Dossier de candidature PHC SAKURA 2012

Transcription

1 Dosser de canddature PHC SAKURA Partenares France Japon Chef de projet 1 Nom : M. Coln Therry M. Suzuk Takash Laboratore 1 Nom - Sgle : Insttut de Mathématques de Bordeaux (IMB) Dvson of Mathematcal Scence Adresse : Unversté Bordeaux cours de la lberaton Department of Systems Innovaton Machkaneyamacho 1-3, Toyonakash, , Japan, Code Postal : Vlle : Talence Osaka Téléphone : Télécope : Emal : coln@math.u-bordeaux1.fr suzuk@sgmath.es.osaka-u.ac.jp Ste Web : Insttuton de Unversté Unversté Bordeaux 1 (UB1) Osaka Unversty rattachement : Adresse : 351 cours de la lberaton Code Postal : Vlle : Talence Pays : France Japon Ste Web : 2 - Projet Ttre : Domane : Secton : MATHEMATICAL MODELS OF TUMOR GROWTH Mathématques et leurs nteractons MATHEMATIQUES APPLIQUEES ET APPLICATIONS DES MATHEMATIQUES 3 - Moyens demandés en 2012 France : Voyages et séjours chercheur francas Nombre total de personnes 2 Nombre total de voyages 2 Durée total des séjours (en jours) 30 Japon : Voyages et séjours chercheur japonas Nombre total de personnes 4 Nombre total de voyages 0 Durée total des séjours (en jours) Moyens demandés en 2013 France : Voyages et séjours chercheur francas Nombre total de personnes 3 Nombre total de voyages 2 Durée total des séjours (en jours) 30 Japon : Voyages et séjours chercheur japonas Nombre total de personnes 4 Nombre total de voyages 0 Durée total des séjours (en jours) 0 DOSSIER N Page 1

2 5 - Autres fnancements reçus Autres fnancements reçus ou demandés? Avez vous déjà bénéfcé d'un fnancement pour ce PHC? Autres demandes déposées pour 2012? aucun Non Non DOSSIER N Page 2

3 6 - Descrpton du projet Objectf scentfque et/ou technologque de la collaboraton The am of ths project s to propose a comprehensve study of the modelng of tumor growth, ncludng mcroscopc (cell level) and macroscopc (tssues and organ level) elements and to apply these modelng tools to therapeutc nnovaton n oncology. The longterm goal s to mprove drug delvery protocols as well as optmzed treatment plannng for clncal tral. Ths crucal am requres as a frst step the mathematcal analyss and control of complex models of tumor growth.tumor growth models have been developed snce a few years by varous teams (Prezos, Bellomo, Chaplan, Rbba, Coln, Fredman ). These models focus on partcular aspects of cancer: Cell cycle models: recent work by Perthame and co-authors on the underlyng knetc equatons, related entropes, and ther control. Immune reacton: modeled by knetc equatons followng Bellomo s works. Tumor growth: usng mechancal models and Darcy law or Stokes equatons, or even more refned elastc models. Angogeness: neovascularsaton of the tumor, whch plays a crucal role for large tumor (see Chaplan and co-authors). Radotherapy. Such models descrbe very accurate behavor n qualtatve descrpton of tumor growth, however, due to ther complexty, straghtforward clncal applcatons of such a modelng seem unreachable. Data assmlaton s clearly a key pont for clncal applcatons. Ths task s almost mpossble for the complex models that we have quoted above and smplfed models that present a good equlbrum between complexty and accuracy have to be consdered. The Bordeaux team has started a whole program concernng data assmlaton for lung metastases of dstant tumor wth smplfed models. A lot of techncal problems have to be overcome. However, the understandng of the mathematcal propertes of the system that are used for data assmlaton s a crucal step: the qualty of the predctons s determned by the mathematcal propertes of the model. Another nterestng s the therapy modelng. The effcacy of a cancer treatment s detected at the macroscopc level (tumor shrnkng measurement). But the delvered drugs for systemc treatments, as well as the local treatments such as radotherapy or thermotherapy occur at the cell level. Therefore accurate modelng of the treatment at the cell level can help n understandng the phenomena that are detected at the tssue scale. Through a 2-years collaboraton wth the Insttut Gustave Roussy, the Bordeaux team has developed a knowledge n the modelng of the electrochemothearpy, a local treatment that take advantage of the drug delvery wth poratng the cancer cells membrane by hgh voltage pulses, ths s descrbed n the attached fle localtreatment.pdf. Osaka group has been engaged n top down and bottom up modelng at cell level. They have been mostly concerned wth chemcal reacton, dffuson, and chemotaxs. Therefore, t s a real challenge to clarfy the role of other factors pcked up n the model by French group usng mathematcal analyss. For all these reasons, one needs to understand the mathematcal propertes of the PDE systems that are used and ths s the goal of the proposal. Durng ths collaboraton, we wll focus on the mathematcal propertes of the PDE model at both cellular and macroscopc level. A more detaled scentfc descrpton can be found n the attached fle project.pdf. A descrpton of the data assmlaton process s n dataassmlaton.pdf whle local treatments are detaled n localtreatment.pdf. Lste des fchers téléchargés par le canddat (cf. annexe) Descrpton du projet (research_program.pdf) Data assmlaton (dataassmlaton.pdf) Local treatments (localtreatment.pdf) Programme de traval proposé et calendrer The frst year wll be devoted to the exchange of nformaton between the teams n order to defne the class of models that have to be studed and the natural problems that have to be solved. The dea s to start wth very smplfed models that are currently used for data assmlaton by the Bordeaux team. A frst step wll be to obtan some toy models n order to answer the followng natural questons : ) Global exstence of the solutons? ) Boundedness of the soluton: for example, the densty of cells have to be between 0 and 1, whch s not ensured a pror by all the systems of the class that we consder. ) Stablty of the solutons wth respect to parameters. Ths pont s crucal for solvng some nverse problem. Indeed, f one wants to parametrze the system usng medcal magng, one has to use data that are ncomplete or not precse. The dependence of the DOSSIER N Page 3

4 soluton wth respect to the parameters (and of the ntal data) s therefore very mportant. The second year wll be devoted to the study of more complex problems wth treatment modelng lke the one that s descrbed n the attached fle. Ths part s clearly prospectve. It wll strongly depend on the result of the frst year as well as of the dscusson between both teams. Intérêt de la collaboraton et complémentarté des équpes The Bordeaux team has a strong expertse n terms of data assmlaton for modelng of tumor growth. Numerous relatonshps have been establshed the last years between Bordeaux and some hosptals specalzed on cancer treatments : Insttut Bergoné n Bordeaux and Insttut Gustave Roussy n Vllejuf near Pars. The Bordeaux team has therefore access to a large panel of data concernng patent, medcal magng, treatments, bopsy In order to be able to take these data nto account, the team has developed some models that are more or less accurate as well as data assmlaton procedure based on nverse problem n order to be able to make some predctons. However, the mathematcal study of these models has not been addressed by the team at the tme beng. The collaboraton wth the team of prof. Suzuk s therefore very nterestng n ths context. The am of usng mathematcal model n the CREST project of Osaka team s two-fold; provdng mathematcal tools for the cell molecular bology and understandng the tumor growth event as a system. We have, on the other hand, developed mathematcal methods for dagnoss; automatc cancer tssue check usng homology and nverse source dentfcaton Thus what we want to do are, frst, to provde several mechancal factors wth the model such as cell-cell adheson and flud moton n tssue level, and second, to set up adaptve smulatons for pathology predcton. Tradtonally, there was a lot of lnks between the French and Japanese school n PDE. Th. Coln was nvted several tmes by Prof. Hayash n Osaka or Prof. Ohta n Satama. The present collaboraton s clearly a new opportunty for the two unverstes: the am goes far beyond pure mathematcal problems. Takash Suzuk acts as a prmary host of Mare Cure Internatonal Research Staff Exchange Scheme (IRSES) between Unversty of Naples II (Italy) and Crete Unversty (Greece). He accepts usually one or two PhD students and Post-Doctor Fellows for the study of nonlnear partal dfferental equatons. He s collaboratng wth M. Chaplan n Dundee Unversty (Scotland) and A. Stevens n Müster Unversty (Germany) for modelng and analyss for nonlnear systems concernng tumor growth events. He s also communcatng wth physcs group n Toulouse (France), partcularly wth P.H.-Chavans and C. Sre for the study of knetc mean feld theores arsng n several areas. Avantages de la collaboraton pour le laboratore franças L équpe de Bordeaux a une bonne expertse en smulaton numérque et assmlaton de données des modèles de crossance tumorale sur des applcatons clnques. En partculer, des lens ont été ms en place avec l Insttut Bergoné à Bordeaux ans que l Insttut Gustave Roussy à Vllejuf. Ces collaboratons nous donnent accès à beaucoup de données médcales qu nous amènent à développer contnuellement nos modèles afn de pouvor tenr compte des dfférentes pathologes et thérapes dsponbles. C est le cas pour les métastases au foe, au poumon ans que pour les tratements ant-angogénques et les nhbteurs de tyrosne knase ans que pour les tratements locaux comme l électrochmothérape. Cependant, l aspect théorque de l étude des systèmes a été ms de côté (devant l ampleur de la tâche!). cette collaboraton va nous permettre d nter, avec une vson nouvelle, toute une branche de notre recherche. En partculer, l dée est de mettre en évdence les proprétés prncpales sur les systèmes afn de trer parte au meux des données dont nous dsposons. Il ya tradtonnellement beaucoup de len entre les écoles françase et japonase en EDP. Nous pensons que cette collaboraton est plus orgnale. 7 - Présentaton des équpes Composton des équpes - (*): personne partcpante au projet. France : Japon : M. Coln Therry*, Professeur M. Lefebvre Gullaume*, Etudant master M. Leguebe Mchael*, Doctorant M. Pognard Clar*, Chargé de recherches M. Saut Olver*, Chargé de recherches M. Rouzmamat Muhamet*, Doctorant M. Sato Makoto*, Doctorant M. Suzuk Takash*, Professor DOSSIER N Page 4

