Modeling Glioma Growth and Personalizing Growth Models in Medical Images

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Models in Medical Images

Ender Konukoglu

To cite this version:

Ender Konukoglu. Modeling Glioma Growth and Personalizing Growth Models in Medical Images.

Human-Computer Interaction [cs.HC]. Université Nice Sophia Antipolis, 2009.

English.

�NNT :

2009NICE4000�. �tel-00633697�

UNIVERSITY OF NICE - SOPHIA ANTIPOLIS

GRADUATE SCHOOL STIC

INFORMATION AND COMMUNICATION TECHNOLOGIES AND SCIENCES

T H E S I S

to fulfill the requirements for the degree of

Ph.D. in Computer Science

from the University of Nice - Sophia Antipolis

Specialized in : C

ONTROL

, S

IGNAL AND

I

MAGE

P

ROCESSING

presented by

Ender K

ONUKO ˘

GLU

Modeling Glioma Growth and

Personalizing Growth Models in Medical

Images

Thesis supervised by Nicholas A

YACHE

prepared at the INRIA Sophia Antipolis, A

SCLEPIOS

Project

defended the February 17, 2009.

Jury :

M. Nicholas A

YACHE

,

Ph.D.

-

Director

M. Pierre-Yves B

ONDIAU

,

M.D./Ph.D.

-

Invited member

M. Olivier C

LATZ

,

Ph.D.

-

Examiner

M. Hervé D

ELINGETTE

,

Ph.D.

-

Examiner

M. Guido G

ERIG

,

Professor

-

Reviewer

M. Emmanuel M

ANDONNET

,

M.D./Ph.D.

-

Invited member

M. Gábor J. S

ZÉKELY

,

Professor

-

Reviewer

UNIVERSITÉ DE NICE - SOPHIA ANTIPOLIS

ECOLE DOCTORALE STIC

SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA

COMMUNICATION

T H È S E

pour obtenir le titre de

Docteur en Sciences

de l’Université de Nice - Sophia Antipolis

Mention : I

NFORMATIQUE

présentée par

Ender K

ONUKO ˘

GLU

Modélisation de la Croissance des

Gliomes et Personnalisation des

Modéles de Croissance à l’aide

d’Images Médicales

Thèse dirigée par Nicholas A

YACHE

préparée à l’INRIA Sophia Antipolis, Projet A

SCLEPIOS

soutenue le 17 Février 2009.

Jury :

M. Nicholas A

YACHE

,

Ph.D.

-

Directeur

M. Pierre-Yves B

ONDIAU

,

M.D./Ph.D.

-

Invité

M. Olivier C

LATZ

,

Ph.D.

-

Examinateur

M. Hervé D

ELINGETTE

,

Ph.D.

-

Examinateur

M. Guido G

ERIG

,

Professor

-

Rapporteur

M. Emmanuel M

ANDONNET

,

M.D./Ph.D.

-

Invité

M. Gábor J. S

ZÉKELY

,

Professor

-

Rapparteur

First of all I would like to thank Ni holas Aya he, for supervising my resear h

and for giving me the opportunity to work in his group for the last 3 and a half

years. HehasbeenalwaystheresupportingmyworkevenattimesIdoubtedmyself.

IthankOlivier Clatzforbeingthereallthewayduring thisthesis. He hasbeen

a olleague, a o-advisor and most of all a friend giving me advi e, dire tion and

ourage. Thisthesisis mostlyaresult ofour day-longdis ussions.

I would like to thank Hervé Delingette and Maxime Sermesant for all their

support and help from whi h I beneted a lot during my resear h. They have

beared my bursts into their o es whi h ended in dirty and unintillegible white

boardsmost ofthetime.

Itakethisopportunitytoexpressmygratitudetothemembersofthe ommittee:

1. GuidoGerigandGáborSzékelyfora eptingtobethereviewersofthisthesis.

Thankyoufor thetimeandattentionyouhavedevotedtothis thesisandalso

for the ommentsand thevery en ouraging ompliments.

2. William Wellsfor a epting to presidethe jury and also for showing interest

in this work both during and at the end of the thesis. Thank you for your

ompliments, valuable advi esand most ofall for your help.

3. Pierre-Yves Bondiau and Emmanuel Mandonnet for being a part of the jury

andmoreimportantlyfordemonstratinginterestinthiswork. Theyhavebeen

very onstru tive throughout thisresear h. Thank you foryour supportboth

interms of knowledge and images, whi h madethis thesis possible, and also

intermsof en ouragement.

I should not forgetto thank BjoernMenzewhose help added immense valueto

this thesis. Thank you for supporting theideas proposed inthis thesis, for sharing

an important database, for sharing knowledge and for devoting your time. I also

thank Mar -André Weber and Bram Stieltjes from the German Can er Resear h

Center (DKFZ) forsharing theirdatabases andtheir lini alknowledge.

I would also like to take this opportunity to thank Kilian M. Pohl who has

helpedmeinallaspe tsofthePhD.Thankyoufordevotingyourtimeandtea hing

meboth about s ien e andabout life.

The ones who have the biggest role in all what I have done are undoubtedly

my family. I thank my parents for providing me all the possibilities. I thank my

brother Ali Emre Konuko§lu for all his support and friendship. I thank my un le

Iwould like to thank two spe ialpeopleJean-Mar Peyrat and Heike Hufnagel

for being there listeningto mewhen I neededto talk, sharing this di ult period,

making the PhD experien e more fun and taking part in my life. Thank you my

friends.

Lastly and mostof all I wouldlike express mygratitude to my friendsand

ol-leagueswithoutwhomIwouldnotbeabletonishthisthesis. ThankyouTommaso

Mansi, Erik Pernod, François Hebert, Ni olas Toussaint, Pierre Fillard, Romain

Fernandez,Jean-ChristopheSouplet,StanleyDurrleman,DanielBarbeau,Floren e

Billet, Radu Stephanes u, Guillaume Dugas-Pho ion, Jimena Costa, Olivier

Com-mowi k, Aurélie Canale, Floren e Dru, Melek Önen, Cedri Roux, Engin Krda,

Leyla Bilge,Grégoire Malandain, XavierPenne andIsabelleStrobant. Iapologize

Mathemati al models and more spe i ally rea tion-diusion based models have

been widely used in the literature for modeling the growth of brain gliomas and

tumorsingeneral. Besides thevastamount ofresear hfo usedon mi ros opi and

biologi al experiments, re ently models have started integrating medi al images

in their formulations. By in luding the geometry of the brain and the tumor, the

dierent tissuestru turesand thediusionimages, modelsareable tosimulate the

ma ros opi growth observable inthe images. Although generi models have been

proposed, methods for adapting these models to individual patient images remain

an unexploredarea.

In this thesis we address the problem of personalizing mathemati al tumor

growth models. We fo us on rea tion-diusion models and their appli ations

on modeling the growth of brain gliomas. As a rst step, we propose a method

for automati identi ation of patient-spe i model parameters from series of

medi al images. Observing the dis repan ies between the visualization of gliomas

in MR images and the rea tion-diusion models, we derive a novel formulation

for explaining the evolution of the tumor delineation. This modied anisotropi

Eikonal model is later used for estimating the model parameters from images.

Thoroughanalysisonsyntheti dataset validatestheproposedmethodtheoreti ally

and also gives us insights on the nature of the underlying problem. Preliminary

resultsonreal asesshowpromising potentialsof theparameter estimationmethod

and the rea tion-diusion models both for quantifying tumor growth and also for

predi ting futureevolutionofthepathology.

Following the personalization, we fo us on the lini al appli ation of su h

patient-spe i models. Spe i ally,we ta kle theproblem of limitedvisualization

ofgliomainltrationinMRimages. Theimagesonlyshowapartofthetumorand

maskthe lowdensityinvasion. Thismissinginformation is ru ialfor radiotherapy

and othertypesof treatment. We proposea formulation for this problembasedon

the patient-spe i models. In the analysis we also show the potential benets of

su h the proposedmethodfor radiotherapy planning.

