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Mathematical modelling of soft tissues

Periodic Reporting for period 1 - SOFT-TISSUES (Mathematical modelling of soft tissues)

Reporting period: 2017-03-01 to 2019-02-28

The last two decades have witnessed an explosion of scientific, medical and industrial activity in the area of “living tissues modelling”, with a particular focus on biological soft tissues. This recent tremendous effort can be explained by (i) the availability of a mature mathematical framework for writing down the equations governing their behaviour: the theory of nonlinear elasticity; (ii) ever-increasing computational power and techniques, making sophisticated and intensive calculations (both algebraic and numerical) less costly and more accurate; (iii) a strong need from medical doctors, surgeons and biomedical engineers to improve episodes of care for patients through computer simulations of events and procedures based on sound physical principles and reproducible experimental protocols.

Soft biological tissues (e.g. skin, internal organs, cardiovascular system. etc.) have many common mechanical features: their response to applied forces is markedly nonlinear; they allow for large strains; and they all exhibit the presence of residual stresses (i.e. they are still under stress when external loads are removed, because of a build-up of forces due to growth and remodelling). The theory of nonlinear elasticity can account for all these effects and in principle, the equations of equilibrium and of motion can be written down for any given model. In practice, they are extremely difficult to solve, even numerically. Moreover, the material parameters of a model must be determined experimentally with a reasonable accuracy in order to determine the magnitude of forces involved in, for instance, the reconstruction of an impact accident. Several testing protocols have been proposed to this end, but still lack authority essentially because, compared to common engineering materials, the properties of soft tissues are highly variable according to site, age, gender, health, etc.

In this context, the SOFT-TISSUES project aimed at:
• improving our understanding of the mathematics and the mechanics of living soft tissues;
• confronting theoretical predictions to experimental results, with a view to propose computer code and simulations, to be used in biomedical science;
• attempting to improve medical diagnosis methods and simulations and to developing medical strategies for disease prevention.
"The main results of the SOFT-TISSUES projects are new mathematical models that predict the mechanical behaviour of different soft tissues such as the brain and tissues with direction- dependent properties (skin, muscles, tendons) under large deformations. I developed:

•Novel, robust and validated modelling approaches to determine the elastic and visco- elastic properties of the brain in the large deformation regime. Based on our mechanical torsion tests on porcine brain samples, we observed and quantified a typical nonlinear property, the Poynting effect, i.e. the tendency of soft matter to expand in the direction perpendicular to the twisting or shearing plane;
• Computer simulations of traumatic head impacts (car and sport accidents) leading to brain damage. We generated coloured maps of the brain showing the regions of highest stress and strain, associated with tissue damage;
• A viscoelastic model for soft tissues with a family of fibres (transversely isotropic) which predicts their time-dependent response under large deformations;
• A mathematical model for the early development of human brain organoids. Organoids are prototypes of real brains, used to study the physiological and pathological development of the organ. This model can be used to identify any deviation from a physiological developing process, helping doctors in identifying at very early stage, the presence of malformations in the brain.

I completed three scientific journal papers: one published in the Proceedings of the Royal Society of London A, one accepted in Soft Matter and one currently under review in Physical Review Letters. All are high impact factor journals in the Applied Mathematics, Engineering and Physics communities. I also edited a special issue on “Constitutive modelling in Biomechanics” for the International Journal of Non-Linear Mechanics and organized a mini- symposium on this topic at the British Applied Mathematics Colloquium (BAMC) in St Andrews in March 2018, one of the biggest UK applied maths conferences

I disseminated my results at several international and national conferences:
• Oberwolfach miniworkshop on Mathematical Aspects of Nonlinear Wave Propagation in Solid Mechanics (invited talk), Mathematisches Forschungsinstitut Oberwolfach 03/03/19 – 09/03/19, Germany;
• 10th European Solid Mechanics Conference (invited talk to the simposium Nonlinear Elasticity), Bologna 02/07/18 – 06/07/18, Italy;
• Indam meeting: Mathematical Physics of living systems (invited talk), Cortona 27/08/17 - 01/09/17, Italy;
• International conference on modeling nonlinear continua (invited talk), Castro Urdiales 26/06/17 - 30/06/17, Spain;
• Third Workshop on Soft Tissue Modelling, Glasgow 07/06/17 - 9/06/17, UK.

To further communicate my research results to a wider audience, I delivered several talks and seminars and participated in the following outreach activities:
• FameLab Galway, with a 3-minute talk entitled “the invisible universe”;
• Maths Lessons in Middle School: I taught 9 lessons to 11 to 13 years old pupils, on four different maths topics: mathematics of the bicycle, the mechanics of the brain, the shape of the gut and how to ""watch"" the sound;
• Open Days at the host institution NUI Galway;
• An article published on RTE Brainstorm, an online section of the Irish television channel RTE) on “Exploring the hidden black holes at the centre of our galaxy”.
• Contribution to an article “Artificial skin could allow robots to feel like we do” published in Horizon, the EU Research & Innovation Magazine, which was also posted on Printed Electronics World."
The project was aligned with the objectives of the Horizon 2020 societal challenge “Health, demographic change and wellbeing”. By focusing on medical applications, by integrating theoretical, numerical and experimental approaches, and by promoting interdisciplinary and international exchanges, the SOFT-TISSUES project has improved our understanding of healthy and pathological soft tissues.
The results of the computer simulations of head impacts provide an invaluable information for medical doctors to validate other clinical methods to identify brain damage following rotational impacts, such as those experienced by the brain during a car or a sport accident.
Furthermore, this project helps strengthen healthy and active ageing, starting from the early stages of development, which is one of the objectives of the European program Health 2020. My research addressed this fundamental European economic challenge by investigating the normal development and growth of living tissues in order to provide insights into the prevention of developmental diseases.
vertical stress levels in a brian subject to rotational acceleration
Modeling the evolution of brain organoids
Testing protocols for brain tissues.