Optimization of Size of Nanosensitizers for Antitumor Radiotherapy Using Mathematical Modeling
Abstract
:1. Introduction
2. Results
2.1. Mathematical Model
2.2. Free Growth and Irradiation of Tumor
2.3. Injection of Nanoparticles
2.4. Optimization of Nanoparticle Size
3. Discussion
4. Materials and Methods
4.1. Model Equations
proliferating tumor cells: | ||
quiescent tumor cells: | ||
normal cells: | ||
damaged cells: | ||
interstitial fluid: | ||
VEGF: | ||
normal capillaries: | ||
abnormal capillaries: | ||
glucose: | (1) | |
free nanoparticles: | ||
bound nanoparticles: | ||
nanoparticles in blood: | ||
solid stress: | ||
irradiation: | ||
where | ||
Diffusion in tissue: clearance from blood: | |
fraction of available pore cross-section area: | |
diffusive permeability: | |
(3) | |
where | |
4.2. Parameters
4.3. Numerical Solving
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Parameter | Description | Value | Based on |
---|---|---|---|
Cells: | |||
B | maximum rate of cell proliferation | 0.01 | [26] |
critical stress for cell proliferation | 15 | [22] | |
smoothing parameter of Heaviside function | 500 | see text | |
M | the rate of death of damaged cells | 0.01 | see text |
Stress: | |||
k | solid stress coefficient | 500 | see text |
minimum fraction of interacting cells | 0.3 | [27] | |
initial fraction of cells | 0.8 | [27] | |
Interstitial fluid: | |||
hydraulic conductivity of normal capillaries | 0.1 | [13] | |
hydraulic conductivity of abnormal capillaries | 0.22 | see text | |
fluid pressure in capillaries | 4 | [13] | |
hydraulic conductivity of lymphatic capillaries | 1300 | [13] | |
lymph pressure | 0 | [13] | |
K | tissue hydraulic conductivity | 0.1 | [28] |
VEGF: | |||
secretion rate | 1 | [29] | |
internalization rate | 1 | [30] | |
degradation rate | 0.01 | [31] | |
diffusion coefficient | 21 | [31] | |
Capillaries: | |||
R | maximum rate of angiogenesis | 0.008 | [32] |
maximum surface area density | 5 | [32] | |
characteristic degradation rate | 0.03 | [6,32] | |
coefficient of degradation in the tumor core | 2 | [6,32] | |
normalization rate | 0.1 | [33] | |
denormalization rate | 0.1 | [33] | |
Michaelis constant for VEGF action | 0.001 | see text | |
coefficient of active movement | 0.03 | [6,32] | |
X | maximum pore radius | 100 | see text |
Glucose: | |||
Michaelis constant for consumption | 0.01 | [34] | |
permeability of normal capillaries | 4 | [35] | |
permeability of abnormal capillaries | 10 | [18] | |
parameter of consumption by proliferating cells | 1200 | [26] | |
rate of consumption by normal tissue | 0.5 | [36] | |
diffusion coefficient | 100 | [37] | |
hydrodynamic radius | 0.36 | [38] | |
Irradiation: | |||
linear parameter of cell radiosensitivity | 0.1 | see text | |
quadratic parameter of cell radiosensitivity | 0.01 | [4] | |
coefficients for proliferating and quiescent cells | 1, 0.2 | [39] | |
D | irradiation dose | 2 | see text |
factor of dose enhancement by radiosensitizer | 100 | see text | |
Nanoparticles: | |||
coefficient of binding with tumor cells | 0.5 | see text | |
width of polymer coating | 7 | see text | |
parameter for diffusion coefficient | 65 | [18] | |
parameter for clearance rate | 0.003 | see text |
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Kuznetsov, M.; Kolobov, A. Optimization of Size of Nanosensitizers for Antitumor Radiotherapy Using Mathematical Modeling. Int. J. Mol. Sci. 2023, 24, 11806. https://doi.org/10.3390/ijms241411806
Kuznetsov M, Kolobov A. Optimization of Size of Nanosensitizers for Antitumor Radiotherapy Using Mathematical Modeling. International Journal of Molecular Sciences. 2023; 24(14):11806. https://doi.org/10.3390/ijms241411806
Chicago/Turabian StyleKuznetsov, Maxim, and Andrey Kolobov. 2023. "Optimization of Size of Nanosensitizers for Antitumor Radiotherapy Using Mathematical Modeling" International Journal of Molecular Sciences 24, no. 14: 11806. https://doi.org/10.3390/ijms241411806