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Abstract

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

To support future studies of tibial bending in a murine model of senile osteoporosis (SAMP6), we sought to determine the relationship between applied external bending force and peak endocortical strain in the tibiae of SAMP6 and control SAMR1 mice. The lower hindlimbs of mice were loaded by three-point bending in the lateral-medial plane with a support length of 10 mm. Force-periosteal strain relations were first determined using standard strain gauge methods. Finite-element analysis (FEA) models of the tibia-fibula were generated based on microcomputed tomography images. After choosing appropriate boundary conditions, FEA predictions of periosteal strains were within 15% of measured values. FEA revealed a narrow (3–4 mm) region of the central tibia with well-developed bending strains (tension medially, compression laterally); outside this region, we observed high shear strains. Both the strain gauge data and the finite-element simulations indicated that the tibia of the SAMP6 mouse was 20–25% stiffer than the SAMR1 tibia, consistent with a larger moment of inertia and higher cortical bone modulus. Thus, higher levels of force are required to produce the same target values of strain in the SAMP6 tibia. The ratio of periosteal to endocortical strain in the region of interest was similar for the two mouse strains (1.5–1.6). Based on these ratios, we scaled the strain gauge data to estimate the force-endocortical strain relations for the two mouse strains. In conclusion, FEA, with supporting strain gauge measurements, has provided unique insight regarding the strain environment throughout the tibia during three-point bending in mice. © 2005 Wiley-Liss, Inc.

During development and in adulthood, the skeleton responds to alterations in its mechanical environment through the processes of modeling and remodeling. Experimentalists wishing to examine relations between altered mechanical loading and bone formation (modeling) have developed several animal models that allow controlled loading histories to be imposed at targeted skeletal sites. In rats and mice, two widely used noninvasive methods for skeletal overloading are tibial bending (Turner et al., 1991; Akhter et al., 1998) and forelimb compression (Torrance et al., 1994; Lee et al., 2002). Peak strain magnitude is typically used as an index of the severity of loading in these models, and the magnitude of bone formation has been shown to correlate with strain magnitude (Turner et al., 1994; Forwood and Turner, 1995; Mosley et al., 1997; Cullen et al., 2001; Lee et al., 2002). Using these models, investigators have focused primarily (but not exclusively) on periosteal responses. This is probably for several reasons: surface strains are highest on the periosteal surface due to long bone bending; strain magnitude and direction can be measured periosteally with use of surface strain gauges; and bone formation at the periosteal surface has the greatest influence on bone cross-sectional properties and thus the resistance of a bone to bending and torsion.

Despite the obvious importance of the periosteal surface, we have a particular interest in the endocortical response to mechanical loading because this surface is in direct contact with the bone marrow. The bone marrow is host to a population of multipotent cells that can give rise to bone, cartilage, tendon, ligament, and muscle (Dennis et al., 1999; Pittenger et al., 1999). These so-called mesenchymal stem cells (MSCs) are the source of osteoprogenitor cells that in turn are the source of osteoblasts. With aging, there may be a decrease in the number of marrow MSCs or a decrease in their ability to support bone formation (Friedenstein, 1976; Bergman et al., 1996; D'ippolito et al., 1999; Mueller and Glowacki, 2001; Stenderup et al., 2003), which may contribute importantly to bone loss in senile osteoporosis. A related issue is that an age-related reduction in MSCs might make the skeleton less sensitive to mechanical loading and thus contribute to the findings of age-related reductions in mechanoresponsiveness (Rubin et al., 1992; Turner et al., 1995; Hoshi et al., 1998; Srinivasan et al., 2003).

To address the issue of skeletal responsiveness to loading and the role of the bone marrow in supporting loading-induced osteogenesis, we have turned to a mouse model of osteoporosis. Strain P6 of the senescence-accelerated mouse (SAMP6) has many relevant features of senile osteoporosis, including reduced trabecular bone volume (Jilka et al., 1996; Silva et al., 2004b), reduced marrow osteogenic potential (Jilka et al., 1996), low rates of endosteal and endocortical bone formation (Jilka et al., 1996; Silva et al., 2005), and increased bone fragility (Silva et al., 2002). [These relative deficits are in comparison to the control strain, SAMR1. There are several phenotypically distinct strains of the senescence-accelerated mouse (SAM). SAMP strains are prone to early senescence, while SAMR strains are resistant to early senescence and serve as controls. The SAM strains were developed by selective inbreeding, starting from AKR/J mice (Takeda et al., 1997).]

For future in vivo studies, we plan to use the tibial bending model adapted from Turner et al. (1991) and Akhter et al. (1998) because the mouse tibia (in contrast to the ulna) has a marrow cavity large enough to yield sufficient MSCs for cell culture studies and an endocortical bone surface large enough to enable histomorphometric quantitation of bone formation in response to increased loading.

