Hyperlipidemia affects multiscale structure and strength of murine femur

Abstract

To improve bone strength prediction beyond limitations of assessment founded solely on the bone mineral component, we investigated the effect of hyperlipidemia, present in more than 40% of osteoporotic patients, on multiscale structure of murine bone. Our overarching purpose is to estimate bone strength accurately, to facilitate mitigating fracture morbidity and mortality in patients. Because i) orientation of collagen type I affects, independently of degree of mineralization, cortical bone’s micro-structural strength; and, ii) hyperlipidemia affects collagen orientation and µCT volumetric tissue mineral density (vTMD) in murine cortical bone, we have constructed the first multiscale finite element (mFE), mouse-specific femoral model to study the effect of collagen orientation and vTMD on strength in Ldlr^−/−, a mouse model of hyperlipidemia, and its control wild type, on either high fat diet or normal diet. Each µCT scan-based mFE model included either element-specific elastic orthotropic properties calculated from collagen orientation and vTMD (collagen-density model) by experimentally validated formulation, or usual element-specific elastic isotropic material properties dependent on vTMD-only (density-only model). We found that collagen orientation, assessed by circularly polarized light and confocal microscopies, and vTMD, differed among groups; and that microindentation results strongly correlate with elastic modulus of collagen-density models (r²=0.85, p=10⁻⁵). Collagen-density models yielded 1) larger strains, and therefore lower strength, in simulations of 3-point bending and physiological loading; and 2) higher correlation between mFE-predicted strength and 3-point bending experimental strength, than density-only models. This novel method supports ongoing translational research to achieve the as yet elusive goal of accurate bone strength prediction.

Introduction

Areal bone mineral density (aBMD) is the current clinical “gold standard” employed to diagnose patients’ osteoporosis. Such diagnosis gates opportunity for early intervention to prevent fracture. The insufficiency of low aBMD as a patient-specific facture predictor, notwithstanding its association with increased population-based fracture risk, motivated development of the FRAX index and methods that extract information from clinical imaging (Keyak et al., 1998; Boutroy et al., 2008; Li et al., 2009; Krug et al., 2010; Kazakia et al., 2011) to identify additional parameters that can aid prediction of fracture.

As a complementary path, we investigate the effect of genetic and environmental factors at the tissue level of compact bone, increasingly recognized as a major contributor of bone strength (Rockoff et al., 1969; Bonnick, 1998). We focused on factors of bone strength at tissue-level that can lead to micro-crack elongation and coalescence, and eventually to macroscopic fracture. We had previously found that changes in orientation of collagen type I, of distinct genetic and environmental origin, and reduced mineralization, can reduce cortical strength (Ascenzi et al., 2000 and 2012; Sage et al., 2011; Pirih et al., 2012). Further, collagen orientation at specific locations of the proximal tibia was found to differ in a genetic mouse model of hyperlipidemia Ldlr^−/− mutant (MUT) in dependence on diet – normal diet (ND) vs. high fat diet (HFD). Recent micro-level structural analysis of the compact component of human and animal femora has established that the orientation of collagen type I (locally parallel to carbonated hydroxyapatite crystals) and the degree of calcification vary independently from each other in dependence of location, loading, and presence/absence of disease (Ascenzi, 1988; Riggs et al., 1993; Power et al., 2003; Goldman et al., 2005; Ascenzi and Lomovtsev, 2006; Ramasamya and Akkusb, 2007; Cristofolini et al., 2008; Beraudi et al., 2009 and 2010).

To predict bone strength in relation to altered parameters at bone tissue level, we present here a multiscale finite element (mFE), mouse-specific femoral model. The multiscale nature of the model enables appreciation of the effect of macroscopic mechanical testing at the bone tissue-level, to simulate experimental loading conditions ex vivo and in vivo. We apply the mFE model to femora from the hyperlipidemic MUT, its control (WT for wild type), on either ND or HFD. Both genetic and diet-induced hyperlipidemia were previously found to increase generation of lipid oxidation that promotes bone loss and reduces bone strength (Pirih et al., 2012). Motivating this study were: i) the effect of HFD on collagen orientation of mice (Sage et al., 2011); and, ii) the hypothesis that two independent parameters, orientation of collagen and degree of calcification, affect the strength of bone in mice as they have been hypothesized to do so for humans (Ascenzi et al., 1997).

Materials and Methods

The Institutional Animal Care and Use Committee, University of California at Los Angeles, approved the experimental protocols. We used 16 right femora of 25-week-old male mice: 8 WT and 8 MUT which were for 18 weeks on either ND or HFD (TD 9022)1, Harlan Teklad), i.e. 4 groups of 4 mice per group (Pirih et al., 2012). We used µCT Skyscan 1172 (Aartselaar, Belgium) with a calibration phantom (Aartselaar, Belgium) containing two cells of 0.25 g/cm³ and 0.75 g/cm³ equivalent concentration of calcium hydroxyapatite, to scan the femora with isotropic voxel size of 13µm, 55kVp, 72 µA, and a 0.5 mm aluminum filter. We used the µCT axial slices (1024 × 1024 pix) and volumetric tissue mineral density (vTMD).

