334
H. Courtecuisse et al.
applied on the FE model (H i is a sparse matrix whose dimension is the number of
constraints times the number of degrees of freedom of the FE model). Moreover,
even for a relatively coarse FE mesh, computing the inverse of K i −1 at each
simulation step would be too expensive, preventing an intra-operative use of the
method. Instead we used the compliance warping method introduces by [36] where
the Delassus operator is approximated using the stiffness matrix inverted at the
initial step and nodal rotations: W = H i R i K 0 −1 R i T H i T , where R i is the current
nodal rotation derived from the corotational formulation [34].
Finally a modified Gauss-Seidel solver provides λ satisfying the KKT conditions
(see [37] for details). At equilibrium, i.e. u FE = 0, the displacement field of the
FE model is used to warp the pre-operative MRI image.
13.3.4 Constraint-Based Iterative Registration
Let M be the pre-operative model, P be the collision models (obtained with the
barycentric mapping J) and I the intra operative data. Three types of constraints are
used to match M to I:
Boundary conditions: Bilateral constraints are applied between the external
surface of the brain model M and P dura . Tissue is then prevented from exiting
the skull cavity and can slide in contact with the dura mater. These constraints
are removed in the craniotomy area and identified in the ultrasound B-mode
images, so that tissues can sag. Finally, a few nodes are fixed in the cerebellum
area.
Cortical surface: Unilateral constraints force M to remain under the US probe
footprint I probe . However, tissue can sag and is not necessarily in contact.
Vessels registration: The pre- and intra-operative vascular trees P vessels and
I vessels are registered using bilateral constraints.
All these constraints are integrated and solved with the same formalism,
described in Eq. (13.6). At the end of each simulation step, a λ force is computed
allowing to impose displacements on the model satisfying the constraints. Yet, a
major difficulty of this constraint based lies in the uncertainty during the pairing
step. Indeed, although pre- and intra-operative data represent the same anatomical
structures, their characteristics and quality may differ greatly due to noisy data and
calibration errors, inherent in clinical applications. For this reason, we introduce the
notion of sliding constraints. We voluntary under sample the constraints, allowing
this way to partially solve the registration problem while letting the mechanical
model deform and refine pairing of data over the iterations. The overall iterative
process is depicted in Fig. 13.4 and in Algorithm 13.1.
H. Courtecuisse et al.
applied on the FE model (H i is a sparse matrix whose dimension is the number of
constraints times the number of degrees of freedom of the FE model). Moreover,
even for a relatively coarse FE mesh, computing the inverse of K i −1 at each
simulation step would be too expensive, preventing an intra-operative use of the
method. Instead we used the compliance warping method introduces by [36] where
the Delassus operator is approximated using the stiffness matrix inverted at the
initial step and nodal rotations: W = H i R i K 0 −1 R i T H i T , where R i is the current
nodal rotation derived from the corotational formulation [34].
Finally a modified Gauss-Seidel solver provides λ satisfying the KKT conditions
(see [37] for details). At equilibrium, i.e. u FE = 0, the displacement field of the
FE model is used to warp the pre-operative MRI image.
13.3.4 Constraint-Based Iterative Registration
Let M be the pre-operative model, P be the collision models (obtained with the
barycentric mapping J) and I the intra operative data. Three types of constraints are
used to match M to I:
Boundary conditions: Bilateral constraints are applied between the external
surface of the brain model M and P dura . Tissue is then prevented from exiting
the skull cavity and can slide in contact with the dura mater. These constraints
are removed in the craniotomy area and identified in the ultrasound B-mode
images, so that tissues can sag. Finally, a few nodes are fixed in the cerebellum
area.
Cortical surface: Unilateral constraints force M to remain under the US probe
footprint I probe . However, tissue can sag and is not necessarily in contact.
Vessels registration: The pre- and intra-operative vascular trees P vessels and
I vessels are registered using bilateral constraints.
All these constraints are integrated and solved with the same formalism,
described in Eq. (13.6). At the end of each simulation step, a λ force is computed
allowing to impose displacements on the model satisfying the constraints. Yet, a
major difficulty of this constraint based lies in the uncertainty during the pairing
step. Indeed, although pre- and intra-operative data represent the same anatomical
structures, their characteristics and quality may differ greatly due to noisy data and
calibration errors, inherent in clinical applications. For this reason, we introduce the
notion of sliding constraints. We voluntary under sample the constraints, allowing
this way to partially solve the registration problem while letting the mechanical
model deform and refine pairing of data over the iterations. The overall iterative
process is depicted in Fig. 13.4 and in Algorithm 13.1.