# Adaptive finite element computations of nonlinear elasticity problems

1st October 2006 to 30th September 2010
We propose to use mathematics to help doctors working in hospitals provide better care for two groups of patients. these groups of patients are women with breast cancer, and patients with heart disease. We begin this summary by describing how mathematics may help provide better care for these groups of patients, and then brifly describe the mathematics we use. Approximately 10% of women in developed countries will develop breast cancer during the course of their lives. A cancerous tumour may be located using one or more clinical imaging techniques from a range of clinical technique, one of these techniques, called magnetic resonance imaging, the woman will be lying on her front. For another technique, called X-ray mammography, the woman will be standing with her breast compressed between two plates. For another technique, called ultrasound, the woman will be lying on her side. The breast will take a very differnt shape for each of these techniques. Should the woman require surgery to remove a tumour, she will lie on her back during surgery and the breast will take yet another shape. This will make hard for the surgeon to locate the tumour in the breast. We intend to use mathematics to help the surgeon locate the tumour. Heart disease is the major cause of death in the western world. The forces needed for a heartbeat are generated by biochemical reactions inside cells in the heart. In a healthy heart the cells will contract at roughly the same time, which enables the efficient pumping of blood around the body. This doesn't always happen in an unhealthy heart. We propose to use mathematics to explain what happens instead. The mathematics we use relates the dformation of the breast or the heart to the forces that are applied. Body organs change shape when a force is applied to them, or when a person changes prosition, e.g. stands up rather than lies down. This change in shape is described by equations that may be solved by programming a computer to use a mathematical algorithm. As body organs can be very complex, a computer may take a long time to solve the problem using these algorithms. The work described in this proposal intends to establish the most suitable algorithm, i.e. the alorithm that will execute on a computer in the shortest time whilst ensuring that the answer is correct.