An Oblique Cutting Based Mechanical Model For Insertion Torque of Dental Implant

 Abstract: The insertion torque of a dental implant is an important indicator for the primary stability of dental implants. Thus, the preoperative prediction for the insertion torque is crucial to improve the success rate of implantation surgery. In this present research, an alternative method for prediction of implant torque was proposed. First, the mechanical model for the insertion torque was established based on oblique cutting process. In the proposed mechanical model, three factors, including bone quality, implant geometry and surgical methods were considered by defined bone-quality coefficients, chip load and insertion speeds, respectively. Then, the defined bone-quality coefficients for cancellous bone with the computed tomography (CT) value of 235~245, 345~355 and 415~425 Hu were obtained by a series of insertion experiments of IS and ITI implants. Finally, the insertion experiments of DIO implants were carried out to verify the accuracy of developed model. The predicted insertion torques calculated by the mechanical model were compared with that acquired by insertion experiments, which were agreed match with the relative error less than 15%. This method reduces the time consumption on establishing the fitting equations for different implants and enhance the predicted accuracy by considering the effects of implants’ geometries and surgical methods.


Introduction
Implant dentures have been one of the most popular options for teeth loss in the last decade [1].After the implant socket is prepared by a series of processes such as drilling, reaming and tapping, an implant is inserted in alveolar bone with a certain torque, named the insertion torque.Many clinical data have shown that 30~70 N•cm is a reasonable range of insertion torque for the great primary stability of implants, and therefore it has been accepted to evaluate surgical success [2][3][4].Although the reasonable range of insertion torque are determined by many factors such as shape and diameter of implants [5][6], loading condition [7], or age, gender and height of patients [8][9], it would not be discussed in the present research.As the insertion torque could not be obtained until the implant was fully inserted, if the implant was fully inserted and the insertion torque was not in the reasonable range, the implantation surgery would most likely fail and the patient have to endure a second surgery.Therefore, to improve success rate and avoid the second surgery, preoperative prediction for the insertion torque of implants have been the focus in clinical.
In current clinical researches, computed tomography (CT) value has been used to evaluate the condition of bone quality and proved positively correlated with the insertion torque [10,11].In addition, the effects of implant geometries [12] and surgical methods [13][14] on insertion torque have been realized.For example, the larger insertion torque will be obtained by a conical [15], large-diameter implant [16] or a small-diameter implant socket [17,18].As there are no quantitative models to describe the effects of implant geometry and surgical methods on insertion torques, many researchers focus on the empirical formulas fitted by CT value to predict the insertion torque [19][20][21].Although the accuracy of these fitting formulas is mostly more than 80%, however, they would not work once the implant or the surgery method was changed.Moreover, it is a time and •3• money consuming project to establish fitted formulas for all implants and surgical methods.
In this present research, an alternative method was provided by establishing the mechanical model for the insertion torque of a dental implant.In the proposed mechanical model, three factors, including bone quality, implant geometry and surgical methods were considered by defined bone-quality coefficients, chip load and insertion speeds, respectively.Two kinds of bone-quality coefficients for different forming methods were obtained for bone with the CT value of 235~245, 345~355 and 415~425 Hu by a series of insertion experiments of IS and ITI implants, respectively.And the accuracy of developed model was verified by DIO implants with the relative error less than 15%.

A mechanical model based on oblique cutting theory
The insertion processes of dental implants involves two forming methods, i.e., the thread-cutting process [22] for implants with cutting edges and the thread-forming process for implants without cutting edges [23].In this section, the implant typed DIO SFR5010 (DIO Innovation Health Care, Busan City, Korea) with 4 cutting edges in the apical part and continuous threads in the tail part were selected to establish the mechanical model for insertion torque.

Forming process of matching threads
Figure 1 describes the geometry of DIO SFR5010 and its insertion process.In the insertion process, each thread would process a layer of bone material as the implant fed into the predrilled implant socket.When the implant was inserted into the implant socket, the matching threads were firstly cut by the cutting edges in the apical part.This is the thread-cutting process with physical displacement and chip separate of bone material [24].Until the apical part was fully inserted into the implant socket, the processed mating threads were further formed by continuous threads in the tail part.This is the thread-forming process with only physical displacement but no chip separate of bone material.Meanwhile, the thread-cutting process still continuous and the torque were increasing.Until the implant was fully inserted into the implant socket, the torque reached peak point-this is the insertion torque discussed in this research.In this process, the shape of matching threads experienced both thread-cutting and thread-forming process were defined by the last contacted continuous thread while that of threads only experienced the thread-cutting process were defined by the last contacted cut edge.In order to detail the shape of matching thread, DIO SFR5010 was cut into 12 thin slices with one thread involved in and in the apical part, each slice was further separated into 4 cutting elements by 4 cutting edges.Two coordinate systems were defined.The global coordinate system {C:OXYZ} was attached to the implant with the Z-axis along the rotation axis.The cutting element coordinate system {c:oxiyizi} was attached to each cutting element with the x-axis paralleling the helical path, where i indicates the ith thread.
Supposing the starting point as M(b,0,0) in {C:OXYZ}, the helical path of threads can be expressed as follows: ( , ) ( / 2) cos where, x, y, z are the point coordinates of the helical path, θ is the angle position of the helical path, b, hd, P and β is initial radius, tooth height, pitch and taper angle of DIO SFR5010, respectively.Particularly, in the tail part, there is β=0.
The radial distance ri from the Z-axis to outer geometry of ith thread can be expressed as: The radial engagement hi of ith cutting element can be calculated as follows: where, rk is the first contacted cutting element, Hi is the diameter of the implant socket connecting ith cutting element.
The inclination angle of threads γ can be calculated as follows: ) where, ξ and λ is the thread lead angle and the flute helix angle, respectively.

