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Determination of E-modulus of cancellous bone derived from human humeri and validation of plotted single trabeculae: Development of a standardized humerus bone model
∗Corresponding author: Alexander Jahnke. alexander.jahnke@ortho.med.uni-giessen.de
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Received: ,
Accepted: ,
This article was originally published by Reed Elsevier India Pvt. Ltd. and was migrated to Scientific Scholar after the change of Publisher.
Abstract
Abstract
Evaluation of the mechanical behavior of the microstructure of cancellous bone seems important for the understanding of the mechanical behavior of bone. Prevention and treatment of fragility fractures due to osteoporosis is a major challenge according to ageing population. A bone model might help to assess fracture risk. Measurement of single trabeculae of bone should give further information compared with bone densitometry alone. This study measures the mechanical properties of single cancellous trabeculae derived from human proximal humerus.
34 single trabeculae dissected from human humeral heads were measured and evaluated mechanically. Trabeculae were fixed on microscope slides and geometrical data were reported during axial rotation of the specimens to measure the transverse section using computer aided design (CAD). The samples were subjected to a two-point bending test and were loaded with a measure-stamp at a defined distance. Force and deflection were measured by high-resolution sensors. The E-modulus was then calculated in combination with finite elements method simulation (FEM), using the previously obtained CAD-Data.
The average E-modulus from 34 valid measurements of human humeral trabeculae was 1678 MPa with a range from 829 to 3396 MPa, which is consistent with existing literature. The planned additional validation of the measurement method using manufactured three-dimensional synthetic trabeculae with known mechanical properties showed an average elastic modulus of single trabeculae of 51.5 MPa, being two dimensions lower than the value reported in the datasheet of the plastic.
This newly developed, time and cost-efficient procedure allows the measurement of E-modulus in single trabeculae. Measurement of mechanic parameters of single trabeculae might give insights on mechanic behavior of bone and be relevant for the research of systemic bone diseases, complementing the existing data on bone-mineral-density. Further examination of single trabeculae of human cancellous bone should give an insight on the mechanical behavior of bone also considering systemic bone diseases.
Keywords
E-modulus
Single bone trabecula
Mechanic properties
Synthetic cancellous bone
Osteoporosis
Fracture
1 Introduction
Fractures of proximal humerus are common in the elderly population and account for up to 20% of osteoporotic fractures.1–3 Bone densitometry often underestimates actual bone density, leading to a high prevalence of osteoporotic fractures in the elderly population1 and showing the need of an additional assessment parameter. The management of those fractures remains controversial.3 A further increasing number of osteoporotic fractures at different locations might be expected.1,2,4 Open reduction and internal fixation might be necessary depending on the type of fracture.2,5 Locked plate fixation of osteoporotic fractures of the proximal humerus often fails due to loosening and cutting-out of screws, according to a significant difference in stiffness between bone and implant.5 At present, no suitable bone model exists to examine the load transfer at the implant-bone interface. The mechanical properties of the microstructure of cancellous bone needs further research since the prevention and management of osteoporotic fragility fractures seems more important in the future.6 The examination of single trabeculae of bone might give more information on the fracture risk7 and help to develop a suitable model to predict fracture risk. Beside the bone mineral density, the microstructure of the bone has an important influence on the mechanical behavior of bone since the risk of fracture is not solely dependent on bone mineral density.8 This points out the necessity for a suitable bone model to allow the early treatment of osteoporosis.9 Studying the smallest unit of bone, the single trabeculae might be useful for that purpose.
The E-modulus seems the most important material characteristic of a single trabecula of cancellous bone. Literature reports a wide range of values of elastic modulus of single trabeculae from only small numbers of specimens.10–14 Other studies were based on strongly simplified assumptions of the geometry of cancellous bone, with only two orthogonal measurements in the middle of the trabeculae.15 Examinations of animal bones are limited regarding differences in morphology.13,15 This study examined single trabeculae from human cancellous bone derived from five proximal humeri. Following individual measurements of the geometry, bending tests were performed on 34 trabeculae.
