Building 3D Surface Model of the Human Hip Bone from 2D Radiographic Images Using Parameter-Based Approach

Abstract

Human hip bone results from the fusion of three primarily stand-alone bones, which makes it a complex structure whose accurate 3D model is hard to obtain, given that a sufficiently dense cloud point is provided by dedicated Computer Tomography (CT) scanning process. Therefore, it was necessary to develop a parameter-based approach that enables reverse engineering of the human hip bone in the cases when it is impossible to get a CT scan of the whole bone or when the only available data are 2D X-ray images in anteroposterior and lateral projection. Parameter-based approach, developed on the wing of ilium bone, is based on anatomical landmarks which define parameters of the bone used to predict point's coordinates on the surface. Between the parameters, ten parametric regions that represent parts of the bone are separated. By constructing all regions, it is possible to obtain a complete polygonal model of wing of ilium at the right or left bone, measuring only 10 parameters’ values. Parameter-based approach served as a base for method of parametric regions, which allows combining individual regions (enlarging) or dividing into smaller regions, called sub-regions. It is also applicable to the ischium and pubic bone.

1 Introduction

Human hip bone represents a very complex morphological structure of irregular shape, resulting from the fusion of three primarily stand-alone bones: ilium, ischium and pubic bone. Being a part of the skeletal system, the hip bone can be significantly damaged due to various traumas, tumors and other pathological conditions. Hip bone fractures are fractures of any constitutive bone, such as pubic fractures, partial fractures of the ischium and ilium bone or pubic dislocations which involve acetabulum. Some of these traumas require the use of osteofixation materials (reconstructive plates, fixators, screws, clamps). Tumor resections have to be conducted with great precision and are followed by bone reconstruction process, which requires the existence of some personalized prosthesis and implants. They are constructed to completely match the lost part of the bone after tumor removal. In such cases, having a high-quality 3D model of the bones is necessary in order to simulate the correct placement of osteofixation materials and implants in a virtual environment, and to plan and simulate the surgical procedure.

Direct procedures for creating 3D models of hip bone are almost impossible to apply due to complex bone shapes, as well as lack of explicit knowledge about surface shape. The procedures used to reconstruct the 3D geometric model of individual bones are based on free-form technologies with a certain degree of approximation and reverse engineering technologies. Initial data for reverse engineering procedures are obtained from volumetric medical images, mostly by using procedures like CT or Magnetic Resonance Imaging (MRI). These digital images have different resolutions and therefore consist of point clouds of different densities. These point clouds need to be converted into usable 3D geometric models that can be translated into a digital representation of a physical object in reverse engineering procedures.

3D polygonal models are used for visualization purposes in the cases when high accuracy of the model is not required, since they have a certain degree of approximation of rough surfaces. However, for planning and simulating operations, as well as in the processes of osteofixation materials design, and similar applications, it is necessary to have a quality 3D volume model. This requirement is particularly pronounced in production of personalized implants. 3D polygonal and volume hip bone models can be acquired by reverse engineering methods only if there is a sufficiently dense point cloud of complete bone. Otherwise, in the case of incomplete data, it is not possible to obtain high-quality 3D model and to conduct accurate patient-specific bone reconstruction. For such cases, a new method of parametric regions was developed, using information about specific anatomical points and parameters, which represent a prerequisite for complete morphometry of the hip bone. Parameter-based approach, with the aid of statistical tools, allows for determining regression models for predicting the position of points on the hip bone surface. These positions are used for creating a surface model of the hip bone and/or its parts in the cases when all available input data are taken from two biplanar X-ray projections. The resulting 3D surface point models of the wing of ilium bone and its part as examples, are accurate enough for application in systems for planning and simulation of operative flow, as well as creating 3D models of missing parts of bone for the purpose of making implants and constructing personalized osteofixation material and similar applications.

1.1 Methods for 3D Models Reconstruction

Statistical modeling and analysis of anatomical shapes is the subject of research in the field of medical images (Styner et al. 2003; Chintalapani et al. 2007). Statistical shape analysis is important for 3D reconstruction of anatomical structures and improvement of shape prediction from incomplete input, while multivariate statistical analysis helps to determine the relevant shape variation in the population (Aguirre et al. 2007).

The anatomical structures of different individuals show large but limited natural variability that can be statistically represented. Appropriate parametric description of surfaces is needed to preserve variability and given characteristics. In methods that use parametric description of surfaces, deformable super-quadratics can be used in combination with local deformation modeling, a series of extensions, or coordinates of points on the surface (Sierra et al. 2006).

Similar to the representation of surfaces of any object, 3D bone surface models can be presented in the form of meshes or Non-uniform Rational B-Splines (NURBS) (Su et al. 2013). It is also possible to represent the surface of an object (like the bone) with the set of unique points in 3D space which have the same positions on each object (so-called “landmarks”). Each item belongs to a particular part of the object. Point Distribution Model (PDM) in 2D or 3D is obtained by statistical examination of the landmarks’ positions (Zsemlye 2005). Nonlinear models such as polynomial regression point distribution model, are based on polynomial regression, where the variation modes in linear model are replaced by polynomial curves (Kirschner 2013). In order to simplify and reduce the number of parameters describing the 3D object, modal analysis can be used, with the help of Delaunay methods in the reconstruction process (Angelopoulou et al. 2015).

Finding correspondent points is essential for the automatic production of statistical surface models from an initial set of 3D surfaces. In order to solve the problem of point correspondence, minimum description length function can be used for error reduction and generalization (Zsemlye 2005; Kirschner 2013; Chen and Shapiro 2009). The iterative closest point method can be used to establish correspondences (Zsemlye 2005; Blanc et al. 2012).

Predicting the shape of a particular organ or its part in practice, especially in the case of incomplete surface shape information, is performed on the basis of various predictors, using three methods: Principal Component Analysis (PCA), which is also known as principal component regression, partial least squares and canonical correlation analysis. Scalar predictors, such as morphometric and anthropometric parameters, can be directly included in the aforementioned regression models (Blanc et al. 2012). Prediction techniques, based on observing the population, are used when shapes need to be predicted based on partial information. Minimizing the Mahalonobis distance is an iterative technique for shape prediction. The surface is controlled by selected points on the surface that the user can directly identify in a given data set. The positions of these landmarks represent the boundary conditions for shapes (Zsemlye 2005). Some parametric shape description methods use deformable superquadrics combined with local deformable modeling or point coordinates of the vertices of the surface. It is possible to reduce the number of parameters to a certain degree using procedures such as PCA (Sierra et al. 2006). Spherical harmonic description allows parametric shape description, where shapes can be represented by an object that has a spherical topology (Styner et al. 2003; Zsemlye 2005; Besbes 2010).

A great number of anatomical researches of the human hip bone were conducted with the aim of studying its morphology and morphological variations that depend on age, sex, characteristics of a certain population, etc. In some previous studies, 3D polygonal models of the human hip bone have been created in specialized programs for processing medical images or statistical models for estimating shapes and variations, but the applied methods give results only when there is a complete and high-quality volumetric image of the whole bone.

Lamecker et al. (2004) have created a statistical surface model for semi-automatic segmentation of the pelvic bone, based on its polygonal model (mesh). A statistical surface model was generated from 23 CT images of the male pelvic bones. Statistical shape and intensity models of the pelvic bone were generated from a set of initial polygonal models, specific for given patient, using the PCA method and by generating virtual X-rays from deformable anatomical models (Ehlke et al. 2013). Statistical shape model was obtained from 50 CT scans which were manually segmented and converted into polygonal models (Seim et al. 2008). Statistical atlases of anatomical bone shapes from 110 CT images of individual patients were obtained using PCA and PDM, through procedures for identifying landmarks, establishing point-to-point correspondence and conducting statistical analysis to study shape variations (Chintalapani et al. 2007).

