Bone remodelling analysis of the tibia after a total knee arthroplasty

Transcription

1 Bone remodellng analyss of the tba after a total knee arthroplasty Ana Rta Vlhena Tavares Martns Rbero MSc n Bomedcal Engneerng Insttuto Superor Técnco rtarbero@tecnco.ulsboa.pt Abstract: Total Knee Arthroplasty s a surgcal procedure that consttutes one of the soluton treatments for osteoarthrts of the knee. It s an extremely common procedure, wth a very hgh rate of success, especally durng the frst years after mplantaton. However, the man concern regardng ths surgery s the aseptc loosenng of the components, whch can occur later on, due to the reducton n the proxmal tba s bone mass. The tbal components currently used nclude the metallc tbal tray, the polyethylene nsert and the standard stem (polyethylene tp). Addtonally, other components can also be used, such as long stems, that provde better prosthess fxaton and algnment, and metaphyseal sleeves, whch promote the mechancal support when the tbal defects are more sgnfcant. The am of ths study was to analyse the bone remodellng process after a total knee arthroplasty, usng sx dfferent mplant confguratons, whch nclude three dfferent stem lengths, wth or wthout the presence of a metaphyseal sleeve. A threedmensonal model of the tba was obtaned from CT mages and the prosthess components were modelled n Soldworks. The bone remodellng model used maxmzes the bone s structural stffness, takng nto account the metabolc cost assocated wth bone mantenance. The appled loads ncluded the knee jont reacton forces and muscle tendon forces at sx dfferent tme nstances of the walkng gat. The results obtaned after the bone remodellng smulatons were analysed and ther densty dstrbutons were qualtatvely and quanttatvely compared. The results showed that the densty dstrbutons of the mplanted tba depend on the prosthess geometry. The models wth the standard stem showed relatvely low values of bone mass change, whereas long stems lead to bone apposton n the regons surroundng the stem and sgnfcant bone resorpton n the proxmal regons. The presence of the metaphyseal sleeve ncreased the values of bone resorpton. Although short stems may be the better opton for long-term success of the total knee arthroplasty, a longer stem and a metaphyseal sleeve should be used when the clncal status of the patent requres t. Keywords: Knee jont, total knee arthroplasty, fnte element method, stress sheldng, bone remodellng. 1. Introducton The knee jont has an extremely ntrcate three-dmensonal geometry and, consequently, a very complex mechancal response under physologcal loads. The stablty of ths jont s provded by the mensc, lgaments and muscle forces, and t s essental for daly actvtes, such as walkng, runnng, star clmbng, among others (Bendjaballah et al., 1995). The knee s very susceptble to njures, wth osteoarthrts beng the most common pathology that leads to total knee arthroplasty (TKA), a surgcal procedure n whch the damaged cartlage and bone tssues are removed and replaced by artfcal (femoral, tbal and patellar) components. TKA s one of the most common orthopaedc procedures performed and ts number s set to grow by 673% between 2005 and 2030, n the Unted States of Amerca (Kurtz, 2007). Arthroplasty regsters over the years have noted aseptc loosenng as the most common cause of falure. One of the causes of the component loosenng may be the fact that component mplantaton leads to alteratons n the tbal load transfer (Cawley et al., 2012). More specfcally, the load s partally transferred through the prosthess, unloadng the perprosthetc bone and resultng n stress sheldng (Jaroma et al., 2016). Bone loss n the proxmal tba after mplantaton s one of the concerns regardng the TKA prosthess and, therefore, studyng the mechancal behavor of the assembly of the tba wth the mplant s extremely mportant, n order to predct the prosthess performance. That s why many clncal studes have been conducted over the years, n whch dfferent mplantaton technques have been assessed, as well as the nfluence of the mplant desgn and materal on the tbal bone densty. Addtonally, fnte element method (FEM) studes have wdely been used to analyse the bomechancs of the ntact and mplanted bone. By ncludng bone remodellng smulatons n the FEM, t became possble to predct bone adaptaton after a prosthetc mplantaton and consequent loadng varaton (Garcıá et al., 2002). In lght of ths, Cawley et al., (2012) and Chong et al. (2011) both nvestgated bone remodellng n the proxmal tba through computatonal fnte element smulatons, usng dfferent cemented and uncemented tbal fxaton cases. Quílez et al. (2015) take on a wder varety of mplant components, ncludng metaphyseal sleeves, whch are used n revson TKA. Zhang et al. (2016) study the effect of mplant materal, desgn and algnment on the stress sheldng felt n the tba and Ja et al. (2017) test dfferent stem lengths and materals, takng bone remodellng nto account. Ths work focuses on the analyss of the densty dstrbuton of the tba, after beng mplanted wth sx dfferent confguratons of prosthess (three dfferent stem lengths wth and wthout sleeve), usng a bone remodellng model to smulate the dynamc behavour of the bone tssue, and make conclusons about the optmal desgn of the TKA prosthess. 2. Methods and Materals 2.1. Bone Remodellng Model The bone remodellng model used n ths work was developed n 1999 by Fernandes et al. (1999), n whch bone s 1

