<%BANNER%>

Hepatitis C Infection in the Human Liver

Permanent Link: http://ufdc.ufl.edu/UFE0043674/00001

Material Information

Title: Hepatitis C Infection in the Human Liver Viral Dynamics, Treatment and Related Side-Effects
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Debroy, Swati
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: anemia -- bistability -- hepatitisc -- interferon -- ribavirin -- svr
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Treatment of Hepatitis C Virus (HCV) infection is fraught with side-effects and successful in only 50% of the treated patients. Our goal in this dissertation is to use mathematical models to estimate individualized optimal treatment. Given a realistic patient parameter set, our model (which includes the dynamics of the side-effect of anemia) is able to estimate the amount of anti-anemia drug that is necessary to eliminate HCV using antiviral therapy without causing anemia in the process. Dynamical properties of another model for HCV infection can simulate the phenomenon of cure with biologically realistic parameters. Moreover, for a subset of patients we are able to estimate the minimum drug efficacy and minimum time of treatment necessary to achieve cure.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Swati Debroy.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Martcheva-Drashanska, Maia.
Local: Co-adviser: Bolker, Benjamin M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043674:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043674/00001

Material Information

Title: Hepatitis C Infection in the Human Liver Viral Dynamics, Treatment and Related Side-Effects
Physical Description: 1 online resource (112 p.)
Language: english
Creator: Debroy, Swati
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: anemia -- bistability -- hepatitisc -- interferon -- ribavirin -- svr
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Treatment of Hepatitis C Virus (HCV) infection is fraught with side-effects and successful in only 50% of the treated patients. Our goal in this dissertation is to use mathematical models to estimate individualized optimal treatment. Given a realistic patient parameter set, our model (which includes the dynamics of the side-effect of anemia) is able to estimate the amount of anti-anemia drug that is necessary to eliminate HCV using antiviral therapy without causing anemia in the process. Dynamical properties of another model for HCV infection can simulate the phenomenon of cure with biologically realistic parameters. Moreover, for a subset of patients we are able to estimate the minimum drug efficacy and minimum time of treatment necessary to achieve cure.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Swati Debroy.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Martcheva-Drashanska, Maia.
Local: Co-adviser: Bolker, Benjamin M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043674:00001


This item has the following downloads:


Full Text

PAGE 2

2

PAGE 3

3

PAGE 4

Itismypleasuretotakethisopportunitytothankthepeoplewhomadethisdissertationpossible.IowemydeepestgratitudetomyadvisorDr.MaiaMartchevafortheknowledgesheimpartedintome.Herrelentlesssupport,utmostencouragementandunfadingfaithinmehasmadeitpossibleformetodotheresearchIbelieveinandcompletethisdissertation.Iamimmenselygratefultomyco-advisor,Dr.BenBolker,forstructuringthebiologicalandnumericalpartofthehumblemathematicalbiologistIamtoday.Iappreciatehisabundantpatience(whichIsometimesabused),andalwaysgivingmorethanthreewaystohandleeveryresearchproblemhastaughtmetothinkindependently.IwouldliketothankalltheprofessorsatUFwhohavetaughtmethroughtheircoursesandmycommitteemembersfortheirtimeandinvaluablesuggestionsthathelpedtoimprovemydissertation.ItisanhonorformetothankDr.CarlosCastillo-ChavezforgivingmetheopportunitytobeapartofMathematicalandTheoreticalBiologyInstituteinArizonaStateUniversityfortwosummerworkshopsandmentoringme.Theresearchexperiencethereenrichedmyunderstandingofmathematicalbiologyandalsodirectedmetochoosemytopicfordissertation.IamgratefultoDr.ChristopherKribs-Zaletawhomadehishelpandadviceavailableinresearchandprofessionaldecisions.Ithankthesupportofafriend,philosopherandguideIfoundinDr.AnujMubayi.Iamindebtedtomyclosestfriendsandcolleagues,namely(inalphabeticalorder):ShibdasBandyopadhyay,forhelpingwithallsoftwareissueseveninprogramminglanguageshedidnotknowandbeingapatientlistenertomyrantsthroughthejobsearchperiod;SouvikBhattacharya,forbeingacompanionthroughthecoursework,qualiersandalargechunkoflifeatUFingeneral;MiriamCastillo-Gil,forbeingatruefriendandalsoteachingmealotofthewaysoflivinginthiscountry;SujataHalder,inwhomIfoundthesisterIneverknewIhad;OliviaProsperforbeinganon-judgementalcondanteandalike-mindedcolleague. 4

PAGE 5

5

PAGE 6

page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 11 1.1BackgroundandtheProblem ......................... 11 1.2TheImmunologicalBackdrop ......................... 13 1.3TheCombinedDrugAction .......................... 14 1.4Side-effectofHemolyticAnemia ....................... 16 1.5TheCureThreshold ............................. 17 2REVIEWOFTHERELEVANTLITERATURE ................... 21 2.1ContributionofMathematicalModelingtoUnderstandingofwithin-hostHepatitis-C ................................... 21 2.2MathematicalInvestigationoftheHCV/HIVModelingsystems ....... 26 3EVALUATINGTREATMENTFORHEMOLYTICANEMIAMANAGEMENT-SIMPLEVIRALMODEL .................................... 31 3.1IntroductoryRemarks ............................. 31 3.2Model ...................................... 34 3.3AnalyticalWork ................................. 39 3.3.1Analysisof(R,C)system ....................... 39 3.3.2Analysisof(T,I,V) 43 3.3.3CriticalDosages ............................ 53 3.4NumericalSimulation ............................. 55 3.4.1ModelParameterEstimation ...................... 55 3.4.2NumericalResults ........................... 61 3.5ConcludingRemarks .............................. 64 4BACKWARDBIFURCATIONANDLONG-TERMCUREINANODEMODELOFHCV ........................................ 67 4.1IntroductoryRemarks ............................. 67 4.2Modeldenition ................................. 71 4.3Bistability:physicalinterpretation ....................... 73 4.4MathematicalAnalysis ............................. 76 4.4.1TheEquilibriums ............................ 76 6

PAGE 7

........................ 78 4.4.3CalculatingRc 80 4.4.4StabilityAnalysisofEndemicEquilibria ................ 81 4.5NumericalExploration ............................. 85 4.5.1RealisticParameterSets ........................ 85 4.5.2ObservedPatientProles ....................... 86 4.5.3TreatmentEstimation .......................... 88 4.6ConcludingRemarks .............................. 91 5VIRALDYNAMICSINHCVPATIENTS-PDEWITHAGE-SINCE-INFECTION 94 5.1AgeofInfectionModel ............................. 94 5.2AnalyticResults ................................ 97 6CONCLUSION .................................... 103 REFERENCES ....................................... 106 BIOGRAPHICALSKETCH ................................ 112 7

PAGE 8

Table page 3-1ThisTablegivesabriefinterpretationoftheparametersusedinthemodel. .. 39 3-2Basevaluesusedtodeterminedosingparameters ................ 57 3-3ParameterValueTable ................................ 60 3-4InitialConditionsofStateVariables ......................... 61 4-1Realisticrangeformodelparameters ........................ 85 8

PAGE 9

Figure page 1-1Qualitativeviralload(HCV-RNA)dynamicsintreatedpatients .......... 18 3-1SchematicrepresentationofsensitivityofRBCconcentrationinthebodyandviralloadtotheamountofdruginthebody. .................... 33 3-2CompartmentalModel ................................ 34 3-3Plotoftheparametriccurve. ............................ 42 3-4PlotsCAvsCSVRtondbc. ............................. 62 3-5InthisgraphweobservethetrajectoriesoftheRBCconcentrationatdifferentlevelsofEPOadministration. ............................ 64 3-6Viralloaddynamics. ................................. 65 3-7Anillustrationofthedrugamountinthebody,RBClevel,andvirioncountresultingfromcontinuous(smoothcurve)vs.discrete(sawtooth)dailyadministrationofHCVdrugsduringtreatment. ........................... 65 4-1Anexampleofthestabilityproleofthemodelwithrealisticparameters .... 74 4-2DistinctPatientProles ............................... 86 4-3StreamPlotofPatientI. ............................... 89 4-4UnderstandingtreatmentstrategyforPatientI ................... 90 5-1Showstheschematicmodel. ............................ 94 9

PAGE 10

TreatmentofHepatitisCVirus(HCV)infectionisfraughtwithside-effectsandsuccessfulinonly50%ofthetreatedpatients.Ourgoalinthisdissertationistousemathematicalmodelstoestimateindividualizedoptimaltreatment.ThecombinationtherapywithInterferon-(IFN)andRibavirin(RBV),alsodepletesredbloodcellscausingthereversibleside-effectofhemolyticanemia.Weuseanordinarydifferentialequations(ODE)tomodelthisdynamicsandgivenarealisticpatientparametersetweareabletoestimatetheamountofanti-anemiadrugthatisnecessarytoeliminateHCVusingIFNandRBVwithoutcausinganemiaintheprocess.Also,weusethebackwardbifurcationpropertyofanODEmodelforHCVinfectiontosimulatethephenomenonofcurewithbiologicallyrealisticparameters,withoutimposinganyexternalcriteria.Moreover,forpatientswithparametersetsatisfyingtheconditionsforbistability,weareabletoestimatetheminimumdrugefcacyandminimumtimeoftreatmentnecessarytoachievecure.Toenhanceourunderstandingoftheimplicationsofbistabilityforawiderrangeofpatientpopulation,wegeneralizethepreviousODEmodeltoformulateanage-since-infectionpartialdifferentialequationmodel.Thismodelprovidestheframe-workfortheuseofnon-exponentiallydistributedinfectionparameterstobetterunderstandtheimplicationsofthebistabilityproperty. 10

PAGE 11

44 ].Currently,thereisnovaccineforHCVinfection.ThemajormodeoftransmissionofHCVisbyexposuretoinfectedblood.SexualandverticaltransmissionofHCVhasbeenreported;however,itisrare[ 34 ].HepatitisC(Hep-C)causeschronicdiseasesoftheliverlikecirrhosisandhepatocellularcarcinoma[ 38 ].TheHCVinfectshepatocyteswhichformamajorportionofthecytoplasmicmassoftheliver.AlthoughHCVpredominantlyreplicatesinhepatocytes,tracesofithavebeendetectedinothercelltypes[ 2 61 ]. SomepatientswithHep-Cinfectionwillnaturallycleartheviruswithoutmedicalintervention.However,amajorproportionofHCVinfectedindividualsdevelopchronicHCVinfectioninwhichthebody'simmunesystemdoesnotnaturallyclearthevirus.About55to85%ofHCVpatientsdonotclearthevirusthemselvesanddevelopchronicHep-Cinfection[ 34 ].Theprogressiontochronic-stageHCVinfectionisaresultofweakimmuneresponseagainstHCV(reviewedin[ 32 ]).Currently,thestandardprotocolforthetreatmentofHep-Cinvolvestwoantiviraldrugs,Interferon-(IFN)andribavirin(RBV),givenincombination[ 11 21 ].IthasbeenclinicallyobservedthatthecombinationofRBVandIFNdemonstratesasynergisticpharmacologicaleffect.Thatistosay,thetreatmentefcacyofthiscombinationtherapygoeswellbeyondtheefcaciesoftheindividualtreatmentsaddedtogether[ 44 ].AlthoughtheexactmechanismoftheactionofeitherofthesedrugsintreatmentofHep-Cinfectionisnotclear,directantiviral,antiproliferative,andimmunomodulatoryactivitiesofIFNarewellknown[ 44 ].InthecaseofRBV,manyscientistsseemtofavoritsimmunomodulatoryactionasanexplanationofitsefcacyinHCVtreatment;however,itsroleasaviralRNAmutagencannotbeignored,asdiscussedinLauetal..andreferencestherein[ 44 ]. 11

PAGE 12

21 ].IfapatientdoesnotshowtracesofHep-Cviralloadinthebodyaftersixmonthsoftherapycessation,thepatientissaidtohaveachievedSustainedVirologicalResponse(SVR),implyingclinicallycured.ThegoalofthistreatmentistolowertheviralloadandeventuallyachieveSVR.IfSVRisnotachieved,apatientisconsideredachronicHep-Cpatient.TherateofachievementofSVRinHep-Cpatientsvariesaccordingtothegenotypeoftheparticularinfectingvirus.Thetreatmentforgenotype1and4areusuallycarriedoutfor48weeksandthatforgenotype2and3typicallylastsfor24weekswithlesserdosingofRBV[ 21 ].Underthistreatment,SVRratesof46%areobservedinpatientswithHCVgenotype1,whereas76%and82%ofpatientswithHCVgenotypes2or3achieveSVR[ 60 ].Although,thisisafairlysuccessfultreatmentforHep-Cpatients,PEG-RBVcombinationtherapyisexpensiveandassociatedwithtreatment-limitingside-effectsincludinganemia,neutropenia,thrombocytopenia,u-likesymptoms,depressionandageneral'ontreatment'poorqualityoflife[ 31 ]. InthecontextofHep-C,mathematicalmodelinghasbeenextensivelyusedtodeterminetheefcacyofIFNasmonotherapyandincombinationwithRBV[ 11 22 ].Differencesinresponsedependingongenotype[ 48 ],andtheconceptofearlyvirologicalresponse(EVR)toestimatepossibilityofachievingSVR[ 17 ]havebeenreinforcedwithmathematicalmodels.Viralanddrugkineticsstudiesusingmathematicalmodelshaveshedlightontheunderstandingofthisvirusanditstreatmentinseveraldirections[ 53 ].Allthesestudieshavecontributedenormouslytotheimprovementofthetreatmentprocedurethatdoctorspresentlyapplytopatients.Wefurtherthisefforttodevelopmathematicalmethodologiesthatcanbeusedtoproduceseveralobservedpatientproles,estimateseverityofside-effects,specicallyhemolyticanemia,inagivenpatient.ThisstudywillhelpthedevelopmentofindividualizedtreatmentofHepatitis-Cinthefuture. 12

PAGE 13

Mostoftheimmuneresponsesaremediatedbyavarietyof'cellsoftheimmunesystem'.Thecellsoftheimmunesystemarecategorizedaslymphocytesandantigenpresentingcells.ThelymphocytesaretheB-lymphocytes(B-cells)andtheT-lymphocytes(T-cells)andtheantigenpresentingcellsarecellsthatinternalizetheantigensbyphagocytosisorendocytosisandexhibitappropriateantigen-proteinstoactivatetheB-andT-cells.Uponactivation,B-cellsreproduceswiftlyandtheoff-springsseparateintomemorycellsandtheplasmacells.Theplasmacellssecreteanenormousamountofantibodieswhicharetheleadingeffectorsofthehumoralimmunesystem.WhenT-cellsinteractwithantigensboundtotheproteinmajorhistocompatibilitycomplex(MHC)molecules,theygetactivatedproliferatefastandsetapartasmemoryandvariouseffectorcells.ActivatedT-cellsaregenerallyT-helper(TH)lymphocytesandT-cytotoxic(TC)lymphocytes,displayingCD4andCD8respectively.TheTHcellsrecognizeandinteractwithMHCclassIImoleculesandsecreteseveralcytokines.SomeofthecytokinessecretedbytheTHcellshelpactivatetheTClymphocyteswhichendupproducingthecytotoxicTlymphocytes(CTL)whichactuallyeliminatesanyantigenpresentingcell,e.g.virusinfectedcells,cancercellsetc.[ 29 ]. IFNsarealsocytokinesproducednaturallybyamammalianbodyinresponsetoviralinfection.ItisknownfromstudiesdoneintissueculturesofmurinethatIFNsplayakeyroleinmountingtheinnateantiviralimmuneresponsebytargetingtheuninfectedcells.Ingeneral,IFNsareoftwocategories,viz.TypeI(IFN-=)andTypeII(IFN-). 13

PAGE 14

29 ],[ 58 ]. 29 ].Thisproducestheneedformedicalintervention. Interestinglyenough,IFNandvirusesmaintaina'dynamicequilibrium'innature.Thisequilibriumcomesfromatrade-offbetweentheviruswantingtoreplicatemore,eveninthepresenceofIFNandnotreproducingsomuchastokillthehost,therebyeliminatingahugereservoirofitsownkind.IthasbeenobservedinlaboratoryexperimentsthatvirusesareverysensitivetohighIFNdosesandconversely,limitedquantityofIFNcanbemadequiteimpotentinpresenceofahugeviralload[ 58 ].ThisisoneofthereasonsthatIFN,morespecicallyIFN-issoeffectiveinHepatitisCpatients.ThereisaremarkabledecreaseinviralloadintheinitialdaysofIFN-mono-therapy[ 17 ],althoughSVRisachievedonlyin16%ofthetreatedpatients. 14

