LectureNotesinBioengineering
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HaraldKirchsteiger • JohnBagterpJørgensen
EricRenard • LuigidelRe
Editors
PredictionMethodsforBlood
GlucoseConcentration
Design,UseandEvaluation
123
Editors
HaraldKirchsteiger
InstituteforDesignandControlof MechatronicalSystems
JohannesKeplerUniversityLinz Linz
Austria
JohnBagterpJørgensen DepartmentofAppliedMathematics TechnicalUniversityofDenmark KongensLyngby
Denmark
EricRenard
InstitutdeGénomiqueFonctionnelle deMontpellier Montpellier France
LuigidelRe
InstituteforDesignandControlof MechatronicalSystems JohannesKeplerUniversityLinz Linz
Austria
ISSN2195-271X
LectureNotesinBioengineering
ISSN2195-2728(electronic)
ISBN978-3-319-25911-6ISBN978-3-319-25913-0(eBook) DOI10.1007/978-3-319-25913-0
LibraryofCongressControlNumber:2015953261
SpringerChamHeidelbergNewYorkDordrechtLondon © SpringerInternationalPublishingSwitzerland2016
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Preface
Standarddiabetesinsulintherapyfortype1diabetesandlatestagesoftype2 isbasedontheexpecteddevelopmentofbloodglucose(BG)bothasaconsequence ofthemetabolicglucoseconsumptionaswellasofmealsandexogenousinsulin intake.Traditionally,thisisnotdoneexplicitly,buttheinsulinamountischosen usingfactorsthataccountforthisexpectation.
Theincreasingavailabilityofmoreaccuratecontinuousbloodglucosemeasurement(CGM)systemsisattractingmuchinteresttothepossibilitiesofexplicit predictionoffutureBGvalues.Againstthisbackground,in2014atwo-day workshoponthedesign,useandevaluationofpredictionmethodsforbloodglucoseconcentrationwasheldattheJohannesKeplerUniversityLinz,Austria.One intentionoftheworkshopwastobringtogetherexpertsworkinginvarious fieldson thesametopic,inordertoshedlightfromdifferentanglesontheunderlying problemofmodelingtheglucoseinsulindynamicsoftype1diabetespatients. Amongtheinternationalparticipantswerecontinuousglucosemonitoringdevelopers,diabetologists,mathematiciansandcontrolengineers,both,fromacademia andindustry.Intotal18talksweregivenfollowedbypaneldiscussionswhich allowedtoreceivedirectfeedbackfromthepointofviewofdifferentdisciplines. Thisbookisbasedonthecontributionsofthatworkshopandisintendedto conveyanoverviewofthedifferentaspectsinvolvedintheprediction.Theindividualchaptersarebasedonthepresentationsgivenbytheauthorsattheworkshop butwerewrittenafterwardwhichallowedtoincludethe findingsandconclusions ofthevariousdiscussionsandofcourseupdates.
Thechapter “AlternativeFrameworksforPersonalizedInsulin–GlucoseModels” byHaraldKirchsteigeretal.asksthequestionwhethermoreandmoredetailed physiologicaldescriptionsoftheglucosemetabolismwithanever-increasing degreeofsophisticationandnumberofmodeledphenomenaarereallywhatis neededforpushingtheboundariesinglucosepredictionforcontrol.Asanalternative,thechapterintroducestwodata-basedapproachesthatfocusnotonthe predictionofexactfuturebloodglucosevalues,butratheronthepredictionof changesinthepatients’ bloodglucoserange.
v
Thechapter “AccuracyofBGMetersandCGMSystems:PossibleInfluence FactorsfortheGlucosePredictionBasedonTissueGlucoseConcentrations” by GuidoFreckmannetal.discussesperformancemetricsusedtocharacterizethe accuracyofcontinuousglucosemeasurementdevices.Thistopicishighlyrelevant forpredictionmodelssincemanyofthemrelyonthedatagivenbythecontinuous sensorswhicharepreviouslycalibratedwithbloodglucosemetermeasurements whicharealsosubjecttomeasurementerrors.Inaccuratemeasurementswill directlyaffecttheperformanceofthecorrespondingpredictions.
Thechapter “CGM HowGoodIsGoodEnough?” byMichaelSchoemaker andChristopherG.Parkinalsotacklestheproblemofcontinuousglucosemonitor performanceevaluation.Severalperformancemetricsusedindifferentpublished studiesarecomparedandtheirindividualcharacteristicsanalyzed.Thechapter revealswhythecomparisonofasensorevaluatedintwodifferentclinicalstudiesis notalwaysstraightforward.
Thechapter “CanWeUseMeasurementstoClassifyPatientsSufferingfrom Type1DiabetesintoSubcategoriesandDoesItMakeSense? ” byFlorianReiterer etal.makesuseofcontinuoustimepredictionmodelstodescribetheinteraction betweeningestedcarbohydrates,subcutaneouslyinjectedinsulin,andcontinuously measuredglucoseconcentration.Theidentifiedmodelparametersof12subjects wereanalyzedandstatisticallysignificantcorrelationsbetweentheparametersand patientcharacteristicssuchasweightandagecouldbefound.
Thechapter “PreventionofSevereHypoglycemiabyContinuousEEG Monitoring” byClausBorgJuhletal.showshowtouseEEGsignalstopredict upcominghypoglycemicsituationsinreal-timebyemployingartificialneuralnetworks.Theresultsofa30-daylongclinicalstudywiththeimplanteddeviceandthe developedalgorithmarepresented.
Thechapter “Meta-LearningBasedBloodGlucosePredictorforDiabetic SmartphoneApp” byValeriyaNaumovaetal.demonstrateshowahighly sophisticatedglucosepredictionmodelcanbeportedfromadevelopmentlanguage runningonaPCtoaformatsuchthatitcanbeusedconvenientlybythepatients. Auniquefeatureofthealgorithmisitsindependenceofanyuserinputotherthan historicCGMdatawhichisautomaticallytransmittedfromaCGMdevice.No parameterestimationnorpredictionmodelindividualizationisrequired.
Thechapter “PredictingGlycemiainType1DiabetesMellituswithSubspaceBasedLinearMultistepPredictors ” byMarziaCesconetal.usesdata-based methodstodevelopindividualizedpredictionmodels.Themodelcanbeconsidered asacombinationofphysiologicalmodelstoprecomputetherateofappearanceof injectedinsulinandingestedcarbohydratesinthebloodstreamandofdata-based modelstocombinethisinformationandcomputepredictionsupto120mininthe future.Theresultsshowtheperformanceondatafrom14type1diabetespatientsin aclinicaltrial.
Thechapter “EmpiricalRepresentationofBloodGlucoseVariabilityina CompartmentalModel” byStephenD.Pateketal.showsamodelingtechnique designedtoextracttheinformationontheneteffectofmealsonthebloodglucose concentration.Byassumingthatallmajorunexplainedglycemicexcursionscanbe
vi Preface
attributedtooralglucoseingestion,amealvectorisestimatedwhichsignifi cantly improvesthemathematicalmodel.Resultsareshownonthreepatientsduringa clinicaltrialandonvirtualpatientswhereitisshownhowthemethodcanbeused foradjustmentsofthebasalinsulinrate.
