Modelling Vasculogenesis Modelling Vasculogenesis

Dept. Mathematics

Politecnico di Torino

Division of Molecular Angiogenesis

Inst. Cancer Research and Treatment

Candiolo (TO)

F. Bussolino

E. Giraudo

G. Serini

D. Ambrosi

A. Gamba

R. Kowalczyk

L. Preziosi

V. Lanza

A. Tosin

tumour cells

Tissue level Cellular level Sub-cellular level

macrophages

Endothelial cells

lymphocytes T helper

lymphocytes T killer

plasma cells

Dal punto di vista fisiologico la descrizione degli aspetti che giocano un ruolo inportante nello sviluppo e nella crescita dei tumori e’ molto complicato. Molto dipende dall’ingrandimento utilizzato dal biologo nel descrivere i fenomeni o da chi vuole sviluppare i modelli matematici. Ci si puo’ infatti focalizzare sugli aspetti macroscopici e descrivere- la crescita dello sferoide multicellulare nella fase avascolare (ossia quando non si e’ ancora circondato di una propria rete di capillari) - o il processo di angiogenesi (i.e. la crescita di questa rete), - o la fase vascolare, - o il distacco di metastasi ed i meccanismi di diffusione ed adesione nei siti secondari.Tutto cio’ pero’ dipende da quanto succede ad un scala ancora piu’ piccola, la scala cellulare. Bisogna tener conto che le cellule tumorali interagiscono con altre cellule dell’organismo (cellule endoteliali, del sistema immunitario) e che esse stesse, come dei Pokemon, evolvono.Infine, il risultato di queste interazioni dipende da cosa succede ad una scala ancora piu’ piccola: la scala cellulare (degradazione del DNA, espressione dei geni, trasduzione dei segnali, adesione cellulare). Quindi il problema matematico viene ad essere intrinsecamente multi-scala.

Dal punto di vista fisiologico la descrizione degli aspetti che giocano un ruolo inportante nello sviluppo e nella crescita dei tumori e’ molto complicato. Molto dipende dall’ingrandimento utilizzato dal biologo nel descrivere i fenomeni o da chi vuole sviluppare i modelli matematici. Ci si puo’ infatti focalizzare sugli aspetti macroscopici e descrivere- la crescita dello sferoide multicellulare nella fase avascolare (ossia quando non si e’ ancora circondato di una propria rete di capillari) - o il processo di angiogenesi (i.e. la crescita di questa rete), - o la fase vascolare, - o il distacco di metastasi ed i meccanismi di diffusione ed adesione nei siti secondari.Tutto cio’ pero’ dipende da quanto succede ad un scala ancora piu’ piccola, la scala cellulare. Bisogna tener conto che le cellule tumorali interagiscono con altre cellule dell’organismo (cellule endoteliali, del sistema immunitario) e che esse stesse, come dei Pokemon, evolvono.Infine, il risultato di queste interazioni dipende da cosa succede ad una scala ancora piu’ piccola: la scala cellulare (degradazione del DNA, espressione dei geni, trasduzione dei segnali, adesione cellulare). Quindi il problema matematico viene ad essere intrinsecamente multi-scala.

Tumour ProgressionTumour ProgressionThe progression of a normal cell into a tumor cell implies several key steps

The progression of a normal cell into a tumor cell implies several key steps

VASCULOGENESIS ON “MATRIGEL”

Stimulation Migration OrganisationProliferation

Angiogenesis

VASCULOGENESIS ON “MATRIGEL”

2h30’ 1h 4h

10h8h 12h 14h

6h

100 M10 M1 M

0.1 M0.01 M 0.001 MControl

Cords: Dose Response

(Courtesy: Pharmaceutical Institute Mario Negri - Bergamo)

Let me mention that vasculogenesis in vitro is a standard test used by pharmaceutical companies and research centres to test the validity of antiangiogenic drugs

Let me mention that vasculogenesis in vitro is a standard test used by pharmaceutical companies and research centres to test the validity of antiangiogenic drugs

Questions

• What are the mechanisms driving the generation of the patterns?

• What is the explanation of the transition obtained for low and high densities?

• Is it possible to “manipulate” the formation of patterns?

