# Bone geometry and mechanical properties of the human scapula using computed tomography data.

Abstract

Mechanical properties of scapular trabecular bone are assumed to be similar to those of other trabecular bone of different anatomical regions, like tibia, femur, humerus. The goal of this study was to develop a technique that may be useful to detect bone geometry and pixel gray value from contour data of CT-scan slice of the scapula. In this paper an attempt has been made to relate quantitative Computed Tomography (CT) gray values with apparent density, and apparent density with elastic modulus. A contour detection algorithm has been developed that finds the optimised bone contour by connecting the points with high derivatives of CT gray value on lines perpendicular to an initial contour. The number of points in a contour were stored as keypoints which were useful for generating a three-dimensional model of the scapula. A linear regression, generalised for all CT-scan slices defining the whole scapula, was derived from two reference points (one nobone condition, i.e. air, another cortical bone). Based on structural and analytical models of trabecular bone, power law relations were fitted for two ranges of apparent density. Powers of 2 and 3 (E ~ [rho]2, E ~ [rho]3) have been used for open cell rod-like structure and closed cell plate-like structure, respectively. The transition from open to closed structure was assumed to occur at an apparent density of 350 kg [m.sup.-3]. The theoretical relationships were fitted to experimental data of glenoid cancellous bone specimens. The above-mentioned relationships for scapular trabecular bone are meant to be used for a finite element model of a scapula, with or without an implant, based on CT-scan images.

Keywords : Shoulder, glenoid, bone geometry, apparent density, elastic modulus, finite element model.

Introduction :

Characterisation of mechanical properties of trabecular bone is essential for the design of prosthesis. Finite Element (FE) models of glenoid component were mostly based on assumptions that the material properties of the glenoid were similar to those of the tibial plateau [1,2]. Research directed towards investigating the causes of mechanical loosening of the glenoid component and its improvement in design, calls for reliable data on mechanical properties of scapular trabecular bone.

A unique quantitative relationship between the apparent density and the elastic modulus is necessary in order to assign material properties to each element of the FE model of the scapula. A relation between quantitative Computed Tomography (CT) gray values of pixels in Hounsfield units and the apparent density of bone is of prime importance to start with. McBroom et al. [3] found a good correlation (R2 =0.89, p<0.0001) and a linear relationship between the CT gray values and apparent density of vertebral trabecular bone. Bentzen et al. [4] and Hvid et al. [5] reported a correlation coefficient of 0.93 and established a linear relationship between CT gray values and apparent density of tibial trabecular bone. A statistically significant linear relation between CT gray values and apparent density (R2=0.82) was observed by Ciarelli et al. [6] for human trabecular bone from major metaphyseal regions, like tibia, femur, radius and humerus.

Although none of the existing reports predicts a relation between CT gray value and apparent density for scapular trabecular bone, a similar linear relationship can be expected for scapula trabecular bone. Many authors have reported trabecular bone mechanical properties and relationships between the apparent density and bone stiffness [7-11]. The results of these studies predicted linear as well as power law relationships. The variation was large and was primarily due to differences in anatomic location, anisotropy of material properties, direction of testing and testing protocol [6,7,10,12]. Currey [13], while criticising Hvid et al. [5], preferred power law models to linear regression models for mechanical properties of cancellous bone. He argued that there are three advantages: (1) theoretically, a power law relation is predicted, which was confirmed by experiments; (2) the equation will produce marked improvements towards extrapolation to compact bone on one side and non-bone on the other; (3) the distribution of the residuals will be improved for power law model as compared to linear model.

The goal of this study was to develop a technique that may be useful to detect bone geometry and pixel gray value from contour data of CT-scan slice of the scapula. The pixel gray value can be calibrated in terms of bone density, which can be in turn related to bone elastic modulus and strength. Using the experimental data of density and elastic modulus for scapular bone [14], the purpose of this study was to formulate a relationship between CT gray value (H) in Hounsfield Units and apparent density ([rho]), and between apparent density ([rho]) and elastic modulus (E), by regression analyses.

Materials and Method :

The CT-scan images of the scapula were stored in 512X512 pixels, with a pixel size of 0.5 mm X 0.5 mm and slice thickness of 3.0 mm in Digital Imaging and Communications in Medicine (DICOM) format. A single DICOM file contains a header that stores information about the patient's names the type of scan, image dimensions, and the image data, which contains information in three dimensions. A total number of 69 slices was obtained to define the whole scapula. The CT images were used for subsequent image processing and analysis. CT gray values in Hounsfield units corresponding to specific positions within the scanned image were obtained. The range of the CT gray value of bone (trabecular and compact) is between -1200 (i.e. nonbone) and 2895 (i.e. cortical bone).

In another study, Frich [14] reported experimental data of apparent density and elastic modulus of trabecular bone samples taken from twenty-five fresh human scapulae. The bone samples, one from each scapula, had all been taken between 3 and 11 mm from the articular surface at the "bare areas" (i.e. approximately in the middle of the articular surface). The cylindrical bone samples 8 mm long and 6.5 mm in diameter, were taken from an area, just below the subchondral plate of the glenoid. They were subjected to compression test in order to obtain stress-strain curves. The elastic modulus was determined as the maximum slope of the stress-strain curve in compression up to 1.5% strain. The strain rate of the bone cylinder compression test was 0.1% per second. After these tests the bone samples were cleaned with compressed air to remove bone marrow, degreased in alcohol and acetone, again cleaned with compressed air, and allowed to dry. Apparent densities of the samples were calculated from measurement of weight and volume.

