In Situ Tooth Replica Custom Implant: A 3-Dimensional Finite Element Stress and Strain Analysis

In a previous study, the author proposed the concept of in situ tooth replication.¹  It is to restore a missed tooth to its anatomy, radicularly and coronally. The concept was motivated by the awareness of the importance of teeth anatomy and configurations that are allied to function.^2–5 

As a first part of the research program, the previous study focused on the technique of application and the prediction of the biologic acceptance of the suggested bioactive glass, bone substitute Cortoss. To the best of knowledge, it was the first reported attempt of such an initiative. Therefore, it is worth mentioning here some of the features of this relatively recent material and to review the proposed technique. Cortoss is a high degree of 3-dimensional, cross-linked resin that is reinforced by ceramic particles. The high molecular weight of the monomers (286–640 g/mol) is the causal of the high degree of conversion (76% to 86% experimentally and 95% theoretically) and indicates minimal leaching at body temperature and limited chemical trauma.⁶  The material is packaged sterile in a dual cartridge as a paste-paste system. It has a compressive strength of 146 ± 18 MPa, a 4-point bending strength of 57 ± 10 MPa, and a bending modulus of 5505 ± 509 MPa.⁷  Cortoss has been described by being strong enough to bear the load of body weight,⁸  and hence, augmentation of load-bearing bones is one of its applications.^9,10  It is also used in vertebral augmentation,^11,12  cranioplasty,^13,14  femoroplasty,⁹  and screw augmentation.^15,16  The material is classified as osteoconductive bioactive glass that induces osteosynthesis.¹⁷  Direct contact with host bone is achieved via a carbonated apatite bond.¹⁷  The pull-out strength is nearly 140 N after 6 and 12 weeks, 340 N after 24 weeks, and 900 N after 1 year.⁸  These values surpass the 500 g (≈50 N) anchorage required for dental implants.³  The state of mutual clinical rigid attachment alongside histological direct bone contact is the basic definition of osseointegration.^2,18,19 

What's more, periodontal ligaments have the ability to proliferate from their formative fibroblasts, regenerate, and attach to replanted teeth as well as to bioactive glass ceramics.^20–25  Therefore, if remnants of periodontal ligaments (Sharpey's fibers) are reserved after extraction, the ligaments will have the potential to regenerate onto replicas made of Cortoss.

Based on the background mentioned above, a tooth replica made of Cortoss may be brought about, with 1 of 2 possibilities: (1) attachment via periodontal ligament or (2) osseointegration. There will be a chance for the first if simple nontraumatic extraction is performed and periodontal ligaments are preserved. Some cases may, however, require osteotomy. Surgical widening and/or deepening of the socket may be required to increase the volume and strength of the root replica or to increase the replica-bone surface area. Surgical osteotomies will deprive the socket of the Sharpey's fibers and minimize the chance for regeneration of periodontal ligament. In these cases, osseointegration will remain the alternative possibility.

As an early stage of study, the technique was demonstrated on a model.¹  It considered the importance of occlusal harmony and progressive loading protocol. Attaining a smooth cervical region was also emphasized. It addressed the potentials of a substantial permucosal transgingival seal, a barrier between intrabony and extrabony environments. The technique may be summarized by a preplanned injection of the material into an extraction socket. A delivery system was established, and the armamentarium was described. The material was self-set in about 5 minutes. Immediate duplication and implantation was achieved. Initial stabilization could have been augmented by splinting to adjacent teeth. The advantages of the new concept were restoring normal tooth morphology and its spatial orientation, immediate implantation, socket grafting, and elimination of the problems of galvanism and corrosion.

