Evasion of tumours from the control of the immune system: consequences of brief encounters

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1 Al-Tameem et al. Bology Drect 22, 7:3 RESEARCH Open Access Evason of tumours from the control of the mmune system: consequences of bref encounters Mohannad Al-Tameem, Mark Chaplan * and Alberto d Onofro 2* Abstract Background: In ths work a mathematcal model descrbng the growth of a sold tumour n the presence of an mmune system response s presented. Specfcally, attenton s focused on the nteractons between cytotoxc T-lymphocytes (CTLs) and tumour cells n a small, avascular multcellular tumour. At ths stage of the dsease the CTLs and the tumour cells are consdered to be n a state of dynamc equlbrum or cancer dormancy. The precse bochemcal and cellular mechansms by whch CTLs can control a cancer and keep t n a dormant state are stll not completely understood from a bologcal and mmunologcal pont of vew. The mathematcal model focuses on the spato-temporal dynamcs of tumour cells, mmune cells, chemoknes and chemorepellents n an mmunogenc tumour. The CTLs and tumour cells are assumed to mgrate and nteract wth each other n such a way that lymphocyte-tumour cell complexes are formed. These complexes result n ether the death of the tumour cells (the normal stuaton) or the nactvaton of the lymphocytes and consequently the survval of the tumour cells. In the latter case, we assume that each tumour cell that survves ts bref encounter wth the CTLs undergoes certan benefcal phenotypc changes. Results: We explore the dynamcs of the model under these assumptons and show that the process of mmunoevason can arse as a consequence of these encounters. We show that the proposed mechansm not only shape the dynamcs of the total number of tumor cells and of CTLs, but also the dynamcs of ther spatal dstrbuton. We also brefly dscuss the evolutonary features of our model, by framng them n the recent quas-lamarckan theores. Conclusons: Our fndngs mght have some nterestng mplcaton of nterest for clncal practce. Indeed, mmuno-edtng process can be seen as an nvoluntary antagonstc process actng aganst mmunotherapes, whch am at mantanng a tumor n a dormant state, or at suppressng t. Revewers: Ths artcle was revewed by G. Bocharov (nomnated by V. Kuznetsov, member of the Edtoral Board of Bology Drect), M. Kmmel and A. Marcnak-Czochra. Keywords: Tumour growth, Immune response, Cytotoxc T-lymphocytes, Immuno-evason, Mathematcal models, Chemotaxs, Dffuson, Immuno-edtng Background Cancer research, both expermental and theoretcal, n last 2 years has been deeply nfluenced by the paper The Hallmarks of Cancer by []. In ths paper sx key aspects ( hallmarks ) of cancer development and growth were dentfed and examned. Ther nterplay wth other *Correspondence: chaplan@maths.dundee.ac.uk; alberto.donofro@eo.eu Dvson of Mathematcs, Unversty of Dundee, Dundee, Scotland, UK 2 Department of Expermental Oncology, European Insttute of Oncology, Va Rpamont 435, Mlano, I-24, Italy cellular populatons was manly seen as cooperatve, and thus postve for the tumour. Recently, the same authors have publshed an nterestng follow-up paper, Hallmarks of Cancer: The Next Generaton [2], where ther descrpton of cancer development and growth was modfed and up-dated. In partcular, among the varous new topcs dscussed, two mportant new aspects were added: the role of epgenetc phenomena and the possblty of compettve nterplay wth the nnate and adaptve mmune systems. In partcular, evason from mmune destructon s explctly lsted as a new hallmark. 22 Al-Tameem et al.; lcensee BoMed Central Ltd. Ths s an Open Access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

