Relationship between the shape and density distribution of the femur and its natural frequencies of vibration.

Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Relationship between the shape and density distribution of the femur and its natural frequencies of vibration G. Campoli a, N. Baka b, B.L. Kaptein b, E.R. Valstar a,b, S. Zachow c, H. Weinans a,d, A.A. Zadpoor a,n a Department of Biomechanical Engineering, Faculty of Mechanical, Maritime, and Materials Engineering, Delft University of Technology (TU Delft), Mekelweg 2, Delft 2628 CD, The Netherlands b Biomechanics and Imaging Group, Department of Orthopaedics, Leiden University Medical Center, P.O. Box 9600, 2300 RC Leiden, The Netherlands c Visualization & Data Analysis Medical Planning & Computational Medicine Groups, Zuse Institute Berlin (ZIB), Berlin, Germany d Departments of Orthopedics and Rheumatology, Utrecht University Medical Center, Utrecht, The Netherlands

art ic l e i nf o

a b s t r a c t

Article history: Accepted 8 August 2014

It has been recently suggested that mechanical loads applied at frequencies close to the natural frequencies of bone could enhance bone apposition due to the resonance phenomenon. Other applications of bone modal analysis are also suggested. For the above-mentioned applications, it is important to understand how patient-specific bone shape and density distribution influence the natural frequencies of bones. We used finite element models to study the effects of bone shape and density distribution on the natural frequencies of the femur in free boundary conditions. A statistical shape and appearance model that describes shape and density distribution independently was created, based on a training set of 27 femora. The natural frequencies were then calculated for different shape modes varied around the mean shape while keeping the mean density distribution, for different appearance modes around the mean density distribution while keeping the mean bone shape, and for the 27 training femora. Single shape or appearance modes could cause up to 15% variations in the natural frequencies with certain modes having the greatest impact. For the actual femora, shape and density distribution changed the natural frequencies by up to 38%. First appearance mode that describes the general cortical bone thickness and trabecular bone density had one of the strongest impacts. The first appearance mode could therefore provide a sensitive measure of general bone health and disease progression. Since shape and density could cause large variations in the calculated natural frequencies, patient-specific FE models are needed for accurate estimation of bone natural frequencies. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Femur Bone shape Bone appearance Modal analysis Finite element method

1. Introduction It is well established that bone apposition and resorption are partially controlled by the mechanical loading that bone experiences (Campoli et al., 2012; Carter, 1984, 1987; Grimston et al., 1993; Pearson and Lieberman, 2004; Price et al., 2011; Sugiyama et al., 2010; Turner et al., 1994; Zadpoor et al., 2013). The characteristics of mechanical loading including the frequency and amplitude of dynamic mechanical loads are found to be particularly important for regulation of bone turnover (Hsieh and Turner, 2001; Rubin et al., 2002; Turner, 1998; Werner et al., 2009). It has been recently (Zhao et al., 2014) suggested that dynamic mechanical loads with a frequency close to the bone resonance frequencies could enhance bone apposition. This finding has

n

Corresponding author. Tel.: þ 31 15 2781021; fax: þ31 15 2784717. E-mail address: [email protected] (A.A. Zadpoor).

potentially important practical implications. For example, the exercise regimes (Kerr et al., 1996; Wallace and Cumming, 2000; Wolff et al., 1999) or vibration therapies (Cardinale and Rittweger, 2006; Flieger et al., 1998) designed to increase bone mass and/or to prevent bone mass loss could be optimized such that the musculoskeletal system is loaded with a frequency close to the natural frequencies of the targeted bone. Furthermore, the exercise regimes could be made subject-specific taking account of the natural frequencies of every specific subject. It is therefore important to understand the effects of bone shape and density distribution on the natural frequencies of bones. In addition to regulating the bone regeneration process, modal analysis of bones could be useful in several other applications. It has been shown that the natural frequencies of long bones such as tibia and femur are correlated with their mechanical properties including stiffness and strength (Arpinar et al., 2005; Christensen et al., 1986; Cornelissen et al., 1986, 1987; Lowet et al., 1993; Van der Perre and Lowet, 1996). This correlation has been used for

http://dx.doi.org/10.1016/j.jbiomech.2014.08.008 0021-9290/& 2014 Elsevier Ltd. All rights reserved.

