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MODELING AND COMPUTER SIMULATIONS:
TOOLS TO SUPPORT EXPERIMENTAL
RESEARCH IN BIOPHYSICS
APPLICATIONS TO TUMOR GROWTH
M.Scalerandi, P.P.Delsanto, M.Griffa
INFM - Dip. Fisica, Politecnico di Torino, Italy
e-mail: [email protected]
Also with:
Istituto Nazionale
di Fisica della Materia
G.P.Pescarmona, Università di Torino, Italy
C.A.Condat, University of Puerto Rico at Mayaguetz, US
M.Magnano, Ospedale Umberto I, Torino, Italy
B.Capogrosso Sansone, University of Massachusets, US
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GOALS of MODELING
Support in the interpretation of data
Optimization of experiments
Predictive power
• Prediction of the evolution of a tumor “in vivo” (???)
• Suggest new experiments
• Preliminary validation and formulation of hypotheses
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MODELING
Formulation of a problem into mathematical terms
(equations), which allows to obtain predictions
Ingredients
• basic knowledge (biological, physical, biochemical, etc.
• phenomenology (in vivo and in vitro observations)
• hypotheses (to bridge the gap !)
Simplification: impossible to describe entirely the real
system (mathematical complexity)
 Specific problem identification
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Validation: rejection or acceptance of the hypotheses
through comparison with data
 Design of new experiments
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COMPUTER SIMULATIONS
The tool to obtain predictions from the model
• computers are capable to solve a problem regardless of the
mathematical difficulty
• computers are fast (parallel computing) and cheaper than real
experiments
• computers may describe the spatio-temporal evolution of a
given system
• nevertheless the computational time may increase
dramatically with the complexity of the problem (keep it
simple to avoid computational complexity !)
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MODELING AND COMPUTER SIMULATIONS
Determination of the problem
Selection of few mechanisms (eventually aggregation of
biological properties into a single mechanism)
Restriction of the field of validity
!
Additional hypotheses
New mechanisms
!
Math. inconsistency
Simulations
Different hypotheses
Failure
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No problem to restart!
Comparison with data
!
• prediction of new results not yet
observed: suggest new experiments
• confirmation of biological assumptions
• optimization of existing experiments
• performing experiments not feasible in
reality (e.g. prediction of the growth
outcome without any therapy in a patient)
• application to a different problem
OUR MODEL. I specific problem
Radiotherapy,
chemiotherapy, etc.
!
The problem: tumor growth depends upon
the intrinsic neoplastic properties, the host
properties and the action of drugs
Regulation of cells
behavior according to
the environment.
Regulation of
apoptotic inhibition
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!
Cellular growth
is controlled by
nutrients availability
Apoptosis is regulated
by adhesion properties
which are modulated by
pressure constraints on
the neoplasm
Antiangiogenetic
therapies
!
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OUR MODEL. II biological mechanisms
CELLS
 Absorption (energy storing)
 Metabolism (energy consumption)
 Mitosis, necrosis and apoptosis
(depending on the absorbed signals)
 Adhesion, metastasis and invasion
(diffusion)
Emission
Inhibition or
activation
SIGNALS
 Growth factors,
nutrients, tumor
angiogenetic factors,
apoptosis inhibitors
 “Fast” diffusion
ENVIRONMENT
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OUR MODEL. III hypotheses

