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Immunology |
Institute for Medical Biomathematics, Bene-Ataroth, Israel
Requests for reprints: Zvia Agur, Institute for Medical Biomathematics, P.O. Box 282, 10 HaTe'ena Street, Bene-Ataroth 60991, Israel. Phone: 972-3-9733075; Fax: 972-3-9733410; E-mail: agur{at}imbm.org.
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Abstract |
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Introduction |
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The interleukin (IL)-2 family of cytokines is well known for its central involvement in the regulation of acquired immunity. A recently identified member of this family, sharing homology with IL-2, IL-4, and IL-15, is IL-21, a product of activated CD4+ T-helper cells (2). In both in vitro and in vivo studies, IL-21 directly modulates number and function of natural killer (NK) cells, dendritic cells, and lymphocytes (3, 4). Importantly, IL-21 enhances effector cell-mediated lysis of tumor cells, induces efficient antitumor immune memory, and significantly minimizes angiogenic and metastatic processes in numerous tumors (2, 5–10). The potent antitumor effect of IL-21 is attributed to its role in facilitating the transition from innate NK-cell responses to acquired cytotoxic CD8+ T-cell responses. This is presumably achieved by inhibiting the former and stimulating the latter and by enhancing the cytotoxic activity of both cell-types (2, 3, 11, 12).
The immunostimulatory effects of IL-21 motivated its development as an immunotherapeutic agent for the treatment of cancer. The minimal toxicity of IL-21 to mice compared with IL-2, IL-15, and other immunotherapeutic factors (2, 5, 13, 14) further supports its clinical applicability. Yet, theoretically, the stimulatory effect of IL-21 on CD8+ T cells can be counterbalanced by its inhibitory influence on NK cells and other factors of cell-mediated immunity (2, 11). These contradicting effects may limit the IL-21-induced antitumor response. Therefore, it seems mandatory to analyze the IL-21 net immunotherapeutic power in different oncological scenarios before the establishment of an immunotherapeutic policy for this molecule.
Such analysis is enabled by biomathematically modeling IL-21 interactions with the involved immunologic and pathologic processes. Biomathematical models have been employed previously for analyzing the effect of environmental disturbances on population survival, in general, and the effect of the interdosing intervals of the drug on chemotherapy efficacy/toxicity ratio, in particular (15). Analysis of individual tumor growth patterns in xenografted human ovary carcinoma spheroids was also possible by biomathematical modeling of complex angiogenesis-related processes (16). Another mathematical model describing detailed thrombopoiesis was employed for optimizing treatment strategies of thrombopoietin. The accuracy of the qualitative and quantitative predictions of the model was prospectively validated in the preclinical setting (17). Recent mathematical models of tumor-immune interactions emphasize the role of certain effectors in anticancer responses and evaluate efficacy of immunotherapy in the context of tumor challenge (18, 19).
In the current study, the antitumor effects of IL-21 under different oncological settings are evaluated in a mathematical model of the underlying biological processes. Following the calibration of the model using published experimental data, we investigate IL-21 treatment strategies and suggest efficacious regimens for eradicating or substantially attenuating tumor mass. A beneficial treatment should maximize IL-21-stimulated activation of adaptive immunity and, concurrently, minimize its negative effects on innate immunity. These conflicting aims for different tumor dynamics and immunogenicity levels will be investigated in this work.
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Materials and Methods |
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Biological Assumptions and Mathematical Model
As IL-21 has a central role in mediating the transition from initial NK-cell immunity to secondary T-cell immunity, this bipolar mechanism is highlighted in our model, where the effects of IL-21 on NK and CD8+ T cells are taken to be direct and independent of other factors (4–10, 13, 14, 20–25). Importantly, the model focuses on the IL-21 contribution to cellular immunity and implicitly refers to the naturally induced responses (in the non-IL-21 setting) by subsuming them into the baseline (control) tumor growth. Thus, factors normally influencing NK and CD8+ T cells (such as dendritic cells, secondary chemokines, and reciprocal interactions between the effectors) are not explicitly incorporated in this stage. Due to their minor contribution in the IL-21-associated anticancer response and their transient and low secretion of IL-21 compared with therapeutic doses (2, 5, 6, 13), CD4+ T helpers are neglected as well.
