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Imagerie oncologique et modélisation mathématique :développement, optimisation et perspectives

Francois Cornelis

To cite this version:Francois Cornelis. Imagerie oncologique et modélisation mathématique : développement, optimisationet perspectives. Analyse numérique [math.NA]. Université de Bordeaux, 2015. Français. <NNT :2015BORD0121>. <tel-01270590>

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THÈSE PRÉSENTÉE

POUR OBTENIR LE GRADE DE

DOCTEUR DE

L’UNIVERSITÉ DE BORDEAUX

ÉCOLE DOCTORALE de Mathématiques et Informatique de Bordeaux

SPÉCIALITÉ Mathématiques Appliquées

Par François CORNELIS

Imagerie Oncologique et Modélisation Mathématique :

Développement, Optimisation et Perspectives

Sous la direction de : Thierry COLIN (co-directeur : Olivier SAUT)

Soutenue le 01/10/2015 Membres du jury : M. Emmanuel GRENIER Professeur, ENS, Lyon Président

M. Nicolas GRENIER Professeur, CHU, Bordeaux Examinateur

Mme Florence HUBERT Maitre de Conférence, Marseille Examinateur

M. Marc GARBEY Professeur, Houston, USA Rapporteur

M. Philippe FERNANDEZ Professeur, CHU, Bordeaux Invité

M. Pierre SOUBEYRAN Professeur, Institut Bergonié, Bordeaux Invité

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Titre : Imagerie Oncologique et Modélisation Mathématique : Développement, Optimisation et Perspectives

Résumé : Ce travail de thèse, réalisé à l'Institut Mathématiques de Bordeaux (IMB) de 2010 à 2015 sous la direction de Thierry Colin et Olivier Saut, décrit la création et le développement progressif d'un ensemble de théories, de techniques et d'outils liant l'imagerie médicale aux mathématiques appliquées dans le but d'envisager leur application clinique à courte échéance en oncologie. Cette thèse a tout d'abord consisté à optimiser les modèles spatiaux de croissance tumorale développés à l'IMB incluant des éléments microscopiques et macroscopiques obtenus par analyse des informations disponibles des examens d’imagerie. Plusieurs étapes ont été réalisées permettant de mieux appréhender la modélisation in vivo. Différents organes et types tumoraux ont été explorés, en particulier au niveau du poumon, du foie, et du rein. Ces localisations ont été successivement étudiées afin d’enrichir progressivement les modèles par les réponses qu'elles apportaient et répondre ainsi à la réalité clinique. De façon concomitante, des outils ont été intégrés au fur et à mesure afin de standardiser la démarche de recueil de données et permettre d'affiner l'évaluation thérapeutique par l'imagerie à l'aide de marqueurs numériques. L'implémentation de l'imagerie fonctionnelle dans une pratique clinique est ainsi devenue une réalité. Le but est à terme d’appliquer de façon prospective ces outils d'assistance en pratique quotidienne. La modélisation a été aussi appliquée en oncologie interventionnelle par l'étude de la distribution du champ électrique lors des électroporations de prostate et bientôt du foie. Ceci permettra de mieux contrôler les zones d'ablation et ainsi améliorer la sécurité et l'efficacité de ces traitements. Tout cela a permis d'envisager des projets cliniques combinant une part exploratoire impliquant la modélisation. Ces développements et leurs perspectives sont rapportés successivement dans ce manuscrit.

Mots clés : ONCOLOGIE; MATHEMATIQUES; RADIOLOGIE

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Title : Imaging in Oncology and Applied Mathematics: Development, Optimisation and future

Abstract : [1700 caractères maximun]

This work performed at the Institute of Mathematics of Bordeaux (IMB) from 2010 to 2015 under the direction of Thierry Colin and Olivier Saut describes the creation and gradual development of a set of theories, techniques and tools linking medical imaging and applied mathematics in order to consider their clinical application in the short term in oncology. The first goal was to optimize the spatial models of tumor growth developed at the IMB including microscopic and macroscopic elements obtained by analyzing the information available on imaging explorations. Several steps were performed to better understand the in vivo modeling. Various organs and tumor types were investigated, especially in the lung, liver, and kidney. These locations were studied successively to progressively enrich the model by the answers they brought and thus respond to clinical reality. Concomitantly, tools were integrated to standardize the data collection process and help to refine the therapeutic evaluation by imaging with digital markers. The implementation of functional imaging in clinical practice has become a reality. The goal is ultimately to apply prospectively these support tools in a daily practice. Modelling was also applied in interventional oncology for the study of the electric field distribution after percutaneous irreversible electroporation in the prostate and soon in the liver. This will allow a better control of the ablation areas and thereby improve the safety and efficacy of these treatments.

Keywords : ONCOLOGY; MATHEMATIQUES; RADIOLOGY

Unité de recherche

Institut de Mathématiques de Bordeaux,

INRIA Sud-Ouest (équipe MONC)

351 cours de la Libération, 33405 Talence Cedex, France

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Remerciements:

Je remercie les rapporteurs et les membres du jury de m'avoir fait l'honneur de juger mon

travail de thèse. Je ne prétends pas être devenu mathématicien compte tenu de mon passif de

médecin mais j'espère par ce travail avoir modestement contribué à l'essor des mathématiques

appliquées en médecine, et tout particulièrement en oncologie interventionnelle.

Je tiens aussi à remercier profondément Thierry Colin et Olivier Saut, mathématiciens

appliqués, et toute l'équipe MONC. Ils ont réussi la gageure de me faire oublier les mauvais

souvenirs de ma professeur de mathématiques revêche du lycée. Etant partagé entre mon

activité clinique quotidienne au CHU (qui représente 200% de mon temps de travail) et le

laboratoire (qui ne représente malheureusement pas assez), il m'a bien fallu 5 ans pour mener

à bien cette tâche grâce à vous. Ce travail n'est clairement pas fini mais je pense sincèrement

qu'ensemble, nous avons bien avancé sur ces projets depuis tout ce temps.

Je remercie Jean Palussière par qui tout a commencé et qui m'a donné les moyens de

progresser à l'Institut Bergonié; Guy Kantor qui a perçu très vite l'impact potentiel de ces

modèles; ainsi que Josy Reiffers, Binh Bui, Pierre Soubeyran, Nicolas Grenier et Alain

Ravaud qui d'une façon ou d'une autre nous ont aidés dans notre démarche.

Je remercie également l'équipe du Memorial Sloan Kettering Cancer Center mais aussi Clair

Poignard qui m'ont permis de découvrir la technique et les perspectives de l'électroporation.

Je remercie toutes les équipes de mathématiciens qui m'ont invité pour leur chaleureux accueil

(à Guidel, Houston ou Paris) et les précieux échanges à ces occasions même si mes topos ne

collaient pas forcément avec leur quotidien.

Et bien sûr je remercie ma petite famille nombreuse: Emilie, Arthur, Camille et Oscar. Avec

tout mon amour. Sans oublier mes parents et mes grands parents; mes beaux-parents; mes

soeurs et leurs familles pour leur soutien inconditionnel. Encore Merci!

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Table des matières

Table des matières...........................................................................................................................................51. Introduction..............................................................................................................................................61.4.1. Modélisation mathématique de la croissance tumorale basée sur l'imagerie: généralités et perspectives.................................................................................................................................................161.4.2. Modélisation de la croissance des nodules pulmonaires.........................................................191.4.3. Modélisation de la croissance des métastases de tumeurs stromales de l’intestin grêle..241.4.4. Analyse de l'erreur de contourage...............................................................................................271.4.5. Intégration de l'imagerie fonctionnelle.......................................................................................281.4.5.1. Développement de l'imagerie fonctionnelle............................................................................281.4.5.2. Développement d’outils de traitement de l'imagerie fonctionnelle...................................311.4.6. Modélisation du champs électrique dans les procédures d'électroporation irréversible342. Modélisation mathématique de la croissance tumorale basée sur l'imagerie: généralités et perspectives.....................................................................................................................................................433. Modélisation de la croissance des nodules pulmonaires................................................................524. Modélisation de la croissance des métastases de tumeurs stromales de l’intestin grêle.........785. Analyse de l'erreur de contourage..................................................................................................1026. Intégration de l'imagerie fonctionnelle..........................................................................................1196.1. Développement de l'imagerie fonctionnelle...............................................................................1196.2. Développement d’outils de traitement de l'imagerie fonctionnelle.......................................1347. Modélisation du champs électrique dans les procédures d'électroporation irréversible....1518. Conclusion............................................................................................................................................170

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1. Introduction

Au delà de la perpétuelle nécessité d'améliorer le temps du dépistage et du diagnostic,

les enjeux futurs de l’imagerie cancérologique sont d’évaluer encore plus précocement la

réponse thérapeutique, mais aussi de proposer au meilleur moment et de guider de façon

encore plus fiable, les procédures de radiologie interventionnelle. Les mathématiques

appliquées via la modélisation des phénomènes physiques et biologiques, et notamment grâce

aux données obtenues par l'imagerie, peuvent représenter un axe prometteur de

développement alternatif et transversal permettant de répondre à ces objectifs.

Ce travail de thèse, réalisé à l'Institut Mathématiques de Bordeaux de 2010 à 2015

sous la direction de Thierry Colin et Olivier Saut, décrit la création et le développement

progressif d'un ensemble d'éléments liant l'imagerie médicale aux mathématiques appliquées

dans le but d'envisager leur application clinique à courte échéance en oncologie. Ce travail a

été réalisé parallèlement à notre activité clinique en imagerie diagnostique et interventionnelle

oncologique (plus communément appelée "oncologie interventionnelle"). La coexistence de

ces activités a autorisé un perpétuel transfert ("bench-to-bedside") permettant de progresser

sur les deux versants tant en imagerie (en IRM fonctionnelle tout particulièrement) qu'en

modélisation. Tout ceci a permis d'envisager des projets cliniques, combinant une part

exploratoire impliquant la modélisation, permettant d'étudier l'impact des traitements

proposés au niveau de différents organes (par exemple le projet de modélisation MOD financé

dans le cadre du Labex TRAIL en 2013). Dernièrement, l'utilisation clinique de

l'électroporation irréversible a permis d'envisager des applications directes des modèles

mathématiques en radiologie interventionnelle.

Après quelques rappels et une présentation générale du plan de cette thèse, tous ces

développements et leurs perspectives sont rapportés successivement dans ce manuscrit.

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1.1. Place de l'imagerie en oncologie

De nos jours, l’imagerie appliquée en cancérologie s’est considérablement développée

en raison des évolutions des techniques et des pratiques. Elle est maintenant totalement

intégrée au parcours de soin des patients. Ainsi, un patient bénéficiera surement d’un examen

d’imagerie aux différentes étapes de sa maladie que ce soit au moment du diagnostic, du bilan

d’extension de la maladie, du bilan pré-thérapeutique (pour éliminer des contre-indications au

traitement : infections, morbidités), de l’évaluation de la réponse thérapeutique ou de la

surveillance (à la recherche de signes de récidives). Le radiologue est devenu un acteur

central des réunions de concertations multidisciplinaires et il est impliqué dans toutes les

prises de décision en raison de l’expertise qu’il détient.

Toutefois, la quantité d’images et d’informations disponibles est importante mais sans

aucune homogénéité. Les modalités varient profondément en fonction des pathologies et des

équipes impliquées, rendant difficile l'uniformisation de l'analyse de ces données obtenues par

l'imagerie. Il existe une grande variabilité interindividuelle en terme d'exploration, non

seulement en raison de la sévérité ou des caractéristiques phénotypiques et/ou génétiques des

tumeurs; mais aussi en raison des habitudes des structures d’accueil, dont les moyens

disponibles d'imagerie varient. De plus, en se focalisant uniquement sur l'imagerie

morphologique, l’apport de cette imagerie restera limité à un sous type de population. En effet,

l'étude de la simple croissance tumorale par exemple n'est pas possible pour tous les patients,

car de nombreuses tumeurs primitives sont opérées rapidement, ne permettant pas une

observation longitudinale de la croissance tumorale (tumeurs du sein par exemple), alors que

des patients présentant des métastases dans le cadre d’une maladie évolutive sont traités et

surveillés plus régulièrement (métastases pulmonaires ou hépatiques). Basé sur les simples

critères actuels morphologiques, les patients atteints d'une maladie métastatique, peu

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évolutive ou en situation palliative bénéficieront tout de même de l’application des critères

morphologiques tel le RECIST, ou l'OMS ou le CHOI même si ceux-ci restent imparfaits (1).

En effet, outre leur multiplication, les critères de réponse fondés sur l'imagerie des tumeurs

ont montré leurs limites notamment depuis l’introduction des thérapies ciblées en complément

de la chimio et/ou radiothérapie. Les critères OMS, publiés en 1981, ont été les premiers

utilisés et sont basés sur des mesures bidimensionnelles (plus grands diamètres

perpendiculaires). Par la suite, des critères RECIST (Response Criteria for Solid Tumor) ont

été créés en 2000, révisés en 2009 (RECIST 1.1) et modifiés (mRECIST). Ainsi, des mesures

unidimensionnelles (plus grand diamètre) sont réalisées et ont permis de palier aux carences

des critères OMS. Néanmoins, l’introduction de traitements ciblés, jouant plus sur la

dévascularisation et la nécrose, a démontré également les limites de ce système et a nécessité

la révision de ces critères pour prendre en compte le rehaussement des lésions et non plus

uniquement la taille. Ainsi sont apparus les critères Choi pour les tumeurs stromales (GIST),

les critères RECIST modifiés pour le carcinome hépatocellulaire, et les critères de réponse

immunitaire lors de mélanomes. Plus récemment les critères Cheson et PERCIST (PET

Response Criteria for Solid Tumors) ont été proposés pour évaluer la réponse des tumeurs

solides par TEP permettant de fournir des informations supplémentaires sur le métabolisme à

ces critères morphologiques.

Pour surmonter cela et afin de proposer une plus grande homogénéité et un retour

d’information plus précoce qui apparaissent indispensables à la bonne conduite des essais

cliniques, de nouveaux marqueurs en imagerie ont été proposés, assimilés aux « bio-

marqueurs » classiquement décrits en oncologie. En effet, en parallèle du développement de

l’imagerie, la recherche de biomarqueurs d’une maladie a révolutionné l’approche de la

pathologie cancéreuse. Lorsqu’ils sont identifiés, ces marqueurs influencent la prise en charge

et le devenir d’un malade. Il en existe 2 types : les biomarqueurs prédictifs et ceux

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pronostiques (2). Ils peuvent être d'origines variées : biologiques, génomiques,

morphologiques ou fonctionnels par exemple.

Parmi eux, la variance génétique spécifique des différents cancers focalise

actuellement l’attention. Elle est au cœur de la « pharmaco-génomique ». Il a ainsi été montré

que les mutations sont de 100 fois à 500 fois plus fréquentes dans le génome des cellules

cancéreuses par rapport aux cellules normales (3) et chacune d’elles pourraient être un

indicateur d’une maladie. Néanmoins, il existe une grande variabilité au sein de ces génomes

à la fois au niveau inter mais également intra-individuel. De ce fait, il a été proposé une

différentiation entre mutations causales et acquises (3, 4) qui ont un impact différent sur la

maladie. Les premières vont provoquer la maladie par dérégulation de la croissance et par

l’invasion des tissus environnants, alors que les secondes seront plutôt à l’origine de la

résistance aux traitements. Néanmoins toutes ces mutations peuvent aider au diagnostic, au

pronostic, à apprécier la réponse aux traitements ciblés, et éventuellement à la prévention,

quand elles sont identifiées. Cela reste difficile car l’information est noyée à l’échelle

cellulaire et demande également un recueil épidémiologique très large au sein des populations

étudiées. Ce recueil suit les évolutions des pratiques et des techniques, notamment de

radiologie interventionnelle. De plus, une fois obtenues, ces informations ne sont pas encore

toutes exploitables car l’identification de tous les mécanismes impliqués est encore délicate

(5). Il est difficile de les retrouver ou de les suivre dans le temps pour un patient donné, en

raison du caractère encore invasif des prélèvements nécessaires à leur obtention. A terme le

but est néanmoins d’obtenir un traitement le plus spécifique possible d’une voie de

signalisation anormalement active offrant la perspective d'une médecine personnalisée.

Même si de nombreuses et diverses voies de signalisation existent, certains

biomarqueurs prédictifs sont néanmoins déjà des cibles de médicaments, comme ceux

impliqués dans les voies moléculaires de l’angiogenèse (VEGF), de la réparation de l'ADN,

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dans l’invasion (tyrosine kinase) ou dans le métabolisme des médicaments (6). Il apparait

donc essentiel de développer des corrélations entre ces biomarqueurs et des méthodes

d’identification et/ou d’évaluation non invasive. L’imagerie pourrait apporter un certain

nombre de réponses.

Figure 1.2– les différents mécanismes impliqués dans la croissance tumorale et les molécules

actives sur ces voies. D’après Hanahan D, Weinberg RA. Hallmarks of cancer: the next

generation. Cell 2011; 144:646–674.

1.2. L'imagerie fonctionnelle

Pour l’instant, l'imagerie des voies de signalisation repose principalement sur

l’imagerie fonctionnelle qui est en plein essor (7). Les différentes techniques accessibles en

pratique clinique quasi quotidienne sont rapportées dans ce manuscrit au niveau du chapitre 6.

Cet aspect de l’imagerie peut apporter de nombreuses informations sur les activités cellulaires

EGFR is a family of tyrosine kinase receptors thatincludes HER-2 (1,16). After ligand binding, EGFRdownstream signal activates major cellular pathways,including the RAS/RAF/MAPK pathway responsiblefor cell proliferation and metastasis and the PI3K/AKT/mTor pathway responsible for cell survival (16).Other cellular and receptor kinases also have beentargeted successfully, including platelet-derived growthfactor receptor, vascular endothelial growth factorreceptor-2 (VEGFR2), mammalian target of rapamycin(mTOR), and tyrosine-protein kinase kit (c-KIT) (16).VEGF is a family of six related glycoproteins, VEGF-

A through VEGF-F, and placenta growth factor 1 and2. VEGF family members are expressed by tumor cellsin response to hypoxia, nutritional stress, and acidosis(17). They stimulate members of the VEGF receptorfamily, VEGFR1, VEGFR2, and VEGFR3, found onhematopoietic, lymphoendothelial, and endothelial cellsand some cancer cells, and result in angiogenesis andlymphangiogenesis (17). VEGF may also recruitendothelial progenitor cells.An evolving appreciation of the underlying molecular

tumorigenic mechanisms of cancer outlines the complex-ities and oversimplifications associated with targeted

therapies. For example, EGFR expression by immuno-histochemistry did not predict response to anti-EGFRtherapy in colon cancer (18). Drugs may target myriadcellular, molecular, and genetic or epigenetic pathways.The activation status of these pathways or subpathwaysmay provide the explanation for either treatment sen-sitivity or treatment resistance. The hallmarks of canceroutline pathways and targets for drug interventions (Fig 1)(19). Factoring imaging biomarkers and using image-guided biopsy may help address the tumor heterogeneity,which is inherent to many cancers (20). Prospective cor-relation of specific imaging characteristics and specific tissuefindings can build a library of information potentiallyleading in the long-term to the replacement of direct tumorsampling by noninvasive imaging.

TARGETED THERAPIESTargeted anticancer therapies is a generic term with avariety of potential meanings, mostly related to drugstargeting specific biologic pathways that cause regressionof malignant processes (1,3,9). Targeted therapies candirectly alter molecular pathways of tumor cells or

Figure 1. Targeting the hallmarks of cancer. Drugs that interfere with, or target, the hallmarks of cancer growth, progression, orresistance have been developed. (From Hanahan D, Weinberg RA. Hallmarks of cancer: the next generation. Cell 2011; 144:646–674.Copyright © 2011 Elsevier. Reprinted with permission from Elsevier.)

Volume 24 ’ Number 8 ’ August ’ 2013 1085

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(ex: étude des voies métaboliques en TEP), la consommation de nutriments (8), l’angiogénèse

ou la perméabilité capillaire (IRM de perfusion) (9) mais également la micro-architecture

tissulaire (IRM de diffusion) (7) par exemple.

Parmi ces biomarqueurs actuellement identifiés, certains ont focalisé l'attention. Les

marqueurs biologiques favorisant l'angiogenèse, tels que le VEGF, semble être une cible

potentielle à l’évaluation en imagerie. Le VEGF est une famille de 6 glycoprotéines, VEGF-A

à F, et le placenta growth factor 1 et 2. Ces protéines sont exprimées par les cellules tumorales

en réponse à l'hypoxie, au stress nutritionnel, ou une acidose (10). Elles stimulent les

membres de la famille des récepteurs de VEGF (VEGFR1-3) retrouvés sur les cellules

hématopoïétiques, lympho-endotheliales, endothéliales et cancéreuses. L’activation de cette

voie conduit à l'angiogenèse et la lymphangiogenèse nécessaire à la croissance tumorale.

L’expression du VEGF peut également permettre de recruter des cellules souches

endothéliales. Ce biomarqueur est très répandu dans les lignées cancéreuses car pour grossir

au delà de 2mm3, une tumeur doit acquérir un phénotype angiogénique (11). Il est donc

intéressant de privilégier cette voie. Plusieurs techniques d’imagerie sont capables d’apprécier

la vascularisation et le contenu en oxygène des tissus dans une pratique clinique. Elles font

appel en grande majorité à l’utilisation de produits de contraste ou d’équivalent biologique ;

néanmoins d’autres techniques existent mais sont en cours d'exploration. Toutes ne sont pas

encore utilisées en routine clinique et sont réservées seulement à quelques centres. Toutefois

de nombreux arguments laissent penser à croire que leur impact pourrait être majeur dans

l'évaluation thérapeutique précoce.

En lien avec les mathématiques appliquées, l’étude de la distribution d’un agent

observable (produit de contraste par exemple) a aboutit à deux approches différentes (12). La

première, plus simple, consiste en une description de la morphologie de la cinétique de

rehaussement en fonction du temps. Néanmoins cela ne repose sur aucune hypothèse

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physiologique. Cette approche permet d’extraire des paramètres semi-quantitatifs (pente de

rehaussement, temps au pic [TTP], aire sous la courbe [AUC], etc.) et présente des variations

considérables selon la méthode d’acquisition employée pour chaque examen individuel et

selon la méthode d'imagerie utilisée. La deuxième méthode, plus complexe, utilise les

modèles pharmacocinétiques qui tentent plus ou moins d’intégrer les contraintes spécifiques à

la perfusion du milieu étudié. Ces modèles reposent généralement sur la détermination d’une

fonction d’entrée artérielle obtenue par la courbe de rehaussement des vaisseaux afférents

(aorte, artère afférente). Elle permet de prendre en compte les paramètres hémodynamiques à

un moment donné et d’ajuster au mieux le modèle. Mais ceci peut parfois s'avérer délicat et

des alternatives ont été proposées (autre tissu, moyenne). Par la suite l’application

d’algorithmes de déconvolution et/ou de modèles compartimentaux nécessite de définir le

nombre de compartiments adaptés aux tissus explorés. De nombreux modèles

compartimentaux existent, dont le plus connu est le modèle dit de Tofts (13, 14). Cette

analyse requiert plusieurs hypothèses : l’agent de contraste utilisé doit être mélangé en

concentration uniforme à travers tout le compartiment, le flux sanguin doit être linéaire entre

les compartiments (échange passif uniquement) et enfin les paramètres décrivant les

compartiments doivent être invariants durant l’acquisition des données. En fonction de la

modélisation, on extrait alors les paramètres d’intérêt qui restent dépendant du modèle choisi.

Ainsi, le choix du modèle dépendant de l’organe étudié et de l’expérience des utilisateurs, les

paramètres obtenus dépendant du modèle utilisé, tout cela rend difficile la généralisation des

données. Toutefois, comme plusieurs molécules ciblent cette voie (bevacizumab, inhibiteurs

kinases comme le sunitinib), l’intégration de l’imagerie dans les essais thérapeutiques devrait

aider à valider ce concept à grande échelle.

Par ailleurs, comme les protéines issues des gènes mutés interagissent les unes avec les

autres selon les voies de signalisation avantageant les cellules cancéreuses (5), les traitements

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donnés auront tendance à bloquer cela. L’imagerie, par l’observation des conséquences de

l’effet, sera capable potentiellement d’observer cela par les effets induits avec plus ou moins

de spécificité. Par exemple, les récepteurs aux tyrosines kinases (ex : EGFR) jouent un rôle

majeur dans la prolifération. Ainsi les inhibiteurs de ce type de récepteurs à tyrosine kinase

(15) agissent sur un grand nombre de voies (PI3K/ AKT/mTOR/RAS /RAF/MAPK/platelet-

derived growth factor/VEGFR2/c-KIT) et induisent une nécrose cellulaire, observable en

imagerie même si elle apparaît peu spécifique d’une voie donnée. Ceci est la base des critères

d’évaluation de Choi. Afin de faciliter ces analyses, le développement d’outils d’analyse des

images semi-automatiques pourrait aboutir à une meilleure appréciation.

Mais il est possible d'aller encore plus loin grâce aux mathématiques appliquées. En

effet l'utilisation de modèles mathématiques pourrait aider à mieux comprendre le vivant.

Traduire une observation clinique, obtenue par l'imagerie par exemple, par un modèle

descriptif voire prédictif pourrait permettre de développer des outils, des techniques et des

théories mathématiques capables d'approfondir nos connaissances dans le monde réel une fois

tout cela appliqué inversement. De nouvelles perspectives en oncologie mais aussi en

imagerie sont ainsi envisageables.

1.3. Développement de la modélisation en oncologie

La modélisation mathématique a été développée en cancérologie dans le but de

comprendre et de prédire la croissance tumorale mais également d’anticiper les effets des

thérapies ciblées ou non (ex : anti-angiogéniques). Ces modèles sont très variés et concernent

de nombreuses étapes impliquées dans l’évolution des tumeurs. Ainsi, ils peuvent par

exemple aider à comprendre l'influence de la régulation génétique (16) ou à prédire un

14

phénomène observé expérimentalement (17). Cependant, dans la plupart des cas, ces modèles

sont mieux adaptés aux études in vitro, car elles sont souvent focalisées sur un faible

contingent cellulaire dont l’évolution est plus facilement observable. In vivo, la problématique

devient plus complexe car elle doit prendre en compte une tumeur à part entière contenant un

grand nombre de cellules et donc de mécanismes à décrypter. De plus, contrairement aux

études in vitro, la quantité d'informations disponibles sans modélisation sur une tumeur in

vivo est très limitée car « noyée », en raison de ce grand nombre de cellules et donc de

signaux. Il apparait donc essentiel de déterminer les techniques permettant de recueillir le

maximum d'informations afin d'optimiser ces modèles.

Hormis le recours aux biopsies qui permettent d’obtenir des informations à l’échelle

microscopique néanmoins échantillonnées, l’utilisation des examens d’imagerie s’est imposée

car elle permet de récupérer de façon non invasive des informations répétées dans le temps.

La taille (ou volume), la forme, la localisation, le rehaussement à l’injection de produit de

contraste sont des informations assez faciles à obtenir alors que des informations de micro-

architectures ou de perfusion apparaissent toujours délicates en fonction de l’organe ciblé.

Ainsi les informations obtenues peuvent permettre d’affiner les modèles et à terme de les

appliquer en pratique clinique.

Pour surmonter cela, l’apport de l’imagerie morphologique intégrée aux modèles dits

« spatiaux » semble déterminant. L’association de l’imagerie et de la modélisation permet de

résoudre des problèmes complexes et ainsi de décrire de nombreuses facettes de la croissance

tumorale ou de la réponse thérapeutique. L’imagerie fonctionnelle pourrait encore apporter

des informations supplémentaires sur le métabolisme ou la micro-architecture et permettre

ainsi d'aller plus loin dans la complexité des modèles, toutefois il apparait encore nécessaire

de l'optimiser pour son application au quotidien.

15

Il est maintenant possible d’envisager à moyenne échéance une utilisation clinique de

description mais aussi de prédiction de la croissance ou de la réponse pour un patient donné.

Cela pourrait permettre d'optimiser au mieux la prise en charge thérapeutique. Les

développements que nous avons progressivement réalisés sont exposés successivement dans

ce manuscrit.

1.4. Plan du travail

Cette thèse a consisté à optimiser les modèles de croissance tumorale développés au

sein de l’Institut Mathématiques de Bordeaux, sous la direction du Pr T. Colin et du Dr O.

Saut, en incluant des éléments microscopiques (niveau cellulaire) et macroscopiques (niveau

tissulaire et de l’organe) obtenu par analyse des informations disponibles des examens

d’imagerie. Plusieurs localisations comme le poumon, le foie, le rein et la prostate ont été

successivement étudiées afin d’enrichir progressivement les modèles et répondre ainsi à la

réalité clinique. De façon concomitante de nombreux outils ont été intégrés au fur et à mesure

afin de standardiser la démarche de recueil de données. Ainsi l'implémentation de l'imagerie

fonctionnelle dans une pratique clinique a été réalisée en parallèle. Le but est à terme

d’appliquer de façon prospective ces outils, basés sur l’imagerie et les mathématiques

appliquées en pratique quotidienne.

L'objectif de ce chapitre est de faire une présentation synthétique des ces différents

travaux afin de donner un aperçu global de cette thèse. Les articles scientifiques publiés ou en

cours de soumission sont colligés par la suite pour plus de détails sur la méthodologie et les

résultats.

16

1.4.1. Modélisation mathématique de la croissance

tumorale basée sur l'imagerie: généralités et perspectives

De façon volontairement schématique, plusieurs étapes successives peuvent être

identifiées dans la croissance d’une tumeur solide due à la dérégulation de la division

cellulaire: la croissance avasculaire initiale, l’angiogénèse et finalement la diffusion

métastatique.

Durant la phase de croissance avasculaire, la tumeur (selon l’hypothèse monoclonale

du cancer, cette tumeur est initialement composée d’une cellule anormale) grossit aux dépens

de son environnement proche sur un modèle caractéristique dit « en couche » (18). Au centre

de la tumeur, l’apport en nutriments n’est pas suffisant et une nécrose peut être observée.

Entre les deux couches, on trouve une couche de cellules quiescentes, qui survivent sans se

diviser, avec un métabolisme restreint et lent. En suivant ce type de modèle, la croissance de

la tumeur est alors sévèrement limitée et pour poursuivre son développement, il lui faut

trouver des sources additionnelles de nutriments. La tumeur rentre alors dans le deuxième

stade de sa croissance : la croissance vascularisée. Sous la pression hypoxique, certains types

de phénotypes peuvent être favorisés (au sens de la sélection darwinienne). Certaines cellules

se mettent alors à produire des signaux (VEGF par exemple) qui encouragent l’organisme à

vasculariser la tumeur et donc à lui fournir de nouveaux nutriments (c'est le processus

d’angiogénèse). La prise en compte de cette étape est importante pour plusieurs raisons. En

effet, une fois vascularisée la tumeur peut atteindre des tailles beaucoup plus significatives

que dans le stade précédent et peut alors être visible en imagerie. Finalement, les phénomènes

d’effraction vasculaire aboutissent alors à la dissémination métastatique.

17

Il est ainsi aisément compréhensible que du fait de la complexité des processus

engagés entre toutes ces étapes, les modèles mathématiques sont en grande partie

phénoménologiques et très simplifiés par rapport à la réalité biologique. Il est possible de

distinguer deux types d’approches principales de cette modélisation. Elles dépendent de

l’intégration ou non de la tumeur dans l’espace avoisinant.

Les modèles non spatiaux calculent l’évolution d’une ou plusieurs grandeurs scalaires

adéquates pour suivre la croissance tumorale. Les exemples classiques sont les modèles basés

sur des équations différentielles ordinaires (EDO) qui décrivent l’évolution du volume ou du

rayon de la tumeur. Ils ne prennent pas en compte l’environnement de la tumeur et ses

interactions mais simplement des grandeurs mesurables (19). Toutefois, ces modèles sont trop

simples pour fournir une information intéressante, fiable et reproductible sur la tumeur à part

l’évolution phénoménologique de son volume qui est finalement assez aléatoire d’une tumeur

ou d’un patient à l’autre. Ils peuvent être néanmoins complexifiés pour faire intervenir

plusieurs compartiments (fonction du cycle cellulaire) ou pour prendre en compte différents

processus comme l’angiogénèse par exemple.

A l’opposé, l’environnement tumoral est pris en compte dans les modèles spatiaux.

Plusieurs méthodes ont été proposées pour modéliser ainsi la croissance tumorale (20, 21).

L’une consiste à décrire l’évolution des cellules individuellement (22) et donne lieu aux

modèles "discrets" de type automates cellulaires (23) ou modèles d’agents (24). Ces modèles

relativement simples sont extrêmement efficaces pour simuler des croissances in vitro, pour

décrire des phénomènes microscopiques et effectuer des couplages au niveau moléculaire (25).

En revanche, il ne semble pas possible à l’heure actuelle -car trop difficile et coûteux

numériquement- de modéliser ainsi un ensemble complexe de cellules dans son

environnement, c'est-à-dire une tumeur dans son ensemble.

18

Une autre approche de modélisation spatiale consiste à travailler sur plusieurs cellules

ou populations de cellules en même temps et donc à décrire l’évolution de densités cellulaires

(ou des frontières des tumeurs par exemple). Elle permet d’étudier des tumeurs de tailles

réalistes et rend bien compte des effets macroscopiques (interactions entre les cellules et une

membrane extra-cellulaire ou l’adhésion cellulaire par exemple) grâce aux modèles inspirés

de la mécanique des milieux continus (26). Le tissu étudié est alors considéré comme un

continuum régi par une loi de comportement dans cet ensemble. Il est alors facile de tenir

compte d’un environnement complexe et étendu, mais il est en revanche difficile de prendre

en compte tous les phénomènes microscopiques indépendamment ou non (26). Toutefois,

certains modèles prennent en compte l’angiogénèse (25, 27), d’autres s’intéressent plus

particulièrement au chimiotactisme (28). Parallèlement, en ce qui concerne la description des

tissus, des modèles d’interaction de populations de cellules ont été étudiés, de façon mono ou

multiphasique (29, 30). La modélisation du cycle cellulaire a été également explorée (31),

dans le cadre de l’optimisation de traitement (32-34) . Ainsi, ces modèles spatiaux complexes

peuvent intégrer un cycle cellulaire à 2 phases : une à cellules proliférantes, l’autre à cellules

quiescentes. Une phase supplémentaire de nécrose reste envisageable. D'autres éléments

peuvent être pris en compte comme la diffusion de l’oxygène, le processus d’angiogénèse,

l’interaction avec des membranes (35) ou les tissus environnants…

Ces modèles spatiaux, qui ont l’avantage d’intégrer beaucoup de connaissances d’un

point de vue biologique, présentent plusieurs intérêts qui justifient leur utilisation. En effet, ils

sont censés reproduire des comportements réalistes, ce qui est le but recherché, et permettent

également de tester des hypothèses d’interaction (36) entre les cellules dans un ensemble, par

exemple une tumeur. Néanmoins ils sont souvent très complexes car ils contiennent beaucoup

de paramètres souvent inaccessibles à l'expérience, et ils ne sont pas spécifiques d'un patient

19

donné, car les valeurs sont très variables d’une situation à l’autre. Ils ne peuvent donc pas être

appliqués en l’état pour une utilisation clinique.

Pour adapter ces modèles spatiaux à chaque pathologie et/ou patient, il est donc

nécessaire de les simplifier et de les calibrer à l’aide de données cliniques, pouvant être

obtenues par l’imagerie, permettant d'estimer les différents paramètres intervenant dans les

équations et d’obtenir ainsi des résultats quantitatifs. Nous avons donc développé des modèles

pour les métastases pulmonaires ou encore certaines tumeurs stromales (GIST).

Plus de détails se trouvent dans cet article présenté dans ce manuscrit au chapitre 2:

Cornelis F, Saut O, Cumsille P, Lombardi D, Iollo A, Palussiere J, Colin T. In vivo

mathematical modeling of tumor growth from imaging data: soon to come in the future?

Diagn Interv Imaging. 2013 Jun;94(6):593-600.

1.4.2. Modélisation de la croissance des nodules

pulmonaires

Les tumeurs pulmonaires ont été initialement sélectionnées car il est possible de

bénéficier d’un excellent contraste en scanner entre les tumeurs, de densité tissulaire (0-40

UH), et le parenchyme pulmonaire sain environnant rempli d’air de densité plus faible (-1000

UH). Les métastases ont ensuite été privilégiées dans nos modèles car ces tumeurs

secondaires sous leur forme nodulaire peuvent avoir des contours plus nets en raison de leurs

modes de croissance comparées aux tumeurs primitives (ayant pour origine le tissu

pulmonaire), plus infiltrantes.

20

Un avantage d'utiliser ce modèle est que les cancers secondaires pleuro-pulmonaires

sont fréquents. Ils viennent en 3ème position après les métastases ganglionnaires et

hépatiques et ils sont retrouvés dans 30% des autopsies de patients porteurs d’une néoplasie.

La survie moyenne après constatation de métastases pulmonaires non traitées est de 9 à 11

mois mais varie en fonction du nombre, du type histologique, et du traitement proposé

(traitement local chirurgical ou ablatif, radiothérapie ou traitement général comme la

chimiothérapie). Les principales tumeurs associées à des métastases pulmonaires sont le

cancer du sein (19%), le cancer de l’œsophage/estomac et le cancer du colon/rectum (17%), le

cancer des reins et de la vessie (10%), le cancer génital (10%), les sarcomes et en particulier

les ostéosarcomes (9%), les cancers ORL (4%), les cancers de la thyroïde (3%), les cancers de

prostate (2%), puis les autres cancers (6%) ou restant indéterminés (15%). L'origine peut-être

par contiguïté, par voie hématogène, ou par voie lymphatique. Et de ce fait, trois présentations

radiologiques existent. Les nodules pulmonaires correspondent à la dissémination hématogène.

Ils sont en général multiples, de taille variable réalisant l'aspect de lâcher de ballons ou une

miliaire carcinomateuse. Ils sont visibles à la radiographie du thorax mais le scanner

millimétrique est plus performant pour la détection des petits nodules (<5-10mm). Le nodule

unique est également fréquent et pose beaucoup plus de problèmes diagnostiques car il peut

être en rapport avec des lésions bénignes ou primitives du poumon. Un des principaux

arguments en faveur de sa nature secondaire est sa croissance rapide ou son apparition par

rapport à des examens antérieurs dans un contexte bien sûr de néoplasie. La lymphangite

carcinomateuse correspond enfin à une diffusion par voie lymphatique. Radiologiquement il

s’agit d’un syndrome réticulomicronodulaire diffus, souvent bilatéral associé à des

épaississements périlobulaires. De nouveau, le scanner réalisé en coupes fines est bien plus

performant que la simple radiographie thoracique.

21

Dans nos travaux, la forme nodulaire a été privilégié car l'observation d'un net

contraste entre la lésion et le parenchyme semblait plus facile à reproduire avec nos modèles

mathématiques qui intègrent une loi de Darcy et une disposition en couche, qui en général

favorisant ces formes nodulaires. Le caractère infiltrant, par exemple stellaire dans le cadre

d'une tumeur primitive ou présent lors d'une lymphangite carcinomateuse, peut être une limite

aux modèles car nécessite d’intégrer des paramètres supplémentaires.

Toutefois, il existe des limites à l'usage de ces nodules dans le cadre de la

modélisation. Un des principaux problèmes a été de réaliser une segmentation adéquate. Nous

nous sommes aperçus que la segmentation du nodule reste un acte médical qui dépend de

l'opérateur, même si la dépendance au radiologue s'est atténuée avec le temps avec

l'apprentissage. Elle peut varier entre différents intervenants et être une source d’erreur. Une

segmentation complétement automatique est difficile à réaliser en raison de la fréquence des

atélectasies d’aval dues à la compression tumorale sur les voies aériennes. Une analyse semi-

quantitative est néanmoins possible. Le caractère extrêmement variable du volume

pulmonaire et du contenant, en raison des variations d’acquisition sans synchronisation à la

respiration qui complique le recalage d’un examen à l’autre, est également un facteur limitant.

Nous avons donc étudié l'erreur générée par la segmentation. Ce travail est rapporté par la

suite. Par ailleurs, en raison du caractère agressif de la pathologie, il est rare de bénéficier

d’un suivi suffisant sans thérapeutique rendant difficile l’évaluation de la simple croissance

des nodules pulmonaires par la modélisation sur un délai suffisant. Compte tenu de

l’hétérogénéité des thérapeutiques employées, il apparait aussi difficile d’intégrer au modèle

ces traitements pouvant jouer sur différents paramètres.

Des modèles à équations différentielles ordinaires (EDO) et à équations aux dérivées

partielles (EDP) ont été développés intégrant différent paramètres prédéfinis, mais dont les

valeurs sont inconnues. Ces paramètres sont estimés selon deux approches: l'une

22

populationnelle permettant d’intégrer la variabilité inter-patient et l'autre individuelle, basée

sur l’information 2D ou 3D et ayant pour but d’intégrer différentes informations comme la

stratégie thérapeutique.

