Multiple types of data are required to identify the mechanisms influencing the spatial expansion of melanoma cell colonies

Estimating parameters of a stochastic cell invasion model with fluorescent cell cycle labelling using approximate Bayesian computation

We develop a parameter estimation method based on approximate Bayesian computation (ABC) for a stochastic cell invasion model using fluorescent cell cycle labelling with proliferation, migration and crowding effects. Previously, inference has been performed on a deterministic version of the model fitted to cell density data, and not all parameters were identifiable. Considering the stochastic model allows us to harness more features of experimental data, including cell trajectories and cell count data, which we show overcomes the parameter identifiability problem. We demonstrate that, while difficult to collect, cell trajectory data can provide more information about the parameters of the cell invasion model. To handle the intractability of the likelihood function of the stochastic model, we use an efficient ABC algorithm based on sequential Monte Carlo. Rcpp and MATLAB implementations of the simulation model and ABC algorithm used in this study are available at https://github.com/michaelcarr-stats/FUCCI.

Reproduction of patterns in melanocytic proliferations by agent-based simulation and geometric modeling

Spatio-temporal patterns of melanocytic proliferations observed in vivo are important for diagnosis but the mechanisms that produce them are poorly understood. Here we present an agent-based model for simulating the emergence of the main biologic patterns found in melanocytic proliferations. Our model portrays the extracellular matrix of the dermo-epidermal junction as a two-dimensional manifold and we simulate cellular migration in terms of geometric translations driven by adhesive, repulsive and random forces. Abstracted cellular functions and melanocyte-matrix interactions are modeled as stochastic events. For identification and validation we use visual renderings of simulated cell populations in a horizontal perspective that reproduce growth patterns observed in vivo by sequential dermatoscopy and corresponding vertical views that reproduce the arrangement of melanocytes observed in histopathologic sections. Our results show that a balanced interplay of proliferation and migration produces the typical reticular pattern of nevi, whereas the globular pattern involves additional cellular mechanisms. We further demonstrate that slight variations in the three basic cellular properties proliferation, migration, and adhesion are sufficient to produce a large variety of morphological appearances of nevi. We anticipate our model to be a starting point for the reproduction of more complex scenarios that will help to establish functional connections between abstracted microscopic behavior and macroscopic patterns in all types of melanocytic proliferations including melanoma.

Computational models of melanoma

Genes, proteins, or cells influence each other and consequently create patterns, which can be increasingly better observed by experimental biology and medicine. Thereby, descriptive methods of statistics and bioinformatics sharpen and structure our perception. However, additionally considering the interconnectivity between biological elements promises a deeper and more coherent understanding of melanoma. For instance, integrative network-based tools and well-grounded inductive in silico research reveal disease mechanisms, stratify patients, and support treatment individualization. This review gives an overview of different modeling techniques beyond statistics, shows how different strategies align with the respective medical biology, and identifies possible areas of new computational melanoma research.

Practical parameter identifiability for spatio-temporal models of cell invasion

We examine the practical identifiability of parameters in a spatio-temporal reaction-diffusion model of a scratch assay. Experimental data involve fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction-diffusion model by treating the total population as two interacting subpopulations. Practical identifiability is examined using a Bayesian Markov chain Monte Carlo (MCMC) framework, confirming that the parameters are identifiable when we assume the diffusivities of the subpopulations are identical, but that the parameters are practically non-identifiable when we allow the diffusivities to be distinct. We also assess practical identifiability using a profile likelihood approach, providing similar results to MCMC with the advantage of being an order of magnitude faster to compute. Therefore, we suggest that the profile likelihood ought to be adopted as a screening tool to assess practical identifiability before MCMC computations are performed.

Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1 and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.

A Bayesian Sequential Learning Framework to Parameterise Continuum Models of Melanoma Invasion into Human Skin

We present a novel framework to parameterise a mathematical model of cell invasion that describes how a population of melanoma cells invades into human skin tissue. Using simple experimental data extracted from complex experimental images, we estimate three model parameters: (i) the melanoma cell proliferation rate, [Formula: see text]; (ii) the melanoma cell diffusivity, D; and (iii) [Formula: see text], a constant that determines the rate that melanoma cells degrade the skin tissue. The Bayesian sequential learning framework involves a sequence of increasingly sophisticated experimental data from: (i) a spatially uniform cell proliferation assay; (ii) a two-dimensional circular barrier assay; and (iii) a three-dimensional invasion assay. The Bayesian sequential learning approach leads to well-defined parameter estimates. In contrast, taking a naive approach that attempts to estimate all parameters from a single set of images from the same experiment fails to produce meaningful results. Overall, our approach to inference is simple-to-implement, computationally efficient, and well suited for many cell biology phenomena that can be described by low-dimensional continuum models using ordinary differential equations and partial differential equations. We anticipate that this Bayesian sequential learning framework will be relevant in other biological contexts where it is challenging to extract detailed, quantitative biological measurements from experimental images and so we must rely on using relatively simple measurements from complex images.

