- Email: [email protected]
The human proximal femur behaves linearly elastic up to failure under physiological loading conditions
Journal of Biomechanics 44 (2011) 2259–2266
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
The human proximal femur behaves linearly elastic up to failure under physiological loading conditions Mateusz Maria Juszczyk a, Luca Cristofolini a,b,n, Marco Viceconti a a b
Laboratorio di Tecnologia Medica, Istituto Ortopedico Rizzoli, Bologna, Italy Facolta di Ingegneria, Universita di Bologna, Italy
a r t i c l e i n f o
abstract
Article history: Accepted 27 May 2011
It has not been demonstrated whether the human proximal femur behaves linearly elastic when loaded to failure. In the present study we tested to failure 12 cadaveric femurs. Strain was measured (at 5000 Hz) on the bone surface with triaxial strain gages (up to 18 on each femur). High-speed videos (up to 18,000 frames/s) were taken during the destructive test. To assess the effect of tissue preservation, both fresh-frozen and formalin-fixed specimens were tested. Tests were carried out at two strain-rates covering the physiological range experienced during daily motor tasks. All specimens were broken in only two pieces, with a single fracture surface. The high-speed videos showed that failure occurred as a single abrupt event in less than 0.25 ms. In all specimens, fracture started on the lateral side of the neck (tensile stress). The fractured specimens did not show any sign of permanent deformation. The force– displacement curves were highly linear (R2 40.98) up to 99% of the fracture force. When the last 1% of the force–displacement curve was included, linearity slightly decreased (minimum R2 ¼ 0.96). Similarly, all force–strain curves were highly linear (R2 4 0.98 up to 99% of the fracture force). The slope of the first part of the force–displacement curve (up to 70% fracture force) differed from the last part of the curve (from 70% to 100% of the fracture force) by less than 17%. Such a difference was comparable to the fluctuations observed between different parts of the curve. Therefore, it can be concluded that the proximal femur has a linear-elastic behavior up to fracture, for physiological strain-rates. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Proximal femur Femoral neck and head Linear elastic behavior In vitro bone fracture Brittle failure
1. Introduction The strain distribution and strength of the proximal human femur has been extensively investigated in the past, both in vitro (Keyak et al., 2001; Bessho et al., 2007; Cristofolini et al., 2009) and with numerical models (Lotz et al., 1991; Ota et al., 1999; Schileo et al., 2007). One of the open questions concerns the linear (or the lack of linear) behavior of the femur. In fact, assessment of the linear behavior would improve the understanding of the etiology of femoral fractures (Cristofolini et al., 2010). Furthermore, because of the increasing use of Finite Element (FE) models of the human femur, it would be essential to include the most
Abbreviations: BL, biomechanical length of the femur; BW, body weight; CT, computed tomography; DEXA, dual energy X-ray absorptiometry; FE, finite element; HD, diameter of the head of the femur; R2, coefficient of determination for a linear regression; VPH-OP, Virtual Physiological Osteoporotic Human Project (http://www.vphop.eu/) n Corresponding author at: Laboratorio di Tecnologia Medica—Istituto Ortopedico Rizzoli, Via di Barbiano, 1/10, 40136 Bologna, Italy. Tel.: þ 39 051 63 66 864; fax: þ39 051 63 66 863. E-mail addresses: [email protected], [email protected] (L. Cristofolini). 0021-9290/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2011.05.038
appropriate rheological model (whether linear-elastic, or elastoplastic). In many cases, bone is modeled as a linear-elastic material (Carter and Spengler, 1978; Yosibash et al., 2007; Helgason et al., 2008). In fact, it has been reported that in tension cortical bone tissue exhibits a linear-elastic behavior up to failure (Carter and Spengler, 1978; Fung, 1980), while post-yield nonlinearity is observed in compression (Keller, 1994). Kopperdahl and Keaveny (1998) and Keaveny (1994) reported a linear pre-yield behavior for the trabecular bone, followed by a plastic deformation both in tension and compression (such a post-yield nonlinearity was much less pronounced in tension). Yield of bone tissue has sometimes been described as an arbitrary point in the stress–strain curve where nonlinearity appears (Hvid and Jensen, 1984) or as a point at which permanent deformation occurs (An and Draughn, 1999). A mechanistic model (Yeni and Fyhrie, 2003) has shown that bone nonlinearity, which is sometimes referred to as post-yield behavior, can be explained by collagen–fiber-bridged microcracks rather than by real plastic post-yield phenomena, as (Keaveny et al., 1994) previously suggested based on in vitro tests. A combination of tension and compression is present in the femoral neck for most loading conditions. As it is impossible to
2260 Table 1 Summary of the literature reporting in vitro destructive tests where the proximal femur was loaded in a quasi-axial loading configuration. The loading rate (time required to fracture the specimen) and the direction of the applied force (angle between the femoral diaphysis and the force on the femoral head) are reported for each reference. Also indicated is the number of specimens tested and the gender and age of donors. The description of the fracture mechanism is either based on the description provided in the text, or on the pictures and plots available in the manuscript. Load rate/time required to induce fracture
Direction of applied force
Specimens: number (preservation)
Gender, age of donors
Fracture mechanism
Alho et al. (1988)
66–390 s (1)
01 in frontal plane 01 in sagittal plane
36 (preservation not specified)
18 female, 18 male 57–87 years old
In most femurs the load–deflection curve was relatively linear, had a gradual curving without any yielding point. A yielding point was identified only in few cases
Augat et al. (1996)
1.0 mm/s (time to fracture of the order of 10–20 s (1))
01 in frontal plane 01 in sagittal plane
20 (formalin fixed)
13 female, 7 male 66–100 years old
Neck (n¼ 18) and trochanteric fractures (n¼ 2). Presence/lack of linearity was not expressly reported
Bessho et al. (2007)
180 s (1)
201 in frontal plane 01 in sagittal plane
11 (fresh-frozen)
6 female, 5 male 52–85 years old
Fluctuations in the force–displacement curve. The yield point indicated in the plot corresponds to 80% of the failure load. The entire curve fits within a band that is narrower than 10% of the failure load
Dalen et al. (1976)
10 s (1)
01 in frontal plane 01 in sagittal plane (plastic block around the specimen up to lesser trochanter)
61 (fresh)
54 female, 7 male 67–80 years old
Linear-elastic plot with small decrease of slope in the upper 30% of the force–displacement curve
Delaere et al. (1989)
2.0 mm/s (actual time to fracture not indicated)
251 in frontal plane 01 in sagittal plane
20 pairs (dried macerated)
Male þfemale 57–89 years old
Not reported
Keyak (2001), Keyak et al. (2001)
0.5 mm/s (time to fracture of the order of 10–20 s (1))
201 in frontal plane 01 in sagittal plane
18 (fresh-frozen)
10 female, 8 male 52–92 years old
Subcapital and oblique transcervical fractures parallel to the shaft. Presence/lack of linearity not expressly reported. The force–displacement plot in Keyak (2001) is linear, with a two-step fracture
Link et al. (2003)
‘‘stepwise’’ (actual rate not specified)
111 in frontal plane 01 in sagittal plane
31 (fresh-frozen)
14 female, 17 male 29–91 years old
Linear-elastic up to failure
¨ Lochmuller et al. (1998)
1.0 mm/s (load cycles of increasing magnitude)
01 in frontal plane 01 in sagittal plane
58 (formalin fixed)
24 female, 34 male 57–100 years old
Trochanter (n¼ 10) and neck fractures (n¼48). Presence/lack of linearity was not expressly reported
¨ Lochmuller et al. (2002)
6.5 mm/s (time to fracture of the order of 1–3 s (1))
01 in frontal plane 01 in sagittal plane
103 (alcohol/ formalin fixed)
62 female, 41 male 46–97 years old
Trochanter (n¼ 10) and neck fractures (n¼83). Presence/lack of linearity was not expressly reported
Lotz et al. (1991)
Not indicated
01 in frontal plane 01 in sagittal plane
1 (fresh-frozen)
Female 66 years old
The load–displacement plot shows fluctuations, including a slight decrease of slope at 50% of the failure load. The entire curve fits within a band as wide as approximately 10% of the failure load
Ota et al. (1999)
0.5 mm/s (time to fracture of the order of 10–20 s (1))
201 in frontal plane 01 in sagittal plane (2)
1 (formalin fixed)
1 male 76 years old
Base-of-the-neck fracture, parallel to the shaft. Presence/lack of linearity was not expressly reported
Yosibash et al. (2010)
26 s (1)
01 in frontal plane 01 in sagittal plane
1 (fresh-frozen)
Male 30 years old
One step fracture following limited post-elastic nonlinearity (5% decrease of linear slope occurred at 67% of maximum force)
Notes: (1) Time required to fracture the femur specimens was not expressly indicated in the manuscript. The value reported was estimated based on the plots/tables. (2) Direction of the applied force not specified: the angle was estimated based on the published pictures.
