Analytical solutions to Fisher’s equation with time-variable coefficients

Hammond, Jason F.; Bortz, David M.

doi:10.1016/j.amc.2011.03.163

Cited by 14 publications

(9 citation statements)

References 14 publications

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“…More direct experimental measurements of single-cell proliferative and migratory behaviours in their population environment at different locations and time points during the pore bridging experiments would help validate the importance of cell crowding effects for the geometric control of tissue growth. These kinds of experimental data would also be useful to determine whether generalisations of the Porous-Fisher model with time-dependent coefficients are required [61,67], such as a time dependent diffusivity D (t ) and a time dependent proliferation rate, λ(t ), such that the product D (t )λ(t ) is maintained constant. We have not considered this possibility here because there are no obvious trends in our experimental observation that motivate this kind of extension, however, this is a potential avenue for future consideration.…”

Section: Discussionmentioning

confidence: 99%

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size

et al. 2020

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Tissue growth in bioscaffolds is influenced significantly by pore geometry, but how this geometric dependence emerges from dynamic cellular processes such as cell proliferation and cell migration remains poorly understood. Here we investigate the influence of pore size on the time required to bridge pores in thin 3D-printed scaffolds. Experimentally, new tissue infills the pores continually from their perimeter under strong curvature control, which leads the tissue front to round off with time. Despite the varied shapes assumed by the tissue during this evolution, we find that time to bridge a pore simply increases linearly with the overall pore size. To disentangle the biological influence of cell behaviour and the mechanistic influence of geometry in this experimental observation, we propose a simple reaction-diffusion model of tissue growth based on Porous-Fisher invasion of cells into the pores. First, this model provides a good qualitative representation of the evolution of the tissue; new tissue in the model grows at an effective rate that depends on the local curvature of the tissue substrate. Second, the model suggests that a linear dependence of bridging time with pore size arises due to geometric reasons alone, not to differences in cell behaviours across pores of different sizes. Our analysis suggests that tissue growth dynamics in these experimental constructs is dominated by mechanistic crowding effects that influence collective cell proliferation and migration processes, and that can be predicted by simple reaction-diffusion models of cells that have robust, consistent behaviours.

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Section: Discussionmentioning

confidence: 99%

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size

et al. 2020

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“…Similar to other applications of the Fisher-Kolmogorov equation [ 1 – 4 , 21 , 29 , 31 , 34 ], we have made the standard assumption that the parameters in each experiment, D , λ and K , are constants which do not vary with position, time or cell density. Recently, there has been considerable interest in the theoretical physics and applied mathematics literature regarding the analysis of extensions of the Fisher-Kolmogorov equation where D and λ vary with position, time or cell density [ 44 , 45 ]. Although these extensions are mathematically interesting, we have not attempted to apply such an extension here since the precise form of the putative spatial or temporal dependence is unknown, and at this stage, we anticipate that more detailed experimental data would be required to calibrate these more detailed mathematical models.…”

Section: Discussionmentioning

confidence: 99%

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

et al. 2015

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BackgroundStandard 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.ResultsThe 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.ConclusionsOur 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.

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“…While classically concerned with homogeneous environments, interest in the behavior of fronts in heterogeneous environments has increased over the last several years. In particular, traveling waves, or propagating fronts, through certain classes of time [7] and spatially varying environments [3,8] have been well studied. Nonlinear, density-dependent diffusion, where the diffusion coefficient depends on u, has also been thoroughly examined in [1,2] and [6] amongst others.…”

Section: Introductionmentioning

confidence: 99%

Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion

Curtis

Bortz

2012

Phys. Rev. E

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The propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion coefficients is studied. Using coordinate changes, WKB approximations, and multiple scales analysis, we provide an analytic framework that describes propagation of the front up to the minimum of the diffusion coefficient. We also present results showing the behavior of the front after it passes the minimum. In each case, we show that standard traveling coordinate frames do not properly describe front propagation. Last, we provide numerical simulations to support our analysis and to show, that around the minimum, the motion of the front is arrested on asymptotically significant time scales.

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Analytical solutions to Fisher’s equation with time-variable coefficients

Cited by 14 publications

References 14 publications

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size

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

Propagation of fronts in the Fisher-Kolmogorov equation with spatially varying diffusion

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