Prior to conducting a study of in vivo loading, we sought to determine the relationship between applied external bending force and peak endocortical strain in the tibiae of SAMP6 and SAMR1 mice. Because SAMP6 and SAMR1 tibiae have different bone sizes (Silva et al., 2002) and different values of cortical bone modulus (Silva et al., 2004a), and also because there is no way to measure endocortical strains directly, we employed finite-element analysis (FEA). Our objective was to develop a method to estimate the peak endocortical strains generated during three-point bending of the tibia in SAMP6 and control SAMR1 mice in order to estimate the forces needed to produce target endocortical strains of 1,000 and 2,000 microstrain.

OVERVIEW OF STUDY DESIGN

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

To estimate endocortical strain during tibial bending, we used a combined strain gauge-FEA approach (Fig. 1). The first step was to measure periosteal strains at the loading point using standard strain gauge methods in order to establish the relationship between applied force and periosteal strain for SAMP6 and SAMR1 mice. We then used finite-element models of the SAMP6 and SAMR1 tibiae to determine the ratio of peak periosteal to peak endocortical strain at the loading point. We used this ratio to scale the force-strain relationships from the periosteal surface to estimate the force versus endocortical strain relationships. We did not believe it critical to calibrate the model to match exactly the strain data, because the model was used primarily to define the relationship between endocortical and periosteal surface strains in the region of interest. Because this is a ratio, it should not depend critically on predicting the absolute values of strain. Nonetheless, we did choose boundary conditions based partly on achieving a good approximation to the absolute strain data.

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Figure 1. Study design.

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We chose three-point rather than four-point bending because of the short length of the tibia (Fig. 2). Based on preliminary loading experiments, the position of the hindlimb in the loading fixture was chosen so that the loading point was approximately equidistant from the distal tibiofibular junction (TFJ) and the knee joint. Caliper measurements of sample specimens dissected after loading indicated that the loading point was located 5.5 mm from the distal TFJ.

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Figure 2. Sketch of mouse tibia-fibula in three-point bending fixture: anterior view. The distal tibio-fibular junction (TFJ) is used as a reference point; the distal support is centered 0.5 mm proximal to TFJ and the center of the loading surface is 5.5 mm proximal to TFJ. Note that the loading surface contacts both tibia and fibula as it displaces downward (medially).

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STRAIN GAUGE STUDY

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

Four-month-old SAMR1 and SAMP6 mice (n = 6/group) were obtained from our breeding colonies that were started from breeding pairs supplied by the Council for SAM Research (Kyoto, Japan). Animals were housed and treated in accordance with the National Institutes of Health Guidelines for the Care and Use of Laboratory Animals. All experimental procedures were approved by the Washington University Animal Studies Committee. Mice were killed by CO2 asphyxiation and the right hindlimbs dissected immediately for strain gauge placement. An incision was made through the skin, and muscles were retracted to expose the anteromedial surface of the bone. This is the large convex surface that corresponded to the underside of our in vivo loading apparatus. While the bone was exposed, the animal was placed in the loading apparatus and the location on the bone directly beneath the loading surface was marked. A single element strain gauge with an active area of 0.5 × 0.5 mm (EA06-015LA-120; Vishay Measurements Group, Raleigh, NC) was attached at this site per the manufacturer's instructions (instruction bulletin B-127-13). Briefly, the gauge was trimmed to 3 × 2 mm, the periosteum was stripped, the bone surface was sanded and cleaned, and the gauge was applied using a cyanoacrylate adhesive (M-Bond 200). After 5 min, the lead wires were soldered to the gauge and a thin layer of polyurethane (M Coat A) was applied to the site. Saline-soaked gauze was placed over the site and measurements began 1 hr later.

After application of a 1 N compressive preload, a triangle waveform to the target peak force was applied. The waveform was chosen to match the one used for subsequent in vivo loading studies. The load/unload time was 0.5 sec followed by a 10-sec rest interval, which was followed by the next triangle wave. Eight cycles were applied and strains were recorded via a signal conditioning/amplifier system (SCXI-1001; National Instruments, Austin, TX). Peak compressive forces of 4, 8, 12, 16, 20, and 24 N were used with a 160-sec recovery interval between each set of eight cycles. Peak strains were determined for cycles 4–8 and the average was computed. After strain measurements, the tibia was dissected, the lead wires of the gauge were cut, and the region of the bone at the gauge site was scanned using microcomputed tomography (μCT 40; Scanco, Baseldoff, Switzerland) in order to assess the location of the strain gauge (Fig. 3).

