Numerical PDE Analysis of Retinal Neovascularization

139 Pages·2019·3.027 MB·English

by  William E. Schiesser

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
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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 angio-
genesis). 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(circula-
tion)istermedhypoxia.
Theseries ofevents leading toneovascularization ismodeled
with a system of partial differential equations (PDEs). Initially,
three PDEs are formulated with the dependent variables: capil-
lary 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 nu-
trients 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.