ieee transactions on biomedical engineering 1...

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING 1

Constraint-based Soft Tissue Simulation forVirtual Surgical Training

Wen Tang, Tao Ruan Wan Member, IEEE

Abstract—Most of surgical simulators employ a linear elastic model to simulate soft tissue material properties due to its computationalefficiency and the simplicity. However, soft tissues often have elaborate nonlinear material characteristics. Most prominently soft tissuesare soft and compliant to small strains, but after initial deformations they are very resistant to further deformations even under largeforces. Such material characteristic is referred as the nonlinear material incompliant which is computationally expensive and numericallydifficult to simulate. This paper presents a constraint-based finite element algorithm to simulate the nonlinear incompliant tissuematerials efficiently for interactive simulation applications such as virtual surgery. Firstly, the proposed algorithm models the materialstiffness behaviour of soft tissues with a set of three-dimensional strain limit constraints on deformation strain tensors. By enforcinga large number of geometric constraints to achieve the material stiffness, the algorithm reduces the task of solving stiff equationsof motion with a general numerical solver to iteratively resolving a set of constraints with a nonlinear Gauss-Seidel iterative process.Secondly, as a Gauss-Seidel method processing constraints individually, in order to speed up the global convergence of the largeconstrained system a multi-resolution hierarchy structure is also used to accelerate the computation significantly, making interactivesimulations possible at a high level of details. Finally, this paper also presents a simple-to-build data acquisition system to validatesimulation results with ex vivo tissue measurements. An interactive virtual reality-based simulation system is also demonstrated.

Index Terms—Nonlinear soft tissue simulation and modeling, virtual surgical training, Robotic-assisted surgery.

F

1 INTRODUCTIONEfficient and accurate simulation of biological softtissues is a key computational component in virtualreality (VR) based surgical simulators as well asin robotic-assisted surgery and image-guided surgeryfor intraoperative dynamic guidance [Tang et al, 2012].Despite the rapid development in computer simulationtechnologies, achieving accurate soft tissue simulationremains challenging, especially medical applicationsrequire fast and accurate computations in order toaccommodate user interactions and haptic feedbacks.

Modeling material nonlinearity of biological tis-sues often involves setting up complex constitutivemodels to determine deformation behaviors of softtissues [Ogden, 1997], [Bonet and Wood, 2008]. It is wellacknowledged that tuning material parameters for theseconstitutive models to find approximations to their real-world counterparts is a complex and time-consumingtask [Mendis, 1995]. In recent years efforts have beenmade to model the parametrization of complex biome-chanical models as an inverse problem that are solvedby optimization. For examples, the estimation of elasticproperties of soft tissue is solved as a least squaresproblem in [Eskandari, et al., 2011], and an evolutionaryalgorithm is used to estimate the parameters of acomplex organ behavior model taking into account

• W. Tang is with the School of Computing, University of Teesside,Middlesbrough, UK, TS1 3BA.E-mail: [email protected]

• T. R. Wan is with School of Informatics, University of Bradford, Bradford,UK BD7 1DP.E-mail: [email protected]

of various real patient data sets in [Vidal et al., 2012].Previous methods for accelerating nonlinear finiteelement computations include pre-computing variousquantities of a constitutive model [Muller et al., 2001],modal analysis to compute deformations in a reducedsubspace [James and Pai, 2002], as well as GPU exe-cutions [Taylor et al., 2008] and data-driven approach-es [Bickel et al., 2009]. Still, a linear model is mostlyused in surgical simulators due to its computation-al efficiency and simplicity [Chentanez et al., 2009],[Pratt et al., 2010].

The distinctive feature of soft tissues such as skinand relaxed muscles is the so called nonlinear materialincompliant, which are easily deformed under smallforces, but beyond some thresholds soft tissues becomevery stiff and resistant to large deformations evenunder large forces [Brouwer et al., 2001]. The stiffnesscharacteristic is due to the macroscopic behavior ofthe materials that are submissive to small deformationsonly within some thresholds. Such material stiffnessdetermines the nonlinear stress-strain relationship ofsoft tissues and is the key in capturing physicalcharacteristics of the materials. In this paper ,we proposea constraint-based algorithm to simulate the highlyincompliant constitutive regimes of soft tissues throughthe use of a large number geometric constraints to enforcethe material stiffness.