2 Material and methods
2.1 Trabecular preparation
The human humeri were derived from five persons, aged between 81 and 91 years. The bones were commercially available (Science Care, Phoenix, USA). No disease of bone metabolism was known and ethical approval was obtained (ethics committee vote 303/11). First, ten specimens with a size of 5 × 5 mm were macroscopically taken from each humerus. The specimens were then dissected using scalpels and a stereomicroscope with zoom 7–45x to create a single trabecula (Expert Trino, Müller® Optronic Erfurt, Germany). To prevent the disappearance of the small bone pieces the cancellous bone was irrigated with normal saline solution. The trabeculae were fixed with tweezers. After drying the trabeculae on the microscope slides due to the warmth of the microscope illumination, the specimens were fixed to the edge of the glass using cyanoacrylate glue and geometry was determined. For logistic reasons, several specimens were prepared simultaneously. The bone trabeculae were then stored in an airtight fridge with a temperature of 4–8° Celsius and mechanical testing was performed within 48 h later.
2.2 Production of synthetic-trabecula specimens
The synthetic specimens were produced at the University of Applied Sciences of Mittelhessen (THM, Gieβen, Germany) using a three-dimensional printer Stratasys® Connex 350 3D-Printer working with the polyjet procedure (Stratasys Ltd. Eden Prairie, Minnesota, USA). The minimal slice thickness of one printable layer was 16 × 10−6 m. The model of the humerus was supplied from a computer file of a human humerus bone, which was generated from a nano computer tomographic (CT) scan. The cylindrical model consists of 5 million virtual triangles which define the structure of the surface. The material Fotopolymer Durus® (Stratasys Ltd. Eden Prairie, Minnesota, USA) belonging to the material class “PP-like Materials” was chosen according to the elastic modulus mentioned in the datasheet, ranging from 1000 to 1200 MPa. For the printing of cavities an additional support material (SUP 706TM (Stratasys Ltd., Eden Prairie, Minnesota, USA)) is necessary, which can be removed by a jet of water or using a chemical solvent (stratasys.com, 2019). The further preparation of the synthetic trabecula was carried out like that of the human specimens.
2.3 Measurement of the geometry of trabeculae
Measurement of the geometry of the trabeculae to determine the moment of inertia is difficult according to the different three-dimensional geometry of the specimens. Nano-CT-scan and 3D-Photography are time-consuming and not completely reliable for those very small dimensions. Therefore, the geometry was measured approximately using a combination of different projection surfaces. Every single trabecula was axially rotated 180° within the calibrated microscope reaching 13 steps. Within each angle, a photography with 2040 × 1580 pixels was taken using the program Capture® (Fuzhou Xintu Photonics C., Ltd., Fuzhou, China). The pictures were processed using the program ImageJ (Open Source, Wayne Rasband, USA). From every angle of view, the average diameter of the trabeculae can be calculated from the projection surface based on a simple surface formula (diameter d = surface A/length l) since l is the length of the trabecula being the distance of 500 μm from the microscope slide to the measurement device (Fig. 1).
From the obtained thirteen diameters per trabecula, a two-dimensional cross-section can be calculated and CAD software (Autodesk®-Inventor 2012 (Autodesk Corporation, San Rafael, USA)) might compute the related area moment of inertia. (Fig. 2).
2.4 Experimental setup
The measurement of the specimens was done with the universal test device “Inspect Table Blue 20 kN” (Hegewald & Peschke, Nossen, Germany). The test device can measure the position of the specimens and the force as well as the displacement. Within the relevant range of forces, the machine does not provide sufficient resolution and therefore the force acting on the specimens was measured with a S2M sensor (HBM, 64293 Darmstadt, Germany). The range of measurement of the sensor is 0–10 N with an accuracy of±0.02 N. Related to the dimensions of the values, this accounts for a potential deviation between 2 and 20%. Mechanical feed was measured with the universal test device, which ensures a measuring error of less than 1 μm. We found an error of 0.8% at a deflection of 500 μm which is 4 μm in absolute value. The linear range of measurement, which is relevant for the measurement, was maintained up to 300 μm deflection and therefore a deviation should not exceed 2.4 μm. The velocity of the bending force application was 0.05 mm/min.
