閉口型矯正線的設計是矯正治療中一個重要的部分。適當的設計可以產生可預測力和牙齒移動。它不像滑動的機構，在矯正線和矯正器之間不會產生磨擦，因此使用閉口型矯正線產生的力可以直接造成牙齒移動。已經有很多closing loop的型式，像是vertical or teardrop loops, T-loops, L-loops。Closing loop主要的特性為M/F ratio，它是確定在牙齒移動過程中，施力點到牙齒旋轉中心的距離。先前文獻的研究是以簡化模型的去探討closing loop的力學反應。但仍然有許多因素會影響closing loop的力學反應，像是矯正器和矯正線之間的影響、牙列的弧度。本研究的目的是評估在閉口型矯正線真實情況下力學的反應。
本研究依造先前文獻的設定方法，重現它的結果。在先前文獻利用有限元素分析(FEA)，在closing loop的幾何外型和位置的力學影響。模擬三種形狀的closing loop (T-loop, Vertical-loop and L-loop)和調查closing loop在13個不同位置時的力學反應。Closing loop的長度和高度分別為14 mm和10 mm。但先前文獻中沒有顯示出導圓角的尺寸。因此在本研究中以半徑0.5 mm設定，這是臨床上closing loop最小導源角的尺寸。Closing loop的幾何形狀是以Beam element去建立方型截面的不銹鋼矯正線，大小為0.016×0.022 in。兩端各施1 mm的位移量向外水平拉開，量測兩端的力學反應。在本研究中，在相同的設定條件下使用Beam和Solid element去做重現先前文獻的結果與本研究的模型驗證。
然後，本研究在實驗中力用六軸的荷重元去量測closing loop的反應。使用Solidworks和3D-printing去設計和製造荷重元的夾具。荷重元會先以不同重量的砝碼校正，如50克、100克、200克(0.05N、0.1N and 0.2N)。請臨床醫生彎折實驗所需的樣本。本研究會收集七個不同closing loop的位置。每個樣本在每個位置時，會進行三次量測。此外，使用有限元素分析依造實驗的環境下去模擬closing loop的力學反應。利用Solidworks去建立矯正器的模型。本研究在模擬時，網格的大小都為0.1 mm。本研究利用實驗的結果去驗證模擬的結果。
本研究比較closing loop在有無矯正器時的力學反應。在矯正器和closing loop的接觸面是設為無摩擦(frictionless)，在停止裝置和矯正器的接觸面設為結合(bonded)。並且比較矯正線有無弧度時的力學反應。所有設定都跟上續相同。本研究也評估矯正器之間高低差異所帶來的影響。
本研究的結果表示，兩種模型建立方式的結果是相似的，也驗證本研究closing loop的模型。在實驗與模擬的結果有著顯著的差異。因為實驗樣本是請臨床醫生進行彎折，所以尺寸和本研究的設定會有些微差異。根據實驗樣本的尺寸再次模擬，兩者結果差異大為降低。因此，本研究在實驗上有很多因素是難以控制，像是closing loop的幾何形狀和位移的量。實驗樣本的形狀有些微差異就可能導致M/F ratio很顯著的差異。本研究也無法控制每次位移的大小都是2 mm。不論是矯正器或是矯正線有弧度時，M/F ratio的結果都與先前文獻的結果相似。但存在這兩個條件會得到除了原先施力方向以外，其他方向的力和力矩。應該更進一步去評估各種矯正線模型的力學反應。先前文獻指出最佳的M/F ratio為10：1。然而矯正器之間高度有些微差異，就可以造成M/F ratio有很大的變化。因此，矯正器之間高度的因素是會影響closing loop的力學反應。
本研究的結論對於M/F ratio來說，在模擬中一個二維的情況下(簡化的模型與無矯正器)去探討就可以提供足夠的真實性。然而，真實情況下closing loop的幾何形狀有些微差異時，會造成力學反應結果有很大的差異。在模擬中，在有無矯正器或矯正線有無弧度這兩個條件下，對M/F ratio的影響很小。然而，在模擬時有矯正器和矯正線有弧度時，可以得到除了原先施力方向以外，其他方向的力和力矩。不論矯正線有無弧度，只要矯正器之間的高低有些微差異，就會造成M/F ratio有很大的變化。

Abstract
Mechanical Evaluation of Orthodontic Closing Loop：Finite Element Analysis
Ming-Chuan Kuo
Chih-Han Chang
Department of Biomedical Engineering & National Cheng Jung University
SUMMARY
The aim of this study was to determine the mechanical responses of the loop in an actual setting. The results of this study were in accordance with a previous report with their setting. This study compared the mechanical responses of a loop with and without brackets. n addition, the present investigation compared the mechanical responses of the loop with and without an arch wire curvature. This study evaluated the influence of height between the brackets. Furthermore, regardless of the conditions of the bracket or arch wire curvature, the M/F ratio of loop was similar to that described in the previous study. The results of the present study showed small variations in the height difference between the brackets, which in turn could induce a significant change in the M/F ratio. The existence of a bracket or a simple arch in the simulation imparts a minor effect on the M/F ratio. Regardless of the presence or absence of the arch wire, a small variation in height difference between the brackets could cause a greater effect on the M/F ratio.
