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THANKREVIEWOPEN PROBLEMSREQUEST FOR
INTERACTIONNEW THEORIES First of all let me
thank the scientific committee for proposing my
name for this lecture and the organisers for the
invitation. It is a great honour for me to be
here. The aims of my lecture will be - to give an
overview of some mathematical models recently
developed to support cancer research - to
describe some open problems of interest both from
a mathematical and a bio-medical
viewpoint Actually from the bio-medical point of
view there is now a strong request for
interaction on concrete problems. In general
bio/medical problems present a type of difficulty
not encountered in other fields. In fact new
theories need to be developed. In this respect,
Id like to start with a sentence taken from an
article appeared on Nature which puts in evidence
the need for interdisciplinary research and the
hidden difficulties related to the treatment of
complex biological systems. However, this
characteristic also represent a big stimulus for
developing new research areas. Because the
research field is in rapid evolution Ill
restrict citation to the last 5 years. What has
been written more than 10 years ago is in most
cases obsolete.
Modelling Tumour Growth and Progression
Luigi Preziosi - Politecnico di Torino
Although living systems obey the laws of physics
and chemistry, the notion of function or purpose
differentiate biology from other natural
sciences Hartwell, Hopfield, Leibner, Murray,
Nature 402, c47-c52 (1999)
calvino.polito.it/biomat calvino.polito.it/pre
ziosi
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Tissue level
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
dallingrandimento 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 dellorganismo (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 cells
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Modelling Tumour Masses
When dealing with macroscopic models one can
distinguish between two types of actors cells
and the chemical factors that influence their
motion and proliferation. The structure of the
model should consist in general of at least a set
of mass balances and a set of reaction-diffusion
equations for the chemical factors. In the first
class of models I will deal with no force or
momentum balance is taken into account. One
encounters a closure problem as one has to
describe how cells move. You will see in the
literature that I will cite that the most active
scientific communities in the field are italian
and british.
D. Ambrosi L.P., Math. Models Meth. Appl. Sci.
12, 737-754 (2002)
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Only tumor cells in 1D
The first models in this fields were developed
under the following assumptions.... These
hypotheses allow to integrate the mass balance
equation which becomes a geometric condition on
the evolution of the tumour radius (clearly this
depends on chemical factors and nutrients through
Gamma) The problem is then reduced to the
integration of a system of reaction-diffusion
equations on a time dependent domain 1 Necrosis
and apoptosis 2 localised and non GF In the
discussion I will focus on the mass balance
equations because they present a closure problem

• Single population with constant density
• Spherical symmetry
• Chemical factors and nutrients diffuse

- H. Byrne M. Chaplain, J. Theor. Med. 1,
223-235 (1998) - H. Byrne, Math. Models Methods
Appl. Sci. 9, 541-568 (1999) - M. Chaplain L.
Preziosi, Math. Models Methods Appl. Sci. 12,
(2002) (review) - A. Friedman F. Reitich, Math.
Models Methods Appl. Sci. 11, 601-626 (2001)
(analytical)
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Only tumor cells in 3D
This can not be done in 3D problems. In this case
one has to relate the vector velocity field to
the other scalar fields. The easiest way is to
use a potential flow assumption, many times
incorrectly called Darcys law Sometimes it is
said that cells move in response to chemical
gradients (the so-called chemotaxis).

• 1. Constant density
• 2. Potential flow

- H. Byrne, IMA J. Math. Appl. Med. Biol., 14,
305-323 (1997) - H. Byrne M. Chaplain, Eur. J.
Appl. Math. 8, 639-658 (1997) - ...
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More cell populations in 1D
What are the problems encountered when dealing
with n cell populations. Focusing on the mass
balance equations one has the 2n unknown (volume
ratios and velocities) and n equations.
- J. Ward J. King, IMA J. Math. Appl. Med.
Biol. 14, 36-69 (1997) 15, 1-42 (1998)
J. Theor. Med. 1, 171-211 (1999)
- C.
Breward, H. Byrne C. Lewis, Eur. J. Appl. Math.
45, 125-152 (2002)

