XFEM FRAMEWORK FOR CUTTING SOFT TISSUE

Including Topological Changes in a Surgery Simulation

Luis F. Guti

errez and F

elix Ramos

Department of Computer Science, CINVESTAV-Guadalajara, Av. Cient

ıﬁca 1145, Zapopan, Jalisco, 45015, Mexico

Keywords:

XFEM, Cutting soft tissue, Surgery simulation.

Abstract:

Currently, there are many approaches in computer graphics (CG) that deal with topological changes; some of

these are non-interactive animations, unstable or not precise enough to medical applications. It has been found

that the Extended Finite Element Method (XFEM) is stable, accurate, with excellent performance and suitable

for virtual surgery in real time; nevertheless, to maintain the provided advantages, the selection and creation of

a set of CG methods is required that fulﬁll the requirements of the XFEM. We propose an embedded mapping

method that enables the relation of the XFEM elements, with the visual and collision meshes, making the user

interaction more dynamic. Furthermore, based on this new mapping method an interactive cutting algorithm is

suggested considering a geometric analysis. The XFEM, as a core of our framework, efﬁciently simulates the

topological changes consequently, making the interaction in real time possible, which will allow the creation

of more complex simulations of higher impact in the medical area.

1 INTRODUCTION

From the perspective of computer graphics in

medicine, the computer can allow surgeons or med-

ical students to analyze patients before performing

the surgery by simulating the physical behavior of the

body, giving the possibility to train the procedures,

avoiding the risks caused to patients by the lack of

experience. In order to simulate the incisions in a

realistic manner, the physical based methods are re-

quired, these give the adequate accuracy; Neverthe-

less, an interactive and real time simulation is desir-

able; therefore, it is necessary to look for efﬁciency in

the methods and the stability in the simulation. De-

spite many approaches deal with topological changes,

these do not fulﬁll the necessary characteristics previ-

ously described.

We identify the XFEM as a method that ensures a

robust simulation; however, there is no framework to

exploit its beneﬁts to make it applicable to complex

simulations as surgery simulation. Hence, it is neces-

sary to design a framework completely focused on the

XFEM in order to obtain efﬁcient responses.

This paper explains how to introduce the XFEM

in a surgery simulation including a geometric analysis

of our cutting method considering how this affects the

collision algorithm, mapping method and the remesh-

ing process. Moreover, many other aspects have been

considered in order to achieve a real time simulation

such as haptic interaction, handling multiple topolo-

gies, etc.

There are many researches that work with soft tis-

sue deformation without changing the topology, how-

ever, the majority of surgery procedures require inci-

sions and dissections; thus, this paper aims to provide

a framework that is easy to implement and suitable

for an interactive virtual surgery simulation including

topological changes.

This paper is organized as follows: a review of the

related work is analyzed in Section 2. After, in Sec-

tion 3 the design of our framework is explained based

on the XFEM and the embedded mapping method

that works as mediator between the physical origi-

nal mesh and the other meshes (visual and collision),

also the creation of a speciﬁc cutting algorithm that

quickly updates the associations of the meshes. Fur-

thermore, the implementation and testing employing

SOFA framework in 2D and 3D is showed in Section

4. Finally, in Section 5 the main contributions of our

framework are emphasized; moreover, the limitations

and the future work are also mentioned.

275

Gutiérrez L. and Ramos F. (2010).

XFEM FRAMEWORK FOR CUTTING SOFT TISSUE - Including Topological Changes in a Surgery Simulation.

In Proceedings of the International Conference on Computer Graphics Theory and Applications, pages 275-283

DOI: 10.5220/0002836402750283

 SciTePress

2 CUTTING APPROACHES

The majority of the applications oriented to medicine,

model the tissue using physically-based methods (Vi-

dal et al., 2006); the most popular is the FEM because

of its accuracy on the results. An overview of phys-

ically based models and applications can be found in

(Nealen et al., 2005).

