Automated Support Generation for Fixed Partial Dentures and Impact of Bone Loss, Bone Quality and Support Types: Parametric Cad and Finite Element Analysis
Abstract
:1. Introduction
2. Materials and Methods
2.1. CAD Modelling
2.1.1. Manual Preprocessing
2.1.2. Automatic Generation of FPD Support
- (1)
- Merge Mmesial, Mdistal and Msurface into a single mesh, MFPD.
- (2)
- Extract the boundary curves Cdistal and Cmesial from the open surfaces Mdistal and Mmesial using the naked edge curve function in Grasshopper, respectively; see Figure 3a. These boundary curves represent the open curves of the surfaces, which prevent the mesh from being closed.
- (3)
- Find the centre points Pdistal and Pmesial for the curves Cdistal and Cmesial.
- (4)
- Create the line LMD between Pdistal and Pmesial and compute its midpoint PMD.
- (5)
- Compute the smallest bounding box, Bmin, for the mesh MFPD using the bounding box function in Grasshopper.
- (6)
- Deconstruct the bounding box Bmin into 6 faces, fi, and compute the centre point, Pface-i, for each face.
- (7)
- Identify the gingival face, fgingival, which centre point is closest to the point PMD; see Figure 3a.
- (8)
- Create a plane, EFPD, parallel to fgingival at Pdistal.
- (9)
- Define a local coordinate system for MFPD, denoted as SFPD (). This coordinate system is centred at Pdistal. The z-axis () of SFPD is set to be perpendicular to EFPD at Pdistal. The x-axis () is defined to align with the line LMD, and the y-axis () is constructed to be orthogonal to both and ; see Figure 3b.
- (10)
- Translate MFPD so its local coordinate system’s origin Pdistal matches the global coordinate system’s origin P0 via the Grasshopper function move to point.
- (11)
- Rotate MFPD to align its local axes SFPD () with the global coordinate system’s axes Sglobal () using the function Orient; see Figure 3c.
- (1)
- Create a line Lp that connects the centre points, Pface-g and Pface-o, of the gingival face fgingival and the occlusal face focclusal, respectively.
- (2)
- Generate 100 equally spaced planes Ei along the line Lp, each perpendicular to Lp.
- (3)
- Find the intersection curves, Ci-distal, between the planes Ei and the lumen mesh Mdistal (analogue Ci-mesial with Mmesial for the mesial side). Retain only the closed curves and remove the initial and final curves to eliminate any errors; see Figure 3d.
- (4)
- Calculate the circularity index (CI) of the curves Ci-distal using the formula , where n is the number of curves, Ai represents the area of each curve and Pei represents the perimeter of each curve [21].
- (1)
- Generate the interior mesh surface, Minterior, by creating a parallel mesh to the distal lumen mesh surface, Mdistal, at a distance defined by the cement thickness, dcement. This involves creating new vertices parallel to each original vertex of Mdistal inward along its normal vectors by dcement and then using these newly calculated vertices to construct Minterior; see Figure 4a.
- (2)
- Extract the boundary curve, Cinterior, from Minterior.
- (3)
- Create a lofted mesh surface, Mloft, by interpolating between the curves Cdistal and Cinterior; see Figure 4a.
- (4)
- Join the surfaces Mdistal, Mloft and Minterior to create the geometry of the cement layer, MCement.
- (1)
- Divide Cinterior into 100 equal-length segments and identify Pmin (xmin, ymin and zmin), as the point with the lowest z-coordinate.
- (2)
- Create a point – ε, where ε prevents the intersection of Cinterior with the first projected curve to enable the loft function.
- (3)
- Create a line segment Ltooth defined by the start point Ps, a tangent vector in the direction and a user-defined length dtooth.
- (4)
- Divide Ltooth into i equally spaced points (with i = 20 in this work).
- (5)
- Create i planes parallel to the xy-plane at and project Cinterior onto each , yielding curves .
