an improved scheme of an interactive finite element model for 3d soft-tissue cutting and deformation

Visual Comput (2005) 21: 707–716DOI 10.1007/s00371-005-0310-6 O R I G I N A L A R T I C L E

Wen WuPheng Ann Heng

An improved scheme of an interactive finiteelement model for 3D soft-tissue cutting anddeformation

Published online: 24 August 2005© Springer-Verlag 2005

W. Wu (�) · P.A. HengDepartment of Computer Science andEngineering,The Chinese University of Hong Kong,Shatin, Hong Kong{wwu1, pheng}@cse.cuhk.edu.hk

Abstract As a safe and feasible al-ternative to enriching and enhancingtraditional surgical training, virtual-reality-based surgical simulators havebeen investigated for a long time. Butit is still a challenge for researchersto accurately depict the behaviorof human tissue without losing theflexibility of simulation. In this paper,we propose an improved scheme ofan interactive finite element modelfor simulating the surgical process oforgan deformation, cutting, dragging,and poking, which can maximallycompromise the flexibility and realityof soft-tissue models. The schemeis based on our hybrid condensedfinite element model for surgicalsimulation, which consists of theoperational region and nonopera-tional region. Different optimizingmethods applied to these regions

make a contribution to the speedupof the calculation. Considering ina real surgical operation, dragging orpoking operations are also necessaryfor surgeons to examine surroundingtissues of the pathological focus.The calculation within the areanewly applied with forces in thenonoperational region is handledin our new scheme. The algorithmis modified accordingly in order tocope with this aspect. The designand implementation of the approachare presented. Finally, we providetwo models to test our scheme. Theresults are analyzed and discussed toshow the efficiency of our scheme.

Keywords Finite element model ·Condensation · Graphics processingunit · Surgical simulation · Soft-tissuedeformation

1 Introduction

Minimally invasive microsurgery, which offers patientsvarious attractive advantages over traditional surgery, hasbeen widely used in otolaryngology, gastroenterology,gynecology, and neurology in the last two decades. Itshifts surgeons from the situation of the open opera-tion to indirect touch-and-watch. Such a new operatingstyle has imposed extra challenges on surgeons. First, themotion of instruments is restricted in a confined spacebecause they are inserted through a small incision. Sec-ond, the original 3D visual perception of the operationsite is lost since a 2D video is displayed on screen in-

stead. Currently, junior surgeons can practice their skillby performing the procedure under the supervision ofan experienced surgeon on animals, cadavers, and, ul-timately, on actual patients. Studies show that a novicesurgeon usually performs significantly worse on the firstprocedure [10]. Moreover, practising how to handle allkinds of complications and variations that arise duringan operation is hard because that would put patientsat an unacceptable risk. Therefore, virtual-reality-basedsurgical simulators are playing a more and more im-portant role in the medical field and has changed theparadigm of traditional surgical training. Surgical sim-ulators have become a safe and feasible alternative forinexperienced surgeons to learn human anatomy and re-

708 W. Wu, P.A. Heng

peatedly practise operation techniques prior to actualsurgery.

This paper presents our current work on developinga more flexible soft-tissue model necessary for surgicalsimulation. In the next section, we review different modelsused for representing deformable objects in surgical ap-plications. In Sects. 3 and 4, we describe our improvedscheme in detail. Section 3 introduces the structure andthe speedup methods of our model after a brief review ofthe theory of linear elasticity and the finite element model(FEM). Based on the contents of Sect. 3, Sect. 4 presentsthe method for handling the interaction newly applied tothe nonoperational region. This removes the limitation ofour previous model, that no interactive actions can bemanipulated in the nonoperational region. In Sect. 5, weimplement our proposed scheme and test it on two softorgans. The analysis and discussions are given with a con-clusion.

2 Related work

The methods of object deformation can be roughly dividedinto nonphysically based models and physically basedmodels. Early work on nonphysically based models fo-cused on pure geometrical models that were originallyemployed in computer-aided design (CAD). For example,the methods in [7, 15], extended from free-form deforma-tion (FFD), were utilized to generate user-desired defor-mation of objects intuitively in computer animation. Themethod proposed in [16] implemented facial animationdirectly on parameter surfaces. These methods could beused to produce vivid deformable effects in computer an-imation. However, the soft-tissue deformation related tosurgical applications requires models that reveal the ac-curate physical principles of human tissues. As a result,the mass–spring model and the finite element model havebecome the most popular representations for deformableorgans in surgical simulations.

