competition and cooperation: tumoral growth strategies
DESCRIPTION
Competition and cooperation: tumoral growth strategies. Carlos A. Condat Silvia A. Menchón CONICET Fa.M.A.F., Universidad Nacional de Córdoba. Collaborators:. P.P. Delsanto, M. Griffa, C. Guiot, Politecnico di Torino, Italy. R. Ramos, University of Puerto Rico at Mayagüez. - PowerPoint PPT PresentationTRANSCRIPT
LAWNP’05
Competition and cooperation: tumoral Competition and cooperation: tumoral
growth strategies growth strategies
Carlos A. Condat
Silvia A. Menchón
CONICET
Fa.M.A.F., Universidad Nacional de Córdoba
LAWNP’05
Collaborators:
P.P. Delsanto, M. Griffa, C. Guiot, Politecnico di Torino, Italy
R. Ramos, University of Puerto Rico at Mayagüez
T.S. Deisboeck, Harvard University
LAWNP’05
•Cancer growth: Macroscopic and mesoscopic approaches.
•Macroscopic approach: Ontogenetic growth law
•Application to tumors
•Spheroids – Applications of the macroscopic theory
•Mesoscopic approach: Model rules
•Simulations
•Single-species model
•Interspecies competition and tumor evolution
•Conclusions
Outline
LAWNP’05
• Carcinogenic change
• Growth
• Invasion
• Metastasis
Cancer Cancer dynamicsdynamics..
LAWNP’05
Microscopic description
Study of individual cell properties
In vitro experiments
Biological models
Macroscopic description
Tumor development as a single entity
In vivo experimentsClinical results
Mesoscopic approachSimulation of the behavior of cell
clusters and their interactions
effective parameter
s
predictions
LAWNP’05
The growth of all living organisms follows the same master curve, if we suitable rescale the mass and use a dimensionless time .
(West, Brown and Enquist, Nature, 2001)
This statement can be “proved” using two assumptions:
A: Energy is conserved.
B: The nutrient distribution networks are fractal(circulatory system in mammals, tracheal system in insects, xylem in trees).
Note: assumption B is not universally accepted.
Ontogenetic growth law
LAWNP’05
West, Brown and Enquist, Nature, 2001
Universal
growth curve
(m()/M)1/4
Conservation of energy + fractality of distribution network
LAWNP’05
West, quoted in Nature:
“ If Galileo had been a biologist, he would have written a big fat tome on the details of how different objects fall
at different rates.”
J. Niklas, on the work of West, Brown and Enquist:
Enquist is working on a project“as potentially important to biology
as Newton’s contributions are to physics”
In: Trends. Ecol. Evol.
The hype:
LAWNP’05
ONTOGENETIC GROWTH LAW
The growth of an organism is mediated by cell division and fed by metabolism.
Metabolic Energy
Maintenance
Cell reproduction
Maintenance includes cell replacement.
LAWNP’05
Energy conservation equation:
maintenance
creation
B: energy income rate to the organism cells : single cell metabolic rate
: energy to create a single cellN: total cell number
This equation can be easily turned into a simple differential equation.
LAWNP’05
mc:single cell mass
m = Nmc: organism mass
To be modelled: the basal metabolic rate B(m).
bmmBm
dt
dm c
B ~ m3/4 [Kleiber, 1932 (on phenomenological grounds; West, 2001 (fractal distribution networks)].
b = /
B ~ m2/3 [other authors].
Generally accepted: B ~ mp : a power law.
There are hundreds of power laws in biology!
LAWNP’05
Setting a = mc B0/, b=/,
bmamdt
dm p
Maximum body size:
)1/(1
0
1/1 p
c
pBm
b
aM
[Take dm/dt = 0]
LAWNP’05
If m0 is the mass at birth, and
p
M
mbtp
1
01ln1
we obtain the universal solution:
eM
mp
11
e- is the proportion of energy devoted to cell reproduction. It goes to zero as grows.
This is the curve plotted by West et al., with p = 3/4.
LAWNP’05
Does cancer follow a universal growth law?
We would like to understand the kinetics of tumor growth.
At first: avascular growth (p = 2/3 ?)
Later: angiogenic growth (p = 3/4 ?)
Conjecture:
Energy is conserved, but, what is B(p)?
As for living beings , B(p) ~ mp.
LAWNP’05
Molecular diffusion towards a sphere:
Nutrient molecules
Cell, spheroid
B(m) = B0m2/3
p = 2/3 results from simple scaling between surface and volume.
LAWNP’05
Angiogenesis
At later times, angiogenesis changes the tumor feeding patterns.
LAWNP’05
p=3/4 ? B(m) = B0mp
LAWNP’05
Fit with p=3/4 by Guiot et al. J. Theor. Biol. (2003).
