Toward Predicting the Spatio-Temporal Dynamics of Alopecia Areata Lesions Using Partial Differential Equation Analysis

Hair loss in the autoimmune disease, alopecia areata (AA), is characterized by the appearance of circularly spreading alopecic lesions in seemingly healthy skin. The distinct spatial patterns of AA lesions form because the immune system attacks hair follicle cells that are in the process of producing hair shaft, catapults the mini-organs that produce hair from a state of growth (anagen) into an apoptosis-driven regression state (catagen), and causes major hair follicle dystrophy along with rapid hair shaft shedding. In this paper, we develop a model of partial differential equations (PDEs) to describe the spatio-temporal dynamics of immune system components that clinical and experimental studies show are primarily involved in the disease development. Global linear stability analysis reveals there is a most unstable mode giving rise to a pattern. The most unstable mode indicates a spatial scale consistent with results of the humanized AA mouse model of Gilhar et al. (Autoimmun Rev 15(7):726–735, 2016) for experimentally induced AA lesions. Numerical simulations of the PDE system confirm our analytic findings and illustrate the formation of a pattern that is characteristic of the spatio-temporal AA dynamics. We apply marginal linear stability analysis to examine and predict the pattern propagation.