Virtual Deformation of Soft Tissue using Bulk Variables

Virtual Deformation of Soft Tissue using Bulk Variables

K. SundarajUniversity Malaysia Perlis

School of Mechatronic Engineering02600 Jejawi - Perlis

[email protected]

Abstract: We present an alternative online simulation model for human tissue. Online simulation of human tissuedeformation during surgical training or surgical assistance is becoming increasingly important within the medicalcommunity. Unfortunately, even classical simulation models find human tissue to be computationally too costly foronline simulation. In this paper, we simplify the complex biomechanical nature of human tissue within reasonablelimits to develop a mathematical model which can be used for online simulation. This simplification is based ontwo principles; volume conservation and Pascal’s Principle. Volume conservation is inherent to many organs in thehuman body due to the high concentration of blood (almost incompressible liquid) in them. Given an externallyapplied force, we use Pascal’s Principle to obtain the global deformation vector at each time-step during simulation.

Key–Words: Virtual Reality, Soft Tissue Simulation, Surgical Simulators.

1 Introduction

Human tissue is deformable. Hence, modeling thisbehavior is essential for applications that involve softtissue simulation. In a medical simulator for example[1], human tissue is generally represented by a geo-metrical model and a physical model. Often, a combi-nation of the two is referred to as a numerical model.In current surgical simulation systems, the numericalmodel may be subjected to real-time interactive de-formations and haptic force feedback under differentsurgical tool based gestures [2]. So accurate modelsneed to be designed to realize the consistency betweenthem. In other words, there is a need to model thebiomechanical nature of soft tissue by using mathe-matical equations so that consistent visual deforma-tion and haptic force feedback can be provided inter-actively to the user.

Soft tissue, being complex physically and geo-metrically, is often divided into a set of smaller ele-ments to facilitate analysis. Over the past fifteen yearsor so, many mathematical models that are based onthe concept of discrete elements have been proposedfor soft tissue. In practice, most of these models aresimplified for soft tissue simulation. This results in atrade-off between physical accuracy and computationefficiency. Such an adaptive scheme gives us varia-tions in the formulation of the above mentioned mod-els. The main concerns regarding these physical mod-els are described in the following sections.

1.1 Interactive TimeThe first concern is the effective modeling of soft tis-sue to achieve an interactive-time surgical simulation.Up to now, several models have been suggested, butnone of them has been satisfactory from the simula-tion point of view as yet. Since the simulated objectitself is very complicated and computation resourcesare limited, it is natural to consider a trade-off betweenphysical accuracy and computation efficiency. A com-promise can be done by laying more emphasis on theareas of interest, for example, linear elasticity, localdeformation or volume conservation. So this idea hasto be implemented adaptively. This adaptation schemeis now the main topic of research in biomechanicalmodels of soft tissue.

1.2 Numerical StabilityA second concern regarding soft tissue models is thenumerical resolution scheme applied to solve thesesystems. Currently we find several methods; linearstatic, nonlinear static, linear dynamic and nonlineardynamic, each being applied depending on the appli-cation and interactive-time requirements. For exam-ple, simulating cutting and tearing generally requiresa dynamic model to accurately capture the viscoelas-tic properties of soft tissue when topology changes.On the other hand, simulating large deformations orstress-relaxation may only require at most a nonlin-ear static model owing to the well-damped nature ofsoft tissue. In either case, the main issue of interest

WSEAS TRANSACTIONS on INFORMATION SCIENCE and APPLICATIONS K. Sundaraj

ISSN: 1790-0832 1707 Issue 10, Volume 6, October 2009

is always stability and rapidity of the chosen scheme,during the entire history of load application.

1.3 RealismAnother concern regarding soft tissue models is re-alism. This corresponds to identifying the physicalparameters of a model such that the behavior of themodel is close to reality. This part is the most difficultas it requires a lot of expertise and experimentation.Expertise, usually from medical professionals, is re-quired because soft tissue needs to be alive during ex-perimentation so that the results obtained from themare valid and accurate.