The invasion speed of cell migration models with realistic cell cycle time distributions

Cell proliferation is typically incorporated into stochastic mathematical models of cell migration by assuming that cell divisions occur after an exponentially distributed waiting time. Experimental observations, however, show that this assumption is often far from the real cell cycle time distribution (CCTD). Recent studies have suggested an alternative approach to modelling cell proliferation based on a multi-stage representation of the CCTD. In this paper we investigate the connection between the CCTD and the speed of the collective invasion. We first state a result for a general CCTD, which allows the computation of the invasion speed using the Laplace transform of the CCTD. We use this to deduce the range of speeds for the general case. We then focus on the more realistic case of multi-stage models, using both a stochastic agent-based model and a set of reaction-diffusion equations for the cells' average density. By studying the corresponding travelling wave solutions, we obtain an analytical expression for the speed of invasion for a general N-stage model with identical transition rates, in which case the resulting cell cycle times are Erlang distributed. We show that, for a general N-stage model, the Erlang distribution and the exponential distribution lead to the minimum and maximum invasion speed, respectively. This result allows us to determine the range of possible invasion speeds in terms of the average proliferation time for any multi-stage model.

Pair correlation functions for identifying spatial correlation in discrete domains

Identifying and quantifying spatial correlation are important aspects of studying the collective behavior of multiagent systems. Pair correlation functions (PCFs) are powerful statistical tools that can provide qualitative and quantitative information about correlation between pairs of agents. Despite the numerous PCFs defined for off-lattice domains, only a few recent studies have considered a PCF for discrete domains. Our work extends the study of spatial correlation in discrete domains by defining a new set of PCFs using two natural and intuitive definitions of distance for a square lattice: the taxicab and uniform metric. We show how these PCFs improve upon previous attempts and compare between the quantitative data acquired. We also extend our definitions of the PCF to other types of regular tessellation that have not been studied before, including hexagonal, triangular, and cuboidal. Finally, we provide a comprehensive PCF for any tessellation and metric, allowing investigation of spatial correlation in irregular lattices for which recognizing correlation is less intuitive.

Morphology and dynamics of tumor cell colonies propagating in epidermal growth factor supplemented media

The epidermal growth factor (EGF) plays a key role in physiological and pathological processes. This work reports on the influence of EGF concentration (c _EGF) on the modulation of individual cell phenotype and cell colony kinetics with the aim of perturbing the colony front roughness fluctuations. For this purpose, HeLa cell colonies that remain confluent along the whole expansion process with initial quasi-radial geometry and different initial cell populations, as well as colonies with initial quasi-linear geometry and large cell population, are employed. Cell size and morphology as well as its adhesive characteristics depend on c _EGF. Quasi-radial colonies (QRC) expansion kinetics in EGF-containing medium exhibits a complex behavior. Namely, at the first stages of growth, the average QRC radius evolution can be described by a t ^1/2 diffusion term coupled with exponential growth kinetics up to a critical time, and afterwards a growth regime approaching constant velocity. The extension of each regime depends on c _EGF and colony history. In the presence of EGF, the initial expansion of quasi-linear colonies (QLCs) also exhibits morphological changes at both the cell and the colony levels. In these cases, the cell density at the colony border region becomes smaller than in the absence of EGF and consequently, the extension of the effective rim where cell duplication and motility contribute to the colony expansion increases. QLC front displacement velocity increases with c _EGF up to a maximum value in the 2-10 ng ml^-1 range. Individual cell velocity is increased by EGF, and an enhancement in both the persistence and the ballistic characteristics of cell trajectories can be distinguished. For an intermediate c _EGF, collective cell displacements contribute to the roughening of the colony contours. This global dynamics becomes compatible with the standard Kardar-Parisi-Zhang growth model, although a faster colony roughness saturation in EGF-containing medium than in the control medium is observed.

Three-dimensional experiments and individual based simulations show that cell proliferation drives melanoma nest formation in human skin tissue

Background

Melanoma can be diagnosed by identifying nests of cells on the skin surface. Understanding the processes that drive nest formation is important as these processes could be potential targets for new cancer drugs. Cell proliferation and cell migration are two potential mechanisms that could conceivably drive melanoma nest formation. However, it is unclear which one of these two putative mechanisms plays a dominant role in driving nest formation.