A hybrid method originating from combining incomplete data obtained from a CT image and geometric data from a visible human data set was used to generate the pelvis’ finite element model, using higher-order Hermite cubic elements. Corresponding anatomical points were selected automatically from CT images, and the pelvic bone was divided into 4 regions (Shim et al. 2007). An anatomical model of the left half of the pelvic bone was constructed from input data in the form of a point cloud, obtained by a 3D laser scanner. Point cloud is further converted into triangle grid surface with certain assumptions and limitations (Phillips et al. 2007).

The hip bone is generated by a reverse engineering process through several steps. A high-precision replica was digitalized using a 3D laser scanner, resulting in a cloud of high-density dots. This initial point cloud was imported into the reverse engineering software, where the model was cleaned, after which the initial surface model was obtained (Popov and Onuh 2009). Surface model of female hip bone was obtained by reverse engineering in Computer Aided Design (CAD) software through following steps: data acquisition from CT scanner and pre-processing, creating an initial polygonal model, healing and smoothing, identification and defining the anatomical and morphological characteristics which correspond to the Referential Geometric Entities (RGEs), creating sets of B-curves and splines, and creating parts of the surface model using loft and blend function. These parts were further merged into complete surface model of the hip bone (Tufegdžić et al. 2013).

1.2 Rationale for Developing a New Method

Reconstruction of medical images in 3D today is an integral part of biomedical research. Registration of multiple cross-sections is of great importance for correct 3D visualization and morphometric analysis of structures (Angelopoulou et al. 2015). CT and MRI are often used in clinical diagnosis and surgery planning, but their use as an imaging modality in interventions is limited due to lack of space in operating rooms and requirements for real-time operating procedures (Yaoa and Taylor 2012). Typical tomodensitometry methods enable obtaining exact 3D information about the human body anatomy, but high radiation doses which are harmful to the patient, large amount of information which should be collected and processed, as well as their price make them less functional (Benameur et al. 2001; Fattah 2013). X-ray radiography is the golden standard for obtaining medical images in orthopedic diagnostics, and radiographic projections still play an important role in diagnosis, surgery, and planning therapeutic procedures (Benameur et al. 2001; Fattah 2013; Lamecker et al. 2006). Belonging to the group of diagnostic methods that is characterized by low cost, mobility, uniform imaging speed and low doses of ionizing radiation (compared to CT), it bears no risk for patients who have ferromagnetic implants (compared to classic MRI machines). The evaluation of orthopedic trauma in traditional surgery planning is based on a small number of 2D radiographic projections in vast number of cases (Fattah 2013; Ehlke et al. 2013).

A number of open source or commercial software packages have been developed to visualize volume data and obtain 3D polygonal mesh bone models and export them to STereoLithography (STL) format, from medical CT images recorded in DICOM (Digital Imaging and Communications in Medicine) format. Some examples are 3D Slicer, 3D-DOCTOR, Amira, democratiz3D, ImageVis3D, MeVisLab, Mimics, OsiriX, etc. But, regardless of the program, the construction of a 3D polygonal hip bone model must be carried out through the following phases: data collection, image segmentation and surface generation. In cases when the volumetric images of the bones are complete and of high quality, their polygonal model will be meaningful and sophisticated. However, in a large number of cases, when due to the poor quality of the image the boundaries of the bones are insufficiently clear, it is necessary to conduct the so-called model healing. This process takes a long time and requires a lot of effort.

The first step is always collecting CT data which means that the patient must be exposed to radiation doses that are significantly higher than those during imaging by other conventional methods, such as radiography. In many cases, the patient has already been exposed to certain doses of radiation in the previous period and it is not possible to take another CT image. In addition, in case of a high degree of osteoporosis, the quality of the CT image is such that it prevents obtaining clouds of points of the entire bone volume. Sometimes, due to great trauma or bone disease, some parts of the bones no longer physically exist, so it is not possible to obtain data on that part based on the images. In such cases, reconstructing the required 3D bone model is impossible.

Therefore, it was necessary to develop a new method that enables obtaining a 3D surface model of the human hip bone, in the cases when the input data are incomplete and medical images do not contain all bone elements. This especially refers to situations when the only available data are 2D X-ray images in Anterior–Posterior (A-P) and lateral projection, which is of particular importance.

1.3 Reconstruction of 3D Models from 2D Images—State of Art

In order to form a 3D model for a specific patient from incomplete data obtained by ultrasound or X-ray imaging, reconstruction of surfaces is a good technology for preoperative planning or for navigation during operations (Yaoa and Taylor 2012). In recent years, methods for reconstructing a personalized bone shape from incomplete data using statistical shape modeling techniques are applied to a large extent. The main idea for shape reconstruction based on a statistical model is to find statistically probable values of model parameters that minimize the adjustment criteria between the initial model and the available information, which is patient specific. The type of information used for reconstruction is common to all methods. The information contains data on explicit morphological characteristics, such as parts of surfaces, contours or points. These methods are mainly based on multilinear regression (Blanc et al. 2012).

Some problems may arise when reconstructing pathological deformities or missing anatomical structures, when the normal natural appearance is unknown. In a large number of cases, objective criteria for guiding the remodeling and reshaping processes are lacking, and operators are guided by subjective assessment (e.g., aesthetics) (Lamecker 2008). Another group of methods use constraints based on nonlinear functions of point positions, but only explicit landmarks can be treated (Blanc et al. 2012).

Reconstruction of 3D unknown object geometry from 2D images is a problem that needs special attention, since it is necessary to generate a model with only a few (2 or more) radiographic projections (Benameur et al. 2001; Fattah 2013; Lamecker et al. 2006). The lack of information due to the small number of images can be compensated by the inclusion of prior knowledge, formulated to represent the shape of the bones in a way that allows deformation of the model based on the entities taken from the image. Assuming that the entities taken from the image, together with the information from the previously defined form (template), provide sufficient information, the patient’s anatomy could be obtained, with the required accuracy. The template model implies information that is specific to a given bone, and is actually an estimate of the initial shape, in the form of a Statistical Shape Model (SSM) (Fattah 2013).

3D shape reconstruction from a set of incomplete 2D projection images is important for a large number of medical applications. Image-guided intervention systems require a personalized 3D anatomical model, which should be connected to the intra-operating system. Some alternatives suggest 3D model generation from calibrated radiographic projections, but it is necessary to integrate previous knowledge for a successful 3D reconstruction, e.g., using a statistical model shape. In the case of incomplete data, two groups of procedures are applied for the reconstruction of the precise anatomical shape of the bones: statistical deformation model and PDM (Baka et al. 2010).

Hybrid 2D-3D deformable registration was performed by combining a landmark registration from one A-P projection of the hip one with SSM based on a 2D-3D reconstruction scheme. Landmark registration was used to find the initial scale and initial rigid transformation between A-P projection and SSM (Zheng 2009). Similarity criteria, based on surface normal and distance was used for determining landmarks in the process of 3D reconstruction (Baka et al. 2010).

The statistical model of the pelvis was made from a collection of CT images, and a simulation of X-rays (known as Digitally Reconstructed Radiographs—DRRs) from CT data for a specific patient. The model was presented in the form of a network of tetrahedrons. These nets have great flexibility and can adapt to the anatomical shape (Yaoa and Taylor 2012).

