Complex Systems in Biomedicine

298 Pages·2006·6.58 MB·English

by  M. Bertero, M. Piana (auth.), Alfio Quarteroni, Luca Formaggia, Alessandro Veneziani (eds.)

AlfioQuarteroni
LucaFormaggia
AlessandroVeneziani
ComplexSystemsinBiomedicine
A. Quarteroni (Editor)
L. Formaggia (Editor)
A. Veneziani (Editor)
Complex Systems
in Biomedicine
With88Figures
AlfioQuarteroni
MOX,DipartimentodiMatematica
PolitecnicodiMilano
Milan,Italy
and
CMCS-IACS
E´colePolytechniqueFe´de´raledeLausanne
Lausanne,Switzerland
[email protected]
LucaFormaggia
MOX,DipartimentodiMatematica
PolitecnicodiMilano
Milan,Italy
[email protected]
AlessandroVeneziani
MOX,DipartimentodiMatematica
PolitecnicodiMilano
Milan,Italy
[email protected]
Thepictureonthecovershowsanintegrationofasynapsis(bottomright),thecomputational
domainforapulmonaryarterybifurcation(topright),thehumanheart(topleft),andthewall
shearstressinapulmonaryartery(bottomleft).
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ISBN-13 978-88-470-0394-1 SpringerMilanBerlinHeidelbergNewYork
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Preface
Mathematicalmodelingofhumanphysiopathologyisatremendouslyambitioustask.
Itencompassesthemodelingofmostdiversecompartmentssuchasthecardiovascu-
lar,respiratory,skeletalandnervoussystems,aswellasthemechanicalandbiochem-
icalinteractionbetweenbloodflowandarterialwalls,andelectrocardiacprocesses
andelectricconductioninbiologicaltissues.Mathematicalmodelscanbesetupto
simulatebothvasculogenesis(theaggregationandorganizationofendothelialcells
dispersed in a given environment) and angiogenesis (the formation of new vessels
sprouting from an existing vessel) that are relevant to the formation of vascular
networks,andinparticulartothedescriptionoftumorgrowth.
Theintegrationofmodelsaimedatsimulatingthecooperationandinterrelation
of different systems is an even more difficult task. It calls for the setting up of,
for instance, interaction models for the integrated cardio-vascular system and the
interplaybetweenthecentralcirculationandperipheralcompartments,modelsfor
the mid-to-long range cardiovascular adjustments to pathological conditions (e.g.,
to account for surgical interventions, congenital malformations, or tumor growth),
modelsforintegrationamongcirculation,tissueperfusion,biochemicalandthermal
regulation,modelsforparameteridentificationandsensitivityanalysistoparameter
changesordatauncertainty–andmanyothers.
The heart is a complex system in itself, where electrical phenomena are func-
tionallyrelatedtowalldeformation.Initsturn,electricalactivityisrelatedtoheart
physiology.Itinvolvesnonlinearreaction-diffusionprocessesandprovidestheac-
tivation stimulus to heart dynamics and eventually the blood ventricular flow that
drives the haemodynamics of the whole circulatory system. In fact, the influence
is reciprocal, since the circulatory system in turn affects heart dynamics and may
induceanoverloaddependingupontheindividualphysiopathologies(forinstance,
thepresenceofastenoticarteryoravascularprosthesis).
Virtuallyallthefieldsofmathematicshavearoletoplayinthiscontext.Geometry
and approximation theory provide the tools for handling clinical data acquired by
tomographyormagneticresonance,identifyingmeaningfulgeometricalpatternsand
producingthree-dimensionalgeometricmodelsstemmingfromtheoriginalpatient’s
data.Mathematicalanalysis,fluidandsoliddynamics,stochasticanalysisareused
to set up the differential models and predict uncertainty. Numerical analysis and
high performance computing are needed to solve the complex differential models
numerically.Finally,methodsfromstochasticandstatisticalanalysisareexploited
forthemodelingandinterpretationofspace-timepatterns.
