- Email: [email protected]
Mathematical analysis of tooth and restoration contour using image analysis
Dental Materials (2004) 20, 893–899
www.intl.elsevierhealth.com/journals/dema
Mathematical analysis of tooth and restoration contour using image analysis Nabil Alhouria, David C. Wattsb, J. Fraser McCorda, Philip W. Smitha,* a
Unit of Prosthodontics, School of Dentistry, University of Manchester, Manchester M15 6FH, UK Unit of Biomaterials Science, School of Dentistry, University of Manchester, Manchester, UK
b
Received 13 January 2004; accepted 8 June 2004
KEYWORDS Tooth contour; Image analysis; Crown restoration
Summary Objectives. The aim of this study was to develop a methodology for comparison of the contour of artificial crowns in the mid bucco-lingual plane with their equivalent natural teeth on the opposing side of the same arch (antimeres) using a novel application of image analysis software. The objective was to determine whether artificial crowns were overcontoured. Methods. Specimens consisted of thin sections of silicone putty impressions of the buccal and lingual surfaces of 55 full crown restorations and their natural antimeric teeth. A thin slice of the putty was obtained in the mid-tooth bucco-lingual plane and a digital image was captured and this was analysed to produce a data set (x, y) representing the curvature of the tooth surface. Further analysis was performed in order to describe the profile in optimum mathematical terms. Results. The curves were best represented by three equations: yZaCbx(0.5), ln(y)ZaCbx2, and y2ZaCbx. In all equations parameter (b), which expresses the contour curvature, was used as a deciding factor in comparing the degree of contour of the crown restorations with their natural antimeres. Most artificial crowns were found to be either similarly or undercontoured when compared with their natural antimeres. When overcontouring was present in the artificial crowns this tended to occur on the lingual aspects of anterior and posterior crowns. Significance. Simplifying tooth contour into a mathematical model can be useful in determining whether restorations are overcontoured. Clinically, particular attention should be directed towards the lingual aspects of restorations which were more likely to be overcontoured. Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
Introduction * Corresponding author. Tel.: C44-161-275-6629; fax: C44161-275-7822. E-mail address: [email protected] (P.W. Smith).
The contour of a tooth is an important aspect of overall dental aesthetics. It is defined in the Glossary of Prosthodontic Terms as the outline of
0109-5641/$ - see front matter Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.dental.2004.06.003
894 the curving of the tooth, or the line representing this outline [1]. The greatest convexity of tooth contour has been described, in previous literature, according to different positions in the dental arch. The crests of facial contours of all teeth were found in the gingival third of the crown with a projection of no more than 0.5 mm facially beyond the cementoenamel junction (CEJ) [2]. Between the occlusal or incisal surface and the facial crest, the facial surface has a slight convexity. The buccal surfaces of posterior mandibular teeth are convex between the gingival crests of curvature and the occlusal surfaces. Slight facial depressions exist between the occlusal/incisal surface and the gingival crest of curvature in a vertical direction [3]. Similarly, the greatest convexity of the lingual surfaces of some of the teeth is found in the gingival third of the crown. However, on mandibular molars and possibly premolars, the lingual convexity is found in the middle third of the crown. The lingual convexity, it is considered, should protrude lingually only 0.5 mm beyond the CEJ in any tooth except in the mandibular molars and premolars where this maximum convexity may extend to 0.75–1 mm [3,4]. The cervical two thirds of a proximal surface of a tooth has been described as flat or slightly concave. The tooth is flat or slightly concave from the facial to lingual aspect, as well as from the level of proximal contact to the CEJ. The exception is the distal proximal surface of the maxillary first molar, which is convex in shape, but cervically there is a concavity [3,4]. Many theories have been used to explain the need for particular forms of axial contour of full and partial coverage extracoronal restorations. It was argued that many ideas regarding the contours suggested for restorations were not based on scientific evidence, largely because artificial crowns were constructed long before the aetiological role of dental plaque was known [5]. According to the ‘gingival protection theory’, convexities should be created in the cervical third of artificial crowns, to deflect food away from the free gingivae. Food should, theoretically, pass over the gingival crevice and onto the kertinized surface of the attached gingiva [2]. This theory, together with the increased use of full-coverage veneer crowns led to an era of overcontoured restorations [6]. Many authors subsequently questioned the rationale of the concept of the ‘gingival protection theory’, and concluded that crowns constructed according to this concept were likely to be overcontoured. Overcontoured crowns are considered to promote, rather than prevent, gingival inflammation [7–10]. Perel [11] in a study on dogs reflected the same findings, demonstrating
