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Epidermal wound healing requires the coordinated involvement of complex cellular and biochemical processes. In the case of epidermal wounds associated with burns, the healing process may be less than optimal and may take a significant amount of time, possibly resulting in infection and scarring. An innovative method to assist in the repair of the epidermis (the outer layer of skin) is to use an aerosolised apparatus. This method involves taking skin cells from an area of the patient's undamaged skin, culturing the cells in a laboratory, encouraging them to rapidly proliferate, then harvesting and separating the cells from each other. The cells are then sprayed onto the wound surface. We investigate this novel treatment strategy for the healing of epidermal wounds, such as burns. In particular, we model the application of viable cell colonies to the exposed surface of the wound with the intent of identifying key factors that govern the healing process. Details of the evolution of the colony structure are explored in this two-dimensional model of the wound site, including the effect of varying the initial population cluster size and the initial distribution of cell types with different proliferative capacities. During injury, holoclones (which are thought to be stem cells) have a large proliferative capacity while paraclones (which are thought to be transient amplifying cells) have a more limited proliferative capacity. The model predicts the coverage over time for cells that are initially sprayed onto a wound. A detailed analysis of the underlying mathematical models yields novel mathematical results as well as insight into phenomena of healing processes under investigation. Two one-dimensional systems that are simplifications of the full model are investigated. These models are significant extensions of Fisher's equation and incorporate the mixed clonal population of quiescent and active cells. In the first model, an active cell type migrates and proliferates into the wound and undergoes a transition to a quiescent cell type that neither migrates nor proliferates. The analysis yields the identification of the key parameter constraints on the speed of the healing front of the cells on this model and hence the rate of healing of epidermal wounds. Approximations for the maximum cell densities are also obtained, including conditions for a less than optimal final state. The second model involves two active cell types with different proliferative capacity and a quiescent cell type. This model exhibits two distinct behaviours: either both cell types coexist or one of them dies out as the wound healing progresses leaving the other cell type to fill the wound space. Conditions for coexistence are explored.