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Keywords:

  • cortical bone;
  • cranium;
  • femur;
  • finite element modeling;
  • mandible;
  • mechanical properties;
  • microstructure;
  • ultrasound

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References

Elastic structure in cortical bone is usually simplified as orthotropic or transversely isotropic, which allows estimates of three-dimensional technical constants from ultrasonic and density measurements. These elastic property estimates can then be used to study phenotypic changes in cortical bone structure and function, and to create finite element models of skeletal structures for studies of organismal variation and functional adaptation. This study examines assumptions of orthotropic or transversely isotropic material structure in cortical bone through the investigation of off-axis ultrasonic velocities in the cortical plane in 10 samples each from a human femur, mandible and cranium. Longitudinal ultrasonic velocities were measured twice through each bone sample by rotating the perimeter of each sample in 1 ° angular intervals between two ultrasonic transducers. The data were fit to sine curves f(x) = (× sin(B) + C) and the goodness of fit was examined. All the data from the femur fit closely with the ideal sine curve model, and all three coefficients were similar among specimens, indicating similar elastic properties, anisotropies and orientations of the axes of maximum stiffness. Off-axis ultrasonic velocities in the mandible largely fit the sine curve model, although there were regional variations in the coefficients. Off-axis ultrasonic velocities from the cranial vault conformed to the sine curve model in some regions but not in others, which shows an irregular and complex pattern. We hypothesize that these variations in ultrasonic velocities reflect variations in the underlying bulk microstructure of the cortical bone, especially in the three-dimensional patterns of osteonal orientation and structure. Elastic property estimates made with ultrasonic techniques are likely valid in the femur and mandible; errors in estimates from cranial bone need to be evaluated regionally. Approximate orthotropic structure in bulk cortical bone specimens should be assessed if ultrasound is used to estimate three-dimensional elastic properties.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References

Bone tissue material properties and their morphological distribution are important for improving the accuracy of finite element models of skeletal structures, and they hold promise for exploring molecular and cellular mechanisms for the adaptation of those structures. Yet, published characterizations of variation in material properties among anatomical regions, across species, and following transgenic phenotypic alteration are limited.

Some recent work to determine bone material characteristics is attempting to integrate elastic characteristics with skeletal microstructure, osteonal structure and the remodeling process (Ashman et al. 1985; Spatz et al. 1996; Robling & Turner, 2002; Hiller et al. 2003; Lettry et al. 2003; Lucksanasombool et al. 2003; Dechow et al. 2008, 2010). Overall, many of these studies have not focused on three-dimensional elastic and structural properties, and the results are hindered by small sample sizes. More importantly, little consideration has been given to how well cortical bone can be characterized three-dimensionally in terms of axes of symmetry. If most cortical bone structure can be approximated as orthotropic or transversely isometric (Fig. 1), then there is a lack of methodology to determine the fit of that approximation and the primary orientations of material stiffness in cortical bone. Although the database of measurements for the mechanical anisotropy of cortical bone is growing (Rho et al. 2001; Fan et al. 2002; Peterson & Dechow, 2002; Schwartz-Dabney & Dechow, 2002a,b, 2003; Peterson et al. 2006; Wang & Dechow, 2006; Wang et al. 2006, 2010a,b; Rapoff et al. 2008; Daegling et al. 2009; Zapata et al. 2010), three-dimensional and quantitative relationships with possibly influential microstructural features have received little study (Dechow et al. 2008, 2010).

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Figure 1.  Schematic representations of material structure in elastic solids; (A) Two extremes of organization, anisotropy and isotropy. Anisotropy defines material where elastic constants (elastic and shear modulus, and Poisson’s ratio) vary by direction, while isotropy defines material where constants are the same in all directions. (B) Two types of anisotropy, orthotropy and transverse isotropy. Elastic structure in biological tissues like bone can approximate these types of organization at superosteonal levels of size hierarchy. In orthotropy, elastic properties are symmetrical with respect to three orthogonal planes; elastic moduli differ between each of the corresponding three orthogonal axes, but are the same along any one axis. Transverse isotropy is similar to orthotropy, except that within one of the orthogonal planes, elastic moduli are the same in all directions.

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Elastic anisotropy in cortical bone from the craniofacial skeleton of various primates has been shown to vary with anatomical location (Peterson & Dechow, 2002; Schwartz-Dabney & Dechow, 2002a,b, 2003; Peterson et al. 2006; Wang et al. 2006, 2010a,b; Rapoff et al. 2008; Daegling et al. 2009). Assessments of material properties thus should be made with consideration of the region of the skeleton. Although a number of researchers have indicated that cortical bone has orthotropic or transversely isotropic elastic properties (Rho, 1996; Ziv et al. 1996; Liu et al. 1999), details of how variations in microstructure lead to variations in tissue anisotropies is unknown.

An important problem in understanding three-dimensional elastic structure in cortical bone is one of size hierarchy. Cortical bone structure and mechanics can be studied at different scales (Katz, 1971; Martin, 1999), ranging from bone’s molecular constituents of hydroxyapatite and collagen at a nanoscale, to osteonal and lamellar structures at an intermediate or microscale, to complexes of osteons and interstitial bone at a tissue or supraosteonal scale. Although understanding nanostructures like collagen fibers and hydroxyapatite crystals, or microstructures like osteons, are essential for relating bone structure and function (Ascenzi & Bonucci, 1967, 1968, 1976; Ascenzi et al. 2003), it is difficult to relate the intrinsic properties of these structures to directional variations in elastic properties at a supraosteonal level because of the three-dimensional variations in the morphology of these structures at that scale. Some aspects of the mechanical properties of microstructural components such as secondary osteonal bone and interstitial bone have been evaluated (Rho et al. 2001, 2002), but these and similar studies have generally not documented variations in three-dimensional structure at a tissue level. As with any anisotropic composite material, understanding structural and functional relationships in bone at a tissue level requires considerations of hierarchical structure and three-dimensional study of the material at that level.

