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Journal of Structural Biology: X logoLink to Journal of Structural Biology: X
. 2026 Jul 23;14:100154. doi: 10.1016/j.yjsbx.2026.100154

Bipixel rotation method: An algorithm for chirality analysis of biological mesh network

Kanta Kumaki a,1, Katsuhiro Kawaai a,⁎,2, Shinobu Noji a,3, Yukiko Kuroda a,4, Koichi Matsuo a,b,5
PMCID: PMC13446290  PMID: 42564810

Abstract

Vertebrates are bilateral organisms characterized by both symmetrical and asymmetrical structures. While skeletons are grossly symmetrical, microscopic biological networks within the bone—such as cortical canals containing vascular networks—often exhibit deviations from precise mirror-image symmetry. These deviations make it challenging to detect subtle chiral biases amidst random structural variations. We developed the “bipixel rotation method” to quantify the extent of these biases, thereby enabling the characterization of porcine femoral cortical canals. In this approach, binarized 2D images are rotated stepwise and skeletonized, followed by the quantification of vertically adjacent pixel pairs. We defined “center angle” as the rotation angle that yields the peak vertical bipixel count, and calculated “down-right bias” based on the asymmetry of the distribution of vertical bipixels around this center angle. After validating the method using alphabetical fonts and geometric wallpaper patterns, we applied the method to the cortical canal layers in porcine fibrolamellar bone visualized via nano-computed tomography (nano-CT). The method successfully detected potential chirality in the cortical canals, identifying up-right versus down-right biases, demonstrating its capability to capture even subtle structural chirality. These results suggest that the bipixel method is a versatile tool for chirality analysis of diverse biological mesh structures in the body.

Keywords: Chirality, Vasculature, Image analysis, Porcine, Femur, Bone

Graphical abstract

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Highlights

  • •

    X-ray nano-CT visualized 3D plexiform cortical canal networks in porcine femurs.

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    The “bipixel rotation method” quantifies chirality in 2D mesh structures.

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    The method effectively decouples structural chirality from random variations.

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    Subtle chiral bias was detected in the cortical canal networks of femurs.

1. Introduction

The skeletal system is highly bilateral. Each bone has its mirror-image counterpart across the mid-sagittal plane on the other side of the body. In other words, bones on the left and right sides of the body are chiral in that they cannot be superimposed on those on the other side. However, at microscopic levels, apparent deviations from precise mirror-image symmetry blur the distinction between chirality and random structural variations in biological networks such as nerves and blood vessels on the left and right sides of the skeleton. In this study, we developed an algorithm to detect the chirality of mesh-like microstructures in bone quantitatively.

Cortical bones are highly vascularized and innervated not only on the periosteal (outer) and endocortical (inner) bone surfaces but also within the bone tissue (Asghar et al., 2020; Mantyh, 2014; Matsuo et al., 2018; Watson and Adams, 2018). In humans, cortical bone contains cylindrical osteons (also known as Haversian systems) with central canals housing a small bundle consisting of a capillary (arterial), a venule, and often a nerve fiber. Haversian canals are connected through the transverse channels called Volkmann's canals. Whether this longitudinal–transverse classification represents a true bimodal distribution or a simplification of a continuous orientation spectrum in three dimensions has not been formally established with volumetric imaging. By contrast, large animal models such as pig, deer, and sheep, develop a fibrolamellar bone (also known as plexiform bone or laminar bone) characterized by a plexiform vascular network sandwiched between layers of laminar bone matrix, which is then remodeled into osteonal bone structure that resembles human bone morphology (Almany Magal et al., 2014; Barrera et al., 2016; Currey, 1960). In this study, we employed laboratory-based high-resolution nano-computed tomography (nano-CT) to visualize the plexiform cortical canal networks in porcine femurs. Specifically, we aimed to develop a quantitative framework to determine whether subtle structural chirality is a detectable and consistent feature of these complex biological networks, distinguishing it from random structural variations.

Beyond traditional morphometric parameters, we sought to quantify the directional bias of the canal networks—specifically assessing whether the canals exhibit a systematic chiral asymmetry relative to their principal structural axis. While advanced methods for analyzing the geometric patterns of vascular and neural plexuses have evolved to evaluate complex network topologies, existing tools remain limited in their scope of symmetry analysis. For instance, open-source platforms such as Vessel Vio (Bumgarner and Nelson, 2022b), AngioTool (Zudaire et al., 2011), the MATLAB-based REAVER (Corliss et al., 2020), and the Python library Skan (Nunez-Iglesias et al., 2018) provide skeletonization and tracing capabilities. Similarly, commercial products like Vesselucida 360 (MBF Bioscience) offer sophisticated 3D reconstruction. However, as far as we know, none of them has focused on the scoring of chirality or inherent ‘handedness’ of biological network structures, representing a gap in our understanding of cortical bone morphology in bilateral mammals. In this study, we set out to develop an image analysis algorithm to quantify chirality.

Given that cortical canals in the porcine femur exhibit complex, multi-layered, planar mesh-like architectures, we developed a novel algorithm to quantify a chirality score based on 2D pixel-count distributions. To validate the accuracy of this approach, we evaluated its performance using chiral alphabetic characters and chiral wallpaper patterns. The resulting bipixel rotation method (hereafter referred to as the bipixel method) provides a framework for chirality analysis across diverse biological mesh-like structures. While this study focuses more on the computational validation of the algorithm, further investigation is warranted to determine the extent of biologically conserved chirality within the skeletal system.

