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A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing
Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Short communication
A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing Vee San Cheong, Anthony M.J. Bull n Department of Bioengineering, Imperial College London, London, United Kingdom
art ic l e i nf o
a b s t r a c t
Article history: Accepted 12 October 2015
The choice of coordinate system and alignment of bone will affect the quantification of mechanical properties obtained during in-vitro biomechanical testing. Where these are used in predictive models, such as finite element analysis, the fidelic description of these properties is paramount. Currently in bending and torsional tests, bones are aligned on a pre-defined fixed span based on the reference system marked out. However, large inter-specimen differences have been reported. This suggests a need for the development of a specimen-specific alignment system for use in experimental work. Eleven ovine tibiae were used in this study and three-dimensional surface meshes were constructed from micro-Computed Tomography scan images. A novel, semi-automated algorithm was developed and applied to the surface meshes to align the whole bone based on its calculated principal directions. Thereafter, the code isolates the optimised location and length of each bone for experimental testing. This resulted in a lowering of the second moment of area about the chosen bending axis in the central region. More importantly, the optimisation method decreases the irregularity of the shape of the cross-sectional slices as the unbiased estimate of the population coefficient of variation of the second moment of area decreased from a range of (0.210–0.435) to (0.145–0.317) in the longitudinal direction, indicating a minimisation of the product moment, which causes eccentric loading. Thus, this methodology serves as an important pre-step to align the bone for mechanical tests or simulation work, is optimised for each specimen, ensures repeatability, and is general enough to be applied to any long bone. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Bone Geometrical properties Mechanical testing Alignment Anatomical reference frame
1. Introduction When characterising material properties of bone at the tissue level, samples are machined into regular specimens to enable the application of Euler–Bernoulli equations to study the effects of applying loads, producing shear stress in torsion and direct stress in bending. Bending and torsion tests are used to characterise the mechanical behaviour of whole bones. In bending, a load is applied laterally while the bone is supported between two external rollers. In torsion, the epiphyses are mounted to enable a torque to be applied along the diaphysis. To study the mechanical behaviour of bone at the structural level, its complex geometry needs to be taken into account. Currently, anatomical reference frames are used to mount bones for experimental testing in four-point bending and torsion (Cristofolini et al., 1996; Cristofolini, 1997; Cristofoli et al., 2000). These experiments have large standard deviations in the results (Cristofolini and Viceconti, 2000). The Ruff and Hayes (1983) method is the most repeatable method known to assign a standard n
Correspondence to: Department of Bioengineering, Imperial College London, London SW7 2AZ, United Kingdom. Tel.: þ 44 20 7594 2765. E-mail address: [email protected] (A.M.J. Bull).
coordinate system for in-vitro testing (Conti et al., 2008) and has been used in structural testing of bones (Finlay et al., 1995; Ebacher et al., 2007; Varghese et al., 2011). The ability of this system to align bones to optimally resist bending and torsion has never been compared to analytical results calculated from solid mechanics. The aims of this study are to:
develop a method that calculates the principal directions of bone using imaging data,
compare the second moment of area and its coefficient of var-
iation across the cross-section using the anatomical reference frame and the new method in the principal and anatomical directions, and obtain landmarks to define bone alignment for in-vitro testing in four-point bending and torsion.
