Accuracy of 3D radiological skull measurements: reoriented sections and geometric correction versus caliper reference

Abstract

Purpose

Three-dimensional radiological imaging is important for disease diagnosis and surgical planning, yet its accuracy in anatomical length measurements remains challenging. Standard sections calculations often underestimate actual dimensions due to orthographic projection distortions. This study evaluates the accuracy of reoriented section measurements and formula-based calculations compared to standard sections and reference caliper verifications.

Methods

A plastic human skull model was used for standardized length measurements. Five cranial dimensions (Ma-G, Ma-N, Ma-Rh, Ma-Ns, Ma-Me) were measured using a digital caliper as the reference. A computed tomography scan with a slice thickness of 0.625 mm was performed, and measurements were obtained from standard sections (sagittal, transverse, coronal), reoriented section, and formula-based calculations. Measurement agreement was assessed using Bland-Altman analysis and repeated-measures ANOVA.

Results

Reoriented sections values (e.g., Ma-G: 13.18 ± 0.52 cm, Ma-N: 12.59 ± 0.24 cm) and formula-based calculations (e.g., Ma-G: 13.15 ± 0.26 cm, Ma-N: 12.61 ± 0.80 cm) closely matched caliper measurements (p > 0.05). In contrast, standard section parameters were significantly lower across all parameters (e.g., Ma-G sagittal: 12.03 ± 0.48 cm, p = 0.0005; coronal: 7.39 ± 0.56 cm, p < 0.0001). Bland-Altman analysis confirmed high agreement for reoriented sections and formula-based methods, whereas standard sections showed significant deviations (p < 0.05).

Conclusion

Reoriented sections findings and formula-based calculations offer accuracy comparable to caliper measurements, while standard sections results consistently underestimate the reference caliper values. These findings highlight the limitations of standard-section approaches and suggest that alternative methods may improve anatomical assessment in 3D imaging. These techniques may help reduce measurement-related discrepancies in settings requiring precise anatomical assessment. Such precision may be relevant in fields such as orthodontics, orthopedics, and implant planning, where accurate anatomical measurements are important.

Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1186/s12903-026-09969-x.

Background

Three-dimensional (3D) radiological imaging plays an essential role in modern medicine by providing detailed information on anatomical structures, pathological conditions, and anatomical variations. These imaging modalities are widely used in clinical diagnosis, preoperative planning, and research across many disciplines, including orthopedics, neurology, cardiology, and surgical specialties. In particular, 3D imaging enables the evaluation of spatial relationships and anatomical dimensions that are critical for clinical decision-making and procedural planning [1–4].

One important application of 3D radiological imaging is the measurement of anatomical structures, such as bone length and inter-landmark distances. These measurements may directly influence surgical planning in fields such as orthopedic, orthognathic, and reconstructive surgery, where accurate assessment of anatomical dimensions is required for implant selection, deformity evaluation, and procedural guidance. However, obtaining accurate linear measurements from 3D image datasets remains challenging. In routine practice, measurements are often performed on standard orthogonal sections or sections derived from volumetric datasets. Because such measurements are based on orthographic projections of three-dimensional structures onto two-dimensional views, they may underestimate true spatial distances, particularly when the anatomical structure is oriented obliquely relative to the reference Sects [5–8].

To improve standardization, 3D image datasets are frequently aligned according to reference plane such as the Frankfurt Horizontal plane, and measurements are subsequently performed on sagittal, coronal, or transverse Sects [9, 10]. Although this approach facilitates reproducibility and anatomical orientation, it may be suboptimal for structures whose long axis does not coincide with these standard reference directions. In such cases, measurements obtained from standard sections may not fully represent the true anatomical length of the structure [11, 12].

A more representative assessment may be achieved by defining a measurement section within the 3D dataset in which both endpoints of the structure can be visualized simultaneously. When this section is oriented appropriately with respect to the structure of interest, the resulting measurement may better reflect the actual anatomical distance than a value obtained from standard orthogonal sections. In addition, geometric correction using formula-based approaches may provide an alternative means of estimating true length from projected measurements.

