OGSCalc: Mathematical formulae and web-based application to incorporate rotational discrepancies into translational discrepancies for assessment of accuracy in orthognathic surgery

This paper presents a mathematical formula and a corresponding web-based application that integrate rotational discrepancies into translational measurements to provide a more accurate, comprehensive assessment of orthognathic surgical outcomes.

The Gist

Imagine you are trying to hit a bullseye on a dartboard. In the past, when surgeons checked how well they hit the target during jaw surgery, they looked at two separate things:

1. How far off the center you were (Translation: Did you hit left, right, up, or down?).
2. How much the dart was spinning (Rotation: Did the dart tilt forward, backward, or twist?).

The problem is, these two things are secretly best friends. If your dart is spinning (rotating), it changes where the tip of the dart actually lands. But surgeons were usually reporting these as six separate numbers (three for moving, three for spinning). It's like trying to describe a car crash by saying, "The car was 5 miles per hour too fast" and "The steering wheel was turned 10 degrees," without realizing that the turned steering wheel caused the car to go off course.

The "OGSCalc" Solution

The authors of this paper, Jonas Hue and his team, realized that to truly know if a surgery was successful, you can't just look at the "moving" numbers and the "spinning" numbers separately. You have to mix them together to see the real picture.

Here is how they did it, using some simple analogies:

1. The "Twisted Rope" Analogy

Imagine you have a rope tied to a post. If you pull the rope straight, the distance is easy to measure. But if you twist the rope (rotate it) while pulling, the end of the rope doesn't land where you thought it would.

The authors created a mathematical recipe (a formula) that acts like a "twist-correction calculator." It takes the amount of twist (rotation) and figures out exactly how much that twist pushed the end of the rope (the jaw) off its intended path. This allows them to calculate a single, "true" distance of error, rather than six confusing numbers.

2. The "Magic Spreadsheet" (The App)

Math formulas can be scary and hard to do by hand, especially if you have data for 50 or 100 patients. Doing this on a calculator would take forever and be prone to errors.

So, the team built a free, easy-to-use website (called OGSCalc). Think of it as a "magic spreadsheet":

• You bring the ingredients: Surgeons upload a simple Excel file with their patient data (how much the jaw moved, how much it rotated).
• The app does the cooking: The website instantly runs the "twist-correction" math on all the patients at once.
• You get the meal: It spits out a new table showing the corrected accuracy.

Why Does This Matter?

Imagine two surgeons, Dr. A and Dr. B, are competing to see who has the better technique.

• Dr. A is very precise but has a tiny bit of twisting error.
• Dr. B is slightly less precise with moving but has zero twisting error.

If you just look at the "moving" numbers, Dr. B might look better. But if you account for the fact that Dr. A's twisting actually helped land the jaw closer to the target, Dr. A might actually be the winner.

Without this tool, comparing different surgical techniques is like comparing apples and oranges because the "twist" factor is different for everyone. This tool levels the playing field, allowing doctors to say, "Technique X is truly better than Technique Y," with much more confidence.

The Bottom Line

The authors built a translator that turns complex 3D geometry into a simple "Total Error Score."

• For Surgeons: It helps them understand exactly how good their surgery was, accounting for all the twists and turns.
• For Science: It helps compare new, fancy surgical methods against old ones fairly.
• For the Future: The authors joke that as surgery gets perfect, the "twist" will disappear, and they won't need this tool anymore. But until then, it's a vital helper for making jaws fit together perfectly.

In short: They took a confusing 6-part puzzle, figured out how the pieces fit together, and built a free app so anyone can solve it in seconds.

Technical Summary

1. Problem Statement

In orthognathic surgery, assessing surgical accuracy involves comparing post-operative outcomes against a virtual surgical plan. Currently, this assessment relies on six distinct metrics:

• Three translational discrepancies: Anterior-Posterior (AP), Superior-Inferior (SI), and Medial-Lateral (ML).
• Three rotational discrepancies: Pitch, Roll, and Yaw.

The Core Issue: These six values are typically reported as separate, independent numbers. However, translational and rotational discrepancies are inherently linked in 3D space. A rotation about a specific axis physically alters the 3D position of a reference point, thereby influencing the measured translational discrepancy in one or more planes.

• Example: A pitch rotation affects the AP and SI discrepancies; a yaw rotation affects the AP and ML discrepancies.
• Consequence: Reporting these values separately makes it difficult to compare surgical techniques or evaluate overall accuracy, especially when one technique excels in translation but fails in rotation (or vice versa). There is currently no standardized method to quantify how rotational errors "contaminate" translational measurements to derive a unified accuracy metric.

2. Methodology

The authors developed a two-pronged approach: a mathematical derivation to correct the data and a software tool to automate the process.