Most numerical methods perform poorly for largeconstraint systems both in terms of system convergencyand computational efficiency, because the equations ofmotion of a large constraint system are infinitely stiff anda small time integration step must be carefully chosen toavoid the numerical instability. Instead of using a general


numerical solver to solve a set of stiff equations ofmotion, the proposed algorithm employs a Gauss-Seideltype procedure to resolve the constraints iteratively. AsGauss-Seidel methods processing constraints individual-ly, in order to speed up the global convergence of thelarge constrained system, a multi-resolution hierarchymesh structure is developed to accelerate the globalconvergence of the constrained system significantly. Themulti-resolution approach enforces constraints at eachlevel of the hierarchy structure with different meshresolutions, making interactive simulations possible forhighly incompliant tissue properties at a higher levelof detail. We show a simple-to-build acquisition systemto validate the simulation results with ex vivo tissuemeasurements and an interactive virtual reality-basedsimulation system is also demonstrated.

2 MODELLING OF NONLINEAR SOFT TISSUES

2.1 Principal stretches

Classic approaches try to use various constitutivemodels to describe the stress-strain relationship to modelnonlinear soft tissue behaviors [Ogden, 1997]. Instead,we apply constraints directly on the symmetric straintensor of each finite element(tetrahedron or triangle)to modify the stress-strain relationship nonlinearlyto prevent soft tissues from excessive stretching orcompressing. We firstly diagonalize the deformationgradient of each element. The material stiffness ischaracterized by strain ratios with some thresholds alongprincipal directions of deformations. We then applyconstraints directly on the symmetric strain tensor ofeach element.

Most cellular tissues such as livers and skins aretreated as isotropic, hyperelastic, and incompressiblematerials [Fung, 1993], [Humprey, 2003]. The state ofsoft tissue deformation is defined by the deformationgradient F, which is a 3× 3 matrix given by:

F(X) =∂x

∂X(1)

where X is the coordinate of a vertex in the materialspace and x is the coordinate in the world space.We compute the singular value decomposition of thedeformation gradient on each tetrahedral element F =UΣVT , where U and V are orthonormal matrices andΣ is a nonnegative diagonal matrix expressed by:

Σ = diag{λ1, λ2, λ3} (2)

The nonnegative diagonal entries λ1, λ2, and λ3 areeigenvalues representing ratios of stretch and compres-sion in the three principal directions invariant to rigidtransformations. Using eigenvalues of principle strainto model soft tissue material is not new in biomedicalengineering, for example, in the case of uniaxialdeformations, the deformation gradient is modeled asa diagonal tensor given by [Holzapfel, 2004]: Σ =

diag{λ, λ− 12 , λ− 1

2 }, where λ is the (applied) stretchin the principal directions. We employ the modifiedNeo-Hookean material using an energy density functiondefined by [Bonet and Wood, 2008]:

W =µ

2(I1 − 3)− µ log J +

β

2(log J)2 (3)

where µ = E2(1+ν) and β = Eν

(1+ν)(1−2ν) are the Lamecoefficients with E being the Young’s modulus andν being the Poisson ratio. Estimation of the elasticityparameters of E and ν is described in §4.1. The threeprincipal invariants are expressed as I1 = λ2

1 + λ22 +

λ23, I2 = λ4

1 + λ42 + λ4

3 and I3 = λ21λ

22λ

23. In Eq.3, J =

√I3.

We then analytically differentiate energy Eq. 3 withrespect to the invariants to compute forces. Therefore, forstress defined by a strain energy density function, it canbe shown that the diagonal matrix Σ yields a diagonalstress that is invariant under the rotation of the materialspace [Bonet and Wood, 2008].

