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

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

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  May 2019 Early access  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.

Citation: Marek Bodnar, Monika Joanna Piotrowska, Magdalena Urszula Bogdańska. Mathematical analysis of a generalised model of chemotherapy for low grade gliomas. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2149-2167. doi: 10.3934/dcdsb.2019088
References:
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N. AndreD. BarbolosiF. BillyG. ChapuisatF. HubertE. Grenier and A. Rovini, Mathematical model of cancer growth controled by metronomic chemotherapies, CANUM 2012, 41e Congrès National d'Analyse Numérique, 41 (2012), 77-94.  doi: 10.1051/proc/201341004.

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M. BodnarU. Foryś and M. J. Piotrowska, Logistic type equations with discrete delay and quasi-periodic suppression rate, Appl Math Lett, 26 (2013), 607-611.  doi: 10.1016/j.aml.2012.12.023.

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J. BucknerD. J. GesmeJ. O'FallonJ. HammackS. StaffordP. BrownR. HawkinsB. ScheithauerB. EricksonR. LevittE. Shaw and R. Jenkins, Phase II trial of procarbazine, lomustine, and vincristine as initial therapy for patients with low-grade oligodendroglioma or oligoastrocytoma: Efficacy and associations with chromosomal abnormalities, Journal of Clinical Oncology, 21 (2003), 251-255. 

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R. LiuK. SolheimM. PolleyK. LambornM. PageA. FedoroffJ. RabbittN. ButowskiM. 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.

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J. P. MannasD. D. LightnerS. R. DeFratesT. 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.

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F. MarchesiM. TurrizianiG. TortorelliG. AvvisatiF. 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.

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P. MazzoccoC. BarthelemyG. KaloshiM. LavielleD. RicardA. IdbaihD. PsimarasM.-A. RenardA. AlentornJ. HonnoratJ.-Y. DelattreF. 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.

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B. NeynsA. TosoniW.-J. Hwu and D. A. Reardon, Dose-dense temozolomide regimens: Antitumor activity, toxicity, and immunomodulatory effects, Cancer, 116 (2010), 2868-2877.  doi: 10.1002/cncr.25035.

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J. PalludE. Mandonnet and H. Duffau, Prognostic value of initial magnetic resonance imaging growth rates for World Health Organization grade Ⅱ gliomas, Annals of Neurology, 60 (2006), 380-383.  doi: 10.1002/ana.20946.

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J. C. Panetta, A mathematical model of drug resistance: Heterogeneous tumors, Math Biosci, 147 (1998), 41-61.  doi: 10.1016/S0025-5564(97)00080-1.

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V. Pérez-GarcíaM. BogdańskaA. Martínez-GonzálezJ. Belmonte-BeitiaP. Schucht and L. Pérez-Romasanta, Delay effects in the response of low-grade gliomas to radiotherapy: a mathematical model and its therapeutical implications, Math. Med. Biol., 32 (2015), 307-329.  doi: 10.1093/imammb/dqu009.

[38]

M. PeyreS. Cartalat-CarelD. MeyronetD. RicardA. JouvetJ. PalludK. MokhtariJ. GuyotatE. JouanneauM. SunyachD. FrappazJ. Honnorat and D. F., Prolonged response without prolonged chemotherapy: A lesson from PCV chemotherapy in low-grade gliomas, Neuro-Oncology, 12 (2010), 1078-1082. 

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M. J. Piotrowska and M. Bodnar, Logistic equation with treatment function and discrete delays, Mathematical Population Studies, 21 (2014), 166-183.  doi: 10.1080/08898480.2014.921492.

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J. PortnowB. BadieM. ChenA. LiuS. Blanchard and T. Synold, The neuropharmacokinetics of temozolomide in patients with resectable brain tumors: potential implications for the current approach to chemoradiation, Clin Can Res, 15 (2009), 7092-7098.  doi: 10.1158/1078-0432.CCR-09-1349.

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N. PouratianJ. GascoJ. ShermanM. Shaffrey and D. Schiff, Toxicity and efficacy of protracted low dose temozolomide for the treatment of low grade gliomas, J Neurooncol, 82 (2007), 281-288.  doi: 10.1007/s11060-006-9280-4.

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show all references

References:
[1]

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. 

[2]

N. AndreD. BarbolosiF. BillyG. ChapuisatF. HubertE. Grenier and A. Rovini, Mathematical model of cancer growth controled by metronomic chemotherapies, CANUM 2012, 41e Congrès National d'Analyse Numérique, 41 (2012), 77-94.  doi: 10.1051/proc/201341004.

