Autonomous Establishment of CT‐Independent Sections to Obtain Optimum Pedicle‐Screw Axis in Direction, Length, and Safety Margin

ABSTRACT

Background

Accurate perception of pedicle geometry during pedicle‐screw placement surgery is critically important because the margin‐for‐error is small.

Method

An assessment algorithm is developed to provide machine‐independent MultiPlanar Reconstruction (MiMPR) of the pedicle. The reconstruction is independent of the CT‐machine frame and enhances patient data portability. Additionally, the algorithm obtains the pedicle‐screw axis with optimum direction, length, and margin using MPRs. A method for the autonomous identification of four body features to form a CT‐independent vertebral frame, {V}, in the image space is formulated.

Result

Applied to 200 high‐resolution CT images, the approach achieved a 100% success rate in defining the pedicle‐medial axis and maximum screw diameter considering the safety margin of 2 mm.

Conclusions

The method eliminates subjective assessment. It provides objective assessment in determining the pedicle‐medial axis with optimal direction and margin without human annotation. Additionally, it significantly enhances screw placement accuracy in robot‐assisted spinal fusion surgeries, regardless of vertebra orientation.

1. Introduction

Pedicle‐screw placement is one of the most common surgeries. The surgery is performed to restore spinal stability, relieve pain, and prevent further deterioration by preventing the rubbing of vertebral bones. Additionally, this surgery is performed to correct the deformities along the longitudinal axis of the spine [1]. The pedicle is in close proximity to the spinal canal, which stores an integral portion of the central nervous system. Adjacent to the spine are numerous vital organs.

Accuracy in accessing the desired path of the pedicle‐screw, both in direction and length, is of utmost importance in pedicle‐screw placement surgery as the margin‐for‐error is very small [2]. The unguided free‐hand procedure of pedicle‐screw implantation is still a largely prevalent practice. Such practices have recorded a poor success rate, with incidences of pedicle‐screw misplacement ranging from 10% to as high as 40% [3]. It is further recorded that 2%–8% of free hand surgeries leave the patient with critical injuries. There could be multiple reasons for inaccurate placement. One of the issues is presented in [4, 5, 6] emphasises the importance of preoperative planning in aspects of pedicle diameter, pedicle angle, and screw length.

The typical spinal pedicle‐screw placement is as shown in Figure 1a. The transverse length (width) of the spinal pedicle in an axial view varies depending on vertebra orientation (see Figure 1b,c) with respect to the reference coordinates system of CT‐machine. The pedicle volume has an asymmetric structure, and its ‘major axis’ is not aligned with any of the reference axes of a CT‐machine. A surgical planner uses a set of two‐dimensional (2D) multiplanar reconstructed and/or a 3D rendering object to annotate a line. Manual annotation of the path is time‐consuming and also a largely unacceptable addition to the standard workflow. It is difficult to determine precise 3D lines that pass through the mid‐line of the asymmetric pedicle volumes [7]. As a result, the manual annotation of the desired path is prone to human error and inconsistency due to intra‐ and interoperable variability [8].

FIGURE 1.

(a) Region of pedicle in yellow, pedicle‐medial axes and alignment of pedicle‐screw. (b) An ideal case of vertebra orientation with respect to CT frame. (c) A typical vertebra orientation with respect to CT frame due to patient alignment.

In recent years, there has been significant research and innovation to automate preoperative planning to estimate the pedicle axis and reduce the risk of screw misplacement. Machine‐learning‐based methods [9, 10, 11, 12, 13] has been proposed to solve the pedicle segmentation problem and pre‐operative planning of the pedicle‐screw insertion path. The major drawback of the evolving method is that it requires a large amount of CT data, the right balance of age group and gender, and accurate curation of data to predict the pedicle contours. Also, ML does not consider the rotation of vertebrae, in the DICOM axial plane, as shown in Figure 1c, while curating the training data sets.

In this article, a novel method is presented to provide objective‐insight into the pedicle, irrespective of the vertebral orientation in the image space. Additionally, autonomous preoperative planning of pedicle‐screw insertion is devised. The objective is to define a pedicle‐medial axis which serves as optimum pedicle‐screw axis in direction, length and margin for spinal fusion surgery. The novelty of this work begins by establishing the pose of the ROI with respect to the local body coordinate frame for surgery in medical image space. A local coordinate system is uniquely defined for each vertebra based on the body features. Further, a coordinate system is attached to each pedicle of the vertebra to generate coordinate‐based true‐section of the pedicle called machine‐independent MPRs. The approach involves utilising image processing techniques and geometry‐based mathematical algorithms. The pose of the pedicle‐medial axis in relation to the vertebral coordinate system, {V} is established with digital accuracy. This eliminates subjective biases and human error.

The remainder of the paper is organised such that Section 2 provides an overview of data, the framework of the methodology for determination of pedicle‐medial axis and Section 3 outlines the validation and assessment of results. The conclusion and discussion are made in Section 4, giving an account of the benefits of the new approach and the method.