2. Materials and methods 11 Ovine tibiae from 5 months old British Texel lambs weighing approximately 20 kg were harvested immediately after slaughter. The bones were cleaned of soft tissues, wrapped in cloth soaked in 1% Phosphate Buffer solution and frozen at 20 °C in sealed bags. Bones were thawed and scanned at a resolution of 115 μm in a Metris X-Tek 225 CT System (Nikon Metrology, Tring, UK). Manual segmentation was
http://dx.doi.org/10.1016/j.jbiomech.2015.10.011 0021-9290/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
V.S. Cheong, A.M.J. Bull / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
conducted in Mimics 15.0 (Materialse NV, Leuven, Belgium), to create cortical and endocortical masks. Geomagic Studio 12.0 (3D Systems, North Carolina, USA) was used for repair and smoothing to produce approximately 10,000 triangles for export in STL format. Euler–Bernoulli objects have a set of principal moments (I1, I2 and I3, in descending order), and their corresponding principal axes, where they would experience maximum and minimum stresses in the absence of eccentric loading (Ruff and Hayes, 1983): 3 2 I1 0 0 60 I 07 ð1Þ I¼4 5 2 0
0
I3
The volume second moment, I, is a measure of the distribution of the material points with respect to the axis. When defined about cartesian axes, they are: R I xx ¼ A y2 þ z2 dV Z I yy ¼ x2 þ z2 dV A Z I zz ¼ x2 þ y2 dV ð2Þ A
When the load to an object is not applied about the principal axis, or if the object is asymmetrical, a product moment of area is experienced: R I xy ¼ A xy dV Z I yz ¼ yz dV A Z I zx ¼ zx dV ð3Þ A
The product and area moments can be assembled to solve for the principal moments I: I xx I I xy I xz I I I I yx yy yz ¼ 0 ð4Þ Izx I zy I zz I
An area moment is maximised when its product moment is minimised. As bone geometry is irregular, the object will be optimised to experience near pure shear in torsion, and near pure direct stress in bending. Here the principal directions will be referred to as D1, D2 and D3, representing the initial coordinate system used in aligning the bone, and corresponding to I1, I2 and I3. The algorithm (Fig. 1) to align a bone along its principal axes was developed in RhinoScript and carried out in Rhinoceros (Robert McNeel & Associates, Seattle, USA). Firstly, the two masks were converted to a single solid body, using standard CAD procedures. Then, the volume centroid of the body was calculated to act as the origin for the principal axes. The principal moments and their corresponding axes were computed and sorted in descending order. The bones were then rotated so that their first principal directions coincide with the global axes, using the dot and cross product. This was repeated until the error between the current and target axis equals zero (Fig. 1, loop A) and repeated for the second principal axis D2 (Fig. 1, loop B). In bending, only the span of the bone between the external two rollers and in torsional loading only the free region that is not embedded contribute to the strength of the bone in testing (Ebacher et al., 2007). The location of this free region (and span) for experimental testing was found by minimising the variation in the second moment of area along the segment. Firstly, the first cut, which defines the proximal diaphysis, is allowed to be located between 20% and 50% of the bone length. The width at this cross-section was used to locate the distal diaphysis at five times the distance of the initial width. (Fig. 2A). The widest width throughout the produced segment is then calculated to ensure that the segment fulfils the minimum span-to-width ratio of 4:1, shown by Hardy and Pipelzadeh (1991) to be the minimum required to minimise shear. The segment is then passed through the algorithm again for alignment (Fig. 1). In both the second and third sections of the algorithm, cutting planes are used to section the bones and bone segments at 5% interval along the long axis (Fig. 2B). The intersection between neutral axis and each slice is calculated (black dots). These are used as the reference points for the calculation of the second moment of area. Code was written to locate the mesh vertices that were furthest away in the first principal axes, to yield the location of six reference landmarks for aligning the bone during experimental testing, with three on each end of the bone (Fig. 3). Physically, the landmarks are located with the aid of an optical tracker, to avoid misalignment. Data were not normally distributed and so an unbiased estimate of the population coefficient of variation (γv) of the second moment of area for each alignment
Fig. 1. Optimisation of bone segment best suited for bending and torsional tests can be obtained by running this algorithm. The code first processes input STL meshes so that they are suitable for later calculations, before optimising the alignment based on the whole bone and bone segment.
Fig. 2. (A) A segment of a bone calculated based on the minimum length-to-width ratio and aligned to its principal axes, as shown in pink. (B) Slices of bone sectioned at 5% interval along the long axis, after the bone (or bone segment) has been aligned to its principal directions. The blue (or light grey) dots show the position of the centroid of each slice while the black dots show the intersection between the first principal axis and each slice (termed neutral axis point). The difference between the two lines is seen most clearly in the proximal region. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
V.S. Cheong, A.M.J. Bull / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 3. The algorithm calculates and plots the location of six prominent features (in blue) to be used in aligning the bone during experiments. The possible positions on the bone where the roller should make contact are marked in orange with an open dot. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. The second moment of area in its principal directions (N¼11), for optimised tibia segment in the diaphyses, plotted against the distance along the tibia from distal to proximal, and compared against different alignment methods, for I1, I2 and I3 (top to bottom). Solid line: anatomical landmarks. Dash line: optimised whole bone. Dotted line: optimised bone segment. All lines are mean7standard deviation (shaded area).
