How do we understand whether epidemics will spread, or predict the likelihood of extreme meteorological events? How do we optimize nano-particles to make them efficient vehicles for targeted delivery of drugs in medical applications? Can we predict how a cell in a specific state will evolve in time, for example whether it will stay healthy or become diseased? How do we prevent over-heating in fast electronic devices, or understand how energy conversion in the next-generation of solar cells might work? Can insights into extreme events in the sciences be used to detect whether a network of financial institutions is close to a crash? These research challenges all relate to non-equilibrium systems. Such systems are typically irreversible, so that if a movie of the system was played backwards it would look very different. For equilibrium systems, on the other hand, a backwards movie would appear much like the original. This lack of an "arrow of time" makes equilibrium systems relatively easy to understand, and indeed most existing methods for predicting e.g. the behaviour of materials are based on the assumption of equilibrium. For non-equilibrium systems, on the other hand, we know much less. They can "age" towards an equilibrium state that is never reached, or exhibit extreme events. The latter are often the result of cascades of collective failure, as illustrated in phenomena ranging from stock market crashes to environmental disasters. As the examples in the first paragraph show, understanding, predicting and controlling non-equilibrium behaviour is an important challenge in many problems across the physical, mathematical, biological and environmental sciences. The starting point for the proposed interdisciplinary Centre for Doctoral Training is, therefore, that significant progress will require researchers that can exploit and strengthen such links between disciplines. They will need to be trained in how to analyse non-equilibrium systems theoretically, via mathematical models, how to study their behaviour via computer simulations, and how to extract information about them from possibly noisy or incomplete data. They will also need to be aware of the important non-equilibrium problems in different disciplines, to see connections, transfer methods and concepts from one application area to another, and develop new approaches. The Centre for Doctoral Training in Cross-Disciplinary Approaches to Non-Equilibrium Systems (CANES) will provide the training that such a new generation of researchers will need. In a substantial cohort of at least 10 PhD students per year it will make sure that ideas are exchanged systematically; research projects will be designed to build bridges between different disciplines employing similar methods, or to explore the connections between different approaches that are used to study non-equilibrium systems in the same area. Students will acquire transferable communication and presentation skills, and take part in outreach activities to increase the public understanding of non-equilibrium science. In `open questions sandpits', industry engagement events and dedicated careers events they will also obtain a solid understanding of the priorities of industrial partners working on non-equilibrium systems, and of attractive career paths outside of university. Overall, CANES researchers will emerge with a wide variety of skills that are highly sought after in academia and industry. CANES will also put UK research in non-equilibrium systems on the international map, helping the UK to compete against other countries like the U.S.A. where there is already a significant drive to strengthen research in this area.

In the diagnosis and treatment of disease, clinicians base their decisions on understanding of the many factors that contribute to medical conditions, together with the particular circumstances of each patient. This is a "modelling" process, in which the patient's data are matched with an existing conceptual framework to guide selection of a treatment strategy based on experience. Now, after a long gestation, the world of in silico medicine is bringing sophisticated mathematics and computer simulation to this fundamental aspect of healthcare, adding to - and perhaps ultimately replacing - less structured approaches to disease representation. The in silico specialisation is now maturing into a separate engineering discipline, and is establishing sophisticated mathematical frameworks, both to describe the structures and interactions of the human body itself, and to solve the complex equations that represent the evolution of any particular biological process. So far the discipline has established excellent applications, but it has been slower to succeed in the more complex area of soft tissue behaviour, particularly across wide ranges of length scales (subcellular to organ). This EPSRC SoftMech initiative proposes to accelerate the development of multiscale soft-tissue modelling by constructing a generic mathematical multiscale framework. This will be a truly innovative step, as it will provide a common language with which all relevant materials, interactions and evolutions can be portrayed, and it will be designed from a standardised viewpoint to integrate with the totality of the work of the in silico community as a whole. In particular, it will integrate with the EPSRC MultiSim multiscale musculoskeletal simulation framework being developed by SoftMech partner Insigneo, and it will be validated in the two highest-mortality clinical areas of cardiac disease and cancer. The mathematics we will develop will have a vocabulary that is both rich and extensible, meaning that we will equip it for the majority of the known representations required but design it with an open architecture allowing others to contribute additional formulations as the need arises. It will already include novel constructions developed during the SoftMech project itself, and we will provide many detailed examples of usage drawn from our twin validation domains. The project will be seriously collaborative as we establish a strong network of interested parties across the UK. The key elements of the planned scientific advances relate to the feedback loop of the structural adaptations that cells make in response to mechanical and chemical stimuli. A major challenge is the current lack of models that operate across multiple length scales, and it is here that we will focus our developmental activities. Over recent years we have developed mathematical descriptions of the relevant mechanical properties of soft tissues (arteries, myocardium, cancer cells), and we have access to new experimental and statistical techniques (such as atomic force microscopy, MRI, DT-MRI and model selection), meaning that the resulting tools will bring much-need facilities and will be applicable across problems, including wound healing and cancer cell proliferation. The many detailed outputs of the work include, most importantly, the new mathematical framework, which will immediately enable all researchers to participate in fresh modelling activities. Beyond this our new methods of representation will simplify and extend the range of targets that can be modelled and, significantly, we will be devoting major effort to developing complex usage examples across cancer and cardiac domains. The tools will be ready for incorporation in commercial products, and our industrial partners plan extensions to their current systems. The practical results of improved modelling will be a better understanding of how our bodies work, leading to new therapies for cancer and cardiac disease.