The role of initial geometry in experimental models of wound closing

Mechanical Cell Interactions on Curved Interfaces

We propose a simple mathematical model to describe the mechanical relaxation of cells within a curved epithelial tissue layer represented by an arbitrary curve in two-dimensional space. This model generalises previous one-dimensional models of flat epithelia to investigate the influence of curvature for mechanical relaxation. We represent the mechanics of a cell body either by straight springs, or by curved springs that follow the curve's shape. To understand the collective dynamics of the cells, we devise an appropriate continuum limit in which the number of cells and the length of the substrate are constant but the number of springs tends to infinity. In this limit, cell density is governed by a diffusion equation in arc length coordinates, where diffusion may be linear or nonlinear depending on the choice of the spring restoring force law. Our results have important implications about modelling cells on curved geometries: (i) curved and straight springs can lead to different dynamics when there is a finite number of springs, but they both converge quadratically to the dynamics governed by the diffusion equation; (ii) in the continuum limit, the curvature of the tissue does not affect the mechanical relaxation of cells within the layer nor their tangential stress; (iii) a cell's normal stress depends on curvature due to surface tension induced by the tangential forces. Normal stress enables cells to sense substrate curvature at length scales much larger than their cell body, and could induce curvature dependences in experiments.

Cell Migration Assays and Their Application to Wound Healing Assays-A Critical Review

In recent years, cell migration assays (CMAs) have emerged as a tool to study the migration of cells along with their physiological responses under various stimuli, including both mechanical and bio-chemical properties. CMAs are a generic system in that they support various biological applications, such as wound healing assays. In this paper, we review the development of the CMA in the context of its application to wound healing assays. As such, the wound healing assay will be used to derive the requirements on CMAs. This paper will provide a comprehensive and critical review of the development of CMAs along with their application to wound healing assays. One salient feature of our methodology in this paper is the application of the so-called design thinking; namely we define the requirements of CMAs first and then take them as a benchmark for various developments of CMAs in the literature. The state-of-the-art CMAs are compared with this benchmark to derive the knowledge and technological gap with CMAs in the literature. We will also discuss future research directions for the CMA together with its application to wound healing assays.

Wavelength-Dependent Effects of Photobiomodulation for Wound Care in Diabetic Wounds

Photobiomodulation, showing positive effects on wound healing processes, has been performed mainly with lasers in the red/infrared spectrum. Light of shorter wavelengths can significantly influence biological systems. This study aimed to evaluate and compare the therapeutic effects of pulsed LED light of different wavelengths on wound healing in a diabetic (db/db) mouse excision wound model. LED therapy by Repuls was applied at either 470 nm (blue), 540 nm (green) or 635 nm (red), at 40 mW/cm² each. Wound size and wound perfusion were assessed and correlated to wound temperature and light absorption in the tissue. Red and trend-wise green light positively stimulated wound healing, while blue light was ineffective. Light absorption was wavelength-dependent and was associated with significantly increased wound perfusion as measured by laser Doppler imaging. Shorter wavelengths ranging from green to blue significantly increased wound surface temperature, while red light, which penetrates deeper into tissue, led to a significant increase in core body temperature. In summary, wound treatment with pulsed red or green light resulted in improved wound healing in diabetic mice. Since impeded wound healing in diabetic patients poses an ever-increasing socio-economic problem, LED therapy may be an effective, easily applied and cost-efficient supportive treatment for diabetic wound therapy.

How quickly does a wound heal? Bayesian calibration of a mathematical model of venous leg ulcer healing

Chronic wounds, such as venous leg ulcers, are difficult to treat and can reduce the quality of life for patients. Clinical trials have been conducted to identify the most effective venous leg ulcer treatments and the clinical factors that may indicate whether a wound will successfully heal. More recently, mathematical modelling has been used to gain insight into biological factors that may affect treatment success but are difficult to measure clinically, such as the rate of oxygen flow into wounded tissue. In this work, we calibrate an existing mathematical model using a Bayesian approach with clinical data for individual patients to explore which clinical factors may impact the rate of wound healing for individuals. Although the model describes group-level behaviour well, it is not able to capture individual-level responses in all cases. From the individual-level analysis, we propose distributions for coefficients of clinical factors in a linear regression model, but ultimately find that it is difficult to draw conclusions about which factors lead to faster wound healing based on the existing model and data. This work highlights the challenges of using Bayesian methods to calibrate partial differential equation models to individual patient clinical data. However, the methods used in this work may be modified and extended to calibrate spatiotemporal mathematical models to multiple data sets, such as clinical trials with several patients, to extract additional information from the model and answer outstanding biological questions.