Accordingly, the present study aimed to evaluate discrepancies between linear measurements obtained from standard sections of 3D radiological images and reference caliper measurements. It also investigated whether measurements obtained from reoriented sections and values estimated using a formula-based geometric correction more closely approximate reference caliper measurements. The methodological novelty of the present study lies in the systematic evaluation of two alternative approaches—reoriented-section measurement and orthographic projection–based geometric correction—for reducing projection-related discrepancies in the radiological measurement of obliquely oriented anatomical structures, using direct caliper measurements as the reference standard. We hypothesized that: (1) reoriented-section measurements would provide more accurate results than measurements obtained from standard sections, and (2) formula-based correction of standard-section measurements could improve estimation of true anatomical length. By statistically comparing these approaches, the study aimed to determine whether they could improve measurement reliability in 3D radiological evaluation.

Methods

Model formation and study design

A plastic human skull model was used in this study to ensure standardization and eliminate material-related variability, such as surface wear and deformation, which are common issues in dry bone samples. The plastic model was selected to provide consistent reference points and accurate measurements. The skull model used in this study (QSS 7, Somso^®, Coburg, Germany) is widely employed for demonstration purposes in medical education worldwide and is recognized for its anatomical accuracy and dimensional fidelity to the human skull. It is suitable for a methodological study investigating the accuracy of length measurements derived from three-dimensional (3D) radiological images. Five distinct lengths were identified for measurement:

1. Mastoid process to glabella (Ma-G),

2. Mastoid process to nasion (Ma-N),

3. Mastoid process to rhinion (Ma-Rh),

4. Mastoid process to anterior nasal spine (Ma-Ns),

5. Mastoid process to menton (Ma-Me).

In this study, intraobserver reliability was evaluated using the intraclass correlation coefficient (ICC) for caliper measurement, reoriented section measurement, formula-based calculation, and standard section measurements (sagittal, transverse, and coronal). For each of the five predefined lengths, measurements were repeated 10 times on sagittal, transverse, coronal, and reoriented sections. In addition, each length was measured 10 times on the physical model using a digital caliper and calculated 10 times using the formula-based approach. Thus, a total of 300 measurements/calculations were obtained. Repeated measurements were averaged and were not included in the main statistical analyses as independent data points. ICC values were calculated using a two-way mixed-effects model based on absolute agreement and average measures (ICC(3,2)). For the five different lengths assessed, the ICC values were 0.85 for caliper measurements, 0.95 for the reoriented section method, and 0.93 for the formula-based method. The ICC values for the sagittal, transverse, and coronal section measurements were 0.91, 0.91, and 0.96, respectively.

Step 1: reference parameters

A digital caliper obtained reference length parameters from the skull model (Fig. 1). Each identified length was measured 3 times to ensure reliability, and the mean values were calculated and recorded in an Excel spreadsheet. This step established the “gold standard” values against which other measurement methods were compared.

Fig. 1.

Caliper measurement example for Ma-Me length

Step 2: CT imaging and standard section measurements

A computed tomography (CT) scan of the skull model was performed using an Optima CT660 (GE Healthcare, Chicago, IL, USA) scanner with a slice thickness of 0.625 mm. The CT image series was aligned to the Frankfurt Horizontal Plane using Horos software version 3.3.6 (The Horos Project, USA) to generate sagittal, transverse, and coronal views in a multiplanar reconstruction (MPR) format [9]. To align the image with the Frankfurt Horizontal Plane, the sagittal section in the MPR images was first adjusted to pass through the midline, dividing the image into two equal halves. In the second step, the transverse section was aligned to pass through the infraorbital foramen and the external acoustic porus. This procedure standardized the image. After the CT dataset was aligned to the Frankfurt Horizontal Plane, the sagittal, transverse, and coronal orientations were displayed in the multiplanar reconstruction (MPR) environment. The anatomical landmarks defining each measurement were identified using the “Point” tool in Horos. Because the two endpoints of a given distance were often located on different individual slices, direct measurement on a single conventional slice was not possible. Therefore, the Thick Slab reconstruction mode was applied in each of the three orthogonal orientations to include multiple adjacent slices within a thicker reconstructed slab and thereby visualize both landmarks simultaneously. The reconstruction settings were adjusted to enhance visualization of the high-attenuation bony structures. Distances between the two landmarks were then measured on these sagittal-, transverse-, and coronal-oriented thick-slab MPR reconstructions using the “Length” tool. Accordingly, these measurements represented two-dimensional orthographic projections of the actual three-dimensional anatomical length rather than measurements obtained from individual conventional sagittal, transverse, or coronal slices. These measurements essentially represented the orthographic projections of the actual length. These values were recorded as “standard sectional determinations” in the dataset. Figure 2 illustrates the orthographic projections of the desired lengths on each section, highlighting the deviations caused by planar projections.

