Research ArticleThe Parametric Model of the Human Mandible CoronoidProcess Created by Method of Anatomical Features

Nikola VitkoviT1 Jelena MitiT1 Miodrag ManiT1 Miroslav TrajanoviT1 Karim Husain2

SlaZana PetroviT3 and Stojanka ArsiT3

1Faculty ]f Mechanical Engineering University of Nis Aleksandra Medvedeva 14 18000 Nis Serbia2University of Qadisiya Diwaniya Iraq3Faculty of Medicine University of Nis Dr Zorana ETHinđica 81 Boulevard 18000 Nis Serbia

Correspondence should be addressed to Nikola Vitkovic vitkomasfakniacrs

Received 11 January 2015 Accepted 15 April 2015

Academic Editor Ezequiel Lopez-Rubio

Copyright copy 2015 Nikola Vitkovic et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Geometrically accurate and anatomically correct 3D models of the human bones are of great importance for medical research andpractice in orthopedics and surgeryThese geometrical models can be created by the use of techniques which can be based on inputgeometrical data acquired from volumetric methods of scanning (eg Computed Tomography (CT)) or on the 2D images (egX-ray) Geometrical models of human bones created in suchway can be applied for education ofmedical practitioners preoperativeplanning etc In cases when geometrical data about the human bone is incomplete (eg fractures) it may be necessary to createits complete geometrical model The possible solution for this problem is the application of parametric models The geometry ofthese models can be changed and adapted to the specific patient based on the values of parameters acquired from medical images(eg X-ray) In this paperMethod of Anatomical Features (MAF) which enables creation of geometrically precise and anatomicallyaccurate geometrical models of the human bones is implemented for the creation of the parametric model of the HumanMandibleCoronoid Process (HMCP) The obtained results about geometrical accuracy of the model are quite satisfactory as it is stated bythe medical practitioners and confirmed in the literature

1 Introduction

Geometrical models of human bones are of great importancein todayrsquos medicine as well as in anthropology and otherrelated disciplines Computer-Assisted Surgery (CAS) is oneof the most common applications of computer generatedgeometrical models as stated by Adams et al in [1] Theapplication of geometrically precise models enables surgeonsto properly prepare and perform interventions with use ofsuitable computer software tools andor other techniquesand it lessens the possibility of error occurrence The com-parison of conventional methods and CAS is presented in [2]by Bathis et al where the Total Knee Arthroplasty (TKA)process is shown Based on the facts stated in [2] we canconclude that the new technique of performing surgicalprocedures that is surgical interventions may significantlyimprove both the quality of the procedure itself and thepatientsrsquo convalescence

The preoperative planning of surgical procedures andinterventions is an important part of CAS Preoperativeplanning most often implies the use of suitable human organmodels in specific software which enables a surgeon to planthe course of surgical procedure up to a specific level definedby limitations of the applied software The application ofpreoperative planning in the case of mandible reconstructionis presented in [3] by Essig et al and in [4] by Chapius et al

Geometrical models of human bones created as afore-mentionedmay find their use in the area of virtual anthropol-ogy (VA) VA is an area which extends comparativemorphol-ogy but implies introducing and establishing interconnectionamong anthropology mathematics statistics engineeringand other areas of science and technology directed to digi-talization of observed objects fossil specimens (eg bones)Students of anthropology as well as practitioners can learnnecessary information from precise geometrical models ofbones A detailed description of virtual anthropology along

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2015 Article ID 574132 10 pageshttpdxdoiorg1011552015574132

2 Computational and Mathematical Methods in Medicine

with the description of methods and techniques applied inthis area of research is provided in [5] by Weber and in [6]by Benazzi et al in the case of mandible reconstruction

The basic mandible reconstruction can be performedbased on volumetric methods of scanning (ComputedTomography (CT)Magnetic Resonance Imaging (MRI) etc)as presented in [5 6] aswell as by directmethods ofComputerAided Design (CAD) [5]

Volumetricmethods of scanning imply the use of scannerto form volumetric model by the application of differenttechniques and methods described in detail in [6] Basi-cally this is the reverse engineering procedure and containsmultiple actions The first step is to form 2D image (slice)of human body on volumetric scanner By superposition ofprovided slices a volumetric image of scanned object (patient)is formed comprised of volumetric elements (voxels) Byfurther process of segmentation a detailed bonding ofanatomical entities along the whole volume of the scannedmodel is performed as shown in [7] by Archipa et al Seg-mentation can be a very complex process and lots of studieshave been done to solve problems with feature extractionlike it is described in [8] by Huang et al for the automaticextraction of the vertebral column from the SPECT (Single-Photon Emission Computed Tomography) scan of the wholebody Through an adequate process of volumetric renderinga reconstructed 3D model of the scanned object is acquired[9] Volumetric rendering implies shading of projected 3Dscalar field (cloud of points) onto 2D that is the computerscreen and it is applied in various areas of computer graphicsas described in [10] by Li et al Based on created 3D scalar field(initially segmented volumetric model comprised of voxels)by application of an adequate algorithm such as marchingcubes algorithm which is described in [11] by Lorensen andCline a polygonal model (mesh) of scanned object can beconstructed Polygonal model can be further used in CADsoftware packages for creation of surface and volumemodelsas presented in [12] by Tufegdzic et al Such models areconstructed based on geometry of a specific patient andthus they can be used to create implants and fixators adjustedto the patient in preoperative planning intraoperationalnavigation and so forth

Direct modeling implies the creation of models by useof technical elements of CAD software packages This sortof modeling does not use scanned models modeling isperformed based on information in the form of imagesinstructions and presentational models (models of bone andjoint system) The geometrical and anatomical accuracy ofthe models created by the application of these methods isless than the accuracy of models created by the reverseengineering methods Thus created models can be used fortraining students and medical practitioners for the creationof presentational models by use of additive technologieswhich are described in [13] by Salmoria et al and in all otherapplications where there is no need for geometrical models ofgreat precision

Creation of geometrical models of human bonesmandible included can be performed based on predictivemodels Predictive models (most often parametric models)are models whose geometry and topology can be adjusted

to a specific patient based on specific parameters (mostcommonly morphometric but also others such as heightand weight) Morphometric parameters are acquired from2D images (X-ray) or from volumetric models obtained by avolumetric scanning method (CT MRI) [5 6] Such modelscan be very precise if a number of parameters are adequateand the model structure itself is well chosen (eg containingparametric surfaces like NURBS surfaces) These models canbe used for many purposes creation of implants and fixatorspreoperative planning creating geometrical models of themissing parts of bones and so forth

It is important to mention that predictive models arecreated not only for the human bones but also for the otherparts of the human body (or even whole body) In [14] by Liet al prediction of the deformations and movements of bodyorganstissues and skeletal structures using patient-specificnonlinear biomechanical modeling from whole body CTimage registration is presented Besides volumetric internalscanning methods (CT MRI etc) there is a possibility ofcreating predictive human body or parts of the body modelsbased on the various types of the 3D measurements likeit is shown in [15] by Wuhrer and Shu and also by Leonget al in [16] These research studies enable feature extractionand prediction of the shape of the human bodyrsquos anatomicalsection as demonstrated in the example of reconstructionof the human torso in [16] Deformable statistical wholebody model which can be adapted to the single 2D image ispresented in [17] by Chen et al Model presented in [17] canbe applied for the creation of whole body meshes or clothed3D meshes for different people neither of which appears inthe training dataset

In this paper Method of Anatomical Features (MAF)which was introduced in [18] by Vitkovic et al and in [19]by Majstorovic et al is implemented for the creation ofthe parametric (predictive) model of the Human MandibleCoronoid Process (HMCP) The MAF was originally appliedfor the development of the parametric and surface modelsof the human femur and tibia and the results are quitesatisfactory as presented in [18 19] The main objective ofthis research is to show that MAF can be applied for othertypes of human bones not just for the long bones TheHMCP was chosen because of its complex geometric andtopological properties and it is adequate anatomical sectionfor the creation of prototype (test) parametric model MAFwas tested on prototype model and the results are more thanpromising The research will be continued for the creationof the parametric model of the whole human mandible sothat the geometrical and anatomical correctness of the wholemodel can be confirmed

2 Material and Methods

For the geometry analysis of the human mandible ten (10)mandible samples were scanned (input training set) Thesamples were made by 64-slice CT (MSCT) (Aquilion 64Toshiba Japan) according to the standard protocol record-ing radiation of 120 kVp current of 150mA rotation time of05 s exposure time of 500ms rotation time 05 s thicknessof 05mm image resolution 512 times 512 px and pixel size

Computational and Mathematical Methods in Medicine 3

Anatomical modelcreation RGE creation

Creation of spline curves

Creation of anatomical points

Digitized cloud of points

Anatomically defined cloud of points

Measurements of points coor values and

parameters

Creation of parametric equations

Parametric equations(parametric model)

Figure 1 Scheme of the MAF method applied for the creation of parametric model of the human bone

approximately 036ndash042mm 16 bits allocated and storedThe samples came from Serbian adults intentionally includ-ing different gender and age six male samples aged 25ndash67and four women samples aged 22ndash72 of different heightand weight which have been previously scanned (because oftrauma or some disease) It was assumed that this diverse setof samples could present quite a diverse morphology of thevery same bone These samples are used for the creation ofthe parametric model of the human mandible The processof creation of parametric model for femur and tibia by usingMAF is presented in [18 19] in detail but here the shortintroduction of the method is shown

Theprocess of creation of parametricmodel of the humanbone (MAF method) is presented in Figure 1 and it containsseveral steps

(i) Creation of anatomical model morphologically andanatomically defined descriptive model of humanbone This model defines where some anatomicalfeature on the physical model of the bone is and itsmorphometrical and geometrical relations to otheranatomical features

(ii) RGE creation The basic prerequisite for success-ful reverse modelling of a human bonersquos geometryis identification of referential geometrical entities(RGEs) Usually these RGEs include characteristicspoints directions planes and views as presented in[18 19]

(iii) Creation of spline curves Spline curves are createdby the use of RGEs and additional geometry Howcurves are created depends on the shape of anatomicalfeature and its relation to other anatomical features

(iv) Creation of anatomical points Anatomical points canbe created on spline curves andor anatomical land-marks Anatomical points created on spline curvescan be positioned in two distinctive ways First theycan be distributed evenly on the curve or they canbe positioned in correspondence to some anatomical

landmark For example anatomical point can beplaced on gnathion of mandible

(v) Measurement of anatomical points coordinates valuesfor defined number of specimens Values of coor-dinates are measured on each sample of mandiblemodel in 3D Values of morphometric parameters(defined in the step of anatomical model creation) aremeasured on the same 3D models