Mechanical modeling, including finite element modeling, has been one modern approach to understanding structural and functional adaptation of the mandible to altered function (Korioth et al. 1992; Korioth & Versluis, 1997). However, variations in mechanical properties in specific cranial regions can cause significant functional misinterpretations when averaged elastic moduli are used to construct the models (Korioth et al. 1992; Kabel et al. 1999; Dechow & Hylander, 2000; Richmond et al. 2005; Ross et al. 2005; Strait et al. 2005, 2007, 2008, 2009, 2010; Wang et al. 2008; Apicella et al. 2010). The purpose of this study is to characterize elastic anisotropy and approximate tissue material symmetries in cortical bone by studying off-axis ultrasonic velocities. We compare results from bone specimens from a human mandible, crania and femur. A methodology to measure off-axis bulk ultrasonic velocities throughout a 360 ° angular range allows characterization of three-dimensional material properties of cortical bone. This information allows testing of the hypothesis that cortical bone from the human mandible, crania and femur can be characterized as transversely isotropic or orthotropic. (In this article, we will use the term orthotropic to indicate both orthotropy and transverse isotropy, as the latter is a subset of the former. We will specifically note where we wish to indicate differences between transversely isotropic specimens and orthotropic specimens that are not transversely isotropic.)

Our objective is to compare patterns of off-axis ultrasonic velocity measurements of specimens of human cortical bones from different locations. We hypothesize that samples from all locations exhibit patterns of off-axis velocities indicative of elastic orthotropy. In addition, we assess the reliability and validity of using ultrasonic techniques for determining directions of maximum and minimum stiffness in the cortical plane. These directions correspond with the peaks and valleys of the plots of ultrasonic velocities by angle (Schwartz-Dabney & Dechow, 2003; Peterson et al. 2006; Wang et al. 2006, 2010a,b; Dechow et al. 2010), and this information is essential for determining three-dimensional elastic properties in bone, especially where these directions do not correspond regularly with anatomical axes.

Materials and methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References

Ten cortical specimens each were collected from a human mandible, skull and femur. Mandible and skull specimens were prepared from a 62-year-old unembalmed frozen female cadaver head, and the femur specimens were selected from a 71-year-old embalmed male cadaver (Fig. 2). Although embalming has been shown to have a small effect on mechanical properties, such as yield stress, there is even less effect on elastic modulus, especially when measured ultrasonically (Currey et al. 1995; Guo, 2001). In any case, the goal of our study is to compare the variation within each specimen in off-axis ultrasonic velocities in the cortical plane. While embalming in the femur might have minor effects on elastic properties, it is reasonable to assume that these are less than those resulting from variation between individuals due to age, sex and a large variety of other factors, and are unlikely to effect patterns of anisotropy in the bone specimens.

image

Figure 2.  Cortical bone sampling sites.

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Mandible and skull specimens were collected from sites based on the results of our previous investigations in humans (Peterson & Dechow, 2002; Schwartz-Dabney & Dechow, 2002b, 2003; Peterson et al. 2006; Dechow et al. 2010). These investigations quantified intra- and interindividual variation in elastic properties throughout the craniofacial skeleton. Here we selected sites to provide us with the full range of regional elastic property variation in craniofacial cortical bone. Elastic property variation in the femur (Rho, 1996; Fan et al. 2002; Dechow et al. 2008, 2010) is much less than that in the skull. Because of this relative invariance, the femur was selected as a source of cortical bone for comparison with the skull. In the femur, specimens were acquired along the diaphysis on the medial side.

Bone preparation was carried out under a hood with the investigator wearing glasses, mask, gloves and a gown. Sites were marked with a graphite line indicating a reference line. For the skull, three different anatomical reference lines were used depending on the region. These were the occlusal plane, the sagittal suture and a line tangent to the sagittal curvature of the skull. The long axis of the diaphysis was used as a reference axis for the femur.

Bone was cooled continuously with a water drip during preparation. Bone cylinders were harvested from the mandibular cortex using a Dremel 732 Heavy Duty Flex Shaft Rotary Tool and 10.0-mm inner-diameter trephine burrs (Ace Dental Implant System). Cancellous bone on the inner surface of the cortical specimens was removed with grinding wheels on a water-cooled Tormek grinder or by gentle hand sanding. Specimens were stored in a solution of 95% ethanol and isotonic saline in equal proportions. This medium has been shown to maintain the elastic properties of cortical bone over time (Ashman et al. 1984; Zioupos et al. 2000).

Digital calipers were used to measure the specimen diameters and cortical thickness to the nearest 0.01 mm. Cortical thickness was defined as the thickness from the facial surface of the cortical bone to the internal cortical-trabecular interface, and usually ranged between 1.5 and 3.0 mm. For most specimens, the zone of attached trabecular bone was relatively thin, readily visible and easily removed. For a few specimens, especially those from the inferior border of the mandibular corpus, and the frontal and zygomatic regions, the cortical-trabecular interface was more difficult to define. For these sites, ‘trabecular’ bone was removed until the unmagnified endocortical surface had an apparent constant density (or consistent lack of visible porosity).