Complex medical simulations require topological

changes; but, this is a challenging work to simulate in

real time with enough accuracy.

A surgery simulation must show the split of the

elements according to the cutting trajectory. A simple

manner to cut a tetrahedron is subdividing it into more

small sub-tetrahedrons (Bielser et al., 2003). Nev-

ertheless, subdividing can cause simulation instabil-

ity because it generates ill-conditioned elements (sliv-

ers); moreover, the subdivision increments the de-

grees of freedom (DOFs) impacting directly on the

simulation performance. Another alternative con-

sists of removing the elements in contact with the

blade (Forest et al., 2005); despite this method has

the advantage that it does not affect the simulation

stability, it is physically inaccurate and visually un-

even; another option, to avoid the previous meth-

ods, is a successive snapping of nodes to the cut-

ting trajectory (Serby et al., 2001); unfortunately,

this kind of method leads to degenerated elements.

Other approaches merge methods like snapping and

subdivision (Steinemann et al., 2006), but only non-

progressive cutting is enabled.

For fractures simulation, the virtual node algo-

rithm is proposed (Molino et al., 2004); it consists

of replicating cut elements assigning a portion of ma-

terial to each copy, it maintains the initial FEM mesh

conditioning, creating a minimal number of elements;

however, the slivers are not completely avoided; the

limitations of the virtual node algorithm were re-

solved by (Sifakis et al., 2007), avoiding slivers and

allowing arbitrary cutting of tetrahedrons in any num-

ber of sub-elements; nonetheless, both previous ap-

proaches have been tested in an ofﬂine simulation

which is undesirable for a surgery simulation.

(Je

abkov

a and Kuhlen, 2009) employ the XFEM

to physically control the mesh when a cut appears,

while the user is cutting no new elements are cre-

ated (in the original mesh); thus the simulation perfor-

mance is not greatly impacted moreover, the XFEM

avoids ill-conditioned elements. Figure 2 shows how

the XFEM allows the simulation of the discontinuities

maintaining the original mesh.

The XFEM adds local enrichment functions in

sub-regions with discontinuities; this method, pro-

posed in 1999 by (Belytschko and Black, 1999), ex-

Figure 1: The alternative mesh helps to map the visual el-

ements with its corresponding physical elements, this mesh

is displayed on the right side.

ploits the partition of unity property of ﬁnite elements

(Babuska and Melenk, 1997). The XFEM was ini-

tially utilized for fracture mechanics to simulate the

crack growth with stiff materials, this method has

been improved and can be applied to different do-

mains such as material interfaces, 3D elasto-plastic

deformations, ﬂuid mechanics, material-nonmaterial

interfaces and topology optimization considering void

spaces (Nesme et al., 2009). We explain the basics of

the XFEM in the appendix.

The XFEM was introduced to simulation of sur-

gical cuts by (Vigneron et al., 2004), implementing

a simulation of 2D MRI image with small defor-

mations. (Linblad and Turkiyyah, 2007) propose a

framework based on the FEM considering predeﬁned

boundaries for cutting and suturing; to cut arbitrary

surfaces they employ the XFEM; this approach also

allows small deformation as a linear discontinuous

FEM was used. Later, (Turkiyyah et al., 2009) de-

scribe an algorithm for cutting using discontinuous

FEM considering large deformations; nevertheless,

this approach is described considering only the sur-

face triangular mesh.

3 XFEM FRAMEWORK

The fundamentals of the XFEM are described by

(Je

abkov

a and Kuhlen, 2009), they use the XFEM

leading into a stable and accurate simulation. Never-

theless, their approach is very general, omitting how

the XFEM must interact with other methods of CG

without losing its advantages. Thus, our objective is

to create a speciﬁc framework that exploits the ad-

vantages of the XFEM and, by setting the XFEM as a

core of the simulation, create more complex surgery

simulations which we know will be stable, accurate

and interactive in real-time.

