Journal of Biomechanics
Volume 40, Issue 1 , Pages 36-45, 2007

Tensile behavior of cortical bone: Dependence of organic matrix material properties on bone mineral content

  • S.P. Kotha

      Affiliations

    • Biomedical Engineering Department, Rutgers University, 617 Bowser Road, Piscataway, NJ 08854, USA
    • Present Address: Department of Oral Biology, University of Missouri – Kansas City, Rm 432 650 E. 25th St., Kansas City, MO 64108.
    • ,
  • N. Guzelsu

      Affiliations

    • Biomedical Engineering Department, Rutgers University, 617 Bowser Road, Piscataway, NJ 08854, USA
    • University of Medicine and Dentistry of New Jersey − SOM – Biomechanics Laboratory, Science Center, 2 Medical Center Drive, Stratford, NJ 08084-1504, USA
    • Corresponding Author InformationCorresponding author. Department of Osteosciences/Biomechanics Program, UDP-Suite 1700, 42 East Laurel Road, Stratford, NJ 08084-1504, USA. Tel.: +18565662731; fax: +18565662733.

Accepted 26 November 2005. published online 23 January 2006.

Article Outline

Abstract 

A porous composite model is developed to analyze the tensile mechanical properties of cortical bone. The effects of microporosity (volksman's canals, osteocyte lacunae) on the mechanical properties of bone tissue are taken into account. A simple shear lag theory, wherein tensile loads are transferred between overlapped mineral platelets by shearing of the organic matrix, is used to model the reinforcement provided by mineral platelets. It is assumed that the organic matrix is elastic in tension and elastic–perfectly plastic in shear until it fails. When organic matrix shear stresses at the ends of mineral platelets reach their yield values, the stress–strain curve of bone tissue starts to deviate from linear behavior. This is referred as the microscopic yield point. At the point where the stress–strain behavior of bone shows a sharp curvature, the organic phase reaches its shear yield stress value over the entire platelet. This is referred as the macroscopic yield point. It is assumed that after macroscopic yield, mineral platelets cannot contribute to the load bearing capacity of bone and that the mechanical behavior of cortical bone tissue is determined by the organic phase only. Bone fails when the principal stress of the organic matrix is reached. By assuming that mechanical properties of the organic matrix are dependent on bone mineral content below the macroscopic yield point, the model is used to predict the entire tensile mechanical behavior of cortical bone for different mineral contents. It is found that decreased shear yield stresses and organic matrix elastic moduli are required to explain the mechanical behavior of bones with lowered mineral contents. Under these conditions, the predicted values (elastic modulus, 0.002 yield stress and strain, and ultimate stress and strain) are within 15% of experimental data.

Keywords: Cortical bone tissue, Tensile behavior, Bone mineral content (BMC), Organic phase, Composite modeling

 

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1. Introduction 

Examining the tensile stress–strain behavior of bone tissue with different mineral contents is helpful to understand diseased conditions (e.g., osteoarthritis, osteopetrosis), in which the ratio of mineral to organic matrix content is affected (Li and Aspden, 1997). Previous models to explain the mechanical behavior of bone (other than elastic modulus) have used the phenomenon of microscopically observable damage to explain the mechanical properties of bone (Fondrk et al., 1999; Gao et al., 2003). This approach does not provide an insight into why the post-yield slope of the stress–strain curves with different mineral contents should be the same as that of decalcified bone (Kotha and Guzelsu, 2003a).