2.1 Image Smoothing and Edge Detection

Image smoothing was used primarily for diminishing spurious effects that may be present in a digital image due poor sampling system or transmission channel. Smoothing is a process by which data points are averaged with their neighbours in a series, such as a time series, or image. This usually has the effect of blurring the sharp edges in the smoothed data. Smoothing is sometimes referred to as filtering, because smoothing has the effect of suppressing high frequency signal and enhancing low frequency signal. In this study we have adopted boxcar average of specified width.

Edge detection is often the first step in image processing since a lot of information can be obtaind from the edges. In this study the Sobel and the Roberts algorithm have been used for edge detection. The Sobel operator performs a 2-D spatial gradient measurement on an image. Both Sobel and Roberts operators emphasises regions of high spatial frequency that corresponds to edges. Typically it is used to find the approximate absolute gradient magnitude at each point in an input gray-scale image. At least, the operator consists of a pair of 3x3 convolution kernels. The operator consists of a pair of 2x2 convolution kernels. The Sobel operator therefore resulted in better accuracy, but required more computing time as compared to Roberts operator.

As a first step a Graphical User Interface (GUI) was developed to view the DICOM image as shown in Fig. 1. Interactive Data Language (IDL, developed by Research Systems, Inc., USA) has been used as computing environment for visualisation of data and interactive analysis. IDL integrates a powerful, array-oriented language with numerous mathematical analysis and graphical display techniques. The header information was stored separately in a text file for future reference. We trapped the mouse movement and obtained the position co-ordinate of the mouse pointer with respect to the normalised co-ordinate and retrieved the Image value from image matrix. This pixel value was shown simultaneously with the mouse movement in the text box in GUI.

[FIGURE 1 OMITTED]

2.2 Bone Geometry

A semiautomatic contour detection algorithm, based on a mathematical optimisation procedure, was used to find the closed contours in each CT-slice [15,16]. This procedure requires an initial approximate contour (about 10 points per slice) generated manually. The contour detection algorithm optimises the contour by connecting the points with high derivatives of CT gray value on lines perpendicular to the initial contour [16]. The procedure was repeated for all the consecutive CT slices. This algorithm was developed to make the procedure more efficient and accurate. The number of points in a contour was reduced automatically by selecting the keypoints, so that the error between the modified contour and the original points defining the contour remained less than 0.5 mm.

2.3 Statistical analysis

Regression curves are fitted to observations in order to reduce the relationship to a parametric equation. The correlation coefficient (R) and the standard errors (SE) quantify the "goodness of fit" of the dependent variable. The power law relationship, given by E = a [[rho].sup.b], (1) with parameters a and b, is fitted to experimental data points of (and E by means of least square linear regression method. Results are estimates of the parameters of the fitted equation, the correlation coefficient and the standard error. The correlation coefficient indicates the extent of agreement in the data. The SE of an estimate is the sum of the residuals, which is related to multiple correlation coefficient squared (r2). It is a measure of the scatter in the data about the regression line and hence, reflects the variability in the data.

2.4 Results and Discussion

The keypoints with co-ordinate locations defining contour data were stored as text files. These data were transferred to a FE software (ANSYS) for development of a three-dimensional model. The keypoints were connected by cubic B-splines. The splines were connected to form areas within one slice. A skin-like area was generated by connecting the corresponding approximated contours of two consecutive slices. Volume between two slices was enclosed by connecting the upper and lower contour areas, and the skin-like area between them.

3.1 Relation between CT gray values (H) and apparent density ([rho])

The apparent density (() was computed from the CT gray values (H), in Hounsfield units, using a linear calibration derived from two reference points in one of the CT-scan slices. The first point was the CT gray value of air, i.e. -1200, which represents non-bone condition (0 kg [m.sup.-3]). The second point was the CT gray value of cortical bone, i.e. 2895, which was assumed to have an apparent density of 1800 kg m-3 [7,17]. The apparent density at any point in the bone was obtained by linear interpolation of CT gray values.

This linear relationship, generalised for all CT-scan slices and based on these two sets of values is:

[rho] = a + b H .... (2)

where, a = 527 and b = 0.44

It may also be stated that in case of moist (with marrow in situ) bone, the first point in the linear calibration should be the CT gray value of water, i.e. 0, for non-bone (0 kg [m.sup.-3]) condition. The second point can be the CT gray value of compact bone with apparent density of 1800 kg [m.sup.-3].

3.2 Relation between apparent density ([rho]) and elastic modulus (E)

Apart from twenty-five experimental data points, two additional set of values of compact bone have been considered for the power law regression analysis. The density of compact bone is about 1800 kg [m.sup.-3] [7], whereas the elastic modulus varies between 15 GPa and 20 GPa [9,10].

It has been established, both theoretically [17] and experimentally [7,18], that the Young's modulus of cancellous bone is strongly dependent on the bone's apparent density. It was concluded that the power law model was better suited than the linear model for mechanical properties of cancellous bone [13,17]. A statistical analysis of the power law relationships between apparent density and elastic modulus is presented in Table 1. Considering twenty-seven data points (twenty-five cancellous bone + two assumed compact bone) and fitting regression models, the parameters (a and b of E = a [[rho].sup.b]), the SE and the R, were calculated (Table 1); resulting in

E = 9.354 . [10.sup.-7] [[rho].sup.3.15]

Fitting models with predetermined powers of 2 and 3 results in

E = 5.288 . [10.sup.-3] [[rho].sup.2]

and E = 2.998 [.10.sup.-6] [[rho].sup.3]

Results indicate that the cubic relations were better suited to fit the data as compared to the squared relations (Table 1). The predefined exponent value of 3 fits the data slightly better than the exponent value of 3.15, since the SE was marginally higher in case of the latter (Table 1). However, the high correlation coefficient, ranging between 0.979 and 0.988, for all the three models were considered to be artificially high, since these relations were mainly guided by cloud of data points located mostly at the lower and the upper extreme values in the entire range of apparent density. Hence, these relations did not represent the model structure, correctly. To obtain more meaningful relations, the two data points of cortical bone (assumed) were omitted and new regression analyses were performed.

