Comparing the Distribution of Strains with the Distribution of Bone Tissue in a Human Mandible: A Finite Element Study

Authors

  • Flora Gröning,

    Corresponding author
    1. Department of Engineering, University of Hull, Hull HU6 7RX, UK
    2. Department of Biology, University of York, York YO10 5DD, UK
    3. Centre for Anatomical and Human Sciences, Hull York Medical School, University of York, York YO10 5DD, UK
    • Medical and Biological Engineering Research Group, Department of Engineering, University of Hull, Hull HU6 7RX
    Search for more papers by this author
  • Michael Fagan,

    1. Department of Engineering, University of Hull, Hull HU6 7RX, UK
    2. Centre for Anatomical and Human Sciences, Hull York Medical School, University of York, York YO10 5DD, UK
    Search for more papers by this author
  • Paul O'higgins

    1. Department of Engineering, University of Hull, Hull HU6 7RX, UK
    2. Centre for Anatomical and Human Sciences, Hull York Medical School, University of York, York YO10 5DD, UK
    Search for more papers by this author

Abstract

Comparative anatomy and experimental studies suggest that the mass and distribution of tissue within a bone is adapted to the strains the bone experiences during function. Finite element analysis is a powerful tool that can be used to investigate this since it allows the creation of hypothetical models with unadapted morphology. Here we use FE models of a human mandible with modified internal morphology to study the relationships between the gross distribution of bone tissue (i.e., the presence or absence of bone in a certain area), the variation of cortical bone thickness within the mandible and the distribution of strain magnitudes. We created one model in which all internal cavities were filled with cortical bone material and a second, hollow model with constant cortical bone thickness. In both the models, several load cases representing bites at different positions along the tooth row were applied and peak strain magnitudes across these load cases were calculated. The peak strain distributions in both models show striking similarities with the gross distribution of bone tissue and the variation of cortical thickness in the real mandible, but the correlation coefficients are rather low. These low coefficients could be explained by confounding factors and by the limited spectrum of load cases that were simulated. However, the correspondences we find between strain magnitude and bone tissue distribution suggest that models with altered internal geometry are useful in studying the mechanical adaptation of bone, especially in the absence of any in vivo strain data. Anat Rec, 2013. © 2012 Wiley Periodicals, Inc.

It is widely believed that healthy bones are optimized toward maximum strength or stiffness with minimum mass since an increase in mass is costly in terms of the energy required to form and maintain bone tissue and an increase in weight reduces the efficiency of movements (e.g., Currey, 1984, 2002). This idea is supported by empirical data showing that bone tissue is formed in areas where it is needed to resist physiological loads, but resorbed where it is not required for load resistance (e.g., Jones et al., 1977; Uhthoff and Jaworski, 1978; Goodship et al., 1979; Lanyon et al., 1982). These data also suggest that, together with other factors, strain magnitude is important in the formation, maintenance and resorption of bone tissue (e.g., Rubin and Lanyon, 1985; Lanyon, 1987). An influential hypothesis that is based on strain magnitude as a stimulus for the mechanoregulation of bone is Frost's “mechanostat” (Frost, 1987, 2003). It predicts that bone strains above a certain magnitude cause the formation of new bone and thus an increase in bone mass, while strains below another threshold lead to bone resorption (Fig. 1). If this hypothesis is correct, the distribution of bone tissue within a bone should reflect the strain history of the bone.

Figure 1.

Illustration of the assumed relationship between strain magnitudes and bone mass. As an example the strain magnitudes and bone mass at three locations in the cross-section of a long bone are considered (points A–C). If strains in the bone tissue are beyond a certain threshold (upper dashed line), new bone will form, which results in an increase of bone mass until strains fall below the bone formation threshold (locations A and B). The opposite occurs, when strains are below a certain threshold (lower dashed line) so that bone is resorbed and strains increase (location C).