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
Reference
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
predict if the linearity of cortical bone in tension will prevail (or not) over the nonlinearity of the cortical and the trabecular bone in compression, it is also impossible to determine a priori if the structural behavior of a whole femur is linear or nonlinear when loaded to failure. When whole femurs were tested to failure with a quasi-axial load, in some cases a brittle fracture was observed, while in others significant nonlinear post-yield behavior was observed (Table 1). According to a recent review, no study so far has focused on the linear (or nonlinear) behavior of the femur as a structure (Currey, 2009). In a study by Keyak (2001), the femur was modeled as a nonlinear material, based on force–displacement plots obtained in vitro when failure was induced in approximately 10 s (it must be noted that no additional measurement as such from the strain gages was available to confirm such a nonlinearity). The only in vitro plot reported in that paper (Keyak, 2001) can be interpreted as a two-step fracture, rather than as an elasto-plastic curve. In a later study (Bessho et al., 2007) the femur has been reported to behave nonlinearly: when eleven femurs were loaded at a low rate (cross-head speed: 0.5 mm/min, such that failure occurred in minutes) some fluctuations could be observed in the force–displacement curve. However, as ‘‘yield’’ was defined somewhat arbitrarily (Bessho et al., 2007), the amount of nonlinearity cannot be quantified. Other authors reported that the proximal femur behaved linearly (or nearly so) up to failure (Alho et al., 1988; Yosibash et al., 2010). In many cases, although the linear/ nonlinear behavior of the femur was not expressly discussed, linear plots are shown (Dalen et al., 1976; Link et al., 2003). A possible explanation of the discrepancies in the literature concerning linearity of bones (Table 1) lies in the viscoelasticity of the bone. In fact, it has been shown that both the modulus of elasticity and nonlinearity of the stress–strain curve strongly depend on the loading rate (Carter and Spengler, 1978; Lakes and Katz, 1979; Zilch et al., 1980). Viscoelasticity could therefore explain why under certain conditions bone behaves linear-elastic, while in others it shows a strong nonlinearity. Strain-rates in the human tibia during walking and running has a range of 0.002–0.013 s 1 (Lanyon et al., 1975), while during vigorous activity (e.g. uphill and downhill zigzag running) the strain-rate can reach 0.034 s 1 (Burr et al., 1996). In vivo recording from telemetric hip prostheses has shown that the load peak during mild activities (e.g. walking) is reached in 0.3–0.5 s, while the peak during occasional overloading (e.g. stumbling) is reached in 0.1–0.2 s (Bergmann, 2008). It is still
2261
not clear whether for such physiological strain-rates bone behaves linearly or not. The aim of this work was to investigate whether the human proximal femur can be assumed to behave in a linear-elastic way up to failure for clinically relevant simulations.
2. Materials and methods 2.1. Overview In this study we re-analyzed data from destructive tests that were carried out on human femurs in our lab in the last six years. This enabled exploring the effect of two factors on the linearity of destructive tests:
Loading rate: Different loading rates were applied to investigate how linearity is affected within the physiological range.
Tissue preservation: Both fresh-frozen and formalin-fixed femurs were tested.
2.2. Bone specimens A total of 12 human cadaveric femurs were included for this study, which were obtained through different ethically-approved international donation programs (Table 2). Eight femurs were fresh-frozen and four were fixed with 4% ¨ hman et al., 2008). Fresh-frozen femurs were de-frozen 2–3 times (for formalin (O a total of 36–120 h) during the different steps of preparation and testing, and were stored at 25 1C when not in use. During the test, all the femurs were wrapped in clothes soaked with physiological solution at a room temperature of 27 30 1C. All femurs were DEXA-scanned (Excel-Plus, Norland, USA) and CT-scanned (Hi-Speed, General-Electric, USA) to document bone quality and lack of abnormality. The femurs were prepared with a set of anatomical reference frames to allow for reproducible alignment throughout the test (Cristofolini et al., 2007). The femurs were distally potted in a steel box with acrylic cement (Cristofolini et al., 2007). The level of the distal constraint varied between 33% and 80% Biomechanical Length of the femur (BL), measured from proximal, as the specimens were part of different studies.
2.3. Strain measurement Each femur was instrumented with up to 18 triaxial-stacked strain gages following a validated procedure (Cristofolini et al., 2009) (KFG-3–120-D1711L3M2S, Kyowa, Tokyo, Japan). A grid excitation of 0.5 V was selected to avoid heating. Strains and force– displacement data from the testing machine were sampled at 5000 Hz using
Table 2 Details of the specimens investigated. In the first four columns, details of the donors are listed. Bone quality is reported in the 6th column for each femur (T-score of the bone density referred to the young reference population, based on the Norland DEXA scanner reference population). Biomechanical dimensions (biomechanical length, BL, and head diameter, HD) (Cristofolini, 1997) are listed. The level of the distal constraint was measured from the proximal extremity of the femur to the upper surface of the acrylic cement pot, and is expressed as a % of the Biomechanical Length. Loading rate (High-strain-rate or low-strain-rate), and tissue preservation method (fresh-frozen or fixed with 4% formalin) are listed in the last two columns. Specimen
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12
Donors’ details
Femurs’ details
Testing conditions
Gender
Age at death
Donor height (cm)
Donor weight (kg)
Side
T-score
BL (mm)
HD (mm)
Level of distal constraint (%BL)
Loading rate
Tissue preservation
Male Male Male Male Female Female Male Male Female Female Female Female
71 82 73 77 80 74 73 83 72 72 77 77
178 175 183 173 160 180 173 175 166 166 171 171
91 78 86 70 122 170 91 84 61 61 64 64
Right Left Left Right Left Right Left Right Right Left Right Left
1.87 1.49 2.74 3.60 3.80 2.51 3.30 3.31 4.90 n.a. n.a. n.a.
443 445 476 411 396 448 441 468 427 427 415 412
53.3 53.7 48.8 48.5 44.1 51.5 52.6 54.1 47.5 47.5 46.5 47.5
80 80 33 33 33 33 33 33 33 33 33 33
Low Low High High High High High High High High High High
Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Formalin-fixed Formalin-fixed Formalin-fixed Formalin-fixed
2262
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
a 60-channel datalogger (System-6000, Vishay Micro-Measurement, Raleigh, NC, USA). Principal strains were computed based on the three grids of each strain gage.
2.4. Loading conditions The in vitro setup was designed based on previous experience with nondestructive (Cristofolini, 1997) and destructive testing of the femur (Cristofolini et al., 2007). In a previous combined experimental–numerical study (Cristofolini et al., 2007) the range of directions spanned by the hip joint force during daily activities (Bergmann, 2008) was explored to replicate spontaneous fractures of the proximal femur. In the study of Cristofolini et al. (2007), the direction of the hip force (81 in the frontal plane) that generates the highest risk of head–neck fractures was identified. A similar direction (101 in the frontal plane) was reported in
Keyak et al. (2001) as most critical for the neck. Thus, the in vitro loading configuration in the present study did not aim at replicating any specific motor tasks, but the direction of the resultant force that is most likely to induce spontaneous fractures during occasional overloading events. Unfortunately, these two studies (Keyak et al., 2001; Cristofolini et al., 2007) are based on a single femur each. At the same time, to the authors’ knowledge, other configurations in the literature (Table 1) were not based on any sensitivity study capable of identifying the most relevant loading condition. The femur was mounted on top of the load cell of the testing machine with the diaphysis at 81 in the frontal plane. Load was applied to the femoral head through a system of linear bearings to avoid transmission of horizontal force components (Fig. 1). A mould of each femoral head was prepared with acrylic cement (covering 1/5 head diameter) to allow uniform load transfer from the actuator to the head. Muscle forces were not simulated, as it has been shown that they do not significantly alter the stress distribution in the head–neck region (Cristofolini et al., 2007). To cover the range of physiological strain-rates, the femurs were divided into two groups, and force was applied at different speeds:
High-strain-rate group (n ¼10): The crosshead speed (typically 30 mm/s) was
tuned for each specimen so that the strain-rate in the most stressed regions was about 0.05 s 1. As bone tissue fails when strain exceeds 0.007–0.010 (Bayraktar et al., 2004), such strain-rate would generate failure in the order of 0.2 s. This is the typical timescale of physiological and para-physiological loading (Bergmann, 2008), and has been proposed for in vitro testing (Raftopoulos et al., 1993). Low-strain-rate group (n ¼2): A strain-rate ten times lower than high-strainrate was implemented to replicate quasi-static loading events, consistent with Cristofolini et al. (2007) and Cristofolini et al. (2009).
2.5. High-speed movie During the destructive test, the event was filmed by means of a high-speed camera (FastCam-X1024PCI, Photron, UK) at 8000–18,000 frames/s (actual frame rate depended on the size of the field of vision cropped for each specimen). The camera pointed at the superior-lateral part of the neck. Two mirrors were used to
Fig. 1. Testing setup showing a right femur with its distal pot (A) tilted 81 in the frontal plane. The strain gages are visible. Load was delivered through a mould (B) of the femoral head. Load was applied through a system of linear bearings (C) to avoid transmission of horizontal force components (Fig. 1). The two mirrors (D) are visible that enable acquiring the entire head–neck region with the highspeed camera.
Fig. 2. Example of a reassembled femur. A 250-micron needle (A) could not enter the remaining gap. The arrows indicate the fracture (B).
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
2263
film at the same time the anterior-medial and posterior-medial portions of the neck (Cristofolini et al., 2007).
2.6. After-failure inspection In order to evaluate the presence/absence of irreversible residual deformation of the fractured bone, the fractured bone parts were collected and reassembled. The fracture gap was evaluated by means of a hypodermic needle (diameter: 250 mm, Fig. 2).
2.7. Measurement accuracy and repeatability Force and displacement were measured with transducers having an overall precision of better than 0.5% of the readout. The strain gages and its datalogger provided an overall precision of better than 1% of the readout. Intra-specimen repeatability cannot be directly assessed in destructive tests. Based on previous non-destructive tests (Cristofolini et al., 2009), it can be estimated that both strain and displacement would vary by 1–4% if tests could be repeated on the same specimen.
2.8. Statistics Some nonlinearity was observed in the initial part of the force–displacement curves (Fig. 3). As such a toe effect was not observed in the force–strain plots (Fig. 4), the initial force–displacement was possibly due to the cartilage coming in full contact with the acrylic mould of the femoral head. Therefore, the initial part of the force–displacement curves (values below 1 kN) was not taken into account. The fracture force was defined as the largest force value recorded throughout the destructive test for each specimen. The linearity between force and displacement, and between force and strain was measured by the determination coefficient (R2) of the linear regression of the curves. In order to detect slope changes of the force–displacement and force– strain curves, the determination coefficients were calculated including data for increasing portions of the curve (up to 60%, 70%, 80%, 90%, 99% and 100% of the fracture force, for each specimen separately). To enable a comparison with the studies where nonlinearity for the femur has been discussed more in detail (Keyak, 2001; Bessho et al., 2007), the change of slope of the force–displacement was investigated. Because of the high linearity of such curves (see below) it was not possible to identify a yield point based on objective criteria (e.g. Cowin, 2001). Therefore, to enable consistent comparisons, the slope of the force–displacement curve was computed for the first part of the curve (up to 70% of the fracture force) and for the last part (from 70% to 100% of the fracture force). The arbitrary threshold of 70% is consistent with the ‘‘yield’’ point shown in the plots of Bessho et al. (2007). In order to estimate the central value of the distribution of the determination coefficients and of the slopes, the median values were calculated. All statistics were performed with StatView version 5.0.1 (SAS Institute Inc., Cary, NC, USA).
