Anat Clin (1984) 6:45-52 9 Springer-Verlag 1984

New techniques in anatomy A physicomathematical approach to the structure of the mandible JC Ferr~ 1, JY Barbin 2, M Laude 3, and JL Helary 4 1 Laboratoire d'Anatomie, Facult6 de M6decine de Nantes, 1, rue Gaston Veil, F-44035 Nantes Cedex, France 2 Chaire d'Anatomie, Facult6 de M6decine de Nantes, 1, rue Gaston Veil, F-44035 Nantes Cedex, France 3 Chaire d'Anatomie et d'Organog6n6se, Facultb de M6decined'Amiens, 12, rue F. Petit, F-80036 Amiens Cedex, France 4 ECP Nantes, Nantes France

Summary. Classical methods of anatomical study and experimentation have reached an endpoint with respect to the advancement of our knowledge of certain aspects of bone, i.e. its mechanical properties, investigation of the constraints acting on bone and the organization of bone allowing resistance to such mechanical stress. Indeed, current knowledge is rather limited regarding bone as a material. Furthermore, bone from the cadaver cannot be considered a reliable source of study material since its physicochemical composition and mechanical properties are highly different from those of living bone. The types of experimentation used to date, although allowing to study the phenomena occuring on the surface of the bone, do not allow to evaluate those that occur within the bone without modification of its mechanical features. Finally, the number and complexity of the parameters to be taken into account in this respect largely supersede the possibilities of classical study techniques. Accordingly, new types of methodology are required to evaluate the many parameters involved, to perform the corresponding computations and resolve the great number of unknown variables. Such methodology must allow experimentation to be performed without modifying the object of study and to determine the phenomena occuring within the bone itself, i.e. the mandible. A method of computer assisted simulation of a physicomathematical model was used to analyse the structural properties of the mandible. This method was based on that used for the computation and elaboration of large metal structures (offshore drilling platforms), structures submitted to special stress (resistance to force 7 earthquake of the new extension to the radioactive waste disposal OJ]print requests to:

JC Ferr6

factory at the Hague) or aeronautical structures composed of composite material.

Approche physico-math6matique structurale de la mandibule R6sum6. L'~tude des propri6t6s m~caniques de l'os, celle des contraintes qu'il subit, la maniSre dont il est organis6 pour r~sister fi ces contraintes s'avSrent d6boucher sur une impasse si l'on se contente des m6thodes classiques d'6tudes et d'exp6rimentations. En effet, nous n'avons actuellement qu'une connaissance trSs limit6e de l'os en qualit~ de mat6riau. Le mat6riau d'exp~rimentation, l'os de cadavre ne peut ~tre consid6r~ comme fiable, sa constitution physieo-chimique et ses propri6t~s 6tant par trop diff6rentes de celles de l'os vivant. Le type m~me des exp6rimentations utilis6es, s'il permet bien d'6tudier ce qui se passe fi la surface de l'os ne permet pas de prendre en compte ce qui se passe fi l'int6rieur de celui-ci sans en modifier les caract6ristiques m6caniques. Enfin, le nombre et la complexit6 des paramStres fi prendre en compte d6passent largment les possibilit6s des techniques classiques. C'est pourquoi, il est n6cessaire de faire appel ~t des m6thodologies nouvelles capables de prendre en compte un nombre trSs 61ev6 de paramStres, de les calculer et de r6soudre un nombre tr6s important d'inconnues. Ces m6thodologies doivent permettre l'exp6rimentation sans modifier l'objet de l'exp6rience et de calculer ce qui se passe fi l'int6rieur mSme de la mandibule. Cette m6thode de simulation sur modSle physico-math6matique par informatique utilis6e pour le calcul et l'61aboration des grandes structures m6talliques (plate-forme off shore), celui de structures plac6es dans des conditions de contraintes particuliSres (prise en compte des tremblements de terre

46

JC Ferr6 et al. : A physicomathematical approach to the structure of the mandible

de magnitude 7 pour l'extension de Fusine de La Hague) ou bien celui des structures a6ronautiques fi mat6riaux composites a 6t6 ici appliqu6e fi l'6tude structurale de la mandibule.

