SYMOMP 2013 Lsbon, 9-10 September 2013 EOMAS, Portugal OPTIMIZE THE POSITION OF MEHANIAL OMPONENTS OF AN EXTERNAL FIXATOR USING NEURAL NETWORKS AND GENETI ALGORITHMS Mguel Samarra 1,3, Lus Rosero 2,3 *, Vtor Maranha 2, M. Augusta Neto 1,3, João Alves 2 and arlos Alcoba 2 1: Mechancal Engneerng Department Faculty of Scence and Technology, Unversty of ombra Rua Lus Res Santos, Pólo II 3030 788 ombra - Portugal mcs_samarra@hotmal.com ; augusta.neto@dem.uc.pt 2: Polytechnc Insttute of ombra, ISE, DEM Rua Pedro Nunes Qunta da Nora 3030 199 ombra - Portugal lrosero@sec.pt ; jpalves@sec.pt ; calcoba@sec.pt 3: EMU - entre for Mechanc Engneerng Mechancal Engneerng Department Unversty of ombra Rua Lus Res Santos, Pólo II 3030-788 ombra - Portugal Keywords: Expermental mechancs, structural optmzaton, neural networks, genetc algorthms Abstract The external fxators are used to mmoblze bones n order to allow the fractures to heal. An external fxator s a group of mechancal components, workng together to mantan the stablty and rgdty of the bone structure. Accordng to the dscreton of the surgeon, the nstallaton of the components can vary n dstance and angle between them, enablng the constructon of several dfferent confguratons. The bone healng s senstve to the mechancal stablty of the fxator, and the degree of stablty n the focus of fracture can be related to the type of confguraton used. The am of ths work s to optmze the poston of the pns and the beam of the external fxator n relaton to the fracture focus n order to obtan maxmum (or desred) stffness. An external fxator wth the unlateral-unplanar confguraton was consdered n ths study and an expermental setup have been developed n laboratory condtons n order to smulate a fracture n the daphyss regon and obtan the relaton between appled force, dsplacement and force n the fracture focus and stran n the external fxator. Dfferent postons of the mechancal components have been tested and a long seres of expermental data was obtaned and used to tran an artfcal neural network to model the behavour of the fxator. After that t has been developed a genetc algorthm to determne the optmum poston of the components of the fxator. The results are presented and dscussed.
1. INTRODUTION The bone fracture s a common orthopaedc stuaton consstng n the breakng of the contnuty of bone, whch may be partal or total, ncludng the separaton nto varous fragments. Some fractures can be treated even wthout beng dentfed, whle others requre urgent medcal treatment. Although the healng process of a fracture s a phenomenon wth bologcal features, t s drectly related to the medcal treatment carred out, namely how rapd the bone can heal and return to normalty. One of the human bones wth a greater fracture occurrence s the tba. The healng of tba fractures s strongly nfluenced by the magntude and dstrbuton of mechancal stresses wthn the fracture brdgng tssues, collectvely referred as callus [1, 2]. It s the bone callus that re-establshes ntegrty, contnuty and stffness of the bone member, enablng a return to normalty. Usually, medcal treatment ncludes the fxaton of bone or bone fragments whch can be made by nternal or external mechancal systems. The most approprate method depends on the severty of the fracture and the surgeon's choce, normally connected also wth the means avalable for treatment. An external fxator s a group of mechancal components, workng together to mantan the stablty and rgdty of the bone structure. The fxators are attached to the bone wth lnkng pns. The structure and functon of each external fxator depends, essentally, on the shape of ts components. There are several types of external fxators and each one can be used n a certan type of fracture. Accordng to the dscreton of the surgeon, the nstallaton of the components can vary n dstance and angle between them, enablng the constructon of several dfferent confguratons. onsderng a tba fracture externally fxated, the axal load s shared by the fracture callus and the mechancal devce n proporton to the relatve stffness of the fxator and the callus [2, 3]. Therefore, the healng of bone s senstve to the mechancal stablty of fxators, whch depends not only on the materal and geometrc characterstcs of ts components [4], but also on ther geometrc confguratons [5, 6]. One of the smplest and versatle external fxator s the tubular system of type AO [7]. Ths system s composed of pns, sde beams and elements that assure the connecton between pns and sde beams. The poston of the pns nstallaton can vary n dstance and