2 Al-Tameem et al. Bology Drect 22, 7:3 Page 2 of 22 Tumour cells are characterzed by a large number of genetc and epgenetc events leadng to the appearance of specfc antgens (e.g. mutated protens, under/overexpressed normal protens and many others) trggerng reactons by the both the nnate and the adaptve mmune system [3-7]. These observatons have provded a theoretcal bass to the emprcal hypothess of mmune survellance,.e. that the mmune system may act to elmnate tumours [8], only recently expermentally and epdemologcally confrmed [9]. Of course, the compettve nteracton between tumour cells and the mmune system nvolves a consderable number of events and molecules, and as such s extremely complex. Moreover, to descrbe fully these mmuno-oncologcal dynamcs, one has to take nto account a range of spatal phenomena, whch are of outmost relevance n determnng the dynamcs of both mmunogenc and nonmmunogenc tumours [-2]. In partcular, the nterplay between tumour cells and the mmune system s strongly nfluenced by the spatal moblty of both tumour cells and cells of the mmune system.e. effector cells [3]. Apart from the random moton of both types of cell, a promnent role s played by chemoattracton of effector cells towards the tumour cells. Indeed, chemotactc moton of mmune system cells s a hallmark of the defence of the human body aganst non-self agents, ncludng tumours, snce cells belongng to both the nnate mmune system (e.g. Natural Kllers, Macrophages, Dendrtc Cells, etc.. [3]) and adaptve mmune system (e.g. Cytotoxc T Lymphoctes, etc.. [3]) are able to reach ther targets thanks to the gradents of varous knds of chemcals [3,4], e.g. nflammaton-related substances produced by tumour cells. Thus, chemotaxs s of paramount mportance n the nterplay between tumours and the mmune system, snce t nfluences the control of tumour growth and also the mmune survellance. However, besdes temporal and spatal non-lneartes, another mportant pont to stress s that the structure of the above-mentoned nteractons s also characterzed by a seres of evolutonary phenomena. As s self-evdent, the mmune system s not able to elmnate all neoplasms. In other cases, a dynamc equlbrum may also be establshed, such that the tumour may survve n a so-called dormant state [5-8], whch s undetectable (.e. the tumour perssts at a very low, undetectable level of cells but s not completely elmnated by the mmune system). Untl recently ths was largely nferred from clncal data, but [5] have been able to show expermentally, through an ad hoc mouse model, that adaptve mmunty can mantan an occult cancer n an equlbrum state. It s qute ntutve that ths equlbrum can be dsrupted by sudden events affectng the mmune system. If dsease-related mparments of the nnate and adaptve mmune systems, or mmuno-suppressve treatments precedng organ transplantatons occur, then tumour regrowth occurs [9,9]. Ths has been shown both by mouse models and through epdemologcal studes [9,9]. However, there s a major class of causes of dsrupton of the equlbrum that s not related to mmunosuppresson. Over a long perod of tme [9], a neoplasm may develop multple strateges to crcumvent the acton of the mmune system [5,9], whch may allow t to recommence growng [9,8] nto clncally apparent tumours [5], whch theoretcally can reach ther maxmum carryng capacty [8]. From an ecologcal pont of vew, we could say that the tumour has adapted to survve n a hostle envronment, n whch the ant-tumour mmune response s actvated [9,8]. For example, the tumour may develop mechansms to grow and spread by reducng ts mmunogencty [5,9]. In other words, the mmunogenc phenotype of the tumour s nfluenced by the nteracton wth the mmune system of the host. For ths reason, the theory of the nteractons between a tumour and the mmune system has been called mmuno-edtng theory [9]. An mpressve body of research s accumulatng on mmuno-evasve strateges, and a recent monograph [2] has been devoted to some aspects of ths fascnatng subject and to ts close relatonshp wth the effectveness of mmunotherapes. As far as the mathematcal modellng of tumour and mmune system nteractons s concerned, there are many papers n the current lterature whch use determnstc models [3,7,8,2-3] or stochastc models [3-35], as well as models ntroduced by Bellomo based on the knetc theores of nonlnear statstcal mechancs [36,37]. The general approach of Bellomo s theory s based on the concept of changes of actvtes of both the tumour cells and the effector cells of the mmune system after encounters between them. As far as spatal aspects are concerned, [38,39] developed a detaled spato-temporal model focused on the role of macrophages. They showed that the presence of chemoattracton of macrophages towards the tumour cells mples both the onset of travelng waves and a heterogeneous spatal dstrbuton of the tumour cells (see also [4]). Matzavnos, Chaplan and Kuznetsov proposed a spatotemporal model of the nteractons between tumour cells and cytotoxc T-lymphocytes (CTLs) [3,6] by ncludng the spatal motlty of both tumour cells and CTLs, as well as chemotactc moton of the CTLs. They focused manly on the role of the mmune system n determnng dormant states of the tumour, by showng, through a seres of smulatons, that a dormant state s reached, but the tumour cells are spatally dstrbuted n an rregular pattern, whch also temporally oscllates n a non-perodc fashon (see also [4]). In [8,28], the mmuno-edtng phenomenon was emprcally modelled

3 Al-Tameem et al. Bology Drect 22, 7:3 Page 3 of 22 by allowng the presence of slowly tme-varyng generc parameters n determnstc models (wth tme-scales sgnfcantly longer than those typcal of the tumour-mmune system nteracton). Recently, n the framework of the above-mentoned knetc approach, a generc model has been proposed for the learnng ablty of effector cells and for the hdng of tumour cells [4]. In ths paper, based on the concept ntroduced by [9] that the mmune system has the ablty of sculptng the phenotype [9] of tumour cells (.e. promotng the change towards less munogenc and more resstant phenotypes), we propose a cellcentered sem-mechanstc approach amed at descrbng a possble mmunologcally realstc knetc mechansm through whch mmuno-evason begns. Snce there s strong expermental evdence that type, densty and locaton of CTLs are predctve of the clncal outcome of some tumours, such as colorectal tumours [42], and snce we are nterested n the long-tme dynamcs, here we shall deal wth the nterplay of a neoplasm wth CTLs. In our model, we suppose that the tumour cells that survve an attack by CTLs have a probablty of acqurng (through mutatons or even by epgenetc changes) a phenotype that s more resstant to future attacks by CTLs. In turn, at each new encounter wth a CTL, ths resstance can be ncreased further, and after a fnte number of encounters acompleteormaxmal resstance to specfc mmunty s acqured. Moreover, specfc spatal effects may be lnked to the mmuno-evason of neoplasms. Indeed, recently [43] showed expermentally that tumour cells can produce chemcal CXCL2 that, for large concentratons, act as chemorepellent for CTLs, whereas for low concentraton t acts as a normal chemoattractor. We ntegrate - wth some smplfcatons - these expermental fndngs n our model by permttng n the range of features defnng the ncreasngly resstant tumour cell phenotypes an ncreasng ablty to produce such chemorepulsve substances, whereas n a future model we shall consder the above descrbed nonmonotone behavor of the taxs. These two bo-theoretcal hypotheses, although new, are n lne wth the general schema of tumour cell escape from the mmune response. Indeed, as stressed by [9], tumour cells may escape from mmune control through two general mechansms: (a) mechansms that nvolve the secreton of soluble factors; (b) mechansms that are dependent on the contact between the tumour cells and the effectors and that are amed at reducng antgen recognton/adheson and apoptotc resstance. Gven the current expermental knowledge the above mentoned factors are prmarly amed - apart from, n many cases, ther mtogenc acton - at nducng the emergence of mmunosuppressve networks [44]. In our present model, the factors, n lne wth the anmal model by [43], are chemcals that repel CTLs. Fnally, n the concludng remarks, we shall also dscuss, from and evolutonary pont of vew, the dfferences of our model wth the current evolutonary vew of the mmuno-edtng. Methods The Mathematcal Model Followng the knetc scheme employed by [3], n absence of mmuno-edtng mechansms the nterplay between tumour cells and tumour-nfltratng cytotoxc- T-lymphocytes can be modelled as shown n Fgure (see: [3]), where T denotes a tumour cell, E denotes an effector cell (CTL), C denotes the complex formed, T denotes adeadtumourcellande denotes a dead effector cell. The followng assumptons are made: the complexes C consst of a tumour cell and a CTL formng at a rate k +. The parameter k + conssts of the encounter rate between a tumour cell and a CTL, the probablty that the CTL recognzes the tumour cell as a non-self entty, and also the probablty that the tumour cell forms a complex wth the CTLs the break-up of complexes can lead to a stuaton where both the tumour cell and the CTLs are alve wth a rate k the break-up of complexes can lead to a stuaton where ether the mmune cell or the tumour cell survves the encounter wth a rate k the probablty that a tumour cell s klled s p,and correspondngly the probablty that a CTL s klled (.e. the tumour cells survves) s ( p) Usng the Law of Mass Acton, ths leads to the followng system of dfferental equatons descrbng these specfc knetc nteractons: t C = k + ET (k + k)c t T = k + ET + k C + k( p)c () t E = k + ET + k C + kpc The key dea proposed n ths paper whch develops the work of [3], s that a proporton of the tumour cells that survve an encounter wth a CTL are more resstant Fgure Basc local lymphocyte-cancer cell nteractons. Schematc dagram of the basc local lymphocyte-cancer cell nteractons.