Thelastpartofthisthesisdealswithnumeri al methodsforanisotropi Eikonal

equations. This type of equation arises in both of the previous parts of this

the-sis. Moreover, su h equations are also used in dierent modeling problems,

om-puter vision,geometri al opti sand other dierent elds. We proposea numeri al

methodforsolvinganisotropi Eikonalequationsinafastanda urate manner. By

omparing itwitha state-of-the-art methodwe demonstrate theadvantages of our

Lesmodèles mathématiquesetplus spé iquementles modèlesbasés surl'équation

de réa tion-diusionont étéutiliséslargement danslalittérature pour modéliser la

roissan e desgliomes érébraux etdes tumeurs en général. Deplus la grande

lit-tératuredere her hequi on entresurlesexpérien esbiologiquesetmi ros opiques,

ré emment les modèles ont ommen é intégrer l'imagerie médi ale dans ses

formu-lations. In luant la géométrie du erveau et elle de la tumeur, les stru tures des

diérentes tissues et la dire tion de diusion, ils ont montré qu'il est possible de

simuler la roissan ede la tumeur omme 'estobservé dansles images médi ales.

Bien que des modèles génériques ont été proposés, les méthodes pour adapter es

modèles auximages d'unpatient reste un domaine inexploré.

Dans ettethèse nousnousadressonsauproblèmedepersonnalisationde

mod-èlemathématiquedela roissan edetumeurs. Nousnousfo alisonssurlesmodèles

de réa tion-diusion et leurs appli ations sur la roissan e des gliomes érébrales.

Danslapremière étape, nousproposonsune méthode pour l'identi ation

automa-tiquedes paramètres patient-spé iques du modèleà partir d'une série d'images.

En observant la divergen e entre la visualisation des gliomes dansles IRMs et les

modèles réa tion-diusion,nousdéduisonsune nouvelle formulation pourexpliquer

l'évolution de la délinéation de latumeur. Ce modèle Eikonal anistropique

modi-é estutilisé plustardpourl'estimation desparamètres àpartir desimages. Nous

avonsthéoriquement analysé laméthode proposéeà l'aided'unbasedonne

synthé-tiqueetnousavonsmontréla apa itédelaméthodeetaussisalimitation. Enplus,

les résultatspréliminaires, sur les asréels montrent des potentiels prometteursde

laméthode d'estimation desparamètres etdu modèlede réa tion-diusion pour la

quanti ation de la roissan ede tumeur etaussi pour laprédi tion de l'évolution

futurde latumeur.

En suivant lapersonnalisation, nousnous on entrons surles appli ations

lin-iques des modèles patient-spé iques. Spé iquement, nous nous attaquons au

problème de la visualisation limitée d'inltration de gliome dans l'IRM. En eet,

les imagesnemontrent qu'une partie delatumeur et masquent l'inltration

basse-densité. Cetteinformation absente est ru iale pour la radiothérapie etaussipour

d'autre type de traitements. Dans e travail, nous proposons pour e problème

une formulationbaséesurles modèles patient-spé iques. Dansl'analyse de ette

méthode nous montrons également les béné es potentiels pour la plani ation de

laradiothérapie.

La dernière étape de ette thèse se on entre sur les méthodes numériques de

l'équation Eikonal anisotropique. Ce type d'équation est utilisé dans beau oup

de problèmesdiérentstelquelamodélisation,letraitement d'image, lavision par

ordinateur et l'optique géométrique. I i nous proposons une méthode numérique

rapide ete a e pour résoudrel'équation Eikonal anisotropique. En la omparant

ave uneautreméthodeétat-de-l'artnousdémontrons lesavantagesdelate hnique

1 Introdu tion 1

1.1 Context . . . 1

1.2 Problems Investigated . . . 2

1.3 Organizationof theThesis . . . 3

2 Brain Tumorsand Medi al Images 7 2.1 Brain Tumors . . . 7

2.2 Gliomas(Astro ytomas) . . . 9

2.3 Imaging GliomaswithMagneti Resonan e . . . 10

2.3.1 Magneti Resonan e Imaging (MRI) . . . 10

2.3.2 Appearan eof GliomasinMRI . . . 12

3 Literature Review 17 3.1 Introdu tion . . . 17

3.2 Classi ation . . . 19

3.3 Mi ros opi Models. . . 20

3.3.1 Avas ularGrowth/Solid Tumor . . . 21

3.4 Ma ros opi Models . . . 31

3.4.1 Diusive Models . . . 31

3.4.2 Me hani al Models . . . 34

3.5 ImageGuided Tools for Therapy Planning . . . 37

3.6 Appli ationsto Registration andSegmentation . . . 38

3.6.1 Registration . . . 39

3.6.2 Segmentation . . . 42

3.7 Dis ussions . . . 43

4 Parameter Estimation: Method 45 4.1 Introdu tion . . . 46

4.2 Method . . . 47

4.2.1 EikonalApproximationfor Rea tion-Diusion Models . . . . 48

4.2.2 TheParameter EstimationProblem . . . 62

4.3 Theoverallalgorithm. . . 65

5 Parameter Estimation: Results 67 5.1 Resultsfor Syntheti Tumors . . . 67

5.1.1 Comparing Traveling Time withRea tion-Diusion . . . 68

5.1.2 Problemof Non-Uniqueness . . . 69

5.1.3 Fixing

ρ

andthe3 Parameter Case . . . 70

5.1.4 Changingthe xed

ρ

and Speedof Growth . . . 76

5.1.5 Redu ing theNumberof Images Used . . . 77

5.1.6 Forgetting theConvergen e Ee t and

T

0

. . . 78

5.2 PreliminaryResults withRealCases . . . 82

5.2.1 Fitting the Observed Evolution . . . 84

5.2.2 Predi ting Future Evolution Beyond Observed Image Data. . 86

5.3 Con lusions . . . 89

6 Extrapolating Invasion: Method 91 6.1 Introdu tion . . . 91

6.2 Method . . . 93

6.2.1 TumorCell DensityExtrapolation . . . 95

6.2.2 In luding Ee tsof theBoundaryCondition. . . 103

7 Extrapolating Invasion: Results 111 7.1 Experiments . . . 111

7.2 Assessing theEstimation Quality . . . 112

7.3 Comparing Irradiation Margins . . . 118

7.4 Con lusion. . . 124

8 Anisotropi Fast Mar hing 127 8.1 Introdu tion . . . 127

8.2 Method . . . 129

8.2.1 Basi Con epts . . . 129

8.2.2 FastMar hingMethods . . . 131

8.2.3 Re ursive Anisotropi Fast Mar hing . . . 133

8.3 Experiments . . . 139

8.4 Con lusions . . . 141

9 Con lusions and Perspe tives 143 9.1 Con lusions . . . 143

9.1.1 Parameter Estimation . . . 143

9.1.2 Extrapolating InvasionMargins . . . 145

9.1.3 Anisotropi FastMar hing. . . 146

9.1.4 Other Contributions . . . 146

9.2 Perspe tives . . . 147

9.2.1 Te hni al Improvements . . . 147

9.2.2 Appli ation to Clini al Images . . . 148

9.2.3 Validation . . . 149

9.2.4 Future . . . 150

A Hamilton-Ja obi Equations 151 B The Minimization Algorithm 155 C Parameter Estimation Results: Real Cases 159 C.1 Fitting theObserved Evolution: Additional Images . . . 159

Introdu tion

Contents

1.1 Context . . . 1

1.2 Problems Investigated . . . 2

1.3 Organization ofthe Thesis . . . 3

1.1 Context

How an we des ribe theprogression of tumors throughmathemati al models and

omputersimulations? Thisquestionhasbeenkeepings ientistsbusyforthelast30

years. Mathemati ians, lini ians, biologists,physi istsand omputers ientists are

ollaboratingtota klethisproblem. Consideringthe omplexityofthedynami sof

tumor growth and the fa t that most of the underlying phenomena have not been

dis overed yet, these attempts will ontinue for a while. This thesis is a humble

ontribution towardsthese goals.

Can erisoneoftheleading ausesofdeathanditisnotne essarytodes ribeits

graveness. Theimportantpointtonoteisthatitisbe omingmore ommonandwe

have not yettotally understood thereasons for its o urren e and theway to ure

it. Vastamount of experimental resear h inbiology and medi ine enlightens many

dierentaspe tsofthedynami sof an erprogression. Theyprovideinformationin

many dierent s ales fromgeneti s to tissue. Mathemati al modeling is important

inthisrespe tasitprovidesameltingpotforalltheseexperimentalresults. Models

provide a systemati stru ture that brings these results together and shows us the

overallpi ture. Thisgivesustheopportunitytobetterunderstandthetumorgrowth

and intera tion between dierent fa tors and also to help lini ians in diagnosing

and treatingtumors.

