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ListofContributors
AmrM.AbdelAty EngineeringMathematicsandPhysicsDepartment,Facultyof Engineering,FayoumUniversity,ElFayoum,Egypt
SalwaK.Abd-El-Hafiz FacultyofEngineering,CairoUniversity,Giza,Egypt
NorelysAguila-Camacho UniversityofChile,Santiago,Chile
MohsenAlimi UniversityofKairouan,Kairouan,Tunisia
DaliaAllam FayoumUniversity,Fayoum,Egypt
AhmadTaherAzar FacultyofComputersandInformation,BenhaUniversity, Benha,Egypt;SchoolofEngineeringandAppliedSciences,NileUniversity, Giza,Egypt
PagavathigounderBalasubramaniam GandhigramRuralInstitute(Deemedtobe University),Dindigul,TamilNadu,India
YassineBensafia MohandOulhadjUniversityofBouira,Bouira10000,Algeria
MuzaffarA.Bhat JamiaMilliaIslamia,NewDelhi,India
Y.Boukal Universite ´ deLorraine,CosnesetRomain,France;Universite ´ HassanII, Casablanca,Maroc;Universite ´ deValenciennesetduHainaut-Cambre ´ sis, Famars,France
BachirBourouba UniversityofSe ´ tif,Se ´ tif19000,Algeria
DanieleCasagrande UniversityofUdine,Udine,Italy
M.Darouach Universite ´ deLorraine,CosnesetRomain,France
SubirDas IndianInstituteofTechnology(BHU),Varanasi,UttarPradesh,India
ManuelA.Duarte-Mermoud UniversityofChile,Santiago,Chile
MagdyEteiba FayoumUniversity,Fayoum,Egypt
IbiyinkaA.Fuwape FederalUniversityofTechnology,Akure,Nigeria;Michael andCeciliaIbruUniversity,Ughelli,Nigeria
GokulP.M. LodzUniversityofTechnology,Lodz,Poland
HanyN.Hassan BenhaFacultyofEngineering,BenhaUniversity,Benha, Egypt;ImamAbdulrahmanBinFaisalUniversity,Dammam,SaudiArabia
SamarM.Ismail FacultyofInformationEngineeringandTechnology(IET), GermanUniversityinCairo(GUC),Cairo,Egypt
TomaszKapitaniak LodzUniversityofTechnology,Lodz,Poland
KhatirKhettab MohamedBoudiafUniversityofM’sila,M’sila28000,Algeria
xviii ListofContributors
WiesławKrajewski SystemsResearchInstitute,PolishAcademyofSciences, Warsaw,Poland
JitendraKumar InstrumentationandControlEngineeringDivision,Dwarka,New Delhi,India
VineetKumar InstrumentationandControlEngineeringDivision,Dwarka,New Delhi,India
Matı´asG.Mayol-Sua ´ rez UniversityofChile,Santiago,Chile
JoanaP.Neto UniversidadedeLisboa,Lisbon,Portugal
SamuelT.Ogunjo FederalUniversityofTechnology,Akure,Nigeria
KayodeS.Ojo UniversityofLagos,Lagos,Nigeria
AdelOuannas UniversityofLarbiTebessi,Tebessa,Algeria
AdelOunnas UniversityofLarbiTebessi,Tebessa,Algeria
Viet ThanhPham HanoiUniversityofScienceandTechnology,Hanoi, Vietnam;LodzUniversityofTechnology,Lodz,Poland
N.E.Radhy Universite ´ HassanII,Casablanca,Maroc
AhmedG.Radwan FacultyofEngineering,CairoUniversity,Egypt;Nanoelectronics IntegratedSystemsCenter(NISC),NileUniversity,Cairo,Egypt
KamalPalSinghRana InstrumentationandControlEngineeringDivision,Dwarka, NewDelhi,India
AbdelwahebRebai UniversityofSfax,Sfax,Tunisia
AhmedRhif UniversityofCarthage,LaMarsa,Tunisia
LobnaA.Said Nano-ElectronicsIntegratedSystemsCenter(NISC),NileUniversity, Giza,Egypt
T.Sathiyaraj GandhigramRuralInstitute(DeemedtobeUniversity),Dindigul, TamilNadu,India
WafaaS.Sayed FacultyofEngineering,CairoUniversity,Giza,Egypt
MouradS.Semary BenhaFacultyofEngineering,BenhaUniversity,Benha,Egypt
FernandoE.Serranot CentralAmericanTechnicalUniversity(UNITEC), Tegucigalpa,Honduras
BharatB.Sharma NationalInstituteofTechnology,Hamirpur,HimachalPradesh, India
ManojK.Shukla NationalInstituteofTechnology,Hamirpur,HimachalPradesh, India;LovelyProfessionalUniversity,Punjab,India
ShikhaSingh JamiaMilliaIslamia,NewDelhi,India
MayankSrivastava IndianInstituteofTechnology(BHU),Varanasi,UttarPradesh, India
HamedTaghavian SharifUniversityofTechnology,Tehran,Iran
methods,fractionalorderdiscretemaps,multiswitchingsynchronization, metaheuristicalgorithms,backsteppingcontrol,etc.