The potentials of the concept encouraged further evaluations. Biomechanics is one of the recommended investigations and hence constitutes the core of the current study. In this respect, groups of variations should be considered: (1) anatomy and location of the tooth that is going to be replicated and amount and quality of surrounding bone, (2) type of resulting relation with bone whether osseointegration or periodontal ligament attachment,¹  (3) operator judgment regarding whichever modification osteotomies and if needed, (4) design and material of the super structure, (5) the occlusion and whether it can be adjusted, and (6) acting force in terms of magnitude, direction, and whether they can be controlled. It is the interaction of these factors that will play the major role in the developing stresses and strains. Such multiple variables can lead to countless possibilities. Standardization of each of these variables is needed in order to narrow these possibilities smartly and, consequently, draw a groundwork conclusion. After reaching such conclusion, further studies may be carried on to investigate the other possibilities.

To facilitate the analysis, simulation may be an approach. Finite element analysis can be a useful tool. It has been used extensively for assessment of mechanical behavior in many aspects of dentistry and in implantology.

The mandibular second premolar illustrates the least of variations compared with other teeth in terms of root numbers, shape, curvature, and buccal-lingual inclination.²⁶  Furthermore, its buccal cusp tip and buccal incline of the buccal cusp are the occlusal contacts during cusp-fossa apposition and horizontal movement, respectively.²⁷  It may therefore be a suitable candidate for the study and looked upon as a control group.

Regarding occlusal loads, they have shown a great deal of variation among research studies. To conquer this confusion, some information about these loads should be addressed: (1) Nature: static vs dynamic. Masticatory forces are dynamic and intermittent.^2,28–30  According to principles of mechanics, however, equivalent static loads can substitute for the real load in finite element analysis. In that case, caution should be taken that the distribution of stresses and strains will be distorted near the region of applied load.³¹  (2) Duration of application. This will affect the durability and hence is more relevant in studying fatigue properties. (3) Frequency. Masticatory forces are cyclic and intermittent. One stroke lasts for about 1 second.^28,30  (4) Magnitude. It depends on intrinsic factors such as age, gender, and dental state and extrinsic factors such as nature and size of food bollus.³²  It would be more accurate if masticatory loads could be measured individually. For practicality, however, the norms might be the resort. But again, a wide range exists. The recorded average maximum sustainable biting force was reported to be 765 N, with a normal range from 17 to 430 N,²  whereas the maximum voluntary clenching force was reported to be 173.29 ± 15.32 N in the molar region.³³  In fixed prostheses supported by implants as another example, biting force was reported to range from 35 to 330 N, where the second premolars exhibited the highest load mean value, which was 143 N.³⁴  Some researchers have worked with the magnitudes of 200 N,³⁵  178 N,³⁶  100–200 N,³⁷  70 N,^38,39  and so forth. (5) Direction. Studying the vertical load is straightforward, while off-vertical loads will raise the issue of not only at what angle (45,⁴⁰  130,³⁶  or 60, 35, and 14³⁸; or even zero, horizontal), but also what magnitude. The direction of load will definitely affect the resulting stresses and strains, which in turn will be affected in the presence of periodontal ligament.⁴¹  Another question is whether the force direction is innate and/or a product of inclined surfaces,³³  in other words, how much of this load and its direction can be controlled. (6) Location of load application: point force vs area pressure. Pressure equals force/area.⁴²  Thus, it is inversely proportional to area. It would be more appropriate to increase—within limits—the area over which the force is acting and avoid points of stress concentration, a very well-known principle in restorative dentistry. Furthermore, a small occlusal pit or a cusp tip normally has an area of about 4 mm².^2,42  Therefore, it would be more rational to investigate the effects of not only point force but also pressure. Masticatory pressure and pattern showed little variations depending on individuals and food texture. A range from 0.4 to 1.75 kg/m² (≈4–17 MPa) was recorded.²⁹  An interesting review article showed that human dentin could endure stresses resulting from up to 20–30 MPa pressure, which was justified to be the normal range of masticatory pressure.³⁰ 