Thechapter “Physiology-BasedIntervalModels:AFrameworkforGlucose PredictionUnderIntra-patientVariability” byJorgeBondiaandJosepVehitriesto copewiththelargeintrasubjectvariabilitybyusingtheconceptofintervalpredictions.Insteadofpredictingasinglebloodglucosevalueinthefuture,awhole solutionenvelopeisdetermined.Withthepresentedtheoryitcanbeguaranteedthat therealvalueisalwaysinsideoftheenvelopeandmoreovertheenvelopeisnot conservative.Themethodisevaluatedonaphysiologicaldiabetesmodel.
Thechapter “ModelingandPredictionUsingStochasticDifferentialEquations” byRuneJuhletal.considersuncertaintyinthedynamicsbetweendifferentpatients aswellaswithinapatientbymakinguseofstochasticdifferentialequations.Itis shownhowthemixedeffectsmodelingmethodologycanbeappliedsuchthatthe underlyinginformationofseveraldatasetsfromdifferentpatientsisextractedto formthemodel.
Thechapter “UncertaintiesandModelingErrorsofType1DiabetesModels” by LeventeKovácsandPéterSzalayanalyzestheeffectofpredictionmodeluncertaintiesonthecontrolsystemduringadesignprocedureinvolvingthestepsmodel reductionbyeliminationofstatevariables,stateestimationusingextendedKalman FiltersandSigmaPoint filtersandlinearparameter-varyingcontrolsynthesis.
Thechapter “RecentResultsonGlucose–InsulinPredictionsbyMeansofaState ObserverforTime-DelaySystems” byPasqualePalumboetal.introducesapredictionmodelwhichinrealtimepredictstheinsulinconcentrationinbloodwhich inturnisusedinacontrolsystem.Themethodistestedinsimulationona time-delaysystemrepresentingtheglucose–insulinsystem.
Thechapter “PerformanceAssessmentofModel-BasedArti ficialPancreas ControlSystems” byJianyuanFengetal.makesuseofpredictionmodelsto computetreatmentadvices.Thenoveltyoftheproposedalgorithmconsistsin explicitlyconsidering(amongothers)themodelpredictionerrorandmodelerror eliminationspeed.Aretuningoftheadvisorysystemisdoneincasetheprediction modeldoesnotperformwell.Resultson30virtualpatientsshowtheperformance ofthecontrolsystem.
Wewouldliketothankallpeopleinvolvedintheprocessofwritingthisbook: Allauthorsfortheirindividualcontributions,allreviewersofthebookchapters, DanielaHummerfortheentireorganizationoftheworkshop,BorisTasevskifor helpingwiththetypesetting,FlorianReitererforhishelpeditingthebook,aswell asOliverJacksonandKarindeBieforthegoodcooperationwithSpringer.
Linz HaraldKirchsteiger August2015 JohnBagterpJørgensen EricRenard LuigidelRe
Preface vii
HaraldKirchsteiger,HajrudinEfendic,FlorianReiterer andLuigidelRe
Contents
–
....1
AlternativeFrameworksforPersonalizedInsulin
GlucoseModels
1Introduction..........................................1 2AlternativesforModeling................................2 3ModelStructures......................................5 4IntervalModels.......................................9
4.1ContinuousTimeSystemIdentification...................9 4.2IntervalModelResults..............................11 5AProbabilisticApproach................................18
5.3ModelingResults..................................22 6ConclusionandOutlook.................................26 References.............................................27
5.1GaussianandGeneralizedGaussianMixtureModels..........18 5.2ModelingMethodandModelStructure...................20
.........................................31
AccuracyofBGMetersandCGMSystems:PossibleInfluence FactorsfortheGlucosePredictionBasedonTissueGlucose Concentrations
References.............................................40 ix
GuidoFreckmann,StefanPleus,ManuelaLinkandCorneliaHaug 1Introduction..........................................32 2SMBGAccuracyandCGMCalibrationwithSMBGResults........32 2.1SMBGAccuracy..................................32 2.2CGMCalibrationwithSMBGResults....................34 3AccuracyofCGMSystems...............................36 3.1MeanAbsoluteRelativeDifference......................36 3.2PrecisionAbsoluteRelativeDifference...................38 4GlucosePredictionBasedonTissueGlucoseConcentrations........39
MichaelSchoemakerandChristopherG.Parkin
4.1TransientSensorSignalDisruption......................49 4.2TransientSignificantCGMInaccuracies..................50
FlorianReiterer,HaraldKirchsteiger,GuidoFreckmann andLuigidelRe 1Introduction..........................................57
3.1DescriptionoftheModelandSystemIdentification..........61
3.2TrendsandCorrelations..............................64
3.3ClusteringandClassification..........................69
3.4DiscussionofResultsandFurtherOutlook................70 4AnalysisoftheHighFrequencyContentofCGMSSignals.........72
4.1FilteringofCGMSSignals...........................72
4.2TrendsandClassification.............................73
4.3DiscussionofResultsandFurtherOutlook................76
ClausBoghJuhl,JonasDuun-Henriksen,JensAhmSørensen, AnneSophieSejlingandRasmusElsborgMadsen
4QuantitativeEvaluationofEEGRecordedwiththePartly ImplantedEEGRecorder.................................84
5DevelopmentofanAlgorithmforDetectionandWarning ofSevereHypoglycaemiainType1Diabetes..................85 6ClinicalStudies PreliminaryResultswithImplantedDevice.......88
CGM HowGoodIsGoodEnough? .........................43
1Background..........................................43 2CGMPerformanceAssessment............................44 2.1SensorSignal.....................................44 2.2ReferenceMethodology..............................45 2.3AccuracyandPrecision..............................46 3StateoftheArt.......................................48 4UnresolvedIssues.....................................49
5NextStepsinCGMDevelopment..........................51 6Conclusion..........................................51 References.............................................52
...57
CanWeUseMeasurementstoClassifyPatientsSuffering fromType1DiabetesintoSubcategoriesandDoesItMakeSense?