Key words: Mechanical structure of the mandible - Mechanical properties of bone - Physicomathematical modelization - Computer assistance

The study of the form and mechanical structure of the mandible, both of which are independent, as a function of the constraints to which this bone is submitted raises many complex problems. Conventional experimental techniques have reached an endpoint regarding further knowledge that can be gained in this respect. Accordingly, new techniques and methodology are required to increase our understanding of this special bony structure.

Limitations of conventional experimental techniques Certain limitations are related to the type of experimental material employed. Bone obtained post mortem, obviously the only material that can be used in these studies, is a very poor source for experimental investigation. Its physicochemical composition and mechanical properties are very different from those of living bone. Accordingly, the results obtained from the study of such a material cannot be reliably extrapolated to living bone (Jaeger et al. 1980). The extrapolation of results from animal experimentation is also open to considerable debate. Other limitations of conventional experimentation are related to the study techniques per se. The "cracked varnish" technique, which to our knowledge has not been used to study the craniofacial bony structures, only allows a very rough approximation of the structural properties of bone. This criticism also holds true for the technique referred to as "photo stress-sensitive varnish". Subsequent to the initial study by Massa (1957), Boyoud etal. (1975), Champy (1977), Champy and Lodde (1977) and Sonnerbrug et al. (1974) applied the techniques of photoelastimetry to a plexiglass model in order to evaluate the stress forces acting on the normal and fractured mandible consolidated by miniature "bone" plates. Although allowing to visualize the lines of surface stress, this technique does not demonstrate the phenomena occuring within the bone.

Durimetric study of the facial bones was carried out by Combelle et al. (1980) and B6ranger (1979) using an original apparatus derived from Boyer's durimeter. These studies confirmed that the basal mandibular bone is harder than that of the alveolar bone. These authors also demonstrated the high degree of mechanical hardness of the supra- and infraorbital ridges. Subsequent to the initial work of Burny (1968), a group from Strasbourg (Champy, Boyoud, Paty and Jaeger) furnished the biomechanical bases of the technique of miniaturized screw plates. The mandible was studied mathematically using extensiometry gauges. A team of engineers from the "Ecole d'Ing6nieurs de Strasbourg" then calculated the constraints acting on the mandible considered as a weight bearing beam. However, this type of study invalidates the experimental approach by modifying the mechanical conditions of the structure under investigation, since holes must be drilled in the latter to allow the introduction of the gauges. Leung et al. (1980), working in the physiology and biomechanics laboratory of Peugeot-Renault, simulated trauma resulting from automobile accidents using hybrid models made of composite material. These studies, designed with a specific goal in mind (prevention of trauma in the course of automobile accidents), doe not afford new information regarding the craniofacial structures, aside from one important finding, i.e. the mean resistance of facial bone to fracture is 10 daN/cm 2 (1 d a n = 1.02 kg force). Finally, Junier and Spiessl (1981) reproduced with a special apparatus the weight bearing conditions of the mandible in order to study mandibular osteosynthesis. This study was based on the principle of ovalization of holes drilled in the mandible to evaluate the forces of compression or tension acting upon the mandible. However, the criticism of the technique of stress-gauges can also be applied to this technique wherein the object of study is modified, thus invalidating the experimental approach. The limitations of conventional experimental techniques are also related to the mechanical properties of the bone. Using the stress-deformation curves of a test bar under traction, a classical procedure in mechanical evaluation, several authors (Reilly et al. 1974; Jaeger et al. 1980; Bonfield and Datta 1974) have attempted to evaluate Young's modulus of bone (elasticity module). Indeed, it is known that the classical stressdeformation curves of a given crystalline solid (Fig. 1) display an initial zone of so-called elastic deformation, where the

JC Ferr6 et al. : A physicomathernatical approach to the structure of the mandible

g E

....