Please cite this article as: Campoli, G., et al., Relationship between the shape and density distribution of the femur and its natural frequencies of vibration. Journal of Biomechanics (2014), http://dx.doi.org/10.1016/j.jbiomech.2014.08.008i

G. Campoli et al. / Journal of Biomechanics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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several purposes including assessment of the stiffness of bones and monitoring of fracture healing (Alizad et al., 2004; Benirschke et al., 1993; Nakatsuchi et al., 1996a, 1996b; Tower et al., 1993; Wong et al., 2013). In a different line of research, the vibration properties of bones such as natural frequencies and natural modes are used for characterizing the orthotropic properties of long bones such as the femur (Taylor et al., 2002) and validating density–elasticity relationships used in finite element modeling of bones (Scholz et al., 2013). Another application of modal analysis of bones is detection of the loosening of fixation plates (Bull et al., 1991) and measuring/monitoring the structural features of the spine (Van Engelen et al., 2011, 2012). In all abovementioned applications, knowledge about the effects of anatomical variations on natural frequencies could be useful particularly to maximize the accuracy of the proposed technique through integration of patient-specific aspects. Introduction of patient-specific aspects could, for example, be performed using patient-specific finite element models (Poelert et al., 2013). In this study, we examine the effects of shape and density distribution on the natural frequencies of the human femur. A statistical shape and appearance model (SSAM) (Sarkalkan et al., 2014) of the femur is created first. This model consists of the mean femur shape and density as well as their modes of variations found in the training set of 27 CT images. The independent effects of shape and density on the natural frequencies are then studied using the main modes of variation in shape and density and finite element (FE) models. In addition, the combined effects of bone shape and density are studied based on CT images of the femurs used for building the SSAM.

2. Methodology 2.1. SSAM SSAM of the femur was generated according to (Baka et al., 2011; Cootes and Taylor, 2004a, 2004b) using 27 CT scans of the lower extremity. These scans were acquired as a part of a clinical CT angiography routine performed in Leiden University Medical Center. The population included 5 women and 22 men in their fifties or sixties. First, a point distribution model (PDM) of the bony surfaces was constructed. For this, the femora were semi-automatically segmented with the support of a 2D snake algorithm. After segmentation, a triangulated mesh represented each training shape such that the mesh points (in total 1687) were in correspondence throughout the shapes. Correspondence was achieved by registering the distance transform of all shapes to the distance transform of a reference shape using B-spline non-rigid transformation and the advanced mean squares distance metric available in the Elastix toolkit (Klein et al., 2010). A subject with a relatively high resolution CT [0.686 mm 0.686 mm 0.8 mm], about midrange age (56), and no obvious pathology was chosen as the reference subject. The triangulation and landmark positions of this reference shape were then transformed back to each training shape. The shapes were subsequently aligned using Procrustes analysis (translation, rotation and isotropic scaling) to eliminate pose variations from the model. Each shape was then represented as a vector of its concatenated landmark coordinates: si ¼ ½xi1 ; yi1 ; zi1 ; xi2 ; yi2 ; …; zin T

ð1Þ

We modeled the point cloud with a Gaussian distribution and calculated the mean shape through averaging and the variance using principal component analysis (PCA). This resulted in a model of the form: s ¼ s þ Φb

ð2Þ

where s denotes a shape vector, s is the mean shape, Φ is a [3nxnp] matrix containing the directions of largest variance in its columns, and b is the [npx1] shape parameter vector. The number of parameters np (26) was defined such that the model contained 99% of the total variance. For the appearance model, the femur shape differences were eliminated in the training set by deforming all CT images to match the reference femur. The abovementioned deformation field was used together with a b-spline interpolator for the required image resampling. Grid sampling of all voxels inside the femur was performed to create the intensity vectors from all femora. PCA was then applied to create the intensity model. The retained variance for the model creation was set to the value of 99% and required 25 modes.