Fig. 2.

Diagram illustrating measurements of the target lengths on sagittal-, transverse-, and coronal-oriented thick-slab MPR reconstructions

Step 3: formula-based measurements

The formula-based method was derived from the established geometric relationship for the body diagonal of a rectangular prism, as illustrated in Fig. 3. For each anatomical distance, three orthographic projection lengths were obtained from the sagittal-, transverse-, and coronal-oriented reconstructions and were defined as SG, TR, and CR, respectively. The relationship between the actual anatomical length and its three orthographic projections is illustrated in Fig. 4. These three measurements were then entered into the geometric formula to calculate the estimated real length (RL) of the anatomical structure. The calculation was performed in Microsoft Excel [version, Microsoft Corporation, Redmond, WA, USA], where the measured SG, TR, and CR values were entered into a predefined spreadsheet formula and the corresponding RL value was calculated automatically for each measurement. The mathematical principle itself is not novel; rather, the methodological aspect evaluated in this study was the application of this established geometric relationship to compensate for projection-related discrepancies in radiological linear measurements.

Fig. 3.

Illustration demonstrating the diagonal length of a rectangular prism. The green line represents the body diagonal of the prism. The blue line shows the orthographic projection of the body diagonal on the sagittal section, the pink line illustrates its projection on the transverse section, and the orange line indicates its projection on the coronal section

Fig. 4.

Orthographic projection of the desired length on standard sections: The green line represents the desired length to be measured. The blue line indicates the orthographic projection of the desired length on the sagittal section. The pink line shows the orthographic projection on the transverse section, and the orange line represents the orthographic projection on the coronal section

• SG : Orthographic projection length measured in the sagittal section.

• TR : Orthographic projection length measured in the transverse section.

• CR : Orthographic projection length measured in the coronal section.

• RL : The actual length to be measured.

Step 4: reoriented section measurements

To overcome the limitations of standard sections measurements, the measurement window in 3D MPR images was realigned to display the actual length of the structure. For this purpose, landmarks were placed at both endpoints of the length using the “Point” tool. The transverse and coronal sections were then carefully rotated until both landmarks became simultaneously visible in the sagittal window. Image rotation was performed using the “Rotate” tool in the Horos software, which generated a new cross-section parallel to the intended length. As an example, the Ma–Me distance was measured in the reoriented section shown in Fig. 5a, with the adjusted orientation displayed in Fig. 5b. To improve the clarity of this procedural step, a screen-recorded demonstration has been added as supplementary material (Supplementary File 1). This approach was used to reduce errors related to planar misalignment and to enable measurement in a section aligned with the structure of interest.

Fig. 5.

a Diagram showing how the reoriented section intersects the structure for the Ma-Me length. The red section represents the reoriented section. b Appearance of the reoriented section in the 3D MPR window

Data analysis

Values obtained from all methods (caliper, reoriented section, formula-based calculations and standard sections (sagittal, transverse and coronal)) were statistically analyzed. The normality of the data distribution was assessed using the Shapiro–Wilk test. Since the data met the assumption of normality, parametric tests were employed for statistical comparisons. Differences between methods were evaluated using paired t-tests. Method agreement was assessed separately using Bland–Altman analysis, with the mean difference (bias) and 95% limits of agreement (mean bias ± 1.96 SD of the differences) reported for each comparison. Within the Bland–Altman analysis, the hypothesis that the mean bias was equal to zero was also tested; however, this p-value was not considered a measure of agreement. Caliper readings served as the reference, and scatter plots illustrated deviations, with emphasis on the concordance between reoriented section parameters and formula-based results compared with standard section evaluations. Comparisons among the measurement methods were conducted using repeated measures ANOVA, followed by post-hoc analyses with the Bonferroni correction.

