Fig 1.
Histamine dynamics and expanding speed of wheals in experiments vs wheals observed in patients with chronic spontaneous urticaria.
(A) The process of urticarial development. Dermal mast cells are stimulated and degranulate, releasing mediators including histamine (a, b). Released histamine acts on vascular endothelial cells and sensory neurons to induce the formation of intercellular gaps (c) and the release of neuropeptides which activate mast cells in the vicinity (d). The blood plasma exudates through the gap and wheals develop (e). (B) The upper left panel shows photographic images of a wheal induced by the intradermal injection of histamine and the lower left panel shows wheals observed in a patient with chronic spontaneous urticaria (CSU) over a time course. Right panels show the evolution of wheals in terms of their size, showing that wheals in CSU expand much slower than those induced by a bolus injection of histamine. (C) Comparison of the radial expansion velocities of wheals that developed in patients with CSU and of those induced by a bolus intradermal injection of histamine. The maximum radial expanding velocity of wheals induced by intradermal injections of histamine was calculated as the average velocity during the initial 15 minutes after the injection, and that in CSU was calculated as the fastest of the average velocity during the observational time of two adjacent points. The radial expanding velocity of histamine-induced wheals at the indicated concentrations was at least 7 times greater than that of wheals developed in CSU.
Fig 2.
Model assumptions and simulation result.
(A, B) Self-activation and self-inhibition loops of histamine via a mast cell, and the detailed functions of the loops. The example graphs are drawn for γ = 0.008, α1 = 0.4, α2 = 5.0 in model (1). (C) A representative simulation result of the model (1) for wheals and histamine. The parameter is assigned to Table 1 (Set I). The initial condition is shown in the figure at t = 0 for histamine. (D-G) Parameter spaces for wheal development with respect to Pmin are plotted by the color bar. Uniform development is the region where wheals are uniformly developed in the area corresponding to no pattern state. Fine-tuning is the region where wheals developed as very fine dots with diameters less than 3 mm and were widely spread over the area corresponding to eruptions shown in Fig 3. q0, , qμ, qγ, qF, qU are labels of parameter values chosen in the simulation results in (Fig 3A–3F).
Fig 3.
Multifarious eruptions of urticaria and simulation results.
(A-F) Patterns of representative urticaria eruptions developed in patients (lower or right panels), and patterns simulated (upper or left panels) by parameters notated by (q0), ,
,
, (qμ), (qγ), (qF), (qU) in (Fig 2D–2G) were assigned to similar eruptions developed in real patients with urticaria. The right-hand side figure in C was generated with a stronger initial stimulus than that in the left-hand side figure. The time of simulation result is notated in each figure. (G) Wheal patterns for the increase in the strength of the stimulus (Pr). The density of wheals is notably increased. (H) Wheal patterns are shown with respect to two different weight functions of ψ(x) with the fixed Pr. α2 = 5.0,u* = 0.181, and the other parameters chosen were listed in Table 1 (Set I) for (G) and (H).
Fig 4.
Comparison of the expanding speed of wheals developed by intradermal injection with that in urticaria using the mathematical model.
(A) The expanding speed approximated by the Eq (4) for the injection of 10 μg/ml histamine. The maximal speed is determined in the initial stage. (B) Expanding radius of the wheal in the model (1). The parameters chosen were listed in Table 1 (Set I) (C) The expanding speed calculated from (B).
Table 1.
The representative parameter values used in simulations.