To date, the relationship between functional strains and the distribution of bone tissue within a bone has been studied mainly in long bones. For example, computer modeling has been used to study functional adaptations of the trabecular bone tissue in the proximal femur (e.g., Huiskes, 2000; van Rietbergen et al., 2003) and numerous experimental studies have investigated mechanically induced adaptations of cortical bone, some of these by including in vivo strain measurements (e.g., Goodship et al., 1979; Lanyon et al., 1982; Rubin and Lanyon, 1985). In general, experimental studies on long bones have shown that cortical bone thickness decreases when bones that normally bear loads are immobilised (e.g., Uhthoff and Jaworski, 1978; Jaworski et al., 1980), or increases when bones are exposed to higher loads than during their usual function (Jones et al., 1977; Lanyon et al., 1982). The experimental data also show some uniformity in the measured strain magnitudes across different long bones and different taxa when a broad spectrum of activities is considered, which might suggest that these bones adapt to similar levels of strain magnitudes (Fritton et al., 2000).

However, the relationship between the strain history of a bone and its morphology does not seem to be simple. In vivo strain measurements in long bones indicate that it is not straightforward to infer the loading history of a bone from its cross-sectional properties, at least with traditional methods (Lieberman et al., 2004). In addition, individuals who differ in their habitual activities can show similar distributions of cortical bone thickness in their femora (Morimoto et al., 2011).

There are several factors that could confound the relationship between functional strains and the distribution of bone mass within a bone. First, it has to be considered that the functions of bones are not purely mechanical and that their mass and morphology are also compromises between different developmental and phylogenetic constraints and physiological demands such as the protection of internal organs, marrow and mineral storage (Ruff et al., 2006). Second, strain magnitude is not the only mechanical signal that has been shown to be important for the adaptive remodeling of bone, but the number of load cycles, the distribution of loading and the rate of strain also play a role (e.g., Lanyon, 1984; Ozcivici et al., 2010). So, while it is to be expected that there is some correlation between the distribution of bone material within a bone and the peak strain magnitudes that the bone has experienced in its loading history (Rubin and Lanyon, 1985), it is unlikely that a perfect correlation is found because of confounding factors.

All of the above studies investigated mechanical adaptations in long bones. Only a few have investigated how the distribution of bone mass within the mandible is related to functional strains (Demes et al., 1984; Daegling, 1989, 2002; Daegling and Grine, 1991; Fukase, 2007; Fukase and Suwa, 2008). Based on this work, it has been hypothesised that the uneven distribution of cortical thickness in the mandible is associated with unevenly distributed strains in the bone during use (Demes et al., 1984; Daegling and Hotzman, 2003).

One example of an uneven distribution of cortical bone in the mandible is found in the posterior corpora of anthropoid mandibles, where cortical bone is thicker buccally than lingually. This has been explained as a result of the combined effects of the vertically directed bite force and torsion of the mandibular corpora around the anteroposterior axis of the mandible (Demes et al., 1984). Daegling and Hotzman (2003) who conducted in vitro experiments with human mandibles showed that this combination of loads does indeed lead to the strain pattern predicted by Demes et al. (1984). In contrast, Ichim et al. (2007) have suggested, based on finite element analysis (FEA) results, that the asymmetric distribution of cortical bone in the human mandible is not related to masticatory strains but that it may be a retained evolutionary trait.

However, studying the relationship between stress or strain distribution and the distribution of bone tissue with FEA is not straightforward, mainly because of two reasons:

  • 1Bone needs to withstand not only habitual or everyday loads, but also exceptional or aberrant loads that exceed these habitual loads in magnitude. In the mandible such high loads can, for example, occur with fallback foods that are harder or tougher than preferred food. As another example, exceptional high loads can result from accidents or attacks by other animals. Experimental evidence suggests that rare, abnormal high-magnitude loads are important for bone maintenance and the formation of new bone (Biewener and Bertram, 1993; Skerry and Lanyon, 1995). However, it is difficult to include rare loads in FEA studies since time constraints limit the number of load cases that can be simulated, and there are many possible scenarios in which exceptional high loads could occur.
  • 2If a bone is well adapted to loads then this creates an inherent problem in quantatively identifying adaption because the structure is optimized to moderate the impact of its mechanical environment. For example, where the habitual function induces high loads it would be expected that more bone would be deposited subsequently moderating stresses and strains in this region. Consequently, the stress and strain distribution observed in the bone may be expected to be the result of this adaptive process, and thus predicted strain distribution in anatomically realistic FE models may be non-informative in the context of identifying a bone's adaptation to strains.