Mechanical testing

3-point bending was previously reported (Pirih et al, 2012). Briefly, the load was applied at femoral posterior mid-shaft, with a support span set at 6 mm, at a constant displacement rate of 6 mm/min to total failure. Thereafter, we performed 4 microindentations per femur (3 femora per group), starting above the complete transverse fracture at mid-shaft, every 1 to 2 mm at anterior quadrant, towards proximal end, with BioDent 1000 RPI (Active Life Scientific). With one probe fixed at the bone surface, the moving probe cyclically indented the tissue relative to bone surface. The pre-load of the reference probe was 2.7 N ± 0.25 N for all measurements. A BP1 probe (25 µm tip radius with a 90° conical end) was used with 2 N force, 20 cycles of indentation per measurement, 2 Hz, for all indentations. So-called average unloading slope (Avg US (1st-L), N/µm) provides the average unloading slopes of each measurement, from 1st to Last (Randall et al., 2009).

Microscopy

We used circularly polarized light (CPL) microscopy to visualize birefringent signals indicative of collagen orientation (Ascenzi and Lomovtsev, 2006), and scanning confocal microscopy to visualize actual collagen orientation in the cortical component of the femoral shaft. 50µm thick serial anterior-posterior longitudinal sections of the shaft embedded in PMMA were observed at 100× with a Leitz Dialux 20 microscope with CPL (Midland, Ontario, Canada) (Ascenzi et al., 2012). A Leica TCS-SP (Wetzlar, Germany) microscope with krypton laser (567 nm excitation), 20× and 63× Planapochromat lens, 580–700 nm detection range allowed observation of CPL birefringent extinct and bright locations at 200×. CPL and confocal images were analyzed with XaraX (Xara Group Ltd) and Metamorph (Molecular Devices) software. In particular, we compared corresponding locations between microindented femora (3 per group) and non-microindented femora (1 per group). For each quadrant, we combined the data within 1mm bins along the z-axis, and computed the percentage of bright area with respect to total area. Bright birefringence observed on longitudinal sections was set to correspond to an 64% of longitudinal (0°±22.5°), 19% of oblique acute (45°±22.5°), 3% of transverse (90°±22.5°), and 14% of oblique obtuse orientation (135°±22.5°) of collagen; and complement extinct birefringence set to correspond to 4% of longitudinal, 29% of oblique acute, 34% of transverse, and 33% of oblique obtuse orientation of collagen (Ascenzi and Lomovtsev, 2006).

Two-level mFE model

For each of the 16 femora, the macro-geometry of the mFE model was obtained from the quantitative µCT, and meshed with tetrahedral elements (~ 115,000) with mean edge size of 0.17 mm (Lutz and Nackenhorst, 2010). An element was considered bone, not marrow, if its calcium hydroxyapatite density was at least 0.5 g/cm³. We choose an xyz reference with x-axis directed from medial to lateral aspect, y-axis from anterior to posterior, z-axis from distal to proximal along the long axis of shaft.

We prepared 16 so-called collagen-density models whose simulated cortical component consisted of element-specific elastic orthotropic properties calculated from collagen orientation and vTMD, (see entries [1,1] to [6,6], Appendix A). These orthotropic properties follow the observation that murine cortical bone consists mainly of lamellar bone (Bianco and Robey, 2000). Indeed, we modeled the micro-structure as a locally unidirectional fiber composite (Vinson, 1993), where the collagen fibrils represent the composite's matrix and the apatite crystallites represent the fibers. Because collagen and apatite form lamellar bone of both human secondary osteons and mouse bone, we used a lamellar model verified by mechanical testing of single extinct or bright osteons at initial and final degrees of calcification: we extended formulae previously developed to simulate prestress in single fully calcified lamellae and mechanical testing of single human fully-calcified osteons. We wrote an algorithm to distribute the element-specific assignment of collagen orientation so as to respect the percentages of longitudinal, oblique acute, transverse and oblique obtuse orientations of each quadrant and bin. The algorithm chooses randomly among such percentages with the restriction of a smaller than 10° change between orientations in adjacent elements (Ascenzi and Lomovtsev, 2006; Ascenzi et al., 2008). We also prepared 16 so-called density-only models with element-specific elastic isotropic properties computed from µCT vTMD (Lutz and Nackenhorst, 2010). We computed the Young’s modulus

$$\mathrm{E}={\mathrm{E}}_{\mathrm{o}}{(\mathrm{\rho }/{\mathrm{\rho }}_{\mathrm{o}})}^2$$ (1)

with E_o = 6250 MPa, ρ_o=2 g/cm³, ρ as vTMD. The Poisson’s ratio was 0.29.