Fig. 4. The cross-sectional second moment of area in its principal directions (N ¼11), for optimised whole ovine tibiae (top) and optimised tibia diaphyses (bottom), plotted against the distance along the tibia from distal to proximal. All lines are means 7standard deviation (shaded area). method for each bone was determined. The ratio between the standard deviation and the mean value of the second moment of area of the cross section was calculated, correcting for sample size. Friedman ANOVA was used to test the hypothesis that there is no difference in the γv of the second moment of area of the bone among the three alignment methods. When the null hypothesis was rejected, the one-tailed paired sample Wilcoxon signed-rank test with Bonferroni correction was used to test the hypothesis that the new alignment method decreased the median γv, compared to the anatomical alignment.
3. Results The methodology optimises the alignment of long bones by ensuring that I1 4I2 4I3 in the midshaft (Fig. 4). By focussing on the diaphyseal section and optimising the bone segment to be used in the analysis, the spread of values decreased (Fig. 4). The three different alignment methods are compared in Fig. 5 and Table 1. The second moment of area in the D1, D2 and D3 directions was different between the three alignment methods (Friedman ANOVA, χ2(2)¼20.182, po0.001; χ2(2)¼ 9.455, p¼0.009; and χ2(2)¼ 16.545, po0.001 respectively; Table 2).
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
V.S. Cheong, A.M.J. Bull / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Table 1 Range and median (N ¼ 11) of the coefficient of variation of I1, I2 and I3 for three different alignment methods for the ovine tibiae. Alignment method Anatomical Optimised whole Optimised segment
Range Median Range Median Range Median
I1(γv)
I2(γv)
I3(γv)
0.210–0.435 0.305 0.145–0.317 0.224 0.159–0.406 0.239
0.197–0.481 0.289 0.118–0.351 0.195 0.105–0.360 0.166
0.200–0.628 0.351 0.175–0.482 0.292 0.270–0.516 0.375
Table 2 A summary of the p-values of Friedman's analysis of variance on the second moment of area and its coefficient of variation, including further analyses carried out with the Wilcoxon signed rank test. Significant p values are in bold. γv
I Direction
1
2
3
1
2
3
Friedman ANOVA Wilcoxon signed rank Anatomical vs. optimised whole Anatomical vs. optimised segment Optimised whole vs. optimised segment
0.00004 0.009 0.0003 0.0008 0.006 0.004 0.003
0.252 0.084
0.003
0.006 0.028
0.002
0.023 0.002
0.012
0.002 0.718
0.002
0.009 0.002
0.998
0.015
0.998
Fig. 6. The landmarks that can be used to obtain the desired alignment are shown (bottom) for an optimised segment (top). Blue open dots: anatomical. Red dots: optimised whole. Black crosses: optimised segment. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The new methodology resulted in a decrease in γv. for the anatomical vs. optimised whole group in the D1 (p¼0.003) and D2 (p¼0.006) directions (Table 2). The decrease was significant in the anatomical vs. optimised segment group (p¼ 0.012 and p¼0.002). Optimising the segment brought significant improvement only in D2 (p¼0.0015) direction, and resulted in a curve that is flatter and more symmetrical, with a higher value recorded at the ends when compared to the results obtained using the ‘optimised whole’ algorithm (Fig. 5). The intersection between the principal axis and the cortical bone in each alignment method is shown in Fig. 6. Filled dots show the anatomical landmarks that were used (from Finlay et al., 1995). These landmarks are located more anteriorly on the proximal end when the whole bone is aligned based on its geometrical properties and are more medial to the intercondylar eminence when the bone is aligned based on its optimised segment. On the distal end of the tibia, the landmarks shift posteriorly with each alignment method.
4. Discussion A methodology that calculates and uses the initial principal directions of bone as a subject-specific coordinate system, to enable a state of near pure bending and torsion during mechanical
Fig. 7. The second moment of area of whole ovine in its three principal directions, I1 (top), I2 (middle) and I3 (bottom)(N ¼ 11), plotted against the distance along the tibia from distal to proximal, and compared with similar data replotted from Finlay et al. (1995) (N¼ 6). All lines are means 7standard deviation (shaded area).
