Quantitative analysis of tumour spheroid structure

Abstract

Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure; in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time; however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.

Designing and interpreting 4D tumour spheroid experiments

“…However, these simple mathematical models do not provide information about the internal spheroid structure over time. In response, many mathematical models of varying complexity have been developed to explore the internal structure of spheroids 21 – 42 . Here, we revisit the seminal Greenspan mathematical model for avascular tumour spheroid growth 21 and, to the best of our knowledge, quantitatively directly connect it to data for the first time.…”

Parameter estimation and uncertainty quantification using information geometry

“…We apply a probability-based log-likelihood approach when constructing confidence regions for model parameters. From Wilks’ theorem [ 36 ], asymptotically as N → ∞, an approximate α -level confidence region is given by where Δ ν , α is the α th-quantile of the χ 2 ( ν ) distribution, with ν degrees of freedom [ 1 ]. In this work, the degrees of freedom correspond to the number of parameters of interest, i.e.…”

“…To compute the geodesic distance between two specific points in parameter space, as required by equation ( 2.19 ), it is necessary to solve a boundary value problem to obtain the geodesic curve between θ 0 and . Approximate p -values can be computed from these test statistics as and , respectively, where is the cumulative distribution function of χ 2 ( ν ) [ 1 ]. We provide practical examples of each of these approaches to hypothesis testing in §3.…”

“…Computational and mathematical models are versatile tools, providing valuable insight into complex processes in the life sciences. Models can further our understanding of mechanisms and processes, facilitate development and testing of hypotheses, guide experimentation and data collection and aid design of targeted interventions [1][2][3][4][5]. However, there are considerable challenges associated with calibrating these models to data.…”

“…There are a variety of approaches for uncertainty quantification for frequentist inference methods. In profile likelihood, a parameter of interest is varied over a fixed set of values, while re-estimating the other parameters; this provides insight into identifiability and uncertainty [ 1 ]. In asymptotic analysis, confidence regions can be constructed based on local information via a Taylor expansion of the Fisher information about the maximum likelihood estimate (MLE) [ 8 , 25 ].…”