Bellomo, N.

Maini, P. K.

English

Mathematical Institute Eprints Archive

October 24, 2005 13:25 WSPC/103-M3AS 00090
Mathematical Models and Methods in Applied Sciences
Vol. 15, No. 11 (2005) iii–viii
c© World Scientific Publishing Company
PREFACE
The scientiﬁc community is becoming increasingly aware that the great revolution of
this century is going to be the mathematical formalization of phenomena occurring
in the Life Sciences, much as the revolution of the past two centuries was the
development of the above approach in the Physical Sciences. To quote J. E. Cohen:
“Mathematics is Biology’s next microscope, only better; Biology is Mathematics’
next Physics, only better.” This endeavor will be a huge challenge that will require
the intellectual energy of scientists working in the ﬁeld of mathematics and physics
collaborating closely with biologists and clinicians. This essentially means that the
heuristic experimental approach, which is the traditional investigative method in
the Biological Sciences, should be complemented by the mathematical modelling
approach. The latter can be used as a hypothesis-testing and indeed, hypothesis-
generating, tool which can help to direct experimental research. In turn, the results
of experiments help to reﬁne the modelling. The goal of this research is that, by
iterating back and forth between experiment and theory, we eventually arrive at a
deeper conceptual understanding of how the highly nonlinear processes in biology
interact. The ultimate goal in the clinical setting is to use mathematical models to
help design therapeutic strategies.
There has, recently, been a signiﬁcant shift in biological research, with experi-
mentalists now beginning to voice the need for multi-disciplinary research encom-
passing mathematical and computational approaches. For example, Hartwell et al.1
and Greller et al.,2 encourage applied mathematicians to develop mathematical
approaches to mathematical modelling of complex systems in molecular biology.
These papers, only two amongst several others in a similar vein, not only encour-
age research activity, but also suggest methodological approaches. Speciﬁcally, the
ﬁrst paper provides interesting suggestions towards the mathematical formaliza-
tion of the “theory of moduli”, while the second paper motivates the development
of methods of non-equilibrium statistical mechanics to cancer modelling.
Modelling and simulation of tumor growth in competition with the immune
system is certainly one of the challenging frontiers of applied mathematics which
could have a great impact both on the quality of life and the development of mathe-
matical sciences. It is true that mathematics cannot solve problems of immunology
and medicine. However, applied mathematics may be able to provide a framework
in which experimental results can be interpreted, and a quantitative analysis of
iii
October 24, 2005 13:25 WSPC/103-M3AS 00090
iv Preface
external actions to control neoplastic growth can be developed. Speciﬁcally, mod-
els and simulations of particular behaviors of immune system-tumor competition
can reduce the amount of experimentation necessary for drug and therapy devel-
opment. Moreover, the mathematical theory developed might not only provide a
detailed description of the spatiotemporal evolution of the system, but may also
help us in understanding and manipulating aspects of the process that are diﬃcult
to access experimentally.
The interest of mathematicians in the above ﬁeld is rapidly growing and is
clearly evident in the vast literature appearing in journals of applied mathe-
matics. The state of the art is reported in edited books, e.g. Refs. 3 and 4,
where contributions from applied mathematicians to the above-mentioned topics
are presented. The interest of applied mathematicians, and the need for devel-
oping mathematical approaches motivates, as suggested by Gatenby and Maini,5
a new branch of mathematical research which may be called “mathematical
oncology ”.
One of the major problems in modelling biological systems is the multiscale
nature of most biological systems. Hence in a more general setting, the mod-
elling of the tumor-immune system interaction serves as a paradigm that may have
more widespread signiﬁcance. Speciﬁcally, the characterization of the system sug-
gests the identiﬁcation of three natural scales which are also connected to diﬀerent
stages of the disease: processes on the cellular scale are triggered by signals stem-
ming from the sub-cellular level and have an impact on the macroscopic scale,
i.e. on the organism, as tumor cells condense and when tumors grow and spread.
In detail:
The sub-cellular scale: The evolution of a cell is regulated by the genes contained
in its nucleus. Receptors on the cell surface can receive signals which are then
transmitted to the cell nucleus, where the afore-mentioned genes can be acti-
vated or suppressed. In extreme situations, particular signals can induce a cell
to reproduce itself in the form of identical descendants — so-called clonal expan-
sion, or to die and disappear apparently without trace — so-called apoptosis or
programmed cell death. Clonal expansion activates a competitive-cooperative inter-
action between tumor cells and cells of the immune system. If the immune system
is active and able to recognize the tumor cells, then it may be able to develop a
destruction mechanism; otherwise, tumor growth may develop progressively. The