2DatabaseofCGMSRecordings............................60 3ModellingUsingaSimpleTransferFunctionModel..............61
........................................79
References.............................................77 PreventionofSevereHypoglycemiabyContinuous EEGMonitoring
1Background..........................................80 2ClinicalStudies ProofofConcept.........................81 3TheDevice..........................................83
x Contents
7DiscussionandPerspectives...............................89
Meta-LearningBasedBloodGlucosePredictorforDiabetic SmartphoneApp ........................................93
ValeriyaNaumova,LucianNita,JensUlrikPoulsen andSergeiV.Pereverzyev
1Introduction..........................................94
2FullyAdaptiveRegularizedLearningAlgorithm fortheBloodGlucosePrediction...........................96
3AndroidVersionoftheFARLAlgorithm.....................99
3.1TranslationoftheAlgorithmfromMatlabtoAndroidSystem...99
3.2MicroprocessorandPowerConsumptionAnalysis...........100
PredictingGlycemiainType1DiabetesMellitus
withSubspace-BasedLinearMultistepPredictors
MarziaCescon,RolfJohanssonandEricRenard
1Introduction..........................................107
2Subspace-BasedLinearMultistepPredictors...................110
3ExperimentalConditionsandClinicalDataAcquisition............114
4PredictingDiabetesGlycemiawiththeMultistepPredictors.........117
5Results.............................................119
6DiscussionandConclusions...............................129
EmpiricalRepresentationofBloodGlucoseVariability
inaCompartmentalModel ................................133
StephenD.Patek,DayuLv,EdwardA.Ortiz,ColleenHughes-Karvetski, SandipKulkarni,QianZhangandMarcD.Breton
1Introduction..........................................134
2OralCarbohydrate “NetEffect”:ReconcilingCGMandPump DataviaRegularizedDeconvolution.........................136
2.1NetEffectCoreAlgorithm............................138 2.2CGMPreprocessing................................139
2.3Discussion: “NetEffect” Versus “MealEstimation”
References.............................................90
8Conclusion..........................................90
4PerformanceAssessment.................................100 4.1ClinicalAccuracyMetrics............................100 4.2PerformanceAssessment.............................102
References.............................................105
4.3ComparisonoftheMatlabandAndroidVersions............102 5ConclusionsandDiscussion...............................104
................107
2.1Notation........................................111 2.2PredictorsConstruction..............................111
References.............................................130
Contents xi
..........140
4.1NetEffectsandNetEffectSimulationReplay
4.2InSilicoExperiments:UsingNetEffecttoDesign
Physiology-BasedIntervalModels:AFramework
JorgeBondiaandJosepVehi
4.1BergmanModelPredictorBasedonModalIntervalAnalysis....172
4.2BergmanModelPredictorBasedonMonotone
4.3PostprandialGlucosePredictionUsingIntervalModels........175
RuneJuhl,JanKloppenborgMøller,JohnBagterpJørgensen andHenrikMadsen
2.1SingleDataSeries.................................186
2.2IndependentDataSeries.............................192
2.3PopulationExtension...............................193
2.4PriorInformation..................................196
3Example:ModelingtheEffectofExerciseonInsulin Pharmacokineticsin “ContinuousSubcutaneousInsulin Infusion” TreatedType1DiabetesPatients....................197 3.1Data...........................................197
3.2TheGrayBoxInsulinModel..........................198
3.3ExerciseEffects...................................199
3NetEffectSimulation...................................140 3.1
Replay” Simulation................................141 3.2SimulatingModifiedInsulinDelivery....................141 4Results.............................................142
“
fromFieldData...................................143
5Conclusions..........................................151 References.............................................156
BasalRateAdjustments..............................150
............159
forGlucosePredictionUnderIntra-patientVariability
1Introduction..........................................159 2IntervalModels.......................................161 3SimulatingIntervalModels...............................163 3.1IntervalAnalysis..................................164
OutputSystems.......................168
3.2MonotoneInput–
4IntervalGlucosePredictors...............................172
SystemsTheory...................................173
5IntervalModelIdentification..............................175 6Conclusions..........................................178 References.............................................179 ModelingandPredictionUsingStochasticDifferentialEquations .....183
1Introduction..........................................183 2DataandModeling.....................................185
xii Contents
UncertaintiesandModelingErrorsofType1DiabetesModels
LeventeKovácsandPéterSzalay
RecentResultsonGlucose–InsulinPredictionsbyMeans ofaStateObserverforTimeDelaySystems
PasqualePalumbo,PierdomenicoPepe,SimonaPanunzi andAndreaDeGaetano 1Introduction..........................................227
2TheDDEModeloftheGlucose–InsulinSystem................229
3Observer-BasedControlbyMeansofIntravenousInsulinInfusion....230
3.1SynthesisoftheGlucoseControlLaw....................231
3.2EvaluationCriteriaandValidation......................233
4Observer-BasedControlbyMeansofSubcutaneous InsulinInfusion.......................................236 5Conclusions..........................................239
PerformanceAssessmentofModel-BasedArtificialPancreas ControlSystems ........................................243
JianyuanFeng,KamuranTurksoyandAliCinar
3.4ModelComparison.................................200 3.5Predictions.......................................202 4OtherTopics.........................................202 4.1Transformations...................................202 4.2Identification.....................................203 4.3Simulation/PredictionModels..........................203 4.4TestingandConfidenceIntervals.......................203 5Summary...........................................204 References.............................................208
......211
2ModelingDiabetes.....................................212 2.1LinearParameterVaryingModel.......................214 3ModelReduction......................................215 4StateEstimation.......................................218 4.1Sigma-PointSelection...............................219 5ModelUncertainty.....................................221 5.1ErrorWeightingFunction............................221 6Conclusion..........................................223 References.............................................224
1Introduction..........................................211
....................227
References.............................................239
Contents xiii
1Introduction..........................................243 2GPCandControllerErrorDetection.........................245 2.1GPCinAPSystem.................................245
2.2IndexesUsedforCPA..............................246 2.3DetectionandDiagnosisofControllerErrors...............249
3.1ControllerRetuningforModelPredictionError.............252
3.2ControllerRetuningforInsulinDoseConstraintError.........254
3.3ControllerRetuningforObjectiveFunctionWeight
3.4ControllerRetuningforSensor-Noise-Driven MiscalculationError................................255
3ControllerRetuning....................................252
RatioError......................................254
4Results.............................................256 5Conclusions..........................................263 References.............................................263 xiv Contents
AlternativeFrameworksforPersonalized Insulin–GlucoseModels
HaraldKirchsteiger,HajrudinEfendic,FlorianReitererandLuigidelRe
Abstract Thedescriptionoftheinsulin–glucosemetabolismhasattractedmuch attentioninthepastdecades,andseveralmodelsbasedonphysiologyhavebeen proposed.Whilethesemodelsprovideapreciousinsightintheinvolvedprocesses, theyareseldomabletoreplicateandmuchlesstopredictthebloodglucose(BG)value arisingasareactionofthemetabolismofaspecificpatienttoagivenamountofinsulin orfoodatagiventime.Data-basedmodelshaveproventoworkbetterforprediction, butpredictedandmeasuredvaluestendtodivergestronglywithincreasingprediction horizon.Differentapproaches,forinstancetheuseofvitalsigns,havebeenproposed toreducetheuncertainty,albeitwithlimitedsuccess.Thekeyassumptionhidden behindthesemethodsistheexistenceofasingle“correct”modeldisturbedbysome stochasticphenomena.Inthischapter,instead,wesuggestusingadifferentparadigm andtointerpretuncertaintyasanunknownpartoftheprocess.Asaconsequence, weareinterestedinmodelswhichyieldasimilarpredictionperformanceforall measureddataofasinglepatient,eveniftheydonotyieldapreciserepresentationof anyofthem.Thischaptersummarizestwopossibleapproachestothisend:interval models,whichprovideasuitablerange;andprobabilisticmodels,whichprovidethe probabilitythattheBGliesinpredeterminedranges.Bothapproachescanbeused intheframeworkofautomatedpersonalizedinsulindelivery,e.g.,artificialpancreas oradaptiveboluscalculators.
