Abstract
Glioblastoma (GBM), a highly aggressive (WHO grade IV) primary brain tumor, is refractory to traditional treatments, such as surgery, radiation or chemotherapy. This study aims at aiding in the design of more efficacious GBM therapies. We constructed a mathematical model for glioma and the immune system interactions, that may ensue upon direct intratumoral administration of ex vivo activated alloreactive cytotoxicTlymphocytes (aCTL). Our model encompasses considerations of the interactive dynamics of aCTL, tumor cells, major histocompatibility complex (MHC) class I and MHC class II molecules, as well as cytokines, such as TGFβ and IFNγ, which dampen or increase the proinflammatory environment, respectively. Computer simulations were used for model verification and for retrieving putative treatment scenarios. The mathematical model successfully retrieved clinical trial results of efficacious aCTL immunotherapy for recurrent anaplastic oligodendroglioma and anaplastic astrocytoma (WHO grade III). It predicted that cellular adoptive immunotherapy failed in GBM because the administered dose was 20fold lower than required for therapeutic efficacy. Model analysis suggests that GBM may be eradicated by new doseintensive strategies, e.g., 3 × 10^{8} aCTL every 4 days for small tumor burden, or 2 × 10^{9} aCTL, infused every 5 days for larger tumor burden. Further analysis pinpoints crucial biomarkers relating to tumor growth rate, tumor size, and tumor sensitivity to the immune system, whose estimation enables regimen personalization. We propose that adoptive cellular immunotherapy was prematurely abandoned. It may prove efficacious for GBM, if dose intensity is augmented, as prescribed by the mathematical model. Reinitiation of clinical trials, using calculated individualized regimens for grade III–IV malignant glioma, is suggested.
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Acknowledgments
We thank C.A. Kruse and R. Stupp for critical reading of this paper, for suggesting important corrections to the text and for contributing valuable information. We are also grateful to M. Elishmereni and to the referees for valuable revision of the manuscript. This work has been financially supported by an EU Marie Curie grant no. MRTNCT2004503661 to Natalie Kronik, and by the Chai Foundation. Natalie Kronik is supported by EU MarieCurie grant no.MRTNCT2004503661. Yuri Kogan, Vladimir Vainstein, and Zvia Agur are supported by the Chai Foundation.
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Appendices
Appendix: Parameter estimation
In this section we present a list of all evaluated model parameters, the detailed methods and the literature sources for their evaluation (Table 2).
The method for evaluating model parameters
Maximal growth rate of the tumor, r. Swanson et al. [41] assume a MG is diagnosed at 3 cm diameter and when it reaches a 6 cm diameter the patient dies. Assuming a spherical shape, the final to diagnosis initial volume ratio is \(\left(\frac{6}{3}\right)^{3} = 8.\) We assumed that the number of tumor cells is proportional to the tumor volume. Using Eqs. (1–6), r was scaled so an untreated grade III MG (e.g., anaplastic oligodendroglioma) would grow eightfold within 3 years [6]. Thus, for grade III MG we estimated r = 0.00035 h^{−1} . A GBM tumor grows from 3 cm diameter to a 6 cm diameter in about a year [41]. Using Eqs. (1–6), r was scaled to predict eightfold tumor growth within a year. Hence, for grade IV tumor we estimated r = 0.001 h^{−1}.
Tumor carrying capacity (maximal tumor burden), K. Arciero et al. [4] takes the carrying capacity of tumor cells to be 10^{9} cells/ml. Taking a maximal tumor diameter of 6 cm we got a volume of roughly 100 ml, which gave us an estimation of total carrying capacity of 10^{11} cells.
Maximal efficiency of CTL a _{ T }. Wick et al. [60] report that a CTL kills 0.7–3 target cells per day. A mean value of two target cells per day gives the rate of 0.0833 cells/h. The experiment was done with 5 × 10^{5} target cells/ml in 2 ml wells. For this calculation we used h _{ T } value determined by Arciero et al. [4] for mice. This h _{ T } value was smaller than the one we used later in simulations, because in vitro the contact frequency and efficacy of CTLs would be higher. Here we took h _{ T } to be 10^{5} cells/ml and multiplied it by the volume of the well. Substituting the former values into \(a_{T} \cdot \frac{T}{{h_{T} + T}} = 0.0833\,\hbox {h}^{{ 1}},\) we got a _{ T } = 0.12 h^{−1}.
