Geometrically accurate and anatomically correct 3D models of the human bones are of great importance for medical research and practice in orthopedics and surgery. These geometrical models can be created by the use of techniques which can be based on input geometrical data acquired from volumetric methods of scanning (e.g., Computed Tomography (CT)) or on the 2D images (e.g., Xray). Geometrical models of human bones created in such way can be applied for education of medical practitioners, preoperative planning, etc. In cases when geometrical data about the human bone is incomplete (e.g., fractures), it may be necessary to create its complete geometrical model. The possible solution for this problem is the application of parametric models. The geometry of these models can be changed and adapted to the specific patient based on the values of parameters acquired from medical images (e.g., Xray). In this paper, Method of Anatomical Features (MAF) which enables creation of geometrically precise and anatomically accurate geometrical models of the human bones is implemented for the creation of the parametric model of the Human Mandible Coronoid Process (HMCP). The obtained results about geometrical accuracy of the model are quite satisfactory, as it is stated by the medical practitioners and confirmed in the literature.
Geometrical models of human bones are of great importance in today’s medicine, as well as in anthropology and other related disciplines. ComputerAssisted Surgery (CAS) is one of the most common applications of computer generated geometrical models, as stated by Adams et al. in [
The preoperative planning of surgical procedures and interventions is an important part of CAS. Preoperative planning most often implies the use of suitable human organ models in specific software which enables a surgeon to plan the course of surgical procedure up to a specific level defined by limitations of the applied software. The application of preoperative planning in the case of mandible reconstruction is presented in [
Geometrical models of human bones created as aforementioned may find their use in the area of virtual anthropology (VA). VA is an area which extends comparative morphology but implies introducing and establishing interconnection among anthropology, mathematics, statistics, engineering, and other areas of science and technology directed to digitalization of observed objects fossil specimens (e.g., bones). Students of anthropology, as well as practitioners, can learn necessary information from precise geometrical models of bones. A detailed description of virtual anthropology, along with the description of methods and techniques applied in this area of research, is provided in [
The basic mandible reconstruction can be performed based on volumetric methods of scanning (Computed Tomography (CT), Magnetic Resonance Imaging (MRI), etc.) as presented in [
Volumetric methods of scanning imply the use of scanner to form volumetric model by the application of different techniques and methods described in detail in [
Direct modeling implies the creation of models by use of technical elements of CAD software packages. This sort of modeling does not use scanned models; modeling is performed based on information in the form of images, instructions, and presentational models (models of bone and joint system). The geometrical and anatomical accuracy of the models created by the application of these methods is less than the accuracy of models created by the reverse engineering methods. Thus created models can be used for training students and medical practitioners, for the creation of presentational models by use of additive technologies, which are described in [
Creation of geometrical models of human bones, mandible included, can be performed based on predictive models. Predictive models (most often parametric models) are models whose geometry and topology can be adjusted to a specific patient, based on specific parameters (most commonly morphometric, but also others, such as height and weight). Morphometric parameters are acquired from 2D images (Xray) or from volumetric models obtained by a volumetric scanning method (CT, MRI) [
It is important to mention that predictive models are created not only for the human bones, but also for the other parts of the human body (or even whole body). In [
In this paper Method of Anatomical Features (MAF), which was introduced in [
For the geometry analysis of the human mandible, ten (10) mandible samples were scanned (input training set). The samples were made by 64slice CT (MSCT) (Aquilion 64, Toshiba, Japan), according to the standard protocol recording: radiation of 120 kVp, current of 150 mA, rotation time of 0.5 s, exposure time of 500 ms, rotation time 0.5 s, thickness of 0.5 mm, image resolution 512 × 512 px, and pixel size approximately 0.36–0.42 mm, 16 bits allocated and stored. The samples came from Serbian adults, intentionally including different gender and age: six male samples aged 25–67 and four women samples aged 22–72, of different height and weight, which have been previously scanned (because of trauma or some disease). It was assumed that this diverse set of samples could present quite a diverse morphology of the very same bone. These samples are used for the creation of the parametric model of the human mandible. The process of creation of parametric model for femur and tibia by using MAF is presented in [
The process of creation of parametric model of the human bone (MAF method) is presented in Figure
