Biomechanical behaviour of tilted abutment after fixed partial denture restoration of CAD/CAM materials
Dentition defect, referring to tooth loss in the dentition, can lead to the extrusion of the opposite teeth, inclination of the abutment teeth, occlusion disorders, and masticatory disturbances . Under physiological mastication conditions, the force transmission is consistent with the axis of the abutments, which bears the force evenly. However, when the posterior teeth are missing in the dentition defect category Kennedy’s class III due to caries, periodontal disease, tumours, etc., the distal tooth adjacent to the missed tooth will tilt mesially if not restored timely and properly . During mastication, however, the occlusal force cannot be transmitted along the long axis of the tooth. It is first transmitted to the tooth cervix periodontium and concentrated, leading to alveolar bone resorption and periodontal tissue damage. It might give rise to the progressive resorption of the mesial alveolar bone over time, leading to angular alveolar ridges and intrabony pockets . The missing tooth should be restored as soon as possible; otherwise, approximately 98% of molars will tilt mesially . The main methods of rehabilitating missing tooth with tilted adjacent teeth include orthodontic treatment, implantation, fixed partial denture (FPD) and other designs like extracoronal attachment fixed restoration . Among the treatments above, FPDs has higher treating efficiency than orthodontic plan, and little requirements of bone compared to implant restoration, especially fits the patients with inadequate space or bone mass for implant denture restoration . FPDs have a survival rate of 89.7% in 5 years . However, the fixed bridge restoration requires the axial alignment of each unit (abutment and pontic). In the case of a double-ended fixed bridge, both abutment teeth should be prepared during the procedural steps to create a common path of insertion. The complexity of establishing FPD insertion and managing periodontal stress resulting from non-axial forces will increase as the inclination angle of the abutment extends. Previous research reported that the inclination angle of FPD abutment is expected to be less than 24°  or 30° , however, a widely acknowledged clinical index of abutment tilt has not been established. Furthermore, considering varied clinical scenarios, the success of FPDs is mainly derived from design. Material properties determine the resistance of prosthesis to biomechanical burdens in static and dynamic occlusion . Zirconia-based FPDs have exhibited excellent biocompatibility and mechanical features and have been widely used . Lithium disilicate, another popular material for FPDs, has lower mechanical strength but higher translucency than zirconia . In addition, the survival rate of FPDs in zirconia is higher than in reinforced glass ceramics . Moreover, resin-based FPDs have become highly popular because of the rapid development of computer aided design/ computer aided manufacturing (CAD/CAM), with acceptable performance for clinical mid-term applications, offering advantages including low cost and easier fitting . Polymer-infiltrated ceramic network (PICN), manufactured by the ceramic scaffold and three-dimensionally infiltrated resin composite, achieves a balance between the properties of ceramics and resin and has been reported to combine the clinical indications of all-ceramic and resin composite restorations . Furthermore, the choice of material is still debatable, especially for complex clinical scenarios like tilting abutments for FPDs. A variety of engineering techniques can be applied to evaluate the biomechanical behaviour of FPDs, including theoretical mechanical analysis, such as finite element analysis (FEA), and experimental mechanical analysis, such as digital image correlation (DIC), resistance strain methods, photoelastic methods, etc . Among them, FEA is a computer-based numerical analysis method. By constructing a finite element model and specifying material properties, loads, and boundary conditions, the FEA can analyse the stress and strain in any given geometric structure, with the advantages of non-invasiveness and repeatability, without ethical issues . The photoelastic method is an experimental stress analysis approach that uses optical principles to study elastic mechanical problems. Its main principle is that a specific photoelastic material with a birefringence effect can produce interference fringe patterns under a certain load in a polarized light field. The stress distribution state of the model under load can be qualitatively or quantitatively analysed by measuring and analysing the fringe patterns. It has the advantages of non-contact, visualization, high sensitivity, etc., with wide use in various fields of oral medicine . A few studies have explored the stress distribution of periodontal tissues in patients with tilted abutment teeth after fixed bridge restoration via FEA  or a photoelastic method . However, further guidance should be provided on the restorative design and material selection based on the inclination of the abutment teeth. Therefore, this study aims to investigate the biomechanical behaviour of the complex of FPD‒abutments‒periodontal tissue using different FPD materials through in vitro FEA and photoelasticity to guide improvements in tilted abutment restoration materials. 

