Conceived and designed the experiments: ST. Performed the experiments: YJ ST. Analyzed the data: YJ ST. Contributed reagents/materials/analysis tools: YJ ST. Wrote the paper: YJ ST.
The authors have declared that no competing interests exist.
Understanding tumor invasion and metastasis is of crucial importance for both fundamental cancer research and clinical practice.
The goal of the present work is to develop an efficient singlecell based cellular automaton (CA) model that enables one to investigate the growth dynamics and morphology of invasive solid tumors. Recent experiments have shown that highly malignant tumors develop dendritic branches composed of tumor cells that follow each other, which massively invade into the host microenvironment and ultimately lead to cancer metastasis. Previous theoretical/computational cancer modeling neither addressed the question of how such chainlike invasive branches form nor how they interact with the host microenvironment and the primary tumor. Our CA model, which incorporates a variety of microscopicscale tumorhost interactions (e.g., the mechanical interactions between tumor cells and tumor stroma, degradation of the extracellular matrix by the tumor cells and oxygen/nutrient gradient driven cell motions), can robustly reproduce experimentally observed invasive tumor evolution and predict a wide spectrum of invasive tumor growth dynamics and emergent behaviors in various different heterogeneous environments. Further refinement of our CA model could eventually lead to the development of a powerful simulation tool for clinical purposes capable of predicting neoplastic progression and suggesting individualized optimal treatment strategies.
Cancer is not a single disease, but rather a highly complex and heterogeneous set of diseases that can adapt in an opportunistic manner, even under a variety of stresses. It is now well accepted that genome level changes in cells, resulting in the gain of function of oncoproteins or the loss of function of tumor suppressor proteins, initiate the transformation of normal cells into malignant ones and neoplastic progression
The emergence of invasive behavior in cancer is fatal. For example, the malignant cells that invade into the surrounding host tissues can quickly adapt to various environmental stresses and develop resistance to therapies. The invasive cells that are left behind after resection are responsible for tumor recurrence and thus an ultimately fatal outcome. Therefore, significant effort has been expended to understand the mechanisms evolved in the invasive growth of malignant tumors
(a) The invasive branches centrifugal evolve from the central MTS. The linear size of central MTS is approximately
Although recent progress has been made in understanding certain aspects of the complex tumorhost interactions that may be responsible for invasive cancer behaviors
Indeed, cancer modeling has been a very active area of research for the last two decades (see Refs.
In response to the challenge to develop an “Ising” model for cancer growth
The underlying cellular structure is modeled using a Voronoi tessellation of the space into polyhedra
(a) A Voronoi tessellation of the 2D plane into polygons which are the automaton cells in our model. (b) The associated point configuration for the tessellation, generated by randomly placing nonoverlap circular disks in a prescribed region, i.e., the random sequential addition process
Since our new CA model explicitly takes into account the interactions between a single biological cell and its neighbors and microenvironment, each automaton cell here represents either a single tumor cell or a region of tumor stroma. Thus, the linear size of a single automaton cell is approximately
The microenvironment in which tumor grows is usually highly heterogeneous, composed of various types of stromal cells and ECM structures. The ECM is a complex mixture of macromolecules that provides mechanical support for the tissue (such as collagen) and those that play an important role for cell adhesion and motility (such as laminin and fibronectin)
In addition, the tumor in our model is only allowed to grow in a compact growthpermitting region. This is done to mimic the physical confinement of the host microenvironment, such as the boundary of an organ. In other words, only automaton cells within this region can be occupied by the cells of the tumor as it grows. In general, the growthpermitting region can be of any shape that best models the organ shape. Here we simply choose a spherical region to study the effects of the heterogeneous ECM on tumor growth. More sophisticated growthregion shapes have been employed to investigate the effects of physical confinement on tumor growth
For highly malignant tumors, we consider the cells to be of one of the two classes of phenotypes: either invasive or noninvasive. Following Ref.
As a proliferative cell divides, its daughter cell effectively pushes away/degrades the surrounding ECM and occupies the automaton cell originally associated with the ECM
The invasive cells are considered to be mutant daughters of the proliferative cells
We now provide specific details for the CA model to study invasive tumor growth in confined heterogeneous microenvironment. In what follows, we will simply refer to the primary tumor as “the tumor” and explicitly use “invasive” when considering invasive cells. After generating the automaton cells by Voronoi tessellation of RSA sphere centers, an ECM macromolecule density
Each automaton cell is checked for type: invasive, proliferative, quiescent, necrotic or ECM associated. Invasive cells degrade and migrate into the ECM surrounding the tumor. Proliferative cells are actively dividing cancer cells, quiescent cancer cells are those that are alive, but do not have enough oxygen and nutrients to support cellular division and necrotic cells are dead cancer cells.
All ECM associated automaton cells and tumorous necrotic cells are inert (i.e., they do not change type).
Quiescent cells more than a certain distance
Each proliferative cell will attempt to divide with probability
If a proliferative cell divides, it can produce a mutant daughter cell possessing an invasive phenotype with a prescribed probability
A proliferative cell turns quiescent if there is no space available for the placement of a daughter cell within a distance
An invasive cell degrades the surrounding ECM (i.e., those in the neighboring automaton cells of the invasive cell) and can move from one automaton cell to another if the associated ECM in that automaton cell is completely degraded. For an invasive cell with motility