1Introduction
Glucoseisthemainenergysourceforthehumanbody,butinexcessiveamountsit canbedetrimentalaswell.Inhealthypersons,thecorrectglucoselevelisregulated bytheinsulin–glucosemetabolism.Indiabetespatients,thiscontrolsystemdoesnot workproperlyanymore,due,roughlyspeaking,eithertolackofendogenousinsulin
H.Kirchsteiger · H.Efendic · F.Reiterer(B) · L.delRe InstituteforDesignandControlofMechatronicalSystems, JohannesKeplerUniversityLinz,4040Linz,Austria e-mail:florian.reiterer@jku.at URL:http://desreg.jku.at/
©SpringerInternationalPublishingSwitzerland2016 H.Kirchsteigeretal.(eds.), PredictionMethodsforBloodGlucoseConcentration, LectureNotesinBioengineering,DOI10.1007/978-3-319-25913-0_1
1
2H.Kirchsteigeretal.
productionortoaresistancetoinsulinaction[23].Insuchcases,thetherapyofchoice consistsinprovidingthenecessaryinsulinexternally.However,thedelaytimes associatedwiththisprocedureandthedangerofremovingtoomuchglucosefrom thebloodcirculation—withpossiblylethalconsequences—makethedetermination ofthecorrectinsulinquantityquitedifficultanddelicate.
Againstthisbackground,therehasbeenasustainedinterestindescribingthe glucose–insulinmetabolismbyphysiologicalmodelsofincreasingcomplexity[2, 3, 25, 44].AmodelresultingfromthecooperationbetweentheUniversitiesofPadova andVirginia[12]hasbeenacceptedin2008bytheFoodandDrugAdministration (FDA)asasubstituteforpreclinicaltrialsforcertaininsulintreatments.Themodel wasrecentlyextendedwithnewfeatures[11].
And,ofcourse,therehasbeenalargenumberofattemptstousethemodel informationtodeterminetherightquantitiesofexternalinsulinneeded,sotosay toreplacethemissingcontrolaction,to“closetheloop.”Positiveresultshavebeen reportedforovernightoperation[26],butalsoareductionofvariabilityforthedaily usehavebeenreported[33].Newandimprovedsensorshavebeenintroduced,faster insulinisenteringthemarket,andallthisnurturesthehopethatmoreprogresswill come.Unfortunately,inspiteofover40yearsofresearch[1],theresultsofthis “artificial”or“virtualpancreas”arestillnottherewheretheyshouldbeandsimple safetyrules—e.g.,avoidinginsulininfusionduringorneartohypoglycemia—seem tobeabletoofferthelargestpartofthebenefitsofclosed-loopcontrolinamuch simplerwayaswell.
Againstthisbackground,itisnaturaltoleanbackforamomentandwonder whetherweareaskingtherightquestions.Inthischapter,weareaddressingthis veryissuebypresentingtwopromisingalternativestomodeling,namelyinterval modelsandprobabilisticmodelingtechniques.
2AlternativesforModeling
Thekeyassumptionformodelingisthereproducibilityofresults.Tosomeextent,of course,thisisalwaystrue,forinstanceinsulindoesreducethebloodglucose(BG) concentrationandcarbohydrates(Carb)intakeincreasesit.Manyothereffectsare knownbuthardtoquantify—e.g.,somehormonesincreaseitaswell,muscularwork reducesit—buttherearemanycontrolloopsinthehumanbodywhichcanleadtoa complexresponse,e.g.,inthecaseofprolongedphysicalwork[21].Othereffects, likethecircadianvariationofsensitivities,areknownaswell[46].
Physiologicalmodelsdescribeallthesephenomenarelyingondeepunderstandingofphysiologyandfrequentlyonveryspecificmeasurements,e.g.,tracers [24, 42].Itisusuallyimpossibletodetermineallparametersofsuchmodelswith simple“external”measurements,e.g.,takingonlyCarbandinsulinadministration andBGvalues.Indeed,ithasbeenshownthatthe“external”behavior—therelationbetweeninsulinandCarbintakeandBG—ofthesecomplexmodelscanbe approximatedverywellwithverysimpleones[17, 45](seeFig. 1).Thisistheone
Fig.1 Approximationoftheinput–outputrelationshipofmodel[12]asshownin[45]
reasonwhycomparisonsbetweenthevaluespredictedbyphysiologicalmodelsand measurementsareveryseldom,exceptionsbeingforinstance[8, 47].
Indeed,ingeneral,physiologicalmodelsarenotabletoprovideapersonalized description.
Asthecomparisonmakesclear,themeasurementshaveahighdegreeofcomplexitynotreflectedinthephysiologicalmodel,themainreasonbeingthemanyunmodeledeffects,e.g.,relatedtotheemotionalstate,whichmayaffectverystronglythe BGvalues,andwhichcannotbecapturedbythemodelbecausethecriticalquantities, inthiscasetheconcentrationofsomehormones,arenotknown.
Thereareseveralwaystocopewiththisproblem.Ononeside,theattemptcan bemadetofindadditionalmeasurementstoextendthemodel,e.g.,vitalsigns— acceleration,heartfrequency,bodytemperature,andsoon.Similartechniqueshave provedveryusefulintheindustrialframework,e.g.,todetectchangesinmachines [13],buthaveneverreallysucceededinthecaseofdiabetestreatment.
Anotherapproach,relatedtoanotherchapterofthisbook(see“Empirical RepresentationofBloodGlucoseVariabilityinaCompartmentalModel”by S.Pateketal.)consistsessentiallyinestimatinga“corrective”Carbinputtoexplain thedifferencebetweenmeasuredandcomputedvalues.Whilethismethodcannot beusedinrealtime,itallowstostudytheeffectofsomechangesintherapy,e.g., differentamountsofinsulin.
AlternativeFrameworksforPersonalizedInsulin…3 0 1000 2000 3000 4000 5000 6000 7000 8000 80 100 120 140 160 180 200 220 time [min] blood glucose [mg/dL] Approximation of the input−output behaviour of a physiological model Approximation Phys. model
Recursive algorithm using variable gain
Recursive algorithm using constant gain
Fig.2 45minaheadpredictionofglucosefortwodifferentpatientsfrom[6]
Ifweareinterestedinobtainingmodelswhicharesufficientlysimpletoallow theiruseandparameterestimationinrealtime,itmightbebettertolookforother approaches.Averyefficientadaptivemodelwasdevelopedby[5]whichreliesona simplehypothesis,aso-calledARXmodel,anddeterminestheparameterscontinuously,concentratingonthemostcriticalBGranges.Figure 2 showstheperformance ofsuchamodelaspredictor.