2 Parameter-Based Patient Specific 3D Model

Due to the fact that CT became the golden standard for generating 3D geometrical bone models with the high accuracy, we used a sample of 38 male CT scans of the right hip bone, obtained with Toshiba MSCT scanner Aquillion 64 (120 kV, 150 mA, thickness 1 mm, in-plane resolution 0.781 × 0.781 mm (pixel size), acquisition matrix 512 × 512, field of view (FOV) 400 × 400 mm). The bones affected by pathological changes, fractures and deformities are excluded from the sample, so it has been reduced to 32 healthy bones, aged from 20 to 83 years (average 64 years).

Data from CT scans, written in DICOM format are converted to STL format and exported to CAD program with the aim to obtain polygonal models of bones. For some models it was necessary to reconstruct and heal the surfaces, eliminate errors and smooth sharp edges.

Research is conducted through a few steps, similar to those presented in Trajanovic et al. (2018) and Tufegdzic et al. (2015):

• Processing CT scans and generating 3D models from DICOM images

• Model reconstruction in CAD program and obtaining polygonal models

• Determining landmarks and parameters

• Measuring parameter values

• Statistical analysis of data and calculating statistical values

• Determining the correlations between parameters

• Choosing, testing and establishing proper regression models for parameters, and selecting of the proper regression model, according to statistical indicators such as the level of statistical significance and the highest value of variance.

At each polygonal model of the human hip bone, a sufficient number of anatomical points in the form of bilateral landmarks is separated in order to capture the shape of all constitutive bones. These 34 bilateral landmarks, presented on the right and left ilium, ischium and pubic bones, are easily identified and recognized on radiographic projections.

Bilateral landmarks defined and separated at the wing of ilium bone are (Trajanovic et al. 2018; Tufegdzic et al. 2015):

• 1. The most superior point on the iliac crest

• 2. The most lateral iliac crest point

• 3. Anterior superior iliac spine (ASIS)

• 4. Anterior inferior iliac spine (AIIS)

• 5. Posterior superior iliac spine (PSIS)

• 6. Posterior inferior iliac spine (PIIS)

• 7. The most superior point at the acetabular limbus

• 8. The deepest point at greater sciatic notch

• 9. The superior point at the sacroiliac joint

• 10. The inferior point at the sacroiliac joint.

• ZS—the point of intersection between the posterior gluteal line with the outer lip of the iliac crest.

• DS—the point of intersection between the inferior gluteal line with anterior edge of the iliac crest.

• PS—the point of intersection between the anterior gluteal line with anterior end of the iliac crest (at the superior edge of the hip bone).

• NISUA—the deepest point at the anterior iliac notch at the anterior edge.

• NISUP—the deepest point at the anterior iliac notch at the posterior edge.

At the ischium and pubic bone the following bilateral landmarks are defined and separated:

• 11. The deepest point in acetabular fossa (Ruiz 2005; Lubovsky et al. 2010)

• 12. Point at the acetabular rim where ilium and ischium bone are connected (Betti et al. 2013; Ruiz 2005; Lubovsky et al. 2010)

• 13. Point at the acetabular rim where ilium and superior pubis ramus are connected (Betti et al. 2013; Ruiz 2005; Lubovsky et al. 2010)

• 14. The most inferior point of the posterior end of the lunate surface of the acetabulum (Betti et al. 2013; Lubovsky et al. 2010)

• 15. The tip of the ischium spine (Betti et al. 2013; Decker 2010; Margam et al. 2013; O'Connell 2004)

• 16. The deepest point at the lesser sciatic notch

• 17. The most prominent point at the upper part of the ischial tuberosity (Betti et al. 2013; Ruiz 2005; Margam et al. 2013; O'Connell 2004)

• 18. The most lateral point at the posterior edge of the ischial tuberosity (Betti et al. 2013; Ruiz 2005)

• 19. The tip of the ischial tuberosity, the most prominent point at the anterior edge of the ischial tuberosity (Betti et al. 2013)

• 20. The most inferior point at the ischial tuberosity (Betti et al. 2013)

• 21. The most posterior point of the obturator foramen (Betti et al. 2013; Ruiz 2005; O'Connell 2004)

• 22. The most anterior point of the obturator foramen (Ruiz 2005; O'Connell 2004)

• 23. The lowest point at the ischiopubic ramus (Ruiz 2005; Decker 2010; O'Connell 2004; Dhindsa et al. 2013)

• 24. The most superior point of the obturator foramen (Betti et al. 2013; Ruiz 2005; O'Connell 2004)

• 25. The most inferior point where inferior pubis ramus and ischium ramus are connected (Ruiz 2005; O'Connell 2004)

• 26. The most inferior point of the anterior end of the lunate surface of the acetabulum (Betti et al. 2013)

• 27. Pubic tubercle, as the most anterior point at the pubic tubercle (Betti et al. 2013; Lubovsky et al. 2010; Decker 2010)

• 28. The most superior point on the superior edge of the medial aspect of the pubic symphysis (Betti et al. 2013; Ruiz 2005; O'Connell 2004)

• 29. The most inferior point on the inferior edge of the medial aspect of the pubic symphysis (Betti et al. 2013; Ruiz 2005; O'Connell 2004).

Bilateral landmarks in A-P and lateral projection at the ilium, ischium and pubic bone are presented in Fig. 5.1.

Fig. 5.1

Bilateral landmarks at the ilium, ischium and pubic bone a A-P projection, b lateral projection

Anatomical landmarks on ilium, ischium and pubic bone are interconnected by straight lines. These lines, defined by linear distance between chosen anatomical landmarks, represent parameters. From the points listed above, it is possible to create 105 parameters at the wing of ilium bone, and 406 parameters at the ischium and pubic bone. In our research we have used 58 parameters at the hip bone, because that number of parameters is sufficient to fully morphologically define the complex form of the human ilium, ischium and pubic bones. For better preview, the parameters will be listed depending on which bone they belong (26 at the ilium bone, 19 at the ischium bone, and 13 at the pubic bone). This will also make it easier for their practical implementation. Parameters at the ilium bone are presented and described in Table 5.1.

Table 5.1 Parameters at the wing of ilium bone (Trajanovic et al. 2018)

Selected parameters of the human ilium bone are presented in Fig. 5.2.

Fig. 5.2

Modified from Trajanovic et al. (2018)

Parameters at the ilium bone.

At the ischium bone, 18 parameters are separated. One additional parameter is also separated, labeled as d₂₇ and considered common for ilium and ischium bone due to the fact that it connects landmark 8 (as the deepest point at greater sciatic notch) and landmark 12 from different bones. At the pubic bone 10 parameters are separated, as well as 3 additional parameters (d₅₃, d₅₇ and d₅₈) that connect landmarks from different bones (26–14, 11–4 and 13–4, respectively). These parameters are presented and described in Table 5.2.

Table 5.2 Parameters at the ischium and pubic bone

Selected bilateral landmarks and parameters are presented in Fig. 5.3 in A-P and lateral projection of the human ischium and pubic bone. Parameters at the ischium bone are displayed with cyan lines and parameters at the pubic bone with yellow lines. Parameters belonging to different bones appear as dash-dot lines (blue, green and white).

Fig. 5.3

Selected parameters at the ischium and pubic bone a A-P projection, b lateral projection

At each of the constitutive bones, parameters are divided into 2 groups. Parameters which can be easily measured in A-P and lateral projections of the hip bone are treated as independent variables and classified in the first group. The second group of parameters includes dependent variables, with the possibility to set up regression equations (Trajanovic et al. 2018).