3.1 Corotational XFEM

The co-rotational FEM has been used for some ap-

proaches such as (M

uller and Gross, 2004), show-

ing that it is physically accurate and suitable for real-

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276

Figure 2: The XFEM adds new DOFs allowing to simulate

discontinuities on the elements. This ﬁgure shows the orig-

inal physical mesh composed by tetrahedral elements, this

mesh continues even after the elements are split, those orig-

inal elements must be hidden, showing only the cut of the

model.

time applications. This formulation allows us to in-

clude large deformations using a Cauchy’s strain ten-

sor. We employ the the co-rotational XFEM formula-

tion given by (Je

abkov

a and Kuhlen, 2009).

The forces in the co-rotational XFEM, must con-

sider that the rotation of a discontinuous element will

have different behavior in each part of the cutting

plane (above and below). The rotation for the part

above of the cut plane is denoted as R

and the ro-

tation below as R

; the equations of the deformation

forces of the co-rotational XFEM are deﬁned as

∑

j=1

dV (R

− p

0 j

)

}

Above

∑

j=1

dV (R

− p

0 j

)

}

Below

(1)

where B

is the strain matrix deﬁned as B

··· B

···ψ

] where n is the number of

element nodes; c is a matrix of material properties.

From the above equation, the equations of the defor-

mation forces of the standard DOFs and the added

DOFs can be computed separately as follows

∑

j=1

i j

a j

− p

0 j

)

∑

j=1

i j

b j

− p

0 j

)

(2)

∑

j=1

i j

a j

− p

0 j

)

∑

j=1

i j

b j

− p

0 j

)

(3)

Figure 3: When an element is divided, the alternative mesh

is generated with the new virtual elements formed by the

added DOFs, this mesh allows to create a visual mapping to

show the corresponding volume. Note that in the alternative

mesh the virtual elements can overlap, however, this mesh

is only required to make the association of topologies easier.

where the positions of the element nodes are calcu-

lated

s j

= p

0 j

+ u

+ Ψ

s j

where s = a or s = b (4)

where the subscript s indicates that it can be applica-

ble for the parts above and below the cut plane.

In order to directly obtain the physical position of

the DOFs we use the shifted enrichment function (see

eq. 9); as a result the deformation forces of the added

DOFs yield







∑

j=1

−

i j

b j

− p

0 j

) if H

= +1

∑

j=1

i j

a j

− p

0 j

) if H

= −1

(5)

and the positions of the element nodes

a j



0 j

+ u

if H

= +1

0 j

+ u

+ a

if H

= −1

b j



0 j

+ u

− a

if H

= +1

0 j

+ u

if H

= −1

(6)

Employing the framework of the co-rotational

FEM proposed by (Nesme et al., 2005), thus we never

store the whole stiffness matrix, instead, we generate

the displacements by computing the dynamic equa-

tion of the FEM updating the strain matrix and com-

puting directly the forces as is shown in eq. 1. To

obtain the rotation values we use polar decomposition

as is explained by (M

uller and Gross, 2004).

3.2 Mapping Method

Considering topological changes, the XFEM adds

DOFs each time a discontinuity appears remaining the

XFEM FRAMEWORK FOR CUTTING SOFT TISSUE - Including Topological Changes in a Surgery Simulation

277

Figure 4: Added DOFs. The original nodal displacements

(on the left) of a tetrahedron are four, when the tetrahedron

is split new DOFs are added (list on the middle); in order to

use the less memory, the alternative mesh stores in the place

of the added DOFs the real position of the vertexes of the

virtual tetrahedron(on the right).

number of elements of the original mesh; thus, an el-

ement is divided using the values of the added DOFs.

Then, how should the triangular mesh be associated

to the tetrahedrons that do not explicitly exist? If the

association of visual points with the physical points

(mapping) is done directly to the original mesh (initial

FEM mesh), the result will not be correct, because the

discontinuous elements do not show the same physi-

cal meaning of the original.