2. Results

2.1. Cortical canals in porcine femurs

To characterize the cortical canal architecture in porcine femurs (Fig. 1A), cylindrical cortical bone fragments (4 mm in diameter) were excised from the posterior side of the femoral diaphysis using a cork borer (Fig. 1B). Bilateral pairs were collected at approximately the same proximodistal levels: one pair each from Pigs 1 and 2, three pairs from Pig 3, and two pairs each from Pigs 4 and 5. Three-dimensional reconstruction using 3D Slicer software enabled the visualization of the complex canal networks, from which representative horizontal slices were extracted for analysis (Fig. 1C). These networks were further examined in histological sections, which showed blood vessels surrounded by concentric lamellae centered on the canal (Fig. 1D, arrow) or arranged in pseudo-parallel layers (Fig. 1E). Polarized light microscopy visualized characteristic collagen fiber orientations aligned along these canal layers (Fig. 1D and E, yellow areas). In silico analysis revealed high-intensity bone regions (Fig. 1F) and low-intensity pore spaces (Fig. 1G) containing plexiform cortical canal layers (Fig. 1H). For subsequent quantitative analysis (see Fig. 7), two-dimensional representations of these pseudo-parallel cortical canal layers were extracted (Fig. 1I to M).

Fig. 1.

Fig. 1

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Nano-CT imaging of cortical canals in porcine femurs. (A) Posterior view of porcine femurs. Arrows indicate holes where samples were removed. Scale, 5 cm. (B) Sampling of a cylindrical cortical bone fragment (4 mm in diameter). peri, periosteum. endo, endosteum. (C) A virtual transverse section of the cortical bone (gray) and the low-intensity areas including canals and other soft tissues. Histology of areas similar to the red and blue boxed areas are shown in (D) and (E). Scale, 500 μm. (D) An osteon-like concentric structure (arrow) in the porcine femur visualized using polarized light microscopy of H&E-stained sections, at a location corresponding to the red-boxed area in (C). Scale, 20 μm. (E) Pseudo-parallel layers of cortical canals (arrows) in the porcine femur visualized using polarized light microscopy of H&E-stained sections, at a location corresponding to the blue-boxed area in (C). Scale, 20 μm. (F) Binarized high-intensity bone area of right “distal” sample (Pig 5), viewed from the proximal side. Scale, 500 μm. (G) Binarized low-intensity areas representing intracortical canals. Scale, 500 μm. (H) 3D visualization of cortical canal layers. Scale, 500 μm. (I)–(M) Individual cortical canal layers, isolated in silico by removing interconnecting canals between layers. Scale, 500 μm. Images in (C) and (F)–(M) were generated using 3D Slicer software. (A), (B), and (F)–(M): Pig 5; (C)–(E): Pig 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7.

Fig. 7

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Down-right bias and center angles of cortical canal networks in porcine femurs. (A) Processing pipeline for a single cortical canal layer, showing the original 3D Slicer reconstruction, binarization, and skeletonization. (B) A representative vertical bipixel plot of the cortical canal layer shown in (A). The peak indicates the center angle (CA). (C) Scatter plot of down-right bias versus center angle for all analyzed layers. Each data point represents a specific layer from proximal (triangles) or distal (circles) locations in the left (blue, n = 62) and right (red, n = 59) layers. (D) Angular histogram showing the distribution of center angles for the left (blue, n = 62) and right (red, n = 59) layers. Thick arrows represent the circular mean (direction) and resultant vector length (coherency). A significant difference in the CA distribution was observed between the left and right femurs (Kuiper's two-sample test, p = 0.001). (E) Distribution of down-right bias in the left (L, n = 62) and right (R, n = 59) layers. Individual data points are plotted alongside median values (circles) and their 95% confidence intervals (CI: L, [−0.02, 0]. R, [0, 0.02]). Welch's t-test, p = 0.0003. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2.2. The bipixel method and rotation algorithm

In the bipixel method, four types of bipixels, vertical (V), horizontal (H), up-right (UR), and down-right (DR), are systematically counted (Fig. 2A). This is achieved by a forward-only scan of every pixel in the image at each rotation angle (Fig. 2B). To illustrate the algorithm, we employed 30–60-90 triangles as representative chiral structures. These triangles lack an axis of symmetry and are not superimposable on their mirror images by rotation or translation. These chiral pairs, representing the two halves of an equilateral triangle, are hereafter referred to as left- and right-handed triangles (Fig. 2C, top).

Fig. 2.

Fig. 2

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The algorithm of the bipixel method. (A) Four types of bipixels: vertical (V), horizontal (H), up-right (UR), and down-right (DR). (B) Bipixel counting process performed by scanning every pixel throughout the image. (C) (Top) A pair of left- and right-handed 30–60-90 triangles used as representative binarized images. (Bottom) Stepwise image rotation at 15° intervals, ranging from −90° to 90°. Following rotation, structures are thinned to 1-pixel width (skeletonization). (D) A skeletonized portion of the left-handed triangle and its corresponding bipixel counts. (E) Plots showing bipixel counts versus rotation angle for both left- and right-handed triangles.

