Principal stiffness orientation and degree of anisotropy of human osteons based on nanoindentation in three distinct planes

Abstract

Haversian systems or ‘osteons’ are cylindrical structures, formed by bone lamellae, that make up the major part of human cortical bone. Despite their discovery centuries ago in 1691 by Clopton Havers, their mechanical properties are still poorly understood.

The objective of this study is a detailed identification of the anisotropic elastic properties of the secondary osteon in the lamella plane. Additionally, the principal material orientation with respect to the osteon is assessed. Therefore a new nanoindentation method was developed which allows the measurement of indentation data in three distinct planes on a single osteon.

All investigated osteons appeared to be anisotropic with a preferred stiffness alignment along the axial direction with a small average helical winding around the osteon axis. The mean degree of anisotropy was 1.75 ± 0.36 and the mean helix angle was 10.3^°±0.8°.

These findings oppose two well established views of compact bone microstructure: first, the generally clear axial stiffness orientation contradicts a regular ‘twisted plywood’ collagen fibril orientation pattern in lamellar bone that would lead to a more isotropic behavior. Second, the class of transverse osteons were not observed from the mechanical point of view.

Graphical abstract

Highlights

► Stiffness anisotropy of human osteons. ► Nanoindentation in multiple directions in the lamella plane. ► Strong anisotropy with mainly axial alignment is found. ► Helical winding of principal stiffness orientation is plausible.

1. Introduction

Secondary osteons, also called ‘Haversian systems’, represent the basic building block of cortical lamellar bone and therefore determine its macroscopical mechanical properties. An osteon is a cylindrical structure with a diameter of ∼200 μm, aligned along the shaft of the long bones (Rho et al., 1998; Fratzl and Weinkamer, 2007). It consists of a central haversian channel, circumferentially surrounded by an assembly of several layers of bone lamellae. The single bone lamella is again composed of layers of collagen fibrils which are rotated according to a certain fibril orientation pattern, giving rise to a plywood like structure (Giraud-Guille, 1988; Weiner et al., 1997, 1999; Wagermaier et al., 2006). These fibrils are reinforced by bone mineral crystals whose $c$-axis is aligned along the fibril axis (Fratzl et al., 2004).

This striking fibril organization in bone lamellae is supposed to dictate the degree of anisotropy of osteons (Martin and Ishida, 1989; Reisinger et al., 2011; Pidaparti and Burr, 1992; Ziv et al., 1996). Predominantely longitudinally aligned fibrils lead to osteons that are strong in tension and mainly transverse fibrils to good compression capabilities (Ascenzi and Bonucci, 1967, 1968). Oblique fibril angles could result in a main stiffness alignment that possesses a helical winding around the osteon cylinder, inducing a spring-like deformation mechanism under axial load (Fig. 1), (Fratzl and Weinkamer, 2007).

Fig. 1.

Helix shaped alignment of the main principal material axis around the haversian channel in the ideal cylindrical osteon. With $\theta$ being the helix angle and $E_3$ and $E_2$ being the Young’s modules in the major and minor principal axis of the lamella assembly material, respectively.

A completely different point of view suggests that the bone mineral particle orientation is mainly axial and largely independent of the collagen organization (Turner et al., 1995). This mineral alignment is supposed to play the dominant role for bone anisotropy.

To shed more light on this issue, detailed measurements of the anisotropic elastic properties of osteons have to be related to the underlying fibril orientation patterns.

Such measurements are difficult to perform as they must be applied in multiple directions relative to the osteon lamella plane on a lengthscale of several microns. Accordingly, the currently available experimental data about the direction dependent properties of single bone lamellae or an assembly of lamellae is sparse. There were multiple publications on direction dependent properties of cortical bone measured on the tissue level by nanoindentation or acoustic microscopy (Hofmann et al., 2006; Ziv et al., 1996; Roy et al., 1999; Fan et al., 2002; Lakshmanan et al., 2007). To our knowledge only Franzoso and Zysset (2009) assessed lamella assembly properties in the axial and circumferential direction measured precisely in the lamella plane. These two indentation directions were sufficient to estimate the degree of anisotropy of osteons but not a potential helical winding of the main stiffness direction.

The objective of this work is to estimate the orthotropic elastic properties of the osteonal lamella assembly and to measure the angle of the helical alignment of the main principal material axis (Fig. 1). Therefore, a novel sample preparation technique is used to perform nanoindentation on three distinct surfaces in the osteon lamella plane. Then a fabric based orthotropic stiffness model is applied, delivering an estimation of the orthotropic elastic properties of the osteon lamella assembly and the helix angle of the major principal material axis.

2. Materials and methods

2.1. Sample preparation

5 mm thick crosssections were cut out of the midshafts of three fresh frozen human femurs of the left body side using an Exact 310 bandsaw (EXAKT Advanced Technologies GmbH, Norderstedt, DE) with a diamond coated blade under constant water irrigation (Fig. 2(a)). The donors were a 63 year old male, a 68 year old female and a 76 year old female. They were not supposed to suffer from any bone related disease. The transverse surfaces produced in this cut are orthogonal to the femoral axis and are called N(ormal)-faces in this study. Assuming that the major part of the osteons are orientated along the femoral axis, the N-face represents a transverse cut through the osteons.

Fig. 2.

Study design and sample preparation. (a) One mid-diaphysis bone slice per donor was segmented to cubes according to the anatomical quadrant (b) and glued to a sample holder (c). (d) Osteons were selected and indented on a 45° tilted face (O-face) left and right of the Haversian channel (e①), in the circumferential direction (e②) and in the axial direction (e③). In each indentation zone (yellow rectangles) a pattern of 33 indents was placed (f). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Each of the three obtained bone slices was then cut into four cubes related to their anatomical position in the body, using an Isomet low speed diamond blade circular saw (Buehler GmbH, Düsseldorf, DE) under constant water irrigation. This cut exposes (C)ircumferentially orientated surfaces (Fig. 2(b)). Assuming that the major part of the osteons are orientated along the femoral axis, the C-face represents a longitudinal cut through the osteons (Fig. 2(d)). From now on, water contact was avoided to prevent the surface from evolving ultracracks that might alter indentation results (Roschger et al., 1993).

The 12 resulting cubes, originating from the MEDial, LATeral, ANTerior and POSTerior positions with a size of approximately 5×5×5 mm, were dried for several days at room temperature. Then they were glued to L-shaped aluminum sample holders using an epoxy-based 5 min curing glue, exposing their N- and C-face (Fig. 2(c)). The sample holders allow to mount the samples in a 0°- (N-face up), 90°- (C-face up) and a 45°-position (O(blique)-face up).

Mounted in the 45°-position, the sharp sample edge was milled down to a strip shaped O-face of indentation-ready surface quality using the Leica LP 2600 ultra-milling system (Leica Microsystems GmbH, Wetzlar, DE) (Fig. 2(e)①). On these O-faces of the 12 samples, 42 osteons were selected for indentation and assigned to letters (a, b, …) according to the following criteria:

• •First, to maximize the number of osteons that can be halved in a single cut later on, the selected osteons of each sample should have equal distance to the sample edge (Fig. 2(d)).
• •Second, the osteon shape should be circular and clearly segregated from interstitial bone and other osteons.
• •Third, osteons that had a disproportionately large haversian channel were supposed to be in a resorption process and were excluded.
• •Fourth, the osteonal lamellae should be clearly visible and circumferentially surround the haversian channel.

During the whole preparation procedure, the samples were never exposed to any kind of chemicals, to avoid the altering of their mechanical properties.