Independent variables are:

• 1. At the ilium bone: d₁, d₂, d₃, d₄, d₈, d₉, d₁₅, d₁₆, d₁₇ and d₂₂ (10 parameters) (Trajanovic et al. 2018)

• 2. At the ischium bone: d₂₇, d₃₂, d₃₈, d₄₀, d₄₁, d₄₂ and d₄₅ (7 parameters)

• 3. At the pubic bone: d₄₇, d₄₈, d₄₉ and d₅₇ (4 parameters).

Dependent variables are:

• 1. At the ilium bone: d₅, d₆, d₇, d₁₀, d₁₁, d₁₂, d₁₃, d₁₄, d₁₈, d₁₉, d₂₀, d₂₁, d₂₃, d₂₄, d₂₅ and d₂₆ (16 parameters) (Trajanovic et al. 2018)

• 2. At the ischium bone: d₂₈, d₂₉, d₃₀, d₃₁, d₃₃, d₃₄, d₃₅, d₃₆, d₃₇, d₃₉, d₄₃ and d₄₄ (12 parameters)

• 3. At the pubic bone: d₄₆, d₅₀, d₅₁, d₅₂, d₅₃, d₅₄, d₅₅, d₅₆ and d₅₈ (9 parameters).

All parameters’ values are measured over the whole sample and values for descriptive statistics are calculated. Correlation coefficients between parameters from independent and dependent groups are calculated with the aim to determine dependencies between variables. For each of the parameters from the second group, choosing regression models for testing was done by using scatter plots. Linear $(d_j = a + b \cdot d_i )$, squared $(d_j = a \cdot d_i^2 , d_j = a + b \cdot d_i^2 , d_j = a + b \cdot d_i + c \cdot d_i^2 )$, third-degree polynomial $(d_j = a + b \cdot d_i + c \cdot d_i^2 + d \cdot d_i^3 )$, logarithmic ($d_j = a \cdot \ln (d_i )$, $d_j = a + b \cdot \ln (d_i )$), and exponential ($d_j = a \cdot e^{b \cdot d_i }$) models proved to be the best fitted models. Regression coefficients a, b, c and d in regression equation are calculated using least square method, while d_j stands for dependent variable, and d_i stands for independent variable.

The basic condition for the acceptance of the model is established in such way that the value of statistical significance (p-value) must be less than 0.01 for all parameters a, b, c and d. Value of variance R² is also determined. From all the models which satisfy the set condition, the one with the highest R² is selected according to methodology described in Trajanovic et al. (2018) and Tufegdzic et al. (2015).

Adopted regression models for ilium, ischium and pubic bone, so as the values for R² are presented in Table 5.3.

Table 5.3 Regression models for the parameters at the ilium, ischium and pubic bone. Expanded from Tufegdzic et al. (2015)

An exception was made when determining the regression model for the parameter d₂₅, where the correlation coefficient had a higher value for the parameter d₁₁, in relation to the correlation coefficient for the parameter d₂. The values of R² are significantly higher in the case of dependence between the parameters d₂₅ and d₁₁ than in the case of dependence from d₂₅ on d₂. However, due to the fact that the value of the parameter d₁₁ is obtained from the regression equation, which would increase the computational error, a model in which the parameter d₂₅ is calculated as a function of d₂ has been adopted.

Various models presented in Table 5.3 may be associated with the complex shape of the human hip bone. Only non-linear regression is suitable for describing the proper dependencies between the parameters at the bones, such as 11 quadratic, 13 exponential, and 13 logarithmic. For example, at the ilium bone 3 quadratics, 8 exponential and 5 logarithmic, at the ischium bone 6 quadratics, 3 logarithmic and 3 exponential, and at the pubic bone 2 quadratic, 5 logarithmic and 2 exponential models, are adopted.

On the other hand, all regression models can be considered statistically significant based on the values of variance R². The only exception is the parameter d₂₁, since it is difficult to choose the appropriate model because the level of statistical significance for all tested models is greater than 0.01. However, there are few regression models for discussion according to the values of variance R². Some examples are d₁₀, d₁₂ and d₁₄ at the ilium bone, d₃₃ and d₃₇ at the ischium bone, so as d₅₀ at the pubic bone (Table 5.3). Some improvements should be made by increasing the number of bones in the sample. Additional issues to be taken into consideration are the regression models for the parameter d₃₅ at the ischium bone and the parameter d₅₃ at the pubic bone because their values of variance R² are significantly smaller compared to the other value.

There are some interesting conclusions to be drawn, such as dependencies for parameters at the ischium and pubic bone. Parameters like d₂₈ and d₃₃ at the ischium bone are dependent from parameters d₈ and d₁₇, at the ilium bone respectively, while parameters at the pubic bone d₅₄ and d₅₅ show dependencies from parameters d₂ and d₁ respectively, also at the ilium bone.

In order to test the obtained and adopted models, all parameters from both groups are measured at one randomly chosen right male hip bone. Predicted values for dependent values are calculated, according to regression equations from Table 5.3. The differences between the measured and prediction values were calculated and presented in the form of absolute errors. For the purpose of additional comparison, considering the fact that the range of values of the measured parameters is large—from 11.185 to 126.074 mm at the ilium bone, from 11.488 to 87.543 mm at the ischium bone, and 10.584 to 87.331 mm at the pubic bone, the values of relative errors were also calculated.

The values for errors, especially relative errors, could be considered in correlation with some statistical indicators, such as minimum and maximum values for measured parameters, values for variation and in particular, values for standard deviation over the sample. Increasing the number of samples will lead to further improvements.

For the purpose of comparative analysis of the obtained results, the average values of absolute and relative errors are calculated for the entities from which the hip bone is constituted as well as for the complete bone. The results are shown in Table 5.4.

Table 5.4 Average values of absolute and relative errors for the ilium, ischium, pubic and hip bone

The average value of the absolute error between the predictive and measured values of the parameters is the highest for the ilium, and the lowest for the pubic bone. This is understandable because the average values of the measured parameters are the smallest for the pubic bone. The average relative error is the largest for the ischium and the smallest for the pubic bone, while the average value of the relative error for the complete bone is in fact closest to the average value of the relative error for the ilium bone, which is the most complex geometric entity on the human hip bone.

3 Method of Parametric Regions

The process of geometric reconstruction during which polygonal models of human hip bones are formed from point clouds, included in the sample, represents the initial phase in the Method of parametric regions. After healing and smoothing, anatomical landmarks were determined on each of the polygonal models. Parameters were defined as linear distances between the landmarks. After establishing correlations between the measured parameter values, appropriate regression models were tested and selected using tools for mathematical (statistical) modeling.

In order to unambiguously determine the position of anatomical landmarks and all points on the surface of the bones whose coordinates need to be measured, an anatomical coordinate system of the pelvic bone was formed on the basis of anatomical landmarks. The Method of Anatomical Features (described in Chap. 2) applied in this phase of research has been extended and adapted to the complex hip bone anatomy needs, in Geometric morphometry process.

Cutting polygonal models with planes determined by the given parameters, using Quick Plane definition function in CATIA Shape module, results in the set of 26 intersection curves. Another set of intersection curves is obtained by cutting the polygonal models with planes at equal distances of 4 mm. These planes are perpendicular to the given parameters. The intersections of these two sets of curves represent points, whose values of coordinates (X, Y and Z) are measured in right-handed orientation anatomical coordinate system of the hip bone (Trajanovic et al. 2018). These values are taken as the input values for the statistical program. After establishing correlations between coordinates’ and parameters’ values, different linear and nonlinear regression models (quadratic, logarithmic and exponential) are tested for 1195 intersection points. Proper regression models are established using the criteria for the level of statistical significance (p < 0.01) and the highest value for the coefficient of determination (R²) described in Trajanovic et al. (2018).