Moist weights of the specimens were obtained on a Mettler PM460 analytical balance reading to the nearest 0.01 g. Submerged weights were obtained using a Mettler suspension jig. Apparent density calculations were based on Archimedes’ principle of buoyancy (Ashman et al. 1984).

The longitudinal ultrasonic waves were generated by Panametrics transducers (V312-N-SU) resonating at 10 MHz. The transducers were powered with a Hewlett Packard Model 214A pulse generator (2-μs square wave pulses at 500 Hz with an amplitude of 20–50 V to obtain a stable signal). Pulse delays induced by passage of ultrasonic waves through the bone were read on a Tektronix TDS420 digitizing oscilloscope. Velocities were calculated by dividing pulse delays by the specimen dimension: cortical thickness for axis 1; and cortical disk diameter for axes 2 and 3 in the cortical plane. To minimize error, all measurements were repeated twice, and the mean value of those two measurements was used for analysis, unless the readings were inconsistent, and then additional readings were made. If specimens could not be remeasured to within 3%, the specimens were discarded. Most specimens had identical readings (< 0.5% difference) on repeated measurings.

The most important element of our apparatus was a 4″ Rotary Table (P/N 3700, Sherline Products), which allowed accurate rotations to a tenth of a degree. A chuck was mounted to the surface of the rotary table to which was attached a small wooden dowel (3 mm in diameter). The bone specimen was then attached with a dot of cyanoacrylate cement to the flat top of the dowel. The specimen could then be accurately rotated. Tests of ultrasonic velocities through test bone specimens with and without the attachment of the dowel did not show a difference in velocities.

The miniature rotary table can make a 360 ° turn. We set the marked reference line at 0 °. We then measured velocities for each specimen in 1 ° angular intervals up to 180 °. Because the velocities are biaxial, identical readings resulted for the second half of the circles, and these were used in a subset of specimens to validate the technique.

The velocity of longitudinal ultrasonic waves was measured through the cortical bone specimen in the cortical plane at 1 ° intervals around its circumference. A plot of ultrasonic velocity by orientation in an orthotropic material yielded a curve resembling a sinusoid if two of the three material axes were in the plane of measurement. The direction corresponding to the apogee (Axis 3) is that of greatest stiffness in that plane, and the direction corresponding to the nadir (Axis 2) should always be at 90 ° to the apogee and is the direction of least stiffness. The remaining material axis (Axis 1) would be orthogonal to the first two. In these experiments, we have assumed that this third axis is normal or radial to the cortical plane. Based on previous experiments (Yoon & Katz, 1976; Ashman et al. 1984; Schwartz-Dabney & Dechow, 2002a, 2003), this is a reasonable assumption, but it lacks empirical validation. As we suggest in the Discussion of this paper, our results do support this supposition in the femur and in most regions of the skull.

Following the collection of the ultrasonic velocities, the data were fitted to a sinusoidal model using the sinfit function (a least squares fit) in Mathcad Professional, and parameter coefficients for each sine curve were generated. Plots of ultrasonic velocities by angle and the coefficients of the sine functions for each bone specimen were compared. The coefficients describe the relevant parameters of the curves. Coefficient A is the height of curve above and below the mean velocity (coefficient C). The value of A below C relative to the value of A above C defines the magnitude of anisotropy (or orthotropy) in the relevant plane. Coefficient B is the angle of the principal axis from the reference line (Fig. 3) and thus describes the direction of the axis of greatest stiffness. We compared these parameters among all 30 specimens.

image

Figure 3.  The data from each set of ultrasonic velocity measurements were fit to a sinusoidal model f(x) using the sinfit function (a least squares fit) in Mathcad Professional (PTC) to generate parameter coefficients for each sine curve. Each coefficient of the sine function has a different meaning: A: height of curve (degree of orthotropy); B: orientation or the angle of the principal axis in the cortical plane from the reference line; and C: mean velocity.

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We were also interested in the goodness of fit of the data. Correlation coefficients were generated for each plot between measured values and the idealized sine curve as generated by the sinfit function. For specimens that showed the larger deviations from the sine functions, we examined the curves to assess if they were symmetrical, whether they had broad peaks and troughs, and whether the peaks and troughs maintained a 90 ° separation from each other. We consider the morphological and elastic significance of these variations in the Discussion.

We also used the data set to assess the reliability and validity for determining the material axes of cortical bone specimens with different degrees of angular sampling. In a previous investigation (Schwartz-Dabney & Dechow, 2003), data were collected at approximately 22.5 ° intervals. Here, we modified our velocity curves by dropping data to generate data sets collected at 1, 2, 4, 6, 12, 22 and 45 ° intervals for each of the 30 cortical bone specimens. We plotted sine curve coefficients for each specimen for each simulated sampling. This information provided an estimate of the increasing error associated with reduced angular sampling rates in the examined areas of the skeleton. This information allowed a reasonable estimate of what angular intervals should be used to most accurately measure material axes. Because each data set for each specimen was collected twice, we were also able to assess repeatability of our measurements. The combination of the repeatability data and the assessment of the effects of different angular intervals provided estimates of the potential of these factors for generating error.