The trabecular bone was clamped between two slides (as commonly used for microscopy) and loaded for bending at a defined distance (l = 500 μm) from the measuring edge using a measuring punch with a precise edge (Fig. 3).
A Spider 8 measuring amplifier was used for the force sensor and data were recorded with the CatmanEasy program (HBM, 64293 Darmstadt, Germany). For data storage of the measured distances, an individually designed LabView program (Version “LabView 12”, National Instruments, Austin Texas, USA) was employed. Diagrams for force and way were calculated from the data (Fig. 4).
From the chart curve slope of the linear area, the stiffness k could be calculated. Since the stiffness of the system had to be taken into consideration, an empty measurement was performed at different levels of force. The corresponding system stiffnesses could be separated from the stiffness of the specimen using the equation of springs in series.
Compared with the length of the trabeculae the diameter was big and the deflection was high. Therefore, the usage of the math equation of beam was not possible and the calculation of the results was performed with FEM analysis. A virtual two-point bending test, using similar values, was performed with test samples of dimensions comparable with human bone. Measurements of stiffness from these test samples and the calculations from the FEM analysis showed a difference of 10.24%. The proportionality of elastic modulus (E) and stiffness (k) was determined using two different assumed elastic moduli (677 and 706 MPa) for the calculation of the FEM model. Proportionality between elastic modulus and stiffness could be shown with nearly no difference, which was related to mathematical rounding. Though the relation between stiffness and elastic modulus can be expected similar for the model and the bone specimens. This proved that there is a proportionality between the modulus of elasticity and stiffness. The further procedure is based on the consideration that the ratio of stiffness and modulus of elasticity must be the same for both the model and the sample.
Math equation (1) Emodel is assumed with 10.000 MPa and stiffness kmodel results from the simulation of force application in the model with a defined virtual force of 1 N. Stiffness kspecimen is being measured experimentally and though the elastic modulus Especimen can be calculated:(1)Especimenkspecimen=Emodelkmodel
2.5 Statistics
A descriptive statistical analysis was performed since hypotheses concerning correlations between elastic modulus E and age and gender were not the topic and therefore analytical statistical methods did not need to be applied.16 Those would not appear useful for the small number of specimens per individual of n ≤ 10 either. The measurement of arithmetic mean, median, standard deviation, variance, and confidence interval was done. Further, a logarithmic scale was used to transform the data to a log-normal distribution, since the original data showed a shift to the left. The logarithmic scale appears useful for the examination of growth processes which can be expected multiplicative since growth is concerned essentially with the multiplication of living substances.17 In case of logarithmic normal distribution, the logarithmic random variables are within the normal range of values. Represented as equation (2): X is log-normally distributed when Y is normally distributed.(2)Y=ln(X)
Statistical key figures being generated using logarithmic normal distribution need to be calculated into a non-logarithmic scale to provide a clear presentation. According to different curves of distribution for logarithmic and non-logarithmic data, math equations are necessary (μ for expected value, σ for standard deviation of Y).
Median value:(3)μ∗=eμexpected value:(4)E(X)=eμ+σ22standard deviation:(5)S(X)=E(X)∗eσ2−12
Measurement of the confidence interval was performed according to the Cox method.18 Considering the desired confidence interval of 1.96, the calculation includes the number of specimens (n) and multiplication of standard deviation (z):(6)k=Y‾+S²2±zS²n+S42(n−1)
The calculated values need to be transformed back from the logarithmical form with ek to gain the confidence intervals.
