Competition and cooperation: tumoral growth strategies

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 T.S. Deisboeck, Harvard University

Outline • 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

Cancer dynamics. • Carcinogenic change • Growth • Invasion • Metastasis

Macroscopicdescription Tumor development as a single entity Microscopic description Study of individual cell properties effective parameters Mesoscopic approach Simulation of the behavior of cell clusters and their interactions predictions In vitro experiments Biological models In vivo experiments Clinical results

Ontogenetic growth law 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.

(m()/M)1/4 Universal growth curve  West, Brown and Enquist, Nature, 2001 Conservation of energy + fractality of distribution network

The hype: 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.

ONTOGENETIC GROWTH LAW The growth of an organism is mediated by cell division and fed by metabolism. Maintenance Metabolic Energy Cellreproduction Maintenance includes cell replacement.

Energyconservationequation: maintenance creation B: energy income rate to the organism cells : single cell metabolic rate : energy to create a single cell N: total cell number This equation can be easily turned into a simple differential equation.

mc:single cell mass b = / m = Nmc: organism mass To be modelled: the basal metabolic rate B(m). B ~ m3/4 [Kleiber, 1932 (on phenomenological grounds; West, 2001 (fractal distribution networks)]. B ~ m2/3 [other authors]. Generally accepted:B ~ mp: a power law. There are hundreds of power laws in biology!

Setting a = mc B0/, b=/, Maximum body size: [Take dm/dt = 0]

If m0 is the mass at birth, and we obtain theuniversal solution: This is the curve plotted by West et al., with p = 3/4. e- is the proportion of energy devoted to cell reproduction. It goes to zero as  grows.

Does cancer follow a universal growth law? We would like to understand the kinetics of tumor growth. Energy is conserved, but, what is B(p)? Conjecture: As for living beings , B(p) ~ mp. At first: avascular growth (p = 2/3 ?) Later: angiogenic growth (p = 3/4 ?)

Molecular diffusion towards a sphere: B(m) = B0m2/3 Cell, spheroid Nutrientmolecules p = 2/3 results from simple scaling between surface and volume.

At later times, angiogenesis changes the tumor feeding patterns. Angiogenesis

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

Experimental results (m()/M)1/4 Fit with p=3/4 by Guiot et al. J. Theor. Biol. (2003).

(m()/M)1/4 Tumors implanted in rats and mice Fit with p=3/4 by Guiot et al. J. Theor. Biol. (2003).

(m()/M)1/4 Fit with p=3/4 by Guiot et al. J. Theor. Biol. (2003).

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.

Multicellular Tumor spheroid http://www.vet.purdue.edu/cristal/dicspheroid.jpg

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!

The model is defined by, There is a delay in the onset of nutrient absorption, which depends on the cell and the matrix. We replace a by, T: effective accommodation time We applied these ideas to various experimental situations.

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. Asymptotic spheroid mass: m decreases as the nutrient is decreased.

Appl. Phys. Lett., 2004 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. Excellent fit, except for the very starved spheroid (f = 0.8).

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.

Defining, the energy conservation equation is, with: V: volume under conditions of nutrient saturation. R: final cell concentration

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. Ansatz: 0 : initial density. Asymptotic volume:

Appl. Phys. Lett., 2004 Variation of spheroid volume under different mechanical stress conditions [Helmlinger et al., 1997]. Solid curves are model fits. p= 2/3 Final volumes (in cm3) and accommodation timesare, 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). T increases, and final cell density (R) increases by a factor of up to 3.

Experiment III: periodic feeding (proposed) Consider a periodic feeding protocol. Then, After a transient , the live cell mass oscillates, following a hysteretic cycle. Transient length: tT = 1/b(1-p)

tT = 0.1 tT= 1 tT = 0.1 tT = 10 Hysteresis plots m vs sin(t) Maximum remanence:tT = 1 This behavior is peculiar to “non-linear, non-classical” systems (CAC,TSD, 2005).

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.

Growth rules • Feeding:cancer cells absorb free nutrient (concentration p) at a rate This is transformed into bound nutrient. • Consumption:bound nutrient q is consumed by cancer cells at a rate Both rates are proportional to the concentration for low concentrations and then saturate.

Growth rules • Death:A low concentration of bound nutrient leads to cell death. • Mitosis:A high concentration of bound nutrient leads to cell replication. Death Mitosis • Migration:A cell that senses a low nutrient level in its neighborhood tends to migrate. Migration

Simulation • Consider a piece of tissue of arbitrary shape, • which is discretized using a square or cubic grid. • Each node point represents a volume element • that 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.

Simulation 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.

(I) 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: Typical lattice sizes: 300300

Single species Latency Growth These are two phase diagrams, corresponding to different values of . Both data sets are well fitted by a power law with exponent 1/3. Power laws crop up everywhere!

Morphology • Red arrows: • = 0.44 Green arrows: • = 0.22

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.

Two species • 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 – thereis intraspecies competition and there is inter-species competition.

Two species: restarted growth tm=20000 Just latent tumorhas a  = 0.44 mutation t=25000 Observe fast growth of species 2 t=30000 t=35000

Two species: second latency tm=20000 Original cancer well inside latent region leads to second latency t=30000 t=45000 t=40000

Example:  = 1 tm=20000 Restarted growth for cells with restricted mobility t=45000 t=25000 t=35000

Modeling therapy No therapy:cancer cells, dead cells and healthy cells Simultaneous snapshots G.Rivera, MS Thesis, UPR, 2005

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

Modeling therapy With therapy No therapy Therapy favors reproduction of surviving cancer cells, accelerating tissue destruction (!) Modeling therapy can help to determine optimal therapeutic courses.

Conclusions • 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 • FUTURE WORK: • Modeling therapies • Relate macro and meso approaches • Modeling metastasis

Competition and cooperation: tumoral growth strategies