The mass–spring model is a system comprising a set ofsprings connected to a large number of mass points. Thesystem imposes external forces and boundary conditions.Mass points in the system are exerted with mutual springforce. A system of second-order differential equations isderived to depict the final deformation of the object. Earlywork on the mass-spring model of human objects couldbe found in facial animation [18, 19]. Due to its simpli-city and low computation cost for interactive applications,extensive investigation has been conducted to model hu-man tissues with this model in various surgical simula-tors [11, 14, 17].

The finite element method is another choice for re-searchers to construct physically accurate models appliedin surgical simulation. By taking advantage of more pre-cise discretization and different constitutive equations, fi-nite element models are potentially accurate from a phys-

ical point of view. But the cost of accuracy trades offwith a large computing time, which imposes a big chal-lenge to real-time applications. Therefore, various acceler-ation methods have been proposed in recent years to tacklethe problem. Bro-Nielsen et al. [4] discussed the real-timesimulation of deformable objects using a 3D fast finiteelement model with a condensation technique [13]. Withthis technique, computation time can be reduced by con-fining the computation to the surface nodes of the mesh,making possible a solid deformation of relatively largeobjects in video frame rate. Berkley et al. [3] developeda banded matrix technique for a finite element model inreal-time deformation and force feedback. Cotin et al. [8]presented a deformation method for real-time visual andhaptic feedback by using a preprocessing of elementarydeformations derived from the finite element method. Ex-perimental results of biomechanical properties of humantissue are added to the linear physical model in order toenhance the realism.

Those methods significantly improved the computa-tional speed of simulation systems based on finite elementmodels. Their main limitation is that they are not feasiblefor cutting operations since some preprocessing data par-ticipating in the calculation cannot be updated in real timewhen the topology changes by cutting. Cotin et al. [9] pro-posed a new model that combines the tensor-mass modeland the finite element model to allow a volumetric de-formation and cutting. However, in the model, cutting issimulated by successively removing tetrahedra that the in-strument is in contact with. Meanwhile, in order to keepthe mesh conformal, additional tetrahedra may be auto-matically removed after the adjacency of the local vertexand edge is checked. Therefore, it is hard to keep a precisetrack of cutting operations by this model. Our method re-moves that limitation and tracks the exact path of cuttingoperations without losing the simulation of the realisticdeformation behavior.

3 Physical modeling of soft-tissue deformation

Properties of biological tissues have been studied fora long time. In fact, it is not easy to conduct mechanicaltests of tissues in vivo by which complicated properties oftissues can be intensively investigated. However, with thenewly developed testing methods and equipment, the lat-est experimental results have been much more comprehen-sive. Currently, we know that biological tissues are non-linear, time-dependent, and history-dependent viscoelasticmaterials. It is hard to precisely express the general behav-ior of biological tissues since they exhibit very complexbehavior in terms of a set of parameters. As our first stageon the way to physically accurate simulation, we simplyapproximate tissues as an ideal linear elastic material ac-cording to the data from the linear region of experimentalresults (see Fig. 1 adapted from [1]).

An improved scheme of an interactive finite element model for 3D soft-tissue cutting and deformation 709

Fig. 1a,b. Stress–strain relationship. a Swine flexor and extensortendon stress–strain relationship (adapted from [1]); b Ligamentstress–strain relationship (adapted from [1], which was modifiedfrom [5])

3.1 Linear elasticity

Strain is used to describe the deformation of a physicalbody under the action of applied forces. It is a second-order tensor and can be defined by

E = 1

2(FT F − I) = 1

2(C − I), (1)

where E denotes the Green–Lagrange strain tensor andF and C represent, respectively, the deformation gradi-ent and the right Cauchy–Green deformation tensor. In theCartesian coordinates system, the Green–Lagrange straintensor components can be represented as

Exx = ∂u

∂x+ 1

2

[(∂u

∂x

)2

+(

∂v

∂x

)2

+(

∂w

∂x

)2]

, (2)

Exy = 1

2

(∂u

∂y+ ∂v

∂x

)+ 1

2

[∂u

∂x

∂u

∂y+ ∂v

∂x

∂v

∂y+ ∂w

∂x

∂w

∂y

].