3 Results
Each ten humerus trabeculae from five donors were measured. Trabeculae were excluded from the examination when showing less than 0.1 N force resistance because this could be a sign of prior damage. Further measurements with resulting implausible diagrams for correlation of force and distance were also excluded from the study. Therefore, the number of examined specimens was n = 34 (Table 1).
| Group | Sex | Taken out because of damage | Removed for other reasons | Number of valid measurements | Median values of the E-modulus [MPa] |
| H1 | female | 1 | 0 | 9 | 2100 |
| H2 | female | 1 | 3 | 6 | 3939 |
| H3 | male | 1 | 1 | 8 | 306 |
| H4 | male | 5 | 1 | 4 | 337 |
| H5 | male | 3 | 0 | 7 | 199 |
| Arithmetic mean of the median values | 1376.2 | ||||
The results revealed a wide range of values in comparison between different groups and partially within single groups. Therefore, the boxplot diagrams of the elastic modulus E were scaled logarithmically (ln). Within a group, the median value was taken instead of the arithmetic mean, because of the high dispersion of the values (Fig. 5).
![Boxplot of E-modulus [MPa] by trabecular groups H1–H5, log scaled with ln.](/content/220/2022/33/1/img/S0972978X22001532-gr5.jpg)
Examination for normal distribution was performed within the complete number of specimens since single groups with n = 4–9 did not appear sufficient in number. The comparison of the results and the regression line within a QQ diagram and analysis according to Kolmogorov-Smirnov and Shapiro-Wilk, revealed a normal distribution for the complete logarithmic sample. This is shown by the near symmetry within the boxplot diagram (Fig. 6).
The calculation of the results was based on the logarithmic values of the elastic modulus since there was a normal distribution. Conversion in the non-logarithmic range of scale was done according to the described mathematical formula. The mean of the modulus of elasticity of the pooled humerus trabeculae of 1678.52 MPa calculated in this way is slightly higher than the average value of the median values of the individual groups of 1376.2 MPa, which was determined in the course of the work to select a printable plastic (Table 2).
| Statistics | |||
| Emod | Mean | 1678.52 | |
| 95% confidence interval of the mean value | Lower limit | 829.40 | |
| Upper limit | 3396.46 | ||
| Median | 407.27 | ||
| Standard deviation | 4514.33 | ||
The results of the plotted mean values of the trabeculae (51.5 MPa) are below the datasheet values of the plastic Durus White® (1.000–1.200 MPa) (Table 3).
| Statistics | |||
| Emod | Mean | 51.50 | |
| 95% confidence interval of the mean value | Lower limit | 19.64 | |
| Upper limit | 152.66 | ||
| Median | 26.26 | ||
| Standard deviation | 53.65 | ||
4 Discussion
The E-modulus was determined in n = 34 human humerus trabeculae using a bending force and followed by FEM simulation. The overall average of the elastic modulus was 1678 MPa and the 95% confidence interval ranged from 829 to 3396 MPa. The lower range of the elastic modulus value is according to literature10–15 and several studies reported a wide range of distribution.15,19 The existing studies on the subject have only a small number of cases examined10–14 or, if they examined higher numbers of cases, they made highly simplified assumptions when determining the trabecular geometry with only two thickness measurements in the center of the trabeculae that are orthogonal to each other.15 The examination of animal specimen material13,15 in some studies is also not unproblematic for further use of data in human medical research, since conclusions from animals to humans are possible, but not always necessarily correct.
The anisotropy of the properties of trabeculae20,21 further complicates the determination of material properties and is a possible cause for the high scatter of literature values. In addition, the anisotropy limits the generalizability of the data obtained, since, strictly speaking, the values determined only apply to exactly the respective type of force application and loading. In addition to the variance of the measurement methods used, the high variance of the results is also striking. On the one hand, this may be due to the differences in the measurement methods and specimens, but on the other hand it also gives an indication of the high complexity of the problem and the high demands on the measurement methods. Bini et al.,10 for example, were only able to apply the planned loading procedure to three of 26 prepared specimens. The loaded cross-sections were assumed to be ideally symmetrical, as was also the case with Busse et al. which, in turn, harbors an enormous potential for error due to the directional dependence of the moment of inertia of the surface, as shown clearly before.10,22
Generally, the examination of small measured variables increases the proportional measurement error, as the deviation is more significant in percentage terms.23 On the other hand, the preparation of the bone specimens might induce invisible damage to the bone, generating a scattering of the measured values. Small deformations of 3% within plastic behavior lead to a reduction of the elastic modulus of 80%.24 Dispersion of the values might also be related to inaccuracies in the measurement of sample geometry. This might be related to the approximated procedure and especially concerning the synthetic material, it cannot be excluded that some support material remains on the synthetic trabeculae. Within the human specimens remaining soft tissue and fat derived from the medullary space of the bone could alter the measurement of geometry. The enormous influence of the geometry of the trabeculae on the E-modulus is related to the fact that the diameter of the samples is considered with the fourth power in the formula for the area moment of inertia.