2. Methods

Image data preparation involves accusation of CT imaging data sets from individuals aged between 25 and 55 years (mean age 37.2 and median age 30). The data of axial sections were sequentially arranged, with an in‐plane voxel size vary between 0.385 and 0.791 mm and a slice thickness vary between 0.725 and 1.530 mm. Each image plane has 512 × 512 voxels and is acquired under a voltage of 120 kV with radiation ranging from 5 to 151 mAs. The slices that were stacked to form each vertebra in the lumbar region were segmented. Ethical approval for the use of anonymised data was obtained from the hospital. This article does not include any direct experiments involving animals or human participants conducted by the authors.

During pre‐processing, the CT volumes were analysed, with each slice consisting of pixel intensities representing various structures of the human body. One such slice consisting of L2 vertebrae is illustrated in Figure 2a. In this article, such an image is referred to as a raw CT image. The steps of pre‐processing the raw image are shown in Figure 2a–c. The bounding of ROI, as shown in the Figure 2b, is achieved by applying various filters and developing an algorithm. Detailed discussion on the method of bounding of ROI is kept out of the scope of this article. A CT image coordinate system, {I} is attached to the top‐left corner of the image, as shown in Figure 2c, where +u‐axis represents the increasing columns and +v‐axis represents the increasing rows of the image pixels.

FIGURE 2.

(a) A raw CT image of vertebra. (b) A bounding box (BB) enclosing vertebral region (ROI). (c) An image frame {I} fitted on image with edges with illustration of vital regions.

In medical imaging, there is a standard practice of describing image planes in terms that relate to imaging device coordinate systems. In this article, the reference coordinate system of a CT‐machine is termed as the machine coordinate system, denoted by $\left\{\mathrm{C}\right\}$, with $C_0$ representing the origin of the image space of the CT. The unit vectors ${\widehat{X}}_C,{\widehat{Y}}_{C}and{\widehat{Z}}_C$ define the principal directions of $\left\{\mathrm{C}\right\}.$ When discussing the orientation of image planes, the XY plane is commonly understood as axial, the YZ plane as sagittal, and the XZ plane as coronal. However, it’s important to recognise that these descriptions—axial, sagittal, and coronal—originally refer with respect to body of the patient rather than specific orientations within the machine's coordinate system. For instance, what is clinically referred to as a mid‐sagittal corresponds to a YZ plane of the frame $\left\{\mathrm{C}\right\}$, not the medial section of the body. This subtle difference in perception, if not accounted for, can lead to analytical errors.

The challenge lies in reconciling these differing perspectives. While CT axial images may not explicitly account for the orientation of each vertebra, understanding and correcting for this discrepancy is somewhat subjective. It requires a careful consideration of both the anatomical context and the technical orientation of the images. In a pedicle‐screw placement surgery, a body feature‐based coordinate system becomes extremely beneficial for accurate perception. Kaushik et al. illustrated the importance of an ROI‐specific coordinate system in the context of robot assisted neurosurgery. They introduced a body affixed fiducial coordinate system to estimate the pose of the ROI image space [14]. The aim is to make the robot take an optimum posture so as to keep the ROI in its best manipulation zone of the workspace. Therefore, a localised coordinate system for each vertebra and pedicle, based on its body feature will improve the visual perception by providing objectively accurate geometry, which is important for robotic assisted spine surgery and for manual surgery. The cross‐sectional images produced by employing a local body coordinate system, are invariant across different imaging platforms and are said to be Machine‐independent sections.

A framework is formulated to establish pedicle‐medial axis, and create two coordinate systems, (a) vertebral coordinate system, denoted by {V}, and (b) pedicle coordinate system. It is designed in three stages: (i) inputting a stack of axial slices of a vertebra to obtain the Initial Pedicle Region (IPR), (ii) partitioned boundary of the pedicle region as an input and direction of the pedicle‐medial axis as an output, and (iii) generating machine‐independent multiPlanar reconstructions (MiMPRs) of the pedicle region while the pedicle‐medial axis is input.

The first stage of our framework segments vertebral body boundary (VBB) and spinal canal boundary (SCB) using a delineating algorithm (Algorithm 1). It begins with the Hough Transform [15], which identifies a closest circle approximating the centre of the lower vertebral body. A 2D working coordinate frame, labelled BO (${\widehat{U}}_B$, ${\widehat{V}}_B$), is established at the centre of the circle. A distinct working coordinate frame, SC (${\widehat{U}}_S$, ${\widehat{V}}_S$), is set near the centre of the spinal canal, with its origin, ${SC}_0$, positioned by offsetting vector ‘$\bar{D}$’ from ${BO}_0$ along the negative $\widehat{V}$ direction, as shown in the Figure 3a. This detachment of body features from frame $\left\{\mathrm{C}\right\}$ enables the algorithm to pinpoint the extreme boundaries essential for the segmentation of the pedicle region in the local body specific frames. This is highlighted in Figure 3b, showcasing the feature detection and precise separation.

Algorithm 1. Marking of boundary.

1.

FIGURE 3.

(a) Establishment of working coordinate. (b) Illustration of VBB and SCB using different colour and illustration of extreme locations. (c) Centre of incircle fitted to VBB, marked as 1, centre of spinal canal, marked as 2. (d) Segmented left pedicle in green and right pedicle in red with respect to frame {I}. (e) A stack of planes with segmented left pedicle and right pedicle with respect to frame {C}. (f) 3D IPR in frame {C}.

