Precision Measurements and Parametric Models of Vertebral Endplates

Summary

A reverse engineering system is employed to record and obtain detailed and comprehensive geometry data of vertebral endplates. Parametric models of vertebral endplate are then developed, which are beneficial to designing personalized spinal implants, making clinical diagnoses, and developing accurate finite element models.

Abstract

Detailed and comprehensive geometric data of vertebrae endplates is important and necessary to improve the fidelity of finite element models of the spine, design and ameliorate spinal implants, and understand degenerative changes and biomechanics. In this protocol, a high-speed and highly accurate scanner is employed to convert morphology data of endplate surfaces into a digital point cloud. In the software system, the point cloud is further processed and reconstructed into three dimensions. Then, a measurement protocol is performed, involving a 3D coordinate system defined to make each point a 3D coordinate, three sagittal and three frontal surface curves that are symmetrically fitted on the endplate surface, and 11 equidistant points that are selected in each curve. Measurement and spatial analyses are finally performed to obtain geometric data of the endplates. Parametric equations representing the morphology of curves and surfaces are fitted based on the characteristic points. The suggested protocol, which is modular, provides an accurate and reproducible method to obtain geometric data of vertebral endplates and may assist in more sophisticated morphological studies in the future. It will also contribute to designing personalized spinal implants, planning surgical acts, making clinical diagnoses, and developing accurate finite element models.

Introduction

A vertebral endplate is the superior or inferior shell of the vertebral body and serves as a mechanical interface to transfer stress between the disc and vertebral body¹. It consists of the epiphyseal rim, which is a strong and solid bony labrum surrounding the outer rim of the vertebral body, and the central endplate, which is thin and porous².

The spine is subject to a wide array of degenerative, traumatic, and neoplastic disorders, which may warrant surgical intervention. Recently, spinal devices such as artificial discs and cages have been widely used. Accurate and detailed morphometric parameters of endplates are necessary for the design and amelioration of spinal implants with effective prosthesis-vertebra contact and bone ingrowth potential³. Furthermore, information on the exact shape and geometry of vertebral endplates is important for understanding the biomechanics. Although the finite element modeling allows for simulation of the real vertebrae and has been widely used to study physiological responses of the spine to various loading conditions⁴, this technique is patient-specific and not generalizable to all vertebrae. It has been suggested that the intrinsic variability of vertebrae geometry among the general population should be considered when developing the finite element model⁵. Therefore, the geometric parameters of endplates are conducive to the mesh generation and fidelity enhancement in finite element modeling.

Although the importance of the matching of endplate geometry and implant surface has been discussed in previous studies^6,7,8, data on the morphology of vertebral endplates is scarce. Most previous studies have failed to reveal the 3D nature of the endplate^9,10,11. A spatial analysis is required to better and fully depict endplate morphology^12,13,14. In addition, most studies have employed lower precision measurement techniques^10,15,16. Moreover, significant magnification has been reported when geometry parameters are measured by employing radiography or computed tomography (CT)^17,18. Though magnetic resonance imaging (MRI) is considered non-invasive, it is less accurate in defining the precise margins of osseous structures¹¹. Due to a lack of a standardized measurement protocol, there are large differences among existing geometric data.

In recent years, reverse engineering, which can digitize the existing physical parts into computerized solid models, has been increasingly applied to the field of medicine. The technique makes it feasible to develop an accurate representation of the anatomical character of sophisticated vertebrae surfaces. The reverse engineering system includes two subsystems: the instrumentation system and software system. The instrumentation system adopted in this protocol has a non-contact optical 3D range flatbed scanner, which is high-speed and highly accurate (precision 0.02 mm, 1,628 x 1,236 pixels). The scanner can efficiently (input time 3 s) capture surface morphology information of the target object and convert it into digital point cloud. The software system (i.e., reverse engineering software) is a computer application for point cloud data processing (see Table of Materials), 3D surface model reconstruction, free curve and surface editing, and data processing (see Table of Materials).

The purposes of the present report are to (1) devise a measurement protocol and algorithm to obtain quantitative parameters of vertebral endplates based on a reverse engineering technique, (2) develop a mathematical model that allows for a realistic representation of vertebral endplates without digitizing too many landmarks. These methods will be beneficial to surgical act planning and finite element modeling.