Establishment and analysis of finite element model The experiment considered two variables with different values: Four FPD materials and six angles of tilted abutment tooth were applied in the FEA. First, the digital imaging and communications in medicine (DICOM) images from cone beam computer tomography (CBCT) (3D Accuitomo 170, Morita Corp., Japan) of the upper jaw of a 24-year-old female whose maxillary first molar was intact, normal, and without trauma or deformity were imported (Fig. a, b) to Solidworks© (Dassault Systèms Solid Works© Corp., Concord, USA) to perform the modelling phase. By using coordinate system transformation, the central axis of the second molar was tilted mesially by 0°, 6°, 12°, 18°, 24°, and 30° (Fig. c). The maxillary first molar was extracted virtually. The software (Exocad 3.0, Exocad GmbH; Geomagic Studio, 3D System Inc., Germany) was used to design FPDs, whose thickness is 1.5 mm for the occlusal surface, 1 mm for axial surfaces, and 1 mm for the shoulder . The connector area was set as a circle, and the diameters were 10 mm 2 in all models . FPDs were produced by the CAD/CAM system (Upcera x5, China) as monolithic restorations (Fig. d). Moreover, an 0.03-mm cement layer was set up, and FPDs and abutment teeth were assembled with maxilla in Solidworks © (Dassault Systèms Solid Works© Corp., Concord, USA) . The partial maxilla models equipped with titled abutments and FPDs were imported to a finite element analysis software program (ANSYS Workbench 18.2.2, ANSYS Inc., USA) for simulation (Fig. d). Table presents the properties of FPD materials, dental tissues, and cement. All the materials were considered homogenous, linearly elastic, and isotropic. The contact condition of all the interfaces was set as fully bonded. The boundary fixation was along the bone axes. The mesh measured 0.0002 mm, and the sum of meshes was 4,165,066. The number of nodes was qualified to ensure simulation accuracy. Two loading conditions were considered for oral rehabilitation: the vertical forces were respectively applied to the whole FPDs (200 N, three-point loading) (Fig. a) and pontic (120 N, pontic loading) (Fig. b); the loading point comprised 1-mm 2 circles on the occlusal surface and was distributed according to the anatomic features of common occlusion . Production and analysis of photoelastic models Based on FEA simulation, three angles of the maxillary second molar (12°, 24°, and 30°) and two FPD materials (zirconia and resin composite) were selected to manufacture entitative FPDs in the photoelastic experiment. The detailed photoelastic experiment was as follows. According to the anatomic morphology, freshly extracted human second premolars and second molars were paired. Three-dimensional scans and modelling were performed on the paired teeth. The partial maxilla model with maxillary first molar was the same as the modelling data in FEA (Fig. b). The simulated maxillary first molar tooth was fixed to preserve the space, and paired abutments were placed similar to the natural array. Subsequently, the second molar axis was tilted mesially by 12°, 24°, and 30 °. The roots of the abutments were dilated 0.2 mm outward in preparation for the periodontal ligament (PDL) simulation material . The Boolean operation subtracted the abutment teeth; therefore, two customized fossae for placing PDL and abutment teeth were made in the maxillary models. The alveolar fossae‒maxilla models (Fig. e) were printed using a three-dimensional printing machine (3Demax, DMG Medical Devices Co. Ltd., Germany). The moulds of the printed alveolar fossae–maxilla models (Fig. f) were fabricated with silicone rubber (Express™ STD Vinyl Polysiloxane Impression Material Putty, 3 M ESPE, USA). Photoelastic resin (epoxy resin, JN-L, China) was made according to the manufacturer’s instructions, vacuumed in 0 MPa at 25℃ until the bubbles disappeared, and poured into the mould. The resin model was separated from the mould when the resin lost fluidity at the initial solidification. The silicon rubber (Express™ STD Vinyl Polysiloxane Impression Material Light Body, 3 M ESPE, USA) was injected into two tooth fossae to simulate 0.2-mm PDL, and the paired teeth were placed immediately to ensure a tight bond. After the final solidification stage, abuntments–PDL–maxilla photoelastic models were made. The abutments were prepared by an experienced senior clinician. 