Spatial pattern formation in reaction-diffusion models: a computational approach.

Reaction-diffusion equations have been widely used to describe biological pattern formation. Nonuniform steady states of reaction-diffusion models correspond to stationary spatial patterns supported by these models. Frequently these steady states are not unique and correspond to various spatial patterns observed in biology. Traditionally, time-marching methods or steady state solvers based on Newton's method were used to compute such solutions. However, the solutions that these methods converge to highly depend on the initial conditions or guesses. In this paper, we present a systematic method to compute multiple nonuniform steady states for reaction-diffusion models and determine their dependence on model parameters. The method is based on homotopy continuation techniques and involves mesh refinement, which significantly reduces computational cost. The method generates one-parameter steady state bifurcation diagrams that may contain multiple unconnected components, as well as two-parameter solution maps that divide the parameter space into different regions according to the number of steady states. We applied the method to two classic reaction-diffusion models and compared our results with available theoretical analysis in the literature. The first is the Schnakenberg model which has been used to describe biological pattern formation due to diffusion-driven instability. The second is the Gray-Scott model which was proposed in the 1980s to describe autocatalytic glycolysis reactions. In each case, the method uncovers many, if not all, nonuniform steady states and their stabilities.

Multiscale phenomena and patterns in biological systems: special issue in honour of Hans Othmer.

This special issue on "Multiscale phenomena and patterns in biological systems" is an homage to the seminal contributions of Hans Othmer. He has remained at the forefront of multiscale modelling and pattern formation in biology for over half a century, developing models for molecular signalling networks, the mechanics of cellular movements, the interactions between multiple cells and their contributions to tissue patterning and dynamics. The contributions in this special issue follow Hans' legacy in using advanced mathematics to understand complex biological processes.

The role of intracellular signaling in the stripe formation in engineered Escherichia coli populations.

Recent experiments showed that engineered Escherichia coli colonies grow and self-organize into periodic stripes with high and low cell densities in semi-solid agar. The stripes develop sequentially behind a radially propagating colony front, similar to the formation of many other periodic patterns in nature. These bacteria were created by genetically coupling the intracellular chemotaxis pathway of wild-type cells with a quorum sensing module through the protein CheZ. In this paper, we develop multiscale models to investigate how this intracellular pathway affects stripe formation. We first develop a detailed hybrid model that treats each cell as an individual particle and incorporates intracellular signaling via an internal ODE system. To overcome the computational cost of the hybrid model caused by the large number of cells involved, we next derive a mean-field PDE model from the hybrid model using asymptotic analysis. We show that this analysis is justified by the tight agreement between the PDE model and the hybrid model in 1D simulations. Numerical simulations of the PDE model in 2D with radial symmetry agree with experimental data semi-quantitatively. Finally, we use the PDE model to make a number of testable predictions on how the stripe patterns depend on cell-level parameters, including cell speed, cell doubling time and the turnover rate of intracellular CheZ.

A Stochastic Multiscale Model That Explains the Segregation of Axonal Microtubules and Neurofilaments in Neurological Diseases.

The organization of the axonal cytoskeleton is a key determinant of the normal function of an axon, which is a long thin projection of a neuron. Under normal conditions two axonal cytoskeletal polymers, microtubules and neurofilaments, align longitudinally in axons and are interspersed in axonal cross-sections. However, in many neurotoxic and neurodegenerative disorders, microtubules and neurofilaments segregate apart from each other, with microtubules and membranous organelles clustered centrally and neurofilaments displaced to the periphery. This striking segregation precedes the abnormal and excessive neurofilament accumulation in these diseases, which in turn leads to focal axonal swellings. While neurofilament accumulation suggests an impairment of neurofilament transport along axons, the underlying mechanism of their segregation from microtubules remains poorly understood for over 30 years. To address this question, we developed a stochastic multiscale model for the cross-sectional distribution of microtubules and neurofilaments in axons. The model describes microtubules, neurofilaments and organelles as interacting particles in a 2D cross-section, and is built upon molecular processes that occur on a time scale of seconds or shorter. It incorporates the longitudinal transport of neurofilaments and organelles through this domain by allowing stochastic arrival and departure of these cargoes, and integrates the dynamic interactions of these cargoes with microtubules mediated by molecular motors. Simulations of the model demonstrate that organelles can pull nearby microtubules together, and in the absence of neurofilament transport, this mechanism gradually segregates microtubules from neurofilaments on a time scale of hours, similar to that observed in toxic neuropathies. This suggests that the microtubule-neurofilament segregation can be a consequence of the selective impairment of neurofilament transport. The model generates the experimentally testable prediction that the rate and extent of segregation will be dependent on the sizes of the moving organelles as well as the density of their traffic.

