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Three dimensional computer modeling of human mandibular biomechanics Nelson, Gregory J. 1986

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THREE DIMENSIONAL COMPUTER MODELING OF HUMAN MANDIBULAR BIOMECHANICS BY GREGORY J . NELSON B . S c , The U n i v e r s i t y o f B r i t i s h Columbia, 1978 D.M.D., The U n i v e r s i t y o f B r i t i s h Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN THE FACULTY OF GRADUATE STUDIES (Department o f O r a l B i o l o g y ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d Dr. J.M. G o s l i n e Dr. A.G. Hannam ( S u p e r v i s o r ) Dr. A.A. Lowe Dr. C.E. S l o n e c k e r i Dr. W.W. Wood The U n i v e r s i t y o f B r i t i s h Columbia November 1986 ©GREGORY J . NELSON, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Department of DE-6 (3/81) - i i -ABSTRACT Previous analyses of mandibular biomechanics have incorporated a wide v a r i e t y of approaches and v a r i a b l e s in attempts at desc r i b i ng the re la t ionsh ips between the forces generated by the muscle and the forces of resistance at the dent i t ion and temporomandibular j o i n t s . The most d i f f i c u l t element to determine in man has been the role of the j o in t forces which require ind i rec t analyses. A c r i t i c a l l i t e ra tu re review points out the problems associated with previous analyses of mandibular mechanics and predict ions of j o i n t loading and the need for the incorporation of a l l relevant anatomical and physiological parameters in order to r e a l i s t i c a l l y quantify these re la t ionsh ips . A computerized mathematical model of human mandibular biomechanics for s ta t i c functions i s presented which allows the determination of forces occurring at the dent i t ion and the jo in ts due to the indiv idual muscle force cont r ibut ions. U t i l i z i n g the pr inc ip les of s ta t i c equi l ibr ium the model provides for the determination of these forces for any indiv idual for whom the necessary input parameters have been der ived. Anatomical ly, th is model requires the designation of the three dimensional coordinates of the or ig in and inser t ion points of nine pairs of masticatory muscles, any posi t ion of tooth contact, and the temporomandibular j o i n t pos i t ions . Determination of the forces generated by the indiv idual muscle groups, and therefore the overal l muscle force resul tant acting on the system, i s given by the product of a number of physiological parameters. These include the physiological c ross-sec t ion , the i n t r i n s i c force per unit of cross-sect ional area, and the re la t i ve act ivat ion level of each muscle for the spec i f i c s ta t i c funct ion. Also required i s the three dimensional or ientat ion of tooth resistance force at the designated posi t ion of tooth contact, as well as that of the l e f t j o i n t force in the frontal plane. This information reduces the var iables in the equi l ibr ium equations to a determinate number which has a s ingle unique solut ion for each of the tooth and two j o i n t resistance forces. The magnitudes as well as three dimensional or ientat ions of the resul tant vectors of the muscles, the tooth resistance fo rce and the two temporomandibular j o i n t s are thereby determined mathematically. Both b i l a t e r a l l y symmetrical and un i la tera l clenching functions as well as three in terva ls near the intercuspal posi t ion of chewing were tested with th is model using data derived from l i t e ra tu re sources from real subjects. This data was incorporated into a hypothetical average individual data f i l e . Using th is data, der ivat ion of the magnitudes and or ientat ions of muscle and tooth forces were made providing predict ions as to the nature of temporo-mandibular j o i n t loading for th is i nd i v i dua l . The extent of muscle force generated for s ta t i c maximal clenching tasks modeled was a maximum of 1000 to 1200 N during intercuspal c lenching. The or ientat ion of muscle force with respect to the occlusal plane varied from about 90 degrees in the la tera l plane, for more poster ior molar funct ions, to - i v -64 degrees for i nc i sa l funct ions. Maximal tooth resistance forces were around 500 to 600 N at the molars versus only 130 to 140 N at the i nc i so r s . Uni la tera l functions showed the working side j o in t to be more heavily loaded than the balancing side espec ia l ly for a more poster ior function ( i . e . molar). Less muscle and therefore tooth force was produced un i l a te ra l l y but with the benef i t of even less residual j o i n t force. Thus, un i la tera l functions appear to be much more e f f i c i e n t in terms of the d is t r ibu t ion of forces between the dent i t ion and j o i n t s . Var iat ion in tooth or ientat ion produced var ia t ions in both the or ientat ion and magnitudes of the j o i n t forces exh ib i t ing a functional in te r re la t ionsh ip of these forces. Based on the analysis in general, the jo in ts were predicted to be capable of res is t ing up to 300 N of force per side directed anterosuperior ly at about 60 to 100 degrees in the la tera l plane. More divergent forces at the jo in ts were found to be of substant ia l ly lower magnitude in the la te ra l and frontal planes. These f indings are in good agreement with other s tudies. - V -TABLE OF CONTENTS PAGE ABSTRACT i i TABLE OF CONTENTS v LIST OF FIGURES v i i i LIST OF TABLES x ACKNOWLEDGEMENTS xi INTRODUCTION 1 A. MANDIBULAR BIOMECHANICS AND MODELING - LITERATURE REVIEW . . . . 2 1. Two Dimensional Models 5 a . Non-Lever Models 5 b. Lever Models 14 c . Lever-Link Considerations 21 d. Role of Craniofacia l Form 24 e. Role of Muscle Forces 26 2. Three Dimensional Models 33 3. Current Modeling Approaches 48 B. PURPOSE OF STUDY 57 METHODS 59 A. PRINCIPLES OF ANALYSIS 59 B. DATA ENTRY 1. Anatomical Variables a . Coordinate System 62 b. Muscle Attachments 63 c. Tooth Posi t ions 67 d. Tooth Angles of Resistance 68 e. Condylar Pos i t ion 70 f. Condylar Angles of Resistance 70 2. Physiological Variables 71 a. Weighting C r i t e r i a 72 b. Scal ing C r i t e r i a i . Clenching Tasks 77 i i . Chewing 80 C. COMPUTER ANALYSIS 82 1. Lateral Plane - Step 1 83 2. Frontal Plane - Step 2 87 3. Horizontal Plane - Step 3 90 4. Constraints On Muscle Resultants 93 D. COMPUTERIZED ANATOMICAL RECONSTRUCTION 97 E. PROGRAM DESCRIPTION 100 F. PROGRAM USE (IN THIS STUDY) , 107 RESULTS 109 A. DESCRIPTION OF FIGURES AND ABBREVIATIONS 110 1. Figures 10 to 18 "A" 110 2. Figures 10 to 18 "B , C, D", etc 112 a. "MUSCLE RESULTANT PARAMETERS" 113 b. "LAT. PLANE RESULTANT VECTOR ORIENTATIONS" (DEGREES) . 114 c. "RESULTANT VECTOR MAGNITUDES" (NEWTONS) 115 - v i -P A G E d. "FRONT. PLANE RESULTANT VECTOR ORIENTATIONS" (DEGREES) . 115 e. "JF/TF" 115 B. FORCE VECTOR ANALYSIS 1. B i l a t e r a l l y Symmetrical Clenching Tasks a. Intercuspal Clenching (CO) - Figures 10A, B and C i . Muscle Resultant Parameters 117 i i . Tooth Posi t ion Change - Figure 10B 117 i i i . Lateral Tooth Angle (LTA) Changes - Figure IOC . . 122 b. B i la te ra l Molar Clenching (BIMOL) - Figures 11A, B and C i . Muscle Resultant Parameters 125 i i . Lateral Tooth Angle (LTA) Change - Figure 11B . . . 125 i i i . Tooth Posi t ion Change - Figure 11C 129 c. Incisal Clenching With Bi te Stop (INCISS) - Figures 12 A and B i . Muscle Resultant Parameters 132 i i . Lateral Tooth Angle (LTA) Change - Figure 12B . . . 132 d. Incisal Clenching On Natural Contacts (INCISN) - Figures 13A and B i . Muscle Resultant Parameters 137 i i . Lateral Tooth Angle (LTA) Change - Figure 13B . . . 137 2. Uni latera l Clenching Tasks a . Uni latera l Canine Clenching (UNIK9) - Figures 14A, B, C and D i . Muscle Resultant Parameters 142 i i . Lateral Tooth Angle (LTA) Change - Figure 14B . . . 144 i i i . Frontal Tooth Angle (FTA) Change - Figure 14C . . . 147 i v . Left Condyle Frontal Angel (LCFA) Change - Figure 14D 154 b. Uni latera l Molar Clenching (UNIMOL) - Figures 15A, B, C, D and E i . Muscle Resultant Parameters 156 i i . Tooth Posi t ion Change - Figure 15B 157 i i i . Lateral Tooth Angle (LTA) Change - Figure 15C . . . 163 i v . Frontal Tooth Angle (FTA) Change - Figure 15D . . . 166 v. Left Condyle Frontal Angle (LCFA) Change - Figure 15E 170 3. Uni la tera l Chewing Tasks a . Muscle Resultant Parameters - Figures 16A, 17A and 18A . 173 b. Lateral Tooth Angle (LTA) Change - Figures 16B, 17B and 18B 179 c. Frontal Tooth Angle (FTA) Change - Figure 16C, 17C, and 18C 184 C. SUMMARY 1. B i l a t e r a l l y Symmetrical Clenching Tasks 189 2. Uni la tera l Clenching Tasks 191 3. Uni la tera l Chewing Tasks 196 DISCUSSION 199 A. MUSCLE FORCES 200 B. TOOTH FORCES 209 C. JOINT FORCES 225 - vii -PAGE D. STRENGTHS AND WEAKNESSES OF THE CURRENT MODEL 238 E. FUTURE DIRECTIONS 240 REFERENCES 242 APPENDIX I 252 APPENDIX II 257 - v i i i -LIST OF FIGURES PAGE Figure 1 Robinson's Parallelogram of Mandibular Forces 7 Figure 2 "Link" Concept of Mandibular Function as Proposed by Gingerich 13 Figure 3 The Mandible as a Stationary Beam According to Smith (1978) 38 Figure 4A Lateral Plane Attachment Points of the Nine Muscle Groups 65 4B Frontal Plane View of the Muscle Groups of Figure 4A 66 Figure 5 Lateral (yz) Representation of Force Components 86 Figure 6 Frontal (xz) Representation of Force Components 89 Figure 7 Horizontal (xy) Representation of Force Components 92 Figure 8 Determination of the Orientat ion of the Muscle Force (ANG) in Each Plane 96 SYSTEM CHART 105 Figure 9 Computer Reconstruction of Anatomical Var iables I l l Figure 10A Intercuspal Clenching (CO) 118 10B Intercuspal Clenching (CO)/Variable Tooth Posi t ion 121 IOC Intercuspal Clenching (CO)/Variable LTA 123 Figure 11A B i la te ra l Molar Clenching (BIMOL) 126 11B B i l a te ra l Molar Clenching (BIMOLl/Variable LTA at F i r s t Molar 128 11C B i la te ra l Molar Clenching (BIMOL)/Van'able LTA at Second Molar 131 Figure 12A Incisal Clenching with Bi te Stop (INCISS) . 133 12B Incisal Clenching with Bite Stop (INCISS)/Variable LTA • • • 1 3 5 Figure 13A Incisal CIenching-Natural (INCISN) 138 13B Incisal CIenching-Natural (INCISN)/Variable LTA 140 - i x -PAGE Figure 14A U n i l a t e r a l (Right Side) Canine Clenching (UNIK9) 143 14B U n i l a t e r a l (Right Side) Canine Clenching (UNIK9) V a r i a b l e LTA 146 14C U n i l a t e r a l (Right Side) Canine Clenching (UNIK9) V a r i a b l e FTA 151 14D U n i l a t e r a l (Right Side) Canine Clenching (UNIK9) V a r i a b l e LCFA 155 Figure 15A U n i l a t e r a l (Right Side) Molar Clenching (UNIMOL) 158 15B U n i l a t e r a l (Right Side) Molar Clenching (UNIMOL) V a r i a b l e Tooth P o s i t i o n 162 15C U n i l a t e r a l (Right Side) Molar Clenching (UNIMOL) V a r i a b l e LTA 165 15D U n i l a t e r a l (Right Side) Molar Clenching (UNIMOL) Va r i a b l e FTA 169 15E U n i l a t e r a l (Right Side) Molar Clenching (UNIMOL) V a r i a b l e LCFA 171 Figure 16A In t e r v a l 1 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW1) 176 Fig u r e 17A In t e r v a l 2 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW2) 177 Figure 18A Int e r v a l 3 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW3) 178 Figure 16B In t e r v a l 1 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW1)/Variable LTA 180 Figure 17B In t e r v a l 2 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW2)/Variable LTA 181 Figure 18B In t e r v a l 3 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW3)/Variable LTA 182 Figure 16C In t e r v a l 1 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW1)/Variable FTA 185 Figure 17C Int e r v a l 2 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW2)/Variable FTA 186 Figure 18C In t e r v a l 3 of U n i l a t e r a l (Right Side) Chewing Power Stroke (CHEW3)/Variable FTA 187 Figure 19 E f f e c t of Increasing V e r t i c a l Dimension on Tooth Force 216 - X -L I S T OF TABLES PAGE T a b l e I Weighting Factors . . . . 76 T a b l e I I Scal ing Factors (Clenching) 79 T a b l e I I I Scal ing Factors (Chewing) 81 T a b l e IV Tooth and Jo in t Coordinates . . . . . 98 T a b l e V Muscle Attachment Coordinates 99 T a b l e V I Summary of Muscle Resultant Parameters 202 T a b l e V I I Summary of Resultant Tooth Resistance Force Parameters . 211 T a b l e V I I I Previous Incisal Bi te Force Determinations 214 T a b l e IX Previous Uni lateral Molar Bite Force Determinations . . . 220 T a b l e X Previous Uni lateral Canine and Premolar Bite Force Determinations 223 T a b l e X I Summary of Resultant Jo in t Resistance Force Vectors . . . 227 - x i -ACKNOWLEDGEMENTS I am great ly appreciat ive of the e f for ts and expert ise provided by Ms. Joy Scott in a l l aspects of the computer programming and hardware appl icat ion used in th is study. Mr. R.E. DeCou provided very valuable expert ise in numerous ways but espec ia l l y regarding the appl icat ion of s ta t i c equi l ibr ium theory to th is type of ana lys i s . Dr. B i l l Wood's comments and suggestions great ly contr ibuted to the improvement of the text as did those of Ms. Linda Ski bo without whose assistance the smooth completion of the substantial work involved with compiling th is thesis would not have been poss ib le . F i n a l l y , I am great ly indebted to Dr. Alan Hannam for his patience and continued enthusiasm, not to mention his assistance in every aspect of th is work. Thank you a l 1 ! - 1 -INTRODUCTION The basic mechanical components which comprise the masticatory apparatus cons is t of the den t i t i on , the muscles of the jaws, and the temporomandibular j o i n t s . Depending upon the nature of an i nd i v idua l ' s c ran io fac ia l form, and resu l t ing anatomical re la t ionsh ips , i t would seem l i k e l y that cer ta in biomechanical associat ions would also ex is t between the forces generated by the muscles, and the opposing forces of resistance which occur at the jo in ts and teeth. Since the re lat ionships between these components determine the biomechanical nature of the system, they would be expected to be funct iona l ly interdependent. As such, any changes or disturbances ( su rg i ca l , traumatic or pathological) to one component would potent ia l l y inf luence another. Evidence for th is i s provided by c l i n i c a l observations of patients with functional disturbances of the masticatory system. It is apparent, that the mechanical components of the masticatory system are int imately related from a functional and dysfunctional point of view. Hence, cer ta in associat ions of form (which re late the muscles, the j o i n t s , and the dent i t ions) and function must also ex i s t . Determination of the normal functional re la t ionships between these components w i l l great ly aid in our understanding, and treatment of dysfunctional states involv ing abnormal biomechanical re la t ionships of the system. Although some components of th is system are amenable to d i rec t functional study (eg. electromyography, kinesiography, transducer load ing, e t c . ) , others, in par t i cu la r the loads borne by the j o i n t , are not. Here - 2 -i nd i rec t modeling approaches are attempted, at leas t in man, and usual ly involve deductive techniques based upon the pr inc ip les of s ta t i c mechanics. A. MANDIBULAR BIOMECHANICS AND MODELING - LITERATURE REVIEW Over 60 years ago Al f red Gysi (1921) stated that two questions pertaining to mandibular ac t i v i t y which had "been long and sometimes b i t t e r l y discussed" were, (1) which c lass of lever was represented by the mandible during hard c lenching, and, (2) what was the extent of the forces produced at the teeth and temporomandibular condyles. Apparently many invest igators of the time were of the opinion that the mandible was one of nature's engineering f a i l u r e s . Gysi at t r ibuted th is to the persistence of s tudies, up to that t ime, which considered only the working side of the mandible. The balancing side inf luences of muscle a c t i v i t y , "were e i ther overlooked or assumed to be the same as those in the working ha l f . " As such each side of the mandible seemed to act as a separate c lass III lever and thus appeared to be very i n e f f i c i e n t . Gysi (1921) further stated that around 1918 a new school of thought had emerged proposing that the mandible was not a lever at a l l and the ent i re force of the muscle was d is t r ibuted en t i re ly through the teeth. This concept has persisted unt i l f a i r l y recently (eg. Ginger ich, 1971 and 1979) and has become known as the " l i nk " theory whereby the mandible i s suspended in a s l i ng of muscles and merely l i nks the muscle force to the force at the teeth. Gysi a lso considered th is view to be a resu l t of overly s imp l i s t i c assumptions. - 3 -Gys i ' s own analysis (1921) was based on the fol lowing statement, "any a r t i cu la t ing j o in t may be f ixed by muscle action at any stage of i t s possible motion and become jus t as much a fulcrum for lever action as i f i t were a r i g i d anatomical s t ructure. " Leverage analyses had predicted that from one-third to two-thirds of the muscle force exerted was not e f fec t ive at the teeth, which had contributed to the opinion of mandibular i ne f f i c i ency . Gysi at t r ibuted th is " l os t " force as that "required to f i x the condyles, as fulcrums, upon the inc l ined and lubr icated surfaces of the eminentia." U t i l i z i n g both a mathematical der ivat ion and a mechanical model he showed that during the un i la tera l crushing of food, the two sides of the mandible acted as two c lass III levers connected at the symphysis. The resu l t of th is was such that the fulcrum producing ef fect of the balancing side muscles on the condyle would act through the symphysis to contr ibute s i gn i f i can t l y to the crushing force on the working s ide , as well as reduce the force at the working condyle due to the fulcruming action of the muscles pu l l ing on that s ide . Therefore, the resistance forces at the teeth and condyles would be inversely re la ted . This three dimensional analysis produced a much more e f f i c i e n t and complete picture than that of the two dimensional considerat ions. According to his theory, un i la tera l crushing of a hard object more anter ior ly along the dent i t ion would resu l t in a re la t i ve increase in working side condyle force. His analysis also showed that th is force would be "neut ra l ized" at the second molar region. Accordingly, at the th i rd molar posi t ion hard food could not be crushed since the fulcrum - 4 -reducing force acting on the working condyle would be negative and of such magnitude as to tear the ligaments and disclude the j o i n t s . The force at the balancing condyle Gysi determined to be constant, regardless of the posi t ion of the working side b i te point , and equivalent to one th i rd the total force app l ied. In a l l instances th is force was greater than that of the working side condyle. In deriving these re la t ionships Gysi incorporated remarkably deta i led anatomical parameters for his time including the three dimensional posi t ions of the den t i t i on , the condyles and the points of appl icat ion and or ientat ion of the muscles. He derived values for the cross sectional areas of each of the muscle groups and determined the i r maximum force c a p a b i l i t i e s . It i s worth noting here that, according to thei r alignment of f i be r s , he subdivided the muscles into d i s t i n c t groups consist ing of super f ic ia l and deep masseter, medial pterygoid, an ter io r , middle and poster ior temporal is, and superior and i n f e r i o r la te ra l pterygoids. The only information which was unavailable to him were the re la t i ve a c t i v i t i e s of each muscle group which would not be measurable for another t h i r t y years. He therefore assumed a l l muscles to be act ive to the same extent, despite the point of tooth resistance used, since no one knew otherwise at the time. Due to the complexity of factors involved, the i r in te rp lay , and the necessity of i nd i rec t assessments, many of the analyses of mandibular mechanics since Gys i ' s work exh ib i t widely d i f fe r ing approaches to the determination of the functional re la t ionships which e x i s t . By necessity they a l l share the assumption of s ta t i c equ i l ib r ium, but that i s where the s i m i l a r i t y ends. Unfortunately, they have for the most part been very - 5 -Ifmi ted in the i r consideration of the mult ip le factors which can become involved in jaw mechanics depending upon the funct ion. Most have overs impl i f ied the var iables and/or have been res t r i c ted to two dimensions only (sag i t ta l representat ions). As a consequence many erroneous or misleading conclusions and interpretat ions have a r i sen . These d i f fe rent points of view have continued to fuel arguments over which fac to rs , or re la t ionsh ips between fac tors , are most important. Much of the controversy which has persisted concerns whether the mandible acts s t r i c t l y as a l eve r , with some of the functional load produced by the muscles taken up by the j o i n t s , or whether i t functions as a so-ca l led " l i n k " . There has even been some discussion among lever proponents as to what type of leverage system is at work ( i e . c lass I, I I , I I I , or a combination) (Davis, 1955; Turnbul l , 1970; Smith, 1978). 1. Two Dimensional Models a. Non-Lever Models At about the same time as Gysi proposed his models G.H. Wilson (1920 and 1921) was refut ing the lever act ion theory basing his ideas on somewhat simple qua l i ta t i ve analyses of the or ientat ions and combined action of the masseter and temporalis muscles. He concluded that the resul tant muscle force acted at r ight angles to the occlusal plane, and for th is reason, "the whole force of these muscles i s expended upon the bolus of food and not any port ion of i t upon the condyle" (Wilson, 1920). Although he made no d i s t i nc t i on between un i la tera l or b i l a te ra l functions Wilson implies that - 6 -th i s p r inc ip le holds true for a l l cases. However, i t would not be enough for the resul tant to be merely oriented perpendicular to the occlusal plane, as Wilson suggested. I t must pass d i rec t l y through the b i te point as we l l , (Roydhouse, 1955; Hylander, 1975). Some years l a te r Robinson (1946) determined a resul tant force posi t ion for the masseter/medial pterygoid and temporalis muscles in the sag i t ta l plane and concluded that 1t d i d , in f ac t , pass through the molar teeth. Hylander (1975) argued tha t the anatomical r e l a t i o n s h i p s Robinson incorporated in his analyses were erroneous since he considered only selected muscle f i be r alignments and disregarded re la t i ve muscle s i z e , and hence, force c a p a b i l i t i e s (see Figure 1) . Hylander fur ther pointed out that the contr ibut ion of the anter ior temporalis had been underestimated and that of the masseter-medial pterygoid complex posit ioned too far anter ior re la t i ve to. the teeth. The posi t ion of the dental arch, which i s c r i t i c a l to Robinson's argument, also appears to be too far poster ior (see also Page, 1954). The muscle force vector could therefore not pass through the dent i t ion (Roydhouse, 1955). The premise to Robinson's theory was that the t issues comprising the j o i n t s were not suited to r es i s t compressive s t resses. This argument has been one of the main reasons why some invest igators have opposed the lever act ion theory and sought al ternate explanations to account for the seeming incompatabi l i ty between lever- type mechanics and h is to log ica l features. Robinson showed that the a r t i c u l a r disc contains synovial t i s sue , blood vesse l s , nerves and lymphatics and i s composed of f i b roca r t i l age . Since none of these t issues are charac te r i s t i c of the stress-bear ing jo in ts found - 7 -Figure 1. ROBINSON'S PARALLELOGRAM OF MANDIBULAR FORCES. The muscle forces are those due to the temporalis (T) , masseter and medial pterygoid componen ts . R r e p r e s e n t s the resul tant of these components and passes through the tooth row (After Robinson, 1946). between some long bones of the body Robinson argued for the s t ress- f ree function of the temporomandibular j o i n t . He assumed that the point of any a r t i cu la t i on would be within the glenoid fossa of the j o i n t which has a very th in bony roof and would, by i t s e l f , be incapable of bearing any compressive load. However the actual stress bearing region of the j o in t i s not within the fossa, but between the f ib rocar t i lag inous surfaces of the condyle and the opposing surface of the a r t i cu l a r eminence, which i s supported by a r e l a t i ve l y thick cor t i ca l plate of bone (Oberg and Car lsson, 1979; Carlsson and Oberg, 1979). This area of a r t i cu la t i on corresponds with an area of the intervening disc which i s avascular and completely lacks both a synovial - 8 -l ayer , and nerves (Rees, 1954; Hylander, 1975). It i s the poster ior aspect of the disc opposing the roof of the fossa which contains the non-stress elements (Hylander, 1975). The f i b rocar t i l age covering the a r t i cu l a r surfaces of the temporomandibular j o i n t s , as opposed to the a r t i cu la r hyaline ca r t i l age of stress bearing jo in ts of the body, i s considered by many to possess stress bearing qua l i t i es s imi la r to hyal in ca r t i l age under compression. It has also been at t r ibuted with the a b i l i t y to provide superior resistance to both tens i le and shearing forces (see Hylander, 1975). The h is to log ic di f ferences between the temporomandibular j o in t and the other stress bearing jo in ts of the body may be related more to the embryol ogical di f ferences between long bones and the mandible. Long bones are derived from car t i lag inous precursors via epiphyses, whereas the mandible ar ises from membrane bone. Thus the lack of epiphyseal development in the mandible may re f l ec t developmental, rather than functional di f ferences (Barbenel, 1969; Oberg and Car lsson, 1979). Tat te rsa l l (1973) also bel ieved the lever action hypothesis incorrect on the grounds that the t issues of the j o i n t , as well as the condylar neck, were inadequate to support the large reaction forces which would occur. The inherent ine f f i c iency of such a system would not have survived natural se lect ion pressures and would not have evolved according to him. It may be noteworthy, however, that T a t t e r s a l l ' s conclusions were based on analyses of the masticatory apparatus of an ext inct group of primates. Hylander (1975) tested the hypothesis that the condylar neck i s too weak to support the forces ar is ing (shearing and bending) in man which he reasoned to be greatest during inc isa l b i t i n g . Using a dried human mandible, known - 9 -muscle a c t i v i t i e s (Mol ler , 1966) and l i nes of action (Schumacher, 1961), and the mechanical propert ies of bone (Alexander, 1965) Hylander determined the condylar neck to be capable of withstanding an average of at least 238 kg of shearing s t ress , based on the cross sectional area of compact bone at th is l e v e l . He determined the amount of bending force required to break the condyle during inc i sa l b i t i ng to be at leas t 60 kg per condyle which corresponds to inc isa l b i te forces of 70 kg or greater. Maximum inc i sa l b i te recordings in humans have cons is tent ly shown th is force to be of the order 20-25 kg or less (Linderholme and Wennstrom, 1970; Rugh and Solberg, 1972; Ringquist , 1973; Helkimo et a l . 1975 and 1976; Mansour, 1977; F inn , 1978; Helkimo and Ingerva l l , 1978). Thus the condyle of his analysis was c lea r l y of su f f i c ien t strength to support large reaction forces. N e v e r t h e l e s s , Moyers (1950) i n t e r p r e t e d h i s own p ioneer ing electromyographic work on the functional a c t i v i t i e s of the muscles of masticat ion in support of Robinson (1946). Although Moyers did not test the hypothesis, he speculated that the coordination of the muscles he had observed in mandibular movements would also in teract to el iminate forces at the jo in ts (v ia ref lexes ar is ing in the jo in ts themselves). This implies some sort of combined or "coupl ing" action of one muscle (eg. temporalis) to reduce or e l i m i n a t e the s t r e s s - i n d u c i n g ac t i on of another (eg . masseter-medial pterygoid complex) at the j o i n t . A s im i la r e f fect has been ei ther proposed or implied by Scott (1955), based on qua l i ta t i ve anatomical comparisons of sheep, dog and human; by Davis, (1955), using d issect ions of the spectacled bear, Tremarctos ornatus; by Turnbul l , (1970) in a var iety of - 10 -mammalia with funct iona l ly d i f ferent masticatory schemes; and by Roberts and T a t t e r s a l l , (1974) for the mammalia in general . Roberts (1974), and Roberts and Tat tersa l l (1974) believed th is coupling ef fect rotated the jaw, in both elevat ion and depression, and occurred around the mandibular attachment of the spenomandibular ligament. According to them the jaw would rotate around th is fulcrum within the muscular s l ing of the temporalis/masseter-medial pterygoid complex without the necessity of a condylar fulcrum, and thus no resul t ing j o in t force. The unl ikel ihood of a ligamentous fulcrum point had been exhorted very ear ly in the l i t e ra tu re by Wilson himself (1920, 1921). Smith (1978) has since pointed out that forces which form a true couple are not c o l l i n e a r , are equal in magnitude, and are opposite in d i rec t i on . Thus the "couple" formed by the forces of the temporalis and masseter-medial pterygoid muscles of Roberts and Tat tersa l l (1974) do not sa t i s fy the l a t t e r two requirements and therefore do not const i tute a couple. A lso , the i r analysis only accounts for the rotat ional ef fects due to the anteroposterior components of muscle force. The ef fects of the ver t i ca l components of the i r muscle resu l tan ts , which they have neglected, must be accounted for by some other force somewhere in the system. It fol lows that since the "couple" due to the anteroposterior muscle force "components" must be generating the necessary force at the teeth, by v i r tue of the i r rotat ional ef fects the only other area where a force res is t ing the ver t i ca l component of muscle force can occur i s obviously at the j o i n t . The lever analyses of both Davis (1955) and - 11 -Turnbull (1970), also f a i l to consider the ve r t i ca l components of the i r muscle forces and are subject to the same c r i t i c i sms as Roberts (1974), and Roberts and Ta t te rsa l l (1974). F i n a l l y , Smith (1978) has also c r i t i c i z e d such analyses on the grounds that the muscle "vectors" assigned are purely arb i t ra ry and, as such, provide no basis for any conclusions to be drawn. Roberts (1974) had suggested that since the three "vectors" (masseter-medial pterygoid, temporal is, and b i te point) form a closed t r iang le when added geometr ical ly , there was no addi t ional j o i n t reaction force necessary. Since his suppositions appear to be untrue, so must his conclusions. Although the term "couple" maybe inappropriate in these studies the e f fec t these invest igators were t ry ing to describe i s apparent, and the a b i l i t y of the muscles to cooperate, in at leas t reducing some of the resu l t ing j o i n t fo rces, has been described by a number of other workers (Smith and Savage, 1959; Crompton, 1963; Greaves, 1974 and 1978; Noble, 1979). This matter i s discussed l a t e r . The most s imp l i s t i c analysis of the l ink var ie ty was that of Frankel and Burstein (1970). They considered two muscles only (masseter and temporalis) and assigned apparently arb i t rary vectors to both as well as to the tooth res is tance force. They maintain that since these three "vectors" form a closed t r iang le when geometrical ly added a l l forces in the system are accounted fo r . Vector addit ion such as th is does hold, but only when the vectors are r e a l . As already stated no credib le conclusions can be drawn - 12 -from mechanical analyses incorporating arb i t rary vectors. Hylander (1975) a lso leveled the same c r i t i c i s m at the masseter vectors of th is analys is as that of Robinson (1946) since they are posit ioned too far anter ior ly with respect to the teeth. In 1971 Gingerich coined the term when he proposed his " l i nk " action of the jaw based on a previously unknown masticatory movement which he ca l led "orthal re t rac t ion" (upward and backward) and made the questionable assert ion that the temporalis i s the main adductor of the mandible. He maintained that the alignment of the f ibers of the temporalis were such that the i r anter ior project ion past the coronoid inser t ion formed an envelope which included the ent i re tooth row and thus any potent ial b i te point (See Figure 2 ) . Therefore, an upward and backward e f fo r t by the component of the temporalis al igned with the given b i te point would transmit i t s force to the occlusion v ia the coronoid process rather than through the condyles. The t ranslatory c a p a b i l i t i e s of the jo in ts would be most suited to t h i s . Such a system, according to Ginger ich, would be much more e f f i c i e n t than the lever system as no "wasted" force would be necessary at the j o i n t . He also f e l t i t to be more reasonable than the non-lever system of Robinson (1946) since the or ientat ion of the temporalis would not need to be as far back on the cranium ( i e . corresponding to the poster ior port ion of the muscle) as in Robinson's proposal. - 13 -Figure 2 . "LINK" CONCEPT OF MANDIBULAR FUNCTION AS PROPOSED BY GINGERICH. The alignment of the temporalis muscle f ibers projected anter ior ly enclose the ent i re dent i t ion . It was therefore suggested that for any bi te point the temporalis could account for the generated bi te force without necessar i ly invo lv ing the temporomandibular j o i n t s . The mandible would thus act s ingly as a l ink between the two forces (Based on Ginger ich, 1971). The obvious l im i ta t i on with th is system is Ginger ich 's neglect of the other muscles which, he, himself , states "are not al igned with any bi te point" and " the i r force of contract ion i s divided between useful bi te force and wasted react ion force at the jaw j o i n t . " It would therefore seem more appropriate to in terpret th is system as a lever with a means of increasing i t s own e f f i c iency by containing a " l i nk " e f fect which could reduce the "wasted" force occurring at the j o i n t s . However, as Hylander (1975) convincingly shows, neither the premise of "orthal re t rac t ion" type movements, nor the necessary iso la ted a c t i v i t i e s of the temporal is, are supportable. In each of these non- lever t heo r i es the l i m i t a t i o n s of t h e i r general izat ions (eg. two dimensional) and often qua l i ta t i ve and/or erroneous analyses in explaining the actual events involved with mandibular function - 14 -are obvious. More importantly, they serve as examples of the fa l lacy in ascr ib ing too much importance to too few var iab les . b. Lever Models The lever action postulate has generally been more widely accepted (eg. Hylander, 1975; Smith, 1978) as a f i r s t approach to understanding mandibular mechanics. The fulcrum ef fect at t r ibuted to the condyles has formed the basis of a var iety of analyses aimed at explaining these mechanics in man as well as numerous other creatures (eg. Ostrom, 1964). Simple c lass III lever ana lys is , in the sag i t ta l plane, assumes a fulcrum point at the jo in ts and predicts a l i near increase in bi te force as the occlusal contact point moves poster ior ly along the dent i t ion , given a constant applied force (Mainland and H i l t z , 1933; Gosen, 1974). This i s because the length of the moment arm from the fulcrum decreases with more poster ior contacts. Since the applied load of the muscle remains constant larger poster ior tooth loads are necessary to produce the same moment about the fulcrum which must balance, or oppose that due to the muscle force. The morphological di f ferences between the anter ior and poster ior teeth re f l ec t the need for greater d is t r ibu t ion of occlusal loads pos ter io r ly . Based on purely morphological analyses i t has been shown that the consistent s i m i l a r i t i e s in the jaw forms of carnivores and in those of herbivores, seems, in large part , to be based on dif ferences in general lever mechanics which have evolved in response to the nature of the given animal's food type (Becht, 1953; Smith and Savage, 1959; Barghusen, 1972; Scapino, 1972; DuBrul, 1974). Carnivores, for instance, a l l have in common - 15 -r e l a t i ve l y large temporal versus masseter-medial pterygoid muscles, a condyle close to the level of the tooth row, a t a l l coronoid process with respect to the condyle and tooth row, and a v i r t u a l l y hingel ike a r t i cu la t i on of the jaw (Smith and Savage, 1959; Dubrul, 1974). Smith and Savage (1959) have suggested that the alignment and re la t i ve l y large s ize of the temporalis muscle (unl ike the masseter-medial pterygoid complex) would prevent d i sa r t i cu l a t i on of the jo in ts by a struggl ing prey pu l l ing in the opposite d i rec t ion as would the an teropos ter io r^ r i g i d j o i n t a r t i c u l a t i o n . It has also been pointed out that both the t a l l coronoid process (Ostrom, 1964) with a r e l a t i ve l y long moment arm for the large temporal is, and the low posi t ion of the condylar fulcrum re la t i ve to the tooth row* are adaptations which could produce a powerful ( i e . bone-crushing) bi te (Smith and Savage, 1959; Barghuson, 1972; Crompton and Hiiemae, 1969). This arrangement would also produce a large downward and backward force at the condyle tending to cause d i sa r t i cu l a t i on which could be reduced (but not eliminated) by synerg is t ic a c t i v i t y of the masseter muscle pu l l i ng upwards and forwards (Smith and Savage, 1959; Greaves, 1974; Nobel, 1979). *Greaves (1974) has taken exception to the s ign i f icance of low condylar pos i t ion and has argued that adaptive changes in the ver t i ca l posi t ion of the condyle with respect to the tooth row, may not e f fec t any change in mechanical advantage of the system as a whole. He bases th is on the assumption that both the masseter and temporalis muscles are equally important in s ta t i c mechanical function of the organism. According to Greaves, an elevated condyle in the herbivore would reduce the moment arm, and mechanical advantage, of the temporalis and increase that of the masseter. However, since the usual ly un i la tera l nature of herbivorous masticat ion requires that the adaptive changes favor the act ion of the working side masseter and medial pterygoid in order to maximize the t ransfer of muscle force to the teeth (Smith and Savage, 1959) the reduction in temporalis ef fect iveness may be less important. In fact the re la t i ve size of t h i s muscle i s great ly reduced compared to carnivores which have very d i f fe ren t working versus balancing side requirements. - 16 -Further evidence for the l a t t e r point is provided by morphological observations of ancient mammal-like rep t i l es (eg. cynodonts) which exhib i t a p r o g r e s s i v e r e d u c t i o n i n s i z e and s t r e n g t h of the j a w - j o i n t (dentary-squamosal j o i n t at th is point in evo lu t ion) , accompanied by an increase in mass of the adductors and presumably also a subsequent increase in jaw j o in t react ion force (Crompton, 1963; Parkyn, 1963; Barghusen and Hopson, 1970; DuBrul , 1974; Noble, 1979). This apparent paradox, i s bel ieved to be due to the concurrent reor ientat ion of the muscle forces with changes in muscle attachment po ints . Reorientation would produce the moderating e f fec t of the temporalis/masseter-medial pterygoid synergism described above, which reduces j o i n t forces concurrent with increasing muscle and tooth forces. This e f fect has been correlated with the appearance of "molariform" teeth capable of withstanding the increased tooth forces (Crompton, 1963). Bramble (1978) took the two dimensional lever model a step further in his explanation of evolutionary trends and jo in t forms in mammalian feeding complex. He proposed a " b i f u l c r a l " model which considers the b i te point , as well as the jaw j o i n t , to act simultaneously as fu lc ra in the system. Analyzing the rotat ional ef fects of the muscle vectors in th is way Bramble showed that there is a secondary moment produced at the j o i n t due to rotat ional ef fects at the b i te point fulcrum. This i s in addit ion to the primary moment produced at the b i te point due to rotat ion about the j o in t fulcrum. This secondary moment at the j o in t requires a resistance force to maintain equ i l ib r ium. Bramble showed that the magnitude of th is j o i n t force due to the rotat ional ef fects at the b i te fulcrum varies with the posi t ion of the b i te point , and or ientat ion of the muscle vector considered. His model - 17 -suggests that each muscle w i l l have a spec i f i c pos i t ive or negative loading e f fec t at the j o i n t depending on the locat ion of the tooth contact an te ropos te r i o r ^ . A lso , depending on i t s o r ien ta t ion , each muscle has a theoret ical "neutral point" somewhere along the tooth row where i t would not generate any rotat ional forces at the j o i n t by i t s e l f . Only t rans la t iona l forces are present. Rased on t h i s , Bramble suggested that when the temporalis and masseter muscle vectors are considered together the i r net e f fec t may be to minimize the j o in t loads through functional synergism. He concluded that the jaw jo in ts of early mammal-like rep t i l es were not subject to heavy compressive loading but more l i k e l y underwent smal l , neu t ra l , or even s l i gh t l y tens i le loads. Bramble showed how many of the evolutionary changes observed in the jaw form of cynodont therapsids (so-ca l led mammal-like rep t i l es ) such as development of a coronoid process, elaborat ion of the super f ic ia l masseter muscle, rearward growth of the condylar process, and development of the re t roa r t i cu la r process (also present in extant carnivores) could be explained according to th is model. As he also points out, the l i ke l i hood of d i f f e ren t i a l a c t i v i t i e s of the muscle groups and the resu l t ing var iety of muscle l i nes of ac t i on , and subsequent i n te rac t i on , would add to the potential combinations of function and therefore form. Both Craddock (1951) and Roydhouse (1955) attempted to provide a general mathematical solut ion to the analysis of the j o in t function in humans based on essen t ia l l y qua l i ta t i ve analyses using simple c lass III lever mechanics. Craddock l im i ted his analysis to a s ingle ver t i ca l component of assumed muscle force in the sag i t ta l plane and determined the di f ference in - 18 -j o in t force for an inc isor versus molar b i te point . Roydhouse, on the other hand considered both the horizontal and sag i t ta l plane as well as d i f ferent d i rec t ions of pul l between the various muscles (super f i c ia l and deep masseters, medial pterygoid, d i f fe rent portions of the temporal is, and la te ra l pterygoid). He assumed the net force of the muscles to be within a zone somewhere between the two coronoid processes poster ior to the anter ior border of the ramus, and perpendicular to the occlusal plane. He stated that under equi l ibr ium condi t ions, "the l i nes of act ion of the purely ver t i ca l resul tants of muscular action and food resistance [at the teeth] cannot co inc ide , unless food i s chewed on the rami." Since these forces are not co l l i nea r a th i rd ver t i ca l force is required to maintain equi l ibr ium which Roydhouse believed must l i e at the j o i n t s , thus generating a resistance force on one or both condyles. Pr io r to th is both Moyers (1950) and Carlsoo (1952) had applied the new techniques of electromyography to the masticatory muscles and establ ished the i r d i f fe ren t ia l nature of ac t i v i t y and contr ibut ions to various movements. Roydhouse showed very genera l ly , that for d i f f e ren t ia l ac t i v i t y in the r ight and l e f t side muscles the posi t ion of a s ingle ver t i ca l resul tant in the horizontal plane would l i e c loser to the side of greatest e f f o r t . Thus the condyle resistance force would be expected to be unequal on the two s ides . The analyses of both Craddock and Roydhouse have been c r i t i c i z e d for having cons idered only v e r t i c a l f o rces and i gno r ing " h o r i z o n t a l " (anteroposterior) components. Barbenel (1969, 1972, 1974) pointed out that the combination of both ver t i ca l and horizontal components of muscle force could produce a net resul tant which might cross the tooth region or zone - 19 -(Robinson, 1946). Since only the magnitude of tooth forces and not the i r or ientat ions had ever been measured, Barbenel (1969) suggested that the simultaneous muscle and tooth force resul tant vectors, for a spec i f i c funct ion, could theore t i ca l l y be c o l l i n e a r . In such cases l i t t l e or no load would ex is t at the j o i n t , in the sag i t ta l plane at l eas t . He therefore presented a two dimensional ( i . e . b i l a t e r a l l y symmetrical) sag i t ta l analysis of jaw function for which the l i nes of action and moment arms (measured from a fulcrum assumed to be at the condyles*) of the masseter, temporal is, medial and la te ra l pterygoid muscles had been determined by d issec t ion . Assuming s ta t i c equi l ibr ium condit ions to ho ld, Barbenel postulated that three simultaneous l i nea r equations would ex is t such that , (1) the sum of the ve r t i ca l force components; (2) the sum of the anteroposterior force components, and; (3) the sum of the moments, ( forces times the i r moment arms or perpendicular distance to an a r b i t r a r i l y assigned fulcrum point) about the intercondylar axis (fulcrum po in t ) , would a l l equal zero. In order to solve these equations to determine the j o i n t load and i t s or ientat ion both the magnitude and or ientat ion of the occlusal load must be known as well as the magnitudes of the indiv idual muscle forces. Barbenel *Grant (1973) has stated that the moments should be determined about a center of mandibular rotat ion which varied with the posi t ion of the mandible. He concluded that these "instantaneous" centers of rotat ion for given jaw posi t ions ( in the sag i t ta l plane) provided an improved estimate of the jaw c los ing moments of each muscle and correlated better with the i r anatomical alignments and known funct ions. However, because equi l ibr ium condit ions prevai l during s ta t i c functions the point in space about which a l l moments would act i s unimportant, since the net sum of a l l moments about any point i s zero. Grant f a i l ed to include the moments of reaction force at the teeth and j o i n t s (Stern, 1974). As such, the amount of force at the dent i t ion would be the same regardless of where the fulcrum point i s located. Thus, his instantaneous centers of rotat ion are of questionable use in determining s ta t i c mandibular function (Hylander, 1975). - 20 -had determined the s p a t i a l o r i e n t a t i o n s of the l a t t e r . He then used what appears to be a somewhat complex mathematical process (a t l e a s t as f a r as h i s d e s c r i p t i o n of i t goes) namely " l i n e a r programming a n a l y s i s " to determine the t h e o r e t i c a l "minimum" value of j o i n t force compatible with e q u i l i b r i u m . For changes of the tooth force angle (from apparently any s p e c i f i e d i n i t i a l angle of tooth r e s i s t a n c e ) of 15 degrees i n e i t h e r d i r e c t i o n ( r e l a t i v e to the o c c l u s a l plane) p o s i t i v e ( i e . compressive) loading of the j o i n t s was p r e d i c t e d . The extent of t h i s loading increased with more a n t e r i o r p o s i t i o n s of the o c c l u s a l p o i n t . Barbenel then tes t e d the t h e o r e t i c a l minimum p r e d i c t i o n s experimentally, using surface recordings of the electromyographic a c t i v i t y of a l l but the l a t e r a l pterygoid muscle. Barbenel used these to assign force magnitudes to the muscle vectors during measured v e r t i c a l l o a d i n g of the mandible. Since the magnitude of l a t e r a l pterygoid force as well as the magnitude and o r i e n t a t i o n of the j o i n t force remained unknown Barbenel was only able to determine these j o i n t force parameters as a f u n c t i o n of the r a t i o of assumed l a t e r a l pterygoid force to o c c l u s a l l o a d . However the force of a given muscle i s r e a l l y the product of i t s normalized EMG a c t i v i t y and i t s r e l a t i v e maximal force c a p a b i l i t y (Weijs, 1980; Weijs and Dantuma, 1980; Pruim, 1980), which i s r e l a t e d to the muscle's r e l a t i v e s i z e . Barbenel did not consider the l a t t e r . Instead he assumed that since the r e l a t i o n between EMG l e v e l s and i s o m e t r i c muscle force i s l i n e a r , the p r o p o r t i o n a l i t y constant* between the two f o r a given muscle, m u l t i p l i e d by *This can be determined under i s o m e t r i c c o n d i t i o n s from a l e a s t square re g r e s s i o n of the r e l a t i o n between the tooth force and the i n t e g r a t e d EMG (Weijs, 1980). - 21 -i t s i n t e g r a t e d EMG a c t i v i t y , would g i v e the r e l a t i v e f o r c e f o r t h a t muscle. However, W e i j s (1980) has s t a t e d t h a t t h i s method s e r v e s as a f i r s t a p p r o x i m a t i o n o n l y , and depends on the p r o p e r t i e s o f the r e c o r d i n g e l e c t r o d e s , among o t h e r t h i n g s . T h e r e f o r e , B a r b e n e l ' s muscle f o r c e s r e n d e r h i s c o n c l u s i o n s s u s p e c t , even though they are o n l y r e l a t i v e , p r o p o r t i o n a l o b s e r v a t i o n s . N e v e r t h e l e s s , he c o n c l u d e s t h a t the j o i n t i s l o a d e d and t h a t the minimum muscle f o r c e p r i n c i p l e does not a p p l y , i . e . t h e r e i s more f o r c e a p p l i e d to the j o i n t than the "minimum" p r e d i c t e d from h i s a n a l y s e s . A l l i n a l l , t h i s i s not a ver y good r e t u r n of i n f o r m a t i o n f o r the e f f o r t i n v o l v e d and the u s e f u l n e s s o f f u r t h e r a n a l y s e s a l o n g these l i n e s , o t h e r than as an i n t e r e s t i n g a l g e b r a i c e x e r c i s e , i s q u e s t i o n a b l e ( e g . B a r b e n e l , 1983). However the b a s i c approach taken by Barbenel (1969) t o i n c l u d e a l l the r e l e v a n t i n f o r m a t i o n ( i e . l i n e s o f muscle p u l l , t h e i r v e c t o r l e n g t h , and moment arms, as well as the magnitude and o r i e n t a t i o n o f the o c c l u s a l l o a d v e c t o r ) does a l l o w the d e t e r m i n a t i o n o f j o i n t f o r c e magnitude, and o r i e n t a t i o n , i f s t a t i c e q u i l i b r i u m t h e o r y i s used. The o n l y p r e r e q u i s i t e i s knowledge o f the v a l u e s of each v a r i a b l e f o r the task a t hand, i n each p l a n e o r p l a n e s o f r e f e r e n c e . Under such c o n d i t i o n s no movement can take p l a c e and, as d e s c r i b e d above, a l l f o r c e s are c o m p l e t e l y opposed, thus s a t i s f y i n g Newton's t h i r d law, and a l l s t a t i c r e q u i r e m e n t s , c . L e v e r - L i n k C o n s i d e r a t i o n s The i d e a t h a t the components o f the m a n d i b u l a r system may i n t e r a c t , i n c e r t a i n i n s t a n c e s , to reduce the f o r c e t r a n s m i t t e d through the c o n d y l e s seems a p p r o p r i a t e . There i s some e x p e r i m e n t a l e v i d e n c e to suggest t h a t human jaw mechanics are not s t r i c t l y those o f a simple c l a s s I I I l e v e r and t h a t - 22 -other, more complex re lat ionships may e x i s t . For instance, various workers have observed a decrease in maximal bi te force recorded at r e l a t i ve l y more poster ior posi t ions ( i e . the second molar) during both b i l a te ra l (Pruim et a l . , 1980, Tradowsky and Dworkin, 1982) and uni la tera l b i t ing (Mansour and Reynick, 1975). If the mandible rea l l y i s only a lever then poster ior teeth should be capable of producing re l a t i ve l y more force, for the same muscle e f f o r t , than anter ior teeth. However, the a b i l i t y of the system to function in more sophist icated ways than as a lever may not be l im i ted to considerations of poster ior tooth forces. Hylander (1978) determined both the or ientat ion and magnitude of human inc i sa l b i te forces at in to 30 mm of anter ior jaw opening. He concluded that since observed v e r t i c a l l y and anter ior ly directed inc isa l tooth forces could not be res is ted by the muscle forces alone, and thus required concurrent j o i n t resistance forces, the mandible must act as a l eve r . Gingerich (1979) re-evaluated Hylander's resu l ts and concluded that the or ientat ions of the inc isa l forces of resistance were such that the mandible functioned as both a lever and a l i n k . Based on s imi lar assumptions as in his ea r l i e r work (1971) Gingerich surmised that contract ion of middle and poster ior temporalis f i be rs , act ing perpendicular to the inc isa l b i te forces, contr ibuted to the l i nk por t ion . These muscles would not be capable of contr ibut ing to any j o in t forces and could , in f ac t , reduce as well as s t a b i l i z e them. Contraction of the masseter-medial pterygoid muscles would produce j o i n t forces and thus contr ibute to the lever port ion of the mechanics. - 23 -There i s also cer ta in experimental evidence in support of the notion that concurrent lever and l ink ef fects may e x i s t . Ferguson (1977) noted a consistent rocking movement of dental casts from many d i f fe rent pat ients . This occurred about a pivot or fulcrum in the premolar area, and persisted even in older patients with s ign i f i can t occlusal wear. It was only absent in casts of pat ients who had undergone orthodontic, or occlusal equ i l ib ra t ion treatment. Ferguson suggested that the two centr ic occlusion posi t ions observable (one an ter io r , one poster ior) were some sort of tolerance to compensate for j o in t compression. This would imply that the mechanics of the system are d i f fe rent for anter ior tooth contacts as opposed to poster ior contact (Bramble, 1978). Tradowsky and Kubicek (1981) observed the same phenomenon and hypothesized that the resul tant force vector of the mandibular adductor muscles, in b i l a te ra l c lenching, intersected the occlusal plane at the premolars. Such an occurrence requires the sum of the indiv idual vectors in th is plane to in teract in a par t i cu la r way other than as a simple lever mechanism. Consequently these workers concluded that the extent of j o i n t loading would vary depending on the posi t ion of the b i te point with respect to th is "physio logical equi l ibr ium point of the mandible." For b i te points corresponding with th is posi t ion no, or l i t t l e j o in t loading would occur. More poster ior contacts would provide a tooth fulcrum behind the muscle resul tant and thus tens i l e j o in t forces. Anterior contacts would provide for more compressive jo in t forces. These are the same conclusions reached by Bramble (1978). - 24 -However, i m p l i c i t in th is argument i s the assumption that the a c t i v i t i e s of each muscle and the overa l l muscle resu l tan t remain the same for d i f f e r e n t points of tooth contac t , which is not necessar i l y so . Confirmation that both lever and l i n k condi t ions may apply depending upon the circumstances is provided by Hylander (1979c), who measured the in v ivo bone s t r a i n of the condylar necks of macaques under a va r ie ty of c o n d i t i o n s . He found that un i l a te ra l isometr ic b i t i n g at the premolars, or f i r s t two molars produced compressive i p s i l a t e r a l j o i n t f o r c e s , whereas at the t h i r d molar these forces tended to be e i t h e r minor, non-ex is ten t , or t e n s i l e . d. Role of Craniofacial Form The importance of e s t a b l i s h i n g a r e l i a b l e model of the mast icatory system i s not l im i ted to a determination of whether or not the j o i n t s are loaded . Many orthognathic surg ica l procedures (eg. mandiublar advancement, Le F o r t I o s t e o t o m i e s ) are s p e c i f i c a l l y aimed at r e a r r a n g i n g the r e l a t i o n s h i p s of the maxi l la and mandible. Such procedures can e f f e c t i v e l y a lso rearrange the alignments of one muscle to another, and hence, change the morphological r e la t ionsh ips of the en t i re system. A number of inves t iga to rs have observed the d i f fe rences between a v a r i e t y of the morphological r e l a t i o n s h i p s which determine human f a c i a l he igh t , and the r e s u l t i n g b i t e force c a p a b i l i t i e s in i n d i v i d u a l s with s o - c a l l e d " long" versus "short" f a c i a l types (Sassouni , 1969; R i n g q v i s t , 1973 1973; Schendel et a l . , 1976; Ingerval l and Helkimo, 1978; Opdebeek and B e l l , 1978; Finn et a l . , 1980a; P r o f f i t et a l . , 1983a and b ) . Long-faced - 25 -ind iv idua ls with skeletal open b i te are not able to generate the same magnitude of b i te force as short-faced persons with skeletal deep b i t e s . According to Throckmorton, and coworkers (1980) the reasons for th is may include such factors as (1) di f ferences in muscle s ize (2) archi tecture and f i be r type d i s t r i bu t i on , (3) ac t i v i t y l e v e l s , and (4) mechanical advantage. These workers presented a two dimensional model based on c lass III lever mechanics from which they determined the ef fect of changes in mechanical advantage* of the human masseter and temporalis muscles. A l terat ions in maxi l lary height, gonial angle, and ramus height were simulated. It was found that any changes which ef fectve ly decreased the moment arm of the muscles, and/or which increased that of the tooth load, resulted in a reduction of mechanical advantage of the muscles. Subsequent observations of long and short-faced ind iv idua ls suggested that the greater b i te forces reported in the l a t t e r group were due to s imi la r morphological di f ferences which favored improved mechanical advantage in the short-faced ind i v idua ls . Their model suggests that some surgical procedures aimed at correct ing such disharmonies of fac ia l form may s i gn i f i can t l y ef fect the mechanical advantage of cer ta in jaw muscles. Finn and coworkers (1980 a and b) reached s imi lar conclusions when the same modeling scheme was applied to reductions in maxi l lary ve r t i ca l dimension and also to those incorporating correct ion of mandibular de f ic iency . They suggested that the potential for relapse fol lowing surgical correct ion of ve r t i ca l dysplasias of th is type was related to abnormal •Mechanical advantage in th is study was defined as the ra t io of moment arm length from condyle (fulcrum) to muscle, versus that from condyle to tooth load ( f i r s t molar). - 26 -masticatory muscle funct ion. The reasons for th is abnormal function could be due to abnormalit ies of any or a l l of the four factors suggested by Throckmorton et a]_. above. The i n a b i l i t y of a surgical procedure to reduce these abnormal re la t ionsh ips , or the i n a b i l i t y of the muscles to adapt to the changes so produced, were c i ted by Finn et ^]_. (1980a) as s ign i f i can t fac to rs . They concluded that such morphological a l tera t ions af fect the physiology of the muscles as well as the i r mechanical re la t ionsh ips . Observed lower muscle a c t i v i t i e s in long versus short-faced ind iv iduals were suggested to be a re f lec t ion of the re la t i ve l y hypertrophied muscle f ibers in the l a t t e r i nd iv idua ls . Therefore i t would seem that the re lat ionships between muscle, j o i n t , and tooth forces, which are determined in part by the physiology of the muscles, are also capable of inf luencing the physiological development and subsequent a b i l i t i e s of the muscles themselves. Nevertheless, in order to determine the actual re la t ionships between these forces, knowledge of the magnitude and or ientat ions of the tooth and muscle forces involved, are an absolute requirement. e. Role of Muscle Forces There are two determinants of the force produced by a given muscle during a par t i cu la r funct ion. The f i r s t i s the maximum potential force which the muscle i s capable of generating. This i s considered to be proportional to i t s physiological cross section (eg. Weijs and Hi l i e n , 1984a and b) , which can be defined as "the summed cross section of i t s indiv idual f ibers " (Weijs, - 27 -1980*). Values for the maximum force per unit area vary between 2 to 14 kg/cm 2 for humans (see Weijs, 1980 for review). Most recent estimations place them between 30 and 50 N/cm2 ( 3 to 5 kg/cm 2) (Weijs and Hi 11 en, 1984). The second factor i s the extent to which each muscle i s act ive during a pa r t i cu la r funct ional ac t . Various muscles exh ib i t d i f fe rent proportions of t he i r maximal a c t i v i t y and thus exert d i f fe rent forces depending on the pos i t ion of the mandible (which also af fects the alignment of the muscle*) the d i rec t ion of appl ied e f fo r t and/or the pos i t ion of any tooth contacts involved (MacDonald and Hannam, 1984). As such, a muscle force vector i s described by: (1) i t s o r ien ta t ion ; and (2) the product of i t s re la t i ve maximum potent ial force and extent of e f f o r t , or re la t i ve a c t i v i t y . Electromyographic responses have been used to estimate the l a t t e r , using muscle a c t i v i t y during maximum e f fo r t as a yardst ick for comparison. The various modeling analyses mentioned to here, with the exception of Barbenel (1969, 1972, 1974), have a l l neglected to consider one or more of these essent ia l elements in def ining the i r vectors. The human experimental work of Pruim and his coworkers (1978 and 1980) appears to be one of the more complete studies in th is regard. They measured the ve r t i ca l b i te force ( b i l a t e r a l l y ) at three anteroposterior b i te pos i t i ons ; the f i r s t premolar and the f i r s t and second molars. They *A large muscle i s stronger than a re l a t i ve l y smaller one although the degree of pinnation known to ex is t in cer ta in muscles of many species (eg. masseter complex in the p ig , Herring et ^1_., 1979) may lead to an underestimation of t h i s force. + T h i s becomes extremely important in analyses involv ing tooth force measurement which use devices requir ing the jaw to be opened to any great extent (Hylander, 1978). - 28 -correlated th is with the normalized, simultaneous electromyographic ac t i v i t y exhibi ted by the various muscles involved including anter ior and poster ior temporal is, masseter, and the d i g a s t r i c . Maximum responses for each muscle were determined to which a l l other responses for that muscle were normalized for each task modeled. However, they assumed that medial pterygoid ac t i v i t y was represented by that of the masseter and they did not record at a l l from the la te ra l pterygoids. The muscle alignments were determined for each subject from predict ions based on la te ra l cephal ograms. Physiological cross sections were assigned according to Schumacher (1961). From th is information the muscle vectors were determined in two dimensions and the resu l t ing j o i n t forces derived according to s ta t i c equi l ibr ium theory. The resu l ts of Pruim et al_. (1980) showed that under condit ions of s ta t i c equi l ibr ium considerable j o i n t forces could be expected. These increased almost l i n e a r l y , with increasing b i te force, for a l l three b i te pos i t ions . Under a constant b i te force j o in t forces also increased as b i te posi t ions were moved more anter io r ly which might be expected according to lever mechanics (eg. Gosen, 1974). As a consequence of monitoring muscle a c t i v i t y , these workers were also able to show that the overal l muscle force resul tant exhibi ted a more anter ior ly directed or ientat ion for more anter ior b i te pos i t ions . Maximum bi te forces occurred at the f i r s t molar and corresponded with r e l a t i ve l y greater muscle and jo in t forces when compared with those at e i ther the premolar, or second molar pos i t i ons . Pruim et al_. concluded that the mechanics of the system, during these kinds of funct ions, were subject to some form of j o i n t capsule i nh ib i t i on and were therefore determined by an - 29 -upper l i m i t of j o in t force corresponding to that seen during f i r s t molar c lenching. One of the more in terest ing f indings of th is study was a s i gn i f i can t l y reduced bi te force at the second molar. As was previously mentioned, others have also observed a s imi la r phenomenon experimentally during un i la tera l b i t i ng (Mansour and Reynick, 1975) in b i l a te ra l b i t ing (Tradowski and Dworkin 1982) and c l i n i c a l l y (Hawthorn, 1984). The apparent reduction of occlusal load at a more poster ior posi t ion appears to be in d i rec t c o n f l i c t with that expected from simple two dimensional lever mechanics (Gosen, 1974) which predict r e l a t i ve l y greater b i te forces at more poster ior pos i t ions , along with reduced j o in t loading. However, part of the phenomenon may be due to the simultaneous reduction in overal l muscle ac t i v i t y shown to occur by Pruim and coworkers (1978 and 1980) at th is b i te pos i t i on . Pruim et al_.(1980) a t t r ibute the decrease in occlusal load to a cen t ra l l y acting inh ib i t i on due to the possible need for more accurate regulat ion of equi l ibr ium because the muscle force resul tant would l i e close to the b i te point . This too would imply that some sort of synergism between the muscles i s at work to maintain th is equ i l ib r ium. On the other hand, i t should be kept in mind that the alignment of a un id i rect ional force transducer w i l l only reg is ter those forces para l le l to i t s ax i s . It has been shown that tooth resistance forces are quite var iab le in the i r or ientat ion in both b i l a t e r a l l y symmetrical functions (Hylander, 1978) and un i la tera l a c t i v i t i e s (Graf et al_., 1974). Although Pruim et a K (1980) conclude that the muscles are capable of generating maximal b i te forces at the f i r s t molar pos i t ion i t i s also possible that the actual - 30 -alignment of the b i te force produced by the muscles in th is posi t ion were c loser to para l le l with the axis of the measuring device. Experimental measurements of human b i te force or ientat ion indicate that the assumption of purely ver t i ca l b i te forces ( i . e . perpendicular to the occlusal plane) is an overs imp l i f i ca t ion (Weijs, 1980). Incisal b i te or ientat ions have an anter ior as well as ve r t i ca l component (Hylander, 1978) and un i la tera l molar b i te forces have been shown to include the mediolateral component as well (Graf, e_t £ l_ . , 1974). S imi lar f inding have also been reported for other species (Weijs and Dantuma, 1975 and 1981). Thus, at e i ther the second molar or premolar posi t ions (which have already been stated to produce d i f fe ren t alignments of the muscle resul tant force) the p o s s i b i l i t y that the b i te force may be underestimated cannot be discounted. Thus the analysis of Pruim e_t a l . (1980) took only the ver t i ca l components of b i te force into account (Pruim e_t a]_., 1978), although, both the ver t i ca l and anteroposterior components of muscle force were considered. The equi l ibr ium equations of Pruim £ t £]_• (1980) state the fo l lowing: (1) The sum of the rotat ional moments of the various components of the system is given by ,(E EMGm x 0m x r x %) + (F D x a b) = 0. The f i r s t term (in brackets) represents the muscle moments where, EMGm i s the re la t i ve a c t i v i t y of muscle m, 0m_ i s i t s cross sec t ion , r i s the force/ un i t cross sec t ion , and am, is the lever arm length of muscle m. The second term is the moment of the bi te force, F D (which i s the measured ve r t i ca l force only) due to i t s moment arm, a b . The j o in t s are considered the fulcrum and thus have no moment. A l l of the above are known, or assumed, - 31 -except r . The re fo re , an underes t imat ion of b i t e fo rce (F^) w i l l e f f ec t i ve l y push th is value up in order to maintain equi l ibr ium within the equati on. (2) The sum of the ver t ica l (y) components i s wri t ten as T. F m sin am + Fj sin « j + F b = 0 . The f i r s t term, F m , i s the ver t i ca l component of force of muscle m aligned at angle a with respect to the x (anteroposterior) ax i s , and the second term i s that of the j o i n t fo rce, F j . The l a t t e r force i s f ixed at a spec i f i c aliqnment by these workers. Aqain since only Fj i s unknown, and the equation must be s a t i s f i e d , low values of b i te force (F|,) w i l l a lso push the jo in t force value up. The fact that i t s alignment i s f ixed can resu l t in a further increase or decrease in the ver t i ca l component depending upon the assumed or ien ta t ion , and the amount of e r ro r . (3) The sum of the anteroposterior (x) components i s y F m COS nm + Fj COS aj + Fp = 0 . The f i r s t two terms have been descr ibed. Since only the ver t i ca l b i te force i s ever considered in th is ana lys i s , no anteroposterior component ex is ts in th is equation. F p however i s the force of the la te ra l pterygoid muscle. According to Pruim et ol_. (1980) th is muscle has only an anteroposterior component and therefore only appears in th is equation. It i s a lso the only unknown var iab le here. The value of Fj i s derived from the expression of the ver t i ca l components described above and i s subject to the same problems of misalignment. Therefore the values derived for the force of the la te ra l pterygoid muscle are also subject to error due to incorrect b i te force values, and jo in t force or ien ta t ion . - 32 -A more appropriate analysis might have been: (1) to measure both dimensions of b i te force (ver t ica l as well as anteroposterior) to better match the muscle vector ana lys is ; (2) to l e t the re la t i ve proportions of ve r t i ca l versus anteroposterior components of j o in t force be derived from the equi l ibr ium equations, and; (3) to include the electromyographic recordings of both medial and la te ra l pterygoid muscles. 3y l e t t i ng the two components of j o i n t force be derived from the equations both the magnitude and or ientat ion of the resu l t ing vector necessary to maintain equi l ibr ium are establ ished without constraining the r esu l t s . With regard to the l as t point , Pruim et a]_. (1980) acknowledged some of the l im i ta t ions of the i r analysis when they found that the masseter EMG a c t i v i t y was not necessar i ly representative of the medial pterygoid as w e l l . The assumption that medial and la tera l pterygoid function i s dependent upon the ac t i v i t y of other muscles i s a gross overs imp l i f i ca t ion . Nevertheless, the type of approach taken by Prium and coworkers (1980) provides valuable ins ights into the complexity of the problem. Despite the l im i ta t i ons of the i r analysis the conclusions which may be drawn from th is study regarding human jaw function are among the most re l i ab le thus fa r . The p o s s i b i l i t y that a spec i f i c posi t ion of b i te force appl icat ion ( i . e . f i r s t molar) i s more e f f i c i e n t than those predicted from purely mathematical ana lys i s , i s of major s ign i f i cance . - 33 -2. T h r e e D i m e n s i o n a l M o d e l s Most of the analyses discussed to th is point have been l imi ted to the sag i t ta l plane and have assumed b i l a te ra l symmetry. Although two dimensional analyses are su f f i c i en t for inc isor or b i l a te ra l molar b i t i n g , the mandible also functions (and dysfunctions) in three dimensions. Simple two dimensional approaches are therefore i nsu f f i c i en t to model un i la tera l functions (Walker, 1976). As Weijs (1980) has stated, " I f m u s c l e f o r c e e s t i m a t i o n s a r e u s e d i n a t h r e e d i m e n s i o n a l s t a t i c a n a l y s i s , b i t e f o r c e s and j o i n t r e a c t i o n fo rce are found d i f f e r e n t from those r e s u l t i n g from one ( s i m p l e l e v e r ) or two d i m e n s i o n a l s t a t i c a n a l y s i s . The exp lana t ion of tooth and j o i n t morphology i s influenced by these modi f icat ions. " However, previous attempts at modeling the three dimensional biomechanics of the mammalian masticatory system have v i r t u a l l y a l l overs impl i f ied the muscle forces involved. In the i r reviews of biomechanical analyses of the jaw, both Hylander (1975) and Weijs (1980) point out the need for considering the d i f f e ren t ia l a c t i v i t i e s which are known to ex is t between the muscles of the two sides of the jaw (as well as between those of the same side) in un i la tera l functions (eg. Mol le r , 1966). The three dimensional model proposed by Gysi (1921), which was described e a r l i e r , did not incorporate th is information since i t was not avai lab le at the time. Therefore his assumption that each muscle was maximally act ive during a uni la tera l task could lead to possible over-estimations of the force contr ibut ions of the various muscles involved. It was not unt i l some years - 34 -la te r that Mainland and H i l t z (1933) f i r s t speculated that potential di f ferences in muscle force contr ibut ions might ex is t between the two sides of the jaw (a lbe i t for somewhat s imp l i s t i c reasons) by suggesting that the muscles of the r ight side might be stronger than those of the l e f t . They determined absolute values for the three dimensional force capab i l i t i e s of the various jaw muscles from measurements of muscle angulation and cross sect ion from cadavers. They were a lso interested in determining the resu l t ing potential occlusal force generated at the second molar. These they compared to gnathodynamometer recordings reported for un i la tera l clenching at th is tooth on both sides of the jaw. However, in the determinations of the i r predicted tooth forces they neglected the balancing side altogether as well as a l l mediolateral components of force. They simply calculated the force at the r ight second molar, due to the moments produced by the r ight side muscles, and added i t to that for the l e f t s ide. Although th is i s a gross overs impl i f i ca t ion they reported agreement with cer ta in un i la tera l b i te forces recorded from l i v i n g subjects. Overs impl i f icat ions due to a lack of understanding of the functional re la t ionsh ips between the two sides of the jaw, and even complete d issoc ia t ion of the two sides in analyses of un i la tera l funct ion, are not uncommon. Hekneby (1974) used a three dimensional s ta t ic model to determine the j o i n t forces for equal tooth loads applied to ei ther the f i r s t b i cusp id , or the second molar of human mandibles. Although she acknowledged the possible contr ibut ion of the various portions of the muscles, as well as the d i f f e ren t i a l a c t i v i t i e s of the two s ides , her determinations considered only the ver t i ca l forces of the working s ide . Based on an arb i t rary der ivat ion of - 35 -the working side muscle force resu l tan t , Hekneby determined the working side j o i n t force required to balance the applied tooth load. The sum of the working side muscle, tooth and j o i n t forces were assumed to equal zero. The contr ibut ion of the balancing side muscles to force production, and the balancing side j o i n t to force res is tance, were assumed to balance each other and thus not inf luence the greater working side forces to any extent. S imi lar overs imp l i f i ca t ions in an analysis by Nagle and Sears (1958) were c r i t i c i z e d by Hylander (1975) for ignoring working side muscle a c t i v i t i e s . Hylander (1975) noted that since the working side muscles had been shown to be more act ive than the balancing side muscles in un i la tera l molar b i t i ng (Mol le r , 1966) most e f fo r t would be expected to occur nearer the b i te point in the frontal plane. He suggested that in such funct ions, owing to lever act ion e f f ec t s , the working condyle i s subject to less (or perhaps even neg l ig ib le ) loading than the balancing side condyle. He further suggested that th is might explain the c l i n i c a l observations that pat ients with un i la te ra l j o i n t dysfunction tend to f ind contra latera l (balancing or nonaffected side) chewing (and/or i nc i sa l b i t ing) more painful , thereby preferr ing to chew on the i p s i l a t e r a l (working or affected) s ide . Incisor b i t i r \ g , l i kew i se , produces greater j o in t forces according to lever mechanics. For th is to occur there must be some transmission of the forces from one side of the mandible to the other. Hylander (1975) and Beecher (1979) have both proposed that the fused mandibular symphysis of primates i s an adaptation to accommodate greater symphyseal stress a r i s ing from the balancing side muscles during powerful un i la tera l b i t i n g . In subsequent primate studies Hylander provided convincing experimental support of this concept. - 36 -Us ing s i n g l e e lement s t r a i n gauges a n d / o r r o s e t t e s ( i . e . mu l t id i rec t iona l ) bonded d i rec t l y to the lower mandibular border of Gal ago  crassicaudatus Hylander (1977 and 1979a) measured the in v ivo stress patterns a r i s ing during various funct ions, including b i t ing a force transducer. This par t i cu la r species of monkey possess an unfused mandibular symphysis. Not su rp r i s ing ly , he found that very l i t t l e of the balancing side muscle force contributed to the occlusal force generated at the working s ide . The l a t t e r had from f ive to ten times the st ra in of the balancing side during un i la tera l funct ions. Thus the two sides of the mandible in th is species are somewhat f u n c t i o n a l l y independent . Conve rse l y , s i m i l a r a n a l y s i s of Macaca  f a s c i c u l a r i s , which does possess a fused symphysis, show that th is species employs a r e l a t i ve l y greater amount of balancing muscle force to generate a par t i cu la r un i la tera l occlusal force (Hylander 1979a). The di f ference between the working versus the balancing side muscle forces in the two species i s approximately 1.5 to 1 in macaques and 3.5 to 1 in galagos during un i la tera l molar b i t i ng (Hylander 1979b). Thus based on the d i s t r i bu t i on of mandibular bone s t r a i n , the fused symphysis i s an adaptation to maximize the contr ibut ion of balancing side muscle forces in function and thereby increase occlusal forces (Hylander, 1975, 1979a, 1979b; Beecher, 1979). In add i t ion , cer ta in other adaptations in jaw form (eg. v e r t i c a l l y deep and/or t ransversely th ick jaws) which accompany the mechanics involved have been shown to correspond with the ensuing d is t r i bu t ion of force (eg. Badoux, 1965). The close correspondence between primate mandibular form, i t s stress d i s t r i b u t i o n , and i t s function have been thoroughly reviewed by Hylander (1979b). - 37 -Smith (1978) considered the mandible to function according to two d i f fe rent mechanisms depending upon the a c t i v i t y . He suggested that the mandible acts as a c lass III lever during the dynamic c los ing stroke without any occlusal load . When tooth contact i s made durinq mastication for instance, and loading of the system occurs both Smith (1978) and Walker (1978) proposed that the mandible then acts as a stat ionary beam. This beam can be considered to ex is t in a state of instantaneous equi l ibr ium at any given moment during b i te force generation. This l a t t e r s i tuat ion would require more s ign i f i can t adaptive change than those necessary to minimize the r e l a t i ve l y l i g h t forces involved in simple jaw e leva t ion , which rea l l y only act against the weight of the mandible i t s e l f . As such, the morphology of the system would more l i k e l y re f l ec t the mechanics of the beam rather than the c lass III l eve r . Hylander (1979c) has pointed out that whether one considers the jaw as a "beam" or a " lever" is i r re levant when external ly applied forces (such as those of the muscles, teeth and jo in ts ) are considered s ince , mechanically the two act i den t i ca l l y under these cond i t ions. Beam theory becomes more appropriate for considerat ion of internal forces of the mandible such as bending or shearing s t resses. Nevertheless, according to th is model the three areas of contact which can po ten t ia l l y res i s t the applied forces are at the point of tooth contact and at the two j o i n t s . In the sag i t ta l plane Smith (1978) proposed that the beam could be considered as a l i n e drawn between the condylar head and point of tooth contact (see Figure 3 ) . Forces were assumed to be applied by the temporal is, masseter, and medial pterygoid uniformly over the i r lengths of overlap with - 38 -th is l i ne as seen in sag i t ta l or f rontal pro ject ion. Therefore, a posi t ion along the beam behind which occurs the greatest amount of overlap of muscle f i be rs was considered by Smith to be subject to the greatest muscle forces. Smith also assumed the re la t i ve contr ibut ion of each muscle would be proport ional to i t s weight, although he points out that th is serves only as an approximation and that many other factors are also involved. Figure 3. THE MANDIBLE AS A STATIONARY BEAM ACCORDING TO SMITH (1978). CT, total condylar react ion force (balancing and working s ides) ; B, b i te fo rce; FT, muscle force as a vector resul tant of the "d is t r ibu ted" force between points 2 and 5 (balancing and working s ides ) . The extent of "overlap" of the muscles across the beam are considered by Smith to determine the pos i t ion of th is resul tant (After Smith, 1978). ^Smith's analysis proposed that the total muscle forces, viewed in the sag i t ta l plane, could be resolved into a s ingle vector acting at a known pos i t ion along the beam. Assuming that the magnitude of the bi te force was known, both the muscle resul tant magnitude and that of the total condylar react ion force ( r ight plus l e f t ) could then be determined by resolving the - 39 -equi l ibr ium re lat ionships of the s ta t i c forces, and the moments they produce. The der ivat ion of the indiv idual forces at each of the condyles was achieved by applying s imi la r procedures to the beam projected in the frontal plane. According to Smith the posi t ion of the resul tant muscle vector in th is plane was determined by, and calculated from, the re la t i ve magnitudes of the muscle forces of the two s ides . As mentioned, evidence suggests (Mol ler , 1966) the working side muscles of mastication are more act ive than the balancing s ide . However, the magnitude of the di f ference between the s ingle working side muscle vector versus the balancing side muscle vector i s uncertain in th is spec i f i c ana lys is . Therefore, Smith se lec ted, as an upper l i m i t , a ra t i o of 2:1 of working to balancing side muscle force and as a lower l i m i t a ra t i o of 1:1. The d i f fe rence, according to him, a l te rs the distance of the combined resul tant muscle force from the working side condyle. Smith applied th is model to dissected specimens of three species of monkeys and to humans, from which he derived the mean vectors for muscle force and the resu l t ing condylar react ion forces for assumed inc isa l and molar b i te forces. His resul ts show that s i gn i f i can t l y greater condylar react ion forces occur in i n c i s a l , versus molar b i t i n g , in a l l four species. With a 1:1 ra t io of working to balancing side muscle forces Smith also found that the balancing condyle accounts for most (80%) of the total j o i n t load . He further concluded that the muscles probably function close to a 1:1 ra t i o of muscle ac t i v i t y for the two s ides. Smith c l ea r l y intended only to present a s imp l i f ied example of th is approach to three dimensional analyses and i t s appl icat ion to real data. However some of the overs impl i f i ca t ions of th i s analys is bear point ing out as they represent common problems in other three dimensional analyses which - 40 -contr ibute to the i r dubious a p p l i c a b i l i t y as models of the real (absolute) re la t ionsh ips . F i r s t of a l l in determining the indiv idual muscle forces act ing on the beam Smith considered that only the ver t i ca l components, in each plane, contr ibute to the b i te force. As was discussed regarding the studies of Pruim et al_., (1978 and 1980) th is assert ion i s true i f , and only i f , the b i te and j o in t forces are also l im i ted to purely ver t i ca l components, and v ice versa. Smith assumed th is to be the case in his analys is as he neglected any muscle force components not perpendicular to the beam ax i s , as well as a l l mediolateral components. However the l im i ta t ions of such an assumption become c lear when clenching tasks oriented in a d i rec t ion with components other than perpendicular to the beam are considered ( i . e . media l ly , p ro t rus ive ly , or r e t rus i ve l y ) . Secondly, the assumption that the extent of muscle/beam overlap i s an ind icat ion of the posi t ion of the net e f fect of muscle pul l i s c l ea r l y quest ionable. The appropriate posi t ion of any resul tant force is determined by the vector addit ion of i t s component par ts . Thus, force vectors for each muscle would have to be determined, and geometrical ly added. Smith however, determined th is posi t ion by assuming each muscle to generate a force, in proportion to i t s weight, which was evenly d is t r ibuted along i t s width of overlap with the beam. Using his own form of vector add i t ion , he combined these forces, due to the three muscles considered, and picked the middle of t h i s d is t r ibu ted load as the posi t ion of the resul tant so produced. The magnitude of the resul tant was determined, for th is pos i t i on , as that necessary to counter the applied tooth force. A more appropriate, and - 41 -accurate means of der iv ing these var iables would be to determine the indiv idual muscle forces. However th is requires knowledge of indiv idual muscle responses, in turn requir ing l i v e subject data. F i n a l l y , the idea that the three dimensional s ta t i c muscle force vectors can be combined into a s ingle resul tant seems to be widespread among invest igators (eg. Hekneby, 1974; Greaves, 1978), and i s wrong. Force vectors can only be resolved into a s ingle resul tant vector i f the component vectors are c o l l i n e a r , coplanar, or para l le l (Beer and Johnson, 1977). If the mechanics of the system are considered to be b i l a t e r a l l y symmetrical with no net mediolateral components of force then the vectors can be considered to act in a midl ine sag i t ta l plane. Such two dimensional analyses can be considered to have coplanar vectors and the i r resolut ion into a s ingle ver t i ca l and anteroposterior component, (or a s ingle resul tant at a spec i f i c o r ien ta t ion) , i s l aw fu l . However when asymmetric forces are involved, which require the th i rd dimension, vectors which do have various mediolateral components, cannot be considered to be coplanar, and they ce r ta in l y are neither col inear nor p a r a l l e l . Resolution of muscle vectors in a three dimensional system of force vectors can be expressed by e i ther of two equivalent systems of forces. The f i r s t i n v o l v e s s imply the de te rmina t ion of the t o t a l v e r t i c a l , anteroposter ior , and mediolateral components of s ta t i c muscle force from the sum of the same components in indiv idual muscles. In the la te ra l (sag i t ta l ) plane or view, on to which the total ver t i ca l and anteroposterior components can be projected, the axis along which each component acts can be determined from the total rotat ional moment for that plane, a lso obtained from the sum - 42 -of indiv idual muscle moments. In th is view, or plane, any mediolateral force has no ef fect on the system since i t i s oriented perpendicular to th is plane. However, in the frontal plane, which has a d i f ferent total moment, the anteroposterior component would have no ef fect whereas the mediolateral and ver t i ca l would. S im i l a r l y , in the horizontal view the ver t i ca l component would not c o n t r i b u t e to the e q u i l i b r i u m of f o r c e s , whereas the anteroposterior and mediolateral would. A l l three orthogonal components do not necessar i ly pass through the same point in space. As such any resul tant force determined in the la te ra l plane (from the ver t i ca l and anteroposterior components) i s not l i k e l y to be equivalent in pos i t i on , or ientat ion or magnitude, to that determined in the frontal (from the ver t i ca l and mediolateral components) or the horizontal planes (from the anteroposterior and mediolateral components). In other words there w i l l be three separate orthogonal resu l tan ts , one for each plane, which are neither col i near, coplanar nor p a r a l l e l , and which therefore cannot be resolved into a simpler equivalent system of forces. A second system of equivalent forces i s referred to as a "wrench" which, simply s ta ted, i s ac tua l ly a s ingle vector with a par t i cu la r moment about the fulcrum but which also has a secondary twist ing ef fect about the axis of that vector (see METHODS for a more complete d iscuss ion) . The extent of the twist ing e f fec t i s referred to as the "p i t ch" of the wrench and i s perpendicular to the moment e f fec t of the vector to the fulcrum (Beer and Johnson, 1977). As such th is system of forces (which would have the same e f fec t as the three orthogonal resul tant vectors described above for a given - 43 -system of forces) also requires a l l three dimensions to be considered simul taneously. In Smith's (1978) analysis the s ingle resul tant of muscle force in the saq i t ta l plane was also assumed to act in the frontal plane as w e l l . Because Smith assumed only ver t i ca l (para l le l ) forces were ac t ing , i t was possible to derive a s ingle three dimensional resul tant vector for his system. However, h is sweeping assumptions l i m i t the appl icat ion of th is type of ana lys i s , which Smith acknowledges. Simi lar c r i t i c i s m s apply to the assumptions of Hekneby (1978). Greaves (1978) proposed a three dimensional lever model to explain the jaw form of anisognathus ungulates from a mechanical point of view. This model was very s imi la r in many respects to the beam proposal of Smith (1978) and is thus subject to s im i la r l i m i t a t i o n s . Nevertheless, the information gained from s imp l i f ied models such as these has most ce r ta in l y contributed s i g n i f i c a n t l y to the understanding of the workings of the mandibular system. Greave's model points out some of the re lat ionships between the jo in t forces of the working versus balancing s ide , for var iable un i la tera l tooth contact points and muscle force pos i t ions . Like Smith, Greaves (1978) made the assert ion that , "the components of the muscle force that close the jaws can he resolved into a s ingle ver t i ca l vector . " He showed that for un i la tera l b i t i n g , the two j o i n t s , and point of tooth load app l i ca t ion , form a t r iang le of support when viewed in the horizontal plane. The d i s t r i bu t ion of the total forces of resistance between these three points , due to the ver t i ca l muscle resul tant of appl ied fo rce , var ies with the pos i t ion of th i s resu l tan t . Accordingly, for th is t r ipod to be stable the e f fec t i ve ver t i ca l - 44 -muscle pul l must l i e somewhere within th is t r iang le otherwise a d is locat ing force w i l l ar ise at one of the contact points . Using th is s imp l i f ied system Greaves predicted the theoret ical anter ior and poster ior l i m i t s for the posi t ion of the masticatory tooth row in selenodont ar t iodacty l s . The anter ior l i m i t was that point beyond which the tooth force decreased due to (1) a reduced lever e f f i c iency more an te r io r l y , and (2) a smaller muscle resu l tant . The poster ior l i m i t i s that point beyond which the muscle resul tant would come to l i e outside the t r iangle of support described above, and produce d is locat ion of the working side j o i n t . Despite some of the constraints imposed by Greave's assumptions he reports good agreement with the actual locat ion of the tooth row in such ar t iodacty ls and proposed his model as a good working hypothesis of the system in such animals. Subsequently, Druzinski and Greaves (1979) applied th is model to the jaw mechanism of various rep t i l es (assuming symmetry in muscle a c t i v i t y ) . They also found a close cor re la t ion between the observed and expected posi t ions of the most poster ior b i te point as predicted according to th is model. One important point to note with the model is that i t i s not necessari ly l im i ted to s ta t i c condit ions but appears to be compatible with simple dynamic considerat ions. Furthermore, Greaves (1978) points out the need for measurements of the various jaw forces involved, including the need for considering the re la t i ve electromyographic a c t i v i t i e s of the various muscles involved, and/or measuring the actual j o i n t forces themselves. Hylander and Bays (1978) and Hylander (1979c) presented the f i r s t experimental studies measuring _1n vivo forces occurring at the j o i n t s . - 45 -Hylander found that the j o i n t react ion forces predicted from s t ra in deformation at the la te ra l aspect of the condyler necks of monkeys (Macaca f a s c i c u l a r i s and Macaca mulatta), suggested that compressive loading occurred during normal funct ions. These included the power stroke of mastication and i nc i s ion of food, as well as isometric molar and inc isa l b i t i n g . The j o in t forces were directed v e r t i c a l l y and poster ior ly in most instances when tooth contacts occurred anter ior to the th i rd molar, although anter io r ly directed compressive forces were also observed. Furthermore, these forces were greater on the balancinq versus the working sides (c f . Gysi 1921; Hylander, 1975; Smith, 1978; Greaves, 1978). As discussed e a r l i e r , the pattern of loading of the working j o i n t in un i la tera l isometric b i t ing showed a surpr is ing di f ference at the th i rd molar compared to more anter ior b i te po ints . At th is posi t ion the pattern of subcondylar bone s t ra in was reversed from the compressive loading pattern observed at the f i r s t molar, suggesting that at the th i rd molar the working j o i n t was subjected to unloading or tens i l e forces. This supports the predict ions of j o in t function and loading charac te r i s t i cs according to Greave's (1978) model . Models proposed by Gysi (1921), Hylander (1975, 1979b), Hylander and Sicher (1979), and Smith (1978) a lso predict zero or negative working side j o in t loads at more poster ior contacts. S i m i l a r l y , Hylander (1979c) found greater (working side) subcondylar react ion forces durinq premolar than during molar b i t i ng which might be expected from lever p r inc ip les i f muscle force remains the same or s im i la r at both b i te pos i t ions . More recent ly , Hohl and Tucek (1982) implanted an instrumented p r o s t h e s i s i n an anes the t i zed baboon to rep lace the neck of the - 46 -temporomandibular condyle. This prosthesis incorporated ca l ibrated s t ra in gauges which recorded the condylar (neck) forces exerted during simulated i n c i s a l b i t e s . Although the i r data were subject to a var iety of problems, inc luding f a i l u r e of the prosthes is , they found the j o i n t to be loaded a x i a l l y at magnitudes comparable to those measured at the inc isors (up to 8 l bs . ) during these i nc i sa l b i t es . However due to the nature of th is experiment no inferences to natural events could be drawn. S i m i l a r l y , other j_n vivo studies on monkeys (Macaca arctoides) have u t i l i z e d a p iezoe lec t r i c pressure sens i t i ve f o i l implanted d i rec t l y on the a r t i c u l a r surface of the mandibular condyle (Brehnan and Boyd, 1979; Brehnan et jil_. 