Model for tumour growth with treatment by continuous and pulsed chemotherapy

We consider the following initial conditions: ${C}_{0}=0.18$, ${H}_{0}=0.01$, ${R}_{0}=0.48$ and $Z_{0}=0$. These initial conditions are in the limit cycle region of the model solution. The periodic behaviour implies that the tumor levels oscillate around a fixed point, a clinically observed behaviour known as Jeff’s phenomenon. One cancer cell in the model (1) is equal to $66\times 10^{-9}$ in the dimensionless model. Our main aim is to find parameters that make the chemotherapy successfully suppress cancer, but that preserves the lymphocytes population.

Figure 1 displays the time evolution of the dimensionless quantities and variables. Figure 1(a) shows the behaviour of the infusion rate of chemotherapy. For $\Delta=0.02$ [Fig. 1(b)] there is no cancerous suppression and the system presents stable oscillatory behaviour. If we increase the value of $\Delta$ to 0.025 [Fig. 1(c)], the system may present cancerous suppression without the disappearance of lymphocytes. However, for larger $\Delta$, not only the cancer cells, but also lymphocytes disappear. To obtain a global picture of the parameters leading to different behaviour of our model, we construct the parameter space shown in Figure 2, for the parameters $\Delta$ and the predation coefficient of chemotherapy, $p_{1}$. For $p_{1}=0$ the rate of cancer cells proliferation is unaffected by the chemotherapic agent. This case may be interpreted as the use of inappropriated drugs or mechanisms related to drug resistance. When $p_{1}$ is not null, there is drug-induced killing of cancer cells. These parameters are important due to the fact that they are directly related to the chemotherapy used in the treatment of cancer. We can identify three behaviours in this parameter space. In the white region there are cancer cells. The suppression of cancer occurs for the parameters in the black region. The grey region presents an undesired situation, the suppression of lymphocytes. Therefore, we observe that it is possible to achieve cancerous suppression by increasing infusion rate of the chemotherapy to a high enough value, but the threshold depends on the value of $p_{1}$. For the parameters $\Delta$ and $p_{1}$ in the black region the number of malignant tumour cells goes to zero preserving the immune cells. When the number of cancer cells is zero, the treatment can stop and the tumour will not return.

To verify whether the suppression of cancer ($C=0$) is stable, that is, the cancer will not return after its elimination, we calculate the spectra of Lyapunov exponents of our model with time delay. We present only the values of the two largest Lyapunov exponents $\lambda_{i}(t)$ ($i=1,2$) at time $t$. We are interested in the maximal value of $\lambda_{i}(t)$. Firstly, we use a value of $\Delta=0.01$, in that the therapy does not eliminate the cancer. In this case, the system oscillates in a stable limit cycle (Fig. 3a). The largest Lyapunov exponent is about $0$ and the second largest negative. To determine that no cancer is an unstable solution of the model, we calculate the conditional Lyapunov exponents of the whole system but requiring the trajectory to lie in the subspace $C=0$. We obtain one positive Lyapunov exponent [20], which must be associated with the stability in this subspace since the Lyapunov exponents of the 3D reduced version of our model in (3), without the variable for $C$, are all negative. In other words, if $\Delta=0.01$, cancer will certainly not be eliminated.

For $\Delta=0.025$, not only the cancer can be eliminated, but also the dynamics in the subspace $C=0$ is stable. Figure 3(b) shows that the behaviour of the system is a limit cycle. The result of the two largest Lyapunov exponents for the case $\Delta=0.025$ is given in Figure 3(c). The maximum values $\lambda_{1}(t)$ and $\lambda_{2}(t)$, for $t>4400$, are 0.0015 and -0.0001. Notice that this is only an upper bound for the real value and it indicates that the real value, whatever it is, needs necessarily to be smaller than 0.0015, a small number that we interpret as being 0, as required for a limit cycle.

Figure 4(a) exhibits the drug injection pattern for the first protocol. When $\Delta=0.2$, $P=10$, and parameters in Table 3, the tumour cell population does not vanish, as depicted in Figure 4(b). However, considering the same period $P$, it is possible to obtain cancer suppression for $\Delta=0.3$ (Fig. 4c).

The drugs have specific protocols of application according to the type of tumour. For this reason we study the period of the drug injection. Figure 5 exhibits the time interval $P$ of the pulsed chemotherapy, where the points are used to denote the minimum value of the rate $\Delta$ in which the cancer suppression occurs. When $P$ increases, it is necessary to increase the intensity of the chemotherapy to obtain cancer suppression. As a matter of fact, the infusion rate versus the period shows a linear increase with the critical value of $\Delta$ as $P$ grows, $\Delta_{c}(P)\sim P$.