Curvature dependences of wave propagation in reaction–diffusion models

Reaction–diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wavefronts. These waves have important applications in many physical, ecological and biological systems. In this work, we analyse curvature dependences of travelling fronts in a single reaction–diffusion equation with general reaction term. We derive an exact, non-perturbative curvature dependence of the speed of travelling fronts that arises from transverse diffusion occurring parallel to the wavefront. Inward-propagating waves are characterized by three phases: an establishment phase dominated by initial and boundary conditions, a travelling-wave-like phase in which normal velocity matches standard results from singular perturbation theory and a dip-filling phase where the collision and interaction of fronts create additional curvature dependences to their progression rate. We analyse these behaviours and additional curvature dependences using a combination of asymptotic analyses and numerical simulations.

Efficient Bayesian inference for mechanistic modelling with high-throughput data

Bayesian methods are routinely used to combine experimental data with detailed mathematical models to obtain insights into physical phenomena. However, the computational cost of Bayesian computation with detailed models has been a notorious problem. Moreover, while high-throughput data presents opportunities to calibrate sophisticated models, comparing large amounts of data with model simulations quickly becomes computationally prohibitive. Inspired by the method of Stochastic Gradient Descent, we propose a minibatch approach to approximate Bayesian computation. Through a case study of a high-throughput imaging scratch assay experiment, we show that reliable inference can be performed at a fraction of the computational cost of a traditional Bayesian inference scheme. By applying a detailed mathematical model of single cell motility, proliferation and death to a data set of 118 gene knockdowns, we characterise functional subgroups of gene knockdowns, each displaying its own typical combination of local cell density-dependent and -independent motility and proliferation patterns. By comparing these patterns to experimental measurements of cell counts and wound closure, we find that density-dependent interactions play a crucial role in the process of wound healing.

During wound healing, cells work together to close a wound to restore tissue integrity. Thousands of different genes play a role in wound healing, and scratch assay experiments are routinely used to investigate the role of these genes by analysing how a wound closes when each of these is not expressed, i.e. knocked down. So far, the impact of knocking down genes on wound healing has been determined by comparing the size of the wound before and after a given time period, but these measurements do not elucidate the fine-scale mechanisms that determine how cells behave in the presence of their neighbours. By combining a detailed mathematical model with experimental imaging of wound healing, we identify how cells respond to and work together with their neighbours during wound healing. Applying this method to a large number of gene knockdowns, we identify three well-defined functional subgroups of knockdowns, each displaying its own typical behaviours of movement and proliferation to close the wound. These observations explain the role of each of the knockdowns on wound healing and further our understanding of cell-cell interactions in wound healing.

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

Tissue growth in three-dimensional (3D) printed scaffolds enables exploration and control of cell behaviour in more biologically realistic geometries than that allowed by traditional 2D cell culture. Cell proliferation and migration in these experiments have yet to be explicitly characterised, limiting the ability of experimentalists to determine the effects of various experimental conditions, such as scaffold geometry, on cell behaviour. We consider tissue growth by osteoblastic cells in melt electro-written scaffolds that comprise thin square pores with sizes that were deliberately increased between experiments. We collect highly detailed temporal measurements of the average cell density, tissue coverage, and tissue geometry. To quantify tissue growth in terms of the underlying cell proliferation and migration processes, we introduce and calibrate a mechanistic mathematical model based on the Porous-Fisher reaction-diffusion equation. Parameter estimates and uncertainty quantification through profile likelihood analysis reveal consistency in the rate of cell proliferation and steady-state cell density between pore sizes. This analysis also serves as an important model verification tool: while the use of reaction-diffusion models in biology is widespread, the appropriateness of these models to describe tissue growth in 3D scaffolds has yet to be explored. We find that the Porous-Fisher model is able to capture features relating to the cell density and tissue coverage, but is not able to capture geometric features relating to the circularity of the tissue interface. Our analysis identifies two distinct stages of tissue growth, suggests several areas for model refinement, and provides guidance for future experimental work that explores tissue growth in 3D printed scaffolds.

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

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.

