American Institute of Mathematical Sciences

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May  2019, 24(5): 2149-2167. doi: 10.3934/dcdsb.2019088

Mathematical analysis of a generalised model of chemotherapy for low grade gliomas

Marek Bodnar , Monika Joanna Piotrowska , and  Magdalena Urszula Bogdańska

Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

* Corresponding author: monika@mimuw.edu.pl

Received  January 2018 Revised  January 2019 Published  March 2019

We study mathematical properties of a model describing growth of primary brain tumours called low-grade gliomas (LGGs) and their response to chemotherapy. The motivation for considering this particular type of cancer is its large impact on society. LGGs affect mainly young adults and eventually result in death, despite the tumour growth rate being slow. The model studied consists of two non-autonomous ordinary differential equations and is a generalised version of the model proposed by Bogdańska et al. (Math. Biosci. 2017). We discuss the stability of stationary states, prove global stability of tumour-free steady state and, in some cases, justify the existence of periodic solutions. Assuming that chemotherapy effectiveness remains constant in time, we provide analytical estimates and calculate minimal doses of the drug that should eliminate the tumour for particular patients with LGGs.

Keywords: Gliomas, brain tumours, chemotherapy, mathematical model of tumour growth, asymptotic behaviour.
Mathematics Subject Classification: Primary: 34B37, 34B60, 34D20, 34D23, 92C45; Secondary: 34C25, 92C50.
Citation: Marek Bodnar, Monika Joanna Piotrowska, Magdalena Urszula Bogdańska. Mathematical analysis of a generalised model of chemotherapy for low grade gliomas. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2149-2167. doi: 10.3934/dcdsb.2019088
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show all references

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S. Agarwala and J. Kirkwood, Temozolomide, a novel alkylating agent with activity in the central nervous system, may improve the treatment of advanced metastatic melanoma, The Oncologist, 5 (2000), 144-151.   Google Scholar

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M. Khasraw, D. Bell and H. Wheeler, Long-term use of temozolomide: Could you use temozolomide safely for life in gliomas?, Case Reports / Journal of Clinical Neuroscience, 16 (2009), 854-855.  doi: 10.1016/j.jocn.2008.09.005.  Google Scholar

[19]

J. T. Kim, J. Kim, K. W. Ko, D. Kong, C. Kang, M. H. Kim and et al., Metronomic treatment of temozolomide inhibits tumor cell growth through reduction of angiogenesis and augmentation of apoptosis in orthotopic models of gliomas, Oncol Rep, 16 (2006), 33-39.  doi: 10.3892/or.16.1.33.  Google Scholar

[20]

K.-K. Ko, E.-S. Lee, Y.-A. Joe and Y.-K. Hong, Metronomic treatment of temozolomide increases antiangiogenicity accompanied by down-regulated O6-methylguanine-DNA methyltransferase expression in endothelial cells, Exp Ther Med, 2 (2011), 343-348.   Google Scholar

[21]

D.-S. Kong, J.-I. Lee, J. H. Kim, S. T. Kim, W. S. Kim and Y.-L. Suh, Phase Ⅱ trial of low-dose continuous (metronomic) treatment of temozolomide for recurrent glioblastoma, Neuro-Oncology, 12 (2010), 289-296.  doi: 10.1093/neuonc/nop030.  Google Scholar

[22]

H. P. Lashkari, S. Saso, L. Moreno, T. Athanasiou and S. Zacharoulis, Using different schedules of Temozolomide to treat low grade gliomas: systematic review of their efficacy and toxicity, J Neurooncol, 105 (2011), 135-147.  doi: 10.1007/s11060-011-0657-7.  Google Scholar

[23]

U. Ledzewicz and H. M. Schättler, Drug resistance in cancer chemotherapy as an optimal control problem, DCDS-B, 6 (2006), 129-150.  doi: 10.3934/dcdsb.2006.6.129.  Google Scholar

[24]

R. Liu, K. Solheim, M. Polley, K. Lamborn, M. Page, A. Fedoroff, J. Rabbitt, N. Butowski, M. Prados and S. Chang, Quality of life in low-grade glioma patients receiving temozolomide, Neuro-Oncology, 11 (2009), 59-68.  doi: 10.1215/15228517-2008-063.  Google Scholar

[25]

