Mathematical-based morphological classification of skin eruptions corresponding to the pathophysiological state of chronic spontaneous urticaria

The study addresses the challenge that CSU, a human-specific skin disorder characterized by recurrent wheals, lacks an animal model and in vivo pathophysiological clarity. The research question is whether the geometric morphology of wheals can be mathematically linked to underlying pathophysiological networks to classify patients and infer their disease states. By treating visible eruption geometry as a manifestation of dermal molecular dynamics, the authors aim to connect wheal patterns to key players such as histamine (from mast cells and basophils), tissue factor (TF), and coagulation factors, to improve diagnosis and guide patient-specific therapy.

The paper outlines known aspects of CSU pathophysiology, including roles of autoimmunity, cellular infiltrates, coagulation, and complement systems, while noting gaps in understanding due to limited in vivo data and absence of suitable animal models. Prior work suggested histamine’s dual role in TF induction and vascular permeability, and an earlier conceptual mathematical model implicated inhibitory regulation of mast cell histamine release in pattern formation, though lacking experimental validation. The current study builds on these insights by integrating experimental data into a hierarchical mathematical model and formal clinical classification criteria.

• Clinical image collection and classification: Images of wheals from 105 CSU patients at Hiroshima University Hospital (2000–2022; IRB E2020-2388) were analyzed. Six dermatologists classified wheal morphology into annular, broken-annular, geographic, circular, dot, uniform, or NA; uniform and NA were excluded from final analysis, yielding five types across two classes (boundary vs area) per EGe criteria. Vote-counting determined reliability. Sample size was adequate for error <8% at α=0.05.
• Statistical metrics: Defined SB = (# doctors selecting the EGe type)/(6). Defined SS = p(A|B) as the joint reliability of correct class and type selections. Calculated P-values for classifiability via binomial sums comparing classifiable rate against hypothesized probabilities.
• Eruption state function: Defined S([A]) = 1/(1+exp(−β1([A]−[A]t))) mapping dermal concentrations of coagulation factors or histamine to visible eruption presence on the skin (binary state).
• Mathematical model construction: Treated dermis as a 2D sheet with homogeneous microvasculature. Modeled key variables: [HB] (histamine from basophils), [TF] (tissue factor on endothelium), [C] (leaked coagulation factors), [HM] (histamine from mast cells) with ODE/PDE system: d[HB]/dt = δB + γB[TF]X(1−g_inhib^B([HB])) − μB[HB]; d[TF]/dt = δT + γactivation([HB],[HM])(1−g_inhib^T([HB],[HM])) − μT[TF]; ∂[C]/∂t = DC∇²C + f_leakage(x,t) − μC[C]; ∂[HM]/∂t = DM∇²[HM] + δM + γM[C]X(1−g_inhib^M([HM])) − μM[HM]. Heaviside X indicates degranulation state. Inhibitory/activation functions were quantitatively fit from in vitro data.
• In vitro experiments and function estimation: • Adenosine-mediated inhibition of histamine release from human skin mast cells and human peripheral basophils used to fit g_inhib^B([HB]) and g_inhib^M([HM]). • Adenosine inhibition of TF mRNA in HUVECs used to fit g_inhib^T([HB],[HM]). • Histamine±LPS induction of TF mRNA in HUVECs analyzed to estimate f_activation([H],[HM]). • TF-induced endothelial gap formation measured via impedance (iCElligence) to infer switch-like leakage function; observed threshold between 1–3 ng/mL TF; derived f_leakage(x,t) as a switch-like function of [TF].
• Numerical simulations: Generated in silico eruption patterns; compared time-point snapshots to patient images.
• Sensitivity analysis: Applied Extended Fourier Amplitude Sensitivity Test (eFAST) to link five key geometric features (KF1–KF5) to network parameters for TF and mast cell histamine dynamics. Defined sensitivity functions per feature and summarized parameter effects.

• The hierarchical model reproduced five clinically observed wheal patterns: annular, broken-annular (boundary class), and geographic, circular, dot (area class).
• Clinical classification: Among 105 CSU patients, 87.6% of samples were classifiable into one of the five EGe types with SB=100% (full agreement among six dermatologists); 0% were unclassifiable at SB>83%. Using stricter SS reliability (>83%), patterns remained robust, with area patterns (geographic, circular) more frequent than boundary patterns.
• Network signatures by pattern: Sensitivity analysis indicated that TF dynamics predominantly determine boundary vs area distinction—TF decay strongly affected boundary features (KF1), whereas TF activation parameters affected area features (KF3). The divergence between annular and broken-annular patterns was linked to the state of spontaneous histamine release from mast cells. For area patterns, strong involvement of mast cell histamine activation/inhibition drove geographic/circular versus dot outcomes.
• Endothelial gap formation exhibited a TF concentration threshold (~1–3 ng/mL) producing switch-like leakage of coagulation factors, supporting a mechanism for wheal onset only above a stimulus threshold.
• The approach achieved high concordance between in silico and patient eruption geometries, enabling inference of patient pathophysiological states from wheal morphology.

The study demonstrates that wheal geometry can serve as a proxy for underlying dermal molecular dynamics in CSU. By integrating experimental data into a validated mathematical model and establishing EGe criteria, the authors map observable eruption patterns to specific network states, notably the balance between TF activity and mast cell histamine dynamics. This addresses the core question of inferring pathophysiology from morphology and has practical implications: pattern-based insights may predict responsiveness to treatments (e.g., antihistamines versus anticoagulants/protease inhibitors) and guide personalized management. The TF-related determination of boundary/area patterns and mast cell histamine’s role in differentiating within-class types explain heterogeneity in clinical presentations and therapeutic responses. The framework also suggests measurable biomarkers (e.g., D-dimer, PF1+2) to validate predicted coagulation involvement across pattern types.

The work introduces a multidisciplinary pipeline linking wheal morphology to CSU pathophysiology via a hierarchical, experimentally grounded mathematical model and clinically applicable EGe criteria. It enables classification into five eruption types with high inter-rater reliability and connects these types to distinct network signatures (TF activity and mast cell histamine dynamics), supporting patient-specific diagnostic and therapeutic strategies. Future research should: (1) incorporate temporal data to model the disappearance phase; (2) validate predictions with independent clinical cohorts and biomarker assays (e.g., coagulation markers); and (3) explore generalization to other dermatoses with geometric lesions (e.g., psoriasis, tinea corporis, erythema multiforme, autoimmune bullous diseases).

• The model focuses on emergence and development phases; the disappearance phase is not modeled and may involve additional networks.
• Built primarily on in vitro data; requires validation with independent clinical datasets and longitudinal, time-lapse imaging to resolve patterns that are difficult to distinguish in snapshot images.
• Histamine’s early role in TF induction may be partially substitutable by other factors (e.g., LPS, cytokines), potentially affecting generalizability of antihistamine effects across patients.
• Although classification reliability was high, a subset required stricter SS thresholds, and uniform/NA images were excluded from final analysis.