MULTI-SCALE
MATHEMATICAL MODELING OF TUMOR GROWTH

(Last updated August 2009)

(Last updated August 2009)

Motivation

Cancer is a highly complex and heterogeneous set of diseases. Dynamic changes in the genome, epigenome, transcriptome and proteome that result in the gain of function of oncoproteins or the loss of function of tumor suppressor proteins underlie the development of all cancers. While the underlying principles that govern the transformation of normal cells into malignant ones is rather well understood (Hanahan and Weinberg, 2000). a knowledge of the changes that occur in a given cancer is not sufficient to deduce clinical outcome.

The difficulty in predicting clinical outcome arises because many factors other than the mutations responsible for oncogenesis determine tumor growth dynamics. Multidirectional feedback loops occur between tumor cells and the stroma, immune cells, extracellular matrix and vasculature (Kitano, 2004). Given the number and nature of these interactions, it becomes increasingly difficult to reason through the feedback loops and correctly predict tumor behavior. For this reason, a better understanding of tumor growth dynamics can be expected from a computational model.

The holy grail of computational tumor modeling is to develop a simulation took that can be utilized in the clinic to predict neoplastic progression and response to treatment. Not only must such a model incorporate the many feedback loops involved in neoplastic progression, the model must also account for the fact that cancer progression involves events occurring over a variety of time and length scales. By developing individual models that focus on certain tumor-host interactions, validating these models and merging the individual pieces together, I aim to build one algorithm with the potential to address a variety of questions about neoplastic growth and survival.

Model Background

My Ph.D. thesis advisor, Professor Salvatore Torquato, along with several others, has developed a novel cellular automaton (CA) model to simulate the mechanistic complexity of solid tumor growth. The model takes into account four cell types: healthy cells, proliferating tumor cells, non-proliferating tumor cells and necrotic tumor cells. The algorithm they developed was able to successfully predict three-dimensional tumor growth and composition using a simple set of automaton rules and a set of four microscopic parameters that account for the nutritional needs of the tumor, cell-doubling time and an imposed spherical symmetry term (Kansal et al., 2000a). This simple model of tumor growth was applied to consider the effect that an emerging clone of novel genotype has on neoplastic progression (Kansal et al., 2000b, Figure 1) and to study the response of a tumor mass to surgical resection followed by chemotherapy (Schmitz et al., 2002).

Figure 1

Vascular Growth Model

The success of the original CA model is in part related to its simplicity, and one of the simplifying assumptions is that the vasculature is implicitly present and evolves as the tumor grows. In order to incorporate more biological detail in the model, I worked to modify the original algorithm to study the feedback that occurs between the growing tumor and the evolving host blood vessel network (Gevertz and Torquato, 2006).

The computational algorithm is based on the co-option/regression/growth experimental model of tumor vasculature evolution. In this model, as a malignant mass grows, the tumor cells co-opt the mature vessels of the surrounding tissue that express constant levels of bound Angiopoietin-1 (Ang-1). Vessel co-option leads to the upregulation of the antagonist of Ang-1, Angiopoietin-2 (Ang-2). In the absence of the anti-apoptotic signal triggered by vascular endothelial growth factor (VEGF), this shift destabilizes the co-opted vessels within the tumor center and marks them for regression (Holash et al., 1999). Vessel regression in the absence of vessel growth leads to the formation of hypoxic regions in the tumor mass. Hypoxia induces the expression of VEGF, stimulating the growth of new blood vessels.

We developed a system of reaction-diffusion equations to track the spatial and temporal evolution of the aforementioned key factors involved in blood vessel growth and regression. Based on a set of algorithmic rules, the concentration of each protein and bound receptor at a blood vessel determines if a vessel will divide, regress or remain stagnant. The structure of the blood vessel network, in turn, is used to estimate the oxygen concentration at each cell site, and the oxygen concentration determines the proliferative capacity of each automaton cell.

The model proved to quantitatively agree with experimental observations on the growth of tumors when angiogenesis (growth of new blood vessels from existing vessels) is successfully initiated and when angiogenesis is inhibited. In particular, the model exhibits an "angiogenic switch" such that in some parameter regimes the tumor can grow to a macroscopic size whereas in other parameter regimes tumor growth is thwarted beyond a microscopic size (Gevertz and Torquato, 2006, Figure 2).

Figure 2

Growth in Confined, Heterogeneous Environments

Both the original CA model and the vascular growth model limit the effects of mechanical confinement to one parameter that imposes a maximum radius on a spherically symmetric tumor. However, tumors can grow in organs of any shape with nonhomogeneous tissue structure, and in a study performed by Helmlinger et al (1997), it was demonstrated that neoplastic growth is spherically symmetric only when the environment in which the tumor develops imposes no physical boundaries on growth. In particular, it was shown that human adenocarcinoma cells grown in a 0.7% gel inside a cylindrical glass tube develop to take on an ellipsoidal shape, driven by the geometry of the capillary tube. However, when the same cells are grown outside the capillary tube, a spheroidal mass develops. This experiment highlights that the assumption of radially symmetric growth is not valid in complex, non-spherically symmetric environments.

Since many organs, like the brain, impose non-radially symmetric physical confinement on tumor growth, we modified the original CA algorithm to incorporate boundary and heterogeneity effects on neoplastic progression. The CA evolution rules are comparable to the rules used in the original model, except that all assumptions of radial symmetry are removed and replaced with rules that account for tissue geometry and topology (Gevertz et al., 2008). In doing this, we showed that models that do not account for the structure of the confining boundary and organ heterogeneity lead to inaccurate predictions on tumor size, shape and spread. In Figure 3, this is illustrated by studying tumor growth in a 2D representation of the cranium and comparing the size of the tumor as a function of time when the original and modified algorithm is employed.

