1 Introduction

The gliomas are the most common brain tumors and are originated in glial cells [1]. Some gliomas are very aggressive, carrying a rate mortality rate nearing the 100\(\%\) of the cases within 6–12 months after being diagnosed [2, 3]. Usually, the first treatment against gliomas consists of a surgery followed by chemotherapy and radiotherapy, but results are far from being effective [4]. Regarding the chemotherapy, the most common agent against brain tumors is the temozolamide [5, 6]. There are other proposed treatments based on virotherapy [7], and it has been suggested that the combination of different types of therapies could be a successful alternative, in particular, the combination of chemotherapy and virotherapy [8,9,10].

On the other hand, the mathematical modeling of cancer has been an essential tool to understand this problem. Moreover, the mathematical approach also allows to replace laboratory experiments and to simulate them computationally. In this sense, scientists have developed a wide diversity of mathematical models covering various aspects of tumor growth (see [11,12,13] and references therein). In particular case of gliomas, one can find several mathematical proposals [14, 15] focusing on the diagnostic [4, 16], the chemotherapy [17,18,19] or the virotherapy [20,21,22,23].

In the present work, we propose a mathematical model that combines chemotherapy and oncolytic virotherapy as an alternative treatment against gliomas. The paper is organized as follows. In Sect. 2, the mathematical model is proposed in full detail. Moreover, suitable physical constraints and parameter values are provided. In Sect. , a stability analysis of the model associated to the virotherapy is performed. Some simulations are presented in Sect. , and the main results are discussed. Finally, we close the article with a brief section of conclusions.

2 Mathematical model

As we mentioned before, the mathematical model proposed in this work consists of two alternative treatments, namely, chemotherapy and virotherapy. The chemotherapy is based on the model proposed by Iarosz et al. [17]. More precisely, a system of differential equations is used to model a brain tumor (more specifically, a glioma), and the corresponding interactions among glial cells, glioma, neurons and chemotherapy. However, in order to match this model with the virotherapy treatment, it is more convenient to state the variables in terms of the number of cells instead of concentrations. Therefore, it is necessary to rescale the parameters of the model.

The rescaling mentioned above can be performed taking into account that the carrying capacity of tumor cells is \(k_C=1.47 \times 10^{12}\) cells [22]. A population of \(4.2\times 10^9\) cells corresponds approximately to one tumor of 2 cm of diameter. Assuming a mass per cell of \(10^{-12}\) kg, the corresponding carrying capacity in terms of the concentration is equal to \(K_C=125.3\) kg/m\(^3\). In other words, an adequate scale factor is equal to \(\delta =1.17\times 10^{10}\) m\(^3\) cells/kg. As a consequence, we rescale the set of equations 1-4 in [17] as follows

$$\begin{aligned} \frac{dG(t)}{dt}= & {} \varOmega _{1}G(t)\left( 1-\frac{G(t)}{k}\right) -\psi _{1}G(t)C(t)- \frac{p_{1}G(t)Q(t)}{a_{1}+G(t)}, \end{aligned}$$
(1)
$$\begin{aligned} \frac{dC(t)}{dt}= & {} \varOmega _{2}C(t)\left( 1-\frac{C(t)}{k}\right) -\psi _{2}G(t)C(t)- \frac{p_{2}C(t)Q(t)}{a_{2}+C(t)}, \end{aligned}$$
(2)
$$\begin{aligned} \frac{dN(t)}{dt}= & {} \bar{\psi } \dot{G}(t)H(-\dot{G})N(t)-\frac{p_{3}N(t)Q(t)}{ a_{3}+N(t)}, \end{aligned}$$
(3)
$$\begin{aligned} \frac{dQ(t)}{dt}= & {} \varPhi -\zeta Q(t), \end{aligned}$$
(4)

where the new parameters are defined in terms of those used in [17] as

$$\begin{aligned} \bar{\psi }=\frac{\psi }{\delta }, \qquad \psi _i=\frac{\varPsi _i}{\delta }, \qquad p_i=\delta P_i, \qquad a_i=\delta A_i, \qquad k=\delta K. \end{aligned}$$
(5)

Table 1 shows the corresponding values of the new parameters.