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Some time ago, we used an approach which is
somehow different, assuming that cells tend to
move towards the regions where they feel less
pressed. We then used this closure relation,
where Sigma is a measure of the stresses and K is
a measure of the motility of the cells which is
related to the presence of an ECM matrix to crawl
upon. In fact cells move on a network of fibres
moving from adhesive site to adhesive site
towards the most convenient region. I like to use
this picture to describe the motion of cells. A
bunch of cars moving on the streets toward a
region (chemotaxis) avoid traffic jams, or honey
dripping down a net.
- E. De Angelis L. Preziosi, Math. Models
Methods Appl. Sci. 10, 379-407 (2000)
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Stages of tumor development
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
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hyperplasia dysplasia
Misperception of stress
vwn nwa a-u f, fnamn ma
D
D
D
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total volume ratio
tumour
extracellular matrix
normal tissue
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Free boundary value problems
From the approaches I cited it is clear that the
mathematical problems are typically FBP with an
internal domain and an external domain. The two
domains influence each other. For instance, if
one focuses on the process called
metallo-proteasis, one has to describe how the
tumour to grow has to destroy the surrounding
extracellular matrix. So it produces enzymes that
digest it allowing tumour growth. Or in the
process called angiogenesis tumour cells produce
chemical factor which diffuse out where they
stimulate the existing capillaries to produce new
capillaries to bring the tumor more nutrient.
Living tumour cells
Death tumour cells
- H. Byrne M. Chaplain, Eur. J. Appl. Math. 8,
639-658 (1997) - E. De Angelis L. P., Math.
Models Methods Appl. Sci. 10, 379-407 (2000) - A.
Bertuzzi, A. Fasano A. Gandolfi, SIAM J. Math.
Analysis, (2004) - .......
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t10
Tumor cells Death cells Capillaries Nutrients
Tumor Angiogenic Factor
t16
t22
t28
t31
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Multiphase Models
In effetti dal punto di vista fisiologico i
tumori, come tutti gli organi del corpo umano
sono mezzi porosi deformabili per cui per il loro
studio andrebbero utilizzati modelli multifase.
Questo è largomento che abbiamo affrontato negli
ultimi 2 anni al Politecnico di Torino
diffusion of nutrients chemical factors
saturation
tumour as a deformable porous medium
- D. Ambrosi L. P., Math. Models Methods Appl.
Sci. 12, 737-754 (2002) - H. Byrne L. Preziosi,
IMA J. Math. Appl. Med. Biol. (2003)
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Poroelastic Models
in 1D
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Stress response
- G. Helmlinger, P. Netti R. Jain, Nature
Biotech. 15, 778-783 (1997)
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Growing continua
The last thing I want to say is that all this has
been possible thanks to the fact that some times
ago Bellomo was so far-seeing. This is one of his
best virtues having the ability of identifying
the hottest topics, betting on the right horses
and finally the ability of stimulating the
research group.
Evoluzione
- R. Skalak, S. Zargaryan, R. Jain, P. Netti
A. Hoger, J. Math. Biol. 34, 889-914 (1996) - L.
Taber, ASME J. Biomech. Engr. 120, 348-354
(1998) - M. Epstein, G. Maugin, Int. J.
Plasticity 16, 951-978 (2000) - J. Humprey K.
Rajagopal, Math. Models Meth. Appl. Sci. 12,
407-430 (2002) - A. Di Carlo S. Quiligotti,
Mech. Res. Comm. 29, 449-456 (2002) - D. Ambrosi
F. Mollica, Int. J. Engng. Sci. 40, 1297-1316
(2002)
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Interactions with external tissues
This framework is particularly important to study
phenomena involving stresses, e.g. the forces
exerted by the tumor on the external tissues and
viceversa

• Stress response
• - D. Ambrosi, P. Netti F. Mollica, preprint
• Creation and fracture of capsules
• - H. Byrne T. Jackson, preprint
• - A. Perumpanani, J. Sherratt J. Norbury,
  Nonlinearity 10, 1599-1614 (1997)
• - J. Sherratt, SIAM J. Appl. Math., 60, 392-407
  (1999)

• Compression and fracture of ducts
• Capillary collapse

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Tumour Progression
The progression of a normal cell into a tumor
cell implies several key steps
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Select the cell populations involved in the
evolution
Come i tumori, anche le cellule del sistema
immunitario sono soggette a processi di
maturazione ed attivazione. Per questo motivo i
modelli che vogliono descrivere levoluzionedi
tali sistemi necessitano di incorporare una nuova
variabile indipendente che sia capace di
descrivere lo stato (o gli stati) delle cellule e
come il loro comportamento dipenda da tale stato.
Questo puo essere laggressivita, lo stato di
attivazione o maturazione, e cosi via.
Immune System Activation
L. Greller, F. Tobin G. Poste Invasion and
Metastasis 16,177-208 (96)
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Cellular Kinetic Models
In describing this dynamics in collaboration with
an immunologist of the university of Turin, we
have developed some models which might be called.
At present the technology is such that it is
possible to operate at a genetic level or at the
level of signal transduction so that it is
possible to obtain some data on the different
terms, but this is still to be developed further.
The number and the state of the cell can change
because of
Medical treatments
Intrinsic progression
External sources (or sinks)
Proliferative interactions
Destructive interactions
Conservative interactions
N. Bellomo G. Forni Math. Comp. Modelling 20,
107-122 (1994)
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Cell-to-cellInteractions
Of course, the behaviour of the cell depends on
its activation state. In particular the result of
the interaction between cells depends on their
state. Cellular interactions can be

• Proliferative
• Conservative
• Destructive

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On the other hand, from the viewpoint of the
biologists or of the immunologists, the interest
is in determining which is the mechanism on which
it is most convenient to focus to have a better
therapy, or in mathematical terms if there exists
a crucial parameter that triggers a bifurcation
behaviour. You will see in the literature that I
will cite that the most active scientific
communities in the field are Italian and Polish.
D. Ambrosi, N. Bellomo L. P. J. Theor.
Medicine 4, 51-65 (2002)
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On the other hand, from the viewpoint of the
biologists or of the immunologists, the interest
is in determining which is the mechanism on which
it is most convenient to focus to have a better
therapy, or in mathematical terms if there exists
a crucial parameter that triggers a bifurcation
behaviour. You will see in the literature that I
will cite that the most active scientific
communities in the field are Italian and Polish.
- L. Arlotti, N. Bellomo M. Lachowicz,
Transp. Theory Statist. Phys. 29, 125-139
(2000) - L. Arlotti, A. Gamba M. Lachowicz,
J. Theor. Medicine 4, 39-50 (2002) - N. Bellomo
M. Pulvirenti Modelling in Applied
Sciences A Kinetic Theory Approach,
Birkhauser (2001)
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