For an alternate graphical rendering of each specimen, wave velocity maps were generated with the radar graphic function in Microsoft Excel (Fig. 4). The greater the amount of anisotropy, the narrower the waist of the generated ellipse in these graphs. The length and width of the ellipse reveals the maximum and minimum ultrasonic velocities of the specimen, and corresponds to the relative maximum and minimum stiffness, respectively, in the cortical plane. The orientation of the ellipse shows the direction of the material axes.

image

Figure 4.  Radar graphic function in Microsoft Excel showing an orthotropic pattern (hourglass shape) for most cortical bone samples. The length and width of the ellipse reveals the maximum and minimum ultrasonic velocities of the specimen, which correspond to the relative maximum and minimum stiffness, respectively, in the cortical plane. The orientation of the ellipse shows the direction of the principal material axes. Note that the velocity scale of the graph ranges between 3.0 and 3.8 × 103 m s−1. For contrast, the inset shows the data plotted as a sine curve over a 180 ° range. The data are mirrored in the radar graph to show the velocity range throughout a full 360 ° range.

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Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References

Data from all femoral and mandibular specimens approximated the idealized sine curves, and maximum and minimum ultrasonic velocities were oriented at 90 ° to each other (Figs 5 and 6). However, skull samples showed a variable response, with some samples fitting the expected pattern while others showed notable deviations from a sine curve. The correlation coefficients (Table 1) show that all specimens from the femur and mandible are highly significant (> 0.98), whereas half of the skull specimens have coefficients of < 0.9, indicating a poorer fit with the model (Fig. 7;Table 1).

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Figure 5.  Sinfit for ultrasonic velocities of the femoral specimens (y-axis scale is 103 m s−1; x-axis scale is degrees). All femoral specimens show similar sinfit characteristics. The sine peak or the maximum stiffness angle occurs near 0 ° and has the same orientation as the reference line (long axis of the femur) in all specimens.

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image

Figure 6.  Sinfit for ultrasonic velocities of the mandibular specimens (y-axis scale is 103 m s−1; x-axis scale is degrees). Although most mandibular specimens show a good fit with the model, the amplitudes and the orientations of their axes of maximum stiffness vary according to anatomical region. 0 ° is the orientation of the occlusal plane with degrees increasing clockwise.

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Table 1.   Velocities along axes of maximum (3) and minimum (2) stiffness in the cortical plane and normal to that plane (1); and coefficients from the sinfit function (Fig. 2).
 Ultrasonic velocity (103 m s−1)Coefficients of sine curvesR
Axis 1Axis 2Axis 3Coeff ACoeff BCoeff C
  1. *Mean vector and circular standard deviation; Rayleigh test for the femur indicates significant mean orientation at < 0.01. NS for Coefficient B for the mandible and skull indicates no significant mean orientation. R is the correlation coefficient.

Mandible
 M12.6013.3144.0140.2991.4883.6740.990
 M22.7123.3434.1630.3940.8683.7270.992
 M32.7603.2514.1810.4721.7353.7100.997
 M42.1393.4464.3140.4152.7373.8820.996
 M52.2123.5824.0910.2312.5333.8500.979
 M62.3853.1913.9310.3431.5573.5330.993
 M72.1753.3374.2470.4104.3053.7670.992
 M82.4053.4214.1180.3232.4493.7460.993
 M92.7113.3433.9720.2843.6343.6730.988
 M102.7643.5624.0390.1995.8093.7920.989
 Mean2.4863.3794.1070.337NS3.7350.991
 SD0.2530.1250.1220.087NS0.0990.005
Femur
 F13.2303.2623.9220.3201.5733.5750.998
 F23.0163.2073.8740.3261.4143.5250.998
 F32.8853.2683.9060.3111.3323.5710.998
 F42.3883.3193.8820.2681.5313.5900.996
 F53.0213.3373.8740.2471.5013.6010.998
 F62.8633.3663.9720.2791.6383.6200.988
 F73.0863.3663.8590.2371.5073.5980.997
 F83.1403.3083.9300.2901.7733.5970.996
 F92.5903.3023.8900.2771.8333.5800.998
 F102.9493.3023.9300.2981.7233.6130.998
 Mean2.9173.3043.9040.2851.582*3.5870.996
 SD0.2560.0490.0350.0300.152*0.0270.003
Skull
 Z12.5973.2973.7300.1822.0083.5500.950
 Fr12.6243.4833.6270.0725.4583.5550.915
 Fr23.0053.4903.6440.0271.1363.5530.537
 Fr32.7543.2353.4090.0271.1083.3270.561
 P12.9793.2183.5160.1104.7643.3660.978
 P22.9793.5763.8350.0550.8163.7220.571
 T12.8173.4153.8980.2065.7623.6750.989
 T22.3253.4973.7010.0656.093.5800.916
 O12.9293.2403.4710.0540.5123.3930.778
 O22.7973.4093.6230.0672.7523.5120.896
 Mean2.7813.3803.6410.133NS3.5190.810
 SD0.2160.1260.1560.156NS0.1280.185
image

Figure 7.  Sinfit for ultrasonic velocities of the skull specimens (y-axis scale is 103 m s−1; x-axis scale is degrees). Some skull specimens show a poor fit to the model due to irregular curve patterns, as indicated by the low correlation coefficients between experimental results and results predicted from the sine curve model. Coefficient B for many cranial samples is small, indicating less anisotropy in the cortical plane. Only a few specimens show a pronounced orientation of maximum stiffness in the cortical plane (see Fig. 8).