In vitro cell migration quantification method for scratch assays

The scratch assay is an in vitro technique used to assess the contribution of molecular and cellular mechanisms to cell migration. The assay can also be used to evaluate therapeutic compounds before clinical use. Current quantification methods of scratch assays deal poorly with irregular cell-free areas and crooked leading edges which are features typically present in the experimental data. We introduce a new migration quantification method, called 'monolayer edge velocimetry', that permits analysis of low-quality experimental data and better statistical classification of migration rates than standard quantification methods. The new method relies on quantifying the horizontal component of the cell monolayer velocity across the leading edge. By performing a classification test on in silico data, we show that the method exhibits significantly lower statistical errors than standard methods. When applied to in vitro data, our method outperforms standard methods by detecting differences in the migration rates between different cell groups that the other methods could not detect. Application of this new method will enable quantification of migration rates from in vitro scratch assay data that cannot be analysed using existing methods.

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.

A level-set method for the evolution of cells and tissue during curvature-controlled growth

Most biological tissues grow by the synthesis of new material close to the tissue's interface, where spatial interactions can exert strong geometric influences on the local rate of growth. These geometric influences may be mechanistic or cell behavioural in nature. The control of geometry on tissue growth has been evidenced in many in vivo and in vitro experiments, including bone remodelling, wound healing, and tissue engineering scaffolds. In this paper, we propose a generalisation of a mathematical model that captures the mechanistic influence of curvature on the joint evolution of cell density and tissue shape during tissue growth. This generalisation allows us to simulate abrupt topological changes such as tissue fragmentation and tissue fusion, as well as three dimensional cases, through a level-set-based method. The level-set method developed introduces another Eulerian field than the level-set function. This additional field represents the surface density of tissue-synthesising cells, anticipated at future locations of the interface. Numerical tests performed with this level-set-based method show that numerical conservation of cells is a good indicator of simulation accuracy, particularly when cusps develop in the tissue's interface. We apply this new model to several situations of curvature-controlled tissue evolutions that include fragmentation and fusion.

A microfluidics-based wound-healing assay for studying the effects of shear stresses, wound widths, and chemicals on the wound-healing process

Collective cell migration plays important roles in various physiological processes. To investigate this collective cellular movement, various wound-healing assays have been developed. In these assays, a “wound” is created mechanically, chemically, optically, or electrically out of a cellular monolayer. Most of these assays are subject to drawbacks of run-to-run variations in wound size/shape and damages to cells/substrate. Moreover, in all these assays, cells are cultured in open, static (non-circulating) environments. In this study, we reported a microfluidics-based wound-healing assay by using the trypsin flow-focusing technique. Fibroblasts were first cultured inside this chip to a cellular monolayer. Then three parallel fluidic flows (containing normal medium and trypsin solution) were introduced into the channels, and cells exposed to protease trypsin were enzymatically detached from the surface. Wounds of three different widths were generated, and subsequent wound-healing processes were observed. This assay is capable of creating three or more wounds of different widths for investigating the effects of various physical and chemical stimuli on wound-healing speeds. The effects of shear stresses, wound widths, and β-lapachone (a wound healing-promoting chemical) on wound-healing speeds were studied. It was found that the wound-healing speed (total area healed per unit time) increased with increasing shear stress and wound width, but under a shear stress of 0.174 mPa the linear healing speed (percent area healed per unit time) was independent of the wound width. Also, the addition of β-lapachone up to 0.5 μM did not accelerate wound healing. This microfluidics-based assay can definitely help in understanding the mechanisms of the wound-healing process and developing new wound-healing therapies.

Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading-extinction dichotomy

The Fisher-Kolmogorov-Petrovsky-Piskunov model, also known as the Fisher-KPP model, supports travelling wave solutions that are successfully used to model numerous invasive phenomena with applications in biology, ecology and combustion theory. However, there are certain phenomena that the Fisher-KPP model cannot replicate, such as the extinction of invasive populations. The Fisher-Stefan model is an adaptation of the Fisher-KPP model to include a moving boundary whose evolution is governed by a Stefan condition. The Fisher-Stefan model also supports travelling wave solutions; however, a key additional feature of the Fisher-Stefan model is that it is able to simulate population extinction, giving rise to a spreading-extinction dichotomy. In this work, we revisit travelling wave solutions of the Fisher-KPP model and show that these results provide new insight into travelling wave solutions of the Fisher-Stefan model and the spreading-extinction dichotomy. Using a combination of phase plane analysis, perturbation analysis and linearization, we establish a concrete relationship between travelling wave solutions of the Fisher-Stefan model and often-neglected travelling wave solutions of the Fisher-KPP model. Furthermore, we give closed-form approximate expressions for the shape of the travelling wave solutions of the Fisher-Stefan model in the limit of slow travelling wave speeds, c≪1.