Tumor Immune Dynamics Regulated in the Microenvironment ...cancerres.aacrjournals.org/content/73/12/3534.full.pdf · Tumor–Immune Dynamics Regulated in the Microenvironment Inform

Integrated Systems and Technologies: Mathematical Oncology

Tumor–Immune Dynamics Regulated in theMicroenvironment Inform the Transient Nature ofImmune-Induced Tumor Dormancy

Kathleen P. Wilkie and Philip Hahnfeldt

AbstractCancer in a host induces responses that increase the ability of the microenvironment to sustain the growing

mass, for example, angiogenesis, but cancer cells can have varying sensitivities to these sustainability signals.Here, we show that these sensitivities are significant determinants of ultimate tumor fate, especially in response totreatments and immune interactions. We present a mathematical model of cancer–immune interactions thatmodifies generalized logistic growth with both immune-predation and immune-recruitment. The role of agrowing environmental carrying capacity is discussed as a possible regulatory mechanism for tumor growth,and this regulation is shown to modify cancer–immune interactions and the possibility of achieving immune-induced tumor dormancy. This mathematical model qualitatively matches experimental observations ofimmune-induced tumor dormancy as it predicts dormancy as a transient period of growth that necessarilyends in either tumor elimination or tumor escape. As dormant tumors may exist asymptomatically and maybe easier to treat with conventional therapy, an understanding of the mechanisms behind tumor dormancymay lead to new treatments aimed at prolonging the dormant state or converting an aggressive cancer to thedormant state. Cancer Res; 73(12); 3534–44. �2013 AACR.

IntroductionTumor growth is defined by interactions with the local

environment. Immune cells can both promote tumor growththrough inflammatory mechanisms and inhibit growththrough direct cytotoxic effects (7). Other stromal cells con-tribute to a tumor-promoting environment by increasing the

local vasculature and by forming a supportive niche for cancercell proliferation and invasion (8). The intercellular signalingbetween cancer and stromal cells can trigger prosurvivalpathways in both cancer cells and host stroma, includingendothelial cells, fibroblasts, and immune cells (8).

Inherently, there is variability in the intercellular signaling,which may modulate macroscopic measurements such astumor diameter. This variability may arise from many factors,including cell sensitivities to intercellular signals and the ratesat which cells respond to those signals. One example ofenvironmental regulation is shown by dormant and fast-grow-ing tumor clones that exhibit similar growth kinetics in vitrobut display strikingly different growth rates in vivo (9). Here, weshow that intercellular signaling may act to regulate tumorgrowth. We focus on the sensitivity of cancer cells to growthregulatory signals from the tumor microenvironment and thealtered tumor dynamics resulting from treatment-induceddisruption of these signals. We show that the variable sensi-tivities of cancer cells to stromal intercellular signaling mayfundamentally control tumor dynamics, even to the point ofinducing an immune-induced dormant state, with clear impli-cations for therapeutic efficacies.

Immune-induced cancer dormancy is a state of cancerprogression in which the cancer is maintained in a viable, butnonexpanding, state (10), often described as an "equilibrium"phase in immunoediting nomenclature (11). Although thisstate may persist for days to decades, its immunologic reali-zation is one of transience, i.e., the eventual elimination ofthe disease or the development of immune-resistance followed,some time later, by tumor escape.

Authors' Affiliation: Center of Cancer Systems Biology, GeneSysResearch Institute, Tufts University School of Medicine, Boston,Massachusetts

Corresponding Author: Philip Hahnfeldt, Center of Cancer SystemsBiology, GeneSys Research Institute, 736 Cambridge St., CBR 115, Bos-ton, MA 02135. Phone: 617-789-2998; Fax: 617-562-7142; E-mail:[email protected]

doi: 10.1158/0008-5472.CAN-12-4590

�2013 American Association for Cancer Research.

Major FindingsWe show, using a mathematical model, how the sensi-

tivity of tumor cells to immune-mediated environmentalsignals can significantly alter tumor dynamics and, thus,treatment outcomes. Moreover, immune-induced tumordormancy is predicted to be a transient period of tumorgrowth that must necessarily end in either tumor elimina-tion or tumor escape, in agreement with several experi-mental observations.

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In contrast, mathematical models typically describe thedormant state by a stable equilibrium point (or limit cycle)with a basin of attraction (12–17). This implies that thedormant state can attract tumor trajectories and maintainitself for long times. Such analyses neglect the transient natureof the dormant state, however, and require external perturba-tions to the system to explain the eventual escape fromdormancy. Recent mathematical explorations of possibleescape mechanisms include random fluctuations in immunepresence (18), intercellular communication of learned cancercell resistance (19), and immunoediting or evolution in cancercell phenotypes (20).Here, without considering specific escape mechanisms, we

present a formalism that contains one long-term dormancy-associated equilibrium and predicts tumor dormancy that willgenerally end in either tumor elimination or tumor escape. Theequilibrium point is a saddle node with a separatrix thatdivides two attractor regions of ultimate tumor fate. We showthat the duration of dormancy is determined by tumor–

immune dynamics and the proximity of the tumor trajectoryto the separatrix.