2.2. Nanoindentation

Nanoindentation is a powerful technique for measuring nano- and microscale mechanical properties in hierarchically structured materials. It involves the indentation of the sample surface using a tip of a defined shape while monitoring the applied load and the displacement (Ebenstein and Pruitt, 2006; Zysset, 2009). The reduced modulus $E_r$ of the probed material is defined as

$$E_r=\frac{S\left(h_{\max }\right)\sqrt{\pi }}{2\beta \sqrt{A\left(h_{\max }\right)}}$$ (1)

with $S\left(h_{\max }\right)$ and $A\left(h_{\max }\right)$ being the slope of the unloading curve and the projected area of the imprint, respectively, at the point of maximum displacement $h_{\max }$. $\beta$ is the tip shape factor.

The indentation modulus $E_{\mathrm{ind}}$ incorporates the Young’s modulus of the tip $E_{\mathrm{tip}}$ and its Poisson ratio ${\nu }_{\mathrm{tip}}$.

$$E_{\mathrm{ind}}={(\frac{1}{E_r}-\frac{1-{\nu }_{\mathrm{tip}}^2}{E_{\mathrm{tip}}})}^{-1}\text{.}$$ (2)

Eqs. (1) and (2) are valid for anisotropic samples and do not assume an isotropic material configuration (Swadener and Pharr, 2001).

In this study, the indentation modules $E_{\mathrm{ind}}$ of the samples were measured using a diamond Berkovich tip ($E_{\mathrm{tip}}=1440\mathrm{GPa}$, ${\nu }_{\mathrm{tip}}=0.07$) attached to a TriboIndenter nanoindenting system (Hysitron Inc., Minneapolis, MN).

The general indentation strategy of each sample was as follows (Fig. 2(e)):

• 1.Indentation of the O-face, exactly on the bisection line of the osteon on both sides of the Haversian channel. This ensures that the osteon lamellae are indented in-plane (Fig. 2(e)①).
• 2.Milling down the C-face until the osteon is bisected.
• 3.Indenting the C-face of the osteon on both sides of the Haversian channel and thus hitting the lamellae in-plane in the circumferential direction (Fig. 2(e)②).
• 4.Milling down the N-face.
• 5.Indenting the N-face and thus in the axial direction of the osteon (Fig. 2(e)③).

At each of the described sites, 33 indents were performed in a matrix-like pattern as seen in (Fig. 2(f)). This pattern was placed close to the border of the haversian channel. For a particular indentation direction, the patterns were oriented the same way relative to the Haversian channel for all samples. The indents were performed with displacement control to a depth of 250 nm with a loading- and unloading rate of 40 nm/s and a holding time of 20 s.

2.3. Indent filtering

Most of the time, not all 33 indents per pattern were valid, so a filter ladder was applied to remove corrupt indents.

• 1.Due to machine induced inaccuracies in the pattern positioning, some of the 33 indents per pattern were placed either in the Haversian channel, too close to its edge, in lacunae or other pores. Those were filtered out by eye, based on surface images of the indented osteon (Fig. 3).
• 2.Indents, which load–displacement curve contained irregularities from the regular shape were removed manually.
• 3.Indents, whose modulus $E_{\mathrm{ind},i}$ was outside 1.5 times the interquartile range of all indents of the same indentation plane were regarded as outliers and filtered out. This is a standard statistical procedure from Crawley (2007).

Fig. 3.

Surface scans of the N-face-, O-face- and C-face indent sites on an osteon. The indents are visible as small triangles arranged in a 11×3 pattern. Some are close to the Haversian channel or the lacuna and had to be excluded from further processing. Single lamellae are visible. The investigated lamella assembly consists of multiple lamellae that are covered by the indentation pattern.

2.4. Osteon alignment analysis

As osteons are not perfectly parallel aligned hollow cylinders and the sample preparation on this small lengthscale is difficult, the sample’s N- and C-face most probably do not coincide with the ideal osteon’s N- and C-face.

To estimate the relative alignment of the individual osteons relative to the sample surfaces, each indented osteon was investigated in the light microscope. The following distances were measured on the N- and C-face: $d_C$ and $d_N$ is the Haversian channel diameter as appearing on the C- and N-face, respectively. $l_C$ is its length and $\alpha$ is its the tilting angle as seen on the C-face (Fig. 4).

Fig. 4.

Osteon alignment specifications: $d_C$ and $d_N$, Haversian channel diameter as seen on the C- and N-face; $l_C$, Haversian channel length; $\alpha$, tilting angle as seen on the C-face; $\beta$ second tilting angle; $\gamma$, out-of-plane angle of the C-face indents.

The ratio $d_N/d_C$ indicates whether the osteon was accurately bisected. For $d_N/d_C\ne 1.0$ the indents on the C-face are not circumferentially orientated with respect to the osteon and hit the lamellae not exactly inplane. The resulting out-of-plane angle $\gamma$ for indents close to the Haversian channel is then defined by

$$\gamma \approx arccos\left(\frac{d_C}{d_N}\right)\text{.}$$ (3)

The second tilting angle $\beta$ can be estimated by

$$\beta =arctan\left(\frac{d_C}{2l_C}\right)\text{.}$$ (4)

2.5. Lamella assembly material

The lamella assembly-material is considered as the homogenized material of 7–10 osteon lamellae (Fig. 3). This amount of lamellae could be covered by the 50 μm long nanoindentation patterns. As inspected in the light microscope, this span width incorporates in many cases the major part of the lamellae that belong apparently to a single osteon.

The lamella assembly material is related to the 1-2-3-material coordinate system (Fig. 5). Its major principal material axis (3), which indicates the direction of the maximum stiffness, is assumed to be lying in the lamella plane but is arbitrarily rotated around the radial direction (1) by the helix angle $\theta$.

Fig. 5.

The lamella assembly material represents a stack of multiple osteonal lamellae. 1-2-3: material coordinate system; $x–y–z$: osteon coordinate system. The major principal material axis (3) is aligned at the helix angle $\theta$ relative to the osteon longitudinal axis ($z$). First, the in-plane indentation modulus ${\tilde{E}}_{\mathrm{ind}}\left(\phi \right)$ is fitted to the four measured indentation modules ${\overline{E}}_{\mathrm{ind}}$ at 0°, 45°, 90° and 315°. Then the direction dependent Young’s modulus $E\left(\phi \right)$ and the corresponding engineering constants such as $E_2$ and $E_3$ are calculated.

The second coordinate system is fixed to the osteon structure. Its axes are orientated in the radial ($x$), axial ($z$) and circumferential direction ($y$) and the $y–z$-plane is considered as in-plane or the lamella plane, respectively (Fig. 5).

In the following, it is assumed that the lamella assembly properties are constant along the osteon perimeter as well as in some range along the osteon axis.

42 osteons were indented multiple times in three planes (N-, C- and 2×O-face) in the lamella in-plane direction. By calculating the mean of the $n$ valid indentation modules of each indentation pattern, the indentation modulus ${\overline{E}}_{\mathrm{ind}}$ of the lamella assembly in the corresponding direction is gained (Fig. 2(f)).

$${\overline{E}}_{\mathrm{ind}}=\frac{1}{n}\sum _{j=1}^n{E}_{\mathrm{ind},j}\text{.}$$ (5)

Because the lamellae are aligned circumferentially around the Haversian channel, the indentations on the O-face on the left and right side of the channel correspond to two different directions relative to the lamella assembly material. (The same yields theoretically for the C-face indents. However the lamella assembly stiffness tensor is pointwise symmetric for all directions, so all indents on the C-face could be combined.) As a consequence, the indentation modulus ${\overline{E}}_{\mathrm{ind}}$ of the lamella assembly material is known in the $\phi =0°,45°,90°$ and 315° directions in the $y–z$ plane for each osteon (Fig. 5).

