Hindawi Applied Bionics and Biomechanics Volume 2020, Article ID 5786593, 11 pages https://doi.org/10.1155/2020/5786593 Research Article Modeling and Experimentation of the Unidirectional Orthodontic Force of Second Sequential Loop Orthodontic Archwire 1,2 1,2 1,2 1,2 3 1,2 Jin-Gang Jiang , Yi-Hao Chen, Lei Wang, Yong-De Zhang, Yi Liu, and Wei Qian Key Laboratory of Advanced Manufacturing and Intelligent Technology, Ministry of Education, Harbin University of Science and Technology, Harbin 150080, China Robotics and its Engineering Research Center, Harbin University of Science and Technology, Harbin 150080, China Peking University School of Stomatology, Beijing 100081, China Correspondence should be addressed to Jin-Gang Jiang; jiangjingang@hrbust.edu.cn Received 26 November 2019; Revised 18 May 2020; Accepted 25 May 2020; Published 11 June 2020 Academic Editor: Mohammad Rahimi-Gorji Copyright © 2020 Jin-Gang Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The abnormal tooth arrangement is one of the most common clinical features of malocclusion which is mainly caused by the tooth root compression malformation. The second sequential loop is mostly used for the adjusting of the abnormal tooth arrangement. Now, the shape devise of orthodontic archwire depends completely on the doctor’s experience and patients’ feedback, this practice is time-consuming, and the treatment eﬀect is unstable. The orthodontic-force of the diﬀerent parameters of the second sequence loop, including diﬀerent cross-sectional parameters, material parameters, and characteristic parameters, was compared and simulated for the abnormal condition of root compression deformity. In this paper, the analysis and experimental study on the unidirectional orthodontic-force were carried out. The diﬀerent parameters of the second sequential loop are analyzed, and the equivalent beam deﬂection theory is used to analyze the relationship between orthodontic-force and archwire parameters. Based on the structural analysis of the second sequential loop, the device for measuring orthodontic force has been designed. The orthodontic force with diﬀerent structural characteristics of archwire was compared and was measured. Finally, the correction factor was developed in the unidirectional orthodontic-force forecasting model to eliminate the inﬂuence of inherent error. The average relative error rate of the theoretical results of the unidirectional orthodontic-force forecasting model is between 12.6% and 8.75%, which veriﬁes the accuracy of the prediction model. 1. Introduction force of archwire deformation. Its therapeutic eﬀect is related to archwire shape, cross-section, and so on [12–15]. In order to obtain accurate orthodontic force, researchers Malocclusion has been regarded as one of the three major oral diseases by the World Health Organization (WHO). have carried out orthodontic force measurement, statics research, and ﬁnite element analysis. Shima et al. realized And the improper teeth positioning is the most common clinical features [1–6]. It usually causes the uneven tooth the calculation of bending stress by using the hyperelasticity of the archwire in the process of studying the bending prop- alignment and dental arch deformity. And it not only aﬀects erties of hollow nickel-titanium alloy wires in 2010 [16]. The the function of mastication, but also has a certain impact on the pronunciation. The tooth decay, dentin hypersensitivity, measurement and research of orthodontic force are mainly carried out in the laboratory environment. The archwire and more mouth diseases will be accompanied by it. The inci- dence of malocclusion is increasing year by year in adoles- placed on the dental model is used to measure the force, which avoids the limitation of the space in the oral cavity cents [7–11]. The second sequential loop can eﬀectively and ensures the safety of patients [17–20]. Kinzinger et al. treat the tooth root compression malformation; it pulls the misplaced teeth to the correct location through the restoring designed a pendulum force measuring device, which simu- 2 Applied Bionics and Biomechanics experience, which may cause secondary injury to patients lated the oral environment and measured the eﬀect of ortho- dontic force on the crown and root in 2004 [21]. Fuck and and slowing down the treatment time. Firstly, the extraoral Drescher is based on a six-dimensional force sensor to mea- measurement