Numerical approximation and analysis of mathematical models

209 Pages·2017·3.4 MB·English

by  Monika Twarogowska

Numerical approximation and analysis of mathematical
models arising in cells movement
Monika Twarogowska
To cite this version:
Monika Twarogowska. Numerical approximation and analysis of mathematical models arising in cells
movement. Modeling and Simulation. Università degli studi de l’Aquila, 2012. English. ￿NNT: ￿.
￿tel-00804264￿
HAL Id: tel-00804264
https://theses.hal.science/tel-00804264
Submitted on 25 Mar 2013
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est
archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents
entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non,
lished or not. The documents may come from émanant des établissements d’enseignement et de
teaching and research institutions in France or recherche français ou étrangers, des laboratoires
abroad, or from public or private research centers. publics ou privés.
UNIVERSITA` DEGLISTUDIDELL’AQUILA
DipartimentodiMatematicaPuraeApplicata
DottoratoinMatematica-XXIVCiclo
TesidiDottoratoinMatematica
SettoreScientificoDisciplinare: MAT/05
Numericalapproximationandanalysisofmathematicalmodels
arisingincellsmovement
Relatori Candidato
Prof. RobertoNatalini MonikaTwarogowska
Dr. MagaliRibot
CoordinatoredelcorsodiDottorato
Prof. AnnaDeMasi
ANNOACCADEMICO2010/2011
I hereby declare that this Thesis is my own work and effort. Where other sources of
informationhavebeenused,theyhavebeenacknowledged.
Acknowledgments
Looking back over these three years of the PhD studies I see moments which were full
of joy as well as those tough, when everything seemed to be against me and despondency ap-
peared. Iamverygratefulforallofthem. TheycertainlyshapedmeasapersonIamtoday.
Oneofthisjoyfulmomentsisnow, whenIcanthankallthosewithoutwhomthisthesiswould
notbepossible.
I owe my deepest gratitude to the supervisor of my thesis Roberto Natalini, who took
care of my PhD research at a difficult moment time, after the earthquake in L’Aquila. I am
thankful for his time, ideas and support that were irreplaceable, for introducing me to the
field of numerical analysis and for showing interesting and challenging topics. In addition, I
wouldliketothankforhispatience,guidanceanddedicationinthepreparationofthisthesis.
Throughalltheseyears,thejoyandenthusiasmhehasforMathematicshasbeenmotivational
forme.
I would also like to express my sincere thanks to Magali Ribot, who became my co-advisor.
Her presence and contribution to my research enriched my experience and understanding of
numerical methods and implementation of algorithms. Moreover, her openness, kindness and
supportmadethetimeIspentinFranceunforgettable.
I am heartily thankful to Bruno Rubino for his help, care and encouragement since my
firstdayinL’Aquila. IamreallygratefulforthesupportIhavereceivedfromhimduringthese
yearsandfortheceaselesssensationthatinanydifficultyIcanturntohim. Itissocomforting
whilebeingfarfrommyfamily.
I wish to thank Federica Di Michele, my room mate for two years and companion at
University,foragreatfriendshipandbeingalwaysavailabletohelp,giveadviceorjustlisten.
I am grateful for her understanding and encouragement, for fruitful discussions, for making
timeinL’Aquilamoreinteresting,coffeesmoretastyandrunningmoreenjoyable.
I devote special thanks to my best friends in Poland. Especially to Joanna Kamin´ska and
WeronikaPelc-Garskaforirreplaceableencouragementandcompanyinthemomentsofhome-
sickness.
IwouldalsoliketothankProf. P.MarcatiandMarcoDiFrancescoforintroducingme
tothefieldofmathematicalmodellingandforallthesupportduringmyPhDstudies.
I thank the people in the Laboratoire J. A. Dieudonne´ at Universite´ de Nice-Sophia Antipo-
lis for hospitality and for giving me the possibility to broaden my knowledge in the field of
numerical analysis. In addition, I would like to thank Christophe Berthon for an invitation to
LaboratoiredeMathe´matiquesJeanLerayatUniversite´deNantesandforveryinterestingand
fruitfuldiscussions.
IexpressalsomygratitudetoProf. P.Marcati,Prof. L.C.BerselliandProf. D.Benedettofor
acceptingtobemembersofmythesiscommittee.
My time in L’Aquila was made enjoyable in the large part due to many friends that be-
come a part of my life. I am grateful for time spent with Federica Di Michele, Bruno Rubino,
Donatella Donatelli and Danilo Larivera. It was really a pleasure to meet Vincenzo Caracci-
olo,NicolaDeRossiandallthosewithwhomIspentamazingmomentswhileskiingonCampo
ImperatoreandexploringAbruzzoonbicycle.
This thesis would have not been possible unless a constant support of my family, espe-
ciallymymotherandmysister. Iwouldliketothankfortheirloveandencouragement. Despite
thedistance,theirpresenceinmylifehasbeenalwayssignificantandmomentsspenttogether
unforgeable,andforthatIamgrateful. Ifnottheirhelp,moralsupportandcriticismfromtime
totimearrivingtothepointwhereIamnowisdoubtful.
Monika Twarogowska
TomyMother
andmySister
Contents
Introduction I
1 Mathematicalmodellingofcellsmovement 1
1.1 Biologicalbackground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Macroscopicmodelling: mixturetheory . . . . . . . . . . . . . . . . . . . . . 4
1.3 Parabolicmodelsofchemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Hyperbolicmodelsofchemotaxis . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Asymptotic stability of constant stationary states for a 2 × 2 reaction-diffusion
systemarisingincancermodelling 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Preliminariesandresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.1 Constantstationarystates . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Stabilityofalinearisedmodel . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Proofofthemaintheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Backgroundonnumericalmethods 27
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Finitedifferencesmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2.1 Conservativeschemes . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Highresolutionschemes: theflux-limiterapproach . . . . . . . . . . . 32
3.2.3 θ-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.4 Advancedexplicitschemesbasedoftherelaxationtechnique . . . . . . 34