The other four strain tensor components are representedsimilarly. In small deformation and rotation, the second-order portion can be neglected. The Green–Lagrangestrain tensor is approximately equal to the infinitesimalstrain εxx, εyy, εzz, εyz, εzx, εxy used in the linear finite

element analysis

εxx = ∂u

∂x, εyz = 1

2

(∂v

∂z+ ∂w

∂y

),

εyy = ∂v

∂y, εzx = 1

2

(∂w

∂x+ ∂u

∂z

), (3)

εzz = ∂w

∂z, εxy = 1

2

(∂u

∂y+ ∂v

∂x

).

For our future use in the finite element analysis we repre-sent the strain vector ε as the 6-component vector

εT = [εxx εyy εzz εyz εzx εxy

]. (4)

3.2 Finite element method

The finite element method (FEM) is a discrete procedurefor obtaining approximate solutions of many problems en-countered in engineering analysis, such as fluid mechan-ics, heat transfer, solid mechanics, etc. In FEM, a com-plex continuum is discretized into simple and small ge-ometric elements. Properties and governing relationshipsare assumed over these elements. Some nodal points arespecified in an element where the unknown values areexpressed mathematically. The solution is subject to theconstraints at the nodal points and the element boundariesso that the continuity between the elements can be main-tained.

Elements of higher order can improve the calcula-tion precision of the problem, but at the same time theyincrease the computing complexity. In general, we maychoose an element type in a suitable complexity in order toreach a good balance of the accuracy and the computationcost. In our case, a four-node linear tetrahedral element isselected to assemble the soft-tissue models in 3D domains.The displacement variation u = (u, v, w) within an elem-ent can be given by the nodal displacements and the shapefunction as follows:

u = Ne ue

= [INi INj INm INp

] [ue

j uej ue

m uep]T

, (5)

where I is a 3×3 identity matrix and the shape function Nis the linear shape function, which is defined as

Nr = ar +br x + cr y +dr z

6V, r = i , j , m , p , (6)

in which

ai=1 = a1 = (−1)i−1

∣∣∣∣∣xj yj zjxm ym zmxp yp zp

∣∣∣∣∣ ,bi=1 = b1 = (−1)i

∣∣∣∣∣1 yj zj1 ym zm1 yp zp

∣∣∣∣∣ ,


ci=1 = c1 = (−1)i

∣∣∣∣∣xj 1 zjxm 1 zmxp 1 zp

∣∣∣∣∣ ,di=1 = d1 = (−1)i−1

∣∣∣∣∣xj yj 1xm ym 1xp yp 1

∣∣∣∣∣ ,where V represents the volume of the element. The otherconstants are defined by cyclic interchange of the sub-scripts in the order i, j, m, p. From Eq. 1 to Eq. 6, thestrain ε at an arbitrary point in the element can be ex-pressed as

ε = BU , (7)

where B, called the displacement differentiation matrix, isa 6 ×3ne matrix and U is a 3ne ×1 (ne is the number ofnodes in an element) vector of the nodal displacements.

With the assumption of the linear elastic material, thestress σ is linearly related to the strain ε via Young’s Mod-ulus. We obtain the strain energy E as follows:

E = 1

2

∫V

σ TεdV

= 1

2

∫V

εT DεdV

= 1

2

∫V

(BU)T D(BU)dV , (8)

where D is the symmetric elastic matrix that is related tothe material physical properties. The potential energy ofa body is the sum of the total strain energy E and the workW performed on the body by external forces [6]. W can bewritten as

W =∫V

u · f bdV+∫Γ

u · f sdS+∑

i

ui · pi , (9)

where u is the displacement vector, fb are body forcesbeing applied to the object volume dV , fs are surfaceforces being applied to the object surface dS, and pi isthe concentrated load acting at point (xi, yi, zi). In equi-librium, the potential energy gets the minimum. A set oflinear equations can be derived: Keue = f e, where Ke

is the stiffness matrix of a single element. To assembleall of the element stiffness matrices into one using elem-ent connectivity, a large but sparse linear system in thesame form as Ku = f is deduced, where K is the globalstiffness matrix in 3n ×3n, which determines the globalelastic properties of the model, and n is the total num-ber of nodes in the model. Consequently, the problemis transformed into a system of linear equations to besolved.