Protocol

This study was approved by the health research ethics board of the authors’ institute. As cervical vertebral bones have more intricate shapes¹⁹, the protocol uses the cervical vertebrae as an illustration to facilitate relevant research.

1. Preparation of materials, scanning, and image processing

1. Collect a dry cervical vertebra without pathologic deformation or broken parts.
2. Place the vertebra vertically in the platform of the scanner (Figure 1, see Table of Materials), with the endplate facing the camera lens. Use the active light source of the scanner. Then, start the scanning process to obtain point cloud data (.ASC format).
   NOTE: According to the pre-scan images, adjust the scanner and position of the vertebra to capture as much surface morphology information as possible.
3. Open the software specially used for processing point clouds (see Table of Materials). Click Import to import the point cloud data and generate the digital graphic of the vertebra. Set the sample rate to 100%, select Keep Full Data On Sampling, select the unit of data as millimeters, and click Shade Points. Use the Lasso Selection Tool to select redundant points on the graphic, then click Delete to remove them. Click Reduce Noise and set the smoothness level to its maximum to reduce noise and spikes (Figure 2A,B).
   NOTE: There are basic software operation instructions at the bottom of the GUI (graphical user interface). Noise points with obvious sharp spurs laterally or vertically should be removed to reduce error.
4. Click Wrap to package the imaging data into .stl format file to transform the point cloud into mesh, which will convert a point object into a polygon object.
   NOTE: Reverse engineering software usually accepts .stl-style 3D format.
5. Open the software specially used for 3D reconstruction and data processing (see Table of Materials). Click File then New in the submenu. Select Part in the List of Types. Click Start, then Shape in the submenu, then Digitized Shape Editor. Click the Import icon in the toolbar at the right-hand side of the GUI. In the Import window, select the .stl format file, then click Apply > OK. Click Fit All in the icon in the toolbar at the bottom to load the reconstructed image to the main window of the presentation software.
   NOTE: Steps 1.5–2.3.3 are performed with the same software.
6. Click Activate in the toolbar at the right-hand side. In the Activate window, select Trap Mode > Polygonal Type > Inside Trap. Then, select the vertebral endplate on the 3D image to remove unneeded vertebral components, such as the posterior elements and osteophytes (Figure 2C).