Fig. 3. Force–displacement graph for all the femurs. The vertical component of displacement (measured by the transducer of the testing machine) is plotted. The femurs are grouped according to the strain-rate and to the preservation technique. The gray band masks the lower part of the curves, where some toe effect is present due to the mechanical play of the setup. Error bars are not plotted for clarity.
Fig. 4. Typical force–strain plot for two typical femurs: (A) Maximum and minimum principal strains are plotted for each of the 17 strain gages for femur #2. Because of the preservation (fresh-frozen) and loading rate (low-strain-rate), this type of specimen can be expected to show worse linearity. (B) Maximum and minimum principal strains are plotted for each of the 14 strain gages for femur #12. Because of the preservation (embalmed) and loading rate (high-strain-rate), this type of specimen can be expected to show the best linearity. The trend for the other specimens (not reported here for brevity) was quite similar, independent of the preservation (fresh-frozen or embalmed) and strain-rate (high or low).
Fig. 5. Box-and-whisker plot comparing the distribution of the coefficients of determination (R2) for the force–displacement (F–D) and for the force–strain (F–S) curves. The determination coefficients were calculated including data for increasing portions of the curve (up to 60%, 70%, 80%, 90%, 99% and 100% of the fracture force).
2264
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
Table 3 Slope of the force–displacement curve for the first part of the curve (from 1000 N up to 70% of the fracture force) and for the last part (from 70% to 100% of the fracture force). Slope is expressed in kilo-Newton per millimeter (kN/mm) of vertical deflection of the femoral head. Median and range are reported.
Median Range
Slope up to 70% fracture force
Slope from 70% to 100% fracture force
1.64 1.16–2.60
1.36 0.93–2.23
3. Results All force–displacement curves exhibited a single sudden drop when the maximum force value was reached: only in one specimen a fluctuation of 5% of the fracture force was observed (shortly before macroscopic fracture occurred), while in all other specimens fluctuations during the entire load ramp were smaller than 1% of the fracture force (Fig. 3). Similarly, the force–strain curves were highly linear (Fig. 4). The determination coefficients for all force–displacement curves were in all cases larger than R2 ¼ 0.984 up to 99% of the fracture force (Fig. 5). Only when the last 1% of the force–displacement curve was included, the coefficient of determination in some cases slightly decreased (minimum value: R2 ¼0.960). Similarly, the determination coefficients for all force–strain curves were in all cases larger than R2 ¼0.984 up to 99% of the fracture force (Fig. 5). Only when the last 1% of the force–strain curve was included, the coefficient of determination in some cases slightly decreased (minimum value: R2 ¼0.902). No difference of determination coefficients was observed in relation with the preservation (fresh-frozen or embalmed) or the strain-rate (high or low). The slope of the first part of the force–displacement curve (up to 70% fracture force) was similar to the slope of the last part of the curve (from 70% to 100% of the fracture force), Table 3. The differences observed (of the order of 17%) were of the same order of magnitude as those that were observed between different portions of the curve (10–20% variations), and are typical of experimental force–displacement data. High-speed movie was successfully obtained for all specimens. They allowed observing the exact time when fracture initiated, and how it propagated across the bone. In all femurs, fracture occurred in the head–neck region. Fracture consistently started from the lateral side (where bone is subjected to tensile stress). The entire failure event took 0.1–0.25 ms (from the instant when the first sign of fractured was observed to the time when bone was split into two pieces). It was possible to re-assemble all fractured specimens in a way that fractured parts were coming back together leaving a gap of less than 250 mm (Fig. 2).
4. Discussion The aim of this work was to investigate whether the human proximal femur behaves in a linear-elastic way up to failure when loaded in a physiological direction and at physiological strainrates. In the present study, we addressed the structural behavior of the femur in a sort of phenomenological perspective. The present results (and any other results based on structural testing of a whole femur) do not provide any direct evidence of the linear/nonlinear behavior of bone at the tissue-level. Quite the opposite happens: while it is quite well-known to what extent bone tissue behaves linearly and when it behaves nonlinearly, based on the information at the tissue level it is not possible to predict if a whole bone behaves linearly under given loading
conditions. Therefore, this study provides this missing information about the linear behavior of the femur as a structure. The displacements to failure recorded in this study varied from 3 to 5 mm (specimens having the level of the distal pot close to the proximal extremity, with a relatively short part of the femur, 33%BL, that could bend under load) to 5–7 mm (specimens having the distal pot further down, 80%BL). An accurate comparison with previously published results is not possible, as the dimension of the femurs as well as the level of the distal constraint and the direction of the applied load varied. However, the present findings do not conflict with previously published results. A displacement of about 1.5 mm can be estimated in Augat et al. (1996) when the proximal femur was loaded to failure in a vertical configuration. Keyak (2001) reported a displacement at failure 4.1 mm when the level of the distal constraint was more proximal than in any specimen in our study. Yosibash et al. (2010) reported a displacement at failure of 4.3 mm when the level of the distal constraint was similar to the short specimens in our study. Lotz et al. (1991) reported a displacement at failure of 6–7 mm when the level of the distal constraint was similar to the short specimens in our study. Link et al. (2003) reported a displacement at failure exceeding 10 mm when the level of the distal constraint was more distal than in the short specimens in our study. Alho et al. (1988) reported a displacement of 5.5–17.9 mm when full-length femurs were loaded to failure in a vertical configuration. The high-speed video clearly showed the following: (i) fracture occurred abruptly, as a single catastrophic event in 0.1–0.25 ms; (ii) fracture started on the lateral side of the neck, where tensile stress was measured by the strain gages. After failure, the specimens could be reassembled with no visible sign of permanent deformation. The force–displacement curves and force–strain curves were highly linear up to 99% of the fracture force, with coefficients of determination for a linear regression that were close to 1.000 (Table 3). Even including the very last part of such curves, the coefficient of determination only slightly decreased. Neither the preservation (fresh-frozen or embalmed), nor the strain-rate (in the range covered in this study), nor the level of the distal constraint had an effect on linearity: the coefficients of determination varied very little between specimens (Fig. 5), with no obvious trend between the specimen groups. When the fluctuations of the force–displacement curves were investigated, very small decreases of slope were observed in the last part of the curve (17% slope decrease, which was comparable to the changes observed between the different regions of the curves). A consensus generally exists about the linearity of cortical bone tissue in tension up to failure (Carter and Spengler, 1978; Fung, 1980), about post-yield nonlinearity of cortical bone in compression (Keller, 1994), and a nonlinear failure or trabecular bone both in compression and tension (Kopperdahl and Keaveny, 1998; Keaveny, 1994). Very little has been published on the linearity of the femurs (and of bones in general) as a structure. Some authors reported that the femur behaved linearly (or nearly so) up to failure (Alho et al., 1988; Yosibash et al., 2010). Others (Dalen et al., 1976; Link et al., 2003) did not comment on the linear/nonlinear behavior of the femur, but linear plots are published. Keyak (2001) reported some nonlinearity for a femur fractured in vitro (in approximately 10 s), which they interpreted as elasto-plastic behavior. Unfortunately, they did not publish their strain gage data to back up such a hypothesis. In fact, their plot can be better described as a two-step fracture. In a recent study (Bessho et al., 2007) it has been reported that the force– displacement curve exhibited a nonlinear pattern (however, force– strain plots were not reported for the strain gages). However, in that study a criterion for defining ‘‘yield’’ was not indicated. Conversely, a small change of slope is visible in their plots. It is possible that such
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
nonlinearity was due to the extremely low loading rate they implemented (failure occurred in minutes), which is orders of magnitude lower than the physiological one (load peak occurring in 0.1–0.5 s (Bergmann, 2008), which was also implemented in the present study). In a sideways fall in vitro study, a 50-fold strain-rate increase was associated with larger failure load and structural stiffness (statistically significant) (Courtney et al., 1994). The change of slope we found (17% slope decrease) is orders of magnitude lower than the change of slope normally found in plastic materials, where the slope of the stress–strain curves decreases by 2–3 orders of magnitude in the last part (Brady et al., 2002) (as there is no specific recommendation for identifying the end of linearity in force– displacement curves of whole structures, we extend to the structural response of the femur the criteria that are generally accepted at the material level). The present study has some limitations. First of all, in the highspeed videos acquired in the present study (8000–18,000 frames/s) the entire fracture involves a very limited number of frames. Higher frame rates would provide richer information about the fracture mechanism. However, under these circumstances a conclusion can be drawn: as failure occurs in such a short time, it is definitely a brittle event. Supplementary material related to this article can be found online at doi:10.1016/j.jbiomech.2011.05.038. A second limitation lies in the age of the donors, which was biased towards elderly subjects (71–83 years old). This might have biased the results towards the brittle behavior as ageing generally reduces toughness of bone tissue (Fung, 1980). At the same time, fractures in the proximal femoral metaphysis are most frequent in elderly subjects. Therefore, this is the age range that is most interesting to address. A limitation of the experimental setup is that the displacement of the femoral head was measured by the actuator of the testing machine. Therefore, such a displacement includes the actual deflection of the femur, but also any mechanical play of the loading setup (e.g. the articular cartilage coming in contact with the acrylic mould of the femoral head). Such a limitation is probably responsible for the small artifactual initial nonlinearity in the force–displacement curves (Fig. 3). It must be noted that the present findings refer to a specific loading configuration (a force applied to the femoral head at 81 in the frontal plane). A larger angle (Table 1) would increase the compressive force compared to bending. As a consequence, stress would be shifted towards compression, while tensile stress would decrease. As bone tissue exhibits different behavior in tension and in compression (Carter and Spengler, 1978; Fung, 1980; Keller, 1994; Kopperdahl and Keaveny, 1998, Keaveny, 1994; Yosibash et al., 2007; Helgason et al., 2008), this would be associated with different failure modes of the proximal femur. It must be noted that the loading condition chosen for our study is the only one based on a sensitivity study aiming at identifying the most critical loading conditions in the physiological range (Keyak et al., 2001; Cristofolini et al., 2007). In conclusion, the present study has shown that the femur behaves as a linear-elastic structure up to failure when loaded in a physiological direction and with a physiological strain-rate. Indeed, all the observations (lack of plastic deformation, abrupt fracture, highly linear force–displacement curves) allow us to exclude the presence of plastic deformations, for the physiological range of strain-rates that was investigated.
