## Abstract

The results of orthognathic surgery may differ significantly from the planned outcome using dental models. The orientation of dental models mounted on articulators using conventional face bows does not accurately replicate the orientation of the patients’ teeth and jaws, but introduces a systematic error. A mathematical analysis showed that the misalignment of the maxillary model introduces errors in the perioperative wafers, which may lead to the incorrect surgical positioning of the maxilla reported in the literature. The results of the mathematical analysis were validated by image analysis of photographs of mounted maxillary models, used to simulate five orthognathic procedures. No significant difference between the experimental results and the theoretical predictions from the mathematical equations was noted. Planning for maxillary forward and upward movement produced more advancement and only 50% of maxillary impaction. Planning for maxillary forward and downward movement produced less advancement and more inferior displacement in relation to horizontal and vertical reference planes.

Surgical correction of dentofacial deformities requires comprehensive planning for accurate positioning of bone segments during surgery, to ensure the preplanned dental occlusion is obtained. Orthognathic model surgery planning should assist the surgeon to reposition the maxilla and mandible into the desired position.

The procedure requires the surgeon, orthodontist, and laboratory technologist to work together closely with a full understanding of the mechanical and biological principles of jaw repositioning. The surgeon prescribes the magnitude of the necessary horizontal and vertical displacement of the maxilla, based on clinical examination and the evaluation of cephalographs. The technologist mounts dental models, generally on a semi-adjustable articulator using a face bow recording. The upper model is then repositioned, horizontally and vertically, by the prescribed displacements, to produce an inter-occlusal wafer to guide the surgeon perioperatively in repositioning the jaws into the predetermined positions .

The outcome of surgery may differ significantly from the planned outcome . All reports, except one provide some numerical data, generally as mean values of the errors due to the discrepancy between the planned and actual outcomes. The errors incorporate the direction, for example over-advancement of the maxilla is reported as a positive number and under-advancement as a negative number. In calculating the sum of the errors, and hence the mean of the positive and negative values, which eliminate each other, this results in a small mean error with a large standard deviation. As a result, statistical tests lead to the misleading conclusion that the mean error is not significantly different from zero. B amber & H arris reported a mean vertical error of 0.0 mm with a standard deviation (SD) ± 1.0 mm, but a range of errors between −2.3 and 2.4 mm. V an S ickels reported a significant mean horizontal error of 3.6 mm.

S harifi et al. demonstrated that 50% of surgical outcomes showed inaccuracies, defined as values greater than 1 SD from the mean. M cCance et al. described individual outcomes compared with predicted outcomes as ‘disappointing’, with errors of up to 6 mm. P ospisil reported that 60% of outcomes showed ‘significant inaccuracies’ defined as errors greater than 20% of the planned movement. P olido & S inn reported that 48% of vertical surgical movements and 29% of horizontal surgical movements had an error of 2 mm or more; the equivalent values reported by J acobson & S arver were 20–30%.

There is strong evidence of inaccuracies in surgical outcomes, but the causes are unclear, because few papers provide details of prediction and planning procedures. P ospisil reported that 33% of inaccurate outcomes were due to surgery deviating from the treatment plan, in 17% the surgery was satisfactory but the treatment plan was inaccurate and in 50% the surgery was satisfactory, but the outcome was inaccurate for undetermined reasons.

Errors in cephalometric technique, inaccurate prediction of the autorotation of the mandible, face bow recording errors, the difference in mandibular position in upright and supine patients and inaccurate surgery have all been suggested as possible causes of the inaccuracies of orthognathic surgery , but no evidence was presented that these errors occur, nor was there any discussion of the magnitude of the outcome errors that each might produce. Only S harifi et al. identified the well-established discrepancy in the orientation of models mounted on articulators using conventional face bows as a possible source of error in surgical outcome.

The vertical and horizontal displacements prescribed by the surgeon are relative to a set of reference axes. The horizontal axis is represented by the Frankfort plane or the visual axis plane with the head in its natural position and the vertical axis at right angles to the horizontal axis; these axes will be called the horizontal reference ( *H* _{r} ) and vertical reference ( *V* _{r} ). Conventional articulators and face bows have been designed to replicate a few interdental occlusal relations, not to reproduce the position of important anatomical features relating to the maxilla and skull . The horizontal axis defined by the upper cross member of the articulator often differs significantly from the reference horizontal axis and anatomical planes such as the Frankfort plane , which is often assumed to be horizontal.

