Finite element simulation of bone remodelling in human mandible around osseointegrated dental implant

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1 IOP Conference Seres: Materals Scence and Engneerng Fnte element smulaton of bone remodellng n human mandble around osseontegrated dental mplant To cte ths artcle: Z Q Lan et al 2010 IOP Conf. Ser.: Mater. Sc. Eng Vew the artcle onlne for updates and enhancements. Related content - Computatonal bomechancs of bone's responses to dental prostheses osseontegraton, remodelng and resorpton We L, Chay Rungsyakull, Clarce Feld et al. - Force-nduced bone growth and adaptaton: A system theoretcal approach to understandng bone mechanotransducton Solvey Maldonado and Rolf Fndesen - Study on the relatonshp between stress ntensty factor and J ntegral for mxed mode crack wth arbtrary nclnaton based on SBFEM C L Zhu, J B L, G Ln et al. Ths content was downloaded from IP address on 27/07/2018 at 05:29

2 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) Fnte element smulaton of bone remodellng n human mandble around osseontegrated dental mplant Z Q Lan 1, H Guan 2,4, Y C Loo 2, S Ivanovsk 3 and N W Johnson 3 1 State ey Laboratory of Structural Analyss for Industral Equpment, Department of Engneerng Mechancs, Dalan Unversty of Technology, Dalan, , Chna 2 Grffth School of Engneerng, Grffth Unversty Gold Coast Campus, Queensland 4222, Australa 3 School of Dentstry and Oral Health, Grffth Unversty Gold Coast Campus, Queensland 4222, Australa E-mal: h.guan@grffth.edu.au Abstract. Modern dental mplant s a bocompatble ttanum devce surgcally placed nto a jawbone to support a prosthetc tooth crown n order to replace mssng teeth. Implants are superor to conventonal prostheses, n both functon and long-term predctablty. However, placement of an mplant changes the normal mechancal envronment of jawbone, whch causes the bone densty to redstrbute and adapt to the new envronment through a process of remodellng. Ths study ams to predct the densty dstrbuton n human jawbone around osseontegrated dental mplant. Based on two popular, yet dstnctve theores for bone remodellng, a new remodellng algorthm s proposed. The proposed algorthm s verfed by a two-dmensonal (2D) plate model. Then, a 2D fnte element model of mplant and jawbone s studed. The effects of two parameters, vz the reference value of stran energy densty (SED) and lazy zone regon, on densty dstrbuton, are also examned. Ths study has demonstrated that consderaton of the lazy zone, s less mportant than consderaton of the stress and stran (quantfed as SED) nduced wthn the bone. Takng nto account both lazy zone effect and self-organsatonal control process, the proposed bone remodellng algorthm has overcome the shortcomngs of the two exstng theores. 1. Introducton Development of an deal substtute for mssng teeth has been a major am of dentstry for mllenna. For 100 years, mssng teeth have been replaced by conventonal prostheses (fxed or removable), fabrcated from plastc and/or metal alloys and supported by the remanng teeth and/or the soft tssues. These are often unsatsfactory because of lack of retenton and/or psychologcal nablty to accept removable applances - or because of damage to remanng teeth and/or soft tssues. In these cases, prostheses retaned by an mplant are attractve. Implant bone restoratons have become a clncally and scentfcally accepted treatment modalty over the past 30 years. The realzaton that mplants made of commercally pure ttanum attan drect bone to mplant contact ntated ths revoluton n oral rehabltaton. Ths phenomenon of osseontegraton was frst descrbed by Branemark et al. [1] and Schroeder et al. [2]. 4 To whom any correspondence should be addressed. c 2010 Publshed under lcence by Ltd 1