P(Σ) = diag(∂W

∂λ1,∂W

∂λ2,∂W

∂λ3) (4)

Once we have calculated the stress P(F), the forceon a tetrahedral vertex can be found by multiplyingP(F) with an area-weighted normal of the tetrahe-dron [Irving et al., 2006]. Furthermore, we implement amulti-threading algorithm to parallel compute the SVDprocedures on tetrahedron elements on Eq.2 to improvethe computational time significantly, since SVD is acostly computation (see §4.2 for details).

2.2 Strain limiting constraints

Equation (4) revels that the elastic energy is a functionof the singular values λ1, λ2, λ3 of the deformationgradient, which are stretches and compressions corre-sponding to the scaling in the orthogonal principal straindirections. Our constraint-based algorithm enforces thethree-dimensional orthogonal scaling constraints on theprincipal strains by clamping the principal stretch ratiosat defined maximum and minimum values, resultingλmax and λmin for isotropic scaling. For example, λmax =1.25 and λmin = 0.75 specify the maximum of 25 percentof stretching and compressing along the principal straindirections, preventing a tetrahedron from undergoinglarge deformations. This leads to three clamped principalstrains defined by:

λ∗1 = clamp(λ1, λmin

1 , λmax1 ),


2 , λmax2 ), (5)


3 , λmax3 ).

Thus, a new diagonal strain matrix is construct-ed based on the constrained values as: Σ∗ =diag{λ∗

1, λ∗2, λ

∗3}. Substituting Σ∗ into Eq.4 yields a new

constrained stress tensor:


Fig. 1. The multi-resolution hierarchy: A base surface mesh of a liver is extracted from a MRI data set and a fine to coarsehierarchy of tetrahedral meshes is generated from the base mesh, containing 17.1K, 15.3K, 0.6K and 0.23K tetrahedra at eachlevel, respectively.

P(Σ∗) = diag(∂W

∂λ∗1

,∂W

∂λ∗2

,∂W

∂λ∗3

) (6)

Thus, deformation forces on tetrahedron nodes arere-computed based on the constrained Eq.6. Thepresence of constrains in the principal directions isclamping the stress at some maximum values andmodifying the constitutive model near the origin toremove the singularity, which regularizes the systemand enables dynamic simulations remain stable evenfor deformable objects that simultaneously undergoboth free-flying rigid motion and deformations. Moreimportantly, these modifications can be applied toarbitrary constitutive models and our experimentalsetup is aimed at validating the algorithm to be asaccurate as possible for medical simulations.

2.3 Multi-resolution constraint enforcement

The proposed strain limiting algorithm reduces a globalsolution approach of using the classic method of solvingstiff equations of motion to a set of Gauss-Seideliterations. Because a Gauss-Seidel type solver enforcesconstraints individually, it can be slow to convergencefor a system with a large number of constraints. Takingthe view that a hierarchical constraint enforcement cangreatly accelerate the system convergence [Muller, 2008],we construct a multi-level hierarchical structure usingdifferent tetrahedral mesh resolutions at multi-levels andeach level of the hierarchy is solved by a Gauss-Seideliterative process. Therefore, our multi-resolution ap-proach enforces unilateral constraints from fine meshesto coarser meshes, and then translate the solutions ofcoarser meshes back to finer levels. This process istermed as up- and down-sampling processes. Figure 2shows the geometric correspondence between elementsin a tetrahedral hierarchy.

The multi-resolution hierarchy process starts withthe physically-based computation on the bases mesh.Information of the base mesh is down-sampled to carryout the simulation on coarse meshes. At each coarsermesh level, nonlinear Gauss-Seidel iterations are run toenforce limiting constraints on the current coarser level.Results of the nonlinear stain limiting algorithm are thenup-sampled on finer level up the hierarchical chain toenforce constraints on the base mesh. We obtained aMRI data set of a liver from our local hospital with ethic

Fig. 2. Geometric correspondence between elements ina tetrahedral hierarchy: solid lines are a coarse mesh H(j)

containing a subset of higher resolution meshes H(i) withbarycentric coordinates highlighted in dotted red lines.