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Data from each femoral specimen fit with the model with little deviation from the measured velocities, and all three coefficients were similar at each site (Fig. 5). All coefficients from femoral specimens showed a very small range of variation, suggesting nearly identical elastic characteristics (Table 1). In each graph (Fig. 5), the peak of the sine curve represents the maximum velocity and it is located at the starting point of the curve, which is the 0 ° position. This shows that the axis of maximum stiffness (principal materials axis) is in the same direction as the reference line or anatomical long axis of the femur. Wave velocity maps show a similar pattern of principal axis orientations and orthotropic characteristics (Fig. 8).

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Figure 8.  Velocity map of cortical samples using the radar graphic of Microsoft Excel. The principal stiffness axis in the cortical plane or maximum ultrasonic velocity axis is oriented along the long anatomical axis of femur. Mandible specimens show different orientations of principal stiffness axes according to their anatomical region, but similar amounts of anisotropy. Radar graphs of velocity maps from skull specimens are less consistent than in the mandible and femur, showing less anisotropy in the cortical plane.

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The mandibular data sets also fit the sine curve model well (Fig. 6), but the coefficients vary, indicating variations in anisotropy, the orientation of the material axes and stiffness. These variations are more easily visualized on the wave velocity maps from the mandible (Fig. 8). Along the corpus of mandible, degrees of orthotropy and stiffness are similar, while angular positions of the axis of maximum stiffness parallel the occlusal plane (specimens M3, M4, M7 and M8). Specimens M5 and M10 from the ramus show elliptic circles rather than hourglass shapes, which indicate less orthotropy than specimens from the mandibular corpus. The results suggest that specimens from the alveolar and coronoid regions are also orthotropic, although they are less stiff than other mandibular specimens (Fig. 8).

The data sets from the skull specimens show the most variation in the sine curve coefficients and have a much poorer fit to the sine curve model at some sites. P1, Z1 and T1 fit the sine curve model well (R values of 0.98, 0.95 and 0.99, respectively), while T2, Fr1, O2 and, to a lesser extent, O1, have irregular peaks and troughs with high residuals, although these sites basically still conform to the model (R values of 0.92, 0.92, 0.90 and 0.78). P2, Fr2 and Fr3 have irregular patterns with two peaks spaced at < 180 ° and low correlation coefficients (Fig. 7), suggesting that bone from these regions cannot be modeled as orthotropic at a tissue or supraosteonal level of organization. (Fig. 8).

We also validated the reliability of our angular measurements. We computed and compared sine wave coefficients from the velocity data at intervals of 1, 2, 4, 6, 12, 22 and 45 °. The interval data at 1 ° were used as a base comparison to data from all other angular intervals. Overall, as the angular interval of measurement increases, so does the error (Fig. 9). However, there is considerably less error when the sinefit function is used to compute the orientations of the axes of maximum stiffness than when these orientations are determined based on absolute maximum and minimum data values.

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Figure 9.  Error analysis for estimating the angle of maximum or principal stiffness in the cortical plane using the sinfit function. The solid black line with solid circles shows the maximum error vs. angular intervals of sampling, when the axes are determined based on maximum and minimum values alone without modeling the data with a sinfit function. The solid black line with solid squares shows an error reduction of 50% from the solid line with solid circles. The short dash line with open circles shows results from mandibular cortical specimens, while the long dash line with open squares shows results from femoral specimens when the data were modeled with sinfit functions.

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Repeatability errors from each coefficient, A, B and C, were compared between the two sets of measurements for each femoral and mandibular specimen. The data set from skull cortical bone has been excluded as its data were more irregular in that many of the skull specimens did not fit the sine curve model. There was virtually no difference in coefficients A and C between measurements. However, some repeatability error was apparent in coefficient B, which is the angular orientation of the axis of maximum stiffness. Overall, this error increases at the largest angular intervals of measurement.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References

Methodological considerations

To analyze the mechanical properties of cortical bone, we used an ultrasonic technique in which the material stiffness can be determined because it is proportional to ultrasonic velocity. This method has a good theoretical background and provides a strong advantage in the cranial region where it is possible to prepare the needed small-sized and disc-shaped specimens (Ashman et al. 1984; Schwartz-Dabney & Dechow, 2002a).

The shape of these specimens allows measurement of ultrasonic velocities in multiple directions to determine the planes of symmetry. The direction with the highest velocity corresponds to the axis of maximum stiffness because the ultrasonic waves pass more rapidly as stiffness increases. The axis of minimum stiffness corresponds with the lowest velocity. In orthotropic materials, this axis is perpendicular to the axis of maximum stiffness. If the material is orthotropic and correctly oriented, the ultrasonic velocities can be combined with measurements of density to calculate a matrix of elastic coefficients (‘C’ matrix) and technical constants (elastic and shear moduli, and Poisson’s ratios; Ashman et al. 1984). Overall, the magnitude of the ultrasonic velocities in the bone specimens correlates with the magnitude of the elastic moduli.

In our laboratory, we have adapted the pulse transmission ultrasonic technique described by Ashman et al. (1984) for use with specimens of cortical craniofacial bone. As originally described, the technique was validated for the measurement of orthotropic cubic specimens with dimensions > 5.0 mm and known principal axes. However, two important issues were not considered: (i) can bulk specimens of cortical bone be considered orthotropic; and (ii) assuming they can be considered orthotropic, are the principal material axes of the specimens oriented with the long axis or some other morphological feature of the bone?

Can bulk specimens of cortical bone be considered orthotropic?