Moment-flux models for bacterial chemotaxis in large signal gradients.

Chemotaxis is a fundamental process in the life of many prokaryotic and eukaryotic cells. Chemotaxis of bacterial populations has been modeled by both individual-based stochastic models that take into account the biochemistry of intracellular signaling, and continuum PDE models that track the evolution of the cell density in space and time. Continuum models have been derived from individual-based models that describe intracellular signaling by a system of ODEs. The derivations rely on quasi-steady state approximations of the internal ODE system. While this assumption is valid if cell movement is subject to slowly changing signals, it is often violated if cells are exposed to rapidly changing signals. In the latter case current continuum models break down and do not match the underlying individual-based model quantitatively. In this paper, we derive new PDE models for bacterial chemotaxis in large signal gradients that involve not only the cell density and flux, but also moments of the intracellular signals as a measure of the deviation of cell's internal state from its steady state. The derivation is based on a new moment closure method without calling the quasi-steady state assumption of intracellular signaling. Numerical simulations suggest that the resulting model matches the population dynamics quantitatively for a much larger range of signals.

Doxorubicin-loaded glycyrrhetinic acid modified recombinant human serum albumin nanoparticles for targeting liver tumor chemotherapy.

Due to overexpression of glycyrrhetinic acid (GA) receptor in liver cancer cells, glycyrrhetinic acid modified recombinant human serum albumin (rHSA) nanoparticles for targeting liver tumor cells may result in increased therapeutic efficacy and decreased adverse effects of cancer therapy. In this study, doxorubicin (DOX) loaded and glycyrrhetinic acid modified recombinant human serum albumin nanoparticles (DOX/GA-rHSA NPs) were prepared for targeting therapy for liver cancer. GA was covalently coupled to recombinant human serum albumin nanoparticles, which could efficiently deliver DOX into liver cancer cells. The resultant GA-rHSA NPs exhibited uniform spherical shape and high stability in plasma with fixed negative charge (∼-25 mV) and a size about 170 nm. DOX was loaded into GA-rHSA NPs with a maximal encapsulation efficiency of 75.8%. Moreover, the targeted NPs (DOX/GA-rHSA NPs) showed increased cytotoxic activity in liver tumor cells compared to the nontargeted NPs (DOX/rHSA NPs, DOX loaded recombinant human serum albumin nanoparticles without GA conjugating). The targeted NPs exhibited higher cellular uptake in a GA receptor-positive liver cancer cell line than nontargeted NPs as measured by both flow cytometry and confocal laser scanning microscopy. Biodistribution experiments showed that DOX/GA-rHSA NPs exhibited a much higher level of tumor accumulation than nontargeted NPs at 1 h after injection in hepatoma-bearing Balb/c mice. Therefore, the DOX/GA-rHSA NPs could be considered as an efficient nanoplatform for targeting drug delivery system for liver cancer.

Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling.

Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum PDE models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak-Keller-Segel equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli populations. Numerical comparison of the derived PDEs and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy.

Mechanisms of neuroprotection from hypoxia-ischemia (HI) brain injury by up-regulation of cytoglobin (CYGB) in a neonatal rat model.