Conflict of interest statement There is no potential conflict of interest: none of the authors received or will receive direct or indirect benefits from third parties for the performance of this study.
2265
Acknowledgments The authors wish to thank Enrico Schileo and Massimiliano Baleani for the stimulating discussions, Paolo Erani, Francesco Pallini, Elena Varini, Enrico Borgognoni, Giorgia Conti for the technical contribution, Barbara Bordini for her advice with statistics and Luigi Lena for the artwork. This study was funded by the European Community (‘‘The Osteoporotic Virtual Physiological Human—VPHOP’’ Grant FP7-ICT2008-223865) and by Regione Emilia-Romagna (Region-University Research Program 2007–2009).
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2011.05.038. References Alho, A., Husby, T., Hoiseth, A., 1988. Bone mineral content and mechanical strength. An ex vivo study on human femora at autopsy. Clin. Orthop. Relat. Res. 227, 292–297. An, Y.H., Draughn, R.A., 1999. Mechanical Testing of Bone and the Bone–Implant Interface. CRC Press, Boca Raton (0849302668). Augat, P., Reeb, H., Claes, L.E., 1996. Prediction of fracture load at different skeletal sites by geometrical properties of the cortical shell. J. Bone Miner. Res. 11, 1356–1363. Bayraktar, H.H., Morgan, E.F., Niebur, G.L., Morris, G.E., Wong, E.K., Keaveny, T.M., 2004. Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue. J. Biomech. 37, 27–35. Bergmann, G. (Ed.), 2008. Charite – Universitaetsmedizin Berlin, OrthoLoad /http://www.OrthoLoad.comS. Bessho, M., Ohnishi, I., Matsuyama, J., Matsumoto, T., Imai, K., Nakamura, K., 2007. Prediction of strength and strain of the proximal femur by a CT-based finite element method. J. Biomech. 40, 1745–1753. Brady, G.S., Clauser, H.R., Vaccari, J.A., 2002. Materials Handbook 15th ed. McGraw-Hill, New York (0-07-136076-X). Burr, D.B., Milgrom, C., Fyhrie, D., Forwood, M., Nyska, M., Finestone, A., Hoshaw, S., Saiag, E., Simkin, A., 1996. In vivo measurement of human tibial strains during vigorous activity. Bone 18, 405–410. Carter, D.R., Spengler, D.M., 1978. Mechanical properties and composition of cortical bone. Clin. Orthop. Relat. Res. 135, 192–217. Courtney, A.C., Wachtel, E.F., Myers, E.R., Hayes, W.C., 1994. Effects of loading rate on strength of the proximal femur. Calcif. Tissue Int. 55, 53–58. Cowin, S.C., 2001. Bone Mechanics Handbook. CRC Press, Boca Raton, FL. Cristofolini, L., 1997. A critical analysis of stress shielding evaluation of hip prostheses. Crit. Rev. Biomed. Eng. 25 (4&5), 409–483. Cristofolini, L., Juszczyk, M., Martelli, S., Taddei, F., Viceconti, M., 2007. In vitro replication of spontaneous fractures of the proximal femur. J. Biomech. 40, 2837–2845. Cristofolini, L., Juszczyk, M., Taddei, F., Viceconti, M., 2009. Strain distribution in the proximal human femoral metaphysis. Proc. Inst. Mech. Eng. [H] 223, 273–288. Cristofolini, L., Schileo, E., Juszczyk, M., Taddei, F., Martelli, S., Viceconti, M., 2010. Mechanical testing of bones: the positive synergy of finite-element models and in vitro experiments. Philos. Trans. A—Math. Phys. Eng. Sci. 368, 2725–2763. Currey, J., 2009. Measurement of the mechanical properties of bone: a recent history. Clin. Orthop. Relat. Res. 467, 1948–1954. Dalen, N., Hellstrom, L.G., Jacobson, B., 1976. Bone mineral content and mechanical strength of the femoral neck. Acta Orthop. Scand. 47, 503–508. Delaere, O., Dhem, A., Bourgois, R., 1989. Cancellous bone and mechanical strength of the femoral neck. Arch. Orthop. Trauma Surg. 108, 72–75. Fung, Y.C.,1980. Bone and cartilage. In: Biomechanics—Mechanical Properties of Living Tissues. Springer Verlag, New York, pp. 383–415. ˆ lfsson, S., Viceconti, M., 2008. Helgason, B., Perilli, E., Schileo, E., Taddei, F., BrynjU Mathematical relationships between bone density and mechanical properties: a literature review. Clin. Biomech. 23, 135–146. Hvid, I., Jensen, J., 1984. Cancellous bone strength at the proximal human tibia. Archive: Eng. Med. 1971–1988 (vols. 1–17) 13, 21–25. Keaveny, T.M., Guo, X.E., Wachtel, E.F., McMahon, T.A., Hayes, W.C., 1994. Trabecular bone exhibits fully linear elastic behavior and yields at low strains. J. Biomech. 27, 1127–1136. Keller, T.S., 1994. Predicting the compressive mechanical behavior of bone. J. Biomech. 27, 1159–1168. Keyak, J.H., 2001. Improved prediction of proximal femoral fracture load using nonlinear finite element models. Med. Eng. Phys. 23, 165–173. Keyak, J.H., Rossi, S.A., Jones, K.A., Les, C.M., Skinner, H.B., 2001. Prediction of fracture location in the proximal femur using FE models. Med. Eng. Phys. 23, 657–664.
2266
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
Keyak, J.H., Skinner, H.B., Fleming, J.A., 2001. Effect of force direction on femoral fracture load for two types of loading conditions. J. Orthop. Res. 19, 539–544. Kopperdahl, D.L., Keaveny, T.M., 1998. Yield strain behavior of trabecular bone. J. Biomech. 31, 601–608. Lakes, R.S., Katz, J.L., 1979. Viscoelastic properties of wet cortical bone. III. A nonlinear constitutive equation. J. Biomech. 12, 689–698. Lanyon, L.E., Hampson, W.G.J., Goodship, A.E., Shah, J.S., 1975. Bone deformation recorded in vivo from strain gauges attached to the human tibial shaft. Acta Orthop. Scand. 46, 256–268. Link, T.M., Vieth, V., Langenberg, R., Meier, N., Lotter, A., Newitt, D., Majumdar, S., 2003. Structure analysis of high resolution magnetic resonance imaging of the proximal femur: in vitro correlation with biomechanical strength and BMD. Calcif. Tissue Int. 72, 156–165. ¨ Lochmuller, E.-M., Groll, O., Kuhn, V., Eckstein, F., 2002. Mechanical strength of the proximal femur as predicted from geometric and densitometric bone properties at the lower limb versus the distal radius. Bone 30, 207–216. ¨ Lochmuller, E.M., Zeller, J.-B., Kaiser, D., Eckstein, F., Landgraf, J., Putz, R., Steldinger, R., 1998. Correlation of femoral and lumbar DXA and calcaneal ultrasound, with the in vitro failure of the proximal femur. Osteoporos Int. 8, 591–598. Lotz, J.C., Cheal, E.J., Hayes, W.C., 1991. Fracture prediction for the proximal femur using finite element models: Part I—linear analysis. J. Biomech. Eng. 113, 353–360. ¨ hman, C., Dall’Ara, E., Baleani, M., Jan, S.V.S., Viceconti, M., 2008. The effects of O embalming using a 4% formalin solution on the compressive mechanical properties of human cortical bone. Clin. Biomech. 23, 1294–1298.
Ota, T., Yamamoto, I., Morita, R., 1999. Fracture simulation of the femoral bone using the finite-element method: how a fracture initiates and proceeds. J. Bone Miner. Metab. 17, 108–112. Raftopoulos, D., Katsamanis, E., Saul, F., Liu, W., Saddemi, S., 1993. An intermediate loading rate technique for the determination of mechanical properties of human femoral cortical bone. J. Biomed. Eng. 15, 60–66. Schileo, E., Taddei, F., Malandrino, A., Cristofolini, L., Viceconti, M., 2007. Subjectspecific finite element models can accurately predict strain levels in long bones. J. Biomech. 40, 2982–2989. Yeni, Y.N., Fyhrie, D.P., 2003. A rate-dependent microcrack-bridging model that can explain the strain rate dependency of cortical bone apparent yield strength. J. Biomech. 36, 1343–1353. Yosibash, Z., Padan, R., Joskowicz, L., Milgrom, C., 2007. A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments. J. Biomech. Eng. 129, 297–309. Yosibash, Z., Tal, D., Trabelsi, N., 2010. Predicting the yield of the proximal femur using high-order finite-element analysis with inhomogeneous orthotropic material properties. Philos. Trans. A—Math. Phys. Eng. Sci. 368, 2707–2723. Zilch, H., Rohlmann, A., Bergmann, G., Kolbel, R., 1980. Material properties of femural cancellous bone in axial loading. Part II: time dependent properties. Arch. Orthop. Trauma Surg. 97, 257–262.
Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
The human proximal femur behaves linearly elastic up to failure under physiological loading conditions Mateusz Maria Juszczyk a, Luca Cristofolini a,b,n, Marco Viceconti a a b
Laboratorio di Tecnologia Medica, Istituto Ortopedico Rizzoli, Bologna, Italy Facolta di Ingegneria, Universita di Bologna, Italy
a r t i c l e i n f o
abstract
Article history: Accepted 27 May 2011
It has not been demonstrated whether the human proximal femur behaves linearly elastic when loaded to failure. In the present study we tested to failure 12 cadaveric femurs. Strain was measured (at 5000 Hz) on the bone surface with triaxial strain gages (up to 18 on each femur). High-speed videos (up to 18,000 frames/s) were taken during the destructive test. To assess the effect of tissue preservation, both fresh-frozen and formalin-fixed specimens were tested. Tests were carried out at two strain-rates covering the physiological range experienced during daily motor tasks. All specimens were broken in only two pieces, with a single fracture surface. The high-speed videos showed that failure occurred as a single abrupt event in less than 0.25 ms. In all specimens, fracture started on the lateral side of the neck (tensile stress). The fractured specimens did not show any sign of permanent deformation. The force– displacement curves were highly linear (R2 40.98) up to 99% of the fracture force. When the last 1% of the force–displacement curve was included, linearity slightly decreased (minimum R2 ¼ 0.96). Similarly, all force–strain curves were highly linear (R2 4 0.98 up to 99% of the fracture force). The slope of the first part of the force–displacement curve (up to 70% fracture force) differed from the last part of the curve (from 70% to 100% of the fracture force) by less than 17%. Such a difference was comparable to the fluctuations observed between different parts of the curve. Therefore, it can be concluded that the proximal femur has a linear-elastic behavior up to fracture, for physiological strain-rates. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Proximal femur Femoral neck and head Linear elastic behavior In vitro bone fracture Brittle failure
1. Introduction The strain distribution and strength of the proximal human femur has been extensively investigated in the past, both in vitro (Keyak et al., 2001; Bessho et al., 2007; Cristofolini et al., 2009) and with numerical models (Lotz et al., 1991; Ota et al., 1999; Schileo et al., 2007). One of the open questions concerns the linear (or the lack of linear) behavior of the femur. In fact, assessment of the linear behavior would improve the understanding of the etiology of femoral fractures (Cristofolini et al., 2010). Furthermore, because of the increasing use of Finite Element (FE) models of the human femur, it would be essential to include the most
Abbreviations: BL, biomechanical length of the femur; BW, body weight; CT, computed tomography; DEXA, dual energy X-ray absorptiometry; FE, finite element; HD, diameter of the head of the femur; R2, coefficient of determination for a linear regression; VPH-OP, Virtual Physiological Osteoporotic Human Project (http://www.vphop.eu/) n Corresponding author at: Laboratorio di Tecnologia Medica—Istituto Ortopedico Rizzoli, Via di Barbiano, 1/10, 40136 Bologna, Italy. Tel.: þ 39 051 63 66 864; fax: þ39 051 63 66 863. E-mail addresses: [email protected], [email protected] (L. Cristofolini). 0021-9290/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2011.05.038
appropriate rheological model (whether linear-elastic, or elastoplastic). In many cases, bone is modeled as a linear-elastic material (Carter and Spengler, 1978; Yosibash et al., 2007; Helgason et al., 2008). In fact, it has been reported that in tension cortical bone tissue exhibits a linear-elastic behavior up to failure (Carter and Spengler, 1978; Fung, 1980), while post-yield nonlinearity is observed in compression (Keller, 1994). Kopperdahl and Keaveny (1998) and Keaveny (1994) reported a linear pre-yield behavior for the trabecular bone, followed by a plastic deformation both in tension and compression (such a post-yield nonlinearity was much less pronounced in tension). Yield of bone tissue has sometimes been described as an arbitrary point in the stress–strain curve where nonlinearity appears (Hvid and Jensen, 1984) or as a point at which permanent deformation occurs (An and Draughn, 1999). A mechanistic model (Yeni and Fyhrie, 2003) has shown that bone nonlinearity, which is sometimes referred to as post-yield behavior, can be explained by collagen–fiber-bridged microcracks rather than by real plastic post-yield phenomena, as (Keaveny et al., 1994) previously suggested based on in vitro tests. A combination of tension and compression is present in the femoral neck for most loading conditions. As it is impossible to
2260 Table 1 Summary of the literature reporting in vitro destructive tests where the proximal femur was loaded in a quasi-axial loading configuration. The loading rate (time required to fracture the specimen) and the direction of the applied force (angle between the femoral diaphysis and the force on the femoral head) are reported for each reference. Also indicated is the number of specimens tested and the gender and age of donors. The description of the fracture mechanism is either based on the description provided in the text, or on the pictures and plots available in the manuscript. Load rate/time required to induce fracture
Direction of applied force
Specimens: number (preservation)
Gender, age of donors
Fracture mechanism
Alho et al. (1988)
66–390 s (1)
01 in frontal plane 01 in sagittal plane
36 (preservation not specified)
18 female, 18 male 57–87 years old
In most femurs the load–deflection curve was relatively linear, had a gradual curving without any yielding point. A yielding point was identified only in few cases
Augat et al. (1996)
1.0 mm/s (time to fracture of the order of 10–20 s (1))
01 in frontal plane 01 in sagittal plane
20 (formalin fixed)
13 female, 7 male 66–100 years old
Neck (n¼ 18) and trochanteric fractures (n¼ 2). Presence/lack of linearity was not expressly reported
Bessho et al. (2007)
180 s (1)
201 in frontal plane 01 in sagittal plane
11 (fresh-frozen)
6 female, 5 male 52–85 years old
Fluctuations in the force–displacement curve. The yield point indicated in the plot corresponds to 80% of the failure load. The entire curve fits within a band that is narrower than 10% of the failure load
Dalen et al. (1976)
10 s (1)
01 in frontal plane 01 in sagittal plane (plastic block around the specimen up to lesser trochanter)
61 (fresh)
54 female, 7 male 67–80 years old
Linear-elastic plot with small decrease of slope in the upper 30% of the force–displacement curve
Delaere et al. (1989)
2.0 mm/s (actual time to fracture not indicated)
251 in frontal plane 01 in sagittal plane
20 pairs (dried macerated)
Male þfemale 57–89 years old
Not reported
Keyak (2001), Keyak et al. (2001)
0.5 mm/s (time to fracture of the order of 10–20 s (1))
201 in frontal plane 01 in sagittal plane
18 (fresh-frozen)
10 female, 8 male 52–92 years old
Subcapital and oblique transcervical fractures parallel to the shaft. Presence/lack of linearity not expressly reported. The force–displacement plot in Keyak (2001) is linear, with a two-step fracture
Link et al. (2003)
‘‘stepwise’’ (actual rate not specified)
111 in frontal plane 01 in sagittal plane
31 (fresh-frozen)
14 female, 17 male 29–91 years old
Linear-elastic up to failure
¨ Lochmuller et al. (1998)
1.0 mm/s (load cycles of increasing magnitude)
01 in frontal plane 01 in sagittal plane
58 (formalin fixed)
24 female, 34 male 57–100 years old
Trochanter (n¼ 10) and neck fractures (n¼48). Presence/lack of linearity was not expressly reported
¨ Lochmuller et al. (2002)
6.5 mm/s (time to fracture of the order of 1–3 s (1))
01 in frontal plane 01 in sagittal plane
103 (alcohol/ formalin fixed)
62 female, 41 male 46–97 years old
Trochanter (n¼ 10) and neck fractures (n¼83). Presence/lack of linearity was not expressly reported
Lotz et al. (1991)
Not indicated
01 in frontal plane 01 in sagittal plane
1 (fresh-frozen)
Female 66 years old
The load–displacement plot shows fluctuations, including a slight decrease of slope at 50% of the failure load. The entire curve fits within a band as wide as approximately 10% of the failure load
Ota et al. (1999)
0.5 mm/s (time to fracture of the order of 10–20 s (1))
201 in frontal plane 01 in sagittal plane (2)
1 (formalin fixed)
1 male 76 years old
Base-of-the-neck fracture, parallel to the shaft. Presence/lack of linearity was not expressly reported
Yosibash et al. (2010)
26 s (1)
01 in frontal plane 01 in sagittal plane
1 (fresh-frozen)
Male 30 years old
One step fracture following limited post-elastic nonlinearity (5% decrease of linear slope occurred at 67% of maximum force)
Notes: (1) Time required to fracture the femur specimens was not expressly indicated in the manuscript. The value reported was estimated based on the plots/tables. (2) Direction of the applied force not specified: the angle was estimated based on the published pictures.
