Simulation of Surface Mesh Deformation in Orthodontics by Mass-Spring Model

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1 292 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.2 August 2011 Smulaton of Surface Mesh Deformaton n Orthodontcs by Mass-Sprng Model Pathomphong Phannurat 1, Wcht Tharanon 2, and Chanjra Snthanayothn 3, Non-members ABSTRACT The mass-sprng model has been used to descrbe elastcally deformable models such as skn, textles, and soft tssue n computer graphcs. A mass-sprng mesh s composed of a network of masses and sprngs, n whch each edge s a sprng. We apply the masssprng system to mesh deformaton n 3D orthodontc smulaton, the movement of whch s evaluated usng the numercal ntegraton of the fundamental law of dynamcs based on the 4th-order Runge-Kutta method. Computatonal quantty and accuracy s demonstrated on test and dental cast model examples. The expermental results show that t can smulate the deformaton change n real tme and dsplay the results vvdly. Keywords: Mass-Sprng Model, Model Deformaton, Orthodontc Smulaton, STL Dental Cast Model 1. INTRODUCTION Tradtonally, orthodontsts deal wth correctng maloccluson n young patents. The typcal orthodontc patent s 12 to 14 years old whle adults seekng orthodontc treatment for aesthetc reasons alone s very rare. However, t s becomng common nowadays for appearance-conscous adults to place great personal value on the appearance of ther smle, and cosmetc dentstry has become ncreasngly popular among older patents [1]. Orthodontsts use cephalometrc projectons to plan ther treatments [2]. Cephalometrc projectons are x-rays taken of the sde of head. However, t s not convenent to store and compare orthodontc patent records of each patent for each treatment perod. Therefore, computer technology s used n orthodontcs to smulate treatment changes n a 3D space. Soft tssue holds a large proporton of the whole ndvdual structure, ncludng a great deal of mportant organs, for example, the heart, skn, muscles and Manuscrpt receved on January 4, 2009 ; revsed on January 30, ,2 The authors are wth Advanced Dental Technology Center, Thaland, E-mal:pathomphong@nstda.or.th and wthara@yahoo.com 3 The author s wth Natonal Electroncs and Computer Technology Center, Thaland, E-mal: cephsmle@gmal.com so on. The smulaton of soft tssue deformaton relates to the felds of medcne, mechancs, bology, computer graphcs, and robot vson. In addton, the smulaton of soft tssue deformaton s wdely used n the medcnal smulaton system ncludng plastc and musculoskeletal surgery. Modelng of deformable object has two models whch are consdered the most popular methods. The methods are Fnte Element Method (FEM) and Mass-Sprng. However, an deal soft tssue deformng smulaton s stll a challengng task due to the complex nternal structure and surface appearance of the deformable objects. FEM s the most accurate method for smulaton, but t hardly satsfes the real-tme requrement because of ts hgh complexty and large numbers of parameter defnton. Mass-Sprng model could have acceptable response-tme but results n low accuracy. Ths paper develops models of gngval tssue deformaton n orthodontcs based on Mass-Sprng model because we need smulate deformaton of tssue n real-tme. By smulatng physcal propertes such as tenson and rgdty, we can model statc shapes exhbted by a wde range of deformaton objects. Furthermore, by ncludng physcal propertes such as mass and dampng, we can smulate the dynamcs of these objects. The smulaton nvolves numercally solvng the partal dfferental equatons that govern the evolvng shape of the deformable object and ts moton through space. Solvng partal dfferental equatons s mportant to descrbe the behavor of gngval deformaton brought about by tooth movement n the process of rectfcaton that can be smulated by computer. Gngval deformaton s the change of gngval shape (soft tssue) caused by the change of tooth (rgd body) poston under external force n the process of orthodontc treatment. Realstc smulaton of gngval tssue deformaton requres an approprate physcs-based model of soft tssue deformaton. Secton 2 descrbes our model for tssue deformaton. The model s composed of a network of masses and sprngs, whch can be consdered as a varant of elastc models. We gve dfferental equatons of moton descrbng the dynamcs behavor of deformable models under the nfluence of external forces. Secton 3 presents mplementaton mass-sprng model to smulate soft tssue deformaton. We end wth our results, dscusson and conclusons.

