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

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• Carcinogenic change

• Growth

• Invasion

• Metastasis

Cancer Cancer dynamicsdynamics..

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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

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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

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West, Brown and Enquist, Nature, 2001

Universal

growth curve

(m()/M)1/4

Conservation of energy + fractality of distribution network

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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:

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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.

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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.

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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!

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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]

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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.

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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.

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Molecular diffusion towards a sphere:

Nutrient molecules

Cell, spheroid

B(m) = B0m2/3

p = 2/3 results from simple scaling between surface and volume.

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Angiogenesis

At later times, angiogenesis changes the tumor feeding patterns.

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p=3/4 ? B(m) = B0mp

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Fit with p=3/4 by Guiot et al. J. Theor. Biol. (2003).

(m()/M)1/4

Experimental results

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(m()/M)1/4

Fit with p=3/4 by Guiot et al. J. Theor. Biol. (2003).

Tumors implanted in rats and mice

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(m()/M)1/4

Fit with p=3/4 by Guiot et al.

J. Theor. Biol. (2003).

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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.

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http://www.vet.purdue.edu/cristal/dicspheroid.jpg

MulticellularTumor

spheroid

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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!

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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.

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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.

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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

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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

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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

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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:

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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.

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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)

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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).

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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.

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)].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

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•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

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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.

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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

',

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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

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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 .

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Morphology

Red arrows: = 0.44Green

arrows: = 0.22

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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.

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• 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

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t=25000

t=30000t=35000

tm=20000

Two species: restarted growth

Just latent

tumor has a = 0.44 mutation

Observefast growthof species 2

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tm=20000

t=30000

t=40000t=45000

Two species: second latency

Original cancer well inside

latent region leads to second latency

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tm=20000

t=35000

t=45000

Example: = 1

Restarted growth forcells with restricted mobility

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No therapy: cancer cells, dead cells and healthy cells

G.Rivera, MS Thesis, UPR, 2005

Modeling therapy

Simultaneous snapshots

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Cancer treated with immune therapy.Cancer, dead, and healthy cell concentrations.Lymphocyte concentration

Modeling therapy

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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.

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• 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

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• FUTURE WORK:

• Modeling therapies

• Relate macro and meso approaches

• Modeling metastasis

Conclusions