Accurate measurement of a bone surrogate flexural rigidity in three- and four-point bending

The mechanical assessment of long bones through bending is an established preclinical approach to evaluate the effectiveness of treatments for osteoporosis and fractures. Three- and four-point bending (3PB and 4PB) tests are the most common methods for mechanical characterization of long bones with Euler-Bernoulli (EB) theory to calculate of bone flexural rigidity (EI). Previous studies demonstrated that EB theory underestimates the EI of long bones due to its reliance on assumptions that are not entirely applicable to long bones. Therefore, the current study aimed to evaluate the factors that affect the percent error (PE) and bias stemming from the omission of contact and shear deflections in the EI estimation using mechanical testing and finite element analysis (FEA). The true EI of a porcine bone surrogate was used to quantify the percent error and bias of EI estimations from three deflection measurement methods and FEA, in 3PB and 4PB. The analysis confirmed that bending was the main component of total deflection, but only contributed to approximately 50 % and 65 % of the total deflection in 3PB and 4PB, respectively. The combined shear and indentation deflections accounted for the remainder of the total deflection. The FEA aligned with 3PB and 4PB tests with less than 10 % deviation. Underestimation of EI was largest with deflection measurements taken from the machine crosshead (PE 74 % in 3PB and 71 % in 4PB). However, these underestimations improved notably when indentation and shear deflections were considered (PE 42 % in 3PB and 39 % in 4PB). The deflection measurements from extensometer and digital image correlation (DIC) underestimated the EI by 66 % and 71 % in 3PB, and 57 % and 59 % in 4PB. When corrected for shear and indentation deflections, the 3PB PE reduced to 16 % and 14 %, respectively. In 4PB, PE reduced to, 10 % and 7 %, respectively, demonstrating the advantage of the 4PB test configuration over 3PB. The bias resulting from shear deflection was not consistent across deflection measurement methods; and therefore, cannot be generalized with a constant bias correction. The current study highlighted that a slight error in deflection measurement can lead to a significant inaccuracy in EI measurements. This sensitivity comes from the hyperbolic relationship between EI and deflection which not only depends on the ratio of support span to diameter of the specimen but also the test configuration and the ratio of the elastic to shear modulus of specimen. In other words, the PE from neglecting shear effects increases as the specimen EI increases. Accuracy with less than 10 % PE in EI estimations can be achieved by: 1. taking deflection measurements with extensometers, DIC, or FEA; 2. testing in 4PB instead of 3PB; and, 3. correcting for indentation and shear deflections.

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