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A mathematical model of cancer treatment by radiotherapy.

Liu Z, Yang C - Comput Math Methods Med (2014)

Bottom Line: A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper.Some numerical examples are shown to verify the validity of the results.A discussion is presented for further study.

View Article: PubMed Central - PubMed

Affiliation: School of Science, Chongqing Jiaotong University, Chongqing 400074, China ; Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China.

ABSTRACT
A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shown to verify the validity of the results. A discussion is presented for further study.

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(a) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.75. (b) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.8.
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fig7: (a) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.75. (b) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.8.

Mentions: If we take γ = 0.75, it is easy to verify that ɛγL < α1ω and system (1) has a cancer eradication periodic solution. See Figure 7(a). From Figure 7(a) we obtain σ1 = min⁡t∈[0,ω]{x1*(t)}⩾0.43; then condition (26) is satisfied. According to Theorem 10, system (1) has a unique globally asymptotically stable cancer eradication 10-periodic solution. See Figures 8(a) and 8(b). If γ is taken as 0.8, condition ɛγL < α1ω is also satisfied, then system (1) has a cancer eradication periodic solution. See Figure 7(b). From Figure 7(b) we obtain σ1 = min⁡t∈[0,ω]{x1*(t)}⩾0.42, then it is easy to verify that condition (26) is satisfied. According to Theorem 10, system (1) has a unique globally asymptotically stable cancer eradication 10-periodic solution. See Figures 9(a) and 9(b). It can be seen from Figures 5(a) and 5(b) (γ = 0.65), Figures 8(a) and 8(b) (γ = 0.75), and Figures 9(a) and 9(b) (γ = 0.8) that under the circumstance that the system has a cancer eradication periodic solution and all the other conditions are not changed, the increase of the radiation dosage will quicken the extinction of the cancer cells but will also decrease the concentration of the healthy cells at the same time. One can refer to the three figures; in Figure 5 with γ = 0.65, 0.1 ⩽ x1, in Figure 8 with γ = 0.75, 0.05 ⩽ x1 ⩽ 0.1, and in Figure 9 with γ = 0.8, x1 ⩽ 0.05.


A mathematical model of cancer treatment by radiotherapy.

Liu Z, Yang C - Comput Math Methods Med (2014)

(a) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.75. (b) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.8.
© Copyright Policy - open-access
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC4247922&req=5

fig7: (a) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.75. (b) The time series for the cancer eradication periodic solution x1*(t) with L = 8 and γ = 0.8.
Mentions: If we take γ = 0.75, it is easy to verify that ɛγL < α1ω and system (1) has a cancer eradication periodic solution. See Figure 7(a). From Figure 7(a) we obtain σ1 = min⁡t∈[0,ω]{x1*(t)}⩾0.43; then condition (26) is satisfied. According to Theorem 10, system (1) has a unique globally asymptotically stable cancer eradication 10-periodic solution. See Figures 8(a) and 8(b). If γ is taken as 0.8, condition ɛγL < α1ω is also satisfied, then system (1) has a cancer eradication periodic solution. See Figure 7(b). From Figure 7(b) we obtain σ1 = min⁡t∈[0,ω]{x1*(t)}⩾0.42, then it is easy to verify that condition (26) is satisfied. According to Theorem 10, system (1) has a unique globally asymptotically stable cancer eradication 10-periodic solution. See Figures 9(a) and 9(b). It can be seen from Figures 5(a) and 5(b) (γ = 0.65), Figures 8(a) and 8(b) (γ = 0.75), and Figures 9(a) and 9(b) (γ = 0.8) that under the circumstance that the system has a cancer eradication periodic solution and all the other conditions are not changed, the increase of the radiation dosage will quicken the extinction of the cancer cells but will also decrease the concentration of the healthy cells at the same time. One can refer to the three figures; in Figure 5 with γ = 0.65, 0.1 ⩽ x1, in Figure 8 with γ = 0.75, 0.05 ⩽ x1 ⩽ 0.1, and in Figure 9 with γ = 0.8, x1 ⩽ 0.05.

Bottom Line: A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper.Some numerical examples are shown to verify the validity of the results.A discussion is presented for further study.

View Article: PubMed Central - PubMed

Affiliation: School of Science, Chongqing Jiaotong University, Chongqing 400074, China ; Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China.

ABSTRACT
A periodic mathematical model of cancer treatment by radiotherapy is presented and studied in this paper. Conditions on the coexistence of the healthy and cancer cells are obtained. Furthermore, sufficient conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution, and the cancer win periodic solution are established. Some numerical examples are shown to verify the validity of the results. A discussion is presented for further study.

Show MeSH
Related in: MedlinePlus