This model is simple enough to analytically investigate, yetcomplex enough to capture all qualitative behaviors of tumorgrowth, including tumor dormancy. Using parameter setsestimated by aMarkov ChainMonte Carlo (MCMC) algorithm,we show that the sensitivity of cancer cells to environmentalsignals is a prominent factor in determining tumor fate. Wegenerate four parameter sets: one assuming a constant envi-ronmental signal, producing a static carrying capacity, andthree assuming a variable environmental signal, giving rise toa dynamic carrying capacity. These four sets fit the experi-mental tumor growth data equally well, but when the variablesignals are disrupted, the differing cancer cell sensitivitiespredict different tumor growth fates. Interestingly, these foursets predict drastically different results for the same treatment,ranging from rapid tumor elimination to tumor escape.

These results may explain both the high variability oftreatment success among patients as well as the observation

Quick Guide to Equations and AssumptionsAs we focus on cancer dormancy induced by immune predation, the cancer–immune interactions considered are immune

predation of, and immune recruitment by, cancer cells. The cancer, C(t), and immune, I(t), populations are assumed to exhibitgeneralized logistic growth. Without immune predation, C(t) will grow until it reaches the carrying capacity KC. We consider thecarrying capacity to be dictated by the growth regulatory signal emitted by the environment and sensed by the cells. The immunecompartment I(t) impedes tumor growth through cytotoxic actions targeted at the cancer cells. These actions are contained in thepredation term �(I,C), which acts to modulate the growth rate of C(t). The equation governing cancer growth is thus,

dCdt

¼ m

að1þ�ðI ;CÞÞC 1� C

KC

� �a� �; Cð0Þ ¼ C0: ðAÞ

Here, m and a are parameters that capture the growth rate and sensitivity of cancer cells to environmental regulatory signals.The immune population is assumed to consist of cytotoxic cells, for example, CD8þ T cells, NK cells, and macrophages that

are known to directly target cancer cells. The immune response is assumed to grow toward a theoretical limit, or carrying capacityKI through proliferation and recruitment from the blood, spleen, and bone marrow. With parameter r, we allow for both immunecell– and cancer cell–mediated recruitment; the latter occurs through, for example, the production of danger-associatedmoleculesand necrosis (1). The equation governing immune growth is thus,

dIdt

¼ lðI þ rCÞ 1� IKI

� �; Ið0Þ ¼ I0; ðBÞ

where l is the immune growth-rate parameter.Immune predation of cancer cells is assumed to contain both fast and slow dynamics, expressed as

�ðI ;CÞ ¼ ��Ib

�CbþIbþe log10ð1þIÞ

� �; ðCÞ

where�ð0; 0Þ ¼ 0 by definition. The fast saturation behavior observed in some cancer–immune lysis assays (2, 3) is described bythefirst, ratio-dependent term (4). Although this formhas been suggested to better capture the cytotoxic effects of T cells, this termalone neglects innate immunity, which should not exhibit saturation (compare, e.g., the percent lysis curves for cytotoxic T cells vs.NK cells in refs. 2, 3, 5). The second term accounts for innate immunity, and allows for a slow increase in the saturation limit withincreasing immune presence, as seen when the lysis assays are conducted at larger ratios (3, 5, 6). The logarithm transforms thecytotoxic actions of the innate immunity into a range appropriate for�. For small immune populations, we assume that the innateand adaptive effects can be combined into the ratio-dependent term, but for large populations, innate immunity should still have asmall cytotoxic effect. Thus, we assume e ¼ 0:01.

The Transient Nature of Immune-Induced Tumor Dormancy

www.aacrjournals.org Cancer Res; 73(12) June 15, 2013 3535




of accelerated repopulation after treatment (21, 22), a phe-nomenon arising here from disruption of growth regulatorysignals in the microenvironment. These same signals areshown to ultimately determine whether dormancy will bepossible, with higher tumor sensitivities to these signals cor-responding to decreased likelihood of dormancy when thesesignals are disrupted, as may happen from therapy or chronicmicroenvironmental perturbations such as low-dose radiationexposure (23).

Materials and MethodsExperimental data

Basic tumor growth dynamics in the absence of an antican-cer immune response are estimated from experimental mea-surements of a subcutaneous fibrosarcoma induced by 3-methylcholanthrene. Tanooka and colleagues (24) inducedsarcomas in wild-type WBB6F1/J and C3H/He mice. Volumemeasurements were taken when the resulting tumors reacheda palpable size. We note that WBB6F1/J mice are mast celldeficient, which inhibits the host's inflammatory response.

Estimation of model parametersParameters are estimated using a MCMC algorithm (25, 26).