Pour fabriquer un modèle simplifié, nous avons contracté le cycle cellulaire en une

EDP n’ayant plus de structuration en âge et gérant une population de cellules proliférantes

(dont la concentration est notée P). Le taux de croissance est géré par une concentration en

oxygène locale donnée par une équation de diffusion. Si le taux d’oxygène est trop bas

(hypoxie), non seulement les cellules ne prolifèrent plus mais elles passent en phase

quiescente (notée Q) (phase dans laquelle les cellules tumorales restent en vie mais ne

prolifèrent plus). Bien sûr, si le taux d’oxygène augmente, les cellules peuvent faire le chemin

inverse. Le ratio proliférantes/quiescentes est un indicateur de l’agressivité de la tumeur. La

prolifération cellulaire induit une augmentation de volume que l’on explicite comme étant la

divergence d’un champ de vitesse. Ce dernier n’a rien à voir avec la vitesse que pourraient

avoir par exemple des cellules infiltrantes, mais décrit le mouvement global dû à

l’augmentation de volume. On ferme le système en imposant une loi de Darcy sur cette

vitesse avec une perméabilité qui dépend du tissu (sain ou tumoral). Finalement, la

concentration en oxygène est donnée par une équation de diffusion dont le coefficient dépend

du tissu et avec des termes d’absorption dûs à la consommation ces cellules tumorales.

Même si ce système est très simplifié, il contient déjà 7 paramètres à identifier et

l’unique valeur observable est le volume tumoral qui correspond à la somme P+Q de cellules

tumorales, éventuellement associé à de la nécrose. En se basant uniquement sur l’imagerie

morphologique, c’est à dire étudiant la forme et le rehaussement d’une tumeur, l’information

disponible sur les examens d’imagerie est donc très restreinte. Afin de résoudre le problème

inverse, qui correspond à prédire l'évolution à partir de la forme de la tumeur à deux temps

antérieurs, nous avons mis en place une méthode spécifique.

23

De façon très simplifiée, cette méthode consiste à faire plusieurs centaines de calculs

avec le modèle mais avec des jeux de paramètres différents puis à resoudre le problème

inverse. Cette approche à plusieurs avantages :

• elle permet de résoudre le problème inverse sur des équations différentielles ordinaires

ce qui fait un volume moindre de calcul,

• elle permet de se restreindre à un espace de paramètres pertinent (ceux que l’on a

utilisés pour fabriquer la base de données de solutions),

• elle permet une régularisation car on a éliminé les hautes fréquences par le processus

de sélection de valeurs propres.

En parallèle nous nous sommes intéressés aux modèles EDO. La dimension spatiale de

la maladie n’est pas prise en compte dans les modèles EDO puisque la forme ou

l’emplacement de la tumeur ne sont pas calculés par le modèle. Cependant cela n’a pas

empêché leur utilisation efficace pour la mise au point de protocoles thérapeutiques. Leurs

paramètres sont classiquement trouvés avec des méthodes statistiques (ce qui peut empêcher

de travailler sur des patients spécifiques). Néanmoins concernant les modèles EDO, nous

avons développé un modèle semblable au précédent modèle EDP pour analyser des données

longitudinales de taille tumorale. Un modèle simple de type Darcy peut produire les formes

observées expérimentalement.

Les stratégies développées et les résultats de ce travail se trouvent dans ces 2 articles

rapportés dans ce manuscrit au niveau du chapitre 3:

1. Colin T, Cornelis F, Jouganous J, Palussière J, Saut O. Patient specific image driven

evaluation of the aggressiveness of metastases to the lung. Med Image Comput Assist

Interv. 2014;17(Pt 1):553-60.

24

2. Thierry Colin, François Cornelis, Julien Jouganous, Jean Palussière, Olivier Saut.

Patient-specific simulation of tumor growth, response to the treatment, and relapse of a

lung metastasis: a clinical case. Journal of Computational Surgery 2015, 2:1 (4 February

2015)

1.4.3. Modélisation de la croissance des métastases de

tumeurs stromales de l’intestin grêle

Compte tenu des limites de l’utilisation des localisations secondaires pulmonaires en

termes d’observation des effets thérapeutiques, il a été nécessaire de rechercher un modèle

intégrant un schéma de traitement relativement standardisé. Parmi les nombreuses pathologies

cancéreuses, les tumeurs stromales gastro-intestinales (GIST) métastatiques au niveau du foie

semblent répondre à ces critères. En effet, avec la disponibilité des agents thérapeutiques

ciblés très actifs dirigés contre les altérations moléculaires causales et malgré l'importante

variation dans les caractéristiques moléculaires et génétiques qui conduisent la pathogenèse

de ces tumeurs. Les GIST sont devenus les modèles de traitement personnalisé du cancer (37).

Le traitement est ainsi adapté en fonction des mutations génétiques observés mais

généralement est basé sur un traitement initial par anti-tyrosine Kinase (Glivec) puis suivi en

cas d’échappement par antiangiogénique (37).

Les GIST sont les tumeurs mésenchymateuses les plus fréquentes du tractus gastro-

intestinal, avec une incidence de cette maladie de 9-14 cas par million de personnes par an

(38-41). Elles sont assimilées aux sarcomes. Dans la majorité des cas elles se développent au

niveau de l'estomac, du jéjunum, de l'iléon. Toute GIST est potentiellement maligne. Le plus

souvent, elles sont caractérisées par un profil immunohistochimique spécifique (c-kit/CD

117+/DOG.1+) (42). Des altérations moléculaires dans le gène de Kit ont été observées dans

25

ces tumeurs (37) entrainant ainsi un gain de fonction (43). Actuellement, 10 sous-ensembles

moléculaires différents de GIST avec différentes altérations moléculaires ont été rapportés:

les mutations de KIT sont plus fréquemment rapportées au niveau des exons 11 (67%) , 9

(10%) , 13 (1%) , ou 17 (1%) , alors que des mutations de PDGFRA , qui sont moins

fréquentes que les mutations de KIT , se produisent principalement dans les exons 18 ( 6 %) ,

12 ( 0,7 % ) , ou 14 ( 0,1 % ) . En plus de la mutation primaire causale, des mutations acquises

secondaires ont été également identifiées chez des patients atteints de GIST avancé

antérieurement traités avec un inhibiteur de tyrosine kinase. Ceci se traduit par un

échappement au traitement antérieur. Ces mutations secondaires sont généralement situés au

niveau de l’exon 13 (impliqué dans le métabolisme de l’ATP) et de l’exon 17 (44).

De ce fait le pronostic et la sensibilité aux traitements ciblés différents permettent

l'identification de sous-types distincts. L'absence de mutation kit/PDGFR alpha, de mutation

de l’exon 9 de KIT ou de l’exon 18 de PDGFR alpha (résistance au traitement) est un facteur

pronostique défavorable. De plus, une mosaïque tumorale peut être observée à l’échelle non

seulement inter-individuelle bien évidemment mais également intra-individuelle. Ceci semble

se majorer au cours du temps notamment sous la pression de l’environnement et des

traitements réalisés. Cette observation n’est pas spécifique des GIST.

26

Figure 5.1– Représentation de la mosaïque tumorale rencontrée dans la pathologie

métastatique. D’après Vogelstein B, Papadopoulos N, Velculescu VE, Zhou S, Diaz LA Jr,

Kinzler KW. Cancer genome landscapes. Science 2013; 339:1546–1558.

Par ailleurs, dans le cas de ce type de tumeur la segmentation nous apparaissait aisée

car il existe souvent un bon contraste entre la lésion hépatique et le parenchyme adjacent. Un

avantage supplémentaire est la possibilité d’intégrer à terme facilement des données issues de

l’imagerie fonctionnelle compte tenu de la disponibilité de ce type d’examens pour le foie

(échographie de contraste, TEP).

Le but de notre projet était donc de proposer un modèle mathématique basé sur les

évaluations d'imagerie de métastases hépatiques de GIST localement avancé afin de

déterminer le temps de l'émergence de sous-clones, la simulation de la repousse de la tumeur

et de proposer de façon optimales les séquences thérapeutiques ciblés. Afin d’apprécier

l’évolution des localisations secondaires de GIST au niveau hépatique, il a été réalisé dans un

premier temps une étude longitudinale rétrospective de la base de données des patients suivis

with unsuccessful clinical trial results, possibly partlybecause of unselected population or histology ratherthan selective administration to patients based on mu-tation status (ie, EGFR or KRAS status for lung andcolon cancer, respectively). Biomarkers may facilitate,expedite, and validate targeted therapies and drugdevelopment. Biopsy tissue can be used before andafter drug administration to “credential” a drug asproducing its intended effect via an intended mecha-nism (49). Sequential biopsies before and after candidatedrug administration may be faster than waiting forclinical or imaging responses (50). For example, aphase II clinical trial of vandetanib, targeting EGFR,VEGFR2, and ret, in ovarian cancer incorporatedIR-directed biopsies before treatment and 1 month afterdaily drug administration (51). Biopsy specimens of thesame site were obtained on both occasions, and thetissue was examined for proteomic evidence of phar-macodynamic activity. The study demonstrated a lack ofclinical benefit correlated by imaging and IR. Tumorperfusion remained unchanged on dynamic contrast-enhanced magnetic resonance imaging and computedtomography. VEGFR2 activation was also stable fromanalysis of IR-directed biopsies. This example demon-strates the power of multidisciplinary intervention whereIR and imaging specialists are critical to the ability tounderstand the success, or in this case lack of success, ofthe targeted therapy. Clinical trials should consider

sequential biopsies as a tool for validating biomarkersand translational endpoints and speeding developmentof personalized targeted therapies. The success of tar-geted therapies may have been limited by lack ofvalidation of predictive biomarkers, acquired resistance,and molecular site heterogeneity (differences betweenprimary and metastatic tumors) (Fig 4a–d) (3,8). Addi-tional confounding factors explaining limited success oftargeted therapies include intratumoral heterogeneity(8). Although a tumor may appear homogeneous onimaging, expression of mutations may be hetero-geneous (20). Administration of targeted therapiesinduces secondary changes in mutations and molecularprofiling (1). This characteristic can be used to studydrug efficacy but also demonstrates the evolving natureof this process. Local therapies such as chemoembo-lization and radiofrequency have been documented toelicit an immune response that may be used to mo-dulate response to other systemic therapies and in-crease susceptibility to targeted therapies (52). IR inpersonalized medicine is currently limited to tissueprocurement but has the potential to expand further.

CONCLUSIONSThe era of personalized medicine presents a great challengeand opportunity for imaging and biopsy. Radiology and

Figure 4. (a–d) Different types of tumor heterogeneity are illustrated using pancreatic tumor and its metastasis as an example.(a) Mutations during cell growth result in heterogeneity within tumor. (b) As metastases are derived from different clones of primarytumors, intermetastatic heterogeneity is observed. (c) As metastases grow, they derive various mutations resulting in intrametastaticheterogeneity. (d) Finally, differences between patients result in interpatient heterogeneity. (From Vogelstein B, Papadopoulos N,Velculescu VE, Zhou S, Diaz LA Jr, Kinzler KW. Cancer genome landscapes. Science 2013; 339:1546–1558. Copyright © 2013 TheAmerican Society for the Advancement of Science. Reprinted with permission from The American Society for the Advancement ofScience.) (Available in color online at www.jvir.org.)

Abi-Jaoudeh et al ’ JVIR1090 ’ Personalized Oncology in IR

27

à l’Institut Bergonié et dont l’imagerie était disponible sur le réseau d’image (depuis 2005).

Au total, douze patients répondaient aux critères de sélection (sur 130). Chaque dossier

clinique a été revu et l’évolution du produit des 2 grands axes de la localisation ou de la plus

volumineuse a été compilé en fonction du temps. Pour chaque dossier, il a été rapporté le type

de mutation préexistante si l’information était disponible. L’évolution temporelle du produit

des grands axes a montré un profil évolutif différent selon la présence ou non d’une mutation.

Deux approches peuvent être ainsi discutées et modélisées conjointement. La première

hypothèse est de représenter au sein d’un ensemble de cellules observées, une population

mutée, résistante à tel ou tel traitement, en minorité initialement et qui serait au cours du

temps et des traitements sélectionnée par la variation de la pression de l’environnement. Cela

pourrait être en raison de la nécrose des autres clones cellulaires par exemple. L’avantage de

cette approche est qu’elle pourrait facilement expliquer les phases de plateau et de croissance

en fonction des thérapeutiques. L’autre serait d’estimer l’apparition progressive d’une

résistance par mutation au sein d’une seule population cellulaire en se référent à la théorie de

la mutation causale suivie de mutations acquises.

Les détails concernant la méthodologie et nos premiers résultats se trouvent dans cet

article, rapporté au chapitre 4: Lefebvre G, Cornelis F, Cumsille P, Colin T, Poignard C, Saut

O. Spatial modeling of tumor drug resistance: the case of GIST liver metastases.

Mathematical Medicine & Biology 2015. Revision

1.4.4. Analyse de l'erreur de contourage

28

A la lumière des premiers résultats sur le poumon et sur le foie, il a été proposé

d'évaluer l'impact du contourage dans les processus de simulation. En effet il nous est

rapidement apparu que l'erreur générée par la segmentation devait être connue afin de la

prendre en compte dans nos résultats. De plus il nous a semblé souhaitable de connaitre la

variabilité inter-observateur entre des personnes aguerries à cette tache (radiologues) et

d'autres plus novices dans la segmentation au quotidien (chercheurs). En conséquence, il a été

réalisé par 20 opérateurs différents (10 radiologues / 10 chercheurs) une segmentation de

métastases en utilisant le logiciel Osirix. Il y avait au total 7 lésions du poumon et 6 lésions du

foie à segmenter, choisies au préalable par 2 auteurs. Pour chacune de ces lésions, deux

segmentations ont été réalisées consistant simplement à contourer une lésion en 2D selon le

bord externe. Le diamètre était extrait automatiquement. Pour la première, l'operateur devait

choisir la coupe correspondant au critère RECIST : celle pour laquelle la lésion a le plus

grand diamètre. Ensuite, le même opérateur devait segmenter la lésion dans un plan imposé.

Les résultats ont été enregistrés et comparés.

Les résultats de cette étude ont montré que l'erreur de segmentation était plus marquée

quand la taille augmentait ainsi que pour les lésions hépatiques (par rapport à celles du

poumon). Ils suggèrent également que les critères RECIST peuvent être pris en défaut dans le

cas de localisations hépatiques de grandes tailles (>2-3cm). Les résultats ont été récemment

intégrés dans le processus de simulation afin d'améliorer la prédiction. L'ébauche de cet

article rapportant nos résultats se trouve dans le manuscrit au chapitre 5.

1.4.5. Intégration de l'imagerie fonctionnelle

1.4.5.1. Développement de l'imagerie fonctionnelle

29

Comme nous l'avons évoqué auparavant, plusieurs techniques d’imagerie sont

capables d’apprécier la vascularisation et le contenu en oxygène des tissus. En complément

des critères morphologiques, l’imagerie fonctionnelle semble tout particulièrement

intéressante pour l’évaluation précoce des thérapies ciblées. En effet, cette imagerie peut être

au mieux focalisée sur la résultante du mécanisme d’action de la drogue comme c’est le cas

pour les techniques d’évaluation de la perfusion tumorale lors des traitements anti-

angiogéniques. L’enjeu est d’importance afin de mieux appréhender une pathologie, de

limiter la durée des traitements non ou peu actifs, et en conséquence les effets secondaires

ainsi que les coûts.

Les changements du flux sanguin et de la perméabilité microvasculaire (Ktrans) au

sein d’un tissu peuvent précéder la diminution en taille d’une lésion (45) après traitement

ciblé par anti-angiogénique (anti-VEGF) ou par anti-tyrosine-kinase (TKI) (46).

L’identification de caractéristiques fonctionnelles des patients grâce à ces différentes

modalités d’imagerie semble donc une source de développement prometteuse. Cela

permettrait l'identification de patients appropriés pour une thérapie ciblée en effectuant une

évaluation de base unique et prédire par la suite les résultats cliniques à l’échelle de la

population sélectionnée. De plus, Sahani et al (47) a rapporté que l’imagerie fonctionnelle

était plus sensible pour prédire l’évolution tumorale que les simples critères morphologiques.

Toutefois il existe encore peu de séries publiées et de corrélations radio-pathologiques.

En échographie de contraste, la plupart des études publiées se limitent à des analyses

descriptives des modifications des courbes de rehaussement en raison de limites techniques

(48). Les modifications précoces observées (j0 à j3) de plusieurs paramètres sont associées à

la réponse tumorale (notamment l’AUC et l’intensité du pic de rehaussement) et certains ont

même une valeur pronostique en termes de survie sans progression (intensité du pic de

rehaussement) ou de survie globale (l’AUC). En tomodensitométrie, la densité

30

microvasculaire des carcinomes rénaux est corrélée aux paramètres perfusionnels (flux

sanguin, volume sanguin, Ktrans) (49). Ces marqueurs peuvent donc être utilisés pour

identifier initialement les populations à risque d’échec thérapeutique mais également pour

permettre l’évaluation thérapeutique au cours du temps. En IRM, il a été observé des résultats

similaires (50, 51). Ainsi dans le cancer du rein la perméabilité de base (Ktrans) pourrait être

utilisée dés à présent comme un facteur pronostique (46, 52) alors que son impact dans

l’évaluation thérapeutique doit être encore évalué. L’imagerie de diffusion et la mesure de

l’ADC permettent également d’apprécier la réponse thérapeutique par des mécanismes

différents (51)(26, (53). Des résultats similaires ont été rapportés pour la technique d’ASL

(54) avec de faibles valeurs corrélées à une moindre sensibilité au traitement. Sur un modèle

de souris, il a été observé pour les répondeurs des variations significatives du débit sanguin

dans les 30 jours après initiation d’un traitement par sorafenib (55). L’IRM avec effet BOLD

apparaît comme étant une technique sensible permettant l’étude de l’hypoxie tumorale, mais

manquant peut-être de spécificité. Plusieurs études de faisabilité ont déjà proposé cette

technique non invasive pour cartographier l'hypoxie tumorale prostatique par exemple,

permettant de proposer un nouveau marqueur pronostique (56-59). Les résultats des

explorations réalisées en médecine nucléaire montre qu’un SUV élevé semble de plus

mauvais pronostic en FDG-PET (60) tout comme une tumeur hypoxique en [18F]-FMISO (61).

La TEP [18F]-FMISO a été utilisée comme marqueur non invasif de l’hypoxie dans de

nombreuses tumeurs dont les cancers du rein et a été étudiée dans la prédiction de la réponse

thérapeutique et l’évaluation du pronostic notamment dans les cancers des voies aériennes

supérieures. Ces analyses quantitatives restent difficiles car le taux de captation est souvent

faible et la clairance cellulaire est lente mais aussi du fait de la dissémination et de

l’hétérogénéité des zones hypoxiques au sein de la tumeur.

31

L'ensemble des techniques testées par notre équipe sont rapportées dans cet article,

présent dans ce manuscrit au niveau du chapitre 6: Cornelis F, De Clermont H, Ravaud A,

Grenier N. L'imagerie d'évaluation thérapeutique en pratique clinique d'oncologie urologique.

Prog Urol. 2014 Jun;24(7):399-413.

1.4.5.2. Développement d’outils de traitement de

l'imagerie fonctionnelle

Il persiste toujours un manque de standardisation des protocoles pour l’ensemble de

des techniques d'imagerie fonctionnelle. De plus le post-traitement n’est pas homogène et

souvent dépendant des équipes, que ce soit au niveau des modèles de calculs utilisés (n’étant

pas identiques), du recueil de données (région d’intérêt en 2D vs 3D (62)), de la réalisation

d’une normalisation. Afin de proposer une homogénéisation, nous avons travaillé sur une

nouvelle méthode d'analyse des examens d'imagerie.

Ce travail exploratoire a débuté sur le foie et a impliqué Simon Peluchon, en

partenariat avec General Electrics, puis Guillaume Lefebvre. L'objectif était de définir un

indicateur de l’échappement tumoral pour des localisations secondaires de GIST et de le

quantifier. En effet d’un point de vue morphologique et macroscopique, il est couramment

observé des hétérogénéités au cours de l’évolution de cette pathologie : ceci est la base des

critères de Choi. La relecture des dossiers de GIST a montré une éventuelle anticipation des

phénomènes d’échappement aux traitements par ces modifications de densité en scanner.

L’objectif de ce travail était donc d’identifier un indicateur relativement simple permettant

d'anticiper la réponse thérapeutique par rapport aux standards et d’améliorer la modélisation

mathématique. Pour cela un processus de traitement des images a été effectué.

32

Après avoir segmenté et filtré les images CT disponibles, il a été réalisé un calcul de

l’évolution de la variance et de l’histogramme des niveaux de gris, cela a permis de détecter la

présence de ces hétérogénéités dans les tumeurs. En appliquant un algorithme de mélange de

gaussiennes, il est possible de déterminer le nombre de classes contenues dans l’histogramme.

Plus ce nombre est grand, plus la tumeur possède une texture hétéroclite.

Pour compléter ce travail quantitatif, la mise en place d’outils de visualisation a été

effectuée. Ceux-ci offrent la possibilité à l’utilisateur de suivre facilement l’évolution de la

lésion et de superposer les différentes métriques calculées par ailleurs. La visualisation

graphique de l’algorithme de mélange de gaussiennes aide à la visualisation des

hétérogénéités.

Figure – Représentation de la variation de densités au cours du temps de métastases de GIST

(en haut à gauche) par l’application d’un algorithme de mélange de gaussienne (en bas à

gauche). Un codage en échelle de couleur permet la visualisation des différentes composantes

(à droite).

Poursuivant ces travaux, et en raison de la capacité des examens d’imagerie

d’apprécier la vascularisation / l'angiogenèse des tumeurs avant et/ou après traitement, notre

33

unité de recherche a proposé de réaliser une modélisation de la réponse thérapeutique aux

anti-angiogéniques au niveau des lésions rénales (et pulmonaires). Ce projet a été accepté

pour un financement par le Labex TRAIL en 2013. L'enjeu est d'importance, car l’incidence

du cancer du rein en France est estimée à 10 125 cas en 2009 (source INCa). Il représente

environ 3 % des tumeurs malignes de l’adulte. Son incidence est en augmentation depuis une

trentaine d’années, en rapport vraisemblablement avec un nombre plus important de

découvertes fortuites. Il est deux fois plus fréquent chez l’homme. L’âge moyen du diagnostic

se situe à 65 ans.�Le nombre de décès estimés en 2009 est de 3 830. Ce chiffre est en baisse, en

partie liée à une découverte plus précoce de ces cancers. En effet, la survie relative à 5 ans est

globalement de 63 %3. Pour un stade localisé (58 % des diagnostics), elle passe à 90 %. Le

pic de mortalité se situe entre 75 et 85 ans.

Le diagnostic de cancer du rein est le plus souvent suspecté de manière fortuite lors

d’une échographie ou d’une tomodensitométrie abdominale (60 % des cas). Le cancer du rein

peut également être révélé par une hématurie (typiquement totale, indolore, spontanée,

récidivante), une douleur du flanc, la palpation d’une masse lombaire ou une métastase. Ces

symptômes sont souvent constaté dans les formes localement évoluées. Il peut enfin être

découvert au décours d’un dépistage systématique dans le cas des formes familiales. La

tomodensitométrie (TDM) abdominale avec injection est l’examen de première intention

devant une suspicion clinique de tumeur rénale ou une découverte échographique. L'IRM est

actuellement en cours d'évaluation mais semble extrêmement intéressante. Ce diagnostic est

confirmé par l’examen anatomopathologique de la tumeur primitive ou de la métastase. Les

carcinomes à cellules rénales (CCR) représentent 85 % des cancers du rein de l’adulte. Cinq

autres types histologiques et de nombreux sous-types histologiques constituent les 15 %

restants. La néphrectomie, partielle ou élargie selon la taille, la localisation et le stade de la

tumeur, est le standard de prise en charge des formes localisées. �Les traitements ablatifs

34

(radiofrequence, cryoablation, mirco-ondes) peuvent être proposé en alternative. Le bilan

d’extension comprend une TDM thoracique. La radiographie de thorax n’est pas indiquée.

Une imagerie cérébrale et/ou osseuse est réalisée en cas de symptomatologie suspecte. �

Le cancer du rein est généralement résistant à la radiothérapie et à la chimiothérapie

cytostatique classique. �Le développement des thérapies ciblées (traitements anti-

angiogéniques, immunothérapie) a profondément modifié la prise en charge du cancer du rein

métastatique. � Le suivi est fondé sur la clinique, la biologie (évaluation de la fonction rénale),

l’imagerie (TDM abdominothoracique, IRM), selon une fréquence adaptée au stade de la

maladie. �Le développement de nouvelles techniques d'évaluation pourrait aider à mieux

suivre les patients traités par ces thérapies ciblées.

Le projet d'évaluation des métastases de tumeurs rénales, impliquant Agathe Peretti,

comporte plusieurs étapes. La première est l'intégration des séquences d’imagerie par

résonance magnétique et l’obtention de données multiparamétriques conventionnelles

(diffusion, perfusion). En suivant, de nouvelles séquences récemment développées vont être

intégrées telles que la séquence BOLD. Par la suite il sera intégrer des explorations en

imagerie nucléaire. Le but de ce projet est de mieux appréhender le processus impliqué dans

l’angiogenèse afin d’améliorer les simulations impliquant cette voie de signalisation. La

méthodologie est rapportée dans ce manuscrit au niveau du chapitre 6 même si la méthode

demande encore à être approfondie.

1.4.6. Modélisation du champs électrique dans les

procédures d'électroporation irréversible

35

Ce travail a été réalisé au Memorial Sloan Kettering Cancer Center à New York, USA,

avec l'assistance de Govind Srimathveeravalli, PhD sur des données de patients fournies par

le Dr Jonathan Coleman, MD, PhD. Ce projet rapporte une méthode permettant d'évaluer la

distribution du champ électrique lors des traitements par électroporation irréversible de la

prostate.

En effet, la détection du cancer de la prostate a changé entrainant une incidence accrue

de stade précoce et donc de cancers de petits volumes (63, 64). Le cancer de la prostate se

situe en France au premier rang des cancers avec une estimation à plus de 71 000 nouveaux

cas en 2011 (source INCa). Son incidence est en forte augmentation (+ 8,5 % par an entre

2000 et 2005) en raison de l'effet combiné du vieillissement de la population, de

l’amélioration de la sensibilité des techniques diagnostiques et de la diffusion du dépistage

par dosage du PSA. Parallèlement on observe une diminution de son taux de mortalité (en

moyenne - 2,5 % par an sur cette même période) du fait notamment de l'amélioration de

l’efficacité des traitements. Avec une survie relative à 5 ans estimée à près de 80%, c’est un

cancer de très bon pronostic. L’âge moyen au diagnostic est de 71 ans. Il est le plus souvent

découvert sur une élévation de la valeur du PSA sérique total, une anomalie de consistance de

la prostate détectée au toucher rectal ou plus rarement sur un examen anatomopathologique

du tissu prélevé lors du traitement d’une hypertrophie bénigne de la prostate. �Seules les

biopsies (échoguidées) avec examen anatomopathologique permettent de confirmer le

diagnostic. Toutefois, chez un patient asymptomatique, du fait du risque de surdiagnostic et

de surtraitement, l’indication de l'exploration tient compte de l’espérance de vie du patient, de

son état général, de la valeur du PSA et de sa cinétique d’évolution, du rapport

bénéfices/risques attendu par la mise en route d’un traitement et des préférences du patient.

Même si aucune imagerie n’est nécessaire au diagnostic, elle peut être utile pour le bilan

d’extension ou pour planifier une intervention: l’indication d’une imagerie et le cas échéant le

36

choix des examens sont définis par l’équipe de soins spécialisés et requiert de plus en plus la

réalisation d'IRM.

Alors que les traitements de référence sont la chirurgie ou la radiothérapie, les

stratégies de traitement sont maintenant adaptées au fait de l’évolution le plus souvent lente

de la maladie, apparaissant souvent chez des patients asymptomatiques et à risque d’évolution

faible. Dans ces cas, il peut être discuté en réunion de concertation disciplinaire de différer la

mise en route du traitement (y compris à visée curative) ou des traitements focaux qui

épargnent le tissu sain et limitent la morbidité post-opératoire. Le protocole de suivi est

ensuite adapté à chaque patient et à sa prise en charge initiale. �Le suivi repose en première

intention sur l’examen clinique, le dosage du PSA sérique total et éventuellement une IRM. Il

inclut la détection d’éventuels effets indésirables liés au traitement. �Une meilleure

connaissance des effets des thérapies focales permettrait de plus largement proposer ces

traitements.

Les interventions mini-invasives de traitement focal au niveau de la prostate sont

conçues pour fournir un contrôle oncologique local approprié avec des effets négligeables sur

la qualité de vie (65, 66). Ces techniques d'ablation focales entendent préserver les fonctions

érectile, urinaire et rectale en minimisant les dommages en périphérie au niveau des tissus

neurovasculaires, le sphincter urinaire, la vessie et le rectum. Jusqu' à présent, les techniques

d'ablation thermique comme cryoablation (67-69), ou les ultrasons focalisés (HIFU) (70-72)

ont été évalués pour le traitement de patients atteints de cancer de la prostate localisé avec de

bons résultats à court terme (72-75). Cependant, ces techniques d'ablation thermique exigent

une planification attentive pour éviter les dommages liés au traitement à proximité de la zone

d'ablation (74, 75).

L'électroporation irréversible (IRE) est une nouvelle modalité

thérapeutique qui utilise des impulsions courtes mais fortes de champ électrique pour créer

37

des micropores persistants dans les membranes plasmiques des cellules, conduisant à la mort

cellulaire. L'IRE a été évaluée dans l'ablation de la prostate (76-79) mais aussi dans d'autres

localisations comme le foie, les voies biliaires, le pancréas, ou les reins. Même si l'énergie

utilisée au cours de l'ablation IRE peut produire un échauffement dans le voisinage immédiat

des électrodes, cette technique semble moins nocive pour les structures adjacentes. Cela a été

démontré au niveau de structures comme les voies biliaires (80, 81) et le pancréas (82). L'IRE

est effectuée en plaçant des électrodes dans le tissu et en appliquant une tension pour générer

un champ électrique d'ablation in vivo. La distribution in vivo du champ électrique est

déterminée par la géométrie des électrodes, la tension appliquée entre les électrodes et la

conductivité électrique du tissu dans la région ciblée. L'efficacité et les résultats du traitement

sont donc subordonnés à la taille, la forme et la distribution in vivo de ce champ électrique.

Le champ électrique d'ablation utilisé pour induire l'IRE dans les tissus est sensible à des

hétérogénéités de la conductivité électrique dans la région de traitement (83). Il est à craindre

que cette redistribution intrinsèque du champ électrique puisse entraîner des effets

thérapeutiques non intentionnels et modifier également le volume et la forme prévue de la

région à traiter. Des simulations numériques spécifiques peuvent modéliser la distribution du

champ électrique dans la prostate en utilisant les données obtenues à partir de l'imagerie pré et

intra-opératoire, et les paramètres de traitement (77, 84, 85). Le but de cette étude a donc été

de déterminer si ces simulations numériques spécifiques au patient étaient corrélées avec les

résultats cliniquement observés, et la zone traitée visible en IRM.

Ce travail est rapporté dans ce manuscrit au niveau du chapitre 7. Des projets

similaires, impliquant Clair Poignard, PhD, sont en cours afin d'explorer ces possibilités non

seulement au niveau de la prostate mais aussi au niveau du foie.

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2. Modélisation mathématique de la croissance tumorale basée sur l'imagerie: généralités et perspectives

Ce chapitre, volontairement de vulgarisation, vise à expliquer les principes de base de

la modélisation mathématique et donner les avantages, les limites et les perspectives de

l'approche in vivo basée sur les données de l’imagerie que nous avons développée. Cet article

a été publié dans la revue Diagnostic Interventional Imaging en 2013, référence Pubmed:

Cornelis F, Saut O, Cumsille P, Lombardi D, Iollo A, Palussiere J, Colin T. In vivo

mathematical modeling of tumor growth from imaging data: soon to come in the future?

Diagn Interv Imaging. 2013 Jun;94(6):593-600.

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Diagnostic and Interventional Imaging (2013) 94, 593—600

REVIEW / Technical

In vivo mathematical modeling of tumor growth fromimaging data: Soon to come in the future?

F. Cornelis a,b,∗,c, O. Sautd, P. Cumsillea,D. Lombardia, A. Iolloa, J. Palussierec, T. Colina

a Bordeaux University, IMB, UMR 5251, 33400 Talence, Franceb Department of Adult Diagnostic and Interventional Imaging, Pellegrin Hospital, BordeauxUniversity Hospitals, place Amélie-Raba-Léon, 33076 Bordeaux, Francec Department de Radiology, Bergonié Institute, 229, cours de l’Argonne, 33076 Bordeaux,Franced CNRS, IMB, UMR 5251, 33400 Talence, France

KEYWORDSModeling;Tumor;Imaging;Evaluation

Abstract The future challenges in oncology imaging are to assess the response to treatmenteven earlier. As an addition to functional imaging, mathematical modeling based on the imagingis an alternative, cross-disciplinary area of development. Modeling was developed in oncologynot only in order to understand and predict tumor growth, but also to anticipate the effectsof targeted and untargeted therapies. A very wide range of these models exist, involving manystages in the progression of tumors. Few models, however, have been proposed to reproducein vivo tumor growth because of the complexity of the mechanisms involved. Morphologicalimaging combined with ‘‘spatial’’ models appears to perform well although functioning imagingcould still provide further information on metabolism and the micro-architecture. The combina-tion of imaging and modeling can resolve complex problems and describe many facets of tumorgrowth or response to treatment. It is now possible to consider its clinical use in the mediumterm. This review describes the basic principles of mathematical modeling and describes theadvantages, limitations and future prospects for this in vivo approach based on imaging data.© 2013 Éditions françaises de radiologie. Published by Elsevier Masson SAS. All rights reserved.

Nowadays, imaging lies at the heart of patient management, particularly in oncology.Apart from the time when a disease is diagnosed, the different imaging methods are usedto assess the effectiveness of both local (surgery, radiotherapy, heat ablation) and systemic(chemotherapy, targeted therapy) treatments during follow-up. The protocols usually rec-ommend assessment criteria such as the RECIST, WHO or CHOI, although these are lessthan perfect [1].

∗ Corresponding author.E-mail address: francois.cornelis@chu-bordeaux.fr (F. Cornelis).

2211-5684/$ — see front matter © 2013 Éditions françaises de radiologie. Published by Elsevier Masson SAS. All rights reserved.http://dx.doi.org/10.1016/j.diii.2013.03.001

© 2013 Elsevier Masson SAS. Tous droits réservés. - Document téléchargé le 02/06/2013 par SF Radiologie Acces DII (524047)

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594 F. Cornelis et al.

Nevertheless, the future challenges for imaging are toassess response to treatment and to change treatment evenearlier, making it increasingly targeted and appropriate andas soon as possible if necessary. This approach can optimizetreatment and reduce costs. At present it is based mostlyon functional imaging, which is expanding greatly [2]. Thisaspect of imaging provides considerable information aboutcellular activity (PET), nutrient consumption [3], angiogen-esis and capillary permeability (perfusion MRI) [4] and alsotissue micro-architecture (diffusion MRI) [2], to name a fewexamples. In this situation, modeling may be an alternative,cross-disciplinary area of development.

Mathematical modeling has been developed in oncologyin order both to understand and predict tumor growth and toanticipate the effects of targeted or non-targeted therapies(e.g. the anti-angiogenics). There is a very wide range ofthese models involving many stages in tumor progression.They can, for example, help to understand the influence ofgenetic regulation [5] or predict an effect, which has beenobserved experimentally [6].

In most of these cases, however, the models are bettersuited for in vitro studies as they are often focused on asmall cell contingent, progression of which is more straight-forward to observe. The problem in vivo becomes morecomplex as it needs to take account of an entire tumor con-taining a large number of cells and therefore mechanismsto be elucidated. In addition, unlike in vitro studies, theamount of information available without modeling a tumorin vivo is very little as it is ‘‘swamped’’ by the large num-ber of cells and therefore also with signals. It would appearessential therefore to develop appropriate means to col-lect the maximum of information in order to optimize thesemodels.

Apart from biopsies, which provide microscopic infor-mation, albeit from limited samples, imaging examinationshave become essential as they allow repeated, non-invasiveinformation to be recovered over time. The size (or volume),shape, site and uptake after contrast injection are rela-tively easy parameters to obtain whereas micro-architectureor perfusion data are still difficult and depend on the tar-get organ. The information obtained can be used to refinemodels and ultimately be applied to clinical practice.

This review explains the basic principles of mathemati-cal modeling and describes the advantages, limitations andfuture prospect of this in vivo approach based on imagingdata.

Concepts of modeling tumor growth

Entirely schematically, several successive stages can beidentified in the growth of a solid tumor due to deregula-tion of cell division: initial avascular growth, angiogenesis,and finally metastatic spread.

During the avascular growth phase, the tumor (accordingto the monoclonal hypothesis of cancer, the tumor initiallyconsists of a single abnormal cell) grows within the limitsof its local environment in a characteristic ‘‘layer’’ model[7]. Nutrient supply to the centre of the tumor is inadequateand necrosis may occur. A layer of quiescent cells, which sur-vive without dividing with limited slow metabolism, is foundbetween the two layers. According to this type of model,

tumor growth is then severely restricted and in order to con-tinue growing it needs to find additional nutrient sources.The tumor then enters its second stage of growth, the vascu-lar phase. As a result of hypoxic pressure some phenotypescan be favored (through Darwinian selection). Some cellsthen produce signals (such as VEGF), which encourage thebody to vascularize the tumor and therefore provide it withnew nutrients (this is the process of angiogenesis). It isimportant to be aware of this stage for several reasons.Once it is vascularized the tumor can reach significantly farlarger sizes than in the previous stage and then be visible onimaging. Finally, vascular fragmentation leads to metastaticspread.

It is therefore not difficult to understand that becauseof the complexity of the processes involved in all of thesestages, mathematical models are to a large extent phe-nomenologic and extremely simplified compared to whathappens in biological reality. Two main types of modelingcan be distinguished, which depend on whether or not thetumor is integrated into its neighbouring space.

Non-spatial models calculate the change in one or moremajor appropriate scales in order to monitor tumor growth.Classic examples are models based on ordinal differentialequations (ODE), which describe the change in volume orradius of the tumor. These do not take account of thetumor environment or its interactions but just of measurabledimensions [8]. These models are, however, too simple toprovide useful, reliable, reproducible information about thetumor from the phenomenologic change in its volume, whichvaries at random between different tumors and patients.They may, however, be made more complex to incorporateseveral compartments (depending on the cell cycle) or totake account of different processes such as angiogenesis.

Conversely, the spatial model takes into account of thetumor environment. Several methods have been proposed tomodel tumor growth in this way [9,10]. One involves describ-ing the progression of cells individually [11] and produces‘‘discrete’’ automated cell models [12] or agent models[13]. These relatively simple models are very effective insimulating in vitro growth to describe microscopic effectsand link them to the molecular level [14]. However, it doesnot seem currently possible to model an entire complex ofcells in its environment, i.e. an entire tumor, as this is toodifficult and technically complex.

Another approach to spatial modeling involves workingon several cells or population of cells at the same timeand therefore describing the change in cell densities (or theboundaries of tumors for example). This allows tumors ofrealistic size to be studied and takes account of macroscopiceffects (for example, interactions between cells and anextracellular membrane or cellular adhesion) using modelsinspired from continuous medium mechanics [15]. The tissuestudied is then treated as a continuum governed by a behav-ior law in the medium. It is thus straightforward to takeaccount of an extensive complex environment although it isdifficult to take all of the independent and non-independentmicroscopic effects [15] into account. Some models do how-ever, take angiogenesis into account [14,16], and others paymore attention to chemotaxis [17]. In parallel, in terms oftissue description, both mono- and multiphase cell popu-lation interaction models have been studied [18,19]. Cellcycle modeling has also been studied [20], with a view to

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46

In vivo mathematical modeling of tumor growth from imaging data 595

optimizing treatment [21—23]. These spatial models cantherefore incorporate a 2 (or more) phase cell cycle, one ofproliferating cells and the other of quiescent cells, and alsotake account of a necrosis phase. In addition, oxygen diffu-sion, angiogenesis and interactions with membranes [24] orneighbouring tissues can also be taken into account.

These spatial models have the advantage of incorporat-ing far more biological knowledge and have several benefits,which justify their use. They are said to reproduce realisticbehavior, with the desired purpose, and can also test cellinteraction hypotheses [25] in an entity such as a tumor.They are often, however, very complex, as they containexperimentally inaccessible parameters and are not spe-cific for a patient, as parameters vary greatly in differentsituations. They are not therefore suitable for clinical use.

In order to adapt the spatial models to each disease andpatient, they therefore need to be simplified and calibratedwith clinical data which can be obtained for example byimaging in order to estimate the different parameters inthe equations and therefore obtain quantitative results.

Spatial mathematical models based onimagingAll of the information available from morphological imagingcan be used in mathematical modeling. Dimensions, vol-ume, density and intensity are those which are used mostoften. Changes over time provide additional information.The use of functional imaging data is still in the develop-mental phase and the choice of a model will also depend onthe temporal distribution of the available data, as statisticalmodels require more data than determinist models to startthe modeling process.

Statistical models

These models require a large amount of initial data. Informa-tion obtained from imaging investigations often exhibits veryconsiderable inter-individual variability, which has led to thedevelopment of mixed effect regression techniques [26,27].These models take account of two levels of variability:inter-individual variability and intra-individual or residualvariability (as is seen in the classical regression technique).The number of parameters is therefore increased as eachparameter consists of a fixed component (mean parameter)and a random component, hence the name ‘‘mixed effect’’.Tham et al. [26] proposed an empirical model to describethe effect of a combination of two drugs on tumor size inpatients suffering from lung cancer. Wang et al. [27] wentfurther by developing a similar model from a lung cancerdatabase created from different clinical studies and there-fore involving different treatments. The authors showedthat some variables directly derived from model predictions(initial ‘‘baseline’’ tumor size and reduction in tumor size 3weeks after starting treatment) were predictive indicatorsfor survival.