1.5 mm of the occlusal surface and 1 mm of axial surfaces were reduced, and a 1-mm shoulder was shaped (Fig. g). The abuntments–PDL–maxilla models were three-dimensionally scanned and modelled. The design flow of FPDs was in the software (Exocad 3.0; Exocad GmbH), consistent with FEA modelling (Fig. g). FPDs were produced by the CAD/CAM system (Upcera x5, China) (Fig. h). A resin luting agent (3 M ESPE Rely™, U200 Automix, 3 M ESPE, USA) was used for cementing. Six FPD‒abutments‒periodontal tissue models with two materials and three different titled angles were prepared using the same maxilla, and all the elements were tightly bonded (Fig. h). The FPD‒abutments‒periodontal tissue model was placed in a polarized light field in the photoelastic test. A vertical force (120 N) was applied to the central fossa of the pontic by the loading cell, similar to FEA (Fig. d) . The stress distribution diagrams were drawn using a mathematical analysis software (Matlab_R 2023a, The MathWorks, Inc., USA). 

The experiment considered two variables with different values: Four FPD materials and six angles of tilted abutment tooth were applied in the FEA. First, the digital imaging and communications in medicine (DICOM) images from cone beam computer tomography (CBCT) (3D Accuitomo 170, Morita Corp., Japan) of the upper jaw of a 24-year-old female whose maxillary first molar was intact, normal, and without trauma or deformity were imported (Fig. a, b) to Solidworks© (Dassault Systèms Solid Works© Corp., Concord, USA) to perform the modelling phase. By using coordinate system transformation, the central axis of the second molar was tilted mesially by 0°, 6°, 12°, 18°, 24°, and 30° (Fig. c). The maxillary first molar was extracted virtually. The software (Exocad 3.0, Exocad GmbH; Geomagic Studio, 3D System Inc., Germany) was used to design FPDs, whose thickness is 1.5 mm for the occlusal surface, 1 mm for axial surfaces, and 1 mm for the shoulder . The connector area was set as a circle, and the diameters were 10 mm 2 in all models . FPDs were produced by the CAD/CAM system (Upcera x5, China) as monolithic restorations (Fig. d). Moreover, an 0.03-mm cement layer was set up, and FPDs and abutment teeth were assembled with maxilla in Solidworks © (Dassault Systèms Solid Works© Corp., Concord, USA) . The partial maxilla models equipped with titled abutments and FPDs were imported to a finite element analysis software program (ANSYS Workbench 18.2.2, ANSYS Inc., USA) for simulation (Fig. d). Table presents the properties of FPD materials, dental tissues, and cement. All the materials were considered homogenous, linearly elastic, and isotropic. The contact condition of all the interfaces was set as fully bonded. The boundary fixation was along the bone axes. The mesh measured 0.0002 mm, and the sum of meshes was 4,165,066. The number of nodes was qualified to ensure simulation accuracy. Two loading conditions were considered for oral rehabilitation: the vertical forces were respectively applied to the whole FPDs (200 N, three-point loading) (Fig. a) and pontic (120 N, pontic loading) (Fig. b); the loading point comprised 1-mm 2 circles on the occlusal surface and was distributed according to the anatomic features of common occlusion . 