VI Preface
Indeed, the complexity of the problems at hand often stimulates the use of in-
novative mathematical techniques that are able, for instance, to capture accurately
thoseprocessesthatoccuratmultiplescalesintimeandspace(suchascellularand
systemiceffects),andthataregovernedbyheterogeneousphysicallaws.
InthisbookwehavecollectedthecontributionsofseveralItalianresearchgroups
thataresuccessfullyworkinginthisfascinatingandchallengingfield.Eachchapter
dealswithaspecificsubfield,withtheaimofprovidinganoverviewofthesubject
andanaccountofthemostrecentresearchresults.
Chapter1addressesaclassofinversemathematicalproblemsinbiomedicalimag-
ing.Imagingtechniques(suchastomographyormagneticresonance)areapowerful
tool for the analysis of human organs and biological systems. They invariably re-
quireamathematicalmodelfortheacquisitionprocessandnumericalmethodsfor
thesolutionofthecorrespondinginverseproblemswhichrelatetheobservationto
theunknownobject.
Chapter 2 addresses those biochemical processes which are composed of two
phases, generation (nucleation, branching, etc.) and subsequent growth of spatial
structures(cells,vesselnetworks,etc),whichdisplay,ingeneral,astochasticnature
both in time and space.These structures induce a random tessellation as in tumor
growthandtumor-inducedangiogenesis.Predictivemathematicalmodelswhichare
capableofproducingquantitativemorphologicalfeaturesofdevelopingtumorand
bloodvesselsdemandaquantitativedescriptionofthespatialstructureofthetessel-
lationthatisgivenintermsofthemeandensitiesofinterfaces.
A preliminary stochastic geometric model is proposed to relate the geometric
probabilitydistributiontothekineticparametersofbirthandgrowth.Foritsnumeri-
calassessment,methodsofstatisticalanalysisareproposedfortheestimationofthe
geometricdensitiesthatcharacterizethemorphologyofarealsystem.
Chapter 3 presents a review of models of tumor growth and tumor treatment.
Onefamilyofmodelsconcernsbloodvesselscollapsinginvasculartumors,another
isdevotedtothemodelingoftumorcords(growingdirectlyaroundabloodvessel),
highlightingfeaturesthatarerelevantintheevolutionofsolidtumorsinthepresence
ofnecroticregions.Tumorcordsarealsotakenasanexampleofhowtodealwith
certainaspectsoftumortreatment.
TheaimofChapter4isthedescriptionofmodelsthatwererecentlydeveloped
to simulate the formation of vascular networks which occurs mainly through the
twodifferentprocessesofvasculogenesisandangiogenesis.Theresultsobtainedby
mathematical models are compared with in vitro and in vivo experimental results.
Thechapteralsodescribestheeffectsoftheenvironmentonnetworkformationand
investigates the possibility of governing the network structure through the use of
suitablyplacedchemoattractantsandchemorepellents.
Chapter5dealswithmathematicalmodelsofcardiacbioelectricactivityatboth
cellular and tissue levels, their integration and their numerical simulation.The so-
calledmacroscopicbidomainmodelofthemyocardiumtissueisderivedbyatwo-
scale homogenization method, and is coupled with extracardiac medium and ex-
tracardiac potential. These models provide a base for the numerical simulation of
anisotropiccardiacexcitationandrepolarizationprocesses.
Preface VII
InChapter6theauthorsdiscusstheroleofdelaydifferentialequationsforde-
scribingthetimeevolutionofbiologicalsystemswhoserateofchangedependson
theirconfigurationatprevioustimeinstances.AnoticeableexampleistheWaltman
model which describes the mechanisms by which antibodies are produced by the
immunesysteminresponsetoanantigenchallenge.
InthelastChaptertheauthorsillustraterecentadvancesonthemodelingofthe
human circulatory system. More specifically, they present six examples for which
numericalsimulationcanhelptoprovideabetterunderstandingofphysiopathologies
andabetterdesignofmedicaltoolssuchasvascularprosthesesandeventosuggest
possible alternative procedures for surgical implants. Each example provides the
conceptualframeworkforintroducingmathematicalmodelsandnumericalmethods
whoseapplicability,however,goesbeyondthespecificcaseaddressed.