(vi) The measured data which is processed in mathemati-cal software by usingmultilinear regression as the toolfor statistical analysis

(vii) Parametric equations (functions) which define rela-tions between morphometric parameters and coor-dinate values The created parametric model whichconsists of a set of parametric equations is a predictivemodel This means that for every next patient it isenough to measure the same morphometric parame-ters on scannedmandible and to calculate coordinatesof points The resulting model is cloud of calculatedanatomical points which can be imported in anyCADsoftware (eg CATIA)

The whole process of the creation of parametric model ofthe HMCP is presented in the next section of the paper

21 Anatomical Model of Human Mandible Anatomicalmodel is morphologically and anatomically defined descrip-tive model of human bone This model defines where someanatomical feature on the real bone is and itsmorphometricaland geometrical relations to other anatomical features [18]

Lower jaw (mandible) is the biggest and themost massiveface bone which is connected with skull bones through thetemporomandibular joint It represents the biggest odd boneof the face or the viscecranial bone which participates inconstruction of the only mobile head joint It consists ofmandible body and two rami as described in [20 21] byJuodzbalys et al and presented in Figure 2

4 Computational and Mathematical Methods in Medicine

Condylar process

Coronoid process

Submandibular fossa

Sublingual fossa

Alveolar process

Mental foramen

Mentalprotuberance

Body

Ramus

Mandibular foramen

Figure 2 Anatomy of human mandible

Mandible body (Latin corpus mandibulae) is of horse-shoe shape and represents its horizontal part It consists oftwo sides (external and internal) and two edges alveolar partof 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 islocated upward and backward in relation to mandible bodywith which it forms an angle of 90∘ndash140∘ most commonly120∘ndash130∘ It has two sides external and internal and fouredges upper lower anterior and posterior The upper edgehas two processes coronoid process (Latin processus coro-noideus) and condylar process (Latin processus condylaris)

22 Referential Geometrical Entities (RGEs) In reverse mod-eling of geometry of human mandible it is crucial to bothdetermine directions and projections of bone parts or thewhole bone and establish rules for creation of all directionsand views which should be precisely used For better orien-tation we use several orientational lines and planes in dentalmedicine

(i) Medial line a line which passes vertically betweencentral incisors andwhichmainly divides the face intotwo equal parts

(ii) Sagittal (central) plane which passes through thebody and divides it into equal halves right and left

(iii) Frontal plane which passes through the body indirection left-right (parallel with the forehead) anddivides the body into anterior and posterior parts

(iv) Transversal (horizontal) planes positioned horizon-tally which when in basic anatomical position passthrough the body parallel with the ground

The basic prerequisite to successfully perform reversemodeling of human bone geometry is identification ofreferential geometrical entities (RGEs) RGEs include char-acteristic points directions planes and views Other ele-ments of bone curve and surface geometry will be definedwith reference to RGEs To create precise geometry of ahuman bone a set of primary RGEs should be minimizedGeometrical limitations and relations should be based ona minimal set of primary RGEs This is an approach for asuccessful parameterization of human bone geometry Allmentioned planes and lines are RGEs of the humanmandiblegeometrical model

The ability to create anatomical landmarks as geometricalelements on 3Dhuman bonemodels has a significant role anda vast potential for bone reconstruction after innate defectsillnesses and traumas Anatomical landmarks are defined oneach polygonal model of human mandible of the acquiredsamples They can be defined relative to the RGEs they canbe defined as RGEs or they can be created on the supportgeometry which is relative to RGEs (eg spline curves)

The characteristic anatomical landmarks (points in thiscase) defined on the mandible are shown in Table 1 anddescribed in [22] byArsic et alThemandibular cut (MU)wasadded by the authors of this research because it was necessaryto add this point as additional support point for the properdefinition of coronoid process geometryThe points shown in

Computational and Mathematical Methods in Medicine 5

Table 1 Anatomical landmarks (points) defined on the human mandible as RGEs

Anatomical landmark DefinitionMental foramen (BO) Is one of two holes (ldquoforaminardquo) located on the anterior surface of the mandibleGnathion (Gn) It is the most inferior midline point on the mandibleGonion (Go) It is a point along the rounded posteroinferior corner of the mandible between the ramus and the bodyCondylion (Kon) It is the most prominent point on the mandibular condyleMandibular cut (MU) It is the central point on the mandibular notch

Kon

MU

Go

Gn

BO

Figure 3 Anatomical landmarks (points)

Figure 3 and described in Table 1 are defined as RGEs used forthe creation of the parametric model of the HumanMandibleCoronoid Process

23Morphometric Parameters of theHumanMandible Thereare two groups of morphometric parameters linear (linesplanes and points) and angular (defining relative positionof mandible parts) The values of mandible morphometricparameters can define sex some irregularities in the skeletalsystem parametric models and so forth The morphometricparameters were defined as geometrical elements on thepolygonal model of the human mandible As it is statedin [22] ten morphometric parameters are enough for thecomplete definition of mandible geometry and they aredefined in Table 2 and presented in Figure 4

24 The Parametric Model of Human Mandible CoronoidProcess Thefirst step in definition of the parametricmodel isthe creation of themodel coordinate systemwhich is used forthemeasurement of the coordinates of the anatomical pointsThe same coordinate system is created on each polygonalmodel of individual mandible

Origin of the coordinate system is defined as the middleof the distance between mental foreman middle pointsThe constructed planes of the Object Coordinate System(OBC) of mandible are presented in Figure 5 and they areMediosagittal mandibular and coronal The Mediosagittal

(MS) plane was constructed as the plane which contains theorigin of the OBC and it is normal to the line which connectsmental foreman middle points MS is a plane which divideshuman mandible into two halves left and right Horizontalor mandibular plane is the lowest plane normal to the MSplane and it contains the gonion anatomical point (the mostinferior point of the symphysis of the mandible as seen inthe lateral jaw projection) To be used as a plane of OBCthis plane was translated to the origin of OBC Coronal oranterior posterior (AP) plane is a planewhich is normal to themandibular plane and divides mandible into two anatomicalsections anterior (front) and posterior (back) It is placed atthe origin of OBC 119909-axis of the OBC is defined as normal toMS plane 119910-axis is defined as normal to AP plane 119911-axis isnormal to mandibular plane Axes of the OBC are presentedin Figure 5

Coordinate system together with spline curves createdon the polygonal model of mandible is shown in Figure 5Spline curves are created by the intersection of defined planes(mandibular sagittal and coronal) and polygonal model ofmandible These curves are used as the basis for the creationof anatomical points which were created on them Someanatomical points were not created on spline curves yet theywere created directly on the polygonal model of the humanmandible as additional support points

The thirty-nine points were created in the area of theHMCP The anatomical points are labeled so they represent

6 Computational and Mathematical Methods in Medicine

Table 2 Morphometric parameters of the human mandible

Morphometric parameters DefinitionGnathion-interdental distance (Gn-IdD) Direct distance from infradentale (idD) to gnathion (Gn)Bigonial width (Go-GoD) Direct distance between right and left gonion (Go)Bicondylar breadth (Ko-KoD) 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 mandibleperpendicular 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 tothe long axis of the mandibular body

Mandibular length (DTM) Distance of the anterior margin of the chin from a center point on the protectedstraight 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 theramus

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 perpendicularto MVR

Gnathion-condylar distance (Gn-KoD) Distance between Gn and Kon

Ko-KoD KoKo

STM

BO

Go-GoD GoGo

(a)

Kon

MU

Vko

MVRMSR

Gn-KoD

Go DTM

VTM Gn-IdD

Gn

BO

(b)

Figure 4 Morphometric parameters and anatomical points presented on polygonal model of mandible

Computational and Mathematical Methods in Medicine 7

some topological (eg curvature) and anatomical landmarks(eg point distinct fromgonion) on themodelTheir positionwas proposed by the orthodontist and anatomist involved inthis research Created anatomical points on the HMCP arepresented in Figure 5

On each individual model of mandible the values ofcoordinates of these points were measured (distance fromorigin of coordinate system in all three directions 119883 119884and 119885) These values were used as the input vectors forthe multiple linear regression analysis which is well knownand documented statistical function The morphometricparameters were alsomeasured on eachmandible sample andincorporated into regression functions The multiple linearregression algorithm which is applied in this research is

described in detail in [23] by Brown The basic idea is to pre-dict dependent variable 119884which is based on the independentvariable 119883 Using the least squares method the best fittingline can be found by minimizing the sum of the squares ofthe distance from each data point on the line [23] The basicmodel is presented in (1) and defined in [23]

119884 = 119883119861 + 119864 (1)

where 119884 is dependent variable 119883 is independent variable 119861is coefficients and 119864 is error variable

The example of matrix equation for the coordinate of onepoint defined in Matlab is presented in

119883coord = [minus51874 minus47712 minus46269 minus47164 minus5078 minus49684 minus43613]1015840

1198891 = [2855 31902 29342 29225 32697 31727 30733]1015840

1198899 = [175 22375 24286 21533 22684 19454 16193]1015840

11988910 = [122384 136216 131729 124194 130113 116171 115732]1015840

119883 = [ones (size (1198891)) 1198891 1198892 1198893 1198894 1198895 1198896 1198897 1198898 1198899 11988910]

119860 = 1198831015840

lowast 119883

119870 = inv (119860)

119861 = 119860 1198831015840

lowast 119883coord

119872 = 119883 lowast 119861

119864 = 119883coord minus119872

(2)

where 119883 coord is values of 119883 coordinates for definedpoint 1198891 11988910 are measured values of defined parameters(arranged according to the order in Table 2) 119861 is vector ofthe coefficients119872 is values of the calculated coordinate119883 119864is the error vector (measured valuesminus calculated values)and 119860 119870 are helper matrixes

3 Results and Discussion

The calculation was performed for all thirty-nine pointson the Human Mandible Coronoid Process In Table 3coefficients of the multiple linear regression functions arepresented for four chosen anatomical points

For example statistical function for119883 coordinate of Point1 is presented in

119883 = 1198870 + 1198871 lowast 1198891 + 1198872 lowast 1198892 + sdot sdot sdot + 11988710 lowast 11988910 =gt 119883

= 0068 + 1167 lowast 1198891 minus 0001 lowast 1198892 + sdot sdot sdot minus 0549

lowast 11988910

(3)

By using these functions coordinates values were cal-culated and compared with measured values for all thirty-nine points The maximal error for all coordinates for eachindividual patient is presented in Table 4 The maximumdeviation is for119884 coordinate of Point 8 shown in Figure 5Theorthodontists and anatomist suggested that maximal error inthis area should not exceed two (2)mm in defined 119883 119884 and119885 directions Taking into account the orthodontistsrsquo recom-mendations it can be concluded that maximum deviations ofthe values of the coordinates are quite satisfactory and belowrecommended limit