activation and deactivation of immune cells, too, can be regulated by cytokine
signals.
The cellular scale: On the cellular level, models are proposed to simulate the eﬀects
of the failure of programmed cell death and of the loss of cell diﬀerentiation. If and
when a tumor cell is recognized by immune cells a competition starts which may end
up either with the destruction of tumor cells or with the inhibition and depression
of the immune system. Cellular interactions are regulated by signals emitted and
received by cells through complex reception and transduction processes. Therefore,
October 24, 2005 13:25 WSPC/103-M3AS 00090
Preface v
the connection to the afore-mentioned sub-cellular scale is evident. On the other
hand, the development of tumor cells, if not suppressed by the immune system,
tends towards condensation into a solid form so that macroscopic features become
important.
The macroscopic scale: After a suitable maturation time, tumor cells may start to
condense and aggregate into an entity which eventually evolves as a “quasi-fractal
surface” which interacts with the outer environment, for example normal host cells
and the immune system. These interactions usually occur on the surface and within
a layer where angiogenesis (the process of formation of new blood vessels, induced
by factors secreted by the tumor, and vital for tumor growth) takes place. Here, one
has the overlap of phenomena at the cellular level with typical macroscopic behavior
such as diﬀusion or, more generally, phenomena that can be related to the mass
balance or evolution of macroscopic variables such as tumor size. In a later stage, the
tumor can be characterized by three zones: in an external layer environmental cells
penetrate and determine the detachment of tumor cells; an intermediate layer in
which there are clusters of quiescent tumor cells; and an inner zone, which contains
necrotic cells. Due to the heterogeneous nature of tumors, some of these “layers”
may actually occur as “islands”.
Diﬀerent mathematical methods and structures correspond to the above scales.
For instance models at the cellular scale are generally developed in terms of ordi-
nary diﬀerential equations or Boolean networks, while multicellular systems are
modelled by nonlinear integro-diﬀerential equations similar to those of nonlinear
kinetic theory (the Boltzmann equation), or by individual-based models which
give rise to a large set of discrete equations. Macroscopic models lead to moving
boundary problems for systems of nonlinear partial diﬀerential equations. Nonlin-
earity is an intrinsic feature of all models. In short, this system leads to prob-
lems that span the breadth of applied mathematics and stretch our computational
resources.
From the mathematical viewpoint, the main objective of the series of special
issues of which this is the ﬁrst, is to bring to the attention of the wider mathematical
community, the problems involved in multiscale modelling and the methods being
presently applied to the analysis of complex biological systems constituted by large
systems of interacting cell populations. The following topics can be regarded as new
frontiers of applied mathematics:
(i) Modelling and qualitative analysis of systems of nonlinear integro-diﬀerential
equations related to the mesoscopic (kinetic) description of multicellular
systems.6 The above equations deﬁne the evolution over the microscopic state
of the cells and includes both mechanical and biological variables. The struc-
ture of the equation is somehow analogous to that of mathematical kinetic
theory. The substantial diﬀerence (and diﬃculty) is that the dynamics at the
cellular scale has to be modelled taking into account the interaction between
mechanics and biology.
October 24, 2005 13:25 WSPC/103-M3AS 00090
vi Preface
(ii) Mathematical asymptotic theory suitable to recover the macroscopic descrip-
tion of cell mixtures from the mesoscopic description. The theory should pro-
vide a rigorous derivation, rather than a purely phenomenological one, of the
mechanics of the systems in terms of partial diﬀerential equations, generally
nonlinear, of parabolic and hyperbolic type.7
(iii) Qualitative and computational analysis of initial-boundary value problems
with moving boundary, related to the dynamics of solid tumors described by
models derived according to items (i) and (ii).
(iv) Multiscale modelling where a combination of several diﬀerent mathematical
approaches appear, following the conceptual lines given in Refs. 8 and 9, in
the formalized description of the whole system.
In general, simulations may need the use of dedicated computer devices, to
solve systems which include biological variables in the various transport phenom-
ena related to biological systems.10 The common feature of the above mathe-
matical approach is that the equations model living matter, and the ability of
cells to organize their dynamics needs to be an essential feature of these mathe-
matical models. Another key feature is the multiscale approach to modelling and
simulation.8,9 Indeed, we are convinced that the complexity of the system related
to these special issues requires the invention of new mathematical methods, or
at least new ideas, to place into a mathematical context the above mentioned
complexity.
This special issue presents six research papers which deal with diﬀerent aspects
of the mathematical problems which have been outlined above.
