Optimization of Multivariate Inverse Mixing Problems with Application to Neural Metabolite Analysis

A mathematical methodology is presented that optimally solves an inverse mixing problem when both the composition of the source components and the amount of each source component are unknown. The model is useful for situations when the determination of the source compositions is unreliable or infeasible. We apply the model to longitudinal proton magnetic resonance spectroscopy (1H MRS) data gathered from the brains of newborn infants. 1H MRS was used to study changes in five metabolite concentrations in two brain regions of nine healthy term neonates. Measurements were performed three times in each infant over a period of 3 months, starting from birth, for a total of 27 scans. The methodology was then used to translate the metabolite concentration data into measures of relative density for two major brain cell type populations by fitting a matrix of metabolite concentration per unit density to the data. One cell type, reflecting neuronal density, increased over time in both regions studied, but especially in the frontal regions of the brain. The second type, characterized primarily by myoinositol, reflecting glial cell content, was found to decrease in both regions over time. Our new method can provide more specific and accurate assessments of the brain cell types during early brain development in neonates. The methodology is applicable to a wide range of physical systems that involve mixing of unknown source components.