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Figure 3. Section of tibia at site of active area of strain gauge, imaged by microcomputed tomography (μCT) after strain gauge measurements. Gauges were located on the medial aspect of the tibia, just anterior of the posteromedial spine, approximately 5.5 mm proximal to the distal TFJ.

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Force-strain calibration curves for each specimen were assessed and any points past the linear region were eliminated. Linear regression analysis using all the valid data points from all samples (n = 26 for SAMP6; n= 25 for SAMR1) demonstrated highly significant correlations between applied force and measured strain (Fig. 4). The intercept for the SAMP6 regression did not differ significantly from zero (P = 0.28) while the SAMP6 intercept was offset slightly from zero (P = 0.001), which we attribute to experimental error at low strain levels. The intercepts did not differ between groups (P > 0.05). However, the slope of the regression lines did differ significantly; the slope of the SAMP6 regression was 18% less than SAMR1 (P < 0.001), indicating that SAMP6 tibiae were stiffer than SAMR1.

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Figure 4. Periosteal strain versus applied force relationships for SAMR1 and SAMP6 tibiae illustrating greater values of tensile strain for SAMR1 compared to SAMP6 for the same applied force. The slope of the SAMP6 regression was 18% less than SAMR1 (P < 0.001).

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Given the small size of the mouse tibia, we were concerned that application of the strain gauge and polyurethane coating might influence tibial bending stiffness. In a preliminary study, we measured bending stiffness before and after gauge placement and found no significant change in stiffness (data not shown). While this result does not rule out an influence of the gauge on strains at the local level, it suggests that gauge placement did not significantly alter the tibial force-strain relationship.

FINITE-ELEMENT ANALYSIS METHODS

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

Finite-element models of SAMP6 and SAMR1 tibia-fibula (n = 1) were generated from μCT scans and used to simulate three-point bending (Fig. 5). Based on a previous study, the geometry of the tibia shows only modest variance within each of these mouse strains (e.g., coefficient of variation of bone area ∼ 10%, bone length ∼ 2%) (Silva et al., 2002), so we assumed that the samples we modeled were representative of their respective populations. First, the tibia with attached fibula was dissected from four-month-old SAMP6 and SAMR1 mice, embedded in 1.5% agarose gel, and scanned at 16 μm resolution by μCT. Full-length scans were obtained consisting of ∼ 1,100 slices. Tagged image format (TIF; eight-bit) files were created for every 20th CT slice (0.32 mm interval) and exported to a personal computer (Dell Optiplex running Windows 2000; Fig. 6). Image files were opened using Scion Image (Frederick, MD) and boundary points were generated for the outer (periosteal) boundary, the inner (medullary cavity) boundary, and the boundary between cortical and trabecular bone in regions containing trabecular bone. The periosteal boundary points were segmented using the threshold value determined by the software, while at the medullary boundary, a threshold of 30 points lower was used. Prior to boundary detection, some minor image editing was performed on several sections in order to smooth out sharp features near the ends of the bones. The boundary points were then filtered using a custom Excel (Microsoft) macro. Splines of each boundary were generated using I-Deas modeling software (v9.0; UGS, Plano, TX) and then lofted to generate seven separate volumes: tibia proximal to the TFJ; tibial trabecular bone (proximal); tibial marrow cavity proximal to TFJ; fibula; fibular marrow cavity; tibia distal to TFJ; and tibial marrow cavity distal to TFJ. The extent of the trabecular bone region in the proximal tibia was 2.3 mm along the longitudinal direction for SAMP6 and 1.9 mm for SAMR1.

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Figure 5. Method for developing the finite-element models.

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Figure 6. Cross-sections of mouse lower leg obtained by microcomputed tomography (μCT) showing tibia (T) and fibula (F). The sections are spaced 5 mm apart in the longitudinal direction and correspond to the centers of the support and loading surfaces for three-point bending (Fig. 2).

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To create the final bone volumes, the corresponding marrow cavity volumes were subtracted by Boolean operation leaving four separate regions: the tibia proximal to the TFJ (cortical bone), trabecular bone in the proximal region of the tibia, the tibia distal to the TFJ (cortical), and the fibula (cortical). The solid volumes accurately represented the bony anatomy captured by μCT (Fig. 7). Each of the solid regions was meshed separately with solid 10-noded tetrahedron elements using the I-Deas meshing utility. The regions were then joined by merging coincident nodes at their interfaces. Proximally, the tibia and fibula were fused using multipoint constraints to tie together nodes along the top edge of the fibula and the adjacent tibial surface. (In actuality, there is a proximal tibiofibular joint. We ran models comparing the two extremes of rigid fusion versus no constraint, i.e., fibula detached proximally, and observed only a 3% difference in peak strains in the mid-diaphysis region of interest. Thus, we believe that modeling the proximal tibiofibular joint as fused was an acceptable simplification for the purposes of our study.) Two mesh densities for each bone were created to assess mesh convergence. The lower density meshes had ∼ 18,000 elements (Fig. 8) and were meshed using an element size of 0.3–0.45 mm; the higher density meshes had ∼ 28,000 elements for the SAMR1 bone and ∼ 34,000 elements for the SAMP6 bone and were meshed using an element size of 0.2–0.3 mm. Given the large number of elements and the complex geometry, especially near the bone ends, some elements were excessively distorted and had negative volumes. These were fixed by straightening the midsize nodes in I-Deas.