We simulated 3-point bending and physiological loading (Fig. 1) producing strains in the same elastic range. For the 3-point bending, we calculated bone strength (defined as load at yield point) from load-displacement diagrams, as load at 90% of maximum load (Cody et al. 1996). We used the yield point to ensure that models’ load and displacement fell within the elastic range. A force of 3N, directed along the negative direction of the y-axis, was applied as node displacement of an element face. We checked that the force applied to the model produced a stress/strain ratio comparable to the elastic modulus in each specific principal direction. To simulate physiological loading in terms of one hind-legged stance, a force of 4N, forming a 20° angle with the z-axis and parallel to the projection of the femoral neck axis on the xy-plane (Keyak et al, 1998; Voide, 2008), was applied over a 0.35 µm-diameter region of the femoral head as node displacement. We fully restrained the nodes at the distal end. We analyzed the mFE models with Abaqus software (Dassault Systèmes) on UCLA’s Hoffman cluster and UCSD’s Trestles supercomputer. The 20MB file size was due to the highly heterogeneous element-specific material properties. The 10 hours of computing time used 8 cores and 48 GB of RAM.

Figure 1.

mFE model of murine femur (a) under 3-point bending and (b) one-legged stance. 1mm bins 1 to 7 are indicated from mid-shaft to proximal femur in (b) and used for collection of data from microscopy investigations.

We investigated strain along the femoral shaft, ε_zz, at the region of highest absolute value, i.e. at anterior and posterior quadrants of mid-shaft. Because the average diameter of mid-shaft is the same for all femora except for a significant difference between ND MUT and HFD MUT (1.78±0.02 mm vs. 1.62±0.03 mm, p<0.01), we adjusted the computed ε_zz for HFD MUT with respect to the ND MUT (Appendix B). Afterwards, we computed the mFE-predicted strength for 3-point bending by noting that the mFE model behaves linearly and by assuming that yielding begins when the maximum longitudinal strain at the mid-shaft of the mFE model, ε_zz, is 0.6% (Morgan and Keaveny, 2001; Kaneko et al., 2003). Therefore, mFE-predicted strength was computed as 0.006*(3/ε_zz), where 3 (N) is the magnitude of the force applied. Accordingly, higher surface strain ε_zz corresponds to lower strength. We then examined the correlation between the measured bone strength and mFE-predicted strength. Analogous to the use of ε_zz at the bone surface to compute the mFE-predicted bone strength for 3-point bending simulations, we studied the average ε_zz at the bone surface in the physiological loading simulations to infer changes in bone strength among mouse groups.

Statistical analysis

We used the Student t-tests to detect differences of ε_zz between collagen-density and density-only models of each mouse group (4 mice per group); and among the 4 mouse groups for collagen-density and density-only models, separately. We used the r² as measure of correlation, and the p-value of the r² with a t-test of slopes between linear regression and horizontal line, between Avg US (1st-L) and mFE elastic modulus at microindentation site for 48 microindentations, and between experimental strength and mFE-predicted strength of 16 femora. The p-value of single comparisons was considered significant if smaller than 0.05, of m multiple comparisons if smaller than Bonferroni’s 0.05/m, for m. The data are presented as mean ± standard error.

Results

We identified and quantified the effect of murine hyperlipidemia on the multiscale structural properties of the femur, assessed strain with mFE femoral models, and studied the relation between experimental and mFE strength.

At the micro-structural level, the distribution of % birefringent bright area and collagen orientation showed no difference from mid-shaft to proximal femur (bins 1 through 7) and from mid-shaft to distal femur (bins 8 through 14). Hence the data are presented per mouse group for bins 1 to 7. The % birefringent bright area was found to be 1) largest at the anterior quadrant and overall smallest at the posterior quadrant of WT femur (on ND or HFD); 2) increased in all the quadrants from mid-shaft to proximal femur of ND MUT; and 3) smallest at anterior and lateral quadrants, and largest at posterior and medial quadrants, c onsistently from mid-shaft to proximal femur of HFD MUT (Figs. 2, 3a, 3b). The collagen orientation was predominantly longitudinal at 1) anterior quadrant of WT (on ND or HFD); 2) all quadrants of proximal femur of ND MUT; and 3) posterior and medial quadrants of HFD MUT (Table I, Figs. 3c, 3d).

Figure 2.

% of bright area on longitudinal sections of murine femora. The percentages are shown per quadrant (A, anterior; L, lateral; P, posterior; M, medial) and per bin (numbered as 1 to 7, moving up from mid-shaft towards the proximal femur), per mouse group, averaged over the femora of the 4 mice in the group. The data of individual femora follow the trend of the averaged data.

Figure 3.

Birefringent signal by CPL and fluorescent signal by confocal microscopy. We show examples of regions with specific percentages of bright areas: (a) 70% of bright area, occurring e.g. at the anterior quadrant of WT; and (b) 40% of bright area, e.g. at anterior mid-shaft of ND MUT and at anterior of HFD MUT. Confocal microscopy shows matrix and osteocyte lacunae of (c) ND WT and (d) HFD MUT. In enlargements, different collagen orientation patterns are emphasized with white markings.

Table I.

Percentages of prevalent collagen orientation along the femoral shaft.