[3]

S. Benzekry and P. Hahnfeldt, Maximum tolerated dose versus metronomic scheduling in the treatment of metastatic cancers, Journal of Theoretical Biology, 335 (2013), 235-244. 

[4]

M. BodnarU. Foryś and M. J. Piotrowska, Logistic type equations with discrete delay and quasi-periodic suppression rate, Appl Math Lett, 26 (2013), 607-611.  doi: 10.1016/j.aml.2012.12.023.

[5]

M. U. BogdańskaM. BodnarJ. Belmonte-BeitiaM. MurekP. SchuchtJ. Beck and V. M. Pérez-García, A mathematical model of low grade gliomas treated with temozolomide and its therapeutical implications, Mathematical Biosciences, 288 (2017), 1-13.  doi: 10.1016/j.mbs.2017.02.003.

[6]

L. E. J. Brouwer, Über abbildungen von mannigfaltigkeiten, Mathematische Annalen, 71 (1911), 97-115.  doi: 10.1007/BF01456931.

[7]

J. BucknerD. J. GesmeJ. O'FallonJ. HammackS. StaffordP. BrownR. HawkinsB. ScheithauerB. EricksonR. LevittE. Shaw and R. Jenkins, Phase II trial of procarbazine, lomustine, and vincristine as initial therapy for patients with low-grade oligodendroglioma or oligoastrocytoma: Efficacy and associations with chromosomal abnormalities, Journal of Clinical Oncology, 21 (2003), 251-255. 

[8]

M. Chamberlain, Temozolomide for recurrent low-grade spinal cord gliomas in adults, Cancer, 113 (2008), 1019-1024. 

[9]

R. H. ChisholmT. Lorenzi and J. Clairambault, Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation, Biochim Biophys Acta, 1860 (2016), 2627-2645.  doi: 10.1016/j.bbagen.2016.06.009.

[10]

J. Clairambault, Can theorems help treat cancer?, J.Math.Biol., 66 (2013), 1555-1558.  doi: 10.1007/s00285-012-0518-9.

[11]

M. I. S. CostaJ. L. Boldrini and R. C. Bassanezi, Drug kinetics and drug resistance in optimal chemotherapy, Math Biosci, 125 (1995), 191-209.  doi: 10.1016/0025-5564(94)00027-W.

[12]

L. HammondJ. EckardtS. BakerS. EckhardtM. DuganK. ForralP. ReidenbergG. WeissD. RinaldiD. Von Hoff and E. Rowinsky, Phase Ⅰ and pharmacokinetic study of temozolomide on a daily for 5 days schedule in patients with advanced solid malignancies, Journal of Clinical Oncology, 17 (1999), 2604-2604.  doi: 10.1200/JCO.1999.17.8.2604.

[13]

M. e. a. Heng, Can metronomic maintenance with weekly vinblastine prevent early relapse/progression after bevacizumab-irinotecan in children with low-grade glioma?, Cancer Med, 5 (2016), 1542-1545.  doi: 10.1002/cam4.699.

[14]

M. C. Joiner and A. van der Kogel, Basic clinical radiobiology fourth edition, 2009, URL https://www.123library.org.

[15]

M. A. Jordan, Mechanism of action of antitumor drugs that interact with microtubules and tubulin, Current Medicinal Chemistry. Anti-cancer Agents, 2 (2002), 1-17.  doi: 10.2174/1568011023354290.

[16]

G. KelesK. Lamborn and M. Berger, Low-grade hemispheric gliomas in adults: A critical review of extent of resection as a factor influencing outcome, J Neurosurg, 95 (2011), 735-745.  doi: 10.3171/jns.2001.85.5.0735.

[17]

S. KesariD. SchiffJ. DrappatzD. LaFrankieL. DohertyE. MacklinA. MuzikanskyS. SantagataK. LigonA. NordenA. CiampaJ. BradshawB. LevyG. RadakovicN. RamakrishnaP. Black and P. Wen, Phase Ⅱ study of protracted daily temozolomide for low-grade gliomas in adults, Clin Cancer Res, 15 (2009), 330-337.  doi: 10.1158/1078-0432.CCR-08-0888.

[18]

M. KhasrawD. 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.

[19]

J. T. KimJ. KimK. W. KoD. KongC. KangM. 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.

[20]

K.-K. KoE.-S. LeeY.-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. 

[21]

D.-S. KongJ.-I. LeeJ. H. KimS. T. KimW. 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.