3 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) As a lvng tssue, bone undergoes contnuous adaptaton and self-repar by resorpton and formaton, and s capable of optmsng ts nternal structure under the nfluence of external loads. Ths self-adaptaton process, or remodellng, has an enormous effect on the overall behavor of the entre bone tssue throughout an ndvdual s lfetme. Followng an early publcaton of Wolff [3], many theores descrbng the bone-adaptaton process have been proposed. Carter [4] suggested that bone s lazy n terms of poor reacton to mechancal sgnals, and there was recognton of the exstence of a lazy zone representng the range of stmulus wthn whch no remodellng occurs. The concept of lazy zone, recognsng that a threshold stmulus needed to be exceeded for bone adaptaton to occur, was ncorporated by Huskes et al. [5] who used the term stran energy densty (SED) to descrbe the stresses and strans that control bone remodellng through bone formaton and resorpton. Ths theory has been developed further by Wenans et al. [6] and Turner et al. [7]. Mullender et al. [8] and Mullender and Huskes [9] suggested a physologcal approach to smulate bone remodellng assumng that osteocytes are senstve to mechancal loadng and are actve n controllng bone adaptaton. Computer smulaton of bone remodellng generally nvolves fnte element analyss for calculatng stresses, strans and SEDs wthn the bone thus determnng how bone adapts tself n response to mechancal stmul. The fnte element method (FEM) was ntally ntroduced to orthopedc bomechancs by Brekelmans et al. [10] and Rybck et al. [11] to evaluate the stresses n human bone. Snce then, the method has been successfully appled for stress analyses of bone and bone-related tssue. The above-mentoned theores of bone remodellng have been used successfully n conjuncton wth the FEM to predct densty dstrbuton n proxmal femur [8,9,12] and bone adaptaton followng hp prosthetc mplantaton [5,13-16]. Whle these theores were essentally developed for orthopedcs, they are generally applcable to jawbone surroundng a dental mplant [17]. However, work n ths area s lmted. Mellal et al. [18] used three dfferent stmul (equvalent stran, equvalent stress and SED) to predct bone remodellng around a dental mplant based on the exstng theores and the results were compared wth n vvo data. L et al. [19] developed a new bone remodellng algorthm by ntroducng an addtonal quadratc term based on the theory of Wenans et al. [6], whch can smulate both underload and overload resorpton. The algorthm was appled n conjuncton wth the FEM to a practcal case of dental mplant treatment. Under steady condtons, no overall changes n bone would be observable due to balance n resorpton and formaton. However when the mplant s placed nto mandble, the mechancal envronment of jawbone changes accordngly, hence the balance of bone resorpton and formaton s no longer mantaned. From a clncal vewpont, better understandng of the way that functonal forces can affect the remodellng of bone s mportant n terms of dentfyng mnmal and maxmal loadng forces requred for the mantenance of stable osseontegraton durng functon, as well as preventng overload that can lead to mplant falure. Ths paper ams to propose a new algorthm for bone remodellng based on the exstng theores of Huskes et al. [5], Wenans et al. [6], Mullender et al. [8] and Mullender and Huskes [9]. The proposed algorthm s verfed by a 2D plate model studed prevously by Wenans et al. [6] and Mullender et al. [8], whch confrms ts accuracy and relablty. To demonstrate the ablty of the proposed algorthm n predctng the densty dstrbuton of bone surroundng a dental mplant, a 2D FE model of mplant and jawbone s studed. The effects of two parameters vz the reference value of SED and lazy zone regon, on densty dstrbuton are also examned and dscussed n some detal. 2. The exstng bone remodellng theores Among a number of bone remodellng theores [5-9,12,13,20-23], two popular yet dstnctve ones are summarsed heren Huskes and Wenans Theory A popular bone remodellng theory suggested by Huskes et al. [5] and Wenans et al. [6] states that: 2