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Figure 7. Top: Three-dimensional reconstructions of μCT scans of lower leg of SAMP6 mouse: lateral view. Bottom: Corresponding solid model generated in I-Deas. Note that I-Deas model is mirror image of μCT because μCT assumes a toe-to-head view, whereas we stacked the images assuming a head-to-toe view. The distal end of the bone was not modeled as it lies beyond the region of interest.

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Figure 8. Finite-element mesh of SAMR1 tibia-fibula comprised of 18,239 tetrahedral elements. X1 is the distal-proximal direction, X2 the medial-lateral direction, and X3 the anterior-posterior direction.

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Bone was assumed to be an isotropic linearly elastic solid. Cortical bone modulus was assigned a priori based on a previous study (Silva et al., 2004a). In that study, we measured the nanoindentation modulus of dry tibial samples and reported average values of 30 GPa for SAMP6 and 28 GPa for SAMR1. However, for FEA, we needed to assign modulus values for wet bone at the continuum (macroscopic) level. Based on relative comparisons of nanoindentation and continuum modulus values taken from the literature (Reilly and Burstein, 1975; Rho and Pharr, 1999; Rho et al., 1999, 2002; Turner et al., 1999), we estimated that the continuum modulus of wet bone is approximately two-thirds the nanoindentation modulus of dried bone (Silva et al., 2004a). Thus, for the finite- element model, we scaled the measured nanoindentation values and assigned values of 20 and 18.5 GPa for SAMP6 and SAMR1 cortical bone, respectively. Trabecular bone modulus was assumed to be 500 MPa and Poisson's ratio 0.3 for both cortical and trabecular bone.

The convergence study was performed using cantilever bending loading conditions with the proximal end of the tibia fixed and a 10 N transverse force applied to the distal end. We chose cantilever bending because the boundary conditions could be easily matched between models of different mesh densities, whereas the conditions for three-point bending were more difficult to match exactly between models. For the SAMP6 model, an increase in element number from 18,000 to 34,000 resulted in less than 2% difference in the end displacement and total strain energy. The average strain in a longitudinal direction on the endocortical surface at the midshaft differed by only 2% between the two models. Similarly for the SAMR1 model, increasing the element number from 18,000 to 27,000 resulted in a 1% difference in end displacement and strain energy and a difference of less than 1% in average strain on the medial endocortical surface at the midshaft. We concluded that the models containing ∼ 18,000 elements were sufficiently refined for accurate estimation of strains in the region of interest.

Boundary conditions for three-point bending were chosen through an iterative calibration process examining four different cases. The precise in situ boundary conditions at the contact points were not known because the bone is loaded through both soft tissue and the rubber pad. We examined both displacement and pressure boundary conditions at the loading point. Our criterion for final selection was based on how closely the finite-element strains at the gauge site matched the strain gauge data as well as qualitative assessment of the displacement patterns compared to experimental observations. Application of displacement boundary conditions across the loading surface (either 2.5 or 4.5 mm width, i.e., distance along the longitudinal direction X1) produced model behavior that was judged to be unrealistic. In particular, the force-strain relations were poorly matched to the strain gauge data and the location of the peak periosteal strain was not centered beneath the loading point due to the tapered shape of the tibia. Application of uniform pressure at the loading surface was found to be more suitable. Two cases of pressure contact were considered: 2.5 and 4.5 mm width. The 4.5 mm case was observed experimentally to be the width of the loading-pad contact, although because the diameter of the underlying wooden dowel was only 1.8 mm, it is unlikely that there was uniform pressure across the 4.5 mm contact region (Fig. 2). When a 4.5 mm wide uniform pressure was applied to the loading surface, we noted excessive shear deformation at the edges of the contact region and a large amount of axial stretching, which was judged to be inaccurate. We chose a 2.5 mm wide pressure region at the loading point as giving the best match between finite-element and strain gauge behaviors. Note that the ratio of periosteal strain at the gauge site to peak endocortical strain was not sensitive to the width of the pressure region.

