Modelling Vasculogenesis
Dept. Mathematics
Politecnico di Torino
D. Ambrosi
A. Gamba
R. Kowalczyk
L. Preziosi
V. Lanza
A. Tosin
Division of Molecular Angiogenesis
Inst. Cancer Research and Treatment
Candiolo (TO)
F. Bussolino
E. Giraudo
G. Serini
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.
Tissue level
Cellular level
Sub-cellular level
lymphocytes T helper
lymphocytes
T killer
macrophages
tumour cells
plasma cells
Endothelial cells
The progression of a normal
cell into a tumor cell implies
several key steps
Tumour Progression
Angiogenesis
Stimulation
Proliferation
Migration
Organisation
VASCULOGENESIS ON “MATRIGEL”
VASCULOGENESIS ON “MATRIGEL”
30’
2h
1h
8h
10h
6h
4h
12h
14h
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
Cords: Dose Response
Control
0.001 mM
1 mM
0.01 mM
10 mM
(Courtesy: Pharmaceutical Institute Mario Negri - Bergamo)
0.1 mM
100 mM
Questions
• What are the mechanisms driving the generation of the patterns?
• Why is the size of a successful patchwork nearly constant?
• What is the explanation of the transition obtained
for low and high densities?
n = 50 cells/mm2
n = 100 cells/mm2
n = 200 cells/mm2
n = 400 cells/mm2
• Is it possible to “manipulate” the formation of patterns?
Zeldovich model
Assumptions
• 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)
• Cells are accelerated by gradients of soluble mediators
and slowed down by friction (chemotaxis)
• 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
Serini et al., EMBO J. 22, 1771-9, (2003)
calvino.polito.it/~biomat
calvino.polito.it/~preziosi
D. Ambrosi, F. Bussolino, L.P., J. Theor. Med., (2004)
Mathematical Model
a = diffusion coefficient
b = attractive strength
g = rate of release of soluble mediators
t = degradation time of soluble mediators
e = friction coefficient
a = typical dimension of endothelial cells
x = (a t)1/2 ~ 0.1-0.2 mm
D. Ambrosi, A. Gamba, G. Serini,
Bull. Math. Biol., (2004)
a ~ 10-7 cm2/s
t ~ 103 s ~ 20 min
a ~ .02 mm
Mathematical Model
p=0
blow-up
p = ln n
Keller Segel
p = convex
no blow-up
R. Kowalczyk,
J. Math. Anal. Appl., (2005)
Temporal evolution
0h
3h
6h
n = 50 cells/mm2
Temporal evolution
n = 400 cells/mm2
n = 200 cells/mm2
Phase transition
Percolative
transition
Swiss-cheese
transition
A. Gamba et al.,
Phys. Rev. Letters,
90, 118101 (2003)
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)
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.
Percolative transition
This is defined as the mean probability,
Percolative
density of sites belonging to the
percolating cluster, inclosed in
Mean
cluster size,
a box of side r.
This shoud scale asmass,
r^(D-d).
Cluster
For a percolating cluster of
random percolation at the
Sand-box
method
critical point, one expects a
A. Gamba et al.,
Phys. Rev. Letters, 90 (2003)
Percolative transition ~ 90 cells/mm2
fractal dimension D=1.896.
Fractal dimension
We found it.
The value 1.50 may be the
signature of the dynamic
process that lead to the
formation of the clusters
Density of percolating cluster
(driven for r>rc by the rapidly
oscillating components of the
concentration field)
~rD/r2
r
.8
Swiss-cheese transition
Stability of the uniform distribution
R. Kowalczyk, A. Gamba, L. Preziosi
Discrete and Continuous Dynamical Systems
Figure 2. The balanced expression of
heparinbinding VEGF-A versus VEGF120 controls
microvessel branching and vessel caliber.
(A) Schematic representation of hindbrain
vascularization between 10.0 (1) and
10.5 (4) dpc; between 9.5 and 10.0 dpc, the
perineural vascular plexus in the pial
membrane
begins to extend sprouts into the neural
tube (1), which grow perpendicularly toward
the ventricular zone (2), where they
branch out to form the subventricular
vascular
plexus (3,4). (B,C) Microvessel appearance
on the pial and ventricular sides of a
flat-mounted 12.5-dpc hindbrain; the midline
region is indicated with an asterisk; the
pial side of the hindbrain with P, the
ventricular
side with V. (D–F) Visualization of
vascular networks in representative
500-µm2 areas of the 13.5-dpc midbrain of
wt/wt (D), wt/120 (E), and 120/120 (F)
littermates;
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
Saturated
Control
Saturation with VEGF
analysis of ECs plated on Matrigel in
r the presence of saturating amount
stograms of , cos , , and cos (see
trajectories
on Matrigel either in control culture
een) or in the presence a saturating
nt of VEGF-A165 (light blue). The
sities of cos and cos were fitted
ibutions (red lines) by maximum
e observed densities in VEGF-A165
ditions are markedly more symmetric
served in control conditions,
of directionality in EC motility.
indicate that also after extinguishing
ents EC movement on Matrigel
ertain degree of directional
Control
Persistence
ograms of show that in the
saturating amount of VEGF-A165
completely decorrelated from the
imulated VEGF gradients. We
Saturated
s that values in saturating conditions
distributed by performing a
t test (p = 0.397). The same test
values in control conditions gives a
hich allow to reject the hypothesis at
e significance level.
Directionality
Anisotropic case
V. Lanza
Exogenous control
chemoattractant
chemorepellent
L
L
L’
Exogenous chemoattrantant
V. Lanza
Source
in the
center
Source
on the
sides
Exogenous chemorepellent
Line
Source
Point
Source
Exogenous chemorepellent
new characteristic
length
action range of
chemorepellent
• Parameters used give:
l' = 0.31mm
• In dimensionless form:
l'* = 0158
.
*
l'  016
.
Exogenous chemorepellent
new characteristic
length
action range of
chemorepellent
• Parameters used give:
l' = 0.31mm
• In dimensionless form:
l'* = 0158
.
2l'*  0.27
Vascularization:
Tumor vs. Normal
Physiological observations:
• 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
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
Vascularization:
Tumor vs. Tumor
Aim:
Distinguish the
morfological 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
Konerding M. et al
Am J Pathol 152: 1607-1616, 1998