Force-chip load relationship
Before the forces during the insertion process were discussed, three assumptions were made as follows: (i) each cutting element sustained normal and friction forces, and all forces are applied on the centroid of the respective faces; (ii) the effects of elastic recovery for the prediction of insertion torque were ignored; (iii) insertion torques generated by one thread remained constant throughout the whole insertion process.
According to assumptions, the forces applied on all cutting elements can be composed of the normal force Fn and the friction force Ff as follows [25]: where, A is chip load, which equal to the unformed chip area and depend on the implant geometry, Kn and Kf are specific energies, which are related to the tool geometry and work conditions as follows [26]: ) where, V is the insertion speed, h is the radial engagement of each cutting element, a0~a3 and b0~b3 are the specific energy coefficients, which are dependent on materials of cutting tool (i.e., implant) and the workpiece (i.e., cancellous bone).As most of implants' material are titanium or titanium-alloys, therefore, the a0~a3 and b0~b3 were only determined by bone quality and defined as bone-quality coefficients.
Considering the normal force Fn and the friction force Ff are different during thread-cutting and thread-forming processes, they would be discussed in sections 2.2.1 and 2.2.2, respectively.

Forces in thread-cutting process
As shown in Figure 2, the oblique cutting model [27,28] was established based on coordinate system {c:oxiyizi} to define the forces on each elements during thread-cutting process.Two planes, the normal and chip-flow plane, were introduced.The normal plane was defined by the xi-axis and zi-axis and the chip-flow plane was coincident with the rake surface of cut edges.In normal plane, the normal force Fcni was defined perpendicular to the rake surface.In chip-flow plane, the friction force Fcfi was defined collinear with chipflow orientation [29].Meanwhile, the chip-flow angle i was defined equal to the inclination angle γ based on the Stabler's rule.

Figure 2
The oblique cutting process According to Equations ( 5) and ( 6), Fcni and Fcfi can be expressed as follows: ) where, Aci are the chip load of the ith cutting element, Kcn and Kcf, the specific energies in thread-cutting process.According to Equations ( 7) and ( 8), they can be calculated as: ) where, a0~a3 and b0~b3 are the bone-quality coefficients during thread-cutting processes, which would be further determined by insertion experiments.
The chip load Aci can be calculated as follows: ) where, w is the tooth top width of ith cutting element, the radial engagement hi can be calculated according to Equation (3) as: Where, Nt is the number of cutting edges.There is Nt = 4 in DIO SFR5010.By decomposing Fcni and Fcfi into the three axes of {c:oxiyizi}, three axial forces Fxᵢ, Fyᵢ and Fzᵢ can be obtained as follows: Then, the thrust force Fthrᵢ and tangential force Ftanᵢ of each cutting element and the total insertion torque M can be calculated as Equations (16)(17)(18), respectively.si cos n

Forces in thread-forming process
In order to define the forces during thread-forming process, six faces named S1~S6 were introduced as shown in Figure 3.The ridge would be formed as the plastic deformation and flow of bone material during the insertion process.As same as the thread-cutting process, the normal forces Fni were defined proportional to the contact areas, and the friction forces Ffi were defined collinear with chip-flow orientation [30].Ffnᵢ and Fffᵢ can be expressed as follows: where, Kfn and Kff were the specific energies during the thread-forming process.According to the Equations ( 7) and ( 8), they could be calculated as: ln ln ln ln ln ln ln ln ln ln where, c0~c3 and d0~d3 are the bone-quality coefficients in the thread-forming process and they would be further determined by insertion experiments.
The chip load Afi can be calculated as follows: where, y(η) is expressed as: where, hi is the radial engagement of ith thread and it was given as Equation ( 3), α is the thread angle, η1 and η2 is the incident angle and lobe-relief angle of threads, respectively, zi is the z coordinate of point Q and it is given as follows: where, rk is the first-contacting thread.By decomposing Ffni and Fffi of the ith thread into three axis of {c:oxiyizi}, three axial forces can be obtained as follows: Then, the thrust force Fthrᵢ and tangential force Ftanᵢ of each cutting element and the total insertion torque M can be calculated as follows: According to Equations (15-18) and (26-29), it could be observed that the insertion torque was related to the normal and friction force and further determined by a) bonequality coefficients, b) insertion speed V, c) radial engagement hi and d) chip load A. These give a good explanation for the effects of bone quality, surgical methods and the implant geometry, respectively.When the implant and the surgical method were selected, hi, A and V were determined.The only consideration is the bone-quality coefficients, which were given in the section 3.