Maximum surface deviations of the surface model ofHMCP created by the use of parametric functions (calculatedmodel) from the input surface models of original mandiblespecimens are presented in Table 5 The surface modelswere created by the use of spline curves through the inputand calculated points and with the application of technicalfeatures in CATIA software package (multisection surfacefill etc) It can be noticed that these values are also belowthe recommended limit Maximal deviation is 136mm

8 Computational and Mathematical Methods in Medicine

1

2

3 4

5

6

7

8

910

11

12

13

14

15

16

17

18

19 2120 22

23

24

25

26

27

28

29

30

3132

3334

3536

37

38

39

Anatomical points

Anterior posterior plane

Sagittal planeMandibular plane

Figure 5 Coordinate system spline curves and anatomical points defined on human mandible polygonal model

Table 3 Coefficients of the multiple linear regression functions for four anatomical points

Point 1198870 1198871 1198872 1198873 1198874 1198875 1198876 1198877 1198878 1198879 11988710

P1119909 0068 1167 minus0001 minus0685 2443 minus0648 minus0096 0362 minus0102 0317 minus0549119910 24840 0738 minus0002 minus0650 1732 minus0104 0235 0784 0952 minus0471 minus0402119911 18606 0161 0000 minus0325 1247 minus0113 0080 minus0330 0753 minus0582 minus0140

P2119909 12708 1115 minus0001 minus0790 3060 minus0891 minus0012 0181 minus0162 0192 minus0621119910 13340 1226 minus0002 minus0773 1644 0044 0211 0969 1170 minus0565 minus0449119911 18989 0095 0000 minus0318 1393 minus0221 0004 minus0140 0819 minus0387 minus0173

P38119909 minus6762 1098 minus0001 minus0519 1454 minus0591 minus0044 0335 minus0107 0087 minus0378119910 18717 1684 minus0002 minus1030 2287 0181 0386 0764 1158 minus0870 minus0576119911 minus0679 0276 0000 minus0105 0121 0008 0021 minus0405 0657 minus0554 0046

P39119909 minus3095 1215 minus0001 minus0605 1388 minus0470 0077 0536 minus0062 minus0107 minus0409119910 17478 1715 minus0002 minus1018 2109 0210 0413 0773 1160 minus0943 minus0551119911 minus0669 0276 0000 minus0105 0122 0008 0021 minus0405 0657 minus0553 0046

Table 4 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 10119883 0980 0732 0448 0227 0227 0775 0995 092 0216 0897119884 1720 0668 0597 0453 0907 0469 0952 0765 0301 0842119885 0554 0256 0567 0991 0178 0928 0370 0721 0650 0461

Table 5 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 10Max deviation [mm] 136 0526 0661 0654 097 0912 13 097 034 087

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

Computational and Mathematical Methods in Medicine 3

Anatomical modelcreation RGE creation

Creation of spline curves

Creation of anatomical points

Digitized cloud of points

Anatomically defined cloud of points

Measurements of points coor values and

parameters

Creation of parametric equations

Parametric equations(parametric model)

Figure 1 Scheme of the MAF method applied for the creation of parametric model of the human bone

approximately 036ndash042mm 16 bits allocated and storedThe samples came from Serbian adults intentionally includ-ing different gender and age six male samples aged 25ndash67and four women samples aged 22ndash72 of different heightand weight which have been previously scanned (because oftrauma or some disease) It was assumed that this diverse setof samples could present quite a diverse morphology of thevery same bone These samples are used for the creation ofthe parametric model of the human mandible The processof creation of parametric model for femur and tibia by usingMAF is presented in [18 19] in detail but here the shortintroduction of the method is shown

Theprocess of creation of parametricmodel of the humanbone (MAF method) is presented in Figure 1 and it containsseveral steps

(i) Creation of anatomical model morphologically andanatomically defined descriptive model of humanbone This model defines where some anatomicalfeature on the physical model of the bone is and itsmorphometrical and geometrical relations to otheranatomical features

(ii) RGE creation The basic prerequisite for success-ful reverse modelling of a human bonersquos geometryis identification of referential geometrical entities(RGEs) Usually these RGEs include characteristicspoints directions planes and views as presented in[18 19]

(iii) Creation of spline curves Spline curves are createdby the use of RGEs and additional geometry Howcurves are created depends on the shape of anatomicalfeature and its relation to other anatomical features

(iv) Creation of anatomical points Anatomical points canbe created on spline curves andor anatomical land-marks Anatomical points created on spline curvescan be positioned in two distinctive ways First theycan be distributed evenly on the curve or they canbe positioned in correspondence to some anatomical

landmark For example anatomical point can beplaced on gnathion of mandible

(v) Measurement of anatomical points coordinates valuesfor defined number of specimens Values of coor-dinates are measured on each sample of mandiblemodel in 3D Values of morphometric parameters(defined in the step of anatomical model creation) aremeasured on the same 3D models

(vi) The measured data which is processed in mathemati-cal software by usingmultilinear regression as the toolfor statistical analysis

(vii) Parametric equations (functions) which define rela-tions between morphometric parameters and coor-dinate values The created parametric model whichconsists of a set of parametric equations is a predictivemodel This means that for every next patient it isenough to measure the same morphometric parame-ters on scannedmandible and to calculate coordinatesof points The resulting model is cloud of calculatedanatomical points which can be imported in anyCADsoftware (eg CATIA)

The whole process of the creation of parametric model ofthe HMCP is presented in the next section of the paper

21 Anatomical Model of Human Mandible Anatomicalmodel is morphologically and anatomically defined descrip-tive model of human bone This model defines where someanatomical feature on the real bone is and itsmorphometricaland geometrical relations to other anatomical features [18]

Lower jaw (mandible) is the biggest and themost massiveface bone which is connected with skull bones through thetemporomandibular joint It represents the biggest odd boneof the face or the viscecranial bone which participates inconstruction of the only mobile head joint It consists ofmandible body and two rami as described in [20 21] byJuodzbalys et al and presented in Figure 2

4 Computational and Mathematical Methods in Medicine

Condylar process

Coronoid process

Submandibular fossa

Sublingual fossa

Alveolar process

Mental foramen

Mentalprotuberance

Body

Ramus

Mandibular foramen

Figure 2 Anatomy of human mandible

Mandible body (Latin corpus mandibulae) is of horse-shoe shape and represents its horizontal part It consists oftwo sides (external and internal) and two edges alveolar partof 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 islocated upward and backward in relation to mandible bodywith which it forms an angle of 90∘ndash140∘ most commonly120∘ndash130∘ It has two sides external and internal and fouredges upper lower anterior and posterior The upper edgehas two processes coronoid process (Latin processus coro-noideus) and condylar process (Latin processus condylaris)

22 Referential Geometrical Entities (RGEs) In reverse mod-eling of geometry of human mandible it is crucial to bothdetermine directions and projections of bone parts or thewhole bone and establish rules for creation of all directionsand views which should be precisely used For better orien-tation we use several orientational lines and planes in dentalmedicine

(i) Medial line a line which passes vertically betweencentral incisors andwhichmainly divides the face intotwo equal parts

(ii) Sagittal (central) plane which passes through thebody and divides it into equal halves right and left

(iii) Frontal plane which passes through the body indirection left-right (parallel with the forehead) anddivides the body into anterior and posterior parts

(iv) Transversal (horizontal) planes positioned horizon-tally which when in basic anatomical position passthrough the body parallel with the ground

The basic prerequisite to successfully perform reversemodeling of human bone geometry is identification ofreferential geometrical entities (RGEs) RGEs include char-acteristic points directions planes and views Other ele-ments of bone curve and surface geometry will be definedwith reference to RGEs To create precise geometry of ahuman bone a set of primary RGEs should be minimizedGeometrical limitations and relations should be based ona minimal set of primary RGEs This is an approach for asuccessful parameterization of human bone geometry Allmentioned planes and lines are RGEs of the humanmandiblegeometrical model

The ability to create anatomical landmarks as geometricalelements on 3Dhuman bonemodels has a significant role anda vast potential for bone reconstruction after innate defectsillnesses and traumas Anatomical landmarks are defined oneach polygonal model of human mandible of the acquiredsamples They can be defined relative to the RGEs they canbe defined as RGEs or they can be created on the supportgeometry which is relative to RGEs (eg spline curves)

The characteristic anatomical landmarks (points in thiscase) defined on the mandible are shown in Table 1 anddescribed in [22] byArsic et alThemandibular cut (MU)wasadded by the authors of this research because it was necessaryto add this point as additional support point for the properdefinition of coronoid process geometryThe points shown in

Computational and Mathematical Methods in Medicine 5

Table 1 Anatomical landmarks (points) defined on the human mandible as RGEs

Anatomical landmark DefinitionMental foramen (BO) Is one of two holes (ldquoforaminardquo) located on the anterior surface of the mandibleGnathion (Gn) It is the most inferior midline point on the mandibleGonion (Go) It is a point along the rounded posteroinferior corner of the mandible between the ramus and the bodyCondylion (Kon) It is the most prominent point on the mandibular condyleMandibular cut (MU) It is the central point on the mandibular notch

Kon

MU

Go

Gn

BO

Figure 3 Anatomical landmarks (points)

Figure 3 and described in Table 1 are defined as RGEs used forthe creation of the parametric model of the HumanMandibleCoronoid Process

23Morphometric Parameters of theHumanMandible Thereare two groups of morphometric parameters linear (linesplanes and points) and angular (defining relative positionof mandible parts) The values of mandible morphometricparameters can define sex some irregularities in the skeletalsystem parametric models and so forth The morphometricparameters were defined as geometrical elements on thepolygonal model of the human mandible As it is statedin [22] ten morphometric parameters are enough for thecomplete definition of mandible geometry and they aredefined in Table 2 and presented in Figure 4

24 The Parametric Model of Human Mandible CoronoidProcess Thefirst step in definition of the parametricmodel isthe creation of themodel coordinate systemwhich is used forthemeasurement of the coordinates of the anatomical pointsThe same coordinate system is created on each polygonalmodel of individual mandible

Origin of the coordinate system is defined as the middleof the distance between mental foreman middle pointsThe constructed planes of the Object Coordinate System(OBC) of mandible are presented in Figure 5 and they areMediosagittal mandibular and coronal The Mediosagittal