The difference between the axes based on the articulator cross member and on the natural head position is shown in Fig. 1 , in which a maxillary model has been mounted on an articulator cross member that is horizontal. The axis *H* _{a} is parallel to the articulator cross member and *V* _{a} at right angles to *H* _{a} . The reference horizontal and vertical axes, *H* _{r} and *V* _{r} are also shown. Ideally both sets of axes should be the same, but mounting models using a conventional face bow produce the discrepancy of orientation discussed above and there is an angle between the two sets of axes. This is due to the fact that the face bow transfers the maxillary relationship to the upper cross member of the articulator rather than *H* _{r} .

In the laboratory the prescribed displacements of models are made relative to the axes based on the cross member of the articulator, not in relation to the reference horizontal and vertical axes that were used by the surgeon to define the magnitude of deformities of the jaws and the required movement for surgical correction. This produces erroneous and inaccurate occlusal wafers, which are used routinely to reposition the maxillary segment perioperatively.

The magnitude of the inaccuracies depends on the discrepancy between the orientation of the reference horizontal axis, and the horizontal axis based on the articulator upper cross member; differences of up to 20° have been reported .

## Effect of misalignment of axes

Movement of a model relative to axes ( *H *_{a }and *V *_{a }) based on the articulator on which it is mounted also produces a different magnitude of movements of the model relative to the reference axes ( *H *_{r }and *V *_{r }) based on the patient’s natural head position. This is shown in Fig. 2 , in which the reference axes *H *_{r }and *V *_{r }are at an angle to the articulator axes, *H *_{a }and *V *_{a }.

The horizontal axis *H *_{a }is parallel to the articulator cross bar. A horizontal movement of the maxillary model by a distance *h *_{a }(shown as the line AB in Fig. 2 ) produces two movements of the model relative to the reference axes *H *_{r }and *V *_{r }, a movement AC (distance *h *_{r }) parallel to *H *_{r }and a movement CB (distance *v *_{r }) parallel to *V *_{r }; these are horizontal and vertical movements relative to the reference axes. ABC is a right angle triangle with AB as the hypotenuse and hence *h *_{r }is smaller than *h *_{a }; *v *_{r }is greater than *v *_{a }, which is equal to zero for a horizontal movement relative to the articulator axis. Thus, relative to the reference axes the model is under-advanced and over-displaced inferiorly.

Mathematical analysis of the general case of model movements both parallel and at right angles to the articulator axes ( Appendix A ) shows that movements relative to the articulator and to reference axes at an angle to each other are related by the equations

h r = h a cos + v a sin

and

v r = v a cos − h a sin .

The magnitude of the vertical and horizontal errors depends on the magnitude of the movements made relative to the articulator and to the angle between the articulator and reference axes.

The validity of the equations, derived theoretically, were evaluated by an experimental photographic study. A dental model of a normal subject was moved parallel to a horizontal articulator axis and the resulting movements were measured relative to a reference axis determined by image analysis at 20° to the articulator axis. The upper dental model was mounted on an articulator cross bar, that had been set horizontal using a spirit level. A spirit level is a widely used device, defined by the Concise Oxford Dictionary as a ‘device consisting of a sealed glass tube nearly filled with alcohol or other liquid, containing an air bubble whose position is used to test horizontality’.

Five experiments were carried out to simulate impaction, down grafting, advancement, advancement with impaction and advancement with down grafting.

In order to produce defined, accurate and reproducible vertical movements, 10 mm thick washers were used. The model was initially mounted with one washer between the upper model and articulator cross bar. Removal of the washer raised the model relative to the cross bar by 10 mm (impaction); placing a second washer in addition to the initial 10 mm washer lowered the model by 10 mm (down grafting). Horizontal movements were standardized by drilling 3 holes, 10 mm apart, through the cross bar. The model was initially mounted using the central hole, but could be moved 10 mm forwards or backwards by remounting using one of the additional holes.

The photographs for subsequent analysis were taken by a qualified photographer using a Nikon D1x digital camera, with a 105 mm macro lens using NEF files. The positions of the camera and articulator with mounted cast were standardised. The lighting consisted of 2 Broncolor miniplus C80 flash heads placed at 45° angles to ensure even lighting of the subject.

In order to eliminate parallax errors when viewing and photographing the model, a red reference point was marked on the model and two plumb line threads hung vertically in front of the reference mark and aligned so that they were superimposed when viewed at right angles to the model ( Fig. 3 ). A second black reference point was used to define the initial and moved positions of the models.

A photograph was taken of the model in its initial position with the camera aligned such that the two threads were superimposed. The model was moved relative to the articulator cross bar by defined distances and re-photographed, once again eliminating parallax. The digital images were analysed using Adobe Illustrator.