From an initial estimation, a Markov chain of permittedparameter sets is created by randomly perturbing the previousparameter set and accepting this perturbed set with a prob-ability determined by ameasure of goodness offit. Here, we usethe sum of squared deviations between model prediction andexperimental measurements. Each parameter is perturbed andtested for acceptance independently, except parametersm anda, which are handled together because they are inherentlyrelated in Eq. A. The algorithm is repeated 10 times, with eachrun having 20,000 iterations. Final parameters are those thatprovide the best fits.

Parameters are estimated under the assumption of noimmune predation ð� ¼ 0Þ. The results assuming a constantor dynamic carrying capacity are listed in Tables 1 and 2,respectively. Immune size and growth rate values are selectedto approximately match those of the tumor as methods tomeasure immune system checkpoint blockades and other size-determining factors remain elusive.

Mathematical analysisThe nonlinear system given by Eqs. A, B, and C describes a

2-compartment model for cancer–immune interactions. Thegeneral form W ¼ 1þ� allows for immune modulation oftumor growth, which may be stimulatory or inhibitory,depending on W. It should be noted that this form funda-mentally differs from classic predator–prey-type modelswhere immune effectors only inhibit tumor growth and arestimulated directly by tumor presence (4, 13, 17, 27, 28). Incontrast to decaying oscillations or limit cycle behaviors, ourmodel predicts tumor elimination or escape, with only tran-sient periods of dormancy. This is partially due to theassumed form of the immune response, which, once initiated,will progress to a maximal response that is independent ofthe cancer population.

Stability analysis of the system equilibrium points andpossible cancer–immune dynamics. The dynamical systemdescribed by Eqs. A and B is simple enough to allow for analyticinvestigation of system stability. The system has four equilib-rium points: ðI;CÞ ¼ ð0; 0Þ, ðKI ; 0Þ, ðKI ;KCÞ, and ðKI ; �CÞwhere �C satisfying 1þ�ðKI ; �CÞ ¼ 0 is given by

�C ¼ KI1

�

�

1� e� log10ð1þ KI Þ � 1

� �� 1b

; ðDÞ

and represents a cancer equilibrium state that, once obtained,will persist for a long time. We note that �(I, C) is not differ-entiable at (0, 0), but that we can classify this point based onphysical arguments and a topologically equivalent system. Theequilibria stability, based on the Jacobian, are as follows: (0, 0) isan unstable point, (KI, 0) is a stable point if 1þ�ðKI ; 0Þ< 0,otherwise it is a saddle point, and ðKI ; KCÞ is a stable point if1þ�ðKI ;KCÞ > 0, otherwise it is a saddle point. The point

ðKI ; �CÞ with 0< �C<KC is a saddle point if (KI,0) and (KI, KC) areboth stable. The two restrictions guaranteeing the attractorstates bound immune predation strength as follows, ensuringa dormancy-associated equilibrium point:

�min ¼ 1þ e log10ð1þ KI Þ� ��1

< � <K bI

�K bC þ K b

I

þe log10ð1þ KI Þ" #�1

¼ �max:

ðEÞ

Under this condition, a large immune response is strongenough to eliminate small tumors but not large tumors,resulting in a possible dormant tumor of size �C. It guarantees

Table 1. Parameter values for modelsimulations

Parameter Value Interpretation

m 0.16 days�1 Cancer growtha 0.72 Cancer growthl 0.22 days�1 Immune growthr 1�10�3 Immune recruitmentu 2.5 Predation strengthb 0.5 Predation saturationf 50 Predation saturatione 0.01 Innate immunity

predation strengthKC 3.92 � 1010

Cell numCancer-carryingcapacity

KI 3.92 � 1010

Cell numImmune-carryingcapacity

NOTE:CðtÞ and IðtÞmeasure cell number, but their values areplotted in terms of cellular mass diameter in millimeters

according to the formula diameterðXÞ ¼ 2 X106

34p

� �1=3mm.

Wilkie and Hahnfeldt

Cancer Res; 73(12) June 15, 2013 Cancer Research3536




the existence of all possible tumor outcomes: tumor elimina-tion, tumor escape, and long-term tumor dormancy followedby elimination or escape. Three sample cancer–immune phaseportraits are shown in Fig. 1. The long-term dormant stateincreases as immune efficacy increases from umin to umax, andtrajectories that pass near the separatrix have transient dor-mant states.

Bifurcation and sensitivity analysis of tumor dormancy.If � < �min (or 1þ�ðKI ; 0Þ> 0), then the immune response isweak and ðKI ; 0Þ is a saddle point with all tumors escapingimmune control (Fig. 1A). In this case, �C is negative and,thus, unphysical. If �> �max (or 1þ�ðKI ;KCÞ< 0), then theimmune response is strong and ðKI ;KCÞ is a saddle point,implying that all tumors are eliminated (Fig. 1C). In this

Table 2. Tumor growth parameter sets estimated by the MCMC parameter fitting method assuming adynamic carrying capacity

Growth rate andsensitivity

Dynamic carryingcapacity

Constant carryingcapacity

m a p q KCð0Þ KC ¼ pq

� �3=2

Set 1 0.28 1.08 0.90 7.97 � 10�8 1.12 � 108 3.79 � 1010

Set 2 0.23 1.03 34.5 3.14 � 10�6 8.47 � 105 3.64 � 1010

Set 3 0.19 1.44 � 10�5 0.73 5.98 � 10�8 2.90 � 107 4.27 � 1010

NOTE: The last column gives the maximum value of KC, a value used when a constant carrying capacity is forced on the system.