Based on the obtained results some conclusions should be drawn. The largest number of obtained regression models is nonlinear, for X coordinate is 98.83%, for Y coordinate is 99.58%, and for Z coordinate is 96.47%, from overall number of equations. The number of linear models is for X coordinate 1.14% and for Y coordinate 0.42%. These values should be considered as statistical errors, and logarithmic models should be chosen as proper regression models. In the group of nonlinear models, the most common ones are logarithmic models, especially at the Z coordinate (93.00%). The greatest variability is shown for X coordinate, given the fact that the positions of the points, e.g., the values of these coordinates, follow the complex geometry of the hip bone to a great extent.

For polygonal model parts that correspond to the edges of the hip bone (the superior edge and the parts of the anterior and posterior edge), the points are defined in such manner that the surfaces which correspond to the edges are fully described. The superior edge of the hip bone which extends from point 3 (ASIS) to point 5 (PSIS) is described by the parts of the polygonal model above the parameters d₁₃, d₂₀, d₁₉ and d₁₄. At curves obtained by intersecting the polygonal model with the planes perpendicular to the aforementioned parameters, additional points are defined at distances of 5 mm from the intersection points on the curves through the given parameters at both sides of the hip bone (outer and inner). The part of the anterior edge of the hip bones that extends between anterior iliac spines (points 3 and 4) is described by parts of the polygonal model below the parameter d₂₁ and right from the parameter d₂₃. Additional points were obtained in the same way as with the superior edge, at the curves that are perpendicular to the parameters d₂₁ and d₂₃.

The part of the posterior edge of the hip bone between posterior iliac spines (points 5 and 6) was described by the parts of polygonal model left from the parameters d₂₄ and d₁₂. For describing the part of the edge which extends from PIIS (point 6) to the deepest point at greater sciatic notch (point 8), one more measure is introduced. In order to describe the geometry of this part of the posterior edge, it is necessary to measure the distance between point 6 and the point obtained at the intersection of parameters d₃ and d₉ in the plane projection. Based on this distance it is possible to calculate the required number of intersection curves using the expression for the number of curves N_c presented in Trajanovic et al. (2018), where mean value of the measured distance d_3x is used as the average value of the parameters d_sr.

After measuring the values of the coordinates of all selected points on the edges of the hip bone, the methodology used for choosing proper regression models for points at the curves through the parameters was applied. The corresponding mathematical models for the point's coordinates in function of the respective parameter are obtained. The number of points on the curves was chosen so that a particular point covers the coordinate values in at least one half of the sample (N ≥ 16).

For describing the geometry of the edges, e.g., the parts of the edges at the wing of ilium bones, it was necessary to define a total of 454 points. Linear dependences for the point’s coordinates of on the values of parameters cannot describe the geometry on these parts of the hip bone. At the parts of the anterior and posterior edge all regression models for X coordinate are logarithmic, as well as for the Z coordinate at the posterior part of the edge. Variability in regression models is the greatest for the X coordinate at the superior edge of the hip bone bone (82.96% is logarithmic, 15.82% is square), for Y coordinates at the part of the posterior edge (31.52% is square and logarithmic, while 36.96% is exponential), and for Z coordinates at the part of the anterior edge (40.74% is logarithmic, 51.85% is exponential). The variability for the Y coordinates at the part of the posterior edge results from geometric shapes that appear on the part of the edge below the parameter d₃.

The predictive values of the X, Y and Z coordinates of all these points were calculated in the first iteration, and a polygonal model of the wing of ilium bone was constructed. Some additional operations like optimization, smoothing and increasing the neighbor value lead to improving the model, but the surface 3D model’s geometry was disturbed. Bearing in mind that the obtained model has a certain number of holes that have to be filled in the areas between neighboring parameters, some additional points were defined. After thorough analysis, ten parametric regions were defined. Parametric regions are actually the areas between two neighboring parameters. At the wing of ilium bone, ten parametric regions were separated: parametric region d₁₄–d₁₅, parametric region d₁₅–d₁₉, parametric region d₁₇–d₁₅, parametric region d₁₇–d₉, parametric region d₉–d₁₁, parametric region d₁₁–d₁₈, parametric region d₁₈–d₁₆, parametric region d₁₆–d₂₀, parametric region d₁₆–d₁₃ and parametric region d₈–d₂₅–d₂₆. Labels for parametric regions were assigned according to parameters that represent boundaries for the given region (Trajanovic et al. 2018). Parametric regions and the parts of the edges (colored in yellow) at the wing of ilium bone are presented in Fig. 5.4.

Fig. 5.4

Parametric regions at the wing of ilium bone

In each parametric region, additional points on the curves perpendicular to the corresponding parameters were defined. Points were defined at equal distances of 5 mm, starting from the intersection points, at curves obtained by cutting the polygonal model with planes determined by 26 parameters, at the outer and inner side of the bone surface.

To determine additional points in parametric regions the following curves were used:

• 1. For parametric region d₁₅–d₁₄—curves that are perpendicular to d₁₄

• 2. For parametric region d₁₅–d₁₉—curves that are perpendicular to d₁₉

• 3. For parametric region d₁₇–d₁₅—curves that are perpendicular to d₁₅

• 4. For parametric region d₁₇–d₉—curves that are perpendicular to d₉

• 5. For parametric region d₉–d₁₁—curves that are perpendicular to d₁₁

• 6. For parametric region d₁₁–d₁₈—curves that are perpendicular to d₁₈

• 7. For parametric region d₁₈–d₁₆—curves that are perpendicular to d₁₆

• 8. For parametric region d₁₆–d₂₀—curves that are perpendicular to d₂₀

• 9. For parametric region d₁₆–d₁₃—curves that are perpendicular to d₁₃

• 10. For parametric region d₈–d₂₅–d₂₆—curves that are perpendicular to d₈.

Coordinate values regression models of points were obtained using the same methodology as in the cases of points at intersection curves at parameters and points which belong to parts of the surface that correspond to the edges. In order to obtain a regression model for any of the points in the regions, it was necessary to measure the coordinates of the given point in at least one half of the sample (N ≥ 16). The number of additional points depends of the total surface of the region. The number of selected points is directly related to the size of the parametric region, which is limited by the given parameters, e.g., the area of the bone surface that is covered by the given region. For example, for the region d₁₁–d₁₈, number of points was 269, while in the region d₁₆–d₁₃ was only 15 points. The total number of points was 1310, after eliminating the points whose statistically significant results could not be obtained due to insufficient data. Thus, 3930 regression equations for the point's coordinates X, Y and Z in parametric regions were obtained. All obtained equations are logarithmic, because it was the only possible model with statistically significant results which satisfy the set criteria (p < 0.01).

Analyzing results on the number of obtained regression models leads to several conclusions. For building 3D surface model of the wing of the ilium bone it was necessary to choose 8869 regression equations for 2959 point's coordinates X, Y and Z, mostly logarithmic (7693) (Trajanovic et al. 2018). This fact could be explained by the complex surface to be created, with convex gluteal surface at the outer side, and concave iliac fossa and sacropelvic surface at the inner side. Additionally, gluteal surface, which extends from iliac crest to acetabular rim, is divided by gluteal lines to three unequal surfaces, while iliac crest which represents the superior edge is concave in front, rounding inward, and convex in back, rounding outward. These are the main reasons for small number of linear dependences for all coordinates (only 61).