Fig. 3. Multi-resolution mesh hierarchy process pipeline for oursimulation system.

clearance approved by our organizations. Segmentationprocess was carried out manually in the medical imageopen source software GIMIAS [Gimas] to obtain a basesurface mesh of the liver. We then created a hierarchy ofsurface meshes using the base mesh. TetGen [Si, 2010]was used to convert surface meshes into tetrahedralmeshes, containing m levels of meshes H(i), for i =0,m − 1,, from the finest base level H(0) to the coarsestlevel of mesh H(m−1) (see figure 1).

For every tetrahedron in a coarser mesh H(j), allvertices of a finer mesh H(i) inside this tetrahedron aredetermined. For each vertex in H(i) its closest elementin H(j) and its corresponding barycentric coordinatesare computed. The coordinates are used as interpolationweights for sampling a scalar or a vector field definedover the coarser mesh to the finer mesh using linearinterpolation. Mapping values from the finer mesh tothe coarse mesh is done using these same weights.The multi-resolution mesh hierarchy process pipeline issummarized in figure 3. The method described here canbe divided into a number of stages of computations:


• Dynamic simulation and enforce constraints;• Downsampling projections;• Strain limiting on coarse meshes;• Enforce constraints;• Upsampling strain limited results.Given an initial displacement values on the fine

mesh we perform the multi-resolution algorithm on thehierarchical data structure:

• Sample positions on the finest level H(0) to obtain pi onthe coarsest level H(m−1):

• loop:• Start at the coarsest level:H(i) ← H(m−1)

• Save all positions pi of level H(i) in a state vector qi.• Run linear Gauss-Seidel iterations to enforce strain

limiting constraints.• If H(i) = 0 stop else go to the next finer level:H(i) ←

H(i−1):• Sample positions on level H(i)

• end loop.

Colour maps in figure 4 show strain distributions of asoft tissue simulation. In these images, strain values arescaled to between [0, 1], where value 0 representing thegreen colour shows the zero strain of the deformationand value 1 representing the red colour indicates thelargest strain of the deformation. Images (A) and (B)compare the convergency of the constraint enforcementsbetween a multi-resolution hierarchy approach to thatof using only a single mesh with the same finest levelmesh resolution. As can be seen that the multi-resolutionalgorithm archives better strain convergency with only20 constraint iterations in total, whereas the fine onlysingle mesh simulation shows larger strains after 50constraint iterations. This test illustrates the effectivenessof the proposed multi-resolution algorithm for modellinghighly incompliant soft tissue materials. The straindistributions across different levels of a three-levels meshhierarchy are shown in images (C), (D) and (E) with 5iterations on each level.

3 DATA ACQUISITION

We developed a simple-to-build data acquisition systemfor capturing soft tissue deformations ex vivo, whichwas used to record the indentation and stretch testson ex vivo samples. Simulation results were validatedagainst results obtained by these tests. While invivo experiments can measure soft tissue behaviorsin its physiological state [Samur et al., 2007], ex vivotests, however, offer good controls over laboratoryexperimental conditions [Brouwer et al., 2001].

3.1 Experimental setupFor indentation tests, the force measurement wasobtained by a force contact prob (a long thin solidshaft of different tip shapes). A square force sensingresistor of 1.5inch × 1.5inch was attached to the probtip for measuring tissue force responses. Figure 5 showsschematics of the system. The force sensor was connectedto a Phidget Voltage Divider and a USB Interface Kit

Fig. 5. (A): Overview of the data acquisition system and itscomponents: the force probe was inserted into the trocar coverthat was fitted with a linear position sensor. The force sensorwas attached to the tip of the probe and A/D interface kit wasconnected to a computer on Windows operating system, and theforce-displacement measurement was taken from ex vivo tissuesamples. (B): Stretching experiment setup.