Osteons for whose holds ${\overline{E}}_{\mathrm{ind}}\left(45°\right)>{\overline{E}}_{\mathrm{ind}}\left(315°\right)$, are left-hand wound and those with ${\overline{E}}_{\mathrm{ind}}\left(45°\right)<{\overline{E}}_{\mathrm{ind}}\left(315°\right)$ are right-hand wound.

2.6. Fabric-based orthotropic material model

Fabric-based orthotropic material properties are calculated for the lamella assembly material of each of the 42 osteons. Fabric-based orthotropy is a constrained case of orthotropy in which the 9 free parameters are reduced to 6 using a second order fabric tensor $M$ (Zysset, 2003). $M$ describes the influence of the underlying collagen fibril alignment. Evaluated in the material coordinate system (1-2-3) (Fig. 5), $M$ yields

$$M=\left[\begin{array}{ccc}\hfill m_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill m_2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill m_3\hfill \end{array}\right]$$ (6)

with ($m_1$, $m_2$, $m_3$) being the fabric eigenvalues. A fabric-based orthotropic compliance tensor $\mathbb{C}$ may be defined using $M$. In matrix form, $\mathbb{C}$ yields

$$\mathbb{C}\left(M\right)==\frac{1}{{ϵ}_0}\left[\begin{array}{cccccc}\hfill \frac{1}{m_1^2}\hfill & \hfill -\frac{{\nu }_0}{m_1{m}_2}\hfill & \hfill -\frac{{\nu }_0}{m_1{m}_3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{{\nu }_0}{m_2{m}_1}\hfill & \hfill \frac{1}{m_2^2}\hfill & \hfill -\frac{{\nu }_0}{m_2{m}_3}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -\frac{{\nu }_0}{m_3{m}_1}\hfill & \hfill -\frac{{\nu }_0}{m_3{m}_2}\hfill & \hfill \frac{1}{m_3^2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{ϵ}_0}{2{\mu }_0{m}_2{m}_3}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{ϵ}_0}{2{\mu }_0{m}_1{m}_3}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{{ϵ}_0}{2{\mu }_0{m}_1{m}_2}\hfill \end{array}\right]$$ (7)

with

$${\mu }_0={ϵ}_0/\left(2(1+{\nu }_0)\right)\text{.}$$ (8)

This orthotropic model degenerates into isotropy for $M=I$, with $I$ being the identity tensor. Normalizing $M$ with e.g. $\mathrm{tr}\left(M\right)=3$ ensures that the constants ${ϵ}_0$, ${\nu }_0$, ${\mu }_0$ have the physical meaning of the Young’s modulus, Poisson ratio and shear modulus of the isotropic component in $\mathbb{C}$ (Franzoso and Zysset, 2009). The degree of anisotropy of the material model depends on the relation between ($m_1$, $m_2$, $m_3$). In this study, the material orientation is set such that $m_1<m_3$ and $m_2<m_3$. Consequently the 3-axis is pointing into the direction of maximum stiffness. As already pointed out, this major principal direction may be rotated offaxis by the helix angle $\theta$ relative the osteon longitudinal direction ($z$) (Fig. 5).

2.7. Fitting the model to the experiment

For each investigated osteon, the constants $m_1$, $m_2$, $m_3$, ${ϵ}_0$, ${\nu }_0$, ${\mu }_0$ and $\theta$ of the lamella assembly material model, are identified. This is done by applying the condition that virtually indenting this material into the $\phi =0°,45°,90°$ and 315° directions in the $y–z$ plane must give the already known measured indentation values. Taking Eq. (8) and the relation $\mathrm{tr}\left(M\right)=3$ into account, five unknowns oppose four known indentation values. Assuming one unknown would be sufficient to balance the system of equations. Experience has shown that in this case, the solution is very sensitive to small perturbations of the given indentation values. To avoid this, two unknowns were assumed to be constant throughout the study. As the nanoindentation experiments were applied in the lamella plane and the indentation modulus is fairly insensitive to the Poisson ratio, the out-of-plane fabric eigenvalue $m_1$ and ${\nu }_0$ were fixed:

• •$m_1=0.747986$ is a value by Franzoso and Zysset (2009), obtained from 100 indents in transverse isotropic mineralized turkey leg tendon, using the method of Hengsberger et al. (2003).
• •${\nu }_0=0.34$ is an average Poisson ratio for cortical bone found in Cowin (2001)

The influence of the choice of these fixed parameters was evaluated in a sensitivity study (Section 4).

Finally, the fabric-based compliance tensor is dependent on two parameters $\mathbb{C}(\frac{m_3}{m_2},{ϵ}_0)$ and its alignment relative to the osteon contour is defined by $\theta$. To get the indentation modulus ${\tilde{E}}_{\mathrm{ind}}$ for a virtual indent into this material model in a direction defined by $\phi$, the theory of Swadener and Pharr (2001) is used:

$${\tilde{E}}_{\mathrm{ind}}(\phi ,\theta ,\frac{m_3}{m_2},{ϵ}_0)=\mathrm{Sw}(r,\mathbb{C}(\frac{m_3}{m_2},{ϵ}_0))$$ (9)

$$r=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill sin(\phi -\theta )\hfill \\ \hfill cos(\phi -\theta )\hfill \end{array}\right]$$ (10)

with $\mathrm{Sw}$ being a function returning the indentation modulus of a virtual indent into an orthotropic material $\mathbb{C}$ in an arbitrary direction defined by the vector $r$ described in the material coordinate system.

Now a least-squares expression for determining $\theta$, $\frac{m_3}{m_2}$ and ${ϵ}_0$ can be set up. At each indentation direction ${\phi }_i$, the difference between the indentation modulus ${\tilde{E}}_{\mathrm{ind}}$ of the fabric-based orthotropic model and the measured indentation modulus ${\overline{E}}_{\mathrm{ind}}$ must be as small as possible.

$$\sum _{i=1}^4|{\tilde{E}}_{\mathrm{ind}}^2({\phi }_i,\theta ,\frac{m_3}{m_2},{ϵ}_0)-{\overline{E}}_{\mathrm{ind}}^2\left({\phi }_i\right)|\to \min \text{with }\phi =[0°,45°,90°,315°]\text{.}$$ (11)

This expression was minimized for $\theta$, $\frac{m_3}{m_2}$ and ${ϵ}_0$ for each of the 42 osteons.

2.8. Statistics

The fabric elasticity parameters ($m_1$, $m_2$, $m_3$, ${ϵ}_0$, ${\nu }_0$, ${\mu }_0$) were converted to engineering constants ($E_1$, $E_2$, $E_3$, $G_{23}$, $G_{13}$, $G_{12}$, ${\nu }_{23}$, ${\nu }_{13}$, ${\nu }_{12}$) using the relations (7) for the compliance tensor $\mathbb{C}$ (Jones, 1999).

To investigate any significant differences of the engineering constants and the helix angle $\theta$ among the 3 donors or 4 anatomical regions, a two-sided multifactor univariate variance analysis (ANOVA) was performed (Crawley, 2007).