4 Al-Tameem et al. Bology Drect 22, 7:3 Page 4 of 22 to any future attacks by CTLs. Consequently, the phenotypc propertes of these new enhanced tumour cells wll be dfferent from those of the nave tumour cells. Specfcally, we make the addtonal assumptons: ther probablty of beng klled (prevously the parameterp) s smaller ther probablty of beng recognzed and also of formng a complex wth a CTL (embedded n the parameter k + ) s smaller Moreover, we shall also assume that the prolferaton rate of CTLs stmulated by the presence of the complexes s also smaller. We denote the nave tumour cells by T (t) and the non-nave tumour cells by T,where stands for the number of prevous encounters wth the CTLs. We assume that the ftness of tumour cells ncreases up to a maxmum number of encounters N, mplyng that we consder n total + N classes of tumour cells, T, T,..., T N. The new knetc relatonshps of our model are llustrated n Fgure 2 and are characterzed by the followng new groups of parameters: the rate of formaton of complexes [ET ]: k +.We assume that k + s constant or decreasng wth ndex, wth k + N ; the probablty that a tumour cell of the -th class s klled: p. We assume that p s decreasng wth ndex, wth p N ; the probablty of transton T T + to the state : θ. We assume that θ s ncreasng for N. Sncewehaveassumed N classes of tumour cells, θ N =. As far as the temporal dynamcs of the tumour cells, CTLs and complexes s concerned, once agan usng the Law of Mass Acton, the knetc scheme of Fgure 2 can be translated nto the followng system of ordnary dfferental equatons: T T C l E = k + ET + k C + k( θ )( p )C = k + ET + kθ ( p )C + (k + k( θ )( p ))C = k + l ET l (k + k)c l = E k j + T j + k C j + kp j C j j= where =,..., N,andl =,..., N. However, not only the temporal but also the spatotemporal propertes of the ftter tumour cells are lkely to be dfferent from those of the baselne tumour cells. Namely: the producton rates of chemoattractants stmulated by a complex CTL+ non nave tumour cell s assumed to be smaller than that of the nave cells snce recently Vanello et al. [43] showed n an anmal model that tumours produce chemcals that repels the CTLs, here we assume that those chemorepellents are produced by the non-nave cells. For the sake of the precson, Vanello s fndngs showed chemoattracton for low concentratons of chemcal CXCL2 emtted by tumour cells. For the sake of smplcty here we shall only consder chemorepulson. In the followng, we provde the full equatons for all varables, ncludng the spatal components. As mentoned n the prevous secton, we have assumed the model of [3] as our baselne model. j= j= (2) Spatotemporal Dynamcs of the Tumour Cells Followng [3], we assume that the tumour growth may be descrbed by a logstc law, and that the tumour cells mgrate randomly. Thus, t follows that the spatotemporal dynamcs of the nave tumour cells T s as follows: Fgure 2 Extended local lymphocyte-cancer cell nteractons. Schematc dagram of the extended local lymphocyte-cancer cell nteractons. T logstc growth { random moton }} {{}}{ N = D T 2 T + r T β T j local knetcs j= {}}{ k + ET + (k + k( θ )( p ))C

5 Al-Tameem et al. Bology Drect 22, 7:3 Page 5 of 22 and the dynamcs of the non-nave cells T s gven by: T = logstc growth { random moton }} {{}}{ N D T 2 T + r T β T j j= local knetcs {}}{ k + ET + (k + k( θ )( p ))C + kθ ( p )C where =,..., N,andwherer s the baselne exponental growth rate of the tumour (.e. ts theoretcal growth rate when t s small ) and β s the nverse of ts carryng capacty (n absence of mmune reactons). Spatotemporal Dynamcs of the CTLs Consderng the CTLs, as n [3], both random and chemotactc moton of these cells s ncluded. However, as prevously dscussed, an addtonal type of motlty s ncluded due to the postulated onset of negatve taxs due to the producton of a chemorepellent ρ by the nonnave tumour cells. Ths results n the followng equaton: E = random motlty {}}{ D E 2 E chemotaxs {}}{ χ(α).(e α) prolferaton {}}{ chemorepulson supply f q {}}{{}}{ j C j j= + A(ρ).(E ρ)+ sh(x) + g + N T j j= local knetcs { decay }} {{}}{ d E E k j + T j + k C j + kp j C j j= The prolferaton of the CTLs, E, stmulated by the complexes C j s embedded n the rate constant q j, whch, as a consequence, must be decreasng wth the ndex j, wth q N (f, g are constant parameters). Note that n absence of mmuno-edtng ths prolferaton term reads fc/(g +T), and t has been has been ntroduced n [22,45]. It represents the expermentally observed enhanced prolferaton of CTLs n response to the tumour. Ths functonal form s consstent wth a model n whch one assumes that the enhanced prolferaton of CTLs s due to sgnals, such as released nterleukns, generated by effector cells n tumour cell-ctl complexes. We note that the growth factors that are secreted by lymphocytes n complexes (e.g IL-2) act manly n an autocrne fashon. That s to say they act on the cell from whch they have been secreted and thus, n our spatal settng, ther acton can be adequately descrbed by a local knetc term only, j= j= wthout the need to ncorporate any addtonal nformaton concernng dffusvty. The external nflux of CTLs s, for the sake of the smplcty, modelled as sh(x),whereh(x) s a Heavsde functon, taken to be zero over a gven subregon of the doman of nterest cf. [3]. In other words, we assume that there s a subdoman where lymphocytes are not naturally present and whch s penetrated by CTLs only thanks to dffuson and chemotaxs. Chemoattractant The spatotemporal dynamcs of the chemoattractant α produced by the complexes s gven by α producton dffuson decay {}} {{}}{{}}{ = D 2 2 α δ α + π j C j where we assume that the producton rate constant π s decreasng wth ndex, wthπ N, snce we assume that complexes between CTLs and non-nave tumour cells are less and less able to produce such a chemoattractant. Chemorepellent Adaptng the expermental fndngs by Vanello to our framework, we suppose that the non-nave tumour cells produce a chemcal that repels the CTLs and whose concentraton ρ sgovernedbytheequaton ρ j= producton { dffuson decay }} {{}}{{}}{ = D 2 ρ δ 2 ρ + w j T j where the producton rate constants w are such that: w = (absence of producton for nave tumour cells) and w < w 2 < < w N. Tumour cell-ctl Complexes Followng [3], we assume that the motlty of the complexesssosmallthattcanbeneglected: C l = where l =,..., N. j= local knetc {}}{ k + l ET l (k + k)c l