testing, has been developed. This results in a more even distribution of the second moment of area that is accompanied by a lowering of its coefficient of variation in the D1 and D2 directions. Its performance is an improvement on the literature. The alignment of the whole bone using geometrical properties corresponded well with the results obtained by using an anatomical-by-eye reference system in the middle section (Finlay et al., 1995). The location of the lowest second moment of area was in the diaphysis in both studies (Fig. 7). In all three directions, the second moment of area rises more gradually at the epiphyseal regions in the study here than in Finlay et al. (1995). We found that the second moment of area for our whole ovine tibiae ranged (4293–37,874 mm4) in the D2 and (3153–42,461 mm4) in the D3 directions. In contrast, they obtained
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
V.S. Cheong, A.M.J. Bull / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
values of (7000–70,000 mm4) and (4500–68,000 mm4) in the D2 and D3 directions, respectively. In this study the coordinate systems were determined from the volume second moment of the cortex, which is based on its morphology in three-dimensions. However, in all the previous work, the geometrical properties were evaluated on a slice-by-slice basis, a twodimensional property. Therefore, the existing techniques for studying the geometrical properties of bones is more an analysis of how the Imax or Imin axes vary across the cross-section of the bone, which does not yield further information on how the bone behaves as a structural entity. Indeed Finlay et al. (1995) reported a range of 1.76–86.1° of the angle between Imax and the anterior–posterior axis in mature Arcott sheep, with the smallest angle at the midshaft and the largest angle at the proximal and distal ends. Therefore, the second moment of area and standard deviations measured in the midshaft are similar between this study and theirs, whereas these values divert greatly at the ends of the bone as the angle between the reference axes used increases. Other studies used the centroid as the origin for the principle directions (Finlay et al., 1995; Forman et al., 2012; Perz et al., 2015) whereas the neutral axis point was used in this study. The neutral axis refers to the line about which an object experiences no compressive or tensile stress when it is subjected to pure bending, and can be assumed to coincide with the first principal axis. The neutral axis point is obtained by intersecting D1 with each slice. These two points do not always coincide, and thus Imax that is calculated based solely on the slice would deviate from the I2 that is calculated based on the bone cortex. Principal moments can be calculated precisely, but perfect alignment is difficult to achieve experimentally. Although a method that uses reverse digitisation has been suggested to increase repeatability, misalignment might still occur, which would result in shifts in the neutral axis. Moreover, the methodology assumes that the samples have fairly homogenous bone density and so care must be taken in interpreting results obtained from samples with inhomogeneous material properties (e.g. osteoporotic) or geometrical properties (e.g. gene knockout in murine long bones may result in different bone geometries). In general, the testing regions, from the calculated optimised segment, should be fairly similar (e.g. in the diaphysis of the bone) for any comparison to be valid in the first instance. In summary, a new methodology that calculates the alignment of bone based on its principal axes, so that it approximates a state of pure bending and torsion may be achieved in experimental testing, has been developed. This methodology differs from previous work as it accounts for the three-dimensional geometry of
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the bone and isolates the segment of the bone that is most optimised for experimental testing.
Conflict of interest statement The authors declare that no conflict of interest exist in carrying out this study.
Acknowledgements Vee San Cheong held the SIM-You Poh Seng Scholarship and the Department of Bioengineering studentship at Imperial College.
References Conti, G., Cristofolini, L., Juszczyk, M., Leardini, A., Viceconti, M., 2008. Comparison of three standard anatomical reference frames for the tibia–fibula complex. J. Biomech. 41 (16), 3384–3389. Cristofolini, L., 1997. A critical analysis of stress shielding evaluation of hip prostheses. Crit. Rev.™ Biomed. Eng. 25, 4–5. Cristofolini, L., Viceconti, M., 2000. Mechanical validation of whole bone composite tibia models. J. Biomech. 33, 279–288. Cristofolini, L., Viceconti, M., Cappello, A., Toni, A., 1996. Mechanical validation of whole bone composite femur models. J. Biomech. 29 (4), 525–535. Ebacher, V., Tang, C., McKay, H., Oxland, T.R., Guy, P., Wang, R., 2007. Strain redistribution and cracking behavior of human bone during bending. Bone 40 (5), 1265–1275. Finlay, J., Hurtig, M., Hardie, W., Liggins, A., Batte, S., 1995. Geometrical properties of the ovine tibia: a suitable animal model to study the pin–bone interface in fracture fixation? Proc. Inst. Mech. Eng. H: J. Eng. Med. 209 (1), 37–50. Forman, J.L., de Dios, E., Symeonidis, I., Duart, J., Kerrigan, J.R., Salzar, R.S., Balasubramanian, S., Segui-Gomez, M., Kent, R.W., 2012. Fracture tolerance related to skeletal development and aging throughout life: 3-point bending of human femurs. In: Proceedings of the IRCOBI Conference, Dublin, Ireland. Hardy, S., Pipelzadeh, M., 1991. Static analysis of short beams. J. Strain Anal. Eng. Des. 26 (1), 15–29. Perz, R., Toczyski, J., Subit, D., 2015. Variation in the human ribs geometrical properties and mechanical response based on X-ray computed tomography images resolution. J. Mech. Behav. Biomed. Mater. 41, 292–301. Ruff, C.B., Hayes, W.C., 1983. Cross‐sectional geometry of Pecos Pueblo femora and tibiae—a biomechanical investigation: I. Method and general patterns of variation. Am. J. Phys. Anthropol. 60 (3), 359–381. Varghese, B., Short, D., Penmetsa, R., Goswami, T., Hangartner, T., 2011. Computedtomography-based finite-element models of long bones can accurately capture strain response to bending and torsion. J. Biomech. 44 (7), 1374–1379.
