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

Abstract

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.

Formation and growth of co-culture tumour spheroids: new compartment-based mathematical models and experiments

“…In each of the two phases, namely formation and growth, the size of the spheroid is described by an ordinary differential equation. This allows us to estimate spheroid formation time and extends traditional analysis that overlook formation [16, 17, 18]. Early time growth dynamics are reasonably described with a linear model (Fig 1D).…”

“…The model is derived by coupling conservation of volume arguments with geometric constraints that define the boundary of each region. In general, geometric constraints in compartment-based spheroid models may be prescribed [18] or derived by considering additional biological mechanisms. In Greenspan’s model the geometric constraints arise by considering oxygen diffusion and consumption [22].…”

“…[36], McElwain and Ponzo [37], McElwain and Morris [38], Adam [39, 40, 41], Adam and Maggelakis [42, 43], Maggelaskis and Adam [44], Landry [45], Byrne et al [46, 47]; and more recently the radial-death model [18]. However, to the best of our knowledge, such compartment-based models have been restricted to one cell type.…”

Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability

“…For example, one could explore replacing the time-dependent adaptation mechanisms with functional forms incorporating additional biological mechanisms. When developing such alternative models one should take into account challenges with parameter identifiability that can arise as model complexity increases [ 38 ]. We also note that our framework is well suited to explore the role of other changing external conditions on spheroid growth, for example nutrient availability and mechanical confinement [ 4 , 11 ], and can be extended to explore different treatment strategies, for example radiotherapy and chemotherapy [ 34 , 41 ].…”

“…Each mechanism and parameter in Greenspan’s model has a biologically meaningful interpretation. Further, these parameters can be identified and estimated with the experimental data that we collect in this study [ 15 , 16 ], which is not the case for other more complicated mathematical models [ 38 – 40 ]. In addition, here we show that we can extend Greenspan’s model to analyse growth in time-dependent external oxygen partial pressures while retaining physical and biologically insightful interpretations of results.…”