This study was designed to investigate the expression profile of CYGB, its potential neuroprotective function, and underlying molecular mechanisms using a model of neonatal hypoxia-ischemia (HI) brain injury. Cygb mRNA and protein expression were evaluated within the first 36 h after the HI model was induced using RT-PCR and Western blotting. Cygb mRNA expression was increased at 18 h in a time-dependent manner, and its level of protein expression increased progressively in 24 h. To verify the neuroprotective effect of CYGB, a gene transfection technique was employed. Cygb cDNA and shRNA delivery adenovirus systems were established (Cygb-cDNA-ADV and Cygb-shRNA-ADV, respectively) and injected into the brains of 3-day-old rats 4 days before they were induced with HI treatment. Rats from different groups were euthanized 24 h post-HI, and brain samples were harvested. 2,3,5-Triphenyltetrazolium chloride, TUNEL, and Nissl staining indicated that an up-regulation of CYGB resulted in reduced acute brain injury. The superoxide dismutase level was found to be dependent on expression of CYGB. The Morris water maze test in 28-day-old rats demonstrated that CYGB expression was associated with improvement of long term cognitive impairment. Studies also demonstrated that CYGB can up-regulate mRNA and protein levels of VEGF and increase both the density and diameter of the microvessels but inhibits activation of caspase-2 and -3. Thus, this is the first in vivo study focusing on the neuroprotective role of CYGB. The reduction of neonatal HI injury by CYGB may be due in part to antioxidant and antiapoptotic mechanisms and by promoting angiogenesis.

A mathematical model for pancreatic cancer growth and treatments.

Pancreatic cancer is one of the most deadly types of cancer and has extremely poor prognosis. This malignancy typically induces only limited cellular immune responses, the magnitude of which can increase with the number of encountered cancer cells. On the other hand, pancreatic cancer is highly effective at evading immune responses by inducing polarization of pro-inflammatory M1 macrophages into anti-inflammatory M2 macrophages, and promoting expansion of myeloid derived suppressor cells, which block the killing of cancer cells by cytotoxic T cells. These factors allow immune evasion to predominate, promoting metastasis and poor responsiveness to chemotherapies and immunotherapies. In this paper we develop a mathematical model of pancreatic cancer, and use it to qualitatively explain a variety of biomedical and clinical data. The model shows that drugs aimed at suppressing cancer growth are effective only if the immune induced cancer cell death lies within a specific range, that is, the immune system has a specific window of opportunity to effectively suppress cancer under treatment. The model results suggest that tumor growth rate is affected by complex feedback loops between the tumor cells, endothelial cells and the immune response. The relative strength of the different loops determines the cancer growth rate and its response to immunotherapy. The model could serve as a starting point to identify optimal nodes for intervention against pancreatic cancer.

Travelling waves in hybrid chemotaxis models.

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.

Baduanjin is Better Balance Training Compared to Walking: A Cross-Sectional Study Based on Center of Gravity Trajectories.

BACKGROUND:  It has already been demonstrated by previous studies that Baduanjin training can improve the body's balance. However, its biomechanical mechanism remains unknown. Center of gravity (COG) trajectory analysis is an essential biomechanical test to explore the balance ability of the human body. Previous studies have not used the COG trajectory analysis technique to research Baduanjin training. The study utilizes COG trajectory analysis to analyze the trajectory of COG during Baduanjin training and compare it with that of walking, which is a common exercise for improving balance and aerobic ability, to determine if Baduanjin exercises affect the COG more than walking. MATERIALS AND METHODS:  Eight healthy female college students performed the walking and the eight forms of Baduanjin, a total of nine motions. The lower body kinematics were captured by the Vicon Motion Capture and Analysis System, while the kinetic data were acquired by the Kistler 3D Force Platform. The data were imported into Visual 3D to process the trajectory of the COG displacement amplitude, velocity, and acceleration of each motion. The COG horizontal envelope areas were calculated by Origin 9.0 Software (Origin Lab, Northampton, Massachusetts, USA) . RESULTS: Specific motions of Baduanjin provided significantly higher COG displacement amplitude, velocities, and acceleration training than walking. The F2 and F5 motions could provide a larger COG horizontal envelope area than walking. On the x-axis, F2 provided a greater COG displacement amplitude than walking, F1, F2, and F5 provided greater velocities, while all the motions provided greater accelerations. On the y-axis, all the motions except F2 provided greater COG displacement velocities and accelerations than walking. On the z-axis, F1-7 provided a greater COG displacement amplitude than walking, all the motions provided greater velocities, while all the motions except F2 provided greater accelerations. CONCLUSION: Baduanjin training provides a more intense COG perturbation than walking, which may be a more challenging balance training than walking.