Creation of anatomical model, morphologically and anatomically defined descriptive model of human bone. This model defines where some anatomical feature on the physical model of the bone is and its morphometrical and geometrical relations to other anatomical features.
RGE creation. The basic prerequisite for successful reverse modelling of a human bone’s geometry is identification of referential geometrical entities (RGEs). Usually, these RGEs include characteristics, points, directions, planes, and views, as presented in [
Creation of spline curves. Spline curves are created by the use of RGEs and additional geometry. How curves are created depends on the shape of anatomical feature and its relation to other anatomical features.
Creation of anatomical points. Anatomical points can be created on spline curves and/or anatomical landmarks. Anatomical points created on spline curves can be positioned in two distinctive ways. First, they can be distributed evenly on the curve or they can be positioned in correspondence to some anatomical landmark. For example, anatomical point can be placed on gnathion of mandible.
Measurement of anatomical points coordinates values for defined number of specimens. Values of coordinates are measured on each sample of mandible model in 3D. Values of morphometric parameters (defined in the step of anatomical model creation) are measured on the same 3D models.
The measured data which is processed in mathematical software by using multilinear regression as the tool for statistical analysis.
Parametric equations (functions) which define relations between morphometric parameters and coordinate values. The created parametric model which consists of a set of parametric equations is a predictive model. This means that, for every next patient, it is enough to measure the same morphometric parameters on scanned mandible and to calculate coordinates of points. The resulting model is cloud of calculated anatomical points which can be imported in any CAD software (e.g., CATIA).
Scheme of the MAF method applied for the creation of parametric model of the human bone.
The whole process of the creation of parametric model of the HMCP is presented in the next section of the paper.
Anatomical model is morphologically and anatomically defined descriptive model of human bone. This model defines where some anatomical feature on the real bone is and its morphometrical and geometrical relations to other anatomical features [
Lower jaw (mandible) is the biggest and the most massive face bone, which is connected with skull bones through the temporomandibular joint. It represents the biggest odd bone of the face or the viscecranial bone, which participates in construction of the only mobile head joint. It consists of mandible body and two rami as described in [
Anatomy of human mandible.
Mandible body (Latin: corpus mandibulae) is of horseshoe shape and represents its horizontal part. It consists of two sides (external and internal) and two edges, alveolar part of the mandible which corresponds with inferior dental arch (Latin: arcus alveolaris) and a lower edge or mandible basis (Latin: basis mandibulae).
Ramus is approximately of a rectangle shape which is located upward and backward in relation to mandible body with which it forms an angle of 90°–140°, most commonly 120°–130°. It has two sides, external and internal, and four edges, upper, lower, anterior, and posterior. The upper edge has two processes: coronoid process (Latin: processus coronoideus) and condylar process (Latin: processus condylaris).
In reverse modeling of geometry of human mandible, it is crucial to both determine directions and projections of bone parts or the whole bone and establish rules for creation of all directions and views, which should be precisely used. For better orientation, we use several orientational lines and planes in dental medicine.
Medial line, a line which passes vertically between central incisors and which mainly divides the face into two equal parts.
Sagittal (central) plane, which passes through the body and divides it into equal halves right and left.
Frontal plane, which passes through the body in direction leftright (parallel with the forehead) and divides the body into anterior and posterior parts.
Transversal (horizontal) planes, positioned horizontally, which when in basic anatomical position pass through the body parallel with the ground.
The basic prerequisite to successfully perform reverse modeling of human bone geometry is identification of referential geometrical entities (RGEs). RGEs include characteristic points, directions, planes, and views. Other elements of bone curve and surface geometry will be defined with reference to RGEs. To create precise geometry of a human bone, a set of primary RGEs should be minimized. Geometrical limitations and relations should be based on a minimal set of primary RGEs. This is an approach for a successful parameterization of human bone geometry. All mentioned planes and lines are RGEs of the human mandible geometrical model.
The ability to create anatomical landmarks as geometrical elements on 3D human bone models has a significant role and a vast potential for bone reconstruction after innate defects, illnesses, and traumas. Anatomical landmarks are defined on each polygonal model of human mandible of the acquired samples. They can be defined relative to the RGEs, they can be defined as RGEs, or they can be created on the support geometry which is relative to RGEs (e.g., spline curves).
The characteristic anatomical landmarks (points in this case) defined on the mandible are shown in Table
Anatomical landmarks (points) defined on the human mandible as RGEs.
Anatomical landmark  Definition 