In this work, we present another variation ofa soft tissue simulation model based on bulk vari-ables. This model is called the Volume DistributionMethod (VDM) [3]. We are interested in simulatingdeformable objects which have an elastic shell as sur-face and are filled with an incompressible fluid. Manyhuman organs have this characteristic, for example thehuman liver can be considered as such an object. It iscomposed of two major parts; an elastic skin calledCapsule of Glisson as the surface and the Parenchymawhich is the interior that is full of liquid (≈ 95%blood).

2 Previous Works [4]It would be quite impossible to highlight all possiblevariations in soft tissue simulation models that havebeen used. These variations have been tailored for de-formable objects and developed for a certain applica-tion. However, a survey on deformable models whichare used on a virtual reality platform can be found in[5]. In this paper, the conclusion was that physicallybased models are more suitable for computer graphicssimulation. Our work is more related to the formu-lation of physically based models of soft tissue usingdiscrete elements. Within this context, so far finiteelements and particle systems have been used as dis-crete elements. We discuss these models in the fol-lowing section, highlighting the advantages and thedisadvantages which has led us to propose an alterna-tive physical model.

2.1 Particle SystemsThe method of using a particle system network whichconsists of a mesh of nodes connected by elastic linksto model soft objects has been applied by many au-thors in various fields. The mass-spring network is themost common technique used, i.e. the masses are thenodes and the springs are the edges of the mesh. This

mass-spring network is used to discretize the equa-tions of motion. The link connecting pairs of nodes al-lows the local description of elastic properties in softtissue and the masses at the nodes give inertia prop-erties. This particle system model is relatively easyto implement, computationally efficient (fast simula-tion) and numerically stable (no stiffness phenomenaappears in deformable objects).

The mass-spring network is a simple physicalmodel with a solid mathematical foundation and well-understood dynamics. Its computational burden is rel-atively small [4] and is thus suitable for interactive-time applications. Since the mass-spring network hasa simple structure, many operations like large defor-mations and topology modifications can be simulatedeasily. Furthermore, as interactions in this model arelocal between nodes, parallel computations are possi-ble. Hence, it is common that we find this model inmany applications involving soft tissue.

Mass-spring networks has been widely used in 2Dand 3D facial static and dynamic animation [6] [7]. Italso has been used for cloth simulation, video gamesand animation movies. Several methods have beensuggested to avoid numerical instability [8] [9]. Alot of research work has also been done on the mass-spring network to improve various aspects like adap-tive refinement of the parameters [10] and controllingthe isotropy or anisotropy of the material being sim-ulated [11]. [12] developed a simple but efficient al-gorithm based on the mass-spring model for micro-surgery simulation. This algorithm took advantage ofthe locality of the deformations to reduce calculationsby using a wave-propagation technique that automat-ically halts computation when deformations becomeinsignificant. Using this algorithm, they achieved anupdating frequency of 30Hz for the deformations ina suturing vessel surgery, which is compatible withinteractive-time graphic animation.

Unfortunately, this physical model has somedrawbacks. When representing a volume using bi-nary connectors, the model can lead to several prob-lems. Certain constraints like volume conservation arenot easily expressed in the model. Of course, moresprings will improve connectivity and thus producea better approximation of the volume. Thus, a vol-umetric object could perhaps be accurately modeledby an infinite amount of particles and springs, butthis is clearly not an option computationally speaking.To remedy this problem, it has been proposed to addcross springs, thereby connecting opposing corners.However, this implies that the physical behavior of theobject is intrinsically dependent on the connectivity ofthe springs. When aiming for physical realism, this isclearly a handicap. Alternatively [13] proposed theuse of angular and torsion springs, but this again is



another form of topological dependency. Also, propervalues for the constants of the mass-spring networkare not easily specified.

2.2 Finite ElementsThe finite element method is a full continuum model.It gives the equilibrium of a body when subjected toexternal forces. This is obtained by minimizing the to-tal potential energy of the system. In this method, softtissue is represented by an elementary volume. Thedeformation of this volume is often expressed usingthe Green-Lagrange tensor which has the nice prop-erty of being invariant to rotation or translation. Sincethe internal stress of the volume is proportional to thedeformation (or strain), we may obtain the forces atthe nodes if the stress-strain relationship is known.

