Skip to main content

Advertisement

Log in

Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics

  • Original Article
  • Published:
Cancer Immunology, Immunotherapy Aims and scope Submit manuscript

An Erratum to this article was published on 07 December 2007

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 intra-tumoral administration of ex vivo activated alloreactive cytotoxic-T-lymphocytes (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 TGF-β and IFN-γ, which dampen or increase the pro-inflammatory 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 20-fold lower than required for therapeutic efficacy. Model analysis suggests that GBM may be eradicated by new dose-intensive strategies, e.g., 3 × 108 aCTL every 4 days for small tumor burden, or 2 × 109 aCTL, infused every 5 days for larger tumor burden. Further analysis pinpoints crucial bio-markers 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. Re-initiation of clinical trials, using calculated individualized regimens for grade III–IV malignant glioma, is suggested.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Agur Z, Arnon R, Schechter B (1988) Reduction of cytotoxicity to normal tissues by new regimes of phase-specific drugs. Math Biosci 9:1–15

    Article  Google Scholar 

  2. Andaloussi AE, Lesniak MS (2006) An increase in CD4+ CD25 FOXP3+ regulatory T cells in tumor-infiltrating lymphocytes of human glioblastoma multiforme. Neurooncol 8:234–243

    Google Scholar 

  3. Arakelyan L, Merbl Y, Agur Z (2005) Vessel maturation effects on tumor growth: validation of a computer model in implanted human ovarian carcinoma spheroids. Eur J Cancer 41:159–167

    Article  PubMed  CAS  Google Scholar 

  4. Arciero JC, Jackson TL, Kirschner DE (2004) A mathematical model of tumor-immune evasion and siRNA treatment. Discret Contin Dyn S B 4:39–58

    Google Scholar 

  5. Bodmer S, Strommer K, Frei K, Siepl C, de Tribolet N, Heid I, Fontana A (1989) Immunosuppression and transforming growth factor-beta in glioblastoma. Preferential production of transforming growth factor-beta 2. J Immunol 143:3222–3229

    PubMed  CAS  Google Scholar 

  6. Burger PC, Vogel FS, Green SB, Strike TA (1985) Glioblastoma multiforme and anaplastic astrocytoma, pathologic criteria and prognostic implications. Cancer 56:1106–1111

    Article  PubMed  CAS  Google Scholar 

  7. Cappuccio A, Elishmereni M, Agur Z (2006) Cancer immunotherapy by interleukin-21 potential treatment startegies evaluated in a mathematical model. Cancer Res 66:7293–7300

    Article  PubMed  CAS  Google Scholar 

  8. Carpentier PA, Begolka WS, Olson JK, Elhofy A, Karpus WJ, Miller SD (2005) Differential activation of astrocytes by innate and adaptive immune stimuli. Glia 49:360–374

    Article  PubMed  Google Scholar 

  9. Fine HA (2004) Toward a glioblastoma vaccine: promise and potential pitfalls. J Neurooncol 22:4240–4243

    Google Scholar 

  10. Goff BA, Matthews BJ, Wynn M, Muntz HG, Lishner DM, Baldwin LM (2006) Ovarian cancer: patterns of surgical care across the United States. Gyneocol Oncol 103:383–390

    Article  Google Scholar 

  11. Cojocaru L, Agur Z (1992) Theoretical analysis of interval drug dosing for cell-cycle-phase-specific drugs. Math Biosci 109:85–97

    Article  PubMed  CAS  Google Scholar 

  12. Gomez GG, Kruse CA (2006) Mechanisms of malignant glioma immune resistence and sources of immunosupression. Gene Ther Gene Mol Biol 10:133–146

    Google Scholar 

  13. Gomez GG, Kruse CA (2007) Cellular and functional characterization of immunoresistant human glioma cell clones selected with alloreactive cytotoxic T lymphocytes reveals their up regulated synthesis of biologically active TGF-β. J Immunother 30:261–273

    Article  PubMed  CAS  Google Scholar 

  14. Gomez GG, Varella-Garcia M, Kruse CA (2006) Isolation of immunoresistant human glioma cell clones after selection with alloreactive cytotoxic T lymphocytes: cytogenetic and molecular cytogenetic characterization. Cancer Genet Cytogenet 165:121–134