2 consdered a perodc porous materal, wth varable densty, equvalent lnear elastc propertes and orthotropc symmetry. The perodcty of the bone s obtaned by the repetton of an open unt cell, that mmcs the structure of trabecular bone. The unt cell s cubc, wth rectangular holes wth dmensons a 1, a 2 and a 3. At each pont of the mcrostructure, bone s not only defned by these parameters, but also by the unt cell orentaton gven by the Euler angles θ = {θ 1, θ 2, θ 3 } T. The relatve densty of bone s, therefore, gven by ρ = 1 a 1 a 2 a 1 a 3 a 2 a 3 + 2a 1 a 2 a 3, wth a [0,1], = 1,2,3. The extreme values a = 0 and a = 1 correspond to compact bone and vod, respectvely, whereas ntermedate values correspond to trabecular bone (Fernandes et al., 2002; Folgado et al., 2004; Quental et al., 2014). Mathematcally, the bone remodelng process s addressed as the mnmzaton of the combnaton of the structural complance and the metabolc cost of the organsm when mantanng bone tssue. Consderng that the bone occupes a volume Ω, wth boundary Г and s subjected to a set of load condtons, defned by the surface loads t p,the respectve dsplacement felds u p, and consderng as the desgn varables the parameters a and θ, the optmzaton problem can be stated as: NC mn { a p P=1 ( t p Г u p dг) + k ρ(a) m dω} (1) where NC s the number of load cases wth the respectve load weght factors a p, satsfyng NC P=1 a p = 1. There are two terms n the cost functon (equaton 1): the frst one s a weght average of the structural complance for each load case, whereas the second one represents the metabolc cost assocated wth bone mantenance. More specfcally, the parameters k and m control the total amount of bone mass. The former parameter ncludes bologcal factors, lke age, gender, dsease status and hormonal state, whch vary a lot from subject to subject, whereas the latter s a correctve factor for the preservaton of the ntermedate denstes. In qualtatve terms, the hgher the value of k, the lower the bone mass, because the mantenance cost s hgher. Addtonally, the hgher the value of m, the hgher the emphass on ntermedate denstes (Fernandes et al., 1999; Quental et al., 2014) Geometrc modellng The three-dmensonal model of the rght tba was obtaned from medcal CT mages. The freeware program ITK-SNAP was used to perform mage segmentaton, usng the global thresholdng technque to ncrease the mage contrast, then the actve contour method, n whch the sold model starts beng generated and, fnally, the manual segmentaton, to correct nevtable errors from the prevous step. The segmented model was afterwards mported to another freeware program called MeshLab, responsble for flterng and applyng smoothng operatons, essental for surface mesh adjustments. After that, the model was mported to Soldworks, where the ScanTo3D toolbox was used to automatcally create the sold tba and the ntal surface mesh, whch was the startng pont of the fnte element model creaton n ABAQUS. Ω The prosthess components were also created usng the Soldworks software, takng real models under consderaton. The tbal components typcally nclude a tray, an nsert and a stem, whch can dffer n length (n ths work a standard, a 75 mm and a 115 mm stem were used). In some cases, a metaphyseal sleeve can be added to the prosthess. These components were combned nto sx dfferent models, whch am to mmc the sx dfferent combnatons of prosthess that can be used whether n a prmary TKA or n a revson TKA. The prosthess components can be seen n Fgure 1. A B C Fgure 1 Prosthess components. A. Isometrc vew of the model wth the standard stem. B. Frontal vew of the model wth the standard stem and sleeve. C. 75 mm stem (on the left) and 115 mm stem (on the rght). The sx groups of components were assembled nto the ntact tba. In order to do that, to smulate the proxmal resecton of the damaged tba, a horzontal cut was made wth a heght of, approxmately, 10 mm wth reference to the hgh sde of the tba (Schnurr et al., 2011). In addton, cement was modelled under the tbal tray n all sx models. Fgure 2 contans the sx dfferent models assembled nto the tba Fnte element model As mentoned, the model used consders bone a lnearly elastc orthotropc materal. All the remanng materals were consdered to be homogeneous, sotropc and lnear elastc. Compact bone was attrbuted a Young s modulus of 17 GPa (Quental et al., 2014) and a Posson s coeffcent of 0.3. The tbal tray, made of Cobalt-Chromum was attrbuted a Young s modulus of 210 GPa and a Posson s coeffcent of 0.3 (Kluess et al., 2010). The stems and sleeve, made of ttanum, had a Young s modulus of 110 GPa and a Posson s coeffcent of 0.36 (Darwsh, 2008). The UHMWPE nsert was attrbuted 0.5 GPa and 0.3 as the Young s modulus and the Posson s coeffcent, respectvely (Completo et al., 2008). Fnally, the PMMA cement had a 2.5 GPa Young s modulus and a 0.3 Posson s coeffcent cement (Darwsh, 2008). The nteractons between the bone and the cement, the tray and the tbal nsert, the tray and the cement and the bottom of the tray and the stem were consdered as fully bonded, usng the Te opton (Dassault Systèmes Smula, 2012). However, the nteractons bone-stem, tray-bone and sleeve-bone were all modelled as a contact wth a coeffcent of frcton assocated (0.3, 0.4, 0.3/0.6, respectvely) (Rancourt et al., 1990; Shraz-Adl et al., 1993; Taylor et al., 1995), as well as the Small Sldng opton actvated, meanng that lmted sldng between these components was allowed to a certan extent. 2

