In the field of dental study it is most fundamental and necessary to estimate the condition of periodontium. In order to examine a mechanical characteristics of periodontium, the theoretical displacement response to periodontal mechanical model (three elements model) are strictly solved in case of some pulse excitations. Impact excitations (rectangular, triangular and half-cycle sine pulse) are given in physical and mathematical definitions and complete solutions to the impact excitations are provided. The triangular pulse excitation which is obtained by means of a fracture of pencil-lead is most suitable. The mechanical parameters of periodontium are given using this input excitation. This is experimentally confirmed by artificial tooth model. The obtained mechanical characteristic of the periodontal tissues can be applied to clinical diagnosis

Isayama, Yoshiharu

Oka, Hisao

Yamamoto, Tatsuma

English

Okayama University Scientific Achievement Repository

Memoirs of the Faculty of Engineering,Okayama University,Vo1.23, No.1, November 1988Transient Displacement Response to Pulse Excitationson Periodontal TissuesHisao OKA*, Tatsuma YAMAMOTO* and Yoshiharu ISAYAMA*(Received September 14, 1988)SYNOPSISIn the field of dental study it is mostfundamental and necessary to estimate thecondition of periodontium. In order to examine amechanical characteristics of periodontium, thetheoretical displacement response to periodontalmechanical model (three elements model) arestrictly solved in case of some pulseexcitations. Impact excitations (rectangular,triangular and half-cycle sine pulse) are givenin physical and mathematical definitions andcomplete solutions to the impact excitations areprovided. The triangular pulse excitation whichis obtained by means of a fracture of pencil-leadis most sui table. The mechanical parameters ofperiodontium are given using this inputexcitation. This is experimentally confirmed byartificial tooth model. The obtained mechanicalcharacteristic of the periodontal tissues can beapplied to clinical diagnosis.* Department of Electrical and Electronic Engineering4950 Hisao OKA. Tatsuma YAMAMOTO and Yoshiharu ISA YAMAINTRODUCTIONFor dental diagnostic purpose, the tooth mobility test is the actof moving a tooth with fingers or tongue blades in order to determinewhether it is loosely or tightly attached to the alveolus. Teethnormally have a certain range of mobility. Pathologic tooth mobilityis recognized as an alarming sign of periodontal disturbance. Severalvariables contribute ~o pathologic mobility. The most common are lossof alveolar bone, alteration of the width of the periodontal ligamentspace or a combination of these factors (1). Therefore the mobilitytest should be used only as a supplementary form of diagnosis.Measurements of tooth mobility have already been fully studied andsome static(2) and sinusoidal dynamic(3) measurements have beenproposed. The forces applied on the tooth, in occlusion ormastication, are of course basically different from static orsinusoidal loads. They should be considered as impulsive loads. Inrecent dental research there are many reports, which deal with thespectra of physical properties caused by impulsive 10ads(4).An observation of the gingiva surface appearance, a measurement ofthe periodontal pocket depth, a radiogr~ph examination, ahistopathological examination etc. are useful for dental diagnosticpurposes. But their examinations demand skill and experience in thedetermination of mobility in general. In recent dental research, thereare many reports which deal with the displacement response toimpulsive loads applied on the tooth (5). This study gives fullconsideration to the relationship between impulsive loads and physicalproperties based on an analysis of the mechanical elements. Thehitherto tooth mechanical model is examined thoroughly and thedisplacement response is obtained theoretically. They areexperimentally confirmed by artificial tooth model.2 PHYSICAL PROPERTIES OF PERIODONTAL TISSUESThe supporting tissues of