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Finite element modelling of the vibrational behaviour of the human femur using CT-based individualized geometrical and material properties
Journal of Biomechanics 31 (1998) 383—386
Technical Note
Finite element modelling of the vibrational behaviour of the human femur using CT-based individualized geometrical and material properties Be´atrice Couteau!, Marie-Christine Hobatho!,*, Robert Darmana!, Jean-Claude Brignola", Jean-Yves Arlaud" !INSERM U305, Biome& canique, Centre Hospitalier HoL tel Dieu, 31052 Toulouse, France "EUROS, ZE Athelia III, 13600 La Ciotat, France Received in final form 8 January 1998
Abstract Frequency analysis of long bones has been investigated as a tool to assess bone quality or integrity. The objective of the present paper was to develop a three-dimensional finite element model of a fresh human femur with geometrical and mechanical properties derived from quantitative computer tomography images. This model was then exercised and the results were compared to those obtained from a vibration analysis technique. The percent relative error between the numerically and experimentally derived results was found about 4%. Finally, the influence of mechanical properties on the resonant spectre was studied. The results exhibit the limitations of the vibrational technique to detect slight material changes ( 1998 Published by Elsevier Science Ltd. All rights reserved. Keywords: Vibration; Modal analysis; Finite element model; Femur; CT number
1. Introduction The frequency analysis of long bones was investigated to characterize clinically osteoporosis, healing of fractured bones, or process of osteointegration for plates or screws. Main studies of vibrational analysis on the human tibia in vitro (Doherty et al., 1974; Orne et al., 1976; Hobatho et al., 1991) and in vivo (Van der Perre et al., 1992) had been reported. They all found that resonant frequency was a less-sensitive indicator of changes in state than global mass or stiffness. Saha et al. (1977) recorded vibration response of various long bones to the impact of an instrumented hammer. In their study, the effect of soft tissues was evaluated. Experiments with volunteers indicated the difficulty to overcome the influence of soft tissues without involving pain. Only a few studies have been reported on the vibrational analysis of human femur. Campbell et al. (1970)
* Corresponding author. Tel.: (33) 61 77 82 84; fax: (33) 61 59 46 36.
reported an experimental method to measure the in vitro impedance of the femur. Their results showed a capability to estimate the mechanical integrity of the bone. By using experimental and analytical analysis, Khalil et al. (1981) have quantified bending, torsional and longitudinal resonant frequencies. Thomas et al. (1991) observed the response of the human femur to mechanical vibration with fixed end. End loadings and damping seemed to affect the resonant frequencies. The objective of the present paper was to develop a three-dimensional finite element model of a fresh human femur with geometrical and mechanical properties derived from quantitative computed tomography images. The global validation of the model was assessed by means of a vibration analysis technique. This step allowed to validate the modelling method based on CT images. Then the influence of the mechanical properties on the modal parameters was studied. In doing so we emphasized the limit of the vibrational technique to accurately measure the bone quality or integrity. Our long-term goal consisted in finding a criterion capable of quantifying the individual bone strength. So we have had to study some validation techniques more
0021-9290/98/$19.00 ( 1998 Published by Elsevier Science Ltd. All rights reserved. PII S0021-9290(98)00018-9
384
B. Couteau et al. / Journal of Biomechanics 31 (1998) 383—386
sensitive to the materials properties than the vibrational technique.
2. Material and method 2.1. Experimental analysis An experimental modal analysis of a right fresh human femur specimen from a male subject, 70 yr old, devoided of apparent pathology was performed. The experimental modal analysis consisted of the identification of modal parameters. The modal parameters were: f f f
the resonant frequencies, Fk (Hz), the damping ratio fk deduced from the 3 dB bandwidth of the resonant peaks, the mode shapes.
