A three-dimensional nonlinear meshfree algorithm for simulatingmechanical responses of soft tissue

G.Y. Zhang n, A. Wittek, G.R. Joldes, X. Jin, K. MillerIntelligent Systems for Medicine Laboratory, School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway,Crawley, Perth WA 6009, Australia

a r t i c l e i n f o

Article history:Received 8 April 2013Accepted 25 August 2013Available online 17 September 2013

Keywords:Meshfree methodsNonlinear dynamics analysisSoft tissue

a b s t r a c t

This paper presents an effective three-dimensional (3D) nonlinear explicit dynamic meshfree algorithm for thesimulation of soft tissue mechanical responses. In the algorithm soft tissues are considered to be hyperelasticand nearly incompressible materials. The algorithm is based on the element-free Galerkin (EFG) method usingtotal Lagrangian formulation and moving least square (MLS) approximation. This approximation uses arelatively large number of nodes for shape functions creation, which can significantly delay mesh distortion inlarge deformation computations. Essential boundary conditions are imposed exactly by coupling MLS shapefunctions with a finite element (FE) interpolation in the close region of essential boundary. Although volumetricintegration is not exact, the large support domains of the MLS shape functions alleviate some of the keyweaknesses of FE methods such as hour-glassing and volumetric locking. Explicit integration is performed intime domain, using a recently proposed method to calculate the critical time step. Verification against theresults obtained using the established nonlinear finite element procedures available in the ABAQUS codeconfirms the accuracy of the presented meshfree algorithm. Application of the algorithm in modeling of thebrain indentation indicates its ability to facilitate robust and accurate prediction of the organ responsessubjected to large localized deformations consistent with the loading due to surgery.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Surgical simulators present a safe and potentially effective wayfor surgical training, and can also be used in robot and computerassisted surgery for operation planning. Robust, fast and accuratenumerical algorithms for predicting soft tissue deformations havebeen identified as the key technology for the development ofpractical surgical simulators [1]. In the context of computationalbiomechanics, finite element (FE) algorithms accounting for bothgeometric (i.e. large deformations and strains) and material non-linearities have been developed [2–4]. Although the literatureindicates that the models implemented using FE algorithms canfacilitate accurate prediction of the organ responses during sur-gery [5–7], they are still not widely adopted in surgical practice,mainly because of the difficulties in handling very large deforma-tions and discontinuities due to surgical cutting.

With the latest developments, meshfree/meshless methods [8–10]have been suggested as a possible alternative to solve the abovedifficulties. A key feature of meshfree methods is that a predefinedmesh is not necessary for field variable approximation/interpolation.Various meshfree algorithms have shown very good performance inhandling mesh distortion for nonlinear problems [11–19]. However,there are still some open topics for nonlinear explicit dynamic analysis

using meshfree methods, such as handling essential boundary condi-tions effectively, conducting numerical integration, and stable timestep calculation. For the above reasons, only a few published workscan be found on applying meshfree methods for nonlinear analysis ofsoft tissue simulation [20–24]. This work expands the meshfree totalLagrangian explicit dynamics (MTLED) algorithm presented in [20]through the use of coupled MLS and FE shape functions to accuratelyimpose the essential boundary conditions, a more accurate volumetricintegration, and the use of a new stable time step estimation methoddeveloped particularly for the explicit time integration in meshfreemethods. These numerical techniques have been proposed elsewhere,and in this work we integrated them in a consistent verified meshfreealgorithm for soft tissue modeling.

In the next section we briefly introduce the formulations oftotal Lagrangian explicit dynamics meshfree algorithm, and thenpresent three important implementation issues: numerical inte-gration, treatment of essential boundary conditions and calculat-ing stable time step. Verification results are presented next,followed by the example of application in modeling of the bodyorgans (brain indentation), discussions and conclusions.

2. Meshfree total Lagrangian explicit dynamics algorithm

The presented algorithm is based on the meshfree EFG methodusing the total Lagrangian formulation, where stresses and strainsare measured with respect to the original configuration allowing

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Engineering Analysis with Boundary Elements

0955-7997/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enganabound.2013.08.014

n Corresponding author. Tel.: þ61 412612064.E-mail addresses: guiyong@gmail.com, Guiyong.zhang@uwa.edu.au

(G.Y. Zhang).

