In this study, we introduced three morphing techniques and demonstrated their application by parameterizing a specimen-specific model of the load-bearing tissues of the posterior pole of the eye. The parameterizations can be applied independently, or in combination, allowing rapid production of families of models with carefully controlled variations in geometry that are useful for sensitivity analysis, including analysis of factor interactions. This is useful because it harnesses the detail available in specimen-specific models, while leveraging the power of the parameterization techniques common in generic models.
The morphing techniques also enable a more thorough exploration of the biomechanical consequences of changes in the shape of a specimen, such as may occur during aging or disease, for example the age-related changes in the human rib cage and forearm bones (Bouxsein, Myburgh et al., 1994
; Gayzik, Yu et al., 2008
) and the remodeling of the lamina cribrosa in early glaucoma (Roberts, Grau et al., 2009
). We have focused on parameterizing model geometry in this study because methods to parameterize mechanical properties are well developed for the relatively simple materials we used (Brolin and Halldin, 2004
; Anderson, Peters et al., 2005
; Laz, Stowe et al., 2007
; Rissland, Alemu et al., 2009
; Sigal, 2009
). For more complex mechanical properties the techniques for parameterizing geometry and mechanical properties must be generalized, and may have to be considered simultaneously. An example would be the anisotropy of the sclera and lamina cribrosa. As the geometry is morphed, the directions of anisotropy should change as well. Deepening of the lamina cribrosa will require adapting the orientation of the mechanical anisotropy of elements near the edge of the lamina to properly represent the tethering of the peripheral laminar trabeculae into the scleral canal wall (Roberts, Grau et al., 2009
; Roberts, Liang et al., In Press
(Accepted July 2009)). Further, for some systems, accurate predictions of mechanical behavior requires specimen-specific mechanical properties (Humphrey and Na, 2002
; Viceconti, Davinelli et al., 2004
; Vorp and Vande Geest, 2005
; Vande Geest, Sacks et al., 2006
; Girard, Suh et al., 2009
). Generalizing and parameterizing these properties remains a challenge.
A useful property of the morphing techniques in sensitivity analysis is that the models produced are relatively easy to compare with one another. All the models are related to the baseline, sharing nodes and connectivity on the surface, and it is straightforward to transfer the boundary conditions from one model to another. This may be particularly useful for studies with more complex boundary conditions. In addition, models produced through morphing are related to each other by a known transformation. This transformation may be used to identify corresponding regions for analysis, which in turn allows a more direct quantitative comparison of the mechanical response. These can be considerable time savers in pre- and post-processing.
Successful morphing required a smoothing function to reduce discontinuities in the deformation field. The sensitivity of a particular model to these discontinuities depends on the details of the geometry, the quality of the mesh, and the magnitude and direction of the deformation field. For example, when elements pare highly elongated (high aspect ratio), the mesh is more sensitive to deformations that increase the aspect ratio such as compressive deformations on the short axis. Hence, it is not possible to prescribe a priori a degree of smoothness that will be satisfactory in all cases. Therefore, all deformation fields need to be tested over the ranges of parameters and models on which they will be used. We acknowledge that the smoothing functions used in the example are somewhat arbitrary and that their applicability may be limited to the particular cases shown. We explored other functions such as normal distributions, but the long tails were problematic. The trigonometric functions we chose were relatively simple because their magnitudes, and those of their derivatives, were clear and easy to scale. From an implementation standpoint, considerable effort was initially required to identify simple and useful morphing techniques and develop the scripts and modules which support them. Once the scripts were completed, however, the process was almost completely automated, requiring only minimal user intervention. The difficulties and time requirements of applying the techniques to other systems will depend on their complexity, but it is generally much faster than producing new specimen-specific models. As an example, we have applied the morphing techniques to a femur (), which required approximately 3 hours of setup time, after which the time required to generate morphed surfaces was negligible.
Figure 7 The morphing techniques can be readily applied to other structures, as demonstrated by morphing the surface of a femur. Scaling-based morphing was used to enlarge the femoral head while leaving the remaining geometry unaltered (top row). In this case (more ...)
We have demonstrated morphing applied to model surfaces, which required the internal volume of each geometry to be remeshed. When the morphing deformation vector fields are smooth, and the baseline mesh is of high quality it is possible to morph the baseline volume mesh directly (Sigal, Hardisty et al., 2008
). Direct morphing of the volume mesh simplifies pre- and post-processing even further.
The methods described herein have some limitations that deserve consideration. First, there is some degree of arbitrariness in the deformation vector fields, even though our choices for variation were informed by an understanding of the anatomy and biomechanics of the structure of interest. The resulting morphed models agree with the variability of the anatomy and biomechanics of the posterior pole of the eye (Sigal, 2009
; Sigal, Flanagan et al., 2009
; Yang, Downs et al., 2009
). Recent reports have shown that the three geometric properties varied in this study can vary considerably between individuals and also change with aging and disease (Roberts, Grau et al., 2009
; Yang, Downs et al., 2009
; Yang, Downs et al., 2009
). Ideally, the shape variations introduced through morphing would be informed by a population study establishing the nature and magnitude of the physiologic variations (Van Essen, 2005
; Laliberte, Meunier et al., 2007
). However, we believe that the relationship between morphometry and analysis of sensitivity to variations in shape works both ways. Morphometry informs sensitivity analysis to keep the results relevant, and in turn, sensitivity analysis helps focus morphometry by identifying the shape variations of biomechanical consequence. A second potential limitation is that the morphing was applied to the baseline model. Hence, any inherent problems with the original geometry would propagate to the morphed models. In addition, the deformations obtained with morphing were not always ideal. For example, reducing the canal size produced a small distortion of the posterior lamina cribrosa surface near the insertion into the sclera (, middle left panel). Similarly, zero displacement in the Y direction when morphing the size of the canal changes slightly the shape of the peripapillary sclera. Distortions such as these can be avoided using more complex functions. We evaluated several such functions and found their contributions to the biomechanics to be minimal (results not shown). However, these complicate the description of the method, and are therefore not included in this proof-of-concept work. Morphing of independent factors lends itself naturally to multivariate analysis. We are currently developing the scripts to couple the morphing techniques with variations in tissue material properties and loading for use within factorial and response surface experimental designs, which will be the subject of future reports.
Some authors refer to morphing as warping (Zöckler, Stalling et al., 2000
). Variations of these techniques have been used for rapid reconstruction of specimen-specific models (Fernandez, Ho et al., 2005
; Brock, Dawson et al., 2006
; Sigal, Hardisty et al., 2008
), nonlinear strain computation (Veress, Weber et al., 2002
; Phatak, Sun et al., 2007
), medical image registration (Todd-Pokropek, 2002
) and segmentation (Bowden, Rabbitt et al., 1998
), and to make up for the sparsity of data in low quality datasets (Blanz, Mehl et al., 2004
; Shim, Pitto et al., 2007
). Morphing is also popular in the animation and computer graphics community (Sederberg and Parry, 1986
We have introduced morphing methods to parameterize specimen-specific models, and demonstrated their application on models of the posterior pole of the eye and the femur. The originality of this work lies in the application of morphing techniques for parametric analysis suitable for FE modeling and sensitivity analysis. While none of the concepts of morphing, specimen-specific modeling, or sensitivity analysis are novel by themselves, the integration of the three techniques shows promise for the study of biomechanics.