Graphic and geometric tools for analyzing morphology of the human knee

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1 Graphc and geometrc tools for analyzng morphology of the human knee Gabor Renner Computer and Automaton Research Insttute Hungaran Academy of Scences Budapest, Hungary Abstract Medcal mages (X-ray, CT, MR or PET) of human organs are wdely used n the every day clncal praxs. A deeper nsght of the anatomcal propertes can be obtaned by buldng a three dmensonal computer model from the mages. In order to properly evaluate the 2D mages and also the 3D objects physcans need effcent and robust graphcal and geometrcal tools. The paper presents methods that were specfcally developed to nvestgate the morphology and functonalty of the human knee jont. These are manly contour detecton n mages, reconstructon of tessellated and contnuous surface models, geometrcal calculatons n mages and n space. Keywords: Bologcal models, mage analyss, reconstructon 1. INTRODUCTION The knee jont s a very mportant component of the human moton system; t s delcate yet frequently damaged. A wde range of medcal magng technques (MR, CT) s avalable to the clncals to nvestgate the dfferent parts of the knee jont. By makng vsble the shape and morphology of some of the jont dseases these methods offer nvaluable support to the practcng physcan and also to the orthopedc surgeon. The same tools can be appled to the analyss of the knematc behavor as well as the morphology of the artcular cartlage. CT and MR mages are usually evaluated by a human evaluator n the medcal praxs. In case of the human knee, however, ths can be extremely dffcult due to the smlar gray-scale representaton of the synoval flud between the opposte cartlage surfaces of femur and tba, and the partly coverng surfaces. Effcent and senstve computer methods for contour detecton and segmentaton can mprove the qualty of evaluaton. Accurate computaton of dstances, drectons, angles wthn the jont provde sold bass for orthopedc handlng and surgery. Many mportant propertes of the knee jont can only be evaluated n three dmensons. The bone and cartlage surfaces have complcated shapes n 3D, and also the moton takes place n 3D. Consequently, a 3D computer model must be bult, usng nformaton extracted from the 2D mages. Beyond the usual 3D manpulatons (3D transformatons), specfc geometrc tools are needed such as 3D computatons and nterrogatons, correspondng to the structure of the knee and the relevant questons to be answered. Currently avalable software tools (as part of CT/MR magng devces or general purpose systems e.g. 3D SLICER [9]) satsfy needs of the everyday clncal praxs; however, they are not approprate for detaled scentfc nvestgaton of the knee jont and for creatng a computer navgaton system for knee surgery We developed a complete system to nvestgate the human knee jont by computer tools. It conssts of nput, storng and handlng of CT/MR mages of the knee, methods and programs for mage analyss, 3D surface reconstructon, and graphcal and geometrcal tools for the detaled analyss of the shape and moton of the knee. Most mportant elements of the system are reported n the paper. Methods of mage processng, geometrcal modelng and surface reconstructon mplemented n our system are based on fundamental technques of the related felds. Even so, they have to be adjusted and modfed accordng to the specfc features of knee mages and geometry. Also new tools for the precse evaluaton of the geometry have to be added. Integratng of these methods and technques nto one system s also an mportant achevement. 2. ANALYSIS OF 2D IMAGES CT and/or MR mage sequences of the knee are read n DICOM format A computer program was bult for vsualzng the acqured data of the scans, usng commercal programs (IDE, Integrated Developng Envronment by MetroWerks Inc, Houston, USA, Volume Graphcs, GmBH [1]), assembled and modfed accordng to the requrements of the knee nvestgatons. The mage sequences are stored on a 3D grd as a volumetrc model, allowng creatng pctures n the man anatomcal drectons (sagtal, coronal, horzontal). Dfferent mage analyss and graphcal tools have been developed to analyze the morphology of the knee. 2.1 Graphcal tools Study of the anatomcal structure of the knee requres specfc tools beyond the usual graphcal technques (renderng, shft, rotaton, scalng, zoom, etc.) provded by commercal graphcal systems. For the clncals t s mportant to be able to localze ponts exactly n the space and to take measurements n 3D. We added the followng graphcal tool to the basc servces of the above system. - Fast localzaton of anatomcal ponts n three dmensons, ndependently of whch slce s selected currently. The cursor s located and dsplayed at the ntercept pont n the anatomcal planes, whch correspond to the three coordnate planes. - Dstance measurement n 3D. The dstance between a fxed and a movable pont s dsplayed and calculated n 3D. To perform dstance calculatons precsely a resamplng of the

nput data s needed, whch s synchronzed to the locaton of the end ponts - Angle measurement. Angle between two ntersectng straght lnes defned by three ponts arbtrarly selected n the space s calculated.

