A Quick Look at Cellular Automata Models for Solid

Tumor Growth

Philip Crooke

Department of Mathematics

Biomathematics Study Group

Vanderbilt Integrative Cancer Biology Center

Characteristics of a Basic Cellular Automata

• Lattice: A discrete set of points (1D, 2D or 3D) (sites).

• States: Each lattice point takes one of a finite number of states.

• Time: It is discrete and ever site updates simultaneously.

• Neighborhood: A definition what it means for sites to be close to one another.

• Update Rules: Laws that define how states change according to the states of sites in its neighborhood. The update rules can be deterministic or probabilistic.

• Homogeneity: The behavior of all sites are according to the same set of update rules.

Variations on a Basic CA

• Continuous States: Instead of discrete states, they can be continuous.

• Asynchronous Updating: Update of states is not done simultaneously.

• Non-homogeneity: Different update rules for different sites.

von Neumann Neighborhood

Moore Neighborhood

Basic Duchting ModelW. Duchting, A model of disturbed self-reproducing cell systems, in Biomathematics and Cell Kinetics,

A.J. Valleron and R.D.M. MacDonald (editors), Elsevier/North Holland, 1978.

Two-dimensional, finite lattice (10x10, INTEL 8080) von Neumann neighborhood Asynchronous Each lattice site corresponds to a living cell or dead cell (empty

site) Update rules:

o If a living cell is isolated, then it dies immediately.

o A cell is alive only if there are at least one living cell in its neighborhood.

o Cell division only takes place in there is an empty site in the neighborhood. If there are more than one empty sites in the neighborhood, the new site is selected randomly in the neighborhood.

o There are special rules for boundary sites of the lattice.

Some Enhancements

• The growing process for each single cell comes from an initial signal.

• Assign a mean life span for each site. He calls it the life span matrix and determines how long a cell lives before cell division. The smaller the entry in the matrix, the faster the cell grows.

• A cell can be destroyed at any arbitrary time by an external disturbance (by irradiation or by a cytostaticum).

• The model allows two classes of cells which grow together, but with different life span matrices.

Competing Cell Groups

Circles: fast growing. Triangles: slow growing. Werner Duchting, 1978.

Competing Cell Groups with Surgical Removal at T=101

Circles: fast growing. Triangles: slow growing. Werner Duchting, 1978.

Duchting Model (1980)

• Two-dimensional lattice (100x100, CYBER 76)• Each site contains a living or dead (empty) cell.• There are three classes of cells (normal, tumor and dead).• The life span of a cell (normal or tumor) is determined by a random number.• Uses von Neumann neighborhoods.• A normal cell can divide only there is an empty site in the neighborhood of the dividing

cell. If there are more than one empty sites in the neighborhood, then the daughter cell is assigned randomly to one of the empty sites.

• Each normal cell can be deleted at any time by an external signal.• For tumor cells, a division can take place even if there are no neighborhood vacancies. In

this case, a direction (vertical or horizontal) which contains the minimal number of cells that are adjacent and a shift is performed.

• Tumor cells have special rules for the boundary sites.• The probability of the loss of tumor cell is different than a normal cell.• The lattice can be subdivided into sub-lattices with different boundary rules.

Duchting Model (1981)• Lattice is three-dimensional (40x40x40).• Tumor is spherical.• Each site can either occupied by a (tumor) cell or is empty (medium) and when occupied by a cell, the

cell can be in different phases.• The model uses an extended cell cycle. The cell can spend time in the phases (G1, S, G2, M, GO and

N) with phases times (TG1, TS, TG2, TM, TGO, TN) and standard deviations (G1, S, G2, M, GO, N) . These times are generated from a Gaussian distribution. The GO state is fictitious state (resting) for cells and is a check to see if there is sufficient nutrients for a cell to divide. If the nutrient level is too low, then it goes into state N. The N state is a state of dying and TN is the time of death.