Engineering Analysis with Boundary Elements 42 (2014) 60–66

for pre-computing of most spatial derivatives before the time-stepping procedure. The stress (second Piola–Kirchoff stress) andstrain (Green strain) measures used are appropriate for thetreatment of large deformations. Explicit time integration elim-inates the need for solving large systems of equations during thetime-stepping procedure.

2.1. Moving least square approximation

Details of the MLS shape functions construction are availablein the literatures [25,26]. The discrete MLS approximation of afunction uðXÞ, denoted by uhðXÞ, is constructed by a combination ofmonomials using n nodal parameters as follows:

uhðXÞ ¼ ∑m

i ¼ 1piðXÞaiðXÞ ¼ PT ðXÞaðXÞ ð1Þ

where m is the number of terms of monomials (polynomial basis),piðXÞ are basis functions, X is the position of a material point in theinitial configuration, and aiðXÞ are the coefficients of theapproximation.

By introducing a weight function WðX�XIÞ and minimizing theweighted L2 norm (JMLS) as

JMLS ¼ ∑n

I ¼ 1WðX�XIÞ½uhðX;XIÞ�uðXIÞ�2 ð2Þ

where n is the number of nodes within the support of X. Finally bycalculating the coefficients through the minimization, the MLSapproximation is obtained by

uhðXÞ ¼ ∑n

I ¼ 1ϕIðXÞuI ¼ΦðXÞUs ð3Þ

where Us is the vector collecting the nodal parameters of the fieldvariables for all the nodes in the local support domain, and ϕIðXÞ isthe MLS shape function.

In this work, we create the MLS shape functions by using a linearpolynomial basis and quartic spline weight functions, defined as

WðX�XIÞ ¼ 1�6d2þ8d

3�3d4

for dr10 for d41

(; d¼ jX�XIj

dwIð4Þ

where dwI is the radius of the spherical influence domain of node XI .We calculate dwI using the average distance between nodeXI and its neighboring nodes, multiplied by a dilatation parameter.This technique leads to high quality MLS shape functions even forirregularly distributed field nodes.

2.2. Total Lagrangian explicit dynamics algorithm

After introducing MLS approximation into the weak form ofgoverning equations of solid mechanics using the total Lagrangianformulation, the global system of discretized equations describingthe behavior of the analyzed continuum is the following:

M €U¼ Fext�Fint�Fdamp ð5ÞwhereM is the mass matrix, U is the vector of nodal displacementsand Fint is the vector of internal nodal forces assembled as

Fint ¼ ∑NIP

I ¼ 1ðt0Ft

0BTL0

t0SÞjx ¼ xI

WI ð6Þ

where NIP is the total number of integration points (IP) distributed inthe problem domain, t

0S is the second Piola–Kirchoff stress vector attime t, t

0F is the deformation gradient between the undeformedconfiguration and the configuration at time t, WI is the weightcorresponding to integration point xI,

0V is the initial volume of theproblem domain, and t

0BL0 is the matrix of shape functions derivatives

in reference to the initial configuration in the following form:

t0BL0 ¼

ϕ;x 0 00 ϕ;y 00 0 ϕ;z

ϕ;y ϕ;x 00 ϕ;z ϕ;y

ϕ;z 0 ϕ;x

26666666664

37777777775

ð7Þ

inwhich ϕ;x, ϕ;y and ϕ;z are derivatives of shape functions with respectto x, y and z, respectively.

As the purpose of this work is to verify the algorithm rather than toconduct a complete simulation of actual surgery, a simple Neo-Hookean hyperelastic model is a logical choice. The Neo-Hookeanstrain energy potential as used in ABAQUS is adopted in thiswork [27]:

WNeo�Hooken ¼ μ02ðI1�3Þþκ0

2ðJ�1Þ2 ð8Þ

where μ0 is the initial shear modulus, κ0 is the initial bulk modulus, J isthe elastic volume ratio which equals to the determinant of deforma-tion gradient, and I1 ¼ J�ð2=3Þ trðCÞ is the first deviatoric strain invar-iant of the right Cauchy–Green deformation tensor C¼ FTF.