The final boundary conditions used for simulation were as follows (Fig. 9A): the loading point was modeled by applying a normal pressure of 2.4 MPa for SAMR1 and 2.7 MPa for SAMP6. These pressure magnitudes were chosen to produce a 10 N reaction force at the supports. The pressures were applied over element surfaces on the lateral portion of the tibia and fibula over a 2.5 mm wide area centered 5.5 mm proximal to the TFJ. The distal support was centered at 0.5 mm from the TFJ; all nodes on the medial surface over a 2.5 mm wide region were fixed in the X2 (medial-lateral) direction. In addition, 10 nodes along a line at the distal TFJ were constrained in the X1 (proximal-distal) and X3 (anterior-posterior) directions. The proximal support was simulated by fixing all nodes on the medial surface over the proximal 1.25 mm of the tibia in the X2 direction. (Note that the proximal support was centered near the knee joint, and because the finite-element model started just distal to the knee joint, we only modeled one-half of the proximal support. The model predictions near the loading point are not sensitive to the extent of the displacement boundary conditions at the proximal end.) In addition to the X2 constraint at the proximal end, five nodes along a line at the proximal margin of the tibia were fixed in the transverse (X3) direction to prevent anterior/posterior bending, which results from pressure components acting transverse to the longitudinal (X1) axis.

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Figure 9. A: Boundary conditions used to simulate three-point bending. A uniform pressure was applied to the lateral surface of both tibia and fibula over a 2.5 mm wide region centered 5.5 mm from the TFJ. SAMR1 model shown. B: Deformed mesh (red) superimposed on undeformed mesh (green), illustrating medial bending of tibia and fibula in response to lateral pressure. SAMP6 model shown; similar results for SAMR1.

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FINITE-ELEMENT ANALYSIS RESULTS

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

The deformation patterns induced by three-point bending were similar for both models, with medial displacement under the pressure region (Fig. 9B). The peak displacement value in the −X2 direction was 0.023 mm for SAMP6 compared to 0.031 mm for SAMR1. These peak values occurred at the midpoint of the pressure region. Note that because the displacement boundary conditions at the support points prevented translation in both + and −X2 directions, the boundary conditions are similar to fixed wall end conditions and there is negligible displacement of the tibia distal to the distal support point.

The tibial strain distribution was consistent with lateral to medial bending in the central region between the support points (Fig. 10, top). In this region, the values of normal strain in the longitudinal direction (E11) were 98–99% of the values of maximum principal strain, indicating that E11 was the primary strain component in the central region of the tibia. Peak tensile values of E11 occurred on the medial surface near the posteromedial spine opposite the loading surface. The peak tensile strain value averaged over a 0.5 × 0.5 mm region (the size of the active region of a strain gauge) was 1,830 microstrain (μϵ) for SAMP6 and 2,450 μϵ for SAMR1 (Table 1). Note that these values were slightly greater than the strain values at the estimated site of the periosteal strain gauge. The region of well-developed bending strain was limited to the central 3–4 mm of the loading region. Consistent with the short length of the bone, high values of shear strain (E12) were observed in the regions between the support points and the loading points (Fig. 10, bottom). There was excellent agreement between the measured values of periosteal surface strain and the values predicted by the finite-element models (Fig. 11). The SAMP6-to-SAMR1 ratio of periosteal strain predicted by the finite-element models was 0.72, which is nearly equal to the ratio obtained from the regression lines based on the strain gauge data (0.75).

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Figure 10. Contour strain plots of tibia: anterior view. Top: Longitudinal strain (E11; normal strain in the X1 direction) illustrates classic bending strain distribution in the central portion of the tibia. Peak tensile strains occurred on the medial (convex) surface and peak compressive strains on the lateral (concave) surface. Bottom: Shear strain (E12) illustrates large shear deformations in the regions between the support points and the loading point. The E12 component of strain describes shear on planes normal to the X2 axis acting in the ± X1 direction (longitudinal shear) as well as shear on planes normal to the X1 axis acting in the ± X2 direction (transverse shear). The longitudinal shear component can be visualized as parallel longitudinal sheets (in the sagittal plane X1–X3) sliding past each other in the proximal-distal (X1) direction. The magnitude of the peak shear strains was comparable to the peak longitudinal strains. SAMR1 model shown; similar results for SAMP6. Element edges are not shown for better visualization of colors.

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Table 1. Finite-element analysis estimates of tibial bone strain and corresponding values from strain gauge regression lines (Fig. 4) at an arbitrary compressive force of 10 N*
Mouse strainFEAStrain gauge
Peak periosteal strainPeriosteal strain at gauge sitePeak endocortical strainRatio of periosteal strain at gauge site to peak endocortical strainPeriosteal strain at 10 N from regression lines
  • *

    Strains were sampled at their maximum value along the length of the bone (i.e., beneath the loading point) over a 0.5 × 0.5 mm area (equal to the active area of a strain gauge). Strains in the SAMP6 tibia were approximately 75% of the values for the SAMR1 tibia for both FEA and strain gauge methods. The absolute peak values of tensile strain estimated by FEA occurred at the medial apex of the cross-section and were only 6–10% greater than the values estimated at the site of gauge placement.

