A Nonlinear Viscoelastic Tensor-Mass Visual Model for Surgery Simulation

14 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 1, JANUARY 2011

A Nonlinear Viscoelastic Tensor-Mass VisualModel for Surgery Simulation

Shaoping Xu, Xiaoping P. Liu, Senior Member, IEEE, Hua Zhang, Senior Member, IEEE, and Linyan Hu

Abstract—The linear elastic models of soft tissue are widely usedin virtual-reality-based surgery simulation due to their computa-tional efficiency; however, it is well known that these models areonly a coarse approximation of the real biological soft tissue. Toachieve realistic simulation, the deformable model should incorpo-rate many other tissue properties such as nonlinearity, anisotropy,and viscoelasticity. Among these properties, viscoelasticity is avery important one, and it directly determines the behaviors of thetissue when it is cut, deformed, or torn. In this paper, we proposedto incorporate the property of viscoelasticity into the visual modelof soft tissue. One significant advantage of the developed modelis its fast computation. Experiments show that the incorporationof nonlinear viscoelasticity makes the simulated tissue look muchmore realistic than other models, whereas the computation timeis increased by only approximately 5% compared with modelswithout the consideration of viscoelasticity.

Index Terms—Force feedback, soft-tissue modeling, surgerysimulation, tensor-mass model, viscoelasticity.

I. INTRODUCTION

SURGERY simulation allows surgical students and resi-dents to practice and gain skills by operating virtual pa-

tients in a simulation environment before entering the realoperating room [1]–[4]. In addition to haptic feedback, onemajor issue of these systems is to achieve physically realisticvisual effects of the geometric deformation [5], [6]. For thispurpose, the modeling of tissue deformation is of great impor-tance, and several models of different approaches have beenreported, including the nonphysical ones (computationally fastbut less realistic) [7], [8] and the physically-based ones (morerealistic but computation is slow) [4] and [9]. To achieve real-time computation, most physical models incorporate only linearelasticity, i.e., their mathematical formulations assume that thetissue stress–strain response is linear and that the soft tissue

Manuscript received December 13, 2009; revised February 23, 2010;accepted April 10, 2010. Date of publication August 30, 2010; date of currentversion December 8, 2010. This work was supported in part by the NationalNatural Science Foundation of China under Grants 60874020 and 50863003and by the Department of Education, Jiangxi, China, under Grants GJJ09012and GJJ09015. The Associate Editor coordinating the review process for thispaper was Dr. Shervin Shirmohammadi.

S. Xu and L. Hu are with the School of Information Engineering and theSchool of Mechatronics Engineering, Nanchang University, Nanchang 330031,China.

X. P. Liu is with the School of Mechatronics Engineering, NanchangUniversity, Nanchang 330031, China and also with the Department of Systemsand Computer Engineering, Carleton University, Ottawa, ON K1S 5B6, Canada(e-mail:[email protected]).

H. Zhang is with the School of Mechatronics Engineering, Nanchang Uni-versity, Nanchang 330031, China.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2010.2065450

undergoes infinitesimally small deformations. However, thegain of computational efficiency is at the cost of visual realism.In particular, these models are only a rough approximation ofreal biological tissue. To achieve a high degree of realism andaccuracy, many other tissue properties (such as nonlinearity,anisotropy, and viscoelasticity) should be incorporated into thedeformable models. Although a number of such models arereported in the literature [2], [10], [11], these models usuallysuffer from high computational cost because of complicatedconstitutive equations and the intensive calculation for updatingstiffness matrices. So far, it has been very difficult to realize therequired high refresh rate (at least 25 Hz) in surgery simulation,which is essential for achieving a satisfactory degree of visualrealism.