3 The dstal end of the tba was fxed and the appled loads were dvded nto two groups: the loads assocated wth the muscle tendons and the jont reacton forces n the tbofemoral artculaton. The muscles consdered n the fnte element analyss were the bceps femors (BF), the semmembranosus (SM) and the TRIPOD (sartorus, gracals and semtendnosus), as well as the patellar tendon (PT). The forces were computed based on the work of Adoun et al. (2012), who performed a fnte element analyss at sx dfferent tme nstances of the walkng gat: 0%, whch corresponds to heel strke, 5%, 25%, 50%, 75% and 100%, whch corresponds to toe-off. These sx tme nstances correspond to the sx steps used n the fnte element analyss. The muscle forces were computed, usng the orentaton of the muscles obtaned from Aalbersberg et al. (2005) and the rato to body weght values obtaned from Adoun et al. (2012). The reacton forces were collected from a publc database ( Database «OrthoLoad ). In ths case, the chosen subject was male and measurements were made durng approxmately four seconds whle he performed the walkng actvty. The sx walkng gat nstances of nterest were dentfed and selected takng nto account the work of Henlen et al. (2009) and were afterwards converted to new values, consderng the average weght for women of 70Kg. The elements chosen for the mesh were tetrahedral and lnear (four-noded). For the model of the ntact tba, the sze of the elements used was of 2mm, whch was shown n a prevous convergence study to be suffcent to obtan the desred soluton, assocated wth a lower computatonal cost than more refned meshes. For the model of the assembly wth the prosthess, the sze of the elements used for the tba was of 1.8mm, 1mm for the cement 2mm for the tray and nsert, 2.2mm for the stem and 2.5mm for the sleeve, when used Valdaton of the smulatons The parameters k and m, whch correspond to the cost of bone mantenance per volume and the correcton factor for the ntermedate denstes, respectvely, were determned from a quanttatve and qualtatve analyss of the bone remodellng model appled to the ntact tba, n whch the CT densty values were used as an ntal condton. To assess the performance of a model, a set of metrcs s often requred. In ths case, four dfferent types of errors were calculated. The root mean square error (RMSE) and the mean absolute error (MAE) have both been used for years to assess model performance. The RMSE s, by defnton, never smaller than MAE, manly because MAE gves the same weght to all errors, whereas the RMSE gves more weght to errors wth larger absolute value. Therefore, RMSE s better at revealng model performance dfferences (Cha et al., 2014). To relate the absolute errors wth the magntude of the CT densty values, the respectve relatve errors of RMSE and MAE were also calculated. Equatons 2 and 3 represent the absolute and relatve dfferences, referred to as ρ a and ρ r, respectvely, and the formulas for the RMSE and the MAE, whch were used for both of these two dfferences, are shown n equatons 4 and 5, respectvely. ρ a = ρ REM ρ r (%) = ρ REM ρ CT, (2) ρ CT, ρ CT n RMSE = 1 n ρ 2, n =1 MAE = 1 n ρ, =1 where n s the total number of nodes of the tba, ρ REM are the bone denstes for node, whch resulted from the bone remodelng smulaton and ρ CT are the bone denstes for node from the CT scan mages (Quental et al., 2014). After determnng the value of the parameters k and m, the bone remodellng model was appled to the models wth mplant. In all sx models, only the part of the bone s subjected to bone remodellng and ts ntal densty dstrbuton corresponds to the soluton obtaned n the smulatons for the model of the ntact tba. However, snce the meshes of the two models are dfferent, a Matlab routne was used to map the denstes of the nodes of (3) (4) (5) Fgure 2 - Isometrc vews of a coronal cut of the sx dfferent fnal assembles. A. Standard stem. B. 75 mm stem. C. 115 mm stem. D. Standard stem wth sleeve. E. 75 mm stem wth sleeve. F. 115 mm stem wth sleeve. 3

4 one mesh to the other Results analyss method To analyse the results obtaned after the bone remodellng process, two dfferent approaches were taken nto account. The frst s a qualtatve approach, n whch fgures are presented, that nclude a two-dmensonal vew of the tba, where the regons of bone equlbrum, apposton and resorpton are depcted. To complement ths approach, a quanttatve one was also used, n whch the tba was dvded nto 15 regons (Fgure 3) and the change n bone mass (n percentage) was calculated for each one of these regons. In order to do ths, seven transversal planes were used to dvde the tba horzontally and an eghth plane was used to dvde t nto the medal and lateral sdes. Regons 1 and 2 are postoned mmedately below the tbal component, regons 5 and 6, 9 and 10 and 13 and 14 surround the standard, 75 mm