---• 2

t"l

Stress Fig. 1 Stress-deformation curve of a polycrystalline solid, a Zone of elastic deformation; I limit of elasticity, 2 rupture Courbe de contrainte-d6formation d'un solide polycristallin, a Zone ~lastique 1 limite d'61asticit6, 2 rupture

latter disappears as soon as the stress is removed. These curves display a second zone referred to by the term plastic deformation where, beyond a threshold stress force, the deformation persists even after the stress is no longer applied. The curves finally terminate at a point where rupture occurs, this point being characteristic of a given material. In our opinion, calculation of Young's modulus of bone by this technique is not valid. Indeed, living bone most often lies in the plastic, rather than the elastic domain. Furthermore, living bone displays phenomena known as fluage, i.e. when submitted to a mechanical constraint of constant intensity the bone continues to undergo deformation as a function of time. It is important to note that bone is an anisotropic material since its mechanical properties are not identical in all three planes of space. This feature of bone is probably related to its heterogeneity. Indeed, after removal of all fat, dry bone is composed of the following: a) organic material, accounting for 25% of the mass (collagen matrix and mucopotysaccharides; Stack); b) mineral salts, comprising 75% of the bone (mainly calcium in the form of hydroxyapatite crystals). It is thus clear that the maximum resistance varies as a function of the axis of mechanical sollicitation, owing to the anisotropism of the bone and in compact bone, as a function of the orientation of the trabeculae. It is also well known that the resistance of bone to compression is usually greater than its resistance to traction. It must be underlined that the interpretation of numerical results of experiments on bone should be made with utmost caution. Indeed, the results

47

vary as a function of the study technique, owing to the types of instruments used or the mode of conservation of the anatomical material. Furthermore the physiological variations of the mechanical " p e r f o r m a n c e " of bone are such that the establishment of truly useful standard values is a rather hazardous enterprise given our current state of knowledge. Physiological variations can be found within a given bone according to the region under investigation. For example, Evans (1957) calculated that the middle third of the femoral diaphysis is more resistant to fatigue than the upper third and that the posterior quadrant is more resistant than the other quadrants of the diaphysis. According to Blaimont and Burny (1968), the hardness of the cortical bone and Young's modulus would both decrease from the deep layers near the marrow cavity to the periosteum. Variations can also be found as a function of age, the bone showing greater resistance to traction between the ages of 20 to 40 years, whereas resistance to compression is relatively stable with respect to age. It is conceivable that these phenomena result from increased calcification and decreased water content o f bone, thus reflecting a corresponding decrease of viscoelasticity (Pugh et al. 1973). In sum, what is known regarding bone as a material? Bone is a viscoelastic substance, most often working in conditions exceeding the domain of elasticity; it presents phenomena of fluage. Owing to its structure, bone is a composite anisotropic material. It may be reasonable to assume that a sort of "basic" bony material exists, but that this material shows a different mechanical organization according to the type of bone and the mechanical constraints acting upon it. The rather disappointing results obtained from conventional experimental study lead to question whether new techniques might not be able to offer more rigorous experimental conditions based on a model whose features are as close as possible to those of living bone. The use of physicomathematical " m o d e l s " employed in engineering to compute and elaborate complex metal structures or structures made of composite material might offer an answer to this question. Experimentation based on simulation of a physicomathematical model

A study, previously published in this journal (Ferr6 et al. 1982a, 1982b), was based on the simulation of a plane physicomathematical model. This model

48

JC Ferr+ et al. : A physicomathematical approach to the structure of the mandible

