1. Computational biomechanics for image-guidance Why and how? Zeike Taylor
CMIC Image-Guided Interventions workshop series
Workshop I: Biomechanics for IGI

2. Why?

3. Why?
Organs deform between pre- and intra-operative stages, but the deformation cannot be random: they are solid objects, and therefore must conform to physical laws.
 Plausible to restrict their motion using biomechanical models.

4. How?

5. The modelling process in engineering
Surgical scenario, disease process…
REALITY
Physical model
Mathematical model
Numerical solution
Numerical result
Conceptualisation, discarding of irrelevances…
Eqns of continuum mechanics (PDEs), constitutive model, boundary conditions,…
Finite element methods, Boundary element methods, Meshless methods, Finite difference methods…

6. Registration as a solid mechanics problem
EXAMPLE
We have an organ for which we know the initial shape.
We apply some loads/constraints to it.
We would like to know the resulting deformed shape.
REALITY
Physical model g
Mathematical model
Numerical solution
Numerical result

7. Registration as a solid mechanics problem
Body assumed to be a continuum: continuous distribution of matter (as opposed to atomistic approach).
Response is purely mechanical  electrical, chemical, thermal, (quantum!) effects assumed to be insignificant.
REALITY
Physical model g
Mathematical model
Numerical solution
Numerical result

8. Registration as a solid mechanics problem
KINEMATICS
REALITY
Position:
Displacement:
Deformation gradient:
Strain (Green-Lagrange):
Physical model
Mathematical model
Numerical solution
Source: Holzapfel, G.A.: Nonlinear Solid Mechanics. John Wiley & Sons, New York, 2000.
Numerical result

9. Registration as a solid mechanics problem
STRESS
Cauchy (true) stress, σ, is roughly the forces per unit area inside the body.
More useful from computational point of view is the Second Piola-Kirchhoff stress, S:
Stress is related to strain through the constitutive equation:
The strain energy function, W, differentiates one material from another.
REALITY
Physical model
Mathematical model
Numerical solution
Numerical result

10. Registration as a solid mechanics problem
PRINCIPLE OF VIRTUAL WORK
To the body occupying domain Ω, bounded by surface ∂Ω, we apply the following loads/boundary conditions:
Body forces fB
Surface tractions fS acting on surface
Prescribed displacements U=Ū on surface
Deformations within the body must satisfy:
REALITY
Physical model
Mathematical model
Numerical solution
Internal virtual work
External virtual work
Numerical result

11. Registration as a solid mechanics problem
SUMMARY
Want to solve
for deformation field, subject to loads/BCs: fB
fS on
U = Ū on
and assuming the constitutive relation:
REALITY
Physical model
Mathematical model
Numerical solution
Numerical result

12. Registration as a solid mechanics problem
FINITE ELEMENT METHOD
Discretise the body into a mesh of nodes and elements.
Interpolate field variables (displacements) within elements.
Solve for displacements at nodes only.
REALITY
Physical model g
Problem becomes solution of a large system of ODEs:
Mathematical model
Numerical solution
Numerical result

13. Registration as a solid mechanics problem
EXPLICIT TIME INTEGRATION
Central difference method yields a formula for incrementally computing node displacements:
Where
 Well suited to parallel execution, e.g. GPU
REALITY
Physical model
Mathematical model
Numerical solution
Numerical result

14. Example applications BRAIN SHIFT
Wittek, A, et al: Patient-specific model of brain deformation: Application to medical image registration. J. Biomech. 40, , 2007.
Brain shifts when skull is opened.
Corresponding displacements applied on exposed model surface.
Position of internal structures is computed.

15. Example applications LUNG MOTION
Zhang, T, et al: A novel boundary condition using contact elements for finite element based deformable image registration. Med. Phys. 31(9), , 2004.
-ve pressure applied to lung surface. Pleural surface modelled as contact.
Deformation field computed.
CT images warped.

16. LOADS/BOUNDARY CONDITIONS! This is the real research question!
Key differences?
What makes the brain shift simulation different from the lung motion simulation?
LOADS/BOUNDARY CONDITIONS!
Fundamental question when applying biomechanics to image-guidance:
“How do we load/constrain the model?“
Equivalently:
“How do we generate forces/displacements intra-operatively?“
This is the real research question!
Examples: gravity loading, direct displacement, surface matching, landmark matching, contact modelling, intensity-based pseudo forces…

17. Simulating with CMIC TLED
CODE STRUCTURE
Problem-specific loads/BCs
XML file
tledModel
Mesh, materials
tledSolver
tledSolverCPU
Solution data
Constraint data
tledMatlabInteractor
tledSolverGPU
or file, etc…
tledConstraintManager
Constraints at each step

18. Simulating with CMIC TLED
XML MODEL INPUT
<?xml version="1.0" encoding="utf-8"?>
<Model>
<!-- cubehex1  -->
  <Nodes DOF="3" NumNodes="8">
</Nodes>
  <Elements NumEls="1" Type="H8">
</Elements>
<ElementSet Size="1">
<Material Type="NH">
<ElasticParams NumParams="2">  
</ElasticParams>
</Material>
  0
  </ElementSet>
<Constraint DOF="0" LoadShape="RAMP" NumNodes="4" Type="Disp">
  <Nodes> </Nodes>
  <Magnitudes Type="UNIFORM">0.03</Magnitudes>
  </Constraint>
<Constraint DOF="0" NumNodes="4" Type="Fix">
  <Nodes> </Nodes>
<Constraint DOF="1" NumNodes="8" Type="Fix">
  <Nodes> </Nodes>
<Constraint DOF="2" NumNodes="8" Type="Fix">
<SystemParams>
  <TimeStep>0.001</TimeStep>
  <TotalTime>5</TotalTime>
  <DampingCoeff>0</DampingCoeff>
  <HGKappa>1</HGKappa>
  <Density>1000</Density>
  </SystemParams>
</Model>

19. Discussion…

20. Some useful references
SOLID MECHANICS
Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wiley & Sons, New York, 2000.
The best book I’ve come across. Covers essential basics through to some fairly advanced constitutive modelling. Best of all, written by a biomechanist, so it has a distinct slant in this direction.
THE FINITE ELEMENT METHOD
Bathe, K.-J.: Finite Element Procedures. Prentice Hall, Upper Saddle River, N.J., 1996.
Covers nonlinear aspects very thoroughly, although maybe hard work if you’ve no experience on the topic. Also has good coverage of nonlinear continuum mechanics theory.
Zienkiewicz, O.C. and Taylor, R.L.: The Finite Element Method, 5th Ed. Butterworth-Heinemann, Oxford, 2000.
Arguably the standard FEM text. Three volumes covering linear and nonlinear solid mechanics problems, and fluid mechanics applications.
Liu, G.R. and Quek, S.S.: The Finite Element Method: A Practical Course. Butterworth-Heinemann, Oxford, 2003.
More basic text which only covers the linear theory, but provides a simpler introduction to aspects of the method.