Keywords: Orthodontics; Closing loop; Finite element analysis
INTRODUCTION
Loop mechanics utilize closing loops to generate forces for moving teeth. Unlike sliding mechanics, no friction is generated between a wire and a bracket. Several kinds of loop designs have been introduced such as vertical or teardrop loops, T-loops, and L-loops. The major loop property of the closing loop is the M/F ratio that determines the distance of the center of dental rotation during the movement of the teeth. Previous studies have only shown the mechanical responses of the loop using a simple model. However, the mechanical responses of the loop can be affected by various other factors such as the wire, bracket, and curvature of the tooth. The aim of this study was to determine the mechanical responses of the loop in an actual setting.
MATERIALS AND METHODS
The results of this study were in accordance with a previous report with their setting. The aim of that study was to investigate the effect of loop geometry and position on loop mechanical responses using finite element analysis. The responses of loop were simulated for three kinds of loop geometry, T-loop, Vertical-loop, and L-loop, and investigated 13 loop positions. The length and height of the loop were 14 and 10 mm, respectively. However, the dimension of the filet was not described in the previous study. Therefore, the present study employed a filet with a radius of 0.5 mm, which was the minimum size used in a clinical setting. Loop geometry was modeled using stainless steel with a 0.016 × 0.022 inch rectangular cross-section using a beam element. Loop responses on both ends were measured by totally moving both the ends by 2 mm. The Young’s modulus was 157.6 GPa and Poisson’s ratio was 0.3. In the present study, the model was built using beam and solid elements and validated under the same settings.
The present study used a six-axis load cell to determine the date of the experiment. The clamp was designed and manufactured using Solidworks and 3D-printing. The load cell was calibrated using different weights such as 50, 100, and 200 g (0.05 N, 0.1 N, and 0.2 N, respectively). The loop sample was bent by an orthodontist. The present study examined seven loop positions (a/b = 0.07, 0.21, 0.36, 0.50, 0.64, 0.79, and 0.93). Each sample was measured three times at each loop position. In addition, loop responses were simulated under the experimental setting using finite element analysis. The bracket model is built using Solidworks. The loop and bracket were made of stainless steel. The clamp was made of acrylonitrile butadiene styrene, its Young’s modulus was 1.9 GPa, and its Poisson ratio was 0.41. The mesh size of the clamp was 0.1 mm. The results of the previous experiment were validated using the findings of simulation in the present study.
This study compared the mechanical responses of a loop with and without brackets. The interface between the wire and bracket was frictionless and the two components were bonded between a stopper and bracket. In addition, the present investigation compared the mechanical responses of the loop with and without an arch wire curvature. The setting used in the investigation was similar to that previously described. This study evaluated the influence of height between the brackets.
RESULTS ANS DISCUSSION
The present study generated the same results using the two building methods, which is supported by the findings of the previous study. Significant variations between the experiment and simulation were observed. After the sample was measured, the dimensions were not same as the design of the present study. The dimensions indicated the need to perform a second simulation, which showed even lower variations than the first experiment. Several factors could not be controlled in the experiment such as loop geometry and the magnitude of activation. The small variation (10%–30%) in the real loop formation could induce a 50% variation in the M/F ratio. The amount of activation could not be controlled during each experiment. Furthermore, regardless of the conditions of the bracket or arch wire curvature, the M/F ratio of loop was similar to that described in the previous study. However, the results also revealed that the force and moment were moving toward other directions. Therefore, the loop responses using various arch models should be evaluated further. The previous study indicated that the optimum M/F ratio was 10:1. A higher M/F ratio was achieved in the clinical setting by prebending. However, the results of the present study showed small variations in the height difference between the brackets, which in turn could induce a significant change in the M/F ratio. Therefore, the height between the brackets could influence the mechanical responses of the loop.
CONCLUSIONS
Therefore, the present study shows that the M/F ratio generated from 2D simulation is applicable to simple clinical conditions. However, small variations in real loop formation have significant variation in the M/F ratio. The existence of a bracket or a simple arch in the simulation imparts a minor effect on the M/F ratio. However, the force or moment in other directions could only be evaluated using a 3D model. Regardless of the presence or absence of the arch wire, a small variation in height difference between the brackets could cause a greater effect on the M/F ratio.

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