Analytical Models of Intra- and Extratumoral Cell Interactions at Avascular Stage of Growth in the Presence of Targeted Chemotherapy
Abstract
:1. Introduction
2. Mathematical Model and Methodology
2.1. Model Assumptions
- Chemotherapy and innate immune responses decrease proliferation.
- All living cells receive nutrients (consisting of glucose and oxygen) from the underlying tissue, and divide depending on the level of nutrient supply.
- Chemotherapeutic drugs attack proliferating cells, surrounding healthy cells and immune cells.
- One cell population limits the movement of the cell population of another type and vice versa—this phenomenon is called ‘contact inhibition of migration’ [6].
- The effectiveness of the nutrient source term decreases with overall cell density.
- We assume that the nutrients, the immune response, and the drug react, and diffuse over the spatial domain.
- Nutrients diffuse into the tumor space at a diffusion rate that allows for the concentration of nutrient supply to reach a steady state.
- Immune cells are generated through a steady influx into the tumor area, and proliferate within it.
2.2. Modelling Equations
3. Model Results and Validation
3.1. Chemotherapy Effect on Malignant Proliferation and Tumor Regression
3.2. The Immunity Power Effect on Tumor Evolvement under Chemotherapy
3.3. Chemotherapy’s Effect on the Surrounding Healthy Tissue and Immune System
3.4. Model Validation
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
P | Proliferating cell population density, (×109 to give units cells/cm3) |
Q | Quiescent cell population density, (×109 to give units cells/cm3) |
N | Necrotic cell population density, (×109 to give units cells/cm3) |
S | Surrounding tissue cell population density, (×109 to give units cells/cm3) |
C | Nutrient supply intake |
L | Immune cell population density, (×109 to give units cells/cm3) |
D | Chemotherapy drug intake |
kDP | |
kDQ | |
kDS | |
kDL | |
DC | |
DL | |
DD | |
C0 | Nutrient concentration in the absence of abnormal proliferation |
vL | |
k1 | |
k2 | |
k3 | |
k4 | |
k5 | |
k6 | |
kD | |
α | Nutrient coefficient |
Γ | Dimensionless parameter in Equation (4) |
f | Functional form representing the rate of quiescent cells turning to necrotic |
h | Functional form representing the rate of proliferating cells turning quiescent |
g | Functional form representing the mitosis rate of proliferating cells |
vD(t) |
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Parameter | Generalized Tumor Cells | Cell Populations of a Tumor | Surrounding Healthy Tissue Cells | Innate Immune Cells (Varied) | Chemotherapeutic Drug Effect | Nutrients | Immunotherapy Effect | Other | ||
---|---|---|---|---|---|---|---|---|---|---|
Author | Proliferating Cells | Quiescent Cells | Necrotic Cells | |||||||
This study | ⁕ | ⁕ | ⁕ | ⁕ | ⁕ | ⁕ | ⁕ | |||
[4] | ⁕ | ⁕ | ⁕ | |||||||
[19] | ⁕ | ⁕ | ⁕ | |||||||
[17] | ⁕ | ⁕ | ⁕ | ⁕ | ⁕ | |||||
[7] | ⁕ | ⁕ | ⁕ | ⁕ | ||||||
[9] | ⁕ | ⁕ | ⁕ | ⁕ | ⁕ | |||||
[12] | ⁕ | ⁕ | ⁕ | |||||||
[3] | ⁕ | ⁕ | ||||||||
[18] | ⁕ | ⁕ | ⁕ | ⁕ | ⁕ | |||||
[16] | ⁕ | ⁕ | ⁕ | |||||||
[5] | ⁕ | ⁕ | ⁕ | ⁕ | ||||||
[20] | ⁕ | ⁕ | ⁕ | |||||||
[14] | ⁕ | ⁕ | ⁕ | ⁕ | ||||||
[6] | ⁕ | ⁕ | ⁕ | ⁕ | ⁕ | |||||
[10] | ⁕ | ⁕ | ⁕ | ⁕ | ||||||
