Embed Size (px)
Citation preview
ARTICLE IN PRESS
0022-5193/$ - se
doi:10.1016/j.jtb
�CorrespondQuinta Grande
Tel.: +351 214
E-mail addr
Journal of Theoretical Biology 247 (2007) 122–137
www.elsevier.com/locate/yjtbi
How regulatory CD25+CD4+T cells impinge ontumor immunobiology? On the existence oftwo alternative dynamical classes of tumors
Kalet Leona,b,�, Karina Garciaa, Jorge Carneirob, Agustin Lagea
aCentro de Inmunologıa Molecular, Habana, CubabInstituto Gulbenkian de Ciencia, Rua da Quinta Grande, no. 6, Apartado 14, P-2781-901, Oeiras Codex, Portugal
Received 3 October 2006; received in revised form 31 January 2007; accepted 31 January 2007
Available online 11 February 2007
Abstract
Aiming to get a better insight on the impact of regulatory CD25+CD4+ T cells in tumor immunobiology, a simple mathematical
model was formulated and studied. This model is an extension of a previous model for the dynamics of autoreactive regulatory cells and
effector cells that interact upon their co-localized activation at the antigen presenting cells (APCs). It assumes that tumor growth
stimulates the activation and migration to the adjacent lymph node of fresh APCs loaded with tumor antigens. These APCs stimulate the
growth of both effector and regulatory T cells, which may then migrate to the tumor site and induce tumor cell destruction. Our results
predict the existence of two alternative dynamic modes of unbounded tumor growth. In the first mode, the tumor induces the expansion
of effector T cells that outcompete regulatory T cells, but nevertheless fail to control the tumor. In the second mode, the tumor induces a
balanced expansion of both effector and regulatory T cells, which prevents the tumor from being destroyed by the immune cells. Tumors
characterized by a high specific growth rate, low immunogenicity, and that are relatively resistant to T cell destructive functions, will
grow in the first mode; conversely, tumors that have a slow specific growth rate, that are immunogenic, and/or that are more sensitive to
destruction by T cells will grow in the second mode. Overall, this result provides a simple explanation to the fact that the development of
some tumors expands regulatory T cells while others do not, predicting how some key dynamical properties of the tumor determine either
one or the other type of behavior.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Regulatory T cells; Tumor immunology; Tumor growth dynamics; Tumor immune system interaction; Mathematical modelling
1. Introduction
The immune system is regulated at different levels byregulatory T cells, which are contained within theCD4+CD25+Foxp3+ pool and actively suppress theactivity of other cells (Fontenot et al., 2003; Hori et al.,2003; Khattri et al., 2003). Regulatory T cells seem to beengaged in T cell homeostasis (Almeida et al., 2005); in the
e front matter r 2007 Elsevier Ltd. All rights reserved.
i.2007.01.029
ing author. Instituto Gulbenkian de Ciencia, Rua da
, no. 6, Apartado 14, P-2781-901, Oeiras Codex, Portugal.
407 920; fax: +351 214 407 973/970.
esses: [email protected], [email protected],
om (K. Leon).
prevention of autoimmune diseases (Baecher-Allan andHafler, 2006; Sakaguchi and Sakaguchi, 2005); in thecontrol of the immune responses to some chronic and acuteinfections (Coombes et al., 2005; Hori et al., 2002; Suvaset al., 2004); in the prevention of allograft rejection (Bharatet al., 2005; Graca et al., 2002) and graft versus host disease(Edinger et al., 2003); and more recently in tumorimmunobiology (Baecher-Allan and Anderson, 2006; Wolfet al., 2006; Yamaguchi and Sakaguchi, 2006).The involvement of regulatory T cells in tumor
immunobiology is to be expected on theoretical grounds,since regulatory T cells regulate the immune responses tosome self-antigens and tumors are mainly composed ofself-antigens. Recent experimental observations strongly
ARTICLE IN PRESS
E, R, APCs E, R
APCs
ET, RT, Tu
Lymph Node (LN) Tumor compartment (T)
VT = �Tu · Tu + �L · (ET + RT)VLN = �A · A + �L · (E + R)
Scheme 1. Illustration of our two model spatial compartments: the tumor
site (T) and its adjacent lymph node (LN). Compartment T contains
tumor cells (Tu), and infiltrating effector (ET) and regulatory (RT)
lymphocytes. Compartment LN contains effector cells (E), regulatory cells
(R) and antigen presenting cells (APCs), which recognize or present,
respectively, tumor-related antigens. Both compartments have a variable
volume which depends on the number of cells they contain at a given time.
Interaction between compartments results from cell migration. Tumor
growth activates the maturation and migration to the LN of APCs
presenting tumor determinants, while a fraction of the E and R cells
activated at the lymph node migrate into the tumor site.
K. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 123
support the involvement of regulatory T cells in the growthof some malignant tumors. Sakaguchi et al. (Onizuka et al.,1999; Shimizu et al., 1999) have shown that the elimination,with monoclonal antibodies, of the CD4+ CD25+ T cellsleads to syngenic tumor rejection. Although it is still notclearly understood why such effects are observed in somebut not all the studied tumors and why it is only efficientwhen applied before or up to a few days after tumorimplantation. An increased accumulation of regulatory Tcells inside the tumor site or at the adjacent lymph nodesduring tumor progression has also been reported (Curielet al., 2004; Hontsu et al., 2004; Kono et al., 2006; Li et al.,2005; Nishikawa et al., 2005b; Wolf et al., 2005; Woo et al.,2002). But again, such effect is observed in some but not inall known tumors. More recently, and perhaps morerelevant from a therapeutic point of view, the vaccinationagainst some tumor-associated antigen have been shown toinduce the expansion of CD4+CD25+ regulatory T cells,causing an accelerated tumor progression (Nair et al., 2006;Nishikawa et al., 2003, 2005a; Zhou et al., 2006). It is a fairepitome of the situation that regulatory T cells promise abetter understanding and management of tumors, thoughstill many open questions remain. Particularly, we fail tounderstand why some tumors seem to recall the regulatoryT cell activity whereas others do not.
Aiming to get a better insight into the tumor-immunesystem dynamics, in the presence of regulatory T cells, weformulated and studied a simple mathematical model ofthe interaction between a solid tumor and the lymphocytesin its adjacent lymph node. The model is an extension of aprevious model of the dynamics of autoreactive regulatorycells and effector cells (Carneiro et al., 2007; Leon et al.,2000, 2001) that interact upon their co-localized activationat the antigen presenting cells (APCs). It assumes thattumor growth stimulates the activation and migration tothe adjacent lymph node of fresh APCs loaded with tumorantigens. These APCs stimulate the growth of both effectorand regulatory T cells, which may then migrate to thetumor site and induce tumor cell destruction. The efficiencyof tumor elimination is dependent on the balance betweeneach cell population at the tumor site.
Our results predict the existence of two alternativedynamic modes of unbounded tumor growth, whichprovide a simple explanation to the fact that the develop-ment of some tumors expands regulatory T cells whereasothers do not, and further predicting how some keydynamical properties of the tumor determine either one orthe other type of behavior.
2. The model
2.1. Model compartments: lymph node and tumor site
The model includes two spatial compartments ofvariable volume (Scheme 1), representing respectively asolid tumor (T) and its adjacent lymph node (LN). The LNcompartment contains effector cells (E), regulatory cells
(R), and APCs (A), which recognize or present, respec-tively, some tumor-related self-antigens. The tumor com-partment (T) contains tumors cells (Tu), and infiltratingeffector (ET) and regulatory (RT) lymphocytes. Bothcompartments are modeled like a well-mixed bag of cellswith a volume dependent on the number of each cell typeexistent at a given moment in time. These volumes arecomputed respectively for the tumor (VT) and the LN(VLN) compartment as
VT ¼ aTuTuþ aLðET þ RT Þ, (1a)
VLN ¼ aAAþ aLðE þ RÞ, (1b)
where aA, aL,and aTu stand for the specific volume occu-pied per cell for APCs, lymphocytes, and tumor cells,respectively.
2.2. Dynamics in the lymph node
Regulatory (R), effector (E), and APCs (A), in thiscompartment, interact following equations very similar tothose described in our previous model for T-cell-mediatedregulation (Leon et al., 2000). Conjugation and deconjuga-tion of E and R cells at the APC antigenic sites are assumedto be fast processes in quasi-steady-state equilibrium.Activation of both effector and regulatory T cells toperform functions and to progress through the cell cycle isassumed to require interactions with the APCs, and furtherdepends on the interactions these T cells make with eachother. Particularly, E and R cells are assumed to interactindirectly by competition for access to cognate APCs andmore directly by molecular processes that require their co-localization in the physical domains in the vicinity of thesecognate APCs. We have called (Carneiro et al., 2007) thesedomains APC-dependent interaction foci (or foci, for
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137124
short). The simplest form of these foci is the multicellularconjugates studied in Leon et al. (2000), where E and Rcells are assumed to interact while conjugated at the surfaceof single APCs, which contains several independent sitesfor simultaneous T cell conjugation. That is, the extent ofAPC foci is reduced, in that model, to the immediateneighborhood of a given APC. In the model we presenthere, such foci were considered as being composed of acluster of spatially proximal APCs that contains s totalsites for T cell conjugation.