3.3 Systemsofconservationlaws . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.1 Extensionofscalar3-pointschemestosystems . . . . . . . . . . . . . 38
3.3.2 Isentropicgasdynamics . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.3 Godunovtypeschemes: approximateRiemannsolvers . . . . . . . . . 43
3.4 Approximationofthesourceterm . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.1 Upwindingthesources . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4.2 Well-balancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
I
3.5 Boundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4 Numericalapproximation: parabolicmodelswithnonlineardiffusionandchemo-
taxis 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Numericalschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.1 Splittingschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.2 Explicitscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Porousmediumequation . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Angiogenesismodel . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3.3 Keller-Segeltypemodelforchemotaxiswithnonlineardiffusionofthe
porousmediumtype . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Numericalstudyof1DEulerequationsforisentropicgasdynamicswithdamping 86
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Existingmathematicaltheory. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2.1 Waitingtimephenomenon . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.2 Physicalboundarycondition . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 Numericalschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.3.1 Detectionofthewaitingtime . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Numericalsimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6 Numericalapproximation: somehyperbolicmodelsofchemotaxis 116
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.2 Failureofthestandard,explicitscheme . . . . . . . . . . . . . . . . . . . . . 118
6.2.1 Semilinearmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2.2 Quasilinearmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.3 Well-balancedscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Numericalstudyofsomehyperbolicmodelsofchemotaxisin1D 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Stationarysolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2.1 Analyticresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.2.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Linearvs. Nonlinearpressure . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.4 Parabolicvs. Hyperbolicmodel . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.4.1 LogisticSensitivityfunctionandlinearpressureγ = 1 . . . . . . . . . 171
7.4.2 Constantsensitivityfunctionandquadraticpressureγ = 2 . . . . . . . 173
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Bibliography 178
Introduction
Theaimofthisthesisistoinvestigatesomemathematicalmodelsarisingincellsmove-
ment,frombothanalyticalandnumericalpointofview.
Migrationofcellsinthefibrousenvironmentisanessentialfeatureofnormalandpatho-
logical,biologicalphenomenasuchasembryonicmorphogenesis,woundhealing,angiogenesis
ortumourinvasion. Inthesimplestcasewecanconsideratissueasatridimensionalstructure
consistingoffibres(ExtracellularmatrixECM)andcellsattachedtothem. Manyexperimental
studies(seeforinstance[42])giveagoodinsightintohowthecellsmove. Itisknownthatthey
interactwiththetissuematrixaswellaswithothercellsusingdifferentbiologicalandphysical
mechanisms. Theforcesdrivingthecellsmotionaregeneratedbycontrolledremodellingofthe
actinnetworkwithinthecellasaresponsetomechanicalandchemicalsignals. Themovement
itself is strongly influenced by the spatial and temporal configuration of the fibres. Moreover,
it depends on the contact-guidance phenomenon and attached-detached processes. Also their
behaviour can be altered by the presence of some stimulus giving rise to the so called taxis
movements. Basicallyitmeanstochangethedirection,inwhichthecellmoves,forexample,
accordingtothegradientofsomequantitiessuchasadhesionforces(haptotaxis),electricfield
(galvanotaxis), chemical substances (chemotaxis), oxygen concentration (aerotaxis). In this
thesis we focus our attention to the chemotaxis that is a directed movement of mobile species
towards the lower/higher concentration of a chemical substance in the surrounding environ-
ment.
Since the first description of chemotaxis in bacteria by T.W.Engelmann (1881) and
W.F.Pfeffer (1884), our knowledge about this phenomenon has been continuously expanding.
Changes in the behaviour due to the presence of some stimulus have been observed in many
kinds of organisms and are decisive in biological processes. For example, some bacteria can
movetowardshigherconcentrationoffoodsuchasglucosemolecules. Moreover,theycanag-
gregatewhilestarving. Amoebaelikeorganism,DictysteliumDiscoiduum,secretesthecyclic
Adenosine Monophosphate (cAMP), which attracts other amoebae, leading to the formation
of a multicellular organisms (see [117]). White, blood cells, neutrophil granulocytes, migrate
tothecitesofinflammationfollowingchemokinesreleasedbymacrophagesaftertheencoun-
tering an antigen. During tumour invasion the cancerous cells secrete various growth factors,
which attract and stimulate endothelial cells, in order to form a blood network around the tu-
mour(see[40]).
The movement of cells under the effect of a stimulus has been a widely studied topic
in the last decades by biologists as well as by mathematicians and many models have been
proposed [89], [90]. Of course they differ in complexity, that is for example how many con-
stituentsaretakenintoaccountorwhichphenomenaaretobeconsidered,butthemathematical
frameworkofanymodeldependsonthescaleatwhichwedescribetheproblem. Inparticular,
atthemacroscopiclevelweconsidercellsascontinuummediumanddescribetheirbehaviour
viathepopulationdensity. Thisapproachleadstomodelsbuiltintheformofsystemsofpartial
differential equations of reaction-advection-diffusion type. The most classical model describ-
ingthechemotaxisinthisframeworkisduetoPatlak[96]andsubsequentlytoKellerandSegel
I
[70]. Oneofthemodelsweconsiderinthisthesisisofthistype. Itdescribesthedensityρ(x,t)
ofcellsandtheconcentrationofchemoattractantφ(x,t)anditwritesas

