Optimal Strategies for Pathological Epidermis: The Optimal Control of Basal Cell Dynamics

In normal epidermis, the basal layer has the goal of sustaining in a controlled way the epidermis renewal, producing non proliferating differentiated keratinocytes that migrate into the suprabasal zone. However, hyperproliferation and migration of proliferating cells into the supra-basal zone may occur in correspondence of epidermal disorders such as actinic keratosis and psoriasis. In Gandolfi et al. (J. Math. Biol. 73:1595–1626, 2016; SIAM J. Appl. Math. 80:543–571, 2020), we developed and studied an age-structured model of the cell populations dynamics in the basal layer, describing the normal epidermis state together with some pathological states related to hyper-proliferation. Here we approach the problem of controlling such situations through agents that, inducing cell’s death, reduce the effect of proliferation. An optimal control problem is formulated in connection with a cost function that aims to reduce the distance from the steady state of the normal epidermis. Existence of an optimal control is proved and necessary conditions of optimality are derived. This study was intended to continue the work on epidermis modeling done since so many years in collaboration with Alberto Gandolfi [3, 4] (Gandolfi et al., J. Math. Biol. 73:1595–1626, 2016; SIAM J. Appl. Math. 80:543–571, 2020). Indeed it started when Alberto was still with us, building the project. We planned to meet after the summer 2019 to continue working: we miss his contribution and his friendship.