Modelling Tumour Growth and Progression

“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)

Luigi Preziosi - Politecnico di Torino

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, I’d 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 I’ll restrict citation to the last 5 years. What has been written more than 10 years ago is in most cases obsolete.

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, I’d 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 I’ll restrict citation to the last 5 years. What has been written more than 10 years ago is in most cases obsolete.

calvino.polito.it/~biomat

calvino.polito.it/~preziosi

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.

Modelling Tumour Masses

pro lifera ting

q u i escentne c ro t i c

D. Ambrosi & L.P.,Math. Models Meth.

Appl. Sci. 12, 737-754 (2002)

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.

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.

Only tumor cells in 1D

1. Single population with constant density2. Spherical symmetry 3. Chemical factors and nutrients diffuse

R

- 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)

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

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

Only tumor cells in 3D

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)- ...

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 Darcy’s law

Sometimes it is said that cells move in response to chemical gradients (the so-called chemotaxis).

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 Darcy’s law

Sometimes it is said that cells move in response to chemical gradients (the so-called chemotaxis).

More cell populations in 1D

- 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)

1. Saturation

2. Radial Symmetry

3. Proportionality with given j (e.g. j=1) vj = j v

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.

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.

- E. De Angelis & L. Preziosi, Math. Models Methods Appl. Sci. 10, 379-407 (2000)

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.

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.

Stages of tumor development Let’s 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

Let’s 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

Misperception of stresshyperplasia

dysplasia

v=wn n+wa a-u =n+a+mn +ma

total volume ratio

tumour

normal tissue

extracellular matrix

n a

Extracellular matrix

Metallo-proteasis

Free boundary value problems

- 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)- .......

Living tumour cellsLiving tumour cells

Death tumour cellsDeath tumour cells

Angiogenic factors

Nutrients

Endot

helia

l cel

ls

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.

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.

t=10

t=40

t=30

t=31

t=28

t=22

t=16

t=100

Tumor cellsDeath cellsCapillariesNutrients Tumor Angiogenic Factor

Multiphase ModelsMultiphase Models

- D. Ambrosi & L. P., Math. Models Methods Appl. Sci. 12, 737-754 (2002)- H. Byrne & L. Preziosi, IMA J. Math. Appl. Med. Biol. (2003)

+ diffusion of nutrients & chemical factors

+ saturation

tumour as a deformable porous medium

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 l’argomento che abbiamo affrontato negli ultimi 2 anni al Politecnico di Torino

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 l’argomento che abbiamo affrontato negli ultimi 2 anni al Politecnico di Torino

Poroelastic ModelsPoroelastic Models

in 1D

Multicellular spheroid as a • viscous liquid• growing solid

in 3D

M. Dorie et al, Exp. Cell Res. 141

201-209 (1982)

cen

ter

outs

ide

Increasing stress sensitivity

Increasing applied load

Stress response

- G. Helmlinger, P. Netti & R. Jain, Nature Biotech. 15, 778-783 (1997)

Growing continua

- 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)

Evoluzione

Crescita

Deformazione

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.

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.

Interactions with external tissues

• 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

This framework is particularly important to study phenomena involving stresses, e.g. the forces exerted by the tumor on the external tissues and viceversa

This framework is particularly important to study phenomena involving stresses, e.g. the forces exerted by the tumor on the external tissues and viceversa

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

Immune System

Activation

Extra indep. variable:“activation” state

Identify theirspecific activity/ies

+

Select the cell populations involved in the evolution

fi(t,x,u)

=

L. Greller, F. Tobin & G. Poste Invasion and Metastasis 16,177-208 (96)

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 l’evoluzionedi 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 l’aggressivita’, lo stato di attivazione o maturazione, e cosi’ via.

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 l’evoluzionedi 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 l’aggressivita’, lo stato di attivazione o maturazione, e cosi’ via.

Cellular Kinetic Models

Intrinsic progression External sources (or sinks)

Medical treatments

Destructive interactions

Proliferative interactions

Conservative interactions

The number and the state of the cell can change because of

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.

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.

N. Bellomo & G. Forni Math. Comp. Modelling 20, 107-122 (1994)

Cell-to-cellInteractions

• Proliferative • Conservative• Destructive

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

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

D. Ambrosi, N. Bellomo & L. P. J. Theor. Medicine 4, 51-65 (2002)

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.

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)

B. Firmani, L. Guerri & L. P. Math. Models Methods Appl. Sci. 9, 491 (99)

T

I

I

T

t

t

T

I

I

T

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.

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.