A mathematical model of intercellular signaling during epithelial wound healing

Modelling Keloids Dynamics: A Brief Review and New Mathematical Perspectives

Keloids are fibroproliferative disorders described by excessive growth of fibrotic tissue, which also invades adjacent areas (beyond the original wound borders). Since these disorders are specific to humans (no other animal species naturally develop keloid-like tissue), experimental in vivo/in vitro research has not led to significant advances in this field. One possible approach could be to combine in vitro human models with calibrated in silico mathematical approaches (i.e., models and simulations) to generate new testable biological hypotheses related to biological mechanisms and improved treatments. Because these combined approaches do not really exist for keloid disorders, in this brief review we start by summarising the biology of these disorders, then present various types of mathematical and computational approaches used for related disorders (i.e., wound healing and solid tumours), followed by a discussion of the very few mathematical and computational models published so far to study various inflammatory and mechanical aspects of keloids. We conclude this review by discussing some open problems and mathematical opportunities offered in the context of keloid disorders by such combined in vitro/in silico approaches, and the need for multi-disciplinary research to enable clinical progress.

Mathematical modeling of ventilator-induced lung inflammation

Despite the benefits of mechanical ventilators, prolonged or misuse of ventilators may lead to ventilation-associated/ventilation-induced lung injury (VILI). Lung insults, such as respiratory infections and lung injuries, can damage the pulmonary epithelium, with the most severe cases needing mechanical ventilation for effective breathing and survival. Damaged epithelial cells within the alveoli trigger a local immune response. A key immune cell is the macrophage, which can differentiate into a spectrum of phenotypes ranging from pro- to anti-inflammatory. To gain a greater understanding of the mechanisms of the immune response to VILI and post-ventilation outcomes, we developed a mathematical model of interactions between the immune system and site of damage while accounting for macrophage phenotype. Through Latin hypercube sampling we generated a collection of parameter sets that are associated with a numerical steady state. We then simulated ventilation-induced damage using these steady state values as the initial conditions in order to evaluate how baseline immune state and lung health affect outcomes. We used a variety of methods to analyze the resulting parameter sets, transients, and outcomes, including a random forest decision tree algorithm and parameter sensitivity with eFAST. Analysis shows that parameters and properties of transients related to epithelial repair and M1 activation are important factors. Using the results of this analysis, we hypothesized interventions and used these treatment strategies to modulate the response to ventilation for particular parameters sets.

INVESTIGATION OF A STRUCTURED FISHER'S EQUATION WITH APPLICATIONS IN BIOCHEMISTRY

Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.

Modeling keratinocyte wound healing dynamics: Cell-cell adhesion promotes sustained collective migration

The in vitro migration of keratinocyte cell sheets displays behavioral and biochemical similarities to the in vivo wound healing response of keratinocytes in animal model systems. In both cases, ligand-dependent Epidermal Growth Factor Receptor (EGFR) activation is sufficient to elicit collective cell migration into the wound. Previous mathematical modeling studies of in vitro wound healing assays assume that physical connections between cells have a hindering effect on cell migration, but biological literature suggests a more complicated story. By combining mathematical modeling and experimental observations of collectively migrating sheets of keratinocytes, we investigate the role of cell-cell adhesion during in vitro keratinocyte wound healing assays. We develop and compare two nonlinear diffusion models of the wound healing process in which cell-cell adhesion either hinders or promotes migration. Both models can accurately fit the leading edge propagation of cell sheets during wound healing when using a time-dependent rate of cell-cell adhesion strength. The model that assumes a positive role of cell-cell adhesion on migration, however, is robust to changes in the leading edge definition and yields a qualitatively accurate density profile. Using RNAi for the critical adherens junction protein, α-catenin, we demonstrate that cell sheets with wild type cell-cell adhesion expression maintain migration into the wound longer than cell sheets with decreased cell-cell adhesion expression, which fails to exhibit collective migration. Our modeling and experimental data thus suggest that cell-cell adhesion promotes sustained migration as cells pull neighboring cells into the wound during wound healing.

Using an Agent-Based Model to Examine the Role of Dynamic Bacterial Virulence Potential in the Pathogenesis of Surgical Site Infection

OBJECTIVE

Despite clinical advances, surgical site infections (SSIs) remain a problem. The development of SSIs involves a complex interplay between the cellular and molecular mechanisms of wound healing and contaminating bacteria, and here, we utilize an agent-based model (ABM) to investigate the role of bacterial virulence potential in the pathogenesis of SSI.

APPROACH

The Muscle Wound ABM (MWABM) incorporates muscle cells, neutrophils, macrophages, myoblasts, abstracted blood vessels, and avirulent/virulent bacteria to simulate the pathogenesis of SSIs. Simulated bacteria with virulence potential can mutate to possess resistance to reactive oxygen species and increased invasiveness. Simulated experiments (t=7 days) involved parameter sweeps of initial wound size to identify transition zones between healed and nonhealed wounds/SSIs, and to evaluate the effect of avirulent/virulent bacteria.

RESULTS

The MWABM reproduced the dynamics of normal successful healing, including a transition zone in initial wound size beyond which healing was significantly impaired. Parameter sweeps with avirulent bacteria demonstrated that smaller wound sizes were associated with healing failure. This effect was even more pronounced with the addition of virulence potential to the contaminating bacteria.

INNOVATION

The MWABM integrates the myriad factors involved in the healing of a normal wound and the pathogenesis of SSIs. This type of model can serve as a useful framework into which more detailed mechanistic knowledge can be embedded.

CONCLUSION

Future work will involve more comprehensive representation of host factors, and especially the ability of those host factors to activate virulence potential in the microbes involved.

Tracking monotonically advancing boundaries in image sequences using graph cuts and recursive kernel shape priors

We introduce a probabilistic computer vision technique to track monotonically advancing boundaries of objects within image sequences. Our method incorporates a novel technique for including statistical prior shape information into graph-cut based segmentation, with the aid of a majorization-minimization algorithm. Extension of segmentation from single images to image sequences then follows naturally using sequential Bayesian estimation. Our methodology is applied to two unrelated sets of real biomedical imaging data, and a set of synthetic images. Our results are shown to be superior to manual segmentation.

Chemical and mechanical signaling in epithelial spreading

We propose a minimal mathematical model to explain long-range coordination of dynamics of multiple cells in epithelial spreading, which may be induced, under different conditions, by a chemical signal, or mechanically induced strain, or both. The model is based on chemo-mechanical interactions including a chemical effect of strain, chemically induced polarization and active traction, and interaction between polarized cells. The results, showing kinase concentration distribution and cell displacement, velocity, and stress fields, allow us to reproduce qualitatively available experimental data and distinguish between distinct dynamical patterns observed under conditions of injury or unconstraining.