8/8/8 by Trossen Robotics for A/D converting con-trolled from Windows. A force-displacement transducer(linear position sensor) was fitted inside a trocar coverin which the probe was sliding through inside to takethe measurement of force-displacement for indentationtests.

Tests were made along a user defined straight pathwith a constant velocity. The force sensor has thepressure range [1.5psi, 150psi] and the actuation forcerange [100g, 10kg] with a repeatable force reading. Thelinear position sensor had a 6 mm wide active cavity.

The data acquisition system also consisted of a testingboard and a pair of slats with one slat fixed on the boardand another was free to move vertically and horizontally.The stretching measurements were conducted by usingthis novel experimental setup, aimed at capturing tissuestretch behaviors by measuring stretch ratios using thebiaxial tensile method [Brouwer et al., 2001]. The sameforce sensor and the linear position sensor were used fora skin sample of size 80mm × 80mm fixed between thetwo slats on the testing board. Repeated stretching testswere carried out in the two orthogonal stretch directions.

3.2 Experimental procedures

Indentation tests were aimed at validating the force-displacement responses of computer simulation results.These tests also demonstrated tissue relaxation behaviorsdue to constant and periodic step strains. The stretchtests were aimed at evaluating force-stretch behaviors ofthe simulated soft tissue.

Porcine liver and abdominal tissue samples weretested. Tissue samples (previously frozen) were procuredjust slightly over a 24-hour period postmortem. Weprecondition tissues samples as described in researchliteratures [Fung, 1993] to obtain consistent results. Allsamples were tested at room temperature (210C ± 30C).

Three indentation depths were applied with 5 mm(testing data-1), 10 mm (testing data-2) and 15 mm (test-ing data-3) to capture the force-displacement responses,respectively. Deformation examples were taken from aset of different surface points that were well distributedover surface of the sample tissues. In tissue relaxation


Fig. 4. Strain distributions shown in colour map on a soft tissue (red colour indicates large strains and green colour indicatessmall strains). Images (A) and (B) show on the finest mesh level, where using a single mesh resolution requires a large numberof iterations compared with a multi-resolution with only 20 iterations. Images (C), (D) and (E) show a three-level multi-resolutionhierarchy strain distributions, the finer the mesh is, there are less strains and the better material stiffness.

tests, a sample was indented to the pre-defined depths inseparate trials, then the indenter was kept in place untilthe tissue was fully relaxed. For stretch tests, the skintissue was stretched along each of the biaxial directionsto a pre-defined stretch length.

4 RESULTS AND DISCUSSION

4.1 Material parametersIn the context of medical simulations, the finite elementmodel with the modified Neo-Hookean material isshown in Eq.3. It is still a challenging problem toobtain material properties as Young’s modulus andPoission ratio. In this paper the elasticity parameterestimations is carried out as an inverse problemby minimizing the quadratic norm of the differencebetween the measured displacements obtained with thedata acquisition system and the displacements resultedfrom the simulation model [Eskandari, et al., 2011]. Therecovered Young’s modulus and Poisson ratio throughan off-line optimization program are then used in themulti-resolution strain limiting simulations.

4.1.1 Inverse problemAssuming soft tissues are roughly incompressibleisotropic homogeneous materials can significantly re-duce the number of elasticity parameters [Park, 2006].Furthermore, according to [Fung, 1993], Poisson ratio ofsoft tissues is selected in the range of 0.35 < ν < 0.45.During an optimization process, if the value of Poissonration is fixed, it leaves Youngs modulus as the onlyunknown.