So the second Piola–Kirchoff stress in Eq. (6) is calculated as

t0S ¼ μ0J

�2=3 δ�13tr Cð ÞC�1

� �þκ0JðJ�1ÞC�1 ð9Þ

where δ is the 3� 3 identity matrix.In Eq. (5) Fextis the vector of externally applied forces (volu-

metric forces, surface and nodal forces) assembled as

Fext ¼ZΩΦT

I ðxÞb dΩþZΓΦT

I ðxÞτ dΓ ð10Þ

where b is the body force vector and τ is the predefined surfacetraction on the boundary. The damping force Fdamp ¼ cM _U, wherec is the damping coefficient, introduces a mass proportionaldamping which can be used to obtain the steady state solution,as presented in the next subsection.

2.3. Dynamic relaxation solution algorithm

We introduce the damping force to dissipate the kinetic energywhen that the steady state of the deformed continuum needs tobe obtained, such damping does not have to be based on actualphenomena related to e.g. material viscosity. To increase the con-vergence speed towards the deformed state, we use the dynamicrelaxation (DR) technique detailed in [28–30] and solve the dampedequation of motion using the central difference method for integra-tion in time domain. The resulting equation describing the iterationsin terms of displacements, derived from Eq. (5), is

Ukþ1 ¼UkþβðUk�Uk�1ÞþαΜ�1ðFext�FintÞ ð11ÞIn the above equation α¼ 2h2=ð2þchÞ and β¼ ð2�chÞ=ð2þchÞ,where h is a fixed time increment and k indicates the kth timeincrement. The iterative method defined by Eq. (11) is explicit as longas the mass matrix is diagonal. Therefore, we use a lumped massmatrix and a mass scaling algorithm that increases the convergencespeed of the method. The accuracy of the solution is evaluated usinga termination criterion which gives information about the absolutedisplacement error [28]

jjUkþ1�Unjjr ω

1�ωjjUkþ1�Ukjjrλ ð12Þ

where Un is the solution, ω is the spectral radius which approximatesthe reduction in error, and λ is the imposed accuracy. More detailsabout the adaptive computation of these parameters in order toincrease the convergence speed are given in [28].

G.Y. Zhang et al. / Engineering Analysis with Boundary Elements 42 (2014) 60–66 61

3. Numerical implementation

3.1. Numerical integration

Compared to finite element method (FEM), which has well definedrules regarding numerical integration, accurate numerical integrationfor meshfree methods is still an open topic. Meshfree shape functionsare generally not polynomials and are constructed using supportnodes located beyond the boundaries of integration cells, whichmakesthe numerical integration in meshfree methods much more challen-ging [31]. For meshfree methods, Gaussian quadrature over a back-ground mesh is typically used for numerical integration [9,20,31].Some efforts have been made on the numerical integration issue ofmeshfree methods, for example, the stabilized conforming nodalintegration technique proposed by Chen et al. [32–34]. By developingthe generalized gradient smoothing technique and using point inter-polation method for field function approximation [35,36], Liu et al.have developed a series of meshfree smoothed point interpolationmethods (S-PIM) [37–40], which conduct stabilized numerical inte-grations on the generated smoothing domains.

In [20,41] we used regular (equally spaced) hexahedral backgroundgrid to conduct the numerical integration in space. Such integrationgrid can be easily constructed, which offers simplicity and flexibility[20]. However, for complex geometries (such as those of many bodyorgans) it may introduce volume inaccuracy when the hexahedra areintersected by a domain boundary. On possible solution to thisproblem can be to use tetrahedral background grids that can beautomatically generated with integration cells conforming to theboundary of analyzed continuum even for complex geometries.Therefore, the meshfree algorithm we present in this study facilitatesapplication of tetrahedral background integration grids.