Results

In this study, length determinations obtained using calipers on the skull were considered reference caliper measurement/value. The mean reference caliper measurement/value and the mean values obtained from the methods are presented in Table 1. Reference caliper values were compared with measurements obtained from standard sections, reoriented sections, and formula-based calculations using repeated-measures ANOVA. Mauchly’s test indicated violation of the sphericity assumption (χ²(14) = 195.693, p < 0.001); therefore, Greenhouse–Geisser correction was applied (ε = 0.665). Across the five length measurements, the methods yielded significantly different results, F(13.292, 149.540) = 15.042, p < 0.001, partial η² = 0.572 (Table 2). To determine which methods differed from the reference, Bonferroni-adjusted pairwise comparisons based on estimated marginal means were performed (Table 3).

Table 1.

Reference caliper measurement/value of length measurements and measurement averages for each section

	Reference caliper measurement/value (cm)	Reoriented section measurement (cm)	Formula calculation results (cm)	Sagittal section measurement (cm)	Transverse section measurement (cm)	Coronal section measurement (cm)
Ma-G	13.18 ± 0.41	13.18 ± 0.52	13.15 ± 0.26	12.03 ± 0.48	12.08 ± 0.44	7.39 ± 0.56
Ma-N	12.59 ± 0.70	12.59 ± 0.24	12.61 ± 0.80	11.44 ± 0.83	11.48 ± 0.93	7.39 ± 0.72
Ma-Rh	12.92 ± 0.17	12.91 ± 0.21	12.92 ± 0.16	11.77 ± 0.32	12.03 ± 0.11	7.10 ± 0.15
Ma-Ns	11.97 ± 0.16	11.95 ± 0.11	11.92 ± 0.10	10.81 ± 0.16	11.87 ± 0.10	5.16 ± 0.12
Ma-Me	12.41 ± 0.27	12.45 ± 0.15	12.36 ± 0.11	11.34 ± 0.25	10.8 ± 0.13	7.76 ± 0.13

Values are mean ± SD

Table 2.

Repeated-measures ANOVA summary (greenhouse–geisser corrected)

Effect	Sum of Squares	df	Mean Square	F	p	Partial η²
Methods	37.417	13.292	2.815	15.042	< 0.001	0.572
Error	27.984	149.540	0.187	—	—	—

Table 3.

Bonferroni-adjusted pairwise comparisons between the reference caliper measurement and imaging-based methods with Cohen’s dz effect sizes

			Method	Mean Difference (cm)	Cohen’s dz	p	Significant (α = 0.05)
Ma-G	Reference		Reoriented	-0.002	-0.002	1.000	No
			Formula	+ 0.032	0.053	1.000	No
			Sagittal	+ 1.150	1.665	< 0.001	Yes
			Transverse	+ 1.100	1.669	< 0.001	Yes
			Coronal	+ 5.790	6.994	< 0.001	Yes
Ma-N	Reference		Reoriented	+ 0.007	0.008	1.000	No
			Formula	-0.017	-0.023	1.000	No
			Sagittal	+ 1.151	1.361	< 0.001	Yes
			Transverse	+ 1.112	1.481	< 0.001	Yes
			Coronal	+ 5.208	6.000	< 0.001	Yes
Ma-Rh	Reference		Reoriented	-0.011	-0.078	1.000	No
			Formula	-0.009	-0.028	1.000	No
			Sagittal	+ 1.138	3.041	< 0.001	Yes
			Transverse	+ 0.877	3.679	< 0.001	Yes
			Coronal	+ 5.810	31.626	< 0.001	Yes
Ma-Ns	Reference		Reoriented	+ 0.013	0.088	1.000	No
			Formula	+ 0.041	0.261	1.000	No
			Sagittal	+ 1.158	6.492	< 0.001	Yes
			Transverse	+ 0.094	0.503	1.000	No
			Coronal	+ 6.809	44.750	< 0.001	Yes
Ma-Me	Reference		Reoriented	-0.065	-0.268	1.000	No
			Formula	+ 0.033	0.155	1.000	No
			Sagittal	+ 1.049	3.052	< 0.001	Yes
			Transverse	+ 1.594	7.566	< 0.001	Yes
			Coronal	+ 4.631	43.567	< 0.001	Yes

p-values are Bonferroni-adjusted. Cohen’s dz represents the paired-samples effect size, calculated as the mean paired difference divided by the standard deviation of the paired differences. Positive Cohen’s dz values indicate higher reference caliper measurements, whereas negative values indicate higher measurements with the corresponding imaging-based method. “Significant” indicates statistical significance at α = 0.05 after correction

Comparison of Ma-G length measurements

The Ma-G length was measured as 13.18 ± 0.41 cm using calipers as the reference caliper measurement/value. Measurements obtained from the reoriented Sect.  (13.18 ± 0.52 cm) and formula-based calculations (13.15 ± 0.26 cm) closely matched the reference caliper measurement/value, with no significant differences (p > 0.05). In contrast, determinations from standard sections were significantly lower: 12.03 ± 0.48 cm in the sagittal section, 12.08 ± 0.44 cm in the transverse section, and 7.39 ± 0.56 cm in the coronal section (p < 0.05). These differences are detailed in Table 1.