Lately, Picinbono et al. [12] extended Cotin’s linear tensor-mass model [13] by integrating geometrical nonlinearity andanisotropy. Their model is useful in the sense that it well accom-modates large displacements, i.e., it is invariant with respect torotations. This property improves the degree of visual realismand overcomes problems associated with linear elasticity thatis only valid for small displacements. Picinbono et al. also ad-dressed the problem of volumetric variations by adding incom-pressibility constraints into the tensor-mass model. Althoughthe accuracy of Picinbono’s tensor-mass model is improvedcompared with previous models, it does not consider the tissuebehaviors due to viscoelasticity [11], which is a very importantand time-dependant feature of biological soft tissue. The timedependence of soft tissue manifests itself in many aspects of themechanical response. For instance, soft tissue under constantload generally exhibit creep, whereas those under constantdeformation exhibit stress relaxation. In addition, most tissueappears stiffer at higher loading velocities. Many experimentshave revealed that viscous effects due to viscoelasticity cannotbe ignored for an accurate and realistic description of themechanical property of soft tissue [11]. A correct description ofthe mechanical responses of biological tissue therefore requiresincluding its nonlinear viscoelastic behaviors.

In this paper, we propose to incorporate viscoelasticity intothe tensor-mass model of soft tissue (i.e., Picinbono’s tensor-mass model [12]). Although there are a number of viscoelasticconstitutive models, such as the one by Fung [10], it has beenfound that the direct incorporation of viscoelastic models intothe tensor-mass model of soft tissue will lead to a significantincrease of computational cost, because the information mustbe saved at every previous time step [11]. For this purpose,we propose a numerical scheme for achieving a high degree ofvisual realism while not much compromising the computationefficiency of tensor-mass models.

0018-9456/$26.00 © 2010 IEEE

XU et al.: NONLINEAR VISCOELASTIC TENSOR-MASS VISUAL MODEL FOR SURGERY SIMULATION 15

The rest of this paper is organized as follows. Section II givesan overview of Picinbono’s tensor-mass model. Section IIIpresents the new scheme that we developed to incorporate theproperty of viscoelasticity into the tensor-mass model and theapproach to the fast computation of viscoelastic mechanicaldeformations. The efficiency of the constitutive update schemeand implementation is assessed in Section IV, and Section Vpresents a brief discussion on this paper. Section VI concludesthis paper with some conclusions and future work.

II. PICINBONO’S TENSOR-MASS MODEL

For brevity, Picinbono’s tensor-mass model can roughly bedescribed as follows (Readers can refer to [12] for more detailsabout this model). An elastic model can be driven from astrain tensor, which is a quadratic function of the deformationgradient. The St. Venant–Kirchhoff model is a linear modelfor large displacements of hyperelastic materials. The completeGreen–St.Venant strain tensor S can be defined as

S =12(C − I) =

12(∇U + ∇UT + ∇UT∇U) (1)

where ∇U is the 3 × 3 gradient matrix, C is the rightCauchy–Green strain tensor, and I is an identity matrix. Elasticenergy, which is a quadratic function of ∇U in the linear case, isa fourth-order polynomial with respect to ∇U in the nonlinearcase, i.e.,

W (Ti) =∑i,k

U tjB

Ti

i,kUk +∑i,k,l

(Uj · CTi

i,k,l)(Uk · Uj)

+∑

i,k,l,m

DTi

j,k,l,m(Uj · Uk)(Ul · Um) (2)

where BTi

i,k, CTi

i,k,l, and DTi

j,k,l,m are called stiffness parametersand are given by

BTi

i,k =λ

2(αj ⊗ αk) +

μ

2[(αk ⊗ αj) + (αj · αk)ID3]

CTi

i,k,l =λ

2αj(αk · αl) +

μ

2[(αl(αj · αk) + αk(αj · αl)]

DTi

j,k,l,m =λ

8(αj · αk)(αl · αm) +

μ

4[(αj · αm) + (αk · αl)]

(3)

where λ and μ are the Lame coefficients (biomechanical elasticconstants) that characterize the material stiffness, and they aredirectly proportional to the material-dependent Young’s Modu-lus E and Poisson coefficient ν by the following equations:

λ =Eν

(1 + ν)(1 − 2ν)(4)

μ =Eν

2(1 + ν)(5)

where αk(k = 1, 2, 3, and 4) in (3) is a vector defined as afunction of the rest positions Pk of the four vertices of the

tetrahedron Ti by

α0 = (P2 − P1) ∧ (P3 − P1)α1 = (P2 − P3) ∧ (P0 − P2)α2 = (P0 − P3) ∧ (P1 − P3)α3 = (P0 − P1) ∧ (P2 − P0) (6)

where Pa ∧ Pb is the vector product of two vectors. The forceapplied on each vertex Pp(p = 1, 2, 3, and 4) inside a tetra-hedron is obtained by deriving the elastic energy W (Ti) withrespect to the displacement Up, i.e.,

FTip = 2

∑j

BTipj Uj + 2

∑jk

(Uk ⊗ Uj)CTi

jkp

+∑jk

(Uj ⊗ Uk)CTi

pjk + 4∑jkl

DTi

jklpUlUtkUj . (7)

The first term of the elastic force FTip denotes the linear

elastic portion, whereas the rest of the terms describe thenonlinear elastic portion of the elastic force.