Smulaton of Surface Mesh Deformaton n Orthodontcs by Mass-Sprng Model 293 2. MASS-SPRING MODEL 2.

2 Smulaton of Surface Mesh Deformaton n Orthodontcs by Mass-Sprng Model MASS-SPRING MODEL 2. 1 Mesh Representaton In our model, elastcally soft tssue s represented by a trangle mesh of m masses, each mass beng lnked to ts neghbors by mass sprngs of natural length not equal to zero (Fgure 1). Each sprng s natural length s the dstance between masses n the rest poston Sprng Dampng The queston of how to assgn dfferent dampng (c) values to the varous sprngs n a mass-sprng system has been largely gnored n the lterature. Tradtonally, c s treated as a constant throughout the system. We performed the same smulaton as before usng (1) and (2) to calculate m and k. To smulate the best behavor of our models, we compute sprng dampng (c) of each sprng e followed by equaton (3) [4]. c j = 2 k(m + m j ) l j (3) c j s sprng dampng of the sprng lnkng pont and pont j l j s the length between pont and pont j Fg.1: model 2. 2 Pont Mass Mesh of masses and sprngs used for our In our mass-sprng models, masses are allocated at the vertces of the trangle mesh and damped sprngs along the edges. To accurately dstrbute the total mass of the mesh, we compute the mass m of each vertex accordng to the area A N(), where N() s the set of ndces of trangles adjacent to vertex. If D s the materal densty [4], then: 2. 3 Sprng Stffness m = D A N() 3 (1) The easest way to calculate sprng stffness s to use a constant value for the stffness (k). More commonly, k s computed as k = 1/L, where L s the length of the sprng at rest. In [3], Van Gelder suggested a formula to compute sprng stffness for a 3D mesh that s the closest to elastc contnuous representaton. Let E be the Young s modulus, then: k j = E A N() l j 2 (2) k j s the stffness of the sprng lnkng pont and pont j l j s the length between pont and pont j 2. 5 Appled Forces The system under study s the trangle mesh of the m masses, each mass beng postoned at tme t on the pont, where = 1,..., m. The poston of each pont, at any tme t, can be derved through the fundamental law of dynamcs. The nternal forces are due to the tensons of the sprngs [5, 6]. The overall nternal force appled at the pont at tme t s a result of the stffness of all the sprngs lnkng ths pont to ts neghbors: F nt (t) = ( k j P j (t) lj 0 j R ) P j (t) P j (t) (4) R s the set of all couples (j) such as pont j s lnked by a sprng to pont, P j (t) s poston of the sprng lnkng pont and pont j whch P j (t)=p (t)-p j (t), P (t) s poston of pont at tme t, lj 0 s the natural length of the sprng lnkng pont and pont j whch lj 0 = P j(0), k j s the stffness of the sprng lnkng pont and pont j The external forces (F ext ) are of varous natures accordng to the load to whch we wsh the model to be exposed. Omnpresent loads wll nclude gravty, a vscous dampng and a vscous nteracton wth an ar stream (or wnd), but we only appled dampng forces (F ds ) at tme t to smulate our models. The vscous dampng wll be gven by: F ext = F ds (t) = c v (t) (5) c s all dampng coeffcents that are computed by equaton (3) c = j R c j,

294 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.2 August 2011 v (t) s ts velocty of pont at tme t whch can be computed followng equaton (6).

