Chapter 3,4

March 16, 2018 | Author: Barsha Thapa | Category: Tooth, Orthodontics, Mouth, Dentistry Branches, Dental Anatomy


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CHAPTER 3Bracket positioning and case set-up Introduction 57 The need for accuracy 57 Patient management 57 58 Placing molar bands Separation 66 66 66 Upper molar band placement Full or partial set-up? Upper molar bands - rapid maxillary expansion cases 66 Lower molar band placement Direct bonding of brackets bonding of brackets indirect bonding indirect bonding 67 Theory of bracket positioning - avoiding errors 59 Horizontal accuracy during bracket positioning 60 Axial accuracy 61 Vertical accuracy 61 Vertical bracket positioning with gauges and charts 62 Clinical use of gauges 62 Recommended bracket-positioning chart Individualized bracket-positioning charts 63 63 68 Indirect 69 Advantages of 69 Disadvantages of 69 INTRODUCTION Setting up of the case is the most important aspect of the treatment, after correct diagnosis and treatment planning. Banding and bonding should therefore not be delegated and should be managed by the orthodontist, to ensure accuracy of appliance placement. Patient management A calm and unhurried approach to the case set-up helps to minimize patient apprehension and discomfort. This builds early patient confidence, and can raise the level of cooperation later in the treatment.1 Proper post-set-up advice should be given, as discussed in Chapters (p. 112). The use of light-cured systems for bonding brackets and cementing bands is helpful. These reduce time pressure on the orthodontist when setting up cases. The bonding materials should be carefully used exactly to the manufacturer's recommendations, with correct light, to ensure good bond strength and reduce the risk of bond failure. The need for accuracy Accuracy of bracket positioning is essential, so that the builtin features of the bracket system can be fully and efficiently expressed. This helps treatment mechanics and improves the consistency of the results. FULL OR PARTIAL SET-UP? For many patients, it is correct to place all the brackets and bands at the start of treatment so that any disco mfort is limited to one episode, and all the teeth start to be corrected from the outset. However, in some situations, listed below, it may be beneficial to consider partially setting up the case, leaving individual teeth, and in some instances groups of teeth, without attachments. Enamel reduction cases It is normally necessary to carry out enamel reshaping in cases with triangular-shaped incisors (Fig. 3.2). It may be helpful to delay bracketing the incisors, especially in the lo wer arch. If lower incisors are bracketed at the start of treatment, they will inevitably procline a little during tooth alignment, especially in a no n-extraction case. Subsequent enamel reduction, followed by retroclination is a form of round tripping. This undesirable effect can be avoided by not bracketing lower incisors at the outset. Blocked-out teeth If individual teeth are vertically or horizontally displaced from the primary arch form (Fig. 3.1), it is often good technique to delay bracketing the displaced tooth until the other teeth are well aligned, and space has been made available. Deep-bite cases The methods of starting deep-bite cases are shown on pages 134 and 135. In so me cases, when it has been decided not to use a bite plate or occlusal build-up, upper arch treatment should be started first. Later, after the o verbite has started to correct, it will be possible to place the lower incisor brackets without discomfort to the patient or risk of damage to the enamel or the newly placed brackets. Fig. 3.2 Triangular-shaped incisors normall y require reshaping to avoid unest hetic black triangles. It can be helpful to del ay placement of brac kets i n the lower incis or region to reduc e unwant ed procli nation early i n treatm ent. Treatm ent m echanics can be easi er if lower incisors of a tri angular shape are reshaped before bracket placem ent. Sliding jig cases and mixed dentition cases Upper bicuspids and sometimes upper canines are normally not bracketed when starting cases where a sliding jig (Case TC, p. 195) will be used to control or distalize upper molars. In many mixed dentition treatments, only the permanent teeth are included in the set-up. Primary teeth maybe included in some cases, either to impro ve the strength and stability of the appliance, or to influence the position of the primary teeth. Fig. 3.1 This vertically and horizontall y displaced upper ri ght canine was not bracketed at the st art of treatment. It was necessary t o creat e space before att em pting to bri ng it into the line of the arch. 58 63). This helps efficiency in a busy orthodontic practice. and this in turn led to variations in the amount of torque and in-out produced by the brackets. . With the original edgewise appliance. bracket placement was normally carried out using gauges and standard millimeter measurements from the incisal or occlusal edge of each tooth. it is helpful to avoid viewing teeth from the side. patients with large incisors had brackets placed more incisally than patients with small teeth. Ideal positioning can result in cases which show good occlusion with little effort. To properly view the teeth during bonding procedures it will be necessary for the patient to turn the head.2 This overcame the shortcomings of the original edgewise method concerning variations in the amount of torque and in-out produced by the brackets. or from above or below. irrespective of tooth size. However. and will make the finishing stages of the treatment easier. and the authors now advocate the use of gauges. When direct bonding brackets. With this system.3J. because archwire bending was needed in any case. Many vertical errors occurred. and the orthodontist to change seating position from time to time (Fig.THEORY OF BRACKET POSITIONING AVOIDING ERRORS Every effort should be made to achieve accurate bracket positioning. as described below. but with individualized bracket-positioning charts (p. These adhere to Andrews' principle of the middle of the clinical crown but ensure greater vertical accuracy. this system was acceptable with the edgewise appliance. with bracket wings parallel to the long axis of the clinical crown. it is important to view the teeth from the correct perspective. Andrews introduced the concept of the 'middle of the clinical crown'.3 When placing brackets. 3. Fig. 3. it proved difficult to obtain accurate vertical positioning using only the middle of the clinical crown. However. The brackets were positioned at different curvature on the teeth. with less need for re-bracketing. relative to the size of the teeth. as a more reliable theoretical position for use with the SWA. Fig.6) helps bracket positioning relative to the vertical long axis of the crown.7).7 In this case. and molar regions should be checked with a mouth mirror. the lower canine brackets were bonded slightly distal to the vertical midline. . Fig. premolars. premolar. Lower canine brackets should be placed on the vertical midline. molars. 3.Horizontal accuracy during bracket positioning Incisors and molars have relatively flat facial and buccal surfaces. 