We propose a FEM-XFEM mapping that allows us

to easily control the discontinuous elements; it con-

sists in creating an alternative tetrahedral mesh that

inserts elements each time the element is subdivided

as shown in Figure 1 and Figure 3. The alternative

mesh will always have the same number of DOFs

as the original mesh, storing only the real values of

the vertexes of the new tetrahedrons (i.e. virtual el-

ements) as is shown in Figure 4, which are classiﬁed

according to the side of the cut plane (i.e. above or

below). Note that if an element is split, the mapping

method creates two new virtual elements in the same

place, for that reason, it is necesary to identify them

in order to avoid mapping errors.

Another important advantage of the alternative

topology is its speed that takes to reassign the visual

or collision mesh nodes to their corresponding phys-

ical tetrahedron when a cut occurs by recording the

points associated to the tetrahedron of the alternative

mesh as is shown in Figure 5. The alternative mesh

is controlled directly from the physical topology; in

this manner, all the changes in the alternative mesh

are quickly performed and also the search of associ-

ated elements is fairly straightforward. The vertices

of the alternative topology are not DOFs; thereby, the

alternative mesh does not harm the performance of the

simulation.

Figure 5: Initially the association of points is direct, and is

updated when an element is cut. For instance the elements 3

and 4 are cut, then, two new virtual tetrahedrons are created

for each element (4 in total); after, the visual points (only

those connected with the discontinuous elements) must be

re-associated to its tetrahedrons considering the cut plane

(above + or below −).

3.3 Cutting Algorithm

A progressive cut generates many unnecessary faces

of very small sizes, instead we split the tetrahedron

until the tool crosses the whole element obtaining a

semi-progressive cut because there is a delay before

showing the incision. For a well reﬁned mesh the de-

lay in the semi progressive cut can be considered suit-

able in surgical simulations as is shown in Sec.4.3.

When the interactive tool touches a tetrahedron (indi-

cated by the collision detection algorithm), the nearest

tetrahedron to the collision point is sought, right af-

ter, all the neighbors of the tetrahedron are stored in a

list which is used to ﬁnd the next tetrahedron touched

by the cutting tool. Employing the collision detection

and the assignation of points, it is possible to quickly

extract the list of neighbors for the collided region.

If an element has three crossed edges, it can be

cut; however, it is needed to ensure that the three in-

tersections must be in different edges, the error may

happen if the tool trembles while cutting.

L1,L2,L3 //elements with 1,2,3 intersections

collisionEvent (collisionPoint)

{

if the cut starts

FirstPoint = collisionPoint

T=nearest(Mesh, FirstPoint)//nearest tetra

else //while cutting

listNT = neighbors(T); //neighbors of T

Tc = nearest (listNT, fistPoint)

SecondPoint = CollisionPoint

semiProgressiveCut(Tc)

}

semiProgressiveCut(T0)

{

cq=generateQuad(Tool)//collision quad

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278

Figure 6: The collision quad generated by the cutting tool

crosses multiple tetrahedrons in one time, the internal mesh

is created by connecting intersected points on the element

edges.

lT = neighbors(T0)

for each T in lT

for each edgeT in edges(T)

if Intersects(edgeT, cq,intersectionPoint)

if exists T in list L2

L3.add(L2.p1,L2.p2,intersectionPoint)

L2.delete(T)

else

if = exists T in list L1

L2.add(L1.p,intersectionPoint)

L1.delete(T)

else

L1.add(T,intersectionPoint)

for = each tetra j in L3

createXFEM(j)

L3.clear

FirstPoint = SecondPoint

}

The cut is achieved by the intersection of a colli-

sion quad that speciﬁes the trajectory and deepness of

the tool; this quad is updated if the size is higher of a

value speciﬁed by a constant, this size helps to avoid

to re-compute the execution of the algorithm if the

tool continues on the same point. We obtain better re-

sponses if the size of this quad is approximated to the

average sizes of the elements. The cutting algorithm

is presented in Figure 14.