D. N. Louis, A. Perry, G. Reifenberger, A. von Deimling, D. Figarella-Branger, W. K. Cavenee and et al., The 2016 world health organization classification of tumors of the central nervous system: A summary, Acta Neuropathol, 131 (2016), 803-820.  doi: 10.1007/s00401-016-1545-1.  Google Scholar

[26]

A. Mangional, C. Anile, A. Pompucci, G. Capone, L. Rigante and P. De Bonis, Glioblastoma therapy: Going beyond hercules columns, Expert Rev Neurother, 10 (2010), 507-514.   Google Scholar

[27]

J. P. Mannas, D. D. Lightner, S. R. DeFrates, T. Pittman and J. L. Villano, Long-term treatment with temozolomide in malignant glioma, Journal of Clinical Neuroscience, 21 (2014), 121-123.  doi: 10.1016/j.jocn.2013.03.039.  Google Scholar

[28]

F. Marchesi, M. Turriziani, G. Tortorelli, G. Avvisati, F. Torino and L. De Vecchis, Triazene compounds: Mechanism of action and related DNA repair systems, Pharmacological Research, 56 (2007), 275-287.  doi: 10.1016/j.phrs.2007.08.003.  Google Scholar

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W. Mason, G. Krol and L. DeAngelis, Low-grade oligodendroglioma responds to chemotherapy, Neurology, 46 (1996), 203-207.  doi: 10.1212/WNL.46.1.203.  Google Scholar

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P. Mazzocco, C. Barthelemy, G. Kaloshi, M. Lavielle, D. Ricard, A. Idbaih, D. Psimaras, M.-A. Renard, A. Alentorn, J. Honnorat, J.-Y. Delattre, F. Ducray and B. Ribba, Prediction of response to temozolomide in low-grade glioma patients based on tumor size dynamics and genetic characteristics, CPT Pharmacometrics Syst Pharmacol, 4 (2015), 728-737.  doi: 10.1002/psp4.54.  Google Scholar

[31]

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Figure 1.  Sketch presenting the class of steady state $P_3$ depending on considered cases: (left) $z\geq \delta$; (right) $z < \delta$. Blue and white areas represent sets of parameters for which $P_3$ is node and focus, respectively. Dots denote region where $P_3$ is stable, no pattern — region where $P_3$ is unstable
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Figure 2.  Phase portrait of system (1.7) with $z(t)\equiv z$, $f(x+\gamma y) = 1-x-\gamma y$ and $\gamma = 1$ in case when: (left) the positive steady state $P_3$ does not exist, $z = 1.3, \kappa = 2$, (center) $P_3$ is a stable node, $z = 0.4, \kappa = 6.67$, (right) $P_3$ is a stable focus, $z = 0.3, \kappa = 1.11$. Dashed curves represent nullclines
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Figure 3.  Phase portrait of system (1.7) with $z(t)\equiv z <1$, $f(x+\gamma y) = 1-x-\gamma y$ and either $\gamma <1$ (left) or $\gamma>1$ (right). Dashed curves represent nullclines
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Figure 4.  Phase portrait of the system (1.7) with $z(t)\equiv z>1$, $f(x+\gamma y) = 1-x-\gamma y$ and either $\gamma <1$ (left) or $\gamma>1$ (right). Dashed curves represent nullclines
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Figure 5.  Time evolution of $x$ and $y$ due to system (1.7) for $\bar{z} < 1$. The parameters values were: $\gamma = 1$, $\kappa = 0.1$ and $z(t) = 0.98\Bigl(1+\sin(2\pi t/T)\Bigr)$, with $T = 50$
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Figure 6.  Sketch of set $K$ (in red) defined in the proof of Theorem 2.9. Blue points denote points $(x_0, x_0)$, $(x_1, u_1)$, $(x_2, u_2)$, $(x_2, u_3)$ and $(u_3, u_3)$
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Figure 7.  Sketch of an exemplary chemotherapy scheme considered in Proposition 2.11. The parameters $s_j$ are the moments of drug administrations, $C_j$ - drug doses, $T$ - duration of a single cycle. The same colours indicate the same drug administration in each cycle. In the typical TMZ chemotherapy scheme for LGGs we have: $T$ = 28 days; $p = 5$; between $s_p$ and $s_{p+1}$ there is a rest phase (no drug administration) that takes 23 days; all $C_j$ are equal and have value between 150-200 mg of TMZ per m$^2$ of patient body surface area
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Figure 8.  The minimal eradication dose $ d $ per m$ ^2 $ of body surface estimated for the patients' parameters estimated in [5]
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