The end ponts of the straght lnes (measurng trangle) can be localzed n any selected slce by movng the projecton plane to the desred slce.

2 nput data s needed, whch s synchronzed to the locaton of the end ponts - Angle measurement. Angle between two ntersectng straght lnes defned by three ponts arbtrarly selected n the space s calculated. In Fgure 1. the orentaton and the angle between movng knee parts (tba and femur) are shown n two projectons. The end ponts of the straght lnes (measurng trangle) can be localzed n any selected slce by movng the projecton plane to the desred slce. The fgure shows the knee n two orthogonal planes, the sagtal and coronal plane, frequently used n orthopedc nvestgatons. The actual cursor poston s lsted on the upper left corner (n mm) and the actual spatal angle (n degrees) on the upper rght corner. One sde of the measurng trangle crosses the contact pont between the tba and femur. Fgure 1. Measurng angle on the MR mage. 2.2 Contour detecton Segmentaton of contours n mages s based on the fact that regons correspondng to dfferent anatomcal structures are represented wth dfferent ntensty values. Contour detecton algorthms detect the boundares of that regons.e. lnes, where the ntensty gradent has a great value. Actve contour detecton s bascally a marchng process, whch proceeds along ponts wth hgh ntensty gradents n the mage. To mprove the qualty of the contour shape nformaton of the anatomcal structures must be taken nto consderaton. In addton to the gradents, we ncluded geometrcal propertes (contnuty, smoothness) nto the quantty to be optmzed along the contour. The optmzaton s an teratve process. The segmentaton process works n a slce-by-slce manner. In each slce t always starts wth an exstng contour, whch s extracted from the prevous slce. The devaton s measured and mnmzed, n order to mantan smooth transton between consecutve contours. Contour detecton s bascally an automatc process. It starts wth nteractve defnton of a rough contour n the frst slce, than automatcally generates contours n the next slces. All contours appear n ther correct spatal poston, provdng a tool for the medcal nspecton of the contours, and also for the creaton of a complete surface (bone or cartlage) n 3D, descrbed by the contours. Detals of the actve contour method are dscussed n [2], an example of the detected bone contours are gven n Fg. 1. Automatc contour detecton woks relably and effcently n many cases, however n certan cases manual delneaton of the cartlage seemed to be superor to fully automated segmentaton. Medcal Fgure 2. Automatc detecton of bone contours mages regularly contan artfacts, sometme n qute consderable amount, whch produce blurred pctures n consequence of the techncal mperfecton of the equpments, or of the spontaneous moton of the persons; etc. Eventually, when the nose n the MR mages makes a blurred pcture or the synoval flud lmts correct evaluaton of the jont space, vsual control s needed by an experenced physcan. The effcency of hs/her work can be greatly ncreased by sutable computer tools. A fast and flexble way of extractng contours manually from mages and to defne them n an nteractve way has been developed. The physcan has to pck a few characterstc ponts on the mages to establsh the shape of the contours. The ponts are mmedately stored n three coordnates. For dsplayng contours wth good qualty, and also for further computer processng contnuous contour curves are needed. It s expected, that the shape of the contour curve corresponds exactly to the expectatons of the physcan, whch s reflected n the sequence of the specfed contour ponts. In computer graphcs and geometry several methods are known for generatng contnuous curves from dscrete data ponts [3], [4]. One of the most popular, effcent and robust methods are splne curves. A parametrc splne curve s defned by the expresson c( t) = C N n = 0 k ( t) where the curve shape s defned by the C (=0..n) 3D vectors, (so called control ponts), and the N k are polynomal bass () t functons, t s the curve parameter. There s a bult n contnuty (smoothness) n the curve, whch depends on the degree of the curve k. The vector coeffcents C, can be computed wth the condton that the curve passes through,.e. nterpolates the data ponts, specfed by the observer [4]. The above type of nterpolatng splne curve performs well n many graphcal problems; however, t may produce unexpected shapes wth undulatons and waves n medcal applcatons. The reason s that medcal mages always contan nose, and the observer ntroduces subjectve errors. The contnuous curve wll reflect (nterpolate) all nose and error, moreover contnuty condton may even amplfy them along the curve. The dffculty can be elmnated by applyng another condton for defnng the shape parameters C of the curve. If we do not requre that the curve passes exactly through the prescrbed ponts (approxmatng splne), we can gan extra freedom to elmnate the effect of random errors on the curve shape. Of course, devatons between

