GEOMETRY RECONSTRUCTION AND MESH GENERATION USING VARIATIONAL IMPLICIT SURFACES

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1 GEOMETRY RECONSTRUCTION AND MESH GENERATION USING VARIATIONAL IMPLICIT SURFACES S. Gordana *, C. Grffth *, J. Peró *, S. Sherwn * and D. J. Doorly * 1. ABSTRACT Ths paper presents a technque to recover surfaces from Magnetc Resonance Imagng (MRI) and generate volume meshes sutable for Computatonal Flud Dynamcs (CFD). The ultmate am of ths study s the reconstructon of geometry and flow envronment from n-vvo patent data usng MRI, Doppler ultrasound and CFD, partcularly at the dstal anastomoss of arteral perpheral bypass grafts. We apply computer vson technques to automatcally extract medal lnes and ther angles at bfurcatons as a way of characterzng geometry. 2. INTRODUCTION Perpheral arteral dsease descrbes problems that are caused by the obstructon to blood flow n arteres other than the coronary and ntracranal vessels [1]. Specfcally, ths work s focused on chronc arteral dsease n the arteres to the legs and our studes are lmted to patents who requre revascularsaton to avod amputaton. Surgcal revascularsaton s performed by graftng an autologus ven to the dseased artery, reestablshng flow downstream the occluson. Perpheral by-pass grafts present a hgh falure rate at ther dstal anastomoss n the form of hyperplasa or thromboss. There s now clear evdence that such problems are related to local haemodynamcs factors [2]. Geometry determnes flow nsde dstal anastomoses and surgeons desgn an anastomoss whle performng revascularsaton. Studyng dfferent geometrcal confguratons taken from n-vvo measurements and relatng relevant blood flow features to the progress of the dsease may actually gude surgcal practce to determne the most approprate anastomoss confguraton. Keywords: varatonal nterpolaton, mesh generaton, skeletonzaton. * Bomedcal Flow Group Department of Aeronautcs, Imperal College, Prnce Consort Road, London SW7 2BY, UK.

2 To accomplsh ths we requre the ablty to perform CFD nsde anatomcally correct geometres and to correlate CFD results wth relevant geometrc features that surgeons can control. The technques presented n ths paper address reconstructng a CFD mesh from MRI mages and extractng nformaton from the reconstructon. The choce of the tools s geared towards obtanng maxmum automaton of geometrc reconstructon and analyss for reasons of productvty and coherence. Productvty s mportant when a large number of patents are consdered for a study. Coherence s better acheved when user subjectvty s kept to a mnmum. In secton 3 we ntroduce a varatonal approach to surface fttng, followng Turk and O Bren [3], and we present how t can be nterfaced wth our spectral/hp element Naver-Stokes solver. In secton 4 we dscuss some ssues concernng the accuracy of the reconstructon. In secton 5 we outlne the current development of the morphometrc analyss of n-vvo data usng computer vson technques 3. GEOMETRY RECONSTRUCTION MRI produces a set of data consstng of two-dmensonal mages. Each mage s a matrx of pxels coloured n 256 levels of grey. Areas of blood flow appear brghter than surroundng tssue and ther contours can be solated by automatc thresholdng. Ths s accomplshed by means of the freeware package Image ( Contours are jagged lnes, snce they follow pxels patterns, and need to be smoothed by least-squares splne nterpolaton [4]. Ths process s schematcally depcted n fgure 1 where the sequence of nterpolated contours, n fgure 1(c), provdes a sketch of the shape. Fgure 1. Image processng: (a) MRI mage wth area of blood flow marked by crcle (b) Least-squares nterpolaton (c) Sequence of nterpolated contours. The lmted resoluton of MRI results n mssng nformaton at the bfurcaton where geometry changes ts topology from one to two or more branches. Changes of topology are naturally treated by mplct surfaces [3] that defne the geometry as a constant level set of a functon f (x), where x = ( x, y, z) denotes the Cartesan coordnates. Here we seek a functon that assumes a value of zero at a dscrete set of ponts x ; = 1, K, N evaluated on the smooth planar splnes. Followng the approach proposed n [3], we choose f as a lnear combnaton of radal bass functons φ assocated wth the N constrants x as N f ( x) = c φ ( x x ). (1) = 1