and 115 mm stems end, respectvely, and regons 3 and 4, 7 and 8 and 11 and 12 are adjacent to the tbal tray, the 75 mm stem and the 115 mm stem, respectvely. The even regons are on the medal sde of the tba, whereas the odd regons are on the lateral sde, except for regon 15, whch ncludes the dstal end of the tba. It s also relevant to menton that regons 5, 6, 9, 10, 13 and 14, whch are all postoned at the tp of one of the three stems, have the same heght. The change n bone mass, computed for each regon, s gven by: m(%) = n v (ρ fnal ρ ntal (6) =1 ) V n v 100, ntal =1 ρ V where n v s the number of nodes n each regon, ρ ntal and ρ fnal are the tba s denstes before and after the smulaton, respectvely, and V s the volume assocated wth each node. 3. Results 3.1. Bone remodellng parameters k and m Several combnatons of values of k and m were tested: m ranged from 2 to 6 and to each of these values, a k between 2, and was assgned. The desred soluton s the one closest to the orgnal bone densty dstrbuton. Only the results wth the parameter m of 4, 5 and 6 are present n Table 1 for the sake of brefness. Lower values of the parameter m, such as m = 2 and m = 3 were excluded from the soluton because, even though the exteror surface of the tba presented hgher values of densty (ndcatng the presence of cortcal bone), the epphyss of the tba had low densty values. Moreover, the medullary canal of the tba s daphyss was practcally nexstent. The results for m = 4 were also excluded from the soluton because when comparng them wth the results for m = 6 (for the same k = ), even though the exteror surface of the tba and ts medullary canal were very smlar n both cases, there were much lower densty values n the tba s epphyss for m=4. Furthermore, by optng for a soluton wth m=6, the quanttatve analyss was beng respected, beng that MAE a was the only metrc for whch the lowest error value dd not correspond to m = 6. Therefore, the fnal values of the parameters m and k chosen for the bone remodellng model smulatons of the assembly of the tba wth the prosthess, were 6 and (Fgure 4), respectvely. By gong for ths soluton, more mportance s beng attrbuted to the hgh densty values n the epphyss, despte the slghtly less defned medullary canal. In addton, the quanttatve analyss s also beng respected, snce the soluton s assocated wth the lowest error value for the RMSE a. Fgure 4 - Densty dstrbuton resultng from the bone remodellng model smulaton, for m=6, k=0,005. A. Global vew B. Anteror vew of a coronal cut. C. Lateral vew of a sagttal cut Densty dstrbutons Fgure 3 - Dvson of the tba nto regons (1-15). Anteror vew of a coronal cut. A. Standard stem. B. 75 mm stem C. 115 mm stem. The results of the bone remodellng for the cases of the standard, 75 mm and 115 mm stems (wthout sleeve) are shown n fgures 5, 7 and 9 for the case of the standard, 75 mm and 115 mm stems (wthout sleeve) and n fgures 6, 8 and 10 for the case of the standard, 75 mm and 115 mm stems (wth sleeve) respectvely. When compared wth the ntal dstrbuton, the 4

5 bone adaptaton was categorzed nto bone apposton, bone resorpton and equlbrum, whch was defned for a percentage of change n bone mass between -4% and 4% (Quental et al., 2013). In Fgure 5, whch shows the model of the standard stem, t s possble to dentfy a decrease n bone mass (BM) n the proxmal regons (1 and 2) mmedately under the tbal tray (ths s more evdent n Fgure 5C, whch s a top vew of the mplanted tba). On the other hand, an ncrease n BM can be observed around the tbal tray and standard stem, n regons 3 to 6. Regons 7 to 15 show no sgnfcant alteratons n BM. In the case of the 75 mm stem, the decrease n BM n the proxmal regons (1 and 2) s more evdent than n the prevous model, as can be seen especally n Fgure 7A and B. Fgure 7 also shows sgnfcant bone formaton n the regons adjacent to the entre length of the stem ts tp. Regons 11 to 15 show no sgnfcant changes. Fgure 9, whch shows the model of the 115 mm stem, ndcates that the loss of BM n the proxmal regons s even hgher, snce the resorpton (blue colour) can now be seen n the begnnng of the tba s daphyss and not just only n ts epphyss and metaphyss (from regons 1 to 6). Bone apposton around the stem s, once agan, notceable (although less evdent than the apposton around the 75 mm stem n Fgure 7), as well as at ts tp. Table 2 contans the percentage of BM varaton for each of the regons, except for the regon 15, whch dd not undergo any sgnfcant changes n any of the models, and the BM change total for both the medal and the lateral sdes of the tba. Addtonally, From these values, t can be seen that the bone loss was hgher on the lateral sde of the tba and also on the proxmal regons of all the models. Moreover, the loss n the proxmal regons ncreases wth the ncrease of the stem length. The same can be sad about the global BM varaton: a small loss was obtaned for the model of the standard stem (-0,26%), a slghtly hgher loss was obtaned for the model of the 75 mm stem (- 0,93%) and a more sgnfcant loss was obtaned for the model of the 115 mm stem (-2,58%). Fgure 6 contans the results for the model of the standard stem wth sleeve. Smlarly to the three models wthout sleeve, a decrease n BM n the proxmal regons