was an initial step designed to test our methodology and to ensure that it was truly applicable to anatomical research by confirming or refuting our hypotheses regarding the mechanical structure of the mandible. A more elaborate, three-dimensional model was then developed (Ferr6 1980; Ferr6 et al. 1981) allowing a better demonstration of the phenomena occuring within the mandible when the latter is submitted to mechanical stress (Fig. 2). This model was developed from the computer programs used to calculate and elaborate large metal structures (in this case the structures were off-shore drilling platforms). This series of studies was part of a vast program linking engineering techniques to the study of functional osteology. It was found that the problems raised by these engineering studies were similar to those encountered in functional osteotogy and that the solutions found to solve these problems (as shown further on) were directly applicable to our research. The method to be described herein offers the major advantage of being particularly well adapted to the verification of research hypotheses. Indeed, this technique is focused on saving weight. It should be noted that a good program may allow to save as much as 10,000 tons on a platform weighing 30,000 tons. The problem is similar in functional osteology, where the craniofacial structures display maximum rigidity with a minimum of weight. This method also allows to investigate phenomena for which no experimental material is available, e.g. the effects of an earthquake. Accordingly, it can also be applied to the simulation of fracture. Furthermore, access can be gained to the inner parts of the structure to be studied without modifying it, since the use of stress gauges is not required. When pushed to the limit, the physicomathematical mock-up will of course rupture in a given experiment, but the rupture occurs only on paper. The model can thus be used indefinitely. Finally, this method can be used to solve problems of similitude. Without going into details beyond the scope of this review 1, it should be noted that the solution to such problems of similitude allows to take into account a certain number of factors which could not otherwise be considered. This method presents certain disadvantages I The reader with a special interest in this field should consult: JC Ferr6; La mandibule, une structure/t matbriau composite, /t rev6tement travaillant. A p p r o c h e / t l'6tude de la mandibule ~t l'aide de m6thodes modernes utilis6es en Ingbnieurie. M6moire pour le D.E.R.B.H. Section Craniotogie humaine et compar6e, Amiens (Pr. Laude) 1983

which were overcome owing to the computational power of the techniques used. The major drawback to this method is its cost. Indeed, very complex programs are needed, thus requiring the use of elaborate mainframe computers not generally available (the program used in our study required 300,000 instructions for each case of simulation). A second drawback is that the type of problems analysed by this method requires that the component materials be isotropic, i.e. displaying identical mechanical behavior in all directions of space. As previously pointed out, bone is a viscoelastic anisotropic substance. This problem was soh, ed by replacing the bone by a an isotropic material to obtain a so-called "equivalent" model based on the resolution of a problem of "similitude". This procedure is now possible due to recent progress in computer science and to the development of matrix computation allowing the solution to a great number of simple, repetitive equations. All of these programs are only valid in the domain of elasticity (Fig. 1). Likewise, the difficulties encountered when consideration is given to "fluage" of the material should be underlined, i.e. deformation of the material according to time when a constant force is applied to the material. Fluage is encountered in cartilage and bone. This type of problem must be mastered and adapted to the case under investigation. Computational manipulation is required to achieve this. Obtention of an acceptable level of cost requires simplification of the modelization. Such simplification must be reasonable since any error introduced into the modelization will necessarily be reflected in the results obtained. Unfortunately, such hypotheses of simplification cannot be established on an a priori basis because model equivalence is dependent upon the results to be observed. This notion implies the need to verify that all initial hypotheses are in agreement with the results obtained. If such is not the case the computation must be done again based on new hypotheses (with an obvious increase in cost). It is to be concluded from these comments that the initial hypotheses introduced into the program can only be considered valid when compared to the results obtained, hence the requirement of the extremely well developed mechanism of iteration (Fig. 3). Owing to this mechanism our study of the mandible was possible (in the near future a study of the cranio-facio-mandibular bone will be done), despite the current lack of sufficiently precise data regarding the mechanical attributes of bone.

JC Ferr6 et al. : A physicomathematical approach to the structure of the mandible

49

Fig. 4 An element at rest. (Arrows = nodes) E16ment au repos (fiLches: noeuds)

)

Fig. 2 Three dimensional modelization of the mandible (programs CESAM 69 and Norske Veritas) Mod61isation tridimensionnelle de la mandibule (programmes CESAM 69 et Norske Veritas)

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[

~ )

~modificcltion of the ~. J m o d e [ in the tight of the resul.ts /obtained during the experiment.