When an element is divided into two new virtual

elements, the visual and collision mapping methods

are called to be updated. Therefore, the visual and

collision meshes must also divide its element accord-

ing to the cut plane and be assigned to its correspond-

ing virtual element.

Note that if the quad of collision is long, in a well

reﬁned mesh the number of tetrahedrons to split is

higher and there is possibly at least one that is not

a neighbor to the reference tetrahedron; consequently,

we can search in a neighborhood on more levels in

order to obtain the higher list of tetrahedron to com-

pare, this level must be speciﬁed depending on the

Figure 7: The collision model is mapped to the alternative

mesh, not necessarily on the surface of the elements; it can

be in the corresponding part of volume. The external object

can overlap the elements, colliding in the collision triangle.

maximum size of the collision quad (deﬁned by the

tool blade). The precision of the trajectory of the tool

blade will be better as a smaller quad of collision is

chosen.

The physical, visual and collision topologies are

different from each other, commonly the visual mesh

is more reﬁned to obtain a better look of the model,

however, it is very possible that the triangular edges

(visual topology) do not coincide with the edges of

the tetrahedral elements at the time of cutting; given

these facts, we execute the algorithm for the physical

and visual topologies in an independent manner.

3.4 Visual Mapping

The barycentric mapping is used to connect the alter-

native mapping and the visual mesh; this mapping is

only applied to show the portions of material associ-

ated to the virtual elements. In order to show how the

object is opening while cutting, we use subdivision,

despite its creation of points in each cut; these points

do not strongly impact the simulation performance.

In 3D environments the deepness must be consid-

ered, therefore, when a cut happens an internal mesh

must also be created and mapped. To this end, the new

mesh is easily created by connecting the intersection

points of the edges of the virtual tetrahedron as shows

the Figure 6. Note that if the material is very thin, it

is possible that the tool crosses both sides of the body,

requiring the subdivision and the remapping of both

sides.

The connection of visual points give place to two

different possibilities, searching if a point with the

same value and in the same side of the cut plane al-

ready exists, if this point is found it is used to gener-

ate the new triangular faces; if no point is found then

it is required to search in the list of future points to

be created, avoiding the generation of two points in

the same edge and place. The list of future points is

generated if the collision quad cuts more than one ele-

ment at the same time (considering also the deepness).

XFEM FRAMEWORK FOR CUTTING SOFT TISSUE - Including Topological Changes in a Surgery Simulation

279

Figure 8: Cutting the skin in a triangular mesh (2D).

Tetrahedra can be split in different manners as is

shown by (Steinemann et al., 2006), in the case the

element is split with a face of four points, then two

new triangles must be generated using the Delaunay

triangulation.

3.5 Collision Detection

The collision detection algorithm must be mapped to

the alternative mesh; then, if a collision appears, the

values of the forces can be sent directly to the corre-

sponding virtual tetrahedrons and the alternative mesh

will propagate the forces to the original elements, fol-

lowing with the XFEM equations.

The added DOFs of a discontinuous element that

shares an edge of non-discontinuous element, is set

to zero, ﬁxing the point on the corresponding edge,

this means, that the added DOF works only when the

neighbors (of edge) is also discontinuous.

A virtual element is able to collide only on the cor-

responding part of volume, this means that the col-

lision method only manages the forces of that spe-

ciﬁc part and the virtual element helps to obtain the

barycentric coordinates associated to collision ele-

ments in order to be deformed correctly (see Figure

7).

