Development of the mandibular curve of Spee and maxillary ...87 and mandibular movement during function. 88 The purpose of this investigation is to test the hypothesis, using a simplified

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1 Development of the mandibular curve of Spee and maxillary compensating curve: 2 A finite element model

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5 Steven D. Marshall1¶*, Karen Kruger2¶, Robert G. Franciscus3¶, Thomas E. Southard1¶

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9 1Department of Orthodontics, College of Dentistry and Dental Clinics, The University of 10 Iowa, Iowa City, Iowa, United States of America

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12 2Department of Biomedical Engineering, Marquette University, Milwaukee, Wisconsin, 13 United States of America

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15 3Department of Anthropology, The University of Iowa, Iowa City, Iowa, United States of 16 America

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18 *Corresponding author

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20 E-mail: [email protected]

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22 ¶These authors contributed equally to this work.

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.CC-BY 4.0 International licensecertified by peer review) is the author/funder. It is made available under aThe copyright holder for this preprint (which was notthis version posted August 2, 2019. . https://doi.org/10.1101/723429doi: bioRxiv preprint

https://doi.org/10.1101/723429

http://creativecommons.org/licenses/by/4.0/

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31 Abstract

32 The curved planes of the human dentition seen in the sagittal view, the

33 mandibular curve of Spee and the maxillary compensating curve, have clinical

34 importance to modern dentistry and potential relevance to the craniofacial evolution of

35 hominins. However, the mechanism providing the formation of these curved planes is

36 poorly understood. To explore this further, we use a simplified finite element model,

37 consisting of maxillary and mandibular “blocks”, developed to simulate tooth eruption,

38 and forces opposing eruption, during simplified masticatory function. We test our

39 hypothesis that curved occlusal planes develop from interplay between tooth eruption,

40 occlusal load, and mandibular movement.

41 Our results indicate that our simulation of rhythmic chewing movement, tooth eruption,

42 and tooth eruption inhibition, applied concurrently, results in a transformation of the

43 contacting maxillary and mandibular block surfaces from flat to curved. The depth of the

44 curvature appears to be dependent on the radius length of the rotating (chewing)

45 movement of the mandibular block. Our results suggest mandibular function and

46 maxillo-mandibular spatial relationship may contribute to the development of human

47 occlusal curvature.

48 Introduction

49 Viewed in the sagittal plane, the concave occlusal curvature of the mandibular

50 dentition is a naturally occurring phenomenon seen in modern humans and fossil

51 hominins [1,2], Commonly described as the curve of Spee [3,4], this occlusal curvature

52 (Fig 1) is proposed to have functional significance during mastication [5-8] and is


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53 considered an important element of diagnosis and treatment planning in dentistry and

54 orthodontics. [7-18]

55 Despite its clinical importance in extant humans, and its potential relevance to

56 fossil hominin craniofacial evolution, the process by which the concave mandibular

57 curve of Spee develops is not completely understood. Many investigations have

58 analyzed variables thought to be associated with the curve’s development, including the

59 timing of permanent tooth eruption and vertical alveolar development [19,20], variation

60 in craniofacial morphology and various factors associated with malocclusion

61 [2,14,17,20-22], but the underlying mechanisms of the curve’s formation and

62 maintenance remain unknown. In a similar fashion, while intuitively logical, the process

63 by which the corresponding maxillary convex occlusal curvature (i.e. the compensating

64 curve, Fig 1) develops is, nonetheless, also not understood. Unanswered are the

65 following questions: why should a concave curve develop in the mandibular dental arch,

66 why should a convex curve develop in the maxillary arch, why should these two curves

67 develop simultaneously, and why should they persist throughout life?

68 The development of these curves, in part, must be related to tooth eruption.

69 Post-emergent tooth eruption (i.e. tooth eruption after a tooth has emerged through

70 gingival tissue) has been divided into four stages: pre-functional spurt, juvenile

71 equilibrium, adolescent spurt, and adult equilibrium, with the last three stages

72 designating eruptive tooth movement after the tooth has reached the occlusal plane.

