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Chaos and Graphics

An interactive simulation system for training and treatment planning in orthodontics Maria Andre´ia F. Rodriguesa,, Wendel B. Silvaa, Milton E. Barbosa Netoa, Duncan F. Gilliesb, Isabel M.M.P. Ribeiroc a

Mestrado em Informa´tica Aplicada, Universidade de Fortaleza - UNIFOR, Av. Washington Soares 1321, J(30), 60811-905 Fortaleza-CE, Brazil b Department of Computing, Imperial College London, 180 Queen’s Gate, London SW7 2BZ, UK c Centro de Cieˆncias da Sau´de, Universidade de Fortaleza - UNIFOR, Av. Washington Soares 1321, 60811-905 Fortaleza-CE, Brazil Received 1 September 2006; received in revised form 2 February 2007; accepted 15 April 2007

Abstract We have designed and implemented an interactive simulation system for training and treatment planning in orthodontics. Currently, both treatment planning, and the choice of a proper appliance model, are based exclusively on clinician expertise. Most orthodontists work by trial and error when estimating the loading conditions that will achieve the desired tooth movement. There is a strong need for computer methods that will enable them to make realistic visual predictions of the final positions of the teeth and the changes in shape that the dental arch undergoes. To validate our simulator, we used cephalometric measurements and dental cast data taken during oneyear follow-up orthodontic treatments. The results demonstrated a closely degree of fit between experimental orthodontic treatments and simulated case studies. In addition to its use in therapy planning, we expect our simulator to be a useful environment for training, providing a means to explore the temporal evolution of planned treatments. r 2007 Elsevier Ltd. All rights reserved. Keywords: Interactive; Simulation system; Training; Treatment planning; Orthodontics

1. Introduction Recent advances in computer technology are providing better data and methods for addressing a number of medical needs effectively. Currently, the opportunities for developers of medical applications to improve health care are enormous. In particular, innovative work in computer graphics and virtual reality can make a significant contribution to the development of improved visualization, simulation, navigation, and decision support systems for medical training and treatment planning. In the area of orthodontics, for example, there is a strong need for computer methods that will enable students and experienced professionals to make realistic visual predictions of Corresponding author. Tel.: +55 85 3477 3268; fax: +55 85 3477 3061.

E-mail addresses: [email protected] (M.A.F. Rodrigues), [email protected] (W.B. Silva), [email protected] (M.E. Barbosa Neto), [email protected] (D.F. Gillies), [email protected] (I.M.M.P. Ribeiro). 0097-8493/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cag.2007.04.010

the final positions of the teeth and the changes in shape that the dental arch undergoes. Orthodontic treatment is used to adjust the position of the teeth in dental arch in order to obtain the correct occlusion with the best functional and aesthetic features [1–4]. The tooth positions are adjusted by applying forces to the tooth crown by means of elastic deformation of metallic wires. Usually, treatment planning and the choice of a proper appliance model are based exclusively on clinician expertise. Most orthodontists work by trial and error when estimating the loading conditions that will achieve the desired tooth movement. However, unexpected events do occur during the treatment. It is very common to predict that applying a continuous force for a certain time will produce a specific tooth movement, which in practice does not occur. The tooth does not move at all or does not move enough as a response to the applied loading. The main limitation of the present prediction methods is the lack of an interactive 3D system to investigate visually