The Ma-G length was 13.18 ± 0.41 cm using the reference caliper measurement. Measurements obtained from the reoriented Sect.  (13.18 ± 0.52 cm) and formula-based calculations (13.15 ± 0.26 cm) were not significantly different from the reference measurement (p > 0.05). In contrast, measurements obtained from the standard sagittal (12.03 ± 0.48 cm), transverse (12.08 ± 0.44 cm), and coronal (7.39 ± 0.56 cm) sections were significantly lower than the reference values (p < 0.05). Bland–Altman analysis showed mean biases of − 0.006 cm for the reoriented-section method, 0.027 cm for the formula-based method, 1.145 cm for the sagittal section, 1.096 cm for the transverse section, and 5.787 cm for the coronal section. The corresponding 95% limits of agreement were − 1.613 to 1.601 cm, − 1.151 to 1.205 cm, − 0.209 to 2.499 cm, − 0.196 to 2.388 cm, and 4.165 to 7.409 cm, respectively (Fig. 6).

Fig. 6.

Bland-Altman plots show the difference between the caliper Ma-G and the Ma-G measurements obtained from CT images. a Reference caliper measurement/value - reoriented section measurement; (b) Reference caliper measurement/value – formula results measurement; (c) Reference caliper measurement/value - sagittal section measurement; (d) Reference caliper measurement/value - transverse section measurement; (e) Reference caliper measurement/value - coronal section measurement

Comparison of Ma-N length measurements

The reference caliper measurement of the Ma-N length was 12.59 ± 0.70 cm. Measurements obtained from the reoriented Sect.  (12.59 ± 0.24 cm) and formula-based calculations (12.61 ± 0.80 cm) were not significantly different from the reference measurement (p > 0.05). In contrast, measurements obtained from the standard sagittal (11.44 ± 0.83 cm), transverse (11.48 ± 0.93 cm), and coronal (7.39 ± 0.72 cm) sections were significantly lower than the reference measurement (p < 0.05, Table 1).

Bland–Altman analysis showed mean biases of 0.007 cm for the reoriented-section method, − 0.017 cm for the formula-based method, 1.151 cm for the sagittal section, 1.112 cm for the transverse section, and 5.208 cm for the coronal section. The corresponding 95% limits of agreement were − 1.692 to 1.706 cm, − 1.370 to 1.336 cm, − 0.508 to 2.810 cm, − 0.356 to 2.580 cm, and 3.507 to 6.909 cm, respectively (Fig. 7).

Fig. 7.

Bland-Altman plots show the difference between the caliper Ma-N and Ma-N measurements obtained from CT images. a Reference caliper measurement/value - reoriented section measurement; (b) Reference caliper measurement/value – formula results measurement; (c) Reference caliper measurement/value - sagittal section measurement; (d) Reference caliper measurement/value - transverse section measurement; (e) Reference caliper measurement/value - coronal section measurement

Comparison of Ma-Rh length measurements

The reference caliper measurement of the Ma-Rh length was 12.92 ± 0.17 cm. Measurements obtained from the reoriented Sect.  (12.91 ± 0.21 cm) and formula-based calculations (12.92 ± 0.16 cm) were not significantly different from the reference measurement (p > 0.05). In contrast, measurements obtained from the standard sagittal (11.77 ± 0.32 cm), transverse (12.03 ± 0.11 cm), and coronal (7.10 ± 0.15 cm) sections were significantly lower than the reference measurement (p < 0.05, Table 1).

Bland–Altman analysis showed mean biases of − 0.011 cm for the reoriented-section method, − 0.009 cm for the formula-based method, 1.138 cm for the sagittal section, 0.878 cm for the transverse section, and 5.810 cm for the coronal section. The corresponding 95% limits of agreement were − 0.287 to 0.265 cm, − 0.500 to 0.482 cm, 0.405 to 1.871 cm, 0.413 to 1.342 cm, and 5.451 to 6.169 cm, respectively (Fig. 8).