3. and therefore accuracy is important because errors in horizontal bracket positioning cause rotations.4 Errors in horizontal bracket positioning cause rotations. 3.5 Horizontal and vertical accuracy can be checked from the buccal aspect. 3. Viewing canines. and small errors do not significantly affect the position of these teeth (Fig. 3. Fig. Canines and premolars have more rounded facial surfaces. 3. especially on the left side.4). 3. or slightly mesial to it. to ensure good contact with the lateral incisors (Fig.6 Horizontal accuracy in the canine. and rotated incisors occlusally or incisally with a mouth mirror (Fig. The resulting contacts between canines and lateral incisors are less than ideal. Fig. . the bracket can be bonded slightly more mesially or dislally. full correction of the rotation can be achieved. 65). It is helpful to disregard the incisal edges of incisors. and gingival hyperplasia which have been previously reported.9) to achieve accuracy. partly erupted teeth. Axial accuracy It is necessary to accurately visualize the vertical long axis of the clinical crown of each tooth (Pig. 3. In this way. and accuracy is greatly improved by the use of gauges and an individualized bracket-positioning chart (p. 3. sometimes with a very small amount of excess composite under the mesial or distal of the bracket base.10 Vertical accuracy is the most difficult aspect of bracket positioning. On a rotated tooth. 3.10} of bracket positioning. Fig. The bracket wings need to be parallel to the long axis and to evenly straddle it. Fig.-1 Fig.8 On a rotated tooth. Vertical accuracy This is the most difficult aspect (Fig. This will deal with difficulties such as tooth length discrepancies. 3.9 To achive axial accuracy it is necessary to visualize the vertical long axis of the crown of each tooth. 3.Rotated incisors Slight mesial or distal adjustment is helpful when bracketing rotated incisors. 3. because errors will cause incorrect tip position of teeth. the bracket can be bonded slightly more mesially or distally.8). labially and lingually displaced roots. In this way. full correction of the rotation can be achieved with no special measures (Fig. 14). 3. the gauge is placed at 90° to the labial surface (Figs 3.12 In the incisor region.VERTICAL BRACKET POSITIONING WITH GAUGES AND CHARTS Clinical use of gauges The bracket-positioning gauges are used in slightly different ways in different areas of the mouth. the gauge is placed parallel with the occlusal surface of each individual molar (Fig.13). Fig. In the incisor regions. In the molar region.14 In the molar regions.1] & 3. Fig. In the canine and premolar regions. the gauge is placed at 90° to the labial surface. 3. 3. Fig.12). the gauge is placed at 90° to the labial tooth surface. 3. Fig.13 In the canine and premolar regions. 3. 3.11 In the incisor region. the gauge is placed parallel with the occlusal plane (Fig. . the gauge is placed parallel with the occlusal plane. the gauge is placed parallel with the occlusal surface of each individual molar. It is helpful to have an adult and a child version available. and is shown in Table 3. Individualized bracket-positioning charts Fig. and gauges used to position the brackets at the vertical heights shown in ihe chosen row. by measuring either fully erupted teeth in the mouth.4 A recommended bracket-positioning chart was published. because of continuing difficulties with vertical bracket positioning.before completion. A row could then be chosen for the upper arch and a row for the lower arch. It was recommended that ihe tooth size for the patient be determined. 3.15 Individualized bracket-positioning chart .Recommended bracket-positioning chart In the early 1990s.1. . the authors investigated the location of the center of the clinical crown. or teeth on plaster models. especially in cases with pointed teeth. Chart individualization in cases with abnormal incisal edges Some cases may have teeth with wear ot chipping of the incisal edges. Fig. Fig. For such patients. or else the bracket should be placed 0. . 3.5 mm more gingivally than for the other incisors.5 mm more gingivally. Tn others. Tn some cases.17 This patient shows difficult barrel-shaped teeth. 3. or with crowns that are pointed or have developmental irregularities. such as pointed canines. or else the bracket should be placed 0. 3.7. 2. Chart individualization for some upper canines and lower first premolars It is helpful in some cases to place upper canine and lower first premolar brackets 0. the patient may be reluctant to agree to enamel adjustments at the start of treatment. The use of gauges and a standard bracket-positioning chart will not deal with chipped or worn teeth. and amend the individualized bracket-positioning chart accordingly.5 rnm more gingivally.19 This lower right lateral incisor edge should be reshaped before treatment. or teeth of abnormal anatomy. it is necessary to estimate the final shape of the incisal edge and the length of the crowns. The bracket on the upper right central incisor was bonded 0. it will be easier to judge the correct amount of incisal enamel adjustment needed after the teeth have been aligned. in anticipation of the need for reshaping of the incisa! edge.18 This upper right lateral incisor edge should be reshaped before treatment. and these have to be made as treatment progresses.5 mm more gingivally. Fig. 3. 4. Chart individualization in premofar extraction cases In prernolar extraction cases. 3. Chart individualization in deep-bite and open-bite cases It can be helpful to place the incisor and canine brackets 0. This will ensure good vertical relationships between the marginal ridges of first premolars and first molars.20 Individualized bracket-positioning chart for a first prernolar extraction case. 3. . only the height of molar attachments is individualized (Fig. In open-bite cases. for the vertical relationships between the marginal ridges of canines and second premolars.20). they should be 0. ihe adjustment needs lo include the second prernolar bracket positions also (Fig. Fig.5 mm more gingival.21 Individualized bracket-positioning chart for a second prernolar extraction case.3. Fig.5 mm more occlusally in deep-bite cases.21). In first prernolar extraction cases. 3. the height of molar attachments is individualized to avoid vertical steps at the extraction sites. In second prernolar extraction cases. There is a greater risk of separators falling out if they are left in for more than a week. 3. After impression taking. to ensure that they remain in an ideal position during impressio n taking. 3. 3. and then (he distal palatal aspect. bands are selected which are one size too large. Separators should then be replaced until the RME appliance can be cemented a few days later. and metal separators (353-020) from TP (Fig. 3.23). Care is needed to prevent the distal aspect of the band from seating too gingivally.24 When viewed from the buccal. cleaned. the upper m olar tube should straddle the buccal groove. especiall y distal to upper first molars. rather than to the gingival. and sent to the laboratory. After good separation.25).25 When viewed from the occlusal. Good separation is necessary for accurate band placement. 