The biology and mathematical modelling of glioma invasion: a review.

si . oyalsocie ypublishing.o g Re iew Ci e his a icle: Al onso JCL, Talkenbe ge K, Sei e M, Klink B, Hawkins-Daa ud A, Swanson KR, Ha ziki ou H, Deu sch A. 2017 The biology and ma hema ical modelling o glioma in asion: a e iew. J. R. Soc. In e ace 14: 20170490. h p://dx.doi.o g/10.1098/ si .2017.0490 Recei ed: 6 July 2017 Accep ed: 17 Oc obe 2017 Subjec Ca ego y: Li e Sciences–Ma hema ics in e ace Subjec A eas: bioma hema ics, sys ems biology Keywo ds: glioma in asion, cell pheno ypic plas ici y, malignan p og ession, in il a i e umou mo phology, ma hema ical modelling Au ho o co espondence: A. Deu sch e-mail: and eas.deu sch@ u-d esden.de The biology and ma hema ical modelling o glioma in asion: a e iew J. C. L. Al onso1,2, K. Talkenbe ge 2, M. Sei e 3,5, B. Klink4,5,6,7, A. Hawkins-Daa ud8, K. R. Swanson8, H. Ha ziki ou1,2 and A. Deu sch2 1 Depa men o Sys ems Immunology and B aunschweig In eg a ed Cen e o Sys ems Biology, Helmhol z Cen e o In ec ion Resea ch, B aunschweig, Ge many 2 Cen e o In o ma ion Se ices and High Pe o mance Compu ing, 3 Ins i u e o Medical In o ma ics and Biome y, and 4 Ins i u e o Clinical Gene ics, Facul y o Medicine Ca l Gus a Ca us, Technische Uni e si a ¨ D esden, Ge many 5 Na ional Cen e o Tumo Diseases (NCT), D esden, Ge many 6 Ge man Cance Conso ium (DKTK), pa ne si e, D esden, Ge many 7 Ge man Cance Resea ch Cen e (DKFZ), Heidelbe g, Ge many 8 P ecision Neu o he apeu ics Inno a ion P og am, Mayo Clinic, Phoenix, AZ, USA JCLA, 0000-0003-2432-5953; AD, 0000-0002-9005-6897 Adul gliomas a e agg essi e b ain umou s associa ed wi h low pa ien su - i al a es and limi ed li e expec ancy. The mos impo an hallma k o his ype o umou is i s in asi e beha iou , cha ac e ized by a ma kedly pheno- ypic plas ici y, in il a i e umou mo phologies and he abili y o malignan p og ession om low- o high-g ade umou ypes. Indeed, he widesp ead in il a ion o heal hy b ain issue by glioma cells is la gely esponsible o poo p ognosis and he di icul y o inding cu a i e he apies. Meanwhile, ma hema ical models ha e been es ablished o analyse po en ial mechanisms o glioma in asion. In his e iew, we s a wi h a b ie in oduc ion o cu en biological knowledge abou glioma in asion, and hen c i ically e iew and highligh u u e challenges o ma hema ical models o glioma in asion. 1. In oduc ion Gliomas a e he mos common p ima y umou s o he cen al ne ous sys em (CNS) in adul s. They comp ise a clinically, his ologically and gene ically e y he - e ogeneous b ain umou ca ego y. Un il ecen ly, glioma classi ica ion was la gely based on mic oscopic examina ion o his ological sec ions o umou specimens by expe pa hologis s, dis inguishing umou s acco ding o hei mic oscopic simi- la i ies o di e en ypes o glial cells in o as ocy omas, oligodend oglioma o ependymomas, he main sub ypes o gliomas [1]. The cu en 2016 Wo ld Heal h O ganiza ion (WHO) Classi ica ion o Tumou s o he CNS o he i s ime in e- g a es molecula bioma ke s oge he wi h classic his ological ea u es o de ine dis inc glioma en i ies [1]. This pa adigm shi in glioma diagnos ics e lec s he majo p og ess in ou unde s anding o he molecula biology o b ain umou s, which has emendously inc eased in he pas wo decades due o genome-wide molecula -p o iling s udies ha ha e cla i ied he gene ic basis o gliomas. Fo example, di use gliomas wi h his ologically oligodend oglial ea u es a e gene i- cally cha ac e ized by mu a ions in he IDH gene and 1p/19q codele ion, while he diagnosis as ocy oma is usually accompanied by mu a ions in IDH in combi- na ion wi h ATRX and/o TP53 mu a ions bu in ac 1p and 19q. By con as , classical p ima y glioblas omas usually do no show mu a ions in he IDH genes and a e he e o e e e ed o as glioblas oma IDH wild- ype. In addi ion, he WHO classi ica ion dis inguishes ou p ognos ic g ades ha e lec he deg ee o malignancy: WHO g ade I is assigned o he mo e ci cumsc ibed, benign umou s wi h low p oli e a i e po en ial ha mainly occu du ing childhood and in young adul s; WHO g ade II–IV umou s a e di usely in il a i e wi h inc eased cellula abno mali ies (di use gliomas); WHO g ade III umou s also show dedi e en ia- ion and mi o ic cell ac i i y; and WHO g ade IV umou s exhibi , in addi ion o &2017 The Au ho (s) Published by he Royal Socie y. All igh s ese ed. he ea u es p esen in he o he g ades, pa hological p oli e a ion o small essels and/o nec osis. WHO g ade II–IV umou s a e cha ac e ized by ex ensi e, di use in il a ion o glioma cells in o he hos b ain issue, and a e he e o e e e ed o as di use gliomas. These agg essi e b ain umou s a e ypically associa ed wi h a poo p ognosis, sha p de e io a ion in he pa ien s’ quali y o li e and ma kedly low su i al a es. In pa - icula , IDH wild- ype glioblas oma, he mos common (app ox. 45% o all gliomas) and malignan p ima y b ain umou (WHO g ade IV), has a 5-yea su i al o abou 5% om he ime o diagnosis [2,3]. E en o pa ien s wi h di use low-g ade IDH-mu an gliomas (WHO g ade II), al hough su i al canbemo e han10yea s, hep ognosis is un a ou able, as hese umou s e en ually p og ess o a high-g ade malignan lesion (WHO g ade III o IV) [4]. Despi e signi ican ad ances in su gical and medical imaging echniques, as well as in adju an adio-, chemo- and immuno he apy [5–10], he inhe en endency o glioma cells o widely dissemina e wi hin no mal b ain pa enchyma se e ely limi s ea men esponses [11–13]. The e o e, a be e unde s anding o he mechanisms ha igge andgo e ngliomain asioniso highclinicalimpo - ance o he de elopmen o mo e e ec i e and less oxic he apeu ic s a egies. Al hough he e is a conside able amoun o in o ma ion abou he clinical and biological beha iou o gliomas, he high complexi y o he in asion mechanisms emains a majo chal- lenge in clinical neu o-oncology. His ologically, glioma cells closely esemble glial p ogeni o cells, which ha e he abili y o p oli e a e and di e en ia e in o di e en glial cell ypes. These cells ha e a high mig a o y beha iou in he de eloping CNS [11,12,14]. This sugges s ha mechanisms con ibu ing o mig a ion o neu oepi helial cells du ing emb yogenesis a e also ele an o glioma in asion [14]. In addi ion, accumula ing e idence indica es ha spa ial and empo al a ia ions in signal- ling pa hways lead o unc ional and pheno ypic changes in glioma cells, which hen a ec in e ac ions wi h