For optimum loading conditions, bone should be loaded to its physiologic remodeling capacity.⁴³  However, the strength of the tooth or its substitute (replica or implant) should not be exceeded. Regarding cancellous bone, it was reported that it could maintain its health if subjected to stresses in the range from 1.4 to 5 MPa, otherwise resorption might occur.⁴⁴  In terms of strain, physiologic remodeling capacity of cancellous bone was said to range from 1500 to 3000 microstrain (μS),⁴³  while deformation would take place at levels greater than 4000 μS.³⁶  Regarding cortical bone, the 3-point bending strength of cortical bone of the tibia was reported to range from 201 to 230 MPa in one study⁴⁵  and 157 to 181 MPa in another study.⁴⁶  The ultimate compressive strength of femoral diaphyses was said to be 162.2 MPa and ultimate strain to be 0.015 strain (=15 000 μS),⁴⁷  which is in agreement with another reported range of 10 000 to 20 000 μS.⁴⁸  It has also been concluded that bone is liable to deformation at levels from 20% to 40% of its strength.⁴⁸  In the absence of data pertinent to stomatognathic bones, these magnitudes may be considered as an initial indication. By calculations, cortical bone may undergo deformation if subjected to stresses above the range of 32 to 92 MPa [(20% multiplied by 160) to (40% multiplied by 230 MPa)] or if subjected to strains above the range of 2000 to 8000 μS [(20% multiplied by 10 000) to (40% multiplied by 20 000 μS)].

In the lack of consistent information on the exact normal masticatory loads, it may appear reasonable to refer to the golden rule (bone should be loaded to its physiologic remodeling capacity)⁴³  as a tool for better identification of the tolerable loads. In other words, if a tooth (or prosthesis) is subjected to an escalating pattern of loads while the resulting effects are being observed until reaching the maximum tolerable stresses and strains, the causal loads may be considered the tolerable loads. By this way, normal masticatory loads can be figured out. Cross-matching with some of the previously reported ranges may be a source of verification. This methodology can be applied to natural dentition, conventional implants, or in situ replicas.

The purpose of this study was to evaluate tooth replicas made of Cortoss under each of 2 possible circumstances, (1) attachment via periodontal ligament and (2) osseointegration, versus their corresponding natural lower second premolar. Evaluation would be made in terms of tolerability to stresses and strains generated in surrounding cancellous and cortical bone and to stresses generated in the replica (and tooth). Finite element was aimed as the tool of analysis. Each model was aimed to be studied under increments of 2 types of loads separately: (1) pressure acting on the 4 mm² area of buccal incline of the buccal cusp⁴²  and (2) point force acting vertically on the buccal cusp tip. The goal was to find out the loads tolerable by each model. For consistency, the clinical crowns of all models were presumably restored with Empress 2—all ceramic—crowns.

ANSYS Release 11 software (ANSYS Inc, Irvine, Calif) was used for 3-dimensional modeling and analysis. A natural mandibular left second premolar was selected for the study. Manual plotting was approximated from data in the literature for tooth,²⁶  surrounding bone,⁴⁹  and the periodontal ligament.⁴¹  Bone periphery was then extended nearly 1.5 cm both mesially and distally to carry boundary constraints far enough to minimize their influence on the results.⁵⁰  Further modifications in the dimensions and material properties were then applied to produce the models as shown in Figure 1. The models can be described as follows.

• Model 1: Natural tooth (NT): Booleans were applied to produce a tooth prepared for, and restored with, full crown.

• Model 2: Periodontally attached replica (PR): The properties of dentin (elastic modulus and Poisson's ratio) in model 1 were substituted with those of Cortoss.

• Model 3: Osseointegrated replica (OR): The space (volume) occupied by the periodontal ligament in model 2 was added to that of cancellous bone (ie, the replica was in direct contact to bone, with no periodontal ligament in between).

Each volume was glued to adjacent volume(s) to ensure perfect bonding.⁵¹  This would represent the strong bond between periodontal ligaments and surroundings in NT and PR (models 1 and 2)⁴¹  and between Cortoss and bone in OR (model 3).^3,8 

Smart size meshing was executed at the mid of the fine-coarse scale. Grades 4 and 5 were applied as they were shown to be most convenient in terms of ability for reiteration throughout the 3 models and had a reasonable file size and processing time. On the other hand, coarser and finer sizes produced error meshing.