4 Model detail

4.1 Hybrid finite element model and calculationoptimization

We develop the new scheme from our hybrid FEM pro-posed in [20]. The hybrid FEM consists of two regions, anoperational region and a nonoperational one (Fig. 2).

During a surgical operation, the complex resection willproceed to a local pathological area of the organ. Weassume that the topological change occurs only in theoperational part throughout the whole surgery. Differentmethods treat regions with different properties in orderto balance the computational time and the level of simu-lation realism. The complex FEM, which can deal withtopological change, is used to model the small-scale oper-ational region. A linear and topologically fixed FEM withthe limitation of no interactions in the region, where therunning time can be accelerated by precomputation, isadopted to model the large-scale nonoperational region.Since these two regions are connected with each otherthrough common nodes, additional boundary conditionshave to be imposed on them. The same physical law thatthe two models follow makes them behave as one globallinear elastic FEM.

The cutting in the operational region causes an in-stant topological change in the tetrahedral mesh in thispart. Every running step indicates reconstructing the co-efficient matrix and solving the new equations of the newmesh. Obviously, the time spent in this part directly im-pacts the feedback speed of the entire process. The con-jugate gradient iteration is used to solve the equations.The primary operation during an iteration of the conjugategradient algorithm is the matrix-vector multiplication. Ingeneral, matrix-vector multiplication requires O(m) op-erations, where m is the number of nonzero entries inthe matrix. A faster method for matrix-vector multiplica-tion implies faster computation for the cutting simulation.Therefore, we take advantage of the fragment processorof the GPU in its efficient manipulation of local texturememory on mathematical calculation to migrate it fromthe CPU to the contemporary GPU.

The basic principle involved in general computing onthe GPU is to load matrices and vectors as textures into

Fig. 2. Hybrid model


the GPU and then rasterize a proper quad of pixels forinvoking the fragment program, which performs the ac-tual calculation in the fragment processor within the GPUpipeline. The result produced by the GPU can be obtainedas the color value, transferred directly to the next pass forthe execution or readback to the CPU.

The interior nodes of the nonoperational region can beregarded as redundant nodes for the simulation becausethey are unrelated to any action of the surgeon or even in-visible during the simulation. A condensation technique isapplied to remove those nodes from the solving processin order to release any unnecessary burden of the compu-tation caused by them. As a result, the dimension of thecondensed matrix equations is reduced to that of the FEsurface model whereas the original physical character ofthe volumetric model is preserved.

4.2 Processing the interaction in the nonoperationalregion

During a real surgical procedure, in order to perform theoperation safely and efficiently, surgeons usually needprobe organs randomly in order to obtain the currentcondition of the organs or tissues undergoing operation.Therefore, the assumption of no interactive operations inthe nonoperational region in our previous model is unrea-sonable. In this section, we describe the method in ourimproved scheme that can remove this restriction.

Firstly we write the system of equations for the opera-tional region and nonoperational region not in a condensedform with block matrices:[

K11 K1I

K I1 K1II

] [A1AI

]=

[P1−PI

], (10)[

K2II K I2

K2I K22

] [AIA2

]=

[PIP2

], (11)

where superscripts 1 and 2 represent the operational andnonoperational regions and subscript I represents the com-mon nodes shared with these two regions. PI and −PI arethe force and counterforce applied to the common nodeswhen we analyze these two parts separately.

From the above equations we obtain a new matrix sys-tem Eq. 12, from which the displacements of the operationarea A1 and AI can be solved by iterative methods:[

K11 K1I

K I1 K1II +K′

] [A1AI

]=

[P1

−P′]

, (12)

where

K′ = K2II − K I2 · K−1

22 · K2I , (13)

P′ = K I2 · K−122 · P2 . (14)

Now we rewrite Eq. 11 in the condensed form as⎡⎣K2

II K Is K IiKsI Kss KsiKi I Kis Kii

⎤⎦[

AIAsAi

]=

[PIPsPi

]. (15)