neighbou ing malignan and non-malignan cells along wi h o he componen s o he su ounding b ain issue. The exac consequences o he dynamic in e play be ween he e ogeneous cellula en i ies and hei esponse o al e a ions in he ex acellula mic oen i on- men ha e no ye been elucida ed. Mo eo e , i emains unclea why me as ases ou side he CNS a e ex emely a e in di - use gliomas [15,16]. F om a biological and medical pe spec i e, i is di icul o in es iga e he connec ions be ween clinically obse - able glioma beha iou and he unde lying molecula and cellula p ocesses. The challenge is o in eg a e he heo e ically and empi ically acqui ed knowledge o be e unde s and he mechanisms and ac o s ha con ibu e o glioma in asion. In his con ex , ma hema ical models p o ide use ul ools owa ds iden i ying dependencies and a ge s o cance cell mig a ion and in asion. Ma hema ical models and compu- a ional app oaches ha e become inc easingly abundan in cance esea ch o s udy umou dynamics and esponses o ea men modali ies such as chemo- and adio he apy [17–20]. Ma hema ical modelling p o ides a use ul heo e ical amewo k o pe o m in silico expe imen s, as well as o e alua e assump ions and make p edic ions ha can be expe imen ally es ed [21–32]. In he las wo decades, se e al ma hema ical models ha e been de eloped o in es iga e key mechanisms go e ning glioma g ow h and in asion [23,33–35]. Ten yea s ago, se e al o he cu en co-au ho s e iewed ma hema ical models o glioma de elopmen , g ow h and p og ession [34]. Since hen, he ield o glioma esea ch has signi ican ly g own. In his e iew we exclusi ely ocus on ma hema ical models o glioma in asion. We i s in oduce cu en biological knowledge abou glioma in asion. Then, we desc ibe biological model sys ems, in pa icula , in i o expe imen s and in i o animal models o he analy- sis o glioma in asion, and medical imaging echniques. We hen c i ically e iew ma hema ical models o glioma in a- sion, and highligh u u e challenges o ma hema ical and compu a ional modelle s in his esea ch a ea. 2. Biology o glioma in asion In il a ion o he b ain pa enchyma is a p ominen ea u e o di use gliomas, making comple e su gical esec ion almos impossible [36]. Di use gliomas in ade ex ensi ely as single cells anywhe e wi hin he hos b ain issue, wi h some p e e - ence o in il a e along whi e ma e ac s and he pe iphe y o blood essel walls [16]. The in il a ion o he su ounding b ain issue is de e mined by complex in e ac ions be ween glioma cells and he ex acellula mic oen i onmen [37]. He e, we e iew cell in insic mechanisms and ex insic ac o s ha sus ain and os e glioma in asion. 2.1. In insic mechanisms: pheno ypic plas ici y and gene ic a iabili y 2.1.1. Epi helial–mesenchymal ansi ion and mig a ion Glioma cells ha e he abili y o acqui e a mesenchymal pheno- ype in esponse o mic oen i onmen al cues and mig a e h ough he ex acellula ma ix (ECM) exhibi ing an elonga ed, o en wedge-shaped pheno ype [14,38,39]. Mig a ion and in a- sion o glioma cells a e ela ed, mul is ep p ocesses. Mig a ion is de ined as he mo emen o cells om one si e o ano he , o en in esponse o speci ic ex e nal signals such as chemical g adien s o mechanical o ces. Epi helial- o-mesenchymal ansi ion (EMT) is an essen ial p ocess in wound healing, emb yonic de elopmen and issue emodelling, consis ing in he ansdi e en ia ion o pola ized epi helial cells in o mo ile mesenchymal cells (o igina ed om he mesode mal emb yonic issue which de elops in o connec i e and skele al issues). Accumula ing e idence highligh s he c i ical ole o EMT du ing glioma p og ession and i s associa ion wi h inc eased glioma cell mig a ion [40]. Indi idual glioma cells sp ead by ac i e cell mig a ion a he han by passi e mo emen . In asion encompasses glioma cell mig a ion, bu also in ol es deg a- da ion o he ECM [38]. I is a mul i ac o ial p ocess ha consis s o in e ac ions be ween adjacen cance cells wi h he ECM coupled wi h biochemical p ocesses suppo i e o ac i e cell mig a ion. In gene al, glioma cell in asion in ol es ou dis- inc s eps [14,38,39]: (1) de achmen o in ading cells om he p ima y umou mass, (2) adhesion o he ECM, (3) deg ada ion o he ECM and (4) cell mo ili y and con ac ili y (ac i e cell mig a ion) ( igu e 1). A he subcellula le el, sec e ion o p o eases, cell adhesion molecules and ela ed signals play an impo an ole in glioma cell mig a ion [37]. De achmen o glioma cells om he p ima y umou mass in ol es se e al e en s, including des abiliza ion and diso ganiza ion o cell–cell adhesion complexes (cadhe in-media ed junc ions), loss o exp ession o neu al cell adhesion molecules and clea age o CD44, a cell-su ace p o ein which ancho s he p ima y umou mass o he ECM by he me allop o einase ADAM [16,38]. In eg ins si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 2 a e he mos common molecules ha allow glioma cells o adhe e o he ECM, and ma ix-me allop o einases (MMPs) a e he mos common p o eases ha deg ade he ECM c ea ing mig a ion ou es. Se e al glioma-exp essed molecula ac o s, such as ocal adhesion kinase and u okinase- ype plasminogen ac i a o (an enzyme pa icipa ing in ECM deg ada ion), ha e been ound o egula e hei exp ession [38]. Glioma cells mig a e simila ly as non- ans o med neu al p ogeni o cells, wi h myosin II as he majo sou ce o cy oplasmic con ac ili y [41]. In ading glioma cells al e hei shape, ex ending a p omi- nen leading cy oplasmic p o usion ollowed by a bu s o o wa d mo emen o he cell body. The complex molecula mechanisms and changes in signalling pa hways ha occu du ing glioma in asion a e s ill la gely unknown. A majo di - icul y in unde s anding he oncogenomics o glioma cell in asion is o de e mine how and when gene ic al e a ions and signalling cascades in e ac [42]. Only a ew speci ic pa h- ways ha e been consis en ly iden i ied, and he e exis mul iple possible in e ac ions along wi h addi ional unknown ac o s o be elucida ed. 2.1.2. Mig a ion–p oli e a ion dicho omy A he ime o diagnosis, gliomas a e al eady widely dissemi- na ed, as hey ypically g ow and in ade ex ensi ely be o e he pa ien expe iences any symp oms. This hidden dissemina- ion is a majo eason ha makes gliomas di icul o ea success ully and a cu e almos impossible. Cu en ea men s a egies mainly ocus on he highly p oli e a i e umou mass, bu local in asion e en ually leads o ecu ence o he disease. Eno mous e o s ha e been de o ed o iden i