Binary images of these triangles were first rotated CCW at 15° intervals from −90° to 90° (Fig. 2C, bottom), then thinned to 1-pixel width (skeletonization). Fig. 2D shows a magnified view of the skeletonized 60° vertex of the left-handed triangle, along with examples of the four types of bipixels being counted. The resulting bipixel counts were then plotted against the rotation angle for both triangle types (Fig. 2E). Notably, horizontal and vertical bipixel counts are inherently greater than up-right and down-right counts because the diagonal of a pixel is 2 times the length of its side.

Aside from the count differences, the vertical, horizontal, up-right, and down-right bipixel plots exhibit similar periodic profiles, shifted by 45° intervals along the rotation angle (Fig. 3A, left). To determine the peak position accurately, the vertical bipixel plot was interpolated with a spline function incorporating periodic boundary conditions. This approach ensures that the two ends of the plot (−90° and 90°) are joined smoothly, reflecting the circular nature of the rotation angle. The rotation angle at the maximum peak of this interpolated curve was then defined as the center angle (CA) (Fig. 3A, right). Given that the rotation angle is circular every 180°, two distinct areas under the curve were defined: S1, from CA-90° to CA, and S2, from CA to CA + 90°. These areas were used to calculate the area ratio (S1/S2) and the down-right bias (chirality score), defined as log10(S1/S2) (Fig. 3B). Furthermore, an asymmetry score was calculated by quantifying the difference between the left and right sides of the curve relative to the center angle (Fig. 3C). All image processing was performed sequentially using ImageJ macros and custom Python scripts as outlined in the computational pipeline (Fig. 4).

Fig. 3.

Fig. 3

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Definition of center angle, down-right bias, and asymmetry score. (A) A bipixel plot versus rotation angle (from −90° to 90°) for a left-handed 30–60-90 triangle. The vertical bipixel counts were smoothed using spline interpolation. The center angle (CA) is defined as the rotation angle at the maximum peak (red circle). (B) Calculation of chirality bias. The maximum peak (red circle) and the deepest trough (green triangle) are indicated. S1 and S2 represent the areas under the curve from CA-90° to CA and from CA to CA + 90°, respectively. The area ratio S1/S2 (down-right bias or chirality bias) is shown for a pair of left- and right-handed 30–60-90 triangles. The chirality bias is expressed as the log-ratio of S1 to S2 (base 10). (C) Asymmetry score. S3 is the area difference between the original curve and the curve of the image horizontally flipped around the center angle. The asymmetry score is defined as S3 / (S1 + S2). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4.

Fig. 4

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Computational pipeline of the bipixel method. Raw images were binarized using the ImageJ macro “Step1_Make_Binary.ijm.” These binary images were subsequently rotated at 15° intervals and skeletonized using the macro “Step2_Rotate_Skeletonize.ijm.” Numerical analysis was performed using custom Python scripts: “Step3_GUI_Bipixel_Counts.py” was used to extract bipixel counts from the skeletonized images, and “Step4_GUI_CA_DRbias.py” was employed to calculate the center angle, down-right bias, and asymmetry score. All scripts and spreadsheets generated at each step were used for final data integration and graphing.

2.3. Validation of the bipixel method using alphabet fonts

To verify the ability of the bipixel method to detect chirality, we first analyzed 30 asymmetric Sans-Serif characters (11 uppercase and 19 lowercase). Vertical bipixel plots were generated from these rotated and skeletonized fonts to calculate their down-right bias. While the center angles (CA) for 20 of the 30 characters were nearly vertical (0 ± 5°), characters such as “Z”, “z”, “e”, “G”, and “F” exhibited CA values closer to ±90°. This shift in CA correctly classifies “Z” and “z” as down-right oriented (similar to “N”) rather than up-right (Fig. 5A), demonstrating the essential role of CA in orientation normalization. Conversely, for letters such as “r”, “j”, and “f”, the dominant vertical segments largely define the center angle, while the remaining minor curves contribute to an up-right bias, resulting in a negative down-right bias (Fig. 5A). A scatter plot of down-right bias versus asymmetry score identified “Z”, “z”, “N”, and “R” as the most asymmetric down-right characters, contrasting with “r”, “j”, “f”, and “K” as the most asymmetric up-right characters (Fig. 5B). We further compared our method with local dominant orientation (DO) analysis. DO was CCW-rotated in down-right letters (“Z”, “z”, “N”, “R”) and CW-rotated in up-right letters (“r”, “j”, “f”, “K”) (Fig. S1a), yielding a strong positive correlation between down-right bias and CA-adjusted DO (R2 = 0.6789; Fig. S1b). To assess the scale-invariance of the bipixel method, characters were generated at four different scales (144, 72, 36, and 18 pixels in height). The signs of the down-right bias remained consistent across these scales for characters with high chirality. While some instability was observed in characters with lower absolute bias values, the overall chiral trends were preserved regardless of the font size (Fig. 5C). Notably, in mixed-character pools, the collective CA can override the CA of individual characters. For example, the nearly 90° CA of a single “Z” shifted toward 0° when mixed with vertically oriented letters like “R”, “j”, and “r” (Fig. 5D, ZR + jr). In such cases, “Z” contributes to an up-right bias, contrasting with its behavior as an isolated character. This suggests that chiral classification depends on the CA determined for the entire image. Finally, tests on full sentences showed that horizontal flipping reversed the sign of the down-right bias, confirming that the bipixel method reliably distinguishes the orientation of chiral 2D images (Fig. 5E).