Vibrational measurements were performed using the Impulse frequency response technique (Brue¨l and Kjae¨r, 1980) as described by Hobatho et al. (1991). The free-free boundary conditions were obtained using elastic straps at the femoral neck and at the femoral medial condyles. The hammer impact gave the input signal p(t), and the accelerometer provided the output signal a(t). Then, a FFT program analysis calculated the transfer function and other derived functions describing the dynamic behavior of the system. The mode shapes were identified by performing excitation at several points along the length of the femur. The points where the resonant frequency peaks amplitudes were measured were located according to the two vibrational planes. 2.2. Numerical analysis Transverse CT images (Siemens, DRH2) of the femur were performed on the cadaveric specimen. Acquisition parameters for the acceleration voltage and intensity dose were 125 kV, 210 mA s, respectively. Four millimeters thick slices were performed at 4 mm interval for the epiphyses and at 20 mm interval for the diaphyseal region. The acquisition matrix was 256]256 for a field of view of 146.3]146.3 mm. Then, CT images data were transferred to a workstation Indy (Silicon Graphics, Mountain View, CA, USA) by an interface software SIP305 developed in our laboratory. Each image was subjected to an edge detection to separate bone contour lines. The output file of the image processing was a neutral file in an IGES format containing the internal and external contours of the cortical bone. Then the 3D surface model was read via Patran3 Software (MSC Nastran, Los Angeles, CA, USA). The mesh generation gave a finite element model with 3572 3D (hexahedrics and wedges) and 2D (shell) elements and 4068 nodes distributed in different types of element properties.
The mechanical properties of the cortical bone were chosen linear homogeneous and isotropic. Density was 1996 Kg m~3 and Young modulus was 16 700 MPa. Poisson’s ratio was identical for all bone structure and equal to 0.3 taken from literature (Hobatho et al., 1991). The properties of the spongious bone were CT scan datas based. A previously work had established the correlation between the CT number (Hounsfield Unity) and the mechanical properties. This task has been inspired by Rho et al. (1995) study. The transverse CT slices that we used constituted the major difference compared to their protocol. Moreover we considered the axial (superior—inferior direction) modulus E3 available for the two other directions, i.e. the medial-lateral and the posterior-anterior direction. We used our own predictive relationships. This allowed us to use the CT images and the predictive relationships performed on the same CT scan. The predictive relationships concerning the proximal spongious bone were o"1.5CT#17.2 with R2"0.80,
(1)
E3"3.8CT-102 with R2"0.57,
(2)
and for the distal spongious bone, o"1.2CT#303.4 with R2"0.74,
(3)
E3"2.6CT#607.7 with R2"0.59.
(4)
So the measurement of the CT number of regions of interest defined on CT scan slices allowed the threedimensional cartography of the heterogeneity of the trabecular bone to be obtained. In a first case, the proximal and distal spongious materials were represented by two different homogeneous isotropic properties which were issued from the mean value of the CT numbers. According to Eqs. (2) and (4) the proximal Young’s modulus was found to have a value of 609 MPa and the distal one of 1112 MPa. Spongious density was 303 kg m~3 for the proximal region and 540 kg m~3 for the distal one. In a second case, the inhomogeneity of the cancellous bone was approximated. The proximal and distal spongious materials were represented by two different sets of 21 isotropic materials. To simulate the mass marrow, we had to add an epiphyseal non-structural mass. Therefore, the mass and center of gravity were modeled with an error less than 3%. Free—free experimental conditions were simulated by an isostatic load case without applying any force. Numerical resonant frequencies were determined by the Abaqus solver (MSC NASTRAN, Los Angeles, USA). 3. Results 3.1. Experimental analysis The resonant spectrum graphs were shown in Fig. 1. The resonant frequencies F1"301.6 Hz and
B. Couteau et al. / Journal of Biomechanics 31 (1998) 383—386
385
Fig. 1. Resonant spectra graphs of the mobility amplitude in the first plane of vibration (a) and second one (b).
4. Discussion
Fig. 2. Mode shapes of the two first flexural mode. F1 and F2 were in the first plane of vibration, and F@1 and F@2 were in the second one.
