152
K. Miller et al.
6.3.5 Solution Algorithm
A suite of efficient algorithms for integrating the equations of solid mechanics
and its implementation on a graphics processing unit for real-time applications
are described in detail in Chap. 10 and [11, 108]. The computational efficiency
of this algorithm is achieved by using a total Lagrangian (TL) formulation [109]
for updating the calculated variables and an explicit integration in the time
domain combined with mass proportional damping. In the TL formulation, all
the calculated variables (such as displacements and strains) are referred to the
original configuration of the analysed continuum [111]. The decisive advantage of
this formulation is that all derivatives with respect to spatial coordinates can be
precomputed. The total Lagrangian formulation also leads to a simplification of the
material law implementation as these material models can be easily described using
the deformation gradient [108].
The integration of equilibrium equations in the time domain was performed using
an explicit method. When a diagonal (lumped) mass matrix is used, the discretised
equations are decoupled. Therefore, no matrix inversions and iterations are required
when solving non-linear problems. Application of the explicit time integration
scheme reduces the time required to compute the brain deformations by two orders
of magnitude in comparison to implicit integration typically used in commercial
finite element codes like ABAQUS [112]. This algorithm is also implemented on
GPU (NVIDIA Tesla C1060 installed on a PC with Intel Core2 Quad CPU) for realtime computation [11] so that the entire model solution takes less than 4 seconds on
commonly available hardware.
The application of the biomechanics-based approach does not require any
parameter tuning, and the results presented in the next section demonstrate the
predictive (rather than explanatory) power of this method.
6.3.6 Results and Validation
6.3.6.1 Qualitative Evaluation
Deformation field The physical plausibility of the registration results is verified by
examining the computed displacement vector at voxels of the pre-operative image
domain. The deformations are computed at voxel centres only for a region of interest
near the tumour.
Overlap of edges To obtain a qualitative assessment of the degree of alignment
after registration, one must examine the overlap of corresponding anatomical
features of the intra-operative and registered pre-operative image. For this purpose,
tumours and ventricles in both registered pre-operative and intra-operative images
can be segmented, and their surfaces can be compared [97]. Image segmentation
is time-consuming, subjective, not fully automated and not suitable for comparing
K. Miller et al.
6.3.5 Solution Algorithm
A suite of efficient algorithms for integrating the equations of solid mechanics
and its implementation on a graphics processing unit for real-time applications
are described in detail in Chap. 10 and [11, 108]. The computational efficiency
of this algorithm is achieved by using a total Lagrangian (TL) formulation [109]
for updating the calculated variables and an explicit integration in the time
domain combined with mass proportional damping. In the TL formulation, all
the calculated variables (such as displacements and strains) are referred to the
original configuration of the analysed continuum [111]. The decisive advantage of
this formulation is that all derivatives with respect to spatial coordinates can be
precomputed. The total Lagrangian formulation also leads to a simplification of the
material law implementation as these material models can be easily described using
the deformation gradient [108].
The integration of equilibrium equations in the time domain was performed using
an explicit method. When a diagonal (lumped) mass matrix is used, the discretised
equations are decoupled. Therefore, no matrix inversions and iterations are required
when solving non-linear problems. Application of the explicit time integration
scheme reduces the time required to compute the brain deformations by two orders
of magnitude in comparison to implicit integration typically used in commercial
finite element codes like ABAQUS [112]. This algorithm is also implemented on
GPU (NVIDIA Tesla C1060 installed on a PC with Intel Core2 Quad CPU) for realtime computation [11] so that the entire model solution takes less than 4 seconds on
commonly available hardware.
The application of the biomechanics-based approach does not require any
parameter tuning, and the results presented in the next section demonstrate the
predictive (rather than explanatory) power of this method.
6.3.6 Results and Validation
6.3.6.1 Qualitative Evaluation
Deformation field The physical plausibility of the registration results is verified by
examining the computed displacement vector at voxels of the pre-operative image
domain. The deformations are computed at voxel centres only for a region of interest
near the tumour.
Overlap of edges To obtain a qualitative assessment of the degree of alignment
after registration, one must examine the overlap of corresponding anatomical
features of the intra-operative and registered pre-operative image. For this purpose,
tumours and ventricles in both registered pre-operative and intra-operative images
can be segmented, and their surfaces can be compared [97]. Image segmentation
is time-consuming, subjective, not fully automated and not suitable for comparing