ABOUTTHEBOOK
ThenewElsevierbook, MathematicalTechniquesofFractionalOrder Systems,consistsof21contributedchaptersbysubjectexpertswhoarespecializedinthevarioustopicsaddressedinthisbook.Thespecialchapters havebeenbroughtoutinthisbookafterarigorousreviewprocessinthe broadareasofmathematicaltechniquesoffractionalordersystems.Special importancewasgiventochaptersofferingpracticalsolutionsandnovel methodsfortherecentresearchproblemsinthemathematicalmodelingand controlapplicationsoffractionalordersystems.Thisbookdiscussestrends andapplicationsofmathematicalmodelsoffractionalordersystemsin engineering.
OBJECTIVESOFTHEBOOK
Thisvolumepresentsaselectedcollectionofcontributionsonafocused treatmentofmathematicaltechniquesforfractionalordersystems.Thebook alsodiscussesmultidisciplinaryapplicationsinelectricalengineering,control engineering,mechanicalengineering,andcomputerscience.Bothnoviceand expertreadersshouldfindthisbookausefulreferenceinthefieldoffractionalordersystems.
ORGANIZATIONOFTHEBOOK
Thiswell-structuredbookconsistsof21fullchapters.
BOOKFEATURES
● Thebookchaptersdealwiththerecentresearchproblemsintheareasof fractionalordersystems.
● Thebookchapterspresentvariousmathematicaltechniquesforfractional ordersystemssuchasactivecontrol,adaptivecontrol,backsteppingcontrol,nonlinearcontrol,fuzzylogiccontrol,metaheuristicalgorithms,etc.
● Thebookchapterscontainagoodliteraturesurveywithalonglistof references.
● Thebookchaptersarewell-writtenwithagoodexpositionoftheresearch problem,methodology,blockdiagrams,andmathematicaltechniques.
● Thebookchaptersarelucidlyillustratedwithnumericalexamplesand simulations.
● Thebookchaptersdiscussdetailsofengineeringapplicationsandfuture researchareas.