Inthischapter,however,wesuggesttwodifferentapproaches.Thekeyideais nottogetridofuncertainty,assumingoneparticularvaluetobetrue,buttodesign modelsvalidforthewholeregion,implicitlyassumingthatafullrangeofvalues arepossibleandinsomesensetrue.Onepossibleapproachtothisendareinterval models,i.e.,modelswhichcomputeanoutputrangeandnotasinglevalue.The otheralternativeisusingaprobabilisticapproach.Indeed,theexactBGvalueisnot reallyimportantinitself,theclinicianismoreinterestedinkeepingitinsidetheusual (“euglycemic”)rangeandpreventingtoreachadangerousone,e.g.,hypoglycemia. Markovjumpmodelcanhelpindescribingthephysiologythewayitreallyis—i.e., tosomeextentrandom.
Bothmodelscanbeusedforautomatedinsulindeliveryaswell.Inthecaseof intervalmodels,theproblemcanbestatedintermsofamin/maxproblemsuchthat thecarbohydrateamountoptimizesthecostfunctionsforallcasesofuncertainty.In
4H.Kirchsteigeretal. 3000 3200 3400 3600 3800 4000 4200 50 100 150 200 time [min] [mg/dl]
[mg/dl] 3000 3200 3400 3600 3800 4000 4200 50 100 150 200 time [min] y ˆ y
theprobabilisticcase,itbecomestheminimizationoftheprobabilitythat,underthe actionofinsulinandmeals,theBGleavesthegoodrangetoreachadangerousone.
Thefurthersectionsofthischapterareorganizedasfollows:Beforecomingto thetopicofintervalmodelsitself,areviewaboutempiricalcontinuoustimetransferfunctionmodelsisgiveninSect. 3 andthedifferencesbetweenpossiblemodel structuresarediscussed.Themodelspresentedinthatsectionareaspecialform ofcontrol-orienteddata-basedmodelsfordescribingtheinput–outputrelationship oftheglucosemetabolismthathaveprovenquitepowerfulintherecentpast(see e.g.,[37, 38]).Oneofthosepresentedmodelstructuresisthenfurtherusedforthe intervalmodelingintroducedinSect. 4.Afteraquickoverviewaboutthetopicof intervalmodelingingeneral,somedetailsaboutthemethodsusedhereforderiving intervalmodelsfromdataaregiveninSect. 4.1.InfollowingSect. 4.1,resultsfor theintervalmodelingareshown,bothforsimulatedandforrealpatientdata.The subsequentSect. 5 thendescribesaprobabilisticframeworkthatcanbeusedforpredictingchangesfromoneBGrangetoanother.ItstartsinSect. 5.1 withanoverview aboutGaussianmixturemodelsthathavebeenusedinthiscontext.Section 5.2 gives somedetailsabouttheusedmodelstructureandthemethodologyofpredictingtransitionsintheBGrange,whereasactualpredictionresultsforrealpatientdataare presentedinthefollowingSect. 5.3.Thechapterfinishedwithsomefinalconclusions anddiscussiongiveninSect. 6.
3ModelStructures
Amodelconsistsofamathematicalstructureandofparameters.Thusthefirststepin modelingconsistsoffixingthemodelstructure,andthereaftertheparametershave tobetunedtogetthebestcorrespondencebetweenmeasuredandcomputedvalues. Ofcourse,everymodelrepresentsasimplificationoftherealsystem,andnotevery modelstructureisabletocapturethebehaviorofthesystemunderobservationina sufficientlygeneralway.Thisisespeciallytrueinthecaseofasimplifiedmodelwe areinterestedin.
Table 1 summarizespreviouslyproposedmodelstructurestodescribetheblood glucosedynamicswhere BG (s ), Carb (s ),and I (s ) correspondtothebloodglucose concentration,ingestedmealcarbohydrates,andsubcutaneouslyinjectedinsulinbolus,respectively,alltransformedintotheLaplacedomain.Thetablealsoliststhe numberofparameterswhichneedtobeestimatedfromdata.Allmodelsthususethe sameamountofinformation.Atypicaldatasetwhichcouldbeusedforparameter estimationisshowninFig. 3 (datafromtheDIAdvisorproject[14]).
Itisimmediatelyvisiblethatthemodelinputsareimpulse-shapedquantities whicharezeromostofthetime.Thatisbecausesubcutaneousinsulininjectionsare discreteeventsandmealingestions,regardlessofthequantityandtimeittakesto actuallyconsumethefood,arecommonlytreatedasdiscreteeventsalso.Therefore, formodelanalysis,the impulse responsesareofgreatinterestwhereas step responses, commonlyusedincontrolengineering,donotprovidesufficientinformation.Astep
AlternativeFrameworksforPersonalizedInsulin…5
Fig.3 Illustrationoftherelevantmeasurementdataforaspecificpatient,CHU0102(datafrom theDIAdvisorproject[14])
Table1 Selectedmodelstructurespreviouslypublished
a Extendedwithadynamicmodelforcarbohydrates
responsewouldactuallymeanthatfoodorinsulinisaddedtothemetabolismina continuouswayoveranextendedperiodoftime.Whilethismightbetrueforthe caseofinsulin,itisdefinitelyanunrealisticcaseforfood.Inallgivenreferences, continuousinsulindeliveryisnotconsideredformodeling.
Fromaphysiologicalpointofview,theimpulseresponsegivespreciseinformation ontheeffectofonegramofcarbohydrateandoneunitofinsulin,respectively,on thebloodglucoseconcentration.
Theimpulseresponsesforbothinputsofmodelstructure1inTable 1 areshownin Fig. 4 forvariousselectionsoftheparameters T1 and T2 .Itisimmediatelyclearthat theparameters K 1 and K 2 correspondtothesteady-statechangeinBG.Furthermore, theparameters T1 and T2 aretimeconstantswhichdeterminethetimeittakesuntil
6H.Kirchsteigeretal. 0 100 200 300 mg/dl CGM CGM−lin Hemocue 500 1000 1500 2000 2500 3000 3500 4000 0 2 4 6 8 10 12 14 Insulin Bolus [U] time [min] 00 04 08 12 16 20 00 04 08 12 16 20 00 04 08 12 16 20 0 50 100 Carbs [g] time [h after midnight]
No. Modelstructure Parameters Reference 1 BG (s ) = K 1 (1+sT1 )2 s Carb (s ) + K 2 (1+sT2 )2 s I (s ) 4 [29] 2 BG (s ) = K 1 exp ( τ1 s ) (1+sT1 )s Carb (s ) + K 2 exp ( τ2 s ) (1+sT2 )s I (s ) 6 [36] 3 BG (s ) = K 1 (1+sT1 )s Carb (s ) + K 2 (1+sT2 )s I (s ) 4 [9] 4 BG (s ) = K 1 (1+sT1 )2 Carb (s ) + K 2 (1+sT2 )2 I (s ) 4 [4]a 5 BG (s ) = K 1 exp ( τ1 s ) (1+sT1 )(1+sT2 ) Carb (s ) + K 2 exp ( τ2 s ) (1+sT3 )s I (s ) 7 [10]
thissteadystateisreached.Fromthisperspective,thephysiologicalinterpretation ofthemodelparametersisstraightforward.