Based on FEA simulation, three angles of the maxillary second molar (12°, 24°, and 30°) and two FPD materials (zirconia and resin composite) were selected to manufacture entitative FPDs in the photoelastic experiment. The detailed photoelastic experiment was as follows. According to the anatomic morphology, freshly extracted human second premolars and second molars were paired. Three-dimensional scans and modelling were performed on the paired teeth. The partial maxilla model with maxillary first molar was the same as the modelling data in FEA (Fig. b). The simulated maxillary first molar tooth was fixed to preserve the space, and paired abutments were placed similar to the natural array. Subsequently, the second molar axis was tilted mesially by 12°, 24°, and 30 °. The roots of the abutments were dilated 0.2 mm outward in preparation for the periodontal ligament (PDL) simulation material . The Boolean operation subtracted the abutment teeth; therefore, two customized fossae for placing PDL and abutment teeth were made in the maxillary models. The alveolar fossae‒maxilla models (Fig. e) were printed using a three-dimensional printing machine (3Demax, DMG Medical Devices Co. Ltd., Germany). The moulds of the printed alveolar fossae–maxilla models (Fig. f) were fabricated with silicone rubber (Express™ STD Vinyl Polysiloxane Impression Material Putty, 3 M ESPE, USA). Photoelastic resin (epoxy resin, JN-L, China) was made according to the manufacturer’s instructions, vacuumed in 0 MPa at 25℃ until the bubbles disappeared, and poured into the mould. The resin model was separated from the mould when the resin lost fluidity at the initial solidification. The silicon rubber (Express™ STD Vinyl Polysiloxane Impression Material Light Body, 3 M ESPE, USA) was injected into two tooth fossae to simulate 0.2-mm PDL, and the paired teeth were placed immediately to ensure a tight bond. After the final solidification stage, abuntments–PDL–maxilla photoelastic models were made. The abutments were prepared by an experienced senior clinician. 1.5 mm of the occlusal surface and 1 mm of axial surfaces were reduced, and a 1-mm shoulder was shaped (Fig. g). The abuntments–PDL–maxilla models were three-dimensionally scanned and modelled. The design flow of FPDs was in the software (Exocad 3.0; Exocad GmbH), consistent with FEA modelling (Fig. g). FPDs were produced by the CAD/CAM system (Upcera x5, China) (Fig. h). A resin luting agent (3 M ESPE Rely™, U200 Automix, 3 M ESPE, USA) was used for cementing. Six FPD‒abutments‒periodontal tissue models with two materials and three different titled angles were prepared using the same maxilla, and all the elements were tightly bonded (Fig. h). The FPD‒abutments‒periodontal tissue model was placed in a polarized light field in the photoelastic test. A vertical force (120 N) was applied to the central fossa of the pontic by the loading cell, similar to FEA (Fig. d) . The stress distribution diagrams were drawn using a mathematical analysis software (Matlab_R 2023a, The MathWorks, Inc., USA). 