nclnaton, enablng the possblty to apply dfferent confguratons. There are varous hypotheses about the confguraton of these types of fxators: unplanar and unlateral, unplanar and blateral, bplanar or multplanar. Dependng on the type of tba fracture, the degree of stablty of the focus of fracture s drectly related to the type of confguraton used. In a study done by Emam et al. [8] 68 patents were treated wth unlateral external fxator s fractures and, they conclude that the falure results were probably due to weght-bearng beng too hgh n these patents relatve to the mechancal stablty provded by the external fxator system. In another study done by Epar et al. [9], the authors observed the exstence of a relaton between the stablty of fxaton and the resstance and stffness of the bone callus on the fracture focus that s formed after 9 weeks. Rosero and Neto [10] developed a smplfed fnte element model for the whole tba/external fxator n order to determne the dsplacements at the focus of the fracture. After, they used a genetc algorthm to
determne the poston of the mechancal components of the fxator that mnmzes the dsplacement of the focus of the fracture. In ths study an external fxator type AO wth unplanar and unlateral confguraton s used. The am of the work s to optmze the poston of the pns and the beam of the fxator n relaton to the fracture focus n order to obtan maxmum stffness. An expermental setup have been developed n laboratory condtons n order to smulate a tba wth a fracture n the daphyss regon wth the bone callus formed and obtan the relaton between appled force, dsplacement and force n the fracture focus and stran n the external fxator. The study consders the force appled and how t s dvded between the external fxator and the bone fracture. Dfferent postons of the mechancal components have been tested and a long seres of expermental data was obtaned and used to tran an artfcal neural network to model the behavour of the fxator. After that t has been developed a genetc algorthm to determne the optmum poston of the components of the fxator that allows the better stffness of the group. 2. EXPERIMENTAL METHODOLOGY For the purpose of ths study a tba wth a transverse fracture (90 ) n the central daphyss s consdered. It was also consdered that the bone callus was already formed. The model of the tba has been consdered wth rgd fxaton n the foot connecton and free n the contact area wth the knee, where the loads are appled. A smplfed model of the tba, represented n Fgure 1, was used n the optmzaton procedure. The model has constant cross-secton and the focus of fracture n the central area. In order to obtan a better defnton of the tba model, 10 transversal sectons has been made n dfferent postons and the mean value of Ix and Iy was determned. Accordng to the typcal Young Module of cortcal bone, nylon wth 25 mm of cross secton was selected to model the tba. For the callus, the rubber was selected as the materal wth best approxmaton. d 25 mm E 1.5 MPa I 19175 mm 4 E I I ortcal Bone x y 16 GPa 24556 mm 47074 mm 4 4 d 25 mm E 30 GPa I 19175 mm 4 Fgure 1. Natural and artfcal tba model wth tranversal fracture and bone callus.
In order to obtan the necessary data for the mplementaton of the methodology, a mechancal system was developed and bult to ft the artfcal model of the tba bone and one external fxator type AO wth unplanar and unlateral confguraton. The forces were appled n the top of the system and the data montored wth 3 compresson load cells: 2 from AEP Transducers, type TSTM, wth a maxmum rate of 1 kn and one from Vetek, reference 202WA. The pns were attached to the nylon beam through a rng wth 3 fxng and removable bolts, allowng the adjustment of dfferent postons. The rubber block was attached to the nylon beam at one end and to the load cell at the other end. The force actuaton was made wth a controlled pneumatc cylnder and the data were acqured usng a Natonal Instruments acquston board, reference NI USB-9162, wth a LabVew programmng. Fgure 2 shows the schematc model and the expermental setup used. Fgure 2. Schematc model and expermental setup. The poston of the pns and vertcal beam vares accordng to the lmts showed n the fgure 2. Several combnatons of dscrete postons were selected n order to cover the boundares of search space and expermentally tested. For each combnaton the appled force changes from 10 N to 340 N and the results were saved, to be used n the development of an artfcal neural network. A total of 26765 dfferent combnatons of force and dstances were acqured.