SAMR12,4502,3101,4501.602,010
SAMP61,8301,6601,1001.501,500
Ratio P6/R10.750.720.76N/A0.75
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Figure 11. Periosteal strain versus applied force relationships for SAMR1 and SAMP6 tibiae illustrating excellent agreement between finite-element (FE) predictions and strain gauge data. FE lines are based on linear scaling of results for a 10 N applied force, while strain gauge lines are from the best-fit linear regressions (Fig. 4). At 10 N force, the FE predictions differed from the strain gauge values by less than 15%.

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The peak endocortical tensile strains occurred adjacent to the peak periosteal strains near the posteromedial spine of the tibia (Fig. 12). Values of peak endocortical strain averaged over a 0.5 × 0.5 mm area on the endocortical surface were 41% less than peak periosteal values for both SAMP6 and SAMR1 models (Table 1). The ratio of periosteal strain at the estimated gauge site to peak endocortical strain was 1.5 for SAMP6 and 1.6 for SAMR1. Based on these ratios, we scaled the strain gauge results to estimate the force values needed to produce target values of 1,000 and 2,000 peak endocortical microstrain (Table 2). Because of the greater rigidity of the SAMP6 tibia, approximately 20% greater force must be applied to the SAMP6 tibia-fibula to produce the same values of peak endocortical strain as SAMR1.

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Figure 12. Cutaway view of medial half of central tibia (boxed region from Fig. 10, top) illustrating site of peak longitudinal strain (E11) on the endocortical surface near the posteromedial spine. The peak periosteal strains occurred at the same location, on the outside of the cortex. SAMR1 model shown; similar results for SAMP6.

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Table 2. Estimates of applied bending force needed to generate target endocortical strains at the loading point in SAMR1 and SAMP6 tibiae*
Target peak endocortical strainSAMR1SAMP6
FEA estimated periosteal strain at gauge siteApplied force from strain gauge regressionFEA estimated periosteal strain at gauge siteApplied force from strain gauge regression
  • *

    Periosteal strain at the gauge site is determined by scaling the target endocortical strain value using the ratio in Table 1, and the force corresponding to this periosteal strain is determined from the strain gauge regression lines (Fig. 4).

1,0001,6008.21,50010.0
2,0003,20015.13,00017.8

BEAM THEORY

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

To determine the accuracy of using a simpler approach for estimating strain magnitudes and the ratio of peak periosteal to endocortical strain during three-point bending, we performed a beam theory analysis of the tibia. The μCT images corresponding to the loading point (5.5 mm from TFJ) of the SAMP6 and SAMR1 bones were imported into the ImageJ image analysis program (rsb.info.nih.gov/ij/), segmented as above, and the coordinates of the boundary points were exported. From these points, we computed the moment of inertia (I; taken about the A-P axis, corresponding to the axis of bending), maximum distance from centroid to medial periosteal surface (ymax), and medial cortical thickness (thc). To estimate the peak periosteal tensile strain, we used the following equation: ϵPeri = M · ymax/EI, where M = FL/11.37 is the maximum bending moment for a beam with left and right ends fixed and a uniformly distributed force over the central one-third of the span. (Note that this is 30% less than the maximum moment for a single central-point force.) Bending force (F) was 10 N, and beam length (L) was 7.5 mm (the effective length of the beam for our simulated loading conditions). The ratio of peak periosteal to endocortical strains was calculated from the ratio of the distances from the neutral axis: ϵPeriEndo = ymax/(ymax − thc). Beam theory estimates of peak periosteal strain on the medial (tensile) surface of the tibia were within 12% of the values predicted by FEA, indicating that beam theory had relatively good accuracy for estimating absolute values of strain (Table 3). Moreover, beam theory estimates of the ratio of periosteal to endocortical strain were only 2–7% different from values predicted by FEA, indicating excellent accuracy for relative values of strain.

Table 3. Comparisons of strain values estimated by beam theory versus FEA*
Mouse strainymax (mm)thc (mm)I (mm4)Beam theoryFEA
ϵPeriϵPeriEndoϵPeriϵPeriEndo
  • *

    Beam theory values were based on the cross-sectional geometry at the section where the bending force was applied, with the assumption of a uniform cross-section along the length of the beam. For the purposes of this comparison, the FEA values of periosteal strain are based on the single highest nodal value at the section and thus are slightly higher than values shown in Table 1 (which were averaged over a 0.5 × 0.5 mm area).