The IL-21-mediated immunity is evaluated here on a short time-scale of weeks and is restricted to primary tumor challenge. Additionally, possible inflammation-related consequences of IL-21 therapy are neglected in this initial model, as corroborated by the minor toxic effects reported in the murine setting (2, 5–7).
Two classes of administration methods are examined here: drug application associated with tumor mass (Fig. 1A ) and drug application being independent of tumor mass (Fig. 1B). The first method is represented by cytokine gene therapy (CGT), in which genetically modified tumor cells continuously secrete IL-21 (5, 6). The second administration method is represented by hydrodynamics-based gene delivery (HGD) of DNA plasmids encoding large amounts of murine IL-21 or by standard interval dosing (SID) of recombinant murine IL-21 (7, 13).
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IL-21 dynamics. In our model, the concentration of IL-21, u, is measured in units of ng/mL by Eq. A, the initial condition being u0 = u(t = 0). The clearance rate is µ1, and the exogenous IL-21 application route is the function input (Fig. 1): In the first case (CGT), input is taken as proportional to the total number of genetically engineered tumor cells, n. This forms Eq. A as
 | ((I).) |
NK-cell dynamics. The splenic NK population, x, is assumed to grow according to the logistic growth law (Eq. B), where r1 is the growth rate, and the IL-21 effect on NK-cell population size is incorporated in h1(u), the carrying capacity in the logistic function. The latter component (Eq. G) is a linear rational function that satisfies basic biological requirements: Firstly, the mean number of splenic NK cells in normal conditions is constant, given by x0 = h1 (u = 0). This accounts for the high availability of NK cells in stable health conditions (1, 26). Secondly, the IL-21 effect on the growth of mouse NK cells is apparently biphasic: at low doses NK proliferation is stimulated, whereas at higher doses it is inhibited (11, 27). However, as the positive effect on NK expansion is short-lived and observed mainly in the absence or low concentrations of other ILs (11) and hence irrelevant to in vivo conditions, we address only the negative effect of IL-21. Thus, the model considers IL-21-imposed reduction of NK cells in a dose-dependent manner, and h1(u) is selected as a monotonic decreasing function of IL-21. Thirdly, a saturation in the NK population is included, so it cannot decrease below the normal threshold xmin, the steady state for NK dynamics for u = 
. This limit is set within the normative biological range.
The above characteristics serve as constraints for the parameters p1, p2, and q1. In particular, the relations x0 = p2 / q1 and p1 = xmin can be derived. In allowing p2 and q1 to vary while maintaining their ratio constantly equal to x0, a one-parameter family of curves can be generated. A subsequent fit enables to estimate the value of this parameter that interpolates the experimental behavior (see Parameter Estimation).
CD8+ T-cell dynamics. The IL-21-mediated expansion of naive and activated CD8+ T cells in both nonstimulated (4, 28–30) and stimulated (5, 7, 13) conditions suggests a robust influence of the cytokine on the carrying capacity of these cells. CD8+ T cells persist for many weeks after IL-21 application, allowing long-term protection against subsequent tumor inoculation (5, 7, 20, 22, 30). Distinct memory phenotypes observed in T-cell subsets following exposure to IL-21 (31) support this durable effect. Hence, we introduce an indirect factor m as an IL-21 dose-dependent product, which acts on CD8+ T-cell expansion to enable a prolonged adaptive response of these cells even after complete IL-21 clearance. Consequently, the dynamics of m are given by Eq. D, where a is the proportionality constant and µ2 is the clearance rate, the reciprocal of which is a measure of the duration of the CD8+ T-cell response.