Group	Quadrant	Bin 1	Bin 2	Bin 3	Bin 4	Bin 5	Bin 6	Bin 7
ND WT	anterior	48	47	43	47	44	43	44
	lateral	44	49	48	47	46	50	48
	posterior	55	49	53	50	52	51	49
	medial	45	47	45	53	44	48	47
HFD WT	anterior	46	43	44	47	44	44	43
	lateral	46	44	49	44	43	47	44
	posterior	53	56	50	51	52	49	56
	medial	44	47	45	53	45	49	47
ND MUT	anterior	50	50	49	47	43	45	54
	lateral	51	52	50	48	44	47	53
	posterior	51	51	50	47	43	43	52
	medial	53	52	50	47	44	44	50
HFD MUT	anterior	51	52	51	51	52	52	51
	lateral	45	48	48	50	44	48	50
	posterior	46	45	44	48	45	43	44
	medial	47	47	47	55	50	48	51

The data is presented as mean percentages over the 4 mice of each group without standard errors in order to visualize differences in orientation patterns among groups. The bolded entries represent percentages in the longitudinal orientations of collagen (0°±22.5°) measured with respect to the longitudinal axis of the femur. The non-bolded entries are percentages in the oblique orientation of collagen, acute (45°±22.5°) and obtuse (135°±22.5°). The standard error varies up to 4%.

vTMD, averaged over the femoral bone, differed among mouse groups: ND WT: 1.59±0.001 g/cm³; HFD WT: 1.51±0.001 g/cm³; ND MUT: 1.54±0.001 g/cm³; HFD MUT: 1.45±0.001 g/cm³; p<10⁻²⁵. Avg US (1st-L) from microindentation differs between ND MUT and HFD MUT (0.49±0.03 N/µm vs. 0.36±0.02 N/µm; p=0.02). The correlation between Avg US (1st-L) measured with indentation tip penetrating along the anterior-posterior (y) axis (Fig.1), and the elastic modulus computed from both collagen orientation and vTMD along the y-axis was stronger for the collagen-density models than the density-only models (r²=0.85, p=10⁻⁵; vs. r²=0.42, p=0.007; Fig. 4).

Figure 4.

Correlation of microindentation parameter Avg US (1st-L) with elastic modulus in anterior-posterior direction. The correlation was stronger for (a) collagen-density (CD) models than (b) density-only (D) models of 3 femora per group. The p-value of the r² were significant, p<0.01.

For both loading conditions, ε_zz was 1) predominantly positive at anterior, suggesting tension along the femoral z-axis; and predominantly negative at posterior quadrant, suggesting compression along the femoral z-axis (Table II); and 2) predominantly larger in magnitude for collagen-density models than density-only models (Table II). For 3-point bending, 1) ε_zz was significantly larger for collagen-density than density-only models at the anterior quadrant of ND WT, HFD WT, and ND MUT (Table II, Fig. 5); 2) neither collagen-density nor density-only models found significant differences in ε_zz among mouse groups (Table III and Supplemental Table; note that significant differences in #1 of Table III are listed as non significant in Supplemental Table after strain adjustment of HFD MUT); 3) the predominantly larger ε_zz for the collagen-density models yielded lower mFE-predicted strength than the density-only models (Fig. 6); and 4) the correlations between experimental strength and mFE-predicted strength, computed for ND WT, HFD WT and ND MUT at the anterior quadrant, for which ε_zz significantly differed between collagen-density and density-only models (Table II), was stronger for the collagen-density than the density-only models (r²=0.91, p=5×10⁻⁴; vs. 0.73; p=2×10⁻⁶). For physiological loading, 1) in the posterior quadrant, ε_zz was significantly larger in magnitude for collagen-density models than for density-only models, with differences occurring in the ND WT, HFD WT, and HFD MUT groups; 2) in the anterior quadrant, collagen-density models revealed one significant difference in ε_zz among the mouse groups, compared with two differences the density-only models revealed two (Fig. 7); and 3) in the posterior quadrant, collagen-density models demonstrated five significant differences compared with three for the density-only models. To study the implications of such differences on bone strength, we looked at the magnitude of εzz at the bone surface. At the posterior surface, the magnitude of εzz significantly differed between ND WT and HFD WT, and between HFD WT and ND MUT, for density-only models (ND WT: 289 ± 24 vs. HFD WT: 618 ± 25 vs. ND MUT: 387 ± 20; p<0.0005), but not for collagen-density models (ND WT: 469 ±49 vs. HFD WT: 743 ± 98 vs. ND MUT: 386 ± 40; p>0.0277; p is significant if less than 0.0083). Therefore, we infer that the distribution of collagen orientation, that is unchanged by the HFD in WT and is changed by the genetic mutation in ND MUT, compensates for the lower vTMD of the HFD WT, to yield unchanged bone strength among ND WT, HFD WT and ND MUT.

Table II.

Pairwise comparison of strain component ε_zz (expressed in microstrain units) along the longitudinal (z) axis at mid-shaft, between collagen-density (CD) models and density-only (D) models of each mouse group.