Fig. 5.

Fig. 5

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Validation of the bipixel method using asymmetric alphabet fonts.

(A) Vertical bipixel plots of representative asymmetric characters (font size: 144 pt). Characters such as Z, z, and N consistently exhibit positive down-right bias (indicated by a larger S1 area in blue), while characters like j and r show negative down-right bias (larger S2 area in red). (B) Scatter plot showing the relationship between down-right bias and asymmetry score. Note that the sign of the down-right bias was unstable for characters with low asymmetry scores, such as Q and q, across different font sizes. (C) Effects of font size on down-right bias and asymmetry score. Heatmaps visualize the consistency of these metrics across Arial font sizes of 144, 72, 36, and 18 pt. (D) (Left) Representative images showing pools of 32 characters with down-right bias (Z and R), up-right bias (j and r), and a 1:1 mixture of both. (Right) Plot of down-right bias against the number of characters in the pool (2, 4, 8, 16, and 32). (E) Comparison of the down-right bias between a collective alphabet image (Top) and its mirror image (Bottom). The sign of the bias is reversed in the mirror image, confirming the method's sensitivity to chirality. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2.4. Validation of the bipixel method using repetitive wallpaper patterns

Biological networks often exhibit highly repetitive architecture. To evaluate the performance of the bipixel method on such structures, we analyzed “wallpaper patterns,” which were mathematically classified into 17 groups (Pólya, 1924). We selected a pattern from the simplest p1 group, which is characterized by translational symmetry. This specific pattern is chiral due to its lack of any reflective symmetry axes (Fig. 6A). The bipixel method yielded a positive down-right bias of 0.12 for a single unit of this p1 pattern (Fig. 6B). To test whether the sampling area affects the analysis, we cut out 12 distinct regions with varying shapes from the wallpaper (Fig. 6C). Although the center angles for these 12 regions varied slightly (between 80° and 86°), the down-right bias remained consistently positive, ranging from 0.03 to 0.08 (Fig. 6D). This consistency demonstrates the robustness of the method against variations in sampling geometry. Next, we examined the effect of rotation by rotating a square cutout 45° CCW and 45° CW (Fig. 6E). While the center angles shifted in accordance with the rotation, the resultant down-right bias remained nearly constant (Fig. 6F). Finally, to test whether the down-right bias converges with increasing sampling area, we progressively doubled the diameter of a circular mask from 1 to 32 units (Fig. 6G). The down-right bias converged to approximately 0.06 as the diameter increased, consistent with the expectation that larger sampling areas yield more stable estimates of the underlying structural bias. These results confirm that the down-right bias is an intrinsic structural metric that is robust against both sampling shapes and rotational orientation.

Fig. 6.

Fig. 6

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Validation of the bipixel method using repetitive wallpaper patterns. (A) A single unit of the repetitive wallpaper pattern. The arrow indicates the calculated center angle (CA). (B) A vertical bipixel plot of the unit shown in (A), demonstrating a positive down-right bias (0.12). (C) Twelve distinct regions (1−12) sampled from the wallpaper pattern to test the effect of cropping on chirality detection. (D) Scatter plot showing the down-right bias versus center angle for the twelve cutouts shown in (C). The tight clustering confirms that the method is robust against sampling variations. (E) Representative images of a square wallpaper cutout: unrotated (no rotation), rotated 45° counter-clockwise (CCW), and 45° clockwise (CW). (F) Scatter plot showing that while the center angle shifts according to the rotation, the down-right bias remains constant, demonstrating that the chirality bias is an intrinsic structural property. (G) Down-right bias plotted against the diameter of a circular sampling mask (1–32 units, doubled at each step), showing convergence to approximately 0.06 with increasing sampling area.

2.5. Analysis of cortical canal networks in porcine femurs

Having validated the bipixel method, we applied it to analyze the plexiform cortical canal networks in porcine femurs. Following binarization and skeletonization (Fig. 7A), the center angle (CA) and down-right bias for each layer were determined based on the vertical bipixel plot (Fig. 7B). Fig. 7C summarizes the relationship between down-right bias and CA across all analyzed layers. Notably, the CA values of canal layers at proximal locations were generally greater than 45° in absolute terms (Fig. 7C, triangles), whereas those at distal locations were primarily below 45° (Fig. 7C, circles). The circular mean center angles for the left and right femurs were 15.2° (n = 62, r = 0.56) and −14.2° (n = 59, r = 0.36), respectively, suggesting that the macroscopic orientations of the canal networks exhibit a degree of left-right symmetry (Fig. 7D). Regarding structural chirality, the down-right bias for the right femurs was positive on average, with a 95% confidence interval (CI) of [0, 0.02], which was almost entirely above zero (Fig. 7E). Similarly, the 95% CI for the left femurs of [−0.02, 0] was almost entirely below zero (Fig. 7E). Collectively, these findings suggest detectable chiral bias in the canal networks in porcine femurs.

