JBMR
LETTER TO THE EDITOR
Major Source of Error When Calculating Bone Mechanical Properties RJ Wallace,1 P Pankaj,2 and AHRW Simpson1 1
The University of Edinburgh, Department of Orthopaedics The University of Edinburgh, School of Engineering
2
To the Editor: Mechanical testing of bone and fracture callus is performed to assess the functional properties of the tissue. In particular, 3‐ point bending is a commonly performed technique when experimentally evaluating mechanical properties. The Euler‐ Bernoulli equation used to calculate the bending stiffness assumes that the beam is long and slender. In practice, beams that have a span to depth ratio (aspect ratio) of greater than 20:1 are considered to be “slender.” (1) If this method is used on non‐ slender bones an error results as the contribution of the bending due to shear deformation is not taken into account. A review of articles published in the last 2 years in this journal indicated that in the 14 papers that used 3‐point bending to determine elastic modulus none accounted for the effect on deformation due to shear. The error from ignoring the contribution of shear deformation on bones with these properties can be as high as 38%. If the analysis does not take shear deformation into account, large errors in the evaluated material properties will result in an estimated 95% of bones. Three‐point bending is often carried out using long bones from animals such as the mouse, rat, or rabbit that are too small to allow sectioning into smaller samples and must therefore be tested whole. Tests performed on these bones will therefore always be subject to inaccuracies if they assumed to be slender beams. The problem is compounded by the fact that the test length of the specimen is not the length of the bone, but the distance between the supports. In order to provide a suitable location, the supports are generally placed at the metaphyses of the bone. Therefore, the span is less than the total length of the bone, further reducing the aspect ratio, often by 25%, resulting in an increase to the potential error. The problem of not accounting for deflection due to shear is further exacerbated as mechanical testing is often used to determine the influence of medical treatments or physical diseases. In order to study these, animals (usually mice or rats) are bred with genetic deficiencies. In some cases, such as is found in
the FGFR‐3 deficient mouse, this results in musculoskeletal changes.(2) These changes can result in significantly different aspect ratio of limbs in comparison to the wild and the modified specimen as a limb may be longer or thicker without a corresponding change in the complimentary dimension. Assumptions made about the cross‐sectional shape, (ie, whether it is approximately circular as with the shaft of the femur or approximately triangular as for the shaft of the tibia) can also result in errors. However, for whole bones, the major determinant of this error is the aspect ratio rather than the cross‐sectional geometry. It is recognized that small errors may arise due to non‐ prismatic geometry of the bone, ie, changes in cross‐section from proximal to distal, and inhomogeneous material properties. These are best considered through the use of numerical simulation (eg, finite element analysis), which require a full 3D geometry construction via a CT scan and assignment of variable material properties from CT attenuation data. As a consequence it requires considerably more resources: scanning; conversion of images to numerical models; and analysis of models and interpretation of results. As a result, this type of analysis is rarely performed on small animal studies such as those conducted using rat or mouse limbs. Numerical analysis, due to the additional work and resources required, is usually restricted limited to human bones, as the limited supply of these bones can often warrant the use of additional resources. The deflection due to bending and that attributed to shear can be derived following the methods set out in Wang.(3)
References 1. Young WC. Roark’s Formulas for Stress and Strain. New York: McGraw‐ Hill; 1989. 2. Deng C, Wynshaw‐Boris A, Zhou F, Kuo A, Leder P. Fibroblast growth factor receptor 3 is a negative regulator of bone growth. Cell. 1996;84(6):911–21. 3. Wang C. Timoshenko beam‐bending solutions in terms of euler‐ bernoulli solutions. J Eng Mech. 1995;121(6):763–5.
