Table 1.
Estimated memory usage by number of elements.
Figure 1.
Two-dimensional representation of the finite element model, with each point representing a vertex or node in the system.
Parallel computing requires additional memory allocation since vertices located at the interfaces between sub-regions must be stored twice.
Figure 2.
Total simulation time can be significantly decreased by performing initial iterations on coarser grids prior to solving the system on the original (fine) grid.
Since the solution obtained at the coarser scale is already close to the final solution, fewer iterations are required at the finest scale.
Figure 3.
Flow diagram for the processing pipeline.
(a) Mechanical properties for the image-based finite-element model are obtained via the relationship between simulated input strain (applied boundary conditions) and resulting simulated stress, computed from the equilibrium displacement map. (b) The equilibrium displacement map is obtained using a series of parallelized conjugate gradient solvers (c) applied at a series of resolutions (pre-iteration on courser grids).
Figure 4.
Memory usage (GB) versus number of elements (millions) for FE simulations on sub-volumes of a single
CT image using a single thread and eight threads.
The total number of elements in the FE models ranged from 1 to 75 million. Symbols represent experimental data points, straight lines are best fits.
Figure 5.
Average time per iteration (seconds) versus number of elements (millions) for running FE simulations on sub-volumes of a single
CT image using one, two, four, and eight threads respectively.
Symbols represent experimental data points, straight lines are best fits. Linear relationships were found in all cases with R2≥0.998.
Table 2.
Total number of iterations to reach 1% accuracy estimated with and without using the PICG approach in FE simulations on sub-volumes with different number of elements.
Figure 6.
Plots of the “true” relative errors on the finest grid obtained after using four different combinations of pre-iteration on coarser grids: ‘1’, ‘2 1’, ‘4 2 1’, and ‘8 4 2 1’.
E.g., ‘4 2 1’ means running simulations on data sets downsampled from the original data set by a factor of 4 and 2 sequentially, and then running simulations on the original data set. The combination ‘4 2 1’ achieves the same accuracy as the more time-consuming combination, ‘8 4 2 1’.
Figure 7.
Comparisons of the true relative errors on the finest grid obtained from running different number of iterations (200: red; 100: green; 50: purple; 25: blue; 12: orange) on all coarser grids in the combination strategy ‘4 2 1’.
Using 200 iterations on each coarser grid reduced the total number of iterations on the finest grid to around 200 compared to around 400 when using no pre-iteration for a 1% accuracy.
Table 3.
Comparisons in computational performance with literature-reported data.
Figure 8.
Convergence criteria comparisons.
(a) in example 1, the estimated relative error (red) has almost the same trend as the true relative error (blue), whereas the scaled residual (purple) deviates substantially; (b) the estimated relative error (red) in example 2 also better approximates to the true relative error (blue) than the scaled residual (purple).
Figure 9.
Structural MR images and resulting strain energy maps from two 3D data sets taken at 7T field strength at the left distal femur in a patient treated with zoledronic acid (Reclast™) for a period of 12 months.
a, e) cross-sectional images from stack of contiguous slices at baseline and after 12 months of treatment; b, f) volume-renderings of square-shaped sub-regions (7.8×7.8×3.7 mm3) indicated in images a and e, highlighting structural similarity at the two time-points; d, h) longitudinal projections of strain-energy data displayed for a thin slab (1.1×37.7×3.7 mm3) in the anterior region indicated in c and g.
Figure 10.
Sample slice of simulated strain-energy map of a human femur in coronal view.
To simulate a compression along the z-axis, a 1% strain was applied to the top surface of the femoral head. For details see text.