1981; and Boyd et a l . , 1982). This allows for d i rec t measurement of the compressive react ion forces occurring at the condylar head as opposed to the ind i rec t measurements of Hylander (1979c), and Hohl and Tucek (1982). Although the i n i t i a l f indings of these workers (Brehnan and Boyd, 1979) suggested that the condyles were non-stress bearing they subsequently found evidence (eg. Brehnan et_ al_., 1981; Boyd et aj_., 1982) that the temporomandibular j o in ts were, in fac t , stress bearing. However, as Hylander (1985) has pointed out in his review of these studies the in terpretat ion of the masticatory j o i n t forces is d i f f i c u l t as no cor re la t ion was made between the forces recorded and the corresponding jaw pos i t i on . Recently Smith j?t j*l_. (1986) presented a somewhat qua l i ta t i ve numerical model of human condylar loading based on s ta t i c analyses of a hypothetical mandibular system. They mathematically determined the re la t i ve range of condylar loading forces and the i r magnitudes, due to various combinations of - 47 -forces exerted by three muscle groups on each side of the mandible. These were done for un i la tera l points of tooth contact alonq the dental arch. This model was designed to determine only minimal possible j o i n t loads according to re la t i ve changes in the contr ibut ions of these muscle groups. They found the temporomandibular j o in t to be loaded in compression as well as tension "over the normal functional range of b i te force posi t ions and angles." These loads were maximally compressive at i nc i sa l contacts, minimally compressive at the second molars and tens i le at the d is ta l of the th i rd molars. Jo in t loads were found to be re l a t i ve l y small when bi te forces were aligned para l le l to the sag i t ta l plane and corresponded with an or ientat ion perpendicular to the a r t i cu la r eminence. When la te ra l components were added to the b i te forces marked asymmetry between the r ight and l e f t j o i n t forces resu l ted . These also produced a wider range of j o in t force or ien ta t ions . These workers concluded that b i te forces para l le l t o , or within 20 deqrees of the sag i t ta l plane, are mechanically more stable requir ing sma l le r j o i n t loads o r ien ted more favo rab ly wi th respec t to t h e i r fo rce - res is t ing morphology. However, these invest igators also point out that because the re la t i ve magnitude of the muscle forces are unconstrained the i r resu l ts are only representative of the minimal re la t i ve j o in t loads possible with respect to the tooth loads. Mo apparent attempt was made to apply real data to the three pairs of possible muscle forces of th is model and no values for the resu l t ing magnitudes of b i t i ng force are reported in th is work. As such only the re la t i ve magnitudes of j o i n t forces compared with the suspect b i t i ng forces could be descr ibed. This i s unfortunate as the - 48 -appl icat ion of absolute values to th is model may have provided addit ional in teres t ing corre la t ions between form and function and cer ta in ly further c r e d i b i l i t y to th is model. 3. Current Modeling Approaches The most comprehensive modeling analyses of functional jaw mechanics have been the experimental studies by Weijs and Dantuma on the masticatory system of the albino rat (1975) and rabbit (1981). These invest igators considered v i r t u a l l y a l l of the pert inent var iables in the i r s ta t i c analyses inc lud ing : (1) the three dimensional spat ia l coordinates of the condyles and the teeth, as well as the centers of attachment (or ig in and insert ion) of each of the functional muscle groups; (2) the physiological cross sections of each muscle; and (3) the simultaneously recorded electromyographic ac t i v i t y of each muscle for a given task. From th is information the re la t i ve forces of the indiv idual muscles were determined for each of the various stages of the chewing cyc le , as well as during isometric b i t i n g , and vector analysis of the system as a whole was carr ied out. Each of these stages or functions was considered to be e i ther stat ionary (b i t ing) or moving at constant speed (power stroke of chewing) under which condit ions s ta t i c equi l ibr ium was assumed to ex is t (Weijs, 1980). The necessary assumptions involved in the i r determinations were the fol lowing (see Weijs 1980; Weijs and Dantuma 1975; and 1980): - 49 -(1) That muscle groups act or pul l along a l i ne connecting the i r centers of or ig in and i nse r t i on . Here the potential ef fects of over s impl i fy ing the l i ne (s ) of action of a mult i -pinnate muscle are apparent unless detai led determinations of the subgroups are made and the ac t i v i t y of each i s accounted for (Weijs, 1980). (2) That propor t ional i ty ex is ts between the force exerted by a muscle and i t s integrated electromyographic response. (3) That the ra t io between instantaneous (spec i f i c task response) and maximum possible muscle response level i s nroportional to the amount of muscle force per unit of physiological cross sect ion (assumed to be 10 kg/cm 2 in the i r ca l cu la t i ons ) . From th is model Weijs and Oantuma showed that the re lat ionships of the muscle, tooth and jo in t forces for a spec i f i c occlusal task were determined by the spec i f i c simultaneous a c t i v i t i e s exhibi ted by each muscle, in both species. The three muscle resul tant vectors, one for each view or plane, were shown to change the i r pos i t i on , magnitude and or ientat ion depending upon the phase of the chewing stroke or b i t i ng task. As a consequence, these parameters for the resu l t ing forces of resistance at the teeth and j o in t s were also va r iab le . In the rat these workers (1975) found that during mast icat ion* v i r t u a l l y a l l of the applied muscle force was transmitted to the molar teeth leaving the temporomandibular j o in ts unloaded. During inc i sa l b i t i n g , the increase in nape shi f ted the resul tant of the muscle forces more pos te r io r l y , as well as v e r t i c a l l y . This produced s ign i f i can t *According to Weijs and Dantuma (1975) during th is "an te r io r l y directed masticatory grinding stroke, the resul tants of the muscle forces at each side are i den t i ca l " ( i . e . b i l aterally symmetrical). - <=>0 -j o i n t loading because the muscle and tooth forces were not co inc ident , i . e . the tooth force now occurred farther anter io r ly than the muscle resu l tant . It i s for th is reason, they speculated, that the observed ac t i v i t y leve ls of most masticatory muscles are lower in b i t ing than during chewing, in both rat and rabbi t (Weijs, 1980). For un i la tera l chewing in the rabbit Weijs and Dantuma (1981) observed asymmetric ac t i v i t y between the workinq and balancing side muscles. Three dimensional s ta t i c analysis of un i la tera l molar contact showed a d i s t r i bu t ion of j o in t loadinq forces very s im i la r to those recorded by Hylander (1979c) in monkeys. The working side jo in t remained unloaded during the power stroke of chewing while the balancing j o in t remained loaded throughout. Furthermore, manipulations of the model predicted that a decrease in balancing side muscle force increases the load on the working side while decreasing bi te force. Increases in balancing side forces would increase the b i te force but tend to d is locate the working j o i n t due to the creat ion of t ens i l e force there (Hylander, 1979c). This i s to be expected i f the un i la tera l point of tooth contact i s considered as a pivot po int . Increasing balancing side muscle forces tend to balance then surpass those of the working side thus tending to rotate the mandible about th is pivot opposite to that where working exceeds balancing side muscle force. "Hence during natural mastication the muscles of both sides act in a proportion ensuring the largest b i te force possible without pu l l ing the a r t i cu la t i ng surfaces of the working side j o i n t apart" (Weijs, 1980, p. 716). In both animals Weijs and Dantuma found that the resu l t ing tooth forces required to maintain equi l ibr ium in v i r t u a l l y a l l instances were not purely - 51 -ver t i ca l as assumed by so many other analyses. There existed both an anteroposterior as well as a transverse component of tooth force. Based on the i r determinations of these forces and the i r interact ions Weijs and Dantuma were able to account for many of the charac te r is t i cs of form of the two mandibles, and the i r dent i t ions, and concluded that the masticatory apparatus i s adapted to maximize tooth forces, (Walker, 1978; Weijs and Dantuma, 1975 and 1981; Weijs, 1980). The incorporation of a l l simultaneously relevant var iables in the i r ana lys i s , for any given bi te point during s ta t i c masticatory funct ion, lends a great deal of c r e d i b i l i t y to th is type of model. However one point worth noting was the i r i n a b i l i t y to determine the proportions of the tota l transverse (mediolateral) force acting at each of the two jo in ts (Weijs and Dantuma, 1981). According to them th is is determined by the morphology and nature of the a r t i cu la r surfaces of the medial and la te ra l wal ls of the condylar fossae. The d i rect ion of the j o i n t reaction force i s stated by Weijs (1980) to be the determinant of the b i te force d i rec t ion . More than l i k e l y , however, the nature of the re lat ionships between the tooth and j o i n t forces have evolved together in a reciprocal manner depending upon the funct ional requirements of the animal. Weijs and Dantuma (1981) made the assumption that the j o i n t force resul tant in the la te ra l plane acted at a spec i f i c angle (with respect to the occlusal plane) based on morphological observations (eg. 18 degrees). This allowed the ver t i ca l component of j o in t force to be expressed in terms of the anteroposterior component, or vice versa, since the assumed angulation f ixed the i r re la t i ve proportions with respect to each other. Therefore the nine unknown var iables contained in - 52 -the i r s ix simultaneous l i near equi l ibr ium equations reduced to seven. However, since seven unknowns contained in s ix equations i s s t i l l s t a t i c a l l y indeterminate, a further s imp l i f i ca t ion of var iables was required. Weijs and Dantuma accomplished th is by considering the transverse of mediolateral j o in t force components to be represented by a single combined transverse resul tant for both j o i n t s . Since these two j o in t components were in a d i rect l i ne with each other th is was permissible but precluded the determination of the two (working side and balancing side) indiv idual components. The appl icat ion of s imi la r addit ional assumptions, (based upon anatomical or physiological measurements) to d i f ferent var iables in these equations can allow the determination of the two transverse components as well (see METHODS). For example, a three dimensional or ientat ion can be assigned to the tooth resistance forces. This has the potential benefi t of al lowing the incorporation of d i rec t vn vivo measurements of the magnitude, as well as o r ien ta t ion , of tooth force. Such measurements, in man at l eas t , are current ly feas ib le (eg. Graf et ^1_. , 1974). In any case, whether actual values for the three components of tooth force are used, or the re la t i ve proportions between the three (according to the assumed or ientat ion of the resu l tan t ) , other angular assumptions are needed to determine the proportion of transverse force at each j o i n t . If the addit ional assumption is made that the angulation of j o i n t resistance force in the frontal plane i s known for one of the jo in ts (eg. based on morphology) then the extent of the transverse j o i n t react ion force acting at each side can be determined, along with a l l other unknown var iab les . - 53 -Weijs and Dantuma obviously did not have the benefi t of knowledge of any aspect of the j o i n t reaction forces other than the a r t i cu l a r eminence angulation in the la te ra l plane, which they based on morphological evidence. Their analysis was made possible by th is s ingle assumption although a price was paid by the i r i n a b i l i t y to completely describe the nature of each of the two j o i n t reaction force resul tants for un i la tera l a c t i v i t i e s of the mandible. A recent modeling study by Hatcher and Coworkers (1986) incorporated a s im i la r assumption although the i r resu l ts are, on the whole, simply qua l i t a t i ve . They developed both a mechanical and a mathematic model to study human temporomandibular j o i n t loading based on the same basic premises as Weijs and Dantumas' analyses. However they were simply corre la t ing the mathematical analysis with that of the s t ra ight forward mechanical one. The l a t t e r incorporated force transducers at three molar tooth posit ions as well as at the two j o i n t s . Muscle forces due to the anter ior and poster ior temporal is , the super f ic ia l and deep masseters and the medial pterygoid of both sides of the skul l were simulated mechanically along the i r respective o r ien ta t ions . These muscle " forces" were derived simply from cross-sect ional data (Schumacher, 1961) and consisted of proportional values only. Var iat ion in both the occlusal as well as the working and balancing j o i n t forces were corre lated with arb i t rary a l te ra t ions in the balancing side muscle forces, the i r angulation and pos i t ion . They analyzed these same var ia t ions using the i r mathematic model of t he i r mechanical arrangement based on s ta t i c equi l ibr ium condi t ions. In addi t ion however the mathematic model also included estimates of EMG a c t i v i t y - 54 -which they had also derived from the l i t e r a t u r e . Their f indings were, on the whole, only comparative with some var ia t ion between the two models but they showed that the s ta t i c mathematical model approach f a i r l y c lose ly predicted the same var ia t ions in tooth and j o in t loading as the mechanical model. Th is , to some extent, substantiates the appl icat ion of mathematical analysis of the kind used by Weijs and Dantuma to predict actual mechanical events. The e luc idat ion of the nature of these external ly applied forces act ing on the mandible has further s ign i f icance when the internal d i s t r i bu t ion of these forces within the structure of the mandible i s considered. For instance, Hylander (1979c) suggested that the occurrence of var iab le d i rec t ions of macaca subcondylar loading was possibly due to rearrangement of the internal force d is t r ibu t ions within the mandibular condyles of some subjects. As such, more of the compressive load would be d is t r ibuted to more medial aspects of the condyles leaving more tens i le stress and st ra in along the la te ra l aspects during cer ta in funct ions. Therefore the subcondylar s t ra in or stress measured by Hylander at the la te ra l aspect could r e f l ec t th is e f fec t . It fol lows that the mediolateral posi t ion of the point(s) of heaviest resistance of the condylar surfaces and archi tecture of the supporting bone w i l l determine, or at leas t in f luence, th is d i s t r i b u t i o n . A number of papers have s p e c i f i c a l l y dealt with the re la t ionsh ips between external ly applied forces and the resu l t ing internal d is t r ibu t ions of stress within the mandible. Three dimensional photoelast ic stress patterns have been measured d i rec t l y from dried human mandibles (Mongini et al_. 1979). These have shown close cor re la t ion between bone trabecul at ion and jaw arch i tec ture , and surface d i s t r i bu t ion of l i nes of pr inc ipa l s t ress . Two and - 55 -three dimensional models of the human mandible have been developed to simulate these re la t ionships using f i n i t e element analys is (Iwata, et a l . 1979; Gupta e ta l_ . 1972, 1°73; Knoe l l , 1977; Harper, 1982). This i s a method whereby a computerized graphic model of the mandible i s constructed from an assembly of structural elements interconnected at spec i f i c common "nodal" po in ts . Such models have been shown to be a reasonably good representation of the in v i t r o biomechanical response of the mandible to a r t i f i c i a l occlusal loading s i tuat ions as compared to measurement of dried mandibles under the same condit ions (Knoel l , 1977). This information i s of great c l i n i c a l value when i t s app l icat ion to prosthet ic implants and osseous surgical procedures are considered. Knoell (1977) presented a three dimensional model aimed at describing the bone response to occlusal type loading in the areas adjacent to the teeth. Harper (1982) also incorporated muscle vectors acting on his f i n i t e element model due to the masseter, two d iv is ions of the temporal is, and the medial p terygoid.* The muscle loading produces a d is to r t ion of the structural nodes ( i . e . attachment points) according to the assumed modulus of e l a s t i c i t y of the s t ructure. From th is the magnitude of forces occurring at spec i f i c nodes, and hence the types of stresses produced, can be determined. This analys is was used to compare changes in mechanical advantage to the muscles f o r s imu la t i ons of d i f f e r i n g s u r g i c a l a l t e r a t i o n s of the mandible *These were determined according to some arb i t rary and rather s imp l i s t i c assumptions as to the absolute maximum force generation c a p a b i l i t i e s of each muscle, as well as the i r re la t i ve di f ferences in ac t i v i t y between occlusal funct ions. - 56 -used to correct prognathism. Thus the three dimensional descr ipt ion of the reciprocal re la t ionships which appear to ex is t between the external ly applied muscle forces, the internal d is t r ibu t ion of these forces within the mandible, and the resu l t ing external ly occurring resistance forces i s po ten t ia l l y of great c l i n i c a l importance. - 57 -B. PURPOSE OF STUDY A model of any b io log ica l system should reasonably predict the possible outcome of any p lausib le set of condit ions imposed on the system. The more real var iables one i s allowed to incorporate into a model, the more l i k e l y are i t s predict ions to be r e l i a b l e , and therefore usefu l . The wide var iety of possible combinations of these var iables requires a s im i la r degree of la t i tude in the capab i l i t i e s of the model. This i s most important i f real data from a given indiv idual with a par t icu lar c ran io fac ia l form is to be appl ied, biomechanically analyzed, and compared to another indiv idual or morphological group ( i e . c lass I, II or III skeletal bases; short versus long fac ia l types; and combinations thereof) . One of the main object ives in the design of a three dimensional model therefore would be to provide maximum f l e x i b i l i t y in the entry of anatomical as well as physiological parameters, whether real or assumed, and to allow these var iab les to be entered and/or al tered independently. The purpose of the present invest igat ion was, f i r s t l y , the establishment of th is in terac t ive and f l e x i b l e modeling system. The second object ive was the appl icat ion of a r e a l i s t i c and complete set of data to describe and quantify the biomechanical re la t ionships which ex is t between the forces produced by the muscles of mastication and the resu l t ing reac t ion , or resistance forces generated at the point of tooth contact and the two jo in ts in a hypothetical normal indiv idual undergoing d i f ferent s ta t i c tasks. S p e c i f i c a l l y the model and i t s appl icat ion to these par t icu lar tasks was aimed at answering the fol lowing questions: - 58 -What i s the three dimensional nature of the forces produced by the muscles (magnitude and or ientat ions) for a given task and to what extent do these muscle forces change with changes in the posi t ion and type of tooth contact (eg. anter ior versus poster ior ; un i la tera l versus b i l a t e r a l ; natural versus supported tooth contacts)? How do these changes in muscle force/tooth contact ( i . e . task) a f fec t the magnitude and or ientat ion of generated tooth forces in three dimensions? How do these changes in muscle force/tooth force a f fec t the magnitude and o r i e n t a t i o n of generated fo rces at the two temporomandibular jo in ts? - 59 -METHODS A. PRINCIPLES OF ANALYSIS As stated previously the primary purpose of th is study was to estab l ish a useful three dimensional model of the biomechanical re la t ionships between the forces generated by the muscles of mastication and the forces of resistance which occur at the teeth and/or j o i n t s , under s ta t i c ( i . e . isometr ic) condi t ions. As such, the pr inc ip les of s ta t i c mechanics apply which require the sum of a l l forces acting on the mandible to e f fec t i ve l y cancel one another thereby producing no movement of any part of the system, or the system as a whole. In order for th is to be true the sum of the l i near forces acting on the mandible in the three orthogonal d i rect ions (x, y , and z) must equal zero. That i s to say, the sum of a l l mediolateral (x) forces must be zero as must the sums of the anteroposterior (y) and ver t i ca l (z) forces: i e . (1 ) v F x = 0 (2) y. F y = 0 (3) r. F z = 0 In addit ion each l i near vector of force w i l l produce a torque e f fec t , or rotat ional moment, about the center of ro ta t i on , or fulcrum, of the mandible. However s ta t i c mechanics also requires that the total moment, or torque, about any axis passing through the fulcrum point must also equal zero: i e . (4) s M x = 0 (5) v. My = 0 (6) s M z = 0 - 60 -Thus, there can neither be any net l i near forces, nor any rotat ional moments act ing on the mandible under these condit ions and therefore no net movement. For th is reason the posi t ion of the fulcrum point and orthogonal axis of ro tat ion can be assumed to l i e anywhere in space in re la t ion to the mandible. It i s convenient to assign one of the points of resistance (eg. the r ight condyle) as the fulcrum for the system. In doing so the three orthogonal components of r igh t condylar resistance force, which can e x i s t , w i l l pass d i r ec t l y through the fulcrum along the i r respective axes. They therefore produce no torquing e f fec t , or moment, on the system. This el iminates the need to include these three par t i cu la r components in the equations describing the rotat ional moments of the system (equations (4) , (5) and (6) above) and great ly s imp l i f i es the subsequent ca lcu la t ions (see Ana lys is ) . The model mathematically determines the reaction forces ( i f any) which occur at the three points of resistance ( i . e . r igh t and l e f t j o in ts plus assumed tooth contact posi t ion) for any given mandibular system and s ta t i c funct ion, according to the muscle force generated. Since each of these three resistance forces can have an x, y and z component three dimensional ly, there are therefore nine unknown var iables contained in the above six equations (eg. l e f t j o i n t x, y and z ; r igh t j o i n t x, y and z ; and tooth force x, y and z components). As such, the equations can be sa t i s f i ed by an i n f i n i t e number of solut ions and are s t a t i c a l l y indeterminate (Weijs and Dantuma, 1981). However i t i s possible to reduce these unknowns such that a unique solut ion w i l l ex is t by making two important, but reasonable, assumptions regarding the or ientat ions of the resistance forces at (1) the point of tooth resistance and (2) at the l e f t condyle. - 61 -F i r s t of a l l , the three dimensional or ientat ion of the tooth resistance force at the designated point of tooth contact was spec i f i ed . The three orthogonal components of t h i s fo rce cou ld then each be expressed ( t r igonometr ical ly) in terms of the other, as a r a t i o . This e f fec t i ve l y reduced these three unknowns to a s ingle var iab le . It i s noteworthy that the or ientat ion spec i f ied for the tooth resistance force may be assigned a r b i t r a r i l y , or, assumed to correspond with the known or ientat ion of the overal l muscle forces. Conversely, they may be independently assigned according to three dimensional measurements of the actual forces occurring at the given point of tooth contact, for a spec i f i c occlusal funct ion. Such measurements could be made simultaneously with muscle EMG recordinqs in studies s p e c i f i c a l l y designed for th is purpose. Nevertheless th is would s t i l l leave seven unknowns in the six equations, which i s s t i l l s t a t i c a l l y indeterminate. A s im i la r assumption was then applied to the components of the l e f t j o i n t forces by speci fy ing an assumed or ientat ion of j o in t resistance (eg. based on morphological evidence) in one par t i cu la r plane that being the frontal plane (see Ana lys is ) . The two components of j o i n t force in that plane (mediolateral (x) and ver t i ca l (z)) were thus expressed in a s ingle term, also as a r a t i o . This reduced the unknowns to a total of s ix which does have a unique solut ion for a l l orthogonal components of j o in t and tooth force. - 62 -B. DATA ENTRY 1. Anatomical Var iables a . Coordinate System The three dimensional spat ial coordinates of a l l anatomical var iables were entered as distances in mi l l imeters re la t i ve to a common or ig in at the center of the r ight condyle. The medio la tera l , anteroposter ior , and ver t i ca l axes are designated x, y and z respect ive ly . A midsagit tal plane (yz) , a f rontal plane (xz) and a horizontal plane (xy) para l le l to the occlusal plane, thus represent the three orthogonal planes of reference passing through the o r i g i n . The three dimensional coordinates of the centroids of the areas of o r ig in and inser t ion of the muscle groups, the centers of the r igh t and l e f t condyles, and the points of contact of the mandibular dent i t ion were entered using a Hewlett Packard 9874A d i g i t i z e r (see Figure 4A and B) . Because the d i g i t i z e r is capable of handling these coordinates in only two dimensions at a time a l l points were entered from the la te ra l plane for the anteroposterior and ver t i ca l dimensions, then again from the frontal plane for the mediolateral dimension. Since the program deals with the mandible as a b i l a t e r a l l y symmetrical structure only the r ight side of the mandible was d ig i t i zed in the frontal plane. The l e f t side was generated by the computer as a mirror Image of the r igh t in both the frontal and the resu l t ing horizontal plane (see Figures 4A and b) . These coordinates may be entered from any detai led f u l l scale image of a given mandibular system whether i t be anatomical drawings, tracings from cephalometric headfi lms, or d i rec t l y f rom such f i lms or other diagnostic - 63 -imaging reproductions. Although any mandibular "system" ( i . e . from any given ind iv idua l ) can theore t i ca l l y be modeled, the necessity of describing these re lat ionships for a hypothetical "normal" mandibular apparatus required the pooling of mean data or ig inat ing from a var iety of sources in the l i t e r a t u r e . The or ig ins of the various parameters, the assumptions made regarding the i r incorporat ion into a data f i l e representing a hypothetical "normal" i nd i v i dua l , and the appl icat ion of th is information to a var iety of masticatory tasks i s discussed in the fo l lowing. b. Muscle Attachments The coordinates of a l l the muscles except d igas t r i c were derived from the work of Baron and Debussy (1979) which was based on 5 human s k u l l s . No attempt was made to estab l ish the sex, age or ethnic or ig in of these s k u l l s . As these workers s ta te , "In functional anatomy and biomechanics, the average f iber is represented by an average force, each vector of which has two points of anchorage: one mobile, mandibular, and one f i x e d , c r a n i o - f a c i a l " (p. 547). The determination of the respective areas of or ig in and inser t ion for each muscle f asc i c l e were made according to i den t i f i ab le bony landmarks which are due to t rac t ion of the given f a s c i c l e , or group, at i t s attachment s i tes (Van der Klaauw 1963). Baron and Debussy described a total of 24 separate muscle fasc i c les (12 on each s ide , r ight and l e f t ) within the four major masticatory muscles of each side (excluding d i gas t r i c ; see Figure 4A and B ) . According to the i r c r i t e r i a the masseter muscle i s divided into four fasc i c les or groups; a super f i c ia l and intermediate group, and a deep group consis t ing of an - 64 -anter ior and a poster ior por t ion. The medial pterygoid muscle consists of three f a s c i c l e s ; an anter ior part , a super f ic ia l poster ior par t , and a deep poster ior par t . The temporalis muscle consists of an anter ior group which includes the zygomatico-mandibularis portion ( i den t i f i ab le in most mammalian species as a d i s t i nc t muscle i t s e l f , eg. Schumacher, 1961; Turnbu l l , 1970) a middle group of oblique f i be r s , and a poster ior group of horizontal f i be r s . Lateral pterygoid is divided into a superior sphenoidal head, and an i n f e r i o r pterygoidal head. A l i s t of the areas of anatomical or ig in and inser t ion described by these workers i s included in Appendix I. Although the i r study represents one of the most detai led descr ipt ions of the functional d iv is ions of the masticatory muscles to date i t was necessary to combine cer ta in of the subgroups of Baron and Debussy to derive components for which physiological data were ava i l ab le . Thus 9 pai rs of muscles (18 in a l l ) were included and spec i f ied by attachment s i t e ( i . e . or ig in and inser t ion) including the super f ic ia l and deep masseters, medial pterygoid, an te r io r , middle and poster ior temporal is, superior and i n fe r i o r heads of the la te ra l pterygoid, and d i g a s t r i c . Their spec i f i c or ientat ions were ca lcu lated as l i nes representing the middle of the body of each muscle from or ig in to inser t ion (Hiiemae, 1967; Barbenel , 1969 and 1974; Pruim et a l . , 1980). Figure 4A and B depicts both the muscle subgroups described by Baron and Debussy and the functional subgroups incorporated in th is study (Gys i , 1921). Both deep (DM) and super f ic ia l masseter (SM) are each represented by a s ingle l i n e of act ion as opposed to the two fasc i c les of Baron and Debussy. The three anatomical f asc i c les of the medial pterygoid (MP) of these workers are - 65 -MT a b c d e f g Superficial masseter Intermediate " Posterior deep masseter Anterior 11 " Anterior medial pterygoid Posterior » ^ S M . DM -superf. layer -deep layer MP AT PT F i g u r e 4A. LATERAL PLANE ATTACHMENT POINTS OF THE NINE MUSCLE GROUPS. The heavy l ines represent those groups incorporated in this study: SM, superficial masseter; DM, deep masseter; MP, medial pterygoid; AT, MT, PT, anterior, middle and posterior temporalis respectively; IP, inferior head of lateral pterygoid; SP, superior head of lateral pterygoid, and DI, digastr ic. Also depicted (thin lines) are the muscle groupings of Baron and Debussy (1979) from which the SM, DM and MP single l ines of action and attachment were derived. Al l other muscle groups, except DI, correspond with the divisions described by Baron and Debussy (see Appendix I) . The tooth points of contact from incisor to third molar are depicted (dots) as well as those described by Baron and Debussy (small c i rc les - see text for description). The figure assumes the jaw is in the closed posit ion. - 66 -a b c d e f g Superficial masseter 3" SM Intermediate " Posterior deep masseter Anterior » " Anterior medial pterygoid Posterior » » DM -superf. layer -deep layer MP Figure 4B. FRONTAL PLANE VIEW OF THE MUSCLE GROUPS OF FIGURE 4A. On the r igh t of the f igure are the muscle groups and tooth contact points incorporated in the model. On the l e f t are the muscle d iv is ions of Baron and Debussy (1979) from which SM, DM and MP were der ived, and the tooth contact points they used (see text for descr ip t ion) . - 67 -l ikewise represented by a s ingle l i ne of ac t ion . The combination of these subgroups into the i r respective sinqular l i nes of act ion for model analysis was done by determining the geometrical center of the attachment points of Baron and Debussy for the two deep masseter fasc i c les and the two super f ic ia l masseter f a s c i c l e s . The centers of attachment for the s ingle medial pterygoid were derived by f i r s t determining the midpoint of attachment of the super f i c ia l and deep poster ior group and then s im i l a r l y determining the midpoint of th is combined poster ior group and that of the anter ior f a s c i c l e . Except for d igas t r i c the remaining muscle groups ( i . e . anter ior (AT) middle (MT) and poster ior temporalis (PT), and superior (SP) and i n f e r i o r (IP) heads of la te ra l pterygoid) used in the computer modeling system correspond to those described by Baron and Debussy. Digast r ic (DG) or ientat ions were derived from Dubrul (1980) and Pruim et al_. (1978, and 1980). c. Tooth Positions Although any point of the den t i t i on , whether un i la tera l or b i l a t e r a l , may in theory be used, i t i s obvious that a given task which involves a cer ta in occlusal contact w i l l a lso involve a spec i f i c pattern of muscle a c t i v i t i e s . The mathematical ana lys i s , described further below (see Ana lys i s ) , assumes "point" contact at the teeth and condyles, and not necessar i ly bearing "surfaces" as such. It i s therefore important to know exact ly where the applied point of tooth contact l i e s in space. The tasks included here therefore consist of those on spec i f i c occlusal po in ts , those which were b i l a t e r a l l y symmetrical, or at the very leas t car r ied out - 68 -at an assumed single point or tooth ( i . e . chewing). Though not a l l of the muscle data described was idea l l y matched, i t nevertheless represents the best estimates avai lable for man. Figure 4A and B shows the four dental reference points chosen ( a r b i t r a r i l y ) by Baron and Debussy (1979). From anterior/medial to pos te r io r / l a te ra l they are; (1) the contact point between the two central i n c i s o r s ; (2) the contact point between opposing canines; (3) the most i n f e r i o r point of occlusal contact of the mandibular molars and (4) the most d is ta l molar point . Whereas Baron and Debussy (1979) chose an occlusal plane corresponding to a l i ne drawn from point 1 to point 4 ( inc isor to d is ta l most molar po in t ) , for computer model inn purposes an occlusal plane was chosen to correspond to an arb i t rary l i n e drawn intermediate between the above points in the la te ra l plane. Figure 4A and B depicts th is plane of occlusion as well as the indiv idual points of contact for each tooth including th i rd molars. The contact points were arr ived at by comparing re la t i ve tooth morphology and cuspal posi t ions on dried sku l l s and drawing them within the confines of the dental reference points of Baron and Debussy (1979).. Molar contact points correspond to mesio-buccal cusps (of the mandibular teeth) . d. Tooth Angles of Resistance The assumed angle of tooth resistance force in the la te ra l and frontal planes (a and R ; Figures 5 and 6 respect ively) were speci f ied to match the overal l or ientat ion of applied muscle force in these two planes. As was discussed e a r l i e r th is feature allows for the incorporat ion of e i ther a r b i t r a r i l y determined angles of tooth res is tance, or actual or ientat ions - 69 -measured d i rec t l y using three dimensional force transducer arrangements designed for th is purpose. The l a t t e r s i tua t ion would maximize the correspondence between the overal l muscle force alignments and tooth resistance force or ientat ions where both are recorded simultaneously from a given indiv idual . It should be emphasized here that the program analyzes the re lat ionships of various forces in the system from a purely mathematical point of view. Since the system the program is intended to model i s a b io loq ica l one, cer ta in re la t ionships must be borne in mind and some constraints applied to the analysis as a whole. For instance, the program w i l l accept any angular or ientat ion of tooth resistance spec i f ied by the user, whether i t i s appropriate for the given muscle data or not. It i s theore t i ca l l y possib le to specify re t rus ive ly directed muscle e f fo r t with an inappropriate or ientat ion of tooth resistance (would expect the l a t t e r to correspond in nature to res i s t the muscle e f f o r t ) . Unless s p e c i f i c a l l y matched data are ava i l ab le , eg. as a consequence of experiments designed for the purpose, i t i s therefore most appropriate to match the nature of the muscle e f fo r t with an angle of tooth resistance most l i k e l y to occur in the system. The a b i l i t y to minimize the number of unknown var iables in the s ix equi l ibr ium equations for these s ta t i c functions by assigning values to the tooth force or ientat ions has a lso been discussed (see Pr inc ip les of Analysi s ) . - 70 -e. Condylar Position The or ig in of the reference system, as stated lay at the center of the r ight condyle. The vertex or superior-most out l ine of the condyles shown in Figure 4A and B was the basis of the coordinate system used by Baron and Debussy (1979). Their abscissa lay al onq the l i ne jo in ing the ver t ices of the two condyles. Al l other dimensions of the condylar and mandibular out l ines were drawn a r b i t r a r i l y and d i g i t i z e d . The l a t t e r had no bearing upon subsequent modeling procedures, and were used only to enhance graphic portrayal of the data. f. Condylar Angles of Resistance The or ientat ion of the resistance force occurring at the l e f t condyle in the frontal plane only, was also spec i f ied ( y , see Figure 6). This also enabled determination of the s ta t i c equi l ibr ium equations by further reducing the number of unknown var iables (see Pr inc ip les of Ana lys i s ) . The frontal angle of th is resistance force was chosen a r b i t r a r i l y over that in the la te ra l or horizontal plane to minimize constra ints on the condylar resistance forces in the l a t t e r two planes. Since only th is angle was f ixed a l l other condylar resistance force or ientat ions were derived by the computer. - 71 -2. Physiological Variables Force analysis for any simulated task required the determination of the contr ibut ion by each muscle to the overal l forces of the system. These indiv idual forces were determined, along each muscle's spec i f i c l i ne of p u l l , according to two assumptions: F i r s t , large muscles are capable of producing more isometric contract ion force than small ones, the tension in each muscle being proportional to the product of i t s physiological cross section and an assumed force constant per unit of cross sectional area. Second was that various s ta t i c clenching tasks, as well as d i f fe rent phases of the c los ing stroke in chewing, involve d i f fe ren t amounts of ac t iva t ion in a given muscle depending upon the task or phase of the task (MacDonald and Hannam, 1 9 8 2 ; Mo l le r , 1 9 6 6 ) . In other words, the same muscle may exh ib i t 100% a c t i v i t y during one task or phase and only 50% during another. Thus, the resul tant vector of muscle force (Mir) for a par t i cu la r muscle in isometric contract ion at a spec i f i c moment, or during a given task would be given by the product [XMI ' Kl • EMGMl- = Mir where X m j i s the cross-sect ional diameter of muscle Mi in cm 2 , K i s a constant for skeleta l muscle (expressed in N/cm 2), and EMGM,J i s the r a t i o , or scaled value, of the muscle contract ion re la t i ve to i t s maximum response for any task (Pruim et al_., 1 9 8 0 ; Wei js, 1 9 8 0 ) . The product [ X ^ j - K ] i s hereafter referred to as the "Weighting Factor" given to the muscle M i , and the value EMGrvii as i t s "Scal ing Fac tor " . - 72 -a . Weighting C r i t e r i a A v a r i e t y of s tud ies have est imated the fo rce per un i t of cross-sect ional area of skeletal muscle, (K). Although there seems to be considerable var ia t ion in th is value among invest igators [Ralston et a l . , (1949), 1.3-2.4 kg/cm 2 ; Haxton (1944), 3.9 kg/cm 2 ; Hettinger (1961), 4.1 kg/cm 2 ; Ikai and Fukunaga (1968), 7.1 kg/cm 2 ; Gysi (1921), 6.46 kg/cm 2 , Morris (1948), 9.2 kg/cm 2 ; Fick (1910 as c i ted by Pruim et al_. 1980), 10.0 kg/cm 2 ; and Pruim et al_. (1980), 13.7 kg/cm 2] much of th is var ia t ion has been at t r ibuted to di f ferences in methods and errors in force generation and muscle cross sect ion determinations, and the integrat ion of these data (Weijs, 1980; Weijs and H i l l en, 1984a). Such var iables as subject mot ivat ion, discomfort, muscle rest ing length, accuracy of cross sect ion measurement and muscle f iber type ( i . e . fast versus slow-twitch motor units) a l l inf luence these determinations. However, despite these di f ferences and the considerable var ia t ion in the force per unit area known to ex is t between i nd i v idua l s , a mean value of 4.1 kg/cm 2 , or 40 N/cm 2, which i s independent of sex, age and muscle, seems most appropriate (Ganong, 1977; Weijs and Hi 11 en, 1984a) and was chosen as the weighting constant (K) for th is study. Weighting factors assigned to each muscle and the i r der ivat ion from determinations of whole muscle cross-sect ional areas are given in Table I. The whole muscle group cross-sect ions are from the CT scan work of Weijs and H i l len (1984a and b ) , and represent the b i l a te ra l mean cross sectional areas of the four main masticatory muscle groups ( i . e . masseter, medial pterygoid, temporalis and la te ra l pterygoid) of 16 male subjects. This group - 73 -had an average age of 35 years , mean number of missing teeth 1.8 and normal healthy occlusions (Angle c lass I or I I ) . Wei js and Hi l i e n (1984b) took CT scans of t h e i r sub jec ts at approximately r igh t angles to the mean masticatory muscle f iber d i rect ions midway between the or ig in and inser t ion points of each with the jaw in occ lus ion. The or ientat ion of the scan planes had been determined according to previous scans of cadavers such that only three planes were required to study a l l muscles b i l a t e r a l l y (Weijs and H i l l e n , 1984a). These planes were: 1 cm above the zygomatic area and para l le l to the Frankfort horizontal plane (FH) for temporal is; 3 cm anterosuperior of the mandibular anqle at 30 degrees to the FH for masseter and medial pterygoid; and 1 cm anter ior to the la te ra l poles of the condyles perpendicular to FH for the la te ra l pterygoid muscle (Weijs and H i l l e n , 1984a and b) . The cadaver study also provided these workers with spec i f i c l i near regression equations re la t ing the cross sect ions, measured from the scans, with actual d i rec t determinations of the muscle cross-sect ions measured according to two d i f fe rent procedures known in the l i t e ra tu re as the method of Weber (1946) and that of Buchner (1877) (see Weijs and H i l len 1984a for a complete discussion of the two). The former method i s mathematical and i s simply the total f iber weight of a given muscle divided by that muscle's average f iber length. The l a t t e r i s the actual measurement of the total cross sect ion of the teased and stacked f iber bundles of a given muscle. Roth are tedious and require detai led d issect ion and the e l iminat ion of a l l elements of vesse ls , fa t and loose connective t i s sue , and give s l i g h t l y d i f fe rent r e s u l t s . - 74 -Although Weijs and H i l l en (1984b) provide the mean muscle cross-sect ions of the i r subjects predicted according to both Weber's and Buchner's d issect ion methods the predict ions for the l a t t e r were incorporated in th is study. Buchner's method (and thus also the predict ions from the scans) was f e l t to be more accurate and involve less residual error in the l i nea r regression equations when used with scanned cross sections (Weijs and Hi 11 en, 1984a, Table I I I ) . It predicts the ra t io of the cross sections of masseter: medial pterygoid: temporalis to be 1.00 : 0.6 : 1.1 when the masseter i s used as a reference. This agrees f a i r l y well with the same proportions derived from the work of Carl soo (1952) of 1.0:0.5:1.3 and from that of Schumacher (1961) of 1.0:0.5:1.2 (Pruim et a l . , 1980). The same r a t i o , when calcu lated for the dissected cross-sect ions of Wiejs and H i l l en (1984a), shows the Buchner method (1 .0 :0 .6 :1 .1) to be somewhat c loser to th is re la t ion than the Weber method (1 .0 :0 .7 :1 .1 ) . Since the anatomical d iv is ions of the muscle groups of the present study are more deta i led than the whole muscle cross sections provided by Weijs and H i l l en (1984a and b) d iv is ion of the total cross-sect ional areas of each muscle into i t s component groups was necessary. The d i v i s ion of the masseter in to a super f i c ia l and deep qroup in the proportions 0.7 and 0.3 respect ive ly was made a r b i t r a r i l y but was based on the general re la t ionships depicted in standard anatomical texts (eg. Dubrul, 1980). The temporalis muscle was divided into the proportions 0.48, 0.29, and 0.23 for the anter ior (AT) middle (MT) and poster ior (PT) port ions respect ive ly (see Table I ) . This d i v i s ion was based on the ra t i o between AT and PT of 1.0:0.6 from Carlsoo (1952). In order to s p l i t the temporalis in to - 75 -three groups i t was decided, a r b i t r a r i l y , to assign an equal portion of AT and PT to MT. In th is way the ra t io between these three qroups became l .n : 0.6 : 0.5 corresponding to the proportions designated. Although la te ra l pterygoid is divided into an upper (SP) and lower (IP) group anatomical ly, the lack of physiological response data (discussed below)for the SP and the overal l uncertainty surrounding i t s actual function (Grant, 1973; McMamara, 1973; Lipke et al_., 1977; Juniper, 1981; Mahan et a l . , 1983), made i t impractical to include the SP as a d i s t i n c t group in a l l tests at th is t ime. However, the re la t i ve proportions of th is muscle's overal l cross section contributed by each head has been determined, and i s included in Table I. According to the data of Honee (1970) the SP and IP heads can be proportioned at 0.30 and 0.70 respect ive ly whether the i r cross-sect ions are determined according to Buchner's method or Weber's method. Simi lar proportions (0.26 and 0.74) are also found in the la te ra l pterygoid functional analyses of Grant (1973). The assumed d igast r ic cross-sect ion was that of Pruim et al_. (1980) which was based on the d issect ion of four pairs of anter ior d igas t r i c muscles from e lder ly people according to the methods of Buchner and Weber already discussed. These workers ac tua l ly determined a mean cross section of 0.8 cm2 per side (S.D. = 0.2 cm2) but used a value of 1.0 cm2 in the i r biomechanical ca lcu la t ions to account for s ize di f ferences which they f e l t would ex is t between the i r e lder ly d issect ion subjects and thei r young adult biomechanical analysis subjects. - 76 -TABLE I - WEIGHTING FACTORS. Values for the nine functional muscle groups of th is study according to the i r proportions of the whole muscle cross-sect ions of Wiejs and Hi 11 en (1984b), assuming a force capab i l i t y of 40 N/cm2. The determination of the proportioning values of the whole muscles into the i r respective groups i s discussed in the tex t . Whole muscle abbreviat ions: M, masseter; MP, medial pterygoid; T, temporal is; LP, l a te ra l pterygoid. Muscle group abbreviat ions: SM, super f ic ia l masseter; DM, deep masseter; AT, MT and PT, anter io r , middle and poster ior temporalis respec t ive ly ; IP, i n fe r i o r la te ra l pterygoid; SP, superior la te ra l pterygoid; DG, d i gas t r i c . WHOLE MUSCLE MUSCLE MUSCLE MUSCLE X-SECTION (cm2) GROUP GROUP GROUP PROPORTION X-SECT (cm2) WEIGHT (N) M 6.80 1.69 SM 0.70 4.76 190.40 DM 0.30 2.04 81.60 MP 4.37 0.96 MP 1.00 4.37 174.80 AT 0.48 3.95 158.00 T 8.23 1.13 MT 0.29 2.39 95.60 PT 0.23 1.89 75.60 LP 2.39 0.45 IP 0.70 1.67 66.90 SP 0.30 0.72 28.70 DG* 1.00 1.00 40.00 *Cross-sect ion according to Pruim e t . a l . (1980). - 77 -b. Scaling Criteria 1. Clenching Tasks Scaling Factors for each muscle are shown in Table I I . The values for SM, MP, AT and PT were derived from the work of Macdonald (1982). Those for DM were taken from Belser and Hannam (1986), those for IP ac t i v i t y from Wood et al_. (1985), and those for SP and DG (for Tasks 1, 2, 3 and 4) from Mahan et al_. (1983) and Gibbs et al_. (1984). The clenching tasks of th i s study are the same as those used by MacDonald (1982). His electromyographic data for these tasks therefore provided the basis for the Scaling Factors which were used. Scal ing Factors for muscle groups for which no representative data could be found in the l i t e r a t u r e were estimated from avai lab le evidence suggesting the i r l i k e l y contr ibut ions to a given task. The values for the re la t i ve a c t i v i t i e s seen in Table II represent normalized mean EMG data for maximal voluntary clenches for the s ix s ta t i c clenches at the various points of the den t i t i on . MacDonald's recordings were obtained with ac ry l i c "stops" custom made to each subject 's den t i t i on . This produced an increase in ver t i ca l dimension of lmrn at the inc isors in each case (except Task 1 and 4 ) . His data was averaged for ten to twenty subjects (S = 20) (mean age 31-35 years depending on the test ser ies) performing each task at leas t 5 t imes. The DM data of Belser and Hannam (1986) (S = 20; mean age = 33 years) show the responses of SM and DM to be the same (95%) durinq maximal voluntary clenching in the intercuspal pos i t i on . As such DM scal inq values for Task 1 of Table II are the same as SM, both of which agree with the above. DM values for b i l a t e ra l molar clenching (Task 2) were also assumed to match SM - 78 -values. For un i la tera l tasks (Task 5 and 6) the DM values were assigned those of the SM of the same side since Bel ser and Hannam (1^86) found no s ign i f i can t d i f ference between these muscle groups during un i la tera l clenching (although occlusal stops were not incorporated in the i r study). For inc isa l clenchinq Belser and Hannam found SM ac t i v i t y at ~39% which i s s t r i k i ng l y s im i la r to the data of MacDonald (1982, Table I I , Task 3 and 4 ) . Corresponding DM ac t i v i t y of ~26% for inc isa l clenching found by Belser and Hannam was therefore assigned in these tasks. MT a c t i v i t y has been general ly neglected as a d i s t i n c t ent i ty in any form in the l i t e ra tu re despite i t s di f ferences in o r ien ta t ion . To deal with th is void of information i t s scal ing values were derived by assuming ac t i v i t y intermediate between the anter ior and poster ior groups of th i s muscle. The i n fe r i o r pterygoid (IP) a c t i v i t i e s measured by Wood et _aT_. (1985) (S = 9 with s im i la r age p ro f i l es to the above cases) shows mean values of about 27% of maximum for ver t i ca l intercuspal c lenching, and 71% for i nc i sa l clenching on natural contacts (see Table I I , Task 1, 3 and 4 ) . By comparison Mahan et al_. (1983) (S = 9) and Gibbs et al_. (1984) (S = 11) recorded mean a c t i v i t i e s for these same tasks (but with an anter ior sp l i n t or stop for the i nc i sa l e f for t ) of 27% and 33% for the intercuspal clench respect ive ly and 60% and 72% for the inc isa l c lench. Both these l a t t e r workers a lso found IP to be about 30-40% act ive for ver t i ca l clenching on a 1.5 mm fu l l arch sp l i n t (see Table I I , Task 2 ) . The greater a c t i v i t y in SP versus IP during c lenching, to which IP or ientat ion seems less suited (Grant, 1973), has general ly been found by others as well (Juniper, 1981, in man; MacNamara, - 79 -TABLE II - SCALING FACTORS (CLENCHING). Values (mean normalized EMG a c t i v i t i e s ) assigned to the various muscles for the given clenching tasks derived from l i t e ra tu re sources. The r ight side i s assumed to be the working or i p s i l a t e r a l side in a l l cases. Muscle abbreviations are the same as in Table I. Working side (WS) is on the r igh t (R). 