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This work is based on a previous model of Z. Agur, Y. Daniel and Y. Ginosar (2002), retrieving the essential properties of homeostatic tissue development, as reflected by the bone marrow. The original model, represented by cellular automata on a connected, locally finite undirected graph, identifies the minimal basic properties essential for maintaining tissue homeostasis and for guaranteeing the ability of a few stem cells to repopulate the tissue following its depletion. However, this model is too general to ensure a relative “stability” of stem cell numbers in the tissue, a prerequisite for the integrity of biological systems. In the present work, some natural limitations on the model are introduced, under which a formula for the state of a given cell at any given time is obtained, as well as for the proportion of stem cells as a function of model parameters. For tube-like graphs, defined for modeling tissue engineering scaffolds and known tumor geometries, the system obtains a fixed cellular composition, interpreted as homeostasis, thus enabling precise calculation of the necessary conditions for tissue reconstruction. These results also can shed light on conditions for disrupting homeostasis in cancerous tissues.

Resilience in stem cell renewal: development of the Agur--Daniel--Ginosar model