FEA of three-point afterburner The FEA results showed stress concentration areas in the FPD‒abutments‒periodontal tissue complex. As for FPDs, except for the loading sites, the stress was mainly concentrated at the connectors, especially on the lingual side (Figs. a and 5). The stress on the abutment teeth was concentrated on the cervix, i.e., the distal enamel shoulder collar on the distal of the second premolar and the mesial of the second molar (Figs. a and ). In addition to the periapical area, the stress concentration on the PDL was close to the cervix of the abutment tooth (Fig. a). The stress on the alveolar bone was mainly concentrated in the alveolar crest and periapical area (Fig. a). The biomechanical behaviour of the FPD‒abutments‒periodontal tissue complex was influenced by the FPD material type. The average equivalent stress of FPDs increased with an increase in their material strength. For example, when the abutment was not tilted, zirconia exhibited the largest (2.81 MPa), followed by lithium disilicate (2.63 MPa, -6.40%), PICN (2.31 MPa, -17.62%), and resin composite (2.07 MPa, -26.21%) (Fig. b). However, contrary to the changing trend of stress on FPDs, the average equivalent stress on abutments decreased with an increase in the strength of FPDs; for example, the second premolar with a resin composite (1.88 MPa) FPD restoration underwent the greatest stress when the abutment was not tilted, followed by PICN (1.76 MPa, -6.64%), lithium disilicate (1.66 MPa, -12.08%), and zirconia (1.61 MPa, -14.32%) (Fig. b, c). In different FPD materials, the stress on the PDL of the second premolar increased as the elastic modulus of FPDs increased (Fig. b). In contrast, the PDL of the second molar exhibited an opposite pattern (Fig. c). The stress on the alveolar bone decreased with an increase in the elastic modulus of FPDs (Fig. c). The abutment inclination also affected the stress of the FPD‒abutments‒periodontal tissue complex. Taking zirconia as an example, the average equivalent stress on FPDs increased as the mesial inclination angle of the second molar increased significantly by 4.42% (2.93 MPa, 6°), 11.65% (3.13 MPa, 12°), 16.91% (3.28 MPa, 18°), 27.48% (3.58 MPa, 24°), and 38.44% (3.88 MPa, 30°) compared to 0° (2.81 MPa) (Fig. b). Concerning abutment teeth, the stress on both abutments decreased when the inclination angle of the second molar decreased from 0° (1.61 MPa, the second premolar) to 6° (1.75 MPa, + 8.57%, the second premolar). However, when the angle increased from 12° to 30°, the stress on the second premolar tended to increase by 0.92% (1.66 MPa, 18°), 3.88% (1.71 MPa, 24°), and 7.35% (1.77 MPa, 30°) compared to 12° (1.64 MPa), while the second molar showed an opposite trend, decreasing by 0.64% (1.23 MPa, 18°), 2.66% (1.21 MPa, 24°), and 4.83% (1.18 MPa, 30°) compared to 12° (1.24 MPa) (Fig. b, c). With an increase in inclination, the stress on the PDL of the two abutments also showed a similar two-stage trend (Fig. b, c). The average equivalent stress on the alveolar bone increased as the angle increased from 0° to 30° (Fig. c), and the maximum equivalent stress suddenly increased at 30° (Fig. b). FEA of pontic afterburner The results of loading the pontic at 120 N were similar to the three-point loading (Fig. , Fig. , Fig. , Fig. , and Fig. ). However, the PDL of the second premolar behaved differently. Under pontic stresses, when the inclination angle of the second molar increased from 6° to 24°, the stress on the PDL of the second premolar gradually increased first and then decreased, but increased suddenly when the inclination angle reached 0°or 30° (Fig. b). Moreover, the maximum (1754.10 MPa, + 36002.99%) and the average (2.38 MPa, + 729.19%) equivalent stress on the alveolar bone suddenly increased at 30° with the restoration of resin composite (Fig. ). Photoelastic analysis of pontic loading Qualitative photoelastic analysis led to similar conclusions. Under pontic loading, the stress was mainly concentrated near the periapical and root furcation areas. Moreover, as the strength of FPDs decreased and the inclination angle of the abutment tooth increased, the stress concentration increased (Fig. ). 

