu Translated byK Franklin London Oxford University
Press as The anatomy of the heart In Early Science in Oxford R T
Gunther editor Quoted by Acierno page Not in MED Not in
BIP Get Vol or Vol Oxford Colleges and Their Man Science Avail
able Dawsons of Pall Mall ISBN Dillons Quoted
by Grant
LP Aldo A Luisada and Francesco Portaluppi The Heart Sounds New Facts
and Their Clinical Implications Praeger Publishing CBS Inc C
ISBN Have O K
LRC a Paul Peter Lunkenheimer K Redman Colin Walker Cryer P Niederer
and W F Whimster Is there a diastolic organization of myo cardial func
tion Manuscript Have
LRC b Paul Peter Lunkenheimer K Redman Colin Walker Cryer K D Richter
P Niederer W F Whimster and Francisco Torrent Guasp The di cult
access to morphology of the heart clinical applications Technology and
Health Care Have
LS M Lev and C S Simkins Architecture of the human ventricular my
o cardium technique for study using a mo di cation Laboratory Investi
gation Quoted by Greenbaum Ho Gibson Becker and
Anderson Get Order UB
LSE Ian J LeGrice Bruce H Smaill S G Edgar J B Gavin and Peter J
Hunter Laminar structure of the heart I Ventricular myo cyte arangement
and connective tissue architecture in the dog American Journal on Phys
iology Heart and Circulatory Physiology Have preprint XEROX
MED
LSH Ian J LeGrice Bruce H Smaill and Peter J Hunter Consistency of canine
ventricular architecture Is there a platonic heart Circulation Research
Have
LT Chee Shuen Francis Lee and L Talb ot A uid mechanical study on the
closure of heart valves J Fluid Mech Get Order UB
LT I En Lin and Larry A Tab er A mo del for stress induced growth in the de
veloping heart J Biomech and Engineering to appear Have
Lud C Ludwig Ueb er den Bau und die Bewegungen der Herzventrekel
Zeitschr f rationel le Medizin Get
Lui Aldo A Luisada Cardiology volume McGraw Hill New York
Get In PHYSIOLOGIE I MED E
Lui a Aldo A Luisada The Sounds of the DiseasedHeart Green Get LC
ISBN MED E
Lui b Aldo A Luisada The Sounds of the Normal Heart Green LC
ISBN Have UB
LW Paul Peter Lunkenheimer and W F Whimster Heart Muscle Architec
ture and Cardio dynamics Concepts Metho ds Measurements Review and
prosp ect Have Manuscript
Mac John Bruce MacCallum On the muscular architecture and growth of the
ventricles of the heart WelchFestschrift Johns Hopkins Hospital Rep orts
Vol p Quoted byPaul Peter Lunkenheimer A Lunken
heimer and Francisco Torrent Guasp Kardio dynamik Wege zur strukturg
erechten Analyse der Myokardfunktion Have
Mac Richard H MacNeal editor Finite Elements Their Design and Perfor
mance Marcel Dekker Have ISBN
hitecture of the ventricles of the hu Mal Franklin P Mall On the muscular arc
man heart The American Journal of Anatomy Have
O K Continues work of MacCallum Has references to earlier work
Mal Lawrence E Malvern Introduction to the Mechanics of a Continuous
Medium Englewo o d Cli s N J Prentice Hall ISBN
Have O K
Mal P K Mallick Fiber reinforced Composites Materials Manufacturing and
Design Marcel Dekker New York second edition ISBN
Have O K
Mar R Margaria Biomechanics and Energetics of Muscular Exercise Oxford
UK Clarendeon Press Quoted byVogel in Life s Devices Vergri en
lt Kruep er
Mat Claus Mattheck editor Design in der Natur Rombach Oekologie
Have
Mau Gerald A Maugin Material Inhomogeneities in Elasticity Chapman
Hall Have O K ISBN
Maz J Mazumdar An Introduction to Mathematical Physiology and Biology
Cambridge University Press Have O K ISBN Text
b o ok with exercises
McC Andrew D McCullo ch Deformation and Stress in the Passive Heart PhD
thesis University of Auckland Have O K
hael editor Circulation Proc Harvey Tercentenary McM John McMic
Congress London Blackwell Oxford Get In PHYSIOLOGIE
Ivorh Not in BIP UB
McM Thomas A McMahon Muscles Re exes and Locomotion Princeton NJ
Princeton University Press Has problems and solutions ISBN
X Have O K
Mec E A Mechtly Theinternational system of units Physical constants and
conversion factors nd revision Technical rep ort NASA SP
Quoted bySchmidt Nielson Animal Physiology Adaptation and Environ
ment Available rom National Technical Information Service Spring
eld VA Havecwc O K
Mee W Meesmann Nachweis der diastolischen Sogwirkung der Herzkammern
und deren Ein uss auf die intrakardialen Druckablaeufe Z Kreisl Forsch
Have ZD O K
Mei Hans Peter Meinzer Raeumliche Bilder des Ko erp erinneren Spektrum der
Wissenschaft pages July Have
MF Sir Michael Foster Lectures on the History of Physiolo gy During the Six
teenth Seventeent h and Eighteenth Centuries Cambridge University Press
Havecwc O K
MG P A McHale and J C Green eld Evaluation of several geometric mo dels
for estimation of left ventricular circumferential wall stress Circulation
Research Quoted by Arts Reneman and Veenstra
Get
MGRH Andrew D McCullo ch Julius M Guccione Jr Jack M Rogers and Pe
ter J Hunter Three dimensional nite element analysis of stress and ac
tivation in the heart In Mechanics Computing in s and Beyond Pro
ceedings EM Div ASCE pages Columbus OH Have
MGWR Andrew D McCullo ch Julius M Guccione Jr Lewis K Waldman and
Jack M Rogers Large scale nite element analysis of the b eating heart
In Theo C Pilkington Bruce Loftis Jo e F Thompson Savio L Y Woo
Thomas C Palmer and Thomas F Budinger editors High Performance
Computing in Biomedical Research CRC Press Inc Have
MH R R Martin and H Haines Application of Laplace s law to mammalian
hearts Computational Biochemistry and Physiology Get
Quoted by Lin and Taylor Order UB
MH Walter J Mo ore and Dieter O Hummel editors PC Physikalische
Chemie Walter de Gruyter Berlin Have O K
MH Jerrold E Marsden and Thomas J R Hughes Mathematical Foundations
of Elasticity Prentice Hall Dover reprint New York Dover
ISBN Havecwc O K
MHS Andrew D McCullo ch Peter J Hunter and Bruce H Smaill Mechanical
e ects of coronary p erfusion in the passive canine left ventricle American
Journal on Physiology Heart and Circulatory Physiology H
H Have O K
MHW Arthur F T Mak Lidu Hang and Qinque Wang A biphasic p oro elastic
analysis of the ow dep endent sub cutaneous tissue pressure and compaction
due to epidermal loadings issues in pressure sore Transactions of the
ASME Journal of Biomechanical Engineering Have
Mil William R Milnor Hemodynamics Williams and Wilkins Baltimore
nd edition Quoted by ShadwickinBiewener Biomechanics Struc
tures and Systems A Practical Approach Quoted by Nichols and
O Rourke p Have O K
Mil William R Milnor Cardiovascular Physiology Oxford University Press
New York ISBN Have O K
Mir Israel Mirsky Left ventricular stesses in the intact human heart Journ
Biophys Have O K
Mir Israel Mirsky Ventricular and arterial wall stresses based on large defor
mation analyses Journ Biophys Have O K
Mir Israel Mirsky Elastic prop erties of the myo cardium A quantitative ap
proach and clinical applications In R M Berne editor Handbook of Phys
iology Section The Cardiovascular System Vol I The Heart American
Ph ysiological So ciety Bethesda Maryland Have
Miu Rob ert M Miura editor Lectures on Mathematics in the Life Science Vol