    Article  PubMed  CAS  Google Scholar 

  15. Graf MR, Sauer JT, Merchant RE (2005) Tumor infiltration by myeloid suppressor cells in response to T cell activation in rat gliomas. J Neurooncol 73:29–36

    Article  PubMed  Google Scholar 

  16. Gunther N, Hoffman GW (1982) Qualitative dynamics of a network model of regulation of the immune system: a rationale for the IgM to IgG switch. J Theor Biol, pp 815–855

  17. Hickey WF (2001) Basic principles of immunological surveillance of the normal central nervous system. Glia 36:118–124

    Article  PubMed  CAS  Google Scholar 

  18. Hussain SF, Yang D, Suki D, Aldape K, Grimm E, Heimberger AB (2006) The role of human glioma-infiltrating microglia/macrophages in mediating antitumor immune responses. Neuro-oncol 8:261–279

    Article  PubMed  CAS  Google Scholar 

  19. Kim JJ, Nottingham LK, Sin JI, Tsai A, Morrison L, Oh J, Dang K, Hu Y, Kazahaya K, Bennett M, Dentchev T, Wilson DM, Chalian AA, Boyer JD, Agadjanyan MG, Weiner DB (1998) CD8 positive T cells influence antigen-specific immune responses through the expression of chemokines. J Clin Invest 102:1112–1124

    PubMed  CAS  Google Scholar 

  20. Kirschner D, Panetta JC (1998) Modelling immunotherapy of the tumor-immune interaction. J Math Biol 37:235–252

    Article  PubMed  CAS  Google Scholar 

  21. Kleihues P, Soylemazoglu F, Schäuble B, Schniethauer BW, Bruger PC (1995) Histopathology, classification, and grading of gliomas. Glia 15:211–221

    Article  PubMed  CAS  Google Scholar 

  22. Kleihues P, Louis DN, Scheithauer BW, Rorke LB, Reifenberger G, Burger PC, Cavenee WK (2002) The WHO classification of tumors of the nervous system. J Neuropathol Exp Neurol 61:215–225

    PubMed  Google Scholar 

  23. Kreschmer K, Apostolou I, Jaeckel E, Khazaie K, von Boehmer H (2006) Making regulatory T cells with defined antigen specificity: role in autoimmunity and cancer. Immunol Rev 212:163–169

    Article  Google Scholar 

  24. Kruse CA, Cepeda L, Owens B, Johnson SD, Stears J, Lillehei KO (1997) Treatment of recurrent glioma with intracavitary alloreactive cytotoxic T lymphocytes and Interleukin-2. Cancer Immunol Immnother 45:77–87

    Article  CAS  Google Scholar 

  25. Kruse CA, Rubinstein D (2001) Cytotoxic T-lymphcytes reactive to patient major histocompatibility complex proteins for therapy of brain tumors. In: Liau LM, Becker DP, Cloughesy TF, Bigner DD (eds) Brain Tumor Immunotherapy. Humana, Totowa, pp 149–170

    Google Scholar 

  26. Kuznetzov VA, Makalkin IA, Taylor MA, Perelson AS (1994) Nonlinear dynamics of immunologenic tumors: parameters estimation and global bifurcation analysis. Bull Math Biol 56:295–321

    Google Scholar 

  27. Liau LM, Prins RM, Kiertscher SM, Odesa SK, Kremen TJ, Giovannone AJ, Lin JW, Chute DJ, Mischel PS, Cloughesy TF, Roth MD (2005) Dendritic cell vaccination in glioblastoma patients systemic and intracranial T-cell response modulated by the local central nervous system tumor microenvironment. Clin Cancer Res 11:5515–5524

    Article  PubMed  CAS  Google Scholar 

  28. Lopez M, Aguilera R, Perez C, Mendoza-Naranjo A, Pereda C, Ramirez M, Ferrada C, Aguillon JC, Salazar-Onfray F (2006) The role of regulatory T lymphocytes in the induced immune response mediated by biological vaccines. Immunobiology 211:127–136

    Article  PubMed  CAS  Google Scholar 

  29. Marchuk GI, Petrov RV, Romanyukha AA, Bocharov GA (1991) Mathematical model of antiviral immune response. I. Data analysis, generalized picture construction and parameters evaluation for hepatitis B. J Theor Biol 7–151(1):1–40