Glaucoma drainage device implantation and cyclophotocoagulation in the management of refractory glaucoma after Descemet-stripping automated endothelial keratoplasty.

AIM: To compare the surgical outcomes of glaucoma drainage device implantation (GDI) and trans-scleral neodymium:YAG cyclophotocoagulation (CPC) in the management of refractory glaucoma after Descemet-stripping automated endothelial keratoplasty (DSAEK). METHODS: This retrospective study on observational case series enrolled 29 patients who underwent DSAEK and posterior anti-glaucoma surgery (15 with GDI and 14 with CPC). The main outcome measures were intraocular pressure (IOP), glaucoma surgery success rate (defined as IOP of 6-21 mm Hg without additional anti-glaucoma operation), number of glaucoma medications, endothelial graft status, and best-corrected visual acuity (BCVA). RESULTS: The mean follow-up time was 34.1 and 21.0mo for DSAEK or glaucoma surgeries, both for the GDI and CPC groups. Both groups showed significant IOP reduction after glaucoma surgery. The GDI group presented a significantly higher success rate in IOP control than the CPC group (60% vs 21.4%, P=0.03). Both procedures significantly decreased the number of glaucoma medications (P=0.03). Forty percent and 57% of cases in the GDI and the CPC group, respectively, experienced endothelial graft failure during follow-up (P=0.36). Significantly worse BCVA after surgery was observed in the CPC group but not in the GDI group. CONCLUSION: Both GDI and CPC significantly decrease IOP in eyes with glaucoma after DSAEK. GDI is preferable to CPC in refractory glaucoma cases after DSAEK, as it manifests a significantly higher success rate for IOP control, similar endothelial graft failure rate, and relatively preserves BCVA than CPC.

Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation.

The machinery for transduction of chemotactic stimuli in the bacterium E. coli is one of the most completely characterized signal transduction systems, and because of its relative simplicity, quantitative analysis of this system is possible. Here we discuss models which reproduce many of the important behaviors of the system. The important characteristics of the signal transduction system are excitation and adaptation, and the latter implies that the transduction system can function as a "derivative sensor" with respect to the ligand concentration in that the DC component of a signal is ultimately ignored if it is not too large. This temporal sensing mechanism provides the bacterium with a memory of its passage through spatially- or temporally-varying signal fields, and adaptation is essential for successful chemotaxis. We also discuss some of the spatial patterns observed in populations and indicate how cell-level behavior can be embedded in population-level descriptions.

[Effect of xuebijing oral effervescent tablet on endotoxin induced fever and disseminated intravascular coagulation rabbit model].

In order to discover the mechanism of Xuebijing oral effervescent tablet (XBJOET) to treat infectious diseases, the effect of XBJOET on endotoxin induced rabbit fever and disseminated intravascular coagulation (DIC) was investigated. Auricle microcirculation in rabbit was detected by laser speckle blood perfusion imager system; coagulation function was measured by coagulation analyzer, fibrinolytic system was quantified by Elisa assay and micro thrombosis in tissues was observed with HE staining under light microscope. The results demonstrated that the body temperature of rabbit decreased significantly at 1-3 h after administration with 4.8, 2.4 and 1.2 g x kg(-1) XBJOET to endotoxin induced DIC rabbit model, the auricle microcirculation blood flow in model group (54.45 +/- 14.53) PU was lower than that in control group (77.18 +/- 12.32) PU. The auricle microcirculation blood flow increased markedly and there was significant difference between model group and 1.2 g x kg(-1) XBJOET group. There was significant difference between model group and control group in the content of PAI1 and FIB. The PAI1 levels in model and control groups were (30.48 +/- 2.46) ng x mL(-1) and (20.93 +/- 3.25) ng x mL(-1), respectively. The FIB levels in model and control group were (3.34 +/- 1.09) g x L(-1) and (4.84 +/- 1.10) g x L(-1), respectively. The content of PAI1 in rabbit plasma decreased notably, there were significant differences between model group and 4.8, 2.4 g x kg(-1) XBJOET groups. On the contrary the content of FIB increased. XBJOET possessed pharmacological activities of curing infectious fever and DIC, the mechanism of which is related to amelioration of microcirculation disturbance, inhibition of fibrinolytic system activation and coagulation and micro thrombosis elimination.