6 Al-Tameem et al. Bology Drect 22, 7:3 Page 6 of 22 Modelng the transton rates and probabltes Concernng the transtons T T +, we assume that they are a lnear functon of : θ = θ + (θ MAX θ ), =,..., N N θ N =, θ MAX = θ. and that ther baselne value s suffcently small: 5 θ 3. In other words we assume that the probablty of acqurng the less mmunogenc phenotype s small (n analogy wth the smallness of the probablty of survvng to an attack by a CTLs). The probablty p that a tumour cell of class T s lethally ht s gven by: p = p + (p N p ) N, where p N < p. Concernng the rates k +, we assume ether that they are constant or that they are lnearly decreasng wth k N + = : k + = k + ( N ). The producton rate of the chemoattractant s also assumed to vary lnearly: ( π = π ) N wth [6]: π = 2 3 molecules cells mn,we suppose that the chemorepellent s produced va a mechansm of threshold generaton,.e. only after a suffcent number of encounters, yeldng:, f N, w = N (3) w MAX, N < N N N where we assumed: w MAX π (.e. the maxmum producton rate of the chemorepellent s equal to the producton rate of the chemoattractant n absence of mmuno-edtng phenomena). Boundary and ntal condtons We ntally consder the model n a fxed -dmensonal doman [, x a ] and close the system by applyng approprate boundary and ntal condtons. As far as the boundary condtons are concerned, zero-flux boundary condtons are mposed on all state varables (apart from C): E, α, ρ and T, =,..., N. These boundary condtons are approprate for the tumour-mmune dynamcs we are consderng. For example, n BCL lymphomas of the spleen tumour cells are spatally contaned n the lymph tssue of the spleen, an elongated organ that, n mce, s characterzed by a very strong basement membrane, whch s only broken when the tumour cells swtch to an nvasve phenotype. Snce here we are concerned wth earler stage dynamcs of tumour cells n a dormant state evadng the CTLs, t follows that the zeroflux boundary condtons are adequate for our partcular model. As far as the ntal condtons are concerned, we assume an ntal front of nave tumour cells encounterng a front of CTLs, resultng n the formaton of C complexes. We suppose that ntally there are no non-nave tumour cells and hence no complexes nvolvng them. No chemcals are ntally present n the spatal doman. These assumptons yeld: {, f x l, E(x,) = E a ( exp( (x l) 2 )), f l < x x a. { Ta ( exp( (x l) T (x,) = 2 )), f x l, f l < x x a. {, f x / [ l ɛ, l + ɛ], C (x,) = C a exp( (x l) 2 ), f x [ l ɛ, l + ɛ]. T (x,) =, C (x,) =, x [,x a ]. α(x,) =, ρ(x,) =, x [,x a ]. where E a = s, T a =, C a = mn(e a, T a ), <ɛ, d β =,..., N. Note that E a s the baselne homogenous steady-state value of CTLs n the absence of tumour cells, and that T a s the the baselne homogenous steady-state value of the nave cells n absence of CTLs, whereas C a s the maxmum possble densty of complexes resultng from ntal values of E and T. Results and Dscusson We smulated our system after nondmensonalzng t as shown n the Appendx, where one can also fnd the numercal values of the parameters. In addton to the baselne parameter set detaled n the Appendx, n the followng sectons all our smulatons were performed assumng the followng values for key parameters assocated wth the encounter of CTLs and tumour cells: θ = 4, p N {,.5,.75,.9997}. Moreover k + =.3 7 [3] and k + may be ether constant or lnearly decreasng, wth k N + =. Snce the average lfespan of a chmerc mouse s three years, and snce we are nterested n assessng the possblty (and spato-temporal modalty) of the onset of mmunoevason of a tumour, all smulatons (unless stated) represent an nterval of length days 3years. Spatally Homogenous Case In ths frst set of smulatons, we set all the spatal components of the model to zero and consder only the reacton knetcs n order to ascertan whether the prmary mechansm of evason can be purely temporal. All smulatons