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Short communication
A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing Vee San Cheong, Anthony M.J. Bull n Department of Bioengineering, Imperial College London, London, United Kingdom
art ic l e i nf o
a b s t r a c t
Article history: Accepted 12 October 2015
The choice of coordinate system and alignment of bone will affect the quantification of mechanical properties obtained during in-vitro biomechanical testing. Where these are used in predictive models, such as finite element analysis, the fidelic description of these properties is paramount. Currently in bending and torsional tests, bones are aligned on a pre-defined fixed span based on the reference system marked out. However, large inter-specimen differences have been reported. This suggests a need for the development of a specimen-specific alignment system for use in experimental work. Eleven ovine tibiae were used in this study and three-dimensional surface meshes were constructed from micro-Computed Tomography scan images. A novel, semi-automated algorithm was developed and applied to the surface meshes to align the whole bone based on its calculated principal directions. Thereafter, the code isolates the optimised location and length of each bone for experimental testing. This resulted in a lowering of the second moment of area about the chosen bending axis in the central region. More importantly, the optimisation method decreases the irregularity of the shape of the cross-sectional slices as the unbiased estimate of the population coefficient of variation of the second moment of area decreased from a range of (0.210–0.435) to (0.145–0.317) in the longitudinal direction, indicating a minimisation of the product moment, which causes eccentric loading. Thus, this methodology serves as an important pre-step to align the bone for mechanical tests or simulation work, is optimised for each specimen, ensures repeatability, and is general enough to be applied to any long bone. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Bone Geometrical properties Mechanical testing Alignment Anatomical reference frame
1. Introduction When characterising material properties of bone at the tissue level, samples are machined into regular specimens to enable the application of Euler–Bernoulli equations to study the effects of applying loads, producing shear stress in torsion and direct stress in bending. Bending and torsion tests are used to characterise the mechanical behaviour of whole bones. In bending, a load is applied laterally while the bone is supported between two external rollers. In torsion, the epiphyses are mounted to enable a torque to be applied along the diaphysis. To study the mechanical behaviour of bone at the structural level, its complex geometry needs to be taken into account. Currently, anatomical reference frames are used to mount bones for experimental testing in four-point bending and torsion (Cristofolini et al., 1996; Cristofolini, 1997; Cristofoli et al., 2000). These experiments have large standard deviations in the results (Cristofolini and Viceconti, 2000). The Ruff and Hayes (1983) method is the most repeatable method known to assign a standard n
Correspondence to: Department of Bioengineering, Imperial College London, London SW7 2AZ, United Kingdom. Tel.: þ 44 20 7594 2765. E-mail address: [email protected] (A.M.J. Bull).
coordinate system for in-vitro testing (Conti et al., 2008) and has been used in structural testing of bones (Finlay et al., 1995; Ebacher et al., 2007; Varghese et al., 2011). The ability of this system to align bones to optimally resist bending and torsion has never been compared to analytical results calculated from solid mechanics. The aims of this study are to:
develop a method that calculates the principal directions of bone using imaging data,
compare the second moment of area and its coefficient of var-
iation across the cross-section using the anatomical reference frame and the new method in the principal and anatomical directions, and obtain landmarks to define bone alignment for in-vitro testing in four-point bending and torsion.