F2"886.6 Hz (Fig. 1a) were recorded in a first vibration plane while F@1"353.3 Hz and F@2"931.9 Hz (Fig. 1b) were corresponding to a second plane. The damping ratio was around 5% for all frequencies. The mode shapes were illustrated in Fig. 2. They showed two flexural modes in the two planes. The experimental resonant frequencies were reproducible too since the standard deviation did not exceed 4.7 Hz. 3.2. Numerical analysis In the two cases of spongious materials, the calculated natural resonant frequencies and the mode shapes were found close to the experimental results (Table 1). Two first flexural modes were found in the first plane and two others were found in the second plane of inertias. The mean relative error between the experimental and numerical resonant frequencies was about 4% whatever the type of cancellous materials.
Our results demonstrated that the experimental resonant frequencies were two flexural modes in two different planes of vibration. The first plane of vibration corresponded to the minimum bending stiffness (EI)min, at few degrees from the latero-medial plane and slightly twisted near extremities. The second plane, appearing at 90 from the first one, corresponded to the maximum bending stiffness (EI)max. This was in agreement with the results found by Khalil et al. (1981) and described in the Table 2. Our frequency values were 1.2 times higher than Khalil’s values. Differences between Khalil’s results and our results were certainly inherent to the different experimental conditions. Examination of the 3 dB bandwidth of the resonant peaks indicated rather low damping ratios (1.6—8.8%). According to Bru¨el and Kjae¨r (1980), this method allowed to compare the predicted natural frequencies with the damped frequencies. Also, damping ratios in the order of 5% were in agreement with Hobatho et al. (1989) remarks: damping ratios measured on dry tibia were less than 5%, whereas damping ratios relative to fresh tibia were more than 5%. It appeared that our experimental values and those published in the literature are in good agreement. According to Table 1, the heterogeneity of the spongious tissue did not improve the numerical model. This was predictable from remarks of Thomsen J.J. (1990), who concluded that cancellous bone rigidity and pretwist of bone had a minor role on resonant frequency. They found that cancellous bone mass, marrow, shear displacements and section properties had a major role on the resonant spectrum. Now our 3D model with homogeneous bone materials already integrated those preponderant factors therefore it was difficult to improve it by
386
B. Couteau et al. / Journal of Biomechanics 31 (1998) 383—386
Table 1 Experimental and numerical resonant frequencies with relative errors and description of the mode shapes First properties case
Second properties case
F %91 (Hz)
F /6. (Hz)
Relative error(%)
F /6. (Hz)
Relative error(%)
Mode shape
301.6 353.3 886.6 931.6
287.8 364.2 819.0 931.9
4.5 3.1 7.6 0.0
287.7 362.9 814.5 923.5
4.6 2.7 8.1 0.8
1st mode in the 1st plane 1st mode in the 2nd plane 2nd mode in the 1st plane 2nd mode in the 2nd plane
Table 2 Comparison of our resonant spectra with Khalil et al. (1981) results F (Hz) %91 Present study
F (Hz) %91 Khalil et al. (1981)
Description
301.6 353.3 612.0 886.6 931.6
250 315 563 825 879
1st minimal flexural mode 2nd minimal flexural mode 1st torsion mode 1st maximal mode 2nd maximal mode
changing only properties distribution of the spongious tissue. Moreover, according to the Euler Bernoulli beam theory, the ratios of frequencies defined by F2/F1 and F@2/F@1 are constant and equal to 2.75 in free-free boundaries conditions. So, our corresponding ratios equal to 2.93 and 2.63 could justify the simple analytical studies of Viano et al. (1986) and Khalil et al. (1981). Finally, we emphasized the limits of the vibration technique in the assessment of the bone quality. However this dynamical study allows to validate globally the development of finite element models using CT based individualized geometrical and material properties. References Bru¨el and Kjae¨r, 1980. Vibration and Shocks Measurements, edited by B&K Edition, pp. 44—45. Campbell, J.N. and Jurist, J.M., 1971. Mechanical impedance of the femur: a preliminary report. Journal of Biomechanics 4, 319—322.