In our previous studies, we have used a simple shear lag theory with overlapped platelets to model the tensile mechanical behavior of bone tissue (Kotha and Guzelsu, 2000, Kotha and Guzelsu, 2002, Kotha and Guzelsu, 2003a; Kotha et al., 2000). This type of analysis has also been conducted by Jager and Fratzl (2000). The microporosity (volkman's canals, osteocyte lacunae, canaliculi and blood vessels) that affects the mechanical properties of bone tissue (Sevostianov and Kachanov, 2000) has not been considered in these models (Kotha and Guzelsu, 2002, Kotha and Guzelsu, 2003a). In this study, we incorporated two additional refinements to our previous models, namely: (1) the mechanical properties of the organic phase are affected by mineral content, and (2) the mechanical properties of bone tissue are affected by microporosity (Sevostianov and Kachanov, 2000). This approach allowed us to model the mechanical behavior of bones with lower mineral content, which were obtained by fluoride treatments (Kotha and Guzelsu, 2003a; Table 1) with better accuracy than our previous model (Kotha and Guzelsu, 2003a). The entire tensile stress–strain curve, the elastic modulus, the microscopic and macroscopic yield points, the ultimate stresses, and strains of bone tissue are presented as a function of bone mineral content (BMC).

Table 1. Experimental values of control and fluoride treated bone samples: control 12 days in 0.15M KCl at pH=7.5; fluoride 3 days in 1.0M KF at pH=7.5; fluoride 12 days in 1.0M KF at pH=7.5 (Kotha and Guzelsu, 2003b)
GroupNumber of samplesElastic modulus (Gpa)Yield stress (MPa)Yield strain (%)Ultimate stress (MPa)Ultimate strain (%)
Control919.3 (2.9)a100.5 (19.1)a0.698 (0.025)a103.3 (16.6)a2.88 (1.22)a
3 days fluoride913.1 (2.6)b63.8 (9.7)b0.689 (0.032)a73.7 (11.8)b4.64 (1.21)b
12 days fluoride99.3 (1.5)c40.9 (7.3)c0.659 (0.041)a65.5 (12.3)b6.39 (1.17)c

Different superscripts indicate that the properties are significantly different than other groups ().

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2. Theoretical model 

2.1. Basic principles of the model 

Using our previous model (Kotha and Guzelsu, 2003a), predictions of tensile elastic moduli and ultimate strains in bones with lower mineral contents (fluoride treated) were almost 30% off from their corresponding experimental values (Kotha and Guzelsu, 2003a). In our previous model, even after taking into account the microporosity, using mean-field methods similar to those used by Sevostianov and Kachanov (Sevostianov and Kachanov, 2000), predicted values were not improved. In this study, we model the tensile mechanical properties of bone by taking into account: (1) the dependence of organic matrix mechanical properties on mineral content; (2) the load transfer between the overlapped mineral platelets (Jager and Fratzl, 2000; Ji and Gao, 2004; Kotha and Guzelsu, 2003a), and (3) the effect of microporosity.

To model the elastic modulus of organic matrix as a function of mineral content, the following relationships were tried: (i) a constant (independent of mineral content); (ii) a linear relationship (two constants) and (iii) a non-linear relationship (three constants). By matching experimental measurements to model predictions, we show that a non-linear relationship allows us to predict the mechanical properties of bone samples with reduced mineral content much better than a constant or a linear relationship. Presently, a relationship between the organic matrix and mineral content cannot be explicitly modeled, because the following phenomena that determine the extent of straight-jacketing of collagen molecules/microfibrils (McCutchen, 1975) by the bone mineral are not known: (i) the rearrangement and changes in orientation of collagen molecules/microfibrils due to removal of some mineral–organic interactions as mineral content is lowered and (ii) changes in lattice confinement of collagen molecules and/or microfibrils and their interactions in bones with different mineral contents (Lees, 1981; Bonar et al., 1985).

