Introduction

Oral diseases are a kind of common multiple disease. According to the World Health Organization statistics, malocclusion has become one of the three major oral diseases (dental caries, periodontal diseases and malocclusion). For traditional malocclusion treatment, the orthodontic plan mainly relies on a doctor’s experience. After the confirmation of a therapeutic plan, a manual tooth arrangement experiment may help the orthodontic doctor estimate the therapeutic process involved in orthodontic treatment as well as inform the patient of possible tooth movement and final treatment effect. The major defects of manual tooth arrangement consist of low efficiency and large consumption of materials.

In recent years, many personalized virtual orthodontic technologies have become available for tooth arrangement and bracket positioning. Generally, they can be divided into the following types:

  1. (1)

    Virtual measurement and comparative analysis: The main function of a virtual teeth model lies in the measurement and analysis of the model. One approach considers comparative analysis before and after tooth arrangement. Min-Young Cho et al. [1] superimposed the models through Rapidform, and utilized the 3Txer program to measure the angles between the teeth and the occlusal plane or midline. Another approach is to compare between the models before and after manual tooth arrangement. Joon Im et al. [2] measured certain values, such as arch length and width of two models through Rapidform for comparison. These two approaches prove that the result obtained through virtual tooth arrangement is acceptable compared with that obtained through manual tooth arrangement [3, 4]. However, neither approach accounts for the dental root, which leads to the failure to guarantee the relative position between the dental root and alveolar bone.

  2. (2)

    Calculation of tooth movement with mechanical analyses: During orthodontic treatment, tooth movement is the result of the interactions among arch wires, periodontal ligaments and bones. Yukio Kojima and Hisao Fukui [5] adopted finite element analysis to study stress of teeth under arch wires and predict tooth movement. The use of this method while focusing on the movement of a single tooth neglects the stress transfer and displacement change of the whole dentition during orthodontic treatment.

  3. (3)

    Simulation of the virtual tooth arrangement process: Simulation of the tooth arrangement process by virtue of the existing software is a type of research [6–11]. Incognito can split the dental model, and the tooth arrangement process can be completed through curve shape adjustment of the dental arch [12, 13]. Orapix 3D can also split the teeth model and arrange the teeth through interactive adjustment of each tooth’s position [14–16]. Another type of research relates to the development of the tooth arrangement process simulation in a new system. Based on the drawing principle of the Bonwill-Hawley individual ideal dental arch form, Nobuyoshi Motohashi and Takayuki Kuroda [17] drew the dental arch curve and placed the dental crown models, which had been split on the arch curve in order. In the aforementioned studies, the existing software used or those systems developed by the authors considered only dental crown information, which can possibly result in the dental root outside the alveolar bone.

  4. (4)

    Establishment of a complete teeth model: Most orthodontic software still operates only on crown data, and few applications simulate root data with specific algorithms to simulate tooth movement [18]. This kind of simulation is inadequate for dental procedures. Hong-Tzong Yau et al. [19] reconstructed the dental root CT image into point cloud data through a level set algorithm and utilized iterative closest point algorithm to integrate dental root and dental crown data.

  5. (5)

    Tooth-arrangement for complete denture: Yuchun Sun et al. [20], Yong-de Zhang et al. [21–23] and Manabu KANAZAWA et al. [24] significantly contributed to the tooth-arrangement method for complete denture. In particular, Sun and Zhang proposed the principle of Tooth-arrangement. In their research, they considered that the occlusal plane halves the occlusal space, and artificial teeth should be arranged on the alveolar ridge. They also researched the jaw arc configuration and the position relationship of the upper and lower jaw arcs. The curve of the jaw arch and dental arch should be constructed, and an individual tooth is arranged side by side in full denture in close side. Their research provides guidance in the study of tooth-arrangement method.

In conclusion, the existing methods have the following problems. (1) The lack of consideration to the dental root during measurement and analysis may cause the dental root to move outside of the bone cortex during and after treatment. (2) Rough tooth arrangement and inadequate constraints cause low accuracy of tooth arrangement. (3) Low automated tooth arrangement process causes low efficiency of tooth arrangement.

This presentation aims to introduce newly developed software for the computer simulation of the diagnostic cast by demonstrating each clinical processing step. Compared with the manual tooth arrangement, the virtual arrangement does not split the plaster model, and the tooth arrangement process is based on feature constraint driving, without a separate operation for each tooth. With a high automation degree, the tooth arrangement efficiency will be improved accordingly. The relation of position between the dental root and alveolar bone is integrated into the algorithm design, and the dental crown and root information is comprehensively considered. Therefore, a more appropriate and accurate therapeutic plan is designed to prevent the dental root from protruding out of the alveolar bone, and such problems during the growth process of teeth can be avoided.