3. Results

3.1. Osteon alignment errors

The osteon alignment errors of the 42 osteons are summarized in Fig. 6. The mean ratio $d_N/d_C$ of the measured haversian channel diameter as identified on the N-face and C-face, indicating the accuracy of bisection during sample preparation, is 1.03±0.15. This corresponds to an out-of-plane angle of $\gamma =13.8°$, according to (3). The first tilting angle as measured on the C-face is $\alpha =3.3°\pm 2.7°$ and the calculated second tilting angle is $\beta =3.4°\pm 1.8°$. The angles $\alpha$ and $\beta$ indicate the alignment quality of the osteon relative to the reference planes.

Fig. 6.

Osteon alignment results: (a) Histogram of the ratio $d_N/d_C$ of the measured Haversian channel diameter as identified on the N-face ($d_N$) and C-face ($d_C$). (b) Tilting angle $\alpha$ as measured on the C-face. (c) Tilting angle $\beta$ calculated using (4). See also Fig. 4. Vertical blue lines and numbers indicate the mean value. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

3.2. Indentation results

After applying the filter ladder of (Section 2.3), 4773 out of the original 6930 indents could be considered as valid. 26 osteons were found to be left-hand wound and 16 to be right-hand wound.

In this regard, the differentiation between the major and the minor oblique indentation modulus is introduced. ${\overline{E}}_{\mathrm{ind}}^{\mathrm{Omaj}}$ refer to the stiffer and ${\overline{E}}_{\mathrm{ind}}^{\mathrm{Omin}}$ to the softer direction of a single osteon, choosing from either $\phi =45°$ or $\phi =315°$.

$${\overline{E}}_{\mathrm{ind}}^{\mathrm{Omaj}}=\max ({\overline{E}}_{\mathrm{ind}}\left(45°\right),{\overline{E}}_{\mathrm{ind}}\left(315°\right))$$$${\overline{E}}_{\mathrm{ind}}^{\mathrm{Omin}}=\min ({\overline{E}}_{\mathrm{ind}}\left(45°\right),{\overline{E}}_{\mathrm{ind}}\left(315°\right))\text{.}$$ (12)

The overall averaged indentation modules of the lamella assembly material $\overline{{\overline{E}}_{\mathrm{ind}}}$ in the four investigated directions are presented in Table 1. The indentation modules ${\overline{E}}_{\mathrm{ind}}$ for the individual osteons are listed in the Appendix.

Table 1.

Mean indentation modules of the lamella assembly material of all 42 osteons for the 4 indentation directions including their standard deviations. $\overline{{\overline{E}}_{\mathrm{ind}}\left(0°\right)}$ and $\overline{{\overline{E}}_{\mathrm{ind}}\left(90°\right)}$ are the means of all indentation modules measured in the N- and C-face, respectively. $\overline{{\overline{E}}_{\mathrm{ind}}^{\mathrm{Omin}}}$ and $\overline{{\overline{E}}_{\mathrm{ind}}^{\mathrm{Omin}}}$ are the means of all minor and major modules measured on the O-face.

		$n$
$\overline{{\overline{E}}_{\mathrm{ind}}\left(0°\right)}$	27.6 ± 3.3 Gpa	879
$\overline{{\overline{E}}_{\mathrm{ind}}\left(90°\right)}$	20.5 ± 1.9 Gpa	2024
$\overline{{\overline{E}}_{\mathrm{ind}}^{\mathrm{Omin}}}$	21.8 ± 1.9 Gpa	917
$\overline{{\overline{E}}_{\mathrm{ind}}^{\mathrm{Omaj}}}$	24.1 ± 2.3 Gpa	953

The pairwise application of Tukey’s post-hoc test shows that $\overline{{\overline{E}}_{\mathrm{ind}}\left(0^{°}\right)}$, $\overline{{\overline{E}}_{\mathrm{ind}}\left(90^{°}\right)}$, $\overline{{\overline{E}}_{\mathrm{ind}}^{\mathrm{Omin}}}$ and $\overline{{\overline{E}}_{\mathrm{ind}}^{\mathrm{Omaj}}}$ are significantly different ($p\ll 0.05$).

3.3. Fabric elasticity results

The fabric elasticity parameters ($m_1$, $m_2$, $m_3$, ${ϵ}_0$, ${\nu }_0$, ${\mu }_0$) and engineering constants ($E_1$, $E_2$, $E_3$, $G_{23}$, $G_{13}$, $G_{12}$, ${\nu }_{23}$, ${\nu }_{13}$, ${\nu }_{12}$) of the lamella assembly material and the helix angle $\theta$ were identified for each osteon and listed in the Appendix. The overall means and standard deviations of the respective quantities are summarized in Table 2. The shape of the Young’s modulus of this average lamella assembly material is depicted in Fig. 7. The standard deviation range of the in-plane Young’s modulus is presented as a gray area, giving an impression of the prevalent stiffness shape. The mean degree of anisotropy $\overline{E_3}/\overline{E_2}$ is 1.75±0.36.

Table 2.

Mean fabric elasticity parameters, engineering constants, absolute helix angle $\overline{\left|\theta \right|}$ and ratio of anisotropy $\overline{E_3}/\overline{E_2}$ of the average lamella assembly material in the human femur midshaft.

Fabric elasticity		Engineering constants		Orientation, anisotropy
$\overline{m_1}$	0.748 (fixed)	$\overline{E_1}$	10.3±0.79 GPa	$\overline{\left|\theta \right|}$	10.3°±0.8°
$\overline{m_2}$	0.98±0.06	$\overline{E_2}$	17.6±1.9 GPa	$\overline{E_3}/\overline{E_2}$	1.75±0.36
$\overline{m_3}$	1.28±0.06	$\overline{E_3}$	30.2±4.1 GPa
$\overline{{\mu }_0}$	6.9±0.5 GPa	$\overline{G_{23}}$	8.6±0.6 GPa
$\overline{{\nu }_0}$	0.34 (fixed)	$\overline{G_{13}}$	6.6±0.7 GPa
$\overline{{ϵ}_0}$	18.46±1.4 GPa	$\overline{G_{12}}$	5.0±0.4 GPa
		$\overline{{\nu }_{23}}$	0.26±0.03
		$\overline{{\nu }_{13}}$	0.2±0.01
		$\overline{{\nu }_{12}}$	0.26±0.02

Fig. 7.

In-plane- and 3D-representation of the direction dependent Young’s modulus $\overline{E}$ of the average lamella assembly material in GPa. The gray area on the left indicates the standard deviation boundary. $\overline{\left|\theta \right|}=10.3°\pm 0.8°$ is the mean of all absolute individual helix angles.

On average, the main principal axis of the lamella assembly material is rotated by $\overline{\left|\theta \right|}=10.3°\pm 0.8°$ to the osteon axis (Table 2), (Fig. 7). $\overline{\left|\theta \right|}$ is calculated by taking the mean of the absolute angles to cancel out the influence of the individual winding direction.

The individual helix angles $\theta$ are approximately standard distributed with a minimum of −64.8° and a maximum of 32.6° (Fig. 8(a)). The osteon with $\theta =-64.8°$ (76y F POST g) has also the lowest observed degree of anisotropy of 1.1 (Fig. 8(e)). The relevance of its helix angle is therefore small. So all investigated osteons have a more or less axial stiffness orientation. There is no characteristic difference between the left- and right wound osteons.

Fig. 8.