PAGE 15

Mono-therapyofRBVhasonlyminimaleffectontheHepatitisCviralloadbutitspharmacologicalsynergisticeffectincombinationwithIFN-isphenomenal[ 44 ].HenceitmakesmoresensetolookatthemechanismsofactionofRBVmostlyincontexttocombinationtherapywithIFN-ratherthanbyitself.Althoughtheexactmodeofactionofeitherofthedrugsinthecombinationhasnotbeendetermined,thereareseveralexistinghypotheses,noneofwhichcanbedisproved!FourofthemajorprevalenthypothesesregardingactionofRBVarereviewedinLauetal.[ 44 ].Firstly,RBV'spotentialtoincreaseType1immuneresponseanddelayingtheonsetofType2immuneresponsehasbeenestablishedinmurinemodels,whichcanbeaprobablemechanisminhumansalso(detailedexplanationfollows).Secondly,RBVhasalsobeenobservedtoinhibitthehostinosinemonophosphatedehydrogenase(IMPDH)enzymeconcentrationleadingtoimmune-suppression,althoughthisactionmightnotbewhollyresponsibleforitsantiviraleffectagainstHCV.Moreover,intotality,itistheimmunomodulatoryaspectofRBVactionagainstHCVratherthantheimmunosuppressiveangle,whichprovidesabetterexplanationtotheclinicallyobservedeffects.Thirdly,RBVcanalsoweaklyinhibitHCVreplication.Fourthly,instudiesofGBvirus-BRBVincreasedmutationrateoftheRNAtoreducethetnessoftheprogeniesbyagreatextent,aphenomenoncalled'errorcatastrophe'[ 42 ].Progeniesthataregeneticallytness-compromisedarelessproductiveandmoresusceptibletothehostimmunalcombat.ThisisalsoaproposedmechanismofactionofRBVeventhoughitisyettobeexactlydeterminedincontextofHCV.[ 44 ] SincetheimmunomodulatoryactionofRBVismostwidelyobservedinmurineandin-vitromodelsIdirectfurtherexplanationstowardsthat.DetailedelucidationofthismechanismofRBVontheIFN-effectrequiresintroductionofType1andType2T-cellmediatedimmuneresponse.TheType1immuneresponsetakesplaceintheearly 15

PAGE 16

44 ].Asmentionedabove,HCVhasarapidandhighmutationratewhichovercomesthestrain-speciccytokineproducingcapabilityofthebody.Thatiswhy,eveninthepresenceofIFN,severalstrainsofHCVcontinuetoourishwithoutanyhinderance.Therefore,theType2responsehasonlylimitedsuccess(SVRrateforINF-is16%).RBVseemstoshifttheimmuneresponsemoretowardstheType1whichisantiviralbutnon-HCV-specicdefensemechanism,thusinhibitingallkindsofviralparticlesirrespectiveoftheirstrain.Alogicaltheorythatemergesisthat,IFNandRBVpullstheimmuneresponseonoppositedirectionssettlingforanintermediatecombinationofbothresponsestocauseoptimaldeclineinviralloadintherstfewweeksfollowedbysuccessfulachievementofSVR. 44 ][ 39 ].ThereareseveralproposedreasonsforRBVinducedhemolyticanemia.ThemajormechanismisthoughttobetheoxidativestresscausedbyRBVtotheRBC[ 39 ].Oxidativestressoccursinacellwhenthereismorereactiveoxygenpresentthanwhatitcanneutralize.OnceRBVentersdifferentcellsinthebodyitconvertstoitsphosphorylatedderivatives.MostnucleatedcellsintheplasmacanclearoutthisRBV-phosphatesuccessfully.However,erythropoietinlackstheenzymeneededto 16

PAGE 17

25 ]observedconsiderabledeclineinadenosinetriphosphate(ATP)levelswhenerythrocytewasexposedtoRBV.OxidativestresscausedbydepletionofATPisknowntopreventcontrolledapoptosisleadingtobreakdownofthecell. Ingeneralwhenabodyfacesanemia,thebonemarrowstartsproducingmoreerythrocytestocompensatethebloodloss.DeFranceschietal.reportfromtheirclinicaltrials,thatHCVpatientstreatedwithRBValoneshowgreaterproductionofreticulocytes(i.e.theimmatureRBCs),thanwhentreatedwithIFN-also[ 25 ].ThisprovesthebonemarrowsuppressiveactionofIFN.MorethanonemechanismhasbeenproposedtoexplaintheIFN-inducedanemiaintreatedHepatitis-Cpatients.InvitrostudiesofKatoetal..[ 37 ]suggestthatproliferationofdifferentblood-buildingcellsishinderedbyIFNusingintracellularsignalingpathways.Otherstudiessuggestenhancedprogrammeddeathoferythroidcells,andautoimmunehemolyticreactions. Toreducetheseverityofhemolyticanemia,doctorsprescribeathirddrugEpoietin,whichisasyntheticallypreparedhormoneerythropoietintoincreaseRBCproductioninthebonemarrow.EpoietinhasbeenobservedtoincreaseRBCproductionbyuptothreetimesofthepatient'sintrinsicRBCproductionratedependingupontheefciencyofthebody'simmunesystem[ 60 ].EpoietinhasfacilitatedmanyHCVpatientstogothroughtheentirecombinationtherapyprotocol,alsoachievingSVRsuccessfully.However,itistobenotedthat,increasingtheRBCproductionexcessivelyhasledtoautoimmuneRBCaplasiainanegligiblefractionofpatientsreceivingepoietin[ 8 ]. 17

PAGE 18

28 ].However,ithasbeenobservedthatseveralpatientswhoexhibitundetectableviralloadinresponsetotreatmentduringtherapy,oftendonotachievesustainedvirologicalresponseinthelongrun.Thatis,viralloadisdetectedagainatsomepointintheirfuture.Considering,itisreasonabletoexcludethepossibilityofre-infection,thedoctorsarecompelledtosuspectthattheviruswasneverclearedcompletelyintherstplace,inthesepatients.Butthen,thequestionthatevolvesiswhatisthedifferencebetweenapatientthatrelapsesandapatientthatachievesSVR?WediscusspossibleanswerstothisquestionwithrespecttoworkdoneinexistingliteratureinChapter2,andourinterpretationandpossibleanswertothisquestioninChapter4ofthisdissertation. Qualitativeviralload(HCV-RNA)dynamicsintreatedpatients,modiedfrom[ 28 ]. Asaconsequenceofthisambiguity,responseinpatientstotreatmentaremeasuredbysurrogatevirologicalparameters,insteadofaspecicclinicalend-point.Severaltypesofvirologicalresponsethresholdsaredeneddependingontime,relativetotreatmentduration,havingdifferentialdegreeofreliabilityasaprescientoflongtermclinicalcure[ 28 ].Themostimportantamongtheseisthesustainedvirologicalresponse(SVR),whichistheabsenceofHCVviralloadaftersixmonthsbeyond 18

PAGE 19

60 ]. Therapidvirologicalresponse(RVR)isdenedasobservationofundetectableviralloadfourweeksintotreatmentwithalowerlimitof50IU/mL.FulllmentofthisthresholdindicatesahighprobabilityofsuccessfulachievementofSVR[ 35 64 ]. Anothertherapeuticlandmarkisend-of-treatmentresponse(ETR),whichisdenedbyundetectableviralloadattheendof24or48-weekcourseoftherapy.ETRisanecessarybutnotsufcientconditiontoachieveSVR. Anearlyvirologicalresponse(EVR)characterizedby2logreductionorundetectableviralloadatweek12oftherapyisyetanothernecessarybutinsufcientpredictorofachievementofSVR[ 17 26 ].Othertermslike'relapse'denotesthereappearanceofHCVRNAinserumaftertherapyisdiscontinued;'nullresponder'isapatientwhofailstoachieveadecreaseinHCVRNAby2logsafter24weeksoftherapy;'partialnonresponder'isonewhoexhibits2logdecreaseinHCVRNAbutstillHCVRNApositiveatweek24;[ 28 ].SometypicalHCVpatient'sviralprolesfortheabovementionedresponsescanbeseeninFigure 1-1 OnehypothesisthatexplainsthedifferencebetweentherelapsepatientsandpatientswhoachieveSVRisthattherecouldexistacurethresholdbelowtheclinicaldetectionlevelofHCVviralload.Snoecketal.[ 59 ]haverecentlydenedthisthresholdtobetheviralconcentrationcorrespondingtolessthanonevirusintheentireextra-cellularuidofahuman.TheauthorshavebeenabletoproducethephenomenonofSVRusingmathematicalmodelsbyimposingthisthresholdasanexternalcriterion,suchthat,whenthesimulatedviralloadgoesbelowthethresholdcriterion,thenumberofinfectedhepatocytesisimmediatelysetequaltozero,thusachievingtheinfectionfree 19

PAGE 20

20

PAGE 21

9 ].IntheirarticleinSciencein1989,theyre-namedtheNon-A,Non-BHepatitisasHepatitisC.Duetothelackofknowledgeofthenaturalhistoryofthediseaseseveralretrospectiveandlaterretrospective/prospectivestudieswereconducted(reviewedin[ 57 ]).Observationfromthesestudiesraisedmorequestionsthantheyanswered.Foradiseasewithsuchalongprogressionperioditwasnecessarytopredictseverityofoutcomestodetermineneedfortreatment.Also,inefcientinvitromodelshinderedunderstandingofthevirusitselfandtheactionoftheantiviralIFN. Mathematicalmodelingofviraldynamicsin-vivo,underantiviraltherapyatthattime,hadprovedverysuccessfulinunderstandingthepathogenesisandguidingtherapyofhumanimmunodeciencyvirus(HIV)andHepatitisB.TherstmathematicalmodelusedforstudyingthekineticsofHCV-RNAunderIFN-therapyin1996,[ 65 ],consideredaviralmodelwithordinarydifferentialequationsystemwithonlyproductivelyinfectedhepatocytesandviralload(aspecialcaseof 2 below).IntheirtstodatatheyconsideredtheactionofIFN-asblockingdenovoinfectionwith100%efcacyorblockingviralproductionwith100%efcacy.Inthelattercase,theywereabletoproduceonlysingleexponentialviraldecayinresponsetotherapy,whereasmostpatientshowedabiphasicviraldecline.However,withtheassumptionofcompletelyblockingofdenovoinfection,agoodttothedatagivingthebiphasicdeclinewasobserved[ 65 66 ]. 21

PAGE 22

2 and 2 below)[ 41 ].Thismodelallowedforconsideringtheefcacyofthedrugtoactbothtowardsblockingdenovoinfectionandproductionofvirionsfrominfectedcellatthesametime.Themajorcontributionofthispaperwastoestablishthedose-dependenceofHCV-RNAdecline.TheyobservedthatanIFNdoseof10and15mIU(mili-InternationalUnits)gavefarbetterinitialviralresponsethan3mIU.However,theirestimateoffreevirionclearanceratewaslesssuccessfulduetoinfrequentsamplingofpatientserum. Neumannetal.in1998,furtherextendedthepreviousmodelbyincludingaseparatecompartmentforsusceptiblehealthyhepatocytes( 2 ).Thesimpleviralmodel,previouslyusedforHIV,isasfollows. dt=sdT(1)TV dt=(1)TVI dt=(1)pIcV Here,T,IandVrepresentthenumberoftargethealthyhepatocytes,productivelyinfectedhepatocytesandtheviralload.Theparametersistherateofproductionofhealthyhepatocytesfromthebonemarrow,anddistheirnaturaldeathrate.Theparameteristhenumberofinfectionscausedbyonevirionperunittimeandistherateatwhichinfectedhepatocytesareclearedperunitoftime.Therateofreleaseorproductionofvirusisgivenbyp,whilecistheclearanceoffreevirus.Also,istheefcacyofthedruginstoppinginfectionandisitsefcacyinblockingproduction.Notethatthepreviousmodelshadconsideredthetargetcellpopulationtobeconstantduringtherapy,whichwasbiologicallyunrealistic.Withamuchmorethoroughtimeseriesdatasetfrom23patientsNeumannetal.observedthatdailyIFNdosesof5,10and15mIUwascorrelatedtoviralreleaseandproductionblockingefcacyof81,95and96%efcacyrespectively.TheirestimatedHCVvirionhalf-lifeof2.7hoursisarguablythe 22

PAGE 23

41 ]to1012.Theformerunder-estimationwasduetomistakenlyassumingtwoRNAcopiespresentineachvirion.Theyalsoobservedthatthehalf-lifeofinfectedhepatocytecellsvariedwidelyfrom1.7to70days.ThustheproductionrateofHCV-RNAwasfoundtobefasterthanevenHIV(10.3109)[ 52 ]explainingtheobservedviraldiversity.Thefastreplicationratecreatesfavorableconditionsforthevirustodevelopresistancetohostimmuneresponseandantiviraltherapy.DiscoveringthehighreplicationrateofHCVfurtherhelpedtodevelopahypothesisregardingthefuzzysynergisticactivityofIFNandRBVcombinationtherapy. Severalarticleswerepublishedsubsequently,whichusedthismodelanditsslightvariationstobetterunderstandHCVinfectionunderdifferentscenarios,includingHCVcoinfectioninpatientswithHIV.AmongtheliteratureforHCVinfectiononly,thearticlebyHerrmannetal.in2003[ 33 ],ismostrelatedtotheresearchpresentedhere.InaclinicaltrialHermannetal.observedamorerapidthirdphasedecline(roughlyday7to28fromtherapyinitiation)inpatientstreatedwiththecombinationofpeg-IFNandRBVcomparedtotheonestreatedwithpeg-IFNalone.Thisrapiddeclinealsotranslatedtobetterend-of-treatmentresponse(undetectableviralloadatendoftherapy)andmoresustainedvirologicalresponse(SVR).Theyexplainedthisphenomenonwiththeirmodelbyincreasingthedeathrateofinfectedcells.In2004,Pawlotskyetal.[ 51 ]hypothesizedthatthetransienteffectobservedinviraldeclineuponadditionofRBVcouldbeduetoantiviraleffectofRBV.InthemathematicalmodelusedinPawlotsky'spaper,itwasassumedthatRBVrenderedaproportionofnewlyinfectedhepatocytesunproductive,whichmovedbackintothesusceptiblehepatocytecompartment. AmajorqualitativechangetomathematicalHCVmodelswasmadebyDixitetal.in2004[ 22 ]inordertoestimatetheeffectofRBVtomonotherapyofIFN.Contemporarybiologicalevidencein[ 10 ]suggestedthatRBVmadeapartofthenewlyproducedvirionsnon-infectious.Followingthispaper,Dixitetal.incorporatedaproportionof 23

PAGE 24

33 ],butDixitetal.[ 22 ]interpretedtheexcesslossofvirionsinthesecondphase,thiseffectusingthepresenceofnon-infectiousviralparticles. In2005,Daharietal.[ 12 ]analyzeddatafrom10HCVinfectedChimpanzeestobetterunderstandthevirus-hostdynamicsinprimaryinfectionofHCV.TheyconcludedthatthenaturallyoccurringtypeIinterferonhindersproductionofnewvirusintheinitialacutestageofinfection.Also,thatbothcytolyticandnoncytolyticmechanismswereresponsibleinvaryinghalf-lifeofinfectedhepatocytes,whichinturnwascorrelatedtopartialorcompleteviralresponse.In[ 12 ],theauthorsconsideredaseparatedifferentialequationtorepresentthealanineaminotransferase(ALT)concentration.Thenovelfeatureinthismodelwastheintroductionofhomeostaticproliferationofhepatocytes,whichlaterprovedtoplayakeyroleinunderstandingseveralviraldynamics.Theirndingsareisalsoverypivotalintheresultsofmyresearchwork. AnamalgamationofthemodelsinDixitetal.[ 22 ]andDaharietal.[ 12 ]waspublishedbyDaharietal.[ 13 ]whichcouldgeneratethetriphasicdeclineoftheviralload,observedinseveraltreatedpatients,whichthepreviousmodelswereunableto 24

PAGE 25

dt=s+rTT1T+I TmaxdTTVIdI dt=TVI+rII1T+I TmaxI 2 ispartitionedintoinfectiousandnon-infectiousvirions,withthedrugefcacyofIFNandRBVasandrespectively.Otherthanproducingthetriphasicviraldecline,observationsfromthismodelalsosupportedthemutageniceffectofRBVonHCV.Followingthispaper,therewasanotherarticlebyDaharietal.[ 11 ],whichcollapsedthelasttwoequationintheabovemodel 2 intoasingleequationlikethe 2 .Thenewmodelwasalsoabletoproducethetriphasicviraldeclineanddeneacriticaldrugefcacyforviruselimination.Thiscriticaldrugefcacyisderivedbyusingasthetranscriticalbifurcationparameter,suchthatwhenthedrugefcacyisabovethisthreshold,abiphasicortriphasicdeclineisobserved. Ahugerangeofresearchhasbeendonewiththehelpofthemodelintroducedin[ 11 ].Apartfromttingtoclinicaltrialdata,Daharietal.[ 14 16 ]havetestedtheirhypothesistoaround10,000simulatedpatients,whereeachparametersetrepresentsauniquepatient.Daharietal.[ 14 ]havealsostudiedtherelationshipamongdrugefcacy,infectedhepatocytedeathrate,rateofnalphasedeclineandthebaselinefractionofinfectedhepatocytesbeforetreatment.TheotherarticlebyDaharietal.[ 16 ]usedthesamemodeltoshowthatahighbaselineviralloadreducesthechancesofachieving 25