7 Al-Tameem et al. Bology Drect 22, 7:3 Page 7 of 22 suggest that our model, wth the parameter assumptons and values we used, s able to reproduce the onset of mmunoevason n a bologcally realstc tme-frame. Fgure 3 shows the plots of the growth of the tumour cell populaton over tme where the kllng probablty at the last stage s zero,.e. p N =, and k + = constant =.3 7.WeobservethatfN = 4theonsetofevason s at t 2 days,.e. the tumour remans dormant for 2 days, whch s a long perod of tme for a mouse. On the contrary, f N = then the mmunoevason s delayed even further, wth onset at t 5 days. Fgure 4 shows the plots of the growth of the tumour cell populaton over tme where the kllng probablty at the last stage s not zero but t s only halved,.e. p N =.5, and as n the prevous fgure, k + = constant =.3 7. Also n ths case the mmunoevason s reproduced and takes place, respectvely, at t 425 days for N = 4andat t 95 days for N =. Fgure 5 shows the plots of the growth of the tumour cell populaton over tme where the kllng probablty at the last stage s p N =.75, and as n the prevous fgure, k + = constant =.3 7. These results are dfferent from the prevous two cases. Here the mmunoevason takes place n the lfespan of the mouse only for the case N = 4. Ths suggests that n absence of changes n the parameter k + : ) the late stages T are the most mportant to determne the onset of the evason; ) due to the fnte lfespan of chmerc mce and to the slow rate of the transtons, Fgure 3 Growth of the tumour: nonspatal case. Plots showng the growth of the tumour cell populaton over tme n the case where the spatal components of the model (.e. all dffuson, taxs terms) have been set to zero. The plots show that the tumour can evade the mmune system for ether approxmately 2 days or approxmately 5 days dependng on the parameter N. Parameter values: p N = and k + = constant =.3 7 and: N = 4 (sold lne) and N = (dashed lnes). The red lnes represent the populaton T, the blue lnes represent the summed populatons T T N, and the black lnes represent the summed populatons T T N.Tmet s n days Fgure 4 Growth of the tumour cell populaton: nonspatal case 2. Plots showng the growth of the tumour cell populaton over tme n the case where the spatal components of the model (.e. all dffuson, taxs terms) have been set to zero. The plots show that the tumour can evade the mmune system for ether approxmately 4 days or approxmately 95 days dependng on the parameter N. Parameter values: p N =.5 and k + = constant =.3 7 and: N = 4 (sold lne) and N = (dashed lnes). The red lnes represent the populaton T, the blue lnes represent the summed populatons T T N, and the black lnes represent the summed populatons T T N.Tmet s n days. the mmunoevason process requres that the maxmum ablty of genetc or epgenetc changes n a tumour cell upon formng a complex wth a CTL (embedded n the transton probablty whose maxmum, we recall, s at = N ), s reached n a small number of encounters. Fgure 6 shows the growth of the tumour cell populaton over tme where the parameter p N =.75, but n ths case the parameters k + are lnearly decreasng wth k N + =. We notce the followng dfferences from the prevous case: ) here the onset of mmunoevason s for N = 4att 25 days,.e. t s consderably accelerated; ) there s the onset of mmunoevason (at t 55 days)also for N =. Thus, ths smulaton suggests that the role of the decrease of the probablty that a tumour cell s recognzed by a CTL s mportant for the tmng of the onset of mmunoevason. Moreover, the decrease of the parameters k + alone s suffcent to nduce mmunoevason, as suggested n the smulatons shown n Fgure 7, where p = constant =.9997 and k + are lnearly decreasng. However,asshownnFgure8,fp N =, then the addton of the mechansm of decreasng k + does not accelerate the onset of mmunoevason to such a degree wth respect to the baselne case of constant k + shown n the prevous Fgure 3. Fnally, comparng the results of our smulatons, n the cases where k + s decreasng we note that the maxmum sze of the nave tumour cells compartment s smaller

8 Al-Tameem et al. Bology Drect 22, 7:3 Page 8 of Fgure 5 Growth of the tumour cell populaton: nonspatal case 3. Plots showng the growth of the tumour cell populaton over tme n the case where the spatal components of the model (.e. all dffuson, taxs terms) have been set to zero. The plots show that the tumour can evade the mmune system for ether approxmately 85 days or approxmately 9 days dependng on the parameter N. Parameter values: p N =.75 and k + = constant =.3 7 and: N = 4 (sold lne) and N = (dashed lnes). The red lnes represent the populaton T, the blue lnes represent the summed populatons T T N, and the black lnes represent the summed populatons T T N.Tmet s n days Fgure 7 Growth of the tumour cell populaton: nonspatal case 5. Plots showng the growth of the tumour cell populaton over tme n the case where the spatal components of the model (.e. all dffuson, taxs terms) have been set to zero. The plots show that the tumour can evade the mmune system for ether approxmately 35 days or approxmately 85 days dependng on the parameter N.In ths case however, the ntal populaton T s eradcated. Parameter values: p = constant =.9997 and k + are lnearly decreasng functons and: N = 4 (sold lne) and N = (dashed lnes). The red lnes represent the populaton T, the blue lnes represent the summed populatons T T N, and the black lnes represent the summed populatons T T N.Tmet s n days Fgure 6 Growth of the tumour cell populaton: nonspatal case 4. Plots showng the growth of the tumour cell populaton over tme n the case where the spatal components of the model (.e. all dffuson, taxs terms) have been set to zero. The plots show that the tumour can evade the mmune system for ether approxmately 25 days or approxmately 55 days dependng on the parameter N. Parameter values: p N =.75 and k + are lnearly decreasng functons and: N = 4 (sold lne) and N = (dashed lnes). The red lnes represent the populaton T, the blue lnes represent the summed populatons T T N, and the black lnes represent the summed populatons T T N.Tmet s n days Fgure 8 Nonspatal case 6. Plots showng the growth of the tumour cell populaton over tme n the case where the spatal components of the model (.e. all dffuson, taxs terms) have been set to zero. The plots show that the tumour can evade the mmune system for ether approxmately 2 days or approxmately 4 days dependng on the parameter N. Parameter values: p N = andk + are lnearly decreasng functons and: N = 4 (sold lne) and N = (dashed lnes). The red lnes represent the populaton T, the blue lnes represent the summed populatons T T N, and the black lnes represent the summed populatons T T N.Tmet s n days.