Results

We use a suite of three-dimensional (3D) experiments in human skin tissue and a parallel series of 3D individual-based simulations to explore whether cell migration or cell proliferation plays a dominant role in nest formation. In the experiments we measure nest formation in populations of irradiated (non-proliferative) and non-irradiated (proliferative) melanoma cells, cultured together with primary keratinocyte and fibroblast cells on a 3D experimental human skin model. Results show that nest size depends on initial cell number and is driven primarily by cell proliferation rather than cell migration.

Conclusions

Nest size depends on cell number, and is driven primarily by cell proliferation rather than cell migration. All experimental results are consistent with simulation data from a 3D individual based model (IBM) of cell migration and cell proliferation.

Electronic supplementary material

The online version of this article (10.1186/s12918-018-0559-9) contains supplementary material, which is available to authorized users.

Optimal Quantification of Contact Inhibition in Cell Populations

Contact inhibition refers to a reduction in the rate of cell migration and/or cell proliferation in regions of high cell density. Under normal conditions, contact inhibition is associated with the proper functioning tissues, whereas abnormal regulation of contact inhibition is associated with pathological conditions, such as tumor spreading. Unfortunately, standard mathematical modeling practices mask the importance of parameters that control contact inhibition through scaling arguments. Furthermore, standard experimental protocols are insufficient to quantify the effects of contact inhibition because they focus on data describing early time, low-density dynamics only. Here we use the logistic growth equation as a caricature model of contact inhibition to make recommendations as to how to best mitigate these issues. Taking a Bayesian approach, we quantify the trade off between different features of experimental design and estimates of parameter uncertainty so that we can reformulate a standard cell proliferation assay to provide estimates of both the low-density intrinsic growth rate, λ, and the carrying capacity density, K, which is a measure of contact inhibition.

Quantitative comparison of the spreading and invasion of radial growth phase and metastatic melanoma cells in a three-dimensional human skin equivalent model

BACKGROUND

Standard two-dimensional (2D) cell migration assays do not provide information about vertical invasion processes, which are critical for melanoma progression. We provide information about three-dimensional (3D) melanoma cell migration, proliferation and invasion in a 3D melanoma skin equivalent (MSE) model. In particular, we pay careful attention to compare the structure of the tissues in the MSE with similarly-prepared 3D human skin equivalent (HSE) models. The HSE model is identically prepared to the MSE model except that melanoma cells are omitted. Using the MSE model, we examine melanoma migration, proliferation and invasion from two different human melanoma cell lines. One cell line, WM35, is associated with the early phase of the disease where spreading is thought to be confined to the epidermis. The other cell line, SK-MEL-28, is associated with the later phase of the disease where spreading into the dermis is expected.

METHODS

3D MSE and HSE models are constructed using human de-epidermised dermis (DED) prepared from skin tissue. Primary fibroblasts and primary keratinocytes are used in the MSE and HSE models to ensure the formation of a stratified epidermis, with a well-defined basement membrane. Radial spreading of cells across the surface of the HSE and MSE models is observed. Vertical invasion of melanoma cells downward through the skin is observed and measured using immunohistochemistry. All measurements of invasion are made at day 0, 9, 15 and 20, providing detailed time course data.

RESULTS

Both HSE and MSE models are similar to native skin in vivo, with a well-defined stratification of the epidermis that is separated from the dermis by a basement membrane. In the HSE and MSE we find fibroblast cells confined to the dermis, and differentiated keratinocytes in the epidermis. In the MSE, melanoma cells form colonies in the epidermis during the early part of the experiment. In the later stage of the experiment, the melanoma cells in the MSE invade deeper into the tissues. Interestingly, both the WM35 and SK-MEL-28 melanoma cells lead to a breakdown of the basement membrane and eventually enter the dermis. However, these two cell lines invade at different rates, with the SK-MEL-28 melanoma cells invading faster than the WM35 cells.

DISCUSSION

The MSE and HSE models are a reliable platform for studying melanoma invasion in a 3D tissue that is similar to native human skin. Interestingly, we find that the WM35 cell line, that is thought to be associated with radial spreading only, is able to invade into the dermis. The vertical invasion of melanoma cells into the dermal region appears to be associated with a localised disruption of the basement membrane. Presenting our results in terms of time course data, along with images and quantitative measurements of the depth of invasion extends previous 3D work that has often been reported without these details.