The total number of these APC clusters (APC foci) isassumed to depend on tumors size. i.e., the larger the tumorsize, the larger the number of these APC clusters presentingtumor-related antigens. Inside the foci the followinginteractions/processes are considered (see Scheme 2). Effec-tor T cells get activated and proliferate following productiveinteractions with cognate APCs, but such proliferation iscompletely inhibited/suppressed by the presence of a singleregulatory T cell being activated inside the same APCcluster. On the other hand, the interaction between E and Rcells inside a given APC foci promotes the growth of theregulatory population. Such growth is either the conse-quence of an E-dependent proliferation of R cells or aconsequence of an R-dependent differentiation of E cellsthat become regulatory cells.
Note that the model makes no assumptions about theactual molecular mechanism by which R and E cellsinteract with each other inside an APC cluster (APC foci).This mechanism may be mediated by direct cell to cellinteraction, as proposed by Thornton and Shevach (2000);
Possible scenarios for
There is no direct interaction betw
T c
ells
con
jugate
d i
n A
PC
clu
ster
sF
ree
T c
ells
Scheme 2. Illustration of the interactions between E and R cells in the lymph n
APCs) do not interact with each other and die with a constant death rate. Conj
This is while conjugated to APCs which are spatially close to each other a
proliferation; Clusters containing mixtures of E and R cells result in the inhibit
growth is either the consequence of an E-dependent proliferation of R cells (as
that become regulatory cells. Clusters containing only R cells do not lead to p
or by the action of a very short ranged soluble factor; oreven by the temporal conditioning of the activation state ofthe APCs inside the cluster (Cederbom et al., 2000; Taamset al., 2000; Vendetti et al., 2000). The only mechanisticassumption of our model is that the interaction between Rand E cells requires that they get activated by conjugationwith APCs, which are close enough to each other both inspace and time.Following the latter assumptions, the equations for E
and R cells in this compartment are obtained, followingour previous formulation (Leon et al., 2000), as
dE
dt¼ FE þ
Xs
i¼1
Pð1� D1ÞiAi;0ðEc;RcÞ
�Xs
i¼1
Xs�i
j¼1
ZMijAi;jðEc;RcÞ � dEf þ D2ET , ð2aÞ
dR
dt¼ FR þ
Xs
i¼1
Xs�i
j¼1
Mð1� D1ÞijAi;jðEc;RcÞ � dRf þ D2RT ,
(2b)
where:
E, R total number of E and R cells,Ai,j( ) number of APC clusters conjugated to i E and j R
cells,A total number of APC clusters,Ec, Rc number of E and R cells conjugated with APCs,
-E cells activation and proliferation
-R cells activation and proliferation
-Inhibition of E cells proliferation
-R cells do not proliferate
T cells in the model
een T cells in different APC clusters
-Free E and R cells die
ode compartment of the model. Free E and R cells (cells not conjugated to
ugated E and R cells interact between them inside particular APC clusters.
t the LN. Clusters containing only E cells result in cell activation and
ion of E cell proliferation and in the growth of the R cell population. Such
represented) or a consequence of an R-dependent differentiation of E cells
opulation growth.
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 125
Ef, Rf number of free E and R cells, given by Rf ¼ R�Rc
and Ef ¼ E�EC,FE ;FR source term of new E and R cells from the thymus,s maximum number of T cells that can conjugate to
an APC cluster,d death rate of E and R cells,P proliferation rate of E cells,M E-cell-dependent growth rate of the regulatory
population,D1 fractions of the E or R cells that upon activation
differentiate and migrate out of the LN to thetumor compartment,
D2 fractions of the E or R cells recycling back to thelymph node from the tumor site,
Z switch variable controlling the mode of R popula-tion growth: setting it to zero describes themechanism where R cells proliferate stimulatedby E cells and setting it to 1 describes themechanism where E cells are induced to differ-entiate to the R phenotype.
The number of multicellular conjugates of a giventype can be obtained, from our previous study (Leonet al., 2000), as the product of two hypergeometricfunctions (Hyp)
AijðEc;RcÞ ¼ A Hyp½i þ j; ðEc þ RcÞ; sA; s�
�Hyp½i;Ec; ðEc þ RcÞ; i þ j�,
HypðN;N0;M ;LÞ ¼N0
N
!M �N0
L�N
!,M
L
!. ð3Þ
While the number of conjugated E and R cells are obtainedrespectively by resolving in quasi-steady-state assumption,the equilibrium of formation and dissociation of conjugatesof E and R cells is with the APC sites. In practice, thismean to satisfy the following algebraic relations:
ðKE=V LNÞ ¼Ec
ðsA� Ec � RcÞðE � EcÞ,
ðKR=V LNÞ ¼Rc
ðsA� Ec � RcÞðR� RcÞ, ð4Þ
where KE and KR are the conjugation constants of E and Rcells with the APC, respectively, and VLN is the volume ofthe LN compartment given by Eq. (1b).
Equations in (4) indeed resolve analytically to thefollowing expressions in the case KE ¼ KR ¼ K:
Ec ¼E
ðE þ RÞTc; Rc ¼
R
ðE þ RÞTc, (5)
Tc ¼KðsAþ E þ RÞ þ V LN
2K
�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðKðsAþ E þ RÞ þ V LN Þ
2� 4K2sAðE þ RÞ
q2K
The number of APC clusters, presenting tumor-related self-antigens in this compartment (A), is assumed to be linearly
dependent of the tumor size, by the following expression:
A ¼ A0 þ�AMTu
AM þ �Tu, (6)
where A0 is the basal level of antigen presentation in theabsence of tumor; and AM is the maximal possible numberof APC clusters that can fit in the lymph node or that canbe stimulated to migrate to the lymph node by tumorgrowth.Parameter e in the latter equation is of special interest for
this analysis. This parameter modulates how much a giventumor induces the presentation of its antigens in the lymphnode and therefore how much antigen is seen by theimmune system. The larger the value of this quantity, thehigher will be the presentation of tumor-related antigens inthe lymph node for the same given tumor size. Thisparameter in the model gives a precise definition of aproperty that immunologists intuitively refer as tumorimmunogenicity.
2.3. Dynamics in the tumor site
E and R Lymphocytes in this compartment are assumedto increase by migration of activated lymphocytes from theadjacent lymph node and to decrease due to cell death ormigration back into the LN compartment. The equationsfor these cells are:
dET
dt¼Xs
i¼1
PD1iAi;0ðEc;RcÞ � kET � D2ET , (7a)
dRT
dt¼Xs
i¼1
Xs�i
j¼1
MD1ijAi;jðEc;RcÞ � kRT � D2RT , (7b)
where ET, RT is the total number of E and R cellsinfiltrating the tumor; k the death rate for lymphocytes inthe tumor environment; D2 fractions of the E or R cellsmigrating back to the lymph node.The number of tumor cells in the compartment is
assumed to increase with a characteristic growth rate, todecay with some death rate, and to be killed by immuneeffector functions, depending on the proportion of E and Rcells infiltrating it. Thus the equation for the number oftumor cells is
dTu
dt¼ GTu� dT Tu� kill
aLET
V T
1�aLRT
V T
� �g
Tu, (8)
where Tu is the number of tumor cells; G the growthrate for tumor cells; dT the death rate for tumor cells; kill
the maximal rate at which immune cells could destroytumor cells; and g an exponent that controls the efficiencyof the control of tumor cell destruction by regulatoryT cells.Note that in Eq. (8) tumor cell destruction by the
immune cells is represented by a simple function, makingno particular claim about the molecular mechanism which
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137126
mediates this process. The simplest interpretation for thisfunction is that the R cells inhibit locally the killingmediated or induced by the E cells, within a given effectiveregion surrounding them. The parameter g modulates theefficiency of this inhibition. The larger the value of thisquantity, the higher the efficiency of the inhibition exertedby the R cells (i.e., the larger the surrounding region ofeffective control by a single R cell).
2.4. Fast exchange and equilibrium between the lymph node
and tumor site
For the sake of simplicity, the cellular interchangebetween model compartments is assumed to be a very fastprocess. Therefore, a quasi-steady-state approximation isused for the number of R and E cells in the tumorcompartment. In other words, they are assumed to bealways in equilibrium with the lymphocyte composition ofthe lymph node. Thus, setting to zero the derivative inEqs. (7a) and (7b), we obtain
ET ¼PD1
kþ D2
Xs
i¼1
iAi;0ðEc;RcÞ,
RT ¼MD1
kþ D2
Xs
i¼1
Xs�i
j¼1
ijAi;jðEc;RcÞ. ð9Þ
The latter approximation reduces the model to thefollowing set of three ordinary differential equations,where the algebraic relations given by Eqs. (1), (4), (5),and (9) must be substituted
dE
dt¼ FE þP 1� D1 þ
D1D2
kþ D2
� �
� 1�Rc
sA
� �s�1
Ec � ZMRcEc
A� dEf , ð10aÞ
dR
dt¼ FR þ
ðs� 1Þ
sM 1� D1 þ
D1D2
kþ D2
� �RcEc
A� dRf ,
(10b)
dTu
dt¼ GTu� dT Tu� kill
aLET
V T
1�aLRT
V T
� �g
Tu. (10c)
Note that we have just substituted Eq. (9) in Eqs. (2a) and(2b), using the following two identities, which are furtherexplained in Appendix A, under the biologically reasonableconsideration that a large number of every cell type existsin the system (A, E, Rbs)
Xs
i¼1
iAi;0ðEc;RcÞ ¼ Hypð0;Rc; sA; sÞsA
sA� Rc
� 1�Rc
sA
� �s�1
,
Xs
i¼1
Xs�i
j¼1
ijAi;jðEc;RcÞ ¼ðs� 1ÞA
ðsA� 1Þ
RcEc
A�ðs� 1Þ
s
RcEc
A.
