Identifiability of linear compartmental tree models and a general formula for input-output equations

TL;DR

This paper provides a complete combinatorial criterion for the structural identifiability of linear compartmental models with a tree structure, including a general formula for input-output equations and conditions for model enlargement.

Contribution

It introduces a visual method for assessing identifiability in bidirectional tree models and derives a general formula for input-output equations with separate input and output compartments.

Findings

Identifiability can be verified visually for models with a bidirectional tree structure.

A new general formula for input-output equations with separate input and output compartments.

Identifiability is preserved under specific model enlargements involving new compartments.

Abstract

A foundational question in the theory of linear compartmental models is how to assess whether a model is structurally identifiable -- that is, whether parameter values can be inferred from noiseless data -- directly from the combinatorics of the model. Our main result completely answers this question for models (with one input and one output) in which the underlying graph is a bidirectional tree; moreover, identifiability of such models can be verified visually}. Models of this structure include two families of models often appearing in biological applications: catenary and mammillary models. Our analysis of such models is enabled by two supporting results, which are significant in their own right. One result gives the first general formula for the coefficients of input-output equations (certain equations that can be used to determine identifiability) that allows for input and output to be in distinct compartments}. In another supporting result, we prove that identifiability is preserved when a model is enlarged and altered in specific ways involving adding a new compartment with a bidirected edge to an existing compartment.