    Google Scholar 

  30. Morgan RA, Dudley ME, Wunderlich JR, Hughes MS, Yang JC, Sherry RM, Royal RE, Topalian SL, Kammula US, Restifo NP, Zheng Z, Nahvi A, de Vries CR, Rogers-Freezer LJ, Mavroukakis SA, Rosenberg SA (2006) Cancer regression in patients after transfer of genetically engineered lymphocytes. Science 314:126–129

    Article  PubMed  CAS  Google Scholar 

  31. Panek RB,Benveniste EN (1995) Class II MHC gene expression in microglia. J Immunol 154:2846–2854

    PubMed  CAS  Google Scholar 

  32. de Pillis LG, Radunskaya AE, Wiseman CL (2005) A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res 65:7950–7958

    PubMed  Google Scholar 

  33. de Pillis LG, Gu W, Radunskaya AE (2006) Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations. J Theor Biol 238:841–862

    Article  PubMed  CAS  Google Scholar 

  34. Proescholdt MA, Merrill MJ, Ikejiri B, Walbridge S, Akbasak A, Jacobson S, Oldfield EH (2001) Site-specific immune response to implanted gliomas. J Neurosurg 95:1012–1019

    PubMed  CAS  Google Scholar 

  35. Read SB, Kulprathipanja NV, Gomez GG, Paul DB, Winston KR, Robbins JM, Kruse CA (2003) Human alloreactive CTL interactions with gliomas and with those having upregulated HLA expression from exogenous IFN-γ or IFN-γ gene modification. J Interferon Cytokine Res 23:379–393

    Article  PubMed  CAS  Google Scholar 

  36. Skomorovski K, Harpak H, Ianovski A, Vardi M, Visser TP, Hartong SC, van Vliet HH, Wagemaker G, Agur Z (2003) New TPO treatment schedules of increased safety and efficacy: pre clinical validation of a thrombopoiesis simulation model. Br J Haematol 123:683–691

    Article  PubMed  Google Scholar 

  37. Soos JM, Krieger JI, Stuve O, King CL, Patarroyo JC, Aldape K, Wosik K, Slavin AJ, Nelson PA, Antel JP, Zamvil SS (2001) Malignant glioma cells use MHC class II transactivator (CIITA) promoters III and IV to direct IFN-γ-inducible CIITA expression and can function as nonprofessional antigen presenting cells in endocytic processing and CD4+ T-cell activation. Glia 36:391–405

    Article  PubMed  CAS  Google Scholar 

  38. Steiner HH, Bonsanto MM, Beckhove P, Brysch M, Geletneky K, Ahmadi R, Schuele-Freyer R, Kremer P, Ranaie G, Matejic D, Bauer H, Kiessling M, Kunze S, Schirrmacher V, Herold-Mende C (2004) Antitumor vaccination of patients with glioblastoma multiforme: a pilot study to assess feasibility, safety, and clinical benefit. J Clin Oncol 22:4272–4281

    Article  PubMed  Google Scholar 

  39. Strik HM, Stoll M, Meyermann R (2004) Immune cell infiltration of intrinsic and metastatic intracranial tumors. Anticancer Res 24:37–42

    PubMed  Google Scholar 

  40. Suzumura A, Sawada M, Yamamoto H, Marunouchi T (1993) Transforming growth factor-β suppresses activation and proliferation of microglia in vitro. J Immunol 151:2150–2158

    PubMed  CAS  Google Scholar 

  41. Swanson KR, Bridge C, Murray JD, Alvord EC Jr (2003) Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion. J Neurol Sci 216:1–10

    Article  PubMed  Google Scholar 

  42. Thomas DA and Massagué J (2005) TGF-β directly targets cytotoxic T cell functions during tumor evasion of immune surveillance. Cancer Cell 8:369–380

    Article  PubMed  CAS  Google Scholar 

  43. de Vleeschouwer S, Rapp M, Sorg R, Steiger H, van Gool S, Sabel M (2006) Dendritic cell vaccination in patients with malignant gliomas: current status and future directions. Neurosurgery 59:988–999

    PubMed  Google Scholar 

  44. Weller M, Fontana A (1995) The failure of current immunotherapy for malignant glioma. Tumor-derived TGF-beta, T-cell apoptosis, and the immune privilege of the brain. Brain Res Brain Res Rev 21:128–151