2. Quantification of 3D morphology of the endplate

1. Defining the endplate 3D coordinate system
   1. Click Start > Shape in the submenu, then Generative Shape Design. Click the Point icon in the toolbar at the right-hand side. Mark three anatomic landmarks on the epiphyseal rim: the first two are the left and right endpoints of the endplate trailing edge, respectively; the third is the anterior median point.
   2. Click the Line icon in the toolbar at the right-hand side and select the two trailing edge endpoints to define a posterior frontal line. Click the Plane icon, select the plane type to be normal to curve, then select the posterior frontal line and anterior median point to define the mid-sagittal plane.
   3. Click Start > Shape > Quick Surface Reconstruction. Click the Planar Section icon, enter 1 in the number option, then select the endplate image and mid-sagittal plane to generate an intersecting curve. Click Curve from the Scan icon and select the intersection of the intersecting curve and posterior epiphyseal rim. Define the intersection as the posterior median point.
   4. Click Start > Shape > Generative Shape Design. Click the Line icon and select the anterior median point and posterior median point to define a mid-sagittal diameter. Click the Point icon, then Points and Planes Repetition in the submenu. Then, select the mid-sagittal diameter and enter 1 in the Instance(s) option to define the midpoint of the mid-sagittal diameter.
   5. Click the Axis System icon in the toolbar at the bottom. Then, select the midpoint of the mid-sagittal diameter as the origin, the line parallel to the posterior frontal line as the x-axis, the mid-sagittal diameter as the y-axis, and the line pointing forward and perpendicular to the x-y plane as the z-axis (Figure 3).
      NOTE: The two trailing edge endpoints are chosen as reference points because they are consistent and show minimum variation in the presence of osteophytes¹⁰.
2. Fitting characteristic curves and points on the endplate surface (Figure 4A–D)
   1. Click the Point icon, then Points and Planes Repetition in the submenu. Select the mid-sagittal diameter and enter 3 in the Instance(s) option to divide the mid-sagittal diameter equally into four parts.
   2. Click Start > Shape > Quick Surface Reconstruction. Click the Planar Section icon, enter 1 in the Number option, then select the endplate image and x-z plane to generate an intersecting curve. Click Curve from the Scan icon and select the two intersections of the x-z plane and epiphyseal rim.
   3. Define the line between the two intersections as the mid-frontal diameter. In the same way, divide the mid-frontal diameter equally into four parts.
      NOTE: When the endplate is not symmetrical relative to the med-sagittal plane, choose one of the two endpoints of the mid-frontal curve that has a shorter vertical distance to the z-y plane. Then, define the mid-frontal diameter as 2x the length of the shorter, and divide it equally into four parts.
   4. Click the Measure Between icon in the toolbar at the bottom to measure the length of a quarter of the mid-sagittal diameter. Click the Planar Section icon, enter 2 in the Number option, enter the measured value in the Step option, then select the endplate image and x-z plane to generate two fitting curves on one side of the frontal part. Click Swap to generate two fitting curves on the other side. In the same way, obtain the other three fitting curves in the sagittal plane.
      NOTE: The two mid-frontal fitting curves overlap with the two mid-sagittal fitting curves.
   5. Select 11 equidistant points in each curve for subsequent measurements. Specific method is as follows:
      1. Taking the mid-sagittal curve as an example, divide the mid-sagittal diameter equally into 10 parts, resulting in a sum of 11 points, including nine intermediate points and two endpoints (refer to steps 2.1.3 and 2.2.1).
      2. Go through each equidistant point, obtain nine fitting curves on the endplate surface (refer to step 2.2.2). Click Curve from the Scan icon and select the intersection of the fitting curves and the mid-sagittal curve. Finally, obtain a total of 66 points on each endplate (11 points per curve multiplied by six curves). Click the Measure Item icon in the toolbar at the bottom to measure the coordinates of each point.
3. Measurement of endplate morphological parameters
   1. Line parameter:
      1. Click the Measure Between icon to measure the length of line parameter that is the distance between two measured points.
   2. Concavity parameters:
      1. Create a plane parallel to the x-y plane (Figure 5A): click Start > Shape > Generative Shape Design. Click the Sketch icon in the toolbar at the right-hand side, then click the x-y plane. Click the Circle icon, click Origin on the endplate surface, drag the cursor of the mouse to an appropriate distance, then click. Click the Exit Workbench icon, then the Fill icon, and then click.
      2. Click the Offset icon, select the filled plane, and enter an appropriate value in the offset option until it is tangent to the most concave part, and zoom in. Click Start > Shape > Quick Surface Reconstruction. Then, click the 3D curve icon to find and create the most concave point. Click the Measure Item icon to measure the coordinates of the most concave point (Figure 5B).
      3. Click the Measure Between icon, then select the most concave point and x-y plane to measure the whole endplate concavity depth. Similarly, find and create the most concave depth on a particular plane and measure its coordinates.
      4. Click the Projection icon in the toolbar at the right-hand side, then select the most concave point and x-y plane to obtain the projective point. Click the Measure Item icon to measure the coordinates of the projective point, and determine its distribution based on the coordinates.
   3. Surface area parameters:
      1. Click the Measure Inertia icon in the toolbar at the bottom and click endplate surface to measure its area. Click the Activate icon and select the central endplate along the inner margins of the epiphyseal ring (refer to step 1.6), then click the Measure Inertia icon to measure its area (Figure 5C). Click the Activate icon, then the central endplate, and finally the Swap icon in the Activate window to obtain an epiphyseal rim. Then, measure its area.

