  Identifying concentrations of components from an observed mixture is a
fundamental problem in signal processing. It has diverse applications in fields
ranging from hyperspectral imaging to denoising biomedical sensors. This paper
focuses on in-silico deconvolution of signals associated with complex tissues
into their constitutive cell-type specific components, along with a
quantitative characterization of the cell-types. Deconvolving mixed
tissues/cell-types is useful in the removal of contaminants (e.g., surrounding
cells) from tumor biopsies, as well as in monitoring changes in the cell
population in response to treatment or infection. In these contexts, the
observed signal from the mixture of cell-types is assumed to be a linear
combination of the expression levels of genes in constitutive cell-types. The
goal is to use known signals corresponding to individual cell-types along with
a model of the mixing process to cast the deconvolution problem as a suitable
optimization problem.
  In this paper, we present a survey of models, methods, and assumptions
underlying deconvolution techniques. We investigate the choice of the different
loss functions for evaluating estimation error, constraints on solutions,
preprocessing and data filtering, feature selection, and regularization to
enhance the quality of solutions, along with the impact of these choices on the
performance of regression-based methods for deconvolution. We assess different
combinations of these factors and use detailed statistical measures to evaluate
their effectiveness. We identify shortcomings of current methods and avenues
for further investigation. For many of the identified shortcomings, such as
normalization issues and data filtering, we provide new solutions. We summarize
our findings in a prescriptive step-by-step process, which can be applied to a
wide range of deconvolution problems.