Michaelis constant for the dependence of CTL efficiency on MI amount, e _{ T }. Kageyama et al. [51] report the number of MHC I receptors per target cell to be between fewer than ten to several thousands. The value of e _{ T } is the number of M _{I} receptors that brings the CTLs efficacy to half of its maximum value. Taking into account that MHC I receptors expression is suppressed in MGs, we estimated e _{ T } to be 50 rec/cell.
Maximal reduction effect of TGFβ on CTL efficiency, a _{ T,β}. Thomas and Massagué [42] report that under high concentrations of TGFβ CTL efficacy in target cell lysis has dropped to onethird after 3 h. Thus, \(a_{{T,\beta}} = \sqrt[3]{{\frac{1}{3}}}\hbox{h}^{{ 1}} \approx 0.69\,\hbox{h}^{{ 1}}.\)
Michaelis constant for the dependence of CTL efficiency on TGFβ amount, e _{ T,β}. We took this value to be of order of magnitude of the base line found by Peterson et al. [55], multiplied by the volume of the CNS. Thus, \(e_{{T,\beta}} = 60.9\,\hbox{pg}\cdot \hbox{ml}^{1} \cdot 150\,\hbox{ml} \approx 10^{4} \hbox{pg}.\)
Parameter for CTL efficiency saturation due to large tumor size, h _{ T }. We estimated it to be 5 × 10^{8} cells, or 5 × 10^{9} cells by fitting the model predictions to the the results of Kruse et al. [24], Kruse and Rubinstein [25].
Maximal effect of M _{ II } on CTL recruitment, \(a_{{C,M_{{\rm II}}}}.\) To estimate the migration of CD8^{+} cells across the BBB, we used Marcondes et al. [53] reporting that the number of migrating CD4^{+} cells is similar to that of CD8^{+} cells. According to Phillips and Lampson [57], who investigated the migration of CD4^{+} cells, within 2 days about 40 CD4^{+} T cells cross the BBB within a volume of a slide. We calculated the volume of a slide as its cross section area multiplied its depth: 9.2 × 10^{− 6}m^{2}·6 × 10^{− 6}m = 55.2 × 10^{− 6}ml. Therefore, for a 100 ml tumor the maximal number of the CD8^{+} cells recruited per hour is:
To obtain the estimation for \(a_{{C,M_{{\rm II}}}},\) we had to divide the latter number by the estimated number of MHC II receptors, which can be calculated as: (number of M II per cell) × (number of tumor cells).
Bosshart and Jarrett [49] found that the MHC II density on cell surface is about 2 × 10^{3} rec/μm^{2}. We assumed half of that density (because there is poor presentation on tumor cells) and took the surface area of a cell of a diameter of 5 μm to be about 314 μm^{2}. For this calculation, we estimated the number of tumor cells to be 10^{11}, in agreement with the earlier assumption of 100 ml tumor volume. Thus,
Michaelis constant for the effect of M _{II} on CTL recruitment, \(e_{{C,M_{{\rm II}}}}.\) We estimated that number to be 10^{14} rec. This is a rough estimation of the total number of receptors on all the tumor cells, whose number is estimated to be between 10^{10} and 10^{11} cells, while there are hundreds to thousands of receptors on each cell.
