A Well-Balanced Positivity-Preserving Quasi-Lagrange Moving Mesh DG Method for the Shallow Water Equations

A high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method is presented for the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. The strategies in the use of slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties are discussed. Compared to rezoning-type methods, the current method treats mesh movement continuously in time and has the advantages that it does not need to interpolate flow variables from the old mesh to the new one and places no constraint for the choice of a update scheme for the bottom topography on the new mesh. A selection of one- and two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order accuracy of the method and its ability to adapt the mesh according to features in the flow and bottom topography.     

Mechanisms of repigmentation induced by photobiomodulation therapy in vitiligo

Photobiomodulation (PBM) therapy is based on the exposure of biological tissues to low‐level laser light (coherent light) or light‐emitting diodes (LEDs; noncoherent light), leading to the modulation of cellular functions, such as proliferation and migration, which result in tissue regeneration. PBM therapy has important clinical applications in regenerative medicine. Vitiligo is an acquired depigmentary disorder resulting from disappearance of functional melanocytes in the involved skin. Vitiligo repigmentation depends on available melanocytes derived from (a) melanocyte stem cells located in the bulge area of hair follicles and (b) the epidermis at the lesional borders, which contains a pool of functional melanocytes. Since follicular melanoblasts (MBs) are derived from the melanocyte stem cells residing at the bulge area of hair follicle, the process of vitiligo repigmentation presents a research model for studying the regenerative effect of PBM therapy. Previous reports have shown favourable response for treatment of vitiligo with a low‐energy helium‐neon (He‐Ne) laser. This review focuses on the molecular events that took place during the repigmentation process of vitiligo triggered by He‐Ne laser (632.8 nm, red light). Monochromatic radiation in the visible and infrared A (IRA) range sustains matrix metalloproteinase (MMP), improves mitochondrial function, and increases adenosine triphosphate (ATP) synthesis and O₂ consumption, which lead to cellular regenerative pathways. Cytochrome c oxidase in the mitochondria was reported to be the photoacceptor upon which He‐Ne laser exerts its effects. Mitochondrial retrograde signalling is responsible for the cellular events by red light. This review shows that He‐Ne laser initiated mitochondrial retrograde signalling via a Ca²⁺‐dependent cascade. The impact on cytochrome c oxidase within the mitochondria, an event that results in activation of CREB (cyclic‐AMP response element binding protein)‐related cascade, is responsible for the He‐Ne laser promoting functional development at different stages of MBs and boosting functional melanocytes. He‐Ne laser irradiation induced (a) melanocyte stem cell differentiation; (b) immature outer root sheath MB migration; (c) differentiated outer root sheath MB melanogenesis and migration; and (d) perilesional melanocyte migration and proliferation. These photobiomodulation effects result in perifollocular and marginal repigmentation in vitiligo.