Mathemati al modeling of tumor growth has re eived onsiderable attention

duringthe last30years. Dierentmodeling attemptshavebeenproposedspanning

alargerangeofs alesandte hniquesdes ribingdierentdynami sandphenomenon

in thegrowth pro ess. Although lots of eorts have been given to formulate more

realisti and detailed models, little attention has been given to theappli ability of

thesemodelsto lini aldataandtheirpersonalization. Asthemodelsbe omemore

sophisti atedthegapbetweentheinformation lini allyavailableandneededbythe

models widen. As a result adapting mathemati al models to patient data be ome

harder.

The motivation of this thesis is thereforeto study thelink between

lini al settings. Instead ofgoing one step further indetailing existing models and

making them morerealisti , we take a stepba kward and sear hfor ways to apply

these models to spe i patient datausing images. Inthis manner we take a more

pragmati approa hto tumor growth modeling.

Although tumors in dierent parts of the body have ertain ommon

hara -teristi s they also dier in many ways. Therefore, ea h tumor should be studied

separately. Inthisthesiswefo usonthemodelingofaspe i typeofbraintumor,

gliomas. Inattempttolinkmedi alimagesandthegrowthmodels,westartfroman

existing modelexplainingthe growth of gliomas,whi h isbased on thewell known

Rea tion-Diusion(RD)equations,andstudyitslinkwiththeavailableinformation

inMagneti Resonan e Images (MRI).

1.2 Problems Investigated

Inthe previousse tionwehavesetthegeneral motivationofthisthesisasstudying

thelinkbetween tumorgrowth modelsand medi alimages. Thisisavery omplex

problemwithdierent omponentssu hastheoreti alanalysisofthemodels,physi s

of the image a quisition and biologi al analysis of the tissue response to tumor

growth. Of ourse,thisthesisdoesnotaimtoprovidesolutionstoalltheseproblems.

Itisratherintendedtobeapartofa ollaborativeworkta klingallthesementioned

omponents. In this thesis we fo us on thetheoreti al analysis of a type of tumor

growth model, whi h is based on rea tion-diusion equations. In this respe t we

fo uson threedierent problems:

•

Image Guided Personalization of Rea tion-Diusion Type Tumor

Growth Models: The rst problem we ta kle is adapting the

rea tion-diusion tumor growth model to spe i patient images, personalizing the

model. Thisadaptation an also be formulated asestimating theparameters

of the rea tion-diusiontumor growth modelusingtimeseriesof medi al

im-ages taken from the same patient. So the exa t question we try to solve is:

How to estimate these patient-spe i parameters that would best explain the

progression of the tumor observed in the images? How to reate the

patient-spe i model?

•

Extrapolating Extents of Glioma Invasion in MRI: Medi al images

are one of the main sour e of information in diagnosing and treating brain

tumors. Espe iallyinradiotherapy,imagesare ru ialinplanningthetherapy

and outlining thearea whi h will be irradiated. The images however, annot

show the whole extent of gliomas due to the invasive nature of this type of

tumor. The extent of the whole tumor goes beyond the visible part in the

image andthe possible dire tionofthisundete table extensionis important

inoutlining theirradiationarea. These ond question weta kle inthis thesis

is: How an weextrapolate this undete table extension fromthe visible part of

the tumor inthe image usingpatient-spe i models?

tions wederived to solve thersttwo questionsended upto have theformof

modiedanisotropi Eikonal equations. Moreover,after reviewing othertype

of models for dierent organs and pathologies we realized the importan e of

thistypeofequations. Therefore,thethirdquestionweaskisamore

method-ologi al question: How to solve anisotropi Eikonal equations in a fast and

a urate manner?

1.3 Organization of the Thesis

This thesis is organized around the three questions explained in the previous

se tion. We rststart byproviding general information about gliomas andmedi al

images followed by ba kground information on tumor growth modeling. After

the ba kground we present our work on the three main questions making up the

ontributions of this thesis. The detailed des ription of the material overed in

ea h hapter isgiven below.

Chapter 2 gives some general knowledge on brain tumors and more spe i ally

on gliomas. Dierent types of gliomas, the grading onventions and dierent

behavior of these tumors are explained briey. We also give some information

about the appearan e ofgliomas in MRIasthis is ru ial for the understandingof

theremainder ofthethesis.

Chapter 3 provides an overview of the literature on tumor growth modeling.

In this hapter we do not distinguish between brain tumors and tumors in the

other parts of the body as the modeling attempts are linked together. The main

approa hesofmodeling,dierents alesofmodels,dierentte hniquesanddierent

phenomena modeled are overed in this hapter. We dis uss briey about models

fo using on mi ros opi dynami s and models working with information oming

from medi al images. In this hapter we also give a review of dierent image

analysis te hniques whi h use tumor growth modeling to ta kle dierent problems

su h assegmentation andregistration.

Chapter 4 explains our approa h to the problem of personalizing the

rea tion-diusion type tumor growth models. In this hapter we fo us on the dis repan y

betweenthe informationrequiredbytherea tion-diusion modelsandthe

informa-tion available inmedi al images. Rea tion-diusion models des ribe theevolution

of tumor ell density distributions however, in medi al images we only observe

boundaries between the enhan ed/unenhan ed tumoral region and the healthy

tissue. In order to solve this dis repan y, through asymptoti approximations we

derive a formulation whi h des ribes the evolution of tumor delineations in the

images based on the dynami s of rea tion-diusion growth models. Using this

more onsistent mathemati al des ription, we formulate the parameter estimation

problem for rea tion-diusion type tumor growth models using time series of

Chapter 5 analyzes the parameter estimation methodology presentedin Chapter

4. We present experimental results on syntheti and real images. Through

syntheti ally reated data sets we perform theoreti al analysis of the method and

show the feasibility of the parameter estimation problem under the onstraint of

medi al images, spe i ally we show the non-uniqueness of the solution of the

most general ase. We also show that under ertain assumptions the parameter

estimation problem an be solved and ertain values unique to ea h tumor an

be extra ted from medi al images, su h as the speed of progression. Following

this analysis, on real data we present promising results showing the ability of

the method in nding the set of parameters whi h well des ribes the evolution of

the tumor observed in MR images. Moreover, we demonstrate the power of the

estimated parameters and therea tion-diusion models (or rather the formulation

derived from the RD models) in predi ting the future evolution of the tumor in

images.

Chapter 6 explains the problem of limited visualization of gliomas in medi al

images. In this hapter we propose a solution to this problem based on dynami s

des ribed by the rea tion-diusion models. Again through asymptoti appro

xima-tions we derive an extrapolation formulation whi h starting from the visible part

of the glioma in the MR image extrapolates the possible extents of the glioma

undete table in the image. In other words the proposed method onstru ts the

tumor ell densitydistribution beyondthevisible massintheimage.

Chapter 7 presents the syntheti experiments we have performed to test the

extrapolationmethoddes ribedinChapter6. Werstanalyzethemethodto seeif

theextrapolated invasionextent mat hes thea tual tumor ell densitydistribution

of a syntheti ally grown tumor. After verifying this we turn our attention to the

planningofradiotherapy. Wefo usonthephaseofoutliningtheirradiationmargins

starting from the tumor delineation in the image. In onventional radiotherapy a

onstant margin of 1.5-2 m is outlined around the tumor delineation to a ount

for theundete table extent of the glioma. Inthis hapter we showthat a variable

margin onstru ted a ording to the possible extent of the glioma, theoreti ally,

may better target thetumor andharm lesshealthy braintissue.

Chapter 8 fo uses on the numeri al solutions of a type of partial dierential

equation, the anisotropi Eikonal equations. This type of equations arise in the

rsttwoproblems wepresentedinChapters 4and6. Moreover, anisotropi Eikonal

equations also arise inthe modeling of dierent organs and pathologies, espe ially

in ardiovas ular and wound healing models. On the other hand, these equations

are not inherent to biologi al/physiologi al modeling, they also arise in dierent

elds su h asgeophysi s and omputervision. Therefore,fast anda urate solvers

for su hequationsareimportant fordierentdomains. Inthis hapterweproposea

numeri al method for anisotropi Eikonal equations whi h extends thewell known

FastMar hing method to work in anisotropi domains. We detailour method and

Chapter 9 on ludesthis thesisbygoing overthe ontributions we have proposed

inea h hapter and providing theperspe tivesfor thefuture work.