• Incorporates Rcrit (recall talk about diffusion models) for nutrient level. If a site is within the sphere of radius Rcrit, then there the nutrient concentration is not sufficient for a cell to divide and cell passes to the N state.

• Cells may die randomly after some initial period.• In theory, the number of replications of a cell is unlimited. However, replication stops when a site a

prescribed distance from the medium.• For a particular cell at the end of state M, the neighborhood of the cell is checked for any empty site.

If there are more than one side in the neighborhood, then the site that is closest to the medium is selected. If no empty site exists in the neighborhood, then the direction that is closest to the medium is selected and cells are shifted. If there are more than one direction that is is closest to the medium, then the direction is selected randomly.

Duchting Model (continued)

• All cells living a certain distance or more from the medium and in the GO phase, go the N phase.

• Each cell can be removed by an external event (radiotherapy or chemotherapy)

• With radiotherapy, only proliferating cells can be lethally damages with some probability. With chemotherapy, only cells in phases G1, S, G2, and M are destroyed with some other probability.

Brain Tumor Growth ModelA.R. Kansal et al., Simulated brain tumor growth dynamics using a three-dimensional

cellular automaton, J. Thoer. Biol. 203(2000), 367-382. • Lattice is not regular.

• It uses a 3D Voronoi-Delaunay lattice i.e., a collection of space-filling polyhedrons. Each site of lattice is the center of each polyhedron. Each polyhedron represents a cell. The number of cells per unit volume is higher near the center of the lattice. There are 1.5 million lattice sites.

• The neighborhood of a site is the set of lattice sites that share a common surface with the polyhedron of the particular site.

2D Voronoi-Delaunay Lattice

Brain Tumor Growth Model (continued)

• There are two types of cells: cancerous and normal (non-cancerous). Normal cells are represented by empty sites of the lattice.

• Cancerous cells have three discrete states: proliferating, quiescent and necrotic.

• The initial configuration of tumor cells (~1000 cells) are placed at the center of the lattice.

• The update rules are probabilistic. The state of each cell (proliferating, quiescent, or necrotic) transition a probability function.

• The update rules are not all local e.g., the proliferation rule depends on the location of the site and only sites that are sufficiently close to the boundary surface of the tumor can divide.

• There are four important time-dependent parameters: Rt, Dp, Dn and Pd (average tumor radius, proliferating rim thickness, quiescent thickness, and probability of division).

Tumor Cross-section

Brain Tumor Growth Model (continued)

• At each time step:

o Each cell is checked for time: normal, necrotic, quiescent, or proliferating.

o Normal cells and necrotic cells are inert.

o Cancerous, quiescent cells the are more than the distance Dn become necrotic. This incorporates the idea of nutrient starvation.

o Proliferating cells divide if a certain random number that depends on the location of the cell in the tumor is exceeded. This incorporates a mechanical confinement pressure condition.

o Dividing cells searches a sphere of radius Dp for an empty (non-tumor) site. If found, it divides and fills both sites. If unsuccessful, then the site changes to quiescent.

Simulation Over Time: Proliferating, Quiescent and Necrotic Regions (cross-section)

Simulations: Different Cross-sections

Potts Model (2002)(Yi Jiang and Jelena Pjesivac-Grbovic: Cellular Model for Avascular Tumor

Growth, Student Research Symposium 2002, LARC)

Biological Processes

• Nutrient absorption and diffusion

• Waste products

• Cell-cell adhesion

• Chemotaxis

• Cell proliferation

• Mutations

• Geometry and structure of cells

Jiang & Pjesivac-Grbovic Model

• Extended Potts Model in Three-dimensions

• Monte Carlo

• Reaction-diffusion equations (PDEs) for oxygen, nutrients, and waste products

• Incorporates some of the biological processes

Potts Model: Minimize energy of the system of cells

• 3D lattice is partitioned into domains of cells and medium. Cells can occupy more than one lattice point.