Some Mathematical Questions in Biology Muscle Physiology American
Mathematical So ciety Providence Have O K ISBN
MK J L Meriam and L G Kraige Engineering Mechanics Dynamics vol
ume John Wiley Sons Have O K
MMI Jay S Meisner David M McQueen Yoshio Ishida Herb ert O Vetter
Ub erto Bortolotti Jo el A Strom Rob ert W M Frater Charles Samuel
Peskin and Edward L Yellin E ects of timing of atrial systole on LV lling
and mitral valve closure computer and dog studies American Journal on
Physiology Heart and Circulatory Physiology H H
ZD Have O K
MNS Rob ert E Mates Rob ert M Nerem and Paul D Stein editors Mechanics
of the Coronary Circulation American So cietyofMechanical Engineers
New York p ISBN G Havecwc
MO Andrew D McCullo ch and Je rey Howard Omens Non homogeneous anal
ysis of three dimensional transmural nite deformation in canine ventricular
myo cardium Journ Biomech Have O K
MOMC Deidre A MacKenna Je rey Howard Omens Andrew D McCullo ch and
James W Co vell Contribution of collagen matrix to passiveleftventricular
mechanics in isolated rat hearts American Journal on Physiology Heart
and Circulatory Physiology H H Have O K
Mo o Walter J Mo ore Physical Chemistry Longman Scienti c and Technical
Harlow Essex th edition Have O K ISBN
Mor Thomas F Moriarty The law of Laplace Its limitations as a relation for
diastolic pressure volume or wall stress of the left ventricle Circulation
Research Have O K
Mor Frank Morgan Geometric Measure Theory A Beginners Guide Academic
Press ISBN Have O K
Mos Samuel E Moskowitz On the mechanics of left ventricular diastole J
Biomech Get
Mos Samuel E Moskowitz E ects of inertia and visco elasticity in late rapid ll
ing of the left ventricle Journ Biomech Have O K
MP David M McQueen and Charles Samuel Peskin A three dimensional com
putational metho d for blo o d ow in the heart I I contractile b ers Journal
of Computational Physics Get O K In Numerik Bib
Artikel kopieren
MP a Miriam McLean and John Prothero Myo b er orientation in the weanling
mouse heart The American Journal of Anatomy Have
O K
MP b Miriam R McLean and John W Prothero Three dimensional cosntruction
from serial sections Analytical and Quantitative Cytology and Histology
Have O K
MP Miriam R McLean and John W Prothero Determination of relative bre
orientation in heart muscle metho dological problems Anatomical Record
Get Phone conversation
MP Anita A Mayo and Charles Samuel Peskin An implicit numerical metho d
for uid dynamics problems with immersed elastic b oundaries In Angela Y
Cheer and Cornelis P van Dam editors Fluid Dynamics in Biology volume
of Contemporary Mathematics pages American Mathemat
ical So ciety Providence Have O K
MP Miriam R McLean and John W Prothero Get Phone conversa
tion
MPY David M McQueen Charles Samuel Peskin and Edward L Yellin Fluid
dynamics of the mitral valve Physiological asp ects of a mathematical
mo del Am J Physiol Heart and Circulation Physiology Vol
H H Have O K
MRP Miriam R McLean M Alison Ross and John W Prothero Three
dimensional reconstruction of the myo b er pattern in the fetal and neonatal
mouse heart Anatomical Record ZD Have O K
MS Joseph Markwalder and Ernest H Starling On the constancy of the systolic
output under varying conditions J Physiol London
Have O K
MS J C Misra and S I Singh Distribution of stresses in the left ventricular
wall of the intact heart Bul l Math Biol Have
MSC Francois M Melchior R Srini Srinivasan and John B Charles Mathemat
ical mo deling of human cardiovascular system for simulation of orthostatic
resp onse American Journal on Physiology Heart and Circulatory Physi
ology H H Have O K
MSH Andrew D McCullo ch Bruce H Smaill and Peter J Hunter Left ventric
ular epicardial deformation in isolated arrested dog heart Am J Physiol
Heart and Circulation Physiology Vol H H Have
Mur James D Murray Mathematical Biology Springer Verlag New York nd
edition Have ISBN X
NHS Poul Michael F nss Nielsen Peter J Hunter and Bruce H Smaill Bi
axial testing of membrane biomaterials Testing Equipment and Pro ce
dures Transactions of the ASME Journal of Biomechanical Engineering
Have O K
Nie Poul Michael F nss Nielsen The A natomy of the Heart A Finite Element
Model PhD thesis University of Auckland Have O K
NLSH Poul Michael F nss Nielsen Ian J LeGrice Bruce H Smaill and Peter J
Hunter Mathematical mo del of the geometry and brous structure of
the heart Am J Physiol Heart and Circulatory Physiology Vol
H H Have O K
NLW T F Nonnenmacher G A Losa and E R Weib el Fractals in Biology
and Medicine Birkhaeuser Have O K ISBN
NO Wilmer W Nichols and Michael F O Rourke McDonald s Blood Flow in
Arteries Edward Arnold rd edition Have O K ISBN
Nob Mark I M Noble The Cardiac Cycle Blackwell London Have
Xeroxcopy Not in BIP ISBN
Nob Mark I M Noble Energetics of the heart In Henk E D J ter Keurs
and J V Tyb erg editors Mechanics of the Circulation chapter pages
Martinus Nijho Publishers Have
No o Abraham No ordergraaf editor Bal listocardiography and Cardiac Perfor
mance North Holland Amsterdam UB G
No o Abraham No ordergraaf Circulatory System Dynamics Academic Press
Have O K ISBN
W Nowacki Theory of Asymmetric Elasticity Polish Scienti c Publishers Now
Warszawa and Pergamon Press Quoted by Ciarletta and Iesan Non
classical Elastic Solids Pitman Research Notes in Mathematics
Lo ok at
NPS Donald W Novey Marcia Pencak and John M Stang The Guide to Heart
Sounds Normal and Abnormal CRC Press Have O K
NS Abraham No ordergraaf and Isaac Starr editors Bal listocardiography in
Cardiovascular Research North Holland Amsterdam Get UB G
NS a Charles Nash and Siddhartha Sen Topology and Geometry for Physicists
Academic Press Have O K
NS b Mark I M Noble and W A Seed editors The Interval ForceRelation
ship of the Heart Bowditch Revisited Cambridge University Press
Have
NY E Nevo and Y Lanir E ect of residual strain on the diastolic function
of the left ventricle as predicted by a structural mo del Journ Biomech
Get
Ode J Tinsley Oden Finite Elements of Non Linear Continua McGraw Hill
Have O K
uan cheng Fung Residual strain in rat left OF Je rey Howard Omens and Y
ventricle Circulation Research Have O K
Ogd R W Ogden Non linear Elastic Deformations Ellis Horwo o d Chichester
Get Quoted in Bull Not in UB AMS Vol Oct p
Ordered Kruep er
Olv Peter J Olver Applications of Lie Groups to Di erential Equations
Springer Verlag New York second edition Have
Ome Je rey Howard Omens Left Ventricular Strain in the No Load State Due
to the ExistenceofResidual Stress PhD thesis University of California
San Diego CA Have O K
OMM Je rey Howard Omens Karen D May and Andrew D McCullo ch Trans
mural distribution of three dimensional strain in the isolated arrested ca
nine left ventricle American Journal on Physiology Heart and Circulatory
Physiology Have
OMM Je rey Howard Omens Deidre A MacKenna and Andrew D McCullo ch
Measurement of strain and analysis of stress in resting rat left ventricular
myo cardium Journ Biomech Have
Opi Lionel H Opie The Heart Physiology and Metabolism Raven Press New