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
Reference
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
predict if the linearity of cortical bone in tension will prevail (or not) over the nonlinearity of the cortical and the trabecular bone in compression, it is also impossible to determine a priori if the structural behavior of a whole femur is linear or nonlinear when loaded to failure. When whole femurs were tested to failure with a quasi-axial load, in some cases a brittle fracture was observed, while in others significant nonlinear post-yield behavior was observed (Table 1). According to a recent review, no study so far has focused on the linear (or nonlinear) behavior of the femur as a structure (Currey, 2009). In a study by Keyak (2001), the femur was modeled as a nonlinear material, based on force–displacement plots obtained in vitro when failure was induced in approximately 10 s (it must be noted that no additional measurement as such from the strain gages was available to confirm such a nonlinearity). The only in vitro plot reported in that paper (Keyak, 2001) can be interpreted as a two-step fracture, rather than as an elasto-plastic curve. In a later study (Bessho et al., 2007) the femur has been reported to behave nonlinearly: when eleven femurs were loaded at a low rate (cross-head speed: 0.5 mm/min, such that failure occurred in minutes) some fluctuations could be observed in the force–displacement curve. However, as ‘‘yield’’ was defined somewhat arbitrarily (Bessho et al., 2007), the amount of nonlinearity cannot be quantified. Other authors reported that the proximal femur behaved linearly (or nearly so) up to failure (Alho et al., 1988; Yosibash et al., 2010). In many cases, although the linear/ nonlinear behavior of the femur was not expressly discussed, linear plots are shown (Dalen et al., 1976; Link et al., 2003). A possible explanation of the discrepancies in the literature concerning linearity of bones (Table 1) lies in the viscoelasticity of the bone. In fact, it has been shown that both the modulus of elasticity and nonlinearity of the stress–strain curve strongly depend on the loading rate (Carter and Spengler, 1978; Lakes and Katz, 1979; Zilch et al., 1980). Viscoelasticity could therefore explain why under certain conditions bone behaves linear-elastic, while in others it shows a strong nonlinearity. Strain-rates in the human tibia during walking and running has a range of 0.002–0.013 s 1 (Lanyon et al., 1975), while during vigorous activity (e.g. uphill and downhill zigzag running) the strain-rate can reach 0.034 s 1 (Burr et al., 1996). In vivo recording from telemetric hip prostheses has shown that the load peak during mild activities (e.g. walking) is reached in 0.3–0.5 s, while the peak during occasional overloading (e.g. stumbling) is reached in 0.1–0.2 s (Bergmann, 2008). It is still
2261
not clear whether for such physiological strain-rates bone behaves linearly or not. The aim of this work was to investigate whether the human proximal femur can be assumed to behave in a linear-elastic way up to failure for clinically relevant simulations.
2. Materials and methods 2.1. Overview In this study we re-analyzed data from destructive tests that were carried out on human femurs in our lab in the last six years. This enabled exploring the effect of two factors on the linearity of destructive tests:
Loading rate: Different loading rates were applied to investigate how linearity is affected within the physiological range.
Tissue preservation: Both fresh-frozen and formalin-fixed femurs were tested.
2.2. Bone specimens A total of 12 human cadaveric femurs were included for this study, which were obtained through different ethically-approved international donation programs (Table 2). Eight femurs were fresh-frozen and four were fixed with 4% ¨ hman et al., 2008). Fresh-frozen femurs were de-frozen 2–3 times (for formalin (O a total of 36–120 h) during the different steps of preparation and testing, and were stored at 25 1C when not in use. During the test, all the femurs were wrapped in clothes soaked with physiological solution at a room temperature of 27 30 1C. All femurs were DEXA-scanned (Excel-Plus, Norland, USA) and CT-scanned (Hi-Speed, General-Electric, USA) to document bone quality and lack of abnormality. The femurs were prepared with a set of anatomical reference frames to allow for reproducible alignment throughout the test (Cristofolini et al., 2007). The femurs were distally potted in a steel box with acrylic cement (Cristofolini et al., 2007). The level of the distal constraint varied between 33% and 80% Biomechanical Length of the femur (BL), measured from proximal, as the specimens were part of different studies.
2.3. Strain measurement Each femur was instrumented with up to 18 triaxial-stacked strain gages following a validated procedure (Cristofolini et al., 2009) (KFG-3–120-D1711L3M2S, Kyowa, Tokyo, Japan). A grid excitation of 0.5 V was selected to avoid heating. Strains and force– displacement data from the testing machine were sampled at 5000 Hz using
Table 2 Details of the specimens investigated. In the first four columns, details of the donors are listed. Bone quality is reported in the 6th column for each femur (T-score of the bone density referred to the young reference population, based on the Norland DEXA scanner reference population). Biomechanical dimensions (biomechanical length, BL, and head diameter, HD) (Cristofolini, 1997) are listed. The level of the distal constraint was measured from the proximal extremity of the femur to the upper surface of the acrylic cement pot, and is expressed as a % of the Biomechanical Length. Loading rate (High-strain-rate or low-strain-rate), and tissue preservation method (fresh-frozen or fixed with 4% formalin) are listed in the last two columns. Specimen
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 #11 #12
Donors’ details
Femurs’ details
Testing conditions
Gender
Age at death
Donor height (cm)
Donor weight (kg)
Side
T-score
BL (mm)
HD (mm)
Level of distal constraint (%BL)
Loading rate
Tissue preservation
Male Male Male Male Female Female Male Male Female Female Female Female
71 82 73 77 80 74 73 83 72 72 77 77
178 175 183 173 160 180 173 175 166 166 171 171
91 78 86 70 122 170 91 84 61 61 64 64
Right Left Left Right Left Right Left Right Right Left Right Left
1.87 1.49 2.74 3.60 3.80 2.51 3.30 3.31 4.90 n.a. n.a. n.a.
443 445 476 411 396 448 441 468 427 427 415 412
53.3 53.7 48.8 48.5 44.1 51.5 52.6 54.1 47.5 47.5 46.5 47.5
80 80 33 33 33 33 33 33 33 33 33 33
Low Low High High High High High High High High High High
Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Fresh-frozen Formalin-fixed Formalin-fixed Formalin-fixed Formalin-fixed
2262
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
a 60-channel datalogger (System-6000, Vishay Micro-Measurement, Raleigh, NC, USA). Principal strains were computed based on the three grids of each strain gage.
2.4. Loading conditions The in vitro setup was designed based on previous experience with nondestructive (Cristofolini, 1997) and destructive testing of the femur (Cristofolini et al., 2007). In a previous combined experimental–numerical study (Cristofolini et al., 2007) the range of directions spanned by the hip joint force during daily activities (Bergmann, 2008) was explored to replicate spontaneous fractures of the proximal femur. In the study of Cristofolini et al. (2007), the direction of the hip force (81 in the frontal plane) that generates the highest risk of head–neck fractures was identified. A similar direction (101 in the frontal plane) was reported in
Keyak et al. (2001) as most critical for the neck. Thus, the in vitro loading configuration in the present study did not aim at replicating any specific motor tasks, but the direction of the resultant force that is most likely to induce spontaneous fractures during occasional overloading events. Unfortunately, these two studies (Keyak et al., 2001; Cristofolini et al., 2007) are based on a single femur each. At the same time, to the authors’ knowledge, other configurations in the literature (Table 1) were not based on any sensitivity study capable of identifying the most relevant loading condition. The femur was mounted on top of the load cell of the testing machine with the diaphysis at 81 in the frontal plane. Load was applied to the femoral head through a system of linear bearings to avoid transmission of horizontal force components (Fig. 1). A mould of each femoral head was prepared with acrylic cement (covering 1/5 head diameter) to allow uniform load transfer from the actuator to the head. Muscle forces were not simulated, as it has been shown that they do not significantly alter the stress distribution in the head–neck region (Cristofolini et al., 2007). To cover the range of physiological strain-rates, the femurs were divided into two groups, and force was applied at different speeds:
High-strain-rate group (n ¼10): The crosshead speed (typically 30 mm/s) was
tuned for each specimen so that the strain-rate in the most stressed regions was about 0.05 s 1. As bone tissue fails when strain exceeds 0.007–0.010 (Bayraktar et al., 2004), such strain-rate would generate failure in the order of 0.2 s. This is the typical timescale of physiological and para-physiological loading (Bergmann, 2008), and has been proposed for in vitro testing (Raftopoulos et al., 1993). Low-strain-rate group (n ¼2): A strain-rate ten times lower than high-strainrate was implemented to replicate quasi-static loading events, consistent with Cristofolini et al. (2007) and Cristofolini et al. (2009).
2.5. High-speed movie During the destructive test, the event was filmed by means of a high-speed camera (FastCam-X1024PCI, Photron, UK) at 8000–18,000 frames/s (actual frame rate depended on the size of the field of vision cropped for each specimen). The camera pointed at the superior-lateral part of the neck. Two mirrors were used to
Fig. 1. Testing setup showing a right femur with its distal pot (A) tilted 81 in the frontal plane. The strain gages are visible. Load was delivered through a mould (B) of the femoral head. Load was applied through a system of linear bearings (C) to avoid transmission of horizontal force components (Fig. 1). The two mirrors (D) are visible that enable acquiring the entire head–neck region with the highspeed camera.
Fig. 2. Example of a reassembled femur. A 250-micron needle (A) could not enter the remaining gap. The arrows indicate the fracture (B).
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
2263
film at the same time the anterior-medial and posterior-medial portions of the neck (Cristofolini et al., 2007).
2.6. After-failure inspection In order to evaluate the presence/absence of irreversible residual deformation of the fractured bone, the fractured bone parts were collected and reassembled. The fracture gap was evaluated by means of a hypodermic needle (diameter: 250 mm, Fig. 2).
2.7. Measurement accuracy and repeatability Force and displacement were measured with transducers having an overall precision of better than 0.5% of the readout. The strain gages and its datalogger provided an overall precision of better than 1% of the readout. Intra-specimen repeatability cannot be directly assessed in destructive tests. Based on previous non-destructive tests (Cristofolini et al., 2009), it can be estimated that both strain and displacement would vary by 1–4% if tests could be repeated on the same specimen.
2.8. Statistics Some nonlinearity was observed in the initial part of the force–displacement curves (Fig. 3). As such a toe effect was not observed in the force–strain plots (Fig. 4), the initial force–displacement was possibly due to the cartilage coming in full contact with the acrylic mould of the femoral head. Therefore, the initial part of the force–displacement curves (values below 1 kN) was not taken into account. The fracture force was defined as the largest force value recorded throughout the destructive test for each specimen. The linearity between force and displacement, and between force and strain was measured by the determination coefficient (R2) of the linear regression of the curves. In order to detect slope changes of the force–displacement and force– strain curves, the determination coefficients were calculated including data for increasing portions of the curve (up to 60%, 70%, 80%, 90%, 99% and 100% of the fracture force, for each specimen separately). To enable a comparison with the studies where nonlinearity for the femur has been discussed more in detail (Keyak, 2001; Bessho et al., 2007), the change of slope of the force–displacement was investigated. Because of the high linearity of such curves (see below) it was not possible to identify a yield point based on objective criteria (e.g. Cowin, 2001). Therefore, to enable consistent comparisons, the slope of the force–displacement curve was computed for the first part of the curve (up to 70% of the fracture force) and for the last part (from 70% to 100% of the fracture force). The arbitrary threshold of 70% is consistent with the ‘‘yield’’ point shown in the plots of Bessho et al. (2007). In order to estimate the central value of the distribution of the determination coefficients and of the slopes, the median values were calculated. All statistics were performed with StatView version 5.0.1 (SAS Institute Inc., Cary, NC, USA).