1 2 INTERCUSPAL BILATERAL CLENCH MOLAR CLENCH (NATURAL) CO (M2 STOPS) BIMOL 3 4 5 6 INCISAL INCISAL UNILATERAL UNILATERAL CLENCH CLENCH CANINE CLENCH MOLAR CLENCH (WITH STOP) (NATURAL) (CANINE STOP) (M1/M2 STOP) INCISS INCISN UNIK9 R/WS L/BS R/WS L/BS R/WS L/BS R/WS L/BS R/WS L/BS UNIMOL R/WS L/BS SM 1.00 1.00 0.81 0.81 0.43 0.43 0.40 0.40 0.46 0.58 0.72 0.60 DM 1.00 1.00 0.81 0.81 0.26 0.26 0.26 0.26 0.46 0.58 0.72 0.60 MP 0.76 0.76 0.82 0.82 0.59 0.59 0.78 0.78 0.55 0.47 0.84 0.60 AT 0.98 0.98 0.83 0.83 0.26 0.26 0.08 0.08 0.54 0.14 0.73 0.58 MT 0.96 0.96 0.83 0.83 0.18 0.18 0.06 0.06 0.48 0.20 0.66 0.67 PT 0.9.4 0.94 0.84 0.84 0.09 0.09 0.04 0.04 0.42 0.26 0.59 0.39 IP 0.27 0.27 0.36 0.36 0.71 0.71 0.71 0.71 0.30 0.65 0.30 0.65 SP 0.59 0.59 0.61 0.61 0.50 0.50 0.50 0.50 DG 0.28 0.28 0.33 0.33 0.50 0.50 0.50 0.50 SM, MP, AT, PT - from MacDonald (1982) : Figure 17 [Task 1] ; Figure 14 [Task 2 ] ; Figure 10 [Task 3, 4 and 5 ] ; and Figure 15 [Task 6 and 7 ] . DM - from Belser and Hannam (1986) MT - a r b i t r a r i l y assigned IP - from Wood e t . a l . (1985; Figure 3 ) . SP, DG - from Mahan e t . a l . (1983; Figure 5) and Gibbs e t . a l . (1984; Figure 5) . - BO -1973, in monkeys). For un i la tera l clenches (Table II, Task 5 and 6) the data of Wood et al_. (1985) for the working and balancing side IP groups were assigned. Scal ing values are included for SP and DG for b i l a te ra l clenches only (Tasks 1, 2, 3 and 4 ) . No normalized EMG data i s ava i lab le regarding a c t i v i t y in these muscle groups for un i la tera l c lenching, or chewing. 11. Chewing Table III represents the approximate Scaling Factors of the muscle qrouos for un i la tera l gum-chewing. The three time in te rva ls are: (1) 100 msec p r i o r ; and (2) 50 msec pr ior to the onset of in tercuspat ion; and (3) time zero which was about 10-15 msec after i t s onset. Except for DM and MT a l l data were derived from Figure 31 (pp. 80-81) in the work of Mol ler (1966) which presents the average e lec t r i ca l ac t i v i t y recorded for each muscle group (S = 36). The responses were determined at 150, 200 and 250 msec of his time frame which was referenced to the onset of AT a c t i v i t y . The i n i t i a l onset of intercuspat ion occurred s l i g h t l y before the 250 msec interval by about 10-15 msec. The mean a c t i v i t i e s determined from th is f igure are simply the averages of the recorded leve ls and are not normalized to the i r respect ive potent ial or maximum l e v e l s . Therefore each muscle group's normalized scale of chewing a c t i v i t y for these in te rva ls was determined by f inding the proportion of i t s a c t i v i t y re l a t i ve to a mean maximum recorded during any funct ion. For SM, MP, AT and PT these maximal values were taken from the average mean voltage recorded by Moller (1966) during intercuspal maximum b i t ing (Table 25, p. 144; S = 36). The maximal - 81 -TABLE III - SCALING FACTORS (CHEWING). Values assigned to the muscle groups for un i la tera l gum-chewing (see text for sources). Time zero of Interval 3 occurred about 10-15 msec af ter the onset of in tercuspat ion. Abbreviations correspond to Tables I and I I . INTERVAL 1 100 msec INTERVAL 2 50 msec INTERVAL 3 Time 0 R/WS L/BS R/WS L/BS R/WS L/BS SM DM MP AT MT PT IP SP DG 0.33 0.33 0.77 0.45 0.43 0.41 0.25 0.00 0.23 0.23 0.63 0.31 0.32 0.33 0.18 0.00 0.56 0.56 0.97 0.65 0.60 0.54 0.35 0.00 0.20 0.20 0.47 0.51 0.53 0.54 0.25 0.00 0.36 0.36 0.75 0.45 0.37 0.29 0.15 0.00 0.09 0.09 0.29 0.36 0.36 0.35 0.15 0.00 - 82 -mean a c t i v i t y f o r IP occurred during bread chewing (Table VI, p. 222). DG i s not a c t i v e during t h i s phase of chewing ( M o l l e r , 1966; Munro, 1973). Although B e l s e r and Hannam (1986) describe a d i f f e r e n c e i n mean peak a c t i v i t i e s between DM and SM on the balancing side f o r u n i l a t e r a l chewing of 51% versus 24% r e s p e c t i v e l y , they provide no time frame f o r t h i s d i f f e r e n c e r e l a t i v e to i n t e r c u s p a t i o n . As such, f o r the purposes of a n a l y s i s of u n i l a t e r a l chewing, the deep p o r t i o n of the masseter muscle was a r b i t r a r i l y assigned the same scale values as SM. SP a c t i v i t i e s are not a v a i l a b l e f o r chewing and MT was assumed to e x h i b i t a c t i v i t i e s intermediate between those of AT and PT. C. COMPUTER ANALYSIS With a l l the n e c e s s a r y i n p u t f o r a ta s k i n q u e s t i o n e n t e r e d biomechanical a n a l y s i s by the computer was c a r r i e d out. From the angular o r i e n t a t i o n of each muscle group, i t s Weighting Factor (maximum force c a p a b i l i t i e s ) and i t s r e l a t i v e S c a l i n g Factor value (task s p e c i f i c p r o p o r t i o n a l a c t i v i t y ) the program determined the r e s u l t a n t vector of force generated by each muscle along i t s l i n e of a c t i o n f o r the s p e c i f i c task being analyzed. From t h i s r e s u l t a n t the computer program then determined the corresponding x, y and z components of force f o r each muscle group f o r which the tooth and condylar r e s i s t a n c e s forces were c a l c u l a t e d . As discussed the mechanics of the program were based on the laws of s t a t i c e q u i l i b r i u m and c o n s i s t e d of the f o l l o w i n g steps which are summarized i n Figures 5, 6 and 7. - S3 -1. L a t e r a l P l a n e - S t e p 1 I n i t i a l l y , as shown i n F i g u r e 5, the system was a n a l y z e d i n the l a t e r a l p l a n e . The c o n d y l e s were assumed to be c o a x i a l , w i t h the r i g h t one a c t i n g as the f u l c r u m . The computer determined the l e n g t h of each moment arm f o r the muscle f o r c e components ( e g . dMy and dMz) as well as those f o r the t o o t h f o r c e components (dTy and d T z ) . The moment of f o r c e about the f u l c r u m i s g i v e n by the f o r c e v e c t o r component a l o n g each a x i s m u l t i p l i e d by i t s moment arm ( p e r p e n d i c u l a r d i s t a n c e from the f u l c r u m ) . F o r the s i t u a t i o n i n F i g u r e 5, which f o r reasons of c l a r i t y i n c l u d e s o n l y one muscle p a i r ( s u p e r i m p o s e d ) , the moments of f o r c e due t o the muscles are the p r o d u c t s of My (dMy) = Y Moment of Muscle F o r c e Mz(dMz) = Z Moment of Muscle F o r c e Both of th e s e a c t a n t i c l o c k w i s e about the f u l c r u m (the c o n d y l e s ) . S i m i l a r l y the moments about the f u l c r u m due t o the components of t o o t h r e s i s t a n c e f o r c e are the p r o d u c t s Ty(dTy) = Y Moment of Tooth F o r c e T z ( d T z ) = Z Moment of Tooth F o r c e both of which are c l o c k w i s e i n d i r e c t i o n . There a re no moments c o n t r i b u t e d by any j o i n t f o r c e s i n t h i s p l a n e as a l l j o i n t f o r c e s pass thr o u g h the f u l c r u m . The net moment of r o t a t i o n of the system i s g i v e n by the d i f f e r e n c e between the sums of the t o t a l c l o c k w i s e moments and the t o t a l a n t i c l o c k w i s e moments. By d e f i n i t i o n , the sum of the moments must equal z e r o , s i n c e no jaw movement i s assumed t o take p l a c e . i . e . v, (TOTAL CLOCKWISE MOMENTS) + r. (TOTAL ANTICLOCKWISE MOMENTS) = 0 t h e r e f o r e , - 84 -(1) y (My(dMY) + Mz(dMz)l + (Ty(dTy) + T z ( d T z)l = 0 From equation 1, a l l va r iab les have known values except Ty and T z . Since both the angles of the tooth resu l tan t [ a , l a t e r a l angle; and 0, f ronta l angle) are s p e c i f i e d , the r e l a t i v e r a t i o s of the vectors Ty and Tz can be determined from: tan a = Tz/Ty and Ty = Tz / tan a (See Figure 5) where T R L A T 1 S t n e r e s u l t a n t tooth force projected onto the l a t e r a l p lane . Rewrit inq equation (1), and expressing Ty in terms of Tz t h i s equation becomes: y. CMy(dMy) + Mz(dMz)] + [ (Tz / tan a) (dTy) + Tz(dTz)] = 0 or TzfdTz + (dTy) / tan a) = z (Mz(dMz) + My(dMy)l Since only Tz of t h i s equation i s unknown, i t s value i s e a s i l y determined. Subs t i tu t ion of the r e l a t i o n Ty = Tz / tan a y i e l d s a value for Ty. Since the f ronta l tooth angle p i s a lso s p e c i f i e d (see Figure 6), Tx = Tz / tan p. Thus Tx, the l a t e r a l tooth force component viewed in the f ronta l p lane, i s determined. In other words, s ince the total moments due to the muscle forces are known and must be completely opposed only by the sum of the moments of tooth force in t h i s p lane , the procedure determines the r e l a t i v e amounts of tooth force at the designated angular or ien ta t ions (a and R ; see F igures 5 and 6 r e s p e c t i v e l y ) necessary along each axis at the s p e c i f i e d p o s i t i o n of tooth c o n t a c t . It has already been stated that because the condyles are considered - 85 -to be coaxial in th is step, and act as the fulcrum, there are no contr ibut ing condylar moments acting in th is plane. . However, the sum of the s ta t i c l i near anteroposterior muscle (My) tooth (Ty) and j o in t force components (CRy + CLy) must egual zero. L ikewise, the ver t i ca l components of a l l forces must sum to zero, i . e . ?My + Ty + (CRy + CLy) = 0 and sMz + Tz + (CRz + CLz) = 0 therefore (2) CRy + CLy = -(*My + Ty) (3) CRz + CLz = -(sMz + Tz) Thus the resul tant vector of th is total j o in t resistance force (Cr; Figure 4) and i t s angular or ientat ion in the la te ra l plane can be determined t r igonometr ica l ly since Cr = [(CRy + CLy ) 2 + (CRz + C L z ) 2 ] " 1 / 2 and the la te ra l plane j o in t force angle i s given by arctan (CRz + CLz)/(CRy + CLy) (see Figure 5) . For a b i l a t e r a l l y symmetrical s i tuat ion the analysis needs to go no fur ther , (Weijs and Dantuma, 1981). The component of anteroposterior (y) and ver t i ca l (z) condyle resistance forces on each condyle ( r ight and l e f t ) w i l l be simply ha l f of the sum of the values found in equations 2 and 3 respect ive ly . Any mediolateral forces acting on the system w i l l be equal and opposite on the two sides and w i l l cancel each other completely. Figure 5. LATERAL (yz) REPRESENTATION OF FORCE COMPONENTS. In th is view the coaxial condyles act as the fulcrum. Tooth resistance forces are represented by Ty (anteroposterior) and Tz ( v e r t i c a l ) , combined j o i n t forces as CRy + CLy ( r ight and l e f t anteroposter ior) , CRz + CLz ( r ight and l e f t v e r t i c a l ) and muscle fo rces as My and Mz. TRLAT represents the project ion onto the la te ra l (sag i t ta l ) plane of the total 3-dimensional tooth resul tant force TR. Respective moment arms are prefixed by d. Only one muscle i s shown as the r ight and l e f t sides are superimposed. i s the angular or ientat ion of the resul tant vector of resistance force on the tooth in th is plane ( i . e . la te ra l tooth angle). - 87 -2. Frontal Plane - Step 2 As can be seen in Fiqure 6, the system is next analyzed in the frontal plane. Again the condyles are assumed to be coaxial with the r ight condyle acting as the fulcrum. Apparently, therefore, no mediolateral j o i n t force is determinable in th is step as any occurring at the l e f t condyle would not produce any net moment about the r ight condyle fulcrum. The value of Tx has already been derived from the re la t ion Tx = Tz/tan R. The total resul tant force of resistance (react ion force) at the given point of tooth contact along the spec i f ied angles a and B in the la te ra l and frontal planes respect ive ly i s given by TR = T T X2 + Ty2 + T z2 ] - l/2. The lengths of the moment arms for each muscle and tooth vector component, x and z , as well as for the z component of force at the l e f t condyle are determined. Incorporating moments ar is ing at both the l e f t and r ight sides of th is f igu re , the total anticlockwise moment i s the sum of the moments (4) MRz(dMRz) + MLz(dMLz) + MLx(MLx) = K 0 and the total clockwise moment is the sum of the moments MRx(dMRx) + Tx(dTx) + Tz(dTz) + Clz(dClz) = Ki . Only CLz i s unknown in these equations thus, the l a t t e r may be rewr i t ten, (5) Ki = K-ji + Clz(dClz) where K-ji i s subst i tuted for the known muscle and tooth va r iab les . Since the s ta t i c condit ions require the sum of the clockwise and ant ic lockwise moments to equal zero, - 88 -K 0 + Ki = 0 and K 0 + [Kj -j + CLz(dCLz)] = 0 or (6) CLz = r^/dCLz where K x = - ( K 0 + K ^ ) . Since K-j and dClz are both known var iab les , the ver t i ca l component of force on the l e f t condyle (CLz) i s solved. The sum of the s ta t i c l i near ver t i ca l (z) force components and the mediolateral (x) components must both a lso equal zero. Therefore MRz + MLz + Tz + CRz + CLz = 0 and MRx + MLx + Tx + CRx + CLx = 0 Again only the condylar force var iables are unknown and the equations may be rewrit ten (7) CRz + CLz = -(7,Mz + Tz) (8) CRx + CLx = -(r,Mx + Tx) Equation 7 i s the same as equation 3 derived in step 1 and i s therefore redundant. Equation 8 expresses the net total mediolateral reaction force of the r ight plus l e f t condyles. Since they are coaxial the re la t i ve amount of t h i s force on e i ther the r ight or the l e f t condyle cannot be determined unless the angle at which the resistance force acts at one of the condyles i s known. It i s for th is reason that the angle of l e f t condylar react ion force (v , of Figure 6) previously mentioned i s speci f ied in the i n i t i a l data entry for a given task. - 89 -Figure 6. FRONTAL (xz) REPRESENTATION OF FORCE COMPONENTS. In th is view the r ight condyle acts as the fulcrum. Tx represents the mediolateral component of tooth resistance force; TRPRQNT the project ion onto the frontal plane of the total three dimensional resul tant of tooth resistance force TR; and the angular or ientat ion of TRPRQNT• M R X A N D M R Z A R E the r ight mediolateral and ver t i ca l muscle force components; MLx and MLz are those for the l e f t s ide . i s the assumed or ientat ion of the resistance force at the l e f t condyle. A l l other abbreviations as for Figure 2. - 90 -As was the case for the tooth resistance forces the ra t io between the two l e f t condylar reaction forces i s tan v = CLz/CLx or CLx = CLz/tan v Since the value of CLz was determined in equation 6 and anqle Y i s spec i f i ed , subst i tu t ion into the above gives the value of the l e f t condylar mediolateral component of resistance force, CLx. Subst i tut ion of th is value into equation 8 gives the value for the r ight condyle, CRx, since i t remains the sole unknown in that equation. Summarizing to th is point , then, vector magnitudes can be derived fo r ; (a) the components of tooth react ion or resistance force (Tx, Ty and Tz) and the resul tant vector of tooth force (TR) at the speci f ied point of resistance at the designated angular or ienta t ions; (b) the ver t i ca l components of l e f t and r ight condylar react ion forces (CLz and CRz) (c) the mediolateral components of l e f t and r ight condylar react ion force (CLx and CRx) The only va lues remaining to be determined are those of the anteroposterior components of condylar reaction force occurring at the r ight (CRy) and l e f t (CLy) s ides . 3. Horizontal Plane - Step 3 In the horizontal plane (Figure 7) the lengths of a l l moment arms are derived and a value for the l e f t condylar anteroposterior reaction force, CLy, i s determined as fo l lows: - 91 -The sum of the anticlockwise moments i s given by (9) MRx(dMRx) + MRy(dMRy) + MLy(dMLy) + Tx(dTx) = K-f i and the cl ockwi se moments by (10) CLy(dCLy) + MLx(dMLx) + Ty(dTy) = K j v or CLy(dCLy) + Kv = i q v where K v represents the known var iables on the l e f t of equation 10. Combining these clockwise and anticlockwise moments from Figure 7 (which also must completely oppose one another) equations 9 and 10 become K j j j +[CLy(dCLy) +KV] = 0 or (11) CLy = K 2/dCLy where K 2 = - ( K ^ - + K v ) . Since K 2 and dCLy are both known var iab les , the anterposterior component of force on the l e f t condyle (CLy) i s solved. Subst i tut ing th is value of CLy in to equation 2 of Step 1 gives the corresponding anteroposterior component of force on the r ight condyle, CRy. In th is plane,summation of the s ta t i c l i near x and the y force components s imi la r to the x and z components in step 2, can be expressed by MRx + MLx + Tx + CRx + CLx = 0 and MRy + MLy + Ty + CRy + CLy = 0 which may also be wri t ten CRx + CLx = -( sMx + Tx) CRy + CLy = -(j;My + Ty) - 92 -d C L y Figure 7. HORIZONTAL (xy) REPRESENTATION OF FORCE COMPONENTS. The coaxial condyles again have the r ight j o i n t acting as the fulcrum. The f igure appears as though viewed from below with the r ight side on the viewer 's l e f t . A l l abbreviations as for Figures 5 and 6. - 93 -However, both of these values were derived previously and are therefore also redundant (see equations 8 and 2 respec t i ve ly ) . The calculated three dimensional components of muscle, tooth and j o i n t force are stored on computer f i l e and are ret r ievable on e i ther a screen or hard-copy pr in tout . However the data presented in the plots of the RESULTS sect ion are the resu l tan ts , or s ingle net vectors of resistance derived from these components at the point of tooth contact, and condyles. Representation of the resul tants in th is way also gives the i r angular or ientat ion projected onto the respective plane of view. 4. Constraints On Muscle Resultants It would seem appropriate and convenient to d isp lay , in add i t ion , a net s ingle three dimensional resul tant muscle vector and i t s or ientat ion in each plane (Weijs and Dantuma, 1975; Wei js, 1980; Weijs and Dantuma, 1981) in the plots of the RESULTS which fo l low. This would enable comparisons of muscle force or ientat ions to the assumed d i rec t ion of the resu l t ing tooth resistance force in three dimensions. However, t h i s i s theore t i ca l l y not possible because a system of forces acting on a r i q i d body in space can be reduced to a s ingle force or resul tant only i f those forces are (1) concurrent (a l l passing through the same point in space; in th is case three dimensional ly) , (2) coplanar, or (3) para l le l (Beer and Johnston, 1977). The muscle force vectors acting on the mandible sa t i s f y none of these condi t ions. However in each indiv idual plane the muscle forces can be reduced s u f f i c i e n t l y to a resul tant which gives an ind icat ion of the or ientat ion of the e f fec t i ve muscle pul l in that plane only. It must be borne in mind, - 94 -though, that considerat ion of each plane in th is way i s incomplete without a lso considering the other two at the same time. The resul tant muscle forces for a given plane in the respective x, y or z d i rec t ion are the sum of the indiv idual muscle components of the r ight and l e f t s ides. These summed s ta t i c l inear forces (eg. MLR, MAR, MVR respect ive ly , see Figure 8) are also responsible for a muscle moment about the fulcrum in a given plane. I f , in the la te ra l plane shown in Figure 8 , i t i s assumed that a l l of the muscle moment i s due to the total anteroposterior muscle component, MAR, alone ( i . e . MVR passes through the fulcrum and thereby has zero moment) then i t must have the length r z as i t s moment arm. Therefore, for th is plane, MAR • r z = 7, MUSCLE MOMENTS. I f , on the other hand, only the ver t i ca l component MVR i s to produce a l l of the moment in th is plane then i t s moment arm w i l l have to be of a length such that MVR • r y = y MUSCLE MOMENTS. MAR and MVR can be combined to give a " resul tant" of muscle force in th is plane, which ex is ts only in th is view. It i s worth noting that M|_^T can l i e anywhere along the l i n e ind icat inq i t s or ientat ion and that the product of i t s magnitude and i t s moment arm (r) w i l l always sa t i s fy the moment requirement, i e . M L A T * r = x MUSCLE MOMENTS I t ' s angle (ANG'L) with respect to the y axis therefore represents the or ientat ion of the overal l muscle force vectors projected onto the la te ra l pi ane. - 95 -A s imi la r der ivat ion can be applied to both the frontal and horizontal planes as shown in Figure 8. In th is way an ind icat ion of the alignment or or ientat ion of the muscle forces acting on the system within each plane (ANG*L, ANG'F, ANG'H) can be determined. However i t i s most important to keep in mind the fact that the " resu l tants" shown in Figure 8 (ML/\T, MFR0MT» MH0R) a P P l y t 0 t h e i r r e s p e c t i v e p lanes of view on l y , and a l l act simultaneously in three dimensions. Although the components (MLR, MAR and MVR) of these " resu l tants" are consistent in a l l three planes the i r ef fect on the equi l ibr ium of the mandible i s unique to each plane, as evidenced by the fact that the lengths of the moment arms rx , ry and rz are d i f fe rent in each plane. In ef fect then, the system of muscle forces can be reduced to three force vectors which can be ca l led " resu l tants" since they cannot be combined any further into a simpler vector scheme*. Depending on the task and the view considered, the components of the muscle resul tant may l i e a considerable distance from the fulcrum, and often l i e outside the l i m i t s of both the computer system's output screen and p lo t te r . As such i t i s not always possible to p lot the i r pos i t ions . The magnitude of the components of these resul tants of muscle force (MLR, MAR, MVR) remain constant for each task and are given in the computer *I t i s possible to replace these three by what i s known as a "wrench" which represents a set of equivalent forces acting at a spec i f i c point in space cons is t ing of a s ingle vector F with a moment arm r to the fulcrum, and a twist M about the axis of F. Thus there i s rea l l y two twists in a wrench; one about the fulcrum, the other about the axis of F. FRONT MLR = Mx = SMRx + SMLx MAR = My = SMRy + SMLy MVR = Mz = SMRz -t-HMLz MLR Figure 8. DETERMINATION OF THE ORIENTATION OF THE MUSCLE FORCE (ANG) IN EACH PLANE. See tex t f o r abbreviations and d iscuss ion. - 97 -plots of the resul tant tooth and condylar resistance forces of the RESULTS. The or ientat ion of the forces (ANG) i s also given for each view. The muscle data (weights and scales) presumably appl ies to spec i f i c acts and/or points of tooth contact. Therefore the angulation of the tooth resistance force in each plane was matched with that of the muscles (as derived above) and comparisons made of the resu l ts obtained with those of a r b i t r a r i l y assigned angles of tooth res is tance. D. COMPUTERIZED ANATOMICAL RECONSTRUCTION The computer printout of the three dimensional (x, y and z) coordinates of the various possible points of tooth contact for the indiv idual used as the modeling "subject" (hypothetical) are given in Table IV. The center of the r ight condyle has been spec i f ied ( a r b i t r a r i l y ) as the center of referencing for the system and the coordinates of the l e f t condyle center are a lso included here. Any, or a l l of these components may be changed by the operator by simply speci fy ing the tooth for which a change i s required and redesignating the respective x, y and/or z component. The x, y and z coordinates of the respective or ig in and inser t ion points for each muscle are qiven in Table V. Table V i s a computer printout of the muscle data (except Scale Factors) avai lab le as another option in the Main Menu of the program. The Weight Factors are a l l of those derived for each muscle of the "sub jec t " . Both the attachment point coordinates and Weight Factors for the indiv idual muscles remain constant throughout th is study. Since each d i f fe ren t task to be analyzed (eg. i n c i s a l , versus unimolar clenching) has a d i f fe rent combination of Scale Factors none are shown in - 98 -LATERAL ANT/POST VERTICAL 1 I N C I S O R 41 45.425 88.000 -33.100 2 I N C I S O R 42 41.700 86.925 -32.800 3 C A N I N E 43 36.425 83.575 -32.275 4 P R E M O L A R 44 30.975 76.875 -32.625 5 P R E M O L A R 45 28.025 71.450 -32.800 G M O L A R 46 25.850 65.525 . -33.200 7 M O L A R 47 23.325 54.625 -33.200 S M O L A R 48 19.900 45.450 -33.100 9 I N C I S O R 31 45.425 88.000 -33.100 10 I N C I S O R 32 49.150 86.925 -32.800 11 C A N I N E 33 54.425 83.575 -32.275 12 P R E M O L A R 34 59.875 76.875 -32.625 13 P R E M O L A R 35 62.825 71.450 -32.800 14 M O L A R 36 65.000 65.525 -33.200 15 M O L A R 37 67.525 54.625 -33.200 16 M O L A R 38 70.950 45.450 -33.100 L E F T C O N D Y L E 90.850 0. 000 0.000 TABLE IV - TOOTH AND JOINT COORDINATES (above). Computer printout of the x (La te ra l ) , y (Ant/post) and z (Ver t i ca l ) posi t ions of the various tooth contact points measured (mm) from the or ig in at the center of the r ight condyle. That of the l e f t condyle i s also ind icated. TABLE V - MUSCLE ATTACHMENT COORDINATES (fol lowing page). Computer pr intout of the physiological and anatomical parameters for each muscle group (depicted in Figure 9) which remain constant for every task. WT refers to the muscle group "Weighting Factors" . Other abbreviations as per Descript ion of F igures. The Maxi l lary Origins and Mandibular Insert ions are those anatomical attachment pos i t ions , for each muscle group measured (mm) along the x (La te ra l ) , y (Ant/Post) and z (Ver t i ca l ) axes from an or ig in at the center point of the r ight condyle (see Figure 9 ) . The sign convention i s such that negative values indicate posi t ions to the r ight of , poster ior to , and/or below the or ig in point . (The "Scale Factors" are var iable depending upon the task and are described with each of these la te r on). - 99 -TABLE V M A X I L L A R Y O R I G I N S RT . S I D E UT S C A L E L A T E R A L A N T / P O S T V E R T I C A L 1 SM 1 9 0 , , 40 0 . , 0 0 - 3 . 3 5 0 41 , , 3 0 0 - 1 , , 6 2 5 DM 81 , , 5 0 0 , , 0 0 - 1 2 . 4 5 0 15 , . 8 0 0 4 . 9 5 0 3 MP 1 7 4 . , 3 0 0 . , 0 0 25 . 2 7 5 2 7 . , 4 7 5 - 1 2 , , 7 0 0 4 AT 1 5 8 , , 0 0 0 , . 0 0 - 3 . 1 0 0 3 8 , . 4 2 5 4 5 , . 9 2 5 5 MT 9 5 , , 50 0 . , 0 0 -1 4 . 3 5 0 , 5 2 5 5 7 . , 3 7 5 5 PT 7 5 , , 50 0 , , 0 0 - 1 6 . 100 - 5 3 , . 5 5 0 3 8 , . 7 0 0 7 I P E B , , 90 0 . . 0 0 2 3 . 0 0 0 2 7 , , 2 5 0 - 3 . , 2 5 3 3 SP 2 8 . , 7 0 0 , . 0 0 -> n . 7 7 5 . 0 7 5 3 , . 4 2 5 g D6 4 0 , , 3 0 0 , , 0 0 33 . 7 0 0 4 5 , , 0 0 0 -71 , . 7 0 0 L T . S I D E •+* + * • » * * • + • • * * • * * • • * * * * ******+***********•*****»#•****••***+***#**• 10 SM 1 9 0 . . 4 0 0, , 0 0 9 9 , , 7 0 0 41 , . 9 0 0 -1 . 6 2 5 11 DM 81 . , 50 0 . 0 0 1 0 3 , , 3 0 0 16 . , 3 0 0 4 . 9 5 0 1 2 MP 1 7 4 , , 30 0, . 0 0 5 5 , , E 7 5 2 7 , . 4 7 5 - 1 2 . 7 0 0 1 3 AT 1 5 8 . , 0 0 0 . 0 0 9 3 . , 9 5 0 3 3 . , 4 2 5 4 5 . 3 2 5 1 4 MT 9 5 . , 5 0 0 , . 0 0 1 0 5 . , 2 0 0 . 5 2 5 5 7 . 3 7 5 1 5 PT 7 5 . , 5 0 0 . , 0 0 1 0 6 , , 9 5 0 - 3 3 . , 5 5 0 3 8 . 7 0 0 1 S I P 5 6 . , 9 0 0 , . 0 0 5 7 . , 8 5 0 . 2 5 0 - 8 . 2 5 0 17 SP 2 8 . , 7 0 0 , , 0 0 5 3 , , 0 7 5 7 "> , 0 7 5 . 3 . 4 2 5 18 DS 4 0 . , 0 0 0 . . 0 0 5 7 , . 150 4 5 , . 0 0 0 - 7 1 . 7 0 0 MANDIBULAR I N S E R T I O N S R T . S I D E UT S C A L E L A T E R A L A N T / P O S T V E R T I C A L **************************************************************** 1 SM 1 9 0 , . 40 0 , . 0 0 1 , , 5 7 5 2 0 , . 5 0 0 - 4 6 , . 5 0 0 DM 31 . 50 0 . 0 0 1 , , 6 2 5 2 5 . , 0 2 5 - 1 4 , , 5 0 0 MP 1 74, . 3 0 0 , . 0 0 5 , , 9 5 0 1 2 , . 5 5 0 -44 , . 175 4 AT 1 5 3 . . 0 0 0 , , 0 0 3 , , 0 5 0 3 2 , , 1 2 5 - 2 7 , . 9 5 0 5 MT 9 5 . , 5 0 0 , . 0 0 , 4 2 5 —— . 3 2 5 1 , . 5 0 0 5 PT 7 5 . , 60 0 , , 0 0 , 2 7 5 —— , E 7 5 1 , . 4 2 5 7 I P 5 5 . , 90 0 , . 0 0 . 0 2 5 3 , . 2 E 0 . 7 2 5 8 SP 2 8 . , 7 0 0 . , 0 0 1 , . 100 3 , , 7 0 0 , 3 2 5 9 DG 4 0 , . 0 0 0 . 0 0 42 . 0 0 0 77 . 0 0 0 - 5 3 . 5 2 5 ******************************************************** 10 SM 1 9 0 , 4 0 0 . . 3 0 3 9 . , 175 2 0 , , 5 0 0 - 4 6 . , 6 3 0 11 DM 31 , , 5 0 3 . , 0 0 3 9 . , 2 2 5 2 6 . , 0 2 5 - 1 4 , , 5 3 0 12 MP 1 7 4 , , 3 0 0 , . 0 0 9 4 , , 9 0 0 1 2 , . 6 5 0 - 4 4 , . 175 13 AT 1 5 8 , , 3 0 0 . , 0 0 3 2 . , 3 0 0 T n ! 2 5 "7 L. f . , 9 5 3 14 MT 9 5 , . 5 0 0 , . 0 0 9 0 , , 4 2 5 — — . 3 2 5 1 , . 5 3 0 15 PT 7 5 , , 5 0 0 . , 0 0 9 0 , , 5 7 5 . 5 7 5 I , . 4 2 5 16 I P 5 5 , . 30 0 , . 0 0 8 7 , , 3 2 5 — . 2 5 3 _ n 7 7 c 17 SP 2 8 , . 7 0 0 . . 0 0 3 9 , . 7 5 0 — . 7 3 0 1 , . 3 2 5 18 DG 4 0 , . 0 0 0 , . 0 0 4 9 , . 3 0 0 77 . 3 0 0 - 5 3 . 3 2 5 - 100 -Table V. These are shown and graphical ly d isplayed, for each respective task, in Figures 10A to 18A of the RESULTS. However, for any given functional task under study the printout of the data shown in Table V would also include the respective Scale Factors for that task (see Appendix II for exampl e ) . The computer-drawn plots of the l i nes of action from or ig in to inser t ion point of each muscle, the re la t i ve posi t ions of the tooth contact points , and the condyle posi t ions in each plane of reference are depicted in Figure 9 of the RESULTS. The abbreviations for each muscle are as described previously . This i s the anatomical basis of the hypothetical subject used by the computer to model the various s ta t i c occlusal functions in th is study. E. PROGRAM DESCRIPTION The computer software program of th is model was writ ten in FORTRAN IV. The computer system used was a Hewlett-Packard (HP) lOOOE-series minicomputer with an HP 7920A DISC DRIVE for storage of data, and an HP 2621A DISPLAY TERMINAL as the major hardware components. Data pr in t ing was f a c i l i t a t e d by an HP 2607A LINE PRINTER and computer-drawn plots were done using an HP 1350A GRAPHICS TRANSLATER and 1311A GRAPICS DISPLAY SCREEN. Hard-copy of these plots was furnished using a 7210A DIGITAL PLOTTER. The System Chart, which fo l lows, diagramatical ly shows the flow of information and how i t was handled for input, ana lys is , storage and re t r ieva l of the data (see SYSTEM CHART). I n i t i a l l y the anatomical parameters for the given i nd i v i dua l , which include the muscle o r ig in and inser t ion po ints , and the tooth and j o i n t - 101 -pos i t ions , were entered using an HP 9874A DIGITIZER. These coordinates were entered f i r s t from a la te ra l project ion for the ver t i ca l and anteroposter dimensions. Then the frontal project ion of the r ight side of the mandible was d iq i t i zed for the th i rd dimension (medio latera l ) . The l e f t side of the mandible was generated by the computer as a mirror image of the r ight thus completing the three dimensional p ic tu re . This data was stored on disc for future re t r ieva l , or used d i rec t l y in the main program. The physiological var iables for the muscle Weight and Scale Factors were entered i n i t i a l l y v ia the main program. The main program begins with a terminal display of the Main Menu which l i s t s the program options ava i lab le to the user at th is point . These include the fol 1 owi ng: - FILE ENTRY/FILE STORAGE - Any ex is t ing data f i l e previously stored (on- l ine) on the disc can be ca l led up or conversely a data f i l e which has just been speci f ied and/or changed can be stored (also on - l i ne ) . - VIEW/CHANGE MUSCLE DATA - The orthogonal (x, y and z) coordinates of the attachment points (or ig ins and inser t ions) for a l l muscles ( l e f t and r ight) as well as the Weighting and Scaling Factors for the data f i l e under considerat ion are displayed on the terminal . This display i s of the format shown in Table V and i s referenced to a zero axis at the r ight condyle as discussed previously . This also provides the means for the i n i t i a l spec i f i ca t ion of the Weight and Scale Factors for each muscle for a par t i cu la r task and/or f i l e . Any or a l l of these parameters may be changed by ident i fy ing the muscle(s) and - 102 -parameter(s) for which a change i s desired and enterinq the new value(s) v ia the terminal keyboard. - PRINT MUSCLE DATA - This command gives a printout of the muscle data under considerat ion v ia the l i ne pr in ter (see Table V) . - VIEW/CHANGE TOOTH OR LEFT CONDYLE POSITION - This command y ie lds a terminal display of the x, y and z coordinates of the tooth posi t ions and the l e f t condyle pos i t i on . The format i s that shown in Table IV of the RESULTS sec t ion . Any or a l l of these parameters are also changeable as per the muscle data. - PRINT TOOTH/CONDYLE DATA - The l i ne pr in ter provides a hard copy of the above tooth and condyle data (see Table IV). - SOFT PLOT ANATOMY: LAT/FRONT/HORIZ - The graphic portrayal of the anatomical re la t ionships for the given indiv idual i s plotted on the graphics display screen. The muscle attachment pos i t ion and aliqnments, tooth contact po in ts , condyle posi t ions and mandibular out l ine for ei ther the l a t e r a l , f rontal or horizontal project ions are included as shown in Fiqure 9 of RESULTS. - HARD COPY DESIRED PROJECTION - The graphics d ig i ta l p lo t ter provides a hard-copy of the desired project ion of the above data (see Fiqure 9 of RESULTS). - TERMINATE - This ends the program at th is point . - CALCULATE - The program enters the main vector analysis of the model. I n i t i a l l y the force vector for each indiv idual muscle along i t s l i n e of a c t i o n i s d e r i v e d . T h i s i s s i m p l y the p r o d u c t of [X m j • K] • EMGmj as d i scussed p r e v i o u s l y . From these - 103 -vectors (one for each muscle) the three orthogonal x , y and z components of force are derived for each muscle and are used in a l l subsequent analyses. At the completion of th is i n i t i a l ca lcu la t ion mode for the muscle vector components a secondary menu, SUBMENU 1, i s displayed on the terminal screen. The options at th is point inc lude: - SOFT PLOT MUSCLE VECTORS: LAT/FRONT/HORIZ - The graphics display screen plots each of the indiv idual muscle vectors along the i r respective l i nes of act ion ( i . e . from or ig in to insert ion) in e i ther the l a t e r a l , f rontal or horizontal project ions accordinq to the plane speci f i e d . - HARD COPY DESIRED PROJECTION - The araphics p lo t te r provides a hard copy of the desired project ion of the above. The format is that shown in Figures 10 to 18A of RESULTS. - END CALCULATE MODE - The program returns to the Main Menu (eg. i f further data changes, or a new f i l e , i s required) . Continuation of the ca lcu la te mode enters the program in a second phase of ca lcu la t ions where a l l of the individual x, y and z components of muscle force are combined into s ingle overal l components for the system as a whole (as per discussion of Figure 8 ) . These are the total mediolateral (MLR), the total anteroposterior (MAR), and the total ver t i ca l (MVR) muscle forces. The general or ientat ion of these overal l muscle forces acting on the system in each project ion or plane (for the given task) i s a lso der ived. Once these determinations are complete another secondary menu, SUBMENU 2 , i s displayed on the terminal screen: - 104 -- SOFT PLOT MUSCLE RESULTANT VECTORS: LAT/FROMT/HOR - The three orthogonal components of the overal l muscle forces acting on the system are plotted on the graphics display screen for the project ion chosen (see Figure 8 and Appendix I I ) . - HARD COPY DESIRED PROJECTION - END CALCULATE MODE - As previous. Further continuation of the ca lcu la te mode requires the operator to specify four var iables pr ior to the determination of the resistance forces at the teeth and j o i n t s ; (1) TOOTH# - the desired posi t ion of tooth contact. (2) LTA - the la te ra l plane tooth angle ( « ; see Figure 5 ) . (3) FTA - the frontal plane tooth angle ( R ; see Figure 6 ) . (4) LCFA - the l e f t condylar frontal angle (y; see Figure 6 ) . Once these var iables are provided the program then determines the react ion force vectors (forces of resistance) at the teeth and r ight and l e f t j o i n t s according to the above spec i f i ca t ions due to the given forces produced by the muscles involved. The procedures involved in these determinations are described in STEPS 1, 2 and 3 of the Analysis sec t ion . A th i rd secondary menu, SUBMENU 3, i s then d isplayed: - SOFT PLOT REACTION VECTORS: LAT/FRONT/HOR - The resu l t ing forces of tooth and j o in t resistance are displayed on the graphics display screen, in the desired pro jec t ion , for each par t i cu la r task and input spec i f i ca t i ons . The actual format of these computer-drawn plots are shown in Appendix I I . - 105 -SYSTEM CHART. Schematic representation of the flow of the modeling program through the computer system. The MAIN MENU and SUBMENUS 1, 2 and 3 are shown as d isplays of the options ava i lab le to the user at each stage of the program. A complete descr ipt ion of the de ta i l s involved with each step are discussed in the tex t . ( START CALCULATE I n d i v i d u a l (x 18) Muscle v e c t o r s and x, y, z components I - 106 -i CALCULATE T o t a l muscle r e s u l t a n t v e c t o r s f o r each p r o j e c t i o n . Magnitudes - MLR, MAR, MVR O r i e n t a t i o n s - ANG-L e t c . - 107 -- HARD-COPY DESIRED PROJECTION. - END CALCULATE MODE - As previous. Continuation of the ca lcu la te mode from th is point allows for reconsiderat ion of the muscle vectors and subsequent redesignation of the four input var iables should th is be des i red. This provides a means for comparisons of the ef fect of changes to one or more of these input var iables for the muscle data at hand. F. PROGRAM USE (IN THIS STUDY) The computer program is useful only for modeling s ta t i c isometric tasks undertaken by the jaw system, or in the case of the chewinq cyc le , where no movement, and hence s ta t i c condit ions are assumed to p r e v a i l . The parameters necessary to derive the force vectors produced by the muscles for each task were derived from data pooled from the l i t e ra tu re for a hypothetical "average" i nd i v i dua l . It should be re i terated that the parameters ascribed to th is hypothetical subject, used in th is study, ( i . e . muscle anatomical re la t ionsh ips , associated j o i n t and tooth pos i t i ons , Weighting Factors , t ask -spec i f i c Scal ing Factors) were derived only to provide a complete f i l e of the necessary information which could then be applied to the model i t s e l f . This was done in view of the fact that a l l of the necessary data from a single real subject does not ex is t at th is t ime. Although i t i s obvious that th is pooled data serves only as a f i r s t approximation to the real re la t ionships which ex is t between the various parameters involved i t i s nevertheless the most comprehensive and complete descr ip t ion of th is data presently ava i l ab le . - 108 -The modeling program was applied to (1) an intercuspal clench (CO); (2) a b i l a te ra l molar clench (BIMOL); (3) a un i la tera l molar clench (UNIMOL): (4) a un i la tera l canine clench (UNIK9); and (5) a b i te supported (INCISS) and ( 6 ) na tura l i n c i s a l c lench ( I N C I S N ) . These were chosen as being representative of the types of s ta t i c clenching functions undergone by the mandibular system. Appl icat ion of the model to the three " s t a t i c " phases of the chewinq cycle (near intercuspation) required a more ind i rec t der ivat ion of th is data but was undertaken as an example of the potential appl icat ion and use of the model. The absence of a means to d i rec t l y measure the three dimensional or ientat ion of tooth force and/or i t s actual magnitude made i t necessary to i n i t i a l l y assume that the angles of tooth resistance force were para l le l to that of the applied muscle force. However each task was also modeled for a var iety of tooth or ientat ions (±10 degrees approximately) in each plane ( la tera l angle a ; f rontal angle B ) , which included th is par t i cu la r o r ien ta t ion . This was done part ly to observe the e f fec t of changes to these angles of tooth res is tance, and part ly to ensure that a range of these angles was modeled which would include that angle most appropriate with respect to the muscle forces. S im i la r l y the ef fect of changing the l e f t condylar angle in the frontal plane was modeled for a var iety of or ientat ions for those (un i la te ra l ) functions where th is alignment may have been other than purely v e r t i c a l . - 109 -RESULTS The biomechanical re lat ionships between the vectors of force produced by the muscles and the resu l t ing forces of resistance which are generated at the point(s) of tooth contact and r ight and l e f t j o in ts were analyzed and are presented in three main categories of occlusal funct ion. In each case attachment coordinates and Weight Factors remained constant. Only Scaling Factors and b i te point were a l te red . The f i r s t category included clenching tasks which were b i l a t e r a l l y symmetrical requir ing equivalent muscle forces on the r ight and l e f t s ides. These were intercuspal clenching (CO), b i l a te ra l molar clenching (BIMOL) and inc isa l clenching with an "occlusal stop" at the inc isors (INCISS), and inc i sa l clenching on the natural dent i t ion (INCISN). "Occlusal stops" refers to a technique of l im i t i ng the point(s) of tooth contact by s l i g h t l y d isc iuding the dent i t ion by means of ac ry l i c separators (see MacDonald and Hannam, 1 9 8 4 ) . The second qroup of tasks was s ta t i c clenches with occlusal contact l im i ted to one side of the dent i t ion only ( r ight s i de ) , includinq un i la tera l canine clenching with a s t a b i l i z i n g occlusal stop (UNIK9) and un i la tera l molar clenching with a stop at the second molar (UNIMOLAR). The th i rd and f ina l group was the three assumed s ta t i c in te rva ls near the intercuspal phase of gum-chewing which were also un i la tera l in the molar region but car r ied out on the natural dent i t ion (CHEW 1, 2 and 3 ) . Unlike the un i la tera l tasks with asymmetric muscle a c t i v i t i e s the b i l a t e r a l l y symmetrical functions were not complicated by mediolateral - 110 -considerat ions. This provided an opportunity to determine an overal l magnitude for the total muscle resul tant since the two muscle force components (ver t i ca l and anteroposterior) were coplanar, l y ing in the midsagit tal plane*. No such determinations were possible for the uni la tera l tasks for reasons discussed previously (see METHODS). The indiv idual s ta t i c clenching tasks involved in each of the f i r s t two groups of function are presented and discussed on an indiv idual bas is . Comparisons between the tasks involved in each group are discussed subsequently. The three in terva ls of the chewing function are presented ind i v idua l l y but are also discussed as a group. A. DESCRIPTION OF FIGURES AND ABBREVIATIONS The f igures which fol low depict the three dimensional re la t ionships between the vectors of force produced by the muscles and the resul t ing forces of resistance generated at the point(s) of tooth contact and at the r ight and l e f t temporomandibular j o i n t condyles. 1. Figures 10 to 18 "AM These are the computer-drawn depict ions of the resul tant vectors of each muscle group in the three orthogonal planes of reference for each respective task. Lines of action of each muscle are described in Figures 6 and 9 , and Table V. The magnitudes of each vector are determined according to the " S c a l e F a c t o r s " s h o w n i n t h e s e f i g u r e s f o r e a c h *However, prec ise ly where th is resul tant vector l i e s within the two dimensions of th is plane cannot be determined. ">31 Figure 9 . COMPUTER RECONSTRUCTION OF ANATOMICAL VARIABLES. This f igure serves as a key to the muscle and tooth force vectors for the three orthogonal planes which fol low for each ser ies of tests (see Figures 10 to 18, A) . The horizontal plane (lower) i s depicted as viewed from below. The s p a t i a l co rd ina tes bf each are g iven in Table V (see tex t f o r abbrev iat ions) . - 112 -muscle, as well as the constant "Weighting Factors" (see Table V) as explained in METHODS. The scale values and drawn vectors are proport ionate. The combined ef fect of a l l the forces generated by the individual muscle groups for each funct ion, whether i t be intercuspal clenching or i nc i sa l c lenching, e t c . , determines the overal l MUSCLE RESULTANT PARAMETERS. These are described in Figures 10 to 18 labeled "B , C, D", e tc . It i s th is overal l force generated by the muscles which must be resisted by the teeth and/or j o i n t s and therefore to which the input data must be speci f ied to match accordingly (see Program Use, of METHODS sect ion) . 