York second edition Have Go o d overall description of metab olism
up to date has historical references
O R Joseph O Rourke Computational Geometry in C Springer Verlag New
York nd edition Have ISBN X
Oth Hans G Othmer editor Some Mathematical Questions in Biology The
Dynamics of Excitable Media American Mathematical So ciety Providence
Have O K ISBN
otRS Symb ols Committee of the Royal So ciety Quantities Units and Sym
bols Technical rep ort The Royal So ciety London Quoted by
Schmidt Nielson Animal physiology Adaptation and environment
Have O K
OvCG C W J Oomens DickH van Camp en and H J Gro otenb o er A mixture
approach to the mechanics of skin Journ Biomech
Get Quoted by Huyghe et al A two phase nite element mo del of the
diastolic left ventricle
PAPR Tim T Prinzen Theo Arts Frits W Prinzen and Rob ert S Reneman
Mapping of epicardial deformation using a video pro cessing technique
Journ Biomech Have O K
Ped T J Pedley editor Scale E ects in Animal Locomotion Academic
Press London Get ISBN Not in MED ESS
VQS BOC UWB
Ped T J Pedley The Fluid Mechanics of Large BloodVessels Cambridge
University Press Cambridge Have O K ISBN
Ped P Pedersen editor Optimal Design with Advanced Materials Proc IU
TAM Symp Lyngby Elsevier Amsterdam Get Quoted by G Roz
vanyetal Layout optimization of structures ISBN
No price in BIP Anfragen b ei Kruep er Preis
Pen C J Pennycuik Newton Rules Biology Oxford University Press
ISBN pap erback Ordered Kruep er
Pes Charles Samuel Peskin Flow Patterns Around Heart Valves A Digital
Computer Method for Solving the Equations of Motion PhD thesis Alb ert
Einstein College of Medicine Yeshiva University Have O K
Pes Charles Samuel Peskin Mathematical Asp ects of Heart Physiology Lecture
Notes Courant Institute of Mathematical Sciences New York Have
O K
Pes Charles Samuel Peskin Partial Di erential Equations in Biology Lecture
Notes Courant Institute of Mathematical Sciences New York Have
O K
Pes Charles Samuel Peskin Numerical analysis of blo o d ow in the heart
Journal of Computational Physics Have O K
Pes Charles Samuel Peskin Fluid Dynamics of the Heart and the Ear Com
puting Metho ds in Applied Sciences and Engineering North Holland Pub
lishing Company Get Ordered Kruep er
Pes Charles Samuel Peskin The uid dynamics of heart valves exp erimental
theoretical and computational metho ds Ann Rev Fluid Mech
Get ZD
Pes Charles Samuel Peskin Fib er architecture of the left ventricular wall An
asymptotic analysis Communications on Pure and Applied Mathematics
Have
Pet James B Pettigrew On the arrangement of the muscular bres of the
ventricular p ortion of the heart of the mammal Cro onian Lecture Ab
stract Proc Roy Soc London Get CHEMIE
Corrensstr
Pet James B Pettigrew On the arrangement of the muscular bres in the
ventricles of the vertebrate heart with physiological remarks Philos Trans
Quoted by Greenbaum Ho Gibson Becker and
Anderson and quoted byTorrent Guasp Get Order UB
Pet Rob ert Henry Peters The Ecological Implications of Body Size Cambridge
Universit y Press Cambridge Havecwc O K ISBN X
PF J G Pinto and Yuan cheng Fung Mechanical prop erties of the heart mus
cle in the passive state J Biomech Get BIOMECH
PHH Andrew E Pollard Nigel Ho oke and Craig S Henriquez Cardiac propa
gation simulation In Theo C Pilkington Bruce Loftis Jo e F Thompson
Savio L Y Wo o Thomas C Palmer and Thomas F Budinger editors
High Performance Computing in Biomedical Research CRC Press Inc
Have
Phi A P W Philip Some observations relating to the p owers of circulation and
the state of the vessels and an in amed part Medicochirurgical transactions
London Have O K
PHS Mordechai Perl Arie Horowitz and Samuel Sideman Comprehensive
mo del for the simulation of left ventricle mechanics Part I Mo del de
scription and simulation pro cedure Med and Biol Eng and Comput
Have For Part I I see Horowitz Perl Sideman and Rit
man
PKG G Pogatsa M Z Koltai and G Grosz The role of myo cardial water
content in heart function Acta PhysiologicaAcad Sci Hungar
Get Order UB
PLT Theo C Pilkington Bruce Loftis Jo e F Thompson Savio L Y Woo
Thomas C Palmer and Thomas F Budinger editors High Performance
Computing in Biomedical Research CRC Press Inc Bo ca Raton Florida
Have O K ISBN
PM Charles Samuel Peskin and David M McQueen Cardiac uid dynamics
CRC Critical Reviews in Biomedical Engineering
Have O K
PM a Charles Samuel Peskin and David M McQueen Cardiac uid dynam
ics In Theo C Pilkington et al editor High Performance Computing in
Biomedical Research CRC Press Have O K
PM b Charles Samuel Peskin and David M McQueen Computational bio uid
dynamics In Angela Y Cheer and Cornelis P van Dam editors Fluid
Dynamics in Biology volume of Contemporary Mathematics pages
American Mathematical So ciety Providence HaveO K
PM Charles Samuel Peskin and David M McQueen Mechanical equilibrium
determines the fractal b er architecture of the aortic heart valve lea ets
American Journal on Physiology Heart and Circulatory Physiology
H H Have O K
Poi J M Poiseuille Recherche exp erimentales sur le mouvement des liquides
dans les tub es de tres p etits diametres Comptes Rendus Acad
Get UB Z Qu Quoted in McDonald p
POOB G Pelle J Ohayon C Oddou and Philip e Brun Theoretical mo dels
in mechanics of the left ventricle Biorheology Have
O K
PP Je rey S Prothero and John W Prothero Three dimensional reconstruc
tion from serial sections IV The reassembly problem Computers and
Biomedical Res HaveO K
PP Charles Samuel Peskin and Beth Feller Printz Improved volume conserva
tion in the computation of ows with immersed elastic b oundaries Journal
of Computational Physics Have O K
PPS S W Patterson H Pip er and Ernest H Starling The regulation of the
heart b eat J Physiol London Have O K
Pri a William G Pritchard Mathematical mo dels of running SIAM Review
Have O K
Pri b William G Pritchard Scaling in the animal kingdom Bul letin of Mathe
matical Biology Have O K
Pro Cli ord Ladd Prosser editor Comparative Animal Physiology Vols En
vironmental and Metabolic Animal Physiology and Neural and Integrative
Animal Physiology Wiley th edition ISBN Set
Have
PS S W Patterson and Ernest H Starling On the mechanical factors which
determine the output of the ventricles J Physiol London
Have
PSKS Thomas C Palmer Edw ard V Simpson Katherine M Kavanagh and
William M Smith Visualization of bio electric phenomena In Theo C
Pilkington Bruce Loftis Jo e F Thompson Savio L Y Wo o Thomas C
Palmer and Thomas F Budinger editors High Performance Computing
in Biomedical Research CRC Press Inc Have
Puf Alexander Pu Der funktionel le Bau der Herzkammern Georg Thieme
Stuttgart HaveXerox copy Also quoted by Grant who comments
on the excellent sektches
PV Howard C Peters and Maurice B Visscher The energy metab olism of the
heart in failure and the in uence of drugs up on it American Heart Journal
Have O K
Rab Sina Y Rabbany The Genesis of Intramyocardial Pressure PhD thesis
UniversityofPennsylvania Have O K
Rac D K Racker Atrioventricular no de and input pathways a correlated gross
