Abstract: Finite element analysis can accurately describe the biomechanical traits of mandibular bone specimens, and obtaining accurate elastic modulus is necessary to ensure the consistency between finite element analysis and experimental data results. This paper reviews the current status of research on the elastic modulus of human mandibular cortical and cancellous bone, and discusses the differences in the elastic modulus of mandibular bone in different species of animals, as well as the factors affecting the elastic modulus of mandibular bone and the measurement of the elastic modulus.
Keywords: mandible, finite element analysis, elastic modulus
The mandible is the largest and most robust of the facial bones and the only movable of the craniofacial bones. It forms a unity with the related muscles, joints and teeth, and a highly developed masticatory system coordinated by nerves; therefore, its biomechanical behavior is important in various clinical states.
Obtaining various kinetic and static parameters of the mandible is a fundamental part of human biomechanics research and is the basis for building a mechanical model of the human body. Numerous studies (Hart 1992, Korioth 1997, Voo 1996, etc.) [1] have shown that finite element analysis (FEA) can describe the biomechanical properties of mandibular specimens more accurately (correlation coefficient up to 0.992 [1]). Therefore, it is particularly important to study the elastic constants of mandibular bone, because if wrong data are quoted, the results can be “lost by a thousand miles”.
1.Elastic constants of mandible
The mandible has a similar stress-strain relationship with engineering materials, and follows Hooke’s law in the elastic limit. The elastic constants required for modeling include the modulus of elasticity E, shear modulus G, and Poisson’s ratio ν. The modulus of elasticity is a measure of hardness. For a composite material such as bone tissue, its elastic constants vary with its degree of symmetry. Anisotropic materials can have 36 elastic constants [2], fully anisotropic with 21 independent elastic components, and orthogonal anisotropy reduces the elastic components to 9. Some scholars believe that this reflects the anisotropy of mandibular materials to some extent and is simpler and more feasible [2]; there are also many scholars who prefer to describe mandibles with transversely isotropic 5 elastic constants; only 3 in isotropic materials The relationship between the three elastic constants is G=1/2×E/ (1+ν), therefore, only two of the three elastic constants are independent at this point; when the elastic constants are not influenced by the orientation, i.e., the perfectly symmetric isotropic material has only two elastic constants: Young’s modulus of elasticity (E) and Poisson’s ratio (ν) [2]. The human mandible is an anisotropic material.
2. Modulus of elasticity of mandibles of different species of animals
The mandibles of many large animals, such as sheep and cattle, are mainly composed of plexiform cortical bone with elastic constants different from those of cortical bone composed of the Harvard’s system [3], and to some extent, human bone differs from bovine bone in that the former is isotropic in a transverse view and the latter is orthotropic and the two have very different ontogenetic relationships [4]. with E = 7.5 Gpa and ν = 0.4. (However, dog femoral cortical bone exhibits orthogonal anisotropy [6].) Kawahara et al [7] measured an E value of 12.8 ± 3.1 Gpa in Beagle dogs.
3. Modulus of elasticity of human mandibular bone
(1) Elastic modulus of human mandibular cortical bone In the oriented structure of mandibular cortical bone, the fiber orientation determines the direction of the combined force and constitutes the force pillar [8]. Many scholars believe that the anisotropic character of mandibular E values is caused by the orientation of collagen fibers.Lettry et al [9] studied five (53-106 years old) fresh human mandibles to observe the E values of cortical bone at different sites when taken in the same direction and at the same (adjacent) site when taken in different directions, and the results showed that the human mandibular cortical bone had E values were significantly lower in the cortical bone near the alveolar bone in the premolar region than in the cortical bone away from the alveolar bone (near the inferior margin); the E values of the cortical bone near the inferior margin in the molar region were higher than those away from the inferior margin (near the alveolar bone), but there was no statistical difference; the E values of the cortical bone in the molar region near the alveolar bone were significantly higher compared with those in the premolar region. The results also showed that the E values of cortical bone tested after cutting at a certain angle to the long axis of the mandibular body (0, 45, and 90 degrees from the long axis, respectively) were different, and the E values decreased with increasing angle. It can be seen that there is a significant anisotropy in the elastic modulus of mandibular cortical bone. In the study Lettry [9] also compared the results of Tamatsu et al [11] using the method described by Bland [10] and found that the storage conditions of the bone specimens used for testing E values had an effect on the results: Lettry et al [9] placed the bone always in saline with a pH of 7.4 (or in a -18°C refrigerator if it needed to be left for a longer period of time) and the E value results The E values ranged from 4732-10077 MPa, whereas Tamatsu et al [11] used mandibles that were not “fresh” wet bone, but dry mandibular bone that had been wetted and measured, which has been shown to have some changes in physical properties [12], with E values ranging from 12,600-21,000 MPa. 21000 Mpa.Of course, the inconsistency in the results of the two sets of experiments is not sufficient to account for the problem, since it was not obtained on the same test bone.