2.5. Model in non-dimensional form
Finally, to effectively reduce the number of freeparameters in the model, we rewrite the model in termsof dimensionless quantities by introducing the followingchanges of variables:
t ¼ x=d; E ¼ eA0; R ¼ rA0,
Tu ¼ tudA0; A ¼ aA0.
This change of variables normalizes the time in the modelto the death rate of lymphocytes and normalizes the cellnumbers for tumor cells, effector and regulatory T cells tothe basal number of APCs in the absence of the tumor. Itresults in the following system of equations:
de
dx¼ fe þ p 1�
rc
sa
� �s�1
ec � Z0mrcec
a� ðe� ecÞ, (11a)
dr
dx¼ fr þ
ðs� 1Þ
sm
rcec
a� ðr� rcÞ, (11b)
dtu
dx¼ g tu� ki
eT
ðtuþ eT þ rT Þ1�
rT
ðtuþ eT þ rT Þ
� �g
tu,
(11c)
a ¼ 1þlaMtu
aM þ ltu; eT ¼ p 1�
rc
sa
� �s�1
ec,
rT ¼ðS � 1Þ
Sm
rcec
a, ð11dÞ
ec ¼e
ðeþ rÞtc; rc ¼
r
ðeþ rÞtc, (11e)
tc ¼kðsaþ eþ rÞ þ ðaaþ eþ rÞ
2k
�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkðsaþ eþ rÞ þ ðaaþ eþ rÞÞ2 � 4k2saðeþ rÞ
q2k
ð11fÞ
where, d ¼ ðaL=aTuÞðdD1=ðkþ D2ÞÞð1=yÞ; y ¼ ð1� D1 þ
ððD1D2Þ=ðkþ D2ÞÞÞ; fe ¼ FE=ðdA0Þ; fr ¼ FR=ðdA0Þ; Z0 ¼Z=y; p ¼ Py=d; m ¼ My=d; g ¼ ðG � dT Þ=d; ki ¼ kill=d;l ¼ �d; a ¼ aA=aL; k ¼ K=aL; aM ¼ AM=A0.Note that with this process our model goes from 22
different dimensional parameters to just 13 different non-dimensional parameters, which need to be set to particularvalues for numerical simulations.
2.6. Parameter values
Parameters, in the model, are grouped into the followingthree classes, according to the way we treat them in ouranalysis.Class 1 contains seven parameters controlling the
dynamics of E and R cell interaction in the LN, even inthe absence of tumors cells. For these parameters, valueswere chosen following our experience and results studyingthis model for the interaction between E and R cells
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 127
recognizing self-antigens at the APC.1 First, since we haveextensively shown (Leon, 2002; Leon et al., 2003, 2000,2001) that this model behaves dynamically quite similarlyin the cases Z ¼ 0 and 1, we just restrict our analysis here tothe simplest case, i.e., Z ¼ 0. Moreover, we also set thevalues of p, m, k, fe, fr, and s, such that the system isbistable; i.e., the values are set inside the parameter regime(Leon et al., 2000) where the system has two stable steadystates, one dominated by E cells and associated to immuneresponses, and one dominated by R cells and thereforeassociated with tolerance. We have previously argued(Leon, 2002; Leon et al., 2000, 2001) that such bistabilityis essential to explain the known phenomenology ofdominant tolerance. As default values in the figures shownin this paper, we used p ¼ 21, m ¼ 2.9, k ¼ 7.0, fe ¼ 0.1,fr ¼ 2.3 and s ¼ 5, but every other parameter sets testedinside the explored bistable regime render similar qualita-tive results.
Class 2 groups three parameters, whose effects on thefinal results were extensively explored, assigning themrandom values inside a large preset range. We found thatthe model properties reported here remain qualitatively thesame for values of these parameters inside its preset range.The parameters in this class are:
1N
reduc
becom
previo
aM
that sets the maximal number of APC clusters that canbe fitted in the LN compartment. Preset range from103 to 1010. Default value used in figures is 104;g
that controls the strength of the interaction between Eand R cells at the tumor site. Preset range from 1 to12. Default value is 8;a
that sets the ratio between the specific volumes oflymphocytes and APCs. Preset range from 1 to 10.Default value is 1.Class 3 contains the three parameters directly controllingtumor dynamics and its interaction with the immunesystem. These parameters could be actually related tointrinsic, perhaps molecular, properties of a given tumor.They are used as control parameters in our analysis, thustheir impact on the model is explicitly shown in the resultssection. These parameters are
g
that sets the specific growth rate of the tumor; l that characterizes the tumor immunogenicity, i.e., thecapacity of the tumor to induce its own antigenpresentation by APCs in the LN. Note that, in thissense, it fully contains the model dependency with thedimensional parameter e (and see explanation inSection 2.2);
ki
that sets the maximal rate at which immune cells coulddestroy tumor cells.ote in this sense that, in the absence of tumors, the equations in 11
e just to Eqs. (11a) and (11b). Moreover, the number of APCs (a)
es a constant equal to 1. This is precisely the model we have
usly studied (Leon et al., 2003, 2000; Leon, 2002).
We have referred above to a set of typical/default values
for non-dimensional parameters in this work. Theseparameter values are used in the numerical simulationsthat generate our figures, unless explicitly indicated in thefigure itself or in the figure legend. This default assumes alife span for T lymphocytes of 7 days and a doubling timeupon activation of 8 h. Therefore, these typical parametervalues are compatible with a time scale in our simulations,where each non-dimensional time unit corresponds to aweek of real time.3. Result
3.1. Two modes of unbounded tumor growth
To assess the dynamics of the interaction betweentumors and the immune system in our model, we simulatedthe effect of introducing a very small number of tumor cellsin a system with effector (E) and regulatory (R) cellsinitially at a steady-state equilibrium derived from theirrecognition of self-antigens shared by the tumor and itsparental normal tissue. Such procedure is just intended tomimic the effect of a spontaneous growing tumor in anindividual with a pre-existent normal immune system.As explained in Section 2.6, class 1 parameters control-
ling the E and R cell dynamics in the model were set tovalues such that in the absence of tumor, the system hastwo stable steady states. This is, one state dominated by theE cells, and thus associated with a sort of autoimmuneresponse, and another state dominated by the R cells, andtherefore associated with tolerance to the peripheral self-antigens. We have previously argued (Leon, 2002; Leon etal., 2000, 2001) that such bistability is essential to explainthe known phenomenology of dominant tolerance.Namely, the bistable outcome of classical adoptive transferexperiments where, depending on the proportions of R andE cells transferred into the immune-deficient animalmodels, they end up being normal or suffering someautoimmune disorders. However, since in the absence ofdisorders the immune system is typically tolerant to self-antigens, the numerical simulations of tumor growth hereare set with initial values for the number of E and R cellsdrawn from the steady state dominated by the R cells in thesystem.The result of our numerical simulations shows three
alternative dynamical outcomes for the tumor developmentin the model. The tumor could just grow indefinitely(examples in Fig. 1a and b); It could be fully eliminated bythe immune reaction (Fig. 1c); or it could just reach a stableequilibrium size (Fig. 1d). The particular realization ofthese dynamical modes depends on the values of theparameters controlling tumor properties in the model.Namely, the parameter g controlling the specific growthrate of the tumor, the parameter ki defining the sensitivityof tumor cells to the immune effector functions, and theparameter l controlling tumor immunogenicity (i.e.,defined as the intrinsic capacity of the tumor to induce
ARTICLE IN PRESS
Cel
l Nu
mb
ers
(x10
4 )
10
107
1010
104
10-2
Time (weeks)
5 10 15 20 25 30
10
107
1010
104
10-2
0
GR+
r
e
tu
5 10 15 20 25 300
GR-tu
r
e
Cel
l Nu
mb
ers
(x10
4 )
Time (weeks)
5 10 15 20 25 300
r
20 40 60 80
10
107
1010
104
10-2
0
tu
eCel
l Nu
mb
ers
(x10
4 )
Time (weeks)
ER
tue
10
107
1010
104
10-2
Cel
l Nu
mb
ers
(x10
4 )
Time (weeks)
r
Fig. 1. Different kinetic outcomes obtained in model numerical simulations of the growth of a spontaneous tumor in an immunocompetent individual.
The graphs show examples of typical change over time of the number of tumor cells (tu), effector (E) and regulatory (R) T cells obtained by numerical
integration of model differential equations. The illustrated simulations differ in the values used of the control parameter (g, ki and l), but use the same
typical values for other model parameters, following the description provided in Section 2.6. Panel a, g ¼ 5, ki ¼ 400 and l ¼ 0.0001; Panel b, g ¼ 5,
ki ¼ 400 and l ¼ 0.01 Panel c, g ¼ 6, ki ¼ 400 and l ¼ 0.001; Panel d, g ¼ 1.3, ki ¼ 400 and l ¼ 0.1. As initial condition in the simulation the number of
E, R and Tu cells were set to the values E ¼ 0.24, R ¼ 6.9 and Tu ¼ 10�4, which correspond, respectively, to the tolerant steady state for T cells of the
model in the absence of tumor and a very small initial tumor seed. Qualitatively, in panels a and b, tumor grows indefinitely while interacting with the
immune T cells; in panel c, tumor is rejected by the induced immune reaction; and in panel d, tumor reaches a stable equilibrium with the interacting
immune T cells.
K. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137128
the presentation of its antigen by the APCs in the adjacentLN). The results of numerically studying such parameterdependency2 are summarized in Figs. 2a–d.
In these figures, the parameter regions leading todifferent dynamical tumor outcomes, are shown as afunction of the pair of control parameters specified in theaxes, for a fixed value of the remaining control parameter,
2We have built a simple function able to give different numerical values
when the numerical simulation of tumor growth results qualitatively like
any one of the four possibilities shown in Fig. 1. This function is based on
simple rules that analyze the values of variables E, R and Tu after long-
term model simulations. For instance, we recognize tumor rejection when
Tuo10�5 (less than the initial tumor burden) after the simulation. But we
identify tumors growing like in Fig. 1a when after simulation, Tu41010
and E4R. With this function in hand, we use standard bisection
procedures to find parameter values leading to change of these function
values, which consequently stand for qualitative changes in model
behavior. We have validated the robustness of our procedure by randomly
selecting hundreds of parameter sets inside each computed region, and
checking by hand the correctness of the automatic classification and
regions delimitation.
which is indicated in the right bottom corner of the figure.Other model parameters were set to the default valuesprovided in Section 2.6. The most outstanding resultobserved in these figures, is the existence of two disjointparameter regions in the model (labeled GR� and GR+)where the tumor can grow indefinitely, just as it is mostfrequently observed in clinical tumors. These regions aretypically separated from each other by a region wheretumor elimination by the immune reaction is observed(labeled as R). The separate character of these regionsindicates the existence of two alternatives dynamicalstrategies (dynamical modes) of tumors to ensure indefinitegrowth, including, of course, a way to escape from theimmune system reaction. Actually, the region labeled GR�corresponds to tumors of high specific growth rate (g), oflow immunogenicity (l), and of low sensitivity to immuneeffectors functions (ki). Whereas the region labeled GR+corresponds to tumors of low growth rate, of intermediate-to-high immunogenicity and of high sensitivity to immuneeffectors functions. Still, as discussed in the next section,
ARTICLE IN PRESS
0.0001 0.001 0.01 0.1 1
1
2
5
10
20
50
100
200
0.0001 0.001 0.01 0.1 1
1
2
5
10
20
50
100
200
g
0 200 400 600 800 1000
1
2
5
10
20
50
100
200 GR-
R
GR+
E
ki
g
g
0 200 400 600 800
λ=0.01 λ=1
1000
1
2
5
10
20
50
100
200
g
ki
GR+
E
R
GR-
GR-
GR+
R
E ki=100
GR-
R
R
GR+
Eki=400
λ λ
Fig. 2. Dependency of possible dynamical outcomes in the model simulations of the growth of a spontaneous tumor, as a function of control parameters
defined in Section 2.6. In these graphs, parameter values pick up anywhere inside the regions labeled as GR�, result in indefinite tumor growth,
qualitatively as the one shown in Fig. 1a; Parameter values pick up inside the regions labeled as GR+, result in indefinite tumor growth, qualitatively as
the one illustrated in Fig. 1b; Parameter values pick up inside the region labeled as R, result in tumor rejection qualitatively as the one shown in Fig. 1c;
and parameter values pick up inside the region labeled as E, result in stable equilibrium of tumor and immune cells qualitatively as the one shown in Fig.
1d. Solid lines delimiting different parameter regions were computed numerically, using a simple procedure (see footnote 2 on previous page) programmed
on the software mathematica 4.0. Panels a and b show model dependency with parameter g, controlling the specific growth rate of the tumor, and the
parameter l controlling tumor immunogenicity, for the indicated values of parameter ki, which defines the sensitivity of tumor cells to the immune effector
functions. Panels c and d show the model dependency with parameter g and ki for the indicated value of parameter l. Other model parameters are fixed to
the typical values defined in Section 2.6.
K. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 129
tumors growing in these regions entail radically differentways of coupling with the immune cell dynamics.
3.2. Alternative modes of unbounded tumor growth differ in
their coupling to the immune system
Figs. 1a and b show, respectively, the characteristicpopulation dynamics observed in the model for tumorsthat belong to regions GR� and GR+ as defined in Fig. 2.In the first case (mode GR�), the tumor only induces aclear preferential expansion of effector T cells. While in thesecond case (mode GR+), the tumor induces a balancedexpansion of both effector and regulatory T cells. Thustumors growing in these regions obviously entail differentways of coupling with the immune cell dynamics.
In mode GR�, tumor growth induces the breakdown ofthe pre-existent control of E cell response by the R cells(i.e., the breakdown of the pre-existent tolerance), although
the immune response elicited does not control tumorprogression. Such breakdown of tolerance becomes evidentfrom the fact that the values for the ratio R to E cells andthe ratio R cells to APCs in the system goes to very lowvalues, after some time of tumor progression (Fig. 3aand b). Furthermore, the inefficiency of the inducedimmune reaction is explained by the fast-growing kineticsof the tumor and its already large size at the time theresponse is elicited. Interestingly, this sort of dynamics istypically referred and understood as an overgrowth of theimmune system response by the tumor and it does notrequire the existence of regulatory T cells.In contrast, mode GR+ tumor does not induce the
breakdown of pre-existent tolerance. This becomes evidentsince the values for the ratio R to E cells and the ratio Rcells to APCs in the system are not reduced to very lowvalues as in the previous case (Fig. 3a and b). Thus in thismode of growth the level of tumor cell destruction
ARTICLE IN PRESS
Rati
o r
/a
Time
0 10 20 30 40
0
1
2
3
4
5
6
Mode GR-
Mode GR+
0 10 20 30 40
0
2.5
5
7.5
10
12.5
15
Time
Mode GR+
Mode GR-
Rati
o r
/e
a b
Fig. 3. Alternative modes of indefinite tumor growth differ in their coupling with the immune cells. Mode GR�, but not mode GR+, leads to a
breakdown of pre-existent tolerance. Panels a and b, show, respectively, the time variation of the ratio of regulatory T cells per effector T cell and
regulatory T cells per APC in the model simulations leading to indefinite tumor growth. Time courses labeled, respectively, as mode GR� and mode GR+
correspond to the population dynamics shown in Fig. 1a and b.
K. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137130
mediated by the E cells is always regulated by thesignificant presence of the R cells. Therefore, this modeof tumor growth is obviously dependent on the existence ofregulatory T cells in the model, and as far as we know it hasnever been shown in previous theoretical work.
3.3. Simulation of tumor transplantation experiments
Most experimental tumor models used nowadays aretransplantable tumor cell lines. These cell lines werederived a long time ago, and they were, in many instances,explicitly selected to lead to fast-growing tumors. Thischoice was natural at the time, since researchers weredeveloping models to study the effect of radio- and chemo-therapies on tumors. Thus, selecting a fast-growing tumorwas quite convenient, on the one hand, to shorten the timeof the experiments, and on the other hand, to make moreevident the effect of these therapies that mainly affect fastreplicating cells. However, according to our previousresults, the relation with the immune system in these fast-growing tumor models could differ significantly from thatof some spontaneous clinical tumors, which are known tohave much slower growing kinetics. Therefore, thesetumors may be quite misleading for the experimental studyof cancer immunotherapies. Particularly, they are mostlikely inadequate to access the role of regulatory T cells intumor development, since they may be expected to growthin our mode GR�, where the R cells play no significantrole in the tumor development dynamics.