2 MathematicalTechniquesofFractionalOrderSystems
fractionalcalculushasseenarapidgrowthinitsapplications.Asitcanbe saidthatrealobjectsaregenerallyfractional,evenifthat fractionality is verylow(Petra ´ s,2009),fractionalcalculusisparticularlywellsuitedto describethenonlinearrelationshipwithtimeofanomalousdiffusion (Margin,2006).Indeed,nonintegerderivativesoccurmostfrequently,and naturally,inphysicalproblemswheretheessentialmechanisms,reactions,or interactionsaregovernedbydiffusionprocesses.Ofcourse,seeingthat manybiologicalprocessesoftenpresentanomalousdiffusion,fractionalcalculusisaneligibleandpowerfultooltodescribethesephenomena(e.g.,subthresholdnerveconduction,viscoelasticity,bioelectrodes)(Magin,2006). Butdiffusion,beingoneofthemostrelevantapplicationsoffractionalcalculus,isfarfrombeingtheonlyone.Fractionalderivativesareusedtoformulateandsolvedifferentphysicalmodelsallowingacontinuoustransition fromrelaxationtooscillationphenomena;topredictthenonlinearsurvival andgrowthcurvesoffood-bornepathogens;toadapttheviscoelasticity equations(Hooke’sLawandtheNewtonianfluidsLaw)(Rahimy,2010; Petra ´ s,2009);fractionalcontrolplaysanimportantroleingeneralphysics, thermodynamics,electricalcircuitstheoryandfractances,mechatronicsystems,signalprocessing,chemicalmixing,chaostheory,andmanyothers (Petra ´ s,2009).Andtherearemanypossibleapplicationsoffractionalderivativesincontrol(Vale ´ rioandSa ´ daCosta,2006,2011a,2012;Azaretal., 2017).
Manyphysicalprocesses,however,alsoappeartoexhibitafractional orderbehaviorthatvarieswithtimeorspace(LorenzoandHartley, 2002b).Inwhatconcernsthefieldofvisc oelasticityofcertainmaterials, thetemperatureeffectinsmallamplitudestrainsisknowntoinduce changesfromanelastictoviscoelastic/viscousbehavior,whererealapplicationsmayrequireatimevaryingtemperaturetobeanalyzed.Therelaxationprocessesand reactionkineticsofprotein s,whicharedescribedby fractionaldifferentialequations,havebeenfoundtohaveanorderwitha temperaturedependence.Thebehaviorofsomediffusionprocessesin responsetotemperaturechangescanbebetterdescribedusingvariable orderelementsratherthantime-vary ingcoefficients,amongothercases (LorenzoandHartley,2002b).Thesearenaturalapplicationswherevariableorderoperatorsasafunctionoftime(t )orsomeothervariable( x) canbeintroducedwithprofitandinaverynaturalway(Lorenzoand Hartley,2002b;Vale ´ rioandSa ´ daCosta,2013)
Thischapterpresentsonesuchapplicationofvariableorderderivatives: thedevelopmentofsimplermathematicalmodelsofboneremodeling,both forhealthybone,andforbonetissueaffectedbycancer.Thereareseveral modelspublishedintheliterature,butthenoveltywhichistheuseofvariableorderderivativeswillresultinsimplermodelsthatareeasierto understand.
Remodeling
Boneisconstantlybeingrenewed,beingdestroyed,andformed,duetocells termedosteoclastsandosteoblasts,respectively.Intheadultskeleton,both processesareinbalanceandtightlycoupledthroughautocrineandparacrine factorsbetweenbonecellsthatallowforaconstantbonedensitytobemaintained.Asthisspecificmicroenvironmentprovidesthenecessaryconditions forthegrowthandproliferationoftumorcells,boneisacommonsitefor thedevelopmentofmetastases,mainlyfromprimarybreastandprostate cancer.
Mathematicalandcomputationalmodels,withdifferentialequationsthat representthecontrolmechanismsinvolved,canreplicatethisremodeling process(Komarovaetal.,2003).Thesemodelshavebeenextendedto includetheeffectsoftumordisruptivepathologiesinthebonedynamics,as metastasescontributetothedecouplingbetweenboneresorptionandformationandtotheself-perpetuatingtumorgrowthcycle(Ayatietal.,2010). Counteractioneffectsofcurrentlyusedtherapieswerealsocontemplated (Ayatietal.,2010),throughthepharmocokinetic(PK)andpharmacodynamic(PD)combinationofanticancer(chemotherapy)withantiresorptive treatments(bisphosphonatesormonoclonalantibodies)(Christetal.,2018; Coelhoetal.,2016,2015).