method and orthodontic force measurement device based on a six-dimensional force sensor are designed, sure the plaster dentition model in 2005. The sensor is con- nected with the crown. The position and posture of the and the straining characteristics of the second sequence loop model can be regulated freely [22, 23]. Lapatki et al. planed are analyzed. Secondly, the diﬀerent parameters of the sec- an integrated bracket, which integrates the pressure pickup ond sequence loop, including diﬀerent cross-sectional into the bracket and can directly measure the pressure parameters, material parameters, and characteristic parame- ters, were compared and simulated for the abnormal between the archwire and the bracket, but it cannot truly reﬂect the actual relationship between the archwire and the condition of root compression deformity. Finally, the ortho- bracket in the clinical state [24]. Badawi et al. have estab- dontic force forecasting model is established after the error lished a measuring device with a three-dimensional ortho- compensation. The establishment of the model can provide dontic force in 2010, which replaces the actual teeth with a a quantitative calculation for orthodontic treatment. The model predicts the relationship between the features of the simpliﬁed cylinder. It can preliminarily measure the ortho- dontic force of all the teeth on the dental arch and distin- orthodontic archwire and the orthodontic force. It can help guishes the diﬀerence between passive ligation and other doctors design customized orthodontic archwire and ligation methods [25]. Chen et al. added a T-type loop to improve the scientiﬁcity of the orthodontic treatment. And the dental moulds of the ﬁrst premolar and incisor extraction it can reduce patients’ pain and improve treatment eﬃciency. in 2010 and analyzed the eﬀect of orthodontic force produced by the T-type loop on two teeth that are in contact with each other [26]. Wei et al. designed the measure model of archwire 2. Methods and bracket in 2012. By discussing the inﬂuence of archwire features on the orthodontic force system, a math means for 2.1. Orthodontic-Force Loading Unit of the Second Sequential assessing the force system in orthodontic treatment was pro- Loop. Figure 1 shows the basic loading units of ortho- posed [27]. Mencattelli et al. built a device to measure the dontic force, which are the functional archwire and the force of the teeth under various orthodontic conditions in sequential loop. The most important factor aﬀecting the 2015. Taking the dental plaster model as the research object, orthodontic force is the shape of orthodontic archwire. the force of teeth under four kinds of hyperelastic archwire In this paper, the orthodontic-force is measured and and two kinds of invisible orthodontic appliances was ana- quantitatively analyzed with sequence curvature as the lyzed [28]. In 2016, Midorikawa et al. designed a set of ortho- research object. dontic force sensing system, which can measure the stress The sequence loop is able to be classiﬁed into three state of 14 teeth at the same time [29]. In 2017, Higa et al. types based on the shape and the function of the archwire, constructed a stress measuring device for small-diameter which are the ﬁrst, the second, and the third sequential nickel-titanium alloy wires to measure the force produced loop. The arch form of the deformed maxillary and man- by the wires. The mechanical characteristics of the traditional dibular after the treatment of the ﬁrst sequential loop is nickel-titanium alloy archwire and thermally activated the same as the natural shape. In the direction of buccal- nickel-titanium alloy wires under diﬀerent ligation modes lingual, the ﬁrst sequential loop can be used to treat the were studied. The results show that the traditional nickel- light dislocation. However, when the dislocation is serious, titanium alloy wires release higher orthodontic force [30]. we will consider to combine the ﬁrst sequential loop with Lai et al. made a dental model with resin and simulated the the functional loop. The posterior teeth and the anterior temperature of the oral environment with a heating rod in ones are raised and depressed, when the tip forward curve 2018. Based on this, a force analysis system for the archwire of the second sequential loop only bents in the vertical was established [31]. Scholars have analyzed and measured plane. Furthermore, the third