Several extensions to the FEM model has beendone by several authors in various forms such thatit can be applied to soft tissue simulation. Thetensor-mass model discretizes the virtual organ withconformal tetrahedras [14]. It is found that whenthere is a topology modification procedure, the tensor-mass model can give more accurate results. The ini-tially proposed tensor-mass model only accommo-dated small displacements, but [15] made modifica-tions to the model for large displacements by usingnonlinear strain tensors and anisotropic material laws.In the hybrid-elastic model [16], there is a combina-tion of a quasi-static precomputed linear elastic modeland a tensor-mass model. Thus, the hybrid-elasticmodel takes advantage of the good properties of thecombined models. Another combination of models toproduce a hybrid model system can be found in [17].In [18] the hybrid-elastic model is used to simulate ahepatectomy surgical procedure.

On the whole, FEM has too heavy a computa-tion burden to achieve accurate and interactive-timeresults. For complicated objects, under linear analy-sis, we may be interested only on the parts that can beseen (the surface). It is then possible to condense thematrix equation and apply some precomputation tech-niques to reduce computation [19]. In spite of this,we believe that FEM is not suitable for interactive-time applications for the present CPU capacity. Butif there is no topology changes, it is possible to ob-tain real-time deformations by using precomputation[20]. Nevertheless this is limited to small deforma-tions which can be a handicap for soft tissue simula-tions. But [21] have proposed an implementation of anexplicit formulation of FEM taking into account largedeformations and topology changes. They managedto perform cutting on a virtual human liver by consid-ering human behavior and limiting stress conditions.

However, the finite element method (FEM) was

originally intended for small deformations. Large de-formations can be simulated but at the loss of accu-racy. Furthermore, the physical behavior of soft tis-sue, particularly volume conservation, which is re-flected in the choice of the deformation tensor and thestress-strain material tensor is arbitrary. It generallydepends on the application intended for. Neverthe-less, the finite element method has the best approxi-mation of deformation of an elementary volume andas such produces the most realistic physical simula-tion for soft tissue.

3 Volume Distribution Method(VDM) [3]

VDM is a surface based method that allows the com-putation of a global deformation vector produced byan external load vector. It only requires the surfaceto be discretized with the inside being transparent tothe model. The interior of the object is assumed to befilled by some incompressible fluid. This fluid acts asthe medium that transfers the change in energy expe-rienced by the deformable object due to a change instate from equilibrium.

Figure 1: A deformable object with the surface dis-cretized and a zoomed view of the surface. The inte-rior of this object is filled with some incompressiblefluid.

Consider a deformable object of volume V andsurface area A that is represented by Q discrete sur-face elements of vectorial surface area S as shown inFig. 1. Each element has M nodes and the entire sur-face has N nodes. Let each node i be shared by kneighboring elements and connected to j neighboringnodes. The following can be obtained,

Ai =∑k

Sk

M(1)

Vi = V‖Ai‖A

(2)



where A and V are the distributed area and volumerespectively for each node. When a force is applied toa node on the surface, deformation is produced. How-ever, this deformation is a result of the applied forceF and skin tension T . They are given as,

Fi = Bi∆Vi

Vi(3)

Ti =∑j

Bij∆Vi −∆Vj

Vi(4)

whereBi andBij are the modulus of elasticity and theconnectivity modulus of elasticity constants of node irespectively. In the VDM model, stress is a functionof strain and strain is related to a change in volume,

εi =∑j

∆Vi −∆Vj (5)

σi =Bi

Viεi (6)

where ε and σ are the strain and stress respectively.Equilibrium of the system at each node is obtainedwhen the external pressure E is equal to the internalpressure I . They are given as,

Ei = Oi + Fi + Ti (7)

Ii = Ui + Ci +Gi (8)

where O is the surrounding environmental pressure,U is the pressure of the incompressible fluid, C is thepressure when contact is applied andG is the pressuredue to the effects of gravity. G is given as,

Gi = ρigiδi (9)

where ρ is the density of the incompressible fluid andδ is the measured hydrostatic distance of the node dueto the contained fluid. By considering equilibrium ofall nodes, the following is obtained using index nota-tions,

Bi∆Vi

Vi+

∑j

Bij∆Vi −∆Vj

Vi−∆Pi

= Ci + ρigiδi ∀ i = 1 . . . N (10)

where ∆P = U − O. By applying Pascal’s Princi-ple which gives constant change in pressure through-out the deformable object, the index i can be removedfrom ∆Pi. We can now add a boundary conditionto our system. The incompressibility of the fluid im-poses the constraint that the volume of the deformableobject is maintained at all times. This can be stated as,