[11] | ⁕ | ⁕ | ⁕ | |||||||
[13] | ⁕ | ⁕ | ⁕ | ⁕ | ||||||
[15] | ⁕ | ⁕ | ⁕ | |||||||
[21] | ⁕ | ⁕ | ⁕ | |||||||
[8] | ⁕ | ⁕ | ⁕ | ⁕ |
Parameter | Value | Unit | Reference | Description |
---|---|---|---|---|
0.3 | [8] | Fraction cell kill rate of proliferating cells under chemotherapeutic drug effect (value used for generalized tumor cells) | ||
0.1 | [8] | Fraction cell kill rate of quiescent cells under chemotherapeutic drug effect (assumed the same as for healthy cells) | ||
0.1 | [8] | Fraction cell kill rate of healthy surrounding tissue cells under chemotherapeutic drug effect | ||
0.2 | [8] | Fraction cell kill rate of immune system cells under chemotherapeutic drug effect | ||
10 | [6] | Diffusion coefficient of the nutrient | ||
5 | estimated | Diffusion coefficient of the immune system cells | ||
8 | estimated | Diffusion coefficient of a chemotherapeutic drug | ||
1 | dimensionless | [6] | Nutrient concentration in the absence of abnormal proliferation | |
0.2 | [22] | Rate of the external immune cell influx | ||
8 | [6] | Decay rate of the nutrient | ||
1 | [6] | Rate of the nutrient consumption by proliferating cells | ||
1 | [6] | Rate of the nutrient consumption by quiescent cells | ||
1 | [8] | Rate of immune cells death after contacting proliferating cells | ||
0.55 | [8] | Rate of proliferating cells death after contacting immune cells | ||
1 | estimated | Rate of the nutrient consumption by immune cells | ||
0.2 | [23] | Decay rate of a chemotherapeutic drug | ||
α | 0.8 | dimensionless | [6] | Nutrient coefficient |
Γ | 0.4 | dimensionless | [6] | Dimensionless parameter |
Parameter | Present Model | Lit. Model [24] | Lit. Model [25] | Experiment [26] |
---|---|---|---|---|
Density of proliferating cells (before chemotherapy) | 0.145 × 109 cells/cm3 | 0.20 × 109 cells/cm3 (normoxic tumor cells) | 0.15 × 109 cells/cm3 (normalized density of cancer cells) | |
Density of necrotic cells | 0.75 × 109 cells/cm3 | 0.95 × 109 cells/cm3 (necrotic tissue) | ||
Density of proliferating cells (after chemotherapy) | 0.13 × 109 cells/cm3 | 0.1 × 109 cells/cm3 (normoxic tumor cells) |
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Lavrenteva, E.; Theodoropoulos, C.; Binns, M. Analytical Models of Intra- and Extratumoral Cell Interactions at Avascular Stage of Growth in the Presence of Targeted Chemotherapy. Bioengineering 2023, 10, 385. https://doi.org/10.3390/bioengineering10030385
Lavrenteva E, Theodoropoulos C, Binns M. Analytical Models of Intra- and Extratumoral Cell Interactions at Avascular Stage of Growth in the Presence of Targeted Chemotherapy. Bioengineering. 2023; 10(3):385. https://doi.org/10.3390/bioengineering10030385
Chicago/Turabian StyleLavrenteva, Evgeniia, Constantinos Theodoropoulos, and Michael Binns. 2023. "Analytical Models of Intra- and Extratumoral Cell Interactions at Avascular Stage of Growth in the Presence of Targeted Chemotherapy" Bioengineering 10, no. 3: 385. https://doi.org/10.3390/bioengineering10030385
APA StyleLavrenteva, E., Theodoropoulos, C., & Binns, M. (2023). Analytical Models of Intra- and Extratumoral Cell Interactions at Avascular Stage of Growth in the Presence of Targeted Chemotherapy. Bioengineering, 10(3), 385. https://doi.org/10.3390/bioengineering10030385