anatomical and histological study of the canine atrioventricuar junctional
region Anatomical Record Get Have Medline Abstract
Rai Erwin J Raisz Principles of Cartography McGraw Hill New York
ISBN Out of print Ordered UB Zurueck Vorh in
MS UB D
Ral Jack A Rall Sense and nonsense ab out the Fenn e ect American Journal
on Physiology Heart and Circulatory Physiology H H
Have
RAR Jerrold E Marsden Ralph Abraham and Tudor Ratiu Manifolds Tensor
Analysis and Applications Springer Verlag second edition Have
ISBN
Ras M P Rast Simultaneous solution of the navier stokes and elastic mem
brane equations by a fem Int J Numer Methods Fluids
Get
RBK Gin Rozvany M P Bendso e and U Kirsch Layout optimization of struc
tures Appl Mech Rev Have
Red John N Reddy An Introduction to the Finite Element Method McGraw
Hill second edition ISBN Have O K
RFN Sina Y Rabbany John T Funai and Abraham No ordergraaf Pressure
generation in a contracting myo cyte Heart Vessels Have
O K
RFS Thomas F Robinson Stephen M Factor and Edmund H Sonnenblick
The heart as a suction pump Scienti c American June
Have ZC O K
RHM Edward K Ro driguez Anne Hoger and Andrew D McCullo ch Stress
dep endent nite growth in soft elastic tissues Journal of Biomechan
ics Have preprint Get Quoted by Lin and Taylor
BIOMECH
Ric Benjamin Ricci Physiological Basis of Human Performance Lea and
Febiger Philadelphia Have ISBN
RK T F Robinson and R K H Kinne editors Cardiac Myocyte Connective
Tissue Interactions in Health and Disease Issues in Biomed volume
Karger Basel Have O K ISBN
RKKB John Ross Jr Francis Klo cke Gerard Kaiser and Eugene Braunwald Ef
fect of alterations of coronary blo o d ow on the oxygen consumption of the
working heart Circulation Research Have O K
RM Werner C Rheinb oldt and Charles K Mesztenyi On a data structure for
adaptive nite element mesh re nements ACM Transactions on Mathe
matical Software Have O K
RM Gordon Rogers and Yon Mayhew Engineering Thermodynamics Work
and Heat Transfer Longman Have O K
RM Jack M Rogers and Andrew D McCullo ch A collo cation Galerkin nite el
ement mo del of cardiac action p otential propagation IEEE Trans Biomed
Eng in press Hav e
RN J A Rumb erger and R M Nerem A metho d of characteristics calculation
of coronary blo o d ow J Fluid Mech Get Quoted by
Pedley in Fluid Mechanics of Large Blo o d Vessels p
Rob Thomas F Robinson Structural arrangementofmyo cytes and brillar
connective tissue in heart muscle In Thomas F Robinson and R K H
Kinne editors Cardiac Myocyte Connective Tissue Interactions in Health
and Disease pages Karger Basel Have O K
Ros R Rosen Dynamic similarity and the theory of biological transformations
Bul l Math Biol Get UB
Row Loring B Rowell Human Circulation Regulation during Physical Stress
Oxford University Press Oxford p ISBN
Have
Row Loring B Rowell Human Cardiovascular Control Oxford University Press
Have O K ISBN
ROWM Edward K Ro driguez Je rey Howard Omens Lewis K Waldman and
Andrew D McCullo ch E ect of residual stress on transmural sarcomere
length distributions in rat left ventricle American Journal of Physiology
H H Have
RP RuchandPatton Physiology and Biophysics volumes Saunders
Philadelphia Get Quoted by Dinnar in b o ok edited by Ghista
RR J S Robb and R C Robb Abnormal distribution of the sup er cial muscle
bundles in the human heart Americ an Heart Journal
Quoted by Greenbaum Ho Gibson Becker and Anderson Get
RR J S Robb and R C Robb The normal heart anatomyandphysiology of
the structural units American Heart Journal Quoted by
Greenbaum Ho Gibson Becker and Anderson Get
RR a T J Reiter and F G Rammersdorfer Simulation of natural adaptation of
b one material and application in optimum comp osite design In P Peder
sen editor Optimal Design with Advanced Materials Proc IUTAM Symp
Lyngby Elsevier Amsterdam Get Quoted byG Rozvany et al
Layout optimization of structures
RR b Rob ert C Renardy and Michael Rogers An Introduction to Partial Dif
ferential Equations Springer Verlag ISBN Have
O K
RS M A Ross and Daniel D Streeter Nonuniform sub endo cardial b er ori
entation in the normal macaque left ventricle Eur J Cardiol
Quoted by Greenbaum Ho Gibson Becker and Anderson Get
RT K B Rob erts and J D W Tomlinson The Fabric of the Body Oxford
University Press ISBN Get
Rub Sol I Rubinow editor Mathematical Problems in the Biological Sciences
Regional Conference Series in Applied Mathematics So ciety for Industrial
and Applied Mathematics Philadelphia Have
Rub Sol I Rubinow Some problems of uid mechanics in biology In Simon A
Levin editor Some Mathematical Questions in Biology VIII pages
American Mathematical So ciety Providence Have O K ISBN
Rum J A Rumb erger A non linear mathematical model of coronary blood ow
PhD thesis Ohio State University Get Quoted byPedley in Fluid
Mechanics of Large Blo o d Vessels p
Rus Rob ert Frazer Rushmer Cardiovascular Dynamics W B Saunders Com
pany Philadelphia Have Xerox Copy Not in MED Not in BIP
O K
Sau Alfons Aloisius Henricus Johannes Sauren The Mechanical Behaviour of
the Aortic Valve PhD thesis Eindhoven Technical University The Nether
lands Have O K
SB Daniel D Streeter Jr and David L Basset An engineering analysis of
myo cardial b er orientation in pig s left ventricle in systole Anat Record
Have O K
SB Samuel Sideman and Rafael Beyar editors Analysis and Simulation of
the Car diac System Ischemia CRC Press Have ISBN
SB B Szab o and Ivan Babuska Finite Element Analysis Wiley Get
Ordered Kruep er
SC W Milton Swanson and Richard E Clark Dimensions and geometric rela
tionships of the human aortic valve as a function of pressure Circulation
Research Have ZC O K
SC Bryan K Slinker and Kenneth B Campb ell Previous b eat contraction
history alters mechanical restitution in the isolated left ventricle Cardio
vascular Research Have O K
Sch Bradley T Scheer Comparative Physiology John Wiley Sons Inc New
York Havecwc O K
Sch Bradley T Scheer Transp ort Circulation of b o dy uids In Animal Phys
iology chapter pages John Wiley Sons Inc New York
London Have In ZOOLOGIE Phys E
Sch a Bernard Schepartz Dimensional Analysis in the Biomedical Sciences
Charles C Thomas Spring eld Illinois Have Xerox copy O K
ISBN
Sch b Hans Rudolf Schwarz Methode der niten Elemente B G Teubner
Stuttgart Have ISBN
Sch Hans Rudolf Schwarz FORTRAN Programme zur Methode der niten El
emente B G Teubner Stuttgart Have ISBN
Sch Daniel J Schneck Mechanics of Muscle New York University Biomedical
Engineering Series New York University Press nd edition Have
O K ISBN
Sch Alb ert S Schwarz Topology for Physicists Springer Verlag Berlin
In Math Bib O K ISBN
Sco Stephen K Scott Oscil lations waves and chaos in chemical kinetics
Oxford University Press ISBN Have O K
SD Harold Sandler and Harold T Do dge Left ventricular tension and stress
in man Circulation Research Have O K
SdAD Rob ert L Spilker Edgard S de Almeida and Peter S Donzelli Finite ele
ment metho ds for the biomechanics of soft hydrated tissues Nonlinear anal
ysis and adaptive control of meshes In Theo C Pilkington Bruce Loftis