Materials and Methods

The dental models in the article were provided by the Department of Orthodontics, School of Stomatology, Peking University. The patient was a young and healthy subject without previous orthodontic or orthopedic treatment histories. The clinical symptom of the patient was mandibular dental crowding. The participant voluntarily joined the study and provided informed consent.

Root information can be extracted from digital imaging and communications in medicine data by the level set method. The crown data were acquired from plaster models and segmented into individual teeth. The plaster model was positioned according to the referencing model, so the point clouds of root and crown were clustered together exactly. Finally, a full dental model was obtained.

The software to automatically align the individual teeth is newly developed for the computed simulation of diagnostic cast using Visual C++. The automatic tooth arrangement algorithm for computer simulation is demonstrated in Fig. 1.

Fig. 1
figure 1

Pipeline of tooth arrangement algorithm

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Construction of Dental Feature Constraint Database

A precise mathematical description of teeth is very difficult because of the irregular shape of teeth. However, from an anatomical perspective, each tooth has some relatively definite feature points. Definitions of feature points should not only embody dental morphological characters fully so that a coordinate system can be structured, but also restrict teeth to arrange teeth. Subsequently, specific positions are formed using a coordinate system. Meanwhile, the boundary conditions of dental restriction comprise feature points. Feature points are marked by an orthodontist.

Teeth are divided into four main groups based on function and characteristics: incisor, canine, premolar and molar. Figure 2 shows the definitions of feature points of four kinds of teeth.

Fig. 2
figure 2

Definitions of feature points on the models of four kinds of teeth: (a) and (b) Definitions of feature points on the model of the incisor: 1, distal occlusal point; 2, mesial occlusal point; 3, mesial point; 4, bracket point; 5, distal point; 6, root point; 7, fossa point. (c) Definitions of feature points on the model of the canine: 1, cusp point; 2, mesial point; 3, bracket point; 4, distal point; 5, root point. (d) Definitions of feature points on the model of the premolar: 1, lingual cusp point; 2, buccal cusp point; 3, mesial point; 4, distal point; 5, bracket point; 6, root point. (e) Definitions of feature points on the model of the molar: 1, lingual cusp point; 2, the mesiobuccal cusp tip; 3, the distalbuccal cusp tip; 4, mesial point; 5, distal point; 6, bracket point; 7, root point 1; 8, root point 2, the middle point of point 7 and point 8 for root point

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Feature constraints of each tooth constitute tooth position, tooth type, coordinates of feature points and index number of feature points. Feature constraints of all teeth are also included into the feature constraint database, as shown in Fig. 3.

Fig. 3
figure 3

Feature constraint database

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Tooth Arrangement Based on Feature Constraint Driving

Mandibular Tooth Arrangement

  1. Step 1

    Location of mandibular central incisors. An object mover function is developed to rotate the object about three axis and slide the object through space along the three axis. With mandible and maxilla models, the positions and orientation of mandibular central incisors are adjusted interactively using the object mover. This step is based on the experience of doctors, as shown in Fig. 4.

    Fig. 4
    figure 4

    Translucent mandible and maxilla and central incisors with the object mover: (a) Front view. (b) Left view

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  2. Step 2

    Adjustment of initial teeth positions. The important slot plane is constructed using the midpoint of the bracket points of mandibular central incisors and bilateral mandibular first molars. The normal component Np of the slot plane is defined. Positions and pose of mandibular teeth are adjusted such that the teeth bracket points are on the slot plane, and the long teeth axes are superposed with Np, as shown in Fig. 5.

    Fig. 5
    figure 5

    Relationship between slot plane and teeth: (a) Original position. (b) Teeth positions along the plane normal component are adjusted to situate bracket points on the slot plane

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  3. Step 3

    Construction of dental arch curve. The projection points in the midpoint of the mesial points of mandibular central incisors, the cusp points of bilateral mandibular canines and the distal points of bilateral mandibular first molars are defined as model points. The dental arch curve is simulated by B-spline curve (Eq. (1)).

    $$ p(t)={\displaystyle \sum_{i=0}^n}{Q}_i{N}_{i,k}(t),\kern0.5em t\in \left[0,1\right] $$
    (1)

    In the formulae, p(t) refers to model point, Q i is control point, t is node value, n is number of control points and N i,k (t) is k-degree B-spline basis function (Eqs. (2) and (3)).

    $$ {N}_{i,1}(1)=\left\{\begin{array}{l}1,\kern0.5em {t}_i\le t\le {t}_{i+1}\hfill \\ {}0,\kern0.5em else\hfill \end{array}\right. $$
    (2)
    $$ {N}_{i,k}(t)=\frac{\left(t-{t}_i\right){N}_{i,k-1}(t)}{t_{i+k-1}-{t}_i}+\frac{\left({t}_{i+k}-t\right){N}_{i+1,k-1}(t)}{t_{i+k}-{t}_{i+1}} $$
    (3)
  4. Step 4

