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

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1 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 Electronics and Computer Technology Center (NECTEC) National Science and Technology Development Agency (NSTDA) Thailand Science Park, Pathumthani, Thailand ABSTRACT Mass-spring system has been used to describe elastically deformable models in computer graphics such as skin, textiles, and soft tissue. A mass-spring mesh composed of a network of masses and springs, in which each edge is spring. We apply mass-spring system to soft tissue deformation in 3D orthodontic simulation, the movement of which is evaluated using the numerical integration of the fundamental law of dynamics. In the method, good data structure is presented based on the feature of STL teeth models. Computational quantity and accuracy is demonstrated on test and teeth model examples. The experimental results show that it can simulate the deformation change in real time and display result is vivid. 1. INTRODUCTION Traditionally, the orthodontist dealt with correcting malocclusions in growing patients. The typical orthodontic patient was a 12- to 14-year-old boy or girl. Up until the late 1970s and early 1980s, adults seeking orthodontic treatment for aesthetic reasons alone were very rare. It is very common nowadays for appearanceconscious adults to place great personal value on the appearance of their smile. Cosmetic dentistry has become a veritable specialty in its own right [1]. The orthodontist use cephalometric projections to plan their treatments [2]. Cephalometric projections are x-rays taken of the side of head. However, it is not convenient to store and compare orthodontic patient record of each patient for each period. Therefore, computer technology is used to simulate in orthodontics on 3-dimention space. Soft tissues hold a large proportion in the whole individual structure, including a great deal of important organs, for example, heart, skin, muscle and so on. The simulation of soft tissue deformation relates to such field as medicine, mechanics, biology, computer graphics, and robot vision. Therefore, the simulation of soft tissue deformation is widely used in the medicinal simulation system.this paper develops models of gingival tissue deformation in orthodontics which are based on simplifications of elasticity theory. By simulating physical properties such as tension and rigidity, we can model static shapes exhibited by a wide range of deformation objects. Furthermore, by including physical properties such as mass and damping, we can simulate the dynamics of these objects. The simulation involves numerically solving the partial differential equations that govern the evolving shape of the deformable object and its motion through space. It is important to describe behavior of gingival deformation brought by tooth move in the process of the rectification can be simulated by using computer simulation in the virtual orthodontic. The gingival deformation is the change of gingival shape (soft tissue) caused by the change of tooth (rigid body) position under external force in the process of orthodontics. In the orthodontic simulation, gingival tissue deformation should satisfy physical third dimension in real-time, to which the key is construction of an appropriate physically-based model of soft tissue deformation. In our approach, we start in section 2 which describes our implementation of deformable models. The model composed of a network of masses and springs, which can be considered as a variant of elastic models. We give differential equation of motion describing the dynamics behavior of deformable models under the influence of external forces. Section 3 presents simulations illustrating the application of soft tissue deformation. 2. THE MASS-SPRING MODEL 2.1. The Triangle Mesh Our elastically soft tissue model is a triangle mesh of m masses, each mass being linked to its neighbors by mass springs of natural length non equal to zero (figure 1). The 3rd International Symposium on Biomedical Engineering (ISBME 2008) 247

2 2.5. Applied Forces mass : spring : The system under study is the triangle mesh of the m masses, each mass being positioned at time t on the point P i (t), where i = 1,...,m. The position of each point, at any time t, can be derived through the fundamental law of dynamics: Figure 1. Regular mesh of masses and spring Point Mass In our Mass-spring models, masses are allocated at the triangle mesh and damped springs along the edges. To accurately distribute the total mass of the mesh, we compute the mass m i of each vertex i according to the area A j of its adjacent triangle j. If D is the material density [4], then: D m i 3 j A 2.3. Spring Stiffness j The easiest way is to use a constant value for the stiffness (k). More commonly, k is computed as k=1/l, where L is the length of the spring at rest. In [3], Van Gelder suggested a formula to compute spring stiffness for a 3D mesh that is the closest to elastic continuous representation. Let E be the Young s modulus, then: (1) E A j j k (2) 2 L 2.4. Spring Damping The question of how to assign different damping (c) values to the various springs in a mass-spring system has been largely ignored in the literature. Traditionally, c is treated as a constant throughout the system. We performed the same simulation as before using (1) and (2) to calculate m and k. To simulate the best behavior of our models, we compute spring damping (c) follow by equation (3) [4], M is the effective end mass m i + m j. Fi ai vi Fi ai v x t x vi x t Where µ is the mass of each point P i, F i is the sum of all forces applied at point P i, a i is its acceleration caused by the force F i, and v i is its velocity at time t. F i can be divided between the internal and external forces. The internal forces are due to the tensions of the springs [5,6]. The overall internal force applied at the point P i is a result of the stiffness of the all springs linking this point to its neighbors: (4) ( ) 0 Fint Pi k j R l l (5) l where: R is the set regrouping all couples (j) such as P j is linked by a spring to P i, l PP, i j 0 l is the natural length of the spring linking P i and P j, k is the stiffness of the spring linking P i and P j The external forces are of various natures according to the king of load to which we wish the model to be exposed. Omnipresent loads will be gravity, a viscous damping and a viscous interaction with an air stream (or wind), but we only applied damping forces (F dis ) to simulation our models. The viscous damping will be given by: l c 2 km L (3) F ( P) C v (6) dis i dis i 248 The 3rd International Symposium on Biomedical Engineering (ISBME 2008)