66 Fig. elastic separating modules should be in place for about a week. Fig. especially for the seco nd molar.23) can be helpful in this area. Normal gray elastomeric modules can sometimes be used between small premolar contact points. Less than a week can cause sensitivity of the teeth during band placement. . Fig. They should then be temporarily cemented in place with small amounts of glass ionomer cement. the tube and band should be parallel with the buccal cusps.22 Blue S2 separators (3M Unitek 406-084) are preferred when possible. 3.PLACING MOLAR BANDS Separation Good separation is necessary (Figs 3-22 & 3. Fig. and this can be checked by viewing from the occlusal (Fig.23 Metal separators (TP 353-020) are sometim es useful in contact areas between m olars. It is helpful if the tube is welded more to the occlusal on the band. Ideally. 3. and band-seating pressure is therefore applied at the mesial palatal aspect initially. It is sometimes difficult to place elastic separators in the second molar regions. It assists accurate band placement and makes the procedure more comfortable for the patient. the bands can be removed. 3. The band should be checked from the buccal to ensure it is parallel with the buccal cusps (Fig.24). Upper molar band placement The upper molar tube should straddle the buccal groove. Upper molar band selection for rapid maxillary expansion (RME) cases A different technique is recommended for RME cases. It is therefore easier to place lower bands with non-convertible tubes (Fig.29 Lower rnolar non-convertible tubes are often preferable to convertible tubes. 3. Lower molar bands should be checked from the buccal to ensure they are parallel with the buccal cusps. as happened in this case. Parallel Fig. and the tendency to place the band too gingivally.28).29). because they are less bulky. 3. more comfortable. Fig. 3. and the lower first rnolar tube should straddle the mesio-buccal groove (Fig.27 The mesial of the lower first rnolar band should not be seated too low.28 It is an error to allow the mesial of the lower rnolar bands to seat too gingivally.0 mm or 2. leading to occlusal interferences. Convertible tubes are more bulky than non-convertible tubes.26 The lower molar tube should straddle the buccal groove.26). and cause fewer interferences. They are stronger. 3.28). With large lower first molars. Fig. 3. it may be helpful to place the tube a little distal of this position. It is helpful if the tube is welded more to the occlusal on the band (ideally at 2. .Lower molar band placement The lower second molar tube should straddle the buccal groove.5 mm). Fig. and also to prevent the mesial aspect of lower molar bands from seating too gingivally (Figs 3. 3. 3. rather than to the gingival.27 & 3. It is an error to allow the mesial aspect of the band to seat too gingivally (Fig. This should be checked by viewing from the occlusal. Care is needed when banding larger lower first molars to ensure thai the tube is not placed too far mesially. Excess bonding agent is then removed (Fig. Rotational and horizontal positioning is re-checked.30A). 3. etching. 3. The bracket is positioned at the estimated mid-point of the clinical crown. Fig.30B Removal of excess bonding agent. 5. The bracket is then pressed threequarters of the way on to the tooth surface at this position (Fig. the positioning and bonding of the bracket are carried out in five stages: 1. 3.30D Re-checking axial and horizontal positioning. . with bracket wings parallel to the long axis of the crown. 3. Vertical position is checked wilh a gauge. 4. with bracket wings parallel to the long axis of the clinical crown. 3. to equal the individualized bracket-positioning chart (Fig. and then the bracket is pressed fully on to the enamel surface (Fig. 3.30E Light-curing after removal of any additional excess bonding material. 3. Fig. Fig.DIRECT BONDING OF BRACKETS After cleaning of the enamel surface. 2. Fig.30E). Any additional excess of bonding material is removed before light-curing (Fig.30C Checking vertical positioning.30B).30C).30A Positioning at the estimated mid-point of the clinical crown. 3. 3. Fig. 3. and application of primer. 3.30D). It may be preferable to band the upper molars if a headgear is to be used. 28-40. if carefully used. and free from contamination. Journal of the Royal College of Surgeons. better tray materials. as they are convenient. Omar J 2000 Bacterial endocarditis and orthodontics. The orthodontist will subsequently position the brackets ideally. especially in the molar regions. Bennett J C 1995 Bracket placement with the preadjusted appliance. specifically designed for indirect bonding. many improvements are being introduced and evaluated and it is beyond the scope of this book to give full details and recommendations concerning indirect bonding technique. American Journal of Orthodontics and Dentofacial Orthopedics 115:352-359 6 Kalange J T 1999 Ideal appliance placement with APC brackets and indirect bonding.INDIRECT BONDING OF BRACKETS There is currently renewed interest in indirect bonding. After the model has been poured. The technician can then proceed with tray construction and the other laboratory procedures. and store it in a dark box. and who should rinse twice daily 7 with chlorhexidine 0. Mclaughlin R P 1997 Orthodontic management of the dentition with the preadjusted appliance. and has the advantage that no separation appointment is needed. Republished in 2002 by Mosby. and curing was complete in 2 minutes. at a convenient time. DISADVANTAGES OF INDIRECT BONDING An extra set of impressions is needed for indirect bonding cases. and also the patient has a shorter appointment for the case set-up. such as the NolaIM retractors used in the Nola™ Dry Field system. it is important to inform the patient that the bracket positioning was carried out by the orthodontist. Journal of Clinical Orthodontics 33:516-526 7 Roberts G J. and also prior to subsequent adjustment visits.7 Indirect bonding is therefore useful ior this small group of patients. The authors find that pre-coated (APC™) brackels are most efficient for laboratory use. Oxford (ISBN 1 899066 91 8) pp. This material has seen widespread acceptance. At the time of bonding. Isis Medical Media. clearly identified (so that mix-ups do not occur). and upgraded design of retractors. Wells Co. the orthodontist draws a pencil line on the crown of each tooth to represent the long axis. owing to the improved adhesives which have been developed. Currently. The viscosity of the Sondhi material was improved by the use of 5% fine particle fumed silica filler.5 He recommended making a light-cured adhesive base for each bracket and then indirect bonding with the new chemical-cured material. This upsurge in interest is partly driven by the acceptance within the orthodontic specialty that accuracy of bracket positioning is vital to success in modern orthodontics. Otherwise there are no bands on the posterior teeth. Sondhi reported on a new resin. who need to maintain a very high level of plaque control. it is likely that indirect bonding will see greater use than in the past. The reader is referred to the publications by Sondhi" and also the technique advocated by Kalange6 using the Sondhi material.2% mouthwash for 2 days prior to the set-up appointment. which also helped to fill any small discrepancies between enamel and the custom base. Journal of Clinical Orthodontics 29:302-311 