Figure 3

Heterogeneous Tumor Cell Population: Genetic Mutations

It is well known that each patient's tumor has its own genetic profile, and that this profile is important in determining growth dynamics and response to treatment. To account for tumor cell heterogeneity, we begin with the assumption that tumors are monoclonal in origin; that is, a tumor mass arises from a single cell that accumulates genetic and epigenetic alterations over time (Fialkow, 1979). In our algorithm, a tumor cell of one genotype/phenotype initiates tumor growth. As a tumor cell undergoes mitosis, we assume that there is a 1% chance that an error occurs in DNA replication. When such an error occurs, the daughter cell will take on a mutant phenotype that differs in one respect from the mother cell. Given the tumor cell properties considered in the model, we allow malignant cells to acquire altered phenotypes related to cell proliferation rates (which can correspond to cells altering the production rate of growth factor such as PDGF and TGF-α or tumor suppressor proteins such as pRb or p53) and the length of time a cell can survive under sustained hypoxia (which can correspond to the expression levels of the X-box binding protein 1). We have implemented the model such that a tumor cell can express one of eight mutant phenotypes (Gevertz and Torquato, 2008), all of which are defined in Figure 4.

Figure 4

Using the genetic mutations model, we have explored the probability of emergence of both beneficial and deleterious mutations in the tumor cell population. We have found that, as expected, mutations which give cells a proliferative advantage arise frequently in our simulations (78% of the time). More surprisingly, we have also observed that the obviously deleterious mutation that decreases the proliferation rate of a cell emerges in 7% of our simulations. Furthermore, the mutation that determines the hypoxic lifespan of a cell is not strongly selected for or against, provided the changes in the lifespan are not too drastic in either direction (Gevertz and Torquato, submiited). Using this model, we can explore how growth dynamics are impacted by the emergence of one or more mutant strains.

Merged Model of Tumor Growth

Each of the previously discussed algorithms were designed to address a particular set of questions and successfully served their purpose. Much can be gained, however, by merging the evolving vasculature, confined growth and genetic mutations models. The resulting multiscale simulation tool will not only consider multiple forms of feedback and heterogeneity, it will also likely have emergent properties not identifiable prior to the integration of the three models.

To pinpoint the effects that the environment has on tumor growth, here I show environment in which the tumor grows (bottom row of Figure 5), along with the shape and spread of a tumor after approximately four months of growth in this environment (top row of Figure 5). This figure allows us to see how and to what extent the structure of the environment alters tumor shape and spread.

Figure 5

Currently, this merged model is being used to test the impact that various vascular-targeting therapies have on tumor progression.

Future Work

- Generalize the model to three dimensions. Currently, the original CA, confined and heterogeneous growth model, and the genetic mutations model have been implemented in both 2D and 3D. However, the vascular algorithm has only been implemented in 2D, limiting the merged model to be two-dimensional as well.
- Develop a more realistic model of the underlying capillary network in tissue. We only focus on modeling the capillaries because this is the level of the vascular tree where oxygen and nutrient exchange occurs. The capillary network, it should be noted, does not exhibit the same branching structure that is seen in higher-order vessels.
- Expand upon the confined-growth algorithm to examine not only how
the host deforms the tumor, but also to consider the reciprocal
deformities induced in the host by the growing neoplasm.

- Incorporate other forms of tumor-host interactions: extracellular matrix, immune system...
- Include the process of single cell invasion in which individual
cancer cells break off the main tumor mass and invade healthy tissue
(particularly important in brain tumor growth dynamics)

References

D. Hanahan and R.A. Weinberg, 2000. The hallmarks of cancer. Cell 100: 57-70.

H. Kitano, 2004. Cancer as a robust system: implications for anticancer therapy. Nature Rev. Cancer 4: 227-235.

A.R. Kansal et al., 2000a. Simulated brain tumor growth dynamics using a three-dimensional cellular automaton. J. Theor. Biol. 203: 367-382.

A.R. Kansal et al., 2000b. Emergence of a subpopulation in a computational model of tumor growth. J. Theor. Biol. 207: 431-441.

A.R. Kansal and S. Torquato, 2001. Globally and locally minimal weight spanning tree networks. Physica A 301: 601-619.

J.E. Schmitz et al., 2002. A cellular automaton model of brain tumor treatment and resistance. J. Theor. Medicine 2002(4): 223-239.

J.L. Gevertz and S. Torquato, 2006. Modeling the effects of vasculature evolution on early brain tumor growth. J. Theor. Biol. 243: 517-531.

J. Holash et al., 1999. Vessel cooption, regression, and growth in tumors mediated by angiopoietins and VEGF. Science 284: 1994-1998.

G. Helmlinger et al., 1997. Solid stress inhibits the growth of multicellular tumor spheroids. Nature Biotech. 15: 778-783.

J.L. Gevertz, G. Gillies and S. Torquato, 2008. Simulating tumor growth in confined heterogeneous environments. Submitted to Physical Biology.

P.J. Fialkow, 1979. Clonal origin of human tumors. Ann. Rev. Med. 30: 135-143.

J.L. Gevertz and S. Torquato. Growing heterogeneous tumors in silico. Submitted for publication.