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### New therapies for cancer: can a mathematician help?

### How did I get into this?

### The Hybrid CA:

### Consumption equations:

### At each point in space and time, the concentration of a nutrient is given by:

### Conclusions:

SPATIAL MODELS

HYBRID CA IMPLEMENTATION

A.E. Radunskaya

Math Dept., Pomona College

with help from others…

My background: dynamical systems, ergodic theory

(how things change in time, probabilistic interpretation.)

And it all started here:

And it was this guy’s fault:

Tom Starbird -

Pomona graduate, Math PhD, now at the Jet Propulsion Laboratory.

ST. VINCENT’S MEDICAL CENTER

Long Term Project Goals

- Goal: Design mathematical tumor models
- Evaluate current mathematical models
- Create more detailed qualitative models
- Determine alternate treatment protocols
- In cooperation with:
- Dr. Charles Wiseman, M.D., Head of Los Angeles Oncology Institute Mathematics of Medicine Group
- Prof. L. dePillis, Harvey Mudd (and other Mudders)
- Pomona College students: Darren Whitwood (‘07), Chris DuBois (‘06), Alison Wise (‘05 - now at NIH) …

(… last summer)

Physiological Questionsthat we would like to answer …

- Pathogenesis: How do tumors start? How and why do they grow and/or metastasize?
- Immune surveillance: under what conditions is the body able to control tumor growth? (Childhood cancer is much more rare than adult cancer.)
- Treatment: how do various therapies work in interaction with the body’s own resources?

Modeling Questions

- A mathematical model is a (set of) formulas (equations) which describe how a system evolves through time.
- When is a model useful?

“Medical progress has been empirical,even accidental.”

- How do we determine which models are `better’ ?
- Can a deterministic model ever be sufficiently realistic?
- Is the model sufficiently accurate to answer the questions:

How much? How often? Where?

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

Spatial Tumor Growth

Deterministic & Probabilistic:2D and 3D

http://www.lbah.com/Rats/ovarian_tumor.htm

http://www.loni.ucla.edu/~thompson/HBM2000/sean_SNO2000abs.html

http://www.lbah.com/Rats/rat_mammary_tumor.htm

Image Courtesy http://www.ssainc.net/images/melanoma_pics.GIF

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

To add spatial variability, need populations at each point in space as well as time.

A CELLULAR AUTOMATA (CA)is a grid ( in 1-d, 2-d, or 2-d), with variables in each grid element, and rules for the evolution of those variables from one time-step to the next.

EXAMPLE:The grid is a discretization of a slice of tissue:

Sample RULE:

All cells divide

Max 100 per grid element - extras move to adjacent grid elements

= 75

= 50

= 25

= 100

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- The modeling process consists of describing (local or global) rules for the growth, removal, and movement of:
- Tumor cells
- Nutrients
- Normal cells
- Immune cells
- Metabolic by-products (lactate)
- Energy (ATP)
- Drugs (or other therapy)

Deterministic cellular automata (CA) model including oxygen, glucose, and hydrogen diffusion, as well as multiple blood vessels which are constricted due to cellular pressure.

Model assumptions:

Growth and maintenance of cells depends on the rate of cellular energy (ATP) metabolized from nearby nutrients.

Nutrient consumption rates depend on pH levels and glucose and oxygen concentrations.

These tumor cells are able to produce ATP glycolitically more easily than normal cells, (so they survive better in an acidic environment).

Oxygen, glucose and lactate diffuse through tissue using an adapted random walk - mimics physiological process.

Parameters can be calibrated to a given tissue, micro-environment.

Immune cell populations and drugs can be added once model is calibrated.

Start with some initial distribution of normal cells, blood vessels, nutrients, and a few tumor cells.

Oxygen, glucose diffuse through the tissue from the blood vessels, and are consumed by the cells.

Hydrogen and ATP (energy) is produced by the cells during metabolism.

If there is not enough ATP for the cells to maintain function, they become necrotic (die).

If there IS enough ATP for maintenance, then the cells live.

If there is enough ATP left over for reproduction they do that.

If tumor cells get crowded, they move.

If the blood vessels get squeezed, nutrient (and drug) delivery is slowed down.

If the blood vessels get squeezed too much, they collapse.

PUT THIS SCENARIO (ALONG WITH KREBS CYCLE) INTO EQUATIONS…

Concentrations Modeled (in mM):

- [O2 ] - concentration of oxygen molecules: O
- [G] - concentration of glucose molecules: G
- [H] - concentration of hydrogen ions from lactate: H,
- pH = -log10(H / 1000)

The Oxygen consumption rates are the same for both tumor and normal cells:

These parameters have been measured experimentally for some tumors and normal cells, at different glucose, pH and oxygen concentrations by, e.g., Casciari.*

Oxygen consumption as a function of [O2] at different

pH levels and glucose concentrations

Oxygen, Hydrogen and Glucose dependent

where the index, i, in the parameters,ci, i , qi, is either T (tumor), or N (normal), indicating the ability of the cell-type to metabolize glycolytically.

[ cT> cN: “tumor gluttony” (Kooijman*) ] , and

prevents glucose consumption from going to infinity as O goes to zero (q is the maximum consumption rate).