N. Alhouri et al. increased levels of gingival inflammation and hyperplastic changes, which were associated with overcontoured restorations. Such changes were observed both at clinical and histological levels. In contrast, undercontoured restorations were not associated with any such pathological change. In the ‘muscle-action theory’, the rationale of muscular molding and cleansing was used to explain the observable clinical phenomena found around the natural and artificial crowns [7,8]. According to this concept, muscle action could be impaired when intimate contact between the lips, cheeks, and tongue against gingivae is prevented by overcontoured crowns. However, it has been demonstrated that, in the absence of oral hygiene, ‘self-cleansing’ mechanisms do nothing to prevent gingivitis [12,13]. In the ‘anatomic (biologic) theory’, it was suggested that artificial crown contour should simulate the morphology of natural, healthy teeth [5]. Thus, duplication of the features of natural teeth while constructing restorations might be considered one of the goals of Restorative Dentistry and of Dental Materials Science. Morris [8] reported that the pre-operative shape of the embrasures should be reproduced provided the embrasure contained a healthy papilla. Overcontouring proximal surfaces apical to contact areas might produce severe gingival inflammation. Parkinson [14] stated that to minimise iatrogenic dental disease, artificial crown form must approximate natural tooth morphology. If the curvature of the restoration exceeds natural curvature, the restoration contradicts the innate defensive capability of nature. The aim of this study was to develop a scientific methodology to mathematically compare the contour between artificial crowns and their natural antimeres.
Materials and methods Sample set and exclusion criteria The sample set was obtained from a commercial Dental Laboratory in the North of England which takes dental work from throughout the UK. It consisted of 55 full crown restorations and their natural antimeric teeth. Full crowns were included in the study only if their antimeres were natural teeth and were excluded if the antimeres exhibited any of the following: † Antimeres partially erupted † Antimeres with caries or restorations affecting buccal or lingual surfaces
Mathematical analysis of tooth contour
895
Figure 1 Thin bucco-lingual sections of silicone putty taken from an impression of a single crown and its natural antimere.
† Antimeres with malformation or abnormal morphology † Antimeres with any defect in the casts resulting from either defects in the impression or subsequent laboratory techniques that might have an effect on the studied area.
which allowed the surface profile to be characterised (Fig. 3). Equations were ranked within the program according to the fit of the curves. The stages of analysing the tooth curvature are represented in Fig. 4.
Results Analysis of tooth contour Impressions of buccal and lingual surfaces of the full crown restorations and their natural antimeric teeth were made using addition cured silicone putty (Vinyl Polysiloxane Provilw novo, putty, soft, fast set, Heraeus, Kulzer, Dormagen). The putty impression was then sectioned in midbuccal mid-lingual coronal plane perpendicular to the mesiodistal diameter of the tooth, and a thin slice of the putty was obtained (Fig. 1). The slice of polyvinylsiloxane putty was then viewed under a microscope with 6.5 magnification (Wild Leitz, Wild Heerbrugg Ltd, 9435 Heerbrugg, Switzerland). The slice of impression putty was positioned with a standardised orientation on the microscope stage. An image was taken (Fig. 2) using a digital camera (PEC3010, Pulnix, Basingstoke, UK), which had been connected to the microscope and to the computer. The image was then analysed using SigmaScan Software (Sigma Scan Pro. Image Analysis, Version 5.0.0 (Build number 3981) Copyrightq 1987–1999 SPSS Inc.). The edges of the buccal and lingual surfaces of the crown were tracked separately to produce a profile curve. The SigmaScan software was used to provide a text file with an orthogonal data set (x, y) representing the curvature of each surface. The text file was then imported to TableCurve Software (TableCurve 2D Windows v4.07, Copyright 1989–1996 AISN Software Inc.). TableCurve was used to produce an algebraic formula (equation),
It was found that the curves (contour) of buccal surfaces for all natural teeth in the study were best represented by the following equation (EqB): y Z a C bx 0:5 The curves (contour) of lingual surfaces for all anterior natural teeth in the study were best represented by the equation (EqLa): lnðyÞ Z a C bx 2
Figure 2 An image showing buccal tooth contour derived from the thin section of silicone impression.
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Figure 3 Tooth contour characterised, for orthogonal coordinates (x,y), by algebraic formula: yZaCbx0.5. In this example: parameter aZ587.25; bZ21.89; r2Z 0.9966.