A relevant question then, is how to identify, in practice,tumor models, whose dynamics actively involve regulatoryT cells (this is our mode GR+ tumors). A first obviousdirect strategy is to assess the population sizes of the E andR lymphocytes infiltrating the tumor or at its adjacent
lymph node, for several candidate models. According toour previous results (Fig. 1), a net increase in the absolutenumber of R cells that correlates with tumor progression oreven better just with tumor size, must be observed for themode GR+ tumors. Nevertheless, this sort of directstrategy, although theoretically plausible, is limited inpractice by the elusive character of our current definitionsfor the regulatory T cell phenotype. Furthermore, currentcell surface markers for regulatory T cells are alsoexpressed typically by activated naıve T cells. Moreover,FOXP3, which is perhaps the best current marker for Rcells, is an intracellular protein thus posing many problemsfor its practical application on in vivo studies.An alternative strategy to the latter issue is the
identification of some dynamical properties of modeGR� and GR+ tumors, which could discriminate betweenthem and are amenable for experimentation. Using themodel, one such property was identified by studying thedependency of tumor growth, with simulations like those inSection 3.2, with the initial number of cells used to initiallyseed the tumor (i.e., the size of the initial tumor implant).Note that the latter quantity mimics the initial implant sizefor a transplantable tumor, and thus amenable for directexperimentation. The results are illustrated in Fig. 4.Fig. 4 shows how, depending on the size of initial tumor
implant (x axes) and tumor growth rate (y axes), tumorswith equal immunogenicity (l) and equal sensitivity toimmune effector function (ki), progress into the differentdynamical modes described in Section 3.1. The dashed lineshighlight the three qualitatively different dependencies oftumor progression with the initial implant size observed inthe model. Case (p1) The tumor always grows in modeGR� independently of the initial tumor size used; Case
(p2) A threshold value for the initial tumor size is
ARTICLE IN PRESS
E
g
Initial implant Size
GR-
GR+ R
0.00001 0.01 10 10000 107 1010
1
2
5
10
20
50
100
p1
p2
p3
Fig. 4. Parameter dependencies of possible dynamical outcomes in model
simulations of the growth of a transplantable tumor. Parameters inside the
regions label as GR�, result in indefinite tumor growth despite tolerance
breakdown (similar to the case in Fig. 1b), Parameters inside the regions
labeled as GR+, result in indefinite tumor growth but preserving
tolerance (similar to the case in Fig. 1b); Parameters insides the region
labeled as R, results in tumor rejection (similar to the case in Fig. 1c); And
parameter values inside the region labeled as E, result in a stable
equilibrium of tumor and immune cells (similar to the case in Fig. 1d). The
figure explicitly shows model dependency with parameter g, controlling
the specific growth rate of the tumor, and the initial number of cells used
to seed the tumor (the initial implant size). Other model parameters are
fixed to the typical value defined in Section 2.6. Dashed lines remark three
possible dependencies with initial implant size observed in the model. (P1)
the tumor always grows in mode GR� regardless of the initial implant
size; (P2) the tumor is rejected for small implants and grows in mode GR�
for large implants; and (P3) the tumor grows in mode GR+ for small
implants, is rejected for intermediate implants and also grows, though in
mode GR� for large implant sizes.
K. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 131
observed, such that, for values below the threshold thetumor is rejected by the immune reaction, whereas forvalues above the threshold the tumor grows in mode GR�;and Case (p3) Where two independent thresholds T1 andT2 (T1oT2) are observed. For values of initial tumorimplant below T1 the tumor grows in mode GR+; forvalues between T1 and T2 the tumor is rejected by theimmune reaction; and for values larger than T2 the tumoralso grows, though in mode GR�.
The latter results clearly show that progression oftransplantable tumors that can grow in mode GR+ hasa qualitatively different dependency on the initial implantsize, compared with those tumors that can only grow inmode GR�. Note that mode GR+ tumors appear in acharacteristic region of the latter dependencies, wherereducing the initial implant size promotes indefinite tumorgrowth. Conversely, mode GR� tumors appear typicallyon the opposite type of region, where increasing implantsize promotes tumor growth. In summary, our model
predicts that assessing experimentally the lately referreddependency provides an indirect way to distinguish modeGR� from mode GR+ tumors in the case of transplan-table tumor models.
3.4. A semi-analytical rational for model numerical behavior
In this section we try to understand the numericalbehavior of the model, presented in previous sections, in asemi-analytic way. Particularly, we show that kinetics likethose illustrated in Fig. 1c and d, corresponding to tumorrejection and disease stabilization, can be easily understoodfrom transitions between the possible steady states of themodel. Moreover we show that the unbounded kineticsobserved in mode GR� and GR+ of tumor growth, canbe understood by looking for the existence of someparticular type of stable unbounded trajectories that worklike attractor dynamics in the model.
3.4.1. Steady-state analysis of the model
The model, in the absence of tumor cells, reduces backto our original model without tumor cells (Leon et al.,2000). Therefore, since the model admits an equilibriumsolution for tu ¼ 0, it inherits the three steady states ofthe original model in the bistable region of modeloperation. That is, a state of the form (e,0,0) and twostates of the form (e,r,0). The stability analysis shows thatthe state (e,r,0) that is an unstable saddle node in theoriginal model (without the tumor equation) remainsalways unstable in the new model. The other two states,which are stable in the original model, may or may notremain so, depending on the sign of the third eigenvalue ofthe Jacobian matrix. The state (e,0,0) becomes stable ifparameter g takes values lower than ki, whereas the state(e,r,0) results in a stable state if g takes values lower than aconstant g1, which can be computed numerically as afunction of all parameters in the model, except for g and l.For the typical parameter values defined in Section 2.6 thevalue of g1 is 1.32� 10�7.Additional to the latter states, the new model also admits
equilibrium states of the form (e,0,tu) and (e,r,tu). There istypically one state of the form (e,0,tu), with biologicalreasonable values (e40 and tu40), but it is alwaysunstable. There are either one or two states of the form(e,r,tu). However, a single stable state of the latter formexists in the system when g is larger than g1 and lower thang2. Parameter g1 is the complex constant referred beforeand g2 is another complex constant, which could becomputed numerically as a function of all parameters,except for g and l. For the typical parameter values setdefined in Section 2.6, the value of g2 is 1.3.As a summary, Fig. 5 shows the parameter regions where
the different stable steady states of the model exist,contrasting them with the regions of different numericalbehavior observed in the model simulations (those de-scribed in Fig. 2b). From this figure, it can be easilyobserved that the numerical behavior of the model
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137132
previously discussed in this paper occurs in the parameterregion where the equilibrium state (e,r,0) becomes unstable(for g4g1). Thus, starting the system dynamics in a point
0.0001 0.001 0.01 0.1 1
γ 1
γ 2
E
GR-
GR+
R
R
ki
λ
g
10-7
10
102
1
Fig. 5. Summary of parameter regions in the model where different stable
steady states exist. For values of parameter g, controlling the specific
growth rate of the tumor, below the dashed-dotted line (goki) a stable
steady state of the from (e,0,0) always exists in the system. For values of g
between the dashed and the dotted line (g1ogog2), a stable equilibrium of
the form (e, r, tu) exists. For values of g below the dotted line (gog1), theequilibrium state of the form (e, r, 0) is stable. Continuum lines demark
the parameter regions leading to different dynamical behavior in
numerical simulations of the growth of spontaneous tumors (see Fig. 2
for detailed description).
T
b
Cell N
um
be
rs
Time
a
10−6
0
108
1011
105
102
tu
r
e
F1 F2 F3
5 10 15 20 2510−6
0 5
108
1011
105
102
tu
F1
Fig. 6. Kinetic phases of tumor-immune system interactions observed in mo
without affecting the E and R cell dynamics (there is no effective interaction)
(strong interaction); and Phase F3: tumor cells keep growing, while the T cell d
model simulations strongly differ in the dynamics observed in phase F2 of tumo
exponential growth of tumor cells, effector T cells and regulatory T cells (re
different dynamical possibilities in this phase (panels b and c). Either the tumor
(panel b, inside dashed circle), while the regulatory cells grow with a much low
effector and regulatory T cell populations (panel c).
close to this unstable state leads the system to evolverapidly away from this point. Disease stabilization, astypically observed in the parameter region E (see Fig. 1c fora kinetic example), arises from the system evolving to thestable steady state of the form (e,r,tu). This type ofbehavior is obtained almost invariably in the model if suchstate exists (for g4g1 and gog2). Tumor rejection, asobtained in region R of Fig. 2 (see Fig. 1d) arises fromthe system evolving into the stable steady state of theform (e,0,0). However, this type of dynamics only occursrarely inside the region where the latter state is the onlystable steady state of the model (for g4g2 and goki).Inside this region, unbounded growth as the one observedin GR� and GR+ regions is also a possible dynamicoutcome.
3.4.2. A qualitative understanding of the kinetics of
unbounded tumor growth
To qualitatively realize the nature of unbounded growthdynamics in the model, we have to look a bit more closelyto the kinetics of cell growth in the modes GR� and GR+.A first important aspect to note, is the existence of threeindependent phases of tumor cell growth, which are rathercommon to both mode GR� and mode GR+ dynamics(see Fig. 6a–c). In an initial phase (F1) the tumor grows,keeping the pre-existent dynamics of the T cells ratherunaffected; In a second phase (F2), there is an obviousinteraction among the T cell populations and the tumorcells, where the dynamical outcome is rather different formode GR� and mode GR+. Finally, in a third phase(F3), the tumor cells keep growing, while the T celldynamics approach a sort of equilibrium point, though this
10−6
0 1 2 3 4
10
103
105
Time
F1 F2 F3
tu
r
e
ime
10 15
r
e
F2 F3
c
de GR� and GR+ of indefinite tumor growth. Phase F1: tumor grows
; Phase F2: tumor grows conditioning a strong change in T cell dynamics
ynamics approach a sort of equilibrium point. Modes GR� and GR+ in
r growth. In mode GR+ (panel a) this phase is dominated by the coupled
gion inside dashed circle). On the contrary, in mode GR� there are two
cells or the effector T cells grow exponentially, being coupled to each other
er rate; or the tumor cells grow exponentially, but much faster than both
ARTICLE IN PRESS
1
10
100
E
GR-
GR+
R
R
0.0001 0.001 0.01 0.1 10.1
g
λ
Fig. 7. Summary of parameter regions where different classes of
exponential tumor growth, while interacting with immune T cells, exist
in the model. For parameter values inside the region demarked by dashed
lines, the coupled exponential growth of tumor cells, effector T cells and
regulatory T cells is dynamically possible. For parameter values inside the
region demarked by dotted lines, the coupled exponential growth of tumor
cells and effector T cells is dynamically possible. Outside the latter regions
only the exponential growth of tumor cells, being uncoupled to the growth
of the T cells is dynamically possible. Continuum lines demark the
parameter regions that lead to different dynamical behaviors in numerical
simulations of the growth of spontaneous tumors (see Fig. 2 for detailed
description).
K. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 133
equilibrium point is clearly different for mode GR� andmode GR+.
The referred phases of the system dynamics can be easilyunderstood at a qualitative level as follows. The initialphase corresponds to the time before the tumor hasreached a sufficiently large size to significantly increase itsantigen presentation at the lymph node (typically whiletub1/l). Therefore, with no change in the level of antigenpresentation there is no alteration in T cell dynamics. Thethird phase corresponds to the time after the tumor is solarge that it saturates the antigen presentation at the lymphnode (typically for tubam/l). In this case, since thenumber of APCs reaches equilibrium, the number of Tcells shortly follows them. But for the mode GR� a statedominated by the effector cells (the number of R cellsgoes to zero) is reached and for mode GR+ a statedominated by regulatory T cells is reached. The secondphase of the system dynamics is the key feature of themodel behavior. It corresponds to the time window wherethe number of APCs presenting tumor antigens in thelymph node increases roughly linearly with tumor size(this is for 1/l5tu5am/l where a ¼ l tu). In both modeGR� and GR+, this phase is dominated by theexponential growth of tumor cells (see Fig. 6 inside thedashed circle), although they radically differ in thebehavior of the T cells.
In the dynamic mode GR+ (Fig. 6a), tumor cells,effector cells, and regulatory T cells grow rather exponen-tially in the referred phase, being coupled to each other.That is, they all grow with the same collective rate. On thecontrary, in mode GR� there are two different dynamicpossibilities (Figs. 6b and c). Either the tumor cells and theeffector T cells grow exponentially, being coupled to eachother (Fig. 6b), while the regulatory cells grow with a muchlower rate; or the tumor cells grow exponentially, but muchfaster than both the effector and regulatory T cellpopulations (Fig. 6c).
The next section shows that the growth patternsobserved in this second phase of tumor growth are theexpected behavior given the structure of our modelequation.
3.4.3. A quantitative understanding of the exponential
growth kinetics in the model
To address the existence of the exponential growth phasein our model behavior, we look for solutions in the systemof the following form:
eðtÞ ¼ CeðtÞEwt; rðtÞ ¼ CrðtÞE
wt; tuðtÞ ¼ CtuðtÞEwt.
(12)
Further, approximate the number of APCs in the lymphnode compartment in the model as aEl � tu. Note that thelatter assumption is justified, since the exponential growthobserved numerically (and described in the previoussection) takes place roughly in an intermediate timewindow where such linear relationships approximatelyhold.
Substituting Eq. (12) in the model equations, one caneasily realize that the exponential factors cancel out,obtaining a set of ordinary differential equations for thecoefficient CeðtÞ; CrðtÞ; CtuðtÞ. This set of equations re-sembles very much the equation in the original model, butit has some extra terms dependent on the new parameter w,which represents the overall exponential growth rate in Eq.(12). This equations supports quasi-stable steady-statesolutions of the form (ðCe;Cr;CtuÞ; ðCe; 0;CtuÞ), whichexplain, respectively, a system dynamics where the E, Rand Tu cells grow exponentially coupled with the samegrowth rate (w) and a system dynamics where E and Tucells grow exponentially coupled to each other, while Rcells grow much slower (thus it tends to zero values inrespect to the other cell population sizes).Interestingly, the parameter regions where the referred
pattern of exponential growth exists can be numericallycomputed. These regions (Fig. 7) compare reasonably wellwith the region obtained for these behaviors in thenumerical simulations of the model. Note that the regionwhere a coupled growth of R, E and Tu cells is possible,fully contains the parameter region GR+, where thisbehavior is observed in model simulations (see Fig. 6a).Moreover, the computed region explains partially well the
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137134
interface between the parameter region GR+ and theparameter region R where tumor rejection is typicallyobserved in the model numerical simulations. Further-more, the region where coupled exponential growth of Eand Tu cells is possible appears as a sub-region inside thenumerical region GR�. Actually, one can easily check thatnumerical behaviors like the one observed in Fig. 6b aretypically observed inside the computed sub-region, whilethe behavior such as the one observed in Fig. 6c is obtainedin the remaining part of region GR�.
Summarizing, the latter analysis shows that the un-bounded kinetics observed in mode GR� and GR+ oftumor growth, can be partially understood by the existenceof some particular stable unbounded trajectories thatdominate the dynamics, at least for some period of timein the tumor evolution.
4. Discussion
In this work we studied/developed a simple model toasses the role of regulatory T cells in tumor immunobiol-ogy. This model predicts the existence of two alternativedynamical modes for the unbounded growth of aspontaneous tumor. These modes differ in the way thetumor interacts with the immune cells. In the first mode(GR�), the tumor only induces a preferential expansion ofeffector T cells, which does not control its growth due to itslarge size and fast-growing kinetics. In the second mode(GR+), the tumor induces a balanced expansion of botheffector and regulatory T cells, which prevents tumorrejection. The model also predicts that whether one modeor another is reached depends on the dynamical propertiesof the tumor. Tumors will spontaneously grow in modeGR� if they have a high intrinsic growth rate, lowimmunogenicity, and are relatively resistant to destructioninduced by effector T cells. Alternatively, tumors will growin mode GR+, and thus lead to a net expansion ofregulatory T cells, if they have low growth rate, relativelyhigh immunogenicity, and are more sensitive to destructionby T cells.
The latter results provide a simple explanation to theexperimental fact that the development of some tumorsexpand regulatory T cells while others do not, predictinghow some key dynamical properties of the tumordetermine either one or the other type of behavior.Therefore, our analysis provides an insight to the intrinsicproperties of a tumor indicating that its growth doesinvolve or not the expansion of regulatory T cells. Ofcourse this is not an absolute precise recipe to identifytumors in each class, though it is a good stepping stone forinitial guesses and to pinpoint some interesting problems.Particularly, the fact that in our model fast-growing tumorsare expected to grow mainly in mode GR� whereasslow-growing tumors are expected to grow in mode GR+,leads us to cast doubts on the relevance of mosttransplantable tumor animal models, which are fast-growing tumors, to study the role of regulatory T cells
in tumor biology or to study the dynamics of clinicalspontaneous tumors, which are typically slow-growingtumors.Concerned with this latter issue, we turn to the question
of distinguishing transplantable tumor models of modeGR� from those of mode GR+. In fact, the modelpredicts an indirect way to achieve this, based on thequalitative dependence of tumor progression with theimplantation size. The model predicts that increasing theimplantation size leads to tumor rejection for mode GR+tumors, whereas the same increase always reinforces tumorprogression for mode GR� tumors. Moreover the modelshows that some transplantable tumors could only grow inmode GR� while others are able to grow both in GR� andGR+ modes. In the latter case, the tumors will grow forlow implantation sizes in mode GR+ and for largeimplantation sizes in mode GR�.Interestingly, most transplantable tumor models are
known from experimental data to show a similar depen-dency with the initial implant size. That is, they need aminimal implant size to take over and grow, but once theydo so, they keep growing for larger implantation size. Thisobservation is consistent with our original expectation forthese tumors to correspond to our mode GR� tumors.Nevertheless, some transplantable tumors have alreadybeen reported to show a dependency with the initialimplant size, resembling very much the one expected in ourmodel for mode GR+ tumors. Actually, the fact that thesetumors were rejected when implanted with large initialsizes, but were tolerated and grew when implanted withlower sizes, lead to a large set of experimental andtheoretical work in the late 70 s, where the latterphenomenon was termed ‘‘tumor sneaking through’’.However, the latter facts can not be taken as proof of theexistence of our mode GR� and GR+ tumors, since it ispossible that such dependencies can be explained by adifferent phenomenon besides tumor interaction withregulatory T cells.One particular difficulty in interpreting the tumor
sneaking through phenomenon according to our model isthat the tumors involved were mainly allogenic tumormodels. Thus, it is difficult to address the question of self-antigen recognition by regulatory T cells in that context,although regulatory T cells have been already documentedto regulate some cases of alloreaction (Bharat et al., 2005;Graca et al., 2002). Further experimentation is definitelyrequired to assess the existence of these tumor growthmodes and their relative incidences among transplantabletumor models, and perhaps more relevantly amongclinically observed tumors.Another key aspect of our model is the fact that in order
to mount an immune response to the tumor, the system hasto break self-tolerance. This would mean that tumors beingrejected or tumors growing in mode GR� would beconcomitant, at least transiently, with specific autoimmunediseases. In fact, tumor-associated autoimmune diseaseshave been extensively documented, namely in situations
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 135
where an effective immune response against the tumor wasinduced by effective therapeutic interventions (Dudleyet al., 2002; Engelhorn et al., 2006; Gogas et al., 2006;Jones et al., 2002; Overwijk et al., 1999; Wei et al., 2005).Despite these observations, it is hard to imagine that asuccessful immune rejection of a tumor is always con-comitant with a transient or chronic autoimmune disease;instead, it might represent a limitation of the model.Particularly, we have limited our model to represent onlythe self-antigens present in the tumors, but in a morecomplete case one should also include tumor antigens,which are not present in the parent self-tissue or that arepresent at such a low concentration that are most likely notpresented by APCs in the absence of the tumor. Includingsuch mixture of self-related and non-self-related tumorpeptides in the model could possibly lead to frequentsituations where tumor rejection takes place withoutconcomitant autoimmunity.