    Article  PubMed  CAS  Google Scholar 

  45. Wheeler RD, Zehntner SP, Kelly LM, Bourbonniere L, Owens T (2006) Elevated interferon gamma expression in the central nervous system of tumor necrosis factor receptor 1-deficient mice with experimental autoimmune encephalomyelitis. Immunology 118:527–538

    PubMed  CAS  Google Scholar 

  46. Wiseman CL, Kharazi A (2006) Objective clinical regression of metastatic breast cancer in disparate sites after use of whole cell vaccine genetically modified to release Sargarmostim. Breast J 12:475–480

    Article  PubMed  Google Scholar 

  47. Yang I, Kremen TJ, Giovannone AJ, Paik E, Odesa SK, Prins RM, Liau LM (2004) Modulation of major histocompatibilitycomplex class I molecules and major histocompatibility complex-bound immunogenic peptides induced by interferon α and interferonγ treatment of human glioblastoma multiforme. J Neurosurg 100:310–319

    Article  PubMed  CAS  Google Scholar 

  48. Zagzag D, Salnikow K, Chiriboga L, Yee H, Lan L, Ali MA, Garcia R, Demaria S, Newcomb EW (2005) Downregulation of major histocompatibility complex antigens in invading glioma cells: stealth invasion of the brain. Lab Invest 85:328–341

    Article  PubMed  CAS  Google Scholar 

  49. Bosshart H and Jarrett RF (1998) Deficient major histocopatibility complex class II antigen presentation in a subset of Hodgkin’s disease tumor cells. Blood 92:2252–2259

    PubMed  CAS  Google Scholar 

  50. Coffey RJ, Kost LJ, Lyons RM, Moses HL, LaRusso NF (1987) Hepatic processing of transforming growth factor β in the rat uptake, metabolism, and biliary excretion. J Clin Invest 80:750–757

    Article  PubMed  CAS  Google Scholar 

  51. Kageyama S, Tsomides TJ, Sykulev Y, Eisen HN (1995) Variations in the number of peptide–MHC class I complexes required to activate cytotoxic T cell responses. J Immunol 154:567–576

    PubMed  CAS  Google Scholar 

  52. Lazarski CA, Chaves FA, Jenks SA, Wu S, Richards KA, Weaver JM, Sant AJ (2005) The Kinetic stability of MHC class II: peptide complexes is a key parameter that dictates immunodominance. Immunity 23:29–40

    Article  PubMed  CAS  Google Scholar 

  53. Marcondes MC, Burudi EM, Huitron-Resendiz S, Sanchez-Alavez M, Watry D, Zandonatti M, Henriksen SJ, Fox HS (2001) Highly activated CD8+T cells in the brain correlate with early central nervous system dysfunction in simian immunodeficiency virus infection. J Immunol 167:5421–5438

    Google Scholar 

  54. Milner E, Barnea E, Beer I, Admon A (2006) The turnover kinetics of MHC peptides of human cancer cells. Mol Cell Proteomics 5:366–378

    Google Scholar 

  55. Peterson PK, Chao CC, Hu S, Thielen K, Shaskan E (1992) Glioblastoma, transforming growth factor-β, and Candida meningitis: a potential link. Am J Med 92:262–264

    Article  PubMed  CAS  Google Scholar 

  56. Phillips LM, Simon PJ, Lampson LA (1999) Site-specific immune regulation in the brain: differential modulation of major histocompatibility complex (MHC) proteins in brainstem vs. hippocampus. J Comp Neurol 405:322–333

    Article  PubMed  CAS  Google Scholar 

  57. Phillips LM, Lampson LA (1999) Site-specific control of T cell traffic in the brain: T cell entry to brainstem vs. hippocampus after local injection of IFN-γ. J Neuroimmunol 96:218–227

    Article  PubMed  CAS  Google Scholar 

  58. Taylor GP, Hall SE, Navarrete S, Michie CA, Davis R, Witkover AD, Rossor M, Nowak MA, Rudge P, Matutes E, Bangham CR, Weber JN (1999) Effect of Lamivudine on human T-cell leukemia virus type 1 (HTLV-1) DNA copy number, T-cell phenotype, and anti-tax cytotoxic T-cell frequency in patients with HTLV-1-associated myelopathy. J Virol 73:10289–10295