73 [27] As teeth erupt, they appear to erupt until they find their position along these

74 curves. In addition, the development of these curves must be related to occlusal forces

75 acting against the teeth. For instance, when a maxillary or mandibular molar is


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76 unopposed by another tooth, the molar will continue to erupt (i.e., super-eruption),

77 irrespective of the curves formed from the remaining teeth. In other words, occlusal

78 loading must be involved in development of the curves. Lastly, it seems likely that the

79 development of these curves must be related to mandibular movement, constrained

80 within a pathway determined by muscular and ligamentous attachments, either during

81 function or during parafunction, the latter including bruxism and other activities not

82 related to eating, drinking, or talking.

83 Considering the application of occlusal forces to the teeth during functional and

84 parafunctional tooth contact, and the equilibrium theory of tooth position [28], it is

85 conceivable that the development and maintenance of the curve of Spee and

86 compensating curve, results from the interplay between tooth eruption, occlusal loading,

87 and mandibular movement during function.

88 The purpose of this investigation is to test the hypothesis, using a simplified finite

89 element model (FE model), that mandibular curve of Spee and maxillary compensatory

90 curve develop from a predictable interplay between tooth eruption, occlusal load, and

91 mandibular movement.

92 Materials and methods

93 The finite element method uses discretization to compute approximations to space- and

94 time-dependent problems. It has been widely-used in studying the functional

95 morphology of the human masticatory system.[29]

96 We based our FE model upon the following assumptions:


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97 ● The curve of Spee, as classically described by Ferdinand Graf von Spee (Fig 1),

98 exists as a concave curve in the mandibular dental arcade, with an average radius of

99 approximately 100 mm [30]. A corresponding convex curve (compensating curve)

100 exists in the maxillary dental arcade.

101 ● Both curves result from an interplay between tooth eruption, inhibition of tooth

102 eruption by axial occlusal loading, and mandibular movement during function.

103 ● Maxillary and mandibular teeth erupt toward the occlusal plane at a continuous rate

104 (EC).

105 ● Occlusal loading occurs during mastication as the mandibular dental arcade rotates

106 around a center of rotation (CROT) that lies above the maxillary dentition. The position of

107 CROT results from the spatial relationship of the condyles to the mandibular dental

108 arcade and the complex interaction between condylar head translation, condylar head

109 rotation, muscle tension, and ligament restriction.

110 ● The inhibition of tooth eruption (a relative apical tooth movement) is a product of the

111 magnitude of occlusal loading at each point of occlusal contact in a direction normal to

112 the occlusal surface (Fn) integrated throughout the time this force is applied (T).

113 ● Total tooth eruption (ETOTAL) at any point along the maxillary and mandibular dental

114 arch is calculated as the difference between tooth eruption toward the occlusal surface

115 and inhibition of tooth eruption or,

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117 , where k is a constant𝐸𝑇𝑂𝑇𝐴𝐿 = 𝐸𝐶 ‒ ∫𝑇0𝑘𝐹𝑛𝑑𝑡

118


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119 ● Our FE model is restricted along the 2 dimensions of the sagittal plane

120 Our FE model was created using ABAQUS v6.9.1 and consisted of two blocks of

121 elements, the upper block representing the maxillary dentition and the lower block

122 representing the mandibular dentition. The block’s left side represents posterior teeth

123 and the block’s right side represents anterior teeth. . Each block was composed of a

124 single layer of brick continuum elements (σ=82.5 GPA, ν=0.33) [31] with 40 elements

125 horizontal, 20 elements vertical, and 1 element deep (Fig 2). That is, 1600 elements

126 are found within each maxillary and mandibular block resulting in block dimensions of

127 20 mm high by 40 mm wide (40 mm is an approximation for posterior-anterior human

128 dental arch length). The maxillary block was held fixed while the mandibular block was

129 rotated about CROT, simulating simplified human chewing motion in 2 dimensions from

130 the sagittal view. Two centers of mandibular block rotation were simulated: one at 100