The previous models were composed of different materials. They were assumed to be elastic linear and isotropic. The ascribed elastic modulus and Poisson's ratio are listed in Table 1.

For the crown material, heat pressable all-ceramic, lithium disilicate (Empress 2) was considered because of its relatively high flexural strength, fracture toughness, and acceptable esthetic in that region of the mouth.^52,53  Furthermore, and in agreement with literature, pilot trials did not show differences in the effect of altering crown material.³⁵ 

Regarding periodontal ligaments, its elastic modulus is known to increase exponentially with the increase of loads.⁶⁰  However, the current study was intended to symbolize masticatory strokes at certain static loads. Therefore, the periodontal ligament in this study was ascribed the isotropic and elastic linear properties.

Regarding bone, mechanical properties of bone depend on several factors, such as micro architecture,^61,62  density,⁴⁸  degree of mineralization,^63,64  distribution of minerals,⁶⁵  volume fraction,⁶⁶  and organization of the collagen fibers.⁶⁷  These factors are influenced by the adaptive capability and loading history.⁶⁸  They may explain the variations in the reported ranges of modulus of elasticity of cancellous and cortical bones from 0.05 to 2 MPa and from 3.4 to 20 Mpa, respectively, and that the elastic modulus of cortical bone is considered to approximate 10% of cortical bone.^39,48  In accordance with the literature, bone in this study was assumed to be isotropic, homogeneous, and linearly elastic.^38,58,59,67  The other options of heterogeneous and anisotropic properties comprise different degrees of anisotropy, indicate multiple probabilities, and are pending documented measurements of the previously mentioned factors that affect bone properties.⁶⁵ 

Boundary displacement constraints were placed on the outer areas of outermost bone volumes (Figure 2). Constraints included all 3 degrees of freedom and were set to zero value.^38,50  Each of the 3 models was subjected to 2 types of loads. The first was pressure load acting on the distobuccal incline of the buccal cusp, which had an area of approximately 4 mm.²  The second was point force load on a single key point at the buccal cusp tip and acting vertically in the apical direction. Pressure was applied at increments of 1 MPa starting from 15 MPa to 50 MPa. Point force was applied at increments of 5 N starting from 70 N up to 405 N.

Solution was prompted for each model, one load increment at a time.

Solution results were plotted in the General Postprocessor module of ANSYS. For each load increment solution, 5 plottings were obtained: (1) Von Mises stress (SEQV) for replica (or tooth), (2) SEQV for cancellous bone, (3) Von Mises total mechanical strain (EPTOEQV) for cancellous bone, (4) SEQV for cortical bone, and (5) EPTOEQV for cortical bone. Figures 3 through 17 show result plotting of pressure loads, while Figures 18 through 32 show result plotting of point force loads. These figures were automatically generated by ANSYS software. Some manipulations were performed, however, in order to gather relevant views sharing the same scale of intensities as for close-up views or views from different sides. From each result plotting (figure), the maximum value was picked and tabulated. Thus, from the result plotting of each load increment solution on each model, 5 corresponding readings were obtained, namely, (1) maximum Von Mises stress (max SEQV) in replica (or tooth), (2) max SEQV in cancellous bone, (3) maximum Von Mises total mechanical strain (max EPTOEQV) in cancellous bone, (4) max SEQV in cortical bone, and (5) max EPTOEQV in cortical bone. Fifteen readings were thus obtained from each load increment solution (5 readings × 3 models). To facilitate the comparison and navigation between the effects of load increments on the 3 models, these readings were grouped into 2 tables. Table 2 shows the effects of pressure, while Table 3 shows the effect of point force loads.

Analyzing Tables 2 and 3, the load tolerable by each model would be determined. It is the load that would fulfill the following group of parametric criteria synchronously: its resulting stresses could be sustained (below tensile strength) by tooth (or replica) and both cancellous and cortical bones, and its resulting strains would be within the physiologic remodeling capacity of both cancellous and cortical bones. These parameters have been explained earlier and can be summarized here: cancellous bone: 1.4–5 MPa or 1500–3000 μS, cortical bone: 32–92 MPa or 2000–8000 μS. The reported tensile strengths of dentin and Cortoss are 52 MPa^30,42  and 57 Mpa,⁷  respectively.

