AbstractAimsAcupuncturewasemployedsince2millenariesbuttheunderlyingmechanismsarenotgloballyhandledThepresentstudyisaimedatproposinganexplanationbypointingoutinvolvedprocessesandaconvincingmodelin ID: 942687
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Commun.Comput.Phys.doi:10.4208/cicp.121214.250515sVol.x,No.x,pp.1-19x2015ABiologicalModelofAcupunctureanditsDerivedMathematicalModelingandSimulationsMarcThiriet1,2,3,,YannickDeleuze1,4andTonyWen-HannSheu4,51SorbonneUniversit´es,UPMCUnivParis06,UMR7598,LaboratoireJacques-LouisLions,F-75005,Paris,France.2CNRS,UMR7598,LaboratoireJacques-LouisLions,F-75005,Paris.3INRIAParisRocquencourt,EPCREO,BP105,F-78153LeChesnayCedex.4DepartmentofEngineeringScienceandOceanEngineering,NationalTaiwanUniversity,Taipei,Taiwan.5CenterforAdvancedStudyinTheoreticalSciences(CASTS),NationalTaiwanUniversity,Taipei,Taiwan.Receivedxxx;Accepted(inrevisedversion)xxx Abstract.(Aims)Acupuncturewasemployedsince2millenaries,buttheunderlyingmechanismsarenotgloballyhandled.Thepresentstudyisaimedatproposinganexplanationbypointingoutinvolvedprocessesandaconvincingmodelingtodemon-strateitsefciencywhencarriedoutbytrainedpractitioners.(Method)Intheabsenceofglobalknowledgeofanymechanismexplainingtheacupunctureprocess,abiologicalmodelisrstdeveloped,basedonstimulationinagivendomainaroundtheneedletipofapropermastocytepopulationbyamechanicalstress,electrical,electromagnetic,orheateld.Whateverthetypeofmechanicalorphysicalstimuli,mastocytesdegranulate.Releasedmessengerseitherfacilitatethetransferofmainmediators,ortargettheircognatereceptorsoflocalnerveterminalsorafterbeingconveyedbybloodtheirreceptorsoncerebralcells.Signalingtothebrainisfastbynervousimpulsesanddelayedbycirculatingmessengersthatneverthelessdistributepreferentiallyinthebrainregionofinterestduetohyperemia.Theprocessisself-sustainedduetomastocytechemotaxisfromthenearbydensemicrocirculatorycircuitandsurroundingmastocytepools,whichareinadequateforacupuncture,butserveasasignalrelay.Asimplemathematicalmodelissolvedanalytically.Numericalsimulationsarealsocarriedoutusingtheniteelementmethodwithmeshadaptivity.(Results)Theanalyticalsolutionofthesimplemathematicalmodeldemonstratestheconditionslledbyamastocytepopulationtooperateefciently.Atheoremgivestheblow-upcondition.Thisanalyticalsolutionservesforvalidationofnumericalex-periments.Numericals
imulationsshowthatwhentheneedleispositionedintheperipheryoftheacupointoroutsideit,theresponseistooweak.Thisexplainswhya Correspondingauthor.Emailaddresses:marc.thiriet@upmc.fr(M.Thiriet),yannick.deleuze@upmc.fr(Y.Deleuze),twhsheu@ntu.edu.tw(T.W.-H.Sheu)http://www.global-sci.com/1c\r2015Global-SciencePress 2M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19longtrainingisnecessaryastheneedleimplantationrequiresaprecisionwithamag-nitudeoftheorderof1mm.(Conclusion)Theacupointmustcontainahighlyconcentratedpopulationofmas-tocytes(e.g.,very-highamplitude,small-widthGaussiandistribution)togetaninitialproperresponse.Permanentsignalingisprovidedbychemotaxisandcontinuousre-cruitmentofmastocytes.Therefore,thedensityanddistributionofmastocytesarecrucialfactorsforefcientacupunctureaswellasavailabilityofcirculatingandneigh-boringpoolsofmastocytes.AMSsubjectclassications:76M10,92C10,92C17,92C50Keywords:Acupoint,chemotaxis,niteelementmethod,mastocyte,mechanotransduction,neu-ralandhormonalcontrol. 1IntroductionTraditionalChinesemedicinedenesacupuncturepoints,oracupoints,fortherapeuticobjectives,morethan2500yearsago.Acupuncturewasaimedatrelievingapathologicalstatebyliberatingthesequesteredenergyandrearrangingthebalanceofyinandyangtoensurehomeostasis.Adiseasewasindeedsupposedtoresultfromanimbalancebe-tweenYinandYang.YangandYinare2fundamentalopposing,complementary,andinterdependentforcesfoundinallthingsintheuniverse,withtracesofoneintheother,thatsupporteachotherandcantransformintooneanother.NothingintheuniverseiscompletelyYinorYang;everythingisamixtureofthesetwo.Inparticular,Yangmaybeconsideredasmentalactivityinitsstrengthaspect,Yinmentalactivityinitsimaginativeaspect;inotherwords,Yangconstructs,Yininstructs,orconversely.Yinisrelatedtostaticandhypoactivephenomena,Yangtodynamicandhyperactiveprocesses,orconversely.Anacupunctureneedleisinsertedintoselectedacupointsonthebody'ssurface,onwhichmechanicalorothertypesofphysicalstimulationsareexerted(e.g.,moxibustion),inparticulartocauseanalgesia.Afferentpathsarestimulatedtoelicitthedeqsensationandsignaltoade