Maximal reduction effect of TGFβ on CTL recruitment, a _{ C,β}. Thomas and Massagué [42] found that excess of TGFβ inhibits the proliferation of CTLs up to 50% within 3 h. Therefore, we estimated the maximal inhibition of CTL recruitment per hour by TGFβ to be \(\sqrt[3]{{\frac{1}{2}}}\hbox{h}^{{ 1}} \approx 0.8\,\hbox{h}^{{ 1}}.\)
Michaelis coefficient for the reduction effect of TGFβ on CTL recruitment, e _{ C,β }. Similarly to e _{ T,β}, we took this value to be of order of magnitude of the base line found by Peterson et al. [55] multiplied by the volume of the CNS. Thus, \(e_{{C,\beta}} = 60.9\frac{{\rm pg}}{{\rm ml}} \cdot 150\,\hbox{ml} \approx 10^{4} \hbox {pg}.\)
Death rate of CTLs, μ _{ C }. Taylor et al. [58] find CTL half life to be 3.9 days so its hourly death rate was estimated to be \(\frac{{\ln\,2}}{{72\,\hbox {h}}} \approx 0.007\,\hbox {h}^{{ 1}}.\)
Degradation rate of TGFβ, μ _{ β }. Coffey et al. [50] find that the hepatic half life of TGFβ is 2.2 min. Because of the distance of the liver from the and because of the necessity to pass the BBB, the actual brain TGFβ breakdown rate will be slower. We estimated it to be 6 min. Thus, the hourly breakdown rate is \(\frac{{\ln\,2}}{{0.1\,\hbox{h}^{{ 1}}}} \approx 7\,\hbox{h}^{{ 1}}.\)
Constant base level production of TGFβ, g _{β}. Peterson et al. [55] found the concentration of TGFβ to be 609 pg/ml in the cerebral spinal fluid (CSF) of a GBM patient, which was tenfold higher than the level found in healthy subjects. We assumed that the volume of the CSF is 150 ml. In a healthy subject there is no tumor production of TGFβ, therefore at steady state we obtained:
Thus, using previously calculated parameter values \(g_{\beta} = 7\,\hbox {h}^{{ 1}} \cdot 60.9\frac{{\rm pg}}{{\rm ml}} \times 150\,\hbox {ml} = 63,945\,\hbox {pg/h}.\)
Production rate of TGFβ by a single tumor cell, a _{ β,T }. Using Peterson et al. [55] we found that for a GBM patient the mean level of TGFβ is \(609\,\hbox{pg}\cdot \hbox{ml}^{1} \cdot 150\,\hbox {ml} = 91,350\,\hbox {pg}.\) We used previously calculated parameter values: μ _{ β } = 7 h^{−1}, T = 10^{11} cells. Using Eq. (3) at steady state, we got
Production rate of IFNγ by a single CTL, a _{ γ,C }. Kim et al. [19] report expression of 200 pg/ml of IFNγ by CTLs. We assumed there were 2·10^{5} CTL/ml and using μ_{γ} = 0.102 h^{−1} we obtained from Eq. (4) at steady state \(a_{{\gamma, C}} = \frac{{0.102\,\hbox {h}^{{ 1}} \cdot 200\,\hbox {pg} \cdot \hbox {ml}^{{ 1}}}}{{2 \cdot 10^{5}\,\hbox {cells} \cdot \hbox {ml}^{{ 1}}}} = 1.02 \cdot 10^{{ 4}}\,\hbox {pg}/(\hbox {cells}\cdot \hbox {h}).\)
Degradation rate of IFNγ, μ _{ γ }. Turner et al. [59] find the median half life of IFNγ to be 6.8 h. Thus, \(\mu_{\gamma} = \frac{{\ln\,2}}{{6.8\,\hbox {h}}} = 0.102\,\hbox {h}^{{ 1}}.\)
Constant base level production of MHC I, \(g_{{M_{\rm I}}}.\) Kageyama et al. [51] find that the number of M _{I} receptors on cell surface varies from less than ten to several thousands. For the purpose of the following calculation we assumed M _{I} = 100 rec/cell. In the absence of IFNγ, taking \(\mu_{{M_{I}}} = 0.0144\,\hbox {h}^{{ 1}}\) and substituting into Eq. (5) at steady state, we obtained: \(g_{{M_{I}}} = 100\,\hbox {rec} \cdot \hbox {cell}^{{ 1}} \cdot \mu_{{M_{\rm I}}} = 1.44\,\hbox {rec}/(\hbox {cells} \cdot \hbox{h}).\)
Maximal production rate of MHC I induced by IFNγ, \(a_{{M_{\rm I}, \gamma}}.\) According to Yang et al. [47] the expression of MHC I receptors on some GBM tumor cells is increased threefold when subjected to excess of IFNγ. This gave us the following ratio: \(a_{{M_{\rm I}, \gamma}} = 2 \times g_{{M_{\rm I}}},\) therefore \(a_{{M_{\rm I}, \gamma}} = 2.88 \,\hbox {rec}/\hbox {h}.\)