If the purpose of modeling is to assess the future progres-sion of a disease in a given patient, statistical approachesare not necessarily the most appropriate. These providean ‘‘average’’ response provided that a sufficient databaseexists containing a large number of similar cases to the case

in question. These models can also be made more complexby combining molecular marker activity with this simpletumor growth method to establish a multi-scale model [28].

Even so, in order to obtain a prognosis for a given tumorin a given patient, statistical modeling contains a large mar-gin of error and does not appear appropriate. The statisticalapproaches may however be useful in validating and select-ing the model and measuring its accuracy and robustness.

Determinist models

As described above, most of the mathematical models usedclinically are based on a group of ordinal differential equa-tions (ODE) and describe the change in tumor volume [8].The spatial dimension (e.g. shape or site of the tumor) isnot taken into account, which limits the use of the dataobtained from imaging and therefore the ability to estimatethe parameters in the equations. In order to compensatefor this, a recent model has been developed based on mor-phological study of different lesions with sufficiently welldefined outlines to extract the images seen after segmen-tation and repositioning (Fig. 1) [29]. Ordinal differentialequation (ODE) and partial derived equation (PDE) modelswere then used, incorporating different pre-determinedparameters whose values, however, were unknown. Theseparameters were estimated using two approaches, a pop-ulation approach allowing inter-patient variability to beincorporated and an individual approach based on 2D or 3Dinformation in order to incorporate different informationsuch as treatment strategy.

From this work and in order to propose a simplified modelwhich could be used in reality with the limited tumor orpatient information generally available, the cell cycle wascontracted into a PDE no longer structured by age, and treat-ing a population of proliferating cells (of concentration P)(Fig. 2). Growth rate is determined by local oxygen concen-tration, which is given by a reaction-diffusion equation. Ifthe oxygen concentration is too low (hypoxia), not only dothe cells not proliferate but they move into the quiescentphase (shown as Q) (the phase in which the tumor cellsremains alive but cells do not proliferate). Conversely, if theoxygen concentration increases, the cells can return to theproliferative phase. The proliferative/quiescent cell ratio isan indicator of tumor aggression. Cell proliferation leads toan increase in volume, which is described as the divergenceof a velocity field, which has to be determined. This veloc-ity field has nothing to do with the velocity, for example,of infiltrating cells, but describes the overall movement ofthe tissue due to its increase in volume limited by the pres-sure of the surrounding environment according to Darcy’slaw, with permeability depending on the tissue (healthy ortumor) [30]. Finally, oxygen concentration is given by a dif-fusion equation which depends on the type of tissue andwhich reflects vascularization with absorption factors dueto consumption by tumor cells.

In summary, the equations which describe changes in den-sity P and Q are shown below:

∂P

∂t+ ∇ · (vP) = (2" − 1)P + "Q,

∂Q

∂t+ ∇ · (vQ ) = (1 − ")P + "Q

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596 F. Cornelis et al.

Figure 1. Segmentation and repositioning of images. a, b: axial computed tomography sections showing a left lower lobe lung nodule inthe pulmonary and mediastinal windows respectively; c: the nodule is segmented in sections and visualized in 3D; d: the lung volume is alsosegmented; e: several contours are shown illustrating the change over a same volume section, the shape and precision of a lung nodule onthe subsequent CT scans after repositioning; f: segmentation, repositioning and 3D visualization of the same nodule on three successiveinvestigations.

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In vivo mathematical modeling of tumor growth from imaging data 597

Figure 2. Basic principle of modeling tumor growth. The pro-liferating phase, P, and the quiescent phase, Q, depend on theconcentration of nutrients and oxygen (linked to !) provided in asimplified way by vascularization.

where the function ! describes the presence or absence ofsufficient nutrients:

! = 1 + tanh(C − Chyp)2

where C represents local oxygen concentration. The hypoxiathreshold Chyp is a model parameter, which describes thesensitivity of cells to hypoxia. As seen previously, in this casecell movement is deemed to be due only to cell division andthe velocity v in the system follows Darcy’s law.

Although this model is greatly simplified, it alreadycontains several parameters, which need to be identifiedwhereas the only finding provided by imaging is tumor vol-ume, which is the sum of variables P + Q in the model. Basedonly on morphological imaging, i.e. by studying volume,the site, shape and uptake of a tumor and any growth,information obtained from imaging examinations is there-fore very limited.

The calculation strategy is as follows: for a given patient,the images must first be processed digitally (Fig. 1) and thenin order to use the PDE model shown schematically in Fig. 2,its parameters must be determined. This is achieved bysolving an inverse problem involving a large number of calcu-lations using several radiological investigations as the inputdata. Under favorable conditions a personalized predictioncan then be obtained.

These models are undergoing preclinical evaluation forgliomas, lung metastases (Fig. 3) and some stromal tumors(gastrointestinal stromal tumor [GIST]) (Fig. 4). Thesetumors have the advantage that it is easy to define their out-lines and can be followed up for relatively long periods oftime with or without treatment. A large amount of informa-tion about their progression is therefore available. As shownin Fig. 3, it is already possible to obtain a quantitative pre-diction for untreated lung metastases. A prediction is not yetpossible for hepatic GIST metastases (Fig. 4), although theresponse to treatment and loss of treatment response phasecan be reproduced. The modeling techniques are describedin detail in reference [30].

Future prospectsAlthough the clinical use of spatial mathematical modelsis far from being actually effective, they have manyshort and medium term future prospects in the clin-ical assessment in surgery or interventional radiologyby determining tumor margins or optimizing needlepositioning (for example, thermal heat ablation orcryotherapy).

Whilst some aspects of tumor growth are not determin-ist (for example, spontaneous genetic mutations and theemergence of phenotypes which are resistant to therapy)and cannot therefore be incorporated into these modelsfor a specific approach to a given patient, this work canenable early changes due to these mutations to be identi-fied from a deviation from the medical findings in relation tothe digital prediction. Treatments can therefore be adjustedearlier.

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598 F. Cornelis et al.

Figure 3. Modeling tumor growth of lung metastases. a, b, c: axial computed tomography sections showing a right lower lobe lungnodule on successive CT scans without any treatment. Only images a and b are used in the calculation. Subsequent investigations aresimply used to validate our method; d: change in volume of the nodule over time: actual observations (round) and those obtained fromcalibrated modeling (curve). The red square is the volume actually measured on an investigation performed after the model was calibrated;e: computed tomography axial section with representation of the tumor calculated at the same time (red) as investigation of the image c.

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In vivo mathematical modeling of tumor growth from imaging data 599

Figure 4. Modeling of tumor growth of hepatic GIST metastases. a, b: computed tomography axial sections with injection showing a hepaticnodule in segment IV on two successive scans; c: change in volume over time on the actual observations. These changes take account ofsuccessive anti-tyrosine kinase treatments (decreased phase) followed by an anti-angiogenic after failure to respond further to treatment;d: change in volume of the nodule over time from modeling. This initial completely spatial model (PDE) takes, at least qualitatively, accountof the control phase with the anti-tyrosine kinase then the loss of response and finally the control by anti-angiogenic. These results are notpredictions as all of the data are used to obtain the simulated curves; e: this second modeling (ODE) takes quantitative account in terms ofthe cell population in the control phase and then the loss of response. These results are not predictions as all of the data are used to obtainthe simulated curves.

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600 F. Cornelis et al.

Conclusion

Whereas many technical challenges remain to be resolvedand cannot be ignored, mathematical modeling of tumorgrowth even if simplistic is now a reality. Imaging appears tobe a valuable aid in solving the different problems raised.The input of further information from functional imagingwill help to facilitate the resolution of current models andeither to predict the development and use of the more pre-cise models in order to come closer to the real life biologicalsituation.

Disclosure of interest

The authors declare that they have no conflicts of interestconcerning this article.

References

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[2] Li SP, Padhani AR. Tumor response assessments with diffusionand perfusion MRI. J Magn Reson Imaging 2012;35:745—63.

[3] Gatenby RA, Gillies RJ. Why do cancers have high aerobic gly-colysis? Nat Rev Cancer 2004;4:891—9.

[4] Desar IM, ter Voert EG, Hambrock T, van Asten JJ, van SpronsenDJ, Mulders PF, et al. techniques demonstrate early vascularchanges in renal cell cancer patients treated with sunitinib: apilot study. Cancer Imaging 2011;11:259—65.

[5] Anderson AR, Chaplain MA. Continuous and discrete mathe-matical models of tumor-induced angiogenesis. Bull Math Biol1998;60:857—99.

[6] Gatenby RA, Gawlinski ET. A reaction-diffusion model of cancerinvasion. Cancer Res 1996;56:5745—53.

[7] Sherratt JA, Chaplain MA. A new mathematical model for avas-cular tumor growth. J Math Biol 2001;43:291—312.

[8] Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E,et al. Predictive pharmacokinetic-pharmacodynamic modelingof tumor growth kinetics in xenograft models after administra-tion of anticancer agents. Cancer Res 2004;64:1094—101.

[9] Modok S, Scott R, Alderden RA, Hall MD, Mellor HR, Bohic S,et al. Transport kinetics of four- and six-coordinate platinumcompounds in the multicell layer tumor model. Br J Cancer2007;97:194—200.

[10] Byrne H, Drasdo D. Individual-based and continuum modelsof growing cell populations: a comparison. J Math Biol2009;58:657—87.

[11] Drasdo D, Hohme S. A single-cell-based model of tumorgrowth in vitro: monolayers and spheroids. Phys Biol 2005;2:133—47.

[12] Alarcon T, Byrne HM, Maini PK. A cellular automaton modelfor tumor growth in inhomogeneous environment. J Theor Biol2003;225:257—74.

[13] Mansury Y, Kimura M, Lobo J, Deisboeck TS. Emerging patternsin tumor systems: simulating the dynamics of multicellularclusters with an agent-based spatial agglomeration model. JTheor Biol 2002;219:343—70.

[14] Ramis-Conde I, Drasdo D, Anderson AR, Chaplain MA. Mod-eling the influence of the E-cadherin-beta-catenin pathwayin cancer cell invasion: a multiscale approach. Biophys J2008;95:155—65.

[15] Macklin P, McDougall S, Anderson AR, Chaplain MA, Cristini V,Lowengrub J. Multiscale modeling and nonlinear simulation ofvascular tumor growth. J Math Biol 2009;58:765—98.

[16] Gerisch A, Chaplain MA. Mathematical modeling of cancer cellinvasion of tissue: local and non-local models and the effect ofadhesion. J Theor Biol 2008;250:684—704.

[17] Hillen T, Painter KJ. A user’s guide to PDE models for chemo-taxis. J Math Biol 2009;58:183—217.

[18] Preziosi L, Ambrosi D, Verdier C. An elasto-visco-plastic modelof cell aggregates. J Theor Biol 2010;262:35—47.

[19] Ambrosi D, Duperray A, Peschetola V, Verdier C. Traction pat-terns of tumor cells. J Math Biol 2009;58:163—81.

[20] Bekkal Brikci F, Clairambault J, Ribba B, Perthame B. An age-and-cyclin-structured cell population model for healthy andtumoral tissues. J Math Biol 2008;57:91—110.

[21] Clairambault J. A step toward optimization of cancer thera-peutics. Physiologically based modeling of circadian control oncell proliferation. IEEE Eng Med Biol Mag 2008;27:20—4.

[22] Levi F, Karaboue A, Gorden L, Innominato PF, Saffroy R,Giacchetti S, et al. Cetuximab and circadian chronomod-ulated chemotherapy as salvage treatment for metastaticcolorectal cancer (mCRC): safety, efficacy and improved sec-ondary surgical resectability. Cancer Chemother Pharmacol2011;67:339—48.

[23] Ribba B, Colin T, Schnell S. A multiscale mathematical modelof cancer, and its use in analyzing irradiation therapies. TheorBiol Med Model 2006;3:7.

[24] Ribba B, Saut O, Colin T, Bresch D, Grenier E, Boissel JP. Amultiscale mathematical model of avascular tumor growth toinvestigate the therapeutic benefit of anti-invasive agents. JTheor Biol 2006;243:532—41.

[25] Billy F, Ribba B, Saut O, Morre-Trouilhet H, Colin T, BreschD, et al. A pharmacologically based multiscale mathemati-cal model of angiogenesis and its use in investigating theefficacy of a new cancer treatment strategy. J Theor Biol2009;260:545—62.

[26] Tham LS, Wang L, Soo RA, Lee SC, Lee HS, Yong WP, et al. Apharmacodynamic model for the time course of tumor shrink-age by gemcitabine + carboplatin in non-small cell lung cancerpatients. Clin Cancer Res 2008;14:4213—8.

[27] Wang Z, Zhang L, Sagotsky J, Deisboeck TS. Simulating non-small cell lung cancer with a multiscale agent-based model.Theor Biol Med Model 2007;4:50.

[28] Bueno L, de Alwis DP, Pitou C, Yingling J, Lahn M, Glatt S, et al.Semi-mechanistic modeling of the tumor growth inhibitoryeffects of LY2157299, a new type I receptor TGF-beta kinaseantagonist, in mice. Eur J Cancer 2008;44:142—50.

[29] Konukoglu E, Clatz O, Menze BH, Stieltjes B, Weber MA,Mandonnet E, et al. Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropiceikonal equations. IEEE Trans Med Imaging 2010;29:77—95.

[30] Ambrosi D, Preziosi L. Cell adhesion mechanisms and stressrelaxation in the mechanics of tumors. Biomech ModelMechanobiol 2009;8:397—413.

© 2013 Elsevier Masson SAS. Tous droits réservés. - Document téléchargé le 02/06/2013 par SF Radiologie Acces DII (524047)

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3. Modélisation de la croissance des nodules pulmonaires

Ce chapitre vise à présenter nos résultats de la modélisation mathématique des nodules

pulmonaires. Ces deux articles ont été publié dans Medical Image Computing and Computer-

Assisted Intervention en 2014, sous la référence Pubmed:

• Colin T, Cornelis F, Jouganous J, Palussière J, Saut O. Patient specific image driven

evaluation of the aggressiveness of metastases to the lung. Med Image Comput Assist

Interv. 2014;17(Pt 1):553-60;

• Thierry Colin, François Cornelis, Julien Jouganous, Jean Palussière, Olivier Saut.

Patient-specific simulation of tumor growth, response to the treatment, and relapse of

a lung metastasis: a clinical case. Journal of Computational Surgery 2015, 2:1.

53

3.1. Patient specific image driven evaluation of the aggressiveness of metastases to the lung

Patient Specific Image Driven Evaluation of the

Aggressiveness of Metastases to the Lung

T. Colin1, F. Cornelis1,2, J. Jouganous1, M. Martin1, O. Saut1

1 Institut de Mathematiques de Bordeaux, Universite de Bordeaux2 Hopital Pellegrin, CHU Bordeaux

Abstract. Metastases to the lung are a therapeutic challenge becausesome are fast-evolving while others evolve slowly. Any insight that canbe provided for which nodule has to be treated first would help clini-cians. In this work, we evaluate the aggressiveness but also the responseto treatment of these nodules using a calibrated mathematical model.This model is a macroscopic model describing tumoral growth througha set of nonlinear partial di↵erential equations. It has to be calibratedto a specific patient and a specific nodule using a temporal sequence ofCT scans. To this end, a new optimization technique based on a reducedorder method is developed. Finally, results on two clinical cases are pre-sented that give satisfactory numerical prognosis of the evolution of anodule during di↵erent phases: growth, treatment and post-treatmentrelapse.Keywords. Tumor growth modeling, Medical imaging, Partial Di↵eren-tial Equations, Clinical data assimilation.

1 Introduction

The behavior of metastases to the lung is di�cult to assess by clinicians. Somemay grow rapidly while some stay stationary for years. This variation makes itdi�cult to decide when to treat especially when elderly and weak patients areconcerned. In those cases, physicians try to restrict treatment to nodules thatmay become malignant. A numerical tool improving the prognosis of each nodulewould be invaluable in this case.

Related works. Currently, most applications of mathematical models inclinical oncology are somehow limited to models that neglect the spatial aspectof the cancer growth like [3]. These models cannot exploit all the information pro-vided by medical imaging devices and must be used with statistical approaches.This prevents their applications for a specific patient as they only provide ”aver-age” answers. Furthermore, these mathematical models are not able to reproducethe observed evolution of a nodule just by using two or three measurements. Asthis is typically the number of images available for each patient, they are notrelevant here. Newer works like [1,4,9] use image data with tumor growth mod-els. They are mostly targeting brain tumors, are simpler from a biological pointof view and the way they are calibrated on patient data uses some very specific

54

features of the model and can not be extended to our case. We built a spatialmodel in order to use, in a more relevant way, the information available fromanatomical imaging. Here, we are concerned with metastases to the lung of adistant tumor. The metastases are not infiltrative and di↵usion-type models arenot well adapted. We introduce a system of nonlinear PDEs based on popula-tions of cells, without di↵usion, but including a micro model of angiogenesis,process by which the tumor drives the emergence of its own neo-vasculature.

Once an accurate model describing tumor growth is derived, its parametershave to be recovered for any patient-specific prognosis. This complex task isusually done by solving an inverse problem using medical images [1,2]. In thiswork, this calibration is solved using classical approaches combining stochasticand deterministic methods. This algorithm is neither model specific, contrary tothe calibration method used in [4], nor computationally expensive like solvingadjoint problems [2,9].

2 Mathematical model

The model we use in this work is derived from the one described in [5]. Weconsider here only one kind of cancer cells. The tumor microenvironment, andin particular the quantity of nutrients available, is essential to explain its evo-lution. Consequently, instead of directly modeling the nutrient density, we usea very simplified angiogenesis model to take into account the process by whichthe tumor escapes the avascular stage.

Cell behavior. The tumor cell density is denoted by P and evolves by

@P

@t+r · (vP ) = (�+ � ��)P, (1)

where v is the velocity corresponding to the growth of volume created by thecellular division. Coe�cients for proliferation and death by hypoxia, �+ and ��,are detailed in (2) and depend on the local vascularization denoted by M . Abovea given threshold of nutrient supplyMth, cancer cells tend to proliferate whereas,below this threshold, they starve to death. The hyperbolic tangent in both �+and �� expressions is used to smooth and regularize the threshold functions,and K is a fixed smoothing constant. These functions are given by:

�+,�(M) = �0,11± tanh(K(M �Mth))

2, (2)

where �0 is the proliferation rate of non hypoxic tumor cells and �1 is the deathby hypoxia rate. We consider that the tissue is saturated, which gives us (see[8]) an equation on v (3)

r · v = (�+ � ��)P. (3)

To close the system of equations (see [8]), we consider that the velocity v isobtained through a Darcy law in Eq.(4): v is derived from a pressure or potential⇡ in the tissue.

v = �r⇡. (4)

55

Angiogenesis. At the end of the avascular stage, the tumor reaches such asize that its direct environment is not able to supply enough nutrients to allowit to keep on growing. At this point, cancer cells emit chemical signals whichmay result in the emergence of a neo-vasculature [6]. It is described by theequations (5) and (6). The scalar variable ⇠ describes the total amount of pro-angiogenic agents which are produced by quiescent cells (given by the expressionR⌦(1�

�+

�0)Pd!, ⌦ being the computing domain), and eventually metabolized.

@⇠

@t= ↵

Z

⌦(1� �+

�0)Pd! � �⇠. (5)

As we assume that the quantity of nutrient is proportional to the density ofblood vessels in the tissue, we collect these two notions in one variable M thatwe shall call ”vasculature”. The vasculature M is damaged by tumor cells andproduced where the quiescent cells are located proportionally to ⇠ by the term�⇠(1� �+

�0)P .

@M

@t= �⌘PM + �⇠(1� �+

�0)P. (6)

Taking therapeutical e↵ects into account. The model architecture makesit easy to include di↵erent types of treatment. Chemotherapy e↵ects can be sim-ulated adding a death term ��P on Eq.(1) which gives:

@P

@t+r · (vP ) = (�+ � ��)P � �P. (7)

To fulfill the saturation assumption Eq.(3) is modified as follows:

r · v = (�+ � �� � �)P. (8)

3 Calibration method

As shown previously, the mathematical model has many parameters, namely ↵,�, �0, �1, ⌘, � and Mth, that must be determined through a complex inverseproblem. Most of these parameters have no physical or biological meaning andcannot be recovered by experimental measurements. Furthermore, the medicalimages (CT scans) have to be processed to be used with this model.

Segmentation, Registration. Lung metastases are particularly interest-ing from a mathematical and technical point of view because of the quality ofthe imagery. Indeed, in CT scans of the lung, the tumor appears as white whilehealthy tissue (full of air) is mainly black. Delineating the tumor is thereforerelatively easy and requires little intervention from clinicians. In practice, thesegmentation is manually performed by the oncologist who choses a representa-tive slice of the tumor. For each exam, this same slice of the tumor is segmented

56

by the clinician. The slice is localized using physiological details such as bloodvessels or bronchi. The patient is not in the same exact position for every exam.The targeted nodule is relocated to have a stationary center of gravity betweenscans. We made the reasonable assumption that the tumor is solid and its vol-ume is not a↵ected by patient’s breath. The rotation of the abdomen betweenscans is also taken into account.

Formulation of the inverse problem. Given a sequence of medical imagesor snapshots of the tumor, we aim at finding a parameter set able to reproduce itsobserved behavior. Our approach is to use an objective function, which basicallyquantifies the di↵erence between the observable data and the model simulation,and try to minimize it. There are di↵erent ways to measure this error and asa criterion we chose a combination of the comparison of the mass and the L2

norm of the images. Mass is measured by integrating the cellular density P .We need at least two images of the metastasis at di↵erent times to have a chanceto personalize the model: the first one at t1 = 0 is the initial condition for thetumor cell density and the other is used to parameterize the model. Whateverthe minimization method, it is necessary to estimate many times the value ofthe objective function, and so to simulate the model for lots of parameter setswhich could be quite expensive.

To make the calibration faster, we have developed a strategy based on a re-duced order method called Proper Orthogonal Decomposition (POD).

Building a reduced order model to speed up computations. POD res-olution method for dynamic systems consists in approaching partial di↵erentialequation systems with ordinary di↵erential equations by decoupling e�cientlythe time and space variables (see [7]). The initial infinite dimension problem isthus replaced by a finite dimension problem.Let us describe the POD use on the tumor cell density variable P . As we wantto decouple space and time variables, we use the following representation forP (or any variable of interest): P (X, t) =

Pdi=1 a

Pi (t)�

Pi (X) + ✏(X, t), where

aPi are scalar functions depending on time and �Pi are spatial functions called

modes and represent the geometry of the variable P . The dimension of the re-duced problem is denoted by d. The approximation error is denoted by ✏(t,X).The goal of POD is to provide us with the best basis of spatial functions �P

i tominimize the error.These functions �P

i are extracted from a database of admissible behaviors of P .To generate this database, we sample the parameter space using a cartesian grid,simulate the direct model for each parameter set thus obtained and keep severalsnapshots (SP

k )k of the variable P . If the sample is correctly chosen, we have arepresentative set of geometrical configurations for the tumor cell density. Thenwe look for the functions �P

i in the d-dimensional vectorial space generated bythe snapshots from the database. They are, in other words, linear combinationsof the snapshots. These functions are taken as an orthonormal basis minimizingthe truncation error of the projection which allows us to use a few modes with-

57

out losing too much precision. The POD approach is used on both the tumorcell density P and the pressure field ⇡ which are the two fields driven by PDEsin our system. Finally, the system of equations is projected along these modesand so approximated by an ODE system on the coe�cients (aPi (t))i and a linearsystem on the coe�cients (a⇡i (t))i.

Complete algorithm used for the inverse problem. Replacing PDEsby ODEs makes the problem simpler and faster to solve so we use this reducedmodel for the inverse problem. Moreover, the modes, and the spatial derivativesassociated, are computed once for all. Then we use a classical optimization strat-egy to minimize the distance between the model simulations and the observable.The first step is to find a reasonable parameter set via a particle swarm algo-rithm. A sensitivity analysis was performed on the model that shows the lowinfluence of parameters ↵ and �. Therefore, these two parameters are fixed anda gradient algorithm is used to refine the set of parameters.

4 Results and discussions

4.1 Trying our method on a first complete test case

Here is a typical case of a patient with lung nodules from a primary bladdertumor.

Fig. 1: Extract from a time sequence of CT scans showing the evolution of onenodule marked in red between 2008/06/07 (on the left) and 2008/09/22 (on theright).

The method described previously is used on this first test case. Six CT scansare available (the first two of which are presented in Fig.1). The first threecorrespond to the tumor growth. Then the nodule reached a critical volumeand the clinicians decided to treat it with a chemotherapy. The two followingscans were used to control the response of the tumor to the treatment. Finally,

58

a last control scan was planned after the end of the chemotherapy that showeda relapse as the tumor started growing again.First, we use the first two scans (see Fig.1) to calibrate the model on the growthphase. Then the model is simulated up to the third scan date to see if theprediction is accurate. The tumor mass thus obtained by the model is comparedto the medical data in Fig.2a.

Fig. 2: Evolution of the tumoral masses as computed by our model after recov-ering its parameters during the growth (on the left) and after the beginning ofthe treatment (on the right). Tumoral masses measured on the CT scans by theclinicians are plotted with +, the reduced model simulation with dotted line andthe direct model simulation with full line.

As the model provides spatial information on the cells distribution, it isalso interesting to evaluate the accuracy in shape of the results obtained withour method. For this, we used shape indicators such as the Volume Concor-dance (given by the expression V C = 100⇥ (1� |P

model

�Pdata

||P

data

| )) and the DICE

(DICE = 100⇥( 2⇤|Pmodel

TP

data

||P

model

|+|Pdata

| )). We also compute a reference DICE betweenthe first scan which is the initial condition of our system and the current scan.This represents the hypothesis of a non evolving tumor and gives a value ofcomparison. Moreover, the temporal prediction error is another significant in-dicator. If we denote by ti the time of the ith exam and t0i the time when thesimulated tumor reaches the size of the real tumor at the ith exam; it is rele-vant to look at the delay between the simulation and the real case ti� t0i and the

normalized delay 100⇥ ti

�t0i

ti

�t0, i = 1, 2. These four indicators are given on Table 1.

Then, we tried to calibrate the treatment parameter � to see if the responseto the chemotherapy is predictable with our tool. Here only one parameter hasto be determined which makes this second inverse problem easier than the firstone. The initial condition we used for P is the last scan before treatment on2008/12/10 and we used the first control scan during chemotherapy to calibrate

59

Date 2008/09/22 2008/12/10DICE 90.96% 87.21%

reference DICE 54.94% 10.25%Volume Concordance 82.54% 77.76%

Delay (days) 0 -6.7Normalized Delay 0% -3.6%

Table 1: Scalar indicators for the tumor growth of the first clinical case: DICE,Volume Concordance and delays.

the treatment parameter. The evolution of the tumor mass during the treatmentprovides a good insight into the therapeutical e�cacy. It is given in Fig.2b andwe can see that here again the model is predictive for this case and provides agood estimation of the response of the patient to this chemotherapy. Moreover,after the end of the treatment, the tumor started growing again and this relapseis also well predicted by the model. The same indicators that were used forthe growth are given in Table 2. For the last exam, on 2009/07/27, the shapeindicators are not relevant as the relapse is located at the periphery of the initialnodule and so the shape and location can not be predicted accurately.

Date 2009/03/21 2009/05/27 2009/07/27DICE 92.26% 87.44% 84.79

reference DICE 57.63% 37.71% 52.78Volume Concordance 84.4% 74.56% 69.9

Delay (days) 0.3 0.6 -6.4Normalized Delay 0.1 % 0.2 % -2.8%

Table 2: Scalar indicators for the tumor under chemotherapy and rebound of thefirst clinical case: DICE, Volume Concordance and delays.

4.2 A second test case

The whole calibration method described previously is used on another case oftumoral growth. Here again, we use two scans at di↵erent time points to calibratethe model and a third image to quantify the accuracy of the prediction. Theindicators are given in Table 3.

In this case, the growth is slower than in the previous one and the model isable to reproduce such a kind of dynamics. Indeed, the time error in predictionis about 6.4 days which, at the time scale we used and considering the tumorregistration uncertainties, is a good result.

60

Date 2010/03/11 2010/07/16DICE 85.41% 88.69%

reference DICE 65.93% 38.09%Volume Concordance 70.59% 76.45%

Delay (days) 0 5.6Normalized Delay 0% 2.3%

Table 3: Scalar indicators for the tumor growth of the second test case: DICE,Volume Concordance and delays.

In each case, we always considered the same slice of the tumor. The sametechnique can be applied on the whole 3D volume reconstructed from the medicalimages which would enable us not to choose a particular slice. The completemethod thus developed was successful to provide us with a relevant prognosis onthe evolution of lung nodules for several clinical cases. A larger study on about20 patients is ongoing to evaluate the quality of the prognosis on a larger scale.

References

1. Olivier Clatz, Maxime Sermesant, Pierre-Yves Bondiau, Herve Delingette, Simon KWarfield, Gregoire Malandain, and Nicholas Ayache. Realistic simulation of the 3-dgrowth of brain tumors in mr images coupling di↵usion with biomechanical deforma-tion. IEEE transactions on medical imaging, 24(10):1334–46, Oct 2005.

2. Cosmina Hogea, Christos Davatzikos, and George Biros. An image-driven parameterestimation problem for a reaction–di↵usion glioma growth model with mass e↵ects.J. Math. Biol., 56(6):793–825, Jun 2008.

3. Monica Simeoni, Paolo Magni, Cristiano Cammia, Giuseppe De Nicolao, ValterCroci, Enrico Pesenti, Massimiliano Germani, Italo Poggesi, and Maurizio Rocchetti.Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics inxenograft models after administration of anticancer agents. Cancer Res, 64(3):1094–101, Feb 2004.

4. K R Swanson, E C Alvord, and J D Murray. Virtual brain tumours (gliomas)enhance the reality of medical imaging and highlight inadequacies of current therapy.Br J Cancer, 86(1):14–8, Jan 2002.

5. T. Colin, A. Iollo, D. Lombardi, O. Saut. System identification in tumor growthmodeling using semi-empirical eigenfunctions. Mathematical Models and Methods in

Applied Sciences, 2012.6. P. Carmeliet, R. Jain. Angiogenesis in cancer and other diseases. Nature, 2000.7. K.Kunisch, S.Volkwein. Galerkin proper orthogonal decomposition methods forparabolic problems. Numer. Math., 117-148, 2001.

8. D. Ambrosi, L. Preziosi. On the closure of mass balance models for tumor growth.Math. Mod. Meth. Appl. Sci, 12(5), 737–754, 2002.

9. Yixun Liu, Samira M. Sadowski, Allison B. Weisbrod, Electron Kebebew, RonaldM. Summers, and Jianhua Yao. Patient Specific Tumor Growth Prediction UsingMultimodal Images. Medical Image Analysis, 2014.

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3.2. Patient-specific simulation of tumor growth, response to the treatment, and relapse of a lung metastasis: a clinical case.

Jouganous et al. Journal of Computational Surgery (2015) 2:1 DOI 10.1186/s40244-014-0014-1

RESEARCH Open Access

Patient-specific simulation of tumor growth,response to the treatment, and relapse of alung metastasis: a clinical caseThierry Colin1,3,4*, François Cornelis1,4,5, Julien Jouganous1,4, Jean Palussière6 and Olivier Saut1,2,4

*Correspondence:colin@math.u-bordeaux1.fr1University of Bordeaux, IMB, UMR5251, F-33400 Talence, France3Bordeaux INP, IMB, UMR 5251,F-33400 Talence, FranceFull list of author information isavailable at the end of the article

AbstractIn this paper, a parametrization strategy based on reduced order methods is presentedfor tumor growth PDE models. This is applied to a new simple spatial model for lungmetastasis including angiogenesis. The goal is to help clinicians monitoring tumors andeventually predicting their evolution or response to a particular kind of treatment. Toillustrate the whole approach, a clinical case including the natural history of the lesion,the response to a chemotherapy, and the relapse before a radiofrequency ablation ispresented.

Keywords: Clinical data assimilation; Medical imaging; Partial differential equations;Tumor growth modeling

BackgroundIntroduction

Themetastatic disease to the lung is frequently encountered in patients with cancer what-ever the primary location, and it has been associated with poor prognosis. The incidenceof such disease in patients who have died of an extrathoracic malignancy is reported to beof 20% to 54% [1]. Nevertheless, limited pulmonary metastatic disease can now be suc-cessfully treated not only for palliative reasons. By controlling the primary tumor and inthe absence of widely disseminated disease in many organs, the resection or the ablationof pulmonary metastases may prolong survival, improve the quality of life, and, in somecases, ensure cure [2]. During the last decade, a better management of the metastatic dis-ease to the lung has been achieved by the evolution of imaging, medical oncology, andsurgical techniques. There has been improvements in CT imaging quality and scan time[3], as well as advances in the field of nuclear medicine and MRI [4] which can give moreprecise information on the location and extent of the disease. In particular, there has beenwidespread use of PET/CT for evaluating patients with metastatic pulmonary disease,which can early detect metabolically active metastatic disease [5]. The targeted oncologytreatments can achieve improved responsiveness, and the resection of the lesion turnsto be possible now with minimally invasive surgical techniques as well as percutaneousthermal ablations (see [6-8]). In order to continue this trend and to improve the knowl-edge of the pulmonary metastatic disease, some authors advocated the development of

© 2015 Jouganous et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly credited.

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new tools able to explore the first steps of metastatic implantation [9]. The majority ofpulmonary metastases are derived from cancer cells that enter the lungs through the pul-monary arteries and disperse in alveolar capillaries. More rarely, metastatic disease is theconsequence of lymphatic spreading or development directly in the bronchial tree (see[10]). Most of these cancer cells are able to adhere to the endothelium of the capillar-ies. However, the cell survival is ultimately determined by local and oncology factors.Althoughmost of these malignant cells do not survive [2], those that survive penetrate theendothelium of the capillaries and install in the pulmonary parenchyma, where they grow.Therefore, a better understanding of this tumor growth could widely have direct clinicalapplications. Cancer growth modeling aims at describing, understanding, and predict-ing the evolution of tumors using numerical models. This science is constantly evolving,from the first cellular automata adapted to the microscopic scale to ordinary differentialequations describing the global dynamics of the tumor or partial differential equationstaking the spatial distribution of the cancer cells into account. If modeling can be usedto help biologists understand the complex mechanisms of tumor growth, in this work,the motivation is to develop a tool for clinical oncologists to improve patient monitoringand eventually to predict the response to a treatment of some metastases to the lung. Aswe deal with medical images, the spatial information is important and partial differentialequations seem to be the most adapted method to model tumor growth in our context. Atrue difficulty is to include in a model the biological interactions, as complex as they are,responsible for the tumor evolution without neglecting that the model must be relevantfor in vivo applications, and its parameters need to be recovered from clinical images.Among the diversity of PDE models existing on this topic, we chose, as a starting point,the one developed in [11] for the promising results obtained. The diffusion term for nutri-ent is replaced by a micro model of angiogenesis (see [12]) which seems more relevant atthe medical imaging scale. The goal is to simulate the evolution of a tumor from a givenpatient so we had to define a calibration method to personalize the model. This methodmust be a good trade-off between accuracy and computation time. For that purpose, theapproach chosen is based on a reduced order method named proper orthogonal decom-position (POD) (see [13] or [14]). The way we address this problem is similar to that of[11]: we first design a PDE model of tumor growth dealing with the spatial distributionof cancer cells with respect to time T(t, x, y). This PDE model contains some parametersthat are patient specific. The problem is therefore to determine for which parameters thenumerical solution fits the data. Once these parameters have been estimated (we call thisstep the calibration of the model), one can perform a prediction for a longer time scale.The outline of this paper is the following: in the following paragraph, we present a clin-ical case of a single metastasis including natural growth, response to a treatment, andrelapse. The ‘Methods’ section is devoted to the presentation of the PDE model of tumorgrowth that we use and the data assimilation technics. Then, the results are discussed ina dedicated part.

A typical clinical case

In this paper, we will focus on a clinical case of a patient with an isolated metastasis in thelung from a primary bladder tumor. The history of the patient is presented in Figure 1.Figure 1a,b,c shows the CT scans of the natural history of the lesion, i.e. without treat-

ment. The patient has been seen for a radiofrequency ablation on 2008/12/10. At this

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 3 of 17

Figure 1 Extract from a sequence of CT scans showing the evolution of a nodule (a to f).

point, the lesion was too large and the oncologist started a chemotherapy. Figure 1d,eshows the CT scans during the treatment while Figure 1f shows the relapse a month afterthe end of the chemotherapy. A radiofrequency ablation was performed in July 2009. Thequestions that we want to answer in this paper are the following ones:

1. Is it possible with Figure 1a,b to estimate Figure 1c?2. Is it possible with Figure 1a,b,c,d to estimate the efficacy of the treatment Figure 1e

and the amount of the relapse Figure 1f?

MethodsAmathematical model for lungmetastasis

The model we use in this work is derived from the one described in [11]. We here con-sider only one kind of cancer cells. The tumor micro environment, and in particular theamount of nutrients available, is essential to explain its evolution. Consequently, insteadof directly modeling the nutrient density, we use a very simplified angiogenesis modelto take into account the process by which the tumor escapes the avascular stage. Thebiological hypotheses of the model are the following:

• One single type of nutrient is considered. Its concentration controls cellularproliferation and death.

• The amount of nutrients available is proportional to the quantity of blood vessels inthe tissue. The nutrient diffusion is not taken into account as it is not relevant at thescale we consider.

• The only cellular motion we take into account is the passive transport due to volumevariations caused by mitosis or cellular death.

• Cancer cells are continuously switching from the proliferative to the quiescentphenotype.

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 4 of 17

Cell behavior

The tumor cell density is denoted by T and satisfies the following equation:∂T∂t + ∇ · (vT) = (γp − γd)T , (1)

where v is the transport velocity. The left part of this transport equation satisfies themass conservation principle, and the source term drives this evolution (proliferation anddeath) of the cells population. The coefficients for proliferation and death by hypoxia(lack of dioxygen), γp and γd, are detailed in Equations 2 and 3. Above a given thresholdof nutrient supplyMth, cancer cells tend to proliferate whereas below this threshold, theystarve to death. The use of a hyperbolic tangent in both γp and γd expressions amounts tosmoothing and regularizing the threshold functions, and K is a smoothing constant fixed.These functions are given by:

γp(M) = γ01+ tanh(K(M − Mth))

2 , (2)

γd(M) = γ11 − tanh(K(M − Mth))

2 . (3)From the tumor cells density T and the field γp, we can have an insight on the prolifer-

ative and quiescent cells location. Indeed, the cells that are proliferative were γp = γ0 andquiescent elsewhere, so proliferative cells are given by the expression P = γp

γ0T whereas

quiescent cells are given by Q =!1 − γp

γ0

"T . In addition to cancer cells, we also take

into account the healthy tissue whose density is denoted by S. We assume here that thishealthy tissue is transported at the same velocity as the cancer cells but is globally nei-ther proliferating nor dying. More precisely, cellular birth and death counterbalance eachother at the timescales we consider so we have, for S, a transport equation without sourceterm (Equation 4):

∂S∂t + ∇ · (vS) = 0. (4)

As we consider that the tissue is saturated, the sum of the two densities T and S is equalto 1 which gives the value of S (Equation 5) (see [15]):

S = 1 − T . (5)

Summing Equations 1 and 4, we obtain an equation on v (6):∇ · v = (γp − γd)T . (6)

This is not sufficient to determine the velocity. However, to close the system ofequations (see [16]), we consider that the velocity v is obtained using a Darcy law inEquation 7: v is derived from a pressure or potential π in the tissue.

v = −∇π . (7)

This formulation amounts to saying that tumor cells are pushed out if they are prolifer-ating or pulled in if they are dying. Here, we could also use a Stokes equation to describethe velocity (see [15]) but it complicates the model without improving significantly itsaccuracy or biological relevance.

Angiogenesis

At the end of the avascular stage of its development, the tumor reaches such a size thatits direct environment is not able to supply enough nutrients to allow the tumor to keep

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 5 of 17

on growing. At this point, cancer cells emit chemical signals such as vascular endothelialgrowth factor (VEGF) which result in the emergence of a tumoral neovasculature (see[12,17,18]). Research on this phenomenon is very active in particular to develop anti-angiogenic treatments. As most of the metastases visible on medical imaging are in thevascular stage, it was important to take the angiogenesis process into account. However,it is out of reach to model this extremely complex phenomenon in details. We thereforechoose to use a very compact model in order to achieve a good balance between thecomplexity of the model and the feasibility of the data assimilation (see next section). Itis described by the last two equations Equations 8 and 9. The scalar variable ξ is the totalamount of pro-angiogenic agent such as VEGF. It is secreted by quiescent cells given bythe expression

!"