131 mm (an approximation of the human curve of Spee), and one at 400 mm (a value

132 chosen to evaluate the behavior of our FE model with increasing CROT), above the

133 superior surface of the mandibular block (i.e. occlusal surface of the mandibular

134 dentition). The starting form of the maxillary and mandibular blocks was rectangular in

135 shape (i.e. 40w x 20h) before being subjected to motion, simulated tooth eruption and

136 masticatory load. One rotation cycle is described as follows: The cycle begins with the

137 maxillary and mandibular blocks in contact such that the posterior surface of the

138 mandibular block approximately 3 mm posterior to the posterior surface of the maxillary

139 block (Fig 2C). The cycle proceeds with the rotation of the mandibular block, along an

140 arc (i.e. the rotation arc) dictated by the length of the radius between CROT and the

141 midpoint of the occlusal surface of the mandibular block, to a point where the anterior


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142 surface of the mandibular block is approximately 3 mm anterior to the anterior surface of

143 the maxillary block (Fig 2D). Each rotation cycle, the blocks are moved toward each

144 other (maxillary block moves inferiorly, mandibular block moves superiorly) a distance of

145 0.01 mm (our proxy for tooth eruption). Similarly, during each rotation cycle, block

146 material was removed at the occlusal surface (our proxy for tooth eruption inhibition)

147 based on a calculation using the Archard-Lancaster relationship for wear. [32] An

148 ABAQUS UMESHMOTION subroutine was created which allowed for this simulated

149 eruption inhibition on both plates as sliding occurred while they were in contact.

150 Following each rotation cycle, the subroutine computed the new occlusal surface

151 topography (depth for each surface node) and the model was re-meshed to simulate

152 this modified geometry prior to entry into the subsequent cycle. In our study, the blocks

153 were modeled as identical material (which would be the case for tooth composition in

154 both arches) and the subroutine had to be adapted to permit maxillary and mandibular

155 blocks to undergo eruption-inhibition at equal rates (for similar values of normal load

156 and time of load application). The large shape change experienced by the “occlusal”

157 surface of the maxillary and mandibular blocks required intensive remeshing of both the

158 surface and underlying nodes to preserve element quality throughout the simulation.

159 The simulation was run until a steady-state curve was reached. The length of the

160 mandibular block rotation arc was chosen to allow a reasonable time to reach steady-

161 state.

162 Results

163 Simulations of both models (CROT = 100m and CROT = 400 mm, in mp4 movie

164 format) may be observed at the web sites shown in Figure 3. An application that allows


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165 viewing of mp4 files is required (e.g. Windows Media Player®, QuickTime®, iTunes®).

166 To move quickly through the simulation, advance the time slider at the bottom from left

167 to right (see Fig 2A).

168 As observed in the simulations, repetitive rotation cycles cause the surfaces of

169 the maxillary and mandibular blocks (representing the maxillary and mandibular

170 dentitions) to undergo a notable change that eventually reaches equilibrium. Two-

171 dimensional rotation of the mandibular block against the stationary maxillary block

172 during simulated tooth eruption interacting with simulated tooth eruption inhibition,

173 resulted in curvature of the contacting block surfaces (i.e. occlusal surfaces) (Fig 3).

174 Beginning with a flat occlusal surface, as the mandibular dentition undergoes a chewing

175 rotation about CROT, a concave occlusal surface (curve of Spee) develops in the

176 mandible, and a compensating curve develops in the maxilla. When the rotation radius

177 of the mandibular block is smaller, the resulting maxillary and mandibular block curves

178 are deeper (Fig 3A). When the rotation radius of the mandibular block is larger, the

179 resulting upper and lower block curves are shallower (Fig 3B).

180 Discussion

181 Using a FE model we sought to simulate the interplay of forces during

182 mastication: tooth eruption modeled by maxillary FE model brick continuum elements

183 being brought towards mandibular FE model brick continuum elements with each cycle;

184 and tooth eruption inhibition modeled as a function of axial loading during a simplified

185 rotational masticatory movement. The spatial relationships of this simplified FE model

186 approximated human maxillo-mandibular spatial relationship as seen from the sagittal

187 view, with the mandibular rotation radius = 100 mm, and the dental arcade length from


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188 the sagittal view = 40 mm. The rotation arc of approximately 3 mm is large compared to

189 that shown for humans (≈ 0.5 mm) in the sagittal plane [24,25], and was chosen to allow

190 a reasonable length of time for our simulation to reach steady state.