- (6)
- Create Bézier functions Bx(t) and By(t) to scale the curves to mimic the anatomy of a natural tooth, and the non-uniform scaling equation is
- (1)
- Compute Bx(t) and By(t) for all t values, resulting in the arrays Bx and By. Each point becomes a scaling factor in the x and y directions.
- (2)
- Scale the curves using the non-uniform scaling factors Bx for the x-direction and By for the y-direction.
- (3)
- Create a lofted mesh surface Mroot by interpolating through the set of curves and Cinterior.
- (4)
- Create a planar surface Mapex from the last curve .
- (5)
- Join Minterior, Mroot and Mapex to create the geometry of the first premolar tooth, denoted as M44.
- (1)
- Perform the steps until step 8 (First premolar).
- (2)
- Create a Bézier function BT(t) to translate the scaled curves in both the x and –x direction to mimic the anatomy of roots, using the control points PT0, PT1, PT2 and PT3, as defined in Table 1.
- (3)
- Compute BT(t) for all t values, resulting in the arrays BT.
- (4)
- Duplicate the curves , translating one set in the positive x-direction and the other set in the negative x-direction by BT × dR, yielding and . In our work, dR represents the distance between the furcation and the root apex in the x-direction.
- (5)
- Create lofted mesh surfaces Mroot-1 and Mroot-2 through the set of curves and Cinterior and and Cinterior, respectively.
- (6)
- Create planar surfaces Mapex-1 and Mapex-2 from the last curves and , respectively.
- (7)
- Join Minterior, Mroot-1 and Mapex-1 to create the geometry of the first root, denoted as MT1, and join Minterior, Mroot-2 and Mapex-2 to create the geometry of the second root, denoted as MT2.
- (8)
- Perform a mesh union on the meshes MT1 and MT2 to create the geometry of the second molar tooth, denoted as M47.
- (1)
- Join the mesh surfaces Mroot and Mapex to create the surface MPDL-in, which interfaces with the tooth geometry M44.
- (2)
- Construct new vertices parallel to each original vertex of MPDL-in outward along its normal vector by the user-defined periodontal ligament thickness dPDL. Use these newly created vertices to create MPDL-ex, which later interfaces with the bone.
- (3)
- Create a lofted mesh surface, MPDL-loft, using the boundary curves CPDL-in and CPDL-ex obtained from MPDL-in and MPDL-ex.
- (4)
- Join the surfaces MPDL-in, MPDL-loft and MPDL-ex to form the periodontal ligament geometry MPDL.
- (5)
- After insertion into bone (step 4), the part of the PDL that was not in contact with the bone was cut.
- (1)
- Create circles CA1, CA2 and CA3 on planes parallel to the xy-plane with user-defined lengths lA1, lA2 and lA3 and diameters dA1, dA2 and dA3. The plane at the distance lA1 is the implant neck plane Eimplant ().
- (2)
- Compute a lofted surface through the set of curves Cinterior, CA1, CA2 and CA3. Join this surface with Minterior to create the abutment geometry without a borehole, denoted as MA-loft.
- (3)
- Create the borehole geometry by creating circles CBA1 and CBA2, each with user defined-lengths lBA1 and lBA2 and diameters dBA1 and dBA2, respectively. Extrude CBA1 upwards to the cement layer and CBA2 downwards to the lower surface of MA-loft, forming the borehole mesh MA-bore.
- (4)
- Extract MA-bore from MA-loft using a Boolean mesh subtraction operation to create the abutment geometry for implant support, MAbutment.
- (1)
- Create a cylinder MI-loft on the plane Eimplant with user-defined diameters dI1 (upper diameter) and dI2 (bottom diameter) and length lI1. In this work, the implant is designed to be straight with dI1 = dI2.
- (2)
- Chamfer the implant body MI-loft at the upper cylinder edge curve with a radius rC and fillet the bottom edge of the cylinder with a radius rF.