The calculated predictive values of the coordinates for all points for one arbitrarily selected hip bone were used to construct the polygonal model with better quality in the second iteration. In need to manipulate a large amount of data, the implementation of equations for predicting the values of point coordinates at the surface of the wing of ilium and linking to the CAD program was automated. Therefore, during the research, three VBA (Visual Basic for Applications) short macro programs were developed: for entering the measured values of parameters and correction factors depending on the size of the X-ray, for selecting the range of points and for selecting a group of points from outer or from inner side of the bone to be exported to the CAD program. Separate sheets in Excel file for each of the predicted parameters is provided, so as for the each of the parametric regions.

Verification, done through comparative deviation and distance analysis between the initial and obtained polygonal models, confirmed significant improvement of the model. Although the method is developed for the right hip bone, it can be applied on the left ilium bone as well (Trajanovic et al. 2018).

3.1 Creating 3D Surface Model by Combining Parametric Regions

The method of parametric regions is flexible, because individual regions can be combined (enlarged), or divided into smaller regions, called sub-regions, depending on the needs and parts of the polygonal model of the wing of ilium bone that need to be constructed. By constructing all regions, it is possible to obtain a complete polygonal model of the wing of ilium bone.

For this purpose, individual parametric regions are aggregated with each other, and the result is then aggregated with parts of the edges, so that their polygonal models are constructed. This procedure was preceded by calculating the coordinate's predictive values for the intersection points on the curves through the parameters, the points on the selected parts of the edges of the hip bone and the points in the selected parametric regions. The selected values were exported to a CAD program and converted into point clouds, from which polygonal models were constructed.

For example, the following individual polygonal models were obtained (shown in Fig. 5.5a, where the red dots show the boundaries of the resulting polygonal models corresponding to the curves through parameters):

Fig. 5.5

Polygonal models obtained by aggregating parametric regions and parts of edges a outer side, b inner side

• 1. Polygonal model d₁₄–d_{1 5}–d₁₉ + Ed₁₄ + Ed₁₉—created by merging the regions d₁₅–d₁₄ and d₁₅–d₁₉ with the parts which represent the edge above curves through parameters d₁₄ and d₁₉

• 2. Polygonal model d₁₇–d₁₅ + Ed₂₄ + Ed₁₂—created by merging the regions d₁₇–d₁₅ with the parts which represent the part of the posterior edge left from d₂₄ and left from d₁₂

• 3. Polygonal model d₁₇–d₉ + Ed₃—created by merging the regions d₁₇–d₉ with part of the posterior edge below d₃

• 4. Polygonal model d₉–d₁₁–d₂₅—created by merging the regions d₉–d₁₁ with part of the parametric region d₈–d₂₅–d₂₆ (part above d₂₅)

• 5. Polygonal model d₁₁–d₁₈–d₂₆—created by merging the regions d₉–d₁₁ with part of the parametric region d₈–d₂₅–d₂₆ (part above d₂₆)

• 6. Polygonal model d₁₈–d₁₆ + Ed₂₃ + Ed₂₁—created by merging the regions d₁₈–d₁₆ with parts of the anterior edge right from d₂₃ and below d₂₁

• 7. Polygonal model d₁₆–d₂₀–d₁₃ + Ed₂₀ + Ed₁₃—created by merging the regions d₁₆–d₂₀ and d₁₆–d₁₃ with parts of the anterior edge above d₂₀ and above d₁₃.

A similar methodology for aggregating was applied on the inner side of the wing of ilium bone, except for the regions that are near the edges. Instead of polygonal model d₁₄–d₁₅–d₁₉ + Ed₁₄ + Ed₁₉, polygonal model d₁₄–d₁₅–d₁₉ is created, and instead of polygonal model d₁₆–d₂₀–d₁₃ + Ed₂₀ + Ed₁₃ polygonal model d₁₆–d₂₀–d₁₃, since the parts which describe the upper edge of the hip bone were constructed in the previous step. Aggregated regions at the inner side are presented in Fig. 5.5b where the yellow dots show the boundaries of the resulting polygonal models corresponding to the curves through parameters.

Individual polygonal models of aggregated regions were merged into a single polygonal model, which required some additional improvement near the most superior point on the iliac crest (point 1). After cleaning (editing convex and concave triangles, detecting and editing collision triangles) and smoothing, the optimization of the polygonal model was performed. The analysis of the model was performed by comparing the initial polygonal model of the ilium wing (obtained from CT images) with the obtained model, where the deviation and distances were measured.

Deviations were measured between the initial polygonal model (reference model) and the model obtained by aggregating regions and parts of the edges (the model to measure). Maximal positive deviation was 10 mm, maximal negative deviation was −9.97 mm, expressed in the area around the posterior superior iliac spine (PSIS), around the posterior gluteal line (in her upper part) and just above parameter d₂₆, while 88.69% of the model surface had the deviations in the amount from −6.65 to 6.67 mm. Deviation analysis is presented in Fig. 5.6.

Fig. 5.6

Analysis of deviations between the initial and polygonal model obtained by aggregating regions

Distance analysis between the initial polygonal model, obtained from CT images (Source) and the polygonal model obtained by aggregating of the regions (Target) was performed with an accuracy of 0.001 mm and shown in the form of a color coded map in Fig. 5.7 and 5.8. Normal distances were taken for measurement direction. At the outer side of the bone maximum distance was pronounced in the middle part of the wing (below point 1), above parameter d₁₁ (presented by red color in Fig. 5.7), with the value of 11.788 mm.

Fig. 5.7

Distance analysis between the outer sides of initial polygonal model and the polygonal model obtained by aggregating parametric regions

Fig. 5.8

Distance analysis between the inner sides of initial polygonal model and the polygonal model obtained by aggregating parametric regions

At the inner side maximal distance was just below the superior edge, at the auricular surface and in the area which is located left from auricular surface (presented by red color in Fig. 5.8).

3.2 Comparative Analysis of the Obtained Models

The comparison of the obtained polygonal models was performed based on the results acquired by the analysis of geometry, deviations and distances (deviations) between the initial and the obtained models. The first model was created without using the method of parametric regions, the second involved using method of parametric regions and the third using the method of parametric regions with aggregating regions.

The parameters chosen for geometry analysis are Neighborhood values (the maximal edge length of the triangles displayed) before and after model optimization, statistical indicators such as number of points and triangles, the need for editing triangles and the existence of holes which need to be filled. In order to compare the models, optimization was conducted using the same parameter's values in all cases (minimal length 0.5 mm, maximal length 1 mm and the values of dihedral angle 30°). Comparative geometry analysis according to chosen parameters is presented in Table 5.5, where PM stands for polygonal model.

Table 5.5 Comparative geometry analysis. Expanded from Trajanovic et al. (2018)

The second polygonal model gives the best results considering the need for editing triangles and filling holes. This model is the easiest to optimize in one step with the ratio of optimization of 6.067, calculated as the ratio of the maximal edge lengths of the triangles, and very low neighborhood value, which sequentially leads to the largest number of triangles, comparing with the first and the third polygonal model.

The values of positive maximal and negative deviation, mean and standard deviation, positive and negative mean deviation, expressed in millimeters are taken into account for deviation analysis. The part of the model surface expressed in % where the value of deviation is in the range from −1 to 1 mm, used as the common denominator for all three analyzes, is also presented. The above values are presented in Table 5.6, where PM stands for polygonal model.