SAMR10.6520.2300.09962,3301.552,6401.69
SAMP60.6860.2400.1151,9701.541,9601.65

DISCUSSION

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

Our objective was to develop a method to estimate the peak endocortical strains generated during three-point bending of the tibia in SAMP6 and control SAMR1 mice in order to estimate the forces needed to produce target endocortical strains of 1,000 and 2,000 microstrain. We used a combination of strain gauge measurements and FEA to estimate the force-endocortical strain relations for the mouse tibia. Both the strain gauge data and FEA results indicated that the tibia of the SAMP6 mouse was approximately 20–25% stiffer than the SAMR1 tibia. The increased stiffness of the SAMP6 tibia is consistent with the increased bending rigidity and moment of inertia (Silva et al., 2002) and the increased elastic modulus of cortical bone (Silva et al., 2004a) that we have documented previously for the SAMP6 mouse compared to SAMR1. Therefore, the two different mouse strains had unique force-endocortical strain relations and higher levels of force are required to produce the same target values of strain in the SAMP6 tibia compared to SAMR1. However, the ratio of periosteal to endocortical strain in the region of interest was similar for the two mouse strains (1.5–1.6).

Several previous FEA studies of rodent long bones have been done in support of in vivo experiments. Ahkter et al. (1992) developed a finite-element model of the rat tibia from histological sections and used it to complement strain gauge measurements during four-point bending. They reported that simple beam theory worked as well as FEA for estimating longitudinal bending strains (both within 10% of measured strains) and that the peak values of shear strain in the 11 mm central loading region were less than 10% of peak bending strains. We similarly found low shear strains in the central region of our model, although because of the smaller dimensions of the mouse tibia this was limited to a narrow 3–4 mm region beneath the loading point. Steck et al. (2003) developed a model of the rat tibia from μCT sections and used it to predict bending-induced fluid flow and molecular transport. Gross et al. (2002) developed a model of the mouse tibia from histological sections and used it to assess the nonuniform distributions of periosteal surface strain induced by cantilever bending. The same group later generated a strain gauge-validated model of the mouse tibia-fibula based on μCT scans and used it to estimate both periosteal and endocortical strains during cantilever bending (Srinivasan et al., 2003). In a recent study by our group, Kotha et al. (2003) developed a model of the rat ulna using similar methods as were used in the current study; we reported strong correlations between patterns of fatigue loading-induced bone formation and periosteal surface strains.

Our method for generating an accurate finite-element model of the tibia has several strengths. The use of microcomputed tomography (μCT) is ideal for capturing the complex geometry of the tibia and fibula at high spatial resolution. We believe that an accurate geometric representation is perhaps the most essential component of producing an accurate finite-element model of a long bone. The only alternative with resolution equal or better than μCT is optical imaging from histological sections. Two advantages of μCT over optical methods are speed and the elimination of errors in aligning serial sections. We used a commercially available computer-aided engineering (CAE) package to create a solid model from the μCT image-derived surface points and then discretize the smooth model into a finite-element mesh. A strength of this approach is that it utilizes the sophisticated automeshing capability of the CAE software so that once a solid model is created, mesh generation is easily done. An alternative is to use a voxel-based approach. While the voxel-based method has advantages for applications such as specimen-specific modeling of trabecular bone microstructure, we believe it has several disadvantages for modeling a whole bone. Voxel-based methods have rough surfaces that may introduce artificial variations in strain. Also, voxel-based models are comprised of many more element numbers and require use of custom finite-element software. A voxel-based model of the tibia-fibula at 0.016 mm resolution would require 3.5 × 106 elements compared to the approximately 1.7 × 104 elements used in this study.

Another strength of our methods was that we assigned cortical bone modulus a priori based on a previous nanoindentation study in which we reported a small but significant difference of 10% in modulus between SAMP6 and SAMR1 bones (Silva et al., 2004a). Therefore, differences in material stiffness were accounted for in this study and contributed to the differences in stiffness between the two mouse strains. Another strength of our methods was the use of strain gauge data to assess model accuracy. This is an important step in development of any bone model even if the model is to be used only for relative predictions. The combination of FEA with strain gauge measurements should be standard for bone models used to estimate strain values and their distributions in skeletal structures.