Appendix A gives a brief overview on Hamilton-Ja obi equations whi h are

extensively used in this thesis. This overview is by no means omplete and it

just aims to introdu e this topi oarsely to readers who are not familiar with it.

Hamilton-Ja obi equations is a wide lass of partial dierential equations and the

emphasisinthisappendixisgiventothetypeofequationsmentioned inthisthesis.

Appendix B givesthe algorithmi details on theminimization algorithm used in

Chapter 4. This algorithm is proposed by Powell in [Powell 2002 ℄ and the review

in this Appendix goes over the basi steps of the method for ompleteness of the

Brain Tumors and Medi al Images

Contents

2.1 Brain Tumors. . . 7

2.2 Gliomas(Astro ytomas) . . . 9

2.3 Imaging Gliomaswith Magneti Resonan e . . . 10

2.3.1 Magneti Resonan eImaging(MRI) . . . 10

2.3.2 Appearan e ofGliomasinMRI . . . 12

Context

The te hniques and methods presented in this work deal with the mathemati al

modelingofbraintumorsandtheuseofmagneti resonan eimagesforthispurpose.

Therefore, some knowledge about brain tumors, magneti resonan e images and

appearan eof tumorsinthese imagesisne essary. In this hapter we provide brief

information about brain tumors in Se tion 2.1 and spe i ally brain gliomas in

Se tion 2.2 sin e they are the main fo usof this thesis. In Se tion 2.3 we des ribe

shortly the magneti resonan e imaging and the appearan e of brain gliomas on

these images. Detailed information about any of these three topi s is outside the

s ope of this work. Please refer to [Wilson1999 ℄ for information on gliomas and

braintumorsandreferto [Westbrook 1998 ,Liang2000 ℄for detailedinformationon

magneti resonan eimaging.

2.1 Brain Tumors

Thetermtumor originallymeansabnormalswellingoftheeshandisderivedfrom

theLatinwordtumor whi hmeansswelling. Inthe urrentusetumormeansalesion

whi h is formed by abnormal growth and un ontrolled rapid ellular proliferation

thatpossessesnofun tion,aneoplasm. Atumorthatislo atedinthebrainis alled

a brain tumor. Brain tumors are not very ommon pathologies, urrent statisti s

indi ates around100 in iden es peryearin 100,000people inthedeveloped world.

Inthe aseof hildrenthe rateofin iden e isevenlower,4.5in iden es in100,000.

Even though these values are not very high, brain tumors are among the leading

ause of an er-related deathfor all ages[DeAngelis 2001 ,ABTA 2008a℄.

Brain tumors an be oarselydivided into groups using two dierent

lassi a-tions, a ordingto thedegree of their aggressiveness and a ordingto their origin.

Benign brain tumorsproliferates slowly and they rarely spread to the surrounding

tissue. They would have a normal appearan e under the mi ros ope and globally

they would showdistin t bordersbetween thetumor and thebrain tissue. Most of

thetime, ifit an bedone,thetumor an betotally removed bysurgery. Although

most of these tumors are not life threatening, they may be so depending on their

size and their lo ation in the brain. Malignant brain tumors on the other hand,

proliferate rapidlyandinvade thehealthy areasofthebrain. Theirbordersarenot

lear due to their inltrative nature making surgi al removal di ult. Moreover,

they show neoangiogenesis and ne rosis. Their ells an travel and olonize other

partsofthebrainandthespinal ordthrough erebrospinaluid. Thesetumorsare

life-threatening witha lowsurvivalrate[Tovi 1993 , Wilson1999, DeAngelis2001 ℄.

Thedistin tion between benignand malignant brain tumorsisnot obvious and

requiresthe denitionofasetof riteria andgrading systems. Inorderto fa ilitate

the diagnosis and the therapy planning, tumors are graded based on their

aggres-siveness. Themost ommonlyusedgradingsystemistheoneproposedbytheWorld

HealthOrganization(WHO).Gradingofatumortakesintoa ountdierentfa tors

su hasmitoti index,vas ularity,presen eofane roti ore,invasivepotential and

similarityto normal ells. In the WHOsystem4 grades areused to lassify tumor

whi h aresummarized inthe Table2.1

Table2.1: WHOTumor Gradesand Chara teristi s

Grade Chara teristi s

Grade I - slow proliferation - ells look like normal - long survival

rate - e.g.. pilo yti astro ytomas

GradeII - relatively slowproliferation - ells looklike almost normal

- mayinvade - mayre ur asgradeIIor ahigher grade

Grade III - rapidly reprodu ing - ells look abnormal - vas ular

pro-liferation - invade surrounding tissue- tendsto re ur- e.g..

Anaplasti astro ytomas

GradeIV -veryrapidproliferation-veryabnormalappearan eof ells

- invasion of large areas - re urs - ne roti ore- forms new

vas ularization to supportgrowth - e.g. Glioblastoma

Mul-tiforme

The lassi ation of brain tumors interms of their origin also has two groups:

primary andmetastati . Primarytumorsarethe onesthatoriginate fromthebrain

ellsandstayinthebrain. They ano uratanyagehowever,statisti allytheyare

more ommonin hildrenandinolderadults. Thesetumors anbebenignor

malig-nant. Dierenttumorsinthisgrouparenamedbasedonthetypeof ellsthey

origi-natefrom. Examplesofthesetumorsaregliomas,meningiomas, medulloblastomas,

ependymomas and pituitary tumors. Among these the most important ones are

meningiomas astheyformthe biggestpartofallprimary braintumorsand gliomas

be ausetheyrepresentthemajorityofthemalignantbraintumors[DeAngelis2001℄.

nantand they arethe most ommontype of braintumors. Majorityof the an ers

whi h metastasize tothe brain arelungand breast.

2.2 Gliomas (Astro ytomas)

Inthisthesiswemainlyfo usonaspe i typeofbraintumor,thegliomas. Gliomas

are the neoplasms of glial ells whi h support and nourish the brain. These

tu-morsappearmost ommonlyinthe erebralhemisphere butthey analso befound

anywhere else in the brain like the erebellum. They an arise either alone or

as a re urren e of a pre-existing tumor. The fa tors that ause glial tumors is

mostlyunknown but the only identied riskfor these tumorsis theionizing

radia-tion[DeAngelis 2001℄.

Gliomas have varying histopathologi al features and biologi al behavior. They

overalargerangeaggressivenessandgradesfrombenigngradeI,pilo yti

astro y-tomas,tomalignantgradeIV,glioblastomamultiforme(GBM).Thedierentfa tors

analyzedforgradingthesetumorsin ludemitosisrates,mi rovas ularproliferation,

nu lear atypia and ne rosis [Wilson 1999 ℄. The lowest grade gliomas, namely the

pilo yti astro ytomas, stand a little dierent than the other ones. These tumors

donot inltrateandthey growveryslowlybymeans ofmitosis. Although they an

be ome large,they are not life-threatening and most of them are urable. Grade I

gliomas aremost ommonlyseeninpediatri ases. Thehigher gradegliomasfrom

II to IV are alled diuse gliomas and they share ertain hara teristi s. These

tumors inltrate into the surrounding tissue and invade the healthy brain. The

grade II ones, diusive astro ytomas, grow slowly however they show malignant

progression despite therapy. The higher grade ones, anaplasti astro ytomas and

glioblastoma multiforme, growvery rapidlyand invade thebrain intenta les

pene-tratinginto the brainparen hyma. Theyareusually surroundedbyedemaand the

grade IV ones reate extensive network of blood vessels and ontain ne roti ore.

Due to their rapid growth and the edema they exert pressure on the brain tissue

and auselo al massee t[Wilson 1999 ,DeAngelis2001 ℄.

The most important dynami inthegrowthof diusegliomas istheinvasionof

the healthy brain. The inltration into the surrounding tissue is seen in dierent

gradesofdiusegliomasand itisavery omplexmole ular pro ess [Demuth 2004℄.

The tumor ells inltrate mostly through the white matter tra ts but also use

erebrospinal uid and the vas ular onduits [Wilson 1999 ℄. The myelinated ber

tra ts a t as a route of invasion on whi h the migration apabilities of ells

en-han e [Giese1996 ℄. Diuse gliomas also show orti al inltration demonstrating

thatthey an invade thegraymatter aswell. However, thegraymatter inltration

isslowerthan thewhitematter one.

