Wolff's law: an 'MGS' derivation of Gamma in the Three-Way Rule for mechanically controlled lamellar bone modeling drifts.

The Gamma function in a model called the Three-Way Rule can predict qualitatively some responses of lamellar bone modeling drifts to defined longitudinal bone strain patterns. The derivation of Gamma in this article suggests a way to define it from in vivo longitudinal bond strain measurements. It depends on three operators called M, G and S. Let M equal +1 on any part of a bone surface (the 'study surface') where the local Magnitude of the longitudinal mechanical strains exceeds a threshold that can turn adaptive modeling drifts ON; otherwise let it equal zero. For the cross section of a bone or trabecula that intersects the study surface, let G equal +1 when the Greatest strain over the whole cross section is in tension, but -1 if it is in compression. On a study surface where M = +1, let S equal G in uniaxial loading, but +1 if the Subsurface strain gradient emerging from inside the bone to this study surface is positive, and -1 if this gradient is negative. Then the product of M x G x S yields gamma as +1, 0 or -1, and those are its only permitted values. The values would mean a mechanically induced formation drift, or no drift, or a resorption drift, respectively should begin on that study surface. Gamma can predict the drift patterns in five basic or 'principal' structural adaptations of trabeculae, cortex and whole bones to defined mechanical challenges and bone strain patterns.