An image of the model in the displaced position was superimposed over the image of the model in the initial position ( Fig. 4 ). The opacity of the superimposed image was reduced to 60% so that the marks on both models were visible. A line was superimposed on the threads to produce a vertical articulator axis. A line was drawn at 70° to this vertical axis to produce a reference axis at 20° to the articulator horizontal axis. The coordinates of the black reference points A and B were recorded and lines parallel and at right angles to the reference axis were drawn through each of the reference points. The lengths of the sides of the resulting rectangle, which were equal to the horizontal and vertical displacement of the models relative to the reference axis, were calculated from the coordinates of the corners of the rectangle. Two estimates of the displacements (vertical and horizontal) were, therefore, obtained from each rectangle. To minimize the errors of the method each measurement was repeated and the mean was used for the analysis.

## Results

The values of the displacements of the models relative to both the articulator and reference axes are shown in Table 1 . The planned vertical surgical movement was not fully achieved, the magnitude of maxillary impaction and downgraft was less than the predicted movement on the articulator. Planning for 10 mm maxillary forward movement without vertical change produced less advancement than predicted and simultaneous downward movement of 3.3 mm. Planning for maxillary forward and upward movement produced more advancement and only 50% of the maxillary impaction in relation to the horizontal and vertical reference planes. Planning for maxillary forward and downward movement produced less advancement and more inferior displacement in relation to the horizontal and vertical reference planes.

Direction and distance (mm) of movement relative to articulator axis | Direction and distance (mm) of movement relative to reference axis | ||
---|---|---|---|

Movement | Measurement | Theory | |

Up |
|||

h _{a }= 0.0 |
H _{r } |
3.40 | 3.42 |

v _{a }= 10.0 |
V _{r } |
9.43 | 9.40 |

Down |
|||

h _{a }= 0.0 |
H _{r } |
−3.62 | −3.42 |

v _{a }= −10.0 |
V _{r } |
−9.34 | −9.40 |

Forward |
|||

h _{a }= 10.0 |
H _{r } |
9.46 | 9.40 |

v _{a }= 0.0 |
V _{r } |
−3.30 | −3.42 |

Forward + up |
|||

h _{a }= 10.0 |
H _{r } |
12.84 | 12.82 |

v _{a }= 10.0 |
V _{r } |
5.83 | 6.00 |

Forward + down |
|||

h _{a }= 10.0 |
H _{r } |
6.25 | 6.00 |

v _{a }= −10.0 |
V _{r } |
−12.84 | −12.82 |

Figure 5 illustrates the relationship between the theoretical predictions applying the equations in the mathematical section of this article and the experimental measurements. The correlation coefficient *R *^{2 }of 0.998 ( Fig. 5 ) indicates that the there is a strong and highly significant linear relationship between the predicted theoretical values of the displacement and the experimental values obtained by image analysis of the photographs.

Calculation of the 95% confidence limits of the values of the intercept and gradient of the line of best fit in Fig. 5 show that the value of 0.0069 for the intercept is not significantly different from 0 and the value of 1.0069 for the gradient is not significantly different from 1. Thus, the best straight line is not significantly different from the line of identity, which proved that the measured value is equal to the theoretical value ( Fig. 5 ).

## Results

The values of the displacements of the models relative to both the articulator and reference axes are shown in Table 1 . The planned vertical surgical movement was not fully achieved, the magnitude of maxillary impaction and downgraft was less than the predicted movement on the articulator. Planning for 10 mm maxillary forward movement without vertical change produced less advancement than predicted and simultaneous downward movement of 3.3 mm. Planning for maxillary forward and upward movement produced more advancement and only 50% of the maxillary impaction in relation to the horizontal and vertical reference planes. Planning for maxillary forward and downward movement produced less advancement and more inferior displacement in relation to the horizontal and vertical reference planes.

Direction and distance (mm) of movement relative to articulator axis | Direction and distance (mm) of movement relative to reference axis | ||
---|---|---|---|

Movement | Measurement | Theory | |

Up |
|||

h _{a }= 0.0 |
H _{r } |
3.40 | 3.42 |

v _{a }= 10.0 |
V _{r } |
9.43 | 9.40 |

Down |
|||

h _{a }= 0.0 |
H _{r } |
−3.62 | −3.42 |

v _{a }= −10.0 |
V _{r } |
−9.34 | −9.40 |

Forward |
|||

h _{a }= 10.0 |
H _{r } |
9.46 | 9.40 |

v _{a }= 0.0 |
V _{r } |
−3.30 | −3.42 |

Forward + up |
|||

h _{a }= 10.0 |
H _{r } |
12.84 | 12.82 |

v _{a }= 10.0 |
V _{r } |
5.83 | 6.00 |

Forward + down |
|||

h _{a }= 10.0 |
H _{r } |
6.25 | 6.00 |

v _{a }= −10.0 |
V _{r } |
−12.84 | −12.82 |

Figure 5 illustrates the relationship between the theoretical predictions applying the equations in the mathematical section of this article and the experimental measurements. The correlation coefficient *R *^{2 }of 0.998 ( Fig. 5 ) indicates that the there is a strong and highly significant linear relationship between the predicted theoretical values of the displacement and the experimental values obtained by image analysis of the photographs.

