Predicting fracture of the femoral neck

A prediction of the probability of safe loading of the femoral neck, based on queueing theory, is presented. The following methods have been applied: (I) criterion of bone fracture was formulated, taking into consideration the complex state of stress–strain in the porosity zones of the bone; (II) tensile stresses around pores in the stretched zone of the bone were evaluated; (III) the influence of random events of the critical regimes of loading was modeled. The evaluation of the probability of safe loading of bones was obtained based on the levels of the tensile stresses, Youngʹs moduli and ultimate tensile stresses which are affected by the increase in bone porosity and the distribution of the pores. Examples of analysis involving typical mechanical properties of bone in areas of vascular and lacunar–canalicular porosity are demonstrated. The ranges of initial average values of effective Youngʹs moduli and ultimate tensile strengths were taken as 15.8–17.5 GPa and 83–95 MPa, respectively. The present analysis discovers the existence of three levels of safe loading: (1) a relatively safe level of the nominal tensile stresses (smaller than (2.8–3.2) MPa) where the probability of safe loading is of the order of 0.95 for the bone porosity which is less than 0.15; (2) an intermediate level of safety where the nominal tensile stresses are below (4.2–4.8) MPa and the probability of safe loading is 0.89 or higher, for the same level of bone porosity; (3) a critical level of safe loading where the nominal tensile stresses are about (8.3–9.5) MPa; they lead to sharp drop of probabilities of safe loading to 0.85–0.8 if the porosity is about 0.10 and to probabilities of 0.41–0.4 if the porosity is about 0.15.