9 Al-Tameem et al. Bology Drect 22, 7:3 Page 9 of 22 than the maxmum sze of all the non-nave tumour cells compartments summed up: Max(T )<Max(T + +T N ). Moreover, Max(T ) seems to be a decreasng functon of p N. These results mght be explaned as follows: the decrease of the competton between all the tumour cells and the mmune system embedded n the decrease of the parameters k +, mght shft the nternal competton between the nave and the non-nave tumour cells. Tumour cell densty t = Spatotemporal Model Before we present the computatonal smulaton results of the new model n ths paper, n Fgures 9 and we plot the spatal dstrbuton of tumour cells and CTLs, respectvely, n the baselne case of the absence of mmunoevasve mechansms. Fgure 9 shows the spatal dstrbuton of tumour cell densty wthn the tssue at tmes, 4, 7, and days. These results llustrate the basc spatotemporal dynamcs of the tumour cell densty nduced by ts nterplay wth the dstrbuton of CTLs.e. a gradual transton between a front of tumour cells to a tran of soltary-lke travellng waves slowly nvadng the tssue, fnally creatng a spatally heterogeneous and tme-changng (through rregular temporal oscllatons) dstrbuton. Smlarly, Fgure shows the correspondng plots of the CTL densty. Fgure shows the spatal dstrbuton of tumour cell densty wthn the tssue at tmes 7, and days where the parameters p N =.75 and k + = constant. These results show that f we nclude the mmunoevasve mechansm wth a decreased value for the parameter p N,we obtan a process that s dentcal to the baselne case for most of the tme. Indeed, bascally the left plot of ths fgure s dentcal to that of Fgure 9. However, after the onset of the evason, the tumour cell densty dstrbuton changes sgnfcantly as can be seen n the left part of the doman. From these observed dfferences, we may say that n ths modellng framework the effect of mmunoevason on the spato-temporal dynamcs s characterzed by a return to a spatally homogeneous steady-state. Ths transton to the new spatal regmen s llustrated n more detal by the tme-slces shown n Fgure 2. Fnally, we note that the effect of chemorepulson, whch s well detectable at level of the spatal denstes T,sno more detectable at level of the total densty of all tumour cells. However, ths effect s agan notceable f we consder the densty of T versus the densty of T + +T N (see the second plot of Fgure ). Fgures 3 and 4 show the correspondng densty of CTLs n the tssue. Note that after the onset of mmunoevason, correspondng to the newly reformed nvasve front of tumour cells at a hgh densty, the densty of CTLs s close to zero. Tumour cell densty Tumour cell densty Tumour cell densty dstance nto tssu t = dstance nto tssu t = dstance nto tssu t = dstance nto tssu Fgure 9 Tumour Spatal densty n absence of mmunoevasve mechansms. Plots showng the spatal dstrbuton of tumour cells wthn the tssue at tmes correspondng to, 4, 7, and days, respectvely, n the baselne case of absence of the mmunoevasve mechansm descrbed n ths paper. Ths corresponds to the results of [3].

10 Al-Tameem et al. Bology Drect 22, 7:3 Page of 22 Lymphocyte densty Lymphocyte densty t = dstance nto tssu t = 4 Fgure 5 shows the spatal dstrbuton of tumour cell densty wthn the tssue at tmes 4, 7, and days n the case where the parameters p N =.75 and k + aredecreasngsuchthatk N + =. Due to the acceleraton of the mmunoevason caused by the synergy exstng between the varablty of p and k +, the plots n ths fgure are substantally dfferent from those n the baselne case n Fgure 9 and also wth respect to the plots n Fgure. Indeed, n ths case the spato-temporal dstrbuton of the tumour cell densty s far more regular, and by t = days almost all of the tssue has been nvaded by the tumour cells close to ther maxmum densty. Moreover, here n large regons of the doman we have T < T + +T N, whch s the opposte of the prevous case, where the nave tumour cells T were prevalent. Fnally, Fgure 5 llustrates the fact that the dstrbutons of nave vs non-nave tumour cells are mrror-mages of one another and they are complementary, snce ther sum s a homogeneous front. In Fgure 6 we show the dfferental effect of chemorepulson on the varous classes of tumour cells. The plots show the total number A (t) of cells n each classes,.e. Lymphocyte densty Lymphocyte densty dstance nto tssu t = dstance nto tssu t = dstance nto tssu Fgure CTLs Spatal densty n absence of mmunoevasve mechansms. Plots showng the spatal dstrbuton of CTLs wthn the tssue at tmes correspondng to, 4, 7, and days, respectvely, n the baselne case of absence of the mmunoevasve mechansm descrbed n ths paper. Ths corresponds to the results of [3]. A (t) = T (x, t)dx, over tme, as well as, n the last plot, the grand-total A (t) + +A N (t). The effect of the chemorepulson on each sub-populaton A s strkng, although overall t s globally compensated (see the last plot). Fgure 7 shows the dstrbuton of tumour cell densty wthn the tssue at tmes correspondng to 7, and days respectvely wth parameter values p N =.5 and k + = constant = k +. Note that n ths case, although p N =.5, probably due to the constancy of k +,nlarge parts of the space the number of nave cells exceeds the rest of the classes of other tumour cells,.e. T > T + +T N. Note that at the end of the average lfespan of the mouse, the tssue s nvaded to a large extent but to a lesser extent than n the case where p N =.5 and k N + =. Fgure 8 shows a more detaled evoluton of the tumour cell densty by presentng the tme-slces from t = to t =. Fgure 9 shows the correspondng dstrbuton of CTL densty. Fnally, Fgure 2 shows the dstrbuton of tumour cell densty wthn the tssue at tmes correspondng to 4, 7, and days respectvely when the parameters p N =.5 and k N + =. Ths fgure summarzes well the mportant role of the two parameters p N and k N + n shapng the spato-temporal dstrbuton of tumour cells. Indeed, the parameter k N + appears to accelerate the onset and the velocty of propagaton of the nvasve front, and moreover t also dfferentally shapes T and T + +T N.