The orientation of each pedicle is not symmetric to the central line of the vertebra. Therefore, it needs to be segmented and isolated individually from the vertebra. The steps for isolation are explained below. An axis aligned rectangular box (see Figure 3b) passing through the extreme pixels E, and bounding the desired IPR, is placed. Further, a line segment is constructed joining centre of SCB and the centre of the incircle, as depicted in Figure 3c. This line segment divides the region into the left and the right pedicles in accordance with the patient's left and right. The division is as shown in Figure 3d using colour code. This procedure is applied to each of axial section of the vertebrae under consideration to obtain the bilateral partition. The coloured locations in Figure 3e represent a ‘planar point cluster’ of the pedicle in XY plane of frame $\left\{\mathrm{C}\right\}$. These cluster, when stacked together along the ${\widehat{Z}}_c$‐axis, forms a 3D pedicle point cloud, as shown in the Figure 3f. The module of the algorithm independently gives the left IPR and the right IPR, each shown in a distinguished colour code. The first stage of the framework is focused on the assessment and the measurement of the data in the images. It is based on the intensity and count of pixels in 2D image frame {I}. Transiting to the stage 2, the description and assessment will shift focus to the study of geometry.

2.1. Pedicle‐Medial Axis Estimation

Henceforth, the discussion will be on stage 2 of the framework, where the development of a novel Decision‐Tree algorithm is elaborated for estimating the pedicle‐medial axis (P). The pedicle‐medial axis gives a true‐perspective of the pedicle geometry. Stage 2 also details the advanced planning strategy for lumbar and thoracic spinal fusion. To the best of our knowledge, no prior work has addressed the issue of identifying the optimal pedicle‐screw axis in both position on a critical section (pedicle isthmus) and orientation, based on reconstructed geometry. The research works [7, 8] suggested a path based on the safety margin and pull‐out strength, while [16] suggested a method based on a coefficient assigned to different levels to obtain the path. However, the asymmetrical structure of the pedicle allows for more possibilities of unique solutions for an optimum path. Unlike the methods mentioned in [7, 8], and [16], the Decision‐Tree algorithm integrates two distinct techniques and is generally patient‐specific.

The first technique is termed as centroidal distance method (CDM). This technique is designed to mark the axis of symmetry of the pedicle represented in frame $\left\{\mathrm{C}\right\}$ as $\text{PC}={\left[P{C}_{x}P{C}_{y}P{C}_z\right]}^T$ where, $P{C}_x,P{C}_y,P{C}_z$ are direction cosines. The optimization is based on the following criteria (a) pivoting the axis through the area‐centroid of the directional minimal area enclosed by the cortical wall of the pedicle, (b) aligning the axis with the area‐centroid of neighbourhood sections to achieve optimal orientation, and (c) achieving the least degree of unevenness of margin along the axis. The CDM operates on (a) a stacked set of 2‐D point cluster of pedicle region (IPR), (b) seed‐axis, and (c) marked central region of IPR, as inputs. All inputs were obtained as described in the step 1 to step 3. The resulting vector is an optimal candidate for the pedicle‐medial axis (P), as described in step 4. The method for establishing the axis consists of the following sequential steps:

1. establishment of a seed axis in pedicle region and regeneration of the planar point cluster.

2. technique for determination of geometrical properties of the pedicle.

3. a novel approach for defining the pivot point for the axis.

4. provide a solution scheme to discover the pedicle‐medial axis.

Step 1

Establishment of seed axis: The algorithm is initiated by establishing a seed axis. The seed axis is designed to pass through the IPR and is taken parallel to ${\widehat{Y}}_C$‐axis of frame $\left\{\mathrm{C}\right\}$. To analyse the geometric nature of the IPR volume, multiple layers are dissected that are parallel to each other and perpendicular to the axis. Figure 4a illustrates the normal planes along the seed axis (a dashed line in green). The layers are discrete and equidistant by predetermined spacing. Understanding the geometry of the pedicle region is critical in finding the pedicle‐medial axis and accurately identifying its physical and geometrical features. One such dissection is shown in Figure 4b.

FIGURE 4.

(a) Reconstruction of XZ section over IPR. (b) Visualisation of pedicle wall of left (green) and right pedicle (red). (c) Listed data to be stored. (d) Variation of area enclosed by the pedicle wall and the point of minimal area (shown as legend) of left‐right pedicle boundary for seed axis.