Mathematcal analyss of the problem shows that mnmzaton of the above expresson leads to a system of lnear equatons for the unknown curve parameters C, whch can be solved by stable and effcent

3 the curve and the data ponts must be carefully controlled. The mechansm of curve constructon s as follows. Let defne the shape parameters C of the curve by mnmzng the quantty of n ( c( t) ) F( c ) = P = 1 2 whch measures the (squared) dstance of the data ponts P from the curve c(t). Mathematcal analyss of the problem shows that mnmzaton of the above expresson leads to a system of lnear equatons for the unknown curve parameters C, whch can be solved by stable and effcent algorthms [5]. If we assume that there s an error wth normal (Gaussan) dstrbuton n the data (whch s a farly good assumpton for random errors), then the contnuous curve wll reproduce the shape wthout errors. Our experment shows, that ths result can usually be acheved by an average devaton less than 0.09 mm, and a maxmum devaton less then 0.2 mm when reconstructng contours n MR mages of the knee. The above method provdes contnuous and smooth curve n good qualty for nosy data ponts. Even more mportant that usng approxmatng splne geometrc quanttes relevant for knee analyss can be computed n a more stable and robust way compared to the nterpolatng splne. Table 1. gves rad of condyles (left and rght n mm) n sagtal drectons, computed from approxmatng and nterpolatng splnes n dfferent flexon angles (degree) of the femur. It s clear, that real values are obtaned by the approxmatng splne. Degree Left Rght Approx. Interp. Approx. Interp. 0 50,98 45,03 44,26 33, ,74 17,15 38,02 25, ,64 7,40 36,47 11, ,94 23,55 27,78 7, ,85 12,62 24,71 20, ,04 8,25 25,49 19, ,61 46,72 27,44 40,18 Table 1. Rad of condyles The same method was used whenever contnuous curves have to be generated for a sequence of dscrete data ponts (e.g. contact curves n the knee). 3. RECONSTRUCTION OF KNEE SURFACES Many anatomcal propertes of the knee can be evaluated n MR mages that are plane ntersectons of the spatal structure. For detaled nvestgatons, however 3D modelng and vsualzaton of bone and cartlage surfaces are needed. We have developed three types of surface reconstructons; sosurface, trangulated surfaces and contnuous surfaces. Isosurface reconstructon s based on voxel ntensty data segmented by an adjustable gray-scale threshold [6]. The sosurface model represents the dentcal gray level ponts. Before vsualzaton, the program automatcally completes the segmentaton based on ntensty threshold. Standard methods for modelng surface materal, lghtng, llumnaton and transparency are used. The basc purpose s to vsualze surfaces, but graphcal objects lke ponts, markers, paths and polygon meshes can be added to a 3D scene. Surface ponts extracted from the sosurface model represent only rough nformaton on the shape. The advantage of the sosurface representaton s that the physcan gets an dea of 3D morphology developed from planar mages. For exact demonstraton and numercal representaton of 3D parameters, however, sosurfaces are not suffcent.. Contnuous representaton of anatomcal surfaces s often needed for detaled nvestgaton of ther shape and geometry. Trangulated representaton of the knee n the medcal practce s generally suffcent but to delneate fne detals and for moton analyss hgh degree, smooth surfaces are needed. Relevant geometrcal propertes (normal vectors, tangent planes, curvatures, plane ntersectons, etc.) can be computed for both surface representatons, but the accuracy s consderably dfferent. A trangulated surface conssts of very small trangles between data ponts, whch are consecutvely connected to each other. The data ponts come from MRI scans, as the ponts of the correspondng contours. The sequence of contour ponts s postoned n 3D accordng to the spatal poston of a slce contanng the contour. In case of cadaver studes, data ponts can be collected by a (laser) scannng process. A good trangulaton must meet several requrements: t must be topologcally correct (no holes and flyng trangles appear n the trangulaton), must elmnate outler ponts, trangles must have comparable sde lengths and angles, ther sze must reflect the curvatures of the surface, etc. We have developed algorthms and computer programs to trangulate and decmate pont sets of anatomcal structures, whch are able to handle mperfectons comng from contour detecton and measurements [7]. If necessary, topologcal correctons are performed, or the trangulaton s decmated to reduce the sze of the data set. Fgure 3. shows the surface of the tba and femur as trangulated surfaces. Although t looks qute smooth, t s not tangental contnuous because of trangulaton. Fgure 3. Trangulated surfaces of tba and femur Our experments show that accurate and smooth representatons of the functonal surfaces of the knee are needed for detaled bologcal or medcal nvestgatons (especally for studyng the sze and shape of the contactng regons of cartlages and moton of the knee).