Fig.4 Impulseresponseofmodelstructure1fromTable 1 using K 1 = 10, K 2 =−8
Theimpulseresponsesforbothinputsofmodelstructure2inTable 1 areshown inFig. 5 forvariousselectionsoftheparameters T1 and T2 .ComparedtotheimpulseresponsesinFig. 4,modelstructure2involvesatimedelaydeterminedbythe parameters τ1 and τ2 .Furthermore,afterthisdelay,theimpulseresponseshowsa discontinuity(seethemagnifiedplotsinFig. 5)whichhardlyappearsinrealsubjects. However,theoverallresponsesofmodelstructures1and2aresimilar.
2 = 5
Impulseresponseofmodelstructure2fromTable 1 using K 1 = 10, τ1 = 10, K 2 =−8,
Theimpulseresponsesformodelstructure3inTable 1 arenotshownexplicitly sincetheyareverysimilartothoseofmodelstructure2showninFig. 5,exceptfor thetimedelaywhichiszerointhiscase(τ1 = 0, τ2 = 0).
Modelstructure4ofTable 1 givesasignificantlydifferentimpulseresponsethan themodelsdiscussedabove,seeFig. 6.Sincethereisnointegratingbehavior,the
AlternativeFrameworksforPersonalizedInsulin…7
0 200 400 600 800 1000 1200 1400 0 1 2 3 4 5 6 7 8 9 10 Time [min] Δ BG T1=20 T1=40 T1=60 T1=80 T1=100 T1=120 T1=140 T1=160 T1=180 T1=200 0 10 20 30 40 50 0 0.5 1 1.5 0 200 400 600 800 1000 1200 1400 −8 −7 −6 −5 −4 −3 −2 −1 0 Time [min] Δ BG T2=20 T2=40 T2=60 T2=80 T2=100 T2=120 T2=140 T2=160 T2=180 T2=200 0 10 20 30 40 50 −1.5 −1 −0.5 0
0 200 400 600 800 1000 1200 1400 0 1 2 3 4 5 6 7 8 9 10 Time [min] Δ BG T1=20 T1=40 T1=60 T1=80 T1=100 T1=120 T1=140 T1=160 T1=180 T1=200 0 10 20 30 40 50 0 0.5 1 1.5 0 200 400 600 800 1000 1200 1400 −8 −7 −6 −5 −4 −3 −2 −1 0 Time [min] Δ BG T2=20 T2=40 T2=60 T2=80 T2=100 T2=120 T2=140 T2=160 T2=180 T2=200 0 10 20 30 40 50 −1.5 −1 −0.5 0
Fig.5
τ
impulseresponsesreturntothesteadystateofzero.Physiologically,theassumption isthatevenintheabsenceofinsulin,glucosewillberemovedfromthecirculation aftermealingestion.Thisisfundamentallydifferentthantheassumptionsinthe previousstructures,whereglucosewillnotberemoveduntilinsulinissupplied.As itcanbeseenintheexperimentalresultsinthecorrespondingpapers,bothmodel assumptionscanleadtoagoodapproximationofrealdata.Notealsothatmodel parametersarenotdirectlyvisiblefromtheimpulseresponse.
Finally,theimpulseresponserelatedtothecarbohydrateinputfrommodelstructure5inTable 1 isshowninFig. 7.Theresponserelatedtotheinsulininputisthe sameasformodelstructure2andcanbeseeninFig. 5.Themaindifferenceinthis structureistheassumptionoftwodifferenttimeconstants T1 , T2 ,bothforthecarbohydratedynamics.Therefore,thereismoreflexibilityintheresponsecomparedto thecaseofusingonlyonevalueforbothconstants( T1 = T2 ).However,itisquestionablewhetherdistinctivevaluesfor T1 and T2 canreallybeidentifiedfromclinical data[29].
Fig.7 Impulseresponseof modelstructure5from Table 1 using K 1 = 100, T2 = 20, τ1 = 10
8H.Kirchsteigeretal. 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [min] Δ BG T1=20 T1=40 T1=60 T1=80 T1=100 T1=120 T1=140 T1=160 T1=180 T1=200 0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 0 200 400 600 800 1000 1200 1400 −1.5 −1 −0.5 0 Time [min] Δ BG T2=20 T2=40 T2=60 T2=80 T2=100 T2=120 T2=140 T2=160 T2=180 T2=200 0 10 20 30 40 50 −1 −0.8 −0.6 −0.4 −0.2 0
Fig.6 Impulseresponseofmodelstructure4fromTable
1 using K 1 = 100, K 2 =−80
0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Time [min] Δ BG
0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5
T1=20 T1=40 T1=60 T1=80 T1=100 T1=120 T1=140 T1=160 T1=180 T1=200
4IntervalModels
Thereareessentiallytwowaystocharacterizeintervalmodels:oneviewpointis todesignthemodelsuchthatallpossiblerealizationsoftheuncertaintieswillbe consideredandthetruesystemoutputwillbewithinthecomputedmodeloutputall thetime,minimizingatthesametimetheconservativeness.Theotherviewpointisto allowacertainamountorpointstolieoutsideofthecomputedintervalandthereby obtainingamuchsmallerinterval[35].Thefirstapproachistreatedinanotherchapter ofthisbook(see“Physiology-BasedIntervalModels:AFrameworkforGlucose PredictionUnderIntra-PatientVariability”byJ.BondiaandJ.Vehi)whilewefocus onthelatterhere.
Wewillchoosethemodelstructure1fromTable 1
becauseitcontainsonlyfourparameterstobeestimatedanddoesnotleadtoa discontinuityintheimpulseresponse.Thegeneralizeddescriptionofthismodel (assumingonlyoneinputforsimplicityofnotation)isacontinuoustimeprocess modeloftheform
Modelingaspecificpatientmeanstoassignvaluestothevariables ai , bi in(2).This isanidentificationproblem[34, 43]andcanbetackledintwoways.Bytransforming (2)intoanequivalentdiscrete-timeformulation,alltheavailabletoolsofdiscretetimesystemidentificationcanbeapplied.However,suchatransformationmight introduceadditionalparametersandtheinitial(physiological)interpretationofthe continuoustimeparametersgetslost[20].Ontheopposite,therearecontinuoustime identificationmethodswhichdirectlyestimatetheparametersin(2)withoutanyneed fortransformation[19].Thebenefitsofsuchadirectestimationinthecontextofthe humanglucoseinsulinsystemhavebeentreatedin[10, 30].Inthiscontribution,we willthusalsofocusonadirectcontinuoustimeestimationmethod.