Doherty, W.P., Bovill, E. and Wilson, E., 1974. Evaluation of the use of resonant frequencies to characterize physical properties of human long bones. Journal of Biomechanics 7, 559—561. Hobatho, M.C., 1989. Etude du comportement dynamique du tibia humain. Application a` l’e´valuation de la consolidation osseuse. Ph.D. Thesis, Universite´ Paul Sabatier, Toulouse, France. Hobatho, M.C., Darmana, R., Pastor, P., Barrau, J.J., Laroze, S. and Morucci, J.P., 1991. Development of a three dimensional finite element model of a human tibia using experimental modal analysis. Journal of Biomechanics 24, 371—383. Hobatho, M.C., Rho, J.Y. and Ashman, R.B., 1991. Mechanical atlas of human cortical and cancellous bone. Proceedings of the COMACBME II.2.6 Meeting. K.U. Leuven, Belgium. Khalil, T.B., Viano, D.C. and Taber, L.A., 1981. Vibrational characteristics of the embalmed human femur. Journal of Sound and Vibration 75, 417—436. Orne, D. and Young, D., 1976. The effects of variable mass and geometry, pretwist, shear deformation and rotatory inertia on the resonant frequencies of intact long bones: a finite element model analysis. Journal of Biomechanics 9,763—770. Rho, J.Y., Hobatho, M.C. and Ashman, R.B. 1995. Relations of mechanical properties to density and CT numbers in human bone. Medical Engineering of Physics 17, 347—355. Saha S. and Lakes, R.S., 1977. The effect of soft tissue on wave propagation and vibration tests for determining the in vivo properties of bones. Journal of Biomechanics 10, 393—401. Thomas, A.M.C., Luo, D.Z. and Dunn, J.W., 1991. Response of human femur to mechanical vibration. Journal of Biomedical Engineering 13, 58—60. Thomsen, J.J., 1990. Modeling human tibia structural vibrations. Journal of Biomechanics 23, 215—228. Van Der Perre, G., Van Audekerke, R., Martens, M. and Mulier, J.C., 1983. Identification of the in vivo vibration modes of human tibiae by modal analysis. Journal of Biomechanical Engineering 105, 244—249. Viano, D., Helfenstein, U., Anliker, M. and Regsegger, P., 1976. Elastic properties of cortical bone in female human femurs. Journal of Biomechanics 9, 703—716.
Technical Note
Finite element modelling of the vibrational behaviour of the human femur using CT-based individualized geometrical and material properties Be´atrice Couteau!, Marie-Christine Hobatho!,*, Robert Darmana!, Jean-Claude Brignola", Jean-Yves Arlaud" !INSERM U305, Biome& canique, Centre Hospitalier HoL tel Dieu, 31052 Toulouse, France "EUROS, ZE Athelia III, 13600 La Ciotat, France Received in final form 8 January 1998
Abstract Frequency analysis of long bones has been investigated as a tool to assess bone quality or integrity. The objective of the present paper was to develop a three-dimensional finite element model of a fresh human femur with geometrical and mechanical properties derived from quantitative computer tomography images. This model was then exercised and the results were compared to those obtained from a vibration analysis technique. The percent relative error between the numerically and experimentally derived results was found about 4%. Finally, the influence of mechanical properties on the resonant spectre was studied. The results exhibit the limitations of the vibrational technique to detect slight material changes ( 1998 Published by Elsevier Science Ltd. All rights reserved. Keywords: Vibration; Modal analysis; Finite element model; Femur; CT number
1. Introduction The frequency analysis of long bones was investigated to characterize clinically osteoporosis, healing of fractured bones, or process of osteointegration for plates or screws. Main studies of vibrational analysis on the human tibia in vitro (Doherty et al., 1974; Orne et al., 1976; Hobatho et al., 1991) and in vivo (Van der Perre et al., 1992) had been reported. They all found that resonant frequency was a less-sensitive indicator of changes in state than global mass or stiffness. Saha et al. (1977) recorded vibration response of various long bones to the impact of an instrumented hammer. In their study, the effect of soft tissues was evaluated. Experiments with volunteers indicated the difficulty to overcome the influence of soft tissues without involving pain. Only a few studies have been reported on the vibrational analysis of human femur. Campbell et al. (1970)
* Corresponding author. Tel.: (33) 61 77 82 84; fax: (33) 61 59 46 36.