Cortical bone is dense and hard, its E value is generally about two orders of magnitude higher than that of cancellous bone at the corresponding site, and the stress value in the former is 20-30 times greater than that in the latter under load [1]. Therefore, the closer the range of cortical and cancellous bone of the constructed model is to the actual situation, the closer the calculated results will be to the actual measurements. Some scholars have replaced the respective E values of cortical bone and cancellous bone with intermediate values of the E values of both, and regarded the mandible as a completely homogeneous and isotropic material, but of course this method is generally applied to more rough qualitative studies, but it can also reflect some problems easily and intuitively. Most scholars still consider cortical bone and cancellous bone separately.
(2) The elastic modulus of human mandibular cancellous bone Goldstein et al [13] found that the E values of cancellous bone at different locations in the same metaphysis differed by a factor of 100, indicating that cancellous bone is highly inhomogeneous. These findings confirm Wolff’s law that the different functions of cancellous bone at different anatomical sites directly affect the mechanical properties of its own structure, and therefore, the study of cancellous bone has received increasing attention.
Although the main biomechanical characteristics of the mandible are determined by the dense bone, the thickness and number of cancellous bone and bone trabeculae are functionally related, and they are arranged into a dentition track and a muscle track to transmit masticatory forces. The elastic modulus of mandibular cancellous bone is more complex than that of cortical bone, so cancellous bone parameters from other sites have been used to study mandibular bone. For example, in the study of Hart et al [14], the E value of mandibular cancellous bone was derived from fibula (whose material parameters of cancellous bone were taken from Dr. Turner’s 1987 paper). In recent years, with the development of high-resolution CT equipment and other equipment, scholars have been studying it in depth. “O “Mahony et al [17] specifically determined the E value of cancellous bone in a 74-year-old female patient with edentulous jaws and concluded that it was isotropic in cross-section. Some authors have assumed that the mechanical characteristics of cancellous bone under certain conditions are not affected by tissue anisotropy (which can be neglected), and a study by Kabel et al [18] confirmed this assumption, showing that microfinite element analysis with “effectively” isotropic tissue moduli can predict the mechanical characteristics of cancellous bone. ‘Mahony et al [17] obtained Young’s modulus in three orthogonal directions for edentulous mandibular cancellous bone by pressure testing: the Young’s modulus was greatest in the proximal and distal mesial directions with an average of 907 ± 849 MPa, followed by approximately 511 ± 565 MPa in the buccolingual direction and 114 ± 78 MPa in the superior and inferior directions.
(3) Relationship between elastic modulus of human mandible and strain rate and density As part of the human skeleton, the mandible also has more significant viscoelastic properties, so the study of its biomechanical properties must consider the effect of strain rate [19].McElhaney and Byars [4] performed isokinetic compression tests on human bone with strain rates ranging from 0.001/s to 1500/s, and the corresponding E from 2.2 × 106 1b/in2 at low strain rates to 5.9 × 106 1b/in2 at high strain rates.Brown and Ferguson [20] tested E values for similar strain rate intervals (10-4/s to 10-2/s) and found larger E values at high strain rates, but no statistical difference.Carter and Hayes [21] found E values with 0.06 power of strain rate, and Linde et al [22] showed that E values were correlated with 0.05 power of strain rate. The results of Bo Bin et al [23] showed that the E value was statistically significantly correlated with the 0.052nd power of the strain rate. It should also be noted that these reflect the dynamic properties of the mandible, but its strain rate is still not considered high. In China, Yang Guitong et al. made some impact tests of human femur at high strain rates and obtained good experimental information and experience, but there are not many studies for the mandible [4].