5 Sondhi A 1999 Efficient and effective indirect bonding. and the procedure is technique sensitive. Considerable laboratory time is required. and that it is an advantage if there is a technician with suitable laboratory facilities in the practice. Edinburgh (ISBN 07234 32651} 4 McLaughlin R P. Edinburgh 45:141-145 . In 1999. This is due to the need for greater accuracy in bracket positioning and because of the improved techniques and materials which are currently available. ADVANTAGES OF INDIRECT BONDING Indirect bonding is more accurate. Child and Family Behavioural Therapy 12(2) 2 Andrews L F 1989 Straight-Wire-the concept and the appliance. and that indirect bonding techniques. those using indirect bonding confirm that the technique needs to be as perfect as possible. it has been recommended that brackets should always be used in REFERENCES 1 Gross A M 1990 Increasing compliance with orthodontic treatment. Although bonding and tray construction techniques are continuing to be refrned. LA 3 Bennett J. Lucas V S. The technique reduces the amount of chairside time for the orthodontist. as well as the manufacturer's literature. which assists in oral hygiene control. because posterior bands are stronger than bonds. The technician will then be able to place the brackets onto the model in approximately the correct position. preference to bands for patients with a history of bacterial endocarditis. can provide greater accuracy. Although there are disadvantages. determining 1AF for each patient 78 77 77 77 Arch form du ring fini shin g and detaili ng .the need for settling 83 Arch form co nsiderations during retentio n Stock control protocol for archwire s 84 83 Case AL A Class I case with a tapered arch form 86 .CHAPTER 4 Arch form Introduction 72 The search for the ideal arch form where expansion may be stable among human arch forms issues facing the clinician Practical solutions 73 72 Relapse 72 Variation tendency after changing arch form 72 Cases 73 Summary of the Modificatio ns to arch form and archwire co ordinatio n 80 Po sterio r torque considerations After maxillary expansion 80 81 82 80 Upper arch expansion with archwires Asymmetries 82 Upper arch expansion with a jockey wire 74 The use of three arch forms 74 Recommended ratios 75 The tapered arch form 76 The square arch form 76 The ovoid arch form 76 Systemized management of arch form Standardized versus customized wires The use of clear templates at the start of treatment 77 Arch form control early in treatment Arch form control with rectangular HANT wires 78 Arch form control with rectangular steel wires 78 Customizing . the 30% of cases in which buccal uprighting will be stable will probab ly include: Relapse tendency after changing arch form In 1969. Some customizing of the arch form for individual patients is important. This is particularly true of inter-canine width. In order to properly manage arch form in a modern orthodontic practice. mandibular inter-canine width tends to expand during treatment by about 1 or 2 millimeters. the search for an 'ideal' arch form. a short literature review will be presented to support the need for this balance. but emphasized that minimizing treatment change was no guarantee of postretention stability. De La Cruz et al6 reported on long-term changes in arch form of 45 Class I and 42 Class TI/1 treated cases. The search for the ideal arch form for the human dentition Arch form has been discussed in dental and orthodontic publications for over a century. parabolas.4 some of which was for full dentures. in a chapter on retenlion in Graber's text. In-out is built into the preadjusted appliance.INTRODUCTION During the era of standard edgewise. fo llowed by the description of a practical svsteni for arch form management. However.2 Brader3 and others. The paper by Burke et al confirms the overall message from the orthodontic literature. particularly in the mand ibular arch. Ideal arch forms were described by I lawley. and to contract post-retention to approximately the same dimension'. This simp lifies arch form. because of the risk of relapse. Helton et al8 pointed out that buccal uprighting will result in lower anterior relapse in approximately 70% of cases. The authors have previously reviewed this early work. They suggested that the p atient's pre-treatment arch form appeared to be the best guide for future arch form stab ility. Expansion of inter-canine width in treated Class 11/2 cases showed significantly greater stability than Class I or Class TI/1. R iedel5 reviewed previous studies on the stability of arch form. In 1995. a minimum of 10 years post-retention. suitable for every patient. Many of the early attempts to explain and classify the human dental arch form invo lved geometric termino logy such as ellipses. Changes in inter-mo lar width seem to be more stable. When the preadjusted appliance became availab le. Shapiro's interesting find ings could . the lo wer inter-canine width should not be increased during treatment. which avoids the need for first-order b ends. Shap iro9 reported on changes in arch length and inter-mo lar width in 22 nonextraction cases and 58 extraction cases after treatment and post-retention. Post-retention arch length reduction was also less in the Class II/2 group. but feel it has little relevance to modern orthodontics.1 Scott. there seemed to be an unwritten assumption that one arch form was appropriate. In this chapter. He c ited only one author who had reported the stability of a slight increase in mandibular inter-canine width after all retention had been removed for what was termed an 'adequate period'. He concluded that mandibular inter-canine width showed a strong tendency to return to its pre-treatment dimension in all groups. They concluded that 'regardless of patient d iagnostic and treatment modalities. the greater the tendency for post-retention change. the lower canines can be uprighted. Burke et al 7 used meta-analysis to review 26 previous stud ies of mandibular inter-canine width. Riedel postulated that 'arch form. Cases where expansion of lower intercanine width may be stable In most cases. In 1974. and catenary curves. and it could be used for all cases with the preadjusted system. cannot b e permanently altered during appliance therapy. They concluded that arch form tended to return toward the pre-treatment shape after retention and that the greater the treatment change. 