In this study, the mineral–organic composite (moc) without microporosity is referred to as moc. Within the moc, the organic matrix mechanical properties depend on mineral content before yield. In the moc, load transfer between overlapped platelets is due to shear stress, τo, in the organic matrix. Therefore, our previous model, formulas for moc can be obtained by modifying the relationships derived previously (Kotha and Guzelsu, 2003a). In the moc, increasing loads parallel to the x-axis (along the long axis of bone, Fig. 1(a)) are transferred between overlapped platelets by shear. On applying load to bone tissue, increasing organic matrix shear stresses (τo) allows the mineral platelets to bear increased normal stresses in the x direction (Jager and Fratzl, 2000; Gao et al., 2003; Ji and Gao, 2004; Kotha and Guzelsu, 2003a). Under increasing load, organic matrix shear stress reaches its yield value, τoy, at the ends of the platelets (τoy, shear yield stress of organic) (Fig. 1b). This is the microscopic yield point of bone tissue as the stress–strain behavior of bone begins to deviate from linearity at this point. With a further increase of loads in the x direction, the organic phase yields in shear over the entire surface of the platelet. After this point, referred as macroscopic yield point, mineral platelets cannot contribute to the load bearing capacity of bone tissue. The mechanical behavior of the moc, after the macroscopic yield point, is determined by the organic phase only and it is independent of the mineral content (Kotha and Guzelsu, 2003a). It is noted that the theoretically predicted macroscopic yield point is different from the experimentally measured yield point which is determined by the 0.2% strain offset method. The organic matrix is assumed to be elastic in tension and elastic–perfectly plastic in shear until it fails. The moc is assumed to fail when the principal stress in the organic matrix is reached. Under the above assumptions, the whole stress–strain curve of the moc (bone tissue without the microporosity) can be generated.

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  • Fig. 1. 

    (a) A representative unit cell for stress transfer, (b) representative unit cell after the organic matrix at the ends of mineral platelets has yielded due to shear stress. The microporosity (osteocytes, lacunae, Volkmann canals etc.) is not shown. For detail of nomenclature, see Kotha et al. (2001).

Microporosity is introduced into the moc to obtain the mechanical properties of bone tissue. The effect of microporosity on stresses and strains in bone are considered through an approximation of mean-field theories (similar to those conducted by Boccaccini et al. (1996) but using mean-field theories developed by Pan and Weng (1995)). The derived relations are qualitatively similar to those obtained for bone tissue by other researchers (Sevostianov and Kachanov, 2000). The mathematical details of the modeling can be seen in Supplementary data.

The theoretical mechanical behavior of control bovine bone is compared with the mechanical behavior of bones with lower mineral content obtained by using in vitro fluoride treatments. In vitro fluoride treatments have been used to uniformly lower the BMC uniformly (Kotha et al., 1998; DePaula et al., 2002; Kotha and Guzelsu, 2003a). Fluoride ions dissolve the bone mineral and precipitate as calcium fluoride and fluoroapatite like material that acts as fillers. The effect of the filler contents on the mechanical properties of bone is negligible (Kotha and Guzelsu, 2003a) and is neglected in our analysis. The previous experimental data obtained with fluoride treatment are summarized in Table 1.

Unless otherwise stated, the volume fractions used for control bone are: mineral—40%, organic—35% and microporosity—15%. The mineral content of control bone decreases by 18% and 29% by immersion of the bones in fluoride ion solutions for 3 days and 12 days, respectively, i.e. the corresponding mineral volume fractions are 32.8% (0.40–0.40×0.18) and 28.4% (0.40–0.40×0.29), respectively. There is no change in the volume fraction of organic matrix or microporosity after immersion in fluoride ion solutions. The elastic modulus of mineral and fully calcified organic matrix (without any porosity) used are 114GPa (Gilmore and Katz, 1982) and 7.5GPa (Sasaki and Odajima, 1996), respectively.

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3. Results 

The elastic modulus of bone increases with an increase in the aspect ratio of mineral platelets (length/thickness; L/t) and with an increase in organic matrix elastic modulus in the moc (E3moc) (Eqs. (5) and (5a), Supplementary data; Fig. 2a). For the same volume fraction of bone mineral (e.g., 0.40 for control bone), the initial elastic modulus of control bone (Ebone=19.3GPa) can be obtained by using different combinations of mineral aspect ratios and organic matrix elastic moduli. In order to obtain the elastic modulus for control bone with increasing mineral aspect ratio (from 17.7 to 27.7), the organic matrix elastic modulus in control bone must decrease from 5.5 to 3.0GPa, according to our model. The elastic modulus of bone tissue increases with an increase in the volume fraction of bone mineral, and also with increasing organic matrix elastic modulus for a constant aspect ratio (Eqs. (5) and (5a) in Supplementary data; Fig. 2b).