Michaelis constant for the production rate of MHC I induced by IFNγ, \(e_{{M_{\rm I}, \gamma}}.\) Yang et al. [47] find a range of M _{I} values as a result of IFNγ treatment. However, they display their results using a scoring scale of MHC I expression which needs to be rescaled to receptor number. We calibrated M _{I} in the absence of IFNγ to be equivalent to a scoring level of 1.5. Next we took the value of IFNγ to be 100 units/ml for MHC I expression level of 2.5 according to the above score. Substituting into Eq. (5) we obtain: for F _{ γ } = 0
and for F _{ γ } = 100 U
From these two equations we obtain:
As mentioned above, the value of \(\frac{{a_{{M_{\rm I}, \gamma}}}}{{g_{{M_{\rm I}}}}}\) is 2. According to Pharmingen manufacturer information, the relation between the used units and IFNγ quantities is in 0.6 × 10^{8} units/mg. Thus, \(F_{\gamma} = \frac{{100\,\hbox {units}/\hbox {ml}}}{{0.6 \cdot 10^{8} \hbox {units}/\hbox {mg}}} = 1.67 \cdot 10^{{ 6}}\,\hbox {mg}/\hbox {ml}.\) Substituting into the previous and taking into account the volume of 100 ml, we obtain:
Degradation rate of MHC I receptors, \(\mu_{{M_{\rm I}}}.\) Milner et al. [54] find that the half life of MHC I molecules varies between 6 and 96 h. We take a representative value to be 48 h. Therefore, the degradation rate is: \(\frac{{\ln\,2}}{{48\,\hbox {h}}} \approx 0.0144\,\hbox {h}^{{ 1}}.\)
Parameters for the influence of IFNγ on MHC II expression, \(a_{{M_{{\rm II}}, \gamma}}, e_{{M_{{\rm II}}, \gamma}}.\) Phillips et al. [56] use IFNγ injections to the brain and increase expression of MHC class II 5 fold. To scale this immunoreactivity we used data from Bosshart and Jarrett [49] who found a fourfold variation in MHC class II expression. Substituting into Eq. (6) at steady state, we obtained the following equation with two unknown variables \(a_{{M_{{\rm II}}, \gamma}}\) and \(e_{{M_{{\rm II}}, \gamma}}:\)
and with two sets of parameters values:

1.
\(F_{\gamma} = 10,000\,\hbox {U}/\hbox {site},M_{{\rm II}} = 1.9 \cdot 10^{3} \frac{{\hbox {rec}}}{{\hbox {mm}^{2}}} \cdot 314\,\upmu \hbox {m}^{2} (314\, \upmu \hbox {m}^{2}\) being the area of cell surface) and \(\mu_{{M_{{\rm II}}}} = 0.0144\,\hbox {h}^{{ 1}};\)

2.
\(F_{\gamma} = 30\,\hbox {U}/\hbox {site},M_{{\rm II}} = 0.5 \cdot 10^{3} \frac{{\rm rec}}{{{\rm mm}^{2}}} \cdot 314\,\hbox {mm}^{2}\) and \(\mu_{{M_{{\rm II}}}} = 0.0144\,\hbox {h}^{{ 1}}.\)
IFNγ unit is given by 0.6 × 10^{8} u/mg we obtained:
Parameters for the influence of TGFβ on MHC II expression, \(a_{{M_{{\rm II}}, \beta}}, e_{{M_{{\rm II}}, \beta}}.\) Suzumura et al. [40] report a drop of 98.8% in MHC expression when using 100 ng/ml TGFβ. We interpreted this result as maximal inhibition and estimated: \(a_{{M_{{\rm II}}, \beta}} = 0.012.\)
Suzumura et al. [40] report also that a dose of 10 ng/ml of TGFβ we get a drop of 89.8% in MHC expression. This gave the following equation:
Substituting into the above equation \(F_{\beta} = 10\frac{{\rm ng}}{{\rm ml}} \cdot 100\,\hbox {ml}\) we obtained:
Degradation rate of MHC II receptors, \(\mu_{{M_{{\rm II}}}}.\) According to Lazarski et al. [52], MHC class II molecule half life varies between 10 and 150 h. We assumed a representing half life of 48 h and therefore \(\mu_{{M_{{\rm II}}}} = \frac{{\ln\,2}}{{48\,\hbox {h}}} \approx 0.0144\,\hbox {h}^{{ 1}}.\)
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Kronik, N., Kogan, Y., Vainstein, V. et al. Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics. Cancer Immunol Immunother 57, 425–439 (2008). https://doi.org/10.1007/s002620070387z
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DOI: https://doi.org/10.1007/s002620070387z