Mathematical Biology 1: An Introduction

576 Pages·2007·4.556 MB·English

by  James D. Murray

Mathematical Biology:
I. An Introduction,
Third Edition
J.D. Murray
Springer
Interdisciplinary Applied Mathematics
Volume 17
Editors
S.S.Antman J.E.Marsden
L.Sirovich S.Wiggins
GeophysicsandPlanetarySciences
MathematicalBiology
L.Glass,J.D.Murray
MechanicsandMaterials
R.V.Kohn
SystemsandControl
S.S.Sastry,P.S.Krishnaprasad
Problems in engineering, computational science, and the physical and biological sci-
ences are using increasingly sophisticated mathematical techniques. Thus, the bridge
betweenthemathematicalsciencesandotherdisciplinesisheavilytraveled.Thecorre-
spondinglyincreaseddialogbetweenthedisciplineshasledtotheestablishmentofthe
series:InterdisciplinaryAppliedMathematics.
The purpose of this series is to meet the current and future needs for the interaction
betweenvariousscienceandtechnologyareasontheonehandandmathematicsonthe
other.Thisisdone,firstly,byencouragingthewaysthatmathematicsmaybeappliedin
traditionalareas,aswellaspointtowardsnewandinnovativeareasofapplications;and
secondly,byencouragingotherscientificdisciplinestoengageinadialogwithmathe-
maticiansoutliningtheirproblemstobothaccessnewmethodsandsuggestinnovative
developmentswithinmathematicsitself.
Theserieswillconsistofmonographsandhigh-leveltextsfromresearchersworkingon
theinterplaybetweenmathematicsandotherfieldsofscienceandtechnology.
Interdisciplinary Applied Mathematics
Volumespublishedarelistedattheendofthebook.
Springer
NewYork
Berlin
Heidelberg
Barcelona
HongKong
London
Milan
Paris
Singapore
Tokyo
J.D. Murray
Mathematical Biology
I. An Introduction
Third Edition
With189Illustrations
1
Springer
J.D.Murray,FRS
EmeritusProfessor
UniversityofOxfordand
UniversityofWashington
Box352420
DepartmentofAppliedMathematics
Seattle,WA98195-2420USA
Editors
S.S.Antman J.E.Marsden
DepartmentofMathematicsand ControlandDynamicalSystems
InstituteforPhysicalScience MailCode107-81
andTechnology CaliforniaInstituteofTechnology
UniversityofMaryland Pasadena,CA91125
CollegePark,MD20742 USA
USA [email protected]
[email protected]
L.Sirovich S.Wiggins
DivisionofAppliedMathematics ControlandDynamicalSystems
BrownUniversity MailCode107-81
Providence,RI02912 CaliforniaInstituteofTechnology
USA Pasadena,CA91125
[email protected] USA
Coverillustration:(cid:1)c 2001Superstock.
MathematicsSubjectClassification(2000):92B05,92-01,92C05,92D30,34Cxx
LibraryofCongressCataloging-in-PublicationData
Murray,J.D.(JamesDickson)
Mathematicalbiology.I.Anintroduction/J.D.Murray.—3rded.
p. cm.—(Interdisciplinaryappliedmathematics)
Rev.ed.of:Mathematicalbiology.2nded.c1993.
Includesbibliographicalreferences(p.).
ISBN0-387-95223-3(alk.paper)
1. Biology—Mathematicalmodels. I. Murray,J.D.(JamesDickson)Mathematical
biology. II. Title. III. Series.
QH323.5.M882001
(cid:2) (cid:2)
570.15118—dc21 2001020448
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(cid:1)c 2002J.D.Murray,(cid:1)c 1989,1993Springer-VerlagBerlinHeidelberg.
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Tomy wifeSheila,whom Imarriedmore
than fortyyearsago and livedhappily ever
after,and toour childrenMark and Sarah
... que see´lfueradesu consejoaltiempode la
generalcriacio´ndelmundo, ide loque ene´lse
encierra,i sehalla´ racone´l,se huvieranproducido
iformado algunas cosas mejor que fueranhechas,
iotras ni sehicieran,use enmendaran icorrigieran.
—AlphonsoX(AlphonsotheWise),1221–1284
KingofCastileandLeon(attributed)
Ifthe Lord Almightyhad consultedme
beforeembarking oncreationIshould
haverecommended something simpler.