The generated SSAM models were evaluated for compactness, normality, and generalization ability. We performed an Anderson–Darling test to assess normality of the modes (Anderson and Darling, 1954). The leave-one-out test was used to estimate the generalization capability of the SSAM. Compactness requires that most variability present in the population be described by the first few modes of variation. 2.2. FE models Numerical methods such as FE models are often used for calculating the natural frequencies of structures (Azrar et al., 2002; Daya and Potier-Ferry, 2001; Taher et al., 2006). Three types of FE models were created to study the effects of shape and density distribution both separately and in combination with each other. In the first series of FE models, the average density distribution was used for all FE models and only the shape modes were varied around the average shape by up to three standard deviations. Secondly, the average shape was used and the density distribution was varied around the average density distribution by up to three standard deviations. Thirdly, FE models were created for the 27 femora that were used for generation of the SSAM. ZIBAmira 2012.30 (Zuse Institute Berlin (ZIB), Berlin, Germany) was used for generating the geometries and density distributions as well as for mapping the material properties. Tetrahedral elements with linear integration (element type C3D4 in Simulia Abaqus v6.10, Dassault Systèmes) were adopted for all the models generated and analyzed in this study. The ‘effective density’ of the bone was defined such that the minimum Hounsfield Unit (HU) within the bone area corresponded to water mass density, i.e. 1000 kg/m3, and the maximum HU corresponded to the mass density of cortical bone, i.e. 1952 kg/m3 (Snyder and Schneider, 1991). For calculating the ‘apparent density’ of bone, the apparent density corresponding to the minimum HU was considered to equal 0, while the apparent density corresponding to the maximum HU minimum was considered to equal the apparent density of cortical bone. The ‘apparent density’ of bone was related to the elastic modulus of bone using the well-established density–modulus relationship proposed by Morgan et al. (2003). This relationships is shown to be an accurate density–modulus relationship (Schileo et al., 2007) in general and for modal analysis of the femur in particular (Scholz et al., 2013): E ¼ 6:85ρ1:49 app

ð3Þ

where E is Young's modulus (GPa) and ρapp is the apparent density of the bone (g/cm3). Poisson's ratio was assumed to be 0.3 (Turner et al., 1999; Ulrich et al., 1999; Zysset et al., 1999) throughout the bone. A sensitivity analysis showed that the natural frequency in the vast majority of cases changes by less than 1% when Poisson's ratio is varied between 0.2 and 0.4. Finite element modal analysis was then performed based on the Lanczos method within Abaqus to predict the natural frequencies of the femur. Free boundary conditions were used. The convergence of the FE solutions was examined using the mean bone shape and mean density distribution that was discretized at five different level. The choice of the discretization level was based on the ZIB Amira re-meshing module options (Zachow et al., 2007; Zilske et al., 2008). This approach attempts at isotropic vertex placement, and allows for adjusting the percentage of the original surface mesh size that must contain high quality triangles. Once the surface triangulation was completed, volumetric tetrahedral elements were generated using the of ABAQUS element types. Table 1 lists the re-meshing percentage levels as well as the corresponding number of nodes, degrees of freedoms, and elements and distorted elements (in absolute values and percentage on the total amount of elements). The first 10 natural frequencies were then calculated and compared with each other.

3. Results 3.1. SSAM The covariance matrix of the training set is a diagonal matrix with decreasing values. Fig. 1a shows these diagonal values Table 1 Convergence study: re-meshing discretization level, number of tetrahedral elements (type C3D4), number of nodes, number of global Degrees of Freedom (DoF), number of distorted elements, and the percentage of distorted elements. Discretization level ZIB AMIRA %

Elements (C3D4) Nr.

Nodes

1 5 10 20 30

13,285 118,878 288,365 681,309 1,124,628

3234 9702 24,185 72,555 56,361 169,083 129,151 387,453 209,972 629,916

Nr.

Degrees of Freedom Nr.

Distorted elements Nr.

Distorted elements %

238 802 1601 3793 6107

1.79 0.67 0.56 0.56 0.54



3

Fig. 1. The shape variance explained by different shape modes (a), the first three shape modes (b), histogram representations of normality test results for mode 12 (c) and generalization error (d) as well as the spatial distribution of the generalization error (e).


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representing the variance per mode. This plot gives an impression of the compactness of the model. The more variance is contained in the first modes of the model, the better the expected separation between real shape variability and noise. When 16 modes were included (np ¼16), the model could describe 95% of the total variance (Fig. 1a). The number of modes required to describe 99% of total shape variance was 25. The first three modes of shape

variation were respectively found to be length/width difference, femoral neck length and torsion (AP femoral head angle), and shaft bending (Fig. 1b). It is important not to generate shapes that are implausible. The way we check for this is to see if all directions in the space of mode shapes are following the expected Gaussian distribution. All modes, except mode 12, failed to reject the normal hypothesis at the 5% level test, meaning that only mode 12 showed

Fig. 2. The appearance variance explained by different appearance modes (a), the first appearance modes (b), histogram representations of normality test results for mode 10 (c) and generalization error (d).