Jo e F Thompson Savio L Y Wo o Thomas C Palmer and Thomas F
Budinger editors High Performance Computing in Biomedical Research
CRC Press Inc Have
SDHA Bruce H Smaill J Douglas Peter J Hunter and I Anderson Steady
state heat transfer in the left ventricle In Leop old Dintenfass Desmond G
Julian and Geo rey V F Seaman editors Heart Perfusion Energetics
and Ischemia Plenum Publishing Corp oration Ha ve
Sen J B Senac Trait e de la structure du co eur Paris J Vincent Quoted
by Greenbaum Ho Gibson Becker and Anderson Get Not in UB
Not in Livres Del Order UB
SG Bryan K Slinker and Stanton A Glantz End systolic and end diastolic
ventricular interaction American Journal on Physiology Heart and Cir
culatory Physiology H H Have O K
SG a Bruce R Simon and M A Gaballa Finite strain p oro elastic nite element
mo dels for large arterial cross sections In Rob ert L Spilker and Bruce R
Simon editors Computational Methods in Bioengineering pages
American So ciety of Mechanical Engineers New York Quoted by
Yang Tab er and Clark Have
SG b Ian Mo at Smith and D V Gri ths Programming the Finite Element
Method Wiley nd edition Have ISBN X
SGL Bryan K Slinker Y Goto and Martin M LeWinter Systolic direct ven
tricular interaction a ects left ventricular contractionand relaxation in the
intact dog circulation Circulation Research Get
SGS S J Sarno J P Gilmore N S Skinner Jr A G Wallace and J H
Mitchell Relation b etween coronary blo o d ow and m yo cardial oxygen
consumption Circulation Research Have O K
SH Daniel D Streeter Jr and William T Hanna Engineering mechanics for
successive states in canine left ventricular myo cardium I I Fib er angle and
sarcomere length Circulation Research Have O K
SH Ulrich Sigwart and Paul H Heintzen Ventricular Wal l Motion Georg
Thieme Verlag Havelistofcontents In MED Get
SH Bruce H Smaill and Peter J Hunter Structure and function of the dias
tolic heart material prop erties of passivemyo cardium In Peter J Hunter
Leon Glass and Andrew McCullo ch editors Theory of Heart pages
Springer Verlag Have
SH Madison S Spach and J Francis Heidlage A multidimensional mo del of
cellular e ects on the spread of electronic currents and on propagating
action p otentials In Theo C Pilkington Bruce Loftis Jo e F Thompson
Savio L Y Wo o Thomas C Palmer and Thomas F Budinger editors
High Performance Computing in Biomedical Research CRC Press Inc
Have
Sha Ascher H Shapiro Shape and Flow Garden City New York Doubleday
Get Not in BIP Quoted by Steven Vogel Life in Moving
Fluid Order UB
SHdSA Paul A Schweitzer Steven Hurder S J Nathan Moreira dos Santos and
Jose Luis Arraut editors Di erential Topology Foliations and Group Ac
tions volume of Contemporary Mathematics American Mathematical
So ciety Providence ISBN Havecwc
Sil FrederickH Silver Biological Materials Structure Mechanical Properties
and Modeling of Soft Tissues New York University Biomedical Engineering
Series New York University Press Have O K ISBN
Sim William Simon Mathematical Techniques for Biology and Medicine Dover
ISBN Have O K
SK Kathleen K Smith and William M Kier Trunks Tongues and Tentacles
Moving with Skeletons of muscles American Scientist
Have
SK Vito Starc and Thomas Kenner editors Compliance of the Cardiovascu
lar System Proceedings Workshop on Cardiovascular Dynamics Tainach
Austria October Medicinski Razgledi Ljublijana ISSN
Havecwc
Ska R Skalak Approximate formulas for myo cardial b er stresses Transactions
of the ASME Journal of Biomechanical Engineering
Have O K
SKKM M Sugawara F Ka jiya A Kitabatake and H Matsuo editors Blood
Flow in the Heart and Large BloodVessels Springer Verlag p
ISBN Not available HaveXerox copy
Smi Gerald F Smith Constitutive Equations for Anisotropic and Isotropic Ma
terials North Holland Amsterdam ISBN X Get Uses
group theory to generate p ossible equations No price in BIP Nachfragen
b ei Kruep er Preis
SMM RichardS Szwarc Lynda L Micklenb orough Shun Ichi Mizuno Gre
gory J Wilson Peter Liu and Shanas Mohamed Conductance catheter
measurements of left ventricular volume in the intact dog parallel con
ductance is indep endentofleftventricular size Cardiovascular Research
Have O K
SMSS Kichi Sagawa Lowell Maughan Hiroyuki Suga and Kenji Sunagawa Car
diac Contraction and the Pressure Volume Relationship Oxford University
Press New York Have O K ISBN
SN Knut Schmidt Nielsen How Animals Work Cambridge University Press
Cambridge Have ISBN
SN Knut Schmidt Nielsen Scaling Why is Animal Size so Important Cam
bridge University Press Have O K ISBN
SN Knut Schmidt Nielsen Animal Physiology Adaptation and Environment
Cambridge University Press fourth edition Havecwc O K ISBN
SNBM Knut Schmidt Nielsen L Bolis and S H P Maddral editors Compara
tive Physiology Water Ions and Fluid Mechanics Cambridge University
Press Get ZOOL Phys E checked out Not in
WHITAKER
Son Edmund H Sonnenblick Force velo city relations in mammalian heart mus
cle Am J Physiol ZD Have O K
Son Edmund H Sonnenblick The structural basis and imp ortance of restoring
forces and elastic recoil for the lling of the heart European Heart Journal
Supplement A Get Quoted by Robinson Factor and
Sonnenblick ZD not upstairs MED Not in MED Order UB
Sp e Anthony James Merrill Sp encer Continuum Mechanics Longman Scienti c
and Technical Essex Have O K ISBN
SPRG Daniel D Streeter Jr Walter E Powers M Alison Ross and Fran
cisco Torrent Guasp Three dimensional b er orientation in the mammalian
left ventricular wall In Abraham No ordergraaf Jan Baan and J Raines
editors Cardiovascular System Dynamics pages MIT Press
Have O K
SQGMH D Sanchez Quintana V Garcia Martinez and J M Hurle Myo cardial
b er architecture in the human heart Anatomical demonstration of mo d
i cations in the normal pattern of ventricular b er architecture in a mal
formed adult sp ecimen Acta Anatomica Basel Get
Have Medline abstract
SR Daniel D Streeter and Ceon Ramon Muscle pathway geometry in the heart
wall Transactions of the ASME Journal of Biomechanical Engineering
Have O K
SS a Shaul Sorek and Samuel Sideman A p orous medium approach for mo d
elling heart mechanics I I D case Math Biosci Have
O K
SS b Shaul Sorek and Samuel Sideman A p orous medium approach for mo d
elling heart mechanics I Theory Math Biosci Have
abstract Have
SS Rob ert L Spilker and Bruce R Simon editors Computational Methods
in Bioengineering American So ciety of Mechanical Engineers New York
BED Vol Have O K
SSBA Kichi Sagawa Kenji Sunagawa D Burkho and J Alexander Jr Mechan
ics of the interventricular septum In Henk E D J ter Keurs and J V
Tyb erg editors Mechanics of the circulation chapter pages
Martinus Nijho Publishers Have
SSP Daniel D Streeter Jr Henry M Sp otnitz Dali P Patel Ross John Jr
and Edmund H Sonnenblick Fib er orientation in the canine left ventricle
during diastole and systole Circulation Research Have
ZC O K
SSWZ G W Schmid Scho enb ein S L Y Wo o and B W Zweifach editors Fron
tiers in Biomechanics Springer V erlag ISBN Have
O K
ST E R Smith and J V Tyb erg Mechanics of the ventricular septum In Henk