Fig. 8.

Bland-Altman plots show the difference between the caliper Ma-Rh measurement results and the Ma-Rh measurement results obtained from CT images. a Reference caliper measurement/value - reoriented section measurement; (b) Reference caliper measurement/value – formula results measurement; (c) Reference caliper measurement/value - sagittal section measurement; (d) Reference caliper measurement/value - transverse section measurement; (e) Reference caliper measurement/value - coronal section measurement

Comparison of Ma-Ns length measurements

The reference caliper measurement of the Ma-Ns length was 11.97 ± 0.16 cm. Measurements obtained from the reoriented Sect.  (11.95 ± 0.11 cm) and formula-based calculations (11.92 ± 0.10 cm) were not significantly different from the reference measurement (p > 0.05). Among the standard-section measurements, the sagittal (10.81 ± 0.16 cm) and coronal (5.16 ± 0.12 cm) measurements were significantly lower than the reference measurement (p < 0.05), whereas the transverse measurement (11.87 ± 0.10 cm) did not differ significantly from the reference (p > 0.05, Table 1).

Bland–Altman analysis showed mean biases of 0.013 cm for the reoriented-section method, 0.041 cm for the formula-based method, 1.158 cm for the sagittal section, 0.094 cm for the transverse section, and 6.809 cm for the coronal section. The corresponding 95% limits of agreement were − 0.296 to 0.322 cm, − 0.279 to 0.362 cm, 0.808 to 1.508 cm, − 0.281 to 0.469 cm, and 6.510 to 7.108 cm, respectively (Fig. 9).

Fig. 9.

Bland-Altman plots show the difference between the caliper Ma-Ns measurement results and the Ma-Ns measurement results obtained from CT images. a Reference caliper measurement/value - reoriented section measurement; (b) Reference caliper measurement/value – formula results measurement; (c) Reference caliper measurement/value - sagittal section measurement; (d) Reference caliper measurement/value - transverse section measurement; (e) Reference caliper measurement/value - coronal section measurement

Comparison of Ma-Me length measurements

The reference caliper measurement of the Ma-Me length was 12.41 ± 0.27 cm. Measurements obtained from the reoriented Sect.  (12.45 ± 0.15 cm) and formula-based calculations (12.36 ± 0.11 cm) were not significantly different from the reference measurement (p > 0.05). In contrast, measurements obtained from the standard sagittal (11.34 ± 0.25 cm), transverse (10.80 ± 0.13 cm), and coronal (7.76 ± 0.13 cm) sections were significantly lower than the reference measurement (p < 0.05, Table 1).

Bland–Altman analysis showed mean biases of − 0.065 cm for the reoriented-section method, 0.033 cm for the formula-based method, 1.049 cm for the sagittal section, 1.594 cm for the transverse section, and 4.631 cm for the coronal section. The corresponding 95% limits of agreement were − 0.552 to 0.423 cm, − 0.370 to 0.436 cm, 0.378 to 1.721 cm, 1.187 to 2.002 cm, and 4.424 to 4.839 cm, respectively (Fig. 10).

Fig. 10.

Bland-Altman plots show the difference between the caliper Ma-Me measurement results and the Ma-Me measurement results obtained from CT images. a Reference caliper measurement/value - reoriented section measurement; (b) Reference caliper measurement/value – formula results measurement; (c) Reference caliper measurement/value - sagittal section measurement; (d) Reference caliper measurement/value - transverse section measurement; (e) Reference caliper measurement/value - coronal section measurement

Discussion

Accurate linear measurements derived from three-dimensional (3D) radiological images are important in many areas of medicine, particularly in applications requiring precise anatomical assessment [13, 14]. In the present methodological study, standard-section measurements consistently underestimated the reference caliper values across all five cranial lengths, whereas reoriented-section measurements and formula-based calculations showed closer agreement with the reference measurements. These findings suggest that measurement accuracy in 3D radiological imaging is influenced by the spatial relationship between the anatomical structure and the selected imaging section.

