Toward a Mathematical Description of Bone Biology: The Principle of Cellular Accommodation

Mathematical theories for bone biology or more specifically, bone mass regulation, should be viewed with considerable interest because they provide powerful tools for prediction of bone mass changes in response to mechanical or humeral stimuli. Frost [1] put forward one such theory when he postulated that bone mass is a controlled mechanical feedback system called the ``mechanostat.'' 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 that 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 cells react strongly to transients in their environment, but eventually ``accommodate'' 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. Importantly, the cellular accommodation theory can have the property of ``path dependence,'' meaning that final bone mass will be dependent upon the temporal sequence of preceding mechanical loading/hormonal events. Bone tissue demonstrates path dependence in its responses to mechanical loading and anabolic agents. Theoretically, it is possible to exploit the nonlinear character of path dependence to maximize the osteogenic effect of various therapeutic regimens. An experimental approach to test this possibility is described.