Download CA model - Pomona College

New therapies for cancer: can
a mathematician help?
SPATIAL MODELS
HYBRID CA IMPLEMENTATION
A.E. Radunskaya
Math Dept., Pomona College
with help from others…
How did I get into this?
My background: dynamical systems, ergodic theory
(how things change in time, probabilistic interpretation.)
And it all started here:
ST. VINCENT’S
MEDICAL CENTER
And it was this guy’s fault:
Tom Starbird Pomona graduate, Math PhD, now at the Jet
Propulsion Laboratory.
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?
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.lbah.com/Rats/rat_mammary_tumor.htm
Image Courtesy http://www.ssainc.net/images/melanoma_pics.GIF
http://www.loni.ucla.edu/~thompson/HBM2000/sean_SNO2000abs.html
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 timestep to the next.
EXAMPLE: The grid is a discretization of a slice of tissue:
Sample
RULE:
All cells
divide
8
10
8
10
10
Max 100 per
grid element extras move
to adjacent
grid elements
= 100
= 75
= 50
= 25
10
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
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Normal Cellular Metabolism
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Metabolism in cancer
cells: increased glycolysis
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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…
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:
dO 
bO  O 


 aO 
n 


dt
GH 
O

k
O 

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
Consumption equations:
Glucose consumption:
Oxygen, Hydrogen and Glucose dependent

dG
c ibG  G 
m
 H aG 


dt
O   i G  kG 

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
c ibG
i 
qi H m  aG
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):
H  1.56 105 G

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: dATPO 


 dt 
dATPO  dO 
 2
 2.36
dt
 dt 
• ATP produced glycolytically: dATPG 


 dt 


dATPO
dGO
 30
dt
dt
dG 4 dATPG dG
 O


dt 3 dt
dt
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
x
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).

1

x  xT
q(x)  
1

 x Max  xT
x  [0, xT ]
q(x)
x  [xT , x Max ]

x
xT
xMax

At each point in space and time, the
concentration of a nutrient is given by:
u
 (consumption)  (in _ rate)  (out _ rate)
t
u
 (consumption) (uxx  uyy )
t
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 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.
U
L
C R
D
A Margulis neighborhood
with center at row i,
column j is:
x i, j  C
x i, j 1 R
x i, j1 L
x i1, j  U
x i1, j  D
Model Flow:
Pressure from surrounding cells squeezes blood vessels and
restricts flux in and out.
Proliferating
Tumor Cells
Necrotic
Tumor Cells
Flow from blood vessels
is restricted in the tumor
ATP available for tumor cells depends on microenvironment and metabolic activity.
Light blue areas show ATP
levels adequate for growth
Proliferating
Tumor Cells
Necrotic
Tumor Cells
Dark blue areas show ATP
below maintenance level
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.
QuickTime™ and a
H.263 decompressor
are needed to see this picture.
The tumor affects the acidity of the micro-environment:
QuickTime™ and a
DV/DVCPRO - NTSC decompressor
are needed to see this picture.
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
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:
Tumor Growth in Time:
340 Iterations
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
Thanks: Dann Mallet
Simulations 1 and 2: Tumor
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
Thanks: Dann Mallet
Simulation 2: Tumor Pop’n
New Approaches to Tumor-Immune Modeling
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.
Can a mathematician help?
(Thanks for listening!)
Ami Radunskaya
Dept. of Mathematics
Claremont, CA, 91711
USA
[email protected]