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  • Fig. 2. 

    (a) Variation in the elastic modulus of bone as a function of aspect ratio (length to thickness ratio) of bone mineral platelets (bone mineral content, BMC=0.40). The dashed horizontal line shows elastic modulus of control bone (19.3GPa). (b) Variation in the elastic modulus of bone as a function of mineral volume fraction {aspect ratio=20.0}; Bone Mineral Content (BMC=0.40)}. The organic matrix elastic modulus is kept constant in each line and does not vary with mineral content. These figures indicate that an increase in the elastic modulus of the organic matrix, the aspect ratio of the bone mineral, and bone mineral content leads to increases in the elastic modulus of bone tissue.

The relationships between E3moc and the Vm-moc (Eq. (4) or (4a) in Supplementary data) depend on aspect ratio (Fig. 3). Variations in E3moc for five different aspect ratios are plotted against Vm-moc with the corresponding a, b and h parameters (Fig. 3). For the curve where E3moc=1.31GPa (constant) in Fig. 3, the mineral platelet aspect ratio (32.0) is larger than the other four curves. Even in this extreme limiting case, it is observed that the low organic matrix elastic modulus value obtained from decalcified bone experiments can be counter balanced with the reinforcement provided by very thin and long mineral platelets. On the other hand, the dotted perpendicular line marked with Vm-moc=0.471 (which corresponds to BMC of 0.40) shows that the lower aspect ratios require larger values of E3moc in order to match the experimental elastic modulus of control bone. When BMC increases such that it fills in all the spacing between collagen molecules (hypermineralization; Vm-moc=0.588 or BMC=0.50), the elastic modulus of the organic matrix reaches that of a collagen molecule (7.5GPa) for all mineral aspect ratios as dictated by Eq. (4) in Supplementary data.

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  • Fig. 3. 

    Variation in the elastic modulus of the organic matrix with respect to bone mineral content of moc. Four different relationships are considered in Eq. (4) (given in Supplementary data). In all cases, the elastic modulus of the organic matrix without any voids is accepted as 7.5GPa (i.e. when Vm-moc=0.588; Hypermineralization). To obtain the elastic modulus of control bone (19.3GPa) when the mineral volume fraction within the moc is 0.471, elastic modulus of the organic matrix must increase with smaller aspect ratios of mineral platelets (Vm-moc=0.471 is equivalent to BMC=0.40). In the case of very large aspect ratios (32.0), elastic modulus of the organic matrix in moc becomes independent of the volume fraction of the mineral phase in the moc which is shown with the horizontal line passing through the E3moc=1.31GPa point.

For a constant E3moc (4.7GPa), microscopic yield strain decreased (Fig. 4a) and microscopic yield stress increased with increasing BMC (Fig. 4b). Furthermore, increasing mineral aspect ratio decreased the microscopic yield strain while increasing the microscopic yield stress (Fig. 4a and b). Similar trends were observed in the macroscopic yield strains and stresses, respectively, (Fig. 5a and b) with varying BMC for a constant E3moc (4.7GPa).

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  • Fig. 4. 

    (a) The microscopic yield strain in control bone tissue normalized by organic matrix shear yield stress (τoy). (b) The microscopic yield stress of the control bone tissue normalized by organic matrix shear yield stress (τoy). Variations in both parameters are presented with respect to different mineral volume fractions and aspect ratios. Elastic modulus of the organic matrix (E3moc) is constant (4.7GPa) in both figures.

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  • Fig. 5. 

    (a) The macroscopic yield strain in control bone tissue that is normalized by shear yield stress of the organic matrix (τoy). (b) The macroscopic yield stress of control bone tissue that is normalized by shear yield stress of the organic matrix (τoy). Variations in both parameters are presented with respect to different mineral volume fractions and aspect ratios. Elastic modulus of the organic matrix E3moc is constant (4.7GPa) in both figures.

