Growth of a long bone cross section - A 2D phase-field model

An approach to model the effect of exercise on the growth of mammal long bones is described. A Ginzburg-Landau partial differential equation system is utilised to study the change of size and shape of a cross-section caused by mechanically enhanced bone growth. The concept is based on a phase variable that keeps track of the material properties during the evolution of the bone. The relevant free energies are assumed to be elastic strain energy, concentration gradient energy and a double well chemical potential. The equation governing the evolution of the phase is derived from the total free energy and put on a non-dimensional form, which reduces all required information regarding load, material and cross-section size to one single parameter. The partial differential equation is solved numerically for the geometry of a cross-section using a finite element method. Bending in both moving and fixed directions is investigated regarding reshaping and growth rates. A critical non-zero load is found under which the bone is resorbed. The result for bending around a fixed axis can be compared with experiments made on turkeys. Three loading intervals are identified, I) low load giving resorption of bone on the external periosteum and the internal endosteum, II) intermediate load with growth at the periosteum and resorption at endosteum and III) large loads with growth at both periosteum and endosteum. In the latter case the extent of the medullary cavity decreases.