5 M. Takahash Ryo*, Assstant Professor Equpement dsponbles pour la réalsaton du projet France : Japon : A cluster for scentfc computng. A cluster for scentfc computng. Publcatons sgnfcatves en rapport avec le projet France : B. Rbba, Th. Coln, S. Schnell, A multscale mathematcal model of cancer growth and radotherapy effcacy: The role of cell cycle regulaton n response to rradaton, Theoretcal Bology and Medcal Modellng 2006, 3:7 (10 Feb 2006). B. Rbba, O. Saut, T. Coln, D. Bresch, E. Grener, J.P. Bossel, A multscale mathematcal model of avascular tumor growth to nvestgate the therapeutc beneft of ant-nvasve agents, Journal of Theoretcal Bology 243 (2006) F. Blly, B. Rbba, O. Saut, H. Morre-Troulhet, Th. Coln, D. Bresch, J.-P. Bossel, E. Grener, J.-P. Flandros, A pharmacologcally-based multscale mathematcal model of angogeness, and ts use n analysng the effcacy of a new ant-cancer treatment strategy. Journal of Theoretcal Bology, vol. 260, Issue 4, 21 October 2009, Pages D. Bresch, T. Coln, E. Grener, B. Rbba, O. Saut, Computatonal modelng of sold tumor growth: the avascular stage, SIAM J. SCI. COMPUT. Vol. 32, No. 4, pp , M. Duruflé, V. Péron and C. Pognard. "Tme-harmonc Maxwell equatons n bologcal cells - The dfferental form formalsm to treat the thn layer" Publshed n Confluentes Mathematc. Vol. 3, Issue 2, Japon : K. Ichkawa, M. Rouzmamat, T. Suzuk, Reacton dffuson equaton wth non-local term arses as a mean feld lmt of the master equaton, Dscrete and Contnuous Dynamcal Systems S 5-1, Specal Issue (2011) K. Ichkawa, T. Suzuk, T. Murata, Stochastc smulaton of bologcal reactons and ts applcatons for studyng actn polymerzaton, Physcal Bol. 7 (2010) (do: / /7/4/046010) T. Suzuk, R. Takahash, Global n tme soluton to a class of tumor growth systems, Adv. Math. Sc. Appl. 19 (2009) A. Kubo, T. Suzuk, Mathematcal models of tumor angogeness, J. Comp. Appl. Math. 204 (2007) M. Kurokba, T. Suzuk On a perturbed system of chemotaxs, Internatonal J. Mathematcal Analyss 1 (2007) Appus demandés et/ou obtenus pour ce projet, en dehors de ce PHC France : Un projet ANR sur l'assmlaton de données sera déposé en Japon : JST, CREST (Allance for Breakngthrough Between Mathematcs and Scences) DOSSIER N Page 5

6 8 - Perspectves de la coopératon Rappel du contexte de la coopératon et des relatons exstantes Modelng of tumor growth s a feld of hgh nterest at the nternatonal level both from the mathematcal and clncal pont of vew. The am of ths project s to share the know-how of each team. As sad before, there s a strong hstory of relatonshp n PDE between France and Japan. In ths context Th. Coln was nvted n Osaka Unversty, Tokyo unversty of scence, Satama unversty and Sapporo unversty several tmes. The goal of these vsts was the study of dspersve PDE. Durng the last vst, Th. Coln and T. Suzuk met n Osaka and dscussed about possble collaboraton. They also met at the last ICIAM meetng n Vancouver n a mnsymposum organzed by Th. Coln and O. Saut. The dea s now to make ths collaboraton concrete thanks to ths project that nvolve young researchers. The subject of the collaboraton s new. Formaton par la recherche On the French sde, Gullaume Lefebvre (master student) s nvolved n the project. He made an nternshp (at the end of the frst year of master) on ths subject n ths Bordeaux team. He plans to start a Phd on ths subject n the team n September He wll vst Osaka n Hs research subject s macroscopc models. Mchael Leguebe wll start hs Phd under the supervson of Clar Pognard and Th. Coln n September 2011 on local therapes. He wll vst Osaka n Prof. Suzuk wll gve some lecture n the graduate school of mathematcs durng hs vst n France. Muhamet Rouzmamat s a PhD student n Osaka Unversty. He s gettng the degree by March, He vsts Bordeaux n 2012 to promote hs study. Takash Sato s a Post-Doctoral Fellow n Osaka Unversty. He vsts Bordeaux n 2012 to dscuss on several models used n cell bology. Nuanprasert Somcha s a graduate student n master course. Hs PhD course starts at Aprl, He vsts Bordeaux n 2013 to learn new therapy usng mathematcal models. Professor Th. Coln gves some lecture n the dvson of mathematcal scence n Osaka Unversty durng hs vst n Japan. Résultats attendus du projet As descrbed before, we frst expect a better knowledge of the mathematcal propertes of the PDE systems that are used for mathematcal modelng of tumor growth. We can recall here that mathematcal model can be used n two drectons. The frst one s for the understandng of complex bologcal mechansms. To ths am, we need some qualtatve results and n ths drecton, t s very mportant to understand the propertes of systems that are used (global exstence? Boundedness of the solutons? Perodcty? Bfurcaton theory?) The second applcaton concerns help to dagnoss for clncal needs. The systems n ths case have to be smplfed n order to be parametrzed for a partcular patent. Therefore, some nverse problems have to be solved startng manly from medcal magng. Agan, a good knowledge of the mathematcal propertes of the systems s crucal n ths drecton. Therefore, the objectves of the project are twofolds: 1) Obtan mathematcal propertes of the PDE systems that are used for tumor growth. 2) Apply these propertes to the applcatons that are consdered n both teams. Perspectves européennes Angela Stevens n Münster Unversty proposed a compettve system of chemotaxs to understand cell type sortng n moundfrutng body formaton of cellular slme molds. Wth E.E. Espejo and J.J.L. Velazquez, she showed component-wse smultaneous blowup and a formal mass separaton for radally symmetrc soluton. Collaboratng wth Takash Suzuk they showed non-radally symmetrc smultaneous blowup and rgorous mass separaton for radally symmetrc solutons. Takash Suzuk modeled a system of partal dfferental equatons to descrbe the formaton of nvadopoda whch appear n the early stage of tumor cell nvason under the collaboraton wth Mark Chaplan n Dundee Unversty. Tona Rccard n Unversty Naples II s workng wth Takash Suzuk concernng a concentraton behavor of the soluton to the mean feld equaton arsng n Onsager s turbulence theory. Autres perspectves nternatonales The Bordeaux team s n contact wth P. Cumslle from the Unversty del Bo-Bo n Chl on the same subject. P. Cumslle s nvted n Bordeaux for one month next fall. A collaboraton has also started wth H. Fathallah (neuro-oncologst at the unversty hosptal of Brmngham, Alabama, USA) on modelng of bran tumor. P. Cumslle and H. Fathallah were also at the mnsymposum at ICIAM n Vancouver. The Osaka team has collaboraton wth the Insttute of Medcal Scence, The Unversty of Tokyo The dea s to renforce ths network of collaboraton by sharng the nternatonal contacts of both teams. Perspectves ndustrelles actuelles ou attendues Many ndustral applcatons are possble on a long tme scale. The development of some softwares that allow the predcton of DOSSIER N Page 6

7 tumor growth for a partcular patent s very mportant. For the oncologsts the development of such tools could be of nterest n the treatment plannng and n the evaluaton of an ant-tumoral treatment. For example wth slowly evolvng tumors, a predcton of growth could renforce the decson of watng wthout specfc treatment or on the contrary to help n the decson of startng a chemotherapy. The predcton could help also n the evaluaton of the response to chemotherapy, to better predct the tme of neffcacy wthout watng an evdent tumor growth and n ths case to be able to adapt the treatment for a maxmal response. Such a software could be coupled wth Imagng techncs and could thus nterest the frm that are buldng CT-Scans, MRI or TEP- Scans. DOSSIER N Page 7

8 9 - Moyens demandés en 2012 France : Nom des chercheurs Nombre de voyages Durée totale en jours M. Pognard Clar 1 15 M. Leguebe Mchael 1 15 Japon : Nom des chercheurs Nombre de voyages Durée totale en jours M. Rouzmamat Muhamet 0 0 M. Sato Makoto 0 0 M. Suzuk Takash 0 0 M. Takahash Ryo 0 0 DOSSIER N Page 8

9 10 - Moyens demandés en 2013 France : Nom des chercheurs Nombre de voyages Durée totale en jours M. Coln Therry 0 15 M. Lefebvre Gullaume 1 15 M. Saut Olver 1 0 Japon : Nom des chercheurs Nombre de voyages Durée totale en jours M. Rouzmamat Muhamet 0 0 M. Sato Makoto 0 0 M. Suzuk Takash 0 0 M. Takahash Ryo 0 0 DOSSIER N Page 9