For tetrahedral grid, we use four Gauss points per cell, whichprovides exact integration for polynomial up to second order.Unfortunately, the MLS shape functions are not polynomials andthe Gaussian volumetric integration is not exact. Although there isno rigorous proof theory, some authors claimed [13] that theinaccuracy of numerical integration in meshfree methods can havesimilar effects to the reduced integration used in FEM to suppressvolumetric locking for nearly incompressible materials. Althoughin FEM the reduced integration can lead to other numericaldifficulties (hour-glassing), such problems can be generallyavoided in meshfree methods by making influence domains larger.

3.2. Imposing essential boundary conditions

A critical feature of MLS shape functions is that they do notpossess the Kronecker delta property (they are not interpolating).Consequently, essential boundary conditions cannot be imposed inthe same way as in FEM. A number of techniques have beendeveloped to overcome this problem, including the penaltymethod [42], the use of Lagrange multipliers in the Galerkin weakform [43] and the transform method [11].

In the explicit time integration framework, the most efficient andaccurate method for imposing essential boundary conditions appearsto be the coupling of MLS and FEM shape functions near the essentialboundary [44]. Because the modifications are only made at shapefunctions level, the method can be easily applied to all kinds ofproblems and allows direct imposition of prescribed values to the fieldvariables on the essential boundaries, just like in FEM. The couplingbetween FEM and MLS shape functions is accomplished by defining aboundary region that includes elements containing the nodes forwhich the essential boundary conditions are prescribed. The problemdomain is divided into two regions, the boundary region ΩB and therest of the domain called the meshfree region ΩM . The displacement

approximation in the boundary region is calculated as:

uhB ¼

uMLSþRðuFEM�uMLSÞ xAΩB

uMLS xAΩM

(ð13Þ

where uMLS is the displacement field approximated using the MLSshape functions as in Eq. (3), uFEM is the displacement field approxi-mated using the FEM shape functions, and R is a ramp function [44].

3.3. Stable time step calculation

We use explicit time integration in the present numerical algo-rithm because of its efficiency and suitability for parallel implementa-tion. However explicit time integration is only conditionally stable andrequires an estimation of the maximum stable time step. The criticaltime step of a numerical algorithm is directly related to the maximumfrequency of free vibration, which is determined by the mass andstiffness of the system. Belytschko et al. [45] developed critical timestep bounds for meshfree methods, but their results are only valid for2D problems with regularly distributed nodes. Furthermore, thebounds become indefinite, due to division by zero, for interpolatingshape functions or shape functions that are not strictly positive [46].

Recently Joldes et al. [47] developed an effective method ofestimating the stable time step for meshfree methods. This estimationmethod is valid for a specific case of lumped mass matrix construction,where the mass associated with an integration point is distributedequally to the nodes influencing that integration point. This approachworks well for the explicit time integration algorithm based on the EFGand is used in the present work for calculating the critical time step as

Δtcrit �minI

2ffiffiffiffiffiffiffiffiffiλImax

q0B@

1CA ð14Þ

where λImax is the maximum eigenvalue of the stiffness matrixcontribution from integration point I, estimated as λImaxrncw2 UðBL0ÞIjIðBL0ÞIjI , in which n is the number of support nodes for integrationpoint I, cw is the dilatational wave speed and BL0is the matrix of shapefunction derivatives. Following [48], the dilatational wave speed cw wascalculated as cw ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλþ2μ=ρ

p, where λ and μ are Lamé′s constants.

4. Numerical verification

A cylinder of height 0.1 m and radius 0.05 m is studied, as shown inFig. 1. The cylinder is deformed by rigidly constraining one face (z¼0),

Fig. 1. Thy cylinder is of height 0.1 m and radius 0.05 m.