Other important aspect of our model is that it assumesthat tumors grow exponentially in the absence of immuneT cells. This assumption allows us to provide a simplifieddescription of the R cells dependent and independenttumors, which are dynamics behaviors observed, qualita-tively the same, on model simulations assuming morerealistic laws of tumor growth. For instance, with either agonspertian or a logistic growth dynamics in the absence ofT cells. Moreover, this assumption allow a simplify analysisof the dependence of this two modes of tumor growth withsome keys dynamical properties of tumors. However, onemust be aware that to describe real tumor dynamics, wewill have to assume more complex kinetics of tumorsgrowth in the absence of T cells. Such complex dynamicsmight result from tumor physical interaction with othercells, from its competition for nutrients and oxygen andmany other factors not included in our model.
Finally, an obvious consequence of the predicteddynamical modes is that they may require differentstrategies for the development of practical therapies.Particularly, the fact that they imply different ways ofcoupling with the immune system calls for their differ-entiation in terms of response to immunotherapies. ModeGR� tumors intuitively seem quite inappropriate forimmunotherapeutic approaches in general. These tumorsspontaneously induce an immune response, which is notable to control the tumor growth due to its large size andfast-growing kinetics. Thus, the advantage of immunother-apy in this context is not obvious, unless combined withother therapies directed to reduce tumor size (surgery) or toreduce the net tumor growth rate (for instance, deprivationof a particular tumor growth factor like EGF or others).On the contrary, mode GR+ tumors seem quite suitablefor immunotherapy. These tumors entail immune re-sponses that are always regulated by the activity ofregulatory T cells. Thus a breakdown of such tolerancemay release effector functions which may be able to controltumor growth. In future works, we will address specificallythis issue.
Acknowledgments
The work was partially supported by Foundation forScience and Technology (Portugal): fellowships to K.L.(respectively BPD/11575/2002) and grants to J.C. (POCTI/MGI/46477/2002 with FEDER funds). We thank Dr. RuiGardner for kindly revising this manuscript.
Appendix A. Simplifying the expression of Ai,j (E, R) for
large cell numbers in the system
The general expression for the total number of APCclusters conjugated to i E cells and j R cells (Ai,j(E, R))given in Eq. (3) of the main text, can be rewritten as
Ai;jðE;RÞ ¼ A
Ec
i
!Rc
j
!sA� Ec� Rc
s� i � j
!
sA
s
!
¼AðsAÞ!
s!ðsA�sÞ!
Ec!
i!ðEc� iÞ!
Rc!
j!ðRc� jÞ!ðs� i � jÞ!
� xðsA� Ec� RcÞ!
ððsA� Ec� RcÞ � ðs� i � jÞÞ!ðA:1Þ
once the hypergeometric functions have been substitutedby their definitions in terms of combinatorial factors (seeEq. (3) in the main text) and some simplifications have beenproperly carried out.Under the biologically reasonable situation of large cell
numbers in the system (A; E; R very larges ds), severalfactorials in Eq. (A.1) become factorials of large numbers,which can be properly simplified by Stirling’s formula(n! � nn
ffiffiffiffiffiffiffiffi2pnp
), obtaining:
Ai;jðE;RÞ ¼A
ðsAÞsAþð1=2Þ
s!ðsA�sÞsA�sþð1=2Þ
ðEcÞEcþð1=2Þ
i!ðEc� iÞEc�iþð1=2Þ
� xðRcÞRcþð1=2Þ
j!ðRc� jÞRc�jþð1=2Þðs� i � jÞ!
� xðsA� Ec� RcÞsA�Ec�Rcþð1=2Þ
ððsA� Ec� RcÞ � ðs� i � jÞÞðsA�Ec�RcÞ�ðs�i�jÞþð1=2Þ
ðA:2Þ
The expression in (A.2) can be further simplified, since theinequalities Ecbi; Rcbj; sAbs and sA� Ec� Rcbs�
i � j become approximately true under our currentapproximation, obtaining after some algebra
Ai;jðE;RÞ � AðEcÞi
i!
ðRcÞj
j!
ðsA� Ec� RcÞs�i�j
ðs� i � jÞ!
s!
ðsAÞs
¼ As!
i!j!ðs� i � jÞ!
Ec
sA
� �iRc
sA
� �j
1�Ecþ Rc
sA
� �s�i�j
.
ðA:3Þ
This final expression gives a much simple representationfor the values of Ai,j(E, R), which somehow resembles theexpression of a binomial distribution. This expression can
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137136
be further used to simplify the main sums, including valuesof Ai,j(E, R) in our model.
For the sum that typically appears in the model equationfor the dynamics of effector T cells we obtain:
Xs
i¼1
iAi;0ðE;RÞ ¼ AXs
i¼1
is!
i!ðs� iÞ!
Ec
sA
� �i
1�Ecþ Rc
sA
� �s�i
¼ AXs
i¼1
is!
i!ðs� iÞ!
Ec
sA� Rc
� �iðsA� RcÞi
ðsAÞi
� 1�Ec
sA� Rc
� �s�iðsA� RcÞs�i
ðsAÞs�i
¼ A 1�Rc
sA
� �sXs
i¼1
is!
i!ðs� iÞ!
�Ec
sA� Rc
� �i
1�Ec
sA� Rc
� �s�i
¼ Ec 1�Rc
sA
� �s�1
, ðA:4Þ
where we have appropriately used the following identity:Xk
n¼1
nk
n
� �pnð1� pÞk�n
¼ kp (A.5)
with p ¼ Ec=ðsA� RcÞ and k ¼ s, which corresponds to themean of a binomial distribution of n events in k trial, withprobability p that the event occurred in any given trial.
For the second sum that appears in the model equation,for the dynamics of regulatory T cells we getXs
j¼1
Xs�j
i¼1
ijAi;jðE;RÞ ¼ AXs
j¼1
jRc
sA
� �j Xs�j
i¼1
is!
i!j!ðs� i � jÞ!
�Ec
sA
� �i
1�Ecþ Rc
sA
� �s�i�j
¼ AXs
j¼1
jRc
sA
� �js!
j!ðs� jÞ!
� 1�Rc
sA
� �s�jEc
sA� Rc
� �ðs� jÞ
¼ AEc
sA� Rc
� �s2
Rc
sA�
Rc
A�
s� 1
s
Rc
A
� �2" #
¼s� 1
s
EcRc
A, ðA:6Þ
where we have used twice the identity given in Eq. (A.5)and once the following identity related to the mean squaredeviation of a binomial distribution:Xk
n¼1
n2k
n
� �pnð1� pÞk�n
¼ kpþk � 1
kðkpÞ2. (A.7)
References
Almeida, A.R., Rocha, B., Freitas, A.A., Tanchot, C., 2005. Homeostasis
of T cell numbers: from thymus production to peripheral compart-
mentalization and the indexation of regulatory T cells. Semin.
Immunol. 17, 239–249.
Baecher-Allan, C., Anderson, D.E., 2006. Regulatory cells and human
cancer. Semin. Cancer Biol. 16, 98–105.
Baecher-Allan, C., Hafler, D.A., 2006. Human regulatory T cells and their
role in autoimmune disease. Immunol. Rev. 212, 203–216.
Bharat, A., Fields, R.C., Mohanakumar, T., 2005. Regulatory T cell-
mediated transplantation tolerance. Immunol. Res. 33, 195–212.
Carneiro, J., Leon, K., Caramalho, I., Van den Dool, C., Gardner, R.,
Oliveira, V., Bergman, M., Sepulveda, N., Paixao, T., Faro, J., et al.,
2007. When three is not a crowd: a crossregulation model of the
dynamics and repertoire selection of regulatory CD4T cells. Immunol.
Rev. 216, 48–68.
Cederbom, L., Hall, H., Ivars, F., 2000. CD4+CD25+ regulatory T cells
down-regulate co-stimulatory molecules on antigen-presenting cells.
Eur. J. Immunol. 30, 1538–1543.
Coombes, J.L., Robinson, N.J., Maloy, K.J., Uhlig, H.H., Powrie, F.,
2005. Regulatory T cells and intestinal homeostasis. Immunol. Rev.
204, 184–194.
Curiel, T.J., Coukos, G., Zou, L., Alvarez, X., Cheng, P., Mottram, P.,
Evdemon-Hogan, M., Conejo-Garcia, J.R., Zhang, L., Burow,
M.etal., 2004. Specific recruitment of regulatory T cells in ovarian
carcinoma fosters immune privilege and predicts reduced survival.
Nat. Med. 10, 942–949.
Dudley, M.E., Wunderlich, J.R., Robbins, P.F., Yang, J.C., Hwu, P.,
Schwartzentruber, D.J., xTopalian, S.L., Sherry, R., Restifo, N.P.,
Hubicki, A.M.etal., 2002. Cancer regression and autoimmunity in
patients after clonal repopulation with antitumor lymphocytes. Science
298, 850–854.
Edinger, M., Hoffmann, P., Ermann, J., Drago, K., Fathman, C.G.,
Strober, S., Negrin, R.S., 2003. CD4+CD25+ regulatory T cells
preserve graft-versus-tumor activity while inhibiting graft-versus-host
disease after bone marrow transplantation. Nat. Med. 9, 1144–1150.