3 294 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.2 August 2011 v (t) s ts velocty of pont at tme t whch can be computed followng equaton (6). The role of ths dampng s n fact to model the frst approxmate dsspaton of the mechancal energy of our model. 3. IMPLEMENTATION In orthodontc treatment smulaton, we can extract teeth structure from the gngval surface before teeth algnment. To analyze gngval surface deformaton, a deformable model (gngval surface) must be created by anmaton and the dfferental equatons of moton are smulated numercally. The frst step s defnng the pont force at a pont on the gngval surface whch s the appled force when teeth are algned. Fnally, the new poston of ponts at tme are computed by solvng the dfferental equatons Pont Force Pont forces are appled forces when teeth are moved. We can defne pont forces wth boundary ponts of gngval surface connectng to the moved tooth (Fgure 2). Fg.2: Pont Forces (balls) are contrbuted on the gngval surface boundary connectng to the moved tooth Gngval Surface Motvaton The movement of gngval surface s caused by forces of teeth movement appled to each pont of the model. The poston of each pont, at any tme t, can be derved through Newton s second law of moton. All the above formulatons make t possble to compute the force F (t) appled on pont P at any tme t. The fundamental equaton of dynamcs can therefore be explctly ntegrated through tme by the smple Euler s method [5,6]: m 2 P t t 2 a (t) = F nt = F nt v (t) = P (t) t (t) + F ext (t) (t) (t) + F ext m = F nt (t) c v m (6) We can solve moton equaton 6 to evaluate a new poston at any tme P (t). The postons and veloctes of the ponts are computed based on 4 th order Runge-Kutta method as follows: r 1 = hf(p 0, t 0 ) r 2 = hf(p 0 + k 1 /2, t 0 + h/2) r 3 = hf(p 0 + k 1 /2, t 0 + h/2) r 4 = hf(p 0 + k 3, t 0 + h) P (t 0 + h) = P 0 + (r 1 + 2r 2 + 2r 3 + r 4 )/6 r are Runge-Kutta functons, P (t) s poston of pont at any tme t, f(p, t) are the forces at pont at any tme t, h s the mnmum tme step. (7) 4. EXPERIMENTAL RESULTS AND DIS- CUSSION The algorthm of the technque was mplemented under Wndows wth Borland C++ Bulder, usng the OpenGL Lbrary for renderng the 3D mages. A set of patent s pretreatment models from the Orthodontc Clnc, Advanced Dental Technology Center, was selected. Two dfferent sets were used durng the experments - a mandble model (Fgure 3) and a maxlla model (Fgure 5). The trangle meshes of two models were generated from the CT scanner. Frst step, we extract a tooth from gngval surface and then we move the tooth to smulate the gngval deformaton. The smulaton results of the gngval deformaton are shown n the followng fgures. Fgures 4 and 6 are the deformaton smulaton result for translaton and rotaton of tooth by arrow drecton, respectvely. Iteratons of each deformable model are shown by Fgures 7 and 8. Lastly, the smulated tmes that nclude the processng and renderng tmes are shown n Table 1. After smulaton at 100 teratons, large deformaton appears near the pont forces, and constrants are sometme not met. But after smulaton at 500 teratons, we ensure both a more natural look of the gngval surface and the enforcement of constrants. The smulaton takes rather a long tme snce the 4th order Runge-Kutta method requres many teratons n order to acheve the fnal result. 5. CONCLUSION The soft tssue deformaton needs to be consstent wth the physcal propertes. In ths paper, we apply

Smulaton of Surface Mesh Deformaton n Orthodontcs by Mass-Sprng Model 295 Table 1: Smulaton Tme Num. of Frame Rate Num. of Trangles (msec.) Frames/sec 29108 0.56 1769 26198 0.53 1904 23286 0.

The process of dynamc deformaton can be descrbed effectvely after the tme varable s ntroduced.

The expermental results show that ths method satsfes the demand for the computatonal real-tme and the gngval deformaton 3D n vrtual orthodontcs. Fg.4: Gngval deformaton of mandble model.

[2] Chanjra Snthanayothn, Medcal Imagng Aded Orthodontcs, In Proc. TMI2005, 2005. [3] Van Gelder, Approxmate smulaton of elastc membranes by trangulated sprng meshes, Journal of Graphcs Tools, pp.

