New therapies for cancer: can a mathematician help?

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New therapies for cancer can a mathematician
help?

• SPATIAL MODELS
• HYBRID CA IMPLEMENTATION
• A.E. Radunskaya
• Math Dept., Pomona College
• with help from others

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How did I get into this?

• My background dynamical systems, ergodic theory
• (how things change in time, probabilistic
  interpretation.)
• And it all started here
• And it was this guys fault
• Tom Starbird -
• Pomona graduate, Math PhD, now at the Jet
  Propulsion Laboratory.

ST. VINCENTS MEDICAL CENTER
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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)

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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 bodys own resources?

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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?

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Spatial Tumor Growth
Deterministic Probabilistic2D and 3D
http//www.lbah.com/Rats/ovarian_tumor.htm
http//www.loni.ucla.edu/thompson/HBM2000/sean_SN
O2000abs.html
http//www.lbah.com/Rats/rat_mammary_tumor.htm
Image Courtesy http//www.ssainc.net/images/melano
ma_pics.GIF
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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
25
50
100
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• 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)

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• 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
• 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.

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KREBS CYCLE REVIEW
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• Normal Cellular Metabolism

Metabolism in cancer cells increased glycolysis
Treatment under study
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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

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• 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.
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Oxygen consumption as a function of O2 at
different pH levels and glucose concentrations
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Consumption equations
Glucose consumption 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 gt cN tumor
gluttony (Kooijman) , and
prevents glucose consumption from going to
infinity as O goes to zero (q is the maximum
consumption rate).
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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 )
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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.
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Calculation of ATP Production from Oxygen and
Glucose Consumption

• ATP produced aerobically
• ATP produced glycolytically

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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
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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!
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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).

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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 -
  (LRUD) ).

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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
The whole grid is represented by a Matrix. A
Margulis neighborhood with center at row i,
column j is
U
C
L
R
D
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Model Flow
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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
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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
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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
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CA Simulation Movie - a snapshot every 20 days
for 200 days showing tumor growth and necrosis.
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The tumor affects the acidity of the
micro-environment
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Summer, 2005

• CA model
• include adhesion (Chris Dubois)

Validate dirty diffusion (Darren Whitwood)
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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).

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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

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• 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
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• Spatial Tumor Growth
• Immune Resistance Experiments

Decreasing Immune Strength
Lower Left
Upper Left
Lower Right
Upper Right
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• Spatial Tumor Growth
• Chemotherapy Experiments Every Three Weeks

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• Spatial Tumor Growth
• Chemotherapy Experiments Every Two Weeks

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• Spatial Tumor Growth
• No Immune Response

Final Tumor Shape 340 Iterations
Tumor Growth in Time 340 Iterations
Thanks Dann Mallet
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• Spatial Tumor Growth
• NK and CD8 Immune Response

Simulation1 and 2 NK CD8
Simulations 1 and 2 Tumor
Thanks Dann Mallet
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• Spatial Tumor Growth
• NK and CD8 Immune Resistance

Simulation 2 Tumor Popn
Simulation 1 Tumor Popn
Thanks Dann Mallet
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Conclusions
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 hosts
  distribution of energy, and inform holistic
  approaches to treatment.
• Hybrid cellular automata might provide efficient
  and (somewhat) realistic computational
  environments.

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Can a mathematician help? (Thanks for
listening!) Ami Radunskaya
Dept. of Mathematics Claremont, CA,
91711 USA aradunskaya_at_pomona.edu