"1 − γp

γ0

#Tdω, " being the computational domain, and evacuated by

the organism.∂ξ

∂t = α

$

"

%1 − γp

γ0

&Tdω − λξ . (8)

As we assume that the quantity of nutrient is proportional to the density of blood ves-sels in the tissue, we mixed these two notions in the unique variable M we shall call‘vasculature’. It can be seen as the environment ability to meet the needs of the tumor interm of nutrients and could be related with the carrying capacity introduced in the Hah-nfeldt ODE model [19]. The vasculatureM is damaged by proliferating tumor cells whileangiogenesis is promoted by quiescent cells; it therefore obeys

∂M∂t = −ηTM + βξ

%1 − γp

γ0

&T . (9)

The model takes into account some important mechanisms involved in tumor growthsuch as proliferation, death, or angiogenesis. Moreover, it is simpler than the previousone [11] as we just have one sort of cancer cells and there is no transport or diffusion ofthe vasculature. This trade-off is made to keep the model biologically relevant yet simpleenough to be parametrized.

Taking therapeutical effects into account

The model architecture makes easy to include different types of treatment. Chemother-apy effects can be simulated by adding a death term on Equation 1 as −δT . In this case,Equation 1 becomes:

∂T∂t + ∇ · (vT) = (γp − γd)T − δT . (10)

To fullfill the saturation assumption, Equation 6 is modified as follows:

∇ · v = (γp − γd − δ)T . (11)

We can also take into account an anti-angiogenic drug. In this case, we define atherapeutical indicator through a function denoted by f as:

f (t) ='1 without treatment,< 1 under treatment.

If we consider that it inhibits the production of pro-angiogenic agent, Equation 8 canbe modified as follows:

∂ξ

∂t = αf (t)$

"

%1 − γp

γ0

&Tdω − λξ .

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 6 of 17

If the anti-angiogenic drug also inhibits the growth factor membrane receptors on theendothelial cells, we can add a corresponding term to the vasculature equation 9:

∂M∂t = −ηTM + βξ f (t)

!1 − γp

γ0

"T .

We postpone the modeling of this effect of anti-angiogenic drugs to a future work.

Data assimilation technique

The idea is to use the model that has been described in the previous section to obtaina forecast of the evolution of the lung tumor presented in the ‘Background’. This modelcontains numerous parameters that have to be recovered using the two images in orderto perform the prediction. Some appear explicitly in the equations, such as the scalars α,β , γ0, γ1, η, λ, andMth. Others are implicit and imposed as initial conditions, such as thescalar field M(t = 0) and the scalar ξ(t = 0), and in the case where the patient is undertreatment, at least one additional parameter has to be determined. The goal of this sectionis to build a calibration method fast and accurate enough to recover an adequate set ofparameters describing the tumor evolution.

Simplifying assumptions

As mentioned before, we have eight scalars and one spatial field to identify. Thesequantities cannot be estimated by in vitro or in vivo experiments. Furthermore, theparameterization problem is ill posed. We can fix ξ(t = 0) arbitrarily without loss of gen-erality since the variations of ξ(t = 0) can be taken into account with parameters α, β ,and λ. For convenience, concerning the order of magnitude in our numerical code, wetake ξ(t = 0) = 0.1. ForM, the situation is more complex and we have no universal solu-tion to propose. We choose to takeM(t = 0) = 0.8×T +S. This means that the availablequantity of nutrients is lower inside the lesion than outside and that initially these quan-tities are constant. Note that this property is not satisfied for t > 0 because of Equation 9.This is one of the strengths of the model: it is possible to obtain an heterogeneous dis-tribution of nutrients within the tumor during the evolution that accounts for complexevolution and that makes the difference with scalar models dealing only with the volumeof the tumor. Again, the ratio 0.8 in the initialization ofM is arbitrary and has to be relatedtoMth and K .

Sensitivity analysis

In order to restrict the parameter space, we perform a sensitivity analysis on the model.It aims at determining if some parameters have more influence on the results than oth-ers. In this case, it means that the variation ranges of the less meaningful parametersmust be modified or that we can simply fix them to a nominal value. For each parameter(pi)i=1...7 = (α,β , γ0, γ1, λ, η,Mth), we choose a range of variation [ai; bi] given in Table 1.Denoting by T (t, x, y, p1, . . . , p7), the solution given by the PDE system described previ-ously, we compute numerically ∂

∂pi#(

$tf , x, y, p1, . . . , p7

%dxdy for each i = 1 to 7, tf being

the final time.We evaluate the mean valuemi and the standard deviation Si of these quan-tities when the values of the parameters (pi)i=1...7 are uniformly distributed on [ai; bi]. We

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 8 of 17

this distance, and as a criterion, we chose a combination of the comparison of the massand the L2 distance as follows. The cost function is denoted by fobj:

fobj(p1, . . . , p7) = w1 ×ns!

i=2

"#$

!(Tdata(ti) − Tmodel(ti, p1, . . . , p7))2dω

%

+ w2 ×ns!

i=2

&''''

$

!Tdata(ti, x, y)dxdy −

$

!Tmodel(ti, x, y, p1, . . . , p7)dxdy

''''

(,

(12)

where (p1, . . . , pk) is the parameter set used in the model simulation, w1 and w2 areweights to balance the influence of each term, and ns is the number of snapshots we have.The first term in the objective function accounts for the shape while the second one onlyaccounts for its volume. Of course, as initial data for our simulation, we use the ‘exact’value given by the data at time t = t1, therefore the sum starts at i = 2. As a consequence,we need at least two images of the metastasis at different times to personalize the model.The tumor cell density Tdata(ti) is extracted from the ith snapshot (corresponding to timeti). The quantity Tmodel(ti, p1, . . . , pk) is the tumor cell density given by the simulation attime ti and for the parameters (p1, . . . , pk). To compute the L2 distance, a data registra-tion is necessary. We simply translate the images in order that their centers of mass matchwith the center of mass of the first image T0. The masses are computed by integrating thedensity on the domain!. From a computational point of view, whatever the minimizationmethod is, one has to evaluate many times the objective function. It implies to simulatethe model for lots of parameter sets which could be quite expensive. For instance, if weuse a gradient algorithm to estimate seven parameters, at each iteration, the model is sim-ulated eight times. To make the calibration faster, we have developed a strategy based ona reduced order method called proper orthogonal decomposition (POD) (see [21]).

Building a reduced ordermodel to speed up computations

POD resolutionmethod for infinite dynamical systems consists in approaching partial dif-ferential equation systems with ordinary differential equations by decoupling the time andspace variables (see [13] and [14]). The initial infinite dimension problem is replaced bya finite dimension (the smallest as possible) problem. Let us first describe the POD usedon the tumor cell density variable T . As we want to decouple space and time variables, weuse the following representation for T (or any variable of interest):

T(X, t) =d!

i=1aTi (t)#T

i (X)+ ϵ(X, t),

where aTi are scalar functions depending on time and #Ti are spatial functions called

modes and represent the geometry of the variable T . The dimension of the reduced prob-lem is denoted by d. As it is an approximation, an error ϵ(t,X) is committed. The goal ofPOD is to provide us the best basis of spatial functions #T

i to minimize the error.These functions#T

i are extracted from a database of admissible behaviors of T . To gener-ate this database, we sample the parameter space, run the direct model for each parameterset thus obtained, and keep several snapshots

)STk

*k of the variable T . If the sample is cor-

rectly chosen, we have a representative set of geometrical configurations for the tumorcell density. Then, we look for the functions #T

i in the d-dimensional vectorial space

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 7 of 17

Table 1 Parameter space used for the sensitivity analysisParameter Variation range

α [5; 25]

β [5; 25]

η [0.1; 1]

γ0 [0.2; 1.4]

γ1 [0; 0.3]

λ [0.2; 1.2]

Mth [0.7; 1.1]

use Morris method (see [20]) to perform this computation. The corresponding values of(mi, Si) are plotted in Figure 2.The influence of the parameter can be established by the distance to the origin of the

corresponding point (mi, Si). In our case, we can see that the most influent parameter isγ0 which is not surprising as it rules the exponential growth of the tumor. Conversely,the model is almost not sensitive to variations of α and β in the ranges we consider so apart of the inverse problem can be simplified by fixing them thus reducing the degrees offreedom.

Formulation of the inverse problem

Given a sequence ofmedical images or snapshots of the tumor, we aim at finding a suitableset of parameters able to reproduce the observed behavior. In other words, simulationswith this set must fit as well as possible the clinical data. For this, our approach is to use ascalar objective function, which basically quantifies the distance between the observabledata and themodel simulation, and try tominimize it. There are different ways tomeasure

Figure 2 Sensitivity analysis on the tumor growth model using the Morris method. Effect of eachparameter on the final tumor volume. The farther the point is from the origin, the more meaningful thecorresponding parameter is for the model.

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 9 of 17

generated by the snapshots from the database (we denote by Eh = Span!!STk

"k"this

approximation space). These functions are taken as an orthonormal basis verifying:

∀h ∈ {1, . . . , d}, ||T(X, t) −h#

i=1aTi (t)!T

i (X)|| = min(Fi)i⊂Eh

||T(X, t) −h#

i=1aTi (t)Fi(X)||.

This property is very interesting as it allows us to use a small number of modes withoutloosing too much accuracy. The basis solution to this minimization problem is extractedfrom the autocorrelation matrix of the snapshotsMT

$MT

i,j =$STi |STj

%, (.|.) being a scalar

product on Span!!STk

"k""

by the formula (Equation 13):

∀i ∈ {1, . . . , d},!Ti (X) =

1&

λTi

ns#

k=1vTi [k] STk , (13)

where λTi denotes the ith eigenvalue ofMT and vTi [k] the kth component of the ith eigen-vector ofMT .Once we have obtained these POD modes, we compute the projection of the PDE systemon the approximation space Eh using the eigenmodes !T

i (X). We use the POD approachon both the tumor cell density T and the pressure field π which are the two fields drivenby PDEs in our system. Thus, we obtain the approximations T(X, t) = 'nT

i=1 aTi (t)!Ti (X)

and π(X, t) = 'nπi=1 aπ

i (t)!πi (X), nT and nπ being the numbers of modes we use to

represent T and π .

Reduced order model on T: We inject the POD expression of T in Equation 1, wherewe denote $ = γp − γd:

#

i

∂aTi∂t !T

i +#

iaTi ∇ .

$v!T

i%=

#

iaTi $!T

i .

Then, this equation is projected on the orthogonal basis!!T

k"k , which gives:

∀j ∈ {1, . . . , nT },∂aTj∂t +

#

iaTi

$∇ .

$v!T

i%|!T

j%=

#

iaTi

$$!T

i |!Tj%.

After simplification, we have a system of ODE on the POD coefficients!aTi

"i which is

solved with an implicit Euler scheme:

∀j ∈ {1, . . . , nT },∂aTj∂t =

#

iaTi

$$!T

i − ∇ .$v!T

i%|!T

j%. (14)

That can be rewritten as:dAT

dt (t) = NT (t)AT (t),

where NTi,j(t) =

$$(t)!T

j − ∇ .$v(t)!T

j%|!T

i%.

Therefore, we avoid the time interpolation phase that was essential in [11] to approxi-mate the time derivative. Consequently, the hypothesis that two consecutive medical dataare close in time is not necessary in our case and the accuracy of the scheme is improved.

Reduced order model on π : From Equations 7 and 6, we obtain:

−'π = $T .

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 10 of 17

Then, we use the POD expression for π ,

−!

jaπj "#π

j = $T .

Finally, we project this last equation on the orthogonal basis"#π

i#i:

∀i ∈ {1, ...nπ },−!

jaπj

$"#π

j |#πi%=

"$T |#π

i#. (15)

Here again, the system above can be written in the compact form:

NπAπ (t) = −Bπ (t),

where Nπi,j =

$"#π

i |#πj%and does not depend on the time and Bπ

j (t) ="$(t)T(t)|#π

i#.

Initialization: For the simulations, we also need to set initial values for the coefficients"aTi

#i and

"aπi#i). These values are obtained by projecting the initial configurations of

the fields associated on the POD basis: ∀i ∈ {1, ...nT }, aTi (t = 0) ="T(t = 0)|#T

i#and

∀j ∈ {1, ...npi}, aπi (t = 0) =

"π(t = 0)|#π

i#.

Resolution of the inverse problem

The aim of this section is to sum up the algorithm we use in order to solve the inverseproblem associated to the cost function (Equation 12), that is, find (p1, . . . , p7) =argmin(fobj(p1, . . . , p7)). We proceed in two steps.

Step 1: We perform a Monte Carlo method in order to determine a first approximationof the parameters and to avoid a local minimizer. This consists in comparing simulationsperformed with parameter sets randomly chosen in an empirical parameter space. Wekeep the parameters set corresponding to the lower value of the cost function fobj (seeEquation 12).

Step 2: Once we have this first approximation, we start a gradient descent method usingthe result of the first step as initialization. In this second step, the value of parametersα and β are fixed to those obtained in step 1, since the sensitivity analysis performed inparagraph 4.2 shows a low variability of the results with respect to their values.Note that the use of the POD method implies that we deal with the evaluation of solu-

tions of ODEs and not of PDEs during this process, and this fact saves a lot of CPUtime.

Results and discussionTumor growth

The method previously described was applied to our clinical case (Figure 1). The firstscan (on 2008/06/07) represents the initial condition for the tumor cell density T . Theother one (on 2008/09/22) is used in the objective function from Equation 12 as Tdata.The process to extract information from this type of medical images is quite simple. First,the tumor is delineated by a radiologist. The scan measures the permeability to X-rayswhich we assume is proportional to the density of cells. So we consider that the tumor celldensity T is proportional to the normalized intensity of the corresponding voxel on theimage. To further simplify the problem, we assume, as a first approximation, that tumor

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 11 of 17

cells do not die from hypoxia so we fix γ1 = 0. Let us remind that we also set the initialconditions for M and ξ . We still have six parameters to estimate. A parameter space isempirically defined, and we generate a database of simulations on this space. In this case,the database contains 374 simulations; and for each simulation, we save 5 snapshots forboth T and π . From this collection of snapshots, we extract the POD modes and thecalibration process is launched with the reduced model. For the simulation of the PODmodel, we only use 36 modes on T and 10 on π which is here a good balance betweenaccuracy and computation speed. We obtain the set of parameters given in Table 2.These parameters are used to make a simulation on the direct model and try to predict

the evolution of the tumor. A third CT scan on 2008/12/10 is available to compare ourprediction and the real evolution.The result of this simulation is compared with the observed growth of the metastasis in

Figure 3. The prediction from the model is satisfactory. To have a quantitative criterionof evaluation of the prediction, we can consider the mass of the tumor which is drawn inFigure 4.We can see that the horizontal error of prediction of the third scan is less than 10

days. Moreover, the relative error is below 5% of the total duration which is lower thanthe 10% segmentation error made by the clinician. If we look at the curves, we observea small difference between the reduced model simulation and the direct model sim-ulation. This is due to the POD hypothesis and the truncation of the POD basis. Toquantify the accuracy in shape of the simulation, we calculated spatial indicators as theDICE and the volume concordance given by: DICE = 100 × 2 ∗ |Tmodel

!Tdata||Tmodel| + |Tdata|

and

VC = 100 ×"1 − |Tmodel − Tdata|

|Tdata|

#. Moreover, the temporal prediction error is another

significant indicator. Indeed, clinicians often need to know when the tumor will reach acritical size. It could be, for instance, the size over which the tumor cannot be treatedusing radiofrequencies. We denote by ti the time of the ith exam and t′i the time when thesimulated tumor reaches the size of the real tumor at the ith exam. It is relevant to lookat the delay between the simulation and the real case ti − t′i and the normalized delay100 × ti−t′i

ti−t0 , i = 1, 2. The values at the calibration time point (2008/09/22) and the finalscan (2008/12/10) are given on Table 3.

Chemotherapy

Given the fast growth of this metastasis, clinicians decided to treat the patient with achemotherapy. The treatment starts just after the last scan on 2008/12/10 and ends on2009/06/29. To monitor the efficiency of the treatment, three control scans were planned:two during the treatment (on 2009/03/21 and 2009/05/27) and a last one, 1 month after

Table 2 Parameter set obtained by the inverse problemα Angiogenic agent synthesis factor 8.109

β Angiogenic growth factor 7.241

η Vasculature destruction factor 0.673

γ0 Proliferation rate 1.116

γ1 Death by hypoxia rate 0

λ Angiogenic agent destruction rate 0.865

Mth Hypoxia threshold 1.045

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 12 of 17

Figure 3 Comparison between the prediction and the observation. Top row shows the real medicalimage, below the simulation. The dates are on the left 2008/09/22 (calibration) and on the right 2008/12/10(prediction).

Figure 4 Mass evolution comparison between the data and the simulations for the tumor growth.

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 13 of 17

Table 3 Scalar indicators for the tumor growth of the first clinical case: DICE, volumeconcordance, and delays

2008/09/22 2008/12/10

DICE 90.96% 87.21%

Volume concordance 82.54% 77.76%

Delay (days) 0 −6.7

Normalized delay 0% −3.6%

the end of the therapy (on 2009/07/27). This last one shows a relapse as the tumor startsgrowing again from the end of the treatment. It is interesting for us to calibrate the modelon this decreasing phase to try to predict the response to the treatment and possiblythe relapse. As the model has already been parametrized on the growth phase, we keepthe same parameters from this calibration and use the cytotoxic drug modeling given inEquations 10 and 11. This second inverse problem is easier as the only parameter to esti-mate is δ. This time again, we need two images. We use the last one before treatment (on2008/12/10) as the initial condition and the first control (on 2009/03/21) to parametrize.We find δ = 1.05 and then the direct model is simulated up to after the date of the lastscan (2009/07/27).The evolution of the tumor mass during the treatment provides a good insight on the

therapeutical efficacy. It is given in Figure 5 and we can see that here again, the model ispredictive for this case and provides a good estimation of the response of the patient tochemotherapy. The tumor shape during treatment obtained by simulation is compared tothe clinical data in Figure 6, and the shape indicators are given on Table 4.The tumor shape is quite well reproduced by the model for the first control scan which

we use to find the treatment parameter δ. However, on the last control scan made duringthe cure (on 2009/05/27), the tumor is quite hard to delineate due to various physiologicalphenomena that are not taken into account in the model (such as edema or fibrosis). Itresults from the cytotoxic effects of the treatment on the tumor cells. Finally, the model

Figure 5 Mass evolution comparison between the data and the simulations under treatment.

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 14 of 17

Figure 6 Comparison between the prediction and the observation for the tumor decrease.Comparison between the prediction and the observation for the tumor decrease due to treatment and therelapse post treatment: above the real medical image, below the simulation. The dates are on the left2009/03/21 (calibration) in the middle 2009/05/27 (prediction for the response to treatment) and on theright 2009/07/27 (prediction for the relapse).

thus calibrated gives a good prediction of the relapse after the end of the chemotherapy.For this last time point, shape comparison provides a good result.

Another clinical test case

The whole calibration method described previously was used on another case of tumorgrowth. Here again, we use two scans at different time points to calibrate the model anda third image to quantify the accuracy of the prediction. The mass comparison is given inFigure 7.In this case, the growth is slower than in the previous case and the model is able to

reproduce such a kind of dynamics. Indeed, the time error in prediction is about 5.6 dayswhich, at the time scale we work and considering the tumor registration uncertainties, isa satisfactory result. The shape of the simulated tumor is also satisfying according to thespatial distribution of tumor cell densities in Figure 8 and the indicators on Table 5.

ConclusionTo try to answer the questions raised by our clinical case, a new approach was developed.First, a model was written that takes important mechanisms of tumor growth like hypoxia

Table 4 Scalar indicators for the tumor under chemotherapy and rebound of the firstclinical case: DICE, volume concordance, and delays

2009/03/21 2009/05/27 2009/07/27

DICE 92.26% 87.44% 84.79

Volume concordance 84.41% 74.56% 69.9%

Delay (days) 0.5 −0.84 −6.4

Normalized delay 0.5% −0.5% −2.8%

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 15 of 17

Figure 7 Second test case: mass evolution comparison between the data and the simulations.

Figure 8 Comparison between the prediction and the observation for the second test case. Above thereal medical image, below the simulation. The dates are on the left 2010/03/11 (calibration) and on the right2010/07/16 (prediction).

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Jouganous et al. Journal of Computational Surgery (2015) 2:1 Page 16 of 17

Table 5 Scalar indicators for the tumor growth of the second test case: DICE, volumeconcordance, and delays

2010/03/11 2010/07/16

DICE 85.41% 88.69%

Volume concordance 70.59% 76.45%

Delay (days) 0 5.6

Normalized delay 0% 2.3%

or angiogenesis into account. Yet, the model is simple enough to be calibrated for a spe-cific patient. We have described a new calibration method based on proper orthogonaldecomposition to fit the patient data and perform a prediction of the tumor evolution;prediction which is confirmed qualitatively and quantitatively by the medical imaging.As the model contains various kinds of treatments, we could, on the same clinical case,establish a precise quantitative prediction of the response to the treatment at the end ofthe protocol. As shown in Figure 3, the evolution is really quick between 2008/09/22 and2008/12/10 and is difficult to predict by the clinicians only using the scans. In that kindof circumstances, a mathematical approach like ours could have helped the oncologist inhis diagnosis. On this particular case, the interest of the simulation is clear. For the timebeing, we do not use any data on the real vasculature. With the constant progress of med-ical imaging, one can imagine that such data will soon be available. This advancement willvalidate our modeling of the vasculature M. We mainly used here two dimensional dataextracted from scans (that naturally give a 3D view of the tumor). The whole method wascoded for three-dimensional data, from the direct model simulation to the POD projec-tion and the calibration algorithm. Currently, it is computationally too expensive to builda database in 3D so we cannot use the whole 3D process on the clinical case. However,by improving and making the database generation faster, it could be interesting to use a3Dmethod as the reduced model simulation calculation time is almost the same as in 2D.Moreover it would allow us to free ourselves from the choice of a particular slice.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsConceived and designed the experiments: TC FC JJ JP OS. Performed the experiments: TC JJ OS. Analyzed the data: TC FCJJ JP OS. Contributed reagents/materials/analysis tools: TC FC JJ JP OS. Wrote the paper: TC FC JJ OS. All authors read andapproved the final manuscript.

AcknowledgementsThis study has been carried out within the frame of the LABEX TRAIL, ANR-10-LABX-0057 with financial support from theFrench State, managed by the French National Research Agency (ANR) in the frame of the ‘Investments for the future’Programme IdEx (ANR-10-IDEX-03-02). Experiments presented in this paper were carried out using the PLAFRIMexperimental testbed, being developed under the Inria PlaFRIM development action with support from LABRI and IMBand other entities: Conseil Régional d’Aquitaine, FeDER, Université de Bordeaux, and CNRS (see https://plafrim.bordeaux.inria.fr/).

Author details1University of Bordeaux, IMB, UMR 5251, F-33400 Talence, France. 2CNRS, IMB, UMR 5251, F-33400 Talence, France.3Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France. 4INRIA Bordeaux-Sud-Ouest, F-33400 Talence, France. 5HôpitalPellegrin, CHU Bordeaux, Place Amélie Raba-Léon, 33000 Bordeaux, France. 6Institut Bergonié, 229 cours de l’Argonne,33000 Bordeaux, France.

Received: 7 May 2014 Accepted: 13 October 2014

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4. Modélisation de la croissance des métastases de tumeurs stromales de l’intestin grêle Ce projet a fait suite aux premiers résultats de la modélisation des nodules

pulmonaires. Comme aucun geste local n'est souvent proposé dans le cadre des localisations

secondaires de GIST et que le traitement est relativement standardisé (antityrosine kinase:

Glivec, puis antiangiogenique si échappement), la possibilité de tester les modèles sur une

période prolongée tout en prenant compte des différents traitement médicamenteux nous a

permis de mieux appréhender les phénomènes de croissance. A terme, l'impact clinique de

cette modélisation pourrait permettre de mieux prévoir et/ou détecter les échappements, ou

d'adapter la thérapeutique (autre séquence thérapeutique, proposition d'un traitement local).

Cet article a été soumis sous cette référence: Lefebvre G, Cornelis F, Cumsille P, Colin T,

Poignard C, Saut O. Spatial modeling of tumor drug resistance: the case of GIST liver

metastases. Mathematical Medicine & Biology 2015. Revision

RESEARCH CENTREBORDEAUX – SUD-OUEST

200 avenue de la Vieille Tour

33405 Talence Cedex

Spatial Modeling of Tumor Drug Resistance:

the case of GIST Liver Metastases

Guillaume Lefebvre⇤†, François Cornellis⇤‡, Patricio Cumsille§ ¶,Thierry Colin⇤†, Clair Poignard†k, Olivier Sautk†

Project-Team MC2

Research Report n° 8642 — December 2014 — 26 pages

Abstract: This work is devoted to modeling gastrointestinal stromal tumor (GIST) metastasesto the liver, their growth and resistance to therapies. More precisely, resistance to two standardtreatments based on tyrosine kinase inhibitors (imatinib and sunitinib) is observed clinically. Usingobservations from medical images, we build a spatial model consisting in a set of nonlinear partialdifferential equations. After calibration of its parameters with clinical data, this model reproducesqualitatively and quantitatively the spatial tumor evolution of one specific patient. Importantfeatures of the growth such as the appearance of spatial heterogeneities and the therapeutical failuresmay be explained by our model. We then investigate numerically the possibility of optimizingthe treatment in terms of progression free survival time and minimum tumor size reachable byvarying the dose of the first treatment. We find that according to our model, the progression freesurvival time reaches a plateau with respect to this dose. We also demonstrate numerically that thespatial structure of the tumor may provide much more insights on the cancer cell activities thanthe standard RECIST criteria, which only consists in the measurement of the tumor diameter.

Key-words: mathematical modeling, numerical simulations, GIST, metastases, resistance andrelapse treatments

Contact: guillaume.lefebvre@math.u-bordeaux1.fr

⇤ Université de Bordeaux, IMB, UMR CNRS 5251, Talence, France† Inria Bordeaux-Sud-Ouest, Talence, France‡ Service d’imagerie diagnostique et interventionnelle de l’adulte, Hôpital Pellegrin, CHU de Bordeaux, Bordeaux, France§ Group of Applied Mathematics (GMA) and Group of Tumor Angiogenesis (GIANT), Basic Sciences Department,

Faculty of Sciences, Universidad del Bío-Bío, Campus Fernando May, Chillán, Chile¶ Centre for Biotechnology and Bioengineering, University of Chile, Beauchef 850, Santiago, Chilek CNRS, IMB, UMR 5251, Talence, France

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Spatial Modeling of Drug Resistance of GIST Metastases 3

1 Introduction

Gastrointestinal stromal tumors (GIST) are the most common mesenchymal tumors of the gastrointestinaltract, with an incidence of 9-14 cases per million people per year (see [16]). In 25% of cases (see [11]),this type of cancer spreads to the liver. Even though GISTs resist to most of conventional cancerchemotherapies, the discovery of activating mutations of the KIT as well as the role of PDGFR and thenew subsequent therapeutic development have revolutionized GISTs treatments. Thanks to the availabilityof these highly active targeted therapeutic agents, GISTs have become typical models of personalizedtreatment of cancer [6]. In particular, the survival of patients with GIST has been improved with the useof tyrosine kinase inhibitors such as imatinib in first-line setting and a multi-targeted receptor tyrosinekinase inhibitor, such as sunitinib or sorafenib, that inhibits PDGFRs, VEGFRs and KIT, as second-linetreatment. However several limitations in terms of diagnosis and outcomes still remain.

First, an important variability exists in the molecular and genetic characteristics that drive thepathogenesis of these tumors. Hirota et al. have proved that molecular alterations in KIT gene withgain-of-function mutations occurred in these tumors (see [12]). Moreover, in addition to the primarymutation, secondary mutations have been identified in patients with advanced GIST pretreated withtyrosine kinase inhibitor. At this time, 10 different molecular subsets of GIST with different molecularalterations have been reported. For patients with KIT mutations, an imatinib resistance is frequentlyobserved, as reported in [5]. For other patients, imatinib controls the metastatic disease during a more orless long period, around 20-24 months in 85% of cases. Physicians have then to switch to another moleculeor use an alternative therapy. Since the prognosis and the sensitivity to the targeted treatments havebeen reported to be patient-dependent, we aim at developing a patient-dependent mathematical modelbased on medical images of liver metastases. We focus on locally advanced GIST in order to determine,for each patient, the time of emergence of mutations in cancer cells, the relapse times after the first-lineand the second-line treatments, as well as the geometric features of tumor growth.

Second, the new anticancer agents with targeted mechanisms of action as used for the treatment ofGIST have demonstrated the inherent limitation and unsuitability of the usual anatomic tumor evaluation,that only considers the largest diameter of the lesion (i.e. the RECIST criteria, see [20]). For clinicians,the challenge consists in optimizing these cancer treatments and in particular to determine the moreadequate time to switch from the first-line to the second-line treatment, in order to increase the overallsurvival. The estimate of the relapse time is therefore crucial.

Clinical follow-up to monitor the disease evolution is mainly performed with CT-scans. We emphasizethat the effect of these new drugs changes the paradigm according to which the tumor sensitivity tothe treatment is measured (see [19]), since CT-scans have reported other information such as tumorheterogeneity: the RECIST criteria seems no more sufficient.

The aim of this work is to provide a spatial model of standard treated GISTs in order to compare themodel with the images, and possibly to highlight the peculiarities of the tumor growth or regrowth. It isworth noting that this paper is a first step in the modeling of tumor drug resistance based on clinicalimages.

We provide a model, which consists in a non-linear system of partial differential equations (PDEs),in order to account for the spatial aspect of the tumor growth. Actually, the models based on ordinarydifferential equations (ODEs) as the models of Mendelsohn, Gompertz or Bertalanffy make it possible totrack the tumor area growth but they do not consider the spatial aspects of the growth. We refer to thereview by Benzekry et al. for more details on such 1D models [3] Our model is derived in the same vein asRibba et al. [17] – we also refer to [9,10] – in order to describe the evolution of the disease. The main noveltyof the model lies in the description of the treatments. Two treatments are considered: the first treatmentconsists in a cytotoxic effect while the second-line treatment has both cytotoxic and anti-angiogenic effects.3 different proliferative cells are used to describe the resistance to the treatment: one cell population issensitive to the two treatments, another one is only sensitive to the second treatment while the thirdcell population is resistant to both treatments. We also provide a simple model of angiogenesis, which is

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4 G. Lefebvre et. al

crucial in the metastasis growth.

Once the model is built, we provide an appropriate numerical scheme that makes it possible to solve thePDEs. In particular, we present in Section 3 a new WENO5 type scheme that stabilizes the computationby using a combination of the classical WENO5 stencil and a twisted stencil. We then extensively compareour model to the clinical data of one specific patient, for whom we have the whole clinical protocol, inSection 4. Let us mention that it has been necessary to introduce a numerical reconstitution of theCT-scan from the numerical results, in order to compare the grey-levels of CT-scans with our simulations.Once the fitting has been obtained, we then investigate numerically the effect of the treatment dose on thetumor growth progression. The counter-intuitive result lies in the fact that according to the parameterswe use for the fitting with the data, the increase of the dose of the first treatment does not improvethe progression free survival rate. This result is explained in subsection 4.2.2. We then conclude on theconsistency of our model by presenting the different behaviors of the tumor evolution that can be obtained.We also fit the numerical tumor area with another patient, whose tumor is close to the liver boundary. Itis worth noting that for such patients, the shape of the tumor cannot be recovered. Our model does nottake the corresponding mechanical constraints intoo account. However, the tumor area seems to be wellreproduced by our simulations.

Main Insights

On the left part of Figure 1 we provide the sequence of the CT-scans, and on the right part we give thetumor area evolution of the GIST liver metastasis of one specific patient called patient A. On each CT-scanof patient A, we have depicted the darkest region, which corresponds mainly to the necrotic cells. Duringthe tumor evolution, one can see that the heterogeneity of the tumor changes (for instance in Figure 1bthe tumor is homogeneous while in Figure 1f is heterogeneous). We distinguish in the tumor area evolutionthe points corresponding to a homogeneous tumor (in fulfilled circle) from the point corresponding toheterogeneous tumor.

Interestingly, just before the tumor regrowth at Day 776 (Figure 1c) and Day 1116 (Figure 1f), one cansee on the tumor a rim clearer than the dark core, while the response to the treatment is followed by a dark-ening of the tumor (see Figure 1b and 1e). Such successive stages of tumor homogeneities/heterogeneities,are particularly pronounced in GIST liver metastases, and our goal is to provide an explanation of suchbehavior.

The first results of this paper lies in the fact that it makes it possible to describe the tumor evolution interms of tumor area compared with the CT-scans measurements, as presented by Figure 1g: the continuousline corresponds to the numerical results. It is worth noting that we do not provide a 1D-model thatdescribes the tumor volume. We deal with a complex non-linear PDE model, which is phenomenologicaland which describes the behaviour of cancer cells with respect to the space and the time variables.Therefore this first fit with the tumor area is a non trivial insight.

The second main insight is that our model brings new information on the tumor structure, that seemscorroborated with the CT-scans. Actually, as it will be presented in the following, we link the tumorheterogeneity to an increase in the cellular activity, meaning that a resistant phenotype is emerging in theclearer region. According to our modeling, such a behavior can be seen before the treatment failure, whileRECIST criteria has not changed. For instance in Figure 2, we compare the structure of the tumor at twodifferent days: the heterogeneity of the tumors seems to be well captured by the numerical simulations.Therefore our paper can be seen as a first step in developing new tools to evaluate the tumor response totreatement based on tyrosine kinase inhibitors.

Inria

81

Spatial Modeling of Drug Resistance of GIST Metastases 5

(a) Sept 16, 2008 –Day 119

(b) June 30, 2009 –Day 406

(c) July 5, 2010 –Day 776

(d) Oct 25, 2010 –Day 888

(e) Jan 7, 2011 –Day 962

(f) June 10, 2011 –Day 1116

(g) Tumor area. Each point represents the tumor area measured onCT-scans and the line stands for our numerical results.The letter refers to the CT-scans shown on the left. The symbol �stands for heterogeneous tumor, • stands for rather homogeneoustumor and � stands for CT-scan on which it is difficult to detectheterogeneous aspect.The value of parameters used in the numerical simulation are given inTable 2.

Figure 1. Spatial evolution of the liver metastasis of patient A on a series of CT-scans.

(a) Day 406 (b) Day 776

Figure 2. Comparison between CT-scans of patient A and the numerical simulations.

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6 G. Lefebvre et. al

2 Description of the model

Our model is a system of PDEs similar to the model of Bresch et al. [8]. Roughly speaking, the tumor isdescribed thanks to densities of proliferative and necrotic cancer cells denoted by P and N respectively.Cell proliferation leads to an area increase in the tissue, which creates a pressure whose gradient transportsthe surrounding healthy cells S away from the core of the tumor. The cells located at the center of thetumor turn into a necrotic phase, because of a lack of oxygen for instance, except if angiogenesis hasoccured to provide them nutrient supply. Angiogenesis and nutrient supply are taken into account with asimplistic description similar to [4, 18]: since the vascularization drives the nutrient concentration towardsthe tumor, we introduce a variable M that describes both vascularization, neovascularization and nutrientsbrought to the tumor thanks to an advection-diffusion equation. We also introduce growth factor effectsthrough a variable ⇠ that modulates M . The set of quantities used in our model is resumed in Table 1.

The main novelty lies in the modeling of the treatments. Two treatments are considered (one can seein [15], the recent works by Lorz et al. for more elaborated models on drug resistance). The first one is aspecific tyrosine kinase inhibitor, such as imatinib, that has a cytotoxic effect on proliferative cells. Thesecond treatment is a multitargeted kinase inhibitor, such as sunitinib or sorafenib, that has both cytotoxicand anti-angiogenic effects, meaning that in addition to the cytotoxic effect, it blocks the productionof growth factors such as vascular endothelial growth factors (VEGF) and thus decreases the nutrientsupply brought to the tumor. It is well-known that the cytotoxic drugs do not impact similarly all themetastatic cancer cells since resistant phenotype can appear in the proliferative cell population. Moreoverit is well-known that cancer cells can resist differently to hypoxia. Therefore we split, as in [7], the densityP of proliferative cells (P -cells) into 3 subpopulations P

1

, P2

and P

3

, such that P = P

1

+ P

2

+ P

3

, where

• P

1

denotes the fraction of proliferative cells that are sensitive to the first-line treatment T1

, basedon imatinib molecule and also to the second-line treatment T

2

, based on sunitinib or sorafenib, thathas both cytotoxic and antiangiogenic effect,

• P

2

describes the density of proliferative cells that are resistant to the first-line treatment T1

andsensitive to treatment T

2

,

• P

3

stands for proliferative cells that are resistant to both treatments.

It is worth noting that we do not aim at describing the evolution of the tumor from the very beginingof the GIST cancer, but we only focus on the evolution of the metastasis located at the liver. Therefore,according to the clinical observations, it seems relevant to consider that the three cell subpopulations arepresent when the GIST metastasis is detected.

Name Meaning UnitP1(t,x) Fraction of cells that are both sensitive to treatments T1 and T2 -P2(t,x) Fraction of cells that are resistant to treatment T1 and sensitive to treatment T2 -P3(t,x) Fraction of cells that are both resistant to treatments T1 and T2 -N(t,x) Fraction of necrotic cells -S(t,x) Fraction of healthy cells -M(t,x) Fraction of nutrients // Vascularization -⇠(t) Average velocity of nutrients transport in direction of the tumor cm.d

�1

v(t,x) Velocity of the passive movement of the tumor under the pressure cm.d

�1

⇧(t,x) Medium pressure 1kg.cm

�1.d

�2

Table 1. List of quantities computed by the model – d = day

1 The mass unit in the pressure ⇧ and in the permeability k has no importance, since only kr⇧ is relevant, and thisterm is homogeneous to cm.d

�1. Thus k and ⇧ have just to be in the same arbitrary mass unit.

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83

Spatial Modeling of Drug Resistance of GIST Metastases 7

2.1 The mathematical model

Let ⌦ be the domain where the tumor grows, and @⌦ denotes its boundary. Note that the following modelis valid for tumors that never reach the boundary @⌦.

2.1.1 PDEs on proliferative, necrotic and healthy cells

The proliferative cell densities are driven by the following transport equations:

@

t

P

1

+r · (vP1

) =

��

pp

(M)� �

pd

(M)

�P

1

��µ

1

�

1

(t) + µ

2

�

2

(t)

�(1 +M)P

1

in ⌦, (2.1)@

t

P

2

+r · (vP2

) =

��

pp

(M)� �

pd

(M)

�P

2

� µ

2

�

2

(t)(1 +M)P

2

in ⌦, (2.2)@

t

P

3

+r · (vP3

) = (�

pp

(M)� �

pd

(M))P

3

in ⌦, (2.3)

where �

i

(t) = 1[T

iini,T

iend[

(t) is the time-characteristic functions of treatment Ti

and µ

i

stands for the rateof the death2 due to T

i

of the proliferative cells, for i 2 {1, 2}. The term v(t,x) denotes the velocity dueto the tumor area changes and M(t,x) stands for the vascularization and the nutrient supply. The rate ofproliferation (resp. death) of P -cells, denoted by �

pp

(resp. �

pd

), depends on M as follows:

�

pp

(M) = �

0

1 + tanh

�R(M �M

th

)

�

2

, (2.4)

�

pd

(M) = �

1

1� tanh

�R(M �M

th

)

�

2

, (2.5)

where R is a numerical smoothing parameter3, �0

and �

1

are respectively proliferative/decay parametersand M

th

is the hypoxia threshold.We assume that healthy cells are only sensitive to hypoxia, and they are passively transported by the

tumor area changes:@

t

S +r · (vS) = ��

sd

(M)S, (2.6)

where �

sd

is the rate of death of healthy cells due to hypoxia:

�

sd

(M) = C

S

�

1

max

⇣0,� tanh

�R(M �M

th

)

�⌘. (2.7)

Note that �

sd

vanishes exactly if M � M

th

in order to ensure that the S = 1 on the outer boundary atany time. Finally, necrotic cell density satisfies

@

t

N +r · (vN) = �

pd

(M)P + �

sd

(M)S +

�µ

1

�

1

(t)P

1

+ µ

2

�

2

(t)(P

1

+ P

2

)

�(1 +M)� �(1 +M)N, (2.8)

whereP = P

1

+ P

2

+ P

3

, (2.9)

and � is a parameter that controls the elimination rate of the necrotics cells by the immune system.The following Dirichlet conditions are used on the boundary if the velocity is incoming:

P

1

= P

2

= P

3

= N = 1� S = 0 for x 2 @⌦, if v.n < 0, (2.10)

where n is the outgoing normal vector of the domain ⌦.2The death rate due to the treatment is clearly linked to the dose of drug delivered to the patient, but not only. For

instance, the sensitivity of the patient and the dose that really reaches the tumor are also involved.3Note that the functions �

pp

and �

pd

are merely regularized Heaviside functions. For the numerical simulations, wearbitrarily set R to 5.

RR n° 8642

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8 G. Lefebvre et. al

2.1.2 Mechanics

By using the following saturation condition (as in [1])

P +N + S = 1, (2.11)

and summing (2.1),(2.2),(2.3),(2.6) and (2.8), we obtain

r · v = �

pp

P � �(1 +M)N. (2.12)

Darcy’s law ensures the solvability of the the system, similarly to [17]:(

v(t,x) = �kr⇧(t,x) in ⌦,

⇧(t,x) = 0 on @⌦,(2.13)

where ⇧ is the pressure (or potential) of the medium and k its permeability. The homogeneous Dirichletcondition is used since we consider that the domain of interest is not isolated, and the outer medium doesnot impose a pressure on the tumor. This assumption is valid for small tumors that are not mechanicallyconstrained by the extratumoral region.