- (3)
- Create the borehole geometry MI- bore using circles with diameters dBI1(= dA1), dBI2(= dA3) and dBI3 and lengths lBI1, lBI2, lBI3 and lBI4. For more geometric details, see Figure 5b.
- (4)
- Extract MI-bore from MI-loft to create the implant body geometry Mimplant.
- (5)
- Construct a helix curve, Chelix, starting lt below the implant neck plane, with the same diameter dI1 as the implant, a user-defined pitch, pt, and number of turns, nturns, along the z-axis using the helix curve function in Grasshopper.
- (6)
- Extend the curve Chelix by creating tangential segments at its end points, Pstart and Pend, to replicate the toolpath movement during the milling process.
- (7)
- Generate the surface of a revolution, Mthread, using the function sweep rail with the metric thread profile as the profile curve, the helix curve Chelix as the rail curve and the z-axis as the axis of the revolution.
- (8)
- Extract Mthread from the implant body geometry Mimplant.
Component | User Defined Parameters for Tooth Support, in mm. |
---|---|
Cement layer | dcement = 0.1 mm (for mesial and distal side). |
Tooth, first premolar | ε = 0.15 mm, dtooth = 16.3 mm, Px0(0;1), Px1(0.08;0.78), Px2(0.75;0.55), Px3(1;0.05), Py0(0;1), Py1(0.2;0.58), Py2(0.95;0.5), Py3(1;0.05) |
Tooth, second molar | ε = 0.15 mm, dtooth = 12.8 mm, Pxi (same as first premolar), Pyi (same as first premolar), PT0(0;0), PT1(0.35;0.55), PT2(0.4;0.9), PT3(1;1) |
PDL | dPDL = 0.3 mm (for mesial and distal side) |
Component | User Defined Parameters for Implant Support, in mm. |
Cement layer | dcement = 0.1 mm |
Abutment | lA1 = 1.0 mm, lA2 = 1.5 mm, lA3 = 7.0 mm, dA1 = 3.0 mm, dA2 = 2.6 mm, dA3 = 2.6 mm, lBA1 = 2.0 mm, lBA2 = 2.2 mm, dBA1 = 1.6 mm, dBA2 = 1.3 mm, (mesial and distal side). |
Implant body | dI1 = 5.0 mm, dI2 = 5.0 mm, lI1 = 8.0 mm, rC = 1.5 mm, rF = 0.2 mm, lBI1 = 0.5 mm, lBI2 = 4.0 mm, lBI3 = 5.0 mm, lBI4 = 5.3 mm, dBI1 = 3.0 mm, dBI2 2.6 = mm, dBI3 = 1.1 mm, rb = 0.2 mm, lt = 1.5 mm, pt = 0.5 mm, nturns = 10, (mesial and distal side). |
Abutment screw | ls = 4.5 mm. |
- (1)
- Merge the FPD (MFPD) and the support geometries on both the mesial and distal sides. For tooth–tooth support, include Mcement-distal, Mcement-mesial, M44, M47, MPDL-distal and MPDL-mesial. For implant–implant support, include Mcement-distal, Mcement-mesial, Mabutment-distal, Mabutment-mesial, Mimplant-distal, Mimplant-mesial, Mscrew-distal and Mscrew-mesial. This step forms the mesh MFPD+S.
- (2)
- Define the distal and mesial sides of the FPD, PCM-distal and PCM-mesial, on the mandibular centre curve, CM, as shown in Figure 6a. The distance between these points corresponds to the distance between Pmesial and Pdistal of the FPD. Further details about the construction of the centre curve CM can be found in our previous study [16].
- (3)
- Project the points PCM-distal and PCM-mesial onto the crestal mandibular surface, resulting in P’CM-distal and P’CM-mesial. The steps required the projection of a point from CM onto the crestal surface of bone were further explained in a previous work [16].