After applying finite element method, Eq. 7 describesthe external force f and displacement u for all the nodes:

Ku = f , (7)

where K is the stiffness matrix, which depends onthe elasticity parameters (Young’s modulus and Poissonratio and the geometry of the elements; f is a vector ofexternal forces. For a 3D object with N nodes, K is a 3N×3N matrix. The soft tissue material parameters needed inthe simulator are p = [E, ν], where E is Young’s modulusand ν is Poisson ratio. For a set of given indentationforces, displacements of a group points on tested tissue

samples and the simulated soft tissues are recorded. Theobjective function to be minimized is defined as thedifference between the displacements ur(p) in the sampletissue and the displacements us(p) in simulated tissue atthe corresponding locations:

Φ(p) =1

2

∑∥us(p)− ur(p)∥2, (8)

The parameter p = [E, ν] is iteratively updated usinga descent algorithm. We employ the L-BFGS methodbased on the numerical analysis library ALGLIB [Alglib]to minimize the above objective function. At the k − thiteration, a descent direction ∆pk is computed based onthe gradient of Φ, and we search for an optimal step sizeλ along the direction based on the value and slop of thefunction:

Φ(λ) = Φ(pk + λ∆pk), (9)

where pk is the vector containing the current estimateof the elasticity parameters.The gradient of the objectivefunction is given by the chain rule:

∇Φ(p) =∑

JTD∥us(p)− ur(p)∥, (10)

where J is the Jacobian matrix of u(p) and D is amatrix with each row essentially the spatial derivativeof ∥us(p) − ur(p)∥ with respect to the j − th axis. Bydifferentiating both sides of Eq. 7, the derivatives of uwith respect to the elasticities can be computed:[

∂K

∂Ej

]u+K

{ ∂u

∂Ej

}= 0, (11)

Therefore,the Jacobian matrix for Young’s moduluscan be computed by solving Eq. 11 for each column Jj ,K{

∂u∂Ej

}=

[∂K∂Ej

]u, similarly, solving for the Jacobian

matrix for Poisson ratio. The L-BFGS optimizationalgorithm uses the history of the gradient in previoussteps to estimate the curvature and compute a sym-metric positive definite approximation of the Hessianmatrix [Nocedal, 1999]. Therefore, we do not need tosupply the Hessian to the optimizer. To acceleratethe optimisation process, while optimizing Young’smodulus, Poisson ratio is kept constant for a number


of steps, then using the resulting Young’s modulus tooptimise Poisson ratio for a number of steps, and so on.

4.1.2 Parameter estimationInitial values of elasticity parameters for optimisationare chosen based on common values used in medicalsimulation, for Youn’s modulus within the range10 kPa < E < 60 kPa and Poisson ratio between 0.3 <ν < 0.45. Table 1 lists initial material parameters and theestimated parameter values through optimisation.

TABLE 1Estimation of Elasticities

Parameter Initial Iterations Estimate

Porcine-liverE (kPa) 15 1500 16.36

ν 0.45 1500 0.44

Porcine-abdominalE (kPa) 30 2000 32.15

ν 0.40 2000 0.43

Skin-padE (kPa) 50 1500 47.75

ν 0.45 1500 0.46

Fixed number of iterations is used in optimization pro-cess for parameter examinations. The optimization is anoffline process. The elasticity parameters resulting fromthe optimization are then used in the multi-resolutionstrain limiting simulations.

4.2 Simulation of soft tissueWe developed a simulation system in C++ usingOpenGL graphics API. The multi-core architectureof CPUs offers opportunities to the parallelizationof physics computations. In our implementation, theSVD (single value decomposition) computation foreach tetrahedron element is parallel computed withmulti-threading operations. A mesh data set is designedinto tetrahedral batches which are dedicated to a list ofthreads by the multi-threading component. Since a nodecan belong to several tetrahedron elements, care mustbe taken to ensure the memory of same variables is notaccessed by different threads at the same time, otherwisethe result will be undefined. The approach taken inour system implementation is not to update the forcesdirectly on the nodes soon after the thread finishing, butto write its value into a temporary buffer that duplicatesthe shared nodes. This buffer is then accessed after theparallel pass so that each node is updated safely.