Engelhorn, M.E., Guevara-Patino, J.A., Noffz, G., Hooper, A.T., Lou,
O., Gold, J.S., Kappel, B.J., Houghton, A.N., 2006. Autoimmunity
and tumor immunity induced by immune responses to mutations in
self. Nat. Med. 12, 198–206.
Fontenot, J.D., Gavin, M.A., Rudensky, A.Y., 2003. Foxp3 programs the
development and function of CD4+CD25+ regulatory T cells. Nat.
Immunol. 4, 330–336.
Gogas, H., Ioannovich, J., Dafni, U., Stavropoulou-Giokas, C., Frangia,
K., Tsoutsos, D., Panagiotou, P., Polyzos, A., Papadopoulos, O.,
Stratigos, A., Markopoulos, C., Bafaloukos, D., Pectasides, D.,
Fountzilas, G., Kirkwood, J.M., 2006. Prognostic significance of
autoimmunity during treatment of melanoma with interferon. N. Engl.
J. Med. 354 (7), 709–718.
Graca, L., Thompson, S., Lin, C.Y., Adams, E., Cobbold, S.P.,
Waldmann, H., 2002. Both CD4(+)CD25(+) and CD4(+)CD25(-)
regulatory cells mediate dominant transplantation tolerance. J.
Immunol. 168, 5558–5565.
Hontsu, S., Yoneyama, H., Ueha, S., Terashima, Y., Kitabatake, M.,
Nakano, A., Ito, T., Kimura, H., Matsushima, K., 2004. Visualization
of naturally occurring Foxp3+ regulatory T cells in normal and
tumor-bearing mice. Int. Immunopharmacol. 4, 1785–1793.
Hori, S., Carvalho, T.L., Demengeot, J., 2002. CD25+CD4+ regulatory
T cells suppress CD4+ T cell-mediated pulmonary hyperinflammation
driven by Pneumocystis carinii in immunodeficient mice. Eur. J.
Immunol. 32, 1282–1291.
Hori, S., Nomura, T., Sakaguchi, S., 2003. Control of regulatory T cell
development by the transcription factor Foxp3. Science 299,
1057–1061.
Jones, E., Dahm-Vicker, M., Simon, A.K., Green, A., Powrie, F.,
Cerundolo, V., Gallimore, A., 2002. Depletion of CD25+ regulatory
cells results in suppression of melanoma growth and induction of
autoreactivity in mice. Cancer Immunol. 2, 1.
Khattri, R., Cox, T., Yasayko, S.A., Ramsdell, F., 2003. An essential role
for Scurfin in CD4+CD25+ T regulatory cells. Nat. Immunol. 4,
337–342.
Kono, K., Kawaida, H., Takahashi, A., Sugai, H., Mimura, K.,
Miyagawa, N., Omata, H., Fujii, H., 2006. CD4(+)CD25high
ARTICLE IN PRESSK. Leon et al. / Journal of Theoretical Biology 247 (2007) 122–137 137
regulatory T cells increase with tumor stage in patients with gastric and
esophageal cancers. Cancer Immunol. Immunother. 55, 1064–1071.
Leon, K., 2002. A Quantitative Approach to Dominant Tolerance. PhD
Thesis, University of Porto, Porto, Portugal.
Leon, K., Perez, R., Lage, A., Carneiro, J., 2000. Modelling T-cell-
mediated suppression dependent on interactions in multicellular
conjugates. J. Theor. Biol. 207, 231–254.
Leon, K., Perez, R., Lage, A., Carneiro, J., 2001. Three-cell interactions in
T cell-mediated suppression? A mathematical analysis of its quantita-
tive implications. J. Immunol. 166, 5356–5365.
Leon, K., Lage, A., Carneiro, J., 2003. Tolerance and immunity in a
mathematical model of T-cell mediated suppression. J. Theor. Biol.
225, 107–126.
Li, X., Ye, D.F., Xie, X., Chen, H.Z., Lu, W.G., 2005. Proportion of
CD4+CD25+ regulatory T cell is increased in the patients with
ovarian carcinoma. Cancer Invest. 23, 399–403.
Nair, R.E., Kilinc, M.O., Jones, S.A., Egilmez, N.K., 2006. Chronic
immune therapy induces a progressive increase in intratumoral T
suppressor activity and a concurrent loss of tumor-specific CD8+ T
effectors in her-2/neu transgenic mice bearing advanced spontaneous
tumors. J. Immunol. 176, 7325–7334.
Nishikawa, H., Kato, T., Tanida, K., Hiasa, A., Tawara, I., Ikeda, H.,
Ikarashi, Y., Wakasugi, H., Kronenberg, M., Nakayama, T., et al.,
2003. CD4+ CD25+ T cells responding to serologically defined
autoantigens suppress antitumor immune responses. Proc. Natl. Acad.
Sci. USA 100, 10902–10906.
Nishikawa, H., Kato, T., Tawara, I., Saito, K., Ikeda, H., Kuribayashi,
K., Allen, P.M., Schreiber, R.D., Sakaguchi, S., Old, L.J.etal., 2005a.
Definition of target antigens for naturally occurring CD4(+)
CD25(+) regulatory T cells. J. Exp. Med. 201, 681–686.
Nishikawa, H., Kato, T., Tawara, I., Takemitsu, T., Saito, K., Wang, L.,
Ikarashi, Y., Wakasugi, H., Nakayama, T., Taniguchi, M., et al.,
2005b. Accelerated chemically induced tumor development mediated
by CD4+CD25+ regulatory T cells in wild-type hosts. Proc. Natl.
Acad. Sci. USA 102, 9253–9257.
Onizuka, S., Tawara, I., Shimizu, J., Sakaguchi, S., Fujita, T., Nakayama,
E., 1999. Tumor rejection by in vivo administration of anti-CD25
(interleukin-2 receptor alpha) monoclonal antibody. Cancer Res. 59,
3128–3133.
Overwijk, W.W., Lee, D.S., Surman, D.R., Irvine, K.R., Touloukian,
C.E., Chan, C.C., Carroll, M.W., Moss, B., Rosenberg, S.A., Restifo,
N.P., 1999. Vaccination with a recombinant vaccinia virus encoding a
‘‘self’’ antigen induces autoimmune vitiligo and tumor cell destruction
in mice: requirement for CD4(+) T lymphocytes. Proc. Natl. Acad.
Sci. USA 96, 2982–2987.
Sakaguchi, S., Sakaguchi, N., 2005. Regulatory T cells in immunologic
self-tolerance and autoimmune disease. Int. Rev. Immunol. 24,
211–226.
Shimizu, J., Yamazaki, S., Sakaguchi, S., 1999. Induction of tumor
immunity by removing CD25+CD4+ T cells: a common basis
between tumor immunity and autoimmunity. J. Immunol. 163,
5211–5218.
Suvas, S., Azkur, A.K., Kim, B.S., Kumaraguru, U., Rouse, B.T., 2004.
CD4+CD25+ regulatory T cells control the severity of viral
immunoinflammatory lesions. J. Immunol. 172, 4123–4132.
Taams, L.S., Boot, E.P., van Eden, W., Wauben, M.H., 2000. ‘Anergic’ T
cells modulate the T-cell activating capacity of antigen-presenting cells.
J. Autoimmun. 14, 335–341.
Thornton, A.M., Shevach, E.M., 2000. Suppressor effector function of
CD4+CD25+ immunoregulatory T cells is antigen nonspecific.
J. Immunol. 164, 183–190.
Vendetti, S., Chai, J.G., Dyson, J., Simpson, E., Lombardi, G., Lechler,
R., 2000. Anergic T cells inhibit the antigen-presenting function of
dendritic cells. J. Immunol. 165, 1175–1181.
Wei, W.Z., Jacob, J.B., Zielinski, J.F., Flynn, J.C., Shim, K.D., Alsharabi,
G., Giraldo, A.A., Kong, Y.C., 2005. Concurrent induction of
antitumor immunity and autoimmune thyroiditis in CD4+ CD25+
regulatory T cell-depleted mice. Cancer Res. 65, 8471–8478.
Wolf, D., Wolf, A.M., Rumpold, H., Fiegl, H., Zeimet, A.G., Muller-
Holzner, E., Deibl, M., Gastl, G., Gunsilius, E., Marth, C., 2005. The
expression of the regulatory T cell-specific forkhead box transcription
factor FoxP3 is associated with poor prognosis in ovarian cancer. Clin.
Cancer Res. 11, 8326–8331.
Wolf, D., Rumpold, H., Wolf, A.M., 2006. Regulatory T cells in cancer
biology: a possible new target for biochemical therapies. Mini Rev.
Med. Chem. 6, 509–513.
Woo, E.Y., Yeh, H., Chu, C.S., Schlienger, K., Carroll, R.G., Riley, J.L.,
Kaiser, L.R., June, C.H., 2002. Cutting edge: regulatory T cells from
lung cancer patients directly inhibit autologous T cell proliferation.
J. Immunol. 168, 4272–4276.
Yamaguchi, T., Sakaguchi, S., 2006. Regulatory T cells in immune
surveillance and treatment of cancer. Semin. Cancer Biol. 16, 115–123.
Zhou, G., Drake, C.G., Levitsky, H.I., 2006. Amplification of tumor-
specific regulatory T cells following therapeutic cancer vaccines. Blood
107, 628–636.
