Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity
Abstract
:1. Introduction
2. Model
2.1. Equations for Tumor Growth
2.1.1. Dynamics of Cells and Necrotic Tissue
2.1.2. Dynamics of Nutrients
2.1.3. Numerical Solving of Tumor Growth Model
2.2. Equations for Radiotherapy
2.3. Optimization of Radiotherapy Fractionation
Algorithm 1: Optimization of dose fractionation for radiotherapy. |
2.4. Parameters
3. Results
3.1. Simulation of Tumor Growth and Radiotherapy
3.2. Optimization of Radiotherapy Fractionation
3.3. Efficiency of the Optimization Algorithm
4. Discussion
Supplementary Materials
Author Contributions
Funding
Conflicts of Interest
Abbreviations
RT | radiotherapy |
PDE | partial differential equation |
OER | oxygen enhancement ratio |
TCP | tumor cure probability |
HM | high malignant |
IM | intermediate malignant |
LM | low malignant |
Appendix A. Choice of Discretization
References
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Parameter | Description | Model Value | Based on |
---|---|---|---|
Cells: | |||
B | tumor cells’ proliferation rate | HM: 0.01 | [36] + see the text |
IM: 0.005 | |||
LM: 0.0025 | |||
M | normal cells’ death rate parameter | 0.01 | [41] |
ratio of death rates of tumor and normal cells | HM: 0.3 | [41] + see the text | |
due to the lack of oxygen | IM: 0.7 | ||
LM: 1 | |||
tumor cells’ motility | HM: 0.01 | [42] + see the text | |
IM: 0.001 | |||
LM: 0 | |||
Nutrients: | |||
glucose inflow parameter | HM: 20 | [32] | |
IM: 10 | |||
LM: 4 | |||
tumor cells’ glucose consumption rate | HM: 12 | [36] + see the text | |
IM: 6 | |||
LM: 3 | |||
normal cells’ glucose consumption rate | 0.3 | [43] | |
Michaelis constant for glucose consumption rate | 0.007 | [44] | |
glucose diffusion coefficient | 100 | [45] | |
oxygen inflow parameter | HM: 50.8 | [46] + see the text | |
IM: 35.8 | |||
LM: 25.4 | |||
oxygen concentration in artery | 5.87 | [47] | |
oxygen concentration, at which | 1.56 | [48] | |
hemoglobin saturation is 50% | |||
Hill coefficient | 2.55 | [48] | |
for oxygen-hemoglobin dissociation curve | |||
tumor cells’ oxygen consumption rate | HM: 63 | [36] + see the text | |
IM: 31.5 | |||
LM: 15.75 | |||
normal cells’ oxygen consumption rate | 8 | [43] | |
Michaelis constant for oxygen consumption rate | 0.005 | [44] | |
oxygen diffusion coefficient | 720 | [49] | |
Radiotherapy: | |||
tumor cells’ linear radiosensitivity parameter | 0.07–0.21 | see the text | |
tumor cells’ quadratic radiosensitivity parameter | see the text | ||
maximum under aerobic conditions | 2.5 | [39] | |
maximum under aerobic conditions | 3 | [39] | |
Michaelis constant for oxygen enhancement effect | 0.193 | [39] | |
k | ratio of radiosensitivity of quiescent | HM: 1 | see the text |
and proliferating tumor cells | IM: 0.5 | ||
LM: 0.2 | |||
Optimization procedure: | |||
alpha-beta ratio for normal tissue | 3 | [3] | |
maximum fractional dose | 5 | [11] | |
the amount of radiation dose added to each fraction | 0.2 | see the text | |
during the search for the “gradient” | |||
the coefficient of fractions alteration during the “descent” | 4 | see the text | |
minimum parameter of fractions alteration | 0.001 | see the text | |
during the “descent” | |||
the threshold coefficient for determining | 0.98 | see the text | |
the second stage of the scheme |
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Kuznetsov, M.; Kolobov, A. Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity. Mathematics 2020, 8, 1204. https://doi.org/10.3390/math8081204
Kuznetsov M, Kolobov A. Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity. Mathematics. 2020; 8(8):1204. https://doi.org/10.3390/math8081204
Chicago/Turabian StyleKuznetsov, Maxim, and Andrey Kolobov. 2020. "Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity" Mathematics 8, no. 8: 1204. https://doi.org/10.3390/math8081204