Thischapteraimsaswelltoprovideanaccountofsuccessfulinterdisciplinary
researchbetweenmathematicians,bioengineersandmedicaldoctors.
Wearewellawarethatthisissimplyapreliminarycontributiontoamathematical
researchfieldwhichisgrowingimpetuouslyandwillattractincreasingattentionfrom
medicalresearchersintheyearstocome.
WekindlyacknowledgetheItalianInstituteofAdvancedMathematics(INDAM)
whosescientificandfinancialsupporthasmadethisscientificcooperationpossible.
Milan, AlfioQuarteroni
February2006 LucaFormaggia
AlessandroVeneziani
Contents
Inverseproblemsinbiomedicalimaging:modelingand
methodsofsolution
M.Bertero,M.Piana ............................................. 1
1 Introduction ................................................... 1
1.1 X-raytomography........................................ 2
1.2 Fluorescencemicroscopy.................................. 5
2 Linearity ...................................................... 5
3 Ill-posedproblemsanduncertaintyofsolution....................... 10
4 Noisemodeling ................................................ 14
4.1 AdditiveGaussiannoise................................... 16
4.2 Poissonnoise............................................ 16
5 Theuseofpriorinformation...................................... 18
6 Computationalissuesandreconstructionmethods.................... 21
7 Perspectives ................................................... 27
7.1 ElectricalImpedenceTomography .......................... 27
7.2 OpticalTomography...................................... 28
7.3 MicrowaveTomography................................... 28
7.4 Magnetoencephalography ................................. 29
Stochasticgeometryandrelatedstatisticalproblemsinbiomedicine
V.Capasso,A.Micheletti .......................................... 35
1 Introduction ................................................... 36
2 Elementsofstochasticgeometry .................................. 41
2.1 Stochasticgeometricmeasures ............................. 48
3 Hazardfunction ................................................ 50
4 Meandensitiesofstochastictessellations ........................... 51
5 Interactionwithanunderlyingfield................................ 53
6 Fibreandsurfaceprocesses ...................................... 55
6.1 Planarfibreprocesses ..................................... 56
7 Estimateofthemeandensityoflengthofplanarfibreprocesses........ 58
7.1 Localmeandensityoflengthandthesphericalcontact
distributionfunction ...................................... 59
7.2 Estimateofthelocalmeandensityoflength .................. 60
7.3 Numericalresults ........................................ 61
7.4 Applications............................................. 63
X Contents
Mathematicalmodellingoftumourgrowthandtreatment
A.Fasano,A.Bertuzzi,A.Gandolfi .................................. 71
1 Introduction ................................................... 71
1.1 Why ................................................... 71
1.2 How ................................................... 72
1.3 What................................................... 73
2 Modelsincludingtheanalysisofstresses ........................... 74
2.1 Applyingthemechanicsofmixturestotumourgrowth:
the“two-fluid”approach .................................. 75
2.2 Vasculartumours:modelsforvascularcollapse ............... 78
3 Abouttumourmorphologyandasymptoticbehaviour................. 80
3.1 Radially symmetric solutions and their stability under radially
symmetricperturbations................................... 80
3.2 Lookingfornon-radiallysymmetricstationarysolutions........ 81
3.3 Thegeneralproblemofthestabilityofradiallysymmetricsolutions 83
3.4 Asymptoticregimesandvascularisation...................... 84
4 Modelswithcellageorcellmaturitystructurefortumourcords........ 85
4.1 Tumourcords............................................ 85
4.2 Ageandmaturitystructuredmodels ......................... 86
5 Atumourcordmodelincludinginterstitialfluidflow ................. 90
5.1 Cellpopulationsandcordradius............................ 91
5.2 Extracellularfluidflowandthenecroticregion................ 93
6 Modellingtumourtreatment...................................... 97
6.1 Sphericaltumours ........................................ 97
6.2 Tumourcords............................................ 98
6.3 Hyperthermiatreatmentwithgeometricmodelofthepatient ....103
7 Conclusionsandperspectives.....................................104