4 Smulaton of Surface Mesh Deformaton n Orthodontcs by Mass-Sprng Model 295 Table 1: Smulaton Tme Num. of Frame Rate Num. of Trangles (msec.) Frames/sec Fg.3: Orgnal mandble model. mass-sprng system to soft tssue deformaton n 3D orthodontc smulaton. The process of dynamc deformaton can be descrbed effectvely after the tme varable s ntroduced. The dynamc moton rule adopts the dfferental equaton form; the numercal method can be carred out for the real-tme computaton of the dynamc system. The expermental results show that ths method satsfes the demand for the computatonal real-tme and the gngval deformaton 3D n vrtual orthodontcs. Fg.4: Gngval deformaton of mandble model. References [1] Robert G.Kem, Aesthetcs n clncal orthodontc perodontc nteractons, Perodontology 2000, Munksgaard, Denmark, pp , [2] Chanjra Snthanayothn, Medcal Imagng Aded Orthodontcs, In Proc. TMI2005, [3] Van Gelder, Approxmate smulaton of elastc membranes by trangulated sprng meshes, Journal of Graphcs Tools, pp , [4] Paloc C., Bello F., Ktney R.I., and Darz A., Onlne Multresoluton Volumetrc Mass Sprng Model for Real TmeSoft Tssue Deformaton, MACCAI 2002, Sprnger-Verlag, Berln, pp , [5] Provot X., Deformaton Constrants n a Mass- Sprng Model to Descrbe Rgd Cloth Behavor, In Proc. of Graphcs Interface, pp , [6] Metaaphanon, N. and Kanongchayos, P., Realtme cloth smulaton for garment CAD. Proc. 3rd Int. Conf. Computer Graphcs Interactve Technques n Aust. SEA., Dunedn, New Zealand, Fg.5: Orgnal maxlla model. Fg.6: Gngval deformaton of maxlla model.

296 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.2 August 2011 Fg.

8: Smulaton at 100 teratons (left) and 500 teratons (rght) of maxlla. Pathomphong Phannurat was born n 1983. He receved a B.S. Degree n Computer Engneerng (1st Class Honors) and M.

He was awarded a research grant supported by the Thaland Graduate Insttute of Scence and Technology (TGIST).

Now, he s a research assstant n the Advanced Dental Technology Center (ADTEC). Chanjra Sn

She receved her Ph.D. Degree n Physcs (Imagng) from Kng s College, Unversty of London.

5 296 ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.9, NO.2 August 2011 Fg.7: Smulaton at 100 teratons (left) and 500 teratons (rght) of mandble. Fg.8: Smulaton at 100 teratons (left) and 500 teratons (rght) of maxlla. Pathomphong Phannurat was born n He receved a B.S. Degree n Computer Engneerng (1st Class Honors) and M.S. Degree from Suranaree Unversty of Technology. He was awarded a research grant supported by the Thaland Graduate Insttute of Scence and Technology (TGIST). Hs research nterests nclude Image Processng and Geometry Modelng. Now, he s a research assstant n the Advanced Dental Technology Center (ADTEC). Chanjra Snthanayothn receved a B.S. Degree n Physcs (1st Class Honors) from Chang Ma Unversty and M.S. Degree n Physcs of Laser Communcaton. She receved her Ph.D. Degree n Physcs (Imagng) from Kng s College, Unversty of London. At the present, she s a prncpal researcher n the Natonal Electroncs and Computer Technology Center (NECTEC). Wcht tharanon receved a B.S Degree n Medcal Scence (2nd Class Honors) and DDS Degree from Chulalongkorn Unversty. He receved certfcate n Aesthetc Maxllofacal Surgery and certfcate n Pedratrc Crano/Maxllofacal Surgery from The Unversty of Texas Southwestern Medcal Center, Dallas USA. At the present, He s drector of the Advanced Dental Technology Center (ADTEC).

SIMULATION OF ELASTIC SOFT TISSUE DEFORMATION IN ORTHODONTICS BY MASS-SPRING SYSTEM

SIMULATION OF ELASTIC SOFT TISSUE DEFORMATION IN ORTHODONTICS BY MASS-SPRING SYSTEM Pathomphong Phannurat 1, Wichit Tharanon 1, Chanjira Sinthanayothin 2 1 Advanced Dental Technology Center (ADTEC) 2 National