A mathematical model for wound healing in the reef-building coral Pocillopora damicornis.

Coral reefs, among the most diverse ecosystems on Earth, currently face major threats from pollution, unsustainable fishing practices , and perturbations in environmental parameters brought on by climate change. Corals also sustain regular wounding from other sea life and human activity. Recent reef restoration practices have even involved intentional wounding by systematically breaking coral fragments and relocating them to revitalize damaged reefs, a practice known as microfragmentation. Despite its importance, very little research has explored the inner mechanisms of wound healing in corals. Some reef-building corals have been observed to initiate an immunological response to wounding similar to that observed in mammalian species. Utilizing prior models of wound healing in mammalian species as the mathematical basis, we formulated a mechanistic model of wound healing, including observations of the immune response and tissue repair in scleractinian corals for the species Pocillopora damicornis. The model consists of four differential equations which track changes in remaining wound debris, number of cells involved in inflammation, number of cells involved in proliferation, and amount of wound closure through re-epithelialization. The model is fit to experimental wound size data from linear and circular shaped wounds on a live coral fragment. Mathematical methods, including numerical simulations and local sensitivity analysis, were used to analyze the resulting model. The parameter space was also explored to investigate drivers of other possible wound outcomes. This model serves as a first step in generating mathematical models for wound healing in corals that will not only aid in the understanding of wound healing as a whole, but also help optimize reef restoration practices and predict recovery behavior after major wounding events.

Identifying important parameters in the inflammatory process with a mathematical model of immune cell influx and macrophage polarization.

In an inflammatory setting, macrophages can be polarized to an inflammatory M1 phenotype or to an anti-inflammatory M2 phenotype, as well as existing on a spectrum between these two extremes. Dysfunction of this phenotypic switch can result in a population imbalance that leads to chronic wounds or disease due to unresolved inflammation. Therapeutic interventions that target macrophages have therefore been proposed and implemented in diseases that feature chronic inflammation such as diabetes mellitus and atherosclerosis. We have developed a model for the sequential influx of immune cells in the peritoneal cavity in response to a bacterial stimulus that includes macrophage polarization, with the simplifying assumption that macrophages can be classified as M1 or M2. With this model, we were able to reproduce the expected timing of sequential influx of immune cells and mediators in a general inflammatory setting. We then fit this model to in vivo experimental data obtained from a mouse peritonitis model of inflammation, which is widely used to evaluate endogenous processes in response to an inflammatory stimulus. Model robustness is explored with local structural and practical identifiability of the proposed model a posteriori. Additionally, we perform sensitivity analysis that identifies the population of apoptotic neutrophils as a key driver of the inflammatory process. Finally, we simulate a selection of proposed therapies including points of intervention in the case of delayed neutrophil apoptosis, which our model predicts will result in a sustained inflammatory response. Our model can therefore provide hypothesis testing for therapeutic interventions that target macrophage phenotype and predict outcomes to be validated by subsequent experimentation.

Modeling the effects of systemic mediators on the inflammatory phase of wound healing.

The normal wound healing response is characterized by a progression from clot formation, to an inflammatory phase, to a repair phase, and finally, to remodeling. In many chronic wounds there is an extended inflammatory phase that stops this progression. In order to understand the inflammatory phase in more detail, we developed an ordinary differential equation model that accounts for two systemic mediators that are known to modulate this phase, estrogen (a protective hormone during wound healing) and cortisol (a hormone elevated after trauma that slows healing). This model describes the interactions in the wound between wound debris, pathogens, neutrophils and macrophages and the modulation of these interactions by estrogen and cortisol. A collection of parameter sets, which qualitatively match published data on the dynamics of wound healing, was chosen using Latin Hypercube Sampling. This collection of parameter sets represents normal healing in the population as a whole better than one single parameter set. Including the effects of estrogen and cortisol is a necessary step to creating a patient specific model that accounts for gender and trauma. Utilization of math modeling techniques to better understand the wound healing inflammatory phase could lead to new therapeutic strategies for the treatment of chronic wounds. This inflammatory phase model will later become the inflammatory subsystem of our full wound healing model, which includes fibroblast activity, collagen accumulation and remodeling.

A differential equation model of collagen accumulation in a healing wound.

Wound healing is a complex biological process which involves many cell types and biochemical signals and which progresses through multiple, overlapping phases. In this manuscript, we develop a model of collagen accumulation as a marker of wound healing. The mathematical model is a system of ordinary differential equations which tracks fibroblasts, collagen, inflammation and pathogens. The model was validated by comparison to the normal time course of wound healing where appropriate activity for the inflammatory, proliferative and remodeling phases was recorded. Further validation was made by comparison to collagen accumulation experiments by Madden and Peacock (Ann. Surg. 174(3):511-520, 1971). The model was then used to investigate the impact of local oxygen levels on wound healing. Finally, we present a comparison of two wound healing therapies, antibiotics and increased fibroblast proliferation. This model is a step in developing a comprehensive model of wound healing which can be used to develop and test new therapeutic treatments.

An in silico approach to the analysis of acute wound healing.