ying he main signalling e en s ha egula e glioma cell mo ili y and in asion. Howe e , he apeu ically a ge ing in asion dynamics is complica ed, because i has been obse ed ha mig a o y and p oli e a i e beha iou s o glioma cells a e mu ually exclusi e p ocesses and in e sely co ela ed [11,43,44]. In pa icula , highly mig a o y cells ha e a lowe p oli e a ion a e compa ed o ac i ely p oli e a ing cells ha mo e slowly. This ei he –o beha iou o p oli e a i e and in asi e glioma cells is suppo ed by bo h in i o and in i o expe imen s [11] and is e e ed o as he mig a ion– p oli e a ion dicho omy (o ‘Go-o -G ow’ mechanism) [43,44]. The ‘Go-o -G ow’ beha iou has been linked o me abolic s ess by se e al expe imen al indings. Godlewski e al. [45] iden i ied a glioma-exp essed mic oRNA (small non-coding RNAs ha egula e gene exp ession) ha egula es he balance be ween glioma cell p oli e a ion and mig a ion in esponse o changes in he a ailable ene gy. Thei expe imen al da a e ealed ha , in addi ion o inhibi ing glioma cell mig a ion, he mic oRNA exp ession also p omo es cell p oli e a ion. This sugges s ha mig a o y and p oli e a i e e en s sha e common signalling pa hways, de ining a unique in acellula mechanism ha egula es bo h phenomena. Mo e ecen ly, Ho ¨ ing e al. [46] in es iga ed he e ec s o ca boxypep idase E (CPE), a neu opep ide-p ocessing enzyme, on glioma in a- sion by means o in i o and in i o s udies. Thei esul s indica e an oxygen- and nu ien -dependen an i-mig a o y, bu p o-p oli e a i e ole o CPE in glioma in asion. Addi ion- ally, expe imen s wi h glioblas oma-de i ed neu osphe es p o ided u he insigh in o he ‘Go-o -G ow’ mechanism. EphB2, a ecep o o he y osine kinase amily was ound o ha e bo h p o-mig a o y and an i-p oli e a i e e ec s in i o [47]. These no el indings sugges ha glioma in asion could be a acked by a ge ing speci ic cellula and molecula mech- anisms associa ed wi h he mig a ion–p oli e a ion dicho omy. Howe e , u he in es iga ions a e equi ed o un a el he unde lying signalling pa hways egula ing glioma cell mig a ion and p oli e a ion. 2.1.3. Cell me abolic plas ici y A common ea u e o cance cells is hei al e ed glucose me abolism [48,49]. Unlike non-neoplas ic cells ha ely on oxida i e phospho yla ion o gene a e he ene gy needed o cellula p ocesses, cance cells can shi hei me abolism om espi a ion owa ds glycolysis p oducing lac ic acid. Indeed, cance cells end o up egula e ae obic glycolysis e en in he p esence o su icien oxygen, a phenomenon known as ae obic glycolysis o he Wa bu g e ec , which is a cha ac e is ic me abolic hallma k o umou de elopmen [50–52]. Al hough he Wa bu g e ec has ecen ly egained a en ion as a possible he apeu ic a ge [53,54], i s biological basis emains elusi e. Compa ed o mi ochond ial oxida i e me abolism, ae obic glycolysis is an ine icien way o gain ene gy [55]. Howe e , inc eased glycolysis c ea es a hos ile acidic en i onmen , in which cance cells ha e an e olu ion- a y ad an age wi h espec o no mal pa enchyma [56]. The e is accumula ing e idence ha acid-induced oxici y is an essen ial componen equi ed o umou in asion, and he e- o e a hallma k o in asi e cance s [56,57]. Glioma cells a e speci ically cha ac e ized by a high a e o glycolysis and lac- a e ex usion, wi h he abili y o lou ish in a ela i ely (1) (3)(2) (4) Figu e 1. Glioma cell mig a ion. Schema ic o he p ocess o glioma cell in asion in o hos b ain issue. In asion o glioma cells in ol es ou dis inc s eps: (1) de achmen o in ading cells om he p ima y umou mass, a p ocess igge ed by down egula ion o cell–cell adhesion molecules and mic oen i onmen al changes, (2) in eg in-media ed adhesion o he ex acellula ma ix (ECM), (3) sec e ion o p o eases, which locally deg ade ECM componen s c ea ing ou es along which glioma cells in ade he b ain and (4) mig a ion by ex ending a p ominen leading cy oplasmic p o usion, ollowed by a bu s o o wa d mo emen o he cell body. Figu e adap ed om [39]. si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 3 hypoxic en i onmen [58]. In ac , many o he in asi e ea- u es o gliomas may depend on dis o ed me abolic unc ions, which makes he s udy o me abolic al e a ions and hei e ec s on in asion p ocesses a g owing ield in cance esea ch wi h po en ial he apeu ic bene i s. 2.1.4. In a- and in e - umou al he e ogenei y Tumou he e ogenei y con ibu es o disease p og ession and de elopmen o he apy esis ance [59,60]. Ex ensi e gene ic and pheno ypic a ia ions exis among glioma cells wi hin a single umou (in a- umou al he e ogenei y) and be ween pa ien s (in e - umou al he e ogenei y) due o ex ensi e mol- ecula di e si y and mic oen i onmen al he e ogenei y [37,61–68]. I is commonly assumed ha umou he e ogen- ei y a ises ei he om a sel - enewing cance s em cell popula ion o due o clonal compe i ion o common esou ces d i en by he acquisi ion and expansion o mu a ions in cance cells [61,69–71]. Recen clinical and expe imen al indings ha e e ealed ex ensi e gene ic a i- a ions in glioma cells due o in a- umou al e olu ion [63,64,67,72]. Besides he la ge gene ic he e ogenei y, in e - ac ions be ween glioma cells and wi h he su ounding b ain pa enchyma lead o unc ional and pheno ypic di e - si y. E idence indica es ha dis inc clones wi hin a umou may ha bo gene ic and epigene ic al e a ions ha p omo e cance cell mig a ion and in asion. Recen ly, expe imen s wi h mix u es o di e en cell ypes o mimic pheno ypic he - e ogenei y ha e e ealed ha in asion is d i en by he coope a ion o mul iple umou -cell subpopula ions [73,74]. Pa e ns o co-in asion we e obse ed wi h inhe en ly in a- si e cells ac ing as he leade and subpopula ions o poo ly in asi e cells as ollowe s. This ‘di ision o labou ’ may acili a e no only umou in asion, bu also malignan p o- g ession. Al hough he e is inc easing app ecia ion ha in a- umou al he e ogenei y is cen al o glioma beha iou , li le is known abou he empo al sequence o gene ic and mic oen i onmen al changes o how genomic ins abili ies and adap a ion o mic oen i onmen al condi ions con ibu e o glioma in asi eness. 