4 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) U U B (1 + s) f > (1 + s) ρ ρ dρ U = 0 f (1 s ) (1 + s ) dt ρ U U B ( 1 s) f < (1 s) ρ ρ (1) where dρ/dt s the rate of change n densty; ρ s the apparent densty of bone tssue; B s a constant regulatng the rate of the remodellng process; s s n percentage denotng the regon of the lazy zone around the threshold value, whch s a reference value for SED (see fgure 1); the lazy zone descrbes the bone not havng a net change n apparent densty, and s defned as U/ρ; U s the SED whch can be expressed as: 1 U = σ ε (2) 2 where σ and ε are respectvely the stress and stran tensors of the bone tssue. For multple-load cases, the SED can be expressed as: U n 1 = U (3) n = 1 where n s the number of load cases and U s the SED under load case. dρ dt Lazy Zone Resorpton (1 s) (1 + s) Formaton U ρ Fgure 1. The assumed local bone adaptaton as a functon of SED wth lazy zone effect Ths algorthm was appled to predct the densty dstrbuton n normal proxmal femur wth results showng a dscontnuous pattern [6]. As dscussed by Rumerman and Huskes [24], only when the densty dstrbutons were locally averaged from dscontnuous densty patterns n the femoral head where the trabecular bone s located, the resultng densty dstrbutons correlate well wth those n a real proxmal femur. Ths algorthm showed an unstable condton n ts mathematcal operaton. Due to the nature of dfferental equatons used n the algorthm to descrbe the adaptve remodellng process, the smulaton produces dscontnuous confguratons, a phenomenon called checker-board Mullender et al. and Mullender and Huskes Theory Not havng ntrnscally dfferent bologcal nature from the theory of Huskes et al. [5] and Wenans et al. [6], Mullender et al. [8] and Mullender and Huskes [9] suggested a new algorthm whch smulated bone remodellng as a self-organsatonal control process. In ther theory, the bone s assumed to have N sensor cells dstrbuted unformly over ts volume. An arbtrary sensor measures a sgnal S, whch s gven as: U S = (4) ρ 3

5 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) where U s the SED and ρ s the densty at the locaton of the sensor. The densty ρ(x,t) at locaton x s regulated by the stmulus value Ф(x,t), to whch all sensor cells contrbute, relatve to ther dstance from x. Hence, (,) x t N f ()( x - ) S (5) Φ = = 1 where N s the total number of sensors; s a reference sgnal of SED and f (x) s a spatal nfluence functon gven as: [ d ( x)/ D] ( x) = e (6) f where d (x) s the dstance between sensor and locaton x, D s the rate of the spatal nfluence reducton. Note that D represents the dstance from a sensor at whch locaton ts effect s reduced to e -1,.e. 36.8%. Ths s llustrated n fgure 2. 10N/mm 2 1 mm Fgure 2. Relatonshp between f(x) and d(x). Fgure 3. 2D plate model smulatng bone t (Carter and Hayes [26], Wenans et al. [6]). The densty ρ(x,t) s now governed by the rate: dρ( x, t) = τφ ( x, t), wth 0 < ρ( x) ρmax (7) dt where ρ max s the maxmum densty of cortcal bone, and τ s a tme constant regulatng the rate of the process. Ths algorthm was used to predct the densty dstrbuton for a 2D plate model prevously studed by Mullender et al. [8]. The model produced trabecular-lke structures wthout the checker-board patterns. It should be noted that the checker-board effect produced by the phenomenologcal models lke the one of Wenans et al. [6] has been completely solved by dfferent technques ncludng extrapolaton to the nodes [23] or usng non-local stmul [8,9]. The checker-board phenomenon s beleved to be attrbutable to the softenng nduced by resorpton rather than remodellng tself. When usng the non-local defnton of the stmulus as a sum of the sgnal of dfferent osteocytes near the pont under concerned, Mullender et al.'s model [8,9] does not produce checker-board patterns. 3. The proposed algorthm The above-mentoned two exstng theores have both shown to exhbt advantages and dsadvantages. Huskes and Wenans s theory takes nto account the lazy zone effect but t s unstable n ts mathematcal operaton leadng to a dscontnuous and an nevtable checker board pattern. Mullender s theory, on the other hand, s physologcally and mechancally more consstent and the checker board phenomena can be effectvely elmnated. However the effect of lazy zone was not consdered. The lazy zone effect, ntally proposed by Carter [4] based on expermental 4

