  An important problem in biological modeling is choosing the right model.
Given experimental data, one is supposed to find the best mathematical
representation to describe the real-world phenomena. However, there may not be
a unique model representing that real-world phenomena. Two distinct models
could yield the same exact dynamics. In this case, these models are called
indistinguishable. In this work, we consider the indistinguishability problem
for linear compartmental models, which are used in many areas, such as
pharmacokinetics, physiology, cell biology, toxicology, and ecology. We exhibit
sufficient conditions for indistinguishability for models with a certain graph
structure: paths from input to output with "detours". The benefit of applying
our results is that indistinguishability can be proven using only the graph
structure of the models, without the use of any symbolic computation. This can
be very helpful for medium-to-large sized linear compartmental models. These
are the first sufficient conditions for indistinguishability of linear
compartmental models based on graph structure alone, as previously only
necessary conditions for indistinguishability of linear compartmental models
existed based on graph structure alone. We prove our results by showing that
the indistinguishable models are the same up to a renaming of parameters, which
we call permutation indistinguishability.