Group	Quadrant	3-point bending			1-legged stance
		CD	D	p	CD	D	p
ND WT	anterior	488 ± 58	301 ± 26	0.009	194 ± 40	95 ± 24	0.044
	posterior	−348 ± 299	−178 ± 216	0.648	−660 ± 61	−422 ± 34	0.002
HFD WT	anterior	507 ± 61	330 ± 44	0.029	336 ± 56	226 ± 33	0.105
	posterior	−340 ± 267	−158 ± 308	0.658	−977 ± 72	−722 ± 30	0.004
ND MUT	anterior	395 ± 49	262 ± 25	0.029	173 ± 39	105 ± 28	0.167
	posterior	−202 ± 195	−43 ± 242	0.611	−585 ± 51	−464 ± 24	0.043
HFD MUT	anterior	480 ± 50 328 ± 34	434 ± 27 300 ± 19	0.425 0.481	83 ± 22 44 ± 18	78 ± 22 40 ± 22	0.873 0.872
	posterior	−266 ± 296 −78 ± 233	−221 ± 331 −68 ± 253	0.918 0.762	−516 ± 35 −393 ± 30	−539 ± 25 −451 ± 22	0.585 0.135

We show ε_zz (mean ± standard error) and p-value (bolded if significant as less than 0.05), at each of anterior and posterior quadrant of the transverse section at mid-shaft, for each loading condition. Because the diameters of the transverse section at mid-shaft are smaller in HFD MUT than ND MUT, the adjusted ε_zz of HFD MUT is smaller than the original ε_zz. Adjusted strains and relative p-values are entered in italics.

Figure 5.

Comparison of distributions of ε_zz at cortical mid-shaft between collagen-density (CD) and density-only (D) models. For 3-point bending, ε_zz was larger for CD than D at the anterior quadrant of ND WT (488±58 µε vs. 301±26 µε; Table I).

Table III.

Significant differences of strain ε_zz (expressed in microstrain units) among mouse groups.

Loading	Quadrant	Significant differences						p	#
3-point bending	anterior	D	ND WT:	301 ± 26	vs.	HFD MUT:	434 ± 27	0.002	1
			ND MUT:	262 ± 25	vs.	HFD MUT:	434 ± 27	0.001
one-legged stance	anterior	CD	HFD WT:	336 ± 56	vs.	HFD MUT:	83 ± 22 44 ± 18	5 * 10−4 10−5	2
		D	HFD WT:	226 ± 33	vs.	ND WT:	95 ± 24	0.003
					vs.	HFD MUT:	78 ± 22 40 ± 19	0.001 10−4
	posterior	CD	ND WT	−660 ± 61	vs.	HFD MUT:	−393 ± 30	0.001	3
			ND MUT	−585 ± 51	vs.	HFD MUT:	−393 ± 30	0.003
			HFD WT:	−977 ± 72	vs.	ND MUT:	−585 ± 51	10−4	4
					vs.	ND WT:	−660 ± 61	0.002
					vs.	HFD MUT:	−516 ± 35 −393 ± 30	10−5 3 * 10−7
		D	HFD WT:	−722 ± 30	vs.	ND MUT:	−464 ± 24	2 * 10−7	5
					vs.	ND WT:	−422 ± 34	2 * 10−7
					vs.	HFD MUT:	−539 ± 25 −451 ± 22	5 * 10−5 10−7

We show differences for each loading condition, at each of anterior and posterior quadrant of mid-shaft, for collagen-density (CD) models and for density-only (D) models. Strains of HFD MUT, adjusted for smaller diameters at mid-shaft, and relative p-values are shown in italics. Because we compared 4 groups, we used Bonferroni’s coefficient m=6, the p-values are considered significant if smaller than 0.0083. Note that the comparisons for 3-point bending (#1) lose their significance after strain adjustment of HFD MUT due to its smaller mid-shaft diameters than ND MUT (Table III).

Figure 6.

Comparisons of distributions of ε_zz at cortical mid-shaft among collagen-density models. For one-legged stance, the absolute value of ε_zz was larger for (a) HFD WT than (b) HFD MUT at the anterior (336±56 µε vs. 44±18 µε; Table II, #2) and the posterior quadrant (977±72 µε vs. 393±30 µε; Table II, #4). The absolute value of ε_zz was larger for (a) HFD WT than (c) ND MUT at the posterior quadrant (977±72 µε vs. 585±51 µε; Table II, #4).

Figure 7.

Correlations between experimental bone strength and mFE-predicted strength for 3-point bending. The correlations were computed for ND WT, HFD WT and ND MUT at the anterior quadrant, for which ε_zz significantly differed between collagen-density (CD) models and density-only (D) models (Table I). The p-values of the r² were significant, p<10⁻³.

Discussion

We have developed a new methodology to assess mouse-specific bone strength for investigations on the effect of genetic and environment on bone quality. We have made the first application to a mouse model of genetic hyperlipidemia with diet as environmental factor.