Based on different aspect ratios, elastic modulus of bone, τoy, ultimate stress and ultimate strain of bone are plotted in terms of BMC (Fig. 6a–d). The corresponding experimental points are also shown on these figures. For each aspect ratio, the experimental and theoretical elastic moduli of control bone are matched by varying E3moc. To calculate E3moc for fluoride treated samples, Eq. (4a) in Supplementary data is used with the corresponding Vm-moc after fluoride treatment (Vm-moc-3 days F=0.382 and 0.334; for 3- and 12-day fluoride treatment, respectively). The organic matrix shear yield stress, τoy, for control bone is calculated by matching the theoretical and experimental 0.002 yield stress values of control bone. In addition, τoy for fluoride treated samples (3 days and 12 days) are calculated by matching the theoretical and experimental 0.002 yield stress values of the fluoride treated samples. The calculated E3moc and τoy values for the fluoride treated samples are also function of aspect ratios. Mineral aspect ratios were not measured but several discrete values were assumed.

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  • Fig. 6. 

    (a) Elastic moduli of bone (Ebone). (b) Organic matrix shear yield stress (τoy). (c) Ultimate stress of bone (σbone ultimate). (d) Ultimate strain of bone (εultimatestrain). Variations in these parameters are shown with respect to different mineral aspect ratios and corresponding organic matrix elastic modulus E3moc which is a function of mineral volume fraction. Elastic modulus of the organic matrix of control bone varies between 5.5 and 1.31GPa resulting in aspect ratios of bone mineral between 17.7 and 32. For 3-day fluoride treated samples, elastic modulus of the organic matrix varies between 4.2 and 1.31GPa for aspect ratios between 17.7 and 32. For 12-day fluoride treated samples, elastic modulus of the organic matrix varies between 3.57 and 1.31GPa for aspect ratios between 17.7 and 32. Shear yield stress of the organic matrix is assumed so that theoretical and experimental 0.002 yield stress values are matched as summarized in Table 2. The best match for experimental parameters is provided when the elastic modulus of the organic matrix in control bone is 3.0GPa and the aspect ratio is 27.7. It is noted that failure occurs before microscopic yield, when the organic matrix elastic modulus is 5.5GPa. Lines connecting different points show the constant aspect ratio solutions. Bars over experimental points indicate standard deviations.

The elastic modulus of bones with different mineral contents is best predicted with a mineral aspect ratio of 27.7 (Fig. 6a). For an aspect ratio of 27.7, the model predicts the elastic modulus of control and 3-day fluoride treated bone very closely for E3moc values of 3.0 and 2.3GPa, respectively. The results indicate that the experimental elastic modulus of 12-day fluoride treated bone is 8.5% lower than the best-predicted theoretical values (aspect ratio of 27.7, E3moc-12 days F=2.1GPa and Vm-moc-12 days F=0.334).

The theoretically predicted shear yield values for different aspect ratios for control and fluoride treated samples are plotted in Fig. 6b. For an aspect ratio of 27.7, the following values of organic matrix shear yield stresses are obtained (Fig. 6b) (τoy control=17.5MPa, τoy 3 days F=13.3MPa and τoy 12 days F=9.4MPa). Table 2 summarizes the E3moc, and τoy, for control, 3-day and 12-day fluoride treated groups for different aspect ratios. The predicted ultimate stresses are within 10% of the experimental values for all aspect ratios, BMCs and E3moc (Fig. 6c). The only reasonable values for ultimate strains in all groups are obtained if we use a mineral aspect ratio of 27.7 (E3moc=3.0GPa for control) (Fig. 6d). The corresponding a, b and h parameters for Eq. (4a) (in supplementary data) are 4.7, 0.62 and 6.1, respectively (Fig. 3). In the calculations of ultimate strains in fluoride treated bones, E3moc-3 days F=2.3GPa and E3moc-12 days F=2.1GPa are used with the aspect ratio of 27.7 in Eq. (10). For all other values of aspect ratios, the ultimate strains of the three groups are higher or lower than the measured values. Even with the best case fit, the theoretical ultimate strains for the 12-day fluoride treated samples are 12.2% lower than the corresponding experimental value (Fig. 6d). In summary, an aspect ratio of 27.7 provides the best fit to the experimental values of elastic modulus of bone (Fig. 6a), ultimate stress of bone (Fig. 6c) and ultimate strain of bone (Fig. 6d).