    PubMed  CAS  Google Scholar 

  59. Turner PK, Houghton JA, Petak I, Tillman DM, Douglas L, Schwartzberg L, Billups CA, Panetta JC, Stewart CF (2004) Interferon-gamma pharmacokinetics and pharmacodynamics in patients with colorectal cancer. Cancer Chemother Pharmacol 53:253–260

    Article  PubMed  CAS  Google Scholar 

  60. Wick WD, Yang OO, Corey L, Self SG (2005) How many human immunodeficiency virus type 1-infecfted target cells can a cytotoxic T-lymphocyte kill? J Virol 79:13579–13586

    Article  PubMed  CAS  Google Scholar 

Download references

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. MRTN-CT-2004-503661 to Natalie Kronik, and by the Chai Foundation. Natalie Kronik is supported by EU Marie-Curie grant no.MRTN-CT-2004-503661. Yuri Kogan, Vladimir Vainstein, and Zvia Agur are supported by the Chai Foundation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zvia Agur.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s00262-007-0432-y

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).

Table 2 Parameter estimation used in the current model

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. (16), 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. (16), 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 109 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 1011 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 × 105 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 105 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 TGF-β on CTL efficiency, a T. Thomas and Massagué [42] report that under high concentrations of TGF-β CTL efficacy in target cell lysis has dropped to one-third 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 TGFamount, e T. 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 ×  108 cells, or 5 × 109 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− 6m2·6 × 10− 6m = 55.2 × 10− 6ml. Therefore, for a 100 ml tumor the maximal number of the CD8+ cells recruited per hour is:

$$ \frac{{100\,\hbox{ml} \cdot 40\,\hbox{cells}}}{{55.2 \times 10^{{- 6}}\hbox{ml} \cdot 48\,\hbox{h}}} \approx 1.5 \times 10^{6}\,\hbox{cells/h}. $$

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 × 103 rec/μm2. 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 μm2. For this calculation, we estimated the number of tumor cells to be 1011, in agreement with the earlier assumption of 100 ml tumor volume. Thus,

$$ a_{{C,M_{{\rm II}}}} = \frac{{1.5 \times 10^{6} \hbox{cell} \cdot \hbox{h}^{{-1}}}}{{314\,\hbox{mm}^{2} \cdot \hbox{cell}^{{- 1}} \cdot 10^{3}\,\hbox{rec} \cdot \hbox{mm}^{{- 2}} \cdot 10^{{11}} \hbox{cells}}} \approx 4.8 \times 10^{{- 11}} \,\hbox{cell}/(\hbox{h}\cdot \hbox{rec}). $$

Michaelis constant for the effect of M II on CTL recruitment, \(e_{{C,M_{{\rm II}}}}.\) We estimated that number to be 1014 rec. This is a rough estimation of the total number of receptors on all the tumor cells, whose number is estimated to be between 1010 and 1011 cells, while there are hundreds to thousands of receptors on each cell.

Maximal reduction effect of TGFon CTL recruitment, a C. Thomas and Massagué [42] found that excess of TGF-β inhibits the proliferation of CTLs up to 50% within 3 h. Therefore, we estimated the maximal inhibition of CTL recruitment per hour by TGF-β to be \(\sqrt[3]{{\frac{1}{2}}}\hbox{h}^{{- 1}} \approx 0.8\,\hbox{h}^{{- 1}}.\)

Michaelis coefficient for the reduction effect of TGFon CTL recruitment, e C,β . Similarly to e T, 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 CTLsμ 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 TGF-βμ β . Coffey et al. [50] find that the hepatic half life of TGF-β 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 TGF-β 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 TGF-β, g β. Peterson et al. [55] found the concentration of TGF-β 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 TGF-β, therefore at steady state we obtained:

$$ 0 = g_{\beta} - \mu_{\beta} \cdot F_{\beta}. $$

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 TGF-β by a single tumor cell, a β,T . Using Peterson et al. [55] we found that for a GBM patient the mean level of TGF-β 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 = 1011 cells. Using Eq. (3) at steady state, we got

$$ a_{{\beta, T}} = \frac{{91,350\,\hbox {pg} \cdot 7\,\hbox {h}^{{- 1}} - 63,945\,\hbox {pg}\cdot \hbox {h}^{{- 1}}}}{{10^{{11}}\,\hbox {cells}}} \approx 5.75 \times 10^{{- 6}}\,\hbox {pg}/(\hbox {cells}\cdot \hbox {h}). $$