Figure 6 shows the results of computational time inmilliseconds with respect to the number of CPU coresused and the number of tetrahedra, demonstrating anoverall speed up in parallel computing SVD operations.It is worth noting that the speedup using multi-cores inparallel computing is limited by the time needed for thesequential fraction of the program where threads for one

Fig. 6. Computational time with respect to CPU cores used andthe number of tetrahedra computed.

step in the algorithm have to wait for the threads of theprevious step to signal that they are ready. Although thesynchronization and reconstruction processes introduceslight computation overhead as seen in figure 6 wherethe cost of synchronization outweighs the benefit ofparallel when using 3 CPU cores than 2 cores, it isnot a problem because the overall performance gain isdefinitely noticeable.

Tests were carried out in a single processor 8-core3.40GHz Intel Xeon CPU with 16 GB RAM. Eulerintegration scheme was employed with step lengthschosen between 0.001 to 0.004 shown in table 1. Figure 7shows computational speed up by using multi-resolutionapproach. Figure 7 shows computation times on amulti-resolution hierarchy with 3200 elements at thelevel-0, and 1800, 800 and 200 elements at levels1, 2 and 3, respectively. A single mesh with 3200elements and 200 constrained iterations requires anaverage of 40 milliseconds computation time, whereasa multi-resolution 4-levels approach requires an averageof 11 milliseconds for a total of 200 iterations (50iterations on each level). The significant speed updemonstrates that the multi-resolution approach canquickly archive convergence, because only 50 iterationsare applied to the most dense mesh with 3200 elements.The subsequent meshes in the hierarchy each has farfewer constraints that can be enforced with 50 iterations.As can be seen in figure 7, the same multi-resolutionhierarchy with 20 iterations on each level archives aninteractive 6 milliseconds computational time, whereastable 2 demonstrates much higher numbers of tetrahe-drons and iterations.

TABLE 2Time and simulation performance

Porcine-liver Porcine-abdominal Skin-padNo. tets H(0) 17.1K 16.4K 11.6K

levels 4 4 4Time/frame 8.45s 7.0s 6.58sStep length 0.004 0.004 0.001

λmax 1.59 1.5 1.5E(kPa) 16.36 32.15 47.75

ν 0.44 0.43 0.46Iterations 155 165 125

Figure 8 shows comparisons between ex vivo samples


Fig. 7. Computational time comparison of single and multi-resolution simulations: single resolution has 3200 elements;multi-resolution has 4-levels with 3200 elements (level-0), 1800elements (level-1), 800 elements (level-2) and 200 elements(level-3).

Fig. 8. Comparisons between sample tissues (A) andsimulated soft tissues (B), and a simulated heart example (C).

and simulated soft tissues using the multi-resolutionfinite element algorithm. Parameters used for simula-tions are listed in table 2. Images in column (C) showa simulated heart with parameters E = 50.0kPa, ν =0.45, λmax = 1.20.

4.3 Validation of force and displacement relation-shipWe defined a set of corresponding points both in thesample tissue and the simulated models, for whichforce and displacement measurements were taken atthese locations to validate the proposed multi-resolutionnonlinear simulation algorithm. Two ex vivo porcinetissues were tested in indentation tests, which were asize of 8 cm3 abdominal tissue porcine-abdominal and aporcine-liver tissue. A suturing skin pad skin-pad of size8 cm × 12 cm × 2 cm was also used in indentationtests, which was considered as is an ideal testingmedia for iso-tropic soft tissue materials. Nonlinearmaterial properties of these samples were evaluated asforce-displacement relationships as shown in Figure 9 forindentation tests of simulation and tissue samples.

Tissue relaxation tests evaluate the material behaviorin terms of the force-time relationship to show thematerial nonlinearity after an indentation force beingreleased. Figure 9(D) compares relaxation behaviors ofthe simulated data and tested tissue samples after 10

mm indentation depth. The close match between the twodemonstrates the capability of the proposed constrainedfinite element algorithm can effectively capture softtissues’ material properties.