3. Discussion

In this study, we developed an algorithm to quantify the chirality of mesh-like 2D structures. This “bipixel method” is applicable to a wide range of planar biological networks, extending beyond the plexiform canal networks observed in cortical bone of large animals. Over the past decade, intrinsic chirality of immortalized cell lines and primary cells has been increasingly recognized under culture conditions. For example, consistent CCW or CW inclination of cell collectives at the adhesive and non-adhesive boundary reveals cell chirality (Wan et al., 2011; Weng et al., 2025). Given that both osteoblasts and vascular endothelial cells exhibit such handedness, higher-order structures such as bone matrix and capillary networks may similarly possess inherent chirality. Indeed, homochirality in bilateral organisms has been reported at various scales, from unidirectionally-deflected neurites (Tamada and Igarashi, 2017) and left-handed helical myofibrils in Ciona larva (Matsuo et al., 2021) to the right helical ducts of eccrine sweat glands (Takagi and Tagawa, 1957), all of which break left-right symmetry consistently on both sides of the body. Unlike helical structures, which possess self-evident chirality, the chiral nature of complex, plexiform networks has remained elusive and difficult to quantify. Our method addresses this by decoupling the “center angle” (representing the dominant orientation, analogous to a tree trunk) from the “down-right bias” (representing the directional skew of branches relative to that trunk). By calculating the extent of these biases, the bipixel method provides a framework for revealing the oriented nature of chiral mesh structures that would otherwise appear as random asymmetry. Positive and negative down-right bias would reveal the oriented direction of chiral mesh structures.

Previous studies have employed high-resolution synchrotron-based X-ray microscopic CT to analyze human Haversian canal structures in 3D (Maggiano et al., 2016). Curiously, Haversian canals or osteons of opposite helical tendencies in each long bone have been detected (Heřt et al., 1994). While porcine bone contains both Haversian and plexiform vascular networks (Hillier and Bell, 2007), we focused on the latter. Using laboratory-based X-ray nano-CT, we clearly visualized the plexiform cortical canal networks situated between the multiple layers of fibrolamellar bone, which typically form on the periosteal surface (Barrera et al., 2016). Notably, these planar layers were interconnected by relatively thick capillaries oriented perpendicular to the cortical layers, forming a highly organized three-dimensional plexiform canal network.

By leveraging the quasi-2D nature of these individual layers, we applied the bipixel method to quantify their structural characteristics. As expected from the gross bilaterality of bone, the center angles were largely symmetrical between the left and right femurs. At the level of individual layers, however, the down-right bias was highly variable, exhibiting both positive (down-right) and negative (up-right) values. Intriguingly, our results revealed a notable difference in the manifestation of this structural chirality between the two sides. The layers from left and right femurs exhibited predominantly negative and positive down-right bias, respectively, with a wide dispersion of data points around zero (Fig. 7E). This observation raises the question of whether cortical bone chirality is a systemic trait or a mirrored phenomenon. One possibility is that bone canal networks possess an intrinsic homochirality; in this scenario, a consistent chiral bias would be a species-wide property derived from the fundamental chirality of cellular components. Alternatively, the divergent trends observed between the two sides might hint at a latent mirror-image symmetry between the left and right femurs. If further studies confirm that the left and right sides indeed exhibit opposing chiralities (positive vs. negative down-right bias), it would lead to a profound conclusion: even at the scale of complex canal networks, bone morphogenesis adheres to the bilateral symmetry of the body plan. This would imply that the intrinsic homochirality of individual cells—dictated by chiral cytoskeletal dynamics—is integrated into a higher-order developmental program that enforces macroscopic mirror-image organization.

The geometric logic of vascular trees—governed by branching angles, segment lengths, and hierarchical organization—ensures efficient transport (Kassab, 2006), and disruptions in these patterns are often a hallmark of vascular pathologies (Pries and Secomb, 2014). Conventionally, blood flow and circulation are evaluated through 2D morphological parameters, such as vessel density, fractal dimension, vessel diameter, and flow indices. However, blood supply may not be the sole function of these canals. In smaller mammals, cortical canals often penetrate the cortical bone radially between the endosteal and periosteal surfaces, housing transcortical vessels (Grüneboom et al., 2019). In addition to longitudinal canals that run along the long axis of the bone, these transverse canals serve as vital conduits; murine cortical canals facilitate the migration of periosteal osteoprogenitors toward the endocortical surface, a coordinated process termed “trans-pairing” that is critical for endocortical bone formation (Kuroda et al., 2024). Additional functional and morphological characterization of cortical canals in fibrolamellar bone awaits future studies.

In conclusion, the bipixel method presented here provides a versatile framework for quantifying the chirality of 2D images through the calculation of the down-right bias. This approach effectively captures subtle structural asymmetries that are often overlooked by conventional morphological analyses. The bipixel method can in principle be extended to three dimensions as a “bivoxel” method: the four forward counting directions of 2D 8-connectivity, which correspond to horizontal, vertical, and two diagonal bipixel pairs, would generalize to 13 forward directions of 26-connectivity in 3D. Future efforts should focus on developing and validating this 3D bivoxel extension, along with more generalized algorithms capable of quantifying the chirality of complex, non-helical 3D architectures, further advancing our understanding of structural handedness in diverse biological systems.