The curves (contour) of lingual surfaces for all posterior natural teeth in the study were best represented by the equation (EqLp): y 2 Z a C bx These equations were considered to be representative equations of the curvatures (contour) of natural teeth. Fig. 5 represents all equations of the contour of buccal and lingual surfaces of natural antimeric anterior and posterior teeth.
The effects of different values of (a, b) on the curvature of tooth contour In order to know the effects of the different values of (a, b) on a graph representing EqB, a fixed value was given to parameter (b) with three different values to parameter (a) (Fig. 6), and vice versa (Fig. 7). It can be noticed from Fig. 6 that different values of parameter (a) do not have an effect on the shape of the graph (contour of the tooth) representing EqB. It can be noticed from Fig. 7 that different values of parameter (b) have an effect on the shape of the graph (contour of the tooth) representing EqB. The negative values of parameter (b) make it more convex (or concave), whereas positive values make it straighter.
Figure 4 Analysis of buccal and lingual contour of artificial crowns and their natural antimeres.
Figure 5 Graphical representation of the equations of contour of buccal/lingual surfaces for anterior and posterior natural teeth. Curve a, lingual posterior; curve b, lingual anterior; curve c, buccal anterior; curve d, buccal posterior.
The value of the parameter (b) in the relevant equations The mean value and standard deviation of parameter (b) in the equations representing the contour of buccal and lingual surfaces for antimeric natural teeth versus full crown restorations in the study are presented in Table 1.
Figure 6 Graphical representation of the effects of positive and negative values of parameter (a) in EqB yZaCbx0.5. Curve a, parameter a positive; curve b, parameter a; curve c, parameter a negative.
Mathematical analysis of tooth contour
897 Table 2 The 95% limits of agreement of the parameter (b) for natural teeth. 95% limits of agreement of parameter (b) Anterior teeth (buccal) Anterior teeth (lingual) Posterior teeth (buccal) Posterior teeth (lingual)
(K3.07, 3.07) (K0.00000075, 0.00000075) (K7.73, 7.73) (K84.34, 84.34)
Comparison of the tooth contour between full crowns and their natural antimeric teeth
Figure 7 Graphical representation of the effects of positive and negative values of parameter (b) in EqB yZ aCbx0.5. Curve a, parameter b positive; curve b, parameter b; curve c, parameter b negative.
Limits of agreements In order to assess the limits of agreement in the present study, two teeth were selected randomly from the sample. Ten images were taken for buccal/lingual surfaces for each tooth. The images were then analysed and the values of the parameters (b) in the correspondent equations were statistically analysed. The 95% limits of agreements of the parameters (b) in the relevant equations were calculated as G2.26SD (Table 2). The contour of the artificial crown would be considered similar to the contour of the natural antimeric crown, if the differences in the parameter (b) of the relevant equation between them are comparable to those found in the measurement repetition.
The values of the parameter (b) of the equations representing buccal/lingual surfaces of full crown restorations (C)b were subtracted from those of the antimeric natural teeth (A)b as follows: (A)bK(C)bZb*. If the resultant values of parameter (b*) did not exceed the 95% limits of agreement presented in Table 2, the contour of the full crown restoration was regarded to be similar to the contour of the antimeric natural tooth. However, if the resultant value (b*) exceeded the 95% limits presented in Table 2, the contour of the full crown restoration was not regarded to be similar to the contour of the antimeric natural tooth. When value (b*) was positive the restoration was considered to be overcontoured, and conversely when (b*) was negative the crown was undercontoured.
The contour of full crown restorations The values for (b*) were categorised according to whether they were overcontoured or undercontoured. The results are presented in Table 3.
Discussion Table 1 The statistical analysis of parameter (b) in EqB, EqLa and EqLp for full crown restorations and their natural antimeres. b value in equations
Full crown mean (SD)
Natural antimeres mean (SD)
EqB for anterior teeth EqB for posterior teeth Eqla
K23(3)
K24(3)
K27(7)
K28(7)
K0.000004 (0.000001) K422(88)
K0.000005 (0.000002) K441(134)
Elp
Tooth contour has been studied previously using optical observation [2–4] or photographic approaches [15]. The expressions ‘overcontoured and undercontoured’ of the tooth surfaces were derived from observational judgement of the naked eye, or alternatively via metric assessment of the surface prominence [2]. Shillingburg et al. [16] indicated that the corresponding surfaces of the adjacent teeth, if they were in a normal position, make an excellent guide for judging the contours of the facial and lingual surfaces of wax patterns. It is also important to evaluate the contour of the contralateral teeth
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Table 3
The contour of full crown restorations as compared with their natural antimeres.
