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N U M E R I C A L P D E A N A LY S I S O F R E T I N A L N E O VA S C U L A R I Z AT I O N N U M E R I C A L P D E A N A LY S I S O F R E T I N A L N E O VA S C U L A R I Z AT I O N Mathematical Model Computer Implementation in R WILLIAM E. SCHIESSER Lehigh University Bethlehem, PA, United States AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom Copyright©2019ElsevierInc.Allrightsreserved. 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Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusingany information,methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethodstheyshouldbe mindfuloftheirownsafetyandthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliabilityforany injuryand/ordamagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuseor operationofanymethods,products,instructions,orideascontainedinthematerialherein. LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-818452-3 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:MaraConner AcquisitionEditor:ChrisKatsaropoulos EditorialProjectManager:FernandaOliveira ProductionProjectManager:SujathaThirugnanaSambandam Designer:MilesHitchen TypesetbyVTeX CONTENTS v CONTENTS Preface ...................................................... vii Chapter 1 PDE Model Formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction................................................... 1 1.2 Model Specification ........................................... 3 1.2.1 Capillary Tip Density .................................... 4 1.2.2 Blood Capillary Density.................................. 5 1.2.3 VEGF Concentration..................................... 6 Acknowledgment.............................................. 8 References.................................................... 8 Chapter 2 Model Implementation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Introduction.................................................. 11 2.2 Method of Lines Routines..................................... 11 2.2.1 Main Program ......................................... 11 2.2.2 ODE/MOL Routine...................................... 18 2.3 Model Output ................................................ 26 2.4 Summary and Conclusions.................................... 31 References................................................... 31 Chapter 3 Variation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Introduction.................................................. 33 3.2 MOL Analysis ................................................ 33 3.2.1 Main Program, Source Routine.......................... 33 3.2.2 Model Output.......................................... 36 3.3 Summary and Conclusions.................................... 41 References................................................... 41 Chapter 4 Detailed PDE Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Introduction.................................................. 43 4.2 Analysis of the t Derivatives................................... 43 4.2.1 Main Program ......................................... 43 vi CONTENTS 4.2.2 ODE/MOL Routine...................................... 49 4.2.3 Model Output.......................................... 52 4.3 Analysis of PDE RHS Terms ................................... 58 4.3.1 Main Program ......................................... 59 4.4 Summary and Conclusions.................................... 70 Chapter 5 Oxygen Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.1 Introduction.................................................. 71 5.2 Four PDE Model.............................................. 71 5.2.1 Main Program ......................................... 72 5.2.2 ODE/MOL Routine...................................... 79 5.2.3 Model Output, Uncoupled PDEs......................... 86 5.2.4 Model Output, Coupled PDEs ........................... 93 5.3 Summary and Conclusions................................... 100 References.................................................. 101 Chapter 6 Anti-VEGF Drug Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.1 Introduction................................................. 103 6.2 PDE Model.................................................. 103 6.2.1 Main Program ........................................ 104 6.2.2 ODE/MOL Routine..................................... 112 6.2.3 Subordinate Routines ................................. 115 6.2.4 Model Output......................................... 118 6.3 Summary and Conclusions................................... 124 Appendix A1 Functions dss004, dss044. . . . . . . . . . . . . . . . . . . . . . . . . . . 127 A1.1 Function dss004............................................. 127 A1.2 Function dss044............................................. 