A numerical evaluation of state reconstruction methods for heterogeneous cell populations

Abstract

Heterogeneity among cells is a common characteristic of living systems. For mathematical modeling of heterogeneous cell populations, one typically has to reconstruct the underlying heterogeneity from measurements on the population level. Based on recent insights into the mathematical nature of this problem as an inverse problem of tomographic type, we evaluate numerical methods to perform such a reconstruction in basic case studies. We compare a kernel density based optimization approach, filtered back projection, and algebraic reconstruction techniques. The latter two are well established methods in computed tomography.

Estimation methods for heterogeneous cell population models in systems biology

“…with coefficients c i to be determined through the estimation and predefined ansatz functions w (i) (x). In past studies, the so-called hat functions [53] and Gaussian kernels [7,68] have been used for the ansatz functions. For each of the ansatz functions, the PBE is solved with the ansatz function as the initial density, and the corresponding output densities w (i) y(t) (y) are obtained according to the output equation (3.11).…”

“…However, the FBP is quite sensitive to the output equation not covering all possible projection directions, which typically occurs in population models because the system dynamics do not give a sufficiently large 'rotation' of the density function as would be required for the classical tomographic reconstruction. In that case, the reconstruction from the FBP suffers from streak artefacts [68]. A more reliable reconstruction method from tomography is the algebraic reconstruction technique, which is based on a grid discretization of the state space and the approximation of the output equation by a finite-dimensional linear mapping.…”

“…Model parameters are either the IPS functions themselves or real-valued parameters that characterize these functions. According to the distinction between state and parameter estimation, estimation problems for PBE models will thus target the initial cell number density function N 0 (x) to be estimated [7,53,67,68], or the IPS functions such as division or death rates [69 -71]. Both problems are in principle non-parametric estimation problems, since functions have to be estimated, but can be translated into real-valued estimation problems by a suitable parametrization of the target functions.…”

“…In past studies, the L 2 norm has been used for this optimization [53,68], though using other norms is also possible. Another measure to compare density functions from the model and the data is the Kullback-Leibler divergence, which has been used in a recently proposed particle filter for the estimation of cell populations [101].…”

“…The reconstruction then amounts to finding a minimum-error norm inversion of this mapping, commonly via an iterative method such as the Landweber algorithm [104]. The result is a gridwise reconstruction of the initial state density in the model corresponding to the measured output densities [68].…”