The complex interactions that characterize acute wound healing have stymied the development of effective therapeutic modalities. The use of computational models holds the promise to improve our basic approach to understanding the process. By modifying an existing ordinary differential equation model of systemic inflammation to simulate local wound healing, we expect to improve the understanding of the underlying complexities of wound healing and thus allow for the development of novel, targeted therapeutic strategies. The modifications in this local acute wound healing model include: evolution from a systemic model to a local model, the incorporation of fibroblast activity, and the effects of tissue oxygenation. Using these modifications we are able to simulate impaired wound healing in hypoxic wounds with varying levels of contamination. Possible therapeutic targets, such as fibroblast death rate and rate of fibroblast recruitment, have been identified by computational analysis. This model is a step toward constructing an integrative systems biology model of human wound healing.

Dosimetry of nasal uptake of water-soluble and reactive gases: a first study of interhuman variability.

Certain inhaled chemicals, such as reactive, water-soluble gases, are readily absorbed by the nasal mucosa upon inhalation and may cause damage to the nasal epithelium. Comparisons of the spatial distribution of nasal lesions in laboratory animals exposed to formaldehyde with gas uptake rates predicted by computational models reveal that lesions usually occur in regions of the susceptible epithelium where gas absorption is highest. Since the uptake patterns are influenced by air currents in the nose, interindividual variability in nasal anatomy and ventilation rates due to age, body size, and gender will affect the patterns of gas absorption in humans, potentially putting some age groups at higher risk when exposed to toxic gases. In this study, interhuman variability in the nasal dosimetry of reactive, water-soluble gases was investigated by means of computational fluid dynamics (CFD) models in 5 adults and 2 children, aged 7 and 8 years old. Airflow patterns were investigated for allometrically scaled inhalation rates corresponding to resting breathing. The spatial distribution of uptake at the airway walls was predicted to be nonuniform, with most of the gas being absorbed in the anterior portion of the nasal passages. Under the conditions of these simulations, interhuman variability in dose to the whole nose (mass per time per nasal surface area) due to differences in anatomy and ventilation was predicted to be 1.6-fold among the 7 individuals studied. Children and adults displayed very similar patterns of nasal gas uptake; no significant differences were noted between the two age groups.

Effects of differences in nasal anatomy on airflow distribution: a comparison of four individuals at rest.

Differences in nasal anatomy among human subjects may cause significant differences in respiratory airflow patterns and subsequent dosimetry of inhaled gases and particles in the respiratory tract. This study used computational fluid dynamics (CFD) to study inter-individual differences in nasal airflow among four healthy individuals. Magnetic resonance imaging (MRI) scans were digitized and nasal-surface-area-to-volume ratios (SAVR) were calculated for 15 adults. Two males and two females, representative of the range of SAVR values, were selected for flow analysis. Nasal CFD models were constructed for each subject by a semi-automated process that provided input to a commercial mesh generator to generate structured hexahedral meshes (Gambit, Fluent, Inc., Lebanon, NH, USA). Steady-state inspiratory laminar airflow at 15 L/min was calculated using commercial CFD software (FIDAP, Fluent, Inc., Lebanon, NH, USA). Streamline patterns, velocities, and helicity values were compared. In all subjects, the majority of flow passed through the middle and ventral regions of the nasal passages; however, the amount and location of swirling flow differed among individuals. Cross-sectional flow allocation analysis also indicated inter-individual differences. Laboratory water-dye experiments confirmed streamlines and velocity magnitudes predicted by the computational model. These results suggest that significant inter-individual differences exist in bulk airflow patterns in the nose.

Characterization of deposition from nasal spray devices using a computational fluid dynamics model of the human nasal passages.

Many studies suggest limited effectiveness of spray devices for nasal drug delivery due primarily to high deposition and clearance at the front of the nose. Here, nasal spray behavior was studied using experimental measurements and a computational fluid dynamics model of the human nasal passages constructed from magnetic resonance imaging scans of a healthy adult male. Eighteen commercially available nasal sprays were analyzed for spray characteristics using laser diffraction, high-speed video, and high-speed spark photography. Steadystate, inspiratory airflow (15 L/min) and particle transport were simulated under measured spray conditions. Simulated deposition efficiency and spray behavior were consistent with previous experimental studies, two of which used nasal replica molds based on this nasal geometry. Deposition fractions (numbers of deposited particles divided by the number released) of 20- and 50-microm particles exceeded 90% in the anterior part of the nose for most simulated conditions. Predicted particle penetration past the nasal valve improved when (1) the smaller of two particle sizes or the lower of two spray velocities was used, (2) the simulated nozzle was positioned 1.0 rather than 0.5 or 1.5 cm into the nostril, and (3) inspiratory airflow was present rather than absent. Simulations also predicted that delaying the appearance of normal inspiratory airflow more than 1 sec after the release of particles produced results equivalent to cases in which no inspiratory airflow was present. These predictions contribute to more effective design of drug delivery devices through a better understanding of the effects of nasal airflow and spray characteristics on particle transport in the nose.