An ultimate challenge is the development of models that incorporate the behavior of the entire tumor. In the early-stages of tumor development, cancer cells are clustered together into an avascular structure, which has been modeled in vitro
with multi-cellular spheroids (71
) and in silico
with mathematical balances of proliferating and dying cells that are coupled to physical constraints such as nutrient diffusion. In general, these models are limited to describing tumor morphology and distribution of necrotic cells (72
). More recently, multiscale modeling was used to analyze avascular multi-cellular tumors (73
). This model incorporated proliferation, diffusion and consumption of nutrients, diffusion and production of wastes, adhesion and cell–environmental interactions. A notable improvement over previous models that relied on probabilities for cellular decisions was the inclusion of a Boolean logic model for the G1
to S progression, which incorporated molecular components such as transforming growth factor-β, p27, p21 and cyclins. Model simulations closely matched in vitro
spheroid morphology, size and cell cycle distribution over a 20 day period (73
). Further refinement of detailed models such as this should allow investigators to test the impact of molecular interventions on early-stage tumors.
As a tumor increases in size, the center core becomes necrotic and the tumor needs to develop its own vasculature to continue to grow. Therefore, targeting tumor angiogenesis is an attractive strategy to treat cancer (74
). Computational models are beginning to provide tools to address how the tumor microenvironment and growth factor signaling regulate these events. Although numerous models for angiogenesis have been proffered, the complexity of the system presents a challenge regarding how to connect the behavior of the blood vessels to the tumor and how to incorporate molecular control mechanisms. Two recent models link the behavior of the tumor, growing blood vessel and cellular microenvironment. Macklin et al.
) modeled tumor growth in response to the altered oxygen profile resulting from new vascular growth. In this model, mechanical forces directed blood vessel growth, and simulations demonstrated that when heterogenous oxygen profiles were formed from the new vessels, tumor cell proliferation was heterogenous resulting in invasive tumor morphologies. In the second model, blood vessel growth was modeled as a response to local gradients in angiogenic factors such as VEGF (76
). By changing the various rules, these model forms can mimic different environmental or genetic conditions; however, they do not allow for the direct test of molecular interventions on angiogenesis.
As an alternative approach, another recent model linked the cancer cell cycle to sensing of environmental conditions such as oxygen levels, with the assumption that suboptimal oxygen levels result in the production of VEGF (77
). In contrast to more phenomenological models, the impact of VEGF on endothelial cell proliferation and migration was determined by the pharmacological Emax model and additional pro- and anti-angiogenic molecules were included. Simulations of the effect of endostatin induced by gene therapy indicated there is a critical rate of production needed; below this rate, longer treatment times were predicted to ‘rebound’ more quickly, whereas above this rate, the time to rebound increased with the duration of treatment (77
). Expansion of the molecular components of multiscale models of angiogenesis will be essential to utilize them to determine drug targets and optimal treatment regimens.
An alternative method for a tumor to continue its growth is for cells to leave the primary tumor and implant in other tissues—the metastases that result from this process are responsible for the majority of cancer deaths. It is becoming increasingly apparent that metastasis is more than a random process, with additional mutations required for cancer cells to leave the primary tumor and metastatic cells demonstrating ‘preferences’ for target organs (78
). Similar to angiogenesis, models of tumor invasion and metastasis must incorporate multiple scales and are primarily constructed using approximations of cellular signaling. For example, in one virtual tumor model, cells respond to microenvironmental changes (e.g. oxygen level, cell proximity) and decide to proliferate or die (79
). At the same time, cells undergo random inheritable mutations that change the ‘phenotype’ of the cell (e.g. the likelihood to proliferate). Changes to the microenvironment to incorporate harsher conditions such as hypoxia or heterogenous matrix led to selection for cells with more aggressive traits and invasion tumor shapes. The impact of the matrix on cancer cell invasion is supported by a recent model of invadopodia, the cellular extensions believed to degrade extracellular matrix as tumor cells invade (80
). In this model a single cancer cell sits on top of a matrix, which is modeled from known dimensions and characteristics of extracellular matrix proteins. A series of rules describe how invadopodia invade and interact with the extracellular matrix. The simulations suggested that dense matrix (such as basement membrane) forms an effective barrier to prevent invadopodia penetration and matrix degradation, whereas looser matrices (such as stroma) have gaps sufficient to allow invadopodia penetration and matrix degradation.
In order to metastasize to distant organs, cancer cells must invade into the blood or lymph to be transported to other parts of the body. The first stage in this process is for the cancer cell to invade the endothelial layer. To model this process, a multiscale model of transendothelial migration was developed, incorporating both endothelial cells, cancer cells and the cadherin interactions between the distinct cells (81
). In the simulated invasion, cancer cells attach to endothelial cells by N-cadherins, and the endothelial layer is connected by VE-cadherin bonds, which the cancer cell must break in order to squeeze through. The model follows protein concentrations within each cell through a series of ordinary differential equations, cell–cell forces by a modified Hertz model and cell movement according to Langevin equations. As cancer cells contact endothelial cells, a N-cadherin bond forms between the cells, leading to competition for β-catenin and dissolution of the existing VE-cadherin bonds. The observed behaviors are similar to experimental results that noted the loss of N-cadherin slows transendothelial migration (82
Ultimately, models such as these must be built upon to balance multiple signaling pathways and phenotypic processes in order to be used for identification of drug targets. A novel multiscale model of non-small-cell lung cancer provides an example of how such a process can be implemented. In an early version, ErbB activation of ERK and PLCγ in each cell in a tumor was modeled by a series of ordinary differential equations (83
). Phosphorylated ERK and PLCγ are used as readouts for proliferation and migration, respectively, and cells migrated or proliferated in a virtual two-dimensional space. In this early version, overexpression of ErbB1 led to a migration dominant phenotype and accelerated tumor expansion. Building on this initial work, the group expanded the virtual space and tumor to three dimensions and added transforming growth factor-β activation of RAS to each cell's signaling module (84
). By comparing the results of simulations with single and cotreatments, the expanded model demonstrated conditions where targeting a single pathway would be ineffective in deterring tumor expansion.