Preface to the Third Edition
Inthethirteenyearssincethefirsteditionofthisbookappearedthegrowthofmathe-
maticalbiologyandthediversityofapplicationshasbeenastonishing.Itsestablishment
as a distinct discipline is no longer in question. One pragmatic indication is the in-
creasingnumberofadvertisedpositionsinacademia,medicineandindustryaroundthe
world;anotheristheburgeoningmembershipofsocieties.Peopleworkinginthefield
now number in the thousands. Mathematical modelling is being applied in every ma-
jordisciplineinthebiomedicalsciences.Averydifferentapplication,andsurprisingly
successful, is in psychology such as modelling various human interactions, escalation
todaterapeandpredictingdivorce.
The field has become so large that, inevitably, specialised areas have developed
whichare,ineffect,separatedisciplinessuchasbiofluidmechanics,theoreticalecology
andsoon.ItisrelevantthereforetoaskwhyIfelttherewasacaseforaneweditionof
abookcalledsimplyMathematicalBiology.Itisunrealistictothinkthatasinglebook
couldcoverevenasignificantpartofeachsubdisciplineandthisneweditioncertainly
does not even try to do this. I feel, however, that there is still justification for a book
which can demonstrate to the uninitiated some of the exciting problems that arise in
biology and give some indication of the wide spectrum of topics that modelling can
address.
Inmanyareasthebasicsaremoreorlessunchangedbutthedevelopmentsduring
thepastthirteenyearshavemadeitimpossibletogiveascomprehensiveapictureofthe
currentapproachesinandthestateofthefieldaswaspossibleinthelate1980s.Even
thenimportantareaswerenotincludedsuchasstochasticmodelling,biofluidmechanics
andothers.Accordinglyinthisneweditiononlysomeofthebasicmodellingconcepts
arediscussed—suchasinecologyandtoalesserextentepidemiology—butreferences
areprovidedforfurtherreading.Inotherareasrecentadvancesarediscussedtogether
with some new applications of modelling such as in marital interaction (Volume I),
growthofcancertumours(VolumeII),temperature-dependentsexdetermination(Vol-
umeI)and wolfterritoriality(VolumeII).Therehavebeen manynewand fascinating
developments that I would have liked to include but practical space limitations made
it impossible and necessitated difficult choices. I have tried to give some idea of the
diversityofnewdevelopmentsbutthechoiceisinevitablyprejudiced.
Astogeneralapproach,ifanythingitisevenmorepracticalinthatmoreemphasis
is given to the close connection many of the models have with experiment, clinical
data and in estimating real parameter values. In several of the chapters it is not yet
viii PrefacetotheThirdEdition
possible to relate the mathematical models to specific experiments or even biological
entities. Nevertheless such an approach has spawned numerous experiments based as
muchonthemodellingapproachasontheactualmechanismstudied.Someofthemore
mathematical parts in which the biological connection was less immediate have been
excisedwhileothersthathavebeenkepthaveamathematicalandtechnicalpedagogical
aim but all within the context of their application to biomedical problems. I feel even
morestronglyaboutthephilosophyofmathematicalmodellingespousedintheoriginal
prefaceasregardswhatconstitutesgoodmathematicalbiology.Oneofthemostexciting
aspectsregardingthenewchaptershasbeentheirgenuineinterdisciplinarycollaborative
character. Mathematical or theoretical biology is unquestionably an interdisciplinary
scienceparexcellence.
The unifying aim of theoretical modelling and experimental investigation in the
biomedical sciences is the elucidation of the underlying biological processes that re-
sult in a particular observed phenomenon, whether it is pattern formation in develop-
ment, the dynamics of interacting populations in epidemiology, neuronal connectivity
and information processing, the growth of tumours, marital interaction and so on. I
must stress, however, that mathematical descriptions of biological phenomena are not
biologicalexplanations.The principaluse ofany theoryisinitspredictionsand,even
thoughdifferentmodelsmightbeabletocreatesimilarspatiotemporalbehaviours,they
aremainlydistinguishedbythedifferentexperimentstheysuggestand,ofcourse,how
closelytheyrelatetotherealbiology.Therearenumerousexamplesinthebook.
Why use mathematics to study something as intrinsically complicated and ill un-
derstoodasdevelopment,angiogenesis,woundhealing,interactingpopulationdynam-
ics, regulatory networks, marital interaction and so on? We suggest that mathematics,
rathertheoreticalmodelling,mustbeusedifweeverhopetogenuinelyandrealistically
convertanunderstandingoftheunderlyingmechanismsintoapredictivescience.Math-
ematicsisrequiredtobridgethegapbetweenthelevelonwhichmostofourknowledge
isaccumulating(indevelopmentalbiologyitiscellularandbelow)andthemacroscopic
levelofthepatternswesee.Inwoundhealingandscarformation,forexample,amathe-
maticalapproachletsusexplorethelogicoftherepairprocess.Evenifthemechanisms
werewellunderstood(andtheycertainlyarefarfromitatthisstage)mathematicswould
be requiredto explorethe consequencesof manipulatingthe variousparametersasso-
ciated with any particular scenario. In the case of such things as wound healing and
cancergrowth—andnowinangiogensesiswithitsrelationtopossiblecancertherapy—
thenumberofoptionsthatarefastbecomingavailabletowoundandcancermanagers
will become overwhelming unless we can find a way to simulate particular treatment
protocolsbeforeapplyingtheminpractice.Thelatterhasbeenalreadyofuseinunder-
standingtheefficacyofvarioustreatmentscenarioswithbraintumours(glioblastomas)
andnewtwostepregimesforskincancer.
