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An analysis of a mathematical model describing the growth of a tumor treated with chemotherapy

Anderson L. A. de Araujo
Anderson L. A. de Araujo
, 
Artur C. Fassoni
Artur C. Fassoni
 and 
Luís F. Salvino
Luís F. Salvino
   | Oct 28, 2020
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Published Online: Oct 28, 2020
Page range: 185 - 204
Received: Sep 04, 2019
Accepted: Feb 07, 2019
DOI: https://doi.org/10.2478/amns.2020.2.00037
Keywords
© 2020 Anderson L. A. de Araujo et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Results of Simulation 1, for model (1.1) within a two-dimensional domain Ω = [0, L] × [0, L] = [0, 1] × [0, 1]. Plots of model solutions N(x, y, t) (normal cells, blue, top row), A(x, y, t) (cancer cells, red, middle row) and D(x, y, t) (chemotherapeutic drug concentration, green, bottom row) at time points t = 0, 1, 15 (columns 1,2 and 3, respectively). See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced in the vicinity of the blood vessel, while the distant tumor cells persist along time (the shape of the solution at time t = 15 is stationary). Within the vicinity of the blood vessel, the removal of tumor cells allows the normal tissue to recover and grow.
Results of Simulation 1, for model (1.1) within a two-dimensional domain Ω = [0, L] × [0, L] = [0, 1] × [0, 1]. Plots of model solutions N(x, y, t) (normal cells, blue, top row), A(x, y, t) (cancer cells, red, middle row) and D(x, y, t) (chemotherapeutic drug concentration, green, bottom row) at time points t = 0, 1, 15 (columns 1,2 and 3, respectively). See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced in the vicinity of the blood vessel, while the distant tumor cells persist along time (the shape of the solution at time t = 15 is stationary). Within the vicinity of the blood vessel, the removal of tumor cells allows the normal tissue to recover and grow.

Fig. 2

Results of Simulation 2, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and extinct within a given distance from the blood vessel (x < 0.6), but not in the entire tissue (x > 0.6). Within the region of tumor extinction, the removal of tumor cells release the normal tissue to recover and grow.
Results of Simulation 2, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and extinct within a given distance from the blood vessel (x < 0.6), but not in the entire tissue (x > 0.6). Within the region of tumor extinction, the removal of tumor cells release the normal tissue to recover and grow.

Fig. 3

Results of Simulation 3, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug infusion rate μ is increased here. Within the entire tissue, the removal of tumor cells release the normal tissue to recover and grow.
Results of Simulation 3, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug infusion rate μ is increased here. Within the entire tissue, the removal of tumor cells release the normal tissue to recover and grow.

Fig. 4

Results of Simulation 4, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 25. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug diffusion coefficient σ is increased here.
Results of Simulation 4, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 25. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the drug diffusion coefficient σ is increased here.

Fig. 5

Results of Simulation 5, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the chemotherapy toxicity against tumor cells, αA, was increased.
Results of Simulation 5, with Ω = [0, L] = [0, 1]. Plots of model solutions A(x, t) (cancer cells, red), N(x, t) (normal cells, blue) and D(x, t) (chemotherapeutic drug concentration, green) at time points t = 0, 3, 6, 9, 12, 15. See Table 1 for parameter values used here. At time t = 0, the tumor is spread trough the tissue, and as chemotherapy is applied (t > 0), the tumor cells are reduced and in the entire tissue. In comparison with Simulation 2, the tumor extinction is reached because the chemotherapy toxicity against tumor cells, αA, was increased.

Set-up of different simulations an their biological outcomes. Each row indicates the numerical values used for the chemotherapeutic parameters αA (cytotoxicity), σ (diffusion coefficient), μ (infusion rate), and the position of ω ⊂ Ω ⊂ ℝ2. Simulation 1 was performed in a two-dimensional domain Ω = [0, 1] × [0, 1], while simulations 2–5 were performed in a one-dimensional domain Ω = [0, 1].

SimulationFigureOutcomeαAμσω
11tumor persistence530.1[0.45, 0.55] × [0.45, 0.55]
22tumor persistence1030.1[0, 0.1]
33tumor extinction1060.1[0, 0.1]
44tumor extinction1030.2[0, 0.1]
55tumor extinction2030.1[0, 0.1]
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