Principal Investigator/Program Director (Last, First, Middle):
Hayward, Simon W.
Specific Aim 3a. Spatio-temporal mathematical modeling of tumor progression.
We will develop a stochastic cell-based model for the behavior of normal and altered cells during the
tumorigenic process. We propose an “agent-based” model where cells obey rules motivated by experimental
findings that describe cell-cell and cell-matrix interactions. The purpose of our model will be to test hypotheses
regarding gene product interactions and characterize relevant variables, parameters and interactions. We will
coordinate feedback between model predictions and experimental results so that biological experiments will be
used to indicate necessary model modifications and validate the model, and simulation results will be used to
make predictions and indicate interesting experiments.
Rules for cells are selected to describe key processes for each experimental system but may include rules for
random cell motion, directed cell motion (chemotaxis or haptotaxis), the secretion and absorption of diffusible
chemical factors, cell-cell interactions (cell-cell adhesion and contact inhibition) and cell-matrix interactions
(cell-matrix adhesion via substrate adhesion molecules). Cells follow rules as autonomous agents, interacting
with other cells and with the microenvironments cell activities produce. Protein factors and other solutes diffuse
within the matrix surrounding cells, while substrate adhesion molecules remain fixed. Different cell types
including knock-outs and cells at different stages of differentiation are modeled as cells that follow different cell
rules for movement, absorption, secretion, growth and division. Stored cell products and cell secretions for
each cell type are computed with an intracellular network that outputs the rate of secretion of each chemical
factor at the cell boundary as a function of time and as a function of the amount of absorbed chemical factors.
Cell-based models (e.g., cellular automaton (CA) models) are well-suited to modeling cancer since they
describe individual cell behaviors and easily account for spatial cell heterogeneity that are important aspects of
cancer. M Kiskowski-Byrne and co-authors have reviewed CA models in general and for tumor growth in (Alber
et al, 2002). An early CA model for tumor growth was developed by Düchting and Vogelsaenger (1981) that
accounts for cell growth, division and death and nutrient and cell diffusion. Recently, more sophisticated hybrid
models include continuum descriptions of nutrient and waste diffusion (Patel et al, 2001; Dormann and
Deutsch, 2002; Anderson, 2005) and a Boolean network to describe protein expression (Jiang et al, 2005).
Cancer invasion and angiogenesis are important elements of tumorigenesis and have also been modeled
using both continuum (Chaplain, 1996; Perumpanami et al, 1996; Levine et al, 2001; Marchant et al 2001) and
individual/hybrid approaches (Anderson and Chaplain, 1998; Turner and Sherratt, 2002). The most successful
models are multi-scale, encompassing interactions and dynamics at macroscopic, mesoscopic and
microscopic levels (tissues, cells and intra-cellular processes, respectively) (Alarcón et al, 2004; Jiang et al,
2005). Our model will depart from the cell-based model of Kiskowski et al (2004) -- in which cell-matrix
interactions are described with easily modified rules for random cell motion, cell secretion, cell absorption of
diffusible growth factor and cell-substrate adhesion -- by adding components for cell division and cell invasion.
Altered Stromal TGF signal processing
The basis for this aim will be the construction of a mathematical model to look at the protective versus tumor
promoting effects of altered stromal conditioned medium. We will model the interactions of groups of cells with
different knock-out stromal ratios to determine at what point a growth inhibiting effect is overridden by a progrowth/pro-cancer effect. A “cancer progression model” will be developed that is consistent with what is
already known about this tumorigenic process, in particular, with the results of the completed experiments
described in Aim 1. This cancer progression model will represent our current understanding of the nature of
growth promoting and growth inhibitory factors. A computational model will then be developed that implements
the developmental model. Predictions of the computational model will be compared with the results of intended
experiments described in Aim 1. If there is any discrepancy, we will investigate with the computational model
how the cancer progression model must be modified in order to duplicate the experimental results.
Modifications in the developmental model will represent a refinement of our understanding of the nature of
growth promoting and growth inhibitory factors.
Preliminary Results
A. Cancer Progression Model
From hundreds of cell functions, we select a small number of key processes that are thought to be involved in
tumor progression to define a simplified cancer progression model. The following rules for the cancer
progression model are based on 4 cell types: normal stroma, altered stroma (knock-out), normal epithelia, and
transformed epithelia.