73). Similarly. In 1998. there needs to be a balance between efficiency (a single arch form for all patients) and accuracy {the customizing needed for case stability). The overbite corrections must remain post-treatment for this movement to be stable. there was a strong tendency for these teeth to return to their pre-treatment position. He cited numerous authors who had reported that when intercanine and inter-mo lar width had been changed during orthodontic treatment. has been an unrealistic goal because of the wide ind ividual variations (p. most orthodontists customized archwires to each patient's arch form. with the exception of Class II/2 cases. Time has shown that this assumption was not correct. that if arch form is changed during orthodontic treatment. but it does not eliminate the need to use different shapes for different ind ividuals. in many cases there will b e a tendency for relapse to the original d imensions.' Deep-bite cases (such as Class 11/2 cases) in which lower canines have inclined lingually in response to the palatal contour of the upper canines As the bite is opened. the shape becomes in flu enced by the oral mu sculatu re. and 30 Class II non-extraction cases. They found that no particular arch form predominated in any of the th ree samp les. . but rh e apices of the roots of th ese teeth may mov e lin gu ally.n who observed that lower can in es will upright and increase intercan ine width by an average of 1. with lower canines inclined lingually in relation to the palatal surface of th e upper can in es. For example. p. 30 Class I non-extraction cases. They examined the mandibular casts of 30 untreated normal cases (from Andrews' 120 normals study). Wh en the bite is opened. there is a strong tendency (in as much as 70% of c ases) for the arch form to return to its original shape after appliances are removed. which will be help ful to the orthodontist. which can be stable. Genetic and environmental d ifferences produce great variability. • If the patient's original arch form is changed during treatment. and then after eruption of the teeth.1 mm. in 1987 Felton et al8 published a study to find out wh ether an ideal orthodontic arch form could be identified. It is gen erally accepted that the d ental arch form is initially shaped by the form of the underlying bone. w ith the bod ies of the teeth remaining in the same position. there does not seem to be any single arch form that can be used for all orthodontic cases. even though some modificatio ns may be need ed? In the following p ages a systemized approach to arch form management is described and recommended. Braun et al13-' 1 reported that the most popular nick el-titanium archwires so ld by the major orthodontic companies expand the lower inter-can in e width by 5.possibly be due to the fact that C lass II/2 cases normally show a deep bite. • As a result of these variations. Variation among human arch forms Most au thors hav e acknowledged that there is variability in the size and shape of human arch form. The amount of this response w as studied by Sandstrom et aJ. Desp ite the ov erwh elming evidence on the instability of lower arch expansion. which is confirmed in day-to-day clinical observation. Haas12 reported on aggressive upper arch expansion. and found an increase in inter-cuspid width of 3-4 mm in only 'a few cases'. 2. They stated that customizin g arch forms appeared to be necessary in many cases to obtain optimum long-term stability.2 mm on average. because of the great variability in arch form observed in the study.9 mm and th e upper inter-canin e width by 8. and molars w ill upright and increase inter-mo lar wid th on average 2. the in cisal ed ges of the lower can ines may move lab ially (Fig. Summary of the issues facing the clinician Research papers and clinical observations are giving clear messages: • There are extensive variatio ns among human arch forms. 46). How do the above find ings affect the clinical orthodontist? Do they mean that archwires must b e ind ividually customixed for each patient? Or can some form of preformed archwire system be used.9 mm. This effect does not seem to produce an extensiv e amount of additional space in the lower arch. Cases where rapid maxillary expansion is indicated in the upper arch and this expansion is maintained post-treatment Ladner and Muhl10 have reported that the lower arch wil! follow this with buccal uprightin g.46. 8% were square. and it is interesting that the Japanese samp le showed ratios of 12% tapered.1 No jima et al16 used tapered. The results (F ig. normal. and a typical Caucasian caseload would contain fewer C lass III cases. and broad). This shows the opposite ratio of square to tapered arch forms. and ovoid. In an unpublished study in one of the authors' practic es. Fig.3). ' the lower arches o f 200 consecutive cases (predominantly Caucasian) were evaluated with tapered. and square transparent templates. If one classifies the arch forms in the F elton et al8 study into tapered. and 42% ovoid. Class I. Fig. Hence the ratios of 50% tapered.PRACT ICAL SOLUTIONS The use of three arch forms Arch forms were first classified as tapered. Class II. 4. the ratios of these shapes in the Andrews'. square. and eventually orthodontic manufacturers began producing arch forms based on this classification (also referred to as narrow. and 42% ovoid are a more probab le reflection of a predominantly Caucasian practice.3 .1. and Class II samples are approximately as shown in Table 4. The Caucasian samp le showed 44% tapered arch forms. 4 . Numerous authors and clinic ians have used this classification over the years. and 42% were ovoid.1) were that approximately 50% of the lower arch forms were tapered. 46% square. No jima et al inc luded an equal proportion of Class III cases (of which 44% have square arches) in both samp les. This is quite similar to the Felton results. especially in the early archwire stages.2 Fig.1 Table 4. 4. compared with the Caucasian samp le. and ovoid by Chuck15 in 1932. square. ovoid. and 38% ovoid arch forms. and Class III cases in both Japanese and Caucasian samples (F igs 4.2 & 4. 8% square. Such a three arch form approach allo ws for greater ind ividualizatio n than the single arch form approach. and ovoid temp lates to evaluate the arch forms of Class 1. 18% square arch forms. 4 . However. Global differences are clearly significant. square. 4. and ovoid .6 Square Fig.4) of 45% tapered. Thus the recomm ended ratio s (Fig.5 Tapered Fig.7 Ovoid . Later in treatm ent an individual arch form (IAF) is used for each patient (p.4 Fig. 4. Fig. 4. The three shapes . and 45% ovo id (p. 72) that two categories of cases do show post-treatment stab ility after minor lower arch buccal uprighting. 4.tapered. 84) seem practical for a predominantly Caucasian practice.used by the authors early in treatment are shown below (Figs 4. These are p alatal exp an sion cases and deep-bite cases. 4.Recommended ratios it h as b een noted (p. square. 78).5-7). 10% square. The square arch form This arch form is indicated from the start of treatment in cases with broad arch forms (C ase CW. at least in the first part of treatment. It is also helpful. p. no expansion of the treated arch occurs.8 & 9) in this area. 17 for example. 289) has resulted in a majority of cases with good stability. In this way. giving a range of approximately 6 mm (Figs 4. Cases undergoing single arch treatment often require the use o f the tapered arch form. The tapered arch form is often used in combination with inverted canine brackets for these patients. When superimposed. The combined use of this arch fo rm with appropriate finishing. However. 120. The square arch form is useful to maintain expansion in upper arches after rapid maxillary expansion (p. p. for cases that require buccal uprighting of the lower posterior segments and expansion of the upper arch. this has b een the authors' p referred arch form for most of their cases. settling. It is particularly important to use this form for patients with narrow arch forms. it may be beneficial to change to the ovoid arch form in the later stages of treatment. Case IN. After overexpansion has been achieved. 80). 