Distribution of the helix angle $\theta$ and the degree of anisotropy $E_3/E_2$ of the fabric-based lamella assembly material model. (c)–(f) Four examples for extreme cases of $\theta$ and $E_3/E_2$. The indentation modulus shape ${\tilde{E}}_{\mathrm{ind}}\left(\phi \right)$ of the material model is fitted to the indentation measurements at 0°, 45°, 90°, 315°, finally giving a Young’s modulus estimation $E\left(\phi \right)$ for the lamella assembly.

The degree of anisotropy of the in-plane Young’s modules $E_3/E_2$ is always >1.0 and distributed between 1.1 and 2.5 (Fig. 8(b)).

Apparently, the lamella assembly material of the investigated osteons is mainly non-isotropic with the main principal axis laying close to the osteon axis but possessing a certain degree of helical winding.

Detailed results of four osteons, representing extreme cases for $E_3/E_2$ and $\theta$, are depicted in Fig. 8(c)–(f). The indentation modulus shape of the fabric-based model ${\tilde{E}}_{\mathrm{ind}}\left(\phi \right)$ approximates the measured modules ${\overline{E}}_{\mathrm{ind}}$ at 0°, 45°, 90°, 315°. The corresponding Young’s modulus shape $E\left(\phi \right)$ shows always a higher degree of anisotropy than ${\tilde{E}}_{\mathrm{ind}}\left(\phi \right)$.

3.4. Statistical results

Five ANOVA analyzes were performed with the donors and anatomical regions as factors and $E_2$, $E_3$, $E_3/E_2$, $G_{23}$ and the absolute helix angle $\left|\theta \right|$ as response variables (Table 3). The requirements on the data were fulfilled: the variances in the groups are similar, the response values are approximately normally distributed and independent. The level of significance was set to $\alpha =0.05$.

Table 3.

ANOVA results for $E_2$, $E_3$, $E_3/E_2$, $G_{23}$ and the absolute helix angle $\left|\theta \right|$, depending on the donor (factor 1) and the anatomical region (factor 2). Abbreviations: Degrees of Freedom (Df), Sum of Squares (Sum Sq), Mean Squares (Mean Sq).

	Df	Sum Sq	Mean Sq	$F$ value	$p$ value
Response: $E_2$
Donor	2	23.4	11.7	3.78	0.034 ⁎
Anatomical region	3	1.1	0.36	0.12	0.95
Donor↔anat. region	5	34.1	6.8	2.2	0.079
Residuals	31	96.1	3.1
Response: $E_3$
Donor	2	3.5	1.7	0.11	0.89
Anatomical region	3	51.1	17.0	1.12	0.35
Donor↔anat. region	5	171.3	34.3	2.25	0.074
Residuals	31	471.8	15.2
Response: $E_3/E_2$
Donor	2	0.23	0.11	1.04	0.36
Anatomical region	3	0.19	0.064	0.59	0.62
Donor↔anat. region	5	1.4	0.28	2.61	0.044 ⁎
Residuals	31	3.3	0.11
Response: $G_{23}$
Donor	2	1.24	0.62	1.64	0.22
Anatomical region	3	0.87	0.29	0.77	0.52
Donor↔anat. region	5	2.0	0.4	1.06	0.4
Residuals	31	11.7	0.38
Response: $\left|\theta \right|$
donor	2	140.3	70.2	0.60	0.55
Anatomical region	3	1025.1	341.7	2.94	0.0497 ⁎
Donor↔anat. region	5	767.0	153.4	1.32	0.28
Residuals	31	3606.9	116.4

^⁎Significant relations according to a level of significance of $\alpha =0.05$.

Regarding $E_3$ and $G_{23}$, no significant differences among the donors and the anatomical regions were found. For $E_2$ there is a slight difference between the donors ($p=0.034$). Using a post-hoc Tukey test the difference was found to occur between the 63y. male and 76y. female donor.

The ratio of anisotropy $E_3/E_2$ was found to be marginally significant for the coupling of the donor and the anatomical region factors ($p=0.044$). The absolute helix angle $\left|\theta \right|$ was found to depend marginally on the anatomical region with $p=0.049$. However, this significance occurs due to the outlier osteon with the extremely high $\left|\theta \right|$ of 64.8° and may be disregarded.

4. Discussion

A novel method to estimate the orthotropic elastic properties of the human secondary osteonal lamella assembly was proposed. Nanoindentation was performed on three distinct planes in bone lamellae of the same osteon, allowing for calculating osteon specific orthotropic elastic properties and the material alignment.

The sample preparation was performed without using chemicals for cleaning and dehydration. In this way the mechanical properties of the microstructural features were kept as natural as possible. As reported in Bushby et al. (2004) dehydration in ethanol can increase the indentation modulus of bone by 15%–20%. A drawback of the gentle sample treatment was the remaining fat in the porous space which sometimes contaminated the indenter tip when hit. This caused sporadic surface contact recognition problems during the measurements and increased the number of indents that had to be excluded.

Usually, bone surface preparation involves polishing using aqueous solutions or glycol. In this way, the repeated drying and wetting of the sample surface by the cooling agent might introduce ultracracks in the surface (Roschger et al., 1993). As nanoindentation operates on a small lengthscale, those potentially distort the measurements. Additionally, aqueous solutions are known to demineralize the bone matrix by dissolution (Donnelly et al., 2006). Those problems were entirely avoided by smoothing the sample surface at a dry state using an ultramiller system. As verified by an in-house study, the surface quality of cortical bone produced by ultramilling is slightly better than the one of polishing.

As reported in Cheng et al. (2010), the rotation of the triangular Berkovich tip imprint shape relative to the fiber direction in fibrous samples may influence the measurements. In this work, the tip orientation relative to the lamella layers was kept constant for all indentations in a particular plane and should not affect the reported results.

The relative alignment $\alpha$ and $\beta$ of the milled N- and C-surfaces to the axis of the Haversian channel was measured, (Fig. 6). The angles are quite small with mean values just above 3° and a maximum value below 10°. The ratio of the Haversian channel diameters $d_N/d_C$ is a measure for the bisection accuracy and affects the indentation modules of the C-face only. Many osteons were bisected precisely with $d_N/d_C$ around 1.0 with no geometry induced error in ${\overline{E}}_{\mathrm{ind}}\left(90°\right)$ and ${\overline{E}}_{\mathrm{ind}}\left(270°\right)$. However the stiffness of osteons with values close to 1.3–1.4 was underestimated in 90° and 270° as it was not measured exactly in-plane.

So, how do the alignment errors affect the the degree of anisotropy $E_3/E_2$ and helix angle $\theta$ results? To answer this question, the average lamella assembly material of Table 2 was indented virtually in the N-, C- and O-face directions, perturbed by the worst case values of $\alpha$, $\beta$ and $\gamma$. The obtained defective indentation modules were then treated as input values for the standard fitting procedure of Section 2.7. The resulting $E_3/E_2$ and $\theta$ deviate from the ones of the original average lamella assembly material due to the applied alignment errors.

As shown in Table 4, $\beta$ and $\gamma$ mainly affect the prediction of $\theta$ and $E_3/E_2$, respectively. The tilting angle $\alpha$ has a comparatively minor influence. A pure $\beta$-error of 10° leads to an $\theta$-deflection factor of 31% and a pure $\gamma$-error of 45° ($\approx d_N/d_C=1.4$) lead to a change of 22% of $E_3/E_2$. The worst case combination of alignment errors is $\alpha =-10°$, $\beta =-10°$, $\gamma =\pm 45°$, leading to an underestimation of 34% for $\theta$ and an overestimation of 23% for $E_3/E_2$, changing $\theta$ from 10.3° to 6.8° and $E_3/E_2$ from 1.75 to 2.15. However, most of the recorded tilting angles and bisection values are by far smaller. So in general, the errors caused by osteon misalignment are regarded as acceptable and do not reduce the significance of the results of this study.