2.1.3 Vascularization, Nutrient Supply and Angiogenesis

It remains to describe the vascularization/nutrient supply M and the impact of treatment T2

on it. It isworth noting that the second-line treatment does not impact directly M , but it blocks the production ofgrowth factors that drive the quantity M .

We thus introduce a scalar variable ⇠, which is related to the mean concentration of growth factors.It has been reported by [13] that hypoxic cells increase their production of growth factors, while highlyproliferative cells do not need additional nutrient supply. Therefore, if M is below M

th

then ⇠ shouldincrease. Note also that the anti-angiogenic effect of treatment T

2

decreases the production of ⇠, but onlyfor the cells P

1

and P

2

since P

3

is the density of cells that are sensitive neither to T1

nor to T2

. We thusdescribe the evolution of ⇠ as

@

t

⇠ = ↵

Z

⌦

�1 + ✏

⇠

� �

pp

(M)/�

0

�⇣�1� ⌫

2

�

2

(t)

�(P

1

+ P

2

) + P

3

⌘dx� �⇠. (2.14)

The dimensionless parameter ⌫2

2 (0, 1) stands for the anti-angiogenic effect of T2

, assumed to be similarfor P

1

and P

2

, while ✏⇠

reflects the ground production of growth factors by cancer cells.The quantity M is then driven by the following equation:

8><

>:

@

t

M � ⇠

rS

krSkrM = C

0

S

✓1� M

2M

th

◆� ⌘PM + �M in ⌦,

M(t,x) = 2M

th

on @⌦,(2.15)

where C

0

is the angiogenic capacity of healthy cells, ⌘ denotes for the destruction of the vascularizationby proliferative cells, and is a diffusion parameter. The diffusive term describes the infiltration ofblood vessels into the tumor. From the numerical point of view, this term has a regularizing effect on thevascularization M , and thus stabilizes the numerical scheme.

Note that if we have initially

0 kM |t=0

kL

1 2M

th

(2.16)

then at any time t, 0 M(t) 2M

th

. This reflects the fact that the healthy tissue surrounding the tumoris well-vascularized and supplied with enough nutrients.

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85

Spatial Modeling of Drug Resistance of GIST Metastases 9

2.2 Vector formulation of the equations on cancer cellsTo simplify the notations, let us denote the vector of cell populations by W =

t(P

1

, P

2

, P

3

, N). We definethe vector-function

G(M,W) =

0

BBBB@

⇣�

pp

(M)� �

pd

(M)��µ

1

�

1

(t) + µ

2

�

2

(t)

�(1 +M)

⌘P

1⇣�

pp

(M)� �

pd

(M)� µ

2

�

2

(t)(1 +M)

⌘P

2��

pp

(M)� �

pd

(M)

�P

3

�

pd

(M)P + �

sd

(M)(1� P �N) +

�µ

1

�

1

P

1

+ µ

2

�

2

(P

1

+ P

2

)

�(1 +M)� �(1 +M)N

1

CCCCA.

It is also convenient to define F(M,W) as

F(M,W) := �

pp

(M)

3X

i=1

W

i

� �(1 +M)W

4

, (2.17)

so that the set of equations (2.1)-(2.3) and (2.8) and (2.12) closed by the Darcy law read(@

t

W + (rW).v +W(r · v) = G(M,W) on ⌦,

W = 0 on @⌦, if v. n < 0,

(2.18)

and (�r · (kr⇧) = F(M,W), in ⌦,

⇧(t,x) = 0 on @⌦.

(2.19)

Note that the density of healthy cells S is then given by (2.11).

3 Numerical methods

We use a 2D cartesian staggered grid with a finite volume method. For the numerical calculations, thedomain ⌦ is the rectangle [0, L]⇥ [0, D]. The domain is meshed by a cartesian grid with N

x

points alongthe x-axis and N

y

points along the y-axis.The cancer cell densities are discretized at the center of the cell grid and the velocities are discretized

at the middle of each edge as shown in Figure 3.

(i� 1

2

, j)

(i, j) (i+

1

2

, j)

(i, j +

1

2

)

(i, j � 1

2

)

e

y

e

x

P

1

, P

2

, P

3

, N, S,M

v

x

v

y

Figure 3. Discretization of unknown variables on one cell grid.

Note that the equality (2.11) gives straightforwardly S, without solving equation (2.6). We split thecomputation into the following steps.

RR n° 8642

86

10 G. Lefebvre et. al

• Given4W

n, and M

n at the time t

n, we infer F

n

= F(M

n

,W

n

) and G

n

= G(M

n

,W

n

).

• We first compute the pressure ⇧

n solution to (2.19) with F

n as right hand side, from which we inferthe velocity v

n thanks to the equation (2.13).

• Then the new computation time t

n+1

= t

n

+�t is determined using the equation (3.17).

• Afterwards we compute W

n+1 from (2.18), from which we infer S

n+1.

• We end by computing ⇠

n+1 and M

n+1 thanks to (2.14)–(2.15).

Let present precisely the schemes used in the numerical simulations.

3.1 Computation of pressure and velocity

According to (2.12) and (2.13), the pressure ⇧

n is given by(�r · (kr⇧

n

) = F

n

:= F(M

n

,W

n

) on ⌦,

⇧

n

= 0 on @⌦,

(3.1)

where F is defined by (2.17). We solve this equation thanks to a classical 5-points scheme. The componentalong e

x

of the velocity v

x,n

i+1/2j

at x

i+1/2 j

(resp. the component along e

y

, vy,ni,j+1/2

at the point x

i j+1/2

)are given thanks to

v

x,n

i+1/2,j

= �k

⇧

n

i+1,j

�⇧

n

i,j

�x

, v

y,n

i,j+1/2

= �k

⇧

n

i,j+1

�⇧

n

i,j

�y

and the velocity v

n

ij

at the point x

ij

is approached by

v

n

ij

= v

x,n

ij

e

x

+ v

y,n

ij

e

y

=

1

2

⇣v

x,n

i+1/2,j

+ v

x,n

i�1/2,j

⌘e

x

+

1

2

⇣v

y,n

i,j+1/2

+ v

y,n

i,j�1/2

⌘e

y

. (3.2)

3.2 Advection equation

By definition of F given by (2.17), the equation (2.12) leads to r · v = F. Thus, the equation (2.18) canread also as a non-conservative form

@

t

W + (rW).v = G(M,W)� (r · v)W = G(M,W)� F(M,W)W, (3.3)

solved thanks to the following time-splitting scheme

W

⇤ �W

n

�t/2

= G

n � F

n

W

n

, (3.4)

W

# �W

⇤

�t

+ (rW

⇤).v

n

= 0, (3.5)

W

n+1 �W

#

�t/2

= G

n � F

n

W

n

. (3.6)

A WENO5 type method as given by [14] is used to approach the gradient rW involved in (3.5).

4The superscript n stands for the discrete time t

n of the quantity (for instance S

n is the density of healty cells at thetime t

n)

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87

Spatial Modeling of Drug Resistance of GIST Metastases 11

3.3 Computation of the quantities ⇠ and M

The scalar equation (2.14) is solved thanks to the classical forward Euler method:

⇠

n+1 � ⇠

n

�t

= ↵

Z

⌦

�1 + ✏

⇠

� �

pp

(M

n

)/�

0

�⇣(P

n

1

+ P

n

2

)

�1� ⌫

2

�

2

(t

n

)

�+ P

n

3

⌘dx� �⇠

n

, (3.7)

where we use a standard rectangle rule to approach the integral in the right hand side.In order to solve the advection-diffusion equation (2.15) satisfied by M , we use the following time-

splitting schemes:

M

⇤ �M

n

�t/2

= C

0

S

n

✓1� M

n

2M

th

◆� ⌘P

n

M

n

, (3.8)

M

# �M

⇤

�t

�

�M

#

+�M

⇤

2

= ⇠

n

rS

n

krS

nkrM

n

, (3.9)

M

n+1 �M

#

�t/2

= C

0

S

n

✓1� M

n

2M

th

◆� ⌘P

n

M

n

. (3.10)

Equation (3.9) is computed as a heat diffusion equation with a standard 5-points scheme on the grid. Theright hand side is approached by a WENO5 type scheme.

3.4 Modified WENO5 scheme

The standard WENO5 scheme as given by [14] is accurate in most of the cases, however some sets ofparameters5 make us face numerical instabilities. More precisely, starting from irrotational initial data,the simulation can generate a clover-like structure, as reported by Figure 4, while the circular shapeshould be preserved.

Note that such errors on the shape provide error on the evolution of the area of the lesion, since theclover shape increases the contact surface and thus modifies the interaction between the vascularizationand the tumor. These instabilities have to be fixed. The problem is due to the WENO5 stencil thattends to favor the directions of the grid where changes in the velocity direction occur. As we can see onFigure 4d, on the center of the tumor (around x = 5 cm in Figure 4d), there is a compression point: thevelocity has centripetal directions, since v

x

is positive on the right and negative on the left. Moreover,around 1.5 cm from the center of the tumor, there is a rim of proliferative cells that induce a spreadingmovement: the velocity is centripetal close to the center but centrifugal far from it.

More precisely, for the standard WENO5 scheme, at any point x

ij

of the grid, the numerical approxi-mation W

n+1

ij

to equation (3.5) at the time t

n+1 is given by

W

n+1

i,j

= W

n

i,j

+�t

⇣v

x,n

i,j

F��x, (W

n

i+k,j

)

k=�3,··· ,3�+ v

y,n

i,j

F��y, (W

n

i,j+k

)

k=�3,··· ,3�⌘

, (3.11)

where v

x,n

i,j

and v

y,n

i,j

are defined by (3.2) and where F is the WENO5 functional given by [14]. In order toavoid the numerical instabilities, we introduce the following twin-WENO5 scheme, which is a combinationof the standard WENO5 stencil and a rotation at the angle a of the WENO5 stencil (see Figure 5), wherea is defined by the grid steps �x and �y as

a = arctan(�y/�x) 2 (0,⇡/2).

5The set of parameters have been found incidently by fitting the tumor area evolution of patient B, see Section 5.1.

RR n° 8642

88

12 G. Lefebvre et. al

(a) Day 0 (b) Day 543 (c) Day 1053

(d) Profile of vx

along ex

at Day 1053

Figure 4. Numerical simulations with the standard WENO5 stencil for the specific set of parameters.Starting from circular initial data, clover-like structures appear.

We introduce the coefficients (v

r,n

i,j

, v

✓,n

i,j

) and �r defined by6

✓v

r,n

i,j

v

✓,n

i,j

◆=

✓cos a sin a� sin a cos a

◆✓v

x,n

i,j

v

y,n

i,j

◆, �r =

p�x

2

+�y

2

, (3.12)

and we discretize equation (3.5) thanks to our twin-WENO5 scheme:

W

n+1

i,j

= W

n

i,j

+ (1� �)�t

⇣v

x,n

i,j

F��x, (W

n

i+k,j

)

k=�3,··· ,3�+ v

y,n

i,j

F��y, (W

n

i,j+k

)

k=�3,··· ,3�⌘

+ ��t

⇣v

r,n

i,j

F��r, (W

n

i+k,j+k

)

k=�3,··· ,3�+ v

✓,n

i,j

F��r, (W

n

i�k,j+k

)

k=�3,··· ,3�⌘

,

(3.13)

where � 2 (0, 1) is a numerical parameter that has to be chosen. In particular, the standard WENO5scheme holds for � = 0. As we can see on Figure 6, our new scheme keeps the irrotational property incases where WENO5 does not.

6The coefficients v

r,n

i,j

and v

✓,n

i,j

are defined such that

vn

i,j

= v

x,n

i,j

ex

+ v

y,n

i,j

ey

= v

r,n

i,j

er

+ v

✓,n

i,j

e✓

, with er

= cos a ex

+ sin a ey

, e✓

= � sin a ex

+ cos a ey

.

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89

Spatial Modeling of Drug Resistance of GIST Metastases 13

�y

�x

(a) Uniform grid �x = �y

�y

�x

(b) Non-uniform grid

Figure 5. Stencil of the twin-WENO5 scheme for a uniform grid (left) and a non-uniform grid (right).

(a) Day 0 (b) Day 543 (c) Day 1053

Figure 6. Numerical simulations with the twin-WENO5 scheme (� = 0.26). Comparing in Figure 4, theconservation of irrotational invariance is very clearly improved.

3.5 CFL condition

In addition, CFL-type restriction condition is required to preserve numerical stability. First, the WENO5type scheme leads to a CFL condition that writes :

�t < min

✓�x

max |vx

| ,�y

max |vy

| ,min(�x,�y)

max ⇠

◆:= �t

adv

. (3.14)

The forward Euler scheme on equation (3.4) leads to the following inequality (coordinate by coordinate)

W

⇤= W

n

+

�t

2

(G

n � F

n

W

n

) �✓1+

�t

2

(

¯

G

n � F

n

1)

◆�Wn

,

RR n° 8642

90

14 G. Lefebvre et. al

where 1 =

t(1, 1, 1, 1), the symbol � stands for the Hadamard product (the pointwise product of the

vectors, coordinate by coordinate) and where ¯

G

n reads:

¯

G

n

=

0

BB@

�

pp

(M

n

)� �

pd

(M

n

)� (µ

1

�

n

1

+ µ

2

�

n

2

)(1 +M

n

)

�

pp

(M

n

)� �

pd

(M

n

)� µ

2

�

n

2

(1 +M

n

)

�

pp

(M

n

)� �

pd

(M

n

)

��(1 +M

n

)

1

CCA .

A similar relation between W

# and W

n+1 can be read from equation (3.6). Thus, assuming that Wn � 0

and S

n

= 1� ⌃

i

W

n

i

� 0 at the time t

n, the following restriction of the time step

�t < min

✓1

max

i

k ¯Gn

i

� F

nk1,

1

k�sd

(M

n

) + F

nk1

◆:= �t

W

. (3.15)

ensures that W

n+1 � 0 and S

n+1 � 0. Similarly, the forward Euler scheme in equation (3.7) and inequation (3.10) on the vascularization, leads to

�t < min

✓1

⌘

,

1

�

◆:= �t

angio

. (3.16)

Finally, since the velocity might be very small, to prevent too large �t, we arbitrarily choose a minimumvelocity v

min

and our CFL condition reads7

�t = C

CFL

min

✓min(�x,�y)

v

min

,�t

adv

,�t

W

,�t

angio

◆, (3.17)

for a given constant C

CFL

< 1.

4 Numerical Results

4.1 Numerical tools to compare the results with the CT-scansIn order to compare the numerical results to the CT-scans, we have to define the appropriate quantities ofinterest as well as to develop a numerical tool that reproduces the grey scale.

4.1.1 Numerical determination of the tumor area, the necrotic part and the tumor mass

Let the threshold ✏

th

be the minimal fraction of tumor cells above which we define numerically the tumor.The tumor area, that is numerically measured reads

A(t) =

Z

⌦

1{x :P (t,x)+N(t,x)>✏th}(x) dx. (4.1)

We also define the area of each cancer cell population as

AJ

(t) =

Z

⌦

1{x : J(t,x)>✏th}(x) dx, for J 2 {P1

, P

2

, P

3

, N}. (4.2)

We also define the mass of each population, and the total proliferative mass at any time as

MJ

(t) =

Z

⌦

J(t,x) dx, J 2 {P1

, P

2

, P

3

, N}, (4.3)

M(t) =

Z

⌦

P (t,x) dx. (4.4)

7In the simulations, we choose vmin

= 1 cm/month, considering 30 days in one month and CCFL = 0.4.

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Spatial Modeling of Drug Resistance of GIST Metastases 15

4.1.2 Numerical reconstitution of CT-scans

The comparison of the numerical simulations with the CT-scans raises an important difficulty. Actually,unlike the numerics that provide the numerical values at any point of each quantity S, P and N , theCT-scans provide grey-levels that are related to local radiodensities thanks to the Hounsfield scale. SinceHounsfield unit (HU) makes it possible to quantify the tumor area and to detect its location on CT-scans,we introduce a numerical HU, which is a linear combination of the numerical results. More precisely,we consider a linear grey scale ranging from black to white. To each species (P -cells, healthy cells andnecrotic cells), we allocate a coefficient ⌧

P

, ⌧S

and ⌧

N

, we then plot the quantity

⌧

P

P + ⌧

N

N + ⌧

S

S, (4.5)

which is a kind of numerical grey-level. Since for abdominal CT-scans, the Hounsfield scale is limitedfrom -200 to +200, we arbitrarily fix the above coefficients to

⌧

P

= 60, ⌧

S

= 120, ⌧

N

= �140,

and we set the value -200 to black color and +200 to white color.

4.2 Extensive study of one specific patient4.2.1 Comparison of the numerical results with to clinical data

We focus on patient A for whom we have the whole clinical protocol, as well as the clinical data ofthe tumor area evolution and a sequence of CT-scans. The numerical simulations are performed in asquare of side L = D = 6 cm with 120 points in each direction. The time step �t is computed by usingequation (3.17).

We choose the numerical parameters in order to reproduce the evolution of the tumor area. Inparticular, we did not try to fit with the images, we only verify that the spatial evolution is plausiblecompared to the images. The parameters are summarized in Table 2.

The numerical tumor area is then compared to the measured areas on Figure 1g (the circles representthe real data and the solid lines represent the numerical simulation). It is worth noting that according tothis figure, the evolution of the tumor area is well reproduced. We emphasize that the initial conditionsare crucial in the tumor growth. In order to match qualitatively with the shape of the lesion at theinitial time (see Figure 1a), the initial condition is chosen as a perturbed ellipse. More precisely, given 3parameters r

1

, r2

and e, we define in the domain ⌦, d(x) as

d(x) =

s✓| x� L/2 |

e | x� L/2 | +cr

1

◆2

+

✓| y �D/2 |

e | y �D/2 | +cr

2

◆2

, (4.6)

with x = (x, y) and where c =

2⇡

2⇡ � arccos(1� 2✏

th

)

. We then use the function Y defined by8

Y (x) =

8><

>:

1 if d(x) 0.5,

0 if d(x) � 1,

1

2

⇣1� cos

�2⇡d(x)

�⌘else,

(4.7)

in order to impose the initial conditions:

P

1

|t=0

= (1� ⌃

ini

)Y, P

2

|t=0

=

⌃

ini

1 + q

ini

Y, P

3

|t=0

= q

ini

P

2

(t = 0), N |t=0

= 0, (4.8)

M |t=0

= 2M

th

, (4.9)8Note that if e = 0, then Y is rotationally invariant.

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Value for fit Value for fitName Meaning Unit patient A patient B

(Figure 1g) (Figure 10g)�0 Tumor cells growth rate d

�1 2.0e-2 6.33e-3�1 Tumor cells apoptosis rate d

�1 8.0e-3 4.46e-2C

S

Healthy tissue apoptosis rate compared to �1 - 10 10M

th

Hypoxia threshold - 2 2� Elimination rate of the necrotic tissue by the

immune systemd

�1 1.33e-2 8.19e-2

Diffusion rate of the oxygen cm

2.d

�1 1.33e-2 3.33e-3⌘ Consumption rate of tumor cells d

�1 6.67e-2 8.05e-3↵ Angiogenic excitability d

�1 1.11e-3 8.0e-3� Elimination rate of angiogenic growth factor signal d

�1 2.0e-2 0.68C0 Angiogenic capacity of healthy tissue d

�1 3.33e-2 3.33e-2k Tissue permeability kg

�1.cm

3.d 1 1

T1ini Beginning (in days) of the treatment T1

administrationd 119 0

T1end Ending (in days) of the treatment T1 administration d 867 845

T2ini Beginning (in days) of the treatment T2

administrationd 867 1049

T2end Ending (in days) of the treatment T2 administration d 1298 1600

µ1 Proliferative cells death rate due to treatment T1 d

�1 7.17e-3 3.45e-3⌫2 Inhibition rate of the angiogenesis by treatment T2 - 0.8 0.90µ2 Proliferative cells death rate due to treatment T2 d

�1 4.27e-3 3.0e-4✏

th

Minimal proportion of tumor cells that can bedetected on scans – Minimal treshold for thenumerical location of the tumor

- 1.0e-2 0.1

⌃ini

Proportion of cells that are resistant to imatinib atthe time t = 0 – Equivalent to (P2 + P3)t=0

- 3e-06 0.10

q

ini

Proportion of imatinib resistant cells that also re-sist to sunitinib at time t = 0 – Equivalent to(P3/P2)t=0

- 7.5e-3 0.41

⇠

ini

Growth factor signal at time t = 0 cm.d

�1 3.33e-3 0✏

⇠

Residual production of growth factor - 0.1 0.1

L,D Dimensions of the computational domain cm 6 12N

x

, N

y

Number of point for each dimension of the compu-tational domain

- 120 132

r2 Radius along x-axis of the initial condition cm 0.47 0.5r2 Radius along y-axis of the initial condition cm 0.36 0.5e Kind of eccentricity of the initial condition - 0.35 0� Twin-WENO5 weight - 0 0.3

Table 2. List of parameters of the models and their values for the two patients considered – d = day

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Spatial Modeling of Drug Resistance of GIST Metastases 17

where ⌃

ini

denotes the proportion of cells that are resistant to treatment T1

and q

ini

is the proportion ofP

2

cells that resist also to treatment T2

. For patient A, we choose e = 0.35 and r

1

= 0.47; r2

= 0.36.

(a) Day 119 (b) Day 409 (c) Day 778

(d) Day 870 (e) Day 961 (f) Day 1120

Figure 7. Numerical simulations for patient A: spatial evolution of the lesion with numericalreconstitution of CT-scans.

The spatial aspects of the numerical simulations presented in Figure 7, make appear the followingfacts:

i) During the phase without treatment, from Day 0 to Day 119, the tumor grows. Since necrotic andproliferative cells are present, the numerical tumor is heterogeneous, as reported by Figure 7a.

ii) Then treatment T1

is delivered from Day 119 to Day 867. P1

-cells are killed and necrotic cells becomepredominant. Due to the choice of the coefficient ⌧

N

, the numerical tumor becomes homogeneousand darker as shown by Figure 7b.

iii) The rebound of the proliferative activity just before the regrowth of the tumor at Day 776 ischaracterized by an increase in tumor heterogeneity: a proliferative rim appears and gradually fulfillthe necrotic interior of the tumor as illustrated by Figure 7c. It is worth noting that treatment T

1

isstill delivered and thus resistant cells start to be predominant.

iv) Then treatment T2

is delivered from Day 867 to Day 1298. Once again, the necrotic populationincreases, and the numerical tumor is darker, as shown by Figure 7d-7e.

v) Finally, at Day 1116, new therapeutic failure is getting ready. It is characterized once again by aproliferative rim on the tumor boundary (see Figure 7f).

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Interestingly, the numerical spatial results are in accordance with the data, at least up to the lastrelapse around Day 1000. Actually, the five steps that have been numerically observed are reported byCT-scans (see Figure 1a-1f).

After Day 1116, our model is probably no longer valid since the numerical structure of the tumoris very different from the CT-scan. Maybe other phenomena that are not accounted for by our modeloccur, such as micro-environment changes or cell mutations. Some interactions in the 3

rd direction (notaccounted for here) can be also involved.

We emphasize that the model seems to provide important information that clinicians cannot haveaccess to with imaging devices. More precisely, Figure 8, which presents the area and mass evolution ofeach cell population, makes it possible to state that according to our model:

• During the first shrinkage of the tumor caused by treatment T1

, from Day 119 to Day 406, weobserve that

i) Treatment T1

kills P

1

-cells that become necrotic tissue.

ii) The reduction of the area is due to the elimination of necrotic cells by the immune system.

iii) P

2

and P

3

-cells that are not sensitive to treatment T1

keep dividing.

• During few months, from Day 406 to Day 778, the tumor area continues to slowly decrease due tothe death of P

1

-cells, however P

2

and P

3

cells keep growing and replace progressively the eliminatednecrotic cells. Even though the cellular activity of P

2

and P

3

is not affected by the treatment, thisleads in a first time, to a stabilization of the tumor area, before a regrowth of the tumor at Day 778.Actually, when P

2

+ P

3

becomes too high, the tumor growth reoccurs, governed by cells that areresistant to treatment T

1

.

• During treatment T2

, from Day 867 to Day 1298, we can notice that

i) P

1

-cells are still sensitive to the treatment.

ii) P

2

-cells become necrotic since they are sensitive to treatment T2

.

iii) P

3

-cells that are resistant to the two treatments keep growing.

It is worth noting that for each relapse, the proliferative activity occurs on the tumor boundary, wherethere are nutrients. Moreover, our model produces differences in the evolution of the tumor area A andthe tumor mass evolution M given respectively by (4.1) and (4.4).

In particular, the mass of cancer cells decreases right after the drug delivery, while the tumor areadecreases with a delay in each case. This delay may be due to the fact that the killed proliferativecells turn into the necrotic phase. Therefore the total area is still the same until the necrotic cells areeliminated by the immune system. In addition, our model distinguishes the effect on angiogenesis of thetwo treatments. In both cases the angiogenic signal decreases but for different reasons. Actually, whiletreatment T

2

inhibits directly the angiogenic signal, treatment T1

kills P

1

-cells, which implies indirectly adecrease of the production of this signal.

4.2.2 Numerical study of the influence of treatment T1

efficacy

We focus now on the numerical study of different outcomes of treatment T1

, to investigate their influenceon the tumor growth. We take the parameters of Table 2, except that we let vary µ

1

. Let us define twocharacteristic durations:

• T

PFS

, which is the progression free survival time. It is the time duration for which the tumor issmaller (in term of area) than at the beginning of the treatment.

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Spatial Modeling of Drug Resistance of GIST Metastases 19

Figure 8. Evolution of the mass (integral of grey levels, in arbitrary unit) and area (mm

2) of eachcellular population and angiogenic signal evolution (cm.d

�1) given by the numerical simulations.

• T

double

, which is the time duration for which the tumor area is below twice its initial value at thebeginning of the treatment.

Note that this study is purely theoretical and cannot be used for treatment optimization since thewhole evolution of the disease (including the relapse phase) is needed in order to obtain the parametersthat are used for the simulation. It is therefore clear that our approach, for the time being, cannot lead tothe determination of an optimal protocol, but this numerical investigation is a crucial step in addressingthis important challenge.

In Figure 9a, we show the progression free survival time T

PFS

with respect to µ

1

. If µ1

is below athreshold value µ

th

(µth

⇠ 0.0047 for patient A), then the tumor growth is not stopped. For µ1

above thisthreshold, T

PFS

increases rapidly and reaches a plateau, which means that it is not necessary to increasethe dose, since it has no effect on T

PFS

.In Figure 9b, we provide T

double

. As one can see, Tdouble

is not increased by the increase the doseabove the threshold µ

th

, which means once again it is not necessary to increase the drug delivery above acertain threshold value.

In Figure 9c, we have represented the minimum size reached by the lesion with respect to the doseµ

1

. Note that this curve is decreasing: therefore for high values of µ1

, the minimum of the lesion area issmaller. However, as shown in the Figure 9d, the relationship between the minimum size of the lesionand the doubling time is not monotonic. In particular, if the minimum size of the lesion is very small,then the doubling time can be smaller. This could be thought of as some Darwinian selection mechanism:P

1

-cells, that are predominant when the lesion is detected, are killed faster by the treatment and thereforemore room and more nutrients for P

2

-cells are available for their growth. Thus, the control time becomessmaller. These curves show that there exists a threshold µ

th

above which treatment T1

is efficient. Beyondthis threshold, the minimal tumor area decreases again, but the lifetime of the patient is not increased.

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(a) Progression free survival time (TPFS in days) withrespect to µ1.

(b) Time corresponding to the growth of tumor area by afactor 2 (T

double

in days) with respect to µ1.

(c) Minimal area reached (Amin in mm

2) with respect toµ1.

(d) Phase portrait.

Figure 9. Efficacy of treatment µ

1

for patient A. The star corresponds to the parameters used inFigure 1 for the fit of the tumor area.

5 Discussion

In the previous section, we have extensively studied our model on one specific patient, patient A, whosetumor lesion was followed up by a sequence of CT-scans. We found parameters that make it possible tocompare qualitatively the images with the numerics, thanks to our numerical reconstitution of CT-scans,and we also fitted the tumor area. The lesion of this patient is interesting since it is confined in theinner of the liver, and thus the tumor evolution was not constrained by mechanics of the organ. However,in some cases, the tumor metastasis is close to the liver boundary. In such a case, there is no hope ofproviding quantitative results on the spatial evolution of the lesion, but in the next subsection, we showthat the tumor area evolution can be well described.

5.1 Tumor area evolution for patient BIn this subsection, we focus on patient B, whose tumor evolution is quite different from patient A. Actually,the clinical protocol of this patient was the following:

i) Patient B was treated in a first time with a specific tyrosine kinase inhibitor (imatinib), whichstabilizes the increase of the tumor area during more than 10 months before a relapse.

ii) The multi-targeted tyrosine kinase inhibitor sunitinib was started, but unlike for patient A, it was

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Spatial Modeling of Drug Resistance of GIST Metastases 21

totally inefficient.

iii) A third treatment, was then delivered. Sorafenib is another multi-targeted tyrosine kinase inhibitor,that has both antiangiogenic and cytotoxic effects. The tumor seemed to be sensitive to this drugduring several months until the treatment failure occurs.

Unfortunately, as illustrated by Figure 10g, the tumor evolution was so quick between the CT-scan atDay 429 and the CT-scan at Day 845 that the metastasis has reached the boundary of the liver, andthus with our model we have no hope to provide a numerical tumor growth that would be spatially inaccordance with the CT-scans, since mechanical effect of the liver membrane have to be accounted for.Therefore we focus on the tumor area as given by the clinicians, the main challenge being to capture suchquick tumor growth.

(a) May 23, 2007 –Day 0

(b) July 25, 2008 –Day 429

(c) Sept 14, 2009 –Day 845

(d) April 06, 2010 –Day 1049

(e) Sept 28, 2010 –Day 1224

(f) May 20, 2011 –Day 1458

(g) Tumor area. Each point is the tumor area measured on CT-scansand the line stands for our numerical results.The letter refers to the CT-scans present on the left. The symbol �stands for heterogeneous tumor, • stands for rather homogeneoustumor and � stands for CT-scan on which it is difficult to detect theheterogeneous aspect.The value of parameters used in the numerical simulation are given inTable 2.

Figure 10. Spatial evolution of the liver metastasis of patient B on a series of CT-scans. The smallround lesion in the top of Figure 10a is considered.

Since sunitinib is totally inefficient in this case, we consider that treatment T1

is delivered from Day 0to Day 845, while treatment T

2

consists in the sorefinib, delivered from Day 1049 to Day 1600. We find8Do not confuse the metastasis with the gallbladder that are bigger on the first two CT-scans.

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parameters that make it possible to fit qualitatively the tumor area measured by the clinicians and ournumerical tumor area. These parameters are given in Table 2. The initial data has been chosen rotationalyinvariant, in accordance with the data of Figure 10a. As we can see, the tumor evolution is very stiff:between Day 416 and Day 614 the tumor area has been multiplied by more than 9, and once sorafenib isdelivered the tumor area decreases from 4 500 mm

2 to 2 850 mm

2. We manage, at least qualitatively, toobtain such behavior, even though the fit is not perfect.

We have also investigated the efficay of the treatment T1

, as for patient A. It has been observed thatthe progression free survival time T

PFS

, the doubling time T

double

and the minimum area reached by thelesion A

min

have the same profile as the patient studied in the previous section. In particular, there existsa threshold µ

th

below which treatment has no effect on T

PFS

. Then an increase of the dose does notimprove the T

PFS

that reached, as for patient A, a plateau beyond µ

th

. The doubling time T

double

withrespect to µ

1

is also not monotonic, contrary to Amin

. Thus, as previously, beyond the threshold µ

th

, thetumor area decreases again, but the overall survival time of the patient is not increased.

5.2 Consistency of the model

(a) Imatinib from 119th day (b-c-d) Imatinib from 119th day until867th day and sunitinib just after

(e-f-g) Imatinib from 119th day until300th day and sunitinib just after

Figure 11. Different behaviors accounted for by the model.

Our model reproduces the clinical data give for patient A and patient B. Moreover, it is possible toaccount for the following behaviors that have been reported by physicians as shown in the Figure 11 (seeTable 3 in the supplementary informations section, for the different values of the used parameters):

a) The metastasis is controlled by the treatment T1

(imatinib). In this case, there is no clinical need tochange the treatment.

b) The metastasis is controlled by the treatment T1

but then the tumor regrows. The treatment T2

(sunitinib or sorafenib) is then delivered successfully, and the tumor area is controlled.

c) The metastasis is controlled by the treatment T1

before a first relapse. Then, the treatment T2

isefficient before a second relapse.

d) The metastasis is controlled by the treatment T1

before a first relapse. Then, the treatment T2

istotally inefficient.

e) The treatment T1

is totally inefficient. Then, the treatment T2

is efficient and the tumor area iscontrolled.

f) The treatment T1

is totally inefficient. Then, the treatment T2

is efficient before a relapse.

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Spatial Modeling of Drug Resistance of GIST Metastases 23

g) The treatment T1

is totally inefficient. Then, the treatment T2

is totally inefficient. This profileholds for a patient with a genetic mutation EXON, as reported by Andersson et al. [2], or Hirota et

al. [12].

6 Conclusion

In this paper, we have provided a patient-dependent model, based on PDEs, that describes the globalbehavior of GIST metastasis to the liver during the different stages. We have presented the numericalmethods used to solve the PDE system and we introduced a new WENO5 type scheme, called twin-WENO5.Then, this model has been numerically compared with clinical observations concerning patient A, whohave been treated successively by imatinib and sunitinib. As presented by Figure 1g, our model providesresults that are qualitatively in accordance with the clinical data. In particular, our model is able todescribe not only the evolution of the size of the lesion, but also its structure, as illustrated by Figure 1and 2.

Interestingly, it has been reported by our simulations that a rim of proliferative cells appears on thetumor boundary just before the relapse time. This seems to be corroborated in the CT-scans images byan increase of the tumor heterogeneity, in the sense of grey-level, before the regrowth. The more themetastases are heterogeneous, the quicker is the relapse. This result underlines the fact that the RECISTcriteria is not sufficient to evaluate the efficiency of a treatment.

We also investigated numerically the effect of the parameter µ

1

, linked to the efficacy of treatment T1

.We have shown that, according to the numerical model, increasing the value of µ

1

, which can be seen asan increase in the drug delivery, does not provide better results in terms of progression free survival timeas reported by Figure 9.

It is worth noting that our model fits well with the data, but it is not predictive. Indeed, Figure 11b-c-dshows that the knowledge of the first 400 days is not sufficient to determine uniquely the tumor growth.This means that more precise data such as functional imaging could be necessary for a better analysis ofthe inner structure of metastases.

In conclusion, we have provided a model that fits with the clinical CT-scans we had. The forthcomingwork will consist in adding more biological information that cannot be obtained from the CT-scans, inorder to provide a predictive model. Note that functional imaging data (TEP or MRI) or biopsies mightbe crucial in enriching the present model.

Acknowledgement

This study has been carried out within the frame of the LABEX TRAIL, ANR-10-LABX-0057 withfinancial support from the French State, managed by the French National Research Agency (ANR) in theframe of the "Investments for the future" Programme IdEx (ANR-10-IDEX-03-02).

Experiments presented in this paper were carried out using the PlaFRIM experimental testbed,being developed under the Inria PlaFRIM development action with support from LABRI and IMBand other entities: Conseil Régional d’Aquitaine, FeDER, Université de Bordeaux and CNRS (seehttps://plafrim.bordeaux.inria.fr/)

The work of Patricio Cumsille (PC) was partially supported by Associative Research Program (PIA)from Conicyt under grant number FB0001. The work of PC was also partially supported by Universidaddel Bío-Bío under grant DIUBB 121909 GI/C and DIUBB 122109 GI/EF.

The authors thank very warmly Dr H. Fathallah-Shaykh for discussions and advices that significantlycontributed to improving the quality of the paper.

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7 Supplementary informations

The set of parameters used to compute the numerical results given in Figure 11 of section 5.2 is presentedin Table 3.

RR n° 8642

26 G. Lefebvre et. al

Name a) b) c) d) e) f) g)�0 2.0e-2 2.03e-2 2.0e-2 1.97e-2 1.33e-2 1.33e-2 1.33e-2�1 6.67e-3 1.0e-2 1.0e-2 1.0e-2 1.0e-2 1.0e-2 1.0e-2C

S

10 10 10 10 10 10 10M

th

2 2 2 2 2 2 2� 2.67e-2 3.0e-2 5.0e-2 3.0e-2 3.0e-2 3.0e-2 3.0e-2 3.33e-3 3.33e-3 3.33e-3 3.33e-3 3.33e-3 3.33e-3 3.33e-3⌘ 6.67e-2 6.67e-2 6.67e-2 6.67e-2 6.67e-2 6.67e-2 6.67e-2↵ 1.11e-3 1.11e-3 1.11e-3 1.11e-3 1.11e-3 1.11e-3 1.11e-3� 2.0e-2 2.0e-2 2.0e-2 2.0e-2 2.0e-2 2.0e-2 2.0e-2C0 3.33e-2 3.33e-2 3.33e-2 3.33e-2 3.33e-2 3.33e-2 3.33e-2k 1 1 1 1 1 1 1T1

ini 119 119 119 119 119 119 119T1

end 3000 867 867 867 300 300 300T2

ini 3000 867 867 867 300 300 300T2

end 3000 1700 1298 1700 1700 1700 1700µ1 8.33e-3 8.33e-3 8.33e-3 8.33e-3 8.33e-3 8.33e-3 8.33e-3⌫2 0.9 0.9 0.99 0.9 0.9 0.9 0.9µ2 6.0e-4 6.0e-4 6.6e-4 6.0e-4 6.0e-4 6.0e-4 6.0e-4✏

th

1.0e-2 1.0e-2 1.0e-2 1.0e-2 1.0e-2 1.0e-2 1.0e-2⌃

ini

0 4e-06 4e-06 2e-06 1 1 0.9q

ini

0 0 4.5e-2 1 2e-07 3.0e-2 0.9⇠

ini

3.33e-3 3.33e-3 3.33e-3 3.33e-3 3.33e-3 3.33e-3 3.33e-3

L,D 6 6 6 6 6 6 6N

x

, N

y

120 120 120 120 120 120 120r1, r2 0.62 0.62 0.62 0.62 0.62 0.62 0.62e 0 0 0 0 0 0 0� 0 0 0 0 0 0 0

Table 3. Values of the parameters for curves presented in Figure 11.

Inria

102

5. Analyse de l'erreur de contourage

Comme évoqué précédemment, le but de ce travail est d'apprécier l'erreur secondaire

au contourage et à terme de déterminer son impact dans le processus de modélisation. Ce

travail est en cours de soumission.

Impact on RECIST of manual 2D segmentation of lung and liver metastases F. Cornelis, MD; M. Martin, PhD; P-A Linck, MD; O. Saut, PhD; J. Palussiere, MD; T. Colin, PhD

ABSTRACT

Purpose: To establish the impact of manual segmentation of lung and liver metastases on

Response Evaluation Criteria In Solid Tumors (RECIST).

Material and Methods: After random selection on a de-identified database of 13 tumors (6

in liver, 7 in lung) imaged by contrast-enhanced CT-scan, manual 2D segmentations were

independently performed by 20 readers (10 physicians, 10 scientists). Diameter and area were

calculated for each segmentation and mean, min/max, standard deviation (SD) were compiled.

Regression models of SD according to diameter or area were obtained. After implementing

these models, the 95% confidence interval (CI95%) was calculated for diameter or area and

for limits of progressive disease (+20%) and partial response (-30%) according to RECIST

1.1. Overlaps of the CI95% were recorded if identified and cut-off values were obtained.

Results: A total of 247 ROI was finally used. No significant differences were observed

between the results of physicians and scientists. A linear regression was observed for diameter

or area and SD on liver while a sigmoid curve was observed on lung. After implementation of

the regression model, the CI95% of the diameter or area overlapped with the CI95% of the

limits of progressive disease or partial response on liver but not on lung. The corresponding

cut-off values for diameter of liver metastases were 22.74mm and 37.92mm, respectively.

Conclusion: For liver but not for lung metastases, evaluating RECIST 1.1 remains uncertain

for large tumor. Above the thresholds, tumor progression or response may be only related to

the error of 2D segmentation.

103

INTRODUCTION

The clinical management of lung or liver metastasis may be challenging as behavior of

metastases remains difficult to assess by clinicians over the time. Some tumor may grow

rapidly while some stay stationary for years. To date, the evaluation of tumor progression or

response to treatments has been made according to response criteria based on morphologic

imaging assessments such as those proposed firstly by the World Health Organization (WHO)

or by the more widely used Response Evaluation Criteria in Solid Tumors (RECIST) (1).

These two criteria are based on the mensuration of a given tumor along the great axes. In fact,

this corresponds to an assessment of anatomical tumor burden and its change over the time (2).

RECIST proved to be useful in clinical trials where objective response was the

primary study endpoint, as well as in trials where assessment of stable disease, tumor

progression or time to progression analyses were undertaken (2). To date, the contrast-

enhanced CT-scan is considered as the best currently available and reproducible method to

measure lesions selected for tumor response assessment. In RECIST, measurable disease is

defined by the presence of at least one measurable lesion. When more than one measurable

lesion is present at baseline all lesions up to a maximum of five lesions total (and a maximum

of two lesions per organ) representative of all involved organs should be identified as target

lesions (3). Target lesions should be selected on the basis of their size (lesions with the

longest diameter), be representative of all involved organs, but in addition should be those

that lend themselves to reproducible repeated measurements. Thus it may be possible to

define the partial response (PR), which corresponds to at least a 30% decrease in the sum of

diameters of target lesions, taking as reference the baseline sum diameters (2). Progressive

disease (PD) is at least a 20% increase in the sum of diameters of target lesions, taking as

reference the smallest sum on study. In addition to the relative increase of 20%, the sum must

also demonstrate an absolute increase of at least 5 mm.