4.4 Force and stretch ratio

Stretch tests are designed to measure force-deformationresponses along two biaxial directions to show ma-terial nonlinear characteristics. Literature [Fung, 1993]and [Brouwer et al., 2001] show that the material nonlin-earity can be expressed as an exponential function givenby the force-stretch ratio relationship: |f(λ)∥ = γeαλ, df

dλ =αf , λ = l

l0, where f(λ) is the measured force, l and l0

are the stretched and the rest tissue length, respectively.α and γ are coefficients of tissue stiffness which can befound through curve-fittings. Taking the derivative df(λ)

dλ ,the stress due to stretching along two biaxial directionsis dependent on stretching ratio λ. In our stretch tests,λ is the (applied) stretch in the principal directions inEq. 2.

Stretch tests were carried out on a 8 cm2 squareskin sample using the experimental setup shown infigure 5. Triangle meshes were used in our simulationfor modelling the skin deformation using the proposedmulti-resolution algorithm. We build a 2×2 deformationmatrix from dx and dX, where x is the current positionand X the reference position of a vertex. Thus, wecompute two dimensional strain-limiting along the twoprincipal stretch directions using triangular elements.Figure 10 (A) and (B) compare the simulation datawith the testing data, which demonstrate the nonlinearbiphasic behavior of skin tissue under stretch forces withhighly stiff incompliant material behaviors. Figure 10 (C)captures nonlinear characteristics of the skin and showsa simulation example of the proposed algorithm ona three-level multi-resolution hierarchy, and simulationparameters were E = 180, ν = 0.45.

5 CONCLUSION AND FUTURE WORK

We present a novel strain based constraint finite elementmethod for simulating nonlinear homogeneous softtissues efficiently. The algorithm is capable of modelingrich nonlinear deformations in a straightforward finiteelement framework. The proposed algorithm achievesan average of 11 milliseconds computational timeinteractive frame rate on a 4-level multi-resolutionhierarchy with 50 constrained iterations on each level,a 70 percent speed up compared with a single levelwith the same total number of iterations. Therefore,the algorithm offers effective simulations for nonlinearsoft tissue properties with highly incompliant materialcharacteristics. Simulation results were validated byexperimental data captured by a simple-to-build system.Measurements on ex vivo soft tissues were taken andcompared with simulation data. We would like toimplement a virtual reality suturing application for


Fig. 9. (A), (B)and(C): Force-displacement measurements taken from ex vivo porcine-abdominal, porcine-liver, and skin-padtissues and simulation data. (D): comparisons the relaxation processes between simulated and experiment tissues for 10 mmindentation depth.

Fig. 10. (A) and (B): A stretching experiment compares stretching strain-force responses between simulated and skin-pad sample;(C): A simulation example of horizontal stretch of simulated skin using three levels of multi-resolution triangle elements with 2048,968 and 338 triangle elements at level 0, 1 and 2, respectively.

which the proposed algorithm is applicable in terms ofisotropic and incompliant material behaviors.

We aim to improve our tests in the future workincluding the selection of tissue samples, i.e tissuesthat are mixed with glandular and fibrous materialssuch as kidney. These soft tissues will challenge thecurrent assumptions of isotropic material behavior andanisotropic materials will need to be addressed in ourfuture work. It is also worth exploring to what extent theproposed approach could capture anisotropy by usingdifferent stretch ratios with respect to each principaldirections.

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PLACEPHOTOHERE

Wen Tang received her PhD degree in Engi-neering and Computer Science from Universityof Leeds, UK. She is currently an AssociateProfessor in the School of Computing at theUniversity of Teesside, UK. He research inter-ests are in physically-based computer simulationalgorithms, data-driven analysis and interactivetechniques for applications in medicine.

PLACEPHOTOHERE

Tao Ruan Wan received his PhD degreein Engineering and Computer Science fromUniversity of Leeds, UK. He is now working in theSchool of Creative Technology at the Universityof Bradford, UK as a Senior Lecturer. Hisresearch interests are in the area of virtual realitysimulation and modelling, including medicalimage analysis, surgical simulation, interactivetechnologies in human-computer interactionsand haptic interfaces.

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