Accuracy Quantification of the Reverse Engineering and High-Order Finite Element Analysis of Equine MC3 Forelimb

Highlights

• The modeling of equine third metacarpal bones with limited available geometric data was investigated.

• The error map of the reconstructed geometries was plotted.

• Anatomical features causing noticeable errors were identified and discussed.

• A displacement-based error estimation and the relations of finite element convergence analysis were developed to assist future investigators to maximize the accuracy of their models.

• Quantifying errors of the finite element analysis presented in this study is essential before treatment of bone diseases.

Abstract

Shape is a key factor in influencing mechanical responses of bones. Considered to be smart viscoelastic and inhomogeneous materials, bones are stimulated to change shape (model and remodel) when they experience changes in the compressive strain distribution. Using reverse engineering techniques via computer-aided design (CAD) is crucial to create a virtual environment to investigate the significance of shape in biomechanical engineering. Nonetheless, data are lacking to quantify the accuracy of generated models and to address errors in finite element analysis (FEA). In the present study, reverse engineering through extrapolating cross-sectional slices was used to reconstruct the diaphysis of 15 equine third metacarpal bones (MC3). The reconstructed geometry was aligned with, and compared against, computed tomography–based models (reference models) of these bones and then the error map of the generated surfaces was plotted. The minimum error of reconstructed geometry was found to be +0.135 mm and -0.185 mm (0.407 mm ± 0.235, P > .05 and −0.563 mm ± 0.369, P > .05 for outside [convex] and inside [concave] surface position, respectively). Minor reconstructed surface error was observed on the dorsal cortex (0.216 mm ± 0.07, P > .05) for the outside surface and −0.185 mm ± 0.13, P > .05 for the inside surface. In addition, a displacement-based error estimation was used on 10 MC3 to identify poorly shaped elements in FEA, and the relations of finite element convergence analysis were used to present a framework for minimizing stress and strain errors in FEA. Finite element analysis errors of 3%–5% provided in the literature are unfortunate. Our proposed model, which presents an accurate FEA (error of 0.12%) in the smallest number of iterations possible, will assist future investigators to maximize FEA accuracy without the current runtime penalty.

Introduction

Bones are mainly responsible for withstanding and absorbing applied loads. To predict bone fracture and failure, and to investigate reasons for such incidents, comprehensive insight into the responses of bones to loading is crucial. Identifying the strains and stresses to which bones are exposed will assist in elucidating the reasons for fractures and locating their most likely sites. Bones are complex both in their material characteristics and their shape but respond in similar ways to external loads throughout the animal kingdom [1]. Clearly, a large bone that shows a relatively restricted range of normal movements would provide the best chance to develop a model to investigate the normal responses of bones to loading. Horses are large animals with large, elongated, and simplified forelimb bones that are apparently well-adapted for exercise at high speeds. Hence, considerable forces can be exerted on their forelimb bones. These forces are believed to be involved in different kinds of injuries and incidents, and most disastrous injuries in racing horses worldwide are associated with forelimb injuries, especially failures of the third metacarpal bone (MC3) [2], [3], [4], [5], [6], [7], [8]. The third metacarpal bone forms an essential part of the lower forelimb in withstanding loads [9]. Furthermore, due to its large size, minimal muscle attachments, and relatively simple movements, the MC3 is a unique long bone that can assist in investigating the responses of bones when they are exposed to forces. Surface strains on the MC3 midshaft bone can be related to exercise speed [10], locomotion type [11], [12], and, more significantly, the shape of the bone [13], [14]. The shape of the midshaft dorsal cortex (DC) of MC3 alters (expands and thickens) when it undergoes increasing applied forces during racing and training [15], [16]. Many investigations into human bone fatigue, for example, are heavily reliant on the outcome of equine bone research [9].