Results from the model simulation, parameters calibrated so that concentrations to agree with data

Glucose consumption as a function of [O2] at different pH levels

(Glucose concentration is 5.5 mM )

Lactate (Hydrogen) is produced when a glucose molecule is metabolized (either aerobically or anaerobically):

If metabolism occurs glycolytically, more lactate is produced, since more glucose is required to produce the same amount of energy (ATP).

Intracellular competition through metabolic differences:

Tumor cells increase the acidity of the micro-environment through glycolysis. Normal cells show decreased metabolism in an acidic environment, and both cell types consume more oxygen when pH is lower.

Calculation of ATP Production from Oxygen and Glucose Consumption:

- ATP produced aerobically:
- ATP produced glycolytically:

Cells are extremely sensitive to micro-environment:

ATP production by tumor cells as a function of [O2] at different pH levels and at different Glucose concentrations

ATP production as a function of [O2] for two cell types at different pH levels.

% glycolytic differs only at low O2,and then not by much!

Effect of pressure from surrounding tumor cells on blood vessels

- Physical pressure from proliferating tumor and necrotic cells surrounding a blood vessel may compress the vessel and restrict nutrient flow, eventually causing vascular collapse.
- The rate of flow of small molecules through vessel walls is proportional to the gradient across the wall, with qperm, the permeability coefficient the constant of proportionality. Scale as a function of pressure, x, (modeled by number of neighboring cells).

Discretize time and space:

u = t (consumption rate - (u) )

Derivatives (consumption) become differences (O, G, H)

Second derivatives (diffusion) become differences of differences ( (C-L)-(R-C) + (C-U)-(C-D)= 4C - (L+R+U+D) ).

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

A MARGULIS NEIGHBORHOODis a 3-by-3 square. It represents the computatioal neighborhood in this model.

EXAMPLE:The grid is divided into Margulis neighborhoods:

The whole grid is represented by a Matrix.

A Margulis neighborhood with center at row i, column j is:

U

L

C

R

D

Pressure from surrounding cells squeezes blood vessels and restricts flux in and out.

Necrotic

Tumor Cells

Flow from blood vessels

is restricted in the tumor

Proliferating

Tumor Cells

ATP available for tumor cells depends on micro-environment and metabolic activity.

Light blue areas show ATP

levels adequate for growth

Proliferating

Tumor Cells

Dark blue areas show ATP

below maintenance level

Necrotic

Tumor Cells

CA simulation (2) results: two initial tumor colonies of 80 cells each.

Tumor growth shows hypoxic regions after 200 days.

Add cellular automata models here …

CA Simulation: Movie - a snapshot every 20 days for 200 days showing tumor growth and necrosis.

Advantages of DEB approach:

- Cell growth and death are predicted by metabolic efficiency, not by macroscopic size (controversial).

- Competition between cell types is indirect (no need to conjecture complicated formulas describing interactions between tumor and normal tissue).

- The model is naturally able to include the effects of immune response and therapies (delivery and biodistribution of immunotherapy and vaccines).

- Calculation of ATP production can be used to quantify overall health (as opposed to markers from peripheral blood).

Numerical Advantages of Hybrid Cellular Automaton Approach

- parallelizable
- potential for a hierarchical, multi-grid approach
- easily adaptable to specific organs, tumor types and treatment protocols (we are starting with CNS melanoma, peptide vaccine, DC vaccines)
- diffusion modeled locally, incorporating tissue heterogeneity, simplifying computations

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- Spatial Tumor Growth: one nutrient, one blood vessel
- Nutrients diffuse from blood vessel (at top) in a continuous model (PDE).
- Cells proliferate according to a probabilistic model based on available nutrients.

A blood vessel runs along the top of each square

Normal to cancer cell diffusion coefficient

Cancer to normal cell consumption

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- Spatial Tumor Growth
- Immune Resistance Experiments

Decreasing Immune Strength

Lower Left

Upper Left

Lower Right

Upper Right

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- Spatial Tumor Growth
- Chemotherapy Experiments: Every Three Weeks

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- Spatial Tumor Growth
- Chemotherapy Experiments: Every Two Weeks

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- Spatial Tumor Growth
- No Immune Response

Final Tumor Shape:

340 Iterations

Tumor Growth in Time:

340 Iterations

Thanks: Dann Mallet

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- Spatial Tumor Growth
- NK and CD8 Immune Response

Simulation1 and 2: NK & CD8

Simulations 1 and 2: Tumor

Thanks: Dann Mallet

Modeling Tumor Growth and Treatment

L.G. de Pillis & A.E. Radunskaya

- Spatial Tumor Growth
- NK and CD8 Immune Resistance

Simulation 1: Tumor Pop’n

Simulation 2: Tumor Pop’n

Thanks: Dann Mallet

New Approaches to Tumor-Immune Modeling

L.G. de Pillis & A.E. Radunskaya

Spatial heterogeneity in tissue is a common characteristic of cancer growth.

Vasculature (angiogenesis) is a crucial factor in tumor invasion.

The ability of tumor cells to metabolize in an anaerobic environment is also an important factor in tumor invasion.

The metabolic pathway might explain the host’s distribution of energy, and inform “holistic” approaches to treatment.

Hybrid cellular automata might provide efficient and (somewhat) realistic computational environments.

(Thanks for listening!)

Ami Radunskaya

Dept. of Mathematics

Claremont, CA, 91711

USA

aradunskaya@pomona.edu

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