A computational modelling framework to quantify the effects of passaging cell lines

In vitro cell culture is routinely used to grow and supply a sufficiently large number of cells for various types of cell biology experiments. Previous experimental studies report that cell characteristics evolve as the passage number increases, and various cell lines can behave differently at high passage numbers. To provide insight into the putative mechanisms that might give rise to these differences, we perform in silico experiments using a random walk model to mimic the in vitro cell culture process. Our results show that it is possible for the average proliferation rate to either increase or decrease as the passaging process takes place, and this is due to a competition between the initial heterogeneity and the degree to which passaging damages the cells. We also simulate a suite of scratch assays with cells from near–homogeneous and heterogeneous cell lines, at both high and low passage numbers. Although it is common in the literature to report experimental results without disclosing the passage number, our results show that we obtain significantly different closure rates when performing in silico scratch assays using cells with different passage numbers. Therefore, we suggest that the passage number should always be reported to ensure that the experiment is as reproducible as possible. Furthermore, our modelling also suggests some avenues for further experimental examination that could be used to validate or refine our simulation results.

Quantifying rates of cell migration and cell proliferation in co-culture barrier assays reveals how skin and melanoma cells interact during melanoma spreading and invasion

Malignant spreading involves the migration of cancer cells amongst other native cell types. For example, in vivo melanoma invasion involves individual melanoma cells migrating through native skin, which is composed of several distinct subpopulations of cells. Here, we aim to quantify how interactions between melanoma and fibroblast cells affect the collective spreading of a heterogeneous population of these cells in vitro. We perform a suite of circular barrier assays that includes: (i) monoculture assays with fibroblast cells; (ii) monoculture assays with SK-MEL-28 melanoma cells; and (iii) a series of co-culture assays initiated with three different ratios of SK-MEL-28 melanoma cells and fibroblast cells. Using immunostaining, detailed cell density histograms are constructed to illustrate how the two subpopulations of cells are spatially arranged within the spreading heterogeneous population. Calibrating the solution of a continuum partial differential equation to the experimental results from the monoculture assays allows us to estimate the cell diffusivity and the cell proliferation rate for the melanoma and the fibroblast cells, separately. Using the parameter estimates from the monoculture assays, we then make a prediction of the spatial spreading in the co-culture assays. Results show that the parameter estimates obtained from the monoculture assays lead to a reasonably accurate prediction of the spatial arrangement of the two subpopulations in the co-culture assays. Overall, the spatial pattern of spreading of the melanoma cells and the fibroblast cells is very similar in monoculture and co-culture conditions. Therefore, we find no clear evidence of any interactions other than cell-to-cell contact and crowding effects.

Optimization of cell seeding in a 2D bio-scaffold system using computational models

The cell expansion process is a crucial part of generating cells on a large-scale level in a bioreactor system. Hence, it is important to set operating conditions (e.g. initial cell seeding distribution, culture medium flow rate) to an optimal level. Often, the initial cell seeding distribution factor is neglected and/or overlooked in the design of a bioreactor using conventional seeding distribution methods. This paper proposes a novel seeding distribution method that aims to maximize cell growth and minimize production time/cost. The proposed method utilizes two computational models; the first model represents cell growth patterns whereas the second model determines optimal initial cell seeding positions for adherent cell expansions. Cell growth simulation from the first model demonstrates that the model can be a representation of various cell types with known probabilities. The second model involves a combination of combinatorial optimization, Monte Carlo and concepts of the first model, and is used to design a multi-layer 2D bio-scaffold system that increases cell production efficiency in bioreactor applications. Simulation results have shown that the recommended input configurations obtained from the proposed optimization method are the most optimal configurations. The results have also illustrated the effectiveness of the proposed optimization method. The potential of the proposed seeding distribution method as a useful tool to optimize the cell expansion process in modern bioreactor system applications is highlighted.

Continuum approximations for lattice-free multi-species models of collective cell migration

Cell migration within tissues involves the interaction of many cells from distinct subpopulations. In this work, we present a discrete model of collective cell migration where the motion of individual cells is driven by random forces, short range repulsion forces to mimic crowding, and longer range attraction forces to mimic adhesion. This discrete model can be used to simulate a population of cells that is composed of K ≥ 1 distinct subpopulations. To analyse the discrete model we formulate a hierarchy of moment equations that describe the spatial evolution of the density of agents, pairs of agents, triplets of agents, and so forth. To solve the hierarchy of moment equations we introduce two forms of closure: (i) the mean field approximation, which effectively assumes that the distributions of individual agents are independent; and (ii) a moment dynamics description that is based on the Kirkwood superposition approximation. The moment dynamics description provides an approximate way of incorporating spatial patterns, such as agent clustering, into the continuum description. Comparing the performance of the two continuum descriptions confirms that both perform well when adhesive forces are sufficiently weak. In contrast, the moment dynamics description outperforms the mean field model when adhesive forces are sufficiently large. This is a first attempt to provide an accurate continuum description of a lattice-free, multi-species model of collective cell migration.