    Automatic tooth arrangement. B-spline curve is constructed with the control points reversed by five model points. B-spline curve is edited according to expertise so that a suitable arch can be obtained, as shown in Fig. 6. Based on tooth width, the position of the corresponding string of teeth on the arch is obtained by iterative calculation. Subsequently, the teeth can be automatically arranged. In the figure, (xi,yi) is the projection of the distal point of tooth on the slot plane, and (xi − 1,yi − 1) is the projection of the mesial point on the slot plane. Location information of each tooth can be defined as Ti = {Tmesial, Tdistal} (i = 1, 2, …, N). Ti is the location information of the tooth, Tmesial is the location of the mesial point, Tdistal is the location of the distal point and N is the number of teeth. Fig. 7 shows the result of tooth arrangement. According to the orthodontist, the dental arch curve can be adjusted by moving the control points on the slot plane to establish the optimum occlusal relationship. The result of tooth arrangement changes accordingly, as shown in Fig. 8. Written in pseudo-code, the iterative algorithm is shown as follows:

    Fig. 6
    figure 6

    Dental arch and dental model: (a) Dental arch and the positions of teeth before arrangement; (b) Location of teeth and dental arch

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    Fig. 7
    figure 7

    Location of teeth and dental arch after tooth arrangement: (a) Translucent model; (b) Opaque model

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    Fig. 8
    figure 8

    Tooth arrangement after adjusting the arch curve: (a) Adjusted dental arch curve; (b) Result of tooth arrangement

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    figure a
  5. Step 5

    Construction of root plane and root curve. The above method used dental crown information only, and teeth roots cannot be guaranteed in the alveolar bone. According to experts, the root plane is parallel to the slot plane, and the two planes are 15 mm apart. The projections of mandibular teeth root points are designed as model points p(t). Thereafter, the root curve is constructed by the B-spline curve, as shown in Fig. 9.

    Fig. 9
    figure 9

    Root plane, root curve and tooth arrangement with root information: (a) Root plane and root curve. (b) Unadjusted tooth root. (c) Adjusted tooth root

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  6. Step 6

    Modification of root position. The roots should be guaranteed in the alveolar bone, and the above constraints are always maintained. Using the line between Tmesial and Tdistal as the axis, the tooth can be rotated about the axis by moving the point p(t) to ensure root in the alveolar bone, and the root curve is edited in real time simultaneously. Mandibular teeth are completely arranged after editing the teeth roots. The middle plane is then established so that the position information of mandibular central incisors is delivered to the maxillary central incisors (Fig. 10).

    Fig. 10
    figure 10

    Orientation of maxillary central incisors: (a) Situation and posture information of mandibular central incisors delivered by the middle plane. (b) Overjet and overbite relation (A is the mesial occlusal point of right mandibular central incisor, A0 is the fossa point of right maxillary central incisor, B is the mesial occlusal point of right maxillary central incisor). (c) Maxillary central incisor model after adjustment

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Maxillary Tooth Arrangement

  1. Step 1

    Location of maxillary central incisors. The positions of maxillary central incisors are adjusted relative to the middle plane so that Tmesial is on the middle plane. According to overbite and overjet relation, the positions of bilateral maxillary central incisors are confirmed using the projections of mesial occlusal points of mandibular central incisors, fossa points and mesial occlusal points of maxillary central incisors.

  2. Step 2

    Adjustment of initial maxillary teeth positions. Similar to mandibular tooth arrangement, the maxillary slot plane is constructed. The positions and pose of maxillary teeth are adjusted so that the bracket points of teeth are on the slot plane, and the long teeth axes are superposed with the normal slot plane component.

  3. Step 3

    Construction of maxillary dental arch curve and arrangement of maxillary teeth. The occlusion problem must be considered. By projecting the result of expanding mandibular dental arch by 2 mm along its normal direction onto the maxillary slot plane, the maxillary dental arch curve is constructed (Fig. 11). Written in pseudo-code, the maxillary arch algorithm is shown below:

    Fig. 11
    figure 11

    Set up of maxillary teeth: (a) Maxillary dental arch and translucent dental model. (b) Tooth arrangement with root information

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    figure b
  4. Step 4

    Modification of root position. The construction of the maxillary root plane and root curve is the same as the mandibular tooth arrangement. Root curve and pose of maxillary teeth are edited in real time by moving the model points. Consequently, maxillary tooth arrangement is accomplished.

Experimental Results

The visual data show that the tooth arrangement algorithm can achieve better result (Figs. 12 and 13). Combining the translucent skull models, the parts of all teeth roots can be observed in the alveolar bone (Fig. 14). The total tooth arrangement time is about 10 min.