2.2. Cell-ex insic ac o s 2.2.1. Guidance mechanisms The pa icula s uc u es o he b ain such as blood essels, whi e ma e ac s and b ain pa enchyma, and speci ic umou cell–ECM adhe ence mechanisms a e c ucial ac o s in glioma in asion [75,76]. A any s age o in asion, glioma cells a e con on ed wi h non-neoplas ic b ain issue com- posed o mul iple cell ypes and a ious ECM componen s. The ECM in he CNS is di e en om he ECM in o he is- sues in ha i has low ib ous p o ein con en and high ca bohyd a e concen a ions [77]. In pa icula , he b ain ECM is mainly p oduced by as ocy es and oligodend ocy es, comp ises an es ima ed 20% o he b ain olume in adul s and consis s p ima ily o hyalu onic acid, excep a ound blood essels and a he pial su ace ( he bounda y be ween g ey ma e and ce eb ospinal luid) [76]. The in asion o glioma cells in o he adjacen b ain issue is guided by a com- bina ion o mul iple molecula and physical mechanisms along p e-exis ing acks o leas esis ance. The majo in a- sion ou es a e basemen memb anes and in e cellula acks p o ided by myelina ed axons and as ocy e p ocesses [76,78]. Glioma cells mig a e along blood essels by using he ou wa d essel–pa enchyma in e ace and he lumen o he pe i ascula space [76]. Mo e p ecisely, blood essels guide in ading glioma cells ia laminin- and collagen-IV-media ed in eg in engagemen (ECM p o eins media ed), whe eas whi e ma e acks guide by cell–cell con ac s and mechanisms egula ing cell–ECM adhesion o ces [76]. How- e e , he speci ic guidance ac o s and ela ed molecula mechanisms o mos dissemina ion ou es emain unclea . 2.2.2. Hypoxia-induced mig a ion Uncon olled glioma cell p oli e a ion leads o he de elop- men o hypoxic egions. This mic o egional change p oduces a local milieu ha a ou s ce ain glioma cell beha- iou s such as in asion [60]. A commonly held iew is ha cons i u i e up egula ion o glycolysis is likely o be an adap- a ion o he lack o oxygen [57]. Acco ding o he WHO classi ica ion, de ec ion o ascula p oli e a ion wi h highly pa hological blood essels and umou nec osis is essen ial o he diagnosis o g ade IV gliomas [1]. These umou - induced ea u es a e o en spa ially and empo ally ela ed, wi h si es o pa hological neo ascula iza ion indica i e o he o ma ion o hypoxic and nec o ic egions. Quan i a i e immunohis ochemical analysis e ealed ha while high- g ade gliomas may locally show a s ong angiogenic ac i i y, many egions o bo h low- and high-g ade gliomas display ascula densi ies in he ange o no mal ce eb al g ey o whi e ma e , indica ing limi ed angiogenesis [79–81]. This suppo s he obse a ion ha in high-g ade gliomas di e en in asi e and p o-angiogenic umou cell pheno ypes coexis [39]. O e exp ession o p o-angiogenic ac o s by umou cells esul s in local ascula o e g ow h wi h de ec i e blood essels, which ha e signi ican ly la ge diame e s and hicke basemen memb anes han hose in no mal b ain issue [82]. I has been obse ed in i o ha unde oxygen-limi ing condi ions due o ascula abno mali ies, glioma cells ac i ely mig a e away om hypoxic egions [83]. Di e en pa hological and expe imen al obse a ions sugges ha aso-occlusion could eadily explain he apid pe iphe al expansion and di usely in il a i e g ow h beha - iou o high-g ade gliomas [84,85]. Occlusion o ascula u e mainly occu s due o inc eased mechanical p essu e by ei he umou cells o by in a ascula p o- h ombo ic mech- anisms [83,84,86]. Occluded o collapsed blood essels induce pe i ascula umou hypoxia and nec osis in glio- blas oma, which ypically o m lines wi h pe i ocally inc eased cell densi y, e med pseudopalisades [83–85,87]. Pseudopalisades a ound nec o ic oci, a common ea u e o high-g ade gliomas, a e se e ely hypoxic and linked o wa es o glioma cells ac i ely mig a ing away om such oxygen-de icien egions [83–85,87]. Expe imen al s udies u he sugges ha umou hypoxia esul s in inc eased glioma cell mig a ion and in asion, and s ongly co ela es wi h umou malignancy [39,88,89]. The exac pa hophysiolo- gical ac o s and mechanisms unde lying hypoxia-induced cell mig a ion in gliomas a e s ill no known, and u he in es iga ion is equi ed o unde s and he complex molecula pa hways in ol ed in hypoxic esponses and me abolic con ol by glioma cells. 2.2.3. Blood–b ain ba ie An impo an s uc u al componen o he b ain ascula u e is he blood b ain ba ie (BBB), which is essen ial o supplying si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 4 he b ain issue wi h oxygen and glucose, media ing e lux o was e p oduc s, and main aining a p ecisely egula ed mic oen- i onmen o eliable neu onal signalling [90]. The BBB is composed o igh ly bound endo helial cells and pe i ascula as ocy es ha es ic he exchange o molecules om he blood- s eam much mo e han capilla ies anywhe e else in he body. High-g ade gliomas ha e he abili y o dis up he in eg i y o he BBB, which is associa ed wi h inc eased umou g ow h and di use in asion in o he su ounding b ain pa enchyma [16,91–93]. Recen s udies ha e shown ha he BBB can be he - e ogeneously dis up ed in high-g ade gliomas, and he deg ee o BBB dis up ion is ela ed o umou malignancy [92]. How- e e , he p ecise ela ionship be ween glioma-induced BBB dys egula ion and cell in asion is s ill no clea [91]. On he o he hand, expe imen al e idence shows ha one o he majo obs acles o s anda d an i-cance d ug deli e y in he b ain is he BBB, which limi s he e icacy o chemo he apy [16,92]. In ac , he p esence o an almos in ac BBB is one o he main ac o s ha makes he ea men o low-g ade gliomas wi h chemo he - apy challenging. Al hough he BBB may be dis up ed a he co e o high-g ade gliomas, i can be ela i ely in ac a he umou pe iphe y whe e in ading glioma cells a e loca ed [92]. This pe - mi s in il a i e glioma cells o escape chemo he apy-induced dea h, which can esul in umou ecu ence. 2.2.4. Immune sys em engagemen Tumou s ha e long been ecognized as wounds ha do no heal [94]. Bo h ca cinogenesis and wound healing in ol e cell p oli e a ion, mig a ion, in asion, angiogenesis, in lam- ma ion and as ocy e ac i a ion in esponse o inju ies [76,95]. The mic oen i onmen o high-g ade gliomas esembles in many ways a ch onic wound [95]. In pa icula , he ex ensi e cell mig a ion and in asion obse ed in gliomas also accompanies eac i e gliosis, a non-speci ic eac ion whe e as ocy es a e ac i a ed in esponse o inju ies o he CNS as pa o a healing p ocess. The bo de o gliomas exhi- bi s an inc eased numbe o eac i e as ocy es, which oge he wi h glioma cells sec e e a ious p o-mig a o y sig- nalling molecules [76]. Recen ly, a p o ein (connec i e issue g ow h ac o ) p oduced a high le els by eac i e as ocy es has been iden i ied o s imula e mig a ion o glioma cells [95]. Indeed, he