# American Institute of Mathematical Sciences

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  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 & Continuous Dynamical Systems - B, 2019, 24 (5) : 2149-2167. doi: 10.3934/dcdsb.2019088
##### References:

show all references

##### References:
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
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
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
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
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 $T = 50$
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 $(u_3, u_3)$
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
The minimal eradication dose $d$ per m$^2$ of body surface estimated for the patients' parameters estimated in [5]
 [1] Clara Rojas, Juan Belmonte-Beitia, Víctor M. Pérez-García, Helmut Maurer. Dynamics and optimal control of chemotherapy for low grade gliomas: Insights from a mathematical model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1895-1915. doi: 10.3934/dcdsb.2016028 [2] Akisato Kubo, Hiroki Hoshino, Katsutaka Kimura. Global existence and asymptotic behaviour of solutions for nonlinear evolution equations related to a tumour invasion model. Conference Publications, 2015, 2015 (special) : 733-744. doi: 10.3934/proc.2015.0733 [3] Diego Samuel Rodrigues, Paulo Fernando de Arruda Mancera. Mathematical analysis and simulations involving chemotherapy and surgery on large human tumours under a suitable cell-kill functional response. Mathematical Biosciences & Engineering, 2013, 10 (1) : 221-234. doi: 10.3934/mbe.2013.10.221 [4] Carole Guillevin, Rémy Guillevin, Alain Miranville, Angélique Perrillat-Mercerot. Analysis of a mathematical model for brain lactate kinetics. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1225-1242. doi: 10.3934/mbe.2018056 [5] Ismail Abdulrashid, Xiaoying Han. A mathematical model of chemotherapy with variable infusion. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1875-1890. doi: 10.3934/cpaa.2020082 [6] Urszula Ledzewicz, Behrooz Amini, Heinz Schättler. Dynamics and control of a mathematical model for metronomic chemotherapy. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1257-1275. doi: 10.3934/mbe.2015.12.1257 [7] Hsiu-Chuan Wei. Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1279-1295. doi: 10.3934/dcdsb.2016.21.1279 [8] Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040 [9] Giuseppe Viglialoro, Thomas E. Woolley. Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3023-3045. doi: 10.3934/dcdsb.2017199 [10] Toru Sasaki, Takashi Suzuki. Asymptotic behaviour of the solutions to a virus dynamics model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 525-541. doi: 10.3934/dcdsb.2017206 [11] Philip Gerlee, Alexander R. A. Anderson. Diffusion-limited tumour growth: Simulations and analysis. Mathematical Biosciences & Engineering, 2010, 7 (2) : 385-400. doi: 10.3934/mbe.2010.7.385 [12] Eduardo Ibargüen-Mondragón, Lourdes Esteva, Edith Mariela Burbano-Rosero. Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma. Mathematical Biosciences & Engineering, 2018, 15 (2) : 407-428. doi: 10.3934/mbe.2018018 [13] María Rosa, María S. Bruzón, M. L. Gandarias. A model of malignant gliomas throug symmetry reductions. Conference Publications, 2015, 2015 (special) : 974-980. doi: 10.3934/proc.2015.0974 [14] Tinevimbo Shiri, Winston Garira, Senelani D. Musekwa. A two-strain HIV-1 mathematical model to assess the effects of chemotherapy on disease parameters. Mathematical Biosciences & Engineering, 2005, 2 (4) : 811-832. doi: 10.3934/mbe.2005.2.811 [15] Urszula Ledzewicz, Helmut Maurer, Heinz Schättler. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy. Mathematical Biosciences & Engineering, 2011, 8 (2) : 307-323. doi: 10.3934/mbe.2011.8.307 [16] Urszula Ledzewicz, Mozhdeh Sadat Faraji Mosalman, Heinz Schättler. Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 1031-1051. doi: 10.3934/dcdsb.2013.18.1031 [17] Wenxiang Liu, Thomas Hillen, H. I. Freedman. A mathematical model for M-phase specific chemotherapy including the $G_0$-phase and immunoresponse. Mathematical Biosciences & Engineering, 2007, 4 (2) : 239-259. doi: 10.3934/mbe.2007.4.239 [18] Urszula Ledzewicz, Omeiza Olumoye, Heinz Schättler. On optimal chemotherapy with a strongly targeted agent for a model of tumor-immune system interactions with generalized logistic growth. Mathematical Biosciences & Engineering, 2013, 10 (3) : 787-802. doi: 10.3934/mbe.2013.10.787 [19] Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305 [20] Francesca R. Guarguaglini. Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks & Heterogeneous Media, 2018, 13 (1) : 47-67. doi: 10.3934/nhm.2018003

2018 Impact Factor: 1.008