The aim in allthese applicationsisnot toderivea mathematicalmodelthat takes
intoaccounteverysingleprocessbecause,evenifthiswerepossible,theresultingmodel
wouldyieldlittleornoinsightonthecrucialinteractionswithinthesystem.Ratherthe
goal is to develop models which capture the essence of various interactions allowing
their outcome to be more fully understood. As more data emerge from the biological
system,themodelsbecomemoresophisticatedandthemathematicsincreasinglychal-
lenging.
PrefacetotheThirdEdition ix
In development (by way of example) it is true that we are a long way from be-
ing able to reliably simulate actual biological development, in spite of the plethora of
models and theory that abound. Key processes are generally still poorly understood.
Despitethese limitations,Ifeel thatexploringthe logicofpatternformationisworth-
while, or rather essential, even in our present state of knowledge. It allows us to take
a hypothetical mechanism and examine its consequences in the form of a mathemat-
ical model, make predictions and suggest experiments that would verify or invalidate
the model; even the latter casts light on the biology. The very process of constructing
a mathematical model can be useful in its own right. Not only must we commit to a
particularmechanism, but we are also forcedto considerwhat is trulyessential to the
process,thecentralplayers(variables)andmechanismsbywhichtheyevolve.Weare
thusinvolvedinconstructingframeworksonwhichwecanhangourunderstanding.The
modelequations, themathematicalanalysis and thenumericalsimulationsthat follow
serve to reveal quantitatively as well as qualitatively the consequences of that logical
structure.
This new edition is published in two volumes. Volume I is an introduction to the
field; the mathematics mainly involves ordinary differential equations but with some
basicpartialdifferentialequationmodelsandissuitableforundergraduateandgraduate
courses at different levels. Volume II requires more knowledge of partial differential
equationsandismoresuitableforgraduatecoursesandreference.
I would like to acknowledge the encouragement and generosity of the many peo-
plewho have writtento me(includinga prisoninmate in NewEngland) since the ap-
pearance of the first edition of this book, many of whom took the trouble to send me
details of errors, misprints, suggestions for extending some of the models, suggesting
collaborationsandsoon.Theirinputhasresultedinmanysuccessfulinterdisciplinary
research projects several of which are discussed in this new edition. I would like to
thankmycolleaguesMarkKotandHongQian,manyofmyformerstudents,inpartic-
ular Patricia Burgess, Julian Cook, Trace´ Jackson, Mark Lewis, Philip Maini, Patrick
Nelson,JonathanSherratt,KristinSwansonandRebeccaTysonfortheiradviceorcare-
ful reading of parts of the manuscript. I would also like to thank my former secretary
ErikHinkleforthecare,thoughtfulnessanddedicationwithwhichheputmuchofthe
manuscript into LATEX and his general help in tracking down numerousobscure refer-
encesandmaterial.
IamverygratefultoProfessorJohnGottmanofthePsychologyDepartmentatthe
UniversityofWashington,aworldleaderintheclinicalstudyofmaritalandfamilyin-
teractions,withwhomIhave hadthegoodfortunetocollaboratefornearlytenyears.
Withouthisinfectiousenthusiasm,strongbeliefintheuseofmathematicalmodelling,
perseveranceinthefaceofmyinitialscepticismandhispracticalinsightintohumanin-
teractionsIwouldneverhavebecomeinvolvedindevelopingwithhimageneraltheory
ofmaritalinteraction.IwouldalsoliketoacknowledgemydebttoProfessorEllworth
C.Alvord,Jr.,HeadofNeuropathologyintheUniversityofWashingtonwithwhomI
havecollaboratedforthepastsevenyearsonthemodellingofthegrowthandcontrolof
braintumours.Astomygeneral,andIhopepractical,approachtomodellingIammost
indebtedtoProfessorGeorgeF.CarrierwhohadthemajorinfluenceonmewhenIwent
toHarvardonfirstcomingtotheU.S.A.in1956.Hisastonishinginsightandabilityto
extractthekeyelementsfromacomplexproblemandincorporatethemintoarealistic