Considering the set of only twenty-five experimental data points, the power law regression analysis results in,

E = 0.0272 [[rho].sup.1.41]

Models with predefined exponent values,

E = 780 . [10.sup.-6] [[rho].sup.2]

and E = 1.63 [.10.sup.-6] [[rho].sup.3], suggest even weaker correlation (Table 1). A low correlation implies that incorrect model structure had been used, and/or a parameter (e.g. anisotropy) causing the scatter in experimental data could not be estimated. These results indicated that the data might not belong to a single type of structural material.

The mechanical behaviour of cancellous bone, consisting of an interconnected network of rod and plate like trabeculae, is similar to that of other cellular solids, such as polymeric foam [17,19]. Gibson [17], on theoretical considerations, suggested relationships of powers 2 and 3 between Young's modulus and density for cancellous bone depending on cancellous bone architecture. These relationships were fitted to experimental data of Carter and Hayes [7]. At low relative densities, it has rods connecting to form open cells. At higher relative densities, more material is accumulated in the cell walls and the structure transforms into a more closed network of plates. Gibson's analysis showed that the Young's modulus varied with the square of the density for open cell structure and with the cube of density for closed cell structure. Results of Carter and Hayes [7] agree with this prediction and suggest a transition from rod-like to plate-like elements at a density of about 350 kg [m.sup.-3], or a relative density of 0.20. Whitehouse [20] in his scanning electron micrograph study indicated that the cancellous bone structure of the human sternum becomes plate-like for relative densities over 0.20. Following these information we divided the set of data into two parts as shown in Fig. 3.

[FIGURE 3 OMITTED]

The first set consisted of data in the range of 170 - 350 kg [m.sup.-3] (n=15). The second set ranged from 350 kg [m.sup.-3] to 1800 kg [m.sup.-3] (n=13). The apparent density value of 350 kg [m.sup.-3] was common to both the sets, and was assumed to be the transition point from open cell structure to closed cell structure. Based on theoretical analysis [18], powers of 2 and 3 were fitted to the range of data that corresponds to the open cell and the closed cell structure of cancellous bone, respectively.

A power law model, for open cell structure,

E = 1049.25 . 10-6 [[rho].sup.2] , for [rho] [less than or equal to] 350 kg [m.sup.-3] (3)

was fitted to the data (n=15) in the lower range of apparent density (Fig. 2).

[FIGURE 2 OMITTED]

The available number of data above 350 kg [m.sup.-3] density (n=13) was limited to a short range. For closed cell structure, a power law model,

E = 3.00 . [10.sup.-6] [[rho].sup.3], for 350 [less than or equal to] [rho] [less than or equal to] 1800 kg [m.sup.-3] (4)

was fitted to the higher range of apparent density (Fig. 2). Although in real bone some interval of densities between cancellous and compact bone may never occur, one should expect intermediate values as a result of averaging over CT voxel. In order to ascribe matching mechanical properties to finite elements, elastic modulus should be defined for the whole range of densities.

Anisotropy of cancellous bone is a major factor responsible for the scatter in the data. Frich et al. [21] found an average anisotropy ratio of 5.2, which indicated strong anisotropy. Mansat et al. [22] also presented data on anisotropic material properties for glenoid cancellous bone with ratios [E.sub.1]/[E.sub.2] = 1.72, [E.sub.1]/[E.sub.3] = 1.95 and [E.sub.2]/[E.sub.3] = 1.22, which represents the degree of anisotropy. The effect of anisotropy was evident in the experimental data by the presence of high ([E.sub.max]) and low ([E.sub.min]) values of elastic modulus corresponding to a value of apparent density ([rho] = 270, 290, 330, 370 kg [m.sup.-3]). It was also observed that within a very small range of apparent density, an increase in density was associated with a decrease in value of elastic modulus. The other causes that lead to the scatter in the measured data was the presence of pores and other forms of inhomogeneity. This was evident in the experimental data; corresponding to a value of apparent density there was a very high and a very low value of elastic modulus. The ratio of a set of high and low values of elastic modulus was found to abnormally high (greater than 10), which was no way comparable to the values of degree of anisotropy presented in earlier studies [21,22]. Moreover, the degree of anisotropy could not be quantitatively determined, since information on the trabecular orientation of bone specimens was not known. For practical purposes, relations between the effective isotropic tissue modulus and the apparent density has been formed that might be useful for the FE model of the scapula and the glenoid prostheses. Considering the experimental errors, it seemed reasonable to accept the relations of powers 2 and 3 for the lower range ([rho] [less than or equal to] 350 kg [m.sup.-3]) and for the upper range (350 [less than or equal to] [rho] [less than or equal to] 1800 kg [m.sup.-3]), respectively, the purpose of finite element modelling and stress analysis. The power law relations not only take into account a combination of cancellous, trabecular and cortical bone but also the mechanically implicit data point of zero bone mineral content, with associated zero values of elastic modulus. The relationships obtained from experimental data of Frich [14] regarding the apparent density and elastic modulus of human scapula cancellous bone, were consistent with the predictions of Gibson [17] and can be useful in finite element modelling of scapula from CT-scan images.