3 The radal bass functons depend on the dstance between a pont x and the constrant pont x. The evaluaton of the N coeffcents c requres the soluton of a lnear system Ac = h (2) wth A = φ ( x x ) and h = 0. Snce c = 0 s a soluton of (2), extra nformaton s j j requred to close the problem. To ths effect, we also request that h = 1 at a second set of nteror ponts x ; = N +1, K, 2N generated by dsplacng the frst set of ponts along the normal to the boundary. Smoothness of the nterpolated surface depends on the choce of the radal bass functon φ. Here we use the famly proposed by Duchon [5] that mnmzes the deformaton energy of a thn-plate nterpolaton through the set of ponts and s gven by N = 1 3 f ( x) = c x x. (3) The geometry of the vessel can then be reconstructed by extractng the zero level set of f usng an mplct functon polygonzer [6] (fgure 2). Ths method produces trangular meshes that are good enough for computer graphcs. However mesh enhancement operatons are requred to produce a surface trangulaton sutable for generaton of a volume mesh adequate for CFD. A flow soluton, gven boundary condtons from Doppler ultrasound and provded the geometry s adequately extended, can be obtaned by the fnte volume method (fgure 3a). The accurate calculaton of wall shear stress s very senstve to mesh resoluton and hgh-order p-type fnte element technques offer the potental of hgher accuracy per computatonal effort compared to methods based on lnear approxmatons [7]. Graft Proxmal Dstal Fgure 2. Trangulaton of the zero level set of the mplct functon: (a) Full surface. (b) Detal of polygonzer mesh. (c) Detal of enhanced qualty mesh. Our hgh-order mesh generator bulds a boundary-conformng unstructured mesh of hgh-order spectral/hp elements by subdvdng an ntal coarse mesh of lnear elements. The method s descrbed n full detal n [8]. At present our mesh generator needs a parametrc surface representaton as startng nput. Faced wth the non-trval problem of parametrzng an mplct surface, we adopt a sem-automatc procedure where the user nteractvely draws splnes curves on the sosurface trangulaton and specfes a connectvty to buld splne patches. Fgure 3b shows that patches obtaned n ths way are close to the so-surface but they do not cover t exactly and do not show more than poston contnuty through boundares. Therefore, we project these patches on the so-surface usng an algorthm by Hartmann [9]. Ths provdes a correct parametrzaton and acheves a better degree of contnuty across patches. The hgh-order mesh generator produces a surface mesh whose spacng s a functon of

4 Fgure 3. (a) Wall shear stress [Pa] computed by the fnte volume method (FLUENT commercal package, Re 400, occluded proxmal vessel). (b) Snapshot of the patches drawng procedure. (c) Projected patches. the surface curvature. A vscous layer of prsms and tetrahedral elements consttute the volume mesh. Flow solutons are computed wth a spectral/hp fnte element solver [7]. The outlned reconstructon and mesh generaton procedure s completely automatc but stll requres the user to draw patches. Further work can proceed n two drectons: ether to automate the surface parametrzaton or to modfy the hgh-order mesh generator to take a dfferent ntal geometry nput. Fgure 4. (a) Hgh-order mesh. (b) Detal of mesh at nflow. (c) Wall shear stress [Pa] computed by spectral/hp solver (steady flow, Re = 400, occluded proxmal vessel). 4. RECONSTRUCTION ACCURACY We started our error quantfcaton nvestgaton by tryng to reconstruct a known shape. We scanned two plastc tubes joned at an angle of 45 degrees. The shape acqured after segmentaton, mplct functon fttng and so-surface extracton s depcted n fgure 5(a). The reconstructed surface presents a dent near the juncton regon that s not part of the real model. Ths appears to be a feld artefact caused by the magnetc propertes of the glue used to jon the tubes. Furthermore, the two reconstructed tubes do not show unform crcular cross-sectons. A rapd and automatc way of elmnatng these artefacts s treatng the mplct functon as a 3D mage and applyng a Gaussan flter. We do not allow the smoothed and the orgnal surfaces to be further apart than one pxel. Ths s consstent wth the MRI resoluton.