191 The principle finding of our pilot study is that the surfaces of the maxillary and

192 mandibular FE model blocks, in contact during simulation of human chewing in two

193 dimensions during which the forces of tooth eruption and tooth loading are also

194 simulated, transform from their original flat shape to a curved shape in the steady-state.

195 We find steady-state curve shape is dependent on the rotation arc radius of the

196 mandibular FE model block, with a smaller rotation arc radius producing with a deeper

197 curve. These findings have implications for explaining variation in the development and

198 maintenance of the mandibular curve of Spee and maxillary compensating curve in

199 humans. A depiction of the variation in curves generated by our analysis, displayed as

200 maxillary and mandibular dentitions, is shown in Figure 4.

201 Our model is based, in part, on the equilibrium theory of tooth position, the

202 assumption that eruptive forces continue throughout life and are opposed by forces

203 generated during dynamic contact and release of mandibular agonist teeth with their

204 maxillary antagonists. [26,27] Although, the direction, magnitude and duration of force

205 necessary to inhibit eruptive tooth movement is not fully understood [27], we assume

206 that the loading of the teeth during the rhythmic mandibular movements of mastication,

207 and daytime and nocturnal bruxism, results in complex loading of the maxillary and

208 mandibular teeth, including forces directed apically along the long axis of individual

209 teeth (i.e. axial loading). [23,24,25,40-42] There exists ample evidence that tooth

210 position equilibrium is disrupted when axial loading forces are greatly diminished.


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211 Numerous studies of humans and fossil hominids have demonstrated the super-

212 eruption of teeth when occlusal antagonist teeth are lost. [33-39]

213 The influence of the spatial relationship of the mandibular condyles to the dentition is

214 not well understood. Studies have suggested the horizontal distance between the

215 condyle and the dentition is inversely related to the depth of the curve of Spee.

216 [2,5,6,11,14,17] One theoretical explanation for these observations suggests a shorter

217 horizontal distance between the condyle and the dentition necessitates a deeper curve

218 of Spee for greater mechanical advantage during chewing. [5,6] Our FEM simulation

219 did not test this directly. A more complex FE model is required to address this question.

220 Conclusions

221 A finite element model (FEM) was developed to simulate tooth eruption, and forces

222 opposing eruption, during simplified masticatory function. The maxillary arch FE model

223 was held stationary, and the mandibular arch FE model was moved against the

224 maxillary block to simulate two-dimensional chewing motion. The results of our

225 experiment show:

226 1. Two-dimensional simulation of rhythmic chewing movement, tooth eruption, and

227 tooth eruption inhibition applied concurrently, resulted in a transformation of the

228 contacting block surfaces from flat to curved.

229 2. A shorter radius for the rotation arc of the mandibular dental arch results in more

230 deeply curved surfaces of the maxillary and mandibular dental arches. A larger radius

231 for the rotation arch of the mandibular dental arch results in shallower curved surfaces


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232 of the maxillary and mandibular dental arches. This finding may explain, in part, human

233 mandibular curve of Spee and maxillary compensating curve variation.

234 References

235 1. Osborn JW. Orientation of the masseter muscle and the curve of Spee in relation to 236 crushing forces on the molar teeth of primates. Am J Phys Anthropol. 1993;92 237 (1):99-106.

238 2. Laird MF, Holton NE, Scott JE, Franciscus RG, Marshall SD, Southard TE. Spatial 239 determinants of the mandibular curve of Spee in modern and archaic Homo. Am J 240 Phys Anthropol 2016 Oct;161(2):226-36.

241 3. Spee, FG. Die verschiebungsbahn des unterkiefers am schadel. Arch Anat Physiol 242 1890:285-294.