2. Materials and methods 11 Ovine tibiae from 5 months old British Texel lambs weighing approximately 20 kg were harvested immediately after slaughter. The bones were cleaned of soft tissues, wrapped in cloth soaked in 1% Phosphate Buffer solution and frozen at 20 °C in sealed bags. Bones were thawed and scanned at a resolution of 115 μm in a Metris X-Tek 225 CT System (Nikon Metrology, Tring, UK). Manual segmentation was
http://dx.doi.org/10.1016/j.jbiomech.2015.10.011 0021-9290/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
V.S. Cheong, A.M.J. Bull / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
conducted in Mimics 15.0 (Materialse NV, Leuven, Belgium), to create cortical and endocortical masks. Geomagic Studio 12.0 (3D Systems, North Carolina, USA) was used for repair and smoothing to produce approximately 10,000 triangles for export in STL format. Euler–Bernoulli objects have a set of principal moments (I1, I2 and I3, in descending order), and their corresponding principal axes, where they would experience maximum and minimum stresses in the absence of eccentric loading (Ruff and Hayes, 1983): 3 2 I1 0 0 60 I 07 ð1Þ I¼4 5 2 0
0
I3
The volume second moment, I, is a measure of the distribution of the material points with respect to the axis. When defined about cartesian axes, they are: R I xx ¼ A y2 þ z2 dV Z I yy ¼ x2 þ z2 dV A Z I zz ¼ x2 þ y2 dV ð2Þ A
When the load to an object is not applied about the principal axis, or if the object is asymmetrical, a product moment of area is experienced: R I xy ¼ A xy dV Z I yz ¼ yz dV A Z I zx ¼ zx dV ð3Þ A
The product and area moments can be assembled to solve for the principal moments I: I xx I I xy I xz I I I I yx yy yz ¼ 0 ð4Þ Izx I zy I zz I
An area moment is maximised when its product moment is minimised. As bone geometry is irregular, the object will be optimised to experience near pure shear in torsion, and near pure direct stress in bending. Here the principal directions will be referred to as D1, D2 and D3, representing the initial coordinate system used in aligning the bone, and corresponding to I1, I2 and I3. The algorithm (Fig. 1) to align a bone along its principal axes was developed in RhinoScript and carried out in Rhinoceros (Robert McNeel & Associates, Seattle, USA). Firstly, the two masks were converted to a single solid body, using standard CAD procedures. Then, the volume centroid of the body was calculated to act as the origin for the principal axes. The principal moments and their corresponding axes were computed and sorted in descending order. The bones were then rotated so that their first principal directions coincide with the global axes, using the dot and cross product. This was repeated until the error between the current and target axis equals zero (Fig. 1, loop A) and repeated for the second principal axis D2 (Fig. 1, loop B). In bending, only the span of the bone between the external two rollers and in torsional loading only the free region that is not embedded contribute to the strength of the bone in testing (Ebacher et al., 2007). The location of this free region (and span) for experimental testing was found by minimising the variation in the second moment of area along the segment. Firstly, the first cut, which defines the proximal diaphysis, is allowed to be located between 20% and 50% of the bone length. The width at this cross-section was used to locate the distal diaphysis at five times the distance of the initial width. (Fig. 2A). The widest width throughout the produced segment is then calculated to ensure that the segment fulfils the minimum span-to-width ratio of 4:1, shown by Hardy and Pipelzadeh (1991) to be the minimum required to minimise shear. The segment is then passed through the algorithm again for alignment (Fig. 1). In both the second and third sections of the algorithm, cutting planes are used to section the bones and bone segments at 5% interval along the long axis (Fig. 2B). The intersection between neutral axis and each slice is calculated (black dots). These are used as the reference points for the calculation of the second moment of area. Code was written to locate the mesh vertices that were furthest away in the first principal axes, to yield the location of six reference landmarks for aligning the bone during experimental testing, with three on each end of the bone (Fig. 3). Physically, the landmarks are located with the aid of an optical tracker, to avoid misalignment. Data were not normally distributed and so an unbiased estimate of the population coefficient of variation (γv) of the second moment of area for each alignment
Fig. 1. Optimisation of bone segment best suited for bending and torsional tests can be obtained by running this algorithm. The code first processes input STL meshes so that they are suitable for later calculations, before optimising the alignment based on the whole bone and bone segment.