N∑i

∆Vi = 0 (11)

Since we are interested to obtain the global dis-placement vector, ∆V can be rewritten as,

∆Vi = Ai∆Li (12)

where ∆L is displacement vector of a node. We nowhave 3N + 1 equations and 3N + 1 unknowns; ∆Li

for i = 1 . . . N and ∆P . These equations can be as-sembled in the following form,

K∆L = R (13)

where K is the state matrix of the VDM assemblage,∆L is the global deformation vector and R is the loadvector which consists of applied contact pressure Cand the hydrostatic pressure G terms. During run-time, this equation is solved using standard numericalmethods and the geometrical model is updated.

In our experiments, a nonlinear analysis was con-ducted whereby all nonlinear terms are updated ateach time-step. This amounts to simulating large de-formations and large strains. In this case, for largesystems, a simple inversion or preconditioning of thestate matrix at each time-step may be computation-ally expensive for interactive-time applications. How-ever, the rapid increase in computational power haspopularized iterative methods as a resolution scheme.We chose the Bi-Conjugate Gradient (BCG) iterativemethod as our optimal resolution scheme [22] forour experiments. This method is attractive for largesparse systems because only the nonzero terms of thestate matrix is stored; hence minimal memory. Forreal-time solution of very large systems, even a so-lution in n iterations may be too expensive. How-ever, in interactive-time applications the solution ∆Lonly changes minimally from one time-step to an-other. Then, by using the previous result of the dis-placement vector as the starting guess for ∆L, we canachieve dramatic gains in speed after finding the firstsolution. The number of iterations needed to mini-mize the error below a certain tolerance is very muchsmaller than the value of N .

4 Simulation ResultsIn this section, we presents the simulation results ofthe VDM model. This simulation model was testedfor anisotropic behavior, stress distribution and finallya comparison with a classical model like FEM wasdone.

4.1 Stress DistributionTo plot the stress distribution in our VDM model, weused the cube at rest as an example again (see Fig. 2).



This time, a force to compress the cube was appliedand deformation was allowed in all directions withoutpreference. Three test points were used to observe themagnitude of stress of the cube. The results are pre-sented.

Figure 2: A test cube at rest was used as an exam-ple for the stress distribution test. 3 test points werechosen; A, B and C on the xz-plane.

The results in Fig. 3 show that the stress expe-rienced by the cube due to volumic tension is indeeddependent on the displacement and the distributed sur-face area. In other words, they are a function of volu-mic change. The stress in the x-direction is very min-imal for all the test points because these points havedisplacement vectors with small x-components. Onthe other hand, since all the test points have signifi-cant displacement in the other directions, stress is ob-served to increase in the y and z directions. Point Chas almost zero stress in the z-direction because dur-ing compression, this point has minimum distributedarea in this direction. For the stress in the y-direction,point A has minimum stress. This is due to the factthat this point is constrained not to move in this direc-tion. On the other hand, points B and C are displacedin the y-direction but pointB has a higher stress whichis due to the higher net volumic change experienced ascompared to point C.

4.2 Anisotropic BehaviorTo test this behavior, we compared deformationcurves of three points on a cube at rest that wereplaced on the xy, xz and yz planes respectively (seeFig. 4). In this test, a force was applied to com-

(a)

(b)

(c)

Figure 3: (a) Stress distribution along the x-direction.(b) Stress distribution along the y-direction. (c) Stressdistribution along the z-direction.



press the cube at rest. We first allowed deformationin all directions. We then changed the bulk modulusto observe the behavior of the cube. The results fromthe deformation curves show that the VDM model re-spects the imposed anisotropic constraints.

Figure 4: A test cube at rest was used as an examplefor the anisotropic behavior test. 3 test points werechosen; A on the xy-plane, B on the xz-plane and Con the yz-plane.