Table 1 Parameter values used in the equations (1)–(4)
Full size table

The variables G(t), C(t) and N(t) correspond to the number of glial cells, glioma cells and neuron cells, respectively, and Q(t) is the concentration of the chemotherapeutic agent given in mg/m\(^2\). According to [17], the first term in (1) and (2) models a logistic growth of the glial and glioma cells, while the second describes the competition between the two types of cells, and the third models the effect of the chemotherapy on the cells in terms of the Holling type functions [28, 29].

The dynamics of number of neuron cells is described by Eq. (3). In that equation, the first term models the decrease of neurons as a consequence of the glial cells death. Here, H(t) is the well-known Heaviside function, which is defined as

$$\begin{aligned} H(x)=\left\{ \begin{array}{ll} 0 &{} ,x<0, \\ \frac{1}{2} &{} ,x=0, \\ 1 &{} ,x>0. \end{array} \right. \end{aligned}$$
(6)

Meanwhile, the second term is the effect of the chemotherapy on the neurons.

Finally, Eq. (4) describes the evolution of the concentration of the chemotherapeutic agent. The first term is the function \(\varPhi \), which is defined as a periodic step function. More specifically, \(\varPhi \) has a constant value of 400 mg(m\(^2\) day)\(^{-1}\) during 5 days (chemotherapy session), and is switched off for 23 days, and so on (temozolomide is one of the most common chemotherapeutic agents against brain tumors [5, 6]). In turn, the second term of equation (4) models the corresponding decrease of the agent.

The second part of the model corresponds to the oncolytic virotherapy. This therapy is provided on day 21 (16 days after of the ending of the chemotherapy session) in order to reduce the effects of the agent on the virus. Moreover, this is done in view that the effectiveness of the virotherapy lasts around one week. This part of the model is based on the work of [22]. However, we have considered some additional assumptions and modifications. Concretely, the system of equations is provided as follows:

$$\begin{aligned} \frac{dC_{S}(t)}{dt}= & {} \varOmega _2 C_{S}(t)\left( 1-\frac{\left( C_{S}(t)+C_{I}(t)\right) }{ k}\right) -\psi _{2}G(t)C_S(t) \nonumber \\&-\frac{p_{2}C_S(t)Q(t)}{a_{2}+C_S(t)} -C_{S}(t)\beta _{C}v(t), \end{aligned}$$
(7)
$$\begin{aligned} \frac{dC_{I}(t)}{dt}= & {} \beta _{C}C_{S}(t)v(t)-\lambda _{C}C_{I}(t), \end{aligned}$$
(8)
$$\begin{aligned} \frac{dv(t)}{dt}= & {} b_{C}\lambda _{C}C_{I}(t)-\beta _{C}C_{S}v(t)-\omega v(t). \end{aligned}$$
(9)

The glial cells equation (1), the neurons equation (3) and the agent equation (4) remain unchanged. We assume that the virus is specialist, which means that only infects glioma cells. As a consequence, the tumor cells are divided in healthy \(C_S(t)\) and infected \(C_I(t)\) by the viruses v(t). The last term in (7) corresponds to the competition between healthy tumor cells and the viruses. We assume that the infected tumor cells cannot reproduce, so that it is important to change C(t) for \(C_S(t)\) in (1). In (8) the first term corresponds to the infection of healthy cells as a consequence of the competition with the virus, while the last term is the lysing of the infected cells.

The virus equation (9) describes the releasing of the infection by means of the infected cells. The second term is the competition with healthy cells, and the last one is the clearing of the viruses by the immune response. For simplicity we have dropped the chemotherapy agent effect (which it is minimal between day 21 and 28) in Eqs. (8) and (9). Of course at the beginning of the virotherapy we suppose that \(C_S(21)=C(21)\) and \(C_I(21)=0\). The new parameters used in the equations are similar to those used in [22], but written in days units as one can see in the Table 2. Note that three different types of oncolytic viruses are taking account [22]. They are based on some previous works in the literature [7, 20, 30, 31].