reported an experimental method to measure the in vitro impedance of the femur. Their results showed a capability to estimate the mechanical integrity of the bone. By using experimental and analytical analysis, Khalil et al. (1981) have quantified bending, torsional and longitudinal resonant frequencies. Thomas et al. (1991) observed the response of the human femur to mechanical vibration with fixed end. End loadings and damping seemed to affect the resonant frequencies. The objective of the present paper was to develop a three-dimensional finite element model of a fresh human femur with geometrical and mechanical properties derived from quantitative computed tomography images. The global validation of the model was assessed by means of a vibration analysis technique. This step allowed to validate the modelling method based on CT images. Then the influence of the mechanical properties on the modal parameters was studied. In doing so we emphasized the limit of the vibrational technique to accurately measure the bone quality or integrity. Our long-term goal consisted in finding a criterion capable of quantifying the individual bone strength. So we have had to study some validation techniques more
0021-9290/98/$19.00 ( 1998 Published by Elsevier Science Ltd. All rights reserved. PII S0021-9290(98)00018-9
384
B. Couteau et al. / Journal of Biomechanics 31 (1998) 383—386
sensitive to the materials properties than the vibrational technique.
2. Material and method 2.1. Experimental analysis An experimental modal analysis of a right fresh human femur specimen from a male subject, 70 yr old, devoided of apparent pathology was performed. The experimental modal analysis consisted of the identification of modal parameters. The modal parameters were: f f f
the resonant frequencies, Fk (Hz), the damping ratio fk deduced from the 3 dB bandwidth of the resonant peaks, the mode shapes.
Vibrational measurements were performed using the Impulse frequency response technique (Brue¨l and Kjae¨r, 1980) as described by Hobatho et al. (1991). The free-free boundary conditions were obtained using elastic straps at the femoral neck and at the femoral medial condyles. The hammer impact gave the input signal p(t), and the accelerometer provided the output signal a(t). Then, a FFT program analysis calculated the transfer function and other derived functions describing the dynamic behavior of the system. The mode shapes were identified by performing excitation at several points along the length of the femur. The points where the resonant frequency peaks amplitudes were measured were located according to the two vibrational planes. 2.2. Numerical analysis Transverse CT images (Siemens, DRH2) of the femur were performed on the cadaveric specimen. Acquisition parameters for the acceleration voltage and intensity dose were 125 kV, 210 mA s, respectively. Four millimeters thick slices were performed at 4 mm interval for the epiphyses and at 20 mm interval for the diaphyseal region. The acquisition matrix was 256]256 for a field of view of 146.3]146.3 mm. Then, CT images data were transferred to a workstation Indy (Silicon Graphics, Mountain View, CA, USA) by an interface software SIP305 developed in our laboratory. Each image was subjected to an edge detection to separate bone contour lines. The output file of the image processing was a neutral file in an IGES format containing the internal and external contours of the cortical bone. Then the 3D surface model was read via Patran3 Software (MSC Nastran, Los Angeles, CA, USA). The mesh generation gave a finite element model with 3572 3D (hexahedrics and wedges) and 2D (shell) elements and 4068 nodes distributed in different types of element properties.
The mechanical properties of the cortical bone were chosen linear homogeneous and isotropic. Density was 1996 Kg m~3 and Young modulus was 16 700 MPa. Poisson’s ratio was identical for all bone structure and equal to 0.3 taken from literature (Hobatho et al., 1991). The properties of the spongious bone were CT scan datas based. A previously work had established the correlation between the CT number (Hounsfield Unity) and the mechanical properties. This task has been inspired by Rho et al. (1995) study. The transverse CT slices that we used constituted the major difference compared to their protocol. Moreover we considered the axial (superior—inferior direction) modulus E3 available for the two other directions, i.e. the medial-lateral and the posterior-anterior direction. We used our own predictive relationships. This allowed us to use the CT images and the predictive relationships performed on the same CT scan. The predictive relationships concerning the proximal spongious bone were o"1.5CT#17.2 with R2"0.80,
(1)
E3"3.8CT-102 with R2"0.57,
(2)
and for the distal spongious bone, o"1.2CT#303.4 with R2"0.74,
(3)
E3"2.6CT#607.7 with R2"0.59.