Fractionalandvariableorderderivativescanbesuccessfullyusedin modelingthedynamicsofboneremodeling. Christetal.(2018) implemented fractionalderivativesinthedifferentialequationsofboneremodelingpresentedin Ayatietal.(2010),andanalyzeditsdynamicbehaviorinthe absenceandpresenceoftumorandPK/PDappliedtreatment,foradiscretizedsinglepointandaone-dimensionalbone. Vale ´ rioetal.(2016) applied thesamemethodologytothemoreaccuratebiochemicalmodelof Coelho etal.(2016).Variableorderderivativeshavealsobeenintroducedin Neto etal.(2017),asasimplificationtechnique,inthemodelsof Ayatietal. (2010) and Coelhoetal.(2015),inanefforttoreplicatethesamebone microenvironmentresponsebutrecurringtolessparameterstoimposethe knownbonebehavior.Itisthelatterthatishererevisitedandfurther analyzed.
1.1.3ChapterOrganization
Theremainingsectionsofthischapterareorganizedasfollows.Variable orderconceptsanddefinitionsareaddressedin Section1.2.Boneremodeling physiology,PK/PDconcepts,andpublishedintegermathematicalmodelsare presentedin Section1.3.Variableorderbonemodelsareaddressin Section1.4.Finally,conclusionsarepresentedin Section1.5.
memoryofpastvaluesof f ðt Þ (Vale ´ rioandSa ´ daCosta,2013).Recallthat,for constantdifferentiationorders,theGru ¨ nwald Letnikoff(GL)definitionis givenby Eq.(1.8a),andtheRiemann Liouville(RL)definitionisgivenby Eq.(1.8c).
Thefractionalorderofintegralsandderivativescanbeafunctionoftime orsomeothervariable(LorenzoandHartley,2002b).Herewewillexpress itafunctionoftime;theothercaseisastraightforwardgeneralization.Ifno memoryofpastvaluesoftheorderisintended,thedefinitionmustonly includeitscurrentvalue:
TheseGLandRLformulationswithoutmemoryof α areequivalent(if thefunctioniswell-behavedenoughsothatbothformulationscanbe applied).Theyarecalledtype-1variableorderderivativesin Lorenzoand Hartley(2002a); Vale ´ rioandSa ´ daCosta(2011b); Vale ´ rioandSa ´ daCosta (2013),andtype-A variableorderderivativesin Sierociuketal.(2015a,b)
FIGURE1.3 Testingtheeffectsofaccumulatedvaluesoforder α.Obtainedresultsfor α 5 0 5 integratoroutputwithinitialconditions c 5 0 5anddifferentvaluesof T (thelengthofinitial conditionsfunction).
essentialfortheapplicationoftype-D variableorderdefinitiontothebone remodelingmodelspresentednext.
1.3BONEREMODELING
Thissectionbeginsbyintroducingbonephysiologyconcepts,in Section1.3.1. PKandPDmodelsfollow,in Section1.3.2.Finally,aroad-mapthroughpublishedintegermodels,thatreflectbonedynamic,ispresentedin Section1.3.3. Allmodelspresentedhereusedimensionlessvariablesandparameters, includingthecellpopulations,exceptwhenexplicitlysaidotherwisein Table1.1 D1 referstothefirstorderderivativeintime, d dt ,and Dαðt Þ or Dαðt ;xÞ referstotheGrunwald Letnikofftype-D variableorderderivative, D N Dαðt Þ t or D N Dαðt ;xÞ t ,respectively.