E D J ter Keurs and J V Tyb erg editors Mechanics of the circulation
chapter pages Martinus Nijho Publishers Have
ST Rob ert F Schmidt and Gerhard Thews editors Physiologie des Menschen
Au age Springer Verlag Berlin Havecwc O K ISBN
Sta Ernest H Starling editor The LinacreLecture on the Law of the Heart
Longmans and Green London Get UB Va German Das
Gesetz der Herzarb eit
Sto J J Stoker Nonlinear Elasticity Gordon and Breach Have O K
ISBN
Sto J J Stoker Di erential Geometry Wiley Have O K
Str Daniel D Streeter Jr ARational Model for Myocardial Stress Analy
sis BasedonFiber Orientation and Curvature PhD thesis The Catholic
University of America Washington D C Dissertation Abstracts In
ternational Have
Str Dirk J Struik Biomechanics Its Foundations and Objectives Prentice
Hall Get Quoted byPedley in Fluid Mechanics of Large Blo o d
Vessels p
Str Daniel D Streeter Jr Gross morphology and b er geometry of the heart
In R M Berne editor Handbook of Physiology Section The Cardiovas
cular System Vol I The Heart American Physiological So ciety Bethesda
Maryland Have
Str Dirk J Struik Lectures on Classical Di erential Geometry Dover
Have O K
SW Jan Strackee and Nicolaas Westerhof editors The Physics of Heart and
Circulation IOP Pub p ISBN X Have
SW Hal L Smith and Paul Waltman The Theory of The Chemostat Cambridge
University Press Get ISBN Order Kruep er
Swa John M Swan How shall we tell wether or not the myo cardium is comp e
tent Archives of Internal Medicine Have
Sym Ciba Foundation Symp osium editor The Physiological Basis of Starling s
Law of the Heart Elsevier Amsterdam Bo oks on Demand LC
ISBN Have O K
nIntroduction American Math Tam Itiro Tamura Topology of Foliations A
So c Providence Rho de Island ISBN In Math Bib
O K
TDD Wolfgang Trautwein Thomas Do err and Ralf Denger Exp erimente an
isolierten zellen des ventrikelmyokards Video lm von Institut fr den
wissenschaftlichen Film IWF Vorgestellt in der Zeitschrift IWF aktuell
Nummer Mai Best Nr C
TG Stephen P Timoshenko and J N Go o dier Theory of Elasticity McGraw
Hill rd edition Havecwc ISBN
TGH Carlos Thomas Gerfried Geb ert and Vinzenz Hombach Grund lagen
der klinischen Medizin Anatomie Physiologie Pathologie Mikrobiologie
Klinik Band Herz und Gefaesse F K Schattauer ISBN
Have
TH K Takamizawa and K Hayashi Strain energy density function and uniform
strain hyp othesis for arterial mechanics Journ Biomech
Get Quoted by J H Omens p XEROX Biomech
Tho Carolyn Eyster Thomas The muscular architecture of the ventricles of hog
and dog heart Am J Anatomy Quoted byPeskin
ZD Have O K
Tho D Arcy Wentworth Thompson On Growth and Form Cambridge Uni
versity Press Cambridge abridged edition Have O K ISBN
Thu Mano Thubrikar The Aortic Valve CRC Press Bo ca Raton Florida
ISBN Have O K
Tig Rob ert Tigerstedt Die Physiologie des Kreislaufes volume Walter
de Gruyter Berlin nd edition Get In PHYSIOLOGIE I Very
thorough Has copies of pictures of MacCullum Has section on work done
by heart
Tim Stephen P Timoshenko History of Strength of Materials Dover
ISBN Havecwc
tKK Henk E D J ter Keurs and J C Kentish On the force length relation
in myo cardium In Henk E D J ter Keurs and J V Tyb erg editors
Mechanics of the circulation chapter pages Martinus Nijho
Publishers Have
tKN Henk E D J ter Keurs and Mark I M Noble editors Starling s Law of the
Heart Revisited Kluwer Academic Publishers Have ISBN
X Has list of recent texts in the series Development in Cardiovascular
Medicine
tKRvHN Henk E D J ter Keurs W H Rijnsb erger R van Heuningen and M J
Nagelsmit Tension development and sarcomere length in rat cardiac tra
b eculae evidence of length dep endent activities Circulation Research
Get MED
tKT Henk E D J ter Keurs and J V Tyb erg editors Mechanics of the
O K Circulation Martinus Nijho Dordrecht Have Xerox copy
ISBN
TKZ H Tillmanns W Kuebler and H Zeb e Microcirculation of the Heart
Theoretical and Clinical Problems Springer Verlag Have table of
contents ISBN Lo ok at in MED
TMPG John V Tyb erg Gregory A Misbach William W Parmley and Stanton A
Glantz E ects of the p ericardium on left ventricular p erformance In Jan
Baan and Alexander C Arntzenius editors Cardiac Dynamics pages
Martinus Nijho Publishers Have
TN C Truesdell and W Noll The Non Linear Field Theories of Mechanics
Handbuch der Physik volume I I I Springer Verlag Berlin Have
cwc O K ISBN
To e A To ezeren Static analysis of the left ventricle Transactions of the ASME
Journal of Biomechanical Engineering Quoted by Misray
and Singh Uses bres emb edded in a uid matrix Have O K
TOY Tad W Taylor Haruka Okino and Takami Yamaguchi Three dimensional
analysis of left ventricular ejection using computational uid dynamics
Transactions of the ASME Journal of Biomechanical Engineering
Have
TP C Tu and Charles Samuel Peskin Stability and instability in the compu
tation of ows with moving immersed b oundaries A comparison of three
metho ds SIAM Journal Sci Stat Comput Get O K In Numerik Bib
Artikel kopieren
TR Dalin Tang and Samuel Rankin Numerical asymptotic solutions for p eri
staltic motion of nonlinear viscous ows with elastic free b oundaries SIAM
Journal on Scienti c Computing No Have
Tro UlrichTrost Analysis eines raeumlichen Model ls vom Herzmuskel PhD
thesis Inaugural Dissertation zur Erlangung der medizinischen Doktor
wuerde Freie Universitaet Berlin Have O K
Tru Cli ord Ambrose Truesdell The Elements of Rational Mechanics Springer
Verlag Berlin Get Not in BIP
Tru Cli ord Ambrose Truesdell A First Course in Rational Continuum Me
chanics Vol General Concepts Academic Press second edition
ISBN Have O K
TS J V Tyb erg and E R Smith On the interaction b etween the p ericardium
and the heart In Henk E D J ter Keurs and J V Tyb erg editors
Mechanics of the circulation chapter pages Martinus Nijho
Publishers Have
TSB Mona F Taher William P Santamore and Daniel K Bogen Ventricular
interaction is describ ed by three coupling co e cients American Journal on
Physiology Heart and Circulatory Physiology H H
Have O K
TT C Truesdell and R A Toupin The Classical Field The ories Handbuch
der Physik volume I I I Springer Verlag Berlin Have
TW W P Thurston and Je rey R Weeks The mathematics of three
dimensional manifolds Sci Am Quoted by Winfree
Get
TW Jo e F Thompson and Nigel P Weatherill Structured and unstructured
grid generation In Theo C Pilkington Bruce Loftis Jo e F Thompson
Savio L Y Wo o Thomas C Palmer and Thomas F Budinger editors
High Performance Computing in Biomedical Research CRC Press Inc
Have
Tyt Jan Tytgat How to isolate cardiac myo cytes Cardiovascular Research
Have O K
VAB W M Vogel C S Apstein L L Briggs L Gaasch and J Ahn Acute
alterations in left ventricular diastolic chamb er sti ness role of the erec
tile e ect of coronary arterial pressure and ow in normal and damaged
hearts Circulation Research Quoted by Huyghe p
Get
Val JaroslavValenta editor Clinical Aspects of Biomedicine Biomechanics
Elsevier Amsterdam ISBN Have O K
VBSD Jan Verburg Hans Bot Jan Strackee and Ben Delemaire Diastolic ow
in the left v entricle In Jan Strackee and Nicolaas Westerhof editors The
