Fig 1.
Modeling of the expansion of erythema.
(A) Process of the inflammatory response for erythema development. When keratinocytes in the epidermis and resident immune cells in the dermis are stimulated (i), they secrete inflammatory mediators that induce their own production from these mediator-secreting cells (ii). The mediators diffuse in the dermis and cause the dilation of local blood vessels (iii). The dilation appears as redness on the skin surface, forming erythema (iv). (B, C) Photographs of erythema expansion showing well-circumscribed lesion (B) and poorly circumscribed lesion (C) of mRNA COVID-19 vaccine. (D) Range (surrounded by black solid line) of a and b such that the model (Eq 3, d = 0) exhibits bistability. (E) Kinetics of bistability in Eq (3) represented by the production rate of mediators () as a function of the concentration (q). Two filled circles represent stable steady states (SNI, SI), whereas the hollow circle indicates an unstable steady state (ST). This system can switch between the stable states of low (SNI)- and high (SI)- concentration depending on the perturbation, such as initial stimulation or diffused mediators (a = 2.14, b = 0.05).
Fig 2.
Diffusion and bistability can cause expansion via the traveling wave.
(A, E, F, J) Spatiotemporal evolution of inflammatory mediator levels (q; inset at the left) upon initial stimulation in a circular area (A; a = 4.0 and F; a = 2.5) and in three separate areas (E; a = 4.0 and J; a = 2.5). (B, G) Spatial pattern of mediator levels at three different time points, (C, H) spatiotemporal evolution in the inflamed area, and (D, I) temporal evolution of the diameter of the inflamed area (above the unstable steady state ST, q = 0.26 for D, q = 0.69 for I; red), at y = 100 in A (B–D) and F (G–I) (dashed line in the left panel). b = 0.01, d = 0.5 in A–J. The obtained results were almost the same for Δt = 0.05 and Δt = 0.1 in D.
Fig 3.
Expansion velocity is controlled by diffusion and positive feedback for given bistability.
(A–C) Dependence of the expansion velocity on the diffusion coefficient (d; A), maximum production rate (a; B), and basal secretion rate (b; C). (D, E, J) Spatiotemporal changes in the inflamed area, which is above the unstable steady state ST (D) q = 0.55; © q = 0.63; (J) q = 0.72, respectively, for three different values of maximum production rate indicated in B. (F) Simulation results were superimposed on the theoretically calculated range of bistability (surrounded by black solid line) shown in Fig 1D. Symbols represent the expansion (orange circles) and shrinkage (blue diamonds). The dashed line represents the theoretical velocity of zero (v = 0) calculated from Eq 5. d = 0.5. (G–I) Dependence of the spatial pattern on the diffusion coefficient (d; G), maximum production rate (a; H), and basal secretion rate (b; I). a = 4, b = 0.01 in A, G. b = 0.05, d = 0.5 in B, D, E, H, J. a = 2, d = 0.5 in C, I.
Fig 4.
Balance of mediator concentration regulates expansion or shrinkage.
(A) Range of a and b for the velocity v > 0 (calculated from Eq 5), indicating the expansion (orange), v < 0, indicating the shrinkage (light blue), and v = 0 (dashed line). d = 0.5. (B–D) Kinetics of bistability in Eq (3) represented by the production rate of mediators () as a function of the concentration (q) for a = 2.14 (B), a = 2.04 (C), a = 1.96 (D). b = 0.05 in B–D. Two filled circles (SNI, SI) represent stable steady states, whereas the hollow circle (ST) indicates an unstable steady state.
Fig 5.
Traveling wave of inflammatory response regulates the expansion and shrinkage of erythema.