Tooth contour
Similar
Overcontour
Undercontour
Anterior restoration (buccal) Anterior restorations (lingual) Posterior restorations (buccal) Posterior restorations (lingual)a
18 (56.3%) 6 (18.8%) 16 (69.6%) 10 (43.5%)
4 8 3 6
10 (31.3%) 18 (56.3%) 4 (17.4%) 5 (21.7%)
a
(12.5%) (25%) (13%) (26.1%)
Two values are missing.
(if present and sound) on dental casts, when constructing indirect restorations [17]. Furthermore, the use of callipers was suggested to compare tooth dimensions between measurements of the study casts and final restorations [17]. This study used a mathematical method to evaluate and characterise tooth contour using an image analysis package. This novel method was used to compare the surface contours of the artificial crowns and their natural antimeric teeth. Similarly, image analysis systems have been used previously to analyse tooth shape variability of buccal surfaces [18]. The main factor in determining the curvature of the surfaces of teeth was found to be the value of parameter (b) in the relevant equation. Increasing negative values of parameter (b) in EqB, EqLa and EqLp indicates that the surface of the tooth is overcontoured, while increasing positive values of parameter (b) indicates that the surface of the tooth is undercontoured. In the equations determined mathematically from this study: (X) and (Y) are co-ordinates, variables (a) and (b) are mathematical parameters where a is the intersection point between the curve and the Y axis b represents the curvature value at each point of the curve. In clinical terms therefore: parameter (a) becomes the intersection point between the tooth contour and gingival margin. Thus the (a) value depends on choosing the start point of the measurement. Parameter (b) expresses the contour of curvature. It has been suggested that the amount of tooth preparation can have an influence on the contour of the restoration [19]. There is a tendency even for experienced operators to under prepare labial shoulders, both in laboratory and clinically. Seymour et al. [20] explained that this under preparation leaves the technician facing a dilemma: of either restoring the tooth to optimum contour and thus compromise on material or aesthetic qualities. The second option is to use
the material in the recommended thickness and thereby overbuild the crown, thus compromising its final contour and emergence profile. In general, the contour of full crown restorations compared with their antimeric natural teeth had the following trends: † Most of the full crown restorations had either similar contour or were undercontoured compared with their antimeric natural teeth. This result indicates that in this study, dental technicians tended not to overcontour the surfaces of full crown restorations. This result fits in part with the recommendation of Tjan et al. [5] and Becker and Kaldahl [6]. † The buccal contour of full crowns was similar to their antimeric natural teeth in most cases for the posterior crowns and in more than half of the cases for the anterior crowns. † If not similar, the buccal surface of anterior full crown was mostly made undercontoured compared with the contour of their antimeric natural teeth. † The lingual aspects of the anterior full crowns were made undercontoured in more than half of the cases compared with the contour of their antimeric natural teeth. † The lingual contour of the posterior full crowns was similar to their antimeric natural teeth in just less than the half of the cases. It may be concluded that in this study artificial crowns were either similarly contoured or undercontoured when compared with their natural antimeres. This study suggests that it is valid that dental technicians use unprepared antimeres as a guide for developing the contours artificial crowns.
References [1] Academy of Prosthodontics. The glossary of prosthodontic terms. J Prosthet Dent 1999;39–110. [2] Wheeler RC. Complete crown form and the periodontium. J Prosthet Dent 1961;11:722–34. [3] Koidis PT, Burch JG, Melfi RC. Clinical crown contours: contemporary view. J Am Dent Assoc 1987;114:792–5.
Mathematical analysis of tooth contour [4] Burch JG. Ten rules for developing crown contours in restorations. Dent Clin North Am 1971;15:611–8. [5] Tjan AHL, Freed H, Miller GD. Current controversies in axial contour design. J Prosthet Dent 1980;44:636–40. [6] Becker CM, Kaldahl WB. Current theories of crown contour, margin placement and pontic design. J Prosthet Dent 1981; 45:268–77. [7] Herlands RE, Lucca JJ, Morris HL. Forms, contours and extensions of full coverage restorations in occlusal reconstruction. Dent Clin North Am 1962;6:147–62. [8] Morris ML. Artificial crown contours and gingival health. J Prosthet Dent 1962;12:1146. [9] Hagen SP, Osborne JW. Relationship of operative dentistry to periodontal health. Dent Clin North Am 1967;11:245–51. [10] Youdelis RA, Weaver JD, Sapkos S. Facial and lingual contours of artificial complete crown restorations and their effects on the periodontium. J Prosthet Dent 1973; 29:61–6. [11] Perel M. Axial crown contours. J Prosthet Dent 1971;25: 842–9. [12] Lindhe J, Wincen P. The effects on the gingivae of chewing fibrous foods. J Periodontal Res 1969;4:193–200.