For the NT model, cancellous bone sustained Von Mises stress (SEQV) resulting from up to 23 MPa pressure. However, the physiologic remodeling level was reached at Von Mises total mechanical strain (EPTOEQV) resulting from 19 MPa pressure. The latter was also tolerable by cortical bone in terms of strength and physiologic remodeling. Its generated max SEQV in the tooth was also below the tensile strength of dentin. Therefore, 19 MPa masticatory pressure will be considered the tolerable load for NT. Its effect is shown in Figures 3 through 7. It is obvious that both magnitudes of 19 and 23 MPa approximate and hence support the values of masticatory stresses reported in the literature (4–17 MPa²⁹  and 20–30 MPa³⁰).

For the PR model, cancellous bone sustained SEQV resulting from up to 18 MPa pressure. However, the generated max EPTOEQV (3391 μS) was slightly greater than the bone physiologic remodeling level (1500–3000 μS).⁴³  Therefore, 15 MPa masticatory pressure is more justified. The latter was also tolerable by cortical bone in terms of strength and physiologic remodeling. Its generated max SEQV in the replica was also below the tensile strength of Cortoss. Therefore, 15 MPa masticatory pressure will be considered the tolerable load for PR. Its effect is shown in Figures 8 through 12. Again, both values of 15 and 18 MPa approximate the previously mentioned normal masticatory stresses.^29,30 

For the OR model, cancellous bone sustained SEQV and physiologically tolerated EPTOEQV resulting from up to 50 MPa pressure. Tooth replica, however, could sustain SEQV resulting from up to only 23 MPa pressure, where the tensile strength of Cortoss was approximated. Therefore, 23 MPa pressure is more justified thus far. Concerning cortical bone, the pressure of 23 MPa resulted in a tolerable level of 89 MPa max SEQV and 6025 μS max EPTOEQV. Interestingly, as shown in Figures 16 and 17 (and from the “list result” option of ANSYS), only very tiny regions lie above 29 MPa or 2000 μS (ie, the mass lies within a safe and tolerable range). Therefore, 23 MPa masticatory pressure will be considered the tolerable load for OR. Its effect is shown in Figures 13 through 17.

For the NT model, the previous criteria of bone strength, bone remodeling, and tooth strength were met at force magnitudes of 85 N masticatory forces and will be considered the tolerable load for NT (Figures 18 through 22).

For the PR model, the tolerable load would be 80N (Figures 23 through 27).

For the OR model, cancellous bone sustained SEQV and physiologically tolerated EPTOEQV resulting from up to 400 N force. Tooth replica, however, could sustain SEQV resulting from up to only 217 N, at which the tensile strength of Cortoss was approximated. Therefore, 217 N is more justified thus far. Concerning cortical bone, the pressure of 217 N resulted in 72.7 MPa max SEQV and 4910 μS max EPTOEQV. However, as shown in Figures 16 and 17 (and from the “list result” option of ANSYS), only very tiny regions lie above 32.2 MPa or 3200 μS (ie, the mass lies within a tolerable range). Therefore 217 N vertical point force load will be considered the tolerable load for OR. Its effect is shown in Figures 28 through 32.

The current study was based on averages. It might seem easier if the plotting of models could have been obtained from reality by using imaging technologies. However, this would require real measurements of bone properties as well as masticatory loads in order to reach more specific results, and those results would have been valid for only that particular case. Perhaps there could be a motivation to spur research workers and manufacturers to accomplish such methods and tools to suit the daily practice. It would be of major benefit to many fields in dentistry.

The investigation was preceded by tentative trials of magnitudes smaller than the recorded loads. The reason was to cover safely the ranges of the masticatory loads reported in the literature. The significant smallest values were 15 MPa and 80 N, around which tabulation was started.