Effect of adding hip exercises to general rehabilitation treatment of knee osteoarthritis on patients' physical functions: a randomized clinical trial.

BACKGROUND: Hip adductor and abductor strength were both reduced in KOA patients. But to date, most of the researches have only focused on quadriceps combined with hip abductor strengthening versus quadriceps strengthening. OBJECTIVE: The aim of the study is to evaluate the effect of adding hip abductor and adductor strengthening to quadriceps strengthening on lower limb strength, knee pain and physical function in patients with medial compartmental knee osteoarthritis. METHODS: In this study, 42 participants, were randomly divided into two groups: the general treatment group (GT group) and the added-hip-exercise group (AH group). All participants were given a general rehabilitation treatment. The AH group performed hip abductor and adductor strengthening in addition to the general rehabilitation treatment. Knee and hip muscle strength, Five Times Sit-to-Stand Test (FTSST), the Timed Up and Go Test (TUGT), Numerical Rating Scale (NRS), and Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) scores were assessed at baseline and 6 weeks. A two-sided 2-sample unpaired t test was performed to compare the difference in mean change scores between AH and GT groups. RESULTS: Finally, 36 participants completed the study: both groups consist of 18 participants. In the per-protocol analysis, the AH group had a greater improvement in knee extension strength (mean changes, 7.84 versus 36.48; P < 0.001) and hip abduction strength (mean changes, 5.05 versus 26.62; P = 0.001) than the control group. Similarly, the AH group had a greater improvement in the FTSST time (mean changes, 0.40 s versus 3.57 s; P < 0.001) and the TUFT time (mean changes, 0.18 s versus 1.67 s; P = 0.002) than the GH group. No statistical difference was found in the change of WOMAC pain scores and NRS between the 2 groups. CONCLUSIONS: Older adults with knee OA in the AH group had superior muscle strength, symptoms and daily activity performance at the 6th week than those in the GT group. And adding hip exercises could expedite improvement of pain at the 2th week, but not at the 6th week. TRIAL REGISTRATION: Clinical trial registration numbers and date of registration: ChiCTR-IOR-16009124, Registered 30 August 2016.

Case report: A de novo ERBB3 mutation develops in a gallbladder cancer patient carrying BRCA1 mutation after effective treatment with olaparib.

Background: Gallbladder cancer (GBC) is highly lethal and resistant to most chemotherapeutic drugs. GBC was reported to carry multiple genetic mutations such as TP53, K-RAS, and ERBB2/3. Here, we unexpectedly identified a patient with GBC harboring germline BRCA1 p.Arg1325Lys heterozygous mutation. We sought to determine if olaparib, the poly ADP-ribose polymerase inhibitor (PARPi) commonly treated for BRCA mutation, can inhibit cancer development via a therapeutic trial on this patient. Case presentation: The patient received GBC R0 resection after an 8-week olaparib treatment. After surgery and 6-month follow-up treatment with olaparib, the patient's blood carbohydrate antigen 19-9 (CA19-9) level declined from 328 to 23.6 U/ml. No recurrence in CT scanning was observed, indicating a disease-free survival of 6 months with conventional therapy. Two months later, CT examination and CA19-9 level showed cancer relapse. A blood biopsy revealed a new ERBB3 p.Gly337Arg mutation. GBC cell lines ectopically expressing BRCA1 p.Arg1325Lys together with ERBB3 p.Gly337Arg mutations were challenged with olaparib and/or afatinib, an ERBB2/3 inhibitor. The dual mutation cells were more responsive to the combined olaparib with afatinib than a single drug in the cell proliferation assay. Conclusion: Olaparib is effective in a GBC patient with a BRAC1 mutation. The efficacy of olaparib and afatinib in both cultured BRAC1 and ERBB3 mutation cell lines suggests that a combined regimen targeting BRCA1/2 and ERBB2/3 mutations may be an optimal strategy to treat GBC patients who carry both gene mutations.

Improved long-term outcomes after innovative preoperative evaluation and conception of precise surgery for gallbladder cancer.