Step 2

Determination of geometrical properties of the pedicle: A planar surface or volume would be evaluated by connecting the simplices using a general formula (given by Equation 1)

$$V_n\left(P\right)=\sum _{\sigma \mathit{\in }T}\frac{1}{n\mathit{!}}\mathit{det}\left(v_1\left(\sigma \right)\mathit{\dots }\mathit{\dots }\mathit{\dots }\mathit{\dots }{v}_n\left(\sigma \right)\right)$$ (1)

In this context, consider a layer (as described in step‐1) enclosing the boundary of the pedicle. The boundary is divided into smaller pieces called $\left(n-1\right)$ simplices, where n is dimension of space. Now, for a given layer of IPR, boundary points of dissected layer lie in a plane and therefore, simplex will be a line segment, displayed in Figure 4b. The important aspect is that these simplices are oriented to form a closed chain, which in this case, represents the wall of the pedicle. For n = 2, Equation (1) can be modified to a shoe formula [17] given by Equation (2).

$$2{{}^i\mathrm{A}}_j\left({\mathrm{\Pi }}_j\right)=\left(\left|\begin{array}{cc}{\mathrm{x}}_1& {\mathrm{x}}_2\\ {\mathrm{y}}_1& {\mathrm{y}}_2\end{array}\right|\right)+\left(\left|\begin{array}{cc}{\mathrm{x}}_2& {\mathrm{x}}_3\\ {\mathrm{y}}_2& {\mathrm{y}}_3\end{array}\right|\right)+\left(\left|\begin{array}{cc}{\mathrm{x}}_3& {\mathrm{x}}_4\\ {\mathrm{y}}_3& {\mathrm{y}}_4\end{array}\right|\right)+\text{…}.+\left(\left|\begin{array}{cc}{\mathrm{x}}_m& {\mathrm{x}}_1\\ {\mathrm{y}}_m& {\mathrm{y}}_1\end{array}\right|\right)$$ (2)

here, ${{}^i\mathrm{A}}_j$ is enclosed area by boundary of jth dissected layer $\left({\mathrm{\Pi }}_j\right)$ and (X_i, Y_i) are the vertices of the (n − 1)‐simplex that is line segment. A further modification in the Equation (2) leads to the development of a formula for the calculation of area‐centre, displayed in Figure 4b. The area and its centroid are stored, for both, left and right pedicle, separately. The list is illustrated in Figure 4c.

Step 3

Defining anchor‐point for axis: Steps 1 and 2 describe the volume discretisation of IPR into layers and evaluation of the area enclosed by the pedicle boundary and its area‐centroid. An anchor point on an axis is uniquely defined as the directional minimal area (smallest cross‐sectional area normal to the given axis) within a ROI. It is crucial to do that because that minimal area has the least available margin. To achieve this, the area‐centroid, ${{}^i\mathcal{G}}_n$ (x, y, z), is localised as given in Equation (3).

$${{}^i\mathcal{G}}_n\left(x,y,z\right)=\arg \underset{Y\mathit{\in }\left[a,b\right]}{\min }{{}^i\mathrm{A}}_j\left({\mathrm{\Pi }}_j\right)$$ (3)

where,

• — ${{}^i\mathcal{G}}_n$ symbolises the anchored‐point (temporary pivot) for subsequent axis, with left‐superscript indicating ith candidate of PC‐axis and subscript ‘n’ represents that unique plane which encloses directional minimal area.
• — ${{}^i\mathrm{A}}_j\left({\mathrm{\Pi }}_j\right)$ represents the area, as function of temporary coordinate system (x, y, z), embedded in the jth dissected planes
• — $\left({\mathrm{\Pi }}_j\right)$ represent jth dissected plane which is normal to ith axis.

The temporary coordinate system (x, y, z) can be replaced by specific frame as per requirement; for instance, when i = 0, machine frame {C} is selected and the location of the anchored‐point, ${{}^0\mathcal{G}}_n$ is marked,shown in Figure 4b. This is the anchored‐point, which also acts as the origin of a successive frame that is to be used to establish next axis, i = 1. Similarly, for i = 2, a frame fixed at ${{}^1\mathcal{G}}_n$ is used and so forth. It is important to note that the level at which the transverse‐sectional area becomes minimum is not the same for both the pedicles. The Figure 4d shows the variation of the left (in red) and right (in green) pedicle coronal areas as the axis descends into the pedicle. The result suggests the possibilities that: (a) the geometry of left and the right pedicle is not mirror image about ${\widehat{Y}}_c$‐axis, or (b) the vertebra is rotated about axes of frame $\left\{\mathrm{C}\right\}$, (c) combined effect of both (a) and (b). It is critical to identify and account for such differences during surgery.

Step 4

Establishment of pedicle‐medial axis: The pedicle volume has an asymmetric structure, and the difference between the pedicle's anatomical alignment with respect to frame $\left\{\mathrm{C}\right\}$ introduces a challenge in identifying its medial axis. In order to overcome this difficulty and to provide better perspective, an axis of symmetry of pedicle volume is termed the PC‐axis, incorporating local symmetries inherent in the pedicle structure.

To isolate the local symmetry and for a more precise analysis, the IPR is segmented into three distinct regions: (a) the top diverging region, (b) the central region, and (c) the bottom diverging region. The central region, being narrowest, exhibits close symmetry, whereas the other two regions on either side have asymmetric diverging regions. The depth of the central region of IPR about anchored points ranges as follows: T11 and T12 vertebrae: −1.8 mm to +2.4 mm, L1 and L2 vertebrae: −2.1 mm to +2.4 mm, and L3 vertebra: −1.2 mm to +1.8 mm showing vertebral anatomical variations.