(m()/M)1/4
Experimental results
LAWNP’05
(m()/M)1/4
Fit with p=3/4 by Guiot et al. J. Theor. Biol. (2003).
Tumors implanted in rats and mice
LAWNP’05
(m()/M)1/4
Fit with p=3/4 by Guiot et al.
J. Theor. Biol. (2003).
LAWNP’05
Multicellular Tumor Spheroids
MTS: spherical aggregates of proliferating, quiescent, and necrotic cells
•In vitro models for the study of cancer cell biology.
•They can be grown under strictly controlled conditions.
•Spheroid-forming ability is inherent to solid tumor cells.
•Typically, they grow to diameters of up to 1.6 mm.
•A necrotized core appears when the diameter is ~ 0.8 mm.
LAWNP’05
http://www.vet.purdue.edu/cristal/dicspheroid.jpg
MulticellularTumor
spheroid
LAWNP’05
Do MSTs grow as live beings?
•Verify whether or not they grow according to West’s law.
•If so, MST’s can be used as test banks for growth theories:Use large groups of similar specimens, varying the environmental conditions.•Feeding is purely diffusive p = 2/3 (?)
•p = ¾ would suggest that West’s ideas are incorrect.
Unfortunately, both power laws yield similar-quality
fits!
LAWNP’05
The model is defined by,
bmamdt
dm p
There is a delay in the onset of nutrient absorption,which depends on the cell and the matrix.
We replace a by,
Tteata /1 1
T: effective accommodation time
We applied these ideas to various experimental situations.
LAWNP’05
Experiment I: Restrict feeding
(Freyer and Sutherland, Cancer Research, 1986)
The nutrient content of the medium is restricted. We model this by introducing a feeding restriction parameter f. f = 0 for a well-fed spheroid.
bmmtafdt
dm p 11
)1/(11 pfMm
Asymptotic spheroid mass:
m decreases as the nutrient is decreased.
LAWNP’05
Time variation of an undernourished spheroid mass [data: Freyer and Sutherland, 1986].
Solid curves: model fits (p=2/3). y-intercept: m0 = 2×10-6 g.
Final masses m, starting from lowest curve: 4.4 mg, 3.7 mg, 1.95 mg, and 3.56×10-5 g.
Accommodation time: T = 10 h.
p
M
tmty
1
1
Excellent fit, except for the very starved spheroid (f = 0.8).
Appl. Phys. Lett., 2004
LAWNP’05
Experiment II: Increase matrix rigidity
(Helmlinger et al., Nature Biotechnology, 1997)
Because of the increase in mechanical stress, growth is inhibited by increasing gel concentration.
Cells may be compacted, and the density changes.
We use the spheroid volume as the variable of interest.
tm
tv
LAWNP’05
Defining,
p
V
tvtz
1
1
the energy conservation equation is,
Tteg
fz
dt
gd
dt
dz /11
1ln
with:
p
Rttg
1
V: volume under conditions of nutrient saturation. R: final cell concentration
LAWNP’05
We must specify (t)
Note: (i) Nutrient availability and growth are closely related.
(ii) An increase in stress is a result of an increase in volume.
(iii) An increase in stress effectively hampers feeding.
TteRRt /0
)1/(11 pfVv
Ansatz:
0 : initial density.
Asymptotic volume:
LAWNP’05
Variation of spheroid volume under different mechanical stress conditions [Helmlinger et al., 1997]. Solid curves are model fits. p= 2/3Final volumes (in cm3) and accommodation times are, starting from the lowest curve: (6×10-4, 30 h), (3.8×10-5, 100 h), (2.65×10-5, T = 110 h), (4.88 ×10-6, 120 h).
p
V
tvtz
1)(
1)(
Appl. Phys. Lett., 2004
T increases, and final cell density (R) increases by a factor of up to 3.
LAWNP’05
Experiment III: periodic feeding (proposed)
Consider a periodic feeding protocol. Then,
bmmtcadt
dm p sin
After a transient , the live cell mass oscillates, following a hysteretic cycle.
)(sinsin)( tCtAtm
Transient length: tT = 1/b(1-p)
LAWNP’05
Hysteresis plots m vs sin(t)
tT = 0.1
tT = 0.1 tT=
1
tT = 10
Maximum remanence:tT = 1
This behavior is peculiar to “non-linear, non-classical” systems
(CAC,TSD, 2005).
LAWNP’05
Mesoscopic approach
•Instead of analyzing cancer as a whole, we propose a model for the behavior of groups of cells, based on single-cell properties.
•Define the growth rules.
•Perform simulations for tumors containing one or two cancer cell species.
First, we state the model rules.