C—

C—

A

Can

cer

(mm

)

Perc

en

t ch

an

ge

Immune (mm)

KC KC

KI

40

30

20

10

0

C

KC

40

30

20

10

0

Can

cer

(mm

)

40

30

20

10

0

KC

Can

cer

(mm

)

40

30

20

10

0

80

60

40

20

0

–20

–40

–60

–80

B

D E

Cθ = 0.5 < θmin

θmax

θ

θ βφ

θ = 10 > θmaxθmin < θ = 2.5 < θmax

0

0 2 4 6 8 10

10

Stable Unstable

20 30 40Immune (mm)

KI

KI

0 10 20 30 40Immune (mm)

Sensitivity of dormant state

Increase in parameter Decrease in parameter

KI0 10 20 30 40

Transient

dormancy

θmin

Figure 1. The progression of the immune–cancer phase plane as immune efficacy u increases. A, if u < umin, immune predation is weak and all cancers reach thecarrying capacity KC. B, if �min < � < �max, a dormant cancer state exists as a saddle point between two stable equilibria creating two basins of attraction, onerepresenting cancer elimination (bottom, blue region) and one representing cancer escape (top, red region). The separatrix is the black curve separating thetwo basins of attraction, and leads to the dormancy-associated equilibrium point ðKI ; �CÞ. C, finally, if � > �max, all cancers are eventually eliminated. In D,a bifurcation diagram shows the stability of the cancer equilibrium points as immune efficacy u increases. In E, the sensitivity of the size of the dormantstate, �C, to changes of 10% in the immune model parameters KI , u, f, and b, is shown. Analyses were completed with parameter values fromTable 1 unless otherwise stated.






case, �C >KC, and because we only consider tumor growthin the positive quadrant bounded by the carrying capacity,we neglect this dormant equilibrium point. Furthermore,if � ¼ �min (or 1þ�ðKI ; 0Þ ¼ 0), then �C ¼ 0, and if � ¼ �max

(or 1þ�ðKI ;KCÞ ¼ 0), then �C ¼ KC. Thus, u is a bifurca-tion parameter with two transcritical bifurcations. A bifur-cation diagram showing the transitions and stability ofthe equilibrium points is shown in Fig. 1D. Because therange ½�min; �max� guarantees the existence of all possibletumor outcomes, we focus on it for the remainder of thediscussion.

Ranges of immune efficacy guaranteeing tumor dor-mancy. Figure 2 shows the dependence of the range½�min; �max� on parameters KI, KC, b, and f. As theimmune-carrying capacity increases, the upper bound on

immune efficacy decreases because the same amount ofcancer cell predation can be mediated by either a large andweakly effective immune presence or a small but stronglyeffective immune presence. Similarly, as the cancer-carryingcapacity increases, the upper bound on immune efficacyincreases, as stronger immune predation is required toeliminate the larger tumor. The effects of the immunepredation shape parameters, (b, f), on the range½�min; �max� depends on the relative size of the two carryingcapacities. Notice that if KC ¼ KI , then b has no effect on½�min; �max�. If this is not the case, however, then if KI <KC,increasing b increases umax, and if KI >KC, increasing b

decreases umax. This behavior results from the shape of thefast saturation kinetic in the cell-kill term, which is a func-tion of the ratio KI=KC [see Eq. C]. Parameter f also controls

A

B

C

D

E

F

G

H

KI

15

10

5

0

θmax

θ

15

10

5

0

15

10

5

0

θmaxθmax

θmax

θmax

θ θ

θmin

θmin

15

10

5

0

θmax

θ

θmin

15

10

5

0

θmaxθ

θmin

θmin

15

10

5

0

θ

θmin

15

10

5

0

θ

θmin

109 1010 1011

KC

15

10

5

0

θmax

θ

θmin

109 1010 1011

0 0.5 1 1.5 2 0 20 40 60 80 100β

0 0.5 1 1.5 2β

0 0.5 1 1.5 2β

φ

0 20 40 60 80 100φ

0 20 40 60 80 100φ

KI = 3.92 109 < KC = 3.92 1010 KI = 3.92 109 < KC = 3.92 1010

KI = 3.92 1010 = KC = 3.92 1010 KI = 3.92 1010 = KC = 3.92 1010

KI = 3.92 1011 > KC = 3.92 1010 KI = 3.92 1011 > KC = 3.92 1010

Figure 2. The dependence of the range ½�min; �max� on model parameters KI , KC, b, and f. In all figures, the 3 shaded areas represent regions describingdistinct tumor outcomes: all tumors are eliminated in the (light blue) upper regions; tumor elimination, tumor escape, and long-term tumor dormancyare all possible in the (green) middle regions; and all tumors escape in the (pink) lower regions. The range of u permitting a long-term dormant cancerstate decreases as immune carrying capacity increases (A) and increases as cancer carrying capacity increases (B), for fixed (b, u) in the predation term, C–H,the effects of changes in b and f on the range ½�min; �max� depend on both KI and KC. Unless otherwise stated, parameter values are from Table 1.






