We consider how interfaces can influence the early stages of a new phase forming in an existing phase, e.g., diseased or cancerous cells attempting to form in a region of healthy cells, or misfolded protein molecules forming amyloid deposits. We suggest a model where diseased cells can be suppressed by suitable interplay between the kinetics of disease formation, apoptosis and replacement cell creation. Maximising the energetic cost of the interface between cancerous and healthy tissue may be important for inhibiting metastasis, as small clusters attempt to escape from the surface of the parent tumour. Anything that can stabilise the interface between the tumour and healthy tissue, and so inhibit escaping cells, is useful Interfaces associated with neurodegenerative illness form when layers of misfolded protein molecules are deposited around cells. Particular nucleation sites might catalyse or 'seed' misfolding of proteins, or stabilise misfolded conformations, to begin formation of insoluble protein layers. Therapeutic drugs could suppress dangerous deposits of misfolded proteins by binding to, and hence blocking, these sites. Numbers of potential nucleation sites might be reduced if stressed cells are eliminated through apoptosis: it is likely better to remove a few stressed cells than have them remain as nucleation sites for deposits which can grow and damage many cells. Finding compromises that diminish potential nucleation sites without excessive destruction of functioning cells will become more difficult as reserves of healthy cells diminish with increasing age. There is a natural arrow of time at work, and an inverse relation with cancer. Particular combinations of interfacial mobility and deposition geometry may result in amyloid plaques that vary in density and which are thus more or less resistant to being removed by natural processes or medical interventions.