III. PROPOSED MODEL THAT

INCORPORATES VISCOELASTICITY

As indicated in the previous sections, many experimentsrevealed that viscous effects due to tissue viscoelasticity cannotbe ignored for an accurate description and realistic visualizationof the mechanical properties of biological tissue [1], [2], [10],[11]. However, Picinbono’s tensor-mass model introduced inSection II does not incorporate the viscoelastic characteristicsof soft tissue.

In this section, we show how we can incorporate the prop-erties of viscoelasticity into Picinbono’s tensor-mass model.Given the expression of the elastic energy W , the force appliedto a vertex Pi is defined by

FTi

Pi=

∂W (Ti, E)∂Pi

(8)

where E is Young’s modulus constant in Picinbono’s tensor-mass model. According to Fung’s theory, viscoelastic materialscan typically be represented by the generalized Maxwell solid,which is a combination of springs and dampers. In this case,Young’s modulus E(t) of viscoelastic materials is a functionof time t, is also called relaxation functions, and commonlyassumes the form of a Prony series [14]

E(t) = E0 +N∑

j=1

Eje− t

tj (9)

where N is the number of Maxwell elements, Ej is theelastic coefficient (E0 is the long-term elastic modulus thatcorresponds to the system’s steady-state elastic response, i.e.,limt→∞ E(t) = E0), and τj is the relaxation time. We mayrewrite (9) as

E(t) =

⎛⎝1 +

N∑j=1

Ej

E0e− t

tj

⎞⎠ E0 =

⎛⎝1 +

N∑j=1

αje− t

tj

⎞⎠ E0

(10)

where αj is the jth normalized elastic modulus.


Substituting E(t) into (8), we can obtain force FTi

Pi(t), which

contains elastic and viscoelastic portions. We have

FTi

Pi(t) =

∂W (Ti, E(t))∂Pi

=

t∫0

∂

∂s

∂W (Ti, E(t − s))∂Pi

ds

(11)

where

E(t − s) =

⎛⎝1 +

N∑j=1

αje− t−s

tj

⎞⎠ E0. (12)

Splitting the elastic portion and viscoelastic portion from theintegral of (11) leads to

FTi

Pi(t) =

t∫0

∂

∂s

∂W (Ti, E0)∂Pi

ds + αj

×t∫

0

N∑j=1

e− t−s

tj∂

∂s

∂W (Ti, E0)∂Pi

ds. (13)

We may rewrite (13) as

FTi

Pi(t) = FTi

Pi,E0(t) + αj

N∑j=1

hj(t) (14)

where FTi

Pi,E0(t) is the elastic force exerted on the vertex Pi of

tetrahedron Ti at the time t, which can be calculated by using(7) of Picinbono’s tensor-mass model, where hj(t) is calledinternal stress variables and can be defined as

hj(t) =

t∫0

e− t−s

tj∂

∂s

∂W (Ti, E0)∂Pi

ds. (15)

An efficient solution to this hereditary integral is crucialfor numerical implementation. Considering the time interval[t0, tn+1], the discretization of hj(t) yields

hj(tn+1) =

tn+1∫0

e− t−s

tj∂

∂s

∂W (Ti, E0)∂Pi

ds. (16)

Separating the deformation history into two periods, 0 ≤ t ≤tn, where the result is known, and tn ≤ t ≤ tn+1, yields

hj(tn+1) = e−∇t

tj

tn∫0

e− tn−s

tj∂

∂s

∂W (Ti, E0)∂Pi

ds

+

tn+1∫tn

e− tn+1−s

tj∂

∂s

∂W (Ti, E0)∂Pi

ds. (17)

∫ tn

0 e−((tn−s)/tj)(∂/∂s)(∂W (Ti, E0)/∂Pi)ds in the firstterm of (17) is hj(tn), and (∂/∂s)(∂W (Ti, E0)/∂Pi) in the

second term of (17) can be rewritten as (∂/∂s)FTi

Pi,E0(s);

therefore, (17) can be rewritten as

hj(tn+1)=e−∇t

tj hj(tn) +

tn+1∫tn

e− tn+1−s

tj∂

∂sFTi

Pi,E0(s)ds.