the shape of the fast saturation kinetic. If � ¼ 0, then�min ¼ �max, and as f increases, the upper bound alsoincreases. Notice that for all these cases, the lower bound,umin, is essentially constant, suggesting that there exists astrictly positive threshold for immune predation efficacy,below which long-term tumor dormancy cannot exist.

Discussion of Simulations and ResultsNumerical simulations of tumor growth are presented with

and without immune predation. Simulations are computednumerically inMAPLE (www.maplesoft.com) using the param-eter values in Tables 1 and 2 as appropriate.

Tumor growth is regulated by signals from themicroenvironmentWe investigate the effect of environmental signaling on

tumor dormancy. When a tumor grows, it modifies the local

environment, through angiogenesis, to increase the nutrientsupply to that area. In this way, the carrying capacity increaseswith tumor mass. This effect can be modeled by a differentialequation that describes the growth of the cancer-carryingcapacity, KC(t), in response to the growing tumor, C(t). Thefunctional form, as first proposed by Hahnfeldt and colleagues(29), is

dKC

dt¼ pCðtÞ � qKCðtÞCðtÞ

23; ðFÞ

where p and q are growth stimulation and inhibition constants,respectively.

Replacing the constant carrying capacity in Eq. A with adynamic capacity as described earlier, creates an extension ofthe mathematical model given by Eqs. A, B, and C. By com-paring these two models, we show that disturbance of thesupport signals from the tumor environment can significantly

A 50

40

30

20

10

0

C D E F

B

0 10 20

Data Set 0 Set 1 Set 2 Set 3 Data Set 0 Set 1 Set 2 Set 3

30

Time (days)

Constant and dynamic KC

Set 2 with dynamic KC Set 3 with dynamic KC Set 2 with constant KC Set 3 with constant KC

Forced constant KC

Tu

mo

r d

iam

ete

r (m

m)

50

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0

Tu

mo

r d

iam

ete

r (m

m)

40 50 60 70 80 0 10 20 30Time (days)

40 50 60 70 80

Can

cer

(mm

)

Immune (mm)

KI

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00 10 20 30 40

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cer

(mm

)

Immune (mm)

KCKCKC

KI

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cer

(mm

)Immune (mm)

KC

KI

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00 10 20 30 40

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cer

(mm

)

Immune (mm)

KI

40

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00 10 20 30 40

C—

C—

Figure3. Tumormicroenvironmental growth signalsmodulate tumorgrowth.A, tumorgrowth ispredictedbyfittinggrowthparameters to theexperimental dataof Tanooka and colleagues (24). Set 0 (Table 1) assumes a constant carrying capacity, whereas sets 1, 2, and 3 (Table 2) allow for a dynamic carryingcapacity. All curves fit the experimental data equally well. B, the same parameter sets are used, but a constant carrying capacity is forced on sets 1, 2, and 3

after the parameter fitting usingKC ¼ pq

� �32 (values fromTable 2). Note that set 2 still fits the datawell, whereas sets 1 and 3do not. This indicates that the tumor

described by set 2 is not sensitive to the local environmental growth cues, whereas sets 1 and 3 are very sensitive. Cancer–immune phase portraits forsets 2 and 3 show the same general behavior when the dynamic carrying capacity is allowed (C and D) but show significantly different behavior when aconstant carrying capacity is enforced (E and F) simulating a disrupted environmental signal. After disruption, set 2 still allows immune-induced transientperiods of tumor dormancy, whereas set 3 no longer predicts any significant dormant periods.






alter tumor growth dynamics. This formulation attempts tocapture some of the effects stromal cells (including immunecells) can have on the proliferation rate, viability, differentia-tion state, and invasiveness of cancer cells (30).

Parameter estimation for tumor growth in the absence ofimmune predation is conducted with the experimental dataof Tanooka and colleagues (24) using an MCMC parameterestimation algorithm. Parameter sets are estimated (i) forthe tumor growth model assuming a constant carryingcapacity given by Eq. A; and (ii) the tumor growth modelassuming a dynamic carrying capacity given by Eqs. A andF. Table 1 lists the estimated parameter values used for theconstant carrying capacity model, referred to as set 0 (m, a,and KC from Table 1), and Table 2 lists 3 possible parametersets for the dynamic carrying capacity model, referred to assets 1, 2, and 3.