243 4. Spee FG, Beidenbach MA, Hotz M, Hitchcock HP. The gliding path of the mandible 244 along the skull. J Am Dent Assoc 1980;100:670-5.

245 5. Osborn, JW. Relationship between the mandibular condyle and the occlusal plane 246 during hominid evolution: Some effects on jaw mechanics. Am J Phys Anthropol 247 1987;73:193-207.

248 6. Baragar FA, Osborn JW. Efficiency as a predictor of human jaw design in the 249 sagittal plane. J Biomech 1987;73:193-207.

250 7. Ash MM, Stanley JN. Wheeler’s dental anatomy, physiology and occlusion, 8th ed. 251 Philadelphia: Saunders; 2003.

252 8. Okeson, JP. The determinants of occlusal morphology. In: Management of 253 temporomandibular disorders and occlusion. 7th ed. St. Louis: Elsevier Mosby; 254 2013.

255 9. Andrews, FL. The six keys to normal occlusion. Am J Orthod 1972;62:296-309.

256 10. Koyama TA. Comparative analysis of the curve of Spee (lateral aspect) before and 257 after orthodontic treatment--with particular reference to overbite patients. J Nihon 258 Univ Sch Dent. 1979;21(1-4):25-34.

259 11. Orthlieb JD. The curve of Spee: understanding the sagittal organization of 260 mandibular teeth. Cranio 1997;15(4):333-40.

261 12. Carcara S., Preston CB., Jureyda O. The relationship between the curve of Spee, 262 relapse, and the Alexander discipline. Semin Orthod 2001;7(2):90-99.

263 13. De Praeter J, Dermaut L, Martens G, Kuijpers-Jagtman A. Long-term stability of 264 the leveling of the curve of Spee. Am J Orthod Dentofacial Orthop 2002;121:266-72.


https://doi.org/10.1101/723429


12

265 14. Farella M, Michelotti A, Martina, R. The curve of Spee and craniofacial 266 morphology: a multiple regression analysis. Eur J Oral Sci 2002;110:277-281.

267 15. Lynch CD, McConnell. Prosthodontic management of the curve of Spee: Use of 268 the Broadrick flag. J Prosthet Dent 2002;87:593-7.

269 16. Shannon KR, Nanda, R. Changes in the curve of Spee with treatment and at 2 270 years posttreatment. Am J Orthod Dentofacial Orthop 2004;125:589-96.

271 17. Cheon SH, Park YH, Paik KS, et al. Relationship between the curve of Spee and 272 dentofacial morphology evaluated with a 3-dimensional reconstruction method in 273 Korean adults. Am J Orthod Dentofacial Orthop. 2008;133(5):640.e7-14.

274 18. Rozzi M, Mucedero M, Pezzuto C, Cozza P. Leveling the curve of Spee with 275 continuous archwire appliances in different vertical skeletal patterns: A retrospective 276 study. Am J Orthod Dentofacial Orthop. 2017;151(4):758-766.

277 19. Marshall SD, Caspersen M, Hardinger RR, Franciscus RG, Aquilino SA, Southard 278 TE. Development of the curve of Spee. Am J Orthod Dentofacial Orthop. 279 2008;134(3):344-52.

280 20. Veli I, Ozturk MA, Uysal T. Curve of Spee and its relationship to vertical eruption of 281 teeth among different malocclusion groups. Am J Orthod Dentofacial Orthop. 282 2015;147(3):305-12.

283 21. Xu H, Suzuki T, Muronoi M, Ooya K. An evaluation of the curve of Spee in the 284 maxilla and mandible of human permanent healthy dentitions. J Prosthet Dent. 285 2004;92(6):536-9.

286 22. Batham PR, Tandon P, Sharma VP, Singh A. Curve of Spee and its relationship 287 with dentoskeletal morphology. J Indian Orthod Soc, 2013;47(3):28-134.

288 23. Koolstra JH. Dynamics of the human masticatory system. Crit Rev Oral Biol Med. 289 2002;13:366-76.