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the temporal evolution of the treatment. There is a need for a realistic model of tooth motion under different appliance models in the presence of collisions. This would allow simulations to determine the most suitable orthodontic strategies to avoid causing any undesired tooth movement and to overcome dental arch clinical problems. Many commercial systems have been proposed for orthodontic practices, but in no case do they include this functionality. The majority of the existing software used is essentially for clinical management and 2D cephalometric analysis. We have developed an interactive simulation system for training and treatment planning in orthodontics that runs on commonly available personal computers. In particular, we have a special interest to represent orthodontic processes so that the relationship between different mouth measurements, tooth movements, and appliances can be analysed. This includes investigating methods for characterizing tooth movement and 3D dental arch behaviour, when the teeth are subjected to a variety of loading conditions and restricted by neighbouring teeth. To achieve our goals, we had to find a reasonable balance between the processing time, the size, and the complexity of the 3D geometric models and techniques used for real-time simulation, including collision detection and rendering. Initial investigation has proved that we can simulate behaviour that closely replicates the real teeth movements observed in our experimental studies. In addition to its use in therapy planning, we expect our simulator to be a useful environment for training orthodontists, residents and students. It provides a means to explore the temporal evolution of planned treatments rapidly and without involving patients. The rest of the paper is organized as follows. Section 2 introduces the system components. Section 3 describes the method and the detailed results of the case studies investigated. Also, it evaluates the merits of the system by discussing its main features and limitations in light of the case study results. Section 4 covers related work. Finally, Section 6 concludes the paper by summarizing the research and suggesting topics for future work.

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treatment simulator is made up from a tooth movement simulator, a collision detection and response module, and an animation module. The animation module includes specific tools for 2D morphing and 3D animation [5]. The morphing tool reproduces the changes in the mouth geometry based on photographic records of the subject taken during regular clinical checkups. These animation tools allow the clinician to simulate and visualize the temporal evolution of the treatment. 2.1. 3D mesh representations In our system, the user interface panel is implemented based on the geometrical standard references and parameters usually adopted in the design of treatment planning. We use a realistic and standard 3D geometric model of the mandible, maxilla, and dental arch of a virtual patient. The skull anatomy does not influence the simulation results. It is used only to provide a better understanding of the movements. The behaviour and constraints have been designed and implemented. The number of roots of a tooth can vary depending on the bone characteristics of a given patient. In particular, we have individually modelled the teeth of our standard patient by generating a complete 3D mesh representation with crown and three different types of roots [5], as shown in Fig. 1. To represent the 3D model of a specific subject, the standard virtual patient model is modified by 3D interpolation to adapt itself to the main patient data obtained from X-rays and dental cast measurements (Fig. 2). Because the first molar teeth are present in the majority of the population, we chose them as the reference points for the 3D mandible mapping. The length and the width of the mandible are then calculated based on these data measurements and reference points. The 3D geometric data representing the teeth (with already customized root

2. The system components Our orthodontic system, which is written in Java, consists of three basic components for: cephalometric mapping, 3D mesh generation, and orthodontic treatment simulation. The cephalometric mapping component is responsible for determining a mapping from the actual X-ray and dental cast measurements onto the vertices of the volumetric dental arch and mandible in our standard virtual patient. The 3D mesh generator is responsible for building a virtual customized model of the patient. Based on the measurements calculated with the aid of dental casts and X-rays, the 3D mesh generator takes into account the initial position and orientation of the teeth (including their root types, length, and crown dimensions), as well as of the maxilla (upper jaw bone), and mandible. The orthodontic

Fig. 1. The 3D mesh representation of the standard virtual patient.

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Fig. 2. 3D meshes representing the mandible and teeth in the system.

length and crown size dimensions) are then uniformly distributed and positioned across the dental arch, according to their initial orientations. A tooth absence in the dental arch (if there is any) is represented in our system, by removing a given tooth (through the user interface panel), after generating the customized 3D mesh of a patient. The tooth movements and dental arch can be visualized from any point of view chosen from the user interface panel. The generated mesh representations are composed of a sufficient number of vertices and polygons to construct a realistic virtual patient with good degree of accuracy. Approximately 125 vertices (180 polygons) are used to model each tooth to represent its skeletal shape. The skull is composed of 3087 vertices (5426 polygons) and the jaw of 1094 points (2054 polygons). The quality of all geometric meshes generated was kept at a very high level, where neither geometric mesh distortions nor mesh nonlinearities were found. We designed and implemented two types of orthodontic appliance models. First, attached to each tooth crown at right angles, we designed the bracket, represented by a cube element. The brackets are then linked together by an orthodontic wire (with rectangular or spherical shape). Spherical wires are used in our simulator to rotate the tooth in the x and z-directions, while rectangular wires are used to translate the tooth in the x, y, and z-directions, as well as to rotate it. Finally, we used cubic spline curves (whose controls points are the brackets) to generate the appliance models. 2.2. Tooth movement simulator The system simulates 3D tooth movements through time with respect to a fixed Cartesian frame located in the middle of the dental arch. It is designed and implemented to displace and rotate the teeth interactively to obtain the correct dental occlusion, based on the measurements and system of forces defined by the user in the interface panel. Any tooth (or a group of teeth) can be interactively ‘‘extracted’’ or selected through the user interface panel to apply loadings. Using the current and next tooth position in the dental arch, as well as the force system set up by the user, our system automatically calculates those new tooth positions (trajectories) that best fit the geometry of the