899 [13] Loe H, Theilade E, Jensen S. Experimental gingivitis in man. J Periodontol 1965;36:177–82. [14] Parkinson CF. Excessive crown contour facilitate endemic plaque niches. J Prosthet Dent 1976;35:424–9. [15] Croll BM. Emergence profiles in natural tooth contour. Part I: photographic observations. J Prosthet Dent 1989;62: 4–10. [16] Shillingburg HT, Hobo S, Whitsett LD, Jacobi R, Brackett SE. Fundamentals of fixed prosthodontics, 3rd ed. Chigago, IL: Quintessence Publishing Co; 1997 p. 338. [17] Davis MV. The importance of contour on full coverage restorations. Pract Periodontics Aesthet Dent 1992;4: 17–23. [18] Robinson DL, Blackwell PG, Stillman EC, Brook AH. Planar procrustes analysis of tooth shape. Arch Oral Biol 2001;46: 191–9. [19] Seymour K, Zou L, Samarawickrama D, Lynch E. Assessment of shoulder dimensions and angles of porcelain bonded to metal crown preparations. J Prosthet Dent 1996;75:406–11. [20] Seymour K, Samarawickrama D, Lynch E. Metal ceramic crowns—a review of tooth preparation. Eur J Prosthodont Restor Dent 1999;7:79–84.
www.intl.elsevierhealth.com/journals/dema
Mathematical analysis of tooth and restoration contour using image analysis Nabil Alhouria, David C. Wattsb, J. Fraser McCorda, Philip W. Smitha,* a
Unit of Prosthodontics, School of Dentistry, University of Manchester, Manchester M15 6FH, UK Unit of Biomaterials Science, School of Dentistry, University of Manchester, Manchester, UK
b
Received 13 January 2004; accepted 8 June 2004
KEYWORDS Tooth contour; Image analysis; Crown restoration
Summary Objectives. The aim of this study was to develop a methodology for comparison of the contour of artificial crowns in the mid bucco-lingual plane with their equivalent natural teeth on the opposing side of the same arch (antimeres) using a novel application of image analysis software. The objective was to determine whether artificial crowns were overcontoured. Methods. Specimens consisted of thin sections of silicone putty impressions of the buccal and lingual surfaces of 55 full crown restorations and their natural antimeric teeth. A thin slice of the putty was obtained in the mid-tooth bucco-lingual plane and a digital image was captured and this was analysed to produce a data set (x, y) representing the curvature of the tooth surface. Further analysis was performed in order to describe the profile in optimum mathematical terms. Results. The curves were best represented by three equations: yZaCbx(0.5), ln(y)ZaCbx2, and y2ZaCbx. In all equations parameter (b), which expresses the contour curvature, was used as a deciding factor in comparing the degree of contour of the crown restorations with their natural antimeres. Most artificial crowns were found to be either similarly or undercontoured when compared with their natural antimeres. When overcontouring was present in the artificial crowns this tended to occur on the lingual aspects of anterior and posterior crowns. Significance. Simplifying tooth contour into a mathematical model can be useful in determining whether restorations are overcontoured. Clinically, particular attention should be directed towards the lingual aspects of restorations which were more likely to be overcontoured. Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved.
Introduction * Corresponding author. Tel.: C44-161-275-6629; fax: C44161-275-7822. E-mail address: [email protected] (P.W. Smith).