290 24. Xu WL, Bronlund JE, Potgieter J, et al. Review of the human masticatory system 291 and masticatory robotics. Mech Mach Theory 2008;43:1353-75.

292 25. Benazzi S, Nguyen HN, Kullmer O, Kupczik K. Dynamic modelling of tooth 293 deformation using occlusal kinematics and finite element analysis. PLOS One 2016; 294 11(3): e0152663.

295 26. Weinstein S, Haack DC, Morris LY, et al. On an equilibrium theory of tooth position. 296 Angle Orthod 1963;1:1-26.

297 27. Proffit WR, Frazier-Bowers SA. Mechanism and control of tooth eruption: overview 298 and clinical implications. Orthod Craniofac Res 2009;12:59–66.

299 28. Proffit WR. Equilibrium theory revisited: factors influencing the position of teeth. 300 Angle Orthod 1978;48:175-86.


https://doi.org/10.1101/723429


13

301 29. Richmond BG, Wright BW, Grosse IR, Dechow PR, Ross CF, Spencer M. Finite 302 element analysis in functional morphology. Anat Rec. Part A, Discoveries in 303 molecular, cellular, and evolutionary biology. 2005;283. 259-74.

304 30. Ferrario VF, Sforza C, Miani Jr A. Statistical evaluation of Monson's sphere in 305 healthy permanent dentitions in man. Arch oral biol 1997 May 1;42(5):365-9.

306 31. Craig RG, Selected properties of dental composites. J Dent Res 1979;58(5):1544-307 1550.

308 32. Archard JF, Contact and rubbing of flat surfaces. J Appl Phys 1953;24(8):981–988.

309 33. Sicher H. Tooth eruption: The axial movement of continuously growing teeth. J 310 Dent Res 1942;21(2): 201–10.

311 34. Hylander WL. Morphological changes in human teeth and jaws in a high-attrition 312 environment. In: Dahlberg AA, Graber TM, editors. Orofacial growth and 313 development. Paris: Mouton Publishers; 1977. p. 301–333.

314 35. Ainamo A, Ainamo J. The width of attached gingivae on supra-erupted teeth. J 315 Periodontal Res 1978;3:194-98.

316 36. Levers BGH, Darling AI. Continuous eruption of some adult human teeth of ancient 317 populations. Arch Oral Biol 1983;28:401–408.

318 37. Compgnon D, Woda A. Supra-eruption of the unopposed maxillary molar. J 319 Prosthet Dent 1991;66:29-34.

320 38. Danenberg PJ, Hirsch RS, Clarke NG, Leppard PI, Richards LC. Continuous tooth 321 eruption in Australian aboriginal skulls. Am J Phys Anthropol 1991;85:305-312.

322 39. Kiliardis S, Lyka I, Friede H, et al. Vertical position, rotation, and tipping of molars 323 without antagonists. Int J Prosthodont 2000;13:480-86.

324 40. Lavigne GL, Rompre PH, Poirier G, et al. Rhythmic masticatory muscle activity 325 during sleep in humans. J Dent Res 2001b;80:443-48.

326 41. Lavigne GL, Khoury S, Abe S, et al. Bruxism physiology and pathology: An overview 327 for clinicians. J Oral Rehab 2008;35:476-94.

328 42. Po JMC, Gallo LM, Michelotti A, Farella M. Comparison between rhythmic jaw 329 contractions occurring during sleep and while chewing. J Sleep Res 2013;22:593-330 99.

331 Authors’ contribution: S.D.M., K.K., R.G.F., and T.E.S. have contributed equally to 332 this work. S.D.M. contributed conceptualization, supervision, validation, visualization, 333 original draft preparation. K.K. contributed data curation, formal analysis, investigation, 334 software, validation, review & editing. R.G.F. contributed conceptualization, validation, 335 visualization, review & editing. T.E.S. contributed conceptualization, methodology, 336 resources, supervision, validation, visualization, review & editing.


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Development of the mandibular curve of Spee and maxillary ...87 and mandibular movement during function. 88 The purpose of this investigation is to test the hypothesis, using a simplified