PAGE 26

CurrenttreatmentofhepatitisCvirus(HCV)islengthy,expensiveandfraughtwithside-effects,succeedinginonly50%oftreatedpatients.Thishighlightstheneedforpredictingchancesofachievingcureinpatientsverysoonafterinitiationoftherapy,ifnotbeforeanytreatment.Thenull-respondersandpartialresponders(Figure 1-1 )arerelativelyeasiertodetectanddoctorscanmodifyorstoptherapyuponidenticationofsuchpatients.ThedistinctionbetweenthepatientsthatrelapseafterfullcourseoftherapyandthepatientswhogoontoachieveSVRisthemostchallenging.Toexplainthissituationscientistshypothesizetheexistenceofasub-clinicalcurethreshold,asexplainedinChapter1.Snoecket.al.[ 59 ],usetheModel 2 toquantifysuchaviralcount.Theideausedhereisthestochasticextinctionoftheviruswhentheviralloadisreducedbelow1virusintheentireextracellularuidinapatient.TheauthorshaveusedthisthresholdasaexternalcriteriainpreviousmodeltosuccessfullysimulaterealisticviralpatternofrelapseintreatedHCVpatients.ThiscouldpotentiallyresultinabreakthroughinfuturetherapeuticstrategiesofHCVtreatment. 2 .Themodelin 2 hasbeenusedforstudyingHIVbyNowaketal.in1996[ 50 ].Amoregeneralversionofmodel 2 hasbeenextensivelyanalyzedbyDeLeenherandSmith2003[ 20 ].Thegeneralizedmodelisthe 26

PAGE 27

2 .Here,representsthepercapitadeathrateofinfectedcellsthatproducenewvirionsandNrepresentsthenumberofnewvirionsproducedbyeachinfectedcell.Notethat,theabovemodelismathematicallyidenticaltothemodel 2 Theauthorscomputeabasicreproductionnumber,R0whichgivesthetotalnumberofinfectionsthatwillbecausedbyasinglevirusifintroducedintoacompletelysusceptiblepopulation.Mathematically,theR0isathresholdparameterwhichgivesthefollowingbehavior. 1. ForR0<1,thereisonlyasinglesteadystate,thatisthediseasefreeequilibrium(DFE)givenby(T0,0,0).Thisequilibriumpointisgloballystable.Inthiscase,thevirusiscleared. 2. ForR0>1,therearetwoequilibria.TheDFEandauniqueendemicequilibrium(EE),(T,I,V)characterizedbynon-zeroexistenceofthevirusandproductivelyinfectedcells.TheDFEisunstableifR0>1whereastheEEislocallyattractingiff0(T)0. 3. UnderappropriateconditionstheauthorsshowglobalstabilityorsustainedoscillationsoftheEE. Inrelevancetomywork,thefunctionf(T)isrepresentedbyaconstantrecruitmentrateandanaturaldeathrate,viz.,sdT.Againin[ 19 ],DeLeenherandPilyugindenedaLiapunovfunctionsimilarmodels.Iusetheirapproachtoestablishglobalstabilityintherstmodelinthisdissertation. Relugaetal.analyzedageneralversionoftheModel 2 oftheHCVmodelusedbyDaharietal.in[ 11 13 ].Theparametersinthismodelarethesameasin 2 ,and 27

PAGE 28

dt=s+r1T1T+I TmaxdT(1)TV+qIdI dt=(1)TV+r2I1T+I TmaxIqIdV dt=(1)pIcV In[ 55 ],theauthorsconcludedthatthewithin-hostHCVinfectionwillspreaditselfonlyifthebasicreproductionnumber,R0isgreaterthan1.Andinthatcase,theinfectionpersistsatanon-trivialequilibrium,towhichtheviraldynamicsconvergeseithermonotonicallyorthroughdampedoscillations.Thenoveltyofthe 2 modelwastheintroductionofthehomeostaticproliferationoftheuninfectedandinfectedhepatocytes.In[ 55 ]theauthorsexplorethequalitativeandquantitativechangeinthebehaviorofthemodelduetothisinclusion.TheyobservethatwhenrT>rI,thishasonlyalimitedeffectonthereproductionnumber,butreducestheoscillationsandmagniesproductionofhealthyhepatocytes.But,whenrT
PAGE 29

AgestructuredpartialdifferentialequationmodelshavebeenusedforHIVbyNelsonetal.[ 47 ],Rongetal.[ 56 ]amongothers,tostudytheeffectofantiviraltherapyondifferentaspectsofviralreproductionprocess.Inmodelin 2 givenbelow,theydifferentiateamongtheinfectedcellsbasedonhowlongtheyhavebeeninfected.Thebaselinemodelinthepapermentionedaboveisasfollows(parametersrenamedtomaintainconsistency). dt=Z10p(a)I(a,t)dacV Inthismodel,T(t)andV(t)representtheconcentrationofuninfectedtargetTcellsandvirionsattimetrespectively,whereass,d,,crepresentthesameparametersasbefore.ThetermI(a,t)representsthedensityoftheconcentrationofinfectedT-cells,whichhaveinfectionagea(i.e.timeelapsedsincetheHIVvirushadrstpenetratedintotheT-cell)atanygiventimet.Tobemorebiologicallyrealisticthedeathrateandtherateofproductionofnewvirionsfromaninfectedcellvariesdependingontheageofinfection.Thustherates(a)andp(a)arefunctionsoftheageofinfection.Forthismodeltheauthorsprovedtheexistenceofapositivesolutionandperformedstabilityanalysisofthesteadystates.TheyshowedthattheDFEwaslocallyasymptoticallystablewhenthebasicreproductionnumberR0<1andunstablewhenR0>1,infactforR0<1theDFEisaglobalattractor.AlsothelocalasymptoticstabilityoftheEEforR0>1wasproved.IncaseofHIV,Nelsonetal.in[ 47 ]suggesttheuseofthefollowing 29

PAGE 30

TheauthorsassumethatthereisatimedelaybeforetheinfectedT-cellsexpressthesurfaceproteinstobecomesusceptibletokillingbytheimmuneresponse,ordieduetoexhaustioncausedbyviralreproduction.Thusthesuggestthefollowingfunctionalform.(a)=8><>:0,aa2 Wehaveusedasimilarimmunologicalage-of-infectionmodelandperformedlocalstabilityanalysis. 30

PAGE 31

1 ].TheRBVinducesexcessivehemolysis,thatis,thebreakdownofredbloodcells(RBC)andreleaseofhemoglobinintothesurroundingbloodplasma.Thebody'sabilitytoproduceRBCatafasterratetocompensateforthisexcessivelossisstuntedbythesimultaneousbone-marrowsuppressingeffectofIFN[ 60 ].Theadjective'reversible'signiesthefactthatinmostcases,theRBCcountgetsbacktonormallevelsoncethetreatmentisstopped.Tocombatthesituation,doctorsoftenprescribeathirddrug,epoietin-(EPO),whichactslikethehormoneerythropoietinandfacilitatesRBCproductioninthebonemarrow[ 60 ].Epoietin(alsospelledepoetin)isobservedtobequiteefcientatreversingRBV-inducedanemiainmostpatients.AccordingtoSulkowski[ 60 ],successofthecombinationtherapywithIFNandRBViscontingentonmaintainingadequatedosesofbothdrugsthroughoutthetreatmentperiod.Theoccurrenceofsideeffectsleadstoatrade-offbetweencontinuingthetreatmentwithoptimaldosagetoclearthevirusandexacerbatingthesideeffectsversusdecreasingdosagetorelievesevereanemia,whilereducingthechancesofachievingSVR.Withthisinmind,weusemathematicalmodelingtoestimatetheEPO-inducedincreaseinRBCproductionnecessaryforapatienttobeabletoundergothecompletecourseofthecombinationtherapywithoutsufferingfromacuteanemia. 31

PAGE 32

60 ].CertainEPO-initiatedsideeffectslikepurered-cellaplasiaduetopresenceofanti-erythropoietinantibodieshasbeenobservedinsomepatientswithchronicrenalfailure,[ 60 ].ThusdosingonlysufcientEPOisdesirable. InthecontextofHep-C,mathematicalmodelinghasbeenextensivelyusedtodeterminetheefcacyofIFNasmonotherapyandincombinationwithRBV[ 11 22 ].Differencesinresponsedependingongenotype[ 48 ],andtheconceptofearlyvirologicalresponse(EVR)toestimatepossibilityofachievingSVR[ 17 ]havebeenreinforcedwithmathematicalmodels.Viralanddrugkineticsstudiesusingmathematicalmodelshaveshedlightontheunderstandingofthisvirusanditstreatmentinseveraldirections[ 53 ].Allthesestudieshavecontributedenormouslytotheimprovementofthetreatmentprocedurethatdoctorspresentlyapplytopatients.WefurtherthisefforttodevelopmathematicalmethodologiesthatcanbeusedtoestimatethenecessaryusageofEPOtohelpanHCVpatienttosustaintheantiviraltreatment. Weconstructasetofcoupledveordinarydifferentialequationswheretherstthreerepresentuninfectedtargethepatocytes,infectedhepatocytesandfreevirionsbasedontherstmodelofNeumannetal.[ 49 ].TothatweincorporatethesideeffectofhemolyticanemiabyconsideringtheRBCconcentrationasaseparatestatevariableandalsotheamountofdruginthebodyasadynamicquantityratherthanaconstantparameter. WefocusontheinteractionoftheRBClevelandamountofdrugwiththegoalofndinganoptimaldrugtreatmentregimentominimizetheseverityofanemiawhilestillobtainingSVR.UsingtherstthreeequationswedeterminetheminimumamountofdrugnecessaryforthepatienttoachieveSVR.Then,wemathematicallyreplicateahypotheticalsituationwithasensitiveHep-Cpatientundercombinationtherapywho 32

PAGE 33

SchematicrepresentationofsensitivityofRBCconcentrationinthebodyandviralloadtotheamountofdruginthebody.TheboldgreycurverepresentstheRBCconcentrationinthepatientwhichdecreasesastheamountofdruginthebodyincreases.RIistheinitialRBCconcentrationofthepatientwhennodrugispresentandRAistheRBCconcentrationatanemia.Theblackcurverepresentstheviralloadwhichalsodecreasesastheamountofdruginthebodyincreases.CistheamountofdruginthebodywhentheRBCconcentrationreachesequilibrium.CAistheamountofdrugthatcausesanemiaandCSVRistheminimumamountofdrugrequiredbythebodytoachieveSVR.C,CA,CSVRarecalculatedfromthemodel.AclinicalproblemoccurswhenthedrugamountCSVRismuchhigherthanCA.OurgoalistouseEPOtoboosttheRBCproductiontomakeCAgreaterthanCSVR.ThegreydottedlinerepresentstheexpectedeffectofEPOwhichkeepstheRBCconcentrationatahealthylevelevenathigherdosesofcombinationdrugs.FurtherdiscussionofthedrugrelatedquantitiescanbefoundinSection startstakingtheregulardosageprescribedbythedoctorsdependingonbodyweight.Here,'sensitive'meansthatthepatienthasahistoryofencounteringhemolyticanemiawhenonthistreatment.Duringtreatmentthepatient'sRBCconcentrationismonitored,andhisdosageadjustedaccordingly.Fromthisprocesswecalculatetheamountofdrugthepatientcanhandlewithouthittinganemia.ThenweintroducetheeffectofEPOintheequations,i.e.,increasetheRBCproductionbyafactorofthepatient'soriginalRBCproductionlevel,toletthepatientintakegreateramountsofdruginthebodywhilemaintainingahealthyRBCconcentration.ThisallowsustocalculatetheEPO-inducedincrementinRBCproductionnecessarytoletthepatientsustainatleasttheminimumamountofdrugtoachieveSVRforthenecessaryperiodoftime.A 33

PAGE 34

3-1 .Analyticandnumericalmethodsfacilitateevaluationofseveraldosingregimens. InSection 5.1 ,weintroduceourmodelanditsparameters;inSection 3.3 weanalyzethemodelandcalculatedifferentreproductivenumbersforHCVwithandwithouttreatmentundervaryingassumptions.InSection 3.4 wediscusstheestimationofparametersandestablishtheresultsofouranalyticworknumerically.Inaddition,criticaldrugamountsarenumericallycalculatedforthesystemandtheresultsofsimulationsareshown. 49 ]whichincludesthestatevariablesT(t),targethealthyhepatocytes;I(t),infectedhepatocytes;andV(t),theviralloadoffreeHCV.InordertofocusontheinterplaybetweenthedrugsandtheRBCconcentrationinanHCVpatient,weintroducetwofurtherstatevariables,R(t)fortheRBCconcentrationinthebodyandC(t)fortheamountofdrug(IFNandRBV)inthebody.(NotethatC(t)representsthetotalamountofdruginthebodyattimetandnottheconcentrationordosageattimet.)Wedenet=0tocorrespondtothebeginningoftreatment.Figure 3-2 illustratesthekeyelementsofthemodelandtheirinteractions(labeledbyparameters). CompartmentalModel 34

PAGE 35

dt=sTdTT +CTV dt= +CTVdII dt=pk k+CICVdVV dt=sRdRRCR dt=(R)hC where(R)=R2 TheamountofdrugCinthebody,measuredinmg,changescontinuouslyovertimeinresponsetotwofactors:thedrugdosage(inmgday1),denotedby(R),andthebody'sabilitytoclearthedrugsfromthebodyataratehperday.AtypicalHCVpatientisinitiallyprescribedadosageofIFNandRBV,basedonhis/herbodyweight.ThedailyRBVdosageisaround1000mg,andtheIFNdosageis0.1gkg1week1,whichisordersofmagnitudelowerinquantity.AlthoughthepharmacokineticsandpossiblymodesofactionofIFNandRBVaredifferent,themechanismofactionofeachofthedrugsindividuallyhasnotbeenrmlyestablished,leavingtheexactnatureoftheirsynergyevenlessconrmed[ 44 ].Thus,forsimplicity,ratherthanmodelingtheamountofeachdrugseparatelyweconsidertheamountofdrugCtobethequantityof 35

PAGE 36

WeconsiderthedosageasafunctionoftheRBCconcentration,sinceinthisscenariothegoalistoavoidanemiacausedbyHCVtherapy.Ifapatientencountersanemiaatanypointduringthecourseofthetherapy,doctorsreducethedosageofRBVtohalf,whichincreasesthechancesofunsuccessfulachievementofSVR[ 60 ].Weareinterestedinndingoutthemaximumamountofdrugapatientcantakewithoutencounteringanemia.Thus,insteadofwaitingforthepatienttohitanemiaandthenreducingthedosagetohalfwewillconsiderthefollowinghypotheticalsituationtoconstructourdosingterm,(R),inthedC dtequation.SupposeaHep-CpatientoncombinationdrugtherapyundergoesRBCconcentrationmonitoringfollowedbyacontinuousreductionintherelativedose.StudyingthiscorrelationwecancalculatetheamountofdruginthebodywhichkeepstheRBCconcentrationatahealthyequilibrium.Werstconstruct(R),thedosagetermasafunctionofRBCconcentration,R.Inreality,thedosagegivenbyadoctorisbestapproximatedbyaRBCconcentrationdependentstepfunction 1. itshouldbeastrictlyincreasingfunctionofR, 2. attimet=0,(R)shouldequalthevalueofinitialdosageprescribedbythedoctor,and 36

PAGE 37

whentheRBCconcentrationreachesanemiclevels(R=RA)(R)shouldbereducedtohalftheoriginaldosage, ItcanbenotedthatthefunctionR2 Wealsosupposeherethatthepatient'sRBCconcentrationisconstantlymonitoredandhisdosageadjustedcontinuously.Thisassumptionofoptimal(continuousratherthandiscrete)responsetimepermitsthedosageadjustmentstobeincorporatedintothequalitativeanalysisofthemodel,inadditiontothenumericalanalysis.Inpracticethepatient'sconditionislikelytobemonitoredonceeveryseveraldays,althoughinoneclinicaltrialviralloadwasmonitoredseveraltimesperday[ 49 ].ThisassumptionalsoallowsustoaddressthequestionofwhetheritistechnicallypossibletoachieveSVRwhileavoidinganemia. Thisdescriptionoftreatmentascontinuousintimeanddosageisinaccurateintworegards.First,therapyistypicallyadministeredindiscretedosesratherthancontinuouslylikesomeIVdrugs.Second,inpracticethedosagewouldnotbeadjustedonacontinuousbasis,butratherswitchedbetweenalimitednumber(possiblyonlytwo)oflevels,saytheinitiallevelandareducedlevel.However,therstinaccuracyhasminimaleffectonthelong-termpredictionsofthemodel,sincedosesaretypicallygivendailyanyway(theendofSection 3.4.2 foranillustrationofthedifference).ThesecondinaccuracytendstoidealizetheeffectsofmakingdrugdosageresponsivetoRBCconcentration:thatis,themodeldescribesanoptimal(immediateandcontinuous)responsivenesswhereasrealresponsivenessislikelytobemorecoarse(inbothtime 37