3. NEURAL NETWORKS MODEL It has been recognzed snce early that neural networks offer a number of potental benefts for applcaton n the feld of engneerng, partcularly for pattern recognton problems. Some appealng features of neural networks are ts ablty for learnng through examples, they do not requre any a pror knowledge and can approxmate arbtrary well any non-lnear contnuous functon [11]. Among the several archtectures used n practce, feedforward type neural networks, shown n Fg. 3, have been consdered more sutable for the purposes of the sgnature analyss, the problem under nvestgaton n ths work. Fgure 3. Schematc feedforward neural network. A feedforward neural network conssts on several layers; each one of t wth some processng elements, called neurons, lnked each other by weghts. The weghts determne the nature and the strength of the connecton between the neurons. The number of nodes consdered n the nput and output layers depend on the specfcatons of the problem. The number of hdden layers, the number of neurons n each hdden layer as well as the actvaton functon type for each neuron s selected accordng to the experence and some convergence crterons. The applcaton of artfcal neural network mples the followng two stages: tranng and testng. Durng the tranng stage an nput-to-output mappng, usng the avalable sample data, s present to the network. The network evaluates ts own output based on the presented nput and compares ths value wth the target (presented) output. The actual output error s used to adjust the node weghts so that the error can be reduced. The learnng stage stops once a cross valdaton pre-set error threshold s reached and the node weghts are frozen at ths pont. Durng the testng stage the data, whch have not been presented to the network n the learnng stage, s provded as nput and the correspondng output s calculated usng the fxed node weghts. The feedforward neural network used n ths study has been programed n Matlab and t accounts 3 layers. The frst layer ncludes 4 neurons correspondng to the force ntroduced nto the tba model (P) and to the poston of the 3 structural elements of the fxator (D1, D2, D3). The output layer consders a neuron correspondng to the value of the appled force that s transmtted through the artfcal tba model (F). The hdden layer consders 6 neurons (fgure 4). The normalzaton of the data s assured wthn the network by the Matlab mapmnmax functon and, the weghts and bas are random, n order to optmze ther learnng, whch s also done nternally to acheve computatonal performance. The hyperbolc tangent sgmod transfer functon (equaton 1) s used n all the layers and the tranng of the neural network have been performed wth a second order type algorthm, the Levenberg Marquardt [12].
2 (1) 1 e f ( a) 1 2a A total of 26765 patterns obtaned wth the expermental setup were consderng tranng and performng the neural network. The acqure data was randomly assgned to tranng (70%) and testng (15%). A cross valdaton methodology was also used to stop the tranng process, beng consdered the remanng 15% of data. The tran of the neural network stops after 500 epochs, and the relatve error on the tranng set after the test s 0.393%, showng that the network can model the structural behavour of the expermental setup for the ntended purpose. 4. GENETI ALGORITHM Fgure 4. Neural network consdered n the study. Genetc algorthms (GA s) are search and optmzaton technques nspred by Darwn's theory of natural evoluton [13]. GA s starts wth a populaton of chromosomes each one of them representng a soluton n the search space. Each soluton s evaluated usng a ftness functon whch demonstrates the mert of the respectve ndvduals. Based on the ftness, a random selecton s made and the applcaton of genetc operators s made n order to select new ndvduals and generate tendentously more fttng populatons. When the algorthm attans a pre-establshed crteron, t stops. The GA s have a hgh probablty of tendng to the global mnmum; that s, to the best performance. The rsk of the algorthm beng stuck n a local mnmum s relatvely low f the search s made from a large enough random set of solutons and f the populaton dversty