Mathematical Modelling and Biomechanics of the Brain

160 Pages·2019·2.412 MB·English

by  Corina Drapaca, Siv Sivaloganathan

Fields Institute Monographs 37
The Fields Institute for Research in Mathematical Sciences
Corina Drapaca
Siv Sivaloganathan
Mathematical
Modelling and
Biomechanics
of the Brain
Fields Institute Monographs
VOLUME 37
The Fields Institute for Research in Mathematical Sciences
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Corina Drapaca • Siv Sivaloganathan
Mathematical Modelling
and Biomechanics
of the Brain
123
CorinaDrapaca SivSivaloganathan
CollegeofEngineering DepartmentofAppliedMathematics
PennStateUniversity UniversityofWaterloo
UniversityPark,PA,USA Waterloo,ON,Canada
ISSN1069-5273 ISSN2194-3079 (electronic)
FieldsInstituteMonographs
ISBN978-1-4939-9809-8 ISBN978-1-4939-9810-4 (eBook)
https://doi.org/10.1007/978-1-4939-9810-4
Mathematics Subject Classification (2010): 92C10, 74L15, 74Fxx, 78A70, 92C50, 76Z05, 92C05,
35Q35,35Q72
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Preface
The brain is one of the most important organs in the human body and serves as
the control center for the nervous system supporting all the required functions of
the body parts and other physiological sub-systems. In a typical human being, the
cerebral cortex (the largest part of the brain) is estimated to contain 15–33 billion
neurons, each connected by synapses to several thousands of other neurons. Inter-
neuronalcommunicationisfacilitatedbymeansoflongprotoplasmicfiberscalled
axons,whichcarrytrainsofsignalpulsesoractionpotentialstodistantpartsofthe
brainorbodytargetingspecificrecipientcells.
Adevelopmentofafundamentalunderstandingoftheformandfunctionofthe
brainissomethingthathaspreoccupiedscientistsandthinkersformillennia.From
amorepragmaticpointofview,itisalsoquiteclearlynecessaryinordertoprovide
accurate diagnostics and optimal therapies for neurological disorders and proper
protectioninjobsandsports,whichexposeindividualstohighrisksofbraininjuries,
andtodeveloppatternstoenhancelearningabilitiesforthosewhosecognitivebrain
functionhasbeenimpaired.Theenormoustechnologicaladvanceswitnessed,over
the last few decades, in numerous areas of human endeavor have contributed to
unprecedentedprogressinbrainscience,asaresultofmoreaccuratemeasurements
and the ability to carry out very large-scale computer simulations quickly and
efficiently.Despiteallthisdramaticprogress,thereiscurrentlyarealneedtobridge
andsynthesizebraininformationavailablefromdisparate,disjointdisciplinessuch
as medicine, neuroscience, neurobiology, biomechanical engineering, biophysics,
biochemistry, mathematics, and computer science. The main challenge for such
multidisciplinaryresearcheffortsistoestablishacommonlanguagethatfacilitates
fruitful and efficient communication among researchers with different expertise.
In particular, it is our contention that mathematical modeling provides the pivotal,
unifyingframeworkforbrainsciencesinceitiscapableofutilizingthebio-chemo-
physical phenomena that govern brain mechanisms to predict brain responses to
variousstimuliandcanalsoproviderationalguidanceforthedesignofexperimental
toolsandprotocolsforstudyingthebrain.Therefore,theaimofthismonographisto
provideanaccessibleyetrigorouspresentationofsomefundamentalmathematical
conceptsusedinmodelingbrainmechanics.Inthiscontext,wepresentanoverview
vii
viii Preface
ofthemechanicalmodelsforhydrocephalus,traumaticbraininjuries,tumorgrowth,
and aneurisms. The book is intended as a brief introduction to both theoreticians
and experimentalists interested in brain mechanics, with directions and guidance
forfurtherreading,forthosewhowishtopursueparticulartopicsingreaterdepth.
It can also be used as a complementary textbook in a graduate level course for
neuroscientistsandneuroengineers.
Toronto,ON,Canada CorinaDrapaca
Toronto,ON,Canada SivSivaloganathan
June2018
Contents
1 Introduction .................................................................. 1
2 BriefReviewofContinuumMechanicsTheories......................... 5
2.1 Kinematics............................................................... 5
2.2 ConservationLaws...................................................... 10
2.2.1 ConservationofMass........................................... 10
2.2.2 ConservationofLinearMomentum............................ 11
2.2.3 ConservationofAngularMomentum.......................... 13
2.2.4 ConservationofMechanicalEnergy........................... 13
2.3 ConstitutiveLaws....................................................... 14
2.3.1 Noll’sPrinciples ................................................ 15