Computer-aided design (CAD) and finite element analysis (FEA) are indispensable tools in attempts to model and analyze bones and to replicate experimental data. Using different CAD methodologies in biomedical and tissue engineering practices demands considerable effort and diligence to appreciate the value and significance of different methods [17], [18]. Finite element analysis may still be unfamiliar to the readers of equine-related journals, yet it has the potential to provide tremendous benefits [19]. As Erdemir et al. reported, “There has been a 6,000% increase in the number of finite element articles published between 1980 and 2009” [20]. However, mesh quality, model validation, and appropriate energy balance have not been adequately addressed in these studies [21]. Generating a three-dimensional model, which has been done by various methods such as geometry-based and voxel-based [22], [23], [24], [25], is an initial step before performing FEA. In many biomechanical engineering activities [26], [27], [28], alternative methods such as reverse engineering via CAD are frequently required to produce the 3D models of bones. Examples of times when alternative methods are required include when computed tomography (CT)/magnetic resonance imaging (MRI) images are not available, if the images possess poor or bad quality, if bones are shattered, or when modifications are required to the 3D model of bones before implant design or conducting FEA. Furthermore, CT/MRI imaging has drawbacks, being expensive and difficult to perform on live animals [29], and causing a relatively high radiation exposure [30], [31], [32], so CT scanning may not be ethically justified [25]. Neither is it advisable for it to be regularly used in scanning healthy bone volunteers [33]. Reverse engineering techniques can be used either to reconstruct an incomplete or damaged bone or to make a duplicate model from an existing one [34]. In addition, 3D voxel-based surface extraction (CT/MRI-based modeling) requires a substantial computational power, and although they can describe anatomical morphology, a CAD-based solid modeling (vector-based) environment is needed to design, analyze, and simulate anatomical models [25]. This emphasizes the utility of CAD-based solid modeling. Irrespective of many influential factors that alter the mechanical behavior of bones, the importance of shape has been overly neglected in the literature. However, the significance of shape variation in the diaphysis of MC3 has been highlighted in some research [15], [29]. The DC of MC3 midshaft enlarges when the horses are exposed to fast exercise speed [10], [13], [35], [36], and compressive strains exceeding 3000 and 5670 microstrains have been recorded in this region [37], [38]. These findings define bone as a smart material that is able to adjust its mass and structure according to the loads it experiences [39]. The midshaft of MC3 is a region of considerable interest to investigate its response under load and to comprehend the shape-dependent mechanical behavior of bones. To commence addressing such inquiries, reverse engineering and determining accuracy of the 3D models are required to shape the fundamental grounding. Afterward, investigating the relationship among size, shape, and mechanical properties is feasible. The 6 cm segment of the MC3 midshaft which was investigated in this study has thick cortical bone, and this bone material redistributes toward either ends to become predominantly trabecular. The MC3 has variable directions of loads at either ends, which need to focus onto the central shaft (the segment being studied here). Not only does the elliptical cross-sectional shape of MC3 prevent and equalize directional loading but also it serves to enhance the sagittal bending once a horse is racing [9], [40]. In addition, having unique length and cross-sectional properties, MC3 can support relatively large compressive loads without experiencing substantial strains [41]. The objective is to comprehend the importance of different sizes and shapes in this piece of MC3 and to investigate how those shapes change with loads and strains. To accomplish this, reliable 3D models of the MC3 in this region and accurate FEA are essential.

Nonetheless, there is no current evidence of reverse engineering (CAD-based) through an extrapolation process of slices in the literature for reconstructing MC3. Neither has dimensional error analysis been performed in most of the previous studies to verify and assess the accuracy of reconstructed models [29], [42], [43], [44]. In CT imaging (voxel-based representation), a 3D model of an object is reconstructed from a 3D region growing from a series of cross-sectional slices (2D segmentations). In extrapolation via CAD systems, on the other hand, inner and outer cortices (contours) of slices are detected independently for each slice. A multisection surface will then be generated by sweeping the contours (cortex or curves) of each slice along an automatically defined long axis of the bone. This process is separately performed to generate internal and external surfaces. A similar process was discussed as an application of reverse engineering methods to reconstruct the human femur [26].

Several studies have been conducted on 3D modeling and stress distribution in the equine MC3 and surrounding structures [45], [46], [47], [48], [49], [50]. An essential part of an FEA should be dedicated to convergence and error analysis before publishing the results. A limitation to most of these previous studies is the lack of evidence in terms of quantifying dimensional error of reconstructed geometries, performing mesh quality assessment, determining the error of FEA, or assigning linear tetrahedron meshes to the models, which would all potentially lead to misleading stress and strain results. A summary of considerations in FEA error analysis reported in the literature is presented in Table 1. In addition, orthotropic material identification is one of the most crucial contributions toward a reliable FEA. To the best knowledge of the authors, finite element error analysis, as conducted in the present study, has never been discussed in the equine literature. Even in human studies, literature similar to what has formerly been conducted on the computer simulations of human feet [56] is limited in determining proper elements and methods for FEA. The purpose of this study is to present dimensional error analysis of reverse engineering through an extrapolation process of cross-sectional slices. In addition, the relations of stress and strain convergence and the displacement-based error estimation are presented that can be used to evaluate the accuracy of FEA and to minimize the associated errors.