This blog explores space analysis in orthodontics, a summary of the processes essential to orthodontic planning. While the concept of space analysis has guided clinicians for decades, the principles are scattered across research and clinical training. This blog, and subsequent series of blogs, consolidates these principles. This blog will explore space analysis of crowding, Curve of Spee, transitioning primary to secondary dentition and prosthetic tooth space requirements, with key formulae for each topic, based on hallmark orthodontic references.
The blog is part of a series of blogs on the Mathematics of Orthodontics by Daniel Richmond, an orthodontist from Ontario, Canada I came across on social media. Daniel has put together an innovative online resource for orthodontic space analysis, the Ortho Calculator, I found the website interesting and very useful in terms of ensuring a structured and comprehensive orthodontic assessment for space planning (I have no conflict of interest).
Space analysis is a fundamental step to ensure that all teeth can be accommodated within the available arch space. By calculating the Tooth Size Arch Length Discrepancy TSALD, it can be determined whether space needs to be created through techniques such as arch expansion, IPR, or extractions. While crowding and spacing are the most apparent concerns, additional space is often required for leveling the Curve of Spee (COS) and addressing dental protrusion. A precise space assessment serves as the foundation for subsequent treatment decisions.
At the most basic level, crowding and spacing are measured relative to the arch line that reflects the majority of teeth (Figure 1) (Kirschen et al. 2000). To avoid confusion, this blog will define arch perimeter as the curved measurement along the arch form, reflecting space available for alignment, while arch length will refer to the linear distance from molars to incisors along the midline. Several reliable methods are used to quantify crowding and spacing. Traditional approaches include a direct visual assessment of contact points, where overlaps or gaps between teeth are evaluated. A more structured approach involves brass wire measurements, which approximate the arch shape and provide a line of best fit. This measurement is then compared to the cumulative mesiodistal widths of all teeth, measured individually with calipers.
Figure 1. Crowding and spacing are assessed relative to an arch line, serving as a reference for space analysis.
More recently, digital models have provided an alternative method for analyzing space discrepancies (Stevens et al. 2006; Zilberman et al. 2003). AI-driven segmentation allows for precise tooth-width calculations and digital space assessments. Studies suggest that crowding measurements tend to be slightly lower on digital analysis compared to traditional cast models, though these differences are typically not clinically significant (Stevens et al. 2006; Yoon et al. 2018). Regardless of the method used, mathematical calculations remain the foundation of accurate space analysis.
The Curve of Spee (COS) refers to the natural curvature of the mandibular dentition observed in the sagittal plane. Levelling the COS requires space, often attributed to multiple slipped contact points of teeth, which have convex rather than flat surfaces (Kirschen et al. 2000). As these contacts align during levelling, the arch length increases (Figure 2). However, this concept of slipped contacts is debated; Woods (1986) argues that the slippage of contact points is insignificant and that teeth should be able to slide vertically to achieve levelling without affecting arch space if mechanics are planned carefully. Regardless, leveling mechanics frequently lead to proclination of the lower incisors, which further affects space requirements. Pandis et al. (2010) found that for every 1 mm of COS leveling, mandibular incisors proclined by approximately 4 degrees, reinforcing the need to account for secondary space demands.
Figure 2. During leveling of the Curve of Spee, slipped contact points are thought to align, increasing arch length and overall space requirements. Adapted from Kirschen et al., 2000.