The systematic underestimation observed in standard sections can be explained by the fact that these measurements reflect orthographic projections of three-dimensional structures onto two-dimensional views rather than their true spatial length [5, 15]. When the long axis of a structure is not aligned with the selected section, the resulting projected value may be smaller than the actual anatomical distance [5, 15]. In contrast, our findings indicate that reoriented sectional measurements may offer a means of reducing projection-related discrepancies by enabling the structure to be assessed in a section more closely aligned with its spatial orientation. Similarly, the formula-based approach appears to offer a mathematically derived alternative to direct reorientation through the estimation of true spatial length from projection measurements obtained on standard sections.

Many studies have explored the correlation between radiological measurements and actual anatomical dimensions [16–19], yet practical correction approaches for these discrepancies remain limited. The primary issue with standard-section measurements is that they capture projections of true lengths rather than their actual dimensions [6, 7, 20, 21]. This principle helps explain why anatomical measurements obtained from standard sections may be affected by projection-related inaccuracy in radiology [5, 16, 22].

Ardalani et al. evaluated linear measurements identified on the mandible using a caliper and accepted these values as the gold standard. They then assessed the same distances on reoriented CT images according to the mandibular and occlusal sections. Their findings demonstrated that measurements obtained from both sections differed from the gold standard values; however, for the selected lengths, measurements performed on the occlusal section yielded better results than those obtained from the mandibular section. Based on these findings, the authors concluded that measurements derived from CT images may deviate from the true values [23]. Nevertheless, they did not propose any specific method or mathematical formula to correct this discrepancy.

Similarly, Dings et al., in their study on dry skulls, first obtained linear measurements from CT images and subsequently measured the same distances directly with a caliper after physically sectioning the bony structures. They reported differences between radiological measurements and the actual anatomical lengths [24]. However, as in the study by Ardalani et al., no method or formula was presented to address or correct this discrepancy.

The influence of image orientation on linear measurement accuracy has also been emphasized in previous studies. Hayeck et al. suggested that linear measurements obtained from skull CT images may be affected by head rotation. To investigate this, they first performed measurements on images standardized according to the Frankfort horizontal plane and then examined the extent to which rotations of 2°, 4°, 6°, and 8° altered the measured values. Their findings showed that variations in head position affected the linear dimensions intended to be measured. As a practical solution, they recommended reorientation of images in individuals who could not maintain a standardized position during image acquisition [25].

Comparable observations were reported by Mowafy et al., who compared three skull measurements obtained both by caliper and from CT images. They found that CT-derived measurements were lower than the actual measurements, although the difference did not reach statistical significance. This may be explained by the fact that the selected measurement lines were more parallel to the standard sections, thereby reducing projection-related discrepancy [26].

Evidence from the review by Fokas et al. further supports this general pattern. Their findings indicated that, although linear measurements obtained from CT images are generally considered reliable, they may still deviate from the true values. Accordingly, they emphasized that the possible margin of error and the confidence interval of such measurements should be taken into consideration during clinical evaluation [27].

In this context, the findings reported by Baldini et al. are particularly noteworthy. In their study on CT images, linear measurements located on the midsagittal section could be obtained accurately using a lateral approach; however, distortion remained a significant problem for measurements outside the midsagittal plane. The authors further emphasized that the future development of specific algorithms aimed at correcting distortion in such measurements could help clinicians obtain all the information required for orthodontic diagnosis from CBCT scans [28]. Our study addresses this methodological issue by evaluating two alternative approaches reoriented section measurement and formula-based calculation for reducing discrepancies associated with routine standard-section measurements. The purpose was neither to challenge the validity of direct 3D Euclidean point-to-point measurements nor to introduce a new geometric principle, but rather to determine whether these two approaches could provide more representative estimates of anatomical length under controlled conditions.

The two approaches differ in terms of practical applicability, with distinct advantages and limitations. Reoriented-section measurement appears to provide values that are closely aligned with the reference standard, but it requires manual adjustment of the viewing sections and may depend on operator experience in defining the appropriate measurement plane. In the present study, all measurements were performed by a single observer; therefore, although intraobserver reliability was evaluated, interobserver reliability could not be assessed. Consequently, the reproducibility of the reoriented-section method between different operators remains uncertain and should be investigated in future studies involving multiple observers with different levels of experience.