Fig. 12
figure 12

Process of maxillary tooth arrangement: (a) Original teeth. (b) After adjusting the location of the maxillary central incisors. (c) After adjusting the initial pose of maxillary teeth. (d) After arranging the maxillary teeth

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Fig. 13
figure 13

Process of mandibular tooth arrangement: (a) Original teeth. (b) After adjusting the location of the mandibular central incisors. (c) After adjusting the initial pose of the mandibular teeth. (d) After arranging the mandibular teeth

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Discussion

The relative position between teeth roots and alveolar bone, as well as the difference between manual and virtual setup models, is quantitatively analyzed through linear measurement.

Analysis of Arch Form

Figure 15 describes the reference lines and landmarks for the linear measurements, and Table 1 shows the measurement results.

Fig. 14
figure 14

Result of tooth arrangement: (a) Occlusal relationship. (b) Dentition after tooth arrangement. (c) Dentition and translucent skulls

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Table 1 Comparison of arch dimensions between manual and virtual setups
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The measurement results of the manual setup model closely resemble that of the virtual setup model. Results show that the arch width of manual setup model is wider than that of the virtual setup model at about 0.1-0.9 mm. The arch length of the manual setup model is 0.3 % longer than that of the virtual setup model. The arch perimeter of the manual setup model is 1.2 % longer than that of the virtual setup model. Thus, the virtual setup model is tighter than the manual setup model.

Analysis of the Positions of Teeth Roots

An innovation in this study is the application of a complete dental model with information of teeth roots to tooth arrangement, which is hardly reported in the literature. In the course of measuring the relative position between teeth roots and alveolar bone, the concept of root tip distance is introduced. Root tip distance is the minimum distance from the root tip to alveolar bone (Fig. 16). The root tip distance is positive when the root tip is in the alveolar bone, but negative when the root tip is outside the alveolar bone (Table 2).

Fig. 15
figure 15

Measurement values for arch dimensions: ICW, intercanine width, distance between the cusp tips of the right and left canines; IP1W, interfirst premolar width, distance between the cusp tips of the right and left first premolars; IP2W, intersecond premolar width, distance between the cusp tips of the right and left second premolars; IM1W, interfirst molar width, distance between the mesiobuccal cusp tips of the right and left first molars; IM2W, intersecond molar width, distance between the mesiobuccal cusp tips of the right and left second molars; AL, arch length, distance between the midpoint of the occlusal point of the central incisor and the midpoint of the distal point of the second molars; AP, arch perimeter, perimeter between the distal points of the right and left second molars; OJ, overjet, the horizontal distance between the midpoint of the upper and lower central incisors’ occlusal points; OB, overbite, the vertical distance between the midpoint of the upper and lower central incisors’ occlusal points

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Fig. 16
figure 16

Root tip distance (RTD) is the minimum distance between the root tip and alveolar bone: (a) Root tip in the alveolar bone, “+” distance. (b) Root tip outside the alveolar bone, “-” distance

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Table 2 Root tip distance: the minimum distance between root tip and alveolar bone
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Results show that the root tips of the virtual setup model are all in the alveolar bone. Root tip distances of bilateral maxillary central incisors are relatively large. Bilateral maxillary central incisors incline their roots towards the skull. The present tooth arrangement methods are mainly about crown information, whereas root information is not considered. Thus, the negative effect of these methods is that teeth roots are outside of the alveolar bone. This phenomenon tends to be overlooked during therapy.

Efficiency Analysis

In comprehensive consideration of information of the crown and root of the dental model, the tooth arrangement method designed in this paper realizes automation of tooth arrangement. The total tooth arrangement time is 10 min, and manual setup takes about one week. Without handling plaster casts, this method does not only increase the tooth arrangement efficiency, but also lowers cost. With rapid prototyping technology, arch wire and brackets can be rapidly manufactured simultaneously.

Conclusions

Orthodontic treatment features complex tooth arrangement, long treatment cycle and unpredictable outcome. However, in traditional orthodontic treatment, dentists consider only the relationship among dental crowns and neglect the root condition, which might result in the hidden danger of unpredictable long-term treatment. Based on the complete dental model containing information on teeth roots, this paper proposes a novel tooth arrangement system that is applied in an orthodontic treatment plan to design a comprehensive tooth arrangement algorithm with automated teeth crown contact and teeth root pose. This system has several advantages. (1) The system is characterized by highly efficient tooth arrangement. This method of tooth arrangement can achieve automated tooth arrangement, with an operation time of about 10 min. (2) The system considers the positions of teeth roots, thereby avoiding the hidden danger that teeth roots might move to outside of the cortex in or after treatment. (3) The system has a stable quality of tooth arrangement, and the outcome is similar to that of manual tooth arrangement.

However, some problems still exist for this system; for example, the automatic level should be raised. In future studies, many aspects, such as the feature points, should be automatically flagged for the dental feature constraint database, and the relationship between root tip and alveolar bone must be automatically detected.