New methodologies for the filtering and state estimation problems in bio-medical applications

Biomedical engineering (BME) is the application of engineering principles and design concepts to medicine and biology. This field seeks to close the gap between engineering and medicine: it combines the design and problem solving skills of engineering with medical and biological sciences to advance healthcare treatment, including diagnosis, monitoring, treatment and therapy. Biomedical engineering has only recently emerged as its own discipline, compared to many other engineering fields. Such an evolution is common as a new field transitions from being an interdisciplinary specialization among already-established fields, to being considered a field in itself. Much of the work in biomedical engineering consists of research and development, spanning a broad array of subfields (tissue and cellular engineering, bioinformatics, robotics, etc.). In this field, the control theory is becoming increasingly important in medical applications because the use of specific techniques (state observer, parameter estimation, Kalman filter etc.) allows to obtain information not available with traditional engineering methods. For this reason, in this thesis have been developed new techniques for filtering and state estimation for the solution of biomedical problems. In particular in Chapter 1, it has been described a method of discretization and a state observer capable of tackling the problem of tumor growth. A state observer is a mathematical tool that allows tracking the behavior of the unknown state variables of a system starting from the value of a measurable subset. A state observer is based on the knowledge of the dynamics of the system and its evolution from an initial estimate of the state variables converges asymptotically to the true value. In the field of molecular biology, the study of gene expression, meaning by this term the series of events that after the activation of transcription of a gene, leading to the production of the corresponding protein, is of considerable interest. Gene expression is regulated internally to the cell mainly by transcription factors, specific proteins that determine which genes are being expressed and which are not. In this thesis in Chapter 2, has been developed an original mathematical model for the gene expression between co-regulated genes. The dynamics of the adjustment made by the transcription factor was modeled as a white noise and the model parameters were estimated using the Kalman filter and identified through the maximum likelihood estimation. Finally, it was studied two classical problems of robotic field (planar tracking and perspective vision) and their solutions have been addressed through the use of the new concept of virtual measurement map, in the Chapter 3.