Suppose that you are sitting alone at a table in a crowded restaurant where you can hear a mixture of several conversations. This mixed sound is not particularly interesting, but suppose you could record it and then use some sort of algorithm to extract the individual conversations that were going on in the restaurant, that would certainly be much more informative. In statistical jargon this is called a "mixture model", and there is a surprisingly large number of real-life situations where they are very useful. As a motivating application we consider an important biomedical problem called alternative splicing. Although all humans have the same genes encoded in our DNA, it turns out that each of our genes can be expressed in several variations (called splicing variants), and that each of these variations performs different functions in the organism; some may even help cause complex neurodegenerative diseases or cancer. Fortunately, technologies from recent years produce data that allow us for the first time to study this phenomenon in detail. We can now observe the overall expression of the gene from which, similar to the restaurant example, we would like to learn what are the individual contributions of each gene variant (indeed, to learn whether a given variant was even present at all). These technologies are becoming cheaper every year, and one can easily envision a nearby future where they are part of our regular medical check-ups, but solving this mixture problem poses formidable methodological and practical challenges. For instance, the number of possible solutions even when considering a single gene is larger than the number of atoms in the universe, and the required calculations can be prohibitive even on the latest computers. This example highlights some of the most important challenges that are common to many modern applications of mixture models, hence solving them would have positive implications in a much wider range of areas (e.g. technology, industry, public policy, social sciences). In this project we aim to develop a framework that can be used to solve the alternative splicing and other challenging mixture model problems. Our first goal is to propose a novel formulation for general mixture models that has proven highly successful in other complex settings, studying both theoretical and practical aspects. In our example this formulation says that, when identifying different conversations in the restaurant, we cannot have two tables uttering exactly the same words (else these should be regarded as a single conversation). This apparently simple consideration turns out to have important mathematical consequences that greatly simplify the problem. Our second goal is to apply these general principles to solve the alternative splicing problem, where we will also bring to bear scientific considerations to ensure that the solution is useful in practice. Our third goal is to propose and study strategies to make fast and accurate calculations, which can quickly become prohibitive, so that a computer can find the solution in reasonable time. As part of this project we will provide open-source software that others can use freely for their own research or applied data analysis. Given the technical challenges involved the bulk of the research will be carried at the Dept. of Statistics at the University of Warwick by the PI working with other members of the department and several further statistical and biomedical collaborators from prestigious overseas universities and hospitals who will be actively involved in the project, e.g. helping translate our methodology to biomedical research and clinical practice, or ensuring that our statistical predictions are indeed accurate.

Heart failure is a lethal syndrome representing a common 'final pathway' for sufferers of a multitude of cardiac and respiratory diseases. 1 in 5 people will suffer from heart failure during their life time and once diagnosed ~40% of patients die within one year. Heart failure is caused by the heart's inability to perfuse the organs of the body with blood. The energy starvation hypothesis is a new model of heart failure and proposes that the reduced supply of energy is a fundamental cause of heart failure. The energy starvation hypothesis is the result of genetic studies and new experimental methodologies and provides a unifying mechanism to explain the development of cardiac contractile failure, yet the significance of compromised energy supply is debated. This project will investigate the importance of the energy starvation hypothesis by analysing the extent to which decreases in energy supply during heart failure compromise heart function. The cardiac energy supply chain (CESC) spans from the organ to the sub cellular scale. Energy supply decreases during heart failure due to the compromise of independent compounding links of the CESC at the organ, tissue and cellular scale. At the organ scale, blood flow through the arteries supplying blood to the heart decreases. At the tissue scale, oxygen and metabolite flux from the capillaries to the cells is reduced. At the cellular scale, the conversion of oxygen and metabolites to high energy molecules and the transport of these to the points of utilization are inhibited. I propose to investigate the energy supply to heart cells in the failing heart by developing a series of coupled models representing the cellular scale (metabolism, electrical activity, biochemical, contraction), tissue scale (movement of oxygen and metabolites, capillary circulation) and organ scale (blood supply to the heart, mechanics, electrical activation) components of the CESC. Changing model parameters and geometries will then allow the CESC during heart failure to be simulated. The model will be systematically validated against experimental results at each stage in model development. The final integrated multi-scale model will be used to test the energy starvation hypothesis by quantifying how the individual and integrated changes to the CESC during heart failure affect whole heart function.In order to build these models, we will use sophisticated image processing techniques to build an accurate 3D geometrical representation of the heart, arteries supplying blood to the heart and capillary network from high resolution datasets. Advanced numerical methods will be used to formulate mathematical equations for the transduction of energy within the heart. Cutting edge experimental procedures will provide key information on changes in cellular, tissue and organ structure and function during heart failure. Such combinations of mathematical modelling techniques and experimental investigations are vital for elucidating the mechanisms underlying the causes and progression of heart failure and may ultimately lead to improved treatment and prevention.