(4)
So the measurement of the CT number of regions of interest defined on CT scan slices allowed the threedimensional cartography of the heterogeneity of the trabecular bone to be obtained. In a first case, the proximal and distal spongious materials were represented by two different homogeneous isotropic properties which were issued from the mean value of the CT numbers. According to Eqs. (2) and (4) the proximal Young’s modulus was found to have a value of 609 MPa and the distal one of 1112 MPa. Spongious density was 303 kg m~3 for the proximal region and 540 kg m~3 for the distal one. In a second case, the inhomogeneity of the cancellous bone was approximated. The proximal and distal spongious materials were represented by two different sets of 21 isotropic materials. To simulate the mass marrow, we had to add an epiphyseal non-structural mass. Therefore, the mass and center of gravity were modeled with an error less than 3%. Free—free experimental conditions were simulated by an isostatic load case without applying any force. Numerical resonant frequencies were determined by the Abaqus solver (MSC NASTRAN, Los Angeles, USA). 3. Results 3.1. Experimental analysis The resonant spectrum graphs were shown in Fig. 1. The resonant frequencies F1"301.6 Hz and
B. Couteau et al. / Journal of Biomechanics 31 (1998) 383—386
385
Fig. 1. Resonant spectra graphs of the mobility amplitude in the first plane of vibration (a) and second one (b).
4. Discussion
Fig. 2. Mode shapes of the two first flexural mode. F1 and F2 were in the first plane of vibration, and F@1 and F@2 were in the second one.
F2"886.6 Hz (Fig. 1a) were recorded in a first vibration plane while F@1"353.3 Hz and F@2"931.9 Hz (Fig. 1b) were corresponding to a second plane. The damping ratio was around 5% for all frequencies. The mode shapes were illustrated in Fig. 2. They showed two flexural modes in the two planes. The experimental resonant frequencies were reproducible too since the standard deviation did not exceed 4.7 Hz. 3.2. Numerical analysis In the two cases of spongious materials, the calculated natural resonant frequencies and the mode shapes were found close to the experimental results (Table 1). Two first flexural modes were found in the first plane and two others were found in the second plane of inertias. The mean relative error between the experimental and numerical resonant frequencies was about 4% whatever the type of cancellous materials.
Our results demonstrated that the experimental resonant frequencies were two flexural modes in two different planes of vibration. The first plane of vibration corresponded to the minimum bending stiffness (EI)min, at few degrees from the latero-medial plane and slightly twisted near extremities. The second plane, appearing at 90 from the first one, corresponded to the maximum bending stiffness (EI)max. This was in agreement with the results found by Khalil et al. (1981) and described in the Table 2. Our frequency values were 1.2 times higher than Khalil’s values. Differences between Khalil’s results and our results were certainly inherent to the different experimental conditions. Examination of the 3 dB bandwidth of the resonant peaks indicated rather low damping ratios (1.6—8.8%). According to Bru¨el and Kjae¨r (1980), this method allowed to compare the predicted natural frequencies with the damped frequencies. Also, damping ratios in the order of 5% were in agreement with Hobatho et al. (1989) remarks: damping ratios measured on dry tibia were less than 5%, whereas damping ratios relative to fresh tibia were more than 5%. It appeared that our experimental values and those published in the literature are in good agreement. According to Table 1, the heterogeneity of the spongious tissue did not improve the numerical model. This was predictable from remarks of Thomsen J.J. (1990), who concluded that cancellous bone rigidity and pretwist of bone had a minor role on resonant frequency. They found that cancellous bone mass, marrow, shear displacements and section properties had a major role on the resonant spectrum. Now our 3D model with homogeneous bone materials already integrated those preponderant factors therefore it was difficult to improve it by