Physics of Heart and Circulation Institute of Physics Publishing Bristol
Have
VC Julian F V Vincent and John D Currey editors The Mechanical Prop
erties of Biological Materials Symp Soc Exp Biol Cambridge Uni
versity Press Cambridge Have O K
vCHBA DickH van Camp en Jacques Maria R J Huyghe Peter Hermanus Marie
Bovendeerd and Theo Arts Biomechanics of the heart muscle Eur J
Mech A Have O K
Ves a Andreas Vesalius Epitome Translatedfrom the Latin by L R Lind MIT
Press ISBN UB has several parts Have Xerox copy
O K
Ves b Andreas Vesalius The Il lustrations from the Works of Andreas Vesalius of
Brussels Dover ISBN Havecwc O K
Ves Carol Co ckerham Vesier The Role of Papil lary Muscle Mitral Valve Geom
etry in Systolic Anterior Motion of the Mitral Valve PhD thesis Georgia
Institute of Technology Get DA p B Order No DA
Order UMI
Vie R Vieussenius Nouvelles Decouvertes sur le Co eur Paris Get Not
in UB Not in print Not in Livres Del
Vin Julian F V Vincent Structural Biomaterials Princeton University
Press Princeton p pap erback ISBN
Have O K
Vin Julian F V Vincent editor Biomechanics Materials A Practical Ap
proach Oxford University Press Have O K ISBN
VL K S Valanis and R F Landel The strain energy function of a hyp erelastic
material in terms of the extension ratios J Appl Phys
Get ZC XEROX UB
Vog Steven Vogel Life in Moving Fluids The Physical Biology of Flow Prince
ton University Press Princeton Have O K ISBN Get
new edition sec ed ISBN Also have set of instruc
tional material
Vog Steven Vogel Life s Devices The Physical World of Animals and Plants
Princeton University Press Princeton Have O K ISB
Vog Steven Vogel Vital Circuits On Pumps Pipes and the Workings of Cir
culatory Systems Oxford University Press Oxford Have O K ISBN
Vog Steven Vogel Life in Moving Fluids Instructional Material to Accompany
Have O K
VY Carol Co ckerham Vesier and A P Yoganathan Validation of a metho d
to simulate blo o d ow in the heart Journal of Computational Physics
Get O K In Numerik Bibl Artikel kopieren
Wai Stephen A Wainwright Axis and Circumference The Cylindrical Shape
of Plants and Animals Cambridge MA Harvard University Press
ISBN Have O K
Wal LewisK Waldman On the Mechanical Coupling of the Heart to the Circu
lation PhD thesis University of California San Diego Have O K
Wal LewisK Waldman In vivo measurement of regional strains in myo cardium
In Frontiers in Biomechanics G W Schmidt Schoenbein et al editors
Springer Verlag New York Have pp
Wal Lewis K Waldman Mo deling of the coupling of the left ventricle to the
p eripheral circulation In Frank C P Yin editor Ventricular Vascular
Coupling pages Springer Verlag Have
Wal Lewis K Waldman Multidimensional measurement of regional strains in
the intact heart In Leon Glass Peter J Hunter and Andrew D McCul
lo ch editors TheoryofHeart pages Springer Verlag Have
O K
WBCG Stephen A Wainwright W D Biggs John D Currey and J M Gosline
Mechanical Design in Organisms Princeton University Press Quoted
bySchepartz p and p re cartilage tendon and wo ol Quoted by
Schmidt Nielson Bks on Demand Have ISBN
Order Kruep er for cwc
Web E H Web er Ueb er die Anwendung der Wellenlehre auf die Lehre vom
Kreislaufe des blutes und insb esondere auf die Pulslehre Berichte ue
ber die Verhand lungen der Koeniglich Saechsischen Gesel lschaft der Wis
senschaften zu Leipzig Mathematisch physische Klasse Get UB NZ
WEdB Nicolaas Westerhof Gijs Elzinga and G C Van den Bos In uence of cen
tral and p eripheral changes on the hydraulic input imp edance of the sys
temic arterial tree Medical and Biological Engineering
Have
Wee Je rey R Weeks editor The ShapeofSpace Dekker New York
ISBN X Quoted by Adams p Have O K
Wei Ewald R Weib el The Pathway for Oxygen Harvard University Press
Cambridge ISBN X Have O K
Wel Walter Welkowitz Engineering Hemodynamics Application to Cardiac As
sistant Devices New York University Biomedical Engineering Series New
York University Press nd edition From K B Chandran Cardio
vascular Biomechanics Ordered Kruep er
WELGC Sir W E Le Gros Clark The Tissues of the Body Clarendon Press
Oxford th edition Havecwc O K ISBN
WFC Lewis K Waldman Yuan cheng Fung and James W Covell Transmural
myo cardial deformation in the canine left ventricle Normal in vivo three
dimensional nite strains Circulation Research Have
O K
Whi Paul Dudley White Heart Disease Macmillan New York Get
Quoted byParmley and Talb ot p Not in MED Not in BIP In
PHYSIOLOGIE I
Whi E T Whittaker ATreatise on the Analytical Dynamics of Particles Rigid
Bodies Cambridge University Press Have O K
Wig Carl J Wiggers The dominant control of mammalian ventricular dynamics
by initial length Rev Soc arg de Biol Suppl Get Not
available UB
Wig Carl J Wiggers Circulatory Dynamics Grune and Stratton New York
Quoted byParmley and Talb ot p Not in BIP Not in
MED Have Xerox copy
Wil T J Willmore An Introduction to Di erential Topology Clarendon Press
Oxford Havecwc O K
Win J B Winslow Observations sur les bres du co eur et ses valvules Histoire
et Memoires de l Acad emie Royale des Sciences Paris Quoted by
Greenbaum Ho Gibson Becker and Anderson Get
Win Arthur T Winfree Organizing centers for chemical waves in two and three
dimensions In R Field and M Burger editors Oscil lations and Travel
ling Waves in Chemical Systems pages Wiley NewYork
Quoted by Winfree Get
Win Arthur T Winfree When Time Breaks Down The Three Dimensional
Dynamics of Electrochemical Waves and Cardiac Arrhythmias Princeton
University Press Princeton Havecwc ISBN
Win Arthur T Winfree The Geometry of Biological Time Springer Verlag
New York Have O K ISBN
Win Donald F Winter On the Stem curve of a tall palm in a strong wind
SIAM Review Have O K
Win Arthur T Winfree Puzzles ab out excitable media and sudden death In
Simon A Levin editor Frontiers in Mathematical Biolo gy pages
Springer Verlag Have O K
Wit K Witzig Ueber erzwungene Wel lenbewegun ge n zaher inkompressibler
Fluessigkeiten in elastischen Rohren PhD thesis Universitaet Bern
Get
WJ Tom Wickham Jones MathematicaGraphics Techniques and Applications
Springer Verlag Have O K ISBN
WK Fredrick A Willius and Thomas E Keys editors Classics of Cardiology
former title Cardiac Classics volume Krieger Have O K
WKB Howard Weisb erg Louis N Katz and Eugene Boyd In uence of coro
nary owuponoxygen consumption and cardiac p erformance Circulation
Research Have O K
WM Lewis K Waldman and Andrew D McCullo ch Nonhomogeneous ventricu
lar wall strain Analysis of errors and accuracy Transactions of the ASME
Journal of Biomechanical Engineering Have
WNC Lewis K Waldman D Nosan and F Villarreal J W Cowell Relation
between transmural deformation and lo cal myo b er direction in canine left
ventricle Circulation Research Get Quoted byGuc
cione McCullo ch and Waldman
Wol Julius Wol Das Gesetz der Transformation der Knochen Hirschwald
Berlin Reprinted Schattauer Verlag Stuttgart ISBN
An english translation is also available yHave O K
Wom J R Womersley The mathematical analysis of the arterial circulation in a