Our methods also had limitations. Sophisticated methods have been described for utilizing data from multiple gauges to describe spatial strain patterns and to estimate accurate loading conditions for FEA (Coleman et al., 2003). However, the small size of the mouse tibia dictated that only a single uniaxial gauge could be used, thus limiting the data available for model calibration/validation. There are many parameters that could have been varied to match the strain data at the periosteal site, including material properties and boundary conditions. We chose to fix material properties and to load the model using a uniform pressure over the central tibia-fibula after determining that uniform displacement was unrealistic. However, we did not know the precise area or distribution of pressure at the contact regions. A more accurate modeling approach would require simulation of contact between compliant layers of muscle, skin, and the rubber-padded loading surfaces. This would introduce many additional unknowns in terms of material properties and would be computationally difficult to implement. We determined that a 2.5 mm wide uniform pressure loading condition gave a good match to the experimental behavior in the central tibia. Increasing the contact region from 2.5 to 4.5 mm width did not change the primary outcome of interest, i.e., the ratio of periosteal to endocortical strain. Therefore, we believe that use of a more complex scheme to model contact between the loading apparatus and the mouse leg was not essential for our objectives. Another limitation related to the boundary conditions is that the nodes at the support points were constrained from translation in both + and −X2 directions, resulting in a fixed wall end condition at both proximal and distal supports with negligible displacement of the tibia distal to the distal support (Fig. 9). While this may be somewhat overconstrained, we believe these boundary conditions represent reasonably well the conditions that exist during in vivo loading. Proximally the knee joint and distal femur provide constraint on the proximal tibia and prevent it from rotating or displacing, while distally a small restraining clip limits vertical translation of the foot and ankle.

Previous studies of in vivo bending in the mouse lower leg have utilized four-point rather than three-point bending (Akhter et al., 1998). A four-point bending setup in the mouse lower leg requires that the two loading points are essentially side-by-side and that there is very little distance between the support and loading points. When we simulated a 4.5 mm wide contact area and thereby reduced the distance between support and loading points, we observed even greater values of shear strain in the tibia in the region between the support and loading points. In fact, the values of shear strain were approximately two times greater in magnitude than the values of peak periosteal tensile strain. In contrast, using the 2.5 mm wide pressure contacts and thereby simulating three-point bending, the peak value of shear strain was comparable to the peak value of periosteal tensile strain. While we believe it is desirable to minimize shear strains, our model indicates that the center of the loading region experiences primarily bending for either four-point or three-point bending. Thus, either loading condition appears appropriate for generating bending strains in the central region of the mouse tibia.

A comparison between beam theory versus FEA predictions of tibial strains indicated that beam theory was relatively accurate for estimating strain magnitude when the loading conditions were matched (i.e., fixed ends; distributed force over central 2.5 mm). However, we do not conclude that beam theory by itself would be suitable for general use in accurately predicting strains because of differences in boundary conditions and material properties that may exist between different studies. The appropriate boundary conditions in this study were not known a priori; if we calculated peak periosteal strain based on a point force rather than a distributed force, beam theory would overestimate strain magnitude by ∼ 30%. Nevertheless, beam theory estimates of the ratio of periosteal to endocortical strain were within 2–7% of estimates based on FEA, independent of boundary conditions or material properties. Therefore, it may be reasonable to combine strain gauge measurements of periosteal surface strain and beam theory estimates of the ratio of endocortical to periosteal strain in order to estimate force-endocortical strain relations in future studies. The critical issue is whether or not the cross-section of interest is loaded in relatively pure bending. Because this condition appears to be satisfied for the bending loading conditions we used, simple beam theory is appropriate for analysis of strain distribution within the cross-section.

In summary, we generated finite-element models of the mouse tibia-fibula using μCT and public domain and commercially available software. These models used in conjunction with strain gauge measurements enabled estimation of force-endocortical strain relations for the tibia. These models accounted for differences in bone geometry and material properties between SAMP6 and SAMR1 mice and indicated that ∼ 20% higher forces must be applied to the SAMP6 tibia to reach the same levels of strain. We also observed that simple beam theory can provide relatively accurate estimates of the ratio between periosteal to endocortical strain at the midpoint of the loading region in the mouse tibia. Therefore, future studies may not necessarily require development of a finite-element model if the region of interest is loaded primarily in bending. In conclusion, FEA has provided unique insight regarding the strain environment throughout the tibia during three-point bending and is an invaluable complement to understanding models of in vivo loading.

Acknowledgements

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED

The authors thank Brian Uthgenannt for assistance with μCT scanning and Roberta Strigel for developing the Excel macros used to generate points for spline creation.

LITERATURE CITED

  1. Top of page
  2. Abstract
  3. OVERVIEW OF STUDY DESIGN
  4. STRAIN GAUGE STUDY
  5. FINITE-ELEMENT ANALYSIS METHODS
  6. FINITE-ELEMENT ANALYSIS RESULTS
  7. BEAM THEORY
  8. DISCUSSION
  9. Acknowledgements
  10. LITERATURE CITED