Local tumor radius (varies with cell positions) 

Local maximum tumor extent (varies with cell positions) 

Characteristic proliferative rim thickness 

Characteristic livingcell rim thickness (determines necrotic fraction) 

Probability of division (varies with cell positions) 



Base probability of division, linked to celldoubling time (0.192 and 0.384) 

Base necrotic thickness, controlled by nutritional needs ( 

Base proliferative thickness, controlled by nutritional needs ( 



Mutation rate (determines the number of invasive cells, 0.05) 

ECM degradation ability ( 

Cell motility (the number of “jumps” from one automaton cell to another, 



ECM density (determines the ECM rigidity and varies with positions, 
The aforementioned automaton rules are briefly illustrated in
Necrotic cells are black, quiescent cells are yellow, proliferative cells are red and invasive tumor cells are green. The ECM associated automaton cells are white and the degraded ECM is blue. (a) A proliferative cell (dark red) is too far away from the tumor edge to get sufficient nutrients/oxygen and it will turn quiescent in panel (b). A quiescent cell (dark yellow) is too far away from the tumor edge and it will turn necrotic in panel (b). Another proliferative cell (light red) will produce a daughter proliferative cell in panel (b). (b) The dark red proliferative cell and the dark yellow quiescent cell in panel (a) turned quiescent and necrotic, respectively. The light red proliferative cell in panel (a) produced a daughter cell. Another proliferative cell (light red) will produce a mutant invasive daughter cell. (c) The light red proliferative cell in (b) produced an invasive cell. (d) The invasive cell degraded the surrounding ECM and moved to another automaton cell.
To characterize quantitatively the morphology of simulated tumors, we present several scalar metrics that capture the salient geometric features of the primary tumor, dendritic invasive branches or the entire invasive pattern. These metrics include the ratio
Following Ref.
(a) Invasive area
The specific surface
The asphericity
To quantify the degree of anisotropy of the invasive branches, we introduce the angular anisotropy metric
To verify the robustness and predictive capacity of our CA model, we first employ it to reproduce quantitatively the observed invasive growth of a GBM multicellular tumor spheroid (MTS)
(a) A snapshot of the simulated growing MTS at 24 hours after initialization. The region circled is magnified in panel (b). (b) A magnification of the circled region in panel (a). One can clearly see that the invasive cells (green) are following each other to form chains within the dendritic branches (blue), as observed in experiment
The ratio of the invasion area over the primary tumor area
Metrics  Simulated MTS  Experimental data 
Specific surface 
9.24  9.78 
Asphericity 
1.09  1.12 
Angular anisotropy metric 
0.17  0.19 
Having verified the robustness and predictive capacity of our CA model, we now consider three types of distributions of the ECM density, i.e., homogeneous, random and sinusoidallike, to systematically study the effects of microenvironment heterogeneity on invasive tumor growth (see
(a) Uniform distribution. (b) Random distribution, i.e., the value of
In the beginning of the simulation, a proliferative tumor cell is introduced at the center of the growthpermitting region and tumor growth is initiated. The growth parameters for the primary tumors in all cases studied here are the same and are given in
We first simulate the growth of malignant tumors with different degrees of invasiveness in a homogeneous ECM with
For the invasive growth, the mutation rate is
Noninvasive tumor in ECM with 

Metrics  Day 50  Day 80  Day 100  Day 120 
Specific surface 
1.23  1.13  1.09  1.04 
Asphericity 
1.21  1.18  1.08  1.12 
Invasive tumor with 

Metrics  Day 50  Day 80  Day 100  Day 120 

0.66  0.38  0.21  0.19 
Specific surface 
1.76  1.48  1.26  1.18 
Asphericity 
1.23  1.12  1.08  1.06 
Angular anisotropy metric 
0.13  0.29  0.33  0.17 
Invasive tumor with 