To verify the effect of the period on the behaviour of our model, we show the parameter space in Fig. 6, similar to the one shown in Figure 2, but considering pulsed chemotherapy with a period equal to 10. We also see the three behaviours: white regions represent parameters that lead to cancer, black regions represent parameters that lead to the suppression of cancer, and the grey region represent parameters that lead to the suppression of lymphocytes. Therefore, it is still possible to obtain a successful chemotherapy. However, the threshold of values of $p_{1}$ and $\Delta$ leading to a successful chemotherapy are larger in the pulsed chemotherapy with $P=10$, then these threshold values for a continuous chemotherapy. This is a realistic behaviour observed in treatments that prescribe chemotherapic drugs in a continuous or pulsed way.

Another case for the pulsed chemotherapy consists in infusions of drugs with different concentrations and periods. There is recent research about the successful rate of each type of pulsed chemotherapy [10]. For instance, the treatment for colon cancer adding oxaliplatin to bolus, fluorouracil mixed with leucovorin has been used with different infusion rates [10]. Due to clinical treatment described in the literature, we consider $\Delta$ to oscillate between two values.

Figure 7(a) shows the drug injection pattern with $\Delta_{1}=0.3$, $\Delta_{2}=0.1$, and $P=10$. For these values of $\Delta$ the cancer cells do not disappear but they oscillate at regular intervals (Fig. 7b). On the other hand, fixing $\Delta_{1}$ and increasing $\Delta_{2}$ the cancer is suppressed, as can be observed in Figure 7(c).

When the infusion rate is constant and chemotherapy sessions are periodically repeated, the cancer suppression depends on the time interval $P$. For this reason we fix $\Delta_{1}$ and vary $P$ and $\Delta_{2}$ that results in the successful treatment of cancer. As a result, we see in Figure 8, the lines represent the minimal values of $\Delta_{2}$ for a given $P$ that lead to cancer suppression. The circles are for $\Delta_{1}=0.212$ and the squares are for $\Delta_{1}=0.3$. Fixing $\Delta_{2}$ and varying $\Delta_{1}$, we obtain the same result.

Through the Figure 5 and Figure 8, we can see a linear relation between $\Delta$ and $P$. Defining the frequency as $f=1/P$, the infusion rate versus the frequency will show a linear decreases with the critical value of $\Delta$ as $f$ decreases, $\Delta_{c}(f)\sim f^{-1}$. This way, when tuning the frequency, the value of the rate $\Delta$ in that the cancer suppression occurs is altered.

The drug Diethylstilbestrol to prostate cancer can be used in treatment by continuous or pulsed chemotherapy. In a continuous treatment may be administered 50 mg (oral) per day, every day, while in a pulsed treatment this drug may be administered 500 mg (infusion) once per week, many weeks. Comparing, we can see that the value of the pulsed is ten times the value of the continuous. Then, the values that we used are in accordance with realistic values.

Michor and collaborators [12] analysed 169 chronic myeloid leukaemia patients, a cancer of the white blood cell, or leukocytes. They studied the dynamics of different treatment responses to tyrosine kinase inhibitor imatinib, which is also known as glivec. The imatinib is used to treat some types of leukaemia and soft tissue sarcoma. The treatment with this drug causes the death of cancer cells by inhibiting the signals exchanged by the cancer cells responsible to produce the growth and the division of the cancer cells.

In Ref. [12] was showed that a successful therapy using imatinib leads to a biphasis exponential decline of leukaemic cells in time. The value of the first slope (quantifying the exponential time decay rate of cancer cells) is approximately $-0.05$, and represents the death of leukaemic differentiated cells. The second slope is around $-0.008$, and is due to the death of leukaemic progenitors. If the imatinib therapy is interruped the slope is approximately $0.09$. Without imatinib the differentiated leukaemic cells arise from leukaemic stem cells. Figure 9 shows the slopes obtained through the model (2). We consider $P=10$, $\Delta=0.3$, and two values for $p_{1}$. We use for the time interval $0<t\leq 175$ that produces the first slope (black line) $p_{1}=0.001$, and for the time interval $175<t\leq 300$ that produces the second slope (blue line) $p_{1}=0.0009$. For $t>300$, time interval that produces the third slope (green line) there is not chemotherapy. As a result, we obtain the slopes $-0.06$, $-0.005$, and $0.08$, which are remarkably similar with the slopes obtained in Ref. [12].

In our model, the case without therapy can be simulated by setting the parameter $p_{i}$ equal to zero. This parameter can be also used to simulate a case where the cancer cells become resistante to the drug. The leukaemic cells may present resistance to imatinib therapy. Resistance can be modelled decreasing the effect of chemotherapy to cancer cells (i.e., decreasing $p_{1}$), while leaving the effect on immune cells constant (maintaining $p_{2}$ and $p_{3}$).