state of oscillatory motion Wright Air Development Center
Get Micro che
Won Alan Y K Wong Some prop osals in cardiac muscle mechanics and ener
getics Bul letin of Mathematical Biology Get ZD
O K XEROX UB
Wo o Rob ert H Wo o ds A few applications of a physical theorem to membranes
in the human b o dy in a state of tension J Anat Physiol
Have O K
WR Alan Y K Wong and P M Rautaharju Stress distribution within the left
ventricular wall approximated as a thick ellipsoidal shell American Heart
Journal Have ZQu O K
WS Arthur T Winfree and S H Strogatz Singular laments organize chemical
waves in three dimensions WaveTaxonomy Physica D
Quoted by Winfree PHYS Get
WTB Julian J Wu T C T Ting and David M Barnett Modern Theory of
Anisotropic Elasticity and Applications So ciety for Industrial and Applied
Mathematics Philadelphia Have ISBN
WW E H Web er and W Web er Wel lenlehre auf experimente begruendet o der
Ueber die Wel len tropfbarer uessigkeiten mit Anwendung auf die Schal l
und lichtwel l Fleischer Get
Yam Hiroshi Yamada Strength of Biological Materials Williams Wilkins
Baltimore Have Xerox copy O K ISBN
Yan Ming Yang Application of Nonlinear Porosity to the Analysis of Cardiac
Muscle PhD thesis UniversityofRochester DA Diss
Abstr No May Order UMI
YGG A L Yettram B S Grewal and D G Gibson Mo deling the left ventricle
for determination of the elasticity of the myo cardiuml J Eng Med
Get AMR A
Yin Frank C P Yin Applications of the nite element metho d to ventricular
mechanics CRC Critical Reviews in Biomedical Engineering
Have O K
Yin Frank C P Yin editor Ventricular vascular Coupling Clinical Physio
logical and Engineering Aspects Springer Verlag New York Get
WG Out of Print MED
YLK Ajit P Yoganathan Jack D Lemmon Jr Young H Kim Peter G Walker
Rob ert A Levine and Carol C Vesier A computational study of a thin
walled three dimensional left ventricle during early systole Transactions
of the ASME Journal of Biomechanical Engineering
Have
You Thomas Young Hydraulic investigations subservienttoanintended cro o
nian lecture on the motion of the blo o d Phil Trans Roy Soc London
Get Order UB
You Thomas Young On the function of the heart and arteries Philosophical
Transactions Royal So ciety London Get In UB ZC
YSBL Hitoshi Yaku Bryan K Slinker Stephen P Bell and Martin M LeWinter
E ects of free wall ischemia and bundle branch blo ck on systolic ventricular
interaction in dog hearts American Journal on Physiology Heart and
Circulatory Physiology H H Have O K
YSCZ Frank C P Yin Rob ert K Strumpf Paul H Chew and Scott L Zeger
Quanti cation of the mechanical prop erties of noncontracting canine my
o cardium under simultaneous biaxial loading Journ Biomech
Have O K
YSF Edward L Yellin Edmund H Sonnenblick and Rob ert W M Frater Dy
namic determinants of left ventricular lling An overview In Jan Baan
Alexander C Arntzenius and Edward L Yellin editors Cardiac Dynam
ics pages Martinus Nijho Publishers Quoted by Robinson
Factor and Sonnenblick Have O K
YTC Ming Yang Larry A Tab er and E B Clark A nonlinear p oro elastic mo del
for the trab ecular embryonic heart T ransactions of the ASME Journal of
Biomechanical Engineering Have
YVG A L Yettram C A Vinson and D G Gibson E ect of myo cardial
b er architecture on the b ehaviour of the human left ventricle in diastole
Transactions of the ASME Journal of Biomechanical Engineering
Quoted byPerl Horowitz and Sideman Get
ZLL Conrad M Zapanta Edward G Liszka Jr Theo dore C Lamson David R
Stinebring Steve Deutsch David B Geselowitz and John M Tarb ell A
metho d for real time in vitro observation of cavitation on prosthetic heart
valves Transactions of the ASME Journal of Biomechanical Engineering
Have
ZT a Olgierd Cecil Zienkiewicz and R L Taylor The Finite Element Method
Vol Basic Formulation and Linear Problems McGraw Hill London th
edition Have O K
ZT b Olgierd Cecil Zienkiewicz and R L Taylor The Finite Element Method
Vol Solid and Fluid Mechanics Dynamics and Non Linearity McGraw
Hill London th edition Have O K
ZZZ ZZZ Coronary circulation Transactions of the ASME Journal of Biome
chanical Engineering Nov Get AMR A
ZZZ ZZZ In uence of endo cardial epicardial crossover of muscle b ers on left
ventricular wall mechanics Journ Biomech Get
A
Index
endo cardium
afterload
epicardium
allometric equations
erectile prop erties of the coronary microvas
Anrep e ect
culature
aortic regurgitation
external work
aortic stenosis
aortic valve
Feng e ect
applied mechanics
Fenn e ect
ariti cial valves
brillation
arrhythmias
brotendinous ring
arterioventricular valves
eld equations
atrioventricular valves
nite elements
uid bre mo dels
ballistic cardiography
foliation
basal metab olic rate
force frequency relationship
b o dy surface area
force interval relationship
Bowditch
formula of DuBois and DuBois
Buckingham s Theorem
fractal analysis
capillaries
Frank Starling Law
cardiac arrhythmias
garden hose e ect cardiac energetics
geo desics
cardiac tamp onade
geometric measure theory
cardiovascular muscles
Gregg e ect
chaos
growth chordae tendineae
collagen matrix
heart as a mechanical pump
computer graphics
Heart Murmurs
constitutivelaws
Heart Sounds
continuum mechanics
heart sounds
coronary artery
Ho dgkin Huxley equations
coronary circulation
hydraulic mo del
coronary p erfusion
hydrostatic skeleton
coronary p erfusion pressure
coronary sinus
immersed interface problems
immersed b oundary metho d
descending limb
Incomp etence
diastole
inferior vena cava
dicrotic wave
intercalated disk
Dimensional analysis
interval force relationship
dynamic similarity
interval strength relationship
EDP
intramyo cardial pressure
EDV
isometric loading
ejection fraction
elastic similarity
elasticity
embryonic heart
end diastolic pressure
end diastolic volume
isotonic
rightventricle
Rightventricular wall motion
jet
sarcomeres
kinesiology
semilunar valves
knots
septum
Korotko sounds
ts shun
Single unit smo oth muscle
Law of Laplace
skeletal muscles
left atrium
smo oth muscle
left ventricle
Starling curve
Left ventricular wall motion
Starling s Law of the Heart
Lie Group
Stenosis
stenosis
mechanics
striated muscle
mitral valve
strokevolume
Mo ens Korteweg formula
sup erior vena cava
mouse to elephant curve
symmorphosis
Multi unit smo oth muscle
synctial smo oth muscle
Murray s formula
systemic circulation
muscle
systole
muscle mechanics
muscular hydrostat
thermo dynamics
myo cardial oxygen consumption
trab eculae carnae
myo cyte
tricuspid valve
myo cytes
Turbulent ow
new elastic b o dy
valve vibration
numerical metho ds
vaso dilators
venous system
optimal design
ventricular function curve
orthostasis
ventricular interaction
papillary muscles
visceral smo oth muscle
pattern formation
volume pressure work
p erfusion pressure
vortex
p ericardectomy
whirl ow
p ericardial e usion
Wol s Law
p ericarditis acute
Womersley numb er
p ericarditis constrictive
Womersley parameter
p ericardium
phyllotaxis
Poiseuille ow
p oro elastic mo dels
pre load
pulmonary circulation
pulmonary valve
regurgitation
remo delling
Reynolds numb er
right atrium