sequential loop is mainly the orthodontic force by means of experimental simulation utilized to generate the torque to move the tooth root and mechanical analysis through the discussion of research and tongue side, because it is able to only be bent by status. Scholars gain the orthodontic force by measuring the the square section archwire. orthodontic force on the teeth, but the eﬀective orthodontic The main parameters in this paper are as follows: F is the force is delivered by the stent. Therefore, in order to obtain orthodontic force; M stands for the material characteristics of accurate measurement results, orthodontic archwires are the archwire; S represents the section characteristics of the directly measured in orthodontic force sources. Much as archwire; and P is the archwire’s feature parameters. Stainless the relationship between the features of the archwire and steel archwire, Australian archwire as the most commonly the orthodontic force is considered. The relationship used orthodontic archwire. between archwire bending features and orthodontic force Furthermore, the characteristic parameters of the arch- is invalid. These researches cannot guide doctors to accu- wire are as follows: E is a symbol of modulus of elasticity; rately speculate the orthodontic force produced by design- the Poisson’s ratio is μ; S and S represent sectional area A S ing archwire [10]. and sectional shape of archwire’s section characteristics; I This study carries out the experiment and modeling of stands for the inertia moment to the bending axis. The type the second sequential loop to help doctors handle the prob- of archwire curve decides the characteristic parameter of lem of adjusting excessive and small orthodontic force by the archwire. Applied Bionics and Biomechanics 3 Bracket A Bracket C Bracket B Target tooth Anchorage tooth 𝜃a 𝜃b Cv ab Figure 1: Second sequential loop. Thus, the basic form of the prediction model of the Figure 2: The force diagram of root compressing deformity. orthodontic force is as follows. F = FM, S, P : ð1Þ ðÞ archwire, the inertia moment is I = c c /12,where c is the z 1 2 2 side length parallel to the z-axis, and c is the side length per- pendicular to the z-axis [32]. 2.2. Unidirectional Orthodontic-Force Prediction Modeling. The angular equation is represented by θðxÞ,andvðxÞ The orthodontic force is aﬀected by many factors in its gen- stands for the deﬂection equation. Then, the second sequential eration process due to the complexity of the orthodontic loop is able to be represented via the integral of Equation (2). treatment. Therefore, these main parameters should be considered to establish the unidirectional orthodontic force dv Mx ðÞ forecasting model, and the related experiments and experi- θ x = = dx + C, ð3Þ ðÞ mental results are designed and analyzed to correct this dx EI forecasting model. Thus, the basic assumption had been ðð developed as follows to simplify the establishment process Mx ðÞ vx = dxdx + Cx + D, ð4Þ of the prediction model: ðÞ EI (1) The orthodontic archwire and the orthodontic where C and D are constants of the integration which are brackets are ligated tightly, which means the sliding decided by the condition of boundary and continuity [32]. friction is not existing between the orthodontic arch- The bending moment equation of the second sequential loop wire and the orthodontic brackets is as follows. (2) The orthodontic force generated on the tooth is the Pb restoring force generated by the elastic deformation Mx ðÞ = x,0ðÞ ≤ x ≤ a , of the orthodontic archwire l ð5Þ Pb Because of the representativeness of the second sequen- Mx ðÞ = x −PxðÞ − a ,ðÞ a ≤ x ≤ l , tial loop, its orthodontic-force forecasting model will be stud- ied in this paper. The tooth compression generated in the where l represents the length of the archwire between the two root direction may lower the tooth. After the archwire is anchorage teeth which named as the anchorage distance. a is inserted into the bracket, the orthodontic force on the root the distance between the target tooth and the distal anchorage cap direction will be generated on the target tooth, and the tooth which named as the oﬀset distance [12]. The distance anchorage is the adjacent teeth. Figure 2 shows the connec- between the target tooth and the mesial anchorage tooth is tion relation between the bracket and the archwire. represented by b. According to the materials of mechanics, this