Figure 1.
Schematic diagram of the host tissue surrounding a tumor in response to cell invasion.
The tumor cells adhere to the surface of the ECM through integrins. Then, the tumor cells can invade into the ECM or surrounding tissue through the MMPs, which are ECM degradation enzymes secreted by the tumor cells [30], [31].
Table 1.
Reference variables used in the tumor model.
Table 2.
Parameters used in the tumor model.
Table 3.
The main parameters used in each simulation and the resulting calculated power exponent b values.
Figure 2.
Tumor invasion is unaffected by the microenvironment.
(a–d) The time evolution of the n, MMP, ECM, and glucose in a one-dimensional spatial length x at time (a) t = 0, (b) t = 1, (c) t = 10, and (d) t = 20. (e) The cell density n(x,t) when t ranges from 0 to 20. The inset shows t ranging from 0 to 0.08. (f) The curve fitting the half-width of cell density versus evolution time data, xh-t, b = 0.49618±0.00134. Simulation parameters: γf = γG = λ = β = λa = 0. The rests of the parameters were fixed. The ECM is degraded by the MMP; the simulated exponent b in this situation is almost 0.5, the typical Fickian diffusion.
Figure 3.
Tumor invasive diffusion with haptotaxis of the ECM.
(a–c) The time evolution of the n, MMP, ECM, and glucose in a one-dimensional spatial length x at (a) t = 0, (b) t = 0.1, and (c) t = 1. (d) A power function curve is fitted to the time and peak position of n, xm-t, b = 0.68564±0.00518. Simulation parameters: The haptotaxis coefficient γf = 0.01 and γG = λ = β = λa = 0. The rests of the parameters were fixed. The peak location of the n curve shifts considerably with the evolution time when haptotaxis of the ECM is included and the exponent b is larger than 0.5, signifying occurrence of superdiffusion.
Figure 4.
The simulated exponent b with a variety of haptotaxis coefficients γf.
The b increases with the γf, indicating that a tumor's invading is enhanced by haptotaxis of the ECM. Simulation parameters: γG = λ = β = λa = 0. The rests of the parameters were fixed.
Figure 5.
Tumor invasive diffusion with heterogeneity of the ECM.
(a–b) The time evolution of the n, MMP, and ECM in a one-dimensional spatial length x when the ECM is heterogeneous, at times (a) t = 0 and (b) t = 1. The inset shows the enlarging curves of the n and MMP at a different time. (c) The curve fitting the time and peak position of n data, xm-t, at different values of A. The exponent b decreases with the heterogeneity of the ECM. Simulating parameters: P2 in Table 3, where γf = 0.01 and γG = λ = β = λa = 0. The rests of the parameters were fixed.
Figure 6.
The simulated exponent b with a variety of chemotaxis coefficients γG.
The b increases with the γG, implying that a tumor's invading is promoted by glucose chemotaxis. Simulation parameters: γf = 0.01, λ = 1.0, β = 2.2, and λa = 0. The rests of the parameters were fixed.
Figure 7.
The time evolution of the n at different times.
The inset is the curve fit to the time and peak position of the cell density data, xm-t, b = 0.40973±0.00382, where λa = 0.003. Parameters: γf = γG = λ = β = 0, the rests of the parameters were fixed. The exponent b is smaller than 0.5, signifying occurrence of subdiffusion.
Figure 8.
The images are taken from in vitro culturing of cells from the breast line MDA-MB-231.
Upper row: MDA-MB-231 cells without a matrix on the (a) 1st day, (b) 2nd day, (c) 3rd day, and (d) 4th day. Bottom row: MDA-MB-231 cells with a matrix on the (e) 1st day, (f) 2nd day, (g) 3rd day, and (h) 4th day. All cells were maintained in DMEM 10 µg/ml gentamycin at 37°C in 5% CO2, the matrix was derived from murine EHS sarcoma cells and collagen. (a–h) reprinted figures with permission from Trevigen Inc (Cultrex Catalog #: 3500-096-K).
Figure 9.
The average radius of MDA-MB-231 cancer cell cultures versus culture time.
(a) MDA-MB-231 cells without a matrix and (b) MDA-MB-231 cells with a matrix. The exponent b is larger than 0.5 for the cells with a matrix and smaller than 0.5 for the cells without a matrix.
Table 4.
The measured value of the exponent b for different in vitro tumor cells with and without the ECM.
Figure 10.
The relationship between tumor size (r) and time (t) in clinical tumors.
(a) the adrenal tumor, (b) liver tumor A, and (c) liver tumor B. The exponent b of the adrenal tumor is smaller than 0.5; the b is between 0.5 and 1.0 for liver tumor A and larger than 1.0 for liver tumor B.
Table 5.
The calculated values of the exponent b for three tumor patients.
Figure 11.
The border fractal dimensions of the in vitro cultured MDA-MB-231 cells.
The border fractal dimension of the in vitro cells cultured with a matrix is larger than that of the cells cultured without a matrix.
Figure 12.
The border fractal dimensions of the clinical tumors.
(a) The border fractal dimensions of the adrenal tumor. (b) The border fractal dimensions of the liver tumors A and B, analyzed using clinical medical images. The border fractal dimension remains at a low level for the adrenal tumor and maintains a high level for the infiltrative liver tumor A. The liver tumor B is clinically diagnosed as a metastatic tumor that has separated from its primary surrounding host tissue.