Table 2. Organic matrix properties used in Fig. 6
Mineral aspect ratioOrganic matrix elastic modulus (GPa)/Shear yield stress (MPa)
Control3-day fluoride12-day fluoride
17.75.5/25.4a4.2/16.33.57/11.4
22.54.0/20.33.0/15.52.5/11.0
27.73.0/17.52.3/13.32.1/9.4
32.01.3/15.21.3/11.81.3/8.3

aIndicates that failure of bone occurs before yield is reached. However, yield stress would be 25.4MPa if applicable.

In the above calculations for ultimate stresses (Eq. (8) in Supplementary data), stresses perpendicular to the applied loads (σy) has been assumed to be zero. By using experimentally determined ultimate strain values and the principal failure stress of the organic matrix, σy values can be estimated by using Eq. (9) (in Supplementary data). The σx value of the organic phase at failure is obtained by adding the organic stresses up to macroscopic yield (E3moc×εbone macroscopic yield) to the stresses after the yield point {(εultimate−εbone macroscopic yield)E3moc (1.31GPa)}. This approach showed that when the σy's are averaged across the three groups, the smallest average values are obtained for an aspect ratio of 27.7 (Fig. 7). Because of these results, the aspect ratio of 27.7 is used for all groups.

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  • Fig. 7. 

    Theoretically predicted organic matrix normal stresses in the y direction as a function of mineral volume fraction for different aspect ratios. Stresses in the y direction are obtained by matching the theoretically predicted ultimate strains to experimental values for control and fluoride treated bone samples and assuming the principal failure stress criteria (Eq. (13) in Supplementary data). This figure indicates that the variation in the normal stresses in the y direction is minimum in all three groups, when we use the organic matrix elastic modulus and the aspect ratio 3.0GPa and 27.7, respectively, in control bone. The a, b and h parameters are computed from the control bone. The elastic modulus of the organic matrix phase for 3- and 12-day fluoride treated samples were computed from Eq. (4) (given in Supplementary data) with corresponding bone tissue mineral volume fraction by using a, b and h values from the control bone.

The best fit theoretical stress–strain curve for control bone (aspect ratio=27.7, E3moc control=3.0Gpa, τoy control=17.5MPa), and the stress–strain curves for 3- and 12-day fluoride treated samples are shown in Fig. 8.

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  • Fig. 8. 

    Experimental and theoretical stress–strain curves of plexiform cortical bovine bone tissue with different mineral contents. All the theoretically predicted mechanical parameters (elastic modulus, yield stresses and strains, ultimate stresses and strains) are within 15% of the experimental values.

The tensile properties of the organic matrix for different groups predicted from the theoretical calculations are shown in Fig. 9. After macroscopic yield, all stress–strain curves are parallel, indicating that the organic matrix elastic modulus is independent of BMC. The elastic–perfectly plastic, shear stress–strain behavior of the organic matrix in bones with different mineral contents is illustrated in Fig. 10. After the microscopic yield stresses of the control and fluoride treated bone tissues are reached, the organic matrix yields in shear until it fails in tension. Failure occurs when the principal stress in the organic matrix reaches the principal stress of the organic matrix, which is the ultimate stress of the decalcified organic matrix tested in tension.