Address correspondence to: Dr Robert Wallace, Chancellor’s Building, 49 Little France Crescent, Edinburgh, EH164SB. E‐mail: [email protected] Journal of Bone and Mineral Research, Vol. 29, No. 12, December 2014, pp 2697 DOI: 10.1002/jbmr.2304 © 2014 American Society for Bone and Mineral Research
2697
LETTER TO THE EDITOR
Major Source of Error When Calculating Bone Mechanical Properties RJ Wallace,1 P Pankaj,2 and AHRW Simpson1 1
The University of Edinburgh, Department of Orthopaedics The University of Edinburgh, School of Engineering
2
To the Editor: Mechanical testing of bone and fracture callus is performed to assess the functional properties of the tissue. In particular, 3‐ point bending is a commonly performed technique when experimentally evaluating mechanical properties. The Euler‐ Bernoulli equation used to calculate the bending stiffness assumes that the beam is long and slender. In practice, beams that have a span to depth ratio (aspect ratio) of greater than 20:1 are considered to be “slender.” (1) If this method is used on non‐ slender bones an error results as the contribution of the bending due to shear deformation is not taken into account. A review of articles published in the last 2 years in this journal indicated that in the 14 papers that used 3‐point bending to determine elastic modulus none accounted for the effect on deformation due to shear. The error from ignoring the contribution of shear deformation on bones with these properties can be as high as 38%. If the analysis does not take shear deformation into account, large errors in the evaluated material properties will result in an estimated 95% of bones. Three‐point bending is often carried out using long bones from animals such as the mouse, rat, or rabbit that are too small to allow sectioning into smaller samples and must therefore be tested whole. Tests performed on these bones will therefore always be subject to inaccuracies if they assumed to be slender beams. The problem is compounded by the fact that the test length of the specimen is not the length of the bone, but the distance between the supports. In order to provide a suitable location, the supports are generally placed at the metaphyses of the bone. Therefore, the span is less than the total length of the bone, further reducing the aspect ratio, often by 25%, resulting in an increase to the potential error. The problem of not accounting for deflection due to shear is further exacerbated as mechanical testing is often used to determine the influence of medical treatments or physical diseases. In order to study these, animals (usually mice or rats) are bred with genetic deficiencies. In some cases, such as is found in
the FGFR‐3 deficient mouse, this results in musculoskeletal changes.(2) These changes can result in significantly different aspect ratio of limbs in comparison to the wild and the modified specimen as a limb may be longer or thicker without a corresponding change in the complimentary dimension. Assumptions made about the cross‐sectional shape, (ie, whether it is approximately circular as with the shaft of the femur or approximately triangular as for the shaft of the tibia) can also result in errors. However, for whole bones, the major determinant of this error is the aspect ratio rather than the cross‐sectional geometry. It is recognized that small errors may arise due to non‐ prismatic geometry of the bone, ie, changes in cross‐section from proximal to distal, and inhomogeneous material properties. These are best considered through the use of numerical simulation (eg, finite element analysis), which require a full 3D geometry construction via a CT scan and assignment of variable material properties from CT attenuation data. As a consequence it requires considerably more resources: scanning; conversion of images to numerical models; and analysis of models and interpretation of results. As a result, this type of analysis is rarely performed on small animal studies such as those conducted using rat or mouse limbs. Numerical analysis, due to the additional work and resources required, is usually restricted limited to human bones, as the limited supply of these bones can often warrant the use of additional resources. The deflection due to bending and that attributed to shear can be derived following the methods set out in Wang.(3)
References 1. Young WC. Roark’s Formulas for Stress and Strain. New York: McGraw‐ Hill; 1989. 2. Deng C, Wynshaw‐Boris A, Zhou F, Kuo A, Leder P. Fibroblast growth factor receptor 3 is a negative regulator of bone growth. Cell. 1996;84(6):911–21. 3. Wang C. Timoshenko beam‐bending solutions in terms of euler‐ bernoulli solutions. J Eng Mech. 1995;121(6):763–5.
Address correspondence to: Dr Robert Wallace, Chancellor’s Building, 49 Little France Crescent, Edinburgh, EH164SB. E‐mail: [email protected] Journal of Bone and Mineral Research, Vol. 29, No. 12, December 2014, pp 2697 DOI: 10.1002/jbmr.2304 © 2014 American Society for Bone and Mineral Research
2697