virtual patient dental arch through linear interpolations, during the evolution of the orthodontic treatment. Each tooth has its own system of coordinates. A director vector between two neighbouring teeth in the x direction can be found easily. In our simulator, a tooth can be also displaced (translated) into extraction spaces, following the direction of the dental arch, by calculating the director vector. For the z direction, the orthogonal vector is calculated via the cross product. This is useful, for example, when the user wants to perform inclinations on the incisors of the patient. The simplest orthodontic movements [6], such as tipping, occur about the centre of tooth resistance (13 from the root apex). Translations are modelled in our simulator by applying a bodily movement where the whole tooth structure is uniformly loaded. It is expected that during rotations, tipping also may occur. Extrusions and intrusions are both modelled by vertical movements where forces are used to move the tooth down or up, respectively. The centre of resistance and centre of rotation of the tooth are calculated automatically and they help to evaluate the effect of the force system on the tooth movement. The translation, inclination, rotation, torque, extrusion, and intrusion tooth movements have all been implemented in our system. They are shown in Table 1. Collision detection is fundamental to interactive simulation systems and ensures that the properties of the solid real world are maintained. Therefore, we have also implemented collision effects among dental surfaces caused by tooth movements. The computer simulation imposes several restrictions that have a direct impact on our collision detection method. These include how many objects there are in a given orthodontic scenario, their relative sizes, level of detail, and positions, if and how they move, and whether they are rigid or not. The problem of detecting when teeth collide, modelling the contact between them and determining an appropriate response, are all critical operations requiring careful coding to achieve a high and constant frame rate. The whole process has to operate under strict time and size restrictions to guarantee the correct real-time teeth collisions [7]. Given the usual time and space trade-off, the detection accuracy must be balanced against computation time to meet performance requirements. In addition to an efficient collision method,

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Table 1 Tooth movements implemented in the system Translation

Inclination

Rotation

we have also implemented a ‘‘domino’’ collision effect among teeth. After collisions have been detected and the contact area found, a response must be determined. This involves an instantaneous direct correction of the direction and speed of movement. The main characteristics of our pseudo-code used to move the teeth interactively in presence of collisions is presented in Algorithm 1, and is detailed as follows. In each frame of the simulation, we verify whether there is some force being applied to any tooth (the original userdefined force or the one that was propagated along the dental arch due to collisions). In the affirmative case, Algorithm 1 is called. We first verify the orientation of applied force (line 1). Based on the displacement orientation found and current tooth position, we identify the ‘‘next tooth’’ (visible or not) in the dental arch (lines 2–4). Following the orientation of the tooth displacement, we then search for the nearest visible tooth in the dental arch (lines 5–7). The direction vector (line 8) is obtained from the centre of mass of a tooth (which is being subjected to a force) towards the centre of mass of the ‘‘next tooth’’ in the dental arch. While not detected a collision between the bounding spheres of the moving tooth and the next visible tooth (line 9), the moving tooth can still be translated and/or rotated, and the orthodontic wire model used is identified (lines 10 and 15). If we change the orthodontic arch model used, the behaviour of the tooth changes accordingly and this fact is taken into account in our tooth movement simulator. Therefore, if it is necessary to apply a torque, the rectangular wire model should be used and the tooth rotates with the centre of rotation in this case located at its crown (lines 10–12). However, if the wire model is rectangular and the tooth is subjected to a force, this tooth can translate in the x, y, and z-directions (lines 10, 13, and 14). Finally, if the wire model is spherical and the tooth is subjected to a force, it can only rotate (lines 15 and 16). The centre of rotation, in this case, is taken to be a position located at 13 of its root length. In the negative case, the moving tooth stops and the force being applied to it is partially propagated to the next visible tooth (that suffered the collision), by simulating a ‘‘domino’’ collision effect among teeth (lines 17

Torque

Extrusion

Intrusion

and 18) and thus successively, while there is enough force to move any tooth as well as colliding teeth in that direction.