The influence of geometry on measurement errors was found more pronounced in bending force applications than in tension tests since a deviation of diameter of 10% created a variance of the E-modulus of 40%.19 A further source of error related to geometry might be the homogenization of the diameter of the trabeculae during optical measurement. Constrictions and notches are not being considered in measuring the average diameter of trabeculae, but those morphological changes might reduce stiffness significantly. This seems related to a systemic underestimation of the calculated elastic modulus and might induce a diversification of the values in case of stochastic appearance.
Another point that should certainly be considered in future work is the question of the viscoelastic properties of bone, since it has organic and inorganic components. In this study, however, only the elastic properties in the form of Young's modulus were investigated. With a few modifications, however, the experimental setup might be able to determine basic viscoelastic properties, such as whether bone is a Kelvin or Maxwell body.
Compared with other studies in the literature,10–14 the present study examined a larger amount of 34 samples giving evidence for the efficient preparation technique of single trabeculae since the measured values appeared reproducible. Therefore, this method enables the effective processing of questions concerning the elastic modulus of human bone. Intraindividual and interindividual differences of the E-modulus as well as the influence of systemic diseases on bone metabolism can be examined in vitro.
The measurement procedure in this study is an integral method since the specimens are being tested entirely. Other acoustic and mechanic measurement methods also measure complete specimens however the non-integral nanoindenter method11 needs to rely on the mechanical properties of the microscopic stamp impression for the whole sample. This seems to be a relevant factor for the uncertainty of those methods, regarding the anisotropy of bone.
The loading of trabeculae in the bending test leads to stress in the form of tension and pressure, which is physiological for bone tissue. In contrast, acoustic measurements, which do not use physiological application of force, measure systematically higher values of elastic modulus compared with mechanical examination methods. Elastic modulus measurement of single trabeculae using ultrasound appeared 42.3% higher compared with mechanic testing.25
4.1 Conclusion
The newly developed measurement procedure offers advantages concerning the efficacy, the possibility to measure samples as a whole and the application of physiological strain. Further improvement of the method regarding removal of support material from synthetic material and optimization of the preparation technique of single human trabeculae seems possible. Measurement of single trabeculae appears suitable to give further insight into the mechanic behavior of the smallest element of human bone. This might be useful for the assessment of mechanical properties of bone in systemic disease and help to predict fracture risk. Also, it might help to develop standardized humerus bone models.
Funding/sponsorship
The purchase of the fresh frozen humeral heads was supported by the Association for Orthopaedic Research (AFOR Stiftung, Olten, Switzerland).
Institutional Ethical Committee Approval.
Positive ethics vote of the ethics committee of the Justus-Liebig-University (file number 303/1).
Credit authors statement
Florian Kuhn: Methodology; Investigation; Data Curation; Writing- Original draft preparation.
Rasmus Johannes Clausing: Methodology; Investigation; Data Curation.
Alexander Stiller: Software, Validation.
Carlos Alfonso Fonseca Ulloa: Software; Data Curation; Writing - Review & Editing.
Christian Foelsch: Writing - Review & Editing.
Markus Rickert: Resources; Conceptualization; Supervision; Writing - Review & Editing.
Alexander Jahnke: Conceptualization; Supervision; Writing - Review & Editing; Project administration.
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