(MS) plane was constructed as the plane which contains theorigin of the OBC and it is normal to the line which connectsmental foreman middle points MS is a plane which divideshuman mandible into two halves left and right Horizontalor mandibular plane is the lowest plane normal to the MSplane and it contains the gonion anatomical point (the mostinferior point of the symphysis of the mandible as seen inthe lateral jaw projection) To be used as a plane of OBCthis plane was translated to the origin of OBC Coronal oranterior posterior (AP) plane is a planewhich is normal to themandibular plane and divides mandible into two anatomicalsections anterior (front) and posterior (back) It is placed atthe origin of OBC 119909-axis of the OBC is defined as normal toMS plane 119910-axis is defined as normal to AP plane 119911-axis isnormal to mandibular plane Axes of the OBC are presentedin Figure 5

Coordinate system together with spline curves createdon the polygonal model of mandible is shown in Figure 5Spline curves are created by the intersection of defined planes(mandibular sagittal and coronal) and polygonal model ofmandible These curves are used as the basis for the creationof anatomical points which were created on them Someanatomical points were not created on spline curves yet theywere created directly on the polygonal model of the humanmandible as additional support points

The thirty-nine points were created in the area of theHMCP The anatomical points are labeled so they represent

6 Computational and Mathematical Methods in Medicine

Table 2 Morphometric parameters of the human mandible

Morphometric parameters DefinitionGnathion-interdental distance (Gn-IdD) Direct distance from infradentale (idD) to gnathion (Gn)Bigonial width (Go-GoD) Direct distance between right and left gonion (Go)Bicondylar breadth (Ko-KoD) 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 mandibleperpendicular 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 tothe long axis of the mandibular body

Mandibular length (DTM) Distance of the anterior margin of the chin from a center point on the protectedstraight 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 theramus

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 perpendicularto MVR

Gnathion-condylar distance (Gn-KoD) Distance between Gn and Kon

Ko-KoD KoKo

STM

BO

Go-GoD GoGo

(a)

Kon

MU

Vko

MVRMSR

Gn-KoD

Go DTM

VTM Gn-IdD

Gn

BO

(b)

Figure 4 Morphometric parameters and anatomical points presented on polygonal model of mandible

Computational and Mathematical Methods in Medicine 7

some topological (eg curvature) and anatomical landmarks(eg point distinct fromgonion) on themodelTheir positionwas proposed by the orthodontist and anatomist involved inthis research Created anatomical points on the HMCP arepresented in Figure 5

On each individual model of mandible the values ofcoordinates of these points were measured (distance fromorigin of coordinate system in all three directions 119883 119884and 119885) These values were used as the input vectors forthe multiple linear regression analysis which is well knownand documented statistical function The morphometricparameters were alsomeasured on eachmandible sample andincorporated into regression functions The multiple linearregression algorithm which is applied in this research is

described in detail in [23] by Brown The basic idea is to pre-dict dependent variable 119884which is based on the independentvariable 119883 Using the least squares method the best fittingline can be found by minimizing the sum of the squares ofthe distance from each data point on the line [23] The basicmodel is presented in (1) and defined in [23]

119884 = 119883119861 + 119864 (1)

where 119884 is dependent variable 119883 is independent variable 119861is coefficients and 119864 is error variable

The example of matrix equation for the coordinate of onepoint defined in Matlab is presented in

119883coord = [minus51874 minus47712 minus46269 minus47164 minus5078 minus49684 minus43613]1015840

1198891 = [2855 31902 29342 29225 32697 31727 30733]1015840

1198899 = [175 22375 24286 21533 22684 19454 16193]1015840

11988910 = [122384 136216 131729 124194 130113 116171 115732]1015840

119883 = [ones (size (1198891)) 1198891 1198892 1198893 1198894 1198895 1198896 1198897 1198898 1198899 11988910]

119860 = 1198831015840

lowast 119883

119870 = inv (119860)

119861 = 119860 1198831015840

lowast 119883coord

119872 = 119883 lowast 119861

119864 = 119883coord minus119872

(2)

where 119883 coord is values of 119883 coordinates for definedpoint 1198891 11988910 are measured values of defined parameters(arranged according to the order in Table 2) 119861 is vector ofthe coefficients119872 is values of the calculated coordinate119883 119864is the error vector (measured valuesminus calculated values)and 119860 119870 are helper matrixes

3 Results and Discussion

The calculation was performed for all thirty-nine pointson the Human Mandible Coronoid Process In Table 3coefficients of the multiple linear regression functions arepresented for four chosen anatomical points

For example statistical function for119883 coordinate of Point1 is presented in

119883 = 1198870 + 1198871 lowast 1198891 + 1198872 lowast 1198892 + sdot sdot sdot + 11988710 lowast 11988910 =gt 119883

= 0068 + 1167 lowast 1198891 minus 0001 lowast 1198892 + sdot sdot sdot minus 0549

lowast 11988910

(3)

By using these functions coordinates values were cal-culated and compared with measured values for all thirty-nine points The maximal error for all coordinates for eachindividual patient is presented in Table 4 The maximumdeviation is for119884 coordinate of Point 8 shown in Figure 5Theorthodontists and anatomist suggested that maximal error inthis area should not exceed two (2)mm in defined 119883 119884 and119885 directions Taking into account the orthodontistsrsquo recom-mendations it can be concluded that maximum deviations ofthe values of the coordinates are quite satisfactory and belowrecommended limit

Maximum surface deviations of the surface model ofHMCP created by the use of parametric functions (calculatedmodel) from the input surface models of original mandiblespecimens are presented in Table 5 The surface modelswere created by the use of spline curves through the inputand calculated points and with the application of technicalfeatures in CATIA software package (multisection surfacefill etc) It can be noticed that these values are also belowthe recommended limit Maximal deviation is 136mm

8 Computational and Mathematical Methods in Medicine

1

2

3 4

5

6

7

8

910

11

12

13

14

15

16

17

18

19 2120 22

23

24

25

26

27

28

29

30

3132

3334

3536

37

38

39

Anatomical points

Anterior posterior plane

Sagittal planeMandibular plane

Figure 5 Coordinate system spline curves and anatomical points defined on human mandible polygonal model

Table 3 Coefficients of the multiple linear regression functions for four anatomical points

Point 1198870 1198871 1198872 1198873 1198874 1198875 1198876 1198877 1198878 1198879 11988710

P1119909 0068 1167 minus0001 minus0685 2443 minus0648 minus0096 0362 minus0102 0317 minus0549119910 24840 0738 minus0002 minus0650 1732 minus0104 0235 0784 0952 minus0471 minus0402119911 18606 0161 0000 minus0325 1247 minus0113 0080 minus0330 0753 minus0582 minus0140

P2119909 12708 1115 minus0001 minus0790 3060 minus0891 minus0012 0181 minus0162 0192 minus0621119910 13340 1226 minus0002 minus0773 1644 0044 0211 0969 1170 minus0565 minus0449119911 18989 0095 0000 minus0318 1393 minus0221 0004 minus0140 0819 minus0387 minus0173

P38119909 minus6762 1098 minus0001 minus0519 1454 minus0591 minus0044 0335 minus0107 0087 minus0378119910 18717 1684 minus0002 minus1030 2287 0181 0386 0764 1158 minus0870 minus0576119911 minus0679 0276 0000 minus0105 0121 0008 0021 minus0405 0657 minus0554 0046

P39119909 minus3095 1215 minus0001 minus0605 1388 minus0470 0077 0536 minus0062 minus0107 minus0409119910 17478 1715 minus0002 minus1018 2109 0210 0413 0773 1160 minus0943 minus0551119911 minus0669 0276 0000 minus0105 0122 0008 0021 minus0405 0657 minus0553 0046

Table 4 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 10119883 0980 0732 0448 0227 0227 0775 0995 092 0216 0897119884 1720 0668 0597 0453 0907 0469 0952 0765 0301 0842119885 0554 0256 0567 0991 0178 0928 0370 0721 0650 0461

Table 5 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 10Max deviation [mm] 136 0526 0661 0654 097 0912 13 097 034 087

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

4 Computational and Mathematical Methods in Medicine

Condylar process

Coronoid process

Submandibular fossa

Sublingual fossa

Alveolar process

Mental foramen

Mentalprotuberance

Body

Ramus

Mandibular foramen

Figure 2 Anatomy of human mandible

Mandible body (Latin corpus mandibulae) is of horse-shoe shape and represents its horizontal part It consists oftwo sides (external and internal) and two edges alveolar partof 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 islocated upward and backward in relation to mandible bodywith which it forms an angle of 90∘ndash140∘ most commonly120∘ndash130∘ It has two sides external and internal and fouredges upper lower anterior and posterior The upper edgehas two processes coronoid process (Latin processus coro-noideus) and condylar process (Latin processus condylaris)

22 Referential Geometrical Entities (RGEs) In reverse mod-eling of geometry of human mandible it is crucial to bothdetermine directions and projections of bone parts or thewhole bone and establish rules for creation of all directionsand views which should be precisely used For better orien-tation we use several orientational lines and planes in dentalmedicine

(i) Medial line a line which passes vertically betweencentral incisors andwhichmainly divides the face intotwo equal parts

(ii) Sagittal (central) plane which passes through thebody and divides it into equal halves right and left

(iii) Frontal plane which passes through the body indirection left-right (parallel with the forehead) anddivides the body into anterior and posterior parts

(iv) Transversal (horizontal) planes positioned horizon-tally which when in basic anatomical position passthrough the body parallel with the ground

The basic prerequisite to successfully perform reversemodeling of human bone geometry is identification ofreferential geometrical entities (RGEs) RGEs include char-acteristic points directions planes and views Other ele-ments of bone curve and surface geometry will be definedwith reference to RGEs To create precise geometry of ahuman bone a set of primary RGEs should be minimizedGeometrical limitations and relations should be based ona minimal set of primary RGEs This is an approach for asuccessful parameterization of human bone geometry Allmentioned planes and lines are RGEs of the humanmandiblegeometrical model

The ability to create anatomical landmarks as geometricalelements on 3Dhuman bonemodels has a significant role anda vast potential for bone reconstruction after innate defectsillnesses and traumas Anatomical landmarks are defined oneach polygonal model of human mandible of the acquiredsamples They can be defined relative to the RGEs they canbe defined as RGEs or they can be created on the supportgeometry which is relative to RGEs (eg spline curves)

The characteristic anatomical landmarks (points in thiscase) defined on the mandible are shown in Table 1 anddescribed in [22] byArsic et alThemandibular cut (MU)wasadded by the authors of this research because it was necessaryto add this point as additional support point for the properdefinition of coronoid process geometryThe points shown in