104

However RECIST suffers of significant limitations (4). Firstly, tumors have to be

accurately measured in at least one dimension (longest diameter in the plane of measurement

is to be recorded) with a minimum size of 10 mm by CT scan. Poor measurement

reproducibility has been observed (5). Secondly, RECIST offers only a limited assessment of

response as decrease in volume is not always observed after treatment, in particular if

antiangiogenic and biologic targeted agents are used (6, 7). Third, in practice, the maximal

size mensurations or the segmentation (in 2 or 3 dimensions) are manually performed (8).

However, a concern remains on the accuracy of such segmentation. In 1D or 2D, the

radiologist chooses a representative slice of the tumor for a given patient. Same slice of the

tumor must be segmented over the time, whatever the reader. The slice is localized using

anatomical landmarks, as the patient may be not in the same exact position for every exam. In

an effort to reduce inconsistencies from current response evaluation criteria, the aim of our

study was to establish the impact of segmentation of lung and liver metastases on RECIST.

105

MATERIAL AND METHODS

This Health Insurance Portability and Accountability Act – compliant study was approved by

the institutional ethics research board and the requirement for patient informed consent was

waived. The authors had full control of the data and the information submitted for publication.

This study has been carried out within the frame of the LABEX TRAIL, ANR-10-LABX-

0057 with financial support from the French State, managed by the French National Research

Agency (ANR) in the frame of the "Investments for the future" Programme IdEx (ANR-10-

IDEX-03-02). Experiments presented in this paper were carried out using the PlaFRIM

experimental testbed, being developed under the Inria PlaFRIM development action with

support from LABRI and IMB and other entities: Conseil Régional d’Aquitaine, FeDER,

Université de Bordeaux and CNRS (see https://plafrim.bordeaux.inria.fr/).

Study cohort

A data selection was randomly performed in our departmental electronic database of de-

identified CT images. Inclusion criteria were CT explorations performed in patients imaged

for metastasis to the liver and lung whatever the primitive tumor. Two authors selected 13

tumors for this study using random number generation (7 in the lung, 6 in the liver). This

sample size was selected to reflect the power of typical studies in the literature (9). Analyzed

lesions were variable in diameter or in location within the two organs.

CT image acquisition

All patients included in this study underwent CT-scan with contrast injection. Chest and

abdominal CT scans were performed with a 64-detector CT system (GE Light Speed VCT or

GE Discovery CT750 HD, GE Healthcare, Milwaukee, WI, USA) by using the following scan

106

parameters: 1.25-mm section width, 1.25-mm reconstruction interval, pitch of 0.984, 120 kV

and 250 mA (64-detector CT system). Standard high-resolution methods were used for image

reconstruction.

Image analysis

Datasets were imported to OsiriX version 5.9 (OsiriX, Geneva, Switzerland), an open source

DICOM image analysis suite and PACS workstation designed for the Apple Macintosh

platform. Twenty readers analyzed independently CT data of 13 identified non-treated index

lesion using two different methods. Ten readers were radiologists with experience ranged

from 1 to 7 years (group 1). Ten readers were scientists with basic knowledge on image

segmentation (group 2). Method 1 consisted to select the slice where a RECIST mensuration

may be performed for a given tumor and to subsequently contour the tumor on this slice in 2D.

Maximal diameter was automatically extracted from this contour in order to perform a

RECIST evaluation. For patients with multiple tumors, an approximative location of the

tumor was given by a range of slices where the tumor could be located. Method 2 consisted to

perform the same contour but the readers were aware of the number of the slice and tumor

location. Method 2 was performed after Method 1. Region of interest were exported to the

PlaFRIM experimental testbed.

Statistical analysis

A first review of all segmentations was performed by the 2 initial authors who selected only

adequate segmentations and removed aberrant one. Adequate segmentation was defined as a

segmentation performed at worst 2 slices away the mean slice (i.e the slice more often

selected) and on the pre-identified nodule. Other cases led exclusion. All analyses were

conducted using Stata 12.0 (StataCorp, College Station, Texas, USA). First, mean, minimal

107

and maximal values and standard deviation (SD) of the index lesions were calculated for each

reader according to organs and group of reader. Second, to determine interobserver agreement

we calculated between-subject SD and within-subject SD of each variable. The between-

subject SD is a measure reflecting the overall variability between all subjects while the

within-subject SD is a measure of the agreement between readers analyzing each subject.

Intraclass correlation coefficients (ICCs) were calculated based on repeated measures

ANOVA (10, 11). ICC results were interpreted according to the following criteria: poor (ICC

< 0.50), moderate (0.50 < ICC < 0.75), good (0.75<ICC<0.90), and excellent (ICC>0.90).

Third, the mean standard-deviation ratio (SDr) (SD area or diameter / area or diameter) was

calculated after compilation of all data in order to normalize the results. Then a regression

analysis was performed in order to derive the 95% confidence interval (CI95%) according to

each organs and diameter or area (y=x). The CI95% (±σ) of area and the maximal diameter

were obtained for the limits of RECIST 1.1 criteria of progressive disease (PD, +20%)

(y1=x+20%x) and partial response (PR, -30%) (y2=x-30%x). The purpose was to detected

overlap between basic error and standard limits (yx±σ). A cut-of value was determined if

identified.

108

RESULTS

A total of 260 contours were obtained but only 247 were selected for further evaluation. A

total of 13 contours were removed because due to consistent errors of segmentation: 4 of

these were performed by radiologist (3 into the liver, 1 in the lung) and 9 by scientists (3

livers, 9 lungs). No significant differences were observed between the 2 groups.

Inter-reader agreement

For radiologists, the mean standard deviation ratio (mean standard deviation / area or

diameter) was 0.13 and 0.07 on liver, respectively (table 1) and 0.08 and 0.05 on lung. For

scientist, the mean SDr was 0.11 and 0.08 on liver, and 0.13 and 0.08 on lung. No significant

differences were observed between the results of the 2 groups. Inter-observer agreement

remained excellent (ICC>0.90) for all variables. After combining all readings, ICC were

99.1% and 99.4% for diameter and area, respectively.

Table 1: Results for each group (A: physician, B: scientist) and calculation of the mean

ratio of standard deviation

A Physicians Area

Diameter Mean value Std Dev Min Max Mean value Std Dev Min Max

Liver 1 1,45 0,10 1,29 1,61 17,63 1,61 16,22 19,01 Liver 2 13,48 3,23 9,28 18,62 48,55 18,62 43,00 57,06 Liver 3 32,81 8,04 15,47 45,51 79,14 45,51 51,83 105,49 Liver 4 5,01 0,34 4,55 5,78 27,45 5,78 26,12 29,21 Liver 5 7,60 0,63 6,48 8,57 35,52 8,57 31,79 38,46 Liver 6 25,55 1,54 22,69 27,01 65,96 27,01 60,78 69,75 Mean (std dev A/A) : 0,13 Mean (std dev D/D) : 0,07 Lung 1 1,37 0,11 1,19 1,55 14,51 1,55 13,50 15,45 Lung 2 0,30 0,02 0,26 0,33 7,32 0,33 6,93 7,74 Lung 3 0,51 0,08 0,34 0,62 10,52 0,62 8,88 12,18 Lung 4 4,87 0,40 4,33 5,39 29,86 5,39 27,25 32,61 Lung 5 15,01 0,21 14,64 15,39 48,63 15,39 47,33 49,63 Lung 6 2,54 0,20 2,14 2,72 22,32 2,72 18,87 24,55 Lung 7 1,34 0,11 1,18 1,46 15,90 1,46 14,96 17,45 Mean (std dev A/A) : 0,08 Mean (std dev D/D) : 0,05

109

B Scientists Area

Diameter Mean value Std Dev Min Max Mean value Std Dev Min Max

Liver 1 1,47 0,18 1,03 1,59 17,68 1,14 15,23 19,02 Liver 2 9,84 0,83 8,46 10,86 43,49 1,98 40,28 46,00 Liver 3 27,88 3,41 19,40 31,47 74,31 5,14 60,50 80,32 Liver 4 4,75 0,37 3,92 5,27 27,00 1,14 24,68 28,93 Liver 5 7,55 0,91 6,39 9,20 35,92 3,02 33,36 41,74 Liver 6 19,09 3,15 14,55 24,02 57,59 8,77 50,97 77,78 Mean (std dev A/A) : 0,11 Mean (std dev D/D) : 0,08 Lung 1 1,28 0,25 0,89 1,60 14,08 1,22 12,23 15,83 Lung 2 0,32 0,05 0,21 0,36 7,58 0,50 6,51 8,14 Lung 3 0,60 0,11 0,43 0,77 11,04 0,91 9,80 12,61 Lung 4 4,87 0,49 4,06 5,75 29,86 3,04 24,95 34,52 Lung 5 15,37 0,44 14,32 15,93 48,93 0,77 47,36 50,27 Lung 6 2,65 0,30 1,99 2,96 23,02 2,77 17,54 27,16 Lung 7 1,50 0,15 1,24 1,69 16,65 1,16 14,81 18,33 Mean (std dev A/A) : 0,13 Mean (std dev D/D) : 0,08

Comparing mensuration on selected slices and imposed slices, no significant differences were

observed between the groups or after combining the groups. The mean SDr remained similar

on lung and liver. Results are summarized in table 2.

Table 2: Comparison of mean standard deviation ratio selected vs imposed slices

Selected slice Imposed slice Mean (std dev A/A)

Mean (std dev D/D)

Mean (std dev A/A)

Mean (std dev D/D)

Medical Liver 0,13 0,07 0,14 0,07 Lung 0,08 0,05 0,06 0,05

Scientist Liver 0,11 0,08 0,14 0,08 Lung 0,12 0,08 0,13 0,07

All Liver 0,15 0,08 0,16 0,09 Lung 0,11 0,06 0,10 0,06

110

Impact on evaluation of area and maximal diameter

A combination of the results of the 2 groups was performed and is summarized in table 3.

Table 3: Overall results of area or maximal diameter evaluation for lung and liver

All Area Diameter Mean value Std Dev Min Max Mean value Std Dev Min Max

Liver 1 1,46 0,14 1,03 1,61 17,66 0,94 15,23 19,02 Liver 2 11,66 2,96 8,46 18,62 46,02 4,06 40,28 57,06 Liver 3 30,21 6,40 15,47 45,51 76,72 10,14 51,83 105,49 Liver 4 4,88 0,37 3,92 5,78 27,22 1,03 24,68 29,21 Liver 5 7,58 0,75 6,39 9,20 35,71 2,61 31,79 41,74 Liver 6 22,32 4,11 14,55 27,01 61,77 7,67 50,97 77,78 Mean (std dev A/A) : 0,15 Mean (std dev D/D) : 0,08 Lung 1 1,32 0,19 0,89 1,60 14,30 0,07 12,23 15,83 Lung 2 0,31 0,04 0,21 0,36 7,43 0,37 6,51 8,14 Lung 3 0,55 0,10 0,34 0,77 10,76 1,04 8,88 12,61 Lung 4 4,87 0,43 4,06 5,75 29,86 2,43 24,95 34,52 Lung 5 15,19 0,39 14,32 15,93 48,78 0,76 47,33 50,27 Lung 6 2,60 0,26 1,99 2,96 22,67 2,22 17,54 27,16 Lung 7 1,42 0,15 1,18 1,69 16,28 1,03 14,81 18,33 Mean (std dev A/A) : 0,11 Mean (std dev D/D) : 0,06

Using these results, a regression model was obtained after plotting the mean maximal

diameter or mean area of the tumor and the mean standard deviation of the mensuration

(figure 1). A linear correspondence was observed for liver whereas a sigmoid curve was

observed for lung.

111

Figure 1: Mean standard deviation according to mean maximal diameter (A) and mean

area (B) in lung, and mean maximal diameter (C) and mean area (D) in lung for each

segmented tumors (7 in the lung, 6 in the liver).

Based on these models, the 95% confidence interval of area and the maximal diameter were

calculated for each organ in order to take into account the uncertainty due to the error of

manual 2D segmentation performed by readers. For both organs, the 95% confidence interval

increased with the diameter or area. However, the results showed that the CI95% of a manual

2D segmentation remains at worst of [46.83,53.17] for a nodule of 5cm of diameter in the

lung while it was [40.66,59.34] in the liver. CI95% for other size are reported in table 4.

Table 4: 95%CI limits according to diameter of the tumor for lung and liver

Liver Lung Size (mm) Min Max Min Max 10 9,48 10,52 9,06 10,94 20 18,19 21,81 18,41 21,59 30 26,26 33,74 27,85 32,15 40 33,74 46,26 37,32 42,68 50 40,66 59,34 46,83 53,17

112

Impact on tumor response evaluation and RECIST

The 95% confidence intervals (±σ) of diameter or area for the limits of RECIST 1.1 criteria of

progressive disease (+20%) (y1=x+20%x) and partial response (-30%) (y2=x-30%x) were

obtained (figure 2). In the liver but not in the lung, the CI95% of the diameter of the tumor

(y=x) and the CI95% (±σ) obtained for the limits of RECIST 1.1 criteria of progressive

disease and partial response overlapped (figure 3).

Figure 2: CI95% obtained for the limits of RECIST 1.1 criteria of progressive disease

(+20%) and partial response (-30%) in lung according to diameter (A) and area (B) and

in liver according to diameter (C) and area (D). Uncertainty increased with tumor size.

In the liver, the cut-off value was x1=22,74mm at the intersection of CI95% of the diameter of

the tumor (y=x) and the CI95% obtained for the limits of RECIST 1.1 criteria of progressive

113

disease (+20%) (y1=x+20%x) (figure 3C). Similarly, the cut-off was x2=37.92mm at the

intersection of CI95% of the diameter of the tumor (y=x) and the CI95% obtained for the

limits of RECIST 1.1 criteria of partial response (-30%) (y2=x-30%x) (figure 3C).

Figure 3: CI95% of the diameter (A) and area (B) in the lung did not cross the CI95%

obtained for the limits of RECIST 1.1 of progressive disease (+20%) and partial

response (-30%) while in liver an overlap (blue zone) was observed for both the

diameter (C) and the area (D). RECIST may be difficult to assess for tumor size above

the intersections both for the diameter and for the area. Cut-off values were identified

for diameter (dashed lines).

114

DISCUSSION

Our study showed that no significant differences were observed between the results of

a group of radiologists aware of RECIST and of a group of scientists with only basic

knowledge on RECIST. These findings suggested that no particular experience is mandatory

to perform an adequate evaluation of manual 2D segmentation. As proposed recently (12), our

results allow the perspective of RECIST measurements performed by a paramedic which may

impact positively the workflow of radiological units.

Moreover, no significant differences were observed between the groups or after

combining the groups comparing mensuration on selected slices and imposed slices. The

mean standard deviation ratio remained similar either on lung than on liver. Therefore, the 2D

segmentation, even manual, seems to be not affected by a slight variation of the selection of

the slice. These findings justify the recommendations of RECIST 1.1, which proposed to

perform mensuration using the same plane of evaluation but with a maximal diameter of each

target lesion always be measured at subsequent follow-up time points even if this results in

measuring the lesion at a different slice level or in a different orientation or vector compared

with the baseline study (1, 2).

While the error assessed by the measure of standard deviation increased quasi-linearly

with size in liver, the error was higher for small nodules in the lung than for larger tumors.

This finding is consistent with the introduction of a minimum lesion size in lung (>10mm),

which improved intercriteria reproducibility between WHO and RECIST (13). However as

observed after regression, a quasi-linear correspondence was observed for liver segmentation

whereas a sigmoid curve was observed for lung segmentation. Thus the CI95% of a manual

2D segmentation increased with the diameter or area in particular for liver and must be taking

into account for the assessment of large liver tumor.

115

Our results showed that the CI95% of a 2D manual segmentation remains at worst of

[46.83,53.17] for a nodule of 5cm of diameter in the lung while it was [40.66,59.34] in the

liver. These results proved that the identification of the precise boundary definition of a lesion

by CT may be difficult especially when the lesion is embedded in an organ with a similar

contrast such as the liver whereas it may be less problematic in the lung due to the interface

between gas and tissue. In liver, the peritumoral oedema surrounding a lesion may be

confounded with actual tumor renders difficult to distinguish on certain modalities (14).

While further studies are mandatory, we hypothesize that similar conclusion may be drawn in

pancreas, kidney, adrenal or spleen (2).

Due to the wide CI95% on large tumor observed in our study and its potential high

impact on RECIST, a careful interpretation of the segmentation findings must be done when

performing RECIST. In the liver but not in the lung, the CI95% of the diameter or area of a

tumor and the CI95% obtained for the limits of RECIST 1.1 criteria of progressive disease

and of partial response may overlap. We identified cut-off values of diameter at the

intersection of the overlaps. The thresholds were x1=22,74mm for PD and x2=37.92mm for

PR. These size limits must be take into account in the RECIST (1, 2). Evaluating tumor above

these cut-off values renders the RECIST uncertain due to the mismatch between error

secondary to the segmentation and the true tumor progression or response. Such findings may

have affected the results of studies focus on liver metastases. It may justify the development

of alternative to RECIST in the liver such as the recently proposed modified RECIST

(mRECIST) (5) or the introduction of functional imaging in the current evaluation of liver

mets after treatment (15, 16).

Our study has some limitations. The series is retrospective and may have selection bias.

As radiological tumor response evaluation according to RECIST and WHO-criteria are

subject to considerable inter- and intra- observer variability (17), we performed 260

116

segmentations in order to reduce the impact of variability. The segmentation was performed

manually despite we calculate the maximal diameter from an evaluation of the perimeter of

the tumor. Alternatively, software tools may be employed (18, 19). These softwares allow

repeatable and reproducible measurement of both diameter and volume measurements but do

not reduce variability in response classification. Further study may compare the results of

manual vs automatic segmentations. As proposed recently (20), volumetric assessment of the

entire tumor has not been performed. However volumetric assessment and RECIST have been

showed to not be interchangeable; neither technique demonstrated clinical superiority (21, 22).

In summary, the results of our study highlighted the fact that some concern remains

for manual segmentation while performing manual 2D segmentation may be not limited by

the experience of operator. For liver but not for lung metastases, evaluating RECIST 1.1

remains uncertain for large tumor. Above the thresholds, tumor progression or response may

be only related to the error of 2D segmentation. While a prospective validation of these

findings on a larger scale is now needed before drawing definitive conclusions regarding their

true impact in a clinical prospective, these results justify the development of alternative

quantitative assessment of tumor response using functional imaging tools.

117

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comparison of RECIST and volumetric algorithms. Eur J Radiol. 2013 Nov;82(11):1831-9. PubMed PMID: 22717124. 15. Kekelidze M, D'Errico L, Pansini M, Tyndall A, Hohmann J. Colorectal cancer: current imaging methods and future perspectives for the diagnosis, staging and therapeutic response evaluation. World J Gastroenterol. 2013 Dec 14;19(46):8502-14. PubMed PMID: 24379567. Pubmed Central PMCID: 3870495. 16. Bernardin L, O'Flynn EA, Desouza NM. Functional imaging biomarkers for assessing response to treatment in liver and lung metastases. Cancer Imaging. 2013;13(4):482-94. PubMed PMID: 24334562. Pubmed Central PMCID: 3864224. 17. Suzuki C, Torkzad MR, Jacobsson H, Astrom G, Sundin A, Hatschek T, et al. Interobserver and intraobserver variability in the response evaluation of cancer therapy according to RECIST and WHO-criteria. Acta Oncol. 2010 May;49(4):509-14. PubMed PMID: 20397778. 18. van Kessel CS, van Leeuwen MS, Witteveen PO, Kwee TC, Verkooijen HM, van Hillegersberg R. Semi-automatic software increases CT measurement accuracy but not response classification of colorectal liver metastases after chemotherapy. Eur J Radiol. 2012 Oct;81(10):2543-9. PubMed PMID: 22264447. 19. Dinkel J, Khalilzadeh O, Hintze C, Fabel M, Puderbach M, Eichinger M, et al. Inter-observer reproducibility of semi-automatic tumor diameter measurement and volumetric analysis in patients with lung cancer. Lung Cancer. 2013 Oct;82(1):76-82. PubMed PMID: 23932487. 20. Vargas HA, Delaney HG, Delappe EM, Wang Y, Zheng J, Moskowitz CS, et al. Multiphasic contrast-enhanced MRI: single-slice versus volumetric quantification of tumor enhancement for the assessment of renal clear-cell carcinoma fuhrman grade. J Magn Reson Imaging. 2013 May;37(5):1160-7. PubMed PMID: 23152173. 21. Schoot RA, McHugh K, van Rijn RR, Kremer LC, Chisholm JC, Caron HN, et al. Response assessment in pediatric rhabdomyosarcoma: can response evaluation criteria in solid tumors replace three-dimensional volume assessments? Radiology. 2013 Dec;269(3):870-8. PubMed PMID: 23985275. 22. Wulff AM, Fabel M, Freitag-Wolf S, Tepper M, Knabe HM, Schafer JP, et al. Volumetric response classification in metastatic solid tumors on MSCT: initial results in a whole-body setting. Eur J Radiol. 2013 Oct;82(10):e567-73. PubMed PMID: 23827800.

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6. Intégration de l'imagerie fonctionnelle

6.1. Développement de l'imagerie fonctionnelle

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Progrès en urologie (2014) 24, 399—413

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ARTICLE DE REVUE

L’imagerie d’évaluation thérapeutique enpratique clinique d’oncologie urologiqueQuantitative imaging in uro-oncology

F. Cornelis a,∗, H. de Clermontb, J.C. Bernhardd,A. Ravaudc, N. Greniera

a Service d’imagerie diagnostique et interventionnelle de l’adulte, hôpital Pellegrin, CHU deBordeaux, place Amélie-Raba-Léon, 33076 Bordeaux, Franceb Service de médecine nucléaire, hôpital Pellegrin, CHU de Bordeaux, placeAmélie-Raba-Léon, 33076 Bordeaux, Francec Service d’oncologie médicale, hôpital Saint-André, CHU de Bordeaux, 1, placeJean-Burguet, 33076 Bordeaux, Franced Service d’urologie, hôpital Pellegrin, CHU de Bordeaux, place Amélie-Raba-Léon, 33076Bordeaux, France

Recu le 18 decembre 2013 ; accepté le 28 fevrier 2014Disponible sur Internet le 5 avril 2014

MOTS CLÉSCancer ;Imagerie ;IRM ;CT-scan ;Traitement ;Rein ;Prostate ;Imageriefonctionnelle

RésuméIntroduction. — L’imagerie réalisée actuellement en oncologie urologique peut apporter denombreuses informations utiles en plus de la simple description morphologique. L’exploitationde toutes ces informations pourrait aider à mieux appréhender la croissance tumorale et lesréponses aux traitements. Il semblait donc intéressant de faire l’état de lieux des connaissances,de décrire les principales techniques disponibles actuellement et de préciser leurs résultats.Matériel et méthode. — Une revue systématique de la littérature a été effectuée concernantles techniques d’imagerie d’évaluation en uro-oncologie dans la base de données PubMed.Les mots clés utilisés étaient : « cancer », « kidney », « bladder », « prostate », « urology »,« biomarkers », « imaging », « ultrasound », « CT-scan », « MRI », « PET-CT », « RECIST », « BOLD »,« ASL », « Diffusion or DWI », « contrast », « F-miso ». Les premières publications mises en évi-dence ont été analysées pour y rechercher des études non identifiées par les mots cléssélectionnés.Résultats. — Du simple aspect morphologique aux données plus complexes de l’imagerie fonc-tionnelle (TEP, IRM), l’exploitation des données de l’imagerie peut permettre de facon effectivede mieux appréhender la croissance tumorale et les réponses aux traitements. Même si des opti-misations sont à venir, l’ensemble des techniques rapportées apparait réalisable et accessibleà de nombreux centres.

∗ Auteur correspondant.Adresse e-mail : francois.cornelis@chu-bordeaux.fr (F. Cornelis).

http://dx.doi.org/10.1016/j.purol.2014.02.0081166-7087/© 2014 Elsevier Masson SAS. Tous droits réservés.

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Conclusion. — L’imagerie d’évaluation en onco-urologie peut apporter une grande quantitéd’informations. L’intégration aux protocoles de recherche est maintenant indispensable pourpérenniser cette activité.© 2014 Elsevier Masson SAS. Tous droits réservés.

KEYWORDSCancer;Imaging;MRI;CT-scan;Treatment;Kidney;Prostate;Functional imaging

SummaryIntroduction. — Imaging currently performed in uro-oncology could provide useful information.The use of all this information could help to better understand tumor growth and responseto treatment. Therefore, it seems interesting to review the knowledge, to describe the maintechniques currently available in many centers or in process and to clarify their results.Materials and methods. — A systematic literature review was conducted in the PubMed data-base to identify all imaging techniques performed for therapeutic evaluation in uro-oncology.The keywords used were: cancer, kidney, bladder, prostate, urology biomarkers, imaging, ultra-sound, CT-scan, MRI, PET-CT, RECIST, BOLD, ASL, gold DWI Diffusion, contrast, F-miso. The firstpublications identified were analyzed to search unidentified studies by the selected keywords.Results. — From simple to more complex morphology data from functional imaging (PET, MRI),data obtained from imaging helps to better understand tumor growth and response to treat-ment. Although optimizations are coming, all the techniques reported are available in manycenters or going to be.Conclusion. — The imaging evaluation in onco-urology can bring a large amount of information.Integrating to research protocols is now essential to sustain this activity.© 2014 Elsevier Masson SAS. All rights reserved.

Introduction

Ces dernières années, l’imagerie appliquée en cancérologies’est considérablement développée en raison des évolutionsdes techniques et des pratiques. Elle est maintenant tota-lement intégrée au parcours de soin des patients et reposeprincipalement sur l’étude de critères morphologiques.

En parallèle du développement de l’imagerie morpho-logique, la recherche de biomarqueurs d’une maladiea révolutionné l’approche de la pathologie cancéreuse.Lorsqu’ils sont identifiés, ces marqueurs influencent la priseen charge et le devenir d’un malade. Ils peuvent être denombreuses origines : biologiques, génomiques, morpholo-giques ou fonctionnels par exemple. Ce concept peut êtretransposé en imagerie dans un certain nombre de cas sousle terme d’imagerie fonctionnelle et moléculaire. Mais cetteimagerie s’étend de la recherche fondamentale aux applica-tions cliniques et seules quelques techniques sont arrivéesà maturité permettant d’envisager son application en pra-tique clinique quasi-quotidienne.

Parmi ces biomarqueurs identifiés, la variance génétiquespécifique des différents cancers focalise actuellementl’attention et est au cœur de la « pharmaco-génomique ».Récemment ce type d’approche a été proposé en imageriesous le terme de « radio-génomique » pour les tumeurs durein [1]. Les résultats sont très préliminaires et demandentà être validés mais ils semblent prometteurs. Les mar-queurs biologiques favorisant l’angiogenèse, tels que leVEGF, semblent être une cible plus spécifique de l’évaluationpar l’imagerie. Ces protéines sont exprimés par les cellulestumorales en réponse à l’hypoxie, au stress nutritionnel, ouune acidose [2]. L’avantage de cibler cette voie est qu’ila été rapporté que ce biomarqueur est très répandu dans

les lignées cancéreuses car pour grossir au-delà de 2 mm3,une tumeur doit acquérir un phénotype angiogénique [3].Néanmoins, cette évaluation par l’imagerie ne peut-êtrequ’indirecte et requière des moyens et des techniques spé-cifiques.

Dans ce cadre de l’évaluation thérapeutique, il semblaitdonc intéressant de faire l’état de lieux des connaissances,de décrire les principales techniques disponibles actuelle-ment dans de nombreux centres ou en passe de l’être et depréciser leurs résultats.

Matériel et méthode

Une revue systématique de la littérature a été effectuéeconcernant les techniques d’imagerie d’évaluation enuro-oncologie dans la base de données PubMed (Fig. 1).Les mots-clés utilisés étaient : « cancer », « kidney »,« bladder », « prostate », « urology », « biomarkers »,« imaging », « ultrasound », « CT-scan », « MRI » (Magne-tic Resonance Imaging), « PET-CT » (Positron EmissionTomography—Computed Tomography), « RECIST » (Res-ponse Evaluation Criteria In Solid Tumors), « functional »,« perfusion », « BOLD » (Blood-Oxygen Level-Dependent),« ASL » (Arterial Spin Labelling), « Diffusion or DWI »(Diffusion Weighted Imaging), « contrast », « F-miso » ([18F]-fluoromisonidazole). Les premières publications mises enévidence ont été analysées pour y rechercher des étudesnon identifiées par les mots-clés sélectionnés.

Nous avons retenu les études prospectives ou rétrospec-tives décrivant et/ou évaluant une technique d’imagerieidentifiant des biomarqueurs en imagerie et impliquant uneévaluation fonctionnelle pré- ou post-thérapeutique des

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Figure 1. Flowchart de l’étude bibliographique à partir de la base de données PubMed.

tumeurs urogénitales, ainsi que les revues de la littératuresur le sujet. Deux cent six abstracts ont été étudiés. Parmiceux-ci, 34 articles ont été retenus. La recherche a été limi-tée aux publications de langue francaise et anglaise.

Résultats

Concepts et bases biologiques

Les techniques permettant une évaluation morphologiquesont largement connues et reposent principalement surl’échographie, la tomodensitométrie, et l’IRM. Plusieurstechniques d’imagerie sont capables en outre d’apprécierla vascularisation et le contenu en oxygène des tissus etseront détaillées dans les prochains paragraphes. Néanmoinsà ce stade, il est nécessaire de préciser certaines modalitéspermettant d’obtenir ces informations.

L’étude de la perfusion des tissus est la méthode laplus développée actuellement. Elle fait appel principale-ment à l’utilisation de produits de contraste ou d’équivalentbiologique. Une fois les images obtenues, l’étude de la vas-cularisation des tumeurs est donc basée sur l’étude de ladistribution d’un agent observable (produit de contraste parexemple). Cependant elle requiert une évaluation quantita-tive.

L’application de deux approches mathématiques permetd’en dériver ces paramètres quantitatifs [4]. La premièreconsiste en une simple description de la cinétique de rehaus-sement en fonction du temps (Fig. 2). Néanmoins cela nerepose sur aucune hypothèse physiologique et est définicomme « libre » : on observe simplement l’évolution del’aspect des tissus après injection de produit de contraste.Les tissus très vascularisés se rehausseront plus vite que

ceux faiblement perfusés alors que les structures liqui-diennes ou kystiques ne se rehausseront peu ou pas. Cetteapproche permet d’extraire toutefois des indicateurs semi-quantitatifs (pente de rehaussement, temps au pic [TTP],aire sous la courbe [AUC]. . .) qui présentent des varia-tions considérables selon la méthode d’acquisition employéepour chaque examen individuel et selon l’imageur uti-lisé. Une variation des paramètres peut être un indicateurd’une évolution tumorale ou d’une réponse thérapeutiquemais apparaît difficilement reproductible d’un patient à un

Figure 2. Paramètres obtenus après analyse semi-quantitative dela courbe de variation du signal observé au cours du temps. TTP :temps au pic ; IR : intensité maximale de rehaussement ; FWHM :largeur à mi-hauteur ; AUC : aire sous la courbe ; Pente : pente derehaussement.

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autre. Cette méthode a le mérite d’être simple et réali-sable en l’état par le plus grand nombre en échographie,tomodensitométrie ou IRM. La deuxième méthode, pluscomplexe, utilise les modèles mathématiques pharmacoci-nétiques qui tentent d’intégrer plus ou moins les contraintesspécifiques à la perfusion du tissu et/ou du milieu étudié.Des valeurs quantitatives peuvent en être extraites, per-mettant une comparaison intra-individuelle plus fiable maisaussi d’envisager une comparaison inter-individuelle, utilepour préciser au mieux la réponse thérapeutique dans unecohorte. Ces modèles reposent généralement sur la déter-mination d’une fonction d’entrée obtenue par la courbede rehaussement après injection d’un produit de contrastedes vaisseaux afférents (aorte, artère afférente). Cela per-met de prendre en compte les paramètres hémodynamiquesà un moment donné et d’ajuster au mieux le modèle.Mais ceci peut parfois être délicat et des alternatives ontété proposées (autre tissu comme référence, moyennage).Par la suite l’application d’algorithmes de déconvolutionet/ou de modèles compartimentaux nécessite de définirle nombre de compartiments adaptés aux tissus explorés.Ainsi il est possible d’apprécier quantitativement des para-mètres proches de la réalité, inaccessibles sans ces modèlescar invisibles. De nombreux modèles existent dont le plusconnu est le modèle dit de Tofts [5,6] (Fig. 3). Mais, de

Figure 3. Modèle de Tofts : modèle d’echange d’un produit decontraste entre le milieu plasmatique et le milieu interstiel.

nouveau, des limites peuvent être observées car cela néces-site plusieurs hypothèses : l’agent de contraste utilisé doitêtre mélangé en concentration uniforme à travers toutle compartiment, le flux sanguin doit être linéaire entreles compartiments (échange passif uniquement) et enfin

Figure 4. IRM rénale réalisée dans le cadre d’un bilan d’évaluation thérapeutique d’une lésion rénale droite : coupes coronales enpondération T1 avant (A) et après (B) injection centrées sur l’aorte, coupe coronale T1 avant (C) et après injection au niveau de la lésionexplorée (temps artériel [D], médullaire [E] et tardif [F]), courbe de rehaussement au niveau de l’aorte correspondant à la fonction d’entréeartérielle (G).

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Figure 5. IRM rénale réalisée dans le cadre d’un bilan de tumeur rénale droite (flèches) : axial T2 (A), axial T1 après injection de produitde contraste (B, C, D), séquences de diffusion (b : 100-400-800) (E, F, G) et ADC (coefficient apparent de diffusion) calculé montrant unerestriction de diffusion sur la cartographie parametrique codée en échelle de gris (lésion plus noire que le parenchyme environnant) (H).

les paramètres décrivant les compartiments doivent êtreinvariants durant l’acquisition des données. En fonction dela modélisation, on extrait alors les paramètres d’intérêt quirestent dépendant du modèle choisi.

Le choix du modèle dépendant de l’organe étudié et del’expérience des utilisateurs. En absence d’uniformisation,les paramètres obtenus dépendent encore du modèleutilisé, rendant difficile la généralisation des données.Toutefois, l’intégration de l’imagerie dans les essais thé-rapeutiques devrait aider à valider ce concept à grandeéchelle notamment en raison de l’utilisation des traite-ments anti-angiogéniques (bévacizumab, inhibiteurs kinasescomme le sunitinib) dont l’impact peut être évalué par cestechniques.

Échographie et échographie de contraste

L’échographie est une technique non invasive, peu chèreet facile d’utilisation, basée sur les propriétés physiquesdes ultrasons (US). Elle permet d’obtenir des données mor-phologiques des organes et lésions étudiés mais elle estsouvent limitée en oncologie à un bilan de surveillance oud’extension de première intention. Néanmoins, en complé-ment, l’échographie de contraste peut permettre d’obtenirdes données additionnelles par l’observation des struc-tures vasculaires. Elle nécessite l’injection de microbulles(phospholipides contenant un gaz inerte) comme agents decontraste [7] et qui sont ensuite observées par l’opérateurselon les techniques conventionnelles appliquées en

échographie. Son utilisation requiert des sondes et deslogiciels adaptés, toutefois actuellement largement dispo-nibles au plus grand nombre. Les limites sont identiquesà celle de l’échographie conventionnelle. Ces microbullesne diffusent pas en dehors des structures vasculaires etseules des variations d’intensité du signal rétrodiffusé parles microbulles sont observées au cours du temps en rai-son des fluctuations de leur concentration locale. Tout celapeut être ensuite modélisé afin d’obtenir des paramètrescaractérisant la microcirculation selon des algorithmes pro-posés par les constructeurs. Une analyse dynamique dubolus permet d’obtenir des informations fonctionnelles surle tissu exploré soit par l’analyse des courbes d’intensité derehaussement pendant la phase de remplissage (wash-in)et la phase d’élimination (wash-out) du bolus permettantl’estimation de la microcirculation, soit par la mesure dutemps de transit d’organe, soit par l’analyse de la cinétiquede reperfusion des tissus (après destruction et reconstitu-tion des microbulles dans le volume exploré). Il est toutefoisnécessaire généralement de faire la distinction précise entreles signaux de rétrodiffusion de microbulles et du tissu natifalentour permettant ainsi la quantification linéaire du signalpar l’étude de la valeur du signal dans une région d’intérêt(ROI) définie sur les données brutes (raw data) de l’imageavant ou après le traitement du signal vidéo. Ainsi, corréléau temps d’injection, il est possible d’obtenir des para-mètres sur le pic d’intensité en rapport avec la circulationsanguine, la vitesse et l’aire sous la courbe en rapport avecle volume sanguin [8].

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Figure 6. IRM prostatique réalisée dans le cadre d’un bilan de tumeur de l’apex droit en zone périphérique (flèches) : axial T2 (A), ADCcalculé montrant une lésion en restriction par rapport au reste du tissu prostatique (B), valeur de ROI au niveau de la lésion (C) et fusiond’images T2 et diffusion (D) permettant d’obtenir une image « PET-like ».

Tomodensitométrie

La tomodensitométrie est largement usitée dans le cadredu bilan d’extension d’une pathologie et dans une moindremesure pour la caractérisation morphologique des tumeurs.La technique est basée sur l’étude de l’atténuation desrayons X par les tissus au cours du temps. La localisation, lataille, la forme (circonscrite ou invasive) ou la densité (nor-malisée grâce aux unités Hounsfield) des tumeurs sont ainsiétudiées. Mais les informations ainsi obtenues peuvent êtrepartielles. Il est également possible d’affiner ces observa-tions par l’obtention de données fonctionnelles. En effet, letransit d’un agent de contraste iodé à partir du milieu intra-vasculaire vers l’espace extracellulaire peut être appréciépar la répétition des acquisitions et l’application de modèlesd’échange, de type bi-compartimental en général. Ainsi lerehaussement puis le lavage d’une lésion sont reliés auxcaractéristiques propres de la tumeur, mais également dusystème vasculaire, et reflètent le flux sanguin au sein de latumeur, la perméabilité vasculaire et le volume de l’espaceextracellulaire [5,6]. Pour cela, l’aire sous la courbe (AUC)de l’atténuation observée après le passage du produit decontraste est observée au niveau de l’artère afférente

(fonction d’entrée) ou du compartiment vasculaire puis auniveau du tissu cible (modèle de Tofts). Des paramètressont ensuite calculés grâce au produit de convolution, telle Ktrans qui reflète l’échange entre les milieux vascu-laire et interstitiel. Les avantages de cette méthode sontqu’il existe une linéarité entre la concentration tissulaireen contraste et l’intensité de l’image, permettant ainsi unequantification plus simple des paramètres de perfusion. Deplus, ces protocoles peuvent être réalisés par le plus grandnombre de centres en raison de la disponibilité des appa-reils. Toutefois, l’exposition aux rayons X observée lors deces examens est importante et limite la répétabilité intra-individuelle, et donc de ce fait cette technique dans lecadre de l’évaluation thérapeutique. De plus, les agents decontraste iodés peuvent être mal tolérés et/ou source decomplications (insuffisance rénale, hyperthyroïdie induite)[9].

Imagerie par résonance magnétique (IRM)

L’imagerie pas résonance magnétique (IRM) est de plus enplus proposée dans la caractérisation morphologique destumeurs au moment du diagnostic et afin de déterminer

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avec plus de précision l’extension locale [10]. L’IRM reposesur le principe de la résonance magnétique nucléaire (RMN)qui utilise les propriétés quantiques des noyaux atomiques(et notamment des protons). Cependant, la faible disponi-bilité de la technique en raison de son coût d’installationet donc de sa faible implantation sur le territoire, limiteson accès. L’IRM permet d’évaluer les caractéristiquestissulaires des tissus. En complément, une acquisition dyna-mique après injection d’un agent de contraste de petitpoids moléculaire à base de gadolinium peut être réali-sée (Fig. 4 et 5) permettant l’exploration de la perfusiondes tissus. Toutefois, contrairement à la tomodensitométrie,les paramètres de rehaussement sont étudiés sans êtrelimités par l’exposition aux rayonnements. La résolutiontemporelle est généralement plus élevée, de sorte que leflux sanguin au sein de la tumeur, la perméabilité vas-culaire et le volume de l’espace extracellulaire peuventêtre évalués plus précisément [6]. Toutefois, il n’existepas de relation linéaire entre la variation de l’intensité dusignal en IRM et la concentration de l’agent de contrastenotamment lors de concentrations élevées [11], ce quirend plus difficile l’application des modèles mathéma-tiques.

Le développement de la technique de marquage des spinsartériels (ASL) peut être une alternative pour l’étude de laperfusion notamment pour les patients ne tolérant pas lesproduits de contraste (insuffisance rénale) [12]. Des protonssitués dans les artères sont « marqués » dans celles-ci (engénéral l’aorte) et se substituent au produit de contraste.Toutefois, en raison d’un faible rapport signal/bruit, le déve-loppement de ces séquences est délicat pour les organesmobiles.