Epidemiological studies associate hyperlipidemia with low bone mass (Adami et al., 2004; Orozco et al., 2004; Cui et al., 2005). HFD-induced hyperlipidemia reduces bone formation, mineral density and strength in animal studies (Parhami et al., 2001; Turek et al., 2003; Pirih et al. 2012). Hyperlipidemia 1) increases lipid oxidation products that blunt osteoblastic function; and 2) induces osteoclastic differentiation (Greenfield et al., 1993; Parhami et al., 1997; Huang et al., 2007; Hirasawa et al., 2007). In the present study, we found that hyperlipidemic MUT mice, either on a ND or HFD, had altered collagen orientation in long bones. Recent studies also show that lipid oxidation products reduce the anabolic effect of parathyroid hormone PTH(1–34) treatment of osteoporosis (Huang et al., 2007; Sage et al., 2011). One of the beneficial effects of PTH(1–34) treatment includes increased heterogeneity of collagen orientation in osteoporotic patients (Ascenzi et al., 2012), but the present study suggests that hyperlipidemia may also impair PTH(1–34) effects on collagen orientation. Further, because the HFD is also diabetogenic, Pirih et al. (2012) conjectured occurrence of insulin resistance with subsequent induction of advanced glycation end-products, and hypothesized additional involvement of non-canonical pathways of osteoclastogenesis, such as through TNF-a and IGF-I, and of factors affecting osteoprogenitors.

Our results on the ND WT indicate prevalent orientation forming small angle with the longitudinal axis of the femur at anterior quadrant and larger angles with the longitudinal axis of the femur at posterior quadrant, confirming previous findings (Ramasamya and Akkusb, 2007). Also, the effect of HFD on MUT’s collagen orientation, now quantified in the cortical component of the whole shaft, confirms that the HFD alters the orientation of collagen in the MUT (Sage et al., 2011). Over the last fifty years, the experimental findings on the orientation of collagen type I have given rise to models of orientation patterns, that were incorporated in models to analyze the effect of such orientation (for reviews, Ascenzi et al., 1999, 2008 and 2013). Interestingly, we found that the distribution of collagen orientation does not differ between upper and lower shaft of murine femur. This contrasts with the difference found in the human femur (Ascenzi, 1988), perhaps due to presence and location-dependent size variation of secondary osteons (Evans, 1973) interacting with the variation of collagen orientation along the shaft to strengthen the shaft of human long bones, in addition to the load distribution difference between bipeds and quadrupeds.

The higher ε_zz found in the collagen-density models, in comparison to the density-only models, indicates that the strain on osteocytes is higher th an anticipated. Our higher strain increases the possibility that such cells are indeed able to sense the strain exerted by the surrounding bone matrix and signal for bone remodeling (Bonewald, 2011). Further, we found that one of the parameters measured by the BioDent RPI instrument, the average unloading slope Avg US (1st-L), was well predicted by the elastic modulus in the direction of microindentation penetration. We interpret this result in light of the parallelism of orientation between collagen and hydroxyapatite needles (Bonucci, 2000) that would pose a different resistance to the microindenter, as previously hypothesized in regard to nanoindentation (Reisinger et al, 2011).

Murine finite element models usually used to study bone defects are based on µCT vTMD, here called density-only models (Spatz et al., 2013). Because the organization of collagen type I orientation in murine long bone was previously found to depend on genetic and environmental factors (Ramasamya and Akkusb, 2007; Ionova-Martin et al., 2010; Tang et al., 2012), we tested whether the mechanical performance of bone with altered patterns would show reduced strength. To this end, we newly developed a collagen-density mFE femoral model, and compared the new model to usual density-only models. When we adjusted the micro-level stress for bone diameter at mid-shaft of group HFD MUT, we employed a scaling factor for continuum-level stress in a homogeneous isotropic hollow cylinder subjected to bending. This allowed us to correct only for differences in bone diameter without considering material differences. We then used the compliance matrix for the orthotropic heterogeneous micro-structure to compute the adjusted micro-level strain. This approach assumes that the effect of bone diameter on micro-level stress is proportional to that on continuum-level stress, which is reasonable for this comparative study. Nevertheless, this is a limitation of the study.

Strain ε_zz at the femoral mid-shaft differs between loading conditions, between collagen-density and density-only models, and among mouse groups. The different ε_zz between loading conditions confirms that the macro-geometry affects physiological loading’s ε_zz even though femoral neck angles do not differ significantly among the mouse groups. The difference in results between collagen-density and density-only models, at anterior and posterior quadrants (Table III), shows that the effect of collagen orientation is quadrant-dependent. Because the difference in ε_zz between ND WT and HFD WT of density-only models, is missing in collagen-density models, collagen orientation appears to compensate for difference in vTMD (Table III, #2). Collagen orientation also has a differentiating effect on ε_zz between HFD MUT and each of ND WT and ND MUT, which is missing in density-only models (Table III, #3). Further, differences in mean and standard error of vTMD affect ε_zz: vTMD is greater in HFD MUT vs. ND MUT at specific locations within the anterior mid-shaft. We find that the investigated genetic mutation affects the distribution of collagen orientation, but diet affects collagen orientation only in conjunction with the genetic mutation. The unchanged collagen orientation of the HFD WT compensates for its lower vTMD compared with that of ND MUT, to yield the same bone strength of the ND MUT despite a less than optimal collagen orientation. The lack of additional differences in bone strength among the mouse groups is due to the fact that the femur does not break at mid-shaft under physiological loading. The present work provides foundation for further studies on the implications of diet and genetics for organization of elementary components and bone strength at locations of fracture occurrence in humans, even though the mouse groups considered do not fracture.