Calculation of the 95% confidence limits of the values of the intercept and gradient of the line of best fit in Fig. 5 show that the value of 0.0069 for the intercept is not significantly different from 0 and the value of 1.0069 for the gradient is not significantly different from 1. Thus, the best straight line is not significantly different from the line of identity, which proved that the measured value is equal to the theoretical value ( Fig. 5 ).

## Discussion

The angle between the maxillary-occlusal plane angle of the dental model and the horizontal cross bar of the articulator, representing the Frankfort plane, may differ significantly from the angle between the patient’s occlusal plane and the true horizontal plane . This discrepancy has been identified as a source of error when preparing inter-occlusal wafers.

The effect of misorientation of mounted models on the outcome of orthognathic surgery has been identified and discussed by several investigators. E llis et al. assumed that the Frankfort plane was horizontal. The authors reached the qualitative conclusion that impacting the maxilla vertically would produce both vertical and horizontal movements if the maxillary-occlusal plane angle relative to the Frankfort plane was underestimated on the articulator. O’M alley & M ilosovic suggested that the discrepancy between the Frankfort plane and articulator cross member would lead to the incisors appearing either retroclined or proclined.

It was also suggested that for a 10° discrepancy, a 6 mm impaction of the maxilla would produce an unwanted advancement of 1 mm, although it is not clear how this figure was derived and it appears to be produced by a rotation of the upper incisors. The under-impaction that would also occur was not identified. G ateno et al. investigated the magnitude of the error in a single tracing incorporating a 12° discrepancy between axes and a 10 mm maxillary advancement. N atterstad & V edtofte presented a general mathematical analysis of the effect of using different reference planes in model surgery, one being an angular error equivalent to a discrepancy in the orientation of the reference and articulator planes. Only the vertical position was analysed, which underestimated the magnitude of the errors associated with model surgery.

The mathematical analysis in the present study produced equations that are relatively simple to follow. They show that the magnitudes of both the movements of the maxilla relative to the reference axes produced by study model movements relative to the articulator axes, and the resulting errors, depended on three variables: the magnitudes of the horizontal and vertical movements made relative to the articulator and the angle between the upper cross member of the articulator and the reference horizontal. This dependence was not surprising, but the quantification of the errors produced surprisingly large values. When the angular discrepancy is 20° a 10 mm movement can produce a 4 mm error ( Table 1 ; Fig. 4 ) and even a 5 mm movement can produce an error of more than 3 mm. A smaller angular discrepancy will reduce the errors, but even a 10° discrepancy in mounting the maxillary cast on the articulator produces an error of more than 2 mm in a 10 mm forward movement ( Fig. 1 ).

These errors will be incorporated into the inter-occlusal wafer, prepared for the surgeon and used as a template for positioning the maxilla, and may have a seriously deleterious effect on the surgical outcome. In this study, the mathematical analysis has been validated by a photographic study using image analysis of maxillary casts, but simulating orthognathic surgery using model skulls would further validate the analysis and clarify the clinical relevance of the results presented here. It is now apparent that to produce more accurate inter-occlusal wafers requires that models be mounted on an articulator to accurately replicate the orientation of the dental structures seen in the patient. An articulator and face bow system to produce a more accurate replication of the clinical situation has been described , but requires further evaluation.

In conclusion, existing methods for prediction planning of the surgical correction of dentofacial deformities using dental models and currently available articulators are inaccurate. This is mainly due to inaccurate orientation of the maxillary model on the articulator. The extent of inaccuracy is correlated to the magnitude of the movement and the direction of the reference plane from the true horizontal.

## Ethical approval

Not required.

## Acknowledgements

The authors are grateful to James Eyeland, Clinical Photographer and Anne Hanlon-Bucher, Senior Graphic Designer of the Department of Medical Illustration, Southern General Hospital, Glasgow for technical assistance.

## Appendix A

A model is displaced from its initial position O relative to orthogonal axes based on the articulator, with horizontal axis *H *_{a }and vertical axis *V *_{a }. The model is moved by the horizontal and vertical distances *h *_{a }and *v *_{a }, which can be represented in Fig. A1 by the line *r *at an angle to the *H *_{a }axis and:

h a = r cos ε and v a = r sin ε