significant deviation from the assumption of having a Gaussian distribution. As for mode 12, the histogram seems to be uniform rather than Gaussian (Fig. 1c). The leave-one-out test showed a median point-to-point accuracy of 1.05 mm (Fig. 1d). Errors were not uniformly distributed on the shape, the largest errors were found in the linea aspera (posterior shaft) and the anterior intertrochanteric area (Fig. 1e). For the appearance model, the estimated number of modes to describe 95% of the variance is 22 (Fig. 2a). The first mode of appearance mainly affects the overall cortical bone thickness and trabecular bone density (Fig. 2b). All modes, except for mode 10 (Fig. 2c), failed to reject the normal hypothesis at the 5% level test. The leave-one-out experiments with 99% retained variance in the models showed a median intensity difference of 2 HU, with a 25 percentile at 64 HU and a 75 percentile at 61 HU (Fig. 2d). The

Fig. 3. Convergence of the calculated natural frequencies as the degrees of freedom of the FE model increased.

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few extreme values in the histogram were most probably caused by registration errors during model creation (Fig. 2d). 3.2. FE models The first 10 natural frequencies (NF) calculated using the two finest levels of discretization were not more than 1–2 Hz (o1%) different from each other. The second finest mesh with 387,453 degrees of freedom was therefore used for the rest of the modal analyses reported throughout the study (Fig. 3). The first five natural frequencies were mostly composed of bending and/or torsion modes (Fig. 4). While first and fourth modes were almost purely bending modes in the frontal plane, third and fifth modes show torsional modes of vibration (Fig. 4). The second mode was a bending mode in the sagittal plane (Fig. 4). The first ten natural frequencies of the bone were between 238 and 2376 Hz when mean bone shape and mean density distributions were used (Table 2). Varying the femoral shape from the mean shape by up to three standard deviations of first ten shape modes resulted in 7–16% variation in the magnitude of the first ten natural frequencies of vibration (Table 3). The first shape mode (SM) had limited influence on the natural frequencies (Fig. 5) and is mainly representative of the width/length ratio. Mode shapes 2–4 caused the largest variations in the natural frequencies of the bone (Fig. 5). Single appearance modes when varied up to three standard deviations around the mean appearance changed the first natural frequencies of vibration by 6–14% (Table 3). The first appearance mode, mainly describing the overall cortical bone thickness and trabecular bone density, had by far the largest influence on the natural frequencies of the femur particularly when higher modes of natural vibration were considered (Fig. 6). In general, the

Fig. 4. The first five natural modes of vibration when the mean shape and density distribution were used (right femur). Upper figures show the frontal plane view while bottom figures show the saggital plane view.



6

Table 2 First ten natural frequencies (NF) calculated for the mean bone shape and mean density distribution. NF 1

NF 2

NF 3

NF 4

NF 5

NF 6

NF 7

NF 8

NF 9

NF 10

238 Hz

261 Hz

558 Hz

742 Hz

782 Hz

1404 Hz

1483 Hz

1927 Hz

2294 Hz

2376 Hz

Table 3 The variations caused by shape and density distribution in the first ten natural frequencies of the femur. The reference values (Ref) refer to the values calculated for the mean shape and mean density distribution when SSAM are used. When actual femora are used, the reference values (Ref) refer to the mean values calculated for all actual femora (Table 4). Shape

NF 1 NF 2 NF 3 NF 4 NF 5 NF 6 NF 7 NF 8 NF 9 NF 10

Appearance

Ref

Min (Hz)

Max

(Max Min) 100/Ref (%)

238 261 558 742 782 1404 1483 1927 2294 2376

217 241 501 701 722 1322 1396 1840 2168 2289

252 273 588 758 823 1444 1546 1969 2347 2441

15 12 16 8 13 9 10 7 8 6

Actual femora

Ref

Min (Hz)

Max


238 261 558 742 782 1404 1483 1927 2294 2376

226 249 520 696 730 1311 1374 1754 2127 2201

241 264 575 759 807 1444 1538 2027 2374 2462

7 6 10 8 10 10 11 14 11 11

relative influence of other appearance modes tended to decrease as the higher modes of natural vibration were considered (Fig. 6). In particular, when compared with the first mode of appearance, the higher modes of appearance seemed to have very small impact on the fifth natural frequency and above (Fig. 6). The mean values of the first ten natural frequencies calculated for the 27 training femora varied between 234 and 2369 Hz (Table 4). Interestingly, the standard deviation of natural frequencies across all SSAM generated femora were within 7–8% of their corresponding mean values for the first ten modes of natural vibration (Table 4). For the 27 training femora, the calculated natural frequencies were between 25 and 38% different from their corresponding mean values Fig. 7 (Table 3).

