Novel Method for Superposing 3D Digital Models for Monitoring Orthodontic Tooth Movement

Abstract

Quantitative three-dimensional analysis of orthodontic tooth movement (OTM) is possible by superposition of digital jaw models made at different times during treatment. Conventional methods rely on surface alignment at palatal soft-tissue areas, which is applicable to the maxilla only. We introduce two novel numerical methods applicable to both maxilla and mandible. The OTM from the initial phase of multi-bracket appliance treatment of ten pairs of maxillary models were evaluated and compared with four conventional methods. The median range of deviation of OTM for three users was 13–72% smaller for the novel methods than for the conventional methods, indicating greater inter-observer agreement. Total tooth translation and rotation were significantly different (ANOVA, p < 0.01) for OTM determined by use of the two numerical and four conventional methods. Directional decomposition of OTM from the novel methods showed clinically acceptable agreement with reference results except for vertical translations (deviations of medians greater than 0.6 mm). The difference in vertical translational OTM can be explained by maxillary vertical growth during the observation period, which is additionally recorded by conventional methods. The novel approaches are, thus, particularly suitable for evaluation of pure treatment effects, because growth-related changes are ignored.

Appendix: Derivation of equations

In order to derive a set of equations for EFM that approximates actual OTM for individual teeth from dental models obtained at different stages of treatment, on the basis of mechanical and biomechanical principles, mechanical considerations were reduced to the one-dimensional case for each individual direction of movement. Hence, force F_i acting on a tooth because of translational ITM in direction i is related to an effective stresses state $\bar{\sigma }_{{\text{l}}i}$ by:

$$F_{i} = \bar{\sigma }_{{\text{l}}i} \cdot \bar{A}_{{\text{R}}i} .$$

(5)

Here the effective surface areas $\bar{A}_{{\text{R}}i}$ were approximated as the projection areas of the tooth roots normal to the direction of action, and were taken from the literature (Table 1) because patient-specific data are usually unknown. Because these data do not conform with tooth coordinate systems, to estimate adequate effective projection areas we assumed a simplified box-shaped root and introduced corrections for tooth position for rotation about the mesial and vestibular axes:

$$\begin{aligned} \bar{A}_{\text{Rm}} & = 2 \hat{A}_{\text{m}} \cos \hat{\xi } + \hat{A}_{\text{o}} \sin \hat{\xi } \\ \bar{A}_{\text{Rv}} & = 2 \hat{A}_{\text{v}} \cos \hat{\eta } + \hat{A}_{\text{o}} \sin \hat{\eta } \\ \bar{A}_{\text{Ro}} & = \left( { \hat{A}_{\text{o}} \cos \hat{\xi } + 2 \hat{A}_{\text{m}} \sin \hat{\xi }} \right)\cos \hat{\varphi } + 2 \hat{A}_{\text{v}} \sin \hat{\eta } \\ \end{aligned}$$

(6)

Here, indices m, v, and o represent the axis directions (mesial, vestibular, and occlusal) and the hat symbols indicate literature data. The angles $\hat{\xi }$ and $\hat{\eta }$ are the so-called mean tip and torque angles of the long axes of the different teeth relative to the occlusal direction (Fig. 2); these, also, were taken from the literature (Table 1). A factor of 2 for mesial and vestibular projection areas was added to take into account the presence of both a tension side and an opposing compression side in any type of horizontal movement, whereas only one side and, therefore, one mechanism acts for vertical movement.

A linear relationship between ITM and OTM was chosen such that:

$$ D_{i} = R \cdot \bar{\sigma }_{i} \cdot \Delta T, $$

(7)

where D is linear OTM during a time period ΔT and R represents the rate of remodeling. In contrast with the model proposed by Beaupre et al.,6 which assumes a multi-linear rate of remodeling, our simplified approach assumes a constant rate for resorption and apposition, and continuous and linear movement of all teeth from initial to final positions. We also assumed that the effective stress, $\bar{\sigma }$, is a representative stress stimulus affecting bone remodeling. Combining Eqs. (5) and (7), the force on tooth t can be written in vector notation as:

$$ \varvec{F} = \frac{{\varvec{D} \circ \bar{\varvec{A}}_{\text{R}} }}{R \Delta T}, $$

(8)

where ∘ denotes the Schur product.

The moment exerted on a tooth can be represented by a force couple of the same magnitude but in the opposite direction:

$$ \varvec{M} = \bar{\varvec{l}}_{\text{Fc}} \times \bar{\varvec{F}}_{\text{c}} + \bar{\varvec{l}}_{\text{Fa}} \times \bar{\varvec{F}}_{\text{a}} = \left( {\bar{\varvec{l}}_{\text{Fc}} - \bar{\varvec{l}}_{\text{Fa}} } \right) \times \bar{\varvec{F}}_{\text{c}} . $$

(9)