quatezonesinthecentralnervoussystem.Acupointsaretransfersitesofq.Meridians(Chinese:jing)andcollaterals(Chinese:luo)arecommunicationpathsforq,thevitalenergymasterofbodyuids,whichcanbetransportedtoacupoints.Thesespatiallyrestrictedsitesthatdonotcorrespondtoaspecializedbiologicaltissue,buttolocalizedstructuralandfunctionalunits,fromwhichenergypoursandpervadesintothebody'stissues.Thepresentstudyfocusesonthepermanentregimeofacupuncture,thatis,oncetheneedlehasbeenimplantedandthestresseldinthesubcutaneousconnectivetissueisfullyestablished.Thereaderisinvitedtoreadtheaccompanyingpaper[1]togetinformationoneventsoccurringduringthetransientregime. M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-193ContextThebrainisacomplexprocessorthatcansensechemical,physical,andmechanicalsig-nals,treatthem,andtransmitanextremelyquickaswellasdelayedlong-lastingoutputforadaptationusingneuralandvascularroutes.Surgicalinterventionscanbecarriedoutusingeithergeneralanesthesia,thatis,amedicallyinducedcoma,oracupuncture,thatis,performingtasksinconscioussubjectsnaturallyanesthetized.Inthelattercase,thebrainthatiscapableofsynthesizingopioidsandantalgicsisstimulatedfromacupointsthatareknownsince2millenaries.Inadditiontoabettercomfortforthepatientwhoavoidscoma,thecostforthehealthserviceismuchlower.AcupuncturemodalitiesAcupuncturerequiresalong-termtrainingtohandleacupoints.Fourtechniquesexist:(1)developmentofalocalmechanicalstresseldbyneedlemotions(liftingthrustingcycleorrotation)atacupoints;(2)developmentofalocaltemperatureeldbydirectlyapplyingaheatingmoxa(mugwortherb)stickontheskinorindirectlybyapplyingthisstickontheacupunctureneedle(moxibustion)atacupoints;(3)developmentofalocalelectricaleldbyapplyingasmallelectriccurrentbetweenapairofacupunctureneedles(electroacupuncture,orpercutaneouselectricalnervestimulation[PENS])atacupoints;and(4)excitationofputativelight-sensitiveG-proteincoupledreceptorsonthesurfaceofmastocytesbylaserbeam.SubcutaneousconnectivetissueTheskinconsistsofthreemainlayers:theepidermis,dermis,andhypodermis.Be-neaththeskin,thesubcutaneousconnectiv
etissueliesaboveskeletalmuscles.Itspreadsthroughoutthebodyandisincontinuitywithinterstitialconnectivetissuessurroundingallmuscles,organs,andneurovascularbundles.Theconnectivetissuecontainsscatteredcellsimmersedintheextracellularmatrixthroughwhichtheaqueousphase,orintersti-tialuid,canow[2].Celltypesofconnectivetissueincludebothresidentcells,suchasbroblasts,bro-cytes,mastocytes,andmacrophages,andscoutingcellsoftheimmunity(e.g.,monocytes,lymphocytes,andgranulocytes).Theextracellularmatrixismadeupofcollagenandelastinbersandagroundsub-stance(i.e.,mostlyglycoproteinsandproteoglycans),ahydratedmediumthatismoreorlesspolymerizedaccordingtothephysicalenvironmentalconditions.Collagenandelastinbersarethemainelementsresponsibleforthematrixrheologicalproperties.Elastinbersenabletissuedeformationuptoacertainlevel,theenergybeingrestituteduponmechanicalstressremoval.Collagenbersyieldtensileresistance.Theextracellularmatrixtransmitsmechanicalstressestotheembeddedcellaswellasfromthecell,whenthelattercontractsormigrates.Itthenparticipatesinmechanotrans-duction,thatis,theconversionbyanycellofmechanicalstimulisensedatitssurface(plasmamembrane)intochemicalsignaling(i.e.,acascadeofchemicalreactionsleading 4M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19totransferintheextracellularspace[exocytosis]ofstoredmaterialsandgenetranscrip-tion[proteinsynthesis]).Thesubcutaneousconnectivetissuecanbeconsideredasauid-lleddeformableporousmedium.TheDarcyandBrinkmanequationsdescribeinterstitialincompressibleuidowintheframeworkofacontinuum.Inasteadystate,thatis,whenthesubcuta-neousconnectivetissueispermanentlydeformedwithoutuctuationsofstressorstrain,thematrixisinagivenpolymerizationstateandinterstitialuidissupposedtobeinanewequilibrium(resting)state.AcupointInsertionintotheskinofthinneedlesisthemostcommontechnique.Aninitialmanualmanipulationisthenachieved.Anyacupointlocalizedtothevicinityofbones,aponeu-roses,muscles,andtendonsthatcontainneuralunitswithsomatosensoryreceptors.Theacupointischaracterizedbyahighdensityofmastocytes.Thispoolofmastocytesresidecloseto
neurovascularbundles,inaregionwherecapillaries,lymphaticvessels,andnerveendingsabound.Otherfeaturesofacupointcomprisealargeskinelectricalconductanceandhighionicconcentrations(especiallyCa++).Freenerveendingsandcutaneousreceptors(Merkel,Meissner,Rufni,andPaciniancorpuscles),sarcoussensoryreceptors(musclespindlesandtendonorgans),andtheiraf-ferentbers,aswellassomaticefferentbersinnervatingmuscles,smallnervebundles,andplexiareobservedinacupoints[3].ThemechanicalstresseldcanactivateA,A,andAbers,aswellasC-bersofnervousstructuresatacupoints.Anervebercorrespondstoanaxonpossiblyendowedwithamyelinsheath,accord-ingtowhetherthenerveberismyelinatedornot.Threetypesofperipheralnervebersexistaccordingtotheircaliber(AC).TheAtypeconsistsof4subtypesofrelativelylarge(122m),myelinatedberscharacterizedbyahighconductionvelocity(5120m/s):A(afferentorefferent;motorandproprioceptive;bore1322m;conductionvelocity70120m/s);A(afferentorefferent;motorandproprioceptive;caliber813m;con-ductionvelocity3090m/s);A\r(efferent;onlymotorbers;caliber37m;conductionvelocity1540m/s);andA(afferent;onlysensorybers[rapidpainsensation];cal-iber15m;conductionvelocity515m/s)bers.Type-Aberstransmitimpulsesre-latedtomuscle,tendon,andjointmovementandsituation.B-typenervebersarealsomyelinated,albeitthinner(3m).Thesepreganglionicbersoftheautonomicnervoussystemtransmitinvoluntaryimpulseswithaslowerconductionvelocity(315m/s).C-type,thin(0.31.3m),unmyelinatedbershavealowconductionvelocity(0.62.3m/s).Theyincludepostganglionicbersintheautonomicnervoussystemaswellasnervebersofthedorsalrootsthatcarrysensoryinformation(heat,pressure,andslowlypain).Mastocyte,themastercellinacupunctureOthercellslocatedatacupoints,suchasneurons,macrophages,broblasts,andlympho-cytes,cancontributetotheemissionoflocalandendocrinesignals.Mastocytesthatcon-tainnumerousgranulesservingasstoresfortransmittersisthemajorcellularspecies[4]. M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-195Table1:Signalingmediatorsreleasedatacupoints(Sources:[5,6];CGRP:calcitoningene-relatedpeptid
