MATHEMATICAL MODELS IN BIOMEDICINE .

M. A. Herrero: EMS Lecturer 2014

Every year, the EMS appoints an EMS Lecturer. This is an honorary post, bestowed on an internationally renowned researcher, affiliated to a European institution. An EMS Lecturer is asked to deliver several lectures in various European places; one of the lectures is tied to an international conference in Europe. Costs of EMS Lecturers are supported by the EMS. In 2014, the EMS Lecturer is Miguel Angel Herrero, of Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense, Madrid , España.

Lectures at IAC-CNR, via dei Taurini 19, 00185 Rome

Novembre 11 14:30-16:30 - Lecture 1 and 2

November 13 14:30-16:30 - Lecture 3 and 4

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.