Here $\bar{\varvec{F}}$ represents effective forces generated in the cervical and apical regions of the tooth root, denoted by indices c and a, respectively, that are separated by the CR. Assuming, also, linear stress distribution with a root at the CR and the projection areas approximated by a rectangle in the cervical region and a triangle in the apical region, the effective lever arms $\bar{\varvec{l}}_{\text{F}}$ of the forces can be given as:

$$ \bar{\varvec{l}}_{\text{Fc}} = - \frac{2}{3}\varvec{l}_{\text{CR}} \,\,{\text{and}}\,\,\bar{\varvec{l}}_{\text{Fa}} = \frac{1}{2}(\varvec{l}_{\text{R}} - \varvec{l}_{\text{CR}} )\,\,{\text{with}}\,\,\varvec{l}_{\text{CR}} = \hat{k}_{\text{CR}} \varvec{l}_{\text{R}} , $$

(10)

where vector $\varvec{l}_{\text{R}}$ describes the tooth axis in the region of the tooth root and $\varvec{l}_{\text{CR}}$ its portion from the alveolar crest to the CR. Both quantities were derived from literature data (Table 1, Fig. 2). Furthermore, on the basis of the definition of the CR and the assumption of a uniform distribution of the effective stress over the root projection area for exertion of a pure force, the cervical and apical areas were considered to be of the same size.

The total rotational OTM Φ was transformed into average translation in both regions, assuming angular movements are generally small, by use of the relationship:

$$ \bar{\varvec{D}}_{\text{r}} \approx \varvec{l}_{{\bar{\sigma }}} \times {\varvec{\Phi}}\,\,{\text{with}}\,\,\varvec{l}_{{\bar{\sigma }_{\text{c}} }} = - \frac{1}{2}\varvec{l}_{\text{CR}} , $$

(11)

where $\varvec{l}_{{\bar{\sigma }_{\text{c}} }}$ is the distance from the CR to the point in the cervical region where the representative stress $\bar{\sigma }$ is present for all directions. Combining Eqs. (5), (7), (9), (10) and (11) and rearranging, the moment exerted on a single tooth can finally be written as:

$$ \varvec{M}_{\text{CR}} = \frac{{ (\varvec{l}_{\text{R}} \times {\varvec{\Phi}}) \circ \bar{\varvec{A}}_{\text{R}} }}{R \Delta T} \times \varvec{l}_{\text{R}} \left( {\frac{{\hat{k}_{\text{CR}} }}{8} + \frac{{\hat{k}_{\text{CR}}^{2} }}{24}} \right)\,\,{\text{with}}\,\,\bar{\varvec{A}}_{\text{Rc}} = - \frac{1}{2}\bar{\varvec{A}}_{\text{R}} . $$

(12)

Similarly to the previous approach, for MRV, linear and rotational movement were resolved and remodeling volumes were estimated individually. Likewise, distinct volumes were calculated separately for all axial directions. Thus, the remodeling volume for pure

$$\varvec{V}_{\text{l}} = \bar{\varvec{A}}_{\text{R}} \circ \varvec{D}.$$

(13)

translation, $\varvec{V}_{\text{l}}$, was approximated for a single tooth by use of:

By following the simplifications described above for rotational movement about the CR, the remodeling volume, $\varvec{V}_{\text{c}}$, in the cervical region was considered to be wedge-shaped whereas $\varvec{V}_{\text{a}}$ in the apical region was approximated as a tetrahedron, by using:

$$ \varvec{V}_{\text{c}} = \frac{{\varvec{A}_{\text{c}} \circ \varvec{h}_{\text{c}} }}{2}\,\,{\text{and}}\,\,\varvec{V}_{\text{a}} = \frac{{\varvec{A}_{\text{a}} \circ \varvec{h}_{\text{a}} }}{3}, $$

(14)

where $\varvec{A}_{\text{c}}$ and $\varvec{A}_{\text{a}}$ represent the projection areas of the tooth root in the cervical and apical regions. Again, assuming small angular movement of the teeth, vector $\varvec{h}$ containing the heights of the remodeling volumes resulting from rotational OTM was approximated by:

$$ \varvec{h}_{\text{c}} \approx \varvec{l}_{\text{CR}} \times {\varvec{\Phi}}\,\,{\text{and}}\,\,\varvec{h}_{\text{a}} \approx (\varvec{l}_{\text{R}} - \varvec{l}_{\text{CR}} ) \times {\varvec{\Phi}} $$

(15)

for cervical and apical regions, respectively. Keeping in mind the assumption that $\varvec{A}_{\text{c}} = \varvec{A}_{\text{a}} = \bar{\varvec{A}}_{\text{R}} /2$, the volume for pure rotation about the CR for each tooth was finally derived by using the equation:

$$ \varvec{V}_{\text{r}} = \varvec{V}_{\text{c}} + \varvec{V}_{\text{a}} \approx \frac{{\varvec{A}_{\text{c}} \circ \left( {\varvec{l}_{\text{CR}} \times {\varvec{\Phi}}} \right)}}{2} + \frac{{\varvec{A}_{\text{a}} \circ \left( {\left( {\varvec{l}_{\text{R}} - \varvec{l}_{\text{CR}} } \right) \times {\varvec{\Phi}}} \right)}}{3} = \bar{\varvec{A}}_{\text{R}} \circ \left( {\varvec{l}_{\text{R}} \times {\varvec{\Phi}}} \right)\left( {\frac{1}{6} + \frac{{\hat{k}_{\text{CR}} }}{12}} \right). $$

(16)