86). and especially in cases with gingival recession in the canine and premolar regions (most frequently seen in adult cases). 76 . tapered arch forms (Case AL. relative to the untreated arch. the three shapes vary mainly in inter-canine and inter-firstpremolar width. 152). and retention procedures (p. p. The ovoid arch form Over the past 15 years. the recent research (above) ind icates that a greater number of tapered arch forms should also be used. and minimal amounts of post-treatment relapse. The posterior part of this arch form can easily be modified to match the inter-mo lar width of the p atient.The tapered arch form This arch form has the narrowest inter-canine width and is useful early in treatment for patients with narrow. or sometimes .9 Lower arch form superimposition.0175 multistrand. The use of clear templates at the start of treatment Clear templates can be used lo assess the patient's lower model at the start of treatment. . Archwire adaptation will obviously not be needed at this stage for individuals with an ovoid starting arch form.10).015 or round HANT . As the teeth align. and .SYSTEM IZED M ANAGEM ENT OF ARCH FORM Standardized versus customized wires It is nol practical to customize every wire for every patient in a modern orthodontic practice. Their form is often temporarily distorted.020 round steel wires should be adapted as necessary for individuals with a tapered or square form at the start of treatment. and it is not necessary if the system described below is used. Consequently. the ovoid . Arch form control early in treatment It is recommended that all round wires be stocked in ovoid form only (p. square or ovoid form. with no customizing. 4. Fig. Often there will only be an approximate fit at this stage. It is therefore reasonable to use a standard ovoid arch form for these early wires. square. there will be a need to customize some wires.016. archwires gradually have more effect on arch form. to customize wires to an arch form suitab le for each individual patient. 4. Fig. These may all be used in ovoid form. . 84).018. The opening wires will normally be . or ovoid form (Fig.10 Clear templates may be used at the start of treatment to assess whether the patient's lower arch has a tapered. Fig 4. It is therefore beneficial.015 or .014 steel. and these exert light forces. This helps to limit inventory. as previously determined using the clear templates.8 Upper arch form superimpositions. Multistrand . and the treatme nt pro gresses into heavier HANT and then steel wires. This is because of their greater tensile strength and the fact that they are used fo r a longer period of time. d ue to tying into malaligned teeth. As leveling and aligning progress into heavier round wires (pp 111 & 112).016 wires are used early in treatment as the initial leveling and aligning archwires. but it is useful to have an early indication.016 HANT. 4. later in treatment. 77 . They can be expected to have little influence on arch form for the short periods that they are used. to determine whether the lower arch has a tapered. especially in the important canine region. square. or a Xerox copy of the model. The follo wing procedure (Fig.019/. 84). This is a good option if the caselo ad includes mainly children.11A). They rherefore need to be customized to each patient's individual arch form (IAF). b ased on the form of the lower dentition. a wax temp late is molded over the lower arch to record the indentations of the brackets (Fig. When stocking all three shapes. Archwire coordination is impo rtant throughout treatment. and modify as necessary. while accepting higher inventory levels. 4. or ovoid forms. square. and they do influence the patient's arch form. where the square arch form is seldom used. a . a Xerox copy of the wire is made and stored in the patient notes. • The wire is then compared with the starting lower model. because the IAF for many patients will not exactly match the shape of the manufactured wires in the basic tapered.025 stainless steel archwire can be individuali/ed for each patient. 4. To stock ovoid and tap ered shapes.11 A-F) is used: • After the rectangular HANT stage (Fig. • The .019/. . and tap ered shapes. or ovoid shap e.025 steel wires (p. It is therefore necessary to stock them in tapered.019/ . which will reduce the amount of wire modification need ed. and provides fo r correct archwire coordinatio n in the majo rity of cases (Fig.025 rectangular stainless steel wires.Arch form control with rectangular HANT wires The manufactured shape of rectangular HANT wires cannot be customized.019/. and thus minimize the amount of wire adaptation needed. 3. Concerning stock contro l of . 4J IF). and 3 mm wider in all areas. there are three possibilities: f. • The wire is then checked for symmetry on a template. 2. especially with the heavier round wires and the . there will always be a need to customize some wires. To stock ovoid. An upper form can then be made which is coordinated with the lower. To stock ovoid shape only. to ensure that it closely resemb les the overall starting shape. 4. If not used in the appropriate tapered.025 stainless steel archwire is bent to the ind entations in the wax bite (Fig.Q25 working wires have a major influence on arch form.019/. because (like the heavier round wires) they should b e used in the approximate form for the patient. square. The upper wire should superimpose approximately 3 mm outside of the lower wire. Arch form control with rectangular steel wires Rectangular steel . square. Customizing steel rectangular wires determining the IAF for each patient After the rectangular HANT wires have served their purpose. This is representative of the overlap of the upper teeth relative to the lower teeth.11D). as determined using the clear temp lates-Rectangular HANT wires may be in place for several months. 4. they can cause undesirable changes in the patient's starting arch form. and ovoid form. This is the patient's IAF. and shaping c an be delegated and then finally checked by the orthodontist. h is straightforward and quick to adapt working wires to the IAF. Lower rectangular steel wires are then used in the IAF shape and uppers in a form which is 3 mrn wid er.1 IB). • Finally. .11D The . 4.0197.11F After the patient's IAF has been determined for the lower archwire. Fig. Fig. 4.Fig. and then a Xerox copy can be made and used as the patient's IAF for the lower arch. 4.11A The lower rectangular HANT wire has been removed.11C The wax template viewed from the labial. Fig. 4. 4. Fig.11B A wax template is softened in warm water and molded over the lower arch to record indentations of the brackets.11E The steel rectangular wire is checked for symmetry on a template. 4. an upper wire can be created which should superimpose approximately 3 mm outside of the lower wire.025 rectangular steel wire is bent to the indentations. Fig. 13B).13 After upper arch expansion (A). If this is observed. Modification due to posterior torque considerations The additional buccal root torque in the upper molar brackets tends to nattow the upper arch. The combined effect o f these appliance features can be a tendency towards molar crossbite in some cases. the lower arch tends to upright buccally.13A).for example fro m tapered to ovoid) and the upper arch expansio n can be held with a correspondingly wid er arch form. Fig. two things can occur. 