(18)

Using the midpoint rule, the second term∫ tn+1

tn

∑Nj=1 e−(tn+1−s/tj)(∂/∂s)FTi

Pi,E0(s)ds in (18) can

be approximated as follows:

tn+1∫tn

e− tn+1−s

tj∂

∂sFTi

Pi,E0(s)ds

= e− tn+�t−s

tjd

dsFTi

Pi,E0(s)ds

∣∣∣s=

tn+1−tn2

∇t

= e−∇t2tj

(FTi

Pi,E0(tn+1) − FTi

Pi,E0(tn)

). (19)

Substituting (19) into (17), we can obtain

hj(tn+1)=e−∇t

tj hj(tn)+e−∇t2tj

(FTi

Pi,E0(tn+1)−FTi

Pi,E0(tn)

).

(20)

Thus, hj(tn+1) can be determined by (20) in terms of hj(tn)and an integral over the time step[tn, tn+1]. We have

FTi

Pi(tn+1) = FTi

Pi,E0(tn+1) + αj

N∑j=1

hj(tn+1). (21)

The first part of (21) is the nonlinear elastic portion, whichcan be calculated by using Picinbono’s tensor-mass model, andthe second part of (21) corresponds to the viscoelastic portion,which can be calculated by the iterative computation of (20).Because of the aforementioned mathematical manipulation andapproximation, the computation cost is significantly reducedcompared with formulations for which the viscoelastic proper-ties was directly included. This result will be shown and verifiedby the simulation results later on.

IV. NUMERICAL EXAMPLES AND IMPLEMENTATION

A. Benchmark Test

The mechanical response of soft tissue is commonly charac-terized from compression experiments on cylindrical or cubicsamples. Many experimental data have been obtained and com-pared from the previously published literature. In this paper,we used the experimental data reported by Samur et al. [15] asthe benchmark data for our simulations and comparison. Theconfiguration of compression experiments is shown in Fig. 1.

Using the formula given by Lee and Radok [16], the force-displacement response of soft tissue for a small indentation ofan elastic half space by a rigid hemispherical indenter can beobtained by

F =163

Gδ√

Rδ (22)


Fig. 1. Schematic of the indenting experiment.

where G is the effective shear modulus, and δ is the indentationdepth of a rigid sphere with a radius of R.

To find out the parameters of the developed viscoelastictensor-mass model, an inverse tensor-mass solution was devel-oped for the characterization of the material properties of softtissue from the experimental data. First, some initial values ofthe material parameters, which were chosen based on recentresults for the viscoelastic response of soft tissue in vivo,were set as seed values for the model. Then, we employed asimulated annealing algorithm [17] to automatically tune theparameters of the model until the force response approximatesthe reference one measured from real compression experiments.The optimal material parameters were achieved by minimizingthe cost function Θ, which can be defined as

Θ(E∞, E1, E2, τ1, τ2, ν) =N∑

I=0

|Fm(ti) − Fs(ti)| (23)

where Fm, Fs, ti, and N are measured forces from the realexperiments, simulated forces, time, and the total number ofdata, respectively. The cost function is the sum of force errorsbetween experimental and simulated data. As aforementioned,the relaxation function defined by the generalized Maxwellsolid results in a Prony series representation, given by (9).To simplify the calculation, we only consider a short-termrepresentation of the coefficients of the Prony series for N = 2(i.e., the viscoelastic material properties E∞, E1, E2, τ1, τ2).The reference force data for simulation were reproduced byusing (22) using the material parameters for the liver of pig #2reported by Samur [15]. Based on Samur’s data, the materialparameters E∞, ν optimized by inverse tensor-mass solutionare summarized in Table I. We set the parameters E1, E2, τ1,and τ2 to zero, which means that the viscoelasticity was notconsidered in this case. The numerical simulation results ofthe tensor-mass model were compared with the experimentaldata, as shown in Fig. 2. The simulation results suggest that theparameters listed in Table I can capture the experimental datareported in [15].