Computational and Mathematical Methods in Medicine 5

Table 1 Anatomical landmarks (points) defined on the human mandible as RGEs

Anatomical landmark DefinitionMental foramen (BO) Is one of two holes (ldquoforaminardquo) located on the anterior surface of the mandibleGnathion (Gn) It is the most inferior midline point on the mandibleGonion (Go) It is a point along the rounded posteroinferior corner of the mandible between the ramus and the bodyCondylion (Kon) It is the most prominent point on the mandibular condyleMandibular cut (MU) It is the central point on the mandibular notch

Kon

MU

Go

Gn

BO

Figure 3 Anatomical landmarks (points)

Figure 3 and described in Table 1 are defined as RGEs used forthe creation of the parametric model of the HumanMandibleCoronoid Process

23Morphometric Parameters of theHumanMandible Thereare two groups of morphometric parameters linear (linesplanes and points) and angular (defining relative positionof mandible parts) The values of mandible morphometricparameters can define sex some irregularities in the skeletalsystem parametric models and so forth The morphometricparameters were defined as geometrical elements on thepolygonal model of the human mandible As it is statedin [22] ten morphometric parameters are enough for thecomplete definition of mandible geometry and they aredefined in Table 2 and presented in Figure 4

24 The Parametric Model of Human Mandible CoronoidProcess Thefirst step in definition of the parametricmodel isthe creation of themodel coordinate systemwhich is used forthemeasurement of the coordinates of the anatomical pointsThe same coordinate system is created on each polygonalmodel of individual mandible

Origin of the coordinate system is defined as the middleof the distance between mental foreman middle pointsThe constructed planes of the Object Coordinate System(OBC) of mandible are presented in Figure 5 and they areMediosagittal mandibular and coronal The Mediosagittal

(MS) plane was constructed as the plane which contains theorigin of the OBC and it is normal to the line which connectsmental foreman middle points MS is a plane which divideshuman mandible into two halves left and right Horizontalor mandibular plane is the lowest plane normal to the MSplane and it contains the gonion anatomical point (the mostinferior point of the symphysis of the mandible as seen inthe lateral jaw projection) To be used as a plane of OBCthis plane was translated to the origin of OBC Coronal oranterior posterior (AP) plane is a planewhich is normal to themandibular plane and divides mandible into two anatomicalsections anterior (front) and posterior (back) It is placed atthe origin of OBC 119909-axis of the OBC is defined as normal toMS plane 119910-axis is defined as normal to AP plane 119911-axis isnormal to mandibular plane Axes of the OBC are presentedin Figure 5

Coordinate system together with spline curves createdon the polygonal model of mandible is shown in Figure 5Spline curves are created by the intersection of defined planes(mandibular sagittal and coronal) and polygonal model ofmandible These curves are used as the basis for the creationof anatomical points which were created on them Someanatomical points were not created on spline curves yet theywere created directly on the polygonal model of the humanmandible as additional support points

The thirty-nine points were created in the area of theHMCP The anatomical points are labeled so they represent

6 Computational and Mathematical Methods in Medicine

Table 2 Morphometric parameters of the human mandible

Morphometric parameters DefinitionGnathion-interdental distance (Gn-IdD) Direct distance from infradentale (idD) to gnathion (Gn)Bigonial width (Go-GoD) Direct distance between right and left gonion (Go)Bicondylar breadth (Ko-KoD) 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 mandibleperpendicular 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 tothe long axis of the mandibular body

Mandibular length (DTM) Distance of the anterior margin of the chin from a center point on the protectedstraight 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 theramus

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 perpendicularto MVR

Gnathion-condylar distance (Gn-KoD) Distance between Gn and Kon

Ko-KoD KoKo

STM

BO

Go-GoD GoGo

(a)

Kon

MU

Vko

MVRMSR

Gn-KoD

Go DTM

VTM Gn-IdD

Gn

BO

(b)

Figure 4 Morphometric parameters and anatomical points presented on polygonal model of mandible

Computational and Mathematical Methods in Medicine 7

some topological (eg curvature) and anatomical landmarks(eg point distinct fromgonion) on themodelTheir positionwas proposed by the orthodontist and anatomist involved inthis research Created anatomical points on the HMCP arepresented in Figure 5

On each individual model of mandible the values ofcoordinates of these points were measured (distance fromorigin of coordinate system in all three directions 119883 119884and 119885) These values were used as the input vectors forthe multiple linear regression analysis which is well knownand documented statistical function The morphometricparameters were alsomeasured on eachmandible sample andincorporated into regression functions The multiple linearregression algorithm which is applied in this research is

described in detail in [23] by Brown The basic idea is to pre-dict dependent variable 119884which is based on the independentvariable 119883 Using the least squares method the best fittingline can be found by minimizing the sum of the squares ofthe distance from each data point on the line [23] The basicmodel is presented in (1) and defined in [23]

119884 = 119883119861 + 119864 (1)

where 119884 is dependent variable 119883 is independent variable 119861is coefficients and 119864 is error variable

The example of matrix equation for the coordinate of onepoint defined in Matlab is presented in

119883coord = [minus51874 minus47712 minus46269 minus47164 minus5078 minus49684 minus43613]1015840

1198891 = [2855 31902 29342 29225 32697 31727 30733]1015840

1198899 = [175 22375 24286 21533 22684 19454 16193]1015840

11988910 = [122384 136216 131729 124194 130113 116171 115732]1015840

119883 = [ones (size (1198891)) 1198891 1198892 1198893 1198894 1198895 1198896 1198897 1198898 1198899 11988910]

119860 = 1198831015840

lowast 119883

119870 = inv (119860)

119861 = 119860 1198831015840

lowast 119883coord

119872 = 119883 lowast 119861

119864 = 119883coord minus119872

(2)

where 119883 coord is values of 119883 coordinates for definedpoint 1198891 11988910 are measured values of defined parameters(arranged according to the order in Table 2) 119861 is vector ofthe coefficients119872 is values of the calculated coordinate119883 119864is the error vector (measured valuesminus calculated values)and 119860 119870 are helper matrixes

3 Results and Discussion

The calculation was performed for all thirty-nine pointson the Human Mandible Coronoid Process In Table 3coefficients of the multiple linear regression functions arepresented for four chosen anatomical points

For example statistical function for119883 coordinate of Point1 is presented in

119883 = 1198870 + 1198871 lowast 1198891 + 1198872 lowast 1198892 + sdot sdot sdot + 11988710 lowast 11988910 =gt 119883

= 0068 + 1167 lowast 1198891 minus 0001 lowast 1198892 + sdot sdot sdot minus 0549

lowast 11988910

(3)

By using these functions coordinates values were cal-culated and compared with measured values for all thirty-nine points The maximal error for all coordinates for eachindividual patient is presented in Table 4 The maximumdeviation is for119884 coordinate of Point 8 shown in Figure 5Theorthodontists and anatomist suggested that maximal error inthis area should not exceed two (2)mm in defined 119883 119884 and119885 directions Taking into account the orthodontistsrsquo recom-mendations it can be concluded that maximum deviations ofthe values of the coordinates are quite satisfactory and belowrecommended limit

Maximum surface deviations of the surface model ofHMCP created by the use of parametric functions (calculatedmodel) from the input surface models of original mandiblespecimens are presented in Table 5 The surface modelswere created by the use of spline curves through the inputand calculated points and with the application of technicalfeatures in CATIA software package (multisection surfacefill etc) It can be noticed that these values are also belowthe recommended limit Maximal deviation is 136mm

8 Computational and Mathematical Methods in Medicine

1

2

3 4

5

6

7

8

910

11

12

13

14

15

16

17

18

19 2120 22

23

24

25

26

27

28

29

30

3132

3334

3536

37

38

39

Anatomical points

Anterior posterior plane

Sagittal planeMandibular plane

Figure 5 Coordinate system spline curves and anatomical points defined on human mandible polygonal model

Table 3 Coefficients of the multiple linear regression functions for four anatomical points

Point 1198870 1198871 1198872 1198873 1198874 1198875 1198876 1198877 1198878 1198879 11988710

P1119909 0068 1167 minus0001 minus0685 2443 minus0648 minus0096 0362 minus0102 0317 minus0549119910 24840 0738 minus0002 minus0650 1732 minus0104 0235 0784 0952 minus0471 minus0402119911 18606 0161 0000 minus0325 1247 minus0113 0080 minus0330 0753 minus0582 minus0140

P2119909 12708 1115 minus0001 minus0790 3060 minus0891 minus0012 0181 minus0162 0192 minus0621119910 13340 1226 minus0002 minus0773 1644 0044 0211 0969 1170 minus0565 minus0449119911 18989 0095 0000 minus0318 1393 minus0221 0004 minus0140 0819 minus0387 minus0173

P38119909 minus6762 1098 minus0001 minus0519 1454 minus0591 minus0044 0335 minus0107 0087 minus0378119910 18717 1684 minus0002 minus1030 2287 0181 0386 0764 1158 minus0870 minus0576119911 minus0679 0276 0000 minus0105 0121 0008 0021 minus0405 0657 minus0554 0046

P39119909 minus3095 1215 minus0001 minus0605 1388 minus0470 0077 0536 minus0062 minus0107 minus0409119910 17478 1715 minus0002 minus1018 2109 0210 0413 0773 1160 minus0943 minus0551119911 minus0669 0276 0000 minus0105 0122 0008 0021 minus0405 0657 minus0553 0046

Table 4 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 10119883 0980 0732 0448 0227 0227 0775 0995 092 0216 0897119884 1720 0668 0597 0453 0907 0469 0952 0765 0301 0842119885 0554 0256 0567 0991 0178 0928 0370 0721 0650 0461

Table 5 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 10Max deviation [mm] 136 0526 0661 0654 097 0912 13 097 034 087

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

Computational and Mathematical Methods in Medicine 5

Table 1 Anatomical landmarks (points) defined on the human mandible as RGEs

Anatomical landmark DefinitionMental foramen (BO) Is one of two holes (ldquoforaminardquo) located on the anterior surface of the mandibleGnathion (Gn) It is the most inferior midline point on the mandibleGonion (Go) It is a point along the rounded posteroinferior corner of the mandible between the ramus and the bodyCondylion (Kon) It is the most prominent point on the mandibular condyleMandibular cut (MU) It is the central point on the mandibular notch

Kon

MU

Go

Gn

BO

Figure 3 Anatomical landmarks (points)

Figure 3 and described in Table 1 are defined as RGEs used forthe creation of the parametric model of the HumanMandibleCoronoid Process