e;MOR:-opioidreceptor).MechanicalstressescanactivateA,A,andAbers,aswellasC-bersofnervousstructuresatacupointsandaugmentlocallythevascularpermeabilitytoacceleratethetransferofmediatorstothe\rowingblood. MediatorsReleasingcellsReceptors Effects AcetylcholineNeuron,keratinocyteM2 AdenosineA1 ATPEpidermalcellsP2X,P2Y BradykininLocalcellsB1,B2 CGRPEpidermalcells,Tcell, macrophage Cytokines(interleukins,tumor-necrosisfactor- IL1/6/8,LocalcellsEnhancedexcitability TNFofafferentbers IL4/10LocalcellsInhibitionofinammatory signalsinafferentterminals -endorphinFibroblasts,leukocytes,MOR keratinocyte,melanocyte HistamineMastocyteH1,H3 GABAMacrophage,lymphocyteGABAA GlutamateMacrophage,epidermalcells NitricoxideManycelltypesInhibitionofsubstance-P releasefromnerveterminals Stimulationofacetylcholine and-endorphinsecretion NoradrenalineSympatheticnerve2AR SerotoninMastocyte,platelet,5HT1, 5HT3(afferentnerve) SomatostatinMerkelcell,keratinocyteSstR Substance-PMastocyte,broblast, platelet,macrophage, keratinocyte ProstaglandinsLocalcellsEP Mastocytesareactivatedbyamechanicalstresseld(mechanotransduction),heat-ing(thermotransduction),electricaleld(electrotransduction),orlight(especiallylaser;phototransduction).Twomastocytestatesarethusconsideredaccordingtothelocationwithrespecttoacupoint:granulatedclosetothestimulationsourceanddegranulatedoutsidethisregion,thatis,atadistanceofabout5mmfromtheneedleinthecaseofmechanotransduction. 6M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19Allthese4techniquessharethesamemechanism,astheyrelyoncalciumsignaling.Nevertheless,mechanotransductioncorrespondstothemoststudiedoperatingmode,experimentallyornumerically.Needlemanipulationinacupuncturecancausedegranu-lationofmastocytesuponexposuretomechanicalstress[79].Calcium,theprimaryintracellularmessengerinacupunctureWhatevertheoperationmode,calciumveryrapidlyentersthemastocytethroughpropergatedCa2+channelsintheplasmamembrane.Accordingtotheoperatingmode,thetargetCa2+channelismechanosensitive,thermosensitive,voltage-gated,andphotosen-sitive.Inthelast-mentione
dcase,theCa2+channelmaybesupposedtobeintrinsicallyphotosensitiveorbelinkedtoaphotosensitiveG-proteincoupledreceptor.Itthentrig-gersexocytosisofgranulesservingasstoresofmessengersveryrapidlyalongcytoskele-taltracks.ReleaseofautacoidsandmessengersforremotetargetsCalciuminuxinthemastocytecytosolprovokesdegranulationandreleaseofchemoat-tractants,neuralstimulants,vasoactiveagents,andendocrinesubstances.Therefore,acupunctureeffectsresultfromasetofsignalssentfromactivatedmastocytesatgivenacupointstolocalnerveendings,lymphatics,andcapillaries,andfromthesetwolast-mentionedstructurestotheheartandbraintoimprovethebloodcirculationatselectedacupointsandtriggerthecerebralresponse,respectively.Numeroustypesofmoleculesfreedbymastocytesincludecalcitoningene-relatedpeptide(CGRP),heparin,histamine,leukotrienes(LTb4,LTc4,LTd4,andLTe4),platelet-activatingfactor,prostaglandin-E2,serotonin,substance-P,andthromboxane-A2(Table2).Mastocytesalsosecretepeptidases(e.g.,tryptase),growthfactors(e.g.,FGF,CSF2,andNGF),andcytokines(e.g.,interleukinsandtumor-necrosisfactor).Nerveendingsarestimulatedandreleasesubstance-Pthatfurtheractivatesmastocytesandtriggersthepro-ductionofnitricoxide,agaseousvasodilator.Nervesandmastocytesexchangechemicalmessengerssuchassubstance-P.Thelat-terstimulateshistamineandnitricoxide(NO)release.Calcitoningene-relatedpeptide(CGRP)andNOcausevasodilation.NitricoxidecooperateswithCGRPtoincreaseitspositiveinotropiceffectthatraisesthelocalbloodowindilatedvessels.Histamineisquicklycatabolized,therebyactingnearthesiteofrelease.Resultingvasodilationandincreasedvesselwallpermeabilitysupporttransferofchemicalmediatorsintothebloodcirculation.Serotoninhasabiphasiceffect,asittriggersavasoconstrictionandpro-motesNOrelease,hencesubsequentlyvasodilation.Nervegrowthfactor(NGF),tumor-necrosisfactor(TNF),andinterleukins(IL)arepotentmastocytechemoattractants.Mas-tocytechemotaxisissupportedbymatrixdegradationbysecretedpeptidases.ThistraditionalprocedureofChinesemedicinereliesonintra-,auto-,juxta-,para-,andendocrinesignalingaimedattriggeringmastocytechemotaxisandsendingmessagesvia:(