An alternative approach to testing the relationship between strain and bone distributions is to use models in which the bone is not adapted to functional loads. In this way predictions about mechanical adaptation can be tested. This can be achieved by using models with hypothetical internal morphology: for example, models in which all internal cavities have been filled with bone material or in which the cortical thickness has been made constant throughout the mandible (e.g., Reina et al., 2007; Gröning et al., 2011b). Reina et al. (2007) have used such an approach to study the spatial distribution of bone density in the human mandible. By applying an internal bone remodeling algorithm to an FE model of a human mandible they were able to generate a distribution of bone density and elastic properties similar to that observed in the real specimen.

This study will adopt a similar approach by using models with hypothetical “unadapted” internal morphology to study the association between the strain pattern resulting from simulated masticatory loads and the distribution of bone tissue. First, we will use a model of a human mandible in which all internal cavities have been filled to investigate whether the normal gross distribution of bone tissue (e.g., the presence of bone tissue in a certain area) can be predicted based on the strains that occur during masticatory function. Second, we will assess whether the distribution of cortical thickness within the mandible is correlated with the distribution of strain magnitudes by using a model with constant cortical thickness.

MATERIALS AND METHODS

The human mandible investigated in this study was the same as that used in a previous study. Details of the computed tomography (CT) scanning and model creation are therefore described by Gröning et al. (2011a). In addition, details of the validation of our modeling approach against in vitro loading experiments can be found in Gröning et al. (2012a). On the basis of the model used by Gröning et al. (2011a), two hypothetical models were created using the automatic and manual segmentation tools available in Amira 4.1.1 (Fig. 2): one model, in which all internal cavities were filled with bone material (solid model) and a second, in which an arbitrary constant cortical bone thickness of about 1.7 mm (= 7 voxel layers, each 0.24 mm thick) was created (constant cortical thickness model).

Figure 2.

Transverse sections through the original model (A), the model filled with cortical bone material, the so-called solid model (B), and the model with constant cortical bone thickness (C). All models have been sectioned slightly below the mental foramina.

The two hypothetical models were converted into VOX-FE meshes (Fagan et al., 2007) and material properties were defined: Poisson's ratios (ν) of 0.3 and 0.45 were assigned to bone and soft tissue (periodontal ligament and cartilage in the temporomandibular joint (TMJ)) respectively and Young's moduli (E) of 17 GPa, 1 MPa, and 3 MPa to bone, periodontal ligament, and TMJ cartilage respectively (see Gröning et al., 2011a for full details).

Since it is impossible to simulate all load cases that might occur in the living organism (including rare, aberrant loads), we used a selection of load cases in which we simulated forceful biting on different teeth to produce loads that should represent high-magnitude habitual loads. Thus, 12 different load cases were simulated for each model: incision with all four incisors, right and left canine bites (including the lateral incisors and the first premolars), bites on the right and left first molars and bites on the right and left second molars) using the muscle forces listed in Table 1. For all load cases the models were constrained in all directions at the corner nodes of the simplified TMJs and the occlusal surfaces of the teeth (Gröning et al., 2011a).

Table 1. Maximum masticatory muscle forces and scaling factors
MuscleMax. muscle force (N)aScaling factorsb
IncisionCanine biteMolar bite
RightLeftRightLeftRightLeft
  • The scaling factors were multiplied by the maximum muscle forces to replicate the activation of each muscle during the modeled bites.

  • a

    From Gröning (2011a, b), calculated based on measurements by van Eijden et al. (1997).

  • b

    Reina et al. (2007) and Korioth and Hannam (1994).

Superficial masseter218.20.400.400.460.580.720.60
Deep masseter111.70.260.260.460.580.720.60
Anterior temporalis168.00.080.080.540.140.730.58
Middle temporalis137.30.060.060.480.200.660.67
Posterior temporalis118.70.040.040.420.260.590.39
Medial pterygoid192.00.780.780.550.470.840.60
Inferior lateral pterygoid90.20.710.710.430.930.300.65

After solution of the FEA, values for the following parameters were calculated: maximum principal strain (ε1), minimum principal strain (ε3), and von Mises strain (εv). Von Mises strain is a function of all principal strains (ε1, ε2, ε3) and can be used to predict failure in a ductile material under load. To create summary contour plots representing the peak strain pattern over all load cases (Fig. 3), the maximum strain value for each finite element across the different load cases was selected from the exported element strain value files and visualized in a new cumulative contour plot (Witzel and Preuschoft, 2005; Kupczik et al., 2009; Curtis et al., 2011). The creation of peak contour plots is based on the idea that mandibular bone does not adapt to particular bites but to the whole range of loads it experiences during function.

