A single reaction-diffusion equation for the multifarious eruptions of urticaria

Urticaria is a common skin disorder characterized by the rapid appearance and disappearance of local skin edema and flares with itching. It is characterized by various macroscopic skin eruptions unique to patients and/or subtypes of urticaria with respect to shape, size, color, and/or duration of eruptions. Nevertheless, the mechanism underlying multifarious eruptions in urticaria is largely unknown. The eruptions are believed to be evoked by histamine release from mast cells in the skin. However, the majority of visible characteristics of urticaria cannot be explained by a simple injection of histamine to the skin. To explain the multifarious eruptions of urticaria, we developed a single reaction-diffusion model suggesting the self-activation and self-inhibition regulation of histamine release from mast cells. Using the model, we found that various geometrical shapes of eruptions typically observed in patients can be explained by the model parameters and randomness or strength of the initial stimuli to mast cells. Furthermore, we verified that the wheal-expanding speed of urticaria, which is shown to be much smaller than that of the intradermal injection experimental system may be explained by our model and a simple diffusion equation. Our study suggests that the simple reaction-diffusion dynamics, including the independent self-activating and -inhibitory regulation of histamine release, may account for the essential mechanism underlying the formation of multifarious eruptions in urticaria.Author summary: Urticaria is a common skin disease but the mechanism underlying wheal formation is not well understood. Our mathematical model suggests that not only the self-activation of histamine production via mast cells, but also self-inhibition of histamine dynamics plays a critical role in generating the wide-spread wheal patterns observed in urticaria; this has not been previously considered in medicine. The study findings may increase the understanding of the pathogenesis of urticaria and may aid decision-making for appropriate treatments. It may also open an entirely new avenue for mathematical approaches to analyze various skin diseases with geometric eruptions and predict the effectiveness of treatments through in silico experiments.