model is Substituting Equations (2) and (5) into Equations (3) and simpliﬁed as a simply supported beam model [10], and the (4), respectively, the following equations can be obtained. axis of the beam is x-axis. The deﬂection curve diﬀerential equation is given as follows: Pb x θðÞ x = + C ,0ðÞ ≤ x ≤ a , 2 l 2 d v Mx ðÞ ð6Þ = : ð2Þ > 2 > Pb x x − a ðÞ dx EI θðÞ x = − P + C ,ðÞ a ≤ x ≤ l , l 2 2 where v represents the horizontal arm’s moving distance and the vertical arm’s bending deﬂection as well. MðxÞ stands for Pb x vx = + C x + D ,0 ≤ x ≤ a , ðÞ ðÞ 1 1 the x-axis bending moment of the vertical arm, E is the mod- l 6 ulus of elasticity of the archwire, and I is the inertia moment 3 3 > Pb x ðÞ x − a generated by the archwire section to the z-axis. For the round- vx ðÞ = − P + C x + D ,ðÞ a ≤ x ≤ l : 2 2 l 6 6 section archwire, the inertia moment is I = πð2rÞ /64,where ð7Þ r is the radius of the round section. For the rectangular section 4 Applied Bionics and Biomechanics After orthodontic Before orthodontic treatment treatment Target teeth Target teeth Bracket Second sequential loop Figure 4: Simulation of tooth moving process aﬀected by the second sequential loop. Figure 3: Standard oval loop. C , C , D ,and D can be determined by the boundary Hexagon copper cylinder 1 2 1 2 Ligation wire condition and continuity condition. Hence, C = C = ‐ðPb/lÞ 1 2 2 2 ðl − b Þ, D = D =0. 1 2 Archwire Substituting C , C , D , and D into Equations (7), the 1 2 1 2 following equations can be obtained. Pbx > 2 2 vx ðÞ = − l − x − b ,0ðÞ ≤ x ≤ a , Bracket 6EI Pb l 3 2 2 3 vx = − x − a + l − b x − x , a ≤ x ≤ l : : ðÞ ðÞ ðÞ 6EI b Positioning plate Chute ð8Þ Figure 5: Simulative orthodontic environment. Therefore, the v on the position x = a aﬀected by the orthodontic-force is able to be expressed as follows: patient, it is necessary to adjust the shape of the standard Pa 2 2 2 archwire. The orthodontic doctors always bend the ﬁrst, the v = − l − a − b ð9Þ 6EI second, and third sequential loop to adjust the shape of the standard archwire. According to the standard arch map of The orthodontic force F is the counter force of the acting the mandible, the preformed egg shape archwire of the man- force (P) which generated the deformation based on the prin- dibular is shown in Figure 3. ciple of acting force and the counter force [32]. The deformation of the second sequential loop generated the orthodontic-force. The amount and type of deformation 6EId is the determining of the orthodontic force. During the actual F = −P = , ð10Þ 2 2 orthodontic treatment, the patient had worn the standard al − a − b archwire on his misplaced teeth. Thus, the deformation will be generated on the archwire at the misplaced teeth. And where d is the moving distance of the archwire in the y the deformation may generate restoring force. Due to the direction. inﬂuence of the restoring force, the misplaced teeth are driven to the standard position. Thus, it is diﬃcult to mea- 3. Results sure the orthodontic force in the complex force deliver pro- cess. In this paper, we consider the extraoral measurement 3.1. Design of the Measuring Method of the Second Sequential method and use the loading unit to generate the displace- Loop Unidirectional Orthodontic-Force. Sequential loop is one of the basic loading units of orthodontic-force [33–35]. ment load on the standard archwire. The displacement load drives the target tooth to the abnormal position to simulate The archwire is needed to be bent into the shape of a stan- dard arch, which its initial shape is straight. After the arch- the orthodontic process [12]. In order to facilitate subsequent wire is ﬁxed with the bracket, the preformed orthodontic research, the six-dimensional force sensor is used to measure the magnitude of the orthodontic force. The six-dimensional archwire will be deformed to generate the orthodontic force because of the limitation of the bracket position. For the pur- force sensor can simultaneously detect the three-dimensional full force information, that is, three force components and pose of bending the archwire to ﬁt the dentition shape of the Applied Bionics and Biomechanics 5 Universal gripper Upper machine Hexagon pillars Micro slide Sequence curve Six dimensional force transducer Clamping sleeve Six dimensional Positioning plate force collector (a) Composition of measuring device for the unidirectional orthodontic force of the second