The solution step initiates with a seed axis aligning parallel to the ${\widehat{Y}}_c$‐axis, which serves as an initial approximation for the PC‐axis. This setup occurs in the first iteration and serves as a basal step for subsequent refinements, shown as the converging axis (${PC}_i$) in Figure 5a. A converging axis is anchored at the area‐centroid (${{}^i\mathcal{G}}_n$) of minimal area. A new volume of the central region is reconstructed, using discrete layers normal to the‐axis (see Figure 5b), maintaining proximity to the central region. The essence of an iterative solution lies in employing a covariance matrix. This mathematical approach binds the centroidal location within the vicinity of the minimal area while refining the axis by calculating the least perpendicular distances. Solving the covariance matrix yields three principal vectors, with the one exhibiting the highest variance designated as the axis of convergence. Further, the weighted root mean square (wrms) of the shortest Euclidian distance between the axis and the area‐centroid (axis‐centroid offset) is calculated. The centroid is the average point representation of boundary points and the axis with the minimum value is considered PC‐axis.

FIGURE 5.

(a) Iterative solution to obtain PC‐axis on right pedicle. (b) A section normal to converging axis, ${PC}_1$ embedded in blue box and red box is normal to seed axis, ${PC}_0$.

The non‐uniform geometry of the pedicle provides multiple solution towards optimal axis with the same set of critical constraints. We find the other possible optimal solution maintaining critical constraints. The closeness of the solution not only validates the first method but also provides the optimal axis with different geometrical approaches. The second technique is the least area method (LAM), which establishes area‐optimum axis (PA‐axis). The variation of the optimal axis between the two techniques will reveal the difference due to geometrical criteria while serving the critical constraints. The PA‐axis is normal to plane enclosing the narrowest possible area, also termed as omnidirectional‐minimal area, enclosed by a pedicle cortical boundary. This method begins with the establishment of a seed axis, aligned parallel to the ${\widehat{Y}}_C$ ‐axis, serving as the preliminary approximation of the PA‐axis. The IPR is discretised along the seed axis, as outlined in the step 1, and subsequently each plane is rotated within a range specifically tailored to the geometry of the pedicle under examination. A central to this methodology is the identification of the plane orientation that encloses the omnidirectional‐minimal area of the pedicle. Further, an axis normal to this minimal area is considered as PA‐axis. The corresponding flowchart detailing this algorithm is shown in Figure 6a. It is important to note that the PA‐axis marks the critical insertion path along which the area of the pedicle is at its minimum, positioning the PA‐axis as a prominent and critical candidate for pedicle‐medial axis (P). The criticality of the PA‐axis lies in its precision; any linear deviation towards the medial wall may lead to catastrophic damage to the spinal cord. Also, any angular displacement about the anchored‐point increases the volume around the axis, thereby providing a greater pedicle isthmus area, but there should be check on the embedding optimum screw diameter.

FIGURE 6.

(a) Flowchart of LAM for establishment of PA‐axis. (b) A Decision‐tree algorithm.

The decision‐tree algorithm integrates the two methods (CDM and LAM) to select the most suitable candidates between the PC‐axis and PA‐axis for the pedicle‐medial axis (P), based on the calculated values of (a) wrms of axis‐centroid offset, (b) wrms of unevenness of Buffer Margin (uBM) for each section (Π _j ) along both axes. The flowchart is shown in Figure 6b.

2.2. Optimization of Size of Insertion Path of Pedicle‐Screw

In the context of pedicle‐screw placement, the diameter of the screw is calculated considering the designed safety margin (SM), defined at a plane containing the pedicle isthmus, which is the narrowest section of the pedicle. The SM is crucial as it accounts for allowable deviation in orientation and translation during screw insertion into the pedicle without any risk of failure. This concept is mathematically expressed as in Equation (4).

$$SM={{}^{P}D}_n-D_S$$ (4)

where, ${{}^{P}D}_n$ is diameter of pedicle isthmus and $D_s$ is diameter of Pedicle‐Screw. The SM value is pre‐decided by the surgeon as it depends on factors like bone mineral density and bone mass, and the direction of axis chosen.

Therefore, the diameter of the‐screw is dependent on the morphological as well as physical properties of the pedicle, leading to Equation (5):

$$D_S={{}^{P}D}_n-SM$$ (5)

To ascertain ${{}^{P}D}_n$, an approach known as the expanding circle method is formulated. This method starts with the formation of a circle that originates from a predetermined area‐centroid within the pedicle isthmus and expands outward until it tangentially contacts the pedicle walls. This process is visually demonstrated in Figure 7a.

FIGURE 7.

(a) Expanding circle method to obtain optimum pedicle‐isthmus diameter. (b) A dissect plane enclosing the pedicle wall with the parameters illustrating uBM and BM.