the teeth are referred to collectively asthe periodontium. These tissues consist of the gingiva, periodontalligament, cementum, alveolus, and supporting bone. Among them, theperiodontal ligament has direct effects upon the tooth mobility. Thischapter explains a structure of periodontium and its dynamicmechanical model.Transient lJisplaHem('nt RcspmJsc 011 j'eriodontaL Tissues 512.1 structure of periodontiumaLveoLarboneperiodontaLmembranedentineFig.1 Structure ofperiodontium(6).mature periodontal ligament (0.1 'V0.3 mm) is organized into theprincipal and secondary fiberbundles. The principal bundlestraverse the periodontal spaceobliquely and insert into thecementum and alveolar bone asSharpey's fibers. The secondary bundles are randomly oriented betweenthe principal bundles. The periodontal ligament and space contain arich network of blood vessels, lymphatics, and nerves(6). The gingivais the mucous membrane covering the alveolar ridge and forming a cuffaround the neck of each tooth.Fig.1 shows a structure ofperiodontium. Periodontium consistsof the gingiva, periodontalligament, cementum, alveolus, andsupporting bone. The cementumcovers a surface of tooth root. TheTeeth normally have a certain range of mobility. The mobility isprincipally in a horizontal direction. It also occurs axially, but toa much lesser degree. Tooth mobility occurs in two stages: (1) Theini tial or intrasocket stages, in which the tooth moves wi thin theconfines of the periodontal ligament. this is associated withviscoelastic distortion of the ligament and redistribution of theperiodontal fluids, interbundle content, and fibers. (2) The secondarystage, which occurs gradually and entails elastic deformation of thealveolar bone in response to increased horizontal forces. The toothitself is also deformed by the impact of a force applied to the crown,but not to a clinically significant degree. When a force such as thatnormally applied to teeth in occlusion is discontinued, the teethreturn to their original positions in two stages: the first is animmediate, spring-like elastic recoil; the second is a slow,asymptomatic recovery movement. The recovery movement is pulsating andis apparently associated with the normal pulsation of the periodontalvessels, which occurs in synchrony with the cardiac cycle. Pathologicmobility is caused by one or more of the following factors(7):(1) Loss of tooth support (bone loss). The amount of tooth movement52 Hisao OKA. Tatsuma YAMAMOTO and Yoshiharu ISAY AMAdepends upon the severity and distribution of the tissue loss onindividual root surface, the length and shape of the roots, and rootsize compared to that of the crown.(2) The extension of inflammation from the gingiva into theperiodontal ligament results in degenerative changes which increasemobility. The changes usually occur in periodontal disease which hasadvanced beyond the early stages, but pathologic mobility is sometimesobserved in severe gingivitis.2.2 Mechanical model of periodontal tissuesThe movement which results in a human incisor tooth to the crown ofwhich a horizontal force is applied may be divided into threecharacteristic phases: the range of initial tooth mobility, ofintermediate tooth mobility, and of terminal tooth mobility(2). Withinthe initial tooth mobility the resistance which the tooth puts upagainst being moved in its socket is very small. If the force isincreased regularly with a dynamometer, it will be noted that with acertain amount of force (around 100 gf) this resistance suddenlyincreases. Far greater forces are now necessary, in order to push thecrown over a distance similar to the phase of initial tooth mobility.Within the range of approximately 100 to 1500 gf force, the increasein motion stays in linear relation to the increase in force. With anincrease in force over 1500 gf (maximal force), pain is registered.When a quasi static horizontal or axial force is removed abruptly fromthe crown, the tooth will in both cases return towards its originalposition in two phases.The first linear phase