Evaluation of the generality and accuracy of a new mesh morphing procedure for the human femur

Abstract

Various papers described mesh morphing techniques for computational biomechanics, but none of them provided a quantitative assessment of generality, robustness, automation, and accuracy in predicting strains. This study aims to quantitatively evaluate the performance of a novel mesh-morphing algorithm.

A mesh-morphing algorithm based on radial-basis functions and on manual selection of corresponding landmarks on template and target was developed. The periosteal geometries of 100 femurs were derived from a computed tomography scan database and used to test the algorithm generality in producing finite element (FE) morphed meshes. A published benchmark, consisting of eight femurs for which in vitro strain measurements and standard FE model strain prediction accuracy were available, was used to assess the accuracy of morphed FE models in predicting strains. Relevant parameters were identified to test the algorithm robustness to operative conditions. Time and effort needed were evaluated to define the algorithm degree of automation.

Morphing was successful for 95% of the specimens, with mesh quality indicators comparable to those of standard FE meshes. Accuracy of the morphed meshes in predicting strains was good (R² > 0.9, RMSE% < 10%) and not statistically different from the standard meshes (p-value = 0.1083). The algorithm was robust to inter- and intra-operator variability, target geometry refinement (p-value > 0.05) and partially to the number of landmark used. Producing a morphed mesh starting from the triangularized geometry of the specimen requires on average 10 min.

The proposed method is general, robust, automated, and accurate enough to be used in bone FE modelling from diagnostic data, and prospectively in applications such as statistical shape modelling.

Introduction

A current focus of computational biomechanics research is the development of subject-specific models reproducing mechanical and geometrical features of an anatomical region of interest from biomedical images, aimed at diagnostic, therapeutic or surgical planning purposes [1]. Many techniques have already been proposed for developing finite element (FE) models of bones from computed tomography (CT) data. It seems that a mature stage of this technology has been reached with respect to in vitro model accuracy, since several independent groups have achieved very good validation results on the proximal femur loaded in vitro [2], [3], [4], [5]. There is a substantial agreement among these validation works about how to build a model. The standard procedure for producing an accurate FE model of a bone consists first in a segmentation of the CT dataset to obtain a surface tessellation of the bone segment contour. The shape obtained is then mathematically parameterised, usually by Non-Uniform Rational B-Splines (NURBS), and finally meshed using a dedicated software. These modelling procedures, though representing the state of the art, show several limitations with respect to possible applications. A first one is automation, since they are generally user-intensive and time consuming. A second one, perhaps even more important, is flexibility, since they do not permit fast mesh adaptation and transposition between subjects and they cannot be easily used to define an indexation of the population variability in terms of both anatomical parameters and material properties distribution to generate collections of synthetic models and define response surfaces. These issues, which can globally referred to as statistical modelling, are receiving increasing attention in the literature [6], [7], [8]. Morphing (or shape registration) is a technique, originally born for computer graphics purposes, that consists in deforming a template geometry onto a target one [9]. It has been hypothesized to extend this technique to biomechanical applications, that is, adapting a template mesh onto a subject-specific geometry extracted from magnetic resonance (MR) or CT images [10]. Morphing of subject-specific models of bone segments from CT data is a stepping stone to explore the definition of indexation of bone shape or material properties on a population [11], [12], [8]. Moreover, it could be a promising tool to: (i) fast re-mesh when conducting sensitivity studies (e.g. on prosthesis design or positioning) (ii) easily compare results sets from two or more meshes (since morphing generates isotopological meshes, that share the same node numbering and connectivity) (iii) improve the speed and automation of subject-specific FE model generation while keeping surface regularity.