Algorithm 1. The pseudo-code used to move the teeth interactively in presence of collisions

01: 02: 03: 04: 05: 06: 07:

08:

09: 10: 11: 12: 13: 14: 15: 16: 17:

18:

// nextVisibleTooth is used to collision detection // nextTooth is used to calculate the direction vector // Verify the force orientation if (force o 0) inc ¼ +1; elseif inc ¼ 1; // Get the next tooth instance nextTooth ¼ Tooth[currentTooth.id + inc]; if (currentTooth.location ¼ ¼ nextTooth.location) nextTooth ¼ Tooth[nextTooth.id + inc]; // Search the nearest visible tooth in the data struct nextVisibleTooth ¼ Tooth[currentTooth.id + inc]; while (not nextVisibleTooth.isVisible) nextVisibleTooth ¼ Tooth[nextVisibleTooth.id + inc]; // Calculate the direction vector to move the tooth DirectionVector ¼ GetVector(currentTooth,nextTooth); // Check for collision detection if (not Collide(currentTooth, nextVisibleTooth)) // Check orthodontic archwire model if (archtype ¼ ¼ aw_rectangular) if (torque) // Apply tooth torque currentTooth.RotateCrown(force, DirectionVector); elseif // Apply tooth intrusion, extrusion or translation currentTooth.Translate(force, DirectionVector); elseif (archtype ¼ ¼ aw_spherical) // Apply tooth inclination or rotation currentTooth.RotateRoot(force, DirectionVector); elseif // In case of collision, apply force loss to simulate a domino collision effect ApplyForce(force*LOSS, nextVisibleTooth);

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3. Case studies and results In a validation study we investigated a group of patients with malocclusion problems. On the basis of the orthodontist’s clinical experience, we selected four subjects from this group on the basis that their clinical problems represent the most common problems in orthodontics. The initial dental casts and cephalometric measurements that we used are displayed in Fig. 3 and in Table 2 (lines 2 and 3), respectively. As part of the therapy (Fig. 4 and line 1 in Table 2), measurements of dental displacements were taken during clinical checkups of the patients over one

year. The reference teeth on the cast models, and the positions to which the teeth moved, were marked for identification. We used the molar and premolar teeth as the reference teeth in our computer simulations. In planning the biomechanical aspects of orthodontic treatment for a specific patient, it is imperative that the orthodontist consider not only the forces required for the necessary tooth movement to achieve the objectives of the treatment, but also any undesired tooth movement that may occur in response to these forces. What happens using our simulation system depends on the level and duration of force.

Fig. 3. Dental casts of the four subjects. Frontal (a) and lateral (b) views.

Table 2 Measurements and simulated results for the case studies

1. Treatment planning

2. Mandible measurements (cm) (length, width) 3. Cephalometric measurements (1) (upper incisors, lower incisors, mandible) 4. Time interval (in number of frames) to reach the target position for applied forces of (0.02, 0.08, 0.15 N)

Patient 1

Patient 2

Patient 3

Patient 4

Retraction of a group of incisors and canines, filling space in the direction of the molars

Inclination of maxillary incisors forwards

Filling the space between the central upper incisors

Filling the space between maxillary incisors and canine teeth

5.0, 2.9

4.0, 2.6

5.1, 2.6

5.0, 3.0

26, 21.28, 40.84

31.84, 25.09, 40.24

34.51, 17.3, 30.75

34.82, 34.28, 16.60

1; 72; 36

1; 106; 50

1856, 39, 14

239, 108, 27

Fig. 4. Therapy planning for patient 1 (a and b), patient 2 (c), patient 3 (d), and patient 4 (e).