129 References.................................................. 131 Index....................................................... 133 PREFACE vii PREFACE Theintentofthisbookistoprovideamethodologyfortheanalysis of retinal neovascularization. This pathological condition devel- opsasaconsequenceofinadequatebloodflowtotheeyesothat theoxygenavailabletotheretinaisinadequate. Iftheretinalbloodsupplydeclines,possiblywithageorfroma visual(ocular)disease,theeyetypicallyrespondswiththegrowth of additional capillaries (neovascularization, also termed angiogenesis). The additional blood vessels can develop in the retina and eventually interfere with the phototransduction (conversion oflighttoelectricalsignals),leadingtoimpairedvisionfrom,for example,age-related maculardegeneration(AMD),andpossibly blindness.Insufficientoxygenfromadequatebloodflow(circulation)istermedhypoxia. Theseries ofevents leading toneovascularization ismodeled with a system of partial differential equations (PDEs). Initially, three PDEs are formulated with the dependent variables: capillary tipdensity,bloodcapillary density,andvascular endothelial growthfactor(VEGF)concentration.TheVEGFformsinresponse tohypoxia,andthenleadstocapillarytipandbloodcapillaryfor- mation.Laterinthebook,afourthPDEisaddedforoxygencon- centration. The systems of PDEs are expressed in 1D Cartesian coordi- nates:(x,y,z)reducedtox.Thespatialindependentvariablex is thedistancealongtheretina.ThePDEsaredynamicandinclude variationofthePDEdependentvariableswithtimet. ThePDEmodelswithindependentvariables(x,t)areapprox- imatedbythemethodoflines(MOL),aprocedureforconverting PDEs to approximating ordinary differential equations (ODEs). The latter can then be integrated (solved) numerically with a li- braryinitial-valueODEintegrator. Foreachexampleapplication,themodelPDEsarestatedfirst, includingtheinitialconditions(ICs),boundaryconditions(BCs), andmodelparameters.Thecoding(programming)oftheapplica- tionisthendiscussedintermsofdocumentedRroutinesthatare explainedindetail,andareavailablefromadownloadsothatthe reader/analyst/researchercanusethemtoconfirmthesolutions presentedinthebook,thenextendtheroutinestoincludeaddi- tional details and effects that might be of interest. For example, thebookconcludeswithananalysisofanti-VEGFtherapyimple- mentedwithatermincludedintheVEGFPDEthatdecreasesthe VEGFconcentration. viii PREFACE The author would welcome comments about the reported methodologyandhowitmightbeapplied,andpossiblyextended, foranenhancedquantitativeunderstandingofretinalneovascu- larization. W.E.Schiesser Bethlehem,PA,UnitedStates June1,2019 1 PDE MODEL FORMULATION 1.1 Introduction Thisbookdetailsamathematicalmodel(systemofequations) fortheevolutionofretinalneovascularization.Toexplainsomeof theseterms: • Theretinaisthepartoftheeyethatreceiveslightandconverts itintoanelectricalsignal,aprocesstermedphototransduction orsimplytransduction.Theelectricalsignalthentravelsalong theopticnervetothebrainwherethesenseofvisionoriginates. • This remarkable series of events which takes place on a time scale of milliseconds (ms) provides a sense of immediate (in- stantaneous) vision. However, other events can occur on a muchlongertimescale;hereweconsiderseveralmonths. • The energy to power this vision comes from oxygen and nutrients that metabolize (react). The oxygen and nutrients are provided by blood flowing through fine capillaries in the eye (thevasculature).Thus,thefunctioningoftheeyeisdependent onanadequatebloodsupply. • Ifthebloodsupplydeclines,possiblywithageorfromavisual (ocular)disease,theeyetypicallyrespondswiththegrowthof additional capillaries (vascularization). The additional blood vesselscandevelopintheretinaandeventuallyinterferewith the transduction, leading to impaired vision and blindness. Insufficient oxygen from adequate blood flow (circulation) is termedhypoxia. • The vascularization proceeds through the production of sig- nalingproteinsgenericallytermedvascularendothelialgrowth factor(VEGF).Endothelial referstocellsthatlinetheinternal wallsofthecapillaries.Sincetheadditionalcapillariesarenot anormalpartoftheretinaandotherinternalstructureofthe eye, the process of their growth is termed neovascularization (lessfrequently,angiogenesis). • An established treatment (therapy) for neovascularization is the injection of various forms of anti-VEGF drugs into the eye(aninformativeintroductorydiscussionofthisprocedure is given in [6]). The time of effectiveness for this therapy is about one month, so repeated injections are generally used, for at least three months with observation of the patient re- sponse. Impairment of vision can include prenatal neovascu- 1 NumericalPDEAnalysisofRetinalNeovascularization https://doi.org/10.1016/B978-0-12-818452-3.00006-6 Copyright©2019ElsevierInc.Allrightsreserved.