2 . Figures 10 t o 18 " B , C , D " , e t c . These f igures show the corresponding tooth and condylar reaction force vectors resu l t ing from the speci f ied occlusal function and MUSCLE RESULTANT PARAMETERS. The l a t t e r remain constant for a given type of a c t i v i t y (eg. in te rcuspa l , i n c i s a l , or unimolar c lenching) . The parameters within the dashed boxes are the input var iables spec i f ied by the operator for a par t i cu la r task (see also SYSTEM CHART). Thus, for any given a c t i v i t y , which has a unique combination of muscle vectors which produce that a c t i v i t y , the e f fec t of changes to any or a l l of the fol lowing var iables (and the i r e f fect on the magnitude and or ientat ion of the tooth and condylar react ion forces) can be observed: - The point of appl icat ion of the tooth resistance force ( i . e . tooth pos i t i on ) . - The or ientat ion of the tooth resistance force in both the la te ra l (LTA) and frontal (FTA) planes. - 113 -- The l e f t condylar frontal angle (LCA or LCFA) which i s the o r ien ta t ion , in th is pro jec t ion , at which the l e f t j o i n t would be assumed to provide res is tance. * Unless otherwise indicated th is or ientat ion i s a r b i t r a r i l y assumed to be 90 degrees. Each combination depicted in these f igures represents a s ingle run of the computer. Examples of the actual computer-drawn pr intout of data are represented in Appendix I I . These f igures summarize a l l of the runs for each type of ac t i v i t y and provide the range of reaction vectors for that ac t i v i t y according to the changes in the input var iables which are chosen. a . "MUSCLE RESULTANT PARAMETERS" - MLR, MAR, MVR (NEWTONS) - Orthogonal muscle resul tant vector components in the medio la tera l , anteroposter ior, and ve r t i ca l d i rect ions respect ive ly . These do not necessar i ly in tersect but have very spec i f i c posi t ions in three dimensional space. They essen t ia l l y describe the total l a t e r a l , anter ior and ver t i ca l components of muscle force for each ac t i v i t y (see METHODS for complete descr ip t ions) . - ANG-L (DEGREES) - The angle at which a resul tant of the ver t i ca l (MVR) and anter ior (MAR) components of total *A1though the LCA var iab le is not necessar i ly l i k e l y to exhib i t much, i f any, var ia t ion in a par t i cu la r type of a c t i v i t y , at leas t in th is hypothetical i n d i v i d u a l , th is parameter must be spec i f ied by the operator in order to operate the program. The reasons for th is have been discussed in the METHODS. - 114 -muscle force would appear to act i f they intersected. This parameter gives a general idea of the overal l or ientat ion of muscle force when viewed in the la te ra l piane. - ANG-F (DEGREES) - As above but for the frontal plane and muscle components MLR and MVR. ANG-L and ANG-F serve as f i r s t approximations as to the or ientat ion of the overal l muscle forces which must be res is ted by the jo in ts and teeth in each plane. The ef fect of changes to the tooth r e s i s t a n c e / r e a c t i o n fo rce or ientat ion (LTA and FTA) i s observed for a range of angles which include that of ANG'L and ANG-F. S i m i l a r i l y , the e f fec t of changes in tooth posi t ion and j o i n t resistance or ientat ion (LCA/LCFA) can also be observed. b. "LAT. PLANE RESULTANT VECTOR ORIENTATIONS" (DEGREES) - RCR - Right Condylar Angle. The angle formed by the project ion of the r ight condylar reaction vector resul tant on to the la te ra l plane (computer der ived). - LTA - Lateral Plane Tooth Angle. As above but for the tooth react ion vector (operator spec i f i ed ) . - 115 -- LCA - Left Condylar Angle. As above for the l e f t condyle react ion vector resul tant (computer der ived). c . "RESULTANT VECTOR MAGNITUDES" (NEWTONS) - RCR - Right Condyle Reaction Vector Resultant (computer der ived). - TR - Tooth Reaction Vector Resultant (computer der ived). - LCR - Left Condyle Reaction Vector Resultant (computer der ived). d. "FRONT. PLANE RESULTANT VECTOR ORIENTATIONS" (DEGREES) - RCA - Right Condylar Angle. The angle formed by the project ion of the r ight condylar reaction vector resul tant on to the frontal plane (computer der ived). - FTA - Frontal Tooth Angle. As above for the tooth resul tant vector (operator spec i f i ed ) . - LCA - Left Condylar Angle (or l e f t condylar frontal angle). As above but for the l e f t condylar resul tant vector (operator spec i f ied ) . e. "JF /TF" This i s the ra t io of total j o i n t or condylar reaction force (RCR and LCR) to total tooth react ion force (TR). It i s - 116 -assumed that the forces of resistance occurring at the jo in ts are a residual ef fect of the production of useful force at the teeth and the former ar ise only as a s t a b i l i z i n g inf luence on the system. The ra t io between these forces would therefore r e f l ec t some element of the e f f i c iency of the system with respect to the d i s t r i bu t ion of these forces. - 117 -B. FORCE VECTOR ANALYSIS 1. Bilaterally Symmetrical Clenching Tasks a. Intercuspal Clenching (CO) - Figures 10A, B and C I. Muscle Resultant Parameters Figure 10A shows the re la t i ve a c t i v i t i e s of the various muscles to be f a i r l y h igh. The masseters and temporalis groups were at the i r highest l eve l s for a l l tasks of th is study (see Table II and III for comparisons). Figure 108 and C indicate the magnitude of the two coplanar components of muscle force to be almost en t i re l y ver t i ca l with respect to the occlusal plane. The ver t i ca l (MVR) component of 1187.6 N was, in fac t , the highest value of a l l tasks analyzed whereas the anteroposterior component (MAR) of 40.9 N was the lowest. The overal l muscle resul tant force which was oriented at an angle of 88.0 degrees (ANG-L) in the la te ra l plane, due to these two components, was calculated to be 1188.3 N, which was only s l i g h t l y greater than the ver t i ca l component by i t s e l f . In the frontal plane a l l force vectors l i e para l le l to the y (ve r t i ca l ) axis (ANG-F = 90.0 degrees) since no mediolateral components e x i s t . II. Tooth Position Change - Figure 10B Since the posi t ion of the functional center of intercuspal occlusal contact was not known exact ly , although i t would l i e in the mid l ine, various possible posi t ions of th is point were analyzed including the second b icusp id , and the f i r s t , second and th i rd molars. The la tera l (LTA) and frontal (FTA) plane tooth angles were f ixed to match the or ientat ion of the muscle forces (ANG-L and ANG-F) f o r each run ( i . e . 8 8 . 0 and 90 .0 degrees respec t i ve ly ) . - 118 -Figure 10A. INTERCUSPAL CLENCHING (CO). Right and l e f t side scale factor values and three dimensional (computer-drawn) depict ion of indiv idual muscle vectors. - 119 -Most obvious from Figure 10B i s the increase in the compressive tooth resistance force at more poster ior tooth contacts for the same muscle force, the magnitude varying from 556.2 N at the second bicuspid to a maximum value for a l l the tasks studied of 866.4 N at the th i rd molar which is an increase of 56%. At the same time 49% less compressive resistance force i s required at the j o i n t s making the more poster ior contact r e l a t i ve l y more e f f i c i e n t (JF/TF = 0.37) in terms of the ra t io between the j o i n t and tooth forces. The magnitudes of the j o i n t forces at a maximum of 316.0 N per side at the bicuspid and a minimum of 161.0 N per side at the th i rd molar are among the greatest values observed in th is study at each of the respective tooth pos i t i ons . This i s a re f lec t ion of the re l a t i ve l y greater overal l muscle forces involved here. The leas t e f f i c i e n t posi t ion of tooth contact here appears to be the second bicuspid (JF/TF = 1.14) with corresponding increases occurring at more poster ior points of tooth contact (see Figure 10B) . Another way of considering the e f f i c iency of the d is t r i bu t ion of the resistance forces i s to compare the re la t i ve proportion of the total resistance force acting on the system assumed by the dent i t ion with that assumed by the j o i n t s . In each run of Figure 10B the or ientat ion in space of the overal l muscle resul tant and the tooth and j o i n t components (TR, RCR and LCR respect ively) were ident ica l (ANG-L = RCA = LCA = 88.0 degrees in the la te ra l plane and ANG-F = RCA = LCFA = 90.0 degrees in the frontal p lane). Therefore the sum of the tooth and j o i n t forces must equal the overal l magnitude of the resul tant force generated by the muscles which was determined to be 1188.3 N. The proportion of th is force taken up by the - 120 -teeth and jo in ts for each posi t ion of the dent i t ion with respect to the total force generated i s : RUN 1 second bicuspid 47% dent i t ion 53% jo in ts 2 f i r s t molar 51% dent i t ion 49% jo in t s 3 second molar 61% dent i t ion 39% jo in ts 4 th i rd molar 73% dent i t ion 27% jo i n t s I t should be noted that these par t i cu la r proportions can only be determined when a l l three elements of force (muscle, tooth and jo in ts ) are per fect ly al igned (para l le l ) in space as i s the case in Figure 10B. This i s because the i r respective axial vector components are in the same re la t i ve proportions as the resul tant vectors. Otherwise, comparison of the re la t i ve proportion of the respective orthogonal components of these forces must be made, which i s tedious to determine and d i f f i c u l t to in terpre t . The l inear increase in tooth force and respective decl ine in j o i n t force at more poster ior posi t ions i s due to the concurrent reduction in the moment arm length of the tooth forces as the b i te point becomes more poster ior . The moment, or torquing e f fec t of the tooth force at each pos i t i on , must be equivalent to balance the muscle forces. Thus, a tooth contact at the th i rd molar, which l i e s a shorter distance from the fulcrum and therefore has a shorter moment arm, requires a proportional increase in tooth force to match the moment produced by the re l a t i ve l y smaller force occurring at the second b icusp id , but which has a correspondingly longer moment arm. Because the fulcrum for the system is assumed to l i e at the j o in t s and the task i s b i l a t e r a l l y symmetrical neither j o i n t can contr ibute any moment to the system. The forces at the jo in ts in th is case (and the fol lowing b i l a t e r a l l y Figure 10B. INTERCUSPAL CLENCHING (CO)/VARIABLE TOOTH POSITION. Corresponding tooth and condylar reaction force vectors which result for various assumed anteroposterior points of central occlusal stability during an intercuspal clench. This activity is bilaterally symmetrical therefore the frontal plane angles for these forces have no mediolateral components. LTA and FTA are specified according to the overall orientations of muscle force (ANG-L and ANG-H) which likewise have no mediolateral component (eg. MLR = 0; ANG'F = 90°) . The left condyle 1n the lateral plane is shown with dashed outline. MUSCLE RESULTANT PARAMETERS MLR= o.o MAR= 40.9 MVR= H87.6 ANGL= 88.0 ANGF= 90.0 RUN 2 3 4 480 N Tooth Posit" 5 6 7 8 I I LAT. PLANE RESULTANT VECTOR ORIENTATIONS RCA 88.0 RESULTANT VECTOR MAGNITUDES LTA 88.0 LCA 88.0 RCR 316.1 291.4 232.2 161.0 TR. 556.2 605.6 723.9 866.4 LCfL 316.0 291.3 232.2 160.9 FRONT. PLANE RESULTANT VECTOR, OWEIJWIQNS RCA 90.0 _EJA 90.0 LCA* 90.0 L. I I I I * LCFA i i—» ro i—» i - 122 -symmetrical tasks) are simply the residue required to balance the l i near ve r t i ca l and anteroposterior muscle forces, since those that are due to the tooth forces (determined by the moments acting on the system) are i nsu f f i c i en t to do so. As such, the or ientat ions of the resul t ing j o i n t resistance forces in these instances (RCA and LCA) also match those of the muscle and tooth forces in the la te ra l and frontal planes. If even more poster ior posi t ions of tooth contact were assigned the re la t i ve proportional increase in tooth force and decrease in j o i n t forces would continue unt i l the tooth force exact ly matched the muscle force in or ienta t ion and magnitude. At th is point no j o i n t forces would e x i s t , because a l l resistance of the muscle force, both rotat ional moments and l i nea r forces, would be accounted for en t i re ly by the tooth force. At tooth posi t ions more poster ior to th is the (compressive) tooth force would continue to increase whereas the j o i n t forces would become tens i le in nature and thus opposite in or ientat ion to those observed in Figure 10B. 111. Lateral Tooth Angle (LTA) Changes - Figure IOC To assess the e f fec t of changes to the LTA var iable (and FTA which fol lows) the functional center for the tooth contact posi t ion was a r b i t r a r i l y assumed to l i e at the f i r s t molars, which i s s l i g h t l y poster ior to the geometric center of occlusal support. Changes in the or ientat ion of the angle of tooth resistance in th is plane from a protrusive to a retrusive d i rec t ion had two main ef fects on the resistance forces of the system. F i r s t , the magnitude of the tooth resistance force (TR) shows an increase from 574.2 N to 686.8 N, or 20%, due to shortening of the respective lengths of the moment arms for each Fiqure IOC. INTERCUSPAL CLENCHING (COJ/VARIABLE LTA. Corresponding tooth and condylar reaction force vectors resu l t ing from changes in la tera l plane tooth angle. The range of th is change i s approximately ten degrees to e i ther side of ANG-L. Frontal angulations have no mediolateral component due to b i l a te ra l symmetry. MUSCLE RESULTANT PARAMETERS MLR= o.o MAR= 40.9 MVR= 1187.6 ANGL= 88.0 ANG-F = 90.0 480 N .Tooth , Posit n , LAT. PLANE RESULTANT VECTOR ORIENTATIONS RESULTANT VECTOR MAGNITUDES |RUN RCA ! LTA | LCA RCR 312.5 TR 574.2 LUR 312.4 i 6 95.4 80.0 95.4 2 91.0 | 85.0 | 91.0 299.0 592.1 298.9 3 | | 85.9 1 90.0 1 85.9 286.6 615.9 286.5 4 1 1 79.8 ! 95.0 ! 79.8 275.9 647.0 275.8 5 L 1 72.6 Lioo^oj 72.6 267.9 686.8 267.8 FRONT. PLANE RESULTANT VFQTOR , Og^HW'pNS RCA 90.0 FTA 90.0 L C A * 90.0 I I I I * L C F A ro oo - 124 -o r ien ta t ion . This resulted in a reduction of the j o i n t force from 312.5 N to 267.9 N per j o i n t , or 14%. Therefore, in terms of e f f i c iency since the JF/TF ra t io decreases for more retrusive tooth forces, these would appear to be more e f f i c i e n t . Second, there i s a reciprocal e f fect with respect to the j o i n t force or ientat ions which become more an ter io r ly oriented with more poster ior ly oriented tooth forces. This i s due to the change in d i rec t ion of the anteroposterior component of tooth fo rce. In RUN 1 th is component was calcu lated to be 99.7 N in the poster ior d i rec t ion whereas in RUN 5 i t was 119.5 N in the anter ior d i r ec t i on . The s ta t i c l i near components of the muscle, tooth and j o i n t forces must balance. This component of r ight and l e f t muscle, tooth and j o i n t force was 29.4 N anter io r ly per j o i n t in RUN 1 and 80.1 N poster ior ly in RUN 5 which was a total of 58.8 N and 160.2 N respec t ive ly . However the muscle force also contr ibutes an anter ior component of 40.9 N (MAR). Therefore for RUN 1 the sum of anteroposterior components i s 40.9 (muscle) + 58.8 ( jo ints) = 99.7 (tooth) and for RUN 5, 40.9 (muscle) - 160.2 ( jo ints) = 99.7 ( tooth). Thus, more poster ior components of tooth force require more anter ior components of j o i n t force to oppose these forces and maintain equi l ibr ium for a constant muscle force. When the or ientat ion of the tooth force is matched with that of the muscle force as seen in RUN 2 of Figure 10B (LTA and ANG-L = 88.0 degrees; FTA and ANG'F = 90.0 degrees) then the j o i n t force or ientat ions (RCA and LCA) also match. - 125 -In general according to the model the or ientat ions of the j o i n t resistance forces due to intercuspal clenching at the f i r s t molar are such that one would predict a j o i n t morphology with bearing surfaces directed anterosuperior ly from about 70 to 95 degrees and capable of withstanding up to about 320 N of fo rce. b. Bilateral Molar Clenching (BIMOL) - Figures UA, B and C I. Muscle Resultant Parameters In general , the ac t i v i t y leve ls of the main adductors of the mandible shown in Figure 11A for th is task (which has a s t a b i l i z i n g occlusal stop) was approximately 80 to 85% of the i r maximal l e v e l s . Except for MP and IP th i s represents a reduction of roughly 15 to 20% compared to the intercuspal clench (CO) a c t i v i t i e s of Figures 10A (see also Table I I ) . As such there was a corresponding decrease in the overal l muscle force generated. Although the anter ior component of muscle force (MAR) of 57.0 N represented about a 40% increase over that of intercuspal clenching (40.9 N) the ver t i ca l component (MVR) of 1039.1 N was about 13% less than for intercuspation (1187.6 N). This equates to a reduction in the overal l muscle resu l tant , for b i l a t e r a l l y supported molar c lenching, calculated to be 1040.7 N, or 12% less than that of intercuspal clenching (1188.3 N). The ef fect of the increased MAR value for th is task was to a l ign the muscle resul tant s l i g h t l y more anter io r ly at 86.9 degrees (vs . 88.0 degrees for CO). II. Lateral Tooth Angle (LTA) Change - Figure 11B The b i l a te ra l molar clenching task modeled in Figure 11B was i n i t i a l l y assumed to occur at the f i r s t molar tooth to allow for comparisons of the - 126 -Figure 11A. BILATERAL MOLAR CLENCHING (BIMOL). Right and l e f t side scale factors and three dimensional (computer-drawn) depict ion of indiv idual muscle vectors. - 127 -resistance forces with those of intercuspal clenching (see Figure IOC). As can be seen, the magnitudes of both the tooth and j o in t resistance forces are somewhat less for bimolar clenching than intercuspal c lenching, re f lec t ing the decrease in the muscle generated force of th is task. The ef fect of changes in LTA, shown in Figure 11B, on the magnitude of tooth force and j o i n t forces, and on the or ientat ion of the l a t t e r forces are the same as those discussed for the intercuspal c lench. That i s , a more poster io r ly oriented LTA resu l ts in correspondingly greater magnitudes of tooth resistance force but a decrease in j o i n t resistance forces. However, over the range of LTA modeled in Figure 11B from 80.0 degrees to 95.0 degrees there was an increase from 495.7 N to 558.6 N in tooth force or 13%, with only a 10% decrease in j o i n t force from 275.9 N to 247.0 N per j o i n t . Over th is same range of LTA for intercuspal clenching (CO) there was the same (13%) increase in tooth force (574.2 N to 647.0 N) but a 12% decrease in j o i n t forces (312.4 N to 275.8 N per s i de ) . This i s re f lected in the r e l a t i v e l y better e f f i c iency of the intercuspal task of Figure 10C over bimolar function seen here according to the JF/TF ra t i os . Comparison of the proportion of the calculated total muscle resul tant force (1040.7 N) res is ted by the teeth (518.3 N) and j o in t s (261.2 N per side) in RUN 2 with muscle-matched or ientat ions shows the resistance forces to be equally divided (50%) between the teeth and j o i n t s . Hence the JT/TF ra t io of v i r t u a l l y 1.0. Intercuspal clenching at the second molar was s l i g h t l y more e f f i c i e n t with 51% of the force res is ted by the teeth and 49% by the j o i n t s (JF/TF = 0.96; see Figure 10B, RUN 2) . Figure 11B. BILATERAL MOLAR CLENCHING (BIMOL)/VARIABLE LTA AT FIRST MOLAR. Corresponding tooth and condylar react ion force vectors for b i l a t e r a l l y symmetrical clenching with tooth contact assumed to occur at the f i r s t molars only . The range of LTA var ia t ion i s approximately ten degrees e i ther side of ANG-L. MUSCLE RESULTANT PARAMETERS M L R = o.o M A R = 57.0 M V R = 1039.1 A N G - L = 85.9 A N G F = 90.0 480 N _ALTA ["Tooth ~| LAT. PLANE RESULTANT VECTOR ORIENTATIONS RESULTANT VECTOR MAGNITUDES RUN , Posit n - RCA | LTA j LCA RCR TR LCR 1 1 6 1 93.0 1 80.0 1 93.0 275.9 495.7 275.9 2 1 | 86.9 I 86.9 | 86.8 261.2 518.3 261.2 3 1 1 83.6 1 90.0 1 83.6 255.2 531.8 255.2 4 j | 77.7 | 95.0 1 77.7 247.0 558.6 247.0 1 1 1 J FRONT. PLANE RESULTANT VECTOR, ORIENTATIONS RCA 90.0 FTA 90.0 LCA *i 90.0 I J I I * LCFA - 129 -111. Tooth Position Change - Figure 11C With tooth contact assumed to l i e at the second molars as opposed to the f i r s t molars greater magnitudes of tooth resistance force occur for each run. Over the same range of LTA of 80.0 degrees to 95.0 degrees at the second molar (RUN 2 to 5) there i s approximately 90 to 115 N more tooth force generated at the second molar (585.9 to 616.3 N; Figure 11C) compared to those at the f i r s t molar (495.7 to 558.6 N; Figure 11B). Again the greatest increases occurred with a more poster ior ly oriented LTA. The re la t i ve increase in TR over th is range for f i r s t molar contacts, as mentioned e a r l i e r , was 13% whereas th is increase was 16% at the second molars. In add i t ion , however, there i s a concurrent decrease in RCR and LCR of 18% from 232.6 to 191.7 N per side over th is range at the second molars (Figure 11C) compared to only a 10% decrease for f i r s t molar contact discussed previously (see Figure 11B). Because of th is great ly improved re la t ionship between the tooth (TR) and j o i n t forces (RCR and LCR) the JT/TF ra t ios for second molar contacts are considerably less for second molar contact (Figure 11C) than those for f i r s t molar contact (Figure 11B). The e f fec t of improving th is re la t ionship due to changes in LTA at more poster ior tooth contacts becomes even more prominent when Figure 11C of bimolar clenching i s compared to Figure 10C of intercuspal c lenching. Although, as was discussed, the l a t t e r s i tuat ion had a re l a t i ve l y greater magnitude of muscle force acting on the system, comparison of these two f igures with two d i f fe rent occlusal contact posi t ions show very s im i la r magnitudes of tooth force. When the runs with corresponding or ientat ions of LTA are compared, bimolar clenching has s l i g h t l y higher TR magnitudes than - 130 -intercuspal clenching due to the more poster ior tooth pos i t i on . However th is also resul ts in r e l a t i ve l y lower magnitudes of RCR and LCR and thus lower JF/TF ra t ios o v e r a l l . The total re la t i ve increase in TR over the LTA range modeled in Figure IOC (RUNS 1 to 5) for intercuspal clenching at the f i r s t molars was 20% and the decrease in RCR and LCR was 14%. That of bimolar clenching at the second molars was a 19% increase in TR (567.9 to 676.3 N) and a 23% decrease in RCR and LCR (249.3 to 191.6 N). Nevertheless, intercuspal clenching with LTA matched with ANG-L (and FTA with ANG-F) in Figure 10B, shows RUN 3 at the second molar shows TR to be 723.9 N and RCR and LCR 232.2 N each. This amounts to 61% of resistance force at the teeth and 39% at the j o i n t s with respect to the total force generated by the muscle resul tant (1188.3 N). RUN 3 of Figure 11C for bimolar clenching on the other hand, with TR of only 618.5 N, and RCR and LCR of 211.1 N each, had only 59% of the total resistance force at t r ibuted to the teeth and 41% to the jo in ts re la t i ve to the muscle resul tant of 1040.7 N. This i s s im i l a r l y ref lected in the JF/TF ra t ios of 0.64 vs . 0.68 respect ive ly for the two d i f ferent tasks. This i s a function of the lesser total muscle force acting on the system on the one hand and a s l i gh t l y more anter ior LTA on the other for bimolar clenching with matched data. Concerning the range of j o i n t force angles in the la te ra l plane (RCA and LCA) the more anter ior tooth contact of Figure 11B shows a c loser range of 15.3 degrees (from 93.0 to 77.7 degrees) for LTA of 80.0 to 95.0 degrees (RUNS 1 to 4) compared to 23.1 degrees (95.5 to 72.4 degrees; RUNS 2 to 5) for Figure 11C and a more poster ior contact. This i s because the more poster ior contacts resu l t in a greater TR magnitude due to shorter moment arm Figure 11C. BILATERAL MOLAR CLENCHING (BIMOD/VARIABLE LTA AT SECOND MOLAR. Range of LTA var ia t ion as per Figure 1 1 B . MUSCLE RESULTANT PARAMETERS MLR= o.o MAR= 57.0 MVR= 1039.1 ANG-L= 86.9 ANGF= 90.0 480 N ["Tooth | LAT. PLANE RESULTANT VECTOR ORIENTATIONS RESULTANT VECTOR MAGNITUDES RUN , Posii n , RCA j LTA j LCA RCR TR LCR l I 7 I 100.4 1 75.0 | 100.4 249.4 567.9 249.3 2 | | 95.5 I 80.0 | 95.5 232.6 585.0 232.5 3 86.8 1 86.9 1 86.8 211.1 618.5 211.0 4 | j 81.9 1 90.0 | 81.9 202.6 637.9 202.6 5 I I 72.4 L IS^OJ 72.4 191.7 676.3 191.6 FRONT. PLANE RESULTANT VECTOR ORIENTATIONS RCA 90.0 FTA 90.0 LCA* 90.0 I I I I * LCFA i i—• oo JF/TF .88 .79 .68 .64 .57 - 132 -lengths. Thus the anteroposterior components of these forces are re l a t i ve l y greater requir ing correspondingly greater anteroposterior components of j o i n t force to balance the system. Based on the foregoing observations the j o i n t morphology would be predicted to coincide with those a t t r ibutab le to res i s t i ng the same magnitudes and or ientat ions of force as predicted from intercuspal c lenching. c. Incisal Clenching With Bite Stop (INCISS) - Figures 12A and B I. Muscle Resultant Parameters Figure 12A c l ea r l y depicts the re l a t i ve l y small muscle forces involved in th is task compared with those associated with more poster ior funct ion. The total muscle force resul tant calculated to be 461.2 N i s less than one hal f that observed for e i ther the intercuspal (CO) or b i l a te ra l molar (BIMOL) funct ions. A l l Scale Factors are great ly reduced except IP which i s at i t s maximal value for the ent i re study. Considering the much smaller MVR force component of 432.3 N, i t i s not surpr is ing therefore that MAR of 160.8 N i s a s i gn i f i can t component of the overal l muscle resu l tant . The corresponding or ienta t ion of th is force of 69.6 degrees i s much more anter ior ly directed than the b i l a t e r a l l y symmetrical molar functions which were near v e r t i c a l . This muscle force i s only 39% that of intercuspal muscle ac t i v i t y and 44% that of bimolar a c t i v i t y . II. Lateral Tooth Angle (LTA) Change - Figure 12B Although the muscle force for i nc i sa l clenching i s 55 to 60% less than that for the two previous molar tasks , the re la t i ve decrease in TR magnitudes - 133 -Figure 12A. INCISAL CLENCHING WITH BITE STOP (INCISS). Right and l e f t side scale factors and three dimensional (computer-drawn) depict ion of indiv idual muscle vectors) . - 134 -to 141.0 N i s much greater. In th is instance the increase in the length of the moment arm for the tooth resistances forces necessitates re la t i ve l y small TR magnitudes. On the other hand the j o i n t forces are s t i l l r e l a t i ve l y high, being within 55 to 70% of those values for the molar functions (CO and BIMOL) with matched or ientat ions of tooth and muscle forces ( i e . RUN 3 of each). Therefore the JF/TF ra t ios are, in general , very high for these i nc i sa l clenches and are than hal f as high as molar clenches. As such, clenching on the inc i sa l teeth i s less than hal f as e f f i c i e n t as molar clenching in terms of the d is t r ibu t ion of resistance forces. In add i t ion , close inspection of the JF/TF ra t ios for th is task in Figure 12B shows the most i n e f f i c i e n t run to be with matched data (Run 3 ) , which also has the lowest TR magnitude. 31% of the total force of resistance (141.2 N) occurs at the inc isors and 69% at the j o in t s (160.0 N per s i de ) . This i s quite d i f fe rent from the trends seen up to th is point . Prev ious ly , as the tooth force became more poster ior ly al igned (decreasing poster io r ly di rected component and/or increasing anter io r ly directed component along the y axis) the length of the moment arm for each increment became shorter. This required greater magnitudes of tooth force to maintain rotat ional s t a b i l i t y but also increased the l inear components (ver t i ca l and anteroposterior) contr ibuted by the tooth forces. Th is , in turn , l e f t less l inear resistance force unaccounted for and thus less j o i n t forces since RCR and LCR are essen t i a l l y the residual resistance forces needed to balance the l i near s ta t i c equi l ibr ium of the system (see METHODS). Up to now as LTA became more poster ior in or ientat ion TR increased and RCR and LCR decreased. Figure 1 2 B . INCISAL CLENCHING WITH BITE STOP (INCISS)/VARIABLE LTA. Corresponding tooth and condylar react ion force vectors for b i l a t e r a l l y symmetrical clenching on an inc isa l stop ( s t a b i l i z e r ) . The range of LTA var ia t ion i s as per ANG'L and preceding f igures. MUSCLE RESULTANT PARAMETERS MLR= o.o MAR= 160.8 MVR= 4 3 2 . 3 ANGL= 69.6 ANG-F = 90.0 r- 480 N LAT. PLANE RESULTANT LTA RESULTANT VECTOR RUN , Posit n , RCA I LTA ; LCA RCR TR LCR RCA 1 1 41 1 73.8 60.0 73.9 160.5 143.2 160.5 90.0 2 , | 71.6 1 65.0 | 71.6 160.1 141.7 160.1 3 1 1 69.6 1 69.6 1 69.6 160.0 141.2 159.0 4 1 1 67.2 j 75.0 1 67.2 160.1 141.9 160.1 5 L__ J 60.2 L i ° L ° J 60.3 162.1 150.9 162.1 FRONT. PLANE RESULTANT VECTOR ORIENTATIONS r"»/~* ' n - A ' ' I O A jk.1 FT  90.0 I I L C *  90.0 i I I I *LCFA cn JF/TF 2.24 2.25 2.27 2.25 2.15 - 136 -I n c i s a l c lench ing however i s sub jec t to d i f f e r e n t geometr ica l re la t ionships with respect to the LTA increments modeled. This re la t ionship depends on the distance of the point of tooth contact from the fulcrum of the system ( jo ints) both anteroposter ior!y (y - axis) and v e r t i c a l l y (z - a x i s ) . The posi t ion of the inc isa l contact i s such that the length of the moment arm for a tooth resistance force oriented at an LTA of 69.6 degrees (RUN 3) is very s l i gh t l y greater than that for any of the other or ientat ions modeled in Figure 12B. As a consequence, TR at th is or ientat ion i s somewhat less and RCR and LCR correspondingly greater. Increasing or decreasing LTA or ientat ion from 69.6 degrees decreases the respective moment arm lengths thereby requir ing greater TR magnitudes. It would be expected that RCR and LCR values would vary in a reciprocal manner with the highest value corresponding to the lowest TR of RUN 3. Such i s not the case however. Actua l ly the opposite occurs, with the j o i n t forces varying with the TR magnitude change, although the extent of th is change i s very smal l . The reason for th is can be most eas i l y understood by analyzing RUN 5 (LTA at 90.0 degrees) which demonstrates the greatest TR magnitude for th is task (150.9 N). This run also has the greatest RCR and LCR values. At LTA of 90.0 degrees there is no anteroposterior component of force at the tooth contact. However there is an anter ior component of force applied to the system by the muscles of 160.8 N (MAR). This can only be res is ted by the j o i n t s , which means there w i l l be a poster ior component calculated to be 80.4 N per j o i n t . The ver t i ca l component of j o i n t force is the residual of MVR (432.3 N) minus TR (150.9 N, which i s purely v e r t i c a l ) , which i s 281.4 N or 140.7 N per j o i n t . The resul tant of j o i n t force on the r ight and l e f t - 137 -sides (RCR and LCR) due to these axial components of force, according to the Pythagorean re la t ionsh ip , i s therefore 162.1 N per s ide . Thus, the re la t ionsh ips between TR, RCR and LCR become quite d i f ferent for changes to LTA than that previously observed due to d i f ferent geometrical re lat ionships of the components of the system. d. Incisal Clenching On Natural Contacts (INCISN) - Figures 13A and B I. Muscle Resultant Parameters Comparison of Figures 12A and 13A shows that when no s t a b i l i z i n g inc i sa l stops are provided there is a general decrease in ac t i v i t y of the various muscle groups (eg. SM, AT, MT, PT) except that of MP which has increased from 0.59 to 0.78. The ver t i ca l component of muscle force (MVR) shows a corresponding decrease from 432.3 N to 395.9 N whereas the anteroposterior component (MAR) has increased from 160.8 N to 194.2 N. This value of MAR i s the maximal value of th is var iable for th is study. The or ientat ion of muscle force for th is natural i nc i sa l clench i s therefore much more anter io r ly directed at 63.9 degrees (ANG-L). The overal l muscle resul tant for th is task i s determined to be 441.0 N, which i s only 4% less than that of i nc isa l clenching with stops. Thus, with th is (assumed) less stable occlusal contact there is only a s l i gh t decl ine in the overal l muscle force generated, but i t i s more anter ior ly d i rec ted. II. Lateral Tooth Angle (LTA) Change - Figure 13B Over the range of LTA tested, which was more acute than any tested up to th is point , the TR values are s i gn i f i can t l y less than those for the - 138 -Figure 13A. INCISAL CLENCHING-NATURAL (INCISN). Right and l e f t side scale factors and three dimensional (computer-drawn) depict ion of indiv idual muscle vectors. - 139 -s tab i l i zed inc isa l clenching of the previous task. However the magnitudes of RCR and LCR of Figure 13B indicate that very s imi la r forces of j o i n t resistance e x i s t . For RUN 3 of th is task, with LTA and ANG-L matched, 29% of the resistance force is contributed by the teeth and 71% by the j o i n t s , which i s a somewhat less e f f i c i e n t s i tuat ion that the corresponding run (RUN 3) of s t ab i l i zed inc isa l clenching (30 and 70% respect ively) in Figure 12B. The JF/TF ra t ios of Figure 13B re f l ec t the overal l decrease in e f f i c iency of resistance force d is t r ibu t ion for th is task. A more d i rec t comparison of the d i f fe rent e f fect on the d is t r i bu t ion of resistance forces due to the d i f fe rent type of i nc i sa l contact i s given by comparison of RUN 5 in Figures 12B and 13B (LTA = 90.0 degrees in both). Figure 13B shows 134.9 N of force occur at the teeth and 162.7 N per side at the j o i n t s . The same occlusal function but with b i te stops in Figure 12B shows 150.9 N at the teeth but only 162.1 N at each j o i n t . JF/TF ra t ios for these two tasks are 2.41 and 2.15 respect ive ly . This di f ference i s apparently due so le ly to the d i f fe rent a c t i v i t y pattern of the muscles in response to less stable occlusal contact , and thus a less e f f i c i e n t d i s t r i bu t ion of resistance forces. As was described for b i te s tab i l i zed inc isa l clenching the re la t ionship of j o i n t and tooth forces for natural i nc isa l clenching for LTA changes i s not the same as molar c lenching. Figure 13B shows the j o i n t forces to increase over the range of LTA or ientat ions tested in RUNS 1 to 5. TR on the other hand decreases to 126.3 N (RUN 4) then increases as LTA increases to 90.0 degrees. Again th is i s due to d i f fe rent geometry and correspondingly Figure 13B. INCISAL CLENCHING - NATURAL (INCISN)/VARIABLE LTA. As per Figure 12B but with natural i nc i sa l contact . MUSCLE RESULTANT PARAMETERS MLR= o.o MAR= 194.2 MVR= 395.9 ANG-L= 63.9 A N G F = 90.0 , rf VCA 480 N ^ 7 \ A R C A . A F. T A. 41 38 1 Tooth 1 L A T . P L A N E R E S U L T A N T V E C T O R O R I E N T A T I O N S R E S U L T A N T V E C T O R M A G N I T U D E S R U N , Positn , RCA J LTA ; LCA RCR TR LCR 1 1 67.6 55.0 67.6 156.5 130.4 156.4 2 I | 65.5 | 60.0 | 65.4 156.7 128.0 156.7 3 63.9 1 63.9 1 63.8 157.0 126.9 157.0 4 j j 61.5 ! 70.0 j 61.4 157.9 126.3 157.8 5 1 1 53.4 L S P i P J 53.3 162.7 134.9 162.6 F R O N T . P L A N E R E S U L T A N T V E C T O R O R I E N T A T I O N S RCA 90.0 FTA 90.0 LCA*! 90.0 I I I I * LCFA o J F / T F 2.41 2.45 2.47 2.51 2.41 - 141 -d i f fe rent lengths of the respective moment arms for each LTA at th is tooth pos i t i on . According to the predict ions of j o i n t resistance forces of i nc i sa l c lenching, the temporomandibular j o in t s would be predicted to have the i r load bearing surfaces at the anterosuperior aspect of the condyles. Jo in t resistance forces occur from about 50.0 to 70.0 degrees with respect to the occlusal plane during th is type of funct ion, with magnitudes of around 160 N. - 142 -2. Unilateral Clenching Tasks a. Unilateral Canine Clenching (UNIK9) - Figures 14A, B, C and D 1. Muscle Resultant Parameters Figure 14A shows the difference in muscle activity between the two sides of the mandible. The point of tooth contact lies at the right canine. The right, or working side MP, and especially the AT, MT and PT are more active than their left , or balancing side counterparts. Conversely the balancing side SM, DM and especially the IP group are more active than those of the working side. In general the activity levels of the various muscle groups are more or less intermediate between those of the bilateral molar and incisal clenching tasks of the previous section (see also Table II of METHODS). Reliable data regarding the activities of SP and DG were not available for unilateral tasks and these groups were therefore excluded from consideration. The magnitude of the muscle vector components reflect the generally intermediate activity of the muscle groups. MVR of 554.0 N and MAR of 99.7 N are within their corresponding ranges of magnitudes attributed to the bilaterally symmetrical molar clenches on one hand and incisal clenches on the other. ANG«L of 79.8 degrees is less vertical than bimolar or intercuspal clenches with more posterior tooth contact, but more vertical than incisal clenches with more anterior contact. However for this unilateral task there also exists a lateral component of muscle force, MLR, of 15.8 N directed towards the working (right) side, (indicated by the minus sign). As such it is not possible to determine an overall muscle resultant force magnitude because there are, in this instance, - 143 -Figure 14A. UNILATERAL (RIGHT SIDE) CANINE CLENCHING (UNIK9). Right and l e f t side scale factors and three dimensional (computer-drawn) depict ion of indiv idual muscle vectors. - 144 -three muscle vector components (as opposed to two previously) which are n o t coplanar. ANG-F indicates the overal l d i rec t ion o f muscle e f f o r t , due to the la te ra l component, to be 91.6 in the frontal plane. 1 1 . L a t e r a l T o o t h A n g l e ( L T A ) Change - F i g u r e 14B A l l runs of Figure 14B were done with FTA f ixed at 91.6 degrees to match ANG-F of the muscle force data. LTA was matched with ANG-L in RUN 3 at 79.8 degrees. In th is matched data r u n the tooth resistance was 226.8 N and the total j o i n t force was 339.0 N with 171.2 N at the working side and 167.8 N at the balancing side j o i n t s . JF/TF of 1.50 indicates canine clenching to be re l a t i ve l y more e f f i c i e n t than inc isa l (2.27 to 2.47; Figures 12B and 13B) and less e f f i c i e n t than intercuspal (0.96; Figure 10B) or b i l a te ra l (0.68 to 1.01; Figures 11B and C) molar clenching regarding resistance force d i s t r i bu t i on . At more acute LTA or ientat ions there i s a decrease in TR, a concurrent increase in RCR and LCR and hence JF/TF increases. The converse i s true at less acute LTA's and lower JF/TF ra t ios r esu l t . Although both RCR and LCR magnitudes vary rec ip roca l l y with TR over the LTA increments of change, those of RCR do so to a greater extent than LCR. Over th is ser ies of runs in Figure 14B from LTA of 70.0 to 90.0 degrees RCR decreases by about 15 N, from 179.9 to 165.1 N, whereas LCR does so by only 2 N, from 169.5 to 167.5 N. As a consequence, at LTA of 90.0 degrees the l e f t or balancing side j o i n t i s more heavi ly loaded than the r ight working s ide , whereas the converse i s true at more acute LTA or ien ta t ions . - 145 -Two further consequences of increasing LTA seen in th is ser ies i s , f i r s t , the fami l ia r e f fect of producing more acute j o i n t force angles ( in the la tera l plane) also seen in the b i l a t e r a l l y symmetrical functions prev iously . Secondly, however, there i s an obvious di f ference between the r ight and l e f t side j o i n t force or ien ta t ions . The balancing condyle resistance angles (LCA) are more anter io r ly oriented than those of the working side (RCA)-. This i s due to the fact that , in th is task, the l e f t condyle does contr ibute a moment of force to the rotat ional s ta t i cs of the system because the b i te point and muscle e f fo r t are not b i l a t e r a l l y symmetrical. As LTA increases, the tooth force moment arm length decreases, requir ing re la t i ve l y greater TR magnitudes to e f fect the same rotat ional moment. In add i t ion , however, the re la t i ve magnitudes of the poster ior (y) component of tooth force decreases. As such the total poster ior (y) component of j o i n t resistance force (of RCR and LCR) must increase to balance that produced by the muscle forces. The reason the l e f t condyle contr ibutes r e l a t i v e l y more of th is component than the r i gh t , and thus has more acute LCA or ientat ions in the la te ra l plane, i s because neither the tooth force (which has f ixed or ientat ions and therefore spec i f i c x, y and z components) nor the r ight j o i n t (which i s the fulcrum point of the system and can contribute no moments) can o f fset the rotat ional moment due to the muscles in the horizontal (occlusal) plane. As LTA increases and the poster ior component of tooth force decreases, less rotat ional moment due to the tooth force i s ava i lab le to of fset that of the muscle forces in th is plane. This requires Figure 14B. UNILATERAL (RIGHT SIDE) CANINE CLENCHING (UNIK9) VARIABLE LTA. Corresponding tooth and condylar react ion force vectors for un i la tera l canine clenching. The range of LTA var ia t ion i s as per previous f igures but FTA i s also speci f ied to match ANG*F since the muscle vectors also have a mediolateral component ( i e . MRL = -15 . 8 ; ANG-F = 91.6) in the frontal plane. The negative value for MLR indicates that the muscle vectors have a mediolateral component t o the r igh t . LCFA i s a r b i t r a r i l y spec i f ied at 90° . MUSCLE RESULTANT PARAMETERS M L R = -15.8 M A R = 99.7 M V R = 554.0 A N G L = 79.8 A N G - F = 91.6 l- 480 N .Tooth | L A T . P L A N E R E S U L T A N T V E C T O R O R I E N T A T I O N S R E S U L T A N T V E C T O R M A G N I T U D E S R U N . Posit n , RCA LTA LCA RCR TR LCR 1 43 93.9 j 70.0 j 77.8 179.9 222.7 169.5 2 | | 90.4 | 75.0 | 75.2 175.2 223.9 168.5 3 86.8 1 79.8 1 72.7 171.2 226.8 167.8 4 j | 82.6 ! 85.0 69.8 167.7 231.8 167.4 5 1 1 78.1 L2°zPJ 66.9 165.1 238.7 167.5 F R O N T . P L A N E R E S U L T A N T V E C T O R O R I E N T A T I O N S RCA 93.2 FTA 91.6 L C A * 90.0 I I I I * LCFA -p» cn - 147 -an increase in th is component due to the l e f t condyle, causing more acute LCA or ientat ions in the la te ra l plane. I t i s in terest ing to note that in RUN 3, with matched data, the mean of RCA and LCA in the la te ra l plane (86.8 and 72.7 degrees respect ively) is 79.8 degrees, which also matches LTA and ANG-L of that run. S im i l a r l y , in the frontal plane the mean of RCA (93.2 degrees) and LCA (90.0 degrees) i s 91.6 degrees, also matching FTA which was speci f ied at th is angle to correspond with ANG-F of the muscle forces in th is pro jec t ion. H i . Frontal Tooth Angle (FTA) Change - Figure 14C In th is ser ies of runs LTA was speci f ied at 79.8 degrees to match that of ANG-L and the e f fec t of changes to FTA were observed. Note that in Figure 14C there is no run with both LTA and FTA matched to thei r respective muscle force angles since th is run was included in Figure 14B (RUN 3) . In contrast to the ef fects produced by changes to LTA or ientat ions previously observed, changes to FTA produce very d i f ferent d is t r ibu t ions of resistance forces on the system. F i r s t of a l l , increases in FTA from 90.0 to 100.0 degrees (RUNS 3 to 5) resu l t in an increase in TR. As FTA becomes less acute the moment arm lengths of the tooth forces decrease. This previously has meant the magnitude of TR had to continue to increase to maintain the same net rotat ional moment due to the tooth resistance force. However, in t h i s ser ies because LTA i s f ixed at 79.8 degrees the ver t i ca l and poster ior components of tooth force remain constant. As such, in the la te ra l plane project ion the magnitude of TR "appears" to be ident ica l for each run. - 148 -However the increase in FTA ( i . e . less acute) seen in the frontal plane requires a change in the re la t i ve magnitude of the mediolateral component of tooth resistance force. For changes of FTA from 80.0 to 90.0 degrees th is component decreases resu l t ing in the observed decrease in the total (three dimensional) magnitude of TR. From FTA of 90.0 to 100.0 degrees th is component increases, resu l t ing in the increase of TR observed over th is range. Since the increments of FTA change are equal in e i ther d i rec t ion from90.0 degrees (RUN 3) there i s a correspondingly equal increase in magnitude (but opposite in d i rect ion) of the mediolateral component of tooth force at FTA of 80.0 (RUN 1) vs . 100.0 (RUN 5) degrees. As a consequence, equivalent TR magnitudes are observed at these or ientat ions (230.1 N) as well as at 85.0 (RUN 2) vs . 