10 Research program between the departments of mathematcs of the unverstes of Bordeaux and Osaka on mathematcal modelng of tumor growth. 1 Scentfc context The am of ths project s to propose a comprehensve study of the modelng of tumor growth, ncludng mcroscopc (cell level) and macroscopc (tssues and organ level) elements and to apply these modelng tools to therapeutc nnovaton n oncology. The long-term goal s to mprove drug delvery protocols as well as optmzed treatment plannng for clncal tral. Ths crucal am requres as a frst step the mathematcal analyss and control of complex models of tumor growth. Tumor growth models have been developed snce a few years by varous teams (Prezos, Bellomo, Chaplan, see e.g. [1], [4], [2], [3], [5], [6], [7]). These models focus on partcular aspects of cancer: Cell cycle modellng: recent work by Perthame and co-authors on the underlyng knetc equatons, related entropes, and ther control. Immune reacton: modelled by knetc equatons followng Bellomo s works. Tumour growth: usng mechancal models and Darcy law or Stokes equatons, or even more refned elastc models. Angogeness: neovascularsaton of the tumour, whch plays a crucal role for large tumour (see Chaplan and co-authors). Radotherapy. Such models descrbe very accurate behavor n qualtatve descrpton of tumor growth, however, due to ther complexty, straghtforward clncal applcatons of such a modellng seem unreachable. Data assmlaton s clearly a key pont for clncal applcatons. Ths task s almost mpossble for the complex models that we have quoted above and smplfed models that present a good equlbrum between complexty and accuracy have to be consdered. The Bordeaux team has started a whole program concernng data assmlaton for lung metastases of dstant tumor wth smplfed 1

11 models. Such methods are descrbed n the attached fle named dataassmlaton.pdf. As explaned n ths paper, a lot of techncal problems have to be overcome. However, the understandng of the mathematcal propertes of the system that are used for data assmlaton s a crucal step: the qualty of the predctons s determned by the mathematcal propertes of the model. The effcacy of a cancer treatment s detected at the macroscopc level (tumor shrnkng measurement). But the delvered drugs for systemc treatments, as well as the local treatments such as radotherapy or thermotherapy occur at the cell level. Therefore accurate modellng of the treatment as the cell level can help n understandng the phenomena that are detected at the tssue scale. Through a 2-years colloboraton wth the Insttut Gustave Roussy, the Bordeaux team has developed a knowledge n the modellng of the electrochemothearpy, a local treatment that take advantage of the drug delvery wth poratng the cancer cells membrane by hgh voltage pulses, ths s descrbed n the attached fle localtreatment.pdf. Osaka group has been engaged n top down and bottom up modelng at cell level. They have been mostly concerned wth chemcal reacton, dffuson, and chemotaxs. Therefore, t s a real challenge to clarfy the role of other factors pcked up n the model by French group usng mathematcal analyss. For all these reasons, one needs to understand the mathematcal propertes of the PDE systems that are used and ths s the goal of the proposal. Durng ths collaboraton, we wll focus on the mathematcal propertes of the PDE model at both cellular and macroscopc level. 2 Macroscopc level The goal s to study a class of PDE systems descrbng tumor growth wth or wthout treatment: An example of ths system s: t P + (vp ) = γp + γ P + (γ 1 + γ )Q δ 0 f(t)mp, t Q + (vq) = γ P (γ 1 + γ )Q, 1 + tanh(m M T hre ) γ = γ 0, 2 γ tanh(m M T hre 1) = γ 1, 2 2

12 v = γp δ 0 f(t)mp, t + (vm) = (α Q βm(p + Q))(3M T hre M). In ths model, P (t, X) denotes the densty of prolferatve cells, Q that of quescent cells. The quantty v represents the global velocty of the growth of the tumor. Ths velocty s a consequence of the ncrease of volume due to tumor growth. M denotes bascally the densty of vascularzaton of the tumor. Ths system s therefore a very compact descrpton of angogeness. Coeffcent α, β and δ 0 are useful to descrbe the acton of therapes lke chemotherapes, ant-angogenc drugs or tyrosn-knase nhbtors. The system s not closed, one uses a Darcy s law: v = k π. Probably, we wll have to modfy the system dependng on the applcatons. The natural questons that arse are exstence, controllablty of solutons of ths system. 3 Mcroscopc level: electrochemotherapy modellng Electroporaton s a mcroscopc phenomenon that conssts n mposng hgh electrc short pulses on the cell n order to weaken the structure of the plasma membrane. Bologcally actve molecules that otherwse cannot dffuse through the membrane (e.g. hydrophlc compounds such as bleomycn or DNA) may then spread through the cell membrane: ths s the electrochemotherapy (ECT). If the pulses are short enough the process s reversble: the membrane s not destroyed and reseals wthn mnutes. The cell therefore can nternalze external actve molecules wthout losng ts vablty (Weaver and Chzmadzhev [8], Weaver [9], Chen et al. [10]). Such nternalzaton mechansm s a complex mcroscopc phenomenon hdden behnd the term reversble electroporaton (or electropermeablzaton). The Bordeaux team s currently dervng membrane laws that descrbe the non-lnearty of the electrcal membrane propertes. The membrane capacty and the membrane surface conductvty s governed by tme-dependent PDEs 3

13 nvolvng parameters, whch descrbe the tmes for the membrane permeablzaton and resealng and the proporton of the proporton of the permealzed regon. These laws lead to PDE model descrbng the cell electrc potental. Ths leads to the transmembrane potental (the jump of the potental across the membrane), whch s lnked to the cell volume and the on fluxes accordng to [11]. At the end of the cell modelng, we therefore expect to provde a dynamc electrc model that descrbes the membrane electroporaton, the on fluxes as well as the quantty of drugs that enter the cell durng the pulses. Despte the gap between the mcroscopc scale and the tssue level, whch prevents the straghforward couplng between the cell and the macroscopc models, we are confdent that these mcroscopc PDE models wll help n understandng and n dervng new tumor growth models coupled wth local treatments. In partcular we emphasze that the mathematcal analyss of mcroscopc models wll hghlght the specfcty of the model parameters, whch s essental to determne the preponderant propertes of the system, that have to be kept at the macroscopc scale. We also notce that once the ECT s well understood, the modellng of other local treatments such as thermotherapy of radotherapy seems qute easly reachable wth a satsfyng accuracy by adaptng the ECT models. 4 Presentaton of the Bordeaux team 4.1 Therry Coln coln Age : 43 ans Poston : Professor (PR-ex), Insttut polytechnque de Bordeaux. Phd: Ecole Normale Supreure de Cachan 1993, HdR Unverst Bordeaux 1, Research nterests: Partal dfferental equatons Modellng and scentfc computng Applcaton to flud mechancs and bology. Publcatons : 4

14 80 publcatons n nternatonal journals (SIAM, M2AN, PRB, Physca D, JTB, JCP, PRL) 20 nvted conferences 50 Semnars (Unversty of Illnos at Chcago, Seoul, Tokyo, Chle,..) Dstnctons : Junor member of Insttut Unverstare de France ( ). Students: 14 Phd students. Responsabltes: Responsable of the graduate school for mathematcs and computer scence Bordeaux (03-05). Responsable of the INRIA team MC2. (04-now) Member of the CNRS commttee for mathemtcs (04-08). Adjont-drector of the Mathematcs nsttute of Bordeaux (140 permanent researcher). 4.2 Clar Pognard pclar Age : 31 years-old Current Poston: Researcher (CR 1re classe), INRIA. Doctorate: Unversté de Lyon, November Vstng Student : Rutgers Unversty, NJ (USA) Postdoctorate : Ecole Polytechnque X/Ecole Supéreure Physque Chme Industrelle, Research nterests: Partal dfferental equatons (asymptotc analyss) Modelzaton and numercal analyss. Applcatons to electromagnetc felds and cellular bology. Publcatons : 14 publcatons n nternatonal journals (IEEE, Eur.Bophys J, EJAM, M2AS, Appl. Anal.) 5

15 1 nvted summer school (Insttut Paul, Venne (Austra)), about 10 semnars (Rutgers Unversty (USA), Montefore Insttute (Belgum)) 4.3 Olver Saut saut Age : 34 years-old Current Poston: Researcher (CR 1), CNRS. Doctorate: INSA Toulouse Research nterests: Partal dfferental equatons Mathematcal modelng and scentfc computng. Applcatons to flud mechancs, bology, medcne and nonlnear optcs. Publcatons: 15 publcatons n nternatonal journals (JCP, M2AN, JTB, SISC, Eur.Bophys J). 5 nternatonal conferences as an nvted speaker ; 15 semnars. Responsabltes: Head of the PDE Scentfc Computng semnar of the Insttut de Mathématques de Bordeaux ( ). Regonal coordnator (French Socety for Appled and Industral Mathematcs). 6

16 References [1] D. Ambros and L. Prezos, On the closure of mass balance models for tumor growth, Math. Models Methods Appl. Sc., 12 (2002), pp [2] Chaplan, M.A.J., Grazano, L. Prezos, L. (2006), Mathematcal modellng of the loss of tssue compresson responsveness and ts role n sold tumour development, Math. Med. Bol. 23, [3] Chaplan, M.A.J., McDougall, S.R., Anderson, A.R.A. (2006) Mathematcal modellng of tumor-nduced angogeness, Annu. Rev. Bomed. Eng. 8, [4] Lu, W. Hllen, T. and Freedman, H.I., A Mathematcal Model for M-phase Specfc Chemotherapy Includng the G0-Phase and Immuno- Response, Mathematcal Boscences and Eng., 4(2), , [5] B. Rbba, Th. Coln, S. Schnell, A multscale mathematcal model of cancer growth and radotherapy effcacy: The role of cell cycle regulaton n response to rradaton, Theoretcal Bology and Medcal Modellng 2006, 3:7 (10 Feb 2006). [6] B. Rbba, O. Saut, T. Coln, D. Bresch, E. Grener, J.P. Bossel, A multscale mathematcal model of avascular tumor growth to nvestgate the therapeutc beneft of ant-nvasve agents, Journal of Theoretcal Bology 243 (2006) [7] F. Blly, B. Rbba, O. Saut, H. Morre-Troulhet, Th. Coln, D. Bresch, J.-P. Bossel, E. Grener, J.-P. Flandros, A pharmacologcally-based multscale mathematcal model of angogeness, and ts use n analysng the effcacy of a new ant-cancer treatment strategy. Journal of Theoretcal Bology, vol. 260, Issue 4, 21 October 2009, Pages [8] Weaver, J.C and Chzmadzhev, Y.A. TBoelectrochemstry and Boenergetcs. (41), , [9] Weaver, J.C. Electroporaton of cells and tssues. IEEE Trans. on Plasma Sc. (28), 1,