As can be seen from Fig. 3, both models and all parametersets fit the experimental data equally well. When microenvi-ronmental regulation of tumor growth is disturbed, however,sets 1 and 3 no longer fit the data. Microenvironmentalsignaling is disrupted in the model by forcing the dynamiccarrying capacity to be constant and equal to the maximumpossible value for each parameter set (these values are listed asKC in Table 2). Doing so removes early damping of tumorgrowth, providing a strongly protumor environment. Its effectis different for the three parameter sets. Parameter set 2 stillfits the data well, but sets 1 and 3 predict much faster tumorgrowth. More specifically, the rate at which the environmentincreases either its vascularization, the sensitivity of the stro-mal cells to the proangiogenic or other growth factors, or thesensitivity of the tumor cells to the signaled levels of environ-mental support, all contribute to tumor-specific inherentvariabilities that determine basic tumor growth dynamics.

These results highlight the important fact that the sametumor growth curve can be observed for three different tu-mors, each with different cellular proliferation rates and dif-ferent sensitivities to the regulatory signals originating fromtheir different environments. At the same time, when thesesignals are altered, the resulting changes in growth dynamicscan be drastic.

Altering the balance of the tumor and microenvironmentalcompartments, for example, by selective reduction of tumorburden by therapy, leaving the microenvironmental supportnearly intact (or enhanced; ref. 23), may accelerate tumorrepopulation, a problematic phenomenon familiar to cliniciansthat remains poorly understood (21, 22, 31). In this regard, Fig. 3shows how the rate of tumor regrowth depends on the sig-naling sensitivities of the tumor cells.

Another possible interpretation of our disrupted regulatorysignals comes from Hu and colleagues (32). They found thatwhen tumor-associated and arthritis-associated (inflammato-ry) fibroblasts were coinjected with breast cancer cells, tumorweight was increased. This contribution may explain oursimulations for how enhanced capacity leads to acceleratedgrowth in sets 1 and 3. Typically, these stromal cells woulddevelop their protumor associations over time as the tumorgrows (modeled here by the dynamic capacity), but in these invivo experiments, the stromal cells are already protumorigenic

and are thus capable of supporting a tumor faster (modeledhere by the forced-constant capacity).

In addition, the ability of the immune response to inducetumor dormancy may be lost when environmental regulatorysignals are disrupted. Using parameter set 2, simulations oftumor growth with disrupted signals result in a growth curvesimilar to the data, and the resulting tumor–immune phaseportraits, shown in Fig. 3C and E, are similar for both thedynamic and the (disrupted) forced-constant carrying capacitymodels. This suggests that the tumor represented by param-eter set 2 is not very sensitive to regulation signals produced bythe microenvironment and that the cancer cells proliferateat a rate consistent with the growth curves. This allows theimmune response to achieve transient dormant states in boththe dynamic and (disrupted) forced-constant capacity models.

In contrast, the cancer–immune phase portraits for set 3in Fig. 3D and F show a lost ability of the immune response toinduce tumor dormancy, due to the tumor being very sensitiveto growth regulation signals emanating from the local micro-environment. In addition, disrupting the regulatory signals forset 3 causes the tumor to grow much faster than the exper-imental data. These observations suggest that immune-medi-ated tumor dormancy is dependent on the sensitivity of cancercells to environmental regulatory signals.

Success of immunotherapy depends on tumor sensitivityto environmental regulatory signals

To elucidate the role of variability in signaling sensitivity totreatment success, we now consider immunotherapies thatincrease the immune effectiveness of the cytotoxic T cells,achieved conceivably, for instance, by checkpoint blockade-targeted antibodies to cytotoxic T-lymphocyte antigen 4(CTLA-4) and programmed cell death protein 1 (PD-1). Thesetherapies release the immunologic block that limits theimmune response, allowing the development of a large spe-cifically targeted T-cell response to the cancer.We simulate theeffects of these treatments by allowing a large immuneresponse to develop with an increased recruitment ability(r) and an increased cytotoxic efficacy (u). Treatment occursafter the tumor is initially detected (at t ¼ 0) with no appreci-able initial immune presence [Ið0Þ ¼ 0]. In Fig. 4A and B, theexperimental data for tumor growth without predation isincluded as a reference to the initial growth trend.

Including strong immune predation in the tumor growthsimulations of Fig. 4 results in the parameter sets all predictingdifferent outcomes for the same treatment, mimicking real-world interpatient variability. A striking comparison can beseen between the case with no immune predation, where allcurves fit the data equally well (Fig. 3A), and the case withstrong immune predation, where each curve predicts a differ-ent outcome (Fig. 4A). Note also that while set 2 indicateselimination happens faster than in the constant capacitymodel (set 0), set 1 indicates a short dormant period beforeescape, and set 3 indicates escape at only a slightly slower ratethan the experimental reference. Thus, with environmentalgrowth regulation signals intact, these simulations suggest thatthe tumor may escape, be eliminated, or be transiently dor-mant before elimination or escape. Such results may explain






why patient responses to immunotherapies are quite varied,and shed light on the connection between regulatory sensi-tivities and treatment response.With the loss of environmental growth regulation, however,

all tumors are predicted to escape (Fig. 4B). Disrupting theregulatory signals by forcing the dynamic capacity to beconstant results in tumor escape: (i) with a short dormantperiod (set 2); (ii) at a slightly slower rate than the reference (set1); or (iii) at a faster rate than the reference (set 3). Again,forcing the dynamic carrying capacity to be constant simulatesa reduction of tumor burden with little modification of envi-ronmental support. When regulation is disrupted, only tumorescape is predicted, implying that a stronger dose or combina-tions of therapiesmay be required to eliminate the tumor uponrecurrence.