95.0 (RUN 4) degrees (227.5 N). Another obvious di f ference in resistance force d is t r ibu t ion due to FTA changes i s that the or ientat ions of RCA and LCA in the la te ra l plane vary in an opposite manner re la t i ve to one another. As FTA increases from 80.0 degrees RCA decreases, as has been previously observed for LTA changes. LCA in th is pro jec t ion , on the other hand, increases. In the la te ra l plane project ion there can be no rotat ional moments due to e i ther condyle, as they are both coaxial with the fulcrum (see METHODS). The tooth resistance force must therefore account for a l l of the rotat ional moment act ing on the system due to the muscle forces. However, the s ta t i c l i nea r components (ver t i ca l and anteroposterior) of th is force (TR) are i n s u f f i c i e n t to balance those of the muscle leaving the remainder to be - 149 -accounted for by the j o i n t s . Both the or ientat ion of overal l muscle force (ANG-L) and the tooth force (LTA) are matched and are essen t ia l l y para l le l in th is plane. When such has been the case in b i l a t e r a l l y symmetrical tasks (eg. Figure 10B of CO; Figure 11B, RUN 2 and Figure 11C, RUN 3 of BIMOL; Figure 12B, RUN 3 of INCISS; and Figure 13B, RUN 3 of INCISN) the combined ( r igh t plus l e f t ) j o i n t resistance forces have been at the same or ientat ion (RCA and LCA) as w e l l . This was because the re la t i ve proportions of the ve r t i ca l and anteroposterior components of j o i n t force must match those of the muscle and tooth forces to maintain s ta t i c equi l ib r ium. Ind iv idual ly the RCA and LCA of the la te ra l plane of Figure 14C do not do so. Nevertheless, the combined ef fect of the two j o i n t forces of th is Figure do ac tua l ly match the muscle and tooth forces as evidenced by the fact that the means of RCA and LCA of each run are v i r t u a l l y that of ANG-L ( i . e . RUN 1 = 78.7; RUN 2 = 79.3; RUN 3 = 79.7; RUN 4 = 79.8; RUN 5 = 79.7 degrees). In other words, for RUN 1 of Figure 14C for instance, the anter ior ly directed component of r ight condyle force is e f fec t i ve l y reduced by the poster io r ly di rected component of the l e f t such that the i r combined net or ientat ion (eg. mean) is 78.7 degrees. The reason not every instance exact ly matches ANG-L i s due to FTA or ientat ions d i f fe ren t from ANG-F which af fects the mediolateral component of the r ight condyle force (and hence RCA). Th is , in - 150 -turn inf luences the magnitude of RCR, i t s ver t i ca l and anteroposterior components and thus RCA in the la te ra l plane*. In the frontal plane project ion of Figure 14C an increase in FTA from 80.0 to 100.0 degrees resul ts in a decrease of RCA or ientat ions in th is plane from 106.4 to 81.6 degrees. La te ra l l y directed tooth forces (eg. RUN 5) do not contr ibute to s t a b i l i z i n g the l ikewise directed muscle force (ANG'F = 91.6 degrees). RCA becomes more acute due to a requirement for a more medial component somewhere in the system to balance these combined la tera l forces of muscle and tooth. Conversely, medial ly directed tooth forces (eg. RUN 1) require the opposite condi t ions. Since the l e f t side j o i n t force angle (LCFA) i s f ixed at 90.0 degrees i t cannot contr ibute e i ther a medial or a la te ra l component of resistance force to the system. *Another way of considering how the combination of the two j o i n t forces balance the system involves considerat ion of the horizontal plane which i s not depicted here. As FTA increases and the l a t e r a l l y directed (x) component of tooth resistance force decreases (changes d i rec t ion in fact) there is a decrease in the net rotat ional moment on the system in th is plane due to the muscle and tooth forces. Only an increase in the poster io r ly di rected component of LCR can balance the system under the given condit ions ( i e . fulcrum at the r ight condyle and LCFA of 90.0 degrees) as FTA increases. The magnitude of th is component at the l e f t condyle determines both the magnitude and or ientat ion of that at the r ight condyle then required to balance the 1 inear s ta t i cs of the system which i s also observed in the l a te ra l plane pro jec t ion. For instance, in RUN 1 (FTA = 80.0 degrees) a r e l a t i v e l y large poster ior component i s needed at the l e f t j o i n t to balance the rotat ional moments due to MLR (-15.8 N) plus that of TR which essen t ia l l y adds to that of MLR. However th is resu l t ing l e f t j o i n t component i s greater than that needed to balance the residual anter ior component of muscle force (MAR = 99.7 N) not accounted for by the poster ior component of tooth force seen in the la te ra l pro jec t ion. Therefore an anter io r ly directed component of force i s needed at the r ight j o i n t to balance the system. In RUN 5, however, the poster ior component needed to balance the residual rotat ional moment of the muscle and tooth forces for FTA of 100.0 degrees i s much l e s s . So much so, in f ac t , that an addit ional poster ior component i s required at the r ight j o i n t to maintain the l i near s ta t i c balance. Figure 14C. UNILATERAL (RIGHT SIDE) CANINE CLENCHING (UNIK9) VARIABLE FTA. Corresponding tooth and condylar react ion force vectors in th is f igure re f l ec t the speci f ied var ia t ions in FTA which are approximately ten degrees e i ther side of ANG-F. LTA i s speci f ied to match ANG-L. LCFA 1s a r b i t r a r i l y spec i f ied as 90°. As such, no var ia t ion 1n the l e f t condylar vector angle i s seen in the frontal plane whereas RCA 1n th is plane does exh ib i t var ia t ion in i t s or ientat ion as i t 1s computer der ived. MUSCLE RESULTANT PARAMETERS MLR= -15.8 MAR= 99.7 MVR= 554.0 A N G L = 79.8 ANG-F = 91.6 4 8 0 N ^ Tooih LAT. PLANE RESULTANT RESULTANT VECTOR FRONT. PLANE RESULTANT RUN • Pos i t 0 , RCA i L T A i LCA RCR TR LCR RCA i ^ i ', LCA* j 1 1 4 3 1 99.8 1 7 9 * 8 57.5 197.5 230.1 170.8 106.4 | 80.0 j 1 90.0 | 2 I | 94.5 | | 64.0 183.8 227.5 168.0 101.1 1 85.0 1 1 1 3 1 1 88.7 70.6 173.7 226.7 167.5 95.2 j 90.0j | | 4 j j 82.5 1 1 77.1 167.5 227.5 169.2 88.7 I 95.0 | 1 1 5 1 1 75.9 I 1 83.5 165.6 230.1 173.0 81.6 LIQO^OJ 1 1 * LCFA I h—* tn i - 152 -This e f fect also has further consequences on the magnitudes of the r ight and l e f t j o i n t forces. Over the ser ies of runs tested the magnitudes of the r i gh t , or working side condyle resistance force (RCR) exceed those of the l e f t or balancing side (LCR) for FTA or ientat ions of 90.0 degrees or less (RUN 1, 2 and 3 ) . At FTA or ientat ions greater than 90.0 degrees (RUN 4 and 5) the converse i s t rue. This i s due to a progressive (nonlinear) decrease in RCR magnitudes from 197.5 N to 165.6 N as FTA increases from 80.0 to 100.0 degrees (RUN 1 and 5 respec t i ve ly ) . LCR, on the other hand decreases from 170.8 N to 167.5 N for FTA increases from 80.0 to 90.0 degrees but then increases to 173.0 N from FTA or ientat ions of 90.0 to 100.0 degrees. The reasons for th is require careful consideration of the system three dimensionally and the e f fec t of FTA change to LCA or ienta t ions. F i r s t of a l l , in the frontal project ion of Figure 14C, as FTA increases from 80.0 to 100.0 degrees in RUNS 1 to 5 the rotat ional moment due to the tooth resistance force decreases (due to decreasing tooth force moment arm lengths) . Only the l e f t j o i n t force can contribute any more moment to the system to res i s t that of the muscles not accounted for by the rotat ional moment of the tooth resistance force. Therefore an increase in the ver t i ca l component of LCR resu l ts as FTA increases in RUNS 1 to 5. This i s apparent in both the la te ra l and frontal pro ject ions. As was mentioned above, because LCFA i s f ixed at 90.0 degrees only a change in the mediolateral component of RCR i s ava i lab le to accommodate any residual l i near components of muscle force not accounted for by that of the tooth force along th is ax i s . Hence - 153 -the change in frontal plane RCR over the se r i es . A lso , due to the increase in the ve r t i ca l component of LCR as FTA increases, that of RCR decreases. Secondly, in the la te ra l project ion of Figure 14C as FTA increases there i s a concurrent decrease in the poster ior component of LCR due to increasing LCA or ien ta t ions . At same time RCA in th is plane is decreasing to maintain the same overal l or ientat ion of the combined r ight plus l e f t j o i n t resistance forces, as has been discussed. The overal l e f fect of the changes in FTA or ien ta t ion , from 80.0 to 100.0 degrees, to the three orthogonal (x, y and z) components of r ight j o i n t force i s to produce the progressive decrease in RCR magnitudes seen in Figure 14C. The e f fec t of th is change to LCR magnitudes involve only two components, the ve r t i ca l (z) and the anteroposterior (y ) . This change in FTA produces an increase in the ver t i ca l component of th is force as discussed above and a concurrent decrease in the poster ior ly directed anteroposterior component. The magnitude of LCR in RUN 3 of 167.5 N i s at i t s lowest value because the magnitudes of both of these components are re l a t i ve l y less than those of e i ther RUN 1 (maximal anteroposterior component) or RUN 3 (maximal ve r t i ca l component). Therefore the combination of the two respective components of l e f t j o i n t resistance force which are necessary to maintain s ta t i c equi l ibr ium produce LCR magnitudes which increase for FTA of 80.0 to 90.0 degrees and then decrease from 90.0 to 100.0 degrees (RUN 1 to 5 ) . F i n a l l y , as a further consequence of FTA increase over th is ser ies the overa l l e f fect on the magnitudes of RCR, TR, and LCR i s to produce an increase in the e f f i c iency of resistance force d is t r ibu t ion in terms of JF/TF r a t i o s . The more medially directed tooth forces have r e l a t i ve l y lower JF/TF - 154 -r a t i o s . The range of these values are comparable to those observed as at resu l t of LTA change discussed in the previous sect ion. i v . Lef t Condyle Frontal Angle (LCFA) Change - Figure 14D Both LTA and FTA or ientat ions were speci f ied to match ANG-L (79.8 degrees) and ANG-F (91.6 degrees) respect ively in th is ser ies of runs shown in Figure 14D. Only LCFA or ientat ions were changed from 80.0 to 100.0 degrees. As a resu l t there ex is ts a s ingle vector of TR which remains constant for each run at 226.8 N and the above or ientat ions in space. In the l a te ra l project ion the mean of RCA (86.8) and LCA (72.7) i s also 79.8 degrees. As was discussed previously th is occurs in order to balance both the ver t i ca l and anteroposterior forces due to the muscle force acting in th is d i rec t ion (ANG-L) which i s not accounted for by TR also acting in th is d i rec t ion (but opposite to the muscle fo rce) . The total residual resistance force acting at both j o in t s must para l le l these other forces. The d i s t r i bu t i on of th is residue between the two jo in ts i s such that the l e f t side has a re l a t i ve l y greater poster ior component than the r i gh t . In th is plane both the r ight and l e f t j o i n t ver t i ca l components are equal in each run regardless of LCFA or ientat ions as are the poster ior ly directed components. In the frontal project ion as LCFA increases from 80.0 to 90.0 degrees (RUN 1, 2 and 3) there i s a decrease in the medial (rightward) component of force at the l e f t j o i n t . Since th is component of force at th is j o i n t does not reduce any of the rightward component due to the muscle force (MLR = -15.8 N) the r ight j o i n t requires a r e l a t i ve l y greater medial (leftward) component to balance the system. At LCFA of 90.0 degrees, however, the l e f t j o i n t does not add any mediolateral component. Thus the medial component of Figure 14D. UNILATERAL (RIGHT SIDE) CANINE CLENCHING (UNIK9)/VARIABLE LCFA. A l l var iab les f ixed except the l e f t condylar angle in the frontal plane which varies over the same range as the FTA of Figure 14C. LTA and FTA are spec i f ied to match ANG-L and ANG-F respect ive ly . MUSCLE RESULTANT PARAMETERS MLR= -15.8 MAR= 99.7 MVR= 554.0 ANGL = 79.8 ANG-F = 91.6 480 N RUN 1 2 3 4 5 Tooth Posit" 43 LAT. PLANE RESULTANT VECTOR ORIENTATIONS RCA 86.8 LTA 79.8 LCA 72.7 ' ,234 .432,1 RESULTANT VECTOR FRONT. PLANE RESULTANT I I I I cn cn RCR TR LCR RCA 1 FTA j J LCA*} JF/TF 175.1 226.8 170.2 102.5 | 91.6 | | 80.0 | 1.52 172.6 168.4 97.9 1 1 1 85.0 1 1.50 171.2 167.8 93.2 1 1 | 90.0 j 1.50 171.0 168.4 88.5 1 1 1 9 5 ' ° 1 1.50 172.0 170.2 83.7 1 l n a o ^ j j 1.51 * LCFA - 156 -the r ight j o i n t (RCA = 93.0 degrees) i s simply that necessary to balance the corresponding component of muscle force not neutral ized by the medial or leftward component of TR. At LCFA greater than 90.0 degrees a leftward la te ra l component occurs requir ing an opposite balancing component ( la tera l but rightward) at the r ight j o i n t . Since the ver t i ca l component of force at each j o i n t i s ident ica l for each run the magnitude of LCR at 80.0 and 100.0 degrees (170.2 N; RUN 1 and 5) i s the same because they both have equivalent (but opposite) mediolateral components contr ibut ing to the overal l vector . LCFA of 90.0 degrees, with no such added component therefore has the lowest magnitude (167.8 N). The same ef fect causes the var ia t ion in RCR magnitudes but LCFA of 95.0 degrees (RUN 4) produces the lowest value for th is vector since the corresponding RCA or ientat ion (88.5 degrees) i s the c losest to ve r t i ca l of the r ight j o i n t forces. The extent of these changes to RCR and LCR due to LCFA var ia t ions are r e l a t i v e l y s l i gh t and as such there i s l i t t l e change in the JF/TF ra t io from 1.50 in th is se r i es . However, at LCFA or ientat ions much beyond a range of 5.0 degrees e i ther side of ver t i ca l the d is t r ibu t ion of resistance forces would appear to be less favorable according to the s l i gh t increase in th is ra t io observed in RUN 1 and 5. b . U n i l a t e r a l M o l a r C l e n c h i n g (UNIMOL) - F i g u r e s 1 5 A , B , C , D a n d E 1 . M u s c l e R e s u l t a n t P a r a m e t e r s Figure 15A depicts the predominance of r ight (working) side muscle a c t i v i t y during th is task. The Scale Factors of SM, DM, MP, AT and PT are - 157 -s i g n i f i c a n t l y higher on the r i g h t than the l e f t s i d e . MT i s equivalent on the two s i d e s , whereas only IP shows s i g n i f i c a n t l y greater r e l a t i v e a c t i v i t y on the l e f t balancing side than i t s working side counterpart. Except f o r IP, a l l of these Scale Factors represent an increase i n the r e l a t i v e force produced by each of the various muscle groups compared to those of u n i l a t e r a l canine c l e n c h i n g . IP alone i s l e s s a c t i v e during unimolar clenching than unicanine clenching although the balancing side i s more a c t i v e in both t a s k s . SP and DG data are unavailable f o r unimolar clenching and are omitted here. Despite the obvious d i f f e r e n c e s between the Scale Factors of u n i l a t e r a l canine and molar clenching the magnitude of the l a t e r a l component of the o v e r a l l muscle force f o r the l a t t e r task of -15.1 N (MLR) i s v i r t u a l l y the same as that of the canine task. This i n i t s e l f c l e a r l y i n d i c a t e s how various d i f f e r e n t combinations of muscle force can e f f e c t the same r e s u l t s . MAR of molar clenching of 87.9 N, on the other hand i s somewhat l e s s than f o r the unicanine task (99.7 N). Nevertheless, unimolar clenching produces a much greater v e r t i c a l component (MVR) of muscle force at 839.9 N ( v s . 554.0 N of u n i c a n i n e ) . The o v e r a l l o r i e n t a t i o n s of the muscle force produced by the components a t ANG-L of 84.0 and ANG-F of 91.0 degrees are s i m i l a r to canine c l e n c h i n g but i n general are more v e r t i c a l than the l a t t e r and much more so than the i n c i s a l c lenching t a s k s . Only i n t e r c u s p a l and b i l a t e r a l molar cl e n c h i n g tasks produce greater v e r t i c a l ANG-L o r i e n t a t i o n s . 11. Tooth Pos i t ion Change - Figure 15B Although an o c c l u s a l b i t e stop p o s i t i o n e d at tooth 47 was incorporated i n the d e r i v a t i o n of the major muscle groups (eg. SM, MP, AT, PT; see Table - 158 -Figure 15A. UNILATERAL (RIGHT SIDE) MOLAR CLENCHING (UNIMOL). Right and l e f t side scale factors and three dimensional (computer-drawn) depict ion of ind iv idual muscle vectors. - 159 -II of METHODS), the ef fect of changes to the posi t ion of tooth contact for these same muscle a c t i v i t i e s was modeled as shown in Figure 15B. As has been observed previously , more poster ior posi t ions of tooth contact resu l t in greater magnitudes of TR. Comparison of the TR value of 454.6 N for f i r s t molar clenching in RUN 1 of Figure 15B with corresponding runs for intercuspal (515.9 N; RUN 3, Figure IOC) and b i l a te ra l molar clenching (531.8 N; RUN 3, Figure 11B) shows that somewhat less total tooth resistance force occurs during un i la tera l molar c lenching. S im i la r l y the TR magnitude of 545.3 N at the second molar of Figure 15B i s also less than that during b i l a te ra l molar clenching (637.9 N) of the corresponding run (RUN 3) of Figure 11C. This i s , in par t , a function of the re la t i ve l y lower overal l muscle force generated during un i la tera l molar clenches. However i t must be remembered that in both intercuspal and b i la te ra l molar clenches the TR force produced i s d is t r ibu ted to both sides of the den t i t i on . The tooth forces of Figure 15B occur at the single tooth contact designated on the r ight side of the mandible. As such i t would appear that despite r e l a t i ve l y less muscle fo rce , proport ionately more tooth resistance force ( i . e . per tooth) can be produced for un i la tera l molar contacts. In each run of Figure 15B the balancing ( le f t ) side j o i n t i s more heavi ly loaded than the working (r ight) s ide . For f i r s t molar contact (RUN 1) LCR of 258.2 N i s approximately twice that of RCR at 138.9 N. As the pos i t ion of tooth contact becomes more poster ior LCR decreases only s l i g h t l y to 247.8 N at the second molar and 244.3 N at the t h i r d . LCA remains v i r t u a l l y constant for a l l three tooth posi t ions at approximately 78.0 degrees in the la te ra l plane (90 degrees in the f r on ta l ) . The r ight j o i n t - 160 -resistance force, on the other hand, changes dramatical ly at the more poster ior tooth contacts with respect to both magnitude (RCR) and or ientat ion ( i . e . RCA, in both the la te ra l and frontal planes). At the second molar contact RCR of 64.9 N i s only about one quarter that of LCR at 247.8 N and i s directed more poster ior ly (RCA = 56.4 degrees in the la te ra l plane) than for f i r s t molar contact (RCA = 75.3 degrees). Tooth contact at the th i rd molar produces the most dramatic change in r igh t (working) side j o i n t force. At th is tooth posi t ion RCR increases very s l i g h t l y to 66.0 N but has an upwards as well as poster ior component. Hence RCA or ientat ions of 302.9 and 254.3 degrees in the la te ra l and frontal project ions respect ive ly . That i s , for the muscle forces described here for a un i la tera l molar clench and with input var iable or ientat ions (LTA, FTA and LCFA) of 90.0 degrees a th i rd molar contact alone creates a tens i le force at the working side j o i n t . Up to now only compressive j o i n t forces have been observed. This i s due to the fact that more poster ior tooth contacts are also posit ioned more l a t e r a l l y in the frontal plane, which was not the case in the b i l a t e r a l l y symmetrical clenching tasks previously observed. I n i t i a l l y , in the la te ra l plane, more poster ior tooth contacts (with shorter moment arms) require proport ional ly greater tooth forces (TR) to completely balance the moment due to the muscle force (MAR and MVR). In th is plane the moment due to the tooth force alone must account for a l l of the muscle force moment of ro ta t ion . In the frontal plane, however, the rotat ional moment due to TR (RUN 1) at the f i r s t molar (the magnitude of which was determined as that necessary to balance the moments in the la te ra l plane) accounts for less of the muscle force moment than TR of e i ther the - 161 -second (RUN 2) or th i rd molars (RUN 3) . Therefore the ver t i ca l component of LCR seen in th is ( f rontal ) project ion is greater for f i r s t than second or th i rd molar contacts since the f i r s t molar contacts must contr ibute a r e l a t i v e l y greater rotat ional moment to the system than ei ther of the l a t t e r . Although th is now s a t i s f i e s the rotat ional s ta t i cs of the system three dimensional ly, i t does not do so regarding the l inear s t a t i c s . Because the or ientat ions of the input var iable angles ( i . e . LTA, FTA and LCFA) are a l l purely v e r t i c a l , only the r ight j o i n t in the frontal plane i s able to contr ibute a medial component of force to r es i s t the la te ra l component of muscle force (MLR = -15.1 N). Thus, for each tooth posi t ion there i s an equivalent medial component of force at the r ight j o i n t seen in the frontal p ro jec t ion . However the sum of the ver t i ca l components due to the tooth (TR) and l e f t j o i n t (LCR) resistance forces in th is plane are i nsu f f i c i en t to balance MVR of the muscles. In the case of f i r s t molar contact (RUN 1) the necessary addit ional ver t i ca l component must be contributed by the r ight j o i n t , and i t i s compressive. For the second molar contact r e l a t i ve l y less addi t ional ver t i ca l (compressive) force at the r ight j o i n t i s required, due mostly to a proport ional ly greater TR magnitude in th is plane (which was determined as that necessary to balance the rotat ional s ta t i cs of the la te ra l p lane) . Regarding the th i rd molar posi t ion (RUN 3) , the re l a t i ve l y greater ( ve r t i ca l ) TR force combined with the corresponding ver t i ca l component of LCR in th is plane ( f ronta l ) are greater than that necessary to balance MVR. As such a negative, or tens i le s t a b i l i z i n g force resul ts at the r ight j o i n t to balance the l i near s ta t i cs of the system for a uni la tera l th i rd molar contact Figure 15B. UNILATERAL (RIGHT SIDE) MOLAR CLENCHING (UNIMOL)/VARIABLE TOOTH POSITION. Three possible pos i t ions of r ight molar contact are shown. A l l other var iables are a r b i t r a r i l y spec i f ied at 90° to s impl i fy comparisons of tooth pos i t ion . Therefore the mediolateral component of muscle force (MLR) 1s ref lected 1n the frontal plane RCA value only. MUSCLE RESULTANT PARAMETERS MLR= -15.1 MAR= 87.9 MVR= 839.9 ANG-L = 84.0 ANG-F = 91.0 4 8 0 N RUN ["Tooth ~j LAT. PLANE RESULTANT VECTOR ORIENTATIONS , Posit n , RCA | LTA | LCA l 1 4 6 1 75.3 90.0 j 78.1 2 1 47 | 56.4 | | 77.6 3 1 48 1 302.9 1 1 77.5 RESULTANT VECTOR MAGNITUDES _ R C B _ TR LCR FRONT. PLANE RESULTANT VECTOR ORIENTATIONS 138.0 454.6 258.2 64.9 545.3 247.8 66.0 655.4 244.3 RCA 96.5 106.4 254.3 FTA 90.0 LCA* ! 90.0 I I I I I I I I * LCFA CTl PO - 163 -under the given condi t ions. S tab i l i za t i on of the rotat ional and l i near s ta t i c s in the horizontal plane require equivalent poster ior ly directed components of l e f t and r ight j o i n t resistance forces for a l l three tooth contact posi t ions as shown in the la te ra l plane of Figure 15B. In terest ing ly , the JF/TF ra t ios of these runs suggest that un i la tera l molar clenching can generate not only more occlusal force per tooth with r e l a t i v e l y less muscle force than b i l a te ra l clenching tasks, but i t does so with greater e f f i c iency with respect to the d is t r ibu t ion of the remaining resistance forces. Although the second molar pos i t ion i s the most appropriate for the derived muscle data, the f i r s t molar posi t ion had a JF/TF ra t io of 0.87. The corresponding runs of intercuspal clenching (RUN 3; Figure IOC) and bimolar clenching (RUN 3; Figure 11B) had rat ios of 0.93 and 0.96 respect ive ly , which are s i gn i f i can t l y less favorable. Comparison of th is ra t io for the second molar run of th is uni la tera l task of 0.57 with the corresponding run of the bimolar task (RUN 3; Figure 11C) with a ra t io of 0.68 also lends credence to th is p red ic t ion . Most of th is improvement i s derived from the fact that the resistance forces of the working side j o i n t during poster ior ( i e . molar) un i la tera l clenching are s i gn i f i can t l y reduced and those of the balancing side are somewhat less while the tooth resistance remains re l a t i ve l y high. i i i . Lateral Tooth Angle (LTA) Change - Figure 15C The point of tooth contact designated was the r ight second molar for th i s ser ies with FTA f ixed at 91.0 degrees to match ANG-F and LCFA at 90.0 degrees. An increase in LTA produces an increase in TR and a decrease in both the balancing (LCR) and working side (RCR) j o in t force magnitudes. - 164 -As LTA increases from 75.0 to 95.0 degrees in Figure 15C TR also increases from 485.6 N to 578.2 N, again due to shortening of the moment arms of tooth force in each RUN. Consequently LCR decreases a corresponding amount from 265.1 N to 245.3 N due to a requirement for r e l a t i ve l y less ve r t i ca l and s l i g h t l y more poster ior ly directed components of l e f t j o i n t force to balance the moments of the system ( i . e . in the frontal and horizontal p lanes). Regarding RCR, there i s a requirement for reduced ve r t i ca l components as LTA increases, but r e l a t i ve l y more medial (leftward) force i s necessary as evidenced by the increasing RCA or ientat ions in the f rontal plane from 93.7 to 101.0 degrees. This l a t t e r component increase adds re l a t i ve l y l i t t l e to the overal l RCR magnitudes which are a l l s i gn i f i can t l y less than those of the balancing j o i n t . However the anteroposterior component of RCR var ies great ly over th is ser ies and i s i n i t i a l l y (RUN 1) anter io r ly directed but becomes poster ior ly directed at greater LTA values (RUN 5) . The greatest RCR magnitude of 119.3 N also occurs at the more acute LTA but decreases to a greater extent than LCR with increasing LTA. The lowest RCR i s in RUN 4 at 64.0 N. This i s simply a consequence of the ef fects of the changes to the three components of force over the range of LTA which when combined determine the RCR magnitudes. It i s apparent that while the change (decrease) in LCA or ientat ion of the la te ra l plane i s only about 10 degrees for th is ser ies (86.6 to 75.9 degrees; RUN 1 and 5 respect ively) that of RCA i s about 100.0 degrees (116.7 to 18.4 degrees for the same runs). The greater LTA angles with higher values for TR and reduced RCR and LCR magnitudes are more e f f i c i e n t regarding JF/TF ra t ios which has cons is tent ly Figure 15G. UNILATERAL (RIGHT SIDE) MOLAR CLENCHING (UNIMOD/VARIABLE LTA. The second molar 1s spec i f ied as the pos i t ion of tooth contact. LTA i s speci f ied as per previous f igu res . The r igh t condylar angles 1n the f rontal plane (RCA) are not col lnear although the re la t i ve l y small s ize of the vector project ions 1n th is plane makes them almost appear as such. LCFA Is a r b i t r a r i l y speci f ied at 90°. M U S C L E R E S U L T A N T P A R A M E T E R S MLR= -15.1 MAR= 87.9 MVR= 839.9 ANGL= 84.0 ANG-F = 91.0 4 8 0 N LAT. PLANE RESULTANT RESULTANT VECTOR FRONT. PLANE RESULTANT CTl R U N Posit n RCA i L T A i LCA RCR TR LCR RCA I ^ 1 ', L C A * | JF/TF 1 j 47 , 116.7 I 75.0 | 86.6 119 .3 485 .6 265 .1 93.7 I 9 1 - ° 1 | 90.0 | .79 2 1 1 105.4 1 80.0 1 84.4 92 .2 500 .2 260 .0 94.2 1 1 1 1 .70 3 1 1 89.8 1 84.0 1 1 | 82.4 73 .9 515 .5 256 .0 94.8 j 1 j 1 .64 4 | | 50.4 | 90.0 | 79.1 64 .0 545 .4 250 .0 96.5 | | 1 1 .58 5 1 1 18.4 L 2 5 J ) J 75.9 82 .8 578 .2 245 .3 101.0 1 1 1 1 .57 - 166 -been the case in a l l tasks observed. However the fact that un i la tera l molar clenching i s b i o l og i ca l l y more e f f i c i e n t i s again suggested by the JF/TF of 0.64 for RUN 3, with the input var iables matched to those of the MUSCLE RESULTANT PARAMETERS. Bimolar clenching of RUN 3, Figure 11C with s im i l a r l y matched data did produce more absolute occlusal force (618.5 N v s . 515.5 N) although d is t r ibuted over both sides of the mandible but also at a cost of 4% more j o i n t force (JF/TF = 0.68). iv. Frontal Tooth Angle (FTA) Change - Figure 15D For th is ser ies LTA was f ixed at 84.0 degrees and FTA varied 10.0 degrees e i ther side of the ANG-F or ientat ion of 91.0 degrees. The e f fec t of th is change is qua l i t a t i ve l y ident ica l to that observed for the unicanine clench of Figure 14C, although somewhat magnified due to the greater magnitudes of force involved. Var ia t ion of FTA requires equivalent ver t i ca l components of TR but adds a var iab le mediolateral component as FTA increases or decreases from v e r t i c a l . As such the re la t i ve l y greater and equivalent TR magnitudes at FTA of 80.0 and 100.0 degrees (523.3 N; RUN 1 and 5) and s l i g h t l y lower magnitudes at 85.0 and 95.0 degrees resu l t (517.4 N; RUN 2 and 3 ) . RUN 3 (matched) with FTA of 91.0 degrees has the lowest TR magnitude (515.4 N), since i t has the smallest extra mediolateral component adding to i t s resul tant length. This e f fec t also produces the decrease in LCR magnitude and the great change in RCA or ientat ions as FTA increases in the frontal plane. This i s because more acute or ientat ions of tooth force (with longer moment arms in the frontal plane) account for r e l a t i ve l y more of the muscle moment. Thus - 167 -less LCR force at the designated LCFA or ientat ion ( i . e . ve r t i ca l ) i s requiredto make up the di f ference between the muscle and tooth rotat ional moments. However the various added mediolateral components of these TR vectors also introduce addit ional l i near forces to the system in the frontal plane which must be balanced. The muscle force in th is plane i s nearly ve r t i ca l (ANG*F = 90.0 degrees) and thus has a small la te ra l (rightward) component (MLR = -15.1 N). Only the r ight condyle can contr ibute any extra mediolateral force to balance these components of tooth force since LCFA i s f i x e d . Hence the dramatic decrease in frontal RCA or ientat ions from 133.8 degrees of RUN 1 (large la te ra l tooth force component and correspondingly large medial j o i n t force component) to 41.7 degrees of RUN 5 (large medial tooth force component and corresponding la te ra l j o i n t fo rce ) . In each run most of the ver t i ca l components of the muscle force (MVR) act ing on the system are balanced by the sum of the tooth and l e f t j o i n t ve r t i ca l components of force. However the residual l i near ver t i ca l force which i s at t r ibuted to the r ight j o i n t i s much less than that of the l e f t j o i n t forces. A more acute ( l a t e r a l l y directed) FTA also requires a re l a t i ve l y greater poster ior component of LCR to balance the moments in the horizontal plane. This produces more acute la te ra l plane LCA or ien ta t ions . Conversely FTA or ientat ions greater than 90.0 degrees which have a medial ly directed component require less poster io r , and ( in the case of RUN 5) even an anter ior component of LCR. Thus LCA increases from 66.7 to 93.1 degrees in the l a te ra l plane as FTA increases from 80.0 to 100.0 degrees in the frontal (RUN 1 to 5 ) . In RUN 1 to 4 the l e f t j o i n t forces shown in the la te ra l plane have - 168 -a poster ior component (LCA < 90.0 degrees) while RUN 5 (LCA = 93.1 degrees) has an anter ior component. The remaining necessary anteroposterior components contr ibut ing to the resistance force at the r ight condyle (RCR) are those required to balance the remaining l inear s ta t i cs of the system. For instance, in RUN 1, FTA of 80.0 degrees requires a r e l a t i ve l y large poster ior component of l e f t j o i n t force (and a r e l a t i ve l y acute LCA or ientat ion) to balance the moment due to the muscle force in the horizontal (occlusal) plane. However, as i s apparent in the la te ra l plane pro jec t ion, th i s poster ior component due to the l e f t j o i n t force i s greater than that necessary to balance the anter ior component of the muscle force (MAR). Therefore an addit ional anter ior component resul ts at the r ight j o i n t at FTA of 80.0 degrees in RUN 1. In RUN 5 the opposite condit ions prevail such that the anter ior component of force at the l e f t j o i n t adds to that of the l i near component of muscle forces. This requires a poster ior component of force at the r igh t j o i n t at FTA of 100.0 degrees. The combination of the three orthogonal components of RCR at the various FTA or ientat ions are such that RCR decreases from 163.6 N at FTA of 80.0 degrees (RUN 1) to a minimum of 70.9 N at FTA of 95.0 degrees. Further increase in FTA to 100.0 degrees increases RCR to 100.IN. It i s in terest ing that for th is task the most optimum d is t r ibu t ion of resistance forces occurred in RUN 3 with the input var iables matched with the muscle data. Var iat ions in FTA or ientat ions have less favorable consequences on the biomechanics of the system under these condi t ions. Figure 15D. UNILATERAL (RIGHT SIDE) MOLAR CLENCHING (UNIMOL)/VARIABLE FTA. LTA 1s spec i f ied to match ANG-L. FTA var ies as described in previous f igures. LCFA a r b i t r a r i l y spec i f ied as 90°. MUSCLE RESULTANT PARAMETERS M L R = -15.1 M A R = 87.9 M V R = 839.9 A N G L = 84.0 A N G - F = 91.0 4 8 0 N RUN l 2 3 4 5 Tooth Posit" 47 LAT. PLANE RESULTANT VECTOR ORIENTATIONS I I RCA 118.4 109.0 89.8 70.2 41.7 LTA 84.0 RESULTANT VECTOR MAGNITUDES I I LCA RCR. TR 66.7 163.6 523.3 74.2 115.5 517.3 82.4 73.9 515.5 87.4 70.9 517.4 93.1 100.1 523.3 LCR 236.7 243.2 256.0 267.1 283.9 FRONT. PLANE RESULTANT VECTOR ORIENTATIONS RCA 133.8 122.7 94.8 63.9 41.7 FTA 80.0 85.0 91.0 95.0 UQQ .O J L C A * ! 90.0 I I •* LCFA CTl - 170 -v. Lef t Condyle Frontal Angle (LCFA) Change - Figure 15E With LTA and FTA matched with the i r respective or ientat ions of muscle force the var ia t ions of LCFA for th is molar clench have s imi lar overal l e f fec ts on j o i n t force or ientat ion as those seen in Figure 14D of the unicanine task. However both the range of RCA in the frontal plane for LCFA changes from 80.0 to 100.0 degrees and the magnitudes of these resistance forces are re l a t i ve l y greater. This i s due to a s i gn i f i can t l y greater magnitude of muscle force and a tooth contact posi t ion much c loser to the fulcrum in a l l three dimensions for th is molar c lench. Var ia t ion of LCFA above or below 90.0 degrees adds a mediolateral component of l e f t j o i n t resistance to the (constant) ver t i ca l component which increases the net resul tant of th is fo rce. Hence LCR at 80.0 and 100.0 degrees (RUN 1 and 5) is the same at 259.8 N and at 85.0 and 95.0 degrees (RUN 2 and 4) i s 256.9 N. No such component ex is ts at LCFA of 90.0 degrees and thus RUN 3 has to lowest LCR magnitude of 256.0 N. These extra mediolateral components imposed on the l e f t j o i n t force require balancing components at the r ight j o i n t . Therefore in RUN 1 the medial ly (rightward) directed component of LCFA requires a leftward medial ly d i rected component at the r ight condyle (eg. RCA = 124.7 degrees). In RUN 5 the opposite set of circumstances require the l a t e r a l l y directed (rightward) component at the r ight j o i n t (eg. RCA = 62.4 degrees). RUN 3, on the other hand, with no l e f t j o i n t mediolateral component requires no addit ional balancing component at the r ight condyle. The s l i gh t medial (leftward) component which does ex is t in th is run at the r ight j o i n t i s simply that necessary to balance the la te ra l (rightward) component of muscle force (MLR = Figure 15E. UNILATERAL (RIGHT SIDE) MOLAR CLENCHING (UNIMOD/VARIABLE LCA. LTA and FTA are spec i f ied to match ANG-L and ANG-F respect ive ly . LCFA var ies over the same range as FTA of Figure 15D. M U S C L E R E S U L T A N T P A R A M E T E R S MLR = MAR = MVR = A N G L = ANGF = -15.1 87.9 839.9 84.0 91.0 4 8 0 N RUN l 2 3 4 5 Tooth Posit" 47 LAT. PLANE RESULTANT VECTOR ORIENTATIONS RCA 89.8 LTA 84.0 LCA 82.4 L J I I -1321 RESULTANT VECTOR MAGNITUDES _BCEL 89.5 78.9 73.9 75.4 83.1 TR 515.5 LCR 259.8 256.9 256.0 256.9 259.8 FRONT. PLANE RESULTANT VECTOR, QRIEN,TATj0iJ5 RCA 124.7 111.1 94.8 77.7 62.4 FTA 91.0 I I i LCA* j JF/TF 80.0 | .68 85.0 1 .65 90.0 j .64 95.0 | .64 100 JDJ .66 •* LCFA - 172 --15.1 N; ANG*F = 91.0 degrees). The sum of the ver t i ca l and mediolateral components of resistance force so produced at the r ight j o i n t are such that the leas t magnitude of resul tant of th is force occurs in RUN 3 (RCR = 73.9 N)and LCFA at 90.0 degrees. Var ia t ion from th is or ientat ion at the l e f t j o i n t from 90.0 degrees produces increased RCR magnitudes. Since no change in TR magnitude occurs, the JF/TF ra t io i s minimal at the LCFA or ientat ion with the lowest combined j o i n t resistance forces which i s in RUNS 3 and 4 at 0.64. Beyond th is JF/TF increases as shown in Figure 15E. Thus, var ia t ion in LCFA from 90.0 degrees produces a reciprocal e f fec t on RCA or ientat ions but in general do not increase the e f f i c iency of resistance force d is t r ibu t ion of the system. - 173 -3. Unilateral Chewing Tasks a. Muscle Resultant Parameters - Figures 16A, 17A and 18A Comparison of Figures 16 to 18 "A" portrays the change in the a c t i v i t i e s of the various muscle groups (see also Table III of METHODS) during three in te rva ls of the power stroke of r ight side gum-chewing. The f i r s t two in te rva ls are 100 msec (Interval 1 ) , and 50 msec (Interval 2) before Time 0 (Interval 3) which occurred approximately 10 to 15 msec af ter in tercuspat ion. Each of these in terva ls is considered to be in s ta t i c equi l ibr ium with no net movement of the system actual ly occurr ing. In Interval 1 and 2 a l l of the working (r ight) side muscle groups are more act ive than the i r balancing side ( le f t ) counterparts (see Figure 16A and 17A). At Interval 2 a l l of the working side muscle a c t i v i t i e s increase to the i r maximal leve ls of the three phases considered, as do the three temporalis groups and i n fe r i o r head of the la te ra l pterygoid of the balancing side (AT, MT, PT and IP respec t i ve ly ) . Only the balancing side masseters (SM and DM) and medial pterygoid (MP) show a decl ine from Interval 1 to 2 although not to the i r minimal l e v e l s . During Interval 3 (see Figure 18A) a l l muscle groups of both sides show a decrease in ac t i v i t y from that of Interval 2 , with the working side pterygoids (MP and IP) and temporalis groups (AT, MT, PT) at minimal l eve ls of the three phases. The balancing side masseters (SM and DM) and pterygoids (MP and IP) are also least act ive in th is phase. A l l other leve ls are more or less intermediate between those of the f i r s t two in te rva ls (eg. Working side masseters and balancing side temporals). D igast r ic i s inact ive during a l l three phases. - 174 -Comparison of these ac t i v i t y leve ls with those of the preceding s ta t i c clenching tasks (see Tables II and III of METHODS) shows that the a c t i v i t i e s observed during masticatory functions do not necessar i ly coincide with any pa r t i cu la r s ta t i c funct ion. The various leve ls of muscle ac t i v i t y seen during chewing and the i r combinations are unique to each chewing i n t e r v a l . Nevertheless, the combined ef fect of the various muscle groups in each phase of the chewing power stroke produce overal l muscle forces quite s imi la r to those observed in the s ta t i c molar clenches. The "MUSCLE RESULTANT PARAMETERS" of the three power stroke in terva ls from Figures 16, 17 and 18 B, C and D are summarized here as are these var iables from the uni la tera l molar clench for comparison. INTERVAL  1 2 3 UNIMOL MLR -0.4 11.7 15.0 -15.1 (Newtons) MAR 67.6 54.1 40.2 87.9 (Newtons) MVR 522.3 676.8 451.3 839.9 (Newtons) ANG-L 82.6 85.4 84.9 84.0 (Degrees) ANG-F 90.0 89.0 88.1 91.0 (Degrees) The muscle e f fo r t of Interval 1 in the la te ra l plane i s directed anterosuperior ly at 82.6 degrees with a ver t i ca l component of 522.3 N and an anter ior component of 67.6 N. In the frontal plane i t i s essen t ia l l y ve r t i ca l having v i r t u a l l y no mediolateral component (MLR = -0.4 N). The change in a c t i v i t y l eve ls of the muscles in Interval 2 , and the i r combined ef fect produces an overal l muscle e f fo r t more v e r t i c a l l y oriented in the la te ra l plane at 85.4 degrees. This i s due to an increase in the ve r t i ca l component of force (MVR) to a peak level for the three phases to - 175 -676.8 N combined with a s l i gh t decrease in the anter ior component (MAR) to 54.1 N. In the frontal plane the change in muscle a c t i v i t i e s generates a component of mediolateral force of 11.7 N which i s directed leftward at 89.0 degrees. The muscle e f fo r t of Interval 3 in the la te ra l plane i s s l i g h t l y less ve r t i ca l than Interval 2 at 84.9 degrees due to a further decrease in MAR to 40.2 N and a decl ine in MVR to 451.3 N from the maximal level in the l a t t e r phase. In the frontal plane the increase in MLR to 15.0 N combined with the lower ver t i ca l component produces a s l i gh t l y more leftward muscle e f fo r t of the three phases. These changes in d i rec t ion and magnitude of the muscle e f fo r t are consistent with those to be expected for r ight side chewing, i . e . maximal ve r t i ca l e f fo r t at 50 msec pr ior to time zero (or 35-40 msec before intercuspation) with re l a t i ve l y more medial ly directed e f fo r t as the mandible i s brought c loser to in tercuspat ion. I t was previously observed that more anter ior ly posit ioned clenching tasks produced re l a t i ve l y more an ter io r ly directed muscle e f f o r t . The or ientat ions of muscle force of these three chewing in terva ls in the la te ra l plane (ANG-L) from 83.0 to 85.0 degrees coincide with those one would expect from predict ions based on the s ta t i c molar clenching tasks. Unimolar clenching (see Figure 15B, etc.) produced muscle e f fo r t at 84.0 degrees in the la te ra l plane and near ver t i ca l in the frontal (91.0 degrees). S im i la r l y the intercuspal clench and bimolar clench produced muscle forces oriented at about 88.0 and 87.0 degrees respect ively (see Figures 10B and 11B e t c . ) . Unicanine clenching had th is e f fo r t at a more acute angle of around - 176 -Figure 16A. INTERVAL 1 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW1). Right and l e f t side scale factors and three dimensional (computer-drawn) depict ion of indiv idual muscle vectors. - 177 -Figure 17A. INTERVAL 2 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW2). Right and le f t side scale factors and three dimensional (computer-drawn) depiction of individual muscle vectors. - 178 -Figure 1 8 A . INTERVAL 3 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW3). Right and l e f t side scale factors and three dimensional (computer-drawn) depict ion of indiv idual muscle vectors. - 179 -80.0 degrees with the inc isa l clenches even more so at 64.0 to 70.0 degrees (see Figures 14, 12, and 13B e tc . respec t ive ly ) . In a l l instances the d i rec t ion of muscle e f fo r t in the frontal plane (ANG-F) was very near v e r t i c a l , as i s the case for the three chewing in terva ls despite the very d i f fe ren t indiv idual muscle group a c t i v i t i e s seen in a l l tasks. It would seem that many possible combinations of muscle group a c t i v i t i e s are capable of producing s imi la r overal l e f fects on the system. b. Lateral Tooth Angle (LTA) Change - Figures 16B, 17B and 18B In each run of these three f igures the posi t ion of tooth contact with the bolus i s assumed to have been at the r ight second molar (#47). The range of LTA was modeled from 75.0 to 95.0 degrees, which i s approximately ten degrees e i ther side of ANG-L for each interval (82.6, Interval 1; 85.4, Interval 2; and 84.9, Interval 3 ) . The TR magnitudes over the range of LTA modeled in each interval d i r ec t l y re f l ec ts the di f ferences in the ver t i ca l muscle forces (MVR) generated for each i n t e r v a l . Interval 1 with intermediate muscle force produces tooth resistance forces of from around 300 to 355 N. Interval 2 which has the greater muscle forces has tooth forces around 390 to 460 N whereas Interval 3 , with the lowest muscle forces produced the lowest TR magnitudes from approximately 260 to 305 N. The j o i n t forces produced for a l l three in terva ls were greater on the balancing than the working s ide . Comparing the runs with tooth or ientat ions most c lose ly matched with those of the applied muscle force (RUN 3 of Figures 16B and C, 17B and C, and 18B and C) the balancing j o i n t was loaded from 100 Figure 16B. INTERVAL 1 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW1)/VARIABLE LTA. All variables specified as per Figure 15C. MUSCLE RESULTANT PARAMETERS MLR= -0.4 MAR= 67.6 MVR= 522.3 ANGL= 82.6 ANG-F = 90.0 l- 480 N RUN l 2 3 4 5 Tooth Positn 47 I I LAT. PLANE RESULTANT VECTOR ORIENTATIONS RCA ; LTA I LCA 106.3 75.0 85.1 94.4 | 80.0 | 82.7 84.3 1 83.01 81.1 51.5 j 90.0 j 77.0 27.0 |_ _95_.0j 73.5 RESULTANT VECTOR MAGNITUDES RCR 81.6 67.7 61.0 55.5 64.7 TR 298.8 307.8 314.7 335.6 355.8 LCR 155.9 152.9 151.1 147.1 144.4 FRONT. PLANE RESULTANT VECTOR ORIENTATIONS f - W N f t ' I — r - A 1 ' I A RCA "9TJT3 90.4 90.4 90.6 90.9 FTA 90.0 L C A * 90.0 I 1 I I *LCFA Figure 17B. INTERVAL 2 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW2)/VARIABLE LTA. A l l var iab les spec i f ied as per Figures 15C and 16C. MUSCLE RESULTANT PARAMETERS MLR= 11.7 MAR= 54.1 MVR= 676.8 ANG-L= 85.4 A N G F = 89.0 480 N RUN l 2 3 4 5 Tooth Posit" 47 LAT. PLANE VECTOR RESULTANT L J RCA 103.0 93.8 80.9 63.7 43.6 LTA RESULTANT VECTOR FRONT. PLANE RESULTANT i L T A i LCA RCR TR LCR RCA j FTA | LCA* j j 75.0 j 95.3 136.5 388.9 169.1 87.8 | 89.0 | | 90.01 | 80.0 | 92.6 119.3 400.6 163.5 87.7 1 I I 1 1 85.0 1 89.6 105.5 416.3 158.1 87.6 1 I I 1 i I I i j 90.0 ! 86.0 98.0 436.8 152.6 87.4 1 i l I 1 I I 1 LLS .^J 81.8 101.1 463.1 147.3 87.0 1 1 1 1 JF/TF .79 .71 .63 .57 .54 * LCFA Figure 18B. INTERVAL 3 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW3)/LTA. A l l var iables spec i f ied as per Figures 15C, 16C and 17C. MUSCLE RESULTANT PARAMETERS MLR= 15.0 MAR= 40.2 MVR= 451.3 ANG-L= 84.9 ANGF = 88.1 r- 480 N LAT. PLANE RESULTANT ._ALTA RUN , Posit n , RCA W1 \1 I LTA ; LCA RCR 1 47 98.3 75.0 96.8 96.7 2 | | 89.2 | 80.0 | 93.9 86.5 3 77.3 1 85.0 1 90.7 78.6 4 | j 62.1 ! 90.0 j 86.8 74.7 5 1 1 44.8 82.2 76.8 RESULTANT VECTOR MAGNITUDES TR 258.0 265.7 276.2 289.8 307.2 LCR 107.6 103.7 99.8 96.0 92.4 FRONT. PLANE RESULTANT VECTOR _ ORIENTATIONS RCA 86.2 86.1 86.0 85.8 85.4 FTA 88.0 L C A * ! 90.0 I I I I * LCFA co - 183 -to 160 N and the working side from 60 to 105 N. These are less than those of both the un i la tera l molar about (75 N at the working side and 260 N at the balancing side) and canine clenches (about 170 N both s ides ) . The magnitude of the working and balancing side j o i n t forces, however, exh ib i t rather d i f fe rent re la t ionships through the three phases. Although the changes observed in LCR magnitude of the three chewing in te rva ls coincides with the var ia t ion in TR and MVR, those of RCR do not. RCR forces are leas t in Interval 1, intermediate in Interval 3 and maximal in Interval 2. Comparison of RUN 3 of Figures 16B, 17B and 18B (with LTA and ANG-L matched at the appropriate or ientat ions for each in terval ) shows RCR (61.0 N) to be 60% less than LCR (151.1 N) in Interval 1, 34% less in Interval 2 (105.5 versus 158.1 N) and only 21% less in Interval 3 (78.6 versus 99.8 N). There is thus an overal l increase in the proportion of j o i n t resistance force a t t r ibu tab le to the r ight or working s ide . This i s due to the increase in the la te ra l component of muscle force (MLR) from Intervals 1 (-0.4 N) to 3 (15.0 N). Comparison of these resul ts with RUN 3, Figure 15C (matched data) of un i la tera l molar clenching shows that where MLR was opposite in d i rec t ion ( i . e . -15.1 N) RCR was very much less (71%) than LCR. This has a s imi lar e f fec t on the range of RCA or ientat ions seen in the la te ra l plane since these angles vary considerably less in the three chewing in terva ls (approximately 80, 60 and 55 degrees in Intervals 1, 2 and 3 respect ively) than for the unimolar clench (100 degrees). LCA or ien ta t ions , however, are comparable. I t i s also noteworthy that the three chewing in terva ls of Figures 16B, 17B and 18B exhibi ted JF/TF ra t ios very s imi la r to those seen in the unimolar - 184 -clenching runs of Figure 15C despite the great di f ferences in the indiv idual contr ibut ions of muscle a c t i v i t i e s between them. c . Frontal Tooth Angle (FTA) Change - Figure 16C, 17C, and 18C Var ia t ion of the FTA or ientat ions in the three in terva ls of the chewing power stroke from 80.0 to 100.0 degrees produce the same var ia t ion in TR magnitudes seen previously for FTA changes. FTA or ientat ions greater or less than ver t i ca l have an addit ional mediolateral component contr ibut ing to the resul tant of tooth force in the frontal plane (not observed in the la te ra l p lane). Hence, RUN 3 (muscle matched) of each interval seen in Figures 16C, 17C and 18C, which i s c losest to ver t i ca l in each case has the lowest TR magnitudes. Those of RUN 1 and 5 in each f igure are re l a t i ve l y greater and are equivalent to one another since they occur at equivalent divergent angles from ve r t i ca l ( i . e . 10 degrees). This added mediolateral component of tooth resistance force also produces the change in the frontal plane RCA or ientat ions seen in Figure 16C, 17C and 18C. This i s because an addit ional mediolateral component of j o i n t resistance force at the r ight condyle i s also necessary to balance the system where FTA var ies as was explained previously . As the power stroke progresses from Interval 1 to Interval 3 there i s an increase in the leftward component of MLR which resul ts in a decrease in the range of RCA in the frontal plane (approximately 88, 74 and 63 degrees for Intervals 1 to 3 respec t i ve ly ) . This i s also apparent when the muscle-matched RUN 3 of each phase are considered. The frontal plane RCA is 90.4, 87.6 and 86.0 degrees in Figure 16C, 17C and 18C respect ively where MLR increases from -0.4 N in Interval 1, Figure 16C. INTERVAL 1 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW1)/VARIABLE FTA. A l l var iables spec i f ied as per Figure 150. MUSCLE RESULTANT PARAMETERS MLR = -0.4 MAR = 67.6 MVR= 522.3 A N G - L = 82.6 A N G - F = 90.0 480 N -I RUN 1 2 3 4 5 Tooth Posit n 47 LAT. PLANE RESULTANT VECTOR ORIENTATIONS I I RCA 108.5 98.3 84.3 66.1 46.0 LTA 8170" I I LTA LCA RESULTANT VECTOR MAGNITUDES FRONT. PLANE RESULTANT VECTOR ORIENTATIONS LCA RCR TR LCR RCA FTA L C A * ! 