The FEA results showed stress concentration areas in the FPD‒abutments‒periodontal tissue complex. As for FPDs, except for the loading sites, the stress was mainly concentrated at the connectors, especially on the lingual side (Figs. a and 5). The stress on the abutment teeth was concentrated on the cervix, i.e., the distal enamel shoulder collar on the distal of the second premolar and the mesial of the second molar (Figs. a and ). In addition to the periapical area, the stress concentration on the PDL was close to the cervix of the abutment tooth (Fig. a). The stress on the alveolar bone was mainly concentrated in the alveolar crest and periapical area (Fig. a). The biomechanical behaviour of the FPD‒abutments‒periodontal tissue complex was influenced by the FPD material type. The average equivalent stress of FPDs increased with an increase in their material strength. For example, when the abutment was not tilted, zirconia exhibited the largest (2.81 MPa), followed by lithium disilicate (2.63 MPa, -6.40%), PICN (2.31 MPa, -17.62%), and resin composite (2.07 MPa, -26.21%) (Fig. b). However, contrary to the changing trend of stress on FPDs, the average equivalent stress on abutments decreased with an increase in the strength of FPDs; for example, the second premolar with a resin composite (1.88 MPa) FPD restoration underwent the greatest stress when the abutment was not tilted, followed by PICN (1.76 MPa, -6.64%), lithium disilicate (1.66 MPa, -12.08%), and zirconia (1.61 MPa, -14.32%) (Fig. b, c). In different FPD materials, the stress on the PDL of the second premolar increased as the elastic modulus of FPDs increased (Fig. b). In contrast, the PDL of the second molar exhibited an opposite pattern (Fig. c). The stress on the alveolar bone decreased with an increase in the elastic modulus of FPDs (Fig. c). The abutment inclination also affected the stress of the FPD‒abutments‒periodontal tissue complex. Taking zirconia as an example, the average equivalent stress on FPDs increased as the mesial inclination angle of the second molar increased significantly by 4.42% (2.93 MPa, 6°), 11.65% (3.13 MPa, 12°), 16.91% (3.28 MPa, 18°), 27.48% (3.58 MPa, 24°), and 38.44% (3.88 MPa, 30°) compared to 0° (2.81 MPa) (Fig. b). Concerning abutment teeth, the stress on both abutments decreased when the inclination angle of the second molar decreased from 0° (1.61 MPa, the second premolar) to 6° (1.75 MPa, + 8.57%, the second premolar). However, when the angle increased from 12° to 30°, the stress on the second premolar tended to increase by 0.92% (1.66 MPa, 18°), 3.88% (1.71 MPa, 24°), and 7.35% (1.77 MPa, 30°) compared to 12° (1.64 MPa), while the second molar showed an opposite trend, decreasing by 0.64% (1.23 MPa, 18°), 2.66% (1.21 MPa, 24°), and 4.83% (1.18 MPa, 30°) compared to 12° (1.24 MPa) (Fig. b, c). With an increase in inclination, the stress on the PDL of the two abutments also showed a similar two-stage trend (Fig. b, c). The average equivalent stress on the alveolar bone increased as the angle increased from 0° to 30° (Fig. c), and the maximum equivalent stress suddenly increased at 30° (Fig. b). 

The results of loading the pontic at 120 N were similar to the three-point loading (Fig. , Fig. , Fig. , Fig. , and Fig. ). However, the PDL of the second premolar behaved differently. Under pontic stresses, when the inclination angle of the second molar increased from 6° to 24°, the stress on the PDL of the second premolar gradually increased first and then decreased, but increased suddenly when the inclination angle reached 0°or 30° (Fig. b). Moreover, the maximum (1754.10 MPa, + 36002.99%) and the average (2.38 MPa, + 729.19%) equivalent stress on the alveolar bone suddenly increased at 30° with the restoration of resin composite (Fig. ). 

Qualitative photoelastic analysis led to similar conclusions. Under pontic loading, the stress was mainly concentrated near the periapical and root furcation areas. Moreover, as the strength of FPDs decreased and the inclination angle of the abutment tooth increased, the stress concentration increased (Fig. ). 