G.Y. Zhang et al. / Engineering Analysis with Boundary Elements 42 (2014) 60–6662

while the opposite face (z¼0.1 m) is displaced. We use the massdensity of 1000 kg/m3. As there is vast body of evidence [2,49–60] thatsoft tissues can be regarded as hyperelastic (or hyperviscoelastic)nearly incompressible continuum, we used Neo-Hookean materialwith Young′s modulus in undeformed state equal to 3000 Pa, Poisson′s ratio of 0.49, and density of 1000 kg/m3. The loading (displacements)were applied using a smooth curve that ensures zero velocity andacceleration at the start of loading:

dðtÞ ¼ dmax U ð10�15tþ6t2ÞUt3 ð15ÞThree basic experiments are presented: compression, extension

and shear of the cylinder. The computations were first performedusing the proposed meshfree algorithm with a maximum applieddisplacement dmax ¼ 0:02 m. The obtained results were compared tothose from the commercial finite element software ABAQUS [27] usinga fine mesh of 20-node quadratic brick, hybrid, linear pressureelements (C3D20H) and the dynamic implicit solver.

For the meshfree algorithm, the dilatation parameter forinfluence domain was taken as 1.6, the problem domain wasdiscretized using 7619 nodes and time step calculated usingEq. (14) is 8.2e–5 s. The reference solution of this problem wasobtained with ABAQUS using a mesh consisting of 33,897 nodesand 7680 elements. Table 1 lists the displacement in x direction atthe middle node of the right edge of plane y¼0 and the totalreaction force on the displaced face in the direction of the applieddisplacements (along the z-axis for compression and tension, andalong the x-axis for shear). Compared with the reference solution,the maximum relative difference in both displacement and reac-tion force is smaller than 5%, which conforms the accuracy of theresults obtained using the presented algorithm. In Fig. 2, weplotted the deformation shape of plane y¼0, the displacementson the right edge of this plane and time history of the totalreaction force on the displaced face. The results obtained using thepresented meshfree algorithm closely agree with the referencesolution from ABAQUS FE solver.

5. Application in soft organ modeling: brain indentation

To illustrate the performance of the meshfree algorithm intro-duced in this study in simulation of the body organs we applied inmodeling of the experiments on the swine brain indentationconducted by [61].

5.1. Model description

The swine brain used in the experiment had a mass of 89.9 gand was approximately 92.5 mm, 62.5 mm and 28.5 mm in itsmajor axis, minor axis and height respectively. During the experi-ments the bottom brain surface was glued to a glass plate and thebrain was laterally supported using two molds (Fig. 3a) with thebrain surface firmly sticking to the molds. Therefore, the nodesdefining the bottom brain surface and the areas in contact with themolds were rigidly constrained (Fig. 3b). As in this study we focus

on evaluation of the performance of our meshfree algorithm ratherthan on detailed modeling of the interactions between theindentor and brain, the indentor was not directly simulated inthe model. The indentor was an aluminum cylinder of diameter10 mm. The loading was applied by prescribing a constant velocityof 1 mm/s (velocity of the indentor) on the nodes located in thearea that was in contact with the indentor when conducting theexperiments.

Geometry for the brain model was obtained from the magneticresonance image MRIs acquired before the experiments anddiscretized using 21,498 nodes and 115,029 tetrahedral integrationcells as shown in Fig. 3b. In the meshfree algorithm, the dilatationparameter for influence domain was taken as 1.6 and time stepcalculated using Eq. (14) is 4.3e�4 s. Neo-Hookean hyperelasticmodel was used for the brain tissue for verifying the developedalgorithm. As explained in detail in [61], the subject-specific tissueproperties (shear modulus of 210 Pa) were determined fromcompression of the cylindrical tissue samples extracted from thebrain after the indentation experiments. Poisson's ratio was 0.49and the mass density was 103 kg/m3.

5.2. Results

As shown in Fig. 4, the indentor reaction force–indentationdepth relationship predicted by the model implemented using ourmeshfree algorithm agrees very well with experimental resultsthroughout the simulation. These results also confirm that thealgorithm facilitates robust and accurate solution even for largelocal deformation of the analyzed continuum.