23Morphometric Parameters of theHumanMandible Thereare two groups of morphometric parameters linear (linesplanes and points) and angular (defining relative positionof mandible parts) The values of mandible morphometricparameters can define sex some irregularities in the skeletalsystem parametric models and so forth The morphometricparameters were defined as geometrical elements on thepolygonal model of the human mandible As it is statedin [22] ten morphometric parameters are enough for thecomplete definition of mandible geometry and they aredefined in Table 2 and presented in Figure 4

24 The Parametric Model of Human Mandible CoronoidProcess Thefirst step in definition of the parametricmodel isthe creation of themodel coordinate systemwhich is used forthemeasurement of the coordinates of the anatomical pointsThe same coordinate system is created on each polygonalmodel of individual mandible

Origin of the coordinate system is defined as the middleof the distance between mental foreman middle pointsThe constructed planes of the Object Coordinate System(OBC) of mandible are presented in Figure 5 and they areMediosagittal mandibular and coronal The Mediosagittal

(MS) plane was constructed as the plane which contains theorigin of the OBC and it is normal to the line which connectsmental foreman middle points MS is a plane which divideshuman mandible into two halves left and right Horizontalor mandibular plane is the lowest plane normal to the MSplane and it contains the gonion anatomical point (the mostinferior point of the symphysis of the mandible as seen inthe lateral jaw projection) To be used as a plane of OBCthis plane was translated to the origin of OBC Coronal oranterior posterior (AP) plane is a planewhich is normal to themandibular plane and divides mandible into two anatomicalsections anterior (front) and posterior (back) It is placed atthe origin of OBC 119909-axis of the OBC is defined as normal toMS plane 119910-axis is defined as normal to AP plane 119911-axis isnormal to mandibular plane Axes of the OBC are presentedin Figure 5

Coordinate system together with spline curves createdon the polygonal model of mandible is shown in Figure 5Spline curves are created by the intersection of defined planes(mandibular sagittal and coronal) and polygonal model ofmandible These curves are used as the basis for the creationof anatomical points which were created on them Someanatomical points were not created on spline curves yet theywere created directly on the polygonal model of the humanmandible as additional support points

The thirty-nine points were created in the area of theHMCP The anatomical points are labeled so they represent

6 Computational and Mathematical Methods in Medicine

Table 2 Morphometric parameters of the human mandible

Morphometric parameters DefinitionGnathion-interdental distance (Gn-IdD) Direct distance from infradentale (idD) to gnathion (Gn)Bigonial width (Go-GoD) Direct distance between right and left gonion (Go)Bicondylar breadth (Ko-KoD) 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 mandibleperpendicular 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 tothe long axis of the mandibular body

Mandibular length (DTM) Distance of the anterior margin of the chin from a center point on the protectedstraight 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 theramus

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 perpendicularto MVR

Gnathion-condylar distance (Gn-KoD) Distance between Gn and Kon

Ko-KoD KoKo

STM

BO

Go-GoD GoGo

(a)

Kon

MU

Vko

MVRMSR

Gn-KoD

Go DTM

VTM Gn-IdD

Gn

BO

(b)

Figure 4 Morphometric parameters and anatomical points presented on polygonal model of mandible

Computational and Mathematical Methods in Medicine 7

some topological (eg curvature) and anatomical landmarks(eg point distinct fromgonion) on themodelTheir positionwas proposed by the orthodontist and anatomist involved inthis research Created anatomical points on the HMCP arepresented in Figure 5

On each individual model of mandible the values ofcoordinates of these points were measured (distance fromorigin of coordinate system in all three directions 119883 119884and 119885) These values were used as the input vectors forthe multiple linear regression analysis which is well knownand documented statistical function The morphometricparameters were alsomeasured on eachmandible sample andincorporated into regression functions The multiple linearregression algorithm which is applied in this research is

described in detail in [23] by Brown The basic idea is to pre-dict dependent variable 119884which is based on the independentvariable 119883 Using the least squares method the best fittingline can be found by minimizing the sum of the squares ofthe distance from each data point on the line [23] The basicmodel is presented in (1) and defined in [23]

119884 = 119883119861 + 119864 (1)

where 119884 is dependent variable 119883 is independent variable 119861is coefficients and 119864 is error variable

The example of matrix equation for the coordinate of onepoint defined in Matlab is presented in

119883coord = [minus51874 minus47712 minus46269 minus47164 minus5078 minus49684 minus43613]1015840

1198891 = [2855 31902 29342 29225 32697 31727 30733]1015840

1198899 = [175 22375 24286 21533 22684 19454 16193]1015840

11988910 = [122384 136216 131729 124194 130113 116171 115732]1015840

119883 = [ones (size (1198891)) 1198891 1198892 1198893 1198894 1198895 1198896 1198897 1198898 1198899 11988910]

119860 = 1198831015840

lowast 119883

119870 = inv (119860)

119861 = 119860 1198831015840

lowast 119883coord

119872 = 119883 lowast 119861

119864 = 119883coord minus119872

(2)

where 119883 coord is values of 119883 coordinates for definedpoint 1198891 11988910 are measured values of defined parameters(arranged according to the order in Table 2) 119861 is vector ofthe coefficients119872 is values of the calculated coordinate119883 119864is the error vector (measured valuesminus calculated values)and 119860 119870 are helper matrixes

3 Results and Discussion

The calculation was performed for all thirty-nine pointson the Human Mandible Coronoid Process In Table 3coefficients of the multiple linear regression functions arepresented for four chosen anatomical points

For example statistical function for119883 coordinate of Point1 is presented in

119883 = 1198870 + 1198871 lowast 1198891 + 1198872 lowast 1198892 + sdot sdot sdot + 11988710 lowast 11988910 =gt 119883

= 0068 + 1167 lowast 1198891 minus 0001 lowast 1198892 + sdot sdot sdot minus 0549

lowast 11988910

(3)

By using these functions coordinates values were cal-culated and compared with measured values for all thirty-nine points The maximal error for all coordinates for eachindividual patient is presented in Table 4 The maximumdeviation is for119884 coordinate of Point 8 shown in Figure 5Theorthodontists and anatomist suggested that maximal error inthis area should not exceed two (2)mm in defined 119883 119884 and119885 directions Taking into account the orthodontistsrsquo recom-mendations it can be concluded that maximum deviations ofthe values of the coordinates are quite satisfactory and belowrecommended limit

Maximum surface deviations of the surface model ofHMCP created by the use of parametric functions (calculatedmodel) from the input surface models of original mandiblespecimens are presented in Table 5 The surface modelswere created by the use of spline curves through the inputand calculated points and with the application of technicalfeatures in CATIA software package (multisection surfacefill etc) It can be noticed that these values are also belowthe recommended limit Maximal deviation is 136mm

8 Computational and Mathematical Methods in Medicine

1

2

3 4

5

6

7

8

910

11

12

13

14

15

16

17

18

19 2120 22

23

24

25

26

27

28

29

30

3132

3334

3536

37

38

39

Anatomical points

Anterior posterior plane

Sagittal planeMandibular plane

Figure 5 Coordinate system spline curves and anatomical points defined on human mandible polygonal model

Table 3 Coefficients of the multiple linear regression functions for four anatomical points

Point 1198870 1198871 1198872 1198873 1198874 1198875 1198876 1198877 1198878 1198879 11988710

P1119909 0068 1167 minus0001 minus0685 2443 minus0648 minus0096 0362 minus0102 0317 minus0549119910 24840 0738 minus0002 minus0650 1732 minus0104 0235 0784 0952 minus0471 minus0402119911 18606 0161 0000 minus0325 1247 minus0113 0080 minus0330 0753 minus0582 minus0140

P2119909 12708 1115 minus0001 minus0790 3060 minus0891 minus0012 0181 minus0162 0192 minus0621119910 13340 1226 minus0002 minus0773 1644 0044 0211 0969 1170 minus0565 minus0449119911 18989 0095 0000 minus0318 1393 minus0221 0004 minus0140 0819 minus0387 minus0173

P38119909 minus6762 1098 minus0001 minus0519 1454 minus0591 minus0044 0335 minus0107 0087 minus0378119910 18717 1684 minus0002 minus1030 2287 0181 0386 0764 1158 minus0870 minus0576119911 minus0679 0276 0000 minus0105 0121 0008 0021 minus0405 0657 minus0554 0046

P39119909 minus3095 1215 minus0001 minus0605 1388 minus0470 0077 0536 minus0062 minus0107 minus0409119910 17478 1715 minus0002 minus1018 2109 0210 0413 0773 1160 minus0943 minus0551119911 minus0669 0276 0000 minus0105 0122 0008 0021 minus0405 0657 minus0553 0046

Table 4 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 10119883 0980 0732 0448 0227 0227 0775 0995 092 0216 0897119884 1720 0668 0597 0453 0907 0469 0952 0765 0301 0842119885 0554 0256 0567 0991 0178 0928 0370 0721 0650 0461

Table 5 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 10Max deviation [mm] 136 0526 0661 0654 097 0912 13 097 034 087

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

6 Computational and Mathematical Methods in Medicine

Table 2 Morphometric parameters of the human mandible

Morphometric parameters DefinitionGnathion-interdental distance (Gn-IdD) Direct distance from infradentale (idD) to gnathion (Gn)Bigonial width (Go-GoD) Direct distance between right and left gonion (Go)Bicondylar breadth (Ko-KoD) 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 mandibleperpendicular 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 tothe long axis of the mandibular body

Mandibular length (DTM) Distance of the anterior margin of the chin from a center point on the protectedstraight 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 theramus

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 perpendicularto MVR

Gnathion-condylar distance (Gn-KoD) Distance between Gn and Kon

Ko-KoD KoKo

STM

BO

Go-GoD GoGo

(a)

Kon

MU

Vko

MVRMSR

Gn-KoD

Go DTM

VTM Gn-IdD

Gn

BO

(b)

Figure 4 Morphometric parameters and anatomical points presented on polygonal model of mandible

Computational and Mathematical Methods in Medicine 7

some topological (eg curvature) and anatomical landmarks(eg point distinct fromgonion) on themodelTheir positionwas proposed by the orthodontist and anatomist involved inthis research Created anatomical points on the HMCP arepresented in Figure 5

On each individual model of mandible the values ofcoordinates of these points were measured (distance fromorigin of coordinate system in all three directions 119883 119884and 119885) These values were used as the input vectors forthe multiple linear regression analysis which is well knownand documented statistical function The morphometricparameters were alsomeasured on eachmandible sample andincorporated into regression functions The multiple linearregression algorithm which is applied in this research is

described in detail in [23] by Brown The basic idea is to pre-dict dependent variable 119884which is based on the independentvariable 119883 Using the least squares method the best fittingline can be found by minimizing the sum of the squares ofthe distance from each data point on the line [23] The basicmodel is presented in (1) and defined in [23]