Early microscopic studies suggested a mechanical role of osteonal structure in cortical bone (Ascenzi & Bonucci, 1967, 1968, 1976). Because most functional loads like compression and bending on long bones indicate maximum stress along their anatomical long axis, and because osteons are also primarily oriented in this direction, it was reasonable to model the cortical bones as an orthotropic or a transversely isotropic composite material (Katz, 1971; Martin, 1999). However, this line of reasoning is less relevant to cranial bone as the patterns of mechanical loading are different from bone like that of the femur, and the internal osteonal structure of craniofacial cortical bone has not been quantified in a meaningful way.

Some of our previous studies, which showed variation in human cranial elastic properties, were made with an assumption of orthotropic structure (Peterson & Dechow, 2002; Schwartz-Dabney & Dechow, 2002a,b, 2003). Here our attention to off-axis ultrasonic velocities allows an assessment of that assumption. All mandibular and femoral specimens in this study followed the sine curve model in off-axis ultrasonic velocity measurements, while some skull specimens did not fit the model. These results imply that cortical bone in the skull has a different internal structure from that of either the femur or mandible, and that mandibular cortical bone has more variations than that of the femur in its internal structure. These differences should be explained by variations in three-dimensional osteonal structure and, in some cranial regions, by the structure of internal porosities in the bone.

The mechanical characteristics of cortical bone, especially in postcranial bones like the femur, have been studied assuming a symmetric structure such as transverse isotropy or orthotropy (Yoon & Katz, 1976; Ashman et al. 1984). This assumption was initially based on histomorphological observation in two-dimensional sections (Yoon & Katz, 1976), and has resulted in subsequent discussion regarding the orientations of symmetry in the cortex of the diaphyses of various postcranial bones (Van Buskirk et al. 1981; Ashman et al. 1984; Katz & Meunier, 1987). It has been suggested that a mechanical understanding of cortical bone structure is not improved by using an orthotropic model compared with a transversely isotropic model, as there is little difference between transversely isotropic elastic constants and orthotropic elastic constants (Van Buskirk et al. 1981), but this assertion is not without dispute (Ashman et al. 1984). However, this may not be true in cranial bones, like the mandible, where larger differences in elastic properties have been found between tangential and circumferential directions (Schwartz-Dabney & Dechow, 2003), and may reflect greater variation in osteon orientation within the cortical plane (Dechow et al. 2008).

Because all data in this study were made in one plane, the cortical plane, this raises the question of whether two axes of symmetry are actually found within this plane. We made this initial assumption based on previous work (Schwartz-Dabney & Dechow, 2003) that showed lower ultrasonic velocities tangent to the cortical plane than in any direction in that plane for nearly all specimens. However, our findings do provide evidence that for most specimens in the current investigation, two axes are approximately within the cortical plane. If the axes were not in this plane, we would expect to see deviations from our idealized curves in the form of asymmetries between the shoulders of the curves. However, almost all curves derived from cortical bone of the mandible and femur show good symmetry, although a few specimens show a slightly skewed pattern, which may indicate that in these specimens the orthotropic axes deviate from the cortical plane. How consistent such patterns might be in regions of various bones or whether deviations might reflect variations in gross cortical curvature remain questions of interest.

Are the principal material axes in bone tissue oriented with the long axis or some other morphological feature of the skeletal structure?

In previous studies on human bone (Peterson & Dechow, 2003; Schwartz-Dabney & Dechow, 2002a,b, 2003; Peterson et al. 2006), we found that gross skeletal shape could not be used to predict material axes in much of the craniofacial skeleton. Here, with greater emphasis on methodology for showing material orientation, we show a similar result for some skeletal sites. Clearly, femoral sites had great similarity and could be accurately predicted by skeletal shape. In the mandible, some regions have a similar pattern (longitudinal orientation in the corpus for example), while in the cranium, patterns are less well defined.

These variations raise the issue of whether the results here can be generalized to other skeletal organs and to the same organs in other vertebrates. We suggest that the variation found within this sample suggests a need for more widespread sampling of different types of skeletal organs and systems in different organisms at different stages of development.

Using ultrasound to measure bulk bone tissue symmetries

Methodologically, we approached this problem by using cylindrically shaped specimens. These allowed measurement of ultrasonic velocities around the specimen perimeter to determine the direction of maximum and minimum velocities of the longitudinal waves, which then corresponded to the orientations of the axes of maximum and minimum stiffness in the plane parallel to the surface of the cortical bone. This approach gave a good picture of craniofacial regional variation in the material axes in the cortical plane.

The technique was validated by comparing results from cubic and cylindrical specimens (Schwartz-Dabney & Dechow, 2002a), and by repeated measurements on a subset of mandibular samples to demonstrate reproducibility of the identification of the axes of minimum and maximum stiffness. However, there are several limitations to this methodology. Due to difficulties involved in manual rotation and assuring axis reproducibility in 4.0-mm-diameter specimens, velocity readings were only taken in 22.5 ° increments. This limitation means that in any individual specimen, axes had a maximum potential error of 11.25 °. A theoretical analysis of ‘off-axis’ velocity measurements through cubic specimens predicted errors of 1.3% in elastic moduli and 5.0% in shear moduli for measurements taken at 10 ° off-axis. This error exponentially increased to 4% and 18%, respectively, at 20 ° off-axis (Turner & Cowin, 1988). This error should not affect the calculation of mean values, but will contribute to an increase in variance. For measurements of individual samples, so that associations with morphological features can be explored, greater precision can be achieved by greater angular sampling.