4. Conclusion

A semi-automatic contour detection method has been developed to obtain accurate data on bone geometry. A linear relation between CT gray values and apparent density of scapula trabecular bone is formed. Power law relationships between apparent density and elastic modulus have been formulated for two different ranges of apparent density, depending on open cell and closed cell structure of bone.

References

[1] T.E. Orr, D.R. Carter and D.J. Schurman, Stress analyses of glenoid component designs, Clin. Orthop. Rel. Res. 212, 217-224 (1988).

[2] J.R. Friedman, M. LaBerge, R.L. Dooley and A.L. O'Hara, Finite element modelling of the glenoid component: Effect of design parameters on stress distribution. J Shoulder Elbow Surg. 1, 261-270 (1992).

[3] R.J. McBroom, W.C. Hayes, W.T. Edwards, R.P. Goldberg and A.A. White, Prediction of vertebral body compressive fractures using quantitative computer tomography. J. Bone Joint Surg. 67A, 12061214 (1985).

[4] S.M Bentzen, I. Hvid and J. Jorgensen, Mechanical strength of tibial trabecular bone evaluated by X-ray computed tomography. J. Biomechanics 20, 743-752 (1987).

[5] I. Hvid, S.M. Bentzen, F. Linde, L. Mosekilde and B. Pongsoipetch, X-ray quantitative computer tomography: the relations to physical properties of proximal tibial trabecular bone specimens. J. Biomechanics 22, 837-844 (1989).

[6] M.J. Ciarelli, S.A. Goldstein, J.L. Kuhn, D.D. Cody and M.B. Brown, Evaluation of orthogonal mechanical properties and density of human trabecular bone from major metaphyseal regions with material testing and computer tomography. J. Orthop. Res. 9, 674-682 (1991).

[7] D.R. Carter and W.C. Hayes, The compressive behaviour of bone as a two-phase porous structure. J. Bone Joint Surg. 59-A, 954-962 (1977).

[8] R.B. Ashman, J.Y. Rho and C.H. Turner, Anatomical variation of orthotropic elastic modulii of the proximal human tibia. J. Biomechanics 22, 895-900 (1989).

[9] K. Choi and S.A. Goldstein, A comparison of the fatigue behaviour of human trabecular and cortical tissue. J. Biomechanics 25, 1371-1381 (1992).

[10] J.H. Rho, R.B. Ashman and C.H. Turner, Young's modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. J. Biomechanics 26, 111-119 (1993).

[11] B. van Rietbergen, H. Weinans, R. Huiskes and A. Odgaard, A new method to determine trabecular bone elastic properties and loading using micromechanical finite element models. J. Biomechanics 28, 69-81 (1995).

[12] S.A. Goldstein, The mechanical properties of trabecular bone: dependence on anatomical location and fixation. J. Biomechanics 20, 1055-1061 (1987).

[13] J.D. Currey, Power law models for the mechanical properties of cancellous bone. J. Engineering in Medicine 15, No. 3, 153-154 (1986).

[14] L.H. Frich, Glenoid Knoglestyrke og knoglestruktur. Doctoral thesis, University of Aarhus, Denmark (1994).

[15] A. Martelli, An application of heuristic search methods to edge and contour detection," Communications of the ACM, 19, 73-83 (1976).

[16] A.C.M. Dumay, J.J. Gerbrands and J.H.C. Reiber, Automated extraction, labelling and analysis of the coronary vascular from arteriograms. Int. J. Cardiac Imaging, 10, 205-215 (1994).

[17] L.J. Gibson, The mechanical behaviour of cancellous bone. J. Biomechanics 18, 317-328 (1985).

[18] M. Dalstra, R. Huiskes, A. Odgaard and L. van Erning, Mechanical and textural properties of pelvic trabecular bone. J. Biomechanics 26, 523-535 (1993).

[19] L.J. Gibson and M.F. Ashby, Cellular solids: structure & properties, Pergamon Press, Oxford, UK (1988).

[20] W.J. Whitehouse, Scanning electron micrographs of cancellous bone from the human sternum. J. Pathol. 116, 213-223 (1975).

[21] L.H. Frich, N.C. Jensen, A. Odgaard, C.M. Pedersen, J.O. Sojbjerg and M. Dalstra, Bone strength and material properties of the glenoid. J. Shoulder Elbow Surg.; 6: 97-104 (1997).

[22] P. Mansat, C. Barea, M-C. Hobatho, R. Darmana and M. Mansat, Anatomic variation of the mechanical properties of the glenoid. J. Shoulder Elbow Surg. 7, 109-115 (1998).

Sanjay Gupta and Prosenjit Dan

Department of Applied Mechanics, Bengal Engineering College (Deemed University), Howrah 711 103, West Bengal, India.

Mechanical properties of scapular trabecular bone are assumed to be similar to those of other trabecular bone of different anatomical regions, like tibia, femur, humerus. The goal of this study was to develop a technique that may be useful to detect bone geometry and pixel gray value from contour data of CT-scan slice of the scapula. In this paper an attempt has been made to relate quantitative Computed Tomography (CT) gray values with apparent density, and apparent density with elastic modulus. A contour detection algorithm has been developed that finds the optimised bone contour by connecting the points with high derivatives of CT gray value on lines perpendicular to an initial contour. The number of points in a contour were stored as keypoints which were useful for generating a three-dimensional model of the scapula. A linear regression, generalised for all CT-scan slices defining the whole scapula, was derived from two reference points (one nobone condition, i.e. air, another cortical bone). Based on structural and analytical models of trabecular bone, power law relations were fitted for two ranges of apparent density. Powers of 2 and 3 (E ~ [rho]2, E ~ [rho]3) have been used for open cell rod-like structure and closed cell plate-like structure, respectively. The transition from open to closed structure was assumed to occur at an apparent density of 350 kg [m.sup.-3]. The theoretical relationships were fitted to experimental data of glenoid cancellous bone specimens. The above-mentioned relationships for scapular trabecular bone are meant to be used for a finite element model of a scapula, with or without an implant, based on CT-scan images.