PAGE 38

The(T,I,V)systemexplainstheHCVinteractioninthehepatocytecellsoftheliver.TheparametersTistherateofproduction,anddTisthenaturaldeathrateofhealthyhepatocytesfromthebonemarrow.isrelatedtotheefcacyofcombinationtherapy.Itisessentiallytheamountofthedrugatwhichproductionofnewinfectedcellsisreducedtohalf;thatis,whenC=,weget +C=1andthevirusinfectsthehealthyhepatocytesataconstantrate.Ifinsteadthedrugworkswith100%efcacy, +C=0.isthenumberofinfectionscausedbyoneinfectedcellperunittime.Astheamountofdrugincreases, +Cbecomesasmallerfractionandtheproductionofinfectedcellsdecrease.TVisthe[unreduced]concentration(incells/mL)ofhealthyhepatocytesinfectedbyvirus,V,perunitoftime.Thistotalrateofinfectedhepatocytes,TV,goesintothesecondclassofhepatocyteswhichistheinfectedhepatocytes,I.Here,dIIistherateatwhichinfectedhepatocytesareclearedperunitoftime. ThepercapitaproductionrateofHCVispandhencepIaccountsforthevirionsproducedbythetotalpopulationofinfectedhepatocytesperunittime.k,like,isrelatedtotheefcacyofthedrugs.Itisessentiallytheamountofthedruginthebodywhichreducestheproductionofnewvirionstohalfoftheamountproducedinabsenceoftreatment.isaratewithunitamount1time1.CVaccountsforincreasedclearanceoffreevirionsduetotheimmunomodulatoryeffectofbothdrugs.Neumannetal.observedonlyapossiblyminoreffectofdrugsonviralclearanceintherst2daysoftherapy,basedonamodelwhichassumesconstantdrugefcacyoverthewholeperiodoftreatment.Since,inourcase,theefcacyofthedrugsisconstantlychanging 38

PAGE 39

Table3-1. ThisTablegivesabriefinterpretationoftheparametersusedinthemodel. ParameterInterpretation dtanddC dtwithoutregardtothecompletesystem.EquatingdC dt=0weidentifyC= dt=0,wegetC=R2 39

PAGE 40

dt=0,weget0=sRdRRCR=sRdRRR3 a3sR a2+dRR asR AgaindividingthroughoutbysR a3 a2+dR a1=0. a,d=dR a),ea= h).Thenweget Herenotef(0)=1,limr!1f(r)=+1>0asea+d>0,andmostimportantlyf(r)<0ifr0(becauseifr0,alltermsoftheequationwillbe<0). 3 hasexactlyonepositiverealroot. Proof. Nowletusconsiderthecaseof3possibleroots.Usingtherescaledequation,weanalyzethepossiblerootsintheparameterspace.Fromtheprevioussectionwehavef(r)=(ea+d)r3r2+dr1=0.

PAGE 41

Thisisalinearsystemwithrespecttoeaandd.Hencewecanwrite: Now,sincethedeterminantofthecoefcientmatrixisnon-zero(r6=0),wecancalculateitsinversetoget 2r3)2643r2+1r3r3r2r3375264r2+12r375. Therefore, 2r>0. Thusourbifurcationlineliesinthesecondquadrantoftheparameterplane.However,motivatedbybiologicalreasons,ourregionofinterestliesintherstquadrantwhereea>0andd>0.Pluggingin(ea,d)=(1,1)inf(r)wegetf(r)=2r3r2+r1.Inthiscase,f(r)=0hasonlyonepositiverealroot.ThesesolutionsarebiologicallyextraneoussinceallparametervaluesarepositiveandthebifurcationliesonlyinthesecondquadrantasshowninFigure 3-3 .Therefore,wehavedeterminedthatthesystemdoesnothaveabiologicallyrelevantbifurcationpoint. Hence,weconcludethatthesystemcannothavethreepositiverealroots.Wedeterminethatthereisonlyonepositiverealrootbyanalyzingthebifurcationdiagramintherstquadrant. 41

PAGE 42

Thiscurveisplottedusingparametriccurveea=(r2+1)2 2r>0 Proof. Proof. 42

PAGE 43

@R(sR @C(R a2+R2hC R)=sR R<0)thereisnolimitcycle. Weobservebysimplecomparisonthat,dC dt
PAGE 44

7 ]calculatethebasicrepro-ductionnumber(R0). WerstanalyzeEquations 3 3 ,takingCasanasymptoticconstant;inasensewetakeittobeaparameter.Tondtheequilibriaofthereducedsystem,wewrite dt=sTdTT +CTV=0 dt= +CTVdII=0 dt=pk k+CICVdVV=0 Todeterminethediseasefreeequilibrium(DFE),weletI=0andV=0andsolvedT dt=0forT.ItisclearbyinspectionthattheDFEis(sT LinearizingthesystemabouttheDFE,andimposingconditionsforstabilitywecalculatebRwhichwecallthecontrolledreproductionnumber(CRN). +Ck k+CdV +Ck k+CdV TheexpressionofthebRisderivedintheproofofthefollowingobservation.

PAGE 45

k+CCdV377777775. Substituting(sT dT,0,0)=266666664dT0sT dT dT k+CCdV377777775. Thecharacteristicequationis: dT k+CCdV=0. k+C)=0 k+C) dT k+C).Nowweapplythequadraticformula, 45

PAGE 46

Wecancanceloutliketermsfrombothsides,sincetheyarebothpositive. k+C>0 dIdT(+C)(k+C)(C+dV)<1. ThustheDFEislocallyasymptoticallystableifandonlyifbR<1. Thus,theinfectioniseliminatedfromthehepatocytepopulationwhentheDFEisstable. NowourattentionshiftstotheendemicequilibriumsinceourbiologicalinterestistreatmentofchronicallyHCVinfectedindividuals.HeretheendemicequilibriumisdenedastheequilibriumwhenI>0andV>0.Now,wesolvefortheendemicequilibriumpoints,usingV6=0andI6=0. dt= +CTVdII=0)I= +CTV dt=pk k+CICVdVV=00=pk

PAGE 47

dt=0,weseethatsTdTT= +CTV,)V=sT +CdT(+C) dI(dV+C)(k+C)dT(+C) +CTV dI(dV+C)(k+C)dT(+C) ThensimplifyingusingbR,theendemicequilibriumis Proof. k+CCdV3777775

PAGE 48

3 3 ;thefollowingJacobianmatrixresults:bJ=26666664dTdTbR10sT k+CCdV37777775 dT(+C)(k+C)bRdIdT(C+dV)(bR1) NowweapplytheRouth-HurwitzCriteriontodeterminestability.Let +Ck k+CpsT (3) ThenwecanwritebR=D dIdTq. 1.

PAGE 49

EquatingthelefthandsideofEquation 3 tozero,weobtaintworoots,D1,2as Let IfQisnegative,theninequalityofEquation 3 isalwayssatised.Furthermore,since4(q+dI) ThenweproceedtofurtherthisresulttoprovetheglobalstabilityoftheEndemicEquilibrium. Theglobalstabilityofthe(T,I,V)equilibriuminthe(T,I,V)subsystemcanbeprovedusingaLiapunov'sfunctionasinDeLeenherandPilyugin(2008)[ 19 ].Letus 49

PAGE 50

Then, +CTV dI(dV+C)(k+C)dT(+C) +CT=dI(dV+C)(k+C) +CTV=dII k+CI=(C+dV)V Now,let,,beconstantstobechosenlaterandWbetheLiapunovfunction. 50

PAGE 51

+CTV+1I +CTVdII+1V k+CICVdVV(3) Choosing=dIk+C pk,==1andusing 3 3 3 3 simpliedW0becomes +CITV I+dIIdIIV pk(C+dV)V(3) Thenaddingandsubtractingf(T)(1T +CITV I+dII+f(T)1T NowusingEquations 3 3 weget, VI3 VI)1 3=1)isalwayslessthanthearithmeticmeanofthoseterms.Therefore,W00. WenotethatW0=0iffboththerstandthesecondtermiszero.ThesecondtermiszeroifT VI=1,inwhichcasethersttermalsobecomeszero.Thusthelargestinvariantsetis I=V Vg

PAGE 52

43 ]impliesthatallboundedsolutionsinint(<3+)convergetothelargestinvariantsetinM.Nowtoshowtheboundednessofthesystem.Consider,d(T+I) WerstsetU=T+IandU0=T0+I0.Now,dU dtsTd0UdU dt+d0UsT dt+d0ed0tUsTed0td dt[ed0tU]sTed0t d[ed0U]dsTZt0ed0ded0tUU0sTZt0ed0dUed0tU0+sTed0tZt0ed0d=ed0tU0+sTed0t(ed0t1) Again,dV dt=pk k+CI(C+dV)Vpk k+CsT

PAGE 53

Hencethe(T,I,V)systemhasboundedsolutionsanditisclearthatthelargestinvariantsetinMisthesingletonsetf(T,I,V)g.ThisisbecauseMisasimplyconnected1-dimensionaldomain,wheresolutionsarealwaysmonotone.Alternatively,usingT(t)=TinEquation 3 (i.e.,dT dt=0)weconcludethatVisconstantwithrespecttotimeinsideM,whichinturnimplies(withEquation 3 )Iisalsoconstant.Also,thesolutionsstartingontheboundarymoveintotheinteriorofthesetexceptontheTaxis.Hencetheresultholdsforallsolutions.Therefore,(T,I,V)isgloballyasymptoticallystable. 3 tosolveforC,weobtainacubicequationinCoftheformG(C)=C3+A2C2+A1C+A0=0,where WeobservethatG(0)<0ifandonlyifR0>1.InadditionG0(C)>0foreachCandlimC!1G(C)=+1;thereforethereexistsoneuniqueroot,namely 33p 3+B1q 3#, whereB1=2A329A1A0+27A0.B2=A223A1. 49 ]thatforhigherefcacyofthetherapy,theeffectoftreatmentontheinfectionrateTVisnegligible.Inthislimitingcasewehave +C=1. 53

PAGE 54

3 becomesquadratic:C2+k+dV 2. NotethatweintroducetreatmenttoaninfectedpatientonlyifR0>1,implyingthatthebodyisincapableofclearingoutthevirusbyitself.Therefore,(R01)>0makingCSVR>0always. AsmentionedbeforewedeneCAasthemaximumamountofdrugabodycantoleratewithoutencounteringanemia.TocomputeCAwerstdetermineRA,i.e.,theRBCconcentrationatanemiaforthepatient,whichisaclinicallyxedquantityindependentoftheamountofdruginthebody.Equation( 3 )representsthechangeinRBCconcentrationwithrespecttotheamountofdrug,andatequilibriumitgivesarelationshipbetweentheamountofdruginthebodyandtheRBCconcentration.Thus,usingtheRBCconcentrationRAwesolvedR=dt=0forCA,theamountofdrug(IFN+RBV)inthebodythatcausesanemiaintheabsenceofadditionaltreatmentwithEPO: Thus,iftheamountofdruginthebodyismorethanCA,apersonwillgetanemia.IntermsofthebiologyofHepatitisCtreatmentandthesideeffectofanemia,theanalysisofthe(R,C)and(T,I,V)subsystemsallowsustocomparethefollowingcriticaldrugamounts:theequilibriumdrugamountC,whichkeepstheRBCconcentrationatanequilibrium;theminimumamountofdruginthebodysothattheviralloadgoesbelowdetection,CSVR,leadingtosuccessfulachievementofSVR;andthedrugamountCA,thatis,themaximumamountofdrugapatientcantoleratewithoutgettinganemia. 54

PAGE 55

3.5 TheidealscenarioforadoctoristoobserveCSVRCA.Thendependinguponotherfactorsconcerningthepatienthecanprescribeadosingregimesothattheamountofdruginthebodystaysintheinterval(CSVR,CA).ThatwillensurethatSVRisachievedwithoutencounteringanemia.Thepatientsforwhomtheinequalityisreversed,wetreatwithEPOtoboostCAhigherthanCSVRsothatdoctorshaveanopportunitytotreatthemwithoutanemiccomplications. Tointroducetheeffectofepoietinweincrease(multiply)theRBCproductionbyafactorbandrecalculateCA.EpoietinisknowntoincreaseRBCproductionbyafactorofthepatient'soriginalRBCproductionlevelfromclinicaltrials[ 60 ].TheaimforourtreatmentistoachieveCSVRCAbyincreasingtheproductionofRBCinEquation( 3 )byafactorofb(b>1).Solvingforbin calculatesthecriticalfactorbcwhichistheminimum-foldincreaseinRBCproductionusingepoietinforaparticularpatient,tosafelyachieveSVR.Inthefollowingsection,weestimatethevalueofbcfromnumericalsimulations. 3.4.1ModelParameterEstimation 55

PAGE 56

3-3 )[ 11 ],[ 46 ]and[ 3 ].RecallthatourgoalistoprovideoptimaltreatmentforaHCVpatientwhootherwisewoulddevelopanemiaunderstandardtreatment.Inthistheoreticalstudy,wecapturethishypotheticalHCVpatientbyappropriatelyestimatingtherestoftheparameters(exceptb).WeapplytheeffectofEPO(anti-anemicdrug)tothispatient,throughchangeoftheparameterbinthemodel.ThecompletelistofparametersincludingtheestimatedonesistabulatedinTable3. DosesofRBVaretypicallyprescribedbasedonhemoglobinconcentrationatagivenpointintime.Itisknownfrom[ 18 ]thattheRBC-to-hemoglobinratioinapersonisroughlyconstant,asmakesbiologicalsense.Therefore,takingvaluesforRI,theusual(initial)hemoglobinconcentrationHI,andthehemoglobinconcentrationHAatanemiafrom[ 6 ],wecandeterminetheRBCconcentrationRAatanemiabyasimpleproportion: WetakethevalueofI=1200mgday1fortheinitialdosagefrom[ 60 ].Theclinicaldenitionofanemiaappearstovaryintheliterature:paperslikeAfdhaletal.reduce 56

PAGE 57

1 ]whereasSulkowskietal.takethisvaluetobe10gdL1[ 60 ].Inthisdissertation,weuse10gdL1asthehemoglobinlevelatwhichtheRBVdoseisreducedtohalf.Since1gm=100mgand1dL=10mL,wederivethatHA=100mgmL1.FromthesequantitieswecanthenuseEquations( 3 )tocalculateanda(Table 3-4 forallvalues). Table3-2. Basevaluesusedtodeterminedosingparameters SymbolMeaningValueSource 6 ]HAHemoglobinlevelatanemia100mgmL1[ 6 ]RINormal(initial)RBCconcentration6.1103cellsmL1[ 6 ]RARBCconcentrationatanemia3.8103cellsmL1eqn.( 3 )IInitialdosage1.2103mgday1[ 60 ]aHalf-saturationconstantfordosing8.1103cellsmL1eqn.( 3 )Maximumdosage3.3103mgday1eqn.( 3 ) ItshouldbenotedthatEquations( 3 )(andhencethecriteriafromwhichtheywerederived)requirethatRI=RA>p 3 ),thisisequivalenttosayingHI=HA>p dt=sRdRR

PAGE 58

46 ],wegetthatdR=.0231day1.NowusingtheinitialRBCconcentrationR=RI(giveninTable 3-4 ),wecalculatesR=1.4102cellsmL1day1. dtbysubstitutingappropriateestimatesofR,C,dR dtalongwiththeparametervalues,asfollows.=1 dt. dtRIRA Fromtheresultsoftheclinicaltrialin[ 62 ],weknowthatmorethan50%ofpatientsreceivingIFN-andRBVencounteredanemiawithinthe14daysleadingtodosereduction.FromTable 3-4 ,weseethatourhypotheticalpatientwilldevelopanemiaifthereductioninRBCconcentrationisRIRA=2.3103cellsmL1.Sinceourpatientis'sensitive',weassumethisreductiontotakeplacewithin14daysofinitiationoftherapy.Hence,dR dt2.3103 Tocalculatetheamountofdrugintheexpressionfor,weconsidertheeffectofdrugontheRBCconcentrationifaconstantdosingregimewasusedinsteadofthe'self-adjusting'dosing.ThisdynamicsisrepresentedbytheequationdC dt=IhC.Thenatequilibrium,theamountofdruginthebodywouldbe (3) Sincethepatientencountersanemiaatt=14days,wetaketheRBCconcentrationR=RA.Substitutingthesevalueswecalculate=9105mg1day1. 58