The other two highgrade spe i hara teristi s seen inthe growth of gliomas

arethe formationofthene roti oreandthevas ularization [Wilson1999 ℄. When

thetumorgrowsveryrapidly,the ells ompeteforthelimitednutritionandoxygen.

Inthe aseof gliomas thetumorstarts growing asa spheroidgettingthene essary

the enter undergoes ne rosis and a ne roti ore forms. The existen e of ne roti

oreisusedindistinguishingbetween gradeIIIandgradeIVgliomas. Thereforeby

denition itonly existsinthe aseof glioblastoma multiforme. Theother dynami

thattakespla easa resultof theextensive needof nutritionof therapidlygrowing

tumor is the vas ularization. Asthetumor needs more blood owitforms its own

blood vessel systems within the tumor. These systems are either formed through

angiogenesis orremodeling oftheexistingvas ulature. Thevas ularsystemsinthe

lowgradegliomasaresimilartotheoneofthebrainwhileitismu hmoreprominent

inthe ase ofhigher gradegliomas.

The treatment ourse of brain gliomas in ludes surgery, radiation therapy and

hemotherapy. Theexa t planning of the treatment and thetype of therapy to be

applied depend on thegrade and thelo ation of thetumor. The treatment

strate-gies of grade I gliomas and the others dier due to the inltration present in the

higher grade tumors. The grade I gliomas have distin t boundaries therefore

sur-gi al removal when total rese tion is possible might su e. When total rese tion

is not possible,due to the lo ation or the size of thetumor, thenadditional

radio-therapy and/or hemotherapy is applied to the remaining part [ABTA2008b ℄. In

general theaveragesurvivalratesfor patientsofgradeIgliomas isprettyhigh. The

treatment ofgradeIIto gradeIVgliomasonthe otherhand ismu h moredi ult.

The rst step is again surgi al removalwhen it is possible. However, thetotal

re-se tion is not possible due to theinltrative nature of diuse gliomas. Even when

the visible tumor is totally rese ted, removal of mi ros opi inltration into the

brainparen hymaisnotpossible[Wilson 1999,DeAngelis 2001 ℄. Therefore,patient

follow-up withadditionaltreatment intheformofradiotherapyand/or

hemother-apy is applied. The inltration also poses problems for the additional treatments

and as a result the tumor re urs. In the ase of grade II gliomas the tumor may

re urasahighergradegliomashowing malignantprogression. Theaveragesurvival

rates for patientsof thesetumorsis5-10years however,thevariabilityislarge. For

grade III and IV gliomas the applied treatment is mu h more aggressive however

theprogressionofthediseaseismu hfasteraswell. Theprognosisforthese asesis

reallylow, theaveragesurvivalratesremainaround3yearsand1yearforthegrade

IIIandgradeIVgliomasrespe tively. Intheviewofthiss enarioextensiveresear h

is being ondu ted on dierent hemotherapeuti agents and radiation therapy

s hemes [Ri ard2007,Bat helor 2007 ,Fiveash 2003,Mahajan 2005 ,Nandi 2008 ℄.

2.3 Imaging Gliomas with Magneti Resonan e

2.3.1 Magneti Resonan e Imaging (MRI)

Magneti Resonan e(MR)isanimagingte hniquewhi husesthethenu lear

mag-neti resonan e(NMR)signalsemittedfromthe obje tsthemselves. Inthis respe t

itdiersfromtheotherimagingte hniqueslikeX-rayComputedTomography(CT)

or Positron Emission Tomography (PET), where either a beam is irradiated or a

radioa tiveagentisgivento thebody. Theprin ipleofMRIisbasedonthenatural

reatesanatural magneti eldwhen ombined withthespin. Clini alMR fo uses

on the hydrogen whi h is the most abundant atom in thehuman body. The

mag-neti momentsofhydrogen nu lei inthebodyarerandomly orientedundernormal

onditions. In the presen e of an external magneti eld these nu lei align

them-selvesalong theexternal eldand ontinue their pre essing around thedire tionof

eld. Thisrelationshipforms thebasisof MRI.

During MR imaging a onstant base magneti eld

B

0

is applied to the body

aligning the nu lei, whi h keep pre essing at a frequen y alled as pre ession

fre-quen y. When a Radio Frequen y (RF) pulse with thesame frequen y asthe

pre- ession is applied, the nu lei resonate, gain energy, hange their alignment and go

in phase withea h other. As every element has a dierent pre ession frequen y a

spe i RFpulse only resonates with the nu lei of spe i elements. As theRF is

stopped the nu lei relaxandlose theirenergy. Thispro essis alledrelaxationand

theenergy emitted inrelaxationis the MRsignal we dete t. There aretwo

impor-tantpropertiesoftheappliedRFpulse,therepetitiontime(TR)andthee hotime

(TE).Therepetitiontimeisthetimedieren ebetween ea hRFpulseandthee ho

time is the time elapsed between appli ation of an RF pulse and the peak signal

obtained. Therelationship between TRand TE reates the ontrast visiblein the

MRimagesandgivesthe ontextinMR.By hanging thisrelationshipone obtains

dierent images su h asT1-weighted and T2-weighted. Based on similar ideas an

MRimage anbemadetobeaspatialmapofdensityofthespins,oftherelaxation

timesor ofthe water diusion. Asa resultdierent imagessu h asdiusion tensor

(DT)MRI, MRspe tros opy(MRS)or fun tionalMRI(fMRI) an be obtained.

MR is very good insoft tissue dis rimination ompared to other imaging

te h-niques. The two extreme ases in terms of ontrast dieren e in MR are the fat

and the water. InT1-weightedimages thefat tissueis enhan edwhile thewater is

not, showing the uid around the orti al areas and within the ventri les as dark

regions. Ontheotherhand,inT2-weightedimagesfreewaterand waterembedded

in the tissue is strongly enhan ed and appears bright, see Figures 2.1(a) and (b).

Although this high intrinsi ontrast dieren es are very useful in dis riminating

brain tissues, they maynot always be enough to dete t pathologies a urately. In

order to in rease the ontrast between pathologies and the brain tissue,

enhan e-ment agentsmaybe given tothepatient and additional imagesmight be a quired.

One important agent that is widely used for imaging brain tumors is Gadolinium

(Gd). Gadolinium inje tion is followed by a T1-weighted image a quisition and it

helpsin reasing theenhan ement ofwatermole ulesneighboringtissue. IntheMR

imagesthis is espe iallyvisibleinhighly vas ular regions(vessels themselves or

re-gionwith abnormalangiogenesis). Tumors and otherlesions are thereforestrongly

enhan ed due to the inje tion [Westbrook1998 ℄. Another modality whi h is very

useful inthe ase ofpathologies istheFLAIR.The important property oftheair

isthatthe erebrospinaluid(CSF)isnot enhan edasintheT2-weightedimages.

Therefore,thepathologies adja ent to the CSF areseen mu h more learly.

In the methods presentedinthis thesis, besides theanatomi al MR images, we

also fo us on the diusion tensor MR images (DT-MRI). The DT-MRI is not an

(a) (b) ( )

Figure 2.1: (a) An axial sli e of a T1-weighted image of a healthy brain. (b) The

same axial sli e of the T2-weighted image of the same brain. We see that in

T1-weightedimagesthefattissueisenhan edwhilethewaterisnot. Ontheotherhand,

inT2-weighted imagesthe appearen e of fat andwaterare inverse. Thisexibility

inMR imaginggivesus the opportunity to have a very dierent appearan e ofthe

same brainintwo dierent modalities. ( )Anaxialsli eofaDT-MRI imageofthe

samebrainasshowninFigures(a)and(b). Ea htensorisvisualizedasanellipsoid

and theimageis subsampledfor a learervisualization. The olorsof theellipsoids

represent thedire tion of their major axis.

thebraintissue. Usingtheseimageswe anunderstand howmu h awatermole ule

an migrate along ea h dire tion in a given lo ation. Through a quiring DWIs

along dierent dire tions we an onstru t lo al estimates of ovarian e matri es

representing the lo al dire tional diusion information of water. These ovarian e

matri es are alled diusion tensors and the image onsisting of these matri esin

dierent lo ations is alled DT-MRI. In Figure 2.1( ) we show a single sli e of an

example DT-MRI image where ea h point onsistsof a tensor des ribing the lo al

diusivityof the water mole ule.