A. Mathematical Derivation

The authors utilized Euclidean geometry and quadratic functions to model the relationship between rotation and translation.

• Geometric Model: They modeled the discrepancy as a chord length on a circle. The "true" discrepancy ($d + x$) acts as the radius ($r$), where $d$ is the measured translational discrepancy and $x$ is the correction factor needed to account for rotation ($\theta$).
• Derivation Steps:
  1. Used the chord length formula: $c = 2r \sin(\theta/2)$.
  2. Applied the Pythagorean theorem to two right-angled triangles formed by the rotation and translation vectors.
  3. Combined the equations to eliminate the intermediate height variable ($h$).
  4. Reduced the resulting equation into a quadratic form to solve for the correction factor $x$.

The Final Equation:
The correction $x$ is derived by solving the following quadratic equation for $x$, given the translational discrepancy $d$ and rotational discrepancy $\theta$ (in radians):
$$[4(\sin(\theta/2))^2 - 2]x^2 + [8d(\sin(\theta/2))^2 - 2d]x + 4d^2(\sin(\theta/2))^2 = 0$$

• Application: This calculation is performed iteratively for each translational axis, correcting for the two rotational axes that influence it (e.g., AP is corrected for Pitch and Yaw).
• Total Discrepancy: Once the corrected translational values ($d_{AP}, d_{SI}, d_{ML}$) are obtained, a single "Total Discrepancy" value is calculated using the Euclidean distance formula:
  $$Total \ Discrepancy = \sqrt{d_{AP}^2 + d_{SI}^2 + d_{ML}^2}$$

B. Web-Based Application (OGSCalc)

To make these complex calculations accessible to clinicians, the authors developed a web application using the R programming language and the Shiny package.

• Input: Users upload an Excel (.xlsx) file containing 7 columns: Case ID, AP, SI, ML, Pitch, Roll, and Yaw.
• Processing: The app automatically applies the quadratic correction formula to the uploaded data.
• Output: Generates a downloadable CSV file containing the corrected translational discrepancies and the unified Total Discrepancy.
• Accessibility: The source code is open-source (GitHub), and the tool is hosted at a public URL.

3. Key Contributions

1. Novel Mathematical Formula: The first derivation of a mathematical model that quantifies the specific impact of rotational discrepancies on translational measurements in 3D space, converting them into a unified correction factor.
2. Unified Metric: The ability to collapse 6 separate data points (3 translations + 3 rotations) into a single, clinically meaningful "Total Discrepancy" value that accounts for the interdependency of these variables.
3. Open-Source Tool: The creation of OGSCalc, a user-friendly, free, web-based application that automates these calculations, removing the barrier of manual mathematical computation for surgeons and researchers.
4. Standardization: Provides a standardized method for comparing different surgical techniques (e.g., navigation vs. conventional wafer) by neutralizing the confounding variable of rotational error.

4. Results

• Validation: The authors tested the application using a simulated cohort of 10 patients with normally distributed translational (1.5–2.0 mm) and rotational (1.7–2.8°) discrepancies.
• Correlation Analysis: The study calculated the Pearson correlation between the uncorrected translational discrepancies and the OGSCalc-corrected total discrepancy:
  • AP vs. Total: $r = 0.741$
  • SI vs. Total: $r = 0.732$
  • ML vs. Total: $r = 0.938$
• Interpretation: The varying correlation coefficients demonstrate that individual translational axes do not linearly represent the overall 3D error. The ML axis showed the strongest correlation, while AP and SI were weaker, highlighting the necessity of the correction formula to obtain an accurate global assessment.
• Usability: The application successfully processed the data, allowing users to upload files, specify decimal precision, and download results instantly.

5. Significance and Clinical Relevance

• Improved Comparative Analysis: The tool allows researchers to compare surgical techniques more fairly. For instance, if a new navigation system reduces ML translation but increases Yaw rotation, the uncorrected data might show mixed results. OGSCalc can reveal whether the rotational increase artificially inflated the ML error, or if the technique is genuinely superior once rotations are accounted for.
• Future-Proofing: The authors note that as surgical techniques improve and rotational errors approach zero, the correction factor will naturally diminish, eventually making the tool obsolete. However, in the current era of incremental improvements, it is crucial for accurate data interpretation.
• Democratization of Data: By providing open-source code and a free web interface, the authors enable the broader medical community to adopt, adapt, and scrutinize the methodology, fostering transparency and reproducibility in orthognathic research.

In summary, OGSCalc bridges the gap between complex 3D geometric reality and clinical reporting, offering a robust method to assess orthognathic surgical accuracy by mathematically integrating rotational and translational errors.