PAGE 59

dt=0anddC dt=0equationswecalculatetheequilibriallevelsC=233.67mgandR=3.19103cellsmL1. 49 ]foundthattheinitialdropintheviralloadundertherapydependsmostlyontheefcacyofthedrugontheproductionrateofvirus.Infact,iftheefcacyofthedrugonpis100%,thentheeffectoncanbeignored.However,nodrugisperfect,andifk k+C<1,theeffectofdrugonbecomesmoreprominentastreatmentcontinues.OtherliteraturehasmodeledtreatmentofHCVwiththeseantiviralswithoutconsideringanyeffectofdrugsonatall[ 13 22 ]. Hereweperformourprimarycomputationsconsideringthescenariothatthereisnoeffectofdrugson.Thechangesinnalresultduetoconsideringtheeffectofdrugswith50%efcacyinreducinginfection( +C)willbediscussedintheresultssection.Now,k k+Cistheefcacyofthecombinationtherapyinreducingnewvirionproductionininfectedcells.FromHerrmannetal.[ 33 ]wehavethatthemeanefcacyofthetherapywithstandardIFNandRBVis36%andofpeg-IFNis63%.Takingtheaverageweuse=49%,forthecalculationofk.FortheamountofdrugintheexpressionforkweusethevalueofCavgascalculatedinEquation( 3 ). Therefore, k+C=(1) C 11 ]is7.8perdaywhichisestimatedinthepresenceofdrugs.Itisdifculttondtheclearancerateofthevirusinvivointheabsenceoftreatment.MostpatientswhoarediagnosedforHCVinfectiongettreatedimmediately.Here,forillustrationpurposes,weinvestigatethescenariowhere 59

PAGE 60

5.1 .NotethatatthispointwehavealreadyestimatedalltheparametersnecessarytocalculateC.Thisgives=0.009mg1day1.Theeffectonthenumericalresultifweassume=0isdiscussedinsection 3.4.2 Table3-3. ParameterValueTable ParameterValueReference 11 ]sR1.4102cellsmL1day1estimated*dV5.5day1estimated*dT0.0026day1[ 11 ]dI0.26day1[ 11 ]dR.0231day1[ 46 ]p2.9virionscells1day1[ 11 ]2.25107mLvirion1day1[ 11 ]k657mgestimated*.009mg1day1estimated*9105mg1day1estimated*3.3103mgday1estimated*a8.1103cellsmL1estimated*h1.9day1[ 3 ] *Asexplainedinthetext. 11 ]. 60

PAGE 61

Table3-4. InitialConditionsofStateVariables VariableInitialValue dt=0anddC dt=0numericallywecalculatethesteadystateRBCconcentration,R=3.19103cellsmL1,whichismuchbelowtheanemiclevelofRBCconcentration,calculatedaboveas3.8103cellsmL1.FromEquation( 3 ),wegetthatCSVR=691mg.Thatis,theminimumamountofdrugthatshouldbepresentinthebodyonanaverage,toensureachievementofSVRis691mg.Themaximumdrugamountthatthebodycantakeinandstillavoidanemia,calculatedfrom( 3 ),isCA=153mg.Here,theminimumamountofdrugnecessaryforapatienttoachieveSVRisgreaterthanthemaximumamountofdrugapatientcantoleratewithoutencounteringanemia,i.e.,CA
PAGE 62

InthisgraphthethicklineshowstheincreaseofCAasEPOincreasestheproductionofRBCinthebody.ThethinhorizontallinerepresentsthedrugamountCSVR.Thereforethecriticalfactorbcismarkedbytheintersectionofthesetwolines,asindicatedbythearrow. normalRBCconcentrationfortheentireperiodoftreatmentaround48weeks=336days.ItmightappearthatthemoreweincreaseRBCproductionthebettertheresult.However,theabilityofEPOtoincreaseRBCproductionislimited,asmentionedearlier[ 27 ].AHCVpatienthasbeenobservedtoincreaseRBCproductionbyamaximumof3timesonly[ 60 ].ManypatientswithweakimmunesystemdonotrespondwelltoEPOdosing,asexplainedintheintroduction.Thustheminimumincrementnecessarytoendureantiviraltreatmentshouldbetargeted. Wenumericallysolvetheentiremodel( 3 )( 3 )withoutandwithepoietinatdifferentlevels,fortheperiodof48weekstreatment(=336days).Forthepurposeofthisdissertationweareinterestedintheperiodoftreatmentonly,sincetherapy-inducedhemolyticanemiaisreversible.Thatis,RBCconcentrationshootsbacktohealthylevelsassoonastheseantiviralsarediscontinued.WenoteinthegraphsinFigure 3-5 ,whenEPOisnotadministered(denotedbyboldline),theRBCconcentrationdropsbelowanemiclevels.Duetothat,ourhypotheticaldosingregimenstartsdecreasingtheantiviraldosing,leadingtoarelapseinviralloadevenbeforetheendofthetreatmentperiod. 62

PAGE 63

NotethattheRBCcanbemaintainedatnon-anemiclevels(dottedline)whenalighterdoseofEPO(causingapproximately1.5timesincreaseinRBCproduction)isgiven.However,theviralloadstartstorelapsearoundtherapycessation(336days),resultinginunsuccessfulachievementofSVR.(Inpractice,relapseismorecommonaftertreatmentstopsthanbefore.) Again,ahigherdoseofEPOtoincreaseRBCproductionby3times(smalldashedline)canmaintaintheRBCconcentrationathealthypre-treatmentlevels,whichhelpsthebodysustainahigherconstantconcentrationoftheantivirals.Althoughthisappearstobetheperfectscenario,itispracticallyalmostimpossibletoincreaseapersons'sRBCproductionby3timeswithoutsignicantcomplications,asdescribedbefore. Nowifweassumetheefcacyofeffectofdrugonis50%andcomputeexactlylikek,wegetequalto631mg.KeepingalltheotherparameterssamewegetthenewCSVR=405mg,fromEquation 3 .ThenanincreaseinRBCproductionbyaround1.6timesisenoughtoclearthevirusandnotcauseanemiaatthesametime,i.e.,tohaveCA=CSVR.Weobservethatchangesintheseparametervaluescanchangetheresultssignicantlyandthusthenecessitytoconductclinicalstudiestoestimatethemeanandrangeofvaluesoftheseparameterscannotbeover-emphasized. IfweassumenoeffectofdrugontheclearanceoffreeHCVfromtheliver,thatis,=0,withefcacyofdrugeffectonas50%wegetCSVR=726mg.Inthatcase,theRBCproductionratewillrequireanincreaseof2.4timestoachieveCA=CSVR. Finally,wepresentinFigure 3-7 anillustrationofthedifferencebetweenadministrationofHCVdrugscontinuouslyorona(discrete)dailybasis.Itcanbeseenthattheamount 63

PAGE 64

InthisgraphweobservethetrajectoriesoftheRBCconcentrationatdifferentlevelsofEPOadministration.TheboldlineshowsthetrajectorywhennoEPOisgive,i.e.b=1.ThesmalldashedlinesshowthebehaviorwhenonlyEPOenoughtomakeCA=CSVRisgiventoapatient.WhenenoughEPOtoincreaseRBCproductionabout1.5timesisgiven,wenotefromthedottedlinethattheRBCconcentrationismaintainedaboveanemiclevels.ThelargedashedlineshowsthebehaviorofthestatevariablewhensufcientEPOisgiventoachieveb=3.Weobservethattheviralloadhasasteepdeclinebutdoesnotshootbackduringtheperiodoftreatmentof48weeks=336days(Figure 3-6 ),andtheRBCconcentrationismaintainedatpre-treatmentlevels.However,thismightbeclinicallyundesirable. ofdruginthebodyundercontinuousdosingfallswellwithinthesawtoothintervalresultingfromdiscretedailydosing. ItisusefultohaveanestimateoftheminimumincreaseinRBCproductionnecessarytomaintainhealthyRBCconcentrationaboveanemiclevelsandsufcientlyreduceviralloadatthesametime.Thechoiceofdosingstrategy(R)turnedouttobepivotal,sinceotherwise,thesystemwouldmaintainthehighdoseirrespectiveofthepatient'sRBClevel. 49 ]byaddingtwoseparatestatevariablestorepresenttheRBCconcentrationanddrugamountinthebodyasdynamicalsystems.Analysisofthedecoupledequationsseparatelygivestwocriticalvaluesofdrugamount:C,thedrugconcentrationrequired 64

PAGE 65

Intheabovegraph,weshowbytheboldlinethedynamicsoftheviralload.ThesmalldashedlinesshowsthebehaviorwhenonlyEPOenoughtomakeCA=CSVRisgiventoapatient.WhenenoughEPOtoincreaseRBCproductionabout1.5timesisgiven,wenotefromthedottedlinethattheviralloadtendstore-emergebeforetherapycessationalthoughtheRBCconcentrationismaintainedaboveanemiclevels(Figure 3-5 ).ThelargedashedlineshowsthebehavioroftheviralloadwhensufcientEPOisgiventoachieveb=3.Weobservethattheviralloadhasasteepdeclinebutdoesnotshootbackduringtheperiodoftreatmentof48weeks=336days,andtheRBCconcentrationismaintainedatpre-treatmentlevels(Figure 3-5 ).However,thismightbeclinicallyundesirable. Anillustrationofthedrugamountinthebody,RBClevel,andvirioncountresultingfromcontinuous(smoothcurve)vs.discrete(sawtooth)dailyadministrationofHCVdrugsduringtreatment.Thecontinuous(*-)curveisgeneratedfromourmodel.Theothercurve(-)isgeneratedfromamodelwherethelastequationischangedtodC dt=hC,andanimpulsivedosingregimenwhichisadjustedandadministereddailyisconducted.Notetotaldrugadministeredperdayremainsthesameforthetwosetsofcurves,asdoallothermodelparameters. tomaintaintheRBCpopulationatequilibrium,andCSVR,thedrugconcentrationrequiredforpatientrecovery.Also,theRBCconcentrationandamountofdrugequation 65

PAGE 66

IfinapatientCA>CSVR,thedoctorscanprescribeadosagetomaintaintheequilibriumamountofdrugintheinterval(CSVR,CA)whichwillachieveSVRinthepatientwithoutencounteringanemia.Theclinicalproblemariseswheninapatientthisinequalityisreversed.Inthatcase,thedoctorappliesEPOtoboostCAbyincreasingtheRBCproductiontomakeCACSVR.ThefactorbywhichRBCneedstobeincreasedcanbecalculatedfromourmodels.Ourestimatedsetofparameterspresentsa'proofofidea'situationinwhichthisfactorisfoundtobe2.3toachieveCA=CSVR. WealsoshowthroughgraphicalsimulationsthatalowerincreaseinRBCproductionmightcompromisechancesofachievingSVR.Theoretically,veryhighdosesofEPOcanavoidanysignicantdecreaseinRBCconcentrationthroughouttherapyandincreasetolerabilityofdrugthusstrengtheningchancesofachievingSVR.However,thatisnotpracticallyfeasibleinamajorityofpatients.EPOhaslimitedcapabilitiestoincreaseRBCproductionandcomeswithitsshareofpossiblesideeffects.ThustheneedtoestimatethenecessaryincreaseinRBCproductioncannotbeover-emphasized.OurmodelprovidesawaytousemathematicalmodelingtopredictifapatientwillencounteranemiaundertheprescribeddosingofIFN+RBV,ornot.Further,itcanestimatetheincreaseinEPO-inducedRBCproductionnecessarytoachieveSVRwithoutencounteringanemia. 66

PAGE 67

[ 57 ] HepatitisCisaveryslowlyevolvingdisease,wherechronicHCVinfectioncancontinuefordecades,withorwithouttreatment.Duetothisreasonithasnotbeenprovedbeyonddoubtthattreatmentprovidesabsolutecureandhaltsprogressiontoadverseliverinfectionslikecirrhosisandhepatocellularcarcinomaamongothers.ThustherapyofHCVisprimarilytargetedtowardsrestrictingdeteriorationofliverconditionavoidinglivertransplantorcausingpatient'sdeath.Achievingasustainedvirologicalresponse(SVR)conferslong-termviralclearanceandrepresentsacure. Thecurrentstandard-of-caredrugregimenshelpattainthiscriticaltherapeuticmilestoneinonly50%oftreatedpatients.Ithasbeenobservedthatseveralpatientswhoexhibitundetectableviralloadinresponsetotreatmentduringtherapy,oftendonotachievesustainedvirologicalresponseinthelongrun.Asaconsequence,responseinpatientstotreatmentaremeasuredbysurrogatevirologicalparameter,insteadofaspecicclinicalend-point.Severaltypesofvirologicalresponsethresholdsaredeneddependingontime,relativetotreatmentduration,havingdifferentialdegreeofreliability 67

PAGE 68

28 ].ThesehavebeendiscussedindetailsintheSection 1.5 ItisbecomingincreasinglyclearthatxedtreatmentdurationforeveryHCVpatientmaybeimpractical.Toestablish'responseguidedtherapy'[ 45 ],doctorsusetheabovementionedcriteriathatcanbemeasuredduringtreatmenttomodifyongoingtherapy,aswellastopredictthelong-termgoalofSVR(Figure 1-1 ). Oneroleofmathematicalmodelsistotrytounderstandhowvariationamongthegenotypesandphenotypesofpatients,andamongHCVgenotypes,canleadtothisarrayoftreatmentresponses,withtheultimategoalofimprovingpredictionandtreatment.Inparticular,previousimmunologicalmodelsofHCVsuchasthosebyDaharietal.[ 11 ],Dixitetal.[ 22 ]havemadeconsiderableheadway,exploringthefactorsthatdeterminetheefcacyofpeg-IFNandRBVtreatmentofHCV. Aquestiongenerallyneglectedbycurrentmathematicalmodelsiswhentreatmentcanactuallycureapatient,asopposedtosimplyproducingshort-termeffects(whichmayneverthelessbegoodclinicalindicatorsoffuturecure).Clinically,curingapatientmeansthatthepatientremainsdisease-free,andfreeofanydetectablesignsofthepathogen,indenitely;forHCV,SVRappearstobeagoodproxy(orpredictor)forcure.Ausefulmathematicaldenitionofcureisthatafterthecessationoftreatmentthedensityofthepathogen(denedhereasviralload)convergestoastablediseasefreeequilibrium,(DFE).Thatis,whilethemathematicalmodelpredictsthatthepathogenhasnotcompletelydisappeared,itsdensityiscontinuouslydecreasingtowardzero.Incontrast,chronicinfectionorfailuretocureischaracterizedasconvergencetowardastableendemicequilibrium(EE)withpositivepathogendensity. Mostcurrentmodelsofwithin-hostdynamicssharethemathematicalpropertyofhavingasinglestableequilibrium.Inthesemodels,asuccessfultreatment(onetowhichthepatientresponds)ischaracterizedbyaswitchinstabilityfromtheEEtotheDFE.Whenthishappens,thepathogendensitydeclinesduringtreatment(sometimes 68

PAGE 69

Thereareseveralwaystoreconciletheobservationofeffectivetreatmentwiththisbehaviorofmathematicalmodels.First,andprobablymostoftencitedbymathematicalmodelers,istheeffectofdiscretenessandstochasticitynotincorporatedinstandarddeterministicmodels.Thatis,atverylowpathogendensitieswherethereareonlyafewcopiesofthepathogenremaininginthehost'sbody,chanceeventscanleadtoitscompleteextinction.Second,thepost-treatmentreboundtowardsthepreviousEEmaybesoslowthatvirallevelsarelikelytoremainundetectableevenpastthehost'snaturallifespan. Athirdpossibility,knowntobutlessoftenexploredbymodelers(althoughconsider[ 30 ]),isthatthehost-pathogensystemisbistable,meaningthateithertheDFEortheEEcouldbethelong-termstablestates,dependingoninitialconditions.Inthisscenario,thepatientbeginstreatmentatornearthe(stable)EE.WhiletreatmentmayormaynotchangethestabilityoftheEE,itcanpushpathogendensities(andpropertiesofthehost,suchasdensitiesofsusceptiblecellsorimmunefactors)overathreshold,intotheso-calledbasinofattractionoftheDFE,wherethesystemwillnaturallyapproachtheDFEinthelongrun.Whentreatmentceases,thesystemisstillinbasinofattractionoftheDFE,thepathogendensitycontinuestodeclinetowardstheDFE,andthepatientisbothclinicallyandmathematicallycured.Thenthisuniquethresholdforeachindividualpatientcanbeconsideredasa`curethreshold',whichtreatmentshouldaimtoovercome. 69