3 where C dis is a damping coefficient that compute by equation (3), and v i is its velocity of point P i. The role of this damping is in fact to model in first approximation the dissipation of the mechanical energy of our model. 3. IMPLEMENTATION OF DEFORMABLE MODELS 3.2. Point Force Point forces are applied force when teeth are aligned. We can define point forces with boundary points of gingival surface which connected aligned tooth (figure 2). In orthodontic treatment simulation, we can extract teeth structure from gingival surface before teeth alignment. To analysis gingival surface deformation must be create animation with deformable model (gingival surface), the differential equations of motion are simulated numerically. First step is design data structure of massspring model. Next, define point force that is point on gingival surface which are applied force when teeth are aligned. At Last, compute new position of points at time step by solving differential equations Data Structure In the data structure, on the basic of the properties of STL files, the spring mass point system surface is described by mass point, and spring. The complete geometry and topology information is registered in this system, which can satisfy the demand of the computation. The structures are as follows: class MassPoint{ public : float mass; float p[3]; //position vector float v[3]; //velocity vector float f[3]; //force vector int *adjspring; //adjacent spring int numadjspring; }; Figure 2. Point Forces (balls) are contributed on gingival surface boundary which connected aligned tooth Gingival Surface Motivation The movement of gingival surface is caused by forces of teeth movement that applied to each point of model. The position of each point, at any time t, can be derived through the Newton s second law of motion (equation 4). All the above formulations make it possible to compute the force F i (t) applied on point P i at any time t. The fundamental equation of dynamics can therefore be explicitly integrated through time by a simple Euler s method [5,6]: ai( tt) Fi( t)/ v ( tt) v ( t) ta ( t t) (7) i i i Pt ( t) Pt ( ) tv( t t) i i i class Spring{ public : long m_vid; float Kc; //spring stiffness float restl; //rest length float currl; //current length //list of face id that share this edge std::list<face*> m_lstfacets; }; where t is a chosen time-step and F i (t) is different of internal forces and external forces. 4. EXPERIMENTAL RESULTS AND DISCUSSION The algorithm of the technique was implemented under Windows with Borland C++ Builder, using the OpenGL library for rendering the 3D images. Two different sets were used during the experiments a mandible model (figure 3) and a maxilla model (figure 6). The simulation results of the gingival deformation are shown in follow figure. Figure 4 and figure 7 are respectively the deformation simulation result for translation and rotation The 3rd International Symposium on Biomedical Engineering (ISBME 2008) 249

of tooth by arrow direction. Iterations of each deformable model were shown by figure 5 and figure 8.

CONCLUSIONS The soft tissue deformation needs to satisfy the physical properties.

The process of dynamic deformation can be described effectively after the time variable is introduced.

third dimension in virtual orthodontics.

support and the authors would like to thank the Advanced Dental Technology Center (ADTEC) for providing the teeth model and The

Keim, "Aesthetics in clinical orthodonticperiodontic interactions", Periodontology 2000, Munksgaard, Denmark, pp. 59-71, 2001.

[3] Van Gelder, "Approximate simulation of elastic membranes by triangulated spring meshes", Journal of Graphics Tools, pp.

4 of tooth by arrow direction. Iterations of each deformable model were shown by figure 5 and figure 8. Technology Center (NECTEC) for research training grant. 5. CONCLUSIONS The soft tissue deformation needs to satisfy the physical properties. In this paper, we apply mass-spring system to soft tissue deformation in 3D orthodontic simulation. The process of dynamic deformation can be described effectively after the time variable is introduced. The dynamic motion rule adopts the differential equation form; the numerical method can be carried out for the real-time computation of the dynamic system. The experimental results show that this method satisfies the demand for the computational real-time and the gingival deformation third dimension in virtual orthodontics. ACKNOWLEDGEMENT This work is supported by the Thailand Graduate Institute of Science and Technology (TGIST) for their financial support and the authors would like to thank the Advanced Dental Technology Center (ADTEC) for providing the teeth model and The National Electronics and Computer REFERENCES [1] Robert G.Keim, "Aesthetics in clinical orthodonticperiodontic interactions", Periodontology 2000, Munksgaard, Denmark, pp , [2] Chanjira Sinthanayothin, "Medical Imaging Aided Orthodontics", In Proc. TMI2005, [3] Van Gelder, "Approximate simulation of elastic membranes by triangulated spring meshes", Journal of Graphics Tools, pp , [4] Paloc C., Bello F., Kitney R.I., and Darzi A., "Online Multiresolution Volumetric Mass Spring Model for Real TimeSoft Tissue Deformation", MACCAI 2002, Springer- Verlag, Berlin, pp , [5] Provot X., "Deformation Constraints in a Mass-Spring Model to Describe Rigid Cloth Behavior", In Proc. of Graphics Interface, pp , [6] Dochev V., and Vassilev T., "Efficient Super-Elasticity Handling in Mass-Spring Systems", CompSysTech, Figure 3. Original mandible model. Figure 4. Gingival deformation of mandible model. Figure 5. Simulation each iteration of mandible model. 250 The 3rd International Symposium on Biomedical Engineering (ISBME 2008)

5 Figure 6. Original maxilla model. Figure 7. Gingival deformation of maxilla model. Figure 8. Simulation each iteration of maxilla model. The 3rd International Symposium on Biomedical Engineering (ISBME 2008) 251

Cloth Simulation. Tanja Munz. Master of Science Computer Animation and Visual Effects. CGI Techniques Report

Cloth Simulation. Tanja Munz. Master of Science Computer Animation and Visual Effects. CGI Techniques Report Cloth Simulation CGI Techniques Report Tanja Munz Master of Science Computer Animation and Visual Effects 21st November, 2014 Abstract Cloth simulation is a wide and popular area of research. First papers