4. Discussion Based on the results presented in the previous section, it is clear that both femoral shape and appearance significantly influence the natural frequencies of vibration of the femur. Single modes of shape variation as well as single modes of appearance (density) variation result in as much as E15% changes in the calculated values of natural frequencies (Table 3). When combined together, as in the case of the training femora, shape and density distribution could change the natural frequencies of vibration by up to 38% (Table 3). One could therefore conclude that patientspecificity plays an important role when trying to determine the resonance frequencies of the femur. Consequently, it is not be possible to use one single value as the resonance frequency of the femur when one tries maximizing bone apposition by stimulating the bone at its resonance frequency as suggested in (Zhao et al., 2014). Patient-specific FE models could be used for estimating the patient-specific natural frequencies of the femur. Statistical models of shape and appearance similar to the one developed in this study could play important roles in facilitating the generation of patientspecific FE models of different upper-extremity (Campoli et al., 2013, 2014) and lower-extremity (Fernandez and Hunter, 2005; Trabelsi et al., 2011) bones. For example, patient-specific FE models could be generated from sparse imaging data such as 2D images when SSAM of the involved bones is available (Zadpoor, 2013). Moreover, the natural frequencies of different bones could

Ref

Min (Hz)

Max


234 261 550 729 782 1381 1481 1903 2256 2369

195 227 445 640 668 1196 1301 1625 1940 2062

277 309 654 826 923 1630 1735 2109 2682 2795

35 31 38 25 33 31 29 25 33 31

be pre-calculated for different combinations of shape and appearance modes (Zadpoor, 2013). A look-up-table approach could then be used for estimating the natural frequencies of every new femur by relating its shape and appearance modes to the ones precalculated and already available in our database. The obtained results have implications in terms of other applications of bone modal analysis too. For example, the first appearance mode that chiefly describes the cortical bone thickness and trabecular bone density (Fig. 2) is by far the most dominant appearance mode when considering the effects of bone density distribution on the natural frequencies of the femur. Since cortical bone thickness and trabecular bone density are closely related to osteoporosis progression (Dalzell et al., 2009; Rüegsegger et al., 1991), it might be possible to use the first appearance mode together with its associated natural frequencies as surrogate measures of osteoporosis progression. Such an approach is particularly useful when follow-up measurements are performed on the same patient to monitor the disease progression. As the femoral shape of adult osteoporotic patients is not likely to drastically change over time, the changes in the natural frequencies of the bone can be attributed to changes in bone appearance (density). The results presented here suggest that the measured natural frequencies are likely to be very sensitive to changes in cortical bone density and trabecular bone density, as these are essentially described by the first appearance mode (Fig. 6). Since the first appearance mode is particularly dominating the changes in the natural frequencies for the higher natural modes of vibrations, it might be useful to consider also the higher modes of vibration when analyzing the changes in the vibration response of the bone. It should be, however, noted that the higher modes of vibrations die-off much faster due to the damping effects of the soft tissues surrounding the femur. More sensitive probes and more sophisticated signal processing techniques may therefore be needed when extracting the contribution of higher modes from frequency response functions. As opposed to appearance modes where one single mode, i.e. mode 1, clearly has the largest influence on natural frequencies, no such single shape mode could be found. It is therefore difficult to draw qualitative conclusions regarding the shape modes that influence the natural frequencies of the femur the most. The natural frequencies calculated in this study are in the range of experimentally observed natural frequencies of the femur



NF6

Frequency [Hz]

NF1 255

SM1

250

SM2

245

SM3

240

SM4

235

SM5

1400

230

SM6

1380

SM7

1360

225 220

-2

-1

210

1460 1440 1420

SM8

215 -3

1340

SM9 0

1

2

3

1320

SM10

Deviation from the mean (xSD)

1300 -3

-2

-1

275

1560

270

1540

265

1520

1

2

3

-3

-2

-1

-1

2

3

1

2

3

1920 1900 1880 1860 1840 0

1

2

3

-3

-2

-1

1820

2370

760

2340

750

2310

740

2280

730

2250

720

2220

710

2190

700

2160 0

0

NF9

770

1

2

3

-3

-2

-1

2130

0

NF10 2460 2440

820

2420

800

2400

780

2380

760

2360 2340

740

2320

720

2300

700 -1

1

1940

NF5

-2

3

1960

840

-3

2

1980

NF4

-2

0

NF8

NF3

-3

1

1380 0

690

3

1400

235

-1

2

1420

240

-2

1

1440

245

-3

3

1460

250

600 590 580 570 560 550 540 530 520 510 500 490

2

1480

255

-1

1

1500

260

-2

0

NF7

NF2

-3

7

0

1

2

3

-3

-2

-1

2280

0

Fig. 5. The first ten natural frequencies (NF) when different shape modes were varied up to three standard deviations around the mean shape. The mean density distribution was used in all cases.