Stochastic simulation tools and continuum models for describing two-dimensional collective cell spreading with universal growth functions

Two-dimensional collective cell migration assays are used to study cancer and tissue repair. These assays involve combined cell migration and cell proliferation processes, both of which are modulated by cell-to-cell crowding. Previous discrete models of collective cell migration assays involve a nearest-neighbour proliferation mechanism where crowding effects are incorporated by aborting potential proliferation events if the randomly chosen target site is occupied. There are two limitations of this traditional approach: (i) it seems unreasonable to abort a potential proliferation event based on the occupancy of a single, randomly chosen target site; and, (ii) the continuum limit description of this mechanism leads to the standard logistic growth function, but some experimental evidence suggests that cells do not always proliferate logistically. Motivated by these observations, we introduce a generalised proliferation mechanism which allows non-nearest neighbour proliferation events to take place over a template of [Formula: see text] concentric rings of lattice sites. Further, the decision to abort potential proliferation events is made using a crowding function, f(C), which accounts for the density of agents within a group of sites rather than dealing with the occupancy of a single randomly chosen site. Analysing the continuum limit description of the stochastic model shows that the standard logistic source term, [Formula: see text], where λ is the proliferation rate, is generalised to a universal growth function, [Formula: see text]. Comparing the solution of the continuum description with averaged simulation data indicates that the continuum model performs well for many choices of f(C) and r. For nonlinear f(C), the quality of the continuum-discrete match increases with r.

An analytical method for disentangling the roles of adhesion and crowding for random walk models on a crowded lattice

Migration of cells and molecules in vivo is affected by interactions with obstacles. These interactions can include crowding effects, as well as adhesion/repulsion between the motile cell/molecule and the obstacles. Here we present an analytical framework that can be used to separately quantify the roles of crowding and adhesion/repulsion using a lattice-based random walk model. Our method leads to an exact calculation of the long time Fickian diffusivity, and avoids the need for computationally expensive stochastic simulations.

Spatio-temporal morphology changes in and quenching effects on the 2D spreading dynamics of cell colonies in both plain and methylcellulose-containing culture media

To deal with complex systems, microscopic and global approaches become of particular interest. Our previous results from the dynamics of large cell colonies indicated that their 2D front roughness dynamics is compatible with the standard Kardar-Parisi-Zhang (KPZ) or the quenched KPZ equations either in plain or methylcellulose (MC)-containing gel culture media, respectively. In both cases, the influence of a non-uniform distribution of the colony constituents was significant. These results encouraged us to investigate the overall dynamics of those systems considering the morphology and size, the duplication rate, and the motility of single cells. For this purpose, colonies with different cell populations (N) exhibiting quasi-circular and quasi-linear growth fronts in plain and MC-containing culture media are investigated. For small N, the average radial front velocity and its change with time depend on MC concentration. MC in the medium interferes with cell mitosis, contributes to the local enlargement of cells, and increases the distribution of spatio-temporal cell density heterogeneities. Colony spreading in MC-containing media proceeds under two main quenching effects, I and II; the former mainly depending on the culture medium composition and structure and the latter caused by the distribution of enlarged local cell domains. For large N, colony spreading occurs at constant velocity. The characteristics of cell motility, assessed by measuring their trajectories and the corresponding velocity field, reflect the effect of enlarged, slow-moving cells and the structure of the medium. Local average cell size distribution and individual cell motility data from plain and MC-containing media are qualitatively consistent with the predictions of both the extended cellular Potts models and the observed transition of the front roughness dynamics from a standard KPZ to a quenched KPZ. In this case, quenching effects I and II cooperate and give rise to the quenched-KPZ equation. Seemingly, these results show a possible way of linking the cellular Potts models and the 2D colony front roughness dynamics.

Standard melanoma-associated markers do not identify the MM127 metastatic melanoma cell line

Reliable identification of different melanoma cell lines is important for many aspects of melanoma research. Common markers used to identify melanoma cell lines include: S100; HMB-45; and Melan-A. We explore the expression of these three markers in four different melanoma cell lines: WM35; WM793; SK-MEL-28; and MM127. The expression of these markers is examined at both the mRNA and protein level. Our results show that the metastatic cell line, MM127, cannot be detected using any of the commonly used melanoma-associated markers. This implies that it would be very difficult to identify this particular cell line in a heterogeneous sample, and as a result this cell line should be used with care.