4. Materials and methods

4.1. Porcine bone samples

The bilateral femurs of five pigs (Pig 1 to Pig 5) were used in this study. Pig 1 was a female Göttingen minipig (9 months of age, 18.8 kg). Pig 2 was a female microminipig subjected to neonatal thymectomy (5 months of age, 7.6 kg). Pigs 3, 4, and 5 were domestic pigs (approximately 6 months old, 100 to 120 kg). Pigs 1 and 2 were obtained as discarded tissues from approved animal experiments at Keio University School of Medicine. The domestic Pigs 3–5 were purchased from Tokyo Shibaura Organs Company (Tokyo, Japan).

4.2. Nano-CT imaging

The cylindrical cortical bone samples were scanned using a high-resolution 3D X-ray microscope (nano3DX, Rigaku, Tokyo, Japan). Continuous CT datasets were acquired by scanning each cylindrical sample across two to four fields of view (FOVs) along its longitudinal axis. The sample was translated axially (z-axis) between FOVs, and the resulting scans were manually stitched to ensure full coverage of the entire specimen from the periosteum to the endosteum. To facilitate subsequent anatomical analysis, the samples were aligned so that their anatomical axes corresponded with the CT coordinate system. The scans were performed in step scan mode under the following conditions: X-ray source, Mo target (50 kV, 24 mA); X-ray magnification lens, L0540 (0.65 μm/pixel); FOV, 1.331 × 1.331 mm; acquisition resolution, 1.3 μm/voxel (binning 2); X-ray detector position, 2 mm; sample rotation angle range, 180°; number of projections, 600; and exposure time, 25 s/projection. Total acquisition time: 4 h 29 min.

4.3. Bipixel method

Custom macros in Fiji (an ImageJ-based processing package) were used to binarize raw images (Step1_Make_Binary.ijm). Each binary image was then rotated from −90° to 90° in 15° increments, followed by skeletonization using the Analyze Skeleton plugin (Arganda-Carreras et al., 2010), resulting in 13 skeletonized images per sample (Step2_Rotate_Skeletonize.ijm). Adjacent white pixel pairs (bipixels) were counted in four directions (vertical, horizontal, up-right, and down-right) using an in-house Python script (Step3_GUI_Bipixel_Counts.py). To analyze the orientation bias, the vertical bipixel counts were plotted against the rotation angle. Due to the circular and periodic nature of the data, the curves were fitted with a periodic smoothing spline (UnivariateSpline with ±180° tiling) to ensure boundary continuity at ±90° (Step4_GUI_CA_DRbias.py). The “center angle (CA)” was defined as the rotation angle at the global maximum of the fitted spline. The chirality bias was quantified as log10(S1/S2), where S1 and S2 represent the areas under the curve integrated from CA–90° to CA and from CA to CA + 90°, respectively. To assess the structural symmetry around the peak, an asymmetry score was calculated as the mean absolute difference between the two halves of the curve relative to the center angle (α), normalized by the peak amplitude.

Asymmetry Score=meanyα+θ−yα−θmaxy.

where the angular offset (θ) spans from 0° to 90°. The count (y) of vertical bipixels (adjacent pixel pairs) at a given rotation angle, derived from the fitted smoothing spline curve. An asymmetry score near 0 indicates strong mirror symmetry, while larger values (typically >0.01) indicate increasing asymmetry.

Dominant orientation was calculated for each image using the ImageJ plugin, OrientationJ software (Püspöki et al., 2016). Yellow ellipses show dominant orientation and coherence when the sigma is changed from 1.0 to 5.0 (every 0.1).

4.4. Alphabetical fonts

Isolated uppercase and lowercase letters (A–Z, a–z) were generated in a sans-serif font (Arial) at pixel heights of 144, 72, 36, and 18 pixels. To ensure geometric precision, these were rendered without anti-aliasing and centered within square canvases of 512, 256, 128, and 64 pixels per side, respectively. This setup maintained a constant ratio between character size and canvas resolution across all scales. The resulting binary images were saved in Portable Network Graphics (PNG) format. [Note: The image of the quote from Friedrich Schiller on Gustav Klimt's 1899 painting, Nuda Veritas, was obtained from the public domain and is used in accordance with its copyright-free status.]

4.5. Wallpaper images

A wallpaper pattern with C1 symmetry (crystallographic group p1) (Pólya, 1924) was reproduced, multiplied, and manually tiled after translation. Various shapes of the wallpaper pattern were cut out using diverse cropping masks, including diamonds, triangles, circles, ovals, and parallelograms. These cropped segments were saved as PNG files and subsequently subjected to the bipixel method.

4.6. Analysis of cortical canals

Nano-CT images were converted into 8-bit and further reduced by binning 2 using ImageJ, resulting in 2.52 μm/pixel. Stack CT images are oriented so that the top is proximal and the bottom is distal, the right and left are right and left of the animal, respectively. The cortical canals were binarized using the segment editor module of the open-source software 3D Slicer (Fedorov et al., 2012) by applying the threshold effect (Automatic-threshold, Kittler-Illingworth, and threshold below). Small islands (< 5000 voxels) were removed using the Islands tool. Each cortical canal layer was manually cropped out using the scissors tool, and the posterior view of each layer was captured into a PNG file.

4.7. Histology

The cortical bones of the left and right femoral mid-diaphysis were cut in annular sector shape using a diamond saw (Isomet, Buehler). Samples were decalcified in 10% EDTA in 0.1 M Tris (final pH 7) for 20 days at 37 °C, and then dehydrated and paraffin-embedded. Paraffin blocks of Pig 2 posterior sectors were sectioned at 5 μm thickness using a sliding-type microtome (Yamato REM-710). Standard hematoxylin and eosin (H&E) staining was performed. Images were observed under brightfield illumination and polarized light using upright microscopy (BX53, Olympus).