Table 2 Parameter values used in Eqs. (7)–(9) (see [22])
Full size table

3 Local stability

Before we provide some computational simulations, we perform a local stability analysis of the model when the virotherapy is supplied.

Let us suppose that \(E(\bar{G},\bar{C_S},\bar{C_I},\bar{N},\bar{Q},\bar{v})\) is an equilibrium point which is a solution of the following system:

$$\begin{aligned} \frac{dG}{dt}= & {} \varOmega _{1}G(t)\left( 1-\frac{G(t)}{k}\right) -\psi _{1}G(t)C_S(t)-\frac{p_{1}G(t)Q(t)}{a_{1}+G(t)}=0, \end{aligned}$$
(10)
$$\begin{aligned} \frac{dC_{S}}{dt}= & {} \varOmega _{2}C_{S}(t)\left( 1-\frac{\left( C_{S}(t)+C_{I}(t)\right) }{k}\right) -\psi _{2}G(t)C_{S}(t) \nonumber \\&-\frac{p_{2}C_S(t)Q(t)}{a_{2}+C_S(t)}-C_{S}(t)\beta _{C}v(t)=0, \end{aligned}$$
(11)
$$\begin{aligned} \frac{dC_{I}}{dt}= & {} \beta _{C}C_{S}(t)v(t)-\lambda _{C}C_{I}(t)=0, \end{aligned}$$
(12)
$$\begin{aligned} \frac{dN}{dt}= & {} \bar{\psi }\dot{G}(t)H(-\dot{G})N(t)-\frac{p_{3}N(t)Q(t)}{ a_{3}+N(t)}=0, \end{aligned}$$
(13)
$$\begin{aligned} \frac{dQ}{dt}= & {} -\zeta Q(t)=0, \end{aligned}$$
(14)
$$\begin{aligned} \frac{dv}{dt}= & {} b_{C}\lambda _{C}C_{I}-\beta _{C}C_{S}(t)v(t)-\omega v(t)=0. \end{aligned}$$
(15)

Here, we have taken into account that the chemotherapy infusion \(\varPhi \) is null during the virotherapy. It is obvious that \(E_1(0,0,0,0,0,0)\) is an equilibrium point. Moreover, the eigenvalues of the corresponding Jacobian matrix are \(\lambda _1=\varOmega _1\), \(\lambda _2=\varOmega _2\), \(\lambda _3=-\lambda _C\), \(\lambda _4=0\), \(\lambda _5=-\zeta \) and \(\lambda _6=-\omega \). Note that \(\lambda _1\) and \(\lambda _2\) are positive, therefore we have an unstable point. This case is biological unfeasible because we have neither glial cells nor neurons.

A second equilibrium point is \(E_2(k,0,0,\bar{N},0,0)\), which is the case of a healthy patient without cancer and therapies. Here, we set \(\bar{G}=k\) in order to guarantee the equilibrium for the glial cells, and \(\bar{N}\) is any real and positive value. The eigenvalues of the Jacobian matrix are \(\lambda _1=-\varOmega _1\), \(\lambda _2=\varOmega _2-\psi _2 k\), \(\lambda _3=-\lambda _C\), \(\lambda _4=0\), \(\lambda _5=-\zeta \) and \(\lambda _6=-\omega \). Note that \(\lambda _2\) is positive according to the values of Table 1. Therefore, the equilibrium point is unstable. More precisely, if the treatment is ceased with some healthy glioma cells \(C_S\ne 0\) then the glioma would return. A similar result is established in [17] for the chemotherapy treatment.

4 Simulations and results

In this section, we provide some simulations using the proposed model. Firstly, we provide simulations considering only the chemotherapy treatment from day 0 to day 21, that is, we use Eqs. (1)–(4). From from day 21 to day 28, we will incorporate the virotherapy. The initial, intermediate and final values of the variables are shown in Table 3. Figure 1 shows the evolution of the variables. In particular, we note that the tumor cells results for three different types of virotherapies (there was not significant difference for glial cells or neurons with the three types of virotherapy): adenovirus (ONYX-15) [20, 22], HSV [7, 22] and VSV [22, 30, 31].