PAGE 70

23 24 40 ].Amongthefewpreviousmodelstoexplorebistabilityinanimmunological(invivo)model,Gomez-AcevedoandLi[ 30 ]usedbackwardbifurcationtoexplaintheviraldynamicsofHTLV-Iinfectionandencouragetreatmentmeasurestoconsiderthemoredifcultthresholdthatneedstobeovercometoachievecureincaseofbistability.Also,[ 54 ]haveinterpretedbistabilitytoposeaharderthresholdforsuccessfultreatment,whenHCVandHBVinfectionisconsideredinbothbloodandliver.Weutilizethebistabilitypropertyofthismodeltogeneratealltheclinicallyobservedpatientproleandestimatetheeffectoftreatmentnecessaryforagiveninsilicopatient. Awell-establishedmathematicalmodeloftheinteractionbetweentreatment,HCV,andhostcellsisknowntoallowbistability[ 55 ];however,thisbehaviorhasnotbeenpreviouslyexploredinanydetail.Inthisdissertation,weanalyzethesystemofequationstoestablishmathematicalcriteriathatdenewhenthispreviouslyoverlookedbehavioroccursandshowthatitdoesoccurforrealisticparametervalues.Inaddition,weexploretheparameterspacenumericallytoshowthatthemodeliscapableofreproducingallofthepatientproles(nullresponder,partialresponder,SVR,etc.)displayedinFigure 1-1 Inlightofthesendingsweforwardanothersteptoshowthatthismodelalsoallowsustonumerically'treat'apatient,sothatapatientwhowouldnotspontaneouslyclearthevirus,showsundetectableviralloadposttherapycessation.Mathematicalanalysisofthexedpointsinthismodelrevealsaregionofbistabilityforrealisticparametervalues.Forpatientsthatexhibitthisdynamics,wecanreducetheviralloadtomakethestatevariablesconvergetotheDFEaftertherapyisstopped.Particularly,wecanestimateaviralloadvshealthyhepatocytecombinationthatwillensurelongtermcurefromHepatitisC.Usingthatinformation,foragivenpatientandxedefcacywecannumericallyestimatetheminimumrequiredtimeoftreatmenttoachieveSVR.Also, 70

PAGE 71

11 13 ]: dt=s+r1T1T+I TmaxdTTVdI dt=TV+r2I1T+I TmaxIdV dt=(1)pIcV Thethreecontinuousstatevariablesinthemodelrepresentthenumberofhealthyhepatocytes(T),infectedhepatocytes(I)andfreevirus,orviralload(V).Theunitoftimeisindays,thusallratesaredenedtobe'perday'.Themodelassumesabaselinerecruitmentrateofhealthyhepatocytesatratesandabaselinepercapitamortalityrated.Hepatocytesproliferateinadensity-dependent,orhomeostatic,waywithamaximumpossibledensity-dependentproliferationrater1andamaximumhepatocytedensityor`carryingcapacity'Tmax,basedonbothhealthyandinfectedhepatocytes.Healthyhepatocytesareinfectedbyfreevirusataper-virus,per-hepatocyterateof;thus,TVistheabsoluterateofhepatocyteinfection.Infectedhepatocytesarecleared,duetonaturaldeath,immuneresponse,ordrugaction,atarate.Infectedhepatocytesundergoproliferationsimilarlytohealthyhepatocytes,butwithamaximumproliferationrateofr2ratherthanr1(asshownbelow,thisdifferenceiscriticalfortheoccurrenceofbistability).Theclearancerateofinfectedhepatocytesduetonaturaldeath,immuneresponseordrugisgivenby.Finally,thedynamicsoffreevirusparticlesinthe 71

PAGE 72

[ 49 ]studiedtheeffectofIFN+RBVtreatmentonHCVpatientsbyttingasimpleHIV-viraldynamicsmodeltodatafrompatientsundergoingtherapy.ThispioneerarticleinmodelingtreatmentofHCVestimatedthatdrughadnegligibleeffectsontheclearanceratesofinfectedhepatocytes,andfreevirusc.Theauthorsalsoobservedthat,whentheefcacyoftreatmentonpiscloseto100%,theefcacyoftreatmentonisalsoapproximatelyzero.In[ 11 ],althoughtheauthorsincorporatetheeffectofdrugonbothproduction(p)andinfection()formathematicalanalysis,theirnumericalsolutionscombinetheefcaciesintoasingleeffectonpalone.Here,weanalyzethemodelinabsenceoftreatmentandexploretheeffectsoftreatmentnumerically;following[ 11 ]inthiscase,weincludetheeffectofdrugonproduction(p)alone,throughthetreatmentefcacyparameter. Thismodelhasbeenthoroughlyanalyzedandttedtoavailabledataonpatientprolesundertreatment[ 11 14 16 55 ].ThemodelappearstobeappropriateforunderstandingthedynamicsofinvivoHCVinfectionaswellaspossiblyforHepatitisBinfection[ 15 ].Daharietal.[ 14 ]havealsostudiedtherelationshipamongdrugefcacy,infectedhepatocytedeathrate,rateofnalphasedeclineandthebaselinefractionofinfectedhepatocytesbeforetreatment.AnotherarticlebyDaharietal.[ 16 ]usedthesamemodeltoshowthatahighbaselineviralloadreducesthechancesofachievingSVR.Convenientlyforourpurposeshere,thisextensiveefforthasdenedrealisticrangesforalloftheparameterswhichweusehereafter. Thebasicreproductionnumber(R0)isastandardmetricinepidemiology,alsousedforwithin-hostdynamics,thatdescribestheabilityofapathogentospread(inapopulationofcellswithinahost)afteritsinitialintroduction.Inthiscase,R0canbeinterpretedasthetotalnumberofsecondaryhepatocyteinfectionscausedbyasingle 72

PAGE 73

TheR0ofmodel 4 withouttreatmentisgivenby cT0+r2 whereT0istheconcentrationofhealthyhepatocytesattheDFE,(T0,0,0)givenbyT0=Tmax 11 ].TherstterminEquation 4 accountsforthefactthattheinfectedhepatocytecanproduceuptopvirionsinitslifetimeof1 5 ]:forexample,thebifurcationcurveshowninFigure 4-1 depictschangesinthevalueandstabilityofequilibriaasaresultofchangesintheclearanceparameter,holdingallotherparametersxed(equivalenttochangingR0).TheDFEislocallystableforR0<1andunstableforR0>1.ThusR0=1isabifurcationpoint. IfastableEEexistsforsomerangeofR0<1aswellasintheregionwithR0>1,themodelissaidtoexhibitbackwardbifurcation[ 40 ].InthiscaseboththeDFEandtheEEcanbestableinaregionwithR0<1(bistability).Whichequilibriumthesystemapproachesdependsonthestartingvaluesofthestatevariables(T,I,andV).Figure 4-1 showsthebifurcationcurveformodel 4 ,withouttreatment,withbiologicallyrelevantvaluesofparameters,showingtheequilibriumquantitiespermLoffreevirus(V)asafunctionofR0asthebifurcationparametervaries.WeuseRctodenote 73

PAGE 74

Whenabackwardbifurcationoccurs,thereareatleastthreeequilibriaintheregionRcr1,i.e.,iftherateofproliferationofinfectedhepatocytesisgreaterthantheproliferationrateofhealthyhepatocytes.Althoughonemightexpectthatinfectedhepatocyteswouldalwaysproliferateslowerratherthanfasterthanhealthyonesbecausetheirfunctionwouldbeimpairedbyviralinfection,studiesoftissuesfromhepatocellularcarcinomapatientsshowthatHCVcanenhanceinfectedhepatocyteturnoverinordertoreplacecellsdestroyedbyimmunologicalattack[ 4 ]. Anexampleofthestabilityproleofthemodelwithrealisticparameters;s=3661.7,Tmax=6.33106,d=1.4103,=7.4108,p=40.5,c=11.5,r1=0.31,r2=4.4,whileisvariedfrom16.56to1.1correspondingtoR0from0.1to1.5.Therealisticrangeforisdenedtobebetweendand3.0day1,[ 14 ],isshownintheFigure.ThebiologicallyinconsistentequilibriumsareomittedinthisFigure. Aparticularchoiceofmodelparametersdescribesthecharacteristicsofaparticularpatientphenotypeinfectedwithaparticularviralgenotype:wereferinterchangeablytoparametersetsandsimulated(insilico)HCVpatients.Eachpatient/setofparameters 74

PAGE 75

4-1 .Dependingonapatient'sR0andthecorrespondingEE(representedintheFigurebyviralload),apatientcanfallinanyofthefourdistinctregionsshownintheFigure 4-1 1. PatientswhosediseaseparameterswouldcorrespondtoaR0nearzero(region1inFigure 4-1 )wouldhaveaveryhighchanceofspontaneouslyclearingthevirus.Thesepatientswouldprobablyneverrequiremedicalintervention,whichiscommonamonginfectedinfantsandwomen[ 28 ]; 2. Inregion4,whereR01,theModel 4 neverpredictsSVR.InthisregiontheDFEexistsbutisunstable,whiletheEEisstable.Themodelpredictsthatpatientswithparametersinregion4willalwaysreboundtotheiroriginalviralloadoncetreatmentisstopped.Thisregionbehavessimilarlyasintheabsenceofbistability; 3. Inthecentral(blueshaded)region,themodelisbistable.Threeequilibriums(astableDFE,astableEE,andanunstableequilibriumbetweenthem)exist.Ifthepre-treatmentconcentrationofinfectedhepatocytes/viralloadofapatientfallsinregion3,thenthesepatientswillbehavesimilarlyasthoseinregion1andmostlikelyexhibitspontaneouscure.Ifitfallswithinregion2,thenintheabsenceoftreatmenttheywillestablishstableEE,similartothepatientsinregion4.Unlikepatientsinregion4,however,treatmentmaypushtheviralloaddownintoregion3,whichwillthenleadtoSVR.Regions2and3areseparatedbytheunstableendemicequilibriumwhichmaybetreatedasaviralloadthresholdthatneedstobeachievedtoleadtocureposttherapycessation. Werstanalyticallydenetheconditionsthatwilldetermineifaparametersetliesintheregionofbistabilityornot. Bistabilityoccursunderthefollowingconditions(derivedintheSection 4.4 ): T20+r1 cr2 c+r1 Thesecondconditionimpliesthatr2>r1.Thatis,therateofproliferationofinfectedhepatocytesisgreaterthantherateofproliferationofhealthyhepatocytes.Itislikelythatwhenr2issufcientlygreaterthanr1,evenwhenR0<1,thesystemcankeepproducingenoughnewinfectedhepatocytesandinturnenoughfreevirus,thatanon-zeroEEcanbestable.Usingthebistabilitycriteria,wecanalsodeterminetheregionofbistability 75

PAGE 76

30 ],wecandeterminethecriticalvaluesofc,whichcorrespondstoRc,thelowerboundaryofthebistableregion. Inepidemiologicalmodelswithbackwardbifurcation,thecontrolisintroducedtopermanentlychangeapopulationparametersuchthatR0
PAGE 77

4 ,r11T0 T0. Theremainingtwoeigenvaluesaretheeigenvaluesofthematrix264r21T0 77

PAGE 78

c 30 ],weknowthatthemodelhasatmosttwopositiveequilibria. Equation 4 leadstoV=p cI,whichallowsustoreduce 4 4 totwoequations.DeningT=f(I),werewrite 4 as cf(I)I=0. Tocomputethedisease-freeequilibrium(DFE)weassumeI=V=0,givingtheconcentrationofhealthyhepatocyteswithoutinfectionT0:=f(0)=Tmax 4 asafunctionF(I): c Notethatf(0)=T0,andF(0)=p c cT0+r21T0 78

PAGE 79

cr2 cr1 f(I)21. Sincef0(I)isalways<0,thesignofF0(I)dependson(p cr2 1. when(p cr2 2. when(p cr2 cr2 cf(I)+r21f(I)+I 4-1 ).Fromtheaboveweget cr2 cT0+r21T0 79

PAGE 80

4 ,bothR0andthedenominatorarealwayspositive.Hencefortheexpressiontobenegativewerequirer2 cr2 T20+r1 cr2 c+r1 Thisinequalitygivesanecessaryandsufcientconditionforbackwardbifurcation. 30 ].Werstdenethefunctionf1(T)asfollows; TmaxdT Fromthesecondequationofthesystem 4 ,forI6=0wecomputeI=Tmax cr2 4 wegetthefollowing; cr2 cTTmax cr2 Thiscanbewrittenas cTmaxp cr2 NowwenametherighthandsideoftheEquation 4 asf2.Notethatf1andf2areparabolasandachronicEEwillsatisfythecondition,f1(T)=f2(T).Alsonotethatatthetip,thereisauniqueEEandthegraphsoff1andf2touchintherstquadrant.Thathappenswhenboththesefunctionshavethesametangent(oraretangenttoeach 80

PAGE 81

TheCondition 4 canbewrittenasaquadraticinTasfollows, cTmax)p cr2 cTmax)(r2)(r1d)s.(4) AndtheCondition 4 givesalinearinTasfollows, cTmax)p cr2 cTmax)(r2)(r1d)(4) NoweliminatingTfromEquation 4 and 4 wegettheequation cTmax)p cr2 cTmax)(r2)(r1d)2(4) SolvingforfromtheEquation 4 ,givesusthevalueofthec,thevalueofatthelowerendofthebistabilityregion. cTmax)"s cTmax)p cr2 UsingthiscwecancomputetheRcas, c 81

PAGE 82

Tmaxd(T+et) Tmax(T+et) ReducingtheSystem 4 usingtheconditionsfromtheSystem 4 4 andignoringhigherordersofperturbations,weget, Sincewearelookingforexponentialsolutions,wegetthefollowingsystem.NotethatherewearereplacingV=p c. cIetr1T cr2 ConstructingtheJacobianfortheSystem 4 withrespectto(et,i,v),wegetthefollowing.J(T,I,V)=hr11T+I cIir1T cr2 cI=s T,fromEquation 4 ,andr21T+I cT.ThustheJacobiancanbere-writtenas, 82

PAGE 83

Tr1T cr2 cTr2I cr2 Tr1T cTr2I cr2 cr2 cr2 Thisequationisalsothecharacteristicequationofthissystemandweinvestigateitseigenvalues.NotethatA>0,byobservation.NowweconsiderC,whichcanbe 83

PAGE 84

cT+p cTp cr2 cr2 T+r1T cT+r2 cT+p cTp cr2 cr2 T+r1T cr2 c+r1 T+r1T T+r1T cr2 T+r1T cr2 T+r1T cr2 T+r1T cT+r2I T+r1T cT+r2I ccT+r1T cr2 Tp cT+r2I cT+s Tc+r1T cT, Thenwecanseethatifp cr2 84

PAGE 85

cr2 4.5.1RealisticParameterSets 14 16 ]havedoneextensivestudiestodeterminerealisticrangesforallthemodelparametersthatweuse.Themostsignicantcriteriaforthebistabilityisthefactthatwerequiretheparameterstosatisfyr2>r1bythequantityderivedfromanalysis.Notethattheabovementionedarticleshavealsoconsideredthiscondition,inadditiontor2r1intheirinvestigation,butwereunabletondanysignicantdifferenceinqualitativebehavior. Toensurethatthebistabilityconditioncouldbesatisedwitharealisticparameterset,weusedparameterrangesfromtheliteratureastabulatedinTable 4-1 .WerstdrewtheparametersTmax,d,s,r1,,pandcfromanuniformdistribution.Wethendrewr2suchthatthesecondoftheConditions 4 wassatised.Basedontheseparameters,wethencalculatearangeforthebifurcationparameter:R0=1givesus0,andthecalculationof[ 30 ]givesusthevalueofc.Allvaluessuchthat0<
PAGE 86

4.5.3 and 4-3 forinspection.WeselectedaparametersetthatproducedgoodillustrationsasPatientI.Extremecaseswhereveryhighefcacywasrequiredforalongtime,toachieveSVRwerealsoobserved. 4 werstshowthatitispossibletosimulateallthefourdistincttypesofpatientprolesobservedinrealpatients,asseeninFigure 1-1 BThedistinctprolesgeneratedfromthemodelwhenthebasicreproductionnumberisinthebistabilityregion,Rc
PAGE 87

4-2A ;theothercategoryofpatientswhosediseaseparametershaveRc1.Evenifhighlyeffectivetreatmentisintroducedforalongtime,relapseoccursalmostinstantaneouslyaftertherapycessation.Forpatientsinthiscategorywecangenerateprolesfornullandpartialresponders,EVR,andRVR,butneverSVR.Furthermore,eventhepatientswhoareabletoshowRVRdonotachieveSVRfrommodelsimulations,althoughRVRisaverygoodindicatorofSVRinreality.ThusModel 4 neverpredictscureforpatientswithsuchparametercombinations.However,prolongedsuppressionofviralloadforthesepatientsmightbepredictedasaconsequenceofstochasticextinctionorslowrebound.Forexample,ifwecanreducetheviralloadlessthanonevirusperpatient,itislikelytheviruscannotestablishitselfanymore;orincaseofrebound,itmightbepossibletoincreasetheperiodofresurgencesothatapatientcanavoidsignicantliverdamagewithintheirlifetime.Boththesescenariosrequireaverycarefulnumericalcomparisontodrawconclusions. Ifapatient'sparametersetsatisesthecriteriaforbackwardbifurcation,withRc
PAGE 88