Figure 3.

Creation of a peak strain contour plot from different load cases. The two upper rows show strain contour plots for the 12 load cases simulated in this study (left and right incisor bites, canine bites and bites on the first, second, and third molars). The summary plot at the bottom shows the maximum strain value for each element across the twelve load cases.

To assess the relationships between bone morphology and functional strains, these peak contour plots were then compared with the distribution of bone in the real specimen in different ways:

The solid model was used to assess whether the gross distribution of bone can be predicted based on strains. For this purpose, sections through the filled FE model were compared visually with the corresponding CT slices. In addition, the relationship between strain and bone distribution was quantified by calculating the correlation between the peak strain in each element of the solid model and the presence or absence of bone in the corresponding voxel of the segmented CT image stack. The determination of presence or absence of bone in each voxel was based on the X-ray attenuation values or grey levels in the CT scan. If the attenuation value was above the segmentation threshold, the material “bone” was assigned to the respective voxel. If the attenuation value was below the segmentation threshold (e.g., because the voxel represents air or cancellous bone tissue with low density), no material was assigned to the voxel. Since we thus correlated a ratio variable (strain) with a ranked variable (presence or absence of bone), we used Spearman's rank correlation coefficient, which shows how well the relationship between those variables can be described as a monotonic function. Because of the high resolution of the model and thus very high number of elements, the calculation of the correlation coefficient required the processing of about 3 million pairs of variables and, therefore, we calculated the correlation coefficient only for one strain parameter: von Mises strain. The calculation was performed with the statistical computing language and environment R 2.12.2 (R Foundation for Statistical Computing, Austria).

The constant cortical thickness model was used to study the relationship between cortical thickness and strain magnitudes. This was done by visually comparing the surface strain contour plots from this model with a 3D map of cortical thickness variation within the mandible. The 3D cortical thickness map was created by calculating the minimum distances between the endosteal and periosteal bone surfaces and thus the thickness of the cortical bone using the “distance module” in Amira. In addition, the correspondence between surface strains and the cortical thickness variation in the original specimen was quantified by defining 111 evenly distributed points on the bone surface and extracting the strain value as well as the cortical bone thickness for each point (Fig. 4). The association between strain magnitudes and cortical bone thickness (both ratio variables) was then quantified by using Pearson's product-moment correlation coefficient, which describes how strong the linear relationship between the two variables is. The P-values for all correlation coefficients were calculated using two-tailed tests.

Figure 4.

The sampling points used for measuring strain magnitudes and cortical bone thickness on the lingual (left) and buccolabial (right) bone surfaces.

RESULTS

The results for the solid model are shown in Fig. 5. Those for the constant cortical thickness model are presented in Figs. 6 and 7.

Figure 5.

Comparison between sections through the solid model with von Mises strain (εV) contour plots and corresponding CT slices of the original specimen: (A) coronal sections through the first molar; (B) transverse sections through the mental foramen; (C) mid-sagittal sections through the symphysis.

Figure 6.

Comparison between the strain distributions in the constant cortical thickness model and the cortical bone thickness variation in the real mandible. The coronoid processes (in gray) are not included in the comparison because the shape of the processes causes an overestimation of cortical bone thickness with the measurement method used. The teeth and the condylar heads have also been excluded from the comparison.

Figure 7.

Cortical bone thickness in the real mandible plotted against maximum principal strain in the constant cortical thickness model.

In the solid model the highest bone strain magnitudes are found at the bone surface as well as in areas where dense trabecular bone is seen in the corresponding CT slices (Fig. 5). Low strain areas, on the other hand, correspond to areas, where no bone exists or where only few trabeculae are present. Only in some regions are strains relatively low where cortical bone is present in the real specimen. The coronal section through the posterior corpus (Fig. 5a) shows a difference in strain magnitudes between the buccal and lingual sides: especially in the upper half of the section, strains are higher buccally than lingually. This corresponds with the thicker cortical bone found on the buccal side compared to the lingual side (Demes et al., 1984).