17 [10] Chen, C; Smoye, S.W; Robnson, M.P; and Evans, A. Membrane electroporaton theores: a revew. Bol Eng Comput. (44) [11] Pognard, C; Slve, A; Campon, F; Mr, L.M; Saut, O and Schwartz, L. Ion flux, transmembrane potental, and osmotc stablzaton: A new electrophysologcal dynamc model for Eukaryotc cells. Eur. Bophys. Journal. Vol. 40, Issue

18 Some models for the predcton of tumor growth: general framework and applcatons to metastases n the lung Therry Coln, Angelo Iollo, Damano Lombard, Olver Saut, Françose Bonchon, and Jean Palussère 1 Helpng clncal practce wth mathematcal models Mathematcal models of cancer have been extensvely developed wth the am of understandng and predctng tumor growth and the effects of therapes [9]. They may help understandng the nfluence of genetc regulaton [4] or to predct a phenomena that wll later be confrmed by experments as n [19] for the gap between the tumoral front and host tssue. However, n most cases, these models are better adapted to study n-vtro tumors than n-vvo ones. Indeed, contrary to the n-vtro cases, the amount of nformaton avalable on a tumor n-vvo s really lmted. Performng mcroscopc measurements s almost mpossble (asde from bopses that study a lmted number of cells) and cannot be repeated. Hence modelng mathematcally the smallest scales s probably out of reach n an n-vvo context. On the other hand, the evoluton of cancer s montored by clncans mostly thanks to magng devces. Ther assessment of the dsease s thus restrcted by the lmtatons of these devces. In ths work, one example of mprovng the nsght provded wth medcal mages thanks to a mathematcal model wll be presented. In partcular, throughout ths paper, we are nterested n one clncal test case: evaluatng the aggressveness of some metastases n the lung (especally lung metastases from thyrod cancer). Refractory thyrod carcnomas are a therapeutc challenge owng to some beng fast-evolvng whlst others evolve slowly. Ths varaton makes t dffcult to decde when to treat: aggressve nodules are good canddates for trals wth molecular targeted therapes [41] whle the others may be left untreated (but carefully montored) for years or treated by surgery or mnmally nvasve tech- Therry Coln, Angelo Iollo, Damano Lombard, Olver Saut Insttut de Mathématques de Bordeaux UMR 5251, Unversté de Bordeaux and INRIA Bordeaux Sud-Ouest, équpe projet MC2, 351 cours de la Lbératon, Talence, France. e-mal: {coln,ollo,lombard,saut}@math.u-bordeaux1.fr, Franços Bonchon, Jean Palussère Insttut Bergoné, 229 Cours Argonne Bordeaux, France e-mal: {Bonchon, Palussere}@bergone.org 1

19 2 Authors Suppressed Due to Excessve Length ncs lke radofrequency. In ths chapter, we descrbe a mathematcal tool that could help clncans evaluate the aggressveness of the nodules consdered. For clncal applcatons, qualtatve answers are not acceptable: quanttatve results are to be obtaned. From ths work, t s possble to obtan a more accurate estmate of the evoluton of a targeted nodule usng only non-nvasve technques. In the followng, we descrbe the constructon of the tool and the varous challenges we had to tackle. The outlne s as follows. We start by descrbng the typcal clncal case that M.D. are facng: evaluatng the aggressveness of several lung metastases from thyrod carcnoma. The dffcultes to overcome are then lsted. In Secton 2, we present dfferent approaches to model tumoral growth and the model we have fnally settled on for the applcaton. Ths model has parameters that are to be recovered to perform a prognoss: ths s detaled n Secton 3. We hghlght two technques that were successfully used to fnd parameter values, allowng us to reproduce the growth of the nodule as observed on a sequence of CT scans. Fnally, addtonal test cases are dscussed. 1.1 A clncal test case Here s a typcal case of a patent wth lung metastases from thyrod carcnoma. In 2005 the patent was 74 and had many comorbdtes : hstory of hypertenson, renal falure and concomtant prostatc cancer. Durng the work-up of the prostatc cancer many blateral lung nodules were dscovered on a chest x-ray procedure. Because lung metastases from prostatc cancer are very seldom, a transthoracc bopsy was performed and the pathology record concluded to lung metastass from thyrod cancer. Total thyrodectomy was done and ths confrmed there was a poorly dfferentated carcnoma wth nsular and oncocytc components. Subsequent radodne treatment was gven but there was only a fant odne uptake on some nodules but not on all. Ths cancer was therefore consdered refractory but the patent could not be ncluded n a ant-tyrosne knase tral because of the concomtant prostatc cancer. A 18F-FDG Tep/CT was performed and we decded to treat wth radofrequency 2 nodules n the left lung whch had a hgh 18F-18F-FDG uptake synonym of aggressve tumor as shown by [38]. The other nodules were not treated and the patent was followed up only by clncal examnaton, bologcal tests (thyroglobuln) and not contrast enhanced CT scan. The treated nodules are the bass of ths work. By the end of aprl 2011 the patent s stll alve. To evaluate ths aggressveness, clncans typcally have sequences of medcal mages. In ths case, an extract of such a CT scans tme sequence of the patent s shown n Fg. 1. Several nodules are vsble and we are focusng on the one marked n red.

20 Predcton of tumor growth and applcaton to lung metastases 3 Fg. 1 Extract from a tme sequence of CT scans showng the evoluton of one nodule marked n red between 2005/11/15 and 2009/04/09. The dates are from the exams shown are from top left to bottom rght: 2005/11/15, 2007/07/06, 2008/07/09 and 2009/10/19. The queston s then the followng: are we able to evaluate the growth of the red nodule usng some mages of ths sequence? 1.2 Challenges to overcome Cancer growth s a very complex process and one cannot serously pretend to descrbe all ts mechansms. For nstance [23] and the numerous molecular pathways t descrbes clearly shows that there s no hope to descrbe the oncogeness n ts whole complexty. Yet t does not mean that mathematcal modelng s useless for provdng a new nsght n oncology. The role of mathematcal modelng n the resoluton of engneerng and physcs problems s undsputed. In ths paper, we hope to show that t could also helps n understandng and treatng ths dsease. Because of ths complexty, the mathematcal models developed are mostly phenomenologcal but should produce behavors that are n good agreement wth the current bologcal and medcal knowledge. Varous scales are nvolved n the tumor growth process rangng from genetc regulatons and molecular pathways, cellular adheson, metastases to systemc therapes at the scale of the whole body. Choces would have to be made to keep the model as smple as possble yet producng nterestng outputs from a clncal pont of vew. Ths s essental partcularly f we wsh to obtan calbrated models adapted for a patent. Indeed, the task of fndng reasonable values for the parameters of the mathematcal model s another bg challenge that wll have to be addressed. For clear ethcal reasons, we do not want to use any knd of nvasve technque (asde from the one already performed n the routne clncal practce) on the patent

21 4 Authors Suppressed Due to Excessve Length to help calbratng the models. The amount of nformaton on the tumoral evoluton s then really scarce. We can barely rely on a few medcal mages. However, the quantty of nformaton that we can recover from medcal magng devces s reasonable as t, at least, shows the locaton and shape of the tumor. Currently, we are studyng only secondary tumors and n partcular lung metastases. The shapes of such tumors s smoother than prmary tumors and are probably easer to reproduce wth mathematcal models snce they show less nfltratve nature contrary to prmary tumors. One of the man advantage of workng n the lung, s also that the correspondng CT scans are hghly contrasted. It greatly smplfes the segmentaton process. However, we shall note that we consder that segmentng the nodule s a medcal act that s operator dependent. It may vary between dfferent clncans but the outcome of our model should not vary much. The desgn of a robust and relable method s mandatory for a clncal use. 2 Modelng tumor growth To evaluate the aggressveness of a gven nodule, a model has frst to be wrtten, computng the evoluton of a meanngful ndcator that could help clncans decde whether the nodule has to be treated or not. In order to descrbe tumor growth, one could descrbe mathematcally several phenomena among whch how and when are the cancer cells are dvdng? how are they movng? how to descrbe the mutatons they undergo? how to descrbe the nfluence of ther mcroenvronment? A mathematcal model has to take some of these phenomena nto account to be accurate or t has to recover some nsght on the expected behavor of the tumor through statstcal methods. More precsely, we wrte a mathematcal model descrbng tumoral growth. Ths very complex phenomenon s translated (and smplfed) nto equatons. These equatons descrbe the evoluton of quanttes relevant to descrbe the tumoral growth (cellular denstes, nutrents,...). The equatons are coupled and ther nterplay s tuned through varous parameters. Once the model s wrtten, the parameters are stll unknown and for a clncal applcaton, one has to fnd a way to determne ther values. Typcally, we wll have access to medcal mages showng the evoluton of the tumor (and we do not want to neglect any bt of nformaton gven by these mages).

