Role of the Interplay Between the Internal and External Conditions in Invasive Behavior of Tumors

Tumor growth, which plays a central role in cancer evolution, depends on both the internal features of the cells, such as their ability for unlimited duplication, and the external conditions, e.g., supply of nutrients, as well as the dynamic interactions between the two. A stem cell theory of cancer has recently been developed that suggests the existence of a subpopulation of self-renewing tumor cells to be responsible for tumorigenesis, and is able to initiate metastatic spreading. The question of abundance of the cancer stem cells (CSCs) and its relation to tumor malignancy has, however, remained an unsolved problem and has been a subject of recent debates. In this paper we propose a novel model beyond the standard stochastic models of tumor development, in order to explore the effect of the density of the CSCs and oxygen on the tumor’s invasive behavior. The model identifies natural selection as the underlying process for complex morphology of tumors, which has been observed experimentally, and indicates that their invasive behavior depends on both the number of the CSCs and the oxygen density in the microenvironment. The interplay between the external and internal conditions may pave the way for a new cancer therapy.

It appears that such requirements are available for some internal cells, as well as those on the border. Active cells thickness reaches 3 mm in some area. The unit of proliferation activity at each site is the number of devisions in the previous 96 hours at that site, which can be a noninteger number because we consider proliferation as a continuous process.
The nutrient concentration is held fixed in the medium, and a CSC is placed at the central plaquette in the lattice. A lattice site is then randomly chosen and the governing equation, Eq. (4), for the diffusion and consumption of the nutrient for the living cells (if any) are first numerically solved. If there exist living cells at the chosen site (or plaquette), Eq. (1) for the internal energy is also solved numerically. If the internal energy of the cell exceeds the threshold u p , then the cell will proliferate according to the rules summarized in Fig. 3 of the text. In the case of duplication of a CSC, Eq. (2) is numerically solved; otherwise, the CSC concentration is only governed by the first term of Eq. (2). After the first round of the proliferation develops, the first generation of the cancerous cells is produced [the second term in Eq. (3)]. Each "time step" is defined by the completion of 201 × 201 (the total number of sites of the lattice) trials, i.e. every site on average has the chance to be updated at least once. By increasing the internal energy of the cancerous cells, the next generations may emerge, leading ultimately to the production of the dead cells. Repeating the procedure over time, a tumor forms whose morphological properties are the main subject of the paper. The main parameters of the model are, (i) p s , the cancerous stem cell proliferation rate, and (ii) n, the nutrient density in the medium. The competition between the two parameters, representing the internal and external factors, can lead to completely different structures for the tumors, an issue that was addressed in the past. We believe that this might pave the way to devise an experiment to test the CSC hypothesis, as well as help controlling the tumor growth by controlling the external parameters.
Note that the main idea of the paper is based on many experimental observations in which a qualitative relation between the complexity of the morphology of tumors and their malignancy was reported. In this paper we have quantified the relation in terms of two possible main parameters that may play major roles in the tumor growth.
As emphasized in the paper, proliferation is not confined to a tumor's border, as previously was suggested [1, 2]. About 200 layers of cells on the border contribute to proliferative activity that can barely be considered as "surface" growth.
The morphology of the tumors under the effect of various number of the CSCs and nutrient availability: In contrast to the previous studies [3][4][5][6][7], internal features of the tumors (in this case the number of the CSCs) can increase tumor malignancy. In addition to the fractal analysis, Figure 9 depicts the malignancy of tumors as a result of both the internal features and the external environment of the tumor.
Malignancy may be the result of distinct conditions. We chose the tumors with fractal dimension D f ≈ 1.82 as malignant. This type of tumors can arise by various values of p s and n; see Fig. 10.
Circularity is another method for classifying irregular shapes based on their space-filling feature. It is a measure of how close a shape is to a circle, and is defined by Circularity(r) = area which is filled by shape between r and r + dr area of ring wtih radius equal to r with thickness of dr Based on the previous studies, tumors with larger values of the probability p s should have more regular shapes than those with lower values [8]. Our results indicate, however, a completely different behavior. In addition to the fractal analysis, circularity indicates that irregularities increase for each tumor during its growth and for tumors with the same area, those with larger p s have more irregular shapes. Circularity in tumors of different sizes, as well as for different tumors but with the same mass, are shown in Fig. 13 We also present the cells' distribution in various tumors under various conditions. To test the various assumptions of the model and their efect on the main results, we carried out extensive simulations with various scenarios. In what follows we describe the results.
(a) Clearly, alternative boundary conditions and oxygen supply systems can be considered, and the model is fully capable of adapting them. Thus, we carried out simulations in which we varied the density of oxygen at the perimeter of the circle with a radius of 1 cm. Figure 8 shows that the main results for the relation between D f and p s are preserved.
(b) Regarding the radius of the circles (see the text of the paper), we varied the size of the medium in which the tumor grows. The results are shown in Fig. 9, indicating that they do not depend on radius of the circle.
(c) As for the structure of the oxygen supply system, We carried out simulations in which the same disc (circle) was considered, but instead of supplying the oxygen from the outside of the disc, we used a lattice of vessels that were separated by 0.2 mm and supplied the oxygen to the tumor. Then, the density of oxygen at such units was updated to 1 with a fixed certain rate. Equal numbers of such units with a random spatial distribution in the disc were used. We also simulated the case in which each unit directly acquires its oxygen. In all the cases, other rules for  nutrient evolution, such as oxygen diffusion, remained unchanged. But, as Fig. 10 shows, the main results remained unchanged.
(d) In the main simulations we used the reported value of oxygen diffusion cofficient β [3, 10], but the results will not change by lower values of β. Figure 11 shows that our results do not depend on β.
(e) The mobility of different kinds of cells is not the same, of course. We assumed the diffusion coefficient of the cells to be the same, but as Fig. 12 indicates, the main results do not depend on the differences between the diffusion coefficients of various cells.
(f) Regarding the oxygen consumption rate, α: the CSCs and CCs are assumed to have the same rate of oxygen consumption, but when we changed the rates for every kind of cell, the results remained unchanged, as Fig. 13 demonstrates it.
(g) The CSCs and CCs are assumed to have the same internal energy threshold u p for duplication, and equal rates of crossing the S, G 2 and M phases in the cell cycle, R m . But, changing the proliferation activity of the cells does (h) We assume that the dead cells remain inactive in the medium. But, even if we eliminate them after their death, the main results would be unchanged. This is shown in Fig. 15.