1)nervesforanimmediate(O[1s1mn]),fast,andtransientresponseofthecentralnervoussystemresponsibleforhyperemiainagivenlocalregionofthebrain,inwhich M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-197Table2:Releasedmoleculesbythemastocyteandtheireects. AgentEffects CGRPVasodilation, positivechronotropy,inotropy,andlusitropy, mastocytedegranulation HeparinBloodclotprevention HistamineVasodilation(directlyandviaNO), nervestimulation LeukotrienesVasodilation,vascularpermeabilityelevation IL,NGF,TNFChemotaxis Prostaglandin-D2Nervestimulation Prostaglandin-E2Vasodilation, inhibitionofmediatorrelease SerotoninVasoconstrictionfollowedbyNO-mediatedvasodilation Thromboxane-A2Vasoconstriction,plateletaggregation Tryptase,chymaseMatrixdegradationforenhancedcellmigration neuronsthensecreteendocannabinoids,enkephalins,endomorphins,dynorphins,andotheranalgesicsubstances,inparticular,aswellasapermanentresponseensuredbythecontinuousuxofactivators;and(2)bloodandlymphvesselsforadelayedandslowerreactionbasedontransmissionofsubstancesthatareconveyedthroughoutthebrain,butpreferentiallytohighlyperfusedregions.Targetnervouscentersthenreplybyregulatingthebehaviorofproperperipheralorgans.AprolongedprocessTheprocessissustainedbyrecruitmentofmastocytes(chemotaxis).Thetimerequiredforthesynthesisofchemicalmediatorsindegranulatedmastocytesisindeednitewithamagnitudeorderofonehour.Therefore,acontinuoussecretionofmessengersreliesonapermanentarrivalofmastocytesintothestimulatedregion,wheretheybecomeactivatedanddegranulate.Twosourcesofincomingmastocytesincludethelocaldensecapillarynetwork(circulatingmastocytes)andnearbypools,thatis,clustersofmastocytessitu-atedrelativelyclosetothestimulatedregion,butthatcannotbeusedasacupoints,astheirdistributiondoesnotmatchthatatacupointcharacterizedbyaveryconcentratedmastocytepopulation,whichenablesaDirac-likeresponse.MechanotransductionAcupointischaracterizedbyaclusterofmastocytesandotherisolatedcelltypes.Cellsareimmersedinamediumcontainingextracellularuid(i.e.,waterandsmallmoleculesandions)andmacromolecules.Amongthesemacromolecules,so
meformbers(elastinandcollagen),whereasothersformhydratedmeshworkofglycoproteinsandproteogly-cans. 8M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19Onceitissubjectedtoastresseld,thisbiologicalmediumdeforms.Themeshworkconstitutedbyhydratedproteoglycansandglycoproteinscanbedegraded,henceunder-goinguidization.Elastinandcollagenbersarestretchedandbent.Theymayalsodepolymerize,hencecontributingtothemediumuidization.Afterthetransientregime,whenthematrixreachesanewequilibriumstate,thatis,inthepermanentregimebeforethefollowingneedlemanipulation,proteoglycansandglycoproteinsaswellasberscanrepolymerize(gelication).Inthetransientregime,thatis,duringneedlemanipulationandshortlyafteritsend,theinducedmotionoftheextracellularuidisveryslow.Moreover,convectionisstillhinderedbyuidizingmacromolecularcomplexes(mainlybersandproteoglycans).Whenneedlingceases,theuidmotionsoonstopsandrapidregelicationhampersanyfurtherdisplacement.Duetothetransientuidization,convectionlinkedtothemotionoftheextracellularuidisasecondaryphenomenonthatcanbeneglected.Theplasmamembraneofembeddedcellsmakesavirtualboundarybetweentheextra-andintracellularmedia.Theyindeedcontainproteinssuchasintegrinsandad-hesionplaquesthatdeneacontinuumbetweencellandmatrixconstituents.Therefore,anystressimposedbyacelltoitsenvironment,orconverselybytheextracellularmilieutothecell,istransmittedbythesemembranestructures.Intheabsenceoforganizedcellu-lararchitecturesuchasinepithelia(e.g.,theepidermis),needlingdisturbsmainlytheex-tracellularmilieu,inducingalocalmechanicalstresseldthatistransmittedtoimmersedcells.Insidethecell,thecytoskeletonalsoundergoespolymerizationdepolymerizationcycles,especiallywhenitbearsmechanicalstressestoadapttothem.Thecellcytosolbehavesthusliketheextracellularmatrix;bothexperienceuidizationandgelication.Hence,theplasmamembraneofmastocytesisstretchedandsheared.Thesensedme-chanicalstressdeformsplasmalemmalmolecules.Itevenactuatesasetofproteinsem-beddedinthelipidicbilayerconstitutingtheplasmamembrane,theso-calledmechanosen-sitivecomponents,amongwhichsomearetypesofcalc
iumchannels.Activatedchannelsenablecalciuminux.Calciumionisawellknowncellularmessengerthatcantriggerasetofcascadesofchemicalreactionsaswellasvariousphenomenasuchasexocytosisofstoredmaterials.MechanicalstimulationTheclassicalmodeofacupuncturereliesonthefollowingsetofevents:(1)generationofalocalstresseldcausedbyneedlerotationand/ortranslationalmotions;(2)mechan-otransductionofsensedlocaltensionintochemicalsignals,i.e.,increaseincytosolicCa2+concentration,granuleexocytosis,andsubstancerelease;(3)early,quicktriggeringofac-tionpotentialsbynervestimulantsboundtotheircognatereceptorsonlocalnerveter-minalsandactivationofthetargetbrainregionthatisassociatedwithalocalfunctionalhyperemia(localincreaseinbloodowduetonervousactivity);(4)localelevationofvas-cularpermeabilityandvasodilationprimedbyreleasedcompoundswithincreasedlocalbloodowbythesentcardiotonicmessengers,alltheseprocessesenhancingendocrinesignaling,i.e.,delayedstimulationofthetargetbrainregion,asendocrinemessengersare M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-199preferentiallydistributedinactivebrainregions;and(5)chemotaxisofmastocytesfromregionalpoolsandcirculatingbloodanddegranulationofnewlyarrivedmastocytesatacupointsthatenablesasustainedprocess,degranulationbeingeffectiveonlyneartheneedle,wherethemechanicalstressishighenough.2MethodInthepresentpaper,theoperatingmodeisneedlingwithinitialandperiodicneedlerotationand/ortranslationalmotions(needlingwithliftingthrustingcycleortwisting).Theresultinglocalstresseldengendersapermanentlydeformedregion,infactthatexperiencedduringacupunctureaballofabout5-mmcalibercenteredontheneedletipinthesubcutaneoustissue.2.1BiologicalmodelandhypothesesTheacupointisconsideredasahighdensitypoolofmastocytes.Twomastocytestatesareconsideredaccordingtothelocationwithrespecttothestimulationregionaroundtheinsertedneedle:granulatedanddegranulated.Acupunctureeffectsrelyonintra-,auto-,juxta-,para-,andendocrinesignalingaimedattriggeringmastocytechemotaxisandsendingsignalslocallytonerveendingsandlymphandbloodcapillaries.Ligandedreceptorsonthenervesurfaceveryra