BACKGROUND: Three-dimensional visualization preoperative evaluation (3D-VPE) and enhanced recovery after surgery (ERAS) have been suggested to improve outcomes of cancer surgery in patients, yet little is known regarding their clinical benefit in patients with gallbladder cancer (GBC). We hypothesized that the combination of 3D-VPE and ERAS would improve the outcome of patients undergoing surgery for GBC. OBJECTIVE: This study aimed to determine if 3D-VPE and ERAS can improve the outcomes and overall survival in patients with GBC, establishing a novel patient management strategy for GBC. METHODS: A total of 227 patients with GBC were recruited and divided into two groups: those who received traditional treatment between January 2000 and December 2010 (n = 86; the control group) and those who underwent 3D-VPE and ERAS between January 2011 and December 2017 (n = 141). Univariate and multivariate analyses were employed to assess the relationship among disease stages, lymph node invasion, and cell differentiation between the two groups. Cox regression analysis was used to investigate patient survival in these groups. RESULTS: Patients who underwent 3D-VPE and ERAS showed a significantly higher R0 resection rate (67.4% vs. 20.9%, p < 0.001) and dissected lymph node number (26.6 ± 12.6 vs. 16.3 ± 7.6 p < 0.001) compared to the control group. The median survival was 27.4 months, and the 1- and 3-year survival rates were 84.4% and 29.8%, respectively, in patients who received combined management; in the control cohort, the median survival was 12.7 months, and the 1- and 3-year survival rates were 53.5% and 15.1%, respectively. In addition, some postoperative complications and risk factors were diminished relative to the traditionally treated patients. CONCLUSION: The implementation of 3D-VPE and ERAS can significantly improve the prognosis and outcomes of patients with GBC and should be considered for wide use in clinical practice.

Radial and spiral stream formation in Proteus mirabilis colonies.

The enteric bacterium Proteus mirabilis, which is a pathogen that forms biofilms in vivo, can swarm over hard surfaces and form a variety of spatial patterns in colonies. Colony formation involves two distinct cell types: swarmer cells that dominate near the surface and the leading edge, and swimmer cells that prefer a less viscous medium, but the mechanisms underlying pattern formation are not understood. New experimental investigations reported here show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to previously-reported concentric rings. These new observations suggest that swimmers are motile and that indirect interactions between them are essential in the pattern formation. To explain these observations we develop a hybrid model comprising cell-based and continuum components that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams results from a swimming bias of the cells near the surface of the substrate. The spatial patterns generated from the model are in qualitative agreement with the experimental observations.

Propagation of cutaneous thermal injury: a mathematical model.

Cutaneous burn wounds represent a significant public health problem with 500,000 patients per year in the USA seeking medical attention. Immediately after skin burn injury, the volume of the wound burn expands due to a cascade of chemical reactions, including lipid peroxidation chain reactions. Such expansion threatens life and is therefore highly clinically significant. Based on these chemical reactions, the present paper develops for the first time a three-dimensional mathematical model to quantify the propagation of tissue damage within 12 hours post initial burn. We use the model to investigate the effect of supplemental antioxidant vitamin E for intercepting propagation. We show, for example, that if tissue levels of vitamin E tocotrienol are increased, postburn, by five times then this would slow down the lipid peroxide propagation by at least 50%. We chose the alpha-tocotrienol form of vitamin E as it is a potent inhibitor of 12-lipoxygenase, which is known to propagate oxidative lipid damage. Our model is formulated in terms of differential equations, and sensitivity analysis is performed on the parameters to ensure the robustness of the results.

A mathematical model of ischemic cutaneous wounds.

Chronic wounds represent a major public health problem affecting 6.5 million people in the United States. Ischemia, primarily caused by peripheral artery diseases, represents a major complicating factor in cutaneous wound healing. In this work, we sought to develop a mathematical model of ischemic dermal wounds. The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The extracellular matrix (ECM) is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary. The model equations involve the concentration of oxygen, PDGF and VEGF, the densities of macrophages, fibroblasts, capillary tips and sprouts, and the density and velocity of the ECM. Simulations of the model demonstrate how ischemic conditions may limit macrophage recruitment to the wound-site and impair wound closure. The results are in general agreement with experimental findings.