The tumor-specific CD8+ T subset in the lymph nodes, denoted by y, is therefore described by Eq. C, with the initial condition, y0 = h2 (m = 0), representing the number of such cells in an untreated setting. The logistic growth rate is r2, and the carrying capacity, h2, is a function of the memory factor m, given by Eq. H. The coefficient 
relates the growth of the carrying capacity to the memory factor (Eq. D). The actual limitation for specific CD8+ T cells, substantiated by recognized inhibitory functions of T-regulatory cells and Th2 cytokines (32), is enclosed in D. To maintain the system within realistic biological ranges, we assume that this expansion has an upper bound y = h2(0) + D.
Cytotoxic protein dynamics. Our model also accounts for the IL-21-mediated increase in effector-cell killing potential by assuming existence of cytotoxic proteins, such as perforin and IFN-
(2, 3, 10, 12, 33). This effect is assumed to be IL-21 dose-dependent (5, 7, 13, 21, 28), coinciding with the biological function of IL-21 as a ligand for the signaling cascade resulting in cytotoxic protein secretion. Because both CD8+ T and NK cells exert their effects via the same cytotoxic factors, we describe this component as a general protein p, representing the average effector cytotoxicity and given by Eq. E, where µ3 is natural degradation of p within the biological ranges, and b1 and b2 are variables of the chosen function. The initial condition is p0 = p(t = 0).
Tumor dynamics. To allow for in vivo tumor growth as affected by natural immune responses that differ for immunogenically varying cancers (1), the function g(n) (Eq. F), standing for the dynamics of tumor cell number, n, in normal untreated mice, is constructed separately for each tumor type according to the observed growth curves (see Parameter Estimation). Tumor progression is assumed to be affected by the cytotoxic, p-mediated, antitumor activity of CD8+ T and NK cells, affinities of tumor and NK or CD8+ T-cell interactions being represented by k1 and k2, respectively.
As experiments evaluate tumor size by measuring tumor surface, Eq. F is modified to allow a simple transformation between these quantities. Based on observations that tumor cells, inoculated in a solution, develop a defined shape within a few days (5, 6),1 the corresponding simplified evaluation of initial tumor surface is to divide the number of injected cells by the scale of a tumor cell surface, 10–6 mm2, so that, for instance, 105 cells will correspond to a size of 0.1 mm2. Secondly, the tumor is thought to evolve as a homogenous sphere with a constant density volume, allowing tumor volume, and tumor cell number (n), to be proportional to z3/2. Using the scale law, n 
z3/2, and skipping the irrelevant coefficient, Eq. F is transformed into
 | (J) |
Collectively, the above equations constitute an autonomous ODE system. The positivity of the solutions, corresponding to positive initial conditions, is easily verified. Moreover, the immune system components x and y are always confined to the biological relevant strip [xmin, x0] x [y0, y].
Parameter Estimation
IL-21 dynamics. As tumor size is estimated through surface measurements, rather than through cell number, IL-21 dynamics (Eq. I) are modified to
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2 ng/mL IL-21 in 1 day (5), the constant h may be retrieved. Simply, if tumor size is assumed unchanged within the first day, and IL-21 concentration at the initial time is 0, IL-21 is estimated by
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= 1 day) = 2 ng/mL gives
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 | (N) |
10–1 mm2, µ1 = 10 days–1, 1 day, h is estimated at 6.34 x 102 mm–3/2 ng mL–1 days–1. As discussed previously, for HGD and SID, the input function forms a pulse whose amplitude corresponds to the given dose. For SID, the exponential decay rate was only roughly estimated due to unavailability of IL-21 pharmacokinetic data. The homology of IL-21 to IL-2 (4), whose decay rate is a few hours (34), suggested a similar value for IL-21 (µ1 = 10 days–1). Conversely, IL-21 kinetics in HGD (Fig. 2A ) were evaluated by exponentially fitting IL-21 concentrations from the experimental assay (13). Here, µ1 was approximated at a much lower rate (0.58 days–1) due to continuous durable production of IL-21 in this method. In HGD simulations, administration will be delayed by a day with respect to experiments, because plasma IL-21 levels were measured only 24 hours following injection.