Table 5.6 Comparative analysis for deviation parameters. Expanded from Trajanovic et al. (2018)

The third model has the smallest deviation regarding the initial polygonal model, if we consider the values for positive maximum and minimum deviation and positive mean deviation. On the other side, the mean deviation is the lowest in the second model, while the first model shows the best characteristics based on the values of the negative mean deviation and the area of the model that has a small value of deviations.

Statistical data from color coded maps with the same scale gradients were used for comparative distance analysis between the initial polygonal model and obtained polygonal model. Distance values were ranged in four classes and the cumulative values for the parts of the surface model expressed in %, found within the appropriate tolerance limits are presented in Table 5.7 (PM stands for polygonal model).

Table 5.7 Comparative analysis for distance values. Expanded from Trajanovic et al. (2018)

Taking into account the values shown in Table 5.6 by the criteria of deviation values for the given ranges, as well as for the maximum deviation value, the third model should be preferred over the previous two because the values for the model area expressed in % are the largest for all ranges given in Table 5.7. This statement, along with the fact that the third model has the smallest deviation value, leads to the conclusion that the third model is actually the one that deviates the least from the initial model.

Considering the results of comparative analyses of geometry, deviations and distances, and especially taking into account the fact that the second model is the only one that does not require additional editing of the obtained surfaces, has the best mean deviation, and that the difference in surfaces is in the range of 0–10 mm insignificant in relation to the third model (0.72%), as well as the difference in the maximum deviation values between the second and third model 0.635 mm, the second model should be chosen for the construction of the polygonal model of the complete wing of ilium. Additional argument for this is the fact that the difference in the areas of the model where the deviations range from −1 to 1 mm is only 0.42% in favor of the third model, but the time required to obtain the second model is significantly shorter. But, in the cases where there is a need to create some of the individual parts of the wing of ilium the third model should be used as a better solution.

3.3 Application of the Method of Parametric Regions for Implant Creation

An additional aspect of the application of the method of parametric regions lies in the fact that individual regions can be divided into subregions, in accordance with the needs, i.e., the parts of the bone for which it is necessary to make an personalized implant.

These implants can be made of artificial materials (substitutes for bone material) or natural bone in the form of an allograph obtained from a donor or in the form of an autograph of bone taken from a healthy part of the bone or from the patient's skeletal system. Thanks to the development of biomaterials, a large number of different materials, such as metals, polymers, ceramics and composites are successfully used for bone repair. In any case, there are always problems when creating an implant of the appropriate shape from the chosen material. In orthopedics, the geometry of a personalized implant is hand-drawn, and such implants can have geometric errors that make them less effective for a longer period of time. With the development of CAD, CAE (Computer Aided Engineering), CAD/CAM (Computer Aided Design/Computer Aided Manufacturing), new trends in medical technology are emerging that lead to a personalized approach, i.e., enable the construction and production of personalized implants at an affordable price and within a reasonable time (Trajanovic and Tufegdzic 2018).

In the example of applying the method of parametric regions it will be assumed that the bone damage is in the area at the outer side of the wing of ilium, in the parametric region d₉–d₁₁, bounded by parameters d₇ and d₁, which corresponds to the hatched area in Fig. 5.9. The first step was to determine anatomical landmarks 1, 3, 5 and 8, since the values for the parameters d₁ and d₉ need to be measured and inserted into Excel sheet, while the values for the parameters d₇ and d₁₁ were calculated according to the formulas in Table 5.3 (which already exist in the Excel file). Points at the parameters d₁, d_9, d₇ and d₁₁ were selected from special sheets. Point coordinates were calculated according to the algorithm used for the ilium bone, based on the regression equations which also exist in the Excel file, and selected by running a macro for point selection. The selected point coordinates must satisfy the condition from expression for N_c. In this case, only the points at the outer side should be selected (indexed with d_i-j-01). This procedure is repeated after selecting the required region, in this case d₉–d₁₁.

Fig. 5.9

Part of the parametric region d₉–d₁₁

The values for the point coordinates are exported to the CAD program. Points that do not belong to the target area should be removed manually. These are all points outside the area bound by the points at the parameters d₇ and d₁, as well as the points at d₉ and d₁₁. After removing the unnecessary points, the remaining points are converted into a point cloud, and finally the desired subregion polygonal surface is created.

In order to conduct the analysis of deviations and distances the initial subregion polygonal surface is created by cutting the initial polygonal model of wing of ilium, obtained from CT images. Deviations were measured between the initial subregion polygonal model and the obtained subregion polygonal surface. Most of the obtained polygonal surface (71.73%) has deviations from the initial model in the range of −3.31 to 2.67 mm. The value for maximal positive deviation is 4 mm and the maximal negative deviation is −4.96 mm.

A slight maximum deviation of the constructed polygonal surface is noticeable, i.e., 1.36%. Maximal distance value is 4.649 mm. Most of the obtained subregion polygonal surface (89.22%) is within the tolerance limits from 0 to 2.905 mm. The obtained surface can be additionally adjusted in the procedures of successive iterations, which imply the possibility of removing additional points or small corrections of coordinate values.

Methodology applied in creating the part of the area bound by parameters can be applied to each of the regions or subregions. Subregions can be selected depending on the part of the bone that needs to be created, grouped or further divided. It is necessary to determine only the landmarks used to define the parameters whose values are to be measured, from two planar X-ray projections. Created subregions polygonal models can be increased or decreased using the offset function in CAD program.

The required subregion can also be constructed by intersecting a given parametric region. In the presented example, it is necessary to construct a complete parametric region d₉–d₁₁ first, which is further intersected by the curves obtained by cutting polygonal models with planes determined by the given parameters d₇ and d₁.

The results of the deviation analysis between initial subregion polygonal model and obtained subregion polygonal model show that 67.37% of the subregion surface area of the obtained model is in the range of deviations −2.99 to 2.76 mm. On the other hand, the distance analysis gives better results compared to the method of separating points in the area of interest, because 91.39% of the obtained model surface is within the tolerance limits from 0 to 0.512 mm, and the value of the maximum deviation is reduced to 4.092 mm.

The methodology can also be applied to obtain polygonal models with completely arbitrary shape, but it is necessary to have a CT image since the target areas cannot be localized with sufficient accuracy from two plane X-ray projections.

3.4 Parametric Regions at the Ischium and Pubic Bone

The parametric region separation method described above at the example at the wing of ilium bone can be used to create 3D models of the ischium and pubic bones, or its parts. In some areas, it is necessary to use some of the parameters that are previously defined at the ilium bone, such as d₂₅ and d₂₆ (Fig. 5.2), and some of the anatomical landmarks defined for the wing of ilium, 4, 7 and 8 (Fig. 5.1). In such manner, a connection between constitutive entities of the hip bone is established, which ultimately provides complete coverage of the hip bone surface.

Input data in the form of parameters were also acquired from A-P and lateral X-ray projection of the hip bones. From the ischium bone the values for 7 parameters (d₂₇, d₃₂, d₃₈, d₄₀, d₄₁, d₄₂ and d₄₅) and from pubic bone the values for 4 parameters (d₄₇, d₄₈, d₄₉ and d₅₇) were taken. Predicted parameters’ values were calculated using regression equations presented in Table 5.3. Parameters are used to obtain intersections with polygonal bone models from the sample using Quick Plane Definition function in CATIA module. To determine the points at the intersection curves, cross-sections of polygonal models with planes at distances of 4 mm that are perpendicular to the parameters should be used. Two cross-sectional points are to be found on each of these curves. The notation for these points should be the same as in the case of ilium bone, d_i-j-01 and d_i-j-02 (index i refers the number of parameter, index j refers to the ordinal number of the curve obtained by intersecting with the plane perpendicular to the parameter, index 01 shows that the point is on the outer side, index 02 shows that the point is at the inner side of the hip bone). After measuring points’ coordinates, values will be statistically processed. Testing and regression models will be selected for each of the points in the manner described for the points at the surface of the wing of ilium bone.