Table 3 Initial, intermediate and final values for glial cells G(t), neurons N(t), tumor cells C(t) and concentration of chemotherapeutic agent Q(t) for the first simulation
Full size table
Fig. 1
figure 1

Results of the first simulation with the temporary variable t in days: a number of glial cells G, b number of neurons , c number of tumor cells with adenovirus (cyan), HSV (red) and VSV (blue) virotherapy, d concentration of chemotherapeutic agent in \(\mathrm{mg}/\mathrm{m}^2\), e the same graphic of c but in logarithmic scale and f chemotherapy infusion in \(\mathrm{mg}(\mathrm{m}^2 \ \mathrm{day})^{-1}\) (Color figure online)

Full size image

As a second experiment, in order to search for effects of more chemotherapy sessions, we supplied the virotherapy after the second chemotherapy session, more specifically from day 49 to day 56. The initial, intermediate and final values of the variables are shown in Table 4. Figure 2 shows the evolution of the variables. Similarly to the first experiment, we show the tumor cells results for the three different types of virotherapies. In order to see with more detail the differences among the virotherapies, we show the tumor cells results in logarithmic scale for the two experiments in Figs. 1e and 2e. It is important to remark that at the end of both simulations all the tumor cells are infected by the virus, i. e. \(C_S(28)=0\) for the first simulation and \(C_S(56)=0\) for the second one.

After of an analysis of the simulations we can establish that

  • The chemotherapy attacks the tumor very slowly and it would be necessary to supply many sessions to see good results.

  • The virotherapy acts faster on the tumor than the chemotherapy.

  • There is no significant relevance if we incorporate the virotherapy after the first or the second chemotherapy session.

  • Although the result is good for different types of viruses, there is a noticeable difference each other.

  • The simulations present better results for the HSV virus, which is precisely used in glioma treatments [7].

  • After the virotherapy, in principle, it would not be necessary to apply more therapies.

  • The glial cells and neurons practically are not affected for the chemotherapy, in fact, one can notice a slight recovery of the number of glial cells as a consequence of reduction of glioma cells during the virotherapy.

  • The glioma is two or more orders of magnitude smaller at the end of the simulations, in addition, at the same time all the tumor cells are infected by the virus and, as a consequence of this, they do not have capacity of reproduction.

Table 4 Initial, intermediate and final values for glial cells G(t), neurons N(t), tumor cells C(t) and concentration of chemotherapeutic agent Q(t) for the second simulation
Full size table
Fig. 2
figure 2

Results of the second simulation with the temporary variable t in days: a number of glial cells G, b number of neurons, c number of tumor cells with adenovirus (cyan), HSV (red) and VSV (blue) virotherapy, d concentration of chemotherapeutic agent in mg/m\(^2\), e the same graphic of c but in logarithmic scale and f chemotherapy infusion in mg(m\(^2\) day)\(^{-1}\) (Color figure online)

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5 Conclusion

In this paper, we propose a mathematical model of a combination of chemotherapy and oncolytic virotherapy as an alternative treatment against cancer, more specifically a brain tumor called glioma. The system is based on interactions among glial cells, glioma cells and neurons, including the effects of the corresponding concentration of chemotherapeutic agent. The virotherapy is incorporated by means of the supply of a specialist virus that attacks only the tumor cells, in this work we use three different types of viruses. During the virotherapy the tumor cells are divided in healthy and infected, because it is assumed that the infected cells cannot reproduce. The results show that the proposed treatment could reduce the number of chemotherapy sessions and it is recommendable to choose the better kind of virotherapy depending of the type of cancer. Finally, we can conclude that this model strengths the proposals [8,9,10, 32] that claim good results against cancer using a combination of chemotherapy and virotherapy treatments.