  p it
   1  exp t
 i 

t
i
t
i




 signals consumption to perform various
functions;
absorbed signal
 signals absorption by means of receptors
available signal
 it 1   it   it   it  k it
 mitosis (> threshold) and necrosis (<
0)
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PARAMETERS
Important task in modeling is the choice of
reasonable values for the large number of parameters
(which increases dramatically with the problem
complexity: parameter space complexity):
a) parameters with a biological (physical) interpretation
experimentally measured
 estimate, at least, the order of magnitude
b) parameters with a biological interpretation, difficult to
measure or never measured
 suggest experiments or indirect measurements
b) parameters with a purely mathematical meaning
 used to fit the data
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SIMULATIONS AND VALIDATION.
I - AVASCULAR PHASE
Spherical shape
Necrotic core
Latency at a radius of about 200 mm
Gompertzian growth law
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SIMULATIONS AND
VALIDATION.
I - AVASCULAR PHASE
The cord grows around the vessel
and reaches an equilibrium
“dynamical” state
A necrotic core is formed at the
front of the neoplasm
=0.02
=0.05
=0.07
=0.098
The cord radius increases when
the nutrient consumption
decreases
The cord radius (calculated from
the volume) oscillates between
50 and 130 mm, in agreement
with in-vivo data
Experimental data from J. V. Moore, H. A. Hopkins, and W. B. Looney, Eur. J. Cancer Clin.
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Oncol. 19, 73 (1984).
SIMULATIONS AND VALIDATION
II - CT SCANS
COMPARISON WITH CLINICAL DATA
Numerical Results
B
CTScan
B
D
!
B
A
A
C
D
A
C
C
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Temporal
sequence
Clinical data:
Dr. M.Magnano
Head and Neck Division
Ospedale Umberto I
Torino, Italy
• identification of features which might help
a better prediction of the tumor margins
(optimization)
• prediction of the tumor evolution without
intervention (not feasible experiment)
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SIMULATIONS AND
VALIDATION
III - ANGIOGENESIS
MORPHOLOGY
• latency in the avascular phase
• directional vessels growth
• correct profile of the
capillaries distribution
• infiltration of the vascular
system inside the tumor mass
Cancer cells
Cancer cells
(no angiogenesis)
T =2000
T=10000
T=20000
For experimental data, see
e.g. M.I. Koukourakis et al.,
Cancer Res. 60, 3088 (2000)
T=25000
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Vessels
For experimental data see e.g. R. Cao et al.,
Proc. Natl. Acad. Sci. USA 96, 5728 (1999)
z=1
z = 0.4
Angiogenesis may be inhibited when the
affinity of EC for TAF’s is reduced (e.g.
by inhibiting VEGFR2)
Experimentally observed
z = 0.1
SIMULATIONS AND
VALIDATION
III - ANGIOGENESIS
INHIBITION
z = 10
Surprisingly angiogenesis is also inhibited
when affinity is increased.
For experimental evidence see e.g.
H.H.chen et al., Pharmacology 71, 1
(2004)
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t = 20000 t
t = 50000 t
t = 20000 t
t = 70000 t
t = 20000 t
t = 180000 t
t = 20000 t
t = 180000 t
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SIMULATIONS AND VALIDATION
III - ANGIOGENESIS VEGFR2-inhibition
Slight stimulation of VEGF receptor 2
2500
z = 0.4
(a)
(b)
3
Tumor volume (mm )
3000
2000
Simulation
Experiment
1500
z = 0.3
1000
500
0
z = 0.1
1
z = 0.2
2
X
3
4
Action of a monoclonal antibody (2C3) inhibiting VEGFR2
Experimentally has been observed a reduction up to
70% of the VEGF affinity a a function of the drug dose
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Exp. data from Brekken et al., Cancer Research 60, 5117 (2000)
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SIMULATIONS AND VALIDATION
IV - ROLE OF THE ENVIRONMENT RIGIDITY
Multicellular spheroids growth
in a matrix with different
percentuals of diluted agarose
• different agar concentrations are
simulated using different rigidities
of the matrix
•comparison of the average
diameter of the spheroids (in
equilibrium conditions) between
numerical and experimental data at
different concentrations
Experimental data from G.Helmlinger et al., Nature Biotechnology 15 (1997) 778
SIMULATIONS AND VALIDATION
IV - ROLE OF THE ENVIRONMENT RIGIDITY
Cellular density
(Variation with respect
to the 0% agar matrix)
Mitosis rate
(Variation with respect
to the 0% agar matrix)
N.B. both in experiments and simulations
the final pressure on the spheroids is
independent from the agar concentration
Experimental data from
G.Helmlinger et al., Nature
Biotechnology 15 (1997) 778
CONCLUSIONS
• Modeling and simulations = simplified version of a specific real problem
• Hypotheses must always be introduced
• A validation through comparison with experimental data and application of
the model to novel problems is needed
• suggest new experiments and new questions
• optimize existing experiments (in particular for therapies)
• validate preliminary hypotheses
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