Parallel Computation in Simulating Diffusion and Deformation in Human Brain - Parallel Computing for Bioinformatics and Computational Biology - page 139

In Ref. [4], the biomechanical simulation of the volumetric brain deformation
is conducted by modeling the brain as an elastic object with homogeneous linear
elasticity. When there is no initial stresses or strains, the deformation energy of this
elastic body submitted to externally applied forces is modeled as

1
2

σ T d F T u d , E = + (5.17)

where F is the vector denoting the forces applied to the elastic body, u the unknown
displacement vector field, the elastic body, the strain vector, and σ the stress
vector. and σ are related to each other by the constitutive equations of the material,
Eq. (5.15). Strain is linked to displacement by the assumption

L T u , where L is a =

linear operator.
The finite element method is used to generate a volumetric unstructured tetrahedral
mesh in the image domain, on which the biomechanical simulation of deformation
is performed. The volumetric deformation of the brain is obtained by solving the
displacement field that minimizes the deformation energy described in Eq. (5.17).
This yields a linear equation system, which is solved for the displacements due to the
forces applied to the elastic body

Ku =− F , (5.18)

where K is the stiffness matrix. Equation (5.18) is solved in a way such that the
derived deformation field over the entire mesh matches the prescribed displacements
at the boundary surfaces by fixing the displacements at the boundary surface nodes
to match those generated by the active surface model. The entries of the rows in K
corresponding to the nodes at which a displacement is to be imposed are set to zero
and the diagonal elements of these rows are set to 1. The force vector F is set to be
the displacement vector, to be imposed at the boundary nodes.
To achieve the timing constraint set by the real-time situation of neurosurgery
as well as robustness and high accuracy in the matching of brain image data, the
linear equation system (5.18) is implemented and solved in parallel with the PETSc
package using the generalized minimal residual (GMRES) solver with a block Jacobi
preconditioning scheme. The rows of the global stiffness matrix K are divided equally
among the different processors such that each processor has an equal number of rows
to compute and it assembles the local K e matrix for each element in its subdomain.
After the global matrix K is assembled in parallel, it is adjusted to reflect the enforced
boundary conditions determined by the surface matching. The total number of mesh
nodes is 43,584, which specifies 214,035 tetrahedral elements and represents a system
of 130,752 unknown displacements to be identified. The brain deformation simulation
during neurosurgery is carried out on a Sun Microsystems StarFire 6800 symmetric
multi-processor machine with 12 UltraSPARC-III processors running at 750 MHz
and 12 GB of RAM.
A set of parallel scaling experiments are performed to demonstrate the scaling
characteristics of the implementation with the timings for solving the biomechani-
cal model under serial and parallel environment reported as well as good simulation