Fig. 5 Deformation of an element when external force is applied to

it D6formation d'un 616ment sous l'influence de contraintes ext6rieures ~ lui-mSme

~

The experilm e nt~'-epeat ed ty used... ~

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Fig. 3 The phenomenon of iteration Ph6nom~ne d'it6ration

The importance of the hypotheses and the need for great experience render these programs the exclusive domain of specialists. This observation does not, however, minimize the role of the research worker in anatomy, who must intervene at two levels. The first level of intervention is to formulate the hypotheses to be verified, identify the problems of muscular inertia and environment, assist in the selection of the hypotheses of simplification and to determine the type of experimentation and selection of the mechanical constraints to be applied to the model. The research worker also intervenes in the interpretation of the results. A "spatial" modelization of the mandible (Fig. 2) was thus made and included certain hypotheses to be verified. To achieve the modelization the mandible was broken down into small vol-

umes by the computer (Fig. 4), hence the name "the method finite elements", whose form depends on the program used. A limited number of different forms can be used. Each small element is connected to the other elements by its summits, known as "knots ". The next step is to identify, as a function of the different forces one applies, how the mandible undergoes deformation. One can also identify the type of constraints that will be applied to the mandible to achieve a given deformation, with the restriction that permanent deformation be avoided so as to remain in the zone known as the domain of elasticity, i.e. when the constraint is removed the structure under study returns to its initial state. This situation is an example of two "equivalent" problems, the program allowing easy transfer from one to the other. One can now look for the position of an element as well as all of the deformations (Fig. 5) that occur when external forces are applied to it. Only the displacements of the knots (summits of each element), and not that of the element itself, are calculated according to the mechanical constaints applied. The displacement of each knot requires the computation of 3 angles and 3 vectors

50

JC Ferr6 et al. : A physicomathematical approach to the structure of the mandible

I

.~z

/II

/

/

/ a

/

/

-t )

f

/

/I

I

/

I

i

I I

71

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Fig. 6 Displacement of a knot when submitted to a force external to the element (nodes with 6 degrees of freedom: 3 angles and 3 vectors) D6placement d'un noeud sous l'influence d'une contrainte ext6rieure fi l'61+ment (noeuds/t 6 ~ de libert6 : 3 angles et 3 vecteurs)

(3 coordinates and 3 angles are needed to define the rotation of the lines connected to the knot as well as to identify the displacement of the knot). Each knot is thus said to have 6 degrees of freedom (Fig. 6). Accordingly, in a model comprising say 2,000 knots, the computer must calculate 12,000 values for each system of stress applied to the model. Once the modelization has been done the problem remaining to be solved is to verify t h e " equivalence" of the model, i.e. verify that the model behaves exactly like the mandible. To achieve this verification, one applies mechanical constraints to the model, the results of which are entirely known. If the model shows the same results as those seen in the true mandible the model is said to be equivalent and can be considered reliable. The mandible is particularly appropriate to this type of experimentation. Indeed, the anatomoclinical forms of mandibular fracture are well known. Accordingly, we applied a force of 80 kg to the region of the model corresponding to the tip of the chin. The force was applied at two different angles, one at 45 ~, the other horizontally. These angles were used since they were identical to those described in our initial study of the plane model (Ferr~ et al. 1982a, 1982b). When the 80 kg force was applied at 45 ~ the outer and inner surfaces (Figs 7, 8) of the model displayed zones of maximum isoconstraint in the areas where subcondylar and angular fractures normally occur (it was also noted that these isoconstraints were greater in the zones of subcondylar fracture). Zones of isoconstraints were also seen on the upper and lower surfaces of the model (Fig. 9) corresponding to the sites of paramedian (canine zone) and lateral fractures (premolar zone). When the 80 kg force was applied horizontally, the outer and inner surfaces of the model (Figs. 10, 11) displayed zones of maximum isoconstraints

Fig. 7 Outer aspect of the model. Note the zones of maximum isoconstraints corresponding to the sites of subcondylar and angular fractures (force applied obliquely at 45 ~ Vue externe du mod61e. A noter les zones d'isocontraintes maximales correspondant aux fractures sous-condyliennes et angulaires (cas d'un choc oblique ~ 45 ~

/ Fig. 8 Inner aspect of the model. Note the zones of maximum isoconstraints corresponding to the sites of subcondylar and angular fractures (force applied obliquely at 45 ~) Vue interne du mod61e. A noter les zones d'isocontraintes maximales correspondant aux fractures sous-condyliennes et angulaires (cas d'un choc oblique fi 45 ~