The data from this experiment allow a refinement of our technique so that the material axes of the cortical bone specimens can be measured with greater precision. We can determine: (i) the reliability with which each of the sine wave coefficients can be measured; and (ii) what sampling of velocities around the perimeter of a bone cylinder is necessary to get the most accurate measurements. Our goal is to achieve the best three-dimensional elastic property data for comparison with three-dimensional measurements of microstructural features of cortical bone.

For achieving this purpose, we have modified our technique through the use of a miniature rotary table. We have added this table to a larger structure that allows accurate positioning of the ultrasonic transducers above it. Previously, all positioning of our bone specimens was done by hand, with visual alignment of the transducers with nine axes on the surface of the cylinders. Bone cylinders were supported by the pressure of the two transducer faces. With our new technique, we were concerned that attachment of the dowel to the middle of the base of the cortical bone cylinder would alter the transmission speeds of the ultrasound. However, tests comparing results with and without an attached dowel indicated no effect.

We also increased the size of our bone cylinders (10 mm in the current study vs. 4 mm in previous work), primarily to provide larger bone specimens for histological study and mechanical testing. Experiments with rods of mandibular cortical bone from various mandibular sites have shown that we can vary the diameter of the cylinders between 2 and 10 mm and obtain identical velocities (Schwartz-Dabney & Dechow, 2002a).

When specimens of cortical bone are modeled as orthotropic, what level of angular sampling is needed to show the principal axis of the orthotropy most accurately? Our repeatability analysis indicated slightly more repeatability error in the mandibular specimens as the angular interval increased. The femoral specimens showed less than a 5 ° repeatable error, although there were a few outliers. Figure 9 shows the reduction in error for estimating the angle of the principle material axes when the data are fit with a sine function. Our results suggest that velocity measurements are best taken at a 12 ° interval for mandibular cortical bone, and 15 ° intervals for femoral cortical bone (Fig. 9). Practically, 10 ° intervals are suggested. Because femoral specimens reveal very consistent results and closely resemble the sinusoidal model, their principal axes can be found even with a larger angular interval. Conversely, mandibular specimens exhibited more variation, probably because they were taken from different regions of the mandible, with a wider range of functions and loadings. Therefore, we could expect that there might be variations in their internal structure that merit investigation.

Developmental and functional considerations

Cortical bones from skull and femur have different developmental origins. Femoral cortical bone is developed from lateral plate mesoderm and starts its formation with a cartilaginous anlage, which then leads to differentiation and deposition of bone with osteoblasts forming osteoid on the cartilagenous framework. The cranial vault is comprised of several dermal bones that osteoblasts form from a calcified matrix within a collagen framework in a process of intramembranous bone formation. The contribution of neural crest is important in the development of most cranial bones. A small portion of the mandible forms from Meckel’s cartilage, but the majority results from intramembranous bone formation (Carlson, 1999). It is unclear whether developmental differences may affect the mechanical properties, although regional differences in these properties may result from a complex interplay of developmental origin, variations in cell signaling mechanisms, patterns of growth and biomechanical factors.

The different structures and functions of the mandible, crania and femur suggest different mechanical roles for cortical bone within and between these regions. Support of human weight is done by the femur. In the head region, the skull’s basic role is protecting the brain, while mandible and midface provide the necessary structure for masticatory and other orofacial processes.

Results indicated that cortical bone from the femur and mandible approximates an orthotropic structure at the bulk tissue level of organization. The femoral specimens showed the most regularity among sites in all coefficients, but despite greater variation in material orientation, stiffness and anisotropy, mandibular sites showed a similar orthotropic organization to that of the femur. However, the differences between mandibular and femoral cortical bone are sufficient to suggest differences in microstructure, a topic requiring further exploration. The cranium, which performs primarily a protective function for the brain, reveals more irregularities and differences from an orthotropic structural model. Its strains during functions like biting and mastication are low, suggesting that its main role in resisting mechanical loading would be less than other parts of the skeleton, although mastication is not the only function that can be related to the structural variations in the skull. We also have to consider the implications of ontogenetic and interspecific variation (Ravosa et al. 2000).

Very little is known about actual bone strains in the human cranium during function, although there are likely similarities in loading patterns to macaque monkeys, on which relevant experiments have been conducted (Hylander & Johnson, 1992). In macaques, there are higher strains along the zygomatic arch (200 and −70 μЄ for maximum and minimum strain, respectively) and lower strains along orbital regions (50 and −20 μЄ). Our results show differences in elastic properties between the zygomatic arch (Z1) and post orbital area (F4), where Z1 has greater anisotropy (Coeff A: 0.182) and stiffness (Coeff C: 3.55) than F4 (Coeff A: 0.027; Coeff C: 0.0327). Mastication experiments (Hylander & Johnson, 1992) on macaques also show relatively higher strain ranges: 388 μЄ (maximum) and −410 μЄ (minimum) on the working side, compared with the averages of 291 and −414 μЄ on the balancing side. On the balancing side, maximum principal stresses are oriented nearly perpendicular to the lower border of the mandible, which supports the notion that during the power stroke the mandible is both bent and slightly twisted during mastication (Dechow & Hylander, 2000). From the mechanical behavior during mastication, we expect that the orientation of the maximum stiffness is aligned along the lower border of the mandible, which is the principal axis of bending.