• Why is the size of a successful patchwork nearly constant?

n = 50 cells/mm2 n = 400 cells/mm2n = 200 cells/mm2n = 100 cells/mm2

Zeldovich modelZeldovich model

Assumptions

• Cells are accelerated by gradients of soluble mediators and slowed down by friction (chemotaxis)

• Cells move on the Matrigel surface and do not duplicate

• The cell population can be described by a continuous distribution of density n and velocity v

• Cells release chemical mediators (c)

• For low densities (early stages) the cell population can be modeled as a fluid of non directly interacting particles showing a certain degree of persistence in their motion

• Tightly packed cells respond to compression

calvino.polito.it/~biomat

calvino.polito.it/~preziosi

D. Ambrosi, F. Bussolino, L.P., J. Theor. Med., (2004)

Serini et al., EMBO J. 22, 1771-9, (2003)

Mathematical Model

= diffusion coefficient

= attractive strength

= rate of release of soluble mediators

= degradation time of soluble mediators

= friction coefficient

a = typical dimension of endothelial cells

D. Ambrosi, A. Gamba, G. Serini,

Bull. Math. Biol., (2004)

~ 0.1-0.2 mm ~ 10-7 cm2/s ~ 103 s ~ 20 mina ~ .02 mm

Mathematical Model

Keller Segelp = ln n

R. Kowalczyk,

J. Math. Anal. Appl., (2005)

no blow-upp = convex

blow-upp = 0

Temporal evolution

0 h 3 h 6 h

Temporal evolution

n = 200 cells/mm2

n =

50

cell

s/m

m2

n =

400

cel

ls/m

m2

n =

10 0

cel

l s/m

m2

Phase transition

Percolative transition

A. Gamba et al.,

Phys. Rev. Letters,

90, 118101 (2003)

Swiss-cheese transition

R. Kowalczyk, A. Gamba, L.P.

Discr. Cont. Dynam. Sys. B

4 (2004)

Percolative transition

Fong, Zhang, Bryce, and Peng

“Increased hemangioblast commitment,

not vascular disorganization, is the

primary defect in flt-1 knock-out mice”

Development 126, 3015-3025 (99)

Percolative transition ~ 90 cells/mm2Percolative probability, Mean cluster size, Cluster mass, Sand-box method

Percolative transition

Fractal dimension

r

~rD/r2

.8

D=1.87

D=1.5

Density of percolating cluster

A. Gamba et al.,

Phys. Rev. Letters, 90 (2003)

A quantity that can give us information about the structure of the percolating cluster at different scales is the density of the percolating cluster as a fanction of the radius.

This is defined as the mean density of sites belonging to the percolating cluster, inclosed in a box of side r.

This shoud scale as r^(D-d).

For a percolating cluster of random percolation at the critical point, one expects a fractal dimension D=1.896.

We found it.

The value 1.50 may be the signature of the dynamic process that lead to the formation of the clusters

(driven for r>rc by the rapidly oscillating components of the concentration field)

A quantity that can give us information about the structure of the percolating cluster at different scales is the density of the percolating cluster as a fanction of the radius.

This is defined as the mean density of sites belonging to the percolating cluster, inclosed in a box of side r.

This shoud scale as r^(D-d).

For a percolating cluster of random percolation at the critical point, one expects a fractal dimension D=1.896.

We found it.

The value 1.50 may be the signature of the dynamic process that lead to the formation of the clusters

(driven for r>rc by the rapidly oscillating components of the concentration field)

Swiss-cheese transition

R. Kowalczyk, A. Gamba, L. Preziosi

Discrete and Continuous Dynamical Systems

Stability of the uniform distribution

C. Ruhrberg, H. Gerhardt, M. Golding, R. Watson, S. Ioannidou, H. Fujisawa, C. Betsholtz, and D.T. Shima,“Spatially restricted patterning cues provided by heparin-binding VEGF-A control blood vessel branching morphogenesis”, Genes & Development 16, 2684–2698

Figure 2. The balanced expression of heparin-binding VEGF-A versus VEGF120 controlsmicrovessel branching and vessel caliber.(A) Schematic representation of hindbrainvascularization between 10.0 (1) and10.5 (4) dpc; between 9.5 and 10.0 dpc, theperineural vascular plexus in the pial membranebegins to extend sprouts into the neuraltube (1), which grow perpendicularly towardthe ventricular zone (2), where theybranch out to form the subventricular vascularplexus (3,4). (B,C) Microvessel appearanceon the pial and ventricular sides of aflat-mounted 12.5-dpc hindbrain; the midlineregion is indicated with an asterisk; thepial side of the hindbrain with P, the ventricularside with V. (D–F) Visualization ofvascular networks in representative500-µm2 areas of the 13.5-dpc midbrain ofwt/wt (D), wt/120 (E), and 120/120 (F) littermates;