sequential loop Six dimensional force transducer Bracket Simulation of the teeth Chute Positioning plate Sequence curve (b) Measuring process of the unidirectional orthodontic force of the second sequential loop Figure 6: Measuring device for the unidirectional orthodontic force of the second sequential loop. three-moment components. The detection of all the informa- The design of sequential curve measuring device should tion at the same time can eliminate experimental interference meet the following requirements. and conform to the principle of a single variable. As shown in Figure 4, the measured-force of diﬀerent displacement load is (1) The measuring device is required to meet the mea- the orthodontic-force. surement requirements of the six-dimensional force. The force in one-direction can be measured sepa- 3.2. Design of the Measuring Device of the Second Sequential rately, and the superposition of force and force in Loop Unidirectional Orthodontic-Force. In order to avoid multiple directions can be measured simultaneously the patient’ hurt, in this paper, we use an “in vitro” measure- ment due to the narrow space of the human mouth. The posi- (2) The adjustment accuracy of the straight movement of tioning plate is ﬁxed on the table, the hexagonal stud the measuring device should be 0.1 mm. The setting simulates teeth, the chute simulates human gums, and the range should be more than 5 mm. The adjustment friction between the chute track, and hexagonal stud is equiv- accuracy of the rotation should be 1 . The setting alent to the resistance in the orthodontic process. The ortho- range should be more than 10 dontic archwire and bracket are installed according to the (3) The measuring range of the force measuring device real orthodontic process, especially the ligation wire adopts should be greater than the maximum amount of the way of tight ligation, that is, the sequential loop archwire orthodontic-force that can be applied to the sequen- will not generate sliding friction in the bracket. The position tial loop. The measuring range should not be less of the archwire was ﬁxed according to the actual tying than 15 N or 50 N� mm, and the resolution should method to simulate the real in-mouth working environment, not be less than 0.1 N or 0.01 N� mm which is shown in Figure 5. 6 Applied Bionics and Biomechanics Table 1: The unidirectional orthodontic force measuring results of diﬀerent second sequential loops (N). Type of archwire Moving distance (mm) I II III IV V VI VII VII IX X 0.5 0.32 0.25 0.21 0.18 0.33 0.42 0.61 0.63 0.24 0.28 1.0 0.66 0.51 0.41 0.35 0.68 0.85 1.24 1.28 0.49 0.57 1.5 0.99 0.77 0.63 0.54 1.02 1.28 1.87 1.92 0.72 0.85 2.0 1.33 1.04 0.85 0.73 1.35 1.72 2.48 2.55 0.95 1.16 2.5 1.64 1.28 1.07 0.92 1.67 2.15 3.08 3.19 1.21 1.46 3.0 1.96 1.53 1.29 1.10 2.01 2.58 3.67 3.82 1.45 1.75 3.5 2.27 1.78 1.51 1.26 2.35 3.03 4.30 4.46 1.69 2.03 4.0 2.57 2.00 1.72 1.42 2.65 3.44 4.92 5.09 1.91 2.3 4.5 2.87 2.22 1.90 1.58 2.94 3.83 5.53 5.72 2.13 2.56 5.0 3.18 2.43 2.05 1.78 3.22 4.22 6.13 6.35 2.37 2.79 According to the measurement method and requirement, the installation way of measuring devices is as follows. The 3.0 arched chute is made on the positioning plate. The shape of the arched chute is the same as the standard arch type. The 2.5 hexagon copper cylinders are mounted on the arched chute to simulate the position of the actual teeth. The locations of 2.0 the hexagon copper cylinders can be adjusted along the arched chute. The brackets are pasted on the hexagon copper 1.5 cylinders. The archwire is ﬁxed on the brackets by the clinical ligation mode [12]. 1.0 Two hexagonal copper pillars can simulate two nonadja- cent teeth at any tooth position, and their distance is equiva- 0.5 lent to the anchorage distance. The position of the sensor needs to be adjusted ﬂexibly to adapt to diﬀerent measuring positions. When measuring the orthodontic-force, a six- 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Moving distance (mm) dimensional force sensor is installed on the adjusting device. When measuring the unidirectional orthodontic-force, a S16162010 S16162410 micro moving sliding table is used as a displacement adjust- S16162210 S16162610 ment device to adjust the distance from the crown to the root, Figure 7: Orthodontic-force at diﬀerent anchorage distances. as shown in Figure 6. Adjusting device through the ﬁttings installed, in the end of the execution