pidlylaunchactionpotentialsthattraveltothecentralnervoussystem.Ligandedreceptorsonthevas-cularcellsurfaceincreasethevascularpermeabilityandprovokevasodilation,therebyfacilitatingthetransferofmoleculesintothelymphandblood.Onceconveyedinblood,messengerstargetthecentralnervoussystemandthecardiacpumptoenhancebloodowatacupoints.Aquasi-instantaneousreleaseofchemicalmediatorsisassumeduponstimulationbythestresseldresultingfromneedlingandCa++ionentryintomastocytes.Themagni-tudeordersforCa++channelactivationandCa++entryintothecytosolfromtheextra-cellularmilieuareactuallyO(ms)andO(s),respectively.Oncethecontentstoredinintracellularvesiclesisliberated,acupuncturecanbemod-eledbyanimmediateandalateresponsecorrespondingtonervous(actionpotentials)andbloodbornesignals.Chemotaxis,thatis,therecruitmentofmastocytesfromthelocalbloodcirculationandlocoregionalpools,enablesapermanentreleaseofsubstancesduringthedelayedregen-erationofgranulescontent.Thismechanismisassumedtobetheessentialphenomenonofacupuncturethatenablestheprocessmaintenance.Theextracellularspaceconstitutesasinglecompartmentthatconsistsofauid(in-terstitialuid)andsolid(matrix)fraction(phase).Itscompositematerialisconstitutedofbersembeddedinthesemipermeablegroundsubstance.Thisporousmediumiscrossedbytheinterstitialuidthat,undermechanicalstress,owsslowlyalongbers 10M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19andthroughgapsofthemacromolecularmeshworkandbundlesofelastinandcolla-genbers,theviscousdissipationbeingthepredominantterm(DarcyBrinkmanow).TheDarcy'slaw,aphenomenologicallyderivedequationcanbeobtainedfromthelinearStokes'equationsinaperiodicunitcellusingthehomogenizationtheorywithanon-penetrationconditionatthedomainboundary.Matrixconductanceisdenedastheowofinterstitialuidperunitpressuredropacrossunitarea;matrixresistanceistheinverseofconductance.Moreover,theextracellularmatrixevolvesbetweenauidandagel,ac-cordingtolocalmechanical,physical,andchemicalconditions(uidizationgelicationcycle).Inthepresentwork,theStokes-likeowintheextracellularmatrixandhenceconvectionintheinvolved
masstransferprocessisneglected.2.2MathematicalmodelAnymastocyteinthelocalvasculaturemovesalongthechemoattractantgradient,henceundergoesatransmigration(acrossbloodvesselwalltoexitblood,remaininggranulatedoutsidearegionoftriggeringmechanicalstress(x`)andliberatingitsgranulecontent,onceitreachesaregionclosetotheacupoint(0`),whereasignicantmagnitudeofthemechanicalstresscanbesensed.Similarly,anymastocytefromaneighboringpoolcanmigrate.Twomastocytestatesindeedexistaccordingtoitslocalizationwithrespecttotheacupoint(non-degranulatedanddegranulated).Therefore,inthecaseofmechan-otransduction,themechanicalstressismodeledbyacompact-supportedfunction.Chemicalmediatorsaresupposedtobequasi-instantaneouslyreleasedbythemechan-otransductionprocessthatmainlyreliesonasudden,rapid,andcopiouscalciumentryinthemastocytecytosol.Thiscalciumwavethatgushesandpervadesthemastocyteen-ablesittodischargechemoattractants,nervemessengers,cardiovascularstimulants,andendocrinemessengers.Ontheotherhand,theregenerationofgranulescontentinsidethemastocyteisdelayedandslow.Followingchemotaxisfromregionalpoolsandblood,newlyarrivedmastocytesatacupointsexperienceadegranulationtriggeredbythestresseld.Theresultingself-sustainedprocessenablesthelocalelevationofvascularpermeabilityforimprovedcar-diacoutputandenhancedendocrinesignaling,remotecardiaceffectcooperatingwithvasodilationtoraiselocalbloodow,endocrinesignalingtocentralnervoussystembe-ingsupportedbyneuronalactivityinthebrainregionofinterest(hyperemia),thatsimul-taneouslyreceivesetsofactionpotentials.Themathematicalmodelisaninitialboundaryvalueproblemcomposedofasetofvepartialdifferentialequationsforaboundeddomain(x2W)andevolvingtime(t2R+).Twoequationsarerelatedtothepopulationsofgranulatedanddegranu-latedmastocytes.ThismodelofmastocyteresponsetoacupunctureneedlingisbasedontheKeller-Segelmodelforchemotaxis.Threeassociatedequationsdescribethetem-poralevolutionofconcentrationsofchemoattractants((t,x)),liberatednervestimulants(sn(t,x)),andendocrineactivators(se(t,x))ofsomesitesofthecentralnervoussystemin M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comp