Keywords : Shoulder, glenoid, bone geometry, apparent density, elastic modulus, finite element model.

Introduction :

Characterisation of mechanical properties of trabecular bone is essential for the design of prosthesis. Finite Element (FE) models of glenoid component were mostly based on assumptions that the material properties of the glenoid were similar to those of the tibial plateau [1,2]. Research directed towards investigating the causes of mechanical loosening of the glenoid component and its improvement in design, calls for reliable data on mechanical properties of scapular trabecular bone.

A unique quantitative relationship between the apparent density and the elastic modulus is necessary in order to assign material properties to each element of the FE model of the scapula. A relation between quantitative Computed Tomography (CT) gray values of pixels in Hounsfield units and the apparent density of bone is of prime importance to start with. McBroom et al. [3] found a good correlation (R2 =0.89, p<0.0001) and a linear relationship between the CT gray values and apparent density of vertebral trabecular bone. Bentzen et al. [4] and Hvid et al. [5] reported a correlation coefficient of 0.93 and established a linear relationship between CT gray values and apparent density of tibial trabecular bone. A statistically significant linear relation between CT gray values and apparent density (R2=0.82) was observed by Ciarelli et al. [6] for human trabecular bone from major metaphyseal regions, like tibia, femur, radius and humerus.

Although none of the existing reports predicts a relation between CT gray value and apparent density for scapular trabecular bone, a similar linear relationship can be expected for scapula trabecular bone. Many authors have reported trabecular bone mechanical properties and relationships between the apparent density and bone stiffness [7-11]. The results of these studies predicted linear as well as power law relationships. The variation was large and was primarily due to differences in anatomic location, anisotropy of material properties, direction of testing and testing protocol [6,7,10,12]. Currey [13], while criticising Hvid et al. [5], preferred power law models to linear regression models for mechanical properties of cancellous bone. He argued that there are three advantages: (1) theoretically, a power law relation is predicted, which was confirmed by experiments; (2) the equation will produce marked improvements towards extrapolation to compact bone on one side and non-bone on the other; (3) the distribution of the residuals will be improved for power law model as compared to linear model.

The goal of this study was to develop a technique that may be useful to detect bone geometry and pixel gray value from contour data of CT-scan slice of the scapula. The pixel gray value can be calibrated in terms of bone density, which can be in turn related to bone elastic modulus and strength. Using the experimental data of density and elastic modulus for scapular bone [14], the purpose of this study was to formulate a relationship between CT gray value (H) in Hounsfield Units and apparent density ([rho]), and between apparent density ([rho]) and elastic modulus (E), by regression analyses.

Materials and Method :

The CT-scan images of the scapula were stored in 512X512 pixels, with a pixel size of 0.5 mm X 0.5 mm and slice thickness of 3.0 mm in Digital Imaging and Communications in Medicine (DICOM) format. A single DICOM file contains a header that stores information about the patient's names the type of scan, image dimensions, and the image data, which contains information in three dimensions. A total number of 69 slices was obtained to define the whole scapula. The CT images were used for subsequent image processing and analysis. CT gray values in Hounsfield units corresponding to specific positions within the scanned image were obtained. The range of the CT gray value of bone (trabecular and compact) is between -1200 (i.e. nonbone) and 2895 (i.e. cortical bone).

In another study, Frich [14] reported experimental data of apparent density and elastic modulus of trabecular bone samples taken from twenty-five fresh human scapulae. The bone samples, one from each scapula, had all been taken between 3 and 11 mm from the articular surface at the "bare areas" (i.e. approximately in the middle of the articular surface). The cylindrical bone samples 8 mm long and 6.5 mm in diameter, were taken from an area, just below the subchondral plate of the glenoid. They were subjected to compression test in order to obtain stress-strain curves. The elastic modulus was determined as the maximum slope of the stress-strain curve in compression up to 1.5% strain. The strain rate of the bone cylinder compression test was 0.1% per second. After these tests the bone samples were cleaned with compressed air to remove bone marrow, degreased in alcohol and acetone, again cleaned with compressed air, and allowed to dry. Apparent densities of the samples were calculated from measurement of weight and volume.

2.1 Image Smoothing and Edge Detection

Image smoothing was used primarily for diminishing spurious effects that may be present in a digital image due poor sampling system or transmission channel. Smoothing is a process by which data points are averaged with their neighbours in a series, such as a time series, or image. This usually has the effect of blurring the sharp edges in the smoothed data. Smoothing is sometimes referred to as filtering, because smoothing has the effect of suppressing high frequency signal and enhancing low frequency signal. In this study we have adopted boxcar average of specified width.

Edge detection is often the first step in image processing since a lot of information can be obtaind from the edges. In this study the Sobel and the Roberts algorithm have been used for edge detection. The Sobel operator performs a 2-D spatial gradient measurement on an image. Both Sobel and Roberts operators emphasises regions of high spatial frequency that corresponds to edges. Typically it is used to find the approximate absolute gradient magnitude at each point in an input gray-scale image. At least, the operator consists of a pair of 3x3 convolution kernels. The operator consists of a pair of 2x2 convolution kernels. The Sobel operator therefore resulted in better accuracy, but required more computing time as compared to Roberts operator.

