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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, 2108 SAS Hall, 2311 Stinson Drive, Raleigh, NC, 27695, USA

    Atanaska Dobreva

  2. Dr. Phillip Frost Department of Dermatology and Cutaneous Surgery, University of Miami Miller School of Medicine, Miami, FL, 33136, USA

    Ralf Paus

  3. Centre for Dermatology Research, University of Manchester and NIHR Manchester Biomedical Research Centre, Manchester, UK

    Ralf Paus

  4. Monasterium Laboratory, Muenster, Germany

    Ralf Paus

  5. Department of Mathematics, Florida State University, 208 Love Building, 1017 Academic Way, Tallahassee, FL, 32306, USA

    N. G. Cogan

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  1. Atanaska Dobreva
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  2. Ralf Paus
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  3. N. G. Cogan
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Correspondence to Atanaska Dobreva.

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Atanaska Dobreva received support from the Research Training Group in Mathematical Biology, funded by a National Science Foundation Grant RTG/DMS—1246991. Ralf Paus was supported in part by departmental start-up funds. N. G. Cogan received support from NSF CBET 1510743.

Appendix

Appendix

See Table 2 and Figs. 12, 13, 14 and 15.

Table 2 Effects of parameter variations on the dispersion relation
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Fig. 12
figure 12

t—time (in days), x—space (the interval \([0 \,\mathrm{cm},10 \,\mathrm{cm}]\)). \(p_{\mathrm{IFN}}=0.115\), \(s=0.1\). Snapshots of the patterns’ spread at 10, 15, 20, 25, 30, 40 and 50 days

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Fig. 13
figure 13

t—time (in days), x—space (the interval \([0 \,\mathrm{cm},10 \,\mathrm{cm}]\)). \(p_{\mathrm{IFN}}=0.15\), \(s=0.1\). Snapshots of the patterns’ spread at 10, 15, 20, 25, 30, 40 and 50 days

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Fig. 14
figure 14

t—time (in days), x—space (the interval \([0 \,\mathrm{cm},10 \,\mathrm{cm}]\)). \(p_{\mathrm{IFN}}=0.115\), \(s=0.01\). Snapshots of the patterns’ spread at 10, 15, 20, 25, 30, 40 and 50 days

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Fig. 15
figure 15

t—time (in days), x—space (the interval \([0 \,\mathrm{cm},10 \,\mathrm{cm}]\)). \(t_{\mathrm{end}}=50\) days. a\(p_{\mathrm{IFN}}=0.115\), \(s=0.1\); b\(p_{\mathrm{IFN}}=0.15\), \(s=0.1\); c\(p_{\mathrm{IFN}}=0.115\), \(s=0.01\). In each case, the patterns underlying the behavior of T4 and T8 are similar to the pattern exhibited by IFN, i.e., places where the IFN concentration is elevated are also places where the levels of T4 and T8 are increased

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Dobreva, A., Paus, R. & Cogan, N.G. Toward Predicting the Spatio-Temporal Dynamics of Alopecia Areata Lesions Using Partial Differential Equation Analysis. Bull Math Biol 82, 34 (2020). https://doi.org/10.1007/s11538-020-00707-0

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