Determination of bone-quality coefficient and validation of mechanical model
To

Insertion experiments
Three groups of insertion experiments were conducted, where IS implants (IS BIS4510 and IS BIS5010, Neobiotech Co.,Ltd., Seoul, Korea) with cutting edges were used to determine bone-quality coefficients a0~a3 and b0~b3 during thread-cutting process, and ITI implants (ITI RN4510 and ITI RN5010, ITI International Team for Implantology, Basel, Switzerland) with continuous thread typed were used to determine bone-quality coefficients c0~c3 and d0~d3 during thread-forming processes.DIO SFR5010 were used to verify the established model.The geometry parameters of these implants were shown as Table 2. w2=0.12where β1, α1, L1, D1, P1, H1 are the parameters of apical part of implant DIO SFR5010 while β2, α2, α3, L2, D2, P2, H2 the tail part of implant DIO SFR5010.
The insertion experiments, including the drilling process of implant sockets and the insertion process of implants, were conducted on the CNC machine (HAAS OM-2A, Haas Automation Inc., Oxnard, CA, USA) as shown in Figure 5.The parameters of drills, implants, and experiment setting were listed as Table .3. To minimize the coaxiality error between the implant and corresponding implant socket, there was no interruption between the drilling and insertion processes.The high accuracy dynamometer (Kistler9119AA2, Kistler Instruments Ltd., London, UK, sampling rate: 1200 Hz) was used to capture the thrust forces and insertion torques during the insertion process of implants.

Bone-quality coefficients
The results of thrust forces Fthrᵢ and insertion torques of IS and ITI implants were presented as follows: The peak torque and thrust force were used to determine the bone-quality coefficients for thread-cutting and threadforming processes.The obtained bone-quality coefficients were listed in Tables 4 and 5.It was observed that the bone-quality coefficients a0~a3, b0~b3, c0~c3 and d0~d3 were different in 3 group with CT value of 235~245, 345~355 and 415~425 HU, which is the great explanation for the effects of bone quality.

Validation of mechanical model
Substituting obtained bone-quality coefficients into the established model, the predicted insertion torque and measured insertion torque were shown in Figure 7 and Table 6.As shown in Figure 7, the variations of material properties of bone blocks brought a significant fluctuation of the initial insertion torques obtained by experiments.But the trends and predicted peak insertion torques by mechanical models agreed well with that acquired by insertion experiments.The relative errors were calculated as follows.

Conclusions
In this present research, a mechanical model was established for predicting insertion torque of dental implant.The effect of bone quality, the surgical method and the implant geometry were explained by the model parameters: a) bonequality coefficients, b) insertion speed, and c) radial engagement hi, chip load A and implant diameter ri, respectively.The more specific conclusions can be drawn as follows: (1) The bone-quality coefficients were determined by bone CT value and different in implants with or without cutting edges.The reasonable explanation for this phenomenon may be the bone quality depended on not only bone density, i.e., bone CT value, but also the microstructure of trabecular bone.
(2) The error of this mechanical model may result from the effects of local anisotropy of cancellous bone, which were ignored in the present research.
(3) The established mechanical model can help clinicians to make accurate assessment whether the implants and surgical methods are reasonable for individual.Comparing to the fitting formula, this method could avoid plenty of experiments caused by changing implants and surgical method.

Declaration
Figures  The oblique cutting process

Figure 1
Figure 1 DIO SFR5010 and its insertion process: (a) the geometry of DIO SFR5010 and two coordinate systems, (b) geometric parameters of DIO SFR5010 and the insertion process: the initial position was in red and the position of one rotation cycle was in black, (c) the relationship of angle parameters γ, ξ and λ.

Figure 3 A
Figure 3 A typical form tap tooth: (a) the schematic diagram of form tap tooth; (b) three views of ith tooth.
define the bone-quality coefficients, more than 80 bone blocks with the size of 25 × 25 ×40 mm 3 were cut from the epiphysis areas of four bovine femurs with different age, weight and gender as shown in Figure4.The mean CT value of bone material within 1 mm around the predicted implant socket for each bone block were recorded by Planmeca ProMax 3D Mid CT (Planmeca UK Limited, London, UK. scanning time: 13.929 s, tube voltage: 90 kV tube current: 10mA).According to recorded CT value, 36 bone blocks were selected and further classified into 3 groups with CT value of 235~245, 345~355, and 415~425 HU, respectively.

Figure 4
Figure 4 Preparation of bone blocks: (a) bovine femur, (b) A-A cross-section, (c) the bone blocks used in experiments and the CT scan area.

Figure 5
Figure 5 Preparation of bone blocks: (a) bovine femur, (b) A-A cross-section, (c) the bone blocks used in experiments and the CT scan area.

Figure 7
Figure 7 Insertion torques obtained by the mechanical model and experiments

Figure 3 A
Figure 3

Figure 4 Preparation
Figure 4

Figure 5 Preparation
Figure 5

Figure 6 The
Figure 6

Table 1
CT value of 3 group bone blocks

Table 2
Implant parameters

Table 4
Bone-quality coefficients for thread-cutting