ut.Phys.,x(2015),pp.1-1911particular:¶tng Dmr2ng+r(Sngr)= AFng+Rnd;¶tnd Dmr2nd=AFng Rnd;¶t Dcr2=kcAFng Dc;¶tsn Dnr2sn=knAFng Dnsn;¶tse Der2se=keAFng Dese.(2.1)Intheabove,ng(t,x)andnd(t,x)arethedensitiesofgranulatedanddegranulatedmas-tocytes,respectively,F(x)thedimensionlessmagnitudeofmechanicalstress(i.e.,bulkmagnitudeoflocallyexertedpressureandshearstress;0F(x)1,0x`;`:spacerangeoftheappliedstress)inthemechanicalstress-responsiveregion(areain2D[volumein3D]);Dm/c/n/ethediffusioncoefcients;Atheactivationrateofgranu-latedmastocytessubjectedtothemechanicalstressFthatthendegranulate;Rthere-generationrateofdegranulatedmastocytes;Sthemastocytesensitivitytochemoattrac-tant;Dc,e,nthedegradationrate;andkc,e,nthereleasequantitycoefcient.Intheseequa-tions,theparametersSandkc,e,nensurethephysicalhomogeneity.Alltheparameters(S,Dm/c/n/e,A,R,D,andk)aresupposedtobeconstants.Equationvariablesandpa-rametersaregivenwiththeirphysicalunitsinTable3.Table3:Equationvariablesandparameterswithphysicaldimensions(units)expressedinthefundamentalMLT(length,mass,time)system. ParameterDenitionPhysicaldimension AActivationrateofmastocytesT 1 cConcentrationmol.L 3 DDegradationrateT 1 DDiffusioncoefcientL2.T 1 nMastocytedensityL 3 RRegenerationrateofT 1 mastocytegranules SMastocytesensitivityL5.mol 1.T 1 tochemoattractant kReleasequantitycoefcientmol FMechanicalstressfunctionDimensionless Thesystemofequations(2.1)isendowedwithboundaryconditions,thatis,homoge-neousNeumannboundaryconditions:nrnd=0,nrng=0,nr=0,nrsn=0,nrse=0,(2.2)orzerototaluxboundaryconditionsgivenbyn(rng Sr)=0,nrnd=0,nr=0,nrsn=0,nrse=0,(2.3) 12M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19wherendenotestheoutwardunitnormalontheboundaryG.Theequationsystem(2.1)isalsoendowedwiththeprescribedinitialconditions:ng(t=0)=n0g,nd(t=0)=n0d,(t=0)=0,sn(t=0)=s0n,se(t=0)=s0e.(2.4)Inthepresentmodelofchemotaxis,thechemoattractantisemittedfromaasubdo-maincenteredontheneedleinwhichmechanicalstressissufcientlyintensetobede-tectedandtransducedbymastocytes.ThestressfunctionFcanberepresentedasaC¥andcompact
lysupportedfunctionfromR2to[0,1]withthefollowingproperties:8(x,y)2R2,jF(x) F(y)jkFjx yjF(x)1,8x2R2,F(x)=0,8x2R2,jxj`.(2.5)Inthenumericalsimulations,itisrepresentedbyabumpfunctionusingtwopositiveparametersandbthatcontrolamplitudeandshape:FB(x,y)=exp b/(`2 x2 y2),ifx2+y2`2,0,elsewhere.(2.6)Distributionanalysisreliesonstatisticalmoments(mean,variance,skewness,andkurtosis,amongothers).Theinitialmastocytedistributionisrepresentedbyanormal(Gaussian)distribution:n(x,y,t=0)=aexph (x xmax)2/2s2x+(y ymax)2/2s2yi+b,(x,y)2R2,(2.7)whereaisthemastocytedistributionamplitude,btheminimalmastocytedensityinthetissue,sxandsythedispersionsinthex-andy-direction,andf(xmax,ymax)gthecoordi-natesofthesiteofmaximalmastocytedensity.Whenbn=0,sx=sy=s,andxmax=ymax=0,thenumberofmastocyteisgivenbymjR2=Z ¥ ¥Z ¥ ¥n(x,y,t=0)dxdy=2pas2.(2.8)Inthediscofcenter(0,0)withradiusR,thenumberofmastocyteisgivenbymjD(0,R)=ZZD(0,R)n(x,y,t=0)dxdy=Z2p0ZR0aRexp R2/2s2drdq=2pas2(1 e [R2/2s2]).(2.9)Thespatialdistributionofmastocytesisdescribedbythezerothandsecondstatisticalmoments:m0(t):=ZR2ng(t,x)dx;m2(t):=ZR2jxj2 2ng(t,x)dx.(2.10) M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-1913Asmallsecondmomentindicatesthatthecellsaregatheredinthevicinityofthecenterofthedomain.Asimplestformoftheequationsetdescribestheevolutionofthedensityofmasto-cytesandthechemoattractantconcentration,assumingthatthediffusionofthechemoat-tractantisinstantaneousandhenceimplyingtheabsenceofevolving(intime)chemoat-tractantconcentration:¶tng Dmr2ng+r Sngr= AFng,t0,x2W,Dcr2= kcAFng,ngjt=0=n0g.(2.11)Inthedifferentialsystem(2.11),thetotalmassisconserved,thatis,thetotalnumberofcellsinthedomainremainsconstant,buttheirlocalizationchanges:Asingularityappearsinanitetimeandthecelldensityblowsupatthesingularitypoint[11],thatisblow-upsolutionhappens,if:(1)theinitialdensityofmastocytesishighenoughm0(0)=ZR2n0g(x)dx8p ASkcexpAT(2.12)and(2)thesecondmomentissmallenough(smalldispersion)m2(0)=ZR2jx2jn0g(x)dxx,(2.13)wherex=(b2 2ac) bp 4ac+b2 22,(2.14)anda=2m0(0)1 ASkc 8pm0(0)exp LT,b=3p 2 2pASkc lm0(0)
3 2and=1 pASkc lkfm0(0).Inmathematicalphysics,partialdifferentialequationsmostoftenarerendereddi-mensionless.Dimensionlessindependent(x,andt)anddependentvariables(ng,nd,,sn,andse)arehenceintroduced:x=x/`;t=tDm/`2;ng=ng/nref;nd=nd/nref;=/ref;sn=sn/sref;se=se/sref.Thereferencequantitiesaretheinitialdensityofgranulatedmastocytes(nref),concentra-tionofmessengersstoredbeforedegranulation(refandsref). 14M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19Thesetofequations(2.1)thenbecomes,removingforthesakeofsimplicitythetildedenotingdimensionlessvariables:¶tng r2ng+PSr ngr= PAFng+PRnd;¶t PDcr2=PkcFng PDc;¶tnd r2nd=PAFng PRnd;¶tsn PDnr2sn=PknFng PDnsn;¶tse PDer2se=PkeFng PDese.