1.3.1IntroducingBonePhysiology
Theskeletonisanactivemetabolictissue,besidesprovidingsupportand protectiontothevitalorgans(Chenetal.,2010).Itisnot,however,static,as itconstantlyundergoesremodeling.Thisprocessisspatiallyheterogeneous, withregularbutasynchronouscyclesthatcantakeplacein5% 25%ofthe totalbonesurfaceavailable(Crockettetal.,2011).Itisestimatedthatabout 10%oftheboneisrenewedeachyear(Lerner,2006).Corticalboneprovides strengthandprotectionwhiletrabecularisthemostmetabolicallyactive. Consequently,itiswithinthetrabecularbonethatmostofboneturnover occurs,undernormalconditionsandindiseasesofbonelossorformation. ThisreconstructionoccurswithinaBasicMulticellularUnit(BMU),a temporaryanatomicalstructurewhereboneisresorbedbycellstermed osteoclasts andsequentiallyformedduetocellscalled osteoblasts (Parfitt, 1994).Thisprocessisremarkablywellbalanced,asatightlycontrolled
10 MathematicalTechniquesofFractionalOrderSystems
TABLE1.1 SummaryandDescriptionoftheVariablesandParametersof theModelsofEqs.(1.19) (1.22)(inthecaseofnon-integermodels,units ofDay-1 arereplacedbypseudo-unitsDay-α)
Variables Description
t TimeDay x Distance x A½0; 1
C ðt ; x Þ
B ðt ; x Þ
Osteoclastpopulation
Osteoblastpopulation
z ðt ; x Þ Bonemassdensity %
T ðt ; x Þ Bonemetastasesdensity %
αðt Þ=αðt ; x Þ
Variableorderexpression
d1 ðt Þ Effectofdenosumab—
d2 ðt Þ Effectofzoledronicacid—
d3 ðt Þ Effectofpaclitaxel—
Parameters Description Units
αC OCactivationrate Day 1
αB OBactivationrate Day 1
β C OCapoptosisrate Day 1
β B
gCC
gBC
gCB
gBB
OBapoptosisrate Day 1
OCautocrineregulator
OCparacrineregulator
OBparacrineregulator
OBautocrineregulator
κC Boneresorptionrate Day 1
κB Boneformationrate Day 1
σ C
DiffusioncoefficientforOC Day 1
σ B DiffusioncoefficientforOB Day 1
σ z
σ T
rCC
rBC
rCB
DiffusioncoefficientforbonemassDay 1
DiffusioncoefficientformetastasesDay 1
OCtumorousautocrineregulation—
OCtumorousparacrineregulation—
OBtumorousparacrineregulation— (Continued )
VariableOrderFractionalDerivativesandBoneRemodeling Chapter|1 13
Thepresenceofmetastaticcancercells(breast,prostate,lung,renal,and myelomaamongothers)acceleratestheremodelingprocessanddisturbsthe balancebetweenbonecellsbydisruptingitsbiochemicalregulation(Lerner, 2006).Boneintegrityisconsequentlylost.Thesesitesofcancermetastasis areusuallythosewhereboneremodelingratesarehigh,suchasthepelvis, theaxialskeleton,orboneswithabundantbonemarrow(Boyce,2012; Schneideretal.,2005).
Bonemetastasescanbeosteolytic(increasedboneresorption),orosteoblastic(boneformationisstimulatedinanunstructuredway).However,both arestillpresentinanycase,althoughoutofbalance,resultinginlossof boneresistanceandintegrity.Breastcancermetastasesarepronetodevelop osteolyticmetastasisandprostatecanceronesareusuallyosteoblastic(Suva etal.,2011).
For osteolytic metastases,tumorcellsstimulateosteoclastactivityand receive,inreturn,positivefeedbackfromfactorsreleasedbythebonemicroenvironmentduringbonedestruction(Casimiroetal.,2016;Chenetal., 2010).AsTGF-β isreleasedfromthebonematrixduringresorption,itstimulatestumorgrowthandparathyroidhormone-relatedprotein(PTHrP)productioninmetastaticcells.BybindingtoPTHreceptorsoncellsof osteoblasticlineage,RANKLlevelsarethenenhanced.Subsequently,osteoclastsareactivated,leadingtoincreasedboneresorption(Casimiroetal., 2016).Osteoclastsactivity,inturn,willresultinthereleaseofTGF-β from thedegradedbone,whichfurtherstimulatestumorgrowthandPTHrPsecretion,givingrisetotheviciouscycle.