Fig. 9 Overhead view of model. Note the zones of maximum isoconstraints corresponding on the left and right to angular and paramedian fractures, respectively (premolar and canine zones) (oblique force at 45 ~) Vue supbrieure du modble. A noter les zones d'isocontraintes maximales correspondant respectivement de gauche fi droite aux fractures angulaires et param6dianes (zones prbmolaire et canine) (cas d'un choc oblique fi 45 ~

corresponding to the sites where angular fractures occur. A new zone of isoconstraints also arose in the median and paramedian regions, especially on the upper and lower surfaces of the model (Fig. 12).

JC Ferr6 et al. : A physicomathematical approach to the structure of the mandible

\

Fig. 10 Outer aspect of model. Note the lines of maximum isoconstraints corresponding to subcondylar fractures and the absence of these lines in the zone where fractures of the mandibular angle occur (tangential force has been applied to the model) Vue externe du module. A noter l'existence de lignes d'isocontraintes maximales correspondant aux t?actures sous-condyliennes et l'absence de celles-ci dans la zone off se produisent les fractures de l'angle (cas d'un choc tangentiel)

Fig. 1! Inner aspect of model. Note the lines of maximum isoconstraints corresponding to subcondylar and paramedian fractures and the absence of these lines in the zone where fractures of the mandibular angle occur (tangential force has been applied to the model) Vue interne du mod61e. A noter l'existence de lignes d'isocontraintes maximales correspondant aux fraxtures souscondyliennes et param6dianes et l'absence de celles-ci dans la zone oti se produisent Ies fractures de l'angle (cas d'un choc tangentiel)

Fig. 12 Overhead view of model. Note the zones of isoconstraints in the paramedian and median regions (a tangential force has been applied to the model) Vue sup~rieure du mod61e, A noter l'existence de zones d'isocontraintes au niveau des zones param6dianes et m~dianes (cas d'un choc tangentiel)

It can thus be deduced from this study that the model is truly equivalent to the mandible and is reliable, since it showed the same results as those of the mandible when forces with known effects were applied to it.

5t

One can use this model to test hypotheses or theories by incorporating them into the model which would then be submitted to the above described forces. If identical results are obtained the hypotheses or theories would be considered valid; if not, they would o f course be considered invalid. In agreement with Jaeger et al. (I 980), it is our firm belief that the future lies in this type of experimental approach since it is the only method to deal with problems that cannot be solved by a classical experimental approach.

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JC Ferre et al. : A physicomathematical approach to the structure of the mandible

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Koeck B, Sander G (1978) Deformation elastique de la mandibule. Dtsch Zahnarztl Z 33:254-261 Lacroix P (1949) L'organisation des os. Sesoer, Liege Leung YC, Tarriere C, Fayon A, Banzet P (1980) Simulation de Ia face hm~aine st~ module de mannequin. Ann Chir Plast 25 : 311-318 Massa H (1957) Application de la photo+lasticit6 fi t'etude des problemes relatifs ~t la sollicitation de certains os et de certaines dents. Sciences et Lettres ed pp 1-48 Piret N, Dhem A (1975) Le remaniement de l'os compact chez le vieux chien. Acta Anat 93:315 Pope MH, Outwater JO (1974) Mechanical properties of bones as a fonction of position and orientation. J Biomech 7:61 66 Pugh JW, Rose RM, Radin EK (1973) Elastic and viscoelastic properties of trabecular bone: dependance on structure. J Biomech 6:475 485 Reilly DT, Burnstein AH, Frankel VII (1974) The elastic modulus for bone. J Biomech 7:271--275 Robinson M (1946) The temporo-mandibular joint theory of reflex non levers actions of the mandible. J Am Dent Assoc 33:1260 Sonnerbrug M, Hartel J, Hass E (1974) Etude photoelastique des traits de fractures de la mandibule. Zahn Mund Kieferheilk 62:506-512 Vincent J (1955) Recherches sur la constitution de l'os adulte. These Universite de Louvain, Arslia, Bruxelles 153 p