As a first step a Graphical User Interface (GUI) was developed to view the DICOM image as shown in Fig. 1. Interactive Data Language (IDL, developed by Research Systems, Inc., USA) has been used as computing environment for visualisation of data and interactive analysis. IDL integrates a powerful, array-oriented language with numerous mathematical analysis and graphical display techniques. The header information was stored separately in a text file for future reference. We trapped the mouse movement and obtained the position co-ordinate of the mouse pointer with respect to the normalised co-ordinate and retrieved the Image value from image matrix. This pixel value was shown simultaneously with the mouse movement in the text box in GUI.

[FIGURE 1 OMITTED]

2.2 Bone Geometry

A semiautomatic contour detection algorithm, based on a mathematical optimisation procedure, was used to find the closed contours in each CT-slice [15,16]. This procedure requires an initial approximate contour (about 10 points per slice) generated manually. The contour detection algorithm optimises the contour by connecting the points with high derivatives of CT gray value on lines perpendicular to the initial contour [16]. The procedure was repeated for all the consecutive CT slices. This algorithm was developed to make the procedure more efficient and accurate. The number of points in a contour was reduced automatically by selecting the keypoints, so that the error between the modified contour and the original points defining the contour remained less than 0.5 mm.

2.3 Statistical analysis

Regression curves are fitted to observations in order to reduce the relationship to a parametric equation. The correlation coefficient (R) and the standard errors (SE) quantify the "goodness of fit" of the dependent variable. The power law relationship, given by E = a [[rho].sup.b], (1) with parameters a and b, is fitted to experimental data points of (and E by means of least square linear regression method. Results are estimates of the parameters of the fitted equation, the correlation coefficient and the standard error. The correlation coefficient indicates the extent of agreement in the data. The SE of an estimate is the sum of the residuals, which is related to multiple correlation coefficient squared (r2). It is a measure of the scatter in the data about the regression line and hence, reflects the variability in the data.

2.4 Results and Discussion

The keypoints with co-ordinate locations defining contour data were stored as text files. These data were transferred to a FE software (ANSYS) for development of a three-dimensional model. The keypoints were connected by cubic B-splines. The splines were connected to form areas within one slice. A skin-like area was generated by connecting the corresponding approximated contours of two consecutive slices. Volume between two slices was enclosed by connecting the upper and lower contour areas, and the skin-like area between them.

3.1 Relation between CT gray values (H) and apparent density ([rho])

The apparent density (() was computed from the CT gray values (H), in Hounsfield units, using a linear calibration derived from two reference points in one of the CT-scan slices. The first point was the CT gray value of air, i.e. -1200, which represents non-bone condition (0 kg [m.sup.-3]). The second point was the CT gray value of cortical bone, i.e. 2895, which was assumed to have an apparent density of 1800 kg m-3 [7,17]. The apparent density at any point in the bone was obtained by linear interpolation of CT gray values.

This linear relationship, generalised for all CT-scan slices and based on these two sets of values is:

[rho] = a + b H .... (2)

where, a = 527 and b = 0.44

It may also be stated that in case of moist (with marrow in situ) bone, the first point in the linear calibration should be the CT gray value of water, i.e. 0, for non-bone (0 kg [m.sup.-3]) condition. The second point can be the CT gray value of compact bone with apparent density of 1800 kg [m.sup.-3].

3.2 Relation between apparent density ([rho]) and elastic modulus (E)

Apart from twenty-five experimental data points, two additional set of values of compact bone have been considered for the power law regression analysis. The density of compact bone is about 1800 kg [m.sup.-3] [7], whereas the elastic modulus varies between 15 GPa and 20 GPa [9,10].

It has been established, both theoretically [17] and experimentally [7,18], that the Young's modulus of cancellous bone is strongly dependent on the bone's apparent density. It was concluded that the power law model was better suited than the linear model for mechanical properties of cancellous bone [13,17]. A statistical analysis of the power law relationships between apparent density and elastic modulus is presented in Table 1. Considering twenty-seven data points (twenty-five cancellous bone + two assumed compact bone) and fitting regression models, the parameters (a and b of E = a [[rho].sup.b]), the SE and the R, were calculated (Table 1); resulting in

E = 9.354 . [10.sup.-7] [[rho].sup.3.15]

Fitting models with predetermined powers of 2 and 3 results in

E = 5.288 . [10.sup.-3] [[rho].sup.2]

and E = 2.998 [.10.sup.-6] [[rho].sup.3]

Results indicate that the cubic relations were better suited to fit the data as compared to the squared relations (Table 1). The predefined exponent value of 3 fits the data slightly better than the exponent value of 3.15, since the SE was marginally higher in case of the latter (Table 1). However, the high correlation coefficient, ranging between 0.979 and 0.988, for all the three models were considered to be artificially high, since these relations were mainly guided by cloud of data points located mostly at the lower and the upper extreme values in the entire range of apparent density. Hence, these relations did not represent the model structure, correctly. To obtain more meaningful relations, the two data points of cortical bone (assumed) were omitted and new regression analyses were performed.

Considering the set of only twenty-five experimental data points, the power law regression analysis results in,

E = 0.0272 [[rho].sup.1.41]

Models with predefined exponent values,

E = 780 . [10.sup.-6] [[rho].sup.2]

and E = 1.63 [.10.sup.-6] [[rho].sup.3], suggest even weaker correlation (Table 1). A low correlation implies that incorrect model structure had been used, and/or a parameter (e.g. anisotropy) causing the scatter in experimental data could not be estimated. These results indicated that the data might not belong to a single type of structural material.