2.3.2 Appearan e of Gliomas in MRI

TheMRimagesareoneofthemostimportant radiologi alinformation inthe

diag-nosisandgradingofbraingliomasandtumorsingeneral[DeAngelis2001 ,Tovi 1994 ,

Pri e 2007 ℄. Theappearan eofgliomasinMRimagesdierdependingonthegrade

of the tumor and the modality of the image. The most important property of the

tumors that is visualized in the anatomi al MR is the ex essive ontent of free

water. Dueto this gliomas appearashyper-intense regionsintheT2-weighted

im-ages andhypo-intense inthe T1-weighted, asshowninFigures 2.2andFigures 2.3.

When gadolinium is inje ted the highly vas ularity in the tumor gets enhan ed in

theT1-weighted image and we gethyper-intensity regionsinsidethetumor for the

(a) (b) ( ) (d)

Figure 2.2: In the images above we show axial sli es of (a) T1-weighted, (b)

T2-weighted, ( ) T1-weighted post gadolinium inje tion and (d) FLAIR images of a

pediatri patientwitha diusiveastro ytomaofgradeII.Weseethatthetumor is

enhan edinthe T2-weightedand the FLAIRimages.

der MR images. In Figure 2.2 we show axial sli es of MR images of a diusive

astro ytoma (grade II) dete ted in a pediatri patient. We observe that the

tu-mor is enhan ed inthe T2-weighted and theFLAIR images with lear boundaries

separating the tumor from the healthy tissue. Onthe other hand we only observe

hypo-intense regionsin the T1-weighted and the T1-weighted after gadolinium

in-je tion images. The lear boundaries seen in the T2-weighted images in the ase

of grade-II-astro ytoma might be misleading due to the inltrative nature of the

tumor [DeAngelis2001 ℄. Although we see su h lear separation, thetumor might

have penetrated the brain paren hyma beyond the enhan ement of the MR

sig-nal[Wilson1999 , Johnson1989 , Tovi 1994 ℄.

(a) (b) ( )

Figure 2.3: In the images above we show axial sli es of (a) T1-weighted, (b)

T2-weightedand( )T1-weightedpostgadoliniuminje tionimagesofapatientsuering

from agradeIV glioma, glioblastoma multiforme. Theappearan e ofGBM isvery

appearan e of the tumor is very irregular. In the T1-weighted image again we do

not see any enhan ement inthetumor region. InT2-weighted imageson theother

hand we see several ompartments of the tumor whi h arewell enhan ed. Looking

at Figure2.3(b),we observetheveryhighlyenhan edmiddlepartofthepathology

whi h in ludesthe highly a tive partofthe tumorand thene roti ore. However,

the ne roti ore annot be distinguished. Around this part we observe another

highly enhan ed part, whi h orresponds to the edema. The edema region is also

inltrated withtumor ellshowever, thenumberof tumor ells pervolume ismu h

lower than the a tive part [Johnson 1989 , Tovi 1994 ℄. In the T1-weighted image

after gadolinium inje tionwe learly seethemost a tive partof thetumor andthe

ne roti ore. The dark area inside thepathology isthe ne roti ore where there

arenolive ells. Thebrightrim aroundthisareaisthea tivelyproliferating region

of the tumor where the vas ularization is very dense and thetumor ell density is

high. In the ase of grade III gliomas these images look dierent as there is no

ne roti oreand thetheremight not be anyedemaregion.

One of the most ru ial points of MR appearan e of gliomas is the inltration

of the tumor whi h beyond a ertain ore region is not enhan ed in the images.

Dierent experiments omparing histopathologi al analysis with MR images have

shownthattumor ellsexistsbeyondtheenhan edregionintheT1-weightedimage

and theT2-weightedimage [Tovi 1994,Johnson 1989 ℄. Intheimagesthedieren e

between the tumorous region and the brain tissue seems abrupt. However, the

hypotheti al distribution of tumor ell densityis smoother. InFigure 2.4 we show

thehypotheti al rossse tionofaGBM wherethetumor ell densityisrepresented

bytheheight ofthe blue urve. TheT1 andT2 image intensities areshowninthe

gureasdierentthresholdsonthetumor elldensity[Swanson 2008b ℄. Weseethat

hypotheti ally the transition between the enhan ed region in the post gadolinium

T1-weighted image and the enhan ed region in the T2-weighted image is smooth.

Moreover,thetumor elldensity ontinue todrop aftertheT2thresholdsuggesting

inltrationbeyondtheenhan edregionintheimage. Thisdete tionproblemposes

di ultiesforthetreatmentofthetumorespe iallyinthe aseofradiotherapywhere

images guide the irradiation. Inorder to deal withthis problem in radiotherapy a

normal looking band around the tumor is also irradiated [Kantor2001 ℄. However,

theseeortsseemto benot enoughbe ausediusegliomastendtore urduetothe

inltration [Wilson1999 , DeAngelis2001 ℄.

Re ent resear h on other MR modalities su h as DT-MRI and MRS have

shown that these images an also be used to gather information about the

tu-mor hara teristi s and its spatial distribution. As the tumor invades the brain

through white matter it damages the underlying ber stru tures. DT-MRI

im-ages have shown to be useful indete ting this damage by using dierent measures

[Lu 2003 , Lu 2004 , Roberts2005 , Sinha 2002 , Pri e 2003 ℄.. The rst hange that

o urs is that the mean diusivity (MD) in reases in the regions invaded by the

tumor or by edema, see Figure 2.5(b). Moreover, as the ber stru tures are

dam-aged thedire tional organization of the bers is lost and this an be quantied by

the fra tional anisotropy (FA) 2.5( ). MR spe tros opy on the other hand, gives

Figure 2.4: The MR images of grade II to grade IV gliomas are not able to show

the whole inltration of the tumor inside the brain paren hyma. In the plot we

showhypotheti al distributionofthe tumor elldensityandtherelationofittothe

enhan edregion intheMRimages. Weseethat tumor ell densitydropssmoothly

suggesting inltrationbeyond theenhan edregion intheT2-weightedimage.

(a) (b) ( )

Figure2.5: TheDT-MRIimages anshowdierent ee tsofthetumor totheber

stru turesproviding usanother meanto visualize thepathology. (a) FLAIRimage

of a grade II astro ytoma, (b) Mean Diusivity (MD) image of the same patient

derivedfromtheDT-MRI,( )Fra tionalanisotropy(FA)imageofthesamepatient

againderivedfromtheDT-MRI.WeobservethatinthetumorregiontheMDimage

showsextraenhan ementwhiletheFAimageshowsdegradationinthesameregion.

helpus gathersome information aboutthetumor andits extent [Devos2005 ℄. The

not very possible toobtain however,asthe MRte hnology improve these problems

will besolved aswell.

There are also other imaging te hniques whi h provide dierent information

about gliomas than MRI. Positive emission tomography for instan e gives lo al

metaboli al information about the tumor. In PET a radioa tive agent is inje ted

and the uptake of this material is orrelated with the existen e of tumor ells.

Dierent studies have analyzed and shown dis repan ies and similarities between

theappearan eofgliomas inPETandMRI[Ogawa 1993 ,Kra ht 2004,Kato 2008 ,

Miwa 2004 ℄. They have demonstrated that the appearan e in both imaging

te h-niques might be dierent. Therefore, using these images together might be the

Tumor Growth Models: Literature Review Contents 3.1 Introdu tion . . . 17 3.2 Classi ation . . . 19 3.3 Mi ros opi Models . . . 20

3.3.1 Avas ularGrowth/SolidTumor . . . 21

3.4 Ma ros opi Models . . . 31

3.4.1 DiusiveModels . . . 31

3.4.2 Me hani alModels . . . 34

3.5 ImageGuided Toolsfor Therapy Planning . . . 37

3.6 Appli ations to Registrationand Segmentation . . . 38

3.6.1 Registration . . . 39

3.6.2 Segmentation . . . 42

3.7 Dis ussions . . . 43

Context

In this hapter we present an overview of mathemati al tumor growth modeling.

We explain the main approa hes by going through dierent models proposed. We

talkaboutmi ros opi models,dierentstagesoftumorgrowthandtheirmodeling,

ma ros opi modelsand some imageanalysis toolsusing these models.

3.1 Introdu tion

The domain of mathemati al tumor growth modeling in the resear h ommunity

is vast. There is extensive existing resear h both on brain tumors and on tumors

inother parts of the body. In order to situate themethods presented inthis work

a good understanding of the literature is ne essary. In this hapter we provide an

overviewofthe literaturepublishedontumorgrowth models. A ompletereviewof

all theworkson this topi wouldbe too long therefore, we provide the main steps

andtheresear horientations. Forfurtherreviewsonthetopi referto[Araujo2004 ,

Mantzaris2004 ,Sanga 2007℄.