Quantifying the effect of experimental design choices for in vitro scratch assays

Scratch assays are often used to investigate potential drug treatments for chronic wounds and cancer. Interpreting these experiments with a mathematical model allows us to estimate the cell diffusivity, D, and the cell proliferation rate, λ. However, the influence of the experimental design on the estimates of D and λ is unclear. Here we apply an approximate Bayesian computation (ABC) parameter inference method, which produces a posterior distribution of D and λ, to new sets of synthetic data, generated from an idealised mathematical model, and experimental data for a non-adhesive mesenchymal population of fibroblast cells. The posterior distribution allows us to quantify the amount of information obtained about D and λ. We investigate two types of scratch assay, as well as varying the number and timing of the experimental observations captured. Our results show that a scrape assay, involving one cell front, provides more precise estimates of D and λ, and is more computationally efficient to interpret than a wound assay, with two opposingly directed cell fronts. We find that recording two observations, after making the initial observation, is sufficient to estimate D and λ, and that the final observation time should correspond to the time taken for the cell front to move across the field of view. These results provide guidance for estimating D and λ, while simultaneously minimising the time and cost associated with performing and interpreting the experiment.

Melanoma Cell Colony Expansion Parameters Revealed by Approximate Bayesian Computation

In vitro studies and mathematical models are now being widely used to study the underlying mechanisms driving the expansion of cell colonies. This can improve our understanding of cancer formation and progression. Although much progress has been made in terms of developing and analysing mathematical models, far less progress has been made in terms of understanding how to estimate model parameters using experimental in vitro image-based data. To address this issue, a new approximate Bayesian computation (ABC) algorithm is proposed to estimate key parameters governing the expansion of melanoma cell (MM127) colonies, including cell diffusivity, D, cell proliferation rate, λ, and cell-to-cell adhesion, q, in two experimental scenarios, namely with and without a chemical treatment to suppress cell proliferation. Even when little prior biological knowledge about the parameters is assumed, all parameters are precisely inferred with a small posterior coefficient of variation, approximately 2–12%. The ABC analyses reveal that the posterior distributions of D and q depend on the experimental elapsed time, whereas the posterior distribution of λ does not. The posterior mean values of D and q are in the ranges 226–268 µm²h⁻¹, 311–351 µm²h⁻¹ and 0.23–0.39, 0.32–0.61 for the experimental periods of 0–24 h and 24–48 h, respectively. Furthermore, we found that the posterior distribution of q also depends on the initial cell density, whereas the posterior distributions of D and λ do not. The ABC approach also enables information from the two experiments to be combined, resulting in greater precision for all estimates of D and λ.

Quantifying the underlying parameters that drive the expansion of melanoma cell colonies such as the cell diffusivity, cell proliferation rate and cell-to-cell adhesion strength can improve our understanding of melanoma biology and its response to treatment. We combine a simulation-based model of collective cell spreading with a novel Bayesian computational algorithm to estimate these parameters from carefully chosen summaries of collective cell image data and to quantify their associated uncertainty across different experimental conditions. Our summarisation of the image data leads to precise estimates for all parameters. Our analysis reveals that the cell diffusivity and the cell-to-cell adhesion strength estimates depend on experimental elapsed time. Furthermore, the cell-to-cell adhesion strength estimate appears to depend on the initial cell density, whereas the cell proliferation rate estimate is approximately the same over different experimental conditions.

Reconciling transport models across scales: The role of volume exclusion

Diffusive transport is a universal phenomenon, throughout both biological and physical sciences, and models of diffusion are routinely used to interrogate diffusion-driven processes. However, most models neglect to take into account the role of volume exclusion, which can significantly alter diffusive transport, particularly within biological systems where the diffusing particles might occupy a significant fraction of the available space. In this work we use a random walk approach to provide a means to reconcile models that incorporate crowding effects on different spatial scales. Our work demonstrates that coarse-grained models incorporating simplified descriptions of excluded volume can be used in many circumstances, but that care must be taken in pushing the coarse-graining process too far.

Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte ZOOM™ assay data using the Fisher-Kolmogorov model

Background

Standard methods for quantifying IncuCyte ZOOM™ assays involve measurements that quantify how rapidly the initially-vacant area becomes re-colonised with cells as a function of time. Unfortunately, these measurements give no insight into the details of the cellular-level mechanisms acting to close the initially-vacant area. We provide an alternative method enabling us to quantify the role of cell motility and cell proliferation separately. To achieve this we calibrate standard data available from IncuCyte ZOOM™ images to the solution of the Fisher-Kolmogorov model.

Results

The Fisher-Kolmogorov model is a reaction-diffusion equation that has been used to describe collective cell spreading driven by cell migration, characterised by a cell diffusivity, D, and carrying capacity limited proliferation with proliferation rate, λ, and carrying capacity density, K. By analysing temporal changes in cell density in several subregions located well-behind the initial position of the leading edge we estimate λ and K. Given these estimates, we then apply automatic leading edge detection algorithms to the images produced by the IncuCyte ZOOM™ assay and match this data with a numerical solution of the Fisher-Kolmogorov equation to provide an estimate of D. We demonstrate this method by applying it to interpret a suite of IncuCyte ZOOM™ assays using PC-3 prostate cancer cells and obtain estimates of D, λ and K. Comparing estimates of D, λ and K for a control assay with estimates of D, λ and K for assays where epidermal growth factor (EGF) is applied in varying concentrations confirms that EGF enhances the rate of scratch closure and that this stimulation is driven by an increase in D and λ, whereas K is relatively unaffected by EGF.

Conclusions

Our approach for estimating D, λ and K from an IncuCyte ZOOM™ assay provides more detail about cellular-level behaviour than standard methods for analysing these assays. In particular, our approach can be used to quantify the balance of cell migration and cell proliferation and, as we demonstrate, allow us to quantify how the addition of growth factors affects these processes individually.

Interpreting scratch assays using pair density dynamics and approximate Bayesian computation

Quantifying the impact of biochemical compounds on collective cell spreading is an essential element of drug design, with various applications including developing treatments for chronic wounds and cancer. Scratch assays are a technically simple and inexpensive method used to study collective cell spreading; however, most previous interpretations of scratch assays are qualitative and do not provide estimates of the cell diffusivity, D, or the cell proliferation rate, λ. Estimating D and λ is important for investigating the efficacy of a potential treatment and provides insight into the mechanism through which the potential treatment acts. While a few methods for estimating D and λ have been proposed, these previous methods lead to point estimates of D and λ, and provide no insight into the uncertainty in these estimates. Here, we compare various types of information that can be extracted from images of a scratch assay, and quantify D and λ using discrete computational simulations and approximate Bayesian computation. We show that it is possible to robustly recover estimates of D and λ from synthetic data, as well as a new set of experimental data. For the first time, our approach also provides a method to estimate the uncertainty in our estimates of D and λ. We anticipate that our approach can be generalized to deal with more realistic experimental scenarios in which we are interested in estimating D and λ, as well as additional relevant parameters such as the strength of cell-to-cell adhesion or the strength of cell-to-substrate adhesion.

Spectral analysis of pair-correlation bandwidth: application to cell biology images

Images from cell biology experiments often indicate the presence of cell clustering, which can provide insight into the mechanisms driving the collective cell behaviour. Pair-correlation functions provide quantitative information about the presence, or absence, of clustering in a spatial distribution of cells. This is because the pair-correlation function describes the ratio of the abundance of pairs of cells, separated by a particular distance, relative to a randomly distributed reference population. Pair-correlation functions are often presented as a kernel density estimate where the frequency of pairs of objects are grouped using a particular bandwidth (or bin width), Δ>0. The choice of bandwidth has a dramatic impact: choosing Δ too large produces a pair-correlation function that contains insufficient information, whereas choosing Δ too small produces a pair-correlation signal dominated by fluctuations. Presently, there is little guidance available regarding how to make an objective choice of Δ. We present a new technique to choose Δ by analysing the power spectrum of the discrete Fourier transform of the pair-correlation function. Using synthetic simulation data, we confirm that our approach allows us to objectively choose Δ such that the appropriately binned pair-correlation function captures known features in uniform and clustered synthetic images. We also apply our technique to images from two different cell biology assays. The first assay corresponds to an approximately uniform distribution of cells, while the second assay involves a time series of images of a cell population which forms aggregates over time. The appropriately binned pair-correlation function allows us to make quantitative inferences about the average aggregate size, as well as quantifying how the average aggregate size changes with time.

How much information can be obtained from tracking the position of the leading edge in a scratch assay?