Dormancy depends on tumor cell-inherent sensitivitiesand parameters of the immune responseAs discussed above, immune-mediated tumor dormancy is

dependent not only on immune effectiveness but also on the

sensitivity of cancer cells to environmental regulatory signals.To show this, we examine the initial immunepresence requiredto induce a period of dormancy in our simulations. We definedormancy as a period of at least 30 days within the first 150days of growth, wherein the effective tumor growth rate issmall, i.e., 1

CdCdt

� < 0:04, and the tumor size is moderate, i.e.,

between 1 and 30 mm in diameter. This definition excludesdormant regions where the tumor has effectively been elim-inated or where the tumor has grown to the maximal size.

In Fig. 4C, predicted tumor fate for an initial immunepresence is indicated by color for each model (i.e., for "con-stant" and "dynamic" carrying capacities) for parameter sets 0,1, 2, and 3 as appropriate. The dormancy windows, i.e., rangesof initial immune presence guaranteeing tumor dormancy, areindicated by the red (escape afterwards) and blue (eliminationafterwards) regions. Pink regions correspond to tumor escapeand light blue regions to elimination. Parameter values arer ¼ 0:47 and � ¼ 4 as used in the rest of Fig. 4 to simulateimmunotherapy. As can be seen, the constant carrying capacitymodel with parameter set 0 predicts only elimination, with

A

C

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KC Forced constant

0 50 100 150 200

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Effects of immune predation (r, θ) = (0.47, 4) Effects of immune predation (r, θ) = (0.47, 4)

Tu

mo

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iam

ete

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m)

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Set 0 C 1.5 × 106

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5.0 × 106 1.0 × 107 1.5 × 107 2.0 × 107 6.0 × 108 7.0 × 1080

5.1 × 105

4.5 × 103

Set 1 D

Set 2 D

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Set 1 C

Set 2 C

Set 3 C

Data Set 0 Set 1 Set 2 Set 3

Figure 4. Immunotherapy treatment is simulated by an increased immune recruitment rate, r, and an increased predation efficacy, u. Treatment is given at thetime of tumor detection (here, t ¼ 0) and is assumed to be immediately effective with no decay. Any immune presence at detection is ignored [Ið0Þ ¼ 0]. A andB, the data points show the growth trend with no immune interaction (as in Fig. 3). The dynamic carrying capacity model with parameter sets 1, 2, and 3 areshown in (A) alongwith the constant capacitymodel with parameter set 0. InB, thedynamic capacitymodels are forced to have constant capacities, disruptingthe regulatory signal to the tumor. All curves are modified to different extents by immune predation when compared with the data (no predation).C, tumor fate is indicated for increasing initial immune presence, Ið0Þ. From left to right, the pink regions represent tumor escape with no dormancy,the red regions represent tumor escape after dormancy, the blue regions represent tumor elimination after dormancy, and the light blue regions representtumor elimination with no dormancy. Dormancy is defined as a period of non-growth lasting at least 30 days. The width of each dormancy range (redþ blueregion) is labeled to the right of each range. Parameter sets 1, 2, and 3 are used with the dynamic capacity (D) and constant capacity (C) models. Parametervalues are r ¼ 0:47 and � ¼ 4, together with the values from Tables 1 and 2.






dormancy occurring for small initial immunepresence.When adynamic carrying capacity is used, however, the three param-eter sets (sets 1D, 2D, and 3D) predict different outcomes withdifferent dormancy windows. After disrupting the growthregulation signals (sets 1C, 2C, and 3C), these dormancywindows tend to shift to the right, indicating that a higherinitial immune presence is required to obtain dormancy. Forcomparison, the left-most color of Fig. 4C corresponds tooutcomes predicted in Fig. 4A and B. Because parameter set3 requires a much higher initial immune presence to obtaindormancy, and the size of the window is small compared withthose predicted by sets 1 and 2, we conclude that dormancy isharder to achieve in this instance due to high sensitivity toenvironmental regulation.

Dormancy is explored further through the parameter sen-sitivity analysis shown in Fig. 5 using the constant carryingcapacity model and parameters from Table 1. Each immuneparameter is varied, and the resulting tumor fate is indicated bycolor for ranges of initial immune presence. Under the assump-tion that large dormancy windows at low levels of initialimmune presence correspond to more easily achieved dor-

mancy, these sensitivities may suggest treatment strategiestargeted at obtaining and maintaining the dormant state.

