Measuring bone stiffness using spherical indentation

Abstract

Objectives: Bone material properties are a major determinant of bone health in older age, both in terms of fracture risk and implant fixation, in orthopaedics and dentistry. Bone is an anisotropic and hierarchical material so its measured material properties depend upon the scale of metric used. The scale used should reflect the clinical problem, whether it is fracture risk, a whole bone problem, or implant stability, at the millimetre-scale. Indentation, an engineering technique involving pressing a hard-tipped material into another material with a known force, may be able to assess bone stiffness at the millimetre-scale (the apparent elastic modulus). We aimed to investigate whether spherical-tip indentation could reliably measure the apparent elastic modulus of human cortical bone.

Materials and methods: Cortical bone samples were retrieved from the femoral necks of nineteen patients undergoing total hip replacement surgery (10 females, 9 males, mean age: 69 years). The samples underwent indentation using a 1.5 mm diameter, ruby, spherical indenter tip, with sixty indentations per patient sample, across six locations on the bone surfaces, with ten repeated indentations at each of the six locations. The samples then underwent mechanical compression testing. The repeatability of indentation measurements of elastic modulus was assessed using the co-efficient of repeatability and the correlation between the bone elastic modulus measured by indentation and compression testing was analysed by least-squares regression.

Results: In total, 1140 indentations in total were performed. Indentation was found to be repeatable for indentations performed at the same locations on the bone samples with a mean co-efficient of repeatability of 0.4 GigaPascals (GPa), confidence interval (C.I): 0.33-0.42 GPa. There was variation in the indentation modulus results between different locations on the bone samples (mean co-efficient of repeatability: 3.1 GPa, C.I: 2.2-3.90 GPa). No clear correlation was observed between indentation and compression values of bone elastic modulus (r = 0.33, p = 0.17). The mean apparent elastic modulus obtained by spherical indentation was 9.9 GPa, the standard deviation for each indent cycle was 0.11 GPa, and the standard deviation between locations on the same sample was 1.01 GPa. The mean compression apparent elastic modulus was 4.42 GPa, standard deviation 1.02 GPa.

Discussion: Spherical-tip indentation was found to be a repeatable test for measuring the elastic modulus of human cortical bone, demonstrated by a low co-efficient of repeatability in this study. It could not, however, reliably predict cortical bone elastic modulus determined by platens compression testing in this study. This may be due to indentation only probing mechanical properties at the micro-scale while platens compression testing assesses millimetre length-scale properties. Improvements to the testing technique, including the use of a larger diameter spherical indenter tip, may improve the measurement of bone stiffness at the millimetre scale and should be investigated further.

Fig 1. Stress-strain curve.

Stress-strain curve demonstrating the proposed “safe elastic range” of bone deformation during press-fit cementless joint replacement surgery. A “safe elastic range” is demonstrated. A comparison is shown between healthy bone (black curve) and more porous, osteoporotic bone (blue curve).

Fig 2. Sample preparation.

(a) The femoral neck cut from the femoral head using a water-cooled bandsaw. (b) Femoral neck after sectioning with the bandsaw, displaying the thicker cortical bone on the medial, calcar area of the femoral neck. (c) Custom-made clamp (sample grip) with the four screws gripping the cortical bone section, positioned for the 3 mm-apart parallel cuts. The sample was orientated so that the wafering blade cut the bone perpendicular to the arrow marked on the bone. Arrow corresponds to the direction of the osteons. (d) 6x3x3 mm rectangular parallelepiped cortical bone sample from the medial femoral neck cortex. The 6mm length surface is parallel to the direction of the osteons (marked with blue arrow).

Fig 3. Experimental setup.

(a) Cortical bone sample mounted in the Micro Materials NanoTest indentation machine. A 1.5 mm diameter, spherical, ruby tip is being used to indent the bone in the direction of the osteons. (b) 6x3x3 mm rectangular parallelepiped sample being compressed between two stainless steel platens on a screw-driven Instron materials testing machine.

Fig 4. Load-displacement graph of indentation.

Load-displacement graph showing indentation in six different locations on the same sample. The bone was loaded to 10000 mN (10 N), there was a 60 second hold period and then it was unloaded to 5 N, followed by immediate reloading to 10 N, with another 60 second holding period at 10 N. There were 10 cycles of loading and unloading at each indent location, followed by complete unloading. The indenter then moved to a new location on a pre-defined grid and there were a further 10 cycles. With 10 cycles of indentation at 6 different locations, 60 indentations were performed in total for each sample.

Fig 5. Compression testing stress-strain graph.

Graph displaying strain on the x-axis and stress (in GigaPascals) on the y-axis from one of the cortical bone samples that underwent compression testing in the materials testing machine in this study. The elastic modulus was calculated from the slope of a best-fit line at the steepest part of the linear section of the stress-strain curve, over a 1% strain range, which is displayed on the graph.

Fig 6. Repeatability of indentation measurements of bone elastic modulus.

Distribution plot of the co-efficient of repeatability measurements. The co-efficient of repeatability values (GigaPascals) are on the x-axis. The y-axis displays the relative probability. Two graphs are plotted: In blue, the co-efficient of repeatability values for repeated indentations at the same locations are plotted. There were six different locations on each of the 19 patient bone samples so there are 114 data points for repeated, same location, indentation repeatability co-efficient values. In red, the co-efficient of repeatability values for indentations at different locations are plotted. There were 19 data points, corresponding to the 19 patient samples. The mean co-efficient of repeatability was 0.4 GigaPascals (GPa) (confidence interval (C.I): 0.33–0.42 GPa) for indentations at the same bone locations and 3.1 GPa (C.I: 2.2–3.90 GPa), at different locations on the bone samples.

Fig 7. Indentation modulus vs compression modulus scatter plot.

Scatter plot of the compression apparent elastic modulus (GPa) compared to the indentation apparent modulus (GPa). The regression line plotted has an r value of 0.33, p = 0.17.

Fig 8. Schematic of compression testing, compared to indentation testing.

Left: Bone (B) undergoing compression testing between two stainless steel platens (P). The blue arrow indicates the load being applied. Centre: Bone undergoing indentation using the 1.5 mm diameter, ruby, spherical indenter tip (R) used in this study. Right: Bone undergoing indentation using a proposed larger diameter, spherical indenter tip (S).