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4. Discussion 

A micromechanical model for cortical bone tissue has been developed by using an overlapped platelets model. The model suggests that the organic matrix elastic modulus and shear yield stress depend upon BMC, up to the yield point. This model implies that reinforcement of the organic matrix by mineral platelets influences pre-yield mechanical properties of bone tissue while organic matrix behavior alone determines the post-yield mechanical properties. This model supports the theory that ultimate failure in bone results from failure of the organic matrix. By combining the model's (overlapped platelets) prediction of the mechanical behavior of bone tissue up to macroscopic yield point, with the organic matrix behavior after the yield (post-yield), the tensile mechanical behavior of cortical bone is predicted.

The biggest assumption in this study is the modulation of organic matrix material properties by mineral content. In applying this to bones with lower mineral contents: when the numbers of mineral platelets are reduced due to dissolution induced by immersion of bone tissue in fluoride ion solutions, the mechanical behavior of the organic matrix becomes closer to that of bulk collagen or a microfibrillar structure than that of a collagen molecule. As the mechanical behavior of a collagen molecule yields the upper bound for different collagen-based structures (Sasaki and Odajima, 1996), a reduction in mineral content would be expected to reduce the elastic modulus of the organic matrix. Hypermineralization can be thought as the opposite of what is explained above. Increasing mineral content reduces the deformations between collagen structures and forces them to behave like collagen molecules or individual fibrils (Sasaki and Odajima, 1996). Therefore, variations of the mechanical properties of the organic matrix change from a collagen molecule to decalcified bone matrix depending on the mineral content.

The elastic behavior of bone tissue has also been modeled by using only collagen molecule's mechanical behavior in a crystal foam composite structure with different hierarchical structure approximation (Hellmich and Ulm, 2002; Hellmich et al., 2004). In our method as explained above, the properties of the organic matrix differ from Hellmich and his co-workers’ approach, where the mineral content affects the packing of the collagen structure and its mechanical properties (also slightly different from that of Jager and Fratzl, 2000). Our approach allows us to change the mechanical properties of the organic phase from collagen fibril bundles (totally decalcified samples) to collagen molecules (totally mineralized samples—hypermineralization) depending on the mineral content of the bone tissue.

The tensile elastic modulus, ultimate stress and strain of the decalcified organic matrix have been measured by different researchers (Bowman et al., 1996; Catanese et al., 1999; Wang et al., 2002; Kotha and Guzelsu, 2003a). There is no data available on its shear behavior. These researchers showed that after the initial toe region, bone organic matrix behaves elastically in tension (Jager and Fratzl, 2000) and reaches its ultimate stress without yielding. This supports our assumption that bone matrix behaves as an elastic brittle material in tension.

We suggest that the origin of organic matrix shear yield stresses may be from pre-existing mineralization stresses that are compressive in nature and act in the y-axis direction (Fig. 1). These compressive stresses push the collagen molecules together and make them more closely packed in the y direction (Bonar et al., 1985; Lees, 2003) and increase frictional forces inside the organic matrix. Under tension, deformation of collagen fibers involves stretching, and slippage of laterally adjoining elements under shear force transmission (Pins et al., 1997; Sasaki and Odajima, 1996; Sasaki et al., 1989; Christiansen et al., 2000; Kotha and Guzelsu, 2003b). Therefore, the organic matrix, which has more mineral content and larger frictional forces, can sustain larger shear load transfer before slippage. Once the shear forces overcome frictional forces on the collagen molecules and/or collagen fibrils, the collagen molecules and/or fibrils start to slip relative to one other, creating shear yield in bone matrix. Deformations due to applied external loads, and partial demineralization due to fluoride treatment alter the fibrillar orientation and packing of the collagen molecules. These changes take place at the fibrillar level, which alters the number of interactions and their intensity among the collagen molecules (Pins et al., 1997). This is similar in principle to Mohr–Columb frictional interactions that determine the mechanical behavior of soils, except that normal stresses in the y-direction are generated by mineralization in bone.