Subscript i here represents the interior nodes to be con-densed out, and s represents the surface nodes to be re-tained. From Eq. 15 we have

(Kss − Ksi · K−1ii · Kis) · As = Ps − Ksi · K−1

ii · Pi

+ (Ksi · K−1ii · Ki I − KsI ) · AI . (16)

Because the external forces will not be applied to the inte-rior nodes, all terms related to Pi make no contribution tothe calculation and will be ignored. Finally, we calculatethe displacements of the surface nodes by K∗ As = Ps +P∗ · AI , which involves the preprocessing data

K∗ = Kss − Ksi · K−1ii · Kis , (17)

P∗ = Ksi · K−1ii · Ki I − KsI . (18)

From the equations of the nonoperational region dis-cussed above, the limitation of no interactive operationsin this region makes all terms of K′, P′, K∗, and P∗ un-changed during the entire simulation process. Therefore,in our previous model they are calculated in the prepro-cessing stage. Now, to remove this restriction, a new cal-culating procedure for the preprocessing data has to be de-veloped since P2 in Eq. 11 and Ps in Eq. 15 are no longerzero when users freely detect any virtual tissue models byvirtual tools.

Let P′2 represent the interactive force vector being ex-

erted in the nonoperational region produced by user in-teraction; it is changed on-the-fly. We can rewrite Eq. 14as P′ = K I2 · K−1

22 · P′2 and calculate the preprocessing data

K I2 · K−122 instead of K I2 · K−1

22 · P2. From Eqs. 16–18 weobtain

As = K∗−1 · Ps +K∗−1 ·P∗ · AI . (19)

Because such an interaction takes place only on the sur-face of models, the nodes on which interactive forces actshould be among surface nodes of the nonoperational re-gion Ps. We can represent the first term related to externalforces in Eq. 19 by two different terms in Eq. 20:

As = K∗−1 · Ps0 +K∗−1 · Ps_interact +K∗−1 ·P∗ · AI ,(20)

where Ps0 is the external force vector of the initial condi-tion when the simulation starts and Ps_interact is the vec-tor of the applied force that will be updated during thesimulation in real time. We can still calculate K∗−1 · Ps0in the preprocessing stage since it remains unchanged in


Table 1. Comparison of preprocessing data in the two models

Old model New model

K′ K′P′ K I2 · K−1

22Preprocessing Data K∗−1 · Ps K∗−1

K∗−1 ·P∗ K∗−1 · Ps0

K∗−1 ·P∗

the simulation. The interactive operation in the nonopera-tional region causes the changes in calculating the prepro-cessing data, which are related to the surface nodes. Wecompare the differences in Table 1.

The simulation process comprises the following stages:

(1.) Calculate the corresponding preprocessing data listedin Table 1.

(2.) Use Eq. 12 to compute displacements of nodes A1 andAI in the operational region.

(3.) Determine the interactive nodes of the nonoperationalregion and update the applied force vector Ps_interact .

(4.) Compute the new value of K∗−1 · Ps_interact .(5.) Use Eq. 20 to compute the displacements of surface

nodes As in the nonoperational region.

Most entries of Ps_interact are zero because only a verysmall portion of nodes are under user manipulation. If wecarry out the calculation by the standard matrix and vectormultiplication, a large number of useless zero multiplica-tions will be performed. In order to avoid such computa-tion redundancy, we use a selective multiplication methodinstead of the general method. K∗−1 · Ps_interact is com-puted by choosing the nonzero entries of Ps_interact and the

Fig. 3. Update displacements of interactive nodes

corresponding columns in K∗−1. Consequently, the chang-ing of As, caused by any interactive operation, can beeasily updated by adding the new values of the nonzeroentries to the previous values of As (Fig. 3).

5 Implementation and experiments

To validate the approaches described above, we have im-plemented them by VC++ on the PC platform. The tetra-hedral model of the human organ was generated from thesegmented volume, which is the result of segmentation ona series of computer tomography (CT) or magnetic reson-ance imaging (MRI) images [21]. The operational part andthe nonoperational part of the model are determined in-teractively during the preprocessing stage. Moreover, theboundary condition of the model, such as the displacementconstraints, the external forces, and the initial strains, areprescribed before the simulation. The physical parametersthat describe the tissue properties, namely Young’s Modu-lus and Poisson’s ratio, are decided after referring to [12].Submatrices of the new model listed in Table 1 are calcu-lated in advance.