is an immediate spring-like elastic recoilmovement. The second phase is a slow asymptomatic recovery movement(creep, elastic after-effect). Under normal conditions, the distanceof the recoil movement is proportional (1) to the degree of elasticdeformation of the socket and the surrounding bone and (2) to theelastic tooth distortion produced by tooth loading. If the tooth isloa'ded and unloaded with single horizontal forces different inmagni tude, it will in both cases quickly return in a linear form tothe same position at some distance from the original rest position.After rapid recoil the tooth gradually moves towards its original restposition. The slow recovery phase can be observed after horizontal oraxial recoil movements and is similar. It is advantageously studied bymeasuring probes having no contact with the tooth. In teeth withTYllllSICIll IJisphlllcmnll Rt'spmls/' 011 jlaludOlltal Tissues 53Fig.2 Three elementsmodel (two springsand dashpot).( 1 )xx 1 , x2 andin x(t) iscf}X2normal mobility and loaded horizontallycomplete recovery may last up to 90 seconds.Though a number of mechanical models forperiodontal tissues are feasible, threeelements model shown in Fig.2 is suitable forthe expression of the physicalcharacteristics of periodontium. When theperiodontium is necessarily diseased, thevalues of three elements (two springs anddashpot) change, and then the physiologicalchanges of periodontium would be clear. Adisplacement response of the 3 elements modelto impact load is similar to an actual caseof periodontium. Then the three elementsmodel is proper for a mechanical model ofdisplacement response to impact load. When the applied excitationforce on three elements model is fIt), the displacement is x(t) andthe forces and displacements of the mechanical elements as shown inFig.2 are the following respectively.x(t)=x1 (t)+x2(t)f(t)=cdx2/dt+k2x2' f(t)=k 1x 1If the assumption is that the initial conditions of x,dx/dt are zero at the time t=O, the following equationobtained.X(S)={F(S)/k1}{s+(k1+k2 )/c}/(s+k2/c) (2)Where, XIs) and F(s) are the Laplace transforms of x(t) and fIt)respectively.3 DISPLACEMENT RESPONSES TO SOME PULSE EXCITATIONSThe input excitation fIt) applied on the three elements model isexamined in cases of rectangular, triangular and half sine pulseexcitation.3.1 Rectangular pulse excitationWhen the input excitation is a rectangular pulse fIt) with themagnitude FO' fIt) is a superposition f 1+f 2 of two step functions f 1and f 2 • Where f 1 (t) is FOu(t), f 2 (t) is -f1 (t-d), u(t) is a unit step54 Hisao OKA, Tatsuma YAMAMCYfO and Yoshiharu ISAY AMAfunction and d is a weighting time. The displacement response isx(t)=xl (t)+x2(t) }xl (t)=(FO/k1 ){ (kl+k2)/kr(kl/k2)e-(k2/C)t}u(t)x2 (t)=-x l (t-d)xl(t) and x2(t) are step responses to f 1 and f 2 respectively.3.2 Triangular pulse excitation( 3 )( 4 )When the input excitation is a t:dangular pulse fIt), fIt) is asuperposition f 3 +f 4+f 2 of two ramp functions f 3 (t)=tFOu(t)/d, f 4 (t)=-f 3 (t-d) and a step function f 2 as shown in Fig.3. When fIt) in eqn.lis a triangular pulse, a displacement response x(t) isx(t)= x3(t)+x4(t)+x2(t) }X3(t)=(Fo/kld){(-Ckl/k22)+(kl+k2)/k2t+(Ckl/k22)e-(k2/C)t}u(t)x4(t)=-x3(t-d)x2' x3 and x4 are responses to f 2 , f 3 and f 4 respectively.3.3 Half sine pulse excitationWhen the input excitation is a half sine pulse excitation f(t),fIt) is a superposition f S-f6 'of two sine waves fS(t)=FOu(t) sinyt andf 6 (t)=fS (t-d). When fIt) in eqn.l is a half sine pulse, a displacementresponse x(t) isx(t)= xS(t)+x6(t)xS(t)=(FO/k l ){YQe-(k2/C)t+Msin(yt-~)}u(t)x6(t)=-xS(t-d)where y=TT/dQ=k l C/(k2 2+y2 c 2)} ( S)M={1/(k2 2+C 2y2)} ~ k22(kl+k2)2+c2y2(k22+2klk2+2k22)+c4y4~=tan-l [{k2(kl+k2)+c2y2}/CYkl]3.4 Determination of the mechanical model parameterThe recovery of a tooth on removal of a horizontal load is in twophases. Initially, there is a fast return of the tooth towards itsstarting position (in a linear manner with time). As the toothapproached the starting position, a second and logarithmic