Our new mFE model uses animal-specific findings. Namely, animal-specific observations of collagen direction supplement the µCT grey values (interpreted as hydroxyapatite), and water is included indirectly through validation of mechanical testing of wet micro-specimens. Including collagen orientation in the elastic mFE model increases the Young’s modulus in the direction of collagen and slightly reduces the Young’s modulus in the two perpendicular directions, to yield an average Young’s modulus on the order of the isotropic Young’s modulus of the density-only models. For instance, because the majority of collagen forms small a ngles with the longitudinal axis of the femur at the anterior quadrant of the ND WT, and forms large angles with the longitudinal axis of the femur at the posterior quadrant, the strain component ε_zz is larger at the anterior quadrant in collagen-density models in comparison to density-only models, and smaller at the posterior quadrant in collagen-density models in comparison to density-only models (Table II). Recently Blanchard et al. (2013) differently interpreted the µCT grey shade, considering it a fixed combination of densities of collagen, hydroxyapatite and water, where collagen and water would attenuate the grey shade. Their models, incorporating a fixed (generic, not animal-specific) orientation of mineralized collagen, to reinforce the transversely isotropic material properties, yield smaller strains than usual density-only models.

Our novel approach advances the field by identifying and quantifying the effect of specific genetic and environmental factors on multiscale structural parameters of mouse-specific bone quality. Inclusion of both collagen orientation and mineralization in the mFE models, instead of including only mineralization, improves predictions of bone strength. We provide a foundation for translational research to achieve bone strength prediction from patient-specific genetic and environmental factors, with a view to preventive treatment.

Appendix A

The material properties at bone-tissue level were modeled as a locally unidirectional fiber composite (Ascenzi, 1999), where the collagen fibrils represented the composite’s matrix, and the carbonated apatite crystallites represented the fibers. The stiffness matrix relative to a specific collagen orientation followed the composite formulation with small displacements (Vinson, 1993) and used vTMD from μCT for the apatite contained in the elements (Ascenzi, 1999). We used Maple software and Python 1.3 for property assignment to element. First, a stiffness matrix Q_h for the linear orthotropic material with respect to the longitudinal axis of the femur, was written in terms of the nine variables that define such material. Then values of the variables determined by elastic moduli, shear moduli and Poisson’s ratios previously obtained through mechanical testing, were specified. Specifically, 11.6 GPa and 5.5 GPa were used for the respective longitudinal and transverse elastic moduli for extinct birefringence, and 6.3 GPa was used for the longitudinal and transverse elastic moduli. 4 GPa was assigned to the shear modulus and 0.3 to the Poisson’s ratio (Ascenzi and Lomovtsev, 2006; Ascenzi and Bonucci, 1967, 1968, 1972, 1976; Ascenzi et al., 1997; Bonucci, 2000).

Second, a stiffness matrix Q_f that represents the locally unidirectionally-fibered material, transversely isotropic along the circumferential (axis 2) and longitudinal (axis 3) directions, was written in terms of five parameters, of which three were known (Ascenzi, 1999). The unknown parameters of Q_h and of Q_f were then obtained by solving the system of equations resulting by equating the orthotropic matrix Q_h with the matrix that averages radially the contributions of the conjugate matrices (T_γ)⁻¹Q_f T_γ, where Tγ is the rotation matrix around the radial axis 1 relative to the collagen orientation defined by the angle γ (Vinson, 1993). By so doing, the material properties assigned to the elements of the mFE model by means of the matrix (T_γ)⁻¹Q_f T_γ, in combination, yield overall the homogeneous material properties experimentally observed for single osteons. Here d = element-specific vTMD, co = element-specific collagen orientation with respect to the z-axis within the circumferential-axial reference. The 21 [i,j] entries of the symmetric matrix (T_γ)⁻¹Q_f T_γ are:

[1,1]=10^5*(4.17*d^2+88.57*d+470.01)/(477.00*d+4928.00);
[1,2]=10^5*(1.68*d^2-0.25*d*cos(0.017*co)^2*d+35.71*d-2.53*d*cos(0.017*x)^2-
      2.72*cos(0.017*co)^2*d+189.66-27.47*cos(0.017*co)^2)/(477.00*d+4928.00);
[1,3]=10^6*(1.10*d^2-1.65*cos(0.017*co)^2*d^2+0.97*cos(0.017*co)^4*d^2+22.89*d-
      31.86*cos(0.017*co)^2*d+17.83*cos(0.017*co)^4*d+118.81-
      152.77*cos(0.017*co)^2+80.96*cos(0.017*co)^4)/(477.00*d+4928.00);
[1,4]=10^5*(0.25*d*cos(0.017*co)^2*d+2.53*d*cos(0.017*co)^2*d+2.53*cos(0.017*co)^2*d+
      1.43*d^2+30.47*d+2.72*cos(0.017*co)^2*d+27.47*cos(0.017*co)^2+
      162.19)/(477.00*d+4928.00);
[1,5]=10^5*(9.67*cos(0.017*co)^2*d^2-9.67*cos(0.017*co)^4*d^2+178.30*cos(0.017*co)^2*d-
      178.30*cos(0.017*co)^4*d+1.68*d^2+35.71*d+809.64*cos(0.017*co)^2-
      809.64*cos(0.017*co)^4+189.66)/(477.00*d+4928.00);
[1,6]=10^5*(9.67*cos(0.017*co)^4*d^2+178.30*cos(0.017*co)^4*d-2.82*cos(0.017*co)^2*d^2-
      37.96*cos(0.017*co)^2+809.64*cos(0.017*co)^4-91.54*cos(0.017*co)^2+
      4.17*d^2+88.57*d+470.01)/(477.00*d+4928.00);
[2,2]=10^5*(0.25*sin(0.017*co)*cos(0.017*co)*d^2+ 5.24*sin(0.017*co)*cos(0.017*co)*d+
      27.47*sin(0.017*co)*cos(0.017*co))/(477.00*d+4928.00);
[2,3]=10^5*(8.26*sin(0.017*co)*cos(0.017*co)*d^2-9.67*sin(0.017*co)*cos(0.017*co)^3*d^2+
      159.31*sin(0.017*co)*cos(0.017*co)*d-178.30*sin(0.017*co)*cos(0.017*co)^3*d+
      763.87*sin(0.017*co)*cos(0.017*co)-809.64*sin(0.017*co)*cos(0.017*co)^3)/
      (477.00*d+4928.00);
[2,4]=10^5*(9.67*sin(0.017*co)*cos(0.017*co)^3*d^2+178.30*sin(0.017*co)*cos(0.017*co)^3*d+
      809.64*sin(0.017*co)*cos(0.017*co)^3-1.41*sin(0.017*co)*cos(0.017*co)*d^2-
      18.98*sin(0.017*co)*cos(0.017*co)*d-45.77*sin(0.017*co)*cos(0.017*co))/
      (477.00*d+4928.00);
[2,5]=10^6*(1.93*cos(0.017*co)^2*d^2-1.93*cos(0.017*co)^4*d^2+35.66*d*cos(0.017*co)^2-
      35.66*cos(0.017*co)^4*d+161.93*cos(0.017*co)^2-161.93*cos(0.017*co)^4+0.11*d^2+
      3.39*d+23.46)/(477.00*d+4928.00);
[2,6]=0; [3,3]=0; [3,4]=0; [3,5]=0;
[3,6]=10^2*(2.87*d-0.61*cos(0.017*co)^2*d+31.23+16.37*cos(0.017*co)^2);
[4,4]=0; [4,5]=0; [4,6]=0; [5,5]=0;
[5,6]=10^2*(-0.61*sin(0.017*co)*cos(0.017*co)*d+16.37*sin(0.017*co)*cos(0.017*co));
[6,6]=10^2*(0.61*cos(0.017*co)^2*d-16.37*cos(0.017*co)^2+2.27*d+47.60).

Appendix B

Because there is a significant difference between ND MUT and HFD MUT (1.78±0.02 mm vs. 1.62±0.03 mm, p<0.01), we adjusted the computed ε_zz for HFD MUT with respect to the ND MUT. For 3-point bending, we adjusted the stress σ_zz of the femur (f) with a correction factor (cf) equal to the ratio of the estimated stress due to bending at f’s mid-shaft to that of reference femur (rf):

$$\mathrm{c}\mathrm{f}={(\mathrm{M}\mathrm{d}/\mathrm{I})}_{\mathrm{r}\mathrm{f}}/{(\mathrm{M}\mathrm{d}/\mathrm{I})}_{\mathrm{f}}$$ (2)

where M is the moment of the force component, d is the distance between neutral axis and evaluation point, approximated by (r_i+r_o)/2 where r_i is the inner and r_o is the outer radius of the transverse section, and I is the second areal moments of inertia of the transverse section, π(r_o⁴-r_i⁴)/4. Because the bending moment is the same for all femora, equation (1) simplifies to:

$$\mathrm{c}\mathrm{f}={({\mathrm{r}}_{\mathrm{i}}+{\mathrm{r}}_{\mathrm{o}}/\mathrm{I})}_{\mathrm{r}\mathrm{f}}/{({\mathrm{r}}_{\mathrm{i}}+{\mathrm{r}}_{\mathrm{o}}/\mathrm{I})}_{\mathrm{f}}.$$ (3)

For physiological loading, we adjusted the stress due to axial compression and the stress due to bending separately. We adjusted the stress due to axial compression by multiplying by a correction factor equal to the ratio of transverse section areas at mid-shaft. We used Eq. (2) to adjust the stress due to bending. In fact, we noted that the bending moments at mid-shaft were not significantly different in magnitude or direction among the groups (p=0.06). If θ denotes the angle between the neutral axis and the y-axis, d=((r_i+r_o)/2)sin(θ) and Eq. (2) reduces to Eq. (3). The adjusted stress components for axial compression and bending were combined to yield the adjusted σ_zz that yielded the adjusted ε_zz by multiplication with compliance matrix. We dismissed adjustment of σ_xy that would alter ε_zz by less than 1%.