Table 4.

Influence of the alignment errors $\alpha$, $\beta$ and $\gamma$ on the degree of anisotropy $E_3/E_2$ and helix angle $\theta$ results. Left part: combinations of the identified worst case errors (Fig. 6). Right part: resulting relative changes of $E_3/E_2$ and $\theta$.

$\alpha$	$\beta$	$\gamma$	$E_3/E_2$	$\theta$
10°	0	0	−0.046	0.074
−10°	0	0	0.018	−0.081
0	10°	0	−0.093	0.31
0	−10°	0	−0.014	−0.12
0	0	±45°	0.22	−0.17
10°	10°	±45°	0.064	0.068
−10°	10°	±45°	0.12	−0.047
10°	−10°	±45°	0.16	−0.26
−10°	−10°	±45°	0.23	−0.34

As shown by Gupta et al. (2006), the indentation modulus of osteonal lamellae varies depending on the indentation position on the osteon because of the local variation of the collagen fibril orientation. In the current investigation, the indents were positioned in a dense matrix-like pattern to maximize the information gain per osteon zone. This ensures equal weighted measurements across the covered area. However, the indents had then to be quite shallow with a depth of 250 nm to prevent neighboring measurements from interfering. Indents with a low depth are prone to distortions by surface scratches or signal noise. So the quite high intra pattern standard deviation of the measured indentation modules $E_{\mathrm{ind},i}$ may be attributed to these perturbations as well as to the real local variation of the lamella stiffness (Fig. 8(c)–(f), Appendix).

Shallow indents also tend to deliver generally higher indentation modules than deep indents in the same material (Voyiadjis and Peters, 2010). This might be the reason for the generally lower modules reported in Franzoso and Zysset (2009) who had a similar study design. Their indents in human femur cortical bone were 800 nm in depth. They can be perfectly compared with this study as they were placed also on the N- and C-face of osteons and orientated in the lamellar plane. They measured indentation modules of 22.31±2.16 GPa on the N- and 18.06±1.84 GPa on the C-face compared to 27.6±3.28 and 20.05±1.91 GPa of Table 1. To the best of our knowledge, no other studies involve indents in multiple directions exactly in the osteon lamella plane. However, many indentation experiments in the relatively simple measurable osteon axial direction were published. For example Xu et al. (2003) reported an indentation module of 22.48±2.4 GPa for a shallow 150 nm indentation depth in human femur osteons in the axial direction (N-face).

As described in multiple publications, indentation modules of dry tissue are approximately 20%–30% higher than those of wet bone (Rho and Pharr, 1999; Hengsberger et al., 2002; Hoffler et al., 2005). Hence, the values reported here are most probably overestimating the natural wet properties.

The pairwise statistical comparison of the multiaxial indentation results of Table 1 shows a highly significant difference for all combinations. This indicates clearly that the lamella assembly, circumferentially forming the osteon cylinder, is indeed an non-isotropic material. Moreover, as ${\overline{E}}_{\mathrm{ind}}^{\mathrm{Omin}}$ and ${\overline{E}}_{\mathrm{ind}}^{\mathrm{Omax}}$ are significantly different, the principal axes of the lamella assembly material are generally not orientated along the osteon axis. The helical winding of the main stiffness direction as depicted in Fig. 1 is evident.

The amplitude of the helix angle $\theta$ as well as orthotropic elastic constants were estimated for the lamella assembly material (Table 2) by fitting a fabric-based orthotropic stiffness tensor to the indentation outcome. The resulting engineering constants reflect the magnitude of the measured indentation modules. Utilizing an analog fitting method, Franzoso and Zysset (2009) calculated engineering constants for the ($x–y–z$)-directions, while disregarding a potential helical orientation of the stiffness tensor. They report $\overline{E_x}=9.17\pm 0.36$, $\overline{E_y}=17.28\pm 1.89$, $\overline{E_z}=24.66\pm 2.71$ and $\overline{E_z}/\overline{E_y}=1.426$. The corresponding average degree of anisotropy of the current study, is higher with $\overline{E_z}/\overline{E_y}=\overline{E}\left(0°\right)/\overline{E}\left(90°\right)=1.657$.

More interesting than the magnitude of the engineering constants are the distributions of the helix angle $\theta$ and the degree of anisotropy of the lamella assembly material among the investigated osteons (Fig. 8). With an average degree of anisotropy of $\overline{E_3}/\overline{E_2}=1.75\pm 0.36$ and a minimum value of 1.12 the lamella assembly stiffness is obviously always direction dependent in its lamella plane. The dominant direction is close to the osteon axis with $\left|\overline{\theta }\right|=10.3°\pm 0.8°$. This finding is in conflict with the widely known twisted plywood pattern and orthogonal plywood pattern which describe the fibril alignment in bone lamellae (Giraud-Guille, 1988). The twisted plywood pattern proposes a continuous and even rotation of the collagen fibrils in the lamella plane leading to in-plane isotropic behavior of the single lamella (Reisinger et al., 2011). Also the orthogonal plywood pattern, consisting of two orthogonal fibril layers per lamella results in an in-plane degree of anisotropy of 1.0. Consequently, the proposed alternation of twisted lamellae and orthogonal lamellae add up to a ratio of $E_3/E_2$ identical to 1.0 for the lamella assembly material, opposing the results of the current work.

Those rather approve two less known fibril orientation patterns. The pattern proposed by Weiner et al. (1997, 1999) consists of a mainly axial alignment of fibrils with each osteon lamella being separated into five sublamellae. Wagermaier et al. (2006) measured the fibrillar alignment in 8 successive osteon lamellae using small angle X-ray scattering (SAXS) and wide angle X-ray diffraction (WAXD). The associated degrees of anisotropy and helix angles have been calculated by Reisinger et al. (2011) applying a multiscale model. The outcome was $E_3/E_2=1.32$ and $\theta =12.6°$ for the Weiner-pattern and $E_3/E_2=1.21$ and $\theta =23.2°$ for the Wagermaier-pattern. Obviously the mechanical properties derived from these two fibril orientation patterns are at least qualitatively close to the values of the present study.

Ascenzi and Bonucci (1967, 1968) gave distinction to the so called longitudinal, intermediate and transverse secondary osteons, which appear dark, intermediate and bright in polarized light microscopy. This was related to a mainly axial, oblique and transverse fibril orientation inside the respective osteons. Assuming that the fibrillar arrangement is reflected in the osteon’s mechanical properties, obviously only longitudinal osteons were investigated, when looking on the distribution of $\theta$ in Fig. 8(a). The main principal stiffness axis of 41 out of 42 investigated osteons is aligned more or less along the osteons’ longitudinal direction. Not a single osteon possesses a clear transverse orientation with $\theta$ close to 90°, although the medial femoral midshaft was also investigated. In this anatomical region, the shaft is loaded more by compression and therefore contains a higher fraction of transverse osteons, as shown by Bromage et al. (2003). Transverse osteons are supposed to own better compression resistance.

However, the average investigated osteon is not exactly longitudinal but possesses a moderate helical winding as the mean absolute helix angle is $\overline{\left|\theta \right|}=10.3°\pm 0.8°$.