4.8. Statistical analysis

For center angles, circular statistics including Kuiper's two-sample test were performed using the Circular Statistics Toolbox (Directional Statistics, version 1.21.0.0) with Matlab. For down-right bias, Welch's t-test (unequal variances t-test) was performed.

CRediT authorship contribution statement

Kanta Kumaki: Visualization, Software, Methodology, Investigation, Formal analysis, Conceptualization. Katsuhiro Kawaai: Writing – review & editing, Funding acquisition, Conceptualization. Shinobu Noji: Writing – review & editing, Methodology. Yukiko Kuroda: Writing – review & editing, Supervision. Koichi Matsuo: Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Resources, Project administration, Funding acquisition, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Acknowledgments

We thank the Collaborative Research Resources at Keio University School of Medicine for their technical assistance.

Funding

This work was supported by Keio University Fund for the Promotion of Academic Affairs (Interdisciplinary Collaborative Research).

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

During the preparation of this work, the authors used Gemini 3 Flash (April 2026) and Claude code (Anthropic) to improve coding and to assist with editing text. After using this, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

Editor: Dr. Philiswa Nomngongo

Footnotes

Appendix A

Supplementary data to this article can be found online at https://doi.org/10.1016/j.yjsbx.2026.100154.

Contributor Information

Kanta Kumaki, Email: kumaki22@keio.jp.

Katsuhiro Kawaai, Email: k-kawaai@keio.jp.

Shinobu Noji, Email: snoji@keio.jp.

Yukiko Kuroda, Email: kuro.z6@keio.jp.

Koichi Matsuo, Email: kmatsuo@ciem.or.jp.

Appendix A. Supplementary data

Comparison between down-right bias and dominant orientation

mmc1.pdf (228.3KB, pdf)

Data availability

Codes for performing the bipixel method are available at https://github.com/Koichimts/Bipixel_Method and are archived at Zenodo (https://doi.org/10.5281/zenodo.21439490).