Fig. 3. Force–displacement graph for all the femurs. The vertical component of displacement (measured by the transducer of the testing machine) is plotted. The femurs are grouped according to the strain-rate and to the preservation technique. The gray band masks the lower part of the curves, where some toe effect is present due to the mechanical play of the setup. Error bars are not plotted for clarity.
Fig. 4. Typical force–strain plot for two typical femurs: (A) Maximum and minimum principal strains are plotted for each of the 17 strain gages for femur #2. Because of the preservation (fresh-frozen) and loading rate (low-strain-rate), this type of specimen can be expected to show worse linearity. (B) Maximum and minimum principal strains are plotted for each of the 14 strain gages for femur #12. Because of the preservation (embalmed) and loading rate (high-strain-rate), this type of specimen can be expected to show the best linearity. The trend for the other specimens (not reported here for brevity) was quite similar, independent of the preservation (fresh-frozen or embalmed) and strain-rate (high or low).
Fig. 5. Box-and-whisker plot comparing the distribution of the coefficients of determination (R2) for the force–displacement (F–D) and for the force–strain (F–S) curves. The determination coefficients were calculated including data for increasing portions of the curve (up to 60%, 70%, 80%, 90%, 99% and 100% of the fracture force).
2264
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
Table 3 Slope of the force–displacement curve for the first part of the curve (from 1000 N up to 70% of the fracture force) and for the last part (from 70% to 100% of the fracture force). Slope is expressed in kilo-Newton per millimeter (kN/mm) of vertical deflection of the femoral head. Median and range are reported.
Median Range
Slope up to 70% fracture force
Slope from 70% to 100% fracture force
1.64 1.16–2.60
1.36 0.93–2.23
3. Results All force–displacement curves exhibited a single sudden drop when the maximum force value was reached: only in one specimen a fluctuation of 5% of the fracture force was observed (shortly before macroscopic fracture occurred), while in all other specimens fluctuations during the entire load ramp were smaller than 1% of the fracture force (Fig. 3). Similarly, the force–strain curves were highly linear (Fig. 4). The determination coefficients for all force–displacement curves were in all cases larger than R2 ¼ 0.984 up to 99% of the fracture force (Fig. 5). Only when the last 1% of the force–displacement curve was included, the coefficient of determination in some cases slightly decreased (minimum value: R2 ¼0.960). Similarly, the determination coefficients for all force–strain curves were in all cases larger than R2 ¼0.984 up to 99% of the fracture force (Fig. 5). Only when the last 1% of the force–strain curve was included, the coefficient of determination in some cases slightly decreased (minimum value: R2 ¼0.902). No difference of determination coefficients was observed in relation with the preservation (fresh-frozen or embalmed) or the strain-rate (high or low). The slope of the first part of the force–displacement curve (up to 70% fracture force) was similar to the slope of the last part of the curve (from 70% to 100% of the fracture force), Table 3. The differences observed (of the order of 17%) were of the same order of magnitude as those that were observed between different portions of the curve (10–20% variations), and are typical of experimental force–displacement data. High-speed movie was successfully obtained for all specimens. They allowed observing the exact time when fracture initiated, and how it propagated across the bone. In all femurs, fracture occurred in the head–neck region. Fracture consistently started from the lateral side (where bone is subjected to tensile stress). The entire failure event took 0.1–0.25 ms (from the instant when the first sign of fractured was observed to the time when bone was split into two pieces). It was possible to re-assemble all fractured specimens in a way that fractured parts were coming back together leaving a gap of less than 250 mm (Fig. 2).
4. Discussion The aim of this work was to investigate whether the human proximal femur behaves in a linear-elastic way up to failure when loaded in a physiological direction and at physiological strainrates. In the present study, we addressed the structural behavior of the femur in a sort of phenomenological perspective. The present results (and any other results based on structural testing of a whole femur) do not provide any direct evidence of the linear/nonlinear behavior of bone at the tissue-level. Quite the opposite happens: while it is quite well-known to what extent bone tissue behaves linearly and when it behaves nonlinearly, based on the information at the tissue level it is not possible to predict if a whole bone behaves linearly under given loading
conditions. Therefore, this study provides this missing information about the linear behavior of the femur as a structure. The displacements to failure recorded in this study varied from 3 to 5 mm (specimens having the level of the distal pot close to the proximal extremity, with a relatively short part of the femur, 33%BL, that could bend under load) to 5–7 mm (specimens having the distal pot further down, 80%BL). An accurate comparison with previously published results is not possible, as the dimension of the femurs as well as the level of the distal constraint and the direction of the applied load varied. However, the present findings do not conflict with previously published results. A displacement of about 1.5 mm can be estimated in Augat et al. (1996) when the proximal femur was loaded to failure in a vertical configuration. Keyak (2001) reported a displacement at failure 4.1 mm when the level of the distal constraint was more proximal than in any specimen in our study. Yosibash et al. (2010) reported a displacement at failure of 4.3 mm when the level of the distal constraint was similar to the short specimens in our study. Lotz et al. (1991) reported a displacement at failure of 6–7 mm when the level of the distal constraint was similar to the short specimens in our study. Link et al. (2003) reported a displacement at failure exceeding 10 mm when the level of the distal constraint was more distal than in the short specimens in our study. Alho et al. (1988) reported a displacement of 5.5–17.9 mm when full-length femurs were loaded to failure in a vertical configuration. The high-speed video clearly showed the following: (i) fracture occurred abruptly, as a single catastrophic event in 0.1–0.25 ms; (ii) fracture started on the lateral side of the neck, where tensile stress was measured by the strain gages. After failure, the specimens could be reassembled with no visible sign of permanent deformation. The force–displacement curves and force–strain curves were highly linear up to 99% of the fracture force, with coefficients of determination for a linear regression that were close to 1.000 (Table 3). Even including the very last part of such curves, the coefficient of determination only slightly decreased. Neither the preservation (fresh-frozen or embalmed), nor the strain-rate (in the range covered in this study), nor the level of the distal constraint had an effect on linearity: the coefficients of determination varied very little between specimens (Fig. 5), with no obvious trend between the specimen groups. When the fluctuations of the force–displacement curves were investigated, very small decreases of slope were observed in the last part of the curve (17% slope decrease, which was comparable to the changes observed between the different regions of the curves). A consensus generally exists about the linearity of cortical bone tissue in tension up to failure (Carter and Spengler, 1978; Fung, 1980), about post-yield nonlinearity of cortical bone in compression (Keller, 1994), and a nonlinear failure or trabecular bone both in compression and tension (Kopperdahl and Keaveny, 1998; Keaveny, 1994). Very little has been published on the linearity of the femurs (and of bones in general) as a structure. Some authors reported that the femur behaved linearly (or nearly so) up to failure (Alho et al., 1988; Yosibash et al., 2010). Others (Dalen et al., 1976; Link et al., 2003) did not comment on the linear/nonlinear behavior of the femur, but linear plots are published. Keyak (2001) reported some nonlinearity for a femur fractured in vitro (in approximately 10 s), which they interpreted as elasto-plastic behavior. Unfortunately, they did not publish their strain gage data to back up such a hypothesis. In fact, their plot can be better described as a two-step fracture. In a recent study (Bessho et al., 2007) it has been reported that the force– displacement curve exhibited a nonlinear pattern (however, force– strain plots were not reported for the strain gages). However, in that study a criterion for defining ‘‘yield’’ was not indicated. Conversely, a small change of slope is visible in their plots. It is possible that such
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
nonlinearity was due to the extremely low loading rate they implemented (failure occurred in minutes), which is orders of magnitude lower than the physiological one (load peak occurring in 0.1–0.5 s (Bergmann, 2008), which was also implemented in the present study). In a sideways fall in vitro study, a 50-fold strain-rate increase was associated with larger failure load and structural stiffness (statistically significant) (Courtney et al., 1994). The change of slope we found (17% slope decrease) is orders of magnitude lower than the change of slope normally found in plastic materials, where the slope of the stress–strain curves decreases by 2–3 orders of magnitude in the last part (Brady et al., 2002) (as there is no specific recommendation for identifying the end of linearity in force– displacement curves of whole structures, we extend to the structural response of the femur the criteria that are generally accepted at the material level). The present study has some limitations. First of all, in the highspeed videos acquired in the present study (8000–18,000 frames/s) the entire fracture involves a very limited number of frames. Higher frame rates would provide richer information about the fracture mechanism. However, under these circumstances a conclusion can be drawn: as failure occurs in such a short time, it is definitely a brittle event. Supplementary material related to this article can be found online at doi:10.1016/j.jbiomech.2011.05.038. A second limitation lies in the age of the donors, which was biased towards elderly subjects (71–83 years old). This might have biased the results towards the brittle behavior as ageing generally reduces toughness of bone tissue (Fung, 1980). At the same time, fractures in the proximal femoral metaphysis are most frequent in elderly subjects. Therefore, this is the age range that is most interesting to address. A limitation of the experimental setup is that the displacement of the femoral head was measured by the actuator of the testing machine. Therefore, such a displacement includes the actual deflection of the femur, but also any mechanical play of the loading setup (e.g. the articular cartilage coming in contact with the acrylic mould of the femoral head). Such a limitation is probably responsible for the small artifactual initial nonlinearity in the force–displacement curves (Fig. 3). It must be noted that the present findings refer to a specific loading configuration (a force applied to the femoral head at 81 in the frontal plane). A larger angle (Table 1) would increase the compressive force compared to bending. As a consequence, stress would be shifted towards compression, while tensile stress would decrease. As bone tissue exhibits different behavior in tension and in compression (Carter and Spengler, 1978; Fung, 1980; Keller, 1994; Kopperdahl and Keaveny, 1998, Keaveny, 1994; Yosibash et al., 2007; Helgason et al., 2008), this would be associated with different failure modes of the proximal femur. It must be noted that the loading condition chosen for our study is the only one based on a sensitivity study aiming at identifying the most critical loading conditions in the physiological range (Keyak et al., 2001; Cristofolini et al., 2007). In conclusion, the present study has shown that the femur behaves as a linear-elastic structure up to failure when loaded in a physiological direction and with a physiological strain-rate. Indeed, all the observations (lack of plastic deformation, abrupt fracture, highly linear force–displacement curves) allow us to exclude the presence of plastic deformations, for the physiological range of strain-rates that was investigated.