s assured durng the process. Attendng to characterstcs of the problem under study, whch uses 3 real varables, we chose to apply a Floatng Pont Genetc Algorthm, (FPGA), whch decreases the sze of the chromosome and where each one corresponds to a drect varable decson [14]. Ths type of algorthm avods the occurrence of "Hammng lffs" and other phenomena produced by operatng n the bt strng treated as unsgned ntegers n change, requrng a smaller number of generatons to acheve conformty of the populaton. The FPGA algorthm has the man followng steps: 1) ntalze the populaton randomly; 2) evaluate ts ftness; 3) choose the best chromosome among them (eltst model); 4) ross chromosomes randomly selected from the populaton and evaluate them; 5) Introduce mutatons n some chromosomes randomly selected from the populaton and evaluate them; 6) choose a sub-populaton and submts t to
a process of selectng the fttest through stochastc tournament; 7) replace the worst chromosome n the populaton wth the best one; 8) teratng the process untl convergence s attaned. The algorthm can consder dfferent types of crossng and mutaton [14] randomly selected n each generaton. Arthmetc crossng: for all probablty P are created two chld s 1 and 2 from two parents, X and Y usng the followng expressons: 1 2 P. X (1 P). Y P. Y (1 P). X Smple crossng: a random nteger number r s generated from a unform dstrbuton and the parent chromosome s broken at that pont. X, f < r Y, otherwse Heurstc crossng: ths process uses the ftness nformaton of the chromosomes. A lnear extrapolaton s produced between 2 chromosomes and a new one s created accordng to equaton 4 where P=U(0,1) and X have better ftness than Y. If 1 s not admssble another P wll be generated and a new soluton s tested. After n faled attempts (6 n our algorthm), the process stops and the chld s take the same confguraton of the parents. 1 2 X P.( X Y) X U ( a, b) 1, Admssblty 0, f a c b, otherwse Mutaton non-unform: to use the non-unforme mutaton the equaton 5 s appled. If ths operator s used for all the chromosomes the mutaton s desgned as mult-non-unforme. wth: p1, p2 probabltes G actual generaton Gmax maxmum number of generatons b shape parameter x ( b x ). f ( G), f p x ( x a ). f ( G), f p Where: f ( G) p 2 1 G G max 1 1 b < 0,5 0,5 (2) (3) (4) (5)
Boundary mutaton: a varable s selected and transferred to the upper or lower boundary, accordng to equaton 6 and P=U(0,1) a, f P < 0,5 x b, f P 0,5 (6) x, otherwse The parameters used n the genetc algorthm are descrbed n table 1. Genetc Algorthm Parameter Value Dmenson of populaton 80 Number of generatons 500 Arthmetc crossng 12 Smple crossng 4 Heurstc crossng 4 Non-unforme mutatons 5% Mult-non-unforme mutatons 7.5% Boundary mutaton 12.5% Stochastc tournaments 4 Table 1. Parameters of the genetc algorthm. The objectve functon of the genetc algorthm s to mnmze the percentage of the appled force that passes through the artfcal bone (F), e maxmze the rgdty and load capacty of the external fxator. In fact, the smaller force component whch passes through the artfcal bone, the greater rgdty of the external fxator, whch depends on D1, D2 and D3 relatve postons. Thus, the genetc algorthm mnmzes the smulaton results of the neural network defned n secton 3 n order to obtan the optmal dstance for a gven appled force P and accordng to the constrants represented n fgure 2. Mn S. A. 1 2 sm( net, D1, D2, D3, P D D The obtaned results shows that, as expected, the best poston for D3 s as proxmal as possble to the bone (D3=55 mm). For the other 2 varables, the best poston depends on the appled force, but wth low varaton, as shown n fgure 5. For D1 the best poston was 73mm, whch only vares 1 mm between the force lmts (postve slope) and for D2 the best poston was 148mm whch vares 5 mm between the force lmts (negatve slope). For the best poston acheved, the force component n the callus has the mean value of 4.35 N for the forces appled. Fgure 6 shows the evoluton of ths capacty wth the force appled. T ) (7)
Fgure 5. Best poston for varables D 1 and D 2 accordng to appled force. Fgure 6. omponent of the appled force (P) absorbed by callus component (F).