The contour of a tooth is an important aspect of overall dental aesthetics. It is defined in the Glossary of Prosthodontic Terms as the outline of
0109-5641/$ - see front matter Q 2004 Academy of Dental Materials. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.dental.2004.06.003
894 the curving of the tooth, or the line representing this outline [1]. The greatest convexity of tooth contour has been described, in previous literature, according to different positions in the dental arch. The crests of facial contours of all teeth were found in the gingival third of the crown with a projection of no more than 0.5 mm facially beyond the cementoenamel junction (CEJ) [2]. Between the occlusal or incisal surface and the facial crest, the facial surface has a slight convexity. The buccal surfaces of posterior mandibular teeth are convex between the gingival crests of curvature and the occlusal surfaces. Slight facial depressions exist between the occlusal/incisal surface and the gingival crest of curvature in a vertical direction [3]. Similarly, the greatest convexity of the lingual surfaces of some of the teeth is found in the gingival third of the crown. However, on mandibular molars and possibly premolars, the lingual convexity is found in the middle third of the crown. The lingual convexity, it is considered, should protrude lingually only 0.5 mm beyond the CEJ in any tooth except in the mandibular molars and premolars where this maximum convexity may extend to 0.75–1 mm [3,4]. The cervical two thirds of a proximal surface of a tooth has been described as flat or slightly concave. The tooth is flat or slightly concave from the facial to lingual aspect, as well as from the level of proximal contact to the CEJ. The exception is the distal proximal surface of the maxillary first molar, which is convex in shape, but cervically there is a concavity [3,4]. Many theories have been used to explain the need for particular forms of axial contour of full and partial coverage extracoronal restorations. It was argued that many ideas regarding the contours suggested for restorations were not based on scientific evidence, largely because artificial crowns were constructed long before the aetiological role of dental plaque was known [5]. According to the ‘gingival protection theory’, convexities should be created in the cervical third of artificial crowns, to deflect food away from the free gingivae. Food should, theoretically, pass over the gingival crevice and onto the kertinized surface of the attached gingiva [2]. This theory, together with the increased use of full-coverage veneer crowns led to an era of overcontoured restorations [6]. Many authors subsequently questioned the rationale of the concept of the ‘gingival protection theory’, and concluded that crowns constructed according to this concept were likely to be overcontoured. Overcontoured crowns are considered to promote, rather than prevent, gingival inflammation [7–10]. Perel [11] in a study on dogs reflected the same findings, demonstrating
N. Alhouri et al. increased levels of gingival inflammation and hyperplastic changes, which were associated with overcontoured restorations. Such changes were observed both at clinical and histological levels. In contrast, undercontoured restorations were not associated with any such pathological change. In the ‘muscle-action theory’, the rationale of muscular molding and cleansing was used to explain the observable clinical phenomena found around the natural and artificial crowns [7,8]. According to this concept, muscle action could be impaired when intimate contact between the lips, cheeks, and tongue against gingivae is prevented by overcontoured crowns. However, it has been demonstrated that, in the absence of oral hygiene, ‘self-cleansing’ mechanisms do nothing to prevent gingivitis [12,13]. In the ‘anatomic (biologic) theory’, it was suggested that artificial crown contour should simulate the morphology of natural, healthy teeth [5]. Thus, duplication of the features of natural teeth while constructing restorations might be considered one of the goals of Restorative Dentistry and of Dental Materials Science. Morris [8] reported that the pre-operative shape of the embrasures should be reproduced provided the embrasure contained a healthy papilla. Overcontouring proximal surfaces apical to contact areas might produce severe gingival inflammation. Parkinson [14] stated that to minimise iatrogenic dental disease, artificial crown form must approximate natural tooth morphology. If the curvature of the restoration exceeds natural curvature, the restoration contradicts the innate defensive capability of nature. The aim of this study was to develop a scientific methodology to mathematically compare the contour between artificial crowns and their natural antimeres.
Materials and methods Sample set and exclusion criteria The sample set was obtained from a commercial Dental Laboratory in the North of England which takes dental work from throughout the UK. It consisted of 55 full crown restorations and their natural antimeric teeth. Full crowns were included in the study only if their antimeres were natural teeth and were excluded if the antimeres exhibited any of the following: † Antimeres partially erupted † Antimeres with caries or restorations affecting buccal or lingual surfaces
Mathematical analysis of tooth contour
895
Figure 1 Thin bucco-lingual sections of silicone putty taken from an impression of a single crown and its natural antimere.
† Antimeres with malformation or abnormal morphology † Antimeres with any defect in the casts resulting from either defects in the impression or subsequent laboratory techniques that might have an effect on the studied area.
which allowed the surface profile to be characterised (Fig. 3). Equations were ranked within the program according to the fit of the curves. The stages of analysing the tooth curvature are represented in Fig. 4.