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2% to 5% (26, 35), which increases in presence of antigen. Hence, r1 is set so that the NK-cell replacement rate does not exceed 10% per day. Following strong reduction in NK-cell numbers, their dynamics can be approximated by an exponential growth, and the ratio between population sizes of 2 consecutive days is er1. A daily increment of 10% (er1 = 1.1) allows to calculate r1 = log(1.1). Near the carrying capacity, where the approximation of an exponential law is no longer valid and growth is slower, this percentage is generally decreased. For parameter, p1, denoting the lower threshold for NK numbers, a value within a biological range (104 cells) is designated. IL-21-mediated NK-cell decrease was evaluated based on an in vivo murine HGD study where splenic NK cells were counted 1 week after IL-21 administration in a noncancerous setting (13). IL-21 dynamics in HGD were applied to evaluate the parameters under which the function h1(u) interpolates the two experimental points (Fig. 2B). The resulting values for p2 and q1 were 1.054 x 106 and 0.54 x 106 cells, respectively.
CD8+ T-cell dynamics. To determine parameters of IL-21-affected dynamics of CD8+ T cells, we used a study evaluating the response to i.p. injections of murine IL-21 in a thymoma line (7). Here, six 20 µg doses (equivalent to 10 µg/mL for a 2 mL murine blood volume) were given on alternate days, decay rate being µ1 = 10 days–1, and CD8+ T cells were counted in draining lymph nodes (DLN) on days 5, 15, 30, and 45. To evaluate the initial number of specific CD8+ T cells, we assumed an initial exponential growth as suggested by the global behavior. The resulting curve enabled to determine the exponential factor r2 (at a value of 0.26 days–1) and consequently lead to a rough approximation of the initial condition as 0.06% of DLN CD8+ T cells.
A maximum percentage of specific CD8+ T-cell clones in the lymph nodes was set to 10% of DLN cells (y = D = 3 x 106 cells = 10% of 3 x 107 cells). The maximum was configured at 5% to 6% of 3 x 107 cells at day 30,2 allowing separate evaluation of D and . These latter two variables depend on the specific tumor type, as this ability is expected to be more pronounced in more immunogenic types. Due to scarcity of experiments for analyzing the CD8+ T-cell dynamics for a nonimmunogenic model, values are roughly set in the ranges = [0.002, 0.008] cells ng mL–1 and D = [0.19 x 103, 1.4 x 103] ng mL–1. Finally, µ2, the clearance parameter of m, is set to 0.014 days–1, as the duration of the response was 100 days (5, 7), and a is set as 0.57, to fit experimental data (Fig. 2C).
Cytotoxic protein dynamics. Lytic activity is elevated up to several weeks after primary challenge of genetically modified tumor cells in CGT (5, 6) and normal tumor cells in HGD or SID (7, 13). Therefore, parameters were fixed so that µ3–1 is in the range of tens of days. The concentration at saturation, b2, is set to 0.1 ng mL–1. This constant, along with b1 (set as 10–1 days ng mL–1), is evaluated by curve fitting.
Tumor dynamics. To describe the nonimmunogenic B16 melanoma growth function in control mice (5, 13), a logistic growth was assumed for the total number of tumor cells. Using the scaling law, n z3/2, this corresponds to the tumor surface
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Results |
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Model retrieval of CGT experiments. In the first experiment (5), inoculation of an immunogenic MethA IL-21-secreting fibrosarcoma (2 x 106 cells at day 0) resulted in nonmonotonic tumor growth (Fig. 4A, filled circles
). These dynamics seemed comparable with tumor growth in untreated mice during the first week (Fig. 4A, filled squares) yet decreased thereafter toward complete elimination. This immunogenic scenario (5) was reproduced in the model (Fig. 4A, thick line), assuming (a) a high CD8+ T-cell effect, roughly reflected by k2D, the combined affinity and carrying capacity values, and (b) a low affinity for the NK tumor-killing interaction as suggested by the strong NK inhibition in highly immunogenic tumors (1, 37). Simulations showed an initial rapid tumor cell increase due to lack of lysis in early stages of the response. The resulting high IL-21 concentrations lead to CD8+ T-cell proliferation and subsequent tumor eradication. The simulated behavior complied with experimental curves and with the observation that full immune control of the tumor was accomplished within 3 weeks.
