This guest blog explores how transverse expansion can be planned objectively and how it predictably influences the space analysis. Although the relationship between expansion and arch perimeter is crucial for everyday clinical decision-making, the supporting principles remain scattered across the literature and clinical training. This is the fourth blog post in a series called The Mathematics of Orthodontics, which aims to bring those principles together into a consolidated framework.
This series is written by Daniel Richmond, an orthodontist from Ontario, Canada, who I came across on social media. Daniel has put together an innovative online resource for orthodontic space analysis, the Ortho Calculator, which I found interesting and very useful in terms of ensuring a structured and comprehensive orthodontic assessment for space planning (I have no conflict of interest).
Previous blogs from Daniel Richmond on Space Analysis: Part 1 – How To Calculate, Part 2 – Where Should The Incisors Be Positioned, Part 3 – Incisor Movement
The primary objective of transverse correction is arch coordination, enabling predictable posterior occlusal relationships and function in maximum intercuspation (Graber et al., 2012). Arch coordination is not defined solely by compatible transverse widths of the maxillary and mandibular dental arches; it also requires appropriate buccolingual inclination and cusp heights of posterior teeth. In a coordinated occlusion, posterior cusps should be relatively even, with differences between buccal and palatal/lingual cusp heights within approximately 1 mm (Casko et al., 1998). These cusp height differences create a transverse curvature, known as the curve of Wilson, which facilitates stable occlusion (Figure 1).
Figure 1. Normal buccolingual inclination of posterior teeth demonstrating the curve of Wilson, with a concave curvature in the mandible and a convex curvature in the maxilla.
Achieving this occlusal relationship requires the maxillary and mandibular posterior teeth to be upright and centered within their respective alveolar and basal bone foundations without excessive buccal or lingual tipping (Graber et al., 2012). This can only occur when the basal bone dimensions of the maxilla and mandible are compatible. Transverse coordination, therefore, is not simply a matter of widening dental arches, but of ensuring skeletal compatibility that allows posterior teeth to be positioned upright and function without occlusal interferences.
Transverse coordination is planned by first establishing an appropriate skeletal reference. Because the mandible lacks a patent suture and is therefore immutable, the mandibular basal bone serves as the primary reference for transverse planning. Clinical assessment should first focus on whether the mandibular posterior teeth are upright and centered within the alveolar trough or dentally compensating for a narrow maxillary arch. Lingual inclination of the mandibular molars can mask the true transverse dimension of the mandible, presenting as an accentuated curve of Wilson. Mandibular posterior uprighting—centering the central occlusal grooves over the alveolar troughs—is therefore required to reveal the true transverse width of the mandibular skeletal base (Figure 2).
Figure 2. Mandibular posterior dental compensation masking a maxillary transverse deficiency; the discrepancy becomes evident after posterior uprighting (outline).
Once the mandibular reference has been established, transverse coordination of the maxilla can be planned. The objective is not to achieve a predetermined maxillary width, but to expand the maxillary dentition and skeletal base as needed to match the mandibular foundation. Excessive buccal inclination of the maxillary posterior teeth often reflects dental compensation for an underlying skeletal transverse deficiency, indicating that further dental tipping may be insufficient to achieve arch coordination (Figure 3). Clinician judgment is therefore required to determine whether expansion can be achieved through dentoalveolar methods or if a skeletal approach is necessary (Richmond et al., 2023).
Figure 3. Decompensation of buccally inclined maxillary posterior teeth reveals the true transverse discrepancy and increased expansion requirement.
When the widths between the mandibular alveolar troughs, mandibular molar central occlusal grooves, and maxillary molar palatal cusps are equalized, arch coordination is achieved (Figure 4). With the transverse endpoint clearly defined, the amount of correction required can be quantified.
Figure 4. Ideal transverse arch coordination.
The required amount of expansion can be calculated mathematically by comparing dental widths to the fixed transverse reference of the mandibular basal bone, measured between the centers of the left and right alveolar troughs (Figure 5, red line).
Figure 5. Three transverse measurements needed to plan arch coordination.