(1) All cell types secrete high levels of TGFβ.
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Principal Investigator/Program Director (Last, First, Middle): Hayward, Simon W.
(2) Normal stromal cells respond to extracellular TGFβ by inhibiting the production of the growth factor
HGF. Altered stroma cells have defects in the SMAD pathway so that they do not respond to TGFβ and
secrete HGF.
(3) Normal epithelial cells respond to extracellular HGF by transforming and becoming initiated epithelial
cells.
(4) Altered stroma cells and initiated epithelial cells proliferate.
.
B. Computational Model
The computational model is designed to implement and test the developmental model. Described below is a
preliminary 2D model with simplified rules that does not account for cell movement and cell division.
Stromal and the epithelial cells
Although the experimental system is 3D, as a first approximation we model a 2D cross-section of “the
prostate” on a 228x139 square lattice. Lattice size and initial cell positions correspond to image size and
positions of cell nuclei depicted in an experimental picture (Fig 31). The size of a lattice node thus
corresponds to the size of an image pixel that is approximately one-third of a cell diameter. Cells are modeled
as occupied nodes on the lattice; a ‘1’ corresponds to a node that is occupied by a cell and a ‘0’ corresponds to
a node that is not occupied. Since cells are approximately 3 pixels/nodes in diameter, each cell occupies nine
nodes on the lattice (as a 3 node x 3 node square). Thus cells have a single interior node and 8 exterior
nodes. Different cell types are modeled by assigning different states to the occupied nodes that represent a
cell. A lattice node may have multiple states corresponding to overlapping cells at that node. At the beginning
of a simulation, all epithelial cells are normal and stroma cells are randomly assigned type normal or type
knock out according to the fraction of knock-outs fKO.
Fig 3A1: Experimental picture (left) and corresponding position of simulation cells (right) assuming 50% altered
cells. Epithelial cells are shown in black, normal stromal cells are shown in blue and altered stromal cells are
shown in cyan.
Intracellular Network: TGFβ Secretion, Absorption and Diffusion
Since all cells produce TGFβ, we assume that TGFβ is abundant at all nodes and do not explicitly model the
secretion, absorption or diffusion of TGFβ. We assume that normal stroma cells absorb TGFβ and altered
stromal cells do not.
Intracellular Network: Growth Factor Secretion and Diffusion
Altered stromal cells stochastically secrete unspecified units of growth factor (HGF) from 8 exterior nodes that
define their cell boundary. Altered stroma cells produce HGF at a constant rate of k HGF units per time-step.
This is modeled by creating one unit of HGF at rates of k HGF /8 at each exterior pixel of altered stromal cells.
Units of HGF diffuse by random walk as described in (Kiskowski et al, 2004) in the lattice space between cells
(i.e., they cannot diffuse into regions occupied by interior cell pixels or overlapping cells). Restricting diffusion
to the exterior space of cells does not increase computation time and more realistically models the flow of
factors through the extracellular matrix, especially for subsequent models where the cells will be modeled at
higher resolution and will have more interior pixels. We do not model the decay of growth factor since we
assume the simulation/experimental time-scale is short compared to the decay time.
Cell Transformation
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Levels of growth factor (HGF) are measured at the eight exterior cell nodes. In response to threshold levels of
HGF, normal epithelial cells transform to initiated epithelial cells.
C. Results
We simulate the interaction of cells for an initial population of cells in which half the stroma cells are randomly
assigned the altered (knock-out) state (see Figure 3A1). We assume initial zero levels of HGF. Over 10 timesteps, the altered stromal cells produce a local environment of high HGF levels (Figure 3A2 left). These levels
build up over time and eventually transform nearby epithelial cells (Figure 3A2 right).
Fig3A2: Left: HGF levels after 10 simulation time-steps produced by altered stromal cells. Right: Cell types
after 100 time-steps. Normal epithelial cells are shown in black, normal stromal cells are shown in blue and
altered stromal cells are shown in cyan. Transformed epithelial cells are shown in red.