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Experimental CGT results, showing effective cancer regression, suggest that the exact IL-21 concentrations in this method can be of therapeutic significance. A model-independent analysis of the previously reported IL-21 production rates in CGT (5) tested this possibility: In fibrosarcoma, the IL-21 intensity measured by 
was calculated at 30 µg mL-1 days for the first 20 days of therapy. This implies that mean levels of circulating IL-21 during the antitumor activity are in the order of 1 µg/mL. In contrast, evaluation of mean IL-21 levels in B16 melanoma was problematic due to imprecise tumor mass measurements. Nevertheless, a value of 10 ng/mL during the first therapeutic week is reasonable in this tumor line. This discrepancy suggests different requirements for IL-21-mediated elimination of immunogenically varying tumors.
Our model was used to retrieve experiments of immunodeficient mice challenged with IL-21-secreting B16-melanoma (5). In our simulations, each effector type was "knocked out" by setting the corresponding tumor affinity to 0. Model simulations (Fig. 4C and D, thick lines) agreed with experimental curves (Fig. 4C and D, filled squares), where the IL-21-induced antitumor effect was significantly decreased in abrogation of NK or CD8+ T cells. However, different dynamics were observed for each case: Tumor growth under CD8+ T-cell depletion, delayed at first, was accelerated later on, whereas in NK-depleted mice the opposite behavior was seen. Of note, tumor-effector affinity values, derived from both depletion simulations (Fig. 4C and D), yielded tumor growth (data not shown) similar to the experimental melanoma growth curves in normal IL-21-treated mice (Fig. 4B). Thus, in our model, any reciprocal interactions between NK and CD8+ T cells in the IL-21-treated scenario, if such exist, are likely subsumed into the independent effector dynamics, supporting the exclusion of their explicit description in the current model.
Model retrieval of HGD experiments. We retrieved elimination of a nonimmunogenic B16 melanoma and a moderately immunogenic MCA205 fibrosarcoma via i.v. gene delivery of IL-21 (13). In the experiment, two 20 µg IL-21-expressing plasmids were given in days 5 and 12 following tumor inoculation (5 x 105 cells), and IL-21 concentrations followed previously described kinetics (Fig. 2A). Simulation results show that the tumor-specific parameters, derived above for CGT, were sufficient for mimicking melanoma dynamics in HGD (Fig. 5A ) and for leading to the appropriate change in NK-cell dynamics (Fig. 5B). Similarly, retrieval of fibrosarcoma growth (Fig. 5C) was satisfactory with tumor-effector affinity values from CGT simulations in the analogous tumor line. The average concentration of circulating IL-21 produced by the given plasmids during the first 3 weeks did not exceed 1 ng/mL, a lower value than in CGT. This result is corroborated by the limited tumor elimination shown in this HGD method compared with CGT (Fig. 4A and B). Overall, simulated growth fits experimental measurements in both tumor models described.
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In the simulated melanoma (105 tumor cells inoculated), a regimen of 10 daily injections resulted in notable tumor rejection (Fig. 6A ), dose intensities remaining in the range of nanogram per milliliter (Fig. 6B). The IL-21-mediated NK-cell inhibition was lower in the exogenous CGT-resembling therapy than in original CGT, coinciding with the lower total IL-21 levels used in this regimen (data not shown). Notwithstanding, the simulated tumor remained controlled for several weeks as in CGT (Fig. 4B). In contrast, simulating application of the same treatment regimen to immunogenic MethA fibrosarcoma (2 x 106 tumor cells inoculated) resulted in incomplete tumor elimination (Fig. 6C), although the given IL-21 doses reached maximum values of 10 µg/mL (Fig. 6D). These results indicate that both the immunogenic nature of the cancer and its size determine the success of tumor-dependent IL-21 therapy.
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Discussion |
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As in nonimmunogenic tumors, the activity of innate-phase effectors seems crucial; their destruction by the given IL-21 must be prevented. In such circumstances, IL-21 levels that are relatively low (in the range of 10 ng/mL) seem most appropriate for small tumor mass, because IL-21 reduces the number of NK cells in a dose-dependent manner. Indeed, tumor elimination was achieved when simulating CGT-like therapy in nonimmunogenic melanoma, that is, dynamic dosing of IL-21 according to tumor mass at the time of administration.