To demonstrate the capabilities of our model to describe thedifferent nonlinear viscoelastic behaviors of soft tissue, a sec-ond group of simulations were conducted. In these simulations,the reference force response was still reproduced based on (22).

TABLE IMATERIAL PARAMETERS OBTAINED BY INVERSE TENSOR-MASS

SOLUTION THROUGH OPTIMIZATION ALGORITHM FOR

INDENTATION EXPERIMENTS

Fig. 2. Inverse tensor-mass solution simulation: experimental and simulatedforce-time relationship for indentation depths of 2, 4, 6, and 8 mm (viscoelas-ticity is not considered here).

TABLE IIMATERIAL PARAMETERS OBTAINED BY INVERSE TENSOR-MASS

SOLUTION THROUGH OPTIMIZATION ALGORITHM

FOR STRESS RELAXATION EXPERIMENTS

The time dependence of the mechanical response of soft tissueis defined through the relaxation function g(t) [1], i.e.,

g(t) = 1 −2∑

k=1

gk

(1 − exp

(− t

tk

))(24)

The values of the viscoelastic parameters in (24) are taken tobe the same as Nava’s work [1, Table I]. Given the referenceforce response, again, we obtained the material parameters(summarized in Table II) through the inverse tensor-mass solu-tion through optimization iterations. In Fig. 3, good agreementwas found between the reference force-strain curves and thecorresponding prediction based on our model.

B. Liver Organ Simulation

To access the ability of the proposed method to modelthe viscoelasticity of real tissue, a set of experiments havebeen conducted to simulate typical viscoelastic behaviors ofhuman organ. The tensor-mass model for a complete liver is


Fig. 3. Stress relaxation experiments: experimental and simulated force-timerelationship for indentation depths of 2, 4, 6, and 8 mm.

Fig. 4. Rendered view of the undeformed liver.

created, and the corresponding constitutive equation is imple-mented. We use a 3-D volumetric model (about 800 nodes and2400 tetrahedral elements) of a human liver in the tensor-massmodel computations. The rendered view of the undeformedliver is shown in Fig. 4.

The creeping has been verified by applying a constant forceto a node of the liver surface mesh with a virtual probe. Inour creep test, we applied different constant loading forcesto the node 15 for 6 s, and the resulting strain is recordedas a function of time. The displacement response calculatedusing our nonlinear viscoelastic tensor-mass model is plottedin Fig. 5. Fig. 5 shows the variation of displacement againsttime under a constant force. It is clear that a linear increasein the load does not correspond to a linear increase in thedisplacement. The observed responses of our model are similarto the measurements obtained from real tissue.

In the stress-relaxation test, the different nodes of the livermesh were first indented to the predefined depths of 1cm in1 s, and then, the indenter was held for 5 s to record the forcerelaxation response. The force-time curves of these nodes areshown in Fig. 6. We can see that the histories of the forceresponses can be separated into a nonlinear elastic part and atime-dependent reduce relaxation part. This observation is be-cause of the time dependence of the viscoelastic behaviors. Asshown in Fig. 7, the dashed line shows the deformations underthe loading process, and the solid line shows the deformations

Fig. 5. Displacement response of node 15 applied different constant loadingfor 6 s.

Fig. 6. Force-time curve of indentation experiment of node 15, 16, and145(locations in Fig. 4).

Fig. 7. Response of the force to deformation (constant strain rate for oneloading/unloading cycle).

under the unloading process. The plot of strain against forcedemonstrates that the loading and unloading processes followtwo distinct paths, which is in agreement with the hysteresisphenomenon exhibited by soft tissue.