• Each cell has properties assigned: type (proliferating, quiescent, or necrotic), adhesion strength to neighboring cells, and the volume of the cell. For growing cells, the properties also include growth rate, metabolic rate (nutrient uptake and waste production).

• The total energy of the system depends on cell-cell surface interaction due to adhesion, cell elastic bulk energy due to growth, and chemical energy due to cell interaction with local chemical gradients (oxygen, nutrients e.g., glucose, and wastes e.g., lactate).

• The lattice evolves by a Monte Carlo process. At each Monte Carlo step, a random change is made at a lattice site. The change in energy as a result of the perturbation is computed. If the proposed change decreases the total energy of the system, then the change is accepted and the next Monte Carlo step is made. If the change results in an increase of the energy, then the change is accepted with a probability that is dependent a Boltzmann parameter.

• The chemical dynamics are solved using an irregular 3D grid with each point of the this irregular grid being the center of mass of the cell.

(continued)

• Two different time scales are used. For cell dynamics, there are many Monte Carlo steps for each real time step (~cell cycle) and for the chemical dynamics the number of numerical steps are determined by chemical diffusion and metabolic rates. There are approximately 70 Monte Carlo steps per day.

• Cell growth and division are incorporated in the volume constraint in the energy.

• Each cell has its own cell cycle. This cycle is sensitive to its nutrient and waste environment. Different nutrients affect the cell cycle differently. The diffusion rates for different nutrients can be different.

• When the nutrient levels fall below some threshold value or the waste level is above some threshold level for a particular cell, then the cell stops proliferating and becomes quiescent. Depending on nutrients and waste levels, a quiescent cell becomes necrotic.

• Cells divide only when a cell reaches the last stage of the cell cycle and its volume reaches a targeted valued. A cell splits along its longest axis into two daughter cells and these two cells may inherit the state of the mother cell or undergo mutation with some given probability.

Energy

€

E = Jτ (s)τ (s' )(1−δs, s' )lattice sites

∑ + λ v [cells

∑ vs −Vs]2 + μ f C f

cells

∑

S - cell identification number (1,2,3,…)

(S) - cell type (proliferating, quiescent, or necrotic)

J(S)(S’) - coupling energy between cell types (S) and (S’)

v - elasticity

vs - cell Volume

Vs - target Volume

Cf - concentration of chemical (nutrient or waste)

µf - chemical potential

€

Energy = Eadhession + E shape + Echemical

Chemical Diffusion in Tumor

€

∂CO2

∂t= DO2

∇ 2CO2 - a(x, y,z) CO2

|∂Ω = CO2

0

∂Cn

∂t= Dn∇

2Cn - b(x, y,z) Cn |∂Ω = Cn

0

∂Cw

∂t= Dw∇

2Cw + c(x, y,z) Cw |∂Ω = 0

€

CO2= concentration of oxygen

Cn = concentration of nutrient

Cw = concentration of waste

a,b,c = source /decay terms [c = Ψ(a,b)]

DO2,Dn ,Dw = diffusion parameters

∂Ω = boundary of tumor

Nutrient Diffusion

(from Jiang & Pjesivac-Grbovic)

Simulation Example(after 440 Monte Carlo steps)

(from Jiang & Pjesivac-Grbovic)

Waste Distribution

(from Jiang & Pjesivac-Grbovic)

Number of Cells over Time

(from Jiang & Pjesivac-Grbovic)

Growth of Tumor - 12 Hours

(from Jiang & Pjesivac-Grbovic)

Day 3

(from Jiang & Pjesivac-Grbovic)

8.5 Days

(from Jiang & Pjesivac-Grbovic)

Growth in Number of Cellsversus O2 and Nutrient

(from Jiang & Pjesivac-Grbovic)

Volume of Cellsversus O2 and Nutrient

(from Jiang & Pjesivac-Grbovic)