can be seen, rght under the tbal tray. Bone formaton can also be seen n the nterface of the tbal tray and sleeve wth the tba. It s possble to observe sgnfcantly more bone resorpton after ncludng the metaphyseal sleeve n the prosthess. Regons 7 to 15 show no sgnfcant alteratons n BM. In Fgure 8, bone resorpton s evdent not only n the proxmal regons but also n the nterface of the sleeve wth the tba, where t s slghtly more substantal than n the model wth the standard stem. Bone formaton around the 75 mm stem and ts tp occurs, as t also dd n the model wthout sleeve (Fgure 7). There s also more bone resorpton happenng near the exteror surface of the tba s daphyss n the model wth the sleeve. Regons 11 to 15 show no sgnfcant changes. In Fgure 10, bone resorpton n the proxmal regons and at the nterface between the sleeve and the tba can be seen as well, and bone densfcaton contnues to occur around the 115 mm stem and at ts tp, although t s less evdent than the bone formaton around the 75 mm stem. Table 3 contans the percentage of BM varaton for each of the regons, except for the regon 15, whch dd not undergo any sgnfcant changes n any of the models, and the BM change Table 1 - Values of RMSE and MAE for both the relatve and absolute dfferences μ (for m rangng from 4 to 6 and all the values of k). The lower value observed for the results of each metrc s marked green. m k RMSE a MAE a RMSE r MAE r 2, ,2162 0,1585 0,3246 0, ,2155 0,1580 0,3252 0,2435 7, ,2021 0,1469 0,3185 0, ,1873 0,1352 0,3187 0, , ,1750 0,3437 0, ,1866 0,1276 0,4093 0,2444 7, ,1950 0,1321 0,4345 0, ,2081 0,1402 0,4706 0,2809 2, ,2398 0,1801 0,3215 0, ,1960 0,1412 0,2935 0,2134 7, ,1861 0,1340 0,2949 0, ,1763 0,1280 0,3057 0, , ,1721 0,1253 0,3457 0, ,1892 0,1327 0,4205 0,2581 7, ,1981 0,1376 0,4459 0, ,2112 0,1455 0,4809 0,2945 2, ,2174 0,1599 0,2895 0, ,1830 0,1332 7, ,1761 0,1290 0,2915 0, ,1266 0,3098 0, , ,1731 0,1289 0,3581 0, ,1936 0,1385 0,4353 0,2734 7, ,2024 0,1432 0,4596 0, ,2150 0,1504 0,4926 0,3068 5

6 Fgure 5 - Bone densty changes of the mplanted tba wth the standard stem, after the bone remodellng smulaton. A. Posteror vew of a coronal cut. B. Anteror vew of a coronal cut. C. Top axal vew of the tba s surface D. Top axal vew of a cut through regons 3 and 4. Fgure 6 - Bone densty changes of the mplanted tba wth the standard stem and sleeve, after the bone remodellng smulaton. A. Posteror vew of a coronal cut. B. Anteror vew of a coronal cut. C. Top axal vew of the tba s surface D. Top axal vew of a cut through regons 3 and 4. Fgure 7 - Bone densty changes of the mplanted tba wth the 75 mm stem, after the bone remodellng smulaton. A. Posteror vew of a coronal cut. B. Anteror vew of a coronal cut. C. Top axal vew of the tba s surface D. Top axal vew of a cut through regons 3 and 4. Fgure 8 - Bone densty changes of the mplanted tba wth the 75 mm stem and sleeve, after the bone remodellng smulaton A. Posteror vew of a coronal cut. B. Anteror vew of a coronal cut. C. Top axal vew of the tba s surface D. Top axal vew of a cut through regons 3 and 4 Fgure 9 - Bone densty changes of the mplanted tba wth the 115 mm stem, after the bone remodellng smulaton. A. Posteror vew of a coronal cut. B. Anteror vew of a coronal cut. C. Top axal vew of the tba s surface D. Top axal vew of a cut through regons 3 and 4. Fgure 10 - Bone densty changes of the mplanted tba wth the 115 mm stem and sleeve, after the bone remodellng smulaton A. Posteror vew of a coronal cut. B. Anteror vew of a coronal cut. C. Top axal vew of the tba s surface D. Top axal vew of a cut through regons 3 and 4 6

7 total for both the medal and the lateral sdes of the tba. From these values, t can be seen that the bone loss was hgher on the medal sde of the tba and also on the proxmal regons of all the models. Moreover, the loss n the proxmal regons ncreases wth the ncrease of the stem length, except for the 115 mm stem, whch showed a slghtly lower value of BM decrease than the 75 mm stem. The proxmal and global BM varaton was much hgher for the models wth sleeves than for the models wthout sleeves: a -1,04% BM change was obtaned for the model of the standard stem, the 75 mm stem model suffered the hghest global bone loss (-3,35%) and the model of the 115 mm had a BM varaton of -2,90%. 4. Dscusson The objectve of ths work was to examne the densty dstrbutons of the mplanted tba after a TKA, usng a bone remodellng model to mmc the behavour of the tssue, and study the dfferences n bone adaptaton between sx dfferent prosthess, the effects of the stem length and the use of the metaphyseal sleeve. Frstly, all the models experenced a global reducton n bone mass (BM). In all the models, the tbal mplant s manly supported by trabecular bone. Accordng to Wolff s law, bone remodellng s affected n response to changes n the appled loads, whch s what happens when an mplant s nserted nto the tba: the load s partally transferred through the prosthess, unloadng the (less stff) tba. Ths phenomenon s called stress sheldng and t translates nto bone resorpton (Jaroma et al., 2016; Zhang et al., 2016). Accordng to Zhang et al. (2016), under the normal physologcal loadng condtons, the stresses are at ther maxmum mmedately under the