(2.15)wherePS=Scref Dm;PA=`2A Dm;PR=`2R Dm;PDc=Dc Dm;Pkc=kcA`2nref Dmref;PDc=`2Dc Dm;PDn=Dn Dm;Pkn=knA`2nref Dmsref;PDn=`2Dn Dm;PDe=De Dm;Pke=keA`2nref Dmsref;PDe=`2De Dm.2.3SolvingprocedureNumericalsimulationswerecarriedoutusingtheFreeFem++software,anopensourcepartialdifferentialequationsolverbasedonthevariationalformulationusingtheC++languagecodedevelopedintheLaboratoryJacques-LouisLions(LJLL)ofUniversityPierreetMarieCurie(UPMC,Paris,France)[10].TheFreeFem++softwarecomprisesthefollowingmodulus:triangularniteelementspaces;2Dand3Dautomaticmeshgen-eratorandmeshadaptation;fastlinearsolvers;andMPItoolsforparallelcomputing.Neumannboundaryconditionscanbedirectlyimplementedintotheweakformulationwithoutintroducinganyderivationerror.ApenaltymethodisemployedtoimposeDirichletboundaryconditions.Tosolvethelinearsystemassociatedtothediscretizedproblem,FreeFem++providesmanydirectanditerativesolvers.Inthepresentstudy,themultifrontalsolverUMFPACKandparallelsolverMUMPSareutilized.AsintheFreeFem++software,problemsareexpressedintheweakformulation,equationsaremultipliedbyatestfunctionwiinH1(W))5(i=g,d,,n,e)andintegratedoverthedomainW).Inthepresenceofanonlinearterm,theequationisarticiallyde-coupledusingthepreviouslycomputedsolutions.Thesolvingprocedurereliesonasemi-discretizationintime.Meshadaptivityenablestofollowthelocalevolutionofthesolution.Thealgorithmforthedecoupledmeth
odisgivenasfollows:1.Findm+12VhcsuchthatZWm+1 m Dtwc+ZWDcrm+1rwc+ZWDcm+1wc=ZWkcFnmgwc,8wc2Vhc;(2.16) M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19152.Findnm+1d2VhdsuchthatZWnm+1d nnd Dtwd+ZWrnm+1drwd+ZWkrnm+1dwd=ZWAFnmgwd,8wd2Vhd;(2.17)3.Findnm+1g2VhgsuchthatZWnm+1g nng Dtwg+ZWrnm+1grwg ZWSnm+1grm+1rwg+ZWAFnm+1gwg=ZWkrnm+1dwg,8wg2Vhg;(2.18)4.Findsm+1n2VhnsuchthatZWsm+1n smn Dtwn+ZWDsnrsm+1nrwn+ZWDnsm+1nwn=ZWknFnm+1gwn,8wn2Vhn;(2.19)5.Findsm+1e2VhesuchthatZWsm+1e sme Dtwe+ZWDsersm+1erwe+ZWDesm+1ewe=ZWkeFnm+1gwe,8we2Vhe,(2.20)whereVhc,Vhd,Vhg,Vhh,andVheareincludedinH1(W)andcorrespondtothediscretizedniteelementspacesforallvariables.ThechoiceoftheclassicalP1niteelementspacesfornm+1g,nm+1d,sm+1n,andsm+1eisreasonable.However,forn+1,theclassicalP2niteelementspaceisrequired.Neumannboundaryconditions(nocellandchemoattractantuxes)areappliedattheborderofthecomputationaldomain.ThesolutiondependsonthechemotacticsensitivityS,activationrateA,andreleasecoefcientkc.Tothebestofourknowledge,noinformationonthepreviousthreepa-rametersisavailable.Fromtheanalysisofthemodelandthetwoconditionsonthetotalinitialmass(2.12)andtheinitialsecondmoment(2.13),thesetofdimensionlessparame-tersischosentoenlightentheblow-upconditionsstudiedtheoretically.Forthenumericalsimulation,theconstantparametersaresetasfollows:PS=1,PA=10,PR=0.001,PDc=PDn=PDe=5,Pkc=Pkn=Pks=1,PDc=PDn=PDe=1. 16M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-193ResultsAnyacupunctureischaracterizedbythedistributionofmastocytes,themajortissularconstituentassumedtobeinvolvedinacupuncture.Asmallsecondmomentofthemas-tocytedistributionatacupointmeansthatthemastocytedensityishighenoughtoberesponsivetoacupuncturestimulation.NumericalresultsarepresentedfortwoinitialmastocyteGaussiandistributionswithanidenticalcellnumber:(1)acupoint(concentrateddistribution;m0(0)=50,m2(0)=11.12)and(2)nonacupoint(disperseddistribution;m0(0)=50,m2(0)=165.19;Fig.1).Thelocationsofacupointandnon-acupointarecenteredonthepointofcoordinate(0,0)thatalsocorrespondstothepointwheretheneedleinserts.Theexpectedblo
w-upsolutionisobtainedornot(Fig.2)accordingtowhethertheinitialdistributionofthemastocytedistributionissharporblunt(Fig.1).Theexpectedblow-upsolutionisobtainedornotreliesonwhethertheneedleisin-sertedfarorclosetothelocationofthepeakcelldensityoroutsidethemastocytepool.InFig.3,theneedleisinsertedat(0,0)whiletheacupointiscenteredat( 7.5, 7.5).Therefore,asuccessfulacupuncturetargetsasubcutaneousregionwheremastocytesdenselyaggregate.Conversely,acupuncturefailswhentheneedleisimplantedinazoneenrichedinmastocytes,butthesecellsspreadoveratoolargevolume.Inotherwords,blow-upofthesolution(i.e.,thesolutiondoesnotremainbounded)occursinnitetimewhentheinitialdensityofmastocytesaroundtheneedleishighenough.Acupunctureissuccessfulwhentheneedleisimplantedintheacupointandnotinanearbypoolofmas-tocytesthatcanhavethesamecellnumber,butinagreatervolume(i.e.,lessdensemasto-cytepopulation).Blow-upsolutionthusresultsfromastrongaggregationofmastocytesandindicatestheexpertiseofthepracticianandefciencyoftheneedlemanipulationattheselectedacupoint. 0 1 2 3 4 5 6 7 8 9 10 -15 -10 -5 0 5 10 15 Figure1:InitialmastocyteGaussiandistributioninanacupoint(concentrateddistribution)andnon-acupoint(disperseddistribution)withthesamecellnumber. M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-1917 -200 0 200 400 600 800 1000 1200 1400 1600 1800 2000 -15 -10 -5 0 5 10 15 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -15 -10 -5 0 5 10 15 Figure2:Initialandnaldistributionofmastocytesatacupoint(left)andinanon-acupointmastocytepool(right),threeorderofmagnitudedierentiatingtheordinateaxisofthesetwoplots.Expectedblow-upsolutionisonlyachievedatacupoint. 0 1 2 3 4 5 6 7 8 9 10 -14 -12 -10 -8 -6 -4 -2 0 Figure3:Needlingoutsideanacupoint.Meshwithrenementsintheneedleregion(center)andmastocytepool(leftbottomcorner).Absenceofsignicantchangeincellpopulationdistribution.4DiscussionThepresentworkisaimedatinvestigatingthelocaleffectofacupunctureneedlingatasingleacupoint,butneithereffectonthebrainanditsresponsetoagiventargetor-gan,noratmultipleacupoints.