The current findings rather support the bone lamella microstructure proposed by Turner et al. (1995) and Pidaparti et al. (1996), in which not the fibrils but the extrafibrillar mineral particles dominate the resulting elastic properties. This mineral fraction is supposed to make up 75% of the total tissue mineral with its crystals’ $c$-axis being aligned along the osteon axis. In contrast, the collagen fibrils are claimed to be adjusted obliquely at 30°. This would lead to a generally axial stiffness alignment in osteons which is consistent with the results of this study.

Statistically there were no significant differences of the osteon properties between the four tested anatomical regions (Table 3). In contrast, Espinoza-Oras et al. (2009) found lower elastic constants in the posterior quadrant and a lower anisotropy in the medial and lateral quadrants in the mid diaphysis using ultrasound wave propagation. This is consistent with Bensamoun et al. (2004), who found lower longitudinal stiffness in the posterior quadrant. These local differences are supposed to be attributed to the macroscopic loading configuration of the bone. Zones in tension due to bending were reported to show increased number of longitudinal osteons (Ascenzi, 1988) and a higher degree of anisotropy (Takano et al., 1999). On the osteon lengthscale, these findings are not supported by the current study.

Two parameters ($m_1$, ${\nu }_0$) of the utilized fabric-based orthotropic stiffness tensor were set to constant values, see 2.7. A sensitivity analysis was carried out to inspect the influence of $m_1$ and ${\nu }_0$ on the resulting fabric-based stiffness tensor adapted to the average indentation results of Table 1. Modifying $m_1$ and ${\nu }_0$ by 10% from their fixed values would lead to only minor changes of $m_3/m_2$ and ${ϵ}_0$ and to no changes of $\theta$ as listed in Table 5. Consequently, the results of this study are robust regarding the choice of $m_1$ and ${\nu }_0$.

Table 5.

Sensitivity of $m_3/m_2$, ${ϵ}_0$ and $\theta$ on a 10% increase of the fixed parameters $m_1$ and ${\nu }_0$ from the chosen values of 2.7. For example, adding 10% to $m_1$ would decrease $m_3/m_2$ by 0.0054 and increase ${ϵ}_0$ by 0.63.

	$m_3/m_2$	${ϵ}_0$	$\theta$
$s_{m_1}^{\pm 10\%}$	$\mp 0.00054$	±0.63	0.0
$s_{{\nu }_0}^{\pm 10\%}$	0.0	$\mp 0.5$	0.0

In the following, some limitations of this work are pointed out: first, the utilized fabric-elasticity relationship disregards the generally more complex anisotropic properties of osteonal lamellae.

Second, it was assumed that the lamella assembly material parameters do not vary along the osteon perimeter. The indentation measurements on the O-face, positioned left and right of the Haversian channel, were used for the $\phi =45°$ and 315° direction.

Third, the 11×3-indentation pattern (Fig. 2(f)) covers at maximum a 50 μm part of the osteon radial dimension. Possible more remote lamellae were therefore not included.

Fourth, osteons are by no way perfectly aligned cylinders. So regardless of the alignment analysis of the Haversian channel, the lamellae themselves may have been undulated and were not indented exactly in-plane.

5. Conclusion

This work presents a method for assessing the orthotropic elastic properties of the osteonal lamellae assembly and the helical winding of the main stiffness direction. The results show that osteons are generally stiffer in longitudinal than in circumferential direction. The direction of maximum stiffness is slightly rotated relative to the osteon axis leading to a evident but moderate helical winding.

This outcome contradicts the widely known ‘twisted plywood’ collagen fibril orientation pattern in lamellar bone that would lead to a more isotropic behavior. Additionally the often reported transverse osteons, holding a mainly transverse fibril orientation, were not observed from the mechanical point of view.

No differences of the osteon mechanical properties associated to the anatomical quadrants in the human femur midshaft were found.

Appendix. Individual osteon results

See Table A.1

Table A.1.

Individual osteon results. Young’s modules, shear modules and indentation modules in GPa, angles in degrees. Fixed parameters: $m_1=0.747986$, ${\nu }_0=0.34$.