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We classified the applied loadings used in our study as low, medium, and high force values (0.02, 0.08, and 0.15 N, respectively). Different effects of loading acting on the appliance (tooth) are then observed and behaviours are simulated. Our system calculates, in number of frames, the time interval needed for a certain tooth to reach the desired position in the dental arch. All the data generated by our computer simulations were evaluated by comparing the calculated and clinically observed tooth movements. At the start of our computer simulations, customized virtual patient models were generated. We used our standard virtual patient (Fig. 1) as the basis for all of the simulated case studies. We customized the shape of the standard model using medical measurements and advice from orthodontic experts. The 3D results that were obtained for each customized subject are shown in Fig. 5. The treatment that was designed for the first patient required the extraction of the first premolars on both sides of the mandible and maxilla (Fig. 4a). This was followed by the use of a fixed appliance model with brackets linked by a rectangular wire with pre-established torsion in its structure. The intention is that the canines have to be distalized into the extraction space (approximately 6.8 mm) in direction of the molars, and the incisor group retracted and their angle of inclination corrected (Fig. 4b). The second subject (Figs. 3 and 5) had the problem that the upper incisors projected forwards in the maxilla with an extrusion of 8.04 mm. There were only 23 teeth on his dental arch (12 teeth in the mandible, and 11 in the maxilla). The treatment plan involved retraction of his upper incisors (Fig. 4c). Patient 3 (Figs. 3 and 5) had a dental arch with 30 teeth (14 teeth in the mandible, and 16 in the maxilla), with a 3.0 mm space between his upper incisors teeth. In addition to this problem, the upper

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incisors had an inclination of 31.511, with an extrusion of 9.41 mm. For this patient the treatment plan, shown in Fig. 4d, involved closing the gap between the incisors. Finally, patient 4 (Figs. 3 and 5) had 28 teeth on his dental arch (14 teeth on the maxilla and 14 teeth on the mandible), with a space of 4.0 mm between the first premolar and the canine positioned on the right-hand side of the mandible. The treatment plan is shown in Fig. 4e. The numerical and graphical results of the computer simulations for all patients are summarized in Table 2 (line 4), and in Figs. 6, 7, 8 and 9, respectively. The results showed that a low force (0.02 N) was not sufficient to displace the canines of the patient 1 (Fig. 6a). A medium force (0.08 N) needed 72 frames to move the canines by 6.8 mm, when collision detection was found with the second premolar tooth (Fig. 6b). The application of a high force value (0.15 N) needed only 36 frames to move the canines to the planned position. Similarly to what happened with the medium force, the teeth stopped moving when a collision was detected (Fig. 6c). The results for patient 2 (Fig. 7, and line 4 in Table 2) show that a low force value was not enough to change the incline of the upper central incisors (Fig. 7a). A medium force produced a very slow movement and needed 106 frames to produce an inclination of 8.041. This result was very close to our empirical studies, and is shown in Fig. 7b. The application of a high force value produced a relatively fast movement and needed just 50 frames to achieve an inclination of 8.161 (Fig. 7c). Collisions among teeth were not detected. Medium force values were more appropriate, because of the high risk of causing an undesired tooth movement, for example where the upper incisors might incline beyond the angle aesthetically considered as ideal.

Fig. 5. Initial 3D geometric configurations of the four simulated patients. Frontal views (above) and lateral views (below).

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Fig. 6. Graphical results of simulation for patient 1. Inner upper views of the maxilla (above), and lateral views of the lower and upper jaw (below).

Fig. 7. Graphical results of simulation for patient 2 (lateral views).

Fig. 8. Graphical results of simulation for patient 3 (frontal view).