Future Directions
(1) Cell Diffusion and Proliferation Epithelial and stromal cells will be able to diffuse by random walk with a
basement membrane between them so that cells of either type are not allowed to pass the basement
membrane unless the cell type is invasive. Altered stroma, initiated epithelium and CAF stroma will
proliferate at rates of PKO, PT and PCAF cells per time-step respectively. This will be implemented by a
probability of division of PKO, PT and PCAF at each time-step. If a cell divides, the parent cell remains at
its original node but the daughter cell may randomly diffuse with biased motion towards unoccupied
lattice regions. As a first approximation, we estimate that the rate of cell division is low enough so that
cells have time to grow between divisions so that we do not model the cell cycle explicitly and we
assume constant cell growth even in areas of high density. If these approximations yield results that are
inconsistent with experimental data, we will model the cell cycle and nutrient-dependent growth.
(2) Epithelial Invasion and Influence We would like to add the effect of the epithelial cells upon the stroma.
Initiated epithelial cells will be able to dedifferentiate and become invasive, i.e., they will be able to
diffuse beyond the basement membrane. Also, these cells will release a transforming morphogen TM
that changes normal stroma cells into CAF cells.
(3) Application: Testing Hypotheses of Epithelial Protection We would like to identify a mechanism that will
explain the observation that epithelial transformation is most pronounced at intermediate densities of
altered stroma compared to low and high densities of altered stroma. Large populations of cells often
communicate through a quorum response: they communicate by secreting a chemical signal that
coordinates the response of the population (reviewed in Swift et al, 2001). We will test the hypothesis
that in response to high but sub-threshold levels of HGF, normal epithelia cells secrete a diffusive
protective factor. At low densities of altered stromal cells, we predict that few epithelial cells are
transformed because there are low levels of HGF. At intermediate densities of altered cells, local HGF
levels build quickly in small areas and many cells transform in these areas. However, at high densities
of altered cells, the protective quorum is triggered due to the high global levels of HGF. We will analyze
whether this hypothesis is consistent with the experimental system by comparing experimental
observables with similar response variables in silico.
Hypothesis 1a: A diffusible protective factor secreted by epithelial cells in response to low levels of
HGF raises the threshold level of HGF required to make neighboring epithelial cells transform.
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Hypothesis 1b: A diffusible protective factor secreted by epithelial cells in response to low levels of
HGF binds with HGF, making HGF unavailable for absorption by neighboring cells.
Hypothesis 2: A small fraction of epithelial cells (~5%) are less differentiated and respond to subthreshold levels of TGF differently than the rest of the cell population. Epithelial heterogeneity,
rather than a diffusing epithelial protective factor, is responsible for the non-linear response of
epithelial transformation to altered stroma fraction.
(4) Model Validation: Statistical Analysis Our computational model, and the developmental model, will be
validated by quantitative comparison with experimental data. Response variables that can be measured
and compared are the number of initiated, changed and dedifferentiated cells, and the spatial
patterning of altered cells. For example, statistical correlations of disease in adjacent ducts can be
easily tested experimentally and is relevant especially in the context of testing hypotheses regarding
protective epithelial effects, since disease in a duct may increase or decrease the probability of disease
in an adjacent duct.
Anticipated Problems and Solutions
While the spatial scale of the computational model is set by the cell size, an anticipated challenge is
determining the simulation time-scale. Typically, the time-scale of pattern formation is used to approximate the
simulation time-scale. However, this is not ideal since there are many degrees of freedom in choosing
parameters and different sets of parameters can yield different rates of change in the pattern dynamics. One
way we can address this issue is by measuring the rate of cell diffusion and cell division experimentally.
Incorporating these in our computational model will set the temporal scale since we include diffusion rates and
division rates in the computational model.
Another anticipated challenge is that many parameters will not be precisely known during the early modeling
stages (e.g., morphogen diffusion rates) or may never be known (e.g., diffusion rates of putative signaling
factors). We will need to investigate ways to quantitatively estimate the required diffusion rates, either by
mining the literature or systematically surveying the parameter space (global sensitivity analyses). Once a set
of parameters has been selected for analysis of computational results, we will study the extent to which the
computational results are robust to stochastic fluctuations in parameters (local sensitivity analyses).
An important modeling challenge in the context of pattern formation is that more than one hypothesis may yield
results that are quantitatively consistent with the observed pattern. In the spirit of the scientific method, our
approach will be to consider several hypotheses (for example, hypotheses 1a, 1b and 2 above) and try to
eliminate the hypotheses by using simulations to generate testable predictions that can be verified or refuted
with the experimental system.
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