Optimization proceeds in a stepwise manner, with each iteration recalculating the ashes of margin (AOM), as explicated in Equation (6). To do that, in the pedicle isthmus. An expanding circle initially make contact with only one side of the pedicle wall, the centre of circle is displaced in the radially opposite direction (see Figure 7a) by amount $\frac{\mathit{A}\mathit{O}\mathit{M}}{\mathbf{2}}$. This is an iterative process of expansion and translation of centre that is performed until the circle tangentially contacts the pedicle walls on both sides. This process converges if $\mathit{A}\mathit{O}\mathit{M}\mathit{\le }0.05mm$ thus, ensuring an accurate measurement of the isthmus diameter. Moreover, this method facilitates the identification of a pivotal point by ensuring that the expansion of the circle continues symmetrically on both sides and refines the parameters for determining the medial axis (P).

$${}^{P}AOM=\left|\min \left(D\left({{}^P\mathcal{G}}_n^k\left(x,y,z\right),V_n\left(x,y,z\right)\right)\right)-\min \left(D\left({{}^P\mathcal{G}}_n^k\left(x,y,z\right),V_n^{\mathit{\prime }}\left(x,y,z\right)\right)\right)\right|$$ (6)

where,

• — ${{}^P\mathcal{G}}_n^k$ symbolises the $k^{th}$ anchored‐point(pivot), with left superscript P indicating medial path (axis) and subscript ‘n’ represents that unique plane ${\mathrm{\Pi }}_n$.which encloses enclosing pedicle isthmus $\left({{}^{P}A}_{\min }\right)$.
• — $V_n$ (x, y, z) belongs to coordinate of pedicle wall at a nth plane enclosing pedicle isthmus $\left({{}^{P}A}_{\min }\right)$ that is normal to P‐axis.
• — $V_n^{\mathit{\prime }}$ (x, y, z) belongs to coordinate of pedicle wall that are radially opposite to $V_n$ (x, y, z), at a nth plane enclosing pedicle isthmus $\left({{}^{P}A}_{\min }\right)$ that is normal to P‐axis.
• — D is the distance operator that returns the distance between a 3‐D axis and a point.

Once the pivotal point's location is optimised, the computation of optimum ${{}^{P}D}_n$ is given in Equation (7)

$$\frac{{{}^{P}D}_n}{2}=\min \left(D\left(I_{P\mathit{\cap }{\mathrm{\Pi }}_n}\left(x,y,z\right),V_n\left(x,y,z\right)\right)\right)$$ (7)

where,

• — $I_{P\mathit{\cap }{\mathrm{\Pi }}_n}$ belongs to intersection point of P‐axis and nth normal plane (${\mathrm{\Pi }}_n$).

After that, uBM is computed, as detailed in expression‐8.

$${{}^{P}uBM}_j=\left[\left.\min \left(D\left(I_{P\mathit{\cap }{\mathrm{\Pi }}_j}\left(x,y,z\right),V_j^{\mathit{\prime }}\left(x,y,z\right)\right)\right)-\frac{{{}^{P}D}_j}{2}\right)\right]$$ (8)

where, ${{}^{P}D}_j$ symbolise the maximum diameter of jth dissected plane ${\mathrm{\Pi }}_j$.

In Figure 7b, the jth dissected plane (${\mathrm{\Pi }}_j$), represents a section where the condition BM > 0 indicates a positive margin surrounding the screw at its maximum available diameter. This surplus margin is crucial as it enhances pull‐out strength, reinforcing structural stability and ensuring secure anchoring. In figure, the uBM quantifies the degree of unevenness of the margin by measuring the absolute difference between the distances from the pedicle medial axis (P) to the medial cortex wall and the outer lateral cortex wall. The results of this analysis are discussed in Section 4 of the article.

Determination of the insertion depth of the‐screw depends on the path. As of now, there are majorly three ways for insertion of Pedicle‐screws, i.e. TT approach, CBT technique and MC approach [18]. The medially directed traditional trajectory (TT approach) has gained popularity during the past 2 decades [19]. If the surgeon follows the TT approach for screw insertion, the Decision Tree Algorithm (see Figure 6b), optimises the pedicle‐screw insertion depth based on quantitative measures. The measure of the insertion depth of a pedicle‐screw is well explained in [20, 21].

3. Results

The proposed method was implemented in PythonV3.7 and executed on an HP workstation operated under windows 10, 64‐bit operating system. The workstation is embedded with an Intel i7‐9800X processor at 3.80 GHz, 32 GB memory, and graphics processing unit (GPU) acceleration (Nvidia Quadro 4000 with CUDA). As mentioned above, the proposed method consisted of three stages, that is, (i) IPR segmenting, (ii) establishing the pedicle‐medial axis(P), and (iii) MiMPRs of pedicle region. Visualisation of MPRs is implemented using the open 3D library, while other major libraries used are scikit‐image, SciPy and NumPy.

3.1. Validation and Assessment

The proposed framework to formulate CT‐independent sections and to obtain optimum pedicle‐screw axis in direction, length, and safety margin is evaluated by optimising three criteria: guidance based on the narrowest section of the pedicle isthmus, the non‐uniformity of the margin around the axis (referred as ‘unevenness of margin or uBM’), and aligning the axis with the area‐centroid of neighbourhood sections to achieve optimal orientation.

The (a)‐series of figures from Figures 8, 9, 10, 11 provide a holistic visualisation of the results for the left pedicle and the (b)‐series from Figures 8, 9, 10, 11 show the results for the right pedicle through violin plots. The input data are generated from medical images of T11‐L3 vertebrae obtained from a CT scan of the patient in DICOM format. The slice thickness of 0.625 mm along ${\widehat{Z}}_C$, equal Pixel spacing of 0.404297 mm along ${\widehat{X}}_{C}and{\widehat{Y}}_C$. The plots offer a comparative analysis of the central pedicle region, as discussed in step 4 of Section 2.1. Results and comparison of attributes such as the sectional area, the axis‐centroid offset, the degree of uBM and the available screw diameter in central pedicle region are demonstrated through violin plots. This analyses the results of both the CDM and LAM methods on both the left and right pedicles of vertebrae. The axis determined through CDM is referred to as the PC‐axis, while the axis determined through LAM is termed the PA‐axis.