ρ = ∆(P(ρ))−χ∇·(ρ∇φ),
 t
(1)
 φ = D∆φ+aρ−bφ.
t
Thefibrouscharacterofthetissueismodeledbythenonlinearfunction
P(ρ) = εργ, ε > 0,γ > 1. (2)
The positive parameters D,a,b are respectively diffusion coefficient, production and degra-
dation rates of chemoattractant, while χ measures the strength of the response of cells to its
presence. Thismodelhasbeenwidelystudiedintherecentyears. Itisknownthatinthecase
oflineardiffusionthesolutionsexistgloballyintimeinonedimensionandthereisthepossi-
bilityofsomepointwiseblow-upinhigheronesiftheinitialmassisbigenough. Toovercome
this problem, modifications were introduced to prevent from overcrowding and guarantee the
global existence of solutions. For example, it was proposed to use a density dependent diffu-
sion, such as (2), having a volume filling effect (see [95], [23]) or a nonlinear chemosensitive
functionχ = χ(ρ,φ)insteadoftheconstantparameter[99].
In this framework, in Chapter 2 we propose a reaction-diffusion system with a chemo-
taxis term and nutrient-based growth of tumour. The formulation of the model considers also
an influence of tumour and pharmacological factors on the nutrient concentration. It is based
ontheonepresentedin[101]. Denotingbyφthetumourdensityandbycthenutrientconcen-
trationitwritesas

φ = ∇·(φ∇F(φ))−∇·(ωφ∇c)+α(φ,c)
 t
(3)
 c = D(cid:52)c+β(φ,c)
t
with
(cid:16) c (cid:17) (cid:16) c (cid:17)
α(φ,c) = γ1p(cid:98) c∗ −1 φH(cid:98)(φ∗−φ)−γ2p(cid:98) 1− c∗ φ−δφ (4)
β(φ,c) = −dcφ+Gφ−Rc, (5)
whereHˆ(x),pˆ(x)areregularizedapproximationsoftheHeavisideandthe’positivepart’func-
tion(x) respectively. Inthisthesiswestudytheconvergenceofsolutionstoconstantsteady
+
statesinonedimensionalcaseforsmallperturbationsfromtheequilibria. Thenonlinearstabil-
ityresultsareobtainedbymeansoftheclassicalsymmetrizationmethodandenergyestimates.
Macroscopic, parabolic type of models describe very well the aggregation phenomena
butfailwhenthenetworkstructureshavetobereproduced. Thiscanbeobservedwhenstudy-
ing the experiments with human vascular, endothelial cells. Randomly seeded on the plain
gel substratum, these cells migrate and aggregate to finally form a complex network seen as
the beginning of vascularization, that is de novo formation of blood capillaries. The Patlak-
Keller-Segelmodelcannotexplainthisprocessaswellascannotdescribethe’runandtumble’
movement of, for example, Eusterichia Coli. This is because it is mainly directed to the long
II