sec e ed ac o modula es nea ly all aspec s o he signalling mechanisms ha egula e cell in asion, such as modi ica ion o g ow h ac o ac i i ies, ECM composi ion, in eg in ( ansmemb ane cell-ma ix adhesion ecep o s) and E-cadhe in exp ession (a ansmemb ane p o ein ha media es cell–cell adhesion). Thus, he iden i ica ion o p o- cesses in ol ed in glial eac i a ion may con ibu e o a be e unde s anding o glioma cell in asion wi h po en ial he apeu ic implica ions. Accumula ed his opa hological da a ha e es ablished ha issue- esiden mic oglia and mac ophages a e he p edomi- nan in il a ing immune cells in gliomas, accoun ing o up o 30–50% o he o al umou mass [96]. Monocy es ci cula e in he bloods eam and a e con inuously ec ui ed in o umou s in esponse o se e al umou -de i ed chemo- a ac an s. Once inside he umou , monocy es apidly di e en ia e in o umou -associa ed mac ophages (TAMs), and hen accumula e in hypoxic/nec o ic a eas [96]. TAMs p oduce se e al ac o s ha no only s imula e he su i al and p oli e a ion o umou cells, bu also supp ess an i- umou immuni y [96,97]. The e is g owing e idence ha TAMs a e also in ol ed in egula ing glioma cell mig a ion and in asion [96,98–100]. Mo eo e , TAMs con ibu e o malignan p og ession o gliomas h ough sec e ion o a - ious chemical ac o s ha a ec angiogenesis and ECM emodelling, which p omo es glioma in asion [96,97,101]. Mo e p ecisely, TAMs can display wo majo pheno ypes, he classically (M1) and al e na i ely (M2) ac i a ed, which can be iewed as wo ex eme pheno ypes [102,103]. Unlike TAMs displaying an M1-like pheno ype wi h p o- in lamma o y and an i- umou unc ions, M2 mac ophages a e immunosupp essi e, p oduce p o-angiogenic ac o s, con ibu e o he ECM- emodelling, and hus c ea e a a ou - able mic oen i onmen o glioma g ow h and in asion. Al hough he e is g owing accep ance ha he M1 and M2 s a es a e no dicho omous bu a he ep esen ex emes o a con inuum o pheno ypes, a majo i y o mac ophages in gliomas a e mac ophages wi h M2-like pheno ype which exhibi p o-in asi e p ope ies, pa icula ly in la e s ages o disease p og ession [104,105]. Recen e idence indica es ha he colony s imula ing ac o -1 (CSF-1) sec e ed by glioma cells induces TAMs o p omo e in asion [104]. I has been obse ed ha CSF-1 le els a e ele a ed in high-g ade glio- mas, which is associa ed wi h he exp ession o M2 mac ophage ma ke s. In addi ion, he cy okine in e leukin- 10 (IL-10) is also commonly associa ed wi h mac ophages o he M2 pheno ype and has been ound o s imula e glioma cell in asion [104]. Toge he , chemokines, cy okines and g ow h ac o s sec e ed by TAMs ac i a e di e en sig- nalling pa hways ha can swi ch glioma cells owa ds mo e agg essi e beha iou . 3. Biological model sys ems As i is cu en ly impossible o moni o he en i e p ocess o glioma in asion in he human b ain, di e en biological model sys ems ha e been in oduced. In i o cell cul u es p o ide he oppo uni y o gene a e insigh s in o molecula and cellula pa hways ela ed o glioma in asion unde con- olled condi ions. Va ious in i o models ha e been in oduced which allow a mo e ealis ic ep esen a ion and moni o ing o he complex glioma dynamics. 3.1. In i o expe imen s Expe imen al p o ocols in i o a e a ailable o independen ly obse e and con ol a iables o in e es a a ious scales, om single-cell mo emen o mul icellula clonal g ow h and cance cell popula ion dynamics [106]. In asion s udies ha e been pe o med in bo h wo-dimensional (2D) umou monolaye s and h ee-dimensional (3D) mul icellula sphe - oids, combining glioma cell mig a ion and p oli e a ion assays, and conside ing di e en ECM composi ions and subs a e igidi ies [107]. Howe e , he e a e signi ican di e ences be ween umou cell mig a ion on a 2D su ace and in a 3D ma ix [108], and e en mo e be ween in i o and in i o expe imen s [106]. Unde con olled expe imen al condi ions, glioma cells may exhibi a mo e ib oblas ic shape wi h a b oad lamellipodium and an undis o ed nucleus, whe e he o wa d mo emen is con inuous and unimpeded [41]. By con as , glioma cells mig a ing h ough he complex he e ogeneous b ain pa enchyma a e highly pola ized and elonga ed [38,39]. The mechanical cons ain s due o small in e cellula spaces impede he o wa d mo emen o nucleus si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 5 and cell body un il necessa y con ac ion o ces a e p o ided [41]. This abili y o glioma cells o adap hei mo ili y o he pa icula mic oen i onmen ein o ces he need o assays ai h ully ep esen ing he en i onmen al condi ions o he b ain o imp o e ou unde s anding o in asion mechanisms. Using so-called no mal b ain cell agg ega es de i ed om e al a b ains o u iliza ion o oden o o he mammalian b ain slices migh p esen a p omising comp omise o pe - o m 3D in i o s udies ha can be mo e s anda dized and a oid e hical issues a ached o animal s udies [109–112]. 3.2. In i o models Animal models a e essen ial o in es iga ing he in e ac ions be ween glioma cells and he complex b ain mic oen i on- men . The adi ional use o animal models in ol es injec ing es ablished umou cell lines ei he in a enously o a he a ge si e and hen wai ing o a umou o de elop be o e es - ing a he apy o a gi en hypo hesis. While he complex mic oen i onmen s in he animal models mimic he human b ain s uc u e much be e han in i o s udies, e y ew es ablished cell lines a e able o ep esen he his opa hological cha ac e is ics o human gliomas, pa icula ly hei in asi e na u e. Recen ly, me hods ha e been de eloped o ha es cells om pa ien s o use in animal models. These a e e e ed o as pa ien -de i ed umou agg ega es and a e be e able o ecapi ula e pa e ns o umou cell in asion obse ed in pa ien s [113]. Mo eo e , xenog a models ha e been ex ensi ely employed o assess he e icacy o he apies a ge ing glioma cells, such as he in a- umou al adminis- a ion o IL13-PE oxin (a usion p o ein composed o IL-13 and a mu a ed o m o Pseudomonas exo oxin), o moni o glioblas oma angiogenesis and o e alua e an i-angiogenic he apeu ical app oaches [114]. The main d awback o xenog a s is ha he his ology and gene ics o he o iginal umou a e equen ly no main ained. Mo eo e , high- esolu ion imaging o single glioma cell in asion in xenog a s emains labo ious, cos ly and ime-consuming. Tumou s can also be induced in animals using e o- i uses. Two no able cases use cells in ec ed wi h e o i uses enginee ed o o e exp ess ei he cons