5 Fgure 5. Smoothng: (a) Reconstructon of 2 tubes showng a dent at the juncton. (b) Surface after maxmum allowed smoothng. (c) Dstance map [pxels] between smoothed and orgnal surfaces. 5. GEOMETRIC FEATURES EXTRACTION CFD calculatons n anatomcally correct geometres must be accompaned wth dentfcaton of geometrc effects on flow to provde surgeons wth feedback on the anastomoss desgn. For example, surgeons are concerned wth the angle between the grafted ven and the host artery. Snce out-of-plane curvature strongly affects flow [10], some quantfcaton of planarty must also be found. It s mandatory to defne a centre lne of each anastomoss. We adopt a fast three-dmensonal sx subteratons thnnng algorthm to extract the skeleton (medal lne) of reconstructed geometres developed by Palagy and Kuba [11]. The method takes a 3D bnary mage as nput, the space s then dscretzed nto voxels and each voxel s gven an ntensty value of 1 f t s nsde the blood vessel or 0 elsewhere. The algorthm teratvely thns the object by removng voxels on the external surface untl an approxmaton of the medal lne remans, as shown n fgure 6(a). Snce the thnnng algorthm requres each voxel to know whether ts 26 neghbours are part of the background, neghbourhood relatons are bult and updated durng thnnng so that the fnal medal lne s made up of ponts and ther connectvty. It s then natural to extract the juncton pont and the branches n an automatc fashon. Interpolatng the branches wth least squares straght lnes we defne a reference plane contanng the graft and the proxmal host vessel. Angles are measured between graft and proxmal, planar component of the dstal host vessel and graft, dstal host vessel and normal to the plane. Angles change accordng to how much nformaton s used n the nterpolaton n terms of how far nterpolated ponts are from the bfurcaton, fgure 6(c). From the varaton of the angles we deduct that the dstal host vessel possesses a consderable amount of out-of-plane curvature, whle the graft s approxmately straght. Ths smple quantfcaton can be appled n a statstcal fashon to large data sets and hopefully be correlated wth CFD results. Further work s amed at expandng the parameter space by fttng hgher-order curves to the medal lne, for example a helx. Ths wll permt to assocate a value of torson to each branch. We are also nvestgatng effcent and accurate methods for calculatng cross-sectonal areas along the medal lne. By changng the shape of the medal lne and mantanng the same area dstrbuton, we could perform parametrc studes to nvestgate the effects of vessel curvature and torson on flow. 6. CONCLUSIONS We have descrbed a sem-automatc varatonal reconstructon of geometry from MRI data. The reconstructed surfaces are used as nput to CFD solvers and geometrc analyss tools. The ablty to assocate flow features wth parameters representatve of

6 the vessel geometry s fundamental to perform parametrc studes that mght help surgeons understand how the desgn of an anastomoss nfluences ts patency. β α γ Fgure 6. (a) Medal lnes. (b) Angles used for classfcaton. (c) Dependence of angles on the dstance of nterpolated ponts from the bfurcaton, expressed n graft dameters. 7. ACKNOWLEDGEMENTS Ths work has benefted from the fnancal support from the Henry Smth s Kensngton Estate Charty. The Imperal College centres of Bomedcal Vsualzaton and Parallel Computng provded computatonal resources. We also would lke to thank the staff at London St. Mary s hosptal nvolved n ths study, coordnated by Prof. C. Caro and to Y. Papaharlaou for performng the MRI scans. 8. REFERENCES 1. Ourel, K., Perpheral arteral dsease, The Lancet, 358, 2001, Wootton, D. M. and Ku, D., Flud mechancs of vascular systems, dseases and thromboss, Ann. Rev. Bomed. Eng., 1, 1999, Turk, G. and O Bren, J. F., Shape transformaton usng varatonal mplct surfaces, n Proc. SIGGRAPH99, 1999, Derckx, P., Curves and surface fttng usng splnes, Oxford Scence Press, Duchon, J., Splne mnmzng rotaton-nvarant sem-norms n Sobolev spaces, n Constructve Theory of Functons of Several Varables, Lecture Notes on Mathematcs, 571, 1977, Bloomenthal, J., An mplct surface polygonzer, n Graphcs Gems IV, Academc Press, 1994, Karnadaks, G. E. and Sherwn, S. J., Spectral/hp element methods for CFD, Oxford Unversty Press, Sherwn, S. J. and Peró, J., Mesh generaton n curvlnear domans usng hghorder elements, Int. J. Numer. Meth. Engng., 53, 2002, Hartmann, E., Numercal parameterzaton of curves and surfaces, Computer Aded Geometrc Desgn, 17, 3, 2000, Sherwn, S. J., Shah, O., Doorly, D. J., Peró, J., Papaharlaou, Y., Watkns, N., Caro, C. G. and Dumouln, C. L., The nfluence of out-of-plane geometry on the flow wthn a dstal end-to-sde anastomoss, ASME J. Bomech., 122, 2000, Palagy, K. and Kuba, A., A 3D 6-subteraton thnnng algorthm for extractng medal lnes, Pattern Recognton Letters, 19, 7, May 1998.