surface of the jont, whch means that the proxmal regons of the tba are the most affected by the ntroducton of the mplant. Fgures 5 to 10 show bone resorpton n these regons, snce the blue colour s extremely evdent under the top surface of the tba. Addtonally, the values n Tables 2 and 3 confrm that, n fact, regons 1 and 2 of all the models have the most negatve values of percentage of BM change when compared to the values of the other regons for the same model. Regardng the assembles wthout sleeve (Fgures 5, 7 and 9), not only the qualtatve, but also the quanttatve results show an ncrease n bone resorpton n the proxmal regons, that extend to the begnnng of the tba s daphyss when the length of the stem ncreases. Furthermore, the results also ndcate bone formaton n the regons surroundng the three stems. The proxmal bone resorpton and the dstal bone formaton are related, snce the load actng on the tbal tray can be transferred to the bone dstally, along the bone-stem nterfaces. The longer the stem, the hgher the load transfer to the daphyss, resultng n a hgher decrease n BM n the proxmal regons (Cawley et al., 2012; Chong et al., 2011; Munro et al., 2010; Small et al., 2013) and, consequently, n the global tba. On the one hand, the purpose of usng shorter stems s to help centre the mplant n the canal, rather than support the load. On the other hand, n spte of the hgher values of bone resorpton, longer stems are known for provdng addtonal fxaton and ensurng more consstent component algnment. Therefore, they are routnely Table 2- Bone mass varaton, n percentage, for regons 1 to 14 (both from the lateral and medal sdes of the tba) for the cases of the standard, 75 mm and 115 mm stems wthout sleeve, after the bone remodellng smulatons. Lateral sde Medal sde Regons Stem Regons Stem Standard 75 mm 115 mm Standard 75 mm 115 mm 1-3,01-6,18-7,56 2-1,49-2,57-6,03 3 1,29-2,59-5,77 4 2,75 0,59-3,08 5 1,02-0,44-4,09 6 0,79 1,31 0,80 7-0,07 2,18 0,90 8-0,36 2,96 1,32 9-0,14 1,42 1, ,18 2,00 2, ,17 0,13 1, ,11-0,14 1, ,04-0,13 1, , ,54 Sde BM varaton -0,43-1,61-3,06 Sde BM varaton -0,06-0,10-1,98 Table 3 - Bone mass varaton, n percentage, for the regons 1 to 14 (both from the lateral and medal sdes of the tba) for the cases of the standard, 75 mm and 115 mm stems wth sleeve, after the bone remodellng smulatons. Lateral sde Medal sde Regons Stem Regons Stem Standard 75 mm 115 mm Standard 75 mm 115 mm 1-0,24-7,70-5, ,41-13,11-11,79 3 1,55-6,21-4,60 4-0,71-6,42-3,48 5 0,88-2,95-3,62 6 0,99-2,02-1,45 7 0,08 1,71 0,30 8-0,32 2,12 0,73 9-0,08 1,65 1, ,03 2,39 2, ,01 0,21 1, ,25-0,11 1, ,20 0,04 1, ,02 0,33 0,90 Sde BM varaton 0,26-2,72-2,27 Sde BM varaton -2,64-4,15-3,71 7

8 used n revson TKA, because they unload the epphyss of the tba, where the bone qualty s partcularly poor (Barrack et al., 2004; Quílez et al., 2015). However, longer and uncemented stems are commonly assocated wth end-of-stem pan, known to be due to stress concentraton or stress transfer to the tp of the stem (Barrack et al., 1999). Ths stress transfer leads to bone formaton, whch can especally be seen n Fgure 7 and Fgure 9, at the tp of the 75 mm and 115 mm stems. The global quanttatve results were: a small BM loss of 0,26% for the model wth the standard stem; a BM reducton of 0,93% for the model wth the 75 mm stem; and a more sgnfcant BM reducton of 2,58% for the model wth the 115 mm stem. Addtonally, for the proxmal regons, a BM loss of 2,26% was obtaned for the model of the standard stem, a loss of 4,41% for the model of the 75 mm stem and, fnally, a BM reducton of 6,80% for the model of the 115mm stem. These values are wthn the very wde range of results obtaned n clncal studes, n whch the regons of nterest are manly n the epphyss and metaphyss of the tba. Small et al. (2013) regstered a mean reducton of bone mneral densty (BMD) n all regons of nterest of 10,9%, after 10 years, the proxmal regons reachng a loss of 13,9%. Munro et al. (2010) reported an overall BMD loss between 5,7% and 10,5% at 3 years and Petersen et al. (1995) observed a progressve decrease n BMD, as well, reachng 22% at 3 years follow-up. Levtz et al. (1995) reported small varatons durng the frst year after TKA, but overall decrease n BMD of 36,4% at 8 years follow up. Fnte element studes, such as Cawley et al. (2012) and Chong et al. (2011) obtaned a BM reducton of 29% and 11% for surface cementaton models, respectvely. These extremely dvergng values are due to the fact that the studes have dfferent ams and, therefore, use dfferent fxaton technques, prosthess models and materals, and even dfferent ways of analysng the data. In addton, the tba s morphology and bone qualty obvously vares from patent to patent. Regardng the assembles wth sleeve, n all cases, there was a reducton n BM manly n the proxmal regons and there was densfcaton n the daphyss, around the stem and at ts tp. Sleeves are usually a good long-term opton for bologcal fxaton, snce they allow the fllng of severe defects n the tba s epphyss, provdng the necessary mechancal support for the mplant (Quílez et al., 2015). Fgures 6, 8 and 10 and the values n Table 2 show a hgher bone resorpton n the proxmal regons, as well as at the nterface of the sleeve wth the