i u i ely ac i a ed epide mal g ow h ac o ecep o (EGFR) [115] o pla ele -de i ed g ow h ac o (PDGF) [116]. Bo h e o- i uses a e able o ini ia e umou g ow h wi h human glioblas oma cha ac e is ics when injec ed in o a a o mouse b ain. In addi ion o p o iding a good model sys em o d ug he apy, hey also shed ligh on he umou ini ia ing p ocess. Ano he de elopmen in animal models is he c e- a ion o ansgenic mice, whe e gene ic enginee ing echniques a e used o c ea e mice wi h ubiqui ous mu a ions ha a e p edisposed o de eloping gliomas [117]. Many in i o expe imen al echniques exis , he mos common in ol es sac i icing he animal o allow o s aining o he b ain issue, bu his only p o ides a empo al snapsho o umou composi ion. Ano he echnique is e e ed o as ex i o imaging whe e b ain issue is ha es ed and hin slices a e placed on nu ien - illed media. This allows o mic o- scopically obse e cell mo emen o a ime pe iod up o 24 h [116]. A u he op ion ha allows longe obse a ion is he use o bioluminescence (o bio luo escence) a he cell popula ion scale, o mul ipho on mic oscopy a he single- cell le el. While any o he a o emen ioned animal models p o ides expe imen al condi ions close o he human b ain, he abili y o ully moni o ing he cance dynamics is s ill challenging. Addi ionally, he ui ly D osophila melanogas e has been conside ed as an al e na i e in i o glioma model because many molecula pa hways and cellula unc ions a e unda- men ally conse ed [118,119]. Ad an ages include easy handling, a ully sequenced genome, a wide ange o a ailable gene ic echniques and a well-known ana omical si ua ion [118,120]. Model o ganisms such as D. melanogas e ha e been use ul no only o isualize umou cell mig a ion and o in es- iga e he e ec s o induced me as asis, bu also o iden i y glioma signalling cascades ia ad anced gene ic echniques. Howe e , he D. melanogas e model also has some limi a ions. In asion s udies in D. melanogas e lack an accu a e ep esen- a ion o he human b ain pa enchyma, including he absence o blood essels and an adap i e immune sys em. 4. Medical imaging and his opa hology Con en ional compu e omog aphy scanning e eals mo pho- logical in o ma ion o gliomas, bu umou s a ea ly s ages o small me as a ic lesions a e o en no de ec ed. This echnique has been g adually eplaced by magne ic esonance imaging (MRI), which is signi ican ly mo e sensi i e o he p esence o umou s and has become he s anda d imaging modali y in he e alua ion o b ain umou s [121,122]. MRI c ea es non- in asi e images by exploi ing he magne ic p ope ies o wa e molecules in he body. By changing he in ensi y, iming and du a ion o adio equency pulses and di ec ional g adien s, my iad non-in asi e images wi h a ying con as s and in o ma ion can be c ea ed [123]. The mo e common T1- and T2-weigh ed MRI sequences a e mainly used o dis- playing gene al ana omic ea u es o gliomas; howe e , he wo ypes o sequences emphasize ea u es di e en ly (e.g. ce - eb al spinal luid (CSF) is b igh on T2-weigh ed images and da k on T1-weigh ed images). The T1-weigh ed image can be used o highligh he leaky blood essels cha ac e is ic o glioblas oma by acqui ing he image a e adminis a ion o gadolinium, a con as agen ha appea s b igh on T1-weigh ed images. Gadolinium seeps ou om he leaky ascula u e haphaza dly c ea ed by he umou , highligh ing wha is belie ed o be he mos ac i e/agg essi e umou egion on T1-weigh ed gadolinium-enhanced MRI (T1Gd). The T1Gd and T2 sequences a e mos commonly used o deli- nea ing umou egions, bu i is well known ha nei he o hese sequences is able o p o ide a p ecise isualiza ion o umou abno mali y due o he ex ensi e in asion o he umou cells [124]. In ac , in one s udy o high-g ade gliomas, i was demons a ed ha human gliomas g ow in a- si ely, wi h umou cells demons able o e 4 cm om he g oss umou [125]. While T1- and T2-weigh ed MRI sequences emain he dominan images clinically used, he e a e many o he ad anced MRI echniques ha a e being explo ed such as ascula pe usion imaging, di usion-weigh ed imaging (DWI) and p o on magne ic esonance spec oscopy (MRS) [126]. Vascula pe usion imaging highligh s egions o high ascula i y and has been shown o be use ul in p edic ing which pa ien s a e esponding o an i-angiogenic he apies [127,128]. DWI can be used o a ious pu poses, bu o glio- blas oma i is mos commonly used o quan i y he appa en di usion coe icien , which is hough o be in e sely si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 6 co ela ed wi h cell densi y [129,130]. MRS uses he p o on signals o de e mine ela i e concen a ions o a ge me ab- oli es a he han an ana omical image. In a ious s udies, MRS has been shown capable o iden i ying egions o issue en iched wi h s em-like cell-en iched oci [131], de ec - ing umou s wi h mu a ions in he isoci a e dehyd ogenase (IDH) genes [132,133], and assessing esponse o a ious he apies such as adia ion and PI3K/mTOR inhibi o s [134,135]. Fu he , he use o hype pola ized (HP) con as agen s can signi ican ly inc ease he sensi i i y o MRS by enhancing he signal- o-noise a io [136,137]. Mo e gene ally, MRS has opened up he p omising ields o me abolomics ( he s udy o me abolomic signa u es in umou s) [138] and adiomics ( he use o imaging echnology o ex apola e mol- ecula umou da a) [139]. These s udies a e encou aging, bu one mus no e ha he esolu ion o a s anda d MRS is much lowe han on a s anda d MRI, oxel sizes being app oxima ely 10 10 10 mm 3 e sus 1 11mm 3 . Thus, while he e a e many p omising ad anced magne ic esonance me hods o de e mining a ious umou cha ac- e is ics, di e en ia ing be ween no mal and pa hological issue on he basis o MRI indings alone is complica ed. Func ional imaging echniques like posi on emission omog aphy (PET) scans a e use ul in p o iding deepe insigh s in o he biology o gliomas [140,141]. PET imaging is inc easingly implemen ed in neu o-oncology, because i o e s unique da a abou me abolic and physiologic p ocesses such as glucose me abolism, p o ein/DNA syn hesis, cell p oli e a ion and apop osis, as well as angiogenesis and hypoxia ha can e lec he changes in a neoplasm. Assess- men o he s a us o hese p ocesses has been shown help ul in delinea ion o umou ma gins, and co ela es wi h clinical me ics such as umou g ade, pa ien su i al and he apy esponse [140,142,143]. Clinically, his ype o in o ma ion is iewed as complemen a y o he ana omical MRI, as