Complex Systems in Biomedicine PDF
Preview Complex Systems in Biomedicine
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). LibraryofCongressControlNumber:2006923296 ISBN-10 88-470-0394-6 SpringerMilanBerlinHeidelbergNewYork ISBN-13 978-88-470-0394-1 SpringerMilanBerlinHeidelbergNewYork Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthe materialisconcerned, specificallytherightsoftranslation, reprinting, reuseofillustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplicationofthispublicationorpartsthereofispermittedonlyundertheprovisions of the Italian Copyright Law in its current version, and permission for use must always be obtainedfromSpringer.ViolationsareliabletoprosecutionundertheItalianCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:2)c Springer-VerlagItalia,Milano2006 PrintedinItaly Cover-Design:SimonaColombo,Milan Typesetting:PTP-BerlinProtago-TEX-ProductionGmbH,Germany PrintingandBinding:Signum,Bollate(Mi) 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
The list of books you might like