119884 = 119883119861 + 119864 (1)

where 119884 is dependent variable 119883 is independent variable 119861is coefficients and 119864 is error variable

The example of matrix equation for the coordinate of onepoint defined in Matlab is presented in

119883coord = [minus51874 minus47712 minus46269 minus47164 minus5078 minus49684 minus43613]1015840

1198891 = [2855 31902 29342 29225 32697 31727 30733]1015840

1198899 = [175 22375 24286 21533 22684 19454 16193]1015840

11988910 = [122384 136216 131729 124194 130113 116171 115732]1015840

119883 = [ones (size (1198891)) 1198891 1198892 1198893 1198894 1198895 1198896 1198897 1198898 1198899 11988910]

119860 = 1198831015840

lowast 119883

119870 = inv (119860)

119861 = 119860 1198831015840

lowast 119883coord

119872 = 119883 lowast 119861

119864 = 119883coord minus119872

(2)

where 119883 coord is values of 119883 coordinates for definedpoint 1198891 11988910 are measured values of defined parameters(arranged according to the order in Table 2) 119861 is vector ofthe coefficients119872 is values of the calculated coordinate119883 119864is the error vector (measured valuesminus calculated values)and 119860 119870 are helper matrixes

3 Results and Discussion

The calculation was performed for all thirty-nine pointson the Human Mandible Coronoid Process In Table 3coefficients of the multiple linear regression functions arepresented for four chosen anatomical points

For example statistical function for119883 coordinate of Point1 is presented in

119883 = 1198870 + 1198871 lowast 1198891 + 1198872 lowast 1198892 + sdot sdot sdot + 11988710 lowast 11988910 =gt 119883

= 0068 + 1167 lowast 1198891 minus 0001 lowast 1198892 + sdot sdot sdot minus 0549

lowast 11988910

(3)

By using these functions coordinates values were cal-culated and compared with measured values for all thirty-nine points The maximal error for all coordinates for eachindividual patient is presented in Table 4 The maximumdeviation is for119884 coordinate of Point 8 shown in Figure 5Theorthodontists and anatomist suggested that maximal error inthis area should not exceed two (2)mm in defined 119883 119884 and119885 directions Taking into account the orthodontistsrsquo recom-mendations it can be concluded that maximum deviations ofthe values of the coordinates are quite satisfactory and belowrecommended limit

Maximum surface deviations of the surface model ofHMCP created by the use of parametric functions (calculatedmodel) from the input surface models of original mandiblespecimens are presented in Table 5 The surface modelswere created by the use of spline curves through the inputand calculated points and with the application of technicalfeatures in CATIA software package (multisection surfacefill etc) It can be noticed that these values are also belowthe recommended limit Maximal deviation is 136mm

8 Computational and Mathematical Methods in Medicine

1

2

3 4

5

6

7

8

910

11

12

13

14

15

16

17

18

19 2120 22

23

24

25

26

27

28

29

30

3132

3334

3536

37

38

39

Anatomical points

Anterior posterior plane

Sagittal planeMandibular plane

Figure 5 Coordinate system spline curves and anatomical points defined on human mandible polygonal model

Table 3 Coefficients of the multiple linear regression functions for four anatomical points

Point 1198870 1198871 1198872 1198873 1198874 1198875 1198876 1198877 1198878 1198879 11988710

P1119909 0068 1167 minus0001 minus0685 2443 minus0648 minus0096 0362 minus0102 0317 minus0549119910 24840 0738 minus0002 minus0650 1732 minus0104 0235 0784 0952 minus0471 minus0402119911 18606 0161 0000 minus0325 1247 minus0113 0080 minus0330 0753 minus0582 minus0140

P2119909 12708 1115 minus0001 minus0790 3060 minus0891 minus0012 0181 minus0162 0192 minus0621119910 13340 1226 minus0002 minus0773 1644 0044 0211 0969 1170 minus0565 minus0449119911 18989 0095 0000 minus0318 1393 minus0221 0004 minus0140 0819 minus0387 minus0173

P38119909 minus6762 1098 minus0001 minus0519 1454 minus0591 minus0044 0335 minus0107 0087 minus0378119910 18717 1684 minus0002 minus1030 2287 0181 0386 0764 1158 minus0870 minus0576119911 minus0679 0276 0000 minus0105 0121 0008 0021 minus0405 0657 minus0554 0046

P39119909 minus3095 1215 minus0001 minus0605 1388 minus0470 0077 0536 minus0062 minus0107 minus0409119910 17478 1715 minus0002 minus1018 2109 0210 0413 0773 1160 minus0943 minus0551119911 minus0669 0276 0000 minus0105 0122 0008 0021 minus0405 0657 minus0553 0046

Table 4 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 10119883 0980 0732 0448 0227 0227 0775 0995 092 0216 0897119884 1720 0668 0597 0453 0907 0469 0952 0765 0301 0842119885 0554 0256 0567 0991 0178 0928 0370 0721 0650 0461

Table 5 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 10Max deviation [mm] 136 0526 0661 0654 097 0912 13 097 034 087

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

Computational and Mathematical Methods in Medicine 7

some topological (eg curvature) and anatomical landmarks(eg point distinct fromgonion) on themodelTheir positionwas proposed by the orthodontist and anatomist involved inthis research Created anatomical points on the HMCP arepresented in Figure 5

On each individual model of mandible the values ofcoordinates of these points were measured (distance fromorigin of coordinate system in all three directions 119883 119884and 119885) These values were used as the input vectors forthe multiple linear regression analysis which is well knownand documented statistical function The morphometricparameters were alsomeasured on eachmandible sample andincorporated into regression functions The multiple linearregression algorithm which is applied in this research is

described in detail in [23] by Brown The basic idea is to pre-dict dependent variable 119884which is based on the independentvariable 119883 Using the least squares method the best fittingline can be found by minimizing the sum of the squares ofthe distance from each data point on the line [23] The basicmodel is presented in (1) and defined in [23]

119884 = 119883119861 + 119864 (1)

where 119884 is dependent variable 119883 is independent variable 119861is coefficients and 119864 is error variable

The example of matrix equation for the coordinate of onepoint defined in Matlab is presented in

119883coord = [minus51874 minus47712 minus46269 minus47164 minus5078 minus49684 minus43613]1015840

1198891 = [2855 31902 29342 29225 32697 31727 30733]1015840

1198899 = [175 22375 24286 21533 22684 19454 16193]1015840

11988910 = [122384 136216 131729 124194 130113 116171 115732]1015840

119883 = [ones (size (1198891)) 1198891 1198892 1198893 1198894 1198895 1198896 1198897 1198898 1198899 11988910]

119860 = 1198831015840

lowast 119883

119870 = inv (119860)

119861 = 119860 1198831015840

lowast 119883coord

119872 = 119883 lowast 119861

119864 = 119883coord minus119872

(2)

where 119883 coord is values of 119883 coordinates for definedpoint 1198891 11988910 are measured values of defined parameters(arranged according to the order in Table 2) 119861 is vector ofthe coefficients119872 is values of the calculated coordinate119883 119864is the error vector (measured valuesminus calculated values)and 119860 119870 are helper matrixes

3 Results and Discussion

The calculation was performed for all thirty-nine pointson the Human Mandible Coronoid Process In Table 3coefficients of the multiple linear regression functions arepresented for four chosen anatomical points

For example statistical function for119883 coordinate of Point1 is presented in

119883 = 1198870 + 1198871 lowast 1198891 + 1198872 lowast 1198892 + sdot sdot sdot + 11988710 lowast 11988910 =gt 119883

= 0068 + 1167 lowast 1198891 minus 0001 lowast 1198892 + sdot sdot sdot minus 0549

lowast 11988910

(3)

By using these functions coordinates values were cal-culated and compared with measured values for all thirty-nine points The maximal error for all coordinates for eachindividual patient is presented in Table 4 The maximumdeviation is for119884 coordinate of Point 8 shown in Figure 5Theorthodontists and anatomist suggested that maximal error inthis area should not exceed two (2)mm in defined 119883 119884 and119885 directions Taking into account the orthodontistsrsquo recom-mendations it can be concluded that maximum deviations ofthe values of the coordinates are quite satisfactory and belowrecommended limit

Maximum surface deviations of the surface model ofHMCP created by the use of parametric functions (calculatedmodel) from the input surface models of original mandiblespecimens are presented in Table 5 The surface modelswere created by the use of spline curves through the inputand calculated points and with the application of technicalfeatures in CATIA software package (multisection surfacefill etc) It can be noticed that these values are also belowthe recommended limit Maximal deviation is 136mm

8 Computational and Mathematical Methods in Medicine

1

2

3 4

5

6

7

8

910

11

12

13

14

15

16

17

18

19 2120 22

23

24

25

26

27

28

29

30

3132

3334

3536

37

38

39

Anatomical points

Anterior posterior plane

Sagittal planeMandibular plane

Figure 5 Coordinate system spline curves and anatomical points defined on human mandible polygonal model

Table 3 Coefficients of the multiple linear regression functions for four anatomical points

Point 1198870 1198871 1198872 1198873 1198874 1198875 1198876 1198877 1198878 1198879 11988710

P1119909 0068 1167 minus0001 minus0685 2443 minus0648 minus0096 0362 minus0102 0317 minus0549119910 24840 0738 minus0002 minus0650 1732 minus0104 0235 0784 0952 minus0471 minus0402119911 18606 0161 0000 minus0325 1247 minus0113 0080 minus0330 0753 minus0582 minus0140

P2119909 12708 1115 minus0001 minus0790 3060 minus0891 minus0012 0181 minus0162 0192 minus0621119910 13340 1226 minus0002 minus0773 1644 0044 0211 0969 1170 minus0565 minus0449119911 18989 0095 0000 minus0318 1393 minus0221 0004 minus0140 0819 minus0387 minus0173

P38119909 minus6762 1098 minus0001 minus0519 1454 minus0591 minus0044 0335 minus0107 0087 minus0378119910 18717 1684 minus0002 minus1030 2287 0181 0386 0764 1158 minus0870 minus0576119911 minus0679 0276 0000 minus0105 0121 0008 0021 minus0405 0657 minus0554 0046

P39119909 minus3095 1215 minus0001 minus0605 1388 minus0470 0077 0536 minus0062 minus0107 minus0409119910 17478 1715 minus0002 minus1018 2109 0210 0413 0773 1160 minus0943 minus0551119911 minus0669 0276 0000 minus0105 0122 0008 0021 minus0405 0657 minus0553 0046