LAWNP’05
)].exp(1[' p
•Feeding: cancer cells absorb free nutrient
(concentration p) at a rate
•Consumption: bound nutrient q is
consumed by cancer cells at a rate
)]./exp(1[' cqBoth rates are proportional to the concentration for low
concentrations and then saturate.
This is transformed into bound nutrient.
Growth rules
LAWNP’05
•Death: A low concentration of bound nutrient leads to cell
death. •Mitosis: A high concentration of bound nutrient leads to
cell replication.
•Migration: A cell that senses a low nutrient level in its
neighborhood tends to migrate.
DQic
iq
)(
)(
Death
)(
)(
ic
iqQM
Mitosis
MD QQ
Growth rules
DPic
ip
)(
)(Migration
LAWNP’05
Simulation
•Consider a piece of tissue of arbitrary shape, which is discretized using a square or cubic grid.
•Each node point represents a volume elementthat contains many cells and nutrient molecules.•Due to the complexity of the problem, we write all equations directly in their discrete form.•Initially the tissue is composed only of healthy cells (h per node) and nutrients [concentration p(i,t)].
Scalerandi et al., 1999; CAC et al., 2001.
LAWNP’05
The nutrient concentration evolves according to,
NN Diffusion Absorption Sources
• Once the steady-state is reached, a cancer seed is placed somewhere in the lattice.
•Cell populations are modified because of migration, reproduction, and death. Nutrient concentrations are modified through diffusion, absorption, and consumption.
•Discretized iteration equations embodying these rules are written and implemented in a simulation.
Simulation
iSipihipipiptipNN
i
',
LAWNP’05
Simulation
Here we consider a square piece of tissue,with a blood vessel running along the lower edge. There the free nutrient concentration is a constant, P0.
The cancer seed is placed at the center of the tissue.
Initial conditions: Iicic
00,
00, id
Iiqiq
00, (I)
Typical lattice sizes: 300300
LAWNP’05
Growth
Latency
Both data sets are well fitted by a power law with exponent 1/3. Power laws crop up everywhere!
Single species
These are two phase diagrams, corresponding to different values of .
LAWNP’05
Morphology
Red arrows: = 0.44Green
arrows: = 0.22
LAWNP’05
Coming out of latency
Method A: angiogenic development.
Mediates the transition between the spheroid and the vascularized stages.
Method B: cell mutations and emergence of a species having comparative advantages.
Cell mutations lead to the development of acquired resistance to chemotherapy. Chemotherapy may induce latency or remission, but fails when a resistant subspecies develops.
LAWNP’05
• We let a single-species tumor evolve up to a time tm .
• At tm some cells at a localized position mutate (i.e., some of their defining parameters are changed) and begin to compete for nutrients with the original population.
• If the original tumor is either latent or slowly growing, small parameter modifications may drastically alter the tumor evolution.
• The tumor evolution depends not only on the intrinsic properties of the new species, but also on the location of the mutation.
• Main determinants: local nutrient availability and local concentration of competing cells – there is intraspecies competition and there is inter-species competition.
Two species
LAWNP’05
t=25000
t=30000t=35000
tm=20000
Two species: restarted growth
Just latent
tumor has a = 0.44 mutation
Observefast growthof species 2
LAWNP’05
tm=20000
t=30000
t=40000t=45000
Two species: second latency
Original cancer well inside
latent region leads to second latency
LAWNP’05t=25000
tm=20000
t=35000
t=45000
Example: = 1
Restarted growth forcells with restricted mobility
LAWNP’05
No therapy: cancer cells, dead cells and healthy cells
G.Rivera, MS Thesis, UPR, 2005
Modeling therapy
Simultaneous snapshots
LAWNP’05
Cancer treated with immune therapy.Cancer, dead, and healthy cell concentrations.Lymphocyte concentration
Modeling therapy
LAWNP’05
No therapy With therapy
Therapy favors reproduction of surviving cancer cells, accelerating tissue destruction (!)
Modeling therapy
Modeling therapy can help to determine optimal therapeutic courses.
LAWNP’05
• Both macroscopic and mesoscopic techniques are useful to
study tumor growth.
• Macro: Ontogenetic growth laws describe observed
behavior (starving, stress) and lead to predictions.
• Meso: Simulations reproduce morphologies. Phase
diagrams are useful to predict tumor evolution.
• Modeling of subspecies competition can be useful for
therapy design. Mutations leading to an increase in
absorption rates are particularly aggressive.
• The success of a mutation depends not only on its intrinsic
competitive advantages, but also on its location.
Conclusions
LAWNP’05
• FUTURE WORK:
• Modeling therapies
• Relate macro and meso approaches
• Modeling metastasis
Conclusions