Mathematical modelling and analysis of brain tumour immunotherapy.


Each year about 20 thousand new patients are diagnosed with malignant gliomas (MG) in the United states. Adult primary MG are among the most virulent forms of cancer. Median survival for high grade MG varies from 18 months (Stupp, et al., 2005) for GlioBlastoma Multiforme (GBM, Grade IV) to three to five years for grade III MG (Kleihues, Soylemezoglu, Schauble, Scheithauer, & Burger, 1995; Kleihues, et al., 2002). Due to their genomic instability, heterogeneity, and infiltrative behavior in their sequestered location beyond the blood brain barrier (BBB), MG are refractory to conventional treatments, including surgery, radiation, and chemotherapy. Thus, novel therapies are sought, notably immunotherapy, in the hope they offer a survival advantage.

In recent years cancer immunotherapy has shown some success and carried far less risks and side effects to the patients than traditional therapy (de Vleeschouwer, et al., 2006). Clinical trial of Kruse (Kruse, et al., 1997; Liau, 2001) reports six patients treated with alloreactive cytotoxic-T lymphocytes (aCTL), three of whom were recurrent grade III MG patients and the other three were recurrent GBM patients. The six patients were treated with periodic intra-tumoral aCTL infusions 2-3 times within two weeks, followed by a rest period of 6 weeks. This process was repeated for five cycles, total dose of administered aCTL varying between 108 to 5.2•109 cells (Kruse, et al., 1997). Although technically complicated, passive aCTL immunotherapy overcomes two major problems encountered in systemic immunotherapy: it is independent of the often anergic patient’s own immune system, while the use of intracranial infusion bypasses the BBB hurdle. Indeed, the above clinical trials showed success in that two of the grade III patients survived for at least 12 years after treatment and one survived for 40 months post treatment. However, all GBM patients died within several months. This discrepancy between the success in curing grade III MG and the failure of GBM immunotherapy points to a crucial difference between the characteristic system dynamics in the two indications.

Methods and Results

The aforementioned difference may be clarified by analyzing the reaction rates governing the disease and immune processes. The dynamics of tumor-immune system interactions are complex, involving cytotoxic processes, cytokine modulation, migration through extracellular matrix, as well as negative and positive feedbacks by paracrine and autocrine factors. In order to study these dynamics and propose new insights in immunotherapy treatment approach, we have constructed a mathematical model for glioma and the immune system interactions, that may ensue upon direct intra-tumoral administration of ex vivo activated aCTLs (Kronik, Kogan, Vainstein, & Agur, 2008). 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.

The mathematical model successfully retrieved clinical trial results of efficacious aCTL immunotherapy for recurrent grade III MG. 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. Our work shows that, depending on the patient’s disease parameters, a personal infusion rate threshold exists, below which the patient is expected to be refractory to the treatment and above which tumor elimination is predicted (Kogan, Forys, Shukron, Kronik, & Agur, 2010).


We propose that the path of adoptive cellular immunotherapy should be maintained and improved. It may prove efficacious for GBM, if dose intensity is augmented, as prescribed by the mathematical model. Under the currently available T-cell therapy methodologies, individual planning of dosing schedules may contribute to improving success of adoptive T-cell therapy.

For further reading

  1. de Vleeschouwer, S., Rapp, M., Sorg, R. V., Steiger, H. J., Stummer, W., van Gool, S., et al. (2006). Dendritic cell vaccination in patients with malignant gliomas: current status and future directions. Neurosurgery, 59(5), 988-999; discussioin 999-1000.
  2. Kleihues, P., Louis, D. N., Scheithauer, B. W., Rorke, L. B., Reifenberger, G., Burger, P. C., et al. (2002). The WHO classification of tumors of the nervous system. J Neuropathol Exp Neurol, 61(3), 215-225; discussion 226-219.
  3. Kleihues, P., Soylemezoglu, F., Schauble, B., Scheithauer, B. W., & Burger, P. C. (1995). Histopathology, classification, and grading of gliomas. Glia, 15(3), 211-221.
  4. Kogan, Y., Forys, U., Shukron, O., Kronik, N., & Agur, Z. (2010). Cellular immunotherapy for high grade Gliomas: mathematical analysis deriving efficacious infusion rates based on patient requirements. SIAM J. Appl. Math., 70(6), 1953-1976.
  5. Kronik, N., Kogan, Y., Vainstein, V., & Agur, Z. (2008). Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics. Cancer Immunol Immunother, 57(3), 425-439.
  6. Kruse, C. A., Cepeda, L., Owens, B., Johnson, S. D., Stears, J., & Lillehei, K. O. (1997). Treatment of recurrent glioma with intracavitary alloreactive cytotoxic T lymphocytes and interleukin-2. Cancer Immunol Immunother, 45(2), 77-87.
  7. Liau, L. M. (2001). Brain tumor immunotherapy. Totowa, N.J.: Humana Press.
  8. Stupp, R., Mason, W. P., van den Bent, M. J., Weller, M., Fisher, B., Taphoorn, M. J., et al. (2005). Radiotherapy plus concomitant and adjuvant temozolomide for glioblastoma. N Engl J Med, 352(10), 987-996.

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