The results for patient 3 (Fig. 8, and line 4 in Table 2) show that a low force value was sufficient to translate the upper incisor teeth by 3 mm. The simulation needed 1856 frames to displace the teeth by 1.68 mm and a further 1890 frames to complete the movement. With a medium force, the total number of frames required for a displacement of 1.72 mm was 39. High force values produced considerably greater and faster tooth displacements. Only 14 frames were necessary to fill the space between the upper incisors teeth, and to correct the inclination and extrusion problems. Collisions were observed under all loading conditions. When the velocity was slow, the subjective analysis of the collision detection and response seemed more natural but as the velocity increased, the collision was perceived as less realistic. The results for patient 4 (Fig. 9, and line 4 in Table 2) showed that a low force value was enough to displace the first lower premolar tooth by 3 mm towards the lower canine (Fig. 9a). The displacement was slow with a duration time of 239 frames. A medium force was then applied and it was observed that only 108 frames were needed to reproduce the same movement. A domino collision effect was also observed, where a group of teeth was displaced taking 253 frames to complete the movement. However, it was found that the medium force was

too low to cause a final collision with the lower lateral incisor tooth (Fig. 9b). A high force was then applied and produced greater and faster movements (Fig. 9c), causing collision with the lower incisor tooth by way of the propagation of the movement by the domino collision effect. Validation was based on the degree of fit between reality and its model representation. We demonstrate that the results obtained with the computer simulations and geometric models are consistent with the empirical observations. Orthodontists assume, in their routine practice, that tooth movements within the range of [0, 1 mm] can be neglected (this range of movement is not significant enough and is still considered as an inertial tooth state). Considering this fact, in the majority of the investigations performed the degree of fit between experimental results and simulator output is very high. We found that the computer simulated results, when compared to the real experiments, generate no significant error during tooth rotations, inclinations and torques. During tooth translations (Figs. 10a and b) in the x, y, and z-directions, we found a mean error of 1.29343, 0.62972, and 1.09198 mm (due to the calculation of the tangent vector, particularly for the canine and first premolar teeth, which are located in the most curved portion of the curve that represents the dental arch shape), with a standard deviation of 1.06997, 0.58663, and 1.01575 mm, respectively. The median in the x, y, and z directions was 1.08516, 0.42390, 0.81437 mm, respectively. Extensive clinical evaluation will be a slow process because of the long time of orthodontic treatment and the fact that, for ethical reasons, it is not possible to experiment with the treatment regimes. The resulting animations will be of use to clinical researchers or

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3rd Molar

2nd Molar

1st Molar

2nd Pre Molar

Canine

1st Pre Molar

Lateral Incisor

Canine

Central Incisor

Points of the tooth path along the dental arch

Central Incisor

-40

2

nd

y

0 Lateral Incisor

-20

Y= 1

1

Pre Molar

X

2

1st Pre Molar

0

3

1st Molar

z

20

4

3rd Molar

40

5

2nd Molar

Center of mass of the teeth crowns

60

Deviation between the ideal and simulated tooth positions (mm)

Fig. 9. Graphical results of simulation for patient 4 (lateral views).

Points of the tooth path along the dental arch ideal tooth positions

simulated results

axis x

axis y

axis z

Fig. 10. Accuracy of the computer simulations for tooth movement. In (a), the points of tooth path along the dental arch. In (b), the deviation of the empirical and simulated results within the range of ½0; 5 mm for tooth translations in the x, y, and z-directions.

practitioners interested in the 3D motion of the tooth, for training and treatment simulation. A video illustrating our interactive orthodontic simulation system can be found temporarily at http://jortho.project.googlepages.com/ 4. Related work There are some other 3D systems that have proved to be invaluable computer simulation approaches for training and treatment planning in the health area. Some of the most representative systems are discussed below. When pertinent, they are compared or contrasted with our orthodontic system. 4.1. Previous orthodontic systems A few commercial systems that support a certain degree of planning of orthodontic treatments are available in the market. Most of them do not offer tools for valid clinical simulation, nor do they integrate all the necessary functionality into one single tool. They fall into three main groups: clinical management systems, 2D tools for image analysis, and 2D and 3D systems for simulation of orthodontic treatments. The most commonly available systems fall into the first category offering financial management and support for following up orthodontic treatments. Some systems include a cephalometric analysis module as one of the software components. However, none