Conflict of interest statement There is no potential conflict of interest: none of the authors received or will receive direct or indirect benefits from third parties for the performance of this study.
2265
Acknowledgments The authors wish to thank Enrico Schileo and Massimiliano Baleani for the stimulating discussions, Paolo Erani, Francesco Pallini, Elena Varini, Enrico Borgognoni, Giorgia Conti for the technical contribution, Barbara Bordini for her advice with statistics and Luigi Lena for the artwork. This study was funded by the European Community (‘‘The Osteoporotic Virtual Physiological Human—VPHOP’’ Grant FP7-ICT2008-223865) and by Regione Emilia-Romagna (Region-University Research Program 2007–2009).
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jbiomech.2011.05.038. References Alho, A., Husby, T., Hoiseth, A., 1988. Bone mineral content and mechanical strength. An ex vivo study on human femora at autopsy. Clin. Orthop. Relat. Res. 227, 292–297. An, Y.H., Draughn, R.A., 1999. Mechanical Testing of Bone and the Bone–Implant Interface. CRC Press, Boca Raton (0849302668). Augat, P., Reeb, H., Claes, L.E., 1996. Prediction of fracture load at different skeletal sites by geometrical properties of the cortical shell. J. Bone Miner. Res. 11, 1356–1363. Bayraktar, H.H., Morgan, E.F., Niebur, G.L., Morris, G.E., Wong, E.K., Keaveny, T.M., 2004. Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue. J. Biomech. 37, 27–35. Bergmann, G. (Ed.), 2008. Charite – Universitaetsmedizin Berlin, OrthoLoad /http://www.OrthoLoad.comS. Bessho, M., Ohnishi, I., Matsuyama, J., Matsumoto, T., Imai, K., Nakamura, K., 2007. Prediction of strength and strain of the proximal femur by a CT-based finite element method. J. Biomech. 40, 1745–1753. Brady, G.S., Clauser, H.R., Vaccari, J.A., 2002. Materials Handbook 15th ed. McGraw-Hill, New York (0-07-136076-X). Burr, D.B., Milgrom, C., Fyhrie, D., Forwood, M., Nyska, M., Finestone, A., Hoshaw, S., Saiag, E., Simkin, A., 1996. In vivo measurement of human tibial strains during vigorous activity. Bone 18, 405–410. Carter, D.R., Spengler, D.M., 1978. Mechanical properties and composition of cortical bone. Clin. Orthop. Relat. Res. 135, 192–217. Courtney, A.C., Wachtel, E.F., Myers, E.R., Hayes, W.C., 1994. Effects of loading rate on strength of the proximal femur. Calcif. Tissue Int. 55, 53–58. Cowin, S.C., 2001. Bone Mechanics Handbook. CRC Press, Boca Raton, FL. Cristofolini, L., 1997. A critical analysis of stress shielding evaluation of hip prostheses. Crit. Rev. Biomed. Eng. 25 (4&5), 409–483. Cristofolini, L., Juszczyk, M., Martelli, S., Taddei, F., Viceconti, M., 2007. In vitro replication of spontaneous fractures of the proximal femur. J. Biomech. 40, 2837–2845. Cristofolini, L., Juszczyk, M., Taddei, F., Viceconti, M., 2009. Strain distribution in the proximal human femoral metaphysis. Proc. Inst. Mech. Eng. [H] 223, 273–288. Cristofolini, L., Schileo, E., Juszczyk, M., Taddei, F., Martelli, S., Viceconti, M., 2010. Mechanical testing of bones: the positive synergy of finite-element models and in vitro experiments. Philos. Trans. A—Math. Phys. Eng. Sci. 368, 2725–2763. Currey, J., 2009. Measurement of the mechanical properties of bone: a recent history. Clin. Orthop. Relat. Res. 467, 1948–1954. Dalen, N., Hellstrom, L.G., Jacobson, B., 1976. Bone mineral content and mechanical strength of the femoral neck. Acta Orthop. Scand. 47, 503–508. Delaere, O., Dhem, A., Bourgois, R., 1989. Cancellous bone and mechanical strength of the femoral neck. Arch. Orthop. Trauma Surg. 108, 72–75. Fung, Y.C.,1980. Bone and cartilage. In: Biomechanics—Mechanical Properties of Living Tissues. Springer Verlag, New York, pp. 383–415. ˆ lfsson, S., Viceconti, M., 2008. Helgason, B., Perilli, E., Schileo, E., Taddei, F., BrynjU Mathematical relationships between bone density and mechanical properties: a literature review. Clin. Biomech. 23, 135–146. Hvid, I., Jensen, J., 1984. Cancellous bone strength at the proximal human tibia. Archive: Eng. Med. 1971–1988 (vols. 1–17) 13, 21–25. Keaveny, T.M., Guo, X.E., Wachtel, E.F., McMahon, T.A., Hayes, W.C., 1994. Trabecular bone exhibits fully linear elastic behavior and yields at low strains. J. Biomech. 27, 1127–1136. Keller, T.S., 1994. Predicting the compressive mechanical behavior of bone. J. Biomech. 27, 1159–1168. Keyak, J.H., 2001. Improved prediction of proximal femoral fracture load using nonlinear finite element models. Med. Eng. Phys. 23, 165–173. Keyak, J.H., Rossi, S.A., Jones, K.A., Les, C.M., Skinner, H.B., 2001. Prediction of fracture location in the proximal femur using FE models. Med. Eng. Phys. 23, 657–664.
2266
M.M. Juszczyk et al. / Journal of Biomechanics 44 (2011) 2259–2266
Keyak, J.H., Skinner, H.B., Fleming, J.A., 2001. Effect of force direction on femoral fracture load for two types of loading conditions. J. Orthop. Res. 19, 539–544. Kopperdahl, D.L., Keaveny, T.M., 1998. Yield strain behavior of trabecular bone. J. Biomech. 31, 601–608. Lakes, R.S., Katz, J.L., 1979. Viscoelastic properties of wet cortical bone. III. A nonlinear constitutive equation. J. Biomech. 12, 689–698. Lanyon, L.E., Hampson, W.G.J., Goodship, A.E., Shah, J.S., 1975. Bone deformation recorded in vivo from strain gauges attached to the human tibial shaft. Acta Orthop. Scand. 46, 256–268. Link, T.M., Vieth, V., Langenberg, R., Meier, N., Lotter, A., Newitt, D., Majumdar, S., 2003. Structure analysis of high resolution magnetic resonance imaging of the proximal femur: in vitro correlation with biomechanical strength and BMD. Calcif. Tissue Int. 72, 156–165. ¨ Lochmuller, E.-M., Groll, O., Kuhn, V., Eckstein, F., 2002. Mechanical strength of the proximal femur as predicted from geometric and densitometric bone properties at the lower limb versus the distal radius. Bone 30, 207–216. ¨ Lochmuller, E.M., Zeller, J.-B., Kaiser, D., Eckstein, F., Landgraf, J., Putz, R., Steldinger, R., 1998. Correlation of femoral and lumbar DXA and calcaneal ultrasound, with the in vitro failure of the proximal femur. Osteoporos Int. 8, 591–598. Lotz, J.C., Cheal, E.J., Hayes, W.C., 1991. Fracture prediction for the proximal femur using finite element models: Part I—linear analysis. J. Biomech. Eng. 113, 353–360. ¨ hman, C., Dall’Ara, E., Baleani, M., Jan, S.V.S., Viceconti, M., 2008. The effects of O embalming using a 4% formalin solution on the compressive mechanical properties of human cortical bone. Clin. Biomech. 23, 1294–1298.
Ota, T., Yamamoto, I., Morita, R., 1999. Fracture simulation of the femoral bone using the finite-element method: how a fracture initiates and proceeds. J. Bone Miner. Metab. 17, 108–112. Raftopoulos, D., Katsamanis, E., Saul, F., Liu, W., Saddemi, S., 1993. An intermediate loading rate technique for the determination of mechanical properties of human femoral cortical bone. J. Biomed. Eng. 15, 60–66. Schileo, E., Taddei, F., Malandrino, A., Cristofolini, L., Viceconti, M., 2007. Subjectspecific finite element models can accurately predict strain levels in long bones. J. Biomech. 40, 2982–2989. Yeni, Y.N., Fyhrie, D.P., 2003. A rate-dependent microcrack-bridging model that can explain the strain rate dependency of cortical bone apparent yield strength. J. Biomech. 36, 1343–1353. Yosibash, Z., Padan, R., Joskowicz, L., Milgrom, C., 2007. A CT-based high-order finite element analysis of the human proximal femur compared to in-vitro experiments. J. Biomech. Eng. 129, 297–309. Yosibash, Z., Tal, D., Trabelsi, N., 2010. Predicting the yield of the proximal femur using high-order finite-element analysis with inhomogeneous orthotropic material properties. Philos. Trans. A—Math. Phys. Eng. Sci. 368, 2707–2723. Zilch, H., Rohlmann, A., Bergmann, G., Kolbel, R., 1980. Material properties of femural cancellous bone in axial loading. Part II: time dependent properties. Arch. Orthop. Trauma Surg. 97, 257–262.