Themainaim oftumorgrowthmodeling istodevelopmathemati al models

whi h lead to the growth of the tumor, via mathemati al abstra tions. In order

to explain the underlying me hanisms as a urately as possible, su h abstra tions

take into a ount many dierent biologi al fa tors, whi h were observed through

experimentation. Su h fa tors in lude internal dynami s of an erous ells, their

intera tions with ea h other and with healthy tissue, nutrition and oxygen

trans-portfromtheextra ellularmatrix(ECM)andfromthevas ularnetwork, hemi als

se reted by tumor ells, type of the underlying tissue, and many more. Models

aim to ombineallthese fa torsinauniedmathemati al framework,whi h would

agree withthe observed results. Whilemost of the work intheliterature hasbeen

on entratedinmodelingthegrowthpro essinageneralframework,therehasbeen

some re ent attempts to develop patient spe i models. These developments are

dire tedmoreondes ribingthegrowthofthetumorusingtheobservationsobtained

from thepatient.

Benets of des ribing the tumor growth mathemati ally are numerous. First

of all, su h des riptions would help us to ombine experimental ndings made in

many diverse elds of an er resear h in a ommon mathemati al ground. These

models allow us to interpret experimental results and understand the underlying

me hanisms of tumor growth and behavior of an erous ells. Virtualexperiments

and simulations give us the opportunity to observe ee ts of dierent treatments

on an erous ells, and would lead us to improve these treatments or suggest new

ones. Onthe otherhand,patient-spe i models ouldbeusedintreatingpatients.

Su h models ould be used for therapy planning, suggesting radiotherapy margins

adapted to the growth dynami s or helping the on ologist make a hoi e between

dierenttypesofdrugs thatwouldbestsuitthepatient. Virtualrealizations ofthe

tumorandthe brainstru tures ouldhelpneurosurgeonsduringoperations,

provid-ingpre iselo ationsofvitalstru tures. Anotherbenetoftumor growthmodelsis

that they ould allow us to make predi tions. The shape and invasion margins of

an existing tumor ina future time ould be predi ted using su h models and

om-puter simulations. The predi tions would give themedi al do tor theopportunity

to foreseethe problems thepatient might undergo and also wouldhelp him de ide

onthebesttimeofoperationifne essary. In ludingthegeneti informationinsu h

models, one ould even produ e the probability of o urren e of a brain tumor in

thefuture.

As we said the mathemati al work on tumor growth modeling is trying to

de-velopmathemati alabstra tionsthatwouldbestexplaintheobservedphenomenon;

hen e, it is very losely asso iated with the experimental and lini al work being

done in an er resear h. Most growth models use observations oming from

dier-ent sour eslike in-vitro experiments, in-vivo experiments done onanimal subje ts,

biopsy results, autopsy results and medi al images of patients like Computed T

o-mography (CT) s ans or Magneti Resonan e Images (MRIs). Theseexperiments

and images are keys to developing models des ribing the tumor growth pro ess

a urately. Observations used an be lassied in two groups based on the s ale:

ma ros opi andmi ros opi s ales. Experiments on entratedonthe ellular

a tiv-ities anbepla edunderthemi ros opi lass,likein-vitro andin-vivo experiment,

ners, inthis thesiswe make distin tionbasedon theuseof medi alimages.

Therehasbeengreatadvan esintumorgrowthmodeling,thereareseveral

prob-lemson the way of developing more a urate models. Themost ru ial problem is

thela kof knowledge on the behaviorof tumor ells inthe living tissue.

Observa-tions oming fromin-vitro andin-vivo experimentsgivesusinsight onthebehavior

oftumor ellsonlaboratoryset-upslikepetridish oron animalsubje ts. However,

in-vivo observations on humanbeings, whi h is the ase the tumor models aim to

des ribe,ares ar e. Thebestone an doistoproposeassumptionsonthebehavior

of tumor ells in the human brain, using observations available at hand. Another

problem related to observations is limitations in ma ros opi imaging te hniques,

[Tovi 1994℄. Medi al imaging te hniques are able to enhan e and dete t regions

ontainingtumor ells,only ifthenumberof tumor ellsareabovesome threshold.

There areseveralestimatesgivenintheliteratureonthelowestdete tion threshold

of CT images (1-40 % of the maximum number of tumor ells brain paren hyma

an handle),[Tra qui 1995 ,Swanson2008b ℄. Although thereisnowork beingdone

on the dete tion threshold of MRIs for tumor ells, the extent of the tumor

(inva-sion margin) inthese images arevery similar to theone in CT imagesthus, it isa

ommon pra ti e to a eptthesame threshold.

In the restof this hapter we willgive general information abouttumor growth

models,summarizesomeofthemilestonesintumorgrowthmodelingandalsoreview

re ently proposed tumor growth models tryingto give an overviewon the state of

the art. In Se tion 3.2 we will introdu e a lassi ation of tumor growth models

whi h we will use throughout this hapter to analyze dierent models proposed.

Based on this lassi ation we review the orresponding literature of mi ros opi

modelsinSe tion3.3andma ros opi modelsinSe tion3.4. InSe tions3.5and3.6

we fo us on the appli ations on medi al images and explain some of the models

proposedfor therapyplanning and other workswhi h usemodels for segmentation

and registration.

3.2 Classi ation

Resear h being done on tumor growth modeling an be oarsely lassied into two

largegroups. This lassi ationisbasedonthes aleofthemodelandtherearetwo

lasses: mi ros opi models and ma ros opi models. Themain dieren e between

these lasses is the s ale of observations they are trying to explain and formulate.

Mi ros opi models on entrateonobservationsinthemi ros opi s ale,likein-vitro

and in-vivo experiments. Theytryto explain the growthphenomena at the

mi ro-s opi levelbydes ribingtheintera tionsbetweendierent ells,dierent hemi als

se retedby ells, nutrition sour es,oxygenandnearbyvessels. Ma ros opi models

on the other hand, are on entrated on observations at the ma ros opi s ale like

theonesprovidedbymedi alimages. Theyformulatetheaveragebehavioroftumor

ells and their intera tions with underlying tissue stru tures, whi h are visible at

this s ale of observation (gray matter, white matter, bones, ...). These models try

tumorgrowthbeinganalyzedortheee tofthegrowthonthebrain. Classi ation

based on the stage riteria ismore suitable for themi ros opi models and will be

used for those models only. On the other hand the ee t riteria will be used for

thema ros opi models.

The lassi ation basedon stagesof thetumorgrowth onsistsofthree lasses,

whi h are basi ally three dierent phases of the growth: the avas ular growth,

the angiogenesis and the vas ular growth. At this point, we would like to give

verysimpliedexplanationsforthesestagesfor ompleteness. Theavas ulargrowth

orrespondsto the stage where the pro ess ismostly governed bythe proliferation

of tumor ells. In this stage the tumor is onsidered to be a solid mass, whi h is

growing by means of mitosis. Although not ompletely known, it is thought that

there is no invasion of the healthy tissue. The intera tions between tumor ells

and the healthy tissue is also thought to be limited, [Araujo2004 ℄. The tumor

annot grow indenitely in the avas ular stage be ause as the tumor mass grows,

less and less nutrition is available for the ells deep inside the avas ular mass. As

a result ne rosis begins,tumor ells thatare not getting enough nutrition die, and

only ellsonthe outer perimeterofthetumor ontinue toproliferate. Atone point

ne rosis and the proliferation balan esea h other and theavas ular tumor rea hes

a limiting size, whi h isassumed to be around 1-3 mm indiameter, [Orme1996b ℄.

Angiogenesis (vas ularization) isthestage where tumor ells intheavas ularmass

modifythe existingvas ularstru ture,to reate new vessels thatwouldfeed them.

