Surgical Thread Simulation

J. Lenoir, P. Meseure, L. Grisoni, C. Chaillou

Alcove/LIFL INRIA Futurs, University of Lille 1

Outline

• Context• Geometric model• Mechanical model• Physical constraints management• Results• Conclusion and Perspectives

Context

• Surgical Simulators

• Need models of thread [Pai02]

• 3-sided model– Geometric model (rendering)– Mechanical model– Collision model

Mechanical model

Geometric model

Collision model

positions

forcespositions

Outline

ContextGeometric model

• Mechanical Model

• Physical constraints management

• Results

• Conclusion and Perspectives

Geometric model (a)

• Visual model = Axis with a volumetric skinning

• Axis = a spline curveDesired continuity with few control points

0

( , ) ( ) ( )n

i ii

P s t b s q t

s : parametric abscissas [0..1]

t : time

qi : control pointsbi : basis functions

Geometric model (b)

• Implemented splines:– Catmull-Rom (De Casteljau) (C1)– Cubic uniform B-Spline (C2)– NUBS (generic)

• Skinning by a generalized cylinder

Outline

ContextGeometric modelMechanical Model

• Physical constraints management

• Results

• Conclusion and Perspectives

Mechanical model (a)

• Mass-spring model [Provot95]– Discrete models are hard to identify

• Finite Element Model [Picinbono01]– No rest shape for a thread

• Lagrangian model [Rémion99]– Well adapted for curve– Various energies support (including

continuous)Identification is automatic

Mechanical model (b)

• Lagrangian equations :

With:K Kinetic energy,βi Degree of freedom, Qi Work of the external forces,E Deformation and gravitational energy,n Number of degrees of freedom.

( ) for i 0..nii i i

d K K EQ

dt

Mechanical model (c)

• Degrees of freedom = control points positions

• Lagrangian equations applied to splines:

z

y

x

z

y

x

B

B

B

A

A

A

M

M

M

00

00

00 With :

1

0

)()( dssbsbmM jiij

iqAi

B{x,y,z}, terms of potential energies.

z}y,{x, and0..n i with iq

Mechanical model (d)Deformation energies

• Discrete deformation energy:Stretch and bend springs [Provot95], no twisting yet

• Continuous deformation energy [Terzopoulos 87], [Nocent 01]

• Approximation of a continuous stretching energy:– Current length l and rest length l0 , computed by

sampling– Evaluation of by numerical variation of

2200 ))/(1(

2

1llkElE

iq

E

iq

Mechanical model (e)Resolution

• Properties of the matrix M:– symmetric– constant over time– band (thanks to the spline locality property)

• Real-time aspect:System resolved by pre-computing a LU decomposition

• A=M-1B => resolution in O(n)

• A is numerically integrated to get qi(alpha)

1

0

)()( dssbsbmM jiij

Outline

ContextGeometric modelMechanical ModelPhysical constraints management

• Results

• Conclusion and Perspectives

Physical constraints management (a)Unilateral constraints

Collisions and self-collisions

Collision sphere of another object

• The collision model is constrained by the simulation test-bed– Approximation by spheres– Penalty method

Physical constraints management (b)Bilateral constraints

• Constraints by Lagrangian multipliers Extension of the Lagrangian equations:

E

B

B

B

A

A

A

LLL

LM

LM

LM

z

y

x

z

y

x

zyx

Tz

Ty

x

T

0

00

00

00

n

kckck

c

kkki

ii

ii

qqEqL

Lq

EQ

q

K

q

K

dt

d

0

0

),(

.)(

=> extended matrix equation system:

for c=0..nb constraints-1

for i=0..n

Physical constraints management (c) Bilateral constraints

• Some constraints managed by Lagrangian multipliers on a thread :– Fixing 3 degrees of freedom

of a point = a fixed point

– Fixing 2 degrees of freedom of a point =the point can move in 1 direction

– Fixing 1 degree of freedom of a point =the point can move on a plane

Outline

ContextGeometric modelMechanical ModelPhysical constraints managementResults

• Conclusion and Perspectives

Results (a)

Computer: Pentium IV1.7 GhzNumerical integration: Implicit Euler [Hilde01]Energy: Springs

Number of controlpoints (n)

Number ofconstraints

Time computation(ms)

30 6 4.5

30 9 6.1

50 9 10.7

Cost analysis :

Resolution without constraints in O(n)

Resolution with c constraints in O(cn2+c2n+c3)

Results (b)

Some videos :

Collisions The 3 types of implemented constraints

Results (c)

Some videos :

Self-collisions

Conclusion and future works

• Conclusion:– Mechanical simulation of threads in interactive time

• Future works:– Use of a correct continuous deformation energy

including twisting– Manage self-collisions via the Lagrangian

multipliers and implement others constraints– Offer a mechanical multi-resolution for more precise

interaction (knot creation, sewing…)

Thank you !!