Low back pain is the leading cause of disability worldwide and is estimated to cost the NHS £500 million annually. A link has been found between degeneration of the intervertebral discs and low back pain, suggesting degeneration may be a contributing factor. Identifying patients where this is the case is not straight forward, particularly as it is possible to have degenerate discs without experiencing any pain at all. Initially, patients with low back pain are treated conservatively, but for those who require surgery, fusion of the vertebrae is the most common procedure, although increasingly total or partial disc replacement technologies are considered to preserve motion at the joint. Outcomes from these procedures are relatively poor, with revision surgeries required in as many as 20% of patients who undergo a lumbar fusion within 10 years. Improving patient selection is important for good clinical outcomes using these treatments, however the tools currently available (usually Magnetic Resonance Imaging (MRI) or X-ray) provide little information of how the disc is functioning before a clinical decision is made. The ability to assess quantitatively the deformations within discs would provide a unique tool to allow treatments to be targeted towards appropriate patients and therefore improve outcomes. Recent advances have been made in measuring disc deformations in human cadaveric specimens by combining a technique called Digital Volume Correlation (DVC) and MR images captured with a high-resolution research scanner (9.4T). The technique works by obtaining two sets of images of the same specimen, one unloaded, and one loaded. Three dimensional patterns within the images are then tracked between the two sets of images such that deformations and strains can be calculated. Results from this study show huge potential but a real breakthrough will come when the tool can be used clinically, this is not currently possible because the bore of research MRI scanners is less than 10cm in diameter. This study will utilise DVC in clinical MRI scanners (that is, scanners used in every hospital to image patients) to create a non-invasive clinical method of measuring intervertebral disc deformations. This novel diagnostic tool will allow better stratification in treatment. It will also provide fresh insight into the intricate mechanics of healthy and degenerate discs, information that will guide future surgical treatments and medical device designs.

Heart failure is a lethal syndrome representing a common 'final pathway' for sufferers of a multitude of cardiac and respiratory diseases. 1 in 5 people will suffer from heart failure during their life time and once diagnosed ~40% of patients die within one year. Heart failure is caused by the heart's inability to perfuse the organs of the body with blood. The energy starvation hypothesis is a new model of heart failure and proposes that the reduced supply of energy is a fundamental cause of heart failure. The energy starvation hypothesis is the result of genetic studies and new experimental methodologies and provides a unifying mechanism to explain the development of cardiac contractile failure, yet the significance of compromised energy supply is debated. This project will investigate the importance of the energy starvation hypothesis by analysing the extent to which decreases in energy supply during heart failure compromise heart function. The cardiac energy supply chain (CESC) spans from the organ to the sub cellular scale. Energy supply decreases during heart failure due to the compromise of independent compounding links of the CESC at the organ, tissue and cellular scale. At the organ scale, blood flow through the arteries supplying blood to the heart decreases. At the tissue scale, oxygen and metabolite flux from the capillaries to the cells is reduced. At the cellular scale, the conversion of oxygen and metabolites to high energy molecules and the transport of these to the points of utilization are inhibited. I propose to investigate the energy supply to heart cells in the failing heart by developing a series of coupled models representing the cellular scale (metabolism, electrical activity, biochemical, contraction), tissue scale (movement of oxygen and metabolites, capillary circulation) and organ scale (blood supply to the heart, mechanics, electrical activation) components of the CESC. Changing model parameters and geometries will then allow the CESC during heart failure to be simulated. The model will be systematically validated against experimental results at each stage in model development. The final integrated multi-scale model will be used to test the energy starvation hypothesis by quantifying how the individual and integrated changes to the CESC during heart failure affect whole heart function.In order to build these models, we will use sophisticated image processing techniques to build an accurate 3D geometrical representation of the heart, arteries supplying blood to the heart and capillary network from high resolution datasets. Advanced numerical methods will be used to formulate mathematical equations for the transduction of energy within the heart. Cutting edge experimental procedures will provide key information on changes in cellular, tissue and organ structure and function during heart failure. Such combinations of mathematical modelling techniques and experimental investigations are vital for elucidating the mechanisms underlying the causes and progression of heart failure and may ultimately lead to improved treatment and prevention.