Toward a mathematical description of bone biology: The principle of cellular accommodation

Mathematical theories for bone biology or more specifically, bone mass regu lation, should be viewed with considerable interest because they provide po werful tools for prediction of bone mass changes in response to mechanical or humeral stimuli. Frost [1] put forward one such theory when he postulate d that bone mass is a controlled mechanical feedback system called the "mec hanostat." He suggested that certain hormones and biochemical agents act on bone biology by changing the thresholds (or minimum effective strains) of the mechanostat. Critical examination of the mechanostat theory indicates t hat it does not conform well with certain experimental observations. In the present paper, a new theory is presented that addresses some of the flaws in the mechanostat. The new theory is based upon the assumption that bone c ells react strongly to transients in their environment, but eventually "acc ommodate" to steady state signals. This cellular accommodation, represented by a relaxation function, forms the basis for mathematical rate equations that describe bone mass changes in response to external stimuli. importantl y, the cellular accommodation theory can have the property of ''path depend ence," meaning that final bone mass will be dependent upon the temporal seq uence of preceding mechanical loading/hormonal events. Bone tissue demonstr ates path dependence in its responses to mechanical loading and anabolic ag ents, Theoretically, it is possible to exploit: the nonlinear character of path dependence to maximize the osteogenic effect of various therapeutic re gimens. An experimental approach to test this possibility is described.