Fractal dimension and universality in avascular tumor growth

For years, the comprehension of the tumor growth process has been intriguing scientists. New research has been constantly required to better understand the complexity of this phenomenon. In this paper, we propose a mathematical model that describes the properties, already known empirically, of avascular tumor growth. We present, from an individual-level (microscopic) framework, an explanation of some phenomenological (macroscopic) aspects of tumors, such as their spatial form and the way they develop. Our approach is based on competitive interaction between the cells. This simple rule makes the model able to reproduce evidence observed in real tumors, such as exponential growth in their early stage followed by power-law growth. The model also reproduces (i) the fractal-space distribution of tumor cells and (ii) the universal growth behavior observed in both animals and tumors. Our analyses suggest that the universal similarity between tumor and animal growth comes from the fact that both can be described by the same dynamic equation—the Bertalanffy-Richards model—even if they do not necessarily share the same biological properties.

Figure 1

Growth of two types of tumors, (a) EMT6 and (b) KHJJ, implanted in mice and rats. These data show the usual growth behavior in tumors: an exponential rise in the first stage followed by power-law growth. Source: Ref. [40].

Figure 2

The universal growth law of the dimensionless mass ($\mu$) as a function of the dimensionless time ($\tau$). Data from completely different organisms collapse in the same universal curve: animals, tumors, and data from the model proposed (see Sec. 6), regardless of the parameter values. The information about tumor and animal growth was extracted from Refs. [40] and [8], respectively.

Figure 3

Schematic of cellular dynamics. In a competitive environment, each cell tries to move in a direction that minimizes the influence (competitive field) of other cells.

Figure 4

(a) Plot of the field $I(N,D_f)$, given by Eq. (6), as a function of the fractal dimension. The plot was made for fixed values of population size and using $\gamma =5$. Black circles represent the minimum value of $I(N,D_f)$ in relation to the fractal dimension, which leads to $D_f^{\mathrm{opt}}$. When the population is small, the plot suggests that the population tends to be compact, that is, $D_f^{\mathrm{opt}}=3.$ However, when the population becomes sufficiently large, the optimal dimension is fractal. (b) Plot of the fractal dimension which minimizes the interaction field ($D_f^{\mathrm{opt}}$) as a function of the population size. The optimal fractal dimension decreases monotonically as the population increases, and it becomes slower for a sufficiently large population.

Figure 5

Temporal evolution for (a) mass, (b) replication rate, and (c) optimal fractal dimension given by our microscopic model. We used different values of the intrinsic replication rate $k$, and we kept fixed the parameters $J=0.1, \gamma =5$, and $m_c=1$. The curves were obtained by Eq. (11). When $k>k_{\mathrm{power}}$ the population (or its mass) diverges, when $k=k_{\mathrm{power}}$ the population grows according to a power law, and when $k<k_{\mathrm{power}}$ the population saturates. If $k\sim k_{\mathrm{power}}$, the population remains in a power-law growth regime for a long period. The population grows in a power-law regime because the replication rate $R_i$ also decreases as a power law. The optimal dimension of the spatial structure formed by the population is 3 (compacted form) in the initial stage, but then it becomes fractal when the population increases. This optimal fractal dimension saturates to what we call $D_f^{\mathrm{conv}}$.

Figure 6

Predictions of the microscopic model of the population growth (its mass), using two approaches: (i) $D_f$ evolves over time, represented by dots; and (ii) the (fractal) dimension is fixed, represented by the red line (special case of the model, when $D_f=D_f^{\mathrm{conv}}$). In both cases $J=0.1$ and $\gamma =5$. The straight dashed line shows a power-law behavior as a guide for the eyes. The red line is slightly different from the dotted curve during the growth process, but they converge to the same saturation value. Inset: Fractal dimension of the population for these two approaches. For the dotted line, the population starts with $D_f^{\mathrm{opt}}=D=3$ (compacted form) but shortly becomes fractal and converges to a saturation value, $D_f^{\mathrm{opt}}\to D_f^{\mathrm{conv}}$.