FIGURE 8.

(a) Comparison of sectional area normal to PA and PC axes measurement for left pedicle. (b) Comparison of sectional area normal to PA and PC measurement for right pedicle.

FIGURE 9.

(a) Comparison of offset between the axis and area‐centroid of the left pedicle. (b) Comparison of offset between the axis and area‐centroid of the right pedicle.

FIGURE 10.

(a) Comparison of degree of unevenness over margin for left pedicle among vertebrae. (b) Comparison of degree of unevenness over margin for right pedicle among vertebrae.

FIGURE 11.

(a) Comparison of trend between diameter and area for left pedicle of L2 vertebra. (b) Comparison of trend between diameter and area for right pedicle of L2 vertebra.

The violin plots Figure 8a,b highlights significant observation about the sectional areas embedded in the dissected planes. The sectional areas are measured along the depth of pedicle region normal to PC‐axis and PA‐axis and for the left and right pedicle across various vertebral levels. The sectional areas of T11, T12, L1, L2 demonstrate low variance and a strong tendency to cluster within a narrow range, indicating effective identification of central regions. However, it differs for L3 due to the morphology. Therefore, in cases like L3, the analysis of area variations meets the basic objective of increasing diameter but does not conclusively guarantee symmetricity along the length of the pedicle‐medial axis (P). It lays a foundational basis for further evaluation of the offset between the axis and the area‐centroid with the understanding that (i) the axis is medial or close to medial to pedicle volume (ii) uBM should also be least for the axis.

Figure 9a,b provide a comparison of the offset for PC‐axis and PA‐axis along the depth of pedicle region. The graph shows how the spread of offset is distributed along PC‐axis and PA‐axis. Low variance (spread) signifies symmetric bone morphology along the depth of the axis. In general, the variance along the PC‐axis is significantly lower than along the PA‐axis (except T11 vertebra). This is further supported by the wrms value of the offset, where the PC‐axis has a lower value than the PA‐axis (Table 1), indicating that the PC‐axis has closer alignment with the area‐centroid. The ideal medial path should have low variance and low wrms value of the offset.

TABLE 1.

Illustration of statistical data obtained via CDM and LAM, which help to choose the‐medial axis (P).

Distinct algorithm ⟶	Centroid distance method							Least area method
Name of pedicle ↓	Level of vertebra	ROTZ ($\mathit{°}$)	ROTx ($\mathit{°}$)	Wrms of offset (mm)	Wrms of uBM (mm)	Diameter of isthmus (mm)	Safe diameter of pedicle‐ screw (mm)	ROTZ ($\mathit{°}$)	ROTx ($\mathit{°}$)	Wrms of offset (mm)	Wrms of uBM (mm)	Diameter of isthmus (mm)	Safe diameter of pedicle‐ screw (mm)
Left pedicle	T11	−7.659	1.169	0.630	0.395	6.4	6.074	−8	−10	0.647	0.409	6.669	6.27
	T12	−1.093	6.960	0.165	0.322	7.33	7.23	4	−2	0.298	0.578	7.206	7.083
	L1	−13.169	9.162	0.253	0.382	6.76	5.85	−11	−2	1.037	0.336	6.508	6.057
	L2	−14.595	−0.696	0.237	0.256	6.99	6.58	−13	−6	0.458	0.425	6.968	6.548
	L3	−18.406	−0.777	0.313	0.417	9.04	9.024	−10	1	1.342	1.450	9.2207	7.81
Right pedicle	T11	4.218	−15.426	0.523	0.587	5.99	5.95	−5	−14	0.214	0.519	5.89	5.75
	T12	0.729	6.260	0.124	0.313	6.929	6.862	−10	6	0.240	0.466	7.1370	6.263
	L1	3.288	5.419	0.139	0.218	6.665	5.79	4	0	0.506	0.258	6.304	5.692
	L2	2.475	−0.438	0.160	0.225	6.62	6.50	6	0	0.294	0.449	6.351	6.22
	L3	9.931	−12.495	0.479	0.411	8.45	8.03	17	−4	0.581	0.730	8.33	7.465

Note: ROT_Z (°): Rotation of axis about the Z‐axis of frame {C}, representing the lateral angle in the axial plane. ROT_x (°): Rotation of axis about the X‐axis, indicating the cranial angle in the sagittal plane. Wrms of offset (mm): weighted root mean square of the offset between the axis and area‐centroid, where area is embedded in dissected plane through which the axis passes. Wrms of uBM (mm): weighted root mean square of the unevenness of margin, representing the deviation in bone morphology along the axis. Safe Diameter of pedicle‐screw (mm): The maximum diameter of the pedicle‐screw that can be safely inserted.