6 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) nvestgaton, has been verfed by Rubn and Lanyon [25] to be an mportant factor whch should be consdered n the smulaton of the remodellng process. To overcome the shortcomngs of the above two theores, a new algorthm combng these two theores s proposed heren. For an arbtrary sensor, and dρ ( x, t) = τφ ( x, t) (8) dt N U ( x) U f (1 ) + s f > (1 + s) = 1 ρ ρ U Φ ( x, t) = 0 f (1 s) (1 s) + ρ, wth 0 < ρ( x) ρmax (9) N U f ( x) U ( 1 s) f < (1 s) = 1 ρ ρ where f (x) s the spatal nfluence functon n the same form as n equaton (6). In fnte element smulaton, t s assumed that each element contans one sensor whch s located at the center of the element. As the apparent densty of bone tssue changes durng the process of remodellng, the Young s modulus changes accordngly. Some emprcal relatonshps between the apparent densty and the mechancal propertes of bone have been establshed expermentally [26-29]. The Young s modulus E at locaton x s calculated accordng to Currey [27] and Rce et al. [29] as E( x, t) = Cρ( x, t) γ (10) where ρ s the apparent densty of bone tssue; C and γ are the constants. In the teratve fnte element analyss process, the governng equaton (8) s necessary to be rewrtten as: ρ ( t+δ t) = ρ () t +ΔtτΦ () t (11) where Δt s the tme step n the teraton process; Ф(t) s n the same form as n equaton (9). The teraton s contnued untl no more sgnfcant changes occur n densty dstrbuton. 4. Verfcaton The proposed algorthm s verfed aganst a 2D plate model (see fgure 3), studed by Wenans et al. [6] and Mullender et al. [8] to smulate the densty dstrbuton wthn a bone tssue. All the parameters are the same as those adopted by Wenans et al. [6] and Mullender et al. [8]. For the purpose of comparson, the value s s set to zero meanng that the effect of lazy zone s not consdered n the analyss. Fgure 4 compares the densty dstrbuton outcomes due to Mullender et al. s [8] theory and the proposed algorthm. The predcted result shown n fgure 4b bears a strong resemblance to that of Mullender et al. (fgure 4a), confrmng the accuracy of the proposed algorthm. To evaluate the effect of lazy zone, a further analyss s carred out by settng s=0.1 whle other parameters reman unchanged. When the lazy zone effect s consdered (fgure 4c, s=0.1), the pattern of densty dstrbuton becomes smoother n shape as compared to that of s=0 (fgure 4b), although the morphology s almost the same. The total area of grey regons (representng cancellous bone) s larger n fgure 4c than n fgure 4b suggestng that less bone adaptaton (vz resorpton and formaton) occurs f the lazy zone effect s consdered. The predcted densty dstrbutons for the plate model agree well wth the Mullender s results, confrmng the relablty of the proposed algorthm when takng nto account both the lazy zone 5

7 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) effect and mathematcal stablty. The proposed algorthm s hence appled to smulate densty dstrbuton wthn a mandbular bone surroundng a dental mplant g/cm g/cm 3 (a) (b) (c) Fgure 4. Comparson of densty dstrbuton: (a) Mullender et al. s [8] predcton wth s=0; (b) predcted result wth s=0; (c) predcted result wth s= Smulaton of densty dstrbuton n mandbular bone Studed heren s a 2D representaton of an mplant and mandble. The modellng scheme follows that of a prevous study [30]. It should be noted that ths s a prelmnary study dealng wth a new remodellng algorthm amed at overcomng the lmtatons of the exstng methodologes. For the purpose of ntal verfcaton, 2D analyss s undertaken to obtan fundamental understandng of the remodellng process. 3D analyss wll be conducted n the future so that detaled mplant geometry can be consdered and more accurate solutons be obtaned. Dfferent types of bone,.e. cortcal and cancellous bone are dentfed n the model based on CT mages. The mplant dmensons shown n fgure 5a are based on those of Neoss mplant (Neoss Lmted, U). The mplant s concal wth 2 degrees of taperage and has a helcal thread. The load and boundary condtons shown n fgure 5b are based on the work of Chou et al. [31] n whch the mplant system s loaded wth an occlusal load of 100N on the crown at an angle of 11 and a unformly dstrbuted pressure of 500kPa on the outer surface of the cortcal bone. The pressure load s appled to smulate more realstcally the boundary condtons. 4.5mm 100N 11 11mm 21mm 500kPa Crown E=172GPa Abutment E=105GPa Abutment Screw E=93GPa Implant E=105GPa Cancellous Bone E=7.93GPa Cortcal Bone E=13.7GPa (a) (b) Fxed Constrant Fgure 5. Fnte element model of mplant and jawbone: (a) dmensons; (b) load, boundary condtons and materal propertes. 6