Results Analysis of tooth contour Impressions of buccal and lingual surfaces of the full crown restorations and their natural antimeric teeth were made using addition cured silicone putty (Vinyl Polysiloxane Provilw novo, putty, soft, fast set, Heraeus, Kulzer, Dormagen). The putty impression was then sectioned in midbuccal mid-lingual coronal plane perpendicular to the mesiodistal diameter of the tooth, and a thin slice of the putty was obtained (Fig. 1). The slice of polyvinylsiloxane putty was then viewed under a microscope with 6.5 magnification (Wild Leitz, Wild Heerbrugg Ltd, 9435 Heerbrugg, Switzerland). The slice of impression putty was positioned with a standardised orientation on the microscope stage. An image was taken (Fig. 2) using a digital camera (PEC3010, Pulnix, Basingstoke, UK), which had been connected to the microscope and to the computer. The image was then analysed using SigmaScan Software (Sigma Scan Pro. Image Analysis, Version 5.0.0 (Build number 3981) Copyrightq 1987–1999 SPSS Inc.). The edges of the buccal and lingual surfaces of the crown were tracked separately to produce a profile curve. The SigmaScan software was used to provide a text file with an orthogonal data set (x, y) representing the curvature of each surface. The text file was then imported to TableCurve Software (TableCurve 2D Windows v4.07, Copyright 1989–1996 AISN Software Inc.). TableCurve was used to produce an algebraic formula (equation),
It was found that the curves (contour) of buccal surfaces for all natural teeth in the study were best represented by the following equation (EqB): y Z a C bx 0:5 The curves (contour) of lingual surfaces for all anterior natural teeth in the study were best represented by the equation (EqLa): lnðyÞ Z a C bx 2
Figure 2 An image showing buccal tooth contour derived from the thin section of silicone impression.
896
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Figure 3 Tooth contour characterised, for orthogonal coordinates (x,y), by algebraic formula: yZaCbx0.5. In this example: parameter aZ587.25; bZ21.89; r2Z 0.9966.
The curves (contour) of lingual surfaces for all posterior natural teeth in the study were best represented by the equation (EqLp): y 2 Z a C bx These equations were considered to be representative equations of the curvatures (contour) of natural teeth. Fig. 5 represents all equations of the contour of buccal and lingual surfaces of natural antimeric anterior and posterior teeth.
The effects of different values of (a, b) on the curvature of tooth contour In order to know the effects of the different values of (a, b) on a graph representing EqB, a fixed value was given to parameter (b) with three different values to parameter (a) (Fig. 6), and vice versa (Fig. 7). It can be noticed from Fig. 6 that different values of parameter (a) do not have an effect on the shape of the graph (contour of the tooth) representing EqB. It can be noticed from Fig. 7 that different values of parameter (b) have an effect on the shape of the graph (contour of the tooth) representing EqB. The negative values of parameter (b) make it more convex (or concave), whereas positive values make it straighter.
Figure 4 Analysis of buccal and lingual contour of artificial crowns and their natural antimeres.
Figure 5 Graphical representation of the equations of contour of buccal/lingual surfaces for anterior and posterior natural teeth. Curve a, lingual posterior; curve b, lingual anterior; curve c, buccal anterior; curve d, buccal posterior.
The value of the parameter (b) in the relevant equations The mean value and standard deviation of parameter (b) in the equations representing the contour of buccal and lingual surfaces for antimeric natural teeth versus full crown restorations in the study are presented in Table 1.
Figure 6 Graphical representation of the effects of positive and negative values of parameter (a) in EqB yZaCbx0.5. Curve a, parameter a positive; curve b, parameter a; curve c, parameter a negative.
Mathematical analysis of tooth contour
897 Table 2 The 95% limits of agreement of the parameter (b) for natural teeth. 95% limits of agreement of parameter (b) Anterior teeth (buccal) Anterior teeth (lingual) Posterior teeth (buccal) Posterior teeth (lingual)
(K3.07, 3.07) (K0.00000075, 0.00000075) (K7.73, 7.73) (K84.34, 84.34)
Comparison of the tooth contour between full crowns and their natural antimeric teeth
Figure 7 Graphical representation of the effects of positive and negative values of parameter (b) in EqB yZ aCbx0.5. Curve a, parameter b positive; curve b, parameter b; curve c, parameter b negative.
Limits of agreements In order to assess the limits of agreement in the present study, two teeth were selected randomly from the sample. Ten images were taken for buccal/lingual surfaces for each tooth. The images were then analysed and the values of the parameters (b) in the correspondent equations were statistically analysed. The 95% limits of agreements of the parameters (b) in the relevant equations were calculated as G2.26SD (Table 2). The contour of the artificial crown would be considered similar to the contour of the natural antimeric crown, if the differences in the parameter (b) of the relevant equation between them are comparable to those found in the measurement repetition.
The values of the parameter (b) of the equations representing buccal/lingual surfaces of full crown restorations (C)b were subtracted from those of the antimeric natural teeth (A)b as follows: (A)bK(C)bZb*. If the resultant values of parameter (b*) did not exceed the 95% limits of agreement presented in Table 2, the contour of the full crown restoration was regarded to be similar to the contour of the antimeric natural tooth. However, if the resultant value (b*) exceeded the 95% limits presented in Table 2, the contour of the full crown restoration was not regarded to be similar to the contour of the antimeric natural tooth. When value (b*) was positive the restoration was considered to be overcontoured, and conversely when (b*) was negative the crown was undercontoured.
