Mathematics and Alopecia Areata

By Nick G. Cogan and Atanaska Dobreva

Alopecia areata (AA) is an autoimmune disease that is characterized by distinctive, patterned hair loss. This condition has no cure, and only very limited treatments are available — none of which are effective for all patients or throughout the entire disease process [9]. Therefore, disease presentation, disease progression, and patient immune composition guide treatment plans. This arrangement presents a challenge for both practitioners and patients, since treatments that were effective during a previous flare-up may not work for a subsequent one. Furthermore, because this condition is typified by visible symptoms, it is often associated with long-term, psychological stress [5, 6].

There are many reasons for scientists to study this condition beyond the obvious search for a cure. For instance, AA lends insight into how one might possibly control the immune system [5, 7]. Researchers believe that a collapse of immune privilege within hair follicles causes AA, meaning that the immune system (which is typically suppressed by local signals) is allowed to disrupt normal hair follicle dynamics. Understanding the mechanism of suppression—or immune privilege guardianship—will help develop anti-autoimmune disease treatments, lower the risks of transplant rejection, and inform overall understanding of immunological tolerance.

We have developed and analyzed the first mathematical models for AA (see Figure 1). First, we created an ordinary differential equation (ODE) model that neglected the hair follicle cycle [2] and instead focused on the interactions between the key immune AA drivers: CD8+ T-cells, CD4+ T-cells, the pro-inflammatory cytokine interferon-gamma (IFN\(\gamma\)), and immunosuppressive agents called immune privilege guardians (IPG). Cytokine signaling via IFN\(\gamma\) leads to coordinated responses from the T-cell community, which the IPG signal normally suppresses.

Our model can reflect health, disease, and treatment states by varying the level of IPG suppression. Immune cells are at a basal level in the healthy state and reach very high levels in the disease state (in response to elevated production of IFN\(\gamma\)). During the treatment state, AA is suppressed as immunocyte populations diminish in response to increased synthesis of strong IPG, which represents a potential treatment target. Sensitivity analysis using Partial Rank Correlation Coefficients revealed that lymphocyte levels are most sensitive to changes in the parameters that control IFN\(\gamma\) and IPG. These findings agree with the autoimmune hypothesis for AA development [9].

Next, we linked our immune model to hair cycle equations [3]. The coupled model exhibits states of health, disease, and treatment, as well as the transitions between states; it thus represents the hair growth disruption that occurs in response to the autoimmune reaction. While healthy follicles have a hair growth phase with normal length, the disease profile is manifested through a very short growth phase as the immune cell populations expand in response to elevated IFN\(\gamma\). In the treatment state, a sufficiently high amount of IPG suppresses AA and restores the normal duration of hair growth. 

This understanding has led to insights that were not previously accessible, namely that the AA autoimmune response alters the importance of intrinsic hair growth processes. Analyzing the model with Sobol sensitivity measures showed that although some processes impact hair growth duration in both healthy and diseased follicles, mechanisms also exist with strong effects in disease but not health (and vice versa) [3]. Ongoing research is considering these findings in the context of new potential therapeutic targets [8].

Extending the core ODE model to include spatial dynamics and investigate typical spatial patterning yielded a partial differential equation model that describes the spatiotemporal dynamics of the key immune components [4]. Immune cells produce IFN\(\gamma\), which then diffuses and degrades. Immunocyte dynamics comprise activation by IFN\(\gamma\)and proliferation—both of which are suppressed by IPG—as well as death. The cells also exhibit random motion and motion that is directed up the concentration gradient in IFN\(\gamma\). Results from numerical simulation and global linear stability analysis illustrate the development of a pattern that is characteristic of AA, with spatial and temporal scales that are consistent with those in experimental findings [5]. By applying marginal linear stability analysis, we discovered that the pattern spreads through a larger domain with a uniform velocity in the short term. This finding suggests that an emerging hairless patch during disease onset will enlarge at a constant rate [4].

After completing these investigations, we were able to ask questions at a therapeutic level [1]. We had broad understanding of AA dynamics under the assumption that AA is caused by the collapse of the immune privilege suppression. This collapse leads to elevated expression of macrophages that disrupt hair follicle dynamics, resulting in hair loss that can cause distinctive patterns through a combination of cytokine diffusion and directed T-cell motility. Because of the time scales of the dynamics (e.g., a diffusional time scale in seconds and hair follicle cycling in days), one must be able to connect observations with model parameters.