Mental foramen (BO)  Is one of two holes (“foramina”) located on the anterior surface of the mandible? 
Gnathion (Gn)  It is the most inferior midline point on the mandible. 
Gonion (Go)  It is a point along the rounded posteroinferior corner of the mandible between the ramus and the body. 
Condylion (Kon)  It is the most prominent point on the mandibular condyle. 
Mandibular cut (MU)  It is the central point on the mandibular notch. 
Anatomical landmarks (points).
There are two groups of morphometric parameters: linear (lines, planes, and points) and angular (defining relative position of mandible parts). The values of mandible morphometric parameters can define sex, some irregularities in the skeletal system, parametric models, and so forth. The morphometric parameters were defined as geometrical elements on the polygonal model of the human mandible. As it is stated in [
Morphometric parameters of the human mandible.
Morphometric parameters  Definition 

Gnathioninterdental distance (GnIdD)  Direct distance from infradentale (idD) to gnathion (Gn). 
Bigonial width (GoGoD)  Direct distance between right and left gonion (Go). 
Bicondylar breadth (KoKoD)  Direct distance between the most lateral points on the two condyles. 
Height of the mandibular body (VTM)  Direct distance from the alveolar process to the inferior border of the mandible perpendicular to the base at the level of the mental foramen. 
Breadth of the mandibular body (STM)  Maximum breadth measured in the region of the mental foramen perpendicular to the long axis of the mandibular body. 
Mandibular length (DTM)  Distance of the anterior margin of the chin from a center point on the protected straight line placed along the posterior border of the two mandibular angles. 
Minimum ramus breadth (MSR)  Least breadth of the mandibular ramus measured perpendicular to the height of the ramus. 
Maximum ramus height (MVR)  Direct distance from the highest point on the mandibular condyle to gonion (Go). 
Height of the condyles (VKo)  Distance between Kon and axis of the lowest point of mandibular cut perpendicular to MVR. 
Gnathioncondylar distance (GnKoD)  Distance between Gn and Kon. 
Morphometric parameters and anatomical points presented on polygonal model of mandible.
The first step in definition of the parametric model is the creation of the model coordinate system which is used for the measurement of the coordinates of the anatomical points. The same coordinate system is created on each polygonal model of individual mandible.
Origin of the coordinate system is defined as the middle of the distance between mental foreman middle points. The constructed planes of the Object Coordinate System (OBC) of mandible are presented in Figure
Coordinate system, spline curves, and anatomical points defined on human mandible polygonal model.
Coordinate system together with spline curves created on the polygonal model of mandible is shown in Figure
The thirtynine points were created in the area of the HMCP. The anatomical points are labeled so they represent some topological (e.g., curvature) and anatomical landmarks (e.g., point distinct from gonion) on the model. Their position was proposed by the orthodontist and anatomist involved in this research. Created anatomical points on the HMCP are presented in Figure
On each individual model of mandible the values of coordinates of these points were measured (distance from origin of coordinate system in all three directions
The example of matrix equation for the coordinate of one point defined in Matlab is presented in
The calculation was performed for all thirtynine points on the Human Mandible Coronoid Process. In Table
Coefficients of the multiple linear regression functions for four anatomical points.
Point 













P1 

0.068  1.167  −0.001  −0.685  2.443  −0.648  −0.096  0.362  −0.102  0.317  −0.549 

24.840  0.738  −0.002  −0.650  1.732  −0.104  0.235  0.784  0.952  −0.471  −0.402  

18.606  0.161  0.000  −0.325  1.247  −0.113  0.080  −0.330  0.753  −0.582  −0.140  


P2 

12.708  1.115  −0.001  −0.790  3.060  −0.891  −0.012  0.181  −0.162  0.192  −0.621 