As a parameter describing the structural properties of bone, Martens and Ishida et al [24] suggested that bone density varies with the degree of mineralization and porosity of bone, so it also affects the E value.Rho et al [25] established linear and nonlinear equations for anisotropic E value and bone density, and the results showed that the E value is positively correlated by bone density, and in the nonlinear equation it is 1.35 to 1.75 quadratic relationship. In China, Bo Bin et al [23] concluded that E value was correlated with 0.44th power of BMD. Wang Yijin et al [19] also found that the level of bone density tends to decrease with age and the E value also changes.
4. Factors affecting the elastic modulus of the mandible
The elastic constants of mandibular bone are difficult to obtain from in vivo, and isolated tissues are the main source for obtaining data. However, the E value can be influenced by a variety of external factors such as the site of extraction, test environment, method, test conditions, specimen fabrication, load direction, strain rate magnitude, etc. It can also be influenced by internal factors such as the origin of the specimen species, age, gender, body mass, as well as the content and arrangement of collagen, the role of living soft tissues, and the feedback regulation of nerves and body fluids. Therefore, the available data on mandibular E values can vary somewhat, and in some cases, significantly. For example, anatomical structures affect the E value of the human mandible: E values vary near the mandibular foramen, at muscle attachments, at the internal and external oblique lines, and at the sublingual glandular fossa, etc. Generally, E values decrease near the concavity, fossa, and foramen and increase in areas where muscle forces are strengthened. Although it has been suggested that mandibular cortical bone E values are almost similar in people aged between 60 and 90 years [26], most believe that the presence of mandibular teeth affects mandibular cortical bone E values: when no teeth are present the mandible is accompanied by some degree of bone resorption, the cortical bone thins, and the mandibular body is left with 60% of its original size [27], collagen fibers change, and conditions such as mineralization after tooth loss also The bone cancellous density at the base of the mandible will also increase (compensation after tooth loss) [29], all of which may cause changes in the mandibular cortical bone E value.
5, Determination of mandibular bone E values
The elastic constants can be determined by quasi-static mechanical tests or dynamic tests, the latter measuring high data [4]. Mandibular E values are generally obtained by standardized and unified material mechanics experiments so that the results obtained are credible and easy to compare. Specimens are generally made with reference to ASTM (American Society for Testing and Materials) standards [9]. There are various test methods, for example, one author found that when solving the E value by the three-point bending test of cortical bone specimens, the length, width, and height of the specimen are very critical parameters, especially the height h of the specimen, which affects the E value more than other parameters. It was also found that the smaller the specimen, the greater the difference in the obtained E values, and the obtained E values were more constant when the ratio of length to height exceeded 25. Lettry [9] pointed out that the ratio of length to height of the test specimens of Tamatsu et al [11] and others was about 10, which had a greater effect on the results. Recently, other authors [30] used atomic force microscopy techniques to measure mandibular cortical and cancellous bone E values by determining the nanoscale surface deformation curves of the measured tissue with the advantage that no special specimen preparation techniques are required; the differences in their E values were determined without affecting the microstructure or composition of the tissue.
Since the E values of living mandibles cannot be determined by any destructive experiments, scholars have developed CT techniques and ultrasonic techniques for measuring in vivo E values. There is a linear relationship between the CT value of any point in the bone (Hounsfield) and bone density, and many scholars have established a relationship between E value and density [23], such as the Carter-Hayes empirical formula [21], so that the E value of a point in the mandible can be derived from the CT value. However, some scholars hold a different view that the structure of the mandible will change due to age and that bone density will become not an accurate predictor of E value, such as Lettry et al [9] who argued that E value has a weak correlation with CT value (A weak correlation) and that using CT value to accurately predict bone material properties is not sufficient.
Abendschein and Hyatt [4, 31] found a high correlation between ultrasonic velocity and the E value and density of cortical bone specimens, with the ability to propagate both shear-related transverse waves and capacitance (or length)-related longitudinal waves in solids, with the wave velocity of longitudinal waves = (Young’s E value/density)1/2; and the wave velocity of transverse waves = (shear E value/density)1/2. It should be noted that is that this formula has limitations, and Yoon and Katz [4, 31] applied the generalized Cosserat theory to study the mechanism of ultrasound propagation in bone, pointing out that the propagation of ultrasound waves in bone may have other mechanisms, such as dispersion, that are not well understood, in addition to viscoelasticity.
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