Fig. 2. (A) A segment of a bone calculated based on the minimum length-to-width ratio and aligned to its principal axes, as shown in pink. (B) Slices of bone sectioned at 5% interval along the long axis, after the bone (or bone segment) has been aligned to its principal directions. The blue (or light grey) dots show the position of the centroid of each slice while the black dots show the intersection between the first principal axis and each slice (termed neutral axis point). The difference between the two lines is seen most clearly in the proximal region. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
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Fig. 3. The algorithm calculates and plots the location of six prominent features (in blue) to be used in aligning the bone during experiments. The possible positions on the bone where the roller should make contact are marked in orange with an open dot. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. The second moment of area in its principal directions (N¼11), for optimised tibia segment in the diaphyses, plotted against the distance along the tibia from distal to proximal, and compared against different alignment methods, for I1, I2 and I3 (top to bottom). Solid line: anatomical landmarks. Dash line: optimised whole bone. Dotted line: optimised bone segment. All lines are mean7standard deviation (shaded area).
Fig. 4. The cross-sectional second moment of area in its principal directions (N ¼11), for optimised whole ovine tibiae (top) and optimised tibia diaphyses (bottom), plotted against the distance along the tibia from distal to proximal. All lines are means 7standard deviation (shaded area). method for each bone was determined. The ratio between the standard deviation and the mean value of the second moment of area of the cross section was calculated, correcting for sample size. Friedman ANOVA was used to test the hypothesis that there is no difference in the γv of the second moment of area of the bone among the three alignment methods. When the null hypothesis was rejected, the one-tailed paired sample Wilcoxon signed-rank test with Bonferroni correction was used to test the hypothesis that the new alignment method decreased the median γv, compared to the anatomical alignment.
3. Results The methodology optimises the alignment of long bones by ensuring that I1 4I2 4I3 in the midshaft (Fig. 4). By focussing on the diaphyseal section and optimising the bone segment to be used in the analysis, the spread of values decreased (Fig. 4). The three different alignment methods are compared in Fig. 5 and Table 1. The second moment of area in the D1, D2 and D3 directions was different between the three alignment methods (Friedman ANOVA, χ2(2)¼20.182, po0.001; χ2(2)¼ 9.455, p¼0.009; and χ2(2)¼ 16.545, po0.001 respectively; Table 2).
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
V.S. Cheong, A.M.J. Bull / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Table 1 Range and median (N ¼ 11) of the coefficient of variation of I1, I2 and I3 for three different alignment methods for the ovine tibiae. Alignment method Anatomical Optimised whole Optimised segment
Range Median Range Median Range Median
I1(γv)
I2(γv)
I3(γv)
0.210–0.435 0.305 0.145–0.317 0.224 0.159–0.406 0.239
0.197–0.481 0.289 0.118–0.351 0.195 0.105–0.360 0.166
0.200–0.628 0.351 0.175–0.482 0.292 0.270–0.516 0.375
Table 2 A summary of the p-values of Friedman's analysis of variance on the second moment of area and its coefficient of variation, including further analyses carried out with the Wilcoxon signed rank test. Significant p values are in bold. γv
I Direction
1
2
3
1
2
3
Friedman ANOVA Wilcoxon signed rank Anatomical vs. optimised whole Anatomical vs. optimised segment Optimised whole vs. optimised segment
0.00004 0.009 0.0003 0.0008 0.006 0.004 0.003
0.252 0.084
0.003
0.006 0.028
0.002
0.023 0.002
0.012
0.002 0.718
0.002
0.009 0.002
0.998
0.015
0.998
Fig. 6. The landmarks that can be used to obtain the desired alignment are shown (bottom) for an optimised segment (top). Blue open dots: anatomical. Red dots: optimised whole. Black crosses: optimised segment. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
The new methodology resulted in a decrease in γv. for the anatomical vs. optimised whole group in the D1 (p¼0.003) and D2 (p¼0.006) directions (Table 2). The decrease was significant in the anatomical vs. optimised segment group (p¼ 0.012 and p¼0.002). Optimising the segment brought significant improvement only in D2 (p¼0.0015) direction, and resulted in a curve that is flatter and more symmetrical, with a higher value recorded at the ends when compared to the results obtained using the ‘optimised whole’ algorithm (Fig. 5). The intersection between the principal axis and the cortical bone in each alignment method is shown in Fig. 6. Filled dots show the anatomical landmarks that were used (from Finlay et al., 1995). These landmarks are located more anteriorly on the proximal end when the whole bone is aligned based on its geometrical properties and are more medial to the intercondylar eminence when the bone is aligned based on its optimised segment. On the distal end of the tibia, the landmarks shift posteriorly with each alignment method.