Production rate of IFN-γ by a single CTL, a γ,C . Kim et al. [19] report expression of 200 pg/ml of IFN-γ by CTLs. We assumed there were 2·105 CTL/ml and using μγ = 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 IFN-γμ γ . Turner et al. [59] find the median half life of IFN-γ 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 IFN-γ, 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 IFN-γ, \(a_{{M_{\rm I}, \gamma}}.\) 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 IFN-γ. 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 IFN-γ, \(e_{{M_{\rm I}, \gamma}}.\) Yang et al. [47] find a range of M I values as a result of IFN-γ treatment. However, they display their results using a scoring scale of MHC I expression which needs to be re-scaled to receptor number. We calibrated M I in the absence of IFN-γ to be equivalent to a scoring level of 1.5. Next we took the value of IFN-γ 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

$$ \frac{{g_{\rm I}}}{{\mu_{\rm I}}} = 1.5, $$

and for F γ = 100 U

$$ \frac{{g_{\rm I} + \frac{{a_{{M_{\rm I}, \gamma}} \cdot F_{\gamma}}}{{F_{\gamma} + e_{{M_{\rm I}, \gamma}}}}}}{{\mu_{\rm I}}} = 2.5. $$

From these two equations we obtain:

$$ e_{{M_{\rm I}, \gamma}} = F_{\gamma} \cdot \left(\frac{3}{2}\frac{{a_{{M_{\rm I}, \gamma}}}}{{g_{{M_{\rm I}}}}} - 1\right). $$

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 IFN-γ quantities is in 0.6 × 108 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:

$$ e_{{M_{\rm I}, \gamma }} = F_{\gamma} \cdot 2 \cdot 100\,\hbox {ml} = 5 \cdot 10^{{- 4}} \hbox {mg} = 3.38 \cdot 10^{5}\,\hbox {pg}. $$

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 IFN-γ on MHC II expression, \(a_{{M_{{\rm II}}, \gamma}}, e_{{M_{{\rm II}}, \gamma}}.\) Phillips et al. [56] use IFN-γ 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 \(e_{{M_{{\rm II}}, \gamma}}:\)

$$ \frac{{a_{{M_{{\rm II}}, \gamma}} \cdot F_{\gamma}}}{{F_{\gamma} + e_{{M_{{\rm II}}, \gamma}}}} - \mu_{{M_{{\rm II}}}} \cdot M_{{\rm II}} = 0, $$

and with two sets of parameters values:

  1. 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. 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 \(\mu_{{M_{{\rm II}}}} = 0.0144\,\hbox {h}^{{- 1}}.\)

IFN-γ unit is given by 0.6 × 108 u/mg we obtained:

$$ \begin{aligned}a_{{M_{{\rm II}}, g}} \gamma &= 8,660\,\hbox {rec}/(\hbox {cells}\cdot \hbox {h})..\\ e_{{M_{{\rm II}}, \gamma}} &= 1,420\,\hbox {pg}. \end{aligned} $$

Parameters for the influence of TGF-β 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 TGF-β. 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 TGF-β we get a drop of 89.8% in MHC expression. This gave the following equation:

$$ (1 - a_{{M_{{\rm II}}, \beta}})\frac{{e_{{M_{{\rm II}}, \beta}}}}{{F_{b} + e_{{M_{{\rm II}}, \beta}}}} + a_{{M_{{\rm II}}, \beta}} = 0.102 $$

Substituting into the above equation \(F_{\beta} = 10\frac{{\rm ng}}{{\rm ml}} \cdot 100\,\hbox {ml}\) we obtained:

$$ e_{{M_{{\rm II}}, \beta}} = 10^{5}\,\hbox {pg}. $$

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 \(\mu_{{M_{{\rm II}}}} = \frac{{\ln\,2}}{{48\,\hbox {h}}} \approx 0.0144\,\hbox {h}^{{- 1}}.\)

Rights and permissions

Reprints and permissions

About this article

Cite this article

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/s00262-007-0387-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00262-007-0387-z

Keywords

Navigation