of them use a patient specific 3D model, or provide a real simulation of tooth movement over time. In our system, these basic functionalities are provided. There has been significant recent research into the use of cephalometric measurements that has resulted in the creation of interesting and useful analysis software [8]. Cephalometric markers are known anatomical points on the face and skull that can be accurately identified by medical practitioners. They are used to define shape and can be used to plan maxillary-facial and reconstruction surgery. Analysis systems have traditionally used a superimposition of the X-ray and photograph into one manipulable image that shows both hard and soft tissue. Cephalometric landmarks are then identified and anatomical measurements are compared to established norms and displayed for easy reference. Using clinically accurate hard tissue movement, the patient’s profile can be morphed to show the results of the proposed treatment. The relevance and advantage of these systems are demonstrated by the significant increase of patient acceptance of proposed treatment plans after seeing the results of cephalometric analysis. This type of system has proved to be an invaluable support in planning certain dental treatments, and therefore we expect it to play a significant role in orthodontics. Some researchers have concentrated on the development of 2D and 3D computer-based orthodontic treatment tools [9], and on real-time mandible (lower jaw bone) movement

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simulators [10–12]. In these systems, data can be obtained by cephalometric measurements of the patient. The 2D models use elementary parameters, and consequently, represent an excessive oversimplification of the orthodontic problem. They do not include volumetric information about the dental arch or appliance shapes. Some of this work is related to ours in that tooth movement is represented by a functional model composed of geometric restrictions on displacements in three dimensions. However, none of the systems provide advice on the best type of treatment to apply to the patient, they do not include 3D collision detection methods, and they have the further disadvantage that they are expensive commercial tools that are inextensible since there is no open access to the source code. 4.2. Other medical applications Surgical simulation, augmented reality and virtual reality systems are other examples of computer based applications that have made significant advances in treatment planning, computer-assisted diagnosis, and training in medicine [13,14]. The state-of-the-art development in this area has been proposed either in the fields of computer graphics or in domains closer to biomechanics, with the greatest challenge being still the creation of a fully interactive and immersive surgery simulation by working collaboratively [15] or individually. Examples are the work done on skin, with particular application to facial expression representations [16,17]. These models account for a geometric structure which allows an interaction between bones, muscles, fatty tissue, and skin. Other examples include a surgical simulator using finite element analysis in which a realistic 3D computer model of the eye and its surroundings is modelled with a complex behaviour [18]. Systems for computer plastic surgery are also good examples [19]. In the virtual reality and interactive graphics areas, other projects aim at developing simulation systems for laparoscopy [20], endoscopy [21], laryngoscopy [22], epidural needle insertion [23], robotic cardiac procedures [24], etc. In these examples, the anatomy can be viewed on a computer screen and, in some cases, can be manipulated using a force feedback haptics device. As a whole, these systems have the ultimate goal of facilitating the evaluation of the patient’s condition, or the simulation of surgical procedures for training or planning. Achieving these goals, while ensuring patient safety, is important to provide a real significant contribution to improve the health system. 5. Conclusion and future work This paper presented the design and implementation of an interactive simulation system that we have developed for training and treatment planning in orthodontics. It combines and improves the main features of some existing commercial models and fills training gaps in the real world. The inputs to the simulation system are the external forces

