Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates

An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth experiments. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, classify non-identifiable parameter spaces into identifiable parameter combinations that relate to features in the data characterised by parameters in a surrogate model, and overall provide additional biological insight from complex non-identifiable models.Author summary: Mathematical models play important roles in the interpretation of biological data. These models can be made arbitrarily complex, meaning issues related to parameter identifiability are relatively common. However, complex models with non-identifiable parameters can be useful to provide insight into the biological questions of interest, since they contain parameters of direct biological interest. In contrast, simpler identifiable models lack biological granularity and comprise parameters that relate indirectly to the underlying biology through data features. In this work, we study the interrelationship between the non-identifiable parameters in a complex model and the identifiable parameters in a simple surrogate model. We aim to resolve the mismatch between model and data complexity by utilising the simple surrogate model to provide insight in cases where the parameters of interest cannot be determined from the available data. We demonstrate our approach by analysing mathematical models of multicellular tumour spheroid growth, an experimental model of cancerous tumour growth. Using the most fundamental and commonly reported measurements, we predict non-identifiabilities arising from different data collection regimes, and draw additional insight from complex models with non-identifiable parameters.