From the results shown in Fig. 6, we can see thatthe cube behaves differently when the bulk modulus ischanged. In the first test, the displacement of pointsBand C coincide. This is because of the uniform bulkmodulus. Hence, these points should move equally tomaintain conservation of volume. In the second test,point B is on the xz-plane which has bulk modulusset to infinity. From the displacement curves, pointB is seen to have nearly zero displacement. On theother hand, point C has a larger displacement vectorin time. This is due to the constraint of conservationof volume. In the last test, deformation was preferredin the y-direction. This is observed in the displace-ment curve of pointB by comparing with the first test.PointC however, has a smaller displacement vector tomaintain volume conservation. We note that point Ahas a constant displacement curve in all the three tests.This is due to the constraint of type contact applied tothis node. In conclusion, by changing the bulk modu-lus that is associated to the nodes, deformation can bepreferred in a particular direction.

4.3 Comparison with FEMTo investigate the accuracy of the VDM model, we de-cided to compare it with a well known classical modellike the FEM model. In these experiments, two beam

mesh as shown in Fig. 5 with similar rigidity charac-teristics was used. One end of the beam was fixed andthe other end was subjected to a displacement vectordescribing various types of large deformations. Thefinal configuration of the beam was observed and thedisplacement vectors of several points along the beamwas compared.

Figure 5: A test beam with one end fixed was used asan example for the comparison test. 3 test points werechosen; A, B and C along the y-axis.

From the results, we can see that there is a dif-ference in the behavior of the nodes but the generalshape of the beam seem to be identical. We observedthe difference from one tine step to another. In thefirst test for stretching, a systematic increase of about5% in the average error of the curves are observed forall the test points. When a force to bend the beamwas applied, a constant systematic increase of about2% in the average error is observed between all thetest points. Twisting was applied in the last test whereagain a systematic increase of about 5% in the averageerror of the curves are observed for all the test points.

The difference between the models is neverthe-less expected. FEM is a volumic model as comparedto VDM which is surface based. Also, the physicalparameters of FEM and VDM are not easily matched.The error in the rigidity constant is another source oferror in the results. It is unclear how a deformablebeam would behave under externally applied forces.But, we would like to note that realism was rather ob-served in the VDM model. At each time-step, we cal-culated the volume of the beam of the two models andfound that volume conservation was rather observedin the VDM model. In conclusion, there is a differ-ence between the two models, but if we would liketo observe volume conservation, VDM seems to be abetter choice.



(a)

(b)

(c)

Figure 6: (a) Compression with uniform bulk modulus. (b) The bulk modulus along the x-direction was set toinfinity. (c) The bulk modulus of the cube was set such that deformation is preferred in the y-direction.



(a) (b)

(c) (d)

Figure 7: Stretching of a beam with one end fixed and a force applied along the y-axis at the other end. Finalconfiguration of the beam with the, (a) VDM model and (b) FEM model. (c) The variation of the magnitude of thedisplacement vector of the 3 test points. (d) The variation in total object volume during stretching.



(a) (b)

(c) (d)

Figure 8: Bending of a beam with one end fixed and a force applied along the x-axis at the other end. Finalconfiguration of the beam with the, (a) VDM model and (b) FEM model. (c) The variation of the displacementvector of the 3 test points. (d) The variation in total object volume during bending.



(a) (b)

(c) (d)

Figure 9: Twisting of a beam with one end fixed and a force applied to rotate along the xz-plane at the otherend. Final configuration of the beam with the, (a) VDM model and (b) FEM model. (c) The variation of thedisplacement vector of the 3 test points. (d) The variation in total object volume during twisting.



5 ConclusionAn alternative soft tissue simulation model based onbulk variables like pressure, volume and modulus ofelasticity has been presented. This model is surfacebased, hence it’s complexity is in general lower thana volume based simulation model like FEM. A keyfeature of VDM is the absence of discretization of theinterior. The physical properties of soft tissue in theVDM model depends on the organ being simulated.If for example a liver is being simulated, since it is95% irrigated by blood, we require the density ρ andmodulus of elasticity B of blood. Experiments to de-termine these parameters must be conducted. Sim-ulation results show that the behavior of the VDMmodel follows a similar pattern like the FEM modelfor large deformations like stretching, bending andtwisting but volume conservation was observed muchmore in VDM. In addition, these deformation patternscould be obtain much faster as VDM is computation-ally less costly. These results were obtained using asimple beam mesh to simulate soft tissue fibers.

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Virtual Deformation of Soft Tissue using Bulk Variables

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