Figure 2. The balanced expression of heparin-binding VEGF-A versus VEGF120 controlsmicrovessel branching and vessel caliber.(A) Schematic representation of hindbrainvascularization between 10.0 (1) and10.5 (4) dpc; between 9.5 and 10.0 dpc, theperineural vascular plexus in the pial membranebegins to extend sprouts into the neuraltube (1), which grow perpendicularly towardthe ventricular zone (2), where theybranch out to form the subventricular vascularplexus (3,4). (B,C) Microvessel appearanceon the pial and ventricular sides of aflat-mounted 12.5-dpc hindbrain; the midlineregion is indicated with an asterisk; thepial side of the hindbrain with P, the ventricularside with V. (D–F) Visualization ofvascular networks in representative500-µm2 areas of the 13.5-dpc midbrain ofwt/wt (D), wt/120 (E), and 120/120 (F) littermates;

Saturation with VEGFC

ontr

olS

atu

rate

d

Con

trol

Sat

ura

ted

Persistence DirectionalityCell migration analysis of ECs plated on Matrigel in the absence or the presence of saturating amount of VEGF-A. Histograms of , cos , , and cos (see Fig.2D) for the trajectoriesof ECs plated on Matrigel either in control culture conditions (green) or in the presence a saturating(20 nM) amount of VEGF-A165 (light blue). The observed densities of cos and cos were fittedwith Beta distributions (red lines) by maximum likelihood. The observed densities in VEGF-A165saturating conditions are markedly more symmetric than those observed in control conditions,showing loss of directionality in EC motility. Histograms of indicate that also after extinguishingVEGF-A gradients EC movement on Matrigel maintains a certain degree of directional persistence.However, histograms of show that in the presence of saturating amount of VEGF-A165 ECmovement is completely decorrelated from the direction of simulated VEGF gradients. We checkedthe hypothesis that values in saturating conditions are uniformly distributed by performing a goodness-of-fit test (p = 0.397). The same test applied to the values in control conditions gives a p= 3 x 10-8, which allow to reject the hypothesis at any reasonable significance level.

Cell migration analysis of ECs plated on Matrigel in the absence or the presence of saturating amount of VEGF-A. Histograms of , cos , , and cos (see Fig.2D) for the trajectoriesof ECs plated on Matrigel either in control culture conditions (green) or in the presence a saturating(20 nM) amount of VEGF-A165 (light blue). The observed densities of cos and cos were fittedwith Beta distributions (red lines) by maximum likelihood. The observed densities in VEGF-A165saturating conditions are markedly more symmetric than those observed in control conditions,showing loss of directionality in EC motility. Histograms of indicate that also after extinguishingVEGF-A gradients EC movement on Matrigel maintains a certain degree of directional persistence.However, histograms of show that in the presence of saturating amount of VEGF-A165 ECmovement is completely decorrelated from the direction of simulated VEGF gradients. We checkedthe hypothesis that values in saturating conditions are uniformly distributed by performing a goodness-of-fit test (p = 0.397). The same test applied to the values in control conditions gives a p= 3 x 10-8, which allow to reject the hypothesis at any reasonable significance level.

Anisotropic case

V. Lanza

L

L

L’

Exogenous control

chemoattractant

chemorepellent

Exogenous chemoattrantant

Source

in the

center

Source

on the

sides

V. Lanza

Exogenous chemorepellent

Point

Source

Line

Source

new characteristic length

action range of chemorepellent

l' .* 016

l' .0 31mm

• Parameters used give:

l' .* 0158• In dimensionless form:

Exogenous chemorepellent

Exogenous chemorepellent

2 0 27l' .*

new characteristic length

action range of chemorepellent

l' .0 31mm

• Parameters used give:

l' .* 0158• In dimensionless form:

Vascularization:Tumor vs. Normal

• Increased vessel permeability• Increased proliferation of EC• Abnormal blood flow• Swelling (dilatation)

• Increased tortuosity • Abnormal branching• Presence of blind vessels• Loss of hierarchy• Increased disorder

Physiological observations:

Vascularization:Tumor vs. Tumor

Konerding M. et alAm J Pathol 152: 1607-1616, 1998

Aim:

Distinguish themorfological characteristics to quantify the abnormality

• Identify with non invasive techniques the existence of abnormal morfologies• Quantify the progression state of the tumor• Quantify the efficacy of drugs

Not only this but even from tumor to tumor one can identify tumor aggressiveness from the degree of “disorder” of the vascular network sorrounding it. The wish of medical doctors would be to identify the quantities which are important to monitor to quantify the abnormality

Not only this but even from tumor to tumor one can identify tumor aggressiveness from the degree of “disorder” of the vascular network sorrounding it. The wish of medical doctors would be to identify the quantities which are important to monitor to quantify the abnormality