Through this pro ess the tumor an over ome its limit size, growmu h faster and

invade thesurroundingtissue. Dueto the ru ialroleof angiogenesison thetumor

growth, its underlying me hanism has aptured attention and many models have

been proposed trying to explain it. The third stage of thetumor growth,vas ular

growth, hasbeen paid less attention than two previous stages. The omplexity of

thetumorgrowthinthisstageishigherbe ausethereareseveralpro essesgoingon

simultaneously. Inadditionto ellularand hemi alintera tionsgoingonintherst

two stages, tumor ells start to invade the surrounding tissue via me hanisms not

learly knownyet. At thisstage,thetumorbe omesdiusive andisnot onsidered

to be solid anymore. While the dieren e between an erous and healthy regions

are learin the avas ularstage,this dieren e vanishesduring thevas ular growth

be ause tumor ells move towards healthyregions.

Classi ation based on the ee t of tumor growth on thebrain is more

appro-priate for ma ros opi models. We an distinguish two major groups: me hani al

models, whi h on entrate on the mass-ee t of the tumor and diusive models,

whi h on entrate on the inltration of the brain tissue. In following se tions we

will go oversome of important and re ent models thathave been proposed. While

mentioningdierent modelswewilltrytomakeuseof lassi ationtypesexplained

above,whi h issummarizedingure 3.2.

3.3 Mi ros opi Models

TUMOR GROWTH

MICROSCOPIC

MODELS

MACROSCOPIC

MODELS

AVASCULAR

GROWTH

ANGIOGENESIS

VASCULAR

GROWTH

DIFFUSIVE MODELS

MECHANICAL

MODELS

Figure3.1: Classi ationof tumor growthmodels

level. They take into a ount physi al and hemi al intera tions between ellsand

theECM,andbuilda ause-resultrelationshipbetweenthetumorgrowthandthese

intera tionsusingmathemati alformulation. Me hani alphenomenonlike ohesion

for es, adhesion for es and pressures are often in luded to des ribe physi al

inter-a tions between an erous and healthy ells. As for hemi al intera tions, they

in lude pro esses like diusion of nutrition and oxygen, se retion of dierent

fa -tors by tumor ells and their ee ts on the ECM, blood vessels and other ells.

Mathemati al systems obtained are usually very detailed as they try to take into

a ount all the fa torsobserved to ae t the tumor growth. Formulations used in

reating mi ros opi models enjoy a large variety of mathemati al methods. Most

ommonly used methods are partial dierential equation (PDE) systems, ellular

automataand statisti almodels.

3.3.1 Avas ular Growth/Solid Tumor

Most of the modeling work at the mi ros opi s ale hasbeen on entrated on the

avas ularstageofthegrowth. Inthebeginningitwasthoughtthatthewholepro ess

of tumorgrowth was onlygoverned bythe proliferation of ells. Models usingonly

populationgrowth dynami s like exponential growth or Gompertzian growth were

proposed. Intable 3.1we give some ofthe populationgrowth equations ommonly

used.

u

inthese equations is the normalized density of tumor ells(normalized by themaximum tumor ell density the underlying tissue an handle),

∂u/∂t

denotes the hange of

u

in time and

ρ

is the proliferation rate of tumor ells whi h is taken to be

ρ = 1

in the gures. One of the rst papers employing this idea was publishedbyMayneordin1932,[Mayneord1932 ℄. Thisworkexplainedtheee tof

Table 3.1: Commonly usedpopulationgrowthterms

Exponential (green) Gompertz (red) Logisti (blue)

∂u/∂t = ρu

∂u/∂t = ρu ln(1/u)

∂u/∂t = ρu(1

_{− u)}

solidtumor.

Inlightoffurtherexperiments,diusionand onsumptionofoxygenwasin luded

in the model as a fa tor in the mitosis rate and ne rosis. Models proposed by

Thomlinsonetal. andBurton[Thomlinson 1955,Burton 1966 ℄examinedthisee t

and showed that when the blood supply (as a supply of oxygen) was limited to

the perimeter of thetumor, formation ofthe ne roti part ould be explained asa

result of la k of oxygen. These developments showed that theGompertzian model

bettertsthe tumorgrowth. Although thesemodelswereabletomat hthegrowth

rate of the tumor, they were not able to explain its ompa tness. Greenspan in

[Greenspan 1972 ℄, in luded surfa e tension among living ells on the periphery, in

ordertoobtaina ompa ttumor. Inthispaper,heassumedthatne roti ellswere

dissolvingandduetothesurfa etension, ellsontheperipherywerepushedtowards

the ne roti region. He also tried to explain theinhomogeneity in themitosis rate

throughout thetumorviathe se retionofgrowthinhibitingfa tors(GIF)bytumor

ellsinaspatiallyuniformmanner. Thetumorradiusevolutionfollowedan

integro-dierential equation, whi h was oupled to rea tion-diusion equations explaining

thedistributionofnutritionandGIFs. Althoughin ludingtheoxygen onsumption

inthemodelwasabigstep, itwasunabletoexplaintheslowthinningoftheviable

rim following the formation of a ne roti ore. Deakin in [Deakin 1975 ℄ in luded

inhomogeneous onsumption of oxygen in the tumor explaining this phenomenon.

Besides the ell lossinthetumor dueto ne rosis,M Elwainet al.[M Elwain1978 ℄

in luded another ell lossme hanism,apoptosis, following theexperimentsshowing

that tumor ells may dieeven though they do not la k nutrition nor oxygen. The

Besidesthedeterministi modelsofgrowththathasbeenproposedtherehasalso

beensomesto hasti onesemphasizingtheprobabilisti natureofthegrowth. Oneof

therstworksinthis ontext isofWetteetal. in[Wette 1974a ,Wette 1974b ℄. The

mainargument ofsu hmodelsisthatu tuationsaround theaveragebehaviormay

be more important than the average behavioritself, when the population assumes

smallvalues. Thesemodelssimplyaddthepossibilitythataveragevalueslikemitosis

rateor diusion of fa torsmay deviatea lot.

Later on, the ee t of GIFs on the growth pro ess was analyzed by Adam et

al. in[Maggelakis 1990 ℄. Theyshowed thatGIFs indeedplay a ru ial role on the

dorman y of the tumor in theavas ularstage. In their modelthey in luded

inho-mogeneity ofnutrition onsumption and GIF produ tion using spatiallydependent

fun tions, assuming GIFs were produ ed more in the ne roti ore and de rease

linearly towards the perimeter. Even though they did not in lude thevolume loss

in the ne roti ore, they were able to obtain limiting sizes, showing GIFs alone

an also ausedorman yinsolidtumors. Byin ludingbothinhomogeneous

forma-tion of GIFs and onsumption of oxygen they ombined all previous ideas in one

formulation.

Followingexperimentssuggestingthat ellsinsolidtumorstendtogrowtowards

bloodvessels,M Elwainet al.,[M Elwain 1993 ℄introdu ed an a tive migration

to-wards nutrition gradient in their model. Des ribing the motion of ells with two

parts: passivemotiontowardsthene roti ore ausedbypressuresandthe

hemo-taxis towards nutrition sour es. However, experiments showed that not all ells

followed the a tive migration. Pettet et al. [Pettet 2001 ℄ proposed to use the ell

y leto explain this. In this model, ells thatweregoingthrough mitosiswerenot

hemota ti allya tive. Onlyquies ent ellswereae tedbythe hemotaxistowards

thenutritiongradient.

As models des ribing the motion of a single element, tumor ells, are getting

more and more elaborate, some attention has started to be given to multiphase

models to be applied to tumor growth. Please et al. in [Please 1999 ℄ used the

theoryofmultiphase tomodelthetumor growthusingtwo phases: tumor ellsand

extra ellularmatrix. Theymodeledphysi al intera tions between these twophases

and analyzedthe ee t of me hani al stressesinthetissue onthe formationof the

ne roti region. Both phases in this model were assumed to be invis id. Later

on,Landman et al. added the interphasedragfor es inthis formulation and using

the model, showed the ee t of surfa e tension on the formation of the ne roti

ore and also on the stable limiting size of the tumor, [Landman 2001 ℄. Breward

et al. in [Breward 2002 ℄, also used two phases, however, they took into a ount

the intera tions between tumor ells by modeling ellular ohesion between them.

Hen e,thepressureinthetumordierentiatedfromthepressureoftheextra ellular

matrix, due to these intera tions. Byrne et al. in [Byrne 2003℄ took a dierent

approa h to two phase models. Theyused theoryof mixtures to model thetumor

asanorgani balloon reatingasolid- ellularphaseandthesurroundingmediaasa

liquid ontainingnutritionanddierentgrowthfa tors. Thedynami softhegrowth

weredes ribedbymassandmomentumbalan esinadditiontothe onstitutivelaws.