Distal abutment teeth usually tilt toward the missing tooth space if this space is not restored. During mastication, non-axial forces on inclined abutments can easily lead to periodontal trauma. FPDs are one of the mainstream restorative options for inclined abutments. The stress of the inclined abutment can significantly improve after the double-end fixed restoration, making the inclination force of the abutment before restoration an axial force close to the tooth long axis . Therefore, this study used FEA combined with photoelasticity to analyse the stress distribution in abutments with different inclination angles in the restoration and in teeth and periodontal tissues after FPD restoration to guide the selection of prosthetic materials under different abutment inclination angles. The stress transmission patterns and stress concentrations of FPD‒abutments‒periodontal tissue complexes conformed to expectations. FPDs and abutments were subjected to most stresses (Figs. and , Fig. and Fig. ). The stresses concentrated at the connector area of the FPD (Figs. , and and Fig. ), which is an irregularly shaped construction with less support structure, is frequently associated with stress concentration, and is regarded as the most susceptible area responsible for FPD fracture . Since the maxillary dentition is inclined slightly buccally and the tip of the tongue is functionally sharp, stresses are more likely to be concentrated on the lingual side of the FPDs connector (Figs. , and , and Fig. ), which should be appropriately strengthened in the restoration design . The stress is transmitted from FPDs to the abutment teeth and is concentrated in the shoulder platform close to the pontic (Figs. , and , and Fig. ) as another stress concentration area . Concerning the periodontal tissue, the stresses transmitted to the PDL and alveolar bone were already minimal, approximately 10% of the stress experienced by FPDs (Figs. , and , Fig. , Fig. and Fig. ). Since stresses are transmitted to the second premolar in an almost straight line, the second premolar and its PDL bear more stress than the second molar and its PDL (Figs. and , Fig. and Fig. ). The abutments, PDL, and the alveolar bone act symbiotically to dissipate the mechanical loads incurred during mastication. The PDL functions both in tension and compression . FEA is often combined with in vitro mechanical analysis methods such as photoelastic analysis in biomechanical research . Based on the important biomechanical role of PDL, we innovatively use light-body silicone rubber to simulate PDL in the photoelastic modelling. The FEA and photoelastic analysis results reveal that the stresses on the PDL and alveolar bone are concentrated in the tooth neck near the gap, periapical area, and root bifurcation (Figs. and , Fig. and Fig. ), which is consistent with the results of other studies . Low-modulus FPD materials are prone to fracture and lead to restoration failure when subjected to masticatory forces. In contrast, high-modulus materials are prone to abutment and jaw wear, which further causes periodontal trauma . PICN, a novel role in CAD/CAM materials, has an interpenetrating network of polymer and ceramics. Compared to traditional ceramics, PICN has a higher resilient modulus and lower stiffness and is softer. This material meets the basic needs of various clinical scenarios and is a good choice for chairside fixed prostheses. The mechanical performance of PICN overcomes the great brittleness of ceramics and the poor wear resistance of the polymer. In addition, its fracture toughness and wear resistance are close to dentin and enamel, respectively, making PICN more ‘friendly’ to natural teeth . Therefore, in this study, PICN materials were used to compare and analyze traditional ceramic restoration materials (zirconia and lithium disilicate) and resin composite, although Vita has not yet recommended it for FPDs due to the insufficient mechanical strength of PICN. The results of both FEA and photoelastic analyses showed that when external stress was applied, the stress of high-modulus FPD materials was easy to concentrate on itself and not easily transmitted to the abutment teeth and the alveolar bone due to the considerable stiffness and ability to absorb stresses (Figs. , and , Fig. , Fig. , and Fig. ). When the elastic modulus of FPDs increased, the stress on both abutments decreased (Fig. b, c and Fig. b, c), and the stress that should be buffered by PDL (mainly the PDL of the second premolar) was higher (Fig. b and Fig. b). At this time, the PDL of two abutments assumed the task of stress buffering, so the stress on the second molar decreased with an increase in the elastic modulus of FPDs (Fig. c and Fig. c). If the stress concentration exceeds the bearing limit of FPD materials, it can easily cause restoration fracture. On the contrary, low-modulus materials can give rise to periodontal injury due to stress concentration in abutment teeth and periodontal tissues . After