phasedeveloped. Fig.4 illustrates the pattern for human maxillary centralincisor following the sudden application and sudden removal of aTnmsicltt jJispfrwonl'IIt Response 011 jJaiot!rmla[ TissllI's 55Thus, onapplying the load thereis an initial rapiddisplacement away fromthe force, followed by aphase of a more gradualdisplacement, and alsothe bi-phasic response onremoval of the load. Theo d tperiodontal tissues tohorizontal loading cannotbe explained in terms ofa simple linearmechanical model. Thepattern shown in Fig.4might suggest aviscoelastic systemcomposed of Maxwell orVoigt elements (springsand dashpots in series orin parallel).Fig.3 Triangular pulse excitation.resistance of the;10..+"c(I)E(I)u'"0.eno(a)time,s(b)5Fig.4 Relationship between horizontaltooth displacement and time followingthe sudden application and removal(8).( 6 )( 7 )}Fig.S shows a theoreti-cal displacement responsein case of rectangularpulse excitation. Fig.6shows in case of triangular pulse excitation. Ri is an elastic recoilmovement. Re is a slow recovery movement. Fig.7 shows a theoreticalresponse in case of half sine pulse excitation and also has a slowrecovery movement at sudden removal of load.When a maximum of displacement is XO' a damping ratio is a and aslope of tangent is 0,Xo=(FO/k1k2d)e-(k2/C){(Ck1/k2)(1-e(k2d/C)+k1de(k2/C)}0=1 /T=k2/cO=FO(k1+k2)/(k1k2d)These mechanical parameters of the damped curve can be obtainedexperimentally. The values of the mechanical elements can be obtainedfrom three parameters of eqn.6.C=(Fo/Xo){e-ad+ad-1}/(a2d)k -ad -ad1={FO(e +ad-1 ) }/{ (e +ad-1 )dO-XOad}56 Hisao OKA. Tatsuma YAMAMOTO and Yoshiharu ISA YAMAk2=(FO/XO){e-Od+Od-1}/(02d)The input excitation, however, must be measured and FO and d must beknown.4 MEASURING SYSTEM FOR IMPACT RESPONSE13 ms10 ms.---1R. 30jJm1ox (t>_""I 1_1009 ffragment of a pencil-lead which isused in the field of acousticemission(9) are used.Fig.8 shows a block diagram of ameasuring system. The displacementtransducer is held 20mm away fromthe surface of tooth. The opticalaxis of transducer is perpendicularto tooth surface and the range ofIn a dental study, the small-sized piezoelectric accelerometerwhich is responsive to mechanical acceleration, is used and adheres tothe labial surface of tooth. A mechanical excitation is applied bymeans of an impact hammer. In this study, an optical displacementtransducer (PT-A165, KEYENCE) whichis out of contact with tooth and a(strain gauge) of pencil-lead isplaced on to pencil tube and theinput excitation force can bedetection is ±2mm. A load-cell Fig.5 Theoretical displacementand time following in case ofrectangular pulse excitation.rt'f"t) U13 msoO~ :10 ms: 30jJmIIX (t) IIIIx (t) :,,,, Re,0 13 ms~'00'ff(t) U---=-0+: :~1:-:0=--m-s---, ': 30jJm,Fig.6 Theoretical displacementand time following in case oftriangular pulse excitation.Fig.7 Theoretical displacementand time following in case ofhalf sine pulse excitation.TnwsicIIl Ihsp/allemclI( Response 011 fJcriodmtlal Tissues 57opticaldisplacementtransducerstrainamplifier- !I~I---'DdisplacementamplifierFFT spectrumanalyzer1==--2=1Fig.8 Block diagram of measuring system.measured. The digital processing of the excitation force anddisplacement response is performed on a FFT spectrum analyzer (SM2701,IWATSU) and then a data analysis is performed on a general-purposecomputer (PC9801VX2, NEC), or the data time-courses are inscribed on arecorder. The experimental results are easily shown on the monitordisplay of the computer, copied and stored on a floppy disk.A triangular pulse excitation is obtained by means of a pencil-leadfragment(10). The pencil-lead is snapped at once on the incisal edgeand it is possible to obtain a relative stable triangular pulse. Theweighting interval d can be sufficiently varied and unloading intervalis about 1 \lS.In the measurement of human teeth in vivo, the subject must sit onthe dentist-chair without a voluntary immobilization of his head, andopen his upper and lower lips using an angle-wider. The weightinginterval of the input excitation using a pencil-lead is so short thatit is possible to measure the input excitation in the maxillary andmandibular periodontal tissues.5 IMPACT RESPONSE OF ARTIFICIAL TOOTH MODELSome tooth models illustrated in Fig.9 are made. The thickness ofsilicone impression material (Flexicon Regular Type) simulatingperiodontal ligament are changed 0.28 'V O. 84mm. The weight of epoxy58 Hisao OKA. Tatsuma YAMAMOTO and Yoshiharo ISA YAMAgypsum888r~--4-siLicone impressionmateriaL+-----epoxy resin50L0time, msE (a) 0.28mm;J. 50...