332
H. Courtecuisse et al.
13.3.2.2 Probe Footprint from B-Mode Ultrasound Images
The ultrasound probe footprint located at the interface between the probe and brain
tissues is easily identifiable and can thus be extracted from B-mode images in order
to provide additional constraints at the surface of the brain. The visualisation of this
boundary arises from an ultrasound artifact: when the waves enter the tissues of
the brain, a part is reflected on the surface forming this specific band in the image
(see Fig. 13.3b). The footprint being located on the edges of the images, the outer
envelope of the volume is extracted. A low-pass filter is then applied to provide a
binary volume, and finally a point cloud is generated sampling data at 1 point every
10 mm (see Fig. 13.3c).
13.3.3 Biomechanical Model: Formulation
In the context of registration with intra-operative data, a static problem can be
considered. The governing equation is given by:
F(u FE ) + H(u col , p I )λ = 0
(13.4)
where F(u FE ) is a nonlinear function describing the internal forces according to the
displacement u FE of the FE model. H(u col , p I ) is a nonlinear function associating
Lagrangian multipliers λ used to impose constraint forces to the FE model. The
number and directions of Lagrangian multipliers depend on the collision model
positions, u col , and the positions extracted from intra-operative images, p I . In
addition, the solution u FE to this problem must satisfy a set of constraints equations
which can mathematically be represented as follows:
H(u col , p I ) = δ
(13.5)
where δ is the violation of the constraints (e.g. penetration through a boundary
surface).
13.3.3.1 Constitutive Law and Parameters
As seen in Sect. 13.2.1.2, the constitutive behaviour of the brain is known to be complex and should be described using hyper-elastic laws. However, in our constraintbased context, displacements are imposed through Lagrangian multipliers. The solution in displacements is thus weakly sensitive to the constitutive law and parameters
[12, 32]. A linear elastic law is therefore used. Following [33], Young’s modulus and
Poisson’s ratio are, respectively, set to E = 1.5 kPa and ν = 0.45. A higher stiffness
H. Courtecuisse et al.
13.3.2.2 Probe Footprint from B-Mode Ultrasound Images
The ultrasound probe footprint located at the interface between the probe and brain
tissues is easily identifiable and can thus be extracted from B-mode images in order
to provide additional constraints at the surface of the brain. The visualisation of this
boundary arises from an ultrasound artifact: when the waves enter the tissues of
the brain, a part is reflected on the surface forming this specific band in the image
(see Fig. 13.3b). The footprint being located on the edges of the images, the outer
envelope of the volume is extracted. A low-pass filter is then applied to provide a
binary volume, and finally a point cloud is generated sampling data at 1 point every
10 mm (see Fig. 13.3c).
13.3.3 Biomechanical Model: Formulation
In the context of registration with intra-operative data, a static problem can be
considered. The governing equation is given by:
F(u FE ) + H(u col , p I )λ = 0
(13.4)
where F(u FE ) is a nonlinear function describing the internal forces according to the
displacement u FE of the FE model. H(u col , p I ) is a nonlinear function associating
Lagrangian multipliers λ used to impose constraint forces to the FE model. The
number and directions of Lagrangian multipliers depend on the collision model
positions, u col , and the positions extracted from intra-operative images, p I . In
addition, the solution u FE to this problem must satisfy a set of constraints equations
which can mathematically be represented as follows:
H(u col , p I ) = δ
(13.5)
where δ is the violation of the constraints (e.g. penetration through a boundary
surface).
13.3.3.1 Constitutive Law and Parameters
As seen in Sect. 13.2.1.2, the constitutive behaviour of the brain is known to be complex and should be described using hyper-elastic laws. However, in our constraintbased context, displacements are imposed through Lagrangian multipliers. The solution in displacements is thus weakly sensitive to the constitutive law and parameters
[12, 32]. A linear elastic law is therefore used. Following [33], Young’s modulus and
Poisson’s ratio are, respectively, set to E = 1.5 kPa and ν = 0.45. A higher stiffness