IL-21 applied via CGT resulted in tumor extinction in several experiments (5, 6, 20, 22), whereas HGD did not remove tumor completely (13). Our calculations show that mean IL-21 levels in CGT (10-10,000 ng/mL) are significantly higher than IL-21 concentrations in HGD (1 ng/mL). It can be argued, then, that dominating the success of CGT is the large IL-21 concentrations exerted in this method. However, this is not the case, because other administration strategies using similar or even higher IL-21 concentrations show ineffective tumor eradication (10, 23, 24). We believe that successful CGT is due to the given IL-21 dose depending on tumor mass at any moment. In this way, a more natural stimulation of the immune response is achieved, avoiding abrogation of NK cells when innate immunity is indispensable.
Although T-cell responses are dominant in tumors of intensified immunogenicity, NK cells may contribute via indirect mechanisms that aid T-cell function (1). Investigation of fibrosarcoma dynamics in NK-cell-depleted mice treated with IL-21 can further elucidate this possibility. If NK cells are negligible in these circumstances, the use of high-dose (10 µg/mL) IL-21 tumor-independent regimens, already from early disease stages, would seem a more reasonable treatment. Indeed, such drug concentrations were efficacious in a recent murine study of IL-21 therapy in an immunogenic renal cell carcinoma (25). Clinical implementation of such a strategy, however, should still scout for safety limitations as in any intensified treatment. In another work,3 we use optimal control methods for fine-tuning the above-suggested tumor-dependent and tumor-independent immunotherapy regimens, to maximize antitumor efficacy, while minimizing putative IL-21 side effects.
Experimental investigation of cytotoxic protein (perforin and IFN-) elevation in various tumors under IL-21 administration (5, 6, 21, 33, 38) will illuminate possible adverse effects of this cytokine, including IL-21-mediated inflammation, whereas information on synergistic ILs (8, 23, 39) may aid in evaluating monotherapy and combination immunotherapy. This information can be transformed to biomathematical terms for fine-tuning our model, in addition to long-range effects of the drug, which can be incorporated by allowing for humoral, antimetastatic, and antiangiogenic properties of IL-21 (6, 8, 25).
IL-21 continuously exhibits therapeutic promise in its antitumor effects. Similarity of murine IL-21 interactions to the human system will substantiate the clinical implications of our findings. Validating the universality of our suggested model in prospective trials will pave the way for actual recommendations of IL-21 immunotherapy strategies, based on this work.
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Acknowledgments |
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The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked advertisement in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
We thank Drs. Margery Ma, Vladimir Vainstein, Natalie Kalev-Kronik, Yuri Kogan, and Protul A. Shrikant for valuable discussions and comments.
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Footnotes |
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1 Dr. Margery Ma (Wyeth Pharmaceuticals), personal communication. 
2 Dr. Protul A. Shrikant (Roswell Park Cancer Institute), personal communication.
3 A. Cappuccio et al., submitted for publication.
Received 1/23/06. Revised 4/11/06. Accepted 5/11/06.
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References |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionReferences |
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. J Immunol 2003;171:608–15.-dependent CXC chemokines in syngeneic mice. J Immunol 2004;172:1540–7.-
T cells towards distinct memory phenotypes. Cell Immunol 2002;218:1–6.[CrossRef][Medline] and 
ß T lymphocytes with Th2 or Tr1 cytokine profiles. J Interferon Cytokine Res 1999;19:555–61.[CrossRef][Medline] production in human NK and T cells. J Immunol 2003;170:5464–9.This article has been cited by other articles:
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  K. Leon, K. Garcia, J. Carneiro, and A. Lage How Regulatory CD25+CD4+ T Cells Impinge on Tumor Immunobiology: The Differential Response of Tumors to Therapies J. Immunol., November 1, 2007; 179(9): 5659 - 5668. [Abstract] [Full Text] [PDF]
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