The ﬁrst paper, by Gatenby, Vincent and Gillies,11 deals with modelling at the
sub-cellular scale with speciﬁc focus on an evolutionary game theory approach to
somatic evolution to describe how genomic changes result in cancerous cells that
can invade normal tissue.
The second paper, by Bellouquid and Delitala,12 develops methods of mathe-
matical kinetic theory to deal with large systems of active particles, cells whose
microscopic state, in addition to a geometrical and mechanical description, also
includes biological functions.
Approaches from generalized kinetic theory form a framework in which models of
continuum mechanics for biological tissues can be derived and a suitable asymptotic
theory, ideally analogous to the one known for classical particles, developed. This
mathematical approach is proposed in the third paper, by Lachowicz,13 where it is
shown how the continuous model used in the paper by Chaplain and Lolas14 can be
obtained. The latter paper, the fourth in this issue, presents a variety of interesting
phenomena, in the context of cancer invasion of tissues. It is worth pointing out
the conceptual connection between the above two papers where the macroscopic
description is related to the underlying microscopic description.
October 24, 2005 13:25 WSPC/103-M3AS 00090
Preface vii
The ﬁfth paper, by Bertuzzi, Fasano and Gandolﬁ,15 deals with the macro-
scopic description, and speciﬁcally with mathematical aspects of moving boundary
problems, for tumor cords incorporating the ﬂow of interstial ﬂuids. Finally, the
paper by Hatzikirou, Deutsch, Schaller, Simon and Swanson16 provides a review,
and a critical analysis, of the vast literature concerning mathematical modelling
and medical therapies of glioblastoma.
This issue is the ﬁrst of a series which aims to create a forum for applied mathe-
maticians involved in the ﬁeld. Hopefully, the interplay between diﬀerent approaches
will lead to the generation of a new mathematics needed to deal with the above
class of biological systems and phenomena.
This issue is dedicated to the memory of Lee A. Segel, an applied mathemati-
cian who brought mathematics to several areas of the life sciences. His work is an
inspiration to us.
References
1. H. L. Hartwell, J. J. Hopﬁeld, S. Leibner and A. W. Murray, From molecular to
modular cell biology, Nature 402 (1999) c47–c52.
2. L. Greller, F. Tobin and G. Poste, Tumor hetereogenity and progression: Conceptual
foundation for modeling, Inv. Metastasis 16 (1996) 177–208.
3. Eds. J. Adam and N. Bellomo, A Survey of Models on Tumor Immune Systems
Dynamics (Birkha¨user, 1997).
4. L. Preziosi, Modeling Cancer Growth (CRC Press, 2003).
5. R. A. Gatenby and P. K. Maini, Mathematical oncology: Cancer summed up, Nature
421 (2003) 321–323.
6. N. Bellomo, A. Bellouquid and M. Delitala, Mathematical topics in the modelling com-
plex multicellular systems and tumor immune cells competition, Math. Mod. Meth.
Appl. Sci. 14 (2004) 1683–1733.
7. N. Bellomo and A. Bellouquid, From a class of kinetic models to macroscopic equations
for multicellular systems in biology, Cont. Disc. Dyn. Syst. B 4 (2004) 59–80.
8. T. Alarco´n, H. M. Byrne and P. K. Maini, A mathematical model of the eﬀects of
hypoxia on the cell-cycle of normal cancer cells, J. Theor. Biol. 229 (2004) 395–411.
9. T. Alarco´n, H. M. Byrne and P. K. Maini, A multiple scale model for tumour growth,
SIAM J. Multiscale Simu. 3 (2005) 440–475.
10. A. Deutsch and S. Dorfman, Cellular Automaton Modelling of Biological Pattern For-
mation (Birkha¨user, 2004).
11. R. A. Gatemby, T. L. Vincent and R. J. Gillies, Evolutionary dynamics in carcino-
genesis, Math. Mod. Meth. Appl. Sci. 15 (2005) 1619–1638.
12. A. Bellouquid and M. Delitala, Mathematical methods and tools of kinetic theory
towards modelling complex biological systems, Math. Mod. Meth. Appl. Sci. 15 (2005)
1639–1666.
13. M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue
invasion by solid tumors, Math. Mod. Meth. Appl. Sci. 15 (2005) 1667–1683.
14. M. A. J. Chaplain and G. Lolas, Mathematical modelling of cancer cell invasion of
tissue: The role of the urokinase plasminogen activation system, Math. Mod. Meth.
Appl. Sci. 15 (2005) 1685–1734.
October 24, 2005 13:25 WSPC/103-M3AS 00090
viii Preface
15. A. Bertuzzi, A. Fasano and A. Gandolﬁ, A mathematical model for tumor cords
incorporating the ﬂow of interstitial ﬂuids, Math. Mod. Meth. Appl. Sci. 15 (2005)
1735–1777.
16. H. Hatzikirou, A. Deutsch, C. Schaller, M. Simon and K. Swanson, Mathematical
modelling of glioblastoma tumour development: A review, Math. Mod. Meth. Appl.
Sci. 15 (2005) 1779–1794.
Nicola Bellomo
nicola.bellomo@polito.it
and
Philip Maini
maini@maths.ac.ox.uk


A mathematical model of the effects of hypoxia on the cell-cycle of normal cancer cells,

A multiple scale model for tumour growth,

A Survey of Models on Tumor Immune Systems Dynamics (Birkha¨user,

Cellular Automaton Modelling of Biological Pattern Formation (Birkha¨user,

From a class of kinetic models to macroscopic equations for multicellular systems in biology,

From molecular to modular cell biology,

Mathematical oncology: Cancer summed up,

Mathematical topics in the modelling complex multicellular systems and tumor immune cells competition,

Modeling Cancer Growth

Tumor hetereogenity and progression: Conceptual foundation for modeling,

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