Modellingtheformationofcapillaries
L.Preziosi,S.Astanin............................................. 109
1 Vasculogenesisandangiogenesis..................................109
2 Invitrovasculogenesis ..........................................111
3 Modellingvasculogenesis........................................116
3.1 Diffusionequationsforchemicalfactors .....................117
3.2 Persistenceequationfortheendothelialcells..................119
3.3 Substratumequation ......................................120
4 Insilicovasculogenesis..........................................121
4.1 Neglectingsubstratuminteractions..........................122
4.2 Substratuminteractions ...................................127
4.3 Exogenouscontrolofvascularnetworkformation .............130
5 Anangiogenesismodel ..........................................133
6 Futureperspectives .............................................137
Contents XI
Numericalmethodsfordelaymodelsinbiomathematics
A.Bellen,N.Guglielmi,S.Maset.................................... 147
1 Introduction ...................................................147
2 SolvingRFDEsbycontinuousRunge-Kuttamethods.................151
2.1 ContinuousRunge-Kutta(standardapproach)andfunctional
continuousRunge-Kuttamethods ...........................152
3 Athresholdmodelforantibodyproduction:theWaltmanmodel........154
3.1 Thequantitativemodel....................................155
3.2 Theintegrationprocess....................................159
3.3 Trackingthebreakingpoints ...............................161
3.4 SolvingtheRunge–Kuttaequations .........................164
3.5 Localerrorestimationandstepsizecontrol ...................169
3.6 NumericalillustrationfortheWaltmanproblem ...............170
3.7 Software................................................172
4 ThefunctionalcontinuousRunge-Kuttamethod .....................172
4.1 Orderconditions .........................................176
4.2 Explicitmethods .........................................177
4.3 Thequadratureproblem ...................................182
Computationalelectrocardiology:mathematicalandnumerical
modeling
P.ColliFranzone,L.F.Pavarino,G.Savaré............................ 187
1 Introduction ...................................................187
2 Mathematicalmodelsofthebioelectricactivityatcellularlevel ........189
2.1 Ioniccurrentmembranemodels ............................189
2.2 Mathematicalmodelsofcardiaccellarrangements.............192
2.3 Formaltwo-scalehomogenization ..........................196
2.4 Theoreticalresultsforthecellularandaveragedmodels ........199
2.5 Γ-convergenceresultfortheaveragedmodelwith
FHNdynamics...........................................201
2.6 Semidiscrete approximation of the bidomain model with FHN
dynamics ...............................................203
3 Theanisotropicbidomainmodel ..................................205
3.1 BoundaryintegralformulationforECGsimulations ...........208
4 Approximatemodelingofcardiacbioelectricactivitybyreducedmodels 211
4.1 Linearanisotropicmonodomainmodel ......................211
4.2 Eikonalmodels ..........................................213
4.3 Relaxednonlinearanisotropicmonodomainmodel ............216
5 Discretizationandnumericalmethods..............................218
5.1 NumericalapproximationoftheEikonal–Diffusionequation ....218
5.2 Numerical approximations of the monodomain and bidomain
models .................................................219
6 Numericalsimulations...........................................223
7 Conclusions ...................................................230
XII Contents
Thecirculatorysystem:fromcasestudiestomathematicalmodeling
L.Formaggia,A.Quarteroni,A.Veneziani............................. 243
1 Anoverviewofvasculardynamicsanditsmathematicalfeatures .......243
2 Casestudies ...................................................247
2.1 Numericalinvestigationofarterialpulmonarybanding .........247
2.2 Numericalinvestigationofsystemicdynamics ................253
2.3 Thedesignofdrug-elutingstents ...........................261
2.4 Pulmonaryandsystemiccirculationinindividualswith
congenitalheartdefects ...................................267
2.5 Peritonealdialysisoptimization.............................271
2.6 Anastomosisshapeoptimization ............................277
3 Awiderperspective.............................................281
Index.......................................................... 289