PET scans gene ally lack ana omic con ex , and ha e a ela i ely low spa ial esolu ion [144]. Diagnosis and classi ica ion o glioma is based on his o- pa hology, e e ing o he mic oscopic examina ion o issue sec ions by an expe ienced pa hologis , e.g. neu opa hologis . Addi ionally, his opa hology analysis o issue samples has some po en ial o p o ide u he in o ma ion a he single- cell le el ha is ex emely impo an o quan i y and classi y in a- umou al he e ogenei y. His ological and immunohis o- chemical analyses a e ou inely pe o med by pa hologis s o con i ma ion o he p esence o absence o disease, de e - mining glioma g ading and assessing disease p og ession [1,145,146]. Howe e , issue biopsies can be seen as ‘snap- sho ’-like ozen scenes o dynamic biological p ocesses p o iding da a se e ely limi ed in bo h space and ime. Thus, despi e he cons an expansion o medical imaging ech- nology, he iden i ica ion o umou s a an ea ly s age, assessmen o in a- umou al he e ogenei y, educ ion o adi- a ion exposu e and imp o emen o esolu ion a e challenging o ana omic, unc ional and me abolic imaging alike. 5. Ma hema ical modelling o glioma cell mig a ion and in asion A wide a ie y o ma hema ical models ha e been p oposed o in es iga e he mechanisms o glioma in asion, which is cha ac e ized by in asi e cell mig a ion, pheno ypic plas ici y, in il a i e umou mo phologies and he abili y o malignan p og ession. Model ypes include disc e e and con inuous app oaches such as cellula au oma on (CA), la - ice-gas cellula au oma on (LGCA), cellula Po s model (CPM), pa ial di e en ial equa ions (PDE), agen -based models (ABM) and e olu iona y game heo y models (EGT) [17–20,35,147–153]. We subsequen ly e iew ma hema ical models o in asi e cell mig a ion, in asi e e ec s o pheno- ypic plas ici y, in il a i e umou mo phologies and malignan p og ession. Table 1 p o ides an o e iew o he e iewed models. 5.1. In asi e cell mig a ion Based mainly on in i o expe imen s, se e al ma hema ical models ha e been de eloped o in es iga e he e ec s o cell–cell adhesion s eng h unde dis inc mic oen i onmen- al condi ions on he in asi e beha iou o glioma cells. Khain e al. [27] in es iga ed, bo h heo e ically and expe - imen ally, he e ec o cell–cell adhesion on glioma on p opaga ion and he s uc u e o he in asi e in e ace. Mig a ion cha ac e is ics o U87-MG cells we e measu ed using a sc a ch wound-healing assay, and in asion on pa e ns we e simula ed by means o bo h a 2D disc e e la ice-based s ochas ic model and a con inuum app oach. Simula ions o he con inuum model show ha a small e ec- i e cell–cell adhesion change does no in luence he p opaga ing on speed, and clus e s o glioma cells we e no o med ( igu e 2a). By con as , he mic oscopic disc e e model shows ha a cell–cell adhesion s eng h exceeding a c i ical h eshold leads o clus e o ma ion in he in asi e zone esul ing in inge ing-like on p opaga ion pa e ns ( igu e 2b). The expe imen al ime was cha ac e ized as a ansien egime, which coincides wi h he pe iod equi ed o a ela i ely sha p ini ial cell densi y p o ile o de elop in o a p opaga ing on . Al hough simula ions success ully ep oduced he maximal dis ance o mig a ion o glioma cells on a plas ic subs a e, his model unde es ima ed he mig a ion o he main mass o umou cells, sugges ing he p esence o chemo ac ic s imuli. Glioma cell mig a ion on a subs a e o collagen was in es iga ed by Aube e al. [162]. The p oposed 2D CA model indica es ha chemo axis o cell–cell communica ion h ough gap junc ions (specialized in e cellula channels ha pe mi di ec cell–cell ans e o ions and molecules) is necessa y o ep oduce expe imen al densi y p o iles o glioma cell dis ibu ions in umou sphe oids. In a ollow- up s udy, mig a ion pa e ns o glioma cells in he p esence o as ocy es we e s udied by Aube e al. [154]. An ex ended e sion o he model p oposed in [162] was used o analyse he opposi e e ec s o homo ypic (be ween glioma cells) and he e o ypic (be ween glioma cells and su ounding as o- cy es) gap junc ion communica ion on he in asi eness o gliomas. Lowe ing glioma cell–cell in e ac ions on a passi e subs a e o collagen was p edic ed o enhance he mig a o y po en ial, whe eas he simul aneous inhibi ion o glioma cell–cell and glioma cell–no mal as ocy e gap junc ion com- munica ion leads o educed cell mig a ion. This sugges s ha he in e ac ions be ween glioma cells and as ocy es play an impo an ole in glioma in asion, due o he e ec o he e o ypic gap junc ion inhibi ion which domina es ha o homo ypic inhibi ion. Model simula ions a e consis en wi h expe imen al da a o glioma mig a ion pa e ns in bo h si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 7 Table 1. O e iew o e iewed ma hema ical modelling app oaches. Cellula au oma on (CA), la ice-gas cellula au oma on (LGCA), cellula Po s model (CPM), pa ial di e en ial equa ions (PDE), agen -based model (ABM) and e olu iona y game heo y (EGT). ma hema ical modelling app oach Khain e al. [27] Aube e al. [154] Khain e al. [155] Szabo ´ e al. [156] Kim e al. [157] Kim [158] Tek onidis e al. [25] Ha ziki ou e al. [31] Bo ¨ ge e al. [28] Pham e al. [29] Ge lee e al. [30] Al onso e al. [32] Sande e al. [159] Ma ı ´nez- Gonza ´lez e al. [26] F ieboes e al. [21] Zhang e al. [160] F ieboes e al. [161] Jiao e al. [24] Basan a e al. [22] Swanson e al. [23] (2D) CA & (1D) PDE (2D) CA (2D) CA (2D) CPM (2D) PDE (2D) HYBRID (2D) LGCA (2D) LGCA (2D) LGCA (2D) PDE (2D) CA & (1D) PDE (1D) PDE (2D) HYBRID (1D) PDE (2D) PDE (3D) ABM (3D) HYBRID (2D) STATISTICAL EGT (1D) PDE main ocus o ma hema ical models in asi e cell mig a ion 333333 pheno ypic plas ici y 33 3 333 3 3 3 umou mo phology 33 3 33 3 malignan p og ession 333 main biological assump ions cell–cell in e ac ions 3333333 3 333333 cell–ECM in e ac ions 333 33 hypoxia-induced mig a ion 3333333 cell densi y- dependen in asi eness 3333 3 changes in me abolism 33 3 clonal he e ogenei y 3 angiogenesis 33 3 3 si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 8 homo ypic and he e o ypic si ua ions by only in oducing a ac i e con ac be ween mig a ing umou cells. Howe e , he expe imen s conside ed in [154,162] only in ol ed ela i ely small sphe oids wi hou cen al hypoxia o nec osis. Thus, o s udy mig a ion pa e s o glioma cells in la ge sphe oids, Aube e al. [163] assumed a chemo epellen ac o p oduced by cells submi ed o s ess ul condi ions