3. Development of endplate surface mathematical model

1. Determining the fit order of the parametric equation
   1. Open the data analysis and visualization software (see Table of Materials). Input x = [corresponding data] in the command window. Click Enter.
      NOTE: The “corresponding data” refers to x-coordinate data of the 11 characteristic points in one curve that has been measured in the previous steps. Click Enter after inputting each command, with the same applying to subsequent operations. Steps 3.1–5.5 are performed uniformly with the same software.
   2. In the same way, input z = [corresponding data].
   3. Input the code for i=1:5 z2=polyfit(x,z,i); Z=polyval(z2,x); if sum((Z-z).^2)<0.01 C=i break; end; end​.
      NOTE: The protocol sets the error sum of squares below 0.01 to obtain higher precision, the value of which can be readjusted to satisfy various demands.
   4. Click Enter to obtain a C value that is the desired fit order.
2. Parameter equation fitting
   1. Input cftool and click Enter to bring up the Curve Fitting Tool.
   2. Input the coordinates of a curve in the command window (refer to steps 3.1.1 and 3.1.2). In the Curve Fitting Tool, select x-coordinate data when fitting frontal plane curves and y-coordinate data when fitting sagittal plane curves in the x data option, select z-coordinate data in the y data option, select polynomial, and enter the fit order obtained. Then, the software will output the parametric equation and goodness of fit automatically.
      NOTE: As the curve is a 2D image, the default work option is the x and y options in the Curve Fitting Tool when fitting a curve.
   3. In the similar way, input the 3D coordinates of the 66 points and match the coordinate data to the corresponding axis options. Select polynomial and enter the fit order to gain the parametric equation of the endplate surface (Figure 6B).

4. Acquisition of geometric data based on parametric equation

1. Input x- and y-coordinate values of any point on the endplate in the command window.
2. Input P_X1, P_X2, P_X3....
   NOTE: Px is the parameters of the parametric equation that have been fitted using polynomial in the steps above.
3. Input the equation and click Enter to obtain the result (i.e., input format: z = P₀₀ + P₁₀*x + P₀₁*y + P₂₀*x^2 + P₁₁*x*y + P₀₂*y^2 + P₃₀*x^3 + P₂₁*x^2*y + P₁₂*x*y^2 + P₀₃*y^3 + P₄₀*x^4 + P₃₁*x^3*y + P₂₂*x^2*y^2 + P₁₃*x*y^3 + P₀₄*y^4).

5. Representation of the endplate based on parametric equation

1. Input P_X1, P_X2, P_X3....in the command window.
2. Input the code X=N₁:0.01:N₂;.
   NOTE: N₁–N₂ is the range of X-axis data (i.e., the values of the two endpoints of the themid-coronal curve).
3. Input the code “Y=N₃:0.01:N₄;”.
4. Input the equation (i.e., z=@(x,y)P₀₀ + P₁₀.*x + P₀₁.*y + P₂₀.*x.^2 + P₁₁.*x.*y + P₀₂.*y.^2 + P₃₀.*x.^3 + P₂₁.*x.^2.*y+ P₁₂.*x.*y.^2 + P₀₃.*y.^3 + P₄₀.*x.^4 + P₃₁.*x.^3.*y + P₂₂.*x.^2.*y.^2+ P₁₃.*x.*y.^3 + P₀₄.*y.^4;).
5. Input the code ezmesh(z, [N₁,N₂,N₃,N₄]) to obtain 3D simulation graphics (Figure 6C).

Results

Using the highly accurate optical 3D range flatbed scanner, the endplates were converted into more than 45,000 digital points, which adequately characterize the morphology (Figure 2A,B).

In the measurement protocol, the spatial analysis of endplate surfaces was conducted. Representative curves were fitted and quantified on the surface to characterize morphology (Figure 4B). The line...

Discussion

Reverse engineering has been increasingly and successfully applied to the field of medicine, such as cranioplasty²⁰, oral²¹, and maxillofacial implants²¹. Reverse engineering measurements, namely product surface digitization, refers to the conversion of surface information into point cloud data employing specific measuring equipment and methods. On the basis of such data, complex surface modeling, evaluation, improvements, and manufacturing can be pe...

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
Catia	Dassault Systemes, Paris, France	https://www.3ds.com/products-services/catia/	3D surface model reconstruction, free curve and surface editing and data processing
Geomagic Studio	Geomagic Inc., Morrisville, NC	https://cn.3dsystems.com/software?utm_source=geomagic.com&utm_medium=301	point cloud data processing
MATLAB	The MathWorks Inc., Natick,USA	https://www.mathworks.com/	analyze data, develop algorithms, and create models
Optical 3D range flatbed scanner	Xi’an XinTuo 3D Optical Measurement Technology Co.Ltd., Xi’an, Shaanxi, China	http://www.xtop3d.com/	acquire surface geometric parameters and convert into digital points