Modelling Vasculogenesis and Tumour Growth

1
Modelling Vasculogenesis and Tumour Growth

• L. Preziosi
• Dipartimento di Matematica
• Politecnico di Torino

2
Modelling Vasculogenesis

• Dept. Mathematics
• Politecnico di Torino

Division of Molecular Angiogenesis Inst. Cancer
Research and Treatment Candiolo (TO)
3
Angiogenesis
Stimulation
VASCULOGENESIS ON MATRIGEL
4
VASCULOGENESIS ON MATRIGEL
5
Questions

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

• What affects their structure?

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

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

• Is it possible to manipulate the formation of
  patterns?

6
(No Transcript)
7
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

• For low densities (early stages) the cell
  population can be
• modeled as a fluid of non directly interacting
  particles

• Tightly packed cells respond to compression

8
(No Transcript)
9
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
D. Ambrosi, A. Gamba, G. Serini, preprint,
calvino.polito.it/biomat calvino.polito.it/pre
ziosi
10
Dimensionless Form
a 10-7 cm2 s-1 t 103 s 20 min a .02 mm
11
Temporal evolution
0 h
3 h
6 h
12
(No Transcript)
13
Phase transition
Serini et al., EMBO J., 22 (2003)
14
Percolative transition
15
Percolative transition
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
D1.896. 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 rgtrc by the
rapidly oscillating components of the
concentration field)
Percolative probability, Mean cluster size,
Cluster mass, Sand-box method
Density of percolating cluster
D1.5
rD/r2
r
D1.87
.8
16
Swiss-cheese transition
Stability of the uniform distribution
R. Kowalczyk, A. Gamba, L. Preziosi Discrete and
Continuous Dynamical Systems
17
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)
18
Figure 2. The balanced expression of
heparin- binding 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. (DF) 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
19
Anisotropic case
20
Saturation with VEGF
Control
Saturated
21
Persistence
Directionality
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 trajectories of 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 fitted with Beta
distributions (red lines) by maximum likelihood.
The observed densities in VEGF-A165 saturating
conditions are markedly more symmetric than those
observed in control conditions, showing loss of
directionality in EC motility. Histograms of
indicate that also after extinguishing VEGF-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 EC movement is
completely decorrelated from the direction of
simulated VEGF gradients. We checked the
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.
Control
Saturated
22
Cords Dose Response
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
Control
0.1 mM
0.01 mM
0.001 mM
1 mM
100 mM
10 mM
(Courtesy Pharmaceutical Institute Mario Negri -
Bergamo)
23
A similar problem occurs when looking at the
vascular networks sorrounding a normal tissua
anad a tumor
Heart Retina Skin
Tumors
24
VascularizationTumor vs. Normal
Fisiological observations

• Increased vessel permeability
• Increased proliferation of EC
• Increased tortuosity
• Swelling (dilatation)
• Abnormal branching
• Presence of blind vessels
• Abnormal blood flow
• Loss of hierarchy
• Increased disorder

25
VascularizationTumor vs. Tumor
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
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
26
(No Transcript)
27
Phase transition
Serini et al., EMBO J., (2003)
28
Temporal evolution
29
Quantify the efficacy of anti-angiogenic therapies
This is a problem posed to my attention by a
Pharmaceutical research center in Italy but it is
also of great interest for the National Institute
of Health. Their interest is mainly related to
the quantification of the efficacy of drugs and
therefore again to the need of classifying and
ordering
Control
Treated
Giavazzi, Taraboletti et al., J Natl Cancer Inst
85 235-240, 1993
30
Before concluding Id like to open a windows on
the following open problem. In making a
diagnosis a pathologist usually starts from the
morphological characteristics, distinguishing
what is normal and what is not. The diagnosis is
then defined by the perception of the
pathologist. Today there is a need of
substituting this subjective decisional process
with a more objective procedure, through the
identification of quantitative methods. In few
words, substituting the sentence Your
cholesterol level is too high with your
cholesterol level is 300 while its normal value
would be between 130 and 250 (mg/dl). Nowadays
one does not even remember what in which units
colestherol is measured. Meters,
Pounds? Quantifying gives not only an idea of how
pathological is the situation, but also a
criterium easily readable by other medical
doctors.
Diagnosis
Morfological characteristics
diagnosis
perception
31
Stages of tumor development Identification and
quantification
Lets make an example When observing a ductal
carcinoma it is possible to identify different
phases, but it is not always easy to distinguish
one phase from the other
32
Tongue tumor
Take the following example. Usually the
connective tissue and the epithelium of a tongue
are clearly divided by a smooth border. As the
tissue becomes cancerous, this border becomes
irregular. This is related to a change in the
adhesion properties of the cells. The experiment
is the following some pathologists were given a
series of samples and asked to classify them at
sight in normal tissue, dysplasia, and carcinoma.
The classification was known through istological
tests.
1,7
epithelium
1,6
Normal Tissue
connective tissue
1,5
Fractal dimension
1,4
1,3
Dysplasia
1,2
1,1
1
Carcinoma
Dysplasia
Carcinoma
Normal
G. Landini (Birmingham)
33
(No Transcript)
34
Mixture Model