11 Al-Tameem et al. Bology Drect 22, 7:3 Page of 22 t=7 t= Fgure Tumour Spatal densty n presence of mmunoevasve mechansms. Plots showng the dstrbuton of tumour cell densty wthn the tssue at tmes correspondng to 7, and days respectvely. These plots llustrate the spatotemporal onset of mmunoevason. The fnal plot at t = should be compared to the equvalent plot n Fgure 9, whereas the plot n the left panel, referrng to tme t = 7 days,s analogous to the equvalent plot n Fgure 9. These plots suggests that the onset of mmunoevason occurs after t = 7 days. Parameter values p N =.75 and k + = constant. Sold lne wth chemorepellent, dashed lne wthout. The red lnes represent the populaton T, The blue lnes represent the summed populaton T T N, and the black lnes represent the summed populaton T T N. Conclusons In ths paper we have presented a novel mathematcal model of the mmune response to cancer, focusng on the specfc spato-temporal response of cytotoxc T- lymphocytes to tumour cells. We have developed and extended deas orgnally formulated by [3] by proposng a possble knetc mechansm leadng to tumour evason from the mmune control. Our model s based on the key concept that a tumour cell whch survves the formaton of a complex wth a cytotoxc T-lymphocyte can develop, wth a gven probablty, an ncreased probablty of survvng further attacks by CTLs. We do not specfy whether ths so-called ncreased resstance s genetc or epgenetc.indeed,fromaknetcpontofvew,thss mmateral. However, n order to expermentally valdate our hypothess, ths dstncton would be of paramount relevance. Note that the model by [3] s based on a mass-acton law mechansm. The reacton knetcs n a crowded cellular envronment could dffer from that, as studed (n absence of mmunoevason) n [7,27,28]. We shall nvestgate these more realstc settngs n future works, but T +...+T N densty Fgure 2 Tme-slces of Tumour Spatal densty n presence of mmunoevasve mechansms. Plots showng detaled changes n the spatal dstrbuton of all tumour cells N T j wthn the tssue over tme n the case of mmunoevason. Parameter values p N =.75 and k + = constant.ths j= fgure shows the onset of mmunoevason n the nterval (7, ) days.

12 Al-Tameem et al. Bology Drect 22, 7:3 Page 2 of 22 6 t=7 5 t= Lymphocyte Sze 3 2 Lymphocyte Sze Fgure 3 CTLs Spatal densty n presence of mmunoevasve mechansms. Plots showng the dstrbuton of CTLs wthn the tssue at tmes correspondng to, 4, 7, and days respectvely. These plots llustrate the spatotemporal onset of mmunoevason. The fnal plot at t = should be compared to the equvalent plot n Fgure, whereas the plot n the left panel, referrng to tme t = 7 days, s analogous to the equvalent plot n Fgure. These plots suggests that the onset of mmunoevason occurs after t = 7 days. Parameter values p N =.75 and k + = constant. Sold lne wth chemorepellent, dashed lne wthout. we thnk that the basc results showed here should not substantally change. In ths work we have dealt wth the spato-temporal nterplay between tumours and a specfc mmune response from CTLs. We chose ths approach because of the expermental evdence on the relevance of CTLs n determnng tumour dormancy or the evason of many mportant tumours such as melanomas, ovaran carcnomas and colorectal carcnomas, where the presence of nfltratng lymphocytes s a useful prognostc marker [42,46]. However, tumour mmunoevason from dormancy s a mult-faceted phenomenon. We stress here that by no means do we thnk that ours s an exhaustve theoretcal treatment of a such complex phenomenon. Concernng spatal ssues we also brefly menton that also n case of small dormant tumours the next generaton of spatal models of tumour growth should better stress and nvestgate the nterplay of tssue 3D geometry and the tumour vascularzaton wth phenotypc changes n tumour cells. We have bult our model based on the tumour dormancy mathematcal model of [3,6], where parameters were ftted to expermental anmal (mouse) data. However, embeddng the proposed evolutonary mechansm n a more complex settng, where a more 6 TICL densty Fgure 4 Tme-slces of CTLs n presnece of mmunoevason. Plots showng detaled changes n the spatal dstrbuton of CTLs wthn the tssue over tme n the case of mmunoevason. Parmeter values p N =.75 and k + = constant..8

13 Al-Tameem et al. Bology Drect 22, 7:3 Page 3 of 22 t=4 t= t= Fgure 5 Tumour cell densty wthn the tssue n for decreasng k + and p. Plots showng the dstrbuton of tumour cell densty wthn the tssue at tmes correspondng to 4, 7, and days respectvely. These plots llustrate the spatotemporal onset of mmunoevason. Parameter values p N =.75 and k + are decreasng such that k N + =. Sold lne wth chemorepellent, dashed lne wthout. The red lnes represent the populaton T, The blue lnes represent the summed populaton T T N, and the black lnes represent the summed populaton T T N. detaled descrpton of both adaptve and nnate mmunty s ncluded, should lead to results qualtatvely smlar to thoseherellustrated. Our smulatons suggest that the proposed mechansm s able to mmc varous dynamcs of mmunoevason durng the lfespan of a mouse. We have also hghlghted the dfferental spatotemporal contrbutons to evason due, respectvely, to: ) a decrease n the probablty p of beng lethally ht; ) a decrease n the probablty, embedded n k +, that a tumour cell s recognzed by a CTL. In partcular, our model suggests that a decrease n the parameters p s needed to produce evason, whch does not occur n the case where p remans constant at ts baselne level nferred from the expermental data. However, the role of the parameters k + s mportant snce t can greatly accelerate the smulated process. Moreover, our computatonal smulatons also showed that the proposed mechansm can also deeply affect the spatal patternng of the tumour. In partcular, our model suggests that to have a unform nvason profle for the tumour cells necesstates also havng a decrease n the recognton rate, embedded n the parameters k +. These parameters also dfferentally shape the spatal dstrbuton of the varous classes of tumour cells. Concernng the possble chemorepulson of CTLs, our computatonal smulaton results showed that, n our bologcal settngs, although t does not affect the spatotemporal dynamcs of the total number of tumour cells, t has a remarkable nfluence on the spato-temporal dstrbuton of the dfferent ndvdual classes of tumour cells. Further analyss s needed to ascertan f, wth dfferent parameters, the effect of ths factor can be dfferent, and n order to understand the behavour n the current settng. As far as the key mmuno-evason-related parameters such as θ, p,andk + are concerned, we were not able to ft them wth expermental data (apart, of course, from the values for p and k +, from [3,6]) because n the lterature, to the best of our knowledge, mmuno-evason of tumours s only llustrated by means of qualtatve clncal or molecular expermental fndngs. In partcular, no mmuno-evason-related tumour growth data are avalable. Indeed, a complete expermental knetc study of the adaptve evason from tumour dormancy allowng, for example, the plottng of tumour growth curves would currently be very dffcult to undertake. Thus we hope that ths theoretcal work may contrbute to trggerng such expermental nvestgatons, whch would allow us to valdate our model.