Les séquences pondérées en diffusion permettentd’apprécier la microarchitecture des tissus par l’étudedes mouvements des protons liés aux molécules d’eauqu’ils contiennent au cours d’un temps défini [13]. Lesséquences de diffusion reflètent donc la composition dutissu exploré [14]. Le coefficient apparent de diffusion (ADC)est une valeur quantitative correspondant à la dérivée del’atténuation du signal des images obtenues au cours dutemps [15] (Fig. 6). L’ADC est souvent représenté par unecartographie avec un codage des valeurs en échelle de griset ne doit pas être confondu avec l’image de diffusion quireprésente le signal obtenu (Fig. 4 et 7). Coefficient de dif-fusion est dépendant initialement de la perfusion capillaire(les protons se déplacent par le flux sanguin) puis de la dif-fusion « vraie » des protons, qui est spécifique d’un tissuou d’une structure donnés. La diffusion influencée par lamicro-perfusion correspond au modèle dit des « intravoxelincoherent motion » (IVIM) qui dépend de la vitesse du sangcirculant et de l’architecture vasculaire. Une analyse sépa-rée de ces paramètres permettrait la caractérisation encoreplus fine des tissus, ce qui reste à valider cliniquement.

Les séquences BOLD (Blood-Oxygen Level-Dependent)utilisent l’effet paramagnétique de la désoxyhémoglobineet permettent de déterminer en conséquence la teneur enoxygène dans le sang et les tissus [16]. Très simplement, lemilieu extravasculaire sain possède in vivo une faible suscep-tibilité magnétique, tout comme le sang oxygéné (présenceoxyhémoglobine). C’est en revanche l’inverse pour le sangnon oxygéné ou les tissus hypoxiques qui possèdent une fortesusceptibilité magnétique (Fig. 8) en raison notamment de la

Figure 7. Principe de calcul des paramètres de l’imagerie de dif-fusion. Le receuil du signal obtenu par l’imagerie de diffusion (danscet exemple S1 ou S2) en fonction du paramètre de diffusion b (b1 oub2) montre une évolution différente des courbe A et B. La pente deces courbes (dérivée) permet de définir le coefficient apparent dediffusion (ADC) qui diffère entre ces 2 courbes (ADC A > ADC B). Lacourbe B correspond à une lésion en restriction de diffusion parrapport à la courbe A.

présence de désoxyhémoglobine. Les différences de suscep-tibilité magnétique entre différents milieux vont induire desvariations locales de champ magnétique qui vont perturberle temps de relaxation T2* des noyaux d’hydrogène. Le signalBOLD correspond à la capture de ces différences de signalT2* en IRM à l’aide de séquences spécifiques (séquencesechoplanar T2*). On obtient ainsi pour chaque voxel unevaleur du signal BOLD, l’ensemble des voxels correspondantspermet d’obtenir des cartes paramétriques permettant desmesures au sein de régions d’intérêt (Fig. 8), souvent rap-portée aux valeurs de R2* (R2* = 1/T2*). Toutefois, le R2*n’est pas seulement dépendant de la concentration dedésoxyhémoglobine, mais aussi d’autres facteurs tels quele flux sanguin et l’hématocrite ou la microarchitecture[16] rendant l’interprétation délicate. Des mesures semi-quantitatives semblent plus fiables (Fig. 9).

TEP et tomodensitométrie

La tomographie par émission de positons, éventuellementassociée à la tomodensitométrie, est une technique per-mettant de détecter un rayonnement en coïncidence depositons émis par des noyaux radioactifs !+ liés à unemolécule définie et administrés par voie intraveineuse.Elle permet de localiser les zones de concentrations anor-malement élevées de radiotraceurs. Il existe maintenantde nombreux traceurs plus ou moins spécifiques, adaptésaux pathologies et aux organes explorés : le [18F]-fluoro-desoxyglucose (FDG) qui explore le métabolisme du glucose ;[18F]-fluoromisonidazole (F-miso) qui explore l’hypoxie ;[18F]-fluoro-3’-deoxy-3’-L : -fluorothymidine (FLT) qui est unmarqueur de la prolifération ou la [18F]-choline qui marquele métabolisme lipidique, notamment utilisé en pathologieprostatique. L’absorption du traceur est par la suite quanti-fiée par le SUV (standardized uptake value) qui correspond

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Figure 8. IRM rénale réalisée dans le cadre d’un bilan diagnostique d’une lésion rénale polaire supérieure (carcinome à cellules claires)(flèches) : coupe coronale en pondération T2 (A), coupe axiale T2 (B), images BOLD montrant une atténuation plus rapide du signal dans lalésion comparée au reste du rein traduisant un aspect hypoxique (C, D, E, F).

Figure 9. IRM prostatique réalisée dans le cadre d’un bilan de tumeur : axial T2 (A), courbe de decroissance du signal en T2* (B) puiscartographie BOLD (C) avec mesure de ROI au niveau de différents tissus (muscle [D], prostate en zone périphérique [E] et en zoneperiphérique [F]).

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Figure 10. Principe de la tomodensitométrie au [18F]-fluorodesoxyglucose (FDG). Le FDG est internalisé comme le glucose parl’intermédiaire de transporteurs spécifiques (GLUT-1) et est par la suite piégé par action de l’hexokinase. L’accumulation et l’émissiondes rayons y de 0,511 MeV issus de la désintégration !+ permettront l’évaluation du métabolisme du glucose dans les cellules (expressionde GLUT-1, consommation de glucose).

Figure 11. Tomodensitométrie au [18F]-fluorodesoxyglucose (FDG) d’une lésion rénale. La tomodensitométrie avec injection en coupecoronale montre une lésion rénale polaire supérieure gauche (A). La tomodensitométrie sans injection (B), la tomographie au FDG (C) sontfusionnés (D) montrant une lésion hypermétabolique correspondant à un carcinome à cellules claires.

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à une mesure de la moyenne des concentrations en traceursdes tissus observés par rapport à la concentration moyennede l’ensemble du corps.

Pour l’imagerie d’évaluation, le 18-fluoro-desoxyglucose(FDG) est le radiotraceur le plus répandu et permet la carto-graphie du métabolisme du glucose et donc d’identifier deslésions tumorales dont le métabolisme du glucose est aug-menté (Fig. 10). En effet, le traceur est piégé par les cellulesqui consomment du glucose par action de l’hexokinase suiteà l’internalisation par le transporteur GLUT-1 (Fig. 11).

Le [18F]-fluoromisonidazole (F-miso) est également inté-ressant dans l’évaluation thérapeutique car il permet uneévaluation non invasive de l’hypoxie. Comme toutes lesmolécules du groupe des nitro-imidazoles, le [18F]-F-misoest piégé sélectivement dans les cellules viables hypoxiquespar la nitroréductase lorsque la pression partielle d’O2 estinférieure à 10 mmHg. Ce traceur diffuse en effet librementet pénètre dans toutes les cellules par un mécanisme dediffusion dépendant de sa lipophilicité. Dans les cellulesnormoxiques, la nitroréduction n’est pas O2 dépendante etest rapidement réversible. Dans les cellules hypoxiques,

cette réduction aboutit à la formation d’une hydroxylaminedont les produits de dégradation se fixent sur les protéinesintracellulaires. Le degré de fixation de ce traceur, quis’accumule dans les cellules hypoxiques dans lesquelles ilreste séquestré, est inversement corrélé à l’oxygénationtumorale. Ainsi, plus une cellule est hypoxique plus la capta-tion du traceur sera importante. Toutefois, l’accumulationspécifique est lente et faible entraînant un faible rapportsignal sur bruit [17] et donc des difficultés d’exploitationdes données.

Discussion

Critères morphologiques

À partir de ces techniques d’imagerie et en dehors des carac-téristiques morphologiques utiles pour le diagnostic deslésions tumorales, il a été proposé des critères d’évaluationmorphologique des lésions après traitement (Tableau 1).Outre leur multiplication, les critères de réponse fon-

Tableau 1 Comparaison des critères OMS, RECIST 1.1, Choi, mRECIST et PERCIST.

Réponse OMS RECIST 1.1 Choi (GIST) RECISTmodifié (CHC)

PERCIST(FDG-PET)

Réponsecomplète (RC)

Pas de lésionsdétectées à4 semaines

Disparition detoutes lésionsouganglions < 10 mmde petit axe

Disparition detoutes leslésions cibles

Disparition durehaussementau tempsartériel deslésions cibles

Disparition deslésionstumoralesactives

Réponsepartielle (RP)

≥ 50 % dediminution dela somme duproduit desdiamètres(SPD) à4 semaines

> 30 % dediminution dela somme desplus longsdiamètres(SLD) deslésions cibles

≥ 10 % dediminution entaille deslésions ou≥ 15 % dediminution del’atténuationen scanner ;pas denouvelleslésions

> 30 % dediminution dela somme desplus longsdiamètres(SLD) deslésions ciblesviables (serehaussant autempsartériel)

> 30 %(0,8-unité) dediminution duSUL (SUVnormalisé surla massemaigre) entrela lésion laplus intenseavanttraitement etcelle aprèstraitement

Progression dela maladie(PM)

≥ 25 %d’augmentationdu SPD d’aumoins1 lésion ;nouvelleslésions

> 20 %d’augmentationdu SLD deslésions ciblesavec uneaugmentationabsolue≥ 5 mm ;nouvelleslésions

≥ 10 %d’augmentationdu SLD ; nesatisfaisantpas lescritères de RPau niveau del’atténuation ;nouvelles ouaugmentationde lésionsintra-tumorales

> 20 %d’augmentationdu SLD deslésions viables(se rehaussantau tempsartériel)

> 30 %(0,8-unité)d’augmentationdu pic de SULou nouvelleslésionsconfirmées

Maladie stable Tous les autrescas

Tous les autrescas

Tous les autrescas

Tous les autrescas

Tous les autrescas

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Figure 12. Tomodensitométrie avec injection de produit de contraste réalisée dans le cadre du suivi de métastases de tumeur stromalede l’intestin grêle (GIST) : le volume ne varie pas dans le temps alors que la densité de la métastase diminue.

dés sur l’imagerie des tumeurs ont rapporté leurs limitesnotamment depuis l’introduction des thérapies ciblées encomplément de la chimio- et/ou radiothérapie. Les critèresOMS, publiés en 1981, ont été les premiers utilisés et sontbasés sur des mesures bidimensionnelles (plus grands dia-mètres perpendiculaires). Par la suite, des critères RECIST(Response Criteria for Solid Tumor) ont été créés en 2000 etrévisés en 2009 (RECIST 1.1) et modifiés (mRECIST). Ainsi,des mesures unidimensionnelles (plus grand diamètre) sontréalisées et ont permis de palier aux carences des critèresOMS. Néanmoins, l’introduction des traitements ciblés,jouant plus sur la dévascularisation et la nécrose, a démon-tré également les limites de ce système et a nécessité larévision de ces critères pour prendre en compte le rehausse-ment des lésions et non plus uniquement la taille. Ainsi sontapparus les critères Choi pour les tumeurs stromales (GIST)(Fig. 12), les critères RECIST modifiés pour le carcinomehépatocellulaire et les critères de réponse immunitaire lorsde mélanomes.

Plus récemment, les critères Cheson et PERCIST (PETResponse Criteria for Solid Tumors) ont été proposés pourévaluer la réponse des tumeurs solides par TEP permettantde fournir des informations supplémentaires sur le métabo-lisme à ces critères morphologiques.

Imagerie fonctionnelle

En complément de ces critères morphologiques, l’imageriefonctionnelle semble tout particulièrement intéressantepour l’évaluation précoce des thérapies ciblées. En effet,cette imagerie peut être au mieux focalisée sur la résultantedu mécanisme d’action du produit comme c’est le cas pourles techniques d’évaluation de la perfusion tumorale lors destraitements anti-angiogéniques. L’enjeu est d’importanceafin de mieux appréhender une pathologie, de limiter ladurée des traitements non ou peu actifs, et en conséquenceles effets secondaires ainsi que les coûts.

Les changements du flux sanguin et de la perméabi-lité microvasculaire (Ktrans) au sein d’un tissu peuventprécéder la diminution en taille d’une lésion [18] aprèstraitement ciblé par anti-angiogénique (anti-VEGF) ou paranti-tyrosine-kinase (TKI) [19]. L’identification de caracté-

ristiques fonctionnelles des patients grâce à ces différentesmodalités d’imagerie semble donc une source de déve-loppement prometteuse. Cela permettrait l’identificationde patients appropriés pour une thérapie ciblée en effec-tuant une évaluation de base unique et prédire par lasuite les résultats cliniques à l’échelle de la populationsélectionnée. De plus, Sahani et al. [20] ont rapporté quel’imagerie fonctionnelle était plus sensible pour prédirel’évolution tumorale que les simples critères morpholo-giques.

Toutefois, il existe peu de grandes séries publiées etde corrélations radio-pathologiques. En échographie decontraste, la plupart des études publiées se limitent àdes analyses descriptives des modifications des courbesde rehaussement en raison de limites techniques [7]. Lesmodifications précoces observées (j0 à j3) de plusieurs para-mètres sont associées à la réponse tumorale (notammentl’AUC et l’intensité du pic de rehaussement) et certains ontmême une valeur pronostique en termes de survie sans pro-gression (intensité du pic de rehaussement) ou de survieglobale (l’AUC).

En tomodensitométrie, la densité microvasculaire descarcinomes rénaux est corrélée aux paramètres perfusion-nels (flux sanguin, volume sanguin, Ktrans) (r = 0,600 à 0,829,p < 0,05). De plus, pour des stades et des grades de Fuhrmanélevés, ces paramètres semblent plus faibles, alors qu’ilssont plus élevés dans les CCR avec moins de 50 % de nécrose(p < 0,05) [21]. Ces marqueurs peuvent donc être utiliséspour identifier initialement les populations à risque d’échecthérapeutique mais également pour permettre l’évaluationthérapeutique au cours du temps.

En IRM, il a été observé des résultats similaires. D’unpoint de vue morphologique, il existe une augmentationsignificative de l’intensité du signal T1 des carcinomesrénaux (p < 0,0001) après traitement par le sorafenib, etune diminution significative du rehaussement (p < 0,0001)[22], ce qui peut être apprécié par les critères de Choi. Lescritères morphologiques, et notamment le RECIST, ont dimi-nué de manière significative après traitement (p = 0,005)pour les patients répondeurs. Toutefois, il existait danscette étude une maladie stable selon les critères RECIST surles contrôles précoces limitant de ce fait l’impact de ces

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Figure 13. IRM pelvienne réalisée dans le cadre d’un bilan de tumeur de l’urètre : axial T2 (A), axial T1 après injection de produit decontraste (B), cartographie de rehaussement (C), séquences de diffusion (b : 800) (D) montrant un signal élevé à b élevé de la lésiontraduisant une restriction.

critères en termes de modification thérapeutique. D’unpoint de vue fonctionnel, Desar et al. [23] ont évalué leseffets vasculaires précoces de sunitinib chez des patientsatteints de carcinomes rénaux et ont rapporté la diminu-tion significative du volume sanguin relatif des tumeurs(RBV) et du débit sanguin (RBF) à 3 jours de l’introduction(p = 0,037 et p = 0,018) et à 10 jours (p = 0,006 et p = 0,009).Malgré ces résultats prometteurs, Hahn et al. [24] ont rap-porté que les variations après traitement (sorafenib) del’AUC et du Ktrans n’étaient pas associées à la survie sansprogression (PFS) et que seuls les patients ayant une valeurinitiale élevée de Ktrans avaient une meilleure survie sansprogression (p = 0,027) (Fig. 13). Ainsi dans le cancer durein, la perméabilité de base (Ktrans) pourrait être utiliséedès à présent comme un facteur pronostique [19,20,24,25]alors que son impact dans l’évaluation thérapeutique doitêtre encore quantifié. L’imagerie de diffusion et la mesurede l’ADC permettent également d’apprécier la réponsethérapeutique par des mécanismes différents, exposés pré-cédemment (Fig. 12). Desar et al. [23] ont rapporté uneaugmentation significative de l’ADC à 3 jours (p = 0,015)suivie d’une diminution du niveau de base à 10 jours

(p = 0,001) des tumeurs d’origine rénale. Par ailleurs, toutcomme le Ktrans, il a été rapporté qu’une valeur d’ADCpré-thérapeutique élevée des métastases hépatiques estassociée à un mauvais pronostic [26,27].

Des résultats similaires ont été rapportés pour la tech-nique d’ASL [12] avec de faibles valeurs corrélées à unemoindre sensibilité au traitement. Sur un modèle de souris,il a été observé pour les répondeurs des variations significa-tives du débit sanguin dans les 30 jours après initiation d’untraitement par sorafenib [28].

L’IRM avec effet BOLD apparaît comme étant une tech-nique sensible permettant l’étude de l’hypoxie tumorale,mais manquant peut-être de spécificité (Fig. 14). Plusieursétudes de faisabilité ont déjà proposé cette technique noninvasive pour cartographier l’hypoxie tumorale prostatiquepar exemple, permettant de proposer un nouveau marqueurpronostique [29—32]. Hoskin et al. [30] ont rapporté que lasensibilité du paramètre de relaxation R2* (1/T2*) à affirmerl’hypoxie tumorale était élevée (88 %) avec une spécificitéde 36 %. Cette technique présentait aussi une valeur prédic-tive négative de 70 % lorsqu’elle était combinée avec desinformations sur le volume sanguin (rBV). Dans l’étude de

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Figure 14. IRM rénale réalisée dans le cadre d’un bilan d’évaluation thérapeutique d’une lésion rénale (flèche) : coupe coronale enpondération T2 (A), coupe axiale T1 après injection (B), cartographie BOLD (C) et fusion d’image avec le T1 avec injection (D), résultats del’étude fonctionnelle avec calcul des paramètres perfusionnels et représentation sous la forme de cartographies fusionnées avec l’imagerieanatomique (E).

Chopra et al. [29], une corrélation a été observée entre leR2* et HP5 (hypoxic fraction < 5 mmHg) (r = 0,76, p = 0,02)et une tendance a été notée entre R2 * et PaO2 (r = −0,66,p = 0,07).

Les résultats des explorations réalisées en médecinenucléaire montrent qu’un SUV élevé semble de plus mau-vais pronostic en FDG-PET [33] tout comme une tumeurhypoxique en [18F]-F-miso [34]. La TEP [18F]-F-miso a étéutilisée comme marqueur non invasif de l’hypoxie dans denombreuses tumeurs dont les cancers du rein et a été étu-diée dans la prédiction de la réponse thérapeutique etl’évaluation du pronostic notamment dans les cancers desVADS. Ces analyses quantitatives restent difficiles car letaux de captation est souvent faible et la clairance cellu-laire est lente mais aussi du fait de la dissémination et del’hétérogénéité des zones hypoxiques au sein de la tumeur.

Toutefois, il persiste toujours un manque de standardi-sation des protocoles pour l’ensemble de ces techniques.Par exemple, Heye et al. [35] ont rapporté en tes-tant la reproductibilité des mesures quantitatives etsemi-quantitatives des paramètres pharmacocinétiques deperfusion des fibromes utérins sur différents systèmes desdifférences significatives dans les valeurs moyennes exis-taient après normalisation. L’agrément inter-observateurétait de 48,3—68,8 % pour le Ktrans, 37,2—60,3 % pour kep,27,7—74,1 % pour ve et 25,1—61,2 % pour l’aire sous lacourbe initiale (AUC). Le coefficient de corrélation intra-classe était faible à modérée (Ktrans : 0,33—0,65 ; kep :

0,02—0,81 ; ve : −0,03—0,72 ; AUC : 0,47—0,78). De plus,le post-traitement n’est pas homogène et souvent dépen-dant des équipes, que ce soit au niveau des modèles decalculs utilisés (n’étant pas identiques), du recueil de don-nées (région d’intérêt en 2D vs 3D [36]), de la réalisationd’une normalisation.

Conclusion

En dehors des résultats histologiques, l’imagerie réaliséeactuellement en oncologie semble donc capable d’apporterune quantité importante d’informations même si des opti-misations sont à venir. Du simple aspect morphologiqueaux données plus complexes de l’imagerie fonctionnelle,l’exploitation de ces informations pourrait aider à mieuxappréhender la croissance tumorale et les réponses auxtraitements. L’ensemble des techniques rapportées estmaintenant disponible dans de nombreux centres ou enpasse de l’être grâce à des partenariats. L’intégration auxprotocoles de recherche est maintenant indispensable pourpérenniser cette activité.

Déclaration d’intérêts

Les auteurs déclarent ne pas avoir de conflits d’intérêts enrelation avec cet article.

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Remerciements

Les auteurs remercient pour son aide Bastien Perez, ingé-nieur d’application, General Electric, Paris, France.

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[17] Mees G, Dierckx R, Vangestel C, Van de Wiele C. Molecularimaging of hypoxia with radiolabelled agents. Eur J Nucl MedMol Imaging 2009;36(10):1674—86.

[18] Desar IM, van Herpen CM, van Asten JJ, Fiedler W, MarreaudS, Timmer-Bonte JN, et al. Factors affecting the unexpectedfailure of DCE-MRI to determine the optimal biological dose ofthe vascular targeting agent NGR-hTNF in solid cancer patients.Eur J Radiol 2011;80(3):655—61.

[19] Flaherty KT, Rosen MA, Heitjan DF, Gallagher ML, Schwartz B,Schnall MD, et al. Pilot study of DCE-MRI to predict progression-free survival with sorafenib therapy in renal cell carcinoma.Cancer Biol Ther 2008;7(4):496—501.

[20] Sahani DV, Jiang T, Hayano K, Duda DG, Catalano OA, Ancu-kiewicz M, et al. Magnetic resonance imaging biomarkers inhepatocellular carcinoma: association with response and cir-culating biomarkers after sunitinib therapy. J Hematol Oncol2013;6:51.

[21] Reiner CS, Roessle M, Thiesler T, Eberli D, Klotz E, Frauen-felder T, et al. Computed tomography perfusion imaging ofrenal cell carcinoma: systematic comparison with histopa-thological angiogenic and prognostic markers. Invest Radiol2013;48(4):183—91.

[22] Kang HC, Tan KS, Keefe SM, Heitjan DF, Siegelman ES, FlahertyKT, et al. MRI assessment of early tumor response in metastaticrenal cell carcinoma patients treated with sorafenib. AJR AmJ Roentgenol 2013;200(1):120—6.

[23] Meyer JM, Perlewitz KS, Hayden JB, Doung YC, Hung AY,Vetto JT, et al. Phase I trial of preoperative chemoradia-tion plus sorafenib for high-risk extremity soft tissue sarcomaswith dynamic contrast-enhanced MRI correlates. Clin CancerRes 2013;19(24):6902—11, http://dx.doi.org/10.1158/1078-0432.CCR-13-1594 [Epub 2013 Oct 16. PubMed PMID: 24132922;PubMed Central PMCID: PMC3869565].

[24] Desar IM, ter Voert EG, Hambrock T, van Asten JJ, van SpronsenDJ, Mulders PF, et al. Functional MRI techniques demonstrateearly vascular changes in renal cell cancer patients treatedwith sunitinib: a pilot study. Cancer Imaging 2011;11:259—65.

[25] Hahn OM, Yang C, Medved M, Karczmar G, Kistner E, Karrison T,et al. Dynamic contrast-enhanced magnetic resonance imagingpharmacodynamic biomarker study of sorafenib in metastaticrenal carcinoma. J Clin Oncol 2008;26(28):4572—8.

[26] Cui Y, Zhang X-P, Sun Y-S, Tang L, Shen L. Apparent diffu-sion coefficient: potential imaging biomarker for predictionand early detection of response to chemotherapy in hepaticmetastases. Radiology 2008;248(3):894—900.

[27] Kukuk GM, Mürtz P, Träber F, Meyer C, Ullrich J, Gieseke J, et al.Diffusion-weighted imaging with acquisition of three b-valuesfor response evaluation of neuroendocrine liver metas-tases undergoing selective internal radiotherapy. Eur Radiol2014;24(2):267—76, http://dx.doi.org/10.1007/s00330-013-3008-6 [Epub 2013 Oct 1. PubMed PMID: 24081644].

[28] Schor-Bardach R, Alsop DC, Pedrosa I, Solazzo SA, Wang X,Marquis RP, et al. Does arterial spin-labeling MR imaging-measured tumor perfusion correlate with renal cell cancerresponse to antiangiogenic therapy in a mouse model? Radio-logy 2009;251(3):731—42.

[29] Chopra S, Foltz WD, Milosevic MF, Toi A, Bristow RG, Menard C,et al. Comparing oxygen-sensitive MRI (BOLD R2*) with oxygenelectrode measurements: a pilot study in men with prostatecancer. Int J Radiat Biol 2009;85(9):805—13.

[30] Hoskin PJ, Carnell DM, Taylor NJ, Smith RE, Stirling JJ, DaleyFM, et al. Hypoxia in prostate cancer: correlation of BOLD-MRIwith pimonidazole immunohistochemistry-initial observations.Int J Radiat Oncol Biol Phys 2007;68(4):1065—71.

[31] Diergarten T, Martirosian P, Kottke R, Vogel U, Stenzl A,Claussen CD, et al. Functional characterization of prostate

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Author's personal copy

L’imagerie d’évaluation thérapeutique en pratique clinique d’oncologie urologique 413

cancer by integrated magnetic resonance imaging and oxy-genation changes during carbogen breathing. Invest Radiol2005;40(2):102—9.

[32] Jiang L, Zhao D, Constantinescu A, Mason RP. Comparisonof BOLD contrast and Gd-DTPA dynamic contrast-enhancedimaging in rat prostate tumor. Magn Reson Med 2004;51(5):953—60.

[33] Kayani I, Avril N, Bomanji J, Chowdhury S, Rockall A, SahdevA, et al. Sequential FDG-PET/CT as a biomarker of response toSunitinib in metastatic clear cell renal cancer. Clin Cancer Res2011;17(18):6021—8.

[34] Hugonnet F, Fournier L, Medioni J, Smadja C, Hindie E, HuchetV, et al. Metastatic renal cell carcinoma: relationship betweeninitial metastasis hypoxia, change after 1 month’s sunitinib,

and therapeutic response: an 18F-fluoromisonidazole PET/CTstudy. J Nucl Med 2011;52(7):1048—55.

[35] Heye T, Merkle EM, Reiner CS, Davenport MS, Horvath JJ, Feuer-lein S, et al. Reproducibility of dynamic contrast-enhancedMR imaging. Part II. Comparison of intra- and interobservervariability with manual region of interest placement versussemiautomatic lesion segmentation and histogram analysis.Radiology 2013;266(3):812—21.

[36] Vargas HA, Delaney HG, Delappe EM, Wang Y, Zheng J, Mosko-witz CS, et al. Multiphasic contrast-enhanced MRI: single-sliceversus volumetric quantification of tumor enhancement forthe assessment of renal clear-cell carcinoma fuhrman grade. JMagn Reson Imaging 2013;37(5):1160—7.

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6.2. Développement d’outils de traitement de l'imagerie fonctionnelle

Ce travail en cours consiste à identifier des facteurs prédictifs de rechute par la simple analyse

de l'évolution du rehaussement sur des IRM réalisées avec injection de produit de contraste.

Ce post-traitement est réalisé par Agathe Peretti sur des IRM fonctionnelles de patients

présentant des tumeurs du rein métastatiques obtenues prospectivement.

A partir d'un jeu d'IRM morphologiques en pondération T2 et en T1 après injection de produit

de contraste réalisée chez des patients avant la mise en place d'un traitement anti-

angiogénique, à J15 de l'introduction puis à 1 mois et 3 mois, les informations contenues sont

utilisées. Même s'il existe des modifications morphologiques subjectives (cf figures ci-

dessous), seules l'évaluation de la perfusion impliquant des modèles mono/bi-

compartimentaux permettait jusqu'à présent de définir des paramètres capables d'identifier ou

non une réponse au traitement. Toutefois une limite persiste avec cette approche car l'ajout de

ces modèles opacifient l'analyse: ils correspondent à une boite noire souvent déléguée à des

logiciels extérieurs. Le travail réalisé tente donc de simplifier la démarche en proposant un

indicateur simple mais objectif permettant de déterminer l'impact thérapeutique sur des

images morphologiques sans avoir recours à l'imagerie fonctionnelle "conventionnelle" à base

de modèles compartimentaux. Dans un deuxième temps, une comparaison aux données

cliniques mais aussi aux résultats des IRM fonctionnelles sera réalisée pour vérifier la validité

de cet indicateur.

135

IRM en pondération T2 (coupe coronale) montrant une volumineuse tumeur hétérogène

(flèche) et ceci à 4 temps thérapeutiques différents (avant, J15, M1 et M3 de

l'introduction d'un traitement anti-angiogénique). Des modifications visuelles

subjectives sont observées ne permettant pas de conclure sur l'efficacité du traitement.

Images correspondantes en T1 après injection de produit de contraste montrant une

majoration de l'hétérogénéité intra-tumorale au cours du temps (avant, J15, M1 et M3)

pour la même patiente que ci dessus. De nouveau l'analyse visuelle ne reste que

subjective justifiant le développement d'indicateur quantitatif objectif pour permettre

de déterminer s'il existe une réponse thérapeutique objective.

136

137

2 EVALUATING THE HETEROGENEITY OF A TUMOR

1 Introduction

Assessment of tumor response is based on morphological and anatomical as-pects in contrast-enhanced CT or MR imaging. For this purpose, the ResponseEvaluation Criteria in Solid Tumors (RECIST) remains the standard [2] and iswidely used in clinical trials. However such criteria may fail as new therapeuticagents rather induce disease stabilization than tumor size regression [5].

In case of targeted therapies, it has been proved that changes in tumorsize may lag behind physiological measures of response in functional images[6, 4, 1, 7]. To overcome this, development of predictive and pharmacodynamicbio-markers to assess tumor response has been therefore proposed [3]. How-ever, technique evaluating the changes in tumor angiogenesis, vascularizationand vascular permeability, tumor cell proliferation and density, tumor hypoxiaand glucose metabolism remains di�cult to perform in a routine practice be-cause highly complex and time-consuming [5]. Methods able to quantify thetherapeutic response using only the standard CT or MR sequences would behelpful by overcoming such limitations.

The purpose of our study was to evaluate the Dynamic Contrast Enhanced(DCE) MR imaging changes using a new semi-quantitative algorithm.

2 Evaluating the heterogeneity of a tumor

Two ways of evaluating the heterogeneity of a tumor are studied in this paper.The first one deals with the gradient, whereas the second one involves gaussianmixture models.

2.1 Gradient

First, a way of measuring heterogeneity is to compute the gradient on the tumor.Indeed, the higher is the gradient value, the more heterogeneous the tumor isgoing to be.

The gradient magnitude is computed on each pixel of the ROI 1, then oneaverages the values on the tumor.

Let I(i, j) be the intensity of the pixel of coordinates (i, j), and G(i, j) itsgradient value.

One has (1), (2) and (3).

G(i, j) =q

G2i

+G2j

(1)

Gi

= I(i+ 1, j)� I(i, j) (2)

Gj

= I(i, j + 1)� I(i, j) (3)

Then, when a patient has had several medical exams, one can compare thegradient average values over time. The higher the average gradient gets, themore cancer cells are flourishing.

1Region Of Interest

2

138

2 EVALUATING THE HETEROGENEITY OF A TUMOR

2.2 Gaussians

Another way of measuring the heterogeneity is to draw a histogram, representingthe intensities of the pixels in the tumor and their frequency. Then, one canapproximate the histogram with gaussians, in order to have a better idea of thepixels’ spreading.

A gaussian g is a function defined as in (4). Its 3 parameters are its averageµ, its standard deviation � and its weight ⇡.

g(x) = ⇡ ⇥ 1

�p2⇡

⇥ exp�(x� µ)2

2�2(4)

In order to restrain the number of parameters, one uses up to 4 gaussians(that is to say up to 12 parameters).

To avoid having a gaussian similar to a delta function in case a single pixelhas a di↵erent intensity, one applies a mean filter on the image.

As a result, one assigns to each pixel the mean of its neighbors intensities.Another filter that is more accurate is the gaussian filter. It takes 2 argu-

ments: a standard deviation � and a radius factor. The filter assigns to eachpixel an intensity that is the weighted average of the 8 or 24 pixels around(depending on the value of the radius factor). The weights are defined in (5).

1

2⇡�2⇥ exp

�(x2 + y2)

2�2(5)

In this study, the standard deviation is chosen equal to 0.5. It needs to beinferior to 1 to avoid a blurring phenomenon. The radius factor is equal to 1.(8 pixels are taken into account to perform the computation.)

The lower the number of gaussians used is, the more homogeneous is thetumor. Similarly to the gradient, one can compare the evolution of the numberof gaussians over time.

One can plot a composite of the gaussians (their sum), to see if there is anycorrelation between the ROI’s heterogeneity and the clinical data.

2.3 Rebuilding the tumor with the gaussians

2.3.1 First method

In order to visualize whether if the tumor is heterogeneous or not, one canrebuild the tumor by assigning each pixel an intensity corresponding to thegaussian it belongs to. One assigns to each gaussian a shade of grey. The lowerthe gaussian’s mean is, the darker is the shade. First of all, each pixel of theROI is associated with the gaussian whose frequency is the most important.Then, it is associated with the shade of grey corresponding to the gaussian.

For instance, on figure, a pixel with an intensity of 135 will be assigned tothe blue gaussian, and consequently to the second brighter shade of grey.

3

139

2 EVALUATING THE HETEROGENEITY OF A TUMOR

Figure 1: Pixel 135 assigned to the blue gaussian.

On figure 2, one can see the initial tumor on the left, and the tumor rebuiltwith 2 gaussians on the right.

Figure 2: Original Image (left) and rebuilt image (right).

As this method is not particularly subtle, one may want to introduce shadingin the rebuilt.

2.3.2 Second method

This time, instead of assigning a gaussian to the pixels, one assigns them apercentage of belonging. Let G be a gaussian, P its weight and O the occurrenceof a pixel on G. Considering a decomposition of the histogram into n gaussians,one gets (6)

Pi

=O

i

nPi=0

Oi

(6)

4

140

2 EVALUATING THE HETEROGENEITY OF A TUMOR

For instance, on figure 3, the blue gaussian is associated with the intensityI0 and the green one to the intensity I1. In this case, a pixel of intensity 1100will have the intensity computed in (7):

P =O0 ⇥ I0 +O1 ⇥O1

O0 +O1(7)

Figure 3: Example of the pixel 1100.

On figure , one can see the initial tumor on the left, and the tumor rebuiltusing the second method on the right. In that case, the tumor seems quiteheterogeneous.

Figure 4: Original Image (left) and rebuilt image (right).

5

141

3 RESULTS

3 Results

3.1 Technique

The first thing to do is to segment the tumor. It consists in drawing a ROIaround the tumor. In case a contrast agent is injected, one segments the ROIduring portal phase.

In order to approximate the intensities with gaussians, one uses the ScikitLearn library, and more particularly the mixture.GMM 2 function. The gaussianparameters characterising the function are given by the GMM function.

However, the function takes the number of gaussians to be used as an argu-ment. One tries the approximation with 1, 2, 3 and 4 gaussians, to see whichnumber fits the histogram best.

Obviously, using 4 gaussians is going to be more precise than using 1. Butone is willing to avoid using gaussians with weights that are inferiors to 0.1;or whose means are closed to each other. Instead of picking the number ofgaussians that minimize the L2 error between the gaussians and the histogram,one adds both the AIC 3and BIC 4.

Their expressions are given in (8) and (9).Let k be the number of parameters in the model, n the sample size and L

the maximized value of the likelihood function.

AIC = 2k � 2 ln(L) (8)

BIC = �2 ln(L) + k ln(n) (9)

Those criteria take into account the performance of the model but they alsopenalize models that are made of more parameters. The lower the criteria are,the better is the model. A new error is therefore defined in (10)

error = differencehistogram

+1

1000⇥ (AIC +BIC) (10)

The 11000 factor enables us to have the same order of magnitude between

the L2 error and the 2 criteria. On figure 5, one can see the result of theapproximation of a histogram with 4 gaussians. The composite (defined as thegaussians’ sum) is plotted with red dots.

As one is interested in knowing how the tumor evolves over time, one canalso plot the composite of the chosen number of gaussians for each exam. Oneobtains figure 6.

One has to notice that, on figure 6, tumors from time t2 and t5 have higherintensities. (The lower is the intensity the darker is the tumor).

But, both times t2 and t5 match exams that were made in di↵erent medicalinstitutions. As a result, it is necessary to calibrate the intensities on all examsso that data is comparable.

The scaling factor is chosen so as the spleen has a constant intensity in allexams.

2Gaussian Model Mixture3Aikaike Information Criterion4Bayesian Information Criterion

6

142

3 RESULTS

Figure 5: Approximation with gaussians

Figure 6: Evolution of gaussian composite over time.

7

143

3 RESULTS

3.2 Results on kidney metastasis

3.2.1 First patient

The first tumor studied is a kidney metastasis in the liver. The patient under-went 4 exams. The intensities were scaled so as to have a constant value in thespleen.

The gaussians composite are plotted over time on figure 7.

Figure 7: Evolution of gaussians.

One can notice that concerning exams t0 and t1 the composites becomedarker. On the other hand, they get clearer from t1.

Similarly, one observes one figure 8 that the gradient decreases and rises upagain starting from the second exam.

This is in agreement with clinical results since the patient was getting sickerand his treatment had to be readjusted.

3.2.2 Second patient

The same study is carried on another patient. The intensity was also scaledtaking the aorta as a reference. The gaussians composite are plotted over timeon figure 9. They seem to darken slightly.

The gradient value increases between the first two exams and then decreasesagain, as shown in figure 10.

This case was also in agreement with the clinical results since the patientwas responding correctly to his medication.

8

144

3 RESULTS

Figure 8: Gradient evolution.

Figure 9: Evolution of gaussians.

9

145

5 APPENDIX

Figure 10: Gradient evolution.

4 Conclusion

It seems that the study of the gradient and of the gaussian composite on thetumor may be good indicators to foresee the evolution of the disease.

Indeed, darkening gaussians and decreasing gradient might represent thee�ciency of the treatment.

On the opposite, if the gradient increases and the gaussians are lightening itmight be the sign of the cancer cells spreading.

This should be validated by testing this method on other patients in thesame condition.

5 Appendix

5.0.3 Study lead on other patients

The study lead in 3.2, was carried on two other patients.The first patient has a tumor in the sacrum.As the spleen was not visible on the exam, the intensities were re-scaled so

that the intensity in the muscles remains constant. The gaussian evolution isplotted on figure 11. One can notice that the intensities are getting clearer overtime.

The gradient evolution is plotted on figure 12 It remains stable between thefirst two exams but then it increases significantly.

This is consistent with the fact that the treatment has to be readjusted forthis patient.

The second patient has a kidney tumor. The spleen was not clearly visibleon all his exams so the aorta was used to re-scale the intensities.

10

146

5 APPENDIX

Figure 11: Evolution of gaussians.

Figure 12: Gradient evolution.

11

147

5 APPENDIX

The gaussian evolution is plotted on figure 13. The composite darkens overtime.

The gradient evolution is plotted on figure 14. It is continuously decreasing.

Figure 13: Evolution of gaussians.

This is consistent with the clinical data since the patient was getting better.

5.0.4 E↵ect of the gaussian filter parameters on the gradient

One may want to know what is the e↵ect of the radius factor and the standarddeviation on the gradient.

Therefore, one leads the same study that in 3.2 concerning the gradient.Only this time, one computes the di↵erence between the gradient value at thelast exam and at the first one, as shown in 11.

Difference = gradient(final exam)� gradient(first exam) (11)

This computation is made for various values of the standard deviation andthe radius factor.

Concerning the case shown in section 3.2.2, one obtains figure 15.With a standard deviation equal to 0.5 and a radius factor equal to 1 or 2,

one gets a result in the blue part of the grid. The gradient value is low andnegative, which is consistent with the fact that the gradient is decreasing.

One also has to notice that the e↵ect of the parameters is quite low since,all the values of the color-bar are negative.

12

148

5 APPENDIX

Figure 14: Gradient evolution.

Figure 15: Gradient variation.

13

149

REFERENCES

Concerning the first case, in section 3.2.1, the study was carried betweenday 0 and 20 (when the gradient decreases) and between day 20 and 61 (whenthe gradient rises up again). The results are shown on figure 16.

On the left figure, one notices that the gradient values are decreasing. Theparameters were chosen in the blue area and the scale of the color-bar is negativewhich means the parameters do not have a great impact on the result.

On the opposite, the gradient is increasing on the right figure. The parame-ters that were chosen correspond to the red area, and the scale of the color-baris positive.

Figure 16: Gradient variation.

References

[1] Smith AD, Shah SN, Rini BI, Lieber ML, and Remer EM. Morphology,attenuation, size, and structure (mass) criteria: assessing response and pre-dicting clinical outcome in metastatic renal cell carcinoma on antiangiogenictargeted therapy. Ajr., Jun 2010. 194(6):1470-8. PubMed PMID: 20489085.

[2] Eisenhauer EA, Therasse P, Bogaerts J, Schwartz LH, Sargent D, Ford R,and et al. New response evaluation criteria in solid tumours: revised recistguideline (version 1.1). Eur J Cancer, 2009. 45(19097774):228-47.

[3] Cornelis F, de Clermont H, Bernhard JC, Ravaud A, and Grenier N. [quan-titative imaging in uro-oncology]. Prog Urol, Jun 2014. 24(7):399-413.PubMed PMID: 24861679.