The first dental measurement compared to this reference is the intermolar width measured between the central occlusal grooves of the mandibular first molars (Figure 5, blue line). When the mandibular posterior teeth are upright and centered within the alveolar troughs, this measurement should approximate the transverse width of the mandibular basal bone. If the measured width is smaller, lingual dental compensation is present and the discrepancy defines the amount of mandibular posterior uprighting required:
In the maxilla, the relevant dental measurement is the width between the palatal cusps of the maxillary first molars (Figure 5, purple line). In a coordinated transverse relationship, the maxillary molar palatal cusps should align transversely with the planned position of the mandibular molar central occlusal grooves, which correspond to the centers of the mandibular alveolar troughs. Any discrepancy between these measurements defines the amount of maxillary expansion required:
Once the planned transverse correction has been quantified, its contribution to arch perimeter can be incorporated into the space analysis. Clinical studies of palatal expansion consistently demonstrate that maxillary arch perimeter increases by approximately 60–70% of the posterior width change. Adkins et al. (1990) reported an average increase of 0.72 mm of arch perimeter per 1 mm of posterior expansion, with expansion greatest in the molar region and progressively diminishing anteriorly. Akkaya et al. (1998) reported gains of 0.65 mm with rapid maxillary expansion and 0.60 mm with slow expansion, while O’Higgins and Lee (2000) reported a lower estimate of approximately 0.57 mm using a typodont model.
Geometric modeling studies applying Ramanujan’s equation to maxillary dental casts demonstrate a similar relationship, predicting an average increase of approximately 0.73 mm of arch perimeter per 1 mm of intermolar expansion (Aghera et al., 2016; Chung & Wolfgramm, 2015). Collectively, these findings support a clinically useful planning estimate (Figure 6):
Figure 6. Relationship between posterior maxillary expansion and arch perimeter gain.
Unlike the maxilla, the mandible does not undergo true skeletal expansion. Changes in mandibular arch width are achieved primarily through dentoalveolar uprighting of the posterior teeth, resulting in substantially less arch perimeter gain. Three-dimensional biomechanical modeling studies using computer-based simulations demonstrate that a 1 mm increase in posterior mandibular width produces only a modest increase in arch perimeter of 0.30–0.37 mm (Baswaraj et al., 2013; Motoyoshi et al., 2002).
This space gain is concentrated in the canine–premolar region, reflecting the biomechanical constraints of the mandible, where posterior teeth are already in contact and midline separation is not possible the (Figure 7). Clinically, this highlights a key limitation of mandibular arch development: while posterior uprighting can contribute to space creation, its effects are limited and must be weighed against stability considerations when increasing intercanine width.
Figure 7. Relationship between posterior mandibular uprighting and arch perimeter gain.
Expansion should be planned objectively to achieve arch coordination based on each patient’s unique anatomy. Subsequently, its effect on arch perimeter can be incorporated into the space analysis. To support this math-based approach, the author developed the Expansion Calculator (https://www.orthocalculator.org/expansion-calculator) that converts planned expansion into (a) an expander activation prescription and (b) an estimated contribution to arch perimeter.
While transverse correction can generate a measurable increase in arch perimeter, its effects are finite and highly dependent on the arch involved. In the mandible, where transverse change is limited to dentoalveolar uprighting, the amount of space gained is inherently constrained. Accordingly, transverse expansion alone should not be expected to resolve significant crowding or dental protrusion and alternative methods of space creation must be considered. By quantifying the space created through expansion, these decisions can be approached systematically rather than intuitively.
This fourth post explored how transverse expansion can be mathematically planned to achieve arch coordination and how it contributes in a predictable way to the overall space analysis. With both incisor position (Blog post 3) and arch coordination accounted for, any remaining TSALD must be resolved through alternative means of space creation. The subsequent posts in this series will address conservative methods of space creation, followed by extraction-based solutions, completing the mathematical framework needed to ensure that the numbers behind your treatment plan balance out.
Details of expansion can be found at the orthocalculator –
https://www.orthocalculator.org/expansion-calculator.
Contents: Daniel Richmond 
Visit the Ortho Calculator at www.orthocalculator.org/
(https://www.orthocalculator.org/expansion-calculator