386
B. Couteau et al. / Journal of Biomechanics 31 (1998) 383—386
Table 1 Experimental and numerical resonant frequencies with relative errors and description of the mode shapes First properties case
Second properties case
F %91 (Hz)
F /6. (Hz)
Relative error(%)
F /6. (Hz)
Relative error(%)
Mode shape
301.6 353.3 886.6 931.6
287.8 364.2 819.0 931.9
4.5 3.1 7.6 0.0
287.7 362.9 814.5 923.5
4.6 2.7 8.1 0.8
1st mode in the 1st plane 1st mode in the 2nd plane 2nd mode in the 1st plane 2nd mode in the 2nd plane
Table 2 Comparison of our resonant spectra with Khalil et al. (1981) results F (Hz) %91 Present study
F (Hz) %91 Khalil et al. (1981)
Description
301.6 353.3 612.0 886.6 931.6
250 315 563 825 879
1st minimal flexural mode 2nd minimal flexural mode 1st torsion mode 1st maximal mode 2nd maximal mode
changing only properties distribution of the spongious tissue. Moreover, according to the Euler Bernoulli beam theory, the ratios of frequencies defined by F2/F1 and F@2/F@1 are constant and equal to 2.75 in free-free boundaries conditions. So, our corresponding ratios equal to 2.93 and 2.63 could justify the simple analytical studies of Viano et al. (1986) and Khalil et al. (1981). Finally, we emphasized the limits of the vibration technique in the assessment of the bone quality. However this dynamical study allows to validate globally the development of finite element models using CT based individualized geometrical and material properties. References Bru¨el and Kjae¨r, 1980. Vibration and Shocks Measurements, edited by B&K Edition, pp. 44—45. Campbell, J.N. and Jurist, J.M., 1971. Mechanical impedance of the femur: a preliminary report. Journal of Biomechanics 4, 319—322.
Doherty, W.P., Bovill, E. and Wilson, E., 1974. Evaluation of the use of resonant frequencies to characterize physical properties of human long bones. Journal of Biomechanics 7, 559—561. Hobatho, M.C., 1989. Etude du comportement dynamique du tibia humain. Application a` l’e´valuation de la consolidation osseuse. Ph.D. Thesis, Universite´ Paul Sabatier, Toulouse, France. Hobatho, M.C., Darmana, R., Pastor, P., Barrau, J.J., Laroze, S. and Morucci, J.P., 1991. Development of a three dimensional finite element model of a human tibia using experimental modal analysis. Journal of Biomechanics 24, 371—383. Hobatho, M.C., Rho, J.Y. and Ashman, R.B., 1991. Mechanical atlas of human cortical and cancellous bone. Proceedings of the COMACBME II.2.6 Meeting. K.U. Leuven, Belgium. Khalil, T.B., Viano, D.C. and Taber, L.A., 1981. Vibrational characteristics of the embalmed human femur. Journal of Sound and Vibration 75, 417—436. Orne, D. and Young, D., 1976. The effects of variable mass and geometry, pretwist, shear deformation and rotatory inertia on the resonant frequencies of intact long bones: a finite element model analysis. Journal of Biomechanics 9,763—770. Rho, J.Y., Hobatho, M.C. and Ashman, R.B. 1995. Relations of mechanical properties to density and CT numbers in human bone. Medical Engineering of Physics 17, 347—355. Saha S. and Lakes, R.S., 1977. The effect of soft tissue on wave propagation and vibration tests for determining the in vivo properties of bones. Journal of Biomechanics 10, 393—401. Thomas, A.M.C., Luo, D.Z. and Dunn, J.W., 1991. Response of human femur to mechanical vibration. Journal of Biomedical Engineering 13, 58—60. Thomsen, J.J., 1990. Modeling human tibia structural vibrations. Journal of Biomechanics 23, 215—228. Van Der Perre, G., Van Audekerke, R., Martens, M. and Mulier, J.C., 1983. Identification of the in vivo vibration modes of human tibiae by modal analysis. Journal of Biomechanical Engineering 105, 244—249. Viano, D., Helfenstein, U., Anliker, M. and Regsegger, P., 1976. Elastic properties of cortical bone in female human femurs. Journal of Biomechanics 9, 703—716.