[4] Kang HC, Tan KS, Heitjan DF Keefe SM, Siegelman ES, Flaherty KT,and et al. Mri assessment of early tumor response in metastatic renal cellcarcinoma patients treated with sorafenib. 200(1):120-6. PubMed PMID:23255750.

[5] Braunagel M, Graser A, Reiser M, and Notohamiprodjo M. The role offunctional imaging in the era of targeted therapy of renal cell carcinoma.World J Urol., Feb 2014. 32(1):47-58. PubMed PMID: 23588813.

[6] Rosen MA and Schnall MD. Dynamic contrast-enhanced magnetic resonanceimaging for assessing tumor vascularity and vascular e↵ects of targeted ther-

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apies in renal cell carcinoma. Clin Cancer Res., Jan 2007. 13(2 Pt 2):770s-6s.PubMed PMID: 17255308.

[7] Eckhardt SG. Ratain MJ. Phase ii studies of modern drugs directed againstnew targets: if you are fazed, too, then resist RECIST. J Clin Oncol., Nov2004. 22(22):4442-5. PubMed PMID: 15483011.

15

151

7. Modélisation du champs électrique dans les procédures d'électroporation irréversible

Le but de ce travail est de mieux appréhender les traitements réalisés par

électroporation irréversible (IRE). L'IRE est une technique de destruction locale des tumeurs,

là où des applicateurs sont implantés. La connaissance de la distribution du champ électrique

permet de définir les zones de lésions tissulaires. Toutefois à ce jour, ce champ reste non

visible, et doit être en conséquence modélisé. La prostate étant un organe traité par IRE et

avec un suivi par IRM souvent réalisé, ce travail a consisté à recueillir des informations sur

l'ablation et les comparer aux données de l'IRM de suivi. Outre le développement à terme

d'outil spécifique dédié aux ablations de prostate, il a été proposé de réaliser ces simulations

sur le foie avec des données obtenues auprès du Pr Olivier SEROR, Bondy, France. Ce travail

est en cours.

Cet article a été soumis: Comparison of Ablation Defect on MR Imaging with

Computer Simulation Estimated Treatment Zone Following Irreversible Electroporation of

Patient Prostate. Prostate Cancer and Prostatic Diseases 2015

7.1. Introduction

The detection of prostate cancer has shifted to an earlier point in disease development,

resulting in the increased incidence of early-stage small-volume cancers (1, 2).

Consequentially, there has been an emergence of minimally invasive surgical interventions

designed to provide appropriate local oncologic control with negligible treatment related

effects on quality of life (3, 4). Such focal tissue ablation techniques intend to preserve

152

erectile, urinary and rectal function by minimizing damage to anatomical features such as

neurovascular tissues, the urinary sphincter, bladder and rectum that surround the prostate.

Thermal ablation techniques such as cryoablation (5-7), or high intensity focused ultrasound

(HIFU) (8-10) have been evaluated for treatment of patients with localized prostate cancer

with good short term outcomes (10-13). However, even focal ablation requires careful

planning and application to avoid injury to vitally healthy tissue in the treatment vicinity (12,

13).

Irreversible electroporation (IRE) is a new type of focal ablation that uses short high

voltage electric pulses to create persistent micropores in the plasma membranes of cells,

leading to cell death. IRE has been evaluated for ablation of the prostate (14-17) in the

preclinical setting. Even though the energy used during IRE may produce heating in the

immediate vicinity of the electrodes (37), it has been observed to be safe for the focal ablation

of tumors adjacent to heat sensitive structures such as the bile duct (18, 19) and the pancreas

(20) in humans. IRE is performed by placing needle electrodes in tissue and applying a

voltage between the electrodes to generate an in-vivo ablative electric field. The in-vivo

electric field distribution is determined by the geometry of ablation probe placement, the

voltage applied between the electrodes and the electrical conductivity of the tissue receiving

treatment. The therapeutic efficacy and treatment outcomes following IRE ablation is

therefore contingent on the size, shape and consistency of the in-vivo distribution of this

ablative electric field. The ablative electric field used to induce IRE in tissue has been

reported to be susceptible to heterogeneities in electrical conductivity in the treatment region

(21). There is concern that such intrinsic redistribution of the ablative electric field may cause

unintended safety effects and also alter the intended volume and shape of the final treatment

region. Patient specific computer simulations can model the ablative electric field distribution

in the prostate using data obtained from pre and intra-operative imaging, and the treatment

153

parameters (15, 22, 23). Therefore, the purpose of this study was to determine whether

retrospectively constructed patient specific numerical simulations can map the treatment

effect seen on follow-up MR imaging after irreversible electroporation of patient prostate.

7.2. Material and methods

This retrospective single institution study was approved by the institutional review

board and performed in compliance with the Helsinki Declaration.

PATIENTS AND TREATMENT

Data of six consecutive patients treated with IRE in 2013 (mean age: 66.5 years;

range: 61-70 years) with biopsy proven prostate carcinoma was used in this retrospective

study. IRE was performed after insertion of needles spaced 10-15mm apart through a

transperineal approach under TRUS guidance. Voltages used for treatment delivery was

chosen to achieve an effective electric field strength of 1600-1800V/cm between any pair of

ablation probes used for treatment delivery, seventy 90 microsecond pulses were used to

perform the treatment. Patient characteristics and ablation data are summarized in Table 1. All

treatments were performed under general anesthesia with intravenous muscle blockade to

reduce electric pulse induced muscle contraction.

Clinical History IRE Treatment Information

Patient Age (Years) History Stage Gleason

Number of

probes

Mean probe

spacing (mm)

Mean voltage

(V)

Pulse length

(μs)

Number of

ablations

1 68 Active surv

(2009) T1c 6

(3+3) 4 15 (11-19)

2330 (1650-2850)

90 4

2 70 - T1c Apex

6 (3+3) 5 13.7

(11-19)

2103 (1650-2850)

90 7

3 64 - T1c

Apex right

6 (3+3) 3 13.3

(13-14)

2000 (1950-2100)

90 3

4 61 - T1c 6 4 12.2 2075 90 4

154

Apex right

(3+3) (10-14) (1700-2340)

5 70 Radioth 2008

T2a PZ

right

9 (4+5) 5 14

(11-16)

2301 (1760-2720)

90 7

6 66 - T1c Apex

6 (3+3) 3 14

(13-15)

2520 (2340-2700)

90 3

Table 1:Patient characteristics and ablations data

Preoperative MR imaging and intra-operative axial TRUS acquisitions were recorded,

with measurement of the prostate in the axial cross-section. Two patients had tumors

identifiable on pre-operative MR imaging, and the other patients had the target region for

ablation identified on the basis of prior transrectal ultrasound guided biopsies. Follow-up MRI

was performed within 21 days (+/- 9 days) after IRE treatment. Post-treatment, all patients

underwent follow up imaging in a tertiary referral center on a 1.5T system (GE, Milwaukee,

IL, USA) with an endorectal coil and contrast injection. The MR protocol used for image

acquisition is reported in Table 2. Patient reported outcomes regarding urinary and sexual

function were obtained using a questionnaire at their clinic visit.

MRI protocol MRI Sequences

T1 weighted T2 weighted DWI Dynamic SE SE EPI GE

Plane Axial Axial Axial Axial Fat saturation No No Yes No Time to repeat

(ms) 646 6450 4500 3.77

Time to echo (ms) 12 116 91 1.46

Angulation (°) 134 170 10 Thickness (mm) 3.5 3.5 3.5 3.5

FOV (mm) 160 150 180 190 Matrix

(mm x mm) 246x256 246x256 102x128 128x160

155

Scan time (s) 150 390 340 260 Time resolution

(s) - - - 10

Table 2: MR imaging protocol used during follow-up imaging. MRI: Magnetic Resonance Imaging DWI: Diffusion Weighted Imaging FOV: Field Of View SE: Spin Echo EPI: Echo Planar Imaging GE: Gradient Echo IMAGE PROCESSING

The intra-operative ultrasound image in the axial plane corresponding to the midpoint

of the treatment zone (mid-point of ablation volume) was manually registered with the pre-

operative MR images. The registered image set was then annotated to demarcate the ablation

probes, the outline of the prostate, rectum, neurovascular bundles and the tumor (where

observable) using OsiriX DICOM Viewer (Figure 1A and 1B). This annotated image set was

used to generate the 3D models for the simulation.

Subsequently, the follow-up MR images were also registered with the intra-operative

US images in a similar process. The ablation defect was segmented from the follow-up

imaging. The area of the prostate was measured from the intra-operative US images, and the

follow-up MR images using GNU image processing software (GIMP). The area of the

ablation defect was measured from the follow-up MR image and simulation results. In

addition to visual assessment performed by two radiologists, the influence of post-treatment

edema and resulting error in registration were evaluated by comparing the ratio of the

prostate’s area between intra-operative US images and the follow-up MR images. The

ablation area measurements from the follow-up MR image and the simulation results were

compared in a similar fashion.

SIMULATION

A numerical model of the Laplace equation was solved to estimate the electric field

distribution within the tissue due to application of voltage between pairs of needle electrodes

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used for treatment delivery. Techniques previously described by Neal et al. (24, 25) and

Daniels et al. (26) were used to set up and solve the numerical models. The modeling and

simulation techniques were developed and validated partly using data generated from IRE

treatments performed in the kidney and pancreas of a large animal model (30,31).

Finite element computer simulations were used to estimate in-vivo distribution and

magnitude of the ablative electric field within the prostate and surrounding anatomic

structures. The annotated intra-operative US image were imported into Inventor (Autodesk,

San Rafael CA), and the data was manually segmented. The segmented regions were then

used to generate 2.5mm thick 3D geometry for each region identified in the US image. This

specific slice thickness was chosen to correspond to slice thickness and spacing used during

acquisition of the post-operative MR images (Figure 1C). The 3D CAD models created in

Inventor were imported into Comsol (Comsol Inc., MA) and discretized into a finite element

mesh. Material properties of biological tissue used for performing the simulation are

described in Table 3. The simulation was performed for a single pulse applied between pairs

of electrodes used for treatment delivery in patients, and the electric field resulting from each

such application was used to arrive at the cumulative electric field distribution at the end of

treatment. Isoelectric contours were drawn at 100V/cm intervals (1500V/cm - 500V/cm) for

comparison with follow up MR imaging. The isoelectric contours were used to identify the

electric field threshold (Figure 1D) that best matched with the ablation defect seen on follow-

up MR imaging (Figure 1C).

Tissue Type Electrical Conductivity S/m

Reference

Healthy Prostate 0.41 (15) Prostate Tumor 0.3 (28)

Axon 1.44 (26) Fat 0.012 (29)

Blood 0.7 (29)

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Muscle 0.2 (29) Colon 0.01 (29)

Table 3: Tissue electrical properties used in numerical simulation.

Figure 1:

Overview of workflow used to perform patient specific simulation for IRE treatment

performed on a 70-year-old man with Gleason 9 recurrence after radiation therapy (Patient

7). A: T1weighted post contrast MRI showed a tumor (center marked with an asterisk,

boundary with solid line) in the right peripheral zone. B: Intra-operative axial US guided

needle placement to the tumor (5 IRE needles, white arrows). The intra-operative US image

annotated to demarcate the ablation probes (dashed arrows), the outline of the prostate, the

rectum, and the neurovascular bundles (NVB) (solid lines). Clinical treatment planning data

were compiled and the MR images from the corresponding axial plane were used to identify

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critical structures for segmentation. C: Follow-up axial enhanced T1w with fat saturation MR

imaging performed 15 days after ablation was used to demarcate the ablation defect (solid

line) and showed size and shape of the ablative zone (area: 701 mm2). D: Simulation

predicted ablation zone (white with blue boundary) at the electric field strength contour

(700V/cm; area 624mm2, arrowhead). Image plotted using gradient shading with regions of

highest field strength (700 V/cm and stronger) appearing light and lower field strengths

appearing dark. Simulation predicted that ablation encompassed completely the tumor.

STATISTICAL ANALYSIS

Patient data was compiled from a review of all medical, imaging and pathological

reports. Results of simulation studies were compared to outcomes determined by clinical data

(PSA, quality of life survey) as well as follow-up MR imaging. The size of the ablation as

estimated by numerical simulation and the actual ablation defect seen on MR imaging was

compared and statistically analyzed for significance. Statistical analysis was conducted with

Fisher exact test, χ2 for independence test (Excel; Microsoft, Redmond, WA) and p<0.05 was

considered significant. The ablation zones from the simulation and MR imaging were also

evaluated using Pearson product moment correlation.

7.3. Results

IMAGING OUTCOMES

In the immediate post-treatment setting, non-contrast US images indicated presence of

multiple diffuse regions with hyperechoic appearance within the ablation zone. The

appearance and distribution of these imaging findings was consistent with the presence of gas

bubbles that have been previously reported to appear after IRE in solid organs (32). There was

also the occasional appearance of a hyperechoic at the margin of the planned treatment zone.

However, this was not clearly present in the entire treatment region and was not observed in

all patients.

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It was possible to discriminate between treated and untreated regions of the gland on

follow-up MR imaging. The expected treatment zone appeared as a heterogeneous

hypointense zone on T1 imaging, but it was not possible to clearly separate the treated region

from surrounding normal gland (Figure 2A). On T2 imaging, the ablation zone appeared as a

hyperintense region interspersed with low intensity locations that were consistent with

ablation probe tract (Figure 2C). On contrast-enhanced dynamic T1 imaging performed with

fat suppression, the treatment region appeared non-enhancing with limited to no enhancement

of tissue in the periphery (Figure 2B and 2D). The T2 and contrast-enhanced dynamic T1

images were used to identify and segment the expected ablation zone for comparison with

simulation results.

Figure 2:

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Typical findings on follow-up MRI three weeks after IRE of the prostate. From one patient,

(A) we could not clearly discern the expected ablation zone (arrow) from normal gland on

T1w images. The treatment effect was clearly visualized (B) using contrast enhanced T1

imaging as a homogenous hypointense region (arrow). From another patient, (C) the

treatment zone appears as heterogeneous hyperintense region (arrow) with regions of low

signal intensity. (D) The lesion (arrow) is easily visualized on contrast enhanced T1w

imaging.

It was possible to obtain good registration of the intra-operative US and the follow-up

MR imaging. The following results were obtained from computing the ratio of prostate’s area

as measured with the two modalities at the registered slice (mean: 0.99, range: 0.86-1.07).

The measured area of the prostate from the two modalities were not significantly different

(p=0.49, see Table 4).

Patient Axial Cross-sectional Area of

Prostate (mm2) Axial Cross-sectional Area of

Ablation Zone (mm2) MRI US MRI Simulation

1 2055 1916 590 585 2 1966 1877 849 735 3 1420 1633 210 340 4 888 895 322 449 5 1309 1377 701 624 6 1741 1682 569 461

Mean 1563.16 1563.33 540.16 532.33 p Not significant: (0.49) Not signifiant: (0.43)

Table 4: Area of prostate (mm2) observed with MRI and US at the same level. Size of the

prostate ablation observed with MRI and based on simulation (with a threshold level of

sensitivity of 700 V/m2). Axial cross sectional area of the prostate of the post-treatment MRI

and US were compared to validate the accuracy of registration. The US and simulation image

are at 1:1 scaling.

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SIMULATION OUTCOMES

Individualized models were created and simulations performed for every patient

included in this study (for example; Figure 1). Simulation findings suggest that the ablative

electric field was not restricted to the prostate, and was seen penetrating peri-prostatic fat and

muscle in all patients (Figure 3). In all simulations, the maximum electric field strength in

sensitive structures adjacent to the prostate, such as the neurovascular bundle (<300 V/cm) or

the rectum (<600 V/cm), were at levels at which minimal or no IRE induced damage was

expected at these structures. An electric field gradient was developed within the prostate from

application of voltage between electrodes, and the electric field was strongest in the

immediate vicinity of the ablation probes. The electric field was estimated to completely

cover the tumor in the two cases where tumors were identifiable on pre-operative MR images.

Simulations estimated an irregular and non-convex ablation zone in all patients. None of the

ablations performed between any pair of needle electrodes was observed to have a regular

convex ellipsoidal shape in the axial cross section. Metallic objects in the vicinity of the

treatment zone were observed to influence the electric field distribution. This effect largely

appeared with exposed ablation probe tips present in the treatment region, but not actively

used for energy delivery (Figure 4; energy delivery was performed between one pair of

ablation probes at a time).

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Figure 3: Simulation findings suggest that the ablative electric field was not restricted to the

prostate, and was seen penetrating peri-prostatic fat and muscle tissue. Neurovascular bundle

and rectal tissues may be drawing the ablative electric field towards them and thereby

affecting the size and shape of the ablation within the prostate. A: The neurovascular bundle

influenced the shape of the electric field in ablations performed in the periphery of the

prostate (arrow) (patient 1). B: The rectum was seen to be influencing the electric field of

ablations performed centrally in the prostate (arrow).

Figure 4: The effect of exposed but unused ablation probes on ablation outcomes.

A: Simulation representing actual clinical scenario where unused ablation probes are left

exposed in the prostate while ablation is delivered through the other pair of ablations probes

(white arrow indicates current drawn around un-insulated probe not used for ablation)

(patient 4). B: Simulation representing scenario if the unused probe had been insulated prior

to delivering ablation between the other two probes (white arrow).

COMPARISON OF SIMULATION AND FOLLOW-UP IMAGING

The contour plot of electric field strength from the simulation was registered and

compared with follow-up MR images to identify the electric field strength threshold that

matched with the size of the post-ablation defect on imaging. It was found that the electric

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field strength contour at 700V/cm correlated closely with the follow-up MR images

(Simulation predicted ablation area: 532.33mm2 in mean for all patients vs Ablation area

measured on imaging: 540.16 mm2 in mean, p=0.43) for all patient cases reviewed. This

threshold was used to demarcate the expected ablation zone within the prostate and

measurements (area and cross section) were performed using GIMP and presented in Table 4.

Radiologist interpretation from two separate observers suggested that the shape and size of

the ablation predicted by the simulation compared well with all post-operative MR imaging in

all cases. The two measurements had a correlation coefficient of 0.945 when evaluated using

Pearson product moment technique. The ratio of measurements taken from the two modalities

also indicated good correspondence between the techniques (mean: 0.97; range: 0.61-1.2).

PATIENTS REPORTED OUTCOMES

Evaluation of the self reported forms suggested that none of the patients undergoing

IRE therapy developed impotence or urinary incontinence following treatment. A significant

decrease of PSA (p=0.01) was observed in all patients during clinical follow-up. Clinical

outcomes are summarized in Table 5.

Patient

First Follow-up

MRI (days)

Biological follow-up (months)

PSA (Pre)

(ng/ml)

PSA (Post)

(ng/ml) Potent Continent

1 35 7 3.96 3 Yes Yes 2 23 11 1.9 0.24 Yes Yes 3 29 6 5.59 4.44 Yes Yes 4 22 7 1.71 1.12 Yes Yes 5 15 8 5.63 1.1 Yes Yes 6 10 9 5.42 3.15 Yes Yes

Mean 48.1 7.3 3.63 1.94

Table 5: Clinical outcomes following treatment

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7.4. Discussion

Irreversible electroporation for the focal ablation of prostate is a new treatment

technique and has been used at just select centers in the world (33). The ability to perform

non-thermal treatment with IRE makes it a valuable clinical alternative to existing thermal

ablation techniques such as HIFU or Cryotherapy. However, there exist considerable

knowledge gaps on the typical imaging findings, and treatment outcomes following IRE of

prostate, and this has restricted wider use of this technique. Currently, it is not possible to

directly measure the electric field strength generated in the targeted tissue during IRE.

Therefore unlike HIFU, where MR imaging can be employed to monitor temperatures and

subsequently estimate the effective treatment zone, there is no simple way to predict the

expected ablation zone following IRE. This topic is of considerable importance, and has

motivated research into advanced techniques such as electrical impedance tomography to

monitor and predict the expected effects of electroporation in tissue (34). Our results

demonstrate the feasibility of using numerical simulations constructed with pre-operative MR

images and intra-operative US to estimate the treatment zone, which was comparable to what

was observed on follow-up MR imaging. In the future, such simulations could potentially be

used to plan and guide IRE treatment in patients, similar to how MR thermometry is used to

direct HIFU treatment in the prostate.

As a focal ablation technique, post-treatment imaging findings are crucial to

understand tissue ablation with IRE, and also to prognosticate treatment outcomes. MRI

findings have been previously reported for the acute effects and short-term injury evolution

following IRE in different animal models, including work by Zhang et al. in rats with

hepatoma (35), and Wendler et al. in normal swine kidney (36). There is limited published

information on the typical imaging findings following IRE in the prostate and despite not

being the primary focus of this work, we report our imaging findings. Consistent to prior

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reports from animal models studies, MR imaging following IRE presents as a heterogenous

region with both hyper and hypointense regions on T1 and T2w imaging. The ablation zone

was best visualized using contrast enhanced T1w imaging, appearing as a region of non-

enhancement when compared to untreated gland. In our findings we observed that the healing

process following IRE seems to commence around 3 weeks following treatment, and before

this period there seems to be some edema in the treated region. Therefore imaging earlier than

2 weeks following treatment may not be ideal for measurements as tissue edema may

confound accurate measurements. The development of imaging techniques for assessing the

effects of IRE in patient prostate and detailed description of outcomes is a sizable project, and

is beyond the scope of the presented work.

The ablation boundary estimated at the threshold field strength of 700 V/cm in

simulations correlated well with the post-operative MR images from our patients. Prior

reports from in-vivo studies performed on healthy dog prostate suggest irreversible damage to

cells in regions that experienced electric field strength of 600V/cm or higher (22). A study

performed by Qin et al (27) using a LNCaP tumor model suggests that the critical field

strength required for achieving irreversible contingent can range from 600-1300 V/cm, and

will vary based on the number of pulses applied and the pulse duration used for treatment

delivery. Our results are in agreement with these prior studies. However, a study performed

by Neal et al. (22) with simulation-pathology correlation of human prostates resected 3-4

weeks following IRE ablation reported a higher threshold for IRE induced cell death (1072

V/cm). While this study used volumetric modeling of the ablation zone and a different

simulation technique, the study results are restricted by the small dataset (two patients) used

for their simulations. Since we did not have pathology following ablation from the patients

enrolled in this study, 700 V/cm as the critical threshold is drawn purely in correlation to MR

imaging information. A larger study may be required before the differences between imaging

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and pathology measurements can be clarified, and until such time our findings should only be

taken in context of post-IRE MR imaging of the prostate.

Confirming findings from the large animal study reported by Goldberg et al. (21), the

in-vivo distribution of the ablative electric field that induces IRE in the tissue was found

susceptible to regional heterogeneities in electrical conductivity. During IRE treatment all

ablation probes to be used for the treatment are inserted together in a pre-determined

configuration along the long axis of urethra, and subsequently treatment is delivered between

select pairs of ablations probes. Simulation electric field maps suggest that the electric field

may get drawn towards the exposed tips of the unused probes (Figure 4).

Our retrospective patient-specific simulations suggested lack of injury to the

neurovascular bundle and rectum in our patient cohort, and these findings were reflected in

the patient reported outcomes. Our simulation results suggested that IRE mediated tissue

destruction was largely restricted to the prostate. Despite being in proximity to the ablation

zone, simulation indicated the sparing of the rectum, neurovascular bundle and the urethra. In

agreement with simulation findings, no impotency or incontinence has been reported during a

mean follow up time of 7 months following treatment.

Our study has a few limitations, primarily the exploratory nature of the work and the

small number of patients enrolled in this study. These factors may limit the generalization of

our findings. The simulation was performed using data segmented from intra-operative US

images, which has less information than the pre-operative MR images. Complex and

heterogeneous structures such as the neurovascular bundled were represented in the model as

simpler lumped structures with homogenous electrical properties which reduce precision of

the simulation model. Our simulation was not truly volumetric and was restricted to a single

thick slice of tissue and therefore may not estimate the behavior of the entire ablation volume.

Our results are pertinent for comparison of simulation findings with post-treatment imaging,

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and the absence of pathology data limits drawing definitive conclusions on the status of the

treated tissue. Finally, gadolinium-enhanced MRI by itself is not a validated technique for

measuring the effectiveness of IRE in the prostate; therefore, we are unable to arrive at any

conclusion on the value of simulations for estimating true treatment outcomes. This is a major

limitation of this study.

In summary, our results suggest that simulation of IRE ablation matches treatment

zones seen on MRI, and therefore may help in treatment planning of ultrasound-guided IRE in

the prostate. While we provide evidence that simulations can be used to estimate the size and

shape of the expected IRE ablation in patient prostate with good correlation to MR imaging,

further comparison with pathology are required before using simulations to predict treatment

efficacy.

7.5. References

1. Hoeks CM, Schouten MG, Bomers JG, et al. Three-Tesla Magnetic Resonance-Guided Prostate Biopsy in Men With Increased Prostate-Specific Antigen and Repeated, Negative, Random, Systematic, Transrectal Ultrasound Biopsies: Detection of Clinically Significant Prostate Cancers. European urology. 2012. Epub 2012/02/14. 2. Ahmed HU, Hu Y, Carter T, et al. Characterizing clinically significant prostate cancer using template prostate mapping biopsy. J Urol. 2011;186(2):458-64. 3. van den Bos W, Muller BG, Ahmed H, et al. Focal therapy in prostate cancer: international multidisciplinary consensus on trial design. Eur Urol. 2014;65(6):1078-83. 4. Valerio M, Ahmed HU, Emberton M, et al. The Role of Focal Therapy in the Management of Localised Prostate Cancer: A Systematic Review. Eur Urol. 2013. 5. Cytron S, Greene D, Witzsch U, Nylund P, Bjerklund Johansen TE. Cryoablation of the prostate: technical recommendations. Prostate Cancer Prostatic Dis. 2009;12(4):339-46. 6. Bahn D, de Castro Abreu AL, Gill IS, et al. Focal cryotherapy for clinically unilateral, low-intermediate risk prostate cancer in 73 men with a median follow-up of 3.7 years. Eur Urol. 2012;62(1):55-63. 7. Onik G, Vaughan D, Lotenfoe R, Dineen M, Brady J. The "male lumpectomy": focal therapy for prostate cancer using cryoablation results in 48 patients with at least 2-year follow-up. Urol Oncol. 2008;26(5):500-5. 8. Baco E, Gelet A, Crouzet S, et al. Hemi salvage high-intensity focused ultrasound (HIFU) in unilateral radiorecurrent prostate cancer: a prospective two-centre study. BJU Int. 2013.

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9. Crouzet S, Chapelon JY, Rouviere O, et al. Whole-gland Ablation of Localized Prostate Cancer with High-intensity Focused Ultrasound: Oncologic Outcomes and Morbidity in 1002 Patients. Eur Urol. 2013. 10. El Fegoun AB, Barret E, Prapotnich D, et al. Focal therapy with high-intensity focused ultrasound for prostate cancer in the elderly. A feasibility study with 10 years follow-up. Int Braz J Urol. 2011;37(2):213-9. 11. Ahmed HU, Freeman A, Kirkham A, et al. Focal therapy for localized prostate cancer: a phase I/II trial. J Urol. 2011;185(21334018):1246-54. 12. Ahmed HU, Hindley RG, Dickinson L, et al. Focal therapy for localised unifocal and multifocal prostate cancer: a prospective development study. Lancet Oncol. 2012;13(6):622-32. 13. Barret E, Ahallal Y, Sanchez-Salas R, et al. Morbidity of focal therapy in the treatment of localized prostate cancer. Eur Urol. 2013;63(4):618-22. 14. Onik G, Mikus P, Rubinsky B. Irreversible electroporation: implications for prostate ablation. Technol Cancer Res Treat. 2007;6(17668936):295-300. 15. Neal RE, 2nd, Smith RL, Kavnoudias H, et al. The Effects of Metallic Implants on Electroporation Therapies: Feasibility of Irreversible Electroporation for Brachytherapy Salvage. Cardiovascular and interventional radiology. 2013. 16. Davalos RV, Rubinsky B, Mir LM. Theoretical analysis of the thermal effects during in vivo tissue electroporation. Bioelectrochemistry. 2003;61(1-2):99-107. 17. Davalos RV, Mir IL, Rubinsky B. Tissue ablation with irreversible electroporation. Annals of biomedical engineering. 2005;33(2):223-31. 18. Silk MT, Wimmer T, Lee KS, et al. Percutaneous ablation of peribiliary tumors with irreversible electroporation. J Vasc Interv Radiol. 2014;25(1):112-8. 19. Choi JW, Lu DS, Osuagwu F, Raman S, Lassman C. Assessment of Chronological Effects of Irreversible Electroporation on Hilar Bile Ducts in a Porcine Model. Cardiovascular and interventional radiology. 2013. 20. Bower M, Sherwood L, Li Y, Martin R. Irreversible electroporation of the pancreas: definitive local therapy without systemic effects. J Surg Oncol. 2011;104(1):22-8. 21. Ben-David E, Ahmed M, Faroja M, et al. Irreversible electroporation: treatment effect is susceptible to local environment and tissue properties. Radiology. 2013;269(3):738-47. 22. Neal RE, 2nd, Millar JL, Kavnoudias H, et al. In vivo characterization and numerical simulation of prostate properties for non-thermal irreversible electroporation ablation. Prostate. 2014. 23. Neal RE, 2nd, Davalos RV. The feasibility of irreversible electroporation for the treatment of breast cancer and other heterogeneous systems. Annals of biomedical engineering. 2009;37(12):2615-25. 24. Neal RE, 2nd, Garcia PA, Robertson JL, Davalos RV. Experimental characterization and numerical modeling of tissue electrical conductivity during pulsed electric fields for irreversible electroporation treatment planning. IEEE Trans Biomed Eng. 2012;59(4):1076-85. 25. Neal RE, Singh R, Hatcher HC, Kock ND, Torti SV, Davalos RV. Treatment of breast cancer through the application of irreversible electroporation using a novel minimally invasive single needle electrode. Breast Cancer Res Treat. 2010;123(20191380):295-301. 26. Daniels C, Rubinsky B. Electrical field and temperature model of nonthermal irreversible electroporation in heterogeneous tissues. Journal of biomechanical engineering. 2009;131(7):071006. 27. Qin Z, Jiang J, Long G, Lindgren B, Bischof JC. Irreversible electroporation: an in vivo study with dorsal skin fold chamber. Annals of biomedical engineering. 2013;41(3):619-29.

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28. Halter RJ, Schned A, Heaney J, Hartov A, Paulsen KD. Electrical properties of prostatic tissues: I. Single frequency admittivity properties. J Urol. 2009;182(4):1600-7. 29. Andreuccetti D, Fossi R, Petrucci C. An Internet resource for the calculation of the dielectric properties of body tissues in the frequency range 10 Hz - 100 GHz. Website at http://niremf.ifac.cnr.it/tissprop/. IFAC-CNR, Florence (Italy), 1997. Based on data published by C.Gabriel et al. in 1996. 30. Wimmer T, Srimathveeravalli G, Gutta N, Ezell PC, Monette S, Maybody M, Erinjery JP, Durack JC, Coleman JA, Solomon SB. Planning Irreversible Electroporation in the Porcine Kidney: Are Numerical Simulations Reliable for Predicting Empiric Ablation Outcomes? Cardiovasc Intervent Radiol. 2014 May 17. 31. Wimmer T, Srimathveeravalli G, Gutta N, Ezell PC, Monette S, Kingham TP, Maybody M, Durack JC, Fong Y, Solomon SB. Comparison of simulation-based treatment planning with imaging and pathology outcomes for percutaneous CT-guided irreversible electroporation of the porcine pancreas: a pilot study. J Vasc Interv Radiol. 2013 Nov;24(11):1709-18. 32. Schmidt CR, Shires P, Mootoo M. Real-time ultrasound imaging of irreversible electroporation in a porcine liver model adequately characterizes the zone of cellular necrosis. HPB (Oxford). 2012 Feb;14(2):98-102. 33. Valerio M, Stricker PD, Ahmed HU, Dickinson L, Ponsky L, Shnier R, Allen C, Emberton M. Initial assessment of safety and clinical feasibility of irreversible electroporation in the focal treatment of prostate cancer. Prostate Cancer Prostatic Dis. 2014 Dec;17(4):343-7. 34. Kranjc M, Markelc B, Bajd F, Čemažar M, Serša I, Blagus T, Miklavčič D. In Situ Monitoring of Electric Field Distribution in Mouse Tumor during Electroporation. Epub 2014 Aug 19. 35. Zhang Y, White SB, Nicolai JR, Zhang Z, West DL, Kim DH, Goodwin AL, Miller FH, Omary RA, Larson AC. Multimodality imaging to assess immediate response to irreversible electroporation in a rat liver tumor model. Radiology. 2014 Jun;271(3):721-9. 36. Wendler JJ, Porsch M, Hühne S, Baumunk D, Buhtz P, Fischbach F, Pech M, Mahnkopf D, Kropf S, Roessner A, Ricke J, Schostak M, Liehr UB. Short- and mid-term effects of irreversible electroporation on normal renal tissue: an animal model. Cardiovasc Intervent Radiol. 2013 Apr;36(2):512-20. 37. Faroja M, Ahmed M, Appelbaum L, Ben-David E, Moussa M, Sosna J, Nissenbaum I, Goldberg SN. Radiology. Irreversible electroporation ablation: is all the damage nonthermal? 2013 Feb;266(2):462-70.

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8. Conclusion

Ce travail a tenté de montrer que les mathématiques appliquées pouvaient aider à

faciliter l'analyse clinique des images voire même le traitement des patients. En effet

l'utilisation de modèles mathématiques permet de mieux comprendre le vivant. Traduire une

observation clinique obtenue par l'imagerie par un modèle descriptif voire prédictif peut

permettre de développer des outils, des techniques et des théories mathématiques capables

d'approfondir nos connaissances dans le monde réel une fois tout cela appliqué.

Tout d'abord, il est possible de modéliser la croissance tumorale ou la réponse à un

traitement en fonction de paramètres préalablement identifiés obtenus par l'imagerie, ceci de

façon rétrospective ou prospective. Cela est prometteur dans l'amélioration de la

compréhension des processus biologiques. Il est d'ores-et-déjà possible d'identifier une

maladie indolente vs une pathologie plus agressive ou des patients répondeurs vs des patients

non-répondeurs à un traitement par la simple différence par rapport au modèle théorique.

Nous avons conceptualisé cela par le terme "sortie de route". Une lésion traitée (ou non) ne

restant pas "dans les clous" devra être soit considérée comme agressive ou en échappement si

l'évolution s'accentue, soit en réponse ou ralentissement si l'évolution s'infléchie. Ces deux

phénomènes ont été observés régulièrement dans nos travaux sur les nodules pulmonaires

mais également au niveau du foie. Un échappement peut aussi traduire l'apparition d'une néo-

mutation, phénomène observé dans nos travaux sur les métastases de GIST, expliquant

l'acquisition d'une résistance à un traitement et donc observable dans nos modèles si la voie

d'action du traitement est prise en compte. Jusqu'à présent seulement identifiable par la

génétique et donc par la réalisation de prélèvements invasifs, la perspective d'identification de

l'agressivité tumorale par les modèles apparait être une piste intéressante. Nous souhaitons

ainsi proposer à coutre échéance un outil numérique permettant d'assister le médecin dans son

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étude de l'imagerie. Une part non négligeable du travail, non rapporté dans ce manuscrit car

dépassant nos compétences, a été de fluidifier et optimiser le calcul pour obtenir une analyse

quasiment en temps réel.

Ces résultats ouvrent la voie à la réalisation d'essais thérapeutiques modélisés

permettant de réduire le nombre de patient inclus dans les essais thérapeutiques. En proposant

une partie numérique dans les cohortes de patients, une réduction du nombre d'inclusions et

donc des coûts mais aussi des effets secondaires inhérents aux traitements testés pourrait être

observée. Une fois validé, les modèles pourraient offrir la perspective de remplacer les

témoins. Pour un effectif constant, les modèles numériques pourraient également majorer la

puissance des essais en ajoutant une cohorte numérique. De même, une évaluation très

précoce des effets des traitements pourrait être envisagée permettant les mêmes apports en

termes de coûts ou de morbidité. Toutefois les modèles doivent être dans un premier temps

validés cliniquement et bien sûr complexifiés pour être le plus fiable et reproductible possible.

Les outils numériques doivent être développés pour réduire les temps de calculs et permettre

une évaluation rapide.

Cet impact des modèles peut également s'envisager à plus court terme en radiologie

interventionnelle. Grâce à un modèle calibré notamment au niveau du poumon ou du foie,

sièges les plus fréquents des métastases, il pourrait être également possible de proposer une

évolution "moyenne" d'un ensemble de nodules et les indications de traitements locaux

pourraient être discutées sur les résultats des modèles. Un nodule en échappement posera

l'indication d'un traitement palliatif de sauvetage, alors qu'un nodule stable pourra justifier

une simple surveillance ou bien un traitement éventuellement à visée curative si la maladie

apparait contrôlée. Par ailleurs, le profil évolutif moyen obtenu par les modèles pourrait être

considéré comme un des éléments pronostic facilitant la décision médicale en réunion de

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concertation pluridisciplinaire et permettant ainsi de discuter les traitements locaux vs les

traitements généraux en fonction du stade de la maladie.

Afin d'optimiser ces modèles pour répondre adéquatement à ces perspectives, il a été

(et il sera) nécessaire de rechercher de nouveaux paramètres. Comme nous en avons fait

l'expérience, l'incorporation progressive de ces nouveaux éléments issus de l'imagerie

fonctionnelle (IRM, TEP), de la biologie ou de l'immunologie par exemple, loin d'être triviale,

demande une grande maitrise des techniques et un temps de développement long. De plus, les

outils numériques de post-traitement de l'imagerie ne sont pas encore optimisés ce qui

complexifie encore le traitement des données brutes recueillies. Pour limiter les biais, il est

important d'envisager la maitrise des différents éléments de la chaine de recueil de données

(le workflow). Par exemple, une des étapes a été d'estimer l'erreur provenant du processus de

segmentation. Mais il reste encore d'autres éléments à étudier comme l'analyse quantitative de

l'imagerie fonctionnelle par IRM ou TEP.

Cette optimisation va permettre de mieux appréhender les phénomènes impliqués et

donc enrichira les modèles par l'ajout de nouveaux paramètres. Cependant cela se fera au prix

malheureusement d'une complexité accrue. Nous avons déjà observé cela dans les premiers

développements sur les nodules pulmonaires. Nos EDP/EDO initiales étaient trop simples

pour permettre de modéliser de façon générique une croissance tumorale pour un patient

donné. L'ajout de paramètres d'oxygénation ou de nutrition a permis de mieux coller à la

réalité. Par la suite, il a été nécessaire de recourir aux métastases de GIST pour mieux

appréhender la réponse à un traitement. Les résultats ont été transposés au poumon. Afin de

bénéficier de nombreux développements en IRM fonctionnelle mais aussi d'un modèle

impliquant les traitements anti-angiogéniques, nous nous sommes intéressés aux tumeurs du

rein métastatiques qui actuellement bénéficient largement de ce type de traitement. Par les

connaissances que nous sommes en train d'acquérir, nous pourrons peut-être mieux modéliser

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l'angiogenèse. Dans un futur proche nous prévoyons d'ajouter des données cliniques mais

aussi biologiques incluant notamment des marqueurs de l'immunité. Le recueil prospectif des

données cliniques et biologiques à grande échelle permettra à terme de valider ces modèles

statistiquement mais aussi d'affiner leur précision avec le temps. Ainsi ces modèles

génériques appliqués à une population donnée pourront aider à répondre à des questions

cliniques essentielles en oncologie: quand faut-il traiter et/ou comment faut-il le faire?

Quand la décision de traiter a été prise, de nouveau, les mathématiques appliquées

utilisant l'imagerie peuvent aider à mieux appréhender la clinique. Dans nos travaux sur

l'évolution après mise sous traitement par anti-angiogénique, il a été montré que différents

indicateurs simples mais objectifs peuvent être proposés pour détecter au plus tôt une réponse

au traitement et/ou un échappement. Ceci pourrait de nouveau avoir un impact dans la

caractérisation des patients en répondeurs vs non-répondeurs. Plus ce classement aura lieu

précocement et plus le risque d'effet secondaire pour un traitement inutile sera limité, sans

compter les économies réalisées sur la prescription de traitement inefficace. Toutefois ces

indicateurs restent à valider dans une réalité clinique. Mais du fait même de leur simplicité, ils

pourraient contribuer à l'amélioration de l'évaluation thérapeutique précoce sans avoir recours

à une imagerie fonctionnelle complexe.

Les modèles couplés à l'imagerie peuvent également permettre de mieux traiter les

patients comme nos travaux sur l'électroporation de prostate le laisse entrevoir. Une

connaissance accrue de la distribution spatiale du champ électrique en temps réel améliorera

la tolérance au traitement et son efficacité. Cela permettra de diffuser encore plus largement

ces techniques mini-invasives.

Les éléments contenus dans cette thèse nécessitent encore des développements. Déjà

des outils utilisables au quotidien sont prêts à être proposés grâce à ces travaux transversaux.

Une perspective proche est de proposer des outils d'aide en temps réel à l'évaluation

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thérapeutique par l'imagerie, notamment selon le RECIST, d'assistance à la décision

thérapeutique et de rétrocontrôle après électroporation. De nouvelles possibilités en oncologie,

en imagerie et en oncologie interventionnelle sont ainsi envisageables.