Input data for creatng smooth surfaces s a pont set, comng from contours of MR mages, or from scannng. The pont set must be fltered aganst nose and decmated.

Trangulaton provdes neghborhood relatons, whch s mportant for fttng surfaces and for geometrc calculatons. Because we ft parametrc surfaces commonly used n computer graphcs and CAD (e.g. Bézer, B-splne, NURBS) over the pont set, parameter values must be attached to the data ponts.

The shape defnng data of the surface are computed by mnmzng a functonal: F S( N 2 2 2 2 ( Cj ) = ( S( uk, vk ) Pk ) + λ ( Suu + 2Suv + Svv ) k= 1 S u, v) = Bj ( u, v) Cj, j dudv where the contnuous

The functonal contans two terms; one s the squared dstances of data ponts to the surface ponts wth the same parameter value, and reflects accuracy of the surface.

4 Input data for creatng smooth surfaces s a pont set, comng from contours of MR mages, or from scannng. The pont set must be fltered aganst nose and decmated. Contnuous surface ft starts wth topologcal orderng of ponts, whch s usually done by creatng a trangulaton over the surface ponts. Trangulaton provdes neghborhood relatons, whch s mportant for fttng surfaces and for geometrc calculatons. Because we ft parametrc surfaces commonly used n computer graphcs and CAD (e.g. Bézer, B-splne, NURBS) over the pont set, parameter values must be attached to the data ponts. Ths s performed by projectng the data ponts to a smple and rough approxmatng surface, whch s usually the bcubc Coons surface defned by the boundary ponts of the pont set [4]. The shape defnng data of the surface are computed by mnmzng a functonal: F S( N ( Cj ) = ( S( uk, vk ) Pk ) + λ ( Suu + 2Suv + Svv ) k= 1 S u, v) = Bj ( u, v) Cj, j dudv where the contnuous surface S(u,v) s ftted to the N number of data ponts P k.. The surface s defned by the bass functons B j (u.v) and control ponts as shape parameters C j. The functonal contans two terms; one s the squared dstances of data ponts to the surface ponts wth the same parameter value, and reflects accuracy of the surface. The other descrbes some geometrcal quantty responsble for smoothness, here the approxmaton of the surface curvatures (lower ndces denote dervatves). Parameter λ (set by the user) defnes the rato between accuracy and smoothness. Mnmzaton of the functonal defnes control ponts of the surface and thus ts unque mathematcal representaton. We have developed methods for parametrzng pont sets, solvng the functonal mnmzaton problem, and mprovng parameterzaton durng the fttng process; the detals can be found n [7]. In Fgure 4. actve surfaces of femur and tba (where moton takes place) are shown, together wth the underlyng pont sets. 4. SHAPE ANALYSIS We have developed several methods and programs, to evaluate the shape and morphology of knee surfaces. Varous geometrcal propertes such as tangent planes, normal vectors, extreme ponts, lnes of ntersectons, dstrbuton of mean and Gaussan curvatures, feature ponts, characterstc lnes, etc. can be calculated, vsualzed and evaluated. As an example plane ntersectons of the tba plateau perpendcular to ts axs s shown n Fgure. 5. Fgure 5. Plane ntersectons of the tba plateau From pathologcal and clncal pont of vew t s of basc mportance to have precse measures of the thckness of the cartlage layer, coverng the bone surfaces and characterze ts spatal dstrbuton. When bone and cartlage surfaces are precsely reconstructed, theoretcally t would be possble to calculate the cartlage layer between them. The problem s that ths layer s very thn, (1-2 mm, lght layers around bones n Fgure 1.), and naccuracy n surface reconstructon may destroy the geometry of the cartlage layer. In case of cadaver studes correct postonng n space of the surfaces (regstraton) s also a crucal pont. After bones wth and wthout cartlage are scanned ndependently, the pont clouds must be merged. To perform ths, surface part not covered by cartlage can be used, because they have an dentcal shape. For regstraton we appled a modfed verson of the ICP (teratve closest pont) algorthm [8]. Usng the above surface reconstructon, the ICP regstraton, wth a careful adjustment of ther parameters a precse reconstructon of the cartlage layer can be acheved. In Fgure 6. plane ntersectons of the cartlage layer are shown. It fathfully reflects fne varatons of the thckness expected from the anatomy of the knee. Fgure 4. Actve surfaces of femur and tba Fgure. 6. Cartlage layer on the tba plateau Contact propertes of cartlage surfaces (ther shape and extenson) can be best evaluated by a dstance functon defned between the two surfaces. Ths s the length of the shortest path from a surface pont to the other surface. We have developed a procedure to evaluate the dstance functon for the contnuous surface models of the cartlages. Surfaces can be color coded