[22]

H. P. LashkariS. SasoL. MorenoT. 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.

[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.

[24]

R. LiuK. SolheimM. PolleyK. LambornM. PageA. FedoroffJ. RabbittN. ButowskiM. 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.

[25]

D. N. LouisA. PerryG. ReifenbergerA. von DeimlingD. Figarella-BrangerW. 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.

[26]

A. MangionalC. AnileA. PompucciG. CaponeL. Rigante and P. De Bonis, Glioblastoma therapy: Going beyond hercules columns, Expert Rev Neurother, 10 (2010), 507-514. 

[27]

J. P. MannasD. D. LightnerS. R. DeFratesT. 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.

[28]

F. MarchesiM. TurrizianiG. TortorelliG. AvvisatiF. 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.

[29]

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

[30]

P. MazzoccoC. BarthelemyG. KaloshiM. LavielleD. RicardA. IdbaihD. PsimarasM.-A. RenardA. AlentornJ. HonnoratJ.-Y. DelattreF. 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.

[31]

P. Mazzocco, J. Honorat, F. Ducray and B. Ribba, Increasing the time interval between PCV chemotherapy cycles as a strategy to improve duration of response in low-grade gliomas: Results from a model-based clinical trial simulation, Comput Math Methods Med, 2015 (2015), 297903, 7pp. doi: 10.1155/2015/297903.

[32]

S. Nageshwaran, D. Ledingham, H. C. Wilson and A. Dickenson (eds.), Drugs in Neurology, Oxford University Press, 2017. doi: 10.1093/med/9780199664368.001.0001.

[33]

H. B. Newton, Neurological complications of chemotherapy to the central nervous system, Handbook of Clinical Neurology, 105 (2012), 903-916.  doi: 10.1016/B978-0-444-53502-3.00031-8.

[34]

B. NeynsA. TosoniW.-J. Hwu and D. A. Reardon, Dose-dense temozolomide regimens: Antitumor activity, toxicity, and immunomodulatory effects, Cancer, 116 (2010), 2868-2877.  doi: 10.1002/cncr.25035.

[35]

J. PalludE. Mandonnet and H. Duffau, Prognostic value of initial magnetic resonance imaging growth rates for World Health Organization grade Ⅱ gliomas, Annals of Neurology, 60 (2006), 380-383.  doi: 10.1002/ana.20946.

[36]

J. C. Panetta, A mathematical model of drug resistance: Heterogeneous tumors, Math Biosci, 147 (1998), 41-61.  doi: 10.1016/S0025-5564(97)00080-1.

[37]

V. Pérez-GarcíaM. BogdańskaA. Martínez-GonzálezJ. Belmonte-BeitiaP. Schucht and L. Pérez-Romasanta, Delay effects in the response of low-grade gliomas to radiotherapy: a mathematical model and its therapeutical implications, Math. Med. Biol., 32 (2015), 307-329.  doi: 10.1093/imammb/dqu009.

[38]

M. PeyreS. Cartalat-CarelD. MeyronetD. RicardA. JouvetJ. PalludK. MokhtariJ. GuyotatE. JouanneauM. SunyachD. FrappazJ. Honnorat and D. F., Prolonged response without prolonged chemotherapy: A lesson from PCV chemotherapy in low-grade gliomas, Neuro-Oncology, 12 (2010), 1078-1082. 

[39]

M. J. Piotrowska and M. Bodnar, Logistic equation with treatment function and discrete delays, Mathematical Population Studies, 21 (2014), 166-183.  doi: 10.1080/08898480.2014.921492.

[40]

J. PortnowB. BadieM. ChenA. LiuS. Blanchard and T. Synold, The neuropharmacokinetics of temozolomide in patients with resectable brain tumors: potential implications for the current approach to chemoradiation, Clin Can Res, 15 (2009), 7092-7098.  doi: 10.1158/1078-0432.CCR-09-1349.

[41]

N. PouratianJ. GascoJ. ShermanM. Shaffrey and D. Schiff, Toxicity and efficacy of protracted low dose temozolomide for the treatment of low grade gliomas, J Neurooncol, 82 (2007), 281-288.  doi: 10.1007/s11060-006-9280-4.

[42]

N. Pouratian and D. Schiff, Management of low-grade glioma, Curr Neurol Neurosci Rep, 10 (2010), 224-231.  doi: 10.1007/s11910-010-0105-7.

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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
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
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
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
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 $T = 50$
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 $(u_3, u_3)$
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
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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