14 Al-Tameem et al. Bology Drect 22, 7:3 Page 4 of 22 From a theoretcal pont of vew, our model, although detaled and focused on a very specfc aspect of mmunooncology, and on some very specfc mechansms, s conceptually n lne wth the general theores by Bellomo [36,37,4], who consders tumour cells and mmune system effector cells as actve partcles endowed wth actvtes and propertes. Indeed, also n ths paper the changes of actvtes of cells upon encounters between tumour cells and effector cells of the mmune system are central n determnng the dynamcs of the system. To the best of our knowledge, the evolutonary nature of the mmuno-edtng process has been studed untl now under the framework of the so-called modern synthess, followng whch the envronment (n our case the mmune system) s not the causatve agency [47] but a mere selectve force promotng fxaton of adaptve genomc changes [47]. In the case of mmunoevason, a lowly mmunogenc clone may appear spontaneously due to the large random mutaton rate of tumour cells. Ths new phenotype s then nvoluntarly selected by the mmune system, whch klls the other phenotypes that reman strongly mmunogenc. Thus the sculptng of tumour cells phenotypes [9] mentoned n the ntroducton, s nvoluntary, passve. On the contrary, n our model the more mmunoresstant phenotypes may arse because of genetc or epgenetc causes - due to the nteracton between the tumour cells and the mmune system. Ths pont of vew, whch s quas-larmackan [47], s n lne wth a number of recent dscoveres that are leadng to a new theory of extended evolutonary synthess [48], whch ndeed nvestgates the mpact of both genetc and epgenetc nhertance on evolutonary phenomena n order to decpher the complex nterplay between genotypes, epgenotypes, phenotypes and envronment. From a bologcal and bophyscal pont of vew ths mples also ncludng two other key components: tmescales (see, e.g., the revew paper [49]) and the role of randomness. Followng ths extended and more modern perspectve, we thnk that two general classes of evolutonary mechansms mght bologcally underle our knetc model of transtons between phenotypes of tumour cells: stress-nduced mutatons and epgenetc swtches. In stress-nduced mutageness [47,49], some knd of stress nduces genomc changes n a cell that, although probablstc (e.g. n our model θ (, )), are trggered by the cell and ts nterplay wth the external world, and that are benefcal for the cell [47,49]. As a result, the stress-nduced mutatons are adaptve and benefcal, and thus they are natural canddates to explan bologcally the knetc mechansm of our mathematcal model. Moreover, t s mportant to note that stress-nduced genomc changes are a well-known and mportant mechansm n tumours [47,5,5]. Fnally, Koonn hypotheszed A A A 2 A 3 N A 4 A j j = 2 x x 5 (a) x 5 (b) x (c) x 8 (d) x 8 (e) (f) Fgure 6 Effects of chemorepulson on the total number of spatally dstrbuted classes of tumour cells. Plots showng the effects of chemorepulson on the total number of spatally dstrbuted classes of tumour cells. Plots of the total number of cells A (t) = T (x, t)dx. Panels: (a) A, (b) A, (c) A 2, (d) A 3, (e) A 4, and (f) 4 A j (t). Sold lne wth chemorepellent, dashed lne wthout. j= Tme s measured n days.

15 Al-Tameem et al. Bology Drect 22, 7:3 Page 5 of 22 t=7 t= Fgure 7 Dstrbuton of tumour cell densty wthn the tssue for decreasng p and costant k +. Plots showng the dstrbuton of tumour cell densty wthn the tssue at tmes correspondng to 7, and days respectvely. These plots llustrate the spatotemporal onset of mmunoevason. Parameter values p N =.5 and k + are constant. Sold lne wth chemorepellent, dashed lne wthout. The red lnes represent the populaton T, The blue lnes represent the summed populaton T T N, and the black lnes represent the summed populaton T T N. [47] that quas-larmarckan processes are essentally trggered by very strong sgnals (see e.g. Fgure 3 of [47]), whch, we remark, s the case for tumour-ctl nterplays. As far as the epgenetc path s concerned, t s now known that there are nhertable phenotypc changes wthout underlyng genetc varatons [48,49], and that sometmes those changes are unusually rapd [49]. The epgenome s dynamc and t reflects an ndvdual s or a tssue s envronmental exposure durng the whole of ts lfespan [52]. Among the possble epgenetc changes, we menton specfcally the methylaton of DNA bases ( epmutatons ), and also the swtchng between two dfferent equlbrum states n the bochemcal pathways nfluencng a set of nter-related phenotypes (e.g. low mmunogencty and large mmunogencty ), due for example, to strong stress sgnals. Indeed, due to hysteress phenomena nducng memory, also when an external stress sgnal s removed the system may not return to ts orgnal state. Ths memory may reman stable for many generatons thus provdng an example of epgenetc nhertance [49]. T +...+T N densty Fgure 8 Changes n the spatal dstrbuton of all tumour cells N T j wthn the tssue for decreasng p and costant k +. Plots showng j= detaled changes n the spatal dstrbuton of all tumour cells N T j wthn the tssue over tme n the case of mmunoevason. Parameter values j= p N =.5 and k + = constant.