Hence,computationsarecarriedoutusingthesimpliedmodel(2.11).MeshsizeandconvergenceSevenmeshdensities(spatialdiscretizationlengthrange0.0040.3)usingthreetypesofniteelementsweretested.InthecaseofaP1P2niteelement,theslopesforthemasto-cytedensityngandconcentrationareapproximately2.50,2.11,and1.01and2.11,2.10,and2.08intheL2(W),L¥(W),andH1(W)norm,respectively.InthecaseofaP2P2niteelement,theslopesforthevariablesngandareabout2.57,2.70,and2.37and2.06,2.07, 18M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-19and2.07intheL2(W),L¥(W),andH1(W)norm,respectively.TherateofconvergenceforngintheH1(W)normishighlyimproved.InthecaseofaP2P3niteelement,theratesofconvergenceforngandarenotenhanced.SensitivitytothemodelparametersThequalityofthesolutionsofthesimpliedformofthesetofequationsdependsonthechemotacticsensitivity(S),theactivationrate(A),andthereleasecoefcient(kc).Numer-icalsimulationswerecarriedoutusingaP2P2niteelementandthreemeshsizes(1/16,1/64,and1/256)tocheckthesensitivityofthecomputationtothechemotacticsensitivityparameter(range1100).Thechemotacticsensitivityparameteraffectsstronglythesolu-tionofthemastocytedensityngandveryslightlythatoftheconcentration.Therefore,meshrenementismandatory.Inaddition,meshadaptationallowstorenethemeshlocallyaccordingtotheweightofthechemotaxistermSr ngr.5ConcludingremarksDuringacupuncture,thechemotaxismaintainsthetransmissionofpropermessengerstotheorgansofinterest,primarilythebrain.Ablow-upoccursonlywhentheinitialnumberofmastocytesishighenoughandwhenmastocytesarestronglyconcentratedattheacupoint,thatis,theyformaverydensecluster.Ablow-upsolutionofthechemo-taxisproblemmeansthatincomingmastocytesconvergetowardtheneedle,theexpectedbehavior.Accordingtotheinitialdistributionofthemastocytedistribution,thatis,atrueacu-pointoraninappropriatenearbymastocytepopulation,andtotheneedlepositionwithrespecttothelocationofthemaximumofthemastocytedistribution,thatis,accordingtowhetherthepracticianisanexpertornot,theexpectedblow-upsolutionisobtainedornot.AcknowledgmentsThisstudywaspartlysupportedbyFoundationSciencesMath´emat
iquesdeParis(FSMP).Y.D.receivedaPhD.grantfromPierreandMarieCurieUniversitySorbonneUniversityandwasalsosupportedbyCASTSattheNationalTaiwanUniversity.References[1]DeleuzeY,ThirietM,TWHSheu(2015).Modelingandsimulationoflocalphysicalstressonmastocytescreatedbytheneedlemanipulationduringacupuncture.CommunicationsinComputationalPhysics(submitted). M.Thiriet,Y.DeleuzeandT.W.-H.Sheu/Commun.Comput.Phys.,x(2015),pp.1-1919[2]ThirietM(2011).CellandTissueOrganizationintheCirculatoryandVentilatorySystems,SeriesBiomathematicalandBiomechanicalModelingoftheCirculatoryandVentilatorySys-tems(Vol.1),Springer,NewYork.[3]ZhouF,HuangD,XiaY(2010).Neuroanatomicalbasisofacupuncturepoints(Chap.2;32-80),InXiaY,CaoX,WuG,ChengJ(Eds.)AcupunctureTherapyforNeurologicalDiseases:ANeurobiologicalView,TsinghuaUniversityPress,Beijing,ChinaandSpringer,BerlinHei-delberg.[4]ThirietM(2013).TissueFunctioningandRemodelingintheCirculatoryandVentilatorySys-tems,SeriesBiomathematicalandBiomechanicalModelingoftheCirculatoryandVentilatorySystems(Vol.5),Springer,NewYork.[5]ZhangZJ,WangXM,McAlonanGM(2012).Neuralacupunctureunit:anewconceptforinterpretingeffectsandmechanismsofacupuncture.Evidence-BasedComplementaryandAlternativeMedicine2012(2012):123.[6]ThirietM(2012).IntracellularSignalingMediatorsintheCirculatoryandVentilatorySystems.SeriesBiomathematicalandBiomechanicalModelingoftheCirculatoryandVentilatorySys-tems(Vol.4),Springer,NewYork.[7]ZhangD,DingG,ShenX,YaoW,ZhangZ,ZhangY,LinJ,GuQ(2008).Roleofmastcellsinacupunctureeffect:apilotstudy.Explore(NewYork)4:170177.[8]WeiF,ShiX,ChenJ,ZhouL(2012).Fluidshearstress-inducedcytosoliccalciumsignalinganddegranulationdynamicsinmastcells.CellBiologyInternationalReports19,no.2(January1,2012):4551.[9]YaoW,LiY,DingG(2012).Interstitialuidow:Themechanicalenvironmentofcellsandfoundationofmeridians.Evidence-BasedComplementaryandAlternativeMedicine2012:19.[10]HechtF(2013).NewDevelopmentinFreeFem++.JournalofNumericalMathematics20,no.34(2013):251.[11]DeleuzeY(2013).Amathematicalmodelofmastcellresponsetoacupunctureneedling.ComptesRendusMathematique351,no.34(2013):101