4. Fig. Modification after maxillary expansion After the upper arch has been expanded with a rapid maxillary expander or a quadhelix (Fig. .MODIFICATIONS TO ARCH FORM AND ARCHWIRE COORDINATION There are some cases that will require arch form modification from the normal IAF and the usual upper/lower archwire coordination. 4. 4.12). the upper arch tends to relapse (Fig. and second. the posterior segment of the upper archwire can be widened to 5 mm wider than the lower archwire in the rnolar regions. First. 4. there is a tendency for a buccal crossbite to develop. the lower arch can be widened by using a wider arch form (usually one size wid er .12 During correction of molar torque. To manage these effects. it is necessary to widen the posterior segment of the upper archwire. 4. When this is observed. it is often necessary to widen the upper arch form and narrow the lower arch form (B) to counteract unwanted rnolar changes. and the progressive buccal crown torque in the lower posterior brackets tends to upright the lower molar teeth and widen the lower arch (Fig. 4. if the ends of the archwi re Fig. This may be done by expanding the 1AF arch form in the molar regio ns. Fig. the expanded arch will s how correct form .16 & 4. due to one arch (usually the upper arch) being slightly smaller lhan the other arch. 81 . are pressed towards the ideal arch.14 It is important to use a correct technique for archwire expansion.ca nine width.15). 4. If overexpa nded or incorrectly expanded (Figs 4. Fig.019/. 4.Upper arch expansion with archwires In so me cases. There is a correct technique for archwire expansion.14). 108). or by use of the square arch form for a limited period. or to maintain expansion previously obtained by use o f a quadhelix or b y rapid maxillary expansion. Fig.16 Incorrect expansion.025 steel wires can be used Lo help correct this condition and achieve some arch expansion (p. The rectangular .17). 4. and pressed back towards the cho sen arch for m (IAF). it will not match the chosen arch form (IAF) whe n the ends are pressed toward s it.15 After correct e xpansion. When the ends of the exp and ed wire are held. 4. the wire sho uld matc h that shape (Fig.17 Incorrect expansion. and this will cause problems due (o narrowing or widening of the inter. arch form coordination requires special attentio n. 4. it is important to make sure it is not overe xpanded and thus distorted from the arch for m. If the wire is b ent to expand its width (Fig. Fig.19-21).025 wire has buccal root torque in the molar regio n (F ig. 4.19 Occlusal view of an asymmetrical lower arch. the archwires later in the treatment mayb e modified to assist correction of the asymmetry (F igs 4.21 Modification of the lower archwire to counteract and correct the dental asymmetry in Figure 4. pp 238 & 239). 4. Fig. 4.019/. 10. 108) to attempt bodily movement of molars and avoid tipping. p. Asymmetries In cases where it is c lear that the patient has an arch asymmetry. also expanded. Fig. If necessary.18 Occlusal view of a 'jockey arch' in place. The jockey arch may be of .Upper arch expansion with a jockey wire There are limits to the expansion force which can be delivered by one .15. p. 4.20 Asymmetry of the lower arch in Figure 4. tied in place over the normal archwire (Fig.30.0197.025 rectangular steel wire or of heavier round steel wire. This is merely a second archwire. or of heavier round steel wire. it can be convenient to end the jockey archwire in those tubes. particularly near the end of treatment. This may be of .025 rectangular steel. 290). 5. . If the upper first mo lars carry headgear tubes.019/.18). and there are many such cases.19. Fig. a little more expansion force can be achieved by using a 'jockey arch' (Case MS.025 rectangular wire during routine treatment. 4. It is helpful if the no rmal .019/.19. It is important to have adequate bone wid th to achieve upp er molar expansion (Fig. compared with the ovoid lower arch form. the upper teeth are held. Typically. If a vacuumformed upper retainer is used. and Case MOT.22).23). to include only the upper incisors. The authors prefer a full . This may slow the settling. relative to the fully retained upper arch (F ig. A new acrylic retainer can then be made. 307) are recommended to minimize this tend ency. A settling phase is required in almost every case. vertical tooth settling occurs and the upper and lower arch forms are also allowed to settle. to let the upper premolars and mo lars ad just to lo wer arch changes (Fig. The patient is checked at 2weekly intervals for approximately 6 weeks (Case JN.ARCH FORM DURING FINISHING AND DETAILING . to prevent space opening.22 During retention.22. 295). 4. During this period. and an upper . p. p. A new upper removable retainer can then be made and fitted. Fig. ARCH FORM CONSIDERAT IONS DURING RETENTION There is a constant tendency for lower inc isor relapse in the majo rity of cases. The following steps are recommended: • Patients should not progress directly from rectangular wires to retainers without a phase of settling in lighter wires. teeth ad jacent to extraction sites should be lightly tied together. 210). near the completion of treatment. a patient in retention will have a lower bonded retainer and an upper acrylic removab le retainer. .016 nickeltitanium lower archwire.THE NEED FOR SETTLING There are important arch form considerations during ihe closing stages of any treatment. A careful protocol allows the arch form to settle in the later stages of treatment.014 upper archwire is necessary. a full . but lower molars and premolars can move labio-lingually. Some second-order bends can be placed in this wire to encourage proper settling. An upper removable acrylic plate may be used for this (Fig. it may be modified for 2 to 4 weeks and then re-made.014 stainless steel or . It may be necessary to modify or leave out the upper acrylic retainer fo r 2 to 4 weeks. 124. so that a balance between the tongue and perioral musculature can re-establish. Lower bonded retainers from canine to canine (p. 10. the bonded retainer may be extended onto the second premolars. bent back behind the mo lars (Case DO. the expansion needs to be held during the settling phase. • During this settling phase. but it is needed to hold the corrected overjet. in combination with light triangular elastics. 274). • If the maxillary arch has been expanded earlier in the treatment. Fig. p. In first premo lar extraction cases. 4. 