66.4 101.7 319.5 140.9 124.5 80.0 j j 90.0j 74.1 76.7 315.9 144.8 111.4 | 85.0 | 1 | 81.1 61.0 314.7 151.1 90.4 1 90.0 1 87.5 61.7 315.9 159.4 62.1 j 95.0 j | | 93.3 78.5 319.5 169.7 46.0 LlOO^Oj 1 1 * LCFA Figure 1 7 C . INTERVAL 2 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW2)/VARIABLE FTA. A l l var iables spec i f ied as per Figures 15D and 16D. MUSCLE RESULTANT PARAMETERS M L R = 11.7 M A R = 54.1 M V R = 676.8 A N G - L = 85.4 A N G - F = 89.0 480 N RUN 1 2 3 4 5 Tooth Posit n 47 LAT. PLANE RESULTANT VECTOR ORIENTATIONS RESULTANT VECTOR MAGNITUDES I I RCA 100.2 90.4 80.9 64.1 49.0 LTA 85.0 I I LCA RCR 73.1 143.9 82.8 117.3 89.6 105.5 98.2 109.1 104.1 130.3 TR 422.6 417.8 416.3 417.8 422.6 LCR 140.1 148.6 158.1 175.7 193.2 FRONT. PLANE RESULTANT VECTOR T ORIENTATIONS RCA 115.6 102.1 87.6 -61.5 41.4 FTA 80.0 85.0 89.0 95.0 i J C H L O j LCA * 90.0 I I LCFA JF/TF .67 .64 .63 .68 .76 Figure 18C. INTERVAL 3 OF UNILATERAL (RIGHT SIDE) CHEWING POWER STROKE (CHEW3)/FTA. A l l var iables spec i f ied as per Figures 150, 16D and 17D. M U S C L E R E S U L T A N T P A R A M E T E R S M L R = 15.0 M A R = 40.2 M V R - 451.3 A N G - L - 84.9 A N G - F = 88.1 V.32 r- 480 N -i LCA RUN 1 Tooth | LAT. PLANE RESULTANT VECTOR ORIENTATIONS RESULTANT VECTOR MAGNITUDES , Posit n , RCA 1 LTA j LCA RCR TR LCR 1 93.8 85.0 75.5 96.9 280.2 88.5 2 1 1 84.0 | | 85.5 82.8 277.1 94.8 3 1 1 77.3 1 1 90.7 78.6 276.2 99.8 4 I 1 59.7 j j 100.8 84.0 277.1 114.2 5 1 1 46.6 1 j 106.6 99.1 280.2 126.3 FRONT. PLANE RESULTANT VECTOR | ORIENTATIONS RCA 110.3 96.3 86.0 58.7 46.6 FTA 80.0 85.0 88.0 95.0 L 1 0 _ ° i 9 J L C A * 90.0 I I * LCFA CO - 188 -to 11.7 N in Interval 2 , and 15.0 N in Interval 3. Thus as the mandible nears intercuspat ion the r ight j o i n t resistance force becomes more medially d i rected in the frontal plane. In the la te ra l plane the fami l ia r reciprocal re la t ionsh ip between the RCA and LCA or ientat ions of the j o i n t resistance forces occurs for FTA var ia t ion during chewing as w e l l . The r ight or working side condylar forces are less than those of the l e f t balancing side (except at very acute FTA or ien ta t ions ; eg. RUN 1 of Interval 2 and 3 where the reverse is t rue) . Comparison of RUN 3 of Figures 16C, 17C and 18C with muscle-matched data i s the same as Figures 16B and 17B with respect to the s ta t i c clenching tasks previously discussed. Based on the data of RUN 3 with muscle and tooth force or ientat ions matched for each chewing interval th is modeling analysis would predict the load-bearing surfaces of the two j o i n t condyles to be capable of res is t ing up to 160 N of compressive force aligned at 75 to 90 degrees with respect to the occlusal plane. - 189 -C. SUMMARY 1. B i l a t e r a l l y Symmetrical Clenching Tasks Clenching a c t i v i t i e s involv ing the molar teeth generate more overal l muscle force than those at the inc isors due to r e l a t i ve l y greater contr ibut ion by each muscle group to molar tasks. The magnitudes of these forces are 1000 to 1200 N for molar clenching but only around 450 N during inc i sa l tasks. For each of the four tasks modeled the muscle force was directed anterosuperior ly. However, greater magnitudes of the muscle forces at more poster ior tooth contacts involve r e l a t i ve l y smaller anter ior components and thus are more v e r t i c a l l y directed than tasks involving less tota l muscle e f fo r t at the inc isors (eg. ANG-L i s 88.0 degrees for CO, 63.9 degrees for INCISN). Changes in tooth posi t ion for the same muscle force resu l t in greater tooth resistance forces at more poster ior contacts and less corresponding j o i n t res is tance. Poster ior contacts are therefore more e f f i c i e n t in terms of resistance force d is t r ibu t ion since a greater proportion i s taken up by the teeth. Comparison of molar versus inc i sa l clenching also shows th is e f fec t even though they involve d i f fe rent muscle forces. Incisal clenching i s much less e f f i c i e n t than more poster ior ( i . e . molar) clenching and involves a considerably smaller magnitude of tooth resistance force. Molar clenching in these analyses produced tooth resistance forces of about 500 to 600 N overal l but a maximum of nearly 900 N could po ten t ia l l y have been genetrated during intercuspal clenching at the th i rd molar posi t ion (RUN 4; Figure 10B). Incisal clenching on the other hand generated magnitudes of only around 130 N for natural contact and approximately 140 N for more - 190 -s tab i l i zed contact with the stops due to s l i g h t l y greater muscle forces generated during the l a t t e r task. The re la t ionship between the or ientat ion of tooth force and i t s magnitude i s , to a large extent, determined by the geometry of the mandible and the spat ia l juxtaposi t ion of tooth contact and fulcrum pos i t i on . In molar clenching th is re la t ion i s s t ra ight forward as greater LTA values shorten the e f fec t ive moment arm requir ing greater tooth resistance forces to balance the moments acting around the j o i n t fulcrum point . Incisal clenching however has a s l i g h t l y d i f fe rent geometry in th is analysis than molar clenching and consequently a d i f ferent re la t ionship between tooth or ientat ion (LTA) and magnitude e x i s t s . The runs with the intermediate LTA or ientat ions (eg. RUN 3 of both Figures 12B and 13B) had s l i gh t l y longer moment arms and re l a t i ve l y smaller tooth force magnitudes than more extreme LTA or ien ta t ions . This in turn has a reciprocal e f fect on the magnitudes of j o i n t resistance fo rce . Changes in the LTA or ientat ion also a l te r the d is t r i bu t ion of resistance fo rces . More poster ior ly oriented LTA requires greater tooth forces and subsequently less j o i n t force to maintain s ta t i c equ i l ib r ium. In addit ion the change in LTA also causes a reciprocal reor ientat ion of the j o i n t resistance forces. As LTA i s changed from an anter ior to a poster ior or ienta t ion the j o i n t force becomes more anter io r ly oriented for a l l tasks. Molar tasks with more poster ior tooth posi t ions and more v e r t i c a l l y directed muscle force have a s im i l a r l y more ve r t i ca l or ientat ion of j o i n t force than i nc i sa l tasks. - 191 -The most appropriate comparison of the e f f i c iency of resistance force d i s t r i bu t i on (JF/TF ra t io) of these tasks is where the or ientat ion of tooth resistance was matched with that of the applied muscle force (ANG-L). In these instances the nature of the tooth resistance most c lose ly corresponds with that of the applied muscle force. The two molar clenching tasks (CO and BIMOL) were much more e f f i c i e n t with JF/TF ra t ios of approximately 1.00 and 0.65 for f i r s t and second molar contacts respect ive ly . S imi lar runs for i n c i s a l clenching had s i gn i f i can t l y less e f f i c i e n t ra t ios of about 2.30 and 2.50 for s tab i l i zed (INCISS) and natural (INCISN) contacts respect ive ly . The biomechanics of more poster ior clenching is thus about two to four times more e f fec t i ve in terms of the resistance forces generated with proport ional ly more occlusal and less j o in t forces produced. In a l l instances, based on the model predict ions of j o in t force, the bearing surfaces of the jo in ts would be expected to be located at the anterosuperior aspect of the condylar heads and vary between 60.0 and 100.0 degrees re la t i ve to the occlusal plane. These surfaces would be expected to be capable of withstanding forces of up to about 300 N per side due to molar c lenching. Jo in t forces due to i nc i sa l clenching are on the order of 160 N. 2. Un i la tera l Clenching Tasks The ver t i ca l component of muscle force (MVR) for un i la tera l canine clenching of about 550 N i s r e l a t i ve l y greater than the previously described i n c i s a l clenching (around 400 N) but s i gn i f i can t l y less than e i ther the un i la te ra l molar clenching of 840 N, or the b i l a t e r a l l y symmetrical clenches (1000 to 1200 N). This i s consistent with the re la t i ve l y intermediate - 192 -p o s i t i o n of the canine between the i n c i s a l and molar p o s i t i o n s an te ropos te r io r ^ in the la te ra l plane. Likewise the anteroposterior component of muscle force (MAR) of 100 N i s also greater than any of the molar tasks but less than the inc isa l tasks. This produces an or ientat ion of muscle force directed less anter io r ly (at ANG-L of 79.8 degrees) than the i nc i sa l functions (64 to 70 degrees) but more anter ior ly than the un i la tera l molar clench and s i gn i f i can t l y more so than the two b i l a te ra l molar clenches. As such i t seems apparent that as the tooth posi t ion moves more poster io r ly the muscle forces become greater in overal l magnitude as well as more v e r t i c a l l y or iented, in the la te ra l plane at l eas t . In the un i la tera l tasks changing the posi t ion of the tooth contacts more l a t e r a l l y from the midl ine involves an addit ional rightward component of muscle force (MLR) of 15.8 N at the canine and 15.1 N at the molar which are very s im i la r despite the s ign i f i can t di f ferences between the two posi t ions medio la tera l ly . As such both the un i la tera l canine and molar tasks have very s im i la r or ientat ions of the overal l muscle force of about 91 degrees (ANG-L) in the frontal plane although the i r overal l total magnitudes of muscle force d i f f e r great ly . As was the case of the b i l a te ra l molar and intercuspal clenching a more poster ior tooth posi t ion for the un i la tera l molar task resu l ts in greater tooth resistance forces as well as less corresponding j o i n t res is tance. However, the balancing j o i n t accounts for only one hal f to one quarter (80 to 120 N) the resistance force of the balancing side which amounts to around 250 N. This i s very s imi la r to those forces occurring at both jo in ts of the two b i l a t e ra l molar tasks. This reduction in j o i n t loading on the working side - 193 -of un i la tera l molar tasks corresponds with a greater tooth resistance force of about 450 to 650 N. Although th is i s s l i g h t l y less than that of the b i l a t e ra l molar clenches the s i gn i f i can t l y reduced overal l j o i n t loads of unimolar clenching provide for much more favorable JF/TF ra t ios (eg. 0.57 at the second molar), i . e . less total residual j o i n t force i s produced for r e l a t i v e l y more force per tooth during un i la tera l than b i l a te ra l molar c lenching. Var iat ions in the la te ra l plane or ientat ion of tooth force (LTA) for un i la te ra l tasks also produce s imi la r ef fects on the j o i n t forces as seen for b i l a t e ra l tasks. More poster ior ly oriented (eg. more ve r t i ca l ) LTA requires greater tooth forces (480 to 580 N) and subsequently less j o i n t force at both j o i n t s to maintain s ta t i c equi l ibr ium at molar and canine contacts. However, the canine clench d i f fe rs from the molar clench in th is regard in that the var ia t ion in magnitude of tooth force for the same range of angular change (20 degrees) i s only about 15 N. That of the molar i s about 100 N. This i s due to the more anter ior posi t ion of the canine which has less applied muscle force and hence smaller tooth forces of around 225 N. In addit ion the same increment of LTA var ia t ion at the canine produces a r e l a t i ve l y small change in the moment arm length. This e f fect i s great ly magnified c loser to the fulcrum, i . e . molar pos i t ions , which also have addit ional applied muscle force. In general the j o i n t forces themselves also become more anter ior ly oriented with more poster ior LTA or ien ta t ions . However, th is e f fect i s also great ly magnified at the r ight balancing j o i n t of unimolar clenching where the most divergent j o i n t resistance forces ar ise due to added mediolateral - 194 -inf luences of a more la te ra l tooth ( th i rd molar) contact. These resistance forces at th is j o i n t can potent ia l l y become tens i le rather than compressive at the most poster ior contact. Nevertheless, the working side j o i n t loads of the unimolar task remain s i gn i f i can t l y less than those at the balancing j o i n t despite any changes in tooth force o r ien ta t ion . Canine clenching exhib i ts j o i n t forces of only around 170 N which are quite s imi lar on the two sides but have the working j o i n t s l i g h t l y more heavi ly loaded than the balancing s ide , for the most part . The ef fect of varying the tooth resistance or ientat ion in the frontal plane (FTA) produces a reversal of the re la t ion between balancing and working j o i n t force or ientat ions in the la te ra l plane. These j o in t or ientat ions (LCA and RCA) change in a reciprocal manner to one another when the tooth force var ies in mediolateral nature (with constant l e f t j o i n t frontal angle, LCFA). In the frontal plane the working side j o i n t forces also vary mediolateral l y opposite to the change in FTA when th is dimension of the tooth force changes, although the re lat ionships between the magnitudes of loading at the two j o i n t s remains the same as above ( i . e . WS < BS). Nevertheless, the substant ial changes in the re lat ionships of the two j o i n t force or ientat ions for var ia t ions of mediolateral tooth force re f l ec t the d i f ferent mechanics of un i la te ra l compared to b i l a te ra l contacts. Again, the di f ference in extent of th is e f fect on both the tooth forces and the j o i n t forces of un i la tera l molar versus canine clenching i s related to the combination of the magnitude of appl ied muscle force and proximity of tooth contact to the fulcrum as out l ined for LTA changes. - 195 -A l tera t ions to the l e f t or balancing j o in t angle of resistance in the f rontal plane (LCFA) has a reciprocal e f fect on the r ight or working jo in t resistance or ien ta t ion . A l l other var iables being equal more l a t e r a l l y or iented j o i n t force on one side would require a more l a t e r a l l y oriented force on the other to maintain equ i l ib r ium. Only minor changes in j o in t force magnitude occur due to th is type of va r ia t i on . Comparison of the JF/TF ra t ios of a l l clenching data with the tooth force or ientat ions matched with those of the applied muscle force shows un i la te ra l molar clenching to have the most favorable re la t ion of j o i n t versus tooth resistance forces of a l l the clenching tasks analyzed. The unimolar clenching JF/TF ra t io was 0.64 compared to 1.50 of the un i la tera l canine; 2.27 of s tab i l i zed inc i sa l clenching ( i . e . with stop); 2.47 of natural i nc i sa l c lenching; 0.68 at the second molar and 1.01 at the f i r s t molar of b i l a te ra l molar c lenching; and 0.96 at the f i r s t molar of intercuspal c lenching. In other words, th is hypothetical indiv idual i s capable of generating more functional tooth force with re la t i ve l y less residual j o i n t loading when clenching un i l a te ra l l y at a poster ior tooth contact point than e i ther b i l a t e r a l l y and/or at more anter ior occlusal contact pos i t ions . According to the modeling analysis of un i la tera l canine and molar clenching the bearing surfaces of the two jo in ts would be expected to be located anterosuperior ly and capable of res is t ing up to about 300 N of force or iented at 60 to 100 degrees re la t i ve to the occlusal plane ( i . e . 60 to 85 degrees at the- balancing j o i n t and 75 to 100 degrees at the working jo in t ) - 196 -for unilateral canine clenching. For unimolar clenching these expected orientations would fall within the same range ( i .e. 65 to 95 degrees) for the balancing side joint. However, the right or working side joint exhibits, on the one hand, resistance force orientations of about 20 to 120 degrees in this respect which is well outside this range. On the other hand, the magnitudes of these forces are for the most part substantially less than those occurring at the balancing side. Thus the joint morphology would be expected to be capable of resisting more divergent and even tensile forces although at magnitudes less than one half those of maximal joint forces ( i .e . less than 150 N). 3. Unilateral Chewing Tasks Application of this model to the three static intervals near intercuspation of unilateral molar chewing show that the biomechanics involved are very similar to those which would be predicted on the basis of the static molar clenching tasks despite very different contributions from the individual muscle groups. The magnitude of the overall muscle forces generated are less than the unilateral molar clench but greater than unilateral canine clenching. The vertical component varied from 450 N to about 680 N which occurred just prior to intercuspation. The mediolateral component of muscle force was increasingly directed more medially from virtually zero at the f irst interval to 15 N at the third interval. This corresponds with expected directions of applied effort as intercuspation is reached. The same component of force (15 N) was shown to exist during the unimolar clench but directed laterally. Nevertheless, the orientations of - 197 -the muscle forces predicted for the unimolar chewing phases, although based on d i f fe rent data, are also s t r i k i ng l y s im i la r to those of the s ta t i c unimolar c lench. The la tera l plane or ientat ion was approximately 85 degrees and about 90 degrees in the frontal plane (84 and 91 degrees respect ively for the s ta t i c unimolar c lench) . The range of tooth resistance forces of 260 to 460 N i s s i gn i f i can t l y less than that of unimolar (450 to 650 N) but greater the un i la tera l canine clenching (around 225 N). Var iat ion of both the la te ra l (LTA) and frontal plane (FTA) or ientat ions of tooth resistance force on the three chewing in te rva ls produce the same ef fects on tooth force magnitudes seen previously for the un i la tera l s ta t i c clenches. S im i l a r l y , the j o i n t forces are greater on the balancing than the working side in a l l three chewing i n te rva l s . However the greater la te ra l components of muscle force for chewing function reduce the range of RCA or ientat ions for var ia t ions of both LTA or FTA compared with those observed for the un i la tera l molar c lench. The magnitude of j o i n t forces of the chewing in terva ls vary from approximately 60 to 105 N at the working side and from 100 to 195 N at the balancing s ide . The JF/TF ra t ios of 0.63 to 0.67 are very favorable suggesting un i la te ra l chewing i s as e f f i c i e n t as unimolar clenching with respect to the d i s t r i bu t i on of resistance forces. However the mechanics of chewing also indicate less var ia t ion of j o i n t force or ientat ions and magnitudes with a l tered tooth force alignment. The predict ions of load-bearing morphology of the j o i n t condyles for these chewing in terva ls are well within the ranges of the previous s ta t i c - 198 -clenching tasks; s p e c i f i c a l l y 160 N of force aligned at 75 to 90 degrees to the occlusal plane. - 199 -DISCUSSION The r e l i a b i l i t y of the predict ions derived from any model of a biomechanical system are d i r ec t l y related to the number of pert inent var iab les incorporated. Numerous workers have establ ished the importance of the in te r re la t ionsh ips of the anatomical va r iab les , s p e c i f i c a l l y the posi t ion of the j o i n t s and points of tooth contact with respect to presumed muscle force production (Finn et al_., 1980; Bramble, 1978; Smith, 1978; DuBrul, 1980;). However very few studies have incorporated a l l of these var iables simultaneously and in three dimensions (Weijs and Dantuma, 1980) and none of them for humans. This model i s unique in i t s a b i l i t y to incorporate a wide range of the relevant anatomical and physiological ( i . e . muscle) parameters which define a mandibular biomechanical system. This f l e x i b i l i t y allows comparisons of the mechanics produced by changes to the physio logical var iab les for the same, or s i m i l a r , anatomical var iables as in the case of d i f fe ren t tasks of th is study. However, th is f l e x i b i l i t y w i l l also permit comparisons of the ef fects of changes to the anatomical var iables for s imi la r types of functional tasks. The necessity of including real values for the physiological parameters which determine the re la t i ve forces generated by the indiv idual muscles, and therefore the overal l to ta l muscle force resul tant vector , provides meaningful ins ights as to how the mandible functions in real terms. This i s s im i l a r to the studies of Weijs and Dantuma (1980) and Pruim and his coworkers (1980). In these analyses the determinations of tooth and j o i n t f o r ces was not cons t ra ined by l i m i t a t i o n s imposed on muscle fo rce - 200 -c a p a b i l i t i e s because the l a t t e r were measured d i r e c t l y . Other models, some very recent work, have incorporated a r t i f i c i a l contr ibut ions of muscle force by hypothesizing some sort of minimization of muscle and/or j o i n t force to e f fec t occlusal loading (Smith et. al_., 1986; Osborn and Baragar, 1986; Hatcher et al_., 1986; Barbenel, 1969, 1972, 1974, 1983). Although these models represent those few analyses where most of the pert inent muscle groups as well as three dimensional considerat ions were incorporated the information derived provides only genera l i t ies about jaw biomechanics. The model of the present analys is assumes only that the spec i f i c values assigned to the muscles i s correct for each task from which predict ions of j o i n t force and tooth force, and the i r o r ien ta t ions , are determined. The "minimization" theory of these other models requires many more assumptions since the algorithm i t s e l f determines the muscle data. A. MUSCLE FORCES There is l i t t l e data in the l i t e ra tu re from which to compare the overal l muscle force resul tant magnitudes of th is study as very few invest igators have attempted to determine th is force. Schumacher (1961) determined the theoret ical maximum clenching force to be 1528 N (156 kg) according to his cross sectional measurements of cadaver specimens. Although th is value is of the same order of magnitude of th is study Schumacher assumed a total average force potential of 10 kg/cm 2 compared to only 4.1 kg/cm 2 of th is model, correct ion of Schumacher's value according to th is c r i t e r i on would be only about 520 N which i s well below the maximum force observed here. In addit ion however, Weijs and Hi 11 en (1984a) have pointed out that the specimens of - 201 -Schumacher were e lder ly and with l i t t l e remaining natural den t i t i on . As such the average force produced by such a group would be expected to be less than that of the younger dentate subjects from whom the data for th is study were der ived. Van Steenberghe and DeVries determined the maximum muscle force to be 2352 N (240 kg) from the data of Carlsoo (1952). Carlsoo used 11 kg/cm 2) as the force constant for muscle and a s imi lar correct ion reduces th i s to approximately 870 N of force. This i s also in agreement with the model f ind ings . Pruim et al_., (1980) derived the values of muscle force potential per unit area from a number of subjects and found that although the value varied between about 88 and 175 N/cm2 (9.2 and 17.7 kg/cm 2) for d i f fe rent subjects they remained re l a t i ve l y constant for each subject. This consistency has been reported by others as well although the mean values seem to be c loser to about 30 to 50 N/cm2 according to Weijs and H i l len (1984a). As such the value of 40 N/cm2 was used in th is study. A summary of the overal l muscle force resul tant parameters generated during each of the nine functional tasks is presented in Table VI. As can be seen, the tasks with the more poster ior posi t ion of tooth contact have, in genera l , r e l a t i v e l y greater magnitudes of the ver t i ca l component of muscle force (MVR) with correspondingly smaller anter ior components (MVR). Subsequently, the overal l or ientat ion of the applied muscle force becomes more ver t i ca l at tooth posi t ions located more poster io r ly (ANG.L). This re la t ionsh ip holds true despite the fact that each of the groups of tasks has s i g n i f i c a n t l y d i f fe ren t forces from each of the indiv idual muscles contr ibut ing to the overal l force applied to the mandible (see Tables II and TABLE VI - SUMMARY OF MUSCLE RESULTANT PARAMETERS. The values are those from the respective Figure 10 to 18 of RESULTS. BILATERAL CLENCHING UNILATERAL CLENCHING CHEWING Muscle Force Variable Intercuspal (CO) B i la te ra l Mol ar (BIMOL) Inci sal Stop (INCISS) Incisal Natural (INCISN) Uni lateral Mol ar (INIMOL) Uni lateral Canine (UNIK9) Interval (CHEW 1) 1 Interval 2 (CHEW 2) Interval (CHEW 3) MLR 0.0 0.0 0.0 0.0 -15.1 -15.8 -0.4 11.7 15.0 MAR 40.9 57.0 160.8 194.2 87.9 99.7 67.6 54.1 40.2 MVR 1187.6 1039.1 432.3 395.9 839.9 554.0 522.3 676.8 451.3 ANG.L 88.0 86.9 69.6 63.9 84.0 79.8 82.6 85.4 84.9 ANG.F 90.0 90.0 90.0 90.0 91.0 91.6 90.0 89.0 88.1 - 203 -III of METHODS). The la te ra l plane or ientat ions of these muscle forces range from about 64 degrees for natural i nc i sa l b i t ing to 88 degrees for in tercuspat ion. Tasks u t i l i z i n g molar contact posi t ions exh ib i t these or ientat ions at approximately 80 to 90 degrees depending on the task and tooth contact pos i t i on . These f indings are in very good agreement with those of Prium and coworkers (1980). They also found that the overal l muscle force resul tant of the i r two dimensional modeling analysis exhibi ted a more an ter io r ly directed or ientat ion at more anter ior b i te pos i t ions . As th is posi t ion of tooth contact was changed from the f i r s t premolar to the f i r s t molar and the second molar the muscle vector or ientat ion (with respect to the occlusal plane) increased from 79 degrees to about 83 degrees in the la te ra l plane (Figure 5, Prium £t_al_. 1980). Mol ler (1974) has suggested that the number of occlusal contacts i s a s ign i f i can t determinant of jaw muscle ac t i v i t y l e v e l s . MacDonald and Hannam (1984) concluded from the i r studies on various types of occlusal contact patterns and posi t ions that a greater number of contacts and thus contact surface area resulted in a generalized increase in muscle a c t i v i t y . This was espec ia l l y true at anter ior pos i t ions . More pos te r io r l y , however, the muscle a c t i v i t y was not affected by ei ther the number or area of contact to the same extent. Since a s ign i f i can t portion of the muscle a c t i v i t y leve ls used in th is study were derived from the work of MacDonald and Hannam (see Table II of METHODS) the di f ferences in the muscle resul tant parameters for s imi lar tooth posi t ions may be at least par t ly due to the d i f fe rent nature of the occlusal contact . For instance, the greatest magnitude of the total muscle force - 204 -vector of intercuspal clenching (CO) is 1188 N, which would have the greatest number of occlusal contacts of a l l tasks modelled. B i l a te ra l molar clenching (BIMOL) however produced overal l muscle forces of 1040 N. This l a t t e r task was modelled with data derived for clenching on occlusal stops at only two contact points (one on each side of the den t i t i on ) . Likewise the inc i sa l clench with the occlusal stop (INCISS) had a total muscle resul tant of 461 N whereas that of i nc i sa l clenching on natural contacts (INCISN) was 440 N. It would seem therefore that greater s t a b i l i t y of occlusal contact in the system i s conducive to greater overal l muscle forces. This i s ce r ta in ly true with respect to the ver t i ca l component of muscle fo rce. The greater anter ior muscle components (MAR) of the less stable occlusal condit ions for the same task may re f l ec t a combination of a c t i v i t i e s by the indiv idual muscle groups to improve the s t a b i l i t y of these types of maximal clenching a c t i v i t i e s . In th is regard, a number of workers have suggested that d i f fe rent jaw muscles may provide d i f fe ren t functions with respect to the i r e f fects on the resistance forces of the mandible with some, l i k e the la te ra l pterygoid, relegated to only a s t a b i l i z i n g inf luence (Hatcher et al_., 1986; Osborn and Barager, 1985; Pruim et _al_., 1980). According to Osborn and Baragar (1985) the major muscle groups of the jaw such as super f i c ia l masseter, medial pterygoid and some of the temporalis act as "power" muscles which are arranged such that they can maximize the generation of b i te fo rce. However, t he i r act ion creates forces at the j o i n t which would tend to displace the condyle from the a r t i cu la r eminence. To maintain s t a b i l i t y in the system the secondary "cont ro l " muscles of the la te ra l pterygoid and oblique portions of the temporalis (eg. poster ior temporalis) come in to p lay. These workers - 205 -have reasoned that these control muscles are arranged such that they have very poor moment arms for b i te force generation but pr imar i ly function to prevent i n s t a b i l i t y in the system, espec ia l l y at the condyles, in an anteroposter ior d i rec t i on . Thus the more anter ior ly directed muscle e f fo r t of clenches at more anter ior ly posit ioned tooth contacts (see TABLE VI) may be a re f l ec t i on of the need for an addit ional anter ior s t a b i l i z i n g component for these types of contact. The a c t i v i t y leve ls of the la te ra l pterygoid (IP) in Table II of METHODS d i r ec t l y re f l ec t th is trend. The more poster ior and stable intercuspal posi t ion had the lowest level of IP a c t i v i t y whereas i n c i s a l clenching had the highest. The fact that the j o in t forces generated at more anter ior tooth contacts were of substant ia l ly smaller magnitudes than more poster ior contacts would imply that th is extra anter ior e f for t produced by the muscles i s more involved with maintaining occlusal than condylar s t a b i l i t y . The addit ional mediolateral components of muscle force of Table VI i n -volved in the un i la tera l tasks, both clenching and chewing, are very small in magnitude compared to the ver t i ca l components. Therefore the frontal plane or ientat ions (ANG-F) remain near v e r t i c a l . However, i t i s un l ike ly that th i s component contr ibutes much to the s t a b i l i t y of the resistance forces of the system. It i s more probable that i t i s merely a residual of the combined e f fo r t of a l l the muscles involved in an asymmetric occlusal funct ion. None of the muscle groups has an alignment conducive to s t a b i l i z i n g mediolateral forces at e i ther the teeth or the j o i n t s . However, i t i s in terest ing to note the increasingly medial component of muscle force of Table VI ocguring close to intercuspat ion (which occurred between Intervals 2 and 3). This would - 206 -be expected as the mandible progresses through the power phase of chewing towards the more medial posi t ion of intercuspat ion with increasing b i te fo rce. Osborn (1982) has suggested that the alignment, or i n c l i n a t i o n , of the molar teeth in the frontal plane is such as to res i s t the t i l t i n g forces act ing on the teeth during the power stroke of mast icat ion. In the frontal plane these resistance forces on the lower molar teeth of the working side would be directed l a t e r a l l y and i n f e r i o r l y opposite to the d i rec t ion of appl ied muscle force observed in th is study. Therefore more poster ior molar teeth which undergo re la t i ve l y heavier occlusal loads would be expected to have the i r long axes aligned correspondingly more medial which is generally observed in human dent i t ions (Osborn, 1982). Osborn and Baragar (1985) have extended th is argument to suggest that the contr ibut ions of force from each of the jaw muscles is coordinated by the periodontal receptors in such a way as to produce a b i te force al igned with the long axis of the roots of the teeth. Therefore, in order to minimize the torques on the dent i t ion the data would predict the long axes of the respective mandibular teeth to general ly correspond with these alignments. Thus, inc isors would be re la t i ve l y more an te r io r l y oriented with more poster ior teeth being c loser to ver t i ca l as well as more medially a l igned. Analysis of the d i rec t measurements of the d i rec t ion of i nc i sa l b i t ing by Hylander (1978) show that these forces l i e between approximately 60 to 80 degrees with respect to the occlusal plane. I f i t i s assumed from the above discussion that the or ientat ion of tooth resistance force coincides with that of the applied muscle force then the or ientat ions of the l a t t e r from th is study agree well with those which would have been predicted from Hylander's data. Osborn (1982), and Baragar and - 207 -Osborn (1986), impl icate th is e f fect as the reason for the establishment of the curve of Spee which is a consistent feature, in the sag i t ta l plane, of human dent i t ions . The addit ional medial component of muscle force generated during the power stroke of chewing would produce a s imi la r torque on the teeth , espec ia l l y at more poster ior posi t ions where re l a t i ve l y more occlusal force can be generated. This would be great ly minimized by medial al ignments, espec ia l l y pos te r io r l y , of the teeth themselves. Such an adaptation would possibly explain the existance of the curve of Wilson in the f rontal plane, and as Osborn (1982) has suggested, when a l l three dimensions are considered, the curve of Monson. It would seem, therefore, that the or ienta t ions of muscle force generated by the mandibular system observed in th i s study may re f l ec t a re la t i ve l y consistent phenomenon. I t i s l i k e l y that for at least the natural i nc i sa l clenches the a c t i v i t y leve ls of MacDonald and Hannam (1984) may have incorporated some s l i g h t anter ior reposi t ioning of the jaw from the intercuspal posi t ion in order to e f fec t the proper i nc i sa l contact. Although th is may not have had a s i gn i f i can t a f fect on the indiv idual muscle a c t i v i t i e s per se, as concluded by these authors, the i r combined ef fects may have produced some possible var ia t ion in overal l muscle force due to a sh i f t in the working l ines of the muscles (Weijs, 1980). Changes in e i ther the re la t i ve posi t ion of muscle attachment points by as l i t t l e as 6.5 mm and the i r or ientat ions by 5 degrees have been shown by Hatcher and coworkers (1986) to produce s ign i f i can t var ia t ion in both occlusal and j o i n t loads (up to 20 percent in the i r study) derived by s im i la r mathematical ana lys i s . The assumptions used in th is study, as well as by most other invest igators of jaw biomechanics (Weijs and - 208 -Dantuma, 1981), that an indiv idual muscle can be considered as a s ingle s t ra igh t l i ne element from the centers of i t s attachment areas i s a necessary one. However the type of var ia t ion observed by Hatcher ^ t al_. point out the problems associated with th is type of s imp l i f i ca t i on (Throckmorton et a l . , 1980). Large areas of attachment with var ia t ions in density of muscle f iber inser t ion can af fect the posi t ion of the true centroid of attachment (Weijs, 1980). As such, the extent of f l e x i b i l i t y of th is model in accommodating these types of var ia t ions and al lowing analysis of a wide range of such p o s s i b i l i t i e s becomes an important considerat ion. An addi t ional complication not general ly considered in modelling analyses i s the internal archi tecture of the muscles themselves. Pennate muscles with d i ss im i l a r alignments of groups of f ibers within the same muscle can red is t r ibu te the applied force over the area of muscle attachments. Weijs and H i l l en (1984a) determined that a l l four of the major jaw muscle groups exh ib i t some degree of pennation, espec ia l l y the medial pterygoid and temporalis muscles. These workers have also pointed out that pennation can also contr ibute to underestimations of muscle cross sectional areas. Heterogeneity in the f i r i n g behavior of groups of f ibers within a muscle fur ther complicates th is problem (Herring and Grimm, 1979) although to a lesser extent for maximal clenching a c t i v i t i e s than chewing behaviour (Weijs, 1980; Pruim et al_., 1980; Hylander, 1979c). Although such problems can be overcome by deta i led considerat ion of jaw muscle archi tecture (Osborn and Baragar, 1985; Baron and Debussy, 1979) the r e a l i t y of modelling human jaw biomechanics and the problems associated with recording muscle a c t i v i t y - 209 -l eve ls and determining precise attachment points and cross sectional areas l i m i t s the incorporation of a l l of these var iab les . Nevertheless, i t i s noteworthy that the muscle force resul tant parameters of Table VI exh ib i t general ly consistent trends with respect to muscle force or ientat ions and magnitudes for both the clenching tasks and chewing phases. The importance of th is becomes more obvious when i t is remembered that the sources of data used to determine the indiv idual muscle force vectors for these two d i f fe rent types of function were, themselves, very d i f fe ren t . The ac t i v i t y leve ls of the clenching tasks were near]y a l l derived from studies conducted in our lab (see Table II) whereas those of chewing were from the ea r l i e r works of Moller (1966). B. TOOTH FORCES Table VII summarizes the tooth force predict ions of th is study for those instances where the or ientat ions of the tooth resistance force corresponded with the muscle resul tant force for each respective task. I t i s well establ ished that greater occlusal loads generally occur at more poster ior posi t ions along the dental arch where the morphology of the dent i t ion i s more conducive to res i s t i ng larger loads. When the mandible funct ions as a b i l a t e r a l l y symmetrical un i t , as i t does in the CO, BIMOL, INCISS and INCISN tasks of th is study, simple lever mechanics have been assumed by a great many workers to govern the forces of the mandible (Tradowsky and Dworkin, 1982; Finn et al_., 1980a and b; Throckmorton et al_., 1980; Hylander,. 1978; Dubrul, 1974; Gosen, 1974; Turnbul l , 1970; Crompton and Hiiemae, 1969; S e i t l i n , 1968; Davis, 1964; Mainland and H i l t z , 1933; - 210 -Gys i , 1920). Most of these analyses have predicted that for poster ior points of tooth contact the mandible can generate correspondingly greater tooth forces due to shortening of the re la t i ve length of the moment arm at these pos i t i ons . These predict ions have frequently ignored potential changes in muscle act ivat ion leve ls for d i f fe r ing occlusal contacts or funct ions. Such was the case of the CO task of th is analysis where only the posi t ion of the assumed center of tooth contact was var ied . Consequently the TR magnitude shows a continuous increase from 556.2 N at the second premolar to 866.4 N at the th i rd molar. Osborn and Barager (1985) observed a s imi lar trend using the data of Pruim et jil_. (1980) to derive muscle a c t i v i t i e s and force contr ibut ions although the magnitudes of occlusal load seem somewhat high. They were 635 N (70 kg) at the i n c i s o r , 833N (85 kg) at the f i r s t premolar, 1029 N (105 kg) at the f i r s t molar and 1715 N (175 kg) at the t h i r d . The only d i rec t measurements of tooth loads near centr ic occlusion are those of Lundgren and Laurel 1 (1986) who found the total mean maximal bi te force under th is condit ion to be only 320 N (+_ 117). Their study minimized d isc lus ion of the jaws to a mere 1.5 mm at the inc iso rs by incorporating transducers in f ixed bridge pontics at four posi t ions in one arch (max i l l a ry ) . However, the periodontal support provided to the apparatus in these ind iv idua ls was apparently very much compromised. Therefore i t i s not surpr is ing such low values for th is task were observed. S i m i l a r i l y low forces were found for maximal un i la tera l b i t ing forces at anter ior and poster ior pos i t i ons . Comparison of the b i l a te ra l i nc i sa l (INCISS and INCISN) and molar tasks (CO and BIMOL) of th is study have shown the muscle a c t i v i t y l e v e l s , which - 211 -TABLE VII - SUMMARY OF RESULTANT TOOTH RESISTANCE FORCE PARAMETERS. This data i s from the modeling runs where the la te ra l (LTA) and frontal plane (FTA) or ientat ions were designated to correspond with the respective or ientat ions of applied muscle force for each task. The f igures from which t h i s data i s taken are indicated for reference. Abbreviations are as per prev ious ly . TOOTH RESISTANCE PARAMETERS FORCE Task Tooth Pos'n LTA (degrees) TR (N) FTA (degrees) Source Figure CO 5 88.0 556.2 90.0 Figure 10B, Run 1 6 605.6 •I Run 2 7 723.9 n Run 3 8 866.4 Run 4 BIMOL 6 86.9 518.3 90.0 II 11B, Run 2 7 618.5 518.3 90.0 11C, Run 3 INCISS 1 69.6 141.2 90.0 12B, Run 3 INCISN 1 63.9 126.9 90.0 •I 13B, Run 3 UNIK9 43 79.8 226.8 91.6 II 14B, Run 3 UNIMOL 46/47 84.0 515.5 91.0 II 15B, Run 3 CHEW 1 47 83.0 314.7 90.0 •I 16B, Run 3 2 47 . 85.0 416.3 89.0 II 17B, Run 3 3 47 85.0 276.2 88.0 •I 18B, Run 3 - 212 -produce the i r respective overal l muscle resul tant forces, to be very d i f fe ren t as we l l . Inc isal clenching has lower overal l muscle resul tant forces which were also more anter io r ly or iented. It is therefore apparent that the increase in tooth loads more poster ior ly must be due to the combined e f fec ts of more favorable mechanics as well as more favorable muscle ac t iva t ion leve ls depending upon the funct ion. Consequently the magnitudes of i nc i sa l tooth forces are much less than those of these b i l a te ra l molar tasks . Di rect measurements of i nc i sa l b i te forces by other invest igators in Table VIII show notable consistency in the range of mean magnitudes observed from about 100 to 300 N despite a wide range of co r re la tes , subject types and techniques. F i n n ^ t aj_. (1980) compared dif ferences in fac ia l height whereas Helkimo et a1_. (1975 and 1976) observed a s imi la r range of forces depending on the state of den t i t ion , the extent of tooth wear, or the presence of j o i n t dysfunct ion. The two studies which do not coincide with the general range of i n c i s a l b i te forces are those of Osborn and Baragar (1986) and Hylander (1978). Osborn and Baragar however, did not measure the occlusal force d i r e c t l y but mathematically derived i t according to the i r computer assisted model of mandibular biomechanics. They used the muscle ac t i v i t y leve ls of Pruim et a l . (1980) for th is der iva t ion . Hylander was not attempting to determine maximal i n c i s a l fo rces but was more i n t e r e s t e d in t h e i r o r ien ta t ions . These were found to l i e between 60 to 80 degrees with respect to the occlusal plane (Ginger ich, 1979) which agrees very well with the data of th is model ana lys is . The magnitudes of maximal inc isa l force produced by the model in th is study are also consistent with the previous values found in - 213 -the l i t e r a t u r e (see Tables VII and V I I I ) . The most prominent factor contr ibut ing to less tooth resistance force at the inc iso rs than at more poster ior posi t ions was the di f ferences in the overal l muscle resul tant vectors and not simply less favorable lever type mechanics mentioned earl i e r . Pruim £ t al_. (1980) measured the maximal bi te forces b i l a t e r a l l y at the f i r s t premolar and the f i r s t and second molar pos i t i ons . The mean leve ls were determined to be 633 N (+_ 210) at the premolar, 965 N (+_ 276) at the f i r s t molar but only 756 N (+_ 289) at the second molar. The tooth resistance forces of BIMOL and CO of th is study correspond reasonably well with these ranges. However, Pruim and coworkers recorded a substantial and consistent decrease in the a c t i v i t y l eve ls of the muscles (masseter, temporalis and d igas t r i c ) during contact at the second molar producing the observed decrease in tooth load at th is pos i t i on . They have at t r ibuted j o i n t i nh ib i t i on as the con t ro l l i ng inf luence on the reduced tooth force at the second molar. They also proposed that more poster ior occlusal contacts may have the muscle force resul tant very near the bi te posi t ion requir ing more accurate control of the equ i l i b r ium. Hence the reduced muscle a c t i v i t i e s and resu l t ing increase in tooth force at the second molar. Tradowsky and Dworkin (1982) have proposed that an equi l ibr ium point of mandibular t i t l i n g ex is ts along the dent i t ion such that the biomechanics of the mandible are d i f fe rent poster ior to th is posi t ion than anter ior to i t . They found that the mandible t i l t s to produce tension on the jo in ts at the poster ior pos i t i ons . If such events do occur then they would cer ta in ly support, although i n d i r e c t l y , some sort of inh ib i tory e f fec t from the j o i n t s . - 214 -TABLE V I I I - PREVIOUS INCISAL BITE FORCE DETERMINATIONS. A l l units have been converted to Newtons of force. The data for d i f ferent sexes has been combined here and a l l are assumed to be at maximal e f fo r t except where noted. Authors Mean (N) Range (N) Relevant Factors Osborn and Baragar, 1986 686 not given mathematically derived (n = 0) Finn e t . al_., 1980 145 286 not given not given long face/skeleta l open b i te (n = 5) short face/skeleta l deep bi te (n = 6) Helkimo and Ingerva l l , 1978 190 34-459 non-maximal e f fo r t (n = 100) 5 mm opening) Hylander, 1978 53 48 28 28-91 19-91 22-36 10 mm opening non-maximal e f fo r t (n = 10) 30 mm opening non-maximal e f fo r t (n = 10) 20 mm opening non-maximal e f for t (n = 10) Mansour, 1977 209 not given approx. 5 mm opening (n = 6) Helkimo e t . a]_., 1976 172 1-44 approx. 5 mm opening 9 (n = 78) Helkimo et . al_, 1975 196 127 147 not given normal group (n = 36) j o i n t dysfunction pretreatment (n = 30) j o i n t dysfunctions posttreatment (n = 30) Mansour and Reynik, 1975 100 not given 6 mm opening (n = 1) Rugh and Sol berg, 1972 162 not given 7 mm opening Ringquist , 1973 293 200-448 6 mm opening (n = 29) Linderholme and Warnstrom, 1970 206 S.D.=83 15 mm opening (n = 20) Howell and Manly, 1948 178 130-235 (n = 4) (Grand Mean of a l l studies = 170 N) - 215 -Furthermore, the resul ts of th is present study show that i f the or ientat ion of LTA becomes more an ter io r , for whatever reason, the j o in t forces not only become reoriented more poster ior ly but increase in magnitude as well (eg. Figure IOC, 11B and lie). The sp l i n t arrangement housing the force transducers of Pruim ej: aj_. separated the dent i t ion by about 6 mm from intercuspation at the second molars and about 11 mm at the premolars (Prium et al_. 1978). Increasing the ve r t i ca l dimension has a number of e f fec ts . F i r s t of a l l the muscle f ibers become elongated reducing the overlap of thei r f i b r i l s and thereby reducing the act ive tension of the muscle from i t s maximum which is at i t s rest ing length near intercuspat ion (Finn et aj_., 1980). Secondly, as depicted in Figure 19 th i s w i l l reor ient the alignment of the long axes of the mandibular teeth more anter io r ly from A to B. If the system attempts to maintain the same alignment of occlusal load with respect to the mandibular dent i t ion th is w i l l produce a more anter ior or ientat ion of the resu l t ing tooth force (B) and according to the model an increase in j o in t force magnitude. This j o i n t force w i l l also be aligned more pos te r io r l y . Comparison of Figures 11B and 11C for BIMOL at the f i r s t and second molar contacts show that although more poster ior contacts have lower j o i n t force magnitudes, the ef fect of th is type of change on the or ientat ion of j o i n t force may be more s i gn i f i can t . Conversely, i f the system maintains, or attempts to maintain, the same alignment of occlusal load at the maxi l lary dent i t ion then t i l t i n g the occlusal plane at wider opening would cause the occlusal load to become more pos te r io r l y al igned with respect to the mandibular teeth as well as more pos te r io r l y posit ioned (C of Figure 19). This changes the mechanics of the - 216 -Figure 19. EFFECT OF INCREASING VERTICAL DIMENSION ON TOOTH FORCE. The angle 0 denotes the angular change to the system whereas A, B and C ind icate potential changes to the tooth force. - 217 -whole system to a greater extent. Finn jet a]_. (1980) and Throckmorton et a l . (1980) have shown that a l te ra t ions to the ver t i ca l posi t ion of the occlusal plane changes the mechanical advantage of the muscles in addit ion to reducing thei r maximum potential force development due to f iber elongat ion. Indiv iduals referred to by these workers as having short faces and a deep b i te general ly have improved mechanical advantage due to shorter moment arms for b i te force and general ly longer moment arms for the muscles. Thus, greater b i te forces are produced for r e l a t i ve l y less muscle e f f o r t . Individuals with a skeletal open b i te and long fac ia l types have a less e f f i c i e n t mechanical arrangement of the i r mandibular components and cannot generate nearly as much occlusal force as normal or short fac ia l types (Finn et al_., 1980; P r o f f i t et al_., 1983a and b; Sassouni, 1969). Consequently under s ta t i c condit ions any device between the teeth which separates them much beyond the intercuspal or rest posi t ion e f fec t i ve l y a l te rs the mechanical re la t ionships of the system and may reduce the potential tooth force production, espec ia l l y at r e l a t i ve l y more poster ior posi t ions of contact. This ef fect may not be l imi ted to b i l a te ra l mechanics however. Although Mansour and Reynik (1975) noticed a ten percent increase in maximal b i te force at the second compared to the f i r s t molar there was a decl ine in the average moment of f i f teen percent. They therefore concluded that d i f fe rent mechanical re la t ionships existed at tooth posi t ions d i s ta l to the f i r s t molar ( i . e . C lass II compared to C lass II I l e v e r mechan ics ) . However, considerat ion of the resu l ts of th is modelling analysis impl icates other p o s s i b i l i t i e s regarding changes to the mechanics of the system due to - 218 -increasing the ver t i ca l dimension. The apparatus of Mansour and Reynik was 6 nm in ver t i ca l height. Therefore at the second molar the jaws would have been much more widely opened than at the i n c i s o r s . As was jus t described wide opening may cause a reorganization of the mechanics of s ta t i c b i t ing due to a reor ientat ion and reposi t ioning of the occlusal force with respect to the occlusal plane and the j o i n t s . If the overal l muscle resul tant force remained constant, or nearly so, as compared to a more closed mandibular posi t ion the s i tuat ion seen in Figure 19 may again be representative of the mechanics at p lay. Note that the posi t ion of tooth loading has become more poster ior from posi t ion A to C due to the i n fe r i o r jaw pos i t i on . This same type of in terp lay was previously observed in Figure 15B. Changing the posi t ion of tooth contact more poster io r ly in th is task was su f f i c i en t to produce a tens i le force at the i p s i l a t e r a l j o i n t . If the j o i n t has a regulatory role in mandibular biomechanics as proposed by Pruim £ t jil_. (1980) then b i t ing at more poster ior posi t ions with increased ver t i ca l dimension may produce lower tooth loads due to an inh ib i to ry e f fec t on muscle ac t iva t ion l e v e l s . Mansour apparently took th is into account in a subsequent paper (1977) where the ver t i ca l dimension was kept constant for measurements of maximal b i t ing force at each tooth pos i t i on . In th is study the mean maximum ver t i ca l b i t ing force progressively increased from inc i so r to th i rd molar. Averaged here between the r ight and l e f t sides these forces were: Tooth Posi t ion 1 2 3 4 5 6 Force (N) 209 229 291 485 568 690 7 8 735 796 - 219 -It i s of in teres t to note that although the di f ferences were not s t a t i s t i c a l l y s i gn i f i can t P r o f f i t et al_. (1983a and b) found greater mean occlusal forces to occur at 6 mn than at 2.5 mm of molar separation for both normal and long fac ia l types. The mean values for maximal b i te force are given in Table IX although th is f inding was also consistent for both swallowing and chewing in adults as well as ch i l d ren . This i s not consistent with the preceding argument. However, i t has been shown by other workers that the maximal bi te force which an indiv idual can generate increases with pract ice (Linderholme and Wennstrom, 1970; Van Steenberghe and De V r i es , 1978). The resu l ts of maximal b i t ing at the 6 mm posi t ion of P r o f f i t and company were recorded after the ser ies of tests at the 2.5 mm posi t ion which may have played a part . In any event the magnitude of un i la tera l molar clenching of 515.5 H (see Table VII) derived by the model conforms well with the averages and ranges observed by other workers summarized in Table IX. S i m i l a r i l y the un i la tera l canine forces of th is study (226.8 N, see Table VII) compare extremely well with the average magnitudes reported by Mansour (1977) of 291 N and of Van Steenberghe and De Vries (1978) of 279 N at th is tooth pos i t i on . Table X l i s t s the average maximal loads measured by these workers as well as those of other workers at the premolar posi t ions which are included for comparison. The resu l ts of the model predict that tooth resistance forces at the more poster ior premolar posi t ions would be greater than those at the canine due to more favorable mechanics and a r e l a t i ve l y greater overal l muscle force vector . The data of th is study and Table X also bear th is out. - 220 -TABLE IX - PREVIOUS UNILATERAL MOLAR BITE FORCE DETERMINATIONS. A l l measurements described as at the f i r s t molar unless noted. Data from d i f fe rent sexes has been combined and a l l units converted to Newtons of force. Authors Mean (N) Range (N) Relevant Factors Lundgren and Laurel 1, 211 S.D.