Our sensitivity analysis indicates that some processes are more sensitive than others, but a nominal parameter regime is necessary to rank the sensitivities. It is impossible to obtain spatiotemporal data for the immune system—especially during a typical dermatological office visit—so we hypothesized that we could use spatial patterns to estimate the parameters via a Bayesian framework that is based on data assimilation techniques. This method utilizes observational data in conjunction with Bayes’ law to obtain the prior parameter distribution (e.g., the parameter statistics that allow the best fit to the data). In our most recent publication, we focused on three main questions that relate to the parameter estimation and observational data [1]. In short, how is Bayesian filtering affected by:

1. The sensitivity of a parameter? In the context of biological modeling, sensitive parameters are often the most “difficult” to deal with because they require more precise estimates to control the uncertainty of model predictions. Is the same true for the parameter estimation problem? 
2. The noise in the observations? Physicians collect data during office visits using calipers, which are inherently variable. Which processes are most affected by noisy observations?
3. The timing of the observations? The caliper data is collected during routine office visits, which are often irregularly spaced and typically several months apart. How does this timeline affect our parameter estimates?

We used synthetic data by selecting known parameters and solving the forward problem to determine the T-cell and cytokine dynamics, then adding noise. We could then vary the level of added noise and the timing of observations.

We found that estimates of sensitive parameters converged more quickly than estimates for insensitive parameters. While this was surprising to us, it is actually a well-known effect in the data assimilation literature. Because the model fits will be much worse if errors are present in sensitive parameters, inverse problems penalize errors in these estimates much more than errors in insensitive parameters. Therefore, to determine a patient’s disease state, the most important parameters to estimate are those whose estimates converge the quickest. However, the same argument explains our observation that noise has a much stronger effect on sensitive parameters, leading to less accurate estimates for sensitive parameters as the signal-to-noise ratio decreases.

Finally, we found that a similar process indicates that less frequent observations lead to faster convergence of our estimates. This results from a similar mechanism as described above; if we gather too many measurements too close together, the observational data lacks sufficient variation to provide information for the filtering algorithm.

Ultimately, this methodology suggests that it may be possible to design observational studies that can inform our mathematical model, which can then identify the current state of AA within a particular patient. This technique may provide a path to personalized medical treatment and care for patients with AA. More broadly, we believe that this approach can help guide clinicians during the course of a variety of diseases, thereby increasing treatment specificity and bolstering the possibility of dynamically tailored treatments. 

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Nick Cogan presented this research during a minisymposium presentation at the 2021 SIAM Annual Meeting, which took place virtually in July 2021.

References 
[1] Cogan, N.G., Bao, F., Paus, R., & Dobreva, A. (2021). Data assimilation of synthetic data as a novel strategy for predicting disease progression in alopecia areata. Math. Med. Biol., 38(3), 314-332. 
[2] Dobreva, A., Paus, R., & Cogan, N.G. (2015). Mathematical model for alopecia areata. J. Theor. Biol., 380, 332-345.
[3] Dobreva, A., Paus, R., & Cogan, N.G. (2018). Analysing the dynamics of a model for alopecia areata as an autoimmune disorder of hair follicle cycling. Math. Med. Biol., 35(3), 387-407.
[4] Dobreva, A., Paus, R., & Cogan, N.G. (2020). Toward predicting the spatio-temporal dynamics of alopecia areata lesions using partial differential equation analysis. Bull. Math. Biol., 82(3). 
[5] Gilhar, A., Etzioni, A., & Paus, R. (2012). Alopecia areata. N. Engl. J. Med., 366, 1515-1525.
[6] Gilhar, A., Laufer-Britva, R., Keren, A., & Paus, R. (2019). Frontiers in alopecia areata pathobiology research. J. Allergy Clin. Immunol., 144(6), 1478-1489
[7] McElwee, K.J., Gilhar, A., Tobin, D.J., Ramot, Y., Sundberg, J.P., Nakamura, M., …, Paus, R. (2013). What causes alopecia areata? Exp. Dermatol., 22(9), 609-626.
[8] Nicu, C., Wikramanayake, T.C., & Paus, R. (2020). Clues that mitochondria are involved in the hair cycle clock: MPZL3 regulates entry into and progression of murine hair follicle cycling. Exp. Dermatol., 29(12), 1243-1249.
[9] Paus, R., Slominski, A., & Czarnetzki, B.M. (1993). Is alopecia areata an autoimmune-response against melanogenesis-related proteins, exposed by abnormal MHC class I expression in the anagen hair bulb? Yale J. Biol. Med., 66(6), 541-554.
[10] Samuel, A.V., Muthu, M.S., Gurunathan, D., & Sharma, A. (2012). Alopecia areata of dental origin in a child. Indian J. Dent. Res., 23(5), 665-669.

Nick G. Cogan is a mathematician at Florida State University. His research focuses on microbial dynamics, biofluids, disease modeling with sensitivity analysis, optimal control, and numerical methods. Atanaska Dobreva is a postdoctoral researcher at Arizona State University. She earned her Ph.D. in mathematics from Florida State University. Dobreva's research applies a variety of mathematical methods—such as modeling, parameter estimation, uncertainty quantification, bifurcation, and sensitivity analysis—to investigate processes that are involved in retinal metabolism and antioxidant defenses, autoimmunity, immune privilege, and immune-cardiovascular interactions.