of the universal holder by universal gripper driven sensor positioning at any position. The sensor through the In this experiment, the naming method of the used clamping sleeve clamp under test, the archwire after the sequential loop can refer to Reference [10]. For example, positioning of the sensor, by adjusting the micro sliding table- S16162010 indicates that the size of the cross-section of stain- driven clamping set of movement, to exert the archwire dis- less steel square archwire is 0:016 × 0:016 inch, the anchorage placement load, its deformation is equal to the oﬀset distance. distance is 20 mm, and the oﬀset distance is 10 mm. The correction force is collected by the six-dimensional The orthodontic measuring experiment is conducted force collector and sent to the upper computer for display according to the loading characteristics of the tooth root and subsequent processing. compression malformation. The moving distance (Preset tooth deformity) is limited to 5 mm, we divided the range 3.3. Second Sequential Loop Unidirectional Orthodontic-Force of 5 mm into 10 measuring points, and each point is the Measuring Experiment. The unilateral orthodontic forces of the typical root and the buccal tongue were measured by hiatus of 0.5 mm for measuring the orthodontic-force of 10 points. the second sequential loop measurement device; measured objects were the standard preformed egg shape archwire. In We use the method of controlling a single parameter to study the eﬀects on orthodontic-force, which is caused by the the measurement of the same type of archwire, the manual diﬀerent anchorage distances, diﬀerent moving distances, dif- measurement error is less than 5%. In each experimental measurement, three the same types of archwires were utilized ferent oﬀset distances, diﬀerent cross-section characteristics, and diﬀerent material properties, ten diﬀerent types of the arch- to conduct three groups’ repeated experiments. The eﬀective measurement data of one type of the archwire is calculated by wires are used. The second sequential loops used in the experi- ment are listed as follows. ①-S16162010; ②-S16162210; averaging the three measurement values. Orthodontic-force (N) Applied Bionics and Biomechanics 7 6.5 6.0 3.0 5.5 2.5 5.0 4.5 4.0 2.0 3.5 3.0 1.5 2.5 2.0 1.0 1.5 1.0 0.5 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Moving distance (mm) Moving distance (mm) S16162010 S16162008 S16162010 S16162009 S16162007 A16162010 Figure 8: Orthodontic-force at diﬀerent oﬀset distances. Figure 10: Orthodontic-force with diﬀerent materials. 6.5 6.0 2.5 5.5 5.0 2.0 14 4.5 4.0 1.5 3.5 3.0 1.0 8 2.5 2.0 0.5 1.5 1.0 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Moving distance (mm) 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Moving distance (mm) S16162010 S16162210 S16162410 S16162610 S16162010 S00162010 Figure 11: The results of theoretical calculations of orthodontic S18252010 force under diﬀerent anchorage distance. Figure 9: Orthodontic-force at diﬀerent sections. ③-S16162410; ④-S16162610; ⑤-S16162009; ⑥-S16162008; small section. The orthodontic-force is has a positive connec- ⑦-S20102007; ⑧-S18252010; ⑨-S00162010; ⑩-A16162010. tion with the moment of inertia. The measuring results are listed in Table 1. As shown in Figure 10, the load-deﬂection rate of the Figure 7 showed the experimental results. The orthodontic Australian archwire is lower than that of the stainless steel force has a negative connection with the anchorage distances. archwire, and the orthodontic-force is obviously weakened. The orthodontic force decreases when the anchorage distances The orthodontic-force has a positive connection with the become greater. Thus, a longer anchorage distance would be elastic modulus of the archwire. expected to generate a mild orthodontic force. It can be seen from the prediction model of the unidirec- The orthodontic-force has a negative connection with the tional orthodontic-force that the orthodontic-force is pro- oﬀset distance through comparison result of ①, ⑤, ⑥,and portional to the moving distance, the moment of inertia, ⑦. The orthodontic-force increases when the oﬀset distances and the elastic modulus and inversely proportional to the become smaller. The comparison result is as shown in Figure 8. cubic function of the anchorage distance and the quadratic As shown in Figure 9, through the comparison result of function of the oﬀset distance. Therefore, it