4-1 andx=2.8day1.ClearlyitisintheblueregioninFigure 4-1 .WeassumethatachronicpatientwhorequirestreatmenthasthestatevariablesatthestableEE.ThispatientdoesnotclearthevirusspontaneouslyandthuswestudytheregionsofattractionoftheEEandDFE. Wereducethe3dimensionalsystemtoaplanarsystemwiththesamedynamicalproperties,toallowforclearerillustrationofresults.ItisareasonableassumptionbecausefreevirushasrapiddynamicsrelativetoTandI,soatanygivenlevelofTandIitequilibratesrapidly,sotreatingVasthoughitisatitsequilibriumisagoodapproximation.Substituting,I=c pV,theModel 4 reducesto, dt=s+r1T1T+c pV TmaxdTTVdV dt=p cTV+r2V1T+c pV TmaxV. ThereducedModel 4 hasthesameequilibriumpointsandsimilarbifurcationpropertiesastheModel 4 .Werstplotthestableandunstablemanifoldsintheplaneofhealthyhepatocytesandviralload,whichhelpsustodeterminetheregionsofattractionoftheDFEandtheEE.TheFigure 4-3 canbeviewedasalongitudinalsectionoftheblueregioninFig 4-1 at=2.8orR0=0.59,withthreeequilibriumpointsrepresentedwithgreendots.TheonewiththehighestviralloadisthestableEE,theonenextistheunstableEE,andtheonewithzeroviralloadistheDFE.Theredlineistheseparatrix,acurvewhichseparatesthespacesuchthatallthetrajectorieswithinitialconditionstotheleft(inthiscase)ofitconvergetotheEEwhereasallthetrajectories 88

PAGE 89

ShowsthestreamplotforPatientIusingthereducedmodel.Theblacklinegivesitsstablemanifoldandtheredlinegivestheunstablemanifold.Thebluedotsshowthethreeequilibriums.Thechangeinhealthyhepatocytesvsviralloadduringtherapyandaftertreatmentcessationisshownbythepurpleandmagentadashedlinesrespectively.Thetreatmentresponsefor=65%andt1=135dayswassimulatedusingtheoriginalModel 4 IntheFigure 4-3 ,themagentadashedlineshowstheTVcurveofPatientIwhiletreatmentissimulatedusingtheModel 4 ,withanefcacyof65%for135days.ThegreendashedlineshowsthecurveaftertreatmentisstoppedandthestatevariablesconvergetotheDFE.Itisexpectedthatifthispatientistreatedwithhigherefcacyandforalongerperiodoftime,wouldalsoachieveSVR,suchthatthemagentalinewouldconvergetotheDFEdirectly. WeobservethatitisnotonlyimportanttolowertheviralloadtoachieveSVR,butalsoincreasehealthyhepatocytestoacertainextent.Thisnotioncanexplainthe 89

PAGE 90

16 ].Fromthismodelwecanseethat,sincetheconcentrationofhealthyhepatocytesisconsiderablylowerinacirrhoticpatient,treatingtoreachtheattractiveregionofDFEinthiscaseisharder. Figure4-4. UnderstandingtreatmentstrategyforPatientI.TheFigureonleftshowstheviralresponseasefcacyofdrug,isincreasedfrom0to.9,by.1whentreatedforaxedperiodoftime48weeks,andcheckedforSVR6monthslater.TheFigureontherightshowstheTheoptimalperiodoftreatmentnecessarytoachieveSVRforarangeofs. Inpresenceofbistability,themodelallowsustoexplorewithsimulations,theeffectoftreatmentnecessarytobringthecombinationofthestatevariablestothebasinofattractionoftheDFE,sothatthepatientwillspontaneouslyclearoffthevirus.Weshallmodulatetheeffectoftreatmentbytheefcacy,andthedurationoftreatmentt1. TheleftofFigure 4-4 showsthatforlow,theinsilicopatientreturnstopre-treatmentviralloadaftercessationoftherapy.However,asincreasesthepatientiscapableofachievingSVR.Thisreasonablyobviousbiologicalfactthat,anHCVpatient'schancesofachievingSVRincreaseswithincreasedefcacyoftreatmenthasbeencapturedbyamathematicalmodelforthersttime. Furthermore,sometimesadesiredefcacyoftreatmentisnotpossibletoachieveforagivenpatientduetoweakimmuneresponse,andinthatcasethedurationof 90

PAGE 91

36 ],itwasfoundthatincreasingthetreatmentdurationto72weekswhenre-treatingpreviousnon-respondersyieldedmoreSVR.Insuchcases,therelationshipbetweenandt1forachievingSVRforaspecicpatientcanbeexplored.Particularly,weseetheoptimaltreatmentdurationforarangeofdrugefcaciesforPatientIintherightofFigure 4-4 Mostpreviousanalysisofbistabilityhavefocusedatthepopulation,orepidemiologicallevelmodels[ 23 24 40 ].Amongthefewpreviousmodelstoexplorebistabilityinanimmunological(invivo)model,Gomez-AcevedoandLi[ 30 ]usedbackwardbifurcationtoexplaintheviraldynamicsofHTLV-Iinfectionandencouragetreatmentmeasurestoconsiderthemoredifcultthresholdthatneedstobeovercometoachievecureincaseofbistability.Also,[ 54 ]haveinterpretedbistabilitytoposeaharderthresholdforsuccessfultreatment,whenHCVandHBVinfectionisconsideredinbothbloodandliver.Weutilizethebistabilitypropertyofthismodeltogeneratealltheclinicallyobservedpatientproleandestimatetheeffectoftreatmentnecessaryforagiveninsilicopatient. Thebistabilitypropertyofthemodel 4 allowsustosimulateallthedistinctpatientprolesundertreatment,thatareclinicallyobserved(Figure 1-1 ).Forpatientswithdiseaseparameterswhichdonotmathematicallyimplyspontaneousviralclearance,weareabletoreproducethequalitativeoutcomesofnullresponders,partialresponders, 91

PAGE 92

Wendthatanecessaryconditionfortheexistenceofbistabilityinthismodelisthattherateofproliferationofinfectedhepatocytesbegreaterthanthatofthehealthyhepatocytes.Thiscounterintuitiveconditionhasbeenprovedtobeabiologicalpossibility,frominvitrostudiesin[ 4 ].Inthisdissertation,giventhediseaseparametersofapatientwecan(1)establishwhetherbistabilityoccursand(2)establishboundsonefcacyanddurationoftreatment.Fromnumericalexplorationoftheseparatrixinthestatevariablespace,wecandeterminethe'curethreshold'whichneedstobeovercomeforsustainedclearanceofthevirus.Wecanalsosimulateviralresponsetotreatmentforanexhaustivecombinationofefcacyanddurationoftreatment.Inrecentdaystheneedtomodifydurationoftreatmentdependingonpatientresponseisbeingwidelyacceptedinthescienticliterature[ 45 ].Thusthisnumericalinvestigationisdenitelymuchneeded. In[ 40 ]incontextofcontrollingTBatthepopulationlevel,theauthorssuggestreducingthebasicreproductionnumberintotheregionofsingularstabilityoftheDFE(region1inFigure 4-1 )bypermanentlychangingamodelparameterbyvaccinatingpeople.Thissortofmeasurecanbeapplicabletoourscenarioalsoifitwaspossibleto 92

PAGE 93

4-1 ). Model 4 isaverysimplemathematicalrepresentationoftheviralinfectioninthehumanliver,whichdoesnotincludethemultiplelevelsofcomplexitiesinvolvedinthehumanphysiologyortheimmunobiologyofthehepatitisCvirus.Furthermore,sinceestimationofparametershastobedonefromveryshorttime-seriespatientdata,theuncertaintyonparameterrangesareverywide[ 14 16 ].Infact,tomakeanyreasonablepredictionforanindividualpatient,diseaseparametersneedtobeestimatedfromevenshorterperiodsoftreatmentresponse,perhapsbasedonclinicalbiomarkerstopredictlongtermoutcomes.Useofwithin-hostmodelsasadiagnosticandmanagementtoolsiswellknownbynow,butstillpresentsmanypracticalchallengeseveninestablishedareaslikeHIVtreatment.However,wehavedemonstratedanewwaytothinkaboutthelongtermdynamicsofHCVinfectionwithinhosts.Inprinciplethismodelcouldultimatelydevelopintoadiagnosticormanagementtoolforcliniciansinthefuture. 93

PAGE 94

5-1 Showstheschematicmodel. Themomentahepatocytegetsinfectedisat=0.Thus,thetermi(,t)isthedensityofhepatocyteswhichhavebeeninfectedfortimeatchronologicaltimet.Therefore,thetotalnumberofinfectedcellsattimetshouldbegivenbyI(t)=Z10i(,t)d.

PAGE 95

dt=s+r1T1T+I TmaxdTTV@i TmaxdV dt=P(t)cV where,R2(t)=Z10r2()i(,t)dP(t)=Z10p()i(,t)d Thefunction()givestheprobabilitythataninfectedhepatocytewilldieattheageofinfection.Thusthetotalnumberofdeathamongtheinfectedhepatocytecohortofageofinfectionisequalto()i(,t),inthetimeintervalttot+t,whichisthesameasthetimeintervalbetweenand+.HenceweobtainEquation 5 .Thefunctionalformofthisfunctioncanbeconsideredlinearforsimplicity.Forexample,()=d+d1,

PAGE 96

Thenumberofnewinfectionsatatimepointtisgivenbyi(0,t).ThisincludesthehepatocyteswhichgotinfectedattimetrepresentedbytherateofnewinfectionTVwithageofinfection=0,andthenumberofinfectedhepatocytesbornoutofpreviouslyinfectedhepatocytes.Thelatterisrepresentedbyalogisticterm,withtherateofthisproliferationdependingonageofinfectionbeingr2().Therateofchangeofvirusdensitywithrespecttochronologicaltimeisgivenbythenumberofnewvirusparticlesproducedfrominfectedhepatocyteattherateofp()dependingonageofinfectionagain.Theratesr2()andp()havetobeconsideredaspiecewisefunctions,sincetheseprocessesgetarrestedordonotinitiate,respectively,immediatelyafterinfection. Althoughtheactualexpressionofinfectedhepatocyte'sproliferationratewithrespecttoageofinfectionr2,hasnotbeenexperimentallydetermined,itisthoughttobeinitiatedbythevirusinsidethehepatocytetocreatemorereservoirsofitself.So,apossiblefunctioncouldbeoftheform, Theexpressionforp()usedin[ 47 ]forproductionofHIVwhichcanalsobeusedforHCVisasfollows,

PAGE 97

Nowwemoveontoanalyzethismodelatequilibriumwithrespecttochronologicaltime. 5 ,weequateallthetimederivativestozero.Thatgivesusthefollowingsystem. TmaxdTTV@i Tmax0=PcV Notethatatequilibrium,allvariablesareconstantwithrespecttochronologicaltime.So,i(,t)=i(),R2(t)=R2andP(t)=P. Nowwesolvefori()fromEquation 5 toget Usingthisvaluewecandenethefollowingatequilibrium. Theseconstantswhenmultipliedwithi0essentiallyrepresentthetotalnumberofinfectedhepatocytes(ofallagesofinfection),rateofproliferationofinfectedhepatocytes 97

PAGE 98

5 andapplyingtheconditionsfromdisease-freeequilibriumwegetthefollowingsystem. dt=r1ex1T0 Tmax@ey dt=Z10p()ey(,t)dcezey(0,t)=T0ez+1T0

PAGE 99

Tmaxy+dy d=()y()

PAGE 100

5 wegetthefollowingcharacteristicequation,1=T0 Differentiatingwithrespecttoweget,G0()=T0 NowifG(0)<1,G()=1hasauniquesolutionwhen<0.Inthatcase,weexplorethesignoftherealpartofthesolutionofinthecomplexeld.Let=a+ib, 100

PAGE 101

Nowweproceedtoinvestigatetheexistenceofendemicequilibria.Solvingoutwecangettwopossibleendemicequilibria(T1,I1,V1)and(T2,I2,V2)givenbythe 101

PAGE 102

102

PAGE 103

HepatitisCinfectsmillionsofpeopleword-wideandisthecauseformaximumnumberoflivertransplantsintheUSA.Thisdiseasebringsseveralchallengestothecliniciansduringandpostthesub-optimalcombinationtherapycurrentlyused.Inthisdissertation,Ihaveaddressedsomekeyissuesthatareconsideredtocontributegreatlytofailureoftherapy.Theentiremotivationistoprovideaproof-of-conceptstudyofhowmathematicalmodelingcanhelpunderstandsomeofthecomplexitiesofthisinfectioninvivoandpossiblyquantifytreatmentefcacy,lengthoftreatmentandneedfordrugstopreventside-effects. InChapter3weaddresspossiblemanagementoptionfortheside-effectofanemia.Thecombinationtherapyofantiviralpeg-interferonandribavirinhasevolvedasoneofthebettertreatmentsforHepatitis-C.InspiteofitssuccessincontrollingHepatitis-Cinfection,ithasalsobeenassociatedwithtreatment-relatedadversesideeffects.Themostcommonandlifethreateningamongthemishemolyticanemia,necessitatingdosereductionortherapycessation.Thepresenceofthissideeffectleadstoatrade-offbetweencontinuingthetreatmentandexacerbatingtheside-effectsversusdecreasingdosagetorelievesevereside-effectswhileallowingthediseasetoprogress.Thedrugepoietin(epoetin)isoftenadministeredtostimulatetheproductionofredbloodcells(RBC)inthebonemarrow,inordertoallowtreatmentwithoutanemia.Thisdissertationusesmathematicalmodelstostudytheeffectofcombinationtherapyinlightofanemia.InordertoachievethisweintroduceRBCconcentrationandamountofdruginthebodyasstatevariablesintheusualimmunologicalvirusinfectionmodel.Analysisofthismodelprovidesaquanticationoftheamountofdrugabodycantoleratewithoutsuccumbingtohemolyticanemia.IndirectestimationofparametersallowustocalculatethenecessaryincrementinRBCproductiontobe2.3timesthepatient'soriginalRBC 103

PAGE 104

InChapter4wendmathematicalconditionsunderwhich'cure'canbeproducedfromsimulations,andanestimationoftheoptimalcombinationoftreatmentefcacyandtreatmenttimecanbemade.Inclinicalsettings,short-termtreatmentresponse(so-calledsustainedvirologicalresponse[SVR])isusedtopredictprolongedviralsuppression.Althoughordinarydifferentialequation(ODE)modelsforwithin-hostHCVinfectionhaveilluminatedthemechanismsunderlyingtreatmentwithinterferon(IFN)andribavirin(RBV),theyhavedifcultyproducingSVRwithouttheintroductionofanexternalextinctionthreshold.HereweshowthatbistabilityinanexistingODEmodelofHCV,whichoccurswheninfectedhepatocytesproliferatesufcientlymuchfasterthanuninfectedhepatocytes,canproduceSVRwithoutanexternalextinctionthresholdunderbiologicallyrelevantconditions.Themodelcanproduceallclinicallyobservedpatientprolesforrealisticparametervalues;itcanalsobeusedtoestimatetheefcacyand/ordurationoftreatmentthatwillensurepermanentcureforaparticularpatient. Wefurthermoveontoformulateageneralizedpartialdifferentialequation(PDE)modelbyincludingtheage-of-infectionoftheinfectedhepatocytesasasecondaryindependentvariable.Wediscussthebasicreproductionnumberandthestabilityofthediseasefreeequilibriuminthiscase.Moreover,wecomputethepossiblestructureofthetwobiologicallyrealisticendemicequilibrium,whereapossibilityofbackwardbifurcationexists. Theresearchworkpresentedinthisdissertationforthersttimeaddressedtheside-effectofhemolyticanemiaincontexttoHCVinfectionandtreatment.Theapplicationofbackwardbifurcationinimmunologydonehereisalsoanovelapplicationofthisotherwisepopularbehaviorinepidemiologicalmodelsstudiedwhenadiseaseparameterischangedpermanently.ThePDEmodeneedsmoreinvestigationand 104