After that, the regionalization of the surfaces which describe the body of the ischium bone and the ramus of ischium, as well as the body and superior and inferior ramus of the pubic bone should be done, assigning the same notation for the labels according to parameters that represent boundaries for given region, as it was in the case of parametric regions at the wing of ilium bone.

At the ischium bone it is necessary to separate the following parametric regions:

• 1. Parametric region d₂₇–d₂₈—region outlined by the parameters d₂₇, d₂₈ and with the part of the posterior edge between the points 8 and 15

• 2. Parametric region d₂₈–d₂₉–d₃₀—region whose boundaries are parameters d₂₈, d₂₉ and d₃₀

• 3. Parametric region d₃₀–d₃₁—region outlined by the parameters d₃₀, d₃₁ and with a part of the posterior edge between points 16 and 17

• 4. Parametric region d₃₁–d₃₂–d₃₃—region whose boundaries are parameters d₃₁, d₃₂ and d₃₃

• 5. Parametric region d₃₂–d₃₄–d₃₆—region whose boundaries are parameters d₃₂, d₃₄ and d₃₆

• 6. Parametric region d₃₆–d₃₇–LAc_12–14—region outlined by the parameters d₃₆, d₃₇, and with part of acetabular rim between points 12 and 14

• 7. Parametric region d₃₄–d₃₅–d₃₈—region whose boundaries are parameters d₃₄, d₃₅ and d₃₈

• 8. Parametric region d₃₇–d₃₈–d₃₉—region whose boundaries are parameters d₃₇, d₃₈ and d₃₉

• 9. Parametric region d₃₉–d₄₀—region outlined by the parameters d₃₉, d₄₀, and with the part of the rim of the obturator foramen, between the projections of point 14 on the rim of the obturator foramen and point 21

• 10. Parametric region d₄₀–d₄₁–d₄₂—region whose boundaries are parameters d₄₀, d₄₁ and d₄₂

• 11. Parametric region d₄₂–d₄₃—region outlined by the parameters d₄₂, d₄₃, and with the part of rim of the obturator foramen, between the points 21 and 22

• 12. Parametric region d₄₃–d₄₄–d₄₅—region whose boundaries are parameters d₄₃, d₄₄ and d₄₅.

For a detailed description of the ischium bone geometry, it is necessary to define the parts of the polygonal model that describe the edges and part of the ischial tuberosity. The part of the posterior edge which is not covered by above regions is described by the part of the polygonal model left of the parameter d₂₉. The parts of the ischial tuberosity are described by the parts of the polygonal model left from the parameter d₃₃ and left and below the parameter d₃₅, while the part of the inferior edge of the hip bone is described by the parts of the polygonal model below the parameters d₄₁ and d₄₄. These parts are represented by the red color in Fig. 5.10 together with parametric regions at the ischium bone. To determine and predict the coordinates of the points on these parts of the polygonal model, the methodology described for the wing of ilium bone will be used, where the parts of the polygonal model are intersected by curves that are perpendicular to the aforementioned parameters.

Fig. 5.10

Parametric regions and the parts of the hip bone edges at the ischium bone

Since the methodology for creating a surface model of the pubic bone is exactly the same, the parametric regions will be listed below. In order to fully describe the pubic bone geometry, the parts of the polygonal model which correspond to the superior and inferior edge parts will also be presented.

The selected parametric regions at the pubic bone (see Fig. 5.11) are:

Fig. 5.11

Parametric regions and the parts of the hip bone edges at the pubic bone

• 1. Parametric region d₄₅–d₄₆–d₄₇, region whose boundaries are parameters d₄₅, d₄₆ and d₄₇

• 2. Parametric region d₄₇–d₄₈–d₄₉, region whose boundaries are parameters d₄₇, d₄₈ and d₄₉

• 3. Parametric region d₄₉–d₅₀–d₅₁, region whose boundaries are parameters d₄₉, d₅₀ and d₅₁

• 4. Parametric region d₅₂–d₅₄—LAc_13–26, region outlined by the parameters d₅₂, d₅₄ and with part of acetabular rim between points 13 and 26

• 5. Parametric region d₅₄–d₅₅–d₅₆—region outlined by the parameters d₅₄, d₅₅ and d₅₆.

The parts of the inferior edge that are included in following parts of the polygonal model are located below and right from the parameter d₄₆, right from the parameter d₄₈ and above the parameter d₅₀. The surfaces at the part of the superior pubic ramus, so as the part of the corresponding superior edge are described by the points constructed on curves that are perpendicular to the parameter d₅₅. The part of the anterior edge towards point 4, as well as part of the body of the ilium bone, is characterized by the points which belong to the set of curves perpendicular to the parameter d₅₈. At the surface which corresponds to the part of the edge of the obturator foramen, from point 22 toward the superior pubic ramus, it is necessary to use the points situated at the set of curves which are perpendicular to the parameter d₅₁. The surface which corresponds to the part of the edge of the obturator foramen below the acetabular notch is presented by the points at the curves that are perpendicular to the parameter d₅₃. These parts are shown in dark green, blue, orange and yellow in Fig. 5.11.

Beside the parametric regions that include parts of different bones (so called common parametric regions) mentioned and presented above d₂₅–d₂₇–LAc_7–12 (see Fig. 5.10) and d₂₆–d₅₈–Lac_13–26 (see Fig. 5.11 for Lac_13–26), it is necessary to point out the parametric region d₅₆–d₅₇–d₅₈. It is a region outlined by the parameters d₅₆, d₅₇, and d₅₈ and curves perpendicular to the parameter d₅₇ will be used to obtain additional points.

4 Conclusion

The method of parametric regions enables reverse engineering of the hip bone, with sufficient accuracy, in a short time. The method can be applied in case of complete or incomplete volumetric data, which can occur due to trauma, osteoporosis or tumors, but also in cases when the only data sources are 2D X-rays in the A-P and lateral projection. The implementation of parameter-based approach allows subject specific morphometry of the hip bone, using 34 anatomical landmarks, interconnected by 58 straight lines which represent parameters. On the other hand, complete morphometry is provided by 21 measured values of the parameters from 2D X-rays, due to developed statistical regression models which establish proper dependencies between parameters.

Measured and calculated parameters’ values are used to predict numerical values for coordinates of the points on the surface at the wing of ilium bone, based on mathematical models. These points are defined for each of the 10 parametric regions, as well as for the corresponding edges. By selecting all points in the region, or some of them, it is possible to construct a polygonal model of the whole region or its parts. Depending on the needs and parts of the polygonal model of the wing of ilium bone that need to be constructed, it is possible to combine and merge individual parameter regions and parts of the edges, only at the outer or at the inner side of the bone, or on both sides at the same time.

Parametric regions are separated at the other constitutive bones, such as 12 regions at the ischium bone, 5 regions at the pubic bone and 3 common regions, since they are outlined by parameters which connect landmarks at different bones. The parts of the polygonal model which describe corresponding edges, the part of the ischial tuberosity and the part of the pubic body, are also defined. These regions and parts of the bones impose certain directions of further research in order to find the regression equations of the coordinates of points on the parameters, edges and parametric regions.