We have assumed that bone fails when the organic matrix in bone fails in tension. The organic matrix cannot carry any more axial loads when the principal stress reaches the ultimate strength of the organic phase. Others have also suggested that the organic matrix is responsible for the tensile strength of bone tissue (Puustjarvi et al., 1999; Zioupos et al., 1999). A possible reason for the failure to predict ultimate strains in bones with the lowest mineral contents (12-day fluoride treated bones) and in decalcified bones (the linear portion of the stress–strain curve extrapolated to zero stress) is the absence of hierarchical stress transfer mechanisms between overlapped collagen microfibrils/fibrils in this model.

A simple model has been considered wherein the viscoelastic nature, hierarchical organization and local variations in BMC of bone tissue have not been taken into account. The orientation of the mineral and organic has not been considered. We have assumed that the mineral and organic are isotropic. Stresses and strains perpendicular to the axial direction (y) were also ignored in this analysis. We have also neglected the shape, orientation and distribution of the microporosity. Stress concentrations due to microporosity have not been explicitly modeled. The stress–strain behavior of bone has been related to the mechanical properties of the mineral phase, the aspect ratio of the mineral platelets and the mechanical properties of the organic matrix. Assuming that the mechanical properties of the organic matrix up to macroscopic yield point depend on the mineral amount yields more insight about the mechanical behavior of bone tissue with different mineral contents. Combining the overlapped platelet model up to macroscopic yield point with organic matrix mechanical behavior after macroscopic yield predicts the entire tensile mechanical behavior of cortical bone with different mineral contents.

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Acknowledgments 

This work was supported in part by grants from the National Science Foundation (BCS-9210253) and the Foundation of the University of Medicine and Dentistry of New Jersey (27-97).

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Appendix A. Supplementary materials 

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Supplementary data

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Further reading 

  1. Blitz RM, Pellegrino ED. The chemical anatomy of bone. I. A comparative study of bone composition in sixteen vertebrates. Journal of Bone and Joint Surgery. 1969;59A:456–466
  2. Calve S, Dennis RG, Kosnik PE, Baar K, Grosh K, Arruda EM. Engineering of functional tendon. Tissue Engineering. 2004;10:755–761
  3. Cowin SC. Bone poroelasticity. Journal of Biomechanics. 1999;32:217–238
  4. Gibson LJ, Ashby MF. Cellular Solids: Structure and Properties. second ed. Cambride, UK: Cambridge University Press; 1997;
  5. Kastelic J, Baer E. Deformation in tendon collagen. In: The Mechanical Properties of Biological Materials Symposia for the Society for Experimental Biology. vol. XXXIV:Cambridge: Cambridge University Press; 1980;p. 397–435
  6. Katz EP, Wachtel E, Yamanuchi M, Mechanic GL. The structure of mineralized collagen fibrils. Connective Tissue Research. 1989;21:149–158
  7. Landis WJ, Hodgens KJ, Arena J, Song MJ, McEwen BF. Structural relations between collagen and mineral in bone by high voltage electron microscopic tomography. Microscopy Research and Technique. 1996;33:192–202
  8. Neuman WF, Toribara TY, Mulrvan BJ. The surface chemistry of bone. VII The hydration shell. Journal of the American Chemical Society. 1953;75:4239–4242
  9. Posner AS. Bone mineral and mineralization process. In:  Peckm WA editors. Bone and Mineral Research Annual. vol. 5:New York: Elsevier science; 1987;p. 65–116
  10. Raspanti M, Guizzardi S, Strocchi R, Ruggeri A. Collagen fibril patterns in compact bone: preliminary ultrastructural observations. Acta Anatomica. 1996;155:249–256
  11. Rho JY, Zioupos P, Currey JD, Pharr GM. Microstructural elasticity and regional heterogeneity in human femoral bone of various ages examined by nano-indentation. Journal of Biomechanics. 2002;35:189–198

PII: S0021-9290(05)00534-8

doi:10.1016/j.jbiomech.2005.11.016

Journal of Biomechanics
Volume 40, Issue 1 , Pages 36-45, 2007

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