Numerous studies have investigated the amount of space required to level the Curve of Spee. Braun et al. (1996) reviewed the literature, highlighting differences in both the magnitude of space needed and whether the relationship follows a linear or non-linear pattern. Germane et al. (1992) found a non-linear relationship, indicating that less than 1 mm of arch perimeter is needed per 1 mm of COS leveling. The Royal London study reported 1 mm of space required for 3 mm of COS depth, 1.5 mm for 4 mm, and 2 mm for 5 mm (Kirschen et al. 2000). These variations suggest that space requirements depend on multiple factors, including individual tooth positions and specific mechanics used—whether bodily movement, relative intrusion, or a combination. Collectively, studies suggest the following approximation:
As the dentition transitions from mixed to permanent, space becomes available in the buccal segments as larger primary molars are replaced by smaller premolars. This difference, known as leeway space, is primarily attributed to the size discrepancy between primary second molars (Es) and second premolars, leading to its common reference as “E space.” According to Proffit, the average E space per quadrant is 1.5 mm in the maxilla and 2 mm in the mandible (Figure 3) (Proffit et al. 2019). A more precise approach involves comparing primary and permanent tooth sizes on radiographs to determine the actual discrepancy.
Figure 3. Leeway space: the size discrepancy between larger primary molars and smaller premolars. Adapted from Profit et al., 2019.
However, in untreated dentitions, leeway space is rarely preserved due to the late mesial shift; after Es exfoliate, first molars (M1s) tend to drift mesially into the available space, rather than incisors and canines drifting distally to alleviate crowding (Figure 4) (Brennan and Gianelly 2000). To prevent mesial drift and preserve space for alignment, a space maintainer should be placed before exfoliation (Bell and Sonis 2014).
Figure 4. Reduction in arch length and arch perimeter during the transition from mixed (light blue) to permanent (dark blue) dentition. Adapted from Bell and Sonis, 2014.
In the maxilla, the Nance appliance effectively preserves leeway space by holding M1s in place using the palate for anchorage (Terlaje and Donly 2001; Watt et al. 2018). A transpalatal arch can also be considered; however, its effectiveness in cases of bilateral primary molar loss is debated, as both M1s may drift mesially together rather than maintaining their positions (Laing et al. 2009; Terlaje and Donly 2001). In the mandible, the lower lingual arch (LLA) is a well-studied and reliable option to maintain leeway space. It preserves arch length by preventing mesial drift of M1s and limiting lingual tipping of lower incisors (Khalaf et al. 2022; Rebellato et al. 1997). LLAs can preserve up to 4–5 mm of space, facilitating non-extraction treatment in cases of moderate crowding (Chen et al. 2019; Gianelly 1995). The preservation of E space has been suggested as a contributing factor to the decline in extraction rates, which decreased from 35% in 1986 to 18% in 2008 (Keim et al. 2008; Sonis and Ackerman 2011).
When teeth are missing or microdontic (e.g., peg laterals), space may collapse, necessitating proper space redistribution. Several mathematical methods help determine the ideal space for prosthetic work. The contralateral tooth can serve as a guide, or standardized approaches like the Bolton analysis (Figure 5) can be used to ensure proportionality between the maxillary and mandibular dentition (Bolton 1958).
Figure 5. Anterior Bolton analysis comparing maxillary and mandibular tooth size proportions.
Another method, proposed by German et al. (2016) establishes balance among anterior teeth using the following relationships (Figure 6):
Figure 6. Mathematical relationships in the anterior dentition. Adapted from German et al., 2016.
The Golden Proportion, a mathematical ratio commonly observed in nature, has also been applied to dental esthetics. This principle suggests that the width of each anterior tooth follows a predictable reduction pattern when viewed from the anterior (Del Monte et al. 2017):
When planning for implants, a minimum of 1.5 mm of space is required on either side to preserve bone levels, facilitate papilla formation, and optimize esthetic outcomes (Kinzer and Kokich 2005).
This first section has outlined a structured approach to establishing a baseline for crowding and spacing, considering contact point discrepancies, the influence of the COS, the role of leeway space, and space planning for prosthetic restorations. These calculations provide a foundation for diagnosis and guide key treatment planning decisions.
The next blog will focus on incisor positioning—how to determine ideal placement and its impact on space requirements.
Contents: Daniel Richmond 
Visit the Ortho Calculator at www.orthocalculator.org/