- (4)
- Define a local coordinate system for the mandibular bone Mbone, denoted as Sbone (). This system’s origin is positioned at . The z-axis () is aligned with the line connecting PCM-distal and P’CM-distal. The x-axis () aligns with the line connecting and , and the y-axis () is constructed to be orthogonal to both and ; see Figure 6a.
- (5)
- Perform the necessary rotations and translations to align MFPD+S’s local coordinate system, SFPD (), precisely with the mandibular bone’s local coordinate system, Sbone (), using the Grasshopper functions Move to point and Orient.
- (6)
- Translate MFPD+S along the z-axis () by a distance dbone, specified by the user. This distance, dbone (dbone = 2.8 mm in this study), represents the gap between Pdistal and the crestal surface of the mandibular bone; see Figure 6b.
- (7)
- After MFPD+S was positioned and aligned, it was extracted from the mandibular geometry, Mbone, using a Boolean mesh subtraction.
2.2. Finite Element Analysis
2.2.1. Material Properties, Contact Models and Mesh Size
2.2.2. Boundary Conditions and Loading
2.2.3. Evaluation
3. Results
4. Discussion
5. Conclusions
- (1)
- The proposed algorithm allows an automatic parametrised generation of support for a 4-unit fixed partial denture that can be used to predict the effect of bone loss and bone quality on stresses for patient-specific geometries.
- (2)
- The optimal treatment in terms of support type should be adapted to the patient’s specific needs to ensure long-term stability. Decision support involving automatic modelling, e.g., of support, and simulation might improve optimal treatment from a long-term perspective.
- (3)
- Further work for automated CAD and simulation still remains necessary. For example, the root generation algorithm can still be improved to adapt closer to clinically observed structures.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Component | Material | Young’s Modulus E [GPa] | Poisson’s Ratio |
---|---|---|---|
Implant 1 | Titanium grade 4 | 104.5 | 0.37 |
Abutment 1 | Titanium grade 4 | 104.5 | 0.37 |
Implant screw 1 | Titanium grade 5 | 114.0 | 0.33 |
Cement layer 2 | Glass ionomer cement | 15.9 | 0.33 |
Abutment tooth 3 | Polyurethane | 3.525 | 0.33 |
FPD 4 | Zirconium dioxide | 210 | 0.27 |
Cortical bone 5 | - | 13.7 | 0.3 |
Transition zone 6 | - | graded | 0.3 |
Cancellous bone 7 | - | 9.5 (D1 quality) | 0.3 |
- | 5.5 (D2 quality) | 0.3 | |
- | 1.6 (D3 quality) | 0.3 | |
- | 0.69 (D4 quality) | 0.3 |
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Jemaa, H.; Eisenburger, M.; Greuling, A. Automated Support Generation for Fixed Partial Dentures and Impact of Bone Loss, Bone Quality and Support Types: Parametric Cad and Finite Element Analysis. Dent. J. 2024, 12, 394. https://doi.org/10.3390/dj12120394
Jemaa H, Eisenburger M, Greuling A. Automated Support Generation for Fixed Partial Dentures and Impact of Bone Loss, Bone Quality and Support Types: Parametric Cad and Finite Element Analysis. Dentistry Journal. 2024; 12(12):394. https://doi.org/10.3390/dj12120394
Chicago/Turabian StyleJemaa, Hassen, Michael Eisenburger, and Andreas Greuling. 2024. "Automated Support Generation for Fixed Partial Dentures and Impact of Bone Loss, Bone Quality and Support Types: Parametric Cad and Finite Element Analysis" Dentistry Journal 12, no. 12: 394. https://doi.org/10.3390/dj12120394
APA StyleJemaa, H., Eisenburger, M., & Greuling, A. (2024). Automated Support Generation for Fixed Partial Dentures and Impact of Bone Loss, Bone Quality and Support Types: Parametric Cad and Finite Element Analysis. Dentistry Journal, 12(12), 394. https://doi.org/10.3390/dj12120394