4 IMPLEMENTATION

In order to evaluate the approach in a medical con-

text, the responses of the approach is tested in an open

surgery. The approach can be analyzed speciﬁcally

by simulating the cut of the skin; thus, it is possible to

observe the responses of the XFEM exposed to many

elements that are small and thin (i.e. low volume),

and thereby, the robustness of the approach can be

conﬁrmed by the avoidance of sliver. The material

parameters of the skin used to test the approach con-

siders a Young’s modulus as 1 × 10

Pa, Poisson ratio

of 0.3 and the density of 1000kg/m

SOFA framework is used to implement the ap-

proach (Allard et al., 2007). In SOFA, a single object

Figure 9: Generation of DOFs (2D): comparing XFEM and

Snapping and Subdivision. While the second one impacts

the performance, the XFEM simulates an approximated ac-

curacy improving the performance of the simulation.

Figure 10: Generation of DOFs(3D): comparing XFEM and

Snapping and Subdivision. This ﬁgure shows the increase

of DOFs when the blade cuts tetrahedral elements. The test

has been performed using continuous cuts.

can be represented by multiple geometrical models

that are connected through mappings (e.g. barycen-

tric mapping).

4.1 2D: Human Face Skin

A human face model, represented with triangular ele-

ments, is compared with a method already included in

SOFA (i.e. the snapping and subdivision). The Fig-

ure 9 shows that the XFEM generates less DOFs than

snapping and subdivision; this difference increases as

the number of cuts increase. Therefore, the XFEM

allows to simulate physically the discontinuities with-

out harming the performance and the simulation sta-

bility; some images of the simulation are shown in

Figure 8

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280

Figure 11: The tetrahedral model of the hand can be cut

interactively.

Figure 12: The delay of crossing an element considering

different speeds and meshes.If the number of cuts incre-

ments the processing time also increases, impacting the per-

formance of the simulation.

4.2 3D: Hand Skin

The test consists of cutting the skin of the dorsal part

of hand; this procedure can be applicable in differ-

ent surgeries (e.g. lipoma removal). The thickness of

the skin of the hand is from 0.5 mm to 2 mm in soft

regions (Schmidt, 2003), we choose to use 2 mm as

thickness of the skin to observe the deepness of the

object.

In 3D, the physical topology is represented by a

tetrahedral mesh; the visual and collision topologies

are represented by a triangular surface mesh. The

hand model is tetrahedralized employing the mesh

generator called tetgen; the force feedback is obtained

using the phantom omni. The comparative with snap-

ping and subdivision is presented in Figure 10.

4.3 Other Remarks

In order to generate a real time simulation the time

step will be dynamically adapted according to the pro-

cessing charge. Using an implicit method ensures the

stability for all the supported simulation time steps.

The resultant images of the simulation of the skin de-

formed and dissected are shown in Figure 11 and Fig-

ure 13. Moreover, the skin can be used together with

other models to simulate an open surgery as is ex-

pressed in Figure 15.

As we are using a semi-progressive cutting we an-

alyze the Framework in a very conventional PC with

a processor Intel Core 2Duo 2GHz, RAM memory of

2G, the delay depends basically on the reﬁnement of

the physical mesh, the speed of the user while cuts

and the speed of the execution of the algorithms, as is

shown in Figure 12.

5 CONCLUSIONS

In this paper, a framework focused on the efﬁcient ap-

plication of the XFEM has been designed; this frame-

work includes the embedded design of an alternative

mapping method, that allows associating, in an easy

and direct manner, the diverse topologies when the

cut of an element appears. The mapping also stores

the association of point-tetra, which makes the update

of the assignation of visual or collision points faster

(i.e. vertices), with its corresponding virtual element

while the user is cutting; all these, considering the

side of the cut plane. Also, a semi progressive cutting

algorithm has been created. Additionally, a variety

of meshes in 2D and 3D were tested obtaining con-

siderable differences in the creation of nodal DOFs,

in consequence, it is proved that the XFEM does not

strongly impact the simulation performance, allowing

real-time simulations.

Figure 13: Interactive surgery simulation considering topo-

logical changes in real time.