Spearman's rank correlation coefficient, which relates the peak von Mises strain value in each element of the solid model to the presence or absence of bone in the corresponding voxel of the CT scan, gives a value of 0.47 (P < 2.2e-16. Note that this significance level is based on 3 million pairs of variables).

Figure 6 shows the spatial distribution of surface strains for the model with constant cortical bone thickness. The highest strains are found below the molar dentition, at the anterior margin of the mandibular ramus, the base of the corpus and the posterior margin of the ramus just below the condyles. These are also the areas where the highest cortical bone thickness is measured. In addition, most of the mandibular ramus surface shows low strains, which corresponds with the thin cortical bone that is found in this area. However, in some areas low strain magnitudes are found in the FE model although cortical bone thickness at the same locations in the real specimen is high (e.g., at the base of the symphysis) and vice versa.

The association between strain magnitudes and cortical bone thickness is quantified by correlation coefficients for three different strain parameters: 0.35 (P < 0.0005) for maximum principal strain, −0.25 (P < 0.05) for minimum principal strain and 0.27 (P < 0.01) for von Mises strain. Note that the negative correlation coefficient for minimum principal strain is due to the fact that minimum principal strain has a negative sign by convention. In general, the correlation coefficients are quite low, but among these, the correlation for maximum principal strain gives the highest value. Figure 7 shows the relationship between maximum principal strain and cortical bone thickness in a bivariate plot. Interestingly, in some locations cortical bone thickness is considerably greater than that predicted by the regression line.

DISCUSSION AND CONCLUSIONS

The comparison of predicted strains in the two hypothetical models with the distribution of bone in the original mandible reveals many similarities between strain and bone distributions. In the solid model high strain areas correspond well with regions where cortical bone or a dense trabecular network is found. In the model with constant cortical bone thickness some high strain areas correspond to areas where the cortical bone is particularly thick, whereas low strain magnitudes are often found in areas where cortical bone is rather thin. However, some areas do not show this pattern and the correlation coefficients are therefore quite low.

The observation that the highest strains are found at the surfaces of the solid model (Fig. 5) could be explained by the occurrence of bending. During bending, strain magnitudes increase with the distance from the neutral axis, which lies in the center of the bone's cross-section (Currey, 2002).

The peak von Mises strain map in the solid model also reflects the difference in cortical bone thickness between the buccal and lingual sides of the posterior corpus, which is typical for anthropoid mandibles (Demes et al., 1984). A previous FEA study by Ichim and colleagues (Ichim et al., 2007) did not confirm this relationship, most probably because these authors looked only at the strain distributions from single load cases instead of summary contour plots derived from several load cases. In addition, the internal morphology of their model was not altered from the morphology of the original specimen, so that they did not control the variation of cortical thickness within the bone. As discussed above, only the use of hypothetical models, in which this variable is controlled, allows an investigation of whether the distribution of cortical bone is determined by the strain patterns resulting from functional loads.

In addition, the strain distributions observed in the solid model are very similar to the distribution of bone density reported by Reina et al. (2007), who applied an internal bone remodeling algorithm to a filled FE model of a human mandible and found a good match between the resulting density maps and the distribution of bone in the real specimen. The muscle force magnitudes used by these authors are very similar to those used in this study. The constraints at the joints and occlusal surfaces are, however, rather different, which suggests some robusticity of the correspondence found between strain and bone distributions. In addition, our study shows that the distribution of bone can be reasonably well predicted by simple summary strain contour plots even without applying a complex time-dependent remodeling algorithm as used by Reina et al. (2007).

The muscle forces applied to our models almost certainly differ from those that the individual experienced during life and our simplified constraints might have introduced artifacts. However, the correspondences between strain and bone distribution gives us confidence that the modeling approach used is relatively realistic under the assumption that the distribution of bone reflects the distribution of strain magnitudes during physiological loading. The low correspondence between strain patterns and the distribution of cortical bone in some areas and thus the low correlation coefficients could be due to the fact that not all relevant load cases have been modeled. This explanation seems plausible since the most extreme outliers are those data points that show lower strain magnitudes than expected based on the cortical thickness measured at those points (Fig. 7).