tba, when compared to the assembles wthout sleeve. Ths was somewhat expected, snce the prosthess became a lot stffer wth the ntroducton of the metaphyseal sleeve, leadng to an ncrease n the stress sheldng phenomenon and, consequently, an ncrease n load transfer to the prosthess. Comparng the global BM varaton values of the assembles wth sleeve wth the values of the assembles wthout sleeve n all cases, t s possble to see that the bone resorpton felt n the former assembles s much severe than the bone resorpton felt n any of the latter cases. However, unlke the assembles wthout sleeve, n ths case, the bone resorpton dd not ncrease wth the length of stem. More specfcally, from Fgure 6 to Fgure 8, that s, from the standard stem to the 75 mm stem, t s possble to verfy a sgnfcant ncrease n bone resorpton. Nevertheless, Fgure 10,.e. the model wth the 115 mm stem, does not show a hgher bone resorpton than the model wth the 75 mm stem, as would have been expected accordng to Quílez et al. (2015), whch s confrmed by the values n Table 3. Addtonally, Quílez et al. (2015) regstered the lowest bone resorpton values for the prosthess wth sleeve and no stem. Even though the global BM varaton value obtaned n ths work was not lower than the ones obtaned for the prosthess wth the standard and 75 mm stems wthout sleeve, t was, n fact, lower than the BM varaton obtaned for the 115 mm stem wthout sleeve. Although some studes have defended that when usng a sleeve, a longer stem should always be used (Hadukewych & Servce, 2012), the combnaton of a sleeve wth a standard stem has been growng over the last years and t has proven useful when a better metaphyseal fxaton (and not daphyseal) of the mplant s needed. Ths prosthess effectvely flls metaphyseal defects, whle beng consderably smple to place wth freedom of rotaton, wthout exceedng the tbal plate and wthout the need for cement, regardless the daphyseal deformty. By not usng a long stem, there s no need to prepare the canal, reducng the rsk of ntraoperatve fssures. The global quanttatve results for the assembles wth sleeve were: a BM reducton of 1,04% for the model wth the standard stem; a BM reducton of 3,35% for the model wth the 75 mm stem; and a reducton of 2,90% for the model wth the 115 mm stem. Addtonally, for the proxmal regons, a BM loss of 5,19% was obtaned for the model of the standard stem, a loss of 10,33% for the model of the 75 mm stem and, fnally, a BM reducton of 8,67% for the model of the 115mm stem. These values are, once agan, wthn the very wde range of results obtaned n the clncal studes already brought up, though to the author s knowledge there are no clncal or fnte element studes that tested the BMD varaton when usng a metaphyseal sleeve. Accordng to several studes (Cawley et al., 2012; Chong et al., 2011; Ja et al., 2017; Small et al., 2013; Zhang et al., 2016) the medal sde of the tba was expected to experence a larger amount of bone resorpton, manly due to the larger porton of the jont load actng on ths sde, whch after the TKA starts beng shelded by the metallc prosthess. Table 2 ndcates that, the assembles wthout sleeve experenced the opposte of what was expected regardng the medal and the lateral sdes, snce the latter was the one that regstered hgher values of bone resorpton. On the other hand Table 3 shows that the assembles wth sleeve ndeed suffer from hgher bone resorpton on the medal sde than on the lateral sde. These dssmlar results may be due to the few lmtatons of ths work, whch are carefully descrbed further on. To summarze, the results confrmed the occurrence of stress sheldng after prosthess mplantaton, beng that n all cases the largest amount of bone resorpton was observed n the proxmal regons of the tba, under the tbal tray component. Global BM varaton ncreased when the length of the stem ncreased for the assembles wthout sleeve. For the assembles wth sleeve, the model wth the 75 mm showed hgher bone resorpton values than the model wth the 115 mm stem. Bone apposton occurred at the tp of the 75 mm and 115 mm stems n both assembles, due to stress concentraton. The assembles wth sleeves regstered hgher values of bone resorpton than all 8

9 the assembles wthout sleeves, ndcatng that the sleeve ncreased the stress sheldng phenomenon. The assembly of the standard stem wth sleeve suffered less bone resorpton than the assembly of the 115 mm stem wthout sleeve. Fnally, the assembles wthout sleeve had more bone resorpton on the lateral sde, whereas the assembles wth sleeve had more bone resorpton on the medal sde. A part of the results goes n accordance wth the lterature, however some of them do not. Therefore, the lmtatons of ths work must be addressed. Frstly, all of the smulatons were performed on a sngle geometry of the tba and the prosthess materal propertes were always the same. Therefore, the varablty n tba morphology, mplants and materal propertes could not be tested. The appled loads were underestmated n some regons of the tba, because of the lack of bologcal condtons. There s not a sngle study that ncludes all of the jont reacton forces, as well as the load components of all of the muscle nsertons. - In ths case, not all of the muscle nsertons were ncluded, nor was