ConclusionsThe analysis presented here shows that, in most cases,

immune-induced tumor dormancy is a transient state thatnecessarily ends in either elimination or escape. This resultcontrasts with behavior predicted by many predator–prey-typemodels wherein dormancy is described by decaying oscilla-tions converging to a limit cycle or steady-state, that, onceobtained, is maintained for a long time. The transient nature ofthe dormant state described here complements the immuno-editing hypothesis because the equilibrium stage must end ineither tumor elimination or escape, and qualitatively matchesreal-world observations (33).

Cancer and immune cell behaviors are constantly evolvingthroughautocrine andparacrine cytokine feedback loops in themicroenvironment. Tumor escape from immune-induced dor-mancy can involve selective editing of cancer cell immunoge-nicity (34) and accumulated resistance to apoptotic signals

A Width = 2,684

Width = 1,604

Width = 6,759

Width = 1,295

Width = 519

I0

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4

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0.001 0.01 0.1

0.001 0.01 0.1

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th

500,000

250,000

0

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th

10,000

5,000

0

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th

50,000

25,000

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0.5 1 1.5 2

109 1010 1011

109 1010 1011

Figure 5. Parameter sensitivity for immune system parameters [l (A), r (B), KI (C), u (D), f (E), and b (F)] using the model with constant carrying capacities andparameter values from Table 1 (set 0). The lower (pink) regions indicate tumor escape, the upper (light blue) regions indicate tumor elimination, andthe mid-regions indicate that the tumor either escaped (red) or was eliminated (blue) after a period of at least 30 days of dormancy. Vertical axes indicate theinitial immune presence relative to the initial cancer presence. Theoretically, tumor dormancy should be easier for the immune response to achievewhen the required initial presence I0 is small and the width of the dormancy region (red and blue, middle regions) is large. The width of the dormancy regions(measured in initial immune presence) is indicated at the left and right end points of the curves as well as on the inset line graphs.






from immune cells or chemotherapeutics (6, 35). In addition,the local cytokine milieu can determine the antitumor orprotumor polarization of the immune response (5, 36, 37).These phenomena are now being investigated through math-ematical models of both tumor escape mechanisms (18–20)and immune polarization. The model presented here does notexplicitly address these phenomena, yet captures the transientnature of dormancy in a comparatively simple framework.With this model, we investigated the range of immune

efficacy required to guarantee the existence of a dormantstate. Obtaining this state, however, is not guaranteed as wealso showed how growth regulatory signals emanating fromthe tumor microenvironment play a significant role in deter-mining tumor fate. The regulatory mechanism of a carryingcapacity that grows with the tumor may be disturbed afterdebulking treatments that leave the microenvironment unal-tered or possibly enhanced. We found that, for tumors that arehighly sensitive to these regulatory signals, tumor regrowthmay be accelerated, offering one possible rationale for thephenomenon of accelerated repopulation frequently observedin the clinic following treatment.Variability in tumor sensitivities, in turn, may account for

unpredictable outcomes following therapy and thus mayserve as a proxy for patient variability. The mathematicalmodel discussed here is the first, to our knowledge, to incor-porate such variability and predict outcomes that qualitativelymatch those seen in the clinic. Tracking variability in sensi-tivity to environmental regulatory signals may well improve

our understanding of why treatments work for some patientsbut not others.

Disclosure of Potential Conflicts of InterestNo potential conflicts of interest were disclosed.

Authors' ContributionsConception and design: K.P. WilkieDevelopment of methodology: K.P. WilkieAcquisition of data (provided animals, acquired and managed patients,provided facilities, etc.): P. HahnfeldtAnalysis and interpretation of data (e.g., statistical analysis, biostatistics,computational analysis): K.P. Wilkie, P. HahnfeldtWriting, review, and/or revision of the manuscript: K.P. Wilkie, P.HahnfeldtAdministrative, technical, or material support (i.e., reporting or orga-nizing data, constructing databases): K.P. WilkieStudy supervision: K.P. Wilkie, P. Hahnfeldt

AcknowledgmentsThe authors thank Dr. M. La Croix for his help with computation and

illustrations.

Grant SupportThis work was financially supported by the National Cancer Institute

under Award Number U54CA149233 (L. Hlatky) and by the Office of Science(BER), U.S. Department of Energy, under Award Number DE-SC0001434(P. Hahnfeldt).

The costs of publication of this article were defrayed in part by the paymentof page charges. This article must therefore be hereby marked advertisementin accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

Received December 20, 2012; revised March 7, 2013; accepted March 18, 2013;published OnlineFirst March 27, 2013.

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2013;73:3534-3544. Published OnlineFirst March 27, 2013.Cancer Res Kathleen P. Wilkie and Philip Hahnfeldt Inform the Transient Nature of Immune-Induced Tumor Dormancy

Immune Dynamics Regulated in the Microenvironment−Tumor

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