4.23 The upper acrylic retainer may be omitted or modified for 2 to 4 weeks to let upper molars and premolars adjust to lower changes. p. The lo wer premo lars and molars are thus free to narrow. In Class II treatments (where overjet relapse may occur during settling).014 stainless steel sectional wire. 4. .STOCK CONTROL PROTOCOL FOR ARCHWIRES An example of a suitable stock syslem is shown below. It is therefore desirable for the clinical orthodontist to have a system o f custo mizing the arch form for each patient. If a broad arch form is used for an ind ividual with a narrow facial appearance. Iwo. or three shapes. for example. It is possible to stock steel working wires in one. there will be a risk of relapse and an unnatural look to the smile. Customization of archwires reduces the risk of relapse and helps to achieve good esthetics. depending on the size of the practic e and the desire to minimize wire modific ation. This chapter has described a system which the authors use and recommend with confidence. but without having to overstock practice inventory or spend time with needless wire bending. A ngle Orthodontist 4:312-327 16 Nojima K. Angle Orthodontist 68(1}:29-36 14 Braun S. Dental Cosmos 47:541-552 2 Scott J H 1957 The shape of the dental arches. Legan H L 1999 An evaluation of the shape of some popular nickel titanium alloy preformed arch wires. Muhl Z F 1995 Changes concurrent with orthodontic treatment when maxillary expansion is a primary goal. Alexander R G 1987 A computerized analysis of the shape and stability of mandibular arch form. McLaughlin R P 1993 Orthodontic treatment mechanics and the preadjusted appliance. American Journal of Orthodontics 94:296-302 12 Haas A J 1980 Long-term posttreatment evaluation of rapid palatal expansion. Leschinksy R. Bennett J C 1999 Arch form considerations for stability and esthetics. F ender D E. Sinclair P M. Goldsmith L J. American Journal of Orthodontics 61:541-561 4 Mclaughlin R P. A merican Journal of Orthodontics 107:518-530 . Angle Orthodontist 68(1):53-60 8 Felton M J. American Journal of Orthodontics 92:478-483 9 Shapiro P A 1974 Mandibular arch form and dimension. Klapper L. Saunders. 7 Burke S P.REFERENCES 1 Hawley C A 1905 D etermination of the normal arch and its application to orthodontia. American Journal of Orthodontics and D entofacial Orthopedics 116:1-12 15 Chuck G C 1934 Ideal arch form. Angle O rthodontist 71:195-200 17 Bennett J. Journal of Dental I Research 36:996-1003 3 Brader A C 1972 Dental arch form related to intra-oral forces. Little R M. Mosby-Wolfe. Angle Orthodontist 50:189-217 13 Braun S . Shapiro P A 1995 Long-term changes in arch form after orthodontic treatment and retention. Sampson P. Legan H L 1998 T he form of the human dental arch. Mclaughlin R P. Hnat W P. Isshiki Y. London (ISB N 0 7235 1906X) 73 n o . American Journal of Orthodontics and D entofatial Orthopedics 108:184-193 11 Sandstrom R A. Philadelphia 6 De La Cruz A R. P apaconstantinou S 1988 Expansion of the lower arch concurrent with rapid maxillary expansion. Yancey J M. Scarfe WC 1998 A meta-analysis of mandibular intercanine width in treatment and post retention. Revista E spana Ortodontica 29(2):46-63 5 Riedel R A 1969 In: Graber T M (ed) Current orthodontic concepts and techniques. Artun J. American Journal of Orthodontics 66:58-70 10 Ladner P T . Hnat W P. Jones D L. Silveira A M. Van Stewart A. S inclair P M 2001 A comparative study on Caucasian and Japanese mandibular clinical arch forms. Fig. There was mild crowding of the upper and lower inc isor regio ns. with imp actions on the left sid e. The mild crowding would be resolved by torqueing and uprighting of the buccal segments and slight proclinalion in the lower incisor region. There was a lack o f space for the third molars.27 The fac ial appearanc e was reflected in the arch form. which would maintain the b asic form of the patient's dental arches. Lower incisors were retroclined at 78° to the mandibular plane to the mandibular p lane and at APo -1mm.022 metal brackets were placed. It was d ec ided to extract all the third mo lars. Dentally. with a high-angle vertic al pattern. The centric occlusion and centric relation were coincident. The opening wires were .33 . Upper and lower midlines were coincident. All the teeth. and a supplemental upper left third molar. with some mandibular asymmetry to the left. Fig. 4. were banded or bracketed. and notable prominence of the canine roots. with no displacements at terminal closure. which was narrow and tapered. restorations in first and second molars. inc luding second molars. 4. the patient was C lass I in the buccal segments. The selec ted arch form would be tapered.30 Standard .5-year-old female patient was Class I skeletalIy. There was a small amount of enamel damage to the inc isal edge of the upper right central inc isor. Fig. From the frontal aspect.016 HANT to an ovoid arch form. 4. The fac ial p rofile was pleasing and harmonious.The follo wing case shows an example of a patient with a tap ered arch fo rm and prominent canine roots at the start of treatment.24 Fig. 4. she had a narrow facial appearance. There were extensive. This 15. The upper and lower canine brackets were inverted. but not deep. to maintain the canine roots in bone. 32 Fig. 4. 4.Fig.28 Fig. 4.34 Fig. 4. 4.35 .31 Fig.29 Fig. 4. The .025 rectangular HANT wires had the effect of torqueing and uprighiing the buccal segments (Fig. This was fo llowed by repositioning of the inc isor bracket.019/.019/. In the lower arch. 4. This in turn provided additional space for anterior alignment. Fig. Fig. Fig.40) as a result of the reduced torque specification of the appliance system in lower premolars and mo lars. and the . with the selected tapered arch form.42 Late in the treatment. to reduce a small overjet which had developed. and then re-Seveling and aligning using a .025 stainless steel wires were carefully maintained in the tapered form.019/.014 steel wire. 4. The occlusal views at the end of treatment show that this was achieved. The earlier decision to invert the canine brackets allowed good control of the canine roots during the leveling and aligning stages.45 . to narrow the arches slightly.025 rectangular stainless steel wires were maintained during the middle and later part of the treatment. 4.019/.016 HANT wire was used to commence settling of the case. The . the referring dentist requested enamel reshaping of the incisal edge of the upper right central inc isor. The .025 rectangular HANT wires.019/.025 rectangular stainless steel wires. a . with tapered arch form. It was felt that the patient's arches had become a little too ovoid during the alignment stage.016 HANT wires in the ovoid form were fo llowed by .39 The . and with soldered hooks.025 rectangular HANT wires were fo llowed by . 4.019/. The patient wore light Class II elastics for a short period. . but the steel rectangular wires were successfully used to restore and maintain the tapered shape into the final result. and +6° torque and 3° tip in the lower. the patient was similar to the starting appearance. 4 . The light wires allowed settling of the arch form. 4. 4.51 A lower bonded retainer and an upper removable retainer were used.54 Facially. These were +7° torque and 8° tip in the upper. Fig. Fig. with 1mm of proclination towards APo. there was a change in angulation of the lower inc isors. Early in the treatment the arch form became a little too ovoid. A pleasing dental result was achieved. The canine crown and root positio ns result from the bracket specifications used in this case. Fig.57 .Selective up-and-down elastics were used with light wires during the settling stage. Dentally. which was very satisfactory. and this help ed to resolve the lower anterior crowding. The tapered arch form was suitab le for this case because it relates to the patient's starting lower arch form and is appropriate for her narrow facial appearance. 49 Fig. 4.59 Fig. 4.Fig.53 Fig. 4.50 Fig . 4.58 Fig. 4. 4.52 Fig . 4.55 .56 Fig . 4.
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