The mechanical behaviour of cancellous bone, consisting of an interconnected network of rod and plate like trabeculae, is similar to that of other cellular solids, such as polymeric foam [17,19]. Gibson [17], on theoretical considerations, suggested relationships of powers 2 and 3 between Young's modulus and density for cancellous bone depending on cancellous bone architecture. These relationships were fitted to experimental data of Carter and Hayes [7]. At low relative densities, it has rods connecting to form open cells. At higher relative densities, more material is accumulated in the cell walls and the structure transforms into a more closed network of plates. Gibson's analysis showed that the Young's modulus varied with the square of the density for open cell structure and with the cube of density for closed cell structure. Results of Carter and Hayes [7] agree with this prediction and suggest a transition from rod-like to plate-like elements at a density of about 350 kg [m.sup.-3], or a relative density of 0.20. Whitehouse [20] in his scanning electron micrograph study indicated that the cancellous bone structure of the human sternum becomes plate-like for relative densities over 0.20. Following these information we divided the set of data into two parts as shown in Fig. 3.

[FIGURE 3 OMITTED]

The first set consisted of data in the range of 170 - 350 kg [m.sup.-3] (n=15). The second set ranged from 350 kg [m.sup.-3] to 1800 kg [m.sup.-3] (n=13). The apparent density value of 350 kg [m.sup.-3] was common to both the sets, and was assumed to be the transition point from open cell structure to closed cell structure. Based on theoretical analysis [18], powers of 2 and 3 were fitted to the range of data that corresponds to the open cell and the closed cell structure of cancellous bone, respectively.

A power law model, for open cell structure,

E = 1049.25 . 10-6 [[rho].sup.2] , for [rho] [less than or equal to] 350 kg [m.sup.-3] (3)

was fitted to the data (n=15) in the lower range of apparent density (Fig. 2).

[FIGURE 2 OMITTED]

The available number of data above 350 kg [m.sup.-3] density (n=13) was limited to a short range. For closed cell structure, a power law model,

E = 3.00 . [10.sup.-6] [[rho].sup.3], for 350 [less than or equal to] [rho] [less than or equal to] 1800 kg [m.sup.-3] (4)

was fitted to the higher range of apparent density (Fig. 2). Although in real bone some interval of densities between cancellous and compact bone may never occur, one should expect intermediate values as a result of averaging over CT voxel. In order to ascribe matching mechanical properties to finite elements, elastic modulus should be defined for the whole range of densities.

Anisotropy of cancellous bone is a major factor responsible for the scatter in the data. Frich et al. [21] found an average anisotropy ratio of 5.2, which indicated strong anisotropy. Mansat et al. [22] also presented data on anisotropic material properties for glenoid cancellous bone with ratios [E.sub.1]/[E.sub.2] = 1.72, [E.sub.1]/[E.sub.3] = 1.95 and [E.sub.2]/[E.sub.3] = 1.22, which represents the degree of anisotropy. The effect of anisotropy was evident in the experimental data by the presence of high ([E.sub.max]) and low ([E.sub.min]) values of elastic modulus corresponding to a value of apparent density ([rho] = 270, 290, 330, 370 kg [m.sup.-3]). It was also observed that within a very small range of apparent density, an increase in density was associated with a decrease in value of elastic modulus. The other causes that lead to the scatter in the measured data was the presence of pores and other forms of inhomogeneity. This was evident in the experimental data; corresponding to a value of apparent density there was a very high and a very low value of elastic modulus. The ratio of a set of high and low values of elastic modulus was found to abnormally high (greater than 10), which was no way comparable to the values of degree of anisotropy presented in earlier studies [21,22]. Moreover, the degree of anisotropy could not be quantitatively determined, since information on the trabecular orientation of bone specimens was not known. For practical purposes, relations between the effective isotropic tissue modulus and the apparent density has been formed that might be useful for the FE model of the scapula and the glenoid prostheses. Considering the experimental errors, it seemed reasonable to accept the relations of powers 2 and 3 for the lower range ([rho] [less than or equal to] 350 kg [m.sup.-3]) and for the upper range (350 [less than or equal to] [rho] [less than or equal to] 1800 kg [m.sup.-3]), respectively, the purpose of finite element modelling and stress analysis. The power law relations not only take into account a combination of cancellous, trabecular and cortical bone but also the mechanically implicit data point of zero bone mineral content, with associated zero values of elastic modulus. The relationships obtained from experimental data of Frich [14] regarding the apparent density and elastic modulus of human scapula cancellous bone, were consistent with the predictions of Gibson [17] and can be useful in finite element modelling of scapula from CT-scan images.

4. Conclusion

A semi-automatic contour detection method has been developed to obtain accurate data on bone geometry. A linear relation between CT gray values and apparent density of scapula trabecular bone is formed. Power law relationships between apparent density and elastic modulus have been formulated for two different ranges of apparent density, depending on open cell and closed cell structure of bone.

References

[1] T.E. Orr, D.R. Carter and D.J. Schurman, Stress analyses of glenoid component designs, Clin. Orthop. Rel. Res. 212, 217-224 (1988).

[2] J.R. Friedman, M. LaBerge, R.L. Dooley and A.L. O'Hara, Finite element modelling of the glenoid component: Effect of design parameters on stress distribution. J Shoulder Elbow Surg. 1, 261-270 (1992).

[3] R.J. McBroom, W.C. Hayes, W.T. Edwards, R.P. Goldberg and A.A. White, Prediction of vertebral body compressive fractures using quantitative computer tomography. J. Bone Joint Surg. 67A, 12061214 (1985).