8 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) An extensve lterature revew by van Staden et al. [32] ndcates that the assumed range of Young s modulus for cortcal bone vares from 5.57 to 22.8GPa and that of cancellous bone from 0.08 to 7.93GPa. Typcal values of Young s modulus (E) of bone and mplant for the analyss are selected based on the work of Papavaslou et al. [33], whch are also detaled n fgure 5b. The Posson s rato for all components s taken as 0.3. The bone tssue and mplant are assumed to be sotropc materals. Note that n ths study a perfect connecton between the mplant and jawbone s assumed. The remodellng s consdered for cancellous bone only wth a unform ntal densty dstrbuton of ρ=1.4 g/cm 3 whch corresponds to E=7.93 GPa. In ths case, a specfc relatonshp between E and densty ρ s chosen based on Carter and Hayes study [26],.e: 3 E x, t) = 3790 (12) ( ρ where x s the locaton of sensors; t s the remodellng tme. The maxmum and mnmum denstes are respectvely ρ max = g/cm 3 and ρ mn =0.064g/cm3 whch corresponds to Young s modulus of 13.7 GPa and 1 kpa. The lazy zone effect s n equaton (9) s taken as 0.1, the constant τ=1(g/cm 3 ) 2 /(MPa tme-unt). The nfluence parameter D=0.25 mm. (a) (b) (c) (d) Fgure 6. Influence of lazy zone effect on the densty dstrbuton of jawbone surroundng dental mplant: (a) s=0.1; (b) s=0.2; (c) s=0.3; (d) s=0.4. To evaluate the effect of lazy zone on the densty dstrbuton, dfferent values of s are consdered wth a constant SED value = J/g. The predcted results are shown n fgure 6. When parameter s ncreases, the regon of lazy zone expands accordngly - hence more bone tssue remans lazy,.e. nether resorpton nor formaton. Ths corresponds to a larger grey regon on the lngual sde of the mplant (fgure 6d). Although more lazy zone s evdent when s ncreases, the overall morphology does not vary sgnfcantly. Dfferent reference values n equaton (9) are also consdered to evaluate the effects of threshold value of SED on densty dstrbuton outcomes. The results shown n fgure 7 ndcate that when ncreases, more bone tssue s resorbed n the surroundng jawbone, especally on the lngual sde of the mplant due to the oblque loadng condton. Fgure 8 shows the relatonshp between the bone mass (for cancellous bone only) and the two parameters s and. When s ncreases by 300%, the mass only ncreases by 2.08%. On the other hand, when ncreases by 16.7%, the mass decreases by 4.5%. It can be deduced that the bone mass ncreases slghtly wth an ncrease n the lazy zone effect s (fgure 8a) and decreases more notceably wth an ncrease n the reference value (fgure 8b). Hence the parameter has more sgnfcant effect than s on bone remodellng outcomes. 7