In osteoblastic metastases,tumorouscellsgrowasboneexpresses endothelin-1(ET-1).ET-1stimulatesosteoblaststhroughtheendothelinA receptor(ETR),activatingWnt-signaling.Tumor-derivedproteasescontributetothereleaseofosteoblasticfactorsfromtheextracellularmatrix,includingTGF-β andIGF-I.RANKLisincreasedduetotumor-inducedosteoblast activity,leadingtothereleaseofPTHandpromotingosteoclastactivity (Casimiroetal.,2016).Thus,tumormicroenvironmentleadstotheaccumulationofnewformedbone.
Severalapproaches,thattreatprimaryandmetastaticbonetumors,have thepotentialtoaffectbothtumoraffectedandhealthycells.However,strategiescanbeorientedtoeffectivelyinhibittumorgrowthbytargetingthebone anditsmicroenvironmentratherthanthetumoralone.Antiresorptivetherapy targetsosteoclasts,whenanosteolyticmetastaticbonediseaseispresent. Bisphosphonates suchasalendronateorzoledronicacid(ZometasZoledronicAcidforInjection,2017; Chenetal.,2002),and monoclonal antibodies likedenosumab(Sohnetal.,2014;Gibianskyetal.,2012),are effectivetreatmentscurrentlybeingadministrated.Whilebisphosphonates lodgeinboneandpoisonosteoclastsastheydegradebone,monoclonalantibodiesinturnbindexclusivelytoRANKL,increasingtheOPG/RANKL ratioandinhibitingosteoclastformation.Forotherdiseases,suchasmultiple
1.3.3ModelingBoneRemodelingCycles—IntegerModels
Modelsbasedondifferentialequationshavebeenappliedtoanalyzeand simulatebiochemicalandbiomechani calinteractionsbe tweenthetumor cellswiththebonemicroenvironment.Theycanbedividedintolocalor nonlocalconstructions(mainlyone- dimensionalmodels)and,withineach category,threestagesofbonebehaviorareencompassed:healthy bonemicroenvironment,tumordisruptedbonedynamics,andtherapy counteraction.
Thesimplestmodelforboneremodelingwasproposedin Komarova etal.(2003),takinganS-systemformdescribedbyEq.(1.17)(Savageau, 1988).Couplingofosteoclasts, C ðt Þ,andosteoblasts, Bðt Þ,behaviorisdone throughbiochemicalautocrine ðgCC ; gBB Þ andparacrine ðgBC ; gCB Þ factors expressedimplicitlyinthesystem’sexponents.Bonemassdensity, zðt Þ,is determinedthroughtheextentwhichvaluesof C ðt Þ and Bðt Þ populations exceedtheirnontrivialsteadystate, CSS and BSS ,respectively.Consequently, bonemassisbutthereflectionofthebonecellsactivities.Productionand deathrateofthebonecellsareencompassedin αC ;B and β C ;B ,respectively, andconstants κC and κB representtheboneresorptionandformationactivity,respectively.
Thismodeliscapableofrepresentingsingleorperiodicalremodeling cyclesbysettingtheautocrineandparacrineparameterstotheappropriate values,speciallytheosteoblast-derivedosteoclastparacrineregulator gBC . TheRANK/RANKL/OPGpathwayisalsoimplicitlyencodedinthisparameter.Responseamplitudeandfrequencydependontheinitialconditions,triggeredbyadeviationfromthesteadystate,asseenin Fig.1.5 forperiodical cyclesonly.Parametervaluesaregivenin Table1.2.
Disruptivepathologiestothebonemicroenvironmentwerealsoadded.In Ayatietal.(2010) thepreviousmodelwasextendedtoincorporatetheeffect ofMMinthebonedynamics,aspresentedinEq.(1.18). T ðt Þ representsthe tumorcellsdensityattime t ,withaGompertzformofconstantgrowth γ T . 0,andactsthroughtheautocrineandparacrineregulationspathwaysin theformof rij parameters.Thetumoractionisconsideredindependentofthe bonemass,withapossiblemaximumtumorsizeof LT .Periodicremodeling cyclesarederegulatedandbonemassdensitydecreases.Thebonemass equationisthesameasthatof Eq.(1.17c),remainingasaconsequenceof