the FPD restoration of the inclined abutment, the mesial component of the force exerted on the tooth can be transmitted to the entire dental arch through the bridge, changing the stress pattern of the inclined abutment. The stress distribution tends to be uniform, and the periodontal tissue of the abutment is stressed within its physiological limit, which is conducive to periodontal tissue health. However, the abutment tooth’s tilt angle should be less than 24°  or 30° ; otherwise, it is not suitable to FDPs restoration. In this study, the stresses of FPDs increased with increased mesial inclination angle of the second molar, possibly due to the decreased mesial and distal diameters of the pontic of FPDs with the increased abutment tooth inclination angle (Fig. and Fig. ). Physiologically, due to the presence of the compensating curve, there was an angle (6 ‒ 12°) to the long axis of the maxillary second premolar and maxillary second molar . When the mesial inclination angle of the second molars increased by no more than 12°, the long axes of the two abutments tended to be parallel. In such a case, the stress was more likely to be transmitted downward along the long axis of the teeth; therefore, the stress on the two abutments and their PDL increased with increased inclination (Figs. and , Fig. , and Fig. ). However, when the mesial inclination angle of the second molars increased by more than 12°, the intersection angle of the long axis of the two abutments increased, and the direction of external force was closer to the long axis of the second premolar, so the stress on the second premolar and its PDL gradually increased. At this time, the stress on both the two abutments and their PDL were shared as a whole, so the stress on the second molar and its PDL decreased with increased inclination (Figs. , and , Fig. , and Fig. ). The stress on the alveolar bone increased with the inclination of the abutment, possibly due to the increased tendency of stress concentration near the tooth neck, periapical area, and root bifurcation (Fig. and Fig. ). Under pontic loading, the biomechanical behaviour of FPD‒abutments‒periodontal tissue complex was similar to the three-point loading; however, the difference is the PDL of the second premolar. The stress axis was not close to the long axis of the two abutment teeth; therefore, the stress on the PDL of the two abutments showed the same trend with a change in inclination within a certain range, i.e., the stress on the PDL of the two abutments tended to increase first (6 ‒ 12°, the intersection angle of the long axes of the two abutments decreased) and then decreased (18 ‒ 24°, the intersection angle of the long axes of the two abutments increases) (Fig. and Fig. ). However, when the intersection of the long axes of the two abutments was too large (the inclination of the second molars was 0° or 30°), the stress on the PDL of the second premolar increased suddenly (Fig. and Fig. ). Based on the photoelastic analysis and FEA results in this study, we believe that FPDs can be used when the inclination angle of the distal abutment tooth does not exceed 24°, but it is necessary to pay attention to protecting the abutments (especially the inclination angle < 12°) and the periodontal tissue (especially the inclination angle > 12°). When the angle exceeds 30°, the maximum stress concentration on the alveolar bone increases suddenly and surpasses the threshold for a bone overload of 170 MPa (Fig. and Fig. ); thus, FPDs should be used cautiously . To protect the abutments and the periodontal tissue, materials with higher strength should be used for FPDs in clinical practice. But when the PDL of the mesial abutment tooth is poor, choosing FPDs with a low elastic modulus is advisable. Based on the typical clinical scenario of distal abutment tooth inclination, this study provided the applicable conditions and material selection suggestions for FPDs. However, this study still has some limitations, such as not including patient-based in vivo studies and not considering other forms of missing teeth. Therefore, further research is necessary to better manage abutment tooth tilting. 

Based on the results of this study, the following conclusions are drawn: During FPD restorations, the stress was easily concentrated in the weak parts, like FPD connectors, enamel shoulder collar, periapical area, and root bifurcation. Three-unit FPDs can be used for restoration when the inclination of the distal abutment is less than 24°, but it is necessary to pay attention to protecting the abutments (especially the inclination angle < 12°) and the periodontal tissue (especially the inclination angle > 12°). In general, to protect abutments and periodontal tissues, choosing an FPD material with higher strength is recommended. But when the PDL of the mesial abutment tooth is poor, FPDs with a low elastic modulus are recommended. 

Below is the link to the electronic supplementary material. Supplementary Material 1 