[4] S.M Bentzen, I. Hvid and J. Jorgensen, Mechanical strength of tibial trabecular bone evaluated by X-ray computed tomography. J. Biomechanics 20, 743-752 (1987).

[5] I. Hvid, S.M. Bentzen, F. Linde, L. Mosekilde and B. Pongsoipetch, X-ray quantitative computer tomography: the relations to physical properties of proximal tibial trabecular bone specimens. J. Biomechanics 22, 837-844 (1989).

[6] M.J. Ciarelli, S.A. Goldstein, J.L. Kuhn, D.D. Cody and M.B. Brown, Evaluation of orthogonal mechanical properties and density of human trabecular bone from major metaphyseal regions with material testing and computer tomography. J. Orthop. Res. 9, 674-682 (1991).

[7] D.R. Carter and W.C. Hayes, The compressive behaviour of bone as a two-phase porous structure. J. Bone Joint Surg. 59-A, 954-962 (1977).

[8] R.B. Ashman, J.Y. Rho and C.H. Turner, Anatomical variation of orthotropic elastic modulii of the proximal human tibia. J. Biomechanics 22, 895-900 (1989).

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[10] J.H. Rho, R.B. Ashman and C.H. Turner, Young's modulus of trabecular and cortical bone material: ultrasonic and microtensile measurements. J. Biomechanics 26, 111-119 (1993).

[11] B. van Rietbergen, H. Weinans, R. Huiskes and A. Odgaard, A new method to determine trabecular bone elastic properties and loading using micromechanical finite element models. J. Biomechanics 28, 69-81 (1995).

[12] S.A. Goldstein, The mechanical properties of trabecular bone: dependence on anatomical location and fixation. J. Biomechanics 20, 1055-1061 (1987).

[13] J.D. Currey, Power law models for the mechanical properties of cancellous bone. J. Engineering in Medicine 15, No. 3, 153-154 (1986).

[14] L.H. Frich, Glenoid Knoglestyrke og knoglestruktur. Doctoral thesis, University of Aarhus, Denmark (1994).

[15] A. Martelli, An application of heuristic search methods to edge and contour detection," Communications of the ACM, 19, 73-83 (1976).

[16] A.C.M. Dumay, J.J. Gerbrands and J.H.C. Reiber, Automated extraction, labelling and analysis of the coronary vascular from arteriograms. Int. J. Cardiac Imaging, 10, 205-215 (1994).

[17] L.J. Gibson, The mechanical behaviour of cancellous bone. J. Biomechanics 18, 317-328 (1985).

[18] M. Dalstra, R. Huiskes, A. Odgaard and L. van Erning, Mechanical and textural properties of pelvic trabecular bone. J. Biomechanics 26, 523-535 (1993).

[19] L.J. Gibson and M.F. Ashby, Cellular solids: structure & properties, Pergamon Press, Oxford, UK (1988).

[20] W.J. Whitehouse, Scanning electron micrographs of cancellous bone from the human sternum. J. Pathol. 116, 213-223 (1975).

[21] L.H. Frich, N.C. Jensen, A. Odgaard, C.M. Pedersen, J.O. Sojbjerg and M. Dalstra, Bone strength and material properties of the glenoid. J. Shoulder Elbow Surg.; 6: 97-104 (1997).

[22] P. Mansat, C. Barea, M-C. Hobatho, R. Darmana and M. Mansat, Anatomic variation of the mechanical properties of the glenoid. J. Shoulder Elbow Surg. 7, 109-115 (1998).

Sanjay Gupta and Prosenjit Dan

Department of Applied Mechanics, Bengal Engineering College (Deemed University), Howrah 711 103, West Bengal, India.

Table 1. Statistical analysis of power law relationships between apparent density ([rho]) and elastic modulus (E). Total number of data points: 27 (experimental data points: 25, and additional cortical bone data points: 2). The coefficients a and b correspond to E = a [[rho].sup.b]. * excluding two compact bone data points. ([double dagger]) for [rho] [less than or equal to] 350 kg [m.sup.-3]; ([dagger]) for 350 [less than or equal to] [rho] [less than or equal to] 1800 kg [m.sup.-3]; + predefined. Number of A Power fit data points (n) E = a [[rho].sup.b] 27 9.35 x [10.sup.-7] E = a [[rho].sup.b] 27 3.00 x [10.sup.-6] E = a [[rho].sup.b] 27 5.30 x [10.sup.-3] E = a [[rho].sup.b] 25 * 27.21 x [10.sup.-3] E = a [[rho].sup.b] 25 * 779.87 x [10.sup.-6] E = a [[rho].sup.b] 25 * 1.626 x [10.sup.-6] E = a [[rho].sup.b] 15 ([double dagger]) 1049.25 x [10.sup.-6] E = a [[rho].sup.b] 13 ([dagger]) 3.00 x [10.sup.-6] B Standard Correlation Power fit error coefficient (r) (SE) E = a [[rho].sup.b] 3.15 714.14 0.988 E = a [[rho].sup.b] 3+ 703.70 0.987 E = a [[rho].sup.b] 2+ 953.37 0.979 E = a [[rho].sup.b] 1.41 75.40 0.524 E = a [[rho].sup.b] 2+ 75.65 0.488 E = a [[rho].sup.b] 3+ 83.81 0.256 E = a [[rho].sup.b] 2+ 43.31 0.403 E = a [[rho].sup.b] 3+ 1034.63 0.987

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Author: | Gupta, Sanjay; Dan, Prosenjit |
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Publication: | Trends in Biomaterials and Artificial Organs |

Geographic Code: | 9INDI |

Date: | Jan 1, 2004 |

Words: | 4268 |

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