in he hypoxic/nec o- ic mic o egions. A good ag eemen be ween model simula ions and expe imen s allows o conclude he exis ence o epellen oxic cues ha would p omo e glioma cell de achmen and mig a ion, al hough u he wo k is needed o iden i y and cha ac e ize hese chemo epulsi e ac o s. The ole o hypoxia in he egula ion o glioma cell–cell adhesion and cell mig a ion was in es iga ed by Khain e al. [155]. A 2D disc e e s ochas ic model was p oposed o desc ibe in i o expe imen s o U87 glioma cell mig a ion (i) away om umou sphe oids placed on a subs a e and (ii) in ypical sc a ch wound-healing assays. The dis ance mig a ed (i.e. in a- si e adius) by bo h no moxic and hypoxic glioma cells was measu ed. In he sphe oid expe imen s, he o e all mig a ion a e o umou cells unde ei he no moxic o hypoxic condi ions was simila . Howe e , hypoxic glioma cells in he wound-heal- ing assays mig a ed less han cells unde no moxic condi ions. This model sugges s ha lack o oxygen no only supp esses cell mo ili y, bu also subs an ially educes he s eng h o cell– cell adhesion. Al hough oxygen de iciency esul ed in educed cell mo ili y, he dec eased cell–cell adhesion allows hypoxic cells o de ach om he umou mass, leading o enhanced glioma in asion. These model p edic ions a e consis en wi h expe imen al da a showing ha hypoxia induces down egula- ion o E-cadhe in, a ansmemb ane p o ein ha posi i ely egula es ex ension and s eng hening o adhesi e con ac s, and p omo es glioma cell in asion. To explo e he in luence o he ECM on glioma cell mig a ion, Szabo ´e al. [156] conside ed no only cell–cell adhesion bu also cell–ECM in e ac ions. Cell agg ega es we e p epa ed om con luen cul u es o wo di e en glio- blas oma cell lines (GBM1 and U87) placed wi hin a 3D ECM o collagen I gel. The aim was o cha ac e ize he collec- i e, la ge-scale in asion o glioma cells om umou sphe oids in o he su ounding ECM. The in e play be ween hap o axis, ma ix deg ada ion and ac i e cell mo emen was in es iga ed by means o a 2D CPM. Simula ion esul s sugges ha he complex in e play be ween space-con- s ained ac i e cell mo ion, cell–ECM adhesion and deg ada ion o he ECM de e mines he pa e ns o mig a ion and inc eases pe sis ence du ing cell in asion. In pa icula , hap o axis and ECM deg ada ion we e obse ed o des abi- lize mul icellula sp ou s as each cell ies o in ade he su ounding ma ix. By con as , when bo h hap o axis and pola ized mo ion a e p esen , e en a homogeneous glioma cell popula ion may be o ganized in o mul icellula sp ou s wi hin an inhomogeneous ECM en i onmen . Kim e al. [157] p oposed a model ha akes in o accoun cell–cell adhesion, hap o axis and chemo ac ic e ec s o a glu- cose g adien on glioma cell mig a ion in i o.Model simula ions e eal ha depending on he chemo ac ic and hap- o ac ic sensi i i ies, and he s eng h o cell–cell adhesion, di e en mig a ion pa e ns o glioma cells a ise: dispe sion, b anching, island o ma ion and a mix u e o hese pa e ns. In pa icula , his model ep oduced he pa e ns obse ed in a ious in asion assays o in i o sphe oids gene a ed om glioma U87 and mu an U87DEGFR cell lines. Mo eo e , changes o adhesion, hap o ac ic and chemo ac ic pa ame e s esul in a g adual shi om b anching o dispe sion as expe - imen ally obse ed. The main inding was ha he on o cell mig a ion can be slowed down by bo h inc easing cell–cell adhesion and blocking he ECM deg ada ion e ec s o MMPs, a amily o enzymes ha a e capable o b eaking down all kinds o p o eins, such as collagen, no mally ound in spaces be ween issues (ECM p o eins). In a ollow-up s udy, Kim [158] de eloped a hyb id mul i- scale model in which glioma cell mig a ion and p oli e a ion a e egula ed by in acellula mechanisms in esponse o glucose a ailabili y and physical cons ain s in he mic oen i onmen . In pa icula , a co e con ol sys em o a single mic oRNA (miR-451) ha egula es AMPK ( he 50-adenosine monophos- pha e ac i a ed p o ein kinase) signalling [45,164] linked o ex acellula glucose was simula ed. Recen expe imen al e i- dence sugges s ha , in a glucose- ich en i onmen , miR-451 is up- egula ed by umou cells, leading o AMPK pa hway inhi- bi ion and in u n cell p oli e a ion. Con e sely, sus ained AMPK ac i a ion unde low glucose condi ions esul s in sup- p ession o miR-451, which induces pheno ypic changes o glioma cells om a p oli e a i e o a mig a o y pheno ype [165]. Based on he assump ion ha glucose le els may induce pheno ypic changes in glioma cells, luc ua ions o he glucose concen a ion we e p edic ed o igge mig a ion–p oli e a ion cycles, which in u n inc eased gliomacell in asion and esul ed in as e umou g ow h. 5.2. Pheno ypic plas ici y The abili y o glioma cells o swi ch hei pheno ype in esponse o local cell densi y and changes in he mic oen i - onmen allows adap a ion and is belie ed o ha e impo an implica ions o glioma in asion. In pa icula , glioma cells can change om a p oli e a i e o a mig a o y pheno ype depending on mic oen i onmen al condi ions [43,44]. I is hus c ucial o in es iga e he ac o s and condi ions ha d i e he ansi ion om he p oli e a i e o he mo ile pheno- ype. Se e al ma hema ical models ha e been in oduced o analyse implica ions o his ‘Go-o -G ow’ dicho omy on glioma in asion. Tek onidis e al. [25] p oposed a la ice- LGCA model o explain he spa io- empo al e olu ion o U87 umou sphe oids in i o epo ed in [166]. I u ns ou ha he ‘Go-o -G ow’ mechanism combined wi h sel - epulsion and a densi y-dependen pheno ypic swi ch is equi ed o quan i a i ely ep oduce he expe imen al obse a ions ( igu e 3). (a) (b) Figu e 2. In asi e cell mig a ion. F on in e ace o small (a) and high (b) e ec- i e cell–cell adhesion alues. Shown a e he simula ions o a disc e e s ochas ic la ice model; e e y black do ep esen s a cell, and e e y whi e do co esponds o an emp y si e. The sys em size is 400 400 (in uni s o cell diame e ). Figu e ep oduced wi h pe mission om [27]. si . oyalsocie ypublishing.o g J. R. Soc. In e ace 14: 20170490 9 9. Wen PY, Rea don DA. 2016 Neu o-oncology in 2015: p og ess in glioma diagnosis, classi ica ion and ea men . Na . Re . Neu ol. 12, 69–70. (doi:10. 1038/n neu ol.2015.242) 10. Ba ich KA e al. 2017 Long- e m su i al in glioblas oma wi h cy omegalo i us pp65- a ge ed accina ion. Clin. Cance Res. 23, 1898–1909. (doi:10.1158/1078-0432.CCR-16-2057) 11. 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