Von Mises stress has other names and abbreviations (equivalent stress, SEQV, equivalent tensile stress, σ_v). It was selected as a means of evaluation because, as explained by the ANSYS manual, it is used to predict yielding of materials under any loading conditions from uniaxial to multiaxial. It satisfies the property that two stress states with equal distortion energy have equal SEQV.⁶⁹  Therefore, maximum Von Mises stress (max SEQV) has been considered an accurate prediction of failure criteria.^37,70,71  This is particularly important here because this study is still an early phase of investigation and was concerned principally with early warning against failure. The interpretation of the results must take this into consideration.

A lot of information can still be obtained from the figures and may be of interest to research workers in future studies.

Tracking the analysis, it can be observed that each of the result components increased constantly by an incremental increase of the load. Table 4 shows the effect of masticatory pressure on the max SEQV generated in each component of each model. Table 5 shows the effect of masticatory forces. This effect is demonstrated in Figures 33 and 34, respectively.

It can also be observed that, under the same loading conditions, max SEQV and max EPTOEQV are greater in cancellous bone of models with periodontal attachment (NT and PR) than those in OR. The reverse is true in the body of the teeth and replicas. This can be observed in both types of loads: pressure and force point. This is consistent with the well-recognized function of the periodontal ligament, which transfers stresses from tooth to surrounding bone.⁴¹ 

Comparing PR versus NT, max SEQV is greater in cancellous bone surrounding PR than NT. The reverse is true in the body of replica and teeth. This can be explained by the fact that both models have the same structure, with the only difference being in the lower elastic modulus of Cortoss than dentin. Hence, it can be anticipated that, under the same loading conditions, Cortoss will flex a bit more and generate a bit more stresses and strains in bone than dentin would.

Confusion should not exist when comparing PR versus NT under the vertical type of load. The max SEQV of each of the tooth and surrounding bone in NT is greater than that in PR. Nevertheless, max EPTOEQV in cancellous bones shows the reverse. To eliminate this confusion, it should be remembered that the recorded values of the stresses and strains in this study were the maximum values no matter where the location was. This can be observed by watching the regions of maximum stress and strains. For NT (Figures 18 and 19), max SEQV is located at the fundus of the socket, which is supported by thick basal bone, and hence stresses could be buffered, resulting in fewer strains. Conversely, in PR (Figures 23 and 24), regions of maximum stresses are located in the crestal region, which is not as thick as the basal region, and hence, more strains developed. The difference in location of max SEQV between NT and RP can be attributed to the difference in elastic modulus of dentin and Cortoss. The latter might have shown some bending at the cervical level more than dentin could have done. This can be manifested by comparing stresses developed cervically in the body of NT and replica. In the first, it is in the zone of 4.85 MPa (Figure 22), while in the latter, it is in the zone of 6.65 MPa (Figure 27).

The effect of masticatory loads on the cortical bone was more pronounced in the crestal level. The effect was greater in the case of OR, followed by PR and then the NT. In all cases, however, only small areas were overstressed; meanwhile, the bulk was subjected to reasonable stress intensities.⁴⁸  Accordingly, there will be a good chance for stronger bone as a response to remodeling because of the presence of strains with lesser magnitudes beneath the crest.⁴⁸  A progressive loading protocol would enhance such an effect.^1,48 

In conclusion, according to the methodology used in this study and considering the parameters set in terms of alveolar bone strength and physiologic tolerance and the strength of the tooth (or replica), it could be concluded that:

• Natural mandibular second premolar tolerated up to 19 MPa pressure acting on a distal incline of the buccal cusp or 85 N vertical force acting on the buccal cusp tip.

• Periodontally attached replica tolerated up to 15 MPa pressure or 85 N force.

• Osseointegrated replica tolerated up to 23 MPa pressure or 217 N force.

The results of this study encourage further research toward the advancement of the new concept of in situ, custom tooth replication.

Abbreviations

EPTOEQV

Von Mises total mechanical stress

NT

natural tooth

OR

osseointegrated replica

PR

periodontally attached replica

SEQV

Von Mises stress