the artcular cartlage, the mensc or the lgaments, whch may explan a few bone resorpton areas on the exteror surface of the tba. Due to the modelled geometry of the 75 mm and the 115 mm stems, there was a small space between them and the wall of the tba s daphyss, whch can also nfluence the densty dstrbuton of the bone surroundng the stem. Lastly, there could have been user postonng errors when nsertng the prosthess nto the tba and, accordng to Zhang et al. (2016), tbal malalgnment can be assocated wth abnormal medal/lateral force dstrbuton, whch may explan the unusual hgher bone resorpton n the lateral sde of the tba for the assembles wthout sleeve. The same study reported that ths component malalgnment also seemed to reduce stress sheldng n the bone, whch may explan the lower bone resorpton values felt n the model wth the 115 mm stem wth sleeve when compared to the model of the 75 mm stem wth sleeve. All n all, despte the lmtatons assocated wth ths study, the results obtaned ncluded the stress sheldng phenomenon and local stress concentratons, as reported n the lterature. In addton, the knowledge regardng the mechancal behavour and bone adaptaton after the nserton of the prosthess s of the utmost mportance when choosng a certan fxaton technque or prosthess desgn or materal. The bone qualty vares from patent to patent and so the surgeons performng TKA should be able to antcpate the bone changes and make approprate decsons to accommodate for the nevtable loss of bone. 5. Conclusons and future work In ths work, the bone remodellng of the tba after a TKA was analysed, usng the fnte element method and a bone remodellng model frst developed by Fernandes et al. (1999). The goal was to evaluate the bone adaptaton when n the presence of sx dfferent confguratons of prosthess, whch nclude three dfferent stem lengths, wth or wthout the presence of a metaphyseal sleeve. The results obtaned n ths study show that the bone densty dstrbuton after a TKA s clearly dependent on the prosthess geometry. Unlke the unstemmed mplants, whch dd not experence sgnfcant bone loss, the assembles wth 75 mm or 115 mm stems caused more substantal values of BM varaton. Ths suggested that the load transfer to the cancellous bone surroundng the dstal stem, that resulted n bone apposton around t, lead to bone resorpton n the proxmal regons, whch were load deprved. The BM changes n the tba nfluence the outcome of the TKA,.e. patents wth a hgh postoperatve bone loss are expected to have an ncreased rsk of fractures or loosenng of the tbal component, whch s the man cause of falure of the TKA (Petersen et al., 1995). The assembles wth the 75 mm or 115 mm stems also presented bone apposton at the tp, confrmng the load transfer capablty of the stem and ndcatng stress concentraton. The addton of the metaphyseal sleeve to the prosthess resulted n an overall BM decrease n all assembles, due to the ncrease of the stress sheldng phenomenon, stll they are known for provdng mechancal support when the metaphyseal defects of the tba are severe. It can be observed that the two assembles wth the standard stem are assocated wth relatvely low values of BM change, makng t seem lke the best opton for a long-term successful TKA. Nonetheless, the prorty when choosng the prosthess components s not ther bomechancal effect on the tba, but the patent s clncal status, whch ncludes age, level of actvty, daphyseal or metaphyseal defects (qualty of the bone), bone geometry, surgeon s experence and surgcal technque used. To complement ths work, further studes wth dfferent methodologes are needed n order to analyse the bone adaptaton process after a TKA. For nstance, all of the assembles used n ths work were developed usng a hybrd cementng fxaton, meanng that a layer of PMMA was postoned beneath the tbal tray, but not around the stem (Chong et al., 2011). Other fxaton technques, such as a fully cemented or a cementless case, could be modelled, although several studes have reported that both cemented and uncemented fxaton methods are both equally vable (Small et al., 2013). Secondly, another suggeston for future studes would be to nclude the lgaments n the fnte element method. The change n the appled loads would consequently change the stmulus to the bone, possbly nfluencng the densty dstrbutons of the tba after the bone remodellng smulatons. Another opton would be to model a dfferent prosthess desgn or assgn dfferent materals to the components. Ja et al. (2017), besdes ttanum and cobaltchromum alloys, also used functonally graded materals (FGM s), defned as nhomogeneous materals, consstng of two (or more) dfferent materals, engneered to have a contnuously varyng spatal compostons profle (Udupa et al., 2014), and concluded that the effects of the materals had a much more pronounced effect on the tba s stress sheldng than the stem length. Fnally, the use of a greater number of tbas would also ncrease the credblty of the results. References Aalbersberg, S., Kngma, I., Ronsky, J. L., Frayne, R., & Van Deën, J. H. (2005). Orentaton of tendons n vvo wth actve and passve knee muscles. Journal of Bomechancs, 38(9), Adoun, M., Shraz-Adl, A., & Shraz, R. (2012). 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