MATHEMATICAL MODELS IN BIOMEDICINE .
M. A. Herrero
Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad
Complutense, 28040 Madrid , España.
INTRODUCTION.- Mathematical methods are increasingly being used to address
problems in biomedical sciences. In fact, statistical analysis has long since been
acknowledged as crucial to validate experimental results. More recently, computer
simulations have shown their unmatched strength to carry out in silico
experiments, whereby the consequences of clearly stated hypotheses can be
explored. In addition, mathematical modelling is becoming a method of choice
when it comes to identify main agents operating in complex scenarios, where it is
important to tell causes from effects in what appears at first glance as a confuse
heap of experimental data.
In the following lectures some examples will be discussed to show how
mathematical modelling, analysis and simulation, used as a complement to (but
not as a substitute of) mainstream biomedical techniques can be used to gain
insight about complex systems of biological (and social) interest. Lectures will be
mainly of an expository nature and addressed to broad audiences, including
undergraduate and graduate students with different backgrounds. Technicalities
will be avoided , and in each any prerequisite will be introduced when required.
Lecture 1.-Comprehensive vs minimal models: Emergent behaviours in
immune response.
The wide availability of computers and the astonishing efficiency of computer
power have greatly stimulated the formulation and study of complex problems in
biomedicine. Indeed, large systems of equations can now be routinely simulated,
which allows for simultaneous consideration of many biological processes
involved. However, uncertainty about parameter values, and hence about the
comparative importance of their underlying mechanisms, sets serious limitations
to the practical relevance of such comprehensive models .
In this lecture a minimal mathematical model will be discussed to account for key
features of immune response to acute infections. In particular the manner in which
a large population of T –cells (lymphocytes) is first generated (clonal expansion)
and then dispensed with (clonal contraction) will be obtained as an emergent
population effect arising from a limited number of possible individual T-cell
decisions, encoded in a simple deterministic algorithm. Further extensions of the
approach presented will also be discussed.
Lecture 2.- Mathematical problems in radiotherapy: Accounting for tumour
heterogeneity.
Radiotherapy, the use of ionizing radiation to eliminate solid tumours, is a
treatment of choice for more than 50% of cancer patients. A key issue to be
addressed in any radiotherapy plan is how to achieve significant tumour control
with a minimum of induced side effects on neighbouring healthy tissues and
organs at risk. Outstanding scientific and technological advances during the last
century have made radiotherapy a successful treatment in a large percentage of
cases. However, as it happens with any other currently used therapy, it still yields
poor results when applied to tumours at an advanced, disseminated stage.
In this lecture some mathematical methods in radiotherapy will be described.
Particular attention will be paid to modelling tumour heterogeneity, characterised
by the presence of several phenotypes with different resistance to radiation within
a tumour, and to the manner in which efficient dosimetries accounting for such
heterogeneity can be selected.
Lecture 3.- Bone remodelling : How our bones are renewed and small
fractures repaired.
Bone health is essential to ensure high quality of life. Out of the many bone- related
hazards, bone fracture is a particularly relevant one. Indeed, no less 1,5 million
people suffer a bone fracture related disease in the U.S every year, and the
seriousness of the resulting disorders increase with age. Yet the human skeleton
has an impressive capability for self-repair. In fact, our bones are being
continuously remodelled so that at any time old bone is replaced by new one in a
self - regulated manner. As a result, our skeleton is fully renewed approximately
every 15 years. Such remodelling may be triggered by external forces. In
particular, bone micro-fractures generated by physical exercise (including
walking) are routinely repaired without being even noticed.
In this lecture we will describe some aspects of bone self-repairing processes for
sufficiently small fractures. In particular, mathematical models accounting for the
underlying biological processes will be presented, and the role of emergent
behaviours in bone repair regulation will be discussed in detail.
Lecture 4.- Exploring Utopia : Mathematical modelling in social dynamics.
Describing the rules that should govern a human society to achieve its perfection
has been a question of paramount interest since the earliest extant records of
human thought. A widely accepted assumption made by many authors is that an
ideal society could be achieved out of the observance of suitable codes of
individual rules. However, to this day perfect societies are nowhere in sight, and
the hypothesis that any law code will generate its code breakers is steadily gaining
ground.
In this lecture I will briefly discuss on how interactions among individuals in a
society can be described, albeit in a crude way, by means of population dynamics
methods. Only a few examples of this approach will be explored, particularly the
impact of social and economical mobility in the overall dynamics of a society
structured in socio-economical classes.