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

• 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.

Mathematics Subject Classification: Primary: 34B37, 34B60, 34D20, 34D23, 92C45; Secondary: 34C25, 92C50.

 Citation:

• 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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