4. Discussion A methodology that calculates and uses the initial principal directions of bone as a subject-specific coordinate system, to enable a state of near pure bending and torsion during mechanical
Fig. 7. The second moment of area of whole ovine in its three principal directions, I1 (top), I2 (middle) and I3 (bottom)(N ¼ 11), plotted against the distance along the tibia from distal to proximal, and compared with similar data replotted from Finlay et al. (1995) (N¼ 6). All lines are means 7standard deviation (shaded area).
testing, has been developed. This results in a more even distribution of the second moment of area that is accompanied by a lowering of its coefficient of variation in the D1 and D2 directions. Its performance is an improvement on the literature. The alignment of the whole bone using geometrical properties corresponded well with the results obtained by using an anatomical-by-eye reference system in the middle section (Finlay et al., 1995). The location of the lowest second moment of area was in the diaphysis in both studies (Fig. 7). In all three directions, the second moment of area rises more gradually at the epiphyseal regions in the study here than in Finlay et al. (1995). We found that the second moment of area for our whole ovine tibiae ranged (4293–37,874 mm4) in the D2 and (3153–42,461 mm4) in the D3 directions. In contrast, they obtained
Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i
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values of (7000–70,000 mm4) and (4500–68,000 mm4) in the D2 and D3 directions, respectively. In this study the coordinate systems were determined from the volume second moment of the cortex, which is based on its morphology in three-dimensions. However, in all the previous work, the geometrical properties were evaluated on a slice-by-slice basis, a twodimensional property. Therefore, the existing techniques for studying the geometrical properties of bones is more an analysis of how the Imax or Imin axes vary across the cross-section of the bone, which does not yield further information on how the bone behaves as a structural entity. Indeed Finlay et al. (1995) reported a range of 1.76–86.1° of the angle between Imax and the anterior–posterior axis in mature Arcott sheep, with the smallest angle at the midshaft and the largest angle at the proximal and distal ends. Therefore, the second moment of area and standard deviations measured in the midshaft are similar between this study and theirs, whereas these values divert greatly at the ends of the bone as the angle between the reference axes used increases. Other studies used the centroid as the origin for the principle directions (Finlay et al., 1995; Forman et al., 2012; Perz et al., 2015) whereas the neutral axis point was used in this study. The neutral axis refers to the line about which an object experiences no compressive or tensile stress when it is subjected to pure bending, and can be assumed to coincide with the first principal axis. The neutral axis point is obtained by intersecting D1 with each slice. These two points do not always coincide, and thus Imax that is calculated based solely on the slice would deviate from the I2 that is calculated based on the bone cortex. Principal moments can be calculated precisely, but perfect alignment is difficult to achieve experimentally. Although a method that uses reverse digitisation has been suggested to increase repeatability, misalignment might still occur, which would result in shifts in the neutral axis. Moreover, the methodology assumes that the samples have fairly homogenous bone density and so care must be taken in interpreting results obtained from samples with inhomogeneous material properties (e.g. osteoporotic) or geometrical properties (e.g. gene knockout in murine long bones may result in different bone geometries). In general, the testing regions, from the calculated optimised segment, should be fairly similar (e.g. in the diaphysis of the bone) for any comparison to be valid in the first instance. In summary, a new methodology that calculates the alignment of bone based on its principal axes, so that it approximates a state of pure bending and torsion may be achieved in experimental testing, has been developed. This methodology differs from previous work as it accounts for the three-dimensional geometry of
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the bone and isolates the segment of the bone that is most optimised for experimental testing.
Conflict of interest statement The authors declare that no conflict of interest exist in carrying out this study.
Acknowledgements Vee San Cheong held the SIM-You Poh Seng Scholarship and the Department of Bioengineering studentship at Imperial College.
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Please cite this article as: Cheong, V.S., Bull, A.M.J., A novel specimen-specific methodology to optimise the alignment of long bones for experimental testing. Journal of Biomechanics (2015), http://dx.doi.org/10.1016/j.jbiomech.2015.10.011i