Metrics  Day 50  Day 80  Day 100  Day 120 

1.28  2.12  2.67  2.08 
Specific surface 
1.94  3.92  3.67  3.28 
Asphericity 
1.42  1.38  1.16  1.23 
Angular anisotropy metric 
0.86  0.67  0.64  0.45 
Invasive tumor with 

Metrics  Day 50  Day 80  Day 100  Day 120 

2.14  2.43  2.64  2.89 
Specific surface 
1.71  4.28  7.89  9.73 
Asphericity 
1.38  1.27  1.13  1.8 
Angular anisotropy metric 
1.25  0.68  0.41  0.18 
Invasive tumor with 

Metrics  Day 50  Day 80  Day 100  Day 120 

  5.17  3.89  2.63 
Specific surface 
1.21  3.40  3.91  5.79 
Asphericity 
1.33  1.36  1.40  1.56 
Angular anisotropy metric 
  1.32  1.02  0.65 
By contrast, for larger cell motility, long dendritic invasive branches are developed as manifested by the large specific surface (e.g.,
It is not very surprising that isotropic tumor shapes and invasive patterns are developed in a homogeneous ECM with relative low density (i.e., the ECM is soft) compared to the ECM degradation ability of the invasive tumor cells. However, real tumors are rarely isotropic, primarily due to the host microenvironment in which they grow, which we now explore.
Consider the invasive growth of a tumor in a much more rigid ECM than that in the previous section, i.e.,
The mutation rate is
The real host microenvironment for tumors are far from homogeneous in general. To investigate how ECM heterogeneity affects the tumor growth dynamics, we use a random distribution of the ECM density, i.e., for each ECM associated automaton cell, its density
The mutation rate is
Invasive tumor with 

Metrics  Day 50  Day 80  Day 100  Day 120 

2.54  4.14  2.13  2.78 
Specific surface 
2.47  3.97  4.65  8.98 
Asphericity 
1.32  1.34  1.18  1.15 
Angular anisotropy metric 
0.63  0.87  0.64  0.19 
Invasive tumor with 

Metrics  Day 50  Day 80  Day 100  Day 120 

2.78  2.46  1.28  0.86 
Specific surface 
1.89  2.95  2.73  1.92 
Asphericity 
1.42  1.61  1.49  1.26 
Angular anisotropy metric 
1.23  1.18  1.09  0.98 
Invasive tumor with 

Metrics  Day 50  Day 80  Day 100  Day 120 

5.24  3.86  3.13  2.96 
Specific surface 
2.51  4.12  6.13  8.76 
Asphericity 
1.19  1.67  1.48  1.21 
Angular anisotropy metric 
1.36  1.33  0.88  0.23 
To represent largescale heterogeneities in the ECM, we use a sinusoidallike distribution of the ECM density, i.e., for an automaton cell with centroid
The mutation rate is
We have developed a novel cellular automaton (CA) model which, with just a few parameters, can produce a rich spectrum of growth dynamics for invasive tumors in heterogeneous host microenvironment. Besides robustly reproducing the salient features of branched invasive growth, such as leastresistance paths of cells and intrabranch homotype attraction observed in
It is noteworthy that the growth dynamics of tumors in a heterogeneous microenvironment is distinctly different than those in a homogeneous microenvironment. This emphasizes the importance of understanding the effects of physical heterogeneity of the host microenvironment in modeling tumor growth. Here we just make a first attempt to take into account a simple level of host heterogeneity, i.e., by considering the ECM with variable density/rigidity. Currently, the invasion of the malignant cells into the host microenivronment is considered to be a consequence of invasive cell phenotype gained by mutation, and is not triggered by environmental stresses. However, the effects of environmental stresses can be taken into account. For example, a CA rule can be imposed that if the division probability of a malignant cell is significantly reduced by ECM rigidity, i.e., it is extremely difficult to push away/degrade ECM to make room for daughter cells, the malignant cell leaves the primary tumor and invades into soft regions of surrounding ECM. This would lead to reduced tumor invasion (i.e., development of the dendritic invasive branches) in soft microenvironments but enhanced invasion in rigid microenvironments
Moreover, the spatialtemporal evolution of more complicated and realistic nutrient/oxygen fields can be incorporated into our CA model. This can be achieved by solving the coupled nonlinear partial differential equations governing the evolution of the nutrient/oxygen concentrations as was done in Refs.
Such an
In our current CA model, the microscopic parameters governing tumor invasion are variable and can be arbitrarily chosen within a feasible range as given in
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The authors are grateful to Robert Gatenby and Bob Austin for valuable comments on our manuscript. The authors are also grateful to the anonymous reviewers for their valuable comments.