9 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) (a) (b) (c) (d) (e) Fgure 7. Influence of reference SED value on the densty dstrbuton of jawbone surroundng dental mplant: (a) = J/g; (b) =0.005 J/g; (c) = J/g; (d) = J/g; (e) = J/g. Mass (g) Mass (g) 'Lazy zone' effect s (a) Reference stran energy densty (J/g) Fgure 8. Influence of two parameters on bone mass: (a) lazy zone effect s; (b) reference value. 6. Dscusson The proposed algorthm takes nto account both the lazy zone effect and the self-organsatonal control process. It s shown to be accurate and relable upon verfcaton wth a plate model. The algorthm s then used to predct densty dstrbuton n human jawbone surroundng a dental mplant. The nfluence of two parameters vz the reference SED value and the lazy zone effect s on densty dstrbuton s also examned. The results ndcate that has more sgnfcant nfluence than s. Wth an ncreasng value of, a reducton n bone formaton and an ncrease n bone resorpton are evdent, and vce versa. There s no general rule or crteron n publshed lterature on selecton of the reference value. It s found n ths study that a medan value among all the SEDs s approprate to yeld a satsfactory densty dstrbuton outcome. Fgure 9 compares the predcted densty dstrbuton n cancellous bone surroundng a dental mplant wth a publshed clncal observaton by Watzak et al. [34] and a numercal predcton by Chou et al. [31]. Fgure 9a shows Watzak s clncal observaton of densty dstrbuton on baboons and fgure 9b presents Chou et al. s bone remodellng predcton around a dental mplant. The predcted result (wth = J/g) due to the proposed algorthm s re-produced n fgure 9c. The overall densty dstrbutons are qute smlar for three cases n terms of the trabecular-lke pattern and the blank area below the mplant. Ths s partcularly true when comparng fgure 9b and fgure 9c. Ln et al s [35] computatonal predcton ndcates that the average densty of per-mplant cancellous bone ncreases throughout the remodellng perod, whch are n agreement wth our results. However, due to the smplfcaton of the fnte element model, dscrepances n densty dstrbuton do exst between the present computatonal modellng results and the clncal observaton. (b) 8

10 IOP Conf. Seres: Materals Scence and Engneerng 10 (2010) (a) (b) (c) Fgure 9. Comparson of predcted result wth clncal observatons: (a) densty dstrbuton on baboons [34]; (b) Chou et al. s predcton [31]; (c) predcted result wth = J/g. As far as the mplant/bone connecton s concerned, Roberts [36] found that even clncally successful mplants had less than half of the ntraosseous nterface n drect contact wth bone. Block et al. [37] found smlar levels of osseontegraton for loaded mplants n dogs. In ths study, a perfect connecton between the mplant and jawbone s assumed due to the dffcultes n accessng accurate clncal data on contact percentage. Ths smplfcaton on nterface modellng may have some nfluence on the densty dstrbuton whch should be nvestgated further. It should be noted that ths study s a prelmnary attempt n developng a new bone remodellng algorthm sutable for computer smulaton of the hghly complex processes of bone healng and remodellng. In addton to the assumed perfect mplant/bone connecton, ths study also employs a smplfed and fxed loadng condton. Ths s consderably dfferent from the actual mastcatory forces whch always vary n both magntude and drecton. Hence, a dscrepancy between the resultng densty dstrbuton (fgure 7) and true anatomy of jawbone s expected. Further work should be done to evaluate the effects of other parameters whch may also affect the densty dstrbuton n jawbone surroundng a dental mplant. 7. Concluson A new algorthm for the study of bone remodellng n mandbular bone nduced by dental mplant s proposed n ths study, whch s a combnaton and further development of two exstng and wdely accepted bone remodellng theores. Whlst further optmsaton of the proposed algorthm s desrable, our model has been shown to correlate well wth clncal observatons. An mportant consderaton s that the lazy zone s less mportant than consderaton of the stran energy densty. Improved understandng of the factors whch nfluence bone remodellng around osseontegrated dental mplants s mportant n quantfyng the forces compatble wth mantanng successful osseontegraton and preventng overload leadng to mplant falure. To ths end, the newly developed algorthm ncorporatng elements from prevous theores more closely replcates the clncal stuaton. Acknowledgement The fnancal support from the 2006 Grffth Unversty Research Grant scheme s gratefully acknowledged. 9

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