is inversely pro- ①, ⑧, and ⑨, the load-deﬂection rate of round archwire is portional to the moving distance of the sequence loop. The lower than the square archwire. The load-deﬂection rate of eﬀect of each parameter on the orthodontic-force is the same the larger section of the square archwire is higher than the as that reﬂected in the sequence prediction model. Orthodontic-force (N) Orthodontic-force (N) Orthodontic-force (N) Orthodontic-force (N) Deviation rate (%) 8 Applied Bionics and Biomechanics 4.5 4.0 30 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Moving distance (mm) S16162010 S16162009 S16162008 S16162007 Figure 12: The results of theoretical calculations of orthodontic-force under diﬀerent oﬀset distance. 4. Discussion 7.0 35 6.5 In the acting process of the orthodontic archwire, the friction 6.0 5.5 25 force may be also generated because of the cooperation of the 5.0 archwire and the brackets. This part cannot be ignored. And 20 4.5 the archwire is made by the manual and robotic bending pro- 4.0 3.5 cess, and the residual stress could be generated on the bent 3.0 archwire because of the deformation of the bent archwire. 2.5 These factors could inﬂuence the establishment of the predic- 2.0 1.5 tion in this study which may generate the diﬀerence between 1.0 the experimental result and the theoretical result. To solve −5 0.5 this problem, the correction factor, K , is developed to lower −10 the deviation rate between the orthodontic-force prediction 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Moving distance (mm) model and the experimental result. By comparing the exper- imental result with the theoretical result, K can be calculated S18252010 through the following equation. S16162010 S00162010 6EIdK Figure 13: The results of theoretical calculations of orthodontic F = : ð11Þ 2 2 force under diﬀerent sections. al − a − b The theoretical value calculated without developing the cor- There two important parameters in the following analysis rection factor is drawn in solid lines, and the theoretical devi- of the experimental result, which are theoretical deviation ation rate is drawn in the dotted lines. rate and theoretical correction rate. Firstly, the theoretical The comparison experiment between the sequential loop deviation rate can be obtained through the ratio calculating bent by the same archwire but the diﬀerent anchorage dis- between the diﬀerence, which is calculated through the sub- tances are conducted through the archwires coded as ①, ②, traction of the experimental and the theoretical value ③, and ④. Through the results shown in Figure 11, it can obtained without developing the correction factor and the be found that the orthodontic force caused by the sequential experimental value. Secondly, the theoretical correction rate is negatively related to anchorage distance. The theoretical can be calculated through the ration calculating between deviation rate is also a negatively relationship with the the diﬀerence which is calculated through the subtraction anchorage distance. of the experimental results and the theoretical data without Through the ﬁtting of the theoretical correction rate of developing the correction factor and the theoretical value. diﬀerent anchorage, theoretical correction coeﬃcient aﬀected The correctness of the prediction model without considering by anchorage K can be obtained as follows: the correction model compared with the experimental result Fa can be shown through the calculating of the theoretical devi- ation rate. And the correcting inﬂuence of the correction fac- 1+ −2:2245l ×10 +62:561 tor generated to the prediction model can also be shown K = : ð12Þ Fl through the calculating of the theoretical correction rate. Orthodontic-force (N) Orthodontic-force (N) Deviation rate (%) Deviation rate (%) Applied Bionics and Biomechanics 9 2.5 2.0 1.5 1.0 0.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Moving distance (mm) A16162010 S16162010 Figure 14: The results of theoretical calculations of orthodontic force under diﬀerent materials. The comparison experiment between the sequential loop 8.75 8.59 bent by the same archwire but the diﬀerent oﬀset distances are conducted with the archwire shown in Figure 12. Through the comparison between its theoretical and experi- mental result, it can be found that the orthodontic force caused by the sequential is negatively related to oﬀset dis- tance. The theoretical deviation rate is also negatively related to the anchorage distance in a quadratic function. 