4.1ContinuousTimeSystemIdentification
Beginningfirstwithouttakingintoaccounttheuncertainty,estimatesofthemodel parameterscanbefoundbyminimizingaquadraticcriterionoftheform
AlternativeFrameworksforPersonalizedInsulin…9
BG (s ) = K 1 (1 + sT1 )2 s Carb (s ) + K 2 (1 + sT2 )2 s I (s ) (1)
G (s ) = B (s ) A (s ) = b0 + b1 s +···+ bm s m a0 + a1 s +···+ s n . (2)
J1 (θ) = 1 N N k =1 ε 2 (k ,θ) (3)
where ε denotesthedifferencebetweenmeasurementandmodeloutput,
N isthetotalnumberofavailablemeasurementsandthevector θ containsallparameters.Variousoptimizationtechniquesexistforactuallyminimizingthecriterion (3)(seee.g.,[43]).
Now,consideringuncertaintyinthesystemtobemodeledweassumetwoseparate datasetsofthesamesystemwithdifferentunderlyingdynamics.Thecostfunction whichisminimizednowisthesumoftwosimilartermsasabove,againtakinginto accountthedeviationbetweenmodeloutputandmeasurement.Additionally,wewill introduceathirdtermwhichpenalizethestandarddeviationoftheestimatedmodel parametersforthefirstdatasetcomparedtotheseconddataset(5).Forexample,if thisthirdtermisnotpresent,themodelparametersforbothdatasetswillbeestimated independentlyandmightdeviatetoagreatextentfromeachother.Byintroducing andweighting(usingtheweightingmatrix Π whichhasthetuningparametersin themaindiagonal)thethirdterm,acompromisebetweenagoodmodelfitofthe individualdatasetsandcompactparametersetscanbeobtained.
Thefunction σ denotesavectorstandarddeviationoperatorwhichdetermines component-wisethestandarddeviationofthemodelparametersoftheparameter vector θ .
MinimizationcanbedonewithaGauss–Newtonalgorithm[43].Thankstothe definitionofthecostandthemodelstructure,gradientvectorandHessianmatrix canbedeterminedanalytically[30].Nevertheless,theoptimizationisnonlinearand iterativeandthuscarehastobetakentoavoidimproperstartingvaluesfortheparameters.Ageneralizationofthecostfunctionto N exp datasetsandthecorresponding gradientvectorsandtheHessianwasderivedin[30].
Asaresultofthecontinuoustimesystemidentification,thereisoneparameter vectorperexperiment.Thenextstepistoextractparameterintervalswhichthendefinethemodeloutputinterval.Consideringagainthemodelstructure1fromTable 1, therearefourmodelparameterswhichmeanstheparametervectorhastheform
Denotingwithsuperscriptsthecorrespondingdatasetnumber,attheendoftheidentification N exp estimates K 1 1 , K 2 1 ,... K Nexp 1 areavailableandtheirmaximumvalue is K max 1 andminimumvalue K min 1 .Thecomputationoftheintervalmodeloutputis then
10H.Kirchsteigeretal.
ε(t
y (t ) −ˆ y (t ,θ) (4)
,θ) =
J2 (θ1 ,θ2 ) = 1 N N k =1 ε 2 1 (k ,θ1 ) + 1 N N k =1 ε 2 2 (k ,θ2 ) + 1 N σ(θ1 ,θ2 ) 2 Π (5)
θ =[ K 1 , K 2 , T1 , T2 ] (6)
Notethat K 2 describestheeffectofinsulinandisthusnegative.Inthefollowing subsection,wewillpresentresultswhenapplyingtheintervalmodelestimationto shortdatasegmentsrepresentingthebreakfastperiod.Thestartingpointofthedata isbreakfasttime(around8:00inthemorning)anddataendpointsjustbeforelunch weretaken(around12:30).
4.2IntervalModelResults
Figure 8 showsatypicalresultobtainedwiththemethodologydescribedabovefor threeindependentmeasurementsetsofasinglepatient.TheblackstarsarethemeasuredglucoselevelsusingaYellowSpringInstrument YSI2300STATPlus™Glucose Analyzer (YSI)device.Whenmodelingeachdatasetindependently(individualmodels),thegreencurvesapplywhichshowofcoursethebestperformance.Depending onthechoiceofthetuningmatrix Π ,theidentifiedparametersthendependstronger orlessstrongoneachotherandanoutputintervalaccordingto(7)canbecomputed, seethereddashedlinesinFig. 8.Finally,alsothemeanintervalmodelresponseis shown,whichiscomputedwhenusingasparametertheaveragevaluesofallthe estimatedparametersoftheintervalmodel.
Remark1 Againnotethedependencyoftheresultsonthetuningmatrix Π .When choosing Π = 0azeromatrix,thecorrespondingterminthecostfunction(5)will disappearandtheresultisanindependentestimationoftheparametersforeach experiment.Thisalsoresultsinthelargestpossibleoutputinterval.Intheother extreme,choosingverylargevaluesin Π resultsinverysmallstandarddeviations ofthemodelparameters,makingthemequivalent.Thisalsoresultsinanoutput intervalwhichdegeneratestoasinglelineandisthesameresultasforastandard multiexperimentidentificationsetup[34].
Foranin-depthanalysisofthemethod,itwasappliedto10simulatedand10 clinicaldatasets.Thesimulateddatawasobtainedfromatime-varyingmetabolic simulationmodel[28].Inthefollowing,wewillusethetermindividualmodelfor amodelestimatedonasingledataset(experiment),thetermintervalmodelfora modelaccordingto(7),andthetermmeanintervalmodelwhenasimulationwith themeanvalueoftheintervalmodelparametersisdone.
AlternativeFrameworksforPersonalizedInsulin…11 ˆ Y max (s ) = K max 1 (1 + T min 1 s )2 s U (1) (s ) + K max 2 (1 + T max 2 s )2 s U (2) (s ) (7a) ˆ Y min (s ) = K min 1 (1 + T max 1 s )2 s U (1) (s ) + K min 2 (1 + T min 2 s )2 s U (2) (s ) (7b)
Fig.8 Outputresponseoftheintervalmodelcomparedtomeasurementsforthreedatasets
4.2.1PerformanceMetrics
Resultswillbeevaluatedbasedonafitvalue(8)andontheerrorindetectingthe peakBG.ThisisvisualizedinFig. 9: Δ BG isthedifferencebetweenthemeasured andpredictedpeakBGand Δ T isthedifferenceintime.Bothquantitiescanbe positiveornegative,wherepositivemeansthemeasuredBGpeakishigherandlater intimethanthesimulated.Themotivationofusingthosemetricsistheirimportance indiabetestreatment.