=77 1.5 mm opening, s ingle poster ior 1986 point contact (n = 12) Osborn and Baragar, 1029 f i r s t molar, mathematically derived 1986 (n = 0) 1715 - th i rd molar, mathematically derived (n = 0) P r o f f i t e t . a l . , 304 S.D.=196 normal adults at 2.5 mm opening 1983 ~~ (n = 21) 350 S.D.=183 normal adults at 6.0 mm opening (n = 21) 110 S.D.=77 long face adults at 2.5 mm opening (n = 19) 152 S.D=103 long face adults at 6.0 mm opening (n = 19) P r o f f i t and F i e l d s , 171 S.D.=188 normal chi ldren at 2.5 mm opening 1983 (n = 18) 152 S.D.=139 normal chi ldren at 6.0 mm opening (n = 18) 98 S.D.=58 long face chi ldren at 2.5 mm opening (n = 12) 119 S.D.=101 long face chi ldren at 6.0 mm opening (n = 12) Finn e t . a l . , 1983 299 not given long face, 10 mm opening at "molar* (n = 8) 706 not given short face, 10 mm opening at "molar" (n = 6) F inn , 1978 569 not given normal face, 10 mm opening 293 long face, Helkimo and 471 191-802 non-maximal e f f o r t , 5 mm opening Ingerva l l , 1978 (n = 100) 728 617-882 strong group e f f o r t , 5 mm opening (n =25) 380 206-461 weak group e f f o r t , 5 mm opening (n = 25) - 221 -Aut h o r s Mean (N) Range (N) R e l e v a n t F a c t o r s Helkimo e t . a l . , 1976 400 98-715 5 mm opening (n = 44) Helkimo e t . a l . , 1975 411 264 353 not g i v e n not g i v e n not g i v e n normal group, 5 mm opening (n = 23) j o i n t d y s f u n c t i o n , p r e t r e a t m e n t (n = 30) j o i n t d y s f u n c t i o n , p o s t t r e a t m e n t Mansour, 1977 690 735 796 S.D. = 38 S.D. = 46 S.D. = 40 f i r s t molar, approx. 5 mm opening (n = 6) second molar, approx. 5 mm opening (n = 6) t h i r d molar, approx. 5 mm opening (n = 6) Mansour and Reynik, 1975 774 842 not g i v e n f i r s t m o lar, 6 mn opening (n = 1) second molar, 6 mm opening (n = 1) R i n g q u i s t , 1973 467 302-679 4 mm opening at "molars" (n = 29) Rugh and Sol b e r g , 1972 363 not g i v e n 7 mm opening a t "molars" {n = 19) Lin d e r h o l m e and Warnstrom, 1970 450 S.D. =107 15 mm opening a t "molars" (n = 72) White, 1967 731 not g i v e n 4.5 mm opening a t "molars" (n = 83) J e n k i n s , 1966 1372 up to 1568! I n u i t s u b j e c t s Howell and Manly, 1948 - 405-882 (n = 4) (Grand mean o f a l l s t u d i e s = 516 N) - 222 -R e l a t i v e l y l i t t l e data is a v a i l a b l e as to the tooth loads generated during m a s t i c a t i o n . What i s a v a i l a b l e i s not e a s i l y compared owing to d i f f e r e n c e s in food types as well as experimental techniques. Some of the previous analyses suggest that the l e v e l s of magnitude for the chewing phases of t h i s study are quite h igh . Other s t u d i e s , on the other hand, report f i n d i n g s which suggest the force l e v e l s of the model for the data incorporated are appropr ia te . DeBoever _et a]_. (1978) observed less than 20 N of average force during u n i l a t e r a l chewing of a va r ie ty of food types in three s u b j e c t s . Thei r study u t i l i z e d a force transducer arrangement incorporated into removable pa r t i a l dentures. The maximum level observed was about 60 N. Anderson (1956 a and b) used a f ixed in lay with attached s t r a i n gauges and found chewing forces up to about 140 N, averaging around 100 N depending on food type. Lundgren and Laurel 1 (1986) recorded s i m i l a r total force l e v e l s (109 N) from the i r br idge transducer apparatus. These magnitudes were t o t a l s of force occuring at four simultaneous points of contact at or very near the intercuspal p o s i t i o n . A given s ing le point of p o s t e r i o r contact exh ib i ted a mean force of 52 N (+_ 33) which was the greates t of t h e i r study. It has been shown that the a b i l i t y of the mandibular system to produce occ lusa l force decreases with a dec l ine in numbers of teeth and overa l l state of the den t i t ion (Hel kimo et^  al_., 1976). The state of the d e n t i t i o n of subjects in these workers' studies appeared to be s i g n i f i c a n t l y compromised. S imi lar cons idera t ions may have contr ibuted to the f ind ings of DeBoever £ t as they had a p r e r e q u i s i t e for the i r subject group of at l e a s t two adjacent teeth m i s s i n g . Never the less , s i m i l a r magnitudes of chewing force have also been r e p o r t e d by Graf e_t al_. ( 1974) who - 223 -TABLE X - PREVIOUS UNILATERAL CANINE AND PREMOLAR BITE FORCE DETERMINATIONS. As per Tables VIII and X I . Authors Mean (N) Range (N) Relevant Factors Van Steenberghe and DeVries, 1978 279 123-588 Canine, 10 mm opening (n = 9) (mean calculated from thei r data) Mansour, 1977 291 S.D.=25 Canine, approx. 5 mm opening (n = 6) Lundgren and Laurel 1, 1986 126 S.D.=50 "anter ior" (premolar), 1.5mm opening (n = 12) Mansour and Reynik, 1975 347 not given f i r s t premolar, 6 mm opening (n = 1) Rugh and Sol berg, 1972 294 not given f i r s t premolar, 7 mm opening (n = 19) Linderholme and Wennstrom, 1970 372 S.D.=111 "premolars", 15 mm opening (n = 72) - 224 -measured these forces at the f i r s t molar in three dimensions. They found the ax ia l (ve r t i ca l ) loads to be greatest with mediolateral components secondary. There was both an i n i t i a l l a t e r a l l y directed ( fac ia l ) load which became medial ly d i rected as the axia l force increased, agreeing with the FTA or ienta t ions assigned to the three chewing phases of th is study. However, Graf , et a l . noted a pronounced poster io r ly directed component, which is opposite to the anter ior component of LTA of th is study i n i t i a l l y assumed as the tooth force or ientat ion to para l le l that of the overal l muscle vector resu l tan t . This f inding takes on greater s ign i f icance when the ef fect of a l t e r i ng LTA i s considered. As these resul ts have consis tent ly shown, when the tooth force or ientat ion in the la te ra l plane becomes more poster ior the magnitude of th is force increases for the same muscle e f fo r t . At the same time the total magnitude of j o i n t force resistance decreases. In terms of the amount of useful tooth force compared to residual j o in t force (JF/TF r a t i o ) , a more poster ior alignment of the tooth force, as found by Graf et a l . i s more sound in terms of mechanics. However, j us t how i t would be possible for the mandibular system to reorient the angles of tooth force to maximize th is type of mechanical benefi t without reor ient ing the d i rect ion of muscle force i s not c lea r . The only studies which have reported tooth resistance forces of magnitudes comparable to the chewing in terva ls of th is study are those of Helkimo and Ingervall (1978) and Gibbs et al_. (1981). The former invest igators i nd i rec t l y measured these force leve ls for simulated chewing and reported a, mean level of 246 N with a range from 67 to 532 N at the molars. L ikewise, Gibbs et a l . used an ind i rec t technique of sound - 225 -transmission to f ind a mean maximum chewing of up to 360 N for hard foods and 240 N for sof t foods. Unlike Helkimo and Ingervall the l a t t e r workers determined these forces at the intercuspal posi t ion where the greatest forces are general ly agreed to occur in chewing (eg. Mol le r , 1966). These f indings also corroborate the resul ts derived by th is model. However, they re f l ec t the total amount of occlusal force at a l l points of contact in the dent i t ion . The modelling resu l ts are determined according to a single point of tooth contact . Both sides of the dent i t ion have been shown to be involved in d i s t r i bu t i ng the force of occlusal loading during chewing (Lundgren and L a u r e l l , 1986; DeBoever et a K , 1978; Graf et afk, 1974). Therefore the resu l ts of th is study may more accurately re f l ec t the amount of tooth force i f the dent i t ion was only able to contact at a s ingle point . When the resistance forces at the teeth become d is t r ibuted over a surface of contact, or mul t ip le asymmetric contacts, the mechanics may d i f f e r (Bramble, 1978). C. JOINT FORCES At the present time there are no other studies which have determined r e l i a b l e estimates of the actual magnitudes and three dimensional or ientat ion of the j o i n t forces in humans during s ta t i c function (Smith et al_. 1986). The information which is avai lab le i s mostly based on analyses of incomplete var iab les or highly a r t i f i c i a l parameters. Unlike the models which have been presented for the rabbit (Weijs and Dantuma, 1980) and the rat (Weijs and Dantuma, 1975) v i r t u a l l y a l l the previous modelling analyses of human jaw mechanics have incorporated a rb i t ra ry , or at best, a r t i f i c i a l magnitudes of ind iv idual muscle forces and often neglect asymmetrical considerations of jaw - 226 -function (Gys i , 1921; Mainland and H i l t z , 1933; Oaddock, 1951; Roydhouse, 1955; Ginger ich, 1971 and 1979; Barbenel 1969, 1972, 1974 and 1983; Osborn and Baragar, 1986). Even those analyses which do consider un i la tera l funct ions are of l i t t l e comparative usefulness for these same reasons (Hekneby, 1974; Smith, 1978; Smith e t a K , 1986; Hatcher et a]_., 1986). It i s su f f i c i en t to say that there i s general agreement among previous invest igators using modelling approaches that; resistance forces do occur at the j o i n t s . Their studies have general ly shown that in order for the s ta t i cs of the system to balance and remain in equi l ibr ium the jo in ts must contr ibute some degree of loading depending upon the posi t ion of tooth res is tance. Indicat ions from these previous model analyses would also imply that th is i s the case for v i r t u a l l y any applied muscle force. The resul ts of the present study show c lea r l y that for the s ta t i c functions modelled the j o in t s are indeed loaded, and to a greater or lesser extent depending upon the spec i f i c task. Simple lever mechanics have been the basis of most of the previous models of jaw mechanics of humans as well as a var iety of other creatures, at l eas t in the two dimensional sag i t ta l plane. Predict ions from this type of analys is produce greater magnitudes of j o in t force at more anter ior posi t ions of tooth res is tance. However th is i s based on the assumption of a constant muscle force applied to the system. Such a s i tua t ion occurs in the intercuspal (CO) clenching of Table XI which summarizes the j o i n t resistance force data when tooth force and overal l muscle resul tant force were aligned p a r a l l e l . In th is task the muscle force remained a r t i f i c i a l l y constant for changes to tooth pos i t i on . Because the tooth resistance force decreases - 227 -TABLE XI - SUMMARY OF RESULTANT JOINT RESISTANCE FORCE VECTORS. Jo in t resistance force or ientat ions and magnitudes for the runs of each task with tooth and overal l muscle force vectors aligned para l le l as in Table X. Lateral Plane Resultant Frontal Plane Resultant Vector Vector Resultant Vector Orientations Magnitudes Orientat ions (degrees) (N) (degrees) Task Tooth Pos' n RCA LCA RCR LCR RCA LCA Source Figure CO 5 88.0 88.0 316.1 316.0 90.0 90.0 Figure 10B, Run 1 6 219.4 291.3 H Run 2 7 323.2 232.2 II Run 3 8 161.0 160.9 •I Run 4 BIMOL 6 86.9 86.9 261.2 261.2 II 11B, Run 2 7 211.1 211.0 II 11C, Run 3 INCISS 1 69.6 69.6 160.0 160.0 12B, Run 3 INCISS 1 63.9 63.9 157.0 157.0 13B, Run 3 UNIK9 43 86.8 72.7 171.2 167.8 93.2 90.0 II 15C, Run 3 UNIMOL 47 89.8 82.4 73.9 256.0 94.8 90.0 II 14B, Run 3 CHEW 1 84.3 81.1 61.0 151.1 90.4 90.0 II 16B, Run 3 CHEW 2 80.9 89.6 105.5 158.1 87.6 90.0 •I 17B, Run 3 CHEW 3 77.3 90.7 78.6 99.8 86.0 90.0 18B, Run 3 - 228 -an te r io r l y the jo in ts must contr ibute more resistance force themselves in order for the system to remain in equi l ib r ium. Genera l ly , however, the present resul ts have also shown that a change in tooth contact posi t ion produces a change in the overal l muscle resul tant vector . The more anter ior posi t ions of tooth contact produce less rather than more overal l j o i n t res is tance, as comparison of j o i n t forces during b i l a t e r a l molar tasks with those of i nc i sa l functions ind ica tes . This i s due to the decrease in muscle ac t i v i t y leve ls at more anter ior pos i t ions . As a consequence the present resul ts do not support simple lever mechanics as an approximation of mandibular jaw function without adequate considerat ion of the overal l muscle resul tant force as we l l . The work of Pruim et a l . (1980) appears to be the only previous study predic t ing human j o i n t force magnitudes from an analysis which included ind iv idual muscle vector determinations. They found that the maximal j o i n t forces during maximum clenching at the f i r s t premolar were of nearly the same magnitude as when clenching at the f i r s t molar, despite the fact that the muscle forces of the former were reduced. They concluded that a more anter ior pos i t ion of tooth contact can therefore produce greater j o i n t fo rces , s im i la r to the predict ions of simple lever mechanics. However they also concluded that the primary determinants of the j o i n t and tooth res is tance forces are the ac t i v i t y leve ls of the muscle groups. The average magnitudes of these j o i n t forces derived from their data (n = 7) are approximately 540 N per j o i n t for the f i r s t premolar pos i t i on , 545 N for the f i r s t molar, and 300 N for the second molar pos i t i on . These .varied from about 400 to over 1100 N per j o i n t . Correction of the i r i n t r i n s i c muscle - 229 -force value from 13.7 N/cm2 to 4.0 N/cm2 of th is analysis puts the i r range between about 114 to 314 N per j o i n t corresponding well with th is study. As has been mentioned in the previous section Prium est jil_. a t t r ibuted the reduction in muscle a c t i v i t y to j o in t i nh ib i t i on at the f i r s t premolar. Their resu l ts imply that the level of j o in t force during maximal clenching on the i r apparatus at the f i r s t molar may have been the maximum allowable by the regulatory factors con t ro l l i ng jaw biomechanics. The reduction in tooth force which these workers observed at the more poster ior second molar produced a corresponding decrease in j o i n t force as w e l l . Pruim et al_. a t t r ibuted th is to a need for more control over the mechanical equi l ibr ium due to close proximity of the overal l muscle force resul tant to the b i te pos i t i on . Their subjects a l l found b i l a te ra l clenching on the second molar uncomfortable, implying that perhaps a tendency may ex is t for i n s t a b i l i t y in the j o in t s to occur for th is type of funct ion. The reduction in overal l muscle e f fo r t has been suggested as an attempt to lessen th is i n s t a b i l i t y . Osborn and Baragar (1985) have suggested that the "cont ro l " muscles of the i r analysis are responsible for maintaining stable j o i n t loading on the load-bearing surfaces of the condylar head and a r t i cu la r eminence. Reduced overa l l muscle e f f o r t , when th is i n s t a b i l i t y occurs, permits these weaker muscle groups ( i . e . l a te ra l pterygoid and poster ior temporalis) to maintain the posi t ion of the condyle surface on the inc l ined plane of the eninence without displacement. The equi l ibr ium point of Tradowsky et aj_. (1981 and 1982) was essen t i a l l y the same as Prium et a l . However, the former invest igators - 230 -concluded that t i l t i n g of the mandible occurs about the dental equi l ibr ium po in t . This would impl icate the p o s s i b i l i t y of j o i n t unloading during such b i l a t e ra l clenches. Ito et a K , (1986) have recently measured th is change in condyle posi t ion for b i l a te ra l molar c lenching. Their resul ts suggest that the condyle may move s l i gh t l y i n t e r i o r l y during th is task. The j o in t forces of b i l a t e ra l molar clenching derived from the model do not support th is f i nd i ng , however. Both intercuspal and bimolar tasks had j o i n t force or ientat ions such that i f any movement were to occur the jo in ts would tend to move super ior ly at approximately 70 to 100 degrees re la t i ve to the occlusal plane, depending on the angle of tooth force (LTA). The intercuspal clenching tasks of Ito ^ t al_., on the other hand, do suggest th is d i rec t ion of j o i n t loading as we l l . It should be pointed out, however, that Ito et a l . , did not consider the ef fects of bending or d is to r t ion of the mandible under the i r isometric condi t ions. Considering that the amount of j o i n t movement during these functions was only about 0.03 to 0.07 mm i t would seem possible for the displacement they measured, to be merely an a r t i f a c t of bone bending. Although very minor amounts of displacement occurred for the b i l a te ra l poster ior or intercuspal clenches of Ito et al_. the i nc i sa l clench did exh ib i t s i gn i f i can t l y more movement. This condit ion was very s imi lar to the s t a b i l i z e d inc i sa l clench of th is study (INCISN). Their resul ts indicate a movement of about 0.6 mm at an angle of approximately 67 degrees, assuming the i r coordinate system is al igned re la t i ve to the occlusal plane. This compares very favourably with the same or ientat ion of j o in t force from th is study (see Table X I ) . However, Ito and company concluded that temporo - 231 -mandibular j o i n t loading increases when poster ior tooth support is removed. They proposed that where poster ior support is present i t e f fec t i ve ly protects the j o i n t s from heavy loading forces by red is t r ibu t ing more of the total resistance force to the dent i t ion . The resul ts of the present model disagree with th is as re l a t i ve l y greater j o i n t resistance forces occurred for the poster ior points of tooth contact, espec ia l l y CO. However, th is discrepancy again points out the fact that the model only considers s ingle point contacts at the teeth , and not actual surfaces or mult ip le contacts. Such i s ce r ta in ly not the case of intercuspal c lenching, espec ia l l y with an in terocc lusa l sp l i n t (I to tyt a l . , 1986). Therefore the j o i n t resistance forces of intercuspal clenching (CO) from the model analysis are rea l l y predict ions of the j o in t and tooth loads as they would occur i f the muscles were as act ive as intercuspal clenching but only a single posi t ion of tooth contact were present. As has been shown by th is study a change in occlusal contact pattern produces a change in the overal l muscle force resul tant due to d i f f e ren t i a l a c t i v i t i e s of the indiv idual muscle groups (MacDonald and Hannam, 1984a and b; Mo l le r , 1966). Therefore changing the posi t ion of tooth contact in the intercuspal task without also producing a d i f ferent overal l muscle force seems un l i ke l y . The predict ions of j o i n t and tooth resistance forces of the CO task are therefore of l imi ted value. However, the notion that j o i n t i nh ib i t i on may l i m i t muscle ac t i v i t y l eve l s and the overal l muscle resul tant force, which thereby also l im i t s occlusal resistance forces, may be important (Wook and Tobias, 1984). If the dent i t ion does, in f ac t , contr ibute to a reduction of the load at the jo in ts then a very worn occlusal scheme with reduced ver t i ca l dimension would - 232 -repos i t ion the condyles more super ior ly as w e l l . This would be expected to produce increased forces at the j o i n t s . In order to compensate for th is the j o i n t s would be expected to undergo adaptive changes due to th is increased load ing. Observations of tooth a t t r i t i o n and j o i n t morphology suggest th is may occur (Hinton, 1981; Seward, 1976). S im i l a r l y , a reduction in the numbers of teeth, espec ia l ly pos te r io r l y , have been shown to produce the same e f fec t (Osberg et al_., 1970; Moffet et al_., 1964). Pathosis such as that manifested by Temporomandibular Jo in t Dysfunction Syndrome is one well known outcome of a dent i t ion with compromised poster ior support (Roberts, 1974), or occlusal disharmonies (Storey, 1981; S e i t l i n , 1968) which a l t e r the mandibular mechanics from that genet ica l ly dictated as most e f f i c i e n t . Un i la tera l funct ions, or at least those with more l imi ted tooth contact, are perhaps more appropriately analyzed by th is model. Uni latera l canine clenching produced j o i n t resistance forces which were very s imi la r on the two sides despite some di f ferences in muscle a c t i v i t i e s . However the unimolar clenching exhibi ted much more d ivers i t y in muscle ac t iva t ion between the two sides as well as general ly higher l e v e l s , espec ia l l y on the working side (see Table I I ) . The model predict ions cons is tent ly show that the balancing side j o i n t exh ib i ts greater compressive forces than the working j o i n t during the un i la te ra l molar clench as well as during the three in te rva ls of chewing analyzed. The extent of the di f ference varies according to the dif ference In muscle forces. S imi lar f indings have been reported by a number of other workers using theoret ical considerat ions (Hatcher et a]_., 1986; Smith et a l . , 1986; Smith, 1978; Walker, 1978; Hylander, 1975; Gys i , 1921). Unfortunately - 233 -there are no estimates of actual j o i n t force magnitudes for humans anywhere in the l i t e ra tu re for comparison. The work of Ito et (1986) provides the only ind ica t ion of the d i f f e ren t i a l loading of the two temporomandibular jo in ts during uni la tera l functions from d i rec t observation of humans. They found the balancing condyle to move in an anterosuperior d i rec t ion (about 40 degrees from thei r data) whereas the working side condyle moved i n t e r i o r l y as well as poster io r ly (about 215 degrees) for un i la tera l molar contact. This suggests the balancing condyle was loaded under compression s imi la r to UNIMOL and the chewing data of the model. The working condyle, however, was d is t racted or loaded under tens ion. This f inding i s consistent with c l i n i c a l and experimental observations by other workers (Hawthorn, 1984; Koivumaa, 1961). Smith et aj_. (1986) u t i l i z e d a mathematical "minimization" model of human j o i n t loading s imi la r to those previously discussed (Osborn and Baragar, 1985; Barbenel, 1974). Using a r t i f i c i a l muscle forces they also found j o i n t loads to be minimal with v e r t i c a l l y directed bi te forces at the second molar. Posi t ion ing the bi te force at the th i rd molar produced a d is t rac t ing or tens i le force but at the balancing, as compared to the working side j o i n t . Although th is seems confusing in re la t ion to the f indings jus t discussed i t must be understood that Smith et were def ining only the minimum possible j o i n t loads for any combination of a rb i t ra ry muscle forces. When th is i s taken into considerat ion the side of th is type of loading becomes i r re levant in the i r study. In any case, the i r analysis showed that the human temporomandibular jo in ts were loaded in tension by more than 5 percent of the b i te force magnitude. - 234 -Uni latera l b i t ing at the th i rd molar posi t ion of Figure 15B produced th is very s i tua t ion with the working side j o i n t experiencing about 10 percent of the bi te load. These f indings h ighl ight a very important po int . The ra t io of j o i n t to tooth resistance force for th is task at the th i rd molar was the lowest of a l l runs observed in th is study (other than the intercuspal clench at the th i rd molar which, as already s ta ted, may not be a r e a l i s t i c s i t ua t i on ; see Figure 14B). This further suggests that the a b i l i t y to u t i l i z e muscle forces of the opposite side during un i la tera l functions provide maximal mechanical e f f i c iency during mandibular funct ion. Studies of primate jaw mechanics by Hylander (1979b and 1977) and Beecher (1977) have shown that much of the advantage of a fused symphysis i s the a b i l i t y to apply contra latera l muscle force to a un i la tera l b i te pos i t i on . Secondari ly, however, i s the advantage of a concurrent reduction in the resu l t ing residual j o i n t forces, observed here. Simi lar conclusions have been drawn by other workers as w e l l . From experimental analyses of the rabbi t Weijs (1980) and Weijs and Dantuma (1981) concluded that , "during natural mastication the muscles of both sides act in a proportion ensuring the largest b i te force possible without pu l l ing the a r t i cu la t i ng surfaces of the working side j o i n t apart" (Weijs, 1980; p. 716). Hylander (1979c) and Hylander and Bays (1978 and 1979) measured subcondylar bone s t ra in in monkeys and found that compressive react ion forces occurred during isometric molar b i t ing at or anter ior to the second molar. The magnitude of the j o i n t forces was less at molar than premolar contacts. More pos ter io r ly however, at the th i rd molar, the working side j o i n t was e i ther unloaded or loaded under tension (Hylander, 1986). The a b i l i t y of the - 235 -mandible to coordinate the mechanical re lat ionships to maximize b i te force while minimizing j o i n t fo rces, at least on the working side may, therefore, be a universal feature of mandibular mechanics (Greaves, 1978) where a fused symphysis e x i s t s . This would substantiate the e a r l i e r suggestion that perhaps the or ientat ion of tooth resistance force i s ac tua l ly aligned more poster io r ly than that of the overal l muscle forces. If such i s the case then the greater tooth resistance force occurs with reduced j o i n t fo rces. This reduces the JF/TF ra t io implying better biomechanical d i s t r i bu t ion of resistance forces (see Figures 15C). The j o i n t force also becomes reoriented more an te r io r l y . Based on the predict ions of th is model the bearing surfaces of the j o in t s res is t ing compressive loads would be expected to be posit ioned at the anter ior and superior aspects of the head of the condyle in the s a g i t t a l , or la te ra l plane. A reciprocal arrangement would be expected for those surfaces of the a r t i cu la r eminence. There would also be a requirement for the j o i n t morphology to be able to r e s i s t d is t rac t ing or tens i le forces of lesser magnitude, as w e l l . In the frontal plane, over the range of tooth force or ientat ions tested the compressive load- res is t i ng surfaces would be expected to extend from the mediosuperior to the laterosuper ior aspects. It i s general ly agreed that the morphology of the temporomandibular j o i n t s i s a re f l ec t i on of jaw function and therefore loading patterns (Car lsson, 1979; McNamara, 1972; Turnbul, 1970). With regard to compressive forces normal j o in ts have been shown to have the th ickest load-bearing type of t issues located at the anter ior and superior aspects of the condylar head and poster ior and i n fe r i o r surfaces of the a r t i cu la r eminence and tubercle - 236 -(Hansson et a K , 1976; Moffet et a K , 1964). Other surfaces ( i . e . roof of the j o i n t fossa and poster ior aspect of the condyle) are not h i s t o l og i ca l l y adapted to withstand the same extent of loading. Moffet and his coworkers also found that these areas exhibi ted a much greater propensity for remodelling of the osseous components in response to the assumed predominant loads at these surfaces. The temporomandibular ligament as well as the j o in t capsule i t s e l f are regarded as the components l im i t i ng d is t rac t ion or tens i le forces of the j o in t s (Rees, 1954). The predict ions of j o i n t function from the model therefore appear to correspond well with the morphology of the temporomandibular j o i n t s . The only r e l i ab le report of j o i n t force or ientat ion for human s ta t i c funct ion i s that of Pruim et al_. (1980). The average or ientat ion from the i r data for b i l a t e ra l clenching functions was approximately 74 degrees with respect to the occlusal plane which substantiates the resul ts from the model. The range of j o i n t forces predicted was approximately 60 to 100 degrees with a mean or ientat ion for the b i l a t e r a l l y symmetrical tasks of about 77 degrees from Table X I . Those compressive j o i n t forces predicted by the model to have or ientat ions greater than about 90 degrees in the la te ra l plane suggest the poss ib le involvement of the roof of the j o i n t fossa for support. This would be a condi t ion for which the fossa i s not designed according to the information jus t described. The apparent incongruity may be explained by the fac t that for the functions modeled, except for in tercuspat ion, one would expect some amount of jaw opening to be involved. This t ranslates into a rotat ional change in the apposit ion of the condylar head to the a r t i cu la r eminence. The nature of th is change would depend on the extent of opening, - 237 -but an increase in jaw opening would e f fec t i ve l y reor ient any j o in t forces more an te r io r l y . For instance, the resu l ts are a l l depicted as i f the mandible remains stat ionary but the rest of the cranium, including the a r t i c u l a r eminence, would become reoriented more poster io r ly for wider opening. Another c o n s i s t e n t f i n d i n g in morpholog ica l s tud ies of the temporomandibular j o in ts i s the fact that the la te ra l aspect of the j o i n t components, espec ia l l y the d i s c , exh ib i t ind icat ions of greatest wear and a t t r i t i o n (Hansson, 1986; Hansson et al_., 1979 and 1977; Hylander, 1979c). I t has been assumed by most workers that th is aspect of the jo in ts undergo greater compressive loads during functional as well as parafunctional a c t i v i t i e s . The ef fects of varying the frontal plane or ientat ion of tooth force (FTA) in the model have shown that more medial alignments produce more l a te ra l j o i n t forces at the working s ide . Since th is or ientat ion of the balancing side j o i n t forces has usual ly remained f ixed for changes in FTA no obvious reor ientat ions of th is force are evident (see Figure 15D). Nevertheless, the model has also shown that for s ta t i c functions where a l l other factors are equal a change in frontal plane or ientat ion more l a t e r a l l y at one j o i n t produces a s im i la r reor ientat ion at the other, and vice versa (eg. see Figure 15E). Perhaps the nature of the dynamic forces which determine the d is t r i bu t ion of resistance forces in these types of function are such that there i s a tendency for l a t e r a l l y oriented j o i n t forces to be produced at one condyle. If th is i s at the working side there would be a s im i l a r la te ra l , component at the balancing side but of greater magnitude. I t may be that the in terp lay of condylar t r ans la t i on , condyle and disc pos i t i on , - 238 -jaw opening and d i rec t ion of muscle e f fo r t combine to produce the types of loading which have a pred i lec t ion to produce th is type of resistance force or ienta t ion at one or the other j o i n t s . D. STRENGTHS AND WEAKNESSES OF THE CURRENT MODEL Inherent in any model of b io log ica l function i s the need to be able to manipulate and control the po ten t ia l l y vast arid complex interplay of the many var iab les involved. Idea l ly , a b io log ica l model should be f l ex ib le enough to permit the incorporat ion of a l l relevant factors inf luencing the system under study. Unfortunately, the r e a l i t y of the s i tuat ion is such that cer ta in assumptions must be made regarding which var iables are most important and how they should be incorporated in the model. F i r s t l y , the enormous mathematical complexity in analyzing dynamic mechanics d ic tates the l im i ta t i on to s ta t i c funct ions, or assumed near-s ta t ic condit ions during dynamic functions such as intercuspat ion in chewing. Although i t may be possible to apply th is model to incremental analyses of purely dynamic condit ions the va l i d i t y of the assumptions necessary may be questionable. Secondly, the mathematical so lu t ion of resistance force d is t r ibu t ions for each task of th is study is dependent upon designating the l e f t condyle f rontal angle (LCFA) of res is tance. This reduces the resistance force var iables to a s t a t i c a l l y determinant number. Th i rd , the posi t ions of tooth (teeth) and j o in t contact through which the resistance forces act is assumed to be single point rather than surfaces or mult ip le points . Again th is i s to s impl i fy the mathematics involved in s ta t i c ana lys is . - 239 -The muscle parameters are determined according to thei r l ines of action between s ingle points of attachment. The posi t ion of these points may be an overs imp l i f i ca t ion for some groups espec ia l l y where pennation of muscle f i be rs e x i s t . In th is study the nine pairs of muscle vectors were assigned to spec i f i c muscle groups such as the medial pterygoid, or subgroups of other muscles such as the three portions of temporal is, and two each of the masseter and la te ra l pterygoid. There i s no reason, or need, to l i m i t the mumber of muscle subgroups s t r i c t l y for the sake of the model i t s e l f , however. The model w i l l eas i l y accommodate the assignment of the nine pairs of coordinates to any portion of any muscle chosen. Inclusion of addit ional pa i rs of muscle coordinates beyond the nine used in th is study i s also poss ib le . The element which l im i t s the a b i l i t y of the model to incorporate great numbers of muscle subgroups as found in a pennated muscle, i s the a b i l i t y to accurately determine precise points of attachment. Furthermore, the muscle parameters are determined according to the i r weighting value given to the muscle group or subgroup ( i . e . physio logical cross section) as well as t he i r scal ing values ( i . e . ac t i v i t y leve ls) for a given functional task. The a b i l i t y to provide or measure these values also determines the number of muscle subgroups which can be incorporated and reasonably modeled (eg. Osborn and Baragar, 1985). The nine pairs used in th is study are current ly feas ib le for determination of these parameters by means of techniques such as magnetic resonance imaging (MRI), CT scanning, and electromyography. This model was designed for appl icat ion to l i v i n g subjects from whom such data can reasonably be derived as well as estimates of in terocclusal fo rces . A l l that i s necessary is the input of actual values for these - 240 -var iables from given ind i v idua ls . Val idat ion of the predict ions of th is study by experiment are therefore current ly feas ib le at least so far as muscle morphology, muscle use, b i te points and occlusal forces are concerned. E. FUTURE DIRECTIONS Appl icat ion of th is model in i t s present form w i l l al low the ind i rec t determination of the biomechanical re la t ionships which ex is t in any given indiv idual during any s ta t i c function or task. Comparison of jaw biomechanics of d i f fe ren t ind iv idua ls based on quant i tat ive rather than qua l i ta t i ve parameters i s poss ib le . Comparison of d i f fe rent c ran io fac ia l types would provide a reference from which predict ions of functional di f ferences due to a l te ra t ions in the mechanics of the system could be made. For example, the ef fect of changes in these re lat ionships could be predicted for a l terat ions to the dent i t ion v ia occlusal or orthodontic in tervent ion. The e f fec t of surgical procedures to correct morphological abnormalit ies could also be modeled from a functional point of view. In add i t ion , funct ional disturbances to the masticatory system can be simulated. This may shed l i gh t on the biomechanics as causat ive, or contr ibut ing fac to rs , or the i r role in appropriate treatment s t ra teg ies , for example occlusal sp l i n t const ruc t ion. F i n a l l y , the model provides a very powerful tool for comparative analys is of anthropological mater ia l . The model has already been used in comparative studies of the re la t ionships between jaw form of various mammalia - 241 -with very d i f fe rent feeding behaviors and mechanics. Appl icat ion in th is wide f i e l d of study i l l u s t r a t e s the model's diverse potential as an experimental and conceptual aid to research. - 242 -REFERENCES Alexander, R.M. 1968. Animal Mechanics. Universi ty of Washington Press, Seat t le . Anderson, D.J . 1956a. 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Capuccino and S.M. Melfer (eds.) Saunders, Ph i lade lph ia (pp. 277-296). Wei j , W.A. 1980. Biomechanical models and the analysis of form: A study of the mammalian masticatory apparatus. Amer. Zool . 20:707-719. Wei js , W.A. and Dantuma, R. 1975. EMG and mechanics of mastication in the albino ra t . J . Morph. 146:1-34. Wei js , W.A. and Dantuma, R. 1981. Functional anatomy of the masticatory apparatus in the rabbi t (Oryctolagus cuniculus L . ) . Neth. Jour . Zoo l . 31:99-147. Wei js , W.A. and Hi 11 en, B. 1984a. Relat ionship between the physiological cross section of human jaw muscles and the i r cross-sect ional area in computer tomograms. Acta. Anat. 118:129-138. Wei js , W.A. and Hi 11 en, B. 1984b. The physiological cross-sect ion of the human jaw muscles. Acta. Anat. ( in press) . W i l s o n , G.H. 1920 and 1921. The anatomy and phys i cs of the temporomandiublar j o i n t . J . Nat. Dent. Assoc. 7:414-420 and 8:236-241. Wood, W.W., Takada, K., Hannam, A.G. 1985. The electromyographic ac t i v i t y of the i n fe r i o r part of the human la te ra l pterygoid muscle during clenching and chewing. Arch. Oral B i o l . 31:245-253. Wood, W.W. and Tobias, D.L. 1984. EMG response to a l te ra t ion of tooth contacts on occlusal sp l i n t s during maximal c lenching. J . Pros. Dent. 51:394-396. - 252 -APPENDIX I Anatomical points of or ig in and inser t ion for the 12 muscle groups which comprise the masticatory elevator muscles as described by Baron and Debussy (1979, pp. 547-8) and depicted in Figure la and b. Masseter (a) Super f i c ia l Masseter ORIGIN - "at the junct ion of the anter ior and middle th i rds of the postero infer ior (masseteric) border of the zygomatic bone. It occupies the top of an eminence which should not be confused with Pature t 's sub-jugal eminence si tuated more anter ior ly and to which Zlabek's tendon attaches (Paturet, 1951, p. 95) . " INSERTION - "at the pre-angular bony project ion which corresponds to the anter ior l i m i t of Cihak and V lcek 's masseteric eminence (1962)." (b) Intermediate Masseter ORIGIN - "at the small bony spur near the squamoso-zygomatic suture at the poster ior end of the masseteric border of the zygomatic bone." INSERTION - " the geometr ic center of a rhomboidal area corresponding to Weidenreich's masseteric fossa. " - 253 -(c) Poster ior Deep Masseter ORIGIN - "the geometric center of the longi tudinal fossa in the poster ior half of the i n fe r i o r border of the zygomatic process of the squama." INSERTION - "at the geometric center of a t r iangle corresponding to Cihak and V lcek 's (1962) zygomatico-mandibularis musculi fossa. This t r iang le situated between the coronoid process and the condylar process i s l imi ted by a soft ridge forming a V with curved branches ca l led the c r i s t a musculi zygomatico-mandibularis." (d) Anter ior Deep Masseter ORIGIN - "the top of the superior border of the superior surface of the zygoma (almost in the middle) . " INSERTION - "the geometric center of a pentagon centered on the body of the coronoid process." Medial Pterygoid (e) Anter ior Medial Pterygoid ORIGIN - "the center of the i n f e r i o r (palat ine) surface of the pyramidal process of the palat ine bone adjacent to the maxi l lary bone." INSERTION - "the top of the bony project ion in the center of an e l l i p t i c a l space in the preangular region." - 254 -(f) Poster ior Medial Pterygoid (Superf ic ia l Layer) ORIGIN - "deep in the pterygoid fossa, in the center of a ver t ica l segment jo in ing the top of the inter-pterygo-scaphoid ridge to the pterygoid notch." INSERTION - "the center of a crescent-shaped area situated in the preangular region. It corresponds to the second-to- last eminence near the gonian (angle)." (g) Poster ior Medial Pterygoid (Deep Layer) ORIGIN - "the center of the poster ior border of the la te ra l pterygoid plate of the pterygoid process, near the poster ior pterygoid spine ( C i v i n i n i ' s spine, Paturet, 1951, p. 329)." INSERTION - "the center of a 1 cm long r idge, para l le l to the mylohyoid groove and si tuated between the dental notch and the angle." INSERTION - "at the top of the coronoid process." Temporali s (h) Anter ior Temporalis (Zygomatico-Mandibularis) ORIGIN - "the geometric center of a trapezoid made by the fol lowing points (Paturet, 1951, pp. 117, 329): - stephanion (on the superior temporal curved l i ne where i t crosses the coronal suture) - upper point of the i n fe r i o r temporal curved l i ne - 255 -- top of the sphenoidal eminence (tuberculum sphenoidale) - poster ior end of the sub-temporal ridge (at the junct ion of the sphenoid and temporal)" INSERTION - "the center of a horizontal segment passing through the top of the retromolar t r iang le and l ink ing the two ridges to the anter ior border of the coronoid process." Middle Temporalis ORIGIN - "at the geometric center of a large rectangle l imi ted by: - the spheno-occipital suture, anter ior ly - the i n fe r i o r temporal l i ne on the outer surface of the par ieta l bone, super ior ly , - the sub-temporal ridge of the squama, i n t e r i o r l y , - the l i nk ing of the poster ior segments to the superior and i n f e r i o r segments of the muscular o r i g i n , pos te r io r l y . " INSERTION - "at the top of the coronoid process." Poster ior Temporalis ORIGIN - " in the parieto-temporal fossa situated behind the area of [ i ] , at the center of a l i ne l i nk ing the central points of the antero- in fe r io r and postero-superior segments in th is fossa. INSERTION - same as [ i ] . - 256 -Lateral Pterygoid (k) Superior Lateral Pterygoid (Sphenoidal Head) ORIGIN - "the center of the segment between the sub-temporal plane of the greater wing of the sphenoid and the superior th i rd of the la te ra l surface of the la te ra l pterygoid plate of the pterygoid process." INSERTION - "the center of the anter ior border of the a r t i cu la r surface of the condylar process. This point i s meniscal ." (1) In fe r io r Lateral Pterygoid (Pterygoidal Head) ORIGIN - "the center of a rectangle which occupies the i n f e r i o r two-thirds of the la te ra l surface of the la te ra l pterygoid plate of the pterygoid process and the points adjacent to the maxi l lary eminence and the pyramidal process of the palat ine bone." INSERTION - "the center of a small depression occupying the anter ior slope of the a r t i cu la r process of the condylar process." REFERENCES (Cited in the above by Baron and Debussy (1979)). Cihak, 0. and Vlcek, 0. 1962. Cr i s ta et fossa muscali zygamatico-mandibularis. Antropologie 66:503-525. Paturet , G. 1951. Trai te d'Anatomie Humaine. V o l . 1, p. 95, 117 and 329. Masson, Pa r i s . - 257 -APPENDIX II The fol lowing f igures are representative of the actual three dimensional computer pr intout for each run of every task. Each of the fol lowing f ive pages corresponds to RUNS 1 to 5 respect ively of UNILATERAL MOLAR CLENCHING at the second molar posi t ion depicted in Figure 15C of RESULTS. The lower diagram depicts the horizontal view of the mandible viewed from below. LTA and FTA of previous f igures are here designated as simply TA corresponding to the i r appropriate plane or project ion ANG-L and ANG-F values are l ikewise indicated but the muscle vectors themselves are not depicted in these printouts for reasons discussed in METHODS. UNIMOL RUN 5 LCR- 2€S. 1 LCA- Bfi. € RCR- 113. 3 RCA- 11£. 1 TR - ms. 5 TA - IS. 0 MLR- - J 5. J MAR- B7. 3 MVR- B33. 3 ANG- 81. 0 LCR- 2fi5. J LCA- 30. 0 RCR- 113. 3 RCA- 33. 1 TR - 1B5. € TA - 31. 0 MLR- ~15. 1 MAR- 91. 3 MVR- B33. 3 ANG- SJ. 0 LCR- 2€S. 1 LCA- 30. 0 RCR- 113. 3 RCA- 2€2. 6 TR - 1B5. fi TA - 33. 1 MLR- -J5. 1 MAR- B7. 3 MVR- B33. 3 ANG- 33. 8 - 259 -UNIMOL RUN 4 L C R - 2€0. 0 L C A - t l 1 R C R - 32. 2 R C A - JD5. 4 TR - 50 D. 2 TA - BO. 0 M L R - - i S . 1 MAR- B7. 3 MVR- B33. 3 A N G - B i . 0 LCR> L C A ' R C R -R C A -TR TA 2«0. 0 30. 0 32. 2 31. 2 500. 2 31. 0 M L R - - I S . 1 MAR- 87. 3 MVR" 833. 3 A N G - 31. 0 LCR-LCA-2«D. 0 30. 0 R C R - 32. 2 R C A - 255. 1 TR TA 5D0. 2 35. 1 M L R - - J 5 . 1 MAR- 87. 3 MVR- B33. 3 A N G - 33. B - 260 UNIMOL RUN 3 LCR-LCA-flCfl-flCA= TH • TA -256. 0 B2. 4 73. 3 89. B 5] 5. 5 8 4. 0 MLR 3 -3 5. 3 MAR- B7. 3 MVR = B33. 3 A N G B 81. 0 LCR* L C A ' RCR-RCA 1 TR « TA • 256. 0 3D. 0 73. 3 31. B SiS. 5 S L 0 MLR= -IS. 1 MAR= 87. 3 MVR= B33. 3 A N G B S J . D LCR-LCA« 256. 0 30. 0 flCR" 11. 3 RCA- 177. B TR - 5J5. 5 TA - 33. 1 MLR* - 1 5 . \ MAR- 87. 3 MVR- B33. 3 ANG- 33. B UNIMOL RUN 2 L C R - 250. 0 L C A - 13. 1 R C R - 6 1 . 0 R C A - SO. 1 TR - S I S . 1 TA - 30. 0 M L R - - 3 5. 1 MAR- B X 3 MVR- B33. 3 A N G - B l . 0 L C R - 250. 0 L C A - 30. 0 R C R - 61. 0 R C A - 36. 5 TR - 515. 1 TA - 91. 0 M L R - - 1 5 . 1 MAR- tl. 5 MVR- B33. 3 A N G - 91. 0 L C R - 250. 0 L C A - 90. 0 R C R - 6 1 . 0 R C A - 31. 3 TR - 515. 1 TA - 160. 0 M L R - - 15 . 1 MAR- B7. 9 MVR- B33. 9 A N G - 93. B UNIMOL RUN 1 L C R - 215. 3 L C A - IS. 3 R C R - 62. B R C A - IB . 1 TR « 5"7B. 2 TA - 35. 0 M L R - - 15 . 1 MAR- B7. 3 MVR- B33. 3 A N G - B l . 0 L C R - 215. 3 L C A - 3D. 0 R C R - 82. B R C A - 101. 0 TR - SIB. 2 TA - 31. 0 M L R - - 15 . 1 MAR- B7. 3 MVR- B33. 3 A N G - 31. 0 L C R - 215. 3 L C A - 30. 0 R C R - 62. B R C A - 33. "7 TR - 57B. 2 TA - 25B. 1 M L R - - 15 . 1 MAR- 87. 3 MVR- B33. 3 A N G - 33. B 

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