2.3.2 SimpleMaterials................................................ 15
2.3.3 Non-agingMaterials............................................ 16
2.3.4 InvarianceUnderRotationoftheReferenceFrame........... 17
2.3.5 MaterialSymmetries............................................ 17
2.3.6 InternalMaterialConstraints................................... 19
2.3.7 SecondLawofThermodynamics.............................. 20
2.3.8 ExamplesofConstitutiveLaws................................ 21
2.4 Non-localTheories...................................................... 29
References..................................................................... 34
3 MechanicsofHydrocephalus............................................... 39
3.1 BackgroundandSignificance........................................... 39
3.2 MathematicalModels................................................... 42
3.2.1 Pressure-VolumeModels....................................... 42
3.2.2 ConsolidationModels .......................................... 48
References..................................................................... 69
4 ModelingTraumaticBrainInjuries,Aneurysms,andStrokes ......... 75
4.1 BackgroundandSignificance........................................... 75
4.2 MathematicalModels................................................... 79
4.2.1 ModelsofTraumaticBrainInjuries............................ 79
ix
x Contents
4.2.2 ModelsofAneurysms .......................................... 94
4.2.3 ModelsofStrokes............................................... 106
References..................................................................... 117
5 ModelsofTumorGrowth................................................... 127
5.1 BackgroundandSignificance........................................... 127
5.2 MathematicalModels................................................... 130
5.2.1 MicroscopicModels............................................ 136
5.2.2 MultiscaleModels.............................................. 139
5.2.3 MacroscopicModels............................................ 142
References..................................................................... 147
6 ConcludingRemarks ........................................................ 153
Chapter 1
Introduction
Mathematicsisplayinganevermoreimportantroleinthebiomedicalsciencesand
hasbeenthecatalystforablurringofboundariesbetweenscientificdisciplinesinthe
naturalsciencesandaresurgenceofinterestinthemodernaswellasclassicalmeth-
ods of applied mathematics. The development of new disciplines (and new ideas)
is a natural consequence of this highly synergistic interaction of the biomedical
andmathematicalsciences,spurredonbydramaticandfar-reachingdevelopments
on the research frontiers of applied mathematics, as computational disciplines,
dynamical systems, stochastic analysis, chaos (amongst others), reinvigorate and
reinforcethetraditionaldisciplinesofappliedmathematics.
Inthelastfourtofivedecades,thegeneralfieldofBiomechanicshasexpanded
dramatically, and advances have been made on many fronts. Biomechanics is the
applications of mechanics to biology and seeks to understand the mechanics of
living systems. In this monograph, we focus on the biomechanics of the central
nervous system. Thus the purpose of this monograph is to present an overview
of mathematical models of brain conditions and diseases as encountered in the
BiomedicalSciences,andalsotoencourageandstimulateclosersynergyandinter-
action between the Biomedical and Mathematical Sciences. Brain biomechanics
is the study of the brain using mathematical methods and models to help predict
andexplaintheresponseofbraintissueundervariouscircumstances.Theunifying
aim of mathematical modelling and experimental studies in brain biomechanics is
the elucidation of the underlying biological mechanisms and processes that lead
toparticularobservedphenomena(e.g.braintissuecompressioninhydrocephalus,
brain tumours etc.). It is (of course) clear that mathematical descriptions of
biological phenomena are not biological explanations. Apart from its explanatory
power,thetruetestofanymathematicaldescriptionortheoryisinitspredictions.
But why use mathematics to study anything as intrinsically complicated as
the human brain? Our contention is that mathematics, in particular mathematical
modelling, must be used if we are ever to stand a chance of converting an under-
standingofunderlyingmechanismsintoagenuinelyrealisticandpredictivescience.
©SpringerScience+BusinessMedia,LLC,partofSpringerNature2019 1
C.Drapaca,S.Sivaloganathan,MathematicalModellingandBiomechanics
oftheBrain,FieldsInstituteMonographs37,
https://doi.org/10.1007/978-1-4939-9810-4_1