Strain magnitudes might have been underestimated because aberrant high-magnitude loads have not been simulated in this study. Since empirical data suggest that adaptive bone modeling and remodeling is very sensitive to such abnormal high-magnitude loads (Biewener and Bertram, 1993; Skerry and Lanyon, 1995), it is likely that the inclusion of such loads could improve the degree of correspondence between model predictions and bone distribution in the real mandible.

Low-magnitude (e.g., < 10 microstrain) high-frequency (e.g., 10–50 Hz) loads are another type of loads that has not been considered here. Such loads arise from muscle contraction during everyday activities such as maintaining posture (Ozcivici et al., 2010). Experiments have shown that these loads also have an anabolic effect on bone, despite the low strain magnitudes they produce (Rubin et al., 2001). However, while this anabolic effect is significant on trabecular bone, it seems to be negligible in cortical bone (Rubin et al., 2002). It is, therefore, unlikely that the inclusion of such loads would have an effect on our model predictions.

To improve model predictions, sensitivity studies could explore the effect of altering input variables, such as the orientation and magnitudes of the muscle forces, on the summary strain plots. Our previous sensitivity analyses, which were based on single load cases, have shown that changes in these variables can have large effects on the results (Gröning et al., 2011a, 2012 b). However, in this case the solution of numerous FE models for each input value would be required, since several load cases have to be applied in order to obtain a summary contour plot.

In addition, the quantitative analyses used here could be further improved. For example, we considered only strain values extracted from points on the periosteal bone surfaces in our calculation of correlation coefficients between strain magnitudes and cortical thickness. The strains surrounding these points, within the bone and on the endosteal surface were thus ignored. For a more comprehensive analysis, mean values based on the strains on the endosteal surface and within the cortical bone should be included or, to avoid decisions regarding the location of corresponding points and areas, since such decisions are prone to error, a topological method such as statistical parametric mapping could be used in future studies (e.g., Pataky, 2010).

In addition, it should be considered that the cortical thickness measurements are not independent of shape. Around processes and prominences, such as the coronoid process, cortical thickness is overestimated when the minimum distances between endosteal and periosteal surface points are used for determining cortical bone thickness. We placed our landmarks away from the coronoid and condylar processes but need to consider that some other, less prominent convex or concave surface features might have led to some inaccuracies in the cortical thickness measurements.

Interestingly, maximum principal strain shows the highest correlation coefficients with cortical thickness. Since bone is weaker under tension and shear than under compression (Currey, 2002), it could be advantageous for bone to increase its thickness especially in those areas, where high tensile and shear strains occur.

However, it has been suggested that there are mechanisms in living organisms that minimize bending and torsion of bones during physiological loading so that high tensile and shear stresses in the bone are avoided or at least minimized (Sverdlova and Witzel, 2010). According to this idea, bones under physiological loading are preferentially compressive structures.

On the other hand, data from strain-gauging studies of nonhuman primates suggest that bending and torsion of the mandible occurs during biting and chewing (e.g., Hylander, 1984). In addition, the human mandibular corpus, which shows a rather thick cortical shell similar to that of long bone shafts, is, from an engineering point of view, well suited to withstand not only compression but also bending and torsion, contrary to, for example, the mandibular condyles that consist of very thin cortical bone surrounding dense trabecular bone tissue. Therefore, experimental data and the morphology of the mandible suggest that bending and torsion of the mandibular bone occur during function and that it is not unreasonable to simulate load cases that deform the mandible in these ways.

Although the models used in this study have several limitations and the correlation coefficients, especially between cortical bone thickness and strain magnitudes, are relatively low, the use of FE models with altered internal morphology seems to be a promising approach to study relationships between functional strains and the distribution of tissue within bone. This approach seems to be particularly useful when in vivo loading experiments and strain measurements are difficult or impossible such as in humans. In addition, future studies could overcome some of the limitations of this study by including a wider spectrum of load cases, sensitivity analyses and comparisons with experimental data. It is likely that with improved models more accurate predictions of bone distribution could be achieved.

Acknowledgements

The authors would like to thank Lee Page and Jia Liu for programming VOX-FE and Sue Taft for scanning the mandible. They are also grateful to Ulrich Witzel for inspiring discussions and to John Currey and two anonymous reviewers for their helpful comments on the manuscript.