The contour of full crown restorations The values for (b*) were categorised according to whether they were overcontoured or undercontoured. The results are presented in Table 3.
Discussion Table 1 The statistical analysis of parameter (b) in EqB, EqLa and EqLp for full crown restorations and their natural antimeres. b value in equations
Full crown mean (SD)
Natural antimeres mean (SD)
EqB for anterior teeth EqB for posterior teeth Eqla
K23(3)
K24(3)
K27(7)
K28(7)
K0.000004 (0.000001) K422(88)
K0.000005 (0.000002) K441(134)
Elp
Tooth contour has been studied previously using optical observation [2–4] or photographic approaches [15]. The expressions ‘overcontoured and undercontoured’ of the tooth surfaces were derived from observational judgement of the naked eye, or alternatively via metric assessment of the surface prominence [2]. Shillingburg et al. [16] indicated that the corresponding surfaces of the adjacent teeth, if they were in a normal position, make an excellent guide for judging the contours of the facial and lingual surfaces of wax patterns. It is also important to evaluate the contour of the contralateral teeth
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N. Alhouri et al.
Table 3
The contour of full crown restorations as compared with their natural antimeres.
Tooth contour
Similar
Overcontour
Undercontour
Anterior restoration (buccal) Anterior restorations (lingual) Posterior restorations (buccal) Posterior restorations (lingual)a
18 (56.3%) 6 (18.8%) 16 (69.6%) 10 (43.5%)
4 8 3 6
10 (31.3%) 18 (56.3%) 4 (17.4%) 5 (21.7%)
a
(12.5%) (25%) (13%) (26.1%)
Two values are missing.
(if present and sound) on dental casts, when constructing indirect restorations [17]. Furthermore, the use of callipers was suggested to compare tooth dimensions between measurements of the study casts and final restorations [17]. This study used a mathematical method to evaluate and characterise tooth contour using an image analysis package. This novel method was used to compare the surface contours of the artificial crowns and their natural antimeric teeth. Similarly, image analysis systems have been used previously to analyse tooth shape variability of buccal surfaces [18]. The main factor in determining the curvature of the surfaces of teeth was found to be the value of parameter (b) in the relevant equation. Increasing negative values of parameter (b) in EqB, EqLa and EqLp indicates that the surface of the tooth is overcontoured, while increasing positive values of parameter (b) indicates that the surface of the tooth is undercontoured. In the equations determined mathematically from this study: (X) and (Y) are co-ordinates, variables (a) and (b) are mathematical parameters where a is the intersection point between the curve and the Y axis b represents the curvature value at each point of the curve. In clinical terms therefore: parameter (a) becomes the intersection point between the tooth contour and gingival margin. Thus the (a) value depends on choosing the start point of the measurement. Parameter (b) expresses the contour of curvature. It has been suggested that the amount of tooth preparation can have an influence on the contour of the restoration [19]. There is a tendency even for experienced operators to under prepare labial shoulders, both in laboratory and clinically. Seymour et al. [20] explained that this under preparation leaves the technician facing a dilemma: of either restoring the tooth to optimum contour and thus compromise on material or aesthetic qualities. The second option is to use
the material in the recommended thickness and thereby overbuild the crown, thus compromising its final contour and emergence profile. In general, the contour of full crown restorations compared with their antimeric natural teeth had the following trends: † Most of the full crown restorations had either similar contour or were undercontoured compared with their antimeric natural teeth. This result indicates that in this study, dental technicians tended not to overcontour the surfaces of full crown restorations. This result fits in part with the recommendation of Tjan et al. [5] and Becker and Kaldahl [6]. † The buccal contour of full crowns was similar to their antimeric natural teeth in most cases for the posterior crowns and in more than half of the cases for the anterior crowns. † If not similar, the buccal surface of anterior full crown was mostly made undercontoured compared with the contour of their antimeric natural teeth. † The lingual aspects of the anterior full crowns were made undercontoured in more than half of the cases compared with the contour of their antimeric natural teeth. † The lingual contour of the posterior full crowns was similar to their antimeric natural teeth in just less than the half of the cases. It may be concluded that in this study artificial crowns were either similarly contoured or undercontoured when compared with their natural antimeres. This study suggests that it is valid that dental technicians use unprepared antimeres as a guide for developing the contours artificial crowns.
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