PAGE 105

105

PAGE 106

[1] Afdhal,N.H.,Dieterich,D.T.,Pockros,P.J.,Schiffshort,E.R.,Shiffman,M.L.,Sulkowski,M.S.,Wright,T.,Younossi,Z.,Goon,B.L.,Tang,K.L.,Bowers,P.J.,andGroup,TheProactiveStudy.EpoetinalphamaintainsribavirindoseinHCV-infectedpatients:aprospective,double-blind,randomizedcontrolledstudy.Gastroenterology126(2004).5:1302. [2] Bain,C.,Fatmi,A.,Zoulim,F.,Zarski,J.P.,Trpo,C.,andInchausp,G.ImpairedAllostimulatoryFunctionofDentricCellsinChronicHepatitisCInfection.Gas-teroenterology120(2001).2:512. [3] Baxter,L.T.,Yuan,F.,andJain,R.K.PharmacokineticsAnalysisoftheperivasculardistributionofbifunctionalantibodiesandhaptens:Comparisonwithexperimentaldata.CancerRes.52(1992):5838. [4] Block,T.M.,Mehta,A.S.,Fimmel,C.J.,andJordan,R.MolcularViralOncologyofHepatocellularCarcinoma.Oncogene22(2003):5093. [5] Brauer,FredandCastillo-Chavez,Caros.MathematicalModelsinPopulationBiologyandEpidemiology.NewYork:Springer-Verlag,2000,1steditioned. [6] Bricker,S.L.,Langlais,R.P.,andMiller,C.S.OralDiagnosis,OralMedicineandTreatmentPlanning.Ontario,Canada:BCDeckerInc,2002,2ndeditioned. [7] Callaway,D.S.andPerelson,A.S.HIV-1infectionandlowsteadystateviralloads.Bull.MathematicalBiology64(2002).1:29. [8] Casadevall,D.,Eckardt,Kai-Uwe,andRossert,Jerome.Epoetin-inducedautoimmunepureredcellaplasia.JAmSocNephro16(2005):S67S69. [9] Choo,QuiLim,Kuo,G,Weiner,AJ,Overby,LR,Bradley,DW,andHoughton,M.IsolationofacDNAclonederivedfromablood-bornenon-A,non-Bviralhepatitisgenome.Science244(1989).4902:359. [10] Crotty,S.,Cameron,C.E.,andAndino,R.RNAviruserrorcatastrophe:Directmoleculartestbyusingribavirin.Proc.NatlAcad.Sci.USA98(2001):68956900. [11] Dahari,H.,Loa,A.,Ribeiroa,R.M.,andPerelson,A.S.ModelinghepatitisCvirusdynamics:Liverregenerationandcriticaldrugefcacy.JournalofTheoreticalBiology247(2007):371. [12] Dahari,H,Major,M,Zhang,X,Mihalik,K,Rice,CM,Perelson,AS,Feinstone,SM,andNeumann,AU.MathemticalmodelingofprimaryhepatitisCinfection:Noncytolyticclearanceandearlyblockageofvirionproduction.Gastroenterology128(2005):1056. [13] Dahari,H.,Ribeiro,R.M.,andPerelson,A.S.TriphasicDeclineofHepatitisCVirusRNADuringAntiviralDecline.Hepatology46(2007):16. 106

PAGE 107

Dahari,H.,Shudo,Emi,Cotler,ScottJ.,Layden,T.J.,andPerelson,A.S.ModellingHepatitisCVirusKinetics:theRelationshipbetweentheInfectedCellLossRateandtheFinalSlopeofViralDecay.AntiviralTherapy14(2009):459. [15] Dahari,H.,Shudo,Emi,Ribieiro,R.M.,andPerelson,A.S.ModelingComplexDecayProlesofHepatitisBVirusDuringAntiviralTherapy.Hepatology49(2009).32-38. [16] Dahari,Harel,Layden-Almer,J.E.,Kallwitz,Eric,Ribieiro,R.M.,Cotler,S.J.,Thomas,T.J.,andPerelson,A.S.AMathematicalModelofHepatitisCVirusDynamicsinPatientswithHighBaselineViralLoadsorAdvancedLiverDisease.Gastroenterology136(2009):14021409. [17] Davis,G.L.,Wong,J.B.,McHutchison,J.G.,Manns,M.P.,Harvey,J.,andAlbrecht,J.Earlyvirologicresponsetotreatmentwithpeginterferonalfa-2bplusribavirininpatientswithchronichepatitisC.Hepatology38(2003).3:645. [18] Davis,RussellC.,Davies,MichaelK.,andLip,GregoryY.H.ABCofHeartFailure.Wiley-Blackwell,2007,2ndeditioned. [19] DeLeenheer,P.andPilyugin,S.S.Multi-strainvirusdynamicswithmutations:Aglobalanalysis.MathematicalMedicineandBiology25(2008).4:285. [20] DeLeenheer,P.andSmith,H.L.VirusDynamics:AGlobalanalysis.SIAMJ.ofAppl.Math.63(2003).4:13131327. [21] Dienstag,J.L.andMcHutchison,J.G.AmericanGastroenterologicalAssociationmedicalpositionstatementonthemanagementofhepatitisC.Gastroenterology130(2006).1:225. [22] Dixit,N.M.,Layden-Almer,J.E.,Layden,T.J.,andPerelson,A.S.ModellinghowribavirinimprovesinterferonresponseratesinhepatitisCvirusinfection.Nature432(2004):922. [23] Dushoff,Jonathan,Huang,Wenzhang,andCastillo-Chavez,Carlos.Backwardsbifurcationsandcatastropheinsimplemodelsoffataldiseases.J.Math.Biol.36(1998):227248. [24] Feng,Z.,Castillo-Chavez,C.,andCapurro,A.Amodelfortuberculosiswithexogeneousreinfection.Theor.Pop.Biol.57(2000):235. [25] Franceschi,LDe,Fattovich,G,Turrini,F,Ayi,K,Brugnara,C,Manzato,F,Noventa,F,Stanzial,AM,Solero,P,andCorrocher,R.HemolyticanemiainducedbyribavirintherapyinpatientswithchronichepatitisCvirusinfection:Roleofmembraneoxidativedamage.Hepatology31(2000):9971004. [26] Fried,M.W.,Shiffman,M.L.,Reddy,K.R.,Smith,C.,Marinos,G.,GoncalesJr,F.L.,Hussinger,D,Diago,M,Carosi,G,Dhumeaux,D,Craxi,A,Lin,A,Hoffman,J, 107

PAGE 108

[27] Gerbino,PhilipP.TheScienceandPracticeofPharmacy.Philadelphia,PA:LippincottWilliamsandWilkins,2006,21steditioned. [28] Ghany,MarcG.,Strader,DorisB.,Thomas,DavidL.,andSeeff,LeonardB.Diagnosis,Management,andTreatmentofHepatitisC:AnUpdate.Hepatology49(2009).4:1335. [29] Goldsby,R.A.andetal.Immunology.W.H.FreemanandCompany,2000,4ed. [30] Gomez-Acevedo,H.andLi,M.Y.BackwardBifurcationinamodelHTLV-IinfectionofCD+Tcells.BulletinofMathemationBiology67(2005):101. [31] Gonzlez-Peralta,RP,Kelly,DA,Haber,B,Molleston,J,Murray,KF,Jonas,MM,Shelton,M,Mieli-Vergani,G,Lurie,Y,Martin,S,Lang,T,Baczkowski,A,Geffner,M,Gupta,S,Laughlin,M,andInternationalPediatricHepatitisCTherapyGroup.Interferonalfa-2bwithribavirinforthetreatmentofhepatitisCinchildren:efcacy,safety,andpharmacokinetics.Hepatology42(2005):1010. [32] Gremion,C.andCerny,A.HepatitisCVirusandtheImmmuneSystem:aConciseReview.ReviewsinMedicalVirology15(2005).4:235. [33] Herrmann,E.,Lee,J.,Marinos,G.,Modi,M.,andZeuzem,S.EffectofRibavirinonHepatitisCViralKineticsinPatientsTreatedwithPegylatedIntereron.Hepatol-ogy37(2003).6:1351. [34] Hoofnagle,J.H.CourseandOutcomeofHepatitisC.Hepatology36(2002).5Suppl.1:S21. [35] Jensen,D.M.,Morgan,T.R.,Marcellin,P.,Pockros,P.J.,Reddy,K.R.,andetal.,S.J.Hadziyannis.EarlyidenticationofHCVgenotype1patientsrespondingto24weekspeginterferonalpha-2a(40kd)/ribavirintherapy.Hepatology43(2006):954. [36] Jensen,DonaldM.,Marcellin,Patrick,Freilich,Bradley,Andreone,Pietro,Bisceglie,AdrianDi,Brando-Mello,CarlosE.,Reddy,K.Rajender,Craxi,Antonio,Martin,AntonioOlveira,Teuber,Gerlinde,Messinger,Diethelm,Thommes,JamesA.,andTietz,Andreas.Re-treatmentofPatientsWithChronicHepatitisCWhoDoNotRespondtoPeginterferon-2b:ARandomizedTrial.AnnInternMed150(2009):528. [37] Kato,K,Kamezaki,K,Shimoda,K,Numata,A,Haro,T,Aoki,K,Ishikawa,F,Takase,K,Ariyama,H,Matsuda,T,Miyamoto,T,Nagafuji,K,Gondo,H,Nakayama,K,andHarada,M.Intracellularsignaltransductionofinterferononthesuppressionofhaematopoieticprogenitorcellgrowth.BrJHaematol123(2003):528535. 108

PAGE 109

Kato,T.,Date,T.,Miyamoto,M.,Sugiyama,M.,Tanaka,Y.,Orito,E.,Ohno,T.,Sugihara,K.,Hasegawa,I.,Fujiwara,K.,Ito,K.,Ozasa,A.,Mizokami,M.,andWakita,T.DetectionofAnti-HepatitisCVirusEffectsofInterferonandRibavirinbyaSensitiveRepliconSystem.JCM43(2005).11:5679. [39] Kowdley,K.V.HematologicsideeffectsofInterferonandribavirintherapy.JClinGastroenterol39(2005):Suppl. [40] Kribs-Zaleta,C.M.andMartcheva,Maia.Vaccinationstrategiesandbackwardbifurcationinanage-since-infectionstructuredmodel.MathBiosci177&178(2002):317. [41] Lam,NP,Neumann,AU,Gretch,DR,Wiley,TE,Perelson,AS,andLayden,TJ.Dose-dependentacuteclearanceofHepatitisCgenotype1viruswithinterferonalpha.Hepatology26(1997):226. [42] Lanford,RE,Chavez,D,Guerra,B,Lau,JY,Hong,Z,Brasky,KM,andBeames,B.RibavirininduceserrorpronereplicationofGBvirusBinprimarytamarinhepatocytes.JVirol75(2001):8074. [43] LaSalle,J.P.Stabilitytheoryforordinarydifferentialequations.J.Diff.Eqns.4(1968):57. [44] Lau,J.Y.N.,Tam,R.C.,Liang,T.J.,andHong,Z.MechanismofactionofribavirininthecombinationtreatmentofchronicHCVinfection.Hepatology35(2005).5:1002. [45] Lee,SamuelSandFerenci,Peter.OptimizingoutcomesinpatientswithhepatitisCvirusgenotype1or4.AntiviralTherapy13Suppl1(2008):9. [46] Mackey,M.C.Periodicauto-immunehemolyticanemia:Aninduceddynamicaldisease.BullMathBiol41(1979):829. [47] Nelson,P.W.,Gilchrist,M.A.,Coombs,D.,Hyman,J.M.,andPerelson,A.S.Anage-structuredmodelofHIVinfectionthatallowsforvariationsintheproductionrateofviralparticlesandthedeathrateofproductivelyinfectedcells.Math.Biosci.Eng1(2004):267. [48] Neumann,A.U.,Lam,N.P.,Dahari,H.,Davidian,M.,Wiley,T.E.,Mika,B.P.,Perelson,A.S.,andTJ.,Layden.DifferencesinViralDynamicsbetweenGenotypes1and2ofHepatitisCVirus.J.Infect.Dis182(2000):28. [49] Neumann,U.Avidan,Lam,NancyP.,Dahari,Harel,Gretch,DavidR.,Wiley,ThelmaE.,Layden,ThomasJ.,andPerelson,AlanS.HepatitisCViralDynamicsinVivoandtheAntiviralEfcacyofInterferon-alphaTherapy.Science282(1998).2:103. 109

PAGE 110

Nowak,M.A.,Bonhoeffer,S.,Hill,A.M.,Boehme,R.,andThomas,H.C.ViralDynamicsinHepatitisBvirusinfection.Proc.Natl.Acad.Sci.93(1996):43904402. [51] Pawlotsky,JM,Dahari,H,Neumann,AU,Hezode,C,Germanidis,G,Lonjon,I,Castera,L,andDhumeaux,D.AntiviralActionofrivavirininChronichepatitisC.Gastroenterology126(2004):703. [52] Perelson,AlanS.,Neumann,AvidanU.,Markowitz,Martin,Leonard,JohnM.,andHo,DavidD.HIV-1DynamicsinVivo:VirionClearanceRate,InfectedCellLife-Span,andViralGenerationTime.Science271(1996):1582. [53] Powers,K.A.,Dixit,N.M.,Ribeiro,R.M.,Golia,P.,Talal,A.H.,andPerelson,A.S.Modelingviralanddrugkinetics:hepatitisCvirustreatmentwithpegylatedinterferonalfa-2b.Semin.LiverDis.23(2003).Suppl1:13. [54] Qesmi,Redouane,Wub,Jun,Wua,Jianhong,andHeffernan,JaneM.InuenceofbackwardbifurcationinamodelofhepatitisBandCviruses.MathBiosci224(2010):118125. [55] Reluga,T.C.,Dahari,H.,andPerelson,Alan.AnalysisofHepatitisCVirusInfectionModelswithHepatocyteHomeostasis.SIAMJ.Appl.Math.69(2009).4:999. [56] Rong,Libin,Feng,Zhilan,andPerelson,AlanS.Mathematicalanalysisofage-structuredHIV-1dynamicswithcombinationantiretroviraltherapy.SIAMJ.Appl.Math.67(2007).3:731. [57] Seeff,LeonardB.NaturalHistoryofChronicHepatitisC.Hepatology36(2002):S35S46. [58] Sen,G.C.VirusesandInterferons.AnnuRevMicrobiol55(2001):255. [59] Snoeck,E,Chanu,P,Lavielle,M,Jacqmin,P,Jonsson,EN,Jorga,K,Goggin,T,Grippo,J,Jumbe,NL,andFrey,N.AComprehensiveHepatitisCViralKineticModelExplainingCure.Nature87(2010).706-713:6. [60] Sulkowski,M.S.AnemiaintheTreatmentofHepatitisCVirusInfection.CID37(2003).Suppl4:S315. [61] Sung,V.M.,Shimodaira,S.,Doughty,A.L.,Picchio,G.R.,Can,H.,Yen,T.S.,Lindsay,K.L.,Levine,A.M.,andLai,M.M.EstablishmentofB-celllymphomacelllinespersistentlyinfectedwithhepatitisCvirusinvivoandinvitro:theapoptoticeffectsofvirusinfection.J.ofVirol.77(2003).3:2134. [62] Takaki,S.,Tsubota,A.,Hosaka,T.,Akuta,N,Someya,T,Kobayashi,M,Suzuki,F,Suzuki,Y,Saitoh,S,Arase,Y,Ikeda,K,andKumada,H.Factorscontributingtoribavirindosereductionduetoanemiaduringinterferonalfa2bandribavirincombinationtherapyforchronichepatitisC.JGastroenterol39(2004):668673. 110

PAGE 111

Thieme,H.R.ConvergenceresultsandaPoincareBendixsontrichotomyforasymptoticallyautonomousdifferentialequations.JofMath.Biol.30(1992):775. [64] Yu,J.W.,Wang,G.Q.,Sun,L.J.,Li,X.G.,andLi,S.C.PredictivevalueofrapidvirologicalresponseandearlyvirologicalresponseonsustainedvirologicalresponseinHCVpatientstreatedwithpegylatedinterferonalpha-2aandribavirin.J.GastroenterolHepatol22(2007):832. [65] Zeuzem,S,Schimdt,JM,Lee,JH,Ruster,B,andRoth,WK.EffectofInterferonAlphaonthedynamicsofHepatitisCvirusturnoverinvivo.Hepatology23(1996):366371. [66] Zeuzem,S,Schimdt,JM,Lee,JH,Von-Wagner,M,Teuber,G,andRoth,WK.HepatitisCvirusdynamicsinvivo:effectofribavirinandinterferonalphaonviralturnover.Hepatology28(1998):245252. 111

PAGE 112

SwatiDebRoydidherB.Sc.(bachelor'sdegree)withHonorsinMathematicsfromSt.Xavier'sCollegeundertheUniversityofCalcuttain2003.ThenshecompletedherM.Sc.(master'sdegree)withaconcentrationinPureMathematicsundertheUniversityofBurdwan.In2005,shejoinedUniversityofFlorida(UF)atGainesvilleasagraduatestudent.InDecember2007,shecompletedherM.S.fromUF.InDecember2011,SwatireceivedherPh.D.,specializinginMathematicalBiology. 112