The formula-based method, by contrast, avoids direct reorientation and may be useful when standard-section measurements are already available; however, its accuracy depends directly on the correct measurement of the three orthographic projection components (SG, TR, and CR). Accurate calculation therefore requires consistent landmark placement in all three standard sections and correct alignment of the image dataset to a common three-dimensional coordinate system. Any error in landmark identification or misalignment during Frankfurt horizontal plane orientation may affect more than one projection measurement simultaneously and propagate into the final calculated length. In addition, the requirement for measurements from multiple sections for a single anatomical length may reduce efficiency in routine use. Since measurement time was not formally analyzed in the present study, no definitive conclusion can be drawn regarding the relative practicality of these two approaches.

Anatomical variability is another important consideration when interpreting the results. The present study was performed on a single plastic skull model with fixed landmarks and consistent morphology. In real anatomical specimens, structures may vary considerably in size, curvature, asymmetry, orientation, and landmark visibility. In addition, the radiological properties of the plastic model differ from those of cortical and cancellous bone, and these differences may affect image contrast, edge definition, and anatomical landmark identification on CT images. Image resolution is another relevant factor. Voxel dimensions may influence partial-volume effects and the sharpness of anatomical boundaries, thereby affecting landmark placement and linear measurement precision. This may influence both direct section-based measurements and the orthographic projection measurements used in the formula-based approach. Therefore, the present findings should be interpreted within the imaging parameters used in this study, and further validation using different imaging resolutions, multiple specimens, cadaveric material, and in vivo datasets is warranted.

These findings may be particularly relevant to orthodontics and forensic dentistry, where accurate craniofacial linear measurements are important for morphometric assessment, skeletal evaluation, and anatomical comparison [29–33]. However, the present study did not assess clinical outcomes or patient-specific decision-making, and the results should therefore be interpreted primarily as methodological evidence rather than as direct evidence of clinical applicability. Within this context, our findings suggest that reliance solely on standard sections may lead to measurement discrepancies in certain anatomical orientations, whereas alternative measurement strategies may help minimize these discrepancies under controlled conditions.

Despite these limitations, the study may serve as a methodological reference for future investigations. Further studies should examine whether similar patterns are observed in different skeletal regions, in multiple anatomical specimens, and in clinical imaging datasets. Future studies may also investigate whether computational tools can support section reorientation or formula-based correction more efficiently and with greater reproducibility.

This study has several limitations. First, measurements were performed on a single anatomically accurate plastic skull model. Although this allowed standardization and avoided deterioration-related problems that may occur in dry bone specimens, it also limited biological and anatomical variability. Second, only a restricted number of anatomical lengths were evaluated. Third, the study did not include soft tissue, motion, patient-specific imaging variation, or clinically relevant in vivo conditions. Fourth, efficiency was not formally assessed, and therefore the time-related practicality of the two alternative methods remains uncertain. These limitations should be considered when interpreting the findings. Finally, because all measurements were performed by a single observer, interobserver reliability could not be evaluated, and the reproducibility of the proposed methods across different operators remains to be established.

Conclusions

Within the controlled conditions of this methodological study, standard-section measurements derived from 3D radiological images underestimated reference caliper values, whereas reoriented-section measurements and formula-based calculations showed closer agreement with the reference measurements. These findings suggest that both approaches may reduce projection-related discrepancy in the measurement of obliquely oriented anatomical structures.

However, the results should be interpreted within the limitations of the study, particularly the use of a single plastic skull model and the absence of clinical or in vivo validation. Accordingly, the present results should be viewed as methodological findings that require further clinical validation before any direct clinical benefit can be inferred. Further studies involving multiple specimens, cadaveric materials, and patient-based imaging datasets are needed to determine the broader applicability and practical value of these approaches.

Abbreviations

CT

Computed tomography

MPR

Multiplanar reconstruction

MIP

Maximum intensity projection

ICC

Intraclass correlation coefficient

SG

Sagittal projection length

TR

Transverse projection length

CR

Coronal projection length

RL

Real length

Authors’ contributions

All authors contributed to the study concept and design. Material preparation, data collection and data analysis were performed by MN, BS. The first draft of the article was written by MN and all authors contributed to the evaluation of the article. The final version of the article has been read and approved by all authors.

Funding

No funding has been received in this study.

Data availability

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

Declarations

Ethics approval and consent to participate

This study was conducted in accordance with the Helsinki Declaration, and ethical approval was not required as it did not involve any materials obtained from humans or animals.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.