However, there are many important aspects to deal

with that have not been considered such as: self-

collisions of discontinuous elements, multi-resolution

techniques, textures in the internal mesh, creation of a

fully-progressive cutting algorithm and parallelism of

the methods. All these aspects are left as future work.

This work was completely focused on the XFEM,

obtaining a simulation that fulﬁll with stability, accu-

racy, interactivity and real time; which are the proper-

ties required for a virtual surgery simulation.

XFEM FRAMEWORK FOR CUTTING SOFT TISSUE - Including Topological Changes in a Surgery Simulation

281

The inclusion of this framework will make the

generation of more complex simulations possible, in

which the interaction of diverse models (organs) that

act together can be possible and, in this manner, de-

sign simulations with major impact in the medical

area, such as the extraction of a tumor or the fully

physical modeling of one part of the body.

ACKNOWLEDGEMENTS

We would like to thank SOFA Team for the help they

bring. We also Graham Maslin for sharing the scalpel

model.

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APPENDIX A. EXTENDED FINITE

ELEMENT METHOD (XFEM)

The main idea of exploiting the partition of unity

property is to construct basis functions through prod-

ucts of classical shape functions and a local enriched

basis; allowing to generate discontinuous elements.

Hence, the equation of the displacements can be cal-

culated as

u(x) =

∑

i=1

(x)u

{z }

classical

∑

j=1

(x)ψ

(x)a

| {z }

enrichment

(7)

where Φ

(x) are the clasical shape functions; the

discontinuous enrichment functions are denoted by

(x), and the new nodal DOFs as a

. The enrich-

ment function ψ(x) can be any discontinuous func-

tion; commonly, it is the Heaviside function (Eq. 8),

another option is the shifted function deﬁned in Eq. 9.

GRAPP 2010 - International Conference on Computer Graphics Theory and Applications

282

Ψ(x) = H(x) =



+1 above the crack

−1 below the crack

(8)

(x) =

(H(x) − H

) (9)

where H

is the value of Heaviside function at the i-

th node. The shifted function consists in using the

enrichment contribution only inside of the discontin-

uous element and ignores the contribution on the bor-

ders and outside of the element.

To keep the delta property, the displacements of

the enriched nodes has to be computed as the sum

of the components u

+ Ψ

. The shifted function in

contrast with the Heaviside, directly stores the values

of the displacement in u

and the added DOFs a

are

only required to establish the displacements of an en-

riched element.

Figure 14: Cutting algorithm: elements 1-5 are frontal el-

ements and 6-9 internal elements. When the cut starts, the

ﬁrst element found is tetrahedron 1, its neighbors are ob-

tained and stored in list lT (list of tetrahedrons), the neigh-

bors are obtained considering all tetrahedrons that share at

least one vertex. For all neighbors, we search those whose

collide with the collision quad. If an element has one, two

or three edges intersecting are stored respectively in lists

L1,L2,L3. After crossing an element, a new internal mesh

is created considering the intersection points. If an element

is recorded in list L2 or L3 must be deleted from the previ-

ous list. In order to ensure that the element is fully crossed,

the crossed edges of the element must be different from the

others stored in list.

Figure 15: The skin of the hand can be used to simulate an

open surgery.

APPENDIX B. CO-ROTATIONAL

FEM

The corotational method is based on the linear FEM

using Cauchy’s strain tensor; this method stores a ref-

erence state of the elements. To compute the deforma-

tion forces, these have to be translated to the reference

state and rotated back to the current state. Therefore,

the deformation forces are expressed as follows

= R

∑

j=1

i j

− p

0 j

) (10)

where R is the rotation matrix is required to translate

from the current to initial state, p are the positions of

the element nodes in the current deformed state de-

ﬁned as p = p

+ u, in which p

indicates the initial

positions of the element nodes.

XFEM FRAMEWORK FOR CUTTING SOFT TISSUE - Including Topological Changes in a Surgery Simulation

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