The likely reasons for the differences between the simulationand experiment results in Fig. 4 are that the pia mater (a think softtissue layer surrounding the brain) was not included in the brainmodel we used (Fig. 3b). As stated in Section 5.1, the purpose ofthis paper is to evaluate the performance of our meshfree algo-rithm rather than to conduct a complete simulation of actualsurgery. Given the fact that only very limited information aboutthe mechanical properties of the pia mater is available [48] andthat the anatomical structure of the soft tissue layers (meninges)surrounding the brain is still debated in the anatomical literature[62], attempt to include the pia mater in the model would obscurethe algorithm evaluation with modeling uncertainties.

6. Conclusions

To address the need for simulating biological soft tissues underlarge deformations we presented a total Lagrangian explicitdynamics algorithm based on the element-free Galerkin method.The coupling between the moving least square (MLS) and finiteelement (FE) shape functions was used to impose essentialboundary conditions, which is simple and guarantees exact results.The large support domains of the MLS shape functions eliminatesome of the weaknesses of FE method, such as hour-glassing.Although the numerical volumetric integration is not exact, it seems

Table 1Displacement at the node (0.05, 0, 0.05) in x direction and reaction force on the displacement face.

Compression Extension Shear

Displacement (m) Force (N) Displacement (m) Force (N) Displacement (m) Force (N)

Reference 0.00703 �6.9823 �0.00508 4.6673 0.00951 1.1188Meshfree 0.007315 �7.2351 �0.005144 4.7128 0.009614 1.1405Difference 4% 4% 1% 0.9% 1% 2%

G.Y. Zhang et al. / Engineering Analysis with Boundary Elements 42 (2014) 60–66 63

-0.06 -0.04 -0.02 0 0.02 0.04 0.00

0.01

0.02

0.03

0.040.05

0.06

0.07

0.08

0.090.1

x(m)

z(m

)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

1

2

3

4

5

6

7

8 x 10

z(m)

Dis

plac

emen

t in

x di

rect

ion

(m)

Reference solutionMeshfree solution

0 0.5 1 1.5 2 2.5 3-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Time(s)

Rea

ctio

n fo

rce

in z

dire

ctio

n (N

) Reference solutionMeshfree solution

-0.06 -0.04 -0.02 0 0.02 0.04 0.00

0.02

0.04

0.06

0.08

0.1

0.12

0.14

x(m)

z(m

)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-6

-5

-4

-3

-2

-1

0x 10

z(m)

Dis

plac

emen

t in

x di

rect

ion

(m) Reference solution

Meshfree solution

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.55

Reaction force

Time(s)

Rea

ctio

n fo

rce

in z

dire

ctio

n (N

) Reference solutionMeshfree solution

-0.06 -0.04 -0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

x(m)

z(m

)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

z(m)

Dis

plac

emen

t in

x di

rect

ion

(m) Reference solution

Meshfree solution

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

Time(s)

Rea

ctio

n fo

rce

in x

dire

ctio

n (N

) Reference solutionMeshfree solution

Fig. 2. Compression, extension and simple shear of a cylinder. Left column: plane y¼0 of the deformed cylinder. Middle column: displacements on the right edge of theplane y¼0. Right column: time history of the total reaction force on the cylinder's displaced face.

Fig. 3. The swine brain indentation modeled using the presented meshfree algorithm: (a) set-up of the experiments, figure based on [61]; (b) domain discretization withscattered field nodes and tetrahedral integration grid, where the pointed circle area including the nodes under prescribed displacement during the indentation.

G.Y. Zhang et al. / Engineering Analysis with Boundary Elements 42 (2014) 60–6664

to prevent volumetric locking for the studied nearly incompressiblesoft tissues materials. A dynamic relaxation scheme was used for thecomputation of the steady state solution and the critical time stepestimation method ensures the stability of the present meshfreealgorithm. The results of our computations compare very well withreference solutions obtained using the commercial FE softwareABAQUS, confirming the accuracy of the presented algorithm.Application of the algorithm in modeling brain indentation indicatesits ability to facilitate robust and accurate prediction of the organresponses.

Acknowledgments

The financial support of the Australian Research Council (DiscoveryGrant no. DP1092893) and the University of Western AustraliaResearch Development Awards 2012 is gratefully acknowledged.

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