22 Predcton of tumor growth and applcaton to lung metastases Statstcal methods We want to evaluate the aggressveness of dfferent lung nodules of the same patent. They have smlar szes and dffer by ther locatons n the lung. The number of smlar cases s small. For classcal statstcal methods, the dfferent nodules should behave more or less the same way: we have an average answer. Ths s far from beng suffcent or satsfactory for our problem. We want to have a predcton adapted the patent, our mathematcal model has to be calbrated for a partcular nodule of the patent consdered. Statstcal methods cannot really help n our case. Yet, statstcal approaches wll be useful to evaluate the accuracy and robustness of our method over a large number of cases. Furthermore, a statstcal approach s helpful to evaluate the behavor of the models when data nose s taken nto account. 2.2 Descrbng the evoluton of the tumoral volume Nowadays most mathematcal models used for clncal purposes are based on a set of ordnary dfferental equatons (ODE) [43, 40]. Typcally, they are descrbng the evoluton of the tumoral volume. The spatal dmenson of the dsease s therefore not taken nto account, the shape or localzaton of the tumor are not computed by these models. Yet, t has not prevented ther successful use for desgnng clncal protocols. Ther parameters are classcally found usng statstcal methods. Ths statstcal parametrzaton s one of ther shortcomngs that we want to overcome. We want a prognoss for a specfc nodule of one patent, and the statstcal answer would probably have a large error margn and would probably not be adapted for one partcular case whle the parameters are recovered usng a large number of patents. Ther second shortcomng s that by neglectng the spatal dmenson, many aspects of the dsease are mssed. Bascally, n our cases, t would mean to neglect the spatal dstrbuton of nutrents, dfferent tssues behavors (whte, grey matter n the bran... ), the composton and structure of the tumor... Let us also note that the effcacy of cytostatc drugs cannot be evaluated just by lookng at the tumoral volume. Furthermore, gven enough parameters and data one could probably always fnd parameters values to ft a sequence of tumoral volumes. Let us now present such a model computng the evoluton of the tumoral volume. Ths model was obtaned by smplfyng the spacal model [37]. Ths model ams at computng the tumoral volume denoted by V. We tred to take more bologcal phenomena nto account than popular models lke [43]. The cellular dvson occurs f only there s enough nutrents. The varable P denotes the total number of prolferatng cells. The nutrent or growth factor s denoted by C and controls the mtoss through the functon γ. Ths functon s a smoothed Heavsde operator equal to 1 f the nutrent concentraton s above an hypoxa threshold C hyp and 0 otherwse. These quanttes evolve by

23 6 Authors Suppressed Due to Excessve Length Fg. 2 Dfferent prognoses on the case of Sec. 1.1 obtaned from the ODE model presented n Eq. (1) whle usng dfferent number of data ponts as nput for the parameter recovery technque. Plan lnes represent the varous evolutons of the volume predcted by the model after recoverng the parameters usng the correspondng number of measurements. Volume Scans Volume 2 Images Pred. 3 Images Pred. 4 Images Pred. 5 Images Pred months dv dt = γpv, dp dt = (2γ 1)P γp 2, dc dt = (1 C)( V V 0 ) 2 3 αp, γ = 1+tanh(10(C C hyp)) 2. The equaton on the volume V descrbe an exponental growth f there s enough oxygen. In the second equaton on P, the term (2γ 1)P γp 2 can be decomposed nto a logstc term for cellular dvson γp(1 P) and one for cellular death (1 γ)p. In the equaton drvng the nutrent quantty C, one can fnd a producton term (1 C)( V V 0 ) 2 3 proportonal to the area of the tumor (hence the 2 3 power) dvded by a characterstc term V 0 and the uptake by the prolferatve cancer cells αp. We also assume that n healthy tssue, the concentraton of nutrent s equal to 1. To use ths model for clncal applcatons, t has to be calbrated on the targeted nodule: reasonable parameter values are to be recovered that allow the model to ft the observed tumoral volumes. The model can then be used for prognoss by lettng t run a lttle further. In our case, the parameters to evaluate would be P 0 = P(t = 0), C hyp, V 0, α, C 0 = C(t = 0). An example a such a recovery of parameters from realstc data s now presented: the ntal test case detaled n Sec We have a sequence of 5 tumoral volumes measured on a patent. We try to evaluate the parameters usng 2, 3 and 4 measurements. The technque used s based on solvng senstvty equatons (the dervatves of the observable w.r.t. parameters s computed through a set of ODE). The spatal verson of ths technque s presented n Sec To check the accuracy of the prognoss, the evoluton predcted by each run s compared to the later measurements. Ths has to be compared wth the prognoss obtaned thanks to the spatal model (1)

24 Predcton of tumor growth and applcaton to lung metastases 7 Fg. 3 Dfferent prognoses obtaned from the ODE model presented n Eq. (1) (and a comparson wth the one of [43]) whle usng dfferent number of data ponts as nput for the parameter recovery technque. The tumoral volumes observed on scans are shown as small crcles. Plan lnes represent the varous evolutons of the volume predcted by the model after recoverng the parameters usng the correspondng number of measurements. Volume Scans Volume 2 Images Pred. 3 Images Pred. 4 Images Pred. 5 Images Pred. 2 Images Pred. [43] 3 Images Pred. [43] 4 Images Pred. [43] Tme [days] (wth two measurements) n Fg. 5. To further prove our pont, we present on Fg. 3 a new case where the predcted behavor changes wth the number of data ponts. As shown n Fg. 3, ths model s not vald when used wth less than 3 measurements. It cannot be serously used for our clncal applcatons and as a tool helpng clncans wth ther decson. Furthermore, even f ths model were always fttng the observed volumes, the nsght t would offer on the tumor s rather lmted to the tumoral volume. Other ODE models lke [43, 22] gve the same knd of outcomes n our case. As a comparson the tumoral evoluton computed usng the model of [43] s shown n Fg. 3. Wth 4 measures, our algorthm s not able to compute accurately the parameters (the best soluton found s plotted). The model [43] has less parameters (3) than the one presented n ths paper so t has less degrees of freedom to ft the observed volumes. We have to take more nformaton from the avalable mages.e. we am at usng the the spatal nformaton. 2.3 Spatal descrpton of the tumor As shown n the prevous secton, a model that computes only the tumoral volume s probably not suffcent for our case. To take more nformaton nto account, we could try to descrbe the evoluton of the boundary between the cancer and host cells. One can fnd such models n [18, 30]. Some models are derved under the assumpton that the tumor has a crcular shape. That allows the varables defnng the tumor to have a unque dependency on one radal varable. Even f, as we sad, secondary tumors have smoother shapes than prmary ones, the assumpton s not relevant n our case as shown n the CT scans presented n ths paper. Yet ths approach yelds numerous nterestng mathematcal results n partcular on the asymptotcal behavors of tumors.

25 8 Authors Suppressed Due to Excessve Length The shape assumpton s lfted by models lke [30]. The boundary of the tumor s descrbed by a level set functon. Even f n ths context, useful numercal technques are developed, the evoluton of the tumoral shape s descrbed by mathematcal terms that are far from beng valdated from a bologcal pont of vew. The valdaton of these models s often purely graphcal. To mprove the accuracy of the prognoss, the composton of the tumor should be taken nto account Dscrete models In dscrete models, one descrbes the evoluton of each cell ndvdually. Although these models are very accurate to descrbe the mcroscopc scales (genetc regulatons... ), at the end of the avascular stage, a tumor contans mllons of cancer cells and one cannot realstcally pretend to compute the evoluton of every cell. There are bascally two types of dscrete models. In the frst knd (cellular automata), one consders that all cells are stuck on a fxed mesh [1]. After mtoss, new-born cells appear n the nearest free grd pont to the mother cell whch yelds an unrealstc movement. Yet ther smplcty make them the deal tool to study the nterplay between the varous sub-cellular scales nfluence tumoral growth. Agent-based models do not suffer from ths drawback [31]. Cells are freely movng. The nteracton between each couple of cells has to be computed whch make them very computatonally expensve. One can fnd fascnatng use of ths model n works lke [4] Macroscopc and contnuous models In macroscopc models, one typcally descrbes denstes of cells,.e. averages over a large number of cells. A deep comparson of these models wth dscrete models can be found n [8]. Classcally, usng the mass-balance prncple, the partal dfferental equaton (PDE) leadng the evoluton of any cellular densty N(t, x) can be wrtten as t N + J = brth death, (2) where J s the total flux of cells. One has to compute J.e. how the cells move. The source terms are gven by the bologcal model and may also depend on denstes of other speces. In models based on reacton-dffuson equatons such as [19, 42], ths flux s gven by J = D N, (3) where D(t,x) s a dffuson coeffcent. If D s constant, these models are easly justfed from a mcroscopc approach (random walk), see also [34]. The most famous reacton-dffuson model used n clncal applcatons s probably found n [46]. Yet, n ths model, only one type of cancer cell s consdered, no

