Inverse potential problems of bio-electric imaging with measures
Coriolis room, Galois building – April 9, 2024 at 11:00 AM.
Inverse potential problems are central to bio-imaging techniques based on bio-electric phenomenon such as electroencephalography (EEG) and electrocardiography imaging (ECGI). These imaging techniques aim to identify regions of the brain or the heart that contain depolarising or repolarising cell clusters at any given time using measurements of the electric potential outside the body. By mapping the patterns of depolarisation and repolarisation, functional connections of the brain or the heart can be mapped to further understand the mechanisms underlying the function of these organs. These depolarisation and repolarisation patterns can also be used to identify diseased tissues of the brain or the heart in clinical settings. For the brain or the heart, the extracellular potential ϕ that is measured is related to the transmembrane potential v in the brain or the heart, respectively, through the elliptic problem ∇ · (M∇ϕ) = ∇ · (Mi∇v), where M and Mi are conductivity tensors that encode the conduction anisotropy of the organs. The conduction anisotropy is due to the cells in the heart and the brain largely being organised into fibres and the cells are of different types as well. The transmembrane potential which ecodes depolarisation and repolarisation of cells can be considered to be a function of bounded variation hence solving the inverse potential problems of the above elliptic problem is equivalent to identifying a measure supported on curves (fibres) in 3D space given measurements of ϕ. We present a numerical technique that leverages the fibrous organisation of the brain’s and the heart’s tissue cells to recover the vector-valued measure ∇v supported in the brain or the heart. The numerical technique we present is biased towards “deep sources” hence presents an opportunity to recover such sources without the need for special regularisers. This numerical technique can further be developed along with stimulation protocols to device a method for estimating conduction anisotropy in vivo in the brain and the heart.