4.2.2ResultsUsingtheSimulatedData
First,wewillpresentresultsfromasinglesimulatedpatientbeforepresentingstatisticsofall10consideredpatients.Theparametervaluesandperformancemetrics reportedinTable 2 arefromsimulatedpatientnumber2.Fortheintervalmodel, Π (5)wastunedinsuchaway,thatthefitvalueoftheintervalmodelisnotsignificantly lessthan90%ofthefitoftheindividualmodel,andthattheratiobetweenmean valueandstandarddeviation(1/ CV )ofeachofthefourparametersisatleastfive. Inthisway,itisensuredthattheintervalpredictionsarereasonablytight.
12H.Kirchsteigeretal. 0 100 200 100 120 140 160 180 200 220 240 260 280 BG [mg/dl] 0 100 200 50 100 150 200 250 300 350 BG [mg/dl] time [min] 0 100 200 120 140 160 180 200 220 240 260 BG [mg/dl] YSI individual model interval model mean interval model
fit = 100 1 ˆ y y 2 2 y −¯ y 2 2 (8)
Fig.9 Performancemetrics (Δ BG , Δ T )forevaluation oftheresults. Bluestars indicateBGmeasurements, interpolatedwith second-orderpolynomial. Blackdashed arethe min/maxresponsesofthe intervalmodel(colorfigure online)
ForthepresentedresultsinTable 2 andFig. 10,theelementsinthemaindiagonal of Π are [15000, 125, 20, 0.3] resultin1/ CV =[5.02, 5.90, 5.94, 6.62].Theparameterestimationwasdoneonthefirst3days(upto t = 72h).Fortheindividual models,thevalidationisbasedonmeanparametersofthefirst3days.Fortheinterval models,theperformancemetricswerecalculatedbasedontheexactestimationsfor days1–3andvalidationwasdonewiththemeanintervalmodel.
FromTable 2,weseethatthereisarathersmallratiobetweenmeanvalueand standarddeviationoftheestimatedparameters(especiallyfor K 2 )—whichwould meanthattheinsulineffectisverydifferent—whentheexperimentsareidentified independentlyofeachother.Thisratioisgreatlyincreasedwhenapplyingtheproposedintervalmodelestimation,withoutdecreasingtheperformance(onthethree trainingdays).Consideringvalidation,theintervalmodelshowsbetterperformance comparedtotheindividualmodel.Notealsothatinthisparticularcase,theindividuallyestimatedmodelforday3isnotuseful,because K 2 < 0,i.e.,aninsulininjection wouldresultinaglucoseincrease.
Theresultsforall10patientsaresummarizedinTable 3 wherethemeanvalues oftheperformancemetricsoverall5daysareshown.Theintervalmodelshows thehighestfitvaluesineverycase,andtheaverageerror(forall10patients)of correctestimationoftheBGpeakisonly3.79mg/dlcomparedto10.80mg/dlofthe individualmodels.Notethatthisratherhigherroristoalargeextentcausedbythe specificglucoseresponsestomealsbythesimulatorforsomepatients,wherethe responsedoesnothaveasinglemaximalvalue.Inthosecasestherearetwoalmost equivalenthighpeaks,butseparatedintime.Suchashapecannotbereproducedwith thechosenmodelstructure(1),andtheapproximationtypicallyliesinthemiddleof thetwopeaks.Itisofinteresttonotethatfromthe30individualmodels,therewere 8thatshowedtobeincorrectfromaphysiologicalpointofview( K 1 < 0and/or K 2 > 0).Consideringtheintervalmodel,onlyforpatient#6nophysiologically correctmodelscouldbefound,asthispatient’sglucosedynamicshaveahightime
AlternativeFrameworksforPersonalizedInsulin…13
0 20 40 60 80 100 120 140 160 180 200 140 160 180 200 220 240 260 280 BG [mg/dl] time [min] ΔBG ΔT
Table2 Estimatedparametersandperformancemetricsforsimulatedpatient#2
Individualmodel
Intervalmodel
14H.Kirchsteigeretal.
Meal#1 Meal#2 Meal#3 Meal#4 Meal#5 Mean Mean/Std. Fit 66.54 66.85 87.66 12.73 42.02 55.16 1.92 Δ BG 2.96 0.56 1.35 10.72 11.06 5.33 1.03 Δ T 7.00 133.00 99.00 129.00 132.00 100.00 1.86 K 1 1.28( ± 0 . 08) 0.99( ± 0 . 05) 0.77( ± 0 . 04) 1.01 1.01 1.01 3.97 K 2 13.83( ± 0 . 59) 5.56( ± 0 . 41) 7.37( ± 0 . 48) 4.01 4.01 4.01 0.38 T1 24.99( ± 1 . 33) 20.28( ± 0 . 91) 14.69( ± 0 . 60) 19.98 19.98 19.98 3.88 T2 99.77( ± 12 . 02) 92.55( ± 16 . 48) 61.43( ± 4 . 25) 84.58 84.58 84.58 4.15
Fit 72.02 69.68 65.40 33.76 51.93 58.56 3.68 Δ BG 0.13 1.51 2.23 7.20 6.14 3.44 1.12 Δ T 9.00 121.00 99.00 114.00 123.00 93.20 1.94 K 1 1.00 0.96 1.36 1.10 1.10 1.10 5.02 K 2 16.19 13.63 11.53 13.78 13.78 13.78 5.90 T1 19.00 21.16 26.26 22.14 22.14 22.14 5.94 T2 211.86 253.75 287.56 251.05 251.05 251.05 6.62
Fig.10 Simulatedpatient#2:simulatedglucoseprofile(blacksolid )individualmodeloutputsfor thebreakfasts(greensolid )andworst-caseintervalmodeloutputsforbreakfasts(reddashed ).The first3days(upto t = 72h)wereusedforestimationofthemodels,theremaining2daysshow validationresults. Bottompanel showsthescaledimpulsecarbohydrate(blacksolid )andinsulin (redsolid )inputs
Table3 Statisticsforall10virtualpatients,meanvaluesoftheperformancemetrics
constant,andBGisonlyrisingintheobservationalperiod.Tocorrectthis,alonger simulationtimewouldbenecessary.
AlternativeFrameworksforPersonalizedInsulin…15 0 20 40 60 80 100 120 100 150 200 250 300 350 BG [mg/dl] Simulator individual model interval model 0 20 40 60 80 100 120 0 0.5 1 time [h] u1 u2
Individualmodel Intervalmodel Pat Fit Δ BG Δ T Fit Δ BG Δ T P1 46.08 5.08 108.00 61.82 5.68 23.40 P2 55.16 5.33 100.00 58.56 3.44 93.20 P3 86.68 2.72 2.40 87.05 2.73 2.20 P4 32.15 16.36 76.00 74.01 4.64 49.60 P5 8.26 33.73 18.80 89.69 2.67 18.20 P6 63.35 6.63 71.40 67.32 6.74 78.80 P7 65.20 6.76 8.00 71.41 3.68 6.00 P8 65.32 3.83 69.00 67.57 3.22 57.80 P9 70.78 3.08 6.40 71.57 2.48 5.40 P10 15.04 24.48 35.40 82.74 2.62 4.20 Avg. 49.15 10.80 49.54 73.18 3.79 33.88