26 Predcton of tumor growth and applcaton to lung metastases 9 effect of the mcroenvronment (nutrents) s consdered and host-tumor nteractons are lmted. Image-drven smulatons from ths model were nsprng to our work [13]. We beleve than by addng more bologcal nsght n our model, we wll mprove ts accuracy and predctons The balance of complexty and our fnal model We have to fnd the rght balance between complexty and ease to parametrze. The more accurate and complex the model s, the more bologcal nsght we could have on the tumor and dsease evoluton. The dffculty of calbratng the model to patent data s ncreasng greatly wth the number of parameters or unknowns. In the present case, can we use a mathematcal model takng tumor-host nteracton, nutrents, mechancal effects nto account and stll recover ts parameters? Frst let us note that there s no master equaton drvng cancer growth. We are wrtng phenomenologcal models. We use a bottom-up approach: startng wth very smple models, we mnmze the total number of parameters and only add complexty only when t s necessary for accuracy. We choose to consder contnuous models based on PDEs because they are not so computatonally expensve and can render a lot of bologcal phenomena and mechancal effects. The model we proposed s a smplfed Darcy-type model descrbng the evoluton of varous cellular denstes (prolferatve and quescent cancer cells, healthy tssue... ) as well as nutrents dstrbuton or mechancal varables usng partal dfferental equatons. We beleve that ths parametrc model s suffcently accurate to take the man physcal features of tumor growth nto account but smple enough to have ts parameters recovered. As n [37], the dynamcs of two dfferent speces of cancer cells s consdered. These denstes wll be denoted by P(t,x) and Q(t,x). The densty P represents the prolferatng cells (dvdng cells, responsble for tumor growth) and Q s the densty of necrotc cells that de because of lack of oxygen n the tssue. The total number of cancer cells s denoted by Y = P + Q. Ths quantty wll be observed on the medcal mages. We assume that the moton of cells s due to the ncrease of volume caused by the mtoss. Ths passve movement s descrbed by an advecton phenomenon at the velocty denoted by v(t,x). As n the prevous secton, the mass balance equatons for denstes P and Q are wrtten: P t + (vp) = (2γ 1)P, (4) Q + (vq) = (1 γ)p. (5) t where the functon γ s the hypoxa threshold, a scalar functon of the oxygen concentraton that s more precsely defned later on Eq. (13). If enough oxygen s avalable then γ = 1 and Eq. (4) descrbes the prolferaton of tumor cells and the quantty

27 10 Authors Suppressed Due to Excessve Length of necrotc cells s constant thanks to Eq. (5). If there s a lack of oxygen, then γ < 1 and some prolferatng cells de and enter the necrotc phase. From Eq. (4) and (5), an equaton on the observable Y can be nferred Y t + (vy ) = γp, (6) n whch the densty of prolferatve cells P s nvolved. The observable cannot be computed wthout ths dstncton between quescent and prolferatve cells. The densty of healthy cells s denoted by S. Ther dvson s neglected n ths work. The equaton for S reduces to an homogeneous transport equaton, as explaned n [37]: S + (vs) = 0. (7) t We use an hypothess of saturated flow (see [3, 37]) that s P+Q+S = 1 at every pont of the space doman and for every tme. Collectng Eqs. (4), (5) and (7) leads to an equaton for the dvergence of the velocty feld, namely: v = γp. (8) We observe that, from a physcal pont of vew, ths s equvalent to state that the mtoss acts as volume source for the flow. Clearly, ths condton on the dvergence s not suffcent to compute the velocty v. In order to close our system we have to make an addtonal assumpton on ths velocty. Several knds of closures have been proposed n the lterature, see [39, 3, 6]. We chose to use a Darcy-type law, that descrbes quas-steady flows n porous meda, wth a varable porosty: v = k(p,q) Π, (9) ths s almost the smplest closure we could take. The scalar functon Π plays the role of a pressure (or of the potental), and k s a porosty feld, that s a functon of P and Q. The most smple, phenomenologcal law s a lnear mappng of the sum (P + Q), so that we have: k = k 1 + (k 2 k 1 )(P + Q), (10) where k 1 represents the constant porosty of the healthy tssue and k 2 s the porosty of the tumor tssue. After defnng the mechancs of the system, we have to specfy the nutrent evoluton that n ths case reduces to a reacton-dffuson equaton for the oxygen concentraton. We make the assumpton of a quas-steady state: (D(P,Q) C) = α P PC + α S S(1 C), (11) where α P s the oxygen consumpton rate for the prolferatng cells, α S corresponds to the nutrent brought by the vascularzaton (assumed unformly dstrbuted n healthy tssue). We also assume that the concentraton of nutrent s equal to 1 n

28 Predcton of tumor growth and applcaton to lung metastases 11 healthy tssue and the producton term s lmted by a factor (1 C) n Eq. (11). We chose to wrte the dffusvty D(P,Q) can be wrtten as a lnear mappng of P + Q: D = D max K dff (P + Q). (12) Ths phenomenologcal law reflects the fact that the dffuson of oxygen s dfferent n the healthy or tumor tssues: dffuson of oxygen s weaker n the tumor. The hypoxa functon γ smply states that, when the concentraton of oxygen s under a certan threshold the cells become necrotc. The defnton of γ s a regularzaton of the unt step: γ = 1 + tanh(r(c C hyp)), (13) 2 where R s a smoothng coeffcent and C hyp s the hypoxa threshold. Accordng to the physcs of the system, reflectng dfferent clncal cases, Drchlet boundary condtons or Neumann boundary condtons can be mposed for both the oxygen and the pressure felds. Imposng Neumann condtons on the pressure feld s equvalent to mpose that there s no mass leavng our doman. In order to have a well posed problem the equaton for the dvergence of the velocty has to be modfed. In partcular the dvergence must be a zero average scalar quantty, so that we can wrte: Ω γp dω v = γ(c)p S. (14) 1 Y dω From a mechancal pont of vew ths s equvalent to mpose that the growth of the tumor causes a compresson of the healthy tssue. Therefore the healthy tssue equaton can be no longer consdered, n ths case, an homogeneous transport equaton and the second term of the rght-hand sde of Eq. (14) s added to Eq. (7). Ths model has several parameters that are summarzed n Tab. 1. The calbraton of ths model wll consst n fndng reasonable values for these parameters that allow to match wth the observatons. Ω Table 1 Parameters n the Darcy-type model used for metastass n the lung. Two parameters are fxed, the others wll be evaluated by our recovery technque. Name Type Descrpton k 1 Scalar Healthy tssue dffusvty (fxed = 1) k 2 Scalar Cancer cells dffusvty D max Scalar Nutrent dffusvty n healthy tssue (fxed = 2) K dff Scalar Tumoral coeffcent n the nutrent dffusvty α P Scalar Nutrent uptake value of cancer cells α S Scalar Oxygen producton C hyp Scalar Hypoxa threshold The ntal condtons are also unknown varables. In the followng, we have consdered that C(t = 0) = C 0 = 1 (the concentraton n healthy tssue). The ntal dstrbuton of cancer cells s recovered from the medcal mages. Prolferatve cells are

29 12 Authors Suppressed Due to Excessve Length ntalzed wth the tumor observed on the ntal mage (after the segmentaton step, the cellular densty of prolferatve cells s dentfed wth grey levels). Intally, we assume that there s no necrotc or quescent cells. Ths s a smplfyng assumpton and n further works the ntal tumor s ntalzed wth a layered structure (whch s parametrzed wth scalar values). Before proceedng wth calbraton, one has to ensure that the chosen model s able to reproduce the wder range of behavors observable n-vvo. The varous behavors should be trggered by the parameters. 3 Calbratng mathematcal models for applcatons Once the parameters are recovered (for a specfc patent and tumor), we have a model n good agreement wth medcal observatons. That has several advantages. Frst, one could study the evoluton of the tumor predcted by the model smply by lettng the model run a lttle further after the last exam. Ths could be a new dagnoss tool for doctors as the computed evoluton of the tumor should gve a reasonable nsght on the real tumor f the mathematcal model s accurate enough. In partcular, we beleve that the computed growth of a nodule would be a good ndcator of ts aggressveness. We could obtan more nformaton than the drect observables f t s computed by the model e.g. vascularzaton, dfferent phenotypes n a tumor wthout makng bopses... Once agan, ths requres an accurate model. We could try dfferent therapes on the tumor by ncorporatng ther effects n the model: ths s a way to desgn new therapeutcal protocol more adapted to the patent and hs dsease. Moreover the model dstngushes cancer cells (prolferatve or quescent cells). Ths could help evaluatng the effcacy of cytostatc drugs (e.g. through the volume of prolferatve cells). The varous parameters nvolved n the model have to be estmated. Some of these parameters are obtanable from expermental data. Unfortunately, most of them cannot be easly determned and do not even have a physcal meanng. It s not an dentfcaton n a statstcal sense. We do not want to fnd dstrbuton for general cases but rather look for parameters adapted for one partcular patent. In order to fnd parameter values, we need data. Avalable data comes, n general, from medcal mages. We can observe varables that are somehow contnuous n space but dscrete n tme. One way to determne the parameter values s by means of nverse problems, explotng data comng from medcal magery, as acheved for example n [25, 13]. The man dffculty s that the amount of data for system dentfcaton s scarce. Although medcal scans allow a qute accurate localzaton of the tumor n space, lttle nformaton can be nferred about ts cellular nature or nutrent dstrbuton. In addton, usually only two scans are avalable before treatment makng estmaton of tme evoluton a challengng problem. On the other hand retrevng the evoluton