Several attempts have been already tried, based on different rationales, to relate a template to a target geometry. Couteau et al. [13] described a mesh morphing technique based on warping and local displacement functions. The algorithm was tested on 11 proximal femur geometries, one for creating the template mesh and 10 used as target geometries. The results, though described as satisfactory, were not compared to any experimental data. The same method has been applied to entire femurs and also to soft tissues, such as maxillofacial models for computer aided surgery planning and simulation, by Luboz et al. [14], [15]. O’Reilly and Whyne [16] developed morphed subject-specific FE models of spinal motion segments, where mesh adaptation was based on the knowledge of some characteristic biomechanical lengths. In the absence of a comparison with experimental data, a comparison with respect to a standard FE model showed a different strain distribution in the morphed model, due to a cortical shell thicker than in the standard FE model. Tada et al. [17] generated subject-specific FE models of fingertips using a shape morphing technique where the spatial transformation was computed using a volume registration technique based on intensity gradient of MR volume data. The study focused on two specimens and there was no comparison with experimental data. Rajamani et al. [18] presented a morphing tool for patient-specific 3D knee surfaces visualization in computer assisted surgery, based on principal component analysis (PCA). The algorithm robustness was tested on 14 proximal femur specimens by means of the leave-one-out method, however the aim was not the definition of subject-specific finite element models. Hraiech et al. proposed a 3D mesh generation method for proximal femurs, consisting in the surface morphing of a template mesh constrained by manually picked landmarks [19], and the subsequent 3D mesh morphing using radial basis functions (RBF) [20]. The technique was tested on 15 specimens but only in terms of morphed mesh quality and geometrical accuracy of the morphed surface. This morphing method has been extended by using spherical instead of planar parameterisation as intermediate morphing domain [21]. Yoo [22] developed a shape morphing technique based on shape deformation method using an implicit function. This method showed good results, even if it reports local self-intersection problems when morphing complex models. However, its application to morphing bones (long bones as the femur in particular) could be difficult since it relies on the manual placement of many landmarks, that could not be repeatably identified as anatomical repere points. Sigal et al. [10] reported a comparison between two of the most frequently adopted techniques for mesh adaptation: (i) automated wrapping, guided by energy minimization criteria, that consists in mapping both the source and the target mesh on a simple geometry; (ii) manual landmarks selection, where a set of landmarks defining one to one correspondences is identified on the template mesh and on the target geometry, and morphing is achieved by making the landmark pairs coincident and applying some smoothing or non-intersection criterion to nodes other than the manually defined ones. The results showed that manual landmarks selection, though not completely automated, can ensure coincidence between source and target, while automated wrapping methods cannot.

Morphing technique has also been used for volume morphing, i.e. for the direct registration of different 3D volume data. Many algorithms have been proposed, based on different techniques such as wavelet domain-based morphing [23], geometrically constrained sphere mapping optimization [24], warping and blending [25]. To the authors’ knowledge, only one work on volume morphing can be directly related to bone biomechanics modelling: [26] described the use triangularized template geometry to automatically segment a CT volume using a morphing method based on the iterative accumulation of displacement fields. However, a quantitative evaluation of the boundary recovery is lacking.

Very recently, a work of Bryan et al. [8] used morphing within a complete statistical modelling workflow for the creation of a three dimensional, statistical, finite element analysis ready model of the femur. In that work, surface morphing, taking as input a surface segmented from CT data, is based on an elastic registration scheme and volume morphing is obtained solving decoupled three-dimensional Laplace equations to deform the baseline volume mesh. Morphed meshes, after CT based material mapping, are subjected to principal component analysis to define the statistical model.

While all the cited works contributed to demonstrate the applicability of morphing techniques to biomechanics modelling, and [8] also demonstrated the feasibility of population based modelling some improvements in the morphing evaluations may be achieved. In fact, to the authors’ knowledge no study evaluated morphing performances on a database characterised in terms of inter-subject anatomical variability, which can be very large, nor any study tried a comparison of the calculations resulting from morphed mesh with experimental in vitro data.

The aim of this work is to evaluate the performances of a new mesh-morphing algorithm on a large database of human femoral anatomies derived from CT scans. The proposed morphing algorithm will be evaluated in terms of generality, strain prediction accuracy, robustness and degree of automation.