~....c:(l)E(l)uIIIa. 01Il time, ms'U(b) 0.56mm50Lc:::J0time, ms(c> 0.84mmFig.10· Experimental responses totriangular pulse excitation (70gf);(a)O.28mm, (b)O.56mm and (c)O.84mm.Fig.9 Artificial tooth model withsilicone impression material.mathematically. In the usual shockA new impact responsemethod is proposed, whichadopts a relatively longtriangular pulse at thefracture of pencil-lead whichis easily obtained experimen-tally and is easily expressed6 CONCLUSIONresin tooth is O.95g and thelength of tooth root is 15mm.The pencil-lead is fracturedin the lingual direction fromlabial. Table 1 shows the goodlinearity of mechanical modelparameters to magnitude ofexcitation. FO is 56.9 80.7gfand is determined by the typeand protruded length of thepencil-lead (length: 5mm;diameter: O.5mm, degree ofoptical density on paper:HB).The response to triangularpulse excitation (70gf)isshown in Fig.10(a) '" (c). Table2 shows their mechanical modelparameters (average values of6 measurements). (b) and (c)assume pathologic teeth andare characterized by a slopeof tangent D and damping ratioa bigger than those of ahealthy tooth (a). It isadequate that the mechanicalmodel parameters of thepathological teeth are smallerthan those of the healthytooth.Tnmsil'1I1 J)isplallcmnll Respmlsi' rm H'riodmltaL TissuesTable 1 The linearity of mechanical model parametersto input magnitude.input magnitude c k 1 k 2gf Ns/m x10 4N/m x10 4N/m56.9 13.4 21 .7 3.5167.4 12.6 21 .7 3.4780.7 12.7 22.5 4.00Table 2 The mechanical parameters in case of somethickness of silicone impression material;(a)0.28mm,(b)0.56mm and (c)0.84mm.thickness c k1 k 2mm Ns/m x10 4N/m x10 4N/m(a I 0.28 35.1 101 8.41(b) 0.56 1 4 • 1 24.1 4.04(c) 0.84 4.76 6.76 1 .6159measurement of dentistry, a frequency spectrum of the accelerationresponse after input excitation is measured. In that analysis, theinput excitation is not considered and only the acceleration responseis investigated. In this study, the input excitation is strictlydefined physically and mathematically and complete solutions to someof the input excitations are provided for the first time. This methodrequires both the measurements of displacement response and inputexcitation. The model parameters of which the mechanicalcharacteristic of periodontal tissues is expressed, can be obtainedand applied to clinical diagnosis.ACKNOWLEDGMENTS -- H.Oka wishes to thank Professor R.Suzuki of TokyoUniversity for his helpful advice.60 Hisao OKA. Tatsuma YAMAMOTO and Yoshiharu ISA YAMAREFERENCES(l)J.F.Prichard, "Periodontal Disease", Lea&Febiger (1977), 310.(2)H.R.Mlihlemann, O.S.O.M.O.P., !(1951), 1220.(3)D.H.Noyes and C.W.Solt, J.Biomech., ~(1973), 439.(4)B.D'Hoedt, D.Lukas, L.Mlihlbradt, F.Scholz, W.Schulte, F.Quante andA.Topkaya, Dtsch Zahnarztl., Z40(1985), 113.(5)R.S.Manly, J.K.Wiren, K.K.Kapur and E.Loftus, J.Dent.Res., 43(1964), 873.(6)C.L.Halstead, G.G.Blozis, A.J.Drinnan and R.E.Gier, "Physicalevaluation of the dental patient", The C. V.Mosby Company (1982),318.(7)F.A.Carranza, "Glickman's clinical periodontology", W.B. SaundersCompany (1984), 283.(8)B.K.B.Berkovitz, B.J.Moxham and H.N.Newman, "The PeriodontalLigament in Health and Disease", Pergamon Press (1984), 270.(9)N.N.Hsu, J.A.Simmons and S.C.Hardy, Mater.Eval., ~(1977), 100.(10)H.Oka, T.Yamamoto, T.Kawazoe, K.Saratani and Y.Hikida, Med.& Biol.Eng.&Comput., ~ (1988), 260.

Transient Displacement Response to Pulse Excitations on Periodontal Tissues

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