5. ONLUSIONS The poston of the components of an external fxator s usually set by the surgeon accordng to hs experence. Although there are many factors that can determne ther poston based on clncal condtons, a model of optmzaton could allow an ad for a better clncal decson. In ths study we used an expermental model of a tba and an external fxator type AO wth unplanar and unlateral confguraton to smulate the clncal procedure wth a transverse fracture n the central daphyss. The mechancal behavour of the coupled system has been successful acheved wth an artfcal neural network. After that t has been developed a genetc algorthm to determne the optmum poston of the components of the fxator that allows the better stffness of the group tba-external fxator. The obtaned results shows the stablty of the external fxator used and demonstrates that t s possble to move towards the development of an optmzaton tool for postonng the dfferent components of the external fxators. AKNOWLEDGEMENTS Ths research s sponsored by FEDER funds through the program OMPETE Programa Operaconal Factores de ompettvdade and by natonal funds through FT Fundação para a ênca e a Tecnologa, under the project PEst-/EME/UI0285/2011. REFERENES [1] L. E. laes and. A. Hegele. Magntudes of local stress and stran along bony surfaces predct the course and type of fracture healng. Journal of Bomechancs 32 (3), 255-266, 1999. [2] V. Vjayakumar, L. Marks, A. Bremmer-Smth, J. Hardy and T. Gardner. Load transmsson through a healng tbal fracture. lncal Bomechancs 21 (1), 49-53, 2006. [3] T. N. Gardner and M. Evans. Relatve stffness, transverse dsplacement and dynamzaton n comparable external fxators. lncal Bomechancs 7 (4), 231-239, 1992. [4] V. L. aja, W. Km, S. Larsson and E. Y. S. hao. omparson of the mechancal performance of three types of external fxators: lnear, crcular and hybrd. lncal Bomechancs 10 (8), 401-406, 1995. [5] B. T. Brggs and E. Y. S. hao. The mechancal performance of the standard Hoffmann-Vdal external fxaton apparatus. Journal of Bone and Jont Surgery 64 (4), 566-573, 1982. [6] R. Huskes and E. Y. S. hao. Gudelnes for external fxaton frame rgdty and stresses. journal of Orthopaedc Research 4 (1), 68-75, 1986. [7] L. Ramos, I. Rotbande, I. Shehata and I. Knackfuss. ontrbução ao estudo mecânco do fxador externo tubular AO. Rev Bras Ortop 34 (2), 134-138, 1999. [8] A. Emam, B. Mjãberg, G. Karlstrãm and S. Larsson. Treatment of closed tbal shaft fractures wth unlateral external fxaton. Injury 26 (5), 299-303, 1995. [9] D. Epar, J. Kass, H. Schell and G. Duda. Tmely fracture-healng requres
optmzaton of axal fxaton stablty. The Journal of Bone and Jont Surgery 89 (7), 1575-1585, 2007. [10] L. Rosero, L. and A. Neto, - Optmzação da onfguração Unplanar-Unlateral dos Fxadores Externos em Fracturas da Tíba, 4º ongresso Naconal de Bomecânca, L. Rosero, M. Augusta et al. (Eds), ombra, Portugal, 4 e 5 de Feverero, 2011. [11] K. Hornk, H. Stnchcombe and H. Whte, Multlayer Feedforward Networks are Unversal Approxmators, Neural Networks, Vol. 2, 183-192, 1989. [12] M. Hagan and B. Menhaj, Tranng Feedforward Networks wth the Marquardt Algorthm, IEEE Trans. on Neural Networks, 5, 6, 989-993, 1994. [13] D. Goldberg, Genetc Algorthms s search, optmsaton and machne learnng, Addson-Wesley, Readng, M.A., (1989). [14] [A] J. R. Alves,. M. Ferrera, F. P. Barbosa and Isa Qamber, Optmal Power Flow Studes n Electrc Power Systems usng a Floatng Pont Genetc Algorthm, 2nd Internatonal onference on Electrcal Engneerng, ombra, Portugal, 26th-28th November, 2007.