Figure 10a,b examine the degree of uBM for PC‐axis and PA‐axis, along the depth of pedicle region. Unlike offset measurement, uBM measurement is based on the local variation of boundary points within dissected plane. In general cases, the axis obtained is considered as pedicle‐medial axis (P) if the wrms values of offset and uBM are low. However in the cases exception to the general, that is, if the wrms value for offset are lower along the PC‐axis and uBM are lower along the PA‐axis, the decision‐tree algorithm selects the pedicle‐medial axis (P) based on lower unevenness of margin or uBM. The left pedicle of the L1 vertebra is the case in study exhibiting such a selection.

Figure 11a,b presents a graph that compares the trends of diameter and area along the PC‐axis for the left and right pedicles of the L2 vertebrae. The results highlight a critical narrowing region along the axis. Interestingly, the minimum diameter does not coincide with the minimum area (pedicle isthmus), as indicated by the markers in the legend. This suggests that the narrowest diameter does not necessarily correspond to the region with the smallest cross‐sectional area because the irregular geometry of the area may not embed the circular diameter proportionally. The study of the sections of area along the length of the axis is important because it has implications on structural assessments, such as load‐bearing capacity or vulnerability to stress along the depth of the pedicle.

Figure 12a shows the machine‐independent vertebral coordinate system and pedicle coordinate system along with the seed axis and pedicle‐medial axis (P). Figure 12b illustrates machine‐independent coronal section that is defined in the space of unit axes (${\widehat{X}}_{L_i},{\widehat{Z}}_{L_i}$) while third axes, ${\widehat{Y}}_{L_i}$, is cross product of first two axes and along the medial axis. Similarly, the true‐sagittal section is contained by (${\widehat{Y}}_{L_i},{\widehat{Z}}_{L_i}$).

FIGURE 12.

(a) Illustration of machine‐independent Vertebral coordinate system and pedicle coordinate system. (b) Machine‐independent coronal section (green) and CT coronal section (red).

Krag et al. reported that increasing the depth of screw insertion can improve the strength of the transpedicular screw‐vertebra interface [22]. Daemi et al. shows the result that using the conventional path planning method, pull‐out strength is a quadratic function of insertion depth [8]. However, the recent introduction of Cortical Bone Trajectory (CBT)/Medial‐Lateral Screw Trajectory (MLST) method [23, 24] has shown to provide superior stability and grip compared to traditional trajectories, in case of osteoporotic vertebrae.

Therefore, depending on physical properties such as BMD [25] and morphological characteristics of pedicles such as diameter and geometry [26], and availability of technology [27] such as screw augmented technology, clinicians decide the most optimum path for screw insertion. Regardless of the path utilised, the current Decision‐Tree algorithm accurately reconstructs the pedicle walls along the pedicle‐medial axis. The structure of machine independent and accurate reconstruction is demonstrated in Figure 12b. In all the cases, the algorithm is successful in securing the optimal diameter of the screw satisfying the safety margin, uniformity of the margin and closeness to the centroid of the sectional area.

The contribution of this paper is also in the ability to suggest the optimal screw insertion path based on local geometry. The Decision‐Tree Algorithm, as explained in Figure 6b, opts for a longer axis length if the PC‐axis and PA‐axis are close. The MiMPRs facilitate the detection of any potential penetrations to the medial or lateral walls while planning manually.

4. Discussion

The MiMPRs of a pedicle‐geometry are based on the orientation of the pedicle‐medial axis, which provides objective perception. The framework considers the uniformity of margin as primary and area‐centroid as secondary requisite in planning direction, length, and safety margin pedicle‐screw axis. The steps discovered are logic‐based, which rely on the geometry of the local region and make decisions without the need for manual data input. This mathematical and logical approach enhances the interpretability of the results, making it straightforward to understand the solution outcomes.

The proposed frame also serves as an advanced surgical planner for pedicle‐screw placement that aligns with the midline of the pedicle volume, based on various parameters determined from multi‐planar reconstructions along the axis. The methods provide an average success rate of 100% among T11‐L3 vertebrae. The method provides patient‐specific and geometric‐specific results. No, general perception or knowledge were used to formulae the path. Considering that the error chain of the surgical system reaches up to 2 mm, the derived path for the pedicle‐screw can reduce the risk of screw misplacement or pedicle penetration, even when errors occur during registration or actual screw insertion.

The stepwise analytical methodology of this study provides a thorough understanding of the effectiveness of the methods in accurately identifying the medial axis of the pedicle and supports the proposed selection criteria.

Author Contributions

Amit Kumar: conceptualisation, data curation, methodology, visualisation, writing–original draft, writing–review & editing. Dwarakanath Thambihalli Aswath: conceptualisation, data curation, methodology, supervision, validation, writing–review & editing. Gaurav Bhutani: supervision, validation, writing–review & editing. Dwarakanath Srinivas: data curation, supervision, validation.

Ethics Statement

Ethical approval for the use of anonymised data was obtained from the hospital. This article does not include any direct experiments involving animals or human participants conducted by the authors.

Consent

Patient informed consent was not necessary for this investigation.

Conflicts of Interest

The authors declare no conflicts of interest.

Data Availability Statement

The data that support the findings of this study are available upon reasonable request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.