We have generated two different volumetric tetrahe-dral models from the image dataset of the Visible HumanProject [2], namely the human knee ligament (Fig. 4) andthe small portion of the upper leg (Fig. 5) for the ex-periment. The model of the ligament has 615 nodes and1900 tetrahedra in total. Of those, 575 nodes (438 surfacenodes) and 1829 tetrahedra are of the nonoperational re-gion (Table 2). The nonoperational part of the upper legmodel has 633 nodes (416 surface nodes) and 2134 tetra-hedra.

The initial forces are imposed on three surface nodesof the operational region to simulate the forces applied bysurgical tools. These forces also cause the initial deforma-tion of the models. In the model of the upper leg, the holeis the thighbone of the body, which we do not show here

Fig. 4a–c. Model of human knee ligaments. a Shading model ofhuman knee ligaments. b Tetrahedral mesh of ligament with com-mon nodes shown. Light gray mesh: operational part. Dark bluemesh: nonoperational part. c Shading model of ligament with sur-face mesh shown


Fig. 5a–c. Model of human upper leg. a Shading model of smallportion of upper leg. b–c Tetrahedral mesh of operational and non-operational parts of small portion of upper leg are shown withcommon nodes separately. b Light gray mesh: operational part.c Dark blue mesh: nonoperational part

Table 2. Scale of original models

Model Operational Nonoperationalregion region(Nodes/tetra.) (Nodes/sur. Nodes/tetra.)

The Ligament 40 / 71 575 / 438 / 1829The Upper Leg 152 / 416 633 / 416 / 2134

in figures for clarity. It has the specific boundary conditionthat nodes fixed on the thighbone have no displacementsduring the entire process.

The cutting operation occurs when the pressure causedby the collision between the scalpel and the tissue in theoperational region is bigger than the specified threshold(Fig. 6). During the cutting process, the deformation of thetissue is also calculated according to the new boundaryand force constraints (Fig. 7).

Before running the simulation, the mesh data and pre-processing data should be read first before the topologicalinformation, such as lists of the nodes, surfaces, and tetra-hedrons, is constructed. Moreover, the global stiffness ma-trix of the nonoperational region to be used in the de-formation calculation has to be merged in advance. Therunning time of these steps depends on the size of themodel (Table 3). Though these steps take slightly longer tocomplete, they are carried out only once in the preparationphase.

Fig. 6. Cutting of ligament. Light gray mesh: operational part. Darkblue mesh: nonoperational part

Fig. 7. Deformation during cutting process

Table 3. Time for reading in preprocessing data

Preprocessing Ligament Upper legdata (ms) (ms)

Construct 6.5813 (Opt) 155.3061 (Opt)Topology 3615.6218(UNOpt) 5034.3597(UNOpt)

GenerateGlobal 34.9449 46.3165stiffness matrix

K′ 7.9326 106.7937

K I2 · K−122 270.2416 864.8767

K∗−1 6406.2949 3504.5390

K∗−1 · Ps0 6.3491 2.3562

K∗−1 ·P∗ 211.6492 587.5250

The interactive phase includes collision detection, soft-tissue cutting, poking/dragging, and deformation. We havetested different sizes of models on an NVIDIA GeForceFX 5950 Ultra with a 5.2.1.6 driver running on 1.5-GHzPentium 4 CPU. The runtime cost of each primary stagein the entire simulation of the upper leg model with differ-ent degrees of freedoms (DOFs) is illustrated in Fig. 8. Forthe most time-consuming operation, the conjugate gra-dient algorithm solving the linear system of the opera-tional region, we compare the time of the sparse matrix-vector multiplication, which runs by the conjugate gradi-

Fig. 8. Runtime cost of each primary stage during entire simulationof upper-leg model with different degrees of freedom (DOFs)


Table 4. Time for obtaining displacements of nonoperational regionby general model and condensed model

Average time General model 2.369of ligament (ms) Condensed model 2.318

Average time General model 8.899of upper leg (ms) Condensed model 6.556

ent solver of our system, on the GPU and the CPU, re-spectively. Because our model has a compact form of fullutilization of the 4-channel characters to fill nonzero en-tries and coordinate indices into the texture, the pass num-ber required can be remarkably reduced. A considerableamount of time on data transfer between two consecutivepasses can be saved.