Moving cell fronts are an essential feature of wound healing, development and disease. The rate at which a cell front moves is driven, in part, by the cell motility, quantified in terms of the cell diffusivity D, and the cell proliferation rate λ. Scratch assays are a commonly reported procedure used to investigate the motion of cell fronts where an initial cell monolayer is scratched, and the motion of the front is monitored over a short period of time, often less than 24 h. The simplest way of quantifying a scratch assay is to monitor the progression of the leading edge. Use of leading edge data is very convenient because, unlike other methods, it is non-destructive and does not require labelling, tracking or counting individual cells among the population. In this work, we study short-time leading edge data in a scratch assay using a discrete mathematical model and automated image analysis with the aim of investigating whether such data allow us to reliably identify D and λ. Using a naive calibration approach where we simply scan the relevant region of the (D, λ) parameter space, we show that there are many choices of D and λ for which our model produces indistinguishable short-time leading edge data. Therefore, without due care, it is impossible to estimate D and λ from this kind of data. To address this, we present a modified approach accounting for the fact that cell motility occurs over a much shorter time scale than proliferation. Using this information, we divide the duration of the experiment into two periods, and we estimate D using data from the first period, whereas we estimate λ using data from the second period. We confirm the accuracy of our approach using in silico data and a new set of in vitro data, which shows that our method recovers estimates of D and λ that are consistent with previously reported values except that that our approach is fast, inexpensive, non-destructive and avoids the need for cell labelling and cell counting.

Assessing the role of spatial correlations during collective cell spreading

Spreading cell fronts are essential features of development, repair and disease processes. Many mathematical models used to describe the motion of cell fronts, such as Fisher's equation, invoke a mean–field assumption which implies that there is no spatial structure, such as cell clustering, present. Here, we examine the presence of spatial structure using a combination of in vitro circular barrier assays, discrete random walk simulations and pair correlation functions. In particular, we analyse discrete simulation data using pair correlation functions to show that spatial structure can form in a spreading population of cells either through sufficiently strong cell–to–cell adhesion or sufficiently rapid cell proliferation. We analyse images from a circular barrier assay describing the spreading of a population of MM127 melanoma cells using the same pair correlation functions. Our results indicate that the spreading melanoma cell populations remain very close to spatially uniform, suggesting that the strength of cell–to–cell adhesion and the rate of cell proliferation are both sufficiently small so as not to induce any spatial patterning in the spreading populations.

Are in vitro estimates of cell diffusivity and cell proliferation rate sensitive to assay geometry?

Cells respond to various biochemical and physical cues during wound-healing and tumour progression. in vitro assays used to study these processes are typically conducted in one particular geometry and it is unclear how the assay geometry affects the capacity of cell populations to spread, or whether the relevant mechanisms, such as cell motility and cell proliferation, are somehow sensitive to the geometry of the assay. In this work we use a circular barrier assay to characterise the spreading of cell populations in two different geometries. Assay 1 describes a tumour-like geometry where a cell population spreads outwards into an open space. Assay 2 describes a wound-like geometry where a cell population spreads inwards to close a void. We use a combination of discrete and continuum mathematical models and automated image processing methods to obtain independent estimates of the effective cell diffusivity, D, and the effective cell proliferation rate, λ. Using our parameterised mathematical model we confirm that our estimates of D and λ accurately predict the time-evolution of the location of the leading edge and the cell density profiles for both assay 1 and assay 2. Our work suggests that the effective cell diffusivity is up to 50% lower for assay 2 compared to assay 1, whereas the effective cell proliferation rate is up to 30% lower for assay 2 compared to assay 1.

Mechanical properties of growing melanocytic nevi and the progression to melanoma

Melanocytic nevi are benign proliferations that sometimes turn into malignant melanoma in a way that is still unclear from the biochemical and genetic point of view. Diagnostic and prognostic tools are then mostly based on dermoscopic examination and morphological analysis of histological tissues. To investigate the role of mechanics and geometry in the morpholgical dynamics of melanocytic nevi, we study a computation model for cell proliferation in a layered non-linear elastic tissue. Numerical simulations suggest that the morphology of the nevus is correlated to the initial location of the proliferating cell starting the growth process and to the mechanical properties of the tissue. Our results also support that melanocytes are subject to compressive stresses that fluctuate widely in the nevus and depend on the growth stage. Numerical simulations of cells in the epidermis releasing matrix metalloproteinases display an accelerated invasion of the dermis by destroying the basal membrane. Moreover, we suggest experimentally that osmotic stress and collagen inhibit growth in primary melanoma cells while the effect is much weaker in metastatic cells. Knowing that morphological features of nevi might also reflect geometry and mechanics rather than malignancy could be relevant for diagnostic purposes.