30 Inverse problems strateges Predcton of tumor growth and applcaton to lung metastases 13 Fg. 4 The nverse problem uses the evoluton of the tumor as Introducton; shown by medcal magng devces to recover the parameters of an accurate Procedure; mathematcal model. Real case; ty(x; t) = f(y, P, π) Concluson; The model descrbes the evoluton: nonobservables and parameters to be determned! of the tumor shape provdes ndrect nformaton thanks to the fact that the models are spatally dstrbuted. There s a wde lterature of methods to solve nverse problems concernng dffuson and propagaton phenomena. In partcular two man classes of methods were developed: determnstc andim stochastc =Im(x; approaches. t) In the latter a random The process data: n general, medcal mages s consdered and the parameters as well as the varable felds of a gven model are nferred once ther statstcal propertes are gven, see for example [33]. Snce a determnstc approach s used E = to descrbe Im -Y(x; the dynamcs t) of the tumor We growth, want to wemnmze the error beween opt for a determnstc framework. In the cases we wll deal wth, one the ofsmulated the most hstory and measurements challengng problem wll be not only to dentfy the parameters, but also to fnd felds that are not observable and that n general make our problems greatly underdetermned. For example t s clncally meanngful to reconstruct the dstrbuton of the oxygen feldsystem n the tssues, dentfcaton or the dstrbuton n tumour of growth prolferatng modelng cell densty. Ths Houston, has a great nterest from a medcal pont of vew, snce t allows havng nformaton Monday, January 24, 2011 about quanttes that wll determne the tumor evoluton. As a matter of fact, n realstc stuatons the source of relevant data s medcal magery, so the observatons that can be retreved are ndrect, contnuous n space but dscrete n tme. One possble approach to formulate the nverse problem s by optmal control theory. A drect system and an adjont one have to be solved forward and backward n tme, respectvely. We have to fnd a set of parameters that mnmzes the dstance between the numercal smulatons and the medcal mages. The error s computed usng the mages and the values of the observables we compute at the same tmes. 3.1 Senstvty approach to nverse problems In ths secton a senstvty approach (see [47, 17]) s descrbed n order to solve nverse problems. In partcular, let us recall the model structure as seen n Eq. (6) for nstance: t Y = f (Y,H,π j ), j = 1,...,N p, (15) t H = g(y,h,π j ), j = 1,...,N p. Y s the measured quantty (observable), whose evoluton s descrbed n terms of f, whch s n general functon of the observable tself, other felds, called genercally

31 14 Authors Suppressed Due to Excessve Length H, that cannot be drectly measured (whose evoluton s descrbed n terms of g), and π j, the parameters. The set of control (c k, k = 1,...,N c ) for the problem s the sum of the parameters and the ntal condtons for the felds that are descrbed by an evoluton equaton (namely, for the model descrbed n the prevous secton, P). The problem s the followng one: gven a set of Ŷ, a number of measures at tmes t, fnd the set of control such that the quantty E = 1 2 N o Y (t ) Ŷ 2 2 (16) s mnmzed. Ths results n a least square mnmzaton of the error. Let us wrte the gradent for the functon, whch provdes the descent drecton: E = c k N o (Y (t ) Ŷ ) Y = c k N o (Y (t ) Ŷ )Z k, (17) where Z k s called senstvty wth respect to the control c k and quantfes the varaton n the soluton wth respect to small perturbaton of the k th control. There are several ways to compute Z k. The most precse and straghtforward one conssts n dervng the governng equatons for the senstvty and smulate them. Ths technque, coupled to an adjont computaton provde all the ngredents to compute the Hessan of the functon, leadng to a Newton type of algorthm. A smpler technque, whch does not requre the soluton of another system of equatons conssts n usng the drect smulatons only. In partcular Z k can be approxmated by means of fnte dfferences, n the followng manner: Z k (t ) = Y c k (t ) Y (t,c k + δc k ) Y (t,c k ) δc k. (18) Practcally, f the elements of the control set are N c, ths lead to N c + 1 drect smulatons per teraton to compute the descent drecton. In order to speed up the convergence, a BFGS ([36]) algorthm s set up. Ths strategy can be parallelzed. In a gradent descent algorthm, the update of the control set s gven by ) c n+1 k = c n k β ( N0 < Y (t ) Ŷ, Z k (t ) > where β s a constant gradent step and <,> the L 2 scalar product., (19)

32 Predcton of tumor growth and applcaton to lung metastases A reduced technque based on POD In ths secton a reduced order technque s descrbed. The goal of settng up a reduced order technque s twofold: frst, the computatonal cost s decreased and second a regularzaton for the nverse problem s provded (see [27]). In our case the model reducton s performed by Proper Orthogonal Decomposton technque (see [26, 35, 28]). A database of smulatons s bult varyng the elements of the control set nto some reasonable ntervals (.e. controls that allow to represent a dynamcs smlar to the observed one). A certan number of orthonormal modes are bult and the unknown and unmeasurable felds can be decomposed on ths bass: P = C = v = γp = The system of equatons presented above becomes: a (P) φ (P), = 1,...,N P, (20) a (C) φ (C), = 1,...,N C, (21) a (v) φ (v), = 1,...,N v, (22) a (γp) φ (γp), = 1,...,N γp, (23) Ẏ + ( a v φ v a γp Y ) = a γp φ γp, (24) = 1 + tanh(r(ac φ c C hyp )), (25) 2 a v φ v = a γp φ γp, (26) k(y ) a v φ v = k a v φ v, (27) (D(Y )a c φ c ) = αa P j a C φ P j φ C + λa C φ C. (28) The observable Y s not decomposed, snce mages are avalable for a certan number of nstants. The model can be recast n the general form: t Y = f (Y,a,π j ). (29) Thus, the nverse problem can be wrtten n the followng manner: {a k,π h} = arg mn ã k, π h { No Ẏ f (Y,ã k (t ), π h ) }, 2 (30) that s, the resdual of the model s mnmzed when the passage condton through the observable s enforced, the dervatve of Y beng approxmated usng the sequence of mages tself. Ths s partcularly advantageous from a computatonal

33 16 Authors Suppressed Due to Excessve Length standpont when few mages are avalable. In such a case the most dffcult task s to fnd a good estmaton for the tme dervatve of Y. The mnmzaton s carred out by means of a standard Newton-type Levenberg-Marquardt scheme (see, for an applcaton to nverse problems [29]). 3.3 Comparson between the technques Both technques allow to have reasonable results n realstc cases, on a sgnfcant tme scale. In ths context a sgnfcant tme scale s the tme nterval between two subsequent medcal exams. The approxmated senstvty approach s qute smple and straghtforward, t requres the ntegraton of the drect smulaton only. The computaton of the functonal gradent can be easly parallelzed. Moreover, t s general and t allows to test dfferent models as well as several control sets or to nclude more observables. In ths sense t appears as a promsng tool to treat functonal magery data. The senstvty provdes also valuable nformatons concernng the soluton obtaned and t can be used to evaluate the effect of random perturbatons (such as the nose) on the data. Its man drawback s the computatonal cost. For a standard 2D case a parameter dentfcaton can take up to 2 days on a sngle standard CPU. On the other hand, the reduced order model concentrate the computatonal cost n the offlne stage: let us pont out that ths phase s massvely parallel and can be done when only one mage s avalable (ths mage s used to get the geometry and ntal condtons requred to smulate the system). Once the offlne stage has been performed the nverse problems takes only half an hour on a standard laptop. Furthermore, the advantage of the POD approach reles on the regularzng effect of the modes, preventng the hgher frequences to degrade the condtonng of the nverse problem. The dsadvantages of the reduced approach n the resdual formulaton consst n some dffcultes to deal wth more observables (or type of observables) and to ncrease the complexty of the models. The settng up of an hybrd approach potentally combnng the advantages of both s under scrutny. 4 Back to the ntal clncal case Let us now go back to the clncal case detaled n Sec We use the mathematcal model descrbed n Sec The tumor volumes obtaned from the real CTs and from the smulatons after the recovery of parameters are plotted n Fg. 5. Two measurements were used to recover the parameters, the remanng ones are plotted for comparson purpose. Both technques gve smlar results even f the one based on senstvty equatons seems more accurate. The thrd volume s perfectly caught n both cases. However,

34 Predcton of tumor growth and applcaton to lung metastases 17 Fg. 5 Evoluton of the tumoral volume as computed by the model after recoverng ts parameters wth the frst two measurements usng the technque based on reduced order models (ROM) and the one based on solvng senstvty equatons. Tumoral volumes measured on the CT scans by the clncans are plotted wth small crcles. Volume 12 x ROM Scans Senstvty months Fg. 6 Comparson between the observed tumoral shape (left) and the shape computed by the mathematcal model (rght) for the thrd exam (2007/10/17) after calbraton on the two frst scans of the sequence shown n Fg. 1. the nsght brought by such a spatal model s far from beng lmted to the evoluton of the tumoral volume (contrary to the model descrbed n Sec. 2.2). The spatal comparson between the observed shape and the one computed by the model (usng the senstvty approach) s shown n Fg. 6 for the thrd exam (on 2007/10/17). Even f the shapes do not perfectly match they are close. The model does not take much heterogenety nto account and could probably be mproved n ths regard at the expense of addtonal parameters. The error commtted on the shape after calbraton s shown for both technques n Fg. 7. The error s mostly commtted on the close vcnty of the tumoral bound-

35 18 Authors Suppressed Due to Excessve Length Fg. 7 Error commtted on the shape by the mathematcal model calbrated wth the ROM technque (left) or senstvty approach (left). The reference shape s the one presented n Fg. 6. Fg. 8 Snapshot of a 3D smulaton of the mathematcal model usng the parameters recovered by our technque on a slce. ary. In ths test, we only consdered the slce n whch the tumor s the largest n order to perform fast computatons. The same technque can be appled on the whole 3D volume reconstructed from the medcal mages. We can already use the recovered parameters wth a full 3D model as they do not depend on the spatal dmenson as shown n Fg. 8. Rather than searchng n the whole parameter spaces, t would probably be more effcent to always compute an approxmaton of these parameters usng a 2D model on a slce and then tune the parameters wth a full 3D nverse problem. 4.1 Other test cases Exponental growth We now study and apply the same approach to another test case. Ths tme the patent presented one sngle metastass from a bladder cancer. The queston s the same: can we evaluate the aggressveness of ths nodules? In ths case, we have 3 scans at our