Table 4 shows the average execution time for obtainingthe displacements of the nonoperational region. We meas-ure the performance in the noncondensed model and thatin the condensed one. The result of the condensed modelruns faster than that of noncondensed model because it ex-ecutes fewer calculation operations related to the internalnodes. The percentage of surface nodes and the commonnodes shared by these two regions determines the ratio ofthe speed improvement.

Table 5 shows the time spent on the interactive drag-ging or poking operation. Because most calculations in

Fig. 9a–d. Cutting with dragging operation. a and b Shading model. c and d Shading model with mesh. Light grey mesh: operational part.Dark blue mesh: nonoperational part

Fig. 10a–c. Dragging and poking operation in nonoperational part. Light grey mesh: operational part; dark blue mesh: nonoperational part

Table 5. Time for interactive poking or dragging operation

Ligament model Upper-leg modelDOFs Time (ms) DOFs Time (ms)

120 0.0352 456 0.0587336 0.0374 960 0.0838501 0.0469 1137 0.0634564 0.0472 1602 0.0654678 0.0477 1854 0.0777720 0.0399 2100 0.0645

this part can be performed in the preprocessing stage, whatwe need to do is to use the selective multiplication tocompute any affected terms corresponding to the nodessuffered by external forces. It does not introduce a heavyburden into the simulation. At present, the node suffer-ing forces is specified in advance as a simple test. Themagnitude of the applied force is controlled by the mousemovement.

Figures 9–11 show the simulation result in this regard.The green point in the figures is the node manipulatedby users, and the green line, which connects the currentmouse point and the green point, shows the direction ofthe applied force. In future work, as long as any colli-sion detection algorithm is involved in detecting a list ofmanipulated nodes, our solution in this paper could allowusers to poke or drag any part of the soft-tissue model.


Fig. 11a–f. Cutting with dragging and poking operation. a–c Shading model with mesh. Light brown mesh: operational part; dark bluemesh nonoperational part. d–f Shading model

6 Conclusion

We have proposed an improved scheme based on ourmethod for modeling surgical simulation that allows usersto freely perform tissue cutting and detecting such as pok-ing or dragging operations with soft-tissue deformation.The operations of cutting, poking, or dragging have beenimplemented in test examples, and the result demonstratesthat this model provides an effective and efficient means.

Using this new scheme, apart from handling the cutting inthe operational region, the interactive manipulation can beflexibly conducted in any other region.

Acknowledgement The work described in this paper was fullysupported by a grant from the Research Grants Council of theHong Kong Special Administrative Region, China (Project No.CUHK4223/04E).

Thanks to Sun Jian and Dr. Shen Hao for algorithm discus-sion, and thanks to Dr. Yang Xiaosong for providing the tetrahedralmodel of the virtual organs.

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WEN WU is currently a Ph.D. candidate in theDepartment of Computer Science and Engin-eering at the Chinese University of Hong Kong.She received her Master of Science degreein computer application technology from theInstitute of Computing Technology, ChineseAcademy of Sciences, in 2000 and her Bachelorof Engineering degree from Beijing Universityof Aeronautics and Astronautics in 1997. Herresearch interests include surgical simulationand computer graphics.

PHENG ANN HENG received his M.Sc. (CS),M. Art (Applied Math), and Ph.D. (CS) allfrom Indiana University in Bloomington, IN,USA in 1987, 1988, 1992, respectively, and hisB.Sc. from the National University of Singaporein 1985. He is a professor in the Departmentof Computer Science and Engineering at theChinese University of Hong Kong. In 1999,he set up the Virtual Reality, Visualization andImaging Research Centre at CUHK and servesas the director of the centre. He is also thedirector of the CUHK Strategic Research Areain Computer Assisted Medicine, establishedjointly by the Faculty of Engineering and theFaculty of Medicine in 2000. His researchinterests include virtual reality applications inmedicine, scientific visualization, 3D medicalimaging, user interface, rendering and modeling,interactive graphics, and animation.

an improved scheme of an interactive finite element model for 3d soft-tissue cutting and deformation

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