3.43 The theoretical correction coeﬃcient of the inﬂuence of the oﬀset distance is: 3 2.38 2.1 2 1.62 1.49 1.49 hi 1.29 1.26 3 3 1+ 8:555 a ×10 − 153:941a ×10 + 703:031 1 K = : Fa ð13Þ The comparison experiment between the sequential loop bent by the diﬀerent archwires but the same bending param- Archwire type eters are conducted through the archwires coded as ①, ⑧, Figure 15: Relative error rate calculated by the theoretical and ⑨. Figure 13 showed these results; therefore, we found calculation of sequential loop orthodontic force prediction model that the theoretical deviation rate is negatively related to the moment of inertia of the section. Through the ﬁtting of the theoretical correction rate of diﬀerent cross-sections, the theoretical correction coeﬃcient According to the ﬁtting of the theoretical correction rate of aﬀected by diﬀerent cross-sections K can be obtained as Fs diﬀerent materials, the theoretical correction coeﬃcient follows: aﬀected by diﬀerent materials K can be obtained as follows: FM 1+ ‐36:5ln I +53:706 ×10 ½ ðÞ 1+ −0:1068E ×10 +20:5699 K = : ð14Þ FS K = : ð15Þ FM The comparison experiment between sequential loop bent Through the analysis above, the correction factors K by the archwires with diﬀerent materials but the same bending and K could be aﬀected by the anchorage distance, oﬀset parameters are conducted through the archwires coded as ① distance, and the material. Multiplying the correction fac- and⑩. Through the results shown in Figure 14, we found that tors directly may over correcting the prediction model. orthodontic force is actively relationship with elastic modulus. However, the archwire coded as S16162010 is used in every The theoretical deviation of orthodontic-force is negatively calculating process of the correction factors of each param- related to the elastic modulus of materials. eter. Thus, the correction factor of the S16162010 can be Orthodontic-force (N) Relative error (%) S16162010 S16162210 S16162410 S16162610 Deviation rate (%) S16162009 S16162008 S16162007 S162010 S18252010 AO16162010 10 Applied Bionics and Biomechanics used as a base parameter to calculate the general correction Data Availability parameter of the prediction model. The unidirectional The data that support the ﬁndings of this study are available orthodontic-force correction coeﬃcient K can be obtained on request from the corresponding author, Jingang Jiang. as follows: Conflicts of Interest K K K Fa FS FM K = K ⋅ ⋅ ⋅ : ð16Þ F Fl The authors declare that there is no conﬂict of interest. K K K Fa0 FS0 FM0 Acknowledgments Then, the prediction model for the orthodontic force of the sequential loop with the correction factor is as follows: This research was supported by the University Nursing Pro- gram for Young Scholars with Creative Talents in Heilong- jiang Province (Grant No. UNPYSCT-2017082), the China K K K 6EIdK Fa FS FM F = K ⋅ ⋅ ⋅ ⋅ : ð17Þ Postdoctoral Science Foundation Special Funded Project Fl 2 2 K K K al − a − b Fa0 FS0 FM0 (Grant No. 2018T110313), the Fundamental Research Foun- dation for Universities of Heilongjiang Province (Grant No. LGYC2018JQ016), the China Postdoctoral Science Founda- Figure 15 showed that the error rate between the pre- tion Funded Project (Grant No. 2016M591538), and the diction model theoretical data and the experimental value. Heilongjiang Postdoctoral Science Foundation Special The relative error rate of the sequential loop orthodontic Funded Project (Grant No. LBH-TZ1705). force prediction loop was calculated and it is ranging from 1.26% to 8.75%. It ﬁlled with the requirement of the predic- References tion of the orthodontic force. Orthodontic doctors can use the prediction model to calculate the orthodontic force [1] J. Stewart, G. Heo, K. Glover, P. Williamson, E. Lam, and through the bent archwire. P. Major, “Factors that relate to treatment duration for patients with palatally impacted maxillary canines,” American 5. Conclusions Journal of Orthodontics and Dentofacial Orthopedics, vol. 119, no. 3, pp. 216–225, 2001. The selection of orthodontic archwire shape is determined by [2] H. Wu, W. Tsai, Y. Chen, J. Liu, and Y. 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Applied Bionics and Biomechanics – Hindawi Publishing Corporation
Published: Jun 11, 2020
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