applied on the tooth. Responses were calculated depending on the applied loadings, and we found that realistic 3D tooth movements can be simulated. Furthermore, the collision detection method implemented can be used to assess potential risks, for example whether abrasion occurs when the teeth collide during the occlusion. The advantage of the system, when compared to traditional methods, is that teeth movements are dynamically visualized instead of statically, with the ability to modify interactively, through the user interface panel, the loadings on a selected tooth or a group of teeth, and evaluate the results of changes in 3D for different orthodontic setups. The underlying simulation component creates an interactive world where the user can see how the teeth are moving in response the environment. To guarantee a good performance of the system, we had to find a reasonable balance between the processing time and the size, and accuracy of the geometric models. Initial investigations have proved that we have been able to set up our system to demonstrate behaviour that closely replicates real tooth movements, similar to our experimental studies, with a high degree of accuracy for tooth rotations, translations, and torques. Presently, validation is done by subjective inspection and correlated with real teeth movements using dental casts, cephalometric measurements, and photographic records taken on a regular basis during one-year planned case studies. However, the modelling does not yet explore all the mechanical aspects involved in orthodontics. For example, around the teeth root, effects of soft tissue deformation can be incorporated to generate more accurate tooth movement. The stresses and strains caused by tooth movement cause bone reshaping. Finite element analysis, for instance, is a possible tool for simulation, but it will only work in real time if the number of elements is small and linear elastic deformations are used. Under these conditions, the simulated results will be inaccurate since it has been established that the behaviour of soft tissue under deformation is highly non-linear and has a clearly timedependent component. Also, it is known that toothmovement occurs at different rates in different individuals and can sometimes even vary during the course of treatment. Thus, the implementation of an optimal orthodontic force system that meets all these requirements presents major research challenges. A detailed physical model of the biomechanics of tooth movement is required and at the same time must be computed in real time. However, we believe that for the development of training systems, simulating certain aspects of the main orthodontic procedures with a sufficient degree of realism is already of benefit for students and professionals, and will enable them to perform the initial routine work entirely in a virtual environment, thus saving cost and time. Training systems may be used by those seeking a better understanding of the geometric and dynamic factors involved in the control strategies of orthodontic treatments, as well as to investigate the accuracy of the results and whether a specific planned treatment can be detrimental to the patient

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in any circumstance. We hope that this work will provide the basis for the development of a practical but effective graphical, clinical tool for research purposes in orthodontics. It is becoming increasingly evident that the medical domain can offer many opportunities for the application of interactive computer graphics. However, the one aspect that may require a further horizon is the development of high-fidelity patient models for design, testing, and validation [25]. Therefore, the next stage of this work is to evolve the current implementation to improve the geometric and modelling accuracy. Among the research to be performed we plan to use 3D reconstruction methods to replace the manual dental plaster measurements. Dental casts can be scanned three-dimensionally and their volume determined exactly from the data set. This will improve the geometric accuracy of the 3D meshes. Use could be made of statistical shape models of the mandible, maxilla, and teeth to improve the speed and accuracy with which the standard geometric model can be made patient specific. We have previously constructed a statistical shape model of the mandible in a different application that demonstrated that patient specific reconstruction can be achieved using a small number of simple cephalometric measurements [26]. Lastly, we hope to explore the use of tactile feedback for the manipulation of objects. Acknowledgements The research was partly supported by The National Council for Scientific and Technological Development of Brazil (CNPq) under grant No. 303046/2006-6. References [1] Alcan˜iz M, Montserrat C, Grau V, Chinesta F, Ramon A, Albalat S. An advanced system for the simulation and planning of orthodontic treatment. Medical Image Analysis 1998;2(1):61–79. [2] Bisler A, Bockholt U, Voss G. The virtual articulator-applying VR technologies to dentistry. In: Proceedings of the 6th IEEE international conference on informatics and visualisation; 2002. p. 600–2. [3] Ferrario VF, Sforza C, Schmitz JH, Miani A, Serrao GA. 3D computerized mesh diagram analysis and its application in soft tissue facial morphometry. American Journal of Orthodontic and Dentofacial Orthopedics 1998;114:404–13. [4] Soncini M, Pietrabissa R. Quantitative approach for the prediction of tooth movement during orthodontic treatment. Computer Methods Biomechanics and Biomedical Engineering 2002;5(5):361–8. [5] Rodrigues MAF, Silva WB, Barbosa RG, Ribeiro IMMP, Barbosa Neto ME. J-Ortho: an open-source orthodontic treatment simulator. In: Proceedings of the 21st annual ACM symposium on applied computing (ACM SAC), Special track on computer applications in health care 2006. Dijon, France. New York: ACM Press; 2006. p. 245–9. [6] Marcotte M. Biomechanics in orthodontics. BC: Decker; 1990. [7] Ericson C. Real time collision detection. Los Altos, CA, Amsterdam: Morgan Kaufmann, Elsevier; 2005.

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