Model-based data analysis of tissue growth in thin 3D printed scaffolds

Browning, Alexander P; Maclaren, Oliver J.; Buenzli, Pascal R.; Lanaro, Matthew; Allenby, Mark C.; Woodruff, Maria A.; Simpson, Matthew J.

doi:10.1016/j.jtbi.2021.110852

“…We take a likelihood-based approach to estimating parameters ( Lehmann et al, 1998 ) employ profile likelihood analysis to produce approximate confidence intervals ( Raue et al, 2009 ; Pawitan, 2013 ; Browning et al, 2021a ) and develop a likelihood-ratio-based hypothesis test to assess consistency in results between seeding densities. Firstly, we work solely with a statistical model that describes the average sizes of the spheroid, inhibited region and necrotic core at each observation time.…”

Section: Introductionmentioning

confidence: 99%

Quantitative analysis of tumour spheroid structure

Browning

¹

,

Sharp

²

,

Murphy

³

et al. 2021

eLife

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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.

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“…A brief survey of the literature shows that different choices of sigmoid growth functions are routinely used automatically in different areas of application. For example, the logistic growth model is used to study a range of applications, including the dynamics of in vitro cell populations [8,23,29,43,55], as well as various ecological population dynamics across a massive range of scales ranging from jellyfish polyps on oyster shells [32] to continental-scale dynamics of human populations [48]. The Gompertz growth model is routinely used to study in vivo tumour dynamics [27,42] 1 as well as various ecological applications such as reef shark population dynamics [20] and dynamics of coral populations [53].…”

Section: Introductionmentioning

confidence: 99%

Parameter identifiability and model selection for sigmoid population growth models

Simpson

¹

,

Ap

²

,

Dj

³

et al. 2021

Preprint

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Sigmoid growth models, such as the logistic and Gompertz growth models, are widely used to study various population dynamics ranging from microscopic populations of cancer cells, to continental-scale human populations. Fundamental questions about model selection and precise parameter estimation are critical if these models are to be used to make useful inferences about underlying ecological mechanisms. However, the question of parameter identifiability for these models -- whether a data set contains sufficient information to give unique or sufficiently precise parameter estimates for the given model -- is often overlooked; We use a profile-likelihood approach to systematically explore practical parameter identifiability using data describing the re-growth of hard coral cover on a coral reef after some ecological disturbance. The relationship between parameter identifiability and checks of model misspecification is also explored. We work with three standard choices of sigmoid growth models, namely the logistic, Gompertz, and Richards' growth models; We find that the logistic growth model does not suffer identifiability issues for the type of data we consider whereas the Gompertz and Richards' models encounter practical non-identifiability issues, even with relatively-extensive data where we observe the full shape of the sigmoid growth curve. Identifiability issues with the Gompertz model lead us to consider a further model calibration exercise in which we fix the initial density to its observed value, neglecting its uncertainty. This is a common practice, but the results of this exercise suggest that parameter estimates and fundamental statistical assumptions are extremely sensitive under these conditions; Different sigmoid growth models are used within subdisciplines within the biology and ecology literature without necessarily considering whether parameters are identifiable or checking statistical assumptions underlying model family adequacy. Standard practices that do not consider parameter identifiability can lead to unreliable or imprecise parameter estimates and hence potentially misleading interpretations of the underlying mechanisms of interest. While tools in this work focus on three standard sigmoid growth models and one particular data set, our theoretical developments are applicable to any sigmoid growth model and any relevant data set. MATLAB implementations of all software available on GitHub.

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“… 2020 ; Browning et al. 2021 ). In this experiment, cells are seeded onto the perimeter of 3D-printed square shaped pores, where each pore has sides of approximately 300 m in length.…”

Section: Introductionmentioning

confidence: 99%

“… 2020 ; Browning et al. 2021 ). A notable feature of these experiments is that tissue formation involves a well-defined moving front that is very obvious in Fig.…”

Section: Introductionmentioning

confidence: 99%

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

El‐Hachem

¹

,

McCue

²

,

Simpson

³

2022

Self Cite

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We consider a continuum mathematical model of biological tissue formation inspired by recent experiments describing thin tissue growth in 3D-printed bioscaffolds. The continuum model, which we call the substrate model, involves a partial differential equation describing the density of tissue, $${\hat{u}}(\hat{{\mathbf {x}}},{\hat{t}})$$ u ^ ( x ^ , t ^ ) that is coupled to the concentration of an immobile extracellular substrate, $${\hat{s}}(\hat{{\mathbf {x}}},{\hat{t}})$$ s ^ ( x ^ , t ^ ) . Cell migration is modelled with a nonlinear diffusion term, where the diffusive flux is proportional to $${\hat{s}}$$ s ^ , while a logistic growth term models cell proliferation. The extracellular substrate $${\hat{s}}$$ s ^ is produced by cells and undergoes linear decay. Preliminary numerical simulations show that this mathematical model is able to recapitulate key features of recent tissue growth experiments, including the formation of sharp fronts. To provide a deeper understanding of the model we analyse travelling wave solutions of the substrate model, showing that the model supports both sharp-fronted travelling wave solutions that move with a minimum wave speed, $$c = c_{\mathrm{min}}$$ c = c min , as well as smooth-fronted travelling wave solutions that move with a faster travelling wave speed, $$c > c_{\mathrm{min}}$$ c > c min . We provide a geometric interpretation that explains the difference between smooth and sharp-fronted travelling wave solutions that is based on a slow manifold reduction of the desingularised three-dimensional phase space. In addition, we also develop and test a series of useful approximations that describe the shape of the travelling wave solutions in various limits. These approximations apply to both the sharp-fronted and smooth-fronted travelling wave solutions. Software to implement all calculations is available at GitHub.

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Model-based data analysis of tissue growth in thin 3D printed scaffolds

Cited by 26 publications

References 71 publications

Quantitative analysis of tumour spheroid structure

Quantitative analysis of tumour spheroid structure

Parameter identifiability and model selection for sigmoid population growth models

A Continuum Mathematical Model of Substrate-Mediated Tissue Growth

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