Even though an orthotropic pattern is not apparent at many cranial sites, it is still possible that some of these sites can be modeled as orthotropic. P4, Z1 and T3 fit with the sine curve model (R values of 0.98, 0.95 and 0.99, respectively). Although orthotropic patterns in sites Z1, P4 and T3 were not detected in our former research (Peterson & Dechow, 2003), we suggest the larger size of our specimens helps to illustrate the anisotropic variation in those regions. Sites T12, F1 and O2 have broad peaks and troughs with high residuals (R values of 0.92, 0.93 and 0.90), and their orthotropy patterns are less clear. Interestingly, sites P13 and F3 have two peaks (Fig. 7). The two peaks of P13 suggest a combined pattern of those found in sites P6 and P13 from our previous study; the previous study used smaller specimens that were contained roughly within the same area as the larger specimen used here (Peterson & Dechow, 2003).

Most skull specimens, except T3, Z1 and P4, had a low degree of orthotropy, although their average stiffness, as expressed by coefficient C, was within the range of similar values for the mandible and femur (Table 1). Clearly, variation in functional strain between the mandible (high) and the cranial vault (low) does not explain variation in mechanical stiffness. However, variation in the average direction and magnitude of functional strain and its variance could be related to variations in anisotropy, including whether or not bulk cortical tissues can be approximated as having an orthotropic elastic structure.

Some cranial sites showed unique patterns of curves, like broad or steep peaks and troughs or two peaks. Although the degree of orthotropy as expressed by coefficient A is low in most skull samples, it is possible that these represent out-of-plane measurements and not simply specimens that lack an orthotropic organization, or they may be indicative of complex composite characteristics like in a multilayered fibered composite (Zemčík, 2002), which is of greatest relevance if the whole structure of cranial vault bone is considered. A simpler possibility is that many cranial cortical sites might be better modeled as transversely isotropic with similar elastic properties in all directions parallel with the cortical plane, which contrasts with approximate transverse isotropy in bones like the femur where elastic properties are most similar in a cross-sectional plane including radial and tangential orientations. This suggests that approximations of bulk transverse isotropy in the cranium result from very different microstructural organizations than in the femur, where the pattern is thought to reflect the longitudinally oriented osteonal structure (Yoon & Katz, 1976; Katz & Meunier, 1987).

While it seems reasonable to conclude that the pattern of orthotropy in the femur is related to the osteonal structure, few investigation have attempted to quantify variation in three-dimensional off-axis elastic properties in conjunction with osteon orientation in cortical bone from any region (Dechow et al. 2008). Rather, existing references have provided qualitative descriptions of Harversian canal orientations in a few postcranial bones (Stout et al. 1999; Cooper et al. 2003).

Predicting orthotropy from sinusoidal curves

If human cortical bone, especially femur and mandible, are orthotropic or transversely isotropic, the axis of principal stiffness should be 90 ° from the axis of minimum stiffness by definition. Although the assessment of how well data from individual sites fit a sine curve is a useful tool for analyzing possible orthotropic patterns of cortical bone, can we justify the cortical bone as an orthotropic material only by a fit with a sinusoidal model? First, some curves are skewed from a model curve even though the degree of skew is very small (for example, M2, M6 and M7). We assume that there could be minor structural variations that result in this pattern. Second, there are some mismatches between the peak value of the model curve and the velocity data, mainly because of variation in the breadth of the curves close to the peaks and troughs (M3, M10 and L8). Repeated measurements verified these patterns in these particular specimens.

Ultrasonic velocities in orthotropic media may appear to vary as sine functions in planes containing two of three material axes with some caveats. Theoretically, such curves may show shapes that deviate from a sine cure by having relatively narrow peaks and broad troughs, or broad peaks and narrows troughs, and still conform to an orthotropic elastic model (Pearson & Murri, 1986; Kline, 1992; Zemčík, 2002). However, in the case of all specimens examined here, modeling using more complex polynomial functions did not yield a better fit of the data and thus did not improve the results. Conversely, a simple model based on a sine curve provides exponents that are easily interpretable in the context of this investigation.

It is possible that application of different theoretical curves, such as polynomial or exponential curves, might reveal correspondence with other morphological features, such as unique fibered composite characteristics. Hierarchical structures of cortical bone contain different types of composite structures from the nanostructure of collagen fibers and bone crystals to the microstructure of osteons and interstitial bone (Katz, 1971; Martin, 1999). For instance, we expect that bone with low anisotropy might have minimal structural organization (woven bone), or it might have more randomly oriented osteons. Bone with greater anisotropy might have strongly unidirectionally oriented osteons or a multiple structure comprised with major orientations of fiber groups, which determine the mean principal stiffness orientation. Broad peaks or shoulders of the curves may represent a predominant or skewed range of osteon or microfiber orientations in the tissues. Different theoretical approaches might be applied for these patterns. Yoon & Katz (1976) used histomorphological observations of Haversian canal structure in remodeled cortical bone to hypothesize transverse isotropic symmetry. However, the actual osteonal arrangement is not ideally hexagonal as they mentioned (Ashman et al. 1984). There are many structural variations, although orthotropy or transverse isotropy are reasonable approximations of elastic structure for most specimens in this investigation.

Acknowledgements

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References

This work was partially supported by NSF Grant BCS 0240865 and Physical Anthropology HOMINID Program Grant 0523159. We thank Drs Jason Griggs, John Cai and Carina Schwartz-Dabney for their comments regarding this study.

Authors’ contributions

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References

Dong Hwa Chung: laboratory work; data collection; aided in experimental design and analysis; initial draft of manuscript. Paul C. Dechow: designed study and analysis; writing, analysis and final editing of the manuscript. All work was conducted in Dr Dechow’s laboratory at the Texas A&M Health Science Center Baylor College of Dentistry.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. Authors’ contributions
  9. References