Table 4 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 10119883 0980 0732 0448 0227 0227 0775 0995 092 0216 0897119884 1720 0668 0597 0453 0907 0469 0952 0765 0301 0842119885 0554 0256 0567 0991 0178 0928 0370 0721 0650 0461

Table 5 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 10Max deviation [mm] 136 0526 0661 0654 097 0912 13 097 034 087

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

8 Computational and Mathematical Methods in Medicine

1

2

3 4

5

6

7

8

910

11

12

13

14

15

16

17

18

19 2120 22

23

24

25

26

27

28

29

30

3132

3334

3536

37

38

39

Anatomical points

Anterior posterior plane

Sagittal planeMandibular plane

Figure 5 Coordinate system spline curves and anatomical points defined on human mandible polygonal model

Table 3 Coefficients of the multiple linear regression functions for four anatomical points

Point 1198870 1198871 1198872 1198873 1198874 1198875 1198876 1198877 1198878 1198879 11988710

P1119909 0068 1167 minus0001 minus0685 2443 minus0648 minus0096 0362 minus0102 0317 minus0549119910 24840 0738 minus0002 minus0650 1732 minus0104 0235 0784 0952 minus0471 minus0402119911 18606 0161 0000 minus0325 1247 minus0113 0080 minus0330 0753 minus0582 minus0140

P2119909 12708 1115 minus0001 minus0790 3060 minus0891 minus0012 0181 minus0162 0192 minus0621119910 13340 1226 minus0002 minus0773 1644 0044 0211 0969 1170 minus0565 minus0449119911 18989 0095 0000 minus0318 1393 minus0221 0004 minus0140 0819 minus0387 minus0173

P38119909 minus6762 1098 minus0001 minus0519 1454 minus0591 minus0044 0335 minus0107 0087 minus0378119910 18717 1684 minus0002 minus1030 2287 0181 0386 0764 1158 minus0870 minus0576119911 minus0679 0276 0000 minus0105 0121 0008 0021 minus0405 0657 minus0554 0046

P39119909 minus3095 1215 minus0001 minus0605 1388 minus0470 0077 0536 minus0062 minus0107 minus0409119910 17478 1715 minus0002 minus1018 2109 0210 0413 0773 1160 minus0943 minus0551119911 minus0669 0276 0000 minus0105 0122 0008 0021 minus0405 0657 minus0553 0046

Table 4 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 10119883 0980 0732 0448 0227 0227 0775 0995 092 0216 0897119884 1720 0668 0597 0453 0907 0469 0952 0765 0301 0842119885 0554 0256 0567 0991 0178 0928 0370 0721 0650 0461

Table 5 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 10Max deviation [mm] 136 0526 0661 0654 097 0912 13 097 034 087

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

Computational and Mathematical Methods in Medicine 9

The preliminary claim about parametric model geomet-rical accuracy and anatomical correctness can be stated asquite satisfactory for the prototype model It is important tomention that designers can choose more points in the area ofmaximal deviation(s) or choose different points which willenable better geometrical definition of the domain includedand thus improve the accuracy

In order to obtain reliable response of the parametricmodel more detailed analysis must be performed The num-ber of samples should be increased parameters influence onthe individual points should be examined and the parametricmodel for the whole humanmandible must be createdThesetasks will be conducted in the future research

4 Conclusion

The presented Method of Anatomical Features (MAF)enables creation of the geometrically accurate and anatom-ically correct parametric model of the Human MandibleCoronoid Process (HMCP)The presented parametric modelof the HMCP can be considered as a prototype (test) modelfor the parameterization of the whole human mandible

It should be emphasized that parametric model enablescreation of the adequate geometric model of the HMCPcustomized to the specific patient The customization isperformed by the application of the values of the parametersin the parametric functions Values of the parameters canbe acquired from medical images (CT MRI or X-ray)The resulting model(s) can be applied in training of themedical staff implant and fixatormanufacturing CADCAMapplication FEA (Finite Element Analysis) and so forth

The current research results are based on a relatively smallnumber of human mandible samples It is crucial to increasethat number as much as possible Besides the number ofsamples the influence of the involved parameters on theposition of the individual points must be investigated All ofthese tasks are the part of future research and they will beperformed in order to improve the geometric precision andanatomical correctness of the presented parametric model ofthe HMCP and future parametric model of the whole humanmandible One possible application of the parametric modelof the whole human mandible is for the prediction of thedental implants position and orientation For example it canbe used for the proper implantation of the osseointegrateddental implants which are presented in [24] by Vairo andSannino Considering that and all other facts presented in thispaper it can be concluded that further research is advisablebecause it can provide a lot of benefits to the medicalpractitioners

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The presented research was done for the project Vir-tual Human Osteoarticular System and Its Application in

Preclinical and Clinical Practice which is sponsored by theMinistry of Science and Technology of the Republic of SerbiaProject ID III 41017 for the period of 2011ndash2015

References

[1] L Adams W Krybus D Meyer-Ebrecht et al ldquoComputer-assisted surgeryrdquo IEEEComputer Graphics andApplications vol10 no 3 pp 43ndash51 1990

[2] H Bathis L Perlick M Tingart C Luring D Zurakowski andJ Grifka ldquoAlignment in total knee arthroplasty A comparisonof computer-assisted surgery with the conventional techniquerdquoThe Journal of Bone and Joint SurgerymdashBritish Volume vol 86no 5 pp 682ndash687 2004

[3] H Essig M Rana H Kokemueller et al ldquoPre-operative plan-ning for mandibular reconstructionmdasha full digital planningworkflow resulting in a patient specific reconstructionrdquo Headamp Neck Oncology vol 3 no 1 article 45 2011

[4] J Chapuis A Schramm I Pappas et al ldquoA new system forcomputer-aided preoperative planning and intraoperative nav-igation during corrective jaw surgeryrdquo IEEE Transactions onInformation Technology in Biomedicine vol 11 no 3 pp 274ndash287 2007

[5] G W Weber ldquoVirtual anthropologyrdquo American Journal ofPhysical Anthropology vol 156 supplement 59 pp 22ndash42 2014

[6] S Benazzi E Stansfield O Kullmer L Fiorenza and G Grup-pioni ldquoGeometricmorphometricmethods for bone reconstruc-tion the mandibular condylar process of Pico della MirandolardquoAnatomical Record vol 292 no 8 pp 1088ndash1097 2009

[7] N Archipa R Rohlingb V Dessennec P J Erardd and L PNoltee ldquoAnatomical structure modeling from medical imagesrdquoComputer Methods and Programs in Biomedicine vol 82 no 3pp 203ndash215 2006

[8] S-F Huang H-Y Chao P-F KaoW-C Shen Y-H Chou andS-H Liu ldquoAutomatic vertebral column extraction by whole-body bone SPECT scanrdquo Computational and MathematicalMethods inMedicine vol 2013 Article ID 647548 11 pages 2013

[9] P S Calhoun B S Kuszyk D G Heath J C Carley and EK Fishman ldquoThree-dimensional volume rendering of spiral CTdata theory and methodrdquo Radiographics vol 19 no 3 pp 745ndash764 1999

[10] J Li Q Liu andH Su ldquoVirtual reality method of portal slewingcrane based on WPFrdquo Advances in Mechanical Engineering vol5 Article ID 320757 2013

[11] W E Lorensen and H E Cline ldquoMarching Cubes a high reso-lution 3D surface construction algorithmrdquo Computer Graphicsvol 21 no 4 pp 163ndash169 1987

[12] M Tufegdzic M Trajanovic N Vitkovic and S Arsic ldquoReverseengineering of the human fibula by the anatomical featuresmethodrdquo Facta Universitatis Series Mechanical Engineeringvol 11 no 2 pp 133ndash139 2013

[13] G V Salmoria D Hotza P Klauss L A Kanis and C RRoesler ldquoManufacturing of porous polycaprolactone preparedwith different particle sizes and infrared laser sintering condi-tions microstructure and mechanical propertiesrdquo Advances inMechanical Engineering vol 6 Article ID 640496 2014

[14] M Li K Miller G R Joldes et al ldquoPatient-specific biomechan-ical model as whole-body CT image registration toolrdquo MedicalImage Analysis vol 22 no 1 pp 22ndash34 2015

[15] S Wuhrer and C Shu ldquoEstimating 3D human shapes frommeasurementsrdquoMachine Vision and Applications vol 24 no 6pp 1133ndash1147 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013

10 Computational and Mathematical Methods in Medicine

[16] I-F Leong J-J Fang and M-J Tsai ldquoAutomatic body featureextraction fromamarker-less scannedhumanbodyrdquoComputer-Aided Design vol 39 no 7 pp 568ndash582 2007

[17] X Chen Y Guo B Zhou and Q Zhao ldquoDeformable modelfor estimating clothed and naked human shapes from a singleimagerdquoThe Visual Computer vol 29 no 11 pp 1187ndash1196 2013

[18] N Vitkovic J Milovanovic N Korunovic et al ldquoSoftware sys-tem for creation of human femur customized polygonal mod-elsrdquo Computer Science and Information Systems vol 10 no 3pp 1473ndash1497 2013

[19] M Majstorovic M Trajanovic N Vitkovic and M StojkovicldquoReverse engineering of human bones by using method ofanatomical featuresrdquoCIRP AnnalsmdashManufacturing Technologyvol 62 no 1 pp 167ndash170 2013

[20] G Juodzbalys HWang andG Sabalys ldquoAnatomy ofmandibu-lar vital structures Part I mandibular canal and inferior alveo-lar neurovascular bundle in relation with sental implantologyrdquoJournal of Oral amp Maxillofacial Research vol 1 no 1 article e22010

[21] G Juodzbalys H L Wang and G Sabalys ldquoAnatomy ofmandibular vital structures Part II mandibular incisive canalmental foramen and associated neurovascular bundles in rela-tion with dental implantologyrdquo Journal of Oral amp MaxillofacialResearch vol 1 no 1 article e3 2010

[22] S Arsic P Peric M Stojkovic et al ldquoComparative analysisof linear morphometric parameters of the humane mandibulaobtained by direct and indirect measurementrdquo VojnosanitetskiPregled vol 67 no 10 pp 839ndash846 2010

[23] S Brown ldquoMultiple linear regression analysis a matrixapproach with MATLABrdquo Alabama Journal of Mathematics2009 httpajmonlineorg2009brownpdf

[24] G Vairo and G Sannino ldquoComparative evaluation of osseoin-tegrated dental implants based on platform-switching conceptinfluence of diameter length thread shape and in-bone posi-tioning depth on stress-based performancerdquo Computationaland Mathematical Methods in Medicine vol 2013 Article ID250929 15 pages 2013