New therapies for cancer: can a mathematician help? 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. How did I get into this? 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 Questions that 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?

Spatial Tumor Growth Deterministic & Probabilistic:2D and 3D Image Courtesy 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: = 100 = 75= 25= 50 Sample RULE: All cells divide Max 100 per grid element - extras move to adjacent grid elements Modeling Tumor Growth and Treatment L.G. de Pillis & A.E. Radunskaya

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)

MODEL EXTENSIONS, continued… 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: 1.Growth and maintenance of cells depends on the rate of cellular energy (ATP) metabolized from nearby nutrients. 2.Nutrient consumption rates depend on pH levels and glucose and oxygen concentrations. 3.These tumor cells are able to produce ATP glycolitically more easily than normal cells, (so they survive better in an acidic environment). 4.Oxygen, glucose and lactate diffuse through tissue using an adapted random walk - mimics physiological process. 5.Parameters can be calibrated to a given tissue, micro-environment. 6.Immune cell populations and drugs can be added once model is calibrated.

KREBS CYCLE REVIEW

Normal Cellular Metabolism Metabolism in cancer cells: increased glycolysis Treatment under study

The Hybrid CA: 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…

The Oxygen consumption rates are the same for both tumor and normal cells: Concentrations Modeled (in mM): [O 2 ] - concentration of oxygen molecules: O [G] - concentration of glucose molecules: G [H] - concentration of hydrogen ions from lactate: H, pH = -log 10 (H / 1000) 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 [O 2 ] at different pH levels and glucose concentrations

Consumption equations: Glucose consumption: Oxygen, Hydrogen and Glucose dependent where the index, i, in the parameters,c i,  i, q i, is either T (tumor), or N (normal), indicating the ability of the cell-type to metabolize glycolytically. [ c T > c N : “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 concentration is 5.5 mM ) Glucose consumption as a function of [O 2 ] at different pH levels

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:

ATP production by tumor cells as a function of [O 2 ] at different pH levels and at different Glucose concentrations Cells are extremely sensitive to micro-environment:

ATP production as a function of [O 2 ] for two cell types at different pH levels. % glycolytic differs only at low O 2,and then not by much!

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 q perm, the permeability coefficient the constant of proportionality. Scale as a function of pressure, x, (modeled by number of neighboring cells). Effect of pressure from surrounding tumor cells on blood vessels

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

A MARGULIS NEIGHBORHOOD is a 3-by-3 square. It represents the computatioal neighborhood in this model. EXAMPLE: The grid is divided into Margulis neighborhoods: Modeling Tumor Growth and Treatment L.G. de Pillis & A.E. Radunskaya C U LR D The whole grid is represented by a Matrix. A Margulis neighborhood with center at row i, column j is:

Model Flow:

Pressure from surrounding cells squeezes blood vessels and restricts flux in and out. Necrotic Tumor Cells Proliferating Tumor Cells Flow from blood vessels is restricted in the tumor

ATP available for tumor cells depends on micro- environment and metabolic activity. Necrotic Tumor Cells Proliferating Tumor Cells Dark blue areas show ATP below maintenance level Light blue areas show ATP levels adequate for growth

Add cellular automata models here … CA simulation (2) results: two initial tumor colonies of 80 cells each. Tumor growth shows hypoxic regions after 200 days.

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

The tumor affects the acidity of the micro-environment:

Summer, 2005 CA model: include adhesion (Chris Dubois) Validate dirty diffusion (Darren Whitwood)

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

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. Cancer to normal cell consumption Normal to cancer cell diffusion coefficient Modeling Tumor Growth and Treatment L.G. de Pillis & A.E. Radunskaya A blood vessel runs along the top of each square

Lower LeftUpper LeftLower RightUpper Right Spatial Tumor Growth Immune Resistance Experiments Decreasing Immune Strength 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 Thanks: Dann Mallet Final Tumor Shape: 340 Iterations Tumor Growth in Time: 340 Iterations Modeling Tumor Growth and Treatment L.G. de Pillis & A.E. Radunskaya

Spatial Tumor Growth NK and CD8 Immune Response Thanks: Dann Mallet Simulation1 and 2: NK & CD8Simulations 1 and 2: Tumor Modeling Tumor Growth and Treatment L.G. de Pillis & A.E. Radunskaya

Spatial Tumor Growth NK and CD8 Immune Resistance Thanks: Dann Mallet Simulation 2: Tumor Pop’nSimulation 1: Tumor Pop’n Modeling Tumor Growth and Treatment L.G. de Pillis & A.E. Radunskaya

Conclusions: 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. New Approaches to Tumor-Immune Modeling L.G. de Pillis & A.E. Radunskaya

Dept. of Mathematics Claremont, CA, USA Can a mathematician help? (Thanks for listening!) Ami Radunskaya