A geometric representation of the axial transmission apparatus showing the water and test sample, where T and R represent the transmitting and receiving transducers, respectively. , , , , and are the initial transducer separation, transducer separation, test plate thickness, height of transducer face above plate, and fracture gap width, respectively. The dotted line above point represents the starting position of the measurements.
Modeling pictures of the six stages of secondary healing: (a) an intact bone (light gray). (b) Stage 1: The initial transverse fracture. Stages 2–4: A bridging callus with degrees of ossification, where (c) shows the hard callus material (dark gray) forming the bridge with cartilage filling the gap (black), (d) shows the hard callus material connecting to bridge the gap and cartilage remaining within the fracture gap, and (e) shows the hard callus material throughout the fracture site. Stages 5 and 6: Remodeling, where (f) represents the hard callus material replaced by cortical bone within the fracture gap and (g) the reduction in callus size as the bone returns to original state. Water is used to represent the blood in these simulations.
Modeling pictures representing the three fracture geometries used in the in-vitro experiments and 2D simulations. A 6 mm, thick plate, surrounded by water and (a) a 4 mm transverse fracture, (b) bone cement filling/bridging the fracture gap, and (c) a callus is added to the top surface.
Typical experimental (a) and simulated (b) signals.
Numerical modeling results of the SPL versus transducer separation for the six healing stages when Young’s modulus value corresponds to callus materials (a) 1 (5 GPa), (b) 2 (10 GPa), and (c) 3 (15 GPa). FTL calculations where preformed using data taken at 120 mm. The first fracture interface is positioned at a transducer separation of 0.07 m.
Results of the SPL versus transducer separation measurements for (a) in-vitro experiments investigating the effect of fracture geometry and mechanical properties on signal amplitude, and (b) simulations of the in-vitro experiments. A 6 mm thick Sawbones® plate with a 4 mm transverse fracture was used for this work. The first fracture interface is positioned at a transducer separation of 0.07 m.
Numerical simulation snapshots of acoustic pressure for healing stages 1–5. At stage 1 the fracture gap is filled with water, stage 2 represents the initiation of bridging and ossification with a symmetrical noncontinuous callus and cartilage at the fracture site, and stages 3 and 4 represent the continued ossification of callus, the difference being the material filling the fracture gap which is modeled as cartilage in stage 3 and callus in stage 4. The front of the acoustic pulse has just passed the fracture.
Material, elastic, and acoustic properties used in numerical simulations. Data taken from the literature are referenced. The remaining values were estimated or calculated from referenced data using standard equations.
A comparison between the FTL calculated from numerical modeling data of the signal loss produced by healing stages 1–6 and increasing Young’s modulus of the callus material. The change in signal amplitude (compared with the baseline data) is also expressed as a percentage and given in parentheses.
A comparison between the FTL measured from in-vitro experiments and predicted by numerical modeling of the signal loss produced by three different fracture geometries simulated on a Sawbones® plate. The change in signal amplitude (compared with the baseline data) is also expressed as a percentage and given in parentheses.
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