References

  1. Almany Magal R., Reznikov N., Shahar R., Weiner S. Three-dimensional structure of minipig fibrolamellar bone: adaptation to axial loading. J. Struct. Biol. 2014;186:253–264. doi: 10.1016/j.jsb.2014.03.007. [DOI] [PubMed] [Google Scholar]
  2. Arganda-Carreras Ignacio, Fernández-González Rodrigo, Muñoz-Barrutia Arrate, Ortiz-De-Solorzano Carlos. 3D reconstruction of histological sections: application to mammary gland tissue. Microsc. Res. Tech. 2010;73(11):1019–1029. doi: 10.1002/jemt.20829. [DOI] [PubMed] [Google Scholar]
  3. Asghar Adil, Kumar Ashutosh, Narayan Ravi Kant, Naaz Shagufta. Is the cortical capillary renamed as the transcortical vessel in Diaphyseal vascularity? Anat. Rec. 2020;303(11):2774–2784. doi: 10.1002/ar.24461. [DOI] [PubMed] [Google Scholar]
  4. Barrera John W., Le Cabec Adeline, Barak Meir M. The orthotropic elastic properties of Fibrolamellar bone tissue in juvenile white-tailed deer femora. J. Anat. 2016;229(4):568–576. doi: 10.1111/joa.12500. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Bumgarner Jacob R., Nelson Randy J. Open-source analysis and visualization of segmented vasculature datasets with VesselVio. Cell Rep. Methods. 2022;2(4) doi: 10.1016/j.crmeth.2022.100189. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Corliss Bruce A., Doty Richard W., Mathews Corbin, Yates Paul A., Zhang Tingting, Peirce Shayn M. REAVER: a program for improved analysis of high-resolution vascular network images. Microcirculation. 2020;27(5) doi: 10.1111/micc.12618. [DOI] [PMC free article] [PubMed] [Google Scholar]
  7. Currey J.D. Differences in the blood-supply of bone of different histological types. J. Cell Sci. 1960;s3-101(55):351–370. doi: 10.1242/jcs.s3-101.55.351. [DOI] [Google Scholar]
  8. Fedorov Andriy, Beichel Reinhard, Kalpathy-Cramer Jayashree, et al. 3D slicer as an image computing platform for the quantitative imaging network. Magn. Reson. Imaging. 2012;30(9):1323–1341. doi: 10.1016/j.mri.2012.05.001. [DOI] [PMC free article] [PubMed] [Google Scholar]
  9. Grüneboom Anika, Hawwari Ibrahim, Weidner Daniela, et al. A network of trans-cortical capillaries as mainstay for blood circulation in long bones. Nat. Metab. 2019;1(2):236–250. doi: 10.1038/s42255-018-0016-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
  10. Heřt J., Fiala P., Petrtýl M. Osteon orientation of the diaphysis of the long bones in man. Bone. 1994;15:269–277. doi: 10.1016/8756-3282(94)90288-7. [DOI] [PubMed] [Google Scholar]
  11. Hillier Maria L., Bell Lynne S. Differentiating human bone from animal bone: a review of histological methods. J. Forensic Sci. 2007;52(2):249–263. doi: 10.1111/j.1556-4029.2006.00368.x. [DOI] [PubMed] [Google Scholar]
  12. Kassab G.S. Scaling laws of vascular trees: of form and function. Am. J. Physiol. Heart Circ. Physiol. 2006;290:H894–H903. doi: 10.1152/ajpheart.00579.2005. [DOI] [PubMed] [Google Scholar]
  13. Kuroda Yukiko, Yoda Masaki, Kawaai Katsuhiro, et al. Developing long bones respond to surrounding tissues by trans-pairing of periosteal osteoclasts and endocortical osteoblasts. Development. 2024;151(17) doi: 10.1242/dev.202194. [DOI] [PMC free article] [PubMed] [Google Scholar]
  14. Maggiano I.S., Maggiano C.M., Clement J.G., Thomas C.D., Carter Y., Cooper D.M. Three-dimensional reconstruction of Haversian Systems in Human Cortical Bone Using Synchrotron Radiation-Based Micro-CT: morphology and quantification of branching and transverse connections across age. J. Anat. 2016;228(5):719–732. doi: 10.1111/joa.12430. [DOI] [PMC free article] [PubMed] [Google Scholar]
  15. Mantyh Patrick W. The neurobiology of skeletal pain. Eur. J. Neurosci. 2014;39(3):508–519. doi: 10.1111/ejn.12462. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Matsuo K., Ji S., Miya A., et al. Innervation of the tibial epiphysis through the intercondylar foramen. Bone. 2018;120:297–304. doi: 10.1016/j.bone.2018.11.007. [DOI] [PubMed] [Google Scholar]
  17. Matsuo K., Tamura R., Hotta K., et al. Bilaterally asymmetric helical myofibrils in ascidian tadpole larvae. Front. Cell Dev. Biol. 2021;9 doi: 10.3389/fcell.2021.800455. [DOI] [PMC free article] [PubMed] [Google Scholar]
  18. Nunez-Iglesias Juan, Blanch Adam J., Looker Oliver, Dixon Matthew W., Tilley Leann. A new Python library to analyse skeleton images confirms malaria parasite remodelling of the red blood cell membrane skeleton. PeerJ. 2018;6 doi: 10.7717/peerj.4312. [DOI] [PMC free article] [PubMed] [Google Scholar]
  19. Pólya Georg. XII. Über Die Analogie Der Kristallsymmetrie in Der Ebene. Zeitschrift Für Kristallographie - Crystalline Materials. 1924;60:278–282. [Google Scholar]
  20. Pries A.R., Secomb T.W. Making microvascular networks work: angiogenesis, remodeling, and pruning. Physiology. 2014;29:446–455. doi: 10.1152/physiol.00012.2014. [DOI] [PMC free article] [PubMed] [Google Scholar]
  21. Püspöki Z., Storath M., Sage D., Unser M. Transforms and operators for directional bioimage analysis: a survey. Adv. Anat. Embryol. Cell Biol. 2016;219:69–93. doi: 10.1007/978-3-319-28549-8_3. [DOI] [PubMed] [Google Scholar]
  22. Takagi Shunzo, Tagawa Masayuki. Predominance of right-handed spirals in the INTRAEPIDERMAL sweat ducts in man and the PRIMATES. Jpn. J. Physiol. 1957;7:113–118. doi: 10.2170/jjphysiol.7.113. [DOI] [PubMed] [Google Scholar]
  23. Tamada A., Igarashi M. Revealing chiral cell motility by 3D Riesz transform-differential interference contrast microscopy and computational kinematic analysis. Nat. Commun. 2017;8(1):2194. doi: 10.1038/s41467-017-02193-w. [DOI] [PMC free article] [PubMed] [Google Scholar]
  24. Wan L.Q., Ronaldson K., Park M., et al. Micropatterned mammalian cells exhibit phenotype-specific left-right asymmetry. Proc. Natl. Acad. Sci. USA. 2011;108(30):12295–12300. doi: 10.1073/pnas.1103834108. [DOI] [PMC free article] [PubMed] [Google Scholar]
  25. Watson Emma C., Adams Ralf H. Biology of bone: the vasculature of the skeletal system. Cold Spring Harb. Perspect. Med. 2018;8(7) doi: 10.1101/cshperspect.a031559. [DOI] [PMC free article] [PubMed] [Google Scholar]
  26. Weng Qingkai, Osaka Takashi, Onoe Hiroaki, et al. Use of a new micropattern tape method to detect chirality shifts in differentiating C2C12 cells. PLoS One. 2025;20(12) doi: 10.1371/journal.pone.0338032. [DOI] [PMC free article] [PubMed] [Google Scholar]
  27. Zudaire Enrique, Gambardella Laure, Kurcz Christopher, Vermeren Sonja. A computational tool for quantitative analysis of vascular networks. PLoS One. 2011;6(11) doi: 10.1371/journal.pone.0027385. [DOI] [PMC free article] [PubMed] [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Supplementary Materials

Comparison between down-right bias and dominant orientation

mmc1.pdf (228.3KB, pdf)

Data Availability Statement

Codes for performing the bipixel method are available at https://github.com/Koichimts/Bipixel_Method and are archived at Zenodo (https://doi.org/10.5281/zenodo.21439490).

Articles from Journal of Structural Biology: X are provided here courtesy of Elsevier

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