Donor	Region	Osteon	$m_2$	$m_3$	${ϵ}_0$	$\theta$	$E_1$	$E_2$	$E_3$	$G_{23}$	$G_{13}$	$G_{12}$	${\nu }_{23}$	${\nu }_{13}$	${\nu }_{12}$	$E_3/E_2$	${\overline{E}}_{\mathrm{ind}}\left(0°\right)$	${\overline{E}}_{\mathrm{ind}}\left(45°\right)$	${\overline{E}}_{\mathrm{ind}}(90°,270°)$	${\overline{E}}_{\mathrm{ind}}\left(315°\right)$
63M	ANT	d	0.96	1.29	16.56	−0.5	9.3	15.3	27.6	7.7	6.0	4.4	0.25	0.20	0.27	1.81	25.4±3.1	20.5±3.0	17.7±3.9	20.6±4.4
63M	LAT	a	0.95	1.30	18.61	4.8	10.4	16.9	31.4	8.6	6.8	4.9	0.25	0.20	0.27	1.87	28.7±3.7	23.8±4.1	19.8±4.0	22.3±4.9
63M	LAT	b	1.01	1.25	17.55	19.1	9.8	17.8	27.2	8.2	6.1	4.9	0.28	0.20	0.25	1.53	25.0±4.2	24.2±3.2	20.4±3.8	20.5±4.9
63M	LAT	c	0.88	1.37	18.61	2.0	10.4	14.5	34.8	8.4	7.1	4.6	0.22	0.19	0.29	2.40	30.5±4.5	22.7±3.0	17.9±2.9	21.8±4.6
63M	LAT	e	0.96	1.30	17.96	7.9	10.0	16.4	30.2	8.3	6.5	4.8	0.25	0.20	0.27	1.84	27.4±3.6	23.7±2.3	19.0±3.6	21.3±4.8
63M	LAT	f	0.94	1.32	18.22	−11.5	10.2	16.0	31.5	8.4	6.7	4.8	0.24	0.19	0.27	1.97	28.2±3.6	20.4±4.2	19.5±3.4	24.2±4.8
63M	MED	b	1.00	1.25	16.86	17.8	9.4	16.9	26.4	7.9	5.9	4.7	0.27	0.20	0.25	1.57	24.5±5.2	22.9±3.0	19.8±4.1	19.3±4.7
63M	MED	f	0.98	1.28	17.66	4.6	9.9	16.8	28.7	8.2	6.3	4.8	0.26	0.20	0.26	1.71	26.7±4.0	22.7±3.6	19.3±3.3	21.5±4.8
63M	MED	g	0.92	1.33	19.90	−5.4	11.1	16.8	35.4	9.1	7.4	5.1	0.23	0.19	0.28	2.11	31.5±1.4	23.2±5.2	20.1±3.1	25.4±5.1
63M	MED	h	1.08	1.17	17.02	−22.6	9.5	19.8	23.5	8.0	5.6	5.1	0.31	0.22	0.24	1.19	22.2±4.2	21.9±3.4	20.5±3.9	23.5±5.1
63M	MED	j	0.95	1.31	18.09	32.7	10.1	16.2	30.9	8.3	6.6	4.8	0.25	0.20	0.27	1.91	23.9±4.3	28.0±5.7	19.8±3.7	20.2±4.8
63M	MED	k	0.97	1.29	19.14	17.7	10.7	17.9	31.7	8.9	6.9	5.2	0.26	0.20	0.26	1.77	27.9±2.7	26.9±3.9	20.7±3.6	22.0±5.2
68F	ANT	b	0.87	1.38	19.45	7.7	10.9	14.8	37.1	8.7	7.5	4.7	0.21	0.18	0.29	2.51	32.1±4.5	24.4±4.3	19.5±3.8	20.7±4.7
68F	ANT	c	1.00	1.26	20.32	12.0	11.4	20.2	32.1	9.5	7.1	5.6	0.27	0.20	0.26	1.59	28.7±5.4	28.5±4.3	21.3±2.9	25.7±5.6
68F	LAT	b	1.08	1.18	16.24	11.6	9.1	18.8	22.5	7.7	5.3	4.9	0.31	0.22	0.24	1.20	22.8±3.8	20.7±4.2	20.8±3.5	19.7±4.9
68F	LAT	c	1.01	1.24	17.12	−3.0	9.6	17.5	26.3	8.0	5.9	4.8	0.28	0.21	0.25	1.50	26.8±3.7	19.0±4.0	22.4±3.1	19.7±4.8
68F	LAT	d	1.09	1.17	17.06	−17.6	9.5	20.1	23.2	8.1	5.6	5.2	0.32	0.22	0.23	1.15	22.5±2.9	21.9±4.7	20.9±4.0	23.0±5.2
68F	LAT	e	1.03	1.22	18.02	−0.2	10.1	19.0	27.0	8.5	6.2	5.2	0.29	0.21	0.25	1.42	26.2±3.0	22.6±4.1	21.4±3.6	22.6±5.2
68F	MED	a	0.90	1.35	18.34	−4.8	10.3	14.9	33.5	8.3	6.9	4.6	0.23	0.19	0.28	2.25	30.3±2.3	19.8±3.6	20.2±3.4	21.9±4.6
68F	MED	b	0.95	1.31	17.34	−2.7	9.7	15.5	29.5	8.0	6.3	4.6	0.25	0.20	0.27	1.90	26.7±2.7	21.3±4.0	17.8±5.0	22.1±4.6
68F	MED	c	0.94	1.31	19.01	5.1	10.6	16.8	32.7	8.8	7.0	5.0	0.24	0.19	0.27	1.94	29.6±3.2	24.4±3.7	19.8±3.8	22.6±5.0
68F	MED	d	0.92	1.33	20.75	0.5	11.6	17.7	36.6	9.5	7.7	5.4	0.24	0.19	0.28	2.07	33.1±4.0	25.2±5.5	21.5±4.0	25.0±5.4
68F	MED	e	0.95	1.30	17.46	−6.2	9.8	15.8	29.5	8.1	6.3	4.6	0.25	0.20	0.27	1.87	27.0±3.9	20.5±3.8	18.9±3.7	22.4±4.6
68F	MED	f	0.95	1.31	20.97	−4.2	11.7	18.8	35.8	9.7	7.6	5.5	0.25	0.20	0.27	1.91	32.7±3.5	24.8±5.0	22.5±4.4	26.4±5.5
68F	POST	d	0.95	1.30	21.15	12.5	11.8	19.2	35.8	9.8	7.7	5.6	0.25	0.20	0.27	1.87	32.7±3.9	27.8±4.1	24.0±3.6	23.1±5.6
68F	POST	e	0.97	1.28	17.74	24.9	9.9	16.6	29.2	8.2	6.4	4.8	0.26	0.20	0.26	1.76	24.8±4.5	26.0±3.6	19.6±3.5	20.1±4.8
68F	POST	h	0.98	1.27	17.28	20.1	9.7	16.7	27.8	8.0	6.1	4.7	0.26	0.20	0.26	1.67	24.8±2.7	24.4±2.3	19.4±3.3	19.9±4.7
68F	POST	i	0.90	1.35	16.93	−2.9	9.5	13.7	30.9	7.7	6.4	4.3	0.23	0.19	0.28	2.26	27.3±4.5	19.9±2.4	16.7±2.1	21.0±4.3
76F	ANT	c	1.01	1.24	19.48	−14.0	10.9	19.9	30.0	9.1	6.7	5.5	0.28	0.21	0.25	1.51	29.6±3.9	21.1±1.7	24.8±3.4	24.6±5.5
76F	LAT	a	1.01	1.24	19.87	6.4	11.1	20.3	30.6	9.3	6.9	5.6	0.28	0.21	0.25	1.51	29.5±4.3	25.4±4.2	23.4±3.7	23.9±5.6
76F	LAT	b	0.98	1.27	19.39	1.3	10.9	18.7	31.3	9.0	6.9	5.3	0.26	0.20	0.26	1.67	30.4±3.1	22.9±4.8	23.4±4.2	22.5±5.3
76F	LAT	d	0.98	1.27	16.83	2.2	9.4	16.2	27.2	7.8	6.0	4.6	0.26	0.20	0.26	1.68	24.1±4.2	23.0±3.4	16.3±3.0	22.5±4.6
76F	LAT	e	1.05	1.20	17.91	−13.8	10.0	19.7	25.9	8.4	6.0	5.3	0.30	0.21	0.24	1.31	25.3±4.2	21.7±4.1	22.0±3.3	23.5±5.3
76F	LAT	f	0.93	1.33	20.25	18.4	11.3	17.3	35.7	9.3	7.5	5.2	0.24	0.19	0.28	2.06	31.1±3.4	27.9±3.9	22.8±3.3	20.5±5.2
76F	LAT	i	1.04	1.21	18.17	17.8	10.2	19.7	26.7	8.5	6.1	5.3	0.29	0.21	0.25	1.36	25.2±3.7	24.8±4.2	21.5±4.1	22.2±5.3
76F	LAT	k	1.04	1.21	17.87	17.1	10.0	19.3	26.3	8.4	6.1	5.2	0.29	0.21	0.25	1.37	25.6±3.0	23.6±4.5	22.0±3.3	20.9±5.2
76F	LAT	l	0.89	1.37	18.25	1.0	10.2	14.3	34.1	8.2	7.0	4.5	0.22	0.19	0.29	2.39	30.3±2.5	21.3±3.0	18.7±3.5	20.9±4.5
76F	MED	a	0.96	1.29	21.02	7.7	11.8	19.3	35.2	9.7	7.6	5.6	0.25	0.20	0.27	1.82	32.3±3.9	27.1±3.6	23.1±3.2	24.4±5.6
76F	MED	b	0.90	1.35	19.65	4.0	11.0	16.0	35.8	8.9	7.4	4.9	0.23	0.19	0.28	2.24	31.7±1.9	24.5±4.1	19.6±3.3	22.8±4.9
76F	MED	c	1.05	1.20	16.18	−1.1	9.1	17.8	23.4	7.6	5.4	4.7	0.30	0.21	0.24	1.31	24.0±4.0	19.2±3.6	20.9±3.9	19.4±4.7
76F	POST	b	0.97	1.28	20.81	2.8	11.6	19.6	34.2	9.7	7.4	5.6	0.26	0.20	0.26	1.75	30.8±4.0	27.5±4.3	21.1±3.8	26.7±5.6
76F	POST	g	1.10	1.16	18.27	−64.8	10.2	21.9	24.5	8.6	5.9	5.6	0.32	0.22	0.23	1.12	21.0±2.7	25.2±4.9	22.2±3.5	26.2±5.6
		MEAN	0.98	1.28	18.46		10.33	17.55	30.23	8.55	6.59	5.02	0.26	0.20	0.26	1.75	27.64	23.51	20.55	22.37
		STD	0.06	0.06	1.41		0.79	1.94	4.13	0.62	0.67	0.38	0.03	0.01	0.02	0.36	3.28	2.58	1.91	2.01

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