C. Real-Time Computational Architecture

The experimental data in Table III show that the proposednew model allows the incorporation of viscoelasticity with as


TABLE IIIONE-STEP EXECUTION TIME OF DIFFERENT NODES CONSIDERING

THE VISCOELASTICITY CASE

little as 5% additional computation cost compared with theoriginal nonlinear elastic models because of the use of anefficient update formulation. With the current implementation,we can reach an update rate of 40 Hz on meshes made ofabout 2400 tetrahedron [on a PC Intel Core 2 Quad with2.40 GHz, 1-GB RAM, and an NVIDIA GeForce 8800GTXgraphics processing unit (GPU)] [4]. Based on the new model,we propose a real-time computational architecture that consistsof three threads running asynchronously. The central-control(main) thread, updated at a frequency of about 100 Hz, per-forms collision detection and calculates the response forcebased on the new tensor-mass model. The response force iscalculated and updated every 10 ms. This force profile is sentto the haptic thread, and the force is then displayed to the userthrough the haptic device until the next central-control threadcycle. The haptic thread, updated at a frequency as high as500 Hz, acquires the haptic probe’s new position and extrap-olates out the feedback force based on prior ones. The visualthread, updated at a frequency of 40 Hz, renders the overallgraphic deformation of the organ caused by the displacementsof all nodes.

V. DISCUSSION

Compared with other tensor-mass models, the proposedmodel has two important advantages: 1) nonlinear viscoelas-ticity (physical nonlinearity) is incorporated, which makes themodel more realistic, whereas the computation is not muchdegraded and 2) the viscous force has a nonlinear relationshipwith the deformation speed, which better reflects the actualsituations of real tissue.

On the other hand, our model also has some limitations.In particular, it needs an elevated number of Prony series.A nonlinear viscoelastic model with N = 2 internal variablesadopted in this paper is the minimum required. An increase inthe number of Prony series can improve the accuracy of themodel. Another possible drawback is related to the number ofparameters employed in the tensor-mass model. To simplify themodel, some material parameters such as Poisson ratio were setto a predefined constant value. This problem needs to further beinvestigated in our future work.

VI. CONCLUSION

By introducing an efficient update formulation, we havesuccessfully incorporated viscoelasticity into the tensor-massmodel of soft tissue. Numerical implementation shows that typ-ical behaviors due to tissue viscoelasticity can be simulated andvisualized at very little additional computational cost for the

presented new model. Our future work is to implement a newGPU-based version of the scheme for surgical simulation usingthe Compute Unified Device Architecture (CUDA) applicationprogramming interface (API) [4] and to apply the scheme to areal-world surgical simulator.

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Shaoping Xu received the M.Sc. degree in computerapplication from China University of Geosciences,Wuhan, China, in 2004. He is currently workingtoward the Ph.D. degree in the School of Mechatron-ics Engineering, Nanchang University, Nanchang,China.

His current research interests include surgery sim-ulation, virtual reality, computer graphics, and digitalimage processing and analysis.


Xiaoping P. Liu (M’02–SM’07) received the Ph.D.degree from the University of Alberta, Edmonton,AB, Canada, in 2002.

Since July 2002, he has been with the Departmentof Systems and Computer Engineering, CarletonUniversity, Ottawa, ON, Canada, where he is cur-rently a Canada Research Chair Professor. He is alsowith the School of Mechatronics Engineering as anAdjunct Professor, Nanchang University, Nanchang,China.

Dr. Liu is a Licensed Member of the ProfessionalEngineers of Ontario (P.Eng). He serves as an Associate Editor for severaljournals, including the IEEE/ASME TRANSACTIONS ON MECHATRONICS

and the IEEE TRANSACTIONS ON AUTOMATION SCIENCE AND

ENGINEERING. He received the 2007 Carleton Research AchievementAward, the 2006 Province of Ontario Early Researcher Award, the 2006 CartyResearch Fellowship, the Best Conference Paper Award at the 2006 IEEEInternational Conference on Mechatronics and Automation, and the 2003Province of Ontario Distinguished Researcher Award.

Hua Zhang (M’03–SM’09) received the Ph.D. de-gree from Tsinghua University, in 1995.

He is currently a Professor and the Dean ofthe School of Mechatronics Engineering, NanchangUniversity and the Director of the Key Laboratory ofRobot and Welding Automation, Jiangxi, China. Hisresearch interests include mobile robot techniques,teleoperation systems, and welding automationtechniques.

Linyan Hu is currently working toward the Ph.D.degree in the School of Mechatronics Engineering,Nanchang University, Nanchang, China.

Her current research interests include teleopera-tion systems and haptic control.

Documents

A Nonlinear Viscoelastic Tensor-Mass Visual Model for Surgery Simulation