5 accordng to the dstance functon, whch provdes an extremely effcent tool for evaluatng the shape and sze of the contact regons. Fgure 7. shows the color coded dstances between for the contactng cartlage surfaces of femur and tba. Our am n the future s to develop a complete computer aded knee surgery and navgatonal system based on the graphc and geometrc components dscussed above. 6. ACKNOWLEDGEMENT The system developed for knee nvestgaton s a jont work of a group of researchers and programmers. The work of Ferenc Pongracz, Dmtr Chetverkov, Levente Hajder, Laszlo Szobonya s gratefully acknowledged. The wok was supported by the Hungaran Government under the project Stereo tactc and navgatonal operatve management of the knee based on threedmensonal geometrc models, NKFP-1B/0009. Fgure 7. Color coded dstance functon of cartlage surfaces 5. CONCLUSION Analyss of the geometrcal propertes of the knee s mportant from many ponts of vew. Based on shape nformaton clncals can draw conclusons on the healthy and pathologcal state of the knee. Surgeons can desgn surgcal nterventon usng geometrcal data of the knee. Better understandng of the morphology and functonalty of the knee may lead to better than exstng prostheses. Accurate geometrcal nformaton facltates preoperatve desgn of knee surgery and computer control durng surgery. In Computer and Automaton Research Insttute, Budapest, Hungary effcent and robust graphcal and geometrcal tools were developed to analyze MR and CT mages, to perform geometrcal calculatons n 2D, to reconstruct medcal/bologcal surfaces from mages and scans, to regstrate and merge them and to evaluate them n 3D. Although the methods and programs were developed to satsfy specfc ams and requrements of knee studes, many elements can be effcently used to nvestgate smlar bologcal structures. 7. REFERENCES [1] VGL Onlne Manual. Documentaton for VGL 3.1(Volume Graphcs GmbH, Hedelberg, 2001). [2] Hajder L, Kardos I, Csetverkov D, Renner G.: Actve contour and fast marchng methods n medcal mage processng, Proc. of IV. Conf. on Image processng and pattern recognton, Mskolc-Tapolca, Hungary, 2004, pp [3] Rogers DF, Adams JA. Mathematcal Elements for Computer Graphcs. Second Edton, McGraw Hll, Boston, SanFrancsco, [4] Hoschek, J., Lasser, D.: Fundamentals of Computer Aded Geometrc Desgn, A. K. Peters, [5] Press, W., Flannery, B., Teukolsky, S., Vetterlng, W.T.: Numercal Recpes n C, Cambrdge Unv. Press, [6] J. Foley, A.van Dam, S.K. Fener and J.F. Hughes. Computer Graphcs: Prncples and Practce. Addson-Wesley, [7] Wess, V., Andor, L., Renner, G., Varady. T.: Advanced surface fttng technques, Computer Aded Desgn, 19, 2002, pp [8] D. Chetverkov, D. Stepanov, P. Krsek: Robust Eucldean algnement of 3D pont sets: the Trmmed ICP algorthm, Image and Vson Computng 23 (2005) [9] 3D SLICER,

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