13.340  1.226  −0.002  −0.773  1.644  0.044  0.211  0.969  1.170  −0.565  −0.449  

18.989  0.095  0.000  −0.318  1.393  −0.221  0.004  −0.140  0.819  −0.387  −0.173  

















P38 

−6.762  1.098  −0.001  −0.519  1.454  −0.591  −0.044  0.335  −0.107  0.087  −0.378 

18.717  1.684  −0.002  −1.030  2.287  0.181  0.386  0.764  1.158  −0.870  −0.576  

−0.679  0.276  0.000  −0.105  0.121  0.008  0.021  −0.405  0.657  −0.554  0.046  


P39 

−3.095  1.215  −0.001  −0.605  1.388  −0.470  0.077  0.536  −0.062  −0.107  −0.409 

17.478  1.715  −0.002  −1.018  2.109  0.210  0.413  0.773  1.160  −0.943  −0.551  

−0.669  0.276  0.000  −0.105  0.122  0.008  0.021  −0.405  0.657  −0.553  0.046 
For example, statistical function for
By using these functions coordinates values were calculated and compared with measured values for all thirtynine points. The maximal error for all coordinates for each individual patient is presented in Table
The maximal error for all coordinates of anatomical points for each individual patient.
Coord. [mm]  Pat. 1  Pat. 2  Pat. 3  Pat. 4  Pat. 5  Pat. 6  Pat. 7  Pat. 8  Pat. 9  Pat. 10 


0.980  0.732  0.448  0.227  0.227  0.775  0.995  0.92  0.216  0.897 


0.668  0.597  0.453  0.907  0.469  0.952  0.765  0.301  0.842 

0.554  0.256  0.567  0.991  0.178  0.928  0.370  0.721  0.650  0.461 
Maximum surface deviations of the surface model of HMCP created by the use of parametric functions (calculated model) from the input surface models of original mandible specimens are presented in Table
Maximum deviations of the calculated surface model of the HMCP from the input HMCP models.
Model  1  2  3  4  5  6  7  8  9  10 

Max. deviation [mm] 

0.526  0.661  0.654  0.97  0.912  1.3  0.97  0.34  0.87 
The preliminary claim about parametric model geometrical accuracy and anatomical correctness can be stated as quite satisfactory for the prototype model. It is important to mention that designers can choose more points in the area of maximal deviation(s) or choose different points, which will enable better geometrical definition of the domain included and thus improve the accuracy.
In order to obtain reliable response of the parametric model, more detailed analysis must be performed. The number of samples should be increased, parameters influence on the individual points should be examined, and the parametric model for the whole human mandible must be created. These tasks will be conducted in the future research.
The presented Method of Anatomical Features (MAF) enables creation of the geometrically accurate and anatomically correct parametric model of the Human Mandible Coronoid Process (HMCP). The presented parametric model of the HMCP can be considered as a prototype (test) model for the parameterization of the whole human mandible.
It should be emphasized that parametric model enables creation of the adequate geometric model of the HMCP customized to the specific patient. The customization is performed by the application of the values of the parameters in the parametric functions. Values of the parameters can be acquired from medical images (CT, MRI, or Xray). The resulting model(s) can be applied in training of the medical staff, implant and fixator manufacturing, CAD/CAM application, FEA (Finite Element Analysis), and so forth.
The current research results are based on a relatively small number of human mandible samples. It is crucial to increase that number as much as possible. Besides the number of samples, the influence of the involved parameters on the position of the individual points must be investigated. All of these tasks are the part of future research and they will be performed in order to improve the geometric precision and anatomical correctness of the presented parametric model of the HMCP and future parametric model of the whole human mandible. One possible application of the parametric model of the whole human mandible is for the prediction of the dental implants position and orientation. For example, it can be used for the proper implantation of the osseointegrated dental implants which are presented in [
The authors declare that there is no conflict of interests regarding the publication of this paper.
The presented research was done for the project Virtual Human Osteoarticular System and Its Application in Preclinical and Clinical Practice which is sponsored by the Ministry of Science and Technology of the Republic of Serbia, Project ID III 41017 for the period of 2011–2015.