36 Real case Predcton of tumor growth and applcaton to lung metastases 19 Fg. 9 Two successve CT scans showng the evoluton of a nodule. The top pctures shows the evoluton computed by the model, the bottom Procedure; ones, the nodule as shown on the medcal mages. Introducton; Real case; Smulatons of the second and thrd scan Concluson; second and thrd scan System dentfcaton n tumour growth modelng Lyon, 9-10 avrl dsposal. We wll use the frst two scans to calbrate the mathematcal model and Tuesday, Aprl 7, 2009 compares ts prognoss for the date of the exam correspondng to the last one. The tumoral dynamcs are dfferent from the prevous case. The growth s exponental, owng to the very fast evoluton, the patent was treated wth chemotherapy after the last scanner. After a good response to chemotherapy a thermo-ablatve local treatment of the nodule was decded. These two consecutve treatments lmted our analyss, nevertheless t s nterestng to check f the method s stll workng for fast growths. For hgh grade tumors, the mathematcal prognoss of a model wthout treatment s probably less nterestng for clncans: n such cases, they wll not wat for a mathematcal nsght and the patent wll be treated as soon as possble. In fact, the patent was treated wth a chemotherapy just after the last scan. In Fg. 9, we shown the evoluton computed by the calbrated model (top row) and the evoluton of the real tumor. The ntal scan s not shown. In ths case, the calbrated model gves an accurate prognoss. The volume and locaton of the tumor predcted are n good agreement wth the observaton. There are small dfferences on the shape, whch can probably be explaned by the lack of heterogenety of the model Two nodules of the same patent We are now consderng two dfferent nodules of the same patent. Ths tme, 2 lung metastases from a kdney cancer were dscovered on a systematc CT scanner follow up. No other metastatc stes are present. Ths s typcally the clncal setup we am at. The clncan may need help for decdng whch nodule has to be resected frst. For our approach t s also nterestng to see whch parameters are dfferent between the nodules (or whch parameters are patent-specfc or nodule-specfc). The evoluton of the frst nodule s shown n Fg. 10, the second one n 12.

37 20 Authors Suppressed Due to Excessve Length Fg. 10 Three successve CT scans showng the evoluton of a nodule on the left lobe. Fg. 11 Evoluton of the tumoral volume computed by the model after recoverng ts parameters wth the frst two CT scans. The volume measurements plotted as crcles were performed by the clncans. volume (a) (b) (c) Fgure 11: Scan for the thrd nverse problem, frst nodule: a) ,b) , 0.6 c) Fg. 12 Three successve CT scans showng the evoluton of a nodule on the rght lobe (a) (b) (c) 0.3 Fgure 11: 12: Scan for the thrd nverse problem, second frst nodule: a) ,b) , c) months 21 Fgure 14: Volume curve as functon of tme ng n the slope of ths functon. Ths phenomenon seems to be systematc, and so, a good canddate as prognoss ndcator. Ths s an ntrnsc feature of the model. For ths frst nodule the results s n accordance wthwhat found n the second problem. composton s reevant for a practcal pont of vew. Furthermore the nformaton about tumor (a) (b) (c) Fgure 12: Scan for the thrd nverse problem, second nodule: a) ,b) , c) On ths frst nodule, the technque gves a accurate result as shown on Fg Nodule 2: We now study the second nodule on the rght lobe. 21 In ths case, the algorthm The nverse gave two problem dfferent assocated solutons to thethat second have nodule approxmately was more dffcult than the same resdual. One soluton the frst one. s correct The same andnumercal s able tongredents catch correctly njected the n the thrd prevous cases volume. The other soluton behaves lke an exponental growth and does not ft the leaded to a bad behavor n terms of resdual: a reparametrzaton was necessary. Furthermore several ntalzatons were done before fndng a good thrd measurement. Between the frst and second exam, the sze of the tumor has almost be multpled by 4. Ths s probably too much to be able to dscrmnate evoluton. between these two possble behavors. Unfortunately, there s no exam avalable The numercal experments showed that there s a soluton correspondng to a local mnmum that s an exponental type soluton, whose basn between these two dates to check s ths hypothess s rght. It shall also be noted that n ths cases, the only two parameters that are really of attracton s qute large. Other combnatons of parameters revealed the dfferent between the two nodules are C hyp and α P.e. the parameters descrbng the behavor of the nodule. The parameters descrbng the mechancal aspects of healthy tssue are almost dentcal. 24

38 Predcton of tumor growth and applcaton to lung metastases 21 Fg. 13 Evoluton of the tumoral volume computed by the model after recoverng ts parameters wth the frst two CT scans. The volume measurements plotted as crcles were performed by the clncans. Ths tme, the algorthm gave 2 dfferent solutons: one s correct shown as black and one not accurate plotted as a dotted lne. volume months 5 Dscusson and perspectves Fgure 16: Volume curve as functon of tme As consequence a hgh volume rato between the mages makes the dentfcaton dffcult (2 mages on a very long tme scale) and suggests that t The smple model descrbed n Sec coupled wth the dentfcaton algorthm s qute mportant to place the exams n such a way that the rato of the gave an nterestng result on the ntal patent. Currently, the technques volumesfor observed recoverng s notparameters too hgh, let areus used say, not on untreated greater that patents. For these (old and clearly unwell) a dffcult patents, problem, the physcans snce t depends wsh strongly to mnmze on the theaggressvty. use The two. Ths s of nvasve technques nodule or unnecessary 1 belongng surgery to theorsame chemotherapy. patent, exhbts Obvously a slower ths growth. wll not be the majorty of clncal In Fg.16 cases: the patents tme behavor are treated of the as soon volume as s possble showed. tothe curedashed blue the dsease. We have tolne add represents the effects the of optmal a therapy exponental n the growth model and whlealso n contnuous recover black lne the correspondng parameters. the logstc Byone takng s showed. therapythe ntolatter account, turned we out wlltobebe allowed the more realstc to have a prognoss ofsoluton. the effcacy Two of elements the therapy have toor bethe underlned. sze, shape of the recurrent tumor. Furthermore, t could also be used to optmze the therapy and develop The frst s that, for ths value of growth rate, the model have self-smlar new clncal protocols. Indeed, the models gve us access to more nformaton than solutons of both type. The second s that, snce the logstc growthsprac- tcally saturated, the observables on standard medcal mages. But n the last years the ntroducton of functonal magng parameters gves the the opportunty dervatve of at a the better very understandng begnnng s almost of the zero. Thus, evoluton of a tumor beyond the problem the sngle of usng measurement an estmate of ths the tumoral dervatve volume. determnng Ths the logstc analyss becomes morecoeffcent basc ands crucal a verywth bad the condtoned use of targeted problem. therapes. A lot of dfferent solutons The problem wth systemc have logstc therapes behavors lkewth chemotherapy a dervatvessmall that one nearcannot the orgn havebut dfferent easly access to the map saturated of the doses states. ofths therapeutcal gves us anagents explanaton n the of tssue the wde (the vascular network s not easly vsble on magery). Therefore, the nformaton on the number of local spatal dstrbuton of cancer cells after treatment s lost and 26 one cannot evaluate the shape of the recurrent tumors. Ths would add addtonal unknown felds (at least the dose dstrbuton) whch could make the nverse problem much underdetermned. Ths should not be the case wth localzed therapes (radotherapy, radofrequency ablaton... ) where the operator has more control on the spatal dstrbuton of the treatment. Hence, dose map s more or less controlled by the operator. Unknown are mostly sensblty of cells to the dose, death rates, decay rates of the

39 22 Authors Suppressed Due to Excessve Length Fg. 14 A typcal setup of electrochemotherapy. The tumor s surrounded by electrodes. By applyng an electrc feld, cells may become permeablzed to a chemotherapeutcal drug lke the bleomycn. drug....e. scalars. Ths does not add much complexty to the nverse problem and t s our logcal next step concernng therapes. In ths paper, we were nterested n metastases n the lung. Clearly, there are a lot more targets of nterest. We are currently workng on bran tumors. We developed a generc model that we wll try to parametrze from patent data. A statstcal map of the bran and ts varous structures has to be used as for nstance n [13]. For the tme beng, we are usng anatomcal magng. Anatomcal magng gves us access to a map of the cancerous cells wthout any nformaton of ther actvtes (are they dvdng, hypoxc?). Functonal magng could help us to better determne the actve part of a tumor. Furthermore, n some case functonal magng can also offer hnts on the prognoss of a tumor (for nstance n the case of chlorne and spectroscopy wth bran tumors). We wll use functonal magng and bomarkers as an addtonal source of nformaton on the tumor evoluton to mprove the accuracy of the nverse problem. Yet, these mages are almost the only source of nformaton when tryng to recover the parameters of the mathematcal models. More generally we want to tap nto the wder range of addtonal sources avalable to help solvng the nverse problem. For each type of mage (PET scan, perfuson, dffuson MRI, spectroscopy) or measurements (concentraton of bomarkers...), one has to descrbe the connecton between the quanttes computed by the tumoral model (cellular denstes essentally) and the observed quanttes on ths mage. At the end of the avascular phase, a large part of the tumor s hypoxc and the angogeness process starts thus allowng the tumor to keep on growng. Tumor angogeness s the process by whch new blood vessels are formed and enhance the oxygenaton and growth of tumors. A complete revew of the mathematcal modelng of ths process can be found n [32]. However, most of these models cannot be used for clncal applcatons as they have too many parameters to determne. Several therapeutcal agents am at preventng the angogeness to occur. The study of ther effcacy requres couplng a tumor growth model wth a mathematcal model of angogeness. Based on the complex model presented n [5], a smpler model for angogeness s currently desgned. Usng functonal magng, we hope to be able to use the recovery technques n order to evaluate or optmze the ant-angogenc drugs.