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Fig. 6. The first ten natural frequencies (NF) when different appearance modes were varied up to three standard deviations around the mean appearance. The mean shape was used in all cases.



9

Table 4 The mean and standard deviation (SD) of the first ten natural frequencies of the actual femora used for creating the SSAM. The same values are presented also for the femora that were created by varying single modes of shape or appearance around the mean femur (in the SSAM).

CT: mean (Hz) SD (Hz) SD/mean 100 (%) Shape modes: mean (Hz) SD (Hz) SD/mean 100 (%) Appearance modes: mean (Hz) SD (Hz) SD/mean 100 (%)

3000

NF 1

NF 2

NF 3

NF 4

NF 5

NF 6

NF 7

NF 8

NF 9

NF 10

234 19 8 235 6 3 235 3 1

261 19 7 259 6 2 258 3 1

550 45 8 551 19 3 552 7 1

729 50 7 733 12 2 733 9 1

782 57 7 774 19 2 773 10 1

1381 96 7 1388 23 2 1388 17 1

1481 103 7 1470 27 2 1467 20 1

1903 125 7 1907 27 1 1904 32 2

2256 155 7 2266 36 2 2268 31 1

2369 165 7 2357 30 1 2349 31 1

Natural Frequencies of the femurs from CT

Frequency [Hz]

2500 2000 1500 1000

0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

500

Femurs

Fig. 7. The first ten natural frequencies (NF) of the 27 actual femora used for creating the SSAM.

(Couteau et al., 1998; Huang et al., 2012; Khalil et al., 1981; Taylor et al., 2002). It is, nevertheless, important to corroborate the presented SSAM and FE models in the future. As for the SSAM, the study of its generalization capability (Figs. 1d, e and 2d) shows that the shape and density distribution of new femora could be described with and average shape error of 1.05 mm (Fig. 1e), provided that the femora are not vastly different from the ones used in generation of the SSAM. For some areas of the femur, the shape errors are larger, up to around 1.6 mm (Fig. 1e). As for the calculated natural frequencies, they are similar to the natural frequencies measured experimentally (Scholz et al., 2013). This study has certain limitations that need to be rectified in future studies. In addition to the general limitations of patientspecific FE models discussed before (Campoli et al., 2012), the effects of soft tissues (Cornelissen et al., 1986; Tsuchikane et al., 1995) on the natural frequencies of vibration need to be taken into account if the vibration analysis is to be performed in vivo. Moreover, only free-free boundary conditions were considered here. Free–free boundary conditions can only be realized ex vivo as in (Taylor et al., 2002). The boundary conditions experienced by the bones in vivo need to be implemented in order to use patientspecific FE models for estimating the natural frequencies of the femur and other bones in vivo. In addition, the shape and density distribution are described independently in the SSAM presented here. In principle, it is possible to extract the combined shape and appearance modes of the femur provided that a large enough dataset with sufficient number of actual femora is available. It is suggested that such a SSAM be created in the future studies to facilitate the application of SSAM in generation of patient-specific FE models. Finally, bone is known to be anisotropic at different scales (Daugschies et al., 2014; Katz, 1980; Reisinger et al., 2011). The current SSAM does not captures bone anisotropy and the FE models used in this study are all isotropic. In principle, it is possible to incorporate anisotropy into SSAM, but microstructural

information and, thus, high-resolution images may be needed for that. High-resolution images could also enable further developments. For example, it might be possible to obtain appearance modes that specific to cortical and trabecular bone structures. In summary, the effects of bone shape and density distribution on the natural frequencies of the femur were studied using an SSAM and FE models of the femur. Single modes of shape and appearance were found to cause as much as E15% variation in calculated natural frequencies. When combined together as in the case of actual femora, shape and density distribution could change the natural frequencies of the femur by up to 38%. Specific shape modes (modes 2–4) and one specific appearance mode (mode 1) influenced the calculated natural frequencies to the greatest extent. It is concluded that the variations caused by shape and density distributions are large enough to warrant application of patient-specific FE models for estimating the natural frequencies of the femur in most applications of bone modal analysis.

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